Advances in applied mechanics. volume 13

Advances in Applied Mechanics Volume 13

Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN L. HOWARTH WILLIAM PRAGER T. Y. Wu HANSZIEGLER

Contributors to Volume 13 H. L. Kuo GEORGE VERONIS JOHN

V. WEHAUSEN

ADVANCES IN

APPLIED MECHANICS Edited by Chia-Shun Yih COLLEGE OF ENGINEERING THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

VOLUME 13

1973

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY F OR M OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION I N WRITING FROM T HE PUBLISHER.

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United Kingdom Edition published by

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CONGRESS CATALOG CARD

NUMBER:48-8503

PRINTED I N TH E UNITED STATES O F AMERICA

Contents

vii

LISTOF CONTRIBUTORS

PREFACE

ix

Large Scale Ocean Circulation George Veronis Introduction I. The Equilibrium Figure of a Self-Gravitating, Rotating, Homogeneous Mass of Fluid 11. Transformations of the Equations of Motion of a Fluid 111. The Coriolis Acceleration IV. Thermodynamic Simplifications-the Boussinesq Approximation V. Scaling of ,the Equations VI. Geostrophic Flow VII. Frictional Dissipation VIII. Modeling of Current Systems IX. The Thermohaline Circulation X. Abyssal Circulation XI. Laboratory Simulation of Large Scale Circulation (with C. C. Yang) References

2 3 6 14 18 28 33 36 42 56 72 75 90

The Wave Resistance of Ships John V. Wehausen I. Introduction 11. The Measurement of Wave Resistance 111. The Analytical Theory of Wave Resistance Bibliography References V

93 96 131 229 230

Contents

vi

Dynamics of Quasigeostrophic Flows and Instability

Theory El. L . Kuo I. Introduction 11. Tendency Toward Geostrophic Balance in Rotating Fluids 111. Simplified Hydrodynamic Equations for Large Scale Quasigeostrophic Flow IV. Permanent-Wave Solutions of the Nonlinear Potential Vorticity Equation in Spherical Coordinates V. Stability of Zonal Currents for Small Amplitude Quasigeostrophic Disturbances VI. General Stability Theory-Integral Relations and Necessary Conditions for Instability VII. Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter VIII. Pure Baroclinic Disturbances IX. Finite Amplitude Unstable Disturbances X. Instability Theory of Frontal Waves XI. Concluding Remarks References

AUTHORINDEX SUBJECT INDEX

248 250 257 265 272 276 281 291 306 316 327 328

331 336

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

H. L. Kuo, Department of Geophysical Sciences, The University of Chicago, Chicago, Illinois (247) GEORGE VERONIS, Department of Geology and Geophysics, Yale University, New Haven, Connecticut (1)

V. WEHAUSEN, Department of Naval Architecture, University of California, Berkeley, California (93)

JOHN

vi i

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Preface

I n this volume Professor H. L. Kuo treats instabilities of the atmosphere, Professor George Veronis discusses the dynamics of the ocean, and Professor John Wehausen reviews wave resistance of ships. Thus two of the three authoritative articles are studies of our fluid environment, and the third is a testimony of one aspect of man’s successful adaptation to it. This volume should therefore appeal to meteorologists, oceanographers, and naval architects, as well as to fluid dynamicists in general. I n view of recent concerns with the environment and with the relevance of scientific work to human activities, the three articles presented herein are perhaps timely. From another point of view, it can be persuasively argued that scientific work itself is an important element of the quality of life, because it bears upon the human spirit. It is hoped that the excellence and the degree of permanence achieved in these articles will lend support to this now somewhat forgotten point of view.

CHIA-SHUN YIH

ix

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Large Scale Ocean Circulation GEORGE VERONIS Department of Geology and Geophysics Yale University. New Haven. Connecticut

Introduction

......................................................

I . The Equilibrium Figure of a Se1f.Gravitating. Rotating. Homogeneous

Mass of Fluid ................................................... I1 . Transformations of the Equations of Motion of a Fluid .................. I11. The Coriolis Acceleration ........................................... A . Conservation of Potential Vorticity ................................ B. An Angular Momentum Argument for the Neglect of 2 a cos 4 ........ IV . Thermodynamic Simplifications-the Boussinesq Approximation .......... A . The Adiabatic Hydrostatic Field .................................. B. The Boussinesq Equations ....................................... C . Use and Limitations of the Boussinesq Approximation ............... V . Scaling of the Equations ............................................ A . Small Scale Motions-the f-plane ................................. B. Motions of Intermediate Scale-the /%Plane ........................ C . Large Scale Motions ............................................ VI . Geostrophic Flow .................................................. VII . Frictional Dissipation ............................................... A . Ekman Layers .................................................. B . Ekman Layers in the Ocean ...................................... VIII . Modeling of Current Systems ........................................ A. Wind-Driven Ocean Circulation ................................... B . Extensions of Stommel’s Model ................................... C . Inertial Effects .................................................. IX . The Thermohaline Circulation ....................................... A. The Pressure Equation .......................................... B. Boundary Conditions at the Top and Bottom ....................... C . Solutions by Means of a Similarity Transformation .................. D . Generalization of the Similarity Solutions .......................... E . Solutions to the Ideal-Fluid Thermocline ........................... F. The Effect of a Barotropic Mode .................................. G . The Role of Diffusion ........................................... H . Remarks about Thermohaline Circulation Models ...................

1

2 3 6 14 15 17 18 19 21 23 28 30 31 32 33 36 36 40 42 43 49 53 56 57 58 58 60 63 65 68 71

2

George Veronis X. Abyssal Circulation.. ............................................... XI. Laboratory Simulation of Large Scale Circulation (with C. C. Yang) . . . . . . A. The Basis for the Simulation ..................................... B. Discussion of More Complete Solutions.. .......................... C. The Flow Due to a Source of Dense Water.. ....................... References ........................................................

72 75 75 78 81 90

Introduction Our theoretical picture of large scale ocean circulation has grown mostly out of the development of simple models which isolate the particular phenomenon to be analyzed. In this sense dynamical oceanography differs substantially from dynamical meteorology which has progressed hand-in-hand with the amount and types of observational data that have been accumulated. The difficulties and costs of gathering oceanographic data preclude the same type of development of oceanographic theories. The present paper contains a discussion of some of these simple theoretical models together with an attempt to extend a few of them to take into account additional features which are not normally included in the models. The presentation is necessarily selective and another author would no doubt have emphasized other models or other approaches. At the outset the plan was to discuss steady state models as well as those which include transient behavior. However, as the work progressed it became necessary to restrict attention to steady models only. The opening section includes a simple model for deriving the ellipticity of the earth. I t is followed by the derivation of the equations of fluid motion in elliptical coordinates and the approximation involved in the use of a spherical coordinate system to analyze oceanographic motions. The latter is included because the errors associated with the use of spherical coordinates are normally referred to in a casual fashion and no real attempt is made to quantify them. The approximations encountered in theoretical studies of large scale flows are then discussed and the stage is set for introducing theoretical modeling. Simple geostrophic flows and their significance are presented next because they form the basis for the remainder of the paper. The study of Ekman layers and the role that they play in large scale circulation are followed by a brief discussion of turbulent transport.

Large Scale Ocean Circulation

3

T he remaining portion of the article is devoted to a discussion of three types of studies. Th e first is wind-driven ocean circulation beginning with Stommel’s (1948) model exhibiting westward intensification of ocean circulation. Some of the difficulties in extending his model are pointed out in the subsequent development which includes an attempt to incorporate density variations into the model. Theories of the thermohaline circulation (the flows driven by sources and sinks of heat and salt) are then reviewed. T h e paper ends with a section, authored jointly with C. C. Yang, on laboratory modeling of ocean circulation. T he presentation concentrates on the evolution of theoretical oceanography by means of simple models. Because this philosophy was adopted here, the important contributions of numerical studies of ocean circulation have been referred to only in passing. It is obvious that future work in the field will probably rely more and more heavily on numerical computations, particularly as the complex phenomena associated with nonlinear interactions become the focus. Th e reason for restricting attention here to simple analytical and laboratory models is that the latter still provide the theoretician with the clearest understanding of the state of the art and what steps one must take to extend that understanding.

I. The Equilibrium Figure of a Self-Gravitating, Rotating, Homogeneous Mass of Fluid Consider a mass of incompressible, homogeneous fluid which is rotating with a constant angular velocity, S2, about a given axis. T h e velocity, v, of a fluid element at distance, R,from the axis of rotation is

and the acceleration is

and is directed inward toward the axis of rotation. At a point within the fluid the equation for the conservation of momentum is

where p is the (constant) density, p is the pressure, and F is the body force

George Veronis

4

per unit mass. Taking F = -V @ where 0 is the Newtonian gravitational potential and making use of (1.2)in (1.3) yields S2 X (S2

x R)= -V(Q2R2)/2 = -V(p/p) - V@

or

+

V[(PIP)

@ - (Q2R2/2)] = 0.

(1.4)

The first integral of (1.4) yields

( p / p )+ 0 - (Q2R2/2)= const.

(1.5)

For a finite mass such as the earth the pressure at the outer (equilibrium) surface must be a constant and (1.5) becomes

CD - (Q2R2/2)= const.

(1.6)

on the outer surface. Since the gravitational potential, @, depends on the shape of the body, the problem is an implicit one. The earth is very nearly spherical and it is convenient to express the radius as the sum of the radius, a, of the smallest inscribable sphere plus the deviation, 5, due to the bulge which results from the rotation (Fig. 1) r=a+c.

(1.7)

Then the gravitational potential, @, on the surface can be written as the sum of the potential due to the mass within the inscribed sphere and the potential due to the mass in the bulge. The potential at an external point due to the spherical mass can be expressed as the potential due to a point with all of the mass at the center. Thus, @=-

where 4 is the potential associated with the bulge and G = 6.67 x dyn cm2 gm-2 is the gravitational constant. Then (1.6) becomes 1 2

- Q2R2= const.

or, making use of (1.7),

+---4

mpa3

1

G - - Q2(a+ 5)" sin2 8 = const. 3a+5 2

where 6 is the colatitude (Fig. 1).

Large Scale Ocean Circulation

5

A first approximation to the figure of the earth is to neglect 4 in (1.9) and make use of the fact that 5 < a. Since the term involving the rotation in (1.9) is small, the lowest order expression involving 5 is $rrpa2G(5/a) - +Q2a2sin2 % = const.

(1.10)

FIG.1. A sketch of a cross section (through the polar axis) of an elliptical earth with an inner inscribed sphere of radius a. The bulge due to the rotation is denoted by 5.

The difference between ( / a evaluated at the equator (0 = ~ 1 2 )and the pole (0 = 0) yields the ellipticity (C/a)eq.-

( 5 / a ) p o l e = +Q2/9,~G

= iCPa/g M

1/580,

(1.11)

where g = 4rrGpa/3 = 980 cm sec-2 is the gravitational acceleration at the earth’s surface. The value of the ellipticity given by (1.11) is about half the known value. Higher order approximations in ( / a cannot alter the result substantially and the only correction that may yield an improved estimate must come from 4. A second approximation may be obtained by retaining 4 in (1.9) while still retaining the approximation 5 < a. Then (1.9) becomes

4 + Q rrpaG[ - ;Q2a2 sin2 % = const.

(1.12)

Now the term involving 5 incorporates the geometrical effect of the bulge. On the surface the potential due to the mass of material in the inscribed sphere will be smaller where 5 > 0 and larger where M < 0. However, it is more convenient to choose an alternative manner of accounting for this geometrical effect. The potential is evaluated on the spherical surface Y = a and the geometrical effect is taken into account by assuming a surface distribution of mass at Y = a with negative mass where 5 > 0 and positive mass where 5 < 0. The surface distribution of mass at Y = a due to the bulge associated with 5 can be written in terms of a jump condition at Y = a in the radial gradient of the potential (Ramsey, 1964) as

(1.13)

6

George Veronis

where a + and a - correspond to the value of a approached, respectively, from outside and from inside the spherical boundary. Substituting (1.13) into (1.12) yields

$

+ Qa[a4/ar]:zg? = @Pa2

sin2 0

+ const.

(1.14)

The potential due to the spherical mass is included in the constant term and will not contribute to the ellipticity. If the constant term is neglected, the problem reduces to one in which there is no mass except for the surface distribution on r = a so that Laplace's equation is satisfied inside and outside the surface, i.e.

V+=O

for r < a

and r > a .

(1.15)

Equation (1.15) can be solved by separation of variables, and the pertinent solution for present purposes is

+

= K(r/a)2(1 - 3

cos2 0) = K ( ~ / Y1)~ (3 C O S ~0)

Y

< a,

r > a.

(1.16)

The application of condition (1.14) yields

K = -R2a2/4,

(1.17)

and (1.13) then gives

[a$/a~];rgT = - 5Q2a(l - 3 cos2 0)/4=

--47TGp5.

(1.18)

Hence,

C/U = 5Q2(1- 3 COS'

0)/1&Gp = 5Q2a(1- 3 cos2 0)/12g.

(1.19)

The ellipticity is therefore evaluated as

(1.20) [(/a]& = 5Q2a/4gM 1/232, i.e. the gravitational attraction of the material in the bulge serves to increase the ellipticity by a factor of 2.5. The result given in (1.20) was derived by Newton by another method. Much more exhaustive treatments of the problem have been presented by various authors (e.g., Jeffreys, 1962). The most accurate estimate of the ellipticity (1/298) is based on measurements of orbits of artificial satellites.

II. Transformations of the Equations of Motion of a Fluid T o a very good approximation the shape of the earth can be taken to be an oblate spheroid (sometimes called a planetary ellipsoid) with the minor axis of the ellipsoid along the axis of rotation. The equations of motion

7

Large Scale Ocean Circulation

can be written in terms of oblate spheroidal coordinates with gravity taken as a constant on the surface of the ellipsoid. However, oblate, spheroidal coordinates are never used for the analysis of oceanic or atmospheric currents. T he usual procedure is to work with the equations on a sphere. We shall go through the derivation of the equations in oblate spheroidal coordinates and then show the approximations involved in the use of spherical coordinates. T h e vector form of the equations for the conservation of momentum is aV

at

+V

1 vv f 2 a x v = - - vp - v@, P

(2.1)

where the centripetal acceleration terms are incorporated into the gravitational potential 0.Th e conservation of mass can be written as

5 + v . v p f p v. v = at

0.

T o write the equations in terms of any curvilinear coordinate system we introduce the generalized coordinates (ql, q 2 , q3)where the qi can be related to the rectangular Cartesian coordinates by the relations qi

= qi(X1,

~2

9

~

3

)

or

Xi

= xi(q1,

q2

43).

(2.3)

Then (Margenau and Murphy, 1949) the derivative along any direction, si , can be written as

where the Q i are defined by 3

Qi2=

C

j=l

ax (2)

In general, the following relations will be used 3

v= xvjij,

(2.6a)

j=1

V=

C

l a ij--,

]=I

Q j a q j

(2.6b) (2 .6 ~ )

8

George Veronis

where ii are the unit vectors along the three coordinate directions and the last relation follows from the first two. From ( 2 . 6 ~ )we have

where the last term in (2.8) is necessary because the unit vectors can change direction. Now the surface of the earth is, to a good approximation, an oblate spheroid for which the orthogonal surfaces are (Fig. 2) (1) oblate spheroids,

FIG.2. Oblate spheroidalcoordinates with orthogonal surfaces given by oblate spheroids with q1 = constant, hyperboloids of one sheet with q2 = constant, and planes (one is the plane of the figure) through the xp axis with q 3 = constant. Also shown are the polar coordinates in the equatorial plane.

q1 =const., (2) hyperboloids of one sheet, q2=const., and (3) planes through the x3 axis, q3 = const., where the x3 axis is taken as the minor axis of the ellipse and the xl, x2 plane is orthogonal to the x3 axis. I n the following the coordinates (x,y , z ) will replace (xl, x 2 , x3). In addition, it is useful to define the polar coordinates R, X in the xy plane. Note that A-q3. T h e intersection of the surface of the earth and a plane through the x axis is the ellipse given by

where the values of a and q1 on the surface of the earth are determined by noting that at z = 0, re = R = a cosh ql, where re is the earth’s equatorial

9

Large Scale Ocean Circulation

radius, and at R = 0, rp = z = a sinh ql, where rp is the earth’s polar radius. The curves orthogonal to the ellipses are the hyperbolas R2 a2 sin2 q2

-

2 2

a2 cos2 q2

= 1.

(2.10)

The relations between the coordinates of the different systems are x = a cosh q1 sin q2 cos q 3 ,

y = a cosh q1 sin q2 sin q 3 , z = a sinh q1 cos q 2 ,

(2.11)

R = a cosh q1 sin q2 , y/x = tan X = tan q 3 , x 2 + y 2 = R2.

(2.12)

The values of the Q, in (2.5) take the form

Q1 = Q2= a(sinh2 q1 + cos2 q2)1/2, Q3 = a cosh q1 sin q2 .

(2.13)

Furthermore, the derivatives of the unit vectors can be calculated.

.

ai, - a2 cos q2 sin q2 -_ 12, 841

ail

- tc2 sinh

-392

Qi2

-ail _ - a sinh q1 sin q2 1.3 ,

Qi2

ai,

8 1

ai, -Go,

-ai,_ -0,

-ai,_ -

841

842

a43

.,

12

-a91 _-

393

q1 - cosh q1

M

u2 cos

q2 sin q2 .

(2.14)

11,

Qi2

sinh q1 sin q2 Qi

.

u

11 -

cosh q1 cos q2 . 12 * Qi

The equations of motion expressed in terms of oblate spheroidal coordinates become dv, v1v2 - _ _ _ _ a cos q2 sin q2 -

dt

Qi

VZ2

81

u2 sinh

2Qa . 1 ap 1 a@ sinh q1 sin q2 v3 = - --- --, Qi PQi Qi 841

-__

v z

q1 cosh q1 - 3a sinh q1 sin q2

81Q3

(2.15)

George Veronis

10

v32

91--

a cosh q1 cos q2

Q1Q3

(2.16)

and the conservation of mass equation is

(2.18) where

d dt

- -= _

a +I-3 v i a

at

j = l

(2.19)

Q j aqj

The gravity potential @ appears only in Eq. (2.15) since we have assumed that the equipotential surface is an ellipsoid. Next suppose that the equipotential surface of the earth is taken to be a spherical surface. For spherical coordinates we have (91, q z , 43) = (r, 8, A),

Qi =

sin 8,

(2.20)

z = r sin 8.

(2.21)

1, Q z = r, Q3

=r

and x = r cos 8 cos A,

y = Y cos 8 sin A,

The equations of motion take the form

11

Large Scale Ocean Circulation

where d/dt is defined by (2.19) and the velocity components (v,., v a , vA) are along the directions (r, 8, A). Now, what error is involved in replacing the set (2.15) to (2.18) by the set (2.22) to (2.25)? We answer this question by relating: (a) the velocities (vI, v a , vA) in spherical coordinates to (vl, c u 2 , v 3 ) in oblate spheroidal coordinates; (b) the total time derivatives in the two systems; (c) the respective pressure gradients in the two systems; and (d) the velocity divergences in the two systems. It has already been assumed that @ is constant on a particular q1 surface in the oblate spheroidal system and on the spherical surface (r = a, say) in the spherical system. Furthermore, we make the following identification i.e. the radius, u, of our spherical earth is the mean of the polar and equatorial radii for the ellipsoid. With the foregoing relations the only difference between the two systems lies in the different metric terms associated with the variations of the unit vectors. From Eq. (2.11) we derive r = (x2 + y 2

+ z2)112= a(cosh2 q1 sin2q2 + sinh2 q1 cos2 q 2 ) 1 / 2 = a(sinh2 ql

+ sin2 q2)1/2

= a(Cosh2 q1 - cos2 q2)1/2,

(2.27)

and from (2.12) and (2.27) sin 6' = R/r = cosh q1 sin q2/(sinh2q1

+ sin2q,)lI2.

(2.28)

Equation (2.28) can be solved for sin q2 in terms of sin 8 sin q2 = sin 6'

(2.29)

1 - sin2 8/cosh2 q1

Now Eq. (2.9) for the ellipse may be rewritten as

R2 a2 cosh2 q1

+ 2 cosh2 ql[lz2-l/cosh2

-

ql] -

(2.30)

But the general expression for an ellipse with ellipticity e is written as 2 2 R2 = 1, b2 + b2(1 - e)2

(2.31)

where b is the major radius. Thus from (2.30) and (2.31) we can make the following identification between cosh q1 and the ellipticity l/cosh2 q1 = 2e - e2.

(2.32)

12

George Veronis

At the earth's surface the ellipticity is 1/298 so that a good approximation to (2.32) is 1/cosh2 q1 w 2e. (2.33) Making use of (2.32) yields (2.29) in the form sin q2 = sin 8[l - e cos2 O

+ - -1,

(2.34)

where terms of order e2 are omitted. In a similar fashion we may write cos q2 = cos 8[l

+ e sin2 8 + .

sinh q1 = -(1 (2e)u2

5

(2et'!2+):

cash q1 = -(1

.I,

.)

+ .. ' ,

(2.35)

Q1 = Q2= u cosh ql[l - e sin2 O + * * .I, Q3= u cosh q1 sin 8[l - e cos2 0 + . -1. Straightforward substitution of the foregoing into the metric terms of Eqs. (2.15) to (2.17) yields, to first order in e, a2 cos

q2 sin q2 - e cos 8 sin 8 +... -

a

Q13

a2sinh q1 cosh Q13

q1 - 1 - $e cos 28 + a

u sinh q1 sin q2 - 1 - e(1 -

a

+ e(1 - 4 cos 28) +

81 u cosh q1 cos

=sin O[l - e(1 - cos 20)

q2

= cos

Qi

Qi

*

8[l

+

.], (2.36)

*

+ e(1 + cos 28) + - - .],

2 - e(1- 2 cos 28) + . . . , a

Qi

q1 - u2 sin qz cos q2 Q13

QlQ3

-

cot 0[l

. -1

a

u sinh q1 sin q2

u cos q2 cosh

9

*.

QlQ3

+

--

+ + cos 28) + -

Q1Q3

u cosh q1 cos q2 - cot 8[1

3

+ e(l a

cos 26)

+ . . -1- _e _ sin 28 _ 2a

+..a.

13

Large Scale Ocean Circulation Hence, (2.15) to (2.18) become dt

1 - 2 e cos 20) -

a

$ [I

- e(1

-y)] 5 1 ap

dv2 -+ v1v2 (I - e cos 28) dt a 2 ~

- 252

cos 8 v3[l

e sin 20

-

1

a@

(2.37)

e vI2 e sin 28 a [ l + e(1 - cos 28)] + 2a v32

1 aP , + e( 1 - cos 28)] = - -

(2.38)

PQZ 842

+ 252 cos 8 v2[1 + e(1 - cos 28)] + 252 sin 8 v,[l

- e(l

+ cos 28)] (2.39)

3 1 avj dP -dt+ P I Cj = 1 Q j a q j

--+

~

a

e

2-41 - 2 ~ 0 ~ 2 8 ) Vl a

[l + e (1-- co;2e)1

v2-e-

2a

(2.40)

The set (2.37) to (2.40) differs from the set (2.22) to (2.25) only by terms of order e, once the identifications between velocity components, gradients, divergences, and total rates of change have been made. I t is instructive to observe that the largest discrepancies are 3e/2 so that neglecting the metric terms arising from the elliptical correction involves an error of 1/200. This error is substantially larger than the error made in oceanographic studies when the spherical radius r is replaced by its mean value, r = a, in the spherical equations of motion. Furthermore, it is worth noting that the metric terms proportional to sin28 in Eqs. (2.37), (2.38), and (2.40) have no counterpart in the spherical system. Since the first two are associated with the radial velocity, it is difficult to visualize a physical situation when they could contribute materially. The last term in Eq. (2.40) would simply add a small contribution to the preceding term. And finally, one should note that differentiated forms of the equations, such as the vorticity equation or the divergence equation, will involve larger errors when the elliptical metric terms are neglected.

14

George Veronis

I n oceanographic studies the equations are more often written in terms of spherical coordinates with latitude instead of colatitude and with the mean radius, a, replacing r in the coefficients. These equations take the form du dt

uw

uvtan+

a

a

-+---

+2Q cos

+ w - 2Q sin q5 v = - pa cos + ax’ ~

wv u2tan+ -+a++2!2 sin + u =

dv dt

a

-dw _u 2 + v~2 dt

a

--+--+ 1

1 dp pdt

- cos 2!2

au

acosq581

+ u=

1 av a

a+

a

--1 aP Pa a+ ’

1 aP

-- --g, P

(2.42) (2.43)

at-

t an

(2.41)

aw 2w + ++= 0, ar a

(2.44)

where

d --_a _

dt-at

u a +--+--+w-

acosq5 ah

v a

a

a+

a7

(2.45)

and gravitational acceleration has been assumed constant. T h e velocities u, v, w are now in the directions of increasing h, r.

+,

111. The Coriolis Acceleration In almost all studies of oceanographic phenomena the horizontal component of the Coriolis acceleration can be neglected. A crude but simple way of showing that this assumption is justified for large scale flows is to make use of the fact that the hydrostatic equation is known to be a very good approximation in this case. Then (2.43) becomes

+

and the term 2Q cos u is neglected. Now the kinetic energy equation of the system does not contain a term involving the rotation Q because v (2S2 x v) vanishes. Hence, for consistency, if 2Q cos u is neglected in (2.43), it is necessary to neglect 2!2 cos w in (2.41) since the latter term would otherwise contribute to the kinetic energy. Two additional justifications of the neglect of 2Q cos in the equations are given in this section. First the conservation of potential vorticity is

+

+

+

15

Large Scale Ocean Circulation

derived and from scale analysis the neglect of 2 Q c o s 4 is shown to be justified except in the immediate vicinity of the equator. Then the same conclusion is drawn from an angular momentum argument proposed by Phillips (1966). I t is important to observe that the neglect of the horizontal component of the Coriolis acceleration means that the direction determined by gravity, i.e., the local vertical direction, is singled out as a dominant direction in large scale flows. If seawater were homogeneous, the direction parallel to the axis of rotation would be dominant. Hence, the neglect of 2Q cos 4 automatically includes the strong constraint of stratification. In homogeneous models of ocean circulation only the vertical component of the Coriolis acceleration is kept. Therefore, the effect of stratification is included even though it may never appear in the actual model. This point will be mentioned again later.

A. CONSERVATION OF POTENTIAL VORTICITY I n large scale oceanic flows the vorticity of the fluid is an important variable because of the inherent vorticity which each fluid particle has as a result of the earth’s rotation. A quantity, called potential vorticity, is conserved in the absence of dissipation. Its derivation follows. Consider the equation of motion in vector form av -+(VXV)XV+V at

1 +252~~=--Vp-V@,

(3.1)

P

9 +pv dt

v = 0,

and suppose that some state variable, s, is conserved so that dv dt

-=-

as

at

+v.

vs=o,

(3.3)

where s = s(p, p). T he curl of (3.1) yields do

_- o.vv+wv.v=-v dt

where 0

= 2s2

+ v x v.

(3 xvp, -

16

George Veronis

Eliminating V . v from (3.2) and (3.4) yields

T h e gradient of (3.3) yields d

- vs f V v . vs= 0. dt

(3.7)

Then the scalar product of (3.6) with Vs plus the scalar product o f o / p with (3.7) leads to the expression

f

(Y) =O,

where use has been made of the fact that s is a state variable so that Vs- Vp x V p vanishes. T h e quantity, Vs-w/p, which is conserved along a trajectory, is the potential vorticity. Th e derivation given here is that of Ertel (1942). T h e principal use of conservation of potential vorticity for present purposes is to note that in spherical coordinates (Vs. w)/p takes the form

and that the component of the Coriolis term parallel to the earth's surface, 2!2 cos 4, is negligible compared to the normal component, 2!2 sin 4, if

(3.10) Denoting characteristic vertical and horizontal scales of variation of s by D and L , respectively, we can rewrite (3.10) in terms of D and L as

DIL 4 tan 4.

(3.11)

For large scale oceanic motions (i.e. when potential vorticity is a useful variable) the ratio D / L is not larger than Hence, the horizontal component of the earth's rotation is negligible at latitudes beyond a fraction of a degree from the equator.

17

Large Scale Ocean Circulation

B. AN ANGULAR MOMENTUM ARGUMENT FOR

THE

NEGLECT

OF 2 Q C O S 4

A more general argument for the neglect of the horizontal component of the Coriolis term has been proposed by Phillips (1966), who writes the set (2.41)-(2.43)before the “ shallow ” approximation is invoked as

du dt

-= F ,

+ (2Q +A)(. sin + - w cos +) r cos +

dv - = F B - (2Q+ 5 ) u sin dt Y cos 4 dw dt

2Q +-)u Y

U

cos

+

+

(3.12)

wv

-

-

(3.13)

Y

cos

+ +-

V2

(3.14)

Y

where

dh ~=h,-, dt h, = Y cos 4,

d4 ~=h,;Zi, h,

=I ,

dr ~ = h , dt ’ h, = 1 ,

and F,, F , , F, include pressure gradients and frictional forces per unit mass. These equations satisfy the angular momentum balance d -dM __ -

dt

dt

[Y

cos

+ (u + Qr cos +)]

=Y

cos+F,

.

