LARGE-SCALE INHOMOGENOUS THERMODYNAMICS – AND APPLICATION FOR ATMOSPHERIC ENERGETICS
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LARGE-SCALE INHOMOGENOUS THERMODYNAMICS – AND APPLICATION FOR ATMOSPHERIC ENERGETICS
LARGE-SCALE INHOMOGENOUS THERMODYNAMICS – AND APPLICATION FOR ATMOSPHERIC ENERGETICS
Yong Zhu Department of Earth, Atmospheric and Planetary Sciences, The Massachusetts Institute of Technology, Cambridge, USA
CISP
CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING
Published by Cambridge International Science Publishing 7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK http://www.cisp-publishing.com First published March 2004 © Yong Zhu © Cambridge International Science Publishing
Conditions of sale All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library
ISBN 1 898326 3 12
Printed by Antony Rowe Ltd, Chippenham, Wiltshire, Great Britain
i
Preface The work in this book started twelve years ago when I was studying for a Ph.D degree in the University of Edinburgh. At the beginning, I was only attempting to apply the developed meteorological dynamics and classical thermodynamics to study the large-scale processes of energy conversion in the Earth’s atmosphere. As the work went on, it was found that the atmosphere was essentially different from the classical dynamic or thermodynamic systems, and we had no a particular thermodynamic theory to deal with the large-scale inhomogeneous and compressible fluid system in the gravitational field of the Earth. The studies on the particular features of the atmosphere lead to establishment of the new science, called the largescale inhomogeneous thermodynamics here. This new thermodynamics may have many applications. The atmospheric thermodynamics and energetics discussed in this book is one of them. A science, such as the classical dynamics, can be illustrated by different theories based on different definitions or expressions. One may use complicated mathematical symbols to represent simple ideals, or introduce new terminologies to answer questions as: “He was unable to sleep at night because he was suffering from insomnia”. I am more interested in understand the world than creating a new one on paper. Thus, the theories in this book are aimed mainly to reveal the new facts and explain them in the large-scale inhomogeneous thermodynamic systems, but not to make hypothesis for unknown processes and then derive mathematically the theoretical consequences. To find is more difficult than to create sometime. Many facts in front of us may not be seen if we do not intend to find them. They could be found by mind before by eyes. It is always interesting for me to discuss the physical philosophies behind the phenomena, while the mathematics is given in the form of easiest understanding. Some discussions may be challenging to widely accepted theories, but may also be inspiring. It was eight years ago, the author published his first book “Geostrophic Wave Circulations” (1993) in the pen name of Yong L. McHall, introducing a simple and effective method to solve the complete primitive equations in atmospheric dynamics. The previous book was focused on the large-scale dynamic processes in the extratropical troposphere and stratosphere without considering the thermodynamic processes. Since the atmosphere is able to produce kinetic energy by itself, the dynamics cannot be perfect if without the theory of kinetic energy generation. Now, I am glad to forward my new contribution in the first year of the new millenium. The two books tell us that the meteorology is no longer the applications of hydrodynamics, classical thermodynamics and numerical technology, as it has its own theoretical systems powerful to solve successfully many problems which cannot be solved before. Yong Zhu At MIT Cambridge, MA
Contents 1 Introduction
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2 Two classical physical systems 2.1 Introduction . . . . . . . . . . . . . . . . . 2.2 The Newtonian systems . . . . . . . . . . 2.2.1 Principle of friction . . . . . . . . . 2.2.2 Dynamic entropy . . . . . . . . . . 2.3 Simple thermodynamic systems . . . . . . 2.3.1 Mole-number and molecular mass . 2.3.2 Thermodynamic variables . . . . . 2.3.3 Pressure of monatomic gas . . . . 2.4 The first law of thermodynamics . . . . . 2.5 State equation of gases . . . . . . . . . . . 2.6 State equation of ideal gases . . . . . . . . 2.6.1 Ideal-gas equation . . . . . . . . . 2.6.2 More features of ideal gases . . . . 2.6.3 Kelvin temperature . . . . . . . . 2.6.4 Mixing ratio of water vapor . . . . 2.7 Thermodynamic energy law of ideal gases 2.8 Internal energy and heat exchange . . . . 2.9 Polytropic process . . . . . . . . . . . . .
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3 Molecular transport processes 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Diffusion velocity and partial velocities . . . . . . . . 3.2.1 Diffusion element and diffusion velocity . . . 3.2.2 Partial velocities . . . . . . . . . . . . . . . . 3.2.3 Diffusion velocity in non-uniform ideal gases . 3.3 Self-diffusion of ideal gases . . . . . . . . . . . . . . . 3.3.1 Diffusive mass flux . . . . . . . . . . . . . . . 3.3.2 Coefficient of self-diffusion . . . . . . . . . . . 3.4 Viscosity of ideal gases . . . . . . . . . . . . . . . . . 3.4.1 Diffusive momentum flux . . . . . . . . . . . 3.4.2 Momentum conduction . . . . . . . . . . . . 3.4.3 Coefficient of viscosity . . . . . . . . . . . . . 3.4.4 Relation to self-diffusion . . . . . . . . . . . . 3.5 Heat conduction of ideal gases . . . . . . . . . . . . . 3.5.1 Conductive heat flux . . . . . . . . . . . . . . 3.5.2 Heat conductivity . . . . . . . . . . . . . . . 3.5.3 Modified Eucken formula . . . . . . . . . . . 3.5.4 Collisional heat capacity . . . . . . . . . . . . 3.5.5 Comparison with experiments . . . . . . . . . ii
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23 23 24 24 25 27 27 28 30 32 33 36 36 38 40 41 41 44 47
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51 51 52 52 54 56 58 58 60 61 61 62 63 64 64 64 65 66 67 68
CONTENTS
iii
4 Predictability and thermodynamic entropy 4.1 Introduction . . . . . . . . . . . . . . . . . . . . 4.2 Change rate in diffusion processes . . . . . . . . 4.3 Mass conservation law . . . . . . . . . . . . . . 4.3.1 Mass diffusion equation . . . . . . . . . 4.3.2 Mass conservation . . . . . . . . . . . . 4.3.3 Diffusive transport equation . . . . . . . 4.4 Unpredictability in classical thermodynamics . 4.5 Thermodynamic entropy law for uniform states 4.6 Thermodynamic entropy change of non-uniform 4.7 Inadditive and scale-dependent features . . . . 4.8 Thermodynamic entropy balance equation . . . 4.9 Relation to dynamic entropy . . . . . . . . . . 4.10 Calculations for ideal gases . . . . . . . . . . . 5
Newtonian-thermodynamic system 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 Field variables . . . . . . . . . . . . . . . . 5.3 Parcel and parcel velocity . . . . . . . . . . 5.4 Mass and heat transport equations . . . . . 5.4.1 Continuity equations . . . . . . . . . 5.4.2 Integrated variations in a system . . 5.4.3 General continuity equation . . . . . 5.4.4 Heat flux equation . . . . . . . . . . 5.4.5 Heat conduction equation . . . . . . 5.5 Inhomogeneous thermodynamic system . . . 5.5.1 Adiabatic and transport processes . 5.5.2 Inhomogeneous thermodynamics . . 5.6 Momentum equation of atmosphere . . . . . 5.6.1 Pressure gradient force . . . . . . . . 5.6.2 Navier-Stokes equation . . . . . . . 5.6.3 Momentum equation of atmosphere 5.7 Shallow water dynamics . . . . . . . . . . . 5.8 Newtonian-thermodynamic system . . . . .
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71 71 72 75 75 76 77 78 79 83 86 89 90 92
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96 96 97 99 102 102 102 103 104 105 106 106 108 108 108 109 111 113 115
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117 117 118 118 119 121 122 124 126 126 128
6 Turbulent entropy and universal principle 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Thermodynamic entropy of turbulent system . . 6.2.1 Simple turbulent process . . . . . . . . . . 6.2.2 Thermodynamic entropy changes . . . . . 6.3 Grid thermometers . . . . . . . . . . . . . . . . . 6.4 Turbulent thermodynamic entropy . . . . . . . . 6.5 Turbulent entropy law . . . . . . . . . . . . . . . 6.6 Difference from classical thermodynamic entropy 6.6.1 General discussion . . . . . . . . . . . . . 6.6.2 Example . . . . . . . . . . . . . . . . . . .
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CONTENTS
iv 6.7 6.8
Turbulent entropy and disorderliness . . . Universal principle . . . . . . . . . . . . . 6.8.1 The principle . . . . . . . . . . . . 6.8.2 Applications . . . . . . . . . . . . 6.9 Partition functions . . . . . . . . . . . . . 6.10 Heat capacity and van der Waals equation 6.10.1 Einstein function . . . . . . . . . 6.10.2 van der Waals equation . . . . . .
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130 131 131 134 136 140 140 142
7 Basic conservation laws 7.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2 Parcel and local energy equations . . . . . . . 7.2.1 Mechanic energy equation . . . . . . . 7.2.2 Bernoulli’s equation . . . . . . . . . . 7.2.3 Principle of kinetic energy degradation 7.2.4 Local energy equation . . . . . . . . . 7.3 System energy equation . . . . . . . . . . . . 7.3.1 From kinetic theory of gases . . . . . . 7.3.2 For the whole atmosphere . . . . . . . 7.3.3 For a part of atmosphere . . . . . . . 7.4 Energy conversions . . . . . . . . . . . . . . . 7.4.1 Conversion functions . . . . . . . . . . 7.4.2 Total potential energy and enthalpy . 7.5 Potential enthalpy conservation . . . . . . . .
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143 143 144 144 145 147 148 149 149 150 151 152 152 154 157
8 Thermodynamic and geopotential entropies 8.1 Introduction . . . . . . . . . . . . . . . . . . . 8.2 Thermodynamic entropy variations . . . . . . 8.2.1 General expression . . . . . . . . . . . 8.2.2 Variation tendencies . . . . . . . . . . 8.3 Baroclinic entropy . . . . . . . . . . . . . . . 8.4 Barotropic entropy . . . . . . . . . . . . . . . 8.5 Thermodynamic entropy level . . . . . . . . . 8.6 Static entropy . . . . . . . . . . . . . . . . . . 8.7 Pseudo- reversible process . . . . . . . . . . . 8.8 The reference state . . . . . . . . . . . . . . . 8.9 Thermo-static entropy level . . . . . . . . . . 8.10 Geopotential entropy . . . . . . . . . . . . . . 8.10.1 For dry air parcels . . . . . . . . . . . 8.10.2 For the dry atmosphere . . . . . . . .
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160 160 162 162 163 165 167 168 169 170 171 173 175 175 177
9 Available enthalpy 9.1 Introduction . . . . . . . 9.2 Available enthalpy . . . 9.3 Constraint relationships 9.4 Variational approach . .
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179 179 183 184 186
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CONTENTS
v
9.5 9.6 9.7
The lowest state . . . . . . . . . . . . . . . . . . Maximum available enthalpy . . . . . . . . . . . Approximate approach . . . . . . . . . . . . . . . 9.7.1 The lowest state . . . . . . . . . . . . . . 9.7.2 Maximum available enthalpy . . . . . . . 9.8 Thermodynamic entropy variation . . . . . . . . 9.9 Geopotential entropy variations . . . . . . . . . . 9.10 Discontinuous examples . . . . . . . . . . . . . . 9.10.1 Baroclinic example . . . . . . . . . . . . . 9.10.2 Barotropic example . . . . . . . . . . . . 9.10.3 Thermodynamic and geopotential entropy 9.10.4 Continuous solutions . . . . . . . . . . . . 10 Dry 10.1 10.2 10.3
processes of energy conversion Introduction . . . . . . . . . . . . . . . . . . Dependence on process . . . . . . . . . . . . Sudden warming and cooling . . . . . . . . 10.3.1 Temperature variation . . . . . . . . 10.3.2 Kinetic energy production . . . . . . 10.4 Change of surface pressure . . . . . . . . . . 10.4.1 Surface pressure and static stability 10.4.2 Surface pressure change . . . . . . . 10.4.3 Change of the thickness . . . . . . . 10.5 Change of static stability . . . . . . . . . . 10.5.1 Partition of available enthalpy . . . 10.5.2 Final mean static stability . . . . . . 10.6 Thermo-static entropy level . . . . . . . . . 10.6.1 Change of barotropic entropy . . . . 10.6.2 Change of thermo-static entropy . .
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189 191 193 193 195 197 198 200 200 202 203 206
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207 207 208 211 211 212 215 215 216 217 220 220 221 222 223 225
11 Available moist enthalpy 11.1 Introduction . . . . . . . . . . . . . . . . . . . 11.2 Available moist enthalpy . . . . . . . . . . . . 11.3 Moist potential enthalpy . . . . . . . . . . . . 11.4 Thermodynamic entropy production . . . . . 11.5 Dry reference state . . . . . . . . . . . . . . . 11.6 Moist reference state . . . . . . . . . . . . . . 11.6.1 The isoperimetric problem . . . . . . . 11.6.2 Approximate approach . . . . . . . . . 11.7 Examples of lowest state . . . . . . . . . . . . 11.8 Available moist enthalpy . . . . . . . . . . . . 11.8.1 General and approximate relationships 11.8.2 Examples of available moist enthalpy .
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226 226 229 230 232 233 235 235 237 239 242 242 244
CONTENTS
vi 12 Moist processes of energy conversion 12.1 Introduction . . . . . . . . . . . . . . . . . . . . 12.2 Saturated reference state . . . . . . . . . . . . . 12.2.1 Saturated humidity profile . . . . . . . . 12.2.2 Minimum precipitation . . . . . . . . . 12.2.3 Temperature profile . . . . . . . . . . . 12.3 Effect of baroclinity . . . . . . . . . . . . . . . 12.4 Effect of horizontal humidity gradient . . . . . 12.5 Surface pressure change . . . . . . . . . . . . . 12.6 Available enthalpy of reference state . . . . . . 12.7 Threshold static instability . . . . . . . . . . . 12.8 Equivalent baroclinic and barotropic entropies . 12.9 Equivalent thermo-static entropy level . . . . .
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247 247 249 249 250 252 253 256 259 260 264 265 267
13 Available enthalpy in the atmosphere 13.1 Introduction . . . . . . . . . . . . . . . . . . 13.2 In the Northern Hemisphere . . . . . . . . . 13.2.1 Distributions in winter and summer 13.2.2 Relation to extratropical cyclones . . 13.2.3 Relation to blocking systems . . . . 13.3 In the Southern Hemisphere . . . . . . . . . 13.4 Development of low system . . . . . . . . . 13.5 Baroclinic entropy . . . . . . . . . . . . . . 13.6 Zonal mean distributions . . . . . . . . . . . 13.7 Least thermodynamic entropy production . 13.8 The highest static stabilities . . . . . . . . .
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269 269 273 273 273 276 276 277 279 282 285 287
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14 Available moist enthalpy in the atmosphere 14.1 Introduction . . . . . . . . . . . . . . . . . . . 14.2 Distribution of moist energy sources . . . . . 14.3 Relation to storm tracks . . . . . . . . . . . . 14.4 Tropical and extratropical tropospheres . . . 14.5 Relation to thunderstorms . . . . . . . . . . . 14.6 Relation to precipitation . . . . . . . . . . . . 14.7 Relation to tropical cyclones . . . . . . . . . .
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289 289 290 294 296 299 302 306
15 A case of typhoon recurvature 15.1 Introduction . . . . . . . . . . . 15.2 Typhoon Orchid recurvature . 15.3 Subtropical cyclones . . . . . . 15.4 Threshold surface temperature 15.5 Energy budget . . . . . . . . . 15.6 Self-feeding mechanism . . . . .
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308 308 310 316 317 319 322
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CONTENTS
16 A case of explosive cyclone 16.1 Introduction . . . . . . . . . . . 16.2 Energy steering mechanism . . 16.3 Baroclinic entropy distribution 16.4 Low-level moist jet . . . . . . . 16.5 Self-feeding mechanism . . . . .
vii
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327 327 328 336 338 340
17 States of maximum thermodynamic entropy 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . 17.2 Heat-death ideal gas . . . . . . . . . . . . . . . . 17.3 Heat-death geophysical air mass . . . . . . . . . 17.4 Heat-death atmosphere . . . . . . . . . . . . . . 17.5 Kinetic-death atmosphere . . . . . . . . . . . . . 17.5.1 Isentropic atmosphere . . . . . . . . . . . 17.5.2 Example . . . . . . . . . . . . . . . . . . . 17.5.3 Comparison with heat-death atmosphere . 17.6 Energy conservation constraint . . . . . . . . . . 17.7 Kinetic equilibrium state . . . . . . . . . . . . . . 17.7.1 General expressions . . . . . . . . . . . . 17.7.2 In statically stable atmosphere . . . . . . 17.7.3 In statically unstable atmosphere . . . . . 17.8 Principle of extremal entropy productions . . . .
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343 343 345 347 349 351 351 352 354 356 357 357 358 361 363
18 Energetics of linear disturbance development 18.1 Introduction . . . . . . . . . . . . . . . . . . . . 18.2 Conversion of available enthalpy . . . . . . . . 18.2.1 Method A . . . . . . . . . . . . . . . . . 18.2.2 Method B . . . . . . . . . . . . . . . . . 18.3 Growth of linear disturbances . . . . . . . . . . 18.3.1 Energy constraint equation . . . . . . . 18.3.2 Time-dependent expression . . . . . . . 18.3.3 Alternative expression . . . . . . . . . . 18.3.4 Numerical procedures . . . . . . . . . . 18.4 Eady wave development . . . . . . . . . . . . . 18.4.1 Evaluation equations . . . . . . . . . . . 18.4.2 Examples . . . . . . . . . . . . . . . . . 18.5 Synoptic geostrophic wave development . . . . 18.6 Development of blocking waves . . . . . . . . . 18.7 Wave development in stratosphere . . . . . . .
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366 366 368 368 371 373 373 374 375 376 377 377 379 382 385 388
19 Energetics of moving parcels 19.1 Introduction . . . . . . . . . . . . 19.2 Linear atmosphere . . . . . . . . 19.2.1 The thermal structure . . 19.2.2 Slope of isentropic surface 19.2.3 Slope of isobaric surface .
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391 391 393 393 395 396
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CONTENTS
viii 19.3 External forces on a parcel . . . . . . . 19.3.1 Adiabatic buoyancy oscillations 19.3.2 Horizontal processes . . . . . . 19.4 Slantwise static instability . . . . . . . 19.5 Slantwise lapse rate . . . . . . . . . . . 19.6 Slantwise adiabatic lapse rate . . . . . 19.7 Slantwise circulation instability . . . . 19.8 Height of slantwise convection . . . . . 19.9 Slantwise buoyancy oscillations . . . .
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398 398 400 402 404 406 408 410 411
20 Primary air engine 20.1 Introduction . . . . . . . . . . . . . . . 20.2 Primary air engine . . . . . . . . . . . 20.2.1 Assumed cycle . . . . . . . . . 20.2.2 General parcel energy equation 20.2.3 Relation to external work . . . 20.3 Adiabatic primary air engine . . . . . 20.3.1 Bernoulli’s equation . . . . . . 20.3.2 Extended parcel theory . . . . 20.4 Kinetic energy created on open paths . 20.4.1 On vertical paths . . . . . . . . 20.4.2 On isentropic surfaces . . . . . 20.4.3 On upward sloping paths . . . 20.4.4 On downward sloping paths . .
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414 414 415 415 416 419 419 420 421 422 422 423 426 428
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430 430 431 431 432 434 436 437 438 438 441 443 444 444 446 447 449 449 452 454
21 Dry air engines 21.1 Introduction . . . . . . . . . . . . . . . . . . . 21.2 Joule air engine . . . . . . . . . . . . . . . . 21.2.1 Joule cycle . . . . . . . . . . . . . . . 21.2.2 Condition of doing positive work . . . 21.2.3 Examples of kinetic energy generation 21.2.4 Entropy productions . . . . . . . . . . 21.2.5 Efficiency of Joule engine . . . . . . . 21.3 Energetics of baroclinic waves . . . . . . . . . 21.3.1 The baroclinic waves . . . . . . . . . . 21.3.2 Kinetic energy generation . . . . . . . 21.4 Kinetic energy generation in a system . . . . 21.5 Carnot air engine . . . . . . . . . . . . . . . 21.5.1 Kinetic energy generation . . . . . . . 21.5.2 Efficiency of Carnot engine . . . . . . 21.5.3 Dependence on working substance . . 21.6 Equilibrium air engine . . . . . . . . . . . . . 21.6.1 Equilibrium cycle . . . . . . . . . . . . 21.6.2 Examples . . . . . . . . . . . . . . . . 21.6.3 Entropy productions and efficiency . .
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CONTENTS
22 Wet air engines 22.1 Introduction . . . . . . . . . . . . . . . . . . . 22.2 Primary wet engine . . . . . . . . . . . . . . . 22.2.1 Kinetic energy generation . . . . . . . 22.2.2 Examples . . . . . . . . . . . . . . . . 22.3 Semi- wet Joule engine . . . . . . . . . . . . . 22.3.1 Kinetic energy generation . . . . . . . 22.3.2 Condition of producing kinetic energy 22.3.3 Efficiency . . . . . . . . . . . . . . . . 22.3.4 Thermodynamic entropy production . 22.4 Perfect storm and negative storm . . . . . . . 22.4.1 Perfect storms . . . . . . . . . . . . . 22.4.2 Negative storms . . . . . . . . . . . . 22.5 Development of negative storm . . . . . . . . 22.5.1 Coupling mechanism . . . . . . . . . . 22.5.2 Cross sections of a tropospheric river . 22.5.3 Height of tropical tropopause . . . . . 22.6 Low- and high-level convection . . . . . . . . 22.6.1 Low-level convection . . . . . . . . . . 22.6.2 High-level convection . . . . . . . . . . 22.7 Multiple semi-wet Joule engine . . . . . . . . 22.8 Wet Joule engine . . . . . . . . . . . . . . . . 22.8.1 Kinetic energy generation . . . . . . . 22.8.2 Efficiency . . . . . . . . . . . . . . . . 22.8.3 Thermodynamic entropy production .
ix
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23 Polytropic mixing processes 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 23.2 Lateral entrainment rate . . . . . . . . . . . . . . . 23.3 Heat capacity of mixing . . . . . . . . . . . . . . . 23.4 Polytropic potential temperature . . . . . . . . . . 23.5 Effect of entrainment on dry engines . . . . . . . . 23.5.1 On Joule air engine . . . . . . . . . . . . . 23.5.2 On baroclinic waves . . . . . . . . . . . . . 23.5.3 On equilibrium air engines . . . . . . . . . . 23.6 Moist polytropic mixing processes . . . . . . . . . 23.6.1 Energy equation of moist air . . . . . . . . 23.6.2 Polytropic equivalent potential temperature 23.6.3 Clausius-Clapeyron equation . . . . . . . . 23.7 Effect of entrainment on wet engines . . . . . . . . 23.7.1 On primary wet air engine . . . . . . . . . . 23.7.2 On semi-wet Joule engine . . . . . . . . . . 23.7.3 On multiple semi-wet Joule engine . . . . . 23.7.4 On wet Joule air engine . . . . . . . . . . .
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456 456 458 458 460 461 461 462 464 465 466 466 468 469 469 471 473 475 475 476 478 481 481 484 485
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487 487 489 490 492 494 494 496 497 499 499 500 501 502 502 503 504 506
CONTENTS
x 24 Limitations on frontogenesis 24.1 Introduction . . . . . . . . . . . . . . . . . . . . 24.2 The theoretical model . . . . . . . . . . . . . . 24.2.1 The basic relationships . . . . . . . . . . 24.2.2 Idealized frontal field . . . . . . . . . . . 24.3 Numerical Iteration . . . . . . . . . . . . . . . . 24.4 Limitations by initial field . . . . . . . . . . . . 24.4.1 Initial fields . . . . . . . . . . . . . . . . 24.4.2 Dependence on initial temperature field 24.4.3 Kinetic energy variation . . . . . . . . . 24.5 Baroclinic entropy . . . . . . . . . . . . . . . . 24.6 Available enthalpy . . . . . . . . . . . . . . . . 24.7 Limitations by other factors . . . . . . . . . . . 24.7.1 Scale of atmosphere . . . . . . . . . . . 24.7.2 Baroclinity of background field . . . . . 24.7.3 Latitudinal position of front . . . . . . . 24.8 Geopotential entropy variation . . . . . . . . .
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508 508 509 509 510 512 514 514 515 517 519 519 523 523 523 524 525
25 Grid-scale prediction equations and uncertainties 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 25.2 Scale-dependent data . . . . . . . . . . . . . . . . . 25.3 Subgrid-scale fields . . . . . . . . . . . . . . . . . . 25.4 Diffusive turbulences and negative diffusions . . . . 25.5 Grid-scale prediction equations . . . . . . . . . . . 25.6 Scale-dependent prediction models . . . . . . . . . 25.7 Errors from finite difference schemes . . . . . . . . 25.7.1 Truncation error . . . . . . . . . . . . . . . 25.7.2 Examples . . . . . . . . . . . . . . . . . . . 25.8 Turbulent diffusion and predictability . . . . . . . 25.9 Thermodynamic entropy produced by diffusions . . 25.10Uncertainties in physics . . . . . . . . . . . . . . .
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528 528 531 532 534 536 538 540 540 542 546 549 551
26 Examinations of model results 26.1 Introduction . . . . . . . . . . . . . . . . . . . 26.2 Scientific tests . . . . . . . . . . . . . . . . . . 26.3 Result-dependent models . . . . . . . . . . . 26.4 Thermodynamic entropy balance . . . . . . . 26.5 Partition of thermodynamic entropy change . 26.6 Examination of parameters . . . . . . . . . . 26.7 Features of thermodynamic entropy variation 26.7.1 The entropy change in a system . . . . 26.7.2 Change by dry air exchange . . . . . . 26.7.3 Change by moisture exchange . . . . .
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555 555 556 558 560 562 565 568 568 569 572
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CONTENTS
xi
A Thermopotential energy of gases A.1 Thermopotential energy . . . . . . . . . A.2 Assumed hard-sphere potential . . . . . A.3 Example of reverse sixth-power potential A.4 Comparisons with experiments . . . . . B Thermodynamics of gas expansions B.1 Energy conversions . . . . . . . . . . . B.2 Joule-Thomson effect . . . . . . . . . B.2.1 The new algorithm . . . . . . . B.2.2 Comparisons with experiments B.3 Joule-Thomson coefficient . . . . . . . B.3.1 The new algorithm . . . . . . . B.3.2 Examples . . . . . . . . . . . . B.4 Temperature inversion curves . . . . . B.5 Free expansion . . . . . . . . . . . . .
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574 574 575 577 581
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583 583 584 584 586 588 588 589 591 593
C Derivation of momentum equation
595
D References
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E List of symbols
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Index
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Chapter 1 Introduction In the study of atmospheric thermodynamics, an air parcel, if it is sufficiently small, may be assumed as a classical thermodynamic system (such as a small piece of ideal gas), of which the thermodynamic equilibrium state is uniform. Variations in the thermodynamic state may be illustrated by the state equation and the first law of thermodynamics (Dufour and van Mieghem, 1975; Iribarne and Godson, 1981). When the atmosphere is considered as a single thermodynamic system, additional assumptions and relationships are applied, since the atmosphere manifests the particular features different from those of a classical thermodynamic system. These features are summarized briefly in the following. Firstly, we never see a uniform state of the atmosphere without parcel motions attained by molecular diffusions. The steady states viewed from the datasets over meteorological scales are inhomogeneous, owing to the gravitation and rotation of the Earth and the heat exchanges between the atmosphere and surroundings. Thus, we may not assume a uniform state to represent an equilibrium state for the study of atmospheric dynamics and thermodynamics. The zeroth law of thermodynamics cannot be applied for the inhomogeneous thermodynamic system. The thermodynamic processes, such as energy conversions, must be studied between inhomogeneous equilibrium states of the atmosphere. Unlike in the classical thermodynamic systems, the inhomogeneous equilibrium state attained with conservation of mass and energy may not be unique. The weather predictions are actually looking for one of the equilibrium states according to the initial conditions and physical processes provided. Secondly, the atmosphere in the gravitational field possesses geopotential energy as well as internal energy and kinetic energy. The energy conservation law of the whole atmosphere is different from the first law of classical thermodynamics, as it includes also the mechanic energy equation derived from the atmospheric momentum equation (Brunt, 1944; Starr, 1951; Hess, 1959). According to this law, the mechanic work and parcel kinetic energy may be converted from the geopotential energy and internal energy through the processes inside the atmosphere. Thus, the atmosphere reduces its height of gravity center or increases its static stability when producing kinetic energy in the dry processes. For a provided initial state, the atmosphere possesses multiple barotropic reference states with different static stabilities and energy partitions. A real reference state attained by energy conversion depends on the process as well as initial state. Thirdly, the development of vertical motions in a fluid is produced by the buoyancy force doing mechanic work for fluid parcels. The buoyancy force on air parcels is zero in the statically stable atmosphere, as the downward gravitational force is balanced by the upward pressure gradient force. Thus, the transfer of geopotential energy and internal energy into kinetic energy may take place only in the statically unstable atmosphere if it is barotropic. In other words, the energy conversions in the atmosphere are restricted by the static stability, which depends on not only the 1
2
1. INTRODUCTION
intensity but also the direction of vertical gradient of potential temperature. While, the thermodynamic entropy is independent of the gradient direction. Therefore, the ability of energy conversion in the atmosphere may not be measured by the classical thermodynamic entropy, which depends on magnitude of the gradient only. Fourthly, the disorderliness of atmosphere depends not only on the distribution of molecules at different microscopic energy levels, but also on the distribution of parcels with different potential temperatures. While, the classical thermodynamic entropy is calculated by integrating the entropy for each element of volume or mass, and so depends on the thermodynamic state of each element instead of the element distribution. In the reversible adiabatic processes with conservation of parcel potential temperature, the classical thermodynamic entropy is conserved, but the thermodynamic disorderliness of atmosphere may nevertheless be increased irreversibly by turbulent parcel motions. Thus, a reversible process predicted by the second law of thermodynamics may not be really reversible in the inhomogeneous fluid system. The thermodynamic irreversibility related to the turbulent diffusions may be independent of that related to molecular diffusions. Also, the extremal state with maximum thermodynamic entropy attained through parcel motions are different from the isothermal state attained by molecular diffusions. As the isothermal heat-death state is seldom observed and is not interested for the study of energy conversion in the atmosphere, some particular physical relationship should be used to filter out the dead reference state. Fifthly, the conversion of geopotential energy in the atmosphere is irreversible in isolation with increasing the static stability or the vertical gradients of air density and potential temperature. This reduction of vertical disorderliness resulting from parcel motions is different from the change of horizontal disorderliness. When the statically unstable atmosphere becomes stable by adiabatic parcel convection, the thermodynamic entropy may not change but the process is irreversible. Thus, there are thermodynamically and geopotentially irreversible processes in the atmosphere related to the molecular or turbulent diffusion and geopotential energy conversion respectively. These processes may be independent of each other, and so the thermodynamic entropy is no longer a perfect discriminate parameter for the variation tendency of atmosphere. The adiabatic parcel exchanges in the vertical direction cannot happen in the statically stable atmosphere, though the thermodynamic entropy is conserved. Also, the statically stable atmosphere cannot reduce its static stability by parcel motions, thought the entropy increases in the process. Thus, we need another discriminate parameter together with the thermodynamic entropy to indicate the variation tendency and irreversibility of the geophysical and thermodynamic system. Sixthly, an isolated small piece of gas may reach the uniform equilibrium state with maximum thermodynamic entropy through molecular diffusions. While, the major processes on meteorological scales in the atmosphere are related to parcel motions driven by external Newtonian forces. Due to the gravity effect, the atmosphere tends to lower its gravity center, or increase its vertical disorderliness in mass density and potential temperature. Thus, the energy conversions in an unstable atmosphere drive the geophysical fluid system towards the equilibrium state possessing maximum kinetic energy, minimum geopotential energy and thermody-
3 namic entropy, compared with other equilibrium states. The isothermal heat-death atmosphere with maximum thermodynamic entropy may not be observed. Thus, there is the principle of minimum thermodynamic entropy production, which has been applied implicitly for long for the studies of maximum kinetic energy generation in the atmosphere. However, owing to some other dynamic equilibrium constraints, such as the geostrophic balance and thermal wind balance, the climatological mean state of the open atmosphere is different from the extremal state of minimum thermodynamic entropy. It possesses negative thermodynamic entropy sources, and so the energy conversions in the atmosphere are always possible. These major differences discussed above make the atmosphere a particular thermodynamic system in the gravitational field, which is different from the classical thermodynamic systems studied by the classical thermodynamics. It is necessary to create a new thermodynamics to study the thermodynamic processes in the largescale compressible geophysical fluid. The atmospheric energetics is an application of the new thermodynamics. To introduce the new thermodynamics, we review simply in the next chapter two basic systems studied in the classical physics. A physical system of which the variations are controlled by Newton’s laws may be called the Newtonian system, which can be studied conveniently in the frame composed of time and space coordinates. The study is concentrated on the motion status of the system. The time and distance are two basic variables. Another variable is mass which describes a steady property of the system. Variations of Newtonian system may be predicted by Newton’s laws together with the conservation laws of mass and energy. According to these laws, changes of the motion status in an inertial coordinate frame are caused by external forces referred to as the Newtonian forces in this study. There is a special force called the frictional force, which converts kinetic energy into internal energy of the system irreversibly. This feature allows us to define a state function, called the dynamic entropy in Chapter 2, to indicate the direction of system variation in frictional processes. As variations of Newtonian system may be represented with measurable variables and are predictable in principle, the dynamic entropy is not used in previous studies. Apart from the Newtonian systems, there are the thermodynamic systems, such as the gases, studied by the thermodynamics. The systems are composed of a great number of discrete microscopic particles, like the molecules, with similar features and structures. The particles usually possess kinetic energy of different motion status and potential energy at different levels. The distributions of these particles and their energies manifest a microscopic state of the system, which may not be detected in a macroscopic manner. While, the statistical features of the microscopic state may be represented by measurable thermodynamic variables, which represent the macroscopic thermodynamic state. The measurements of thermodynamic variables depend on the size of instruments in a non-uniform system. The adjective ‘macroscopic’ specifies the scales which are much larger than a particle. When external macroscopic Newtonian forces, such as the gravitational force of the Earth, are ignored, these variables are independent of space coordinates in equilibrium over a certain scale. The classical thermodynamics studies mainly the idealized equilibrium thermodynamic processes and uniform equilibrium states, without considering the nonequilibrium processes between two equilibrium states. In a fluid system without
4
1. INTRODUCTION
parcel motions, changes in the thermodynamic state are produced by molecular diffusions, which are the statistical effects of random molecular motions and collisions on the measurable thermodynamic variables. The classical thermodynamics does not study the causes of the system variations. These two types of physical systems are reviewed simply in Chapter 2. The molecules in a substance possess the kinetic energy associated with translational motions, rotations and vibrations of molecules. They have potential energy also related to the attractive and repulsive intermolecular forces. According to Zhu (2000), the mean kinetic energy of all molecules in a substance is represented by the heat energy, while the integration of intermolecular potentials over the substance is called the thermodynamic potential energy or simply the thermopotential energy. The heat energy and thermopotential energy construct the internal energy of substance. A new approach in calculating the thermopotential energy is discussed in Appendix A. The energy conversions between the thermopotential energy and heat energy or other energies in gas expansions are discussed in Appendix B. The thermopotential energy is also useful for us to derive the state equation of substance in Chapter 2. Since the thermopotential energy is negligibly small compared with the heat energy in the normal conditions of the atmosphere, the air in the atmosphere will be assumed as an ideal gas in this study which possesses the heat energy but not the thermopotential energy. Also, the heat capacities and gas constant will be taken as constants, and the state equation of ideal gases will be used all the time in the study. The temperature means the absolute temperature in this study except indicated particularly. A link between the Newtonian dynamics and classical thermodynamics is provided by the classical kinetic theory of gases. The first law of thermodynamics for ideal gases will be derived from the kinetic theory in Chapter 2. This theory will be applied again in Chapter 3 to study the transport processes in non-uniform gases produced by molecular diffusions. The transport coefficients, such as the heat conductivity, and the coefficients of self-diffusion and viscosity, will be derived from the new transport theory without solving the Boltzmann conservation equation. The effect of inelastic molecular collisions on heat conductivity is studied by the collisional heat capacity introduced, which can be calculated in terms of molecular structure (Zhu, 1999). Although the current theory is independent of the distribution function of molecular energy and intermolecular potentials or forces, the obtained relations between these coefficients agree well with the experiments for the dilute gases with weak or no polar momenta. In the study of transport properties in Chapter 3, we define the instantaneous diffusion velocity for a non-uniform gas according to the additive feature of momentum for a diffusion fluid element. The diffusion velocity depends on the microscale gradients in a fluid, and the transport carried by the velocity is in the downgradient direction of the variable. This feature in heat conduction gives the Clausius statement for the second law of thermodynamics. In other words, the molecular diffusions are controlled by the law. They may also take place in an isobaric system, and are different from the parcel motions dominated by Newton’s second law. As a result of the downgradient transport caused by molecular diffusions, the gradients are destroyed and so an isolated classical thermodynamic system may eventually
5 reach the uniform equilibrium state. Applying the diffusion velocity, we may derive the mass continuity equation for nonequilibrium diffusion processes without parcel motions, which tells that the mass density may be changed by molecular diffusions. Since neither the diffusion velocity nor the related microscale gradients can be detected or evaluated precisely, the mass continuity equation for the diffusion processes cannot be used in practice, and the molecular diffusions included in other thermodynamic equations cannot be evaluated for predicting the system variations. Thus, variations of the three thermodynamic variables are studied by two time-independent equations without including the diffusions in the classical thermodynamics: the state equation and the first law of thermodynamics. These equations are not prediction equations and can only be used for equilibrium states. For a provided initial equilibrium state, we may use these equations to calculate a new equilibrium state if one variable of the new state is given. The new state evaluated with the time-independent relationships has no explicit time relation to the initial state. It may occur before or after the initial state. However, in an isolated system which has no mass and energy exchanges with the surroundings except the mechanic work created as the system changes value, the downgradient transport resulting from molecular diffusions tends to destroy irreversibly the gradients or increase the disorderliness in the variable distribution, and then convert organized molecular motions into random motions. As discussed in Chapter 3, this feature gives a macroscopic description of the second law of thermodynamics. This law may be derived by introducing the thermodynamic entropy, according to the unidirectional transfer of diffusion kinetic energy into internal energy as discussed in Chapter 4. The irreversible energy transfer provides one of the irreversible entropy sources. Since a change of thermodynamic entropy indicates the direction of system variation in the irreversible processes, whether a new equilibrium state evaluated by the classical thermodynamics may really occur or not can be found by calculating the entropy change in the process. Since the diffusion kinetic energy depends on microscale gradients of thermodynamic variables, its variations cannot be evaluated conveniently. For the ideal gases with constant heat capacities, the thermodynamic entropy may be represented as a function of thermodynamic variables. The obtained algorithm may be used conveniently to calculate the entropy changes between two equilibrium states. Except the dissipation of diffusion kinetic energy, another source of thermodynamic entropy is the irreversible downgradient transport including the heat conduction without mass displacement, which may be independent of the kinetic energy dissipation. We shall discuss in Chapter 4 the entropy change in nonequilibrium system, and prove that the entropy increases as a nonequilibrium system becomes equilibrium through the transport processes. This consequence suggests that thermodynamic entropy may also measure the disorderliness of a system or the ability of system change related to the disorderliness. As a system tends to increase its disorderliness, the thermodynamic states with different degrees of disorderliness possess different probabilities of occurrence. So the entropy may also be represented by statistical probabilities in the statistical thermodynamics. To account for the disorderliness related to the distributions of molecules and
6
1. INTRODUCTION
their energies at different levels, the entropy cannot be evaluated by adding the thermodynamic entropy of each molecule (if it can be defined), since the sum of molecular entropy depends on the entropy of individual molecules and is independent of the distribution. When the entropy of each microscopic particle is conserved, the sum of particle entropy is also conserved, but the system disorderliness may nevertheless be changed by rearranging the particles. In this sense, we say that thermodynamic entropy is inadditive. While, the entropy calculated using thermodynamic variables may represent the system disorderliness, since these variables measure the mean features of the elements which include a number of particles. If the particles and their energies in a substance are not changed, the mean features at local places may still be changed by particle exchanges between elements. This means that the change of local mean features may result from the change of particle distribution only. Although the mean thermodynamic variables of the whole system, such as the mean temperature, evaluated with a linear algorithm are conserved, the entropy calculated with a nonlinear algorithm using the thermodynamic variables is changed by molecular exchanges or diffusions. It is discussed in Chapter 2 that the measurements of thermodynamic variables depend on the sample size especially at nonequilibrium states. Thus, thermodynamic entropy as a function of the variables depends on the size too. The calculations over different scales give different amounts of thermodynamic entropy, which are not errors but the entropy on the corresponding scales. An irreversible process may occur in an isolated system only if the thermodynamic entropy on all scales is not destroyed. This scale-dependent feature was not noticed except in the study of Batchelor (1967), since the entropy changes were considered mostly between the uniform equilibrium states in the previous studies of classical thermodynamics. The entropy changes calculated for uniform states are independent of data scale. The irreversible processes related to molecular diffusions or conductions exist also in the classical Newtonian dynamics, and we have introduced in Chapter 2 the dynamic entropy to account for the mechanic irreversibility. The relation to the thermodynamic irreversibility will be discussed also in Chapter 4. Due to the gravitation and rotation of the Earth, a steady thermodynamic state of the atmosphere viewed from the grid-point data over a certain scale manifests vertical stratifications and horizontal gradients in the thermodynamic fields. This non-uniform steady state will be referred to as the inhomogeneous equilibrium state in Chapter 5. The thermodynamic system characterized by inhomogeneous equilibrium state is called the inhomogeneous thermodynamic system. The mechanic and thermodynamic variables of the system vary with space, and so are referred to as the field variables. The inhomogeneous thermodynamic systems may also be studied with the relationships of classical thermodynamics in the time-space coordinates, by expanding the material differentials into partial differentials with respect to the time and space in the moving medium. The expanded first law of thermodynamics is called the heat flux equation, and the mass conservation law gives the continuity equation. The thermodynamics dealing with the inhomogeneous thermodynamic system will be referred to as the inhomogeneous thermodynamics. It is not called the nonequilibrium thermodynamics, because the system studied may not be in the nonequilibrium states related to molecular diffusions.
7 The differential pressure distribution in an inhomogeneous thermodynamic system may produce the pressure gradient force, which pushes the fluid elements or parcels to move from high pressure to low pressure at a constant height. This mass displacement is different from the mass diffusions discussed in Chapter 3, as it is controlled by Newton’s second law on a macroscopic scale. The parcel motions may change not only the motion status but also the thermodynamic state of atmosphere. The meteorology studies the dynamic and thermodynamic variations over meteorological scales, applying all the relationships used for the studies of Newtonian system and classical thermodynamic system. In an incompressible fluid, such as a shallow water, the mechanic and thermodynamic processes may be considered separately, and no mechanic energy is created through thermodynamic processes. But in the compressible atmosphere, the variations in thermodynamic state and motion status interact with each other. Thus, the compressible inhomogeneous thermodynamic system cannot be considered simply as a Newtonian system or a classical thermodynamic system, but is combination of the both called the Newtonian-thermodynamic system in Chapter 5. The equation of parcel motion referred to as the momentum equation will be discussed in the chapter. In a large-scale fluid system with parcel motions, the migratory particles are the macroscopic parcels as well as the microscopic molecules. The distribution of parcels with different temperatures or pressures manifest the disorderliness on a meteorological data scale. In other words, the disorderliness may also be changed by parcel motions as well as molecular diffusions in an inhomogeneous fluid system, or by the migratory elements in different scales at more than one levels. The classical thermodynamic entropy evaluated by adding the entropy of each fluid element, called the parcel or turbulent entity also, may be conserved in the adiabatic process of parcel exchanges without mixing with each other, since it depends on the entropy of parcels and is independent of parcel distribution. Thus, the classical thermodynamic entropy is unable to measure the changes of disorderliness caused by redistribution of parcels, and so there is an idealized process called the reversible adiabatic process in the atmosphere. In fact, this process is not really reversible. We shall introduce in Chapter 6 the turbulent entropy to measure the disorderliness on the data scale and the irreversibility related to parcel redistribution. Like the classical thermodynamic entropy, the turbulent entropy is also scale-dependent, depending on the data scale which is much larger than a parcel. Since the parcel motions are driven by external Newtonian forces including the pressure gradient force, they tend to destroy the pressure gradient in the system irreversibly. This is similar to the effect of molecular diffusions driven by the microscale thermodynamic gradients. Therefore, we may have a new law called the turbulent entropy law, which is similar to the second law of thermodynamic entropy, to account for the changes of system disorderliness caused by parcel motions. This law tells that the processes with increase of turbulent entropy is irreversible, even if the potential temperature or classical thermodynamic entropy is conserved for individual parcels or the whole system. When the classical thermodynamic entropy is conserved in the reversible adiabatic processes, the turbulent entropy may still change, since the processes are not really reversible. The turbulent entropy law is independent of the classical entropy law, as changes of turbulent entropy and clas-
8
1. INTRODUCTION
sical thermodynamic entropy are caused by different processes related to turbulent and molecular diffusions respectively. These processes on different scales may be independent of each other, as they are controlled by different mechanisms. Increase of disorderliness may be a universal feature for a system including a large number of elements with similar properties. This feature, called the the universal principle, will be illustrated mathematically by introducing the universal entropy in Chapter 6. Both the classical entropy law and turbulent entropy law may be examples of the universal principle applied for the classical and inhomogeneous thermodynamic systems respectively. Only in this sense, we may say that the second law of classical thermodynamics may be regarded as an example of turbulent entropy law applied for the simplest thermodynamic systems, of which the disorderliness is represented by distribution of the elements at a single level. We shall derive also from this principle the partition functions studied in the statistical mechanics, without applying the Maxwell-Boltzmann statistics. The obtained partition functions will be used further to derive the expressions of internal energy and isochoric heat capacity of gases. The van der Waals equation of state may also be obtained theoretically from this approach. Energy conversions in a Newtonian-thermodynamic system may take place between parcel kinetic energy and internal energy. Owing to the gravitational force, parcels in the atmosphere possess geopotential energy which depends on the geographic height. Thus, there are also the exchanges between geopotential energy, kinetic energy and internal energy in the atmosphere. The prediction equations discussed in Chapter 5 may be use to derive the energy equations for either a uniform parcel or the whole inhomogeneous system, called the parcel energy equation or system energy equation in Chapter 7. The parcel energy equation in the steady atmosphere is the well known Bernoulli’s equation. The system energy equation tells that the sum of kinetic energy and thermal enthalpy is conserved in the isolated atmosphere, which has no mass and heat exchanges with the exterior. The enthalpy of the whole atmosphere is identical to the total potential energy, defined as the sum of heat energy and geopotential energy in hydrostatic equilibrium. There is a fixed rate of geopotential energy to heat energy in the dry atmosphere, so the conversion of heat energy into kinetic energy is correlated linearly with conversion of geopotential energy. When only the isobaric molecular diffusions are considered and the temperature changes caused by kinetic energy dissipation are ignored, we obtain in Chapter 7 the conservation law of potential enthalpy for the atmosphere. The potential enthalpy of a parcel is defined as the enthalpy which the parcel will possess when it varies adiabatically and reversibly to a reference pressure. This law gives a weaker constraint for the processes of energy conversion discussed, compared with the adiabatic assumption used in the previous studies. It will be found in Chapters 9 and 11 that the reference state of the atmosphere derived from the energy equation without using this law is isothermal without parcel motions. This state, called the heat-death state, possesses maximum thermodynamic entropy and is reached through molecular diffusions in isolation. However, it is not the normal state attained by energy conversions in the atmosphere on a meteorological scale within a few days, and is not the normal predictions interested by meteorologists. The isothermal atmosphere
9 over a limited area in the lower stratosphere is not produced by molecular diffusions in isolation. The conservation law of potential enthalpy can be used to filter out the heat-death reference state for the study of energy conversion without integrating the prediction equations. The energy equations derived in Chapter 7 are independent of time, so they are diagnostic equations but not prediction equations. The diffusion terms which control the direction of energy conversion do not appear in the energy equations applied for equilibrium states. With these equations, the energy conversions are evaluated by choosing, instead of predicting, a reference state for a provided initial state. The assumed reference state may not be reached if the internal thermodynamic entropy decreases in the process. Thus, the energy equations should be used together with thermodynamic entropy laws which provide an important examination for assumed processes. The simple mathematical manipulations in Chapter 8 show that when anisobaric molecular diffusions are filtered out, the thermodynamic entropy increases with reducing three-dimensional inhomogeneity in the potential temperature field, and is maximum in the atmosphere with constant potential temperature called the isentropic atmosphere. Under the effect of gravity, the isentropic state may be attained only in the statically unstable atmosphere through parcel motions. As thermodynamic entropy increases, the disorderliness or randomness of an isolated classical thermodynamic system viewed in all directions increases also. This may not be the case in the large-scale inhomogeneous fluid system affected by the gravitational force. As materials tend to reduce their geopotential energy, the atmosphere tends to reduce its height of gravity center or increase the static stability. Meanwhile, the horizontal gradients such as the baroclinity may be destroyed in the process of geopotential energy conversion. Thus, variations of the disorderliness are anisotropic in the atmosphere. We introduce in Chapter 8 the baroclinic entropy and barotropic entropy to study the features of thermodynamic entropy changes in the horizontal and vertical directions respectively. The sum of baroclinic entropy and barotropic entropy defines the thermodynamic entropy level relative to the isentropic atmosphere, which is the extremal atmosphere with maximum thermodynamic entropy among all the reference states attained by adiabatic parcel exchanges. With the partition of thermodynamic entropy into horizontal and vertical portions, we may discuss conveniently in Chapter 8 the basic conditions for the dry atmosphere to be a kinetic energy source without solving the system energy equation. According to the change of thermodynamic entropy, the barotropic atmosphere may produce kinetic energy only if it is statically unstable, while a baroclinic atmosphere may be the energy source even if it is statically stable. The production of thermodynamic entropy in energy conversions is equivalent to the sum of baroclinic and barotropic entropy changes. Since the entropy changes in the vertical and horizontal directions may have different signs, the total thermodynamic entropy may be conserved in an irreversible process of energy conversion. This irreversible process with conservation of thermodynamic entropy in a whole system will be referred to as the pseudo- reversible process in Chapter 8. It will be illustrated also in the chapter that the reference state depends not only on initial state but also on production of thermodynamic entropy in the process. An extremal example of the reference state
10
1. INTRODUCTION
will be called the lowest state, which is attained through the process with minimum thermodynamic entropy production and possesses the strongest static stability and minimum thermal enthalpy, compared with the other reference states attained from the same initial state. The thermodynamic entropy, which measures the disorderliness of a classical thermodynamic system, depends on the intensities but not directions of the system gradients. The ability of system change depends on the disorderliness, and so may also be measured by thermodynamic entropy. While, changes of large-scale inhomogeneous fluid system, such as the Earth’s atmosphere and oceans, caused by parcel motions, depends on the static stability, since the fluid in the gravity field tends to lower its gravity center. The statically unstable atmosphere is more capable of producing kinetic energy than the stable atmosphere. Thus, the ability of system change related to conversion of geopotential energy depends on not only the intensity but also the vertical direction of potential temperature gradient. It may not be represented fairly by thermodynamic entropy, since the statically stable and unstable barotropic atmospheres may possess an equal amount of thermodynamic entropy. Thus, we introduce in Chapter 8 the static entropy to estimate the ability of energy conversion in the atmosphere associated with vertical gradient of potential temperature. The sum of the baroclinic entropy and static entropy assumes the thermo-static entropy level of the atmosphere, which decreases with the baroclinity and static instability, and so may represent the ability of energy conversion in the geophysical fluid. The irreversibility of pseudo- reversible process is related to conversion of geopotential energy, and may be independent of thermodynamic entropy law. Thus, there are two independent irreversible processes in the atmosphere: One is thermodynamic and the other is geopotential. The change of thermodynamic entropy accounts for the thermodynamic irreversibility but not the geopotential irreversibility. An example is the development of adiabatic convection in the statically unstable and barotropic atmosphere. This process is geopotentially irreversible, though the thermodynamic entropy or potential temperature of each parcel or the system is conserved. Also, the vertical adiabatic convection with conservation of thermodynamic entropy cannot occur in the stable atmosphere, as it raises the gravity center of atmosphere. To provide a full judgment for the variation tendency of the compressible geophysical fluid, a new variable called the geopotential entropy is forwarded in Chapter 8, which measures the geopotential irreversibility of the changes caused by parcel motions in the dry atmosphere. The meteorological processes in the isolated dry atmosphere are possible only when both the thermodynamic entropy and geopotential entropy are not destroyed. The system energy law discussed in Chapter 7 tells that the kinetic energy generated is equivalent to the destruction of enthalpy in the dry atmosphere. The amount of enthalpy which can be converted into kinetic energy in the process permitted by the thermodynamic entropy and geopotential entropy laws is referred to as available enthalpy in Chapter 9. Unlike the available potential energy of Lorenz (1955), the amount of available enthalpy depends on process. For a classical thermodynamic system which is uniform in equilibrium, the amount of enthalpy is determined uniquely for given mass, energy and potential enthalpy at a constant pressure. This, however,
11 is not true for an inhomogeneous thermodynamic system like the atmosphere. The stratified atmosphere may be in variant thermodynamic states possessing different amounts of thermal enthalpy, when the total mass, energy and potential enthalpy are conserved. Among all the states, there exists the limit state called the lowest state in Chapter 8, which possesses minimum enthalpy and maximum kinetic energy. To meet with the geopotential entropy law, the lowest state is evaluated with the variational approach introduced by McHall (1990a) and possesses the maximum geopotential entropy. The maximum available enthalpy may be calculated by taking the lowest state as the reference state. This variational approach is different from that of Dutton and Johnson (1967), as it applies the following constraint relationships for the isoperimetric problem. The entropy equation discussed in Chapter 8 gives the first constraint relationship, as the energy conversion depends on process which may be identified with thermodynamic entropy production. The second constraint is provided by the conservation law of potential enthalpy to filter out the anisobaric molecular diffusions which lead to the heat-death solution of reference state. Moreover, the potential temperature of an isolated system cannot exceed the maximum and minimum values at an initial state. This feature gives the third constraint for the energy conversion as a particular boundary condition. A reference state derived with the new approach manifests the basic features discussed in Chapter 8. For example, it possesses a stronger static stability, if the initial state is more baroclinic and less statically stable. Particularly, the lowest state, which possesses the highest static stability compared with the other reference states, may be attained through a pseudo- reversible process in the isolated baroclinic atmosphere. The new algorithm may be used to evaluate the energy conversions in either the statically stable or unstable atmosphere. It shows that the maximum available enthalpy with respect to the lowest state increseses with the baroclinity and static instability at initial state. Kinetic energy created in the statically unstable atmosphere is far more than in the stable atmosphere, especially when the baroclinity is low. With the new algorithm we may study the energy conversion with respect to a baroclinic reference state. The developments of atmospheric disturbances are local events in most situations, and the new approach may be used for a local region as it does not apply the zonal mean fields. The lowest state derived by the variational approach is continuous in space. If the initial state is discontinuous, the lowest state evaluated with parcel algorithm may also be discontinuous (Margules, 1904). We shall compare the continuous lowest states with the discontinuous for the examples of Margules. Since the continuous lowest state is attained through molecular and turbulent diffusions, the process creates more thermodynamic entropy and less kinetic energy than the parcel exchange process without the diffusions. As the energy conversions in the real atmosphere are highly irreversible, the lowest state may not be reached and the maximum available enthalpy may not be converted entirely. The dependence of energy conversion on process may be studied by considering the thermodynamic entropy production in developments of various weather systems. It will be found in Chapter 10 that energy conversion in a highly irreversible process is characterized by sudden change. This feature may be adopted to study the energetics of explosive or rapid processes in the atmosphere,
12
1. INTRODUCTION
such as the developments of explosive cyclones and stratospheric sudden warmings or coolings. In a highly irreversible process of a major warming, kinetic energy of mean zonal flow is destroyed. The strong turbulent diffusions in the warming events have been confirmed by the wave breaking observed in the stratosphere (McIntyre, 1982; McIntyre and Palmer, 1983). The warming processes cannot be thermodynamically or mechanically isolated, since both the thermodynamic entropy and geopotential entropy are destroyed in the episodes. We discuss also in Chapter 10 the mechanism of energy conversion according to the thermodynamic entropy balance. The conversion of heat energy is correlated to conversion of geopotential energy. As conversion of geopotential energy is realized by downward motion of cold air and upward motion of warm air, the gravity center of the atmosphere decreases or the static stability increases in the process. If the initial state is statically stable, the vertical disorderliness in the potential temperature field decreases, so does the barotropic entropy. Since thermodynamic entropy cannot be destroyed in the isolated atmosphere, the barotropic stable atmosphere cannot produced kinetic energy in isolation. But the baroclinic atmosphere may increase its baroclinic entropy by reducing the baroclinity to compensate the destruction of barotropic entropy, so that the thermodynamic entropy is not reduced in the whole system. However, kinetic energy may be created in the statically unstable atmosphere even if it it is barotropic, since the barotropic entropy increases with the static stability in the process. These consequences are obtained also in Chapter 8 without solving the energy equation. It will be emphasized that reduction of baroclinity merely may not create kinetic energy, unless the static stability increases. The increase of static stability leads to reduction of total enthalpy of the atmosphere, since the change of geopotential energy is correlated to the change of heat energy. Parcel kinetic energy may also be converted from the latent heat of moisture in the atmosphere. The thermal enthalpy plus latent heat is defined as the moist enthalpy in Chapter 11. The moist enthalpy which can be converted into kinetic energy will be referred to as the available moist enthalpy. Applying the variational approach of McHall (1990b, 1991), we may also study the energy conversion in the moist atmosphere with the three constraints similar to those for the dry processes. A reference state attained by moist process depends on the humidity as well as temperature. Since the behaviors of saturated and unsaturated air are different, and it is impossible for us to track down the individual parcels in the three-dimensional atmosphere, we are unable to predict the realistic moist reference state with the time-independent energy equation. Thus, we discuss two particular reference states in Chapter 11. In the first example, changes in the humidity and temperature are independent of each other, and the reference state obtained is dry and barotropic. In the second example, changes of humidity and temperature are considered together according to the change of equivalent potential temperature, and the reference state may be moist. If represented by equivalent potential temperature, the dry and moist reference states are given by a unified expression, and the available moist enthalpy is approximately equal with respect to the dry and moist reference states attained from a provided initial state. Energy conversions in the moist atmosphere depend highly on three-dimensional gradient in the equivalent potential temperature field. As in the dry atmosphere,
13 the energy conversions depend also on process. The turbulent diffusions increase the thermodynamic entropy production but reduce kinetic energy generation, and a highly irreversible process may also occur abruptly in the moist atmosphere. We shall define in Chapter 12 the equivalent baroclinic entropy, equivalent barotropic entropy and equivalent thermo-static entropy to study the ability of energy conversion in the moist atmosphere. Unlike in the dry atmosphere, kinetic energy may be created in the statically stable and barotropic moist atmosphere. The baroclinity may either increase kinetic energy generation or reduce the threshold humidity and static instability for development of moist disturbance. The horizontal gradient in the humidity field has an effect similar to that of baroclinity on energy conversion, as thermodynamic entropy may also be produced by destroying the gradient. It is impossible to convert all the latent heat released into kinetic energy in the barotropic atmosphere, so the atmosphere may possess more heat energy and geopotential energy after development of moist disturbance. When the static stability increases in a dry process of energy conversion, the temperature decreases in the lower troposphere and increases in the upper troposphere. Since the mass density in the lower troposphere is higher than above, the heat energy in the atmosphere is reduced by changing the temperature stratification. According to the state equation, the change of gas volume is not linearly correlated to the change of temperature. The volume increased by the warming in the upper troposphere is more than the volume reduced by the cooling in the lower troposphere, so that the thickness increases in the process. When the top of air column does not change its height, the pressure in the lower part decreases. The pressure reduction in the dry atmosphere will be calculated in Chapter 10. The increase of thickness may be amplified greatly by release of latent heat as shown in Chapter 12. In general, the surface pressure drops lower as more kinetic energy is created in the dry or moist atmosphere. Examples of the strong depressions produced by moist processes are the tropical cyclones and the explosive cyclones developed over extratropical oceans. A successful study must be able to explain the observed features of atmospheric circulations. The global distributions of available dry and moist enthalpy in the atmosphere will be evaluated, in Chapters 13 and 14 respectively, by the new approaches with the semi-daily meteorological data over six years obtained from the European Centre of Medium-Range Weather Forecasts (ECMWF). The results show that there are the basic correlations between the geographic distributions of the major energy sources and storm tracks, thunderstorms and precipitation centers. The tracks of tropical cyclones occur also in the regions where available moist enthalpy exceeds a certain limit. The climatological distribution of baroclinic entropy will be illustrated also. Contribution of static instability on the baroclinic energy sources may be found from the differences between the distributions of available enthalpy and baroclinic entropy. The minimum thermodynamic entropy produced as the dry atmosphere reaches the lowest state after energy conversion will be demonstrated in Chapter 13. The static stability of the six-year mean lowest state are plotted also, which is different from the climatological stratification of temperature. Since the large-scale circulations are constrained by the geostrophic balance, the atmosphere possesses a large amount of available enthalpy and is not in the lowest state.
14
1. INTRODUCTION
To examine in detail the correlations between the energy sources and atmospheric disturbances, we discuss a case of typhoon recurvature and explosive cyclone in Chapters 15 and 16, respectively. It will be found that Typhoon Orchid occurred in October 1991 moved towards the strongest moist energy source close to it. The northward recurvature from a previous westward track took place as a moist energy source to the west was weakened and a new center of high available moist enthalpy developed to the north. According to the mean temperature lapse rate in the tropical and subtropical regions, we may calculate the threshold temperature of saturated air on the surface for development of deep convection. The result agrees with the minimum sea surface temperature in the regions of tropical cyclones obtained from statistical studies. The different features of tropical and extratropical cyclones may be explained by the differences in their energy sources. We shall classify a new category of cyclones called the subtropical cyclones, which develop or live in the quasi-barotropic and equivalent baroclinic atmosphere in the subtropical regions, and possess the intermediate features between those of tropical and extratropical cyclones. According to the energy budget, the tropical cyclones are not supported merely by existing energy sources, because the moving speed estimated from existing energy sources is higher than the real speed observed. In the viscous barotropic atmosphere over oceans, the cyclonic circulation of tropical storm may produce strong convergence of moisture flux near the low center, which includes the effect of evaporation. The concentration of moisture provides a significant amount of energy for maintaining the cyclones over a warm ocean water. This intensification of kinetic energy source produced by the storm itself is referred to as the self-feeding mechanism. Due to this mechanism the tropical storm depend less on existing energy sources compared with the baroclinic cyclones, and so predictions of tropical storm movement are more difficult. When the self-feeding is not strong enough, a tropical cyclone may nevertheless move towards an existing energy source and convert the moist enthalpy along the track into its kinetic energy. This fact makes it possible for us to predict the tropical storm tracks by analysing the energy sources within a limited time period. Since construction of energy source in the moist tropical and subtropical atmospheres is quicker than in the extratropical atmosphere, tropical cyclones may change the direction of motion dramatically towards a newly established energy source, and so the prediction time is relatively short compared with that for extratropical baroclinic cyclones. The case studied in Chapter 16 is an explosive cyclone, called the tropospheric bomb also, developed in the regions of high available enthalpy over the northwestern Pacific in January 1992. Since the humidity was relatively high over the ocean water compared with that over lands, available moist enthalpy was also high around the storm at the early stage. Unlike in the tropical and subtropical regions, the moist energy source overlapped on the dry energy source and depended highly on the baroclinity. The important effect of moisture on the explosive cyclone could be provided by the water vapor flux in the regions. There was a filamentary moisture flux in the lower troposphere off the eastern edge of the cyclone referred to as the tropospheric river. Maximum moisture flux took place near the leading edge of the river. The flux and its convergence can be intensified by the cyclonic circulation.
15 The cyclone moved towards the leading edge of a river to convert the moist enthalpy into kinetic energy. Thus, there is also a self-feeding mechanism for the extratropical cyclones over oceans, which is different from that of tropical cyclones. This mechanism provides an important complement for the traditional theory of baroclinic cyclone development especially in the moist atmosphere. Also, the vertical thermal structure of the moist cyclone is different from that of dry cyclone. However, development of the cyclones in the extratropical atmosphere depends highly on conversion of available enthalpy and increase of baroclinic entropy, and so they are essentially the baroclinic disturbances. An important feature of inhomogeneous thermodynamic system is existence of the multiple reference states for a provided initial state, when the mass, energy and potential enthalpy are conserved. The lowest state is attained by the extremal process with minimum thermodynamic entropy production and maximum geopotential entropy production. It possesses minimum thermal enthalpy and maximum kinetic energy, compared with other reference states attained from the same initial state. We discuss, in Chapter 17, another extremal state in the atmosphere attained by the processes of maximum thermodynamic entropy production. The state is uniform for ideal gases at rest attained by molecular diffusions without the effect of gravity. While, the maximum entropy state, attained by molecular diffusions including kinetic energy dissipation in the gravitational field, is an motionless isothermal state with vertical stratifications in pressure and mass density. The gravitational force affects the profile of mass but not temperature, since the temperature depends on the mean speed of molecules of which the distribution is independent of the gravity. Apart from the molecular diffusions, there are parcel processes which may be more important than the microscopic processes for the circulation changes in a few days. If the anisobaric molecular diffusions are filtered out by the conservation law of potential enthalpy, the atmosphere possesses multi-equilibrium states which are different from the isothermal dead state. The dry and moist reference states including the lowest states attained by energy conversions over a meteorological scale are examples of the equilibrium states. Apart from the lowest state, another extremal example of the equilibrium state is the isentropic atmosphere, attained by the process of maximum thermodynamic entropy production and minimum geopotential entropy production when the potential enthalpy is conserved. This extremal atmosphere will be called the kinetic-death atmosphere, since no thermodynamic variations and energy transfers may be brought about by either the free or forced parcel motions in the isentropic atmosphere without molecular diffusions. The heat-death state of a gas is a true state in physics. While, the atmosphere can hardly become isothermal, because the changes are produced mainly by parcel motions instead of molecular diffusions, and the system is not isolated or closed. The parcel motions are controlled by the Newtonianforces such as the pressure gradient force, gravitational force and Coriolis force in the Earth. The pressure gradient force pushes the parcels move in the downgradient direction, destroying the gradient eventually in the isolated system. But on the rotational Earth the parcel motions are constrained to be quasi- geostrophic balance and hydrostatic equilibrium. The geostrophic circulations may remain the horizontal gradients of thermodynamic variables. When the balance is broken down in disturbance development, the horizontal
16
1. INTRODUCTION
gradients may be destroyed. While, the downward gravitation is independent of the circulation changes. The competition between gravitational force and vertical pressure gradient force causes the Archimedes’ effect or conversion of geopotential energy and correlated heat energy into kinetic energy in the statically or baroclinically unstable atmosphere. As the result, the atmosphere is stratified in its equilibrium state with a high density below and a low density above. Therefore, the energy conversions driven by the gravity increase the static stability or vertical disorderliness of the atmosphere. The geopotential entropy law discussed in Chapter 8 leads the system to the lowest state with maximum geopotential entropy and minimum thermodynamic entropy, compared with other reference state attained in the same atmosphere. This is different from that predicted by thermodynamic entropy law, since the atmosphere affected by the gravitational force is not really isolated. In other words, increase of geopotential entropy is prior to increase of thermodynamic entropy in the geophysical fluid systems. However, the Archimedes’ effect or geopotential entropy law cannot destroy thermodynamic entropy, but may only constrain the entropy production. We shall introduce the principle of extremal entropy productions in Chapter 17, which tells that the energy conversions resulting from parcel motions tend to follow the extremal process with maximum geopotential entropy production and minimum thermodynamic entropy production. According to this principle, the kinetic-death atmosphere is not recorded often unless in the regions of strong convection. In the traditional linear theories of disturbance development, the unstable waves grow exponentially with time at a constant rate. In fact, the growth rate depends on baroclinity of the background field. As available enthalpy is converted into kinetic energy, the baroclinity decreases and the growth rate changes in disturbance development. This interaction between disturbance and background field cannot be studied with the traditional linear theory. The variational approach discussed in Chapter 9 will be applied to study the interaction in Chapter 18, since we may use a baroclinic reference state for the new approach to evaluate the local energy sources not affected by remote fields. The energy constraint equation for the disturbance development will be derived from the interactions between the disturbances and background energy sources. This relationship may be used to illustrate more realistically the development of baroclinic disturbance, when the environmental baroclinity and growth rate change in the process. It will be found in Chapter 18 that the disturbance development under the interaction is significantly different from the wave growth at a constant growth rate. The Eady wave cannot be a good representation of the extratropical disturbances, since the development is too slow compared with the real processes. Also, it was argued by McHall (1993) that the critical baroclinities for the baroclinic instabilities given by previous studies were below the time-mean in the lower troposphere at middle latitudes. The baroclinic disturbances may be considered more likely as the unstable synoptic geostrophic waves (McHall, 1993). The critical baroclinity for the wave instability is slightly stronger than the time-mean baroclinity in the extratropical troposphere. The geostrophic waves in the varying background grow faster than the Eady waves, and have the time scale close to that of observed extratropical cyclones. This new approach allows us to plot the decay phase as well as the development
17 phase of the process. The planetary wave circulations in the atmosphere may also be destabilized by baroclinic instability. According to McHall (1993), developments of blocking systems in the free extratropical troposphere and stratospheric sudden warmings are associated with the nonlinear instability of planetary geostrophic waves in baroclinic regions, as the basic features of the planetary wave development derived from the theory agree with those observed. The current interaction theory based on energy conversion provides additional examinations for the planetary wave instability. The developments of planetary waves in the troposphere and stratosphere will be demonstrated in this chapter. The maximum amplitude and growth rate evaluated in the varying environments are comparable with those in the real processes. The observed dependence of the growth rate on the wave spectrum is confirmed by the study. For example, the long wave components grow slower than the short components, but may get larger amplitudes at the end. The typical large-scale circulations in the extratropical dry atmosphere are the slantwise convection (Green et al., 1966) around the polar front or in the baroclinic waves. The slantwise convection in the dry baroclinic atmosphere will be studied in Chapter 19 with the Newtonian dynamics for air parcels. It is found that the free slantwise convection may occur in the statically stable and baroclinic atmosphere, when the trajectory slope is smaller than the slope of isentropic surface in the dry processes or equivalent isentropic surface in the moist processes. The static stability for slantwise convection may not be measured by the temperature lapse rate studied previously for vertical convection. To find the static stability of the slantwise circulations, we forward in the chapter the algorithms for calculating the environmental temperature lapse rate and adiabatic lapse rate of parcels along a slantwise trajectory, called the slantwise lapse rate and slantwise adiabatic lapse rate respectively. The slantwise static stability depends on the slope of isentropic surface or the baroclinity, and so may also be represented by the baroclinic instability. From the point of slantwise circulation, the baroclinic instability is a kind of static instability. Both of the instabilities may cause conversion of geopotential energy and heat energy into parcel kinetic energy. Moreover, the large-scale horizontal pressure gradient force in the geophysical fluid depends crucially on the gravity effect. This is also found in Chapter 17. Thus, the horizontal processes may nevertheless be related to conversion of geopotential energy in the atmosphere. Unlike the vertical convection, the slantwise convection may form a perturbation pattern with varying amplitude forced by the slantwise static instability and horizontal pressure gradient force. The induced upward propagating slantwise gravity waves may grow up and cause significant weather changes, and so should not be filtered out in a weather forecasting model. The classical hydrodynamics is an application of Newtonian dynamics for fluids, by introducing a particular Newtonian force called the pressure gradient force. This force depends on pressure gradient inside a fluid. It is an external force for a single fluid parcel but may not be external for the whole system, so it cannot change the total energy of the system when creating parcel kinetic energy. Also, the mechanic work created by the gravitational force may also be represented by conversion of
18
1. INTRODUCTION
geopotential energy. Therefore, the parcel kinetic energy produced by the macroscopic Newtonian forces is actually converted from other energies in the system, such as the internal energy and geopotential energy, and the parcel energetics may also be studied with the pure thermodynamics without using the Newton’s laws. We shall introduce the air engine theory in Chapters 20-23 to study the atmospheric dynamics and energetics for the weather systems with typical circulation patterns. We discuss firstly in Chapter 20 the air engine theory for a single parcel. Since parcel motions in the atmosphere may not form closed thermodynamic cycles, we add auxiliary paths to a parcel trajectory in order to complete a simple cycle, called the primary cycle. A parcel working on this cycle will be referred to as the primary air engine. Change of parcel kinetic energy on the open path is evaluated by the mechanic work created over the primary cycle minus the kinetic energy change in the auxiliary paths. The result in the steady atmosphere leads to the general parcel energy equation. The Bernoulli’s equation derived in Chapter 7 gives the example for adiabatic processes. Unlike the classical heat engines, kinetic energy created by air engines may be converted from the geopotential energy as well as heat energy. The examples of parcel kinetic energy generated on a few selected trajectories are demonstrated in the chapter. The results agree with those derived from the classical dynamics in Chapter 19. On vertical paths, kinetic energy can be created only in the statically unstable atmosphere. If parcels move on the isentropic surfaces, the kinetic energy of vertical velocity is conserved, but of horizontal motions may still be changed by horizontal pressure gradient force. This consequence was not made clearly in previous studies. Owing to the horizontal pressure gradient force also, there is no the recognized negative correlation between the signs of kinetic energy generation and trajectory slope with respected to isentropic surface. Parcel kinetic energy are produced in the slantwise upward motions and destroyed in the downward motions. To remain the slantwise vertical circulations across the polar front in the viscous atmosphere, the downward motions on the cold side must start at the height lower than the top height of upward motions on the warm side, and so the tropopause descends discontinuously across the front from the warm side to the cold side. This feature of kinetic energy generation in the slantwise convection may also explain the observed north-east tilt of troughs and ridges in the large-scale baroclinic perturbations. The air engine theory for a single parcel may be applied to study the energy conversion in a weather system or the whole atmosphere, by introducing the equivalent principle in Chapter 21, according to which a cycle completed by more than one parcels can be considered as by one. The system energy equation discussed in Chapter 7 may also be derived from the air engine theory without using the momentum equation. To study the energetics of a weather system, we may design a particular air engine of which the thermodynamic cycle is similar to that of the typical circulation pattern in the system. To give an example, the energetics of large-scale slantwise convection and baroclinic perturbations will be investigated with the Joule air engine, of which the thermodynamic cycle is the Joule cycle studied in the classical thermodynamics. Also, the mean meridional circulations in the atmosphere will be studied by introducing a new reversible air engine, called the equilibrium air engine. The Carnot engine is a particular example of the new reversible heat engine.
19 The traditional linear theories of baroclinic instability provide the necessary conditions but not the sufficient conditions for the wave instability, since the budget of eddy kinetic energy over the wave spectrum is not considered. A unstable wave cannot develop if no eddy kinetic energy is created in the process. The sufficient conditions may be provided by the air engine theory for the baroclinic perturbations as discussed in Chapter 21. The unstable waves may grow only if the eddy kinetic energy created on a wave cycle is greater than that destroyed by viscosity. The critical baroclinity or wavelength for wave development can be derived from the energy budget. The kinetic energy created over a cycle may be applied further to estimate the growth rate. The baroclinic waves may grow when the critical wavelength or baroclinity obtained from a linear theory agrees with that derived from the energetics. It will be found that only the planetary waves may exist stably in the dry and weak baroclinic atmosphere, and the energy generated over a wave cycle may be maximum at a certain wavelength in the viscous atmosphere. The air engine theory may be applied also for the study of moist convection in the atmosphere. Three basic wet air engines will be discussed in Chapter 22 to study the local airmass storms and the super storms developed in the environments with strong vertical wind shears. It is well known that development of sever storms is favored by some conditions, such as the vertical wind shear and low level temperature inversion. These facts may be explained by the theory of wet air engine. We shall find that the intrusion of cold air from the middle troposphere caused by vertical wind shear may increase the energy generation in a storm. Also, net kinetic energy may be created over a convective cycle constrained in the lower troposphere by the inversion, while the convective cycle extending to a higher level may only destroy kinetic energy in the same environments. According to kinetic energy budget of the moist air engines assumed for the storms, we may provide an algorithm for calculating the low limit of sea surface temperature for development of deep convection, such as the tropical storms in a barotropic atmosphere. The surface temperature calculated may be used to check the air engines assumed. A convective system, including the upward and downward convection in mass balance, may not produce net kinetic energy over the whole circulation cycle. The examples are the convective storms forced by the large-scale baroclinic disturbances. These storms will be called the negative storms in Chapter 22, in contrast with the so called positive storms which produce net kinetic energy by themselves. The possible coupling mechanisms between the negative storms and large-scale baroclinic circulations will be discussed also in the chapter. If the Hadley cells in the tropical troposphere are assumed as heat engines, we may calculate the height of tropical tropopause according to the energy balance in the cells. To give the examples, we introduce also in the chapter the equilibrium Hadley engine and extended Hadley engine to study the relatively weak and strong moist Hadley circulations in summer and winter respectively. These engines may also be applied for a single tropical storm. It will be argued that the coupling between tropical barotropic moist convection and extratropical baroclinic slantwise convection is responsible for the seasonal changes in the intensity of Hadley cell and the height of tropical tropopause. Also, the anomalous circulation patterns in the tropical regions may be associated with the extratropical baroclinic circulations.
20
1. INTRODUCTION
For the study of kinetic energy generation in convective motions, entrainment and detrainment, or the mixing between convective clouds and environmental clear air should be considered. Instead of giving a precise algorithm for evaluating the mixing processes, a simple approach is forwarded in Chapter 23 for the study of kinetic energy generation under the effect of entrainment. The entrainment and detrainment will be assumed as polytropic processes by defining the heat capacity of mixing. We shall introduce also the polytropic potential temperature and polytropic equivalent potential temperature, which are the conservative quantities in the dry and moist polytropic convective processes with constant heat capacities of mixing. The static stability of the vertical convection with mixing should be estimated by the vertical gradient of polytropic potential temperature or polytropic equivalent potential temperature of the atmosphere. The previous studies on the effects of entrainment were generally concentrated on the upward convective processes. It will be found that the entrainment or detrainment in the downward convection of a storm may also have an important effect on the energy balance. The large-scale circulations in the rotational atmosphere are constrained by geostrophic balance, so that the equilibrium state of the atmosphere viewed over a meteorological scale may remain horizontal gradients of temperature and mass density. These gradients may be intensified in the process called frontogenesis. Since thermodynamic entropy in a frontal zone decreases in frontogenesis, the produced large-scale temperature gradient cannot exceed a certain limit in an isolated atmosphere. This limit depends on the thermal structures of initial and final states, and is studied in Chapter 24. The results emphasize the importance of using thermodynamic entropy law to examine the processes simulated by simplified models or with unrealistic boundary conditions. For example, some previous model studies showed that an infinitely large temperature contrast over a limited distance could be attained adiabatically from an initial field with a weak baroclinity. This simulated process of frontogenesis may be spurious, since the atmosphere is not infinitely large and the thermodynamic entropy of the isolated atmosphere cannot be reduced infinitely. It is discussed in Chapter 4 that changes of classical thermodynamic system are generally unpredictable with the classical thermodynamics, due to the uncertainty of molecular diffusions, of which the statistical effect does not follow the Newton’s second law. This molecular diffusions may also have an effect on numerical weather predictions. However, they may not be the major error source in general. Unlike in the classical thermodynamics, the primitive prediction equations of the atmosphere, including the state equation, continuity equation, heat flux equation and momentum equation, may still form a complete set for the three independent thermodynamic variables and parcel velocity included, even if the molecular diffusions are ignored entirely. So, the weather changes are predictable with the prediction equations for provided initial and boundary conditions. However, the predictions are not a simple initial data problem. In other words, the prediction errors depend not only on the errors in the initial fields provided, since there are the intrinsic error sources in the prediction equations and the numerical models constructed using these equations. We discuss in Chapter 25 the intrinsic error sources in current numerical models. The variables in the primitive equations are the functions covering continuously each
21 point at the space and time. But in practice, the meteorological fields are represented by discrete grid-point data over a certain scale of time and space. This means that the variables in the equations are scale-dependent, and the differentials and integrations calculated approximately with these date depend on data scale too. The primitive equations including scale-dependent variables are scale-dependent equations, and will be called the grid-scale prediction equations in Chapter 25. Solutions of the scale-dependent equations are also scale-dependent. Prediction errors may then be produced by the differences between the grid-scale prediction equations and the continuous primitive equations. The grid-scale prediction equations predict the circulations over the grid scale. The subgrid-scale circulations can neither be represented by the grid-point data nor predicted by the grid-scale equations. They will be called the meteorological turbulences in Chapter 25, which may be classified into two categories. One is the small- and meso-scale eddies and organized unstable convective systems, which may cascade subgrid-scale perturbation momentum into or produce kinetic energy for grid-scale circulations. These effects will be called the negative diffusions including the negative viscosity on the subgrid scales. The other one is the stable tiny-scale irregular turbulences similar to the classical turbulences studied by classical fluid dynamics. This kind of turbulences, referred to as the diffusive turbulences, cascade kinetic energy from grid-scale circulations into smaller and smaller scales irreversibly. Apart from the negative and positive diffusions, the physical and chemical processes, such as the moist processes and greenhouse effects included in the subgrid-scale circulations, may also have significant effects on the predicted fields after a certain time. In the quantum physics and classical thermodynamics, the effects of uncertain electron and molecule motions on the measurable variables may be illustrated statistically by the probability wave function and the state equation of substance respectively. While, the effects of meteorological turbulences on the grid-scale circulations cannot be illustrated generally by a statistical function with the datasets available, since the subgrid-scale circulations are different from the random motions of fluid parcels. But the meteorological turbulences are treated as a random process in current numerical models with the parameterizations using the grid-point data which do not really include the subgrid-scale information. The effects of subgrid-scale physical or chemical processes are represented also by parameterizations approximately in the numerical models. Thus, the meteorological turbulences form the major error source for the numerical predictions. It is generally difficult to estimate the strength of this error source, as the errors depend essentially on the unknown circulation patterns. This error source may be weakened by increasing data and model resolutions, as more energy conversion and transport processes may be represented by grid-point data. Like the molecular diffusions, the turbulent diffusions may also produce thermodynamic entropy as discussed in Chapter 6. We discuss in Chapter 25 the error source related to the turbulent diffusions in the atmosphere, by estimating the effects of thermodynamic entropy production on the energy conversions and Eady wave development. The results show that the effects of turbulent diffusions on energy conversions in the baroclinic or moist atmosphere may be comparable with the
22
1. INTRODUCTION
effects on kinetic energy dissipation. So, the disturbance developments predicted by a numerical model may be sensitive to the errors in parameterizations of turbulent diffusions, which may change from time to time and from place to place in the atmosphere. Prediction errors may also be produced by the finite difference schemes used for calculating the differentials and integrations in a numerical model. It is illustrated in Chapter 25 that the errors cannot be reduced continuously by increasing the truncation order of a difference scheme without increasing the data and model resolutions. Since the errors produced by deficiency of numerical models cannot be ignored, the study of predictability cannot be completed by estimating the growth of initial data errors only. The parameters included in the parameterizations for the meteorological turbulences and some other subgrid-scale processes are actually the unknown variables which change with time and space. Usually, these subgrid-scale processes depend on each other in the atmosphere, so it is difficult to test a single parameter in a particular process. These parameters are then determined by running the whole model and comparing the model results with the observations. The solutions so obtained may not be unique and may not be all physically meaningful or correct when considered individually. As the numerical models possess several error sources as discussed in Chapter 25, the comparisons of model results with real processes have a limited significance. A similarity between a model result and reality may be achieved functionally but not always principally, and does not mean necessarily that the model and simulated process are physically true. It will be argued in Chapter 26 that the current numerical models are the result-dependent models, as the parameters included are determined functionally by the request of getting ‘realistic’ results. When the real mechanism is not incorporated perfectly in a model, the ‘realistic’ result may be produced artificially. The meteorological turbulences affect largely the thermodynamic entropy balances in the real and model atmospheres. As discussed in some previous chapters, the processes related to frontogenesis or energy conversions are very sensitive to the errors in the entropy balance. The collapse of front and some model instability may be caused by underestimating the thermodynamic entropy production. To suppress the model instability, the entropy production may be exaggerated in a model. Thus, to calculating the thermodynamic entropy budget in the model atmosphere may provide an useful examination for some of the subgrid-scale processes and the parameters related. We study in Chapter 26 the entropy budget equation for the open atmosphere. The basic features of the entropy variations will be discussed. Particularly, we shall reveal the several sources of negative thermodynamic entropy input, which may influence greatly the kinetic energy generation in the atmosphere and are indispensable for maintaining the circulations. As some other relationships, such as the mass, energy and moisture balances in the local or whole atmosphere, may be broken by the intrinsic or technic errors in a model, the balance equations may also be applied to test the parameters in the related processes.
Chapter 2 Two classical physical systems 2.1
Introduction
There are two basic types of physical systems studied in the classical physics. The system of the first type is represented by its kinetic state or motion status, such as the position, velocity and acceleration, measured and studied in a frame of time-space coordinates usually. The kinetic state tells us also the mechanic energy situation including the mechanic potential energy and kinetic energy. This system, called the Newtonian dynamic system here, is studied with Newton’s laws together with mass and energy conservation laws. While, the system of the second type is specified with its macroscopic thermodynamic state represented by thermodynamic variables. This system is called the thermodynamic system, as we are interested mainly in variations of its thermodynamic state which may lead to exchanges of heat energy and mechanic energy in both directions. An example of classical thermodynamic system is ideal gases. Variations in the thermodynamic states of ideal gases are controlled by thermodynamic energy and entropy laws, state equation and mass conservation law. Usually, changes of kinetic state and thermodynamic state may relate to each other, and there may be exchanges between kinetic energy and heat energy in a real physical system. But in idealized situations, the two types of systems may be studied separately, using different sets of physical relationships described above. However, in a physical system of the third kind, such as the Earth’s atmosphere, changes in the kinetic and thermodynamic states cannot be separated at all, and conversion of mechanic energy is correlated essentially to conversion of heat energy. So, we have to adopt all the relationships used for the two basic systems to study the third kind system, which possesses many new features compared with the two systems. To understand these new features, we review simply in this chapter the two types of the simplest systems without discussing in detail the classical dynamics and thermodynamics. The dynamic forces, which may change the motion status of a Newtonian system may be called the Newtonian forces, as they can be applied for the Newton’s laws. A particular example is the frictional force, which always reduces the macroscopic kinetic energy of a system and increases the heat energy. So the energy conversion resulting from friction is unidirectional. In other words, the dynamic processes affected by friction are irreversible if without thermodynamic processes. This feature may be applied to establish the principle of friction for Newtonian systems as discussed in this chapter, which is analogous to the second law of thermodynamics. According to this principle, we introduce a state function, called the dynamic entropy, to illustrate the irreversibility of mechanic processes. The study of classical thermodynamics is based on energy conversions between equilibrium states. There is a particular energy called the internal energy of sub23
2. TWO CLASSICAL PHYSICAL SYSTEMS
24
stance studied in thermodynamics, which depends on microscopic features of a system and is not studied in Newtonian mechanics. When other minor energies are ignored, the internal energy is partitioned, in the current study, into the heat energy related to molecular motions and the thermopotential energy related to intermolecular forces. These energies will be studied in detail in Chapter 6 and in Appendices A and B. In this chapter, we apply the heat energy and thermopotential energy to illustrate the first law of thermodynamics. This energy relationship is then used to derive theoretically the general state equation of substance, which can be simplified to give the ideal-gas equation upon the assumptions provided for ideal gases. The features of ideal gases will be discussed also. Without using the first law of thermodynamics, the energy conservation law of ideal gases may also be derived using the kinetic theory of gases. Changes in the thermodynamic state of gas may be brought about by molecular diffusions. Motions of individual molecules are affected by intermolecular forces as well as macroscopic Newtonian forces. While, the statistical effect of random molecular motions and collisions, called the molecular diffusions, may not be controlled by these forces in a general way. The molecular diffusions may be studied by introducing the diffusion velocity discussed in the next chapter. We extend in this chapter the first law of thermodynamics for equilibrium states to that for nonequilibrium states, in order to account for the irreversible conversion of diffusion kinetic energy into heat energy. However, as the diffusion velocity depends on microscale gradients in the thermodynamic fields, which may not be measured or calculated accurately in practice, the state equation and energy law are usually used for the equilibrium states of thermodynamics systems and do not include the effects of diffusions.
2.2 2.2.1
The Newtonian systems Principle of friction
Variations in the kinetic state of a classical mechanic system are dominated by Newton’s laws together with the principles of mass and energy conservations. A dynamic system is referred to as the Newtonian system if only the variations of kinetic state are studied with Newton’s laws. The study of Newtonian system is carried out in a frame composed of time and space coordinates. The mass, time and distance are the basic variables in Newton’s laws. The simplest Newtonian system is a solid body, which may be viewed as a point mass at a given time and position in the coordinates. The external cause which leads to changes of the motion status is referred to as the external force. The force can be measured by or defined as the rate of momentum variation produced. The classical dynamics deals with the relations between the external forces (or inertial forces) and variations of motion status in the time-space coordinates. In the study of Newtonian system, variations of thermodynamic state are usually considered as a result of mechanic processes. Except in the energy conservation law, the effect of thermodynamic processes on changes of motion status is not discussed. To explain the conversion of kinetic energy to heat energy, Newtonian dynamics
2.2. THE NEWTONIAN SYSTEMS
25
introduced a special force called the frictional force, which is always in the direction opposite to the motion direction. According to Newton’s second law, the frictional force can be evaluated in terms of the deceleration produced. Since frictional force may only destroy kinetic energy and increase heat energy of a system, the resultant exchange between kinetic energy and heat energy is unidirectional. This may be regarded as the principle of friction. The Newton’s laws are time-dependent relationships. All the variables together with the external forces included are measurable on a macroscopic scale. When these relationships form a closed equation set, variations of Newtonian system may be predicted in principle for provided initial conditions. Since the prediction equations include implicitly the principle of friction which gives the restriction on the energy transfer, the solutions do not violate the second law of thermodynamics and the predicted processes may really happen under realistic conditions. Thus, we need not to examine the predictions additionally using the principle of friction. To see more clearly the restriction of friction on variations of Newtonian system, we introduce the concept of dynamic entropy in the next section. Here, we define the reversible and irreversible processes for Newtonian dynamics. A variation of Newtonian system without exchange of kinetic energy and heat energy caused by friction is a reversible process. A Newtonian system without friction may be called the perfect mechanic system. The change of mechanic energy in a reversible process equals the work done by external forces for the system. While, a process with kinetic energy dissipation caused by friction is irreversible. Since kinetic energy dissipation takes place in all Newtonian systems in the real would, we cannot design a permanent working system without energy exchange with environments.
2.2.2
Dynamic entropy
The processes in a frictional dynamic system are irreversible because the energy transfer produced is in the specific direction from kinetic energy to heat energy. We may, at least in principle, find a state function to account for the unidirectional energy conversion. This is discussed in the following. The Newton’s second law for a moving solid body of unit mass gives d2 r =F+F, d2 t where, r indicates the trajectory; t represents the time, and F and F are external and frictional forces, respectively. The external force is along the direction of trajectory, while the frictional force is always in the direction against the motion. Integrating the previous equation from r0 to r1 yields
v1 =
v02
+2
r1 r0
F · dr − WF ,
(2.1)
in which v denotes the moving velocity in the direction of external force, and WF = −
r1 r0
F · dr
(2.2)
2. TWO CLASSICAL PHYSICAL SYSTEMS
26
signifies the work done by frictional force along the trajectory. This relationship manifests the energy conservation in the process, that is k1 = k0 + WF − WF , r1
where WF =
r0
F · dr
(2.3)
is the work done by external force, and k=
v2 2
is the kinetic energy of unit mass called the specific kinetic energy. The energy conservation equation can be rewritten as ∆k = WF − WF .
(2.4)
The kinetic energy is generated by external force doing work and is destroyed by friction. Now, we define a state function as sd = ln
2
r r0
F · dr + v02 . v2
Its variation is calculated from ∆sd = ln
2
r1 r0
F · dr + v02 v12
as the solid body moves from r0 to r1 . Applying (2.1) yields
WF ∆sd = ln 1 − 2 v1
.
Since the frictional force is opposite to the motion direction, we have v · F ≤ 0 and WF ≥ 0. Therefore, the previous equation gives ∆sd ≥ 0 .
(2.5)
Variations of a frictional mechanic system are in the direction with increasing sd , and ∆sd = 0 occurs only in the reversible processes without friction. The state function sd gives an example of dynamic entropy. The existence of an entropy function in any physical system was proved generally by Giles (1964). Equation (2.5) is a mathematical expression of the principle of friction. When a solid body is pushed by a constant external force and moves from r0 to r1 , it will have velocity v1 at r1 . If it returns back from r1 against the same external force, the solid body will possess the speed less than |v0 | as it reaches r0 again under the effect of friction. The dissipated kinetic energy cannot be converted back into kinetic energy without thermodynamic processes. The processes with ∆sd > 0 are then irreversible.
2.3. SIMPLE THERMODYNAMIC SYSTEMS
27
For the simple mechanic system discussed previously, there is little advantage to introduce the dynamic entropy, as motions of a solid body are predictable principally. Introducing the dynamic entropy can neither add more understandings of the Newtonian dynamics, nor make it easier for us to predict the system variations. For a system composed of n independent Newtonian sub-systems, its dynamic entropy may be defined as the sum of dynamic entropy of each sub-systems, that is Sd =
n
Sdj =
j=1
n
mj ln
∆rj
Fj · drj + kj0 kj
j=1
,
where Fj is the external force acting on unit mass of sub-system j; kj is the kinetic energy density and mj is the mass. The dynamic entropy is additive, since it depends on the entropy of each sub-system but not a particular relation between these subsystems. Now, we assume an isotropic system, in which the work done by external force for each sub-system is identical, and the sub-systems have the same mass m and initial kinetic energy k0 . These features may only be statistically meaningful. The dynamic entropy of the isotropic system is evaluated from
Sd = M ln
∆r F
· dr + k0 . k
(2.6)
Here, M = nm is total mass of the system. This equation will be used to study the relation between the dynamic entropy and thermodynamic entropy in Chapter 4.
2.3 2.3.1
Simple thermodynamic systems Mole-number and molecular mass
The substances studied in the classical physics are composed of molecules. The mass of a molecule is referred to as the molecular mass in units of kg/molecule or g/molecule. It is different from the molecular weight ε = 16
m , mo
which is the ratio of molecular mass m to one sixteenth of the mass of an oxygen atom denoted by mo . In fact, this ratio is a number instead of weight. It will be called the mole-number in this study. Obviously, the mole-number of an oxygen atom is 16, or εo = 16. The molecular mass is then evaluated from m=
ε mo . εo
If the measurement of mass is accurate enough to the level of molecular mass, we may find that changes in the mass of a substance are discontinuous, and the features related to the mass may also change discontinuously. The continuous changes of physical variables are a preconceived idea obtained from old sciences dealing with the macroscopic world.
2. TWO CLASSICAL PHYSICAL SYSTEMS
28
The number of molecule in a substance with mass M is calculated from N= or
M , m
M εo . ε mo
N=
For a mole mass MO = ε(M ) ((M ) indicates unit mass such as 1 kg/mole in the mkgs system or 1 g/mole in the cmgs system), we have N=
εo (M ) . mo
It tells that a mole of any substance has an equal number of molecule. This number is called the Avogadro’s number, given by NA = 6.022 × 1023 (molecules/mole, or
NA = 6.022 × 1026 (molecules/mole,
in cmgs)
in mkgs) .
Now, a molecular mass may be written as m=
ε MO = (M ) . NA NA
This equation is useful as the mole-numbers instead of molecular masses can be found easily in the text books.
2.3.2
Thermodynamic variables
In a dissipation mechanic system discussed previously, kinetic energy is converted irreversibly into heat energy by frictional force. But why the winds in the Earth’s atmosphere never stop? From a mechanic point of view, one may say that there is conversion of geopotential energy of air mass into the kinetic energy caused by the gravitational force of the Earth and the pressure gradient force in the atmosphere. However, the conversion of geopotential energy is also irreversible in a mechanic system. Kinetic energy and geopotential energy can be created continuously in the ceaseless atmosphere, because the atmosphere is not only a mechanic system. When the geopotential energy is converted into kinetic energy in the atmosphere, a part of the heat energy is also converted into kinetic energy. The ratio of the heat energy and geopotential energy converted will be studied in Chapter 7. Usually, the Newtonian dynamics studies the changes of mechanic state, and does not discuss how the heat energy is converted into kinetic energy. This energy conversion is studied by the thermodynamic theories of equilibrium and nonequilibrium thermodynamic systems, which are examples of the non-Newtonian systems as their changes cannot be described by Newton’s laws on a macroscopic scale. A simple example of thermodynamic system is a small piece of gas in rest, which includes a number of molecules in random motions (called the Brownian motions also) toward different directions at different speeds. The difference in the speed is
2.3. SIMPLE THERMODYNAMIC SYSTEMS
29
important for remaining the random motions observed in an inertial frame. If the molecules move at an identical speed toward one direction, they may not collide with each other and look in rest from the inertial frame following the movement. The molecules in random motions possess translational kinetic energy. Usually, the atoms in a molecule are not tied up firmly or have their fixed positions relative to the molecular center, so the molecular collisions may cause rotations or vibrations of atoms and molecules in different directions and at various frequencies. Thus, the diatomic and polyatomic molecules may also possess rotational kinetic energy and vibrational kinetic energy at different levels. Moreover, to cause the molecular collisions and remain the fixed volume of a substance, the molecules must possess an attractive intermolecular force. However, the molecules are not pulled together by the attractive force, since there is also a repulsive force between molecules. It is clear now that the intermolecular forces are attractive at a long distance but repulsive at a short distance. These forces are also important for remaining the random molecular motions, against the inelastic collisions caused by transfers between the translational kinetic energy and rotational or vibrational kinetic energy. The different energy levels and motion status of microscopic particles define a microscopic state of the system, which may not be detected with macroscopic methods usually. However, some statistical features of the states may be indicated macroscopically by three basic measurable quantities: the pressure p, temperature T and specific volume α defined as the volume of unit mass. These quantities are called the thermodynamic variables, which define uniquely a macroscopic thermodynamic state of a gas with given mass and constituents in equilibrium. In general, two of the three variables are independent. Thus, we may write p = p(T, V ) ,
or
T = T (p, V ) ,
or
V = V (T, p) .
The changes of thermodynamic state are constrained on a p-α-T surface in the three-dimensional thermodynamic space. A macroscopic thermodynamic state is called the equilibrium state, if no changes in the thermodynamic variables can be found over a certain scale. However, fluctuations of these variables may still exist in a microscale. This implies that the identification of equilibrium state depends on the scale of observations or the size of instruments. Since the thermodynamic variables represent the statistical features of a microscopic state and the statistics depends on the number of samples, the measurements of thermodynamic variables depend also on the instrument size especially at a non-equilibrium state. For a small piece of gas in a laboratory, we assume explicitly or implicitly that, except the pressure force, stress tensors and intermolecular forces inside, the gas is free of external macroscopic Newtonian forces, such as the gravitational force and Coriolis inertial force on the rotating Earth. In this situation, the equilibrium thermodynamic state is independent of time and space (Referring to Chapter 17), and is assumed expediently or practically to be microscopically uniform too, so that the scale-dependent features of thermodynamic variables may be ignored.
2. TWO CLASSICAL PHYSICAL SYSTEMS
30
2.3.3
Pressure of monatomic gas
The pressure of gas is defined as the force asserting by the gas on a surface with unit area in the normal direction. When the external forces are ignored, the pressure in a fluid is equal at each point towards each direction in equilibrium. Thus, the pressure is defined as a scaler. It may not cause motions of mass in a fluid, unless the distribution is non-uniform. The pressure force can be calculated according to the changes of molecular momentum as the molecules encounter a surface of gas container. We consider here only the monatomic gases which possess the translational kinetic energy but not the rotational and vibrational kinetic energies. Let a square container of volume V in rest to be filled with a monatomic gas at a normal condition. If molecules of the gas are assumed as elastic balls with velocity (j = 1, 2, 3, · · · , N ) ,
ˆ + vyj y ˆ + vzj ˆz vj = vxj x
ˆ, y ˆ and z ˆ denote the unit space coordinates, we may derive the relation where x between the pressure and translational kinetic energy of molecules in an equilibrium state. The container has six square surfaces with the normal directions outward. If ˆ , the surface is labeled as +x-surface. When molecule the normal direction towards x j encounters elastically the +x-surface, the impulse on the molecule produced by the collisions within time ∆t may be represented by its momentum change along the x-coordinate, that is fj ∆t = cm(−vxj − vxj ) = −2cmvxj , in which fj is the impulsive force accepted by the molecule in the negative directive ˆ , and c is the number of collisions in ∆t. Since the distance in the x-direction of x covered by the moving molecule is vxj ∆t, the number of collisions is c=
∆t vx , 2l j
where l = V 1/3 is the length of the container. The force on the molecule reads fj = −
mvx2j l
.
According to Newton’s third law, the force acting by the molecule on the +x-surface is equal in magnitude to the force accepted by the molecule but in the opposite direction. If there are N molecules in the gas, the mean force produced by all the molecules on the +x-surface gives F = where vx2j =
N mvx2 , l N 1 v2 N j=1 xj
2.3. SIMPLE THERMODYNAMIC SYSTEMS
31
is the x-component of mean-square speed of molecules. Since N = nV , in which n is number density of the gas, we see F =
V nmvx2 = Anmvx2 , l
where A = l2 is the area of +x-surface. The pressure on the surface is defined as the force over a unit area, that is p = nmvx2 . In general, the rotation and vibration of diatomic and polyatomic molecules may also have an effect on the pressure, especially at very low temperatures or very high pressures. But this effect is ignored here for the study of atmospheric thermodynamics. In this situation, the pressure is proportional to the number density and translational kinetic energy of random molecular motions. The mean molecular velocity has an equal speed in all directions in an equilibrium state, that is v2 . vx2 = vy2 = vz2 = 3 Applying it for the previous equation yields p=
n mv 2 , 3
(2.7)
For a mole gas with volume V , the number density is n=
NA . V
We have now
2NA mv 2 . (2.8) 3 2 It tells that a mole of any gas in a container with fixed volume possesses the same pressure, if the mean molecular kinetic energy is the same. If the container contains I different gases, the pressure of a gas is called the partial pressure given by 2 pi = ni k¯i , 3 with mi 2 v k¯i = 2 in equilibrium. If the mean molecular kinetic energy is defined as pV =
I 1 ni k¯i , k¯ = n i=1
2. TWO CLASSICAL PHYSICAL SYSTEMS
32 the sum of all the partial pressures gives I
pi =
i=1
2 ¯ nk . 3
Comparing with (2.7) finds p=
I
pi .
(2.9)
i=1
This is the Dalton’s law, which tells that the pressure of a gas mixture in equilibrium is the sum of the partial pressures of each gas included. Equation (2.8) represents a state of the gas. It includes pressure and volume but not explicitly the temperature. As a thermodynamic state can be determined by the three independent thermodynamic variables, the temperature must be a function of the mean kinetic energy. This is discussed later on in the chapter.
2.4
The first law of thermodynamics
As discussed earlier, the molecules in a gas move irregularly and collide with each other. They may spin also in the monatomic gases, and rotate and vibrate in the diatomic and polyatomic gases. Since the molecules do not move in one direction at an equal speed, the translational kinetic energy is not identical to the macroscopic kinetic energy of parcel motion. The sum of the translational, rotational and vibrational kinetic energies of each molecule in a substance at rest may be defined as the heat energy. Meanwhile, the intermolecular forces may be represented by intermolecular potentials. The sum of the potential energy of each molecule gives the thermodynamic potential energy of substance, called also the thermopotential energy. The heat energy and thermopotential energy form the major part of internal energy U of substance. The internal energy may change as the substance does work at the cost of its heat energy or thermopotential energy. It may be changed also by energy exchanges with the surroundings. Without mass exchanges with the surroundings, the energy exchanges of a system with zero kinetic energy density are usually the heat exchanges resulting from radiation, heat conduction or latent heating denoted by dQ. So, the energy conservation law of a substance at rest gives dU + dW = dQ ,
(2.10)
where dW is the work done by the substance. This is also called the first law of thermodynamics. This relationship can be supported by physical experiments, such as the earliest experiments of Count Rumford (1798) and Joule (1850) to find the mechanic equivalent of heat. It will be seen in this chapter that this energy law can be derived theoretically from the kinetic theory of ideal gases. The energy law (2.10) is used for thermodynamic equilibrium states, which is the steady uniform state without mass displacement and macroscopic changes in thermodynamic variables. In a non-uniform gas however, there may be mass displacement across the gradient of number density called the mass diffusion. The
2.5. STATE EQUATION OF GASES
33
velocity of mass displacement caused by molecular diffusion is referred to as the diffusion velocity vd , which may be derived from the additive feature of momentum as illustrated in the next chapter. Thus, the molecular assemblies in a non-uniform gas possess the diffusion kinetic energy Kd =
1 2
vd2 dm ,
which changes with time and space. The integration with mass is used since the distribution of diffusion velocity is non-uniform at nonequilibrium states. The first law for nonequilibrium processes is then given by dU + dW + dKd = dQ .
(2.11)
There may be exchanges between the internal energy and mechanic work in an isolated system, for example, in adiabatic expansion and contraction of a gas. While, the transfer of diffusion kinetic energy to internal energy is unidirectional according to the feature of molecular diffusions discussed in the next chapter. Thus, all the diffusion kinetic energy will be converted into internal energy by molecular diffusions as the gas reaches an equilibrium state. The nonequilibrium thermodynamic energy law should be adopted in the nonequilibrium thermodynamics. However, since it is difficult to measure the diffusion velocity and calculate the diffusion kinetic energy, this law is not used in practice. The nonequilibrium energy law is nevertheless theoretically important, as conversion of the diffusion kinetic energy is irreversible in isolation. This feature can be applied to study the thermodynamics entropy law or the second law of thermodynamics in Chapter 4. The energy law for equilibrium states may also be applied for small air parcels in the atmosphere. As a moving parcel possesses kinetic energy and geopotential energy in the gravitational field, the first law of thermodynamics alone may not be enough to illustrate the energy conversions of the parcel. The equation of parcel kinetic energy and geopotential energy obtained from the Newton’s second law will be used also to derive, in Chapter 7, the energy equation of a moving parcel in the atmosphere.
2.5
State equation of gases
The relationship between three thermodynamic variables of a substance at equilibrium states is called the state equation. The exact expression of state equation for real gases, whether it can be used practically or not, has not been found yet. Many state equations of gases and other substances have been constructed theoretically or experimentally, such as the van der Waals equation, Clausius’ equation, BeattieBridgman equation and Dieterici equation (Sears, 1959; Burshtein, 1996). The best state equations of gases used currently are the virial equations, instead of the theoretical equations, obtained by curve-fitting using experimental data. Applications of these equations are given in Appendices A and B. On the theoretical studies then, different state equations have been forwarded by using various assumptions and approximations. The most well known examples are the ideal-gas equation and van der
2. TWO CLASSICAL PHYSICAL SYSTEMS
34
Waals equation. After development of statistical mechanics and quantum statistics, more complicated state equations were derived in terms of the intermolecular forces assumed. The ideal-gas equation and van der Waals equation may be considered as the approximations of these equations. The van der Waals equation of state will be derived in Chapter 6 using the partition function obtained from the maximum entropy principle in classical thermodynamics. Here, we discuss the general expression of the state equation using the thermodynamic energy law. Let a cylinder with a piston at one end to be filled by c mole of a single gas. As the gas expands slowly, it accepts heat from outside in order to keep a constant pressure. After the temperature increases from T0 to T and the volume increases from V0 to V , the heat absorbed in the isobaric process is c∆Qp = c∆Up + ∆Wp , derived from the first law of thermodynamics (2.10). Now, let the gas return back to the original state, and then we input heat to the gas again without moving the piston this time, until the temperature increases to T as before. No mechanic work is produced in the isochoric process, and the heat c∆Qv = c∆Uv is absorbed. Subtracting the isochoric process from the isobaric process yields c(∆Qp − ∆Qv ) = ∆Wp + c(∆Up − ∆Uv ) . Since ∆Up − ∆Uv = U (T, V ) − U (T0 , V0 ) − [U (T, V0 ) − U (T0 , V0 )] = U (T, V ) − U (T, V0 ) , we obtain c(dQp − dQv ) = dWp + c[U (T, V ) − U (T, V0 )] . for an infinitesimal variation. This equation tells that the heat absorbed in the isobaric process is different from that in the isochoric process, because dQp includes the mechanic work dWp and the internal energy difference between the two final states at the same temperature T . The latter is also the internal energy change as the gas changes its volume in an isothermal process. Provide that Cv is the molar heat capacity of a gas at constant volume defined as (2.12) dQv = Cv dT and Cp is the molar heat capacity of a gas at constant pressure given by dQp = Cp dT , the previous equation reads c(Cp − Cv )dT = dWp + c[U (T, V ) − U (T, V0 )] .
2.5. STATE EQUATION OF GASES
35
It shows that the difference between the isobaric and isochoric heat capacities is contributed by the mechanic work created in the isobaric process together with the change of internal energy in an isothermal process. If the piston moves dx in distance, the work done by the external force F is F dx = p0 Adx = p0 dV , where A is the cross-section area of the cylinder. Now, we have T
c T0
R∗ dT = p0
V V0
dV + c[U (T, V ) − U (T, V0 )] ,
(2.13)
in which we have used the Mayer formula R∗ = Cp − Cv .
(2.14)
When Cp and Cv are not constant, R∗ may also not be constant. Changes of the heat capacities and their difference may be very large across the critical point of a gas. The initial condition is represented by (T0 , p0 , V0 ). Since one thermodynamic variable depends on the other two, the initial state at pressure p0 is also represented by (T0 , V0 (p0 )). If the initial state is chosen at any pressure p, it may be given generally by (2.15) V0 = V0 (T0 , p) . Applying it to eliminate p0 in (2.13) gives T
∗
R dT = p
c T0
V V0 (p)
dV + c[U (T, V ) − U (T, V0 (p))] .
Integrating it by parts for the left-hand side yields ∗
∗
cR T = p(V − V0 ) + cR (T0 )T0 + c
T
T T0
where ∆ΠT = U (T, V ) − U (T, V0 ) =
dR∗ dT + c∆ΠT (V ) , dT V ∂U V0
∂V
(2.16)
dV
T
is the change of internal energy at constant temperature. If the heat energy is independent of volume at constant temperature, ∆ΠT gives the change of thermopotential energy. If the substance has volume b at the lowest temperature which can be set to zero for a temperature scale selected if without phase transitions in the process, we have cR∗ T = p(V − b) + c
T 0
T
dR∗ dT + c∆ΠT (V ) . dT
(2.17)
Here b may depend on pressure, and the difference of thermopotential energy is calculated between the volume V and volume b at temperature T . If the substance at the zero point has a hard surface on which the pressure is in equilibrium with the environments, the extremal volume b is constant. Equations (2.16) or (2.17) includes
2. TWO CLASSICAL PHYSICAL SYSTEMS
36
all the three thermodynamic variables which determine uniquely the thermodynamic state in equilibrium. They give the general expressions of the state equation of substance. The parameters R∗ and b and the thermopotential energy in (2.17) have clear physical meanings, though they may not be determined theoretically especially at the zero point. If the true intermolecular potentials can be given analytically, the thermopotential energy of dilute gases may be calculated with the approach of Zhu (2000, see Appendix A also). To provide the initial volume in (2.16) theoretically, we have to know the state equation (2.15) which is independent of (2.16). If we do not have the independent expression of (2.15), the initial volume may be considered as a parameter depending on pressure for provided T0 . The data can be provided by experiments. When a substance changes phase, the change of volume with temperature may be discontinuous, and so the state equation may give several expressions with different parameters for the different phases respectively. For most gases at normal conditions of the atmosphere, this state equation may be simplified to give the ideal-gas equation, with the assumptions discussed in the next section.
2.6 2.6.1
State equation of ideal gases Ideal-gas equation
The simplest state equation of gases was derived using the relationships obtained from physical experiments, and may be applied approximately for dilute gases at normal conditions. This equation is referred to as the ideal-gas equation, since the gases satisfying this equation are called the ideal gases. When the ideal-gas equation is derived from experimental data, the particular features of ideal gases are not made clear on theory. The assumptions of ideal gases proposed afterwards for the study on the kinetic theory of ideal gases were not used for deriving the state equation. The features of ideal gases different from those of real gases can be revealed by the following theoretical approach for deriving the ideal-gas equation. Since the change of internal energy at constant temperature is small at the atmospheric pressure and temperature (Appendix A), we have the first assumption: The internal energy of ideal gases is constant at constant temperature. This means U (T, V ) − U (T, V0 ) = 0 or Π=0. The ideal gases do not possess thermopotential energy. The state equation (2.17) gives T dR∗ dT . T cR∗ T = p(V − b) + c dT 0 The substance without thermopotential energy may not change phase, and so the volume or pressure may change continuously with temperature in all processes. The
2.6. STATE EQUATION OF IDEAL GASES
37
second assumption reads that R∗ is constant for ideal gases. The previous state equation becomes now cR∗ T = p(V − b) . The random molecular motions and inelastic collisions are induced and maintained by intermolecular forces, which are attractive in a long distance but repulsive in a short distance. When these forces are ignored for ideal gases, the third assumption comes as: Molecules of ideal gases are point masses without size and forces acting on each other. Thus, the volume of ideal gases at the zero point disappears. With the three assumptions, the state equation have the simplest form cR∗ T = pV ,
(2.18)
which is just the ideal-gas equation. Here, we do not assume that the pressure of ideal gases vanishes at the zero point. The ideal-gas equation is good enough for studying the meteorological-scale processes in the atmosphere. Since the mass of c mole gas is M = cMO , (2.18) is replaced by M ∗ R T = pV . MO When (2.14) is divided by a mole mass, we gain the Mayer formula R = cp − cv , where R=
R∗ MO
(2.19)
(2.20)
is the specific gas constant, and cp = Cp /MO and cv = Cv /MO are the specific heat capacities at constant pressure and constant volume respectively. We shall use them to represent the heat capacities of the dry air to study the atmospheric thermodynamic. The state equation of ideal gas gives now p = ρRT ,
(2.21)
p α = RT ,
(2.22)
or where ρ = M/V = 1/α measures the gas density. The units of α are different from those of V , so the units of R and R∗ are different as found by comparing (2.22) with (2.18). The units of R∗ is J/(mol.K) and of R is J/(kg.K). With the assumptions of ideal gases, the first law of thermodynamics may be represented in terms of thermodynamic variables. For example, (2.10) gives c(Cp − Cv )dT = p0 dV for c mole ideal gases in the isobaric processes with p = p0 . It tells that the difference between the isobaric and isochoric heat capacities of a substance without thermopotential energy is attributed simply to the mechanic work created in the isobaric process. Integrating the previous equation yields cR∗ (T − T0 ) = p0 (V − V0 ) .
2. TWO CLASSICAL PHYSICAL SYSTEMS
38
As the initial state may be represented generally by (2.15) as discussed earlier, the previous equation is replaced by cR∗ (T − T0 ) = p(V − V0 (p)) . Since we assumed that the volume disappears at the zero point, we gain immediately the ideal-gas equation (2.18). The state equations of real and ideal gases may be derived directly from the energy equations, so they have a clear relation to the energy laws used. For example, the ideal-gas equation (2.18) or (2.22) tells that as an ideal gas at the zero point with pressure p is heated to get temperature T in an isobaric process or isochoric process, the difference in the heat accepted by the gas in the two processes is equivalent to the mechanic work created in the isobaric process. For a real gas, the difference of thermopotential energy in the two final states with temperature T is added also. However, these state equations are independent of the energy equations, because we have used the independent relationship (2.15) which tells that the thermodynamic state of a substance in rest can be determined uniquely by three thermodynamic variables. Equation (2.15) is actually an implicit state equation. When it is used to diminish a variable in the energy equations, we are rewarded by the explicit state equations. Since the state equation (2.15) is unknown, the V0 in (2.16) is also unknown and so is provided by experiments or other assumptions. The ideal-gas equation derived for the particular processes can be applied for any process. For example, we have p1 T2 = p2 T1 in isochoric processes. This is the Gay-Lussac’s law which agrees with the physical experiments in the normal conditions. We may also have the Boyle’s law p1 V1 = p2 V2 for isothermal processes and the Charles’ law T1 V2 = T2 V1 for isobaric processes obtained from physical experiments. In fact, the ideal-gas equation was derived from these laws in earlier studies (e.g., Planck, 1922).
2.6.2
More features of ideal gases
The energy equation (2.10) gives dUv = dQv for an isochoric process with dW = 0. It follows, from (2.12), that dUv = Cv dT , or
∂U ∂T
= Cv . v
2.6. STATE EQUATION OF IDEAL GASES
39
As the internal energy of ideal gases depends explicitly on temperature only, we have ∂U ∂U = =0 ∂V T ∂p T for ideal gases. These equations give
∂Cv ∂V
= T
∂Cv ∂p
=0. T
The isochoric heat capacity is independent of volume and pressure at a constant temperature. Since R∗ is constant, (2.14) shows that the isobaric heat capacity Cp depends explicitly on temperature only. In general, these heat capacities may change with temperature. Since the changes are small in the atmosphere, they will be taken as constants in the study of atmospheric thermodynamic processes. Different ideal gases are distinguished only by their molecular masses or molenumbers. The Avogadro’s law for ideal gases states: Ideal gases with an equal volume possess an equal number of molecules at a provided pressure and temperature. According to this law, a mole of ideal gases a and b satisfy pb Vb pa Va = = R∗ , Ta Tb as they possess an equal number of molecules. It tells immediately that R∗ is identical for all ideal gases. This constant is then called the universal gas constant. The experimental volume gives R∗ ≈ 8.31 J/(mol.K). Reviewing the three assumptions used for the ideal gases finds that the gas assumed for the discussions of gas pressure in Section 2.4 is also an ideal gas. Comparing (2.18) with (2.8) reveals T =
2NA mv 2 3R∗ 2
for monatomic ideal gases in equilibrium. It is proportional to the translational kinetic energy of molecules in random motions. The gas constant of one molecule given by R∗ ζ= NA is also a constant, called the Boltzmann constant, of which the value is ζ = 1.381 × 10−23 J/(molecule.K). The temperature gives T =
mv 2 3ζ
now. Like the pressure, the temperature depends on the translational kinetic energy of molecules only, when the effect of molecular rotation and vibration are ignored. So, this expression may not be applied for the substances at very low temperatures or very high pressures. The pressure of the ideal gas in (2.7) reads p = nζT .
2. TWO CLASSICAL PHYSICAL SYSTEMS
40
This equation may be used to estimate the number density of a gas according to the pressure and temperature. Again, the thermodynamic variables derived from the kinetic theory represents statistically the macroscopic features of microscopic motion status of molecules. The temperature and pressure for a single molecule may be meaningless in physics. The thermodynamic states represented by the statistical variables are regarded to be continuous in space, and the discrete microscopic features are ignored. Since the mean values depend on the number of samples in general, the thermodynamic variables depend on the size of molecule ensemble or the element of volume in a non-uniform system. This scale-dependent feature of thermodynamic variables are not discussed fully in the previous studies, since only the uniform equilibrium states are studied by the classical thermodynamics, while the microscopic structures and fluctuations of the equilibrium states are ignored.
2.6.3
Kelvin temperature
The state equation tells that temperature of ideal gas is proportional to the volume at constant pressure, or proportional to the pressure at constant volume. It is assumed already that T = 0 K as V0 = 0. If we assume again that T1 = 273.15 K at the freezing point of water under the pressure of one atmosphere (atm), the temperature increases 1 K as an ideal gas which has no phase transitions expands each V1 ∆V = 273.15 where V1 is the volume at the freezing temperature. The temperature is then represented by V V (K) . (2.23) = T = 273.15(K) V1 ∆V It is found from experiments that the volume of one mole kilogram gas at the standard condition with T = 273.15 K and p = 1 atm or 101325 Pa is about V1 = 22.4m3 . So we have ∆V ≈ 8.2 × 10−5 m3 , and (2.23) gives T ≈ 1.2194 × 104
V K, m3
where V is the molar volume at 1 atm in units m3 . The temperature so defined is called the Kelvin temperature, and the temperature scale is called the Kelvin scale, since the temperature agrees with that defined by Kelvin according to the efficiency of reversible heat engines as discussed in Chapter 21. The Kelvin temperature may also be given by pressure change at constant volume. For this case, we have to assume that the pressure is zero as T = 0 K. The temperature may then be calculated from T = 273.15(K)
p p (K) , = p1 ∆p
where p1 is the pressure at the freezing temperature, and ∆p = p1 /273.15.
2.7. THERMODYNAMIC ENERGY LAW OF IDEAL GASES
2.6.4
41
Mixing ratio of water vapor
A variable used in meteorology is the mixing ratio of water vapor defined as the rate of the mass of water vapor Mv to the mass of dry air Md in the moist atmosphere, that is Mv . w= Md If the densities of water vapor and dry air are denoted by ρv and ρd respectively, we see ρv . w= ρd When the dry air and water vapor are assumed as ideal gases, we apply the ideal-gas equation (2.21), giving Rd p v , w= Rv pd where Rd and Rv are the specific gas constants of dry air and water vapor respectively, and pd and pv are the partial pressures of dry air and water vapor respectively. According to Dalton’s law (2.9), we have pd = p − pv , in which p is the pressure of moist air. Since pv p usually, we may use pd = p. Also, (2.20) gives Rd ≈ 287 J/(kg.K) and Rv ≈ 461.5 J/(kg.K). Thus, we gain w ≈ 0.622
pv . p
(2.24)
This equation is used frequently in meteorology.
2.7
Thermodynamic energy law of ideal gases
The first law of thermodynamics discussed previously is given directly by the principle of energy conservation. This law for ideal gases may also be derived from the kinetic theory of gases. According to the assumptions applied for ideal gases, a molecule of ideal gas may be considered as a solid body. If the N molecules in an ideal gas of unit mass possess the translational velocity ˆ + vyj y ˆ + vzj ˆz vj = vxj x
(j = 1, 2, 3, · · · , N ) ,
the translational kinetic energy of these molecules is given by Kj =
mj 2 2 2 (v + vyj + vzj ) + Krj + Kvj , 2 xj
in which mj indicates the mass of molecule j, and Krj and Kvj are the rotational and vibrational kinetic energy. The total amount of molecular kinetic energy gives k=
N
Kj .
j=1
This may not be the kinetic energy of gas motion, since the molecules move in different directions at different speeds. Provided that the barycenter of the gas
2. TWO CLASSICAL PHYSICAL SYSTEMS
42
moves at velocity vx , and the molecular velocity relative to the moving barycenter , we have in the x direction is vxj ˆ + vzj zˆ . )ˆ x + vyj y vj = (vx + vxj
Since
N
mj vxj =0,
j=1
there is
k = k∗ + kH ,
(2.25)
where
vx2 2 may be called the ‘ ordered kinetic energy’ or mass kinetic energy, and k∗ =
kH =
N mj j=1
2
2 2 2 (vxj + vyj + vzj ) + Krj + Kvj
may be regarded as the ‘ disordered kinetic energy’, which is also the heat energy or internal energy of ideal gases. The amount of mass kinetic energy depends on choice of the inertial coordinates, while the disordered kinetic energy does not. A thermometer reads the same temperature of a provided sample in any coordinates. When a macroscopic force acts on a gas towards the x direction, we may think that the force acts equally on each molecule in the gas. Thus, we have dkH − Fx dx = −dk∗
(2.26)
derived from (2.4) and (2.25). It is the conservation law of macroscopic mechanic energy of the gas. Although the total amount of molecular kinetic energy is conserved, the mass kinetic energy may be reduced by converting into heat energy. This process of energy conversion is called the molecular dissipation of mass kinetic energy. The ordered kinetic energy may still not be the kinetic energy of gas motion. The displacement of mass may take place inside a rest gas caused by microscale gradients in the pressure or density. The microscale mass displacement may be called the mass diffusion, which is distinguished with the molecular dissipation, as the former may still be an ordered process but on a scale much smaller than that of the gas, and so may cause the mass displacement towards a certain direction within the gas. The mass diffusion may not increase the heat energy immediately until the molecular dissipation takes place afterwards. For a gas at rest, the ordered kinetic energy is the kinetic energy of mass diffusion at diffusion velocity vd , called the diffusion kinetic energy, that is k∗ =
vd · vd = kd . 2
(2.27)
Now, we have dkH − Fx dx = −dkd .
(2.28)
2.7. THERMODYNAMIC ENERGY LAW OF IDEAL GASES
43
The right-hand side gives the dissipation of diffusion kinetic energy. From the theories of classical probability and statistical mechanics, molecular motions tend to be more disordered through collisions, since a more disordered motion status possesses a higher probability of occurrence. Without external influences, the ordered kinetic energy will eventually become disordered kinetic energy through molecular dissipation, and so dkd < 0. Although there are the pressure force and stress tensors inside a gas, this inequality holds nevertheless. This implies that the statistical effect of molecular dissipation and collision is independent of macroscopic Newtonian forces. This is an important difference between molecular diffusion and parcel motion. Equation (2.28) is similar to (2.4). The effect of friction or molecular dissipation always reduces the ordered kinetic energy and increases disordered kinetic energy. For an monatomic ideal gas which possesses translational kinetic energy only, we have 3 v2 = RT . (2.29) kH = 2 2 As the specific heat capacity at constant volume is given by (Reynolds, 1965) cv =
3 R 2
for monatomic ideal gases, we have dkH = cv dT .
(2.30)
It may also be applied for diatomic or polyatomic gases, when the cv includes rotational and vibrational heats. Moreover, the work done by an external force Fx for the gas in a container is represented by Fx dx = −p Adx = −pdα ,
(2.31)
where A denotes the section area perpendicular to dx, and p is the pressure force of the gas acting on surface A. Since the pressure force is identical in magnitude to the external force, but in the reversed direction, a negative sign is added. Now, we have the energy conservation law cv dT + pdα = −dkd for an isolated ideal gas at rest, which has no mass and energy (including heat energy) exchanges with the surroundings. The first term on the left-hand side gives the change of heat energy or internal energy, and the second term measures the mechanic work done by the gas as its volume changes. Adding the specific heat exchange dq between a gas and the exterior yields cv dT + pdα = dq − dkd .
(2.32)
It is the first law of thermodynamics in (2.11) for nonequilibrium processes of ideal gases, called the thermodynamic energy law of ideal gases in this study. With the ideal-gas equation (2.22), the energy law is also represented by cp dT − αdp = dq − dkd .
(2.33)
2. TWO CLASSICAL PHYSICAL SYSTEMS
44
This expression is less accurate than (2.32) when used for real gases. Since the ordered kinetic energy dissipated by molecular diffusion is converted into heat energy as soon as the system reaches an equilibrium state, the diffusion term may be omitted if used for equilibrium states and we have cv dT + pdα = dq
(2.34)
cp dT − αdp = dq .
(2.35)
or In an isolated system with dq = 0, we see cv dT + pdα = 0
(2.36)
or cp dT − αdp = 0 . The last two equations are also called the adiabatic equations of ideal gases. The second term on the left-hand side of (2.34) or (2.36) represents the mechanic work done by the gas against environmental pressure p at the cost of heat energy when αdp > 0. The negative work is created as the gas contracts under the environmental pressure. The mechanic work created by the environment for the gas is then converted into heat energy. Unlike the energy transfer produced by friction in a Newtonian system, exchanges between mechanic energy and heat energy may take place in both directions in a compressible thermodynamic system. Although there are the frictional force and viscous force in Earth’s atmosphere, the winds never die as the energy conversions take place all the time. The generation of kinetic energy in the atmosphere is a major subject of the current study.
2.8
Internal energy and heat exchange
The thermodynamic energy law of ideal gases may be represented by thermodynamic variables. This may not be the case for ideal gases. The internal energy in the energy equation includes the heat energy H and thermopotential energy Π, that is U =H +Π.
(2.37)
The thermopotential energy and heat energy may be converted into each other or to another energy, so the temperature may change in gas expansions even if the internal energy is conserved. The well known processes of gas expansions are the free expansion and Joule-Thomson expansion. The energy exchanges in these processes are discussed in Appendix B. The precise expression of the heat energy at all temperatures and pressures is not clear so far. For the ideal gases, the heat energy can be given theoretically in the quantum mechanics. A different approach in calculating the heat energy derived from the classical thermodynamics will be forwarded in Chapter 6. According to the study of Zhu (Zhu, 2000, referring to Appendix A also), the thermodynamic energy may be calculated according to the intermolecular forces. However, the
2.8. INTERNAL ENERGY AND HEAT EXCHANGE
45
exact expressions of intermolecular forces are unknown in general, so the forces are usually represented by hypothesis in theoretical studies. Thus, we have no a general expression of internal energy for all substances represented by thermodynamic variables. To use the energy law, we have to calculate the change of internal energy usually. The internal energy is a state function, as it is determined uniquely by the thermodynamic state in equilibrium. Since a thermodynamic state can be given by the three thermodynamic variables of which two are independent, the internal energy may be represented as a function of two thermodynamic variables. If we take T and V as the basic variable, the first law (2.10) is rewritten as ∂U ∂U dT + + p dV . dQ = ∂T v ∂V T Here only the mechanic work pdV = dW created as the system changes volume is considered. This work is positive if the gas expands against the environmental pressure, and is negative as the gas is compressed by the environment. For the isochoric processes of a mole substance, the first law gives
Cv =
∂U ∂T
. v
Inserting it into the first law again produces
dQ = Cv dT +
∂U ∂V
+ p dV .
(2.38)
T
Apply this equation for the isobaric processes yields
Cp dTp = Cv dTp + It follows that
∂U ∂V
∂U ∂V
+ p dVp . T
T
∂T = (Cp − Cv ) ∂V
−p .
(2.39)
p
Now, the change of internal energy is given by
∂T dU = Cv dT + (Cp − Cv ) ∂V
− p dV .
(2.40)
p
The change of internal energy may be integrated from an initial equilibrium state to another equilibrium state, if we know the heat capacities and state equation. We may also choose p and V as the basic variable, and have
dQ =
∂U ∂V
It gives
+ p dV + p
Cp dTp =
∂U ∂V
∂U ∂p
+ p dVp p
dp . v
2. TWO CLASSICAL PHYSICAL SYSTEMS
46 in the isobaric processes, and
Cv dTv =
∂U ∂p
dpv v
in the isochoric processes. Solving the last two equations yields
∂U ∂V
and
Thus, we gain
dU = Cp
= Cp p
∂U ∂p
∂T ∂V
= Cv v
∂T ∂p
−p p
. v
∂T ∂V
− p dV + Cv p
∂T ∂p
dp .
(2.41)
v
The change of internal energy may also be solved from this relationship using the state equation. Since the internal energy is a state function and so dU is a perfect differential, we have
∂Cv ∂V
= T
∂(Cp − Cv ) ∂T
v
∂T ∂V
∂p − ∂T p
∂ + (Cp − Cv ) ∂T v
∂T ∂V
(2.42) p v
derived from (2.40) and
∂Cp ∂p
v
∂T ∂V
+ Cp p
∂ ∂p
∂T ∂V
−1= p v
∂Cv ∂V
p
∂T ∂p
+ Cv v
∂ ∂V
∂T ∂p
v p
(2.43) from (2.41). These relationships can be applied to determine the parameters included in a state equation using the experimental data of heat capacities . Certainly, we may also choose T and p as the basic variables to calculate the internal energy. For a mole ideal gases of which the state equation is given by (2.18), we see
∂T ∂V
= p
p . R∗
We have discussed also that the heat capacities of ideal gases depend explicitly on temperature only. So (2.43) gives Cp − Cv = R∗ which is the mayer formula (2.14) used for deriving the state equation earlier. Now, we gain dU = Cv dT from (2.40). The internal energy is T
U = U0 +
Cv dT . T0
This result is consistent with the assumption used for deriving the ideal-gas equation, that the ideal gases possess no thermopotential energy.
2.9. POLYTROPIC PROCESS
47
Unlike the internal energy, the heat exchange depends on process as the work created is not a state function and depends on process also. The heat exchanges in the isobaric and isochoric processes may be evaluated by integrating the isobaric and isochoric heat capacities with temperature respectively. To calculate the heat exchange in isothermal processes of gases, we may define the molar heat capacity at constant temperature according to the volume change, that is dQT = CT dVT . Applying it for (2.38) yields
CT =
∂U ∂V
+p. T
It follows, from (2.39), that
CT = (Cp − Cv )
∂T ∂V
. p
Thus, the heat exchanges are evaluated from
V
∂T (Cp − Cv ) dQT = ∂V V0
dVT .
(2.44)
p
If we know the state equation, the heat exchange may be calculated conveniently. For ideal gases, (∂T /∂V )p = p/R∗ and we have V
dQT =
pdVT . V0
It tells that the heat accepted is converted into the mechanic work created, as the internal energy is unchanged in the isothermal processes of ideal gases. Applying the ideal-gas equation yields dQT = R∗ T ln
V . V0
(2.45)
As the real gases possess thermopotential energy, the heat accepted is converted also to the thermopotential energy. However, if we have the state equation, the heat exchanges in the isothermal processes may be estimated without knowing the change of internal energy or thermopotential energy.
2.9
Polytropic process
Most of the thermodynamic processes in the real world may not be at constant temperature or pressure or volume. The heat capacity in an arbitrary process is called the polytropic heat capacity. The molar polytropic heat capacity may be defined as the heat absorbed by a mole substance, as its temperature increases a unit degree, that is dQ = Cπ dT .
2. TWO CLASSICAL PHYSICAL SYSTEMS
48
The energy equation (2.38) for the polytropic processes gives
Cπ dT = Cv dT +
∂U ∂V
+ p dV . T
It follows, from (2.39), that
∂T ∂V
Cπ dT = Cv dT + (Cp − Cv )
or Cπ = Cv + (Cp − Cv )
∂T ∂V
p
dV dT
dV , p
p=p(T )
,
(2.46)
where p(T ) is the pressure in the process represented as a function of temperature. For example, if the temperature and pressure change with time, that is T = T (t),
p = p(t) ,
as recorded in an experiment, we may get the reverse function t = t(T ) on theory and so the pressure can be given according to the temperature change in the process. For isochoric and isobaric processes, (2.46) gives Cπ = Cv and Cπ = Cp respectively. For isothermal processes, (2.46) gives Cπ = ∞ as dT = 0. In adiabatic processes, we have dQ = 0 and so Cπ = 0 or
Cv + (Cp − Cv )
∂T ∂V
p
dV dT
=0. p=pa (T )
where pa represents the pressure in an adiabatic process. It follows that
Cp = Cv 1 −
∂V ∂T
p
dT dV
p=pa (V )
.
It gives the mayer formula (2.14) for the ideal gases with constant heat capacities. For ideal gases we may use the energy equations (2.35) and (2.34) giving (cp − cπ )dT − αdp = 0 ,
(2.47)
and (cv − cπ )dT + pdα = 0 , respectively, where cπ is the specific heat capacity of polytropic process. These two relationships give dp cp − cπ dα + =0. p cv − cπ α If all the heat capacities cp cv and cπ are constant, this equation is integrated to gain (2.48) pαγ = p0 αγ0 , where γ=
cp − cπ cv − cπ
2.9. POLYTROPIC PROCESS
49
is called the polytropic index. Applying the ideal-gas equation (2.22) for (2.48) yields
p = p0
or α = α0
µ
T T0
T0 T
(2.49)
,
(2.50)
where µ and - are obtained from cπ = cp − µR ,
(2.51)
cπ = cv − -R .
(2.52)
or Now, the heat exchange may be given by dq = (cp − µR)dT , or dq = (cv − -R)dT . We have µ = 0 or - = −1 in isobaric processes, µ = 1 or - = 0 in isochoric processes, µ = - = −∞ in isothermal processes, and µ = cp /R or - = cv /R in adiabatic processes. In the adiabatic processes with cπ = 0, (2.48), (2.49) and (2.50) give pα
cp cv
cp cv
= p0 α0 ,
p = p0 and
T T0
α = α0
cp
T0 T
R
,
cv R
(2.53)
respectively. These equations show that none of the three thermodynamic variables may remain constant in the adiabatic processes of ideal gases. This is different from the polytropic processes. Equation (2.49) may be rewritten as
θπ = T0 = T
p0 p
R cp −cπ
.
(2.54)
The θπ is the temperature at an initial state. It may also be viewed as the polytropic potential temperature with respect to the reference pressure p0 . Since we have dθπ = 0 in a polytropic process, the polytropic potential temperature is conserved in the process. The potential temperature used in meteorology, given by
θ=T
pθ p
κ
,
κ=
R cp
(2.55)
50
2. TWO CLASSICAL PHYSICAL SYSTEMS
is a particular example of the polytropic potential temperature in the adiabatic processes without phase transitions of air. It is the temperature as the parcel pressure changes reversibly and adiabatically to the reference pressure pθ , taken as 1000 hPa usually. This potential temperature is conserved in reversible adiabatic processes with cπ = 0, but not in polytropic processes when cπ = 0. The polytropic processes and polytropic potential temperature will be applied in Chapter 23 for the study of mixing processes in convective processes.
Chapter 3 Molecular transport processes 3.1
Introduction
From the statistical mechanics or kinetic theory of gases, some microscopic features of gases, such as the mean moving speed of molecules, may be represented statistically by macroscopic thermodynamic variables. Equation (2.29) shows that the local mean molecular speed is inhomogeneous in a non-uniform system. The anisotropic molecular motions and molecular collisions may cause transport of mass, momentum and heat across the gradients in related fields, and produce local changes of thermodynamic variables at nonequilibrium states. The transport processes, called the molecular diffusions, are the major causes of thermodynamic variations in a non-uniform system. In the nonequilibrium thermodynamics which deals with the nonequilibrium processes in a classical thermodynamic system, the molecular diffusions are studied by introducing the hypothesis of linear law. However, the linear law fails to give the reasonable predictions for the transport properties. Thus, different theories have been developed to study the transport processes in non-uniform gases. In the classical transport theory for uniform gases, the relation between the coefficient of viscosity η and the coefficient of self-diffusion D assumed D = η/ρ. However, the experimental data showed that the ratio ρD/η was generally in the range between 1.2 and 1.6 for gases at a normal temperature and pressure (Jeans, 1967; Chapman and Cowling, 1970). Also, the classical theory predicted that heat conductivity of ideal gases could be represented by λ = cv η. While the Eucken ratio λ/(cv η) obtained from experiments varied from 1.4 to 2.6 at a normal condition (Jeans, 1967; Kennard, 1938; Sears and Salinger, 1975). It was believed that these large errors were produced by the uniform gas assumption. In fact, the gases studied by the classical theories are not uniform, since there may be gradients in the mean molecular speed and thermodynamic state of a simple gas studied. Several approaches were made either to provide corrections (Eucken, 1913), or to create different theories by solving the Boltzmann conservation equation (Chapman and Cowling, 1970, Wang Chang et al., 1964), the nonequilibrium dynamic equations (Evans and Morriss, 1990) or the nonequilibrium transfer equations (Woods, 1993). The derived algorithms are generally complex and not all the parameters included can be determined theoretically, so that the theoretical calculations of the transport coefficients with good results may be practically available only for dilute noble gases. Most theoretical treatments used the intermolecular potential assumed for hard sphere, such as the Lennard-Jones potential (Jones, 1924), which could be inappropriate for diatomic and polyatomic molecules. The calculations for the dense and moderately dense gases were made mainly by the modified Enskog theory. The results were also not good enough in general (Hanley et al., 1972; Michels and Trappeniers, 1980). The limitations of the classical theories may be referred to the studies in a book of Millat et al., (1996). 51
52
3. MOLECULAR TRANSPORT PROCESSES
Many statistical features of gases may be studied with the classical thermodynamics or kinetic theory of gases without considering the intermolecular forces. The transport processes are also a kind of statistical effect resulting from random molecular motions at nonequilibrium states. Thus, they are studied in this chapter with the diffusion velocity and partial velocities introduced by Zhu (1999) according to the additive feature of momentum. The transport properties derived from this theory show better agreements with experiments, compared with the previous theories which are much more complicated. The effect of inelastic collisions will be represented by introducing the collisional heat capacity, which can be evaluated easily in terms of molecular structure. This study provides an additional approach for estimating or predicting the transport properties, especially when the information of intermolecular forces is not available. It will be found that the diffusion velocity depends on the gradients of thermodynamic variables, and the transport carried by diffusion velocity is in the downgradient directions and destroys the gradients eventually. As the diffusions disappear in a uniform state, an isolated system in equilibrium cannot establish the gradients again by the diffusions, and so the diffusion processes are irreversible. This feature in heat conduction agrees with the Clausius statement of the second law. This means that the second law for the classical thermodynamic systems may be obtained from the current theory of molecular diffusions. Although the molecular diffusions may not be important for the meteorological processes in the atmosphere within a few days compared with the parcel motions, they are substantial for the irreversible processes studied in the classical thermodynamics.
3.2 3.2.1
Diffusion velocity and partial velocities Diffusion element and diffusion velocity
The mean distance between two molecules in dilute gases is much larger than the size of a molecule (about 10 times of the diameter of a molecule) at a normal temperature and pressure. Also, the mean speed of a molecule in a gas is very large, about 102−3 m/s. The molecules in random motions possess different velocities, which keep changing as the molecules collide with each other. The mean number of collisions for a hydrogen molecule is about 1010 per second at 0◦ C and 1 atmosphere. Consider an element of volume in a gas, which contains a large number of molecules, but is small enough so that the molecules in it may move across the boundaries within a short time. This small volume is referred to as a diffusion element. The random molecular motions in the element may cause transport of mass, heat and momentum at a nonequilibrium state. Although molecules in a diffusion element keep exchanging with the exterior, an element possesses certain amounts of mass, momentum and energy at a given time. We may still evaluate the mass density of a diffusion element. The momentum of a diffusion element may be defined by introducing the instantaneous diffusion velocity vd . If the diffusion element has instantaneous mass M , the instantaneous momentum is M vd . When a substance is broken down into pieces
3.2. DIFFUSION VELOCITY AND PARTIAL VELOCITIES
53
by internal forces, such as in explosion of a bomb, the momentum of the substance may be given by summing up the momentum of each piece. This means that the momentum is additive. Thus, the momentum of a diffusion element may be given by the sum of molecular momentum, that is
vd
N
mj =
j=1
N
mj vj ,
j=1
where vj is the velocity of molecule j which possesses mass mj . This equation gives N j=1
vd = N
j=1
mj vj mj
.
The diffusion velocity is also called the mass average velocity (Hirschfelder et al., 1964). The momentum of a bomb is unchanged by explosion, if no external forces act on the bomb or its pieces. Also, the molecules may change their velocities through collisions, but the momentum of a diffusion element is conserved in the collisions. In other words, the momentum is not changed by intermolecular forces and elastic or inelastic molecular collisions in the element. However, the momentum of diffusion element may be changed by external forces acting on the element, or by exchanging molecules inside and outside the element. The diffusion velocity changes as the momentum changes. The word of instantaneous emphasizes the quick changes in both the direction and speed of diffusion velocity. As molecules in a small diffusion element exchange quickly with those in the surroundings, a definition of trajectory for the diffusion element along with the diffusion velocity may not be physically significant. In general, the diffusion velocity depends on the gradients of thermodynamic variables. A non-zero diffusion velocity does not mean necessarily that the diffusion element moves at the velocity. When the mass center moves inside a non-uniform element, the element itself may not move. Also, the diffusion velocity is different from the velocity of a solid body, since its variation may not have a clear relation to external Newtonian forces in a macroscopic scale including the pressure gradient force. The diffusions may also take place at constant pressure, if we put together two pieces of a gas with different temperatures at a given pressure. Although the molecules in each piece move randomly at an equal mean speed towards all directions, the net effect of the random motions is the mass transport from high density to low density, because the molecules across the boundary to the side of low density are more than those to the side of high density in a unit time. This mass displacement caused by molecular diffusion may also take place if without the external forces such as the gravitational force and pressure gradient force, and is different from the mass movement driven by these forces.
