Lecture Notes in Physics New Series m: Monographs Editorial Board H. Araki Research Institute for Mathematical Sciences Kyoto University, Kitashirakawa Sakyo-ku, Kyoto 606, Japan J. Ehlers Max-Planck-Institut fiir Physik und Astrophysik, Institut ftir Astrophysik Karl-Schwarzschild-Strasse 1, Wo8046 Garching, FRG K. Hepp Institut ftir Theoretische Physik, ETH H6nggerberg, CH-8093 ZUrich, Switzerland R. L. Jaffe Massachusetts Institute of Technology, Department of Physics Center for Theoretical Physics Cambridge, MA 02139, USA R. Kippenhahn Rautenbreite 2, W-3400 G6ttingen, FRG D. Ruelle Institut des Etudes Scientifiques 35, Route de Chartres, F-91440 Bures-sur-Yvette, France .
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R. K. Zeytounian
..Fluid Dyna_____mmics Asymptotic Modelling, Stability and Chaotic Atmospheric Motion
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Author
Radyadour K. Zeytounian Universit6 de Lille I, Laboratoire de M6canique de Lille F-59655 ViUeneuve d'Ascq Cedex, France
ISBN 3-540-54446-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54446-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr. Bookbinding: J. Sch~ffer GmbH & Co. KG., Grfinstadt 2153/3140-543210 - Printed on acid-free paper
PREFACE
This short c o u r s e on M e t e o r o l o g i c a l Fluid D y n a m i c s (MFD) is strongly influenced by the author's own conception of meteorology as a fluid mechanics discipline, which is a p r i v i l e g e d area for applied mathematics techniques. One of the key features of MFD is the need to c o m b i n e m o d e l equations of the basic "exact" Navier-Stokes (N-S) equations for a t m o s p h e r i c m o t i o n s with a careful and rational fluid dynamics analysis. T h e r e f o r e , m u c h of the d i s c u s s i o n of this c o u r s e is d i r e c t e d t o w a r d s t h e s u b s e q u e n t d e r i v a t i o n a n d a n a l y s i s of systematic approximations and consistent model equations to the exact N-S equations for atmospheric phenomena. Obviously,
the
process
of
research
a p p r o p r i a t e fluid dynamics models,
towards
developing
rooted in a rational use of
modeling, is very important for a basic approach to the difficult problem of inserting atmospheric flows in a meteorological context. A complete consistent rational modeling of atmospheric phenomena is a long way in the future, but fairly sophisticated fluid dynamics m o d e l s of v a r i o u s a s p e c t s of t h e i n d i v i d u a l m o t i o n s of the atmosphere are available today. Unfortunately, at the present time a considerable gap still exists between fluid dynamics modeling of various atmospheric motions and the application to the problem of numerical weather prediction. N a t u r a l l y , the d e v e l o p m e n t of a t m o s p h e r i c a l - m e t e o r o l o g i c a l m o d e l s f r o m the p o i n t of v i e w of f l u i d d y n a m i c s p r o c e e d s by c o n s i d e r i n g m o d e l s of submotions, which, when they prove to be successful, can be linked together. It may well be that in the next ten years it will be this aspect of MFD which makes the greatest advances. I feel that, parallel to a "practical" meteorology, whose goal is mainly to
(numerically)
predict the weather,
we should
develop a fluid dynamics meteorology, which would be considered one of the branches of theoretical fluid dynamics. In my opinion, this
VI return of meteorology to the family of fluid mechanics will be of value to both meteorologists and fluid mechanics specialists. It is important to understand that, in the majority of cases, the establishment of models is an intuitive, heuristic matter and so it is not clear how to insert the model under consideration into a hierarchy of rational approximations which in turn result from the general equations chosen at the beginning (either the N-S or Euler equations). It seems obvious that an improvement in weather f o r e c a s t i n g depends largely on the o b t a i n i n g of more e f f i c i e n t models and not only on the development of numerical techniques of analysis and calculation as is thought by certain specialists in the field of numerical weather forecasting. The science of m e t e o r o l o g y and, more particularly, n u m e r i c a l weather prediction is seen to be suffering today from an excess of "experimentation".