(3.15)

If the “ shallow ” approximation is made in the form h, = a cos +,

h, = a

(3.16)

then (3.12) to (3.14) are written with a replacing Y, i.e., the set (2.41) to (2.43) is obtained. However, as Phillips points out, the set (2.41) to (2.43) does not have the angular momentum balance which is obtained by replacing r in (3.15) by a. In fact, there is no function A(+,Y) such that A F , = dM/dt, in general. If the terms 2Q cos w + (uw/a) were absent from (2.41), the equation would give the proper (approximate) angular momentum balance. If the momentum equations are recast into the vector invariant form

+

av _ F +g - V ( i at -

v2)

+ v x curl(v +Rr

cos

+ i,)

(3.17)

18

George Veronis

and if the relations (3.16) are now used in all curvilinear operators, the final set of component equations is du

-dt= F A

+ (22 +La cos) 4 v sin 4

(3.18)

--

) u sin 4

(3.19)

dt

dw

-=

dt

F, -g .

(3.20)

Hence, the horizontal component of the Coriolis parameter as well as the metric terms which gave rise to the angular momentum difficulty are now absent. Consistent with this approximation,the term 2w/a in (2.44) should be neglected. Although Phillips’ argument is correct, there are obvious cases where the final set (3.18) to (3.20) is inapplicable. Even with the “shallow” approximation, flow between two concentric spheres in a laboratory cannot be analyzed by this set of equations. In other words, some justification for the use of the simplified set of equations must be provided by the particular physical system under consideration. For large scale oceanic flows, the set (3.18) to (3.20) is consistent and represents a good approximation.

IV. Thermodynamic Simplifications-The Boussinesq Approximation The equations of motion inherently contain descriptions of a wide variety of processes ranging from high frequency, small scale phenomena such as sound waves, to low frequency, large scale motions which describe the general circulation. For the study of a particular class of phenomena it is very helpful to “filter ” the equations so that a simplified set of equation is available. For example, for investigations of large scale flows it is common practice to consider seawater as essentially incompressible. The ensuing simplification is an enormous help for it reduces the equations to a much more tractable form. In the present section the analysis for deriving this so-called Boussinesq set of equations is given and some of the limitations of the simplified equations are pointed out. The starting point is a brief discussion of the thermodynamic properties of seawater. Seawater contains many dissolved salts (Sverdrup et al., 1942) the most abundant of which is sodium chloride. Because these salts appear in ap-

Large Scale Ocean Circulation

19

proximately the same relative concentrations in seawater,* for dynamical purposes it suffices to lump them together and to define the salinity, s, as a composite measure of the concentrations of the salts. T h e density of seawater is, therefore, a function of salinity as well as of temperature and pressure. p = p(s,

T , PI.

Empirical formulas for determining p in terms of s, T , and p are summarized by Sverdrup et al. (1942) and by Fofonoff (1962a). Surface water in regions of low salinity and high temperature has a density as low as 1.02 gm/cm3. Water in the deepest trenches is subjected to the highest pressures and has a density as high as 1.07 gm/cm3. Hence, the maximum range of density is only about 5% so that where p appears as a coefficient it can be replaced by its mean value, pm, and a maximum error of only 2.5% is incurred. However, when variations in density are important as driving terms for the motion field, replacing p by pm means that the dynamic effects of density variations are ignored. T h e Boussinesq approximation, which is developed below, allows one to ignore the density variation when it contributes only a small quantitative correction and to keep the variation where it is dynamically significant.

A. THEADIABATIC HYDROSTATIC FIELD I n the absence of motion the conservation of momentum reduces to the hydrostatic equation

vp = -gp. Hence, surfaces of constant density and pressure are level (horizontal) and the only admissible variations in p and p are in the direction of gravity,t z. To complete the description of the motionless state it is necessary to specify the equation of state, the thermodynamic process which controls the state variables, and known values of the state variables at some level. Even in this case, however, there will be an infinite number of systems which can satisfy these constraints because seawater is a multicomponent fluid (salt and temperature can vary). I n order to specify a unique static state the salinity is here taken to be constant, with the value s = 34.85%. Hence,

* Carritt and Carpenter (1958) discuss the variability of relative concentrations. t If diffusion of momentum and heat are included, there can be no relatively static state for a rotating system because the equations describing eyuipotential surfaces are, in general, not solutions to the steady diffusion equations. The slow flow which results is called Sweet-Eddington flow. The implicit assumption here is that it is negligible compared to the motions generated by driving forces of the system.

20

George Veronis

variations in s will occur only via interactions with the bounding media (the atmosphere, the bottom etc.) and will normally be associated with motions of the fluid. With s a known constant the density is a function of T and p only. Now, the first law of thermodynamics is

6 g = T dq = C , dT

+ T(aq/ap),dp,

(4.3)

where q is specific entropy, c, is specific heat at constant pressure and Sq is the specific quantity of heat added. For an adiabatic process S q = 0 and it follows from (4.3) that the vertical variations of the state variables are related by

where the hydrostatic equation has been used to give the right-hand term. T h e thermodynamic relation (%IaP)T = - ( a v / a T ) ~ 7

(4.5)

where v is specific volume, can be used to write (4.4)in the form

( a T / a ~= ) , -g a TIC, (4.6) where o! = I/v(av/aT), is the coefficient of thermal expansion. Values of o! and c, as functions of s, T , and p are tabulated by Fofonoff (1962a) who also gives an empirical formula for (aT/ap),as determined by Fofonoff and Froese (1958). For present purposes it suffices to note that (aT/az),for s = 34.85 % has values ranging from O.O16"C/1000m at the surface with T = -2°C to 0.209"C/1000m at 10,000 m depth with T = 4°C. A typical range for the adiabatic temperature is 0.6"Cfrom the surface to 4000 m depth. Hence, the adiabatic temperature gradient is small relative to observed vertical temperature gradients in the upper layers of the ocean but the two are comparable in deep water. Formulas for the calculation of density as a function of s, T , and p are summarized by Fofonoff (1962a). Hence, from the hydrostatic relation (4.2), the adiabatic temperature gradient (4.6), and the formula for p as a function of T and p (s = 34.85 yo),one can calculate the adiabatic density field given p and T at one level. As stated earlier, the observed range of values of p is from 1.02 gm/cm3 at the surface to 1.07 gm/cm3 at a depth of 10,000 m. T h e range of adiabatic density variations is therefore comparable to the observed range. Hence, in contrast to temperature variations the density variations do not differ much from the adiabatic density variations, i.e. the principal contribution to changes in density comes from the pressure field. T h e adiabatic, hydrostatic state will be denoted by variables with subscript a. 9

21

Large Scale Ocean Circulation

B. THEBOUSSINESQ EQUATIONS T he conservation of momentum for a fluid is now expressed as

dv dt

p-+2pQ

x

v = -0j-gj5,

(4.7)

where the pressure and density have each been divided into an adiabatic part and a perturbed (tilde) part, and p = p a ri; and p = p a +fi. T h e adiabatic, hydrostatic field has been subtracted from the right-hand side. On the left-hand side the density appears as a coefficient. Using the fact that the total variation of density is small relative to the mean value enables one to approximate (4.7) by

+

dv -++a )( v = - -1 v p - g - ,P (4.8) dt Pm Pm where pm rn 1.035 is a mean value of the density for the ocean. T h e formal requirement for (4.8) to be valid is

< 1.

(4.9) T h e equation of state is linearized about the adiabatic state, as discussed below. Thus, (IP-Pm(/Pm)=6

P = Pa

+

(g)*,s T + ($) b + ($) T. s

P.T

5

+-

*

..

(4.10)

Hence, the perturbation density can be written as P/pa M P/Pm = - aT

+K$ +Y?,

(4.11)

where

If (4.11) is substituted into the right-hand side of (4.Q the vertical component of (4.8) has the terms

(4.13) on the right-hand side. For oceanic motions the vertical scale of variation,

H , is lo5 cm so that the first term has magnitude 1) I / H . T h e isothermal compressibility, K , has a magnitude of 4.5 x 10-l1 in cgs units (Sverdrup et al., 1942) so that the t e r m g K j has magnitude 1fi1 / H swhere H , = l/gK is the scale height for seawater. Its value is 2 x lo7cm, i.e., substantially larger

22

George Veronis

than the vertical scale H , so that the pressurefluctuation term in the equation of state is dynamically insigniJcant. This means that the equation of state (4.11) can be approximated by

+

p"/pm Fz: 03 ys".

(4.14)

T h e equation for the conservation of mass takes the form (4.15)

T h e first and third terms are O(6)compared to the last term so that both can be neglected. T h e second term involves a time scale which must be specified. If velocity and length scales, V and L, respectively, are chosen, and if a/& is taken to be O( V / L )(i.e., local changes are due to convection), the second term is the same order as the third. An alternative procedure is to specify the time scale on the basis of some other physical process, such as buoyancy oscillations. I n either case the second term can be neglected. T h e reason why +/at must be considered separately is that it is the term which is associated with acoustic phenomena and one must specify the physical process in order to filter out acoustic waves. Hence, to lowest order (4.15) takes the form v*v=o. (4.16) T h e equation for the conservation of salinity may be used in its primitive form dfldt = 0. (4.17) T h e first law of thermodynamics is p (deldt)= -pV

v,

(4.18)

where e is specific internal energy. For the hydrostatic, adiabatic state the first law was expressed in the form (4.3) to give the adiabatic temperature gradient (4.6). It is convenient at this point to express the change in internal energy in terms of increments in specific volume, temperature, and salinity for a process at constant pressure as de dt

dF dt

p - = c, --p v

.v +

ds" c, -, dt

(4.19)

where c, is the specific heat per unit change of salinity. Even though the actual value of c, is not available for seawater, the form (4.19) suffices for the purpose of this development. Substituting (4.19) into (4.18) yields dp dt

ds" dt

c -+cc,--0.

(4.20)

Large Scale Ocean Circulation

23

In view of the conservation of salinity (4.17), Eq. (4.20) reduces to

dF/dt = 0.

(4.21)

Since variations in salinity occur only in the perturbed state and since the adiabatic temperature gradient (4.6)has already been evaluated, Eqs. (4.17) and (4.21) can be used for the perturbation salinity and temperature. In summary, the equations take the following form when the Boussinesq approximation is employed. For the adiabatic, hydrostatic state,

S = const.,

aT/az = -gaT/c,, For the perturbed state

V p = -gp, p = p( T , p , S).

dv 1 -+2Q x v = - - v p - g e , dt Pm V . v 0, Ps/pmz= -a8 =z

dsldt = 0,

d8ldt = 0,

(4.22)

(4.23) Pm

+ YS,

dps/dt = 0,

where 8 is the potential temperature, the adiabatic temperature gradient having been accounted for in the last of Eqs. (4.22);s is the deviation of the salinity from some constant value; and is the potential density, expressed as a linear combination of 8 and s.

C. USE AND LIMITATIONS OF THE BOUSSINESQ APPROXIMATION The set of equations (4.22) and (4.23) is sometimes called a Boussinesq set because it was derived with the use of the Boussinesq approximation. In laboratory experiments and other very shallow layers of fluid the Boussinesq set is a very good approximation to the primitive set of equations for certain classes of motions. For shallow layers, molecular processes (conduction, diffusion, etc.) can be added to the system in a straightforward fashion. For oceanic flows of even moderate scale it is necessary to treat mixing processes by some approximate technique (eddy coefficients, mixing length theory, or some analogously crude method). The great advantages of the Boussinesq system are (a) the filtering out of acoustic waves and other phenomena associated directly with compressibility, (b) the linearization of the equation of state for the perturbation variables, and (c) the relatively simple form of the energy equation. Potential density is the dynamically significant part of the density field, the remaining part being assumed to adjust adiabatically when an element is displaced vertically. The widespread use of the Boussinesq approximation

24

George Veronis

in analyses for single-component fluids and the straightforward interpretation of the adiabatic temperature and density for such fluids would lead one to expect an equivalent simplification for the oceanic case. However, there are limitations on the use of the Boussinesq system for large scale ocean circulation problems. T h e problem is immediately apparent upon examination of observed Auid properties in the deep ocean. Table 1 shows typical values of T and S TABLE 1 CALCULATED PROPERTIES OF TWO SAMPLES OF SEAWATER PRESSURE OF 4000 dB WITH THE SAME DENSITY BUT DIFFERENT TEMPERATURES AND SALINITIES

AT A

T("C) ~ ~ O i o o )

u(gm/liter) &"C) u@(gm/liter)

Sample 1

Sample 2

2.30 34.90 45.93 1.95 27.92

2.09 34.85 45.93 1.74 27.89

and calculated properties for two elements of seawater at 4400 m depth. The two elements have a common value of density" of 1.04593 but different temperatures and salinities. Observe that the two elements have different valuest of a, but the same value of a. Since the in situ density determination states that the two elements have the same density, study of the potential density alone would lead one to the incorrect conclusion that sample 1 is heavier than sample 2 and that it would, therefore, seek a lower level according to inviscid gravitational stability theory. Table 2 contains typical values of T and S and calculated properties for two elements of water, also at 4000 m depth, but with slightly different densities and substantially different values of T and S. In this case, sample 2 is actually heavier than sample 1 according to the in situ density determination but sample 1 appears heavier according to a, values. On the basis of

* To avoid unnecessary decimal digits oceanographers customarily use the quantity sigma, defined by u = 100(p - 1) where p is expressed in cgs units. For example, seawater with a density of 1.04593 has a value of u equal to 45.93. t T h e quantities u, 8, and were calculated with the computer subroutines of the Woods Hole Oceanographic Institution. Density (or a) is obtained from Ekman's (1908) empirical formula. Potential temperature (8) is calculated from the polynomial expression by Fofonoff and Froese (1958). Potential density (no)is derived from the Knudsen (1901) formula for u with p = 0 and with the temperature replaced by 8.

25

Large Scale Ocean Circulation

in situ density, inviscid gravitational stability theory predicts that sample 2 should sink and sample 1 should rise (relative motions) whereas the same criterion, based on potential density, indicates a sinking of sample 1 and a rising of sample 2. The difference between the densities of the fluid parcels in this example is small (a difference of 0.01 in ue is close to the limit of reliability) but it is typical of observed differences in abyssal waters. TABLE 2

CALCULATED PROPERTIES OF T w o SAMPLES OF SEAWATER PRESSURE OF 4000 dB WITH DIFFERENT DENSITIES, TEMPERATURES, AND SALINITIES

AT A

WC) S(O/OO)

u(gm/liter) wc) ae(gm/liter)

Sample 1

Sample 2

2.21 34.90 45.95 1.86 27.92

1.89 34.85 45.97 1.55 27.91

Figure 3 shows the distribution of 0 0 for a vertical section in the western Atlantic Ocean as calculated by Lynn and Reid (1968) from observed data. According to the distribution of potential density the bottom kilometer or so of water is unstably stratified over a large range of latitudes south of the equator. I n situ stability calculations show that the water is, in fact, gravitationally stable. One must conclude, therefore, that the use of potential density for determining the gravitational stability of water in the deep ocean can lead to incorrect results. This apparently paradoxical instability of abyssal water can be explained as follows. In Fig. 4 isopycnals for seawater are shown for depths of 0, 2000, and 4000m. The curves slope more steeply upward to the right as the depth (or pressure) is decreased, i.e. the entire pattern of isopycnals rotates counterclockwise as p is decreased. Hence, a layer of fluid which is neutrally stable or even slightly stable near 4000 m depth appears to be unstable if the reference surface is the top surface as it is for potential density. It should be observed that the apparent instability is present even when the equation of state is highly nonlinear (the complete expression for u at each level was used to generate the curves shown in Fig. 4).T h e difficulty can be avoided only by allowing the various coefficients, whether they be the coefficients in the complete expression for (T or in the linearized Boussinesq form, to be functions of pressure. Allowing a and y in (4.14) to be

8p0S

I

60° I

I

40' I

I

2po

I

00 I

I

2p0

I

4P0

I

I

8p0N

FIG.3. A plot of potential density, ug, as a function of latitude and depth in a longitudinal section west of the Mid-Atlantic Ridge, according to Lynn and Reid (1968). (Courtesy of Deep-sea Research.)

Large Scale Ocean Circulation

27

functions of pressure would make the equations analytically intractable. However, with negligible error, one could approximate the pressure effect by taking u and y to be functions of depth. T h e equation of state would then be more complicated than the form with u and y constant but it would at least be linear.

S FIG.4. Curves of constant density plotted in the potential temperature-salinity plane for depths of 0 (solid curves), 2000 m (dashed curves), and 4000 m (dash-dot curves).

The issue discussed in this section is not important for phenomena such as internal waves where the particle motions are restricted to local regions. However, for large scale circulation problems, where fluid at the surface in polar regions may sink and eventually make its way to the bottom, the use of the Boussinesq approximation in its simplest form would preclude the possibility of direct verification of particle trajectories by comparing theory with observation. For example, water mass analysis shows that the water near the bottom between 30"s and the equator in Fig. 3 originates in the Antarctic Ocean (Weddell Sea) and the tongue of water with ua > 27.9 comes from the Greenland Sea. At the points of origin, water from the Greenland Sea is denser than the water from the Antarctic. This feature is reflected in the values of a,. In the abyss these waters stratify with Antarctic water lying below the water from the Greenland Sea. This inversion is a direct consequence of the effect of pressure on the densities of water of different temperatures and salinities. T h e simple Boussinesq system could not lead to the observed distribution because, according to the Boussinesq system, water which is densest at the point of origin would, in the absence of mixing, end up in the deepest part of the ocean.

28

George Veronis

V. Scaling of the Equations The equations of motion can be scaled to exhibit important balances for large scale flows. By “large scale” we mean flows whose characteristic horizontal scales are substantially larger than the vertical scale (or depth). The scaling procedure makes use of observed (or perhaps only plausible) magnitudes of quantities to reduce the equations to a simpler form. Results from subsequent analyses of these equations can be compared to observations of appropriate phenomena or features in order to obtain an a posteriori check on the validity of the model equations. For this purpose the variables and operators in the Boussinesq equations of motion will be represented as follows :

a

i ----a cos

-=TQ6,, at

u = VU’,

a

i

l a

+ aA - L aa,

v = VV’,

a

w = WW’,

a

1

1 H”,

a+-

-=-

p = (Ap)p’,

p = (AP)p’,

ar

where the prime quantities and the 6 operators are nondimensional and of order unity and the scales T , L, H , V , W, Ap, and AP are to be determined by restricting attention to certain chosen magnitudes which reflect processes of interest. T h e Boussinesq equations on a sphere then take the following nondimensional form : 7St u

T

+ R[u 6, u + v 6, u +(p/s)w 6, u + vpuw - ~ u tan v $1

6,v

+2p cos 4 w - 2 sin 4 v = - PS,p,

(5.1)

+ R[u6,v + z, 6, w + ( ~ / E ) 6,v w + qpvw +7u2 tan 41 + 2 sin 4 u = - P 6,p,

(54

+

+

6 , ~R(u 6 , ~ Z, 6 , +~( P I & ) 6,w - r p 2 - qv2)]- 2 s cos = -P 6,p - (Fs2/R)p,

F ~ [ T

7atp

S,u

+ R[u a a p + v +

+ S,v

6

- qv tan

6

~ ( ~ 1 s S,P]= ) ~

0,

+ + 2rpw + ( p / s ) 6,w = 0,

+u (5.3)

(5.4) (5.5)

where the primes have been dropped and the following nondimensional parameters appear

R = V/QL, p = W/V, 7 = L/a, E =H / L , F =g HQ2 Ap/pm N 2 / Q 2 , P = AP/QVLpm.

(5.6)

T h e parameter R is called the Rossby number and F is a ratio of frequencies.

Large Scale Ocean Circulation

29

Formal expansions of the variables in powers of each of the small parameters in (5.6) would be messy. It is possible to make some simplification by means of two assertions which lead to balances of primary interest. The first assertion is that the horizontal scale is always much larger than the vertical scale so that E < 1. This assertion is taken to be a dominant one in the sense that all conclusions which result from imposing it as a restriction are valid for the flows to be considered. Since the parameter p = W/V is divided by E in several places, it is necessary to make a statement about the amplitude of p. Hence, the second assertion is that the vertical divergence is always upper bounded by individual horizontal divergence so that p 5 E or, equivalently, W < VH/L. Several important consequences follow from these two assertions. First, in the vertical equation of motion the Coriolis term is O ( E )and the remaining acceleration terms are O(2). The bouyancy term is ([Fe2/R]).I n the ocean F 9 1 and R < 1 almost everywhere so that the parameter F@/R may be O(1). Since large scale flow is is known to be essentially hydrostatic, the parameters P and Fe2/Rmust be of the same order. In the following, the parameter P will be set equal to one and F 2 / R must then be of order unity also. An important consequence of this argument is that the lowest order $ow will be hydrostatic and geographic. T h e latter balance follows directly when P = 1 and both T and R are small. Second, the vertical convection term in each of the first four equations is at most of the same magnitude as the horizontal convection terms. Third, the metric terms involving the radial velocity, w, are at most O ( E )and hence can be neglected to lowest order. Fourth, consistent with the hydrostatic relation and the second assertion ( p 5 E ) above, the horizontal component of the Coriolis terms, 2p cos 4 w, in (5.1) can be neglected to lowest order. Therefore, the equations simplify to the following set 7

6,u

+ R(u 6,u + v 6,u + w 6,u

T

6,v

+ R(u 6,v + v 6,v + w 6 , v +qu2 tan 4) -+ 2 sin 4 u = -6,p, 7- 6,p

+ R ( u 6,p

6,u

+ 6,v

-

~ u tan v 4) - 2 sin 4 v = -S,p,

+ v 6,P

- ~v tan

+ w 6,P) = 0.

4 + 6,w = 0,

(5.7) (5.8)

(5.10) (5.11)

where p has been set equal to its upper bound, e, and Q = F 9 / R . Even though Q is order unity it will be retained as a tracer. The last set of equations is the conventional starting point for almost all studies of large scale flows (with frictional and diffusive processes normally introduced by means of parameterizations of small scale phenomena).

30

George Veronis

Further simplification of the set (5.7)-(5.11) requires additional restrictions imposed by the particular phenomenon under investigation. For example, for flows whose horizontal length scale is much smaller than the radius of the earth the condition 7 < 1 leads to a simplification because some of the geometrical distortion terms associated with the spherical geometry can be ignored. Another type of simplification results for relatively weak flows for which the condition R < 1 is valid. For flows which satisfy both of these latter conditions, it is necessary to compare the relative magnitude of R and q in order to derive the appropriate ordering of the equations. Large scale flows in the ocean can be divided into several types, depending on the horizontal scales which are involved. Phillips (1963) has designated two of these by analogy to atmospheric motions as: (a) motions of type 1, in which q 5 R and R < 1, and (b) motions of type 2, in which R < 1 and q 1. T h e specific conditions and simplifications are developed below and a set of equations is presented for three different scales of motion.

-

A. SMALLSCALEMOTIONS-THEPLANE T h e Gulf Stream meanders and the eddies observed by Swallow and Hamon (1960) are examples of type 1 motion. The horizontal length scale, L, of these motions is of the order of 100 km; hence, it is large compared to the depth ( E < 1) but small compared to the scale of the oceanic basins, or equivalently, of the radius of the earth (6 1). Other typical magnitudes for Gulf Stream meanders are

H-105 cm,

Ap/pm-10-3,

V-102 cm sec-'.

(5.12)

Hence, the following magnitudes obtain for the parameters in Eqs. (5.6) ~=10-',

Q-1,

P-1,

R-IO-l,

~-10-',

(5.13)

where AP, and therefore P, is determined by the hydrostatic balance condition. An additional simplification is obtained by making use of the fact that q is small and expanding the trigonometric coefficients about the latitude near which the motion is to be studied. Thus choose a (mid-) latitude, and write 4 = d o 4' = d o ( Y l 4 = d o qy', (5.14)

+

+

where the linear distance, y , has its origin at yields sin 4 and cos 4 about sin 4 = sin do

+

+ = 4,.

Then expanding

+ qy' cos do + . . . , cos 4 = cos do- qy' sin do4 . . . . (5.15)

31

Large Scale Ocean Circulation From the original east-west derivative we can write

1

a

cos40

a

1

-

cos4,

a (5.16)

so that the nondimensional rectangular Cartesian coordinates, defined to replace A. Analogously,

XI,

has been

(5.17) so that x, y , and z form a local rectangular coordinate system. T o lowest order in 7 the equations are those for a fluid in a uniformly rotating system

au + RV

r-

at

T

aP VU-f v = -ax ’

(5.18)

av + RV VV f f u = --,aP -

(5.19)

aPlaz = -Qp,

(5.20)

at

aY

a, + RV -

(5.21)

- + +-

(5.22)

Vp = 0, at au av aw = 0. ax ay az Here, the primes have been dropped T

.

a ax

a +w ii az

v V -siU - f v ay

and f = 2 s i n + , = c o n s t . T h e set (5.18) to (5.22) is sometimes called an f-plane system and is appropriate for the study of smaller scale properties of large scale phenomena but it is inadequate to describe larger scale flows where the sphericity of the earth is important.

B. MOTIONS OF INTERMEDIATE SCALE-THEP-PLANE Theoretical studies of wind-driven ocean circulation of intermediate scale or larger must take into account the nonuniform vertical component of the Coriolis parameter. A set of equations for ocean basins which lie on the equatorial side of 45”latitude can be derived with geometrical considerations. Again choose a latitude 4, as in (5.15) and assume that 7 = L / a 1 so that sin 4,cos 4,and 8, can be approximated as in Eqs. (5.15) and (5.16).

+

George Veronis

32

Now compare the terms 6,v and r]v tan q5 in the continuity equation (5.11) 6,V=-

av

7vtan+=qtanq5,v

"'

sin q50 cos q50

aY '

+-

-1

. (5.23)

T he conditions 7 < 1 and do< 45" require 7 tan q5, < 1 and the metric term can be neglected to lowest order. Similarly the metric terms in the substantial derivatives can be neglected. Th e Coriolis term in (5.7) can be written as 2 sin q5 v = 2 sinq4, v(l qy' cot q5, + -). (5.24)

+

-

Now even though one assumes 7 < 1, the term qy'cot 4, cannot be neglected because q50 < 45" and cot q5, > 1. Indeed, with decreasing q50 the term 7y' cot 4,may dominate. Th e equations in this case reduce to the 2q set (5.18)-(5.22) with the Coriolis parameter written as 2 sin q5, cos d o y -fo 8 y and they are called the 8-plane equations. They are used quite extensively in theoretical oceanography and almost exclusively in studies of wind-driven ocean circulation (Stommel, 1948; Munk, 1950; Fofonoff, 1954; Bryan, 1963). In cases where the 8-plane is used for basins poleward of 45" latitude an obvious error is incurred in neglecting the metric term in the continuity equation while the 8 term is retained.

+

+

C. LARGE SCALE MOTIONS When the motion to be investigated has truly large length scale, e.g., the global circulation of the oceans brought about by differential incident solar radiation between equatorial and polar regions, the equations take on a different form. Here, L = a = 6 x1O8crn, V-1 cmsec-', 7-1, and R If gravity waves or other high frequency phenomena are to be incorporated into the analysis, the time scale is small and the time derivatives must be retained, i.e., T 1. For long period quasi-steady motions, T < 1 and the use of R < 1 simplifies the equations at lowest order to the form 2 sin 4v = (l/cos q5) ap/aA, (5.25a)

-

-

2 sin q5 u = -ap/+,

(5.25b) (5.25~)

dpldt 1 cos q5

= 0,

aua/\+ a42 (v cos q5) ] + -

:

- = 0.

(5.26a) (5.26b)

Large Scale Ocean Circulation

33

When dissipation processes are included, either via parameterization of boundary layer effects or heuristic treatments of internal dissipation processes, Eqs. (5.25) and (5.26) are used for general circulation models of the ocean. Many of the so-called thermohaline circulation studies in which large scale circulation is driven by imposed surface density conditions (Welander, 1959, Robinson and Welander, 1963 ; Needler, 1967; Kozlov, 1966; Veronis, 1969) make use of amodifiedor extended formof (5.25)and

(5.26). Although the three sets of equations discussed above are applicable for the.study of suitably restricted phenomena, it is obvious that the phenomena are not isolated from each other and that some interaction will occur and indeed may be of great importance. Thus, even though some properties of the motions of type 1 may be studied by the equations (5.18) to (5.22) the generation of such motions depends on the overall circulation of the ocean. For example, the Gulf Stream is a necessary part of the overall circulation of the oceans and the dynamical balances which describe this smaller scale feature must be appended to the equations (5.25) and (5.26) via either a boundary layer or some other asymptotic procedure when large scale circulation is studied.

VI. Geostrophic Flow Steady, linear, frictionless flow in a rotating system is geostrophic and hydrostatic. Because of the fundamental nature of geostrophic flow in rotating fluid theory, a simple example of geostrophic balance is given here. This example is essentially that which was first presented by Phillips (1963) but a simple geometrical argument is used. Consider a right cylindrical basin partially filled with water and rotating about its axis of symmetry with constant angular velocity, R. The parboloidal free surface (Fig. 5a) represents a constant pressure surface and can be calculated as a consequence of the hydrostatic relation and the exact balance between the radial pressure force and the centripetal acceleration

( l h ) ap/az= -g ,

(1/p) ap/ar = n2r.