3. MOLECULAR TRANSPORT PROCESSES
54
T1
T2
v+x1 ✲
v−x 1 ✛
✛
✲v+x2
v−x 2
p1
p2 ✲ x
Figure 3.1: Diffusion element and partial velocities (after Zhu, 1999)
3.2.2
Partial velocities
The x-component of diffusion velocity derived from the previous equation is given by N+x N−x j=1 m+xj v+xj + j=1 m−xj v−xj vdx = , N j=1 mj in which the signs + and − indicate the variables related to molecular motions in the +x and −x directions respectively. The two terms on the right-hand side will be called the partial velocities which are mass weighted mean velocities. If the molecules in the element are alike, the partial velocities are represented by = v+x
N+x 1 v+xj N j=1
(3.1)
N−x 1 v−xj . N j=1
(3.2)
and = v−x
They may be rewritten as = v+x
N+x v¯+x N
and
N−x v¯−x , N where v¯ denotes mean molecular velocity. Now, the diffusion velocity along xcoordinate reads 1 vdx = (N+x v¯+x + N−x v¯−x ) . N Since v−x (3.3) v¯+x = −¯ = v−x
and N+x = N−x at an equilibrium state, the diffusion velocity is zero and so no mass diffusion takes place in equilibrium. But the diffusion velocity may not vanish at a nonequilibrium state. To study the diffusions at a nonequilibrium state, we assume that there are two uniform pieces of a simple gas standing side by side as shown in Fig.3.1. One has
3.2. DIFFUSION VELOCITY AND PARTIAL VELOCITIES
55
temperatures T1 and pressure p1 , and the other has temperature T2 and pressure p2 . There is no fixed wall between the two pieces. Now we choose a diffusion element over the discontinuous boundary as shown by the small light box in the figure. If in the element there are N+x1 molecules of pieces 1 which have velocity components in the +x direction, and N−x2 molecules of piece 2 which have velocity components in the −x direction, the partial velocities at the boundary are given by = v+x 1
and v−x = 2
N+x1 v¯+x1 N+x1 + N−x1 N−x2 v¯−x1 , N+x1 + N−x1
derived from (3.1) and (3.2). They are related to the motions of different molecules in the element, but not the velocity components of a mass. Their contribution to the diffusion depends not only on mean molecular velocity, but also on the portion of mass. Since each piece in the element is in equilibrium at the very beginning, we have N+x1 = N−x1 ∼
N1 2
and
N2 , 2 where N1 and N2 are the numbers of molecules in the two pieces respectively. Applying these relationships yields N+x2 = N−x2 ∼
N1 v¯+x1 N1 + N2
= v+x 1
and = v−x 2
N2 v¯−x2 . N1 + N2
= −v+x or Moreover, we may use (3.3) to gain v−x 2 2 =− v−x 2
N2 v¯+x2 . N1 + N2
The mean molecular speed included may be assumed as the arithmetic mean speed derived from Maxwellian distribution function at an equilibrium state, that is (Sears and Salinger, 1975) 8ζT , (3.4) v¯2 = πm where ζ is Boltzmann constant. Since R∗ ζ = =R, m NA m where NA is Avogadro’s number; R∗ is the universal gas constant and R is the specific gas constant, we have 8 v¯2 = RT . π
3. MOLECULAR TRANSPORT PROCESSES
56
According to the principle of energy equipartition, the distribution of molecular kinetic energy is identical in each direction at an equilibrium state, that is v¯x2 = v¯y2 = v¯z2 =
8 v¯2 = RT . 3 3π
Now, the partial velocities give √ N1 RT1 =c N1 + N2
(3.5)
√ cN2 RT2 = −c , N1 + N2
(3.6)
v+x 1
and v−x 2
where c = 2(2/3π)1/2 ≈ 0.92. The ratios of number densities included in partial velocities provide the weights of the fluxes carried by partial velocities, and are similar to the Maxwellian-Boltzmann distribution function in the Boltzmann equation (Chapman and Cowling, 1970). The number of molecules is evaluated from N = nV , where V is the volume of gas, and n is the number density of molecules which is evaluated from p (3.7) n= ζT for ideal gases. If the two pieces of gas in the element have an equal volume, we apply pV N= ζT for (3.5) and (3.6) giving =c v+x 1
and v−x 2
√ p1 T2 RT1 p1 T2 + p2 T1
√ p2 T1 RT2 = −c p1 T2 + p2 T1
(3.8)
(3.9)
respectively. The partial velocities have different speeds when the two pieces are not in thermodynamic equilibrium.
3.2.3
Diffusion velocity in non-uniform ideal gases
The local diffusion velocity at the boundary is the sum of the partial velocities, that is + v−x . vdx = v+x 1 2 It follows, from (3.8) and (3.9), that √
c RT1 T2 p1 T2 − p2 T1 . vdx = p1 T2 + p2 T1
(3.10)
3.2. DIFFUSION VELOCITY AND PARTIAL VELOCITIES
57
The diffusion velocity of gases does not vanish if the two pieces are in different thermodynamic states, and may cause transport across the boundary. These partial velocities are sketched in Fig.3.1. The last equation gives, from ideal-gas equation (2.21), that √ √ ρ1 p1 − ρ2 p2 vdx = c (3.11) ρ1 + ρ2 or √ √ √ ρ1 T1 − ρ2 T2 (3.12) vdx = c R ρ1 + ρ2 respectively. For the molecular diffusion at constant pressure, (3.10) gives √
RT1 T2 T2 − T1 . vdx |p = c T1 + T2
(3.13)
If T1 < T2 , we have vdx > 0. The mass moves from the cold side to the warm side, because the density on the cold side is higher than on the warm side. This can be seen more clearly if (3.11) is rewritten as √ √ √ ρ1 − ρ2 . vdx |p = c p ρ1 + ρ2 If the thermodynamic variables vary continuously in the system, the previous two equations may be replaced by
c vdx |p = 4 and vdx |p = −
R ∂T δx T ∂x
c √ ∂ρ δx , RT 4ρ ∂x
where T = (T1 + T2 )/2 is the mean temperature; ρ = (ρ1 + ρ2 )/2 is the mean density and T1 T2 ≈ T 2 . In the classical kinetic theory of gases, δx is represented by the average collision distance in a given direction (Holman, 1980), that is 2 δx = l , 3
(3.14)
where l is called the mean free path of molecules. Now, we have
c vdx |p = l 6 or vdx |p = −
R ∂T T ∂x
c √ ∂ρ l RT . 6ρ ∂x
(3.15)
(3.16)
If the diffusion is caused by pressure gradient only, and the initial temperatures on the two sides are same, we have, from (3.10), √ p1 − p2 vdx |T = c RT p1 + p2
3. MOLECULAR TRANSPORT PROCESSES
58 or
cl √ ∂p , RT 3p ∂x
vdx |T = −
(3.17)
where p = (p1 + p2 )/2 is the mean pressure. The diffusion velocity is from high pressure to low pressure in an isothermal system. We see also, from (3.12), that √ ρ1 − ρ2 vdx |T = c RT ρ1 + ρ2 or vdx |T = −
cl √ ∂ρ . RT 3ρ ∂x
It is also from high density to low density. The speed is higher than at constant pressure compared with (3.16), because the diffusion is accelerated by pressure gradient force. The pressure gradient force will be discussed in Chapter 5. It is noted that the diffusion velocity may not vanish if mass densities on the two sides are identical, since (3.11) and (3.12) give c √ √ vdx |ρ = √ ( p1 − p2 ) 2 ρ and
c√ R( T1 − T2 ) , 2 respectively. As long as the system is inhomogeneous, there may be mass diffusion and so vd = 0. If thermodynamic variables of the system vary continuously, the last two equations may be replaced by
vdx |ρ =
vdx = −
cl √ ∂p RT 6p ∂x
and (3.15). This diffusion velocity is comparable with that in an isobaric system, and is a half of that in an isothermal system as shown by comparing with (3.17).
3.3 3.3.1
Self-diffusion of ideal gases Diffusive mass flux
The statistical effect of random molecular motions and collisions may produce a net transport of mass towards a certain direction in a nonequilibrium system. The diffusive mass flux in the x direction across a surface with unit area normal to the flux direction may be evaluated from the two-flux approach: + ρ2 v−x2 . Fρx = ρ1 v+x1
For the example discussed earlier, we apply (3.8) and (3.9) and gain 3
Fρx
3
p21 T22 − p22 T12 , =c √ RT1 T2 (p1 T2 + p2 T1 )
(3.18)
3.3. SELF-DIFFUSION OF IDEAL GASES
59
where ideal-gas equation (2.21) is used also. This equation can be rewritten, with the ideal-gas equation again, as √
c R 2 Fρx = ρ1 T1 − ρ22 T2 (3.19) ρ1 + ρ2
or
3√ 3√ c ρ12 p1 − ρ22 p2 . (3.20) Fρx = ρ1 + ρ2 The mass diffusion depends on the gradients of all thermodynamic variables. If the system has constant pressure p, the mass diffusion across the boundary gives 3 3 cp 2 2 T2 − T1 (3.21) Fρx |p = √ RT1 T2 (T2 + T1 )
derived from (3.18), or
√ 3 3 c p ρ12 − ρ22 Fρx |p = ρ1 + ρ2
(3.22)
from (3.20). The diffusion in a system without pressure gradient is from the cold side to the warm side or from high density to low density, and is in the same direction as the diffusion velocity. Equation (3.21) shows that the diffusion may also be represented by the gradient of temperature. If the system has constant temperature T , we gain
derived from (3.18), or
p1 − p2 Fρx |T = c √ RT
(3.23)
√ Fρx |T = c RT (ρ1 − ρ2 )
(3.24)
from (3.19). The mass transport is from high pressure to low pressure or from high density to low density. The mass diffusion may also take place in a system with constant density. From (3.20) and (3.19), we have Fρx |ρ =
c√ √ √ ρ ( p1 − p2 ) 2
(3.25)
and
c √ T1 − T2 , (3.26) Fρx |ρ = ρ R 2 respectively. This diffusion in an isochoric system may be called the thermal diffusion, as it results from the temperature or pressure gradient instead of the density gradient and results in changes in the temperature and pressure instead of density distributions. It is along the downgradient of temperature or from high pressure to low pressure. Since 3
3
T22 − T12 =
T2 −
T1
T2 +
T1 T2 + T1
,
we see by comparing with (3.21) that the thermal diffusion is weaker than the diffusion caused by a density gradient, when the temperature gradient is the same as that in the isobaric system.
3. MOLECULAR TRANSPORT PROCESSES
60
Since the mass transport takes place from high density to low density and depends essentially on the gradient of mass density, the diffusion is weakened by the mass displacement produced and disappears eventually as the system reaches an equilibrium state. The temperature and pressure gradients may also not exist after the diffusion vanishes, since it may be eliminated by thermal diffusion afterwards at constant density. All the diffusion kinetic energy is dissipated into heat energy at the uniform equilibrium state. As the gradients cannot be produced again by molecular motions in equilibrium, the mass diffusion, thermal diffusion and kinetic energy dissipation are irreversible in an isolated system. It can be found in the following sections that the heat and momentum transport resulting from molecular diffusions including the effect of molecular collisions are also irreversible in isolation. The downgradient transport implies that occurrence of a more uniform state possesses a higher probability than that of an ordered distribution. The irreversible energy conversion or increase of randomness is a fundamental feature of molecular diffusions, which can be applied to derive the second law of thermodynamics as shown in the next chapter.
3.3.2
Coefficient of self-diffusion
When the thermodynamic variables vary continuously in the system, the differences of thermodynamic variables may be replaced by differentials. The diffusive mass flux at constant pressure, given by (3.22), is then rewritten as 3 √ ∂ρ δx . Fρx |p = − c RT 4 ∂x If δx is replaced by the average collision distance in (3.14), this equation reads Fρx |p = −D|p
∂ρ , ∂x
(3.27)
where
2RT (3.28) 3π is the coefficient of self-diffusion for non-uniform ideal gases at constant pressure. Similarly, the diffusive mass flux in an isothermal system shows, from (3.24), D|p = l
Fρx |T = −D|T where
∂ρ , ∂x
4l 2RT (3.29) D|T = 3 3π is the coefficient of self-diffusion at constant temperature. It is larger than the diffusion coefficient at constant pressure since D|T = 4D|p /3. We may also define the coefficient of thermal diffusion for an isochoric system, as mass diffusion may also take place if the pressure or temperature is not uniform. The diffusion equation (3.26) may be replaced by Fρx |ρ = −D|ρ
∂T , ∂x
3.4. VISCOSITY OF IDEAL GASES
61
where lρ D|ρ = 3
2R 3πT
gives the coefficient of thermal diffusion. The thermal diffusion is from warm place to cold place, and causes fluctuation in the system until the system gradients are destroyed or an equilibrium state is attained.
3.4
Viscosity of ideal gases
3.4.1
Diffusive momentum flux
Molecular diffusion may also cause momentum transport in the system. If the system shown in Fig.3.1 has mass speed w in the z direction, the diffusive momentum flux in the x direction across a surface with unit area normal to the flux direction may also be evaluated from the two-flux approach: + ρ2 w2 v−x2 . Fwx = ρ1 w1 vx1
For the same element assumed previously, we apply (3.8) and (3.9) for it, giving Fwd = √
cT1 T2 −3 −3 p21 T1 2 w1 − p22 T2 2 w2 R(p1 T2 + p2 T1 )
.
This equation is equivalent to Fwd
√
c R 2 = ρ1 w1 T1 − ρ22 w2 T2 ρ1 + ρ2
or Fwd
3 3 c √ √ = ρ12 w1 p1 − ρ22 w2 p2 ρ1 + ρ2
derived by using the state equation (2.21). The last three equations give
cpT1 T2 −3 −3 T1 2 w1 − T2 2 w2 Fwd |p = √ R(T2 + T1 ) or
at constant pressure,
or
√ 3 3 c p 2 2 ρ1 w1 − ρ2 w2 Fwd |p = ρ1 + ρ2 p2 w1 − p22 w2 Fwd |T = c √ 1 RT (p1 + p2 ) √ c RT 2 ρ1 w1 − ρ22 w2 Fwd |T = ρ1 + ρ2
at constant temperature, and Fwd |ρ =
c √ ρ R w1 T1 − w2 T2 2
3. MOLECULAR TRANSPORT PROCESSES
62 or
c√ √ √ ρ (w1 p1 − w2 p2 ) 2 at constant density. These relationships show that momentum diffusion depends on the gradient of mass velocity as well as the gradients of thermodynamic variables. If without the shear of mass velocity, momentum transport may still be produced by diffusive mass transport, as we have Fwd = wFρd . Fwd |ρ =
3.4.2
Momentum conduction
Molecular collisions may cause exchanges of molecular momentum and kinetic energy. In a non-uniform system, molecules on the cold side increase their kinetic energy after collisions, while molecules on the warm side lose kinetic energy. In the assumption of elastic one-dimensional (or head-on) collisions, momentum and kinetic energy of two colliding molecules are exchanged. The processes of momentum and energy transfer resulting from the collisions are called the momentum and heat conductions respectively. To study the momentum and heat conductions, we assume 1) Only binary collisions take place in the system, and 2) the collisions are elastic. As a matter of fact, the collisions are not elastic at least for diatomic and polyatomic molecules, as there may be exchanges between translational kinetic energy and rotational or vibrational kinetic energy of molecules. The correction for inelastic collisions will be made by introducing collisional heat capacity in a later discussion. The conductive momentum flux across the yp-plane with unit area evaluated from the two-flux approach gives + w2 v−x ), Fwc = nc m(w1 v+x 1 2
where nc is the number density of the molecules which collide with other molecules within a mean free path, and the partial velocities are the weighted mean velocities averaged over the molecules in collisions. In the assumption of binary collisions, the numbers of molecules which come from each side of the boundary and collide near the boundary are equal. Thus, we assume N1 = N2 in (3.5) and (3.6), and insert the results into the previous momentum conduction equation yield √
c (3.30) Fwc = nc m R w1 T1 − w2 T2 2 for our previous example. If the momentum conduction is produced by molecular collisions only without the effect of mass diffusion in a thermodynamically equilibrium system, that is T1 = T2 = T , nc may be replaced by n except at a very low pressure. Applying (3.7), we have c √ Fwc = ρ RT (w1 − w2 ) 2 where ρ = nm may be represented by the mean density of the system. Equation (3.10) shows that the diffusion velocity vanishes in equilibrium. However, molecular momentum may still be transferred by collisions. This is similar to the energy transport in a mechanic wave, as the mass is not delivered in the process of energy propagation. However, the physical mechanism of momentum or heat conduction is different from the wave energy propagation in the classical kinetic theory.
3.4. VISCOSITY OF IDEAL GASES
3.4.3
63
Coefficient of viscosity
Except at a very low pressure, (3.30) is rewritten as
c √ Fwc = ρ R w1 T1 − w2 T2 . 2 If the mass speeds and temperatures are compared over the average collision distance δx, this equation gives √ w ∂T c √ ∂w + √ Fwc = − ρ R T . 3 ∂x 2 T ∂x The momentum flux depends on temperature gradient. According to Newton’s second law, the change rate of momentum is a measure of the external force which causes the change. The external force produced by the momentum diffusion or conduction in a fluid system is called the viscosity or viscous force. The viscosity on an element of unit volume is represented by ∂Fw . (3.31) ∂x It tells that the moving speed of a fluid element is reduced in the place with divergence of momentum flux, and is increased in the place with the convergence. If the temperature gradient is ignored as in an isothermal system, the momentum conduction gives ∂w , (3.32) Fwc = −η ∂x where 2ρl 2RT (3.33) η= 3 3π is called the coefficient of viscosity. It may also be called the momentum conductivity. The mean free path derived from the classical kinetic theory gives (Hirschfelder et al., 1964) 1 . l= √ 2nπσ 2 The effective molecular diameter σ in this equation may be evaluated by inserting the mean free path into (3.33), giving Fz = −
2m η= 3πσ 2
RT . 3π
The coefficient of viscosity varies with temperature alone in this expression. However, experiments showed that the viscosity changes with pressure or density at a given temperature, especially when the pressure is very high or very low. Equation (3.32) shows an important effect of molecular diffusions on mass motions, as they destroy the gradient of mass velocity. Under the influence of boundary friction also, the mass velocity will be eliminated eventually and the organized mass motions will be converted into random molecular motions in an isolated system. From the energy point of view, kinetic energy of mass motion is converted irreversibly into heat energy by molecular diffusions.
3. MOLECULAR TRANSPORT PROCESSES
64
3.4.4
Relation to self-diffusion
For a system of constant pressure, the relation between the coefficient of self-diffusion shown in (3.28) and the coefficient of viscosity in (3.33) is given by D=
3η , 2ρ
(3.34)
in which the diffusion coefficient D|p is denoted simply by D. Table 3.1 displays the coefficients of viscosity and self-diffusion obtained by experiments at 0◦ C and 1 atmosphere (Atm). The ratios in the last column of the table are close to unity. The errors, which can be produced by the ideal-gas approximation, are less than those of the classical theory for uniform gases. The new algorithm may also be used to calculate the molecular diffusion in gaseous mixtures. Table 3.1: Comparison of the coefficients of self-diffusion and viscosity, data sources: 1) Chapman and Cowling (1970), 2) Winn (1950), 3) Hutchinson (1949), 4) Jeans (1967), 5) Kaye and Laby (1995), 6) Loeb (1961) and 7) Trautz and Sorg (1931). (after Zhu, 1999)
Gases
3.5 3.5.1
η
D
0 C, 1 Atm
10−6 N.s.m−2
10−6 m2 .s−1
H2 O2 CO CO2 CH4 N2 Ne Ar Kr Xe
8.45 1 19.26 6 16.35 1 13.70 4 10.20 7 16.56 1 29.80 5 21.00 5 23.40 5 21.20 5
131.0 4 18.9 4 19.0 1 10.9 4 20.6 2 18.5 2 45.2 2 15.8 3 7.95 1 4.8 1
◦
ρ kg.m−3 0.090 1.428 1.250 1.964 0.716 1.250 0.901 1.783 3.740 5.860
2Dρ/(3η) 0.930 0.934 0.968 1.042 0.964 0.931 0.911 0.894 0.847 0.885
Heat conduction of ideal gases Conductive heat flux
The heat energy of ideal gases is proportional to the mean kinetic energy of molecules. So, heat flux is also the flux of molecular kinetic energy. The transfer of molecular kinetic energy produced by molecular collisions is called the heat conduction or thermal conduction. The conductive heat flux across the yp-plane with unit area evaluated from the two-flux approach gives + T2 v−x ), Fhc = cπ nc m(T1 v+x 1 2
3.5. HEAT CONDUCTION OF IDEAL GASES
65
where cπ is polytropic heat capacity given by (2.51) or (2.52), and the partial velocities vx1 and vx2 are again the weighted mean velocities averaged over the molecules in collisions. The cπ is used because the transport process is generally polytropic, so both the pressure and density may change simultaneously. This is different from the heat transport in a solid or liquid. For the binary collisions assumed, we may use N1 = N2 for (3.5) and (3.6), and apply the derived partial velocities for the previous heat conduction equation, giving 3 √ 3 c Fhc = cπ nc m R T12 − T22 . 2 The heat conduction is from high temperature to low temperature, and is in the direction opposite to that of molecular diffusion in the same system. Except at a very low pressure, nc may be replaced by n in (3.7). So, the previous equation is rewritten as 3 √ 3 c Fhc = cπ ρ R T12 − T22 , (3.35) 2 where ρ may be given by the mean density of the system. Applying (2.51) for (3.35) yields 3 √ 3 c (3.36) Fhc = ρ(cp − µR) R T12 − T22 . 2 If the experiments for heat conductivities are made at a constant pressure, we have µ = 0 or cπ = cp and so the previous equation gives 3 √ 3 c 2 2 (3.37) Fhc = ρcp R T1 − T2 . 2 The heat conduction is proportional to the gradient of T 3/2 at a given density. Like the mass diffusion, the momentum and heat conduction caused by molecular collisions depend crucially on the gradients of thermodynamic variables, and transport the momentum and heat along the downgradient of the fields. As a result, the transport destroys the gradients and lead to an equilibrium state of the system. Thus, the momentum and heat conductions are also irreversible in an isolated system. The downgradient heat conduction agrees with the second law of thermodynamics described by the Clausius statement (Sears and Salinger, 1975). This implies that the second law is related to the feature of molecular diffusions and can be derived from the current theory of the diffusions. The mass diffusion and momentum conduction show also the downgradient transport in a non-uniform gas, which may also be related to the entropy law in a more general way discussed in the next chapter. This law was illustrated particularly in terms of heat conduction, because heat conduction is most familiar to us and has a straightforward relation to the Kelvin statement of this law in the theory of heat engines.
3.5.2
Heat conductivity
In a non-uniform system, (3.36) may be replaced by Fhc = −λ
∂T , ∂x
(3.38)
66
3. MOLECULAR TRANSPORT PROCESSES
in which
λ = (cp − µR)ρl
2RT 3π
(3.39)
is called the heat conductivity or thermal conductivity of non-uniform gases. Here, the average collision distance given by (3.14) is used. Comparing with the coefficient of viscosity in (3.33) finds 3 (3.40) λ = (cp − µR)η . 2 It gives 3 (3.41) λ = cp η 2 when the heat conduction takes place at constant pressure. This equation may also be obtained by using (3.34) for the expression λ=
1 4
15 − 6
ρD ρD cp − 15 − 10 cv η η η
derived from the classical non-uniform gas theory (Hirschfelder et al., 1964, Eq.(7.623)). Equation (3.40) allows us to calculate the heat conductivity simply from the data of viscosity. The ratios of heat conductivity to the coefficient of viscosity obtained from experiments are not constant for a gas, but change slightly with temperature (Jeans, 1967). For the monatomic ideal gases, we may use cv = 3R/2 for (3.41) and gain 15 5 λ = cv η = Rη . 2 4 These relationships may also be derived from a classical theory of non-uniform gases in terms of the Maxwellian-Boltzmann distribution function and intermolecular forces. Usually, the gradients in thermodynamic variables may produce mass diffusion in the system, so the heat transport includes the effect of molecular diffusion. Although the heat conductivity is not derived using the peculiar velocity defined by other authors (e.g., Woods, 1993), the results are generally good as shown later on in this section.
3.5.3
Modified Eucken formula
In the classical thermodynamics, the degrees of freedom of molecular motions are defined as the dimensions to which molecular kinetic energy is equally partitioned. A monatomic molecule has three translational degrees of freedom, since the molecule may move in three independent directions. Two rotational degrees of freedom are added to the diatomic molecules which have the dumbbell structure, as the rotation about the molecular gravity center has two independent directions. The polyatomic molecules with linear structure have also two rotational degrees of freedom. But nonlinear polyatomic molecules possess three rotational degrees of freedom. The diatomic and polyatomic molecules may also have vibrational degrees of freedom. If a molecular has n atoms, the vibrational degrees of freedom are 3n − 6 for a nonlinear molecule and are 3n − 5 for a linear molecule.
3.5. HEAT CONDUCTION OF IDEAL GASES
67
In the classical thermodynamics also, each translational or rotational degree of freedom contributes R/2 to the heat capacity at constant volume, and a vibrational degree gives R because a vibrational system possesses kinetic and potential energies. In fact, the energies of rotation and vibration are quantized, so contribution of each rotational or vibrational degree may be different from the classical values, especially at low temperatures. If the heat capacity is divided into three components contributed by translational, rotational and vibrational motions of molecules respectively, we have cv = ctr + crot + cvib . Inserting it and (3.34) into (3.41) yields the modified Eucken formula (Chapman and Cowling, 1970) 5 λ = cv η + (crot + cvib )ρD 2 for monatomic gases. We may assume that ctr = 32 R, as a molecule has three translational degrees of freedom. For diatomic and polyatomic molecules, the vibrational heat capacity is calculated from the Einstein function (Einstein, 1907; Loeb, 1961)
cvib = R
νj h ¯
nv νj ¯h 2
ζT
j=1
e ζT νj h ¯
,
(3.42)
(e ζT − 1)2
¯ (= 6.262 × 10−34 J.s) is where νj is the j-th wavenumber of molecular vibration; h Planck’s constant, and nv is the vibrational degree of freedom. Derivation of this equation may be referred to Chapter 6. In the case of degeneracy, one vibrational frequency may be applied for more than one degrees of freedom. The rotational heat capacity may also be calculated theoretically, if we know the moment about each axis of rotation.
3.5.4
Collisional heat capacity
Equation (3.41) is derived from the assumption of elastic collisions. However, the real collisions between diatomic or polyatomic molecules are not elastic, due to the exchanges of molecular translational kinetic energy and rotational or vibrational kinetic energy. The specific heat capacity in (3.41) is the heat which can be transferred by molecular collisions. When the heat transfer is affected by the inelasticity, the heat capacity may be different from that in elastic processes. To provide a correction for inelastic collisions, we define the collisional heat capacity as the heat which can be transferred by molecular collisions along a given direction. In general, the translational kinetic energy in each direction can be transferred by the binary collisions assumed, so we have cˆtr = ctr , whereˆindicates the collisional heat capacity. The rotational kinetic energy can be transferred when the collisions are not in the direction parallel to the axis of rotation
3. MOLECULAR TRANSPORT PROCESSES
68
in a molecule. For diatomic molecules or linear polyatomic molecules, the energy transfer may take place when the axis is in two independent directions perpendicular to the direction of conductive flux. If distribution of the rotation axis in the three independent directions has an equal probability, the collisional heat capacity of rotation is 2 cˆrot = crot . 3 Meanwhile, we have cˆrot = crot for nonlinear polyatomic molecules with a simple structure, since the rotational kinetic energy can be transferred by collisions from all directions. Moreover, we assume that the vibrational kinetic energy may be transferred only by collisions in the direction of vibration. So the collisional heat capacity of vibration is 1 cˆvib = cvib . 3 The heat which can be transferred in a given direction by collisions reads cˆv = cˆtr + cˆrot + cˆvib now. It follows that
cˆvib =
ctr + 23 crot + 13 cvib for linear molecules for nonlinear molecules ctr + crot + 13 cvib
.
Also, we may write cˆp = cˆv + R for ideal gases. The isochoric and isobaric collisional heat capacities for monatomic molecules are given by cˆv = cv and cˆp = cp respectively. These relationships imply that collisions between monatomic molecules are assumed elastic in the current theory.
3.5.5
Comparison with experiments
Although the classical theories for the gases at non-uniform states are very complicated, the results are good only for monatomic gases. The relative errors, defined as − κhexperiment κh , Er = theory κhexperiment are generally large for diatomic and polyatomic gases. The accuracies of the classical theories may be viewed from the study of Reid et al. (1987), which showed that the mean relative error averaged over six or five classical methods was 8.7% for acetylene at 273 K, and 10.1% for carbon dioxide at 200 K.
3.5. HEAT CONDUCTION OF IDEAL GASES
69
Table 3.2: Heat conductivity of monatomic molecules. The data sources are: 1) Kennard (1938), 2) Kannuluik and Carman (1952), 3) Chapman and Cowling (1970) and 4) Keyes (1955). (after Zhu, 1999)
Gas
η
0◦ C, 1 Atm
10−6 N.s.m−2
J.kg−1 .K−1
cv
He Ne Ar Kr Xe
18.65 3 29.80 21.00 23.40 21.20
3122 2 628 2 312 2 149 2 95 2
λexp
λ
14.39 1 4.60 1 1.64 4 0.87 4 0.51 4
14.49 4.60 1.64 0.87 0.50
10−2 W.m−1 .K−1
Er % 0.7 -0.1 0.0 -0.2 -0.6
For monatomic molecules which possess translational kinetic energy only, (3.41) gives 3 (3.43) λ = cˆp η . 2 The heat conductivities evaluated from this equation are compared with experiments in Table 3.2. The results are just as good as those obtained from the classical theory for non-uniform gases. For diatomic and polyatomic molecules, the vibrational heat capacity is calculated with (3.42). The vibrational spectra of diatomic molecules used are taken from Allen and Pritchard (1974), except those for chlorine from Sears and Salinger (1975), and deuterium from Huber and Herzberg (1979). The vibrational spectra of polyatomic molecules are from Shimanouchi (1972). The rotational heat capacity is evaluated with 3 crot = cv − R − cvib 2 using the experimental heat capacity data reported in other literatures. The heat conductivities evaluated for some diatomic and polyatomic molecules with weak or no polar momenta are listed in Table 3.3, which are compared with experiments. The agreements with experimental data are generally good. For real gases, the ratio cp − cv b= R may not be unity. The deviations are very small in general. But for chlorine, the ratio is about 1.07 (Chapman and Cowling, 1970). If we use this ratio, the collisional heat capacity at constant pressure may be represented by cˆp = cˆv + bR . The heat conductivity evaluated for chlorine becomes, then, 0.76×10−2 W.m−1 .K−1 , and the relative error reduces to -0.2%. For the polyatomic molecules with internal degrees of rotational freedom, the collisional heat capacity of rotation may be less than what is assumed previously. The accuracy may be increased by evaluating more precisely the collisional heat capacity for complicated polyatomic molecules.
3. MOLECULAR TRANSPORT PROCESSES
70
Table 3.3: Heat conductivity of diatomic and polyatomic molecules with no or weak polar momenta. The data sources are: 1) Kennard (1938), 2) Eucken (1913), 3) Chapman and Cowling (1970), 4) Johnston and Grilly (1946), 5) Eucken (1911), 6) Senftleben (1964), 7) Misic and Thodos (1966), 8) Kaye and Laby (1995), 9) Jeans (1967), 10) De Rocco and Halford (1958), 11) Vogel (1914), 12) Beeck (1936), 13) Muller (1968), 14) Touloukian et al. (1975), 15) Eucken & Hoffmann (1920) and 16) Touloukian and Makita (1970). (after Zhu, 1999)
Gas
η
0◦ C, 1 Atm
10−6 N.s.m−2
H2 D2 O2 N2 Cl2 CO NO Air CO2 N2 O C2 H2 CH4 C2 H4 C2 H6
8.45 11.92 14 19.26 16.56 12.50 14 16.35 17.94 2 17.30 8 13.70 13.70 8 9.43 11 10.20 9.07 2 8.69 10
cv
cˆv
J.kg−1 .K−1
10125 13 5179 16 653 1 745 1,9 371 15 741 2 693 2 717 628 2 659 2 1335 12 1665 12 1166 12 1392 12
8758 4487 564 645 271 642 600 623 525 532 1001 1595 982 1050
λexp
λ
10−2 W.m−1 .K−1
16.59 4 11.72 13 2.38 2,4,5 2.37 1,2,9 0.77 2 2.31 4 2.38 5 2.37 2 1.46 3 1.47 2 1.89 6 3.15 7 1.75 6 1.77 6
16.40 11.71 2.38 2.34 0.76 2.30 2.36 2.36 1.47 1.48 1.87 3.24 1.75 1.75
Er %
-1.2 -0.1 -0.3 -1.2 -1.1 -0.3 -0.7 -0.2 0.7 0.6 -1.1 3.0 0.2 -1.3
Chapter 4 Predictability and thermodynamic entropy 4.1
Introduction
It is discussed in Chapter 2 that the Newtonian systems may be studied with a complete set of prediction equations in the time-space coordinates, so the variations are predictable in principle. While in the classical thermodynamics, variations of three thermodynamic variables are studied with two thermodynamic relationships: the state equation and the first law of thermodynamics for equilibrium states. For a compressible fluid parcel, such as a gas at rest, another relationship may be provided by mass conservation law, such as the mass diffusion equation discussed in this chapter, which tells that a local mass density may be changed by molecular diffusions in a fluid system without parcel motions. The mass velocity included in the mass diffusion equation is the instantaneous diffusion velocity discussed in the preceding chapter, which is different from the parcel velocity controlled by macroscopic Newtonian forces. It depends on the microscale gradients of non-uniform system, and may not be measured directly and evaluated precisely in a macroscopic scale. The hypothesis of linear law or other diffusion theories used for the previous studies of transport properties may not be applied for the mass diffusion equation to predict either conveniently or successfully the system variations. Thus, molecular diffusions are not studied in the classical thermodynamics which deals with the equilibrium processes only. When the mass diffusion equation is ignored as we do usually, the two thermodynamic equations cannot form a complete set for the three thermodynamic variables. They may be used to evaluate a new equilibrium state for a provided equilibrium initial state, assuming at least one variable of the new state is given already. The two equations used for the equilibrium states are independent of time, and the evaluations of a new equilibrium state are different from predictions. It is discussed in the previous chapters that changes of thermodynamic states caused by molecular diffusions are irreversible in isolation. So, the new state obtained may not really occur due to irreversibility of diffusion processes. To find whether the state evaluated may really occur afterwards, we may introduce a state function, called the thermodynamic entropy, according to the irreversible downgradient transport and energy conversion in the diffusion processes. A microscopic approach for deriving the thermodynamic entropy law was given by Prigogine et al. (1977) using the theory of quantum field. We derive in this chapter the entropy law for ideal gases from the first law of thermodynamics at nonequilibrium states discussed in Chapter 2, which tells that thermodynamic entropy is created by conversion of diffusion kinetic energy into heat energy. A new equilibrium state may 71
72
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
occur afterwards, only if the thermodynamic entropy is not destroyed in isolation. However, the entropy may not be used to predict when and how the new state occurs. However, conversion of mechanic energy into hear energy is not the only source of thermodynamic entropy. From a microscopic point of view, thermodynamic entropy variation is related to the changes in distribution and partition of molecular energies. As the system changes irreversibly in isolation and the ability of system change depends on the system disorderliness, the microscopic thermodynamic states with different degrees of disorderliness possess different probabilities of occurrence. Thus, the entropy variation may also be studied, in the statistical thermodynamics, with the probability theory based on the statistics of molecular distributions against molecular energies. Without using the theory of statistical thermodynamics, we may also prove in this chapter that the entropy increases with the system disorderliness even if no energy conversions take place. We shall provide also the general algorithm for evaluating the entropy change in a non-uniform system, which shows that the entropy of a system depends not only on the entropy of each parcel in the system, but also on the parcel distribution. The entropy of a system may still be changed if the entropy of each parcel in the system is conserved. Thus, thermodynamic entropy is different from many other quantities, and is not entirely transportable and additive. Changes of thermodynamic entropy are caused by molecular diffusions in the classical thermodynamic systems. However, to integrate the diffusions over a nonequilibrium process is not the best way for evaluating the entropy changes, because we are unable to predict the nonequilibrium processes in general. For the ideal gases with constant heat capacities, the state function of thermodynamic entropy may be represented macroscopically in terms of the measurable thermodynamic variables. The entropy change in an irreversible process may be calculated by assuming a reversible process to connect the initial and final equilibrium states provided. It is noted that the expression thermodynamic entropy as a state function for a real gases or another real substances may be different from that of ideal gases, because the state equation is different from that of ideal gases and the heat capacities are not constant. Since the thermodynamic variables represent the statistical features of molecular assemblies and so depend on the size of samples in a non-uniform system, the thermodynamic entropy calculated with these variables depend on the size too. The irreversible energy conversion exists also in the classical mechanic systems, and may be studied by introducing the viscosity or friction to account for the irreversible energy conversion. The dynamic entropy discussed in Chapter 2 may be viewed as an application of the entropy law for Newtonian dynamics. The relation and difference between thermodynamic entropy and dynamic entropy will be discussed in this chapter.
4.2
Change rate in diffusion processes
In a nonequilibrium thermodynamic system, the thermodynamic state varies with time and space. For example, the temperature may be represented by T = T (r, t) ,
(4.1)
4.2. CHANGE RATE IN DIFFUSION PROCESSES
73
where ˆ + zˆ r = xˆ x + yy z indicates the spatial position of a particular point in the temperature field, represented in Cartesian coordinates of which the unit vectors along the axes are denoted ˆ, y ˆ and ˆ by x z. It is discussed in the preceding chapter that the temperature or heat energy can be transported by heat conduction without mass transport accompanied, especially in a solid substance. Thus, the particular point in a variable field may not be an element of volume or mass. As the temperature field moves or changes in heat conduction, the element of volume or mass does not move. For convenience, a particular point in a variable field at a given time and place may be referred to as the field point. A field point driven by molecular diffusion or conduction may change its position continuously with time in the direction of diffusion velocity or conduction velocity, that is r = r(t). The change rate of the field point is given by the derivative of (4.1), that is ∂T ∂T dr dT = + · . dt ∂t ∂r dt We may write ∂T ∂r
∆T ∆r ∆T ∆r = lim ∆r→0 ∆r∆r ∆T ∆r∆r = lim ∆r→0 ∆r ∆r · ∆r ∆T ∆r . = lim ∆r→0 ∆r ∆r =
lim
∆r→0
Here, ∂T ∆T = ∆r→0 ∆r ∂r is the directional derivative of temperature in the direction of r, and lim
r ∆r = ∆r→0 ∆r r lim
is the unit vector along r. Now, we have ∂T r ∂T = . ∂r ∂r r
(4.2)
This expression is independent of coordinates, and is not changed by rotation of coordinates proved by McHall (1993). Moreover, r dr = vgl = vgl dt r
(4.3)
may be referred to as the general velocity, indicating the displacement velocity of the field point. If the field displacement is carried by mass displacement, the general velocity is a mass velocity such as the diffusion velocity. The general velocity may
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
74
also be a conduction velocity of heat or momentum conduction, or a phase velocity or group velocity (The both may be called the propagation velocity) for propagation of perturbation energy in mechanic waves. A conduction velocity or propagation velocity may not be a mass velocity. In a fluid system without macroscopic parcel motions, the variations may be caused by molecular diffusions including molecular collisions. The sum of instantaneous diffusion velocity vd and conduction velocity vc gives the general velocity vgl = vd + vc . It may also be represented by ˆ + vgly y ˆ + vglz ˆz vgl = vglx x in Cartesian coordinates, in which vglx , vgly and vglz are the components in the directions of x, y and z, respectively. If the angles of r to the x, y and z axes are given by αx , αy and αz respectively, the velocity components are vglx = vgl cos αx ,
vgly = vgl cos αy ,
vglz = vgl cos αz .
(4.4)
If the system changes can also be produced by wave propagations, the general velocity may include the propagation velocity. Combining (4.2) and (4.3) gives ∂T dr · ∂r dt
∂T r · r ∂r r 2 ∂T . = vgl ∂r = vgl
The directional derivative included may be expanded as ∂T ∂T ∂T ∂T = cos αx + cos αy + cos αz . ∂r ∂x ∂y ∂z Applying (4.4) for it yields vgl
∂T ∂r
∂T ∂T ∂T vgl cos αx + vgl cos αy + vgl cos αz ∂x ∂y ∂z ∂T ∂T ∂T + vgly + vglz . = vglx ∂x ∂y ∂z =
(4.5)
Now, the rate of temperature change is replaced by ∂T ∂T ∂T ∂T dT = + vglx + vgly + vglz dt ∂t ∂x ∂y ∂z or
∂T dT = + vgl · ∇T , dt ∂t
where ∇=
∂ ∂ ∂ ˆ+ ˆ+ x y zˆ ∂x ∂y ∂z
(4.6)
4.3. MASS CONSERVATION LAW
75
indicates the three dimensional gradient. The left-hand side of (4.6) is a perfect derivative called the material derivative usually, which is the change rate of a field point with unit mass (if it depends on mass) and is evaluated or measured by following the moving field point. The field point may be physically or symbolically independent of mass in some particular situations, such as a packet of waves. The first term on the right-hand side is called the local derivative, which measures the change rate at the local region occupied by the unit mass or specified field point. The other term, called the advection, manifests the transport effect of mass diffusion or conduction or propagation. This equation tells that the change of field variable at a given time and place may be represented by local change plus diffusion or conduction or propagation transport across the region at the time.
4.3 4.3.1
Mass conservation law Mass diffusion equation
Two thermodynamic equations, the state equation and first law of thermodynamics, have been discussed in Chapter 2. To have a complete set of governing equations for the three thermodynamic variables, we need one more equation which can be obtained from mass conservation law. When the pressure decreases, a gas expands through molecular diffusion. The temperature and mass density decrease also in the expansion. Without heat exchange with the surroundings, the process gives an example of adiabatic variation of thermodynamic system. The variation of density produced by diffusion is discussed in the following. If (4.6) is applied for mass density ρ instead of temperature, it gives dρ dt
= =
∂ρ + vd · ∇ρ ∂t ∂ρ + ∇ · (ρvd ) − ρ∇ · vd , ∂t
where ∇·v =
(4.7)
∂vz ∂vx ∂vy + + ∂x ∂y ∂z
denotes the three dimensional divergence. The conduction velocity is ignored in the equation as the contribution to mass displacement is relatively small. Also, ∇ · vd =
∂vdy ∂vdz ∂vdx + + ∂x ∂y ∂z
is a three dimensional divergence of diffusion velocity, which measures the change rate of the volume of diffusion element as discussed later on. The ρvd is a density flux Fρ produced by mass diffusion, and ∇ · (ρvd ) is divergence of the flux which represents the local concentration rate of mass. If there are no internal sources or sinks of mass at the local place, we have ∂ρ + ∇ · (ρvd ) = 0 , ∂t
(4.8)
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
76 and so (4.7) gives
dρ = −ρ∇ · vd . (4.9) dt The last equation may be called the mass diffusion equation, which is the continuity equation of diffusion system without the parcel motions driven by macroscopic Newtonian forces. The relative change rate of density is proportional to the convergence of diffusion velocity. The flux =v is physically meaningful, only if the quantity = can be transported by the velocity v. When the mass is not transported by heat conduction in a solid medium, we have dρ/dt = ∂ρ/∂t = 0, and (4.7) gives ∇ · (ρvhc ) − ρ∇ · vhc = 0 , where vhc denotes the heat conduction velocity. It follows that vhc · ∇ρ = 0 , which may not be true if vhc = 0 and ρ is not constant. This example emphasize that the velocity in the continuity equation must be mass velocity.
4.3.2
Mass conservation
The mass diffusion equation is equivalent to the conservation law of mass. If M = ρV represents the mass in volume V , we have dV dρ dM =ρ +V . dt dt dt It follows, from (4.9), that
dV dM =ρ − V ∇ · vd dt dt
.
(4.10)
From V = δxδyδz, we obtain 1 dδx 1 dδy 1 dδz 1 dV = + + . V dt δx dt δy dt δz dt Since
d(δy) = δvdy , dt
d(δx) = δvdx , dt as δx, δy, δz → 0, we see 1 dV V dt
d(δz) = δvdz dt
δvdy δvdx δvdz + lim + lim δy→0 δy δz→0 δz δx ∂vdy ∂vdz ∂vdx + + = ∂x ∂y ∂z = ∇ · vd . =
lim
δx→0
Therefore, (4.10) gives dM/dt = 0. The velocity divergence in the last equation measures the relative change rate of the volume. When the volume increases, the
4.3. MASS CONSERVATION LAW
77
divergence is positive or ∇ · vd > 0, otherwise ∇ · vd < 0. The negative divergence is also called convergence. Applying the last equation for (4.8) yields 1 dV 1 dρ =− . ρ dt V dt It gives V0 , V which shows the density change in the process of gas expansions. Since ρV = M is the mass of gas, the last equation gives the mass conservation law M = M0 . ρ = ρ0
4.3.3
Diffusive transport equation
Equation (4.8) may be called the diffusive transport equation. It tells that the local variation is produced by divergence of the diffusion flux, if without sources or sinks in a system. As discussed in Chapter 3, the diffusion velocity depends on microscale gradients of thermodynamic variable, and so can hardly be measured precisely in practice. Thus, the classical equilibrium thermodynamics does not discuss either the diffusive transport equation (4.8) or the mass diffusion equation (4.9). The nonequilibrium thermodynamics and hydrodynamics introduced the alternative algorithms to evaluate the diffusion transport based on the hypothesis of linear law (such as the Fickian law for heat transport and Fick’s law for mass diffusion) (Batchelor, 1967; de Groot and Mazur, 1962). When the transport coefficients are constant, (3.27), (3.32) and (3.38) give examples of the linear law for mass, momentum and heat transport, respectively. But in general these flux equations do not show the linear relationships between the fluxes and gradients, as the coefficients are not constant. This can be seen clearly from the flux equations (3.22), (3.30) and (3.36). Applying the diffusive mass flux equation (3.27) for (4.8) yields ∂ρ = D∇2 ρ + ∇D · ∇ρ , ∂t where ∇2 ρ = ∇ · ∇ρ =
∂2ρ ∂2ρ ∂2ρ + + . ∂x2 ∂y 2 ∂z 2
The diffusion coefficient D is not constant as shown by (3.29). Since it is generally difficult to calculate the variation of this coefficient with real data, the coefficient is usually taken as constant. In this case, we gain the mass diffusion equation ∂ρ = D∇2 ρ . ∂t
(4.11)
Use of this equation needs not to know the diffusion velocity. The solution tells that the density decreases while fluctuates with time. However, limitation of this equation is obvious, as it cannot predict the variation of a system with constant gradients. Lorenz (1955) pointed out that the skin friction and surface heating
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
78
evaluated using the hypothesis of linear law have nearly infinite values throughout an infinitesimal depths near the boundaries. Since ∇2 ρ > 0 in a low center of density and ∇2 ρ < 0 in a high center, (4.11) shows that the diffusion destroys the inhomogeneity of the field. This feature of molecular diffusion, discussed also in the preceding chapter, leads to the second law of thermodynamics.
4.4
Unpredictability in classical thermodynamics
The thermodynamic energy law of ideal gases given by (2.32) may be written as the prediction equation dα dq dT +p = + DTd (4.12) cv dt dt dt for a continuous process, where DTd = −
dkd dt
(4.13)
may be called the heat diffusion, which is the rate of heat production resulting from dissipation of diffusion kinetic energy. For ideal gases we may also have cp
dp dq dT −α = + DTd dt dt dt
(4.14)
derived from (2.33). Since dkd < 0 as discussed earlier, (4.13) shows DTd > 0. The molecular dissipation always converts kinetic energy into heat energy. Now, we have three independent relationships for ideal gases: the ideal-gas equation (2.21) for equilibrium state, the mass diffusion equation (4.9) and the energy conservation law (4.12) or (4.14) for nonequilibrium processes. We may also use the transport equation (4.8) which is not independent of the mass diffusion equation. The three equations have five macroscopic variables: three thermodynamic variables T , p, ρ and diabatic heating rate dq/dt together with the dynamic variable vd . The diabatic heating rate may be provided at a laboratory condition, and the diffusion velocity may be evaluated from the transport theory as shown by (3.10)-(3.12). So the three equations can be used to predict the three thermodynamic variables. However, to measure the non-uniform thermodynamic state on a microscale is generally impossible with an accuracy good enough for the predictions. To solve the thermodynamic equations, we may assume that the system varies slowly, so that it is in quasi-equilibrium at any time and the thermodynamic variables are independent of space. The molecular diffusion may be ignored in this situation, and the mass diffusion equation (4.9) or the diffusive transport equation (4.8) gives dρ = 0 or ∂ρ/∂t = 0. This, however, is not true as the density may change from one equilibrium state to another. Thus, the mass conservation equations are ignored in the study of classical thermodynamics. The three thermodynamic variables of ideal gases are then solved from two independent diagnostic equations: the idealgas equation and the thermodynamic energy law (2.34) or (2.35) for equilibrium processes. These equations can be used to evaluate a new equilibrium state, but not to predict the system variations.
4.5. THERMODYNAMIC ENTROPY LAW FOR UNIFORM STATES
79
If the evaluations are carried out between two equilibrium states, the energy equation gives cv ∆Tequ + pequ ∆αequ = q (4.15) or cp ∆Tequ − αequ ∆pequ = q
(4.16)
when the heat capacities are taken as constant in the current study. The subscript ‘equ’ indicates the variables at equilibrium states. For the adiabatic processes of ideal gases, we use the ideal-gas equation (2.22) for (4.12) and (4.14) giving cv
d ln α d ln T +R =0 dt dt
and
d ln p d ln T −R =0, dt dt respectively. If the heat capacities are constant, the solutions are the adiabatic equations of ideal gases R α0 cv (4.17) T = T0 α cp
and
T = T0
p p0
R
cp
.
(4.18)
.
(4.19)
The two equations give another expression
p = p0
α0 α
cp cv
These three equations may also be obtained from (2.49), (2.54) and (2.48) with cπ = 0. To find the solutions of thermodynamic variables, we have to provide not only the initial conditions but also a part of the solutions or to specify a particular process, such as the isobaric process dp = 0, isothermal process dT = 0, isochoric process dα = 0 or adiabatic process dq = 0. The obtained solutions are not the predictions, because they may not tell whether and when the new state may occur. For this reason, the thermodynamic processes are unpredictable in classical thermodynamics.
4.5
Thermodynamic entropy law for uniform states
The state equation and energy conservation equation for equilibrium states used in the classical thermodynamics are diagnostic equations but not prediction equations. A new equilibrium state derived using these equations may occur before or after a provided initial state in time. Since the diffusion process is irreversible in isolation, the new state, which may occur before the initial state, may not take place afterwards. Thus, the two thermodynamic equations may not be enough to give the solutions which are physically significant.
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
80
To know whether the new state may occur after the initial state, we try to integrate the prediction equation (4.12) for ideal gases from an initial time t0 to final time t1 , giving t1 t0
DTd dt =
T1
cv dT +
α1 RT
T0
α0
α
dα − q .
Since DTd > 0 for a diffusion process, both sides of the equation should not be less than zero if the process may really occur. If the both sides are negative, we have DTd < 0. This process cannot happen as a part of heat energy is converted into diffusion kinetic energy at the end of the process. Since the information of molecular diffusion on a microscale is not available, we are unable to integrate the left-hand side of the previous equation. Also, the thermodynamic variables in the integrations on the right-hand side change with time and place in a non-uniform system, and these variations may not be measured or predicted precisely. Except for some particular situations, it is generally impossible for us to integrate the nonequilibrium thermodynamic processes. Thus, the previous equation cannot be used in practice. If (4.12) is rewritten as d ln α dq d ln T DTd = cv +R − T dt dt T dt
(4.20)
for the ideal gases with constant heat capacities, the integration gives t1 DTd t0
T1 α1 dt = cv ln + R ln − T T0 α0
dq . T
(4.21)
The difficulty in calculating the right-hand side occurs only in the heating term now. For an isolated system without heat and mass exchanges with the surroundings, this equation becomes t1 T1 α1 DTd dt = cv ln + R ln . (4.22) T T0 α0 t0 Now, the right-hand side can be calculated according to the initial and final states without considering the variations in the process. If setting ds = d(cv ln T + R ln α) . we gain
(4.23)
t1 DTd
dt dm (4.24) T for an isolated system with mass M . The S is called the thermodynamic entropy, and s is the specific thermodynamic entropy of unit mass. For the open systems, the thermodynamic entropy change is represented generally by ∆S =
M
ds =
t0
dq DTd + . T T
(4.25)
This expression is equivalent to (4.23) only for the ideal gases with constant heat capacities. In many text books, the entropy change is given by ds =
dq . T
(4.26)
4.5. THERMODYNAMIC ENTROPY LAW FOR UNIFORM STATES
81
It is emphasized that this equation may only be applied for reversible processes with DTd = 0. The differential equations, such as the Maxwell equations, derived using this definition of thermodynamic entropy change may also be used for reversible processes only. The left- and right-hand sides of (4.22) give the microscopic and macroscopic descriptions of the entropy changes. This equation shows that the entropy changes of the ideal gases with constant heat capacities may be calculated with the macroscopic thermodynamic variables at the initial and final equilibrium states provided. With the ideal-gas equation (2.22), the entropy variation may also be represented by ds = d(cp ln T − R ln p) .
(4.27)
As in (4.23), ds is a perfect differential for the ideal gases, and so the thermodynamic entropy is a state function. Using the definition of potential temperature for ideal gases given by (2.55), (4.23) and (4.27) can be rewritten as ds = cp d ln θ or
(4.28)
θ2 . θ1 These two equations tell that potential temperature of the ideal gases is also a state function. Since the heat energy may change in the reversible adiabatic process with conservation of thermodynamic entropy, the entropy change may not be found simply by comparing the heat energy between two equilibrium states of a system at different pressures. There are no the simple relations between increases of the entropy and temperature or volume as claimed in some text books (e.g., Young and Freedman, 1996). To compare the heat energy, the system should be justified to a constant reference pressure in order to exclude the change of heat energy in the reversible adiabatic process. Thus, the right-hand side of (4.28) includes potential temperature instead of temperature. According to the energy equation of ideal gases (2.35), the heat exchange at constant pressure may be measured by the change of enthalpy. Since the state equation for real gases is different from ideal-gas equation and the heat capacities of real gases are not constant, the thermodynamic entropy defined for ideal gases may not be a state function again for real gases. If we are looking for a state function to represent the thermodynamic entropy of real gases or real substances, the expression may be different from that of ideal gases. The different thermodynamic entropies for different physical systems are represented by different probability functions in the statistical thermodynamics or statistical mechanics. The thermodynamic entropy of classical thermodynamic system is represented by the Maxwell-Boltzmann statistics. In the quantum world, the entropies of photon gas and electron gas are represented by the Bose-Einstein statistics and Fermi-Dirac statistics, respectively (Sears and Salinger, 1975). Since T > 0 and DTd ≥ 0, the integration on the right-hand side of (4.24) cannot be negative, and (4.24) tells that molecular diffusions are a source of thermodynamic entropy. The entropy increases as ordered kinetic energy is converted into disordered kinetic energy or heat energy. Thus, we have ∆S ≥ 0 for the diffusion processes s2 − s1 = cp ln
82
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
in isolated systems. Since molecular diffusions are irreversible in any substance isolated, the previous consequence is generally true for all substances, and gives the mathematical expression of the thermodynamic entropy law: Thermodynamic entropy cannot be destroyed in an isolated thermodynamic system. A process with ∆S > 0 is irreversible, because the reversed process has ∆S < 0 and so cannot occur in isolation. The processes without molecular diffusions are called the reversible processes since ∆S = 0. Although it is impossible for us to predict the system change in a nonequilibrium process, the entropy law tells us at least the tendency of system variation. A new equilibrium state evaluated may really occur in isolation only if the thermodynamic entropy does not decrease in the process. The thermodynamic irreversibility is associated with the irreversible conversion of diffusion kinetic energy into heat energy and downgradient transport discussed in Chapter 3. Thermodynamic entropy change caused by downgradient transport only without diffusion kinetic energy dissipation will be discussed in the next section. Not just the thermodynamic entropy but also the thermodynamic state or all the thermodynamic variables change irreversibly in a single irreversible process. The value of a thermodynamic variables may either increase or decrease in the process, and only the entropy increases always in an isolated system. Also, the entropy is a state function, but the other thermodynamic variables may be not. So, the irreversibility can be indicated conveniently by the entropy change. While, another single thermodynamic variable (except the potential temperature for ideal gases) may be unable to tell the variation direction of a system. This does not mean that only the thermodynamic entropy changes irreversibly in an irreversible process. In mathematics, changes of a state function depend only on the initial and final states provided and are independent of the processes. Thus, the change of thermodynamic entropy in a continuous process whether reversible or irreversible is the difference of the entropy at the two ends of the process, that is B A
It gives
ds = sB − sA .
ds = 0 . in a continuous cycles of thermodynamic state. The continuous process usually means the reversible process, since an irreversible process is generally discontinuous. It is learned that thermodynamic entropy is conserved in isolated processes. That the entropy of an isolated system is conserved over a cycle may not confirm that the entropy is a state function. For calculating thermodynamic entropy of ideal gases, the thermodynamic state may be represented by the potential temperature only as shown by (4.28). It is discussed in Chapter 2 that ideal gases do not possess thermopotential energy and so have no latent heat. In other words, they may not change the gas phase. The water vapor in the atmosphere may also be considered as an ideal gas unless in the processes of phase transition. When the vapor changes phase, the changes of volume and pressure with temperature are discontinuous, and the ideal-gas equation cannot be applied. In this situation, the thermodynamic state cannot be represented by potential temperature only, and the thermodynamic entropy in a cycle of potential
4.6. THERMODYNAMIC ENTROPY CHANGE OF NON-UNIFORM STATE
83
temperature change may have a net production. The examples can be found in the studies of wet air engines in Chapter 22. It is not proved in this study that the thermodynamic entropy defined as dQ/T is also a state function for the substances other than ideal gases. The proof may be found in the text books of classical thermodynamics, which is based on the two consequences (i.e., Sears and Salinger, 1975): i) All Carnot engines working between two provided heat reservoirs have an equal efficiency represented by the ratio of the net heat accepted to the Kelvin temperature of the warmer reservoir, and ii) the ratio dQ/T vanishes over any reversible cycle as it may be represented by infinite number of Carnot cycles. It will be argued in Chapter 21 that the classical proof is not convincing, since it applied implicitly the assumption that all Carnot engines are reversible. This assumption is not true, as the heat exchanges may take place in nonequilibrium processes, and the entropy changes in the environments should be considered also for an open system. When the working substance is in thermodynamic equilibrium with the reservoirs over a Carnot cycles, different substances working over the same pressure ranges may not produce an equal amount of mechanic work.
4.6
Thermodynamic entropy change of non-uniform state
The thermodynamic entropy law is derived in the preceding section according to the irreversible conversion of diffusion kinetic energy into internal energy. The entropy increases as the internal energy increases in this situation. However, the diffusion kinetic energy is not the only source of thermodynamic entropy in an isolated system, since the irreversible changes may also be produced by the downgradient transport discussed in the preceding chapter while the internal energy is conserved. For example, when a non-uniform gas reaches the uniform equilibrium state by heat conduction or scattering radiation without mass displacement, no diffusion kinetic energy is transfered to internal energy but the downgradient heat conduction is irreversible. It will be found in Chapter 9 that the entropy may also increase as the atmosphere produces kinetic energy. To calculate the entropy for a non-uniform state, we divide the system into J elements of volume. When the mean potential temperature over the element with mass m changes from θ¯ to θ¯ , the change of thermodynamic entropy of the system may be calculated by adding the entropy change of each element, giving ∆S = cp
θ¯j mj ln ¯ . θj j=1 J
(4.29)
The mean potential temperature is used since the potential temperature of an element may not be uniform at a nonequilibrium state. The downgradient diffusive or conductive fluxes of mass, momentum and heat may destroy the thermodynamic gradients eventually as discussed in Chapter 3. According to the Stefan-Boltzmann law (Brunt, 1944; Hess, 1959), the radiation over the whole spectrum is proportional to the fourth power of temperature, and also
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
84
destroys the temperature gradient at the radiation equilibrium. Thus, the heat conduction or scattering radiation leads the system to the thermodynamic equilibrium state viewed over a certain scale, in which the macroscopic thermodynamic state cannot be changed again by molecular diffusions or collisions at constant pressure. It is noted that the microscopic fluctuations of thermodynamic variables may still exist in the equilibrium state, as the system is not microscopically continuous. The number of collisions for a nonequilibrium system becomes equilibrium is studied in the non-uniform gas theory. The molecular motions at an equilibrium state may be considered isotropic, as they do not produce detectable net transport. The equilibrium state of gases, which are free of external Newtonian forces, is uniform in thermodynamic variables. We consider here the entropy change caused by the change of system disorderliness only without dissipation of diffusion kinetic energy. So, we assume that the initial non-uniform state is uniform in mass density but not in temperature. The mean temperature of a single element may be changed by heat conduction, scattering radiation and random molecular exchanges with other elements. While, the volume of whole system does not change in this process, and so the heat energy or mean temperature or potential temperature of the system is conserved. The potential temperature of the equilibrium state may be calculated from J 1 mj θ¯j , θ¯j = M j=1
where M = yields
J
j=1 mj
is mass of the system. Applying it for the previous equation J
∆S = cp M ln M
¯
j=1 mj θj J M ¯mj θj j=1
,
(4.30)
in which Jj=1 denotes the product of J quantities. It may be proved in mathematics (Beckenbach and Bellman, 1965) that J J mj 1 mj θ¯j ≥ θ¯jM M j=1 j=1
(4.31)
for θ¯j > 0. The equality may be applied if all θ¯j are equal. Thus, thermodynamic entropy is created as the non-uniform system reaches the equilibrium state, though no diffusion kinetic energy is dissipated and the heat energy or mean potential temperature of the system is conserved. This entropy change is caused simply by the change of macroscopic disorderliness in the thermodynamic state. Therefore, the entropy may also be regarded as a measure of the disorderliness or randomness. It may also represent the ability of system variation, as the diffusions which are the causes of system variation depend essentially on the disorderliness. At the first glance, the disorderliness is similar to the inhomogeneity. But usually, the disorderliness increases with decreasing the inhomogeneity, and a high inhomogeneity implies a low disorderliness of the system. The disorderliness reaches the highest level as an isolated system becomes equilibrium. In this process, the thermodynamic entropy rises to the maximum when the mass and energy are conserved.
4.6. THERMODYNAMIC ENTROPY CHANGE OF NON-UNIFORM STATE
85
The change of disorderliness may also be applied to explain the entropy change as a system changes between two equilibrium states. When the system is in thermodynamic equilibrium with environments, the reversible adiabatic process of the system does not change the entropy of the system and environments, and so the disorderliness of the system or environments does not change. While in the nonequilibrium processes, such as the isobaric heating or free expansion of a gas, the thermodynamic state changes discontinuously across the system boundary. Thus, we may take the system and surroundings as an expanded isolated system. The changes of disorderliness in the expanded system can be found immediately by calculating the thermodynamic entropy changes. The study of thermodynamic entropy change in a non-uniform system is particularly important as most physical and non-physical systems including the Earth’s atmosphere and oceans are not uniform in an equilibrium state. The entropy of these inhomogeneous systems may be change by redistribution of a conservative quantity or parameter resulting from transport or communications. This is discussed in Chapter 6. The entropy changes between two equilibrium states may also be calculated from (4.30) with J = 1. When the mean potential temperature in this equation is replaced by a statistical parameter related to the features of microscopic particles, the entropy may be applied to study the microscopic processes such as the quantum statistics. The heat conduction is a transport process discussed in the preceding Chapter. The local change rate of thermodynamic entropy caused by heat conduction in a non-uniform system may be calculated from 1 ∂s = − vhc · ∇T , ∂t T
(4.32)
where vhc is called the heat conduction velocity and can be estimated from Fhc = cπ ρT vhc , where Fhc is the conductive heat flux of which the x-component is given by (3.38). The velocity in the x-direction gives
vhcx = −l
2R ∂T . 3πT ∂x
√ It is proportional to the gradient of T and in the downgradient direction. Thus, the entropy produced is positive as shown by (4.32). Analogously, the entropy created by molecular diffusion is evaluated from 1 ∂s = − vd · ∇T . ∂t T
(4.33)
It is also positive since the heat diffusion is also in the downgradient direction of temperature. The thermodynamic entropy is different from other physical parameters, as it depends on the distribution as well as the local quantity. It is not entirely transportable. The downgradient transport increases the entropy. This will be discussed
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
86
again in Chapter 26. The entropy changes in the internal transport processes discussed are caused by destruction of the gradients, or by the local heating and cooling at different temperatures in the process instead of by entropy transport. Thus, the entropy change caused by heat conduction is included in the heat exchange term of (4.25) together with the scattering radiation. While, the entropy change caused by molecular diffusion is in the diffusion term of (4.25). The reason will be given in the next chapter. When the local changes produced are integrated over a closed system, the integrated entropy of the system increases as discussed earlier, even though no transport takes place across the system boundary. If viewed microscopically, molecules in a fluid do not possess an equal amount of kinetic energy for each motion status even if in the equilibrium state, as shown by the Maxwell-Boltzmann distribution function (Sears and Salinger, 1975). The disorderliness of a system may also be represented by the microscopic state or the non-uniform distributions of particles and their energies. Since a thermodynamic system leads to increase its disorderliness, the states of different degrees of microscopic disorderliness possess different probabilities of occurrence. Thus, the entropy may also be studied with the probability theories based on the statistics of parcel distributions and energy partitions. For the classical thermodynamic systems, the thermodynamic probability and the entropy are illustrated by the Maxwell-Boltzmann distribution of particles at different energy levels. While in the quantum mechanics, the entropy is represented by the Bose-Einstein statistics for the systems in which the particles are indistinguishable, or by the Fermi-Dirac statistics for the systems in which the particles follow the Pauli exclusion principle (Sears and Salinger, 1975). It can be found in Chapter 6 that the thermodynamic entropy of ideal gases defined as a thermodynamic probability is consistent with that defined by the classical thermodynamics, for both of them can be applied to derive the same partition functions for the distributions of particles with different energies.
4.7
Inadditive and scale-dependent features
The dynamic entropy discussed in Chapter 2 is additive, as it depends on the dynamic entropy of each sub-system, but not on the relations between different subsystems. If we can define the thermodynamic entropy for a single molecule, the molecular entropy represents the feature of a particle and is independent of other particles. Thus, the total molecular entropy of a system is not changed by redistribution of the particles, if the entropy of each molecule is conserved. In other words, the sum of molecular entropy does not represent the disorderliness of a system. This can be shown by the following relationship I i=1
ln θi =
I
ln θI−i+1 ,
(4.34)
i=1
in which θ denotes the potential temperature of a molecule (if defined) with mass m, and the prime indicates the variable at the final state. This equation tells clearly that the sum of molecular entropy is independent of molecular distribution. Particularly,
4.7. INADDITIVE AND SCALE-DEPENDENT FEATURES
87
the previous equation gives cp Im
I
ln
i=1
θi =0 θi
(4.35)
for θi = θi . This means that the sum of molecular entropy is conserved if molecular potential temperature of each molecule is unchanged, although the disorderliness of the system is changed by redistribution of the molecules at different energy levels. To measure the disorderliness of a system, the entropy is calculated with the thermodynamic variables, which represent the mean features of microscopic molecular thermodynamic state averaged over an element of volume. Thus, the potential temperature in (4.28) is given by the mean value of the element, and we have θ¯j d ln θ¯j = ln ¯ θj for element j. If the element has mass δM , its thermodynamic entropy change is given by δMj δsj = cp δMj d ln θ¯j or
θ¯j δMj δsj = cp δMj ln ¯ θj
(4.36)
for ideal gases. This equation shows clearly the statistical feature of thermodynamic entropy. In general, the molecules do not in an equal energy level even if in the equilibrium state, so the entropy evaluated with the mean potential temperature is different from the sum of molecular entropy since δM ln θ¯ = m
ln θi .
i=1
We may find from the inequalities given in the preceding section that the entropy is greater than the sum of molecular entropy, because the element is assumed uniform for the calculation. This does not mean that the entropy evaluated from (4.36) is an approximation for the sum of molecular entropy, as they are different physical quantities illustrating different properties of the system. The sum of molecular entropy represents only an integrated quantity, while the thermodynamic entropy express the feature of the quantity distribution. Therefore, unlike the dynamic entropy, thermodynamic entropy is not additive and so should not be defined as a sum of the entropy of each molecule. The downgradient transport caused by molecular diffusions may destroy the gradients between the elements in a fluid system, so the disorderliness manifested by the mean thermodynamic variables over each element is increased by particle exchanges across the boundaries of elements. Since the mean potential temperatures of these elements are changed by the particle exchanges, the entropy change of the system evaluated from (4.29) or ∆S = cp
θ¯j δMj ln ¯ θj j=1 J
(4.37)
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
88
is different from zero, even if potential temperatures of single molecules are conserved. However, the total molecular entropy is conserved in the process as shown by (4.35) or Ij J
ln
j=1 i=1
θi =0. θi
Thus, the sum of molecular entropy cannot measure the disorderliness of a system. The system randomness may be illustrated graphically by the molecular features, but the quantitative measurement in the classical thermodynamics depends on the macroscopic variables representing the mean features of molecular assemblies. This is a limitation of the classical thermodynamic entropy represented by thermodynamic variables, as discussed in Chapter 6. The statistical thermodynamics re-defines the thermodynamic entropy in terms of the molecular features, so it is able to reveal the microscopic features of the system. When δM is infinitely small, the sum in (4.37) may be replaced by integration, giving
∆S = M
δS = cp
θ¯ ln ¯ δM . θ M
Here, δM is the mass of fluid element instead of a single molecule. Since the temperature measured in practice is actually the mean temperature in a scale as small as that of thermometer, and the temperature of a single molecule is meaningless, the overbar for the mean value is omitted usually. For the systems with uniform potential temperature in all scales, evaluations of thermodynamic entropy are independent of the element size, since the mean potential temperatures are equal in the elements of different sizes. But for the non-uniform states, the statistical features, such as the mean potential temperature, depends on the number of samples or the size of element, and so does the thermodynamic entropy evaluated. This agrees with the fact that the randomness of a non-uniform state depends on the scale of measurements, and the samples in a certain scale may not reveal the inhomogeneities or gradients in smaller scales. The different values obtained for different sizes are not errors, but are the entropy over the different sizes respectively. In general, the larger the size is, the higher the entropy will be, because the randomness or disorderliness increases with the size on which the mean potential temperature is obtained. If expressed mathematically, we have S2 > S1 , if δV2 > δV1 ,
or
δm2 > δm1 .
So the sum of molecular entropy gives the lower limit of the thermodynamic entropy and the thermodynamic entropy of the uniform state gives the upper limit. The in-additivity of thermodynamic entropy is referring to, in this study, the entropy of the basic elements of which the distribution manifests the system disorderliness. The entropy was also considered to be additive by some authors, for example Gyftopoulos and Beretta (1991), according to (4.29) or (4.37). However, the entropy so evaluated for nonequilibrium states depends on the size of element, and so different results can be obtained when a non-uniform system is divided into
4.8. THERMODYNAMIC ENTROPY BALANCE EQUATION
89
different sets of subsystems. The scale-dependent feature of thermodynamic entropy was ignored, because the statistical features of thermodynamic variables were not emphasized in calculating the entropy for macroscopic thermodynamic systems, and only the entropy changes between equilibrium states were considered in the classical thermodynamics, while the microscale gradients and fluctuations in the equilibrium states were ignored. If the processes between nonequilibrium states are considered, the classical thermodynamic entropy law discussed previously should be stated more generally as: The thermodynamic entropy in all microscales cannot be destroyed in an isolated thermodynamic system. For the gases studied in the classical thermodynamics, of which the disorderliness may only be represented by the distribution of molecules, the entropy changes in different scales may not be independent of each other.
4.8
Thermodynamic entropy balance equation
The heat exchange given by the last integration in (4.21) can be applied for open systems. We discuss in this section only the open systems with heat exchanges but not mass exchanges with the surroundings. The entropy variations of the open systems are given by (4.25). The discussions here give actually the entropy balance equation of the open systems. The balance equation for the open systems with mass exchanges will be discussed in Chapter 26. It is discussed earlier that diffusion term in (4.25) includes the irreversible entropy sources related to diffusion kinetic energy dissipation and molecular diffusion of heat, that is DTd dm , ∆Skd + dShd = M T where 1 ∂kd dt dm ∆Skd = − M ∆t T ∂t is called the kinetic energy dissipation entropy derived from (4.13), and ∆Shd = −
∆t
M
vd · ∇ ln T dt dm
(4.38)
is called the heat diffusion entropy derived from (4.33). The integrations are used because the variables depend on position in the nonequilibrium states. Now, (4.25) is replaced by δq dm , ∆S = ∆Skd + ∆Shd + M T where δ indicates the change at a local position. It is discussed also that the entropy change caused by heat exchange includes also the irreversible component ∆Shc = −
M
∆t
vhc · ∇ ln T dt dm
(4.39)
derived from (4.32). It is produced by heat conduction and called the heat conduction entropy. Resolving this irreversible entropy source from the heat exchange term yields (4.40) ∆S = ∆Si + ∆Sh ,
90
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
where ∆Si = ∆Skd + ∆Shd + ∆Shc
(4.41)
is called the internal thermodynamic entropy production, and
∆Sh =
M
δq dm T
is the entropy change caused by heat exchanges without heat conduction. When mass exchanges take place in an open system, the internal entropy changes may not mean the entropy change inside the system mass. It represents the entropy changes related to irreversible microscale processes which may not be represented directly by thermodynamic variables. Applying (4.28) for it produces
∆Si = cp M
δ ln θ dm − ∆Sh
for the ideal gases with constant heat capacities. Since the kinetic energy dissipation, heat conduction and heat diffusion always create thermodynamic entropy, we have ∆Si ≥ 0 . The variation tendency of an open system may also be indicated by thermodynamic entropy changes. When heat exchanges take place, the system may not be in thermodynamic equilibrium with the environments, and so the exchanges across system boundary are irreversible. If the open system and environment are viewed as a new isolated system, the sum of the entropy changes in the system and environment should not be less than zero, that is ∆Ssys + ∆Senv ≥ 0 , where subscript ‘env’ indicates a variable of environment. In this equation, only the environmental entropy change caused by the heat exchanges is accounted, that is ∆Senv = −
M
δq dm . Tenv
The entropy changes in the system and environment may be calculated separately by assuming different reversible processes for the system and environment respectively. Some examples of calculating the entropy changes for the air engines working in the atmosphere will be given in Chapters 21 and 22. When the heat is conserved, the heat ejected or accepted by the system is equal in amount to that accepted or ejected by the environment. While, the entropy changes produced in the system and environment may be different, as the system and environment have different temperatures.
4.9
Relation to dynamic entropy
If molecules in an ideal gas are viewed as independent Newtonian systems, they form the isotropic system discussed in Section 2.3. The dynamic entropy of the system is
4.9. RELATION TO DYNAMIC ENTROPY
91
represented by (2.6), of which the perfect differential gives
dsd = d ln
∆r Fdr +
k0
k
for unit mass. It follows that Fdr dk . − k ∆r Fdr + k0
dsd =
As discussed in Chapter 2, the friction or inelasticity of molecular collision may cause conversion of translational kinetic energy of molecules into the rotational or vibrational kinetic energy, but not change the total amount of molecular kinetic energy. Thus, we have ∆r
Fdr + k0 = k
and so
Fdr − dk , k where Fdr = dWF is the work done by the external force. From (2.4), we see dsd = 0 or Fdr − dk = 0 . dsd =
The dynamic entropy variation is zero in the ideal gases. Inserting k = kd + kH into the previous equation yields dkH − Fdr = −dkd .
(4.42)
It is also the equation (2.28). If the change of thermodynamic entropy for unit mass is defined as ds = cv
dkH − Fdr , kH
(4.43)
we gain ds = −cv
dkd . kH
Applying (2.30) for it produces ds = −
dkd . T
(4.44)
As dkd ≤ 0, we have ds ≥ 0. Inserting (2.30) and (2.31) into (4.43) gives ds =
cv dT + pdα T
for isolated ideal gases. It follows that ds = d(cv ln T + R ln α) . This equation is the same as (4.23).
(4.45)
92
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
The right-hand side of (4.42) represents the macroscopic effect of molecular diffusions. This equation is similar to (2.4). However, the thermodynamic entropy changes of ideal gases are caused by redistributions of molecules and their momentum and energies, but not by friction of molecules. Thus, the thermodynamic entropy possesses different features from those of dynamic entropy. For example, the dynamic entropy is additive but the thermodynamic entropy is not. However, the redistribution of molecular momentum has a macroscopic effect similar to that of friction in the mechanic processes, that is to convert irreversibly the ordered kinetic energy (parcel and diffusion kinetic energy) into heat energy. Since the macroscopic changes of momentum may be illustrated as an effect of external forces in the mechanic dynamics, we introduce the viscous force in the fluid dynamics to account for the macroscopic kinetic energy dissipation. The frictional force provides a phenomenal link between the thermodynamic entropy and dynamic entropy.
4.10
Calculations for ideal gases
It is generally difficult to detect and predict precisely the molecular diffusions, so we do not use (4.25) to calculate the thermodynamic entropy change in a diffusion process. Since the thermodynamic entropy of ideal gases is a state function, its variations depend on initial and final states only and is independent of process for the initial and final states provided. The processes with different diffusions may lead to different final states. As long as the initial and final states are given, the entropy change integrated along any process gives an equal result. Thus, we may choose a set of successive reversible processes to connect the initial and final states. However, to know whether a process is reversible or not, we may need to calculate the entropy change first. Thus, the reversible processes are chosen empirically as the slow continuous process in which the system is in quasi-equilibrium with the environments. For convenience, these processes are usually selected under special conditions, such as no pressure change or no temperature change. In many cases, more than one reversible processes are applied. In the reversible processes, we have dkd = 0 and (4.25) gives
∆srev =
N δqrevj j=1
Trevj
(4.46)
for the N reversible processes assumed. Here Trevj is the system temperature in the j-th reversible process which is independent of space, and δqrevj is heat exchange of the system with the exterior in the process assumed. This heat exchange may be different from that in the real process. An isolated system may not be isolated again for assumed reversible processes. The previous equation shows that thermodynamic entropy is conserved in reversible adiabatic processes. It can be used conveniently for isothermal processes, if the heat exchange is known. But in general, the temperature changes as a system exchanges heat with the exterior. If the temperature and heat exchanges may be represented by Trevj (t) and δqrevj (t) respectively, the entropy
4.10. CALCULATIONS FOR IDEAL GASES
93
variation can be evaluated by integrating (4.46) with time, that is ∆srev =
N
∆srevj
j=1
with ∆srevj =
tj dqrevj tj−1
Trevj
dt .
However as discussed earlier, the classical thermodynamics does not provide the predictions generally. Since the analytical expression of internal energy of real substances is unknown, we have no the general expression of thermodynamic entropy change represented by thermodynamic variables for all substances. The entropy equations (4.23) and (4.27) are derived for the ideal gases with constant heat capacities. If we know the equilibrium initial and final states, the entropy change for the ideal gases in a continuous process can be calculated easily with (4.23) or (4.27), which is rewritten as T2 α2 + R ln (4.47) ∆s = cv ln T1 α1 or T2 p2 − R ln . (4.48) ∆s = cp ln T1 p1 If a process is discontinuous, we may assume N reversible processes to connect the same initial and final states. The entropy change in these reversible processes is calculated from N Trevj αrevj cv ln + R ln ∆srev = Trevj−1 αrevj−1 j=1 or ∆srev =
N j=1
Trevj prevj cp ln − R ln Trevj−1 prevj−1
The 0 and N of j indicate the initial and final states, respectively. These equations can be used conveniently if the reversible processes are assumed at constant pressure or temperature or volume. The entropy changes evaluated from the last two equations are equivalent to those obtained from the previous two equations for continuous processes. For example, when an ideal gas of unit mass expands adiabatically to the environment with constant pressure pe notably lower than the initial gas pressure p1 , it accepts no heat or dq = 0. The thermodynamic entropy change in the irreversible adiabatic process cannot be evaluated from (4.26), as it is actually given by (4.25) or (4.44). Since the diffusion is unknown, the entropy change may be calculated from (4.47) or (4.48). We may solve firstly the final thermodynamic state, since the pressure of the state is already known as pe . The thermodynamic energy law (4.15) gives (4.49) cv ∆Tequ + pe ∆αequ = 0
94
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
for the irreversible expansion. Applying the ideal-gas equation (2.22) for it yields pe cv (T2 − T1 ) + RT2 − RT1 = 0 . p1
It follows that
T1 pe cv + R cp p1 The entropy change evaluated from (4.48) gives
T2 =
∆s = cp ln
.
T2 pe − R ln T1 p1
= cp ln
cv pe +κ cp p1
p1 pe
κ
.
(4.50)
It is greater than zero as p1 > pe . We may also assume the following two reversible processes for the calculation: The gas expands reversibly and adiabatically from the initial pressure to pressure pe , then it is heated at the constant pressure to reach the final temperature T2 . In the process of isobaric heating, the gas is in thermodynamic equilibrium with the heat source in the surroundings. The temperature at the end of reversible expansion is κ R pe , κ= T1 = T1 p1 cp obtained from (4.18). The entropy created is zero in the adiabatic process, and is ∆srev = cp ln
T2 T1
in the reversible process of isobaric heating. The result is identical to (4.50) or ∆s = ∆srev . For T1 = 300 K, p1 = 1020 hPa and pe = 1000 hPa, we see T2 = 298.31849 K, T1 = 298.30658 K and ∆s = 0.04 J/(K.kg). The real final temperature T2 is higher than T1 as less mechanic work is created in the irreversible adiabatic expansion than in the reversible adiabatic expansion. The second process assumed is not adiabatic again. The thermodynamic entropy is produced by heat exchange between the gas and environment in the heating process assumed, which is equivalent to the entropy produced by molecular diffusion in the real adiabatic process. The gas accepts the heat cp (T2 − T1 ) in the reversible processes assumed, as it produces more mechanic work than in the real process. The entropy change may also be calculated from (4.28). The potential temperature of the gas is θ1 = T1 at the initial state, but increases to θ2 = T2 at the final state. It is noted that to choose a reversible process for calculating the entropy change may only be used for equilibrium systems. The changes of thermodynamic entropy in non-uniform systems cannot be calculated in this way, but may be calculated by using (4.30). As change of thermodynamic entropy is independent of process for provided initial and final states, the ∆S in (4.40) may be replaced by dsrev for unit mass. This equation may be rewritten, from (4.46), as ∆si =
N δqrev j=1
Trev
−
dq . T
(4.51)
4.10. CALCULATIONS FOR IDEAL GASES
From ∆si ≥ 0, we have
95
N δqrev j=1
Trev
≥
dq . T
(4.52)
It is an important inequality in the classical thermodynamics. The ratio of heat exchange (which does not include the dissipation of ordered kinetic energy) to the system temperature is maximum in reversible processes. In other words, the entropy created by heat exchange is maximum in reversible processes for provided initial and final equilibrium states, as no contribution is made by kinetic energy dissipation. The Earth’s atmosphere is a kind of thermodynamic system. A fundamental feature of the atmospheric variations is irreversibility. The atmosphere never repeats a circulation pattern over any scale in an isolated circumstance. This irreversibility is not simply due to molecular diffusions. The large-scale circulations are also irreversible, although the classical thermodynamic entropy, represented by the potential temperature in meteorology, may be conserved in adiabatic processes. Usually, we do not choose reversible processes for calculating thermodynamic entropy changes in the irreversible processes over a large scale, since the large-scale atmospheric variations are generally slow. The slow adiabatic processes may be considered approximately to be reversible from the point of classical thermodynamics. This implies that the irreversibility of atmospheric circulations may not be interpreted by the classical thermodynamic entropy variations. This irreversibility will be discussed by introducing a new thermodynamic entropy called the turbulent entropy in Chapter 6.
Chapter 5 Newtonian-thermodynamic system 5.1
Introduction
We have discussed in Chapter 2 two types of the simplest physical systems, the Newtonian dynamic system and classical thermodynamic system. Changes in the motion status of Newtonian system are produced by external Newtonian forces. The conversion of kinetic energy into heat energy caused by frictional force is unidirectional, so the kinetic energy lost to friction may not be recovered again in a mechanic system without mechanic energy exchange with the exterior. Meanwhile, changes of thermodynamic state of gases are caused by molecular diffusions, which are the macroscopic effect of microscopic random molecular motions at a nonequilibrium state. Motions of individual molecules may be affected by external forces, while the statistical effect of molecular motions and collisions may not be associated with macroscopic Newtonian forces, and is controlled by the second law of thermodynamics. To study the nonequilibrium thermodynamic processes, the nonequilibrium thermodynamics introduced the generalized forces related to the microscale gradients of thermodynamic variables. The generalized forces may not be applied for Newton’s laws, and so are not regarded as the Newtonian forces. The air in the Earth’s atmosphere will be assumed as an ideal gas in this study, since changes of the heat capacities and thermopotential energy are negligibly small at the normal conditions. The ideal-gas equation and thermodynamic energy law of ideal gases may also be applied to study the atmospheric processes. As in the classical thermodynamics, the molecular diffusions in the atmosphere are also unpredictable. However, they are not as important as the parcel processes for changes of large-scale atmospheric circulations. Thus, the atmospheric dynamics and energetics deal with mainly the processes related to the parcel motions on meteorological scales, which may include the moist effects resulting from phase transitions of water in the atmosphere. Since the atmosphere is compressible, parcel motions in the vertical direction or across the isobaric surfaces may lead to changes in the thermodynamic states. Owing to the thermodynamic processes, exchanges of mechanic energy and internal energy in the atmosphere take place in both directions through expansion and contraction of air parcels, so the parcel kinetic energy lost to boundary friction and fluid viscosity can be recovered again through thermodynamic processes. Therefore, the atmosphere is neither a simple Newtonian system nor a classical thermodynamic system, but is a combination of the both. The changes of kinetic status can be studied by Newton’s laws, and the changes of thermodynamic state are studied by thermodynamic laws. So, the governing equations of atmospheric processes, called the primitive equations, include all the physical relationships adopted for the studies of Newtonian systems and classical thermodynamic systems. The energy equations for the individual air parcels and whole atmosphere may be derived using all the equations as shown in Chapter 7. They include the mechanic energies 96
5.2. FIELD VARIABLES
97
and thermodynamic energies, and so are different from the energy conservation laws used for a mechanic system or classical thermodynamic system. Under the effects of Earth’s gravitation and rotation, the atmosphere, as a thermodynamic system, is different from the classical thermodynamic systems. These differences have not been revealed adequately in the previous studies of atmospheric thermodynamics and energetics. The steady states of rotational atmosphere manifest the inhomogeneities in the vertical and horizontal directions. The inhomogeneous steady states will be referred to as the inhomogeneous equilibrium states. The thermodynamic systems exhibiting the inhomogeneous equilibrium states will be called the inhomogeneous thermodynamic systems. The thermodynamics dealing with these systems is then referred to as the inhomogeneous thermodynamics. It is different from the nonequilibrium thermodynamics, since nonequilibrium thermodynamics deals with mainly the nonequilibrium processes of classical thermodynamic systems of which the equilibrium states are uniform. In the gravitational field, the gradient of pressure or mass density produces a particular internal force in the fluid called the pressure gradient force. This force is also an external force for a fluid parcel. The resultant variations in fluid motions can be studied with Newtonian dynamics. An incompressible fluid like the ocean water is similar to a simple Newtonian system, except that the external forces for parcels may change with thermodynamic state of the system. Motions of incompressible fluid are independent of thermodynamic processes, so the parcel kinetic energy cannot be created inside the fluid if without affected by external mechanic processes. This is the major difference from the processes in a compressible fluid such as the Earth’s atmosphere. The atmospheric processes, in which kinetic energy generation is critical, should not be studied with incompressible approximation.
5.2
Field variables
Due to the gravitational force of the Earth, the atmospheric pressure at a geographic height z indicates also the weight of an steady air column of unit horizontal section area above the height, represented by the hydrostatic equation Z
ρ dz ,
p=g
(5.1)
z
where Z is the top height of the atmosphere. It tells that the local pressure changes with height. The density of a large-scale air mass in equilibrium decreases exponentially with height in the gravitational fields as discussed in Chapter 17. Owing to the input of long wave radiations from the Earth’s surface and the output from the top of atmosphere, the atmosphere exhibits a vertical gradient in the temperature profile also. These vertical gradients cannot be destroyed entirely by molecular diffusions, and do not suggest a nonequilibrium state of the atmosphere. The solar radiation and Earth’s longwave radiation accepted by the atmosphere vary also with the latitude, cloud coverage, surface temperature and albedo, etc.. The temperature, humidity, friction, albedo and evaporation on the surface together with the surface fluxes of sensible and latent heats may change greatly from ocean surfaces to continents. The air masses over oceans and continents or at low and
5.
98
NEWTONIAN-THERMODYNAMIC SYSTEM
high latitudes may be significantly different from each other in their mechanic and thermodynamic features. There are the boundaries with relatively strong contrast of thermodynamic states between the large-scale air masses. Due to rotation of the Earth, the large-scale atmospheric circulations are constrained by geostrophic balance (See Section 5.6), and the ageostrophic component is usually one order smaller than the geostrophic component. Mixing of air masses across the pressure gradient and boundaries resulting from ageostrophic motions is then prevented by the geostrophic balance, so that the boundaries of air masses, like the temperature fronts, may sustain for many days in the geostrophic circulations. Therefore, the atmosphere manifests the large-scale inhomogeneities in the horizontal directions even in the climatological mean fields. The theoretical studies of Paltridge (1975, 1978; Rodgers, 1967; Noda and Tokioka, 1983) show that the climatologically equilibrium state of the open atmosphere with minimum meridional exchanges of heat or thermodynamic entropy manifests the vertical and meridional gradients of thermodynamic variables. To study an inhomogeneous fluid system, we use the time-space coordinates to illustrate the variable distributions and variations or to locate the individual parcels. A thermodynamic variable in the time-space coordinates is a function of time and space, and is called the field variable. In general, the geographic height z of a isobaric surface in (5.1) changes horizontally, and so the pressure depends on the horizontal coordinates also. The variable p represents not only the pressure, but also the vertical position of an air parcel in the pressure coordinate. The parcels at different pressure surfaces possess different amounts of geopotential energy. The variation in geographic height of a parcel may lead to the changes of its thermodynamic state and motion status or the exchanges between its geopotential energy, kinetic energy and internal energy. In an inhomogeneous fluid system with fluid motions, most physical quantities are carried by fluid parcels. Assume that the quantity = is carried by a parcel of unit mass. The change of = in the region occupied by the parcel at a give time is denoted by ∂= δt . ∂t It may be produced by change of the parcel itself d= δt , dt and by replacement of the fluid ∂= δr ∂r across the region. Here, ∂/∂r is the directional derivative along the velocity at the time and place. The negative sign is added, so that the term is positive if the local value is increased by transport. Thus, the local change rate is given by −
∂= ∂t
= =
d= ∂= δr − lim dt ∂r δt→0 δt ∂= d= − vgl . dt ∂r
(5.2)
5.3. PARCEL AND PARCEL VELOCITY
99
Here, dr dt is the general velocity which include the mass velocity, conduction velocity or propagation velocity on all scales. The local change rate is measured at a given place, while the material change rate on the right-hand side of (5.2) is measured by following the general velocity. In a flowing fluid, the local changes are generally different from material changes, unless the fluid is homogeneous so that a particular parcel may not be distinguished from others. The second term on the right-hand side is called the advection produced by fluid motion, conduction or propagation across the gradient. Referring to (4.5) finds ∂= = vgl · ∇= . vgl ∂r The material change rate gives now vgl =
∂= d= = + vgl · ∇= . dt ∂t
(5.3)
It is similar to (4.6) except that the general velocity includes the moving velocity of macroscopic fluid parcels, of which the variations are controlled by Newton’s second law as discussed in the next section. The previous equation may be rewritten as ∂= d= = + ∇ · (=vgl ) − =∇ · vgl . dt ∂t The second term on the right-hand side indicates the gain or loss caused by convergence of = conduction or propagation in the fixed region, which does not include the effect of mass convergence. The last term represents the correction by the volume or density change in the region. For example, when the density decreases resulting from the velocity divergence, a mount of the quantity is removed from the region. If the transport process is independent of mass displacement, the convergence of conduction or propagation velocity in the last term represents a kind of dispersion. For the nondispersive propagation or conduction velocity v which is independent of space, we have ∂= d= = + ∇ · (=v) . dt ∂t This equation may be found in many studies.
5.3
Parcel and parcel velocity
We discussed in Chapter 3 the instantaneous diffusion velocity of a diffusion element. The diffusion element is a small volume of gas. Molecules in a diffusion element keep exchanging with the surroundings. The diffusion velocity represents the statistical effect of molecular motions influenced by molecular collisions, which may not be the moving velocity of a fixed molecular ensemble. It may be changed by exchanging the molecules across the boundary when the effect of external Newtonian forces is
5.
100
NEWTONIAN-THERMODYNAMIC SYSTEM
ignored. We may define mathematically the trajectory of a diffusion element with the instantaneous velocity. However, the definition may not be physically significant, as the constituents of the element keep changing along the trajectory. Although motions of a single molecule are affected by external forces, the macroscopic effect of molecular diffusion may not have a general relation to macroscopic Newtonian forces, or meet with Newton’s second law. The fluid element of which the motion status is controlled by macroscopic Newtonian forces is called the parcel in this study. A parcel is large enough so that the mass and momentum variations caused by molecular diffusions across the boundaries or molecular collisions at the boundaries is negligibly small within a certain time. It is also small enough so that its thermodynamic state is considered to be uniform. Since molecular diffusions exist everywhere in an inhomogeneous fluid, a small parcel is a time-mean feature only. The fluctuations of density and other variables caused by molecular diffusions may be ruled out by making averages over the parcel scale. Motions in a fluid may then be represented by the motions of parcels, and may be studied by using the classical Newtonian dynamics and thermodynamics for individual parcels as in the Lagrangian dynamics. In the Eulerian dynamics, fluid motions are represented by the field of mass velocity covering each point continuously in the three-dimensional space. In general, the total mass velocity includes parcel velocity vp and instantaneous diffusion velocity vd , that is (5.4) v∗ = vp + vd . It is the velocity of mass displacement which may be caused by different mechanisms. The horizontal components of parcel velocity in the atmosphere are called the winds, and can be measured by meteorological instruments. Changes of parcel velocity are caused by Newtonian forces, such as the gravitational fore, pressure gradient force (referring to Section 5.5) and frictional force. While, changes of diffusion velocity follow the second law of thermodynamics and may be independent of macroscopic Newtonian forces. The effect of molecular diffusion on the local changes in a tiny-scale may be different from that of parcel motion. Assuming that an airflow in geostrophic balance is along the isobars in the rotational atmosphere, temperature variations at a local place may be brought about by parcel motions if there is a temperature gradient along isobars. While, the heat conduction resulting from molecular collisions takes place in the direction down to the microscale temperature gradient (seeing Chapter 3). Thus, the heat fluxes resulting from parcel motion and heat conduction may be in different directions. Since the microscale gradient of pressure can not be detected in practice, the identifications of mass diffusion and parcel motion may only be technic in many situations, or depend on the scale resolutions of data and calculation models. If only the mass velocity and conduction velocity are considered, the general velocity is given by vgl = vp + vd + vc , and the material change rate reads ∂= d= = + vp · ∇= + (vd + vc ) · ∇= , dt ∂t
(5.5)
5.3. PARCEL AND PARCEL VELOCITY
101
where (vd + vc ) · ∇= = D1d represents the macroscopic effect of the microscopic processes of molecular diffusions or collisions. When the microscopic processes are ignored, we have ∂= d= = + vp · ∇= . dt ∂t
(5.6)
The material change rates in the thermodynamic equations discussed in Chapter 4 may be expanded with (5.6) or (5.5). The expanded thermodynamic energy law is called the heat flux equation, and the expanded mass conservation law is called the continuity equation. The kinetic energy density of a fluid, called the specific kinetic energy usually, is defined for a moving fluid element in unit mass. This is also true for the kinetic energy of mechanic waves. Thus, only the mass velocities, but not the conduction velocities and propagation velocities, are used for calculating the kinetic energy. When the total mass velocity is resolved into components, the instantaneous kinetic energy is not simply the sum of kinetic energy of each component. This can be seen from m (vp + vd ) · (vp + vd ) 2 vp2 vd2 + + vp · vd , = m 2 2
K∗ =
where m is mass of the fluid element. The last term is different from zero if the parcel velocity and diffusion velocity are not perpendicular to each other. As the parcel is a mean feature, the parcel velocity may be considered as the mean mass velocity over the typical time-scale of parcel denoted by τ , that is 1 vp = τ
v∗ dt .
τ
The instantaneous diffusion velocity is the departure from the mean velocity, and we have vd dt = 0 . τ
Usually, the parcel velocity means the mean velocity in most studies, and so the diffusion velocity is ignored. The mean kinetic energy over the scale of parcel motion is given by vp2 v2 (5.7) K = m+ dm . 2 2 The first term on the right-hand side may be called the parcel kinetic energy, and the last term may be referred to as the diffusion kinetic energy. This equation shows that the effect of molecular diffusion on the kinetic energy cannot be ignored, even if the diffusion is anisotropic in a parcel. As conversion of kinetic energy into heat energy resulting from molecular diffusions is irreversible as discussed in the previous chapters, parcel kinetic energy cannot be conserved in a dynamically isolated system due to molecular diffusions.
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5.4 5.4.1
NEWTONIAN-THERMODYNAMIC SYSTEM
Mass and heat transport equations Continuity equations
If there are parcel motions in a fluid, we may use the total mass velocity v∗ for (4.8) and (4.9) to obtain the mass conservation equations: ∂ρ + ∇ · (ρvp ) + ∇ · (ρvd ) = 0 , ∂t or
dρ = −ρ∇ · vp − ρ∇ · vd . (5.8) dt They are called the continuity equations. The first equation shows that the local change of density is produced by divergences of mass fluxes carried by parcel motion and mass diffusion. The second equation tells that the change of parcel density is produced by divergences of parcel velocity and diffusion velocity. When the effect of diffusion is negligibly small compared with that of parcel motions, the continuity equations are replaced by ∂ρ + ∇ · (ρvp ) = 0 (5.9) ∂t and dρ = −ρ∇ · vp . (5.10) dt The latter is usually written as dα = α∇ · vp , dt
(5.11)
where α is the specific volume of the fluid. Unlike in the study of molecular diffusions in Chapter 4, the continuity equation is still meaningful if molecular diffusions are ignored. This equation together with the heat flux equation, state equation and momentum equation forms a complete set of prediction equations, called the primitive equations, for three thermodynamic variables and parcel velocity. The continuity equation is also used in the nonequilibrium thermodynamics to obtain a complete equation set. The fluids studied by nonequilibrium thermodynamics may not necessarily exhibit parcel motions even at a nonequilibrium state. In this case, the continuity equation should be replaced by the mass diffusion equation (4.9) or diffusive transport equation (4.8).
5.4.2
Integrated variations in a system
If only the transport carried by macroscopic mass velocity or parcel velocity is considered, we multiply (5.6) by mass density ρ and gain ρ
∂= d= =ρ + ρvp · ∇= . dt ∂t
Next, multiplying (5.9) by = gives =
∂ρ + =∇ · (ρvp ) = 0 . ∂t
5.4. MASS AND HEAT TRANSPORT EQUATIONS
103
Adding these two equations together produces
d= ∂(ρ=) dρ ρ = + ∇ · (ρ=vp ) − = ρ∇ · vp + dt ∂t dt
.
(5.12)
When the mass is conserved, the last term disappears according to continuity equation (5.10). Thus, we gain ρ
∂(ρ=) d= = + ∇ · (ρ=vp ) . dt ∂t
(5.13)
The left-hand side is the change of a unit volume. The mass in the moving volume may change. The first term on the right-hand side is the local change of = density in the volume at a given time including the change in mass or density, and the last term is the flux convergence in the element of volume. Integrating it over an atmospheric domain yields 1 g
ps d= A pt
dt
dpdA =
zt ∂(ρ=) A zs
∂t
zt
∇ · (ρ=vp ) dzdA ,
dzdA + A zs
where zs and zt are the heights of the bottom pressure surface ps and top pressure surface pt , respectively. The Gauss equation gives zt zs
∇ · (ρ=vp ) dzdA =
✵
ρ=vn d✵ ,
in which vn is the outward velocity component normal to the outer surface ✵ of the domain integrated in Cartesian coordinates. Thus, we have 1 g
ps d=
dt
A pt
dpdA =
zt ∂(ρ=)
∂t
A zs
dzdA +
✵
ρ=vn d✵ .
(5.14)
For the whole atmosphere or a domain with fixed walls, we see 1 g
ps d= A pt
dt
dpdA =
zt ∂(ρ=) A zs
∂t
dzdA
(5.15)
since vn = 0. The parcel changes integrated over the system may be evaluated by integrating the local changes, as no transport across the boundaries.
5.4.3
General continuity equation
For convenience, the transport process with conservation of the quantity = transported is referred to as the simple transport process. The density of the quantity ρ= may change in the transport, but the total amount m= in mass m is conserved when the mass is conserved. Since there are no the sources or sinks of the quantity in the simple transport process, we have ∂(ρ=) = −∇ · (ρ=v) . ∂t
(5.16)
It tells that the change of local density of = is caused by the flux convergence. The velocity in this equation may be mass velocity or other velocity, which carries
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NEWTONIAN-THERMODYNAMIC SYSTEM
the quantity to form the flux. In other words, the flux may also be produced by the conduction or propagation without mass displacement accompanied. If the = is temperature and the velocity is heat conduction velocity, (5.16) gives the heat conduction equation used widely in physics. It is discussed in Section 4.3 that the velocity in the continuity equation can only be the mass velocity. If the transport is caused by conduction without mass transport, the continuity equation (5.10) cannot be used and (5.12) gives
ρ
dρ d= = −= ρ∇ · v + dt dt
,
where v is the conduction velocity. Adding the first and last terms together yields d(ρ=) = −ρ=∇ · v . dt It tells that the density of = may be changed by dispersion or concentration of the transport velocity, called also the divergence or convergence. This equation may be referred to as the general continuity equation and may be applied for other transport. For mass transport, we set = = 1 and the velocity is a mass velocity. Thus, the continuity equation of mass is an application of this general continuity equation. The discussions above show that the general continuity equation is equivalent to (5.16). It is emphasized that these two equations are only used for simple transport processes.
5.4.4
Heat flux equation
Using the expansion equation (5.5) for the thermodynamic energy law of ideal gases given by (4.12), we obtain the heat flux equation cv
dα dq ∂T = −cv vp · ∇T − p + − cv (vd + vhc ) · ∇T + DTd , ∂t dt dt
where vhc denotes the heat conduction velocity which is not a mass velocity. The first term on the right-hand side is the advection of heat energy caused by parcel motions. The local temperature increases as a flow moves downgradient in the temperature field, or the local mass is replaced by a warmer mass. The second term is the rate of doing mechanic work by a parcel as it changes volume against external pressure force. The third term is the diabatic heating rate including the radiation and latent heating. The fourth term is the heat transport produced by molecular heat diffusion and conduction. The heat diffusion entropy and heat conduction entropy given by (4.38) and (4.39) respectively are derived from this term. The last term is the rate of kinetic energy dissipation resulting from molecular diffusion called the molecular dissipation previously. The heat conduction in the previous equation may be included in the diabatic heating, and the heat diffusion may be added to the last term. Thus, the heat flux equation becomes cv
dα dq ∂T = −cv vp · ∇T − p + + DTd , ∂t dt dt
(5.17)
5.4. MASS AND HEAT TRANSPORT EQUATIONS
105
When the microscopic processes of molecular diffusion and conduction are ignored, the heat flux equation becomes cv
dα dq ∂T = −cv vp · ∇T − p + . ∂t dt dt
(5.18)
dp dq ∂T = −cp vp · ∇T − α + ∂t dt dt
(5.19)
It can be rewritten as cp
derived using (4.14) for ideal gases, where dp/dt may be considered as the vertical velocity in pressure coordinates. The heat flux equation shows only a part of energy transfers in the compressible fluid, and does not account for the energy transfer between heat energy and kinetic energy. The density change in (5.18) and the pressure change in (5.19) provide the physical connections to the kinetic energy transfer. The energy conversions of all kinds in the atmosphere and the energy conservation equation will be studied in Chapter 7.
5.4.5
Heat conduction equation
In the simple transport process caused by heat conduction, the flux equation (5.16) gives ∂(cπ ρT ) + ∇ · (cπ ρT vhc ) = 0 , ∂t in which cπ is a polytropic heat capacity and cπ ρT vhc = Fhc is the conductive heat flux, of which the x-component is given by (3.38). If the flux is represented by Fhc = −λ∇T , where λ is the heat conductivity which is not constant in general as shown by (3.39). we have ∂(cπ ρT ) − ∇ · (λ∇T ) = 0 . ∂t When the heat conductivity, heat capacity and density are taken as constant for a solid substance, we gain the heat conduction equation cv ρ
∂T − λ∇2 T = 0 ∂t
(5.20)
in three dimensions. This equation is similar to mass diffusion equation (4.11). The solution can be found in many text books of mathematical physics. Since ∇2 T is positive at a low center and negative at a high center, the heat conduction destroys the temperature gradient and leads to the isothermal state.
5.
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5.5 5.5.1
NEWTONIAN-THERMODYNAMIC SYSTEM
Inhomogeneous thermodynamic system Adiabatic and transport processes
Adiabatic processes The thermodynamic states of atmosphere are represented by the field variables varying with space at a given time. The parcel and local variations may be produced not only by molecular diffusions, but also by parcel motions. The effect of parcel motions is more important than that of molecular diffusions at least in the dry atmosphere. There are two types of basic processes, adiabatic process and transport process, related to parcel motions and local variations in the atmosphere. In a compressible fluid, the temperature and density of a parcel may change adiabatically as it moves vertically or crosses isobaric surface. If the motion is slow enough, the adiabatic variations may be considered approximately as a thermodynamic reversible process. The parcels are assumed as ideal gases in this study. When ds = 0 in (4.27), we have R . (5.21) d ln T = κd ln p , κ= cp It may be integrated to gain the potential temperature (2.55). Since ds = 0 in reversible adiabatic processes, the potential temperature is conserved as shown by (4.28). It is emphasized that the potential temperature may change in an irreversible adiabatic process. An example of irreversible adiabatic expansion is given in Section 4.10. When the potential temperature is conserved, the temperature may change. To evaluate the change rate as a parcel moves adiabatically in the vertical direction, (5.21) is rewritten as d ln T =κ. d ln p In an adiabatic process of parcel motion, the temperature may be different from environmental temperature. If the variables of a parcel are denoted by´, (5.1) may be replaced by ´ dp (5.22) cp dT´ = α for the parcel. It is assumed usually that the parcel pressure is adjusted to the environment in the movement, so the pressure change in the last equation may be evaluated from hydrostatic equation (5.1), or dp = −gρdz .
(5.23)
In the inhomogeneous atmosphere, the isobaric surfaces may not be horizontal. For a parcel moving on a sloping isobaric surface, the geographic height may change but the pressure may not. So, this equation can only be used in the vertical direction. Inserting (5.23) into (5.22) yields dT = −
gα ´ dz cp α
5.5. INHOMOGENEOUS THERMODYNAMIC SYSTEM
or −
dT = Γd , dz
107
Γd =
gα ´ . cp α
(5.24)
The Γd is called the dry adiabatic lapse rate, which is the change rate of temperature of unsaturated parcels, as the parcels move reversibly and adiabatically in the vertical direction. If the difference between α ´ and α is small, we write Γd ≈
g . cp
(5.25)
The lapse rate is nearly constant. Errors may be produced if this equation is used for the convection with a sloping trajectory, called the slantwise convection, in the atmosphere with strong horizontal gradient of pressure. The adiabatic lapse rate for the slantwise convection will be discussed in Chapter 19. Macroscopic transport processes Apart from the adiabatic processes, the local changes may also be brought about by replacement of air masses. The moving air carrying a physical quantity forms the flux which delivers the quantity from one place to another. A flux of quantity = carried by a flow at a parcel velocity across the plane of unit area perpendicular to the velocity is given by F1 = =vp . The local changes may be produced by convergence of the flux, as shown by continuity equation (5.9). The convergence is evaluated from ∇ · F1 =
∂F1y ∂F1z ∂F1x + + , ∂x ∂y ∂z
which measures the local concentration caused by the flux. To see the adiabatic and transport processes more clearly, the advection of temperature is rewritten as vp · ∇T = ∇ · (T vp ) − T ∇ · vp . Applying continuity equation (5.11) for it yields vp · ∇T = ∇ · (T vp ) −
p dα . R dt
Inserting it into heat flux equation (5.18) gives
cv ∂T = −cv ∇ · (T vp ) + 1 − cv ∂t R
p
dα dq + . dt dt
The first term on the right-hand side is the convergence of heat flux, the second represents the adiabatic expansion or contraction, and the last is the diabatic heating rate. This equation shows that the local changes are produced by the three basic processes, transport process, adiabatic process and heat exchange process.
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5.5.2
NEWTONIAN-THERMODYNAMIC SYSTEM
Inhomogeneous thermodynamics
If the changes caused by molecular diffusions are negligible compared with that resulting from parcel motions, an inhomogeneous fluid system may be in a steady state viewed from a meteorological scale. This steady state, when it is inhomogeneous, will be referred to as the inhomogeneous equilibrium state. The molecular diffusions and produced microscale or tiny-scale variations at an inhomogeneous equilibrium state may not be expressed by the data over a certain scale. A thermodynamic system characterized by the inhomogeneous equilibrium state will be referred to as the inhomogeneous thermodynamic system. The Earth’s atmosphere gives an example of the system. The thermodynamics which studies the inhomogeneous thermodynamic system is referred to as the inhomogeneous thermodynamics. It is not called the nonequilibrium thermodynamics, because the latter deals with the nonequilibrium processes in classical thermodynamic systems. The nonequilibrium states of a classical thermodynamic system are also inhomogeneous, but the equilibrium states are homogeneous. Variations of classical thermodynamic system may be brought about by molecular diffusions only. While, variations of inhomogeneous thermodynamic system depend highly on parcel motions. The mechanic energies, such as the parcel kinetic energy and geopotential energy, are also included in the energy budget equation for individual parcels or the whole system. The inhomogeneous thermodynamic systems possess the properties different from those of classical thermodynamic systems. For example, there is the zeroth law for classical thermodynamic systems which states that when two separate thermodynamic systems are in thermal equilibrium with the third one respectively, the two systems are in thermal equilibrium with each other. This law is no longer available for inhomogeneous thermodynamic systems. The inhomogeneous thermodynamic systems are studied also in the time-space coordinates. In the Lagrangian representation, a parcel may be viewed as a classical thermodynamic system when it is small enough, of which thermodynamic variations may be studied with the classical thermodynamics. In the Eulerian representation then, the thermodynamic state at a given point of the coordinates depend on time only, so a local fluid medium may be considered as an open classical thermodynamic system. The mass and heat exchanges at the point are evaluated with the advection terms, derived by expanding the material derivatives in the thermodynamic equations as discussed earlier.
5.6 5.6.1
Momentum equation of atmosphere Pressure gradient force
To study fluid motions with the classical dynamics, the hydrodynamics has introduced a Newtonian force called the pressure gradient force, which is the net pressure of surroundings acting on the parcel surface in the normal direction. According to Newton’s second law, the external Newtonian force acting on a fluid element may be measured by the resultant change rate of momentum of the element. The mo-
5.6. MOMENTUM EQUATION OF ATMOSPHERE
109
mentum is defined with mass velocity in the classical dynamics. In the quantum physics, the momentum may also be defined with propagation velocity like the light velocity, since a wave packet which propagates at a propagation velocity or group velocity, may be regarded also as a quantum with a certain amount of mass. When the mass of a fluid element is conserved, the change rate of momentum gives the acceleration of fluid motion. The acceleration produced by pressure gradient force is given by dvp p d✵ , =− ρδV dt ✵ where ✵ is the outer surface of a parcel with density ρ and volume δV . The parcel velocity vp instead of general velocity is included in this equation, since variations of the microscopic velocities, such as diffusion velocity and conduction velocity, may be independent of the macroscopic pressure gradient force or Newton’s second law. The environmental pressure acting on the surface is identical in magnitude to the surface pressure of the parcel but in an opposite direction, so it is represented by the parcel pressure p with a negative sign in the equation. From the Gauss equation V
∇p dV =
we obtain
✵
p d✵
dvp =− ∇p dV . ρδV dt V As the volume of parcel tends to zero, we gain dvp = −α lim δV →0 dt
or
∇pdV = −α∇p δV
V
∂p ∂p ∂p dvp ˆ−α y ˆ−α z ˆ = −α x dt ∂x ∂y ∂z
(5.26)
for unit mass. The pressure gradient force is in the downgradient direction of pressure field. Its value is proportional to the intensity of pressure gradient. So, this equation may be derived also in terms of pressure gradient, as shown in Appendix C.
5.6.2
Navier-Stokes equation
Adding the external friction near fixed boundaries F, such as the surface drag, to (5.26), we gain dvp = −α∇p + F . (5.27) dt The derivative on the left-hand side of the previous equation may be expanded as ∂vp dvp = + vp · ∇vp + (vd + vmc ) · ∇vp , dt ∂t where vd and vmc are the diffusion velocity and momentum conduction velocity, and may be derived respectively from the diffusive momentum flux and conductive
5.
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NEWTONIAN-THERMODYNAMIC SYSTEM
momentum flux discussed in Chapter 3. The last term tells that the parcel velocity may be changed by molecular diffusion and momentum conduction. The effects are called the internal friction or viscosity. For the simple transport process caused by momentum conduction without mass displacement, the change of local density of vertical momentum denoted by ρvz may be given by ∂(ρvz ) = −∇ · (ρvz vmc ) ∂t derived from (5.16), where ρvz vmc = Fzc is the conductive flux of z-momentum. Referring to (3.32), the three-dimensional flux may be written as Fzc = −η∇vz , in which η is the coefficient of viscosity. Now, we have ∂(ρvz ) = ∇ · (η∇vz ) . ∂t It follows that
∂vz = ∇η ∇ · vz + η ∇2 vz , ∂t when the local density is unchanged in the transport process, where η = η/ρ is called the kinematic coefficient of viscosity. From (3.33), we see
2l η = 3
2RT . 3π
The local change of three dimensional velocity caused by momentum conduction gives ∂v = ∇η ∇ · v + η ∇2 v (5.28) ∂t with ˆ + ∇ · vy y ˆ + ∇ · vz zˆ ∇ · v = ∇ · vx x and
ˆ + ∇2 vy y ˆ + ∇2 vz zˆ . ∇2 v = ∇2 vx x
The right-hand side of the previous equation gives the expression of viscosity which is similar to the Navier-Stokes viscosity. The velocity equation including this viscosity gives dvp = −α∇p + ∇η ∇ · v + η ∇2 v . dt It is called the Navier-Stokes equation. Apart from the momentum conduction, the viscosity or momentum exchange may also be produced by molecular exchanges resulting from molecular diffusion. This part of viscosity may be evaluated from the diffusive momentum flux discussed in Chapter 3. The viscosity may also be represented by integrating the stress tensors over the surface in the classical hydrodynamics (e.g., Milne-Thomson 1960; Batchelor, 1967). The fluid of which the viscosity may be given approximately by (3.31)
5.6. MOMENTUM EQUATION OF ATMOSPHERE
111
is called the Newton’s fluid. Examples of the steady two-dimensional flows with the balance between pressure gradient force and Newton’s viscosity are the Couette flow and Poiseuille flow (Tritton, 1988). Equation (5.28) may be called the momentum conduction equation which, like the heat conduction equation (5.20) and mass diffusion equation (4.11), is a transport equation. We may also derive the heat diffusion and momentum diffusion equations. These transport processes destroy the system gradients and lead to the uniform state. It is noted that the total momentum of the system may not be changed by the particle momentum exchanges inside the system, if the effects of external forces including the pressure gradient force are ignored. So on theory, the momentum conduction may lead to a uniform velocity of the fluid if without external friction. However, it will be proved in Chapter 7 that the kinetic energy is reduced by the momentum conduction, although the total momentum is conserved in the process of velocity assimilation. Usually, the viscosity and boundary friction are represented together by F in (5.27), and so the derivative on the left-hand side of this equation is replaced by ∂vp dvp = + vp · ∇vp . dt ∂t The momentum equation (5.27) shows that the parcels driven by pressure gradient force move from high pressure to low pressure or from high density to low density at a constant temperature. As a result, the gradients may be destroyed by the mass redistribution. The pressure gradient force depends on thermodynamic states of the parcel and environments, and so may change from time to time and from place to place with the change of thermodynamic states in parcel motions. After the pressure gradient disappears, parcel kinetic energy will be dissipated by internal viscosity and boundary friction. In this process, the molecular diffusions become substantially important. We have learned in Chapter 3 that the diffusive mass flux is also from high density to low density and destroys the gradients. The similarity between the parcel motions and molecular diffusions allows us to extend the classical thermodynamic entropy law to the large-scale inhomogeneous fluid system as discussed in the next Chapter. However, the physical mechanism of molecular diffusions is different from that of parcel motions. The direction of molecular diffusion depends on the gradients of number density of molecules and mean molecular speed. The diffusion follows the second law of thermodynamics and may be independent of Newtonian forces including the gravitational force. An example is the mixing of two gases of different temperatures at a constant pressure. This downgradient mass diffusion may also occur if without the effect of gravity. While, the parcel motions are controlled by Newton’s laws, in which the gravitational force plays an important role. This will be discussed again in the last section of this chapter.
5.6.3
Momentum equation of atmosphere
The moving bodies on the Earth including the air parcels in the atmosphere may remain their velocities in the absolute space, if they are free from external forces.
5.
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NEWTONIAN-THERMODYNAMIC SYSTEM
Since the rotational Earth is not an inertial system with respect to the absolute coordinates, the free motions change their directions when viewed on the Earth. To explain the direction changes in the coordinates of Earth, it is assumed that the moving bodies are affected by an inertial force called the Coriolis force given by (Holton, 1992) Fc = −2Ω × vp , where Ω is the angle acceleration of the Earth with the positive direction to the north. The Coriolis force increases linearly with the speed. Since Ω is positive in the Northern Hemisphere and negative in the Southern Hemisphere, this force is normal to the velocity towards the right if following the moving body in the Northern Hemisphere, and is towards the left in the southern. If viewed in the Cartesian coordinates on the Earth, the Earth’s angle acceleration has the components upward and northward, that is ˆ + Ω sin ϕ zˆ , Ω = Ω cos ϕ y in which ϕ indicates the latitude. The Coriolis force becomes ˆ + 2Ωvx cos ϕ ˆz . x − 2Ωvx sin ϕ y Fc = −2Ω(vz cos ϕ − vy sin ϕ)ˆ The direction of Coriolis force can be found also in this equation. Adding the Coriolis force to the momentum equation (5.27) yields dvp = −α∇p − 2Ω × vp + F . dt When the parcels move from high pressure to low pressure at an increasing speed driven by pressure gradient force in the atmosphere, they are pushed by the Coriolis force towards one side. Since the force increases with the speed, the motion direction keep changing until it is parallel to the isobars if the viscosity is ignored. The steady state, called the geostrophic balance, may be reached as the parcel moves at the horizontal velocity α vg = zˆ × ∇h p f in the statically stable atmosphere. Here, f = 2Ω sin ϕ is the Coriolis parameter, and ∇h indicates a horizontal gradient. It tells that the velocity, called the geostrophic velocity, is perpendicular to the pressure gradient in a geostrophic balance. The velocity components are α ∂p , (5.29) vxg = − f ∂y and α ∂p . vyg = f ∂x The idealized geostrophic balance occur as the air moves along an isobar at a constant latitude in the environment with constant pressure gradient without viscosity. The steady geostrophic motions on the f -plane in the viscous atmosphere have the velocity 1 ˆz × (α∇h p − F) f 1 ˆ×F, = vg − z f
v =
5.7. SHALLOW WATER DYNAMICS
113
in which F depends on the velocity in general. The velocity deflects to the side of low pressure, so that the component of pressure gradient force in the motion direction may be balanced by the viscosity, and the component perpendicular to the velocity may be balanced by the Coriolis force. The departure of velocity from the geostrophic velocity is called the ageostrophic velocity, represented by 1 ˆ×F. va = v − vg = − z f Since the friction is in the opposite direction of the velocity, the ageostrophic component is perpendicular to the real flow towards the low pressure. The previous two equations give v = vg + va . The three velocities form a right triangle. The intensity of ageostrophic component increases with the friction, and so the angle between the real flow and isobars increases also while the flow speed decreases. When the surface friction decreases upward, the wind in the boundary layer turns right or left with increasing height in the Northern or Southern Hemisphere, and the speed increases if the pressure gradient is unchanged. The fluid motions affected by the vertical reduction of viscosity form the Ekman flow (Holton, 1992) in the oceans and atmosphere. Since the friction and viscosity convert the kinetic energy into heat energy, a steady velocity field in the viscous atmosphere must be supported by kinetic energy sources. For the atmosphere in the gravitational field, we may add the gravitational force g (which includes the centrifugal force and is downward in the Cartesian coordinates) to gain the final expression dvp = −α∇p + g − 2Ω × vp + F . dt
(5.30)
It is called the momentum equation of the atmosphere in meteorology, and may also be used to study ocean flows. At the first glance, the gravitational force affect only the vertical motions. This is not true, since the horizontal pressure gradient force depends substantially on the gravity as discussed in the following sections.
5.7
Shallow water dynamics
The shallow water assumption is used frequently for the studies of atmospheric processes. However, the dynamics of shallow water is substantially different from that of the atmosphere. As discussed earlier, there are three basic processes in the atmosphere: the transport process, adiabatic process and diabatic process. The adiabatic process is particularly important, as it leads to exchanges of heat energy and mechanic energy in both directions through the volume change in a compressible fluid. As in the atmosphere, the pressure in a shallow water increases with the depth. But the pressure change alone may not cause the changes of temperature and volume as the water is nearly incompressible. So the shallow water does not possess
5.
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NEWTONIAN-THERMODYNAMIC SYSTEM
the adiabatic processes as in the atmosphere. No mechanic work is created by thermodynamic process when the volume remains constant. Thus, there is no the conversion of internal energy into mechanic energy in water, and the adiabatic processes are purely mechanic characterized by exchanges of the geopotential energy and kinetic energy. The thermodynamic relationships, such as the state equation of gases and the first law of thermodynamics may not be used for the study of incompressible fluid. In other words, the shallow water is a mechanic system dominated by Newton’s laws together with mass and mechanic energy conservation laws. The energy conservation law of shallow water may be derived independently of thermodynamic processes. When ρ is constant, hydrostatic equation (5.1) gives p = gρ(Z − z) . Here, Z = Z(x, y, t) indicates the surface height of the water, and the atmospheric pressure over the surface is ignored for simplicity. Inserting it into (5.30) yields dvh = −g∇h Z + F , dt
(5.31)
where vh is the horizontal component of parcel velocity. The first term on the righthand side shows that the horizontal pressure gradient force in pressure coordinates is proportional to the slope of isobaric surface. The Coriolis force does not appear, as it does not affect the energy conversions discussed in the following. The momentum equation in the vertical direction is not discussed here, since the water is assumed to be hydrostatic equilibrium. The previous equation shows that the horizontal pressure gradient force depends on horizontal gradient of the surface height if without friction. Moreover, the continuity equation (5.11) gives ∂vz = −∇h · vh ∂z for the incompressible fluid. It is considered usually (Gill, 1982; Holton, 1992) that the horizontal velocity in shallow water does not change with height if without friction. In this case, the previous equation may be integrated from the fixed lower boundary at z = 0 up to the water surface, to gain the vertical velocity at the surface: vz|z=Z = −Z∇h · vh . Applying vz|z=Z =
dZ dt
for the previous equation yields ∂Z + vh ∇h Z + Z∇h · vh = 0 . ∂t
(5.32)
The two equations (5.31) and (5.32) form a set of prediction equations for two dynamic variables vh and Z. No thermodynamic processes and the relationships related need to be considered for the shallow water dynamics.
5.8.
NEWTONIAN-THERMODYNAMIC SYSTEM
115
To derive the energy equation of shallow water, (5.31) is multiplied by vh and (5.32) is multiplied by g. Adding the two resultant equations together gives dk dφ + = −∇h · (φvh ) + vh · F , dt dt
(5.33)
where φ = gZ is the specific geopotential energy at the surface height. It tells that changes of the kinetic energy and geopotential energy are caused by convergence of surface geopotential energy flux and kinetic energy dissipation resulting from friction including the surface drag and internal viscosity. Since the frictional force is in the direction against the velocity, we have vh · F < 0 . The kinetic energy destroyed is converted into heat energy eventually by molecular diffusions. However, the heat energy is not included in the energy equation, as the mechanic processes may be independent of thermodynamic processes. So the incompressible fluid may be considered as a mechanic system, which does not convert heat energy into kinetic energy by itself and is driven by external mechanic processes. To keep the motions in a viscous water without changing the surface height at a local region, there must be the flux of surface geopotential energy to the region or the external turbulent forcing on the surface. The positive input of surface geopotential energy may be brought about by the flows come from geographically higher places. The geopotential energy input may then be converted into kinetic energy in the process. Thus, water may flow continuously from a high place to a low place in the gravitational field. The gravitational force of the Moon, run off of inland rivers, precipitation and evaporation over different regions may change the height of sea surface at a local region. Moreover, the winds and turbulences in the atmosphere may produce surface drag or change the surface height also. The ocean waters driven by these external mechanic processes form particular flow patterns, which are studied by the oceanography, oceanophysics and ocean-atmosphere dynamics. If integrated over the whole water with fixed walls, the first term on the righthand side of (5.33) disappears, and so the isolated water without external mechanic forcing will become rest upon the effect of viscosity and boundary friction. The mechanic processes inside the water cannot produce kinetic energy to maintain the circulations.
5.8
Newtonian-thermodynamic system
The Earth’s atmosphere is compressible. The parcel motions in the vertical direction or across the pressure surfaces may cause thermodynamic variations in the atmosphere, through the adiabatic processes which may not occur in an incompressible fluid. The air engine theory developed in Chapters 20-23 reveals that the thermodynamic processes related to parcel motions are the major cause of kinetic energy generation in the atmosphere under the gravity effect. The atmospheric circulations may be maintained thermodynamically without exchanging mechanic energies with the exterior. This is the most important difference between the atmosphere and an incompressible fluid, such as the ocean water.
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NEWTONIAN-THERMODYNAMIC SYSTEM
In some studies, the atmosphere is assumed to be incompressible. An important shortage of using incompressible approximation is excluding the energy generation, as a result of suppressing the thermodynamic processes by this assumption. McHall (1993) pointed out that the incompressible Rossby wave can neither propagate nor follow the quasi- geostrophic balance observed in the real large-scale perturbations. Now, we see also that this classical Rossby wave cannot survive in the viscous atmosphere without external mechanic forcing. The atmospheric processes in which kinetic energy generation is important should not be studied with incompressible approximation. To study the compressible atmosphere, we apply all the physical relationships used for the studies of classical Newtonian systems and thermodynamic systems. These relationships, called the primitive equations, include the Newton’s second law, state equation, continuity equation and heat flux equation. The inhomogeneous fluid system controlled by these equations may be referred to particularly as a Newtonian-thermodynamic system. Variations in the motion status of this system are associated with variations in the thermodynamic state. The momentum equation includes thermodynamic variables, and the thermodynamic equations have dynamic variables. So, these equations form one instead of two complete set if without particular assumptions, such as the incompressible approximation. There is a particular force called the pressure gradient force in the geophysical fluids. Since the fluid pressure depends on the geographic height and acts as a force in all directions, the horizontal component of pressure gradient force is also associated with the gravity, as shown by the first term on the right-hand side of (5.31). In the pressure coordinates, the pressure gradient force is given simply by the gradient of geopotential −∇φ (Holton, 1992). Thus, the vertical gravitational force may change horizontal motions of fluids. The downward gravitational force may also produce upward motions in the compressible atmosphere, when the parcel density is lower than that of the surroundings. This can be seen from the vertical momentum equation or Archimedes’ principle
ρ dvpz =g −1 dt ρ´
derived from (5.30), where ρ´ and ρ are the densities of the parcel and environmental air, respectively. The primitive equations include the variable time and so are prediction equations. In mathematics, the predictions may be obtained by solving these equations for provided initial and boundary conditions. We have discussed in the preceding chapter that variations of classical thermodynamic systems possess the uncertainty to predictions, as we cannot measure and evaluate precisely the molecular diffusions. Apart from the molecular diffusions, there are many other physical and chemical processes in the atmosphere which cannot be represented perfectly by our prediction models and datasets. Also, the current grid-point numerical procedures including the spectral technology have an intrinsic limitation on representing the continuous fields and their variations. Thus, the weather predictions and atmospheric simulations are not a mathematical problem only, and the predictions are acceptable only within a limit time period. The predictability of the atmospheric processes will be discussed in Chapter 25.
Chapter 6 Turbulent entropy and universal principle 6.1
Introduction
The thermodynamic entropy has been mystified to many of us due to its unique macroscopic and microscopic characteristics and wide applications. Like the energy conservation law, the second law of thermodynamics is usually considered as an axiom, which reveals some universal features of the world changes. According to our experiences, the differences, or so called the gradients in a physical or aphysical system, may be weakened or removed by communications. An example of the communications in a thermodynamic system is the mass, heat and momentum transport resulting from molecular diffusions and conductions. A major purpose of the current study is to bring back the clear relations between the entropy law and our previous knowledge and theories. So far, there have been no the general relationship for the changes of inhomogeneities or disorderliness in the various systems, but only the second law of classical thermodynamics has been provided quantitatively in terms of thermodynamic entropy. This law illustrates the basic properties of downgradient molecular transport in a diffusive system, and may be derived using the nonequilibrium energy equation and kinetic theory for ideal gases as shown in Chapter 4. Since the unidirectional transport destroys the gradients irreversibly, the entropy may be used to measure the randomness or disorderliness of the microscopic and macroscopic thermodynamic states. In the large-scale moving fluid, the macroscopic disorderliness in the thermodynamic fields may also be changed by redistribution of fluid parcels with different potential temperatures. The thermodynamic entropy of a non-uniform system may be calculated from the algorithm provided in Chapter 4. In the idealized process of turbulent mixing without molecular diffusions, the system entropy evaluated by adding the entropy of each parcel or fluid element is conserved, but the disorderliness may be changed irreversibly by turbulent motions. To account for the irreversible changes in the disorderliness, we introduce in this chapter the turbulent thermodynamic entropy and the law of turbulent entropy. The turbulent thermodynamic entropy, called simply the turbulent entropy, is also a thermodynamic entropy, since it measures the thermodynamic disorderliness or irreversibility of an inhomogeneous thermodynamic system. However, the turbulent entropy law is independent of the law of classical thermodynamic entropy discussed in Chapter 4, since changes of turbulent entropy are caused by the macroscopic processes of parcel motions which are independent of microscopic molecular diffusions. To the change of turbulent entropy in the idealized turbulent process, 117
118
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
the basic elements manifesting the system disorderliness are the fluid parcels instead of molecules. Thus, the turbulent entropy and classical thermodynamic entropy represent the system disorderliness at different levels and different scales. The gridpoint data used in meteorology may only be used to estimate the turbulent entropy changes which may include the effect of molecular diffusions. The increase of entropy in an isolated system is usually considered as a universal feature of system changes. However, we only have the relationship of entropy increase for the classical thermodynamic systems before. The algorithm of entropy change in non-uniform systems given in this study may be applied also for some other physical and aphysical systems. We introduce in this chapter the universal entropy and the universal principle to manifest the general features of the system changes resulting from communications and exchanges within a system. The classical thermodynamic entropy law and turbulent entropy law are examples of the universal principle. This principle will be applied to derive the partition functions of the microscopic particles at different energy levels without using the concept of thermodynamic probability. These partition functions may then be used to obtain the expressions of internal energy, heat capacity and state equations. The van der Waals equation will be derived in this chapter.
6.2 6.2.1
Thermodynamic entropy of turbulent system Simple turbulent process
Molecular diffusions are the major cause for the change of classical thermodynamic system. While, changes of atmospheric fields viewed in a meteorological scale are produced mainly by parcel motions instead of molecular diffusions. The large-scale circulation changes in the atmosphere are generally irreversible, even if through the reversible adiabatic processes in which the potential temperature or thermodynamic entropy of each fluid parcel is conserved. Apart from the large-scale circulations, there are small-scale turbulent motions in the atmosphere, especially in the boundary layer and near the tropospheric westerly jets such as the clear-air turbulences. According to Pierrehumbert (1991), the chaotic turbulent mixing is a general and basic feature of large-scale circulations in the atmosphere. The turbulent processes may be related to some instabilities (Killworth and McIntyre, 1985; Haynes, 1985; Warn and Gauthier, 1989) or the large-scale deformation field (Welander, 1955). A kind of turbulent mixing in the stratosphere is called particularly the wave breaking (McIntyre, 1982; McIntyre and Palmer, 1983), by which the large-scale potential vorticity perturbations are broken down into turbulent patterns on smaller scales. The turbulences in the atmosphere convert large-scale kinetic energy to smaller scales continuously, and weaken or destroy the horizontal gradients in the atmosphere. The kinetic energy cascade may be considered as the turbulent dissipation of large-scale kinetic energy, caused by diffusive turbulences or turbulent viscosity. To consider the irreversible change of system disorderliness caused by parcel motions instead of molecular diffusions, we assume an idealized turbulent process, called the
6.2. THERMODYNAMIC ENTROPY OF TURBULENT SYSTEM
119
simple turbulent process, in which fluid parcels move slowly and adiabatically and no parcel kinetic energy dissipation and among-parcel molecular exchanges take place. This process may cause rearrangement of the entities or tear an entity into smaller pieces, but may not change the mean potential temperature of the system. The parcels in the simple turbulent process are isolated systems, and called particularly the turbulent entities. They are small enough so that may be considered in their uniform equilibrium states all the time. A close example is the turbulent entity found in the cumulus clouds (Telford et al., 1984). If a system has I turbulent entities, each of which possesses mass mi and potential temperature θi at the beginning, the mean potential temperature of the system reads I
θm =
i=1 mi θi
M
,
(6.1)
where M = Ii=1 mi is total mass of the system. When the molecular diffusions are ignored, an isolated fluid system may reach the turbulent equilibrium state eventually, which is statically stable and horizontally homogeneous viewed over the scale much larger than the size of turbulent entity. The assumption of simple process is not necessary for us to calculate the mean potential temperature or thermodynamic entropy in a system. However, the calculations for this assumed process enable us to see the features of these state functions more clearly. After the system reaches the turbulent equilibrium state by turbulent mixing, the mean potential temperature is unchanged, as potential temperature of each entity is conserved. The position exchanges of turbulent entities may change the disorderliness of the system, and the mixing process is generally irreversible. The changes of disorderliness cannot be represented by variation of classical thermodynamic entropy as discussed in the following.
6.2.2
Thermodynamic entropy changes
For the system of I entities assumed previously, the variation of classical thermodynamic entropy may be calculated by adding the entropy of each entity, giving ∆Sc =
I
mi dsi ,
i=1
in which ds represents the difference between equilibrium states of an entity. When an entity is assumed as an ideal gas, we apply (4.28) for it and gain ∆Sc = cp
I
mi d ln θi ,
(6.2)
i=1
where θi is the mean potential temperature averaged over entity i. This equation is similar to (4.29), except that the overbar is omitted. The mean potential temperature emphasizes the statistical features of thermodynamic variables, especially in a non-uniform state. If the measurement for each entity is available, the entropy may also be evaluated with the local data. Fig.6.1 shows a simple example. When the four entities
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
120
θ1
θ2
y
y
θ3
θ4
θ2
θ4
y
y
❍ ❍✟✟
θ1
θ3
y
y
y
y
Figure 6.1: Evaluation of classical thermodynamic entropy. The block circles indicate thermometers.
exchange their positions, their potential temperatures change from θi to θi . If each entity has unit mass, the entropy variation is evaluated from
θ1 θ θ θ + ln 3 + ln 2 + ln 4 θ1 θ2 θ3 θ4 θ1 θ2 θ3 θ4 = cp ln + ln + ln + ln . θ1 θ2 θ3 θ4
∆Sc = cp ln
(6.3)
The last equation shows that the evaluation with local measurements on a scale not larger than the entity size is identical to the evaluation following individual entities. In general, we have I I θj θ ln i = ln . (6.4) θi i=1,j=i θi i=1 It is similar to (4.34) except that θ represents the mean potential temperature of an entity instead of a molecule. Thus, the entropy calculated with (6.3) depends only on the potential temperature of each entity, and is independent of the entity distribution. If the potential temperatures do not change after position exchanges, the system entropy is conserved though the disorderliness is changed irreversibly by parcel motions. In the theoretical studies, the thermodynamic states are usually represented by continuous functions covering each point in a system, so (6.2) may be replaced by the integration ∆Sc = cp
ρd ln θ δV ,
(6.5)
V
where ρ is mass density of the volume element δV . On theory, the element δV is infinitesimal. But in fact, it cannot be as small as a single molecule. If the thermodynamic entropy can be defined for a single molecule, the sum of molecular entropy
6.3. GRID THERMOMETERS
121
cannot tell the disorderliness of the system, and so loses its physical meanings as discussed in Chapter 4. To account for the changes of disorderliness, we may assume a δV which includes a large number of molecules. In this case, the density and potential temperature in this equation represent the mean values over the element. The turbulent entity may be as small as the element of volume for calculating the thermodynamic entropy. The obtained results depend on the disorderliness of individual entities manifested by the distribution of molecules at different energy levels, but is independent of the disorderliness of the whole fluid system manifested by the entity distribution as shown by (6.4). These two kinds of disorderliness at different levels may be independent of each other, since they are associated with the independent processes of molecular diffusions and parcel motions respectively. The classical thermodynamic entropy evaluated from (6.2) or (6.5) by summing up the entropy of each entity is conserved, as the potential temperature of entity is unchanged in the simple turbulent process without molecular diffusions. Thus, the entropy is unable to tell us the changes of disorderliness caused by parcel exchanges in the reversible adiabatic process. This reversible process is not a purely thermodynamic process, and may not be really reversible due to the unidirectional transport caused by pressure gradient force and gravitational force.
6.3
Grid thermometers
To account for the changes of molecular disorderliness, we apply the thermodynamic variables, which represent the mean features of the fluid element including a large number of molecules, to evaluate the thermodynamic entropy. We do not calculate the mean thermodynamic variables, since the means are made by temperature measurements, which may only read the mean temperature of the element around a thermometer. Analogously, to account for the changes of entity disorderliness in a large-scale fluid system, we may choose a huge element, called the grid, which include a large number of entities to evaluate the thermodynamic entropy. The mean temperature over the grid may be regarded as the temperature measured by a huge thermometer in the size of the grid. The examples of the mean measurements are the grid-point data used in meteorology and oceanography, of which the grid scale is much larger than the size of turbulent entity. A variable measured at a given time and place in the atmosphere or ocean is considered theoretically as the mean over a certain area and time period. So, we may assume that the grid-point temperatures are measured by the huge thermometers at each grid point. The huge thermometer may be called the grid thermometer. Let the atmosphere to be divided into I domains, each of which includes Ji (i = 1, 2, · · · , I) entities. The central position of each domain is a grid point. The potential temperature measured by a grid thermometer at grid point i is the mean potential temperature of Ji entities, that is Ji
Θi = where Mi =
J i
j=1 mij
j=1 mij θij
Mi
,
(6.6)
is total mass of Ji entities. The mean potential temperature
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
122 of the atmosphere gives
Θm =
Ji I 1 mij θij , M i=1 j=1
where M = Ii=1 Mi is total mass of the atmosphere over I grid points. Applying (6.6) for this equation yields Θm =
I 1 Mi Θi . M i=1
(6.7)
It is similar to (6.1) in the mathematical form. This similarity is based on the assumption given by (6.6). In practice however, the grid-point data may not be the means of all entities around a grid point. Equation (6.7) tells that the mean potential temperature of the simple turbulent system may also be represented by grid-point data. The grid-point data are the only data which we can get from the real measurements for meteorological studies and weather predictions. A uniform field represented by grid-point data may not be uniform on a subgrid scale. The gradients and integrations evaluated with grid-point data give also the smoothed values. The errors depend on grid scale of the data. Thus, the feature of uniform or the gradients and integrations calculated with grid-point data are scale-dependent.
6.4
Turbulent thermodynamic entropy
The classical thermodynamic entropy provides little help for us to understand the turbulent processes in the atmosphere, since the disorderliness of atmosphere can be changed by not only molecular diffusions but also turbulent diffusions. Conservation of classical thermodynamic entropy in the turbulent mixing process does not mean that the process is reversible or the disorderliness is unchanged. For example, as the baroclinic atmosphere becomes barotropic (no temperature gradient on the isobaric surfaces) in development of baroclinic disturbance, the geopotential energy and heat energy are converted into parcel kinetic energy. If the energy conversion is produced by reversible adiabatic parcel motions without molecular diffusions, the classical thermodynamic entropy is conserved. But, the process is obviously irreversible and disorderliness of the temperature or potential temperature field is changed. The change of disorderliness resulting from rearrangement of turbulent entities cannot be explained by the classical thermodynamic entropy, because the macroscopic processes are generally independent of microscopic process of molecular diffusion. Parcel motions and circulation changes in the atmosphere may be studied without considering molecular diffusions within a certain time scale. These changes are irreversible in an isolated atmosphere, even if the molecular diffusions are ignored. To explain the large-scale thermodynamic irreversibility, we introduce a new thermodynamic entropy in this section. The thermodynamic entropy evaluated using grid-point data is also a grid-point variable denoted by SG . When a turbulent system changes from an initial state (with subscript 0) to a final state (with subscript r), the grid-scale thermodynamic
6.4. TURBULENT THERMODYNAMIC ENTROPY
123
entropy obtained from (4.28) gives ∆SG = cp
I
M ri
ln
i=1
Θ ri
M0i
.
(6.8)
Θ0i
If there are entity exchanges between the domains, Mri may be different from M0i . This equation gives ρr I Θr i ∆SG = cp ln ρi0i ∆xi ∆yi ∆zi (6.9) Θ0i i=1 in Cartesian coordinates, where ρ is the density represented by grid-point data. In the pressure coordinates, we insert Mi =
1 ∆xi ∆yi ∆pi g
into (6.8) and gain ∆SG =
I 1 Θr ln i ∆xi ∆yi ∆pi . Γd i=1 Θ0i
(6.10)
If the calculations are represented by integration, we have
∆SG = cp V
(ρr ln Θr δV − ρ0 ln Θ0 δV )
in Cartesian coordinates or 1 ∆SG = Γd
ln p A
Θr δAδp Θ0
(6.11)
in pressure coordinates, where δV = δAδz = δxδyδz is a domain in the grid scale. Like the classical thermodynamic entropy, the grid-scale thermodynamic entropy is also scale-dependent, as the grid-scale mean potential temperature depends on the number of entities. The different values obtained for different scales are not errors, but are the entropy over the different scales respectively. A change of the grid-scale entropy may be caused by changes of the entities in a grid or the entity exchanges across the grid. In the simple turbulent process with conservation of potential temperature for each entity, the classical entropy is conserved as discussed earlier, while the gridscale entropy may change, unless the system has constant potential temperature or no entity exchanges across the domain boundaries. Thus, the grid-scale thermodynamic entropy is different from the classical thermodynamic entropy, and is called the turbulent thermodynamic entropy or simply the turbulent entropy. Changes of turbulent entropy may not be a pure thermodynamic process, as turbulent motions are driven by macroscopic Newtonian forces. The turbulent entropy is regarded as a thermodynamic entropy, since only the effect of turbulent motion on change of thermodynamic state is considered. The turbulent motions may also cause irreversible conversion of geopotential energy. This will be studied in Chapter 8. It is generally unable for us to measure potential temperature of each entity in the atmosphere. So, changes of thermodynamic entropy in the atmosphere are calculated from (6.9) or (6.10) instead of (6.2) or (6.5) in meteorology and oceanography, and the results give turbulent entropy variations.
124
6.5
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
Turbulent entropy law
Our experiences tell us that the communications in an isolated system may reduce the inhomogeneities of the system. The communications between the molecules with different energies and number densities lead to the uniform state of a thermodynamic system with maximum thermodynamic entropy, viewed over the scale much greater than the molecules. It is discussed in the preceding chapter that the parcel motions driven by pressure gradient forcemove from high pressure to low pressure or from high density to low density at constant temperature. This is similar to the mass diffusion caused by random molecular motions. As a result of the mass redistribution produced, the large-scale pressure gradient and so the pressure gradient force are destroyed in the isolated fluid system without affected by external forces. When molecular diffusions are ignored, the communications between fluid parcels in an isolated system may lead to the turbulent equilibrium state, viewed over the scale much greater than the parcels. The random molecular motions and irreversible molecular diffusion produced are caused by the intermolecular forces, when the external forces in a scale much greater than a molecule or the mean molecular distance is negligibly small. In the turbulent equilibrium state which is statically stable, the gravitational force is balanced by the large-scale vertical pressure gradient force, and the large-scale horizontal gradients are destroyed. So, the turbulent equilibrium state manifests the gradients in a scale of parcels only, and the turbulent motions are driven by the turbulent pressure gradient force in the parcel scale also. Both the direction and intensity of turbulent pressure gradient force change fast with time too, so the turbulent motions are characterized by randomness. The irreversible downgradient transport caused by either molecular diffusions or turbulent diffusions suggest a tendency of system change or the different probabilities for occurrences of the distributions with different degrees of disorderliness. The probability theory used for molecular diffusions may also be applied for the turbulent diffusions. The variations of probability related to the turbulent randomness, but not just the molecular randomness may be explained by the turbulent entropy law: The turbulent entropy evaluated on any scale cannot be destroyed in an isolated system. The system disorderliness increases irreversibly as it reaches the turbulent equilibrium state, through the classical thermodynamic entropy or parcel potential temperature is conserved in the simple turbulent process discussed earlier. The extremal example of the turbulent equilibrium state is so called the isentropic state in which the system possesses a constant potential temperature. To give a quantitative description of the turbulent entropy law, we consider the change of disorderliness as a system reaches the isentropic state. The constant potential temperature of the isentropic state is given by (6.7), and the turbulent entropy variation gives I
∆sG = cp
i=1 Mi
M
ln ΘΘmi
.