Thus the r e a l i s t i c m o d e l i n g of a t m o s p h e r i c
phenomena is lagging behind. I am of the opinion, however, that only conceptually coherent theoretical modeling can bring to light the time problems to be solved in order to achieve a significant improvement in the reliability of predictions. Of course, it must not be
forgotten
that
such modeling
must
be a m a t h e m a t i c a l
e x p r e s s i o n of real a t m o s p h e r i c p h e n o m e n a that p e r m i t s t h e i r i n t e r p r e t a t i o n . Thus it is n e c e s s a r y f r o m the start to c h o o s e sufficiently realistic equations and conditions which reflect the essential characteristics of atmospheric phenomena such as gravity, compressibility, stratification, viscosity, rotation and b a r o c l i n i t y . The f l u i d m e c h a n i c s t h e o r i s t n o w has a v a i b l e conceptual tools which permit the modeling of atmospheric phenomena above all. I naturally have in mind the a s y m p t o t i c techniques which have proven so decisive in fluid mechanics. I believe that -
these asymptotic techniques should find new applications in the special field of meteorology - a meaningful illustration of this t e n d e n c y can be found in my recent book Asymptotic Modeling of
Atmospheric Flows (Springer-Verlag, Heidelberg 1990). The present "short course" is a good preparation for the reading of this latter book,
which
presents
various
rational
asymptotic
a p p l i c a t i o n in m e t e o r o l o g y and, especially, local weather predictions.
models
for
for s h o r t - t e r m and
VII Meteorological
fluid dynamics
I hope that the p r e s e n t dynamical
studies
selective
in my choice
is a relatively young
course w i l l
in meteorology. of topics
science and
aid the d e v e l o p m e n t
of fluid
In this course I have been highly and in many cases
the choice
of
topics for analysis is based on my own interest and judgement. In fact, the p u r p o s e of this short c o u r s e is only to g i v e a fluid mechanics description of a certain class of atmospheric phenomena. To that extent the text is a personal expression of my view of the subject and is constituted Note notions
that of
this
course
fluid
by ten chapters presupposes
dynamics;
and two appendices.
familiarity
nevertheless,
with
they
the basic
are
briefly
su/-~arized, primarily to introduce suitable notation. I am m o s t this book.
grateful
to S p r i n g e r - V e r l a g
I ask for the i n d u l g e n c e
for the p u b l i c a t i o n
of E n g l i s h - s p e a k i n g
of
readers,
thinking that they might prefer a text in not quite perfect English rather
than
in
Prof.Dr.W.Beiglb6ck
"perfect"
February
1991
Finally
f o r offering me the possibility
these ideas on meteorological
Villeneuve d'Ascq
French.
I
thank
of presenting
fluid dynamics.
Radyadour Kh. ZEYTOUNIAN
CONTENTS
CHAPTER
I. T H E R O T A T I N G
EARTH AND
i. The g r a v i t a t i o n a l 2. The C o r i o l i s
ITS ATMOSPHERE ................
1
acceleration .......................... 1
acceleration ............................... 3
3. The a t m o s p h e r e as a c o n t i n u u m . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 BACKGROUND CHAPTER
R E A D I N G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
II. D Y N A M I C A L FOR
AND
THERMODYNAMICAL
ATMOSPHERIC
MOTIONS
EQUATIONS
............................
12
4. The b a s i c e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. The f o - p l a n e and ~ - p l a n e a p p r o x i m a t i o n s . . . . . . . . . . . . . . . . 20 6. The e q u a t i o n s
for large s y n o p t i c
scale
a t m o s p h e r i c p r o c e s s e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7. The c l a s s i c a l p r i m i t i v e e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . 25 8. The B o u s s i n e s q m o d e l e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . 28 9. The q u a s i - g e o s t r o p h i c m o d e l e q u a t i o n . . . . . . . . . . . . . . . . . . . 30 BACKGROUND CHAPTER
III.
R E A D I N G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
WAVE PHENOMENA
IN THE
ATMOSPHERE
..................
36
10. The w a v e e q u a t i o n for i n t e r n a l w a v e s . . . . . . . . . . . . . . . . . . 36 Ii. The w i n d d i v e r g e n c e e q u a t i o n
for t w o - d i m e n s i o n a l
i n t e r n a l w a v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 12. B o u s s i n e s q g r a v i t y w a v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 13. R o s s b y w a v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 53 14. The i s o c h o r i c n o n l i n e a r w a v e e q u a t i o n (Long's e q u a t i o n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 15. B o u s s i n e s q ' s t h r e e - d i m e n s i o n a l
linearized wave
e q u a t i o n and r e s u l t s of t h e c a l c u l a t i o n s . . . . . . . . . . . . . . 66 BACKGROUND
R E A D I N G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
REFERENCES
TO WORKS
CHAPTER
IV.