(6.1)

The solution, subject to the condition that p = 0 at z = h, is

where h is the height of the free surface above the bottom of the vessel and h, is the height at Y = 0. If the container were rotating with constant

34

George Veronis

FIG.5 . The free surface height, h, is shown as a function of the radial coordinate, Y, for rotation rates of (a) R, (b) R IARl, and (c) R everywhere except 0 lhnl between the dashed lines.

+

+

+

angular velocity Q AQ (Fig. 5b) the same equations with Q replaced by Q As2 would hold. The free surface would have a steeper slope for AQ >O. Now consider a thought experiment in which the container rotates with angular velocity Q but a portion of the fluid, shown by the region between dashed lines in Fig. 5c, rotates with angular velocity Q AQ. The free surface would have the shape given by (6.2) everywhere except above the special region where S2 must be replaced by AQ in (6.2). Now subtract Eqs. (6.1) from the corresponding equations with Q replaced by Q AQ to obtain the following balance of forces of the second system (the fluid in the special region) relative to the rotating frame of the first system

+

+

+

+

(l/p) 8 A p / & = 2Q AQr +(AQ)%.

(6.3)

Large Scale Ocean Circulation

35

<< Q, (6.3) can be approximated by ( l i p ) api/ar = 2 ~ , where p' E Ap and A!& = v'. Th e relative velocity, v', and the perturbation pressure, p', are said to be in geostrophic balance. T h e flow is parallel If AQ

to the isobars. Large scale flows in the ocean are in quasigeostrophic balance, i.e., the balance is geostrophic only to lowest order and higher order terms are required to form a closed problem. However, geostrophic flow at lowest order introduces such a simplification that otherwise intractable or extremely complicated systems can be reduced to a much mqre tractable form. T he conditions for quasigeostrophic flow are that: (a) the time scale be substantially larger than the rotation time (T << 1 ) ; (b) the flow be relatively weak (R < 1 ) ; (c) the length scale be large so that the hydrostatic relation is satisfied; and (d) the flow be outside frictional boundary layers (to be discussed below). Under the foregoing conditions the equations of lowest order are

fk X v = -Vp v

*

(6.5a)

- QPk,

(6.5b)

(pv)= 0,

+

where k is the unit vector in the vertical direction and f =fo , f,, /?y,or 2 sin 4 depending on whether the scale of the flow is small, intermediate, or large in the sense used in the preceding section. When the Boussinesq approximation is used, p refers to the fluctuation density and the equation for the conservation of mass can be divided into the continuity equation and the incompressibility condition. When the pressure is eliminated by cross-differentiation of the vertical equation and each horizontal equation of motion, the results in spherical coordinates are (6.6a) (6.6b) and the radial (or locally vertical) shear of the density-weighted velocity can be obtained from horizontal variations in the density field. Equations (6.6) are the so-called thermal wind equations and are used extensively for obtaining vertical shears from measured horizontal density variations. In theoretical studies they form the primary balance for flow in the interior of basins. T o determine the flow explicitly it is necessary to specify the arbitrary functions which result when Eqs. (6.6) are integrated vertically.

36

George Veronis

In field studies such a determination may be made by obtaining measured values of u and v at a single depth for each location. For theoretical work it is necessary to incorporate boundary layer processes to evaluate the arbitrary functions. An additional relation which emerges from the geostrophic-hydrostatic system is obtained by eliminating the pressure field from the horizontal equations of motion to obtain

a ax

- (fpu)

a +(fpv cos 4) = 0. a4

Use of the equation for conservation of mass yields

where /3 = 2 cos 4. Hence, the meriodional velocity, v , is proportional to the vertical divergence of the density-weighted vertical velocity. If (6.8) is integrated betwen two levels where pw is known, the meridional mass transport is determined. Finally, the vertical derivative of (6.8) combined with (6.6a) yields

which expresses the vertical curvative of the density-weighted vertical velocity in terms of the change of the density along a latitude circle. This relation, as well as (6.8), is important for deducing properties of the vertical velocity since the latter cannot be measured. T h e geostrophic relations given in this section are good approximations for large scale flows. Those relations which involve differences between horizontal derivatives and subsequent cancelling of terms are less reliable because the individual terms which cancel are large and only relatively small residuals remain. Hence, it is necessary to exercise some care in the ordering procedure of the more complete equations in order to determine the validity of the derived relations.

VII. Frictional Dissipation A. EKMANLAYERS Transfer of momentum between neighboring fluid particles in the ocean takes place by means of turbulent processes. T h e inadequate development of the theory of turbulence has resulted in approximate treat-

37

Large Scale Ocean Circulation

ments of turbulent transfer. It is customary to draw an analogy between molecular and turbulent processes and to assume that the latter are similar to the former, the only difference being that eddy or Austansch coefficients are used instead of molecular coefficients. This procedure is still by far the most common even though it is known that substantial errors, qualitative as well as quantitative, can result. However, in spite of the inadequacies of the detailed theory it is possible to avoid some of the errors by making use of certain integral features of the processes. It is helpful to consider an exact model first because it provides a useful guide for subsequent considerations. Consider a homogeneous, rotating fluid in the region -co 5 x 5 0 which is driven by steady, imposed stresses or velocities at x = 0 and suppose that the boundary values, hence all quantities, are independent of the horizontal coordinates. The exact equations for such a flow can be expressed in nondimensional form as 26 = E

a2u"/az2,

(7.la)

2u" = E

a2qaz2,

(7.lb)

where the velocities for this model are denoted by a tilde, E = v/fiDais called the Ekman number, D is a vertical scale length, and v is the kinematic viscosity. The solution, subject to the conditions u", d -+0 as x + - co and (u", d),= = ( C 0 , do) is

5 - do sin OeC, d = (do cos 5 + Go sin 5)ec,

(7.2a)

C = (Go cos

(7.2b)

where 5 = Z / E " ~ This . solution was first derived by Ekman (1905). The e - l depth (dimensional) is x, = z/v/fi = DE1J2and is called the Ekman layer depth. In the hodograph (u, v)-plane the solution traces out a spiral so that the velocity vector decreases in magnitude and rotates with increasing distance from the boundary. Hence, the effect of surface processes is confined by the rotation to a limited region near the boundary. It is evident from Eq. (7.2) that the nondimensional stress, To(=aV/az), at the boundary is O ( E - 1 / 2 )If . the stress instead of the velocity vector Go is prescribed at x = 0, the solution to (7.1) is

+ 5+ v" = + E 1 / 2 [ ( ~+ o zTOY) sin 5 +

u" = Q E 1 / 2 [ ( ~ 0 z-r0Y) cos

(T~' - T ~ Y )sin

{let,

(7.3a)

cos 5]e(,

(7.3b)

( T ~ Y- T ~ ' )

where T~~ and -ray are O(E-Il2)quantities. If the region of interest is -1 5 z 5 CO, the above solutions are valid provided that x is replaced by -( 1 x). The choice of x = - 1 for the lower boundary is convenient in this case as will be seen shortly.

+

38

George Veronis

There are several points to be emphasized here. First, the Ekman layer solution is an exact solution to the equations of motion. Second, the surface velocity in solution (7.3) is directed at an angle of 45" to the right of the prescribed surface stress for C2 > 0. Third, the net transport of fluid is to the right of the applied surface stress. The latter result can be verified by integrating Eq. (7.3) from z = - 00 to z = 0 but it is simpler and more straightforward to integrate Eqs. (7.1) directly. The result is (7.4a) (7.4b) and it is easily seen that the transport is to the right of the surface stress. Fourth, the transport given by (7.4) is independent of the details of the flow. I n anticipation of the situation for oceanic flow (7.1) may be written in the more general form (7.5a)

a

(7.5b)

2ii= - ( ETY), az

where E may be a function of z (because of V) and T is the vertical stress in the surface layer. Th e result (7.4) follows directly, the only requirement being that cc vanish at great depth. An important consequence of the foregoing considerations emerges when the prescribed horizontal velocities (or stresses) have horizontal variations of a scale much larger than the Ekman layer depth (Charney, 1955a). Then the Ekman layer solutions are locally valid near the surface and a vertical flow out of the Ekman layer results when the continuity equation is integrated from z = - 00 to z = 0 with the boundary condition zZo=Oatz=O. Thus

or t%-m=

[-

i a (&$) 2 ax

-

a

--

aY

(ET,')]= 1k . v 2

X

(ET,),

(7.6)

where Eqs. (7.4) have been used to evaluate the vertical integrals of u and v. T he parameter E is included with T~ within the del operator in anticipation of oceanic considerations.

Large Scale Ocean Circulation

39

An extension of the theory of pure Ekman layers is necessary in order to make the theory useful for oceanic (and laboratory) flows. T h e interior region, i.e. the fluid outside the Ekman layer, is normally in motion. T h e usual procedure for linear theory (Greenspan, 1968) is to write the dependent variables as the sum of an Ekman layer contribution (denoted by a tilde) and an interior contribution (denoted by subscript I) which does not decay at the edge of the Ekman layer. This approach, or an analogous one, is necessary for treating the flow in the vicinity of a rigid boundary where the total velocity must vanish. For example, consider the flow in the vicinity of a bottom boundary, z = - 1, where the total velocity must vanish. As noted earlier the solution (7.2) is valid there with t; = -( 1 z)E- 112 but the boundary values of the Ekman layer components are (7.7a)

+ -=

,uO

-%b

1

zjo = -q,, ,

(7.7b)

where subscript b corresponds to interior velocities evaluated at the bottom. Hence, the solution in the bottom Ekman layer can be written in terms of the interior velocity as

ii = -(uIb cos 5 - q,,sin t;)et,

(7.8a)

5 + GI,, sin <)er,

(7.8b)

I5 = -(GIb

cos

where t; = -(z + l)E-1’2. Now the tangential stresses at the bottom can be expressed as T ~ = aG/azl,=-,and the use of (7.8) yields

E1”T,”

(7.9a)

= u,, - Z.’Ib 1

Ell2 71, Y~

UIb

f

(7.9b)

OIb?

so that the stress at the bottom is expressed directly in terms of the boundary values of interior velocities. Another important modification of pure Ekman layer theory emerges when the boundary condition on the vertical velocity at the upper surface is written as toI +G = 0 at z = 0. Here, it is necessary to consider the Ekman layer contribution to vanish with increasing distance from z = 0 so that Eq. (7.6) takes the form Go= - &k V x

(ET,).

(7.10)

Then the upper boundary condition on wr is wIo = &k V x (ET,) at

x

= 0.

(7.11)

Greenspan and Howard (1963) call this condition on the interior vertical velocity “ Ekman pumping (or suction)”. It should be noted that w -El”.

40

George Veronis

Physically, the origin of Ekman pumping is clear as can be seen by referring to Fig. 6. Consider the flow generated by the surface stress shown as single arrows in Fig. 6. The transport in the Ekman layer is to the right of the applied stress as shown. Thus, in a vertical section the horizontal Ekman layer convergence requires a vertical divergence of flow out of the Ekman layer and a “pumping” into the interior results.

1

I

FIG.6 . A schematic picture illustrating the mechanism of Ekman pumping. The transport in the Ekman layer is to the right of the surface stress, T. With the distribution of T shown here fluid will converge in the Ekman layer and must flow downward into the interior (broad arrow).

At the bottom the total velocity field must vanish. Integrating the continuity equation for the Ekman layer variables from z = - 1 to z = 03 yields

and the use of Eq. (7.8) and the boundary condition w, = -wIb yields WIb =

+[v,

(J?”~V,~)

+ k v X (E1”VIb)].

(7.12)

Hence, horizontal divergence and vertical vorticity of the interior field evaluated at the bottom boundary determine the Ekman pumping into the interior.

B. EKMANLAYERS IN

THE

OCEAN

Even though the foregoing simple models are not directly applicable to the ocean, some of the consequences of the theory have been observed in nature. Indeed, Ekman’s (1905) analysis was undertaken to explain the observation by Fridtjof Nansen that ice floes on the surface of the Arctic Ocean moved to the right of the wind. Although it is fairly easy to generate Ekman spirals in the laboratory, almost all attempts to observe them in the ocean have failed. An exception is the observation by Hunkins (1966) of an

Large Scale Ocean Circulation

41

Ekman spiral beneath the ice in the Arctic Ocean. T h e observed spiral was consistent with laminar theory when an eddy coefficient of viscosity was used. I n general, the surface layers of the ocean are turbulent (because of thermal convection, wave motions, and/or various instabilities) and the resultant detailed motions can be expected to differ substantially from those predicted by laminar theory (Deardorff, 1972). In oceanography it is common practice to write T = pvv av/az where v, is defined as the vertical component of a highly anisotropic eddy viscosity and has the range of values 1 cm2 sec- 5 v 5 200 cm2 sec- l. Hence, the Ekman layer depth near the surface takes on values between 10 and 100 m at mid-latitudes and frictional effects are normally considered to be confined to a boundary layer in the top 100 m. Just as in laboratory studies the Ekman layer can then be treated as a true boundary layer and the integrations over an infinite range of x can be replaced by an integration over several Ekman depths. Even though the details of the Ekman layer are not applicable to oceanic flows, the integrated results, such as the transport to the right or left of the wind and Ekman pumping, can be used. However, some modification of ideal fluid theory in a uniformly rotating system is necessary in order to make the foregoing results applicable to the ocean. Since the vertical component of the Coriolis parameter is a function of latitude for large scale flows, Eqs. (7.1) through (7.5) are locally valid if E is defined as E = ./(asin do2),i.e. if SZ is replaced by SZ sin +. T h e equations for Ekman layer pumping involve horizontal. derivatives of the horizontal velocities and this requires that E be treated as avariable. That is the reason for including the parameter E in the horizontal derivatives of Eqs. (7.6), (7.10), (7.11), and (7.12). Th e significance of variable rotation can be seen more clearly if the results are written in dimensional form ZE-

OD

WIO

.

= -(l/pln)k

= (l/pnl)k

WIb = [OH (VIb(V/f)1’2

+

v

x (T31f),

v x (Wn, x (vIb(vlf)1’2)],

(7.6‘) (7.11‘) (7.12‘)

where the variation o ff( =2SZ sin +) appears explicitly. In some studies of wind-driven circulation (Munk, 1950; Bryan, 1963) frictional effects are assumed to take place primarily through lateral transfer of momentum. In this case, it is customary to use a horizontal eddy coefficient of friction, pvH, and to draw an analogy between the statistics of eddy and molecular motions to write the frictional term in the horizontal equations of motion as p v H V H 2 v where OHz is the horizontal Laplacian. Typical values of vH for large scale flows range from lo6 to lo8 cm2 sec-’ (Stommel, 1955; Munk, 1950). Although the use of a horizontal eddy coefficient of viscosity admits the existence of a boundary layer which is

42

George Veronis

interpreted as the western boundary current (Gulf Stream, Kuroshio, etc.) there is some evidence that the traditional use of an eddy viscosity to model momentum flux from mean to perturbation flows may not be appropriate for oceanic flows. Indeed, Webster (1961) has cast doubt on the sign of v, when he showed from an analysis of data taken near the Gulf Stream that the flux of momentum may be from the fluctuations to the mean flow. T he fluctuations in this case would have to draw their energy from the potential energy associated with the statification in the ocean. T h e mechanism for transferring heat and salt (or density when the appropriate combination of T and S is used) into the ocean from the boundaries is also assocated with turbulent processes. Anisotropic eddy coefficients are used for the state variables just as they are used for momentum. T he difference between momentum and heat transfer is that the lowest order equations for momentum balance do not include eddy transfer processes whereas for heat, salt etc. the diffusive processes may be basically important. Even though the use of an eddy viscosity is based on heuristic reasoning, the fact that v, can be related to processes associated with an Ekman layer puts a firmer base under the concept of a vertical component of eddy viscosity. Turbulent processes and variable density alter the detailed structure of the Ekman layer but some quantitative measure of integral effects of the Ekman layer cn still be made. For this reason the following development will make use of v, in preference to v, when dissipation must be included. It will be seen, however, that it is not always possible to make such a restriction and that under certain conditions a horizontal eddy viscosity must be introduced.

VIII. Modeling of Current Systems There is no obvious ordering for deducing a hierarchy of oceanic models which successively incorporate processes and features in order of decreasing importance. I n fact, even the forces which give rise to currents in the ocean are not known sufficiently well to be so ordered. Differential heating and cooling of the surface waters lead to vertical density stratification of the ocean. Winds apply stresses to the surface and as a result the current systems tend to separate into more or less well-defined gyres associated with these wind systems (Munk, 1950). Yet the interaction of motions driven by thermal differences and those resulting from wind stresses is sufficiently complex that very few studies have been undertaken to understand the interaction. Even in those cases where such studies have been

Large Scale Ocean Circulation

43

made the models are heuristic ones with no real attempt to establish the relative importance of the mechanisms. Knowledge of dynamical processes has developed primarily through the use of simple, heuristic models which isolate one or another feature of the circulation relating it to a known or measurable process. Stommel (1965) has been particularly successful in developing simple models to explain important large scale features.

A. WIND-DRIVEN OCEANCIRCULATION Stommel’s (1948) elegantly simple explanation of the westward intensification of the wind-driven ocean circulation provides an excellent example of the value of the simple-model approach as well as an example of the difficulty in using a heuristic model as a base for extending the study to include more complex processes. Beginning with Eqs. (5.18) to (5.22) with f = f o ,6y, Stommel made the additional assumptions that the fluid is homogeneous, the depth of the ocean is constant, the flow is steady, and R = 0. He then wrote down the following equations which correspond roughly to a linear, vertically averaged system

+

au

av

ax

ay

-+--0.

Here, the terms EV are dissipation terms which are assumed to be proportional to the average velocity and E is a nondimensional drag coefficient. The surface stress terms T enter as apparent body forces. Equation (8.3) admits the definition of a stream function

and if h is eliminated from (8.1) and (8.2), the following vorticity equation is obtained

+ ,qa+/ax) = k. v x T.

&v2$

The ocean basin is rectangular with boundaries at x = 0, T and y where # = 0. T h e form of the stress term is

k.V

XT=

-sin x siny.

(8.5) = 0,

T

(8.6)

44

George Veronis

The solution of (8.5) and (8.6) with the given boundary conditions is

D,=

- 1 - (1

+ 4&2)1’2 1

2E

D,=

-

1 +(1 2E

,

(8.7)

and it is characterized by the following features (Fig. 7):

1. For E < 1 the meridional flow is southward everywhere except in a thin boundary layer near x = 0 where it is large, O ( E - ~and ) , northward. 2. T h e interior region of southward flow represents a balance between the p term and the curl of the wind stress. This is called the Sverdrup (1947) transport balance. 3. I n the boundary layer region on the west the frictional term (actually, E a2i,b/ax2or E av/ax and the B term are large and balance each other and the wind stress term is negligible. 4. T h e stream function pattern is symmetric about the mid-latitude of the basin. It is particularly informative to write the term as df/dt. Hence, the Sverdrup balance df/dt= k * V x T means that the rate of vorticity input by the wind stress is balanced by the rate of change of planetary vorticity, f,brought about by the meridional flow. In the boundary layer the balance E &/ax = - df/dt means that the rate of change of the planetary vorticity is balanced by the rate of dissipation of vorticity. Also, comparison of the streamline and h (or pressure) patterns shows that the flow is nearly geostrophic, although there is a discernible Ekman wind drift in the vicinity of the boundaries. For the purpose of extending Stommel’s model it is more instructive to derive the results in terms of the physical balances described earlier. The analysis will be done for a sphere. Again start with Eqs. (5.18)-(5.22) and consider the fluid to be homogeneous,* the depth of the ocean to be constant, the flow to be steady, and R = 0. Under these conditions the equations of motion for the interior flow are

fk x

VI

= -Qp,

- Qk

(8.8)

* Wote that a great deal of oceanic physics has already gone into deriving the approximate set of equations (5.18)-(5.22). The fact that the ocean is stratified is implicit. The assumption of a homogeneous ocean at this point is an analytical convenience.

Large Scale Ocean Circulation

f

45

02

Y

77

X

FIG 7. Calculated contours of the stream function (top) and the height of the free surface (bottom) for the linear, frictional model of wind-driven ocean circulation ( E = 0.05).

where subscript I specifies that the variables are interior ones, i.e. the boundary layer contributions are not included. On a spherical earth the ocean is assumed to be bounded on the east and west by parallels of longitude denoted by h = 0 and h = A,. For a homogeneous system the hydrostatic equation ap,/az = -Q can be integrated from the top surface, z = ~ ( xy,) , where p , = 0 downward to give (8.9) PI = arl - 4. Substituting (8.9) into the horizontal components of (8.8) and integrating from the bottom ( z = - 1) to the free surface yields

fk x VI = -QVW

+.I),

(8.10)

46

George Veronis

where V, is the horizontal volume (or mass) transport. T h e curl of (8.10) yields

fvH*v,+pv,=o.

(8.11)

A vertical integration of the continuity equation gives v€l * vI

+

wIO-

WIb=

O,

(8.12)

which, when substituted into (8.1l), yields

BVI =f (WIO - WIb).

(8.13)

Now since the interior flow is hydrostatic and geostrophic, it follows immediately that uI and vI are independent of depth for a homogeneous fluid. Hence V, can be replaced by (1 7)vI w vIand Eq. (8.13) becomes

+

BvI = f ( W I O

- WIb).

(8.14)

Equation (8.14) contains an important part of the physics of all models for steady, large scale ocean circulation. Each particle of fluid has a vorticity, f, called planetary vorticity, because it is on the rotating earth. Since oceanic flow in the interior of the ocean is slow and its scale is large, the planetary vorticity is much larger than the relative vorticity. (This point is directly reflected in the linearity of the equations of motion.) Sincef depends only on latitude, Eq. (8.14) can be written in the alternative form

d f P t =f (WIO - WIb).

(8.15)

Hence, the rate of vertical divergence of a column of fluid is accompanied by a rate of change in the planetary vorticity of the column of fluid, i.e., the fluid will move to the north or south at such a rate as to comply with the rate of change of vertical divergence or convergence of the column. Hence, the meridional motion of the column of fluid is completely determined by the vertical divergence. T h e vertical divergence can be evaluated by substituting wIo and wIb from Eqs. (7.11) and (7.12) to give

( 2 P / f ) ~ , = k * V ~ ( 2 3 ~ ~V )H-* ( ~ I E 1 ‘ 2 ) - k . xV( v , E ~ ” ) , (8.16) where q b and vIb have been replaced by uI and vI because the latter are independent of depth, as noted earlier. Consider first the case for /3 = 0. Here, it follows directly from the geostrophic equations that V,. vI= 0 and that k . V x V, = - &VH2pI. Hence, (8.16) becomes

-

VH2pI= -2k V x ( T ~

(8.17)

Large Scale Ocean Circulation

47

Therefore, if the wind-stress curl is symmetric about a mid-longitude, the solution to the Poisson equation gives a corresponding symmetry for p , and, consequently, for the interior streamline field. An additional consequence of @ = 0 comes from Eq. (8.13), viz., wIo = WIb so that Ekman suction near the bottom is adsorbed by Ekman pumping near the top surface and vice versa. T h e case p # O yields substantially different results. T h e lowest order balance in Eq. (8.16) is vI= (f/ZP)k * V x

(7,

El ').

(8.18)

Thus, as noted below Eq. (8.15), the interior flow is everywhere determined by the rate of curl of the wind stress and it is O(E1I2)of the flow for the case with @ = 0. Comparison of (8.18) and (8,lZ) also shows that wIb is negligible under the present conditions. Therefore, at lowest order the wind stress at the surface generates an interior vertical velocity whose magnitude must decrease linearly to zero at the bottom. The total meridional transport, vT, at lowest order is given by the interior transport (8.18) plus the transport (7.4a) in the top Ekman layer, and is easily verified to be VT =

(fE/2@)k . V x 7 0 ,

(8.19)

a result which can be derived directly without the involved Ekman layer argument. The present method of derivation makes it easier to look at additional details of the flow and to derive corresponding features for the stratified system as will be seen shortly. Equation (8.19) takes the dimensional form VT

= (k

'

X

TO)/k$,

(8.20)

where V , is the Sverdrup transport, referred to earlier. Since VT vanishes at the latitude where k * V x T~ vanishes, the Sverdrup transport satisfies the condition that there be no normal transport at latitudes of zero windstress curl. This is the justification for treating isolated " basins " whose north-south extent is determined by latitudes where the wind-stress curl vanishes. The meridional flow derived above cannot be the complete solution because a region of the ocean in which the wind-stress curl is of one sign would have a undirectional meridional flow everywhere. T h e simplest argument to complete the flow pattern is to note that if the wind-stress curl imparts vorticity of one sign to the ocean the overall circulation must exhibit that vorticity. Thus, if k .V x T~ is negative the total meridional transport is everywhere to the south. An overall negative vorticity of the circulation exists if an intense northward current exists near the western

48

George Veronis

boundary. Analogously, a positive wind-stress curl requires a northward flow everywhere and an accompanying, intense, southward flow near the western boundary. Hence, intense currents will exist near the western boundary. T h e simplest, formal procedure for deriving the intense current is to incorporate bottom frictional processes from Eq. (8.16) by boundary layer techniques. I n this case, derivatives with respect to the longitudinal coordinate must be large to offset the small coefficient Furthermore, it is evident that of the last two terms on the right-hand side of Eq. (8.16) the dominant one contains the longitudinal derivative of the meridional velocity since a/ah and vI are both large near h = 0. Hence, the lowest order equation in the western boundary layer is

(8.21) where 6 denotes the boundary layer contribution to vI . T h e solution is

6 = A(+)exp [(-2/3 cos +lfE1/2)h].

(8.22)

+

T h e quantity A(+)can be evaluated by satisfying the condition J:(e vT) dh = 0, where vT is given by (8.19). [The Ekman layer contribution to 6 is O(E1I2)and does not enter at this order.] Hence,

I (k. A0

A(+)=

V x ro) cos

0

+ dX.

(8.23)

For the wind stress which drives the gyre containing the Gulf Stream the simplified form

k -V x

so= - Wsin r(+ - +o)/+l

+o

2 4 <+o

+

+1,

(8.24)

is often chosen and A(+)can be evaluated explicitly. Since in this case -c0 is independent of A, the complete meridional transport u = uT 6 can be written as

+

k -V x

(8.25) It is interesting to note that thickness of the western boundary layer I n the increases northward as sin1l24 coss2 4 between +o and +o + /3-plane solution the thickness is constant. T h e continuity equation for the total mass transport (including the Ekman layer contribution) is, au a (8.26) - +- (v cos +) = 0.

ax a+

49

Large Scale Ocean Circulation

S ~ V

Since dh vanishes at each latitude it is necessary to satisfy the condition u = 0 on only the east or the west side because the other boundary condition automatically follows. Hence, integrating westward from the eastern boundary, h = A,, where u = 0, gives (8.27) where v is given by Eq. (8.25) and may be replaced by vT except in the vicinity of the western boundary. Also,

W E sin + sin (4 += - 2 where E sin + is independent of + and it is easily seen that the solution in vT cos

+ O b

1

dl

the interior is (8.28) i.e. uT is antisymmetric about 4 = (+o + +J/2. An important aspect of the present solution can be deduced from the east-west structure of the northward velocity. In the interior v is small and there is no Ekman layer contribution near the bottom. However, in the western boundary layer v is large and decreases eastward. Since adlax is negative and large, to lowest order there is an Ekman suction into the bottom boundary layer in this region. I t is by this mechanism that the vorticity input in the interior is dissipated in the boundary layer. In order to satisfy the no-slip condition at h = 0 and A,, it is necessary to add lateral friction to the model. It is easy to see that the effect of such a subboundary layer would be to generate an additional Ekman pumping. In this region 6 is positive and increases eastward, thereby forcing an Ekman pumping from the bottom boundary layer upward.

B. EXTENSIONS OF STOMMEL’S MODEL Given the results of the foregoing model it is natural to ask whether the model can be expanded to take into account additional mechanisms, processes, and features characteristic of the ocean. The analyses of Fofonoff (1954), Morgan (1956), Carrier and Robinson (1962), Bryan (1963), and Veronis (1963, 1966) are attempts to extend the model to take into account nonlinear processes but Stommel’s model still forms the basis for the extension. In particular, stratification is ignored and the ocean is considered to have constant depth. Charney (1955b), Fofonoff (1962b), and

50

George Veronis

Welander (1968) idealized the stratification with a two-layer system and studied extensions of Stommel’s model. Niiler et al. (1965) used an idealized, basically linear stratification to determine the resultant wind-driven ocean circulation. Bryan and Cox (1967) and Bryan (1969) have carried out extensive numerical calculations of a more complete system, including thermal driving forces, and have deducedZhe resulting stratification as part of the analysis. T h e approach to be taken here is to determine whether the results of the simple, idealized model already discussed can be used as a basis for a more extended model. I n particular, the assumptions R = 0 and slat = 0 are retained but the ocean is taken to be stratified and topographic variations may be present. T h e hydrostatic equation includes density variations and the interior flow is now described by the thermal wind relations (6.6). Ekman layers near the top and bottom boundaries are assumed to provide the required momentum transfer to the surface and bottom. T h e introduction of variable density requires either an additional equation for the determination of p or a specification of some part of the density field with an appropriate equation to determine the residual field. Since the present purpose is to make use of the results of the simple wind-driven model, it is clear that the density cannot be calculated ab initio. The geographical range of the model is specified by the wind-stress field and one cannot hope to obtain a realistic density distribution without including the thermal driving forces which will, in general, have larger scales and a different geographical distribution from that of the wind field. Hence, the density field must be specified in some sense. Since the degree of specification depends on other assumptions which can be made, this point will be postponed until it is clear which part of p is required. Now from the thermal wind equations, the hydrostatic equation, and the continuity equation for a Boussinesq fluid, one can derive the planetary vorticity equation pvI =f aw,laz (8.29) and the equation for the vertical curvature of w , (8.30) Equation (8.29) can be integrated in the vertical from z to the surface to give (8.31) V , = (wIo- w,) tan 4. T h e difference between Eqs. (8.31) and (7.6) is that the present integration is taken from an arbitrary level rather than from the bottom. First, consider the case w, -+ 0 as z decreases so that at some depth the vertical velocity is reduced to zero by the stratification. If this depth does

Large Scale Ocean Circulation

51

not intersect any ridges or seamounts, the interior meridional transport is given by wIo or, equivalently, by the wind stress. Then VI is determined as in the homogeneous problem, the principal difference being that the stratification confines V, to a layer bounded below by the depth where W, essentially vanishes. From (8.30) it can be seen that ap,/aA must vanish at great depth ; hence the thermal wind relation

+ av,/az

+) ap,/aA (8.32) allows an integration upward for the determination of the vertical structure of v , once pI is known. Therefore, in this case a knowledge of p, is necessary for the vertical structure of ZI, as well as for w I , but not for the transport

2 sin

= -(Q/cos

v,.