(6.12)
6.5. TURBULENT ENTROPY LAW
125
Applying (6.7) for it yields
I
i=1 Mi Θi
∆sG = cp ln
M
−
i=1
I
i=1 Mi Θi
= cp ln M
I
I Mi
Mi
M
.
ln Θi (6.13)
M i=1 Θi
If each domain has equal mass Mi = M/I, we gain I
Θi ∆sG = cp ln i=1 1 . I Ii=1 ΘiI The right-hand side is generally different from zero, unless the system has constant potential temperature at the initial state. We have discussed earlier that the classical thermodynamic entropy is unchanged in the turbulent mixing process. The inequality (4.31) gives I I 1 M i Mi Θi ≥ ΘM i M i=1 i=1
and
I I 1 I Θi ≥ Θi I i=1 i=1
for Θi > 0. The equalities exist only if all Θi are identical. Thus, we have ∆sG ≥ 0, and so thermodynamic entropy may be created by turbulent mixing even if without molecular diffusions. The equality is used only for the isentropic system. This implies that the isentropic system has maximum turbulent entropy, as the potential temperature is conserved. When the gravitational force is ignored, the gases in equilibrium have a uniform pressure and so the isentropic state is also the isothermal state. The previous inequalities are the mathematical expressions of turbulent entropy law. An isolated process is irreversible if the turbulent entropy increases, since the reversed process with reducing turbulent entropy cannot happen. A process on a certain scale is thermodynamically reversible on theory, only if the turbulent entropy is conserved on the scale. That turbulent entropy may change in the simple turbulent process with conservation of parcel potential temperature implies that the process may not be adiabatic again when viewed on the grid scale. The grid-scale diabatic variations are produced by entity exchanges between domains. Thus, the feature of adiabat is scale-dependent, and the conservation of turbulent entropy evaluated on a particular scale may not ensure the entropy conservation on all scales. A process in an isolated system is thermodynamically reversible only if the turbulent entropy is conserved on all scales. Like the classical thermodynamic entropy law, the turbulent entropy law is also applied for isolated systems. It is learned that the thermodynamic entropy of an ideal gas is conserved in the reversible adiabatic process, though the gas exchanges mechanic energy with the exterior in the process. Thus, the isolated system which
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
126
has no mass and heat exchanges with the surroundings and may still exchange energy with the surroundings by changing its volume reversibly and adiabatically. While, the isolated system has no energy exchanges with the surroundings will be referred to as the closed system in this study. An example of closed system is a gas in fixed and isolated container. According to the thermodynamic entropy law, thermodynamic entropy may also not be destroyed in a closed system. The Earth’s gravitational force is an external force to the Earth’s atmosphere, which may cause not only the volume changes of the system, but also the parcel motions in the system. Since the heat conduction caused by molecular collisions, which creates thermodynamic entropy, is independent of the external force, and the adiabatic processes without molecular diffusions driven by the force are isentropic, the atmosphere or a part of the atmosphere may also be an isolated system without ignoring the gravitational effects. The inviscid atmosphere may remain the geostrophic circulations as an idealized equilibrium state in the gravitational field. However, the external macroscopic mechanic forces cannot disable the thermodynamic entropy law or destroy thermodynamic entropy of the isolated atmosphere. The turbulent entropy law may be considered as an extension of the classical entropy law to the processes of turbulent diffusions. When the element of volume is large enough in a large-scale inhomogeneous thermodynamic system including parcel motions, the classical thermodynamic entropy law becomes the turbulent entropy law. The turbulent entropy law may also be considered as an independent law, as it is based on the features of random turbulent motions which may be independent of molecular diffusions. These two laws have similar mathematical expressions and are different only in the scales, because they are both the examples of the universal principle discussed later on in this chapter. However, the data in different scales represent different processes, so the turbulent entropy law may be applied to deal with the problems which cannot be solved by the classical entropy law without using additional assumptions. Supposing there are two extremal thermodynamic processes, molecular diffusions without parcel motions and parcel motions without molecular diffusions, the classical entropy law can be applied for the first process only and the turbulent entropy law is for the both, since the molecular diffusions may also have an effect on the macroscopic potential temperature field.
6.6 6.6.1
Difference from classical thermodynamic entropy General discussion
Applying (6.6) for (6.8) yields
∆SG = cp
I i=1
ln
Mr i Ji j=1 mij θrij /Mri M0 i Ji m θ /M 0i j=1 ij 0ij
.
(6.14)
The turbulent entropy variation given by this equation is generally different from variation of classical thermodynamic entropy represented by (6.2). For a nonuniform system, the potential temperature θ in the previous equation is the mean
6.6. DIFFERENCE FROM CLASSICAL THERMODYNAMIC ENTROPY
127
over an entity. Unless the system possesses constant potential temperature, the thermodynamic entropy calculated from (6.10) or (6.11) using the grid-point data is different from the classical thermodynamic entropy represented by the sum of the entropy of each entity, since we have M ln Θ =
ln θ dm .
(6.15)
M
This does not mean that so calculated thermodynamic entropy is an approximation of the classical thermodynamic entropy, because they are different physical quantities. When the grid scale tends to the size of an entity, we have Ji = 1, and the grid-point potential temperature gives the measurement for an entity, that is lim Θi = θi
∆V →δV
obtained from (6.6). Moreover, the turbulent entropy given by (6.14) reads lim
∆V →δV
∆SG = cp
I
mi ln
i=1
θ ri . θ0i
The right-hand side is the sum of classical thermodynamic entropy of each entity. Comparing with (6.2) finds lim
∆V →δV
∆SG = ∆Sc .
The turbulent entropy becomes the classical thermodynamic entropy as the grid scale tends to as small as that of an entity. From the mathematical point of view, classical thermodynamic entropy is the limit of turbulent entropy. For this reason, perhaps, the turbulent entropy was not discussed particularly, as one might think that it is an approximation of classical thermodynamic entropy. However, the molecular diffusions related to changes of classical entropy cannot be considered as the limit of turbulent diffusions which lead to changes of turbulent entropy. These two diffusions are different physical processes associated with different forces on different scales, and may be independent of each other. Thus, turbulent entropy possesses some important features different from those of classical thermodynamic entropy. Similarly, the sum of molecular entropy is the limit of classical thermodynamic entropy and turbulent entropy, but it cannot be applied as the classical or turbulent entropy. The classical thermodynamic entropy of a turbulent entity measures the microscale disorderliness of a single entity, which may have no relation with other entities. The sum of thermodynamic entropy of these separated entities is independent of entity distribution. When the entities are viewed independently of each other and the thermodynamic state of each entity does not change, the different states represented by different arrangements of these entities should have an equal probability of occurrence. Thus, this sum is not changed by rearrangement of the entities as in the simple turbulent process without molecular diffusions. This is shown clearly by the example of (6.3). For this reason, the classical thermodynamic entropy of
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
128
a system, evaluated from (6.2) by adding the entropy of each entity together, may not represent the disorderliness of the system, viewed from a scale much larger than the size of entity. Like the classical thermodynamic entropy which is not the sum of molecular entropy, turbulent entropy is not the sum of classical thermodynamic entropy of each entity. Again, a change of classical thermodynamic entropy accounts for the irreversibility related to variation of individual entities in a fluid system. While, the irreversible changes of the system including a large number of the entities are studied with turbulent entropy, which can be changed by rearrangement of the entities as well as entity variations. Changes of the system caused by reversible and adiabatic processes of each entity may not really be reversible. The classical thermodynamic entropy and classical entropy law may be regarded as the particular examples of the turbulent entropy and turbulent entropy law, applied for the simplest thermodynamic systems which possess only one level of migratory elements and disorderliness. For the nonphysical systems, such as the human societies, which possess different elements or features at different levels, the variation tendency may be studied with the extended turbulent entropy law called the universal principle. This is discussed later on in the chapter.
6.6.2
Example
The difference between turbulent entropy and classical thermodynamic entropy may be seen more clearly from the following example shown by Fig.6.2(a). The initial and final states of the system with four entities are the same as in Fig.6.1. But now, we use grid thermometers located between two entities to read the mean temperatures. As the entities have the same mass, the grid-scale potential temperatures are θ1 + θ2 , 2 for the initial state, and
Θ2 =
Θ1 =
θ2 + θ3 , 2
Θ3 =
θ3 + θ4 2
(6.16)
θ1 + θ3 θ + θ2 θ + θ4 , Θ2 = 3 , Θ3 = 2 (6.17) 2 2 2 for the final state. The turbulent entropy evaluated with the grid-point data gives Θ1 =
Θ1 Θ Θ + ln 2 + ln 3 Θ1 Θ2 Θ3 (θ + θ3 )(θ3 + θ2 )(θ2 + θ4 ) . = cp ln 1 (θ1 + θ2 )(θ2 + θ3 )(θ3 + θ4 )
∆SG = cp ln
When potential temperature of each parcel is unchanged, we see ∆SG = cp ln
(θ1 + θ3 )(θ2 + θ4 ) . (θ1 + θ2 )(θ3 + θ4 )
The turbulent entropy is conserved only if θ2 = θ3 or θ1 = θ4 . This is because the one dimensional distribution of potential temperature is unchanged in the first case, and becomes symmetric to the initial distribution in the second case. Provided that θ2 = θ1 + ∆θ ,
θ3 = θ1 + 2∆θ ,
θ4 = θ1 + 3∆θ ,
6.6. DIFFERENCE FROM CLASSICAL THERMODYNAMIC ENTROPY
θ1 ✇
θ2 ✇
θ3 ✇
θ4
✇
θ3
✇
θ4
✏r θ3 θ3 ✏✏✑ r r✏ ✑ ✑PP✏✏ ✑ ✑✏r✏✏PP✑ r ✑✏ r✏✏ ✑
❍ ✟ ❍✟
θ1
129
θ2
θ2
✇
θ4
θ2
θ1
(a)
(b)
Figure 6.2: Evaluation of turbulent entropy. The block circles in (a) are grid thermometers. The initial and final potential temperature distributions are given by the heavy and light lines in (b), respectively.
we have (θ1 + θ3 )(θ2 + θ4 ) − (θ1 + θ2 )(θ3 + θ4 ) = 4(θ1 + ∆θ)(θ1 + 2∆θ) − (2θ1 + ∆θ)(2θ1 + 5∆θ) . = 3(∆θ)2 > 0 . Thus, we see (θ1 + θ3 )(θ2 + θ4 ) >1 (θ1 + θ2 )(θ3 + θ4 ) and ∆SG > 0 in the previous example whether ∆θ is positive or negative. While, the classical thermodynamic entropy is conserved in the process of parcel position exchanges. In this one-dimensional example with Ji = 2 the grid scale is also the scale of the entities. But in meteorology and oceanography, the data scale is much larger than the size of entity. The turbulent entropy evaluated with the data of different scales gives different results. These differences are not the errors, but just the turbulent entropy in the corresponding different scales. The grid-point data of potential temperature are the mean values of these entities included. The potential temperature of an entity depends on the mean speed of the molecules included. These relationships provide a link between the turbulent entropy to the elements at different levels. In other words, variation of turbulent entropy may include the contribution of classical thermodynamic entropy variation. For example, when parcel kinetic energy is converted into heat energy by molecular dissipation, both the classical thermodynamic entropy and turbulent entropy increase. While, the classical thermodynamic entropy variation may not
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
130
include the contribution of turbulent entropy. For the study of large-scale inhomogeneous thermodynamic system, turbulent entropy offers a better measurement of the disorderliness or irreversibility if compared with classical thermodynamic entropy. Conservation of classical thermodynamic entropy or potential temperature does not mean that the process is reversible, if turbulent entropy increases.
6.7
Turbulent entropy and disorderliness
The heavy line in Fig.6.2(b) shows the initial distribution of potential temperature for ∆θ > 0, meanwhile the final distribution is shown by the light line. The potential temperatures measured by the three grid thermometers are 1 Θ1 = θ1 + ∆θ , 2
3 Θ2 = θ1 + ∆θ , 2
5 Θ3 = θ1 + ∆θ 2
3 Θ2 = θ1 + ∆θ , 2
Θ3 = θ1 + 2∆θ
at the beginning, and Θ1 = θ1 + ∆θ ,
at the end, derived from (6.16) and (6.17) respectively. The standard deviations are evaluated from
σ=
(Θ1 − Θm )2 + (Θ2 − Θm )2 + (Θ3 − Θm )2 , 3
and σ =
(Θ1 − Θm )2 + (Θ2 − Θm )2 + (Θ3 − Θm )2 3
for the initial and final states respectively. Since the mean potential temperature is unchanged after mixing, we have 3 Θm = Θm = θ1 + ∆θ . 2 The standard deviation decreases from the initial value √ 6 |∆θ| σ= 3 √
to the final value
σ 6 |∆θ| = . 6 2 This implies that the system becomes more uniform as turbulent entropy increases. This change of disorderliness cannot be diagnosed by classical thermodynamic entropy. The disorderliness of an inhomogeneous thermodynamic system may be changed by parcel motions as well as molecular diffusions. In other words, there may be different kinds of disorderliness on different scales or levels in an large-scale inhomogeneous thermodynamic system. When parcel motions are independent of molecular
σ =
6.8. UNIVERSAL PRINCIPLE
131
diffusions, changes of disorderliness at different levels may also be independent from each other. The classical thermodynamic entropy accounts only for the change of disorderliness resulting from rearrangement of the primary elements. The entropy which represents the change of disorderliness caused by rearrangement of the secondary elements is the turbulent entropy. In the simple turbulent process without molecular diffusions, turbulent entropy may still change while the classical thermodynamic entropy is conserved. Batchelor (1967) realized the scale-dependent feature of thermodynamic entropy, as he used the word of size-dependent when introducing thermodynamic entropy in his book. This idea was not widely accepted and developed in more than 30 years afterwards, as only the thermodynamic entropy changes between equilibrium states were considered and the theory of turbulent entropy for inhomogeneous thermodynamic systems was not established. Since we are used to consider the entropy for equilibrium states only, the different results obtained from the calculations with low and high resolution data for inhomogeneous thermodynamic systems may be regarded simply as errors. However, errors cannot provide a better diagnostics than the correct calculations. Interestingly, none raised the question: Why the processes simulated with an adiabatic model are not reversible? As discussed earlier, the feature of adiabat depends on scale. An adiabatic process simulated on a scale may not be adiabatic again on other scales, so the turbulent entropy may not be conserved on the other scales and the processes may not be reversible. There is an important difference between thermodynamic entropy and some other scale-dependent variables. The mean values of some variables may not be changed greatly by turbulent diffusions, as the variables change quasi-periodically in a turbulent system. For example, the mean potential temperature is conserved in the process of turbulent mixing as discussed earlier. While, as the diffusion processes are highly irreversible, turbulent entropy of an isolated system always increases. The increment of entropy cannot be removed by taking average over the time or space.
6.8 6.8.1
Universal principle The principle
The thermodynamic entropy laws discussed earlier can be applied for the physical and chemical processes in various thermodynamic systems. The statistical interpretation of these laws based on changes of disorderliness may have more applications out of physical sciences. For example, it can be found in human history, that the differences between the various cultures are disappearing through communications and exchanges. If a living begin is considered as a thermodynamic system, it is highly organized as each part has a special structure and function. The more advanced life has more specific and refined organs. These highly distinguished structures and functions are maintained by exchange of materials and energies with the exterior. The life will die in isolation, and the major differences between organs of a dead body disappear after degeneration. If a system possesses irreversible processes, these processes will lead to a dead
132
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
state of the system, as no further changes can take place at the extremal state in isolation. This system may be referred to as the convergent system. Since the conversion of a quantity to another is irreversible in a convergent system, we may find a quantity in the system which may only increase in the total amount. This non-decreasing quantity may be called the convergent quantity. If this quantity may be represented by a variable, the variable may be called the convergent variable, such as the temperature of an closed system with constant pressure or the potential temperature of a compressible geophysical fluid. The increase of disorderliness in the physical systems or human societies gives the hypothesis called the universal principle which states: All closed systems are convergent, in which the universal entropy for a convergent variable cannot decrease. This principle can be illustrated mathematically in the following. Let Ψ be a convergent quantity which is proportional to the mass or population or space or another quantity of the system denoted by M . To represent a nonuniform state, Ψ may have fine structures or different features. According to the structures and features, Ψ is divided into n groups. The randomness of the system is then expressed by the Ψ-distribution against the structures, and is represented by Mn 1 M2 M1 ln =1 + M2 ln =2 + · · · + Mn ln =n = ln(=M 1 =2 · · · =n ) ,
where =i = Ψi /Mi is the density of Ψ in group i with generalized mass Mi (i = 1, 2, · · · , n), and we have M1 + M2 + · · · + Mn = M . From the inequality (Beckenbach and Bellman, 1965) Mn 1 M2 =M 1 =2 · · · =n ≤
M1 =1 + M2 =2 + · · · + Mn =n M
M
we obtain
M1 ln =1 + M2 ln =2 + · · · + Mn ln =n ≤ M ln
M1 =1 + M2 =2 + · · · + Mn =n M
Since
. (6.18)
Ψ 1 (M1 =1 + M2 =2 + · · · + Mn =n ) = = =m M M is the mean density of Ψ in the system, the right-hand side of (6.18) represents the randomness of the system with uniform distribution of Ψ. The equality sign may be applied only if =i are identical in each group. This equation tells that a uniform Ψ-field exhibits maximum randomness. The two sides of (6.18) may be regarded as two distributions of Ψ in the amount Ψ=
n
=i Mi
i=1
or
= δm .
Ψ= M
6.8. UNIVERSAL PRINCIPLE
133
They may also be considered as the different distributions at different times, when the total amount of Ψ is conserved as the system changes. If Ψ is not conserved but increases from state 0 to state r, we have =mr ≥ =m0 , and so (6.18) gives M1 ln =1 + M2 ln =2 + · · · + Mn ln =n ≤ M ln =mr . According to the universal principle, a convergent system may only increase its randomness towards the extremal state with uniform distribution of its convergent variables. It is emphasized that the universal principle does not suggest that the system must be able to reach the final equilibrium state with uniform distribution of the convergent variable. It tells only that the entropy cannot be reduced in an isolated system. If the convergent quantity is not conserved but increases in the process, there must be the source of the quantity for the system to reach the uniform state. Also, the system variations may be constrained by other relationships like the mass and energy conservations. If the source is not strong enough or the other relationships cannot be broken down, a system variation may stop at an inhomogeneous equilibrium state. The inhomogeneities may be remained by dynamic or thermodynamic mechanisms. For example, the vertical stratifications of the atmosphere may be remained by the hydrostatic equilibrium, and the horizontal gradients may be remained by the geostrophic balance in the isolated atmosphere or by the differential boundary conditions in the open atmosphere. In general, the inhomogeneous equilibrium state depends on the source intensity of convergent quantity and other initial conditions, and so may not be unique. The multi-equilibrium states are more significant than the extremal state with uniform distribution of the convergent variable for the study of a large-scale inhomogeneous thermodynamic system, because the system can hardly reach the uniform state. The numerical simulations and weather predictions are aimed to find one of the inhomogeneous equilibrium states, according to the initial conditions and physical processes provided. The universal principle itself may not be enough for predicting the inhomogeneous equilibrium state. This does not mean that the principle can be violated in isolation. An equilibrium state which is non-uniform in an inconstant convergent quantity may be uniform in another convergent quantity which is conservative in the process. The different convergent variables of the atmosphere and the different equilibrium states attained by different processes will be discussed in Chapter 17. The increase of randomness in a convergent system may be represented generally by n
(Mir ln =ir − Mi0 ln =i0 ) ≥ 0 .
(6.19)
i=1
When a distribution of convergent variable is continuous, (6.19) is replaced by
ln M
=r dm ≥ 0 . =0
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
134
If we introduce the universal entropy defined as S=c
Mi ln =i =
i
Mi si ,
(6.20)
i
where c is a constant and s is called the entropy density or specific entropy, the law of randomness increase is rewritten as Sr − S0 ≥ 0 ,
or
M
sr dm −
M
s0 dm ≥ 0 .
The universal entropy increases in convergent processes. For convenience in theoretical studies, we often assume a limit process in which the entropy is conserved. This idealized process is called the reversible process, as the entropy is not destroyed in a reversed process. In general, the entropy can be produced in two ways which may not be independent of each other. One is by increasing total amount of convergent quantity through irreversible conversions, such as increasing heat energy by friction, and the other one is by redistribution of a conservative convergent variable, such as destruction of gradients. To ensure that the entropy variation is determined uniquely by initial and final state provided, the convergent variable is usually chosen as a state function. In mathematics, the differential of a state function is a perfect differential, so the variations are independent of process for provided initial and reference states.
6.8.2
Applications
The thermodynamic entropy laws for ideal gases and turbulent systems discussed earlier are applications of the universal principle. If the convergent variable in this law is chosen as the potential temperature of entity, or =i = θi , the universal principle gives the classical thermodynamic entropy law (6.2) for ideal gases. If the potential temperature is provided by grid-point data Θ on a scale much greater than that of entity, this principle gives the turbulent entropy law (6.12). For the isochoric processes, we may also use the temperature, but so defined entropy may not be a state function for other processes. In the statistical mechanics, the convergent quantity is chosen as the thermodynamic probability, which is the number of the possible distributions of molecular energy status among the molecules. In the saturated moist atmosphere, we may use the equivalent potential temperature for the principle to evaluate the moist thermodynamic entropy. This will be discussed in Chapter 11. We shall define in the next chapter the convergent quantity called the potential enthalpy, which depends on potential temperature only. For the processes in which the potential enthalpy is conserved, the equilibrium state with maximum thermodynamic entropy is isentropic according to the universal principle. When the gases have a constant pressure, the isentropic state is also isothermal. But for a large-scale inhomogeneous thermodynamic system, such as the Earth’s atmosphere, the isentropic state is not isothermal again. The isentropic atmosphere is not recorded often
6.8. UNIVERSAL PRINCIPLE
135
in a large domain, because the change of atmosphere is also constrained by other relationships such as the energy conservation law. It will be discussed in Chapters 8, 9 and 17 that increase of vertical disorderliness in the mass density and potential temperature profiles is constrained by conversion of geopotential energy into kinetic energy, since the atmosphere cannot be isolated from the external gravitational force which tends to lower the gravity center of atmosphere. Thus, the isentropic state cannot be reached frequently in the atmosphere through parcel motions. The large-scale isentropic state may also not be established by the radiations from the Sun and Earth. According to the Stefan-Boltzmann law (Brunt, 1944; Hess, 1959), the radiation integrated over the whole wave spectrum is proportional to the fourth power of absolute temperature. As the temperature decreases quickly with height (about 10 K/km) in the isentropic atmosphere, the adiabatic temperature profile cannot be maintained for a long period by radiation balance. However, the adiabatic lapse rate may still be observed occasionally in a shallow boundary layer over a dry and hot land surface during day time especially over a desert or in summer. This isentropic atmosphere in a limited time and space provides the evidence of the extremal state remained by dry convection. The evidence of kineticdeath atmosphere to moist processes may be found in the tropical regions or in convective clouds. Xu and Emanuel (1989) reported that the tropical atmospheres experiencing deep convection are nearly neutral for moist adiabatic convection over the subcloud layer. It is discussed in Chapter 4 that there are two different sources of thermodynamic entropy related respectively to conversion of mechanic energy or other energies into heat energy and increase of thermodynamic disorderliness with energy conversions. Thus, thermodynamic entropy may be used to account for the irreversible changes between uniform equilibrium states. To compare the amounts of thermodynamic entropy of two equilibrium states, the convergent variable must be a state function such as the potential temperature of ideal gases. While, the other three thermodynamic variables are not the state functions unless in a particular process. For example, we may also use temperature as the convergent variable for the isobaric or isochoric process. The potential enthalpy defined using potential temperature represents the heat energy at the reference pressure. Although the particular state function of potential temperature cannot define a unique thermodynamic state, it allows us to compare the heat energy between different equilibrium states excluding the effect of reversible adiabatic changes. It is noted earlier that a system may not reach a uniform equilibrium state, if the convergent quantity define by the uniform variable is not conserved in the process. For example, after all kinetic energy is dissipated by viscosity, the atmosphere possesses only the heat energy and geopotential energy. The mean potential temperature or potential enthalpy is no longer conserved in the further changes resulting from molecular diffusions, but the heat energy or geopotential energy is due to the fixed ratio between these energies as discussed in the next chapter. Thus, we may apply the temperature as the convergent variable for the diffusion processes. So derived universal principle is more simple than the thermodynamic entropy law, which tells that the equilibrium state is an isothermal state. However, the entropy law given in this form may only be used for the processes with conservation of heat
136
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
energy. The isothermal equilibrium state may also be derived from the entropy law using potential temperature as shown in Chapters 9 and 17, because the process of molecular diffusion may be considered isobaric, so the potential temperature in the entropy equation may be replaced by temperature for the isobaric process. If the convergent quantity is given by the action defined in the classical mechanics, the equilibrium state agrees with Hamilton’s principle of least action. The universal principle tells that the equilibrium state has constant distribution of action along the direction of momentum P . If the distance is represented by wavelength λ of wave-like particles such as electrons, we obtain the de Broglie’s relationship Pλ = h ¯ for the system in equilibrium, where ¯h is the constant action called also the Planck’s constant. The density of wave action with respect to time is energy. Thus, the constant action at equilibrium state gives also the Einstein equation Eτ = h ¯, or E=h ¯ν , in which τ is wave period and ν is wave frequency. The universal principle does not suggest that the future of the world is determined by an inequality. A living organism may grow, reproduce and evolution in a long life by collecting food and fresh air from the environments. A man may update his mind by learning from the world and his new experiences. The entropy of these open systems may not increase, as the universal entropy may be given by Sr = S0 + ∆S + ∆Se for open systems. The last term in this equation indicates the entropy exchanges between the system and exterior. It will be discussed in Chapter 26, that there are negative entropy fluxes to the Earth’s atmosphere. For example, the atmosphere accepts radiative heat from the Sun and Earth near the warm lower boundary and eject the same amount of heat to the outer space from the cold top. The negative entropy fluxes allow the atmosphere to convert heat energy and geopotential energy into kinetic energy continuously in order to maintain the circulations on all scales (Chapters 8-12). In the complicated systems, the materials, such as the food and oxygen, input directly may not be negative entropy, but may be converted into negative entropy or support the system to reduce its entropy. Thus, maintenance and development of the systems depend also on the ability of entropy conversion. A new scientific book usually includes a large amount of highly organized characters which possess the negative entropy produced by the work of others. However, people may get different understandings from it, as they have different backgrounds and experiences which are the ferment for the entropy conversion in their minds.
6.9
Partition functions
The thermodynamic entropy defined in the classical thermodynamics applies the potential temperature as the convergent variable for the universal principle. Since
6.9. PARTITION FUNCTIONS
137
the temperature represents a mean feature of molecular ensemble, the equilibrium states with the maximum classical entropy are uniform without fine structures. To reveal the fine structures, we may choose the convergent variable which represents the feature of a single molecule. According to quantum mechanics, change of energy in a physical process is not continuous from a microscopic point of view. It may be considered that the energies of microscopic particles, such as the molecules, are at different levels. Moreover, the particles at one energy level may possess different motion status including the potential energies related. The number of energy status is conserved in an isolated system, and is proportional to the number of particles. It is generally unable for us to give the number of energy status for a provided macroscopic state. But, we may calculate, in the statistical mechanics, the number of all possible distributions of the status among the molecules, with the algorithms of permutation or combination according to the physical features of particles (Sears and Salinger, 1975). Here, we use the universal principle to derive some relationships such as the partition functions, which are usually obtained from the statistical mechanics. If the convergent quantity Ψ is chosen as the total number of microscopic energy status denoted by d, the thermodynamic entropy derived from (6.20) gives
S=ζ
Mi ln
i
di . Mi
Here, the c in (6.20) is replaced by Boltzmann constant ζ, and di is the energy status at level i called the degeneracy of the energy level. If the system is composed of identical particles with mass m for each, we have Mi = mNi , where Ni is the number of particles at the i-th level. The last equation is replaced by
S=ζ
Ni ln
i
di . Ni
(6.21)
The thermodynamic entropy in this expression may not be evaluated conveniently using the macroscopic thermodynamic variables, but may be used to derive the relationships between the molecule numbers and energy levels under the condition of mass conservation Ni = N (6.22) i
and energy conservation
Ni ui = U ,
(6.23)
i
where ui is the molecular energy at level i, and U is internal energy of the system. It can be assumed that the entropy becomes maximum as the system reaches the equilibrium state. To find the maximum entropy in conservations of mass and energy is an isoperimetric problem. The auxiliary function can be made as Y =
[Ni ln di − Ni ln Ni + (ln α + 1)Ni − βui Ni ] ,
i
in which ln α + 1 and β are constant Lagrangian multipliers. The Euler equation ∂Y =0 ∂Ni
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
138 gives
(ln di − ln Ni + ln α − βui ) = 0 .
i
It follows that
Ni = αdi e−βui .
Applying it for (6.22) produces α= where Z=
N , Z di e−βui
(6.24)
i
is called the partition function. Thus, we have Ni =
N di e−βui . Z
(6.25)
It shows the mass distribution against microscopic energy levels. The molecules may be in the different levels in the equilibrium state with maximum thermodynamic entropy. So the equilibrium state is not uniform in a microscopic scale. Inserting the last equation into (6.21) and using (6.23) yield S = ζ(βU − N ln α) .
(6.26)
The entropy is maximum in the distribution since 1 ∂2Y =− <0. ∂Ni2 Ni If the thermodynamic entropy is represented by (4.26) for reversible processes, we have the relationship 1 ∂S = ∂U V,N T in the classical thermodynamics (Sears and Salinger, 1975). Inserting it into (6.26) gives 1 . β= ζT Applying it for (6.25), (6.26) and (6.24) yields Ni =
u N − i di e ζT , Z
S = ζN ln Z − ζN ln N + and Z=
i
respectively.
ui
di e− ζT
U T
(6.27)
(6.28)
6.9. PARTITION FUNCTIONS
139
If molecular energy includes the translational kinetic energy ktr , rotational kinetic energy kr , vibrational kinetic energy kv and thermopotential energy φ, we have ui = ktri + kri + kvi + φ . Here, we have assumed that the thermopotential energy of individual molecules is on the same level in equilibrium. The partition function is replaced by Z = Ztr Zr Zv Zφ , in which Ztr =
−
dtri e
ktr
i ζT
,
i
Zr =
kr i
dri e− ζT ,
i
Zv =
kvi
dvi e− ζT ,
i
and
φ − ζT
Zφ = N dφ e
.
(6.29)
Here, dtri , dri and dvi are respectively the degeneracies of translational, rotational, vibrational kinetic energy on the i-th level, and dφ is the degeneracy of thermopotential energy. In the quantum mechanics, the translational kinetic energy of a single molecule is given by (Sears and Salinger, 1975) ktri =
¯ 2 i2 h
,
2
8m(V − b) 3
where ¯h is Planck’s constant, and b is the volume of total molecules. The free space for de Beroy waves in the system is V − b instead of V . So the partition function of translational kinetic energy reads Ztr =
2
e−rix
i
i
with r=
2
e−riy
2
e−riz ,
i
¯2 h 2
8mζT (V − b) 3
,
where the degeneracy dtri is taken as 1, and ix , iy , and iz indicate energy levels of the kinetic energy in the three independent directions. Applying the integration
2
e−ri ≈
i
∞ 0
2
e−rx dx =
1 2
for the last equation yields 3
Ztr =
(2πmζT ) 2 (V − b) . ¯h3
π r
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
140
For a quantized linear oscillator, the vibrational kinetic energy is given by (Sears and Salinger, 1975) 1 ¯hν , kvi = i + 2 in which ν is the vibrational frequency. The partition function obtained for dvi = 1 is τ e− 2T Zv = τ , 1 − e− T where τ = h ¯ ν/ζ. The rational partition function of diatomic gases may be represented by (Fowler, 1966) Zr =
−
(2i + 1)e
i(i+1)¯ h2 8π 2 ιζT
,
i
where ι is the moment of inertia of a molecule about its center of mass, and 2i + 1 = dri is the degeneracy. Except at low temperatures, the sum in this equation may be replaced by Zr ≈
∞ x(x+1)¯ h2 − 8π 2 ιζT 0
e
d[x(x + 1)] .
It follows that
8π 2 ιζT . ¯h2 Again, thermopotential energy of a single molecule derived from (A.9) and (A.10) (referring to Zhu (2000) also) is a , φ=− NV where a is assumed constant. The partition function of thermopotential energy gives Zr =
a
Zφ = N e ζNT V derived from (6.29). The degeneracy of thermopotential energy is also one. Now, the partition function reads √ 5 e ζNT V − 2T 8 . Z = 5 π 2 N ι 2πm(ζT ) 2 (V − b) τ h ¯ 1 − eT a
τ
This equation will be used to derive the state equation of van der Waals gases in the next section.
6.10 6.10.1
Heat capacity and van der Waals equation Einstein function
The partition function derived in the preceding section gives τ a τ 5 ln T + ln(V − b) + − − ln(1 − e− T ) 2 ζN T V 2T 5√ 8 2 π N ιζ 2 2πm . + ln h5 ¯
ln Z =
6.10. HEAT CAPACITY AND VAN DER WAALS EQUATION
141
The internal energy may then be evaluated from (6.23), or U=
ui N di ui e− ζT . Z i
When ui , di and ζ do not depend on T explicitly, the partial derivative of (6.28) with T gives ui 1 ∂Z = di ui e− ζT . 2 ∂T ζT i Applying it for the previous equation yields U = ζN T 2 It follows that 5 U = R∗ T + R∗ τ 2
∂ ln Z . ∂T
1 1 + τ 2 eT − 1
−
a V
(6.30)
for a mole diatomic gases, where R∗ = ζNA is the universal gas constant and NA is Avogadro’s number. The first term on the right-hand side represents the contribution from translational and rotational kinetic energy of molecules, and the second term gives the kinetic energy of molecular vibrations. These two terms form the heat energy. The last term represents the thermopotential energy. The heat capacity at constant volume is evaluated from
Cv =
∂U ∂T
. v
When thermopotential energy is independent of temperature or is zero as in the ideal gases, we see τ
5 τ2 eT Cv = R ∗ + R ∗ 2 τ 2 T (e T − 1)2 for diatomic gases. The last term on the right-hand side is vibrational heat capacity, as shown by Einstein function (3.42). For the monatomic gases which have no rotational and vibrational kinetic energy, the internal energy in (6.30) is replaced by 3 U = R∗ T , 2 and the heat capacity becomes 3 Cv = R ∗ . 2 In general, the molecular parameters of real gases depend on temperature, especially at low temperatures. So the heat capacity discussed previously is for ideal gases. The contribution of thermopotential energy may be found by experiments.
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
142
6.10.2
van der Waals equation
We discussed in Chapter 2 the state equation of substance from the energy conversion point of view. The state equation may also be studied with the partition function derived from the principle of maximum entropy. For the thermodynamic entropy given by (4.26) in reversible processes, we have the relationship
∂S ∂V
= T
1 p+ T
∂U ∂V
(6.31) T
in the classical thermodynamics (Sears and Salinger, 1975). The thermodynamic entropy derived from (6.27) shows S = +
τ 5 ∗ R (ln T + 1) + R∗ ln(V − b) − R∗ ln(1 − e− T ) 2 8 2 5√ R∗ τ ∗ 2 + R ln π ιζ 2πm τ ¯h5 T (e T − 1)
(6.32)
for a mole diatomic gas. Equations (6.30) and (6.32) give
and
∂U ∂V
∂S ∂V
= T
= T
a , V2
R∗ , V −b
respectively, when ι is independent of volume. Applying them for (6.31) yields p=
a R∗ T − 2 . V −b V
It is just the van der Waals equation. This equation can also be derived from (Sears and Salinger, 1975) ∂Υ , (6.33) p=− ∂V T in which,
Υ = −R∗ T ln Z
is the Helmholtz free energy of a mole gas. The van der Waals parameter a depends on intermolecular processes as shown by Appendix A. It depends also on temperature of real gases. Since the partition functions are derived using (4.26) to represent thermodynamic entropy, the entropy given by (4.26) is also a state function for van der Waals gases. A better state equation may be derived by using the more realistic intermolecular forces for evaluating the thermopotential energy included.
Chapter 7 Basic conservation laws 7.1
Introduction
The prediction equations discussed in the previous chapters may be used to predict the atmospheric variations for provided initial and boundary conditions. These equations may also be used to derive some conservation relationships, which are independent of time and so can be used conveniently as diagnostic equations. We derived in Chapter 2 from the kinetic theory of gases the thermodynamic energy equation of ideal gases, which can be applied for a small parcel or turbulent entity in the atmosphere. We shall derive in this chapter the mechanic energy equation of parcels using the momentum equation discussed in Chapter 5. In the compressible atmosphere, thermodynamic processes may not be independent of mechanic processes, and so the two energy equations can be combined together to form the parcel energy equation, such as the Bernoulli’s equation in the steady atmosphere. Variables in the parcel energy equation for equilibrium states are not the field variables, as they changes with time only and are independent of space. So, the parcel energy equation cannot be used directly for inhomogeneous thermodynamic systems. However, the differentials included may be expanded into partial derivatives with respect to time and space. The result gives the local energy equation for an inhomogeneous moving fluid. The energy equation of a whole inhomogeneous system, called the system energy equation, can be obtained by integrating the local energy equation over the system. This energy equation derived without approximations is very simple, as it tells that variation of kinetic energy and enthalpy in the atmosphere is balanced by heat exchange with the exterior. In an isolated system, kinetic energy is created at the cost of enthalpy. For the whole or closed atmosphere, the enthalpy is identical to the total potential energy defined as the sum of heat energy and geopotential energy. But in the atmosphere below a certain height, the total potential energy and enthalpy may not be identical again. The relation between thermodynamic energy equation and mechanic energy equation may be shown clearly by the energy conversion functions discussed in this chapter, which reveal the dynamic and thermodynamic mechanisms of energy exchanges in the atmosphere. For example, when the environmental pressure gradient force does work for a parcel, the parcel gains kinetic energy but loses heat energy simultaneously. So, the pressure gradient force converts the heat energy into kinetic energy. As discussed in Chapter 5, the pressure gradient force is related to the gravitational force, so the conversion of heat energy is associated with conversion of geopotential energy in the atmosphere. The ratio between the two energies converted is nearly constant, and equals the ratio of heat energy and geopotential energy in the whole atmosphere. There are two special conversion functions, which convert the kinetic energy on different scales irreversibly and so are particularly important. One converts parcel 143
7. BASIC CONSERVATION LAWS
144
kinetic energy into microscale diffusion kinetic energy in the mechanic processes, and the other converts the kinetic energy on different scales into heat energy. These unidirectional energy conversions are produced by molecular diffusion and conduction of heat and momentum, called the molecular dissipation, and may occur simultaneously. In the turbulent atmosphere, the large-scale kinetic energy may also be dissipated by turbulent diffusions. The combined effect is conversion of parcel kinetic energy into heat energy. Owing to the molecular and turbulent dissipations, the meteorological processes in the atmosphere are thermodynamically irreversible in isolation. The prediction equations used in meteorology are usually simplified by scale analyses to filter out the small scale processes called the meteorological noises, which may be amplified by numerical integrations and then destroy the acceptable predictions. Particularly, there is the microscopic noise related to molecular diffusions which leads to the isothermal state of an isolated system. Although the heat-death state is theoretically true, it may not be interested by meteorologists. The molecular diffusions may not have a great effect on the numerical predictions for the open atmosphere, since the effect is generally smaller than those of parcel motions. However, it will be found in Chapter 9 that the heat-death state is the only solution of the reference state attained by energy conversions derived from a variational approach dealing with the energy equation, as the diagnostic equation cannot predict the inhomogeneous states at various intermediate stages. This dead state resulting from molecular diffusions can hardly be observed in the real atmosphere over a large scale. To filter out the microscopic process which leads to the dead atmosphere, meteorologists used to assume that parcel motions are adiabatic and so the potential temperature is conserved in the inviscid atmosphere. This adiabatic process is also the simple turbulent process discussed in the preceding chapter, in which the mean potential temperature of the system is conserved. This feature gives the conservation law of potential enthalpy in this chapter, which holds also when molecular diffusions are constrained at constant pressure in the inviscid atmosphere. It will be discussed in Chapter 9 that the dead reference state can be filtered out by using this law in study of energy conversions, so the energy equations may give the useful solutions of meteorological processes associated with parcel motions instead of molecular diffusions.
7.2 7.2.1
Parcel and local energy equations Mechanic energy equation
A small air parcel or turbulent entity may be viewed as a moving classical thermodynamic system in the Lagrangian dynamics. To derive the mechanic energy equation for a parcel, we multiply the momentum equation (5.30) by vp , giving d(kp + φ) = −αvp · ∇p − Dvd , dt
(7.1)
7.2. PARCEL AND LOCAL ENERGY EQUATIONS
in which
145
∂ d = + vp · ∇ dt ∂t
and
vp · vp 2 is specific parcel kinetic energy on all scales; kp =
φ = gz is the specific geopotential energy, and Dvd = −vp · F
(7.2)
represents the kinetic energy dissipation produced by internal and external frictions, as it converts parcel kinetic energy into diffusion kinetic energy and heat energy irreversibly. According to the principle of kinetic energy degradation discussed later on in this section, parcel kinetic energy is destroyed by viscosity. Also, the external friction is in the direction opposite to the direction of parcel velocity and so destroy the kinetic energy too. Thus, we have Dvd ≥ 0 .
(7.3)
The Coriolis force in (5.30) does not affect the energy conversion, since it is normal to parcel motions. The first integration on the right-hand side of (7.1) represents the rate of pressure gradient force doing work, that is dwp = −αvp · ∇p . dt
(7.4)
Parcel kinetic energy may be produced as the parcel moves from high pressure to low pressure at a constant height. Applying the previous equation for (7.1) gives the mechanic energy equation ∆(kp + φ) = wp −
t
Dvd dt .
It tells that mechanic energy of parcel, including the kinetic energy and geopotential energy, is increased by pressure gradient force doing mechanic work for the parcel, and destroyed by viscosity.
7.2.2
Bernoulli’s equation
We have derived the thermodynamic energy equation for ideal gases in Chapter 2 and the mechanic energy equation in this chapter. Since thermodynamic processes are associated with mechanic processes in the compressible atmosphere, these two independent energy equations may be combined together to give a single energy equation for the parcels. Equation (7.1) may be replaced by dp ∂p d(kp + φ) = −α + α − Dvd . dt dt ∂t
7. BASIC CONSERVATION LAWS
146
When the air is assumed as an ideal gas, we substitute the thermodynamic energy equation (4.14) for it and gain dq ∂p d(kp + φ + ψ) = +α + DTd − Dvd , dt dt ∂t
(7.5)
ψ = cp T
(7.6)
where is the specific enthalpy of the ideal gases with constant heat capacity, and DTd represents the heat diffusion which converts diffusion kinetic energy into heat energy irreversibly. When the parcel reaches a new equilibrium state, the parcel kinetic energy dissipated by molecular diffusion is converted into heat energy. It may be considered that dissipation of parcel kinetic energy and conversion of diffusion kinetic energy into heat energy take place simultaneously in the slow processes, so that we have (7.7) DTd − Dvd = 0 . Consequently, parcel energy equation (7.5) reads dq ∂p d(kp + φ + ψ) = +α . dt dt ∂t
(7.8)
It leads to the Bernoulli’s equation dq d(kp + φ + ψ) = dt dt or ∆(kp + φ + ψ) = ∆q
(7.9)
in the steady atmosphere without local pressure changes. In this situation, parcel pressure is adjusted to the environment. The Bernoulli’s equation tells that kinetic energy, geopotential energy and thermal enthalpy of a parcel is conserved as it moves in the steady atmosphere, unless heat exchange takes place between the parcel and environment. When the parcel is small enough, the variables in the energy equation are independent of space at a given time, but may change with time in a moving medium. Thus, the energy partition changes when the total energy is conserved. According to Newtonian dynamics, parcel kinetic energy is created by external forces doing mechanic work. The external forces for a parcel are the pressure gradient force and gravitational force in the inviscid atmosphere. The work down by gravitational force equals the increase of geopotential energy. Thus, the contribution of pressure gradient force is represented by change of parcel enthalpy in the isolated steady atmosphere. This is discussed again in Chapters 19 and 20. The parcel energy equation may be used conveniently for parcels but not for inhomogeneous thermodynamic systems, such as the atmosphere. The pressure gradient force in (7.1) is an external force to a parcel. But to the atmosphere, the force inside the system is not external again. The work done by pressure gradient force, which is transferred into kinetic energy, must pre-exist as another energy within the system, since kinetic energy cannot be created from nothing. In the
7.2. PARCEL AND LOCAL ENERGY EQUATIONS
147
classical Newtonian dynamics and thermodynamics, there is no the energy associated explicitly with the pressure gradient force. So, it may be considered that the pressure gradient force together with the gravitational force causes the conversion of another energy into kinetic energy. We have had several names for this energy, in which available potential energy is the most well known. The potential energy of the whole atmosphere is different from the non-kinetic energy of a single parcel shown by (7.9). This is discussed in the next section.
7.2.3
Principle of kinetic energy degradation
According to the momentum conduction equation (5.28), the momentum exchanges between particles in a moving fluid tend to uniform the moving velocity. The macroscopic momentum of the whole system is conserved in the conduction process if without external forces, while the kinetic energy is not conserved. This can be proved in the following. Assume that a fluid system includes J parcels with the same mass m and velocity vj j = 1, 2, · · · , J in a certain direction. When the system reaches the state of uniform velocity through momentum conduction, the total momentum is conserved and the uniform velocity in the direction is given by the momentum conservation law Jmv = m
J
vj .
j=1
It follows that v=
J 1 vj . J j=1
The system kinetic energy in the direction is K0 =
J m v2 2 j=1 j
at the initial state, and
2
J Jm 2 m v = vj Kr = 2 2J j=1
at the uniform state. It has been proved in mathematics (Beckenbach and Bellman, 1965) that
2
J J vj 1 vj2 ≥ . J j=1 J j=1
The equality may be applied only if vj of all parcel are equal at the initial state. This inequality tells that the parcel kinetic energy at the final uniform state is less than that at the non-uniform initial state, or Kr ≤ K0 .
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It gives the principle of kinetic energy degradation which states that the kinetic energy of a fluid system is reduced by momentum exchanges between the particles in the system, when the total momentum is conserved in the momentum conduction process. The kinetic energy destroyed is converted into heat energy related to random particle motions, vibrations and rotations. The destruction of kinetic energy by momentum conduction is also called the kinetic energy dissipation. Inserting (4.13) into (7.7) yields Dvd = −
dkpd , dt
where kpd is the parcel kinetic energy which is being converted into instantaneous diffusion kinetic energy by molecular dissipation Dvd . The diffusion kinetic energy is then converted into heat energy by heat diffusion DTd . From (7.2), we see dkpd = vpd · F , dt in which vpd is the parcel velocity which will be transferred into diffusion velocity by viscosity. This equation gives dvpd =F. dt The viscosity and boundary friction are the converter of parcel velocity into diffusion velocity. Certainly, the viscosity and external friction may also convert parcel kinetic energy into heat energy directly through molecular collisions in momentum conduction.
7.2.4
Local energy equation
The variables of inhomogeneous thermodynamic system change with space. If only the macroscopic transport processes resulting from parcel motions are considered, the parcel energy equation (7.5) is expand to give ∂p dq ∂(kp + ψ) = −vp · ∇(kp + φ + ψ) + α + + DTd − Dvd . ∂t ∂t dt
(7.10)
Here, we have used ∂φ ∂φ dφ = gvpz + vpx + vpy ∂t ∂x ∂y as
∂φ ∂φ =0, and =g ∂t ∂z in Cartesian coordinates. The geopotential energy does not appear on the left-hand side since its local change is zero at a given place in Cartesian coordinates. This equation may be called the local energy equation, as it gives the energy budget at a local place. The local change rate of pressure may be replaced by α
d(pα) dα ∂p = − αvp · ∇p − p . ∂t dt dt
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149
Applying ideal-gas equation (2.22) for the first term on the right-hand side gives α
dα ∂p dT =R − αvp · ∇p − p . ∂t dt dt
The second term on the right-hand side of this equation is the rate of doing mechanic work by the pressure gradient force as shown in (7.4), and the last term is the rate of doing mechanic work by the local air changing volume, that is p
dwv dα = . dt dt
Now, we have α
dT dwp dwv ∂p =R + − . ∂t dt dt dt
Inserting it into (7.10) gives dwv dq dwp ∂(kp + u) = −vp · ∇(kp + φ + u) + − + + DTd − Dvd . ∂t dt dt dt
(7.11)
Here, u = cv T is specific heat energy or internal energy of ideal gases. We shall use the ideal-gases equation (2.22) and the Mayer formula (2.19) from time to time in this study. When molecular diffusions are ignored, changes in heat, kinetic and geopotential energies of an open system are contributed by energy exchanges together with the mechanic work created by the system and environments.
7.3
System energy equation
7.3.1
From kinetic theory of gases
In deriving the thermodynamic energy law of ideal gases in Chapter 2, the ordered kinetic energy is assumed as diffusion kinetic energy only. In a large-scale inhomogeneous fluid system, the parcel kinetic energy is also the ordered kinetic energy as shown by (5.7) or k∗ = kp + kd . Inserting it into (2.26) gives dkH − Fx dx = −dkp − dkd , where dkH = du denotes the variation of heat energy represented by (2.30), and −Fx dx = pdα = wv measures the mechanic work done by the system (referring to (2.31)). The energy conservation relationship of an inhomogeneous thermodynamic system with mass M reads dKp + dU + Wv = −
M
dkd dm ,
where Kp =
kp dm M
7. BASIC CONSERVATION LAWS
150 is total kinetic energy of the system;
u dm
U= M
is the heat energy, and Wv is mechanic work done by the system as it changes volume. If molecular diffusions are ignored in equilibrium, the system energy law gives (7.12) dKp + dU + Wv = 0 . This equation is derived without considering the gravity effect and heat exchange between the system and surroundings. It tells that the work done by an isolated system is balanced by variations of the heat and kinetic energies.
7.3.2
For the whole atmosphere
The system energy equation for the atmosphere may also be obtained by integrating the local energy equation (7.10). The work done by changing volume in this equation may be rewritten, from the continuity equation (5.11), as dα dwv =p = αp∇ · vp . dt dt Inserting it and (7.4) into (7.11) yields d(kp + φ + u − q) = −α∇ · (pvp ) + DTd − Dvd . dt The integration over a domain V gives V
d(kp + φ + u − q) dV = − ρ dt
V
∇ · (pvp )dV +
V
ρ(DTd − Dvd )dV .
If we consider the energy conversions between two equilibrium states, the kinetic energy dissipated by molecular diffusion is transferred into heat energy, so V
ρ(DTd − Dvd )dV = 0 .
(7.13)
Furthermore, the Gauss equation gives V
∇ · (pvp ) dV =
✵
pvn d✵ ,
(7.14)
where vn is the outward velocity normal to the surrounding surface ✵ of the domain. The right-hand side represents the rate of doing mechanic work by the system, that is dWv . (7.15) pvn d✵ = dt ✵ Now, the system energy equation becomes 1 g
ps d(kp + φ + u − q) A pt
dt
dpdA = −
dWv dt
(7.16)
7.3. SYSTEM ENERGY EQUATION
151
in the pressure coordinates. Here, A is horizontal section area of the atmosphere which extends from the bottom isobaric surface ps up to the top surface pt . This equation gives (7.12) when the geopotential energy and heat exchange are ignored. Comparing with the local or parcel energy equation (7.11) finds that the work done by the pressure gradient force disappears in the system energy equation, because it is involved on the right-hand side of (7.16). If the atmosphere has fixed walls as the boundaries or has a free boundary at the top where pt = 0, we have Wv = 0 and ps d
dt
A pt
(kp + φ + u − q) dpdA = 0 .
It follows that Kp + Φ + U = Kp0 + Φ0 + U0
(7.17)
in an isolated atmosphere. The sum of kinetic energy, heat energy and geopotential energy is conserved in the isolated atmosphere with fixed boundaries.
7.3.3
For a part of atmosphere
In general, the major circulations in the troposphere may be independent of the circulations above. Thus, we have to calculate the energy conversions in the troposphere. The system energy equation in the atmosphere below a limited height or above is discussed in the following. Integrating (7.16) by the time yields 1 g
ps A pt
(∆kp + ∆u − ∆q) dpdA = −Wv −
ps
∆z dpdA ,
(7.18)
A pt
in which the difference ∆ is evaluated between the final (subscript r) and initial (subscript 0) equilibrium states. The last integration reads −
ps
∆z dpdA = A pt
A
(∆zt pt − ∆zs ps )dA +
ps
pd(∆z) dA . A pt
Here, the first integration on the right-hand side gives also the mechanic work Wv done by the atmosphere, that is A
(∆zt pt − ∆zs ps )dA = Wv .
The work is positive when the top pressure surface rises or the bottom pressure surface descends. Moreover, we have 1 R d(∆z) = dzr − dz0 = − (αr dp − α0 dp) = − (Tr − T0 )dp . g gp Using ideal-gas equation (2.22) for it yields ps A pt
pd(∆z) dA = −
ps R A pt
gp
(Tr − T0 ) dpdA
7. BASIC CONSERVATION LAWS
152 in hydrostatic equilibrium. Therefore, we see ps A pt
∆z dpdA = −Wv +
1 g
ps A pt
(∆ψ − ∆u)dpdA .
(7.19)
Here, we have used the specific enthalpy ψ = u + RT
(7.20)
for ideal gases derived from (7.6), where u is the specific internal energy. Inserting it into (7.18) gives another expression of the system energy equation ps A pt
(∆kp + ∆ψ − ∆q) dpdA = 0
(7.21)
for a part of the atmosphere without approximation. It tells that the sum of parcel kinetic energy and enthalpy in the atmosphere with mass conservation can be changed only by diabatic heating. For the isolated atmosphere, we have ps A pt
∆kp dpdA = −
ps
∆ψ dpdA A pt
or ∆Kp = −∆Ψ .
(7.22)
The kinetic energy generated equals reduction of the total enthalpy Ψ. Obviously, not all the thermal enthalpy can be converted into parcel kinetic energy. The enthalpy which may be converted into kinetic energy will be referred to as the available enthalpy. The system energy equation is equivalent to (7.17) for the whole or closed atmosphere as discussed again in the next section. The energy equation of inhomogeneous thermodynamic system given by (7.17) or (7.22) is different from the local energy equation or parcel energy equation discussed in the preceding section. Without heat exchanges, a parcel moving adiabatically in the atmosphere is a moving isolated system. Thus, the Bernoulli’s equation (7.9) is similar to system energy equation (7.22), except the moving parcel may change its geopotential. When energy conversions in a whole atmosphere are studied in Chapters 9-12, the system energy equation is used. The parcel energy equation or Bernoulli’s equation will be applied to discuss the energetics of individual parcels in Chapters 20-23.
7.4
Energy conversions
7.4.1
Conversion functions
When the total energy is conserved in the atmosphere, exchanges between the different energies may still take place. The energy exchanges are illustrated simply in the following. Kinetic energy equation (7.1) may be rewritten as 1 ∂Kp =− ∂t g
ps A pt
vp · ∇p dpdA −
ps A pt
vpz
1 dpdA − g
ps A pt
Dvd dpdA (7.23)
7.4. ENERGY CONVERSIONS
153
in the whole atmosphere or a domain with fixed walls, where ∂Kp = ∂t
zt ∂(ρkp )
∂t
A zs
dzdA
is the change rate of parcel kinetic energy derived with (5.15). The first term on the right-hand side of (7.23) represents the rate of doing mechanic work by the pressure gradient force; the second term gives the destruction of geopotential energy ∂Φ =− − ∂t
ps A pt
vpz dpdA
in vertical motions, and the last term is irreversible conversion of parcel kinetic energy into diffusion kinetic energy and heat energy through molecular dissipation. Next, applying continuity equation (5.11) for heat flux equation (5.17) yields ∂U ∂t
=
zt ∂(ρcv T )
∂t
A zs
= −
1 g
= +
ps A pt ps
dzdA
p∇ · vp dpdA +
1 g
ps dq
dt
A pt
dpdA +
1 g
ps A pt
DTd dpdA
1 vp · ∇pdpdA − ∇ · (pvp )dV g A pt V ps ps 1 dq 1 dpdA + DTd dpdA g A pt dt g A pt
for the same atmosphere. Inserting (7.15) into it gives 1 ∂U = ∂t g
ps A pt
∂Q 1 dWv + + vp · ∇pdpdA − dt ∂t g
ps A pt
DTd dpdA ,
(7.24)
in which ∂Q ∂t
1 g
=
=
ps dq
dpdA
dt A pt zt ∂(ρq)
A zs
∂t
dzdA .
The last equation is derived from (5.15) for the whole atmosphere. The first term on the right-hand side of (7.24) appears also in (7.23) with an opposite sign, and represents conversion of kinetic energy into thermal enthalpy. The second term is the mechanic work done by the system. The third term is the diabatic heating rate including radiative and latent heatings, and the last term is conversion of diffusion kinetic energy into heat energy produced by molecular diffusion. We define the energy conversion functions in the following as: ps
Ckφ = Cuk
1 =− g
A pt
ps A pt
vpz dpdA vp · ∇p dpdA ,
7. BASIC CONSERVATION LAWS
154 Dku
1 = g
ps A pt
Dvd
1 dpdA = g
ps A pt
DTd dpdA ,
and ∂Q . Q˙ = ∂t Here, Ckφ accounts for conversion of parcel kinetic energy into geopotential energy in vertical motions. Cuk indicates conversion of heat energy into kinetic energy resulting from pressure gradient force pushing the parcel from high pressure to low pressure. Dku represents conversion of parcel kinetic energy into heat energy through molecular dissipation Dvd and heat diffusion DTd (referring to (7.13)). Q˙ is the rate of diabatic heating produced by radiations or phase transitions of water vapor. With the conversion functions, the energy equations give ∂Kp = Cuk − Ckφ − Dku , ∂t
(7.25)
∂Φ = Ckφ , ∂t
(7.26)
dWv ∂U = −Cuk + Q˙ + Dku − . ∂t dt
(7.27)
and
Equations (7.25) and (7.27) show that the heat energy is converted into kinetic energy when Cuk > 0. The reversed conversion takes place as Cuk < 0. The heat energy may also be increased by diabatic heating, or reduced by the volume change. Moreover, (7.25) and (7.26) tell that parcel kinetic energy is converted into geopotential energy if Ckφ > 0. Since the processes of molecular diffusions are irreversible, we have DTd > 0 . (7.28) Dvd > 0 , The second inequality can be obtained by using (7.3) for (7.7). These inequalities suggest that molecular diffusions and conductions can only reduce parcel kinetic energy and increase heat energy, that is Dku > 0. However, the viscosity is not the only sink of parcel kinetic energy, as the kinetic energy may also be converted into geopotential energy and heat energy through vertical motions in the statically stable atmosphere.
7.4.2
Total potential energy and enthalpy
Adding (7.25)-(7.27) together yields dWv ∂(Kp + Φ + U ) = Q˙ − . ∂t dt
(7.29)
This is the system energy equation (7.16). Moreover, (7.19) may be replaced by R ∂Φ = ∂t g
ps ∂T A pt
∂t
dpdA −
∂Wv . ∂t
(7.30)
7.4. ENERGY CONVERSIONS
155
It gives another expression of conversion function Ckφ in (7.26). Inserting (7.30) into (7.29) yields ∂(Kp + Ψ) = Q˙ . (7.31) ∂t This is the system energy equation (7.21). The geopotential energy is not shown in this equation, because change of geopotential energy may be represented as a part of enthalpy variation. This can be seen by adding the variation of heat energy to both sides of (7.30), giving ∂Ψ ∂Wv ∂(U + Φ) = − . ∂t ∂t ∂t It may also be obtained from (7.29) and (7.31). The sum of the heat energy and geopotential energy is called the total potential energy, and is identical to the total enthalpy in the whole atmosphere where Wv = 0 (Margules, 1904; Lorenz, 1955). So this equation may be called the total potential energy equation. Equation (7.29) tells that kinetic energy generated in the whole isolated atmosphere is equivalent to reduction of total potential energy. Also, the system energy equation (7.22) is equivalent to (7.17) for the whole atmosphere. But in the atmosphere below a limited height, the total potential energy is different from the enthalpy, and the kinetic energy generated is equal to reduction of enthalpy instead of total potential energy. Eliminating the mechanic work in (7.27) and (7.30) gives ∂Ψ = −Cuk + Ckφ + Q˙ + Dku . ∂t This equation illustrates the relationship of all the conversion functions. For the whole atmosphere, (7.30) gives ∂U ∂Φ =χ , ∂t ∂t
(7.32)
in which χ = R/cv may be referred to as the energy conversion ratio. This equation tells that the ratio of geopotential energy change to heat energy change in the whole atmosphere is constant for ideal gases. If we take Cv = 2.5R for the air, the ratio is ∆U = 2.5 , ∆Φ or
∆Φ = 0.4 . ∆U The last relationship is found also in the studies of Brunt (1944) and Belinskii (1948). The total potential energy or enthalpy which is converted into kinetic energy includes about 71 percent heat energy and 29 percent geopotential energy. This ratio is for energy conversion in the whole atmosphere, and may be not for the atmosphere below a certain height or a single parcel. We can prove by integrating the heat energy and geopotential energy that the heat energy over the geopotential energy in the whole atmosphere gives the same ratio, that is Φ =χ. U
(7.33)
7. BASIC CONSERVATION LAWS
156
The total potential energy includes about 71 percent heat energy and 29 percent geopotential energy. When the heat energy of the atmosphere is increased by the radiations from the Sun and Earth, the geopotential energy is increased simultaneously, according to the discussions above, by the resultant changes in the temperature and volume of the atmosphere. The geopotential energy increased may then be converted into kinetic energy simply as the atmosphere reduces its height of gravity center or increases its static stability through adiabatic parcel displacement. In this process a part of heat energy is also converted into kinetic energy. The energy conversions associated with changes of static stability in the dry and moist atmospheres will be studied in Chapters 9-12. The amount of heat energy or geopotential energy of the atmosphere does not always change with the static stability. After all the parcel kinetic energy is dissipated into heat energy by the friction and viscosity in an isolated atmosphere, the atmosphere may still be changed further by molecular diffusions. Since the total macroscopic energy and energy conversion ratio do not change in the microscopic process, either the heat energy or geopotential energy will not change again, though the profiles of temperature and mass density can be changed. According to the universal principle discussed in the previous chapter, the molecular diffusions with conservation of heat energy lead to the isothermal state which possesses maximum thermodynamic entropy. Applying the energy conversion ratio for the total potential energy equation yields ∂U ∂Ψ = (1 + χ) , ∂t ∂t or 1 ∂Φ ∂Ψ = 1+ . ∂t χ ∂t For the isolated atmosphere, system energy equation (7.31) reads ∂U ∂Kp = −(1 + χ) , ∂t ∂t or
1 ∂Kp =− 1+ ∂t χ
∂Φ . ∂t
Conversion of kinetic energy in the atmosphere is associated simultaneously with changes of heat energy and geopotential energy by constant ratios. When the mass is conserved, reduction of geopotential energy of the atmosphere implies that the atmosphere reduces the height of its gravity center or increases the static stability. This feature may be observed in developments of atmospheric disturbances. With the small perturbation theory of Reynolds (1895), energy conversions in the atmosphere may be divided mathematically into the time and zonal mean fields and perturbations. The early studies on the mean and perturbation energy conversions may be found in the publications of Miller (1950), van Mieghem (1952, 1973), Lorenz (1955), Saltzman (1957) and Wiin-Nielsen et al (1964). The method may also be used to study the energy conversions between the grid-scale and turbulent energies.We have discussed that conversion of kinetic energy into heat energy by
7.5. POTENTIAL ENTHALPY CONSERVATION
157
molecular diffusions is irreversible. This feature is shown explicitly in the prediction equations (7.23) and (7.24) but not in the parcel or system energy equation. Thus, the system energy equation (7.21) cannot tell the direction of energy conversion, and we need additional relationships to check if an assumed process of energy conversion may really occur or not. It has been pointed out in Chapter 4 that the unidirectional energy conversions shown in (7.28) may also be represented by the thermodynamic entropy law for classical thermodynamic systems. Evaluations of thermodynamic entropy variation in the atmosphere will be discussed in the next chapter.
7.5
Potential enthalpy conservation
One of the assumptions used frequently in the studies of atmospheric dynamics and energetics is adiabatic variation of air parcel. The potential temperature and thermodynamic entropy are conserved in the reversible adiabatic process. But in the real atmosphere, potential temperature of dry parcels can be changed by mixing between parcels or dissipation of parcel kinetic energy resulting from molecular diffusions, even if the radiation and latent heating are ignored. It will be discussed with the theory of air engines introduced in Chapters 20-23 that the mixing process is substantially important for the energy conversions, as the potential temperature of the atmosphere changes after development of dry disturbance. If molecular diffusions take place at constant pressure in the isolated dry atmosphere, potential temperature of a parcel may change but the mean potential temperature of the system is conserved, when the thermal effect of kinetic energy dissipation is ignored. The conservation of mean potential temperature gives the conservation law of potential enthalpy discussed in the following. The enthalpy of unit mass called the specific enthalpy is defined as ψp = u + pα in thermodynamics, where u is the specific internal energy. If the air is assumed as an ideal gas, the enthalpy is also given by (7.20) or (7.6). The potential enthalpy of unit air mass is defined as ψp = cp θ , in which θ is potential temperature given by (7.7). It is the enthalpy of a parcel at reference pressure pθ . If parcel pressure is different from the reference pressure, the potential enthalpy is the enthalpy which the parcel will possess as it expands or contracts slowly and adiabatically to the reference pressure. Suppose that there are Ji turbulent entities on isobaric surface i (i = 1, 2, · · · , I) in the atmosphere, each of which possesses mass mij and temperature Tij (j = 1, 2, · · · , Ji ). The potential enthalpy of all entities on surface i gives Ψpi = cp
Ji
θij mij .
j=1
After the isobaric mixing produced by molecular diffusion, the uniform temperature
7. BASIC CONSERVATION LAWS
158 reads
J i
mij Tij
j=1
Ji
Tfi =
mij
j=1
at the final equilibrium state, and the potential temperature is Ji j=1
θ fi = J i
mij θij mij
j=1
.
(7.34)
When the kinetic energy dissipation is not considered in the dry processes, the total enthalpy and potential enthalpy of the surface are evaluated from Ψi = cp
Ji
Ji
mij Tij = cp Tfi
j=1
and Ψpi = cp
Ji
mij
j=1
mij θij = cp θfi
j=1
Ji
mij ,
j=1
respectively. They are unchanged by isobaric molecular diffusion, so the molecular diffusion at constant pressure is referred to as the isenthalpic molecular diffusion. This diffusion process is irreversible since the thermodynamic entropy increases. As potential enthalpy on each isobaric surface is conserved in the process, potential enthalpy of the atmosphere represented by Ψp = cp
Ji I
θij mij
i=1 j=1
is conserved too. If mij is sufficiently small, this equation may be replaced by
Ψp = cp
ρθ dV V
in Cartesian coordinates, or Ψp =
cp g
ps
θ dpdA
(7.35)
A pt
in pressure coordinates. Similarly, we may define the potential heat energy of ideal gases as Up = cv
ρθ dV , V
and prove that it has the same conservative property as that of potential enthalpy. When a large-scale inhomogeneous fluid system is represented by grid-point data, the isobaric mixing may also be brought about by entity exchanges on a isobaric surface over a grid scale, caused by microscale pressure gradient forces. We may prove that the total enthalpy and potential enthalpy of the system are also unchanged by the isobaric turbulent mixing, called the isenthalpic turbulent diffusion. The isenthalpic diffusions in this study refer to either the isenthalpic molecular diffusion or isenthalpic turbulent diffusion or the both. Since potential temperature of parcels
7.5. POTENTIAL ENTHALPY CONSERVATION
159
or the system is conserved in adiabatic processes, the potential enthalpy is also conserved. The isolated process with adiabatic motions and isenthalpic diffusions may be referred to as the quasi-adiabatic process. The potential enthalpy is conserved in the process. The potential enthalpy is particularly important as it is just the convergent quantity corresponding to the parcel potential temperature or grid-scale potential temperature included in the classical thermodynamic entropy law (6.2) or turbulent entropy law (6.12). According to the universal principle discussed in the preceding chapter, the extremal state of maximum thermodynamic entropy attained through the quasi-adiabatic process may be isentropic. This will be proved again in following chapters. While, under theconstraints of the conservation laws, the isentropic state may not be reached. We shall introduce, in Chapters 9-12 and 17 the different algorithms for evaluating the inhomogeneous equilibrium states of the atmosphere, attained from the initial states with various sources of negative thermodynamic entropy or turbulent kinetic energy.
Chapter 8 Thermodynamic and geopotential entropies 8.1
Introduction
It is discussed earlier that the thermodynamic entropy may be used to measure the disorderliness of a thermodynamic system and potential possibility of system change. We discuss firstly in this chapter the general features of thermodynamic entropy change in the dry atmosphere. When the large-scale variations in the atmosphere are brought about mainly by parcel motions instead of molecular diffusions, we may use the conservation law of potential enthalpy, discussed in the previous chapter, to study the entropy changes. It will be found that thermodynamic entropy increases as the baroclinic atmosphere becomes barotropic, and the atmosphere with constant potential temperature has maximum turbulent entropy when the potential enthalpy is conserved. This extremal equilibrium atmosphere is different from the isothermal heat-death atmosphere with maximum thermodynamic entropy attained by molecular diffusions. Although thermodynamic entropy increases as the atmosphere becomes isentropic, the atmosphere does not reach the isentropic state very often. As the materials tend to reduce their geopotential energy in the gravity field, the atmosphere tends to reduce its height of gravity center. For example, when cold air moves downward and warm air moves upward in development of baroclinic disturbances, the static stability increases or the vertical gradients of air density and potential temperature enhance. In this process, a large amount of geopotential energy and heat energy is converted irreversibly into kinetic energy. The reduction of vertical disorderliness may not be predicted by thermodynamic entropy laws, since increase of thermodynamic disorderliness in the vertical direction is constrained by geopotential energy conversion. Unlike in the classical thermodynamic systems, the change of disorderliness is anisotropic in the large-scale geophysical fluid. As the changes of disorderliness in the horizontal and vertical directions exhibit different features owing to the gravity effect, they will be studied separately in this chapter by introducing the baroclinic entropy and barotropic entropy, defined as the portions of thermodynamic entropy associated respectively with horizontal and vertical gradients in the potential temperature field. The sum of baroclinic entropy and barotropic entropy gives the thermodynamic entropy level relative to the isentropic atmosphere. In the development of baroclinic disturbance through the quasi-adiabatic processes discussed in the preceding chapter, the baroclinic entropy increases and the barotropic entropy decreases, but the entropy is not destroyed in an isolated system. Since the changes of baroclinic entropy and barotropic entropy may have different signs, the total amount of entropy production may be zero. But 160
8.1. INTRODUCTION
161
the processes of baroclinic disturbance development are generally irreversible. The irreversible processes with conservation of thermodynamic entropy will be referred to as the pseudo-reversible processes. The process with minimum thermodynamic entropy production may create maximum mechanic energy, so the reference state attained by a pseudo-reversible process possesses the least enthalpy, compared with other reference states attained form the same initial state in the isolated atmosphere. This extremal reference state will be called the lowest state. While, the isentropic state attained with maximum thermodynamic entropy production is another extremal state possessing maximum enthalpy. From the features of baroclinic and barotropic entropy variations, we may discuss also the basic conditions for kinetic energy generation in the atmosphere, and the properties of the reference states attained after energy conversion. It will be made clear that the reference state and kinetic energy generation depend not only on initial state but also on process. The amount of thermodynamic entropy depends on inhomogeneity of thermodynamic system, but not on the direction of system gradient. Thus, the statically stable and unstable atmospheres may have an equal amount of thermodynamic entropy. However, the stable and unstable atmospheres manifest different capacities of energy conversion in the gravitational field. The ability of energy conversion related to the static stability cannot be measured by thermodynamic entropy, as the entropy defined by the classical thermodynamics does not include the effect of gravity. We introduce in this chapter the static entropy to measure the ability of energy conversions in the geophysical fluid. The static entropy depends not only on the intensity of vertical potential temperature gradient but also on the direction. It increases with the static stability. The atmospheres possess an equal amount of thermodynamic entropy may have different amounts of static entropy. When the statically unstable atmosphere becomes stable after creating a large amount of kinetic energy irreversibly, the thermodynamic entropy may not change but the static entropy increases. Meanwhile, the ability of energy conversion associated with horizontal inhomogeneity in the potential temperature field may be represented by the baroclinic entropy. The static entropy together with the baroclinic entropy defines the thermostatic entropy level, which serves as the general assessment for the ability of energy conversion. The atmosphere at a lower thermo-static entropy level may be more capable of producing kinetic energy. The thermo-static entropy level rises after energy conversions in developments of disturbances. However, the thermo-static entropy cannot be applied to account for the irreversibility of a process, as it increases when the statically stable atmosphere becomes more stable. The increase of stability cannot happen, since the thermodynamic entropy decreases in the process. The statically stable and barotropic atmosphere cannot reduce its static stability, though the thermodynamic entropy increases in the dry process. Thus, the thermodynamic entropy is no longer a perfect discriminate parameter for the variation tendency of a large-scale geophysical fluid including parcel motions. We introduce in this chapter another discriminate parameter called the geopotential entropy, which measures the irreversibility of the processes related to conversion of geopotential energy. The energy conversion cannot destroy thermodynamic entropy in an
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
162
isolated system, and the thermodynamic processes cannot change the direction of geopotential energy conversion. So the geopotential irreversibility is independent of thermodynamic irreversibility. When thermodynamic entropy increases, the process of parcel motions may still not happen if geopotential entropy decreases. Also, a meteorological process with increase of geopotential entropy is irreversible, though the thermodynamic entropy is conserved in the system.
8.2 8.2.1
Thermodynamic entropy variations General expression
In a large-scale inhomogeneous thermodynamic system, variations of thermodynamic entropy may be evaluated between inhomogeneous equilibrium states. The process between provided initial and reference states may be assumed as the largescale circulations constrained by the geostrophic balance and hydrostatic equilibrium, that is (8.1) vg · (α∇p + ∇φ) = 0 , where vg represents the geostrophic flow discussed in Chapter 5. Moreover, we assume that the change of thermodynamic state of the atmosphere in the layer from ps to pt is caused by isobaric heating or cooling in the equilibrium process. Thermodynamic entropy variation in this process gives cp ∆S = g or
ps
ln A pt
Tr dpdA , T0
(8.2)
ps θr 1 ln dpdA , (8.3) Γd A p t θ0 where Γd is the adiabatic lapse rate of temperature represented by (5.25). This equation can also be derived by integrating (4.28) without the assumption of isobaric heating or cooling. Since thermodynamic entropy is a state function, the entropy change in the assumed equilibrium process is equivalent to that in the real process which may be nonequilibrium and ageostrophic. In general, the meteorological measurements do not provide the fields continuous in time and space. Thermodynamic entropy of the atmosphere is then evaluated with grid-point data using (6.9) or (6.10), instead of (8.2) or (8.3). The potential temperature in the entropy equations represents the mean value around the gridpoint. So evaluated thermodynamic entropy is the turbulent entropy studied in Chapter 6. As discussed earlier, a variation of turbulent entropy may include the contribution of classical thermodynamic entropy change, and provide a better diagnostics for the irreversibility of a processes in an inhomogeneous thermodynamic system. When the baroclinic atmosphere reaches a barotropic reference state, we may insert an intermediate barotropic state θ0 between the initial and reference states, and (8.3) is rewritten as
∆S =
cp ∆S = g
ps A pt
θ0 θr ln + ln dpdA θ0 θ0
8.2. THERMODYNAMIC ENTROPY VARIATIONS
163
in which
1 dA (8.4) = A A denotes the horizontal mean on an isobaric surface. The previous equation is replaced by ∆S = ∆S1 + ∆S2 ,
in which
ps
cp g
∆S1 =
θ0 dpdA θ0
ln A pt
(8.5)
represents the entropy change as the atmosphere becomes barotropic without changing the horizontal mean static stability, and ∆S2 =
cp A g
ps
ln pt
θr dp θ0
(8.6)
gives the entropy change as the barotropic atmosphere changes its static stability.
8.2.2
Variation tendencies
To evaluate the entropy change ∆S1 in the baroclinic process, we set
θˇ0 θ0 = − ln 1 + ln θ0 θ0
≈−
θˇ02 θˇ0 + . θ0 2θ0 2
Since θ0 = θ0 + θˇ0 ,
(8.7)
where ˇ denotes the departure from the horizontal mean, we see A
θˇ0 dA = 0 .
(8.8)
Applying it for (8.5) yields ∆S1 ≈
cp 2g
ps pt
1 θ0 2
A
θˇ02 dA dp .
(8.9)
The both sides are equal when θˇ0 = 0. This equation shows that the entropy increases when the atmosphere reduces the baroclinity or the horizontal inhomogeneity of temperature on an isobaric surface. Thus, the isolated baroclinic atmosphere tends to become barotropic under the effect of molecular and turbulent diffusions. To calculate the entropy change ∆S2 in the barotropic process, we assume
and in which
θr = θr +θr∗
(8.10)
θ0 = θr +θ0 ∗ ,
(8.11)
1 θ = A(ps − pt )
ps
θ dpdA A pt
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
164
is the ensemble mean over the atmosphere. Applying these relationships produces ln
θr θ0
θr∗ θ0 ∗ − ln 1 + θr θr ∗ ∗ ∗2 ∗2 θr − θ0 θ − θ0 . − r θr 2 θr 2
= ln 1 + ≈
(8.12)
Since the reference state is barotropic and so θr depends on height only, we have θr = Inserting it into (8.10) gives
1 ps − pt
ps pt
ps
θr dp . pt
θr∗ dp = 0 .
(8.13)
For the large-scale processes in the atmosphere, the effect of parcel motions is generally larger than that of molecular diffusions. When anisobaric molecular diffusion (excluding adiabatic process) and kinetic energy dissipation are ignored, the potential enthalpy is conserved in the quasi-adiabatic process, that is ps A pt
or
(θr − θ0 ) dpdA = 0 ,
1 θr = ps − pt
ps pt
θ0 dp = θ0 .
(8.14)
(8.15)
The last equation gives the conservation of the mean potential temperature. Inserting (8.7) and (8.8) into (8.14) yields ps pt
(θr − θ0 ) dp = 0 .
We apply the definitions in (8.10) and (8.11) for it and gain ps pt
(θr∗ − θ0 ∗ ) dp = 0 .
Comparing it with (8.13) finds ps pt
θ0 ∗ dp = 0 .
(8.16)
Now, substituting (8.13), (8.15), and (8.16) into (8.12) and integrating the resultant equation produce Acp ∆S2 ≈ 2g θ0 2
ps pt
(θ0 ∗2 − θr∗2 ) dp .
(8.17)
The equality holds if θr∗2 = θ0 ∗2 . This equation tells that the entropy variation in a barotropic atmosphere depends on the change of potential temperature departure from the adiabatic stratification. As thermodynamic entropy in an isolated system
8.3. BAROCLINIC ENTROPY
165
cannot be destroyed, the isolated barotropic atmosphere may only change its vertical profile toward the adiabatic stratification by parcel motions or turbulent diffusions. The entropy variation in the whole process becomes ∆S ≈
cp 2g
ps 1 pt
θ0 2
θˇ02 dA +
A
A (θ0 ∗2 − θr∗2 ) dp. θ0 2
(8.18)
In the quasi-adiabatic process with limited molecular diffusion, the atmosphere reduces the inhomogeneity in potential temperature field instead of temperature field. The isentropic atmosphere attained possesses maximum thermodynamic entropy. This consequence agrees with the universal principle discussed in Chapter 6, and will be proved again from a variational approach in Chapter 17. As the entropy production is independent of process for provided initial and final states, (8.18) may be obtained also without inserting the intermediate state. If we set θ0 (x, y, p) = θ0 +θ0∗ (x, y, p) and
θr (p) = θ0 +θr∗ (p) ,
(8.3) may be replaced by θr ln θ0
θr∗ θ0∗ = ln 1 + − ln 1 + θ0 θ0
2
≈
θr∗ − θ0∗ θr∗ − θ0∗2 − . θ0 2 θ0 2
(8.19)
From (8.13) and (8.16), integration of the first term on the right-hand side of the last equation is zero. Moreover, θ0∗2 = (θ0 − θ0 )2 = (θ0 + θˇ0 )2 − 2 θ0 θ0 + θ0 2 = (θ0 2 − 2θ0 θr + θ0 2 ) + θˇ02 − 2 θ0 (θ0 − θ0 ) = θ0 ∗2 + θˇ2 − 2 θ0 θˇ0 . 0
Integration of the last term on the right-hand side of the last equation is zero, too. Thus, we obtain ∆S ≈
cp 2g θ0 2
ps pt
A
2 2 θˇ02 dA + A(θ0 ∗ − θr∗ )
dp ,
(8.20)
which is similar to (8.18).
8.3
Baroclinic entropy
The change of thermodynamic entropy given by (8.18) includes two terms: ∆S1 and ∆S2 related to changes of potential temperature inhomogeneities in the horizontal and vertical directions respectively. Thus, thermodynamic entropy of the atmosphere may be classified into two portions, associated with horizontal and vertical
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
166
gradients in the potential temperature field. If the barotropic intermediate state θ is taken as a reference state, the entropy relative to the reference state, represented by cp ps θ dAdp (8.21) ln Sbc = g pt A θ is defined as the baroclinic entropy. It depends on initial state only. Referring to (8.5) and (8.9) finds cp Sbc ≈ − 2g
ps pt
1 θ2
θˇ2 dA dp .
(8.22)
A
It tells that Sbc ≤ 0 . The baroclinic entropy is negative in a baroclinic atmosphere, and is zero in the barotropic atmosphere. The entropy variation ∆S1 is just the change of baroclinic entropy: ∆S1 = ∆Sbc = Sbcr − Sbc0 . When the reference state is barotropic, it gives ∆Sbc = −Sbc0 . The change of baroclinic entropy is the negative baroclinic entropy at initial state. Examples of baroclinic entropy may be given in the assumed atmosphere, called the linear atmosphere, which possesses constant temperature lapse rate Γ and horizontal temperature gradient Ty . The structure of linear atmosphere will be discussed in Chapter 19. Potential temperature distribution of the linear atmosphere is represented by b R ps , b = (Γd − Γ) . (8.23) θ = (Ts0 + Ty y) p g It is assumed that ps = 1000 hPa here. The intermediate barotropic state reads
θ = Ts0
ps p
b
.
(8.24)
The baroclinic entropy is evaluated from cp sbc = − gA(ps − pt )
= −cp
ps
ln A pt
Ts0 dpdA Ts0 + Ty y
1 Ts0 Ts + Ty Y +1 ln Ts0 − ln(Ts20 − Ty2 Y 2 ) − ln 0 2 2Y Ty Ts0 − Ty Y
.
(8.25)
It is independent of the static stability and is less than zero. The calculations for Y = 1000 km and Ts0 = 290 K are plotted by the heavy dashed curve in Fig.8.1. The magnitude increases with the baroclinity represented by horizontal temperature gradient, so the baroclinic entropy gives a measure of the baroclinity from the entropy point of view. In a process of frontogenesis in the atmosphere, the negative baroclinic entropy increases greatly in the produced frontal zone.
8.4. BAROTROPIC ENTROPY
167
Figure 8.1: Baroclinic (heavy dashed) and barotropic (light dashed) entropies of the linear atmosphere with the Γ indicated. The light solid demonstrates the thermodynamic entropy level.
8.4
Barotropic entropy
The baroclinic entropy measures the horizontal inhomogeneity of potential temperature field. The vertical inhomogeneity may be represented by the barotropic entropy defined as the thermodynamic entropy of a barotropic atmosphere relative to the isentropic state with the same potential enthalpy. For the baroclinic atmosphere, the barotropic state may be represented by θ. The barotropic entropy is then evaluated from cp ps θ dp . (8.26) ln Sbt = A g pt θ Since θ ln θ
θ∗ = ln 1 + θ ∗ θ∗2 θ − ≈ , θ 2 θ 2
we gain Acp Sbt ≈ − 2g θ 2
ps pt
θ∗2 dp ,
(8.27)
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
168 which shows
Sbt ≤ 0 . The barotropic entropy is negative in the statically stable or unstable atmosphere, and is zero in the atmosphere with adiabatic stratification. When the static stability increases in the statically stable atmosphere through the quasi-adiabatic process, we have θr∗2 > θ0 ∗2 and so the barotropic entropy decreases. For the linear atmosphere assumed previously, the mean potential temperature θ in (8.26) is given by θ =
Ts0 pbs (ps1−b − pt1−b ) . (1 − b)(ps − pt )
(8.28)
Also, we have ps pt
lnθ0 dp = (ps − pt ) (ln Ts0 + b ln ps + b) − b(ps ln ps − pt ln pt ) .
(8.29)
The barotropic entropy evaluated is plotted by the horizontal light dashed lines in Fig.8.1, with the values -0.88, -0.42, -0.13, 0.0 and -0.04 J/(K.kg) for the temperature lapse rate 0.65, 0.75, 0.85, 0.95 and 1.05 K/100m respectively. It is noted that the barotropic entropy is negative in either the statically stable or unstable atmosphere. However, the stable and unstable atmospheres possess different dynamic features and variation tendencies. Thermodynamic entropy cannot tell the differences. This will be discussed again in the next section. Thermodynamic entropy variation ∆S2 in (8.6) may be represented by the change of barotropic entropy: ∆S2 = ∆Sbt = Sbtr − Sbt0 . It follows that ∆Sbt ≈
Acp 2g θ0 2
ps pt
(θ0∗2 − θr ∗2 ) dp .
(8.30)
In the statically stable atmosphere, the change of barotropic entropy in the quasiadiabatic processes is negative when the static stability increases, and is positive as the atmosphere becomes more unstable. The reverse is true in the statically unstable atmosphere. If the unstable atmosphere becomes stable, the barotropic entropy may increase when the static stability is not too high at the reference state, otherwise the barotropic entropy may also be destroyed if the static stability exceeds a limit.
8.5
Thermodynamic entropy level
The sum of baroclinic entropy and barotropic entropy defines the thermodynamic entropy level of the atmosphere. The mathematical expression is given by Sel = Sbc + Sbt ,
8.6. STATIC ENTROPY
169
or
cp Sel = g
ps
ln pt
A
θ dAdp . θ
It is the thermodynamic entropy relative to the isentropic atmosphere with the same potential enthalpy. The last equation is replaced by Sel = or
cp Sel = − 2g
cp g
ps pt
ps pt
1 θ2
A
ln θ dAdp − A(ps − pt ) ln θ
A
θˇ2 dA dp +
A θ 2
ps pt
∗2
θ
,
dp
.
(8.31)
The thermodynamic entropy level depends on three-dimensional departure of potential temperature from the isentropic state. It is negative, since the isentropic atmosphere has maximum thermodynamic entropy when the potential enthalpy is conserved. For the linear atmosphere assumed previously, the thermodynamic entropy level is plotted by the light solid in Fig.8.1. The change of thermodynamic entropy as the atmosphere varies from an initial state to a reference state is given by (8.32) ∆S = ∆Sbc + ∆Sbt . It may also be represented by the difference of thermodynamic entropy level ∆S = Selr − Sel0 = ∆Sel . This entropy change does not include the entropy exchange between the system and exterior, and cannot be less than zero in the isolated atmosphere. The disorderliness measured by thermodynamic entropy depends on the intensity of potential temperature gradient, but is independent of the gradient direction. If the entropy is used to represent the ability of system variation in the geopotential field, the limitation may be found immediately. For example, (8.27) shows that the statically stable and unstable barotropic atmospheres may have equal amount of barotropic entropy or at the same thermodynamic entropy level. However, the unstable atmosphere may produce kinetic energy immediately but the stable atmosphere cannot. When the unstable atmosphere becomes stable, the height of gravity center decreases and so the geopotential energy together with the heat energy (called the total potential energy) is converted into kinetic energy. This process of energy conversion is irreversible, but the thermodynamic entropy may be conserved in the process. So, thermodynamic entropy cannot account for the effect of gravity in the irreversible process.
8.6
Static entropy
The thermodynamic entropy is introduced by the classical thermodynamics to measure the disorderliness of thermodynamic state or the irreversibility of thermodynamic process. The effect of gravity on changes of classical thermodynamic system, such as ideal gases, is ignored usually. In this situation, changes of non-uniform ideal gases are caused by molecular diffusions.
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
170
While, the disorderliness of a large-scale inhomogeneous thermodynamic system may also be changed by displacement of turbulent entities. Motions of the entities depend on the external forces such as the pressure gradient force and gravitational force. The conversion of geopotential energy in the gravity field is substantial for development or maintenance of atmospheric circulations. As discussed earlier, the barotropic entropy cannot tell the difference between statically stable and unstable atmospheres, so the thermodynamic entropy is unable to measure the ability of energy conversions in the barotropic atmosphere produced by parcel motions. Thus, we introduce a new entropy called the static entropy to account for the ability. The static entropy is identical to the barotropic entropy in the statically unstable atmosphere, and is the negative barotropic entropy in the stable atmosphere, that is Acp p s Statically stable pt ln( θ /θ) dp g , (8.33) Sse = Acp ps − g pt ln( θ /θ) dp Statically unstable
or Sse =
Acp ps θ∗2 dp 2gθ2 pt Acp ps ∗2 dp − 2gθ 2 pt θ
Statically stable Statically unstable
.
(8.34)
It is positive and negative in the stable and unstable atmospheres respectively, and is zero in the isentropic atmosphere. Since the static entropy increases with the static stability, it provides a measure of the static stability from the entropy point of view. From (8.26) and (8.33), a change of barotropic entropy may be given by
∆Sbt =
−Sscr + Ssc0 Statically stable −Sscr − Ssc0 Statically unstable
.
(8.35)
If a statically stable atmosphere becomes more stable, the entropy change is negative. The barotropic entropy may increase when a statically unstable atmosphere becomes stable. As Ssc0 < 0 in the unstable atmosphere, the entropy change may be zero as the unstable atmosphere becomes stable or Sscr = −Ssc0 . This means that the stable and unstable atmospheres may possess an equal amount of thermodynamic entropy, since thermodynamic entropy or thermodynamic disorderliness depends on the strength of potential temperature gradient, but not on the gradient direction. However, this process is irreversible, and the change of static entropy given by ∆Sse = Sscr − Ssc0 is not zero.
8.7
Pseudo- reversible process
The irreversible process with conservation of thermodynamic entropy discussed previously may be referred to as the pseudo- reversible process. Thermodynamic entropy of individual parcels or at local places may still change in the process. This process may occur in the statically stable and baroclinic atmosphere. In the irreversible process of energy conversion, cold air moves downward and warm air moves upward. As a result, the baroclinic entropy increases while the barotropic entropy decreases as the static stability increases, so that the entropy of the whole system
8.8. THE REFERENCE STATE
171
may not change. With the definitions of baroclinic entropy (8.21) and static entropy (8.33), change of thermodynamic entropy in (8.3) may be replaced by
∆S =
−Sbc0 + Ssc0 − Sscr −Sbc0 − Ssc0 − Sscr
Statically stable Statically unstable
.
(8.36)
Since Sbc0 < 0 in the baroclinic atmosphere, the net production of thermodynamic entropy may be zero in the statically stable atmosphere. In the barotropic atmosphere with Sbc0 = 0, the previous equation gives the change of barotropic entropy in (8.35), and the pseudo-reversible processes may occur only in the statically unstable situation. Since available mechanic energy created in a pseudo- reversible process assumes maximum, the kinetic energy generation is maximum in the process. For the static entropy in the stable reference state cannot be less than zero as shown by (8.36), we have
∆Smax =
−Sbc + Sse Statically stable . −Sbc − Sse Statically unstable
(8.37)
The maximum production of thermodynamic entropy increases with the departure of initial potential temperature from the isentropic atmosphere. It cannot be negative in the isolated atmosphere whether the initial state is baroclinic or barotropic, as learned from (8.22) and (8.34). As Sbc = 0, Sse = 0 and ∆Smax = 0, the isentropic atmosphere possesses maximum thermodynamic entropy. In other words, a maximum amount of thermodynamic entropy is created as the stable or unstable atmospheres reaches the isentropic state through quasi-adiabatic processes. The pseudo-reversible process and the process with maximum thermodynamic entropy production are two extremal processes in an inhomogeneous thermodynamic system.
8.8
The reference state
Equation (8.36) may be rewritten as
Sscr =
−Sbc0 + Ssc0 − ∆S Statically stable −Sbc0 − Ssc0 − ∆S Statically unstable
.
(8.38)
The reference state cannot be determined uniquely by initial state, as the static stability depends on thermodynamic entropy production in the process. Since Sbc0 ≤ 0 and Ssc0 ≥ 0 in a statically stable atmosphere but Ssc0 ≤ 0 in an unstable atmosphere, the reference state attained with provided thermodynamic entropy production will be more stable, if the baroclinity and static instability at the initial state are higher. As the static entropy increases with the static stability, this equation tells also that the final static stability may be reduced by molecular and turbulent diffusions which create thermodynamic entropy. The reference state attained by pseudo- reversible process with ∆S = 0 gives
Sscr |max =
−Sbc0 + Ssc0 Statically stable −Sbc0 − Ssc0 Statically unstable
.
(8.39)
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
172
The atmosphere with maximum static entropy possesses the highest stability. Since kinetic energy generated in the pseudo- reversible process assumes maximum, the reference state attained possesses the least enthalpy compared with other reference states attained from the same initial state. The reference state attained with minimum thermodynamic entropy production will be referred to the lowest state in this study, which gives another extremal state compared with the isentropic state possessing maximum thermodynamic entropy. Since energy conversions in the real atmosphere are highly irreversible, the lowest state may not be attained and the kinetic energy generated is less than the maximum amount. In general, kinetic energy generation in the atmosphere is associated with conversion of geopotential energy related to downward displacement of cold air and upward displacement of warm air, so the reference state is more statically stable than the initial state. If we think that kinetic energy may be created as Sscr > Ssc0 , (8.38) gives −∆S Statically stable (8.40) Sbc < −2Sse − ∆S Statically unstable as the condition of kinetic energy generation. To be a kinetic energy source, the stable atmosphere must be baroclinic as ∆S > 0. The critical baroclinity for development of disturbances is given by the right-hand side of the previous equation. It depends on process. If the process is highly irreversible, such as in the energy conversions achieved by breaking a strong geostrophic flow on the steep isobaric surfaces or in the highly stable stratosphere, the critical baroclinity is high. While, the kinetic energy may always be created in the unstable atmosphere (Ssc0 < 0) even if it is barotropic. To find the threshold value of thermodynamic entropy production, over which the reference state attained is more stable than the initial state, we replace Ssc0 for Sscr in (8.36), and gain
∆Sth =
−Sbc Statically stable −(Sbc + 2Sse ) Statically unstable
.
When ∆S < ∆Sth , net kinetic energy is created and the reference state attained is more stable than the initial state. If a sufficiently large amount of thermodynamic entropy is produced by strong molecular and turbulent diffusions, the reference state may be less stable and possesses more geopotential energy and heat energy (or total potential energy) than the initial state. This implies that kinetic energy is destroyed in the highly irreversible process. The ∆Smax in (8.37) is the entropy production as the atmosphere reaches the isentropic atmosphere. Substituting it for ∆S in (8.40) yields Ssc0 < 0. It is true only for the unstable atmosphere. So the stable atmosphere cannot reach the isentropic state without destroying kinetic energy, even if it is strongly baroclinic and the thermodynamic entropy production is positive. But at least on theory, the isentropic state may be attained from a statically unstable atmosphere through the process with the highest irreversibility. This isentropic profile is not observed frequently, since the dry atmosphere can hardly become unstable in a large domain. However, the moist atmosphere may become statically unstable frequently for moist convection. It is discussed in Chapter 11 that the isentropic state of the moist
8.9. THERMO-STATIC ENTROPY LEVEL
173
atmosphere saturated with water vapor is called the moist isentropic state, which possesses constant equivalent potential temperature. The moist isentropic profile produced by moist convection may be found near the center of a moist storm (DaviesJones, 1974; Heymsfield et al., 1978). While, the isothermal profile attained by molecular diffusions is not recorded near a storm center. This fact confirms that parcel motions are more significant than the molecular diffusions in the study of meteorological processes, and so using the conservation law of potential enthalpy to filter out the heat-death state is practically important.
8.9
Thermo-static entropy level
The maintenance of atmospheric circulations depends not only on the total energy but also on energy conversion. When all the kinetic energy is converted into heat energy and geopotential energy, the total energy is unchanged but the circulations die. To maintain the circulations, the atmosphere must possess the ability of kinetic energy generation. Since thermodynamic entropy increases in energy conversions in an isolated atmosphere, we may use the entropy to represent the ability of energy conversion. It can be considered generally that the atmosphere is more capable of energy conversion, if it has less thermodynamic entropy compared with another atmosphere with the same mass and enthalpy. It is discussed earlier that the thermodynamic entropy depends on the inhomogeneity of potential temperature field, but is independent of the gradient direction. While, conversion of geopotential energy in the atmosphere depends on the static stability or the gradient direction. Thus, the ability of geopotential energy conversion in a barotropic atmosphere cannot be measured by thermodynamic entropy, but may be by the static entropy introduced in the preceding section. Meanwhile, the ability of energy conversion related to horizontal gradients of thermodynamic variables may be represented by the baroclinic entropy, as energy conversions in the atmosphere destroy the gradients and increase disorderliness in the potential temperature field. This feature of energy conversion is generally independent of horizontal direction of the gradients, if no particular dynamic conditions are adopted. The sum of baroclinic entropy and static entropy given by Ssl = Sbc + Sse
(8.41)
is defined as the thermo-static entropy level of the atmosphere. It is the thermostatic entropy with respect to the isentropic state. Referring to (8.21) and (8.34) finds
Ssl = =
ps ps cp θ θ dAdp − A dp ln ln g θ θ pt A pt ps ps cp ln θ dAdp − 2A lnθdp + A(ps − pt ) ln θ g pt A pt
in the statically stable atmosphere, and Ssl = Sel
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
174
in the unstable atmosphere. This entropy level in the stable atmosphere may also be represented by Ssl = −
cp 2g
ps pt
1 θ2
θˇ2 dA dp +
A
Acp 2g θ 2
ps pt
θ∗2 dp
derived using (8.22)and (8.34). It is identical to the thermodynamic entropy level for a statically unstable atmosphere, but is not for a stable atmosphere. The atmosphere at a lower thermo-static entropy level possesses a higher baroclinity or static instability and so a higher ability of energy conversion. The reference state used for evaluation of thermo-static entropy level is the isentropic state, which possesses maximum turbulent entropy when the molecular diffusions at different pressures are filtered out. Dutton (1973) selected the isothermal state with maximum classical thermodynamic entropy as the reference state. The isothermal atmosphere can only be reached by molecular diffusions in isolation, which are less important than parcel motions for the meteorological processes. While, the kinetic energy is created in the parcel motions driven by the external Newtonian forces which depend essentially on the gravity as discussed in Chapter 5. Using the isothermal atmosphere as the reference state, we are unable to consider the effect of gravity on the energy conversions, since the heat conduction caused by molecular collisions may be independent of the gravity. To find the critical entropy level below which the energy conversion is possible, we assume that Sscr > Ssc0 and insert (8.40) into (8.41), giving
Sslc =
Sse − ∆S Statically stable −Sse − ∆S Statically unstable
.
If thermo-static entropy level is lower than the critical level, the atmosphere is at an excited state. Kinetic energy generation and disturbances development may occur as soon as the excited state becomes mechanically unstable. We may define the excitation degree as ∆Sex = Sslc − Ssl . It follows that ∆Sex =
−Sbc − ∆S Statically stable . −Sbc − 2Sse − ∆S Statically unstable
(8.42)
The energy conversion may not happen if the excitation degree is negative. This equation tells that the excited state in the statically stable atmosphere must be baroclinic, as the baroclinic atmosphere possesses negative baroclinic entropy. The statically unstable atmosphere is also excited whether it is baroclinic or barotropic. A high excitation level may be achieved in the atmosphere with high baroclinity and static instability. In the process of energy conversion, thermo-static entropy level rises as the atmosphere reaches a barotropic and stable reference state. When the baroclinic atmosphere becomes barotropic, a change of the entropy level may be evaluated from (8.41), giving ∆Ssl = Sslr − Ssl0 = Sscr − Ssc0 − Sbc0 .
(8.43)
8.10. GEOPOTENTIAL ENTROPY
175
Applying (8.38) for it yields
∆Ssl =
−2Sbc − ∆S Statically stable −2Sbc − 2Sse − ∆S Statically unstable
.
(8.44)
The entropy level rises higher in the atmosphere with higher baroclinity and static instability, or in the process with less thermodynamic entropy production. The increase of thermo-static entropy is different from increase of thermodynamic entropy, as shown by comparing (8.43) with (8.36):
∆Ssl − ∆Sel =
2∆Sse Statically stable 2Sscr Statically unstable
.
Since the right-hand side is greater than zero, increase of thermo-static entropy is greater than increase of thermodynamic entropy. When thermodynamic entropy does not change in a pseudo- reversible process, the thermo-static entropy reaches the highest level, given by
∆Sslmax =
−2Sbc Statically stable . −2Sbc − 2Sse Statically unstable
As thermo-static entropy increases, the ability of energy conversion decreases simultaneously. This process is irreversible though the thermodynamic entropy is conserved. In the process of maximum thermodynamic entropy production, we apply (8.37) for (8.44) and gain ∆Sslmin = −(Sbc + Sse ) .
(8.45)
This is just the difference of thermo-static entropy between the given atmosphere and isentropic atmosphere as shown by (8.41).
8.10
Geopotential entropy
8.10.1
For dry air parcels
The irreversible energy conversions in the atmosphere may take place through the pseudo- reversible process without creating thermodynamic entropy in the whole system. This irreversibility cannot be explained by thermodynamic entropy laws. The limitation on applications of thermodynamic entropy is produced by the gravity. The atmosphere in the gravitational field tends to reduce its geopotential energy. This mechanic feature is not involved in the classical entropy law or turbulent entropy law, since the irreversibility of geopotential energy conversion may be independent of irreversible thermodynamic processes. To reveal furthermore the effect of gravity on the system variations, we introduce in the following a new entropy, called the geopotential entropy, of which the changes are independent of thermodynamic entropy variations. The mechanic energy conversion of a solid body in the gravitational field is in the direction from geopotential energy to kinetic energy, if without other forces acting
176
8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
on it. For a parcel in the compressible atmosphere, the direction of vertical motion or energy conversion depends not only on the gravitational force, but also on the vertical pressure gradient force. The latter depends further on thermodynamic state of the parcel itself. The specific geopotential entropy of a dry parcel with unit mass may be defined as φ (8.46) sg = − . θ We use potential temperature in stead of temperature in the definition, since the potential temperature is conserved in the quasi-adiabatic process but the temperature is not. Thus, the change of geopotential entropy in the process represents the weighted change of geopotential. The so defined geopotential entropy may be used conveniently to study the unidirectional conversion of geopotential energy in a pure mechanic process without thermodynamic forcing. The previous equation gives g dsg = − dz , θ
(8.47)
in adiabatic vertical convection. When a parcel ascends adiabatically from the bottom height zs to the top height zt , variation of its geopotential entropy evaluated from (8.47) is represented by ∆sga = −
g (zt − zs ) , θs
where θs denotes the environmental potential temperature at the bottom height. Since the upward motion is against the gravitational force, geopotential entropy variation of the rising parcel is negative. After the parcel changes its height, the position is filled up by another parcel. To consider an isolated process, we assume that the rising parcel starting at height zs exchanges position with another parcel of the same mass at height zt . The geopotential entropy variation of the adiabatic descending parcel gives g ∆sgd = − (zs − zt ) , θt where θt is environmental potential temperature at the top height. This geopotential entropy variation is positive. The change in geopotential entropy of a single parcel is independent of static stability of the atmosphere. The change of geopotential entropy in the dry process of parcel exchange is given by 1 1 − , zt > zs . (8.48) ∆Sg = g(zt − zs ) θt θs In the statically stable atmosphere, θt > θs and so ∆sg < 0. While, the geopotential entropy increases when the parcel exchange takes place in the statically unstable atmosphere. These examples show that the change of geopotential entropy does not simply represent the geopotential energy change, as the geopotential energy is conserved in the parcel exchanges. The adiabatic parcel exchange does not change the geopotential entropy and thermodynamic entropy in the isentropic atmosphere where θt = θs , so the process is reversible in the neutral atmosphere. The reversible parcel exchange process may
8.10. GEOPOTENTIAL ENTROPY
177
also happen in the atmosphere with a temperature stratification different from the adiabatic profile, if thermodynamic state of a parcel is adjusted to the environment as it moves slowly upward or downward. This process, called the equilibrium process here, is highly idealized. The heat exchange may be brought about by the isenthalpic diffusions discussed in the preceding chapter. Geopotential entropy variation in the equilibrium process may be evaluated from
∆sga = g
zs zt − θs θt
,
∆sgd = −∆sga
for the rising and descending parcels respectively. The total variation is zero. The thermodynamic entropy is also conserved in the process, since the entropy accepted by parcels is identical to that ejected from the environments. So the idealized equilibrium process is reversible. Obviously, these two reversible processes do not produce parcel kinetic energy. The changes of geopotential entropy may also be produced by isenthalpic diffusions. Supposing there are two parcels at height z and with potential temperature θ1 and θ2 respectively, geopotential entropy of them gives
Sg 0
m1 m2 = −gz + θ1 θ2
,
where m is the mass of a parcel. After the parcels mix together through isenthalpic diffusions, potential temperature of the mixture is θm =
m1 θ 1 + m2 θ 2 . m1 + m2
If the mixing does not change their height, the geopotential entropy becomes Sgr = −gz
(m1 + m2 )2 . m1 θ 1 + m2 θ 2
So the geopotential entropy variation is given by ∆Sg = gz
m1 m2 (θ1 − θ2 )2 . (m1 θ2 + m2 θ1 )θ1 θ2
It is greater than zero and so the entropy is created by the mixing. This implies that geopotential entropy is also subject to thermodynamic processes.
8.10.2
For the dry atmosphere
Geopotential entropy may be evaluated by following individual parcels in a fluid. If we do not know the parcel trajectories, it may be evaluated in terms of the system variations. For the parcel exchange process discussed earlier, equation (8.48) is replaced by zs zt zs zt + −g + . ∆Sg = g θt θs θs θt If the initial and final states are denoted by subscripts 0 and r respectively, we have
zs zt + ∆Sg = g θt0 θs0
zs zt −g + θtr θsr
= Sg r − Sg 0 .
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8. THERMODYNAMIC AND GEOPOTENTIAL ENTROPIES
This gives geopotential entropy variation of the two-parcel system. Free vertical convection may happen only if the atmosphere is statically unstable. Geopotential entropy of the dry atmosphere may be defined as 1 Sg = − g
ps φ pt
A
θ
dAdp .
(8.49)
So, variation in geopotential entropy of the atmosphere is calculated from 1 ∆Sg = g
ps φ0 pt
A
φr − θ0 θr
dAdp .
(8.50)
Although geopotential entropy variation may include thermodynamic influence, it measures mainly the effect of gravity on atmospheric variations. The geopotential entropy law may be stated as: An isolated dry atmosphere may not reduce its geopotential entropy through parcel motions if without conversion of kinetic energy into geopotential energy. This law is useful for examining the variation tendency of the atmosphere or the irreversibility of a process related to exchange of geopotential energy and kinetic energy. A process related to parcel motions may happen in the isolated dry atmosphere, if the thermodynamic entropy and geopotential entropy are not destroyed. It is reversible when these entropies are both conserved. For example, the parcel exchange process cannot occur in the statically stable atmosphere, since the geopotential entropy decreases; it may occur irreversibly in the statically unstable atmosphere. While, the thermodynamic entropy is conserved over these two processes. For convenience, the irreversibility associated with thermodynamic entropy generation will be referred to as the thermodynamic irreversibility, while the irreversibility associated with geopotential entropy generation may be referred to as the geopotential irreversibility. It will be discussed in Chapter 17 that the geopotential entropy law may not be applied for the processes of molecular diffusions without parcel motions. For example, the gas expansion resulting from radiative heating in the lower troposphere lifts the gravity center of the atmosphere and increases the geopotential energy, while the geopotential entropy is destroyed by the thermodynamic forcing in isolation. Variations of thermodynamic entropy, including the classical thermodynamic entropy and turbulent entropy, are produced by molecular and turbulent diffusions. The state attained by the process with minimum thermodynamic entropy production processes maximum kinetic energy, since the least mechanic energy is destroyed by turbulent and molecular diffusions. While, variations of geopotential entropy are produced by the gravitational force doing work against the vertical pressure gradient force. Equation (8.50) shows that increase of geopotential entropy is related to decrease of geopotential energy. Thus, a process with more geopotential entropy production may created more kinetic energy in the atmosphere, as more geopotential energy is converted into kinetic energy. Since variations of the different entropies are independent of each other, they may be considered separately and have different applications.
Chapter 9 Available enthalpy 9.1
Introduction
We have discussed that the atmosphere is neither a simple Newtonian system nor a classical thermodynamic system, but a combination of them in its nature. It possesses all the major energies of the two systems: the internal energy (or heat energy when the air is assumed as an ideal gas), geopotential energy, parcel kinetic energy and latent heat energy. The sum of the heat energy and geopotential energy in the dry atmosphere is usually called the total potential energy. The maximum amount of the non-kinetic energy which may be converted into kinetic energy was referred to as the available energy or available kinetic energy by Margules (1904). The pioneer studies of available energy were based on conservation of certain quantity of individual parcels, such as the potential temperature. For example, the study of Margules (1904) on the energy conversion was made by considering the adiabatic position exchanges of two air masses without mixing of them. The classical parcel algorithm showed that kinetic energy is created as warm air ascends and cold air descends, and the horizontal mean static stability is changed by energy conversion (See Belinskii also, 1948). This feature agrees with the consequence discussed in Chapter 7, that conversion of heat energy or total potential energy in the atmosphere is associated with conversion of geopotential energy. Thus, the static stability must be changed by the energy conversion at least in the dry atmosphere. Margules emphasized the importance of changing static stability in his barotropic unstable examples. Although the basic mechanism of energy conversion is intuitively simple, it is principally correct. Unfortunately, Margules left meteorology a few years after his study on the energy conversions, and few meteorologists afterwards could really understand his studies near a century ago. In many text books published, only the baroclinic but not the barotropic example of Margules was cited. Since application of the parcel algorithm for the three-dimensional atmosphere is limited by the data resolution and the difficulty in tracking down the individual parcels, the study of available energy, called the available potential energy later on, in the real atmosphere was carried out with a different method forwarded by Lorenz (1955). His initial atmosphere is baroclinic, and his reference state is barotropic of which an isentropic surface is on the mean height of initial state in pressure coordinates. The horizontal mean static stability is nearly unchanged after energy conversion in his study, and the kinetic energy is produced by reducing the baroclinity only (referring to Randall and Wang, 1992, also). However, since the static stability is unchanged, no geopotential energy is converted into kinetic energy. As conversion of geopotential energy is associated with conversion of heat energy in the atmosphere, no heat energy is converted into kinetic energy too. According to the definition of available potential energy, a barotropic and stat179
180
9. AVAILABLE ENTHALPY
ically unstable atmosphere possesses available potential energy too. It is learned in the preceding chapter that this unstable atmosphere is in an excited state, and kinetic energy may be created as the unstable atmosphere becomes stable. A baroclinic state may be viewed as an intermediate state between an unstable barotropic state and a stable reference state, as pointed out by Normand (1946). So, the unstable barotropic atmosphere may possess more available potential energy than a stable baroclinic atmosphere. This was shown by the simple examples of Margules. But, the algorithm of Lorenz (1955) cannot be used to evaluate the strong energy source in the barotropic atmosphere. Although his formula is still referred to in recent studies, it gives actually a measure of atmospheric baroclinity. To consider the contribution of static stability variation to the energy conversion, Lorenz (1960) proposed the concept of ‘ gross static stability’ to discuss the relation between available potential energy and static stability. This relation was not given quantitatively in an applicable form. A further effort was made by Dutton and Johnson (1967). But, the physics adopted by them was similar to that of Lorenz, so the effect of static instability in their new formulation gave a small modification only which depends nevertheless on presence of baroclinity. Pearce challenged the Lorenz’s philosophy in 1978. He divided the available potential energy into two components associated with variations of baroclinity and static stability respectively, and pointed out that Lorenz ignored the latter. He argued that there was an arbitrariness in choosing the static stability of reference state. However, he underestimated the contribution of changing static stability, by ignoring the related change in the mean temperature of a system which provides the largest source of kinetic energy. This will be discussed in the next chapter. Lorenz’s algorithm may only give one reference state, because his reference state depends on initial state only, and is independent of process. That is why his available potential energy was assumed as the upper limit of kinetic energy generation. If the reference state of Lorenz is really the only destination, the baroclinic atmosphere should produce approximately the evaluated maximum kinetic energy as it becomes barotropic, or the baroclinic atmosphere can never become barotropic through adiabatic processes. However, these cannot be proved by theory or observations. Lorenz agreed that kinetic energy generated in the atmosphere is considerably less than his available potential energy. It is unlikely that the large difference between the real and evaluated amounts is produced by the adiabatic and hydrostatic equilibrium assumptions used in his algorithm. The energy conversions in the atmosphere depend on the physical processes including the molecular and turbulent diffusions which may be parameterized in the prediction equations. However, the system energy equation derived from the prediction equations depends on initial and final states only, and so may not illustrate the processes of energy conversion. Although the energy equation is independent of process, it does not mean that the energy conversions are independent of process, as different processes lead to different reference states. For example, the barotropic reference states attained by different processes possess different static stabilities. Different amounts of kinetic energy are produced as a baroclinic atmosphere becomes barotropic, depending on the strength of diffusions which may destroy the kinetic energy source and increase system thermodynamic entropy. While, Lorenz’s
9.1. INTRODUCTION
181
reference state is independent of process, and so his algorithm cannot account for the energy conversions in various processes with different irreversibilities. As pointed out by Pearce (1978) and Marquet (1991), the study of available potential energy is of little significance if the evaluation is independent of process, since the atmosphere can hardly reach the state of minimum total potential energy. To consider the dependence of energy conversion on process, Pearce suggested to give up the reference state, and evaluate the available potential energy in the timedependent energy equation which includes explicitly the dissipation term of kinetic energy. He introduced the unavailable potential energy related to the kinetic energy dissipation. However, as evaluating the diffusions is not easy, Pearce’s method can hardly be used in practice. The time-dependent thermodynamic energy equation has been integrated in Chapter 7, together with the momentum equation, to obtain the time-independent energy equation of the atmosphere shown by (7.21). In this equation, kinetic energy dissipation and heat creation produced by molecular diffusions are canceled out with each other, and so the diffusion terms do not appear in equilibrium. This system energy equation is just the relationship used by Lorenz to derive his algorithm. Instead of integrating the diffusions as suggested by Pearce, the effects may be considered by finding the reference states attained through different diffusion processes. As discussed in the preceding chapter, the processes with different diffusions lead to different reference states. Since Lorenz’s available potential energy was considered as the upper limit of the total potential energy which may be converted into kinetic energy, the reference state must have two basic properties: i) It can be approached actually in the atmosphere; ii) among all the states attained through the actual processes, it possesses minimum total potential energy. The first property suggests that we should evaluate the budgets of thermodynamic entropy and geopotential entropy for assumed processes of energy conversion. A reference state may be acceptable only if these entropy productions are both positive in isolation. This ideal was never considered until the studies of McHall (1990a, b; 1991), because the theories of turbulent entropy and geopotential entropy were not developed before. When the process of energy conversion was assumed to be reversibly adiabatic as we did, the classical thermodynamic entropy was conserved. So, it was thought that the process was reversible and the reference state attained was the limit state with minimum total potential energy. However, energy conversions in the real atmosphere are never reversible. This irreversibility may be explained by the turbulent entropy law and geopotential entropy law. The second property implies that the reference state of Lorenz is assumed as the lowest state discussed in the preceding chapter. It will be illustrated in this and the next chapter that when mass and potential enthalpy is conserved, the barotropic atmospheres may have different static stabilities. The atmosphere with a higher stability possesses less total potential energy. It was not proved that the reference state of Lorenz has the highest static stability, compared with other reference state which may also be attained in the same atmosphere. We know at least that Lorenz’s algorithm fails to give the lowest state for the statically unstable atmosphere. Although there were different opinions on choice of reference state (e.g., Margules,
182
9. AVAILABLE ENTHALPY
1904; Lorenz, 1955; van Mieghem, 1956; Dutton and Johnson, 1967, Pearce, 1978), the reference states assumed by different authors were not proved to have these properties. As the reference state depends on process and is not unique, the lowest state is viewed as an extremal state in the current study. Since the process leading to the lowest state produces maximum geopotential entropy, we apply the variational approach forwarded by McHall (1990a) to solve the lowest state and corresponding maximum kinetic energy generation from the system energy equation derived in Chapter 7. When the processes are identified statistically with thermodynamic entropy productions as discussed in the preceding chapter, we use thermodynamic entropy equation as a constraint relationship for the variational algorithm. To filter out the heat-death solution of reference state, we use the conservation law of potential enthalpy discussed in Chapter 7 as another constraint relationship. When the potential enthalpy is conserved, the potential temperature at a reference state cannot exceed the upper and lower extremal values at the initial state. This fact gives the third constraint called the thermal constraint for the new approach. This variational approach is different from that of Dutton and Johnson (1967), as they did not use these constraint relationships to solve the reference states. Corresponding to the different processes represented by different thermodynamic entropy productions, we may find different reference states with different static stabilities for a provided initial state. The lowest state obtained from the current study possesses the two basic properties discussed previously. The new algorithm may be used to evaluate the energy conversions in either the statically stable or unstable atmosphere. It can also be used to study the energy conversion in a local region, since it, unlike the algorithm of Lorenz, needs not apply the zonal mean fields. Developments of disturbances in the real atmosphere are also the local events in most situations. The reference state derived from the new variational approach is continuous in space. For discontinuous initial states, the reference states obtained by the classical parcel algorithm of Margules (1904) are also discontinuous. For comparison, we evaluate in the last section the continuous reference states for Margules’ examples with the variational approach. It will be found that the maximum available enthalpy with respect to the continuous lowest state is less than that with respect to the discontinuous lowest state, because the continuous lowest states are attained by turbulent and molecular diffusions which destroy kinetic energy and the energy source. Moreover, comparison of the solutions with and without the thermal constraint emphasizes the importance of using the constraint. It will be found that kinetic energy generation is different from the reversible adiabatic process through which air masses of different densities exchange their positions. The potential temperature of a parcel or at a grid point may change in energy conversion, though the whole system has no heat exchange with the surroundings. This agrees with the observations of disturbance development in the atmosphere. Changes of parcel potential temperature and grid-point potential temperature are brought about by molecular and turbulent diffusions. Thus, the large-scale circulations associated with energy conversions may not be studied simply by large-scale dynamics only, as the energy conversions depend on subgrid-scale processes. This
9.2. AVAILABLE ENTHALPY
183
fact increases the difficulty in weather forecasts and numerical simulations using the grid-point data.
9.2
Available enthalpy
According to Lorenz (1955), the non-kinetic energy in the atmosphere (or the total potential energy) which can be converted into parcel kinetic energy is the sum of heat energy and geopotential energy called the available potential energy. It is discussed in Appendix A that the internal energy includes heat energy and thermopotential energy. Since thermopotential energy of air is negligibly small compared with the heat energy in the normal conditions of Earth’s atmosphere, it may be ignored for meteorological studies. The variation in heat energy of the atmosphere is given by cv ∆U = g
ps pt
A
(Tr − T0 ) dAdp .
(9.1)
Moreover, the variation of geopotential energy related to Earth’s gravitational force is represented by ps
∆Φ = pt
A
(zr − z0 ) dAdp ,
(9.2)
where the initial and reference states are denoted by subscripts 0 and r respectively. Integrating (9.2) by parts produces ∆Φ = −Wv +
in which Wv =
A
R g
ps pt
A
(Tr − T0 ) dAdp ,
(9.3)
[ps (zs0 − zsr ) − pt (zt0 − ztr )] dA
is the mechanic work done as the atmosphere changes volume. When energy is conserved in the isolated atmosphere, we have ∆U + ∆Φ + W = 0 ,
(9.4)
where W is the total mechanic work created by the atmosphere. This work includes the kinetic energy generation ∆K and mechanic work Wv , so (9.4) is rewritten as ∆K = −∆U − ∆Φ − Wv . The kinetic energy is in all scales including turbulent kinetic energy. Using (9.1) and (9.3), we gain cp ps (T0 − Tr ) dAdp . (9.5) ∆K = g pt A It is also the system energy equation (7.22). The non-kinetic energy which can be converted into kinetic energy is cp Λ= g
ps pt
A
(T0 − Tr ) dAdp ,
(9.6)
9. AVAILABLE ENTHALPY
184 or
cp Λ= κ gpθ
ps pt
A
pκ (θ0 − θr ) dAdp .
(9.7)
It is identical to the change of atmospheric enthalpy, and so is called the available enthalpy in this study. As discussed in Chapter 7, available enthalpy in the whole atmosphere is equivalent to the total potential energy which can be converted into kinetic energy. But in the atmosphere below a certain height, available enthalpy is different from available potential energy. This equation of available enthalpy will be derived again in Chapter 20 from a new theory of atmospheric heat engines. The available enthalpy depends on initial and final states only, and so can be evaluated conveniently by assuming a reference state for a provided initial state. Although there are no diffusion terms included in the energy equation, the energy conversions evaluated with the equation depend on process as well as initial state because the reference states depend on process. The extremal reference states, such as the isothermal heat-death state, isentropic state and the lowest state discussed in the preceding chapter, may be solved according to the initial state only. However, they still depend on the specified extremal processes, as they are attained with maximum or minimum thermodynamic entropy production under certain conditions. Usually, the reference state is assume barotropic. The thermal structure of a barotropic state depends on the static stability and surface temperature. To consider the relation between available enthalpy and thermal structure of reference state, we assume that the barotropic reference state has a constant lapse rate of temperature Γr , and so is represented by
Tr = Tsr
pt ps
γ
,
γ=
R Γr . g
Applying it for (9.6) gives
Tsr pt ps − pt Λ = Ψ0 − Γd (b + 1) ps
γ
,
where Ψ0 is the total enthalpy at initial state evaluated from cp Ψ= g or
cp Ψ= κ gpθ
ps
T dAdp pt
A
ps pt
pκ θ dAdp ,
(9.8)
A
and the surface temperature Tsr at the reference state may be determined by the conservation law of potential enthalpy. This equation shows that available enthalpy increases with the final static stability Γr for a provided initial state. In other words, the reference state will be more statically stable if more kinetic energy is created.
9.3
Constraint relationships
In mathematics, a weather prediction at any future time can be made with the complete equation set representing perfectly all the physical processes in the atmosphere
9.3. CONSTRAINT RELATIONSHIPS
185
for provided initial and boundary conditions. But the predictions in practice do not depend on initial data only, since the primitive equations cannot be solved without making additional assumptions and simplifications (including using the finite difference schemes to replace the differentials in numerical models), and many physical processes including the molecular and turbulent diffusions cannot be represented accurately in numerical prediction models. Thus, the predictions depend also on the physical processes assumed, such as the parameterizations for the diffusions and other subgrid-scale processes. The system energy equation (9.8) derived from the prediction equations does not include the differentials and diffusions, and so can be used conveniently to evaluate the energy conversions between equilibrium states. However, it is not a prediction equation, and so cannot be used to determine a future state for provided initial conditions only. The energy conversions in the atmosphere are then evaluated by choosing a reference state for provided initial state. The choice can be made by additional relationships which manifest the features of energy conversion. In this sense, we say that the energy conversion and reference state depend on the process as well as initial conditions. In the study of Lorenz (1955), the reference state attained by energy conversion is assumed as the barotropic state with the horizontal mean static stability of the initial field. Since the energy created in a real process is different from that calculated with Lorenz’s algorithm, different reference states may also be assumed. The timeindependent energy equations cannot tell us whether a reference state assumed may really occur after the initial state. Thus, we have to evaluate the variations of thermodynamic entropy and geopotential entropy in the process. According to the turbulent entropy law and geopotential entropy law discussed earlier, an assumed reference state may occur through parcel motions only if these entropies do not decrease in an isolated system. Thus, these entropy laws may be used in addition to the energy equation to identify the processes for the study of energy conversion. The energy equation, turbulent entropy law and geopotential entropy law are all the integration equations. For the initial state together with thermodynamic and geopotential entropy productions provided, the three equations include only one variable: the reference state represented by either the temperature or potential temperature. Since the variable is a field variable instead of a constant value, the right solution of these integration equations may still not be determined uniquely, since the same integration over a given interval may be obtained for different functions. For example, the statically stable and unstable temperature profile may give an equal amount of thermodynamic entropy, as discussed in the previous chapter. It is discussed in the preceding chapter that the geopotential entropy is created by conversion of geopotential energy into kinetic energy, which is correlated linearly with conversion of heat energy in the dry atmosphere. To meet with the geopotential entropy law, we may assume an extremal state as the reference state attained through the process with maximum production of geopotential entropy. Obviously, this extremal state possesses maximum kinetic energy and minimum enthalpy, compared with other reference states attained from the same initial state for a provided thermodynamic entropy production in the isolated atmosphere. This extremal reference state may be solved using the variational approach of McHall
9. AVAILABLE ENTHALPY
186
(1990a). Although the solution may be different from the real state after energy conversion, it gives the limit of the reference states which are physically meaningful. In this approach, the turbulent entropy law is applied as a constrained relationship. The reference state is acceptable only if the turbulent entropy is not destroyed in isolation. It is found in the next section that the reference state of an isolated atmosphere attained through molecular diffusions is an isothermal state without parcel kinetic energy. This heat-death state is not interested by meteorologists, and can be filtered out by the conservation law of potential enthalpy (8.14). This law will be used as the second constraint to study the energy conversion. In the quasi-adiabatic processes including adiabatic parcel motions and isenthalpic diffusions only, parcel kinetic energy may still be destroyed by converting into heat energy and geopotential energy through vertical motions in the statically stable atmosphere. For the energy conversions in the quasi-adiabatic processes without, potential temperature of reference state cannot be less than the minimum value or greater than the maximum value at initial state. These features give the third constraint, called the thermal constraint and denoted by min(θr ) ≥ min(θ0 ) ,
max(θr ) ≤ max(θ0 ) .
(9.9)
This constraint will be used as the boundary conditions to determine the thermodynamic entropy production, since increasing the entropy may reduce the vertical gradient of potential temperature or the difference between the maximum and minimum potential temperatures at the reference state. We shall see in the last section of this chapter that a reference state solved without using this constraint may be false, as the extremal potential temperatures exceed the limits.
9.4
Variational approach
To find out the extremal reference state with maximum production of geopotential entropy or kinetic energy is a simple isoperimetric problem in calculus of variations: From all the curves described by θr (p) (p ∈ [pt , ps ]), which have at least the first class continuous derivative in [pt , ps ] and satisfy the potential enthalpy conservation equation (8.14) and thermal constraint (9.9), to find the curve on which available enthalpy assumes the extremum for provided thermodynamic entropy production ∆S. The geopotential entropy law ∆Sg ≥ 0 is also satisfied when net kinetic energy is generated in the process. We consider in this chapter only the barotropic reference states, while the baroclinic reference states will be studied in Chapter 18. To solve this problem, we make the auxiliary function from (9.7), (8.3) and (8.14), giving (9.10) X = −pκ θr + λ1 ln θr + λ2 θr , where λ1 and λ2 are two constant Lagrangian multiplies. The necessary condition for the extremum is provided by the Euler equation d ∂X − ∂θr dp
∂X ∂θrp
=0
(9.11)
9.4. VARIATIONAL APPROACH
187
with θ rp =
∂θr . ∂p
It gives the potential temperature of reference state θr =
pκ
λ1 . − λ2
(9.12)
To determine the constants λ1 and λ2 , we insert (9.12) into (8.14) and (8.3), producing ps 1 θ0 dAdp (9.13) λ1 = AZ pt A and ps pt
1 ln(p − λ2 ) dp = (ps − pt ) ln λ1 − Γd ∆S˜ − A κ
ps A pt
ln θ0 dpdA ,
(9.14)
where the tilde˜indicates a value in the atmosphere of unit horizontal cross section, and ps dp . (9.15) Z= κ−λ p pt 2 This constant parameter Z can be calculated analytically as shown in the following. If setting p = xcp , we see xs
Z = cp xt
xcp −1 dx xR − λ2
xs
cv −1
x
= cp xt 1/c
cv −R−1
+ λ2 x
+
λ22 xcv −2R−1
+
xcv −2R−1 λ32 R x − λ2
dx ,
1/c
where xs = ps p , xt = pt p . Assuming that R = 286.9 J/K and cp = 1004.15 J/K, we have cp = 3.5R, cv = cp − R = 2.5R and
2 xcv −2R−1 dx = xR − λ2 R
dy , y 2 − λ2
in which y = xR/2 . It follows that
R x 2 − √λ 1 xcv −2R−1 2 dx = √ ln R √ xR − λ2 R λ2 x 2 + λ2
(λ2 > 0) ,
R
2 x2 xcv −2R−1 dx = √ arctan √ R x − λ2 R −λ2 −λ2
(λ2 < 0) .
Now, we gain cp Z = 2 R
5R 5R 3R 3R λ2 2c 1 2c 2c 2c (ps p − pt p ) + (ps p − pt p ) 5 3
R
+
2c λ22 (ps p
R 2cp
− pt
)+
λ32 Z
,
(9.16)
9. AVAILABLE ENTHALPY
188 where 1 arctan Z = √ −λ2
2cR (p p 1 s Z = √ ln R 2 λ2 2cp (ps
√
R 2cp
−λ2 (ps
R 2cp
− pt
)
R 2cp
(ps pt ) − λ2 R √ √ 2cp − λ2 )(pt + λ2 ) R √ √ 2cp + λ2 )(pt − λ2 )
(λ2 < 0) ,
(λ2 > 0) .
For λ2 = 0, we see
ps1−κ − pt1−κ . 1−κ Thus, we have three expressions of Z for different λ2 . Furthermore, it can be proved that Z=
ps pt
ln(pκ − λ2 ) dp = ps ln(pκs − λ2 ) − pt ln(pκt − λ2 ) − κ(ps − pt ) − κλ2 Z .
(9.17)
Applying it and (9.13) for (9.14) yields ps ln(pκs − λ2 ) − pt ln(pκt − λ2 ) − (ps − pt )(ln λ1 + κ) + Γd ∆S˜ ps 1 ln θ0 dpdA = 0 . − κλ2 Z + A A pt
(9.18)
For a given initial state, the constants λ1 can be evaluated from (9.13). Inserting it together with assumed ∆S˜ into (9.18), we solve λ2 by iteration. The reference state is then computed from (9.12). The result is examined with the thermal constraint (9.9). If the constraint does not meet, we increase ∆S˜ to solve λ2 again until the constraint is satisfied. Since λ1 ∂2X =− <0, 2 ∂θr θr where λ1 is positive as shows by (9.13), the available enthalpy with respect to the reference state is maximum. In general, the kinetic energy generation depends on thermodynamic entropy production. The energy generated reaches the upper limit when the entropy produced is minimum. The reference state attained through the extremal process is then called the lowest state as it possesses the least enthalpy. If the conservation law of potential temperature is not used, we have λ2 = 0 in (9.10), and (9.12) gives λ1 θr = κ . p It tells that the reference state is an isothermal state with temperature Tr =
λ1 . pκθ
Although the reference state is true and can be reached by molecular diffusions after a long while in isolation, this heat-death state may not provide useful information for the study of energy conversions within the time limit of weather prediction. Thus, the conservation law is important for us to filter out the heat-death reference state produced by anisobaric molecular diffusion in the study of energy conversions.
9.5. THE LOWEST STATE
9.5
189
The lowest state
Since kinetic energy and the energy source may be destroyed by molecular and turbulent diffusions, the lowest state cannot be determined only by the geopotential entropy law as it depends also on thermodynamic entropy production. Obviously, the maximum kinetic energy is created in the process with minimum thermodynamic entropy production as discussed in Chapter 8. An example of this process is the pseudo- reversible process discussed in the preceding chapter. Thus, the lowest state is the geopotentially and thermodynamically extremal state, which possesses the strongest static stability and minimum enthalpy if compared with other reference states attained from the same initial field in isolation. It is an idealized state and can be solved for provided initial state only as shown by (8.39). This does not mean that the lowest state is independent of process, as the process is specified as the extremal process with minimum thermodynamic entropy production and maximum geopotential entropy production. To give examples of the lowest state, the initial atmosphere is taken as the linear atmosphere of which potential temperature field is represented by (8.23). When the surface pressure is different from pθ , we see θ0 = (Ts0 + Ty y) with γ=
RΓ , g
pκθ pγs pb
b =κ−γ =
(9.19)
R (Γd − Γ) , g
in which Ts0 is a constant temperature on the bottom pressure surface ps and at y = 0. For the atmosphere over the area from −X to X in the x direction and from −Y to Y in the y direction, we gain ps A pt
θ0 dpdA = 4
XY Ts0 pκθ 1−b ps − pt1−b . γ ps (1 − b)
Inserting it into (9.13) yields λ1 =
Ts0 pκθ γ ps (1 − b)Z
ps1−b − pt1−b
.
(9.20)
Moreover, we may apply ps pt
A
ln θ0 dAdp = 2XY (ps − pt )
+
Ts0 Ts + Ty Y 1 ln(Ts20 − Ty2 Y 2 ) + ln 0 2 2Y Ty Ts0 − Ty Y
pκ ln θγ + b − 1 − 2XY b(ps ln ps − pt ln pt ) . ps
(9.21)
˜ for (9.18) to solve the constant λ2 for the minimum ∆S. To find the minimum thermodynamic entropy production, we assume firstly ∆S = 0. The obtained reference state is examined by the thermal constraint (9.9). If the constraint is not satisfied, we increase the entropy production until the constraint is satisfied. The reference state attained through the process with the threshold
9. AVAILABLE ENTHALPY
190
Figure 9.1: The lowest states (solid) corresponding to the baroclinic initial states of which the profile at y = 0 is shown by the dashed curve. The |Ty | varies in the range 4 - 32 K/1000km with interval of 4 K/1000km from left to right at the top of the diagram.
entropy production gives the lowest state. The examples evaluated in the assumed atmosphere from 1000 hPa up to 200 hPa for Ts0 = 290 K ,
Γ = 0.65 K/100m
are sketched in Fig.9.1. The minimum entropy productions are zero in these baroclinic examples and so the processes are pseudo-reversible. This figure shows that the vertical mean static stability of the lowest state is greater than that of the initial state. The final static stability is higher if the initial baroclinity is stronger. This consequence agrees with (8.38), which tells that the static entropy of the reference state increases with the initial baroclinity but is reduced by thermodynamic entropy production. The height at which the temperature is unchanged is independent of the baroclinity in these examples. When the geopotential energy is converted into kinetic energy, the warm air moves upward and cold air moves downward to reduce the height of gravity center. It will be illustrated in Chapter 19 that the energy conversion is achieved by the slantwise convection called also the sloping convection (Palm´en and Newton, 1969 ;Ludlam, 1980; Houghton, 1986) with a trajectory slope less than the slope of isentropic surface. Although the parcel temperature increases (or decreases) adiabatically in the downward (or upward) convection, the local temperature decreases (or increases) in the lower (or upper) troposphere, as the atmosphere reaches the lowest state shown in Fig.9.1. The surface cooling over oceans may reduce the sea surface temperature and form the cold pools.
9.6. MAXIMUM AVAILABLE ENTHALPY
191
Figure 9.2: Cross-section of temperature (long dashed, ◦ C) and vertical velocity (solid and short dashed, 10−2 Ps/s) at 00Z on 10 January 1992, drawn by ECMWF data. The cross marks indicate the cyclone center.
The change of temperature profile may be confirmed by Fig.9.2, which is a cross section of a baroclinic cyclone over European continent at 00Z on January 10, 1992 depicted using ECMWF data. The cyclone center, indicated by the crosses, tilts westward with height and is colder than the surroundings on the surface but warmer near the tropopause. The height where the central temperature is nearly unchanged is higher than that in Fig.9.1, since the disturbance is not isolated and so cold air fills the storm center continuously in the development. As a result of upper tropospheric warming over the cyclone center, the isentropic surfaces and tropopause descend as shown also by other examples in the studies of Palm´en (1951) and Peltonen (1963).
9.6
Maximum available enthalpy
The total enthalpy of the lowest state is calculated by inserting (9.12) into (9.8), giving ˜ r = λ1 (ps − pt + λ2 Z) . Ψ Γd pκθ When the lowest state is taken as the reference state, the maximum available enthalpy in the isolated atmosphere of unit horizontal section area is obtained by
9. AVAILABLE ENTHALPY
192
Figure 9.3: Maximum available enthalpy. The temperature lapse rates of initial states are denoted by Γ. Dashed curves display the ratio of maximum available enthalpy to initial total enthalpy.
applying the previous expression for (9.7), giving ˜ 0 − λ1 (ps − pt + λ2 Z) , ˜ max = Ψ Λ Γd pκθ
(9.22)
where the total enthalpy is calculated by integration from 1000 hPa to 200 hPa. The temperature profile of the initial linear atmosphere is given by
T0 = (Ts0
p + Ty y) ps
γ
,
γ=
RΓ . g
For comparison, we assume that the initial atmospheres have the same amount of total enthalpy as ˜0 = Ψ =
ps Ts00 pγ0 dp γ0 Γd p s p t γ0 pt Ts00 ps − pt , Γd (γ0 + 1) ps
in which Γ0 = 0.65 K/100m ,
Ts00 = 270 K ,
γ0 =
RΓ0 . g
9.7. APPROXIMATE APPROACH
193
The surface temperatures of the initial states are then given by Ts0 =
˜ 0 (γ + 1)pγs Γd Ψ pγ+1 − pγ+1 s t
˜ 0 and λ1 into (9.22) yields at y = 0. Inserting the expressions of Ψ ˜ max = 1 Λ Γd
Ts00 pt ps − pt γ0 + 1 ps
γ0
Ts (p1−b − pt1−b ) − 0 γs (ps − pt + λ2 Z) ps (1 − b)Z
.
The evaluated results are displayed in Fig.9.3. In general, the maximum available enthalpy increases with the baroclinity and static instability of initial atmosphere. It varies almost linearly with the strong horizontal temperature gradient. A statically stable and barotropic atmosphere has no positive available enthalpy. The maximum available enthalpy jumps up highly as the atmosphere becomes statically unstable. Comparing the initial and final states finds that the extremal potential temperatures change after the energy conversion. The changes can be produced by mixing of parcels due to molecular and turbulent diffusions. Thus, the energy conversion is not the simple process by which the grid-scale warm and cold air masses exchange their positions adiabatically, and so depends on interactions between different scales. The ratio of maximum available enthalpy to the initial total enthalpy is given by ˜ max Λ λ1 =1− (p − pt + λ2 Z) . Rh = ˜0 ˜ 0 pκ s Γd Ψ Ψ θ The dependence on the baroclinity is shown by the dashed curves in Fig.9.3. The ratio is less than one percent in the statically stable atmosphere even if the baroclinity is strong, but increases greatly in the statically unstable atmosphere
9.7 9.7.1
Approximate approach The lowest state
If thermodynamic entropy variation in the atmosphere is calculated approximately from (8.20), the lowest state and maximum available enthalpy may be given analytically. With the definitions (8.10) and (8.11), (9.7) is rewritten as A Λ= Γd pκθ
ps pt
pκ (θ0 ∗ − θr∗ ) dp .
(9.23)
Meanwhile, the condition (8.14) is replaced by (8.13). Now, we set 2
X = pκ θr∗ − λ1 θr∗ − λ2 θr∗ , where λ1 and λ2 are two constant Lagrangian multiplies. Its Euler equation reads d ∂X − ∂θr∗ dp
∂X ∂θr∗p
=0,
θr∗p =
∂θr∗ . ∂p
9. AVAILABLE ENTHALPY
194 It follow that θr∗ =
pκ − λ2 . 2λ1
(9.24)
Inserting it into (8.13) and (8.20) produces pκ+1 − pκ+1 s t (κ + 1)(ps − pt )
λ2 = and
(9.25)
1 λ1 = ± 2
C G
(9.26)
respectively, in which C= and
(pκ+1 − p2κ+1 − pκ+1 )2 p2κ+1 s s t t − 2κ + 1 (κ + 1)2 (ps − pt )
ps
2
θ0 ∗ +
G= pt
1 A
A
θˇ02 dA
(9.27)
dp − 2Γd ∆S˜ θ0 2 .
(9.28)
Consequently, (9.24) becomes
θr∗
G C
=
pκ+1 − pκ+1 s t − pκ (κ + 1)(ps − pt )
.
(9.29)
As the lowest state is statically stable, the negative sign in (9.26) is chosen. Finally, from (9.29) we obtain the reference state:
θr = θ0 +
G C
pκ+1 − pκ+1 t s − pκ (κ + 1)(ps − pt )
.
To evaluate the parameter G, we use A
θˇ02 dA =
A
= A
(θ02 − 2θ0 θ0 )dA + Aθ0 2 θ02 dA − Aθ0 2
together with ps pt
θ0 ∗2 dp =
ps p
tps
= pt
(θ0 2 − 2 θ0 θ0 ) dp + (ps − pt ) θ0 2 θ0 2 dp − (ps − pt ) θ0 2
for (9.28) and gain 1 G= A
ps pt
A
˜ θ0 2 . θ02 dAdp − (ps − pt + 2Γd ∆S)
(9.30)
9.7. APPROXIMATE APPROACH
195
Figure 9.4: As Fig.9.1, but for the approximate solutions. It depends only on initial state and assumed thermodynamic entropy production. Since ∂2X = 2λ1 ∂θr∗2 where λ1 is negative, the available enthalpy with respect to the reference state (9.30) is maximum for provided initial state and thermodynamic entropy production. For the initial atmosphere assumed by (9.19), we have 1 G= 1 − 2b
Ts20
Ty2 2 Y + 3
ps − pt
ps pt
2b
2
˜ θ0 − (ps − pt + 2Γd ∆S)
,
where θ0 is evaluated from (8.28). The lowest states attained through pseudoreversible processes are displayed in Fig.9.4. The potential temperature profiles are more bend than those shown in Fig.9.3.
9.7.2
Maximum available enthalpy
The maximum available enthalpy over the atmosphere with unit horizontal section area derived from (9.23) gives ˜ max = cp Λ gpκθ
√
ps
CG + pt
∗
p θ0 dp κ
.
(9.31)
It increases with initial baroclinity. For the initial states assumed previously, the maximum kinetic energy generated is displayed in Fig.9.5, which is similar to Fig.9.3 plotted with the precise algorithm.
196
9. AVAILABLE ENTHALPY
Figure 9.5: As Fig.9.3, but for the approximate solutions.
Inspecting (9.31) and (9.28) finds that available enthalpy of a provided initial atmosphere decreases with increasing thermodynamic entropy production in the process. The available enthalpy is maximum if the entropy production is minimum in the isolated atmosphere. In an open system, local thermodynamic entropy production may be less than zero if there is negative entropy input from the surroundings. In this case, kinetic energy created may be more than the maximum available enthalpy evaluated in isolation. It is noted that the maximum available enthalpy gives only the maximum possible amount of kinetic energy generation in a dry atmosphere. This amount may not be entirely converted into kinetic energy in an highly irreversible process. Also, the atmosphere possesses a certain amount of available enthalpy may not be in the process of energy conversion. The energy conversion may depend on other mechanic conditions. For example, when air motions are constrained by geostrophic and hydrostatic balances in the baroclinic atmosphere, the energy conversion may
9.8. THERMODYNAMIC ENTROPY VARIATION
197
not take place until some perturbation forcings have approached and the previous balances are destroyed.
9.8
Thermodynamic entropy variation
The equilibrium state of a classical thermodynamic system is uniform, so thermodynamic entropy variation of a unit mass between two uniform equilibrium states is identical everywhere in the system. This is not the case in the stratified atmosphere of which thermodynamic variables vary with space at an inhomogeneous equilibrium state. The local entropy variation may change from place to place, and the entropy variation of a parcel may be different from that of another one. The local entropy variation of unit air mass at a given point in the dry atmosphere may be evaluated from θr δs = cp ln . (9.32) θ0 It includes the contribution of thermodynamic entropy flux, which may be greater than the internal entropy production at the local place. We have discussed earlier that for a provided initial state, the kinetic energy created decreases with increasing thermodynamic entropy production. The relation between available enthalpy and the entropy variation may be seen more clearly from Λ=
cp gpκθ
ps pt
δs
p κ θ 0 1 − e cp
A
dAdp ,
(9.33)
obtained by inserting (9.32) or δs
θ r = θ 0 e cp
(9.34)
into (9.7). For a given initial state, available enthalpy depends on the entropy variation only. The lowest state and maximum available enthalpy can be derived also with the previous expression. Inserting (9.34) into (8.14) and (8.3) yields
ps A pt
θ0 e
and ∆S =
cp g
δs cp
−1
dpdA = 0 ,
(9.35)
ps
δs dpdA ,
(9.36)
A pt
respectively. For a provided thermodynamic entropy production, the atmosphere may attain different reference states. Among them, we may find an extremal state which possesses minimum enthalpy. In particular, the extremal state attained by the processes with minimum thermodynamic entropy production is the lowest state. To evaluate the extremal reference state and maximum available enthalpy, we make the auxiliary function δs
X = −cp θ0 (pκ − λ2 )e cp + λ1 δs
9. AVAILABLE ENTHALPY
198
using (9.33), (9.35) and (9.36). Its Euler equation with respect to δs gives δs
θ0 (pκ − λ2 )e cp − λ1 = 0 and follows that δs = cp ln
θ0
λ1 . − λ2 )
(9.37)
(pκ
Comparing it with (9.32) finds θr =
λ1 . pκ − λ2
It is identical to (9.12). Inserting (9.37) into (9.35) and (9.36) gives (9.13) and (9.18). So, the obtained reference state and extremal available enthalpy are the same as discussed previously.
9.9
Geopotential entropy variations
In the process of energy conversion, cold air moves downward while warm air moves upward. This process is geopotentially irreversible in the gravitational field. When more kinetic energy is created, more cold air and warm air exchange their positions and so the process is more irreversible. However, the previous discussions show that kinetic energy generation increases with reducing thermodynamic entropy production. Thus, the thermodynamic entropy itself cannot explain the irreversibility of atmospheric variations. Geopotential entropy of the atmosphere is given by (8.49). For the initial state represented by (9.19), the geographic height is evaluated from
Ts Ts − T = 1− z= Γ Γ
p ps
γ
,
Ts = Ts0 + Ty y
when Γ = 0. Applying it and (9.19) for (8.49) yields the geopotential entropy of the initial state
ps A (pκ − pγs pκ−γ )dp κ Γpθ pt pγs 1 A κ−γ+1 κ+1 κ+1 κ−γ+1 (p (p − pt ) − − pt ) . Γpκθ κ + 1 s κ−γ+1 s
Sg 0 = =
(9.38)
This expression is independent of surface temperature, but depends on the static stability. The hydrostatic equation (5.23) may be rewritten as dz = −
RT dp gp
(9.39)
for ideal gases. When Γ = 0, the geographic height of the isothermal initial field gives ps RT ln . z= g p
9.9. GEOPOTENTIAL ENTROPY VARIATIONS
199
Figure 9.6: Geopotential entropy production in the process of producing maximum available enthalpy. The temperature lapse rates at initial states are denoted by Γ.