FILTERING
16. H y d r o s t a t i c
CITED
IN THE T E X T . . . . . . . . . . . . . . . . . . . . . 83
OF INTERNAL
WAVES ........................
85
f i l t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
X 17. B o u s s i n e s q f i l t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 18. G e o s t r o p h i c
f i l t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
BACKGROUND
R E A D I N G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
REFERENCES
TO W O R K S
CHAPTER
V.
UNSTEADY
CITED
ADJUSTMENT
IN THE T E X T . . . . . . . . . . . . . . . . . . . . . 89 PROBLEMS
........................
90
19. A d j u s t m e n t to h y d r o s t a t i c b a l a n c e . . . . . . . . . . . . . . . . . . . . . 91 20. A d j u s t m e n t to a B o u s s i n e s q s t a t e . . . . . . . . . . . . . . . . . . . . . i01 21. A d j u s t m e n t to g e o s t r o p h y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 BACKGROUND REFERENCE CHAPTER
R E A D I N G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 TO W O R K S
VI. L E E W A V E
CITED
LOCAL
IN THE T E X T . . . . . . . . . . . . . . . . . . . . . 112
DYNAMIC
PROBLEMS
...................
114
22. E u l e r ' s local d y n a m i c m o d e l e q u a t i o n s . . . . . . . . . . . . . . . . 114 23. M o d e l e q u a t i o n s
for the t w o - d i m e n s i o n a l
s t e a d y Lee w a v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 24. B o u s s i n e s q ' s 25. Outer,
i n n e r s o l u t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Guiraud's
and Zeytounian's
s o l u t i o n ........... 129
26. L o n g ' s c l a s s i c a l p r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 27. M o d e l s REFERENCE CHAPTER
VII.
for L e e w a v e s t h r o u g h o u t t h e t r o p o s p h e r e ...... 149 TO W O R K S
BOUNDARY
CITED
LAYER
IN THE T E X T . . . . . . . . . . . . . . . . . . . . . 153
PROBLEMS
..........................
155
28. The E k m a n l a y e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 29. M o d e l e q u a t i o n s
for b r e e z e s . . . . . . . . . . . . . . . . . . . . . . . . . . 161
30. M o d e l e q u a t i o n s of the slope w i n d . . . . . . . . . . . . . . . . . . . . 170 31. M o d e l p r o b l e m for the local t h e r m a l p r e d i c t i o n (the t r i p l e d e c k v i e w p o i n t ) . . . . . . . . . . . . . . . . . . . . . . . . . . 176 REFERENCE CHAPTER
VIII.
TO W O R K S
CITED
METEODYNAMIC
32. W h a t is s t a b i l i t y
IN THE T E X T . . . . . . . . . . . . . . . . . . . . . 186
STABILITY
..........................
187
? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
33. The c l a s s i c a l E a d y p r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 1 34. The E a d y p r o b l e m for a s l i g h t l y v i s c o u s a t m o s p h e r e . . . 1 9 8 35. M o r e on b a r o c l i n i c 36. B a r o t r o p i c
i n s t a b i l i t y . . . . . . . . . . . . . . . . . . . . . . . 200
i n s t a b i l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
×I 37.
The T a y l o r - G o l d s t e i n of s t r a t i f i e d
38.
shear
The c o n v e c t i v e
equation isochoric
instability
and stability f l o w . . . . . . . . . . . . . . . . . . . 205
p r o b l e m . . . . . . . . . . . . . . . . . . . 212
BACKGROUND
READING .......................................
REFERENCES
TO WORKS
CHAPTER
IX.
CITED
DETERMINISTIC
CHAOTIC
OF ATMOSPHERIC
39. A t m o s p h e r i c dynamical
IN THE
BEHAVIOUR
MOTIONS ............................
equations
232
T E X T . . . . . . . . . . . . . . . . . . . . 232
234
as a f i n i t e - d i m e n s i o n a l
system .....................................
234
40.
Scenarios ............................................