A vertical integral of the continuity equation from z to the surface yields

a (VCOS +) = 0, +ah a+

au

-

(8.33)

where U and V incorporate the Ekman fluxes as well as the geostrophic interior flow and U can be determined to within an arbitrary function of +. Anticipating that a boundary layer will exist on the western side only suggests an integration westward from the eastern boundary where U vanishes, and U is then known explicitly. T h e corresponding simplification in the boundary layer, viz. & + 0 as z decreases, means that bottom frictional processes are not important in the western boundary layer. Hence, the system cannot be closed as in the homogeneous model and it is necessary to incorporate lateral frictional processes near X = 0 in order to satisfy the condition of no net meridional transport. A vertical integration of the continuity equation is again possible and the entire analysis can be carried out in terms of transports rather than velocities. T h e solution depends on the parameterization of the lateral frictional processes and specifically on uH if lateral friction is assumed to have the form uH VH2v.This is essentially the approach taken by Hidaka (1949) and Munk (1950). T h e results of Stommel’s model and that of Hidaka and Munk differ only in the details of the western boundary layer, although it is important to keep in mind that the physical processes in the boundary layers are very different. If the vertical velocity in the western boundary layer does not vanish near the bottom, the problem becomes substantially more complicated. Indeed, if bottom topography is included, the details of the density distribution must also be determined and one is then faced with the general circulation problem, i.e., thermal forcing must also enter and the geographical range of the domain must include the sources and sinks of heat and salt. However, with a few simplifying assumptions it is possible to

George Veronis

52

obtain results for the case where z6 does not vanish at great depth near A = 0. T o derive these results note that under the present assumptions the flow in the western boundary layer is geostrophic (except in the Ekman layers); hence (8.29) is valid. Now neglect bottom topography. Then a vertical integration of (8.29) from bottom to top yields 2(/3/f)8=k*

v

X ( 7 0 E ) - v V H . ( ~ b E 1 ’ 2 ) - k * VX(0bE1’2),

(8.34)

where 8 is the transport in the western boundary layer and 0, is the horizontal velocity in the boundary layer evaluated at the bottom. As in the homogeneous case derivatives with respect to longitude are large and at lowest order the contribution of the wind-stress term can be neglected. T h e principal balance is (2/3p/f)

= -((E”2/Cos

$)(a&/aA).

(8.35)

Because of stratification 6 varies in the vertical and the stratification must be known in order to relate 6, to 8.Suppose that the mean density field in this region is known, either by observation or from a numerical model. Then 6, can be related to V . T h e particular assumption made here is that ag/aA decays exponentially from the surface downward. (Better functional approximations to a$/aA in the western boundary layer can be obtained from observed data but the simple exponential form serves present purposes.) Then, from the geostrophic relation one can write

6 = 6,(1 + cekz),

(8.36)

where, in keeping with observations, K is substantially greater than 1 (so that the stratification does not extend to the bottom) and K may depend on 4. The bottom velocity, 6,,may be either positive or negative. For the Gulf Stream 6 must be positive and large near the surface so that c > O if u, > 0 and c 4 - 1 if u, < 0. T h e transport P can now be evaluated from (8.26) as

P=/

0

6dzmd6,(1+c/k).

(8.37)

-1

Then Eq. (8.35) becomes

y6,

= -2

where y = 2/3 cos 4 (1

a.;,/aA,

+

(8.38)

C/K)/fE”2,

and the solution is

6,= 6,,e-yi,

(8.39)

53

Large Scale Ocean Circulation

where Gbo( =6,) o) can be evaluated as before so as to yield zero net meridional flux across the entire basin. For 6, > 0 the thickness of the boundary layer i ~ f E l ’ ~ / cos2 ( 2 q5( 1 c/K)), i.e. it is thinner than the corresponding boundary layer of the homogeneous system. The reason for this is that the entire vorticity of the interior must be dissipated in the bottom Ekman layer by means of a relatively weaker meridional flow, 6,. Hence, the longitudinal gradient of the meridional flow must be larger, or equivalently, the boundary layer must be thinner. It is interesting to note that the thinner the western boundary layer is near the bottom the more likely it is that some other process such as lateral friction or nonlinearity may become more important and eventually dominate the flow. For example, it was pointed out above that 6,= 0 (this is equivalent to zero boundary layer thickness) requires that lateral friction be introduced. Of course, effects of bottom topography or of nonlinearity or of a more realistic density distribution could alter the argument dramatically. For 6, < 0 the present solution is not valid because it grows exponentially with increasing longitude. T h e reason is that negative bottom velocity cannot lead to the required dissipation of negative vorticity unless 6, becomes even more negative; hence the solution does not boundary layer. Observational evidence (Swallow and Worthington, 1961) indicates that south of Cape Hatteras the deep western boundary layer flow is indeed southward. This southward flow appears to be related to the thermally forced circulation (Stommel, 1965) and because of bottom topography it may be displaced eastward relative to the Gulf Stream. Hence, it is possible that the bottom frictionalmodel may not beinapplicable even in this case. T h e issue is by no means clear, therefore, and will probably not be resolved until success is achieved with models which incorporate both bottom topography and stratification. If wI does not vanish with depth in the interior, effects of bottom topography and stratification must be incorporated everywhere and the winddriven circulation cannot really be separated from the general circulation of the oceans.

+

C. INERTIAL EFFECTS Several different methods have been employed to study nonlinear effects in wind-driven ocean circulation. These include perturbation analyses (Munk et al., 1950; Veronis, 1966), numerical studies (e.g., Bryan, 1963 ; Veronis, 1966), and studies which accept the interior flow as given and which treat nonlinear effects in the boundary layers (Fofonoff, 1954; Charney, 1955b; Morgan, 1956; Carrier and Robinson, 1962).

54

George Veronis

Perhaps the principal benefit from these investigations has been an increased understanding of the nonlinear process rather than an advance in our knowledge of the structure of the wind-driven gyres. The reason is that the models tend to be too idealized to admit incorporation of additional fcatures ; consequently, one cannot build an ordered hierarchy of successively more complicated models. T h e present discussion will be restricted to nonlinear effects in Stommel’s original model on the ,&plane although some remarks will be made about variations on the model. T h e nondimensional vorticity equation for the vertically averaged mass field is

R v . V C $ / ~ V + E C = ~ *XV T,

(8.40)

where

or, in terms of the stream function,

T h e nonlinear terms here are only crude approximations to the actual nonlinear terms because the vertical average of the product v . V( is not generally given by the product of the vertically averaged quantities. The Rossby number, R , can be defined in terms of the amplitude of the windstress since the velocity amplitudes are determined by 7. However, for present purposes it suffices to note that R is so small that nonlinear effects in the interior of the ocean are negligible unless the boundary current penetrates the interior. T h e effect of the inertial terms can be treated by perturbation methods (Veronis, 1966) in a straightforward manner. However, it is easy to see the qualitative effect of nonlinearity by rewriting (8.40) for the western boundary layer region as v * V(R5

+f)=

-E<

(8.42)

since the wind stress can be neglected there. Now the linear ( R = 0) solution states that the rate of dissipation of the intense negative vorticity in the boundary layer is balanced by the advection of planetary vorticity, f, northward. When nonlinear effects are included in the boundary layer south of mid-latitude there is also an advection of negative vorticity, R(, northward and that will tend to decrease the rate of dissipation. The latter is possible if EC is reduced, i.e. if aa/ax is decreased. Hence, the boundary layer must thicken in the formation region of the Gulf Stream. At some

55

Large Scale Ocean Circulation

point north of mid-latitude the relative vorticity must begin to increase from its large negative value (it must essentially vanish in the interior). Therefore, v . V ( R [ )must be positive and the result is that - F [ or - E &/ax is increased. Hence, the boundary layer becomes narrower in this region, a north-south asymmetry is introduced by the inertial terms, and more of the dissipation takes place in the northern half of the boundary layer. As R is increased this north-south asymmetry is intensified, the point of maximum 151 is shifted northward, and the dissipation of vorticity takes place closer to the northwest corner of the basin. For sufficiently large R the western boundary current impinges on the northern boundary and must turn and flow eastward near the northern boundary. In this region the variation of planetary vorticity is small (the eastward current is narrow) and the balance must be between nonlinear advection and dissipation. As it turns out, the boundary layer thickens in this region and the intense downstream flow cannot be maintained. Hence, streamlines leave the northern boundary region and return to the interior. Inertial effects thereby penetrate into the interior. An example of the flow described above is given in Fig. 8 where the streamline pattern shown was derived from a numerical integration of (8.41) (Veronis, 1966) with E = 0.05 and R = 0.01 and k - V x T = -sin x sin y. If lateral friction is included as well, the tangential velocities must vanish at the boundaries and the possibility of a shear flow instability arises as the Rossby number is increased. Blandford (1971) integrated the system (8.41) with sV4$ included as well and with the added conditions a$lax = 0 TI

Y

n X

FIG.8. Calculated contours of the stream function for the nonlinear model of winddriven ocean circulation (Veronis, 1966). E = 0.05, R = 0.01.

56

George Veronis

on x = 0, rr and a2#/ay2= 0 on y = 0, T. He found that for R above a certain value the western boundary layer becomes unstable, presumably to a Rayleigh-type of shear flow instability, and transient eddy motions are generated in the western boundary current north of mid-latitude. If the wind-stress curl is assumed to vanish at a latitude south of the northern boundary, the linear solution looks much like the one that would be obtained by placing a rigid boundary at the latitude where the windstress curl vanishes, i.e., the northern “ boundary” is determined by the Sverdrup balance. However, the nonlinear solution shows that the western boundary current penetrates into the region of zero wind-stress curl. Hence, the presence of boundaries to the north and south is required to contain the oceanic gyre. Presumably in nature these “ boundaries ” are provided by fluid masses of substantially different density so that the intense currents can flow to sea along the region of sharp density gradient. Variations in bottom topography also seem to steer intense currents, at least partially, so that they penetrate into the open ocean. Warren (1963) has shown that the meanders of the Gulf Stream after it leaves the coast are associated with bottom topography. In a numerical calculation for an idealized model Holland (1967) exhibited the same steering mechanism for intense currents.

IX. The Thermohaline Circulation When sources and sinks of density are included as forcing terms, the resulting circulation is called the thermohaline circulation. Boundary conditions for the density are, of course, associated with heat and salt transfer at the boundaries. T h e problem is simplified substantially by the assumption that exchanges in heat and salt are described by identical eddy transport processes. For a Boussinesq fluid temperature and salinity can then be combined to describe the density and the problem can be formulated in terms of the density. (Sometimes an “ effective ’’ temperature, into which temperature and salinity are incorporated, is used in place of the density.) Regions of mean evaporation and precipitation are associated with the wind-driven gyres and must be at least quantitatively significant in the thermohaline circulation of the oceans. However, these specific boundary processes have been ignored and the usual procedure has been to force the circulation by thermal or density exchanges which have global scales at the surface. The problem is described by geostrophic and hydrostatic balance with vertical (eddy) diffusion terms incorporated into the density equation.

Large Scale Ocean Circulation

57

In dimensional form the Boussinesq equations, with p corresponding to the potential density, are 2Q sin +v = (1/a cos +)PA, (9.la)

2 0 sin +u = -( 1/a)P, , p, = -gp,

(9.lb) (9.1~)

v'VP=Kp,z,

(9.2a)

v.v=o,

(9.2b)

where p = po/pm, P = -PIPm, and subscripts A, 4,and z correspond to partial derivatives. Th e assumed global scaling is included in the complete latitudinal dependence of the Coriolis term. Since no diffusion of momentum is included in (9.1) the equations describe only the interior (geostrophic-hydrostatic) flow and the effect of Ekman layers at the top and bottom must be introduced via boundary conditions. Lateral boundary conditions on velocity can be satisfied only along a single boundary. For a basin with lateral boundaries one must append boundary layers at the sides and require the boundary layer and interior solutions to match in the region of overlap. The importance of lateral boundary layers is obvious because cold or warm boundary currents will alter the surface boundary conditions in the interior and the effect can, and probably will, be very important. Bottom topographic variations are also important because of the strong constraint that they impose on the motions of the nearly homogeneous abyssal waters. These, too, have customarily been ignored, or at least not incorporated into a complete quantitative analysis. Although one cannot really justify the use of an open system in which the various features mentioned above have been neglected, some progress has been made on the problem without these features. The discussion will be confined to the analysis of the simplified system (9.1) and (9.2). A discussion of various aspects of the model (the use of vertical diffusion, the boundary conditions which can be satisfied, etc.) will be given at the end of this section.

A. THEPRESSURE EQUATION The system (9.1) and (9.2) can be expressed in terms of the pressure by writing z, = (1/2Qa cos 4 sin 4)PA, (9.3a) u = -(1/2Qa sin +)P,, (9.3b) p = -(l/g)Pa , (9.3c)

George Veronis

58

using the expression for the vertical velocity obtained from (9.2a), i.e., w = [KPZZ

and finally, substituting divergence relation

z,

-

(u/acos + ) P A - ( ~ / ~ ) P d l / P Z 7

(9.4)

from (9.3a) and w from (9.4) into the planetary w, = cot

4 v/a.

(9.5)

T h e result is the pressure, or P, equation derived by Needler (1967): K

sin 4cos 4 ( P , Z p,,,, - p,,,P,,,)

+

=

+

PdPZ,, PA, - p,,PA,,) PA(P,, PdBZ - p,,,PdZ cot 4 p,,P,,), (9.6) where K = 2QKu2. It is a generalization of Welander’s (1959) M equation, where M is essentially the vertical integral of P when the barotropic mode is included in (9.6).

B.BOUNDARY CONDITIONS AT

THE

TOPAND BOTTOM

Even a cursory examination of the density structure of the oceans shows that there is a distinct variation of the density with latitude at any given level. Furthermore, the subsurface density is related to the density near the surface and, in particular, the effect of wind-driven gyres is reflected in the density field. Hence, for surface boundary conditions it is common to specify the density as known at the base of the Ekman layer and to assume that the vertical velocity (Ekman pumping) is given by the known, mean wind-stress distribution. At the bottom the obvious condition that there be no normal velocity should be satisfied. I n addition, since the bottom is essentially a nonconductor (geothermal heat flux is ignored), the bottom is treated as insulated. As will be seen below, it is not possible to satisfy all of these conditions with the solutions presently available, even though the model described by (9.6) would appear to contain sufficient arbitrariness to do so. This point will be taken up again after the solutions have been presented.

C. SOLUTIONS BY MEANSOF

A

SIMILARITY TRANSFORMATION

Some solutions to (9.6) can be obtained by a similarity transformation proposed by Robinson and Welander (1963). Although such a procedure imposes restrictions on the form of the solution, it turns out that the solutions can be generalized so that some of the restrictions can be removed.

Large Scale Ocean Circulation

59

Since the prineipal boundary conditions are to be satisfied on the surface z = 0, this suggests that the following form for P be tried

PO, 4 = q ( k $)G(7), $7

7 =zkO,+).

(9.7)

When (9.7) is substituted into (9.6), it is found to be a solution provided that the following conditions are satisfied

k = (sin $)"[A

+ E($)]",

+ E($)I2"+

,[A

q = (sin $),"+

l,

where m and n are real constants and E is an arbitrary function of G(7)must satisfy the following equation

(9.8)

+. Then

K(G'"G"- GfrfGtff) = (2n - m)GG'G"' + (2m + 1 - 2n)GG"G"

- (2n - m)7G'G'G"' + (3n - m)rlG'G"G", (9.9) where primes denote differentiation with respect to 7. Hence, the nonlinear partial differential equation has been reduced to a nonlinear ordinary differential equation. Boundary conditions on P which are consistent with the form (9.7) can be transformed into boundary conditions on G. Equation (9.9) contains three parameters, K , m, and n, and can be integrated numerically for ranges of values of these parameters. It turns out, however, that a few exact solutions of (9.9) can be obtained by dividing the equation into two parts each of which vanishes separately. This method was proposed by Kozlov (1966) and it gives the only exact solutions which have been derived so far. T h e method of solution of (9.9) is to let the terms multiplied by 7 balance each other. Then (9.10) The remaining terms then yield

K(G'"G"- G"'G"')- (2n - m)GG'G'"

+ (2m + 1

-

2n)GG"G" = 0. (9.1 1)

One solution, given by G' = 0, corresponds to a barotropic fluid and is of no interest in the present context. If G' # 0, and if (3n - m)/(2n- m) # 0, 1, or 00, the general solution of (9.10) is

G = y(7 -

)(min) - 1

+ 6,

(9.12)

where y , q 0 , and 6 are constants. Substituting (9.12) into (9.11) yields

6=0,

m=-n ,

y=4K.

(9.13)

60

George Vuonis

[Another possibility is (3n - m)/(2n - rn) = 0 or 00 but then (9.12) is not valid and one must treat these special cases by going back to (9.9).] Then (9.12) takes the form (9.14) G = 4K/(77 f 70)~. This solution, with q o = 0 and E ( 4 ) = constant in (9.8), was first derived by Fofonoff (1962b) by inspection. The complete solution for P is

q = (sin +)2m+2(h+ E)1-2m,

k = [sin +/(A

+ E)]",

(9.15) (9.16)

When (3n - m)/(2n - m)= 1, i.e. n = 0, the bracketed term of (9.10) gives the solution G = d been, (9.17)

+

where b, c, and d are constants. Substituting (9.17) into (9.11) yields m = -1. Hence, (9.17) is a solution to (9.9) and the complete solution for P is k = l/sin 4, q = h E, (9.18)

+

P = ( A + E)(d

+ beczisin@).

(9.19)

Welander (1959) first presented an exponential solution to the thermocline problem for K = 0. He did not make use of the similarity transformation but, as Needler (1967) has pointed out, the exponential solution even with K # O need not be restricted by the similarity transformation. Blandford (1965) derived a simpler version of (9.19).

D. GENERALIZATION OF THE SIMILARITY SOLUTIONS Since the similarity solutions can be generalized, it is preferable to present the more general forms before discussing the properties of these solutions. Needler (1967) has observed that the exponential form deduced from the similarity transformation can be made more useful by substituting the form P = A ( h , 4) M(A, +)eak(l. 6) (9.20)

+

into (9.6) and determining the conditions for (9.20) to be a solution. The resulting restrictions are k , = 0, (9.21)

A,(k,

+ k cot +) = 0,

(9.22)

M,(kd

+ k cot 4) = 0.

(9.23)

61

Large Scale Ocean Circulation

These cbnditions are all satisfied if k , A, and M are independent of A. Although such a restriction leads to current and density distributions which are zonally uniform, a first approximation to observed patterns can be described in this fashion and a correction, incorporating dependence on longitude, can be sought. However, this approach has not been pursued. Instead, the alternative solution to (9.21)-(9.23)) viz.

k = c sin 4)

(9.24)

where c is a constant, has been investigated. The form (9.24) provides a z dependence in (9.20) which is identical to the one given by Welander (1959) and Blandford (1965). Welander considered a conservative system whereas Blandford restricted his attention to similarity forms. Once (2.24) is satisfied, there are no restrictions on either A or M and (9.20) becomes the most general exponential solution for the thermocline problem with diffusion. A crude picture of the density structure can be obtained from (9.20) by choosing c in (9.24) to give the observed depth for the temperature to decay to e - l of its surface value at some particular location. Needler chose c = (1500 m)-l and C$ = 30"N which gives an e-l depth of 750 m. This is associated with the thermocline depth in the ocean. Then choosing a surface temperature proportional to cos(4 10') and a (bottom) reference temperature of 2.45'C, Needler constructed the isotherm pattern shown in Fig. 9. (His isotherm pattern is equivalent to the density pattern in the present analysis.) T h e qualitative agreement with the observed potential density in the western Atlantic (Fig. 3) is good although there are

+

O - -220

5000

I

10

20

30

40

50

60

ON LATITUDE

FIG.9. Calculated isotherms (or isopycnals) for thermohaline circulation according to Needler (1967).

62

George Veronis

differences. No wind-stress effect is included in the pattern of Fig. 9 so that the gyre (between 20"N and 45"N)associated with the Bermuda High is not brought out so sharply in the calculated profile. An additional feature of the exponential solution can be seen by observing that with A = 0 the vertical velocity takes the form

(9.25)

+=

where p o / ( h + E ) is the surface value of p at 30". T h e vertical velocity, therefore, approaches the value w , = cK/sin q5 at great depth. Here, the contribution of w , exactly balances the vertical diffusion term in the density equation. Hence, if one were to work with w - w, instead of w , the original model could be reduced to a conservative one (the terms w, i3Plaz and K i32P/az2balance exactly). This accounts for the fact that the exponential solution has the same form as the one obtained by Welander for the case where K = 0. A similar generalization can be carried out for the inverse power law solution (9.14). Thus, choose

(9.26) Th e conditions to be satisfied by B, C, and q for P to be a solution to (9.6) reduce to

B = 4 K h sin2 4

qJ3B cot d - Bml

+ E(d),

+ qm B,

= 0,

q A c@ - q@c,4 = O.

(9.27) (9.28) (9.29)

Equation (9.28) requires that surfaces of constant q be given by

dd 4 K sin2d - 4 ~ sin h 4 cos 4

dh 3 cot +E - dE/d+ '

+

(9.30)

T he more general solution in this case is nearly as constrained as the similarity solution. Th e particular advantage of the power law solution is that it provides a slower decay of density with depth and one can match the observed vertical profile of density better than with the exponential solution which yields too large a decay with depth. In this solution the entire vertical structure of P,hence of p and v, is proportional to K whereas in the exponential solution the value of vertical diffusion is not critical.

Large Scale Ocean Circulation

63

E. SOLUTIONS TO THE IDEAL-FLUID THERMOCLINE Welander (1971) has extended his earlier solutions to the ideal-fluid problem by making use of three conservative properties. The latter can be derived from the general steady equations of motion for an ideal Boussinesq fluid p ( v - VV+2Q x v ) = -vp-pvo, (9.31)

v , v=o,

(9.32)

Vp=Q,

(9.33)

V.

where Q, is the force potential. It has already been shown (in Section 111) that the potential vorticity H=(2Q+V

xv>.vp

(9.34)

is conserved. Furthermore, it is a straightforward procedure to show that the Bernoulli function

+ +

(9.35) B = p pQ, BV * v is also conserved. Now, since p is also conserved (by assumption in 9.33), B, p must be related (they are all constant along trajecthe quantities tories) and we can write

n,

IT= F ( p, B),

(9.36)

where F is a general function of p and B. It should be observed that (9.36) represents a first integral of a very general, conservative system. However, the result is rather too general to be informative and it is necessary to turn to more specific models together with suitable, simplifying assumptions on F in order to obtain more explicit information. For a geostrophic, hydrostatic system with Q, =gz the quantities II and B reduce to = 2R sin q5 ap/az, B = p +gpz, (9.37a,b) and (9.36) becomes sin

+

ap/az = F(f,p +gp+

(9.38)

One simple solution for this problem can be derived by assuming that F has the linear form (9.39) F = ap b(P + g P ) c,

+

+

where a, b, and c are constants. Then (9.38) takes the form sin 4 ap/az = ap

+ b(p + g p ) + c

(9.40)

64

George Veronis

and differentiation with respect to x together with the hydrostatic equation yields sin 4 a2p/az2= ( a bgx) +/ax.

+

The latter can be integrated twice to yield

where po and C are arbitrary functions of h and 4,and zo= a/bg, D = (-2/bg)lI2. The constant b must be negative for the solution to decay with depth. If the constant a is positive the profile is monotonic with depth. For a < 0, an inflection point appears in the density profile at the constant depth, x = z,, . A second depth, D sin1’24,varies with latitude and is interpreted as the depth of the thermocline since it describes the decrease of density variations with depth. Hence, the solution allows for a two-scale description of the density with depth and, in particular, there is a sharp thermocline at the equator. With appropriate choices of a, b, and D, Welander was able to reproduce some of the overall features in the meridional density distribution in the South Pacific as shown by Reid (1965). Th e theoretically deduced distribution is shown in Fig. 10. For present purposes it suffices to compare it with the corresponding ranges in latitude and longitude shown in Fig. 3. Th e inflection point in Welander’s solution provides a better fit with the data than does the exponential profile which he had derived earlier and which is much like Needler’s, shown in Fig. 9. Neither solution shows the rapid meridional variations which are observed

Fig. 10. Calculated isopycnals from the ideal-fluid thermocline model of Welander (1971). (Courtesy of J . Mar. Res.)

Large Scale Ocean Circulation

65

at higher latitudes. Th e latter are associated with intense currents that are at least partially generated by wind stresses and the simpler dynamics of the thermocline models do not reproduce these features. Welander presents other special solutions to the ideal-fluid model. He points out in his concluding section that the solutions which emerge from such a model are restricted in the sense that they cannot satisfy all of the obvious boundary conditions required by a satisfactory model of the thermocline, a point to which we shall return presently. However, subject to the limits set by the model itself, Welander’s analysis is quite general and further study of these ideal fluid thermocline solutions should serve to provide us with a clearer picture of the possibilities and the shortcomings of this approach.

F. THEEFFECTOF

A

BAROTROPIC MODE

T he model described by (9.1) and (9.2) contains vertical diffusive as well as convective processes in the equation for the conservation of density. The presence of vertical diffusion should allow for the possibility of satisfying boundary conditions on density at the top and the bottom surfaces (a density flux or a fixed density at each of the two). I n addition, the continuity equation provides an additional degree of freedom which is normally used to specify the effect of wind stress in the form of a given vertical velocity (Ekman pumping) at the base of the Ekman layer. A fourth condition is available from the hydrostatic pressure equation because a barotropic current, independent of the vertical coordinate, can be included by specifying the pressure field at some level. Because of the constrained nature of the analytic solutions proposed thus far, it has not been possible to satisfy all of these conditions. A similarity solution is inherently constrained, of course, because once the form is chosen the boundary conditions must conform to the choice. Thus, satisfying one or two boundary conditions normally suffices to specify the behavior of the system throughout. An example of this constraint was given in Section IX, D where it was shown that the exponential solution proposed by Needler (1967) satisfies the diffusive model only because the vertical velocity is separated into two parts, one part balancing the diffusion term exactly and the other part contributing to the convection of density. In this sense, the exponential solution is simply a superposition of a solution for an ideal-fluid model and a solution for a model in which vertical diffusion and vertical convection balance exactly. Th e constraint enters because the density field for the two must be the same.

66

George Veronis

An informative alternative to the similarity-type solutions has been proposed recently by Needler (1972). Th e approach which he has taken is to ask: under what conditions can a solution P(h, 4, x), of the pressure equation be generalized to include a barotropic mode, A(h, +), so that P(h, +,x) A(h, +) is also a solution for any A(h, +)? The generalizations of the similarity solutions discussed earlier show that barotropic modes are indeed possible ; moreover, the exponential solution is not appreciably restricted by adding an arbitrary barotropic component to the pressure field. Physically, of course, the addition of a barotropic mode will be an added constraint on the possible solutions. At any given location it is possible to divide the pressure field into an internal part, which depends explicitly on depth, and an external part which is independent of depth. However, requiring such a division to be valid for an entire region is to impose a restriction on the fields which must satisfy the nonlinear density equation. Nevertheless, the results are very instructive. If P(h, 4, z ) is a solution to (9.6), under what conditions is

+

P(h, +,z> 3- B ( A 41,

(9.42)

with B arbitrary, also a solution? Substituting (9.42) into (9.6) and making use of the fact the P is a solution yields

Bb[PzzaP,,

-

P,a P,m] - Ba[Pzzrn P,,

- P, 6 P,,,

+ cot 4P.a P z z ]

= 0.

(9.43) Now divide by (PZJ2and make use of ( 9 . 3 ~ to ) obtain

B,-a

a,

["I

- B A -a P + x c o t + ] = O . ax p z

Pz

(9.44)

But p = -g aP/az so p is independent of B(h, 4) and the terms involving a/& in (9.44) must vanish individually. This requires that PaIpz = K a 9

( P @ / f Z )+ z c o t 4 = s i n 4 L , ,

(9.45a)

(9.45b)

where K and L are arbitrary functions whose forms in (9.45) have been chosen for convenience. Th e first equation requires that surfaces of constant p satisfy the equation or that p = constant along surfaces satisfying the condition

z

+ K(X,4) = const.