So, we gain
Sg 0
AR ps 1 (pκ+1 − pκ+1 = κ pκ+1 ln − ) t t gpθ (κ + 1) pt κ+1 s
.
(9.40)
Geopotential entropy of the isothermal atmosphere depends only on the height of the atmosphere, and is independent of the temperature. Moreover, the geographic height of the reference state may be computed by inserting (9.12) into (9.39) giving z=
cp λ1 pκs − λ2 ln κ . gpκθ p − λ2
Applying it and (9.12) for (8.49) yields the geopotential entropy Sg r
Acp ps κ pκs − λ2 (p − λ ) ln dp 2 gpκθ pt pκ − λ2 1 Acp (pκ+1 − pκ+1 ) ln(pκs − λ2 ) − λ2 (ps − pt ) ln(pκs − λ2 ) = − κ t gpθ κ + 1 s = −
9. AVAILABLE ENTHALPY
200 −
ps pt
p ln(p − λ2 )dp + λ2 κ
κ
ps pt
ln(p − λ2 )dp κ
for the reference state. Inserting (9.17) and ps pt
p ln(p − λ2 )dp = κ
κ
−
1 pκ+1 ln(pκs − λ2 ) − pκ+1 ln(pκt − λ2 ) t κ+1 s 1 κ+1 κ+1 2 (p − pt ) + λ2 (ps − pt ) + λ2 Z , κ κ+1 s
into the previous equation yields Sg r
pκ − λ2 κ pκt − λ2 pt ln sκ − (pκ+1 − pκ+1 ) t κ+1 pt − λ2 (κ + 1)2 s
=
Acp gpκθ
+
λ2 κ2 (ps − pt + λ2 )Z κ+1
.
The geopotential entropy change is then given by ∆sg =
g (Sg − Sg0 ) . A(ps − pt ) r
The examples for the energy conversions discussed previously are displayed in Fig.9.6. In the baroclinic examples the kinetic energy generated is positive. Fig.9.6 shows that the geopotential entropy grows in the energy conversion. The geopotential entropy created increases with initial baroclinity and static instability. As discussed earlier, the tendency of geopotential entropy change may be opposite to that of barotropic entropy change. When geopotential entropy increases through quasi-adiabatic process in a statically stable atmosphere, the barotropic entropy decreases. Thus, the kinetic energy created increases with reducing thermodynamic entropy production. When more kinetic energy is produced, the process is more geopotentially irreversible though thermodynamic entropy is unchanged in a pseudo- reversible process. Since the atmosphere in the gravitational field tends to reduce its geopotential energy and lower its gravity center, increase of geopotential entropy is prior to increase of thermodynamic entropy in the atmosphere, and the atmosphere changes towards to the lowest state instead of isentropic state. This will be discussed again in Chapter 17.
9.10
Discontinuous examples
9.10.1
Baroclinic example
Margules (1904) studied the energy conversion in the baroclinic and barotropic systems of two discontinuous air masses. The obtained lowest states are discontinuous. While, the reference state evaluated with the variational approach is continuous in space.This new approach may also be used to derive the continuous lowest states for the discontinuous examples. The first example is demonstrated in Fig.9.7. Two air masses with potential temperature θ1 and θ2 (θ2 > θ1 ) sit side by side initially, as shown by diagram (a).
9.10. DISCONTINUOUS EXAMPLES
201
pt
pt
pt
θ2 θ1
θ2
pm θ1
ps
ps
ps
(a)
(b)
Figure 9.7: Margules’ example 1 It is discussed in Chapter 7 that the geopotential energy and heat energy of an isolated system must decrease in the process of kinetic energy generation. Thus, the lowest state is the idealized discontinuous reference state shown in (b) with the warm air mass over the cold. This reference state is attained by adiabatic displacement of the masses without molecular and turbulent diffusions, and possesses minimum geopotential energy and heat energy compared with other reference states. Supposing each of the air masses has unit horizontal section area at the initial state, the total enthalpy is evaluated from ˜ 0 = Ψ01 + Ψ02 , Ψ 2 where
κ
κ
Ψ01
θ1 pκs pt = ps − pt κ Γd pθ (κ + 1) ps
Ψ02
θ2 pκs pt = ps − pt κ Γd pθ (κ + 1) ps
and
are the enthalpies of air masses 1 and 2 respectively. Moreover, the enthalpies at the reference state are given by
Ψ r1 =
and Ψ r2
2θ1 pκs pm ps − pm Γd pκθ (κ + 1) ps
2θ2 pκm pt = pm − pt κ Γd pθ (κ + 1) pm
respectively, where pm =
ps + pt . 2
The total enthalpy at the lowest state is ˜ r = Ψ r1 + Ψ r2 . Ψ 2
κ
κ
,
9. AVAILABLE ENTHALPY
202
Figure 9.8: Maximum available enthalpy of Margules’ examples. Heavy and light solid lines are plotted with respect to the discontinuous and continuous lowest states respectively. The dashed lines are the false results without using the thermal constraint.
The maximum available enthalpy is then given by ˜r . ˜ =Ψ ˜0 − Ψ Λ
(9.41)
For ps = 1000 hPa, pt = 500 hPa, θ1 = 270 K and θ2 = 280 K, the evaluated available enthalpy is depicted by one of the heavy solid lines in Fig.9.8. It increases almost linearly with the potential temperature difference between two air masses.
9.10.2
Barotropic example
Now, we consider the second example: The cold air mass is over the warm one initially as shown in Fig.9.9. The enthalpies of these air masses at the barotropic initial state are 2θ1 (pκ+1 − pκ+1 Ψ01 = h ) Γd (κ + 1)pκθ m
9.10. DISCONTINUOUS EXAMPLES
203
pt
θ1 pm θ2 ps
Figure 9.9: Margules’ example 2
and Ψ02 =
2θ2 pκs pm ps − pm Γd pκθ (κ + 1) ps
κ
,
respectively. The lowest state is the same as that in Fig.9.7(b) attained by adiabatic position exchanges of the two masses. The maximum available enthalpy evaluated from (9.41) is illustrated by the other heavy line in Fig.9.8, and is larger than that of the first example. The lowest state of Margules is depicted by the two heavy lines in Fig.9.10, which draw the boundaries for potential temperature of the system. This discontinuous lowest state attained by adiabatic mass exchange is obviously the limit of other reference states including those derived by the variational approach. So, the other reference states must sit inside the two heavy lines according to the thermal constraint.
9.10.3
Thermodynamic and geopotential entropy variations
As potential temperature of each air mass is conserved, no molecular and turbulent diffusions take place in the processes. So, either the classical thermodynamic entropy or turbulent entropy, evaluated from (6.5) or (6.9), is conserved. This can be proved by ps
pt
and
ps pt
ln θ0 dp = (ps − pt ) ln(θ1 θ2 ) ,
ln θr dp = 2 [(ps − pm ) ln θ1 + (pm − pt ) ln θ2 ] = (ps − pt ) ln(θ1 θ2 )
for the baroclinic example. Similarly, we may prove that thermodynamic entropy is also conserved for the barotropic example. However, conservation of thermodynamic entropy in the gravitational field does not mean that the energy conversions are reversible, because the geopotential entropy increases in the processes.
9. AVAILABLE ENTHALPY
204
Figure 9.10: The lowest states evaluated by the variational approach with ∆s = 0 and 0.12 J/(K·kg) for the Margules’ examples, plotted by dashed and solid lines respectively. The two vertical heavy lines give the lowest state of Margules.
Geopotential entropy of the system may be evaluated from (8.49). For the isentropic air masses, we use κ p T = Ts ps and T = Ts − Γd (z − zs ) , giving
z = zs +
Ts 1− Γd
p ps
κ
.
(9.42)
For the baroclinic initial state, we apply zs = 0 ,
pθ = ps ,
Ts = θ
to gain the geopotential entropy
Sg 0
2 pκ+1 − pκ+1 t =− ps − pt − s κ Γd ps (κ + 1)
.
Moreover, the geopotential entropy at the lowest state is represented by Sgr = −2
ps z pm
θ1
dp − 2
pm z pt
θ2
dp .
(9.43)
9.10. DISCONTINUOUS EXAMPLES
205
The first integration reads
ps z
1 pκ+1 − pκ+1 m dp = ps − pm − s κ θ1 Γd ps (κ + 1)
pm
.
The geographic height in the second integration may be evaluated from
z = zm +
where zm =
Tm 1− Γd
θ1 1− Γd
and Tm = θ2
pm ps
p pm
pm ps
κ
,
κ
κ
.
Applying these relationships yields pm z pt
θ2
dp =
θ1 1− Γd θ 2
+
1 Γd
=
1 Γd
pm ps
pm ps
κ
(pm − pt )
κ
pκ+1 − pκ+1 t pm − pt − mκ pm (κ + 1)
θ1 θ1 + 1− θ2 θ2
pm ps
κ
pκ+1 − pκ+1 t (pm − pt ) − mκ ps (κ + 1)
Thus, we obtain
Sg r
2 pκ+1 θ1 − pκ+1 θ1 t s − = + 1− κ Γd ps (κ + 1) θ2 θ2
pm ps
κ
(pm − pt ) − ps + pm
.
The change of geopotential entropy reads ∆sg |Example 1 =
cp g(Sgr − Sg0 ) = 2(ps − pt ) 2
1−
θ1 θ2
1−
ps + pt 2ps
κ
.
It is greater than zero if θ1 < θ2 and less than zero if θ1 > θ2 . Since the geopotential entropy increases, the process of energy conversion is geopotentially irreversible. For Example 2, the barotropic initial state possesses geopotential entropy Sg 0
2 = Γd
θ2 pκ+1 − pκ+1 θ2 s t − + 1− κ ps (κ + 1) θ1 θ1
pm ps
κ
(pm − pt ) − ps + pm
.
While, the lowest state is the same as for Example 1. The geopotential entropy variation gives
∆sg |Example 2
cp = (θ 2 − θ12 ) 1 − 2θ1 θ2 2
ps + pt 2ps
κ
.
It increases also in the barotropic example. The radio of the geopotential entropy variations is ∆sg |Example 2 θ2 =1+ >1. ∆sg |Example 1 θ1
9. AVAILABLE ENTHALPY
206
So, the barotropic example is more irreversible than the baroclinic example. These examples show that the geopotential irreversibility may not be manifested by thermodynamic entropy variation, and so is different from thermodynamic irreversibility. The processes in the atmosphere without producing net thermodynamic entropy in the whole system may nevertheless be irreversible if the geopotential entropy increases.
9.10.4
Continuous solutions
For the two Margules’ examples discussed earlier, the continuous lowest states derived by the variational approach are the same as shown by the light solid line in Fig.9.10. Comparing it with the heavy solid lines finds that the potential temperature may be changed by irreversible isenthalpic diffusions. The minimum entropy produced in the processes is 0.12 J/(K·kg) obtained by using the thermal constraint (9.9), so that the potential temperature of the lowest state is within the range given by the initial states. The maximum available enthalpy with respect to the continuous lowest states is depicted by the two light solid lines in Fig.9.8. It is less than that with respect to Margules’ discontinuous lowest states shown by the heavy solid lines, due to the isenthalpic diffusions which destroy the discontinuity between the air masses. For comparison, the continuous reference states attained by pseudo-reversible processes without using the thermal constraint is displayed by the dashed line in Fig.9.10. The thermal constraint is violated in this example, and so the reference state is false as it cannot be attained really in isolation. The available enthalpy with respect to the false reference state is depicted by two dashed lines in Fig.9.8. The amount is even greater than that evaluated from the classical parcel algorithm with respect to the discontinuous lowest state. These examples emphasize the importance of using the thermal constraint in the variational approach. A lowest state may be discontinuous on theory, if the initial state is discontinuous. The variational approach may only give the continuous reference state attained by isenthalpic diffusions producing a certain amount of thermodynamic entropy. A continuous reference state may be more realistic, as the clear-cut discontinuity can be destroyed quickly by molecular and turbulent diffusions in the atmosphere.
Chapter 10 Dry processes of energy conversion 10.1
Introduction
We have introduced in the preceding chapter the variational approach for the study of kinetic energy generation in the atmosphere. The reference state discussed earlier is the state possessing the lower limit of enthalpy and upper limits of geopotential entropy and kinetic energy, attained through the process with provided thermodynamic entropy production. When the thermodynamic entropy production is minimum, the reference state gives the lowest state which can be solved from the initial state only with the variational approach. The kinetic energy created in the extremal process is maximum. While, energy conversion in the real atmosphere is highly irreversible, and so the maximum kinetic energy may not be obtained. If we have the statistical data of thermodynamic entropy productions in various weather systems, we may evaluate more realistically the kinetic energy generated in developments of these systems. As discussed in Chapter 8, the reference state attained by a process of higher thermodynamic irreversibility is less statically stable, since less barotropic entropy is destroyed in the process. As less geopotential energy and related heat energy are converted also in the process, the kinetic energy generated is reduced by molecular and turbulent diffusions as found in this chapter. We will see also that a highly diffusive process of energy conversion is characterized by sudden changes. The abrupt process of energy conversion in the dry atmosphere may explain the observed planetary wave breaking and sudden warmings in the stratosphere. From the linear theory of McHall (1993), the planetary stratospheric waves may be destabilized in the breaking layer when the baroclinity is sufficiently strong. The energy conversion in an extremely stable stratosphere may produce a large amount of thermodynamic entropy and negative kinetic energy, since the process is highly irreversible. This turbulent process in the stratosphere was referred to as wave breaking (McIntyre, 1982; McIntyre and Palmer, 1983). The mean westerly circulations may be weakened by the strong turbulent diffusions as observed in the events of stratospheric warmings. When a part of heat energy is converted into kinetic energy, the atmosphere increases rather than decreases its thickness in the process. If the isobaric surfaces over a certain altitude do not change heights in energy conversion, the increase of thickness leads to reduction of pressure at low levels. It is found in this chapter that the surface pressure drops more if more kinetic energy is created, and the maximum pressure reduction occurs in the lower troposphere. The pressure change may take place suddenly in a highly irreversible process. According to the hydrostatic relationship, the pressure reduction over a fixed boundary is accomplished by mass output from a local system. The mass exchange together with the resultant energy and entropy exchanges may have an effect on the energy conversion. 207
208
10. DRY PROCESSES OF ENERGY CONVERSION
The heat energy which can be converted into kinetic energy comes from radiative energy of the Earth and Sun. When the lower troposphere is heated, the volume increases and so the atmosphere is lifted up against the gravity. The geopotential energy and heat energy increased are then converted into parcel kinetic energy by the gravitational force and pressure gradient force doing work. The baroclinity decreases and the static stability increases after the energy conversion. To understand the mechanism of energy conversion, we re-consider the energy partition of Pearce (1978). It will be found that the change of baroclinity itself may have no contribution to kinetic energy generation, if the horizontal mean static stability is unchanged. In other words, the mean static stability must increase in order to convert the geopotential energy into kinetic energy. Since conversion of geopotential energy is associated with conversion of heat energy as discussed in Chapter 7, the change of static stability may lead to the change of mean temperature of the system. The major part of available enthalpy is contributed by the change of mean temperature as shown by (7.32), which was ignored by Pearce (1978). The baroclinity is nevertheless indispensable for kinetic energy generation in a statically stable atmosphere. As discussed in Chapter 8, increase of static stability reduces the barotropic entropy in the statically stable atmosphere. This process cannot happen in the isolated barotropic atmosphere when it is statically stable, because the thermodynamic entropy cannot be destroyed. For the stable atmosphere to be excited as a kinetic energy source, it must be baroclinic as shown by (8.42). In the process of energy conversion, the baroclinic entropy created offsets the barotropic entropy destroyed. While in the statically unstable atmosphere, increase of static stability may produce barotropic entropy, so kinetic energy may nevertheless be generated even if the unstable atmosphere is barotropic. Moreover, the examples given in this chapter show that the thermo-static entropy introduced in Chapter 8 increases in kinetic energy generation even if in the quasi-adiabatic processes. The amount increased is different from the thermodynamic entropy created in the process.
10.2
Dependence on process
The lowest states may be attained by the processes with zero thermodynamic entropy production in the strongly baroclinic atmosphere. This does not mean that the processes are reversible, since the horizontal disorderliness and geopotential entropy increase. In the stratified and baroclinic atmosphere, local variation of thermodynamic entropy between two inhomogeneous equilibrium states may change from place to place. Although thermodynamic entropy of the system is unchanged, the entropy of local air mass or individual parcels may nevertheless be changed by the transport or mixing resulting from turbulent and molecular diffusions. This irreversible process without thermodynamic entropy production is referred to as the pseudo- reversible process in Chapter 8. The real processes in the atmosphere cannot produce the maximum kinetic energy, because they are generally turbulent and highly irreversible. As to the study of energetics, the main feature of turbulent process may be manifested statistically by thermodynamic entropy production. Since the large-scale equilibrium state of
10.2. DEPENDENCE ON PROCESS
209
Figure 10.1: The reference states attained by ∆s = 0.5 J/(K·kg). The initial state at y = 0 is depicted by the heavy line, of which temperature gradient |Ty | varies from 4 - 32 K/1000km with interval of 4 K/1000km from left to right on the top of the diagram. The solid and dashed lines indicate positive and negative available enthalpy, respectively.
atmosphere is not uniform and the thermodynamic entropy is independent of the direction of vertical potential temperature gradient, a reference state cannot be determined uniquely by provided thermodynamic entropy production. However, using the variational approach we may find an extremal state, which possesses minimum enthalpy and maximum geopotential entropy among all the reference states attained through the process with the specified thermodynamic entropy production. When the extremal state is taken as the reference state, we may evaluate furthermore the extremal value of available enthalpy in the specified irreversible process. For the initial linear atmosphere assumed in the preceding chapter, the reference states attained by the process with ∆s = 0.5 J/(K·kg) are displayed in Fig.10.1. The static stabilities are lower than those of corresponding lowest states shown in Fig.9.1. Kinetic energy generated in the highly diffusive processes is less than the maximum available enthalpy. The decrease of available enthalpy with increasing thermodynamic entropy production can be seen clearly from Fig.10.2. A curve in the figure is steeper when the initial static stability is lower. For provided initial baroclinity and static stability, there is the maximum thermodynamic entropy production, over which kinetic energy may be destroyed by turbulences. In this case the process, if occur, will convert kinetic energy into enthalpy. Fig.10.3 compares the available enthalpy solved for the highly diffusive process with the maximum available enthalpy. In this diagram, the kinetic energy generated
10. DRY PROCESSES OF ENERGY CONVERSION
210
Figure 10.2: Dependence of available enthalpy on thermodynamic entropy production. Solid and dashed lines are for |Ty | = 15 and 20 K/1000km, respectively. The Γ indicates the temperature lapse rate of initial state.
has been transferred into the mean wind speed with v =
v0 2 +2∆k ,
∆k =
˜ max gΛ . ps − pt
(10.1)
For a given kinetic energy generation ∆k, the increase of wind speed depends on initial speed, that is ∆v = v − v0
= =
v02 + 2∆k − v0
2∆k
v02 + 2∆k + v0
.
It is smaller if the initial speed is higher. For an easy comparison, the initial wind speed v0 is taken as zero in the figure. Fig.10.3 shows again that available enthalpy is reduced greatly by increasing thermodynamic entropy production, especially in the atmosphere with low static
10.3. SUDDEN WARMING AND COOLING
211
Figure 10.3: Maximum available enthalpy (dashed) and extremal available enthalpy with respect to ∆s = 0.5 J/(K·kg) (solid). The initial temperature lapse rate is denoted by Γ.
stability. Development of a weather system in a highly diffusive process is characterized by sudden change. The rapidity of sudden change increases with the entropy production and initial static instability. The atmosphere may retain a certain degree of baroclinity before the energy conversion takes place in a highly diffusive process. Since energy conversion depends highly on thermodynamic entropy production, there is the limitation on weather predictions provided by the uncertainty of turbulent diffusions incorporated in prediction models.
10.3
Sudden warming and cooling
10.3.1
Temperature variation
Since the energy conversion in the baroclinic disturbances is realized by the slantwise convection with a trajectory slope less than the slope of isentropic surface, the temperature increases in the upper troposphere but decreases in the lower troposphere
212
10. DRY PROCESSES OF ENERGY CONVERSION
after energy conversion as shown by Fig.9.1. So the surface air may be cooled suddenly by development of strong baroclinic cyclones, especially in the highly diffusive processes. For the mean temperature gradient |Ty | = 16 K/1000km averaged over 2000 km in y direction, the largest temperature change in the central area is less than 6 K on the top or bottom of the air column. The surface cooling on the warm side and the top warming on the cold side may be over 20 K. Since the processes of disturbance development are highly irreversible in the real atmosphere, the observed temperature changes are less than depicted. This can be seen by comparing Fig.9.1 with Fig.10.1. An example of the strong surface cooling is the cold-outburst over northeastern Asia continent in winter. The surface wind can be unusually strong in the events of cold-outburst. The energy conversion may produce even larger local temperature change in the planetary stratospheric disturbances. Fig.10.4(a) shows some examples of the lowest state in the atmosphere from ps = 100 hPa up to pt = 10 hPa with the half width Y = 2000 km. The initial states are represented also by (9.19) with Γ = −0.1 K/100m and Ts0 = 210 K at 100 hPa. If the initial temperature gradient is 20 K/1000km, the upper stratosphere at the lowest state is over 20 K warmer than the initial state near y = 0. The warming is even stronger at higher latitudes, as the initial temperature is lower there. While, the low latitudes may be cooled down as the initial baroclinic stratosphere becomes barotropic. The phase of warming and cooling in the vertical direction may be reversed by intensified turbulent diffusions. Fig.10.4(b) shows the reference states attained by the process with ∆s = 5 J/(K·kg). The static stability decreases rather than increases if the initial baroclinity is not sufficiently strong. In this case, the temperature increases in the lower stratosphere and decreases in the upper stratosphere. If |Ty | = 10 K/1000km, the temperature rises about 6 K at y = 0 on 100 hPa shown in Fig.10.4(b). The warming on the cold side may be over 20 K. The strong warming in the stratosphere was observed firstly by Scherhag (1952) in 1952. As the rate may be over 10 K/day, the warming is called the stratospheric sudden warming. McHall (1993) illustrated that the stratospheric warmings occur only in winter hemisphere where the temperature decreases poleward. The summer example of the warming has not been found yet. When the horizontal temperature gradient is weakened by energy conversion in the warming events, abrupt cooling may occur at low latitudes in the lower stratosphere, as reported by Fritz and Soules (1972).
10.3.2
Kinetic energy production
When the baroclinity becomes sufficiently strong in the winter stratosphere, the planetary perturbations therein may be destabilized. The energy conversion with low thermodynamic entropy production may produce net kinetic energy. This process may explain the development of planetary stratospheric perturbations before sudden warmings. The change of initial baroclinic atmosphere to a barotropic reference state is accomplished by heat flux from warm place to cold place (McHall, 1993). Fig.10.4(a) shows that the temperature decreases in the lower stratosphere and increases in the upper stratosphere. The cooling at high latitudes may be offset
10.3. SUDDEN WARMING AND COOLING
213
(a)
more or less by reducing horizontal temperature gradient, so it is most remarkable at low latitudes. If developed disturbances in the stratosphere are strong enough, the induced strong direct meridional circulation at high latitudes may heat the lower stratosphere adiabatically near the pole. In this process, parcel kinetic energy is converted into heat energy. Fig.10.5 shows the kinetic energy generated in the stratosphere with respect to the reference states shown in Fig.10.4(b). The energy generation is negative unless the baroclinity is very strong. For the assumed thermodynamic entropy production, the zero point of kinetic energy generation occurs around |Ty | = 18 K/1000km for all the initial temperature lapse rates plotted. If this is true, there will be the critical baroclinity for kinetic energy generation, which depends on intensity of turbulence and is independent of the static stability. As the subsidence warming is strongest at high latitudes near the polar cap, the equatorward temperature gradient may be reversed by the warming. To meet with the new thermal wind balance, a mean easterly circulation may be established in the major warmings. Meanwhile, the upward branch of the vertical circulation produces adiabatic cooling at low latitudes. The forced warming and cooling are shown by Fig.10.4(b). The cooling up to the mesosphere and the resultant descent of stratopause were reported by many authors (Quiroz, 1969; Labitzke, 1972a, b, 1981). The phase change of the warming form the early stage to late stage looks like downward propagation of the phase. The strong turbulent motions may cause the strong lateral turbulent mixing of potential vorticity, and destroy the normal perturbation patterns of potential vorticity in the stratosphere. This process was referred to as planetary wave breaking (McIntyre, 1982; McIntyre and Palmer, 1983), and is highly nonlinear as shown by the time series of temperature recorded in the warming events (Labitzke, 1977). In
214
10. DRY PROCESSES OF ENERGY CONVERSION
(b)
Figure 10.4: Stratospheric examples of the lowest states in (a) and the reference states attained by the processes with ∆s = 5 J/(K·kg) in (b). The profile of the initial states at y = 0 is shown by the dashed curves.
fact, the wave breaking may occur in other events with irreversible energy conversion. It is found firstly in a stratospheric warming, because the irreversible turbulent activity is extremely strong in the highly stable stratosphere. Fig.10.6 shows the geopotential entropy variation in the diffusion processes. The geopotential entropy is destroyed when kinetic energy is converted into geopotential energy and heat energy. More geopotential entropy and kinetic energy are destroyed in the atmosphere with lower static stability. When the highly diffusive process with destruction of kinetic energy occurs, the stratosphere cannot be thermodynamically or mechanically isolated. The dynamic forcing with input of kinetic energy may be important for occurrence of the major warmings. The forced energy conversion may not be so remarkable in the troposphere. Synoptic baroclinic disturbances in the lower troposphere usually form closed cyclones. The warm air at a cyclone center is cooled down adiabatically as it is lifted up by the cold front. Net kinetic energy is then created and the strong baroclinic zone is destroyed in the disturbance development. The planetary perturbations may also develop in the middle and upper troposphere when the baroclinity is strong enough (McHall, 1993). However, the perturbations are usually weaker than those in the stratosphere, and the forced vertical circulation is limited by the fixed lower boundary. So a large warming rate in the lower troposphere can hardly be observed.
10.4. CHANGE OF SURFACE PRESSURE
215
Figure 10.5: Stratospheric examples of available enthalpy with ∆s = 5 J/(K.kg). The Γ (varying from -0.3 to 0 K/100m with interval of 0.1 K/100m) denotes the temperature lapse rates of initial fields. Dashed curves display the ratio of maximum available enthalpy to initial total enthalpy.
10.4
Change of surface pressure
10.4.1
Surface pressure and static stability
After energy conversion in a tropospheric weather system, the pressure may change in the system. Usually, the pressure change is maximum in the lower troposphere and decreases upward. We discuss simply in this section the surface pressure change, assuming that the height of 200 hPa surface is unchanged in energy conversion. For a barotropic atmosphere with constant temperature lapse rate, we insert T = Ts − Γz into hydrostatic equation (9.39), giving g dz dp =− . p R Ts − Γz It is assumed here that the geographic height is 0 at the surface. The surface pressure of the barotropic linear atmosphere is then given by
Ts ps = p Ts − Γz
g RΓ
.
(10.2)
10. DRY PROCESSES OF ENERGY CONVERSION
216
Figure 10.6: Geopotential entropy variations in the stratosphere over the processes with ∆s ≥ 5 J/(K·kg)
If the geopotential height of 200 hPa surface is taken as 12 km, the dependence of surface pressure on the static stability is displayed in Fig.10.7 for the assumed linear atmosphere with Ts = 290 K. The surface pressure decreases with increasing static stability. For example, it drops 20 hPa as the lapse rate changes from 0.7K/100m to 0.65K/100m. When the static stability decreases before development of convective disturbance, the surface pressure increases. A pressure nose can be found from the time series of surface pressure recorded at the station in the path of a coming storm.
10.4.2
Surface pressure change
We have discussed the dependence of surface pressure on static stability of the atmosphere. As the static stability increases in energy conversion, the surface pressure decreases. The pressure changes in the energy conversion processes discussed earlier are studied in the following. The static equilibrium equation may be rewritten as dp = −
gp Rθ
p pθ
κ
dz ,
where pθ = 1000 hPa. Inserting (9.12) into it yields pt ps
gpκθ pκ−1 dp = − zt . pκ − λ2 Rλ1
10.4. CHANGE OF SURFACE PRESSURE
217
Figure 10.7: Dependence of surface pressure on the static stability of the linear atmosphere with Ts = 290 K
It gives the surface pressure after energy conversion Γd
pκsr = (pκt − λ2 )e λ1 where
Ts 1− zt = Γ
pκ θ zt
pt ps 0
+ λ2 ,
(10.3)
RΓ
g
(10.4)
represents the constant height of pressure surface pt evaluated from (10.2). The ps0 in (10.4) indicates the surface pressure of assumed initial field, and is taken as 1000 hPa here. For a provided surface temperature, the height decreases at higher static stability, since the temperature at high levels decreases with increasing static stability. If the pressure at the constant height zt is assumed as pt = 200 hPa, decrease of surface pressure given by ∆ps = ps0 − psr is plotted by the heavy solid curves in Fig.10.8. In these plotted examples, the surface pressure drops down after energy conversion. The pressure reduction increases with initial baroclinity and static instability, but is reduced by strong diffusions. The heavy dashed curves in Fig.10.8 show the surface pressure reduction in the process with ∆s = 0.5 J/(K·kg). The surface pressure may not change until the baroclinity increases to a certain extend. Then the low system near the surface develops suddenly, especially when the static stability is low. This feature may be found also from the energy conversion in the moist atmosphere as discussed in the next chapter, and may explain the development of explosive cyclones over northwestern Pacific and northwestern Atlantic oceans.
10.4.3
Change of the thickness
Fig.10.9 shows the pressure and temperature profiles of an initial and lowest states discussed. For cleanness, the relatively large temperature lapse rate and baroclinity
218
10. DRY PROCESSES OF ENERGY CONVERSION
Figure 10.8: Decrease of surface pressure as the initial states reach the lowest states (heavy solid) or the reference states by the processes of ∆s = 0.5 J/(K·kg) (heavy dashed). The increase of thickness in the two processes are plotted by the light solid and light dashed respectively. The lapse rates of the initial states are indicated by Γ for the heavy solid.
at the initial state are assumed for plotting this figure. The temperature at the lowest state varies nearly linearly with geographic height, and the lapse rate is smaller than that at the initial state. As shown also by Fig.9.1, the temperature increases over the middle troposphere after energy conversion, but decreases below. The maximum pressure change is found near 2.5 km height. This height is below the level at which the temperature is unchanged, because the pressure and density just below the level are reduced by energy conversion. The heat energy in an air column reduces after energy conversion. Also, a part of air mass in the upper troposphere moves downward to the lower troposphere as the geopotential energy is converted into kinetic energy, so that the gravity center of the column decreases in height. The air reduces its volume as it increases its pressure in subsidence. However, the air column increases rather than decreases its thickness after the heat and geopotential energies are converted into kinetic energy. The reason may be found from the state equation (2.22), which is rewritten as α=R
T . p
The volume and temperature are not correlated linearly as the pressure is not constant in the vertical direction. The volume increased by warming in the upper
10.4. CHANGE OF SURFACE PRESSURE
219
Figure 10.9: Profiles of pressure (heavy) and temperature (light). The initial state with Γ = 0.85 K/100m and |Ty | = 30 K/1000km is plotted by dashed, and the corresponding lowest state is by solid.
troposphere exceeds the volume reduced by cooling in the lower troposphere, since the pressure decreases upwards. Although no latent heat is released in the dry process, the thickness may still be changed by adiabatic displacement of air parcels. This can be studied quantitatively in the following. The thickness between ps and pt at the initial state is evaluated from (10.4), giving RΓ
Ts pt g 1− . z0 = Γ ps The thickness at the reference state is given by (10.3), that is zr =
cp λ1 pκs − λ2 ln κ . gpκθ pt − λ2
Change of the thickness is then calculated from ∆z = zr − z0 . The calculations for the previous examples are plotted by the light solid and light dashed curves in Fig.10.8. Although the atmosphere possesses less heat energy at the reference state than at the initial state, the thickness increases after energy conversion. As the top level stays at a constant height, the increase of thickness lowers the gravity center and so is accomplished by converting the geopotential energy into kinetic energy. When the lower boundary is fixed, the change of thickness
10. DRY PROCESSES OF ENERGY CONVERSION
220
Figure 10.10: Ratio of total enthalpy and reduction of surface pressure is produced by mass divergence at high levels. In this case the local system is not closed, and the transport of mass, thermodynamic entropy and energies should be considered in evaluation of energy conversion.
10.5
Change of static stability
10.5.1
Partition of available enthalpy
It is showed in the previous discussions that both the baroclinity and static stability are changed by the energy conversions. One may like to consider the individual contribution of the changes to kinetic energy generation. Pearce (1978) partitioned the available potential energy into the components related to the static stability and baroclinity respectively. Since conversion of geopotential energy is correlated to conversion of heat energy, the change of static stability may cause the change of mean temperature T . This can be shown in the following example. We consider a simple adiabatic process by which the constant temperature lapse rate changes as a baroclinic atmosphere becomes barotropic. The initial temperature field of assumed linear atmosphere is represented by
T0 = (Ts0 − Ty y)
p ps
γ0
,
γ0 =
RΓ0 . g
The mean temperature gives Ts0 pt ps − pt T0 = (γ0 + 1)(ps − pt ) ps
γ0
.
10.5. CHANGE OF STATIC STABILITY
221
If the lapse rate changes to Γr after energy conversion, we have
Tsr pt ps − pt Tr = (γr + 1)(ps − pt ) ps
γr
.
The surface temperature Tsr may be determined by the conservation law of potential enthalpy (8.14), giving
(1 − br ) ps − pt Tsr = Ts0
(1 − b0 ) ps − pt
ps b0 pt
ps br pt
,
where b = κ − γ. The ratio of the mean temperatures reads Re =
Tsr (1 − γ0 ) ps − pt
Tr = T0 Ts0 (1 − γr ) ps − pt
! ps γr pt γ0 ! p
.
s pt
This is also the ratio of total enthalpy of the final state to that of initial state. It is greater than 1 when Γr > Γ0 but is less if Γr < Γ0 as shown in Fig.10.10. Thus, total enthalpy of the atmosphere decreases when the static stability increases. Referring to Fig.9.3 finds that the ratio of energy conversion Rh is in the order of 1 − Re . A small change in the static stability may lead to a great change in the available enthalpy. While, change of baroclinity may have no contribution to kinetic energy generation, if the mean temperature is unchanged. As the baroclinic atmosphere changes to a barotropic state with the same horizontal mean static stability, we have ps pt
A
pκ θdAdp =
ps pt
ˇ pκ (θ + θ)dAdp =A
A
ps pt
pκ θdp .
The total enthalpy is unchanged and so no kinetic energy is produced. As discussed in Chapter 7, generation of kinetic energy in the dry atmosphere is contributed partially by conversion of geopotential energy, so the static stability increases after the energy conversion. However if evaluated using Lorenz’s formula, the obtained available potential energy may be different from zero and so the conservation of energy is violated.
10.5.2
Final mean static stability
The reference state obtained with the new variational approach is represented generally by (9.12). Its temperature lapse rate is given by Γr = −
Γd λ2 ∂Tr =− κ . ∂z p − λ2
The final static stability decreases upward. Using (9.12) yields Γr = −
λ2 Γd θ r , λ1
10. DRY PROCESSES OF ENERGY CONVERSION
222
Figure 10.11: Dependence of final mean static stability [Γr ] on initial baroclinity. The static stabilities of the initial fields are denoted by Γ.
or
λ1 Γr , λ2 Γd where λ1 and λ2 are two constants depending on initial state and thermodynamic irreversibility; λ1 is positive but λ2 is negative usually. The vertical mean static stability of the reference state may be calculated from θr = −
1 [Γr ] = ps − pt
ps pt
Γr dp = −
Γd λ2 Z . ps − pt
(10.5)
The evaluated examples for the initial states assumed previously are displayed in Fig.10.11. The static stability increases as the atmosphere varies towards a reference state, and the final stability is stronger when the initial state is more baroclinic. Comparing with Fig.9.3 finds that more kinetic energy is created if the reference state becomes more stable. As discussed earlier, the reference state depends not only on initial state but also on process. Figs.9.1 and 10.1 show that the static stability decreases when more thermodynamic entropy is created in the process. This is shown also by (8.38) which tells that the static entropy of the reference state is reduced by thermodynamic entropy production. As a result, less kinetic energy is created in highly irreversible processes.
10.6
Thermo-static entropy level
10.6. THERMO-STATIC ENTROPY LEVEL
10.6.1
223
Change of barotropic entropy
It is discussed in Chapter 8 that the barotropic entropy decreases as the static stability increases in energy conversion. The barotropic entropy of assumed initial state is plotted by the light dashed horizontal lines in Fig.8.1. The barotropic entropy of the reference state (9.12) is evaluated from cp S˜btr = g
ps
ln pt
λ1 dp . θ0 (pκ − λ2 )
where a tilde indicates the value in the air column over a unit horizontal area. Applying (9.17) for it gives S˜btr
= +
cp [(ps − pt )(ln λ1 − ln θ0 +κ) − ps ln(pκs − λ2 ) + pt ln(pκt − λ2 ) g κλ2 Z] .
For the initial state represented by (9.19), the mean potential temperature θ0 is given by (8.28). The change of barotropic entropy in the process reads ∆S˜bt = +
cp g
(ps − pt )(ln λ1 + κ) − ps ln(pκs − λ2 ) + pt ln(pκt − λ2 )
κλ2 Z −
ps pt
lnθ0 dp .
The last integration is evaluated from (8.29). As the linear atmosphere reaches the lowest state, changes of barotropic entropy are displayed by the heavy dashed curves in Fig.10.12. Since the static stability of the lowest state is the highest compared with that of other reference states, the figure shows the maximum reduction of barotropic entropy. Although the reference state is more close to the isothermal state, (8.30) shows that the barotropic entropy decreases as the static stability increases in the statically stable atmosphere. This is because the extremal state attained by parcel motions with conservation of potential enthalpy is isentropic instead of isothermal, and the reference state departs more from the isentropic state compared with the initial field. Fig.10.12 shows that more barotropic entropy is destroyed in the more baroclinic atmosphere, because the reference state attained is more stable. Since the heavy dashed curves in the figure are close to each other, they are not labeled with the initial temperature lapse rate. Otherwise, we may see also that less barotropic entropy is destroyed in a less statically stable atmosphere. However, the difference related to the stability is small in the stable atmosphere. Kinetic energy may be created in an isolated stable atmosphere only if it is baroclinic. In the process of energy conversion, the baroclinic entropy increases to offset the destruction of barotropic entropy. The sum of baroclinic and barotropic entropy changes gives the net thermodynamic entropy created in the process as shown by (8.32), and the thermodynamic entropy is not destroyed in isolation. As discussed earlier, the destruction of baroclinity itself may not produce kinetic energy. However, the baroclinity makes the energy conversion possible in the statically stable atmosphere.
224
10. DRY PROCESSES OF ENERGY CONVERSION
Figure 10.12: Thermo-static entropy level of the initial state (light solid) with the Γ indicated and corresponding lowest state (light dashed). The heavy curves illustrate thermodynamic entropy production (solid) and barotropic entropy change (dashed).
While, a statically unstable atmosphere is an energy source whether it is baroclinic or barotropic, since increasing static stability resulting from conversion of geopotential energy in the unstable atmosphere may produce barotropic entropy. This can also be seen from Fig.10.12. The heavy dashed curve with Γ = 1.05 K/100m is over the zero line if the initial state is barotropic. When the baroclinity is strong in the unstable atmosphere, the reference state attained may possess a high static stability, so that the barotropic entropy is reduced rather than increased. This can be seen from (8.32), which is rewritten as ∆S = −Sbc0 + ∆Sbt
(10.6)
for the barotropic reference state. When ∆S ≥ 0, we have Sbc0 ≤ ∆Sbt . The barotropic entropy may decrease if the baroclinity is strong enough. However, the total thermodynamic entropy does not decrease in this case, since a large amount of baroclinic entropy is created.
10.6. THERMO-STATIC ENTROPY LEVEL
225
After the linear atmosphere reaches the lowest state, the entropy productions are depicted by the heavy solid curves in Fig.10.12. Unless the static instability is high at an initial state, the lowest states may be attained by a pseudo- reversible process. When the potential temperature profile is close to the adiabatic stratification at the initial state, increase of static stability is limited by the thermal constraint in (9.9). Thus, the barotropic entropy destroyed may be less than the baroclinic entropy increased, and so net thermodynamic entropy is produced by reducing the baroclinity.
10.6.2
Change of thermo-static entropy
It is discussed in Chapter 8 that the ability of energy conversion in the atmosphere depends on three-dimensional inhomogeneity of the potential temperature field. As parcel kinetic energy may be created by conversion of geopotential energy in the gravitational field, the ability depends also on vertical direction of potential temperature gradient. The baroclinic entropy may represent the ability associated with the horizontal inhomogeneity. Meanwhile, the ability associated with the vertical gradient may be represented by the static entropy introduced in Chapter 8. The sum of baroclinic entropy and static entropy shown by (8.41) is called the thermostatic entropy, which decreases with increasing the baroclinity and static instability. The atmosphere possessing less thermo-static entropy is more capable of producing kinetic energy. The thermo-static entropy level of the initial state assumed previously with Ts0 = 290 K is plotted by the light solid curves in Fig.10.12. It is lower in the atmosphere with stronger baroclinity and static instability. The light dashed curves in the figure show the thermo-static entropy level of the lowest state. Since the lowest state is barotropic, the thermo-static entropy is identical to the static entropy. This entropy level rises in the process of energy conversion. The height of rising depends on the initial baroclinity, static stability and irreversibility of process. The entropy level may rise higher if the initial baroclinity is stronger. Since the molecular and turbulent diffusions reduce the entropy level at the reference state as shown by (8.44), the thermo-static entropy reaches the highest level at the lowest states attained by the extremal process with minimum thermodynamic entropy production and maximum geopotential entropy production. If the temperature profile is close to the adiabatic profile at initial state, rising of thermo-static entropy level is limited by thermal constraint (9.9). As discussed in Chapter 8, increase of thermo-static entropy is different from the thermodynamic entropy production plotted by the heavy solid curves in Fig.10.12. The processes with conservation of thermodynamic entropy is still irreversible, as the thermo-static entropy and geopotential entropy increase.
Chapter 11 Available moist enthalpy 11.1
Introduction
It has been made clear since the studies of Smagorinsky (1956) and Aubert (1957) that the latent heat is an important source of kinetic energy in the moist atmosphere. However, evaluating the contribution of latent heat is not so straightforward as calculating the contribution of enthalpy in the dry atmosphere. Kinetic energy may be created in a dry process as parcels of high and low potential temperatures move upward and downward respectively. The changes of non-kinetic energies in the dry atmosphere are related simply to the change of vertical profile of potential temperature, since increase of the static stability reduces the total enthalpy when mass and potential enthalpy are conserved. While, change of vertical profile of equivalent potential temperature or another moist parameter may not have a simple relation to the changes of non-kinetic energies in the system, especially in the unsaturated atmosphere. Rising of the parcels with higher equivalent potential temperatures together with descending of the parcels with lower equivalent potential temperatures does not always produce kinetic energy. The static stability may decrease rather than increase after the energy conversion as shown in this chapter. Margules (1904) was the first who studied quantitatively the kinetic energy generation in the systems of moist air masses, applying his parcel algorithm and assuming that equivalent potential temperature of air mass was conserved in the process of energy conversion. He found that the kinetic energy created by redistribution of air masses in stably equilibrium was negligibly affected by release of latent heat. But, the effect of moisture condensation became significant when the moist air was potentially unstable (Margules, 1906). In a baroclinic atmosphere, the latent heat released may also have a great effect on kinetic energy generation even if the atmosphere is conditionally stable (Danard, 1966), because release of latent heat reduces the slantwise adiabatic lapse rate of saturated parcels in the large-scale slantwise convection. Although the latent heat may have an effect on maintaining the horizontal temperature gradient (Clapp and Winninghoff, 1963), the contribution of latent heat to energy conversion is achieved mainly by development of vertical convection, which may be independent of the baroclinity. The algorithm of Lorenz (1955) used for the study of energy conversion in the dry atmosphere depends crucially on the baroclinity, and cannot be used to evaluate the kinetic energy sources in the barotropic and statically unstable atmosphere. Thus, the available potential energy in the moist unstable atmosphere, referred to as the moist available energy by Lorenz, was calculated with the traditional parcel algorithm either graphically or digitally for the real atmosphere (Lorenz, 1965, 1978, 1979). The moist available energy was defined as the difference in the total amount of non-kinetic energy between a moist initial and reference state. As in the dry atmosphere, the reference state was also assumed 226
11.1. INTRODUCTION
227
possessing the least non-kinetic energy. In passing from the initial state to the reference state, each parcel varies dry- or moist-adiabatically, so that the equivalent potential temperature is conserved. Since applications of the parcel algorithm for the three-dimensional atmosphere are difficult in practice, only the “vertical component” of moist energy sources for development of vertical convection is considered mostly in the studies afterwards. This component of moist available energy was called the generalized convective available potential energy ( GCAPE), and was evaluated with an one-dimensional parcel algorithm, assuming that the conditionally unstable atmosphere must possess GCAPE, which could be converted into kinetic energy in vertical parcel exchange processes with conservations of thermodynamic entropy, total mixing ratio and saturation temperature or pressure of parcels, while the equivalent potential temperature may not be conserved (Randall and Wang, 1992; Wang and Randall, 1994). These assumptions are inconsistent with the physics of Lorenz, and cannot be proved either by theory or by observations. In fact, a conditionally unstable atmosphere may not always be a kinetic energy source when it is unsaturated. This can be shown in this chapter and will be discussed further with the parcel energetics in Chapter 21. Moreover, the one-dimensional algorithm cannot be applied for the energy conversions in the slantwise convection, which is the prevailing circulation pattern in the extratropical troposphere. The kinetic energy source for the vertical convection of a single parcel is called the convective available potential energy ( CAPE, Moncrieff and Miller, 1976; Williams and Renn´ o, 1993; Bhat et al., 1996), defined as the mechanic work done by the buoyancy for the parcel in the upward convection (Referring to Chapter 19 also). The CAPE was also called the energy of instability at an earlier time (Belinskii, 1948; Tverskoi, 1962). Since the buoyancy may be given by (19.14), CAPE is similar to the area surrounded by the temperature profiles of a convective parcel and steady environment. A negative CAPE is also called the convective inhibition energy ( CINE, Bohren and Albrecht, 1998). Certainly, the whole air column changes in development of convection, and there are downward motions in a storm to remain the local mass balance. The mechanic work done by the buoyancy for a downdraft is defined as the downdraft convective available potential energy ( DCAPE, Emanuel, 1994). The sum of CAPE and DCAPE gives the total convective available potential energy ( TCAPE) of the parcels in position exchanges (Renn´ o and Ingersoll, 1996). To consider the TCAPE for a whole column, we may add together the CAPE or DCAPE of each parcel in the column. According to the study in Chapter 7, the mechanic work down by buoyancy is equivalent to the change of enthalpy in the atmosphere. This will be discussed again in Chapter 20. Thus, the TCAPE for a whole atmosphere is equivalent to GCAPE. In a precipitation system, the total mixing ratio and saturation temperature or pressure of a parcel are not conserved. Without mixing between parcels and kinetic energy dissipation, the equivalent potential temperature or virtual potential temperature is a better conservative quantity. The algorithms based on conservation of certain quantities of a single parcel used by Margules (1904), Lorenz (1978) and other authors are highly idealized. When potential temperature or equivalent potential temperature of each parcel or air mass is conserved, the surfaces of poten-
228
11. AVAILABLE MOIST ENTHALPY
tial temperature or equivalent potential temperature may only be rearranged, while the extremal values do not change. This does not agree with observed processes of disturbance development in the atmosphere. Variations of potential temperature or equivalent potential temperature may result from the mixing and kinetic energy dissipation produced by molecular and turbulent diffusions. The diffusive processes are not adiabatic again for parcels or air masses, and the thermodynamic entropy is changed by diffusions. Since the gas constant of dry air is different from that of moist air, we usually use virtual temperature instead of temperature to study the moist processes. Calculation of virtual temperature may be referred to the text books of meteorology or to the study of Xu and Emanuel (1989). To concentrate on the basic principle of energy conversion, we use temperature in this study for convenience only. When a saturated parcel rises adiabatically, the water vapor included may condense as the temperature and saturated mixing ratio of water vapor decrease. If the condensed water remains in the parcel, the process is called the moist-adiabatic process. This process may occur also in a downward parcel motion, when the water included evaporates as the temperature increases adiabatically. If the condensed water leaves the parcel at the very beginning, the process is referred to as the pseudo-adiabatic process. Obviously, the adiabatic lapse rate of parcel temperature in the moist-adiabatic and pseudo-adiabatic processes is reduced by latent heating and so is lower than the dry adiabatic lapse rate. The atmosphere which is statically stable to dry convection may not be stable again to the moist convection. The total amount of non-kinetic energies, which includes latent heat energy and can be converted into kinetic energy, is referred to as the available moist enthalpy in the current study. Like the available enthalpy in the dry atmosphere, available moist enthalpy depends not only on initial state but also on process. For a provided initial field and thermodynamic entropy production, the reference state is evaluated with the variational approach of McHall (1990b, 1991), which is similar to that applied for the dry processes to meet with the geopotential entropy law. Since the potential enthalpy is not conserved in the moist processes, we introduce in this chapter the conservation law of moist potential enthalpy to filter out insignificant solutions of reference state. In the processes with conservation of moist potential enthalpy, equivalent potential temperature at a reference state cannot exceed the lower and upper limits at a provided initial state. This fact gives the thermal constraint for the moist processes. When the thermodynamic entropy production is minimum and geopotential entropy production is maximum, the obtained reference state is also called the lowest state which can be solved for provided initial fields. The available moist enthalpy in an isolated system is maximum with respect to the lowest state, and less kinetic energy is created in the process with higher thermodynamic irreversibility. The thermodynamic states of the moist atmosphere depend on the humidity as well as temperature. Variations of these variables may not be independent of each other in the saturated atmosphere, and may not always be correlated to each other especially in the unsaturated atmosphere. The relation between these variations is uncertain at least for the large-scale evaluations. When the equivalent potential temperature is conserved, a parcel may still have different temperatures and hu-
11.2. AVAILABLE MOIST ENTHALPY
229
midities. This fact makes it difficult for us to choose a realistic reference state in the three-dimensional moist atmosphere. We discuss two simplest cases in this chapter, which may be considered as two extremal examples of the reference state. In the first case, variation of humidity is independent of temperature variation, and the reference state obtained is dry. In the second case, variations of temperature and humidity are considered together by evaluating the equivalent potential temperature variation. The reference state so derived may be moist. It will be found that more kinetic energy can be created in the moist atmosphere than in the dry atmosphere. A moist barotropic atmosphere may be an energy source if the humidity is sufficiently high. The moist disturbances developed in the quasi-barotropic regions especially at low latitudes may be more intense than those in the dry atmosphere, and the intensity and lifetime depend essentially on the supply of water vapor. A typical example of the barotropic moist disturbances is the tropical cyclones. A strong hurricane may produce much more kinetic energy than many other disturbances in the troposphere. Like energy conversions in the dry atmosphere, the highly irreversible moist processes may occur and develop suddenly and rapidly. Examples of the moist disturbance developing rapidly are the explosive cyclones, called also the tropospheric bombs (Sanders and Gyakum, 1980; Roebber, 1984), found in some particular regions over extratropical oceans.
11.2
Available moist enthalpy
Generation of kinetic energy in a precipitation system depends not only on variation of total enthalpy but also on release of latent heat. Here, we define the sum of heat energy, geopotential energy and latent heat Ql in the atmosphere as the moist total potential energy, namely, Pm = U + Φ + Ql . It is also the sum of total potential energy and latent heat. Generally, variation of moist potential energy in an atmosphere of limited height may contribute to the kinetic energy created and mechanic work done as the atmosphere changes volume, that is, Pmr − Pm0 + ∆K + Wv = 0 . The amount of moist total potential energy, which may be converted into kinetic energy is given by Λm = Pm0 − Pmr − Wv . Referring to Section 9.2 finds Λm =
cp gpκθ
ps pt
A
pκ (θ0 − θr ) dA dp + Ql0 − Qlr .
In deriving this equation, variation of air mass due to precipitation is not considered. The right-hand side is the sum of available enthalpy and latent heat released. The amount of non-kinetic energies, which include the available enthalpy and latent heat and can be converted into kinetic energy, is referred to as the available moist
11. AVAILABLE MOIST ENTHALPY
230
enthalpy. The latent heat released by water vapor condensation is evaluated from Ql0 − Qlr =
Lc g
ps pt
A
(w0 − wr ) dA dp ,
where Lc is the latent heat of condensation and is assumed constant with the magnitude of 2.5 × 106 J/kg. Applying this equation yields Λm =
cp g
ps pt
A
(T0 − Tr ) dA dp +
Lc g
ps pt
A
(w0 − wr ) dA dp .
(11.1)
It can be obtained also from the system energy equation (7.21) with Q = Ql .
11.3
Moist potential enthalpy
In the study of available enthalpy in the dry atmosphere, we have introduced the potential enthalpy to filter out the heat-death reference state. This quantity is not conserved in the moist processes, if the water vapor included changes phase. Here, we introduce another useful conservative quantity called the moist potential enthalpy and defined as ps
Ψpe = cp
θe dA dp pt
A
for the atmosphere, where Lc w
θe = θe cp Tc
(11.2)
measures the equivalent potential temperature, of which the derivation will be given in Chapter 23. The Tc in this equation indicates the condensation temperature at the lifting condensation level, where a rising moist air becomes saturated. It is also the parcel temperature T in a saturated atmosphere. There were different definitions of equivalent potential temperature in traditional meteorology (Hess, 1959). Here, the equivalent potential temperature is interpreted as the potential temperature which a parcel has as it is lifted adiabatically until all the water vapor included condenses. It is nearly conserved in the dry- and moist-adiabatic processes. Strictly speaking, the θ and cp in (11.2) are different from those for the dry atmosphere, as they depend on the total water included (Emanuel, 1994; Bohren and Albrecht, 1998, referring to Chapter 23 also). If the θ and cp represent respectively the potential temperature and heat capacity of dry air, (11.2) gives the pseudoadiabatic equivalent potential temperature, as it is the potential temperature of a parcel after it expands pseudo-adiabatically until all its water vapor condenses. The pseudo-adiabatic equivalent potential temperature is almost conserved in the dry- and pseudo-adiabatic processes. In general, the real processes are in between the moist-adiabatic and pseudo-adiabatic processes. Since the difference between them are not large except in deep convective clouds, (11.2) is used to represent either the equivalent potential temperature or pseudo-adiabatic equivalent potential temperature, and is called for simplicity the equivalent potential temperature in this study.
11.3. MOIST POTENTIAL ENTHALPY
231
The equivalent potential temperature gives the saturation equivalent potential temperature in the saturated atmosphere, that is θes = θe
Lc ws cp T
.
It may also be defined for unsaturated atmosphere. In this case, the ws is different from the real mixing ratio. Unlike the equivalent potential temperature, the saturation equivalent potential temperature is not conserved in dry adiabatic processes. It may be used conveniently to diagnose the static stability of the atmosphere, either saturated or unsaturated, for saturated parcels. When ∂θes /∂z < 0, a saturated parcel may be warmer than the surroundings as it rises pseudo-adiabatically or moist-adiabatically in the moist atmosphere. The atmosphere is conditionally unstable, if it is statically unstable to saturated parcels but stable to unsaturated parcels. When ∂θes /∂z > 0, the atmosphere is statically stable to either dry or saturated vertical convection. It is noted that the atmosphere, which is unstable to saturated rising motions, may be stable to downward motions without strong evaporation. This feature makes the study of energy conversion in the moist atmosphere more sophisticated than the study of the dry processes. When the equivalent potential temperature is nearly unchanged in adiabatic, moist-adiabatic and pseudo-adiabatic variations, the moist potential enthalpy is almost conserved in these processes. This conservation may not hold when mixing between air parcels of different humidifies takes place. To see how the moist potential enthalpy is changed by mixing, we consider two parcels of dry air mass m1 and m2 , temperature T1 and T2 , and mixing ratio w1 and w2 , respectively. The moist potential enthalpy reads Ψpe0 = cp (1 + w1 )m1 θe1 + cp (1 + w2 )m2 θe2 . If no water vapor condensation takes place after they have been mixed with each other at constant pressure, the final temperature and mixing ratio give Tr =
(1 + w1 )m1 T1 + (1 + w2 )m2 T2 , (1 + w1 )m1 + (1 + w2 )m2
and
m1 w1 + m2 w2 , m1 + m2 respectively. Moreover, the final moist potential enthalpy becomes wr =
Ψper = cp [(1 + w1 )m1 θ1 + (1 + w2 )m2 θ2 ]e
Lc (m1 w1 +m2 w2 ) cp (m1 +m2 )Tcr
.
If the two parcels have a same mixing ratio, the humidity is not changed by mixing without phase transitions of water vapor, and the final temperature is independent of humidity. In this case, the total amount of moist potential enthalpy of the parcels is unchanged by mixing. For simplicity, we make the approximation Tcr = Tc01 = Tc02 = Tc . The previous equation is replaced by Lc m2 (w2 −w1 )
Lc m1 (w1 −w2 )
Ψper = cp (1 + w1 )m1 θe1 e cp (m1 +m2 )Tc + cp (1 + w2 )m2 θe2 e cp (m1 +m2 )Tc .
11. AVAILABLE MOIST ENTHALPY
232 Applying ex ≈ 1 + x for the previous equation yields
Ψper ≈ Ψpe0 + ∆Ψpe , where ∆Ψpe =
Lc m 1 m 2 [(1 + w1 )θe1 − (1 + w2 )θe2 ](w2 − w1 ) (m1 + m2 )Tc
gives the change of moist potential enthalpy in the mixing process. This change takes a small fraction of the moist potential enthalpy in general. If m1 ∼ m2 ∼ 1, we see Ψpe0 ∼ mcp θe ∼ 105 m J. While, for w2 − w1 ∼ 10−3 and
θ e1 − θ e2 ∼ 10−1 , Tc
we have ∆Ψpe ∼ 101 m J which is 4 orders smaller than Ψpe0 . So, the moist potential enthalpy may be considered approximately as a conservative quantity, that is ps pt
A
(θer − θe0 ) dA dp = 0 .
(11.3)
This conservation law may be violated by molecular diffusions at different pressures, which lead to the moist heat-death state in an isolated atmosphere. We shall use this law to filter out the anisobaric molecular diffusion and the dead reference state for the study of moist energy conversion on the scale over a few days.
11.4
Thermodynamic entropy production
To calculate the available moist enthalpy with energy equation (11.1), we have to assume a reference state. As discussed in the previous chapter, a reference state in the dry atmosphere depends on process. This is also true in the moist atmosphere. For the study of energy conversion, the main feature of moist process may also be identified statistically with thermodynamic irreversibility or thermodynamic entropy production. We discuss in this section the entropy variation as the atmosphere changes between two moist equilibrium states. For simplicity, we may assume the following process to connect an initial and final equilibrium states: The parcels with water vapor mixing ratio w0 rise pseudoadiabatically (or adiabatically before they become saturated) until the mixing ratio reaches the final values wr . Then, they descend adiabatically to the original isobaric surfaces, where the parcels are heated or cooled isobarically to get to the final states. Speaking strictly, the pseudo-adiabatic process is not reversible. However, as difference between the pseudo-adiabatic and reversible moist-adiabatic processes is negligible (Holmboe et al. 1945), the variation of thermodynamic entropy in the pseudo-adiabatic process may be ignored. Therefore, the entropy variation in the
11.5. DRY REFERENCE STATE
233
assumed processes is produced only in the stage of isobaric heating or cooling. Since the equivalent potential temperature is conserved in the dry- and moist-adiabatic processes, the potential temperature becomes θ0 = θ0 e
Lc cp
w0 Tc0
− Twr
cr
when a parcel returns back to the original isobaric surface after the pseudo-adiabatic rising. Here, θ0 is potential temperature of initial state. Only the latent heat released by water vapor condensation is considered in this study. The entropy production in the isobaric heating process gives cp ∆S = g
ps
ln pt
A
θr dA dp . θ0
It follows that cp ∆S = g
ps pt
A
Tr Lc ln − T0 cp
wr w0 − Tc0 Tcr
dAdp .
(11.4)
This equation shows that the contribution of moisture reduction resulting from condensation is negative. While, water vapor condensation raises the temperature of reference state, which gives the positive contribution. It is emphasized that thermodynamic entropy production depends only on provided initial and final equilibrium states regardless of the process by which the atmosphere varies. The assumed reversible process is applied only for calculating the entropy change conveniently. The obtained result is equivalent to the real change between the same initial and final states. It is unnecessary for the real process to be pseudo-adiabatic or reversible.
11.5
Dry reference state
In the moist convection with conservation of equivalent potential temperature of parcels, the geopotential irreversibility may be represented by the change of moist geopotential entropy, defined as Sgw
1 =− g
ps pt
A
φ dAdp θe
for the moist atmosphere. The moist convection in an isolated system may take place, only if the moist geopotential entropy is not destroyed when the thermodynamic entropy is also not destroyed. This may be regarded as the moist geopotential entropy law. As the atmosphere tends to reduce its geopotential energy, we may apply the variational approach introduced in Chapter 9 to find the extremal reference state, attained with maximum production of moist geopotential entropy for provided initial state and thermodynamic entropy production. The moist energy conservation equation is given by (11.1). The constraint relationships for the moist isoperimetric
11. AVAILABLE MOIST ENTHALPY
234
problem include the moist thermodynamic entropy equation (11.4) and the conservation law of moist potential enthalpy (11.3). When moist potential enthalpy is conserved, the equivalent potential temperature of reference state is between the minimum and maximum values of initial state, that is min(θer ) ≥ min(θe0 ) ,
max(θer ) ≤ max(θe0 ) .
(11.5)
They give the thermal constraint for the moist processes. The available moist enthalpy represented by (11.1) depends on two variables: the temperature and humidity. Their variations in the atmosphere may be related to each other, especially in the saturated processes. However, the relation cannot be given generally with the large-scale data. We assume in this section that variation of moisture is independent of temperature variation. To find the extremal reference state with specified thermodynamic entropy production and conservation of moist potential enthalpy, we make the auxiliary function from (11.1), (11.3) and (11.4), giving
X = −cp Tr − Lc wr + µ1 cp ln Tr + µ1 Lc
wr ps + µ2 Tr p
κ
Lc wr
Tr e cp Tr ,
where µ1 and µ2 are two non-zero Lagrangian multiplies. For simplicity, the condensation temperature at the reference state Tcr is replaced by temperature Tr approximately. This approximation implies that the reference state is saturated. We shall see that the reference state derived is dry with wr = 0 and so is not affected by the approximation. When the reference temperature Tr and mixing ratio wr are considered as two independent variables, the corresponding Euler equations are ∂X =0, ∂wr
∂X =0. ∂Tr
and
They give µ1 µ2 − 1− Tr cp and µ2 Lc wr µ1 + µ1 − 1− Tr cp Tr2 cp
ps p
ps p
κ
Lc wr
e cp Tr = 0
κ
Lc wr 1− cp Tr
(11.6)
Lc wr
e cp Tr = 0 .
Inserting (11.6) into (11.7) yields
µ1 µ2 + Tr cp
ps p
κ
e
Lc wr cp Tr
Applying (11.6) again gives Lc wr =0. cp Tr It follows that wr = 0 .
Lc wr =0. cp Tr
(11.7)
11.6. MOIST REFERENCE STATE
235
This reference state is dry. Temperature profile of the dry reference state is obtained from (11.6), that is µ1 κ . Tr = 1 − µcp2 pps The potential temperature reads θr =
pκ
λ1 , − λ2
in which λ1 = µ1 pκθ ,
λ2 =
(11.8) µ2 pκs . cp
These two constants are then computed by inserting (11.8) into (11.3) and (11.4), giving ps 1 θe dA dp (11.9) λ1 = AZ pt A 0 and ps ln(pκs − λ2 ) − pt ln(pκt − λ2 ) − (ps − pt )(ln λ1 + κ) + Γd ∆S˜ ps λ2 1 ln θe0 − κ θe0 dpdA = 0 . + A A pt λ1
(11.10)
Here, ∆S˜ gives the column abundance of thermodynamic entropy variation over unit horizontal area, and Z is given by (9.15) or (9.16). The expression of reference state in (11.8) is the same as that for the dry atmosphere given by (9.12). However, the reference state attained in the moist atmosphere may possess more enthalpy than the initial field, since a part of latent heat is converted into heat energy. This will be discussed again in the next chapter. The dry reference state is derived assuming that changes of atmospheric temperature and humidity are independent of each other. It may be considered as the limit of the reference state with minimum humidity and maximum precipitation. In general, the atmosphere does not become dry after development of moist disturbances, and the changes of temperature and humidity are not independent of each other. The variational algorithm may also be used to find the moist reference state. This is discussed in the next section.
11.6
Moist reference state
11.6.1
The isoperimetric problem
If we use the relationship Lc w
e cp Tc = 1 +
Lc w + ··· , cp Tc
(11.11)
the equation of available moist enthalpy (11.1) may be rewritten approximately as Λm
cp = g
ps pt
A
Lc w0
Lc wr
(Tc0 e cp Tc0 − Tr e cp Tcr ) dA dp .
11. AVAILABLE MOIST ENTHALPY
236 It follows that Λm
ps
cp = κ gpθ
pt
If the quantity Ψe =
cp gpκθ
pκ (θe0 − θer ) dAdp .
A
ps pt
pκ θe dAdp
(11.12)
(11.13)
A
is defined as the moist enthalpy, available moist enthalpy is the difference of moist enthalpy between provided initial and reference states. The available enthalpy discussed in Chapter 9 may be considered as the particular example of available moist enthalpy in the dry atmosphere. Moreover, thermodynamic entropy change in the moist processes given by (11.4) may be replaced by θe cp ps ln r dAdp . (11.14) ∆S = g pt A θ e0 For convenience, we may define the moist thermodynamic entropy as sw = cp ln θe . So the thermodynamic entropy change in the moist process is the change of moist thermodynamic entropy. The isentropic state of a saturated moist atmosphere is called the moist isentropic state, which possesses the constant equivalent potential temperature. It can be proved that when the moist potential enthalpy is conserved, the moist isentropic state has maximum thermodynamic entropy. The variational approach for the current problem is to solve the variational equation ps
δ pt
A
pκ (θe0 − θer ) dAdp = 0
under the two constraints of (11.14) and (11.3), which are the equation of thermodynamic entropy production and conservation law of moist potential enthalpy respectively. Making the auxiliary function X = −pκ θer + λ1 ln θer + λ2 θer for this problem, we have the Euler equation λ1 ∂X = −pκ + λ2 + =0. ∂θer θ er It follows that θ er =
pκ
λ1 . − λ2
(11.15)
This is similar to the dry reference state given by (11.8), except that the equivalent potential temperature includes contribution of moisture. The dry reference state may be considered as a limit of the moist state with wr = 0 and so θer = θr . The two constants λ1 and λ2 are computed by substituting (11.15) into (11.3) and (11.14), giving also (11.9) and (11.10) respectively. Thus, they have the same values as those of the dry reference state, and so the dry and moist reference states possess a same profile of equivalent potential temperature.
11.6. MOIST REFERENCE STATE
237
For provided initial temperature and humidity fields, (11.9) and (11.10) can be solved numerically. The reference state is given by θ er =
ps
1 AZ(pκ − λ2 )
pt
A
θe0 dAdp .
(11.16)
The dry and moist reference states must be examined by the thermal constraint (11.5). If the equivalent potential temperature exceeds the constraint, we may increase the entropy production ∆S to meet with the constraint. Since λ1 ∂ 2X =− 2 2 ∂θer θ er where λ1 is positive, the available moist enthalpy with respect to the reference state (11.16) is the upper limit for provided thermodynamic entropy production. The reference state attained by the process with minimum thermodynamic entropy production is also called the lowest state. When the conservation law of moist potential enthalpy is not used, we haveλ2 = 0 in (11.8) and (11.15), and the reference state gives λ1 . pκ
θ er = The temperature profile reads
λ1 − cLpcTwcr r . e pκθ
Tr =
If the humidity decreases with height as usual, the temperature increases upwards. However, there is no possible process which leads to the inversion in isolation. Since the reference state is attained by molecular diffusions which lead to the heat-death state, the reference state is isothermal with uniform distribution of humidity. Although the moist heat-death state is physically true, it may not be interesting to our studies on the large-scale energy conversions or circulation changes within a few days, and so is filtered out by using the conservation law of moist potential enthalpy.
11.6.2
Approximate approach
The equivalent potential temperature fields may be represented generally by θˇe0 = θe0 − θe0 , ∗
θe0 = θe0 − θe0 ,
1 θe0 = A θ e0
1 = ps − pt
A
θe0 dA ;
ps pt
θe0 dp ,
(11.17) (11.18)
where the symbols ,ˇ, and ∗ are explained in Chapter 9. Applying these definitions for thermodynamic entropy equation (11.14) yields approximately cp ∆S ≈ 2g θe0 2
ps pt
A
θˇe20 dA + A(θe0 ∗2 − θe∗2r ) dp ,
(11.19)
11. AVAILABLE MOIST ENTHALPY
238 in which we have used θe∗r = θer − θe0
(11.20)
and the expansion similar to (8.19). If θe∗r = 0, the atmosphere has constant equivalent potential temperature and is called the moist isentropic atmosphere. The thermodynamic entropy produced is maximum as the atmosphere reaches the moist isentropic state. In other words, this moist extremal state possesses maximum thermodynamic entropy when the moist potential enthalpy is conserved.Moreover, applying (11.17), (11.18) and (11.20) for (11.12) yields Λm =
Acp gpκθ
ps
pκ (θe0 ∗ − θe∗r ) dp .
pt
(11.21)
The conservation law of moist potential enthalpy (11.3) is rewritten as ps pt
θe∗r dp = 0 .
(11.22)
The three equations (11.19), (11.21) and (11.22) give the auxiliary function X = −pκ θe∗r + λ1 θe∗2r + λ2 θe∗r for the isoperimetric problem. The Euler equation reads ∂X = −pκ + 2λ1 θe∗r + λ2 = 0 . ∂θe∗r It follows that θe∗r =
pκ − λ2 . 2λ1
(11.23)
Inserting it into the two conditions (11.19) and (11.22) yields λ2 =
pκ+1 − pκ+1 s t (κ + 1)(ps − pt )
and
2λ1 = ±
C , Gm
(11.24)
where C is evaluated from (9.27) and Gm =
ps 1 pt
A
A
θˇe20 dA + θe0 ∗2 dp − 2Γd ∆S˜ θe0 2 .
(11.25)
Now, (11.23) becomes
θe∗r
=
Gm C
pκ+1 − pκ+1 s t − pκ (κ + 1)(ps − pt )
.
As the reference state is statically stable, the negative sign in (11.24) is chosen.
11.7. EXAMPLES OF LOWEST STATE
239
The reference state is then given by
Gm C
θer = θe0 + where
pκ+1 − pκ+1 t s − pκ (κ + 1)(ps − pt )
1 A(ps − pt )
θe0 =
,
(11.26)
ps pt
θe0 dAdp
A
derived from (11.18). If using A
and ps pt
1 θe0 dp = 2 A ∗2
θˇe20 dA =
θe20 dA −
A
ps
2
θe0 dA
pt
A
ps
1 A
2
A
θe0 dA
1 dp − ps − pt
, 2
ps pt
A
θe0 dAdp
,
we gain Gm = −
1 A A2
1 ps − pt
pt
A
θe20 dAdp
2Γd ∆S˜ 1+ ps − pt
ps
2
pt
A
θe0 dAdp
.
(11.27)
The obtained moist reference state is examined by the thermal constraint in (11.5). Moreover, we have ∂2X = 2λ1 . ∂θe∗2r Since λ1 is negative, the available moist enthalpy with respect to the reference state (11.26) is maximum. The dry or moist reference state evaluated with the variational approach is continuous in space. It may be different from the discontinuous reference state derived by a parcel algorithm, because isenthalpic diffusions are allowed in this approach. If the statistical data of entropy production are available for a typical moist process, we may predict the reference state for provided initial fields. The moist lowest state is attained through the extremal process with minimum thermodynamic entropy production and maximum geopotential entropy production. Available moist enthalpy with respect to the lowest state gives the upper limit of kinetic energy generated in a moist atmosphere.
11.7
Examples of lowest state
We assume, for simplicity, that the initial distribution of water vapor depends on pressure only and is represented by
w0 = ws
2
1 − e−(ps −p) /σ 1− 1 − e−p2s /σ2
2
,
(11.28)
11. AVAILABLE MOIST ENTHALPY
240
Figure 11.1: Initial distribution of water vapor mixing ratio with σ = 250 and 300 hPa depicted by solid and dashed curves, respectively.
in which ws signifies saturated mixing ratio of water vapor on the surface ps , and σ 2 indicates the vertical variance of humidity. At the top of atmosphere where p = 0, there is w0 = 0. Examples of this profile are sketched in Fig.11.1. A larger σ indicates more moisture in the atmosphere, as the vertical gradient of humidity is weaker found by comparing the dashed and solid curves in the figure. We assume furthermore that the initial state is barotropic and has constant lapse rate of temperature. The potential temperature profile is represented by
θ0 = Ts
ps p
b
,
b=
R (Γd − Γ) , g
ps = 1000 hPa .
The equivalent potential temperature reads
θe0 = Ts
ps p
b
Lc w0
e cp Tc0 .
(11.29)
The profiles with Γ = 0.75 K/100m and σ = 250 hPa are plotted by the light curves in Fig.11.2. The energy conversion will be evaluated in the assumed atmosphere from 1000 hPa to 200 hPa.
11.7. EXAMPLES OF LOWEST STATE
241
For the horizontally homogenous humidity field assumed, we have ps pt
Also,
A
ps pt
ps
A
θe0 dA dp =
pt
A
ps
κ
p θe0 dA dp =
pt
A
Lc w0
θ0 e cp Tc0 dA dp .
(11.30)
Lc w0
pκ θ0 e cp Tc0 dA dp .
Moreover, we can write ps pt
A
ps
ln θe0 dA dp =
pt
A
ps
ln θ0 dA dp +
pt
Lc w0 dA dp . cp Tc0
A
(11.31)
The first integration on the right-hand side is evaluated from (9.21). The other integrations in the last three equations can be made numerically. The condensation temperature Tc in the equivalent potential temperature is computed from (2.24), or es (11.32) w = 0.622 , p in which, w denotes a saturated mixing ratio of water vapor, and es measures the saturated vapor pressure. The vapor pressure can be calculated from (23.17). But here, it is evaluated from the empiric equation es = es0 10
9.5(T −273.15) T −8
,
(11.33)
where es0 = 6.11 hPa is the saturated vapor pressure at 0 C◦ with respect to a flat surface. These two relationships give Tc = 273.15 +
265 log10 8 − log10
pw 0.622e0 pw 0.622e0
.
(11.34)
It depends on the temperature and pressure. Inserting (11.30) and (11.31) into (11.9) and (11.10), respectively, obtain the two constants λ1 and λ2 . Subsequently, the dry and moist lowest states can be evaluated from (11.8) and (11.15) respective. According to the balance equation of thermodynamic entropy discussed in Chapter 26, the entropy in an isolate atmosphere including the dry air and moisture but not the liquid water and ice may be created by molecular or turbulent diffusions, and may be destroyed by the moisture exchange resulting from moisture condensation and water evaporation under the local balance of moisture. In general, the contribution of turbulences is greater than that of moisture exchange. Thus, we assume ∆S = 0 at the first to calculate the reference state. The solution is examined by the thermal constraint (11.5). If the constraint is broken down, we increase the entropy production until the constraint is satisfied. The reference state so obtained gives the lowest state. For the assumed initial fields with σ = 250 hPa, the lowest states evaluated are plotted in Fig.11.2. The minimum thermodynamic entropy productions in these barotropic examples are generally greater than zero. When the maximum available moist enthalpy is positive, the lowest states obtained with the precise and
11. AVAILABLE MOIST ENTHALPY
242
Figure 11.2: The lowest states (heavy) plotted from the precise and approximate solutions by the solid and dashed respectively. The initial profiles are depicted by light curves. The initial saturated mixing ratios on 1000 hPa surface are denoted on the curves.
approximate algorithms are displayed by the heavy curves in Fig.11.2. These profiles represent either the moist or dry lowest states. The approximate solutions illustrated by the dashed curves are very close to the precise solutions in the figure. These profiles are examined by the thermal constraint in evaluations, so that the equivalent potential temperature at the reference state does not excess the limits given by initial profiles. This can be found from Fig.11.2. The initial states assumed may be conditionally unstable for saturated parcels in the lower troposphere, when the humidity is sufficiently high. The barotropic moist atmosphere may be a kinetic energy source only if the humidity exceeds a certain limit. The conditional stability increases in the lower troposphere but decreases over the middle troposphere after energy conversion. The lowest state is conditionally stable at any height. The dependence of the static stability on the initial distribution of moisture can be studied by the experiments with different values of σ.
11.8
Available moist enthalpy
11.8.1
General and approximate relationships
Moist enthalpy of the moist reference state discussed earlier is evaluated by applying (11.15) for (11.13), giving Ψer
=
cp λ1 A gpκθ
ps pt
pκ dp pκ − λ2
11.8. AVAILABLE MOIST ENTHALPY
243 cp λ1 A (ps − pt + λ2 Z) . gpκθ
=
(11.35)
Inserting it into (11.12) yields the available moist enthalpy Λm =
cp gpκθ
ps pt
A
pκ θe0 dAdp − λ1 A(ps − pt + λ2 Z) .
(11.36)
It follows from (11.9) that cp gpκθ
Λm =
ps pt
A
pκ θe0 dAdp −
ps − pt + λ2 Z
ps pt
A
θe0 dAdp .
The first and second terms on the right-hand side give the moist enthalpy at the initial and reference states, respectively. The available moist enthalpy is maximum if evaluated with respect to the lowest sate. The approximate expression of available moist enthalpy is derived by inserting (11.25) into (11.21), giving ps Acp κ ∗ CG + p θ dp . m e0 gpκs pt
Λm = Here, ps
1 p θe0 dp = A ∗
κ
pt
ps pt
κ
A
p θ e0
pκ+1 − pκ+1 t s dAdp − A(κ + 1)(ps − pt )
ps pt
θe0 dAdp .
A
Applying it together with (9.27) and (11.27) produces Λm
+
pt
A
pκ θe0 dAdp −
pκ+1 − pκ+1 s t (κ + 1)(ps − pt )
ps pt
A
θe0 dAdp
(pκ+1 p2κ+1 − p2κ+1 − pκ+1 )2 s s t t − 2κ + 1 (κ + 1)2 (ps − pt )
ps
A pt
ps
cp = κ gpθ
A
θe20
2 ˜ s − pt ) ps 1 + 2Γd ∆S/(p dAdp − θe0 dAdp . ps − pt pt A
(11.37)
It can be calculated directly with provided initial fields and thermodynamic irreversibility, without solving the reference state firstly. The available moist enthalpy discussed above is for the moist reference state. To the dry reference state, the right-hand side of (11.35) gives the enthalpy Ψr =
cp λ1 A (ps − pt + λ2 Z) . gpκθ
Inserting it together with wr = 0 into (11.1) yields Λm
cp = κ gpθ
ps pt
A
Lc p θ0 dAdp − λ1 A(ps − pt + λ2 Z) + g κ
ps pt
A
w0 dAdp ,
where θ0 is the potential temperature at initial state. Equation (11.36) is the approximation of this equation derived with expansion (11.11). Although the dry and
244
11. AVAILABLE MOIST ENTHALPY
moist reference states possess different amounts of heat energy and moisture, they manifest the same profile of equivalent potential temperature, and so the available moist enthalpies with respect to the different reference states are approximately equal in amount. In other words, the latent heat in the moist reference state is approximately equivalent to the different amounts of heat energy between the dry and moist reference states. We shall use (11.36) and (11.37) to represent the precise and approximate solutions of available moist enthalpy respectively. The thermodynamic entropy in an isolated atmosphere cannot be destroyed. This, however, may not be the case for an open system. We may use a negative value for the entropy variation to calculate the reference state, if there is positive thermodynamic entropy output or negative input in the system. The negative entropy input can be produced by diabatic heating, mass displacement or moisture condensation and water evaporation as discussed in Chapter 26. The kinetic energy created in the process may be greater than the maximum available moist enthalpy evaluated in isolation.
11.8.2
Examples of available moist enthalpy
Examples of maximum available moist enthalpy with respect to the initial and reference states depicted in Fig.11.2 are demonstrated by the heavy curves in Fig.11.3. The approximate solutions are plotted by dashed curves, which are close to the solid representing the precise solutions. Available moist enthalpy in the barotropic atmosphere increases greatly with initial static instability. The statically unstable atmosphere may produce much more kinetic energy than the stable atmospheres, especially when the humidity is low. The deep convection may not develop in the stable atmosphere with low humidity, even if the lower troposphere is conditionally unstable. For a given variance σ in the assumed moisture field, the energy source is also strengthened by increasing the humidity. For comparison, the calculations for different values of σ are plotted by the heavy and light curves respectively in the figure. The atmosphere with more moisture may produce more kinetic energy when the static stability is relatively low. In a highly stable atmosphere, less kinetic energy may be produced if σ is larger, since the conditional stability is reduced by increasing σ. As in the dry atmosphere, the energy conversion in the moist atmosphere depends on process. For the same initial state assumed previously with σ = 250 hPa, we have depicted by the solid curves in Fig.11.4 the available moist enthalpy for the highly diffusive process with whatever is the largest between ∆s = 0.25 J/(K·kg) and the minimum entropy production. The created kinetic energy is converted into mean wind speed with ˜m 2gΛ , v = ps − pt assuming that initial wind speed is zero. For comparison, the maximum available moist enthalpy is displayed by dashed curves in the figure. The available moist enthalpy is reduced greatly by increasing the entropy production especially when the humidity is low, so the reference states are generally
11.8. AVAILABLE MOIST ENTHALPY
245
Figure 11.3: Maximum available moist enthalpy with respect to the initial state with σ = 250 hPa (heavy) and 300 hPa (light). Solid and dashed are drawn for the precise and approximate solutions, respectively. The initial temperature lapse rates are denoted by Γ.
less stable than the moist lowest states. The critical humidity or static instability for disturbance development increases with the entropy production. Fig.11.4 shows that the highly diffusive processes in the moist atmosphere are also characterized by sudden change as in the dry atmosphere, and the suddenness increases with thermodynamic irreversibility. This feature may explain the development of moist explosive disturbances, such as the tropospheric bombs (Sanders and Gyakum, 1980; Roebber, 1984) in the troposphere over extratropical oceans. The calculations show that energy conversion in the moist atmosphere is more sensible to the entropy production if compared with the dry processes. Thus, predictions of moist processes are more difficult than predictions of dry processes. Like available enthalpy in the dry atmosphere, available moist enthalpy measures the potential strength of kinetic energy source. The expected energy conversion may take place under certain conditions. For example, the surface convergence together with upper lever divergence in the wind field may trigger development of convection in the potentially unstable atmosphere. Some of these conditions, such as the temperature inversion and vertical shear of wind, will be studied further by
246
11. AVAILABLE MOIST ENTHALPY
Figure 11.4: Available moist enthalpy for the processes with ∆s = 0 and ≥ 0.25 J/(K·kg), plotted with dashed and solid respectively.
introducing the air engine theory in Chapter 21. Thermodynamic entropy variation in the moist processes is measured by change of equivalent potential temperature as show in (11.14). Since the equivalent potential temperature field is horizontally homogeneous at the initial state assumed, the entropy is produced by reducing the conditional stability. Fig.11.2 shows that except in the lower troposphere the conditional stability decreases after energy conversion. Owing to the effect of water vapor condensation, a barotropic atmosphere may still produce kinetic energy even if it is statically stable, especially when the humidity is high.
Chapter 12 Moist processes of energy conversion 12.1
Introduction
It is discussed in the preceding chapter that the dry and moist reference states attained from a provided initial state and thermodynamic entropy production in the moist atmosphere possess a same profile of equivalent potential temperature. In other words the temperature profile is equivalent to that of the dry reference state, when the water vapor in the moist reference state is condensed through psuedoadiabatic process. An equal amount of moist geopotential entropy is created as the atmosphere reaches the dry and moist reference states. The amounts of kinetic energy created in the different processes are also approximately identical. The moist reference state represented by equivalent potential temperature depends on the temperature and humidity or the partition of latent and sensible heats in the atmosphere. It may not be saturated in general as found in observations. The heat energy and geopotential energy of the atmosphere can be partitioned according to (7.33), while there is no constant correlation between the latent heat and sensible heat or geopotential energy in the unsaturated moist atmosphere. The conservation laws and constraint relationships applied for deriving the reference state in the previous chapter are not enough for us to find the energy partition in the reference state. According to the variational approach applied in the preceding chapter, all the reference states attained from a given initial field through the process with provided thermodynamic entropy production and corresponding maximum moist geopotential entropy production are in between the two extremal states: the dry reference state and saturated moist state. The dry reference state is attained with maximum precipitation and possesses maximum thermal enthalpy, while the saturated reference state possesses minimum enthalpy attained with minimum precipitation. The profiles of temperature and humidity of the saturated state will be demonstrated in the next section using the Clausius-Clapeyron relationship between the saturation temperature and humidity. The minimum precipitation and its dependence on the baroclinity and static stability will be illustrated also. The examples of energy conversion studied in the preceding chapters are independent of the baroclinity, as kinetic energy may be created through moist processes in the barotropic and statically stable atmosphere. It can be found in this chapter that the baroclinity may strengthen the moist energy source and reduce the threshold humidity or static instability for development of moist disturbances. Thus, the barotropic disturbances in the tropical regions may be triggered and intensified by intrusion of baroclinic systems. As in the dry atmosphere, the contribution of baroclinity is also made by increasing baroclinic entropy to compensate the loss of barotropic entropy in the moist processes, so that more geopotential energy and correlated heat energy may be converted into kinetic energy. As thermodynamic entropy in the moist atmosphere is evaluated with the equiv247
248
12. MOIST PROCESSES OF ENERGY CONVERSION
alent potential temperature which depends on the humidity as well as temperature, the role of baroclinity in the energy conversion may also be played by horizontal inhomogeneity of humidity field. This can be confirmed by observations, as development of moist disturbances are preferred in the regions with strong horizontal gradient of humidity or equivalent potential temperature. As in the study of the dry processes, the energy conversions in the moist processes may also be studied according to the changes of horizontal and vertical disorderliness in the thermodynamic fields. We shall introduce the equivalent baroclinic entropy and equivalent barotropic entropy to explain the mechanism of energy conversion in the moist atmosphere. They are similar to the baroclinic and barotropic entropies, except using equivalent potential temperature for the potential temperature. It is found in this chapter that the latent heat released may not be converted entirely into kinetic energy in the barotropic atmosphere. Thus, a part of latent heat is transfered into geopotential energy and heat energy, so that the static stability may decrease. This is different from the energy conversion in the dry atmosphere. Since the humidity is low in the reference state, the decrease of static stability reduces the vertical gradient of equivalent potential temperature and so increases the equivalent barotropic entropy. As a result, the thermodynamic entropy or moist thermodynamic entropy is not destroyed in the isolated atmosphere. However, if the initial baroclinity is strong, the static stability may increase as a part of enthalpy at the initial state is converted into kinetic energy. In this situation, the kinetic energy created may be more than the latent heat released, while the precipitation may not increase greatly. There were the discussions on whether the development of cyclones over warm ocean water, like the explosive cyclones and polar lows, was driven by the baroclinic or moist convective processes (e.g., Harrold and Browning, 1969; Tracton, 1973; Mansfield, 1974; Read, 1979; Bosart, 1981; Gyakum, 1983a, b). We have evaluated, in Chapter 10, the surface pressure changes produced by energy conversion in the dry and baroclinic atmosphere. The pressure reduced is relatively small compared with the processes in the moist atmosphere. For example, the surface pressure may decrease nearly 60 hPa within a day in an explosive cyclone (Gyakum, 1983a) and about 100 hPa in a typhoon (Holliday and Thompson, 1979). The study in this chapter shows that reduction of surface pressure in the moist processes may be much larger than in the dry atmosphere. It is discussed in Chapter 8 that the thermodynamic entropy may not be able to measure the ability of kinetic energy generation related to conversion of geopotential energy in the dry atmosphere. This is also the case for energy conversion in the moist atmosphere. When the potential temperature in the expression of thermostatic entropy defined for the dry atmosphere is replaced by equivalent potential temperature, we gain the definition of equivalent thermo-static entropy which measures the ability of energy conversion in the moist processes. The conditionally stable and unstable atmospheres may possess an equal amount of thermodynamic entropy, but cannot have a same amount of equivalent thermo-static entropy. The equivalent thermo-static entropy increases in energy conversion, even if the thermodynamic entropy is conserved. Thus, the energy conversions in the moist atmosphere are irreversible in isolation.
12.2. SATURATED REFERENCE STATE
249
Figure 12.1: Profiles of water vapor mixing ratio at the initial (light, σ = 250 hPa) and reference (heavy) states, calculated by the precise (solid) and approximate (dashed) algorithms. The saturated mixing ration of water vapor on the surface at the initial field is denoted by ws .
12.2
Saturated reference state
12.2.1
Saturated humidity profile
The moist reference state given by (11.15) is represented by the equivalent potential temperature evaluated from
θer = Tr
pθ p
κ
Lc wr
e cp Tcr .
(12.1)
Here, the condensation temperature Tc is computed from (11.32) and (11.33). The moist reference state depends on humidity as well as temperature. It is generally impossible for us to tell the individual distributions of humidity and temperature without solving the prediction equations. However, we may assume that all the reference states attained for provided initial state and thermodynamic entropy production are between the dry reference state and saturated moist reference state. The dry reference state is attained by the process of maximum precipitation and possesses maximum enthalpy, compared with the corresponding moist reference states. However, kinetic energy created in the process is approximately the same as that with respect to the moist reference state, since the available moist
12. MOIST PROCESSES OF ENERGY CONVERSION
250
enthalpy depends basically on the change of equivalent potential temperature and is independent of the partition between the sensible and latent heats at a reference state. We discuss in the following the saturated reference state. To find the humidity profile, we insert (11.16) into (12.1), giving AZ(pκ
ps
1 − λ2 )
pt
A
θe0 dAdp = Tr
pθ p
κ
Lc wr
e cp Tr .
(12.2)
Here, we have used Tcr = Tr for the saturated atmosphere. For the approximate expression of the reference state, we have
θ e0 +
Gm C
pκ+1 − pκ+1 s t − pκ (κ + 1)(ps − pt )
= Tr
pθ p
κ
Lc wr
e cp Tr
(12.3)
derived from (11.26). The profiles of saturated water vapor can be evaluated by solving (11.34) and (12.2) or (12.3) numerically. For the lowest states displayed in Fig.11.2, the profiles calculated are displayed in Fig.12.1, which are compared with the initial fields depicted by the light curves. This figure shows that the column content of water vapor decreases after energy conversion. The water vapor in the boundary layer is transported upward in a moist disturbance, so that the vertical gradient of humidity decreases and the conditional stability increases at low levels but decreases aloft.
12.2.2
Minimum precipitation
The precipitation is minimum as the atmosphere reaches the saturated reference state through a process with assumed thermodynamic entropy production. The minimum precipitation in the isolated atmosphere may be estimated from 1 P = g
ps pt
(w0 − wr ) dp .
The evaluations for σ = 250 hPa and with respect to the lowest states plotted in Fig.11.2 are demonstrated graphically in Fig.12.2. The dashed line in the figure illustrates the full precipitation evaluated from 1 Pf = g
ps pt
w0 dp .
The real precipitation is between the full and minimum amounts. This figure shows that the precipitation increases with the static instability and humidity of the initial field. The negative precipitation corresponding to a high static stability in the figure implies that the process may not occur unless more water vapor is supplied. This agrees with the observations. The mixing ratio on the surface may be over 20 g/kg in the atmosphere especially at low latitudes or at middle latitudes in summer. The ratio of precipitation to the water vapor content at initial state, called the precipitation ratio in this study, is given by 1 Rp = g
ps w0 − wr pt
w0
dp .
12.2. SATURATED REFERENCE STATE
251
Table 12.1: Minimum precipitation P , precipitation ratio Rp and kinetic energy generation rate Rk shown as the barotropic moist atmosphere with temperature lapse rate Γ reaches the saturated lowest states
ws (g/kg) P (mm) Rp Rk P (mm) Rp Rk P (mm) Rp Rk P (mm) Rp Rk
(Γ = 0.75 K/100m) (Γ = 0.85 K/100m) (Γ = 0.95 K/100m) (Γ = 1.05 K/100m)
3 2.1 0.31 0.41 3.9 0.57 0.76
6 2.4 0.17 0.37 4.8 0.35 0.43 8.0 0.59 0.58
9 0.9 0.05 0.69 4.4 0.22 0.46 7.8 0.38 0.43 12.2 0.60 0.51
12 2.3 0.09 0.85 6.7 0.25 0.48 10.8 0.40 0.43 16.0 0.59 0.47
15 4.1 0.12 0.79 9.1 0.27 0.49 13.9 0.41 0.43 20.0 0.59 0.45
The examples evaluated with respect to the saturated lowest states are displayed in Table 12.1. The Rk in the table gives the energy generation ratio, which is the ratio of generated kinetic energy to the latent heat released, evaluated from Rk =
Λ . Lc P
With this ratio, we may estimate approximately the kinetic energy generation according to the precipitation. Table 12.1 shows that the precipitation ratio increases with initial static instability and humidity in general. More than a half amount of the precipitable water may fall down when the stability is low. The energy generation ratio fluctuates around 50%, so about half latent heat can be converted into kinetic energy in the extremal process with minimum thermodynamic entropy production. The atmosphere has more heat energy after development of moist disturbances, as a part of latent heat released is converted into enthalpy. It is noted that the saturated reference state and precipitation amount depend on process or thermodynamic entropy production. The examples plotted in Fig.12.2 are evaluated with respect to the lowest states. The precipitation in a highly irreversible process is even less than the minimum precipitation. While, it may be more than the plotted values if the lowest state is unsaturated. The evaluations may not be used directly for predictions, since the reference state attained in the real atmosphere may not be saturated, and the local convergence of water vapor flux may have an important effect on local precipitation.
12. MOIST PROCESSES OF ENERGY CONVERSION
252
Figure 12.2: Minimum precipitation in the extremal process leading to the lowest states. The Γ indicates the temperature lapse rate at initial state.
12.2.3
Temperature profile
Temperature profiles of the saturated reference state may be evaluated from (12.2) or (12.3) with the profiles of saturated water vapor obtained previously. For the lowest state plotted in Fig.11.2, the profiles are displayed by the heavy solid in Fig.12.3. The temperature falls down in the lower troposphere, but rises up in the middle atmosphere due to release of latent heat. So the static stability increases in the lower troposphere, and decreases in the upper troposphere. The cooling of surface air may be confirmed by the cold domes observed in the regions of moist disturbances. It may also be found along the tracks of typhoon or hurricane. For the low systems over the oceans, reduction of surface temperature may be enhanced by up welling of oceanic water near the low centers. When the surface air is cooled down in the moist processes, the potential temperature may exceed the lower limit of initial state. This cooling cannot be achieved by the quasi-adiabatic processes discussed in Chapter 7. One of the most efficient process for reducing parcel potential temperature is evaporation. The equivalent potential temperature may still be conserved when evaporation takes place in the downdraft. Thus, we used the thermal constraint for equivalent potential temperature only, but not for potential temperature to study the moist energy source in the preceding chapter.
12.3. EFFECT OF BAROCLINITY
253
Figure 12.3: Profiles of saturated temperature at the moist (solid) and dry (dashed) lowest states. The initial temperature profiles are indicated by the light curves.
For comparison, the temperature profiles for the corresponding dry reference states are plotted by the dashed in the figure. The dry reference states are warmer than the saturated in the lower troposphere, and are even warmer than the initial states. The vertical mean static stability may be reduced by energy conversion with the full precipitation in the whole troposphere. The strong heating of the surface air and the decrease of static stability are not observed usually. This suggests that the atmosphere may not be dehydrated entirely in moist process. Also, the observed surface cooling may not be as strong as that shown by the heavy solid curves Fig.12.3. Temperature profiles of the dry and saturated reference states give the upper and lower limits of the field attained by development of moist systems. Fig.12.3 shows that the temperature decreases at high levels as the moist atmosphere reaches the saturated reference states after energy conversion, unless the humidity is very high. In this case, the isentropic surfaces and tropopause concave upward. This feature can be seen from the cross section of an explosive cyclone developed over the North Pacific shown in Fig.22.7, and is different from the energy conversion in the dry atmosphere shown by Fig.9.2. Without condensation of water vapor, kinetic energy is created simply as cold air descends and warm air ascends. So, the temperature in the upper troposphere increases after disturbance development (referring to Fig.9.1 also). The typical example is occlusion of frontal cyclone. As a result of upper tropospheric warming, the tropopause descends over the baroclinic cyclone.
12.3
Effect of baroclinity
It is discussed in Chapter 10 that horizontal temperature gradient in the dry and statically stable atmosphere is substantial for kinetic energy generation. Although
12. MOIST PROCESSES OF ENERGY CONVERSION
254
Figure 12.4: Moist lowest states with respect to the initial baroclinity Ty = 5, 10 K/1000km, depicted by heavy solid and dashed respectively. The light curves represent the initial states with Γ = 0.75 K/100m at y = 0. The surface mixing ratio at the initial state is denoted by ws .
the baroclinity is not necessary for energy conversion in the moist atmosphere as discussed in the preceding chapter, it may increase kinetic energy generation. This is because destruction of baroclinity makes reference state more statically stable, so that more geopotential energy together with correlated heat energy may be converted. The convective storms developed near a temperature front on the warm side, such as the squall lines, are generally more intense than the air-mass storms produced in a barotropic environment with a similar humidity distribution. The initial humidity field is also given by (11.28) with σ = 250 hPa, while the initial equivalent potential temperature in the linear atmosphere is represented by
ps θ0 = (Ts + Ty y) p
b
Lc w0
e cp Tc0 ,
b=
R (Γd − Γ) g
with ps = 1000 hPa. The profiles for Γ = 0.75 K/100m at y = 0 are plotted by the light curves in Fig.12.4, which are the same as those in Fig.11.2. Strictly speaking, the assumed field with constant humidity on the pressure surfaces may be oversaturated on the cold side. For simplicity, the over-saturation is ignored. The moist reference state is evaluated from (11.15). The two constants in this equation are calculated from (11.9) and (11.10). The lowest states attained from the baroclinic fields are depicted in Fig.12.4. Comparing the heavy solid and dashed curves in Fig.12.4 finds that the baroclinity increases the conditional stability of reference state, as seen also by comparing Figs.12.4 and 11.2. The attained static stability increases also in the baroclinic processes, even in the upper troposphere. This can be found if we plot the temperature profiles of saturated reference states. When the reference state is more stable than initial state, a part of heat energy and geopotential energy in the initial atmosphere is converted into kinetic energy.
12.3. EFFECT OF BAROCLINITY
255
Figure 12.5: Effect of baroclinity on the maximum available moist enthalpy. The dashed and solid curves are plotted for ws = 2.5, 5 g/kg, respectively. The Γ indicates initial temperature lapse rate.
Fig.12.5 demonstrates the maximum kinetic energy generated as the baroclinic atmosphere reaches the lowest states displayed in Fig.12.4. The amount increases with the initial baroclinity for a given humidity field. The evaluations show that the energy generation ratio Rk is generally larger than that in the barotropic atmosphere with the same humidity. The increase of kinetic energy generated is contributed mainly by conversion of geopotential and heat energy in the baroclinic moist atmosphere, since the precipitation does not increase largely. The effect of baroclinity may be larger than plotted, if the over-saturation is considered in the saturated atmosphere according to the Clausius-Clapeyron relationship between temperature and humidity given by (23.16). Fig.12.5 tells also that development of moist disturbances in highly irreversible processes may happen explosively, especially when the baroclinity and static instability are relatively low. Miller (1958) reported that about 75 percent of deep central pressure in northwestern Pacific typhoons were attained through rapid deepening process. The sudden changes are found also in the extratropical moist cyclones. Bosart (1981) noted that the explosive deepening phase of cyclone coincided with outbreak of convection near the storm center.
12. MOIST PROCESSES OF ENERGY CONVERSION
256
Fig.11.3 or 11.4 shows that the moist disturbances may not develop in a barotropic atmosphere until the humidity becomes sufficiently high. The threshold humidity for the development is reduced by increasing baroclinity as shown in Fig.12.5. Thus, the barotropic disturbances may be triggered or intensified by intrusion of baroclinic troughs from the extratropical atmosphere, as observed frequently in the subtropical regions. The effect of baroclinic circulation on development of tropical storm will be discussed again in Chapter 22 with the air engine theory developed in this study. Comparing the solid curves in Fig.12.5 with the dashed in the figure or Fig.9.3 finds that the energy source in the baroclinic atmosphere is strengthened by adding moisture. However, the effect is nonlinear if the humidity is low and the static stability is high. For example, when the temperature lapse rate at the initial state is 0.65 K/100m, a slight increase of humidity may hold in the energy conversion until the baroclinity becomes sufficiently strong or the humidity increases further. This is because release of latent heat in the lower middle troposphere reduces the static instability below, so that the kinetic energy source may be weakened if the conditional instability and baroclinity is low.
12.4
Effect of horizontal humidity gradient
Many convective storms with heavy rains and strong winds are found in the zone of equivalent potential temperature front. As the temperature field is nearly barotropic there, this front manifests a high gradient in the humidity field. Fawbush et al. (1951) pointed out that one of the synoptic conditions for formation of tornado in the United States is that the horizontal moisture distribution within the moist layer must exhibit a distinct maximum along a relatively narrow band. The contribution of humidity front to kinetic energy generation is discussed below. We assume that the initial inhomogeneous field of water vapor mixing ratio is represented by 2 2 1 − e−(ps −p) /σ , w0 = (ws + wys y) 1 − 1 − e−p2s /σ2 where wys measures the horizontal gradient of humidity on the surface, and y varies from −Y to Y . The initial equivalent potential temperature field is then given by Lc wy y
θe0 = θe0 e cp Tc0 ,
in which
2
1 − e−(ps −p) /σ 1− 1 − e−p2s /σ2
wy = wys
2
denotes the horizontal gradient of humidity on isobaric surfaces, and θe0 = θ0 e
Lc w0 cp Tc0
measures the horizontal mean equivalent potential temperature, where
w0 = ws
2
1 − e−(ps −p) /σ 1− 1 − e−p2s /σ2
2
.
12.4. EFFECT OF HORIZONTAL HUMIDITY GRADIENT
257
Figure 12.6: The moist lowest state with respect to the initial humidity field for ε = 0.25 and 0.5, depicted by heavy solid and dashed curves respectively. The light solid represents the equivalent potential temperature of the initial states at y = 0.
We may find ps pt
A
ps
ln θe0 dA dp =
pt
A
Lc wy y lnθe0 + cp Tc0
dA dp .
(12.4)
When the humidity varies in the horizontal direction, the condensation temperature Tc0 changes also in the horizontal direction at the barotropic initial state, but the effect is small. If the horizontal gradient of humidity is constant, the last term in the previous equation is negligible compared with the first term on the right-hand side. When the small term is ignored, we have ps pt
ps A
ln θe0 dA dp =
pt
A
lnθe0 dA dp .
(12.5)
Moreover, when wys is constant, we see 1 A
ps
Lc wy y 1 ps θe0 dA dp = θe0 e cp Tc0 dA dp A pt A pt A ps Lc wy Y Tc cp θe0 0 sinh dp , = Lc Y p t wy cp Tc0
(12.6)
and 1 A
ps pt
A
pκ θe0 dA dp =
1 A
ps pt
Lc wy y
A
pκ θe0 e cp Tc0 dA dp
ps Lc wy Y Tc cp pκ θe0 0 sinh dp . =
Lc Y
pt
wy
cp Tc0
(12.7)
258
12. MOIST PROCESSES OF ENERGY CONVERSION
Figure 12.7: Maximum available moist enthalpy with respect to the horizontally inhomogeneous humidity field with ε = 0.25 and 0.5, depicted by the dashed and solid respectively. The initial temperature lapse rate is represented by Γ.
The integrations in these equations may be carried out numerically. Inserting (12.5) and (12.6) into (11.9) and (11.10), we may calculate the two constants λ1 and λ2 respectively. The reference state is then obtained from (11.15). Assuming that wy = εw0 /1000km (ε = 0.25 and 0.5) and σ = 250 hPa, the evaluated moist lowest states for Y = 1000 km are show in Fig.12.6. They are more statically stable than the lowest state shown by Fig.11.2. The corresponding available moist enthalpy is shown in Fig.12.7. This figure is made without ignoring the last term in (12.4). It is little different from the calculations ignoring the small term. We discussed in Chapter 10 that the horizontal temperature gradient is crucial for the energy generation in the statically stable and dry atmosphere, since increase of baroclinic entropy in the process may compensate the loss of barotropic entropy in the process of energy conversion. While, the thermodynamic entropy changes in the moist atmosphere are evaluated with the equivalent potential temperature instead of potential temperature, as shown by (11.14). Thus, horizontal gradient in the equivalent potential temperature field has an effect similar to that of baro-
12.5. SURFACE PRESSURE CHANGE
259
clinity in the dry atmosphere. This gradient may be contributed by inhomogeneous distribution of humidity only. Fig.12.6 shows that the moist energy source is strengthened by increasing horizontal inhomogeneity in the humidity field. The influence is not very large unless the static stability and humidity are low, since the total amount of moisture does not change with the horizontal gradient, and the inhomogeneity has only a small effect on thermodynamic entropy production as show by (12.4). Fig.12.7 shows that the threshold humidity and static stability for development of disturbance decreases are reduced by increasing the humidity gradient. Thus, development of moist disturbance is favoured near a front of equivalent potential temperature, while the final intensity is not influenced greatly by the front.
12.5
Surface pressure change
As in the dry atmosphere, energy conversion in the moist atmosphere may also reduce the surface pressure. For the assumed initial state with constant temperature lapse rate, the surface pressure is given by (10.2). After energy conversion, the surface pressure at the reference state is also evaluated from (10.3) but the constants λ1 and λ2 are different from those of try processes. If the initial surface pressure is 1000 hPa and the height of 200 hPa surface is unchanged, the pressure reduction in the moist barotropic atmosphere is displayed in Fig.12.8. The height of 200 hPa surface is computed from (10.4). The solid curves in Fig.12.8 show the pressure reduction in the process with maximum kinetic energy generation. The initial states with σ = 250 hPa and the corresponding lowest states are plotted in Fig.11.2. The reduction of surface pressure increases rapidly with initial humidity and static instability. So a low system may be intensified largely by water vapor condensation. The pressure drop takes place more abruptly in a process with more thermodynamic entropy production, as shown by the dashed curves in the figure. As discussed by Miller (1958), the minimum surface pressure in a low system depends on sea surface temperature, humidity of surface air, the static stability and constant height of upper pressure surface. Since the surface temperature and humidity are relatively high in the tropical regions, the surface pressure is generally lower in the tropical storms than in the extratropical cyclones. Usually, the tropical storms may extend over 100 hPa, so the surface pressure drops more than that shown in Fig.12.8 when other conditions are the same. The low pressure depends also on the baroclinity of initial field. Fig.12.9 illustrates the changes of surface pressure in baroclinic possesses, which can be compared with the changes in the barotropic examples plotted by the solid in Fig.12.8. The surface pressure is lower in a system with higher baroclinity, since the lowest state attained is more statically stable. The effect of baroclinity in the dry atmosphere may be found from Fig.10.8. The dry baroclinic disturbances are much weaker than the moist disturbances. Fig.12.10 shows the effect of horizontal inhomogeneity in the initial humidity field. Since the amount of moisture does not change greatly with the horizontal gradient, the effect on surface pressure change is generally small, as it is also small
260
12. MOIST PROCESSES OF ENERGY CONVERSION
Figure 12.8: Reduction of surface pressure as the atmosphere assumed changes to the moist lowest state (solid) or the reference state through the process with ∆s = 0.25 J/(K·kg) (dashed). The initial temperature lapse rates are indicated by Γ.
on kinetic energy generation. However, the humidity gradient reduces the critical humidity for development of moist disturbance as discussed earlier, especially when the static stability is relatively high.
12.6
Available enthalpy of reference state
The dry or moist reference state attained by either the pseudo-reversible or irreversible process in the dry or moist atmosphere may be represented generally by θ er =
λ1 . pκ − λ2
It is barotropic and statically stable, and so cannot be changed again by the quasiadiabatic process in an isolated atmosphere without eliminating kinetic energy. This is discussed in the following. If the reference state is taken as an initial state, that is, θ e0 =
pκ
λ1 , − λ2
12.6. AVAILABLE ENTHALPY OF REFERENCE STATE
261
Figure 12.9: Reduction of surface pressure as the baroclinic atmosphere reaches the moist lowest state. Solid and dashed are drawn for the initial temperature gradients |Ty | = 5 and 10 K/1000km respectively.
the new reference state is given by θ er =
λ1 , pκ − λ2
in which λ1 and λ2 are determined by λ1 = λ1
Z Z
(12.8)
and ps ln
pκs − λ2 pκt − λ2 Z + Γd ∆S˜ + κ(λ2 Z − λ2 Z ) = 0 (12.9) − p ln + (p − p ) ln t s t pκs − λ2 pκt − λ2 Z
derived using (11.3) and (11.14). Here, Z is referred to (9.16) and
ps
Z = pt
pκ
dp . − λ2
12. MOIST PROCESSES OF ENERGY CONVERSION
262
Figure 12.10: Reduction of surface pressure as the barotropic atmosphere reaches the moist lowest states. Solid and dashed are drawn for the initial humidity field with ε = 0.25 and 0.5 respectively.
The moist enthalpy of the initial state is given by ˜ e0 = Ψ =
cp λ1 ps pκ dp gpκθ pt pκ − λ2 cp λ1 A (ps − pt + λ2 Z) gpκθ
derived using (11.13). Applying it together with (11.35) for (11.12) gives the extremal value of available moist enthalpy, that is $ % ˜ m = cp (ps − pt )(λ1 − λ1 ) + λ1 λ2 Z − λ1 λ2 Z . Λ κ gpθ The solutions of (12.8) and (12.9) with ∆S = 0 are λ1 = λ1 and λ2 = λ2 . Thus, the lowest state of the reference state is itself. It is noted that the reference state attained by a highly diffusive process is also the lowest state of itself, although it possesses more non-kinetic energy than the lowest state attained from the same initial state. As discussed in Chapter 8 the ability of energy conversion or the strength of energy source depends on the magnitude and direction of field gradient instead of total energy of the system.
12.6. AVAILABLE ENTHALPY OF REFERENCE STATE
263
Figure 12.11: Available moist enthalpy with respect to the initial states or the reference states attained from the moist linear atmospheres with the temperature lapse rate Γ indicated.
For the linear atmosphere with ws = 9 g/kg and σ = 250 hPa, as shown by a light curve in Fig.11.2 for Γ = 0.75 K/100m, the moist lowest state is plotted by the heavy solid or dashed curve on the left in the same figure. If this moist lowest state is taken as an initial state, the available moist enthalpy visas thermodynamic entropy production is depicted in Fig.12.11. Two other examples with different initial static stabilities are also plotted in the figure. The kinetic energy generation will be negative if the entropy produced is greater than zero. Positive kinetic energy cannot be produced in the isolated system because thermodynamic entropy decreases in the process. The equilibrium state of an ideal gas is the uniform state with maximum thermodynamic entropy. While, the atmosphere, as a large-scale inhomogeneous thermodynamic system, tends to reach the lowest state with minimum thermodynamic entropy instead of the isentropic state with maximum thermodynamic entropy through parcel motions, since the system tends to reduce its geopotential energy. In this sense, we have argued in Chapter 9 that increase of geopotential entropy is prior to increase of thermodynamic entropy in the meteorological processes. Now, we can see more clearly that the process to the isentropic state is constrained by energy conservation. Fig.11.2 shows that parcel kinetic energy will be converted into heat energy and geopotential energy as a stable atmosphere increases thermodynamic entropy. Since only the turbulent kinetic energy with vertical component may be converted
264
12. MOIST PROCESSES OF ENERGY CONVERSION
Figure 12.12: Threshold static instability for development of moist disturbances. The ε denotes the horizontal inhomogeneity of the humidity field.
in the quasi-adiabatic process, and the atmosphere may not possess enough turbulent kinetic energy, the isentropic state may not be reached from the stable atmosphere. The extremal states with maximum thermodynamic entropy will be studied in Chapter 17.
12.7
Threshold static instability
Fig.11.3 shows that energy conversion in the moist atmosphere may not happen until the humidity becomes sufficiently high. The threshold humidity for development of moist disturbance decreases with the static stability. On the other hand, the threshold static instability for the disturbance development depends on humidity, and may be evaluated by setting ∆k = 0 for the process with minimum thermodynamic entropy production. The threshold instability depends on three-dimensional inhomogeneities in the temperature and humidity fields. For a barotropic atmosphere of which the potential temperature profile is given by (11.29) and the humidity profile with σ = 250 hPa is represented by (11.28), the threshold instability is displayed in Fig.12.12. The threshold static instability decreases rapidly with increasing humidity and its horizontal inhomogeneity. Thus, we see again that development of moist disturbances is preferred in the regions with a high humidity and equivalent potential temperature front. According to the discussions in the previous sections, the threshold static instability or humidity decreases with the baroclinity, but increases with thermodynamic irreversibility of the process.
12.8. EQUIVALENT BAROCLINIC AND BAROTROPIC ENTROPIES
12.8
265
Equivalent baroclinic and barotropic entropies
Thermodynamic entropy change in the moist atmosphere depends on change of equivalent potential temperature as shown by (11.14). This equation may be rewritten as (12.10) ∆S = ∆Sec + ∆Set , in which Sec
cp = g
ps
ln pt
A
θe dAdp θe
may be called the equivalent baroclinic entropy, and cp Set = g
ps
ln pt
A
θe dAdp θe
may be referred to as the equivalent barotropic entropy. Here, we have used the definitions given by (11.17) and (11.18). Since
θˇe0 θe = ln 1 + ln θe θe0
≈
θˇe2 θˇe − , θe 2θe 2
the equivalent baroclinic entropy gives Sec ≈ −
cp 2g
ps pt
1 θe 2
A
θˇe2 dA dp .
The horizontal inhomogeneity of equivalent potential temperature on the isobaric surfaces will be referred to as the equivalent baroclinity. The atmosphere, which manifests the equivalent baroclinity and so possesses negative equivalent baroclinic entropy, may be called the equivalent baroclinic atmosphere. An equality sign in the previous equation is used for the equivalent barotropic atmosphere, which has zero equivalent baroclinic entropy. This equation tells that thermodynamic entropy increases as equivalent baroclinity decreases. The negative equivalent baroclinic entropy may be contributed by the horizontal inhomogeneity in either the temperature or humidity field. The baroclinic entropy defined for the dry atmosphere may be considered as a particular example of the equivalent baroclinic entropy, which depends on horizontal gradient of temperature only. The moist barotropic atmosphere may still have negative equivalent baroclinic entropy, if the humidity field is horizontally inhomogeneous. To calculate the equivalent barotropic entropy, we apply (11.18) and (11.20) and gain θe ln θe
θe ∗ = ln 1 + θe ∗ θe ∗2 θe − ≈ . θe 2 θe 2
When the anisobaric molecular diffusion is filtered out, we may use the conservation law of equivalent potential enthalpy (11.3), which can be rewritten as θer = θe0 .
12. MOIST PROCESSES OF ENERGY CONVERSION
266 Inserting it into (11.18) yields ps pt
θe0 ∗ dp = 0 .
The equivalent barotropic entropy gives now Acp Set ≈ − 2g θe 2
ps pt
θe∗2 dp .
It is negative unless in the moist isentropic atmosphere which has constant equivalent potential temperature. The change of equivalent barotropic entropy is evaluated from ps Acp (θe∗20 − θer ∗2 ) dp . ∆Set = 2g θe0 2 pt The equality sign is used if θe∗2r = θe0 ∗2 . The θe∗ in this equation depends on the strength of vertical gradient of equivalent potential temperature and is independent of the gradient direction. The conditionally stable and unstable atmospheres may possess equal amount of equivalent barotropic entropy. In other words, the thermodynamic entropy may be conserved as the conditionally unstable atmosphere becomes stable, although kinetic energy is created irreversibly in the process. We have discussed in Chapter 10 that kinetic energy may be produced in an isolated dry atmosphere if it is either baroclinic or statically unstable. The ability of energy conversion may be explained simply by the changes of thermodynamic entropy in the horizontal and vertical directions. While, the moist processes are more complicated compared with the dry processes, although the entropy changes may be calculated simply by replacing the equivalent potential temperature for the temperature as shown previously. For example, the kinetic energy may be created in the slantwise convection with the trajectory slope less than the slope of the isentropic surfaces crossed in the baroclinic atmosphere, since the parcels are statically unstable in the slantwise convection even if the atmosphere is statically stable. This will be discussed in detail in Chapter 19. While, the equivalent baroclinity may be contributed by humidity gradient only, so the equivalent baroclinic atmosphere may not be an energy source if it is unsaturated and statically stable. Even in a saturated atmosphere, the downward slantwise convection with a trajectory slope less than the slope of equivalent potential temperature surface (called the moist isentropic surface also) may still be statically stable and so destroy the kinetic energy related to vertical motions. Moreover, the kinetic energy may be converted from the latent heat released, and there is no a general relation between the changes of moisture and heat energy or geopotential energy in the moist atmosphere. Thus, the change of static stability or conditional stability may have different signs in the moist processes. When a moist system develops in the troposphere which is conditionally unstable at lower levels and stable at higher levels, the conditional stability increases at lower levels and decreases at higher levels, and the vertical mean conditional stability may decrease or increase. When parcel kinetic energy is converted from a part of latent heat, about 29% of the rest latent heat released is converted into geopotential energy according to the linear correlation between heat and geopotential energies in (7.33).
12.9. EQUIVALENT THERMO-STATIC ENTROPY LEVEL
267
So, the vertical mean static stability may also decrease after energy conversion. The change of static stability or conditional stability is different from that in the dry atmosphere, since the stability can only increase for creating kinetic energy in the dry processes. However, if the equivalent baroclinity is strong, the conditional stability or static stability may increase after development of moist disturbances, and so a part of total potential energy together with the latent heat released is converted into kinetic energy. The destruction of equivalent barotropic entropy in the process is then recovered by increasing the equivalent baroclinic entropy.
12.9
Equivalent thermo-static entropy level
Like the baroclinic entropy in the dry atmosphere, the equivalent baroclinic entropy measures the ability of mechanic energy generation related to the horizontal inhomogeneity in the equivalent potential temperature field. Since the equivalent potential temperature depends on humidity as well as temperature, a barotropic atmosphere may also possess negative equivalent baroclinic entropy, if there is horizontal gradient in the humidity field. More kinetic energy may be created by destroying the gradients in humidity and temperature fields as discussed earlier. The humidity front is similar to the temperature from for energy conversion in the moist atmosphere, except that the effect is generally smaller. It is argued in Chapter 8 that the ability of energy conversion related to vertical disorderliness in the atmosphere may not be represented by thermodynamic entropy, since change of thermodynamic entropy may not include the effect of geopotential energy conversion in the gravitational field. Thus, we introduced the static entropy to measure the ability for the dry atmosphere. Similarly, the conditionally stable and unstable atmospheres possessing different abilities of energy conversion may have an equal amount of thermodynamic entropy, and so the thermodynamic entropy may also not measure the strength of energy source in the moist atmosphere. Thus, we introduce the equivalent static entropy defined as
Sese =
−Set Conditionally stable Conditionally unstable Set
to represent the ability of energy conversion in the moist vertical convection. To illustrate the gravity effect on energy conversion, the equivalent static entropy changes sign from the conditionally stable atmosphere to unstable atmosphere. It is identical to the equivalent barotropic entropy in the conditionally unstable atmosphere, but is negative equivalent barotropic entropy in the conditionally stable atmosphere. Thus, the equivalent static entropy is positive in the stable atmosphere and is negative in the unstable atmosphere. It increases with the conditional stability. The energy conversion ability of the saturated atmosphere may be measured by the equivalent thermo-static entropy level defined as Sesl = Sec + Sese . The change of equivalent thermo-static entropy level is given by ∆Ses = ∆Sec + ∆Sese ,
12. MOIST PROCESSES OF ENERGY CONVERSION
268 or
∆Sesl = −Sec0 + Sesr − Ses0 ,
(12.11)
assuming that the reference state is equivalently barotropic. The thermodynamic entropy change in the process reads
∆S =
−Sec0 + Ses0 − Sesr −Sec0 − Ses0 − Sesr
Conditionally stable . Conditionally unstable
(12.12)
Using this equation to eliminate Sesr yields
∆Sesl =
−2Sec − ∆S Conditionally stable −2Sec − 2Ses0 − ∆S Conditionally unstable
.
(12.13)
If no water vapor condensation takes place in a conditionally stable atmosphere, the dry processes may be studied with the change of thermo-static entropy discussed in Chapter 8. In the conditionally unstable and saturated atmosphere, the change of equivalent thermo-static entropy is larger than the change of thermodynamic entropy, since ∆Sesl − ∆S = 2Sesr derived from (12.11) and (12.12). Thus, equivalent thermo-static entropy increases in the moist processes. While, the thermodynamic entropy may be conserved, especially in the moist baroclinic processes.
Chapter 13 Available enthalpy in the atmosphere 13.1
Introduction
The horizontal fields of temperature, pressure and humidity in the atmosphere are inhomogeneous and zonally asymmetric in general. So, the atmospheric disturbances occur preferentially in some particular regions. For example, the baroclinic cyclones displace along certain paths called the storm tracks in the extratropical regions, where the fronts of temperature occur most frequently. The regional properties of cyclonic activities were studied by Petterssen (1956), Astapenko (1964) and many others (e.g., Zishka and Smith, 1980; Hanson and Long, 1985; Lau, 1988; Bell and Bosart, 1989). Fig.13.1 shows the climatological distributions of extratropical cyclones and anticyclones in the Northern Hemisphere winter. In general, the cyclonic activities are concentrated at middle latitudes. Evaluations of Wiin-nielsen and Chen (1993) showed that kinetic energy generation in the Northern Hemisphere is maximum around the midlatitudes. The low, middle and high latitudes used in this study refer to the magnitude of latitude, ignoring the negative sign for the Southern Hemisphere. The traditional linear theories of baroclinic instability have proved that the stable baroclinic waves may be destabilized by increasing the three-dimensional gradients of temperature in the atmosphere. Since the available enthalpy studied in Chapters 9 and 10 increases with baroclinity and static instability, it can be assumed that development of large-scale extratropical cyclones is supported by the baroclinic energy source, and there may be a positive correlation between the climatological distributions of baroclinic storm tracks and high available enthalpy in the troposphere. Development of baroclinic disturbances is a normal way to convert the available enthalpy into parcel kinetic energy and destroy the air mass boundaries in the atmosphere. The intensity and lifetime of baroclinic disturbance depend on the strength of baroclinic kinetic energy source. The distribution of blocking system in the atmosphere manifests regional properties too (Rex, 1950; van Loon, 1956; Blackmon, 1976; Lejen¨ as and ∅kland, 1983; Lejen¨ as, 1984; Trenberth and Mo, 1985; Dole, 1986). Fig.13.2 shows the longitudinal variation of blocking frequency in the Northern Hemisphere winter. The relatively high frequencies occur over the centers of North Atlantic and North Pacific in winter. As to the meridional distribution, the frequency is peaked at latitudes 54◦ -56◦ N in winter and 64◦ -66◦ N in summer reported by Treidl et al. (1981). According to McHall (1993), development of blocking system depends on baroclinic instability too, so there may also be the climatological correlation between the blocking highs and energy sources. Up to date, there are only a few studies on the global distribution of kinetic energy sources evaluated with the algorithms different from the variational approaches discussed in this study. Wiin-nielsen and Chen (1993) calculated the time mean 269
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13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
Figure 13.1: Distributions of mean maximum wind (solid curve), maximum occurrence frequencies of cyclone (short arrows) and anticyclone (double-shifted arrows) in winter. Rectangles show mean sea-level pressures in major semipermanent low and high centers. Ellipses show regions of maximum upper winds. (After Palm´en and Newton, 1969)
eddy available potential energy for the Northern Hemisphere. The result showed the strongest maximum over eastern Asia together with two secondary maxima over the North Atlantic and Gulf of Alaska respectively. The zonal mean position of the maxima is around 75◦ N. This picture does not agree with the geographic positions of major baroclinic storm tracks at middle latitudes. The global distribution of maximum available enthalpy in the atmosphere will be evaluated using the six-year data collected twice a day at 00Z and 12Z, from June 1991 to May 1997, provided by the European Centre of Medium-Range Weather Forecasts (ECMWF). The lowest states applied for the evaluations are derived with the precise variational algorithm discussed in Chapter 9. The seasonal mean maps illustrated in this chapter are compared with the climatological distributions of baroclinic storm tracks and blocking systems. The depicted values should not be used directly to estimate the kinetic energy generation, since the energy conversions
13.1. INTRODUCTION
271
Figure 13.2: Frequencies of block in the Northern Hemisphere, (a) winter and (b) summer. (After Treidl et al., 1981)
in the atmosphere are not isolated processes and depend on thermodynamic entropy production. We discussed in Chapter 10 the pressure change in the lower troposphere after energy conversion, according to the change of thickness of the atmosphere below a certain height. For a closed cyclonic system, the pressure change may also be estimated according to the gradient wind balance. The algorithm will be discussed in this chapter, assuming that the kinetic energy created is transferred into horizontal wind. Since there are kinetic energy dissipation and export in the local atmosphere, the prediction of surface pressure change is complicated. None of the algorithms discussed in Chapter 10 and in this chapter is perfect. It is discussed in Chapter 8 that the ability of energy conversion in the atmosphere can be measured by the negative thermodynamic entropy, called the thermostatic entropy, which is the sum of baroclinic entropy and static entropy. We plot
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13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
also the six-year climatology of the baroclinic entropy in this chapter. Although change of the baroclinity only may not produce kinetic energy as discussed in Chapter 10, the energy source depends highly on baroclinic entropy. Thus, the general pattern of baroclinic entropy distribution is correlated negatively to that of available enthalpy distribution. Since the energy source depends also on the static stability, the major differences between the fields of available enthalpy and baroclinic entropy may be found in the regions with extremely low or high static stability. The distributions of baroclinic energy source are zonally asymmetric especially in the Northern Hemisphere, as the temperature and its gradient in the lower troposphere depend highly on the topography and boundary forcing. We shall see also the latitudinal dependence of zonal mean available enthalpy and baroclinic entropy. The baroclinic energy sources are generally weak in the tropical and subtropical regions, and is strangest at middle latitudes in each hemisphere except in the Southern Hemisphere winter. The lack of energy peak in the southern winter suggests the significance of topography for construction of available enthalpy at middle latitudes. Owing to the difference of heat capacity over lands and oceans and exchange of permanent day and night near the antarctic, the seasonal variations of baroclinic entropy is maximum there and are more remarkable than the variations of available enthalpy. The maximum available enthalpy is calculated with the lowest state attained through the extremal process with minimum production of thermodynamic entropy and maximum production of geopotential entropy. The minimum thermodynamic entropy produced may be zero in the baroclinic regions at middle and high latitudes, but may be not at low latitudes or near the Antarctic, where the baroclinity is low or static stability is high. The maximum available enthalpy in the regions with high thermodynamic entropy production may be relatively low, and the processes of energy conversion are characterized by sudden changes and strong turbulences. The maps of minimum thermodynamic entropy production are also demonstrated in this chapter. To see the main features of the lowest states in the atmosphere, we plot the maps of vertical mean static stability of the lowest states. The climatological mean atmosphere is significantly different from the lowest states except in the tropical regions, since the atmosphere possesses a large amount available enthalpy in the normal situations. The energy conversion in the rotational atmosphere is restricted by the geostrophic balance and hydrostatic equilibrium. Thus, the development of disturbance accompanied by energy conversion in the atmosphere depends on the mechanic instabilities which may break down the dynamic equilibriums. The most well know mechanic instabilities are the baroclinic instability and static instability, which may cause conversion of available enthalpy into kinetic energy but not only change a previous flow pattern. It will be discussed in Chapter 19 that these two instabilities are related to each other in the baroclinic atmosphere. According to McHall (1993), the time and zonal mean horizontal temperature gradient is slightly stronger than the threshold temperature gradient for the baroclinic instability of synoptic geostrophic waves. Thus, the extratropical circulations are characterized by the kinematic perturbations called the geostrophic wave by McHall, and the climatology of baroclinic disturbance development manifests the zonally asymmetric regional features.
13.2. IN THE NORTHERN HEMISPHERE
273
13.2
In the Northern Hemisphere
13.2.1
Distributions in winter and summer
Fig.13.3 shows the six-year climatology of maximum available enthalpy in Northern Hemisphere winter (December-February) and summer (June-August). These maps are plotted using the semi-daily data of available enthalpy evaluated in the air columns centered at each grid point. The columns extend from the surface up to 400 hPa with horizontal section area of 20 degrees latitude by 2000 kilometers in the zonal direction. The applied temperature data are provided by ECMWF on the standard pressure surfaces of 1000, 925, 850, 700, 500 and 400 hPa, and at the grid points of every 2.5◦ of latitude and longitude. The surface pressure is used to give the lower boundary conditions for the integrations. Assuming that the lapse rate of temperature between two standard pressure surfaces i and i + 1 is constant, we interpolated the temperature into every 25 hPa level with
T = Ti
p pi
γ
,
γ=
ln TTi+1 i ln pi+1 pi
,
(pi+1 ≤ p ≤ pi ) .
So, the integrations in (9.13), (9.7) and (9.18) are given by pj+1
θ dp = pj
pj+1
Tj pκθ γ−κ+1 − pγ−κ+1 , γ pj+1 j (γ − κ + 1)pj
pκ θ dp =
pj
Tj pκθ γ+1 γ+1 , γ pj+1 − pj (γ + 1)pj
and
pj+1 pj
ln θ dp = (pj+1 − pj ) ln
Tj pκθ + (γ − κ)(pj+1 ln pj+1 − pj ln pj − pj+1 + pj ) , pγj
in which pj+1 − pj = 25 hPa. The depicted specific available enthalpy is the mean value over the air column. Fig.13.3 shows that the baroclinic energy sources in the troposphere are generally elongated along a temperature front. There are two major bands of high available enthalpy across the Eurasia continent and North America respectively in winter. The former extends eastward to the central region of North Pacific, and the latter is in a ‘U’ shape with the right arm stretching along the east coasts of North America and Iceland. It is interesting to note that the high centers in the winter map are located on lee sides of North America and Asia continents, the Green land, the Rocky and Tienshan mountains, respectively. The first two off the major continents are the strongest. The high available enthalpy over oceans rides on the maximum gradient of climatological mean sea surface temperature resulting from confluent of warm and cold currents (referring to Peixoto and Oort, 1992).
13.2.2
Relation to extratropical cyclones
The large-scale wave-like stable perturbations in the atmosphere may become unstable if the baroclinity is sufficiently strong. The typical example of the baroclinically
13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
274
(a)
(b) Figure 13.3: Six-year climatology of maximum available enthalpy (m2 /s2 ) in the Northern Hemisphere, (a) December-February and (b) June-August
13.2. IN THE NORTHERN HEMISPHERE
275
destabilized large-scale circulations in the lower troposphere is the extratropical cyclones developed in frontal zones. The linear theory of baroclinic instability has been studied by McHall (1993), and the dynamic mechanism of the instability will be studied in Chapter 19 with the theory of slantwise convection introduced. From the energy conversion point of view, there must be a correlation between baroclinic storm tracks and the energy sources in the troposphere, as development of baroclinic disturbance is accompanied by conversion of available enthalpy into kinetic energy. This correlation can be found by comparing Figs.13.3 and 13.1. As the energy source off the east coast of Asia is stronger than that off the east coast of North America, the wintertime cyclonic activities are more frequent over northwestern Pacific (Petterssen, 1956). The temperature fronts in the troposphere tilt upward and poleward, so the surface fronts are usually on the equatorside of high available enthalpy. Also, the humidity is generally higher on the equatorside than on the polarside. Figs.13.3 and 13.1 show that the storm tracks over the two continents and North Atlantic at middle and high latitudes lie roughly on the equatorside of the energy sources. This can be seen also from the maps of cyclogenesis or cyclone center frequency plotted by Petterssen (1956). Since the equatorside of the energy source over the North Africa deserts is relatively dry, the storm track therein is on the polarside over the Mediterranean Sea. This fact suggests the importance of moisture for development of extratropical disturbances. In Fig.13.6.1 and Fig.13.6.2 of Petterssen (1956), there are the centers of high frequencies of cyclogenesis and cyclone centers on the lee side of Rocky Mountains in winter. This high frequency may not only be a response to the mechanic influence of topography, since baroclinic energy source in the place is strong as shown in Fig.13.3(a). However, the mechanic influence may also be important, as the frequency of lee cyclogenesis is higher than that off the east coast of America according to the statistics of Petterssen, while the energy source on the lee side is weaker than that off the east coast. The dry energy sources depend highly on thermal forcing in the lower troposphere. The surface temperature over oceans is generally colder than over continents at given latitudes in summer. If this temperature contrast decreases at high levels, the static stability over continents below these levels is lower than over oceans. Also, the strong gradients of sea surface temperature decrease in summer. Fig.13.3(b) shows that the baroclinic energy sources at middle latitudes of Northern Hemisphere are weakened in summer especially over oceans, so that the maxima of available enthalpy remain mainly over continents with reduced intensities. According to the climatology of Haurwitz and Austin (1944), there is a polar front over the North Pacific in winter, which retreats to the summer position at higher latitudes over East Asia and North Europe. Also, the arctic front, disappearing from its winter position over Gulf of Alaska, is found in European continent at high latitudes in summer. The polar front over the North Atlantic at lower middle latitudes displaces backward to middle latitudes over North America. Since the seasonal change does not simply reverse the phase of available enthalpy distribution over oceans and continents, the annual mean distribution exhibits high values over major continents and along the east coasts (maps are not shown).
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13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
From the statistics of Petterssen (1956), the most active regions for summertime cyclogenesis over the West Europe and west coast of North America are correlated to the strongest energy sources. Meanwhile, the frequency of cyclogenesis over the North Pacific and North Atlantic decreases from winter to summer, as the energy source is weakened there. However, if compared with the map of cyclone center frequency in summer (Petterssen, 1956), the high frequencies at lower middle latitudes, such as over the South Asia, North Africa and Mexico, do not coincide with high available enthalpy. This implies that development of summertime disturbances at lower middle latitudes does not only depend on the baroclinity. We shall see in the next chapter that they are related more likely to the moist energy sources.
13.2.3
Relation to blocking systems
The baroclinic instability may also destabilize the planetary perturbations in the middle and upper troposphere. The typical circulation pattern of the unstable perturbations over the middle troposphere is the blocking highs which are different from the moving cyclones below. According to McHall (1993), the geographic position of blocking depends highly on the topography, and the systems are not baroclinically active in energy conversion and may be maintained by weak baroclinic flows. Also, the scale of a blocking wave is comparable with that of a storm track, and is larger than a single cyclone. Thus in general, the blocking systems do not travel as far as the frontal cyclones do in the lifetime. For these reasons, the blocking regions may not be in the centers of energy source. Comparing Figs.13.3 and 13.2 finds that the climatological positions of high blocking frequency over the North Atlantic and North Pacific are generally on the edges of high available enthalpy centers. Moreover, Fig.13.2 shows that the major energy source in the blocking region over the North Atlantic is stronger than that over the North Pacific. This may be the reason for the higher blocking frequency over the North Atlantic. From winter to summer, the blocking positions shift north-eastward as shown by Fig.13.2 and other studies (e. g., Lejen¨ as and ∅kland, 1983). This position change is related to the eastward displacement of planetary stationary wave phases (McHall, 1993) and poleward displacement of the fronts, so that the blocking regions are still close to the major energy sources in summer. For example, the one over the North Pacific moved eastward to the west coast of North America near an energy source there in summer. As the strength of energy source off the east coast of Green Land decreases greatly, the blocking region over the North Atlantic displaced to North Europe in summer. In general, the energy sources are weaker in summer than in winter especially over the European continent, so the summertime blocking frequency is lower than the wintertime frequency .
13.3
In the Southern Hemisphere
The Earth’s surface in the Southern Hemisphere is covered mostly by ocean water. Meteorological data collected from observations may not be as abundant as in the Northern Hemisphere. However, the distributions of maximum available
13.4. DEVELOPMENT OF LOW SYSTEM
277
enthalpy in the Southern Hemisphere displayed in Fig.13.4 also exhibit the basic relations to storm tracks and blocking systems discussed. This figure shows that the energy sources are less zonally asymmetric than in the Northern Hemisphere. A band of maximum available enthalpy surrounds the globe discontinuously at middle latitudes. The central latitude is higher over the South Atlantic but lower in the Australian regions. The seasonal changes are smaller than in the Northern Hemisphere, with a slightly poleward displacement of the energy source from winter (June-August) to summer (December-February) over some regions. These different features imply that the energy sources depend highly on surface conditions. As in the Northern Hemisphere, the climatological position of polar front on the surface plotted by Haurwitz and Austin (1944) lies generally on the equatorside of the potential energy band. Since the energy sources are nearly zonal at the southern middle latitudes, the cyclones therein displace eastward further away from their origins compared with the cyclones in the Northern Hemisphere, and draw the long inter-ocean tracks (Astopenko, 1964; Physick, 1981). For example, the baroclinic cyclones may move from the east coast of South America eastward across the South Atlantic and Indian Ocean to the antarctic coast south of Australia, or from the east coast of South Africa eastward across Indian Ocean and southwestern Pacific through to the region of Amundsen Sea. The regions of high blocking frequency in the Southern Hemisphere are also in the vicinity of energy centers. For example, a maximum frequency of blocking occurs in the South Pacific Ocean near New Zealand. Since the energy source in the region is stronger in winter than in summer, blocking events take place more frequently in the winter season. Another two blocking regions are found, respectively, near the southern tips of South Africa and South America in the southwest Indian Ocean and southwest Atlantic or southeast Pacific regions (Trenberth, 1980; Coughlan, 1983; Lejen¨ as, 1984). Following the displacement of maximum available enthalpy from winter to summer, the regions of anticyclone displace poleward too, especially in the Pacific and Indian Oceans (Lejen¨ as, 1984).
13.4
Development of low system
It is discussed in Chapter 10 that energy conversion in the troposphere enhances the thickness of the atmosphere and so reduces the pressure in the lower troposphere. The decrease of low level pressure may not be uniform over the area of energy conversion. We discuss in the following the surface pressure change over a closed cyclonic system according to the gradient wind balance. The gradient wind is represented by
fr ± v=− 2
∂p f 2r2 − rα , 4 ∂n
where n is a vector normal to the flow with positive direction towards the left, and r is the radius of flow curvature. If the center of the curvature is in the positive direction of n, r is positive as for the cyclonic circulations in the Northern Hemisphere. For simplicity, we assume that the pressure gradient is constant, so we have
13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
278
(a)
(b) Figure 13.4: Six-year climatology of maximum available enthalpy (m2 /s2 ) in the Southern Hemisphere, (a) June-August and (b) December-February
13.5. BAROCLINIC ENTROPY
279
r∂p/∂n = ∆p, where ∆p measures the pressure difference at a given height over the distance r in the direction of n. The previous equation is replaced by
fr ± v=− 2
f 2r2 − α∆p . 4
It follows that
v ∆p = − (f r + v) . α If all kinetic energy created is converted into horizontal wind in a closed cyclone, we have also v = 2∆k + v02 . The last two equations give
∆p = −
2∆k + v02 α
(f r +
2∆k + v02 ) ,
(13.1)
in which the initial wind speed is evaluated from
fr ± v0 = − 2
f 2r2 − α∆p0 . 4
If r represents the radius of cyclone, this relationship tells that the pressure variation is proportional to the size of cyclone in the current example. For r = 1000 km, and α=R
280K T =R , p 1000hPa
the evaluated central pressure variation at latitude 45◦ is shown in Fig.13.5. In this figure, the pressure difference ∆p assumed at an initial time is found at ∆k = 0. Fig.13.5 shows that the central pressure decreases in the process of kinetic energy generation. To get a realistic prediction, the calculations should be corrected by kinetic energy export and dissipation. Since the energy conversion may produce angular momentum around the low center, the angular momentum or vorticity may not be constant. The potential vorticity may also be changed by mixing of air parcels in an isolated system.
13.5
Baroclinic entropy
We introduced in Chapter 8 the baroclinic entropy to represent the ability of energy conversion associated with horizontal gradient in the potential temperature field. It is negative in the baroclinic atmosphere and is zero in the barotropic atmosphere. The lower the baroclinic entropy is, the higher the ability will be in the atmosphere with a provided static stability. The baroclinic entropy increases, when the baroclinic atmosphere becomes barotropic after energy conversion. The amount increased in the process is the negative baroclinic entropy at initial state, that is ∆Sbc = −Sbc0 . The baroclinic entropy of the linear atmosphere assumed earlier is
280
13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
Figure 13.5: Pressure variations at cyclone center plotted in Fig. 8.1. It depends on initial state only and is independent of the static stability in the linear atmosphere. We discussed in Chapter 10 that the change of baroclinity may not produce kinetic energy, if the static stability is not changed. However, the negative baroclinic entropy is substantial for the energy conversion as discussed in Chapter 8. It is proved in Chapter 9 that the amount of available enthalpy increases greatly with the baroclinity. Using the six-year ECMWF data again, we have evaluated, from (8.21), the baroclinic entropy in the same air columns as for the evaluation of available enthalpy. The winter and summer patterns are displayed in Figs.13.6 and 13.7 for the Northern and Southern Hemispheres, respectively. The low centers occur generally in the regions where available enthalpy is high, especially in winter or in the Southern Hemisphere. The band of low baroclinic entropy around midlatitudes indicates the mean position of polar front. The minimum baroclinic entropy off the east coasts of North America and Asia is correlated negatively to the maximum sea surface temperature gradient there. According to McHall (1993), there are the troughs of orographic stationary waves off east coasts of major continents at middle latitudes in the Northern Hemisphere, while on the equatorside of the major troughs there are the ridges over subtropical regions. Thus, a strong westerly jet may occur between the high and low systems. In a thermal wind balance, there must be a strong horizontal temperature gradient below the jet core maintained by strong geostrophic flows. The ‘U’ shape zone with large amount of negative baroclinic entropy and positive available enthalpy across
13.5. BAROCLINIC ENTROPY
281
(a)
(b) Figure 13.6: Six-year climatology of baroclinic entropy (J/(K·kg)) in the Northern Hemisphere, a) December-February and b) June-August
282
13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
the North America and North Pacific is correlated to the smaller ‘U’ shape storm tracks there shown in Fig.13.1. Comparing Fig.13.6(a) with Fig.13.1 finds that the cyclones over lands at high latitudes are mainly the baroclinic disturbances. The negative correlation between baroclinic entropy and available enthalpy is found also in the Southern Hemisphere. Since the available enthalpy depends on the static stability as well as on the baroclinity, a distribution of baroclinic entropy, which is independent of static stability, may not be entirely correlated to distributions of available enthalpy. The main difference is found over the subtropical regions of negative available enthalpy, where the baroclinic entropy is not particularly high (referring to Fig.13.7). The low available enthalpy is related to the relatively high static stability. This will be discussed in Section 13.7. The major kinetic energy sources are weakened greatly from winter to summer in the Northern Hemisphere. The destruction of baroclinic entropy is more striking than decrease of available enthalpy, because the regions of high available enthalpy in summer are generally over continents and less statically stable than in winter. However, exceptions can be found over oceans. For example, there is a band of low baroclinic entropy across the North Pacific at middle latitudes in summer, while the energy source is relatively weak off the east coast of Japan, since the static stability over summer oceans is generally higher than over continents. The summertime cyclogenesis over the North America and off the east coast of Japan (Petterssen, 1956, Figs.13.6.3 and 13.6.4) occurs most frequently in the regions of low baroclinic entropy, although the occurrence of cyclone center may not be. This implies that the baroclinity required for baroclinic instability is stronger than that for development of destabilized perturbations. McHall (1993) elucidated that the baroclinic disturbances may develop continuously after the linear adjustment, when the baroclinity is lower than the threshold value for the baroclinic instability at the beginning of development.
13.6
Zonal mean distributions
Fig.13.8 displays the six-year zonal mean climatology of the maximum available enthalpy discussed previously. This figure shows that the available enthalpy can be the energy source of baroclinic disturbances at middle and high latitudes. While, the values in the tropical regions within 20 degrees of latitude are generally low. The zonal mean available enthalpy in the Northern Hemisphere winter is peaked at the lower middle latitudes. The seasonal mean position of polar front on the surface is on the equatorside of the peak. The strongest mean energy source shifts poleward about 10 degrees from winter to summer, following the poleward displacement of Hadley cell. The winter peak is more than twice as high as the summer peak in the Northern Hemisphere. Thus, the baroclinic disturbances are most active near the middle latitudes, and occur more frequently in winter than in summer. The summertime Northern Hemisphere has a secondary maximum at high latitudes, related to the energy sources over northeastern Asia and Canada as shown by Fig.13.3(b). It is interesting to compare the zonal mean baroclinic entropy shown in Fig.13.8(b) with the zonal mean available enthalpy. For an easy comparison, the
13.6. ZONAL MEAN DISTRIBUTIONS
283
(a)
(b) Figure 13.7: Six-year climatology of baroclinic entropy (J/(K·kg)) in the Southern Hemisphere, (a) June-August, and (b) December-February
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13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
(a)
(b) Figure 13.8: Zonal means of maximum available enthalpy (a) and negative baroclinic entropy (b) in December-February (light solid) and June-August (dashed). The six-year means are depicted by the heavy curves.
figure shows the negative baroclinic entropy. The negative baroclinic entropy is also low in the tropical regions, so the circulations at low latitudes are barotropic in general. The basic pattern in the Northern Hemisphere is similar to that of available enthalpy. However, the seasonal variation is larger than that of available enthalpy in the Northern Hemisphere, because the available enthalpy includes the important contribution from reducing the static stability over major continents in summer. So the disturbances manifest more convective features if compared with the wintertime disturbances. The zonal cross section of baroclinic entropy or available enthalpy in the Southern Hemisphere is different from that in the Northern, especially in the winter season. The baroclinity is nearly unchanged from 30◦ S to 60◦ S, so does the available enthalpy. The midlatitude peak found in Northern Hemisphere winter does not occur in the southern winter, since the strong baroclinity or baroclinic energy source at middle latitudes depends highly on the topography. As discussed earlier, the strong baroclinity at northern middle latitudes is necessary for thermal wind balance in the jet regions between the midlatitude topographic troughs and subtropical ridges. While, the nearly constant baroclinity at southern middle latitudes implies that the polar fronts in winter travel continuously across these latitudes. The largest difference between the zonal patterns of negative baroclinic entropy and available enthalpy is found near the south pole in winter. Owing to the high static stability over the antarctic regions, the peak and its seasonal change in the energy field are
13.7. LEAST THERMODYNAMIC ENTROPY PRODUCTION
285
much weaker than in the entropy field. When the mean Hadley cell over the Southern Hemisphere is pushed southward in summer, the baroclinity at lower middle latitudes is weakened. This seasonal change is similar to that in the Northern Hemisphere. Also, the equatorward temperature gradient in the boundary layer at the high latitudes reduces in summer, due to the difference of heat capacities over water and land across the antarctic coast. Thus, the baroclinity and baroclinic energy source in summer are strongest at middle latitudes, 20◦ north of the winter peak. For the same reason, the antarctic atmosphere is strongly baroclinic in winter. The summer peak of available enthalpy at middle latitudes is not as high as that of baroclinic entropy, because the static stability in the lower troposphere is slightly higher in summer than in winter over the middle latitude oceans. It can be expected that the baroclinic disturbances in the Southern Hemisphere are most active at middle latitudes in summer but at high latitudes in winter.
13.7
Least thermodynamic entropy production
Fig.13.8(a) shows that the available enthalpy is generally low in the tropical and subtropical regions, and so may not be crucially important for development of tropical disturbances. The values over oceans are even lower than over continents as shown by Figs.13.3 and 13.4. In general, the low available enthalpy in the subtropical regions appears off the west coasts of South Africa and South America all the year. The study of Peixoto and Oort (1992) shows that the sea surface temperature off the west coasts of major continents is generally lower than off the east coasts in the subtropical regions. So, the static stability is higher off the west coasts. Figs.13.6 and 13.7 show that the baroclinity is not very low in the regions of minimum available enthalpy. So the low or negative available enthalpy is related to the high static stability. Due to the thermal constraint (9.9), energy conversion in the atmosphere with high static stability may produce large amount of thermodynamic entropy. It is discussed in Chapter 10 that for a provided initial state, kinetic energy created is reduced by thermodynamic entropy production. Fig.13.9 gives the minimum thermodynamic entropy production as the atmosphere reaches the lowest state, averaged over the six years in the same air columns as those for the calculations of available enthalpy and baroclinic entropy discussed earlier. The high centers of the entropy production and the corresponding low available enthalpy occur just in the regions of minimum annual mean sea surface temperature (Peixoto and Oort, 1992) where the static stability is relatively high. The regions of high entropy production over oceans is not centred over the equator, as the energy conversion depends highly on the local properties of topography. The low available enthalpy depicted in Figs.13.3 and 13.4 gives the mean value averaged in the columns from the surface up to 400 hPa. In a weak energy source, positive available enthalpy may be found only in the lower troposphere or near the surface over the subtropical regions, while the atmosphere above is not a baroclinic energy source. We also evaluated the maximum available enthalpy in the atmosphere from the surface up to 200 hPa. The positive values appear only at middle and high latitudes, and the minimum entropy production is generally high at low latitudes.
13. AVAILABLE ENTHALPY IN THE ATMOSPHERE
286
(a)
(b) Figure 13.9: Six-year mean least thermodynamic entropy production (J/(K·kg)) in the atmosphere from the surface up to 400 hPa for the (a) Northern Hemisphere and (b) Southern Hemisphere. The dashed curves indicate the regions of negative available enthalpy.
13.8. THE HIGHEST STATIC STABILITIES
287
The dry baroclinic disturbances developed in the regions with low available enthalpy cannot extend over a certain height. The blocking systems which may extend to the upper troposphere may only occur at middle and high latitudes.
13.8
The highest static stabilities
Kinetic energy generation depends on ageostrophic and vertical motions as shown by energy equation (7.1). While, the large-scale circulations in the rotational atmosphere are constrained to be quasi-geostrophic and hydrostatic equilibrium at middle and high latitudes. Conversion of available enthalpy in the rotational atmosphere is therefore prevented by geostrophic balance and hydrostatic equilibrium, and so the available enthalpy is only a potential energy source in the dry atmosphere. In this situation, the extratropical troposphere remains a large amount of available enthalpy, and the vertical stratification is different from that of the lowest state. Only in the tropical and subtropical regions where the Coriolis force is weak and so the ageostrophic motions are relatively strong, the baroclinity is generally low and the statically stable temperature profile is close to that of the lowest state. These features can be seen from Fig.13.10 which displays the six-year mean temperature lapse rate of the lowest states in the atmosphere below 400 hPa, evaluated from (10.5) using the two Lagrangian multiplies λ1 and λ2 obtained from the previous calculations of available enthalpy. Since the static stability of reference state increases with kinetic energy generation, the lowest states possess the highest static stability compared with the other reference states attained from the same initial states. It is noted that the lowest states are evaluated in the local air columns of limited horizontal section area. So, they are barotropic in the local areas but not on the globe. In fact, developments of baroclinic disturbances in the troposphere are local processes. Fig.13.10 shows that the six-year mean lowest state achieved by development of local disturbance is not zonally symmetric, especially in the Northern Hemisphere. A common feature in the both hemispheres is that the static stability increases towards the poles, due to increase of baroclinity at higher latitudes. While, the lapse rate of real troposphere is close to 6.5 K/km at all latitude. Since development of baroclinic disturbance is highly irreversible in the atmosphere, the static stability attained after disturbance development is lower than that of the lowest state plotted. In general, the lowest states are more statically stable near the centers of low baroclinic entropy or baroclinic energy source. This agrees with the discussions in Chapters 8 and 9. The stability of reference state depends also on the stability of initial state. For provided baroclinity at an initial state, the lowest state attained is also less stable if the initial state is less stable. This is the reason for the relatively low static stabilities of the lowest states occurring over the deserts and continents, such as in the North and South Africa and west coast of North America. In the southern subtropical regions with negative available enthalpy and highest thermodynamic entropy production off the west coasts of major continents, the static stability decreases rather than increases after energy conversion, as the geopotential energy increases when the potential enthalpy is conserved.
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(b) Figure 13.10: The least temperature laps rate (K/km) for the six-year mean lowest state of the troposphere in the (a) Northern Hemisphere and (b) Southern Hemisphere
Chapter 14 Available moist enthalpy in the atmosphere 14.1
Introduction
We have discussed in the preceding chapter that the geographic positions of baroclinic storm tracks at middle and high latitudes especially in winter may be explained by the climatological distributions of dry available enthalpy and baroclinic entropy. While, the storm tracks at lower middle latitudes in summer may not correlate to the baroclinic energy sources. Also, the meso-scale thunderstorms and tropical cyclones occur mostly in the tropical and subtropical regions, where available enthalpy is generally low. To find the energy sources of these moist disturbances, we calculate in this chapter the maximum available moist enthalpy in the atmosphere with the precise variational algorithm introduced in Chapter 11, using the six-year ECMWF data from June 1991 to May 1997. The available moist enthalpy includes the important contribution of baroclinity in the baroclinic atmosphere. When the humidity is relatively low and baroclinity is relatively high, the moist enthalpy depends highly on the baroclinity instead of the humidity. Thus, the distribution of large-scale moist energy source is similar to the distribution of large-scale baroclinic energy source at middle and high latitudes in winter. In this situation, the moist energy source may be regarded as the moist baroclinic energy source, which is the baroclinic energy source strengthened by the moisture included and its three-dimensional gradient. In the tropical and subtropical regions where the humidity is high and baroclinity is low, distributions of dry and moist available enthalpy are substantially different especially in the conditionally unstable atmosphere. The difference is also found clearly at middle latitudes in the summer hemisphere. As a small change in temperatures may lead to a large change in humidity in the saturated atmosphere, according to the Clausius-Clapeyron relationship discussed in Chapter 23, the field of moist energy source is more zonally asymmetric than the field of dry energy source, and manifests larger seasonal changes in both the intensities and positions. As the large-scale distributions of the moist and dry energy sources are similar in the extratropical regions, there is also a positive correlation between the baroclinic storm tracks and moist energy sources. The baroclinic cyclones developed over warm ocean waters may be more intense than those over lands. It is discussed in Chapter 12 that the irreversible moist processes may be more abrupt than the dry processes in the atmosphere. We shall see in this chapter that the explosive cyclones, or called the bombs (Sanders and Gyakum, 1980), occur mostly in the regions where the large-scale moist and dry energy sources overlap together almost. The amounts of dry or moist available enthalpy evaluated with the variational 289
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approaches depends on the horizontal scale of air columns. According to McHall (1993), the scales of quasi-barotropic disturbances at low latitudes are smaller than those of extratropical baroclinic cyclones. The typical time and space scales of the tropical and extratropical disturbances may also be explained by their energy sources. The meso-scale moist energy sources can be represented by the available moist enthalpy evaluated in meso-scale air columns. It will be found that the concentration of moist enthalpy are responsible for the thunderstorms and heavy precipitations occurred in the tropical and subtropical regions, and there is also the positive correlation between climatological positions of tropical storm tracks and meso-scale moist energy sources. According to the studies of McHall (1993), the heat and momentum may be transported by the planetary and large-scale geostrophic wave circulations with a small Rossby number in the statically stable extratropical atmosphere. While, the geostrophic circulations at low latitudes are weak and so cannot accomplish the poleward transport in order to maintain the balances there. Accumulation of heat and moisture in the lower troposphere reduces the static stability for moist convection, and makes the tropical circulations essentially different from the extratropical circulations. The heat, moisture and energy are then transported mainly by the direct Hadley cells with strong ageostrophic components induced by thermal forcing in the tropical regions. The difference between the tropical and extratropical tropospheres may also be represented by the typical strengths of the dry and moist energy sources. The zonal mean distribution of available moist enthalpy is near opposite in phase to that of dry available enthalpy, with a maximum in the tropical regions, while the zonal mean distributions of equivalent baroclinic entropy and baroclinic entropy are in the same phases. Thus, the strong moist energy sources in the tropical regions are contributed by relatively low conditional stability. The barotropic moist disturbances, triggered by conditional instability and characterized by strong vertical convection, occur mostly in the tropical and subtropical regions. While, the baroclinic disturbances, corresponding to the baroclinic instability and exhibiting the slantwise circulations, are found mainly at middle and high latitudes.
14.2
Distribution of moist energy sources
The lowest states derived from the variational approach discussed in Chapter 11 depends on vertical and horizontal scales of air columns, so the intensityof energy source, represented by the energy density, depends on the scales also. This implies that the atmosphere with a given baroclinity and static stability may provide different energy sources for the disturbances on different scales. Thus, a smallscale disturbance is generally weaker than a large-scale disturbance developed in the same isolated environment. The dry or moist available enthalpy evaluated in the air columns with a provided horizontal scales may be applied as the energy source of the disturbances on the particular scale. For convenience, this scale will be referred to as the scale of the energy source in this study. The six-year climatologies of the large-scale moist energy sources are demonstrated in Figs.14.1 and 14.2 for the Northern and Southern hemispheres respec-
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tively. The maximum available moist enthalpy are evaluated in the same air columns as for the evaluations of available enthalpy in the preceding chapter, except that the columns extend from 1000 hPa up to 200 hPa. The additional ECMWF data on 300, 250 and 200 hPa pressure surfaces are used, of which the time and space resolutions are described earlier. The values shown in these figures may be overestimated over highlands, since the surface pressure there may be less than 1000 hPa. The available moist enthalpy in extratropical regions of Northern Hemisphere winter is relatively high off the east coasts of major continents, and is relatively low off the west coasts (Fig.14.1(a)). The high centers on the east coasts extend westward over the continents, and form the elongated bands in various shapes. These features are similar to those of dry available enthalpy or baroclinic entropy shown in Fig.13.3 or Fig.13.6. However, the low centers off the west coasts are more striking, since the humidity and static instability are low in the regions with low sea surface temperature. Comparing with Fig.13.3 finds that the meridional gradient of humidity makes the moist energy sources shift more or less equatorward from the positions of dry energy sources. The large-scale moist energy sources are also very strong in the tropical and subtropical regions. For example, the maximum over the Caribbean Sea in winter is produced mainly by the moisture, since the available enthalpy and negative baroclinic entropy are not high as shown by Figs.13.3(a) and 13.6(a). The moist energy source depends highly on humidity at low latitudes, and is weak in a dry region. For instance, the available moist enthalpy exhibits a low center over the deserts in North Africa in winter, though the baroclinity and available enthalpy is not particularly low. The effect of baroclinity on moist energy sources is also clear at middle and high latitudes in the Southern Hemisphere winter (referring to Figs.14.2(a) and 13.4(a)). As in the Northern Hemisphere, the moist energy sources are very strong at low latitudes. However, distribution of available moist enthalpy in the subtropical and tropical regions is more zonally asymmetric and depends more on the features of the surface, if compared with the distribution of dry available enthalpy. For example, the strongest energy centers occur on South America and South Africa continents, because the surface air over continents is warmer and damper than over oceans and so the conditional stability is relatively low. The low centers off the west coasts are also obvious. While, the wintertime moist enthalpy distribution around the continent of Australia is an exception, as the island is not large enough and the latitudes are relatively high. The high center over the Central Pacific shown in Fig.14.2(a) is the tail of the moist enthalpy band extending southeastward from Asia continent along the South Pacific Convergence Zone ( SPCZ) in the northern summer (Fig.14.1(b)). Usually, the western part of SPCZ is merged into the Intertropical Convergence Zone ( ITCZ) over the West Pacific. Although there are convective disturbances in the ITCZ and SPCZ all the time, the strong energy sources may exist for long in the local regions. This fact suggests that construction of moist energy sources by collecting water vapor in the surface convergence zones is as efficient as destruction of these sources in moist disturbance developments. Thus, the moist cyclonic systems over oceans at low latitudes may have a lifetime much longer than that of dry systems living on
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(b) Figure 14.1: Six-year climatology of available moist enthalpy (m2 /s2 ) in the Northern Hemisphere, (a) December-February and (b) June-August
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(b) Figure 14.2: Six-year climatology of maximum available moist enthalpy (m2 /s2 ) in the Southern Hemisphere, (a) June-August and (b) December-February
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the baroclinic energy sources in extratropical regions. The summertime distribution at low and middle latitudes plotted in Figs.14.1(b) and 14.2(b) exhibits a more clear cut between oceans and continents, compared with the wintertime distribution, and is largely different from the distribution of dry available enthalpy in the both hemispheres. The available moist enthalpy is concentrated over major continents with low centers off the west coasts. However, the moist energy sources are also strong in the ITCZ over the West Pacific in the summer hemisphere as shown by Figs.14.1(b) and 14.2(b). The maximum values over continents at middle latitudes are doubled from winter to summer. Meanwhile the energy sources over oceans are not weakened. The increase of available moist enthalpy over continents is contributed mainly by increasing the humidity and conditional instability in the lower troposphere. The band of high moist enthalpy over the SPCZ is also clear in the season from December to February. According to the intensities of moist energy sources, more moist disturbances may occur over continents in the summer hemisphere than in winter especially at low and middle latitudes, while more baroclinic disturbances in extratropical regions may be found in winter. Although the humidity is higher in summer than in winter at middle and high latitudes, the available moist enthalpy off the east coasts of North America and Japan is higher in winter than in summer. Thus, the moist energy sources over the extratropical oceans depends highly on the baroclinity instead of humidity. The large-scale moist-baroclinic disturbances are more active over the northwestern Pacific and northwestern Atlantic in winter, but over East Asia and North America continents in summer. According to the observations, the explosive baroclinic cyclones over oceans occur most frequently in winter. Usually, development of disturbances in the atmosphere are related to certain dynamic instability, which leads to conversion of non-kinetic energies into kinetic energy. As the ability of energy conversion depends on disorderliness of the atmosphere, the disorderliness may also be applied to measure the dynamic instabilities. The instability related to the horizontal disorderliness is called the baroclinic instability studied by McHall (1993). The instability related to the vertical disorderliness is the static instability in the dry atmosphere and conditional instability in the moist atmosphere, including the Conditional Instability of the Second Kind (CISK, Charney and Eliassen, 1964). Some other instabilities, such as the barotropic instability (Kuo, 1949) and shear instability (Charney and Stern, 1962) studied in traditional meteorology, may also destroy original flow patterns but may not create kinetic energy, if the baroclinic instability or static instability does not happen simultaneously or subsequently. The baroclinic instability and static instability including the conditional instability are the most important instabilities for the meteorological processes in the atmosphere. The cloud-top entrainment instability, which is important in cloud entrainment (Emanuel, 1994), is a kind of static instability.
14.3
Relation to storm tracks
We have discussed in the preceding chapter that the extratropical cyclonic activities in the Northern Hemisphere winter are mainly the baroclinic disturbances. Since the moist enthalpy depends highly on the baroclinity in the extratropical atmosphere,
14.3. RELATION TO STORM TRACKS
295
Figure 14.3: Distribution of explosive cyclogenesis during three cold seasons (1976-1979) (After Davis and Emanuel, 1988)
these storm tracks are also correlated to the moist energy sources. For example, along with the moist energy sources, the storm tracks over Eurasia continent shown in Fig.13.1 start from the north coast of Mediterranean Sea extending eastwards to the South China Sea, then recurved north-eastward across the North Pacific to the Gulf of Alaska. The similar correlation is also found between the moist energy sources and storm tracks across the central United States and along the southeast coast of North America in winter. The moisture in the energy sources may intensify the baroclinic disturbances by reducing the static stability for moist convection. While, the storm tracks over Canada and on the west coast are related more likely to the baroclinic energy source around shown in Fig.13.3(a). In the regions of weak energy source over the North Africa, North Atlantic, Arabian Sea and west coast of Mexico, the frequency of cyclonic activity is generally low as shown by the statistical study of Petterssen (1956, Fig.13.6.2). From the Clausius-Clapeyron relationship of temperature and humidity given by (23.16), a temperature front may be accompanied with a strong humidity gradient in the regions of abundant water vapor, such as over a warm ocean water.
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According to the studies in Chapter 12, the three-dimensional inhomogeneities in the temperature and humidity fields may strengthen the energy sources and reduce the threshold static instability and humidity for development of moist disturbance. The strongest dry and moist energy sources off the east coast of Japan or east coast of North America in winter are in similar places and coincident with the regions of explosive cyclones (called the tropospheric bombs also) or explosive deepening positions reported in many studies (e. g., Sanders and Gyakum, 1980; Roebber, 1984). This can be seen by comparing Fig.14.1(a) with Fig.14.3 which gives the distribution of explosive cyclogenesis in the Northern Hemisphere. The important effect of latent heat on the bombs has been studied by many authors (e. g., Liou and Elsberry, 1987; Davis and Emanuel, 1988; Fosdick and Smith, 1991). The latent heat may have a greater effect on the storm tracks in the lower middle latitudes in summer than in winter, since the moist disturbances may only be supported by moist energy sources in the conditionally unstable atmosphere. For example, the summertime storm tracks over North America and South Asia reported by Petterssen (1956, Fig.13.6.4) occur generally in the regions of high available moist enthalpy, where available enthalpy is not particularly high (Fig.13.3(a)). This suggests that the baroclinic instability may not be substantial for development of moist disturbances in these regions. A relatively high frequency of cyclone centers in North Africa is found in summer but not in winter, since the moist energy source over Africa is centered in the summer hemisphere. The correlation between the moist energy sources and precipitation regions or tropical storm tracks will be discussed in the last two sections of this chapter. In the areas of low available moist enthalpy, such as in the North Pacific and North Atlantic in summer, the disturbance frequencies are also low (below 0.1/106 km2 in Fig.13.6.4 of Petterssen).
14.4
Tropical and extratropical tropospheres
An important difference between tropical and extratropical circulations is the Rossby number defined as V , Ro = fL where V is the typical wind speed over horizontal scale L, and f = 2Ω sin ϕ is the Coriolis parameter. The Rossby number is much less than unity in the extratropical large-scale circulations constrained by quasi- geostrophic balance. The typical planetary and large-scale circulation patterns in the extratropical regions, such as the planetary stationary waves and low frequency variability or blocking systems, can be studied with the theory of geostrophic wave circulations (McHall, 1993). The heat and momentum transport requested by local or global balances may be accomplished by the geostrophic circulations including large-scale cyclones in the extratropical atmosphere. The Coriolis parameter f is one order smaller in the tropics than in the extratropical regions. The Rossby number shows that the quasi-geostrophic circulations in the tropical regions are characterized by weak winds over a planetary scale. The slow planetary circulations cannot accomplish efficiently the transport of heat, moisture and momentum requested for local and global balances, and so cannot be the
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(b) Figure 14.4: Zonal means of maximum available moist enthalpy (a) and negative equivalent baroclinic entropy (b) in December-February (light solid) and June-August (dashed). The six-year mean is depicted by the heavy curve.
prevailing circulation pattern at low latitudes. It is discussed in the preceding chapter that the tropical circulations are barotropic in general and not supported by the baroclinic available enthalpy. The energy source is the available moist enthalpy plotted in Fig.14.4(a) which displays the zonal mean of maximum available moist enthalpy in the six years. Comparing with Fig.13.8(a) finds that the water vapor strengthens the energy source at all latitudes. The available moist enthalpy is peaked at low latitudes in the summer hemisphere, where the available enthalpy is minimum. It decreases rapidly towards the two poles, unless at lower middle latitudes in the Northern Hemisphere or in the tropical and subtropical regions of summertime Southern Hemisphere where the meridional change is small. The summer or winter patterns in the two hemispheres are significantly different, and the annual mean is highly asymmetric to the equator with a peak around 10◦ N. Although the ocean area in the Southern Hemisphere is larger than in the Northern, the Northern Hemisphere possesses more available moist enthalpy. The moist energy source depends on three-dimensional inhomogeneities in the temperature and humidity fields. The horizontal inhomogeneities may be measured by the equivalent baroclinic entropy discussed in Chapter 12. Fig.14.4(b) illustrates the zonal means of the negative equivalent baroclinic entropy calculated in the same air columns as for the evaluations of available moist enthalpy. The difference from the mean baroclinic entropy plotted in Fig.13.8(b) is produced by horizontal gradient of water vapor. The magnitude of equivalent baroclinic entropy is generally larger
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than that of baroclinic entropy except near the poles in winter. From this figure, the tropical troposphere is almost equivalently barotropic. However, the equivalent baroclinity in the tropics is nevertheless stronger than the baroclinity due to the gradient of humidity. The peak around 25◦ S in winter (Fig.14.4(a) dashed) is raised sharply by the moisture, The other peaks at middle latitudes are also higher than those of negative baroclinic entropy. Unlike the comparison between the negative baroclinic entropy and available enthalpy shown in Fig.13.8, the zonal mean phase of negative equivalent baroclinic entropy are nearly opposite to that of available moist enthalpy at low latitudes. This suggests that the strong moist energy source in the tropical regions depends mainly on vertical gradient of the equivalent potential temperature represented by conditional instability, instead of horizontal inhomogeneities in the temperature and humidity fields. Rasmussen (1979) pointed out that a division between tropical meteorology and mid-latitude meteorology was often based on the fact that the low tropical troposphere was usually conditionally unstable, whereas the extratropical troposphere is generally conditionally stable. The conditional instability is constructed by the strong vertical gradient of humidity in the lower troposphere, while the vertical temperature stratification has less influence as shown by comparing Fig.13.8(a) with Fig.13.8(b). Both the available enthalpy and negative baroclinic entropy are low in the tropical regions. The Rasmussen’s criterion implies that the tropical circulations can be independent of extratropical circulations from the energy point of view. The dynamic instability which leads to conversion of available moist enthalpy is the conditional instability in the vertical moist convection. Meanwhile, the extratropical circulations have different energy sources, associated with the baroclinic entropy which is usually one or two orders higher in amount than that in the tropical atmosphere. The baroclinic instability which leads to conversion of available enthalpy in the dry processes is also a major difference between the tropical and extratropical meteorologies. Owing to the differences between the dry and moist energy sources, the extratropical and tropical disturbances are driven by different dynamics, and are different in their space scales, lifetimes, flow patterns and thermal structures. For example, conversion of available enthalpy in the dry atmosphere may take place in the large-scale slantwise convection, of which the trajectory slope is less than that of isentropic surface crossed. Since the slope of isentropic surface is generally small, the horizontal scale of geostrophic disturbance is large and the vertical motion is weak compared with those in meso- and small-scale convective storms. The Rossby number is small in the large-scale circulations in the extratropical atmosphere, so the air motions are constrained by geostrophic balance and thermal wind balance. As conversion of dry available enthalpy is much quicker than the formation, the lifetime of baroclinic cyclones is relatively short, if compared with that of hurricanes developing over oceans. When a baroclinic cyclone moves along with a strong energy source such as a front, the moving speed increases with the speed of energy conversion (referring to Section 15.5). Thus, the baroclinic cyclones may move quickly at the developing stage, and slowly after the mature stage. While, conversion of available moist enthalpy in a conditionally unstable atmosphere is brought about by vertical convection, so the horizontal scale of quasi-
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barotropic disturbances at low latitudes is smaller than that of baroclinic cyclones in the extratropical regions (McHall, 1993). Since the moist energy sources at low latitudes are generally stronger than the dry energy sources in the extratropical regions, the wind speed and acceleration in tropical and subtropical moist disturbances are higher than those in the extratropical cyclones. Moreover, since the Coriolis parameter f is small at low latitudes, the Rossby number of low latitude disturbances is relatively large, and the tropical circulations may possess a strong ageostrophic component. It will be discussed in the next chapter that the moist energy source may be strengthened by collecting moisture in the lower troposphere. The collection can be very efficient over warm ocean water, so that the cyclonic systems over the tropical oceans like the hurricanes may sustain for weeks and move slowly or be quasi-stationary for a few days.
14.5
Relation to thunderstorms
We have discussed in Chapter 11 that the available moist enthalpy increases greatly with the conditional instability. So development of strong convective systems is favoured in the areas with high moist enthalpy. This can be confirmed by comparing the six-year mean available moist enthalpy in Fig.14.5 with Fig.14.6 which shows the global distribution of thunderstorm frequency. Since the horizontal scale of thunderstorms is smaller than that of baroclinic cyclones, the moist enthalpy shown in Fig.14.5 is evaluated in the air columns from 1000 hPa to 200 hPa with the horizontal area 10 degrees of latitude by 1000 km of zonal distance. The thunderstorms occur most frequently over the tropical and subtropical continents. The major regions with a frequency higher than 40/year are generally within the areas with the moist enthalpy over 300 m2 /s2 , such as in the North and South America, South Africa and South Asia. The positive correlation is also found over the SPCZ at low latitudes. As the meso-scale energy sources along the SPCZ are stronger in the southern summer than in the winter (referring to Fig.14.8), the frequency in SPCZ is higher in summer. Due to the Coriolis circulation and upwelling in the oceans, the annual mean sea surface temperature is relatively low off the west coasts of South America and South Africa (Levitus, 1982). The air temperature and humidity over the surface are also relatively low in these regions. These negative temperature and humidity deviations from the zonal means disappear at higher levels (Newell et al., 1972), and so the conditional stability is relatively high off the west coasts. As a result, the available moist enthalpy is low and thunderstorms are few there. While, the sea surface temperature off the west coast of Australia is not considerably lower than the zonal mean. The relatively low available moist enthalpy in this region is found in summer (referring Fig.14.8(a)), which could be produced by atmospheric circulations. As discussed in the study of McHall (1993), there are stationary waves in the subtropical regions associated with linear response to the thermal forcing on the lower boundary. These waves manifest poleward flows at low levels and equatorward flows at high levels over the west coasts of continents in summer. This vertical wind shear can be confirmed by the statistical survey of Newell et al. (1970). It may reduce vertical gradient of the temperature or increase the
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14.5. RELATION TO THUNDERSTORMS
Figure 14.6: Global distributions of thunderstorm frequency (After Ayoade, 1983)
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static stability, and so reduce the available moist enthalpy. Since this flow pattern disappears in winter, the asymmetric distribution over the west coast of Australia is not remarkable in Fig.14.8(b). The same mechanism may also weaken the moist energy sources over the west coasts of other continents in the subtropical regions during summer.
14.6
Relation to precipitation
As the moist disturbances may produce a great amount of precipitation, there may be the correlation between the climatological distributions of heavy precipitation and high available moist enthalpy. Fig.14.7 gives the maps of precipitation rates in winter and summer. The climatology of precipitation in the tropical and subtropical regions can also be referred to the study of Newell et al. (1974). The available moist enthalpy evaluated in the meso-scale air columns is displayed in Fig.14.8. This figure is similar to Fig.14.5 except showing the six-year climatologies in winter and summer. Distributions of the meso-scale moist energy sources are different from those of large-scale displayed in Figs.14.1 and 14.2. For example, the large-scale energy sources around middle latitudes disappear in Fig.14.8, and the meso-scale energy sources are concentrated in the tropical and subtropical regions, which may extend poleward to lower middle latitudes over the continents in summer. Thus, the storm tracks in the extratropical regions are mainly the large-scale baroclinic disturbances, while the meso-scale moist disturbances occur mostly in the tropical and subtropical regions, which may extend northward to the Asia monsoon area, southern China, North Africa and southern United States in summer. In general, the largest precipitations in the tropical and subtropical regions occur in the strong convergent zones of surface wind, such as in the ITCZ and SPCZ over oceans. As the surface temperature, humidity and so the conditional instability over land are higher than over ocean in summer, the precipitation centers may also occur over continents. Fig.14.8 shows that the meso-scale moist energy sources are very strong in these precipitation regions. This implies that the heaviest precipitations are produced by meso-scale convective disturbances. Meanwhile, the dry areas off the west coasts of major continents in the subtropics and over the deserts of North Africa are correlated to the low moist enthalpy there especially in the summer hemisphere. For an equal precipitation rate, the moist enthalpy over continents is approximately twice as much as over oceans. This suggests that the water vapor transport and sea surface evaporation are important for disturbance development and heavy precipitation. The wet regions with a precipitation rate greater than 2.5 dm/year at middle and high latitudes are found over the North Pacific and North Atlantic in winter. They are associated more likely with the large-scale moist energy sources shown by Fig.14.1(a), which depends mainly on the baroclinity at middle and high latitudes. The baroclinic instability and negative baroclinic entropy are then important for development of precipitation systems at middle and high latitudes. For example, the precipitation center over the land between the Mediterranean Sea and Caspian Sea is on the northern edge of high moist enthalpy shown in Fig.13.3(a). Since the baroclinity off the east coasts of Asia and North America is stronger in win-
14.6. RELATION TO PRECIPITATION
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Figure 14.7: Global distributions of precipitation rate (dm/year) for (a) December-February and (b) June-August (After Peixoto and Oort, 1992)
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.0
100.0
0
20E 40E 60E 80E 100E 120E 140E 160E
0.0
300.0
0.0 40
200.0
.0
100
20E 40E 60E 80E 100E 120E 140E 160E
100
80N
200.
0
400 700.0
560000.0.0
80S
70S
60S
50S
40S
30S
20S
10S
0
10N
20N
30N
40N
50N
60N
70N
80N
304 14. AVAILABLE MOIST ENTHALPY IN THE ATMOSPHERE
1000km. (a) December-February and (b) June-August. 500.0
400. 0
300.0
400.0
300.0
.0
100.0
160W 140W 120W 100W 80W 60W 40W 20W
0
400.
300.0 200.0
200.0
100.0
0
0
200.0
600.0
300.0
10 0.0
200.0
.0
100.0
400.0
300
200.0
300. 0
100
400.0
.0
20E 40E 60E 80E 100E 120E 140E 160E
100.0
0 100.
300.0
0
400.
100.0
20E 40E 60E 80E 100E 120E 140E 160E
.0 50.00 600
80S
70S
60S
50S
40S
30S
20S
10S
0
10N
20N
30N
40N
50N
.0
100
60N
200
.0
70N
100.0
100
.0 300
160W 140W 120W 100W 80W 60W 40W 20W
500.0 600.0
0
300.0
200.
0.0
.0
70
500
80N
80S
70S
60S
50S
40S
30S
20S
10S
0
10N
20N
30N
40N
50N
60N
70N
80N
14.6. RELATION TO PRECIPITATION
305
(b)
Figure 14.8: Six-year climatology of maximum available moist enthalpy (m2 /s2 ) over 10◦ lat.×
306
14. AVAILABLE MOIST ENTHALPY IN THE ATMOSPHERE
ter than in summer, the explosive cyclones in the areas occur mostly in winter though the humidity is lower in winter than in summer. The positive correlation between precipitation and large-scale baroclinic energy source in the extratropical regions in Southern Hemisphere winter may also be found by comparing Fig.14.7 with Fig.14.8(b). In the Northern Hemisphere, the conditional instability over lands increases at middle latitudes from winter to summer, and so the large-scale moist energy sources are concentrated over major continents (Fig.14.1(b)) in the warm season. While, the baroclinic activities over oceans decrease in summer as the baroclinic energy sources are weakened. Therefore, the precipitation decreases over oceans but increases over lands. In the Southern Hemisphere, however, the seasonal change of baroclinic energy source in the extratropical regions is relatively small except near the antarctic. So the seasonal change of precipitation is also small compared with that in the Northern Hemisphere. Following the poleward displacement of the energy sources, the precipitation bands shift about 10 degrees southward from winter to summer. Also, the maximum precipitation rate increases in summer due to increase of moisture at the southern high latitudes.
14.7
Relation to tropical cyclones
The tropical cyclones, such as hurricanes or typhoons, may produce a huge amount of kinetic energy. It can be expected that there must be climatological correlation between the tropical storm tracks and moist energy sources. Fig.14.9 displays the global distribution of tropical cyclone origins in 20 years produced by Gray (1979). The tropical cyclones occur generally over the tropical areas where sea surface temperature is higher than 26.5◦ C (Palm´en, 1948). Since development of cyclonic circulations depends on Coriolis force, strong tropical cyclones are not found frequently near the equator within 5 degrees of latitude. Usually, the tropical cyclones occur in summer hemisphere, except over the northwestern Pacific in the tropics. Fig.14.8 shows that the summertime available moist enthalpy for meso-scale circulations in the atmosphere below 200 hPa is generally over 300 m2 /s2 in the regions of tropical storm tracks. Since available moist enthalpy is low off the west coasts of major continents in the Southern Hemisphere, tropical cyclones cannot develop in these regions. There are also no tropical cyclones off the east coast of South America, because the moist energy source is weak south of 5◦ S all the year. As the band of high moist enthalpy along with the SPCZ extends eastward to the central South Pacific in the Southern Hemisphere summer, the storm tracks over tropical Pacific may reach further east in the Southern Hemisphere than in the Northern.
14.7. RELATION TO TROPICAL CYCLONES
307
Figure 14.9: Global distributions of the origins of tropical cyclones in a 20-year period (After Gray, 1979)
Chapter 15 A case of typhoon recurvature 15.1
Introduction
Apart from the frictional and viscous dissipation of kinetic energy, a tropical cyclone outputs a huge amount of kinetic energy to the environments. As shown by the statistical survey of Frank (1977a, b), a mean tropical cyclone may export kinetic energy at the rate about 4.8 W·m−2 averaged in the cyclone with radius of 10 latitudinal degrees. The storm must be able to produce this amount of kinetic energy every second in the air column over a unit area. It has been beyond any doubt that the tropical cyclones are the thermally driven disturbances, whose primary energy source is latent heat (Malkus and Riehl, 1960; Ooyama, 1969; Tuleya and Kurihara, 1981). The process of kinetic energy generation has been studied by Emanuel (1988, 1991) with the theory of Carnot engine. The previous studies on the energetics were focused on the intensity of tropical cyclones. While, the theoretical studies on displacement of tropical cyclone were concentrated mostly on the mechanic processes independent of thermodynamic processes. In a homogeneous and stationary atmosphere, a tropical cyclone driven by the βeffect moves westward and slightly northward (Kasahara, 1957; Adem and Lezama, 1960; Holland, 1983, 1984). A self-induced secondary circulation may produce vorticity advection and horizontal divergence, which in turn drive the cyclone poleward (Anthes and Hoke, 1975; Chan and Williams, 1987; Fiorino and Elsberry, 1989). The storm motions in a varying and inhomogeneous environment affected by other circulation systems are complicated. One of the basic interaction mechanisms is the steering concept studied by many authors (e. g., George and Gray, 1976; Neumann, 1979; Brand et al., 1981; Chan and Gray, 1982). Some other effects were also considered, such as the vorticity gradient (Carr III and Elsberry, 1990; Franklin, 1990; Evans et al., 1991) and vortex size or structure (Chan, 1982; DeMaria, 1985; Fiorino and Elsberry, 1989). In those studies, a tropical cyclone was assumed as a symmetric or asymmetric integral entity of a vortex, of which the displacement was steered by the mechanic or kinematic processes in which the thermodynamic effects associated with energy conversions were ignored. An important feature of the storm motion over tropical oceans is the sudden change in the direction, called the recurvature of storm. It is unlikely that this abrupt change in the moving direction is caused by the β-effect or induced secondary circulation, since the storm intensity may not change so dramatically in the episode. It is also difficult to predict the erratic tracks in terms of large-scale background steering flow, as the background circulations may not change suddenly during the storm recurvature. These facts imply that the movement of tropical storm may also be driven by an important mechanism different from the β-effect and steering concept, which manifests a relatively short time scale. We have discussed in the preceding chapter that the tropical cyclones occur generally in the tropical and 308
15.1. INTRODUCTION
309
subtropical regions where the moist energy sources represented by available moist enthalpy are relatively strong. This climatological correlation may also be found in the individual cases. We discuss in this chapter an example of observed typhoon recurvature to reveal the important effect of moist energy source on displacement of tropical cyclone. If tropical cyclones may live on the existing energy sources only, we may evaluate the minimum speed of storm movement according to the rate of energy conversion in the storm. It will be found that the mean moving speed of tropical cyclones observed is generally lower than the minimum speed evaluated with the statistical data of tropical storms. This implies that the tropical cyclones are not supported by existing energy sources only. The additional energy source is constructed by the storm circulation itself, as tropical cyclones produce strong convergent flows in the lower troposphere which may collect moisture efficiently over tropical oceans. The construction or maintenance of energy source resulting from the circulations induced by the disturbance itself may be referred to as the self-feeding mechanism. For a tropical cyclones developed in the relatively homogeneous temperature and humidity fields, the maximum water vapor fluxes and their convergence induced by the storm occur around the low center, and move along with the cyclone. The self-feeding mechanism works efficiently over oceans, so that the tropical cyclones may have a long lifetime in the moist equivalent barotropic environments. As a new energy source may be established in a short time over a warm ocean water, the development of tropical disturbances depends less on particular preconditions compared with development of extratropical cyclones. Also, the tropical cyclones may move erratically or slowly. The typhoon recurvature discussed was caused by a new energy source developed within 24 hours to the north of the storm. This feature makes it difficult to predict the displacement of tropical cyclone for a few days. Parcel kinetic energy may be created by rising of saturated parcels in the conditionally unstable atmosphere. The height of free convection depends on the temperature and humidity at the starting height as well as the static stability. The threshold temperature of surface air for development of deep convection can be evaluated for a provided static stability. The obtained magnitude agrees with the lower limit of sea surface temperature for development of hurricane, obtained statistically by Palm´en (1948). The energy source in the equivalent barotropic atmosphere depends also on the conditional stability and surface temperature and humidity. Thus, there is a positive correlation between sea surface temperature and tropical storm tracks in the barotropic atmosphere. According to the different energy sources, cyclonic disturbances in the troposphere may be classified into several categories, such as the baroclinic cyclones and barotropic cyclones which live on the baroclinic and barotropic energy sources respectively. As discussed in the preceding chapter, the typical space scale, lifetime and circulation features may be explained by the differences in the energy sources. Since the inhomogeneities of temperature and humidity fields change greatly with latitude, the energy sources depend highly on geographic positions, and the cyclonic disturbances may also be termed according to their original regions, such as the tropical cyclones and extratropical cyclones. The analyses in this chapter may lead to classifying a particular category of the storms, called the subtropical cyclones, which
310
15. A CASE OF TYPHOON RECURVATURE
possess the intermediate features between those of tropical barotropic and extratropical baroclinic cyclones. These features may also be interpreted in terms of the energy source. Since the basic properties of a cyclone depends on the energy source, a storm may transfer between the different categories as it crosses different regions.
15.2
Typhoon Orchid recurvature
Fig.15.1 displays the maps of mean sea level pressure and surface temperature over the region and period of Typhoon Orchid in October 1991. The storm was found near 18N, 138E at 00Z on October 4, and moved northwestward to 20N 135E at 00Z on October 6. Then it displaced slowly heading to the west, and recurvated suddenly to the north after 00Z October 8 at an increased speed. As the storm moved at a speed lower than that of environmental easterlies, the eastern part was pressed by the wind and so the pressure gradient was slightly stronger than in the western part. The moist energy sources are represented by the maximum available moist enthalpy evaluated using ECMWF data for the meso-scale air columns described in the preceding chapter. The distributions are plotted by the solid contours in Fig.15.2. There was a high center of available moist enthalpy to the north of Orchid. The central value exceeded 500 m2 /s2 on October 4. Meanwhile, another energy source appeared to the west over Philippine island. The distance to the storm center was longer than that of the former one, but the central value was higher (700 m2 /s2 ). The storm moved northwestward in the following two days driven dynamically by the β-effect and steering current in the mean easterlies. As the energy source to the north retreated northeastward on October 6, the moving direction of the storm was adjusted towards the westward or the second energy source, while the dynamic driving force could also be the β-effect and steering current. As discussed in Chapter 11, the temperature of surface air decreased after conversion of moist enthalpy. This cooling on a sea surface could be aided by upwelling and evaporation of ocean water along the track. From October 4 to October 6, the central pressure of Orchid reduced and the horizontal scale enlarged. Although the available moist enthalpy in the typhoon was relatively low, it was generally above 350 m2 /s2 . After 00Z on October 7, the energy source over Philippine was weakened substantially and the westward movement was slowed down. The available moist enthalpy in the slowly moving storm decreased 200 m2 /s2 in 24 hours, and the storm started to reduce its intensity. At 00Z on October 8, a new energy source developed on the southeast coast of Japan right to the north of Orchid, with a central value over 600 m2 /s2 . The energy source could become saturated quickly as the strong northwestward water vapor fluxes come from the eastern side of the typhoon crossed it. As the energy conversion took place in the energy source, the surface pressure decreased and so the wind convergence enhanced. The convergent wind to the south of the energy source was northward. The typhoon was affected by the pressure reduction and convergent wind, and jumped northward abruptly towards the energy source. The establishment of the energy source which led to the storm recurvature was completed within 24 hours. It is then difficult for us to predict the sudden change in
15.2.
TYPHOON ORCHID RECURVATURE
311
(a)
(b)
15. A CASE OF TYPHOON RECURVATURE
312
(c)
(d) Figure 15.1: Mean sea level pressure (solid, hPa) and surface air temperature (dashed, K)
15.2.
TYPHOON ORCHID RECURVATURE
313
(a)
(b)
15. A CASE OF TYPHOON RECURVATURE
314
(c)
(d)
15.2.
TYPHOON ORCHID RECURVATURE
315
(e)
(f) Figure 15.2: Maximum available moist enthalpy (solid, m2 /s2 ) and equivalent baroclinic entropy (0.01 J/(K·kg)) evaluated using ECMWF data. The typhoon center is marked by X
.
15. A CASE OF TYPHOON RECURVATURE
316
the moving direction using the data 24 hours ago. The successful prediction depends highly on the right prediction of meso-scale moist processes, which influence the energy sources in the next day and may not be extrapolated from the information 12 hours ago.
15.3
Subtropical cyclones
The atmospheric disturbances may be classified into two types: the baroclinic and barotropic disturbances. The extratropical cyclones occurring in the relatively dry and statically stable frontal zones are baroclinic disturbances, in which kinetic energy is generated by increasing the static stability. Since the barotropic entropy decreases as the static stability increases, the dry atmosphere must possess negative baroclinic entropy which is converted into barotropic entropy in the energy conversion, so that the thermodynamic entropy is not destroyed in the isolated baroclinic atmosphere. While, the typical example of barotropic disturbances is the tropical cyclones developed in the conditionally unstable atmosphere, where the baroclinic entropy is close to null. Development and maintenance of tropical cyclones may be independent of baroclinic activities, as the thermodynamic entropy can be produced by increasing the conditional stability in the unstable barotropic atmosphere. The dashed contours in Fig.15.2 show the distribution of equivalent baroclinic entropy in the typhoon region, evaluated in the same air columns applied for the evaluations of available moist enthalpy. The tropical regions were equivalently barotropic in general. The contour of −10 J/(K.kg) was roughly along the east coasts of Asia and Japan. The strong energy source over Philippine island in the equivalent barotropic environment was produced by low conditional stability. Tropical cyclones tend to move towards the unstable regions in the equivalent barotropic atmosphere, if only the thermodynamic effect is considered. The subtropical regions north of Orchid was equivalently baroclinic. The relatively large equivalent baroclinity was contributed mainly by the humidity gradient in the quasi-barotropic environments, where the amount of baroclinic entropy was one order less than that at middle and high latitudes. After October 6, a low center of equivalent baroclinic entropy developed over southern Japan. It was strongest at 12Z on October 7 with a central value of −40 J/(K·kg). The strong energy source which led to the northward recurvature was associated with the negative equivalent baroclinic entropy, as it formed just on the warm and moist side of the low center (Fig.15.2(e)) and so was different from the tropical energy source in the equivalent barotropic regions. The storm supported by the subtropical energy source became a subtropical cyclone, which could have a larger horizontal scale and weaker convection compared with the previous tropical cyclone. The subtropical cyclones are an intermediate cyclonic system between the tropical cyclones and extratropical cyclones. They may also occur over subtropical continent in summer on the warm and moist side of a temperature front, where horizontal gradient of equivalent potential temperature is strong. The development may be related to weak baroclinic activity. Some extremely strong subtropical cyclones over oceans may have convective cloud walls and a hurricane-like eye at the center. They produce a large amount kinetic energy and strong precipitation usually. The
15.4. THRESHOLD SURFACE TEMPERATURE
317
equivalent baroclinic entropy in the subtropical atmosphere has an effect similar to that of baroclinic entropy on energy conversion discussed in Chapters 8 and 9. As discussed in the preceding chapter, the basic features of cyclones in the troposphere depend on the energy sources. The tropical cyclones occur in the equivalent barotropic regions over warm ocean waters. Their energy source is the available moist enthalpy concentrated in the regions of low conditional stability. The moist convection in the unstable atmosphere is vertical, and so the horizontal scale of the storm is small and the vertical velocity is high. The extratropical baroclinic cyclones are those illustrated by the classical Norwegian cyclone model. They form over a surface temperature front and are supported by baroclinic available enthalpy. Since the kinetic energy is created in the slantwise convection with a small trajectory slope, the horizontal scale is large and the vertical motion is weak. The subtropical cyclones may develop in the quasi-barotropic and equivalent baroclinic atmosphere. The energy source is also the available moist enthalpy but in the regions of relatively strong equivalent baroclinity. As the equivalent potential temperature surfaces in the tropical regions are steeper than the isentropic surfaces in the extratropical atmosphere in general, the horizontal scale and vertical air speed of subtropical storm are in between those of tropical and extratropical storms. When a tropical cyclone in the Northern Hemisphere moves northward or northeastward after recurvature, it crosses the regions with different energy sources and so may change to a subtropical storm and then an extratropical baroclinic cyclone. The subtropical cyclones provide the link between the tropical and extratropical cyclones, so the transfer does not occur suddenly. In general, the energy source is weaker at higher latitudes, so the storm moves faster and faster towards high latitudes, and the horizontal scale becomes larger and larger while the vertical convection gets weaker and weaker. Since a tropical and subtropical energy source of a storm can be built within the lifetime of the storm or can be maintained by the storm circulations, a tropical or subtropical cyclone may survive continuously after it is matured, and the central pressure and wind speed may fluctuate before they become a baroclinic cyclone. While, as soon as a baroclinic cyclone gets matured, it is filled up quickly by cold air, as energy conversion in the dry baroclinic atmosphere depends crucially on increase of the static stability.
15.4
Threshold surface temperature
In the moist free convective motions, the temperature of a saturated parcel (or plume or thermal) is higher than that of surroundings as it rises pseudo-adiabatically. The free upward motion in which parcel kinetic energy increases may continue until the parcel reaches pressure level pt (called the level of neutral buoyancy), where its temperature equals that of the surroundings. For a provided pt , we may evaluate the parcel temperature on the surface according to the temperature profile of the surroundings. If a saturated parcel at ps = 1000 hPa has temperature Ts , its temperature will be κ Lc ws wt pt c e , c= − Tt = Ts ps cp Ts Tt
15. A CASE OF TYPHOON RECURVATURE
318
Figure 15.3: Threshold temperature of saturated surface air for development of deep convection. The entrainment effect assumed is represented by dashed.
when it arrives pseudo-adiabatically at the top level pt . Here, ws and wt are saturated mixing ratios of the parcel at the bottom and top levels respectively, and may be evaluated from (11.32) and (11.33). Supposing the atmosphere has constant lapse rate of temperature Γ, the environmental temperature at pt is given by
T t = Ts
pt ps
γ
,
γ=
R Γ. g
When the parcel reaches the temperature of the surroundings at the top level, we have Tt = T t , or R Γ ps ws wt − = 1− ln . Ts Tt Lc Γd pt For provided pt and Γ, this equation is a function of surface air temperature Ts only. According to the statistical study of Oort and Rasmusson (1971), the annual mean temperature lapse rate is about 0.65 K/100m in the tropical and subtropical regions. The dependence of free convection height on the surface temperature is demonstrated by the solid curve in Fig.15.3. The threshold temperature of surface air for development of convection depends on the depth of convection. As reported by Simpson (1947) that the core of hurricane
15.5. ENERGY BUDGET
319
is warmer that the surroundings from the surface up to the height over 11 km. The ECMWF data analyses show that the upward motions inside a hurricane may extend to 100 hPa level or above. The solid curve in Fig.15.3 shows that for the free convection up to 100 hPa without entrainment of clear air from the surroundings, the saturated air temperature on the surface is about 27 C◦ . If sea surface temperature is similar to surface air temperature, it gives the lowest sea surface temperature for development of deep convection. This threshold value is close to the lower limit of sea surface temperature for occurrence of hurricanes obtained by observational statistics (Palm´en, 1948). The surface temperature required for deep convection will be reduced if the temperature lapse rate is greater than 0.65 K/100m. A high sea surface temperature is not a sufficient condition for producing strong tropical storms. Development of deep convection depends also on humidity of surface air and convergence of surface wind. However, Simpson and Riehl (1981) pointed out that hurricanes weaken markedly and even disappear when moving over the water with temperature substantially below the threshold value. The fact that tropical cyclones do not develop off the west coasts of South Africa and South America (referring to Fig.14.9) may be related to the low sea surface temperature there. Available moist enthalpy may be low in the regions with relatively low sea surface temperature as shown in the preceding chapter, since the conditional stability is relatively high. The threshold sea surface temperature for development of deep convection will be discussed again in Chapter 21 from the point of kinetic energy generation in moist convective process. If entrainment of dry air into saturated warm plumes in convection is considered, the minimum surface temperature calculated for development of deep convection will be higher. Supposing the saturated mixing ratio of parcels decreases 20% due to entrainment, the obtained surface temperature is displayed by the dashed in Fig.15.3. In this example, the free convection over the place with sea surface temperature 27.5 ◦ C may still reach 150 hPa with the static stability and entrainment assumed. A strong entrainment may be found in the relatively weak convection at an early stage of development. If the surface temperature is not sufficiently high, moist convection may only reach the low and middle troposphere.
15.5
Energy budget
In general, the lifetime of barotropic cyclone is much longer than that of extratropical cyclone. To find the reason, we consider firstly the energy budget in the tropical cyclones. The energy budgets in northwestern Pacific typhoons were studied by Frank (1977a, b) using 10 years rawinsond data from 1961 to 1970. These data included 248 tropical storms, in which only 143 reached typhoon intensity. The mean rates of kinetic energy generation, averaged from 1000 hPa to 100 hPa over the different rings around the low center, are listed in Table 15.1. The degree of redii in the table is identical to the latitudinal degree. Frank did not provide the data within 2◦ of the radius. Thus, the data of Palm´en and Jordan (1955) over 0.25◦ -2◦ are used for the mean tropical storm, represented by the data of 84 aircraft flights in 28 Pacific tropical cyclones obtained from 1945 to 1947 (Hughes, 1952) plus the rain-reports
15. A CASE OF TYPHOON RECURVATURE
320
Table 15.1: The rate of kinetic energy (KE) generation in the mean northwestern Pacific typhoon (After Frank, 1977b and Palm´en and Jordan 1955)
KE generation rate k˙ (W/m2 ) By mean flows By eddies Total
0.25◦ -2◦ 31.0
Band radii (γi − γi+1 ) 2◦ -4◦ 4◦ -6◦ 6◦ -8◦ 11.9 8.0 3.2 14.6 7.9 4.8 26.5 15.9 8.0 25.5 11.6 -
8◦ -10◦ 2.5 3.4 5.9 -
data from 1945 to 1951 in Pacific and Atlantic storms (Jordan, 1952). Their data were available up to 6◦ as shown in the second line of the Total in Table 15.1. The energy generation is maximum near the center, where vertical motion is strong and static instability is high. The rate of kinetic energy generation in the mean typhoon from 0.25◦ -10◦ radius is evaluated from K˙ =
5
K˙ i ,
i=1
in which K˙ i = Ai k˙ i ,
2 Ai = π(ri+1 − ri2 ) ,
(i = 1, · · · , 5) ,
and
aπγ , 180 where γ represents the radius in latitudinal degree. Using the data of Frank and Palm´en and Jordan for 0.25◦ -2◦ listed in Table 15.1, we obtain K˙ = 46.4 × 1012 W. While, the available moist enthalpy in the same region reads r=
Λ=
A (ps − pt )Λs , g
A=
5
Ai = π(r62 − r12 ) ,
i=1
where Λs is the part of specific available moist enthalpy, which will be converted into kinetic energy. If kinetic energy source of the mean typhoon is existing available moist enthalpy only, we have ˙ K∆t =Λ, where ∆t is the time taken as the available moist enthalpy is converted into kinetic energy. Now, we consider a cyclone moving from x1 to x2 as shown in Fig.15.4. The new area swept by the storm track within the time ∆t is shown by the shaded area. Supposing the shaded area over the path equals the horizontal section area of the cyclone with radius r, the moving distance is given by x2 − x1 =
πr . 2
15.5. ENERGY BUDGET
321
r
x1
x2
Figure 15.4: Displacement of a cyclone When available moist enthalpy along the storm path is converted into kinetic energy, the mean moving speed gives vc =
gr6 K˙ x2 − x1 = ∆t 2Λs (r62 − r12 )(ps − pt )
with r = r6 in the current example. It follows that vc =
90gγ6 K˙ . − γ12 )(ps − pt )
aπΛs (γ62
The speed is proportional to the kinetic energy generation and inversely proportional to the available moist enthalpy. If the moist enthalpy density at a local place in a typhoon track reduces from 550 m2 /s2 to 350 m2 /s2 after the storm passes over the ˙ we gain vc = 11.4 place, we have Λs = 200 m2 /s2 . With the previous value of K, ◦ m/s when the energy generation within 0.25 is ignored. It is equivalent to the displacement of 9.4 degrees of longitude a day at latitude 20◦ . This speed is larger than the mean speed of tropical storms. If the specific moist enthalpy converted into kinetic energy increases to 300 m2 /s2 , the moving speed calculated reduces to 6.2 degrees a day and is still higher than the mean speed observed. When Frank’s data over 2◦ -6◦ are replaced by the data of Palm´en and Jordan shown in the bottom of Table 15.1, the moving speed for Λs = 200 m2 /s2 is 8.6 degrees of longitude a day at latitude 20◦ . The speed of typhoon movement is evaluated assuming that the typhoon lives on existing energy sources only. Since the data of mean tropical storm used for the previous calculations are not all obtained from hurricanes, the moving speed derived for a single hurricane may be larger than the current value, as more kinetic energy can be created in a hurricane. This can be seen from the example of Hurricane Hazel in October 1954, analysed by Palm´en (1958). When a tropical cyclone moves at a speed lower than vc in the atmosphere with the assumed energy source, the existing energy source may not be sufficient for maintaining the large kinetic energy output and dissipation. In order to remain the energy budget, the storm must collect moist enthalpy through convergence of moisture flux in the lower troposphere, which includes the contribution of evaporation from the sea surface. Fig.15.2 shows that the local energy source along the storm track was not weakened greatly from 00Z on October 4 to 00Z on October 7, so the cyclone was maintained largely by the moist enthalpy collected by itself in the 72 hours.
15. A CASE OF TYPHOON RECURVATURE
322
If the mean typhoon moves at a half of the speed evaluated previously for energy budget, it will convert the latent heat collected from the surroundings into kinetic energy at the rate K˙ l = 46.4 × 1012 W. The energy conversion is estimated from K˙ l = Rk Lc A∇ · Fw , where Fw denotes the vertically integrated water vapor flux in a column, and Rk is the ratio of the kinetic energy produced to the latent heat released. This equation gives 1802 K˙ l . ∇ · Fw = Rk a2 π 3 Lc (γ62 − γ12 ) For the K˙ l given before, we have ∇ · Fw = 4.78/Rk × 10−6 kg/(m2 ·s). It is discussed in Section 12.2 that the ratio Rk has the order of 10−1 . Thus, the water vapor convergence in the mean typhoon is in the order of 10−5 -10−4 kg/(m2 ·s). This value is typical over oceans in the tropical regions as shown in the next section. Usually, the horizontal distribution of available moist enthalpy is not homogeneous in the tropics and subtropics. If the energy budgets may be evaluated for the two halves of a cyclone divided by a wall through the track in the last few hours, we may get different moving speeds for the two parts. The storm track may deflect to the side with the lower speed or stronger kinetic energy source. However, owing to the β-effect perhaps, it is not very often for us to see a tropical cyclone has a southward moving component in the Northern Hemisphere. For example, Fig.15.2 shows that an energy source west of Orchid was slightly to the south on October 5 and 6, while the storm moved westward in the two days.
15.6
Self-feeding mechanism
We have discussed that moisture flux from the surroundings into a tropical cyclone is important for development and maintenance of the storm, especially in a quasistationary or slowly moving period. Fig.15.5 shows the integrated water vapor flux and its divergence over the typhoon area. In these maps, there were the strong moisture fluxes and their convergence around the storm, although the maximum convergence of moisture fluxes could not occur in the storm center. The concentration of moisture in the lower troposphere may increase the conditional instability and available moist enthalpy. The maintenance of moist energy source for a storm resulting from the circulation of the storm itself is referred to as the self-feeding mechanism. The moist enthalpy collected by this mechanism is comparable with that converted into kinetic energy in a tropical cyclone as discussed earlier. When the existing energy source is not strong enough or the process is highly irreversible, the self-feeding becomes crucial for tropical cyclones to survive. The important effect of this air-sea interaction has been emphasized by Emanuel (1986). As noted in the preceding chapter, the convective disturbances in the convergent zones of surface wind, such as the ITCZ and SPCZ, convert the moist enthalpy continuously. The loss of water vapor in the boundary layer may be recharged back by the self-feeding, and the thermodynamic entropy produced by latent heating and irreversible processes may be removed by mass and moisture exchanges between the
15.6. SELF-FEEDING MECHANISM
323
(a)
(b)
15. A CASE OF TYPHOON RECURVATURE
324
(c)
(d)
15.6. SELF-FEEDING MECHANISM
325
(e)
(f) Figure 15.5: Water vapor flux integrated from 1000 hPa to 300 hPa (in unit kg/(m·s)) and its divergence (contour interval 2 × 10−4 kg/(m2 ·s)). The typhoon center is marked by X
.
326
15. A CASE OF TYPHOON RECURVATURE
system and environments (referring to Chapter 26). These processes are important for maintaining the moist energy sources or the low conditional stability in the convergent zones. If the kinetic energy created in the SPCZ and ITCZ is sufficient to lift the boundary air above the lifting condensation level and compensate the loss to internal and external frictions, the permanent energy sources may sustain in these zones together with convective disturbances. These effects of moisture and thermodynamic entropy balances on maintenance of convection may be found also in a single storm. When the distribution of humidity is inhomogeneous, the moisture fluxes and their convergence may be asymmetric to the cyclonic circulation, and so the strongest moisture concentration produced by self-feeding may occur a distance away from the storm center. Since the humidity decreases poleward, the maximum convergence center occurs to the north usually as shown by Fig.15.5. The strongest convergence may strengthen the energy source, and so may also influent the movement of tropical cyclone. For example, when a maximum convergence occurred to the north of Orchid on October 4, the typhoon moved north-westward in the next 24 hours. In the following two days when the typhoon moved westward slowly, a weak convergence center was to the west and close to the low center. After October 7, a convergence center north of Orchid developed quickly, so that the northward recurvature took place on the next day. Inspecting Fig.15.5 day by day, we may find an important cause of typhoon recurvature near the east coast of Asia. When a tropical cyclone moves westward towards the continent, the induced strong northwestward moisture flux to the north of the storm may produce a convergence center near the coast, as the westward flux is relatively weak near the continent. This convergence may be intensified in some cases by approach of a trough in the large-scale perturbations from middle latitudes. A moist energy source may be established and strengthened by the moisture concentration off the coast. The increases of available moist enthalpy and negative equivalent baroclinic entropy to the north of Orchid after October 7 (Fig.15.2(e)) were the responses to the intensification of moisture concentration. When energy conversion takes place in the energy source, the surface pressure decreases and so the wind convergence enhances. As a result, moist convection may develops quickly over the northern edge of the typhoon, and the storm recurvates suddenly at an increased speed towards the new energy source. The influence of moisture concentration to the movement of extratropical cyclones is even larger as shown in the next chapter.
Chapter 16 A case of explosive cyclone 16.1
Introduction
The theory of cyclone development in the extratropical regions was based on the classical Norwegian cyclone model forwarded by Bjerknes and his colleagues (e. g., Bjerknes, 1919, 1951; Bjerknes and Solberg, 1921, 1922). The reviews of the classical theory are referred to the work of Reed (1990). According to this theory, extratropical cyclogenesis occurs over a temperature front, which is a sloping boundary between warm and cold air masses with the warm air above. This boundary can be maintained by large-scale geostrophic circulations, and the slope can be calculated according to the changes of geostrophic wind and temperature cross the boundary (Petterssen, 1956). When a surface front is disturbed, for example, as a trough at 500 hPa level approaches the front, the poleward flows before the upper-level trough and the equatorward flows behind may break down the geostrophic balance in the lower troposphere and induce strong cyclonic circulations across the front. So, the eastern part of the front moves poleward and becomes a warm front, while the western part moves equatorward as a cold front. The moving speed of cold front is usually greater than that of warm front. As the warm air is lifted upward by the fronts moving at different speeds, kinetic energy is produced and the vortex is strengthened at the early stage. The classical theory captured the most important features of cyclogenesis in the extratropical regions, except that the effect of moisture on the dynamic processes was not considered. In fact, most extratropical cyclones may produce precipitation. It is discussed in Chapter 13 that the storm tracks at middle latitudes occur generally on the moist side of high available enthalpy, where available moist enthalpy is relatively high. We study, in this chapter, a case of explosive cyclone development and displacement over the Pacific Ocean to reveal the important effect of moisture on the baroclinic disturbances. The example shows that the extratropical cyclone developed in a band of high available enthalpy and moved along with the energy source. The available moist enthalpy was also high near the dry energy source over ocean water and depended highly on the baroclinity.The circulations induced by an extratropical cyclone, especially over oceans, may increase moisture flux on the eastern side of the storm. The maximum fluxes usually form a filament in the horizontal field called the tropospheric river by Newell et al. (1992), which is strongest in the lower troposphere. The related strong winds along the river are called the low-level moist jet. The convergence of moisture fluxes in the lower troposphere may strengthen the energy source, especially near the top or leading edge of the river. Meanwhile, the convergence center of surface wind occurs also near the river top, which may trigger the development of moist convection and reduce the surface pressure. As a result, the explosive cyclone moves towards the river top. This selffeeding mechanism provides a significant complementary to the classical Norwegian 327
328
16. A CASE OF EXPLOSIVE CYCLONE
cyclone model especially for the extratropical cyclones over oceans. The self-feeding mechanism of extratropical cyclone is similar to that of tropical cyclone at an earlier stage, as the strongest moisture fluxes and their convergence occur near the low center. However, for the cyclones developed on the boundary of air masses, it is usually unable for them to remain the strong fluxes near the center after the lower part of the storm is filled by the dry and cold air. Thus, the self-feeding is weakened by the system development, and the distance between moisture fluxes and low center increases as bigger and bigger area in the center is occupied by cold air. Also, the moisture fluxes cannot construct the baroclinic energy source and negative entropy source, which are essential for development of baroclinic disturbances. Thus, the lifetime of baroclinic cyclones is relatively short. Unlike in the tropical moist atmospheres, there are no the long-life baroclinic energy sources with active baroclinic processes in the extratropical atmosphere. When the humidity is relatively high, for example over ocean waters, small- or meso-scale moist convection may develop within an extratropical cyclone. Since the surface temperature and humidity are generally lower than in the tropical regions, the induced sub-synoptic convection may not produce enough kinetic energy for maintenance and development of the storm. In this situation, the moist convection must be coupled with large-scale baroclinic circulations which produce net kinetic energy. The coupling mechanism will be discussed in Chapter 22. In the coupled system, the life of extratropical cyclone depends nevertheless on the baroclinic processes. Kinetic energy generation in the baroclinic disturbances is characterized by increasing the static stability while reducing the baroclinity, or by converting the negative baroclinic entropy into static entropy. Since constructing a large-scale baroclinic energy source is slower than energy destruction in the disturbance development, and the induced convergence of water vapor flux occurs a certain distance away from the cyclone center and is weakened by the storm development, the lifetime of extratropical cyclone is generally shorter than that of tropical cyclone.
16.2
Energy steering mechanism
Fig.16.1 shows the mean sea level pressure in the process of extratropical cyclone development over the North Pacific. The cyclone was generated in a frontal zone on January 2, 1992, around 35◦ N 156◦ E, where available enthalpy was high as shown by the solid curves in Fig.16.2. The displayed available enthalpy is evaluated in the air columns from the surface up to 400 hPa with the horizontal area 2000 km of zonal distance by 20 degrees of latitude. The dashed contours in Fig.16.2 show that the cyclogenesis occurred also in the band of high available moist enthalpy over ocean water. The moist enthalpy is evaluated in the same columns but from 1000 hPa up to 200 hPa. After 12Z on January 2, the available enthalpy in the cyclone was over 200 m2 /s2 and the available moist enthalpy was over 400 m2 /s2 . The moist energy source overlapped on the dry one, as it depended highly on the baroclinity. The cyclone moved along the strong energy source in the stage of development. As both the dry and moist energy sources became sufficiently strong and overlapped together at 00Z on January 3, the explosive episode started through which the surface pressure dropped down more than 24sin ϕ/ sin 60◦ hPa in 24 hours. The
16.2. ENERGY STEERING MECHANISM
(a), 12Z, 01/02/92
(b), 00Z, 01/03/92
329
330
16. A CASE OF EXPLOSIVE CYCLONE
(c), 12Z, 01/03/92
(d), 00Z, 01/04/92
16.2. ENERGY STEERING MECHANISM
(e), 12Z, 01/04/92
(f ), 00Z, 01/05/92
331
332
16. A CASE OF EXPLOSIVE CYCLONE
(g) 00Z, 01/06/92 Figure 16.1: Mean sea level pressure (hPa) and water vapor flux integrated from 1000 hPa to 300 hPa (in units kg/(m·s)) 12 hours before the time denoted for the pressure field
central values of the dry and moist available enthalpy were reduced by the rapid development. The fastest deepening process occurred from 00Z to 12Z on January 4, while the rate of energy conversion was highest in the period. At 00Z on January 5 the energy sources are weakened greatly, so that the dry and moist available enthalpy in the cyclone dropped below 150 and 300 m2 /s2 respectively. The cyclone was filled up thereafter by cold air, so that it was cut off from the decaying energy sources and drifted away by the environmental flows. In this period, a new cyclone was developing to the southwest in another zone of maximum available moist enthalpy. It moved also along with the energy sources in a similar track. To see more clearly the important effect of moisture, we display, in Fig.16.1, the water vapor flux integrated from 1000 hPa to 300 hPa, 12 hours before the pressure field shown in the same diagram. The flux field manifests a filamentary structure to the east of the cyclone. There was strong convergence of the flux in the lower troposphere on the top of the filament. The moist energy source traveled following the maximum convergence center, and the cyclone moved towards the center too. The mean pattern of the explosive cyclones and maximum moisture fluxes over the North Pacific and North Atlantic in January 1992, obtained from fourteen cases can be found in the study of Zhu and Newell (1994). Since conversion of dry and moist enthalpy was faster than construction of the energy sources in the baroclinic process, the energy sources weakened quickly and the cyclone decayed after the mature stage soon.
16.2. ENERGY STEERING MECHANISM
333
(a)
(b)
16. A CASE OF EXPLOSIVE CYCLONE
334
(c)
(d)
16.2. ENERGY STEERING MECHANISM
335
(e)
(f)
16. A CASE OF EXPLOSIVE CYCLONE
336
(g) Figure 16.2: Maximum available enthalpy (solid) and maximum moist available enthalpy (dashed) in units (m2 /s2 ). The cyclone center is marked by X.
16.3
Baroclinic entropy distribution
Although the moisture has an important effect on development and maintenance of large-scale extratropical cyclones, the cyclones developed in the frontal zones are a kind of baroclinic disturbance, because the energy generation depends crucially on conversions of available enthalpy and negative baroclinic entropy. The sub-synoptic scale convection may intensify the vertical circulations in the storm, but may not produce enough kinetic energy for the development independent of the baroclinic processes. The specific baroclinic entropy evaluated in the same columns as those for evaluations of available enthalpy is displayed in Fig.16.3. The patterns were similar to those of available enthalpy demonstrated in Fig.16.2. The cyclone formed in a zone of low baroclinic entropy at 12Z on January 2. When it moved northeastward along with the low baroclinic entropy, the cyclone converted the dry and moist enthalpy into kinetic energy and so the wind speed increased. As discussed already, the energy conversion was accomplished by increasing the baroclinic entropy or reducing the baroclinity. At the beginning of January 3, right before the explosive development, the baroclinic entropy in the cyclone dropped down to the lowest level at −0.45 J/(K·kg) in the period (see Fig.16.3(a)). After the explosive deepening episode, the cyclone got the lowest pressure on the surface at 00Z on January 5, while the baroclinic entropy rose to −0.2 J/(K·kg) as shown in Fig.16.3(b). If the baroclinic entropy is small in magnitude and close to the total thermodynamic entropy production in an isolated atmosphere, the static stability may not increase remarkably in the dry processes as shown by (8.38), and so no large amount of kinetic energy is pro-
16.3. BAROCLINIC ENTROPY DISTRIBUTION
(a)
(b) Figure 16.3: Baroclinic entropy in units J/(K·kg). The cyclone center is marked by X.
337
16. A CASE OF EXPLOSIVE CYCLONE
338
duced in the statically stable atmosphere. That the life of baroclinic extratropical cyclone depends on baroclinic entropy is a major feature of baroclinic disturbance.
16.4
Low-level moist jet
When the large-scale perturbations occur in the environments with equatorward gradient in a humidity field, the poleward flows in front of a trough carry more water vapor than the equatorward flows, so that the wind shear may produce a sharp horizontal gradient of humidity across the trough. Usually, there are the largescale upward motions before the trough as the secondary circulation associated with wind convergence and temperature and vorticity advection (Holton, 1992, McHall, 1993). The secondary circulation may trigger meso- and small-scale convection in the moist air mass especially over oceans, which, in turn, enhances the low-level wind convergence. This interaction between the large and small scales increases further the moisture concentration along a trough on the warm and moist side. A long and narrow filament in the field of moisture flux, called the tropospheric river, may be produced eventually in the surface convergence zone. Since the large-scale circulations in the extratropical atmosphere are constrained almost on the isentropic or moist isentropic surfaces which tilt upward and poleward, the rivers flow from low levels to high levels and so transport water vapor upward. Unlike the rivers on the lands, the head or top of a tropospheric river is on the leading edge instead of the source end called the upper reach for the surface rivers. Examples of the rivers can be seen from Fig.16.1. When the river region is conditionally stable, free convection may occur only if the saturated air moves slantwisely with a slope less than that of moist isentropic surface in the same direction, supposing the equivalent potential temperature of parcel is conserved (referring to Section 20.3). Thus, to produce kinetic energy in the river where vertical motion is relatively strong, the parcels must have a large horizontal wind speed. As the vertical velocity increases in the conditionally unstable slantwise convection, the slope of trajectory increases consequently. If the conditional instability does not increases upward, the trajectory slope must be reduced by converting the vertical momentum produced into the horizontal momentum. A jet of horizontal wind may then be produced in the river below the top of free slantwise convection, which in turn increases the water vapor flux. The distribution of water vapor flux and low-level moist jet associated with the tropospheric bomb at 12Z on January 4 are illustrated by Fig.16.4. The strongest fluxes were northward and northeastward, and occurred on the warm and moist side of cold front. In other words, the cold front was off the western edge of the maximum fluxes, surrounded by the heavy contour in Fig.16.4(a). Usually, convection is strong in the region of largest fluxes, and so a cloud band can be observed from satellite. The low-level moist jet occurred in the regions of maximum moisture fluxes. As the jet moves along with the river and creates kinetic energy continuously, it is strongest at the last stage of cyclone development. However, both the river and moist jet may occur independently of a well developed cyclone.
16.4. LOW-LEVEL MOIST JET
339
(a)
(b) Figure 16.4: Low-level moist jet at 12Z on January 4. The dashed contours plot mean sea level pressure. Vectors show the water vapor flux (m/s) in (a) and wind (m/s) in (b) at 850 hPa level. The heavy contour is for 0.25 m/s in (a) and 35 m/s in (b).
16. A CASE OF EXPLOSIVE CYCLONE
340
16.5
Self-feeding mechanism
We have discussed in the preceding chapter that the cyclonic circulation of a tropical cyclone may produce strong convergence of moisture flux around the low center. The self-induced concentrations of moisture and moist enthalpy may strengthen the energy source for maintenance and development of the system. The self-feeding mechanism exists also in the extratropical cyclones, but manifests different features. In general, a river at middle latitudes is not inside a frontal zone, but lies on the warm and moist side. When a cyclone is developing over a front, the poleward wind speed increases to the east and so the flux is strengthened. As the cyclone center is occupied partially or fully by warm and moist air on the surface at an early stage, the distance from the river to the low center is not too long and may be traveled in 12 hours as shown in Fig.16.1. The enhanced maximum convergence of moisture flux took place near the leading edge of the river. Some examples of the flux convergence are shown in Fig.16.5. Fig.16.4(b) shows that the convergence of surface wind is also maximum near the river top. The strong wind convergence may force the meso- and small-scale moist convection in the river. As a result, the surface pressure drops down and the convergence enhances. The negative pressure tendency pulls the cyclone toward the strong convective area on the leading edge of the river. The fastest deepening episode in the previous example occurred after 00Z on January 4 as the moisture convergence near the low center became strongest in the period (Fig.16.5(a)). The moist process provides a significant complementary to the classical Norwegian cyclone model, in which the dynamic effect of moisture was ignored. The interactions between the large-scale cyclonic activity and subsynoptic scale convection is similar to the Conditional Instability of the Second Kind (CISK) proposed by Charney and Eliassen (1964), except in a baroclinic environment. Since sea surface temperature and atmospheric humidity at middle and high latitudes are generally lower than in the tropical regions, the moist convection may only extend to a low height as discussed in the preceding chapter. It will be shown in Chapter 22 that a local convective system within a limited height may produce negative kinetic energy over the whole circulation cycle with mass conservation. Thus, the convection is usually coupled with large-scale baroclinic disturbances which produce net kinetic energy. Unlike the energy conversion in the moist atmosphere, the kinetic energy in the dry atmosphere is converted from heat and geopotential energies as the cold air sinks and warm air rises. After an extratropical cyclone is occluded, it is filled up by dry and cold air in the lower troposphere, and is unable to produce strong moisture fluxes in the cold air mass. As a larger and larger area in the cyclone is occupied by cold air, the distance from the low center to the river off the eastern edge of the cold mass increases. This can be seen from Fig.16.5. So the river has little influence on the occluded cyclone and the cyclone decays quickly. The strong moisture fluxes may still survive when over oceans after the previous cyclone decays. Since the Coriolis force is relatively large at the high latitudes, the horizontal scale of cyclone, such as the polar low, is small, about 200-500 km in diameter (Businger, 1985). So the river and moist jet strengthened by the cyclonic circulation is close to the low center. According to the case study of Rasmussen (1985), the
16.5. SELF-FEEDING MECHANISM
341
(a)
(b)
16. A CASE OF EXPLOSIVE CYCLONE
342
(c) Figure 16.5: Divergence of integrated water vapor flux in units 10−4 kg/(m−4 ·s). The cyclone center is marked by X.
low-level moist jet is about 50 km to the east of a polar law at 75◦ N over Barents Sea. This distance is one order shorter than that in a baroclinic cyclone at midlatitudes, and is also shorter than that in a tropical cyclone. Thus, the self-feeding mechanism over a warm ocean water may turn an baroclinic cyclones or polar low to an extratropical hurricane or arctic hurricane in some particular situations, which possesses a warm and moist core and manifests a clear eye surrounded by deep convective cloud walls (Shapiro et al., 1987; Emanuel, 1989).
Chapter 17 States of maximum thermodynamic entropy 17.1
Introduction
When the gravity effect is ignored, a laboratory sample of gas in thermodynamic equilibrium is uniform in all thermodynamic variables. The uniform state is attained through molecular diffusions, and called the heat-death state as it may not be changed again by molecular diffusions at a constant pressure in isolation. In other words, an isolated gas at a given pressure has only one equilibrium state. It can be proved in the next section that the equilibrium state possesses maximum thermodynamic entropy. The Earth’s atmosphere is different from an ideal gas. It will be illustrated in this chapter that the equilibrium state of the atmosphere in the gravitational field attained by molecular diffusions is uniform only in temperature. While, the pressure and mass density decreases exponentially with height. For a provided initial state and constant environmental conditions, the non-uniform isothermal state attained through the microscopic processes in isolation is also a unique equilibrium state, and is also a heat-death state possessing maximum thermodynamic entropy. The heat-death atmosphere attained by molecular diffusions may not be observed in a large scale, and meteorologists are more interested in the processes over meteorological scales in which the microscopic processes may be ignored. We introduced in Chapter 7 the conservation law of potential enthalpy to filter out the heat-death state for the study of energy conversion in the atmosphere. When the anisobaric molecular diffusion is ignored, a steady state of the atmosphere represented by meteorological measurements may be considered as an equilibrium state which is generally inhomogeneous in the vertical direction, though the microscopic processes take place actually at the inhomogeneous equilibrium state. It is learned from Chapters 9 and 11 that the atmosphere possesses multiple inhomogeneous equilibrium states with different energy partitions attained after energy conversions in the dry or moist atmosphere. These reference states depend on thermodynamic entropy and geopotential entropy productions in the process for a provided initial state. Particularly, there is an extremal state called the lowest state, which is attained through the process with maximum kinetic energy and geopotential entropy generations and minimum thermodynamic entropy production. We discuss in this chapter another extremal equilibrium state in the dry atmosphere attained through quasi-adiabatic process (defined in Chapter 7) with minimum kinetic energy and geopotential entropy generations and maximum thermodynamic entropy production. This equilibrium state is called the kinetic equilibrium state, since energy transfers may not occur without vertical turbulences and molecular 343
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
344 diffusions in the state.
A particular example of kinetic equilibrium state is the isentropic state with constant potential temperature, which may be attained on theory from the statically unstable atmosphere. The isentropic atmosphere is also called the kinetic-death atmosphere. Air parcels may move at their initial velocities towards any direction, but no further changes in the thermodynamic state and energy partition may be brought about without molecular diffusions in isolation. Although the thermodynamic entropy increases as the statically stable atmosphere reaches the isentropic state, this process may not happen if there is no enough turbulent kinetic energy at initial state. Since the dry atmosphere can hardly become statically unstable, the kinetic-death state may be found most frequently in the tropical moist atmospheres or in convective clouds, which are neutral to moist convection (Betts, 1982a; Xu and Emanuel, 1989). We calculate also in this chapter the geopotential entropy produced as the atmosphere gets to the different extremal states. The results confirm that the statically stable atmosphere cannot reach the kinetic-death state without dynamic forcing though the thermodynamic entropy increases, since kinetic energy and geopotential entropy are destroyed in the process. While, the isothermal atmosphere may be attained by molecular diffusions, although the geopotential entropy decreases also, as the heat conduction which leads to the heat-death state may be independent of the gravitational force and the molecular diffusions do not cause conversion of geopotential energy. This result emphasizes that the geopotential entropy may only be applied for the processes associated with parcel motion and geopotential energy conversion. The molecular diffusions lead the atmosphere to the heat-death state with maximum thermodynamic entropy, destroying parcel kinetic energy in the pure thermodynamic process. While, the parcel motions driven by the external forces including the gravitational force tends to lower the gravity center of atmosphere or increase the static stability by converting the geopotential energy into kinetic energy. The geopotential entropy increases in the process while the barotropic entropy, a part of thermodynamic entropy, decreases. Since the parcel motions are the major cause of atmospheric variations in meteorological scales, the increase of geopotential entropy evaluated on these scales shows the precedence over the increase of thermodynamic entropy. Therefore, the atmosphere driven by the gravitational force tends to reach the corresponding lowest state instead of the kinetic-death state. This property may be illustrated as the principle of extremal entropy productions for the large-scale inhomogeneous thermodynamic systems. The lowest states attained through the processes with minimum thermodynamic entropy production and maximum geopotential entropy production may be derived from the variational approaches discussed in Chapter 9 and 11. To prove this principle, we may apply the turbulent entropy law and geopotential entropy law, which show that both the thermodynamic entropy and geopotential entropy do not decrease as the isolated atmosphere excited reaches the lowest state. While, the meteorological process leading to the isentropic state with maximum thermodynamic entropy destroys geopotential entropy and so cannot happen. The principle of extremal entropy productions makes the studies on the maxi-
17.2. HEAT-DEATH IDEAL GAS
345
mum kinetic energy generation practically meaningful. In fact, it has been applied implicitly since long time ago to study the kinetic energy sources in the atmosphere (Margules, 1904; Lorenz, 1955). This principle is different from the minimum entropy exchange principle of Paltridge (1975). The latter, like the principle of minimum thermodynamic entropy production in an stationary nonequilibrium state studied by nonequilibrium thermodynamics for linear systems (Prigogine, 1955; Groot and Mazur, 1962; Kreuzer, 1981), may be considered as the necessary condition of an equilibrium climatological state. Apart from the conservations of mass, energy, and potential enthalpy, some other physical relationships, such as the geostrophic balance in the large-scale circulations, may also have influences on the equilibrium states. Also, an inhomogeneous steady state of a system may be remained by exchanges across the boundaries. So, the extremal equilibrium states discussed without considering these physical relationships and boundary conditions in this chapter may be significantly different from the climatological mean fields of atmosphere. The more realistic features of the mean atmosphere may be simulated by incorporating more physical constraints and boundary conditions. Applications of the extremal principles for climatological models have been made successfully, to some extent, by many authors (e.g., Paltridge, 1975, 1978; Shutts, 1981; Mobbs, 1982; Noda and Tokioka, 1983).
17.2
Heat-death ideal gas
The heat energy is conserved as an closed ideal gas in nonequilibrium reaches the equilibrium state through molecular diffusion. We show in the following that the volume of the gas is unchanged also. Suppose two pieces of an ideal gas have temperature T1 and T2 at a given pressure, the total volume of them is V =
R (m1 T1 + m2 T2 ) , p
where m1 and m2 indicate the mass of the pieces. After they mix together at the constant pressure, the mixture has the temperature T =
m1 T1 + m2 T2 m1 + m2
in equilibrium. This equation may be derived using the thermodynamic energy law of ideal gases. The final volume is evaluated from V =
R R (m1 + m2 )T = (m1 T1 + m2 T2 ) , p p
and is unchanged after isenthalpic diffusions. This consequence may also be derived simply from the energy conservation law. As total enthalpy or heat energy of the gas is conserved, no mechanic work is done in the mixing process. Thus, the isolated ideal gas in equilibrium or nonequilibrium at constant pressure may be viewed as a closed system.
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
346
From (4.27), thermodynamic entropy variation of an ideal gas at constant pressure is given by
∆S = cp V
= p
cp R
ρr ln Tr dV − S0
V
1 ln Tr dV − S0 , Tr
(17.1)
where the subscripts 0 and r indicate the initial and equilibrium states, respectively. The entropy variation is produced by molecular diffusion. The conservation of mass gives p 1 M0 = ρr dV = dV . (17.2) R V Tr V Since the volume is unchanged by isenthalpic diffusions, the heat energy is also conserved, that is cv ρr Tr dV = pV . U0 = cv R V To evaluate the maximum thermodynamic entropy production, we make the auxiliary function λ p cp ln Tr + X= R Tr Tr from (17.1) and (17.2). The Euler equation ∂X =0 ∂Tr gives ln Tr = 1 −
λ . cp
(17.3)
So, the equilibrium state has constant temperature and is uniform in all the thermodynamic variables including the potential temperature. Now, the mass conservation equation reads pV = RTr M0 . Applying it for the energy conservation equation gives Tr = Moreover, we have
cp p ∂2X =− 2 ∂Tr RTr3
U0 . cv M0
λ 3− − ln Tr cp
.
Inserting (17.3) into it yields cp p ∂2X = −2 . 2 ∂Tr RTr3 It is less than zero, and so the equilibrium state has maximum thermodynamic entropy. The uniform equilibrium state is called the heat-death state, since it cannot be changed again by molecular diffusions in an isolated container with fixed walls.
17.3. HEAT-DEATH GEOPHYSICAL AIR MASS
17.3
347
Heat-death geophysical air mass
The equilibrium state of an ideal gas in a small volume is uniform, because the gravity effect on the gas is ignored, so that the pressure is identical everywhere within the gas in equilibrium. If the volume is very large such as that of the Earth’s atmosphere, the gravity effect cannot be ignored as it causes vertical gradients in the pressure and mass density. Thus, the equilibrium state of a large-scale geophysical fluid with maximum thermodynamic entropy may not be uniform. This is discussed in the following. The thermodynamic entropy variation between two equilibrium states of the closed atmosphere in a fixed domain V is given by
Tr ρr − R∆ ln T0 ρ0
ρr cv ln
∆S = V
dV
derived from in Cartesian coordinates (4.23), when the air is assumed as an ideal gas. The law of mass conservation is represented by
M0 =
ρr dV .
(17.4)
V
Meanwhile, the energy conservation law of the whole atmosphere or a closed domain with fixed walls is provided by (7.17), or
ρr V
where
v2 cv Tr + gz + r 2
dV = E0 ,
E0 =
V
ρ0
v2 cv T0 + gz + 0 2
(17.5)
dV .
These three equations give the auxiliary function
X = ρr (cv ln Tr − R ln ρr ) − λ1 ρr − λ2 ρr
v2 cv Tr + gz + r 2
for the isoperimetric problem of maximum thermodynamic entropy production, where λ1 and λ2 are constant Lagrangian multiplies. The Euler equations are ∂X = cv ρr ∂Tr
1 − λ2 = 0 , Tr
∂X = λ2 ρr vr = 0 , ∂vr and
v2 ∂X = cv ln Tr − R ln ρr − λ1 − λ2 cv Tr + gz + r ∂ρr 2
They give Tr =
1 , λ2
−R=0.
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
348
vr = 0 , and
gz + cv ln Tr − cp − λ1 . Tr Thus, the equilibrium state has constant temperature and zero parcel kinetic energy. The last equation gives gz (17.6) ρr = ρs e− RTr , R ln ρr = −
where ρs = e
cv Tr −cp −λ1 R
is the density at z = 0. Using the ideal-gas equation yields gz
pr = ps e− RTr ,
ps = ρs RTr .
(17.7)
The pressure and mass density change exponentially with geographic height, so the equilibrium state is non-uniform. These vertical stratifications are produced by the gravitational force, which can be seen from the expressions of ρr and pr . If g = 0, we have ρr = ρs and pr = ps , so the equilibrium state becomes uniform and isentropic. The last two equations give dpr = −ρr g . dz The thermodynamic equilibrium state is in hydrostatic equilibrium also. From the auxiliary function at vr = 0, we have ρr ∂2X = −cv 2 , 2 ∂Tr Tr R ∂2X =− , 2 ∂ρr ρr and
They give
∂2X =0. ∂Tr ∂ρr
∂2X ∂Tr ∂ρr
2
−
∂2X ∂2X cv R =− 2 . 2 2 ∂Tr ∂ρr Tr
It is less than zero, so the equilibrium state exists. Moreover, since ∂2X <0, ∂Tr2 the equilibrium state has maximum thermodynamic entropy. The stratified isothermal state is also called the heat-death state, as it cannot be changed again by itself in isolation. The above discussions are similar to the study of Dutton (1973) or Livezey and Dutton (1976). Replacing (17.6) for ρr in mass conservation law (17.4), and integrating over the column from the surface z = 0 up to z = Z yield ρs =
gM0 gZ
A(1 − e− RTr )
,
17.4. HEAT-DEATH ATMOSPHERE
349
where A is horizontal cross section of the column. Now, we have pr =
gM0 A(1 − e
and ρr =
gz
e− RTr
gZ − RT r
)
gM0
gz
gZ − RT r
ARTr (1 − e
e− RTr )
obtained from (17.7) and (17.6) respectively. The former equation shows that the local pressure may change as the compressible geophysical fluid approaches the equilibrium state. If the column is a part of the atmosphere, the pressure change may cause the discontinuity passing through the upper boundary of the column. Inserting the last equation into the energy conservation law (17.5) yields
M0 cp Tr −
gZ gZ − RT
1−e
gZ − RT
e
r
= E0 .
r
The constant temperature of the equilibrium state may be solved from this equation. If Z → ∞, we see E0 . (17.8) Tr = cp M0 All the kinetic energy at an initial state is converted into thermal enthalpy, as the geophysical fluid reaches the heat-death state. The constant temperature depends on the total energy and mass, and is independent of the pressure or density profiles. This suggests that the heat conduction resulting from molecular collisions is independent of the gravitation. While, the pressure and density change with height in the gravitational field. The changes depend crucially on the gravity as they disappear if g = 0.
17.4
Heat-death atmosphere
The equilibrium state of the atmosphere with maximum thermodynamic entropy below a certain height may be studied conveniently in the pressure coordinates, assuming that the atmosphere is in hydrostatic equilibrium, so no vertical accelerations take place. Thermodynamic entropy of the dry atmosphere is evaluated from (8.2), that is ∆S =
cp g
ps A pt
ln Tr dpdA − S0 .
(17.9)
When the mass is conserved, we may choose a domain with constant A, ps and pt . The isolated domain is fixed in the pressure coordinates but is not closed since the upper boundary pt may change its geographic height. The energy conservation law of the atmosphere with the free upper boundary in geographic height is given by (7.22), or ps vr2 1 cp Tr + dpdA = E0 , g A pt 2
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
350 where 1 E0 = g
ps A pt
v2 cp T0 + 0 2
dpdA
(17.10)
The enthalpy cp T in this energy equation includes heat energy, geopotential energy and the energy exchange caused by the volume change, as discussed in Chapter 7 or 9. So, this energy equation is different from (17.5) for a closed atmosphere. Changes of the atmosphere in the free domain do not cause the discontinuity across the upper boundary. The auxiliary function for evaluating the equilibrium state is X = cp ln Tr − λcp Tr + λ
vr2 2
obtained from the previous two equations, where λ is a constant. The Euler equations give ∂X 1 = cp ( − λ) = 0 , ∂Tr Tr and
∂X = λvr = 0 . ∂vr
They lead to Tr =
1 , λ
and vr = 0 . Inserting the reference state into the energy equation yields λ=
cp A(ps − pt ) . gE0
So, the equilibrium state is also a rest and isothermal state. Usually, the free domain may change its volume as it changes the height and cross section adiabatically in the process, so the heat energy may not be conserved exactly. When the small effect of volume change is ignored, the constant temperature reads Tr =
gE0 . cp A(ps − pt )
(17.11)
This is the energy equation of the equilibrium state, and is the same as (17.8). Since cp ∂ 2X =− 2 <0, 2 ∂Tr Tr the equilibrium state has maximum thermodynamic entropy. An isolated atmosphere may eventually become heat-death due to molecular diffusion. The pressure profile of the atmosphere is gz (17.12) pr = ps e− RTr ,
17.5. KINETIC-DEATH ATMOSPHERE
351
derived from the hydrostatic equation assuming that z = 0 at the bottom surface ps . The mass distribution may then be obtained from the state equation, giving gz ps − RT r . e ρr = RTr For ideal gases, ps /(RTr ) = ρsr . The heat-death state of the non-closed atmosphere represented in the pressure coordinates is similar to that of the closed atmosphere discussed in Cartesian coordinates. It is non-uniform with constant temperature and vertical stratifications of pressure and mass density. It is discussed in Chapter 6 that the convergent variable in the thermodynamic entropy equation is the potential temperature when the potential enthalpy is conserved. The heat-death atmosphere is isothermal but not isentropic, because the potential enthalpy is not conserved but the heat energy is in the process of molecular diffusion. In this situation, we may choose the temperature to be the convergent variable of the universal principle discussed in Chapter 6. The thermodynamic entropy law (17.9) gives the principle in the pressure coordinates. According to the hydrostatic equation, we have ∂T ∂T = −gρ . ∂p ∂z So, a variable which is uniform in the pressure coordinates is also uniform in Cartesian coordinates.
17.5
Kinetic-death atmosphere
17.5.1
Isentropic atmosphere
It has been discussed also in Chapter 9 that the reference state attained after energy conversion is the heat-death state, if anisobaric molecular diffusion is not filtered out. The heat-death state is also the only equilibrium state with maximum thermodynamic entropy. Although the heat-death atmosphere is theoretically true, it is not observed frequently or interested by meteorologists, since the atmospheric variations over meteorological scales are produced mainly by parcel motions instead of molecular diffusions and the atmosphere is not closed or isolated. To study the energy conversions in the atmosphere, we have to filter out the microscopic processes using the conservation law of potential enthalpy derived in Chapter 7. The extremal state, attained through the quasi-adiabatic process, possesses constant potential temperature as discussed in Chapter 8. This can be proved further in the following. The thermodynamic entropy variation is also represented by (8.3). As we use the conservation law of system potential enthalpy (8.14) as a filter, the auxiliary function is given by (17.13) X = ln θr − λθr in which λ is a constant. The necessary condition for existence of the extremum is given by the Euler equation ∂X =0. ∂θr
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
352 It follows that
θr =
1 . λ
Inserting it into (17.9) yields 1 1 = λ A(ps − pt )
ps
θ0 dp dA .
A pt
Thus θr = θ0 . Moreover, since 1 ∂2X =− 2 <0, 2 ∂θr θr the isentropic atmosphere has maximum thermodynamic entropy. As the potential enthalpy is conserved, the constant potential temperature is the ensemble mean at the initial field. For the geophysical fluids with vertical gradient of pressure, the isentropic state is different from isothermal state. When the atmosphere is saturated with water vapor, we may prove from (11.14) that the equilibrium state with maximum thermodynamic entropy attained through the moist convection with conservation of moist potential enthalpy is the moist isentropic state with constant equivalent potential temperature.
17.5.2
Example
To give examples of the thermodynamic entropy produced as an dry atmosphere becomes isentropic, we assume that the initial state is barotropic possessing constant temperature lapse rate Γ. The potential temperature profile is given by
θ0 = Ts
ps p
b
,
b=
R (Γd − Γ) . g
(17.14)
When it reaches the isentropic state, thermodynamic entropy produced in a column of unit horizontal section area reads ∆S˜k−d =
cp g
ps
θ0 dp θ0
ln pt
(17.15)
From (8.28), we have ps pt
θ0 dp = (ps − pt ) ln Ts ps − pt
ps pt
b
− ln[(1 − b)(ps − pt )] − b
.
Again from (17.14), we see ps pt
θ0 dp = (ps − pt ) ln Ts − bpt ln
ps . pt
(17.16)
Inserting the last two equations into (17.15) yields ∆S˜k−d = +
cp g
(ps − pt ) ln ps − pt
ps bpt ln pt
&
,
ps pt
b
− ln[(1 − b)(ps − pt )] − b
17.5. KINETIC-DEATH ATMOSPHERE
353
Figure 17.1: Generations of kinetic energy (solid) and thermodynamic entropy (dashed) as the assumed linear atmosphere reaches the isentropic state through quasi-adiabatic process
which is independent of the surface temperature at the initial field. We may prove that ∂ 2 ∆Sk−d ∂∆Sk−d =0 and >0 ∂b ∂b2 at Γ = Γd . So, the entropy production has a minimum value ∆Sk−d = 0 in the atmosphere with the adiabatic lapse rate. The dependence of thermodynamic entropy production on temperature lapse rate of initial field is displayed by the dashed curve in Fig.17.1. The entropy production is positive in the statically stable and unstable atmospheres. It is symmetric to the adiabatic stratification, since the entropy depends only on the magnitude but not the direction of potential temperature gradient. This does not mean that the stable and unstable atmospheres possess the same ability to reach the isenthalpic state. This will be discussed in the next section. The isentropic state is a constant state for parcel motions. Except the anisobaric molecular diffusion including the dissipation of kinetic energy, adiabatic motions and isenthalpic diffusions cannot change thermodynamic states of the parcels or system, so no exchanges between thermal enthalpy and kinetic energy take place in the quasi-adiabatic processes. The parcel kinetic energy is conserved in the isentropic atmosphere if without friction and viscosity. The isentropic state may be called the
354
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
kinetic-death state, as it is unable to produce more kinetic energy or change the thermodynamic state through parcel motions in isolation. The further changes may only be produced by molecular diffusions which destroy the kinetic energy.
17.5.3
Comparison with heat-death atmosphere
We have found that the equilibrium state of the atmosphere with maximum thermodynamic entropy depends on the features of physical process. It is homogeneous in potential temperature or temperature, depending on whether the anisotropic molecular diffusions are filtered out or not. In other words, the conservative convergent quantity used for the universal principle depends on the migratory elements, of which the distribution manifests the system disorderliness. For the thermodynamic system in the gravitational field of which the changes are produced by adiabatic parcel motions or quasi-adiabatic process, the elements are the turbulent entities called also the parcels for convenience. The parcels are identified with their potential temperatures instead of temperatures, because the potential temperatures are conserved while the temperatures may be changed in the process. Thus, thermodynamic disorderliness of the parcel ensemble is represented by the potential temperature instead of temperature. If a certain number of entities increase their potential temperature at the equilibrium state toward one direction, other entities must reduce the potential temperature to remain the potential enthalpy of the system. The different changes in parcel potential temperature reduce the disorderliness and so may not happen in isolation. The variation of thermodynamic entropy in the process is consistent with change of disorderliness in the potential temperature field, as no change in the amount of potential enthalpy is contributed to the entropy variation. Also, the partition of macroscopic energies in the isentropic system is unique, and so the kinetic-death state is unique for a provided initial state, which cannot change without reducing the disorderliness. If molecular diffusions are the only process of system variation, the basic migratory elements are molecules. The mean potential temperature or potential enthalpy of the system with vertical stratifications of pressure and mass density in equilibrium is not conserved again in the process. After all parcel kinetic energy is dissipated by viscosity, the internal energy (or the heat energy of ideal gases) is conserved in the further processes of molecular diffusion in a closed system. As the molecular collisions tend to unify the distribution of molecular kinetic energy in the isolated system, the disorderliness of the system is manifested by the distribution of molecular kinetic energy or the temperature, and the equilibrium state with maximum disorderliness attained by molecular diffusions is the isothermal state. Since the heat energy is conserved, the rest and isothermal state is the unique heat-death state. The heat conduction resulting from molecular collisions may be independent of macroscopic Newtonian forces, so the gravitational force cannot change the temperature distribution in equilibrium. For the pressure changes with height in the atmosphere, the potential temperature changes in the vertical direction at the heat-death state. The kinetic-death state is a conditional equilibrium states. Further changes may still be produced by molecular diffusions which lead to the heat-death state in
17.5. KINETIC-DEATH ATMOSPHERE
355
isolation. The thermodynamic entropy produced in this process is calculated from ∆S˜h−d =
cp g
ps
ln pt
θh−d dp , θk−d
where θk−d = θ0 is the potential temperature of the isentropic atmosphere, and
θh−d = Th−d
ps p
κ
is the potential temperature of the isothermal atmosphere with temperature Th−d . It follows that cp ∆S˜h−d = g
Th−d ps ln + κ (ps − pt ) − κpt ln θk−d pt
.
The constant temperature of the isothermal atmosphere included in this equation is calculated from (17.11), that is Th−d =
gE˜k−d , cp (ps − pt )
˜k−d is the total macroscopic energy of the isentropic atmosphere over a unit where E area, given by (17.10) or ˜k−d E
1 = g
ps A pt
v2 cp Tk−d + 0 2
dpdA .
Here, v0 indicates the wind speed at the isentropic state, and Tk−d is the temperature profile of the isentropic atmosphere. Since
Tk−d = θk−d we obtain
p ps
˜k−d = cp θk−d ps − pt pt E g(κ + 1) ps and so
Th−d =
κ
,
κ
+
v02 (ps − pt ) , 2g
θk−d pt ps − pt (κ + 1)(ps − pt ) ps
κ
+
v02 . 2cp
Applying the last equation for the expression of ∆S˜h−d finds that the entropy created as the isentropic atmosphere becomes isothermal depends only on the initial kinetic energy which will be dissipated by viscosity. If v0 = 0, we have ∆S˜h−d = 7.62 J/(K.kg) for the previous example discussed. The isothermal heat-death state possesses more thermodynamic entropy than the isentropic state.
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
356
17.6
Energy conservation constraint
In getting to the isentropic state discussed in the preceding section, the mass and potential enthalpy are conserved, but the total energy may be not. So, the kineticdeath state may not be attained if without exchanging energy with the surroundings. In other words, increase of thermodynamic entropy produced by parcel motions may not ensure that the process will happen necessarily in the atmosphere. This is different from the process of thermodynamic entropy generation resulting from molecular diffusions in a classical thermodynamic system, since the total energy may be unchanged in the microscopic processes and the diffusion process may be independent of the macroscopic external forces. So a process of molecular diffusion with increase of thermodynamic entropy may always happen in an isolated nonequilibrium system. Now, we discuss the kinetic energy change as the atmosphere reaches the isentropic state. The energy conservation equation for a dry isolated atmosphere is represented by (9.7), which is rewritten as 1 2g
ps A pt
(vr2
−
v02 )
1 dp dA = Γd pκs
ps A pt
pκ (θ0 − θr ) dp dA
(17.17)
or ∆K = Ψr − Ψ0 , where Ψ is the enthalpy evaluated from (9.8). For the barotropic initial field assumed in the preceding section, the total enthalpy in a column of unit horizontal section area gives b−κ ˜ 0 = cp Ts0 ps (pκ−b+1 − pκ−b+1 ), Ψ t g(κ − b + 1) s derived using (17.14). Meanwhile, the enthalpy in an isentropic air column is calculated by b
c T p ps p s s ˜r = ps − pt , Ψ g(1 + κ)(1 − b)(ps − pt ) pt using θ in (8.28) for θr . Thus, the kinetic energy created in the column reads ˜ = ∆K −
cp Ts ps g
1 1− κ−b+1
ps pt
1 1− (1 + κ)(1 − b)(ps − pt )
κ−b+1
ps pt
κ+1
ps − pt
ps pt
b
.
The dependence of kinetic energy generation on the initial lapse rate of temperature is demonstrated by the solid curve in Fig.17.1. The kinetic energy is destroyed as the barotropic and statically stable atmosphere becomes isentropic. Net kinetic energy may be produced only if the barotropic atmosphere is statically unstable. Obviously, we have Kr = K0 + ∆K. Since there must be Kr ≥ 0 in an isolated system, the system variations are constrained by K0 + ∆K ≥ 0 . As the kinetic energy generation is negative, the statically stable barotropic atmosphere may be changed only if the initial state possesses enough turbulent kinetic energy which can be converted into enthalpy by convection.
17.7. KINETIC EQUILIBRIUM STATE
357
17.7
Kinetic equilibrium state
17.7.1
General expressions
The isothermal or isentropic equilibrium state may be considered as the normal state of an ideal gas in a small volume. The classical thermodynamics is built on this equilibrium state. While, the atmospheres observed, simulated and forecasted every day by us are different from the heat-death or kinetic-death atmosphere, because the meteorological processes in the atmosphere are related to parcel motions which are constrained by mass and energy conservations and dynamic equilibrium relationships. Thus, not all the processes with increase of thermodynamic entropy may really happen in the geophysical fluid. The upper limit of thermodynamic entropy produced in a real processes may be evaluated by using the energy conservation law (17.17). In this case, the auxiliary function (17.13) is replaced by
X = ln θr + λ1 θr − λ2
vr2 pκ θ r + Γd pκs 2g
.
(17.18)
The extremal state of maximum thermodynamic entropy exists when ∂X =0 ∂vr
∂X =0. ∂θr
and
They lead to − λ2 vr = 0
(17.19)
and θr =
Γd pκs , λ2 pκ − λ
λ1 Γd pκs , λ2
λ=
(17.20)
respectively. If the initial state is statically stable, (17.19) gives vr = 0. The two constants λ2 and λ involved in the expression of θr are then solved from the equations λ2 = AΓd pκs Zr
' ps A pt
ps
θ0 dp dA ,
Zr =
pt
dp pκ − λ
(17.21)
and ps − pt Zr
ps A pt
ps
θ0 dp dA = +
A pt
Γd pκs 2g
(pκ − λ)θ0 dp dA
ps A pt
v02 dp dA ,
(17.22)
obtained by inserting (17.20) into (17.9) and (17.17), respectively. The derived extremal state constrained by energy conservation is referred to as the kinetic equilibrium state. The kinetic-death state discussed earlier is a particular example of the kinetic equilibrium state.
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
358
17.7.2
In statically stable atmosphere
We assume, now, an initial barotropic atmosphere of which the potential temperature is represented by ε1 θ0 = κ . (17.23) p − ε2 The temperature lapse rate gives Γ=−
Γd ε2 . pκ − ε2
Supposing θs0 = Ts at p = ps and Γ = Γm at pm = 500 hPa surface, we gain ε1 = Ts (pκs − ε2 ) ,
ε2 = −
pκm Γm . Γd − Γm
For the assumed initial state, (17.21) and (17.22) are replaced by λ2 = and (ps − pt )
Γd pκs Zr ε1 Z0
Z0 Γd pκs = ps − pt + (ε2 − λ)Z0 + Zr 2Agε1
respectively, where
ps
Z0 =
pt
pκ
ps A pt
v02 dp dA ,
dp . − ε2
The calculations of Z0 and Zr may be referred to Section 9.3. The two constants λ2 and λ can be solved from the previous three equations. Applying the results for (17.20) gives the kinetic equilibrium state θr =
ε1 Z0 , (pκ − λ)Zr
which possesses maximum thermodynamic entropy as the total energy is conserved. For convenience, we assume that the initial turbulent wind velocity v0 is constant over the domain. For Γm = 0.65 K/100m, the calculated two examples of kinetic equilibrium state with v0 = 25 and 50 m/s are depicted by the light and heavy solid curves respectively in Fig.17.2. The statically stable initial state is displayed by the dashed. The final static stability decreases with increasing initial turbulent kinetic energy. The thermodynamic entropy produced in the process leading to the kinetic equilibrium state is given, from (8.3), by ∆S˜ke = = = +
cp ps θr ln dp g pt θ0 ps Z0 (pκ − ε2 ) cp dp ln g pt Zr (pκ − λ) Z0 cp (ps − pt ) ln + κ(λZr − ε2 Z0 ) g Zr κ κ ps − ε2 pt − ε2 , − pt ln κ ps ln κ ps − λ pt − λ
17.7. KINETIC EQUILIBRIUM STATE
359
Figure 17.2: Potential temperature profiles of the kinetic equilibrium states attained through quasi-adiabatic process from the assumed initial state with Γm = 0.65 K/100m (dashed). The light and heavy solid curves are drawn for v0 = 25 and 50 m/s, respectively.
in a column of unit horizontal section area. For the assumed initial state, the maximum entropy production is depicted by the short dashed curves in Fig.17.3. The thermodynamic entropy increases as the turbulent kinetic energy is converted into heat and geopotential energies adiabatically through parcel motions in the statically stable atmosphere. This energy conversion provides an important sink of kinetic energy in the atmosphere which is different from friction or viscosity. Variations of the atmosphere with higher static stability and more turbulent kinetic energy at an initial state are generally more irreversible, as more thermodynamic entropy is created. The entropy production decreases to zero as the initial temperature lapse rate increases towards the adiabatic lapse rate. For comparison, we evaluated from (17.15) the increase of thermodynamic entropy as the same atmosphere reaches the isentropic state, in which ps pt
ln θ0 dp = (ps − pt ) ln ε1 − ps ln |pκs − ε2 | + pt ln |pκt − ε2 | + κ(ps − pt + ε2 Z0 ) (17.24)
and θ0 =
ε1 Z0 . ps − pt
(17.25)
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
360
Figure 17.3: Maximum thermodynamic entropy productions in the processes getting to kinetic equilibrium state (short dashed) and isentropic state (long dashed). The solid curve plots the change of air speed as the statically unstable atmosphere gets to the isentropic state.
The result gives ∆S˜k−d = −
cp g
(ps − pt ) ln
|Z0 | + ps ln |pκs − ε2 | − pt ln |pκt − ε2 | ps − pt
κ(ps − pt + ε2 Z0 )
.
(17.26)
and is depicted also in the figure by the long dashed curve. The entropy created as the stable atmosphere reaches the isentropic state is higher than that created as the same atmosphere reaches a kinetic equilibrium state. The kinetic-death state may not be attained if the stable atmosphere has no enough turbulent kinetic energy which can be converted into thermal enthalpy through parcel motions. An example of increasing thermodynamic entropy resulting from turbulent parcel motions in the statically stable atmosphere is the major stratospheric warmings, in which a large amount of kinetic energy is converted into heat energy through the vertical circulations induced. When the static stability is not very high, the short and long dashed curves in Fig.17.3 may overlap on each other. This means that on theory, the atmosphere
17.7. KINETIC EQUILIBRIUM STATE
361
may become isentropic after converting the turbulent kinetic energy at the low static stabilities. The amount of turbulent kinetic energy destroyed increases quickly with the static stability. When the turbulent kinetic energy is converted into enthalpy in parcel convection, the potential enthalpy is still conserved, so the potential temperature of the isentropic state attained is the mean of the initial state. Since kinetic energy in the atmosphere are not all the turbulent kinetic energy, a kinetic equilibrium state may possess the kinetic energy of horizontal motions, which cannot be converted into geopotential energy and heat energy without friction or viscosity. If Fig.17.3 is extended to Γ = 0, we may see that a great amount thermodynamic entropy may be produced as the isothermal state reaches the isentropic state. However, the isothermal atmosphere cannot become isentropic through the quasiadiabatic process without destroying a large amount of kinetic energy. On theory, the isothermal atmosphere with a large amount of turbulent kinetic energy may not be the final equilibrium state. The final equilibrium state is the motionless isothermal state.
17.7.3
In statically unstable atmosphere
When the initial atmosphere is statically unstable, we may find by experiments that there are no solutions of the extremal state for vr = 0. The solution may exist only if λ2 = 0 derived from (17.19). In this case, the kinetic equilibrium state of maximum thermodynamic entropy is the kinetic-death state with the constant potential temperature ε1 Z0 . (17.27) θr = θ0 = ps − pt derived using (17.25). Since the static stability increases as the unstable atmosphere reaches the kinetic-death state, available enthalpy is converted into parcel kinetic energy in the system. The kinetic energy in the isentropic atmosphere is evaluated from (17.20), giving Kr =
ps κ p θ0 A pt
v02 + Γd pκs 2g
dp dA − ε1 Z0 A
pκ+1 − pκ+1 s t . (κ + 1)(ps − pt )
For the current example, it follows that κ+1 κ+1 1 ˜ r = ε1 (ps − pt ) + ε2 Z0 − ε1 Z0 ps − pt + K (κ + 1)(ps − pt ) 2Ag
ps A pt
v02 dp dA
in the column over a unit horizontal section area. The solid curve in Fig.17.3 shows the change of air speed in the process as the unstable atmosphere gets to the isentropic state, assuming that the initial speed is zero. The kinetic energy at the kinetic-death state increases with initial static instability, and is less than that plotted in Fig.17.1 since the initial states of the two examples are different. The maximum thermodynamic entropy produced as the assumed unstable atmosphere reaches the isentropic state is evaluated from (17.26). The dependence on initial temperature lapse rate is displayed by the long dashed curve in Fig.17.3, which is similar to the dashed curve in Fig.17.1. The entropy production is zero
362
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
in the isentropic atmosphere and increases with initial static instability. It is symmetric to that in the statically stable atmosphere, since the entropy depends on the inhomogeneity of potential temperature field instead of the gradient direction. It is discussed that the heat-death or kinetic-death equilibrium state is unique for a provided initial state. It does not mean that the atmosphere may only reach the isentropic or isothermal state. The meteorological processes in both the statically stable and unstable atmospheres do to lead to the isentropic state in general. The atmosphere, especially the baroclinic or statically unstable atmosphere, may reach different equilibrium states depending on initial state and process. Comparing (17.20) with (9.12) or (11.8) finds that the expression of kinetic equilibrium state is the same as that of the reference states attained after energy conversions in the dry or moist atmosphere. These reference states with zero turbulent kinetic energy are also the examples of kinetic equilibrium state, although they are not attained through the extremal process with maximum thermodynamic entropy production from an excited initial state. The study of maximum available enthalpy in Chapter 9 gives the lower limit of thermodynamic entropy production and upper limit of kinetic energy generation in an isolated atmosphere. While, the study in this section presents the upper limit of thermodynamic entropy production and lower limit of kinetic energy generation. The entropy production and kinetic energy generation in the isolated real atmosphere are in between these limits. When the entropy production is smaller than the maximum entropy production discussed above, kinetic energy generation is greater than that shown in Figs.17.1 and 17.3. This can be seen by comparing Fig.9.3 with Fig.17.1. The kinetic energy generation in the barotropic atmosphere with Γ = 1.05 K/100m shown in Fig.9.3 is about 475 m2 /s2 , but is about 300 m2 /s2 in Fig.17.1. The difference between the minimum and maximum generations of kinetic energy may be very large, unless the static stability is close to the adiabatic lapse rate so that the maximum thermodynamic entropy produced in this situation is small as shown by Fig.17.3. A baroclinic atmosphere may produce a large amount kinetic energy in a pseudo- reversible process even if it is statically stable. While, the minimum kinetic energy produced as the same atmosphere becomes isentropic may be negative. The isothermal state and kinetic equilibrium state including the kinetic-death state are attained through different processes. The motionless heat-death state possesses more thermodynamic entropy in an isolated system, and is certainly important on theory as the isolated atmosphere may become isothermal actually. While, the kinetic equilibrium state is not a final equilibrium state in isolation. However, the changes from a kinetic equilibrium state to heat-death state may only be produced by molecular diffusions in isolation. For most meteorological processes on the scale of a few days, the parcel motions may be more significant than molecular diffusions, and so the kinetic equilibrium state and related features may be more interested than the heat-death state by meteorologists. It is the purpose of weather predictions and atmospheric simulations to find a right one among the multi-equilibrium states.
17.8. PRINCIPLE OF EXTREMAL ENTROPY PRODUCTIONS
363
Figure 17.4: Geopotential entropy changes as the assumed barotropic linear atmosphere reaches the kinetic-death state (solid) and heat-death state (dashed)
17.8
Principle of extremal entropy productions
Figs.17.1 and 17.3 show that thermodynamic entropy variations in the statically stable and unstable atmospheres are symmetric with respect to the isentropic atmosphere. However, the unstable atmosphere may change its thermodynamic state producing a large amount kinetic energy, while the statically stable and barotropic atmosphere may not change in isolation through quasi-adiabatic process, if the system has no enough turbulent kinetic energy at the initial state which can be converted into geopotential energy and heat energy. Thus, the change of thermodynamic entropy in an assumed process is not enough to tells us if the process may really occur or not in the atmosphere. We have introduced the geopotential entropy in Chapter 8 to study the possibility and irreversibility of an assumed variation in the atmosphere. The geopotential irreversibility measured by change of geopotential entropy may be independent of the thermodynamic irreversibility represented by thermodynamic entropy change. We evaluate first the geopotential entropy created as the linear barotropic atmosphere represented by (17.14) reaches the kinetic-death state, which is the extremal example of kinetic equilibrium state. The geopotential entropy of the initial state assumed is given by (9.38). Moreover, the geographic height of a given isobaric
17. STATES OF MAXIMUM THERMODYNAMIC ENTROPY
364
surface in the isentropic atmosphere is calculated from
cp θr 1− z= g
p ps
κ
,
obtained from (9.42) with zs = 0 and Ts = θr at ps = 1000 hPa. Inserting it into (8.49) yields the geopotential entropy of the atmosphere Sg r
cp =A g
pκ+1 − pκ+1 t s − ps + pt pκs (κ + 1)
.
It is the same as the right-hand side of (9.43) with A = 2. The geopotential entropy produced is plotted by the solid curve in Fig.17.4. We see again that a statically stable atmosphere may not reach the kinetic-death state without mechanic forcing, since geopotential entropy decreases in the process. Thus, the change of statically stable and barotropic atmosphere through quasi-adiabatic process may not happen in isolation, even if thermodynamic entropy increases in the process. This example shows the importance of using geopotential entropy as an additional discriminate parameter. The geopotential entropy created as the linear atmosphere assumed becomes isothermal through molecular diffusions is shown by the dashed in Fig.17.4. The geopotential entropy of the isothermal atmosphere is evaluated from (9.40). The entropy production has the same sign as temperature lapse rate of the initial state, so the entropy is destroyed by molecular diffusions in the inversion atmosphere. However, the heat-death state may still be attained, since the temperature change caused by heat conduction is independent of macroscopic Newtonian forces including the gravitational force and so no conversion of geopotential energy is produced. Although the static stability changes, the total energy is conserved and the partition between heat energy and geopotential energy is unchanged due to the fixed ratio between them. Thus, the geopotential entropy law may not be applied for the processes of molecular diffusion even if in the gravitational field, and the tendency of system variation caused by molecular diffusions without parcel motions can be examined with the thermodynamic entropy only. As geopotential entropy may be destroyed by molecular diffusions in an isolated system, it is possible for the atmosphere to increase its geopotential energy through a pure thermodynamic process without mechanic forcing. When the atmosphere is heated near the surface by the long wave radiation from the Earth, the static stability decreases and the gravity center rises as the atmosphere expands near the bottom. In other words, the increase of heat energy is linearly correlated to the increase of geopotential energy as discussed in Chapter 7. Thus, a part of radiation energy accepted is storied as the geopotential energy, a kind of mechanic energy. This process may happen in the statically stable atmosphere as it is not constrained by destruction of geopotential entropy. When the atmosphere becomes statically unstable by the heating again, parcel convection may take place and the geopotential energy increased is converted into kinetic energy together with a part of heat energy. The energy conversion in the baroclinic and statically stable atmosphere may be achieved by large-scale slantwise convection discussed in Chapter 19. From this point of view, the open atmosphere works like a heat engine, converting the radiation energy into kinetic energy.
17.8. PRINCIPLE OF EXTREMAL ENTROPY PRODUCTIONS
365
It is discussed in Chapter 9 that more kinetic energy is created in a process with less thermodynamic entropy production, and the reference state attained is more statically stable. Thus, more geopotential entropy is produced in the process with less thermodynamic entropy production. When the gravity effect is ignored for a small piece of ideal gas, the gas tends to reach the uniform state of maximum thermodynamic entropy. While in the gravitational field, the atmosphere tends to lower its gravity center and reach the kinetic equilibrium state through parcel motions, which possesses least thermodynamic entropy and maximum geopotential entropy. If molecular diffusions including kinetic energy dissipation are ignored, the baroclinic or statically unstable atmosphere may get to the lowest state discussed in Chapters 8 and 9. This implies that increasing geopotential entropy is prior to increasing thermodynamic entropy in the quasi-adiabatic process. Although the geopotential entropy law cannot disable the thermodynamic entropy law, it may constrain the thermodynamic processes in the atmosphere and drive the system towards the lowest state. Thus, we have the principle of extremal entropy productions, which tells that the parcel motions without anisobaric molecular diffusion in a large-scale inhomogeneous thermodynamic system lead to the lowest state with minimum thermodynamic entropy and maximum geopotential entropy. In other words, the idealized quasiadiabatic processes driven by the gravitational force produce minimum thermodynamic entropy and maximum geopotential entropy, compared with other processes with anisobaric molecular diffusion. This principle gives the reason for us to apply the variational approaches to study the energy conversions in Chapters 9-12. Although we had no this principle before, it was already used implicitly for the studies of energy conversions in the last century. Since the geopotential entropy law has no applications on molecular diffusions, this principle may not be applied for the processes effected by anisobaric molecular diffusion. For example, the atmosphere does not reach the lowest states after energy conversions, though both thermodynamic entropy and geopotential entropy increase. This is because there are the effects of molecular diffusions on the process of energy conversion. When parcel motions are the major cause of atmospheric variations, the principle suggests that the atmosphere tends to the equilibrium state of relatively low thermodynamic entropy and high geopotential entropy.
Chapter 18 Energetics of linear disturbance development 18.1
Introduction
In the analytical studies, development of baroclinic disturbance in the atmosphere was studied with the linear theories initiated by Charney (1947) and Eady (1949). It is assumed usually that the large-scale perturbations in the atmosphere becomes unstable when the imaginary part of the propagation velocity is not zero. The wave instability may happen under certain conditions, for example, if horizontal gradient of atmospheric temperature exceeds a certain limit. This instability is called the baroclinic instability. According to the study of McHall (1993), the critical baroclinity for baroclinic instability depends on the static stability. In other words, the wave instability depends on three-dimensional temperature gradient in the atmosphere. The traditional linear theories of wave instability were created by ignoring the interactions between the disturbances and their energy sources in the background fields. Otherwise, the linear disturbances can never develop, because the destabilized waves become stable again as soon as the background baroclinity is reduced by the interactions. When the disturbances in the linear theories grow exponentially with time at a constant rate, there must exist an infinite energy source in the perturbation field for the development. Development of baroclinic disturbance in the atmosphere depends on the sources of kinetic energy and negative thermodynamic entropy. The climatological studies in Chapters 13 and 14 show that the extratropical baroclinic and tropical barotropic storm tracks are correlated to the excited regions with abundant dry and moist available enthalpy respectively. The case analysis in Chapter 16 shows that available enthalpy and negative baroclinic entropy in the local atmosphere decrease in development of baroclinic disturbance. When the meteorological variables are resolved into the mean background fields and perturbations, changes in the background fields caused by development of the perturbations manifest the interactions between the eddies and background fields, which are in general controlled by nonlinear mechanisms. Due to the mathematical difficulties, studies of the nonlinear interactions were achieved mostly by numerical procedures. The numerical experiments of Simmons and Hoskins (1978) suggested that the linear theory was of significance because the explanation for disturbance growth may extend over most of the intensification period, though nonlinear effects were important in the mature and decay phases of cyclone life cycles. The study on the interactions between perturbations and zonal mean fields is significant, because the perturbation fields could not provide the energy source strong enough for development of baroclinic disturbances. As pointed out by Ludlam 366
18.1. INTRODUCTION
367
(1980), the theory of cyclogenesis as a manifestation of a dynamic instability of polar front was not successful, because the disturbance development studied by linear theory was independent of energy conversion. When the interactions are ignored, the development may only be supported by the energy sources in the perturbation fields. This cannot be confirmed by observations. In fact, the meridional gradient of zonal mean temperature decreases in development of extratropical cyclones. If the energy conversion is not considered, the destabilized waves may either grow or decay and we may not know whether the perturbations may really develop. The traditional linear theories could not illustrate the decay phase, as it was assumed that all unstable waves could develop. In fact, unstable waves may grow only if the perturbation kinetic energy can be created in the process. To demonstrate truthfully the baroclinic disturbance development in the atmosphere, the interactions between the disturbances and background fields cannot be ignored. As the development depends on the background energy sources, the interactions can be studied using the energy conservation law discussed earlier. The dynamic mechanism of kinetic energy generation in baroclinic waves will be revealed in detail in Chapter 21 with the air engine theory introduced. In this chapter, only the energy budget is considered. Development of disturbance weakens the background energy sources, and the decrease of background baroclinity lowers the growth rate in turn. The change of growth rate related to the changes of background fields was considered once by Hart (1971). For the mathematical difficulties, his study was confined to a simplified Eady type disturbance. Also, the changes of background fields were independent of disturbance development in his study, so the feedback of perturbation to the background fields was not considered. The interactions between baroclinic disturbances and background fields may be studied with the variational approach discussed in Chapter 9, assuming a baroclinic state to be the reference state (McHall1, 1992). Developments of baroclinic disturbance in the troposphere are usually a local process, and energy conversions in the atmosphere are generally nonlinear. The partition of energy source into zonal mean and eddy components for the local processes, as in the algorithm of Lorenz (1955), may be artificial or misleading, as the zonal means depend on the features of remote atmosphere. For example, conversion of available enthalpy in the dry atmosphere is accomplished by subsidence of cold air and rise of warm air in a local region. The vertical velocities in a local disturbance may be one order higher than those in the stable regions, and may be underestimated seriously if represented by the departures from the zonal mean. The current variational algorithm may evaluate a local energy source using local data only. The disturbance development studied with local energy sources may not have the artificial influences of remote atmospheres. The growth rate and wave spectra of unstable perturbations derived from some previous linear theories will be applied for the current study of wave and mean field interactions. The restriction on disturbance development resulting from reducing the baroclinity in the background fields was studied theoretically by McHall (1993). He found that development of baroclinic linear disturbances could continue through the linear adjustment which removes the restriction. Applying the data of energy conversion obtained from a statistical survey, we may illustrate the evolution of baroclinic disturbance on the phases of development and decay. The corresponding
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
368
changes of the background baroclinity may be demonstrated also. The developments of Eady waves, synoptic geostrophic waves of McHall (McHall, 1993), blocking waves and planetary geostrophic waves in the stratosphere (McHall, 1993) will be discussed with the interaction theory. The results obtained may be applied to examine the linear theories, by comparing with the real processes of disturbance development.
18.2
Conversion of available enthalpy
18.2.1
Method A
The available enthalpy discussed in Chapter 9 is evaluated with respect to a barotropic reference state. To study the process of disturbance development step by step, we have to evaluate the energy conversion between two baroclinic states. The baroclinic reference state may be represented generally by θr = θr + θˇr ,
(18.1)
where θˇ is the departure of potential temperature from the horizontal mean θr . With this relationship and (8.7), the equation of available enthalpy (9.7) is rewritten as ps A pκ (θ0 − θr ) dp . (18.2) Λ= Γd pκs pt It is noted in Chapter 9 that when molecular diffusions across pressure surfaces are filtered out, air motions in an adiabatic atmosphere satisfy the conservation law of system potential enthalpy (17.9), that is ps pt
(θr − θ0 ) dp = 0 .
(18.3)
Moreover, the thermodynamic entropy produced in the process of energy conversion is calculated from (8.3), or 1 ∆S = Γd
ps pt
θr 1 + A ln θ0 2
ˇ2 θ
θˇr2 − θ0 2 θr 2 0
A
dA dp .
(18.4)
The last two equations give the constraints for evaluation of available enthalpy with the variational approach discussed in Chapter 9. To find the baroclinic reference state which possesses minimum enthalpy, we make the auxiliary function X = pκ θr − λ1 lnθr − λ2 θr , where λ1 and λ2 are two constant Lagrangian multiplies. Its Euler equation gives d ∂X − ∂θr dp
∂X ∂θr p
=0,
It follows that θr =
pκ
λ1 . − λ2
θr p =
∂θr . ∂p
(18.5)
18.2. CONVERSION OF AVAILABLE ENTHALPY
369
We discussed in Chapter 9 that the change of potential temperature in an isolated atmosphere is limited by the thermal constraint (9.9) in the quasi-adiabatic process with conservation of potential enthalpy. Thus, the evaluated reference states will be examined by the thermal constraint. To solve the two constants λ1 and λ2 , we insert (18.5) into (18.3) and (18.4) giving ps ps dp 1 θ0 dA dp , Z= (18.6) λ1 = κ AZ pt A pt p − λ2 and ps ln(pκs − λ2 ) − pt ln(pκt − λ2 ) − (ps − pt )(ln λ1 + κ) + Γd ∆S˜ − κλ2 Z ps
1 lnθ0 − 2A
+ pt
ˇ2 θ
θˇ2 − r2 (pκ − λ2 )2 dA dp . 2 θ0 λ1 0
A
(18.7)
Here, evaluation of Z is referred to (9.16). The available enthalpy with respect to the baroclinic reference state is given by (18.2), which is rewritten as 1 Λ= Γd pκs
ps A pt
p θ0 dp dA − Aλ1 (ps − pt + λ2 Z) κ
(18.8)
using (18.5). This equation gives the dependence on the baroclinity of reference state, that is Λ = Λ(θˇr ). To give an example, we assume that the atmosphere possesses constant temperature lapse rate Γ at an initial state. The potential temperature is given by θ0 = θ0 + θˇ0 , where
ps θ0 = Ts p For this linear atmosphere, we have ps pt
A
b
,
b=
R (Γd − Γ) . g
ATs ps θ0 dA dp = ps − pt 1−b pt
Applying it for (18.6) gives
Ts ps ps − pt λ1 = (1 − b)Z pt
(18.9)
b
.
b
.
(18.10)
Moreover, using ps A pt
pκ θ0 dp dA =
yields Λ = −
A Γd pκs
Ts0 pbs (pκ−b+1 − pκ−b+1 ), t (κ − b + 1) s
Ts0 pbs (pκ−b+1 − pκ−b+1 ) t (κ − b + 1) s
ps Ts ps − pt (1 − b)Z pt
b
(ps − pt + λ2 Z)
.
(18.11)
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
370
Figure 18.1: Dependence of kinetic energy created on reduction of baroclinity. The temperature lapse rate of initial state is denoted by Γ. Solid and dashed curves are drawn for ∆s∗ = 0, 0.01 J/(K2 ·kg), respectively.
When the baroclinity isrepresented by the root-mean-square of potential temperature deviation θR = θˇ2 at pressure surface pc (pt < pc < ps ), we gain approximately ps pt
A
2b 2 AθR pt θˇ02 0 dA dp = p − p s t 2 2 θ0 (2b + 1)Ts ps ps
and 2 AθR 0
pt
p2κ+1 s
p2κ+1 t
− 2κ + 1
A
,
θˇr2 (pκ − λ2 )2 dA dp =
2λ2 κ+1 − (p − pκ+1 ) + λ22 (ps − pt ) t κ+1 s
.
Furthermore, we have ps pt
lnθ0 dp = (ps − pt ) ln(Ts pbs ) − Ab(ps ln ps − pt ln pt − ps + pt ) .
(18.12)
Using these equations for (18.7) may give the constant λ2 . The reference state and available enthalpy may then be obtained from (18.5) and (18.11), respectively. For Ts = 280 K and θR0 = 6 K, we have evaluated the maximum kinetic energy created in the process with minimumthermodynamic entropy production in the dry
18.2. CONVERSION OF AVAILABLE ENTHALPY
371
atmosphere from 1000 hPa to 200 hPa. The dependence of kinetic energy generation on destruction of baroclinity, represented by ∆θR = θR0 − θRr , is displayed by the solid curves in Fig.18.1. The kinetic energy created in a highly irreversible process is also plotted by dashed in the figure. The entropy production ∆s∗ assumed in this figure indicates the ensemble mean produced in the process of reducing one degree of θR . Fig.18.1 shows that the baroclinity is weakened by energy conversion. More kinetic energy is produced in the atmosphere with higher baroclinity and static instability. For a given initial field, kinetic energy increases continuously as the baroclinity decreases in the extremal process with minimum thermodynamic entropy production. However, it may not be the case in the highly diffusive process since the turbulent and molecular diffusions may destroy kinetic energy and the energy source. The dashed curves show that kinetic energy generated may be less than that destroyed by the diffusions when the baroclinity is low.
18.2.2
Method B
The thermodynamic entropy produced in the atmosphere may also be evaluated from (8.3) approximately. In general a baroclinic initial and reference fields may be represented by θ0 = θ0 +θ0∗ + θˇ0 and
θr = θr +θr∗ + θˇr
(18.13)
respectively, in which θ ∗ = θ− θr = θ− θr depends on pressure only. Applying these expressions yields θr ln θ0
θ ∗ + θˇr θ ∗ + θˇ0 = ln 1 + r − ln 1 + 0 θ0 θ0 1 (θ ∗ − θ0∗ + θˇr − θˇ0 ) ≈ θ0 r 1 2 2 (θr∗ − θ0∗ + θˇr2 − θˇ02 + 2θr∗ θˇr − 2θ0∗ θˇ0 ) . − 2 2 θ0
Inserting it into (8.3) gives ∆S ≈
cp 2g θ0 2
ps pt
2
2
A(θ0∗ − θr∗ ) +
A
(θˇ02 − θˇr2 ) dA
dp .
(18.14)
The thermodynamic entropy production cannot be negative in an isolated atmosphere. For the baroclinic reference state, available enthalpy is evaluated from (9.23) with the constraints (8.13) and (18.14). To find the reference state with the variational approach, we make the auxiliary function 2
X = pκ θr∗ − λ1 θr∗ − λ2 θr∗ .
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
372
Figure 18.2: The same as Fig.18.1, but evaluated with Method B. The Euler equation gives θr∗ =
pκ − λ2 . 2λ1
(18.15)
Inserting it into (8.13) and (18.14) yields λ2 =
pκ+1 − pκ+1 t s (κ + 1)(ps − pt )
and
(18.16)
1 λ1 = − 2
C , G
(18.17)
dp − 2Γd θ0 2 ∆S˜ .
(18.18)
in which C is given by (9.27), and ps
G= pt
2
θ0∗ +
1 A
A
(θˇ02 − θˇr2 ) dA
The evaluation of G is further referred to Section 9.5. Consequently, the reference state is rewritten as
θr = θ0 +
G C
pκ+1 − pκ+1 t s − pκ + θˇr , (κ + 1)(ps − pt )
(18.19)
18.3. GROWTH OF LINEAR DISTURBANCES
373
derived by inserting (18.15)-(18.17) into (18.13). The maximum available enthalpy with respect to the reference state is √ ps A κ ∗ Λ= CG + p θ0 dp . Γd pκs pt
(18.20)
It gives the dependence of kinetic energy generation on the three-dimensional thermal structure of a baroclinic reference state. For the initial field and θR0 assumed previously, the evaluated available enthalpy is displayed in Fig.18.2, which is almost the same as Fig.18.1. The approximate approach can be used easily in practice. The energy conservation equation (18.2) provides a test for the linear theories on baroclinic disturbance development. This is discussed in the following sections.
18.3
Growth of linear disturbances
18.3.1
Energy constraint equation
When a three-dimensional wind velocity is resolved into the zonal mean (denoted by an overbar) and eddy (denoted by a prime) components, kinetic energy variation of unit air mass gives ∆k =
v¯r2 − v¯02 vr2 − v02 + + v¯r vr − v¯0 v0 , 2 2
as the atmosphere changes from an initial state to a reference state. Integrating it over the atmosphere from ps to pt with horizontal area A produces ∆K = ∆K + ∆K with ∆K =
1 g
and 1 ∆K = g
So, (17.17) is replaced by
ps 2 v¯r − v¯02 A pt
2
ps 2 vr − v02 A pt
2
(18.21)
dp dA
dp dA .
(18.22)
∆K + ∆K = Λ .
(18.23)
Λ = −∆Ψ
(18.24)
Here, is available enthalpy given by the difference of total enthalpy in the atmosphere between initial and reference states. In the linear theories, growth of perturbation is represented usually by v = v0 e±µt , and where v0 = v|t=0
ˇ aj ) µ = µ(θ,
(18.25)
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
374
measures the growth rate which depends on baroclinity in general (aj are other parameters involved). Equation (18.23) may be called the energy constraint equation of disturbance development. The available enthalpy on the right-hand side is not divided into the zonal mean and eddy components, as energy conversion in the atmosphere is nonlinear and there are interactions between the perturbations and mean fields in the process.
18.3.2
Time-dependent expression
To consider the change of growth rate with the background temperature fields, the growth time is divided into n intervals, which are sufficiently small so that the growth rate over an interval may be considered approximately as constant. Consequently, the growth of perturbation velocity is rewritten as
vn
= v0 exp ±
n
µi ∆ti
.
(18.26)
i=1
The subscript i in this and following equations except for t denotes the time mean over the interval ∆ti = ti − ti−1 . Inserting (18.26) into (18.22) yields 1 ∆K = 2g
ps A pt
v02
exp ±2
n
µi ∆ti − 1 dp dA .
(18.27)
i=1
Moreover, variation of zonal mean kinetic energy is estimated from (Lorenz, 1955) dK = {Ψ · K} + {K · K} − D , (18.28) dt where {Ψ · K} indicates the conversion rate of available enthalpy into zonal mean kinetic energy; {K · K} is the conversion rate of eddy kinetic energy into zonal mean kinetic energy, and D=−
1 g
ps
v · F dp dA
A pt
represents the dissipation rate of zonal mean kinetic energy. Again, the energy source Ψ is not resolved into zonal mean and eddy components. If v¯y = v¯z = 0, there is {Ψ · K} = 0 and so ∆ti ∆K i = g
ps
vxi v ∂¯ vxi yi ∂y
A pt
+
vxi v ∂¯ vxi zi ∂p
dp dA − Di ∆ti ,
where vx , vy and vz are the velocity components in the x, y and z directions respectively. The energy constraint equation derived by inserting the previous equation together with (18.27) into (18.23) gives ps A pt
vxi v ∂¯ vxi yi ∂y
+
vxi v ∂¯ vxi zi ∂p
v 2 ∆ti + i−1 (e±2µi ∆ti − 1) dpdA 2
18.3. GROWTH OF LINEAR DISTURBANCES
375
= −g∆Ψi − gDi ∆ti in which Di = −
ps
1 g
A pt
(18.29)
vi · Fi dp dA
represents the dissipation rate of perturbation kinetic energy, and − ∆Ψi =
1 Γd pκs
ps A pt
pκ (θi−1 − θi ) dp dA .
(18.30)
It can be proved that −∆Ψi = Ψi−1 − Ψi . If using Method B, we have, from (18.20), ∆Ψi =
A √ C( Gi − Gi−1 ) , κ Γd p s
(18.31)
where C and G are evaluated from (9.27) and (18.18) respectively. The energy constraint equation (18.29) includes perturbation velocities. If the perturbation velocity components are unknown, the process may still be studied with the energy transfer equations given by previous studies. This is discussed in the following.
18.3.3
Alternative expression
Equation (18.29) includes perturbation velocity components. If we discuss the energetics in baroclinic disturbance development without considering the structure of perturbations, we may use an alternative relationship instead of (18.29). According to Lorenz (1955), eddy kinetic energy generation in a linear theory may also be estimated from dK = −{K · K} + {Ψ · K } − D . (18.32) dt Here, {Ψ · K } denotes the conversion rate of available enthalpy into eddy kinetic energy. When {Ψ · K } = 0, we define energy conversion ratios as ηe =
{K · K} , {Ψ · K }
ηz =
{Ψ · K} , {Ψ · K }
ηd =
D . {Ψ · K }
(18.33)
Applying them for (18.32) produces {Ψ · K } =
1 dK . 1 − ηe − ηd dt
{Ψ · K} =
ηz dK . 1 − ηe − ηd dt
Thus,
While, conversion of available enthalpy may be divided into the two components (Lorenz, 1955) dΨ = −{Ψ · K} − {Ψ · K } dt
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
376
in an isolated atmosphere. This equation can also be obtained by substituting (18.21), (18.32) and (18.28) into (17.17), giving dΨ dK = −η , dt dt
η=
1 − ηe − ηd . 1 + ηz
It follows, from (18.27), that ps 2 v0 A pt
2g
exp ±2
n
µi ∆ti − 1 dp dA +
i=1
n
ηi ∆Ψi = 0 .
(18.34)
i=1
Here, −∆Ψi is the available enthalpy Λi between the reference states θi−1 and θi as shown by (18.24). When {Ψ · K } = 0, (18.32) gives the kinetic energy dissipation equation ps A pt
2 vi−1 (e±2µi ∆ti − 1) dp dA + 2g({K · K}i + Di )∆ti = 0 .
(18.35)
The perturbation kinetic energy is destroyed by dissipation and conversion into the mean kinetic energy. An advantage of using (18.34) instead of (18.29) is that we may evaluate the development of perturbation without particular information on perturbation velocity field.
18.3.4
Numerical procedures
The initial baroclinity θˇ0 is divided into n segments, for example
i θˇi = θˇ0 1 − n
(i = 1, 2, · · · n) .
Applying the baroclinity parameter θˇi and its gradient at time ti for (18.25) yields the growth rate µi . The i-th reference state is given by θi = θi + θˇi obtained from (18.1) for Method A. Here we have, from (18.5), θi =
λ1i , pκ − λ2i
where the two constants are evaluated from (18.6) and (18.7), that is λ1i
1 = AZi
ps pt
A
ps
θ0 dA dp ,
Zi = pt
dp , pκ − λ2i
and ps ln(pκs − λ2i ) − pt ln(pκt − λ2i ) − (ps − pt )(ln λ1i + κ) + Γd ∆S˜i − κλ2i Zi
ps ˇ2 1 θˇi2 κ θ0 2 + lnθ0 − − (p − λ2i ) dA dp . 2A A θ0 2 λ21i pt
18.4.
EADY WAVE DEVELOPMENT
377
The amount of available enthalpy depends on process or thermodynamic entropy production. For the following examples in this chapter, the maximum available enthalpy attained in the process with minimum thermodynamic entropy production will be applied. When the background baroclinity decreases from θi−1 to θi , the available enthalpy is calculated from (18.30). Inserting the growth rate µi and available enthalpy Ψi into the energy constraint equation (18.34), we may solve numerically the time step ∆ti in the process with specified thermodynamic entropy production. Applying the results for (18.26) acquires further the perturbation velocity at t = ti . Comparing the obtained results with observations allows us to examine the growth rate obtained theoretically. For the approximate approach in Method B, The i-th reference state is, from (18.19), Gi pκ+1 − pκ+1 s t κ θi = θ0 + − p + θˇi , C (κ + 1)(ps − pt ) ps
where Gi =
pt
2 θ0∗
1 + A
A
(θˇ02 − θˇi2 ) dA
dp − 2Γd θ0 2 ∆S˜i .
derived from (18.18). The available enthalpy is then given by (18.31).
18.4
Eady wave development
18.4.1
Evaluation equations
The growth rate of the most unstable Eady wave was given approximately by (Lindzen and Farrell, 1980; Hoskins, 1983) µ = 0.31
g ∂ θ¯ , νz θ¯ ∂y
(18.36)
where θ¯ is zonal mean potential temperature, and
νz =
g ∂ θ¯ θ¯ ∂z
¯ is called the Brunt-Vaisalla frequency in the next chapter. It is assumed that ∂ θ/∂y in the previous equation is independent of pressure. This expression may be applied only if the baroclinity exceeds the critical limit of wave instability. The critical baroclinity can be found from the original study of Eady (1949). In the statically stable atmosphere, we may write gp ∂θ =− ∂z Rθ
ps p
κ
∂θ . ∂p
If using Method A discussed previously, we apply (18.1) and (18.5), giving Γd p κ ∂θ = κ s , ∂z p − λ2
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
378
where λ2 is solved from (18.7). Here, the vertical gradient of θˇr is ignored for simplicity. The growth rate of Eady wave is given by
µ = 0.31(p − λ2 ) κ
cp ∂θ , λ1 pκs ∂y
in which λ1 is evaluated from (18.6). Also, we may use (13.1) and (18.15) for Method B and gain gpκs ∂ θ¯ =− , ∂z cp (2λ1 θ0 −λ2 + pκ ) where λ1 and λ2 are computed from (18.17) and (18.16) respectively. The growth rate gives then λ1 ∂θ . µ = 0.31 −2cp κ ps ∂y It depends on the background baroclinity. We assume that this expression is unchanged at the stage when the development is dominated by linear process. Supposing the half width of disturbance along the zonal mean temperature gradient is Y , we define 1 θR2 = 2Y
Y ¯ 2 ∂θ
∂y
−Y
It follows that
y
Y2 dy = 3
∂ θ¯ ∂y
2
.
(18.37)
∂ θ¯ √ θR = 3 . ∂y Y
For Y = 1000 km and θR = 6 K, it gives |∂T /∂y| ≈ 10.4 K/1000km on the surface. This temperature gradient is typical for development of synoptic baroclinic disturbances. The energy conversion in a weather system varies with time and space generally. Usually, only the ensemble means of energy conversion rates were available from the statistical analyses using real data (e. g., Oort, 1964; Oort and Peixoto, 1974; Tenenbaum, 1976; Tomatsu, 1979). If we are interested in the gross picture other than the details of disturbance development, we may use these mean rates to estimate the ratios in (18.33). With (18.20), (18.34) is replaced by ps A pt
2 vi−1 (e±2µi ∆ti − 1) dp dA = −2g ηi ∆Ψi .