(9.47)

Large Scale Ocean Circulation Hence, p = p(z

67

+ K , +).

(9.48)

Proceeding in a similar fashion with (9.45b) and making use of (9.48) yields the condition that p be constant along surfaces for which (z/sin +)

+ L = const.

(9.49)

Hence, the two conditions are consistent if p varies along surfaces q = constant where K = L sin +. (9.50) 7 = (z/sin +) L(h, +),

+

For convenience choose (9.51)

P = -(1/g)Q'(7L

where prime denotes d/d7. Then

P = sin 4Q(7)

+ A(h, +),

(9.52)

where A ( h , +) is an arbitrary function. For the ideal-fluid model ( K = 0), (9.52) is the most general solution that includes an arbitrary barotropic mode. T h e restricted conditions (9.45) for an arbitrary barotropic mode can be manipulated to yield p d s i n 4 pz)m

=pdsin

4P J A

= pz/(sin

4

Pz)

.

(9.53)

Hence, surfaces of constant density and surfaces of potential vorticity coincide. Thus the requirement that the solution for P include an arbitrary barotropic mode imposes an additional restriction to the class of idealfluid solutions found by Welander and (9.52) is included as a subset of Welander's solutions. For the diffusive system ( K # 0) the left-hand side of (9.6) must vanish separately and the requirement on Q(7) becomes (9.54) so that

Q = aecn+ b7

+ d,

(9.55)

where a, b, and c are constants. This form for Q is equivalent to the exponnential solution given by Needler (1967). It was shown earlier (Veronis, 1969) that if zonal diffusion of density is included as a term, K,p,,/a2 cos2 4, the exponential solution is still valid. Hence, (9.55) is a solution for this expanded system. Needler points out that the more general solution (9.52) is valid if the density field is time-dependent so that ap/at is included in (9.2a). In this case L and A in (9.50) and (9.52) are arbitrary functions of time as well as of and A.

+

68

George Vwonis

Needler's solution (9.52) is limited in its usefulness as are the other analytical solutions found for the thermohaline circulation because it cannot satisfy all of the physical boundary conditions. T h e restrictive character of a solution of the type given by (9.52) is that it ties the vertical and horizontal dependences to a particular form. In general, one would hope to be able to satisfy conditions on w and T at the surface, z = 0, as well as the condition of zero normal velocity at the bottom, z = --h(X, 4). However, once any two of these conditions are chosen the other is essentially determined. Needler (1967) and Veronis (1969) have discussed this inadequacy of the analytical solution in some detail and in his latest paper Needler shows that the same inadequacy is still present inthesolution given by (9.52). Nevertheless, the value of such models as an aid for increasing our understanding of oceanic processes cannot be denied. I n the process of trying to fit the results to observed distributions new insights into the more general system are generated and, even though these models may eventually be discarded, they serve as an important step toward a more satisfactory model.

G. THEROLEOF DIFFUSION In developing analytical thermocline models it is possible to obtain density distributions which exhibit gross features of the observed field. The results obtained by Welander (1971) and by Needler (1972) exhibit many similar features even though the former restricted his attention to ideal-fluid models. In the formulation of the problem with diffusion it would appear that one ought to be able to satisfy a more complete set of boundary conditions than with the ideal-fluid model. Yet the restricted character of the analytical solutions does not admit this possibility. The question which is naturally raised at this point is: what role does diffusion play in the thermohaline circulation? Welander (1972) has attempted to answer this question by means of a scale analysis of (9.1) and (9.2). He defines a variable M(X, 4, z ) as

M

=s:P

dz

+ 2Qu2 sin2+ /:wo dh,

(9.56)

where wo = w(h, 4,0) is the (presumed known) vertical velocity at the upper surface. Then u = -(1/2Qa sin w = (1/252u2sinz + ) M A ,

'u =

(1/2Qu sin

+ cos +)MAZ (9.57)

P = M , ,p = - (l / g ) M Z 2 ,

Large Scale Ocean Circulation

69

and the following equation in M can be derived by substituting (9.57) into (9.2a) and (9.2b) cot +MAM,,,

= 0.

(9.58)

Welander’s M equation (9.58) is simpler in form than Needler’s P Eq. (9.6), the difference being that Welander has essentially incorporated the barotropic mode into his definition of M . Th e boundary conditions can be written as

M

=

zcw

sin2

+ fw od ~ ,

M,,

= -gpo(h,

+)

at z = o (9.59)

0

and cot +MA= a(h, M,)/a(A, +),

M,,

=0

at z = -h(h,

4).

(9.60)

If the equations are scaled so that the momentum and mass equations contain no parameters, an advective vertical scale depth defined by 6,

= (2fla2W/gpo)1’2

(9.61)

emerges, where W is the amplitude of w o. A (diffusive) vertical scale depth defined by

6,

= K /W

(9.62)

also enters into the problem, and the M equation takes the form

where E =S,/S,. Welander makes the implicit assumption that E is small. If M is now scaled to be O( 1) at the top surface and if an undetermined scale depth 6 is introduced, then M , AM16 (Welander incorporated a barotropic velocity into his discussion but his assumptions relegate it to an unimportant role in his argument) and the boundary conditions at the top surface require M w 1, A M / P w 1. (9.64)

-

Keeping the diffusion term in order to meet the diffusive boundary conditions requires that some other term must balance the diffusion term in (9.63). It follows readily that if the first two terms balance, then E=

P,

A M = 6’.

(9.65)

But then the last term in (9.63) is larger than either of the two terms kept so this balance is inconsistent.

George Veronis

70 Balancing

EM^^^^ and cot +MAM,,, (with cot 6 w 8,

AM

+

1) yields (9.66)

M E ~ .

The neglected term is O ( P )compared to the terms kept and this balance is therefore consistent. In this case the scale depth is E , or 6, in dimensional terms, and the change in M is small. Hence, the vertical velocity is essentially undiminished over the depth of the diffusive region. I n downwelling regions (w < 0) the balance between vertical diffusion and vertical advection would mean that the density increases with depth. (With constant w and K , it is easily seen that p grows exponentially with depth.) Hence, one is forced to abandon the emphasis on vertical diffusion and to seek a new balance. It is easy to verify that an ideal-fluid region is consistent and that the conditions on 6 and AM are 87-1,

AMml.

(9.67)

Here, the dimensional scale depth is 8, the advective depth. T he conclusion is that in downwelling regions the surface density is simply convected downwards and laterally and that a particle of fluid which sinks in this fashion is eventually advected to a region where upwelling occurs. Near the surface of the upwelling region diffusive processes are important and serve to adjust the density to the surface value. Because of the planetary vorticity relation, aw/ax = (v cot +)/a, the vertical convergence generated in a downwelling region is associated with an equatorward transport. Hence, a flow pattern, shown qualitatively in Fig. 11, is set up. Below the advective region diffusive processes may again be important as indicated in the figure. However, the important consequence of Welander's argument is that the thermocline layer may be an ideal fluid regime bounded above and below by diffusive layers.

\

Equator

Polar region

FIG. 11. Qualitative flow pattern for thermohaline circulation according to the scale analysis given by Welander (1972). In regions of upwelling near the surface (denoted by D) diffusive processes are important. Elsewhere, the density is determined by an advective balance (denoted by A). In deep water diffusion may be important.

Large Scale Ocean Circulation

71

H. REMARKS ABOUT THERMOHALINE CIRCULATION MODELS There are several features of thermohaline circulation models that merit more detailed study. First, the use of a constant eddy diffusion coefficient in the density equation surely requires a thorough examination. Although such constant eddy coefficients have been used to advantage in many inquiries, the experimental and theoretical framework on which eddy coefficients are based is shaky. We know, for example, that vertical mixing in a stratified fluid is less intense in more stably stratified regions than it is in less stably stratified regions. Even in the case of homogeneous, rotating fluids, numerical construction of a macroscopic eddy diffusion coefficient shows that the latter varies with distance from the boundary (Deardorff, 1972). Furthermore, the physical processes that are responsible for mixing in surface waters are quite different from those which induce mixing in deep and abyssal waters. If Welander’s conclusion is correct, i.e., if the thermocline is an ideal-fluid layer sandwiched between upper and lower diffusive regions, there is additional evidence that the use of a single, constant, eddy coefficient is suspect. Recent observations of the distribution of tritium in the oceans indicate that mixing takes place more or less along surfaces of constant density (Rooth and Ostlund, 1970). Although this idea has long been accepted by observational oceanographers, it has been disregarded to a large extent in modeling the thermohaline circulation. If the observed vertical distribution of tracers is attributed primarily to vertical mixing, the importance of the latter process is grossly exaggerated and even qualitative results which emerge from the use of an exaggerated vertical mixing coefficient must be viewed with suspicion. Observations of chemical tracers near the bottom of the ocean (Craig et al., 1970) show that a constant eddy mixing coefficient in that region is sometimes remarkably accurate. However, the same authors have shown that in some cases it is not possible to model the mixing process in that fashion. T h e mixing coefficient that arises from these studies may be larger than 200 cm2 sec- l. Since the values used for studies of the thermocline cluster around 1 cm2 sec-l, it is evident that a constant K is not appropriate if it is to account for diffusion in both boundary layers. Even if the bottom boundary layer is disregarded, it is difficult to justify the use of a constant K for near surface waters and for deeper water (Veronis, 1969). A second feature that merits further study has to do with the boundary conditions to be satisfied by the variables. It was stated earlier that surface and bottom conditions on density plus a condition of zero normal flow at

72

George Veronis

the bottom are to be satisfied. A fourth condition is available from the hydrostatic pressure equation since the pressure is determined only to within an arbitrary function of X and 4. Most investigators have tried to specify the vertical velocity at the upper surface as the fourth condition. Even though the restrictive nature of the similarity-type solutions has made it impossible to satisfy all four of the boundary conditions, it should be possible to do so for the general problem. As the problem is formulated, one looks for solutions which are forced by specified conditions of global scale. T h e vertical velocity below the Ekman layer generated by the wind stress is not of global scale, as we have seen. Therefore, one may ask if the thermohaline model should be expected to include the effects of the winds. I t is perfectly possible to specify the density (or density flux) at the surface, the density and the density flux at the bottom, and the condition of zero normal flow at the bottom. Hence, the fourth condition in this case would be associated with the density at the bottom rather than the vertical velocity at the top. Specifying the density at the bottom is equivalent to specifying the velocity shear at the bottom (through the thermal wind equations). The problem thus becomes one in which the fields are completely determined by specified conditions on density and a reasonable kinematic condition. T h e wind stress must then be taken into account by means of a perturbation (or other process) of this basic state. This approach has not been tried but it represents a sensible alternative to the models that have been worked out.

X. Abyssal Circulation T h e circulation driven by wind stress at the surface appears to be confined largely to waters in the top kilometer (or less). T h e thermohaline circulation discussed in the preceding section is oriented toward determining the structure of the thermocline although the associated flow pattern reaches from the surface to the bottom. Given the existence of the thermocline it is a straightforward matter to derive a flow for the deep waters in the open ocean if one makes two plausible assumptions. Since the thermocline is a region in which the mean density decreases upward, any mechanical process which induces vertical exchanges will serve to bring water of lower density to greater depth. In order to maintain the density of the water at a fixed value, it is necessary that dense water be brought upward. Hence, the first assumption for an abyssal model is that across some level (say at 2000 m depth) there is an upward flux of water, w = woo.If this upward flux is taken to be uniform everywhere in

73

Large Scale Ocean Circulation

the open ocean, it generates a horizontal circulation in the deep, homogeneous water which can be readily calculated (Stommel, 1957; Stommel and Arons, 1959). The simplest model will be considered here. T h e ocean is taken to have constant depth, the bottom is at z = 0 and the top of the deep homogeneous layer is at z = r ) + H where r ) @ H . The amplitudes of the velocities are small so the equations for a steady, inviscid flow reduce to the geostrophic, hydrostatic equation (9.1) together with the continuity equation (9.2b). An integration of the hydrostatic equation from z to r ) yields

P =gPm(r) - 4,

(10.1)

and the geostrophic equations become 252v sin 4 = (g/a cos +)ar)/aA,

2Qu sin # = -(g/a)

(10.2a) (10.2b)

ar)/a#.

Eliminating the pressure from (10.2) gives the planetary vorticity equation aw/az = (cot +/+.

(10.3)

Finally, a vertical integral of (10.3) and use of the conditions ~ I , = ~ + ~ = w , ( = c o n s t ) , wI,=,=O,

r)

+HwH

(10.4)

yield the relation

V H= aw, tan 4.

(10.5)

Hence, the meridional velocity is determined. For a positive upward flow across the surface, z = r ) H , the meridional velocity is poleward. The vertically integrated continuity equation (9.2b) together with (10.5) and (10.4) provides an equation for u,

+

au/aA = - ( 2 ~ , a / H ) cos 4.

(10.6)

Hence, if these equations are assumed to be valid up to an eastern boundary, say a meridion, A= A,, where u vanishes, (10.6) can be integrated to give u = (2w, a/H)(A,- A)

cos

4.

(10.7)

Stommel (1957) assumed that a western boundary could be added to provide total mass continuity. With this single condition, the flow for a basin can be specified completely although only the total transport in the western boundary layer (not the detailed velocity distribution) is given explicitly. Since a source of deep water must be added in order to supply the upward flux of water through the surface z = r ) H , it is necessary to specify

+

George Veronis

74

the location of the source and to equate the flux from the source to the total upward flux through the surface. The major oceans (Atlantic, Indian, and Pacific) are connected by the Antarctic Circumpolar Current. The latter is not bounded by meridional boundaries in an essential manner so that the (observed) eastward transport of this current must be specified as an added, imposed feature in the present model. Particle trajectories for an idealized basin-configuration of the oceans of the world are shown in Fig. 12. Here, sources of equal intensity were taken to be at the north pole and at 60"S, 40"W. These correspond to the observed sources for deep water in the Greenland Sea and the Weddell Sea, respectively. The transports in the western boundary layers, specified as T / w ,u2, are shown to the left of each basin. There are stagnation points of transport in the western boundary layer at approximately 5"N in the Indian Ocean and 22.5"N in the Pacific Ocean. The flow in the interior of each of the basins is poleward, consistent with (10.5). An arbitrary recirculation, R, around Antarctica can be added. The deterministic part of the transport around Antarctica is shown at the bottom of the figure. Sources for both the Indian and Pacific abyssal circulations are provided via western boundary currents which receive the necessary transport from the Circumpolar Current. The abyssal circulation in Fig. 12 is taken from Kuo and Veronis (1971) and forms the basis for a study of the distribution of tracers in the ATLANTIC OCEAN

1

ANTARCTIC ClRCUYPOLAR CURRENT R + T wh... shown on tn.

..~YoI.

T

IS

0 right

INDIAN OCEAN

I

PACIFIC OCEAN

I

-1

-2

FIG.12. Particle trajectories in abyssal waters for an idealized world ocean with equal sources of water at the North Pole and at 605,40"W, including a recirculation in the Antarctic Circumpolar Current of arbitrary transport R .

Large Scale Ocean Circulation

75

deep oceans of the world. Tracer distributions calculated with the given abyssal circulation show overall agreement with observed abyssal tracer distributions in the world's oceans. However, the uniform upward velocity through the base of the thermocline is an assumption and should be deduced as part of a thermohaline circulation model. Furthermore, the effects of bottom topography should show up strongly in the circulation of abyssal waters. Attempts to include such effects are presently being made.

XI. Laboratory Simulation of Large Scale Circulation" Certain features of large scale ocean circulation can be simulated in laboratory models with containers of fluid on a rotating turntable. Qualitative features of the circulation were reproduced in the laboratory in some early studies by von Arx (1952). A more quantitative basis for laboratory simulations where sources and sinks were used to drive the fluid was given by Stommel et ul. (1958). As the theory of rotating fluids has evolved, more detailed features of laboratory models have been analyzed. Pedlosky and Greenspan (1967) presented a linear analysis of a laboratory model and Beardsley (1969) has substantiated those results with a series of careful experiments and a more comprehensive analysis. Baker and Robinson (1969) used a different laboratory model to exhibit similar features. Kuo and Veronis (1970) and Veronis and Yang (1972) have provided a firmer quantitative basis for the study of the flow driven by sources and sinks (the model of Stommel et ul.). A discussion of the latter will be given here and some recent results from laboratory experiments will be shown. A. THEBASISFOR

THE

SIMULATION

It was shown earlier for models of both wind-driven and abyssal circulation that the principal balance in the interior of the ocean is geostrophic. If the effect of wind stress is incorporated as a vertical flux below the Ekman layer, both models can be interpreted as being driven by sources or sinks at the surface. Thus, the planetary vorticity equation for a homogeneous fluid is awlaz = (cot (6/a)v, (11.1)

and a vertical integration from the bottom of a basin with uniform depth (where w = 0) to the base of the Ekman layer yields z, = (aw,/H)tan

* Jointly authored with C. C. Yang.

(6,

(11.2)

76

George Veronis

where H is the (constant) depth of the fluid and wo is the Ekman flux. Hence, an upward suction into the Ekman layer has the same effect as an upward flux through the base of the thermocline and gives rise to a poleward transport. Now consider a pie-shaped basin rotating about a vertical axis through the apex (Fig. 13). Th e interior, geostrophic equations for steady flow (in nondimensional form) are

2k

XV=

-Vp.

(11.3)

Here, the dimensional variables, v, p , and V have been scaled by means of the dimensional quantities V , P, and 1/a and the order-of-magnitude balance P = paVu has been used.

FIG.13. The source flow experiment in a pie-shaped basin of radius a, rotating with ! about the axis through the apex. angular velocity 2

T he height of the equilibrium free surface is given by (6.2). The equation for the disturbed free surface is z = h(r)

+ 5(r, 6, 9,

(11.4)

where 5 is the change in height due to a rising or sinking of the free surface because of a source or sink. Th e linear condition that the top be a free surface is at ah (11.5) w=-+uat ar

Large Scale Ocean Circulation

77

If u and w are nondimensionalized with respect to V and r with respect to (1 1.5) takes the nondimensional form

a,

w=

[ + Fru,

(11.6)

where [ = (1/V) al/at and F = Q2a/gis the Froude number. T h e incompressibility condition is as before,

v.v=o.

(11.7)

Hence, the interior flow is subject to the constraints of the Taylor-Proudman theorem and, in particular, awI/az = 0.

(11.8)

Since wI vanishes at the bottom (for an inviscid fluid), the vertical velocity at the top must also vanish. Therefore, from (11.6) uI = - [ / F r .

(11.9)

This relation is the laboratory counterpart of the meridional flow (1 1.2) which results from the planetary divergence equation (11.1). Thus, for a source-driven flow ([ > 0) the radial velocity, uI, is negative, i.e. the interior flow is toward the apex (irrespective of the location of the source). For a sink-driven flow the radial velocity is toward the rim of the basin. Hence, the apex models the northern boundary (pole) and the rim the equator. There are two points to be made in connection with (11.9). T h e first is that for source-driven flow the free surface rises locally because a column of fluid moves radially inward without changing its height. Since the equilibrium height increases with radius, such an inward flow raises the level of the fluid at any location. There must be boundary layers near the rim and near the apex since the radial flow must vanish at those extremes. For sink-driven flow a column of fluid moves radially outward thereby decreasing the level of the fluid. The second point has to do with how the change of height affects the local vorticity. Conservation of mass requires that a local increase of height of a column of fluid be accompanied by a horizontal convergence. For an inviscid fluid the angular velocity of inward-moving particles must increase in order to conserve angular momentum. T h e result is an increase in the magnitude of the vorticity. Thus, when S2 is positive (counterclockwise rotation) a source-driven flow will cause the relative vorticity to increase locally throughout the interior. Hence, the overall vorticity pattern must exhibit positive vorticity, i.e., there must be a flow toward the rim in a boundary layer on the left side looking downward. In an analogous manner it is easy to verify that a boundary current on the left side and toward the

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apex is associated with sink-driven flow. Hence, a western boundary current is generated in the pie-shaped basin when the flow is driven by a source or a sink.

B. DISCUSSION OF MORECOMPLETE SOLUTIONS A complete linear analysis of the source-sink flows is not difficult to carry out (Kuo and Veronis, 1971). However, the details require an analysis making use of Stewartson shear layers in a rotating fluid and such a treatment would take us too far afield. Consequently, the present discussion will be confined to a qualitative description of the flow and the reader is referred to the Kuo-Veronis paper for details. Figure 14 illustrates the circulation driven by a source located at the junction between the western boundary and the rim for a pie-shaped basin with a semicircular cross section. T h e fluid in the experiment is a thymal blue solution and the flow is visualized by dye streaks generated by pulsing wire electrodes in the middle layers of the fluid (Baker, 1966). This example is especially instructive because both the eastern and western boundaries lie along the diameter, the western boundary along the left half of the diameter and the eastern boundary along the right half. Hence, the eastern and western boundaries meet at the apex. T h e flow is essentially radially inward everywhere in the interior so that the apex serves as a sink for the interior flow, irrespective of where the actual source is. T h e intense western boundary layer flow is obvious in the figure. Flow along the eastern boundary is slow, consistent with the partial analysis given in the previous section. Hence, at the apex there is a divergent flow parallel to the boundary because the slower moving eastern boundary flow cannot supply the amount of fluid required by the intense western boundary layer so fluid is sucked into the vicinity of the apex from the interior. This convergent flow from the interior is evident in the figure. I n a linear flow the structure of the western boundary layer is controlled by frictional processes. Consequently Stewartson. layers of thickness Ell3 and Ell4 ( E = v/fiu2is the Ekman number introduced earlier) serve to redistribute the fluid from the source and also to satisfy the conditions of zero flow at the solid side boundaries. T h e simplest equations for the western boundary layer admit a solution which satisfies only the condition of zero normal flow at the wall. It is an exact laboratory analog of Stommel’s theoretical model for wind-driven ocean circulation. When a more complete mathematical model is formulated so that tangential velocities also vanish at the side walls, the model is the analog of the HidakaMunk lateral friction model of wind-driven ocean circulation.

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FIG. 14. T h e circulation generated by a source at the southwest corner of a p'ie-shaped basin with semicircular cross section (radius = 45 cm, R = 1.9 cm sec-l, Reyiiolds numbc:r = 1.33, mean height = 9 cm).

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Near Y = a a boundary layer must be included in order to adjust the radial flow to zero at the rim and also to bring the tangential flow to zero there. This boundary layer turns out to be thicker than that near the western boundary. T h e Ekman layer near the bottom serves no function in the lowest order equations except within the boundary layers at the sides and at the rim. Hence, the solution found in the previous section is the exact lowest order solution for the interior flow. T h e analysis for the more complete problem requires including nonlinear as well as frictional terms in the equations. The nondimensional form of the momentum equations for this steady flow is

Rv* VV + 2 k

XV=

-Vp

f

EV'V,

and the free surface condition takes the form w =[

+ Fru +F R v . V[.

Here, the velocity scale is chosen as V = Q/AE1/',where Q is the volume flux per unit time from the source and A is the area of the basin ; the length scale is a, the radius of the basin; E = v / Q a 2 ;the dimensional a[/at is set equal to V [ = Q / A so tha [ = Ell2; the [ scale is chosen as QVa/g= RFa; and the Rossby number is defined as R = V / Q a= Q/QaAE'/'. T h e specific choice [ = E l l 2 is made because for the linear problem the velocities are all O(1) as a result. When the Rossby number is E1/', inertial effects are important. T h e ratio R/E1l2is the Reynolds number, denoted here by y , so linear flow occurs for y < 1 and nonlinear effects are important for y 2 1. A study of nonlinear flows by perturbation methods is possible for y-1 and theory and experiment agree for these moderate Reynolds numbers (Veronis and Yang, 1972). A more direct simulation of the North Atlantic gyre including the Gulf Stream is obtained when the flow is driven by a sink, because the flow in the western boundary layer is toward the north (apex). T h e laboratory Gulf Stream is more intense than the flow some distance from the western boundary and, as a result, the apex serves as a point of convergent flow parallel to the boundary. Hence, the boundarycurrent must fanout at the apexto provide fluid for theinterior flow. As the strength of the sink is increased, the flow in the western boundary layer intensifies. T h e perturbation analysis for y-1 can no longer be 1 and it is used to study the nonlinear flow patterns that occur for y necessary to consider a numerical analysis for the latter cases. A numerical study has been made by Beardsley (1972) for the nonlinear flow in the sliced cylinder model of oceanic circulation. Several features emerge in our experiments with the flows driven by a sink at the southwest corner of the basin. T h e western boundary layer is

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formed near the rim and then flows to the apex before it couples again to the interior flow. Between the rim and the apex the western boundary layer flow is extremely stable. In none of the experiments, even with Reynolds numbers as high as 70, has the western boundary layer shown an indication of becoming unstable. A second feature is that as the flow becomes nonlinear a high pressure region forms in the interior near the apex. Th e western boundary jet leaves the boundary in the vicinity of the apex and much of the fluid is swept around the high pressure region and flows toward the rim on the offshore side of the western boundary layer. Hence, a recirculating gyre is generated near the western boundary. Such a feature appears also in numerical analyses of wind-driven ocean circulation (Veronis, 1966) as well as in other experimental studies (Beardsley, 1969). A photograph of the flow pattern for an experiment with y = 5 is shown in Fig. 15 and the high pressure region at the center of the recirculating gyre is clearly present just to the southwest of the apex. A third feature is that sufficiently intense flows become unstable. As noted earlier, the western boundary layer remains stable. However, as the western boundary jet leaves the vicinity of the apex and penetrates into the interior, swirls formon either side and the flow becomes transient. Figure 16 exhibits both the stability of the western boundery layer and the instability of the jet after it leaves the apex and flows into the interior. For this flow the value of the Reynolds number is 20.

C. THEFLOWDUE TO

A

SOURCE OF DENSEWATER

The experiments reported above have all been carried out with working fluids of constant density. When the density of the fluid varies, it is not possible to simulate the beta effect in the simple manner outlined above. Nevertheless, since abyssal circulation is driven by sinking of dense water in polar regions, it is of some interest to determine what the effect is when the source fluid has higher density than the ambient fluid of the basin. Such experiments are basically transient because the relative amounts of water of different densities change monotonically with time. Even so, however, we were surprised to observe the magnitude of the change from the homogeneous case to the two-fluid case. Photographs from three experiments are shown below. I n all three cases the flow is in the linear range (Reynolds number = 0.75), the rotation rate is 1.9 rad sec-l, the mean depth of the fluid initially is 8 cm, and the ~ . the first density of the fluid in the basin initially is 0.998 gm ~ r n - In experiment the density of the source water is also 0.998gmcm-3. For

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FIG.15. Same as Fig. 14 except that the flow is sink-driven and Reynolds number = 5. For the nonlinear flow shown here a high pressure region forms near the western boundary south of the apex.

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FIG.16. Same as Fig. 15 except that Reynolds number = 20. T h e jet bifurcates as it leaves the apex and forms swirls on either side. These swirls eventually separate and become self-contained eddies which are swept around the high pressure region southwest of the apex.

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the second and third experiments the densities of the source water are 1.001 and 1.075 gm ~ m - respectively. ~ , The motion was observed by two different methods. In the first, illustrated in Figs. 17-19, the source fluid is colored and the penetration of the colored fluid is shown shortly after the experiment was started and then toward the final stage of the experiment. Figures 17a and 17b show the penetration of homogeneous source fluid after 2 min and after 90 min, respectively. The ball of fluid at the apex reflects the disturbance created by the source itself. Since the transport in the bottom Ekman layer is to the left of the flow above the Ekman layer, the increased thickness of the dark band of fluid around the sides and rim of the tank in Fig. 17b shows

FIG.17. Dyed fluid with the same density, p = 0.998, as the ambient fluid is introduced through a source at the apex and makes its way around the sides of the basin. T h e penetration of the dyed fluid is shown (a) 2 min and (b) 90 min after source was turned on.

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the inward penetration associated with the Ekman layer. Note especially that the ball of fluid near the apex penetrates very slowly into the interior and even after 90 min it is confined to a sector within 13 cm of the apex. Figures 18a and 18b illustrate the change that takes place when the source fluid is slightly more dense than the fluid in the basin. Here, it is important to note that the flow pattern in Fig. 18b was photographed after a delay of only 21 min. The denser fluid was generally observed to make the circuit around the sides in a time substantially shorter than that required by the homogeneous fluid. The most striking difference between Figs. 17b and 18b is in the penetration of the ball of fluid from the source near the apex. The penetration into the interior is much greater for the dense fluid and, in fact, it works its way to the rim boundary current near the western

FIG.18. Same as Fig. 17 except source fluid has density 1.001. (a) After 2 min and (b) after 21 min.

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side. This penetrative character of the dense fluid is important in connection with the observed high concentrations of tritium in the bottom 100 m of the ocean at mid-latitudes (Rooth, 1971). Tritium is a bomb-produced radioactive tracer which has a relatively high concentration in the surface waters and throughout the depth in polar regions but its presence is barely observable in deep water at mid-latitude. The present experimental result suggests that the high concentration near the bottom is associated with the Ekman layer flux of dense water which sinks to the bottom in the Greenland Sea and then spreads southward in the Ekman layer. The source fluid in the experiment stays close to the bottom whereas in the homogeneous case it tends to distribute vertically. When the density of the source water is large, as in Figs. 19a and 19b, the circuit around the sides of the tank is even faster (there is a ten-minute

FIG.19. Same as Fig. 17 except source fluid has density 1.075. (a) After 2 min and (b) after 10 min.