(18.38)
Here, ∆Ψi is calculated from (18.31). When −∆Ψi ≥ 0, we choose the positive sign, otherwise the negative sign is used. To examine the linear theories of disturbance development, we evaluate the perturbation velocity v with (18.38), giving 2 (e±2µi ∆ti − 1)A(ps − pt ) = −2g ηi ∆Ψi . vi−1
It follows that ±µi ∆ti
e
=
1−
2g ηi ∆Ψi . A(ps − pt ) vi−1 2
(18.39)
18.4.
EADY WAVE DEVELOPMENT
379
Then we obtain, from (18.26),
vi
=
vi−1
1−
2g ηi ∆Ψi . A(ps − pt ) vi−1 2
(18.40)
Inserting the baroclinity parameter θˇi into this equation gives the perturbation velocity varying with background baroclinic field. The time variation of the background ˇ i ), where baroclinity is given by θˇi = θ(t ti = ti−1 ±
1 vi . ln µi vi−1
Furthermore, (18.25) and (18.31) give, respectively, the time variations of growth rate and available enthalpy, namely µi = µ(ti ) and ∆Ψi = ∆Ψ(ti ). Thus, (18.40) tells also the time dependence of the perturbation. When available enthalpy is exhausted, the kinetic energy dissipation equation (18.35) becomes ( ) ) ±µi ∆ti e = *1 − 2g∆ti
and so vi = vi−1
{K · K}i + Di , A(ps − pt ) vi−1 2
( ) ) *1 − 2g∆t
i
{K · K}i + Di . A(ps − pt ) vi−1 2
(18.41)
This equation illustrates the decay phase of unstable waves.
18.4.2
Examples
The rates of energy conversions may be obtained from previous studies. Song (1971) investigated, with a linear quasi-geostrophic model, the energetics of unstable disturbances in several zonal currents containing vertical and horizontal shears. The mean energy conversion rate {Ψ, K } averaged from his three baroclinic experiments BC, AS and P B was 1.34 W/m2 for wavelength 4000 km. The energy conversion ratio ηe obtained from his energy cycle is about 0.37, and may be negative in barotropic disturbances (Song, 1971; Brennan and Vincent, 1980). The ratio ηz is zero in either the Song’s model or the observational analysis of Saltzman (1970). The dissipation of eddy kinetic energy was not provided by Song. The evaluations by other authors were made for a steady state (e. g., Oort, 1964; Saltzman, 1970; Sheng and Hayashi, 1990a, b). The obtained result varies greatly and cannot be used directly to evaluate the energy conversion in disturbance development. For simplicity, we assume ηd = 0, and so have η = 0.63. For the initial field assumed previously, development of Eady wave is calculated with Method A. The available enthalpy is evaluated in an isolated air column from 1000 hPa to 200 hPa. The width along the temperature gradient is taken as 2Y = 2000 km. The variation of background baroclinity at 1000 hPa is displayed by the solid curves in Fig.18.3(a). The dashed curves present the e-folding time τ = 1/µ. The growth of mean perturbation speed calculated from (18.40) for different initial
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
380
values is demonstrated in Fig.18.3(b). For comparison, the waves growing at the constant initial rate are illustrated by the dashed in the figure. They are significantly different from those constrained by energy conservation, especially near the mature stage. For the quasi-geostrophic disturbances, we have vx ∼ −
ny 1 ∂φ ∼ φ , f0 ∂y f0
vy ∼
nx 1 ∂φ ∼ φ , f0 ∂x f0
(18.42)
where nx and ny are zonal and meridional wavenumbers respectively. Supposing nx and ny are similar in cyclonic disturbances, and the perturbation kinetic energy is measured mainly by horizontal motions, we see f0 φ ∼ √ v . 2nx If wavelength of the unstable wave is twice the width of disturbance, we have 2π/nx = 4Y and √ v (18.43) φ ∼ 2f0 Y . π The later gives φ = φ0 exp ±
n
µi ∆ti
,
i=1
where φ0 denotes the initial perturbation geopotential represented by √ v0 . φ0 = 2f0 Y π The growth of perturbation height at latitude 45◦ is sketched by the solid curves in Fig.18.3(c). Again, the wave development constrained by energy conservation is different from the growth at a constant initial rate plotted by the dashed. Usually, development of large-scale disturbances in the extratropical troposphere takes one or two days to get to the mature stage. In the explosive episode of tropospheric bombs, surface pressure may decreases more than 24 hPa in 24 hours at latitude 60◦ . In contrast, growth of Eady waves depicted in Fig. 18.3 is relatively slow. The wave amplitude increases less than 50 meters in two days. Palm´en and Newton (1969) pointed out that only about four or five developing cyclones of typical size and intensity were required to account for the entire kinetic energy generated in the extratropical cap of the hemisphere. The kinetic energy budget in the extratropical cyclones has been estimated by many authors (referring to the book of Wiin-Nielsen and Chen, 1993). The evaluated kinetic energy generation varied from case to case in a large range as reviewed by Smith (1980). The mean value obtained by Kung and Baker (1975) for 780 cases in 5 years was 8.84 W/m2 and 9.55 W/m2 for developing and mature cyclones, respectively, integrated from 1000 hPa to 100 hPa. The mean of the two numbers is 9.2 W/m2 , which is equivalent to the creation of wind speed about 13.2 m/s per day. Comparing it with Fig.18.3(c) finds that the Eady wave grows considerably slower than the real processes. The growth rate is even lower if the kinetic energy dissipation is not ignored. That the slow developments of Eady (1949) and Charney (1947) waves were also noted by other authors (e. g., Whitaker and Barcilon, 1992). A faster model of synoptic disturbance development is discussed in the next section.
18.4.
EADY WAVE DEVELOPMENT
381
(a)
(b)
382
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
(c) Figure 18.3: Development of Eady wave in the irreversible process with ( ∆s∗ = 2 × 10−3 J/(K2 ·kg)), (a) background baroclinity at 1000 hPa (solid) and e-folding time (dashed), (b) perturbation speed, and (c) perturbation height
18.5
Synoptic geostrophic wave development
The baroclinic disturbances studied in the traditional linear theories cannot grow actually, because there is a lower limit of baroclinity for the linear instability. As soon as the temperature gradient exceeds the limit, the disturbance develops and then the baroclinity decreases. As a result, the wave becomes stable again. According to the study of McHall (1993), the limit of baroclinity on wave instability may be removed by a linear adjustment after the wave is destabilized. The disturbances may then grow continuously to the mature stage until the atmosphere becomes barotropic. This can be seen from the e-folding time of synoptic geostrophic wave development after the linear adjustment (McHall, 1993):
2 cot ϕ τ= |T y |ny (1 + 2λny )
T [(1 + 2λny )C + λny cos2 ϕ] (Γd − Γ) g [(1 + 2λny )C − λny cos2 ϕ]
with C = 1 + λny (ny cos ϕ + sin ϕ) sin ϕ . Here, λ is the ageostrophic coefficient of synoptic geostrophic waves and is assumed constant, and ny is the non-dimensional meridional wavenumber. The e-folding time increases as the baroclinity decreases. The development of disturbance after the adjustment is not restricted again by reducing the baroclinity. Applying λ = 0.02 (McHall, 1993) together with the same initial atmosphere and energy conversion ratios assumed for the Eady wave, development of synoptic geostrophic disturbance calculated using Method A is demonstrated in Fig.18.4.
18.5. SYNOPTIC GEOSTROPHIC WAVE DEVELOPMENT
(a)
(b)
383
384
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
(c)
Figure 18.4: Development of synoptic geostrophic disturbances after linear adjustment with ( ∆s∗ = 2 × 10−3 J/(K2 ·kg)), (a) background baroclinity at 1000 hPa (solid) and e-folding time (dashed), (b) perturbation speed, and (c) perturbation height
The time variations of baroclinity and e-folding time at 1000 hPa and latitude 45◦ are depicted, respectively, by the solid and dashed curves in (a). When the available enthalpy converted is transferred into perturbation kinetic energy, growth of perturbation speed estimated from (18.40) and (18.41) is illustrated in (b). Meanwhile, amplification of height perturbation evaluated from (18.43) is displayed in (c). This figure shows that development of the geostrophic disturbance is faster than development of Eady wave discussed in the preceding section. It is found that the background baroclinity decreases at an increasing rate in wave development. In a process of provided thermodynamic entropy production, conversion of available enthalpy is more efficient and the e-folding time increases more quickly in a stronger disturbance. The fastest deepening episode occurs at a late stage of development. This can be proved by the explosive cyclone discussed in Chapter 16. If other conditions are the same, the system which is stronger at initial time grows more rapidly and decays earlier. In the examples discussed, the baroclinic disturbances get the limited amplitudes as plotted in Fig.18.4, because the evaluations are made in a local area without considering the displacement along a baroclinic energy source. When a cyclone moves in a frontal zone, it may develop longer in the time and get a larger amplitude than those plotted. Fig.18.4 shows also that the baroclinic disturbances decay quickly after the mature stage at a rate similar to that in development. The kinetic energy is dissipated by turbulent and molecular diffusions in the decay process.
18.6. DEVELOPMENT OF BLOCKING WAVES
18.6
385
Development of blocking waves
The large- and planetary-scale circulations in the extratropical atmosphere are characterized by stationary and transient waves embedded in the zonal mean fields. The planetary wave circulation in the troposphere is frequently broken down and replaced by the anomalous circulation pattern called the blocking system. According to McHall (1993) the blocking highs are the alternative of the normal perturbation pattern, and are necessary for maintenance of the time-mean zonal momentum balance in the troposphere under the effects of topographic forcing. It was discussed also in that study that occurrence of blocking is associated with nonlinear planetary wave instability. When the baroclinity enhances in the middle troposphere, a layer called the breaking layer may occur, where a planetary geostrophic wave is split into two single components. According to wave energy conservation or heat and momentum balances, the amplitude of superimposed height perturbation is given by z 2 = z12 + z22 , in which z1 and z2 are the amplitudes of the split perturbations. The superimposed wave in the breaking layer may be destabilized if the baroclinity is strong enough. This nonlinear process of planetary wave development in the troposphere is characterized by the main features of observed blocking events. The blocking theory of McHall (1993) based on the nonlinear planetary wave instability may also be examined by the energy constraint equation discussed earlier. The growth rate of the unstable planetary waves derived by McHall is µ=− with kT2 = −4
u ¯ 2 kT (4f 2 Q˙ 2 − ε2 -2 g4 kT2 n2y z 4 /a2 ) 2f Q˙ ε2 cv Ω2 n2y
a(1 + ε)4 cp RT y
sin3 ϕ cos ϕ ,
-=
cv . cp
Here, Q˙ ≈ RT˙ d is the zonal mean rate of diabatic heating corresponding the the mean temperature change rate T˙ d ; a is the radius of the Earth, and ε is the ageostrophic coefficient of planetary geostrophic waves. As the growth rate depends on wavenumber, the development is inhomogeneous in blocking wave spectrum. Also, the growth rate decreases with increasing wave amplitude, so the planetary wave development is self-limited. To give examples, we use the thermal wind relationship u ¯=u ¯s −
R ps T y ln f p
(18.44)
to evaluate the mean zonal flow, where T y is assumed to be independent of pressure. If the temperature departure from the zonal mean is given by (18.37), we have √ θ ∂T = 3 R ∂y Y
p ps
κ
.
386
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
(a)
(b)
18.6. DEVELOPMENT OF BLOCKING WAVES
387
(c) Figure 18.5: Development of blocking waves with ( ∆s∗ = 1 × 10−3 J/(K2 ·kg)), (a) background baroclinity (solid) at 500 hPa and e-folding time (dashed), (b) perturbation speed, and (c) perturbation height
Inserting it into (18.44) yields √ RθR p κ ps . ln u ¯ = us − 3 f Y ps p The three baroclinic modes BC, AS and P B studied by Song (1971) gave {Ψ, K } = 1.23 W/m2 for the baroclinic perturbations with wavelength 6000 km. The energy conversion ratios of the planetary waves obtained from his study are ηe = 0.545 and ηz = 0. If kinetic energy dissipation is ignored or ηd = 0, we gain η = 0.455. Applying T˙ d = 1 K/day at 500 hPa (referring to Newell et al., 1972), ε = 0.21 (McHall, 1993), u ¯s = 0, θR = 3.5 K and Y = 1000 km, we have calculated the time variations of the growth rate and temperature gradient using Method A. The initial field is given by (18.9) with Γ = 0.65 K/100m and Ts = 280 K. The maximum available enthalpy is calculated in the atmosphere from 1000 hPa to 200 hPa. For the perturbation with meridional wavenumber 2 and 3 and initial amplitude 100 gpm on 500 hPa at latitude 55◦ , the process of development is displayed in Fig.18.5. The figure shows that the baroclinity and growth rate decrease quickly in wave development, especially for short components, so the growth rate cannot be assumed constant. As energy conversion is more efficient in shorter waves, the plotted short component grows faster than the long component at the early stage. However, the long component has a larger amplitude eventually. For the quasi-geostrophic disturbances, the amplitude of geopotential perturbation may also be estimated from (18.43). The results are demonstrated in Fig.18.5(c).
388
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
At the end of development, the baroclinity is nearly destroyed and the blocking waves become stable. The wave decay and kinetic energy dissipation can be produced by molecular and turbulent diffusions. Unlike the occluded cyclones on the surface, the turbulent and molecular viscosity and boundary friction are relatively weak in the free troposphere, and a blocking system is not filled by cold air mass. Thus, a blocking circulation pattern may be remained for more than a week by inputing weak energy source along with the temperature advection from the baroclinic environments, as discussed in the book of McHall (1993). The wave development shown in the previous examples are evaluated as if the disturbance has one wave component only. When more wave components grow simultaneously, the growth rate and baroclinity decrease more quickly, and the developed disturbances at an isolated local place are weaker than those plotted in the figures. However, the atmosphere cannot be isolated for development of baroclinic disturbance, and the exchanges of energy and thermodynamic entropy may have significant influences on the processes.
18.7
Wave development in stratosphere
The breaking layer and nonlinear planetary wave instability may also occur in the stratosphere when the baroclinity is strong enough. The associated wave amplification may be observed just before occurrence of stratospheric sudden warmings. During or before a warming episode, there is usually a blocking event in the troposphere below (Craig and Hering, 1959; Teweles, 1963; Finger and Teweles, 1964; Labitzke, 1965). The growth rate of blocking waves discussed in the preceding section may also be applied for development of planetary perturbation in the stratosphere (McHall, 1993). Applying the energy conversion ratio η and the parameters assumed in the previous section, except for θR = 10K , Γ = −0.3K/100m ,
Y = 2000km , T˙ d = −0.6K/day ,
the development of planetary geostrophic wave in the stratosphere is illustrated in Fig.18.6. The available enthalpy is calculated in the column from 200 hPa to 20 hPa. The horizontal mean temperature is assumed 210 K and the zonal mean wind is 15 m/s on the bottom level. This figure shows the development of planetary stratospheric perturbations with meridional wavenumber 1 and 2 and initial amplitude 100 gpm at 50 hPa and latitude 60◦ . As in the blocking examples, the e-folding time increases rapidly when the planetary waves grow in the stratosphere. The depicted short wave is more baroclinically active than the long one, so its e-folding time increases more quickly. The increase of perturbation speed is displayed in Fig.18.6(b), and the development of geopotential wave is sketched in Fig.18.6(c). Although the short wave is initially more active, the long component will have a larger amplitude. As in the block situation discussed in the preceding section, the baroclinity is nearly destroyed at the end of development,
18.7. WAVE DEVELOPMENT IN STRATOSPHERE
(a)
(b)
389
390
18. ENERGETICS OF LINEAR DISTURBANCE DEVELOPMENT
(c) Figure 18.6: Development of stratospheric planetary geostrophic waves with ( ∆s∗ = 4 × 10−3 J/(K2 ·kg)), (a) background baroclinity (solid) and e-folding time (dashed) at 50 hPa, (b) perturbation speed, and (c) perturbation height
and the planetary waves may become stable afterwards if the turbulent diffusions are weak as in the tropospheric blocking events. In the major warmings, both the baroclinity and mean zonal momentum decrease before the reversal of mean zonal circulation. This implies that conversion of available enthalpy is rather inefficient in the warming events, due to the high static stability. As discussed in Chapter 9, these processes are highly diffusive and characterized by sudden changes. The major part of available enthalpy may be destroyed by the lateral turbulent mixing or entrainment, producing a large amount of thermodynamic entropy. This highly irreversible diffusion process in the stratosphere was referred to as the planetary wave breaking (McIntyre and Palmer, 1983). For ∆s = 0.5 J/(K·kg) (one order higher than that in Fig.18.6) in the process as the baroclinic stratosphere become barotropic, the amplified planetary waves decay quickly before the baroclinity is destroyed entirely (The figures are not shown).
Chapter 19 Energetics of moving parcels 19.1
Introduction
We have discussed in the previous chapters the energy conversions in a whole atmosphere using the system energy equation derived in Chapter 7. The results show that more kinetic energy is created in the more baroclinic and statically unstable atmosphere. The dependence on the baroclinity and static stability has been explained with conversion of negative baroclinic entropy into static entropy in the process. In the Newtonian dynamics, the kinetic energy generation may also be studied with the momentum equation applied for individual air parcels, which tells that the parcel kinetic energy is created by external forces doing mechanic work for the parcels. In this chapter, we study the basic features of the external forces and resultant parcel motions and energy conversions in the dry baroclinic atmosphere. The stratified baroclinic atmosphere is represented by the linear atmosphere assumed in Chapters 8 and 9, of which the fundamental structures, such as the slopes of isentropic and isobaric surfaces, are illustrated in this chapter. This study reveals more clearly the dependence of energy conversion on the baroclinity and static stability. The external Newtonian forces which may create parcel kinetic energy are the buoyancy force in the vertical direction and the pressure gradient force in the horizontal direction. The ability of kinetic energy generation in the vertical convection is usually represented by the static instability, which increases with vertical gradient of environmental temperature. The statically unstable atmosphere is the excited atmosphere in which the buoyancy force is in the direction of vertical displacement of a parcel. The vertical integration of the buoyancy is called the convective available potential energy ( CAPE), which is positive in the statically unstable atmosphere and is negative in the stable atmosphere. The kinetic energy of vertical motions, referred to as the vertical kinetic energy in this study, may be created from the heat and geopotential energies in the unstable atmosphere only. While in the stable atmosphere, the buoyancy is a restoring force for vertical motions causing the Brunt-Vaisalla oscillations of parcels about an isentropic surface. There is no net buoyancy in the neutral atmosphere. The extratropical troposphere is statically stable in general. However, the disturbances with enhanced vertical circulations may often develop in the stable atmosphere when the baroclinity is strong enough. The baroclinic disturbances are characterized by large-scale slantwise convection around the polar front or in the baroclinic perturbations, such as the geostrophic waves (McHall, 1993). The slantwise convection may also be found in the moist atmosphere, especially when the conditional instability is low. So far, we have no theories on development of the slantwise circulations related to energy conversions, as the traditional linear theories of wave instability do not consider the energetics. A new approach for the study on the energy conversions in development of dry baroclinic disturbance may 391
392
19. ENERGETICS OF MOVING PARCELS
be referred to Chapter 18. Since the energy conversions are the nonlinear processes in the atmosphere, the traditional linear theories are unable to reveal clearly the physical mechanism of wave instability. This mechanism will be discussed in this chapter. In the baroclinic atmosphere, the surfaces of temperature, pressure and potential temperature are not horizontal in general. Since the pressure in a geophysical fluid is isotropic in all directions and depends on the gravity as shown by the hydrostatic equation (5.1), the horizontal pressure gradient and the gradient force depend crucially on the gravity, though the gravitational force is vertical. When a fluid parcel remains hydrostatically equilibrium in a horizontal motion, the horizontal pressure gradient force accepted by the parcel is proportional to the slope of isobaric surface with the coefficient g. In other words, the energy conversion in the horizontal processes is nevertheless related to conversion of geopotential energy, and it is impossible to convert the heat energy alone into kinetic energy in the atmosphere as discussed in Chapter 7. Therefore, the pressure gradient force inside a fluid system may change the total momentum of the system, as it depends on the gravitational force which is external to the system. Since the horizontal force and buoyancy force may be intensified by increasing the baroclinity and static instability respectively, more kinetic energy is produced in the atmosphere with stronger baroclinity and lower static stability. The traditional static instability of the dry atmosphere is represented by the difference between environmental temperature lapse rate and adiabatic lapse rate. Analogously, the conditional instability for moist convection is measured by vertical gradient of saturation equivalent potential temperature in the atmosphere. These criteria of static instability in the dry and moist convection cannot be applied for the slantwise convection in the baroclinic or equivalent baroclinic atmospheres, since there are the horizontal gradients of environmental temperature and humidity along a sloping path, and the parcel temperature may be changed by horizontal pressure gradient. We discuss in this chapter only the slantwise convection in the dry atmosphere. To measure the static stability of slantwise circulation, we introduce the algorithms for calculating the environmental temperature lapse rate and parcel adiabatic lapse rate along a slantwise trajectory, called the slantwise lapse rate and slantwise adiabatic lapse rate respectively. The temperature lapse rate and adiabatic lapse rate used in the traditional meteorology are examples of the slantwise lapse rates for the vertical trajectories. The slantwise convection in the baroclinic and statically stable atmosphere may become statically unstable if the trajectory slope is smaller than the slope of isentropic surfaces crossed. The static instability in the vertical convection is independent of the baroclinity, while the slantwise static stability depends on the baroclinity. So the instability of slantwise circulation may also be represented by the baroclinic instability. Unlike the unstable vertical convection, the unstable slantwise convection tends to oscillate about an isentropic surface. The vertical buoyancy oscillations in the viscous atmosphere are damped processes, and the Brunt-Vaisalla frequency is independent of the baroclinity. While, the slantwise oscillations include stable oscillating phase and unstable non-oscillating phase occurring alternatively. The circulation instability and horizontal pressure gradient may force the baroclinic os-
19.2. LINEAR ATMOSPHERE
393
cillations continuously, and the oscillations in the upward slantwise convection may grow up. The slantwise Brunt-Vaisalla frequency depends on the baroclinity, trajectory slope as well as the static stability. The slantwise gravity waves induced in the baroclinic atmosphere may be strong enough to produce significant weather phenomena, and may be worth to be incorporated in the weather forecasting models.
19.2
Linear atmosphere
19.2.1
The thermal structure
The examples in this chapter will be given in the ideal atmosphere which possesses constant vertical and horizontal temperature gradient in Cartesian coordinates. This atmosphere called the linear atmosphere has been applied for the study of energy conversions in Chapters 9 and 10. The isothermal surface is a plane surface in the coordinates, represented by T = TA + Ty y − Γz ,
(19.1)
where TA is the temperature at y = 0 and z = 0, and Ty indicates the horizontal temperature gradient. For convenience, we assume that the bottom isobaric surface of the linear atmosphere is geographically horizontal. Integrating the hydrostatic equilibrium equation (9.39) in the linear atmosphere yields
T = (TA + Ty y)
p pA
γ
,
γ=
RΓ . g
(19.2)
It may be viewed as the temperature represented in pressure coordinates. The pA in this equation denotes the pressure at z = 0, and TA is the temperature at y = 0 and p = pA . The horizontal temperature gradient in pressure coordinates reads
∂T ∂y
= Ty p
p pA
γ
.
It is constant on an isobaric surface, but may change with height. The isothermal and isobaric surfaces tend to be parallel near the top of the linear atmosphere. The relation between pressure and geographic height may be derived from (19.1) and (19.2), giving
p = pA
Γz 1− TA + Ty y
g RT
.
(19.3)
As discussed in Chapter 17, the atmosphere will have a constant pressure if without the gravitation. Thus, the pressure gradient and pressure gradient force in all directions, including the horizontal direction, result from the vertical gravitational force. This consequence derived from the linear atmosphere assumed is generally true, and will be discussed again in this chapter. The isobaric surface is not flat in Cartesian coordinates. The potential temperature is represented by
p θ = (TA + Ty y) pA
γ
pθ p
κ
(19.4)
19. ENERGETICS OF MOVING PARCELS
394
Figure 19.1: Linear atmosphere with Ty = −10 K/1000km and Γ = 0.65 K/100m. The light dashed curves show the temperature (K). The heavy solid curves show the pressure (hPa). The potential temperature (K) is represented by the light solid.
in pressure coordinates or
TA + Ty y θ = (TA + Ty y − Γz) TA + Ty y − Γz
Γd Γ
pθ pA
κ
in Cartesian coordinates. The potential temperature depends on pressure and temperature. In the statically stable atmosphere, the effect of vertical pressure change on the potential temperature is greater than the effect of temperature change. Thus, the potential temperature increases with height, although the temperature decreases. While, the horizontal gradient of potential temperature is in the same direction as the gradient of temperature in most part of the atmosphere. When the horizontal pressure gradient is sufficiently large, the gradient of potential temperature may be reversed and opposite to the temperature gradient. But this situation seldom happen in the atmosphere. An example of the linear atmosphere is displayed in Fig.19.1 for pA = 1000 hPa. When a parcel moves on an isentropic surface in the linear atmosphere, the vertical pressure gradient force accepted by the parcel is balanced by the gravitational force, while the horizontal pressure gradient force may still change the parcel kinetic energy. This will be discussed soon. Here, we see that in the most part of the linear atmosphere, the horizontal pressure gradient force produces parcel kinetic energy as the parcel moves upward on an isentropic surface, and destroy kinetic energy in the downward motions on an isentropic surface.
19.2. LINEAR ATMOSPHERE
19.2.2
395
Slope of isentropic surface
The slope of isentropic surface in the Cartesian coordinates may be calculated from
tan αθ =
∂z ∂y
=−
θ=constant
∂θ ∂y
'
∂θ ∂z
(19.5)
The potential temperature of the linear atmosphere discussed previously is represented by (19.4), which may be rewritten as
κ
pθ θ = (TA + Ty y − Γz) p
.
The partial derivative with respect to y provides ∂θ = Ty ∂y
pθ p
κ
−
κT p
pθ p
κ
∂p . ∂y
(19.6)
For the linear atmosphere assumed, (19.3) gives ∂p gpTy z = . ∂y RT (TA + Ty y) From (19.3) again, we gain TA + Ty y (1 − r γ ) , Γ
z=
(19.7)
in which r = p/pA . Inserting it into the previous equation produces gpTy ∂p = (1 − r γ ) , ∂y RT Γ
(19.8)
Applying this equation for (19.6) yields
Γd ∂θ = Ty 1 − (1 − r γ ) ∂y Γ Moreover, we have
pθ ∂θ = −Γ ∂z p
κ
κT − p
pθ p
pθ p
κ
κ
.
∂p . ∂z
Using the static equilibrium assumption gives
pθ ∂θ = −(Γ − Γd ) ∂z p Now, (19.5) reads
κ
.
Ty Γd (1 − r γ ) 1− tan αθ = Γ − Γd Γ
.
(19.9)
The isentropic surfaces are not flat in the linear atmosphere. The slope is plotted in Fig.19.2. It decrease upward but increases with the intensity of horizontal temperature gradient and static instability.
19. ENERGETICS OF MOVING PARCELS
396
Figure 19.2: Slope of isentropic surface in the linear atmosphere. Solid and dashed curves are for Γ = 0.65 and 0.75 K/100m, respectively
In general, a front surface is nearly parallel to isentropic surfaces in the frontal zone. So, (19.9) may be used to estimate the front slope. Another expression of front slope is represented by the differences in the temperatures and geostrophic winds on both sides of a front, derived according to geostrophic balance (Petterssen, 1956). The slope varied from 1:50 to 1:300 with the average value around 1:150 or 0.0067 (Petterssen, 1956). Fig.19.2 shows that the absolute value of temperature gradient is slightly below 25 K/1000km in the boundary layer with Γ = 0.65 K/100m. This value is typical in the polar fronts. According to (19.9), a front is steeper in the atmosphere with stronger baroclinity and lower static stability. Usually, the strongest baroclinity occur in the boundary layer of the Earth, so the front near the surface is steeper than at high levels. If the horizontal temperature gradient does not change with height, the fronts are also steeper near the surface as shown by this figure.
19.2.3
Slope of isobaric surface
The slope of isobaric surface is given by
tan αp =
∂z ∂y
p
∂p =− ∂y
From (19.3) we see gρTy z ∂p = . ∂y TA + Ty y
'
∂p . ∂z
(19.10)
19.2. LINEAR ATMOSPHERE
397
Figure 19.3: Slope of isobaric surface in the linear atmosphere with Ty = −10 K/1000km . The curves are drawn for every increase of 0.1 K/100m static stability.
Here, we have used the ideal-gas equation. The denominator on the right-hand side is the surface temperature at z = 0 in the linear atmosphere assumed. Applying it together with the hydrostatic equation (9.39) for (19.10) yields tan αp =
Ty z . TA + Ty y
(19.11)
The slope is linearly proportional to the baroclinity and geographic height. It may also be given by Ty (1 − r γ ) , (19.12) tan αp = Γ derived using (19.7). The slope of isobaric surface in the linear atmosphere with Ty = −10 K/1000km is displayed in Fig.19.3. The slope increases slightly with the static stability. If the 1000 hPa surface is geographically horizontal, the mean slope of 500 hPa surface is about 1.876 × 10−4 in the linear atmosphere with Γ = 0.65 K/100m. Since the slope is proportional to the horizontal temperature gradient, the figure may also be applied for the atmospheres with different baroclinities. For example, the slope is doubled for Ty = −20 K/1000km. Comparing with Fig.19.2 finds that the slope of isobaric surface is one order less than the slope of isentropic surface in the linear
19. ENERGETICS OF MOVING PARCELS
398
atmosphere. The large difference between these slopes may be seen from Fig.19.1 also. Unlike the isentropic surfaces, the isobaric surfaces change their slopes slightly with the static stability.
19.3
External forces on a parcel
19.3.1
Adiabatic buoyancy oscillations
The parcel kinetic energy may be created by the external forces doing mechanic work for the parcel. Except the frictional force, the external forces in the momentum equation (5.30) which may change the parcel kinetic energy are the pressure gradient force and gravitational force, that is
F = −α ´
∂p ∂p ˆ− α y + g ˆz ´ ∂y ∂z
(19.13)
in the linear atmosphere discussed earlier with ∂p/∂x = 0. The vertical component, called the buoyancy force, includes the downward gravitational force and upward vertical pressure gradient force. When the environment is in hydrostatic equilibrium, the buoyancy force gives the Archimedes’ principle:
Fz = g or Fz = g
α ´ −1 α
.
ρ − ρ´ . ρ´
If the parcel pressure is justified to the environment in adiabatic motions, we may use the ideal-gas equation to gain Fz =
g ´ (T − T ) . T
(19.14)
g ´ (θ − θ) . θ
(19.15)
It may be rewritten as Fz =
In the linear atmosphere discussed previously, we have Fz =
g (Γ − Γd )(z − zA ) , T
(19.16)
where zA is the equilibrium height at which T´ = T and Fz = 0. This equation shows that the sign of buoyancy force depends on the static stability Γ − Γd in the dry atmosphere. This force is a restoring force in the statically stable atmosphere with Γ < Γd , as it is in the direction opposite to the direction of parcel motion. But in the statically unstable atmosphere with Γ > Γd , the parcel is accelerated by the force.
19.3. EXTERNAL FORCES ON A PARCEL
399
The vertical kinetic energy created by the buoyancy force gives z
∆kz =
Fz dz zA
= g
z ´ T
T
zA
= g
z ´ T zA
T
−1
dz
dz − ∆φ .
It is called the energy of instability (Belinskii, 1948) or convective available potential energy ( CAPE, Moncrieff and Miller, 1976) of a saturated parcel, when the upper limit of integration is the level of neutral buoyancy. If the parcel temperature is represented by T´ = TA − Γd z, the adiabatic lapse rate Γd may be given by (5.24) in the hydrostatic nonequilibrium processes. When the environmental atmosphere is in hydrostatic equilibrium, we use the hydrostatic equation and ideal-gas equation to obtain g
z ´ T zA
T
dz = − = −
Thus, we see ∆kz = −
p pA
p RT´
p
p A p
dp
α ´ dp . pA
α ´ dp − ∆φ .
(19.17)
The two terms on the right-hand side represent the mechanic work created by the vertical pressure gradient force and gravitational force respectively. This equation may be derived also from the air engine theory introduced in the next chapter. For the vertical adiabatic processes, we may use (2.48) for this equation. The result will be derived and illustrated in the next chapter. Using the potential temperature in (2.55), we gain 1 ∂θ 1 (Γd − Γ) = T θ ∂z
(19.18)
in the hydrostatically equilibrium atmosphere. So, (19.16) is replaced by g ∂θ d2 z (z − zA ) =− 2 dt θ ∂z ´ If the static stability in the inviscid atmosphere. This is also (19.15) as θA = θ. represented by vertical gradient of potential temperature is constant, the solution in the stable atmosphere gives the adiabatic buoyancy oscillations z = (Z − zA )eiνz t + zA at the Brunt-Vaisalla frequency
νz =
g ∂θ , θ ∂z
(19.19)
19. ENERGETICS OF MOVING PARCELS
400
where Z indicates the parcel position at the initial time t = 0. This frequency depends on the gravity crucially and decreases with the static stability. The maximum amplitude is also the initial displacement of the parcel from the equilibrium position |Z − zA |. The perturbations caused by propagation of the oscillations are called the internal gravity waves, as they occur inside the stable atmosphere due to the gravity effect. The oscillations or gravity waves become unstable in the statically unstable atmosphere, since the frequency becomes imaginary. In a damping medium with damping rate µ, the slantwise oscillation equation becomes dz d2 z = −νz2 (z − zA ) . − 2µ dt2 dt The damped oscillations are represented by z = (Z − zA )e(−µ+iν)t + zA with ν 2 = νz2 − µ2 . The frequency is reduced by the viscous damping.
19.3.2
Horizontal processes
The horizontal pressure gradient force in (19.13) is given by ´ Fh = −α
∂p ∂y
in the y-direction. When the parcel is in thermodynamic equilibrium with the surroundings, we have α ´ = α. Applying the hydrostatic equilibrium equation α=− yields Fh = g
∂p ∂y
g ∂p ∂z '
∂p . ∂z
From (19.10), we have Fh = −g tan αp .
(19.20)
The horizontal pressure gradient force is proportional to the slope of isobaric surface in Cartesian coordinates and depends on gravitation of the Earth. Although the gravitational force is always vertical, it is also responsible for the horizontal pressure gradient force. Without the gravitation, there will be no pressure gradient in the linear atmosphere as shown by (19.3). The kinetic energy of horizontal motions may be called the horizontal kinetic energy. Generation of horizontal kinetic energy is estimated from δkh = −Fh δy .
19.3. EXTERNAL FORCES ON A PARCEL
401
Inserting (19.20) into it yields δkh = −gδy tan αp = −gδzp .
(19.21)
Here, δzp is the change of geographic height of isobaric surface. This relationship implies that the parcel kinetic energy created by the horizontal pressure gradient force is equivalent to the drop of geopotential as the parcel moves on an slopping isobaric surface to the same horizontal coordinates. To find the relation between the horizontal pressure gradient force and atmospheric thermal structure, we apply (19.11) for (19.20) and gain Fh = −
gTy z . TA + Ty y
(19.22)
We may also use (19.12) to obtain g Fh = − Ty (1 − r γ ) . Γ
(19.23)
The force increases with the baroclinity and height in the linear atmosphere, and is maximum on the top of temperature front. The zonal geostrophic wind speed given by (5.29) is rewritten as g vxg = − Ty (1 − r γ ) fΓ in the linear atmosphere assumed. It is proportional to the meridional temperature gradient increases with height. Thus, the wind is strongest over a deep front and beneath the tropopause. The westerly jet is created and maintained not only by zonal momentum concentration in the poleward branch of Hadley cell under the conservation of absolute angular momentum, but also by kinetic energy generation in the slantwise upward motions close to the front surface. The meridional wind speed increased may be transfered into zonal speed in the rotational atmosphere. As the geostrophic wind speed increases with the static stability for a provided baroclinity, the winds in the extratropical stratosphere is very strong. The mechanic work done by horizontal pressure gradient force is evaluated from y
∆kh =
y
Fh dy A
g = − Ty Γ
y yA
(1 − r γ ) dy
in the linear atmosphere with constant horizontal and vertical temperature gradients. The horizontal kinetic energy created increases with the baroclinity, while increase of static stability has a negative effect on the energy production. In general, the effect of static stability on generation of vertical kinetic energy is greater than on generation of horizontal kinetic energy. Thus, the atmosphere with higher baroclinity and lower static stability may produce more kinetic energy. If a parcel moves adiabatically upward from A to B on an isentropic surface, the environmental temperature along the trajectory is given by T = TA r κ .
19. ENERGETICS OF MOVING PARCELS
402
The temperature may also be evaluated from (19.2), that is T = (TA + Ty y)r γ . These two equations give y= It follows that dy =
TA κ−γ (r − 1) . Ty
TA (κ − γ)r κ−γ−1 dr . Ty
Applying this equation for (19.23) yields g ∆kh = − TA (κ − γ) Γ
r AB 1
(r κ−γ−1 − r κ−1 )dr ,
in which rAB = pB /pA . Thus, the mechanic work produced by horizontal pressure gradient force gives
∆kh = cp TA 1 − rAB κ
Γd κ −γ r (1 − rAB + ) . Γ AB
(19.24)
No vertical kinetic energy is created in the adiabatic motions on the isentropic surface. The energy created on an isentropic surface will be discussed again in the next chapter.
19.4
Slantwise static instability
In meteorology, we use the static stability instead of buoyancy force to measure the ability of vertical kinetic energy generation or the possibility of vertical convection development in the atmosphere. The atmosphere is statically stable when it is in the hydrostatic equilibrium represented by the hydrostatic equation (5.23). This equation tells that the upward pressure gradient force on a parcel, which is in hydrostatic equilibrium with the surroundings, equals the downward gravitational force in magnitude so the buoyancy force is zero. A parcel moving vertically and adiabatically in the stable atmosphere may only produce damped buoyancy oscillations, at the Brunt-Vaisalla frequency about the isentropic surface with the potential temperature of the parcel (called the equilibrium surface afterwards). This is because the buoyancy force is a restoring force of the moving parcel. But in the statically unstable atmosphere, the buoyancy force is in the direction of vertical parcel motion, and produces the vertical kinetic energy. According to the observations, vertical circulations may also develop in the statically stable atmosphere. A well known example is the baroclinic cyclones developed on the polar front in the extratropical troposphere. A cross-section of the temperature and vertical velocity in a mature baroclinic cyclone is shown by Fig.9.2. The development of vertical circulation in the statically stable atmosphere may be studied with the concept of slantwise static instability, which is the static instability along the trajectories of slantwise circulation in the baroclinic atmosphere.
19.4. SLANTWISE STATIC INSTABILITY
403
Z
O
✻ ❍ ❍❍ ❍❍ ❍ ❍❍ ❍❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍ ❍ B ❍ δl ✟ ❍ ✯❍ ❍❍ ❍ ❍ ❍❍ ✟ ✟❍ αl ❍ δz ❍❍ ❍❍ ❍ ✟ ✆ ❍ ❍ A ❍❍ ❍ ❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍
✲
Y
Figure 19.4: Slantwise lapse rate of temperature. The isothermal surfaces are represented by light lines.
Usually, the static stability of the atmosphere is represented by the lapse rate of environmental temperature. The traditional lapse rate Γ can only be used for vertical convection but not the slantwise convection, since both the environmental temperature and pressure may change in the horizontal direction also, and so the change rate of environmental or parcel temperature with height depends on the horizontal displacement. For convenience, the Cartesian coordinates are set with y-axis parallel to the horizontal gradient in the temperature field as shown in Fig.19.4. Provided that projection of parcel trajectory on the yz-plane, denoted by l, has an angle αl to the positive direction of y-axis, the directional derivative of environmental potential temperature along l is represented by ∂θ ∂θ ∂θ = cos αl + sin αl . ∂l ∂y ∂z It follows that
∂θ tan αθ ∂θ = sin αl 1 − ∂l ∂z tan αl
,
(19.25)
in which the slope of isentropic surface is given by (19.5), or by (19.9) for the linear atmosphere. If the trajectory is steeper than the sloping equilibrium surface in the same direction, ∂θ/∂l has the same sign as that of sin αl . So, ∂θ/∂l > 0 in the upward slantwise convection and ∂θ/∂l < 0 in the downward slantwise convection. When the parcel potential temperature is conserved in the adiabatic motion, the parcel is colder than the environments in the upward convection and is warmer in the downward convection. As the result, the parcel is pulled back to the equilibrium surface by the restoring buoyancy force. This means that the slantwise convection is statically stable, as the parcel is constrained around the equilibrium surface. The unstable slantwise convection takes place when the trajectory slope is smaller than the slope of equilibrium surface in the same direction. The vertical kinetic energy may be created along the unstable trajectories by the buoyancy force in the
19. ENERGETICS OF MOVING PARCELS
404
statically stable atmosphere. While, the barotropic and statically stable atmosphere with tan αθ = 0 is stable for the adiabatic motions in any direction. As the parcel is accelerated by the buoyancy force towards the equilibrium surface, the vertical velocity increases until the trajectory becomes steeper than the equilibrium surface. In this case, the parcel becomes statically stable and so is pulled back to the surface as discussed previously. Therefore, unlike the vertical convection in the unstable atmosphere, the parcels in the unstable slantwise convection tend to oscillate about the equilibrium surfaces. On the time average, the large-scale adiabatic trajectories are constrained on the corresponding equilibrium surfaces, and may be represented by the two-dimensional winds on the surfaces. However, to evaluate the trajectories over a certain days, the diabatic cooling and frictional dissipation cannot be ignored. When the large-scale winds measured in situ are on an isentropic surface, the parcels may still cross the surfaces in a diabatic process as the surface may change its position in the varying atmosphere.
19.5
Slantwise lapse rate
To provide the quantitative measurement of the slantwise static instability, we study the variations of environmental temperature and parcel temperature along a slantwise trajectory in this and the next sections respectively. The variation of environmental temperature along a slantwise trajectory is called the slantwise lapse rate of temperature, defined as Γ∗ = lim − δz→0
δT , δz
(δz = 0) ,
where δ indicates the variation along the trajectory. If the trajectory is sufficiently short, Γ∗ may be considered constant, given by Γ∗ = −
δT . δz
(19.26)
When a parcel moves along the trajectory l from A to B shown in Fig.19.4, we have δT =
∂T δl . ∂l
Applying it yields ∂T δl ∂l δz ∂T 1 ≈ − ∂l sin αl ∂T ∂T 1 cos αl + sin αl = − ∂y ∂z sin αl ∂T cot αl , = Γ− (αl =
0) , ∂y
Γ∗ = −
in which αl denotes the angle of the trajectory with respect to a geographically horizontal surface. For a vertical trajectory, we have αl = π/2 and so Γ∗ = Γ.
19.5. SLANTWISE LAPSE RATE
405
When environmental temperature decreases along the horizontal direction of the trajectory (as ∂T /∂y < 0 in Fig.19.4), the slantwise lapse rate is greater than the vertical lapse rate. If the trajectory slope is sufficiently small, the slantwise lapse rate may become statically unstable in the statically stable and baroclinic atmosphere. Using the slantwise lapse rate, the slope of trajectory may be represented by tan αl =
Ty Γ − Γ∗
(19.27)
in the linear atmosphere. Here Γ∗ is the lapse rate along the trajectory with the angle αl . The environmental temperature variation along the trajectory may be estimated from z B TB − TA = − Γ∗ dz . zA
It follows, from (19.27), that TB = TA −
z B zA
(Γ − Ty cot αl )dz .
For a constant Γ and αl , we gain TB = TA − Γ(zB − zA ) + Ty (yB − yA ) .
(19.28)
If the position change in the Cartesian coordinates is unknown, the temperature change along a slopping path may be evaluated in the following way. The environmental temperature at y = yB on pressure surface pB in the linear atmosphere is evaluated from (19.2), that is
p TB = (TA + Ty δyAB ) B pA
γ
,
(19.29)
where TA is environmental temperature at y = yA on pressure surface pA . Since B is on the trajectory, the temperature may also be evaluated from (19.26), giving TB = TA − Γ∗ δzAB . The vertical distance between A and B in this equation is given by δzAB = δyAB tan αl . It follows, from (19.27), that δzAB =
Ty δyAB . Γ − Γ∗
(19.30)
Thus, we obtain
Γ∗ Ty δyAB . Γ − Γ∗ Applying it for (19.29) gives the horizontal distance between A and B: TB = TA −
δyAB =
γ ) TA (1 − rAB γ Ty (rAB +
Γ∗ Γ−Γ∗ )
,
(Ty = 0) .
(19.31)
(19.32)
19. ENERGETICS OF MOVING PARCELS
406 Inserting it into (19.29) yields γ
TB = TA rAB
1+
γ 1 − rAB γ rAB +
Γ∗ Γ−Γ∗
.
The first term in the brackets gives the temperature at the position on pressure surface pB right over pA vertically, and the other term is the temperature change over the horizontal distance on surface pB . This equation is replaced by TB =
Γ
ΓTA ∗ + Γ (r −γ AB
− 1)
.
(19.33)
It may also be obtained by inserting (19.30) and (19.32) into (19.28). For provided trajectory and static stability, the slantwise lapse rate Γ∗ depends on the baroclinity, and so the TB depends on the baroclinity. If the path is vertical or the atmosphere is barotropic, we have Γ∗ = Γ, and so this equation gives γ . TB = TA rAB
(19.34)
Inserting (19.32) into (19.30) yields the vertical distance, that is δzAB =
γ ) TA (1 − rAB . γ (Γ − Γ∗ )rAB + Γ∗
If TA = 280 K, Γ = 0.65 K/100m and Ty = −20 K/1000km in the linear atmosphere, the environmental temperature on the trajectory starting from 1000 hPa with Γ∗ = 1 K/100m is 202 K at 300 hPa. The vertical distance is 7.7 km and the horizontal distance is 1363 km. The trajectory in this example is close to an isentropic surface.
19.6
Slantwise adiabatic lapse rate
The static stability of slantwise convection may be examined by comparing the temperatures of convective parcels and environments. We have discussed that the change of environmental temperature with height depends on the trajectory in the baroclinic atmosphere. This is also true for the change of parcel temperature. In the pressure coordinates, the change of parcel temperature may be calculated simply from the adiabatic equation (4.17). The change rate with geographic height is discussed in the following. The thermodynamic energy law for adiabatic processes is given by (5.22). For a path which is not vertical, we cannot use the hydrostatic equation (5.23) for the right-hand side of (5.22). Applying (5.22) for the sloping path AB in Fig.19.4 yields cp
δp´ δT´AB =α ´ AB , δzAB δzAB
(δzAB = 0) .
If the parcel pressure is identical to the environmental pressure, this equation is replace by α ´ δp δT´ = . δz cp δz
19.6. SLANTWISE ADIABATIC LAPSE RATE
407
The left-hand side is the change rate of parcel temperature along the path. The slantwise adiabatic lapse rate may be defined as Γ∗d = lim − δz→0
δT´ . δz
Applying δp =
∂p δl ∂l
yields δp δz
∂p δl ∂l δz ∂p 1 ≈ ∂l sin αl ∂p ∂p 1 cos αl + sin αl = ∂y ∂z sin αl ∂p cot αl , = −ρg + (αl = 0) . ∂y =
Thus, the slantwise adiabatic lapse rate is given by Γ∗d = Γd −
α ´ ∂p cot αl , cp ∂y
where Γd is referred to (5.24). In the linear atmosphere, we may insert (19.27) into it, giving α ´ (Γ − Γ∗ ) ∂p . Γ∗d = Γd − cp Ty ∂y Applying (19.8) for it yields Γ∗d = Γd
∗ Γ
Γ
(1 − r γ ) + r γ
.
(19.35)
For a given trajectory, the slantwise lapse rate may be evaluated from (19.27). If the path is vertical, we have Γ∗ = Γ, and so the slantwise adiabatic lapse rate becomes the traditional adiabatic lapse rate. The slantwise convection in the dry atmosphere is statically stable when Γ∗ < Γ∗d . If Γ∗ > Γ∗d , the slantwise convection is statically unstable. On a statically neutral path, we have Γ∗ = Γ∗d and so Γ∗ = Γd It follows that Γ∗ =
∗ Γ
Γ
(1 − r γ ) + r γ
.
(19.36)
Γr γ Γd . Γ − Γd (1 − r γ )
This equation gives the neutral lapse rate of environmental temperature. The slope of the neutral path may be evaluated from (19.27), giving
tan αln =
Ty Γd (1 − r γ ) 1− Γ − Γd Γ
.
19. ENERGETICS OF MOVING PARCELS
408
The right-hand side is also the slope of isentropic surface given by (19.9), that is tan αln = tan αθ . So the neutral path lies on isentropic surface. The slope of isentropic surface provides the upper limit of the trajectory slope in the slantwise free convection. For the trajectories with a constant slope, we manipulate (19.9), (19.35) and (19.27) giving tan αθ ∗ ∗ . (19.37) Γ − Γd = (Γ − Γd ) 1 − tan αl Here, tan αθ is the slope of the equilibrium surface of the parcel in the adiabatic motion with trajectory slope tan αl . This equation may be used conveniently for a straight trajectory about a plane equilibrium surface. It tells that a moving parcel is statically unstable in the dry and stable atmosphere if tan αl < tan αθ .
(19.38)
and the two slopes have the same sign. The left-hand side of (19.37) may be used to measure the static stability of slantwise convection, called the slantwise static stability. The slantwise free convection in the statically stable atmosphere may occur when the adiabatic trajectory slope is smaller than the slope of isentropic surface.
19.7
Slantwise circulation instability
The slantwise circulation instability illustrated by (19.37) is a kind of static instability. This instability may lead to development of disturbance as it causes conversion of non-kinetic energy into kinetic energy. From the point of Newtonian dynamics, the energy conversion is produced by the buoyancy force doing mechanic work for the parcels. The buoyancy force in the inviscid atmosphere may be derived from the vertical component of momentum equation (19.14) for unit mass. This equation gives z 1 ∗ (Γ − Γ∗d )dz Fz = g zA T in slantwise circulations. Here, zA is the height of the parcel on its corresponding equilibrium surface at an initial time. This equation shows clearly the relation between the buoyancy force and slantwise static stability. The force is in the direction of vertical displacement when the slantwise convection is unstable, but is in the opposite direction if the convection is stable. Applying (19.37) for it yields Fz = g
z 1 zA
T
(Γd − Γ)
tan αθ − 1 dz . tan αl
(19.39)
The slope of isentropic surface is given by (19.9). If the trajectory is more flat than the isentropic surface, the buoyancy force is in the same direction as that of vertical displacement of the parcel, and the vertical kinetic energy increases in the unstable slantwise convection.
19.7. SLANTWISE CIRCULATION INSTABILITY
409
For a given trajectory, the intensity of buoyancy force in (19.39) is independent of vertical direction of parcel motion in the linear atmosphere. Fig.9.2 shows that the strongest upward and downward speeds were similar near the cyclone center. The slightly stronger upward motion may be associated with lower static stability in the lower troposphere. While, the ECMWF data show (no figures presented here) that the horizontal winds in the eastern part of the cyclone were remarkably stronger than in the western part, since horizontal kinetic energy was created in the upward motions, but destroyed in the downward motions. The static instability is independent of the baroclinity in the vertical convection, as it may occur in the barotropic atmosphere. Development of moist convective storm in the tropical regions is associated with the static instability. While, the convection in the extratropical troposphere is mostly the slantwise convection. Equation (19.37) shows that the static instability of slantwise circulation depends on the horizontal as well as vertical temperature gradient, as the slope of isentropic surface in this equation depends on the baroclinity. For a provided trajectory slope, there is a lower limit of either the vertical or horizontal temperature gradient, over which the slantwise convection becomes statically unstable. In other words, the slantwise static instability may also be represented by the baroclinic instability. The horizontal temperature gradient for the instability may be obtained by applying (19.9) for (19.38), giving Ty <
Γ(Γ − Γd ) tan αl , Γ − Γd (1 − r γ )
where Ty is less than zero in the linear atmosphere shown by Fig.19.4. This baroclinic instability is related to generation of vertical kinetic energy. If production of horizontal kinetic energy is considered also, we may have different criteria for the baroclinic instability. The baroclinic instability is usually studied with a theory of wave perturbation (McHall, 1993) without considering the energy conversion. In the next chapter, we introduce a new approach for the study of baroclinic wave instability from an approach of energy conversion. Since the baroclinic disturbances develop mostly in the statically stable atmosphere, people may think that the baroclinic instability has its particular mechanism independent of the static instability. The previous discussions show that the baroclinic instability is associated with the static instability in the baroclinic slantwise circulations, and is also important as it may cause conversion of non-kinetic energy into kinetic energy. The vertical kinetic energy created by buoyancy force on path AB may be estimated by integrating (19.39), giving ∆kz = g
z z B 1 zA
zA
tan αθ (Γd − Γ) −1 T tan αl
dz dz .
Unlike in the vertical processes discussed earlier, the energy generation in the unstable slantwise circulations increases with the baroclinity as well as the static instability. The dependence on the trajectory is shown also by this expression.
19. ENERGETICS OF MOVING PARCELS
410
Figure 19.5: Maximum height of adiabatic slantwise convection in the linear atmosphere with temperature lapse rate indicated
19.8
Height of slantwise convection
According to the previous discussions, the slantwise convection takes place as the trajectory slope is less than the slope of isentropic surface. Since the isentropic surfaces tend to reduce the slope at higher levels especially in the spherical atmosphere, the slantwise convection with a given trajectory slope may only extend to the height, where the trajectory is parallel to an isentropic surface. As the slantwise lapse rate equals the slantwise adiabatic lapse rate on the isentropic surface, we apply (19.36) for Γ∗ = Γ∗d to give the maximum height of slantwise convection: p ≥ pmin where pmin = pA
∗ g Γ (Γd − Γ) RΓ
. Γd (Γ∗ − Γ) The dependence of the maximum height on the slantwise lapse rate is displayed in Fig.19.5 for pA = 1000 hPa. The height increases with the static instability or the slantwise static instability. The upper limit of slantwise convection may be slightly higher than the plotted, if the overshooting flow penetrates upward until the vertical kinetic energy is entirely converted into heat energy and geopotential energy. The convective flow may still have horizontal kinetic energy at the top height. Applying the pmin derived previously for (19.9) gives the maximum slope of the slantwise convection: Ty . tan αlmax = Γ − Γ∗d It is also the trajectory slope on an isentropic surface, or tan αlmax = tan αln = tan αθ .
19.9. SLANTWISE BUOYANCY OSCILLATIONS
411
Examples of the maximum trajectory slope are displayed in Fig.19.2 for provided baroclinity. The slope decreases upward, but increases with the baroclinity and static instability. It is noted that the convection height observed may not agree with that plotted here, since the convection may be related to moist process and may change the slope; also there is the entrainment of environmental air into the convective flows in the atmosphere. The slantwise convection of saturated parcels in the moist atmosphere is close to an equivalent potential temperature surface, called the moist isentropic surface which is generally steeper than isentropic surface especially in the lower troposphere. So the moist convection may have a steep slope and meso or small horizontal scale. A relatively large amount of kinetic energy may be produced in the moist atmosphere, so that the slantwise convection may form a moist jet. Since the static stability increases rapidly upward, the slope of moist isentropic surface decrease quickly at higher levels, and the moist jets are usually constrained in the lower troposphere, as discussed in Chapter 16. Also, since the horizontal kinetic energy is created in the upward slantwise convection, but destroyed in the downward convection, the moist jets occur only in the poleward flows.
19.9
Slantwise buoyancy oscillations
The vertical buoyancy oscillations superimposed on the slantwise convection form the slantwise buoyancy oscillations in the atmosphere. The vertical momentum equation (19.39) gives the slantwise oscillation equation d2 z =− dt2
z g ∂θ zA
θ ∂z
1−
tan αθ tan αl
dz
derived using (19.18) also. Here, tan αθ is the slope of the equilibrium surface possessing the potential temperature of the parcel in the adiabatic oscillations. Applying (19.25) yields g ∂θ d2 z dl . = − 2 dt A θ ∂l ´ In a Here we used dz = dl sin αl . This equation is identical to (19.15) as θ = θ. A
damping medium with damping rate µ, the slantwise oscillation equation gives dz d2 z =− − 2µ dt2 dt where
νl = νz 1 −
z zA
νl2 dz ,
tan αθ . tan αl
(19.40)
The νz in this equation is the Brunt-Vaisalla frequency given by (19.19). As the parcel potential temperature is conserved in the adiabatic oscillations, the buoyancy on the parcel depends only on the parcel position relative to the equilibrium surface. Thus, the trajectory slope in the previous equation and (19.39) may be considered as the slope of a line from the constant position zA to the current
412
19. ENERGETICS OF MOVING PARCELS
position of the parcel. When the parcel is above the equilibrium surface in the upward slantwise convection with ∆z > 0, the trajectory is steeper than the surface, and so the parcel is pulled downward by the restoring buoyancy force until it is below the surface (The direction of parcel motion depends also on the initial velocity). Now, the assumed trajectory is more flat than the equilibrium surface, and so the parcel gets an upward acceleration until the parcel crosses the surface again. Thus, the trajectory slope changes periodically around the slope of equilibrium surface at the oscillation frequency. It is unable for us to solve the previous equation, as the oscillations depend on the horizontal processes and the trajectory slope changes with time. Here, we may only have a qualitative discussion. If νl is taken as a constant, the previous equation reads d2 z dz = −νl2 (z − zA ) , − 2µ dt2 dt It gives z = (Z − zA )e(−µ+iν)t + zA , in which Z is the departure of vertical distance from zA at the initial time t = 0. When | tan αl | ≥ | tan αθ | or the two slopes have different signs, νl is real and so the solution represents a damped oscillation with the slantwise Brunt-Vaisalla frequency ν 2 = νl2 − µ2 . This frequency of adiabatic slantwise oscillations is decreased by increasing the slope of isentropic surface or the baroclinity, and increases with the slope of slantwise convection. It depends also on the static stability, and is identical to the BruntVaisalla frequency in the barotropic atmosphere with αθ = 0 or in the vertical oscillations with αl = ±π/2. In the atmosphere with provided baroclinity and static stability, the frequency decreases with the slantwise static stability. The parameter νl becomes imaginary if the parcel is below or above the equilibrium surface in the upward or downward slantwise convection. In this situation, | tan αl | < | tan αθ | and the oscillations may become unstable as the parcel is pushed by the buoyancy force toward the equilibrium surface. If the horizontal speed is not strong enough, the parcel may cross the equilibrium surface and start a new oscillation again. Otherwise, the parcel travels continuously on one side of the surface. When the parcel cross the equilibrium surface again to form a new oscillation eventually in the spherical atmosphere, the amplitude is changed. The slantwise oscillations may also be affected by horizontal pressure gradient force. The horizontal kinetic energy created in the upward convection may then be transfered into vertical kinetic energy by the Coriolis effect on zonal motions or by the Earth’s sphericity for either zonal or meridional motions (referring to the momentum equation in the spherical coordinates, e.g., McHall, 1993). In a certain environment with strong baroclinity and horizontal pressure gradient, such as near a high level westerly jet, the slantwise oscillations or gravity waves induced may grow up to produce the cloud streets or other weather phenomena. Thus, the slantwise gravity waves may not be a kind of meteorological noise. However, they are filtered out in the current large-scale numerical prediction models. The unstable oscillations may occur also in the downward slantwise convection about the equilibrium surface.
19.9. SLANTWISE BUOYANCY OSCILLATIONS
413
The downward slantwise oscillations may be damped quickly by the viscosity and horizontal pressure gradient force.
Chapter 20 Primary air engine 20.1
Introduction
The studies on kinetic energy generations in the dry and moist atmospheres in Chapters 9-12 have revealed that the reference state attained and the amount of kinetic energy created depend not only on initial state but also on process. Thus, the weather systems developing in different environments and possessing different circulation patterns may have different efficiencies of energy conversion. However, in the previous studies the processes are identified with thermodynamic entropy production instead of circulation pattern. So, the system energy equations for a whole atmosphere may not be used to investigate the features and efficiencies of energy conversions in different circulation systems. The energetics related to the flow patterns can be studied with the energy equation of air parcels. According to the momentum equation discussed in Chapter 5, the kinetic energy is produced by external forces doing work for the parcels, such as the pressure gradient force and gravitational force. This approach requires the knowledge of pressure, mass density and air velocity distributions in the atmosphere. Unlike in the classical dynamics of solid body, the external forces on a fluid parcel, such as the pressure gradient force, may not be external for the whole fluid system. Thus, parcel kinetic energy created by the external forces is actually converted from other energies, such as the heat energy and geopotential energy. The energy conversions related to thermodynamic processes in the compressible atmosphere must be studied with the thermodynamic equations also. The Bernoulli’s equation discussed in Chapter 7 gives an example of the parcels energy equation, which can be used to study the energy conversions in a particular flow pattern without knowing the distributions of dynamic and thermodynamic variables. The energetics of air parcels may also be studied with the theory of heat engines discussed in this and following chapters without using the momentum equation. The parcels doing mechanic work in the atmosphere will be referred to as air engines, and the energetics will be called the air engine theory. The earlier applications of heat engine theory was given by Emanuel (1988, 1991), who used the efficiency of Carnot engine to estimate the maximum intensity of hurricanes. The energy generated in a man-made engine is evaluated in terms of the mechanic work created by a gas in the cylinder over a complete thermodynamic cycle. According to the first law of thermodynamics (2.10), a change in the internal energy of a given mass disappears over a continuous cycle, since the internal energy is a state function. Thus, the mechanic energy created is equivalent to the net heat absorbed by the gas on the closed cycle. While, a parcel in the atmosphere usually does not draw a complete cycle, and so it is difficult for us to tell how much heat of the parcel is converted into kinetic energy over an open path. The difficulty is overcome in this chapter by adding 414
20.2. PRIMARY AIR ENGINE
415
auxiliary paths assumed to the open path to form the simplest thermodynamic cycle. A parcel working over the cycle may be viewed as the simplest air engine, called the primary air engine. The mechanic work produced by a primary air engine may be calculated in terms of heat exchange over the cycle, assuming the air to be an ideal gas. Kinetic energy created on the open path is the mechanic work produced over the cycle minus the kinetic energy produced in the auxiliary paths. The consequence gives the general form of parcel energy equation. The particular example for the adiabatic processes is just the Bernoulli’s equation. The examples of parcel kinetic energy generation in adiabatic processes will be given in the linear atmosphere, as a parcel moves vertically, or on an isentropic surface or across the surface. The results derived from the air engine theory agree with those obtained in the preceding chapter from the Newtonian dynamics applied for air parcels. In the vertical convection, parcel kinetic energy is created and destroyed in the statically unstable and stable atmospheres respectively. When a parcel moves on an isentropic surface in an inviscid baroclinic atmosphere, it is free of the buoyancy in the vertical direction, but is nevertheless influenced by the horizontal pressure gradient force. So the parcel kinetic energy may still be changed, and there is no the simple correlation between the signs of energy generation and parcel trajectory slope with respect to isentropic surfaces. In general, the large-scale slantwise circulations in the baroclinic atmosphere with trajectories close to the isentropic surfaces may produce kinetic energy in the upward branches, and destroy kinetic energy in the downward branches. As the result, the wind speed increases upward in the free atmosphere. The thermal wind balance over a front in the viscous atmosphere may be maintained by the slantwise circulation across the pressure surfaces. Also, to remain the vertical circulations in the viscous atmosphere, the downward flows start at the height lower than the top of upward flows. This feature may be applied to explain the tropopause folding across the polar front, and the effect of cold air intrusion from the middle troposphere downward into a sever storm. More applications of the air engine theory for some typical weather systems will be discussed in the following chapters.
20.2
Primary air engine
20.2.1
Assumed cycle
In the classical thermodynamic, mechanic work produced by heat engine may be calculated over a closed cycle in a p − α diagram. When a gas in a cylinder returns back to the original thermodynamic state after a cycle, we suppose that mechanic work done over the cycle is converted into kinetic energy except the loss to friction. But, a parcel in the atmosphere does not necessarily vary in a closed p-α cycle, and not all the mechanic work done by a parcel is converted into kinetic energy. To calculate the change of parcel kinetic energy in an open path with the theory of heat engine, we may choose some auxiliary paths to complete the cycle. These auxiliary paths are physically meaningful, though they may not happen in the real atmosphere. Kinetic energy created in the open path is equal to that over the closed
20. PRIMARY AIR ENGINE
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cycle minus that in the auxiliary paths. A simple way to form a closed cycle is shown in Fig.20.1(a). When a parcel at an equilibrium position A on pressure surface pA has moved to B on pressure surface pB where its thermodynamic state may be different from that of the surroundings, a closed cycle may be designed by connecting the open path AB with three auxiliary paths BC, CD and DA. The path BC represents isobaric heating or cooling for the parcel at position B to reach the environmental thermodynamic state, while the parcel kinetic energy is conserved. On path CD the parcel moves downward vertically back to the original pressure surface pA . This path is called the static equilibrium path, as the parcel varies continuously to remain the same thermodynamic state as that of the surroundings. Since the work done by the gravitational force for the parcel is cancelled out by the work resulting from the vertical pressure gradient force on the statically equilibrium path, the parcel kinetic energy is also unchanged. Finally, it returns back to the starting position A along isobaric surface pA and reaches the initial temperature at A through isobaric heating or cooling. As the parcel motion on an isobaric surface is free of the pressure gradient force, the kinetic energy created on DA is converted from geopotential energy of the parcel. If we assume that the parcel pressure is justified to the ambience, no pressure changes occur in the surroundings after the parcel completes a cycle. The p-α diagram of the cycle is illustrated by Fig.20.1(b). It is assumed in this diagram that the potential temperature is 280 K at position A and pA = 1000 hPa. The thermodynamic variables along the p-α cycle pertain to the parcels, and may be different from those of environments. The equilibrium path CD represents also the profile of environment. We have assumed that Γ = 0.65 K/100m for this diagram. As the environmental temperature increases from D to A in the linear atmosphere (referring Fig.19.1 also), the parcel accepts heat on pressure surface pA as it returns back to the initial state at A. The cycle may also be completed by different parcels. For example, as a parcel on pA rises from A to B, another parcel on pB descends from C to D. The two convective paths may be connected by the auxiliary paths BC and DA to form the same cycle discussed. Kinetic energy created over this cycle is then shared by the parcels.
20.2.2
General parcel energy equation
According to the first law of thermodynamics (2.10), the mechanic work created by a parcel is equivalent to the internal energy destroyed plus the net heat absorbed on the cycle assumed. Since the internal energy is a state function, the change disappears over the cycle, that is
dU = 0 . Thus, the mechanic work created equals the net heat absorbed, or
W =
dQ .
20.2. PRIMARY AIR ENGINE
417
(a)
(b) Figure 20.1: Closed cycles of a primary air engine in (a) physical space and (b) p-α diagram
20. PRIMARY AIR ENGINE
418
If the heats accepted and ejected by the parcel are denoted by Q+ and Q− respectively, we have (20.1) W = Q+ − Q− on a cycle. This relationship may not be applied for an open path of a parcel. The work created may also be evaluated in terms of the changes in the thermodynamic state of a given parcel, as discussed in the following. The mechanic work done by the parcel of unit mass in the four paths is evaluated from αB pdα , (20.2) WAB = αC
WBC = αD
WCD =
αB
αA
pB dα = R∆T´BC ,
pdα = R∆T´CD +
p B
αC
(20.3) αdp ,
pA
αA
and WDA =
αD
pA dα = R∆T´DA ,
(20.4)
where T´ denotes the parcel temperature, which may be different from environmental temperature T at the same place, and ∆T´BC = T´C − T´B . As CD is in hydrostatic equilibrium, we have (20.5) WCD = R∆T´CD − ∆φDB with ∆φDB = φB − φD . The net work done over the cycle reads αB
W = αA
pdα − R∆T´AB − ∆φDB .
(20.6)
In the gravitational field, the mechanic work produced depends on change of geopotential as well as thermodynamic state of the parcel. When the environmental pressure field is not disturbed, the parcel does not change its volume or geopotential height over the cycle. So, the previous equation gives also the variation of parcel kinetic energy over the cycle. We have noted that the parcel kinetic energy changes only on paths AB and DA. The kinetic energy generated on AB is equivalent to the total work done by the parcel over the closed cycle minus the energy produced on DA which is equal to ∆φAD , Therefore, we have ∆kAB = W − ∆φAD . It follows, from (20.6), that ∆kAB =
α B αA
pdα − R∆T´AB − ∆φAB .
(20.7)
(20.8)
This equation gives a general form of parcel energy equation in the steady atmosphere, which tells the kinetic energy created as a parcel of unit mass moves from pA to pB . We prove in the following that the first two terms on the right-hand side represent the work produced by environmental pressure gradient force, and the last term is the work done by the gravitational force.
20.3. ADIABATIC PRIMARY AIR ENGINE
20.2.3
419
Relation to external work
The work created by the pressure gradient force for the parcel moving along the open path l from A to B is evaluated from Wp = −
B
α ´ A
∂p dl . ∂l
In a steady pressure field, we have p = p(l) and so Wp = −
p B
α ´ dp =
pA
α B αA
pdα − R∆T´AB .
(20.9)
Meanwhile, the work done by the gravitational force gives Wg = −∆φAB . Thus, the kinetic energy created is equivalent to the total work done by the external forces, or (20.10) ∆kAB = Wp + Wg = WF . The work produced by the surroundings is called the external work, and is different from the work done by the parcel itself on the same path represented by (20.2). Comparing (20.8) with (20.9) finds ∆kAB =
p A pB
α ´ dp − ∆φAB .
(20.11)
This expression gives another form of the general parcel energy equation in the steady environments, and is equivalent to (19.17) derived for vertical motions. The parcel energy equation can also be derived by choosing different auxiliary paths. For example, after the parcel has arrived at B, it moves on an isobaric surface to the position above A, and reaches the environmental thermodynamic state by isobaric heating or cooling. Then, it descends vertically to the starting position A remaining its thermodynamic state identical to the surroundings by diabatic heating or cooling again. The total work done over this cycle is different from that of the previous example, but the kinetic energy variation evaluated on AB is the same. The discussions above show that kinetic energy generation of a parcel can be considered as the work done by a heat engine. The working parcel may then be regarded as an air engine. The cycle composed of an open path together with auxiliary paths assumed gives an example of the simplest air engines, called the primary air engine in this study. It is noted that to produce kinetic energy, a parcel needs not to vary in a closed cycle. To assume a closed cycle may allow us to evaluate conveniently the kinetic energy generation with the theory of heat engine.
20.3
Adiabatic primary air engine
20. PRIMARY AIR ENGINE
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20.3.1
Bernoulli’s equation
If the path AB is adiabatic, kinetic energy produced by the adiabatic primary engine of unit mass can be derived from (20.11). Applying (2.48) yields p B pA
1−κ
αdp = αA pA
p B pA
pκ−1 dp = cp T´A (r κ − 1) .
(20.12)
where r signifies the pressure ratio pB /pA . We assume in the following discussions that parcel temperature at the initial position A is identical to the environmental temperature, that is T´A = TA . Inserting the previous equation into (20.11) gives ∆kAB = cp TA (1 − r κ ) − ∆φAB .
(20.13)
The energy produced on the adiabatic path AB depends only on initial and final thermodynamic states of the parcel. The previous equation may be derived also from (20.8) with α B αA
pdα = −cv ∆T´AB
in an adiabatic process. We have discussed earlier that the left-hand side of (20.12) represents the work done by pressure gradient force along trajectory l, that is Wp = −
B A
∂p α ´ dl = −R ∂l
Applying (2.49) or
T´ = T´A
p pA
B ´ T ∂p A
p ∂l
dl .
κ
for the previous equation produces Wp = −
cp T´A pκA
B ∂pκ A
∂l
dl = cp (T´A − T´B ) .
(20.14)
The work created by pressure gradient force equals the destruction of parcel enthalpy. In other words, the pressure gradient force converts the parcel enthalpy into the kinetic energy. Adding the kinetic energy created by the pressure gradient force for each parcels in a whole system, we obtain the system energy equation (7.22). The total external work achieved by the pressure force and gravitational force gives WF = Wp + Wg = cp (T´A − T´B ) − ∆φAB .
(20.15)
It depends on initial and final states of the partial, and is independent of parcel trajectory. While, the work done by horizontal or vertical component of pressure gradient force depends on trajectory, so the parcel may have the same kinetic energy, but different vertical and horizontal speeds after it reaches the same position adiabatically through different paths. Since T´B = TA r κ
(20.16)
20.3. ADIABATIC PRIMARY AIR ENGINE
421
derived from (2.49), we see that the right-hand side of (20.15) is identical to that of (20.13). From the point of Newtonian dynamics, parcel kinetic energy is created by external forces doing mechanic work, represented by (20.10). The energy equation (20.13) or (20.15) may be written as k + ψ + φ = constant , where ψ is the specific enthalpy given by (7.6). It is the Bernoulli’s equation (7.9) which is an example of the parcel energy equation for adiabatic processes. This equation is derived here from the engine theory without using the momentum equation, since the work down by pressure gradient force and gravitational force may be represented by energy conversions.
20.3.2
Extended parcel theory
In an isentropic atmosphere which possesses constant potential temperature everywhere, the kinetic energy does not change as a parcel moves adiabatically and slowly in equilibrium with the surroundings. The external work is zero in the kinetic-death atmosphere, that is Wp = −Wg = ∆a φAB , where ∆a φAB represents the geopotential difference between pressure surfaces pB and pA in the isentropic atmosphere for provided parcel temperature T´A = TA . Inserting (20.16) into (20.14) yields Wp = cp TA (1 − r κ ) . It tells that the external work done by the pressure gradient force in adiabatic processes depends on pressure change only, and is independent of the path and atmospheric thermal structure. Thus, the Wp in (20.15) can be represented by ∆a φAB , and this equation is rewritten as ∆kAB = ∆a φAB − ∆φAB . It can also be obtained by inserting dp = −ρdφ into (20.11) in the adiabatic equilibrium process. Green et al. (1966) called it the extended parcel theory. It tells that kinetic energy produced in adiabatic motions is identical to the difference between geopotential variations in the real and isentropic atmospheres. Moreover, (20.13) shows that the parcel kinetic energy created equals the decrease of geopotential energy, when the parcel moves adiabatically on an isobaric surface. Thermodynamic state of the parcel is unchanged in the isobaric and adiabatic process, so the parcel does not change its heat energy and produce no mechanic work. The pressure gradient force acting on the parcel is perpendicular to parcel trajectory. Thus, the parcel moving adiabatically on an isobaric surface is like a solid body, and the isobaric and adiabatic process may be referred to as the solid process. No heat energy is converted into kinetic energy of the parcel in the solid process. The linear relationship between conversions of the heat energy and geopotential energy in the whole atmosphere discussed in Chapter 7 may not be applied for a single moving parcel.
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Figure 20.2: Kinetic energy created as a parcel moves upward (solid) and downward (dashed) vertically in the atmosphere with the temperature lapse rate indicated
20.4
Kinetic energy created on open paths
20.4.1
On vertical paths
We consider firstly a simplest example: A parcel rises from pressure surface pA adiabatically and vertically up to pB in the linear atmosphere. The increase of geopotential height is calculated from g Γ = 0 , φAB = (TA − TB ) , Γ where TA and TB are the environmental temperatures at positions A and B, respectively. For all the examples in this chapter, we assume pA = 1000 hPa. Applying (19.34) for the previous equation yields g φAB = TA (1 − r γ ) , Γ where r = pB /pA . Applying it for (20.13) gives
∆kAB = cp TA 1 − r κ −
Γd (1 − r γ ) . Γ
(20.17)
20.4. KINETIC ENERGY CREATED ON OPEN PATHS
423
In the isothermal atmosphere with Γ = 0, we have φAB = RTA ln and
pA , pB
∆kAB = cp TA 1 − r κ − κ ln
pA pB
.
(20.18)
Variation of parcel kinetic energy in the upward motions depends on the external work done by vertical pressure gradient force Wpv = cp (TA − T´B ) and gravitational force φAB = Their ratio gives
g (TA − TB ) . Γ
Wpv Γ 1 − rκ = . φAB Γd 1 − r γ
The energy generated is zero in the isentropic atmosphere where Γ = Γd or γ = κ, and is negative or positive in the statically stable or unstable atmospheres. This is seen from the solid curves in Fig.20.2, which shows examples of the generation in vertical upward paths with TA = 280 K. When a parcel descends adiabatically from pB to pA , the kinetic energy variation is represented by
∆kBA = cp TB 1 − r −κ + and
∆kBA = cp TB 1 − r
−κ
Γd −γ (r − 1) , Γ p + κ ln A pB
Γ = 0 ,
(20.19)
,
Γ=0.
These equations may be obtained by exchanging A and B in the expressions of ∆kAB . It is noted that in general ∆kBA = −∆kAB unless in an isentropic atmosphere, because either T´A or T´B has different values in the upward and downward examples. This is shown clearly by the dashed curves in Fig.20.2. The condition of producing positive kinetic energy is the same as in the upward motions discussed previously.
20.4.2
On isentropic surfaces
Now, we consider another particular example: The adiabatic path AB in Fig.20.1 lies on an isentropic surface in the linear atmosphere. From Fig.20.1(a), we see ∆φAB = ∆φDC + ∆φAD , in which ∆φAD = gzAD ,
(20.20)
20. PRIMARY AIR ENGINE
424 and ∆φDC =
g (TD − TC ) Γ
(20.21)
if Γ is not zero. Inserting TC = TA r κ = TB
(20.22)
TD = TC r −γ
(20.23)
and into (20.21) yields
g TA r κ (r −γ − 1) . Γ The specific kinetic energy produced by a parcel evaluated from (20.13) reads ∆φDC =
∆kAB = ∆kAB0 − g∆zAD ,
(20.24)
where ∆kAB = cp TA
Γd κ r (1 − r −γ ) . 1−r + Γ
for Γ = 0 and
κ
∆kAB0 = cp TA
p 1 − r − κ ln A pB κ
.
for Γ = 0. The previous equations are the same as (20.15), which tells that the kinetic energy created is identical to the reduction of parcel enthalpy and geopotential energy. The expression for Γ = 0 is also given by (19.24) for ∆zAD = 0. Provided that TA = 280 K, the dependence of ∆kAB on the temperature stratification is displayed in Fig.20.3. The two curves for ∆zAD = 0 in Fig.20.3 represent ∆kAB0 . The differences of other curves from these two are given by g∆zAD ≈ 10zAD . If Γ = Γd , we have γ = κ and zA = zD . The isentropic surfaces are vertical, and (20.24) gives ∆kAB = 0 as the parcel motion is free of the buoyancy and horizontal pressure gradient force in the vertical paths. When Γ = Γd and the isentropic surfaces are not horizontal, the atmosphere is baroclinic in general. The temperature decreases from A to D (see Fig.20.1(a)) in the linear atmosphere as plotted in Fig.19.1. Thus, the altitude of isobaric surface over a certain height decreases from left to right (referring to Figs.20.1(a) and 19.1 again), and so the rightward component of horizontal pressure gradient force increases with height. When a parcel moves upward and rightward from A to B on an isentropic surface in Fig.19.1, it is in the static equilibrium and so no vertical acceleration is produced. However, its horizontal velocity may be increased by horizontal pressure gradient force. Therefore, the kinetic energy may nevertheless change as parcels move adiabatically on isentropic surfaces in the baroclinic atmosphere. A simplest example is a parcel moves on a horizontal isentropic surface across isobaric lines. The produced kinetic energy is not zero and may be computed by (20.13) with ∆φAB = 0. When the parcel moves downward and leftward from B to A on an isentropic surface in Fig.19.1, the energy produced is ∆kBA = −∆kAB . The parcel moves against horizontal pressure force and so may lose kinetic energy especially when zAD ≤ 0. Owing to the energy generation and destruction in the upward and downward slantwise convection respectively, the wind speed increases with height in the troposphere. This vertical wind shear is not simply due to the high friction in the
20.4. KINETIC ENERGY CREATED ON OPEN PATHS
425
Figure 20.3: Kinetic energy produced as a parcel moves upward on an isentropic surface in the linear atmosphere. Solid and dashed curves are drawn for Γ = 0.65 and 0.75 K/100m, respectively.
boundary layer. Thus, a weak cross-front circulation embedded on the geostrophic winds in a frontal zone is helpful for maintaining the thermal wind balance in the viscous atmosphere. Since the horizontal pressure force depends on the slope of isobaric surface which increases with height in a baroclinic atmosphere, the energy generated or destroyed increases at a lower pressure ratio r. Moreover, the slope of isobaric surface at high levels increases with static stability in the linear atmosphere, and decreases as the bottom pressure surface pA at position D rises. Thus, the energy generated in the isentropic motions increases with the static stability, but decreases as the height of pA increases at place D. These features are demonstrated in Fig.20.3. The algorithm (20.24) does not depend on the baroclinity explicitly. In other words, the parcel energetics in the isentropic motions is irrelevant to the slope of isentropic surface as long as Γ = Γd . When the slope is larger, the horizontal pressure gradient force in the direction of parcel motion is greater, while the horizontal distance of parcel displacement is shorter. So, the work done by horizontal pressure gradient force in the linear atmosphere is constant as the parcel moves on an isentropic surface between the provided pressures. When a parcel is accelerated by the horizontal pressure gradient force only, it tends to reduce its trajectory slope so that it is difficult for the parcel to remain on an isentropic surface. The energy created as a parcel crosses the isentropic surfaces is discussed in the following.
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20.4.3
On upward sloping paths
When a parcel moves on an isentropic surface, its kinetic energy created is independent of the baroclinity, and decreases with the static stability. This does not agree with the energy conversions in the whole atmosphere studied in Chapter 9, because air motions in the energy conversions are not contained on isentropic surfaces. If the upward trajectory AB is not on an isentropic surface, the parcel temperature is different from that of the surroundings after it arrives at B, and we have (20.25) T´B = TB + δT . For the atmosphere with constant lapse rate, we apply (20.16) and TB = TC giving TC = TA r κ − δT . Inserting it together with (20.23) into (20.21) yields ∆φDC =
g (TA r κ − δT )(r −γ − 1) , Γ
Γ = 0 .
(20.26)
The energy generated can be evaluated from (20.13) using (20.20), giving ∗ + ∆kAB = ∆kAB
g δT (r −γ − 1) , Γ
Γ = 0 ,
(20.27)
∗ is the energy generation as the parcel moves on the isentropic surface where ∆kAB θA from pA to pB represented by (20.24). As discussed earlier, the work done by pressure gradient force in an adiabatic motion is independent of path, while the work created by the gravitational force depends on path in pressure coordinates. The last term in (20.27) represents the difference in the amounts of geopotential energy increased as the parcel moves on and across the isentropic surface. The last equation tells that the energy created will ∗ , if the parcel is warmer or cooler than the surroundings be more or less than ∆kAB at the end of the path. The slope of parcel trajectory is smaller than the slope of isentropic surface in the former case, but is larger in the latter. Equation (20.27) may be rewritten as
∆kAB = ∆kAB0 − g∆zAD ,
(20.28)
in which
∆kAB0 = cp TA
Γd κ g r (1 − r −γ ) + δT (r −γ − 1) . 1−r + Γ Γ κ
The ∆kAB0 evaluated for Γ = 0.65 K/100m and TA = 280 K are shown by the heavy solid and dashed curves in the upper part of Fig.20.4 for δT = 2 and -2 K respectively. If the bottom pressure surface pA is not geographically horizontal, the kinetic energy generated is the value plotted minus the conversion of geopotential energy in the amount of g∆zAD . For comparison, the kinetic energy created as a parcel moves on an isentropic surface is drawn by the light dashed curve, which is the same as the solid curve with ∆zAD = 0 in Fig.20.3. The increase of kinetic
20.4. KINETIC ENERGY CREATED ON OPEN PATHS
427
Figure 20.4: Kinetic energy produced as a parcel moves upward (upper part) and downward (lower part) in the baroclinic atmosphere with Γ = 0.65 K/100m. The heavy solid and dashed curves are drawn for δT = 2 and -2 K, respectively. The light dashed is for δT = 0.
energy generation with reduction of trajectory slope may be seen by comparing the three curves. It was proposed (e.g., Petterssen, 1956; Holton, 1992) that the sign of kinetic energy generation in a baroclinic atmosphere depends on whether the slope of air trajectory is greater or smaller than the slope of isentropic surface. We have found that the vertical kinetic energy is conserved if parcels move on isentropic surfaces in the inviscid atmosphere, while the horizontal kinetic energy may be changed by horizontal pressure gradient force. The heavy dashed curve in the upper part of Fig.20.4 shows now that parcel kinetic energy may still be produced even if the trajectory slope is larger than the slope of isentropic surface. The contribution of horizontal pressure gradient force cannot be ignored especially at high levels. It is learned in the preceding section that the slope of isentropic surface increases with the baroclinity. Thus, for a provided trajectory slope and static stability Γ, the departure of parcel temperature from the environmental temperature δT is larger in the atmosphere with a stronger horizontal temperature gradient. So more parcel kinetic energy is created by the slantwise upward motion in the stronger baroclinic atmosphere. Moreover, the slope of isentropic surface decreases at higher static stability in the stable atmosphere. Thus, less kinetic energy is created on the same trajectory if the static stability is higher.
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20.4.4
On downward sloping paths
If a parcel moves adiabatically downward from B to A, the kinetic energy generation is evaluated by exchanging A and B in (20.13), giving ∆kBA = cp TB (1 − r −κ ) + ∆φAB .
(20.29)
For comparison, we assume that the parcel moves downward along the upward path discussed earlier. The parcel temperature at the starting position B is the environmental temperature TB given by (20.25), or TB = TA r κ − δT derived using (20.16) also. Inserting this equation together with (20.26) and (20.20) into (20.29) yields ∆kBA = ∆kBA0 + g∆zAD with
∆kBA0 = cp TA r κ − 1 +
Γd κ −γ g r (r − 1) − δT (r −γ − 1) + cp δT (r −κ − 1) , Γ Γ
where ∆zAD is the same as that in (20.28), since the two paths overlap on each other. This equation and (20.28) do not include explicitly the baroclinity. Since the slope of isobaric surface depends on the baroclinity, the δT included in these equations depend on the baroclinity for a given trajectory. The last equation is rewritten as ∆kBA + ∆kAB = cp δT (r −κ − 1) . It gives the net energy created in a complete cycle with local mass balance. The δT in the previous equations represents the temperature difference between the upward moving parcel and environment at position B. The temperature of the downward parcel at the end position A is evaluated from T´A = TB r −κ = TA − δT r −κ
(20.30)
derived using (20.25). The temperature difference between the parcel and environment at A is greater than that at the end of upward path B. Applying the last equation for the previous yields ∆kBA + ∆kAB = cp (TA − T´A − δT ) = qd + qu .
(20.31)
The right-hand side of the last equation gives the heats accepted on the isobaric heating (or cooling) processes at the ends of downward and upward paths respectively. The last equation tells that the net kinetic energy created is identical to the net heat absorbed in the cycle. The evaluated examples of ∆kBA0 are demonstrated in the lower part of Fig.20.4. The process on isentropic surfaces is depicted for comparison by the light dashed curve in the same figure, which may not be seen clearly. When δT > 0, the upward parcel trajectory has a slope less than the slope of isentropic surface in the statically stable atmosphere. Since T´A < TA in this situation as shown by (20.30), the
20.4. KINETIC ENERGY CREATED ON OPEN PATHS
429
trajectory of downward parcel is also more flat than the isentropic surfaces crossed. When δT < 0, both the upward and downward paths are steeper than the isentropic surfaces crossed. This figure shows that the parcel kinetic energy is destroyed on the downward path close to an isentropic surface, regardless of the trajectory slope. Again, this consequence does not agree with the traditional studies, which declared that the energy should be created in the more flat trajectory. The kinetic energy destroyed by horizontal pressure gradient force increases slightly with trajectory slope. The dependence on trajectory slope is less than that in the upward motion, since the vertical kinetic energy created or destroyed is canceled out partially by the change of horizontal kinetic energy. Net kinetic energy is created or destroyed in a cycle of the slantwise circulation, when the trajectory slopes are less or greater than the slopes of isentropic surfaces. Usually, the isobaric surfaces are not flat near the Earth’s surface. It is higher on the cold side of polar front or inside a cold high near the surface, where the cold subsidence cross the isobaric surfaces towards the lower pressures. In this situation, ∆zAD > 0 and the kinetic energy destroyed in the downward flows is less than that plotted in Fig.20.4. The strong cold wind on the surface may be experienced as a cold front passes away. In the slantwise vertical circulations across a front or on both sides of a front, warm air climbs over the front surface with a slope close to that of the front. Parcel kinetic energy is produced in the upward flows as shown by Figs.20.3 and 20.4. The accelerated flows may form a westerly jet at the top of the front. Meanwhile, the slantwise downward motions on the cold side of the front destroy kinetic energy. When the trajectory slope is less than the slope of isentropic surface, the kinetic energy created in the upward motion is more than that destroyed in the downward motion. The reverse is also true. This can be seen by comparing the heavy solid or dashed curves in the upper and lower parts of Fig.20.4. Since the net energy produced in the cycle is so small and may not be able to compensate the loss to friction and viscosity, the downward flows may start at the level lower than the top level of the slantwise upward motions. Thus, the tropopause descends discontinuously from the warm side to the cold side called the tropopause folding. The slantwise convection extending to different heights in the upward and downward branches is also found in the super storms, which produce a large amount kinetic energy as discussed in Chapter 22. Starr (1948) claimed that the northward angular momentum transport in the atmosphere is favored by the large-scale perturbations with north-east tilt of troughs and ridges. The theoretical proof may be referred to the study of McHall (1993). He elucidated that these waves may also transport heat poleward. It is argued here that the asymmetric wave structure can be explained by the kinetic energy balance in the viscous slantwise convection. Since the downward and southward wave path is shorter than the upward and northward path, net kinetic energy may be created over a cycle to maintain the circulations against the kinetic energy dissipation.
Chapter 21 Dry air engines 21.1
Introduction
The energy conversions in the dry and moist isolated atmospheres have been studied in Chapters 9-12, using the variational approaches to solve the system energy equations. The energy conversion evaluated depends on choice of reference state. The reference state depends on the process which may be identified with thermodynamic entropy production. There are two basic difficulties for us to use this approach for the study of energy conversion in a particular weather system: 1) So far, we have no accurate statistical data of thermodynamic entropy produced as a weather system develops, and 2) the weather systems are not isolated in general. A particular process of energy conversion may also be identified with the circulation pattern, which can be illustrated straightforwardly by the distributions of dynamic variables. We may then use the primitive equations discussed in Chapter 5 to predict these variables and energy budget at a future time. The predictions may only be accomplished by numerical procedures usually. To avoid the mathematical difficulties in solving these equations, we have introduced in the preceding chapter the air engine theory, which can be applied to study the energy conversions in different circulation systems. The key of this study is to find the typical thermodynamic cycles in a particular weather system in order to calculate the energy conversion using the theory of a heat engine. In general, the thermodynamic cycles in the weather systems are more complicated than those of industrial engines studied in the classical thermodynamics. We shall introduce in this chapter different air engines to study the energetics of large-scale baroclinic waves and mean meridional circulations in the dry atmosphere. The air engines designed for the study of moist convective storms will be discussed in the next chapter. The air engine theory for a system is based on the equivalent principle: If two parcels or entities at a given geographic height are in an identical thermodynamic state and possess the same humidity, they may be considered identical in the study of energy conversion. Thus, a thermodynamic cycle completed by different parcels can be considered equivalently as by one parcel. The produced kinetic energy is then shared by these parcels. For example, if a parcel rises from 1000 hPa surface up to 300 hPa in a convective system and eventually reaches the thermodynamic state of the surroundings there. Meanwhile, another parcel descends from 300 hPa down to 1000 hPa and reaches the environmental thermodynamic state at the low level, the thermodynamic cycle may be considered to be completed by one parcel. The most fundamental large-scale circulation patterns in the dry atmosphere are the large-scale geostrophic perturbation and slantwise convection around the polar front. These circulations will be studied by introducing the Joule air engine, of which the thermodynamic cycle is composed of two adiabatic and two isenthalpic diffusion processes. It can be found that kinetic energy may be created in the Joule 430
21.2.
JOULE AIR ENGINE
431
circulations if potential temperature of slantwise rising flow is higher than that of slantwise subsidence, or the trajectory slope is less or larger than the slope of isentropic surface in the statically stable or unstable atmosphere. A typical example is the vertical convection in the statically unstable atmosphere. As energy conversions in the atmosphere is nonlinear in general, the unstable wavelengths derived from traditional linear theories should be examined by the energy study. It is learned in Chapter 18 that developments of baroclinic disturbances may be illustrated more realistically by considering the interactions between the disturbances and background fields. The study on the interactions did not tell us how to find the developing and decaying wavenumbers in the atmosphere with provided baroclinity. When the Joule convection is applied for the large-scale perturbations at middle and high latitudes, we may tell the unstable wave spectrum supported by baroclinic energy sources. It is also possible to design different air engines for the study of baroclinic instabilities, according to the typical circulation patterns of the waves. It is discussed in Chapter 9 that kinetic energy generated in an isolated system is maximum in the extremal process with minimum thermodynamic entropy production. The minimum thermodynamic entropy may be zero in a baroclinic system. As an extremal example, the work done in a reversible process may be studied by designing a reversible air engine, such as the Carnot engine which absorbs and ejects heat in isothermal processes. Since air motions in the atmosphere are not constrained on isothermal surfaces in general, a new kind of reversible air engine, called the equilibrium air engine, is introduced to discuss the reversible processes in the atmosphere. This air engine may be adopted to deal with the large-scale mean meridional circulations in the atmosphere. We have pointed out in the study of available enthalpy that energy conversion in the atmosphere is not produced simply by exchanging parcel positions, as mixing of parcels may be produced by molecular and turbulent diffusions in the process. The mixing process provides the challenges to the classical definitions of, for example, the adiabatic process, Lagrangian fluid particles and their trajectories. For simplicity, the mixing will be considered only in certain paths in the thermodynamic cycles of air engines. The effects on the adiabatic convective processes, called the entrainment or detrainment, will be studied in Chapter 23.
21.2
Joule air engine
21.2.1
Joule cycle
A simple process of parcel kinetic energy generation is the position exchange of two parcels at different pressure heights. A parcel of unit mass and temperature TA at position A moves from pressure surface pA adiabatically to B on pressure surface pB (referring to Fig.21.1(a)). Meanwhile, another parcel of unit mass and temperature TB at position B on pB moves adiabatically toward A on pA . Kinetic energy created on the two adiabatic paths AB and B A is given by (20.13) and (20.29)
21. DRY AIR ENGINES
432 respectively. Adding them together yields
∆kAB + ∆kB A = cp (r −κ − 1)(TA r κ − TB ) + ∆φA B − ∆φAB . The thermodynamic variables in this equation pertain to the parcels, and may be different from those of environments at the same places. When the mass is conserved at a local place, we assume that a parcel moves from A to A on pressure surface pA , and reaches the environmental temperature TA . Meanwhile, another parcel moves from B to B on pressure surface pB , and gets to the environmental temperature TB . Now, we have a complete cycle called the Joule cycle in the classical thermodynamics. Kinetic energy variation in the two solid processes may be evaluated from (19.21), giving ∆kA A + ∆kBB = φA − φA + φB − φB = ∆φAB − ∆φA B . Thus, the energy generation on the whole cycle reads ∆k = cp (r −κ − 1)(TA r κ − TB ) .
(21.1)
The parcels working on the Joule cycle form a Joule air engine. The previous equation is the same as (20.31). It shows that kinetic energy produced by Joule air engine depends on the environmental pressure and temperature at A and B , and is independent of parcel trajectories between the two positions. Thus, it may be considered that the kinetic energy is created as the two parcels at A and B exchange their positions. The kinetic energy created by a Joule engine is equivalent to the enthalpy absorbed as shown by (20.1). To produce positive mechanic work, the heat engine absorbs heat on the isobaric path A A in the amount Q+ = cp (TA − TA ) = cp (TA − TB r −κ ) , and ejects the heat Q− = cp (TB − TB ) = cp (TA r κ − TB ) on the other isobaric path BB . The net heat absorbed is Q+ − Q− and equals the mechanic work done over the cycle. In the process of kinetic energy generation, the Joule engines transport heat energy from one place to another.
21.2.2
Condition of doing positive work
Using the definition of potential temperature (2.55) for (21.1) yields κ κ ∆k = cp p−κ θ (pA − pB )(θA − θB ) ,
(21.2)
where θB is environmental potential temperature at position B on pressure surface pB . Mechanic work may be created by Joule air engine if the circulations between two pressure levels are not constrained on an isentropic surface. If ambient potential
21.2.
JOULE AIR ENGINE
433
(a)
(b) Figure 21.1: Joule cycle producing positive work in (a) and negative work in (b). The light curves represent environmental profile along AB .
21. DRY AIR ENGINES
434
temperature at the high-pressure end of path AB is higher than that at the lowpressure end, positive work will be produced in the cycle, and the parcels in the circulation form a heat engine. Otherwise, they work like a refrigerator. For the vertical convection, kinetic energy may be created only in the statically unstable atmosphere. In the statically stable atmosphere, the Joule cycles may produce kinetic energy only in the slantwise convection with a trajectory slope less than the slope of isentropic surface. This agrees with the study in the preceding chapter. Thus, the stable atmosphere must be baroclinic for the energy conversion. This consequence has been obtained also in Chapters 8 and 10 from the point of thermodynamic entropy production. Thus, there must be a general relation between the kinetic energy generation and thermodynamic entropy production. This relation will be discussed later on in this section. The p-α diagrams of Joule engines producing positive and negative kinetic energy are illustrated graphically in Fig.21.1. The amount of kinetic energy generated or destroyed equals the area enclosed by the p-α cycle. The environmental profile along path AB is represented by the light curves in the figure. It is plotted using the slantwise lapse rate along the path. The volume of the parcel is evaluated from the ideal-gas equation. The temperature along the path given by (19.33) reads ΓTA ∗ Γ (r −γ
.
(21.3)
RΓTA . p[Γ + Γ∗ (r −γ − 1)]
(21.4)
TB =
Γ+
− 1)
Applying it yields α=
It is assumed that Γ∗ = 1.3 and 0.7 K/100m for the positive and negative examples, respectively, with TA = 280 K at pA = 1000 hPa. The slope of AB is evaluated from (19.27). If Γ = 0.65 K/100m,
and
Ty = −30 K/100m ,
the slopes are tan α = 0.0046 and 0.06 for Γ∗ = 1.3 and 0.7 K/100m, respectively. Fig.21.1(a) shows that the local temperature is increased at high levels but decreased at low levels by the Joule engines creating kinetic energy. So, the vertical temperature gradient is reduced or the static stability is increased after the energy generation. This agrees with the study of energy conversion in Chapter 9 (referring to Fig.9.1). The Earth’s surface which accepts solar radiation more efficiently than the atmosphere above is a heat source of air engines. The heat in the lower troposphere may be transported to high levels by the positive Joule convection. While, the cycle producing negative work shown in Fig.21.1(b) transport heat downwards and so reduce the static stability.
21.2.3
Examples of kinetic energy generation
If the slantwise lapse rate along path AB denoted by Γ∗ is constant, the environmental temperature at position B is evaluated from (21.3). Applying it for (21.1) yields Γ −κ κ . (21.5) ∆k = cp TA (r − 1) r − Γ + Γ∗ (r −γ − 1)
21.2.
JOULE AIR ENGINE
435
Figure 21.2: Kinetic energy produced by Joule air engine. The Γ∗ indicates the slantwise lapse rate along AB .
Fig.21.2 illustrates the examples of kinetic energy generated by Joule air engine with TA = 280 K. The energy is created when the slantwise lapse rate is greater than the slantwise adiabatic lapse rate, or the slope of path AB is less than the slope of the isentropic surfaces crossed. The positive energy generation in the figure may turn to negative at high levels, since the slope of isentropic surface decreases upward and becomes smaller than the trajectory slope in the linear atmosphere. In the slantwise Joule convection along path AB in the statically stable atmosphere, kinetic energy is created in the upward motion and destroyed in the downward motion as discussed in Chapter 20. The energy creation or destruction over a complete cycle of Joule convection depends simply on the slope of path AB with respect to isentropic surface, if the viscosity is ignored. This consequence is obtained also in the previous chapter. Thus, comparing the slopes of trajectory and isentropic surface is still significant for estimating the energy conversion in a system over a whole circulation cycle. The cross section of a mature cyclone developed in the extratropical baroclinic atmosphere is shown in Fig.9.2. A remarkable feature is the highly asymmetric distribution of vertical motions near the cyclone center. Downward and upward motions occurred on the western and eastern sides of the center, respectively. The region of downward flow was colder than the surroundings below 300 hPa, while the region of upward flow was warmer than the surroundings. This fact implies that the
21. DRY AIR ENGINES
436
slantwise convection in the cyclone took place along the paths with slopes less than those of isentropic surfaces. It is well known that the extratropical cyclones may develop as soon as a trough of large-scale perturbation over the middle troposphere catches the polar front. For a provided trajectory, the atmosphere in a frontal zone has the strongest slantwise static instability, which in turn forces the slantwise convection to reach the largest slope in the zone. We see now from (21.1) that kinetic energy generated in the Joule convection is increased by reducing the environmental temperature at high levels. When the temperature decreases at high levels, the slantwise static stability involved in (21.5) decreases too. The reduction of temperature or increase of slantwise static instability may be caused by the cold advection behind the troughs of large-scale baroclinic perturbations over the middle troposphere.
21.2.4
Entropy productions
The condition of doing positive work can be revealed also by evaluating the variation of thermodynamic entropy or geopotential entropy on a Joule cycle. If the two adiabatic processes AB and B A are reversible, thermodynamic entropy is produced only on the two paths of isobaric heating or cooling, given by ∆´ sBB = cp
T ´ B dT TB
and ∆´ sA A = cp ln Applying
TB = TA
pB pA
= cp ln
T´
TB TB
TA . TA
κ
,
TA = TB
pB pA
−κ
(21.6)
for the two equations and adding them together, we see that the entropy production is zero over the cycle. This consequence may also be obtained from (4.28), since the potential temperature does not change in the two adiabatic processes. Meanwhile, the specific entropy produced in the surroundings reads ∆s = −
cp cp (TA − TA ) − (TB − TB ) . TA TB
Applying (21.6) for it yields ∆s =
cp (C − 1)2 , C
in which C=
TB −κ r TA
or C=
θB . θA
(21.7)
Here, θA and θB represent the environmental potential temperatures at A and B , respectively.
21.2.
JOULE AIR ENGINE
437
As thermodynamic entropy of the parcel is conserved over a cycle, the total entropy production is identical to that in the surroundings. Since C > 0, the entropy production is not less than zero. Thus, the process along the p-α cycle shown in Fig.21.1(a) is thermodynamically possible. Kinetic energy generation in the cycle is positive. While, the process in an opposite direction as shown by Fig.21.1(b) produces negative thermodynamic entropy and kinetic energy, and so cannot occur in isolation. Particularly, in the reversible process with zero thermodynamic entropy production, we have C = 1, that is
TB = TA
pB pA
κ
= TB .
The atmosphere with C = 1 is an isentropic atmosphere, or the kinetic-death atmosphere discussed in Chapter 17. No mechanic work is created in this atmosphere over the Joule circulations, and no kinetic energy is produced. It is discussed earlier in this section that the statically stable atmosphere must be baroclinic in order to generate kinetic energy. Now, we see that the entropy is produced only in the process of kinetic energy generation. Thus, the entropy can only be created in the baroclinic atmosphere if it is statically stable. This agrees with the result obtained in Chapters 8 and 10 from the entropy balance in the whole isolated system. It is interesting to point out the Joule engine may be a reversible engine if the isobaric heat exchanges take place as the parcel is in equilibrium with the heat source and sink. In this situation, thermodynamic entropy created in the environments is identical to that produced by the engine. The heat reservoirs of reversible Joule engine change the temperatures continuously at constant pressures. The atmospheric processes are also controlled by the geopotential entropy law discussed in Chapter 8. Equation (21.1) shows that parcel kinetic energy created in the Joule convection depends only on the starting positions. So, a Joule cycle may be viewed simply as two parcels at A and B exchange their positions. According to (8.48), geopotential entropy of the atmosphere is increased by the position exchange, when potential temperature of the rising parcel is greater than that of the sinking parcel. In addition, the mean potential temperature or potential enthalpy of the system is conserved in the quasi-adiabatic process as learned in Chapter 7. Thus, the geopotential entropy increases over the Joule cycle which produces net kinetic energy. While, the Joule convection producing negative kinetic energy will destroy geopotential entropy and thermodynamic entropy, and so may not occur independently of external forcings.
21.2.5
Efficiency of Joule engine
The efficiency of heat engine is defined as the ratio of produced net work to the heat absorbed over a cycle, that is, W Ef f = + . Q Since the work is equivalent to the net heat absorbed over the cycle, we have Ef f =
Q+ − Q− Q+
(21.8)
21. DRY AIR ENGINES
438
from (20.1). Here, Q− is positive. The work or the net heat absorbed is also identical to the kinetic energy generated, so the efficiency is rewritten as Ef f =
∆k , Q+
(21.9)
The heat accepted over a Joule cycle is Q+ = cp (TA − TA ) = cp (TA − TB r −κ ) . Applying it together with (21.1) yields EJ = 1 − r κ .
(21.10)
The efficiency of Joule engine depends on the pressure ratio only. It tends to unity when r → 0. Usually, the slope of isentropic surface is small in the atmosphere without a temperature front, so the Joule convection therein can hardly extend to high levels and the efficiency is generally low. But in the unstable slantwise convection around a polar front, the parcels may rise to a high altitude and so the efficiency is relatively high. If pA = 900 hPa and pB = 300 hPa, we have EJ = 0.27 approximately.
21.3
Energetics of baroclinic waves
21.3.1
The baroclinic waves
The Joule air engine discussed previously may be applied to study the energetics of baroclinic waves in the atmosphere. A simple example of the atmospheric perturbations embedded in zonal flows may be represented by vy = Vy cos(νt − nx x + ψ) ,
ω = Vp cos(νt − nx x)
in pressure coordinates. Here, vy = dy/dt is the perturbation wind velocity in the meridional direction y; ω = dp/dt is the perturbation vertical velocity; ν and nx are the frequency and zonal wavenumber respectively, and ψ is the phase difference between the two perturbation velocities. For simplicity, all the wave parameters are assumed constant, and the zonal wind velocity vx = dx/dt is taken as constant, too. Inserting x = vx t into the previous equations and integrating them yield Vy sin [(ν − nx vx )t + ψ] + y0 ν − nx vx
y= and
p=
Vp sin(ν − nx vx )t + p0 . ν − nx vx
The parcel trajectory obtained from the perturbation equations gives Vy y = y0 + Vp
(p − p0 ) cot ψ ±
∆p2max
− (p − p0
)2
sin ψ ,
21.3. ENERGETICS OF BAROCLINIC WAVES
439
(a)
(b)
(c) Figure 21.3: Joule cycles of baroclinic waves. (a) ψ = 0, (b) ψ = π/4 and (c) ψ = π/2. The light lines represent the potential temperature.
21. DRY AIR ENGINES
440
where y0 and p0 indicate the central position of the perturbations, and Vp ∆pmax = ν − nx vx
is the maximum height deviation from p0 . For the following parameters Vy = −10 m/s ,
vx = 15 m/s ,
Lx = 4000 km ,
Vp = 0.12 Pa/s , y0 = 0 ,
(21.11) (21.12)
the perturbation trajectories are plotted by the heavy curves in Fig.21.3. In this figure, Lx = 2π/nx measures the zonal wavelength. The wave frequency is evaluated from ν = cnx , where c denotes the phase speed, taken as the Rossby wave speed c = vx −
β , n2x
β = 2Ω cos 45◦
in this and following examples. The a and Ω here are the radius and angular speed of the Earth, respectively. In an idealized situation, the parcel rises adiabatically from A to B; while another parcel descends adiabatically from B to A . The positions A and A or B and B may be at different longitudes. If we add two isenthalpic diffusion processes BB and A A to connect the previous two adiabatic paths, the heavy curves in this figure represent also the complete Joule cycles. Kinetic energy generation in the cycles depends on the wavelength and environmental temperature distribution. Supposing pA = pθ = 1000 hPa, potential temperature distribution in the linear atmosphere is given by
pθ θ = (θ0 + Ty y) p
b
,
b=
R (Γd − Γ) , g
(21.13)
where θ0 is a constant. For Ty = −10 K/1000km ,
Γ = 0.65 K/100m ,
the potential temperature profile is depicted by the light lines in Fig.21.3. When ψ = 0, the meridional and vertical perturbation velocities are opposite in phase in the pressure coordinates, so the wave particles move northward and upward or southward and downward. When the mean temperature decreases northward, the trajectory tilts in the same direction as an isentropic surface does. Fig.21.3 shows that the projection of wave trajectory on the yp-plane draws a straight line segment, of which the slope is less than that of isentropic surface. The trajectory B A is not a simple reversal of AB in the p-α cycle, since T´A = T´A and T´B = T´B in general. Another extremal example is given by ψ = π (the figure is not shown). In this case, slopes of the trajectory and isentropic surface have different signs, and the wave destroys kinetic energy over a cycle. For 0 < ψ < π, the trajectory draws the ellipse on the yp-plane as shown by (b) and (c) in Fig.21.3.
21.3. ENERGETICS OF BAROCLINIC WAVES
21.3.2
441
Kinetic energy generation
Kinetic energy generated by the baroclinic waves over a Joule cycle is evaluated from (21.13), the potential temperatures at A and B in the linear atmosphere are given by b pθ , θA = (θ0 + Ty yA ) pA and
θB = (θ0 + Ty yB )
pθ pB
b
,
respectively, in which pB = p0 − ∆pmax ,
pA = p0 + ∆pmax , and yA =
Vy ∆pmax cos ψ , Vp
yB = −
Vy ∆pmax cos ψ . Vp
Thus, (21.2) gives
∆k = +
κ cp p−κ s (pA
− pB ) θ 0
Vy Ty ∆pmax Vp
κ
ps pA
b
ps pA
+
b
ps pB
− b
ps pB
b
cos ψ
.
For the parameters assumed previously, the kinetic energy created is maximum as ψ = 0 and minimum as ψ = π. In the former case, warm air moves upward to the north, while cold air moves downward to the south (referring to Fig.21.3(a)). The reverse is the latter case. Table 21.1: Kinetic energy generation (m2 /s2 ) in the baroclinic waves for Γ = 0.65 K/100m |Ty | Zonal wavelength Lx (km) (K/1000km) 1000 2000 4000 8000 6 -19398 -1752 -358 -85 8 -13011 -500 -57 -11 10 -6624 751 245 64 12 -237 2003 546 139 14 6150 3255 847 213 Period (h) 19 42 132 -198
Table 21.1 illustrates the dependence of kinetic energy generation on the perturbation wavelength and background baroclinity for ψ = π/4. The parameters are given by (21.11) and (21.12), except for those provided additionally in the table. The perturbation trajectory of this case is shown in Fig.21.3(b).
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442
Table 21.2: Kinetic energy generation (m2 /s2 ) in baroclinic waves with Lx = 4000 km |Ty | (K/1000km) 6 8 10 12 14
Static 0.65 -358 -57 245 546 847
stability Γ (K/100m) 0.70 0.75 0.80 -194 30 240 156 369 567 506 707 894 855 1045 1221 1205 1383 1548
The perturbation decreases if the energy is destroyed over a cycle. So only the waves with sufficiently long wavelengths may exist in the atmosphere of low baroclinity. For a provided baroclinity, there is the critical wavelength beyond which the perturbation kinetic energy increases. The critical wavelength decreases with increasing the baroclinity. For a provided wavelength, there is the minimum baroclinity, called the critical baroclinity, over which the baroclinic waves may be destabilized. The critical baroclinity decreases at a longer wavelength. The baroclinic waves may become unstable as they propagate into a region of increased baroclinity. The bottom line in the table gives the periods of eastward waves in units of hour, evaluated from 2π . τ= cnx The negative period means that the wave propagate westward. The rate of energy conversion decreases quickly with increasing wavelength in the progresive waves. If the rate equals that of frictional dissipation, the waves are stable. The wave instability depends also on the static stability. Table 21.2 shows that the critical baroclinity decreases at a lower static stability. More perturbation kinetic energy will be produced in a region with higher baroclinity or static instability. The baroclinic waves discussed are one of the simplest example. The large-scale baroclinic perturbations in the atmosphere may tilt with height and latitude, and the zonal wind or temperature field may have perturbation components, too. Kinetic energy generated in these waves may also be studied with the air engine theory. The critical conditions of baroclinic wave instability have been studied extensively since Charney (1947) and Eady (1949). In the traditional linear theories of disturbance development, only the kinematic processes were discussed without considering the energy conversion. The conditions obtained by solving the simplified perturbation equations may not be sufficient, as they cannot tell us whether the unstable waves will develop or decay. It is pointed out by McHall (1993) that the critical baroclinities for disturbance development derived by previous authors were below the climatological mean baroclinity in the lower troposphere at middle latitudes. The energy source in the weak baroclinic atmosphere may not be strong enough for supporting the perturbation development. The linear wave instability may also be studied with the air engine theory. If the dissipation rate of kinetic energy over a cycle is given by Dv , the unstable wave spectrum and background
21.4. KINETIC ENERGY GENERATION IN A SYSTEM
443
baroclinity may be evaluated from the energy budget equation ∆k −
Dv dt > 0 .
The development may take place only if the unstable wavelength derived by the linear perturbation theory satisfies the energy budget equation. Otherwise, the unstable waves will decay.
21.4
Kinetic energy generation in a system
It is discussed earlier that the pressure gradient force is an external force for a single parcel in the atmosphere, but is not really related to external process for the whole system. Thus, the production of mechanic work resulting from the pressure gradient force may be considered as the process of energy conversion within the system caused by the force. This is seen clear from (20.14), which tells that the work created by pressure gradient force for a parcel is equivalent to the reduction of its enthalpy. The work done by the gravitational force may also be regarded as conversion of the geopotential energy. From this point of view, the energetics of the atmosphere may also be illustrated by the air engine theory without considering the dynamic processes. We have derived in the preceding chapter the parcel energy equation from the air engine theory without using the momentum equation. It can be seen in this section that the system energy equation discussed in Chapter 7 may also be derived from the air engine theory without the help of Newton’s laws. When a system changes from state 1 to state 2, we may choose the simplest process to connect the state 2 and state 1, and complete the p-α cycle for the system. Kinetic energy produced in the real process is the mechanic work created on the cycle minus the kinetic energy variation in the auxiliary process. Usually, the parcels change their pressures as the atmosphere varies from state 1 to state 2. For simplicity, we assume that a parcel on a given pressure surface of state 1 will get to the same pressure surface at state 2. If this is not real, we may assume that the parcel exchanges position with another one on the original pressure surfaces through the static equilibrium process discussed earlier. Temperatures of these two parcels are adjusted to the environments by isenthalpic diffusions in the process. Variations of the parcels from state 1 to 2 may not form closed cycles generally. We may complete the cycle by adding auxiliary process of isobaric heating or cooling for each parcel. Assume that the system has N parcels or entities of mass δmj and temperature Tj (j = 1, 2, · · · , N ). The parcels are sufficiently small so that each of them may be considered to be uniform in equilibrium. The heat absorbed by these parcels, as they vary isobarically from state 2 back to state 1, is given by ∆Ψ = cp
N
∆Tj δmj ,
j=1
where ∆Tj = Tj1 − Tj2 . The total work done by the N parcels depends on the difference of absorbed and ejected heat in the isenthalpic diffusion processes of the
21. DRY AIR ENGINES
444 Joule cycles, namely, W =
N
(cp ∆Tj + ∆m Qj )δmj ,
j=1
in which ∆m Qj is the heat exchange of parcel j with other parcels within the system, as it changes from state 1 to state 2. It is obvious that in an isolated system we have N
∆m Qj δmj = 0 ,
j=1
and so W = cp
N
(T1j − T2j )δmj .
(21.14)
j=1
The total work done by all parcels over a Joule cycle equals the reduction of total enthalpy. Since total energy is conserved, the only way for the isolated system to return back to state 1 is converting the mechanic work W into enthalpy to complete the Joule cycles. This implies that the state 2 has less enthalpy but possesses the mechanic work W . As no variations in the volume and geopotential energy take place after the system has returned back to the original state, the work is represented by kinetic energy. Therefore, (21.14) gives the kinetic energy generated as the isolated atmosphere changes from state 1 to state 2. If the parcels are sufficiently small, the kinetic energy generation may be calculated with the integration ∆K =
cp g
ps pt
(T1 − T2 )dAdp
A
in pressure coordinates. It is just the system energy equation derived in Chapter 7 using the momentum equation and thermodynamic energy equation, and obtained again empirically in Chapter 9 from the principle of energy conservation.
21.5
Carnot air engine
21.5.1
Kinetic energy generation
We have discussed in the previous chapters that kinetic energy generation is maximum in the processes with minimum thermodynamic entropy production. The minimum entropy production may be zero in the baroclinic atmosphere. The processes with zero thermodynamic entropy production in an inhomogeneous thermodynamic system may not really be reversible, since thermodynamic entropy of individual parcels or geopotential entropy of the system may change. These irreversible processes are referred to as the pseudo-reversible processes in Chapters 8 and 10. As an extremal case, a reversible process may also be studied with the air engine theory. The well known example of reversible engine is the Carnot engine studied in classical thermodynamics. The p-α cycle of Carnot engine is composed of two adiabatic processes and two isothermal heating or cooling processes, as shown in Fig.21.4.
21.5.
CARNOT AIR ENGINE
445
Figure 21.4: p-α diagram of Carnot air engine A physical cycle of Carnot convection in the dry atmosphere may not be completed by a single parcel. But for convenience, we use the one parcel description in the following. A parcel at position A on pressure surface pA moves on an isothermal surface to B on pressure surface pB , absorbing heat from the surface to remain its temperature unchanged. Then, it rises adiabatically along an isentropic surface to C on pressure surface pC , where its temperature equals that of the surroundings. From C to D on pressure surface pD , the parcel moves downward on another isothermal surface TC . In this path, the parcel ejects heat to the surroundings so that its temperature remains constant. Finally, the parcel descents adiabatically from D back to the starting position A. The mechanic work created on the isothermal surface AB is equivalent to the heat accepted, or +
αB
WAB = Q = αA
pdα = −
p B pA
αdp = RTA ln
pA pB
(21.15)
since the internal energy is conserved. The work done on the other isothermal surface CD is equivalent to the heat ejected, that is p (21.16) WCD = Q− = RTC ln C . pD The thermodynamic variables in these equations pertain to either the parcel or environments, as the parcel is in equilibrium with the environments over the whole cycle.
21. DRY AIR ENGINES
446 The total work created is given by W = WAB + WCD from (20.1). Here, we have
TC = TB
pC pB
κ
= TA
and TD = TC = TA
pD pA
pC pB
κ
(21.17)
κ
(21.18)
in the two adiabatic processes BC and DA respectively. Comparing these two equations finds pD p = A . pC pB The total work is then given by W = Q+ − Q− = R(TA − TC ) ln
pA . pB
(21.19)
If the friction and viscosity are ignored, the work created is converted into parcel kinetic energy. Positive kinetic energy is produced when the parcel accepts heat from the warm isothermal surface and ejects heat to the cold isothermal surface.
21.5.2
Efficiency of Carnot engine
The efficiency of Carnot engine using an ideal gas as the work medium is evaluated from (21.9), giving TA − TC , EC = TA or κ pC . (21.20) EC = 1 − pB This efficiency depends on the cycle in the atmosphere. It increases as the parcel reaches a higher level in pressure. The Carnot engines working between two heat reservoirs at constant temperatures may produce different amounts of work, depending on the pressures of the reservoirs as shown by (21.19). While, they have the same efficiency when the working gas is an ideal gas. Equation (21.8) tells us that the ratio of the heats accepted and ejected by these reversible engines, given by Q− =1−E Q+ is constant. For ideal gases, the ratio of heats equals the ratio of reservoir temperatures, that is TC Q− = . + Q TA An attempt was made to prove, according to the second law of thermodynamics, that all reversible engines operating between two reservoirs at constant temperatures have
21.5.
CARNOT AIR ENGINE
447
an equal efficiency which is independent of working substance (Sears and Salinger, 1975). In this circumstance, it was believed that the ratios on both sides of the previous equation was independent of substance. Thus, Kelvin suggested to define the temperature scale according to the ratio of heats ejected and accepted over a Carnot cycle. The temperature so defined is called the Kelvin temperature, which is equivalent to the temperature defined using ideal gases discussed in Chapter 2. The efficiency of Carnot engine may not be compared directly with that of another air engine which works on a different cycle. If the convection is constrained within the layer from pA to pC , efficiency of Joule air engine may be higher than that of Carnot air engine. This can be seen by comparing (21.20) with (21.10). However, the ability of kinetic energy generation over a cycle depends also on the heat accepted. The area surrounded by a Carnot cycle on the p-α diagram depends on the pressures of heat reservoirs, which is in thermodynamic equilibrium with the working substance in the Carnot engine. If a Carnot air engine accepts more heat than a Joule air engine, it may produce more kinetic energy on a cycle.
21.5.3
Dependence on working substance
The isothermal processes in a Carnot cycle are reversible only if the working substance is in thermodynamic equilibrium with the heat reservoirs of which the pressure and temperature change continuously at constant temperatures. The thermodynamic states of reservoir are defined by three thermodynamic variables but not just the temperature. Thus, not all Carnot engines are reversible at least on theory. The efficiencies of reversible and irreversible Carnot engines are the same. While, an irreversible Carnot engine may leave net effect in the environments after a cycle, but a reversible Carnot engine does not. When the reservoirs are in equilibrium with an ideal gas, they may not be in equilibrium with another substance if the same amount of heat exchange takes place at the constant temperature provided. For example, the pressure and density changes of a substance on isothermal surface AB may be different from those of an ideal gas. So the Carnot cycles of different substances between the given temperatures may be different from that in Fig.21.4. It may be impossible for different substances to work on the same heat reservoirs without leaving net effect on the environments. Also, when different substances work on the Carnot cycles between the same temperatures and pressure ranges, different amounts of mechanic work may be produced. Thus, it is hard to prove that all substances working on the Carnot cycles between two provided temperatures give an equal efficiency. Reversible Carnot engine is not the only reversible engine. All the engines of which the working substance is in equilibrium with the environments on the cycle are reversible. One example is the reversible Joule engine, of which the efficiency is also given by (21.10), or pκ . EJ = 1 − B pκA We may also argue that the efficiency of reversible Joule engine is independent of substance, when the nonequilibrium between the engine and environments is
21. DRY AIR ENGINES
448 ignored. Comparing with (21.8) finds pκB Q− = . Q+ pκA
If we introduce the pressure scale according to the efficiency, we may declare that the ratio of dQ/pκA is a constant for all substances. The previous equation may be rewritten as Q1 Q2 + κ =0, pκ1 p2 in which p1 and p2 are the pressures of heat reservoir 1 and 2 respectively. The heat ejected is negative in this equation. If any cycle in p-α diagram can be represented by a number of Joule cycle, we have Qj j=1
or
pκj
=0,
dQ =0. pκ
It may suggest that the parameter dQ/pκ is a state function. This is true for ideal gases. The thermodynamic energy equation (2.35) may be rewritten as dp dq dT cp κ − RT κ+1 = κ . p p p When the heat capacity is constant, the left-hand side is a perfect differential since this equation can be replaced by d(cp T p−κ ) =
dq . pκ
If the thermodynamic entropy is defined as ds = we have
dq , pκ
ds = d(cp T p−κ ) .
The entropy so defined is also a state function for the ideal gases with constant heat capacity. For the reversible Otto engine of which the cycle is composed of two adiabatic paths and two isochoric paths (Roy, 1995), the efficiency is given by EO = 1 −
−R/cv
α2
−R/cv
α1
,
α1 < α2 .
Heat is accepted at α1 and ejected at α2 on an Otto cycle. If the thermodynamic entropy is defined as ds = αR/cv dq ,
21.6. EQUILIBRIUM AIR ENGINE
449
it is also a state function for the ideal gases with constant heat capacity. To prove this, we apply cv αR/cv dT + RT αR/cv −1 dα = ds derived from (2.34). It follows that ds = d(cv T αR/cv ) when cv is constant. The previous examples show that the entropy as a state function may be defined in various ways. A linear relationship of these state functions may also be a state function. For example, we may define another potential temperature at a reference volume, that is R/cv α ϑ=T αθ using (2.53). The classical thermodynamic entropy may be evaluated from ds = cv ln ϑ for ideal gases. The entropy may also be given by ds = c1 cp ln θ + c2 cv ln ϑ ,
c1 + c2 = 1 .
It seems unlikely that these expressions of thermodynamic entropy discussed previously may be applied for all substances.
21.6
Equilibrium air engine
21.6.1
Equilibrium cycle
The heat exchanges of Carnot air engine with the surroundings are constrained on isothermal surfaces in the atmosphere. This may not be the real situation, because the temperature changes as a parcel crosses the pressure surfaces. Also, the slopes of isothermal and isentropic surfaces in the troposphere are generally small except in the frontal zones. So, the Carnot convection may not extend to high levels in the dry and statically stable atmosphere. To study the energy conversion in the largescale meridional circulations, we discuss in this section another kind of reversible air engine. Heat exchanges of the new engine are carried out in the static equilibrium paths discussed earlier. The p-α diagram of the new reversible air engine is displayed in Fig.21.5. A parcel rises slowly upward from position A at the cross line of isobaric surface pA and isentropic surface θA . The parcel pressure and temperature are justified to the environments by expanding and mixing with the surroundings. As the parcel absorbs heat in the static equilibrium path, its potential temperature increases to θB after it arrives at B on surface pB . The work done on this path is αB
WAB = αA
pdα = R(TB − TA ) −
p B pA
αdp .
21. DRY AIR ENGINES
450
Figure 21.5: p-α diagram of equilibrium reversible air engine in a statically stable atmosphere. The light curves represent the profiles along the parcel trajectories
On BC, it moves adiabatically on an isentropic surface until hits pressure surface pC , producing the work (21.21) WBC = −cv (TC − TB ) . Then, it descends down to D on θA surface through another static equilibrium path. The work done in this process gives WCD = R(TD − TC ) +
p D
αdp .
pC
Finally, the parcel returns back along the isentropic surface DA, creating the work WDA = −cv (TA − TD ) .
(21.22)
The total work produced over the cycle gives the kinetic energy generation ∆k = cp (TB − TA + TD − TC ) −
p B pA
αdp +
p D
αdp .
pC
As in the Carnot convection, a parcel in the equilibrium cycle remains thermodynamic and hydrostatic equilibrium with the surroundings over the whole cycle, so its thermodynamic variables are identical to those of environments. This cycle may also be completed by more than one parcels. The parcels working on the equilibrium cycle will be referred to as the equilibrium air engine.
21.6. EQUILIBRIUM AIR ENGINE
451
For simplicity, we assume that the static equilibrium paths are vertical. In this situation, we may use the hydrostatic equilibrium equation (5.23) to obtain −
p B
αdp = ∆φAB ,
pA
−
p D
αdp = ∆φCD .
pC
Applying these results gives WAB = R(TB − TA ) + ∆φAB ;
(21.23)
WCD = R(TD − TC ) − ∆φDC
(21.24)
∆k = cp (TB − TA + TD − TC ) + ∆φAB − ∆φDC .
(21.25)
and
The last equation can be obtained also in terms of the external work done for the parcel evaluated from (20.15). On the vertical static equilibrium paths AB and CD, the net external work done by vertical pressure gradient force and gravitational force is zero. While in the two adiabatic processes, the external work is given (referring to Section 20.3) by WBCext = cp (TB − TC ) − ∆φBC and WDAext = cp (TD − TA ) − ∆φDA respectively. The produced kinetic energy is equivalent to the sum of the external work, and is equivalent to the heat absorbed over a cycle. If the static equilibrium paths AB and CD are constrained on isothermal surfaces, we have Γ = 0 and (20.24) gives WAB = RTA ln
pA pB
WCD = RTC ln
pB . pA
and
If the paths AB and CD are not vertical, the mechanic work produced may be evaluated from (21.15) and (21.16) giving the same results. From (21.21) and (21.22), the work done on the two adiabatic paths BC and DA gives WBC = −WDA . Thus, the total work reads W = R(TA − TC ) ln
pA . pB
This is just the work done over a Carnot cycle. The Carnot engine is a particular example of the equilibrium engine.
21. DRY AIR ENGINES
452
21.6.2
Examples
We assume that temperature lapse rate of the atmosphere may change horizontally but remains constant in the vertical direction. In this situation, we have
TB = TA
pB pA
γ1
,
TC = TB
pC pB
κ
,
TD = TC
pD pC
γ2
,
(21.26)
and
∆φAB
g = TA 1 − Γ1
pB pA
γ1
,
where
∆φDC
RΓ1 , g
γ1 =
g = − TC 1 − Γ2
γ2 =
and
pD = pC
pA pB
pD pC
γ2
,
(21.27)
RΓ2 , g
κ−γ1 κ−γ2
derived from the last equation in (21.26) and
TD = TA
pD pA
κ
.
Applying these relationships for (21.25) yields +
Γd pB γ1 −1 1− Γ1 p γ1 A & Γd pB pC κ pD γ2 −1 1− . Γ2 pA pB pC
∆k = cp TA +
No kinetic energy is produced in the kinetic-death atmosphere with Γ1 = Γ2 = Γd , or in the barotropic atmosphere with Γ1 = Γ2 and pC = pB . If we assume that pA = 1000hPa ,
TA = 280K ,
Γ2 = 0.65K/100m ,
the dependence of kinetic energy generation on pB − pC and Γ1 is demonstrated in Fig.21.6. The energy generation is favoured in the atmosphere where the relatively low static stability occurs in the regions of rising motion. For example, when warm air rises slowly in the tropical regions, and cold air descends slowly at middle and high latitudes where the static stability is higher than in the tropics, a great amount of kinetic energy may be produced if without frictional dissipation. The generated kinetic energy increases with the slope of isentropic surface BC in pressure coordinates. The slope of isentropic surface increases with the baroclinity of atmosphere (referring to (19.9)). In the process of kinetic energy generation, heat energy is transported from the tropical source regions poleward to higher latitudes. It is likely that the equilibrium air engine may be applied for the study of the time and zonal mean meridional circulations in the atmosphere. When the parcel motions are in static equilibrium with the environments over the cells, kinetic energy may still
21.6. EQUILIBRIUM AIR ENGINE
453
Figure 21.6: Kinetic energy generated by equilibrium air engine. Solid and dashed curves are drawn for pB = 400 and 300 hPa respectively.
be produced by horizontal pressure gradient force. Usually, the pressure increases poleward on the Earth’s surface but decreases in the upper troposphere. Net kinetic energy is produced when the upward and poleward motions take place in the upper troposphere with the downward and equatorward motions near the surface. The direct meridional circulations at low and high latitudes may produce kinetic energy. While, the indirect Ferrel cells at middle latitudes may be maintained by eddy kinetic energy created in the baroclinic disturbances. It is noted that in the largescale mean meridional circulations, the dissipation of parcel kinetic energy caused by viscosity and boundary layer friction together with the diabatic heating caused by radiation and mixing cannot be ignored. Moreover, owing to the free upward convection in the moist tropical regions, the kinetic energy created in the moist Hadley cells may be more than that in the mean meridional circulation discussed. The energy generation in moist convection will be studied in the next chapter with the theory of wet air engines. If the horizontal pressure gradient force is only in the meridional direction in the pressure coordinates, the absolute angular momentum of parcels, evaluated from R = (a + z) cos ϕ[vx + (a + z)Ω cos ϕ]
(21.28)
21. DRY AIR ENGINES
454
Figure 21.7: Efficiency of equilibrium air engine. Solid and dashed curves are drawn for pB = 400 and 300 hPa respectively.
is conserved in an inviscid atmosphere (Gill, 1982), where a and Ω are the radius and angular velocity of the Earth respectively; z is geographic height of the parcel from Earth’s surface, and vx is zonal wind velocity. However, parcel kinetic energy relative to the Earth’s surface may be changed by meridional pressure gradient force and buoyancy, since the meridional motion produced may either change the height of the parcel or affect the zonal motion by the horizontal Coriolis force.
21.6.3
Entropy productions and efficiency
Since heat accepted and ejected by the parcels in the static equilibrium paths is identical to the heat ejected and accepted from the surroundings respectively, and parcel temperature remains the same as environmental temperature, thermodynamic entropy of the atmosphere does not change over the static equilibrium paths. Also, no thermodynamic entropy is created in the reversible adiabatic processes. Thus, thermodynamic entropy of the system including the parcel and environments is conserved over the cycle. The geopotential entropy variations on the two static equilibrium paths AB and
21.6. EQUILIBRIUM AIR ENGINE
455
CD are evaluated from ∆sgAB =
φA φB − θA θB
∆sgCD =
φC φD − θB θA
and
respectively. On the two adiabatic paths BC and DA, we have ∆sgBC =
φB − φC θB
∆sgDA =
φD − φA . θA
and
The geopotential entropy is also conserved over the cycle. So the equilibrium air engine is a reversible engine. It is possible for us to design more complicated reversible air engines for different circulation patterns in the atmosphere. The efficiency of equilibrium air engine is estimated from (21.9). The heat absorbed on path AB is evaluated from Q+ = cv (TB − TA ) + WAB = cp (TB − TA ) + ∆φAB . Here, we have used (21.23). For the atmosphere assumed previously, we have, from (21.26) and (21.27), +
Q = cp TA
Γd −1 Γ1
1−
pB pA
γ1
.
When the temperature deference between the heat source and sink is not large, the efficiency is generally low. Some examples are shown in Fig.21.7. The efficiency increases with the static instability and pressure difference pB − pC on the isentropic surface BC. It is noted that the efficiency itself may not give the information on the amount and rate of kinetic energy generation. Although the efficiency may be less than that of Joule air engine, the equilibrium air engine may create more kinetic energy over each cycle, as it accepts more heat compared with Joule air engine.
Chapter 22 Wet air engines 22.1
Introduction
The energy conversions in the moist atmosphere have been investigated in Chapters 11 and 12, without considering the circulation features in the moist systems. The dependence of energy conversion on the circulation pattern will be studied in this chapter, by designing the wet air engines to represent the typical circulations in the convective storms. These natural heat engines create kinetic energy from latent heat and solar radiative energy without producing pollution. For simplicity, we use temperature instead of virtual temperature to study the moist processes in this chapter. Usually, there are different thermodynamic cycles in the moist storms. The tropical cyclones were considered as a simple engine by Riehl (1954) or a moist Carnot engine by Emanuel (1988). We discuss in this chapter three types of wet air engines: semi-wet Joule engine, multiple semi-wet Joule engine and wet Joule engine. The thermodynamic cycles simulated by these air engines may exist simultaneously in a storm. For simplicity, we adopt them to represent three typical moist systems in the atmosphere. The semi-wet Joule engine is similar to the Joule engine, except that the upward convection is a pseudo-adiabatic process. A semi-wet Joule storm producing net kinetic energy is a deep system extending up to the tropopause, if there is no temperature inversion in the troposphere. The storm center is warmer than the surroundings in the whole troposphere. An example of the storm is the tropical cyclones. The deep convective system may only develop in the regions with sufficiently high temperature and humidity in the boundary layer. It may occur also in the subtropical and extratropical regions when the conditional stability is low. The convective storms in the extratropical regions usually occur near the polar front. The thermal wind or vertical shear of wind is relatively strong in the baroclinic regions. As a result, the dry and cold air over the middle troposphere may intrude into a storm from the back. The effect on kinetic energy generation will be studied with the theory of multiple semi-wet Joule engine. It is found that intrusion of cold air from the middle troposphere reduces the critical values of surface temperature and humidity for development of extratropical super storm. A large amount of kinetic energy may be produced by the semi-wet Joule storm even if it does not extend high enough to the tropopause. A semi-wet Joule storm becomes a wet Joule storm, if evaporation of raindrops takes place in the downdraft. The kinetic energy created over the cycle may be increased by evaporation. Thus, the wet Joule storm, such as the local air-mass storms or thunderstorms, may produce net kinetic energy. The strong surface wind and downrush or downburst may be experienced in the regions of air-mass thunderstorms. Except the three typical convective cycles discussed, the moist convection in the atmosphere exhibit more different and complicated patterns. The methods introduced in this chapter can be applied also 456
22.1. INTRODUCTION
457
to study other wet air engines according to the convective patterns in the moist storms. A storm will be referred to as the positive or negative storm, if it creates or destroys kinetic energy over a convective cycle. The study on the critical conditions of forming a positive storm may help us to explain the well known facts: The temperature inversion in the lower troposphere may facilitate the development of convective systems (Carlson and Ludlam, 1968; Carlson et al., 1983; Farrell and Carlson, 1989); the vertical shear of mean flow plays a critical role in development of sever storms (Fawbush and Miller 1954; Wallace and Hobbs, 1977), and the tropical cyclones may only develop in the regions with sea surface temperature greater than 26◦ C (Palm´en, 1948). We shall see in this chapter that formation of positive storm is favoured in these environments. Development of negative storms usually occur in a baroclinic environment, and is coupled with development of large-scale baroclinic disturbances. Rosenblum and Sanders (1974) reported that the latent heat released by convection in a polar low was not the primary energy source of the cyclogenesis. Some extratropical cyclones and polar lows which include sub-synoptic scale convection are the examples of the coupled system. Bosart (1981), Gyakum (1983a, b) and Roebber (1984) suggested that the explosive deepening in tropospheric bombs is a combination of the baroclinic and convective processes in the moist atmosphere. We discuss in this chapter the possible mechanisms for development of negative storm. If the mean tropospheric circulations at different latitudes are assumed as different air engines, we may evaluate the heights of tropopause at these latitudes according to the surface temperature and humidity and static stability observed. The height of tropical tropopause calculated from the semi-wet Joule engine is slightly lower than observed. Thus, we introduce simply in this chapter the static equilibrium Hadley engine and extended Hadley engine to study the summertime and wintertime Hadley circulations respectively. The equilibrium Hadley engine is similar to the semi-wet Joule engine, except that the dry subsidence is replaced by the equilibrium process. The extended Hadley circulation depends on the thermal structure of extratropical atmosphere, and the theory may explain the seasonal changes of tropopause height. It is noted that the different Hadley cycles may be found at the same time, and the real Hadley circulations may be more complicated. The studies of air engines in this and previous chapters provide only the typical examples for the simplified or idealized circulation patterns. The circulations in the atmosphere may be the polytropic processes discussed in Chapter 2, which are not isentropic, isobaric or isothermal. For example, the dry and moist convective processes in the real atmosphere can be affected by the mixing of convective flows with the surroundings. The principle of air engine may also be applied to study the energy conversions over the polytropic thermodynamic cycles in a weather system. The effect of environmental air entrainment on kinetic energy generation is not considered in this and the preceding chapters, and will be studied in the next chapter by introducing the polytropic processes to represent heat and humidity changes in the mixing processes.
22. WET AIR ENGINES
458
22.2
Primary wet engine
22.2.1
Kinetic energy generation
We consider firstly the kinetic energy generated by a saturated parcel moving from one pressure surface to another. As for the study of primary dry engine, we introduce auxiliary paths together with the moist path AB to form a complete cycle in a moist p-α diagram. A parcel working on this cycle is called the primary wet air engine, which is similar to the primary dry engine discussed in Chapter 20. The cycle in the physical space and thermodynamic diagram may also be represented by Fig.20.1, except that the path AB is a moist process. If AB in Fig.20.1(b) is assumed as a pseudo-adiabatic process, the work done by the parcel with unit mass and saturated mixing ratio w on the path equals the heat energy absorbed minus the internal energy increased, namely, WAB = dq − du = −Lc ∆w − cv ∆(TB − TA ) ,
(22.1)
where ∆w = wB −wA . The work produced on the other three paths is also evaluated from (20.3)-(20.4), that is WBB = R(TB − TB ) ,
(22.2)
WB A = R(TA − TB ) − ∆φA B ,
(22.3)
WA A = R(TA − TA ) .
(22.4)
and Thus, the net work done over the cycle reads W = cp (TA − TB ) − Lc ∆w − ∆φA B . When the surroundings are not disturbed, it gives the variation of parcel kinetic energy in the cycle. There is no kinetic energy generation in the static equilibrium path B A , and kinetic energy produced on path A A equals the geopotential variation ∆φAA . So, kinetic energy created on the moist path AB gives ∆kAB = W − ∆φAA . It follows that ∆kAB = cp (TA − TB ) − Lc ∆w − ∆φAB .
(22.5)
This is the Bernoulli’s integration for moist process. The first term on the righthand side is the work done by pressure gradient force; the second term gives the contribution of latent heat, and the last term represents conversion of geopotential energy. To evaluate TB , the moist heat flux equation cp dT − αdp = −Lc dw is rewritten as
R dp Lc dw dT − =− . T cp p cp T
(22.6)
22.2. PRIMARY WET ENGINE
459
Integrating this equation from A to B yields TB = TA r κ ec ,
(22.7)
Lc ∆w , cp Tc
(22.8)
with c=−
where Tc is the condensation temperature of water vapor, which may change in convection. But for convenience, the condensation temperature is taken as a constant. Now, the energy created is rewritten as ∆kAB = cp TA (1 − r κ ec ) − Lc ∆w − ∆φAB .
(22.9)
Kinetic energy variation of a saturated parcel may be calculated according to the changes in the thermodynamic state, geopotential height and humidity. If the path AB is vertical, the energy produced may also be derived from the vertical momentum equation (19.14), giving T´ − T dvz =g , dt T where vz is vertical velocity of the parcel. Applying dvz dz dkz dvz dvz = = vz = dt dz dt dz dz yields ∆kz = g
zB ´ T −T
T
zA
It follows that ∆kAB = R
p ´ A T pB
p
dz .
dp − ∆φAB .
This equation is the same as (20.11) derived from the air engine theory without using the momentum equation. In the moist-adiabatic process AB, we may apply (22.6) for it to give (22.5). If we write approximately ec ≈ 1 −
Lc ∆w , cp Tc
(22.10)
(22.9) may be replaced by ∆kAB ≈ (1 − r κ )(cp TA −
TA Lc ∆w) − ∆φAB . Tc
Comparing with the kinetic energy generated by a dry parcel moving adiabatically on the same path represented by (20.13), the saturated air of the same initial temperature produces more kinetic energy as ∆w < 0.
22. WET AIR ENGINES
460
22.2.2
Examples
We consider the barotropic moist atmosphere which is conditionally unstable and is saturated on the surface pA = 1000 hPa. When a parcel rises from the surface, it is warmer than the surroundings due to water vapor condensation. As the parcel reaches the level of neutral buoyancy ( LNB), its temperature is identical to the temperature of the surroundings. If the path AB is vertical, the environmental temperature TB is given by (19.34), that is TB = TA r γ ,
γ=
RΓ g
(22.11)
in the barotropic linear atmosphere. From (22.7), r γ − r κ ec = 0 .
(22.12)
The saturated mixing ratio wB at LNB pB is evaluated from (11.32), giving wB = 0.622
es , pB
(22.13)
where es measures the saturated water vapor evaluated from (11.33). From (22.12), we obtain pB for provided mixing ratio wA on pA . If the evaluated pB is less than 200 hPa, we set the LNB at 200 hPa, the assumed height of tropopause at middle latitudes. The solid curves in Fig.22.1 show the height of LNB which depends on the saturated mixing ratio wA . For example, a parcel with wA = 10 g/kg may reach 450 hPa in the atmosphere with Γ = 0.65 K/100m. The height of LNB increases with the static instability and the mixing ratio at the surface. The parcel kinetic energy created is then computed by inserting these solutions into (22.9). The geographic height between pressure surfaces pA and pB is given by zAB =
TA (1 − r γ ) . Γ
Applying it for (22.9) yields the maximum kinetic energy produced in the process without friction and mixing. The calculations are shown by the dashed curves in Fig.22.1. A saturated parcel with unit mass and water vapor mixing ratio of 10 g/kg at pA may produce about 1700 m2 /s2 kinetic energy as it reaches the LNB around 450 hPa. More kinetic energy can be produced if the atmosphere is more humid and statically unstable. It is noted that the height of LNB and kinetic energy generation displayed in Fig.22.1 are evaluated without considering the mixing of rising parcels with the surroundings caused by entrainment of environmental clear air. The LNB and kinetic energy generated may be lower than those shown in this figure, as the humidity and temperature of rising parcels are reduced by the mixing. The entrainment and the effect will be discussed in the next chapter. The kinetic energy generation discussed is produced in rising motions only. While, there are downward motions also in the convective systems to remain local balance of air mass. The downward motions may bring kinetic energy down to
22.3. SEMI- WET JOULE ENGINE
461
Figure 22.1: The height of LNB (solid) and the kinetic energy generated (dashed) in the free convection.
the ground, but they do not produce kinetic energy in a statically stable atmosphere except in some special situations. Net kinetic energy generation in a convective system should be evaluated over the whole convective cycle including the upward and downward branches. This is discussed in the following sections.
22.3
Semi- wet Joule engine
22.3.1
Kinetic energy generation
To consider the kinetic energy generated over a whole cycle of convection, we suppose that a saturated parcel of temperature TA at position A rises from pressure surface pA pseudo-adiabatically to B on pressure surface pB ; meanwhile, another parcel of temperature TB on pB descends adiabatically from B toward A on pA . Kinetic energy produced for the two parcels is represented by (22.9) and (20.29), respectively.
22. WET AIR ENGINES
462 Adding them together gives ∆kAB + ∆kB A
$
%
= cp TA (1 − r κ ec ) + TB (1 − r −κ ) − Lc ∆w − ∆φAB + ∆φA B .
(22.14)
For maintenance of local mass balance, we assume that a parcel moves from A to A on pressure surface pA and reaches the environmental temperature TA ; another parcel moves from B to B on pressure surface pB getting to the environmental temperature TB . Kinetic energy variations in the two solid processes are evaluated from (19.21), giving (22.15) ∆kA A = φA − φA = ∆φAB and ∆kBB = φB − φB = −∆φA B .