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FIG.20. Experiment of Fig. 17 repeated. The flow of fluid originally in the basin is visualized by the thymol blue technique. Shows flow after 5 min.

interval between the two photographs). The penetration of the ball of fluid from the source is now erratic but very strong. Earlier photographs show that the ball extends to the rim within four minutes after the beginning of the experiment. Figure 19b shows another feature of the highdensity source flow. There is a strong flux of fluid out of the western

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boundary current about a third of the distance from the apex. This feature was observed in all of the experiments when the density of the source fluid was 1.075. The source fluid in this experiment also tends to stay close to the bottom. The second method of observation is shown in Figs. 20-22, where the thymol blue technique mentioned earlier was used to mark the fluid which is initially in the basin before the experiment was started. Figure 20 shows a flow similar to that of Fig. 14 although the source is located at the apex in the former and at the junction between the western boundary and the rim in Fig. 14. The flow here is steady and consistent with what is expected from linear theory. The ball of fluid resulting from the source near the apex is visible in Fig. 20 and a steady eddy occurs near the rim as the intense western jet impinges on the boundary.

FIG. 21. Experiment of Fig. 18 repeated. The flow of fluid originally in the basin is visualized by the thyrnol blue technique. (a) After 5 min and (b) after 7 min.

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The startling contrast introduced when the source fluid is slightly denser ( p = 1.001 gm cm-3) is shown in Fig. 21. The photographs of the fluid pattern in the homogeneous fluid were taken two minutes apart, and the disordered character of the flow is apparent in each photograph. The western boundary current is toward the rim only in the southern half of the region in Fig. 21a but even there it has reversed and exhibits eddy-like structure in Fig. 21b. Near the rim the eastward jet of homogeneous fluid reverses as the source flow works its way around the rim. The latter behavior is seen in the marked fluid near the rim on the westernmost radial marker. Later photographs show that this reversal progresses eastward with the source fluid. The chaotic behavior associated with a source of dense fluid is exhibited in Fig. 22a where just after the start of the experiment even the

FIG.22. Experiment of Fig. 19 repeated. The flow of fluid originally in the basin is visualized by the thymol blue technique. (a) After 2 min and (b) after 12 rnin.

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interior flow no longer exhibits behavior expected for the homogeneous experiment. T he boundary layer near the rim is completely reversed and there is no longer a southward western boundary current. T h e flow pattern after 12 min is shown in Fig. 22b. Th e intense flow out of the western boundary is again apparent and the interior flow is a collection of completely disordered eddies. There is not much hope of obtaining a complete analysis of the flows generated by a dense fluid source. However, some features, such as the rapid penetration of the dense source fluid via the Ekman layer and the diverging fluid from the western boundary layer, can be obtained in a model more suited for the particular purpose.

ACKNOWLEDGMENT Support from the National Science Foundation under Grant GA 25723 is gratefully acknowledged. REFERENCES BAKER, D. J. (1966). A technique for the precise measurement of small fluid velocities. J. Fluid Mech. 26, 573-575. BAKER,D. J. and ROBINSON, A. R. (1969). A laboratory model for the general ocean circulation. Phil. Trans. Roy. SOC.London Ser. A 265, 533-565. R. C. (1969). A laboratory model of the wind-driven ocean circulation. J. Fluid BEARDSLEY, Mech. 38(2), 255-271. BEARDSLEY, R. C. (1972). A numerical model of the wind-driven ocean circulation in a circular basin. J. Geophys. Fluid Dynam. (in press). BLANDFORD, R. R. (1965). Notes on the theory of the thermocline. J. Mar. Res. 23, 18-29. BLANDFORD, R. R. (1971). Boundary conditions in homogeneous ocean models. Deep-sea Res. 18, 739-751. BRYAN, K. (1963). A numerical investigation of a non-linear model of a wind-driven ocean. J . Atmos. Sci. 20, 594-606. BRYAN, K. (1969). Climate and the ocean circulation. 111. The ocean model. Mon. Weather Rev. 97, 806-827. BRYAN,K., and Cox, M. D. (1967). A numerical investigation of the oceanic general circulation. Tellus 19, 54-80. CARRIER, G. F., and ROBINSON, A. R. (1962). On the theory of the wind-driven ocean circulation. J . Fluid Mech. 12, 49-80. CARRITT, D. E., and CARPENTER, J. H. (1958). The composition of sea water and the salinity-chlorinity-density problems. Physical and chemical properties of sea water. Nut. Acad. Sci.-Nut. Res. Counc., Publ. 600, 67-86. CHARNEY, J. G. (1955a). The generation of ocean currents by wind. J . Mar. Res. 14, 477-498. CHARNEY, J. G. (1955b). The Gulf Stream as an inertial boundary layer. Proc. Nut. Acad. Sci. U S . 41, 731-740. CRAIG,H., SCHLATER, J. G., CHUNG, Y., EDMOND, J. M., KROPNICK, P. M., and WEISS, R. F. (1970). Geochemical and temperature profiles in equatorial Pacific bottom water. Trans. Amer. Geophys. Union 51, 326 (abstr.).

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DEARDORFF, J. W. (1972). Numercial investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. (in press). V. W. (1905). On the influence of the earth’s rotation on ocean currents. Ark. Mat., EKMAN, Astron. Fys. 2, 1-53. EKMAN, V . W. (1908). Die Zusammendruckbarkeit des Meerwassers. Publ. Circ. Cons. Explor. Mer 43, 1-47. ERTEL,A. (1942). Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z. 59, 277-281. FOFONOFF, N. P. (1954). Steady flow in a frictionless homogeneous ocean. J. Mar. Res. 13, 254-262. FOFONOFF, N. P. (1962a). Physical Properties of sea water. “ T h e Sea,” Vol. 1, pp. 3-30. Wiley (Interscience), New York. FOFONOFF, N. P. (196213). Dynamics of ocean currents. “ T h e Sea,” Vol. 1 , (Interscience) pp. 323-395. Wiley FOFONOFF, N. P., and FROESE, C. (1958). Program for oceanographic computations and data processing on the electronic digital computer ALWAC 111-E. PSW-1 Programs for properties of sea water. Fish. Res. Ed. Can., Manuscript Rep. Ser. No. 27 (unpublished manuscript). H. (1968). “ T h e Theory of Rotating Fluids.” Cambridge Univ. Press, GREENSPAN, London and New York. L. N. (1963). On a time-dependent motion of a rotating GREENSPAN, H. P. and HOWARD, fluid. J. Fluid Mech. 17, 385-404. HIDAKA, K. (1949). Mass transport in ocean currents and lateral mixing. J . Mar. Res. 8, 1 932-1 936. Holland, W. (1967). On the wind-driven circulation in an ocean with bottom topography. Tellus 19, 582-600. K. (1966). Ekman drift currents in the Arctic Ocean. Deep-sea Res. 13, 607HUNKINS, 620. JEFFREYS, H. (1962). “ T h e Earth.” Cambridge Univ. Press, London and New York. Knudsen, M. (1901). “ Hydrographical Tables.” Williams & Norgate, London. Kozlov, V. F. (1966). Certain exact solutions of the non-linear equation for density advection in the ocean. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okean 2, 1205-1207; see Atmos. Oceanic Phys. 2, 742-744 (1 100). Kuo, H. H. and VERONIS, G. (1970). Distribution of tracers in the deep oceans of the world. Deep-sea Res. 17, 29-46. Kuo, H. H., and VERONIS, G. (1971). The source-sink flow in a rotating system and its oceanic analogy. J. Fluid Mech. 45, 441-464. LYNN,R. J . , and REID,J. L. (1968). Characteristic and circulation of deep and abyssal waters. Deep-sea Res. 15, 577-598. MARGENAU, H. and MURPHY, G. M. (1949). “ T h e Mathematics of Physics and Chemistry.” Van Nostrand-Reinhold, Princeton, New Jersey. MORGAN, G. W. (1956). On the wind-driven ocean circulation. Tellus 8, 301-320. MUNK,W. H. (1950). On the wind-driven ocean circulation. J. Meteorol. 7 , 79-93. MUNK,W. H. (1966). Abyssal recipes. Deep-sea Res. 13, 707-730. MUNK,W. H., GROVES,G., and CARRIER, G. F. (1950). Note on the dynamics of the Gulf Stream. J . Mar. Res. 9, 218-238. NEEDLER, G. T . (1967). A model for thermohaline circulation in an ocean of finite depth. J . Mar. Res. 25, 329-342. NEEDLER, G. T. (1972). Thermocline models with arbitrary barotropic flow. Deep-sea Res. 18, 895-903. NIILER,P. A,, ROBINSON, A. R., and SPIEGEL,S. L. (1965). On thermally maintained circulation in a closed ocean basin. J. Mar. Res. 23, 222-230. PEDLOSKY, J. and GREENSPAN, H. P. (1967). A simple laboratory model for the ocean circulation. J. Fluid Mech. 27, 291-304.

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PHILLIPS,N. A. (1963). Geostrophic motion. Rew. Geophys. 1, 123-176. PHILLIPS,N. A. (1966). The equations of motion for a shallow rotating atmosphere and the “ traditional approximation.” J . Atrnos. Sci. 23, 626-628. RAMSEY,A. S. (1964). “ Newtonian Gravitation.” Cambridge Univ. Press, London and New York. REID, J. L. (1965). “ Intermediate Waters of the Pacific Ocean.” Johns Hopkins Press, Baltimore, Maryland. ROBINSON, A. R., and WELANDER, P. (1963). Thermal circulation on a rotating sphere; with application to the oceanic thermocline. J. Mar. Res. 21, 25-38. ROOTH,C. (1971). Private communication. ROOTH,C. and ~ S T L U N D ,G. (1972). Penetration of tritium into the Atlantic thermocline. Deep-sea Res. 19,481-492. STOMMEL, H. (1948). The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Union 29, 202-206. STOMMEL, H. (1955). Lateral eddy viscosity in the Gulf Stream Systems. Deep-sea Res. 3,88-90. STOMMEL, H. (1957). A survey of ocean current theory. Deep-sea Res. 4, 149-184. STOMMEL, H. (1965). “ The Gulf Stream.” Univ. of California Press, Berkeley. STOMMEL, H., and ARONS, A. B. (1959). On the abyssal circulation of the world ocean. I. Stationary flow patterns on a sphere. Deep-sea Res. 6, 140-154. STOMMEL, H., ARONS,A. B.,and FALLER,A. J. (1958). Some examples of stationary planetary flows. Tellus 10, 179-187. SVEFUIRUP,H. U. (1947). Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Nut. Acad. Sci. U S . 33, 318326. SVERDRUP, H. U., JOHNSON, M. W., and FLEMING, R. H. (1942). “ T h e Oceans.” PrenticeHall, Englewood Cliffs, New Jersey. J. C., and HAMON, B. V. (1960). Some measurements of deep currents in the SWALLOW, eastern North Atlantic. Deep-sea Res. 6, 155-168. SWALLOW, J. C., and WORTHINGTON, L. V. (1961). An observation of a deep countercurrent in the western North Atlantic. Deep-sea Res. 8, 1-19. VERONIS, G. (1963). On inertially-controlled flow patterns in a ,%plane ocean. Tellus 15, 59-66. VERONIS, G. (1966). Wind-driven ocean circulation. Deep-sea Res. 17, 17-55. VERONIS,G. (1969). On theoretical models of the therrnohaline circulation. Deep-sea Res. 16, Suppl. 301-323. VERONIS, G., and YANG,C. C. (1972). Non-linear source-sink flow in a rotating pie-shaped basin. J. Fluid Mech. 51, 513-528. VONARX,W. S. (1952). A laboratory study of the wind-driven ocean circulation. Tellus 4, 311-318. WARREN, B. (1963). Topographic influences on the Gulf Stream. Tellus 15, 167-183. WEBSTER, F. (1961). The effect of meanders on the kinetic energy balance of the Gulf Stream. Tellus 13, 392-401. WELANDER, P. (1959). An advective model of the ocean thermocline. Tellus 11, 309-318. WELANDER, P. (1968). Wind-driven circulation in one- and two-layer oceans of variable depth. Tellus 20, 1-16. WELANDER, P. (1971a). Some exact solutions to the equations describing an ideal-fluid thermocline. J . Mar. Res. 29, 60-68. WELANDER, P. (1971b). The thermocline problem. Phil. Trans. Roy. SOC.London Ser. A 270.69-73.

The Wave Resistance of Ships JOHN V. WEHAUSEN Department of Naval Architecture University of California. Berkeley. California

I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 . The Measurement of Wave Resistance . . . . . . . . . . . A . Residuary Resistance and Refinements-Froude’s Method . B. Momentum Considerations . . . . . . . . . . . . . . C. Wave-Pattern Analysis . . . . . . . . . . . . . . . . 111. The Analytical Theory of Wave Resistance . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . B. The Exact Problem in an Inviscid Fluid . . . . . . . . . C . Perturbation Expansions . . . . . . . . . . . . . . . . D . Methods of Solution . . . . . . . . . . . . . . . . . E. Further Results, Variations. and Extensions . . . . . . . F‘. Numerical Methods . . . . . . . . . . . . . . . . . . G . Comparison of Theory and Experiment . . . . . . . . . H . Applications of the Theory . . . . . . . . . . . . . . I . Higher-Order Theories . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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For Georg Weinblum for his 75th birthday

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I Introduction A newcomer to the subject would probably ask the following questions . What is wave resistance? How do you measure it? How do you compute it? We shall discuss in the following pages all three questions and try to describe how they have been answered up to now . The answer to the question. “What is wave resistance?”. is not as elementary as it might seem at first glance. One observes that waves follow a moving ship. supposes that some expenditure of energy is required for 93

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their generation, and probably recognizes that the propagation of these waves is associated with the presence of a gravitational field. However, wave formation is not the only expense of energy. One also knows that water is endowed with viscosity and that any body moving through a viscous fluid experiences a resistance, partly due to tangentially acting stresses on the body, but partly also because boundary-layer growth and, of course, separation if it occurs yield a resistance resulting from integrating normal components of the stress over the body. A new question then is, “Can we clearly separate resistances due to gravity alone and to viscosity alone?” That is, is there a “gravitational resistance ” and a “ viscous resistance ”? A moment’s thought shows that this is not likely. T h e frictional resistance, i.e., the part due to tangential stress, will certainly depend upon the wave profile along the ship. On the other hand, the wave pattern itself is going to depend in some fashion upon the ship’s boundary layer and wake. T h e effects of gravity and viscosity interact in essential ways. There seems to be no neat practicable definition of gravitational or wave resistance without introducing assumptions or approximations. Since any discussion of measurement or calculation of wave resistance necessarily touches on its definition, we shall have to return to this question subsequently. Finally, one may ask, “Why make any decomposition at all?” The answer is primarily economic. One would like to be able to predict the power requirements of a full-scale ship from its designed lines. T o do this by calculation from the Navier-Stokes equations is beyond our powers. T o do it by model tests leads to dilemmas that can be resolved, at least in part, if we can find the components of resistance, especially as they depend upon the effects of gravity and viscosity. This has usually been simplified further into trying to separate the resistance into parts due to wave making and to friction. As we shall see below, the separation has been moderately successful, in fact, enough so that in this account of the subject we shall be able to avoid dealing with viscous resistance except in a superficial way. A further reason for trying to separate wave resistance and to understand its relationship with hull shape is ship design. For conventionally shaped ship hulls one cannot reduce the frictional resistance significantly by redesign of the hull. Eddy resistance can be kept small by properly designing the afterbody. Added resistance resulting from bilge vortices can also be avoided by proper design. (Wave-breaking resistance, although it shows up in certain experiments as viscous resistance, will be lumped here with wave resistance). This still leaves great freedom in designing the hull, and experience and observation have shown that optimum designs (from thestandpoint of resistance) are those that make the smallest waves. Rational

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hull design requires an understanding of how a suitably defined wave resistance is connected with hull geometry. Although apparently not the first person to try to examine in a systematic and scientific way the resistance of a ship, William Froude (1810-1879) seems to have been the first to appreciate fully the differing roles played by friction and wave making in ship resistance and the significance of this difference in trying to project data from model tests to full-scale size. This is already evident in a memorandum of December 1868 from Froude to the Chief Constructor of the Navy in which he outlines the advantages of model tests and proposes construction of a model tank. In a later paper (1876) Froude discusses the components of ship resistance and their implications for ship design with such insight that this paper can still be read with profit. Froude’s analysis laid solid foundations for future tests of ship hulls and his ideas still dominate the subject. However, Froude’s method, which will be briefly described later on, has one important disadvantage, explicitly recognized by him, namely, that it did not allow one to predict by purely analytical means the resistance of a ship. T h e first significant step in this direction was taken by J. H. Michell(l863-1940) who in a paper published in 1898 derived an analytic expression for the wave resistance of a ship moving in a calm, inviscid fluid. I n much the same way that Froude’s ideas have dominated the field of ship model testing, Michell’s results have dominated much of the subsequent analytical study of wave resistance. T h e present account of the subject is divided as follows. T h e first part treats the measurement of wave resistance. This in turn is divided into a brief exposition of Froude’s method followed by a more lengthy treatment of the determination of wave resistance from wave-pattern measurements. The second part treats analytical methods for calculating wave resistance and some applications. There exist already a number of expository accounts of the separate parts of this article. For a discussion of the problem of separation of wave and viscous resistance one can hardly do better than to read Sharma’s (1964, 1965) account of this subject. For questions concerning the determination of wave resistance from direct wave measurements a paper by Eggers, Sharma, and Ward (1967) gives not only an excellent survey of the subject but also a valuable bibliography. For the analytical theory of wave resistance there are a number of expository accounts directed at different audiences. There is a fairly comprehensive treatise by Kostyukov (1959, 1968). Of the others listed below perhaps the most comprehensive are by Wigley (1949), Lunde (1951a), and Sabuncu (1962b), the last not being really very accessible because of language. T h e others are more restricted

John V . Wehausen

96

in scope or in prospective audience. See the following references: Bessho (1957b), Gadd (1968), Havelock (1926b, 1951), Inui (1954,1957), Kostyukov (1959, 1968), Lunde (1951a, 1957, 1969), Maruo (1957), Sabuncu (1962b), Wehausen (1956), Weinblum (1950, 1959, 1963), Wigley (1930b, 1935, 1949).

11. The Measurement of Wave Resistance A. RESIDUARY RESISTANCE AND REFINEMENTSFROUDE’S METHOD Determination of the wave resistance by experiment has proceeded along two rather different paths. I n the older one, to be described here, wave resistance is usually identified with a component extracted from the total resistance by a method similar to that originally devised by Froude. This component is the “residuary resistance,” defined below, or some part of it. It is easy to criticize the method and to propose modifications, and many persons have done so. Although some of the analyses that have been proposed in recent years are ingenious and throw considerable light on the components of ship resistance, I believe that it will be clear from the brief description below of Froude’s method that, in fact, the effects of gravity and viscosity are “ inextricably interwoven,” to borrow a phrase from Froude himself. The purpose of this section is then not to expound all the recent developments in this field but rather to give some perspective concerning this method of trying to assess the contribution of wave making to the resistance of a ship. Consider a ship moving steadily on a constant course in a calm sea. Let its length, say a t the waterline, be L and its velocity U , and let the density of the water be p and its kinematic viscosity v. T h e ship moves at some expense of energy. Let E be the rate of energy expenditure. We may call R = E / U the resistance of the ship. If we neglect certain other physical parameters, especially the vapor pressure of the water and surface tension, and consider our ship as one of a family of geometrically and dynamically similar ships, we may write R = F ( L , U , p, v, g). We must include g, for we know that the gravitational force plays an important role in the formation of the wave pattern behind the ship. We need not dwell upon the ways of making this dimensionless, but write down immediately

C, =

R zp U2L2

=f( U/(Lg)1‘2, UL/v)= f ( F n , Rn),

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where F n is the Froude number and Rn the Reynolds number. C, is evidently a function of the two variables Fn and Rn and can be represented by a surface over the (Fn , Rn)-plane that will be characteristic for this family of ship hulls. If one experiments with just one of these hulls, how much of the surface can one construct! Experiments at one place and during a restricted time interval will allow variation only in U, i.e., variation along a ray in the (Fn , &)-plane defined by

With a given hull very little variation in A is possible in practice. T h e value of g is practically constant, although one can make thought experiments with dropping elevators, merry-go-rounds, or towing tanks on the moon ; and v can be varied rather little for practically useful fluids, perhaps by a factor of two. We shall call this result the “ ship-model tester’s dilemma ” (although it is only one of several), for it shows why tests on a model cannot in principle give information about the prototype (which we assume to be much larger). Since Ap $ A,, the model experiments simply explore the wrong part of the surface C,(Fn , Rn), as shown schematically on Fig. 1. If one wishes to allow for some variation in v for the model test, one may replace the ray labeled with A, by a narrow sector. It is not customary to use three-dimensional representations. Instead, one projects the curves cut out of the surface C,(Fn, Rn) onto the (Rn, C,)-plane, and, in order to bring the curves conveniently onto one graph, uses a logarithmic scale for

FIG.1. Sketch showing parts of surface CT(Rn,Fn) determined by a model test and needed for prototype ship.

98

John V . Wehausen

0 0

30.00 30.00 24.00

6.350 6.350 7.938

+ (1

X 0

12.00 9.00 6.00 4.00

15.875 21.167 31.750 47.625

REYNOLDS NUMBER

FIG.2. Values of resistance coefficient CT obtained from tests with models of Lucy Ashton. [From Hinterthan, 1956 (Fig. 18).]

R n . Figure 2 shows a typical plot of such resistance curves for a family of models. T h e constant-A curves are labeled by the model lengths, which may not be consistent in a dimensionless plot but is hardly confusing since in practice v changes but little. T h e curves cut out of the surface C,(Fn, Rn) by constant-Fn planes are also labeled, but again not dimensionlessly. T h e prototype in this case, Lucy Ashton, is only 190 f t long. I n order to give some idea of the magnitudes of F n and Rn , we remark that usually 0.1 < F n < 0.5 and that for the usual merchant ship 0.15 < F n (0.3. T h e values of Rn depend then upon ship size. For five-foot models (the smallest usually tested because of laminar-flow difficulties) the Rn range corresponding to 0.15 < F n < 0.3 is approximately 1 x lo6 < Rn < 2 x lo6, for a 20-foot model, 7.5 x lo6 < Rn < 1.5 x lo7, and for a 500-foot ship, 1 x lo9 < Rn < 2 x lo9. T h e dilemma mentioned above was resolved after a fashion by a bold assumption of William Froude. T h e assumption as we shall state it below is not really in the form given by Froude. Froude did not know about Froude or Reynolds numbers, or about dimensional analysis. He discovered were by experimenting with geometrically similar models that, if UL kept constant, the wave patterns of models of different sizes were similar.

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99

However, the equation given below represents (almost) in our notation his assumption, and in any case is usually called Froude’s hypothesis.

C,(F?Z, Rn) = C,(Fn)

+ CF(R7.2).

It follows immediately from this assumption that the curves Fn = const. are all at a constant vertical distance from one another in the ( C , , Rn)-plane and that the curves A = const. are all congruent, one being obtained from the other by sliding along the curves Fn = const. Hence, in applying Froude’s assumption to model testing, one tests the model at the same Froude number as the prototype, which in any case is more practical than testing at the same Reynolds number, and then finds the total resistance coefficient of the prototype from cTp

cT?n

- cFm -k

cFp

*

What is necessary now, of course, is to know the curve C,(Rn) for the given hull form In fact, it would be sufficient to know the function CT(Fn, Rn) for any fixed value of Fn . Any such curve is known as an “ extrapolator.” One could consider Froude’s assumption as purely an empirical one to be supported by the examination of data like those shown in Fig. 2. However, there was a rationale behind Froude’s assumption. Let us decompose C , into parts obtained by integrating over the ship’s wetted surface the components in the direction of motion of the normal and tangential components of the stress vector:

CT(Fn Rn) == cnoi-m(Fn> Rn) f Ctang(Fn Rn). This decomposition is, of course, precisely defined. Now one argues that the tangential force is primarily determined by the viscosity of the water and the normal force primarily by the effect of gravity as a result of the = waves produced by the ship, or, in dimensionless variables, that Cnorm Cnor,(Fn)and C,,,, = Ctang(Rn).In a final bold step one replaces the tangential force by the tangential force on a flat plate of the same water-line length and wetted-surface area as the ship at rest. If S is this area, one defines a frictional-resistance coefficient by C,(Rn) = (flat-plate resistance)/ipUU2S. (Note that L2has been replaced by S . Other coefficients will now be similarly redefined). Th e difference C, - C, = C, is now assumed to be a function of Fn only and is called the “ residuary-resistance ’’ coefficient. The advantages and weaknesses of Froude’s assumption are nearly obvious. T he choice of C, has yielded a function of Rn that is independent of hull form. T he only problem is to measure it with adequate accuracy over a sufficiently wide range of Reynolds numbers. This problem, for years a lively topic among naval architects, and still not a dead one, will not be discussed here. The curve in Fig. 2 labeled “ Schoenherr Line ” is one of

100

John V . Wehausen

the standard formulas for C, , but there are others. T h e residuary resistance contains everything else : wave resistance, eddy resistance, curvature effects, effects of trim and sinkage, the difference between the true wetted area and the wetted area at rest, etc. T h e weakness of the assumption is, of course, to assume that C , is a function of Fn alone. A frequent refinement is to estimate an eddy or form resistance as a function of the form: C, = a bC, , where a and b are empirical coefficients depending upon the hull form. I t is not uncommon to find a = 0 and b g 0.16. T h e constants are chosen so that at the lower end of the measured total-resistance curve, where the C , curve becomes approximately parallel to C,, the curves C , and C, a bCF run together. T h e curve C , -- CF- C , is often identified with the wave resistance, although not without awareness on the part of naval architects of the shortcomings of the definition. A more sophisticated version of Froude’s method, but also one requiring more effort in application, is to define the viscous resistance as half the resistance of a double model tested at appropriate Reynolds numbers in a wind tunnel or deeply submerged in a towing tank. If this is denoted by C,(Rn), then C , - C, will still depend upon Rn as well as Fn inasmuch as the double-model experiment does not include the effect of varying wave profile and sinkage and trim upon frictional resistance, or even upon such phenomena as separation. Even at low Froude numbers when there are almost no waves, the wakes of the single and double model may be different because of the lack of turbulent interchange across the free surface with the single model. On the other hand, C, comes much closer to including all the effects of viscosity associated with a given hull form than do the methods based upon flat-plate frictional resistance. Many further refinements of Froude’s assumption are possible and, since any improvement makes more accurate the prediction of prototype performance, there are frequent proposals. Since it is not our main purpose to discuss such matters, we cite only a few recent papers concerning this topic: Lackenby (1965), Shearer and Cross (1965), Weinblum (1970), Shearer and Steele (1970). As mentioned earlier, the purpose of this somewhat lengthy but still superficial introduction to the phenomenological theory of ship resistance is to try to provide some perspective about the nature of the resistance derived from such experiments, for a theoretically computed wave resistance is often compared with them.

+

+ +

B. MOMENTUM CONSIDERATIONS In the following paragraphs we shall derive some general expressions, in terms of certain surface integrals, for the force acting upon a body in steady translational motion. The derivation will be carried out for a viscous fluid

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101

satisfying the Navier-Stokes equations and the boundary conditions appropriate to them. However, the final formulas are also correct for an inviscid fluid if one sets v = 0. It is not obvious that this will be so, for the boundary conditions appropriate to an inviscid fluid are different from those for a viscous fluid. However, one may verify the correctness of the formulas for an inviscid fluid by examining, as one proceeds through the proof, the effect of altering the boundary conditions on the body. We begin with a coordinate system fixed in the fluid, 07 directed upward (against gravity), & in the direction of motion, and 0 5 to the starboard; &% coincides with the undisturbed water surface. We recall that for an incompressible Navier-Stokes fluid the stress tensor is given by Tij

+p(ui, +

== -@ij

j

(2.1)

uj. i),

where (&, Z 2 , 2,) = (3,7, Z), the velocity components of the absolute . shall use the conmotion are (u, w , w ) = (ul, u2, u,) and ui, = a ~ , / & ~We vention that repeated indices denote summation. The velocity components must then satisfy the equations

pauilat

+

ui, i = 0, puk u i , k

= -psi - pg6i2

+

pu, k k

i=

i

3*

2i

(2.2) (2*3)

If the velocity at any point on the surface S of the body is V = (U,, U,, U3) = ( U , V , W ) ,then the boundary condition on the body is u i = Ui on S. For an inviscid fluid it is only the kinematic condition u 1. n1.= U.n. I I'

(2.4)

(2.5)

Since we shall not be considering the effects of cavitation, there is no loss in generality in taking the atmospheric pressure to be zero. We shall also neglect surface tension, for it plays practically no role for bodies of the size of concern to us. The dynamical boundary condition on the free surface is then simply Ti,

n, = -pni

+ p ( u i ,,+ u,,i)nk= 0,

i = 1, 2, 3.

(2.6)

If the fluid is inviscid, one need only set p = 0 above, to obtain the usual

p

= 0.

(2.7)

In addition, the kinematic condition must be satisfied. If we express the free surface by jj = Y(2,5,t ) , this takes the well-known form

Y$,jj, t)u - v

+ Yrw + Y t = 0.

(2.8)

John V . Wehausen

102

Consider now the fluid bounded by the wetted surface S of the body and a (possibly moving) control surface t; enclosing the body. Part of t; may consist of free surface but the combined surface S and C must bound only fluid. Figure 3 shows schematically two possibilities. The total momentum of the fluid in the volume V is Qi

pui dV,

=

and the rate of change of momentum is

If we now use the Navier-Stokes equations to replace aui/at and use the continuity equation ui,{= 0, we obtain

=

s,

bui(Uknk-Uknk) -tTiknk]dS-pgl VIsiZ,

(2.10)

where I VI is the volume of V. On any part of I; that is the free surface both terms in the integrand vanish, so that the integral over Z extends only over its submerged portions. In the integration over S the first part of the integrand vanishes because of the kinematic condition (2.5). The remainder is just the force acting on the body:

Fi = - J j T i k nk dS. S

If we let Zc stand for the submerged part of C, we finally obtain

FIG.3.

Sketches illustrating surfaces S, C, , and Z;,

(2.11)

The Wave Resistance of Ships

103

We now add a further restriction to the surface C. I t is to move together with the body. If we now assume that the motion has continued for such a long time that the mean motion of the velocity field within V no longer changes with time, then dQi/dt = 0. (Note that we are allowing turbulent motion in the wake as long as the mean motion is steady.) This now gives us a formula for the mean force acting on the body in terms of an integral over the control surface Zc on which

U, nk = Un,,

(2.13)

where U is the velocity of the body in the direction of 62. I n finding the mean value of (2.12) one must take especial care with the quadratic term pui uk nk . If we write temporarily u i = G i uit, where C i gives the mean velocity field, then

+

pui uknk = pGi zik nk

+pui’uk’nk.

We now drop the bars over the mean velocity and denote it by ui . We then find for the mean resistance R = -Fl, the only component of Fi in which we are interested here, the following expression :

R =-

1

[pul(un, - ukn k )-pn, - p u i n k

+

ZC

+

dS. (2.14) T h e quantities u i ,p , R are now all mean quantities, and -Pzdltz&‘nk is a Reynolds stress, which may be much more than the viscous stress. Although we have used only one component from (2.12), one should be aware that much of the ensuing analysis could also be carried through for side forces and sinkage, and, with a further extension of the calculations, for trim. This is an exact formula in the sense that no mathematical approximations have been made in deriving it from the original assumptions. As was mentioned earlier, this is also an exact formula for an inviscid fluid if one sets p = 0. Even if the viscosity of water is taken into account, the term with p as coefficient is usually quite small compared with the other terms. This does not mean, however, that the values of the integral will be nearly the same in either case, for the velocity uk and the pressure p satisfy different differential equations and boundary conditions in the two cases. I t will be convenient to make here a slight change in notation. Since the motion has been assumed steady in the mean in the frame of reference of the ship (taken here as Oxyz), we may write p(ul,k

uk,l)nk]

7,z, t ) = Ci(R - ut,7, 2 ) = C i ( X , y, z), P(%7,%t ) =$(a - ut,7 , 2 )= y , z),

Ui(X,

m,

Y(2,x, t ) = F(X - ut,X) = F(x, z),

(2.15)

John V . Wehausen

104 so that au, -=-u-

at

as,

au, -=-

azk

ax

asi,

etc.

ax,

With this understanding we shall drop the tildes. Formula (2.14) is not altered by this. It is convenient to make some special choices for 6,. Let us first suppose that the ship is moving down a rectangular canal and let V consist of the fluid bounded by the side walls, bottom, and planes ahead of and behind the body perpendicular to the walls and bottom. Of the plane areas bounding this volume we denote the one on the starboard wall by C,, on the port wall by X p ,on the bottom by X H ,ahead of the body by C A ,and behind by C B. Taking account of boundary conditions on the solid surfaces and of the simple form of the normal vectors (always directed out of the fluid) to C,, we find

R=-J

puz dx dy

+ J puZdx dy + J =P

ZS

-

LA

[pu(U-~)-~-pU'2+2pu,]

+J

[pu( U - U) - p

pu,, dx dx

ZH

- pU12

dydx

+2 p , ] dy dz.

(2.16)

We now modify this formula by taking X A far enough ahead of the body so that there is no disturbance of the fluid, a possibility that we assume as one of our boundary conditions. Then only p = pgy remains in the integrand and the upper limit of the y-integral is 0. We may imbed this integral in the Z A integral if we take account of the fact that the upper limit in the y-integral over Z B is Y (xB, x). We then have R=-J

p u, dx dy ZS

+J ZP

pu, dx dy

+J

pug dx dx

ZH

(2.17) where xBis the x-coordinate of the moving plane C B, and z p and xs are the x-coordinates of the walls X P and Cs, respectively. Finally we make one more change suggested by Landweber and Jin Wu

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105

(1963) and, in the form used below, by Sharma (1965). Define a “total head ” as follows : P g H h y , z ) =P

+pgy +

*PP[(U

+ +4

- y2 v2

(2.18)

and let pgH0 = &pU2.

I t follows from Euler’s integral that for irrotational motion of an inviscid fluid H = Ho everywhere. Hence H - H,, represents a kind of measure of the effect of viscosity. A straightforward manipulation allows one to put (2.17) into the form

R

=-

J

p ~ dx ,

dy

+J

pu,

dx dy

ZP

ZS

+21 p J

(- u2 + v2

+J

pug dx

dz

ZH

+ w2) dy dz

ZB

(2.19) I n the case of irrotational flow of an inviscid fluid, one is left with only the last two terms:

1 R = - p J ( - u2 v2 w 2 )dy dz

+ +

ZB

+ -21 pg J

eS

Y2(xB, Z ) dZ.

(2.20)

ZP

This is, of course, a genuine “ wave resistance,” but, as mentioned earlier, its value will not be the same as that of the corresponding terms in (2.19). T h e integrals over C s , C p ,and C H and the contribution of the term 2pu, in the integral over X B are generally negligible in comparison with the other terms. However, this is not the same as neglecting viscosity. If the body is moving in an infinitely deep fluid unbounded horizontally, one may choose as the control surface Xc a rectangular box as we did above for the canal, noting, however, that Xs, C p, and C, are no longer physical surfaces, so that the integrands in the integrals over these surfaces are more complicated :

Z’ H

[puv

+PUT - p(uY+ v,)] dx dx.

(2.21)

John V . Wehausen

106

If one lets the planes, Xs, X p , X H recede to infinity, these integrals converge to zero and one is left with the last three integrals of (2.19) with zp=-a, xs =00. If one sets p = 0, one obtains again (2.20). On the other hand, if one lets X B recede to infinity and sets p = 0, one obtains R = J puw dx dy - J puw dx dy - J puv dx dx.

(2.22)

LY

BP

&S

Whenever we have set p = 0 above, we have also supposed the Reynolds stress to vanish.

1. Separation of Wave and Viscous Resistance Since the last two terms of (2.19) coincide in appearance with (2.20), a true wave resistance in an inviscid fluid, one might be tempted to simply define these two terms as the “ wave resistance ” and

as the “viscous resistance.” However, if one applies (2.19) to a flow without a free surface and with no gravity, one still has the term 1 2

-p

”!

(-u 2 + v 2

+w”)dydz,

EB

even though one now has a true viscous resistance. This integral must evidently play a role in each component in any attempt to separate viscous from wave resistance. The first attempt to base a separation of viscous and wave resistance upon momentum considerations goes back to Tulin (1951). Since then the ideas have been further developed and refined by Jin Wu (1962), Landweber and Wu (1963), Sharma (1964, 1965), Tzou and Landweber (1968), Baba (1969a), Brard (1970a,b), and others. The main idea in these attempts is that there is a boundary-layer plus wake region (hereafter BLW) where the velocity field is rotational, but that outside this region it is irrotational. Then, from a well-known theorem concerning vector fields, we may decompose v as follows : v=v,+v,

(2.23)

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107

where

1 v --curl

S,,

~-+rr

curl v -dv, r

curl v = curl vR, curl vI= 0,

v I ~ v - v R ,

div vR= 0,

(2.24)

div vI= 0.

Since vI is irrotational, it can be generated from a velocity potential vI such that grad rpl = v I . T h e function v1 is by its definition defined everywhere in the fluid, and in the region exterior to BLW. However, v R is also irrotational exterior to BLW and can be generated there by a velocity potential v R .Hence, exterior to BLW = grad(vl

+

YR).

Within BLW this representation does not hold, for v, is not irrotational there. However, v1 within BLW is a harmonic extension of cpI outside BLW, and since it seems reasonable to assume that the field v is continuous with continuous first and second derivatives, the field vRand hence vI may be presumed to have the same properties. I n particular, the function y I will be continuous everywhere in the fluid. Now we should like to assume that vRcan be extended harmonically into BLW. However, we cannot expect that this extension can be made without there being a discontinuity in v R .It will be convenient to define the extended q R so that the discontinuity is on the plane x = 0. Let us now define a potential flow vP

= grad(?l

+

P)R)

= grad Y

P>

which is defined everywhere, but with a possible discontinuity within BLW on z = 0. We further define a wake velocity field by

vw = v - vp. Although we know the boundary condition satisfied by v on the ship's wetted surface, namely v = ( U , 0,O), we cannot easily deduce the boundary behavior there of v, or vw . In particular, we cannot expect that vp.n= U n l , the boundary condition that would ordinarily be imposed for an irrotational flow in an inviscid fluid, for, to speak loosely, the function ' p p exterior to BLW does not know that there is a solid boundary at S. Presumably there does exist a stream surface S' starting at the ship's bow, perhaps not closed, upon which this condition will be satisfied. We now substitute this decomposition into (2.19), discarding the terms in p as being negligible, but retaining the Reynolds stress. T h e possible

John V. Wehausen

108

discontinuity of y p on x = 0 requires an easy modification of (2.19), for the control surface C, must now include the two sides of the surface of discontinuity in order to exclude any discontinuity within the fluid volume V . After some small manipulations one obtains the following formula:

1

I-zpS

[vw2+ww2 t 2 v ~ v w +2wPww] dy dx

ZW

1 + j p g / [ Y w z +2YPYw] dz, where C, is that part of C, within BLW. We now define a "wave resistance " by the first three integrals in (2.25):

1 R w = 2 p Jz,[-upz

+ up2 + wpz]dy dz (2.26)

Corresponding to this we define a

"

viscous resistance by "

R,=R-Rw

(2.27)

The definition of R, is sometimes simplified by discarding the last two integrals of (2.25), although it has not been really established that they are negligible. T h e definitions are precise. Whether they are useful depends to some extent upon how well they lend themselves to independent measurement. Measurement of R, will not be discussed, but various practical approximation procedures are discussed by Sharma (1964, 1965). Measurement of Rw will be discussed in the next section, but the underlying assumption

The Wave Resistance of Ships

109

of the measurement methods excludes the integral over the surface of discontinuity. An empirical proof of the usefulness of the definitions would consist in showing that the sum of the independently measured resistances adds up to the measured total resistance. We note that it is not true that, as defined above, Rw is a function of Fn alone and R , of Rn alone. Consequently we do not anticipate that measurement of these two components will necessarily be helpful in predicting full-scale resistance from model tests. We emphasize that Rw as defined here is not the same as the wave resistance that would be obtained if one were to find the velocity potential that satisfies rp, = Un, on S. As mentioned above, there presumably exists a surface S‘, containing S in its interior and coinciding with S at the bow, upon which rpPn = Un,. If S’ is not closed, but has a ‘ I wake ” trailing off to infinity, one must exclude from the integrals over C, that part occupied by the tail of the body. Th e surface S‘ is, however, irrelevant to the proposed definition of R , for S. On the other hand, it is useful to be aware of S’, for it suggests methods of making numerical experiments for estimating the effect of the wake upon wave resistance. 2. Wave-Breaking Resistance Recently Baba (196913) has pointed out that the rate of energy loss in breaking of ship-generated waves can be treated as a separate component of the total resistance. Th e occurrence of wave breaking is clearly a phenomenon in which gravity, i.e., the Froude number, plays the dominant role. On the other hand, energy lost in wave breaking will not be recorded in the “ wave-resistance ” measurements as defined above, but rather in the measurement of the “ viscous resistance.” This has been confirmed by Baba’s experiments. C. WAVE-PATTERN ANALYSIS Formulas like (2.17) and (2.19) can obviously serve as the basis for an experimental method of determining the resistance of a steadily moving body. However, the measurements would clearly be tedious, even after neglecting the terms in p, for measurements of H , u, v , w are necessary over the whole plane Z B . If one assumes irrotational flow, however, and hence (2.20), Eggers (1962, 1963) has shown that one may calculate R from surface-profile measurements alone. There are two versions of the method. I n one, transverse profiles are measured and R is derived from (2.20), in the other, longitudinal profiles are measured and R is calculated from (2.22). Both methods rely upon linearization of the boundary conditions at the free surface and hence upon measuring profiles far enough

John V. Wehausen

110

behind or to the side of the ship (or other moving body) that no serious error is introduced by this approximation. On the other hand, it puts no restriction upon the form of the ship or the cause of the disturbance, as will be done in the next section when wave resistance is calculated analytically. T he effect of the neglect of viscosity will be discussed later.

1. Transverse Profiles in a Canal We begin with the case of transverse profiles. Suppose, as we did in deriving (2.17),that the ship is moving steadily in a canal. Let its bottom be in the plane y =-h and its side walls in the planes z = &b. The undisturbed free surface is contained in y=O. As before, the coordinate system is moving with the velocity U of the ship. It will be convenient to suppose that the ( y , z)-plane intersects the ship amidships. Let x = x B be a plane sufficiently far behind the ship so that for x < x B it will be an acceptable approximation to replace the exact free-surface boundary conditions for irrotational flow, namely,

by the linearized conditions

+g Y(x,). =o, + UY,(x, z ) 0,

- Uyz(x, 0, 2)

yy(x, 0, ).

(2.29)

=

or, after eliminating Y , by yzz(x,0, z)

+ Kyy

= 0,

K

=g / U 2 .

(2.30)

In addition to Laplace’s equation, y must also satisfy the boundary conditions

(2.31) A complete set of solutions of Laplace’s equation that satisfy these three boundary conditions is given by the functions

I

cos k,x sin k,x cash pn(Y

nr + h)cos 2b ( z - b),

p,

> 0,

k n > 0,

(2.32)

111

The Wave Resistance of Ships where kn2= p n 2- (nT/2b)2, =pzp

(2.33)

+ (nrr/26)2.

Th e p,, must satisfy (2.34) and the p n pmust satisfy (2.35)

If n 2 1, there is a single solution for p, for each value of U2/gh. If n =0, there is one positive solution for po if U2/gh < 1, but none if U2/gh > 1. There are infinitely many solutions for p n p ,which we shall number with p = 0, 1, 2, . . . in order of increasing value. If n = 0, there is no solution pooif U2/gh < 1, but pooexists if U2/gh > 1. There is no steady-state solution if U2/gh = 1. The functions {cos(n~r/26)(z- 6)) form an orthogonal set of functions. However, the families (cosh p n ( y h), cos p n p ( y h), p = 0, 1, 2, . . .} do not, contrary to the behavior of such families in some similar situations. Since these form a complete set of functions for 'p in the region x 5 x B , we may express 'p as follows:

+

x (-a, cos k,x g

+& uk,,x

+

nrr + 6 , sin k,x)cosh p,(y + h)cos (z 2b

1 h

+

-

b)

nrr 2b

cnpexp k,, x cos p n p ( y h)cos - ( z - b).

(2.36) T he associated free surface is than given according to the linearized boundary condition by

Y(x,z ) = C (a, sin k , x n

+ b, cos k,x)cos nrr -( z - b ) 26

nnexp k,, x cos - ( z +b). (2.37) P 2b Th e first summation starts with n = 0 or n = 1 according as U2/gh < 1 or U2/gh> 1, respectively. I n the second summation the term corresponding to n = p = 0 is absent in the first case, but present in the second. f

1 c,, n.

112

John V . Wehausen

The coefficients a,, b, , cnPhave not as yet been determined, but they are uniquely defined by q in the region x 2 x B. Before substituting the expression for q into (2.20), the latter may be simplified a little by taking advantage of linearization in the double integral and integrating up to the surface y = 0 instead of y = Y(x, x). After a tedious but straightforward calculation one obtains the following expression for R. m 1 R = -pgb C e,-l(an2 2 n=Oorl

+ bn2)

where e0 = 4, E , = 1 for n 2 1. Note that the coefficients cnP do not enter into the determination of R, as indeed they should not inasmuch as (2.20) must be valid for any plane behind the ship and the contribution from the second summation in the expression for q must vanish as x B-+ -a.The first summation in q will be said to be the “ free-wave potential ” and the second to represent the “ local disturbance.” There now remains the problem of determining the coefficients a,,, b, needed in R. Multiply the expression for Y ( x , x ) by b-’ cos(mrr/2b)(x- b) and integrate from -b to b. This yields the equation

+ b, cos k,x + C cmPexp k,, x

a , sin k , x

P

=-Ib

mrr

Em

-b

Y(x,X)COS -(Z- b) d X 2b

= Y,(x).

(2.39)

Since x < 0, it is evident that the exponential terms will become negligible if I X I is large enough. Let us suppose that profiles have been measured at N locations x l , . . . , x N where this is the case. We then obtain N equations to determine the two unknowns a , , b, :

a , sin k m x i

+ b, cos k m x i= Y,(x,),

i = 1, . . . , N .

(2.40)

Any two of them will suffice provided sin k,(x, - x,) # 0. However, if N > 2, it is appropriate to use the method of least squares to obtain a more reliable estimate of a , and b, . T h e necessary formulas are as follows:

a, = - A - l

C Y,(x,)cos k,xi

sin k,(xi - xi),

i. j

b, = A - l

C Y,(x,)sin

k m x isin k,(xi - xi),

i. i

1 A = - C sin2 km(xi- x,). 2 i.j

(2.41)

The Wave Resistance of Ships

113

I n planning an experiment it is obviously advantageous to maximize A in order not to amplify measurement errors in Y ( x i , z). For N = 2, A = 4 sin2 k,(xl - x z ) is obviously a maximum for k , 1 x1 - x21 = 7r/2. However, since k , varies with m, an optimum spacing for one value of k , may be poor for another one. Since one usually plans to determine all the necessary values of a , and b, from one set of measurements, Y(xi,x), i= 1, . . . , N , this is a disadvantage of taking N = 2. This difficulty is ameliorated by taking a greater number of cuts and using least squares, for the function

A,(k,S)

1 N-1

1

C sin2 k,(xi - x j ) = C 2

=-

i,i

q=

1

( N - q)sin2 qk,6,

(2.42)

where 6 = xi + 1 - x i has an increasingly wider flat part about k,6 = 7712 as N increases. Figure 4 shows 2 W 2 A , ( k , 6 ) for N = 2, 4, 6 , 8, and 10.

FIG.4.

Graph of function showing effect of spacing and number of cuts upon A.

It is possible to avoid this difficulty with A if one can measure the slope Y , ( x i ,z) at the same time as Y ( x i ,z). T h e double measurement on a single profile then suffices in principle to determine the coefficients, for the equations become a, sin k,x

+ b, cos k,x = Y,(x),

a,,,k , cos k , x - b, k , sin k, x = Y,'(x),

(2.43)

114

John V . Wehausen

with solution

a,=-

1

[Y,(x)k, sin k,x

+ YA(x)cos k,x],

km

(2.44)

1

b,

=

knl

[Y,(x)k, cos k,x

-

YA(x)sin k,x].

If double measurements are made on N cuts x,, . . . , x N , a least-squares fit gives simply the average of the values obtained from each single profile. There remains the question of how far behind the ship the profiles must be measured before we can neglect the exponential terms. Since k,, < k,, and k,, < k, ,, it suffices to examine k,, , KO,, and k,, . It is not difficult to establish that

,-,

k,, > rr[@/h)2

+ (n/2b)2]1’z.

(2.45)

Hence k,, > r / h and k,, >.rr/2b. We recall that k,, does not exist if the speed is subcritical, U2/gh< 1. We assume this for the moment. If XI > # max(h, Zb), then both exp k,,x and exp k,,x are less than 0.009. This is an unsatisfactory result, for it does not take account of the size of the c,, and is overconservative for large h or b. Th e most satisfactory investigations have been numerical experiments by Kobus (1967) and Landweber and Tzou (1968), both carried out for channels of infinite depth but finite width. By introducing a mathematical disturbance approximating that caused by a ship, they are able to estimate the effect of the local disturbance upon Eggers’ method. Th e results of Landweber and Tzou are too elaborate to summarize briefly. However, the conclusion of both papers for shiplike forms is that profiles taken more than a ship length behind the ship will result in errors of less than 1 o/o for the range of practical Froude numbers. This problem has also been discussed in Eggers (1966, p. 667), where a computed curve displays for various distances behind the ship the ratio of the resistance computed from transverse-cut analysis neglecting the effect of the local disturbance to the true resistance. The Froude number was 0.36. Th e results conform to those of Landweber and Tzou. If U2/gh> 1, KO, appears. This may be very small if the flow is only slightly supercritical and is bounded above by .rr/2h. It seems likely that this exponential term may have to be taken into account if measurements are made at the usual distances behind a ship model.

I

2 . Transverse Profiles in Unbounded Fluid Several situations can be analyzed in a manner similar to that above: motion in an infinitely deep canal, motion in horizontally unbounded fluid of finite depth, and in horizontally unbounded fluid of infinite depth.

The Wave Resistance of Ships

115

We choose the last case, for it is most different from that above. However, we shall treat it in approximately the same way. Consider a region x < X, sufficiently far behind the ship that linearized free-surface boundary conditions can be used. Let a(k)= (KP)1/2,

P(k) = &[K+ ( K 2 + 4R2)112],

+ k2.

P2 = a'

(2.46)

One may show that the most general potential function satisfying the linearized free-surface condition and vanishing as y + - 03 is given by g w 1 p,(x, y , z ) = - dk -exp ~(k)y{[-F,(k)cos ax

U o

a(k)

+ [-F2(k)cos

ax

+ Gl(k)sin axlcos kz

+ Gz(k)sin axlsin kz}

+

x { [ K pcos py - ( k 2 p2)sin pyI x [F(k,p)cos kz G(R,p)sin kz]),

+

x < x, ,

(2.47)

where the F i , G i , F(R, p), G(R, p ) are general, but must satisfy certain conditions to ensure proper behavior of the integrals. T h e free surface is then given by ~ ( xz,) = Jo

m

~A{[F, sin ax + G, cos altos k z

+[F,sin ax + G, cos axlsin kz} + JmdpImdkexp(k2 +

p2)1/2x[~(~, p)cos

0

~z

+~

( kp)sin , kz].

0

(2.48) Formula (2.20) for the resistance takes the following form :

1 0 m R =p dyJ dz(-qx2 2 .--m -m

1

+ qy2+ y a 2 )+ 21 pg J m Y 2dz.

(2.49)

-m

Substitution of q into this expression for any fixed x 5 x B is simplified by the fact that the second double integral can be discarded and by use of the following Fourier-integral indentities. If

f n ( x )= Sm[/3,(k)cos kx 0

+ y,(k)sin kx] dk,

n = 1, 2,

(2.50)

John V . Wehausen

116 then

(2.51) m

Jm -m

n, m = l , 2.

fnfrndx=nJo (pnBrn+YnYrn)dk,

After simplification one eventually finds

(2.52) There now remains the problem of determining the Fi and G i , the analogs of the an and b, in the canal. We take Fourier transforms of Y(x,z):

1 *

57

j

Y(x,x)cos kz dz = Y,(x, k)

-m

+

1

7,

m

+ 1 d p ~ ( kp)exp(k2 , + W

= F,(k)sin a ( ~ ) x ~ , ( ~ ) c a(k)x os

p2)1/2x

0

(2.53)

I-,

Y(x,z)sin kz dz E Y2(x,k) = F,(k)sin

a(k)x

+ G, ( ~ ) c oa(k)x s +j

W

+G(K, p)exp(k2

+

p2)1/2x.

If profile measurements Y(xi, z) are made far enough aft that one can neglect the integrals in p, one obtains essentially the same formulas as were found for the a, and b,: N

F,(k) = - A - l

Y , ( x , , k)cos a(k)x, sin a(k)(xi- x i ) , i. i = 1 N

G,(k) = A - l

1 Y,(xj,k)sin a(k)x, sin a(k)(xi

- xj),

(2.54)

i.1=1

1

A = - 1 sin2 a(k)(xi- x j ) , 2 i. i

p = 1, 2.

The remarks about choosing 1 xi -xi 1 made for the canal carry over here without change. If one has simultaneously measured slope Yz(x,,z ) and amplitude Y ( x i ,z), then as before one may derive estimates of F , and G, for each x i and average for greater accuracy:

G,(k)= 2[ Y,(x,k)a(k)cos ax - YDz sin a x ] .

The Wave Resistance of Ships

117

In practice one will, of course, compute the F, G for a discrete set of k's sufficiently close to one another to allow calculation of R with the required accuracy. Under what conditions may the local disturbance be neglected? T h e only numerical experiments that are relevant are those of Kobus (1967) and Landweber and Tzou (1968) discussed earlier, and these were carried out for infinitely deep canals. It seems reasonable to suppose that the same results hold here. In fact, as pointed out by Sharma (1966), a numerical quadrature of (2.52) in which one takes Ak = r/2b and k, = nr/2b yields exactly (2.38) with appropriate correspondences between the a,, b, and the functions F, , G, , F,,G,. Consequently one may expect the two formulas to be valid under the same circumstances.

3. Longitudinal Profiles in Unbounded Fluid From the viewpoint of convenience the measurement of longitudinal profiles has important advantages over transverse profiles, the most important being that a wave gauge can be fixed in position while a ship or model passes by it and that measurements do not have to be made in the ship's own wake. We suppose that there exist values x, > 0 and z p < 0 such that when x 2 z , or z 5 z p it is allowable to use the linearized free-surface boundary condition. If zs' 2 z, and zp' 5 z p , then from (2.22)

-

R(S)+ R(P).

(2.56)

The conceivable contribution from a closing portion of C at, say, x = x B < 0 vanishes as x B-f - co in an unbounded fluid because the disturbance decreases as I x I -1/2, as we shall show later. We consider first only the part R"). In the region z 2 z,, y 5 0 any potential q P ) ( x , y , z) satisfying the linearized free-surface condition and vanishing as y-f - 00 can be represented as follows: (~(s)(x,y,

x) =

k2 f d k exp[--y(k)z]exp - y[-f(k)cos K

UO

kx +g(k)sin kx]

+ 5 Jwdkexp Kk2 - y{[-f,(k)cos kx +g,(k)sin kxlcos y(k)z + [-f,(K)cos kx +g,(k)sin kxlsin y(k)z} k2 +5 JwdpSadk exp[-(k2 +p2)1~2x][cospy + sin py] KP K

0

0

John V . Wehausen

118 where

k K

y(k) = - ( k 2 - KZ)l’Z,

k 2K ,

k y ( k ) = - ( K 2- kZ)l”,

O
K

< K.

The functions, although general, must, of course, give convergent integrals. We can, however, immediately put one restriction upon the functions because we know from the nature of the problem that waves must be outgoing from the ship. In particular, in the second integral only combinations like cos[kx

+ y(k)z],

sin[kx

+y(k)z]

are allowable for the starboard side. This requires that fz = -g1,

The free surface in the region z

YCs)(x,z ) =

gz =f1.

2 x, is then given by

dk 12 exp[-y(k)z][f(k)sin kx +g(k)cos kx] JoK

+ Smdkk([fl(k)sin kx +gl cos kxlcos y(k)z + [-gl sin kx + cos kxlsin y ( k ) z } K

fl

+ ~ m d p ~ m d k k e x p [ - ( k+pz)1’2x] 2 0

0

x u ( k , p)sin kx + g ( k , p)cos kx].

(2.58)

We now substitute the expression for p?(s) into the formula for W“. The calculation is again facilitated by using the theorems about Fourier integrals given in (2.51). After some simplification one obtains the following :

1 7ipgKJKmdk(k2- K 2 ) 1 ’ Z V 1 Z +gl”].