(22.16)
Thus, the energy generated over the cycle reads ∆k = cp (r −κ − 1)(TA r κ − TB ) + cp TA r κ (1 − ec ) − Lc ∆w ,
(22.17)
or, using potential temperature, κ κ −κ (1 − ec ) − Lc ∆w . ∆k = cp p−κ θ (pA − pB )(θA − θB ) + cp TA r
It gives also the specific work done by the parcel over the p-α cycle shown in Fig.22.2. Equation (22.17) shows that the kinetic energy produced depends on the height of free convection and environmental temperature therein. It increases with condensed water vapor. The light curves in the figure represent the vertical profiles of environmental atmosphere with Γ = 0.65 K/100m. This cycle is composed of two subcycles ABB and B A A, which represent a primary wet engine and primary dry engine, respectively. A parcel working on this cycle is called the semi-wet Joule air engine.
22.3.2
Condition of producing kinetic energy
The p-α diagrams in Fig.22.2 are composed of two triangles representing the positive and negative mechanic work respectively. The semi-wet Joule convection produces kinetic energy at high levels and destroy the energy at low levels. As a result, the energy increases with height in the system. To produce net kinetic energy on the cycle, the humidity must be sufficiently high and environmental temperature at high levels is sufficiently low, so that the conditional stability is low. Fawbush et al. (1951) pointed out that development of tornado in the United States may occur in the regions where the upper dry tongue crosses over a lower moisture ridge when other criteria are satisfied. If the downdraft comes from the level of neutral buoyancy, that is TB = TB (where TB is referred to (22.7)), (22.17) gives ∆k = cp TA (1 − ec ) − Lc ∆w .
22.3. SEMI- WET JOULE ENGINE
463
(a)
(b) Figure 22.2: The p-α diagrams of semi-wet Joule engine doing (a) positive work and (b) negative work. The light curves represent the environmental profiles along AB . The positive and negative signs indicate, respectively, the positive and negative mechanic work created.
22. WET AIR ENGINES
464
Figure 22.3: A super storm moving from left to right. (a) Profile of equivalent potential temperature in prestorm environment (solid) and behind the storm (dashed); (b) Profile of wind component in the direction of storm motion, which is relative to the storm; (c) Cloud outline and air motions relative to the storm. (After Wallace and Hobbs, 1977)
As the assumed atmosphere is saturated on pA , we write approximately Lc ∆w 1 c2 =1− + e ≈1+c+ 2 cp TA 2 c
Lc ∆w cp TA
2
(22.18)
by replacing TA for Tc . The kinetic energy produced gives now ∆k ≈ −
(Lc ∆w)2 . 2cp TA
(22.19)
It is negative when ∆w = 0. To produce net kinetic energy, the downdraft must start from the height lower than the LNB. This vertical flow pattern may be observed in the storms shown by Fig.22.3. The rising motion in the storm may extend up to the lower stratosphere, while the downdraft in the rear of the storm comes from the middle troposphere over the 0 ◦ C level.
22.3.3
Efficiency
If the last term in (22.18) is ignored, we have ec ≈ 1 −
Lc ∆w cp TA
(22.20)
Applying it for (22.17) yields $
%
∆k = (1 − r κ ) cp (TA − TB r −κ ) − Lc ∆w . For the cycle depicted in Fig.22.2(a), the parcel absorbs the latent heat Q+ l = −Lc ∆w
(22.21)
22.3. SEMI- WET JOULE ENGINE
465
released in path AB. So, the efficiency of a semi-wet Joule engine gives Es1 =
∆k Q+ l
=
cp (TA − TB r −κ ) (1 − r κ ) . 1− Lc ∆w
(22.22)
When TA − TB r −κ = TA − TA < 0, the efficiency increases with condensed water vapor. If without water vapor, kinetic energy will be destroyed in the Joule cycle. In the moist atmosphere, the semi-wet Joule engine may ‘burn’ the water vapor to produce kinetic energy. If TA > TA , the parcel will absorb sensitive heat −κ ) Q+ s = cp (TA − TA ) = cp (TA − TB r
on path A A. The efficiency is then given by Es2 =
∆k κ + ≈1−r . + Qs
(22.23)
Q+ l
It is larger than Es1 , as more kinetic energy is created in the cycle. The last expression is similar to the efficiency of Joule engine and depends mainly on the pressure ratio. As saturated parcels may rise higher than dry parcels do, the efficiency may be higher than that of Joule engines in the dry atmosphere with the same thermal structure. Even if the efficiency is identical to that of Joule engine, more kinetic energy may be produced by the semi-wet Joule engine, as more heat is absorbed over a semi-wet Joule cycle.
22.3.4
Thermodynamic entropy production
If the paths AB and A B in Fig.22.2 represent approximately a reversible pseudoadiabatic and adiabatic processes, respectively, the thermodynamic entropy is produced mainly in the processes of isenthalpic heating and cooling. The entropy produced for a parcel of unit mass on BB gives
∆´ s
BB
TB TB = cp ln = cp ln + κ ln r + c TB TA
Here, we have used (22.7). On path AA we have ∆´ sA A = cp ln
TA . TA
Applying TA = TB r −κ for it yields
∆´ sA A = cp ln
TA − κ ln r TB
.
.
(22.24)
22. WET AIR ENGINES
466 The total entropy production of the parcel gives, ∆´ s = cp c = −Lc
∆w . Tc
It is produced by water vapor condensation, and is generally greater than zero. Meanwhile, thermodynamic entropy generated in the surroundings is evaluated from cp cp cp ∆s = (TA − TA ) + (TB − TB ) = (C 2 − 2C + ec ) , TA TB C where C is referred to (21.7). The total entropy produced by the parcel and surroundings over a cycle gives
∆(´ s + s) = cp
1 2 (C − 2C + ec ) + c . C
Using (22.10) yields
∆(´ s + s) = cp
1 (C − 1)2 + C
1 +1 c . C
It is generally greater than zero. Due to water vapor condensation, the entropy production is greater than that of Joule air engine. So, the Joule cycle in the moist atmosphere is more thermodynamically irreversible than in the dry atmosphere.
22.4
Perfect storm and negative storm
22.4.1
Perfect storms
Kinetic energy generated by a convective parcel of unit mass in a semi-wet Joule storm is computed from (22.17). It is assumed again that the moist atmosphere is saturated at 1000 hPa, and the parcel temperature is identical to environmental temperature at the level of neutral buoyancy. The dependence of kinetic energy generation on the static stability and surface mixing ratio is shown in Fig.22.4. If the top level of free convection is over 200 hPa, we set pB = 200 hPa in the calculations. The energy produced increases with parcel humidity and static instability. Referring to Fig.22.1 finds that net kinetic energy is created when the free upward convection is higher than the tropopause assumed at 200 hPa, from which the downward convection starts. The energy generated in a whole convective system is less than that produced in the rising motion only, if compared with Fig.22.1. A convective system which does not lose kinetic energy on a whole cycle including upward and downward motions may be called the perfect convective system or perfect storm. In particular, a storm which produces net kinetic energy may be called the positive storm. For a provided tropopause height, the minimum surface temperature and humidity for perfect semi-wet Joule convection may be evaluated from ∆k = D, where D denotes kinetic energy dissipation. If the lapse rate is 0.65 K/100m in the inviscid atmosphere with D = 0, the minimum mixing ratio on the surface is slightly less than 25 g/kg as shown by Fig.22.4. This saturated
22.4. PERFECT STORM AND NEGATIVE STORM
467
Figure 22.4: Kinetic energy generated by semi-wet Joule engine humidity corresponds to the air temperature about 26.5◦ C at 1000 hPa, evaluated from (11.34). Referring to Fig.15.3 finds that the moist upward convection over the regions with this surface temperature may extend to near 100 hPa, and is higher than the depth of semi-wet Joule storm. The minimum surface temperature and humidity for development of positive storm at middle and high latitudes are lower than those in the tropical regions, since the tropopause in the tropics is also lower. The tropical cyclones are perhaps an example of the positive semi-wet Joule storm in the troposphere. The agreement in the calculated and observed minimum sea surface temperature for development of strong tropical cyclones suggests that the upward convection in the cyclone may extend over 200 hPa, while the major downward drafts come from the height below 200 hPa. Due to water vapor condensation in the deep convective system, the positive semi-wet Joule storms have a warm core, and the tropopause over the storm center is lifted upward by the heating. This structure is different from that of the dry and baroclinic cyclone in the extratropical atmosphere. The perfect semi-wet Joule storms may also occur over lands in the subtropical regions, if the surface temperature and humidity are high enough, or the conditional stability is low. The local air-mass thunderstorms developed in the moist and quasibarotropic regions provide an example of the continental positive storms, which have relatively short lifetime if compared with tropical hurricanes. An air-mass storm decays as soon as it converts all the available moist enthalpy into kinetic energy
22. WET AIR ENGINES
468
Figure 22.5: Kinetic energy destroyed by negative semi-wet Joule storm. Solid and dashed curves are for the convective motions up to 250 and 400 hPa, respectively.
in a local place or moves to a region with a higher static stability. If the warm upward motions are not very strong, net kinetic energy may still be produced over a convective cycle by evaporation of raindrops in the downdrafts. This is discussed later in this chapter.
22.4.2
Negative storms
When the temperature and humidity are not sufficiently high, a convective system may not produce net kinetic energy, and so is referred to as the negative storm. Development of negative storm is coupled with other systems in the environments, such as the large-scale baroclinic disturbances which convert available enthalpy into kinetic energy. The free rising motions in negative storms create kinetic energy, while the downward motions destroy more kinetic energy than created. The downdrafts may be forced by horizontal wind convergence in the upper troposphere, as shown by the continuity equation ps
ω = ωs + p
∇h · v dp ,
22.5. DEVELOPMENT OF NEGATIVE STORM
469
where ω = dp/dt is the vertical velocity in pressure coordinates, and ∇h ·v represents horizontal divergence of wind. Using ω ≈ −ρgvz yields vz ≈ vzs −
z 0
∇h · v dz .
The typical value of large-scale convergence is 10−5 /s. If the convergent layer is a few kilometers in thickness, the induced vertical motion is in the scale of 10−2 m/s. The forced vertical motions may occur in the statically stable atmosphere by destroying horizontal kinetic energy in the convergence. The minimum kinetic energy cascade from a large-scale baroclinic circulation into a negative semi-wet Joule storm may be evaluated from (22.9). The results are displayed in Fig.22.5, which illustrates the kinetic energy destroyed by negative storm without including the effect of friction. The solid and dashed curves are plotted for the systems up to 250 and 400 hPa, respectively. A large amount of kinetic energy may be destroyed if the surface temperature and humidity is low. Even more kinetic energy is destroyed when the negative storm extends to higher levels. The humidity field is highly inhomogeneous in the low troposphere. In the largescale ascending motions, the dampest air becomes saturated firstly and may form the moist convective systems embedded in the large-scale slantwise circulations. The trajectories of the warm plumes are steeper than those of unsaturated environmental flows. When the surface temperature and humidity are not sufficiently high in the extratropical troposphere, the convective systems in a meso- or small-scale form the negative storms, which are coupled with the large-scale baroclinic disturbances creating positive kinetic energy. The possible coupling mechanism is discussed in the next section.
22.5
Development of negative storm
22.5.1
Coupling mechanism
The positive storms may develop without being supported by the kinetic energy of large-scale environmental circulations. While, development of negative storms depends crucially on the environments. Two examples of the coupling between negative storm and large-scale baroclinic disturbance are shown schematically by Fig.22.6. The ABB A in Example A represents the Joule cycle of large-scale slantwise circulations producing net kinetic energy. Variation of environmental atmosphere along the slantwise convection path AB with Γ∗ = 1.3 K/100m is demonstrated by the light solid curve AB . The dampest air in the large-scale ascending motions becomes saturated over the condensation level denoted by C. The free rising motion of saturated parcels is represented by the solid curve CB ∗ . The path AC is plotted thickly indicating that the ascending air mass on AC is the sum of those over the moist convection CB ∗ and the large-scale slantwise convection CB. Since the saturated parcels are warmer than the environmental clear air in the upward convection, the path CB ∗ departs from CB towards the warmer side. The profile
22. WET AIR ENGINES
470
(a) Example A
(b) Example B Figure 22.6: Two examples of coupled p-α cycles of baroclinic disturbance and negative storm. The light solid and dashed curves represent the environmental thermodynamic profiles along the parcel trajectories in a baroclinic disturbance and negative storm, respectively.
22.5. DEVELOPMENT OF NEGATIVE STORM
471
of environment along the moist convection is displayed by the light dashed curve CB with Γ = 0.65 K/100m. It is noted that the two positions B and B or B and B ∗ in the example are not at one place, and may be hundreds kilometers away from each other. The dashed curve B A∗ represents the adiabatic downdraft in the negative storm. When the humidity is not sufficiently high, the semi-wet Joule convection over AB ∗ B A∗ produces kinetic energy at high levels and destroy more kinetic energy at low levels. When the negative storm is coupled with the large-scale baroclinic disturbance, the downward convection in the negative storm is forced by the large-scale convergence at high levels. In this process, the large-scale horizontal kinetic energy is transfered into the vertical kinetic energy and then the enthalpy of negative storm. If the mass of large-scale slantwise circulation over ABB A is M1 , and M2 is the mass of the negative storm over AB ∗ B A∗ , the created kinetic energy is estimated from ∆K = M1 ∆kABB A + M2 ∆kAB∗ B A∗ . The forced downdrafts in the negative storm may carry less air mass than the free upward convection does. The rest part is carried by the large-scale slantwise subsidence along B A . The mass transfer from a convective storm into the large-scale environmental circulations may be produced by detrainment from the downdrafts of the storm. As the leg B A is less steep than B A∗ , less kinetic energy is destroyed in the large-scale downward motions. The cycle AB ∗ B A is also a semi-wet Joule cycle. The area surrounded by this cycle is larger than that of Joule cycle ABB A , since water vapor condensation increases the energy generation. The thermodynamic cycles of the coupled storms may be very complex in the real atmosphere. We discuss only the simplest examples in this study. If the humidity is low and the condensation level C is high, the downward convection in the negative storm may not reach the bottom surface pA but stops at the middle, saying at the condensation height. The p-α cycle of this negative storm is CB ∗ B C illustrated by Example B in Fig.22.6(b). The downward convection below the condensation level C merges partially into the large-scale slantwise convection and is represented by C A . The large-scale circulation is again demonstrated by the cycle ABB A , in which the slantwise subsidence is colder than the air in the storm at the condensation level. The negative storm in Example B destroys less kinetic energy than that in Example A.
22.5.2
Cross sections of a tropospheric river
The small-scale or meso-scale moist convective storms are usually embedded in the large-scale baroclinic circulations. They may arrange in a row along a temperature front to form a squall line. Some weak storms in the line may be negative. Fig.22.7 gives the cross sections of vertical velocity and temperature across a moist explosive cyclone and a tropospheric river over northeastern Pacific, plotted using ECMWF data. The explosive cyclone has been discussed in Chapter 16. The mean sea level pressure at the same time is plotted in Fig.16.1(d), which shows that the cyclone center was at about 45◦ N and 178◦ E. The tropospheric river accompanied crossed 45◦ N around 176◦ E as illustrated by Fig.16.1(e).
22. WET AIR ENGINES
472
(a)
(b) Figure 22.7: Cross-sections of an explosive cyclone over the North Pacific at 00Z, 4 January, 1992. The heavy dashed curves are isothermal (◦ C); vertical velocity is represented by solid and short dashed (10−2 Ps/s).
22.5. DEVELOPMENT OF NEGATIVE STORM
473
Fig.22.7(a) shows that there are upward motions from the cyclone center to the river. The strongest upward convection appears over the river rather than the cyclone center, because the humidity in the river is higher than in the cyclone, and the horizontal wind convergence on the surface is strong along the river. The downward motions around are much weaker than the upward motions. If the tropopause is on 300 hPa at the middle latitudes, the surface temperature must be over 21◦ C for the semi-wet Joule storm to produce kinetic energy over a cycle. This surface temperature is much higher than that shown by the cross section. Thus, a part of the strongest upward flows over the river together with the downward motions to the north in Fig.22.7(b) forms a negative storm, of which the downward convection is forced by large scale convergence at high levels. The rest part of the downward mass transport may be accomplished by the large-scale slantwise circulation, indicated by the secondary center of upward motion over the cyclone and the downward motion center to the west in Fig.22.7(a). Certainly, the slantwise downward motions may not occur only to the west of the cyclone. Due to the effect of moist convection, the upward motions in the large-scale circulation are stronger than the downward motions. This is different from the vertical circulations in the dry baroclinic cyclone, plotted in Fig.9.2. In the dry cyclone shown in Fig.9.2, kinetic energy is produced by increasing the static stability. So, the upper troposphere gets warmer, but the lower troposphere gets colder after disturbance development. As a result, the height of tropopause was reduced. The local heating and cooling may be brought about adiabatically by large-scale slantwise convection with a slope less than that of isentropic surface. While in the moist convection, rising parcels are heated by moisture condensation, and so a warm core may form at the storm center in the lower troposphere. When the upper troposphere is relatively dry over a negative storm, the upward motions may lead to adiabatic cooling above the warm core. The vertical phase change of horizontal temperature deviation from the surroundings in the negative storm was just opposite to that in the dry cyclone.
22.5.3
Height of tropical tropopause
We have calculated the surface temperature and humidity for development of a semiwet Joule storm which extends to a given height. On the other hand, the surface temperature and humidity together with the static stability may be applied to determine the height of tropopause in the tropical regions, assuming that the tropopause sits at the mean level of neutral buoyancy, if we know the net kinetic energy output from the tropical storms. This means that the height of tropical tropopause depends basically on the energy budget of the moist convective cells. In the real atmosphere, the downward branch of Hadley cells may have heat exchange with the surroundings and so destroy less kinetic energy than that in the semi-wet Joule cycle. Thus, we may design an equilibrium Hadley engine to study the Hadley circulation, which is similar to the semi-wet Joule engine except that the downward branch is assumed as the static equilibrium process discussed in the preceding chapters. The tropical tropopause evaluated from the energy budget of equilibrium Hadley cycle may be higher than that obtained from the semi-wet Joule cycle. Since the equilibrium pro-
474
22. WET AIR ENGINES
cess may only take place in a slow subsidence, the equilibrium Hadley engine may be applied for the weak Hadley cells in summer. As the surface temperature and humidity are higher in summer than in winter, the Hadley circulation simulated by the semi-wet Joule engine or equilibrium Hadley engine is stronger in summer than in winter, and the height of tropical tropopause calculated from the energy budgets is higher in summer. However, the observational analyses of meridional mass transport and zonal mean temperature (Newell et al., 1972) shows clearly that the Hadley circulation is stronger in winter than in summer, and the tropical tropopause is higher in the winter hemisphere which tilts downward to summer hemisphere. The strong wintertime Hadley cells may be studied with the extended Hadley engine discussed in the following. Not only the negative storms are coupled with large-scale baroclinic circulations, the positive storms may also be coupled with the large-scale slantwise convection. The example may be found in the Hadley circulations. A part of air mass come from the upward convection in Hadley circulation descends in the subtropical regions to form the cycles similar to the semi-wet Joule cycle or equilibrium Hadley cycle discussed earlier; the rest stretches poleward in the upper troposphere to the middle latitudes, and then descends slantwisely equatorward about the steep isentropic surfaces around the polar front. In the rotating atmosphere, the extended Hadley cells are superimposed on zonal circulations, and so the poleward branch and equatorward slantwise branch may be at different longitudes. This high asymmetry of the time mean Hadley circulation is manifested by the study of Newell et al. (1972). As the slantwise downward convection destroy less kinetic energy than the adiabatic subsidence in the subtropical regions, the extended Hadley cells may produce more kinetic energy than the classical Hadley cells over a cycle, or the tropical tropopause calculated from the energy budget of extended Hadley cell is higher than that of classical Hadley cell. Therefore, the tropical tropopause observed may be as high as 100 hPa or above, and may depend on the baroclinity of extratropical atmosphere especially in the winter season. The tropical tropopause is higher in the winter hemisphere, because the zonal mean meridional temperature gradient is stronger or the isentropic surfaces are steeper than in summer hemisphere at middle latitudes, and so less kinetic energy is destroyed in the slantwise downward convection along a given path. The slantwise downward convection may be seen most clearly from the meridional cross-section of potential vorticity. While, the slantwise upward convection on the equatorside may be relatively weak. The energy budget in the baroclinic slantwise dry or quasidry convection may be applied also to study the height of extratropical tropopause, which is generally lower than the tropical tropopause. The dependence of Hadley circulation on the extratropical baroclinic circulation suggests that the anomalous circulation pattern in the tropical regions, such as the El Ni˜ no, La Ni˜ na, southern oscillations and equatorial quasi-biennial oscillations, may also be associated with the abnormal circulations at middle and high latitudes. The equilibrium Hadley cycle and extended Hadley cycle may also be applied for a single tropical storm which may extend up to 100 hPa height. If the warm plumes or thermals in tropical cyclones rise over 150 hPa, and the downdrafts come from 150 hPa, the minimum surface temperature calculated from the semi-wet Joule
22.6. LOW- AND HIGH-LEVEL CONVECTION
475
engine is over 29 ◦ C. This is higher than the threshold sea surface temperature for development of tropical cyclones (Palm´en, 1948). Also, Ito (1963) and Holliday and Thompson (1979) reported that the rapid deepening of tropical cyclone occurred usually over the warm waters with temperature not less than 28 ◦ C. Thus, the heat exchange in the downdrafts or the coupling with the baroclinic circulation may be significant for development of tropical storms, as they may reduce the threshold sea surface temperature and humidity for the development.
22.6
Low- and high-level convection
22.6.1
Low-level convection
It is well known that development of strong convective systems are favoured by a thin layer of temperature inversion at a low altitude (e. g., Fawbush and Miller, 1952, 1954; Carlson et al., 1980; Benjamin and Carlson, 1986; Farrell and Carlson, 1989). The low-level inversion is illustrated in Fig.22.3 which shows an idealized squall line. The inversion is helpful for accumulating convective energy below and producing discontinuous decrease of relative humidity across the inversion. However, the traditional theories cannot explain the kinetic energy concentration, which is necessary for the convection to break through the highly stable layer. The initial vertical velocity for a parcel to pass through an inversion may be estimated from (20.13) or (20.18). Some results are demonstrated in Fig.22.8. For example, to break through from the bottom of an inversion with Γ = −1 K/km and the thickness from 850 hPa to 750 hPa, a parcel of unit mass loses kinetic energy about 200 m2 /s2 . To break the inversion from the top, a slightly larger amount of parcel kinetic energy is destroyed. The condition of forming a perfect convective system discussed previously shows that the downward motions must come from the height below the level of neutral buoyancy. If a highly stable layer occurs below the LNB, the perfect convection may develop below the stable layer. The vertical velocity of parcels may become sufficiently high after undergoing a few cycles in the conditionally unstable boundary layer. The semi-wet Joule convection below 500 hPa, shown by the solid cycle in Fig.22.9, produces kinetic energy. While, the convection extending to 200 hPa in the same atmosphere shown by the dashed cycle in the figure may only destroy kinetic energy. Therefore, the temperature inversion reduces the threshold static instability or humidity in the boundary layer for development of perfect convection below. Some examples are illustrated in Fig.22.10, which is the same as Fig.22.4 except that the convection extends up to 850 hPa instead of 200 hPa. Comparing this figure with Fig.22.4 finds that the perfect convection may develop in the boundary layer with relatively low humidity. If the saturated mixing ratio is 17 g/kg over the surface, a deep convective system up to 200 hPa may not develop in the atmosphere with Γ = 0.65 K/100m, without input of kinetic energy from the surroundings. However, if the convection is constrained by an inversion at 850 hPa, about 50 m2 /s2 kinetic energy may be produced over a cycle when the effects of mixing and
22. WET AIR ENGINES
476
Figure 22.8: Kinetic energy generation of a parcel moving upward (solid) and downward (dashed) in the inversion atmosphere
friction are ignored. The inversion may then be broken by the successive shoots of upward motion. This effect of low-level inversion is especially important for storm development in the winter season, because the temperature and humidity on the surface are generally low.
22.6.2
High-level convection
The previous study shows that the positive convection may develop more easily in a relatively thin layer near the surface. This is also true in the middle and upper troposphere. Fig.22.2 shows that the semi-wet Joule convection produces positive and negative kinetic energy at high and low levels respectively. If the humidity or static instability is not sufficiently high, the convection may destroy kinetic energy over a cycle. The negative storm may become positive in the same environments, when the convection starts above a certain height. The critical bottom height pA
22.6. LOW- AND HIGH-LEVEL CONVECTION
477
Figure 22.9: The p-α diagrams of a semi-wet Joule cycles below 500 hPa (heavy solid) and up to 200 hPa (dashed). The light curve represents the environmental profile along the parcel trajectory.
can be solved, if without friction, by setting ∆k = 0 in (22.17), that is $
%
cp TA (1 − r κ ec ) + TB (1 − r −κ ) − Lc ∆w = 0 . In the barotropic atmosphere with constant lapse rate of temperature Γ, environmental temperature at the level of neutral buoyancy is given by (19.34), or TB = TA r γ . Thus, the previous equation is replaced by $
%
cp TA 1 − r κ ec + r γ (1 − r −κ ) − Lc ∆w = 0 . Here, the pressure pB and mixing ratio wB on the LNB are evaluated from (22.12) and (22.13), respectively. Some examples of the bottom height pA are depicted in Fig.22.11. When the static instability or humidity is low, the convection may only develop above the surface. This height is reduced by increasing humidity and static instability of the atmosphere. The high-level convection may be indicated by some special clouds in the middle troposphere, such as the castellanus and floccus, which may be a presage of coming bad weather. Also, the local convective storms can be very active on a high plateau in summer. The high-level convection may extend downward to the surface by cold air intrusion at the top or increase of humidity below. As cold air may be brought by the north-west wind behind a trough in the Northern Hemisphere, development of convective disturbances can be triggered by the troughs in the largescale baroclinic perturbations. Many air-mass thunderstorms occur during night when the upper atmosphere is cooled by radiation.
22. WET AIR ENGINES
478
Figure 22.10: Kinetic energy generated by the semi-wet Joule engine constrained below 850 hPa.
22.7
Multiple semi-wet Joule engine
It is discussed that the deep semi-wet Joule convection in the troposphere may produce kinetic energy if the humidity and temperature on the surface is high enough. These critical conditions may not be met at middle latitudes especially in winter. However, extremely strong surface winds may be observed at middle and high latitudes as a squall line passes away. The kinetic energy can be produced by another kind of wet air engine in the super storm. Fig.22.3 shows that there is a vertical wind shear in the environment of super storm. This feature has been recognized as an important condition for development of strong convective system. In Fig.22.3, the cold and dry environmental air intrudes into the storm over a certain height, where wind speed in the direction of storm movement is higher than the moving speed of the storm. To consider the effect of cold air intrusion, we design a multiple semi-wet Joule engine, of which the p-α diagram is plotted in Fig.22.12. The cycle ABB E in this figure represents the semi-wet Joule convection discussed previously. If the downward cold flows intrude from the mid-troposphere on pressure surface pm represented by CD, the path CE is replaced by the adiabatic process DA , and the cycle is demonstrated by ABB CDA in Fig.22.12. The work done by the parcel on AB is also given by (22.1). On the other paths, we have WBB = R(TB − TB ) , WB C = −cv (TC − TB ) , WCD = R(TD − TC ) , WDA = −cv (TA − TD ) ,
22.7. MULTIPLE SEMI-WET JOULE ENGINE
479
Figure 22.11: Critical bottom height of semi-wet Joule convection at high levels and WA A = R(TA − TA ) . The total work over a cycle reads W = cp (TA + TB + TD − TA − TB − TC ) − Lc ∆w . As TB is given by (22.7) and
TC = TB
pm pB
κ
,
we see +
−κ ) + TB 1 − W = cp TA (1 − r κ ec ) + TD (1 − rm
where rm =
pm pB
κ &
pm . pA
If the temperature lapse rate Γ is constant, we have γ , TD = TA rm
−κ γ−κ TA = TD rm = TA rm .
− Lc ∆w ,
22. WET AIR ENGINES
480
Figure 22.12: The p-α diagram of multiple semi-wet Joule engine. The light curve represents the environmental profile along the path of convection.
The produced total work reads, now, $
%
γ −κ (1 − rm ) + cp TB 1 − W = cp TA 1 − r κ ec + rm
rm r
κ
− Lc ∆w .
(22.25)
This equation can be rewritten as W = WABB E + WECDA , in which
$
%
WABB E = cp TA (1 − r κ ec ) + TB (1 − r −κ ) − Lc ∆w
is the work done by a semi-wet Joule engine, and $
%
κ −κ ) + TD (1 − rm ) , WECDA = cp TE (1 − rm
TE = TB r −κ
is the work done on the Joule cycle. The work created on the multiple cycle is equivalent to the sum of the work produced by the semi-wet Joule engine and Joule engine. A parcel working on this cycle is referred to as the multiple semi-wet Joule engine. For pm = 500 hPa, the generated kinetic energy is displayed in Fig.22.13. Comparing with Fig.22.4 finds that the cold air intrusion from the middle troposphere
22.8. WET JOULE ENGINE
481
Figure 22.13: Kinetic energy produced by multiple semi-wet Joule engine increases the energy generation, especially when the static stability is relatively high. The critical instability or humidity for development of positive convection is reduced greatly by the intrusion. Referring to Fig.22.1 finds that net kinetic energy may still be created in the multiple semi-wet Joule storm, which does not extend to the tropopause. The convective storms with intrusion of cold air from the back, developed in the baroclinic extratropical troposphere with strong vertical wind shear, are usually characterized by the comma cloud observed from a satellite. While, a tropical storm developed in the barotropic region may have a symmetric cloud wall surrounding the central eye.
22.8
Wet Joule engine
22.8.1
Kinetic energy generation
When the saturated downdrafts in a storm contain raindrops, the water may evaporate as the temperature increases adiabatically, and so the moist downdrafts are cooled by evaporation. If the dry adiabatic subsidence in a semi-wet Joule cycle is
22. WET AIR ENGINES
482
Figure 22.14: p-α diagram of wet Joule engine. The light curve represents the environmental profile along the path of convection.
substituted by a moist-adiabatic process of evaporation, the p-α diagram is shown in Fig.22.14. A parcel working on this cycle will be referred to as the wet Joule engine. The kinetic energy produced as a parcel of unit mass descends from pB to pA can be derived from (22.9). If the parcel temperature is TB at the LNB pB , we see
∆kB A = cp TB (1 − r −κ ec ) − Lc ∆w + ∆φA B , in which c = −
Lc ∆w , cp Tc
(22.26)
(22.27)
and ∆w = wA − wB . The energy produced in the two moist adiabatic processes is then given by the sum of (22.9) and (22.26), namely,
!
∆kAB + ∆kB A = cp TA (1 − r κ ec ) + TB (1 − r −κ ec ) − Lc (∆w + ∆w ) − ∆φAB + ∆φA B . In the local balance of air mass, we assume two solid processes to connect the paths A A on pressure surface pA and BB on pressure surface pB . Kinetic energy created
22.8. WET JOULE ENGINE
483
Figure 22.15: Kinetic energy generated by wet Joule engine. on the solid processes is also evaluated from (22.15) and (22.16). Thus, the energy generated over the cycle gives
!
∆k = cp TA (1 − r κ ec ) + TB (1 − r −κ ec ) − Lc (∆w + ∆w ) .
(22.28)
It may also be viewed as the total work done by two primary wet engines over cycles ABB and B A A respectively. The wB on the LNB involved in c is given by the saturated mixing ratio at temperature TB . The height of the top level is determined by TB = TB . When pB < 200 hPa, we set pB = 200 hPa. In the evaluation, we may use
TA = TB r −κ ec .
(22.29)
The saturated mixing ratio on the surface involved in c is calculated from (11.32), or es wA = 0.622 pA in which es is given by (11.33). For the initial moist atmosphere assumed for Fig.22.4, the kinetic energy produced by wet Joule engine is depicted in Fig.22.15. Evaporation in the downdrafts increases the energy generation, especially when the
22. WET AIR ENGINES
484
static stability is relatively high. This can be seen by comparing with Fig.22.4. The minimum surface temperature and humidity for development of positive wet Joule storm is lower than those for development of positive semi-wet Joule storm, as proved in the following. If the downdrafts come from the LNB in the storm, we insert (22.7) into (22.28), giving ∆k = cp TA (1 − ec+c ) − Lc (∆w + ∆w ) . Since ec+c
∆w ∆w + TA TB
L2c 2c2p
∆w ∆w + TA TB
2
≈ 1−
Lc cp
≈ 1−
Lc L2 (∆w + ∆w ) + 2 c 2 (∆w + ∆w )2 , cp TA 2cp TA
we gain ∆k ≈ −
+
L2c (∆w + ∆w )2 . 2cp TA
It is negative as the condensed water vapor (∆w < 0) is generally more than the water evaporated (∆w > 0) in a precipitation system. Thus, like the semi-wet Joule engine, net kinetic energy is produced as the downdrafts come from the height below the level of neutral buoyancy in the storm. For example, the kinetic energy may be created over the whole wet Joule convection if the warm plumes rise higher than the tropopause where the downward convection starts. If the downdrafts come from the LNB, the kinetic energy can be destroyed over the whole cycle. Comparing with (22.19) finds that the energy destruction in a negative wet Joule storm is less than that in a negative semi-wet Joule storm. In the real atmosphere, the downward convection may start at all levels but not just from the LNB. Thus, the wet Joule storms, such as the air-mass thunderstorms, may produce net kinetic energy even if they are not deep enough. Strong surface wind or gust may be recorded even if there are only the cumulus congestus and young cumulonimbus in the sky. The wet Joule storm has a short lifetime in the environments without a strong vertical shear of wind, since the moist downdrafts kill the storm after the mature stage.
22.8.2
Efficiency
If the downdraft in the wet Joule convection is saturated at the starting level pB , (22.27) may be rewritten approximately as c = −
Lc ∆w , cp TB
and we have
ec ≈ 1 −
Lc ∆w . cp TB
(22.30)
Substituting this relationship together with (22.20) into (22.28) yields $
%
∆k ≈ (1 − r κ ) cp (TA − TB r −κ ) − Lc (∆w − r −κ ∆w ) .
22.8. WET JOULE ENGINE
485
When TA < TA , the parcel absorbs latent heat on the moist-adiabatic path AB only, and the efficiency of wet Joule engine gives
Ew1
cp ∆w (TA − TB r −κ ) + r −κ = 1− (1 − r κ ) Lc ∆w ∆w ∆w = Es1 + (1 − r −κ ) ∆w
derived from (21.9), where Q+ l is referred to (22.21), and Es1 denotes the efficiency of sime- wet Joule engine given by (22.22). Since ∆w < 0 and ∆w > 0, the efficiency is greater than that of sime-wet Joule engine. If TA > TA , the parcel absorbs also the sensitive heat = cp (TA − TA ) Q+ s
= cp (TA − TB r −κ ec ) = cp (TA − TB r −κ ) + Lc r −κ ∆w on A A. The efficiency is then given by Ew2 =
∆k ≈ Es2 . + Q+ s
Q+ l
Here, Es2 is the efficiency of sime- wet Joule engine in (22.23). It is also the efficiency of Joule engine. Since the wet Joule engine absorbs more heat than the semi-wet Joule engine or Joule engine does due to evaporation in the downdraft, it creates more kinetic energy over a cycle though the efficiency is the same.
22.8.3
Thermodynamic entropy production
If paths AB and A B in Fig.22.14 represent approximately two reversible pseudoadiabatic processes, thermodynamic entropy is produced only in the processes of isenthalpic heating and cooling. Considering the parcel firstly, we have
∆´ sBB = cp ln
TB + κ ln r + c TA
,
where c is represented by (22.8). This is also the entropy produced by the semi-wet Joule engine in the same path shown in (22.24). Meanwhile, ∆´ sA A = cp ln
TA . TA
Using (22.29) yields
∆´ ssA A = cp ln
TA − κ ln r + c TB
.
The total thermodynamic entropy produced for the parcel is ∆´ s = cp (c + c ) .
22. WET AIR ENGINES
486
In general, c > |c | in the precipitation systems. The entropy production in the surroundings is evaluated from ∆s =
cp cp cp (TA − TA ) + (TB − TB ) = (C 2 ec − 2C + ec ) , TA TB C
where C is given by (21.7). The entropy produced for the parcel and environments gives 1 2 c c (C e − 2C + e ) + c + c . ∆(´ s + s) = cp C From (22.20) and (22.30), we have approximately
∆(´ s + s) = cp
1 c (C − 1)2 + (C + 1) + c C C
.
It is greater than zero, but is less than the entropy produced by semi-wet Joule engine. The evaporation reduces the entropy production by bringing a part of water vapor back to the low levels.
Chapter 23 Polytropic mixing processes 23.1
Introduction
In the previous studies on the air engines, mixing of convective air (such as the clouds, plumes and thermals in the upward convection and the downdrafts in a storm) with the environmental clear air has not been considered. The definitions of plume and thermal and their differences may be referred to the study of Emanuel (1994). Without the mixing, a rising plume over a heat source may be considered as a simple plume. The rising of a simple plume is independent of the environmental humidity, and the temperature profile within the plume is close to the moistadiabatic lapse rate. In this idealized moist convection, the liquid water condensed and vertical velocity produced can be evaluated with the moist adiabatic theory discussed in the preceding chapter. However, these features and evaluations may not be confirmed in the observed clouds where mixing with the clear air takes place. The mixing within a cloud may be brought about by entrainment of environmental air into the cloud. There may also be the detrainment from the cloud to the surroundings which affects the cloud temperature and humidity too. In general, the effect of detrainment may not be as large as that of entrainment in a cloud at the developing stage. The entrainment can be produced by turbulent motions including the eddy activity in the scale of cloud tower radius (Simpson et al., 1982), called the turbulent entrainment. The entrainment produced by the shear of large-scale horizontal wind or the motion of cloud relative to the environmental flows may be referred to as the shear entrainment. Moreover, there is the dynamic entrainment from the side of cloud produced by the differential vertical mass flux. The vertical growth of cloud and the hydrostatic instability of cloud top may cause entrainment from the top of cloud. Due to the mixing with the dry and cold environmental air, the outer skin of a warm plume becomes colder than the center. The cooling can be intensified by evaporation if the air becomes saturated after mixing. The warm core with a nearly uniform pressure breaks upward through the old top to form a new spheric top, called the head of warm bubble (Scorer and Ludlam, 1953). The outer skin is then shed into disturbed wake below, in which development of further bubbles is favored. Since the upward motions in a cloud are inhomogeneous in general, the top of cumulus cloud is composed of many bubbles at different levels, and looks like a cauliflower. The cloud top is a particularly important region of entrainment. Each bubble has its own lifetime. When the cloud grows rapidly, the entrainment may take place even if the clear air does not sink. The mixed air in the cloud top is cooled further by evaporation of cloud droplets, so that negative buoyancy may be produced. The static instability which causes the collapse of cloud top is called particularly as the cloud-top entrainment instability (Cotton and Anthes, 1989; Emanuel, 1994), 487
488
23. POLYTROPIC MIXING PROCESSES
studied initially by Squires (1958a, b) and then by other authors (e. g., Telford, 1975; Rogers et al., 1985; Gardiner and Rogers, 1987). The resulting downward entrainment may be referred to as the gravity entrainment, which may occur in stratiform clouds also. Owing to the rapid growth of cloud and cloud-top entrainment instability, the gravity entrainment may be more efficient than the lateral entrainment from the side in certain clouds. The earlier aircraft measurements in relatively weak cumulus showed the remarkable upward increase in the departure of liquid water content from the moist adiabatic profile, and small horizontal variations in the water content (Warner, 1955, Squires, 1958a). These results emphasize the importance of cloudtop entrainment, which is almost horizontally homogeneous (Sloss, 1967; Grandia and Marwitz, 1975). On the other side, the observational analyses of Paluch (1979) and other authors (e. g., Boatman and Auer, 1983; Jensen et al., 1985; Blyth and Latham, 1985) showed that the entrained air in the cumulus, which had a small vertical depth and was capped by dry air, generally came from two levels: the cloud base and the other one above or below the cloud top. While, Gardiner and Rogers (1987) found, with the method of Paluch (1979), that the entrainment occurred at all levels. Betts (1982b) illustrated that both lateral and vertical entrainment processes were important. The different entrainment mechanisms described previously may be relate to each other in the real storms. So, the mixing processes can be very complex and depend on regions in a cloud. It is found that in general the clear air entrainment may not be homogeneous everywhere and undergo varying degrees in a storm (Newton, 1966). In particular, Davies-Jones (1974) reported the existence of moist adiabatic ascent in severe thunderstorms undiluted by environmental air. The core of moist adiabatic ascent was also found by Heymsfield et al. (1978) in nonprecipitation cumulus congestus developed in a vertical shear environment over continent. The updraft acts as an obstacle to the horizontal wind protecting the upshear part of the cloud form the entrainment. The previous studies on the entrainment were focused mainly on the upward convection including the slantwise convection. Mixing of the convective flow and environmental air may also take place in the downdrafts of a storm by the detrainment of downward flow to the environments and the entrainment of environmental air to the downdrafts. Except the cloud-top entrainment in the upward convection, all kinds of the lateral entrainment discussed previously may take place also from the side of downdrafts. The effect on energy conversion may be different from that in the updrafts. For example, when the downdrafts are warmer than the surroundings in the statically stable atmosphere, the kinetic energy is destroyed by the upward buoyancy. The entrainment of environmental cold air may reduce the temperature of downdrafts and so the buoyancy. As a result, the kinetic energy destruction is reduced too. We consider in this chapter only the thermodynamic effect of entrainment on energy conversion in the convective systems, by introducing the concept of polytropic mixing process. The change of cloud temperature produced by entrainment depends on the entrainment rate, as well as the temperature and humidity differences between cloud and clear air. The entrainment rate is studied by cloud dynamics and not
23.2. LATERAL ENTRAINMENT RATE
489
discussed here in detail. Although the applied method may not be used directly for the precise calculations and predictions, the results may give the qualitative conclusions of the influences on the energy generation made by entrainment.
23.2
Lateral entrainment rate
The effects of mass entrainment on the energy generation in a storm depend on the rate of entrainment. Since the entrainment processes are related to different physical mechanisms, the rates can be evaluated by different methods, and there is no universally accepted algorithm for all kinds of cloud and entrainment. Schmidt (1947) tried to give a general expression of entrainment rate according to the temperature and humidity inside and outside a cloud, assuming that the mixing is taken at a constant pressure. The heat conservation was evaluated with two independent equations by extracting the condensation process from the mixing process, and the dynamic effect on the vertical momentum variation caused by entrainment was not considered. Malkus (1949) studied the entrainment rate in terms of horizontal momentum conservation in the mixing process. Houghton and Cramer (1951) discussed the dynamic entrainment resulting from vertical stretching of an accelerated convective column. The evaluation was carried out by adding a source of mass and heat into the continuity equations of mass and heat, respectively. The obtained mixing rate is proportional to the temperature (or virtual temperature in moist processes) difference between the cloud and clear air. Although the cloud-top entrainment is important, the lateral entrainment rate for the upward convection En =
µ 1 dM = , M dz r
(dz > 0)
(23.1)
is used most frequently. Here M is cloud mass (Levine, 1959) or mass flux by convective updrafts (Morton et al., 1956); µ is entrainment coefficient, and r is cloud radius. If µ/r is constant, the change of cloud mass with height is represented by µ M = M0 e r z . When µ > 0, the cloud mass increases with height. Some experiments have be done for evaluating the entrainment coefficient (Turner, 1962, 1963). The test of this relationship is still carrying on. McCarthy (1974) reported that the inverse dependence of entrainment rate on cloud radius may be verified by his cloud data. While, disagreement was shared by many others, for example Sloss (1967), Warner (1970) and Cotton (1975a, b). Their calculations of lateral entrainment showed that the predicted liquid water content was higher than observations if the real height of cloud top was simulated. In fact, the assumption that the entrainment is proportional to the velocity of updraft (Morton et al., 1956; Squires and Turner, 1962) used for deriving the relationship is not always true. A simple example is the moist adiabatic ascent without strong entrainment found in the cumulus cloud.
23. POLYTROPIC MIXING PROCESSES
490
23.3
Heat capacity of mixing
We assume for simplicity that the heat absorbed by the convective air of unit mass in mixing processes is proportional to its temperature variation, that is dqm = cm dT .
(23.2)
Here, cm is called the specific heat capacity of mixing. This name is rather exclusive, as the mixing is regarded as the only diabatic process here. This heat capacity may be negative or positive, and the sign does not mean whether the system is heated or cooled. The general polytropic processes have been discussed in the last section of Chapter 2. For the polytropic entrainment processes, we may replace the heat capacity of mixing for the polytropic heat capacity cπ in those equations derived. As the mixing is the only diabatic process, (2.48) is rewritten as pας = constant , where ς=
(23.3)
cp − cm cv − cm
may be called the polytropic mixing index. Applying the static equilibrium assumption for (2.47) yields (cp − cm )dT − gdz = 0 . It gives the polytropic lapse rate of plume temperature Γπ = −
g dT = dz cp − cm
(23.4)
in vertical convection. Under the effect of entrainment, the static stability of convective air is measured by comparing the environmental lapse rate with the polytropic lapse rate Γπ rather than the adiabatic lapse rate Γd . The atmosphere is statically stable for polytropic dry convection if Γ < Γπ , and is unstable if Γ > Γπ . When Γ = Γπ , the atmosphere is neutral for the assumed entrainment. Applying (23.2) for the previous equation gives gcm dqm =− . dz cp − cm
(23.5)
If a cloud of mass M and temperature T´0 is mixed at constant pressure with entrained mass δm of which the temperature is T , the cloud temperature T´r after the mixing may be evaluated from cp (M T´0 + δM T ) = cp (M + δM )T´r . It gives M T´0 + δM T . T´r = M + δM
23.3. HEAT CAPACITY OF MIXING
491
The cloud temperature change reads T´0 − T . δT´ = T´r − T´0 = −δM M + δM The specific heat absorbed by the cloud mass is T´0 − T δqm = cp δT´ = −cp δM . M + δM In the upward motions, we may insert (23.1) into the last equation and gain M + δM dqm µ =− , r cp M (T´0 − T ) dz
(dz > 0) .
It follows, from (23.5), that gcm (M + δM ) µ = , r cp M (cp − cm )∆T where ∆T = T´0 − T is the temperature difference between cloud and environment. In general, δM << M in a thin layer δz, and so we have gcm µ = . r cp (cp − cm )∆T Solving this equation yields cm =
cp µ∆T . Γd r + µ∆T
It is suggested that µ = 0.6 (Turner, 1963). Also, we may assume that ∆T = 1 K in general (Holton, 1992). Thus, Γd r >> µ∆T for r ≥ 1000 m, and the specific heat capacity of mixing is represented approximately by cm = cp µ
∆T . Γd r
(23.6)
It is proportional to the temperature difference and inversely proportional to cloud radius. This relationship may be used to estimate the heat capacity in the upward convection. If r = 1000 m, it gives cm ≈ 60 J/(K·kg) for the µ and ∆T assumed previously. In general, the cloud radius increases with height (Scorer, 1957). Meanwhile, the temperature difference may also increase with growth of cloud. If the change of temperature difference is less than the change of cloud radius, the specific heat of mixing decreases as the cloud grows up. When a parcel rises in the statically stable atmosphere, it is colder than the environment or ∆T < 0, and (23.6) tells that the heat capacity of mixing is negative. The parcel temperature is increased by the mixing according to (23.2). A positive heat capacity is expected in the statically unstable environments as ∆T > 0 in rising motions. These consequences are summarized by
cm
< 0 Statically stable . > 0 Statically unstable
(23.7)
23. POLYTROPIC MIXING PROCESSES
492
The mixing with environmental air may occur also in the downdrafts of convective storms, produced by the detrainment near the leading edge of downdrafts and the entrainment from the side, so that the downward flow cannot be much warmer than the environments as it reaches the surface. As in the upward convection, the entrainment and detrainment in the downdrafts may change not only the mass but also the mean temperature and humidity of the downward flow. The rates of these changes depend on the entrainment rate. However, we have no a general relationship of entrainment rate for the downdrafts. It may be considered qualitatively that the entrainment coefficient µ is negative for the downward convection. As ∆T > 0 for the adiabatic downdrafts in the statically stable environments, but ∆T < 0 in the statically unstable environments, (23.6) is also true for the entrainment in the downward flows. Applying the previous equation for (23.4) finds that the polytropic lapse rate is less than the adiabatic lapse rate in the statically stable environments, but is greater in the statically unstable environments, that is
Γπ
< Γd Statically stable . > Γd Statically unstable
(23.8)
This means that the entrainment reduces the static stability of stable atmosphere or the static instability of unstable atmosphere. In other words, the entrainment tends to make the atmosphere neutral for polytropic convection. The last inequality shows that super- adiabatic lapse rate (Γ > Γd ) may be observed in the unstable atmosphere with strong mixing, such as in the boundary layer over a rough surface.
23.4
Polytropic potential temperature
When the polytropic heat capacity cπ is replaced by cm , (2.47) is rewritten as κm
dp dT = , T p
κm =
R . cp − cm
If cm is constant, this equation may be integrated to gain
T = T0
p p0
κm
.
When a parcel of unit mass rises from pressure surface pA through pB , its temperature becomes κm pB , (23.9) TB = TA pA and the produced mechanic work reads αB
pdα = αA
cm − cv (pB αB − pA αA ) = (cm − cv )(TB − TA ) . R
Here, we have used (23.3).
23.4. POLYTROPIC POTENTIAL TEMPERATURE
493
For a constant cm , we may define the polytropic potential temperature as
θπ = T
pθ p
κm
(23.10)
derived from (2.54). This is the temperature of a parcel with initial temperature T and pressure p varies polytropically to the reference pressure pθ chosen at 1000 hPa usually. The adiabatic potential temperature θ is a special case of the polytropic potential temperature with cm = 0. In the idealized process of mixing, the polytropic potential temperature, instead of potential temperature, is conserved. To see how the potential temperature changes in the mixing process, we write
θB = TB
pθ pB
κ
.
Applying (23.9) for it yields
θB = TA
pB pA
κm
pθ pB
κ
.
If pA = pθ , we have θA = TA and
θB = θA
pθ pB
−cm R cp (cp −cm )
.
Referring to (23.7) finds that the potential temperature increases if a parcel rises in the statically stable environments, and decreases in the unstable environments. For the downward convection, the last equation gives
θA = θB
pθ pB
cm R cp (cp −cm )
.
The potential temperature decreases in the statically stable environments, and increases in the unstable environments as the parcel moves downward. The polytropic potential temperature is defined for parcels. If (23.10) is applied for the atmosphere, the static stability for polytropic convection may be represented by vertical gradient of polytropic potential temperature in the environments. When cm is independent of pressure, we have θπ ∂θπ = (Γπ − Γ) . ∂z T The left-hand side may be called the polytropic static stability or polytropic static instability denoted by ∂θπ . σπ = ∂z The dry atmosphere is statically stable or unstable for polytropic convection if σπ is greater or less than zero. Since the entrainment makes the convection neutral, the polytropic static stability or polytropic static instability is lower than the static
23. POLYTROPIC MIXING PROCESSES
494
stability or static instability in the statically stable or unstable atmosphere. When entrainment takes place in the buoyancy oscillations, (19.19) is rewritten as νπ2 =
g ∂θπ . θπ ∂z
(23.11)
This polytropic Brunt-Vaisalla frequency is generally smaller than the frequency without entrainment, because the polytropic buoyancy oscillations are damped by entrainment. The slope of polytropic potential temperature surface may be evaluated from (19.9), giving tan αθπ
=
Ty Γ − Γπ
= Ty
1−
Γπ (1 − r γ ) Γ
rγ Γ+ 1 Γπ −
1 Γ
for the linear atmosphere. Since Γπ ≤ Γd in the statically stable atmosphere, the slope of polytropic potential temperature surface is less than that of isentropic surface. The polytropic slantwise adiabatic lapse rate may be represented by Γ∗π
= Γπ
∗ Γ
Γ
(1 − r ) + r γ
γ
.
The polytropic slantwise convection is unstable if Γ∗ > Γ∗π , where Γ∗ is the slantwise lapse rate of environmental temperature. Comparing with (19.35) finds that Γ∗π > Γ∗d in the unstable slantwise convection, so the instability is reduced by entrainment. Moreover, (19.37) gives Γ∗π
tan αθπ − Γ = (Γπ − Γ) 1 − tan αl ∗
.
The polytropic slantwise convection may become unstable in the statically stable atmosphere, if the slope of trajectory is less than the slope of polytropic potential temperature surface. The convection is neutral when Γ∗π = Γ∗ or the trajectory slope equals the slope of polytropic potential temperature surface. When the slantwise oscillations are damped by entrainment, (19.40) gives the polytropic slantwise BruntVaisalla frequency tan αθπ , νlπ = νπ 1 − tan αl where νπ is the frequency of polytropic buoyancy oscillations represented by (23.11).
23.5
Effect of entrainment on dry engines
23.5.1
On Joule air engine
The Joule air engine in the dry atmosphere has been studied in Chapter 21 without considering the effect of entrainment. The Joule cycles are shown in Fig.21.1. If
23.5. EFFECT OF ENTRAINMENT ON DRY ENGINES
495
Figure 23.1: Kinetic energy produced by polytropic Joule air engine with cm = −50 J/(K·kg) for Γ∗ < Γ∗π and cm = 50 J/(K·kg) for Γ∗ > Γ∗π
entrainment of environmental air takes place in the two adiabatic processes over the cycle, the work done by the parcel in paths AB and B A is given by WAB = dqm − du = (cm − cv )(TB − TA ) and WB A = (cm − cv )(TA − TB ) , respectively. Here, we have assumed that cm is constant. Meanwhile, the work produced in the two isenthalpic heating or cooling processes are evaluated from WBB = R(TB − TB ) and WA A = R(TA − TA ) , respectively. The total work done over a cycle reads W = (cp − cm )(TA − TB + TB − TA ) .
23. POLYTROPIC MIXING PROCESSES
496 For a constant cm , we have TB = TA r κm and
TA = TB r −κm .
Applying them yields W = (cp − cm )(r −κm − 1)(TA r κm − TB ) . It is also the kinetic energy created over the polytropic Joule cycle, that is ∆k = (cp − cm )(r −κm − 1)(TA r κm − TB ) .
(23.12)
This expression is similar to (21.1) except that cp is replace by cp − cm . When cm is constant, kinetic energy generated by polytropic Joule air engine depends on environmental thermal structure as well as the initial and final parcel positions, and is independent of parcel trajectory. The energy equation is rewritten as κm κm m (pA − pB )(θπA − θπB ) . (23.13) ∆k = (cp − cm )p−κ θ It tells that net kinetic energy may be produced if environmental polytropic potential temperature at the higher pressure is greater than that at the lower pressure. The heat capacity of mixing changes in the real processes generally. Errors can be produced if we use a constant value for the heat capacity. The errors may be substantial when the temperature difference between the cloud and environment changes sign, or the slantwise lapse rate is nearly neutral. Thus, only the examples of kinetic energy generation in the highly stable or unstable paths are plotted in Fig.23.1. The sign of cm is chosen according to (23.7). Comparing this figure with Fig.21.2 finds that entrainment reduces the energy generated in the slantwise unstable Joule convection. While, in the stable circulations, less kinetic energy is destroyed when entrainment takes place.
23.5.2
On baroclinic waves
The Joule air engine is applied for the study of kinetic energy generation in the large-scale baroclinic waves in Chapter 21. The effect of entrainment on kinetic energy generation in the waves may also be discussed. In general, the relative slope of parcel trajectory to the isentropic surface in the baroclinic waves may change sign (referring to Fig.21.3), and so the sign of cm may vary over the cycle. In this situation, evaluation of the energy generation is complicated, and equations (23.12) and (23.13) may not be used. To give an example, we still use a constant cm to estimate the energy generation with (23.13). The polytropic potential temperatures at positions A and B are given by
θπA = (θ0 + Ty yA ) and
θπ B
pθ pA
pθ = (θ0 + Ty yB ) pB
b
, b
,
23.5. EFFECT OF ENTRAINMENT ON DRY ENGINES
497
where
R (Γπ − Γ) . g It is noted that the entrainment affects wave trajectory also, so the parameters Vp , Vy and ψ of the wave may be changed. But here, we still use the wave parameters assumed in Section 21.3. b=
Table 23.1: The same as Table 21.1, except for cm = 50 J/(K·kg). |Ty | (K/1000km) 6 8 10 12 14 Period (h)
Zonal wavelength Lx (km) 1000 2000 4000 8000 -25495 -2617 -555 -134 -19078 -1364 -255 -59 -12661 -111 47 16 -6244 1141 348 90 173 2394 649 164 19 42 132 -198
In the statically unstable slantwise convection, we choose a positive cm according to (23.7). For cm = 50 J/(K·kg), the evaluated examples are illustrated in Table 23.1. Comparing it with Table 21.1 finds that the kinetic energy created is reduced by entrainment, and shows a peak at a particular wavelength. This wavelength with maximum kinetic energy generation reduces with increasing the baroclinity, from a planetary scale down to about 4000-2000 km for Ty = −10 and -14 K/1000km for the current examples. The rate of kinetic energy generation is also maximum at the wavelength and baroclinity. Only the planetary waves may exist in the dry and weak baroclinic atmosphere. The most unstable wavelength derived from the simplest wave pattern does not agree with that obtained from a linear theory. A better result may be acquired for a more realistic wave trajectory and mixing heat capacity. McHall (1993) found theoretically that the most unstable wavenumber is 7 at the middle latitudes. The corresponding wavelength is about 4000 km. Table 23.1 shows that the period of the most unstable wave calculated from the propagation speed of Rossby wave is about 5.5 days. If calculated with the propagation speed of geostrophic wave (McHall, 1993), the period is more close to 5 days. This is the typical period of local weather changes at the middle latitudes.
23.5.3
On equilibrium air engines
We have, in Chapter 21, introduced a new kind of reversible engine called the equilibrium air engine to study the energy conversion in the large-scale mean meridional circulation. Here, we discuss the effect of entrainment on the equilibrium air engines, which takes place in the two adiabatic processes BC and DA in Fig.21.5.
23. POLYTROPIC MIXING PROCESSES
498
Figure 23.2: Kinetic energy produced by polytropic equilibrium engine with cm = −100 J/(K·kg). Solid and dashed lines are drawn for pB = 400 and 300 hPa respectively.
The parcel now moves on the polytropic isentropic surfaces in the two polytropic processes, and is also in equilibrium with the environments. Meanwhile, the two static equilibrium processes AB and CD do not change. For a constant cm , the work done in the polytropic processes is given by WBC = (cm − cv )(TC − TB ) and WDA = (cm − cv )(TA − TD ) . Adding them together with the work WAB and WCD given by (21.23) and (21.24) yields the total work produced over the polytropic equilibrium cycle ∆k = (cp − cm )(TB − TA + TD − TC ) + ∆φAB − ∆φDC . In the polytropic mixing processes, we have
TC = TB
pC pB
κm
23.6. MOIST POLYTROPIC MIXING PROCESSES
and TD = TA
499 κm
pD pA
.
For the atmosphere with constant lapse rates Γ1 and Γ2 along the static equilibrium paths AB and CD respectively, TB and TD are evaluated from (21.26). The last two equations together with the TB and TD in (21.26) give
pD = pC
pA pB
κm −γ1 κm −γ2
.
The geopotential variations are given by (21.27). In the statically stable convection, we use a negative cm . The evaluated effect of entrainment is illustrated in Fig.23.2. Comparing with Fig.21.6 finds that the entrainment reduces the energy generation.
23.6
Moist polytropic mixing processes
23.6.1
Energy equation of moist air
Due to the phase transitions of water in the atmosphere, effect of entrainment on energy conversion in a moist convective system is more complicated than in the dry circulations. A review on the studies of the entrainment in moist processes was given by Cotton and Anthes (1989). Evaluations of the entrainment effect have been given by many authors (e. g., Levine, 1959; Malkus, 1960; Squires and Turner, 1962). The earlier quantitative studies of Schmidt (1947), Stommel (1947) and Austin (1948) showed that the mixing reduces the temperature and humidity in cumulus clouds, and so reduces the buoyancy acting on rising plumes. Austin and Fleischer (1948) derived analytically the temperature lapse rate within a cumulus cloud affected by entrainment, which is greater than the moist- adiabatic lapse rate. In this chapter, the effect is studied simply with the moist polytropic processes discussed below. When a saturated warm plume (or thermal) rises freely in the conditionally unstable environment, the temperature and humidity in the outer part are reduced by mixing with environmental clear air. If the inner part is not affected, water vapor condensation may take place continuously therein as the plume rises. However, the released latent heat averaged in the plume is reduced by the entrainment. Let w denotes the saturated water vapor mixing ratio of an isolated plume without effected by entrainment or detrainment. After the mass exchange between the plume and dry environment resulting from entrainment, the mean mixing ratio is reduced and is represented by æw, where æ (0 ≤ æ ≤ 1) is the mixing coefficient. From (11.32), the original saturated vapor pressure es is reduced by the same rate. If the dry air and moisture are assumed as ideal gases, the thermodynamic energy law for the moist plume including unit mass of dry air and condensed liquid water is given by [cp + æwcpv + (wh − æw)cl − (1 + æw)cm ]dT des d(p − æes ) − æwRv T = −Lc ædw , − RT p − æes es
(23.14)
23. POLYTROPIC MIXING PROCESSES
500
in a moist polytropic process with constant æ. Here, cl is the specific heat capacity of liquid water; cpv is the specific isobaric heat capacity of water vapor; Rv is the specific gas constant of water vapor, and wh is the mixing ratio of total water including liquid water and water vapor. The thermodynamic entropy produced in the water vapor condensation is evaluated from sv − sl =
Lc , T
where sv and sl indicate the specific thermodynamic entropy of water vapor and liquid water respectively. Differentiating this equation gives d dsv − dsl = dT
Lc dT . T
It follows that T
dsl Rv T des d dsv −T = cpv − − cl = T dT dT es dT dT
Lc T
.
(23.15)
Inserting the last equation into (23.14) yields
Lc æw [cp + cl wh − (1 + æw)cm ]d ln T − Rd ln(p − æes ) = −d T
.
This is the energy equation of moist air for the reversible polytropic process, when the condensed liquid water is remained in the plume.
23.6.2
Polytropic equivalent potential temperature
Since æw 1, the energy equation of moist air become
(cp − cm + cl wh )d ln T − Rd ln(p − æes ) = −d
Lc æw T
.
When cp − cm + cl wh is taken as constant, we integrate this equation giving
T = T0
pd pd0
κm
eξ ,
where κm =
R , cp − cm + cl wh
ξ=−
æLc cp − cm + cl wh
w w0 − T T0
,
and pd is the partial pressure of dry air. If the plume is not saturated at the beginning, T0 is replaced by the temperature Tc at the lifting condensation level. When all water vapor is condensed or w = 0, the plume has temperature
T = T0 at pressure p = pd .
p pd0
κm
æLc w0
e (cp −cm +cl wh )T0
23.6. MOIST POLYTROPIC MIXING PROCESSES
501
If the plume including dry air and condensed water descends back to the isobaric surface pd0 through the dry polytropic process with the same heat capacity of mixing, its temperature reads æLc w
Tπe = T e (cp −cm +cl wh )T . For simplicity, the subscript ‘0’ for the initial temperature and mixing ratio is omitted. It may be called the polytropic equivalent temperature. It is noted that the heat capacities of mixing may be different in the moist upward and dry downward processes. If the plume descends continuously to the reference pressure surface pθ , it will have the temperature κm æLc w pθ e (cp −cm )T , θπ e = T pd called the polytropic equivalent potential temperature for saturated air. If condensed water leaves the rising plume at the very beginning, the energy equation for the irreversible polytropic process gives
Lc æw [cp − cm + æw(cl − cm )]d ln T − Rd ln(p − æes ) = −d T
.
Since æes p, the dry air pressure may be replaced approximately by the total pressure of moist air, and we gain æLc w
θπe = θπ e (cp −cm )T . It may be considered as the polytropic equivalent potential temperature in the irreversible polytropic process. We have the polytropic potential temperature θπ when w = 0, or the equivalent potential temperature in (11.2) when æ = 1 and cm = 0 for a saturated parcel. In the polytropic processes with constant heat capacity of mixing cm and constant condensation coefficient æ, the polytropic equivalent potential temperature is conserved, but the equivalent potential temperature may be not when cm = 0 and æ = 0.
23.6.3
Clausius-Clapeyron equation
According to the definition of latent heat, we have Lc = uv − ul + es (αv − αl ) , where uv and ul are the specific internal energy of water vapor and liquid water respectively, and αv and αl are the specific volumes of water vapor and liquid water respectively. Since αl αv , the last term may be omitted. Thus, we use the ideal-gas equation es αv = Rv T to gain Lc = uv − ul + Rv T
23. POLYTROPIC MIXING PROCESSES
502 approximately. It follows that
dLc = cvv − cl + Rv , dT in which cvv is the specific isochoric heat capacity of water valor. Applying the Mayer formula (2.19) for water vapor yields dLc = cpv − cl . dT Inserting it into (23.15) produces the Clausius-Clapeyron equation Lc dT des = . es Rv T 2
(23.16)
If Lc is constant, this equation can be integrated to give es = es0 e
Lc Rv
1 − T1 T0
.
(23.17)
where es0 is the saturated vapor pressure at T0 with respect to a flat surface. If the surface is water or ice, Lc is given by the latent heat of condensation or sublimation. In practice, we may also use an empiric equation, such as (11.33), to evaluate the saturated water vapor.
23.7
Effect of entrainment on wet engines
23.7.1
On primary wet air engine
If the polytropic moist process is applied for a warm plume rising from pA to pB , the diagrams in Fig.20.1 illustrate the cycles of the polytropic primary wet engine. The mechanic work done by the plume on path AB is given by WAB = dqm − du = (cm − cv )(TB − TA ) − æLc ∆w . Here, cm is taken as constant. Meanwhile, the work done in the other three paths of the polytropic primary wet cycle are given by (22.2)-(22.4). The total work created on the cycle gives W = (cp − cm )(TA − TB ) − æLc ∆w − ∆φA B . From (22.24), we see TB = TA r κm eξ . Inserting it into the previous equation and using (20.7) gain the kinetic energy generated on path AB: ∆kAB = (cp − cm )TA (1 − r κm eξ ) − æLc ∆w − ∆φAB .
(23.18)
In a free rising process, the plume temperature decreases or dT < 0. Also, the entrainment cools the warm plume and so dqm < 0. Thus, (23.2) tells cm > 0, which may be obtained also from (23.7) as ∆T > 0 in the rising process.
23.7. EFFECT OF ENTRAINMENT ON WET ENGINES
503
Figure 23.3: The height of LNB (Solid) in the polytropic moist convection with cm = 100 J/(K·kg) and æ = 0.9. The kinetic energy created is shown by the dashed curves.
The previous equation tells that the kinetic energy generated is reduced by entrainment. This consequence is illustrated by Fig.23.3 which is the same as Fig.22.1 except for the moist polytropic processes with cm = 100 J/(K·kg) and æ = 0.9. Since the entrainment reduces the temperature of rising plume, the level of neutral buoyancy is lowered by the polytropic process. Consequently, the work created by the buoyancy force on the plume is reduced, and the energy created is reduced too. These can be seen by comparing the figure with Fig.22.1.
23.7.2
On semi-wet Joule engine
The mixing between convective flow and environmental air may not always reduce kinetic energy generated in a convective system. This can be seen in the following discussions. If effects of entrainment in the adiabatic and pseudo-adiabatic processes of the semi-wet Joule cycle discussed previously are represented by the dry and moist polytropic processes respectively, the energy generated in the polytropic semi-wet Joule cycle is evaluated from ∆k = (cp − cw )TA (1 − r κw eξ ) + (cp − cd )TB (1 − r −κd ) − æLc ∆w
(23.19)
23. POLYTROPIC MIXING PROCESSES
504 with κw =
R , cp − cw
κd =
R , cp − cd
ξ=−
æLc cp − cw
wA wB − TB TA
,
where cd and cw denote the constant heat capacities of mixing for the dry and moist polytropic processes, respectively. In a free rising motion of saturated plume, (23.7) tells cw > 0. Meanwhile, we have cd < 0 in the statically stable downward convection. Thus from (23.19), the mixing in the dry and moist processes has opposite effects on the energy generation. This can be seen also from the p-α diagram in Fig.22.2. When the rising plume is cooled down by entrainment on the moist process AB, the area enclosed by the cycle decreases. While, the cooling produced by entrainment to the dry subsidence on path A B increases the work done on the cycle. Some examples are demonstrated in Fig.23.4. Fig.23.4(a) shows the effect of entrainment in the dry downdraft only. The energy generated in the whole cycle is increased as compared with that without the entrainment shown in Fig.22.4, since the entrainment reduces the buoyancy and so less kinetic energy is destroyed. This effect is similar to the intrusion of cold air from the middle troposphere to form the downdraft of a super storm, studied by the multiple semi-wet Joule engine in the preceding chapter. When entrainment in the moist process is added, Fig.23.4(b) shows that the kinetic energy created decreases at a higher humidity. As the magnitudes of heat capacities of mixing in the dry and moist polytropic processes are assumed equal in these examples plotted, the energy generated may be increased by the entrainment at a low humidity. This can be seen by comparing with Fig.22.4. If this is the true situation, the surface wind may also be intensified in the regions of relatively weak convection without deep cumulonimbus and precipitation. This consequence can be changed if the entrainment into the warm upward convection is stronger than that into the warm downdraft.
23.7.3
On multiple semi-wet Joule engine
The multiple semi-wet Joule engine is introduced to study the kinetic energy generation in a sever storm with intrusion of dry and cold air from the middle troposphere into the rear of the storm. When the entrainment effects are represented by the dry and moist polytropic mixing processes with constant heat capacities of mixing, the energy generation is given by, referring to (22.25), ∆k = (cp − cw )TA (1 − r κw eξ ) +
+ (cp − cd )
γ TA rm (1
−
κd rm )
+ TB 1 −
rm r
κd &
− æLc ∆w .
With the same assumptions used for Fig.22.13, the effect of entrainment is illustrated graphically in Fig.23.5. As discussed previously, the moist mixing process reduces the kinetic energy generated, but the dry mixing process has a reversed effect. In these examples, the moist effect is generally greater than the dry effect when the humidity is high, so
23.7. EFFECT OF ENTRAINMENT ON WET ENGINES
505
(a)
(b) Figure 23.4: Kinetic energy generated by polytropic semi-wet Joule engines, (a) cw = 0, æ = 1 and cd = −100 and - 200 J/(K·kg) in solid and dashed respectively, (b) Solid: cw = −cd = 100 J/(K·kg) and æ = 0.9; dashed: cw = −cd = 200 J/(K·kg) and æ = 0.8
23. POLYTROPIC MIXING PROCESSES
506
Figure 23.5: Kinetic energy generated by polytropic multiple semi-wet Joule engine, solid: cw = −cd = 100 J/(K·kg) and æ = 0.9, dashed: cw = −cd = 200 J/(K·kg) and æ = 0.8
the energy generated is reduced by entrainment (referring to Fig.22.13). When the humidity is low, the dry effect may be more important, so the minimum surface temperature and humidity for development of perfect convection are reduced by entrainment.
23.7.4
On wet Joule air engine
The moist polytropic mixing process may also be applied for the study of wet Joule air engines, in which evaporation takes place in the moist downdrafts. Since evaporation reduces temperature of downward flow in a storm, a wet Joule engine may produce more kinetic energy than a semi-wet Joule engine does. If the effects of entrainment on the updrafts and downdrafts are represented by moist polytropic processes with constant heat capacities of mixing, the kinetic energy generated is given by ∆k = (cp − cwu )TA (1 − r κwu eξu ) + (cp − cwd )TB (1 − r −κwd eξd ) − Lc æ(∆w + ∆w ) with κwu =
R , cp − cwu
κwd =
R cp − cwd
and æLc ξu = − cp − cwu
wA wB − TB TA
,
Lc æ ξd = − cp − cwd
wB wA − TA TB
,
23.7. EFFECT OF ENTRAINMENT ON WET ENGINES
507
Figure 23.6: Kinetic energy generated by polytropic wet Joule engine, solid: cwu = −cwd = 100 J/(K·kg) and æ = 0.9, dashed: cwu = −cwd = 200 J/(K·kg) and æ = 0.8
where cwu and cwd denote, respectively, the specific heat capacities of mixing in the upward and downward moist processes. In general, we have cwu > 0 for the free rising plumes and cwd < 0 for the moist downdrafts in the statically stable environment. With the same conditions used for Fig.22.15, the kinetic energy generation in the wet Joule storms affected by entrainment is displayed in Fig.23.6. As in the previous examples, the entrainment with assumed heat capacities of mixing reduces the minimum humidity on the surface for development of perfect storm. Since the effects of entrainment in the upward and downward flows are opposite, the total effect on kinetic energy generation depends on comparison of the entrainment rates, or the magnitudes of the dry and moist heat capacities of mixing.
Chapter 24 Limitations on frontogenesis 24.1
Introduction
The differential radiative and latent heating at low and high latitudes produces the meridional gradients of temperature, pressure and mass density, which force the meridional transport in the troposphere. The direct circulation at low latitudes, such as the Hadley cells, is an example of the transport circulations. Owing to the Coriolis force at middle and high latitudes, the direct meridional circulation can only extend to the lower middle latitudes. The large-scale circulations in the extratropical atmosphere are characterized by the wave-like perturbations, resulting from the mass advection of which the zonal speed is greater than meridional speed. These large-scale perturbations manifest some features of linear wave propagation, and are recognized as the Rossby wave in the previous studies. McHall (1993) argued that the classical Rossby wave assumed in the non-divergent atmosphere cannot be applied for the atmospheric perturbations, since its ageostrophic component is too large on the β-plane, and the non-divergent wave is lack of a propagation mechanism. He proposed that the observed large-scale perturbations in the extratropical atmosphere may be represented by the geostrophic waves derived using the small-oscillation approximation introduced in his study. The Rossby wave or geostrophic wave is a kind of kinematic waves, and different from the classical mechanic waves. The propagation of kinematic wave is caused by mass advection and is toward to one direction only. The process of advection is nonlinear. The linear waves are only the approximation of the nonlinear process. It can be proved (McHall, 1993) that the large-scale perturbations, represented by geostrophic waves, may transport heat, momentum and energy meridionally and vertically across the wave rays. Since the transport may not be efficient enough for establishing the balances in the rotational atmosphere, especially in the lower troposphere, a strong temperature gradient may occur in the extratropical troposphere to form the slopping boundary between a warm and cold air masses, called the front by meteorologists. The temperature front, especially the polar front, is one of the major synoptic systems at middle and high latitudes. The formation of temperature front is called the frontogenesis. If the diabatic heating in a hemisphere and cross-equatorial transport of mass and heat are ignored, the hemisphere may be considered as an isolated system for the study of frontogenesis. The frontogenesis produced by air motions in an isolated domain without heat exchange with the exterior will be referred to as the kinematic frontogenesis. The kinematic process over a limited area is controlled by the primitive equations discussed in Chapter 5, together with the second law of thermodynamics including the turbulent entropy law introduced in Chapter 6. It will be illustrated in the next chapter that the entropy law is a useful complement to the primitive equations which use the discrete data or parameterizations for the 508
24.2. THE THEORETICAL MODEL
509
diffusions included. The frontogenesis has been simulated numerically by many authors. The earliest studies of the limitations on frontogenesis were focussed mainly on the turbulent diffusions in numerical experiments (Williams, 1967, 1972). When the energy dissipation in William’s models was omitted, the simulated horizontal temperature gradient increased infinitely with time. The similar results came also from other theoretical and numerical models (Hoskins, 1971; Hoskins and Bretherton, 1972; Gall et al., 1987). The experiments of William and Kurth (1976) with an adiabatic and frictionless model showed that formation of the discontinuity would occur when the initial Rossby and Frouds numbers exceeded certain limits, which were independent of the initial thermal structure. According to the study in Chapter 8, the thermodynamic entropy decreases continuously as the temperature gradient increases infinitely in a frontal zone. Meanwhile, the wind speed becomes infinitely large over the front according to the thermal wind balance. These consequences are not acceptable, and the front with infinitely large temperature gradient and wind speed is never observed. These unrealistic results can be suppressed by applying the turbulent entropy law. In the process of kinematic frontogenesis, the thermodynamic entropy destroyed must be compensated by the entropy created as the baroclinity or static stability decreases outside the frontal zone. This implies that the upper limit of the front intensity measured by the temperature gradient depends on the thermal structure of initial field in the isolated atmosphere. The effects of initial conditions on frontogenesis have not been discussed adequately. Except the diffusions, Orlanski et al. (1985) proposed that frontogenesis may be restricted also by development of ageostrophic motions. Some other possible limitations are the shear instability and gravity wave generation (Ley and Peltier, 1978). Meanwhile, according to Eliassen and Kleinschmidt (1957), the condition of symmetric instability in the frontal regions is rarely satisfied in the dry atmosphere. These limitations are associated mostly with temperature gradient in a generated frontal zone, and are independent of initial conditions. The limitations on frontogenesis provided by initial conditions and some other affects are studied in this chapter, in terms of the energy and thermodynamic entropy balances in the process of kinematic frontogenesis.
24.2
The theoretical model
24.2.1
The basic relationships
It is assumed that formation of a frontal field θf from an initial field θ0 is constrained by the relationships including the thermodynamic entropy law cp ∆S = g
ps
ln A pt
θf dp dA ≥ 0 , θ0
(24.1)
and the conservation laws of system potential enthalpy ps A pt
(θf − θ0 ) dp dA = 0
(24.2)
24. LIMITATIONS ON FRONTOGENESIS
510 and total energy
Ψ0 − Ψf = Kf − K0 . Here, cp Ψ= κ gpθ
(24.3)
ps
R κ= cp
κ
p θ dp dA A pt
(24.4)
is the total enthalpy in the atmosphere extending from the surface ps up to pt with horizontal section area A, and K=
1 2g
ps
v 2 dp dA
(24.5)
A pt
is the total kinetic energy. The subscripts 0 and f in these equations indicate the initial and frontal fields respectively. Derivations of these equations are referred to Chapters 7 and 9. It has been noted that conservation of system potential enthalpy in an isolated atmosphere is not affected by isenthalpic diffusions, although the potential temperature of a parcel may change in the mixing process. It is discussed in Chapter 9 that variations of potential temperature in the quasiadiabatic processes are also constrained by the thermal constraint (9.9), or min(θf ) ≥ min(θ0 )
and
max(θf ) ≤ max(θ0 ) .
(24.6)
This constraint is used again for the following study of frontogenesis.
24.2.2
Idealized frontal field
The relationships given previously depend on initial and reference fields, and are independent of time. They cannot be used to predict the genesis of front without using other time-dependent relationships. We use them to discuss the dependence of a front generated kinematically on the initial field provided. To give an example, we assume an idealized front along a latitude. The potential temperature profile is depicted in Fig.24.1. The meridional scale or the width of front is minimum on the surface and increases with height. The mathematical expression of this cross section is given by
p κ − µ1 y − y0 − c(pκs − pκ )/(pκs − pκt ) θc + εf y − µ2 tanh θf = sκ p − µ1 y1
(24.7)
in pressure coordinates. In this figure, we have set y1 (km) = 400 + 100
ps − p , ps − pt
y0 = 0
and θc = 275 K ,
εf = −2 K/1000km , µ1 = −30
c = 2000 km .
µ2 = 10 K .
This front is embedded on the baroclinic field represented by θb =
pκs − µ1 (θc + εf y) . p κ − µ1
(24.8)
24.2. THE THEORETICAL MODEL
511
Figure 24.1: Cross-section of the front assumed. The solid and dashed curves represent isanemones and isotherms respectively.
The rest part in (24.7) defines the frontal zone. The εf in (24.8) represents the horizontal temperature gradient of the background field. The static stability of the field is evaluated from Γb = −
µ1 Γd , − µ1
pκ
(24.9)
which depends on µ1 only at a given height. The background field is statically stable when µ1 < pκ , but is unstable as µ1 > pκ . A temperature inversion occurs if 0 < µ1 < pκ . In the frontal zone then, y0 indicates the central position of the front on the surface; c represents the horizontal extend of the front with height, and µ2 denotes the front intensity which is explained in the following. Due to turbulent processes in the atmosphere, there are small- and tiny-scale discontinuities in the temperature field. The temperature gradient evaluated with discrete data depends on the scale of measurement. As we are not interested in the small-scale discontinuity, we use the temperature difference over a certain horizontal distance comparable with the width of front to measure the temperature gradient or the front intensity, that is ∆T , Ty = ∆y where ∆y is a large-scale distance along the gradient. In Fig.24.1, the maximum horizontal temperature gradient is on the surface, where the potential temperature is given by y − y0 . θfs = θc + εf y − µ2 tanh y1 The difference of surface potential temperature between y = y0 + L and y = y0 − L shows L . ∆θfs = 2Lεf − 2µ2 tanh y1
24. LIMITATIONS ON FRONTOGENESIS
512 Since tanh(L/y1 ) ≤ 1, we have
∆θfs ≤ 2Lεf − 2µ2 . The first term on the right-hand side is the baroclinity of background field. The other term represents the intensity of the front embedded on the background field. The maximum front intensity depends on µ2 only. This is also true at higher levels. The assumed frontal field may be regarded as an example of the inhomogeneous equilibrium state, if air motions are constrained by geostrophic balance and static equilibrium. For this equilibrium front, the wind is zonal and is evaluated by the thermal wind relationship v = vs +
p
R f0 pκθ
pκ−1
ps
∂θ dp ∂y
(24.10)
on the f0 -plane at latitude ϕ0 , where f0 = 2Ω sin ϕ0 and vs measures the surface wind in the zonal direction. The last equation gives vf
R(pκs − µ1 ) = vfs + f0 pκθ − µ2
εf p κ − µ1 ln κ κ p s − µ1
p ps
dp y1 p(1 − µ1 p−κ ) cosh2
κ κ κ y−y0 −c(pκ s −p )/(ps −pt ) y1
.
(24.11)
The zonal wind velocity is displayed by the solid cures in Fig.24.1.
24.3
Numerical Iteration
For provided εf , y0 and c, µ1 , µ2 and θc are three constant parameters which can be solved from (24.1)-(24.3) numerically by iteration. On the sphere, we have dA = a2 cos ϕdλdϕ , where a is Earth’s radius; λ and ϕ indicate the longitude and latitude respectively. If the coordinate origin is at latitude ϕ0 , we gain dA = a cos(ϕ0 + y/a) dλdy , where 0 ≤ λ ≤ 2π and −Y ≤ y ≤ Y . Now, we insert (24.7) into (24.2) and integrate it, giving 2Zθc cos ϕ0 sin(Y /a) + 2Zεf [Y sin ϕ0 cos(Y /a) − a sin ϕ0 sin(Y /a)] − N =
2π Y ps
1 2aπ(pκs − µ1 )
0
−Y
pt
θ0 cos(ϕ0 + y/a) dp dy dλ
(24.12)
in which µ2 N= a
Y ps cos(ϕ0 + y/a) −Y
pt
p κ − µ1
tanh
y − y0 − c(pκs − pκ )/(pκs − pκt ) dp dy , y1
24.3. NUMERICAL ITERATION
513
and
ps
Z= pt
pκ
dp . − µ1
Evaluation of Z is further referred to (9.16). Moreover, (24.1) is rewritten as
pκt − µ1 + (ps − pt + µ1 Z)κ − Γd ∆S˜ cos ϕ0 sin(Y /a) pκs − µ1 y − y0 − c(pκs − pκ )/(pκs − pκt ) 1 Y ps ln θc + εf y − µ2 tanh + a −Y pt y1 × cos(ϕ0 + y/a) dp dy 2π Y ps 1 ln θ0 cos(ϕ0 + y/a) dp dy dλ , (24.13) = 2aπ 0 −Y pt 2 pt ln
where the tilde indicates the value in a column of unit horizontal section area. Finally, (24.3) gives
˜0 − K ˜f ˜0 + K 1 Ψ θc = + gpκθ κ p s − p t + µ1 Z cp (ps − µ1 )
−
Y ps cos(ϕ + y/a) y − y0 − c(pκs − pκ )/(pκs − pκt ) µ2 0 tanh dp dy −κ aC0 −Y pt 1 − µ1 p y1 2εf [Y sin ϕ0 cos(Y /a) − a sin ϕ0 sin(Y /a)] , (24.14) C0
where C0 = 2 cos ϕ0 sin(Y /a) ; Ψ and K are evaluated from (24.4) and (24.5), respectively. To calculate Kf we have to know vfs . Usually, the surface fronts are located in the pressure troughs, and there is horizontal wind shear across a front. For simplicity, we assume a constant vfs for the calculations. In the zonally symmetric process of frontogenesis, the absolute angular momentum of air motion is conserved, or from (21.28) 2π Y p 0
−Y
ps
(vf − v0 ) cos2 (ϕ0 + y/a) dp dy dλ = 0 .
Applying the assumed front fields for it gives
vfs − R
pκs − µ1 p κ − µ1 = C1 cp εf pt ln tκ + κ(ps − pt + µ1 Z) κ 2ΩC2 pθ (ps − pt ) p s − µ1
µ2 p s Y
a
pt
p
−Y
ps
dp y1 p(1 −
×
cos2 (ϕ0 + y/a) dydp sin(ϕ0 + y/a)
+
1 2aπC2 (ps − pt )
κ κ κ y−y0 −c(pκ s −p )/(ps −pt ) µ1 p−κ ) cosh2 y1
2π Y ps 0
−Y
pt
v0 cos2 (ϕ0 + y/a) dp dy dλ ,
(24.15)
24. LIMITATIONS ON FRONTOGENESIS
514 where C1 = ln
tan[ϕ0 /2 + Y /(2a)] − 2 sin ϕ0 sin(Y /a) , tan[ϕ0 /2 − Y /(2a)]
(24.16)
1 Y cos 2ϕ0 sin(2Y /a) + , 2 a and Ω is the angular velocity of the Earth.Firstly, we assume a magnitude for θc . The µ1 and µ2 are solved from (24.12) and (24.13). Inserting the solutions into above equation yields vfs . Then from (24.14) we gain θc again. The final solutions are obtained by repeating the process until the accuracy required is obtained. C2 =
24.4
Limitations by initial field
24.4.1
Initial fields
We assume that the frontogenesis takes place in the linear atmosphere, of which the temperature field is represented by
θ0 = (T0 + ε0 y)
ps p
b
,
b=
R (Γd − Γ0 ) . g
(24.17)
Here, we have assumed that ps = pθ = 1000 hPa. This initial field has constant lapse rate of temperature Γ0 and constant meridional temperature gradient ε0 on isobaric surface. The geostrophic wind of the initial field is evaluated from (24.10) on the f0 -plane, that is
v0 = v0s −
gε0 1− f 0 Γ0
p ps
RΓ0 g
.
We shall use T0 = 280 K and v0s = 0 in the following discussions. The integrations involved in (24.1)-(24.3) are given in the following: ps A pt
4a2 π ps θ0 dpdA = C3 ps − pt 1−b pt
b
with C3 = T0 cos ϕ0 sin(Y /a) + ε0 [Y sin ϕ0 cos(Y /a) − a sin ϕ0 sin(Y /a)] ; ps A pt
pt ln θ0 dpdA = 4a πb pt ln + ps − pt cos ϕ0 sin(Y /a) ps 2
+ 2aπ(ps − pt )
˜ 0 = 2C3 cp ps 1− Ψ gC0 (γ0 + 1)
pt ps
Y
−Y
ln(T0 + ε0 y) cos(ϕ0 + y/a) dy ;
γ0 +1
,
γ0 =
RΓ0 g
24.4. LIMITATIONS BY INITIAL FIELD
515
and ˜0 = K
gε20 ps 8Ω2 Γ20 C0
pt 2 1− − 1− ps γ0 + 1
2γ0 +1
+
1 1− 2γ0 + 1
−
ε0 v0s ps 4ΩΓ0 C0
+
v02s (ps − pt ) cos ϕ0 sin(Y /a) . 2gC0
pt ps
pt ps
γ0 +1
1 1 − sin(ϕ0 − Y /a) sin(ϕ0 + Y /a)
pt 1 1− − 1− ps γ0 + 1
pt ps
γ0 +1
ln
sin(ϕ0 + Y /a) sin(ϕ0 − Y /a)
Moreover, the last integration in (24.15) is given by p A ps
ε0 ps = −a πgC1 ΩΓ0
2
v0 cos2 (ϕ0 + y/a)dp dA
pt 1 1− − 1− ps γ0 + 1
pt ps
γ0 +1
,
where C1 is referred to (24.16).
24.4.2
Dependence on initial temperature field
In the following examples, the extratropical troposphere in a hemisphere within 1000 hPa and 200 hPa will be considered as an isolated atmosphere. The origin of the coordinates is set at y0 = 0 or ϕ0 = 52.5◦ . The meridional scale of the atmosphere is given by 2Y = aπ(90 − ϕ0 )/180. Since thermodynamic entropy production in the isolated atmosphere cannot be negative, destruction of the entropy over a frontal zone must be filled in by the entropy generated in other areas resulting from reducing the baroclinity. From this point of view, kinematic frontogenesis may be viewed as a concentration process of baroclinic entropy. This process cannot occur in the barotropic atmosphere, and is limited when the background field become barotropic. The dependence of front intensity, represented by µ2 , on initial temperature field and the thermodynamic entropy produced is depicted in Fig.24.2. It is assumed that εf = −2K/1000km
and
c = 2000km
for this figure. The obtained results are examined by the thermal constraint (24.6). The solid curves show clearly that the front intensity represented by µ2 increases with the initial baroclinity ε0 , and is reduced by thermodynamic entropy production in the process. Fig.24.2 shows also that the maximum temperature gradient may be attained by the processes without net thermodynamic entropy production. It is discussed in Chapter 9 that a process with conservation of thermodynamic entropy in an inhomogeneous thermodynamic system may not really be thermodynamically reversible, and so is called the pseudo- reversible process, since the entropy at a local place or
24. LIMITATIONS ON FRONTOGENESIS
516
Figure 24.2: Dependence of front intensity on thermodynamic entropy production and initial baroclinity ε0 . The solid and dashed curves are drawn for Γ0 = 0.65 and 0.75 K/100m respectively.
for a given parcel may change. As thermodynamic entropy production is a measure of molecular and turbulent diffusions, the dependence of front intensity on the entropy production presents the effect of these diffusions. An important result obtained is that the maximum temperature gradient attained by a pseudo- reversible process without any dissipation and diffusion does not tend to infinite. Formation of an infinitely large temperature contrast requires infinite destruction of thermodynamic entropy, and so cannot happen in an isolated system or in a limited processes. This consequence could not be drawn from others studies (e.g., Williams, 1974; Hoskins and Bretherton 1972; Gall et al., 1987). In the previous numerical models, the entropy budget was not considered and the temperature gradient in a simulated frontal zone became infinitely large within a limited time period, when effects of diffusions were ignored. In the present examples without considering the dynamic processes, the static stability of initial field has little effect on the front intensity. This can be found by comparing the dashed curves with the solid in Fig.24.2. If the vertical mean static stability of the background field is given by 1 [Γb ] = ps − pt
ps pt
Γb dp ,
24.4. LIMITATIONS BY INITIAL FIELD
517
Figure 24.3: Mean static stabilities of the background field. Solid and dashed curves are drawn for Γ0 = 0.65 and 0.75 K/100m respectively. The initial temperature gradients are denoted on the curves with interval of 0.5 K/1000km.
we gain, from (24.9), [Γb ] = −
µ1 Γd Z . ps − pt
The evaluated examples are illustrated in Fig.24.3. This figure shows that the static stability is reduced slightly after frontogenesis. As discussed in Chapter 9, decrease of static stability may increase thermodynamic entropy in the atmosphere. The thermodynamic entropy produced may in turn compensate the entropy destroyed in the frontal zone. The final static stability is also low when the initial stability is low, and decreases with increasing the initial baroclinity.
24.4.3
Kinetic energy variation
As the static stability does not increase greatly in frontogenesis, no great amount of kinetic energy is produced in the process. The energy conversion in the kinematic frontogenesis depends highly on space, and is evaluated from ∆k = kf − k0 , where gK k = 4XY (ps − pt )
24. LIMITATIONS ON FRONTOGENESIS
518
Figure 24.4: The same as Fig.24.2 but for kinetic energy generation. denotes the assemble mean kinetic energy, and K is the total kinetic energy of the atmosphere. When energy is conserved in an isolated system, total kinetic energy of the frontal field is evaluated from (24.3). Change of the mean geostrophic wind speed is then given by ∆v =
2 ∆k ,
assuming that the initial wind speed is zero. The evaluations for the previous examples are demonstrated in Fig.24.4. The energy generated increases with initial baroclinity and decreases with increasing thermodynamic entropy production. Although the frontogenesis does not create a large amount kinetic energy even in a pseudo- reversible process, the distribution of kinetic energy may be changed. The strong wind required by thermal wind balance in a frontal zone results from momentum concentration and large-scale slantwise convection across the front as discussed in Chapter 20. In a highly irreversible process, the kinetic energy is destroyed especially when the initial baroclinity is low. So, the evaluated front intensity corresponding to a relatively large production of thermodynamic entropy may not be reached actually, if the initial field has no enough kinetic energy. This implies that the frontogenesis may be limited not only by the thermodynamic entropy law, but also by the energy conversion law.
24.5. BAROCLINIC ENTROPY
24.5
519
Baroclinic entropy
It is discussed in Chapter 8 that the ability of kinetic energy generation in the atmosphere is represented by the static entropy and baroclinic entropy, measured by the vertical and horizontal gradients of potential temperature, respectively. The static stability does not change largely in the frontogenesis as discussed in the previous section. While, the negative baroclinic entropy increases greatly in a frontal zone. The mean baroclinic entropy in the frontal zone assumed is evaluated from sbc =
1 2L(ps − pt )
ps L pt
−L
ln
θf dydp , θf
where L is the half-width of the frontal zone, taken as 1000 km in the following examples, and
θf = −
pκs − µ1 µ2 y 1 L − c(pκs − pκ )/(pκs − pκt ) θ − ln cosh c p κ − µ1 2L y1 κ κ κ κ L + c(ps − p )/(ps − pt ) ln cosh y1
is the horizontal mean potential temperature in the frontal zone. Since the width 2L is relatively small, the Earth’s sphericity is ignored in the integrations with y. For the front depicted in Fig.24.1, the dependence of baroclinic entropy on front intensity is illustrated in Fig.24.5. For comparison, the baroclinic entropy of assumed initial linear atmosphere is displayed by the dashed in the figure. The relation between the front intensity and initial baroclinity may be referred to Fig.24.2. For example, a front with µ2 = 15 K may be attained from the initial field of ε0 = −8 K/1000km by the process with ∆s = 0.34 J/(K.kg). Fig.24.5 shows that the negative baroclinic entropy in the frontal zone is three times more than that in the initial field. Meanwhile, the amount of negative baroclinic entropy outside the frontal zone reduces in frontogenesis, so that the internal thermodynamic entropy is not destroyed.
24.6
Available enthalpy
It is discussed in Chapter 8 that concentration of negative baroclinic entropy in a frontal zonal enhances the local ability of energy conversion, since the available enthalpy increases with the negative baroclinic entropy. The evaluation of available enthalpy with the new variational approach has been discussed in Chapter 9. The lowest state with respect to the frontal field assumed is still represented by (9.12). The maximum available enthalpy is given by the difference of total enthalpy between the frontal field and the lowest state. The total enthalpy in the frontal zone with half-width L is evaluated from
Ψf
L ps cp pκ θf dp dy κ gpθ −L pt p κ − µ1 = cp s κ (2Lθc (ps − pt + µ1 Z) − µ2 C4 ) , gpθ
=
(24.18)
24. LIMITATIONS ON FRONTOGENESIS
520
Figure 24.5: Baroclinic entropy of the frontal zone (solid) and assumed initial field (dashed) where
ps
Z= pt
pκ
dp − µ1
and ps L
C4 =
pt
ps
= pt
−L
y − y0 − c(pκs − pκ )/(pκs − pκt ) pκ tanh dy dp p κ − µ1 y1 L−y −c(pκ −pκ )/(pκ −pκ )
0 s s t cosh y 1 pκ y1 ln dp . κ pκ − µ1 cosh L+y0 +c(pκs −pκ )/(pκs −pt ) y1
The maximum available enthalpy evaluated from (9.22) gives ˜ f − λ1 (ps − pt + λ2 Z) . ˜ max = Ψ Λ Γd pκθ Here, λ1 and λ2 are derived from (9.13) and (9.18) respectively, by replacing θ0 with θf . The following relationships are used for the evaluations:
L ps −L pt
θf dp dy =
(pκs
θc − µ1 ) 2L − µ2 C5 Z
,
24.6. AVAILABLE ENTHALPY
521
Figure 24.6: Maximum available enthalpy of the front attained from the initial fields with static stabilities Γ0 = 0.65 (solid) and 0.75 (dashed) K/100m
in which ps L
C5 =
−L
pt
ps
= pt
y − y0 − c(pκs − pκ )/(pκs − pκt ) 1 tanh dy dp p κ − µ1 y1 L−y −c(pκ −pκ )/(pκ −pκ )
0 s s t cosh y1 y1 ln dp , κ pκ − µ1 cosh L+y0 +c(pκs −pκ )/(pκs −pt ) y1
and
L ps −L pt
ln θf dp dy = 2L
pκt ln
pκt − µ1 + κ(pκs − pκt + µ1 Z) + C6 , pκs − µ1
where L ps
C6 =
−L pt
y − y0 − c(pκs − pκ )/(pκs − pκt ) ln θc + εf y − µ2 tanh y1
dp dy .
The integrations with y in these equations are also made on a flat surface. The examples of maximum available enthalpy in the domain from the center of the surface front (y = y0 ) to y = 2000 km on the cold side (referring to Fig.24.1) and from 1000 hPa up to 200 hPa is depicted in Fig.24.6. The solid and dashed curves in the figure represent the front attained from the linear atmosphere with Γ0 = 0.65 and 0.75 respectively. The maximum available enthalpy of assumed initial field may be referred to Fig.9.3. Changes of maximum available enthalpy in the frontogenesis may be found by comparing Fig.24.6 with Fig.9.3 using Fig.24.2. For example, the linear atmosphere with Γ0 = 0.65 K/100m and ε0 = −8 K/1000km possesses the maximum available enthalpy about 100 m2 /s2 according to Fig.9.3. Figs.24.2 and
522
24. LIMITATIONS ON FRONTOGENESIS
Figure 24.7: Local variation of thermodynamic entropy (units: J/(K·kg)) in the process of energy conversion in the frontal zone.
24.6 show respectively that the front attained through the process with ∆s = 0.34 J/(K.kg) has µ2 = 15 K and maximum available enthalpy about 1300 m2 /s2 . More available enthalpy is found if the initial atmosphere is more baroclinic and less statically stable. The local variation of thermodynamic entropy in the process of energy conversion is evaluated from θr . ds = cp ln θf Assuming that the front is attained though a pseudo- reversible process from the initial field with ε0 = −8 K/1000km and Γ = 0.65 K/100m, the front intensity is referred to Fig.24.2. When this front reaches the lowest state after conversion of maximum available enthalpy through a new pseudo- reversible process, the local thermodynamic entropy change is displayed in Fig.24.7. The center of surface front is at y = 0. This figure shows that the thermodynamic entropy decreases on the warm side of the front and increases on the cold side after energy conversion. The maximum gradient of the entropy variation occurs in the frontal zone. This local entropy variation is produced mainly by transport. When the frontal circulation is geostrophic or quasi-geostrophic, the heat transport across front may not be efficient enough, and the energy conversion is inactive. In this situation, the differential heating at low and high latitudes may strengthen the gradients of mass density, pressure and temperature around the front, and so the process of frontogenesis continues. Meanwhile, the surface drag and viscosity may destroy the geostrophic kinetic energy. To remain the geostrophic wind balance over the front, kinetic energy is created by the cross-front ageostrophic circulation, such as the direct slantwise convection with a slop less than that of the isentropic surface.
24.7. LIMITATIONS BY OTHER FACTORS
523
This slantwise circulation may also transport heat, momentum and thermodynamic entropy across the front and so weaken the front. If the stable quasi- geostrophic balance is established in the frontal circulation, the front may exist for many days. However, as the slope of isentropic surface in a front is large, the slantwise static stability of the frontal circulation is relatively low especially around a strong front. Moreover, the kinetic energy destroyed by the surface drag and viscosity increases with wind speed or front intensity, so the relatively strong slantwise circulation around the front is required by the kinetic energy balance. This frontal circulation may be broken down easily by approach of a cyclonic perturbation in the middle troposphere, such as a trough of the large-scale baroclinic waves. As the front is in a pressure trough usually, the strengthened convergent flows near the surface form the closed cyclonic circulation. When the cold air descends equatoward and warm air rises poleward in the development of disturbance, a great amount of kinetic energy is created as the front collapse, and the heat is transported from the source region to the upper troposphere at high latitudes. The process from frontogenesis to baroclinic disturbance development manifests a cycle of discontinuous transport at the middle and high latitudes, which may be accompanied by the periodical weather changes in a local place.
24.7
Limitations by other factors
24.7.1
Scale of atmosphere
It is noted that in the process of kinematic frontogenesis, thermodynamic entropy destruction in a frontal zone must be compensated by the entropy production in other places. The maximum amount of thermodynamic entropy production depends on the scale of background field. A front generated in a local atmosphere bound in a smaller area may be less intense. This is seen in Fig.24.8, which shows the dependence of front intensity on the meridional scale of the atmosphere represented by Y = µY (µ ≤ 1). The initial field assumed for the figure possesses the horizontal temperature gradient ε0 = −8 K/1000km and the static stability Γ0 = 0.65 K/100m. The front intensity varies greatly with the scale of atmosphere in the direction of temperature gradient. This fact suggests that the process of frontogenesis cannot be simulated in an infinite deformation field. Although mass and heat balances can be maintained in a local frontal zone, thermodynamic entropy in the frontal zone may decrease infinitely as the baroclinity decreases in the infinitely large background field. As a result, an infinitely large temperature gradient is produced artificially in the model atmosphere. This kind of discontinuity cannot be proved by observations.
24.7.2
Baroclinity of background field
Since the frontogenesis in the atmosphere depends on the thermodynamic entropy production, the front intensity depends on the changes of baroclinity in the background field. If the background field remains a high baroclinity after frontogenesis, low thermodynamic entropy is created in the process and the front generated is
24. LIMITATIONS ON FRONTOGENESIS
524
Figure 24.8: Dependence of front intensity on the width of atmosphere weaker. This can be proved by the experiments with different εf in the hemisphere with µ = 1. The obtained results are demonstrated in Fig.24.9. In this figure, the initial horizontal temperature gradient is assumed as ε0 = −7 K/1000km and the initial static stability is Γ0 = 0.65 K/100m.
24.7.3
Latitudinal position of front
In general, the intensity of front decreases as the front moves from high latitudes toward the equator. This fact may be attributed to air mass transformation and relatively strong ageostrophic motions at lower latitudes. The process of frontolysis is also facilitated by Earth’s sphericity. Applying the initial fields with ε0 = −7 K/1000km and Γ0 = 0.65 K/100m, the dependence of front intensity on the latitudinal position is shown in Fig.24.10. The front generated (with εf = −2 K/1000km) at higher latitudes is stronger, though the initial conditions are the same. The decrease of front intensity at lower latitudes may be related to the thermal wind balance. Equation (24.10) tells that the thermal wind is stronger at lower latitudes if the surface wind and baroclinity do not change. Thus, the density of kinetic energy in a frontal zone increases as the front moves equatorward without changing the horizontal temperature gradient. When the energy conversion is limited by the energy conservation law and thermodynamic entropy law in an isolated
24.8. GEOPOTENTIAL ENTROPY VARIATION
525
Figure 24.9: Dependence of front intensity on the baroclinity of background field system, the increase of wind speed requested by thermal wind balance may not be obtained. Thus, the front becomes weaker as it moves to the lower latitudes to remain the thermal wind balance. The weakening of the front may be realized by the sub-geostrophic mass and heat transport across the front. In the real atmosphere, the baroclinity is not constant and decreases equatorward from the middle latitudes, so a front generated at lower latitudes may also be weaker.
24.8
Geopotential entropy variation
We have discussed earlier that variations of the atmosphere are controlled also by the geopotential entropy law. The geopotential entropy of the assumed initial linear atmosphere is evaluated from (9.38), in which the geopotential height is calculated from the hydrostatic equilibrium equation Rθf dz = − gp
p pθ
κ
dp .
Inserting (24.7) into it and integrating the result yield z =
R(pκs − µ1 ) gpκθ
θc + εf y pκs − µ1 ln κ κ p − µ1
24. LIMITATIONS ON FRONTOGENESIS
526
Figure 24.10: Dependence of front intensity on the latitudinal position −
ps
µ2
p
y − y0 − c(pκs − pκ )/(pκs − pκt ) pκ−1 tanh dp p κ − µ1 y1
.
Here, it is assumed that the geopotential height is zero at the bottom pressure surface ps . Now, geopotential entropy of the frontal field assumed may be evaluated from (8.49). Fig.24.11 shows the examples of geopotential entropy changes in frontogenesis. The static stability of the initial field is set as Γ0 = 0.65 K/100m, and the parameters of the front generated are taken as εf = −2 (K/1000km) ,
c = 2000 (km) ,
µ=1
y0 = 0 .
The dependence of front intensity on the initial baroclinity and thermodynamic entropy production is displayed in Fig.24.2. The geopotential entropy may increase if the initial field is strongly baroclinic or the turbulent motion is relatively weak. In a highly irreversible process of frontogenesis, a large amount of thermodynamic entropy is created and the front generated is relatively weak. Also, the geopotential entropy may be destroyed, though the thermodynamic entropy increases. The frontogenesis with destruction of geopotential entropy may not happen if without external mechanic forcing. Thus, variation of thermodynamic entropy alone may not tell us whether the evaluated or predicted frontogenesis may really occur or not. The geopotential entropy gives an additional examination for the process.
24.8. GEOPOTENTIAL ENTROPY VARIATION
Figure 24.11: Geopotential entropy changes in frontogenesis
527
Chapter 25 Grid-scale prediction equations and uncertainties 25.1
Introduction
A basic purpose of sciences is to establish the qualitative or quantitative relationships between the measurable variables studied. If the variables include time, these relationships may be represented as the functions of time, and the other variables solved from these relationships for provided initial and boundary conditions may also depend on time. This means that these time-dependent equations may be used to predict the changes of the variables included and so are called the prediction equations. The prediction equations are time-stable, if the predictions are equally available over a sufficiently long time period for the realistic initial conditions provided. The time-stablepredictions are only possible for some simple processes assumed. While for a complicated process which is not the linear combinations of the simple processes, the prediction may have errors which increase with prediction time, and so there is the time-limit beyond which the errors become large enough and destroy the predictions.The state equation and the first and second laws of thermodynamics, derived in Chapters 2 and 4 respectively for ideal gas, are not prediction equations, since they do not include the molecular diffusions which cause the system variations. They may only be applied for calculating the equilibrium states of classical thermodynamic systems. To explain the system changes through nonequilibrium processes, the nonequilibrium thermodynamics introduced the ‘thermodynamic forces’ based on the hypotheses of linear law. These forces are not the Newtonian forces, as the system variations produced do not follow the Newton’s laws in general. The linear law may only be used to predict approximately the system changes in simplified or idealized theoretical processes, so that the transport properties related to molecular diffusions are studied with other complex theories. Thus, the predictability of nonequilibrium processes in a classical thermodynamic system is destroyed by molecular diffusions. We derived in the previous chapters the primitive equations of the atmosphere which are prediction equations. The uncertainty caused by random molecular motions exists also in predictions of meteorological processes. While, there are some other uncertainties in either the atmosphere or prediction equations which are more important than molecular diffusions. The variables in these equations represent the fields which are continuous in time and space. However, it is generally impossible for us to provide the continuous fields of the real atmosphere. The equations used for weather predictions are different from the continuous primitive equations, since the derivatives in the equations are replaced by the differences calculated using the discrete data provided at grid points. These equations are called the grid-scale 528
25.1. INTRODUCTION
529
prediction equations in the study. The variables in the grid-scale equations are scale-dependent, because their values depend on scale. The datasets used for these equations are also scale-dependent, since they are generally the mean values around the grid points. The predictions made by grid-scale equations are different from those produced with the continuous primitive equations, since they predict approximately only a part of the circulations over the grid scale. The predictions and errors are also scale-dependent. The subgrid-scale circulations, which cannot be represented and predicted by the grid-scale equations, are called the meteorological turbulences in this chapter. They are classified into two categories. One is the irregular turbulent motions on small or tiny scales called the diffusive turbulences which, like the classical turbulences, destroy irreversibly the grid-scale kinetic energy and field gradients. The other will be called the negative diffusions, which are the organized subgrid-scale circulation systems and may convert the available enthalpy into gridscale kinetic energy in the unstable atmospheres, or cascade the momentum from subgrid-scale perturbations into grid-scale circulations. Thus, the meteorological turbulences may not be treated by the classical turbulence theories, such as the mixing length theory. However, they are represented parametrically as the diffusive turbulences in current numerical prediction models using the grid-point data only, and so provide a major error source for the predictions. When the parameterizations suppress the model instabilities, the negative diffusions which provide kinetic energy for grid-scale circulations are suppressed too. Moreover, the energy sources in the atmosphere depends essentially on the atmospheric gradients, which can be changed greatly by turbulent diffusions. While, the parameterizations for the diffusive turbulences using grid-point data may not represent the real subgrid-scale processes. The deficiency in evaluating the turbulent diffusions may also lead to important prediction errors for the system with strong energy conversions. It will be found in this chapter that the effect of turbulent diffusion on kinetic energy generation is comparable with that on kinetic energy dissipation. Apart from the meteorological turbulences, there are other physical subgridscale processes in the atmosphere, such as the release of latent heat, the effect of clouds on the radiation and humidity, the details of heat and momentum transport in the boundary layer and small-scale topographic forcings (Leith, 1983). There are also the atmospheric chemical and photochemical processes which may affect the dynamic and thermodynamic processes in the atmosphere, such as the green house effect. These physical and chemical processes are represented by parameterizations using grid-scale datasets in the prediction models. As the subgrid-scales processes may be independent of grid-scale processes, these parameterizations may produce prediction errors too. Moreover, as the grid-scale equations use truncated numerical methods to evaluate approximately the differentials and integrations included, prediction errors may be produced by the time stepping and spatial discretization. It is argued in this chapter that the errors resulting from use of the centred finite difference schemes in current grid-point prediction models may be larger than expected. These uncertainties and errors resulting from use of grid-scale prediction equa-
530
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
tions and truncated numerical schemes are not just technic and cannot be removed entirely, as use of the continuous primitive equations for weather predictions is practically impossible. Another source of intrinsic prediction errors is provided by the hypotheses applied for simplifying the prediction equations or parameterizations. The technic prediction errors are those produced by the significant technic errors in numerical procedures or data measurements, such as the errors in the initial fields and boundary conditions provided, and can be reduced or removed at least on theory by improving the technologies used. Although the technic errors are produced randomly or systematically, they may grow up with prediction time in the nonlinear processes and destroy the predictions eventually. In general, the prediction errors of the grid-scale prediction equation depend not only on the prediction time, but also on the data and model scales, the subgridscale circulations, the truncation method and the assumptions used for simplifying the equations and constructing the parameterizations. The processes which may be influenced more greatly by subgrid-scale circulations may be more difficult for prediction. It is found that many field and circulation features, such as the baroclinity, β-effect (Basdevant et al., 1981; Vallis, 1983), depth of the atmosphere (Holloway, 1983) and moist convection (Daley, 1981), have significant effects on the predictability. Only for the perfect models which do not include the intrinsic uncertainties described above, the prediction errors may be considered as an initial data problem in mathematics. For convenience, the prediction limit related to the defects of model, will be referred to as the model limit. In comparison, the prediction limit resulting from initial data errors will be called the data limit. Although there are the errors caused inevitably from the uncertainties and technic limitations, weather predictions can be made successfully to a certain extent for a few days by current numerical models. It is found also that the grid-scale prediction equations are not time-stable, and cannot become time-stable by correcting the technic errors only. As most errors can be accumulated by time integrations in the models, the prediction time is limited. The earliest studies of the predictability (Lorenz, 1963; Charney et al., 1966; Lorenz, 1969; Herring et al., 1973; Leith, 1971; Leith and Kraichnan, 1972; Lorenz, 1982, 1985; Roads, 1987; Benzi and Carnevale, 1989) were focussed on the technic errors, such as growth of the data errors in initial fields, in order to find the intrinsic uncertainty of atmospheric circulations like the Heisenberg’s uncertainty principle in quantum physics. Little has been done on estimating the strength of the intrinsic error source related to meteorological turbulences. If the errors caused by model limit is comparable with those produced by data limit, the problem of predictability cannot be regarded simply as growth of initial data errors. The available potential energy of Lorenz (1955) depends on initial field only. Now, the prediction errors studied by Lorenz are mainly the amplified errors in initial datasets or boundary conditions. These meteorological theories illustrated simply as solving a perfect equation set fit the traditional philosophy of mathematical physics. They may be referred to as the Lorenz’s meteorology, which depends on initial data only or has a unique set of solutions for provided initial fields and boundary conditions. According to this meteorology, the unlimited growth of an initial error is regarded as the nature of the atmosphere called the chaos, because the effects of
25.2. SCALE-DEPENDENT DATA
531
intrinsic error sources in the prediction models are ignored and the growth of initial error in the models is considered as a real process. It is not argued that the Lorenz’s meteorology is not correct. Unfortunately, it may not be possible for us to establish and solve the perfect prediction questions for the atmosphere and oceans. Many physical processes are either unknown or not represented precisely in the primitive equations, and so the current prediction models are not perfect. None of the current models represents the real atmosphere, and so the meteorology represented by the imperfect prediction models is different from the Lorenz’s meteorology. The predictions using the imperfect models depend on the algorithms as well as initial data, and the chaos in the model atmosphere may also be produced for correct initial fields. This implies that the model, not only the atmosphere, is a chaotic system. The chaos in a prediction model is the major cause of errors in weather predictions. The predictability studied with the imperfect models gives actually the prediction limit for the model atmosphere, and may not represent the intrinsic uncertainty of the real atmosphere. The prediction with an incorrect initial field may not be compared directly with the real process which is unknown, and so we do not really know how the initial error develops in the atmosphere. For example, a small perturbation in the viscous stable atmosphere may not really grow up, and the butterfly effect may be created artificially by a model defect. Also, the chaotic fields simulated with incorrect initial data are not really related to chaotic processes, since they are controlled by known relationships and are the predictions for the particular initial fields provided.
25.2
Scale-dependent data
Usually, the atmospheric variables are represented by discrete datasets measured at or interpolated to the grid points. The typical distance between the grid points (called the grid scale) represents the space scale or space resolution of the data. The data between the grid points are not provided by the direct measurements, but are given approximately by interpolation or extrapolation. Moreover, the atmospheric variables fluctuate with the time at various frequencies. A measurement at a time and place is usually given by the mean value over a certain time period. In practice, the time scale is regarded as the time interval of measurements, that may be different from the period on which the mean values are obtained. This time scale of data is also referred to as the time resolution. In the theoretical studies however, the discrete data are considered as the mean data over the domain V around the grid point and the time period τ about the observation time. For example, the grid-point velocity is assumed as vG
1 = τ
τ
1 dt V
vp dV ,
(25.1)
V
where the subscript G indicates the grid-point data. Obviously, the data of an inhomogeneous field depend on the time and space scales over which the averages are made. When the grid-point data are used to evaluate the time and spatial integrations or differentials, the evaluated quantities are also scale-dependent.
532
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
The parcel velocity vp in (25.1) is given at each point continuously in the coordinates, and τ and V are not greater than the time and grid scales of the data respectively. Thus, this equation shows that the data depend on the subgrid-scale information. Without the subgrid-scale measurements, the grid-point data are usually given by the local samples averaged over a short time period. The data between grid points estimated by interpolation or extrapolation may be different from the real measurements. Therefore, the grid-point data include the uncertainty which may produce errors in their applications. There are some typical circulation systems in the atmosphere, such as the planetary stationary waves, blocking highs, extratropical cyclones, hurricanes and thunderstorms, which manifest the particular circulation patterns over the typical scales. The datasets used for the study of a weather system should have the scales less than those of the system. If development of a system depends highly on the circulations on a smaller scale, the data scale should not be larger than that of the smaller-scale circulations. Thus, selection of data resolution depends on the understanding of studied circulations. As the prediction equations include the scale-dependent quantities, the predictions using grid-point data are generally different from the predictions with continuous variable fields. The difference is a kind of error which increases with grid scale in general. This error source is related to data resolutions, but not to the errors of measurements. It may still exist, when the data are correct at each grid point. To reduce the error effect, we may reduce the grid and time scales. However, the errors cannot be removed entirely as long as we use the grid-point data. To the current numerical technology, the data given by the instant values read directly from meteorological instruments may not be the best for predicting the large-scale weather systems, since they include instrumental errors, measurement errors, and meteorological noises on the scales much smaller than the grid scale. As these errors and noises may cause instabilities in a numerical prediction model, the data obtained from observations and measurements are usually initialized with some balance equations before they are used for the predictions. The procedure of initialization, like a filter, depends on the scale of predicted circulations. The initialized fields may then be considered as the part of circulations over the grid scale. This is the second meaning of the scale-dependent data. It is not possible and necessary to have the data including the information on all scales.
25.3
Subgrid-scale fields
The circulations represented by grid-point data are a part of the circulations over the grid scale. The mechanic and thermodynamic processes on the scales less than the grid scale (called the subgrid scales) may not be represented by the data. In general, the subgrid-scale circulations are different from the turbulences on a tinyscale about a few centimeters or meters studied in the classical hydrodynamics, and so may be regarded as the subgrid-scale eddies of which the scale is much larger than the size of classical turbulences. These subgrid-scale circulations or eddies are called in the present study the meteorological turbulences, and are distinguished with the classical turbulences.
25.3. SUBGRID-SCALE FIELDS
533
When the meteorological turbulences are considered, the parcel velocity at a grid point may be represented by vp = vG + vt , (25.2) where vG and vt are called, respectively, the grid-scale velocity and turbulent velocity on a subgrid scale. In order to use the small perturbation theory to study meteorological turbulences, the grid-scale velocity is defined by (25.1). Consequently, we have vt dV dt = 0 . τ
V
In fact, the grid-scale data are obtained from observations but not from (25.1), since using this equation needs subgrid-scale information. It is noted that the meteorological turbulences are different from the transient perturbations, defined as the departures from the means over the long time period on a scale greater than the grid scale. They are also different from the eddies defined as the departures from the space mean on a scale much greater than the grid scale. However in current theoretical and numerical models, the meteorological turbulences are represented generally by the transient perturbations and eddies evaluated with the grid-point data. Analogously, we may write T = TG + Tt , p = pG + pt and α = αG + αt . Applying (25.2) for (5.30) gives ∂vG ∂t
= −(vG · ∇)vG − αG ∇pG + g − 2Ω × vG ∂vt − (vt · ∇)vG − (vG · ∇)vt − (vt · ∇)vt ∂t − α∇pt − αt ∇pG − 2Ω × vt , −
where the gradient is defined as ˆ + ∇vy y ˆ + ∇vz ˆz . ∇v = ∇vx x The terms with the product of a grid-scale and turbulent variables disappear after taking average over the grid-scale again, so this equation gives ∂vG ∂t
= −(vG · ∇)vG − αG ∇pG + g − 2Ω × vG − αt ∇pt − (vt · ∇)vt .
(25.3)
Here, the viscous effect related to molecular diffusion is ignored. The first line of (25.3) gives the prediction equation on the grid scale. The rest gives the influences of meteorological turbulences on the grid-scale circulations.
534
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
Since the grid-scale variables are not really the averages evaluated from (25.1), the turbulent variables in (25.3) are actually new variables, which may be mathematically independent of grid-scale variables. For example, the atmospheres with similar grid-scale fields may have entirely different subgrid-scale circulations. In the classical hydrodynamics however, the tiny-scale random turbulences are evaluated from the grid-scale gradients, based on a bunch of assumptions such as the mixing length theory and small perturbation theory. The obtained turbulence terms are similar to the expressions of molecular viscosity, as the effect of turbulences is similar to that of the viscosity, destroying the grid-scale gradients and parcel kinetic energy. However, they may not really include any definite subgrid-scale information. The classical turbulences may not be adopted for meteorological studies or numerical predictions, because meteorological turbulences are different from the classical turbulences, and the grid-point data obtained from observations are different from the means averaged over the measurements on subgrid scales. It will be discussed in the next section, that the organized unstable meteorological turbulences may produce negative viscosity for the grid-scale circulations.
25.4
Diffusive turbulences and negative diffusions
The classical turbulences are defined as irregular fluid motions on a tiny scale, which are dynamically stable so that may be studied with mixing-length theory or small perturbation theory. These stable turbulences convert irreversibly the kinetic energy of laminar flow into turbulent kinetic energy, and reduce or destroy the gradients in a fluid. While, the meteorological turbulences represent undetected circulations in the compressible atmosphere, of which the scales may be larger than that of well organized convective cells. They may be dynamically unstable as in the free convection or slantwise convection, and so may be a kinetic energy source of grid-scale circulations especially in the tropical regions. These unstable turbulences cannot be studied with the mixing-length theory or small perturbation theory. The kinetic energy generation resulting from meteorological turbulences may be studied with the turbulence terms included in the second line of (25.3). We discuss firstly the physical meaning of the first term −αt ∇pt . Suppose the barotropic atmosphere has constant lapse rate of temperature Γ on the grid scale, the temperature profile reads T = Ts − Γz , where Ts indicates the surface temperature at z = 0. The grid-scale pressure profile can be evaluated from (10.2) giving
p = ps
Γz 1− Ts
g RΓ
.
Increase of Γ means decrease of static stability or increase of static instability. For a provided surface pressure and temperature, the pressure at high levels decreases with static stability. The departure of turbulent pressure from the large-scale hydrostatic equilibrium is pt at a local place. When the static stability on a subgrid scale
25.4. DIFFUSIVE TURBULENCES AND NEGATIVE DIFFUSIONS
535
decreases upward, there is ∂pt /∂z < 0. Warm parcels (αt > 0) ascending in this circumstance may get upward acceleration relative to the grid-scale circulation, since − αt
∂pt >0. ∂z
(25.4)
When the static stability decrease downward, there is ∂pt /∂z > 0. So, the cold parcels (αt < 0) descending in the atmosphere will be accelerated too. The previous discussions may be carried out in an alternative way. When the parcel pressure is adjusted to the environment, the vertical gradient of turbulent parcel pressure gives the turbulent hydrostatic equation ∂pt ∂p = = −gρ . ∂z ∂z It tells that ρ > 0 or α < 0 when ∂pt /∂z < 0. If the parcel is warmer than the surroundings or (αt > 0), we gain (25.4) again which is rewritten as αt >0. α The kinetic energy is created in this situation and so the small-scale upward free convection may develop in the turbulent pressure field according to the Archimedes’ principle. The last term in (25.3) may be rewritten as x − ∇ · (vty vt )ˆ y − ∇ · (vtz vt )ˆ z + vt ∇ · vt . −(vt · ∇)vt = −∇ · (vtx vt )ˆ The first three terms on the right-hand side provide the sink or source of grid-scale momentum related to the convergence of turbulent momentum flux. The last term is the contribution of turbulent wind divergence, which disappears when the vertical velocity is given in the pressure coordinates. If the meteorological turbulences are replaced by the large-scale eddies, the eddy flux of momentum in certain perturbation patterns may strengthen the mean flow (Kuo, 1951a, 1951b; Mieghem, 1973). Starr (1953, 1966) and Kuo (1951a) introduced the concept of negative viscosity to explain the momentum cascade from eddies to mean flows. The negative viscosity or backward momentum cascade may also be produced by meteorological turbulences or subgrid-scale eddies, when the grid scale is not too small. Unlike the momentum exchanges caused by molecular diffusion and conduction, the turbulent motions are driven by external forces including the pressure gradient force. Thus, we have the first term in the second line of (25.3). The pressure gradient force acting on the turbulent entities may either destroy or create kinetic energy, so the momentum of the system is not conserved in turbulent diffusions. When the gradients are destroyed by the turbulences, kinetic energy may be produced in the excited atmosphereas discussed in Chapters 8-12. However in the numerical prediction models, only the effect of turbulent momentum transport parameterized as the turbulent viscosity, instead of the first term, is included. Therefore, the meteorological turbulences may be classified into two categories. One is the organized circulations, such as the subgrid-scale convective cells or storms, wave perturbations, wind shears and convergent lines, which may increase gridscale momentum, kinetic energy or local inhomogeneities. These grid-scale effects
536
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
produced by subgrid-scale unstable turbulences or organized eddies may be regarded as the negative diffusions. The other one is the random turbulent motions on a small or tiny scale, and is similar to the irregular turbulences studied in the classical fluid dynamics. The random turbulences convert irreversibly the grid-scale kinetic energy into turbulent kinetic energy on a smaller and smaller scale. The turbulent kinetic energy will eventually be transferred into heat energy by molecular diffusions. These random turbulences may also weaken or destroy the gradients or discontinuous boundaries, and so the kinetic energy sources in the atmosphere. They may be called the diffusive turbulences. There are some basic differences between the turbulent and molecular diffusions. Molecular diffusions are related to random motions of molecules, which increase classical thermodynamic entropy in an isolated system. The kinetic energy dissipated by molecular diffusion becomes heat energy as the system reaches equilibrium. Variations of the diffusion velocity may be independent of macroscopic Newtonian forces. While, the turbulences are a part of parcel motions and possess parcel kinetic energy. They increase the disorderliness or the turbulent entropy of isolated atmosphere, when classical thermodynamic entropy may still be conserved without molecular diffusions. The kinetic energy dissipated through turbulent diffusions may not be converted into heat energy immediately at an inhomogeneous equilibrium state. In general, turbulent diffusions have greater influences than the molecular diffusions on the large-scale processes.
25.5
Grid-scale prediction equations
When the molecular viscosity is ignored, the momentum equation is given by (25.3), that is ∂v = −(v · ∇)v − α∇p + g − 2Ω × v − Dvt , (25.5) ∂t where (25.6) Dvt = −αt ∇pt − (vt · ∇)vt represents the meteorological turbulences, which include the diffusive turbulences and negative diffusions discussed earlier. All other variables in this equation are grid-scale variables, of which the subscript G is omitted. As no turbulent data are available usually, the diffusion term is not calculated from (25.6), but from the parameterization using grid-point data based on several hypothesis and approximations, such as the extended linear law and mixing-length theory. The obtained expressions are similar to the viscous stress tensor and may be found in the text books of fluid dynamics. The meteorological turbulences may also cause mass transport and exchanges. Applying the partitioned fields for continuity equation (5.11) yields ∂α = −v · ∇α + α∇ · v + Dαt , ∂t
(25.7)
Dαt = −vt · ∇αt + αt ∇ · vt
(25.8)
where
25.5. GRID-SCALE PREDICTION EQUATIONS
537
is called the turbulent diffusion of mass, which is the divergence of turbulent mass flux. The terms with the product of a grid-scale and turbulent variable are ignored in (25.7), and the subscript G for grid-scale variables is omitted. The two terms on the right-hand side of (25.8) represent convergence of turbulent mass flux. Generally, the effect of turbulences on the grid-scale circulations is smaller than that of grid-scale wind convergence except in the subgrid-scale convective storms, so the continuity equation used in practice does not have the diffusion term. If the heat conduction caused by molecular collisions are ignored for meteorological processes, the grid-scale heat flux equation may be derived by inserting the partitioned fields into (4.14), giving cp in which
dG p dq ∂T = −v · ∇T + α + + DTd + DTt , ∂t dt dt
(25.9)
∂pG dG p = + vG · ∇pG dt ∂t
and dp dG p − αG (25.10) dt dt is the turbulent diffusion of heat. The molecular diffusion DTd is remained in the equation, as it is an important source of thermodynamic entropy. The variables in (25.9), except in the molecular and turbulent diffusion terms, are represented by grid-point data, and the subscript G is omitted. The first term on the righthand side of (25.10) gives the turbulent advection of heat. The other two terms represent the turbulent adiabatic variation related to pressure change. As dp/dt is the vertical velocity in pressure coordinates denoted by ω, these terms may be replaced by αt ωt . In practice, the turbulent diffusion is also evaluated using gridpoint data with some hypotheses such as the linear law and mixing-length theory, but not using the previous equation. Finally, the state equation (2.21) may be replaced by DTt = −vt · ∇Tt + α
RT = pα + pt αt ,
(25.11)
or p = R(ρT + ρt Tt ) . As ρt Tt ρT and pt αt pα in general, the effects of turbulent fluctuations on the grid-scale fields are ignored. If we know the turbulent fields, the four equations (25.5), (25.7), (25.9) and (25.11) form a complete set for the four grid-scale variables included, assuming that the heat exchange with the exterior is known. These equations are then called the grid-scale prediction equations, as they predict only the circulations over the grid scale. The derivatives in the equations are replaced by the differences evaluated using the grid-point data. Since we do not have the subgrid-scale data to evaluate the turbulent fields which may be independent of grid-scale variables, the turbulent terms are usually ignored as in the state equation and continuity equation, or represented parametrically using the grid-scale data as in the heat flux equation and momentum equation.
538
25.6
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
Scale-dependent prediction models
Changes of classical thermodynamic system, such as the ideal gases, may be produced by molecular diffusions. Calculations of molecular diffusions at a nonequilibrium state is discussed in Chapter 3. The diffusion processes without parcel motions may be evaluated with the three prediction equations: the state equation and thermodynamic energy law for equilibrium states together with the continuity equation (5.8). Since it is generally difficult for us to measure the microscopic gradients and diffusion velocity in a real fluid, the molecular diffusions are usually evaluated in the nonequilibrium thermodynamics by introducing the generalized thermodynamic forces which, according to the linear law or phenomenological law assumed, are proportional to the gradients of thermodynamic variables (Batchelor, 1967; de Groot and Mazur, 1962). However, the density changes caused by molecular diffusions cannot be evaluated with the linear law. So, the continuity equation used in the current nonequilibrium thermodynamics is given by (5.11), which accounts for the density changes caused by parcel motions only. In fact, the mass distribution may nevertheless be changed by molecular diffusions without parcel motions. For the pure molecular diffusions without parcel motions, continuity equation (5.11) may not be used. When the molecular diffusions cannot be represented precisely and so are ignored, variations of an ideal gas may only be evaluated with two diagnostic equations: the state equation and thermodynamic energy law for equilibrium states. For example, the three thermodynamic variables in a new equilibrium state may be solved from the two time-independent equations, by assuming a particular process or assigning in prior the value for at least one thermodynamic variable. Although the solutions may represent a real physical state of the system, it is unknown whether or when the evaluated state may really occur afterwards and what the system looks like before it reaches the new equilibrium state. In this sense, the classical thermodynamic systems are unpredictable in the classical thermodynamics, even if the final equilibrium state can be assumed independently according to our experiences. In the inhomogeneous thermodynamic system such as the Earth’s atmosphere, the variations are produced by parcel motions as well as molecular diffusions. When molecular diffusions are ignored, we may still have a closed set of prediction equations called the primitive equations, which include the momentum equation (5.30), heat flux equation (5.18), state equation (2.21) and continuity equation (5.11). If molecular diffusions have no big effects, parcel motions and produced atmospheric variations may be predicted for provided initial conditions, if without difficulties in mathematics and data measurements. However, it is not that easy to make predictions with the primitive equations, even if the molecular diffusions are negligible. Variables in the primitive equations are continuous in the time and space, represented generally by Xj = Xj (x, t) ,
j = 1, 2, · · · , J ,
where x indicates spatial positions in the spatial coordinates applied. The primitive equations may then be written as dXj = Gj (X1 , X2 , · · · , XJ ) . dt
(25.12)
25.6. SCALE-DEPENDENT PREDICTION MODELS
539
Since it is impossible for us to provide the continuous initial fields of the real atmosphere, the data used for weather forecasts and numerical simulations are the grid-point data over a certain scale, which are scale-dependent as discussed earlier. The differentials and integrations evaluated with the scale-dependent data depend also on scale. The grid-point data denoted by X j may be considered conveniently as the ‘mean’ value of Xj over the grid scale, that is 1 Xj = τV
τ
V
Xj dV dt ,
in which V represents a domain of the date scale and τ is the time scale of the data. In practice, the mean values are actually collected at a local place and given time. The prediction equations of scale-dependent data are represented generally by ∆X j = Gj (X 1 , X 2 , · · · , X J ) + Ξj . ∆t
(25.13)
Here, Ξj represents the effects of subgrid-scales circulations on the grid-scale fields. These equations are called the grid-scale prediction equations in the preceding section, given by (25.5), (25.7), (25.9) and (25.11) when the molecular viscosity is ignored. The primitive equations may be considered as established scientific relationships, if all the physical processes on all scales are incorporated correctly. While, the grid-scale equations are uncertain, since the physical processes on subgrid-scales cannot be represented precisely by the grid-scale variables, and there are no the general expressions for the statistical effects of various subgrid-scale processes on the grid-scale circulations. Thus, the turbulent fields included in these equations are independent variables as pointed out previously. But in the current numerical models, they are represented parametrically by the grid-point data, that is Ξj = Ξj (X 1 , X 2 , · · · , X J ) . Thus, the grid-scale equations form a closed set for provided heat exchange in the heat flux equation. The obtained solutions depend on data resolution and the parameterizations. In many studies, the scale-dependent equations are simplified according to the scale analysis or particular assumptions provided additionally. The numerical models can only be made from the scale-dependent equations, and so are called the scale-dependent models. Predictions with the scale-dependent models are generally different from the mean fields predicted by the continuous primitive equations (25.12), that is !
Gj (X 1i , X 2i , · · · , X Ji ) + Ξj (X 1i , X 2i , · · · , X Ji ) τ =
τ 0
Gj (X1 , X2 , · · · , XJ )dt .
The most important error source is the uncertain terms Ξj , which represents the effect of meteorological turbulences on the grid-scale circulations predicted. In the current numerical models of which the grid scale is hundreds of kilometers and may be larger than the scale of a convective storm, the meteorological turbulences are
540
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
usually treated with the mixing length theory using the last term in (25.6) to represent the turbulent diffusion of momentum, while the previous term was thrown away which could be a grid-scale kinetic energy source in the unstable moist atmosphere. Thus, the meteorological turbulences and their effects evaluated parametrically with grid-point data represent the diffusive turbulences only, but not the subgrid-scale energy sources. The diffusive turbulences simulated by the mixing-length theory is similar to the molecular viscosity, except that the turbulent viscosity coefficient is a few orders larger than the molecular viscosity coefficient, and may change greatly with time and space. While, the backward momentum cascade resulting from the negative viscosity cannot be represented by the mixing-length theory. So, the diffusion terms in the numerical models are actually a smooth operator used to suppress the instabilities in numerical integrations. One way to incorporate the negative diffusions in the prediction models is to increase data resolution and model resolution. If the resolutions are high enough and all the circulation patterns over the grid scale may be represented rightly, the subgrid-scale turbulences may be the diffusive turbulences only, and may be represented by the parameterizations approximately. Thus at least on theory, increasing data resolution may reduce prediction errors, but adds more difficulties and technic error sources in data collection and modeling procedures. There are many physical processes in the atmosphere which are related to the subgrid-scale circulations, such as the condensation and evaporation of water vapor, small-scale variations in the surface drag, cloud and surface albedo, and heat and momentum diffusions or conduction. These subgrid-scale processes may intensify the error source provided by meteorological turbulences. Another important error source is produced by the truncations in the numerical methods used to evaluate the derivatives and integrations in the grid-scale equations. Prediction errors may also come from the uncertainties of the boundary conditions, or assumptions used for simplifying the prediction equations. The prediction models possessing these error sources may only produce the predictions available within a time limit, over which the errors may have sufficiently large effects on predicted weather systems.
25.7
Errors from finite difference schemes
25.7.1
Truncation error
Except in the spectral models or the models using orthogonal functions to represent the atmospheric fields, the differentials and integrations in the scale-dependent prediction models are usually evaluated approximated with the finite differences derived from the Taylor expansion Xi±1 = Xi ± Xi ∆x +
Xi X ∆x2 ± i ∆x3 + · · · , 2 6
where ∆x = xi+1 − xi ; the prime indicates the derivative with respect to x (The number of primes tells the derivative order); the subscript i or i ± 1 indicates the
25.7. ERRORS FROM FINITE DIFFERENCE SCHEMES
541
grid position along x, and the upper and lower signs are for forward and backward differences respectively. The previous equation gives Xi = ±
Xi±1 − Xi Xi X ∓ ∆x − i ∆x2 ∓ · · · , ∆x 2 6
or Xi =
(5)
(2n+1)
Xi+1 − Xi−1 Xi X X − ∆x2 − i ∆x4 − · · · − i ∆x2n − · · · 2∆x 3! 5! (2n + 1)!
(25.14)
obtained by adding the forward and backward differences together. The number in superscript () indicates the derivative order. This equation does not include even-th order derivatives. One may also use the forward or backward difference only to derive a difference scheme. If the right-hand side of (25.14) is truncated before the second derivative (referred to as the first-order truncation in this study), we gain Xi [1] =
Xi+1 − Xi−1 , 2∆x
(25.15)
in which the number in superscript [] denotes the truncation order. This is the three-point centred difference scheme used most frequently in current numerical models. It is derived here from the first-order truncation (referring also to Haltiner, 1971), and may also be obtained from a second-order or third-order truncation, if the forward difference in the Taylor expansion is used. The first derivative derived from the third-order truncation (including the first two terms in (25.14)) gives Xi [3] =
−Xi+2 + 8Xi+1 − 8Xi−1 + Xi−2 . 12∆x
(25.16)
Here, we have employed Xi
= = =
Xi+ 1 − X i− 1 2
2
∆x − 2Xi + Xi−1 Xi+1 ∆x2 Xi+2 − 2Xi+1 + 2Xi−1 − Xi−2 2∆x3
derived from the centred difference scheme. Equation (25.16) is a five-point difference scheme, and is generally more accurate than the three-point scheme (25.15), especially for the patterns on a scale greater than five-grid distance. A scheme of a higher order needs more calculation time, and may not be used near the boundaries. The truncation error of the first derivative, evaluated with a difference scheme denoted by ∆X/∆x, is ∆X . (25.17) Rx = Xx − ∆x It follows, from (25.14), that [l]
Ri = −
(n) ∞ Xi n=l+2
n!
∆xn−1
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
542
for the l-th truncated difference scheme. If the high order derivatives in (25.14) can be evaluated analytically with local information at xi , the error is smaller at a higher truncation. For convenience, we define the relative truncation error as Erx[l]
[l]
Rx = . Xx
Since the gradients in the atmospheric fields are not constant, the truncation error may not be ignored as shown by the following examples. The differences in (25.15) and (25.16) were also referred to as the second- and fourth-order differences, as it was believed that their truncation errors were in the order of O(∆x)2 and O(∆x)4 respectively. In fact, the error may not be reduced continuously by increasing truncation order, when evaluated with grid-point data. If without subgrid-scale information, a higher order difference scheme covers more grid points and longer distance, and is unable to improve the accuracy without providing additional subgrid information. This implies that the accuracy is limited by the data resolutions and the truncation is an intrinsic error source. For a provided grid scale, there is a lower limit of truncation error which cannot be reduced further by using a higher-order difference scheme. The scheme with the lowest truncation order and smallest truncation error may be the best, if without considering the computational efficiency and ability.
25.7.2
Examples
For a field assumed simply as the harmonic wave X = X0 + A sin
2π x Lx
where Lx is the wavelength, the first derivative reads Xx = A
2π 2π cos x. Lx Lx
(25.18)
The mean relative error over I grid points is calculated from Er
[l]
=
in which [l]
Eri =
I 1 [l] |Eri | , I i
Xi [l] − Xi , Xi
and Xi is evaluated from (25.18) at grid point xi . For ∆x = 100 km and A = 10, [f,b] in dependence of the mean error on wavelength is listed in Table 25.1. The Er this table indicates the mean relative error produced by the first-order forward or backward difference scheme, averaged over the distance of 10,000 km or 101 grid points. The errors are the largest compared with those of other schemes in the table. [c] The widely used three-point centred scheme has a relative error Er > 10% if the wavelength is less than 8∆x.
25.7. ERRORS FROM FINITE DIFFERENCE SCHEMES
543
Table 25.1: Relative truncation errors Error (%) Er
[f,b]
Er Er Er
[c] [3]
(2)
Lx /∆x 10 20
1
2
4
8
50
100
100
100
36.3
28.2.3
47.5
20.0
16.9
7.2
100
100
36.3
10.0
6.4
1.64
0.26
6.6×10−2
100
100
15.1
1.2
0.5
0.03
8.6×10−4
9.1×10−5
100
100
18.9
5.0
3.2
0.82
0.13
0.03
The relative errors of the forward and backward schemes are extremely high at the points where the first derivative is zero. For example, the forward scheme gives Xi =
∆x 2A ∆x 2π cos cos x+ ∆x 2 Lx 2
for the wave assumed previously. From (25.18) we obtain
∆x ∆x 2π Lx ∆x Xi cos − sin tan = cos x Xx π∆x 2 2 2 Lx
.
The last term becomes infinity at x = (2n + 1)Lx /4. If Lx = 4∆x and x = i∆x, we have iπ 2 Xi 2 ∆x − sin ∆x tan = 2 cos . Xx π 2 2 The infinitely large errors occur at the grid points with odd numbers of i. The large errors are not shown in Table 25.1, since they are filtered out at these points. If the time integrations in a prediction model are carried out with the forward difference scheme, growth of perturbations may be misrepresented near the turning points where the time variations is zero or small.
The truncation error for the assumed harmonic perturbation can be reduced by using the third-order five-point difference scheme. The improvement is significant for the wavelengths over 8∆x. If the wavelength is shorter than 4∆x, none of these schemes in the table may give a good result. The experiments show that a scheme of a higher truncation order is unable to reduce the errors furthermore, as it cannot reduce the grid scale of data. For the convective storms covering 100 km horizontally, the data and model resolutions should not be larger than 25 km for the errors to be constrained below 10% using the five-point scheme. The bottom line in the table shows the relative error of the second derivative evaluated from Xi =
Xi+1 − 2Xi + Xi−1 , ∆x2
and Xx = −A
2π Lx
2
sin
2π x. Lx
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
544
Table 25.2: Relative truncation errors for ECMWF 500 hPa fields with 2.5◦ long. × 2.5◦ lat. resolution in March 1997. The mean errors over total grid points are shown in the upper lines. The lower lines give the mean errors with filter.
Error (%) Er
[f ]
Er Er
[b] [c]
Ratio (%)
ECMWF fields vx vy
g
T
ω
q
140.4 57.1
232.7 79.1
304.0 94.7
302.5 97.2
456.9 142.7
2750 128.1
139.5 57.0
231.2 79.1
305.6 94.6
303.6 97.2
457.3 142.7
4007 128.1
29.0 9.7 3.1
50.1 14.4 5.2
57.2 16.2 5.5
56.9 16.5 5.4
66.6 21.8 6.3
700.4 20.4 6.6
An error more than 19% can be produced for the wave with wavelength shorter than 4∆x, which is smaller than the error of three-point centred scheme but larger than the error of the third-order scheme. This table tells that not only the subgridscale circulations but also the eddies on a scale over a few grid points may not be predicted by the scale-dependent models. The errors may be accumulated by time integrations, and destroy the predictions eventually.
Table 25.3: The same as Table 25.2 except for 1◦ long. × 1◦ lat. resolution in March 1999 Error (%) Er
[f ]
Er Er
[b] [c]
Ratio (%)
T
ECMWF fields vx vy ω
q
226.3 79.5
186.4 67.3
193.2 70.6
375.1 87.0
78837 127.7
223.9 79.5
186.0 67.3
192.8 70.6
375.1 87.0
90439 127.6
48.3 13.7 5.0
38.6 11.8 3.9
38.6 12.0 4.0
59.1 15.9 5.9
27160 19.3 6.1
Now, we compare the difference schemes using the initialized ECMWF (European Centre of Medium-Range Weather Forecasts) data with a resolution of 2.5◦ longitude × 2.5◦ latitude available twice a day at 00Z and 12Z. The monthly mean
25.7. ERRORS FROM FINITE DIFFERENCE SCHEMES
545
errors are evaluated from Er
[l]
J N I 1 = 2N × J × I n j i
|Xi [l] − Xi [3] | |Xj [l] − Xj [3] | + |Xi [3] | |Xj [3] |
,
(l ≤ 3) ,
n
where i and j indicate the grid positions along the latitude and longitude respectively; N is the time of measurement in the month, and Xj [3] is the third-order truncated derivatives given by (25.16). The results are shown by Table 25.2. The variables g, T , vx , vy , ω and q represent the geopotential, temperature, zonal wind, meridional wind, vertical velocity and relative humidity respectively. Although the relative errors in the table may not be identical to real errors, the large values may suggest the high uncertainties in using these truncated difference schemes, especially the forward and backward schemes. The uncertainties are higher if the field possesses more smaller-scale features, such as the humidity field and vertical velocity field. Experiments show that the relative errors in the tropical and subtropical regions are larger than at middle and high latitudes. These large uncertainties are produced by extremely large errors at a few grid points. If the points at which |Er [c] | > 1 are filtered out, the numbers are listed in the lower lines in the table, and the ratios of the number of filtered points to the number of total grid points are displayed in the bottom line. It is found that the mean relative errors may be tripled by the extremely large errors at about 5% of the grid points. Table 25.3 shows the errors evaluated using the ECMWF data with the resolution of 1◦ longitude and 1◦ latitude measured every 6 hours. The high resolution datasets, which we have, do not include the geopotential field. Comparing with Table 25.2 finds that the uncertainty resulting from the errors of difference schemes is reduced slightly by increasing model resolution. The reduction rate is much lower than that of data scale. The ratios of the points with large errors are reduced slightly too. This result does not mean that the high resolution data are only slightly better than the low resolution data, since the previous tables are not made by comparing with real differentials. It is believed that the global spectral models have some advantages over the gridpoint models. However, the spectral algorithm has truncation errors also, which cannot be ignored (Ritchie and Beaudoin, 1994) especially near the poles. The vertical derivatives in current spectral models are still calculated with finite differences. As in the grid-point models, errors in the spectral models may also be produced by time stepping. The time truncation errors may become increasingly important as other error sources are weakened, especially for the large-scale processes (Simmons, 1987; Naughton et al., 1993). There are some studies on the comparisons between spectral models and grid-point models (Doron et al., 1974; Browning and Kreiss, 1989; Grumm, 1993). The comparison cannot be straightforward as pointed out by Browning et al. (1989). Although the spectral models over the whole globe are preferred for the high efficiency, there have been no the conclusion that the prediction time can be expanded significantly by replacing the spectral algorithms for the finite difference method in numerical models.