R‘S’ = -

An analogous analysis for z Combining them, we obtain

1

R = 7ipgK

< zp yields

a similar formula for

S, dk(k2- K 2 ) 1 ’ 2 [ f ~+giS)’ S)2 +

(2.59) R“).

m

fip)2

+g\p)z]. (2.60)

119

The Wave Resistance of Ships

If the ship is symmetric about its centerplane section, the usual case, of course, f i p )=f:"),g(:) =g$") and the formula collapses to

1 dk(k2 m

R = TpgK

- K2)1'2[f,2 +g12],

(2.61)

K

where.f,, gl may be from either side. Newman (1963) has given a rather elegant formula for R directly in terms of Y for a symmetric disturbance. Although his derivation seems to rely upon a special form of disturbance, the result does not. For fixed z he finds

where the kernal N ( x ) is given by ni2

N(x) =

sin28 sec 8 cos(x sec 8) d8

0

= - $(l

+x"Yo(x)

+ *77xY1(x)

+ t.rr2x2[YO(X)Hl(X)- Y1(x)ffo(x)l, where Yoand Yl are Bessel functions of the second kind and Ho and H I are Struve functions. A table of N(x) is provided. We must now determine the functions f(") and g(s'. As before, we take Fourier transforms of Y(x,x), but this time with respect to x instead of 2. This yields the following equations after neglecting the contribution from the double integral : 1

m

77

-m

-

Y(x, z)cos kx dx = Y,(k, z )

0
=i

+

>K , (2.63)

1 7-r -

m

J"

-a,

Y(x,z)sin Kx d x - Ys(k, z )

120

John V . Wehausen

Although we can determine the functions f ( k ) ,g(k) from profile measurements Y(x, xi)along a cut z = x i ,it is the functions fl, g,, that are of especial interest. The equations

1 fi(k)sin r(k)zi +gl(k)cos Y(~)z. - - Y,(k, x i ) ,

’- k

fr(k)co~r(k)zi-g,(k)sin

1

(2.64)

r(k)zi= k Ys(k,zi),

for i = 1, . . . , N can be used in the same way as the similar transversecut equations were used. However, in principle only one longitudinal cut is necessary, for with i fixed we find

1

fl(k) =

- 7; [Y,(R, zi)sin r(k)zi

+ Ys(k, xi)cos ~ ( k ) z i ] , (2.65)

1

gi(k) = k [ Yc(k zi)cos r(k)zi - Ys(k xi)sin r ( k ) ~ i ] , always, of course, for k > K. If multiple cuts have been measured, the appropriate use of the data is simply to average the results from each cut alone. We have not yet exhausted what is known about the behavior of Y(x,z), behavior that has important consequences for the behavior of Yc and Y , and hence for f,(k) and gl(k). If the disturbance causing the waves is properly restricted in extent, then far behind the disturbance, the free surface along any longitudinal cut will behave asymptotically like

Y(x, x) = [Asin Kx

+ B cos K X ] ( - - X ) - ~ / ~ .

This is shown in a special case in (3.35). As a consequence both Y, and Y , will behave like 1 K - k I for k near K. On the other hand, since in practice one deals always with only a finite length or record, this behavior will not show up in the Y , and Y , computed from the record. Two ways have been suggested for compensating for the finite length of record. In one, proposed by Newman (1963) and Sharma (1963, 1966), the tail of Y(x, z) is approximated by a function of the form

[Asin Kx

+ B cos Kx](C- K X ) - ~ ’ ~ ,

where the constants A, B, and C are determined by fitting this function to the trailing portion of Y(x, z). They give the necessary formulas for using this procedure. Th e other way of avoiding the difficulty with record length is to measure Yz(x,z ) instead of Y(x,z), for Y , decays very rapidly. Th e formulas corresponding to (2.65) are easily derived. There is some

121

The Wave Resistance of Ships

advantage in solving for y(k)f,(k) and yg,, for then the singularity at k = K is removed. For further discussion of the singularity and truncation formulas see Eggers, Sharma, and Ward (1967, pp. 123-124) and Eggers and Kajitani (1969). Since the regions x < xB and z > zs overlap in an octant of space, there must be some connection between the two forms for q~ used in the transverse and longitudinal wave profiles. This connection is provided by the integrals in the two expressions without exponential terms in either x or z. If in the first integral of (2.47) one makes a change of variable so that a becomes the independent variable, this integral becomes

5 lKWdxexp K y CC’

-

2a2-K2

{[-Fl(k)cos ax ka(az- K2)’”

+ [-F,(k)cos

ax

Comparing with the second integral of to be identical if one identifies

+ G,(k)sin ~ X]C Ok(a)z S

+ G,(k)sin axlsin k(cx)z}. P ) ( ~ ) , one

(2.66)

sees that they appear

This must be false, however, for we have seen that fis’ = -gl( s ) and gis)= fjs), whereas F,, G1, F, , G, are independent. What one must compare is 2a2- K 2 [-$(F, Ka(a2- KZ)l”

+ Gz)cos(ax +kz) + $(Gl - Fz)sin(ax + kz)] (2.68)

with -f:”)

cos(kx

+

’yz)

+g(s)sin(kx

+ yz).

Then one finds

where a

k(a) = - (az - K Z ) l J 2 . K Similarly one finds

(2.69)

122

John V . Wehausen

Hence either set of functions determines the other. If the wave field is symmetric about z = 0, then, as we have seen, f(") = f P ) and g(")=g") so that F z = Gz=O, as one can also deduce directly. In this case one longitudinal profile determines all functions. I n deriving the formulas for fi and gl we have assumed that z > zs was large enough so that the double integral could be neglected. Sharma (1966) has made a rather thorough study of this question as well as of the effect of truncation by means of numerical experiments [the results are also summarized in Eggers, Sharma, and Ward (1967)l. Briefly, he produces a disturbance by means of a known distribution of sources and sinks satisfying the linearized free-surface condition and computes the values of Y(x, x) and of F,(k),G,(K),and R. He then uses the computed values of Y in the various formulas above in order to test the effect of varying z , of using finite lengths of Y(x, z), and of applying the truncation correction mentioned above. I n the case considered the optimum value of z was about SUz/g. It may seem strange that there is an optimum value of 2. Suppose that 20,

I

I I

EXACT THEORETICAL VALUE

9

FROMY -CUT WITH

k

@ TRUNCATlON CORRECTION FROM Yz-CUT WITHOUT

v)

905 4

T

T 2

2

F

d

'

@ TRUNCATION CORRECTION FROM Y -CUT WITHOUT @TRUNCATION CORRECTION

The Wave Resistance of Ships

123

measurements terminate at x, and that L(z) is the useful length of profile at z. Then because of the nature of ship wave patterns, which will be discussed later,

xt?- L(z)N tan 70"32' =2.8. z

Hence increasing z shortens L(z), while decreasing x increases the unwanted effect of the local disturbance. Figure 5 from Sharma (1966, p. 772) shows the effect of truncation upon R , the importance of a truncation correction and the superiority of measurements of Y , instead of Y if no correction is made. Figure 6

THEORETICAL VALUE K2G ( k k O

-"tiL -20

I

I

1 3 TRANSVERSE WAVE NUMBER k / K +

1

I 4

FIG.6. Comparison of free-wave spectra derived directly from knowledge of disturbance and from theoretically computed Y. [From Eggers, Sharma, and Ward, 1967 (Fig. 6, p. 128), by permission of the Society of Naval Architects and Marine Engineers.]

John V . Wehausen

124

from Eggers, Sharma, and Ward (1967) shows the results of another numerical experiment where F(k) and G(k) are computed directly and also from the theoretical values of Y . This is a more discriminating test than a computation of R, and the agreement is impressive. For further discussion of such numerical experiments one should consult the cited papers.

4. Longitudinal Profiles in a Canal It is not possible in a canal to carry through a development analogous to that for an unbounded fluid. First of all, since all the wave energy is channeled down the canal, no pair of longitudinal cuts on each side of the ship can be long enough so that the contribution from a closing cut at x = x B < 0 can be neglected in computing R. Also, since all the energy passing through a longitudinal cut is (ideally) reflected from the walls, there is no net flux. However, as shown by Eggers (1962), it is in principle possible to evaluate the coefficients a, and b, from longitudinal profile data by means of the following formulas: mrr +2 sec (zs2b

a,

=

b,

= 2 sec - (z, - b) lim

mrr

2b

T-m

Y(x,y)sin k,x dx, 1

-

(2.72) 50

Y(x,z)cos k, x dx,

TJx0-T

where z = z, is the plane in which the profile is measured. In practice this procedure has not been successful, partly because tank walls are not perfectly reflecting, partly because it is not possible to get a long enough profile to approximate the necessary limits. Recently, however, Moran and Landweber (1971, 1972), using a modified procedure, have made both numerical and physical experiments with such profiles. T h e former show the theoretical feasibility of the procedure. Results of the latter are consistent with data obtained independently by other means. Perfect reflection is assumed. There are other ways of making use of longitudinal-profile measurements in a towing tank. In most tanks model dimensions are usually limited to a size that avoids the necessity of making wall corrections. Consequently, if a measured longitudinal profile is truncated before the first reflected waves from the tank walls affect it, that section of the profile should approximate well the profile that would be measured in an unbounded fluid. Th e truncated tail must, however, be supplied by a theoretical extrapolation, as proposed by Newman (1963) and Sharma (1963) and described in the last section. One should note that the difficulty of making a reliable

The Wave Resistance of Ships

125

extrapolation increases as the Froude number, and hence the wavelength ( s 2 r U 2 / g )increases, for one has fewer and fewer wavelengths upon which to base an extrapolation. Figure 7 shows two typical records up to the first reflection. However, the procedure seems to work satisfactorily in practice, as will be seen when we turn to experimental results.

FIG.7. Two typical records of longitudinal cuts, including the first reflected wave from the side wall.

5. Other Methods of Direct Measurement When the transverse-cut method was used to determine a,,, and b, for the waves in a canal, this led to a separate calculation for each value of m, preceded by taking a Fourier transform. These calculations required a knowledge of Y ( x i ,zj)= Y i jfor i = 1, . . . , P ; j = 1, . . ., Q. I t seems reasonable to ask why one should not bypass the Fourier transform and determine all the a,, b,, n = 0, . . . , N , in one calculation. If we may neglect the local disturbance, each measured value Y i , gives rise to an equation satisfied by the a, and b, : nr

C (a, sin k n x i + b, cos k,xi)cos (z,2b N

h=O

+

b) = Yij,

i = 1, . . ., P,

j = 1, ..., Q. (2.73)

If PQ 2 2 N 2, there are in principle enough measurements to determine the desired a , , b,. If PQ > 2 N 2, one may exploit the method of least squares to find a “best” solution. This procedure was first suggested by Hogben (1964) and further developed in Gadd and Hogben (1965) and Hogben (1970). Obviously there must be imposed further requirements on the grid of measurement points for the method to be applicable (for example, P > 2 N + 2 , Q = 1 will not work). These problems are discussed in the referenced reports. Another ingenious method due to Ward (1963, 1964) called by him the “XY-method,” is based upon measurement of the force acting on

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John V. Wehausen

a long vertical cylinder situated in a position that would also be suitable for a wave probe measuring longitudinal wave profiles. The fundamental assumption underlying the method is that the force exerted on the cylinder by a plane oncoming wave is proportional to the wave amplitude, a prediction of the linearized theory of water-wave diffraction in an inviscid fluid. Ward’s experimental studies of this assumption are reported in Ward and Snyder (1968). It appears to be adequately substantiated for the intended use. The method relies upon combination of the formulas (2.20) and (2.22), the force components on the cylinder providing the data for the integral along the longitudinal cut and a separate wave gauge providing the data for an approximation to the closing transverse cut. For details one should consult the cited references. A recent modification of Ward’s idea has been given by Roy and Millard (1971).

6 . Eflect of the Wake The underlying theories for the determination of wave resistance from profile measurements all require irrotational flow. When the methods are applied practically, what is measured? If there is indeed a boundary-layerplus-wake region BLW surrounded by a region in which the flow differs only insignificantly from an irrotational one, then a longitudinal profile taken far enough to the side so that the wake region is not intersected should determine the velocity potential y p used in (2.26) to define R , in a viscous fluid. On the other hand, a determination of y p from a set of transverse profiles that intersect the wake, which may not be avoidable, may be expected to show an effect of the fact that part of the data used to determine y p is being taken in a region where the flow is not irrotational. This may be an explanation of some evident discrepancies between quantities deduced from the two sorts of measurements. The most important attempt to study the effect of the wake by means of numerical experiments is in a paper by Tatinclaux (1970). He takes as body an infinitely long vertical strut of ogival section in unbounded inviscid fluid. A wake is artificially produced by giving the form of the vorticity o in the fluid behind the midsection of the ogive. This is equivalent to assigning H,-H inasmuch as the Navier-Stokes equations for steady motion in the coordinate system Oxyx can be written in the form

pgV(Ho - H ) = W x

(V-

V) - ~ A fpv’ v

VV’

(2.74)

Let L be the length of the ogive and 2bo its beam. Then Tatinclaux makes the following special choice for o :

U

(2.75)

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for x aft of the ogive. For the region between the midsection and the after end x/L is deleted. For the computations h = 0.1, c = 0.25. A solution of the equations of motion for an inviscid fluid with linearized free-surface conditions is then constructed. The velocity is the gradient of a harmonic function outside the wake to which must be added a term V x A inside the wake. (Outside the wake V x A contributes VpR to the velocity.) T h e form of the surface Y(x,z) can then be calculated. For Fn = 0.25 and for sections X B at one, two, and four model lengths behind the model, Tatinclaux calculates the exact wave profile, an I ‘ asymptotic ” profile that neglects the local effect of thevorticity, and the profile that would exist if there were no wake at all. Differences are substantial even after four model lengths. For the resistance it makes no difference whether the exact or the
7. Experimental Observations T h e results of experiment that will be given here are mostly by way of illustration. In the next section, where it will be of interest to compare analytical predictions of wave-making resistance with measurement, further examples will be given. Figure 8 from Eggers, Sharma, and Ward (1967) shows C , = R,/+pU2S for a mathematically defined ship form, symmetric fore and aft, whose

John V . Wehausen

128

0.10

015

QZO

025

030

FADUDE NUMBER F, = (v/@)

035

ado

---c

FIG.8. Wave resistance for a mathematical ship form determined experimentally according to three methods. The theoretical curve is the Michell resistance (see below) with special corrections. [From Eggers, Sharma, and Ward, 1967 (Fig. 12, p. 133), by permission of the Society of Naval Architects and Marine Engineers.]

precise form need not concern us here. Results are shown for three of the methods discussed above : transverse profiles, longitudinal profiles, and Ward’s XY-method. The results are not inconsistent with one another except in the neighborhood of the Froude number 0.35 where Ward’s method and the transverse-cut method deviate substantially. I t is conjectured by Eggers, Sharma, and Ward that this is a result of separated flow at the stern and an associated more dominant wake. Figure 9 from CaliSal (1972) shows a compilation of results for a standard merchant-ship design, “ Series 60, block 60.” Some of the values were obtained by Ward using either his X Y-method or lateral-slope measurements of longitudinal cuts. Others were obtained by CaliSal from transverse cuts and from amplitudes or lateral slopes of longitudinal cuts. The residuary resistance is also included as well as a theoretically computed curve that will be discussed in the next section. Results from two transverse

129

The Wave Resistance of Ships

2

2o

c

0

A

+ A

x

FOR SERIES 6 0 . C g * 6 0

WARD’S ( 1 9 6 8 ) SLOPE METHOD WARD’S POINTS 11965)

2 STATIONS TRANSVERSE CUT L . S M TRANSVERSE CUT LATERAL SLOPE CUT LATERAL WAVE HEIGHT CUT

FIG.9. Wave resistance for a Series 60, C , = 60 hull determined from various sorts of profile measurements and from residuary resistance. [From CaliSal, 1972 (Fig. lo), by permission of the Society of Naval Architects and Marine Engineers.]

cuts are included with the least-squares value in order to show the spread of values obtainable. Once again the several methods do not seem to be inconsistent, even though at large Froude numbers the spread in values is large. As a test of consistency the wave resistance is not as sensitive as a direct comparison of the functions Fl(K), Gl(K),F,(K), G,(k) in (2.48). These functions, or some modification of them, form what is often called the “free-wave spectrum” of the motion. Figure 10 from Eggers, Sharma, K 2 ( F 2 , G12)1’2, and K2(F12 and Ward (1967) shows K 2 ( F I 2 GZ2)lI2, GZ2 FZ2 G12)’/’ plotted against k/K, as determined by both longitudinal and transverse cuts. Although the agreement is generally good, there is evidently a shifting of phases as k/K increases. Although the determination of wave resistance and of the free-wave spectrum from profile measurements is still undergoing development and refinement, there seems to be little doubt that the method is successful. I n the next section comparisons of computed wave resistance with measured wave resistance from profile measurements will be considered more valid than comparisons with residuary resistance.

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John V . Wehausen

FROM LONGITUDINAL Y -CUT -

TRANSVERSE WAVE NUMBER k / K -D FIG.10. Components of free-wave spectrum determined by transverse and by longitudinal cuts. [From Eggers, Sharma, and Ward, 1967 (Fig. 13, p. 133), by permission of the Society of Naval Architects and Marine Engineers.]

Finally it should be emphasized that these presumably more accurate methods of determining wave resistance or of separating in a fairly precise and rational way viscous from wave resistance do not really help much in the problem of extrapolating model test data to full-scale behavior. On the other hand, the greater precision in allocation of causes, i.e., the effects of viscosity as against the effects of wave making, for the resistance of a particular ship hull will almost certainly lead eventually to more efficient hull designs.

8. Some Further References on Wave-Pattern Analysis Since Eggers, Sharma, and Ward (1967) have given a rather complete bibliography of the subject up to 1967, it has not seemed necessary to duplicate it. We list here only a few papers that have appeared since then

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but are not mentioned in the foregoing text: Hogben (1971, 1972), Ikehata (1969), Ikehata and Nozawa (1968), Maruo and Ikehata (1969), Tanaka, Adachi, and Omata (1970), Tanaka, Yamazaki, Ienaga, Adachi, Ogura, and Omata (1969).

111. The Analytical Theory of Wave Resistance A. INTRODUCTION In the preceding section we were especially concerned with methods of measuring wave resistance without being at all concerned about the device that made the waves. It could be a displacement ship, hovercraft, hydrofoil boat, or what not. T h e goal was to find the rate of energy loss in the waves. T h e goal in this section is different. We now wish to start with the geometry of the ship, or the pressure distribution under the hovercraft, and whatever other dynamical specifications are necessary, and predict the wave resistance of the ship. We shall begin by formulating several problems ‘ I exactly,” or almost so, in order that it will become clear what sacrifices have been made when we begin to approximate later on. T h e formulation will be only “almost exact” because we shall neglect viscosity from the start. It would not be difficult to formulate the more exact problem with a viscous fluid, but since we would not really make use of it, little would be gained by doing so. T h e exact problem will turn out to be much too difficult to solve in closed form and methods of approximation must be sought. The methods that will be described here are systematic schemes that in principle allow improvement. There are, of course, valuable single-shot approximation methods, but ones that may be systematically improved would seem to be preferable. On the other hand, one must be aware that such schemes may also be illusory in some respects, as has been pointed out by Lighthill (1951) and others. T h e first such approximation method that was tied to the geometry of the ship was given by J. H. Michell (1898). T h e approximation will be discussed more fully later on, but essentially it consisted in replacing the exact free-surface boundary condition by the usual linearized one and the boundary condition on the ship’s surface by one on the centerplane section analogous to the ‘(thin-wing ” boundary condition. Although the approximation has since then been imbedded in a systematic perturbation scheme

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by Peters and Stoker (1957), it has not really been surpassed. It has been extended to water of finite depth and to canals by Sretenskii (1936), but it is clear from Michell’s paper that he knew how to do this. Extensions to second order in the perturbation scheme have been proposed by several persons and will be discussed later. However, numerical computations are difficult and most of them incomplete in that the second-order computation is applied to only one of the linearized boundary conditions, the one on the body. This is sometimes defended as being “ physically realistic.” In fact, what is meant is that this “ improvement ” is within relatively easy computational reach whereas a more exact treatment of the freesurface condition is not. Since there is no theorem stating that such approximations lie between the Michell value and the true one, they must be considered as numerical experiments, but valuable as such. Another approximation scheme, a “ slender-body ” approximation, would seem intuitively to be better suited to ship forms than the thin-ship approximation. It was introduced more or less simultaneously by Maruo (1962), Tuck (1963a), and Vossers (1962a,b), Tuck’s being the more systematic approach to the method. Unfortunately, the first-order approximation turns out to be deducible from the Michell approximation and the second-order one never to have been brought to the point of computation (see Newman, 1970, p. 80), although one might anticipate that it would be simpler than the second-order thin-ship approximation. In recompense, the slender-body approximation has turned out to be very successful in the theory of ship motions. I n Michell’s original paper only one numerical calculation was given for a simple mathematical hull form and he expressed the hope that others would compare the result with experimental values. In an obituary notice written by his brother, A. G. M. Michell, it is stated that Michell’s paper “ was the subject of an enormous amount of work, expended not only in checking and rechecking, the theory and symbolic calculations, but also in making detailed arithmetical comparisons of the results of the theory with those of recorded tests of full-sized ships and models. Only a small fraction of this work was published, and the fact that the paper appeared without detailed arithmetical or graphical illustrations of the results may, in part, account for its having apparently remained unnoticed by those who undertake the designing of ships until about thirty years later when W. C. S. Wigley in England, and others in Germany, began to recognize its fundamental importance.” T he neglect of Michell’s paper for so long is curious. Havelock’s (1877-1968) first paper on water waves was published in 1908 and on ship wave resistance in 1909. There followed a series of papers on ship waves and wave resistance with the ship being replaced by a moving pressure

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distribution. These papers were of great interest in themselves, but it is not easy to associate a pressure distribution with the ship’s geometry. Only in 1923 did Havelock finally begin to exploit Michell’s results. I n 1925 (a) Havelock gave an alternative derivation based upon Green functions instead of the modified Fourier-integral theorem developed and used by Michell. Thereafter Michell plays practically no explicit role in Havelock‘s work, although Havelock proceeded to exploit Michell’s result in a series of outstanding papers on the effect of various modifications of ship lines upon the wave resistance. [It was perhaps Michell’s influence that later led Havelock (1929b) to devise a Fourier-integral theorem similar to Michell’s in both spirit and application.] As mentioned above in the quotation from A. G. M. Michell, the systematic exploitation and testing of Michell’s resistance formula started with W. C. S. Wigley (1890-1970), his first paper on this subject being published in 1926 and his next-to-last (with J. K. Lunde) in 1948. A paper in 1949 is a summing u p of his own and other’s work plus a few new results. (I regard his subsequent papers and reports as having a different aim.) In this work Wigley was for the most part engaged in making experiments and associated calculations to test the validity of Michell’s formula for hulls of more or less shiplike shape and to explore its limitations and applications. He was joined in this work in 1930 by G. Weinblum, although the latter was engaged more in exploitation than in testing of Michell’s work. These investigations of Wigley and Weinblum and many of those of Havelock required evaluating Michell’s formula, a quintuple integral, for hull shapes with some resemblance to ships. Although the hull forms were usually kept fairly simple, one must keep in mind in assessing the magnitude of their achievement that a point on a resistance curve whose computation might take 2-3 seconds, or less, for a modern computer, required then 2-3 days’ work with a desk computer. Some of their conclusions, as well as those of others overlooked in this brief history, will be mentioned later. In fact, few conclusions have been changed by high-speed computers. Havelock continued developing his Green-function approach in a series of papers culminating (but not ending) with a paper (1932~)in which he gave resistance formulas (based upon linearized theory) for arbitrary distributions of sources and sinks or of normal dipoles over a submerged surface as well as for arbitrary pressure distributions on the surface (not a new result, but a new derivation). Havelock continued to use this approach to investigate various aspects of wave resistance. T h e subsequent history of the subject will appear as individual results are mentioned in the following sections. Of particular importance is the paper of Peters and Stoker (1957) which clarified the nature of the approximations used in waveresistance theory. In addition, two important conferences on the subject

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John V . Wehausen

should not go unnoted: One in Moscow in 1936, whose proceedings contain important papers by Keldysh, Kochin, Lavrent’ev, Sedov, and others, and a more recent one in Ann Arbor in 1963 whose proceedings in three volumes contain many important papers.

B. THEEXACT PROBLEM IN AN INVISCID FLUID There are several “exact ” problems that could be formulated, but the one that seems most interesting is that of a surface ship free to trim and squat. It is important to include this possibility, for it has sometimes been mistakenly assumed that the thin-ship approximation requires that the ship be fixed in position. I n fact, as first pointed out by Peters and Stoker (1957), calculation of trim and sinkage is as much a part of the first-order theory as is the resistance. I n addition to the coordinate system Oxyx introduced in (2.15), moving with steady velocity U with the ship, we shall use another coordinate system O’x’y’z’ fixed in the ship. This coordinate system is to coincide with Oxyz when the ship is in its equilibrium position. T h e plane O‘y‘z‘ contains the midship section, and the plane O’x‘y‘ the centerplane section [a slight restriction from (2.15)]. Let the angle from Ox to O‘x’ be CY and let 0‘ have coordinates (0, e, 0) in Oxyz. Let (xG’,yG‘,0) be the location of the center of mass of the ship, and let m be its mass and V its displacement at rest, so that m = pV. Figure 11 shows these quantities schematically. The coordinate systems are related by x’ = x cos cc

+ (y - e)sin

y’ = -x sin CY

CL,

+ (y - e)cos a.

FIG.11. Sketch showing relation between coordinate system fixed in the ship and moving coordinate system Oxyz.

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135

Here a and e are unknowns to be determined along with the resistance of the ship. We shall suppose that the ship is propelled by an external thrust T = -R acting along a line parallel to O‘x’ and intersecting O’y’z’ at (0, d‘, 0). For a propeller-driven ship d’ < 0. (For a sailing vessel the formulation would have to be changed.) The hull itself will be described by =

*f(. y’). ’,

(34

We suppose it to be piecewise smooth. The part of the centerplane section below O’x’z’ will be denoted by So, the wetted surface by S and the projection of S on the centerplane section by S , . We shall suppose the motion to be irrotational, so that the free-surface boundary conditions are those given in (2.28). Corresponding to these, there will be kinematic and dynamic boundary conditions on the ship. The kinematic condition on the ship’s wetted surface may be written in the form

where

a

- f(x’, y’) =fi(x’, y’)cos a -fi(x’, y’)sin a,

ax

a

- f ( d ,y’) =fl(x’, y’)sin a

(3.4)

+

fi(x’, y‘)cos a, aY with the notation fi = aflax‘, fz = aflay’. In order to write out the dynamic conditions, we must first compute the force and moment acting on the ship. T h e force components resulting from the action of the water upon the ship and resolved along the axes of Oxyz will be denoted by ( F , , F y , 0 ) and along those of O‘x‘y‘z‘ by (F,’, Fg‘,0). The moment about the center of gravity is denoted by (0, 0, M G ) . If p(x, y , z ) is the pressure in the fluid.

F,’

=

y, zb,’ d S = 2

J p[., Y , f (x’,Y”fi SP

dx‘ dy’,

John V . Wehausen

136

T h e dynamic boundary conditions then consist of the equations for equilibrium of forces :

F, + T cos CL = 0, Fy f Tsin a- mg= 0, MG

(3.7)

+ T(y&- d’) = 0.

A kinematic condition must also be satisfied on the ocean bottom, taken to be a horizontal plane at y = --h:

vY(x,-h,

X)

= 0.

(3.8)

If the depth is taken infinite, we may replace this by lim y,, = 0.

(3.9)

y- --m

Finally we need a condition insuring that waves will only follow the ship : 94x3 y , z ) =

0([x2+z2]-1’2) o(1)

as x2+z2+co as x2 z2+ co

+

for x>O, for x (0.

(3.10)

Various modifications of this problem are possible, depending upon the physical situation. I n particular, in a towing tank a model can be restrained from trimming and squatting so that u = e = 0. Such modifications are not difficult to make, and this one actually simplifies the problems. Modification of the boundary conditions in order to accommodate a submerged vessel is also not difficult. Since U has been fixed in the problem formulated above, the unknown quantities are y , Y , T , a, and e . It is obvious from the boundary conditions that they are hopelessly intertwined with one another and that some approximations will be necessary. Before proceeding to the discussion of approximation, we shall formulate one more “ exact ” problem, that of a moving pressure distribution. Here the pressure above the water surface is given, but the surface profile itself is unknown. T h e kinematic condition (2.2813) remains the same, but the dynamic condition (2.28a) must be replaced by

where P(x, X) is a given function, which may be chosen to vanish outside a certain region in order to model a particular physical situation like a moving hovercraft. Although much of the complexity of the ship problem has disappeared, this is still a nonlinear problem. As in (3.5), one can com-

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137

pute the x-component of the force exerted by the pressure distribution upon the water:

s

F, = /P(x, z)n, ds = P(x, z) Y,(x, x ) dx dz.

(3.12)

C. PERTURBATION EXPANSIONS In order to obtain any solutions of the exact problems formulated above, or of others that one may formulate, some method of approximation is necessary. T he standard procedure has been by way of either regular or singular perturbation expansions. In either case one introduces a dimensionless parameter E > 0 connected with the problem in such a way that as E -+ 0 the disturbance near the free surface becomes smaller and smaller, except possibly at certain isolated points or lines. Each value of E labels one of a family of flows. Many perturbation approximations involving moving bodies have a different aim from this one. For example, in thin-wing theory in an unbounded fluid one wishes primarily to avoid the tedious computations associated with conformal mapping of the given wing section onto a circle. However, it still seems fair to say that in problems involving motion near a free surface the nonlinear boundary conditions at the free surface seem to present the greatest obstacle. Hence one chooses a perturbation parameter associated with vanishing of the surface disturbance. It will still be true, however, that some of the classical perturbation approximations also fulfill this requirement. There seem to be two main ideas behind free-surface perturbation schemes for moving bodies. One is to have the body sufficiently deeply submerged that the free-surface disturbance resulting from its motion is not very great, and, of course, must become smaller the deeper the submergence. I n the other, one introduces a family of bodies whose members can be made to approximate more and more closely to a body whose motion will not disturb the fluid. Th e most familiar example of the latter in fluid mechanics is thin-wing theory. For a moving pressure distribution one simply assumes that the imposed pressure P is " small." For moving submerged bodies the two approaches can be schematized as follows. Let L be the length in the direction of motion, d a typical length perpendicular to this direction (say a