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
546
Figure 25.1: Dependence of available moist enthalpy on thermodynamic entropy production in the moist barotropic atmosphere. The solid and dashed curves are drawn for ws = 8 and 12 g/kg respectively.
25.8
Turbulent diffusion and predictability
It is discussed earlier that the subgrid-scale processes in the atmosphere represented by the uncertain terms Ξj in (25.13) may have significant effects on grid-scale circulations. These meteorological turbulences are different from the tiny-scale turbulences studied in the classical hydrodynamics, as they include important physical processes and small-scale eddies, and may produce negative diffusions by converting the thermal enthalpy into grid-scale kinetic energy. However, the meteorological turbulences are usually treated parametrically as the diffusive turbulences using the grid-scale data in current numerical models. There have been no general answers to the question: To what extent, the grid-scale circulations are affected by meteorological turbulences ignored or misrepresented in the scale-dependent models. Certainly, the negative diffusions produced by organized circulation systems and moist convection may have a great influence on predicted circulations. The small-scale thunderstorms in the statically unstable atmosphere may convert a large amount of latent heat into kinetic energy. If the data and model resolutions are high enough so that all the important phys-
25.8. TURBULENT DIFFUSION AND PREDICTABILITY
547
ical processes may be incorporated, errors may still be produced by misrepresenting the tiny-scale or microscale diffusive turbulences in the atmosphere. The turbulent diffusions may influence not only the velocity fields but also the thermal structure of the atmosphere. They may destroy the kinetic energy and the ability of energy conversion, by reducing the disorderliness or increasing thermodynamic entropy in the atmosphere. It is discussed in Chapters 9-12 that the kinetic energy generated depends highly on three-dimensional inhomogeneities in the potential temperature and humidity fields. Turbulent diffusions may destroy these discontinuities and so the kinetic energy sources efficiently. While, the diffusions represented in the atmospheric models are not related to physical processes but are chosen empirically to keep the numerical solutions smooth. For example, the molecular diffusion in the momentum equation should be balanced by that in the heat flux equation, in order to remain the energy conservation as shown by (7.13). This is also true for turbulent diffusions as the grid-scale kinetic energy dissipated is eventually converted into heat energy. But, this may not be the case in current numerical models. The kinetic energy destroyed by the turbulent diffusion in a model may be greater than the heat energy created by diffusion. In this situation, the turbulent diffusion is an artificial process used for eliminating the spurious kinetic energy created by the model instability. As pointed out by Naughton et al. (1993), if the diffusion is not specified correctly, it will result in large errors as we see from the cross runs with different diffusion specifications. The effect of turbulent diffusion on kinetic energy generation in the dry atmosphere are shown by Fig.10.2. The evaluated available enthalpy decreases quickly with increasing thermodynamic entropy production, especially when the static stability is low. In a highly irreversible process, the available enthalpy may be negative and the energy conversion may take place abruptly. The abrupt energy conversion may result in rapid or explosive developments of weather system, such as the stratospheric sudden warmings and tropospheric bombs. Predictions of these abrupt processes are then very sensitive to the molecular and turbulent diffusions, especially near the threshold of sudden change. An overestimate of the diffusions may destroy falsely the kinetic energy and its sources, while the underestimate may lead to spurious development of disturbance. The kinetic energy budget in a local region may be affected even more seriously by incorrect thermodynamic entropy flux across the boundaries in a model. From the point of energy conversion, the model instability may be related to the errors of local entropy balance in the model atmosphere caused, for example, by amplified negative thermodynamic entropy input. When the turbulent diffusions and subgrid-scale thermodynamic entropy flux change from time to time and from place to place, the predictions are even more difficult. The important effect of thermodynamic entropy production on energy conversion in the moist atmosphere has been investigated in Chapter 11. For the initial fields of humidity and equivalent potential temperature represented by (11.28) and (11.29) respectively, the dependence of available moist enthalpy on thermodynamic entropy production is sketched in Fig.25.1. As in the dry atmosphere, the kinetic energy created in the moist processes varies greatly with the entropy production. The variations are larger, and the suddenness of circulation change is more remarkable than in the dry atmosphere. Predictions of these sudden changes are generally
548
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
(a)
(b)
Figure 25.2: Growths of Eady waves, (a) mean perturbation velocity and (b) mean perturbation height. The solid and dashed curves are for ∆s∗ = 0 and 0.02 J/(K2 · kg), respectively.
25.9. THERMODYNAMIC ENTROPY PRODUCED BY DIFFUSIONS
549
difficult. To improve the prediction accuracy, we have to evaluate accurately the diffusions and their dependence on the instantaneous static stability and humidity. It is discussed earlier that the meteorological turbulences have a great effect on energy conversions in the atmosphere. This effect cannot be revealed by shallow water experiments, as the energy conversions are related to thermodynamic processes in the atmosphere, which do not occur in the incompressible fluid. When development of a system depends essentially on the energy conversion, the meteorological turbulences may produce important errors in the predictions, even though the effect of negative diffusions may be ignored. For the examples of Eady wave development discussed in Chapter 18, the effect of turbulent diffusion can be seen by comparing the dashed curves with the solid in Fig.25.2. We have assumed that Γ = 0.65 K/100m and θR0 = 6 K or |Ty | ∼ 10.4 K/1000km at 1000 hPa for the calculations. The thermodynamic entropy production in 3 days is about 0.12 J/(K· kg) in the dashed examples. This figure can be compared also with Fig.18.3(b) and (c), in which thermodynamic entropy production is one order smaller than the current example. Although only the effect on the energy conversion but not on the energy dissipation is considered, the turbulent diffusions cannot be ignored for studying the wave development. The previous discussions show that the diffusive turbulences may destroy not only the kinetic energy but also the energy sources. Serious prediction errors may be produced by the deficiency in evaluating the diffusive turbulences. Also, neglecting the negative diffusions produced by organized subgrid-scale disturbances may produce even larger errors. If without errors in the initial data provided, prediction errors may still be produced by the grid-scale prediction models. Since the diffusions are highly irreversible, thermodynamic entropy produced cannot be offset by taking averages over the time and space, and so the errors will accumulate with time, and eventually destroy the predictions. There has been no evidence to prove that the prediction errors produced by model limit can be ignored, if compared with the errors resulting from data limit over a certain period. In this sense the problem related to the indeterminism or chaos of the atmosphere is far beyond the growth of initial data errors.
25.9
Thermodynamic entropy produced by diffusions
In the dry atmosphere, heat energy converted from kinetic energy dissipation over a time period τ may be estimated approximately from
Dv dt = T ∆s
∆q = τ
for unit mass. According to Oort and Peixoto (1974), the climatologically averaged kinetic energy dissipation in the Northern Hemisphere is about 2 W/m2 . This is equivalent to the heat generation of 16.9 J/kg per day, assuming that the surface pressure is 1000 hPa. If the prediction limit estimated from the growth of initial data error is two weeks (Lorenz, 1969), thermodynamic entropy produced by kinetic
550
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
energy dissipation is about 0.88 J/(K·kg) in the period, estimated from
∆s = τ
Dv dt .
Fig.10.2 shows that this amount of entropy production may reduce the kinetic energy generation more than 500 m2 /s2 in the linear atmosphere with Γ = 0.65 K/100m and |Ty | = 20 K/1000km. The effect is even larger if the static stability or baroclinity is lower or in a moist atmosphere. While, the kinetic energy dissipation in two weeks is about 237 m2 /s2 . So, the effect of diffusion on the energy generation is twice as large as that on the energy dissipation. Another important diffusion process in the atmosphere is mixing of parcels at different thermodynamic states. Thermodynamic entropy produced by isenthalpic molecular diffusion between two parcels with unit mass and different temperatures T1 and T2 is represented by ∆s1 + ∆s2 ∆s = . 2 Here, ∆s1 and ∆s2 are the thermodynamic entropy changes of the parcels, evaluated from Tm , ∆s1 = cp ln T1 and ∆s2 = cp ln
Tm , T2
where Tm is the temperature after mixing: Tm =
T1 + T2 . 2
These equations give ∆s = =
cp (T1 + T2 )2 ln 2 4T1 T2 cp ∆T 2 ln 1 + , 2 4T1 (T1 + ∆T )
in which ∆T = T2 − T1 > 0. If the mixing takes place when ∆T = 1 K, the entropy produced is about 1.72 × 10−3 J/(K·kg) for T1 = 270 K. Assuming again that the horizontal temperature gradient is 20 K/1000km in a frontal zone, and the relative wind speed of warm and cold flows across the front is 10 m/s, the entropy produced by mixing in two weeks gives 0.42 J/(K·kg). This amount of thermodynamic entropy production may reduce kinetic energy generation about 300 m2 /s2 in the linear atmosphere with Γ = 0.65 K/100m and |Ty | = 15 K/1000km, as shown by Fig.10.2. The effect is also comparable with that of the kinetic energy dissipation discussed previously. The entropy produced by mixing may be rewritten as ∆s ≈
cp ∆T 2 8T1 (T1 + ∆T )
25.10. UNCERTAINTIES IN PHYSICS
551
since ln(1 + x) = x + · · ·. It is proportional to the square of temperature difference between the parcels. If ∆T = 2 K, the entropy produced in two weeks becomes 0.83 J/(K·kg) for the previous example. The effect on kinetic energy generation is then doubled. The mixing may occur also in vertical air motions. Temperature difference between a parcel and the surroundings is estimated from T´ − T = −(Γd − Γ)∆z . To produce one degree of temperature difference in the dry atmosphere with Γ = 0.65 K/100m, a parcel moves vertically about 286 meters in a rest environment. If vertical air speed is 0.05 m/s and the mixing takes place as the temperature difference is one degree, about 0.37 J/(K·kg) of thermodynamic entropy will be produced in two weeks. The effect on the energy conversion is also comparable with that on the energy dissipation produced by turbulent viscosity. In the moist atmosphere, thermodynamic entropy can be produced also by irreversible water evaporation and water vapor condensation. These moist processes are associated with micro-scale physics, and add more uncertainties in the predictions. The different diffusion processes discussed above usually work together in the real atmosphere, so the entropy may be produced more efficiently than the previous estimations based on a single process. Due to the important effect of turbulent diffusion on energy conversion, the circulation systems with active energy conversion and dissipation have a relatively short time of predictability, especially when the diffusions are highly anisotropic in the atmosphere.
25.10
Uncertainties in physics
One of the interesting question in the ancient philosophies and current sciences is whether there are the limits for us to know and understand the world. This question becomes particularly important as our studies approach the limits of measurement and prediction. It was believed that there was the intrinsic uncertainty in the quantum world, which could not be removed by any idealized measurement without limitation in accuracy. This uncertainty was illustrated by Heisenberg’s uncertainty principle (Halliday and Resnick, 1970): ∆Pz =
n¯h , ∆z
(25.19)
where ∆z is the vertical width of a single long horizontal slit on a vertical plate; Pz is the vertical momentum component of the diffracted electrons after they pass through the slit; ¯h is Planck’s constant, and n the number of minima pare in the diffraction pattern. This equation tells that the maximum vertical momentum component of diffracted electrons increases with reducing the slit width. While, it was explained as the uncertainty principle: The momentum and position cannot be measured simultaneously with unlimited accuracy, as ∆z → 0 leads to ∆Pz → ∞. In fact, ∆z cannot tend to zero in the quantum world, as the time and space are
552
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
also quantized. The diffraction relationship gives ∆z =
nL , sin ϕ
where ϕ is the diffraction angle corresponding to the n-th minima, and L is the wavelength of electron beam. Since | sin ϕ| ≤ 1, we have |∆z| ≥ nL .
(25.20)
If |∆z| → nL, (25.19) gives |∆Pz | → P derived from the de Broglie’s relationship L=h ¯ /P . Especially, when ϕ → ±π/2 and n = 1, the uncertainty in the electronic momentum measured diminishes as the width of slit tends to the wavelength. The uncertainty of Heisenberg was produced by the conflict between the classical Newtonian physics and modern quantum physics. The classical physics uses the standard mathematical analysis and Euclidean geometric definitions of the positions in the time and space coordinates: A position in the time or space is a geometric point without size. A line formed by the points has length but no thickness, and a surface formed by the lines has area but no thickness too. In the classical Newtonian dynamics, a solid body may be viewed as a mass point which has mass but no volume. In fact, the mass point does not exist in the world and there is no the instruments which can make the point measurements. An important consequence of the Euclideam geometric physics is the continuous variations in the mass and energy, as all the physical quantities can be divided into infinitely small elements. When the time and space are quantized, the Euclideam geometric points of time, space and mass become meaningless and uncertain. For example, the ∆z in (25.19) cannot be infinitely small as shown by (25.20), and so ∆Pz cannot be infinitely large. When ∆z tends to the lower limit, ∆Pz tends to the upper limit. The uncertainty occurs as one tries to explain the quantum world with the classical geometric physics, assuming that both the spatial distance and electronic momentum can be infinitely small. The real uncertainties in quantum physics may be produced by the limitations on the detection and measurement, which may not suggest an intrinsic uncertainty of the world, but may lead to the intrinsic uncertainty in our knowledge. An optical or electronic microscope which men made has limited resolutions, and there are interactions between survey instruments and quantum world. An example of the intrinsic uncertainty in the quantum physics is the behavior of individual electron, of which the survey is affected by interaction between the electron and observational instrument. The interference caused by survey instruments occurs also in the classical thermodynamics. For example, a nonequilibrium state of gas may be disturbed by thermometers, so that the original state may not be found. According to the mathematical theory of probability, a random process of particles in a great number may show statistically some equilibrium feature with maximum entropy production. So the random quantum processes which cannot be observed or measured directly may be represented by their statistical effects on the measurable variables through the probability wave function. The intrinsic uncertainties exist in the classical sciences too. The classical thermodynamics does not consider the unknown behaviours of individual molecules, but
25.10. UNCERTAINTIES IN PHYSICS
553
introduces the thermodynamic variables, such as the temperature and pressure, to represent the statistical features of the random molecular motions. The temperature of a monatomic gas is proportional to the mean speed of the molecules, and the pressure is proportional to the number of collisions between the molecules and pressure sensor. The mean molecular speed measured by a thermometer with a size smaller than the mean distance between two adjacent molecules may fluctuate discontinuously with time. The amplitude of fluctuation decreases with increasing the size of thermometer. Thus, the measurements depend on the size of instrument, and may be uncertain at a geometric point of time and space coordinates. This is also the case for the pressure measurement, and so the thermodynamic variables cannot be defined for a single molecule. Since the instruments used in practice are much larger than the distance between two molecules, the scale-dependent feature and uncertainty may not be noticed in the classical thermodynamics. The uncertainty caused by the discrete microscopic particle motions exists also in the large-scale inhomogeneous thermodynamics. While, some other uncertainties related to the meteorological turbulences and numerical truncations in the scaledependent models may be more important for weather predictions. These major uncertainties are different from those in the quantum physics and classical thermodynamics. The quantum processes can be illustrated statistically by the probability wave function, and the thermodynamic relationships based on the measurable thermodynamic variables can be applied successfully to calculate the system changes between equilibrium states. However, the statistical representations of meteorological turbulences or other macroscopic uncertainties cannot be provided accurately by the parameterizations with grid-scale variables, since they are different from a random process and so have no preferred statistical features manifested in the gridscale fields. Therefore, the grid-scale prediction equations are neither perfect nor time-stable, and the weather predictions have a limited prediction time. When all the diffusions and physical processes in the atmosphere may be represented precisely in all scales, the continuous primitive equations may be assumed as the time-stable prediction equations. The predictions with these idealized perfect equations depend on the initial fields and boundary conditions only, and there is no time limit of the prediction. We say, in this case, that the predictions with timestable equations are only an initial data problem. If a small error in the initial field leads to the chaotic prediction after a certain time period, the result is called the chaos which exhibits a feature of the atmosphere. However, the prediction equations used in weather forecasts or numerical simulations are not perfect at all, so the prediction errors may also be produced by the models themselves if without initial data errors. The chaos found in predictions may not really be the feature of the atmosphere but a feature of the model. The studies on atmospheric chaos or predictability based on growth of initial data errors only may have a limited significance, if the errors resulting from other sources may also destroy the predictions. Like his chaos, Lorenz’s available potential energy depends also on initial states only and is independent of process. Although the system energy equation may be derived without approximations as shown in Chapter 7, the choice of a reference state including the lowest state depends nevertheless on process, since the reference state cannot be predicted by the energy equation as
554
25. GRID-SCALE PREDICTION EQUATIONS AND UNCERTAINTIES
discussed in Chapter 9. These theories of available potential energy and chaos are the typical examples of the Lorenz’s meteorology, in which the weather predictions and energy conversions are simplified or idealized mathematically as the initial data problems.
Chapter 26 Examinations of model results 26.1
Introduction
The current numerical models used for weather predictions and atmospheric simulations are the grid-scale prediction models, which include several intrinsic error sources as discussed in the preceding chapter. To improve the predictions and extend the predictable time limit, we may increase the model and data resolutions and incorporate as much as possible the physical processes into the prediction models. But, the intrinsic error sources may not be removed entirely by increasing the resolutions or reducing the measurement errors in the initial fields. Also, meteorologists use frequently the simplified models, which include only a few selected physical processes or assumed mechanisms, to simulate a particular circulation pattern or test a theoretical assumption. To know whether the simulated results are produced by an assumed mechanism, we usually perform the control runs with and without the mechanism included and compare the model outputs with the real fields. It is found that a particular circulation pattern can be simulated by simplified models with different mechanisms which may not be all true. This suggests that an artificial process may not be identified by the control runs using a simple model. The fact that a similar response can be produced by different mechanisms provides a fatal limitation on applications of simplified models. The meteorological turbulences and other subgrid-scale processes may be independent of the grid-scale circulations and fields. However, they are usually represented by the parameterizations with grid-scale data in current numerical models. The parameters included are not given theoretically or by the specific physical or numerical experiments for each of them exclusively. In fact, these parameters are new variables of which the values are selected by comparing the model outputs with the real processes observed, assuming implicitly that the other processes are incorporated correctly. As this assumption may not be true especially for the simplified models, the parameters and parameterizations so determined may not be physically correct and may affect the major mechanisms in the model. When the major mechanisms in a model atmosphere are different from those in the real atmosphere, the numerical simulations may not be compared directly with the observations, and a similarity between a model result and real field does not mean necessarily that the model atmosphere is a good representation of the real atmosphere. The models in which the parameters are determined by model results will be referred to as the result-dependent models in this chapter. In general, these parameters are not constant and may change with climate. The result-dependent models may not be applied to predict the climate changes, as the parameters determined for a past climate may not fit a new climate. For the models which possess error sources or assumed processes, the parameters determined by comparing with the model results should be examined again indepen555
26. EXAMINATIONS OF MODEL RESULTS
556
dently of the model results. It is generally difficult to make the particular physical or numerical tests for each of the parameters, as the processes parameterized may depend on each other and so cannot be studied separately. However, it may be possible for us to find the orders of these items parameterized by scale analysis, and compare them with those in the model atmosphere. For example, we may check the scale of Ekman pumping in the stationary wave model of Charney and Eliassen (1949). In many cases, it is difficult to compare straightforwardly the subgrid-scale processes in a model and real atmosphere. Since the diffusion processes may affect greatly the balance of thermodynamic entropy, we may compare the entropy changes produced by the diffusions in the model and real atmospheres. Johnson (1997) found that the temperature field in a climate model is extremely sensitive to aphysical sources of thermodynamic entropy introduced by spurious numerical dispersion and diffusion, Gibbs oscillations, parameterizations and other factors, as a 10 K bias in the mean temperature can be produced by a small error of 4% in the mean heat addition of an isentropic layer. The thermodynamic entropy equations for the local dry and moist atmospheres have been discussed in Chapters 9 and 11 respectively. We study in this chapter the general balance equation of thermodynamic entropy in open systems. The major features of the entropy variation in the atmosphere will be discussed also. The turbulent entropy law is derived from the universal principle given in Chapter 6, and is independent of the primitive equations. So it may be applied to solve the parameters in the thermal diffusions parameterized. To compare the entropy balances in the model and real atmospheres allows us to simulate more realistically the turbulent diffusions. The turbulent diffusions have an important effect on the predictability, since they may destroy the kinetic energy and the energy sources in the atmosphere as discussed in the preceding chapter. Usually, the diffusions are amplified artificially in numerical models in order to suppress the model instability. As a result, the energy sources for disturbance development are weakened. Thus, the models with exaggerated diffusions are unable to predict the second cycle of disturbance development in the real atmosphere, and the maximum prediction time is comparable with the lifetime of the major disturbances predicted. A realistic simulation of the diffusions may enable us to extend the prediction time to the second cycle. Moreover, since the grid-scale prediction models include the uncertainties provided by the parameterizations for the subgrid-scale processes, the other conservation relationships, such as the energy and mass conservation laws including the moisture balance equation, may become independent of the prediction equations, and so may also be used to examine or solve the parameters included. The prediction models using the balance equation of thermodynamic entropy and the other conservation equations may be more consistent in physics.
26.2
Scientific tests
In the common sense of science, the truthfulness of a theory may be tested by comparing its products with the realities. For a simple process represented by a relationship including a single parameter or coefficient, the tests can be made directly by observations or physical experiments to determine the parameter. The test for a
26.2. SCIENTIFIC TESTS
557
single parameter may be referred to as the simple test. The simple test is reliable since the value in a given process under a certain condition may be determined uniquely by experiments. Many of the earliest physical and chemical laws, such as the Newton’s law of universal gravitation and Boyle’s law, were derived from simple tests. Some parameters are constant, but some change with other conditions. For the latter situation, it is requested to find the dependence of the parameter on the conditions. If the relationship of parameter change cannot be given analytically, more experiments are needed to determine the different values of the parameter used for different circumstances. A quantitative relationship of a physical process depends on definitions of the variables included and may not be unique. A scientific revolution may happen when new relationships are found, which are more simple and precise than the old for the same processes. Some revolutions provide only a better algorithm without changing the basic definitions of variables, but some may change the basic definitions. Since the definitions of variables depend on the methods of measurements and observations, a development in the technology of data measurements may lead to a scientific revolution. Usually, the prediction models of atmosphere include several parameters. The parameters can hardly be determined by theory or simple test, as it is difficult to isolate the individual processes in the atmosphere which may depend on each other. Thus, they are actually the new variables in the prediction models. The current way to determine these parameters is to run the whole model for different sets of the parameters and compare the results with real processes. If the model is principally correct or does not include error sources, the possible solutions of the parameters under a certain condition may be found by the numerical experiments. However, the numerical solutions are not unique in mathematics and may be as many as one wants, since the number of equations used in the model is less than the number of variables and parameters, and the parameters may depend on each other in the ways unknown. If one parameter changes, the similar result may also be obtained by changing other parameters. Thus, it is difficult to choose from all the mathematical solutions the only set of solutions which is physically right. More difficulties may be encountered, since the models include inevitably the intrinsic and technic error sources as discussed in the preceding chapter. The simulation or prediction errors may still be produced even if the right parameters are used in the imperfect models. In general, the scientific truthfulness of a theoretical or model result may be achieved principally or functionally. As Kuhn (1970) pointed out, the theories, such as Ptolemy’s astronomy and phlogiston theory, which are considered to be unscientific today, could also find evidence to prove. For the planet motions, Ptolemy’s predictions were as good as Copernicus’, and the phlogiston theory was employed to design and interpret experiments. Most statistical forecasting models look for the truthfulness in function but not in principle. Some typical circulation patterns, such as the tropospheric blockings and stratospheric sudden warmings, have been simulated by the models including different mechanisms (McHall, 1993). It seems unlikely that these circulations can be related to any of the different physical processes in the real atmosphere. The planetary stationary waves at middle latitudes represented by a study of Charney and Eliassen (1949) relied on conservation of potential vorticity over the boundary layer with amplified Ekman pumping. The
558
26. EXAMINATIONS OF MODEL RESULTS
observed phase distribution may be obtained by choosing the pumping coefficient which is independent of the topography. While, a study of McHall (1993) showed that the stationary waves are responses to the zonally asymmetric surface drag and heating. The real phase distribution is determined by the phase of topography. In the current numerical predictions and simulations, to add and change the uncertain terms is a common way for obtaining the results wanted, while the physics of the modifications is never made clear. The difference between the functional and principle truthfulness may not be obvious usually. Thus, a mechanism which leads to an expected result may be misinterpreted as a true process in the atmosphere. To find this kind of error, it may not be enough to compare a model result with the observations, and some additional tests are necessary. It is generally difficult to perform physically the simple tests for the each parameters included. The tests may also not be made numerically, since the individual effects of these parameters may not be separated in the real atmosphere for the comparisons with numerical simulations. But for some simple processes assumed, a single parameter may be checked approximately by scale analysis. In the example of orographic stationary waves, we may compare the Ekman pumping in the model and real atmospheres. If the stationary waves are produced by the Ekman pumping with a magnitude much larger than that in the real atmosphere, the assumption of Ekman pumping should be suspected. The unknown parameters may also be determined by additional independent relationships. When the physical relationships of these parameters are unknown, we may use the basic laws, such as the energy conservation law and balance equation of thermodynamic entropy derived from the primitive equations, to check the predictions. Theses relationships are not used particularly in prediction models, since they may not be independent of the primitive equations or introduce new variables. However, the energy and entropy balances in a model atmosphere may depend on the parameters included in the prediction model. So these relationships may be applied to examine these parameters. In other words, the parameters determined by model results are physically consistent only if they meet with the additional balance relationships and conservation laws.
26.3
Result-dependent models
One of the largest uncertainty in the numerical predictions is related to meteorological turbulences in the atmosphere. The important effect of turbulent diffusions may be demonstrated by numerical experiments. Most elaborated numerical models include parameterized meteorological turbulences and other subgrid-scale processes. These models draw different pictures by changing the diffusion coefficients only. Changes in these coefficients may be as large as one order of magnitude. For example, the horizontal diffusion damping times in ECMWF numerical weather prediction model range from 45 minutes at T213 (Triangle truncation after wavenumber 213) resolution to 48 hours at T63 (Buizza, 1998). Boville (1984) showed that even a simulation of the climate in the winter stratosphere (where the static stability is higher and the baroclinity is lower than in the troposphere) by the NCAR Community Climate Model is nevertheless very sensitive to the horizontal diffusion operator.
26.3. RESULT-DEPENDENT MODELS
559
The high dependence of numerical solutions on the specification of diffusions can be found in many studies (e.g., Browning and Kreiss, 1989; Laursen and Eliasen 1989; Naughton et al., 1993). Apart from the positive and negative turbulent diffusions, it is also difficult to evaluate the physical and chemical processes in the models using the grid-point data. These subgrid-scale processes may be independent of the grid-scale fields or circulations, and may change from time to time and from place to place in the real atmosphere. Without using the subgrid-scale information and correct physical relationships, the subgrid-scale processes simulated by parameterizations with the grid-scale data may only work as the adjustable switches of the model. The parameters are then justified according to the model output instead of the real physics. In this case, the simulations are, to a certain extent, looking for a functional truthfulness. The quality of prediction depends highly on the circulation pattern and scale, since the effectivity of parameterization depends on these circulation features. There are also the simplified numerical models used for testing a particular mechanism or assumed process. These models may not be compared straightforwardly with the observations, as the physics included is not real or perfect. The defects of a model may not only distort a real process, but also produce a false result which looks similar to a real one. So, a model producing the observed features may not mean that the atmosphere can be represented by the model atmosphere. For example, a deformation field with a weak temperature gradient and open boundaries but no diabatic heating may eventually produce a strong and deep temperature front in the model atmosphere similar to an observed example. However, as discussed in Chapter 24, the frontogenesis in the kinematic process is limited by initial conditions and thermodynamic entropy production in the real atmosphere. Also, some simulations of tropospheric blocking systems and stratospheric sudden warmings were made by adding artificial forcings in simplified numerical models. The similarities of the model results to the observations need not imply that the external forcings are the real causes of these disturbances. Since all the parameters included in a prediction model or simplified experimental model are chosen simultaneously by comparing the model results with the fields to be predicted or simulated, but not by the independent simple tests for each of them, the current numerical procedures are actually the result-dependent procedures influenced by the requirement of producing ‘ realistic’ or expected simulations. Unlike in the simple tests, the similarity between a model result and reality may not ensure that all the parameters in the model must be correct or the model has no error sources, as the similarity may also be achieved functionally. If a reference field is changed arbitrarily, the new targeting field may still be simulated with the same model simply by changing the parameters. So a process may not be real even if it may be reproduced by a model. The models which include adjustable parameters may be more capable than one thought for producing the expected results. A climate model may ‘predict’ global warming and cooling by changing the parameters only. The purely objective numerical procedures cannot be developed, unless we are able to incorporate theoretically the true subgrid-scale processes into the model based on the experimental results of simple tests. It is possible that this perfect numerical model may not be made using the grid-point data as the subgrid-scale
560
26. EXAMINATIONS OF MODEL RESULTS
processes may be independent of the grid-scale circulations. It is obvious that the result-dependent models and functional predictions can hardly be time stable, and so have a time limit for the predictions acceptable. Since the parameters are not constant in general, they, when determined under certain conditions, may not be applied for a different climate. Although the climate changes are also controlled by the primitive equations same as those used for weather predictions, the parameters included may change significantly with climate. Thus, the long-term forecasts or climate predictions depend on the correct predictions of the parameters, and are much more difficult than the short-term weather forecasts. A model, which has been tested for some particular circulation patterns during a previous time period, may predict the weather changes within the specific climate mod, but may not predict the climate changes in the future, unless the physics of climate change on all scales has been incorporated accurately.
26.4
Thermodynamic entropy balance
The previous discussions show that to examine whether the model mechanism is true or not, we should not only compare a few aspects of model results with observations. The studies of energy conversions and frontogenesis in the previous chapters suggest that a model made from particular assumptions and simplifications should be examined by the thermodynamic entropy budget, as the processes in the atmosphere are controlled by the classical entropy law and turbulent entropy law. The classical entropy law may be derived from the energy equation at nonequilibrium states for ideal gases as shown in Chapter 4. But it is independent of the primitive equations as the derivation applies the features of molecular diffusion discussed in Chapter 3. The molecular diffusions do not follow Newton’s laws. Moreover, the turbulent entropy law used for the quasi-adiabatic process may be derived from the universal principle introduced in Chapter 6. Although the mathematical expression is similar to that of classical entropy law, the turbulent entropy law may be independent of the classical entropy law. Because increases of turbulent entropy and classical thermodynamic entropy are caused by different mechanisms related to turbulent and molecular diffusions respectively. These two laws together may be used to examine the atmospheric processes of which the molecular diffusions may or may not be ignored. The primitive equations form a closed set only if the molecular diffusions are ignored. When the diffusions cannot be represented by the thermodynamic variables and parcel velocity including their gradients, the diffusion velocity is a new variable and the primitive equation including molecular diffusions are not closed. The physical relationships of the diffusion velocities in various transport processes cannot be calculated from Newton’s second law, but may be derived from the studies in Chapter 3. These complex relationships lead to the second law of thermodynamics. Thus, we may use the classical entropy law (4.20) for ideal gases to form a new complete set of primitive equations. This law was not used before, as the molecular diffusions were ignored in previous studies, and the entropy law was not given explicitly in terms of diffusion velocity as shown by (4.20). When molecular diffusions are ignored, we need not use the entropy law for predictions, but this law may still
26.4. THERMODYNAMIC ENTROPY BALANCE
561
be used to estimate the strength of molecular diffusions according to (4.25) and (7.7) for a single parcel or (7.13) for an inhomogeneous system. Since the diffusions provide an error source for the primitive equations without the diffusions, this law may also be applied to estimate the strength of this error source. When the discrete datasets and numerical procedures are used for the primitive equations, these equations become the grid-scale prediction equations which include the subgrid-scale fields also. The grid-scale equations are not closed since the subgrid-scale processes are generally independent of the grid-scale fields and circulations. Without the subgrid-scale information and corresponding physical relationships, the subgrid-scale processes are usually represented approximately by the parameterizations using grid-scale data. These parameterizations include the parameters, which may be viewed as new variables and so should be determined by the physical relationships independent of the prediction equations. However, it is generally difficult to provide all the relationships for the meteorological turbulences and other subgrid-scale processes in terms of grid-scale variables. Since the molecular and turbulent diffusions and other processes may affect greatly the thermodynamic entropy balance in the grid scale, we may use the balance equation of thermodynamic entropy to solve the parameters included in the balance equation. It is discussed in Chapter 6 that the classical entropy law may not be applied for the processes without molecular diffusions, since the classical thermodynamic entropy is conserved in the processes. Thus, we may use the turbulent entropy law derived from the universal principle as a additional relationship for the prediction models. Changes of turbulent entropy may also include the contribution of molecular diffusions. In fact, the thermodynamic entropy calculated using grid-point data may give only the turbulent entropy. Use of the turbulent entropy law allows us to explain the irreversibility of meteorological processes in the atmosphere. Since this law is independent of the other prediction equations, the entropy balance in the model atmosphere simulated without using this law may be significantly different from that in the real atmosphere. When the production of turbulent entropy is underestimated, the false frontogenesis or disturbance development may turn out which may lead to the model instability. To suppress the model instability, the entropy production is usually overestimated in numerical models. This means that the turbulent diffusions are amplified in the model atmosphere, which in turn destroy more kinetic energy and its sources and suppress the development of new disturbances. As a result, the successful predictions can hardly extend to the next cycle of disturbance developments, and so the current prediction time is limited by the lifetime of the major disturbances. For example, it is about 5 days less than a week for the extratropical precesses, but may be longer for the steady circulation patterns such as the blocking systems. The prediction time of the models with amplified turbulent diffusions is even shorter for the tropical connectivities. The thermodynamic entropy variation in mass M within the time period ∆t may be calculated by (4.40), or
∆S = M
∆t
1 dq dt dm + ∆Si , T dt
(26.1)
where dq/dt represents the heat exchange rate of a parcel with unit mass produced
26. EXAMINATIONS OF MODEL RESULTS
562
by radiative heating, and latent heating but no molecular heat conduction, and ∆Si is the internal thermodynamic entropy produced by diffusion kinetic energy dissipation, heat conduction and heat diffusion as shown by (4.41). If the entropy changes are evaluated with grid-point data, the internal entropy production includes the contribution of subgrid-scale processes. In some other studies (e.g., Peixoto and Oort, 1992), the internal entropy production was also considered as a heating, as the kinetic energy dissipated is converted into heat energy. However, it is discussed in Chapter 4 that the entropy may be produced not only by energy conversion, but also by changing the system disorderliness. When the thermodynamic gradients are destroyed by diffusions, the entropy increases but the heat energy may not change. Also, the entropy change represented by dQ/T may only be applied for reversible processes. It is generally impossible to estimate directly the internal entropy production in the diffusion processes with grid-scale data. As changes of thermodynamic entropy are independent of process for provided initial and reference states in equilibrium, it may be evaluated by choosing inviscid reversible processes to connect these equilibrium states. The reversible circulation pattern may be given by (8.1), which is constrained to be geostrophic balance and hydrostatic equilibrium. For the assumed reversible processes, we have
∆Srev =
∆t
M
∂srev + vrev · ∇srev dt dm , ∂t
where vrev is the velocity in assumed reversible process, and may be different from that in the real process. Applying (4.28) yields
∆Srev = cp
ln M
θr dm + cp θ0
M
∆t
vrev · ∇ ln θrev dt dm ,
where the subscripts 0 and r indicate the initial and reference states respectively, and θrev is the potential temperature in the reversible process assumed. Since ∆Srev = ∆S, (26.1) gives
∆Si = cp
θr ln dm+cp θ0 M
M
∆t
vrev ·∇ ln θrev dt dm−
M
∆t
1 ∂q dt dm . (26.2) T ∂t
This is the entropy balance equation for an open atmosphere, and may be derived directly from (4.51). In practice, the slow meteorological processes in the atmosphere may be assumed as the quasi-equilibrium processes, and so the reversible variables may be replaced approximately by the observed data on a grid-scale. This entropy variation calculated with grid-point data is the turbulent entropy on the grid scale. According to the turbulent entropy law discussed in Chapter 6, there must be ∆Si ≥ 0. The internal entropy is conserved only for reversible processes.
26.5
Partition of thermodynamic entropy change
The balance equation of thermodynamic entropy (26.2) may be rewritten as ∆Si = ∆Sld − ∆Se − ∆She
(26.3)
26.5. PARTITION OF THERMODYNAMIC ENTROPY CHANGE
for the dry atmosphere. Here, cp ∆Sld = g
ps
ln A pt
θr dpdA θ0
563
(26.4)
is the local variation in the dry atmosphere, and is called for convenience the change of dry thermodynamic entropy. It is the same as (8.3) used for calculating the kinetic energy generation in the dry atmosphere. When water vapor condensation takes place in the moist atmosphere, the dry entropy change may include the contribution of latent heat. The second term on the right-hand side of (26.4) gives ps cp v · ∇ ln θdt dpdA , ∆Se = − g A pt ∆t where v is the three-dimensional velocity in pressure coordinates. It is the entropy variation produced by mass exchange and is referred to as the mass exchange entropy. The subscript ‘rev’ is omitted, when the meteorological processes are assumed quasi-equilibrium. The integral may be rewritten as v · ∇ ln θ = ∇ · (v ln θ) − ln θ∇ · v . The last term disappears since ∇ · v = 0 in pressure coordinates. Moreover, the Gauss equation gives ps A pt
∇ · (v ln θ)dpdA =
∆t ✵
vn ln θ d✵ dt ,
where ✵ signifies the outer surface of the mass M in pressure coordinates, and vn is the velocity normal to the surface with the positive direction outward. Now, the mass exchange entropy gives cp vn ln θ d✵ dt , (26.5) ∆Se = − g ∆t ✵ If the mean potential temperatures averaged over the input and output air masses Min and Mout are denoted by θ¯in and θ¯out respectively, we gain (26.6) ∆Se = cp (Min ln θ¯in − Mout ln θ¯out ) . The net entropy change depends on the comparison of the entropy input and output. The last term in (26.4) reads ∆She
1 = g
ps A pt
∆t
1 ∂q dt dpdA . T ∂t
It is the entropy variation produced by diabatic heating and may be called the heat exchange entropy. If the mean temperature averaged over the domain of heat input is T in and the mean temperature over the domain of heat output is T out , this equation gives ps 1 ∂qout 1 1 ∂qin + dt dpdA ∆She = g A pt ∆t T in ∂t T out ∂t ps ps 1 1 1 δqin dpdA + δqout dpdA = g T in A pt T out A pt ∆qin ∆qout = Min + Mout , T in T out
26. EXAMINATIONS OF MODEL RESULTS
564
where ∆qin and ∆q out are the mean heat input and output for unit mass over the domains Min and Mout respectively. The last equation may be replaced by ∆She =
Qin Qout + , T in T out
(26.7)
in which Qin and Qout indicate the total heat input and output, respectively. We have Qin > 0 and Qout < 0. In a long-term heat balance, we may write Qin + Qout = 0 and 1 1 − . ∆She = Qin T in T out This equation shows that the entropy is destroyed if heating occurs in warm regions and cooling in cold regions in heat balance. Although the total energy does not change in this situation, the ability of energy conversion is increased by the negative thermodynamic entropy input. The diabatic heating in the atmosphere includes mainly the radiative heating, heat conduction and latent heating. The contribution of radiative heating may be indicated by ∆Shr and called the radiation entropy. Calculations of radiation entropy are referred to the studies of Rosen (1954) and Li at al. (1994). The contribution of heat conduction ∆Shc is called the heat conduction entropy in Chapter 4 and is calculated by integrating (4.32) with time and mass. The latent heating resulting from water vapor condensation or evaporation gives dql = −Lc dw , where w denotes the water vapor mixing ratio. The produced thermodynamic entropy reads ps 1 Lc dw dt dpdA , ∆Shl = − g A pt ∆t Tc dt where Tc is the condensation temperature. It may be referred to as the latent heat entropy. If Lc is constants and Tc is independent of time, we gain ∆Shl
Lc = g
ps w0 − wr A pt
Tc
dpdA .
(26.8)
Now, the heat exchange entropy gives ∆She = ∆Shr + ∆Shl + ∆Shc . It is discussed in Chapter 4 that the heat conduction can only create the entropy. Thus, the heat conduction entropy is put together with the internal thermodynamic entropy, and the entropy balance equation (26.3) is rewritten as ∆(Si + Shc ) = ∆Sls − ∆Se − ∆Shr , where cp ∆Sls = g
ps A pt
θr Lc (w0 − wr ) ln − θ0 cp Tc
(26.9)
dp dA
is the local variation in the moist atmosphere, called for convenience the change of moist thermodynamic entropy. The last equation is also (11.4) used for evaluating
26.6. EXAMINATION OF PARAMETERS
565
the thermodynamic entropy production in the moist processes of energy conversion, and may be rewritten symbolically as ∆Sls = ∆Sld − ∆Shl . The change of moist thermodynamic entropy equals the change of dry thermodynamic entropy minus the latent heat entropy. As discussed in Chapter 11, the contribution of latent heat release to the entropy change is given not just by the last term in this equation, since the temperature change in ∆Sld includes also the effect of latent heat. There are some other minor sources of internal thermodynamic entropy in the atmosphere which are not considered here, such as the boundary friction including the frictional dissipation resulting from falling precipitation (Pauluis and Held, 2000). The climatological data of these components of thermodynamic entropy productions for the whole atmosphere are provided by some authors (Peixoto at al., 1991; Goody, 2000) using the observational data or experimental data. As commented by Goody (2000) that the accuracy of these data needs improved. Also, we need more information for using these data, such as the regional features, seasonal changes and the dependence on circulation patterns. Comparing the entropy balance equation (26.9) with the continuity equation of mass finds that the local change of thermodynamic entropy may not be balanced by transport only, so the balance equation is not a continuity equation of the entropy. In this sense, we say that the entropy is not entirely transportable. The entropy was considered as transportable (e.g., Gyftopoulos and Beretta, 1991), as the internal entropy production may be ignored if compared with the contribution of transport in many situations.
26.6
Examination of parameters
When the diffusions in the heat flux equation are parameterized in a numerical model by DTd + DTt = F (bj ) , where bj (j = 1, 2, · · ·) are parameters, (4.41) is replaced by
∆Si = M
∆t
F (bj ) dt dm , T
and the balance equation (26.9) becomes
M
∆t
F (bj ) dt dm = ∆Sls − ∆Se − ∆Shr . T
(26.10)
The right-hand side may be evaluated with the initial and predicted fields, and is greater than zero, that is M
∆t
F (bj ) dt dm ≥ 0 . T
26. EXAMINATIONS OF MODEL RESULTS
566
If we have the climatological data of the three components of thermodynamic entropy production, the calculations may be compared with the real data. Thus, this equation and the entropy components may be used to examine the parameters included. The balance equation of thermodynamic entropy (26.2) may also be given at each grid point, that is ∆t
in which
θr F (bj ) dt = cp ln + cp T θ0
∆t
v · ∇ ln θdt −
∆t
1 dq dt , T dt
(26.11)
F (bj ) dt = ∆si T ∆t is the change of internal thermodynamic entropy for unit mass at a grid point. This local balance equation may be applied to determine the parameters at the grid points. The parameters so determined may change with space and time. If the gradient in grid-scale temperature field is increased by negative diffusions, ∆si calculated with grid-point data may be negative at a grid point, as the negative entropy flux on a subgrid scale may not be accounted with the grid-point data. If the negative diffusions cause the energy conversions and produce a large amount of thermodynamic entropy, the ∆si may be significantly greater than that produced by grid-scale circulations only. These features may be applied to examine the data and model resolutions selected. When the grid scale is small enough so that no strong negative diffusions are included in the subgrid-scale circulations, we have ∆si ≥ 0 and the internal entropy productions are comparable with observations at each point. The entropy law and climatological data may also be used to solve the parameters bj at the grid points. If the thermal diffusions are parameterized by a Laplace operator as in many current numerical models, this law may not be satisfied. The negative entropy production may also be produced falsely by meteorological noises. When the maximum negative departure of potential temperature from the grid-scale field caused by a noise occurs near a grid point, the local entropy change given by the first term on the right-hand side of (26.2) may be negative. The local cooling may result from the subgrid-scale heat transport which cannot be evaluated using the grid-scale data. Thus, the local entropy destruction cannot be balanced by the second term on the right-hand side of (26.2), and the internal entropy is destroyed in the model. The heat energy missing in the local region is then transferred falsely into kinetic energy to remain energy conservation in the model atmosphere. This error may not be found by examining the energy balance, but may be found using the entropy balance equation. When an internal entropy production at a grid point is negative or far away from the statistical value, it may be corrected at the point without affecting the balances at other points. The correction may be made by interpolation in the field or extrapolation in the time series or by other methods to filter out the noises. If the unrealistic entropy productions are found at many points, we may consider to increase the data and model resolutions in order to reduce the effects of negative turbulences. To solve the other parameters, we need more physical relationships on the grid scale. The local energy equation, such as (7.10), is derived from the prediction
26.6. EXAMINATION OF PARAMETERS
567
equations. If there are no errors in the model, the conservation law of energy is satisfied automatically for the predictions at each time step. In fact, this may not be true as models possess various intrinsic and technic error sources. In this case, the energy equation becomes independent of the prediction equations for the model atmosphere. Thus, it may also be used to examine or solve the parameters included in the expression of kinetic energy dissipation. The change of local energy is given by
∂p vp · ∇(kp + φ + ψ)dt + α dt + ∆(kp + ψ + q) = − ∆t ∆t ∂t
∆t
F (bj )dt −
∆t
G(ai )dt (26.12)
in Cartesian coordinates, where (i = 1, 2, · · ·)
G(ai ) = Dvd + Dvt ,
is the parameterization of kinetic energy diffusions including parameters ai . The turbulent kinetic energy dissipation may be negative if the subgrid-scale processes include the negative diffusions which convert subgrid-scale available enthalpy or kinetic energy to grid-scale kinetic energy. If the grid-scale is fine enough and so no negative diffusions are included in the meteorological turbulences, we have ∆t
G(ai )dt ≥ 0
at each grid point, and the diffusions calculated are comparable with observations. The previous inequality and real data of kinetic energy diffusions may also be used to select the model resolution and check the diffusion terms parameterized in the model. If the resolution is fine enough, the energy equation (26.12) can be used to solve the parameters ai . To add the entropy balance equation and energy conservation law into the prediction equation set makes the prediction model more physically consistent. These additional relationships cannot be used to confirm that the subgrid-scale processes parameterized are physically correct. They may also not remove or correct principally the error sources in the parameterizations. However, they may reduce the error effects and give us a physically consistent model result. When the prediction errors destroy these relationships, they may be found immediately, and the predictions may be adjusted at the right time by correcting the parameters included. The predictions may be checked further by the integrated entropy law (26.10) and the system energy equation (7.22). The letter is identical to the diffusion equation like (7.13) given by M
∆t
G(ai ) dt dm − T
M
∆t
F (bj ) dt dm = 0 . T
This equation gives the relation between the parameters in different processes. We may also use some other conservation equations to test the parameters in a model. When these conservations depend on subgrid-scale processes, they may be used to solve the parameters included, since the parameterizations may make the conservation relationships independent of the prediction equations. The latent heat entropy ∆Shl included in ∆Sls may also have a parameter, which can be determined
26. EXAMINATIONS OF MODEL RESULTS
568
by using the continuity equation of water vapor. Speaking strictly, the subgridscale circulations may affect local mass balance in the atmosphere. If this effect is parameterized also, the mass conservation equation can be used to calculate the parameter included. Also, the geopotential entropy law introduced in Chapter 8 may be used to examine the processes in a model. If a physically consistent model with realistic thermodynamic entropy balance is also successful for the predictions, it may be expected that the prediction time may extend to the next cycle of disturbance development.
26.7
Features of thermodynamic entropy variation
26.7.1
The entropy change in a system
The thermodynamic entropy production in the dry atmosphere derived from (26.3) gives ∆Sld = ∆She + ∆Se + ∆Si . It may include the contribution of latent heat in the moist atmosphere. In an isolated system including dry air and water vapor or condensed water, we have ∆Shr = ∆Se = 0, and so ∆Sld − ∆Shl = ∆Si . It follows that ∆Sls = ∆Si . When no mass exchange and radiative heating take place, the internal thermodynamic entropy production is identical to the entropy variation in the system. Since the internal entropy can only be increased by diffusions, that is ∆Si ≥ 0, we have ∆Sls ≥ 0 . It tells that the moist thermodynamic entropy cannot be destroyed in the isolated system. When the entropy is conserved in a long-term, we have ∆Sls = 0, that is ∆She + ∆Se + ∆Si = 0 . Since the internal entropy production is positive, the sum of the first two terms must be negative, or ∆She + ∆Se ≤ 0 . If no mass exchanges across the boundary, we have ∆Se = 0 and so ∆She ≤ 0 . Equation (26.7) shows that the negative thermodynamic entropy may be produced by diabatic heating in the warm places and cooling in the cold places. In general, the income of long wave radiation in the atmosphere decreases upward and poleward in the troposphere, and so reduces the thermodynamic entropy of the troposphere.
26.7. FEATURES OF THERMODYNAMIC ENTROPY VARIATION
569
Another important diabatic process in the atmosphere is latent heating or cooling. If the mean condensation temperature averaged over condensed water vapor Mvc is T c and is T e over evaporated water Mve , the latent heat entropy (26.8) gives
∆Shl = Lc
Mvc Mve − Tc Te
It follows that
∆Shl = Lc Mvex
1 1 − Tc Te
.
in a long-term water vapor balance or Mvc = Mve = Mvex . The entropy increases if evaporation takes place in the regions warmer than the regions of condensation. This is the general situation, since evaporation occurs mostly on the surface and the water vapor condensation occurs in the atmosphere aloft. When the dry and moist available enthalpy is studied in Chapters 9-12, only the energy conversion in the isolated atmosphere is considered. The entropy balance equation may be applied to study the energy conversion in open systems. Kinetic energy created in an open system may be greater than the maximum available enthalpy, if there is negative thermodynamic entropy input into the system such that the entropy in a local system decreases or ∆Sls < 0. In this situation, we have ∆She + ∆Se < −∆Si . It means that the thermodynamic entropy destroyed by heat and mass exchanges must exceed a certain limit. The entropy destruction may be produced spuriously in a numerical model to cause the model instability, when the diffusions are not incorporated correctly.
26.7.2
Change by dry air exchange
Equation (26.6) tells that thermodynamic entropy may be destroyed by mass exchange in a local system, if the input air possesses a lower potential temperature than the output air. When the mass is conserved in a system, we may use Mout = Min = Mex for (26.6), giving θ¯in ∆Sea = cp Mex ln ¯ , θout in which ∆Sea does not include the exchange of moisture in the atmosphere caused by the phase transitions of water, and may be referred to as the dry air exchange entropy. For example, Fig.22.3 shows that the environmental air with relatively low thermodynamic entropy moves into a sever storm from the bottom, while the outflow near the top takes the high entropy away. This vertical circulation reduces the entropy and increases the energy generation in the storm. Owing to the entropy output also, the sever storms may have a lifetime longer than the airmass storms. If the entropy variation is produced by the dry air exchange, we have ∆Sld = ∆Sea .
26. EXAMINATIONS OF MODEL RESULTS
570
p
O
✻ ✏ ✏ ✏ ✏ ✏✏ ✏✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏v ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✛ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏✲ ✏ t ✏ ✏ ✏ ✏ ✏ ✏✏ ✏✏
y
Y
Figure 26.1: Local change of potential temperature profile produced by mass displacement, solid curves are the initial field and dashed the final field.
Assume that the barotropic and baroclinic atmospheres meet together at y = Y as shown by the solid lines in Fig.26.1. The two air masses have the same constant lapse rate of temperature Γ, while the baroclinic atmosphere has a constant horizontal temperature gradient Ty . The potential temperature profiles are represented by
θ0 = T0
p ps
γ
and
pθ p
p θ0 = [T0 + Ty (y − Y )] ps
κ
(y ≤ Y )
, γ
pθ p
κ
,
(y ≥ Y )
at the initial time t = 0. The barotropic atmosphere is in the domain of 0 ≤ x ≤ X, 0 ≤ y ≤ Y and pt ≤ p ≤ ps at the initial time, while the atmosphere to the north is baroclinic. If the atmospheres move southward at a constant velocity v(< 0), the local air column is replaced entirely by the baroclinic air mass after the time ∆t = −Y /v, and the thermal structure becomes
p θr = (T0 + Ty y) ps
γ
pθ p
κ
,
(y ≥ 0) .
This new profile is plotted by the dashed lines in Fig.26.1. The dry thermodynamic entropy change is evaluated from
∆Sld = = = =
cp Y X ps θr ln dpdxdy g 0 0 pt θ0 cp X(ps − pt ) Y T0 + Ty y ln dy g T0 0 cp X(ps − pt ) [(T0 + Ty Y ) ln(T0 + Ty Y ) − T0 ln T0 − Ty Y − Ty Y ln T0 ] gTy Ty cp X(ps − pt ) (T0 + Ty Y ) ln 1 + Y − Ty Y . gTy T0
26.7. FEATURES OF THERMODYNAMIC ENTROPY VARIATION
571
The entropy change produced by dry air exchange gives ∆Sea
cp = g
derived using (26.5), where
∆t ps X 0
v(ln θs − ln θn ) dxdp dt
pt
0
p γ pθ κ , (y = Y ) , θn = (T0 − Ty vt) ps p is the potential temperature at the northern boundary, and γ κ p pθ , (y = 0) θs = T0 ps p is the potential temperature at the southern boundary which does not change with time in the process. Using these expressions we gain
∆Sea = = +
cp Xv(ps − pt ) ∆t T0 dt ln g T0 − Ty vt 0 cp X(ps − pt ) [(T0 − Ty v∆t) ln(T0 − Ty v∆t) − T0 ln T0 gTy Ty v∆t + Ty v∆t ln T0 ]
Applying Y = −v∆t yields cp X(ps − pt ) Ty (T0 + Ty Y ) ln 1 + Y − Ty Y . ∆Sea = gTy T0 It is identical to the dry entropy change ∆Sld in the local atmosphere derived earlier. If we write ∆θ¯ θ¯in , ln ¯ = ln 1 + ¯ θout θout where ∆θ¯ = θ¯in − θ¯out is the potential temperature difference between the incoming and outgoing air at the northern and southern boundaries respectively. From x2 + ··· , 2 the dry air exchange entropy created by dry air advection varies almost linearly with the potential temperature difference. When the temperature decreases northward in the baroclinic atmosphere as in the case shown by Fig.26.1, the input flow brings low entropy across the northern boundary into the region, and the output flow takes high entropy away from the southern boundary. So, the local entropy in the column decreases. The reverse occurs when the temperature gradient is reversed. The tropical and subtropical atmospheres are similar to the initial field shown in Fig.26.1. When the mean meridional flows move from extratropical regions into the ITCZ (Intertropical Convergence Zone) in the lower troposphere, thermodynamic entropy is reduced in the lower part of Hadley cell. The disturbances in various scales may be active therein, especially when the available moist enthalpy is high or a baroclinic system approaches. While, the mean meridional circulation may increase the local entropy in the upper part of Hadley cell, and so the efficiency of energy conversion may be reduced there. ln(1 + x) ≈ x −
26. EXAMINATIONS OF MODEL RESULTS
572
26.7.3
Change by moisture exchange
When the mass balance is remained in the whole atmosphere, ∆Se in the atmosphere (which does not include liquid water and ice) may still be produced by incoming and outgoing of water vapor in evaporation, condensation and sublimation across the surfaces of the Earth and liquid water or ice in the atmosphere, that is ∆Se = ∆Sea + ∆Sev . The last term is different from the latent heat entropy, and is evaluated from (26.5), or t cp vdn ln θv d✵ dt ∆Sev = − v g 0 ✵ in pressure coordinates. It may be called the moisture exchange entropy for convenience. Here vdn is the inward diffusion velocity of water vapor normal to water surface ✵; cpv is the specific heat capacity of water vapor at constant pressure, and θv is the vapor potential temperature given by
θv = T
pvθ pv
κv
,
κv =
Rv cpv
where pv is the partial pressure of water vapor which can be evaluated from (11.32), or pv w = 0.622 ; p pvθ is the reference partial pressure of water vapor, and Rv is the gas constant of water vapor. Since ∆Sev is a part of the entropy change in the atmosphere which does not include liquid water and ice, the direction of vdn is toward the water or ice and is also outward from the atmosphere. If the mean condensation potential temperature averaged over condensed water vapor Mvc is denoted by θ¯vc , and the mean condensation potential temperature averaged over evaporated water Mve is denoted by θ¯ve , we obtain ∆Sev = cpv (Mve ln θ¯ve − Mvc ln θ¯vc ) . In the long run, the amount of water vapor remains constant in a local region, that is Mvc = Mve = Mvex . Thus, the moisture exchange entropy gives θ¯ve ∆Sev = cpv Mvex ln ¯ . θvc Since the evaporation takes place mostly on the Earth’s surface, and the condensation takes place mostly in the atmosphere aloft, we have θ¯ve < θ¯vc in general and so the hydrocycle reduces the entropy in the atmosphere. This effect is opposite to that of the latent heating discussed earlier. The total contribution by the moist processes shows
∆Sw = ∆Shl + ∆Sev = Mvex Lc
1 1 − Tc Te
θ¯ve + cpv ln ¯ θvc
.
26.7. FEATURES OF THERMODYNAMIC ENTROPY VARIATION
573
in which Tc and Te are the condensation temperature and evaporation temperature, respectively. Usually, the effect of moisture exchange is larger than that of latent heating in the moist processes or |∆Shl | < |∆Sev |. The moist processes destroy thermodynamic entropy in the climatological mean and so increase the ability of energy conversion. The effect of moisture exchange caused by evaporation may be achieved also by concentration of moisture flux in a local precipitation system. Thus, the precipitation may eat the entropy in the system and increase the kinetic energy created. For the disturbances supported by moist energy source such as the hurricanes or typhoons, they may not die as long as the air and moisture exchanges take place continuously. While, the extratropical baroclinic disturbances cannot live for long, since the thermodynamic entropy increased by reducing the baroclinity cannot be removed by the system circulations. Now, the balance equation of thermodynamic entropy (26.9) is rewritten as ∆Sls = ∆Sea + ∆Sev + ∆Shr + ∆Si . The left-hand side is the moist entropy change in a local open system which include the dry air and moisture but not the liquid water and ice. In the isolated system, we have ∆Sea = ∆Sev = ∆Shr = 0 and so the moist entropy in the local system cannot be destroyed. But in an open system, the moist entropy change produced by radiation, dry air and moisture exchanges may be negative in the local atmosphere, so that more kinetic energy may be created by the negative entropy input.
Appendix A Thermopotential energy of gases A.1
Thermopotential energy
In general, thermodynamic potential energy of substance can be calculated by adding the intermolecular potential of each pair of molecule (Mayer, 1937). It is also called thermopotential energy in this study. The main difficulties for us to calculate precisely the thermopotential energy include: i) The precise expressions of intermolecular forces or potentials are unknown, and ii) there are mathematical difficulties in calculating the thermopotential energy related to intermolecular forces. In the classical thermodynamics, the calculations for a substance of limited volume is made by integrating the intermolecular potential assumed over an infinite space. The obtained van der Waals thermopotential energy of a mole substance gives (Eyring et al., 1982; Mayer and Mayer, 1963; Fowler, 1966) 16π 2 3 N εσ (A.1) Π=− 3V A for the reverse sixth-power hard-sphere intermolecular potential φij =
∞,
−4ε σ lij
6
lij < σ , lij ≥ σ
.
(A.2)
Here, V is the volume; NA is Avogadro’s number; ε is the depth of potential well; σ is the effective diameter of molecule, and lij is the distance from i-th molecule to j-th molecule. Zhu (2000) introduced a new algorithm to evaluate the thermopotential energy of a substance with a limited volume. The major part of this chapter may be referred to his work. For a provided intermolecular potential φij between two molecules i and j, thermopotential energy of the gas including N molecules can be evaluated from Π=
N N 1 φij . 2 j=1 i=j
(A.3)
The factor of 1/2 is applied since the sums in the previous equation are evaluated for each molecule, but the intermolecular potential is between two molecules. The sum with respect to i in (A.3) gives the intermolecular potential between molecule j and all other molecules. It may be viewed as the thermopotential energy of this molecule. The next sum for j gives the thermopotential energy of all molecules in the gas. If the volume V is divided into M elements, each element δV contains nδV molecules in equilibrium, where n = N/V is the number density. The element is sufficiently small so that the distances from each molecule inside to a molecule outside may be considered equal. In this situation, thermopotential energy of molecule j is given by φj =
N
φij = n
i=j
M k=1
574
φjk δVk ,
A.2. ASSUMED HARD-SPHERE POTENTIAL
575
in which φjk is the intermolecular potential between molecule j and another molecule at the center of δVk . The number density is taken as constant, as the microscale density fluctuation and gradient are ignored in equilibrium. Similarly, the next addition for the thermopotential energy of each molecule can be made from N
φj =
j=1
M
Πk ,
k=1
where Πk = φj (nδVk ) is the thermopotential energy of element k. Thus, we gain Π=
M M n2 δVk φjk δVk . 2 k=1 k=1
It is the same as (A.3) if δVk includes one molecule only, as in this case we have δVk =
1 V = , n n
φjk = φij and M = n. When the volume of each element is infinitely small, this equation is replaced by Π = n2
dV V
1 2
V
φjk dV
.
(A.4)
The inner integration gives the thermopotential energy at a geometric point in the substance, and the outer integration accounts the thermopotential energy of the substance.
A.2
Assumed hard-sphere potential
There are different assumptions of intermolecular potentials. To give a simple example for a substance including identical molecules, we assume a central force potential in the form β lij > σ , φij = − m , lij where β is a constant. In general, we have the symmetric relationship φij = φji as lij = lji . The expressions of many other intermolecular potentials assumed, such as the softsphere potential, Sutherland potential (or hard-sphere potential), Lennard-Jones potential (called also the Bireciprocal or Mie potential) and square-well potential
A. THERMOPOTENTIAL ENERGY OF GASES
576
are composed of the terms with reverse powers of intermolecular distance (Mason and Spurling , 1969). If thermopotential energy is additive, thermopotential energy of molecule i with respect to molecule j may be defined as φi =
φij . 2
So, thermopotential energy of the two-molecule system is Π = φi + φj = φij . Suppose there are three identical molecules form a line in a polar coordinates with the origin at the middle of the first and third molecule. The potential energy of the second is φ12 + φ23 2 1 1 β − , = − 2 (r1 + r2 )m (r3 − r2 )m
φ2 =
where ri (i = 1, 2, 3) are the distance coordinates of the three molecules respectively. The extremal value for provided r1 and r3 can be calculated from ∂φ2 =0 ∂r2 or
1 1 − =0, m+1 (r + r2 ) (r − r2 )m+1
in which, we have used r1 = r3 = r. It gives r2 = 0. Since m(m + 1) ∂ 2 φ2 =− β<0 2 r m+2 ∂r2 at r2 = 0 for β > 0, the potential is the weakest if the second molecule sits at the origin. In this situation, total thermopotential energy of the one-dimensional system is the lowest for provided maximum distance between these molecules. If this consequence can be extended to a three-dimensional system including more molecules, we see that the system at a constant volume possesses the lowest potential when it is in equilibrium. This implies that the thermopotential energy depends on the disorderliness of molecular distribution as well as the volume. When a gas changes from a nonequilibrium state to equilibrium at a constant volume, the thermopotential energy decreases and the heat energy increases in isolation. Since the disorderliness can be measured by thermodynamic entropy, the thermopotential energy and heat energy depend on thermodynamic entropy in nonequilibrium states. As a system tends to reduce its potential, the processes of getting to the equilibrium state is irreversible in isolation. The intermolecular force between molecules i and j is evaluated from fij = −
∂φij δij , ∂rij
A.3. EXAMPLE OF REVERSE SIXTH-POWER POTENTIAL
577
where rij is a vector from molecule i to molecule j, and
δij =
1 for molecule i −1 for molecule j
.
The positive force is along the direction of rij . When the second molecule in the three-molecule system sits at the equilibrium position with r2 = 0, the net force on it is zero. It does not mean that this molecule can be free of intermolecular forces or has no thermopotential energy. The equilibrium position is a mean position of a moving molecule in a fluid. As soon as the molecule departs from the equilibrium position, it is forced to move towards another molecule until collide with it. So, the forces make the molecules move ceaselessly towards the equilibrium positions, but not stay at the positions. The most possible distribution of the time-averaged intermolecular distance between two neighboring molecules at an equilibrium state is different from that of molecular speed against molecular number. According to the mathematical statistics (Sveshnikov, 1968; Pugachev, 1984), the distribution of a variable, which possesses maximum entropy for a given variance in an unlimited region, is the normal distribution. An example is the Maxwellian distribution of molecular speed. This distribution may be used as a weighting function for integrating the properties related to molecular speed. While, the uniform distribution of a variable with limited values in a bounded region, such as the mean intermolecular distance in a substance with a provided volume, possesses maximum entropy. No weighting function is requested when the properties associated with a uniformly distributed variable are integrated. When a gas changes volume, the mean distance between molecules changes also and so does the thermopotential energy. The mechanic work produced by intermolecular force for a single molecule is identical to reduction of its thermopotential energy, as shown by −
r−δr ∂φ r
∂r
dr = φ(r) − φ(r − δr) .
The work done by the force for all molecules is then equivalent to reduction of thermopotential energy of the gas. Without energy exchange with the surroundings, the thermopotential energy change is balanced by the change of mean kinetic energy of the molecules or temperature of the gas.
A.3
Example of reverse sixth-power potential
The equation of thermopotential energy (A.4) can be rewritten as
Π=n V
with φj =
n 2
φj dV
V
φjk dV .
A. THERMOPOTENTIAL ENERGY OF GASES
578
In the spherical coordinates show in Fig.A.1, the element of volume is represented by dV = r 2 sin θdλdθdr , and the distance from a molecule to the element is l2 = r 2 + s2 + 2sr cos θ .
(A.5)
Thermopotential energy of molecule j with respect to the element on the sphere with radius s is evaluated from φbj = −nβ
2π 0
s−σ
dλ
π
2
r dr
0
0
(r 2
+
sin θdθ m . + 2sr cos θ) 2
s2
It is not divided by 2, as the integration is not made for all molecules, so the intermolecular potential between a pair of molecule is calculated only once. The upper limit of r is not s since the intermolecular potential is not defined inside a molecule, and the shortest distance between two molecules is σ. If the possible changes of molecular parameters, such as β and σ, with temperature are ignored, the thermopotential energy depends on intermolecular distance or the density only. The last equation gives φbj
2πnβ = (m − 2)s
for m = 2. It follows that φbj
s−σ 0
r r − dr m−2 (r + s) (r − s)m−2
2πnβ 1 1 1 − m−3 m−2 m−3 σ (2s − σ)m−3 1 1 1 − s(m − 4) σ m−4 (2s − σ)m−4
= −
− (A.6)
when m > 4. The previous integration does not include the thermopotential energy associatedwith the interactions between molecule j and other molecules on the shell of radius s. This part of thermopotential energy is evaluated from φsj
ns = 2
A
φjk dA ,
in which ns is the number density of the shell. The element of area on the shell is dA = s2 sin θdλdθ , and the distance from molecule j to the element center is l2 = 2s2 (1 + cos θ) obtained by substituting s for r in (A.5). Thus, we have φsj
= − =
ns β 2 sm−2 m +1 2
2π 0
π
dλ
4πns β m +1 2 2 (m − 2)sm−2
σ s
sin θ m dθ (1 − cos θ) 2
1 2 2 −1 m
1 − m [1 − cos(σ/s)] 2 −1
.
A.3. EXAMPLE OF REVERSE SIXTH-POWER POTENTIAL
579
θ j l dθ
s dλ
θ
dr
r
R λ
Figure A.1: Calculation of thermopotential energy of a single molecule The integration starts from σ/s instead of 0 since the shortest distance between two molecules on the shell is σ/s along coordinate θ. When s σ, we may apply cos
σ2 σ ≈1− 2 , s 2s
giving φsj
πns β =− m−2
1 σ m−2
1 − (2s)m−2
.
(A.7)
The number density on the shell can be evaluated from ns =
Ns , A
where A = 4πs2 is the area of the shell, and Ns is the number of molecules on the shell given by dV , Ns = n Vs
in which Vs indicates the volume of the shell with the thickness of one molecule. This equation gives Ns = 4πn
s+ σ 2 s− σ2
s2 ds
= 4πs2 nσ + πn If the last small term is ignored, we see Ns = Anσ .
σ3 . 3
A. THERMOPOTENTIAL ENERGY OF GASES
580 It gives
ns = nσ , and so (A.7) is replaced by φsj = −
πnβ m−2
1 σ m−3
−
σ (2s)m−2
.
The last term on the right-hand side is much smaller than the first. When it is ignored, we gain πnβ φsj ≈ − . (A.8) (m − 2)σ m−3 This thermopotential energy is independent of molecule position on the shell at an equilibrium state, and so the net intermolecular force on the molecules is zero. For the attractive force between molecules, we may take m = 6. In this case, (A.6) gives
4πnβ s − σ 3σ 3 2s − σ πn ≈ − 3β . 6σ = −
φbj
3
The last equation is just the largest term on the right-hand side of (A.6). We have in general 2πnβ . φbj ≈ − (m − 2)(m − 3)σ m−3 The total thermopotential energy of a molecule is φj = φbj + φsj ≈ −
(m − 1)πnβ . (m − 2)(m − 3)σ m−3
It is independent of molecule position. In other words, thermopotential energy of each molecule is equal in the equilibrium system. The thermopotential energy of the whole substance can be obtained by integrating the thermopotential energy of each molecule, giving 2π
Π=n
0
The result reads Π=−
R
dλ
0
r 2 dr
π 0
φj sin θdθ .
(m − 1)πn2 β V . (m − 2)(m − 3)σ m−3
It follows Π=−
5πn2 β V 12σ 3
for m = 6. For a mole substance, we have n = NA /V in equilibrium. Also, we may write β = 4εσ 6
A.4. COMPARISONS WITH EXPERIMENTS
581
Figure A.2: Thermopotential energy of oxygen and methane at 50 MPa. The data from Sychev et al. are shown by the circles. (after Zhu, 2000)
for the reverse sixth-power hard-sphere potential. Now, thermopotential energy of a mole substance is 5π 2 3 N εσ (A.9) Π=− 3V A for the reverse sixth-power hard-sphere intermolecular potential assumed. Since each molecule has identical thermopotential energy in equilibrium, we see Π = NA φ .
(A.10)
Equation (A.9) tells that the thermopotential energy is proportional to the density, molecular power and the size of molecule.
A.4
Comparisons with experiments
The molecular parameters depend on temperature in general. Since we are unable to express the dependence analytically, the parameters are taken as constants in the following examples. When repulsive intermolecular force is ignored in dilute gases at a temperature higher than the critical temperature, we may apply the molecular data of Lennard-Jones (6, 12) potential to calculate thermopotential energy. The data used are ε/ζ = 117.5 K and σ = 3.51 ˚ A for oxygen, and ε/ζ = 148.2 K and σ = 3.82 ˚ A for methane. Here, ζ is Boltzmann constant. These data were obtained from the experimental second virial coefficients (Hirschfelder et al., 1964). The evaluated thermopotential energy is plotted in Fig.A.2. Sychev et al. (1987b, c) published the thermodynamic data of oxygen and methane computed with the virial equations derived by curve-fitting to the ex-
A. THERMOPOTENTIAL ENERGY OF GASES
582
perimental data. The thermopotential energy may also be calculated from Π = Π0 + ∆Π with ∆Π = Ψ − Ψ0 − p(V − V0 ) − Cvo
T T0
Cvo dT ,
denotes the isochoric heat capacity of ideal where Ψ indicates the enthalpy, and gases between temperature T0 and T . With the data of enthalpy, specific heat and p-V -T data provided by Sychev et al., the thermopotential energy calculated may be considered as experimental data and is plotted by the circles in Fig.A.2. The thermopotential energy is small at low pressures unless the temperature is very low, so only the values at 50 MPa are plotted. The slope to temperature is small at high temperatures, but increases quickly at lower temperatures. When the molecular parameters change with temperature, large errors may occur for using the constant parameters assumed at a low temperature. By comparison, the old expression (A.1) gives the thermopotential energy three times higher than that from the current algorithm.
Appendix B Thermodynamics of gas expansions B.1
Energy conversions
The energy transfers in gas expansions related to intermolecular forces are an important subject in the classical thermodynamics and statistical mechanics. The well known examples associated with the energy transfers are the free expansion and Joule-Thomson ( Joule-Kelvin sometimes) expansion of gases. Although the gas expansions are the simplest processes studied in thermodynamics, there are no general algorithms which can be used practically to calculate the thermodynamic changes in the processes. Only the expansions in certain conditions, such as at constant pressure or enthalpy, were studied with the differential equations derived from the first and second thermodynamic laws. It is noted that the equations, including the Maxwell equations, derived using the second law of thermodynamics may only be applied for reversible processes which do not include the free expansion and Joule-Thomson expansion. The difficulty rises from the internal exchanges of gas energies in these expansions. According to the first law of thermodynamics, there is exchange between heat energy and mechanic work created by the gas or environments. However, the heat energy may also become other energies such as the thermopotential energy. For example, the temperature changes in the Joule-Thomson experiments may not be explained simply by the mechanic work created, and the temperature drops down in a free expansion also. Since the internal energy is conserved in free expansion, the heat energy which is a part of internal energy is converted to another energy included also in the internal energy. The energy transfer in gas expansions may be studied with the energy conservation law. However, the first law of classical thermodynamics is not given explicitly in terms of the heat energy and thermopotential energy. The heat energy of a substance may be defined as the total molecular kinetic energy of a substance, which includes the translational, rotational and vibrational kinetic energies, and can be calculated by adding the molecular kinetic energy of each molecule. The thermopotential energy is the sum of the thermopotential energy of each molecule. The heat energy plus thermopotential energy forms the major part of the internal energy as shown by (2.37). Their expressions are not given separately in the classical thermodynamics, and their thermodynamic effects are usually represented by thermodynamic functions which depend on the derivatives of thermodynamic variables. For the reverse sixth-power intermolecular potential defined by (A.2), the thermopotential energy calculated in Appendix A is given by (A.9). If the molecular parameters in (A.9) are independent of temperature, the thermopotential energy depends on the volume but not the temperature. But in general, the molecular parameters depend on temperature, so does the thermopotential 583
B. THERMODYNAMICS OF GAS EXPANSIONS
584
energy. When a gas is heated at a constant volume, both the heat energy and thermopotential energy increase. This feature makes it difficult to determine the heat energy according to the temperature change only. For simplicity, we assume that the heat energy is represented approximately by T
H=
0
Cvo dT ,
where Cvo is the molar heat capacity of an ideal gas at constant volume. This expression is principally correct for ideal gases only. Since ideal gases have no thermopotential energy, the internal energy is identical to the heat energy and the heat capacity is independent of volume. Now, the first law of thermodynamics in equilibrium may be written as dH + dΠ + dW = dQ ,
(B.1)
where dQ is the heat exchange between the system and surroundings, and dW is the work done by the system. This relationship may be used conveniently to study the thermodynamic processes of energy conversions. The applications for the JouleThomson experiments and free expansions are discussed in this appendix, and may be referred to the study of Zhu (2000).
B.2
Joule-Thomson effect
B.2.1
The new algorithm
The temperature change in an adiabatic Joule-Thomson expansion is represented by the Joule-Thomson coefficient, defined as the derivative of temperature with respect to pressure at constant enthalpy, that is
µ=
∂T ∂p
. ψ
The partial derivative is meaningful only if the temperature changes continuously with pressure in the process. But in a single Joule-Thomson experiment, ∆p is constant but ∆T changes. Thus, this equation cannot be applied to illustrate the whole process of Joule-Thomson expansion, or to predict the continuous temperature change by integrating it with respect to pressure. As pointed out by Kirillin et al. (1981), the enthalpy may not be conserved in the irreversible process of JouleThomson expansion, due to exchanges between kinetic energy and enthalpy. But these energy exchanges are usually ignored in the idealized experiments. For a reversible process, the last equation is replaced by
1 V −T µ= Cp
∂V ∂T
(B.2) p
derived from the first and second laws of thermodynamics. Kirillin et al. (1981) argued that this equation can only be applied for reversible processes, but the process of Joule-Thomson throttling is irreversible.
B.2.
JOULE-THOMSON EFFECT
585
When the Joule-Thomson expansion is considered approximately as a reversible process, it was suggested in previous studies that the temperature change, called the Joule-Thomson effect here, could be calculated by integrating this coefficient, giving T2 − T1 =
p2
µ dp ,
(B.3)
p1
where subscripts 1 and 2 indicate the initial and final values. It is noted that the ‘initial’ and ‘final’ pressures do not change in the experiment. An example for calculating the temperature change was given by Dodge (1944) in his Illustration 16. However, it was not successful, since the final volume v2 in the example was 19% overestimated, when it was calculated using the initial temperature which was 42.8 K higher than the final temperature. The calculation of Joule-Thomson effect is very sensitive to the errors in p-V -T . In practice however, we are unable to integrating the Joule-Thomson coefficient from (B.3), unless we know the relationships T = T (p) , and
V = V (p) ,
F (p) =
∂V ∂T
p
in the isenthalpic process. The first two equations are the projections of the intersection curve of the state equation and constant enthalpy Ψ = Ψ1 on a pT -plane and pV -plane, respectively, in the p-V -T coordinates. If we know T = T (p) in an isenthalpic process, Joule-Thomson effect can be calculated directly from this relationship without using (B.3). The temperature changes in gas expansions may be studied conveniently with the new expression of thermodynamic energy equation (B.1). If the isochoric heat capacity is taken as constant, this energy conservation equation gives dT =
1 (dQ − dΠ − dW ) . Cvo
The mechanic work produced by external force in an adiabatic Joule-Thomson expansion is p1 V1 , and the work produced by the gas is p2 V2 (p2 < p1 ). So the net mechanic work done by the gas is dW = p2 V2 − p1 V1 . Applying it yields T2 − T1 = −
1 (p2 V2 − p1 V1 + Π2 − Π1 ) Cvo
for adiabatic Joule-Thomson expansions. As the thermopotential energy increases in the expansions, there is Π1 − Π2 < 0 .
(B.4)
B. THERMODYNAMICS OF GAS EXPANSIONS
586
The thermopotential energy increased is converted from heat energy. As a result, the temperature decreases. This cooling may be intensified if the system creates positive mechanic work in the case of p2 V2 − p1 V1 > 0 . This situation occurs mostly at low temperatures. In a normal condition, the gas obtains net mechanic work from the environments, or p1 V1 − p2 V2 > 0 . Adiabatic heating may be produced by external force in this situation, and the sign of Joule-Thomson effect depends on comparison of the work produced by the external force and intermolecular force.
B.2.2
Comparisons with experiments
The Joule-Thomson effect given by (B.4) may be rewritten as
T2 − T1 =
1 5 1 1 p1 V1 − p2 V2 − πNA2 εσ 3 − o Cv 3 V1 V2
.
(B.5)
The data of molecular power and size included are taken from Hirschfelder et al. (1964) and shown in Table B.1. These data were obtained from experimental p-V -T and the second virial coefficients for Lennard-Jones (6, 12) potential. Table B.1: Data of molecular power and diameter Gas He N2 O2
ε/ζ K 10.22 95.0 117.5
σ ˚ A 2.56 3.72 3.51
Gas CO2 Air CH4
ε/ζ K 189.0 102.0 148.2
σ ˚ A 4.486 3.62 3.82
For the initial pressure p1 and temperature T1 applied by Roebuck and Osterberg (1935) in their Joule-Thomson experiments for nitrogen, the Joule-Thomson effects calculated from (B.5) are illustrated in Table B.2, and compared with the experimental data of Roebuck and Osterberg in the last column. Since the authors did not provide the data of volume change in their experiments, we use the mean state equation of Sychev et al. (1987a) to calculate the initial and final volumes. The data of heat capacity Cvo is replaced by Cv , and is also taken from their book. In this table, 1 (p1 V1 − p2 V2 ) ∆Tw = Cv is the temperature change contributed by external mechanic work, and ∆Tφ = −
1 (Π2 − Π1 ) Cv
B.2.
JOULE-THOMSON EFFECT
587
Table B.2: Joule-Thomson effect in nitrogen (after Zhu, 2000) p1 atm 201.2 201.2 201.8 201.8 201.2 201.2 201.5 201.5 201.5 201.5 201.8 201.8
T1 K 573.2 573.2 524.2 473.9 423.4 372.8 323.0 297.9 273.6 245.5 222.2 172.7
p2 atm 69.2 4.1 4.8 4.8 5.1 4.7 4.2 3.8 4.5 3.6 1.5 1.5
∆Tw K 14.51 21.43 21.21 20.85 20.24 19.52 18.63 18.14 17.63 17.26 17.30 23.80
∆Tφ K -14.59 -22.58 -24.87 -27.71 -31.12 -35.70 -41.91 -46.10 -50.73 -58.46 -67.80 -99.15
∆T K -0.09 -1.16 -3.70 -6.95 -11.06 -16.51 -23.84 -28.70 -34.12 -42.67 -52.74 -84.42
∆Texp K 1.2 0.9 -1.6 -5.2 -10.4 -16.3 -25.3 -31.2 -37.8 -47.5 -58.0 -91.3
is the cooling caused by transfer of heat energy into thermopotential energy. Obviously, the Joule-Thomson effect reads ∆T = ∆Tw + ∆Tφ . If heat exchange between the gas and surroundings takes place in a diabatic expansion, the temperature change may be represented more generally by ∆T = ∆Tw + ∆Tφ + ∆Th , in which ∆Th = dQ/Cv . In the ranges of pressure and temperature shown in Table B.2, the external mechanic work causes adiabatic heating of the gas, while conversion of heat energy produces cooling only. The cooling effect increases with reducing the initial temperature or increasing the initial density at the same initial pressure. If temperature is not too high, the heating effect may be smaller than the cooling effect. The net changes of temperature calculated with the constant molecular data are close to those observed, except the systematic deviations at low and high temperatures. If the molecular diameter increases at lower temperatures, the systematic deviations may be reduced. The Joule-Thomson effects on non-central heavy gas carbon dioxide are displayed in Table B.3. The experimental data come from Roebuck et al. (1942). The density is calculated from the virial state equation of Katkhe (referring to Vukalovich and Altunin, 1968). The data of isochoric heat capacity are taken from the tables of Vukalovich and Altunin (1968) and Din (1956). Since molecules of carbon dioxide possess a larger size and deeper potential well compared with the other gases in Table B.1, the cooling effect resulting from increasing thermopotential energy is greater than that of nitrogen, so is the Joule-Thomson effect.
B. THERMODYNAMICS OF GAS EXPANSIONS
588
Table B.3: Joule-Thomson effect in carbon dioxide (after Zhu, 2000) p1 atm 193.5 193.5 194.7 194.7 193.5 193.5 193.5 193.5 194.4 194.4
T1 K 573.2 523.7 473.7 422.7 398.3 373.2 347.8 322.5 298.4 273.2
p2 atm 3.1 2.5 3.8 4.6 2.5 1.9 41.1 54.0 55.4 49.4
∆Tw K 5.21 3.65 1.91 -0.06 -1.10 0.09 13.33 13.65 14.27 14.55
∆Tφ K -48.60 -57.23 -70.57 -94.35 -115.10 -151.20 -54.46 -31.43 -19.60 -12.30
B.3
Joule-Thomson coefficient
B.3.1
The new algorithm
∆T K -43.39 -53.58 -68.66 -94.42 -116.19 -151.11 -41.13 -17.78 -5.33 2.26
∆Texp K -41.2 -53.1 -71.6 -102.4 -126.0 -161.6 -40.70 -18.10 -7.60 -2.00
If the enthalpy is unchanged at the end of Joule-Thomson expansion, we may use (B.2) to calculate the Joule-Thomson coefficient. As discussed earlier, this coefficient may not mean the rate of temperature change in the process. For ideal gases, we have R p Cp , T = T0 p0 in which the subscript 0 indicates an initial value. It shows that the temperature is unchanged in the adiabatic expansion or contraction at constant pressure, so the Joule-Thomson coefficient of ideal gases is zero. This may also be seen by applying the state equation of ideal gas for (B.2). The temperatures on both sides of the plug are identical, and we have p1 V1 = p2 V2 . The Joule-Thomson coefficient may also be calculated from T2 − T1 µ= p2 − p1 using (B.5) or
µ=
5 1 1 1 p1 V1 − p2 V2 − πNA2 εσ 3 − Cv (p1 − p2 ) 3 V1 V2
.
(B.6)
The experimental coefficients of Roebuck and his colleagues were obtained in the similar way from the experiments over small differences of pressure. If we know the expressions of heat energy and thermopotential energy, the differential expression of this coefficient may be given by
µ=
∂T ∂p
=− ψ
(∂Ψ/∂p)T , (∂Ψ/∂T )p
B.3.
JOULE-THOMSON COEFFICIENT
589
in which Ψ = H + Π + pV is the molar enthalpy, and (∂Ψ/∂T )p = Cp . An advantage of using (B.6) is that it tells us the individual contribution of external and internal forces to the temperature changes. The change rate related to external mechanic work may be called the working coefficient, and is defined as µw =
∆Tw . p2 − p1
(B.7)
Meanwhile, the change rate related to conversion of thermopotential energy, given by ∆Tφ , (B.8) µφ = p2 − p1 may be called the cohesive coefficient. The Joule-Thomson coefficient is a sum of these two coefficients, that is µ = µw + µφ . In a diabatic Joule-Thomson expansion with heat exchange between the gas and surroundings, the diabatic coefficient µh =
∆Th p2 − p1
is added, that is µ = µw + µφ + µh . Now, the µ may be called the diabatic Joule-Thomson coefficient.
B.3.2
Examples
Using the mean state equation of Sychev et al. (1987d), the coefficients calculated with (B.6)-(B.8) for air are displayed in Table B.4. When the differentials in (B.2) are replaced by finite differences, the calculations using the same state equation are listed also in the table under µv . The heat capacities at constant volume and pressure used for these calculations are also taken from Sychev et al. (1987d). The obtained results are compared with the experimental data of Roebuck (1925, 1930) in the last column. The working coefficients listed are negative except a few cases. So the mechanic work created by pressure force leads to adiabatic heating in general. When temperature is low enough at 1 atm, the coefficient turns to positive. In this situation, the mechanic work produced by expansion at the lower pressure is greater than that produced by contraction at the higher pressure, and the net effect is adiabatic cooling. Meanwhile, the cohesive coefficient is positive and indicate adiabatic cooling of the gas. When the temperature is below −50 C◦ at 100 atm, the heating effect caused by external force increases at lower temperature, and may exceed the cooling effect produced by transfer of thermopotential energy, so that the Joule-Thomson coefficient may become negative.
B. THERMODYNAMICS OF GAS EXPANSIONS
590
Table B.4: Joule-Thomson coefficients of air in units K/atm (after Zhu, 2000) T C◦ 250 200 150 100 50 0 -50 -100
µw -0.09 -0.09 -0.08 -0.07 -0.05 -0.03 0.01 0.07
µφ p 0.14 0.15 0.17 0.20 0.23 0.27 0.34 0.43
µ µv = 1 atm 0.05 0.04 0.06 0.07 0.10 0.10 0.13 0.14 0.18 0.20 0.24 0.29 0.34 0.42 0.51 0.65
µexp
µw
0.04 0.06 0.09 0.13 0.19 0.27 0.38 0.5
-0.10 -0.10 -0.09 -0.09 -0.09 -0.08 -0.08 -0.12
µφ p= 0.12 0.14 0.16 0.18 0.21 0.25 0.31 0.38
µ µv 100 atm 0.03 0.02 0.04 0.04 0.06 0.06 0.09 0.09 0.12 0.13 0.17 0.18 0.23 0.25 0.26 0.28
µexp 0.02 0.03 0.06 0.09 0.13 0.18 0.25 0.28
Table B.5: Joule-Thomson coefficients of helium at 1 atm in units K/atm (after Zhu, 2000) T (C◦ ) 300 250 200 150 100 50 0 -50 -100 -140 -180
µw -0.069 -0.069 -0.069 -0.069 -0.069 -0.070 -0.070 -0.070 -0.071 -0.072 -0.073
µφ 0.008 0.008 0.009 0.010 0.012 0.014 0.016 0.020 0.025 0.033 0.047
µ -0.061 -0.061 -0.060 -0.059 -0.058 -0.056 -0.054 -0.051 -0.046 -0.039 -0.026
µv -0.064 -0.063 -0.063 -0.062 -0.061 -0.060 -0.059 -0.057 -0.053 -0.049 -0.040
µexp -0.057 -0.061 -0.062 -0.063 -0.062 -0.061 -0.060 -0.059 -0.056 -0.051 -0.041
These coefficients of air at 220 atm are demonstrated graphically by Fig.B.1. The positive cohesive coefficient is peaked around 220 K. The peak shifts towards a lower temperature at a lower pressure as shown by Table B.4. Meanwhile, the curve of working coefficient manifests a pit around 170 K. The curve of JouleThomson coefficient crosses the zero line twice at a low and high temperatures, called the inversion temperatures. The circles around the curve represent the numbers obtained from the experiments of Roebuck (1925, 1930). The Joule-Thomson coefficients of light gas helium at 1 atm are collected in Table B.5. The density is calculated from the Beattie-Bridgeman equation (Beattie and Bridgeman, 1928). The heat capacity data are taken from Sychev et al. (1987e). The experimental data of Roebuck and Osterberg (1933) are also tabulated. Since molecular power and size of helium are smaller than those of air as shown by Table B.1, the cohesive coefficients are also much less. While, the working coefficients of the two gases are comparable in magnitude. Thus, the Joule-Thomson coefficients
B.4. TEMPERATURE INVERSION CURVES
591
Figure B.1: Joule-Thomson coefficient, working coefficient and cohesive coefficient of air at 220 atm (after Zhu, 2000)
of helium have the same sign as that of the working coefficients, and the expansion leads to heating instead of cooling unless at very low or high temperatures. Although the calculations using the thermopotential energy are not as good as those using the complicated traditional algorithm, the differences are small. The increase of deviation at low temperatures may be caused by the constant molecular data used.
B.4
Temperature inversion curves
The inversion temperatures may be evaluated by assuming µ = 0 for (B.2), that is
T
∂V ∂T
−V =0.
(B.9)
p
With the thermopotential energy introduced, the inversion points may also be calculated by 5 1 1 − =0 (B.10) p1 V1 − p2 V2 − πNA2 εσ 3 3 V1 V2 derived from (B.6). It tells that the mechanic work created in the process is balanced by conversion of thermopotential energy along an inversion curve. With the p-V -T calculated from the mean state equations of Sychev et al. (1987a, 1987b, 1987c and 1987d) for nitrogen, oxygen, methane and air, the inversion curves of these gases calculated from (B.10) are plotted in Fig.B.2. For comparison, the inversion points calculated by Sychev et al. using (B.9) are shown by the circles in the figure. The asterisks in the nitrogen penal are plotted from the experimental data of Roebuck and Osterberg (1935). The asterisks in the methane penal are
592
B. THERMODYNAMICS OF GAS EXPANSIONS
Figure B.2: Inversion curves of four gases (after Zhu, 2000)
plotted from the calculations of Din (1961). The earlier experimental results of Olszewski (1907) for air and nitrogen are plotted by the crosses. There is µ > 0 or cooling inside the curves and µ < 0 or heating outside, found by comparing with Fig.B.1. The cooling is produced by converting a large amount heat energy into thermopotential energy in the process. When the internal energy conversion is not efficient enough at low and high temperatures, the heating effect of external force becomes dominant. Fig. B.2 shows the large deviations of the new calculations from the experimental data and traditional calculations at high temperatures, produced by overestimating the thermopotential energy conversion. These large deviations can be reduced by reducing the molecular sizes at high temperatures.
B.5. FREE EXPANSION
B.5
593
Free expansion
The thermopotential energy of gases may be applied to study other expansion processes. When dW = 0 in a free adiabatic expansion, (B.4) gives the Joule effect T2 − T1 = −
Π2 − Π1 . Cvo
For most gases at a normal pressure and temperature, the effect is larger than that of Joule-Thomson expansion, since no adiabatic heating is produced by external force doing work. The relation between the coolings of a mole different gases a and b can be represented by C o ∆Πb ∆Tb = voa ∆Ta Cvb ∆Πa or
C o εb σb3 ∆Tb = voa . ∆Ta Cvb εa σa3
From the molecular data in Table B.1, the Joule effect of a mole helium is only two percent of that of carbon dioxide. Tables B.3 and B.4 show that the working coefficients of carbon dioxide and air change sign at certain points. This fact suggests that the free expansion may be studied with the particular Joule-Thomson experiments of which ∆Tw = 0 or µw = 0. Since p2 V2 = p1 V1 in the particular experiments, we have Π2 =
p2 Π1 . p1
The Joule effect produced by conversion of heat energy into thermopotential energy gives Π1 T2 − T1 = o (p1 − p2 ) . Cv p1 This equation can be solved together with the state equation F (T, p) = 0 to obtain T2 at p2 . If calculated using the mean state equation of Sychev et al. (1987d), a mole air at 200 K and 50 atm initially will be 18.34 K cooler as it expands to 6.44 atm. As ∆Tw = 0 at the end, this temperature drop is equivalent to that as the air expands freely from the provided initial state to the same final state. The change rate of temperature in a free expansion is called the Joule coefficient defined as ∂T η= ∂V U usually. It follows that (Kirillin et al., 1981)
1 p−T η= Cv
∂p ∂T
. v
B. THERMODYNAMICS OF GAS EXPANSIONS
594
Table B.6: Joule coefficients of air in units K/atm (after Zhu, 2000) p atm 1 60
-100 0.435 0.455 0.468
100
0.322
-50 0.336 0.365 0.331 0.372 0.297 0.301
0 0.274 0.311 0.253 0.275 0.240 0.244
T (C◦ ) 50 0.230 0.249 0.215 0.228 0.201 0.214
100 0.198 0.198 0.185 0.188 0.173 0.179
150 0.173 0.172 0.161 0.156 0.152 0.146
200 0.154 0.145 0.143 0.133 0.136 0.126
Again, there is no advantages to calculate the Joule effect by integrating this coefficient: V2
T2 − T1 =
η dV .
V1
The T involved in the coefficient is the temperature at the free expansion curve, which is the intersection curve of the state equation and constant internal energy U = U1 . If we know the temperature T = T (V ), the Joule effect can be calculated immediately without making the integration. If we have the expression of internal energy, the Joule coefficient may also be derived from (∂U/∂V )T ∂T =− . ∂V U (∂U/∂T )v When the heat energy is independent of volume, we see η=−
1 Cv
∂Π ∂V
.
(B.11)
T
This equation tells clearly the physics of the temperature change in free expansion. The right-hand side represents the change rate of thermopotential energy. As (∂Π/∂V )T > 0, only cooling takes place in the process produced by converting the heat energy into thermopotential energy. Roebuck (1930) calculated the Joule coefficients using the following relationships
η=
∂T ∂p
= Cp − U
∂(pV ) ∂T
'
Cp µ + p
∂(pV ) ∂p
. T
The Joule coefficients may also be calculated from η=−
Π2 − Π1 . Cv (p2 − p1 )
(B.12)
Some examples for air are shown in Table B.6. The coefficients calculated by Roebuck are list below, which are systematically higher than the current calculations except at high temperatures and pressures.
Appendix C Derivation of momentum equation If there is a pressure gradient in a fluid, the pressure on the surface of a fluid element is not uniform in different directions. So, the element accepts a net pressure force in the downgradient direction. This force is called the pressure gradient force. An element of volume in the fluid may be represented by dV = dAdr , where δA is the surface area normal to the pressure gradient r, and dr is the length of r from p1 to p2 (p1 < p2 ). The pressure gradient force is given by F = −(p2 − p1 )δAˆr , in which ˆr is the unit vector in the direction of pressure gradient. It follows that F=−
∂p δrδA ˆr . ∂r
The acceleration of the parcel produced by the force is then evaluated from ρδrδA
∂p dv = − δrδA ˆr dt ∂r
or
∂p dv = −α ˆr dt ∂r according to the Newton’s second law. In the Cartesian coordinates, we may write
(C.1)
∂p ∂p ∂p ∂p = cos αx + cos αy + cos αz , ∂r ∂x ∂y ∂z and ˆ + cos αy y ˆ + cos αz ˆz , ˆr = cos αx x where αx , αy and αz represent the direction angles of the gradient. Applying these relationships yields
∂p ∂p ∂p ˆ cos2 αx + cos αx cos αy + cos αx cos αz x ∂x ∂y ∂z ∂p ∂p ∂p 2 ˆ cos αx cos αy + cos αy + cos αy cos αz y + ∂x ∂y ∂z ∂p ∂p ∂p cos αx cos αz + cos αy cos αz + cos2 αz zˆ . + ∂x ∂y ∂z
∂p ˆr = ∂r
Since
cos2 αx + cos2 αy + cos2 αz = 1 , 595
C. DERIVATION OF MOMENTUM EQUATION
596 we see
∂p ∂p ∂p ∂p ∂p ˆ − cos2 αy − cos2 αz + cos αx cos αy + cos αx cos αz x ∂x ∂x ∂x ∂y ∂z ∂p ∂p ∂p ∂p ∂p 2 2 ˆ − cos αx − cos αz + cos αx cos αy + cos αy cos αz y + ∂y ∂y ∂y ∂x ∂z ∂p ∂p ∂p ∂p ∂p − cos2 αx − cos2 αy + cos αx cos αz + cos αy cos αz zˆ . + ∂z ∂z ∂z ∂x ∂y
∂p ˆr = ∂r
In general we have ∂p = |∇p| cos αy , ∂y
∂p = |∇p| cos αx , ∂x
where |∇p| =
∂p ∂x
2
+
∂p ∂y
2
∂p = |∇p| cos αz , ∂z
+
∂p ∂z
2
is the mode of the pressure gradient. Using these relationships produces ∂p ∂p ∂p ∂p ˆr = ˆ+ ˆ+ ˆ. x y z ∂r ∂x ∂y ∂z Thus, (C.1) is rewritten as ∂p ∂p ∂p dv ˆ−α y ˆ−α z ˆ. = −α x dt ∂x ∂y ∂z This equation is the same as (5.26).
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Appendix E List of symbols Only the principal symbols are listed. Some of them may have other definitions different from the listed. Symbol CT Ckφ Cp Cuk Cv Cv◦ Cπ cd cl cm cp cˆp cpv crot cˆrot ctr cˆtr cv cˆv cvib cˆvib cvv cw cwd cwu cπ DTd DTt Dku Dv Dvd Dvt Dαt D|T D|p D|ρ dr dtr
Definition molar heat capacity at constant temperature conversion function from kinetic energy to geopotential energy molar heat capacity at constant pressure conversion function from internal energy to kinetic energy molar heat capacity at constant volume molar heat capacity of ideal agses at constant volume molar polytropic heat capacity specific heat capacity of mixing in dry processes specific heat capacity of liquid water specific heat capacity of mixing specific heat capacity of dry air at constant pressure specific collisional heat capacity at constant pressure specific heat capacity of water vapor at constant pressure specific rotational heat capacity specific collisional heat capacity for rotation specific translational heat capacity specific collisional heat capacity for translation specific heat capacity of dry air at constant volume specific collisional heat capacity at constant volume specific vibrational heat capacity specific collisional heat capacity for vibration specific heat capacity of water vapor at constant volume specific heat capacity of mixing in moist processes specific heat capacity of mixing in downdrafts specific heat capacity of mixing in updrafts specific polytropic heat capacity molecular heat diffusion rate turbulent heat diffusion rate molecular dissipation of kinetic energy kinetic energy dissipation rate kinetic energy dissipation rate due to molecular diffusion kinetic energy dissipation rate due to turbulent diffusion turbulent mass diffusion rate coefficient of self-diffusion at constant temperature coefficient of self-diffusion at constant pressure coefficient of thermal diffusion degeneracy of rotational kinetic energy degeneracy of translational kinetic energy 614
615 dv dφ EJ EC EO Ef f Ek−d En Es Ew es F F
Fc Fe f g g H ¯h K K∗ Kd Kp k k∗ kH kd kh kp kr kt ktr kv kz Lc Lx l M Mvex Mo m mo NA nx ny
degeneracy of vibrational kinetic energy degeneracy of termopotential energy efficiency of Joule engine efficiency of Carnot engine efficiency of Otto engine efficiency of heat engine total macroscopic energy of kinetic-death atmosphere lateral entrainment rate efficiency of semi-wet Joule engine efficiency of wet Joule engine saturated water vapor pressure external Newtonian force frictional force Coriolis force Helmholtz free function Coriolis parameter (= 2Ω sin ϕ) effective gravity magnitude of effective gravity heat energy Planck’s constant (parcel) kinetic energy instantaneous kinetic energy diffusion kinetic energy parcel kinetic energy specific (parcel) kinetic energy specific ordered kinetic energy specific disordered kinetic energy specific diffusion kinetic energy specific horizontal kinetic energy specific parcel kinetic energy rotational kinetic energy of molecule specific turbulent kinetic energy translational kinetic energy of molecule vibrational kinetic energy of molecule specific vertical kinetic energy latent heat water vapor condensation zonal wavelength free mean path (Chapter 3) mass or generalized mass (Chapter 6) mass of moisture exchange a mole mass mass mass of an oxygen atom Avogadro’s number zonal wavenumber meridional wavenumber
616 Pm p ps pt pθ Q Q− Q+ Q˙ Ql Q+ l Qs Q+ s q qm R R∗ Ro Rv S S SG Sbc Sbt Sc Sd Sec Sel Sese Sesl Set Sg Sgw Si Sld Sls Sse Ssl s sG sbc sbt sd sg si sl
E. LIST OF SYMBOLS
moist total potential energy pressure atmospheric pressure on the bottom surface atmospheric pressure at the top of air column reference pressure for potential temperature (=1000 hPa) heat (exchange) heat ejected heat absorbed diabatic heating rate latent heat latent heat absorbed sensitive heat sensitive heat absorbed specific heat exchange specific heat exchange produced by mixing specific gas constant of dry air = Cp − Cv , or universal gas constant of ideal gases Rossby number specific gas constant of water vapor thermodynamic entropy universal entropy (Chapter 6) turbulent thermodynamic entropy baroclinic entropy barotropic entropy classical thermodynamic entropy dynamic entropy equivalent baroclinic entropy thermodynamic entropy level equivalent static entropy equivalent thermo-static entropy level equivalent barotropic entropy geopotential entropy moist geopotential entropy internal entropy dry thermodynamic entropy moist thermodynamic entropy static entropy thermo-static entropy level specific thermodynamic entropy specific turbulent thermodynamic entropy specific baroclinic entropy specific barotropic entropy specific dynamic entropy specific geopotential entropy specific internal thermodynamic entropy specific entropy of liquid water
617 ss T Tc Te Th−d Ts Ty Tπe T˙d t U u ul uv V v v∗ vG va vd vc vhc vg vgl vh vp vt v v vd |T vd |p vd |ρ vn vx vy vz W WF WF Wg Wp Wv w wh wp ws
specific entropy of water vapor absolute temperature condensation temperature evaporation temperature temperature of isothermal heat-death state atmospheric temperature on the bottom surface meridional gradient of temperature polytropic equivalent temperature change rate of temperature due to diabatic heating time internal energy specific internal energy specific internal energy of liquid water specific internal energy of water vapor Volume three-dimensional velocity total mass velocity grid-scale velocity ageostrophic velocity diffusion velocity conduction velocity heat conduction velocity geostrophic velocity general velocity horizontal velocity parcel velocity turbulent velocity speed partial velocity diffusion velocity at constant temperature diffusion velocity at constant pressure diffusion velocity at constant density velocity normal to the boundary surface zonal velocity meridional velocity vertical velocity in Cartesian coordinates mechanic work external work done by external forces work done by frictional force work done by the gravitational force work done by pressure gradient force mechanic work produced by changing volume (saturated) mixing ratio of water vapor mixing ratio of total water (liquid water and water vapor) specific work done by pressure gradient force water vapor mixing ratio on the bottom surface
618 wy x ˆ x y ˆ y Z Ztr Zr Zv Zφ z zˆ Γ Γ∗ Γd Γ∗d Γπ ∆Se ∆Sea ∆Sev ∆Sex ∆Sh ∆Shc ∆Shd ∆She ∆Shl ∆Shr ∆Sh−d ∆Skd ∆Ske ∆Sk−d ∆Sw Θ Θm Λ Λm Λs Ξ Π Υ Φ Ψ Ψe Ψp Ψp e ψ
E. LIST OF SYMBOLS
gradient of water vapor mixing ratio zonal coordinate eastward unit vector meridional coordinate northward unit vector patition function (Chapter 6) patition function of translational kinetic energy patition function of rotational kinetic energy patition function of vibrational kinetic energy patition function of thermopotential energy vertical coordinate in Cartesian coordinates upward unit vector in Cartesian coordinates lapse rate of temperature slantwise lapse rate of temperature adiabatic lapse rate of temperature slantwise adiabatic lapse rate of temperature polytropic temperature lapse rate mass exchange entropy dry air exchange entropy moisture exchange entropy thermodynamic excitation degree heat exchange entropy without heat conduction heat conduction entropy heat diffusion entropy heat exchange entropy latent heat entropy radiation entropy thermodynamic entropy relative to heat-death state kinetic energy dissipation entropy thermodynamic entropy relative to kinetic equilibrium state thermodynamic entropy relative to kinetic-death state thermodynamic entropy produced in moist processes grid-scale potential temperature mean grid-scale potential temperature available potential enthalpy available moist potential enthalpy specific available potential enthalpy uncertain effects of subgrid-scale circulations thermopotential energy Helmholtz free energy geopotential energy in a system enthalpy, or convergent quantity (Chapter 6) moist enthalpy potential enthalpy moist potential enthalpy specific enthalpy
619 ψp specific potential enthalpy Ω angular velocity of the Earth α specific volume β meridional variation of f (= 2Ω cos ϕ/a) ε mole-number of molecule (Chapter 2) = 16, mole-number of an oxygen atom εo ζ Boltzmann constant η coefficient of viscosity (Chapter 3), Joule coefficient (Appendix B) η, ηe , ηz , ηd energy conversion ratios kinematic coefficient of viscosity η θ potential temperature departure of θ from vertical mean θ∗ θe equivalent potential temperature potential temperature of heat-death atmosphere θh−d θk−d potential temperature of kinetic-death atmosphere mean potential temperature θm root-mean-square deviation of potential temperature θR polytropic potential temperature θπ polytropic equivalent potential temperature θπ e ϑ an alternative definition of potential temperature ι inertia moment of molecule κ = R/cp κd = R/(cp − cd ) = R/(cp − cm ) κm = Rv /cv κv κw = R/(cp − cw ) λ heat conductivity (Chapter 3) µ Joule-Thomson coefficient (Appendix B) diabatic coefficient µh working coefficient µw cohesive coefficient µφ Brunt-Vaisalla frequency νz slantwise Brunt-Vaisalla frequency νl polytropic slantwise Brunt-Vaisalla frequency νlπ polytropic Brunt-Vaisalla frequency νπ R absolute angular momentum ρ specific density of air σ vertical variance of mixing ratio distribution σ effective molecular diameter (Appendices A and B) polytropic static stability σπ ς polytropic mixing index φ specific geopotential energy, or thermopotential energy of molecle intermolecular potential between molecle i and j φij ϕ latitude χ ratio of energy conversion ω = dp/dt, vertical velocity along pressure coordinate
E. LIST OF SYMBOLS
620 æ A0 Aenv Aequ Ain Aout Ar Arev Asys Avc Ave A´ Aˇ A˜ tan αl tan αln tan αp tan αθ tan αθπ
mixing coefficient A at initial state A of environment A in equilibrium process A of input material A of output material A at reference state A in reversible process A of system A of condensed water vapor A of evapored water horizontal mean ensemble mean over whole mass A of a parcel departure of A from horizontal mean A in an air column of unit horizontal section area trajectory slope neutral trajectory slope slope of isobaric surface slope of isentropic surface slope of polytropic isentropic surface
Index ability of energy conversion, 2, 13, 161, 173, 175, 225, 248, 262, 266, 267, 271, 294, 519, 547, 564, 573 ability of system variation, 84, 169 abrupt cooling, 212 absolute angular momentum, 401, 453, 619 additive, 4, 27, 33, 52, 53, 72, 86–88, 92, 576 adiabatic lapse rate, 17, 107, 135, 162, 228, 353, 359, 362, 392, 399, 406, 407, 435, 487, 490, 492, 494, 499, 618 advection, 75, 99, 104, 107, 108, 308, 338, 388, 436, 508, 537, 571 ageostrophic coefficient, 382, 385 air engine theory, 18, 19, 115, 246, 256, 367, 399, 414, 415, 430, 442– 444, 459 air-mass storm, 467 angular momentum transport, 429 Archimedes’ effect, 16 Archimedes’ principle, 116, 398, 535 arctic hurricane, 342 asymmetric surface drag, 558 available energy, 179, 226, 227, 605, 608, 613 available enthalpy, 10, 11, 13, 14, 16, 152, 182, 184, 186, 191–193, 195–199, 202, 203, 206, 208– 211, 215, 220, 221, 228–230, 245, 269, 270, 272–278, 280, 282, 284–287, 289–291, 294, 296–298, 317, 327, 328, 332, 336, 361, 362, 366–371, 373– 377, 379, 384, 387, 388, 390, 431, 468, 519–522, 529, 547, 567, 569, 606 available potential energy, 179–181, 220, 221, 226, 227, 270, 391, 399, 530, 553, 554, 598, 600, 606, 608 Avogadro’s law, 39
Avogadro’s number, 28, 55, 141, 574, 615 β-effect, 308, 310, 322, 530 β-plane, 508 backward difference, 542 backward momentum cascade, 535, 540 baroclinic cyclone, 15, 191, 253, 298, 317, 342, 473 baroclinic entropy, 9, 10, 13, 15, 160, 161, 166–168, 170, 171, 173, 174, 208, 223–225, 247, 248, 258, 265, 267, 271, 272, 279– 285, 287, 289–291, 297, 298, 302, 315–317, 336, 338, 366, 391, 515, 519, 616 baroclinic instability, 17, 19, 269, 272, 275, 276, 282, 290, 294, 298, 302, 366, 392, 409, 602 baroclinic waves, 17, 19, 269, 430, 438, 439, 441, 442, 496, 523, 610 barotropic cyclone, 319 barotropic entropy, 9, 12, 13, 160, 161, 167–171, 200, 207, 208, 223– 225, 247, 248, 258, 265–267, 316, 344, 616 barotropic instability, 294 Beattie-Bridgman equation, 33 Bernoulli’s, 8, 18, 143, 145, 146, 152, 414, 415, 420, 421, 458 blocking frequency, 276, 277 blocking system, 269, 388 Boltzmann conservation equation, 4, 51 Boltzmann constant, 39, 55, 137, 581, 619 bomb, 14, 53, 338, 609 Bose-Einstein, 81, 86 Boyle’s law, 557 breaking layer, 207, 385, 388 Brunt-Vaisalla frequency, 377, 392, 399, 402, 411, 412, 494, 619 butterfly effect, 531 CAPE, 227, 391, 399, 609 Carnot air engine, 444, 445, 447, 449 621
622 Carnot convection, 445, 449, 450 Carnot cycle, 447, 451 castellanus, 477 centred difference scheme, 541 centrifugal force, 113 chaos, 530, 531, 549, 553, 554 Charles’ law, 38 CINE, 227 classical entropy, 7, 8, 123, 126, 137, 175, 560, 561 classical turbulences, 21, 529, 532, 534 Clausius statement, 4, 52, 65 Clausius’ equation, 33 Clausius-Clapeyron, 247, 255, 289, 295, 501, 502 cloud-top entrainment, 294, 488, 489 cloud-top entrainment instability, 487, 488 coefficient of thermal diffusion, 60, 61, 614 coefficient of viscosity, 51, 63, 64, 66, 110, 619 cohesive coefficient, 589–591, 619 cold air intrusion, 478 cold front, 214, 327, 338, 429 collisional heat capacity, 4, 52, 67–69, 614 condensation temperature, 234, 241, 249, 459, 564, 569, 573, 617 conditional stability, 242, 244, 246, 250, 254, 266, 267, 291, 299, 309, 316, 319, 326, 456, 462, 467 conduction velocity, 73–76, 85, 99, 100, 104, 109, 617 conductive heat flux, 64, 85, 105 constraint relationship, 11, 182 continuity equation, 5, 6, 20, 76, 101– 104, 107, 114, 116, 150, 153, 468, 536–538, 568 continuous primitive equations, 21, 528– 530, 539, 553 convective inhibition energy, 227 convergent quantity, 132–137, 354, 618 convergent system, 132, 133 convergent variable, 132–136, 351 Copernicus, 557
INDEX
Coriolis force, 112–114, 145, 287, 306, 340, 454, 508, 615 Coriolis parameter, 112, 296, 299, 597, 615 Couette flow, 111 coupling mechanism, 328, 469 critical baroclinity, 16, 19, 172, 213, 366, 377, 442 critical entropy level, 174 critical wavelength, 19, 442 cumulonimbus, 484, 504, 605–607 cumulus congestus, 488, 602, 610 cyclogenesis, 275, 276, 282, 295, 296, 327, 328, 367, 457, 600, 602, 613 Dalton’s law, 32, 41 data limit, 530, 549 data resolution, 179, 532, 539, 540 DCAPE, 227 de Broglie, 136, 552 decay phase, 367, 379 deep convection, 14, 19, 244, 318, 319, 598 degeneracy, 67, 137, 139, 140, 614, 615 derivative order, 540, 541 detrainment, 20, 431, 471, 487, 488, 492, 499 diabatic coefficient, 589, 619 diffusion coefficient, 60, 64, 77 diffusion element, 52, 53, 55, 75, 99, 100 diffusion kinetic energy, 5, 24, 33, 43, 60, 78, 80, 82–84, 89, 92, 101, 144–146, 148, 149, 153, 562, 615 diffusion velocity, 4, 5, 24, 33, 42, 52– 54, 56–59, 62, 71, 73–78, 99– 101, 109, 148, 536, 538, 560, 572, 617 diffusive mass flux, 58, 60, 77, 111 diffusive momentum flux, 61, 110 direct meridional circulation, 213, 508 disordered kinetic energy, 42, 43, 615 downburst, 456 downgradient transport, 4, 5, 60, 65, 71, 82, 83, 85, 87, 124
INDEX
downrush, 456 dry air exchange entropy, 569, 571, 618 dry energy source, 289, 327 dry reference state, 235, 236, 243, 247, 249 dry thermodynamic entropy, 563, 570, 616 dynamic entrainment, 487 dynamic entropy, 3, 6, 23, 25–27, 72, 86, 87, 90–92, 616 e-folding time, 379, 382, 384, 387, 388, 390 Eady wave, 16, 21, 377, 379, 380, 382, 384, 549 eddy kinetic energy, 19, 374, 375, 379, 453 effective diameter, 574 efficiency of Carnot engine, 414, 446, 447, 615 efficiency of Joule engine, 438, 465, 485, 615 Einstein function, 67, 140, 141 Ekman flow, 113 Ekman pumping, 556–558 El Ni˜ no, 474 electron gas, 81 energy cascade, 118, 469 energy constraint equation, 16, 374, 375, 377, 385 energy conversion ratio, 155, 156, 379, 388 energy generation ratio, 255 Enskog theory, 51 entrainment, 20, 294, 318, 319, 390, 411, 431, 457, 460, 487–490, 492–497, 499, 502–504, 506, 507, 598, 601, 603, 606, 608, 610, 611, 615 entrainment coefficient, 489, 492 entropy flux, 197, 547, 566 equilibrium air engine, 18, 431, 450, 452–455, 497 equilibrium cycle, 450, 498 equilibrium surface, 402–404, 408, 411, 412
623 equivalent baroclinic atmosphere, 14, 266 equivalent baroclinity, 265–267, 298, 316, 317 equivalent barotropic atmosphere, 265, 309, 316 equivalent potential temperature, 12, 20, 134, 226–231, 233, 234, 236– 238, 240, 241, 244, 246–250, 252, 254, 256, 258, 259, 265– 267, 316, 392, 411, 464, 500, 501, 547, 619 equivalent principle, 18, 430 equivalent thermo-static entropy, 248, 267, 268, 616 Eucken formula, 66 Eulerian dynamics, 100 excitation degree, 174, 618 explosive cyclone, 14, 248, 253, 327, 384, 471, 472 extended parcel theory, 421 external work, 419–421, 423, 451, 617 extratropical tropopause, 474 extratropical troposphere, i, 16, 17, 227, 287, 380, 391, 402, 409, 469, 481, 508, 515 extremal process, 15, 16, 188, 189, 207, 225, 239, 251, 252, 272, 362, 371, 431 Fermi-Dirac, 81, 86 Fick’s law, 77 Fickian law, 77 field point, 73, 75 field variable, 98, 185 filament, 327, 332, 338 finite difference, 22, 185, 529, 540, 545 floccus, 477 forward difference, 541, 543 free convection, 309, 318, 319, 338, 408, 461, 462, 466, 534, 535 free expansion, 44, 85, 583, 593, 594 frontogenesis, 20, 22, 166, 508–510, 513– 515, 517–519, 521–523, 526, 527, 559–561, 603, 613 gas expansion, 178
624 Gauss equation, 103, 109, 150, 563 Gay-Lussac’s law, 38 GCAPE, 227 general parcel energy equation, 18, 419 general velocity, 73, 74, 99, 109, 617 geopotential entropy, 10, 11, 15, 16, 161, 162, 175–178, 181, 182, 185, 186, 189, 198–200, 203– 208, 214, 225, 228, 233, 239, 247, 263, 272, 343, 344, 363– 365, 436, 437, 444, 454, 455, 525, 526, 568, 616 geopotential entropy law, 11, 178, 181, 185, 186, 189, 228, 233, 344, 364, 365, 437, 525, 568 geopotential irreversibility, 10, 162, 178, 206, 233, 363 geostrophic balance, 3, 13, 15, 20, 98, 112, 116, 272, 296, 298, 327, 345, 396, 512, 523, 562 geostrophic wave, 272, 290, 296, 382, 385, 388, 497, 508 geostrophic wave circulations, 290, 296 gradient wind balance, 271, 277 gravitation, 1, 6, 16, 97, 349, 393, 400, 557 gravitational force, 1, 3, 8, 9, 15–17, 28, 29, 53, 97, 111, 116, 121, 125, 126, 135, 143, 146, 147, 170, 176, 178, 183, 208, 344, 348, 354, 364, 365, 392–394, 398–400, 402, 414, 416, 418– 421, 423, 443, 451, 617 gravity waves, 17, 393, 400, 412 grid thermometer, 121 grid-point data, 6, 21, 118, 121–123, 127–129, 134, 162, 183, 529, 531–534, 536, 537, 539, 540, 542, 559, 561, 562, 566 grid-scale prediction equation, 530 gross static stability, 180 group velocity, 74, 109 Hadley, 19, 282, 285, 290, 401, 453, 457, 473, 474, 508, 571 hard-sphere potential, 575, 581 head of warm bubble, 487
INDEX
heat capacity of mixing, 20, 490, 491, 496, 501, 614 heat conduction, 4, 5, 32, 52, 62, 64– 66, 73, 76, 83–86, 89, 90, 100, 104, 105, 111, 126, 174, 344, 349, 354, 537, 562, 564, 618 heat conduction entropy, 89, 104, 564, 618 heat conduction equation, 65, 104, 105, 111 heat conductivity, 4, 51, 66, 69, 105, 612, 619 heat diffusion, 78, 85, 89, 90, 104, 111, 146, 154, 562, 614, 618 heat diffusion entropy, 89, 104, 618 heat energy, 4, 8, 12, 13, 16–18, 23–25, 28, 32, 35, 42–44, 60, 63, 64, 71, 73, 78, 80–82, 84, 92, 96, 101, 105, 113, 115, 122, 129, 134, 135, 141, 143–146, 148– 151, 153–156, 158, 160, 169, 172, 173, 179, 183, 185, 186, 201, 207, 208, 213, 214, 218– 220, 228, 229, 235, 244, 247, 248, 251, 254, 255, 345, 346, 350, 351, 354, 360, 361, 363, 364, 392, 410, 414, 421, 432, 452, 458, 536, 547, 549, 562, 576, 583, 584, 586–588, 592– 594, 615 heat engine, 18, 364, 414, 415, 419, 430, 432, 437, 609, 615 heat exchange entropy, 563, 564, 618 heat flux equation, 6, 20, 101, 102, 104, 105, 107, 116, 153, 458, 537–539, 547, 565 heat-death, 2, 3, 8, 9, 11, 15, 144, 160, 173, 182, 184, 186, 188, 230, 232, 237, 343, 344, 346, 348– 351, 354, 355, 357, 362–364, 617–619 height of tropopause, 460, 473 Heisenberg’s uncertainty principle, 530, 551 Helmholtz free energy, 142, 618 high-level convection, 475, 477
INDEX
horizontal kinetic energy, 400, 401, 409– 412, 427, 429, 469, 615 humidity front, 256, 267 hurricane, 229, 252, 309, 316, 318, 319, 321, 342, 597–599, 601, 603, 605, 610 hydrostatic equation, 106, 114, 198, 215, 351, 392, 397, 402, 406, 535 hydrostatic equilibrium, 8, 15, 114, 152, 162, 180, 287, 348, 349, 393, 398, 399, 402, 418, 450, 451, 525, 534, 562 ideal-gas equation, 24, 33, 34, 36–38, 41, 43, 47, 49, 57, 59, 79, 81, 82, 94, 96, 149, 151, 348, 397– 399 idealized front, 510 incompressible approximation, 116 inhomogeneous equilibrium state, 6, 108, 133, 197, 536 inhomogeneous thermodynamic system, 11, 15, 108, 117, 126, 130, 133, 134, 148, 149, 152, 162, 170, 171, 365, 515, 538 inhomogeneous thermodynamics, i, 6, 97, 108, 553 initial data problem, 20, 530 instantaneous kinetic energy, 101, 615 intermolecular force, 576, 577, 580, 581, 586 intermolecular potential, 51, 574, 575, 578, 581, 583, 619 internal energy, 1, 3–5, 8, 18, 23, 24, 32–36, 39, 42–47, 83, 93, 96, 98, 114, 118, 137, 141, 149, 157, 179, 183, 354, 414, 416, 445, 458, 501, 583, 584, 592, 594, 614, 617 internal thermodynamic entropy, 9, 90, 562, 564–566, 616 intrinsic error source, 20, 530, 542, 555 intrinsic uncertainty, 530, 531, 551, 552 irreversible adiabatic process, 93, 106 irreversible process, 6, 9, 11, 13, 82, 169, 170, 172, 207–209, 251,
625 260, 371, 382, 518, 526, 547, 584 isenthalpic diffusions, 158, 159, 177, 206, 239, 345, 346, 353, 443, 510 isenthalpic molecular diffusion, 158, 550 isenthalpic turbulent diffusion, 158 isentropic atmosphere, 9, 15, 134, 135, 165, 169–172, 175, 176, 238, 266, 344, 352, 353, 355, 361– 364, 421, 423, 437 isentropic state, 9, 124, 125, 134, 135, 159–161, 167, 169, 171–174, 184, 200, 223, 236, 238, 263, 264, 344, 352, 353, 355, 356, 359–362 isoperimetric problem, 137, 186, 233, 235, 238, 347 isothermal atmosphere, 8, 174, 199, 344, 355, 361, 364, 423 isothermal state, 2, 15, 105, 135, 144, 156, 174, 186, 188, 223, 343, 348, 350, 352, 354, 361, 362 isotropic system, 27, 90 ITCZ, 291, 294, 302, 322, 326, 571 Joule air engine, 18, 431, 432, 435, 438, 447, 455, 462, 466, 494– 496, 506 Joule coefficient, 594, 619 Joule convection, 434–438, 462, 466, 471, 475, 476, 478, 479, 484 Joule cycle, 18, 431–433, 436–438, 441, 444, 448, 465, 469, 471, 473, 474, 480, 481, 496, 503 Joule effect, 593, 594 Joule-Kelvin, 583, 607 Joule-Thomson coefficient, 584, 588– 591, 619 Joule-Thomson effect, 584–588, 609 Joule-Thomson expansion, 44, 583–585, 588, 589, 593 Joule-Thomson throttling, 584 Kelvin scale, 40 Kelvin statement, 65 Kelvin temperature, 40, 447
626 kinematic wave, 508 kinetic energy dissipation entropy, 89, 618 kinetic equilibrium state, 343, 344, 357, 358, 360–363, 618 kinetic theory, 4, 32, 40, 41, 51, 52, 57, 62, 63, 117, 143, 149, 603, 605, 610, 613 kinetic-death, 15, 16, 135, 344, 354, 356, 357, 360–364, 421, 437, 452, 615, 618, 619 La Ni˜ na, 474 Lagrangian dynamics, 100, 144 laminar flow, 534 latent heat entropy, 564, 565, 567, 569, 572, 618 latent heat of condensation, 230 lateral turbulent mixing, 213, 390 Lennard-Jones (6, 12) potential, 581, 586 level of neutral buoyancy, 317, 399, 460, 462, 466, 473, 477, 484, 503 lifting condensation level, 230, 326 linear adjustment, 282, 367, 382, 384 linear atmosphere, 166–169, 189, 192, 209, 215–217, 220, 223, 225, 263, 279, 280, 353, 363, 364, 369, 391, 393–398, 400, 401, 403, 405–407, 409, 410, 415, 423–425, 435, 440, 441, 460, 494, 514, 519, 521, 525, 550 linear law, 51, 71, 77, 78, 528, 536, 538 linear molecule, 66 linear theory, 16, 19, 207, 275, 366, 367, 375, 497 LNB, 460, 461, 464, 475, 477, 482–484, 503 local change rate, 85, 99, 148 local derivative, 75 local energy equation, 143, 148, 150, 152, 566 long-term forecasts, 560 low frequency variability, 296 low-level inversion, 475, 476 lowest state, 10, 11, 13, 15, 16, 161, 172, 181, 182, 184, 188–191,
INDEX
193, 194, 197, 200, 201, 203– 207, 212, 218, 219, 223–225, 228, 237, 239, 241, 242, 252, 257–263, 272, 287, 288, 343, 344, 365, 519, 522, 553 macroscopic thermodynamic state, 29, 84 major warming, 12 Margules’ example, 201, 203 mass average velocity, 53 mass diffusion, 42, 54, 59, 60, 62, 65, 66, 71, 75–78, 100, 102, 105, 111, 124, 614 mass diffusion equation, 71, 76–78, 102, 105, 111 mass exchange entropy, 563, 618 mass kinetic energy, 42 mass point, 552 mass velocity, 62, 63, 71, 73, 74, 76, 99–104, 109, 617 material change rate, 99, 100 material derivative, 75 maximum thermodynamic entropy, 2, 15, 124, 134, 156, 160, 161, 169, 171, 175, 238, 343, 344, 346–352, 357, 358, 361, 362, 365 Maxwell equations, 81, 583 Maxwell-Boltzmann, 8, 81, 86 Mayer formula, 35, 37, 149, 502 mean free path, 57, 62, 63 mean meridional circulation, 453, 571 mean typhoon, 320, 322 mechanic energy equation, 1, 143, 145 meteorological noise, 412 meteorological turbulences, 21, 22, 529, 530, 532–536, 539, 540, 546, 549, 553, 555, 558, 561, 567 Mie potential, 575 minimum precipitation, 247, 250 mixing length theory, 529, 534, 540 model atmosphere, 22, 523, 531, 547, 555, 556, 558, 559, 561, 566, 567 model limit, 530, 549 moist adiabatic ascent, 488
INDEX
moist convection, 19, 172, 173, 233, 290, 295, 298, 317, 319, 326– 328, 340, 344, 352, 392, 411, 453, 456, 469, 471, 473, 530, 546 moist cyclone, 15 moist energy source, 14, 247, 252, 259, 289, 291, 296–298, 306, 322, 326, 328, 332, 573 moist isentropic surface, 266, 411 moist jet, 327, 338–340, 342, 411 moist polytropic mixing, 499, 504, 506 moist potential energy, 229 moist potential enthalpy, 228, 230–232, 234, 237, 238, 618 moist reference state, 12, 235, 239, 242– 244, 247, 249, 260 moist thermodynamic entropy, 234, 236, 248, 564, 565, 568, 616 moist-adiabatic, 228, 230–233, 459, 482, 485 moisture exchange entropy, 572, 618 molar heat capacity, 34, 47, 584, 614 mole-number, 27, 619 molecular diffusion, 33, 43, 44, 53, 57, 65, 73, 75, 78, 80, 85, 86, 89, 94, 100, 101, 104, 105, 110, 122, 124, 136, 144, 146, 150, 153, 157, 158, 164, 165, 188, 232, 265, 343, 345, 346, 350, 351, 353, 356, 364, 365, 533, 535–537, 547, 560, 614 molecular dissipation, 42, 43, 78, 104, 129, 144, 148, 154, 614 molecular entropy, 6, 86–88, 120, 127, 128 molecular mass, 27, 28 molecular viscosity, 388, 534, 536, 539, 540 molecular weight, 27 moment of inertia, 140 momentum conduction, 62, 63, 65, 74, 110, 111, 147, 148 momentum conduction equation, 62, 111, 147
627 momentum equation, 1, 7, 18, 20, 111– 114, 116, 144, 181, 391, 398, 408, 411, 412, 414, 421, 443, 444, 459, 536–538, 547, 595 most unstable wavelength, 497 most unstable wavenumber, 497 multiple semi-wet Joule engine, 456, 478, 480, 504, 506 multiple semi-wet Joule storm, 456, 481 Navier-Stokes, 109, 110 negative entropy, 136, 244, 328, 566, 573 negative storm, 457, 466, 468–471, 473, 476 negative viscosity, 21, 534, 535, 540, 611 neutral atmosphere, 176, 391 Newton’s fluid, 111 Newton’s second law, 4, 7, 20, 25, 33, 63, 99, 100, 108, 109, 560, 595 Newtonian dynamics, 4, 6, 17, 24, 25, 27, 28, 97, 100, 146, 147, 391, 408, 415, 421, 552 Newtonian force, 17, 108, 615 Newtonian system, 3, 7, 23–25, 44, 96, 179 Newtonian-thermodynamic system, 7, 8, 96, 115, 116 non-kinetic energy, 147, 179, 183, 226, 227, 262, 408, 409 non-uniform gas, 4, 32, 33, 65, 66, 83, 84 nonequilibrium process, 72, 82 nonequilibrium thermodynamics, 6, 33, 51, 77, 96, 97, 102, 108, 345, 528, 538 nonlinear planetary wave instability, 385, 388 normal distribution, 577 Norwegian cyclone model, 317, 327, 340 number density, 31, 32, 40, 56, 62, 111, 574, 575, 578, 579 open system, 83, 90, 149, 196, 244, 569, 573
628 ordered kinetic energy, 42–44, 81, 92, 149, 615 p-α diagram, 417, 445, 450, 463, 477, 480, 482 p-α diagram, 416, 434, 447–449, 458, 462, 478, 482, 504 p-α-T surface, 29 p-V -T data, 582 parameterization, 536, 559, 567, 599 parcel algorithm, 11, 179, 182, 206, 226, 227, 239 parcel energy equation, 143, 146, 148, 152, 415, 416, 418, 419, 443 parcel kinetic energy, 1, 8, 17, 18, 33, 97, 101, 108, 119, 122, 129, 143–149, 152–154, 156, 157, 177, 179, 183, 186, 208, 225, 263, 266, 269, 317, 344, 353, 391, 394, 398, 401, 414–416, 418, 421, 423, 427, 429, 431, 437, 446, 453, 454, 460, 536, 615 parcel velocity, 71, 99–102, 107, 109, 110, 114, 145, 148, 532, 533, 560, 617 partial pressure, 31, 500, 572 partial velocities, 52, 54–57, 62, 65 partition function, 34, 138–140, 142 partition of Pearce, 208 Pauli exclusion principle, 86 perfect convection, 475, 506 perfect derivative, 75 perfect mechanic system, 25 perfect storm, 466 phenomenological law, 538 phlogiston theory, 557 Planck’s constant, 67, 136, 139, 551, 615 planetary stratospheric waves, 207 planetary wave, 17, 207, 213, 385, 390, 602 Poiseuille flow, 111 polar front, 17, 18, 275, 277, 280, 282, 367, 391, 402, 415, 429, 430, 436, 438, 456, 508, 598
INDEX
polar low, 340, 342, 457, 598, 602, 608, 610 polar momenta, 4, 69, 70 polytropic convection, 492, 493 polytropic equivalent temperature, 501, 617 polytropic heat capacity, 47, 65, 105, 492, 614 polytropic index, 49 polytropic lapse rate, 490 polytropic mixing, 488, 490, 498, 619 polytropic mixing index, 490, 619 polytropic potential temperature, 20, 49, 50, 493, 494, 501, 619 polytropic primary wet cycle, 502 polytropic primary wet engine, 502 polytropic process, 49, 500, 501, 503 polytropic slantwise convection, 494 polytropic static stability, 493, 619 positive convection, 476, 481 positive storm, 457, 466, 467 potential enthalpy, 8–11, 15, 134, 135, 144, 157–160, 164, 167, 169, 173, 181, 182, 184, 186, 221, 226, 228, 230, 236, 238, 265, 287, 343, 345, 351, 354, 356, 361, 368, 369, 437, 509, 510, 618, 619 potential temperature, 2, 7, 9–12, 20, 50, 81–88, 94, 95, 106, 118– 130, 132, 134–136, 144, 157– 162, 164–166, 168–171, 173, 176, 177, 179, 182, 185–189, 195, 200, 202, 203, 206, 209, 225–228, 230, 233, 235, 240, 243, 248, 252, 257, 258, 264, 266, 298, 317, 344, 346, 351, 352, 354, 355, 361, 368, 369, 377, 392–395, 399, 403, 411, 416, 421, 431, 432, 436, 437, 439, 440, 449, 462, 464, 492, 493, 496, 510, 511, 519, 547, 562, 566, 570–572, 616, 618, 619 potential vorticity, 118, 213, 279, 474, 557
INDEX
potential well, 574, 587 precipitation, 13, 115, 227, 229, 235, 247–253, 255, 296, 302, 303, 306, 316, 327, 484, 486, 504, 565, 573 precipitation ratio, 250, 251 predictability, 22, 116, 528, 530, 531, 546, 551, 553, 556, 597, 602, 604, 605, 612 pressure gradient force, 1, 7, 15–18, 53, 58, 97, 100, 108, 109, 111– 114, 116, 124, 143, 145–147, 149, 153, 154, 170, 176, 178, 208, 391, 392, 394, 398–402, 413, 414, 416, 418–421, 423– 427, 429, 443, 453, 454, 458, 535, 595, 617 primary wet engine, 462 primitive equations, i, 20, 21, 96, 116, 185, 430, 508, 528, 531, 538, 539, 556, 558, 560, 561 principle of extremal entropy productions, 344, 365 principle of friction, 23, 25 principle of kinetic energy degradation, 148 probability, 21, 43, 60, 68, 72, 81, 86, 118, 124, 127, 134, 552, 553, 607, 611 probability wave function, 21, 552, 553 propagation velocity, 74, 99, 109 pseudo-adiabatic process, 228, 232, 456, 458 pseudo-reversible, 9–11, 161, 170–172, 175, 189, 190, 195, 200, 206, 208, 225, 260, 362, 444, 515, 516, 518, 522 Ptolemy’s astronomy, 557 quantum mechanics, 44, 86, 137, 139 quantum physics, 21, 109, 530, 552, 553 quasi-adiabatic process, 159, 164, 165, 168, 176, 200, 260, 264, 351, 353, 354, 359, 361, 363–365, 369, 437, 560 quasi-biennial oscillations, 474
629 quasi-geostrophic, 287, 296, 379, 380, 387, 522, 610, 612 radiation entropy, 564, 618 randomness, 9, 60, 84, 88, 117, 124, 132–134 Rasmussen’s criterion, 298 ratio of energy conversion, 221, 619 regional properties, 269 reverse sixth-power, 574, 577, 581, 583 reversible adiabatic process, 7, 81, 85, 121, 125, 157, 182 reversible Carnot engine, 447 reversible Joule engine, 437, 447 reversible Otto engine, 448 reversible process, 2, 9–11, 25, 72, 82, 92, 94, 106, 121, 170–172, 175, 189, 200, 208, 225, 233, 362, 431, 437, 444, 515, 516, 518, 522, 562, 584, 620 Rossby number, 290, 296, 298, 299, 616 Rossby wave, 116, 440, 497, 508 rotational degree, 67 rotational kinetic energy, 29, 68, 139, 614, 615, 618 saturated mixing ratio, 228, 240, 241, 319, 458, 460, 475, 483 saturated reference state, 247, 250–252 saturated vapor pressure, 241, 499, 502 scale-dependent, 6, 7, 21, 29, 40, 86, 89, 122, 123, 125, 131, 529, 531, 532, 539, 540, 544, 546, 553 scale-dependent data, 532, 539 sea surface temperature, 19, 259, 273, 280, 285, 299, 306, 309, 319, 340, 457, 467, 598 self-diffusion, 4, 51, 60, 64, 613, 614 self-feeding mechanism, 14, 15, 309, 322, 327, 328, 340, 342 semi-wet Joule engine, 456, 457, 463, 465, 467, 473, 474, 478, 480, 484, 485, 503, 506, 615 semi-wet Joule storm, 456, 467, 469, 473, 484
630 shallow water, 7, 113–115, 549 shear instability, 294, 509 simple plume, 487 simple test, 557 simple turbulent process, 119, 121, 123– 125, 131, 144 slantwise circulation, 17, 392, 402, 408, 409, 415, 429, 471, 523 slantwise circulation instability, 408 slantwise convection, 17–19, 107, 190, 211, 226, 227, 266, 275, 298, 317, 338, 364, 391–393, 403, 404, 406–412, 424, 429, 430, 434, 436, 438, 469, 471, 473, 474, 488, 494, 497, 518, 522, 534 slantwise lapse rate, 17, 392, 404–407, 410, 434, 435, 494, 496, 618 slantwise oscillation equation, 411 slantwise static instability, 17, 402, 404, 409, 410, 436 slope of isentropic surface, 17, 190, 395, 403, 408–410, 412, 425–429, 431, 434, 435, 438, 452, 523, 620 slope of isobaric surface, 114, 392, 396, 397, 425, 428, 620 small perturbation theory, 156, 534 soft-sphere potential, 575 solid body, 24–27, 41, 53, 175, 414, 421, 552 solid process, 421 southern oscillations, 474 spatial discretization, 529 SPCZ, 291, 294, 299, 302, 306, 322, 326 specific gas constant, 37, 55, 500, 616 specific heat capacity, 43, 48, 67, 500, 572, 614 spectral model, 602, 609, 610 squall line, 471, 475, 478 square-well potential, 575 standard mathematical analysis, 552 state equation of gases, 36, 114 state function, 3, 23, 25, 26, 45–47, 71, 72, 81–83, 92, 134, 135, 142,
INDEX
162, 414, 448, 449 static entropy, 10, 161, 170–173, 190, 222, 225, 267, 328, 519, 616 static equilibrium path, 449, 458 stationary waves, 296, 299, 532, 557, 558 statistical mechanics, 8, 34, 43, 51, 81, 134, 137, 583, 606, 612 steering concept, 308 Stefan-Boltzmann law, 83, 135 storm tracks, 13, 14, 269, 270, 275, 277, 282, 289, 290, 294–296, 302, 306, 309, 327, 366, 604 stratopause, 213 stress tensor, 536 sub-synoptic convection, 328 subgrid-scale eddies, 532, 535 subgrid-scale fields, 561 subgrid-scale perturbation, 21 sublimation, 502, 572 subtropical cyclone, 316, 317 sudden change, 11, 211, 245, 310, 547 sudden warming, 212 super storm, 456, 464, 478, 504 surface albedo, 540 surface cooling, 190, 212, 253 surface drag, 109, 115, 522, 540 surface pressure change, 215, 259, 271 Sutherland potential, 575 symmetric instability, 509 system energy equation, 8, 18, 143, 150–152, 154–157, 180, 181, 183, 185, 230, 391, 420, 443, 444, 553, 567 Taylor expansion, 540, 541 TCAPE, 227 technic prediction errors, 530 temperature front, 254, 256, 264, 273, 295, 317, 327, 401, 438, 471, 508, 559 temperature inversion, 19, 245, 456, 457, 475, 511 the first law of thermodynamics, 1, 24, 33, 34, 37, 43, 71, 114, 414, 416, 583, 584
INDEX
the second law of thermodynamics, 4, 5, 23, 33, 65, 78, 96, 100, 111, 117, 446, 508, 560, 583 thermal, 3, 8, 10–12, 15, 20, 59–61, 64, 66, 108, 146, 152, 153, 157, 182, 184, 186, 188, 189, 202, 203, 206, 213, 225, 228, 234, 237, 239, 241, 242, 247, 252, 275, 280, 284, 285, 290, 298, 299, 317, 349, 353, 360, 369, 373, 385, 393, 401, 415, 421, 425, 456, 457, 465, 487, 496, 499, 509, 510, 512, 515, 518, 524, 525, 546, 547, 570, 603 thermal conduction, 64 thermal diffusion, 59–61, 556, 566 thermal wind, 3, 213, 280, 284, 298, 385, 415, 425, 456, 509, 512, 518, 524, 525 thermal wind relationship, 385, 512 thermo-static entropy, 10, 161, 173– 175, 208, 222, 224, 225, 248, 267, 271, 616 thermodynamic entropy, 2, 3, 5–13, 15, 16, 20–22, 27, 71, 72, 80–89, 91–95, 98, 111, 117–131, 134– 138, 142, 157–162, 164, 165, 167, 169–178, 180–182, 184– 186, 188–190, 193, 195–198, 200, 203, 206–213, 220, 222– 225, 227, 228, 232, 233, 235– 239, 241, 244, 247–251, 258, 259, 263–268, 271, 272, 285, 286, 316, 322, 326, 336, 343– 347, 351–366, 368, 371, 377, 384, 388, 390, 414, 430, 431, 434, 436, 437, 444, 448, 449, 454, 465, 466, 485, 500, 509, 515–519, 522–524, 526, 536, 547, 549–551, 556, 558–562, 564–566, 568, 569, 571, 573, 576, 616, 618 thermodynamic entropy balance, 12, 568 thermodynamic entropy level, 9, 160, 167, 169, 174, 616
631 thermodynamic equilibrium state, 1, 84, 348 thermodynamic irreversibility, 2, 6, 10, 82, 206, 207, 222, 228, 245, 264, 363 thermodynamic potential energy, 574 thermodynamic space, 29 thermodynamic variable, 35, 77, 82, 98, 538 thermopotential energy, 4, 24, 32, 35– 38, 44, 46, 47, 96, 139–142, 183, 574–589, 591–594, 618, 619 threshold entropy production, 189 threshold humidity, 13, 247, 256, 259, 264 threshold static instability, 264, 296, 475 threshold temperature, 14, 309, 318 thunderstorm, 299, 301, 600 time resolution, 531 time stepping, 529, 545 time-stable, 528, 530, 553 tornado, 256, 462, 601 total potential energy, 8, 143, 155, 156, 172, 179, 181, 183, 184, 229, 616 translational kinetic energy, 29–32, 39, 41, 43, 67, 69, 91, 139, 614, 615, 618 tropical cyclone, 14, 306, 308, 309, 316, 317, 321, 322, 326, 328, 340, 342, 475, 598–601, 607 tropical storm, 14, 19, 256, 308, 319, 321, 474, 481, 612 tropical tropopause, 19, 457, 473, 474 tropical troposphere, 19, 298 tropopause folding, 415, 429 tropospheric river, 14, 327, 338, 471 truncation, 22, 530, 541–545, 558 truncation order, 22, 541–543 turbulent diffusion, 2, 537, 540, 547, 549, 551, 614 turbulent entity, 7, 119, 121, 127, 143, 144
INDEX
632 turbulent entropy, 7, 8, 95, 117, 118, 123–131, 134, 159, 160, 162, 174, 175, 178, 181, 185, 186, 203, 344, 508, 509, 536, 556, 560–562 turbulent entropy law, 7, 8, 117, 118, 124–126, 128, 134, 159, 175, 181, 185, 186, 344, 508, 509, 556, 560, 561 turbulent mass flux, 537 turbulent mixing, 117–119, 125, 131, 158 turbulent viscosity, 118, 535, 540, 551 typhoon, 14, 248, 252, 308–310, 315, 316, 319–322, 325, 326, 603 Typhoon Orchid, 14, 310 typhoon recurvature, 14, 308, 309, 326 unavailable potential energy, 181 uncertainty, 20, 116, 211, 528, 532, 545, 551–553, 558, 602 unidirectional transport, 117, 121 universal entropy, 8, 118, 132, 134, 136, 616 universal gas constant, 39, 55, 616 universal principle, 8, 117, 118, 126, 128, 132–134, 136, 137, 156, 165, 351, 354, 556, 561 upper-level trough, 327 upwelling, 299, 310 van der Waals equation, 8, 33, 34, 140, 142 variational approach, 11, 12, 16, 144, 165, 182, 185, 203, 204, 206, 207, 209, 221, 228, 233, 236, 239, 247, 290, 367, 368, 371, 519, 600 vertical kinetic energy, 391, 399, 401– 403, 409, 410, 412, 427, 429, 471, 615 vertical wind shear, 19, 424, 478, 481 vibrational degree, 67 vibrational kinetic energy, 29, 41, 62, 67, 68, 91, 139–141, 615, 618 virial equation, 606 virtual temperature, 228, 489
warm front, 327 wave action, 136 wave breaking, 12, 118, 207, 214 wave packet, 109 wave trajectory, 497 westerly jet, 280, 401, 412 wet air engine, 19, 458, 478, 502 wet Joule engine, 456, 461, 482, 483, 485, 615 wet Joule storm, 456, 484 working coefficient, 590, 591, 619 zeroth law, 1 zonal mean kinetic energy, 374