241
41.
The B ~ n a r d
problem
42.
The L o r e n z
dynamical
s y s t e m . . . . . . . . . . . . . . . . . . . . . . . . . . 265
43.
The L o r e n z
(strange)
a t t r a c t o r . . . . . . . . . . . . . . . . . . . . . . . 271
for i n t e r n a l
free c o n v e c t i o n ...... 257
BACKGROUND
READING .......................................
REFERENCES
TO WORKS
CHAPTER
X.
MISCELLANEA
CITED
IN THE
........................................
44.
Internal
The d e e p c o n v e c t i o n
e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . 292
The m o d e l
for l o w M a c h n u m b e r
equations
atmospheric 47.
Fractals
REFERENCES APPENDIX
REFERENCES APPENDIX
2.
in a t m o s p h e r i c
TO WORKS
TO WORKS
BIBLIOGRAPHY.... AUTHOR SUBJECT
CITED
LAYER
SINGULAR
TWO-VARIABLE
REFERENCES
in an i s o c h o r i c
f l o w ......... 280
flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TO WORKS
1. B O U N D A R Y OF
waves
280
45. 46.
solitary
278
T E X T . . . . . . . . . . . . . . . . . . . . 278
t u r b u l e n c e . . . . . . . . . . . . . . . . . . . 307
IN THE
TECHNIQUES
PERTURBATION
CITED
T E X T . . . . . . . . . . . . . . . . . . . . 314 FOR
THE
STUDY
PROBLEMS .................
IN THE
EXPANSIONS
CITED
299
T E X T . . . . . . . . . . . . . . . . . . . . 326
...........................
IN THE
315
327
T E X T . . . . . . . . . . . . . . . . . . . . 336
.........................................
337
INDEX .............................................
341
INDEX ............................................
344
CHAPTER I THE ROTATING EARTH AND ITS ATMOSPHERE
I THE GRAVITATIONAL ACCELERATION
The earth revolves about its axis once in every 23 h 56 min and 4 s or a total of 86164s. The frequency of rotation or the angular velocity of the earth is:
2 ~ - 7.292 flo - 8616-----4
(1,1) The radius
of the earth
at
and the
b~ae g r a v i t a t i o n a l
surface
and at
and
I~l = therefore,
the geographic acceleration
a geographic
(1,2) it
is
be
the
o f ~ = 45 ° i s
of ~
45",
:
is
a = 6370.1 0
km
pull
of the
earth,
on the
body
force,
p~
in
:
m/s 2,
assumed
true
latitude
owing to the
latitude
9.82357
will
momentum e q u a t i o n ,
x 10 - 5 t a d s
here
that
gravitational
the force,
where p is
and
rotating
the
,
the
atmosphere
density. To
distinguish
subscript
"a"
reference, angular Let
~,
the
experiences
denote
quantities
and a subscript velocity
(1,3) The absolute
"r",
respectively
then
fixed to
quantities
0
upward,
a
referred
~ = ~ ~ with respect
~ and ~ denote
vertically
of
unit
:
~=~sine *3cos~. velocity
is
at
at
absolute,
referred
to the the
an
a
observer, inertial
to a frame
absolute
frame.
Vectors
pointing
let frame
rotating east,
north
a of
whith and
a
and since the rotating observer sees only the change the
~
(~, t)
position vector ~ of a point moving with the atmosphere,
]
m ~r in
the respective
velocities for the two observers are related by t :
cl,4)
?=~+~A~Cg, a
t) .
r
Thus we obtain for the absolute acceleration the following relation:
~=~ a r* andsince
~^~
-
2~A~. r
~ ^c~A~)
~^~±,
where subscript
equatorial component, ~ ^ C~ ^ ~) = - ~ 1 Cl,~)
~a= ~r+2~
, ±denotes
the
and
~r- a ~ .
A
Then to the gravitational pull
we should add vectorially the centrifugical
force per unit mass and obtain a modified gravitational acceleration g, such that :
Cl,6~
~: ~
~L,
÷
and resultant vector is slightly inclined away from the radius of the earth because the order of magnitude of the centrifugical acceleration is smaller; thus
E~I o g = l ~ i -
.~i~ll
m
= 9.823~7 - -
o~aocos2~,
S or
(1,7)
g = 9.8066
The ~ n a a d e ~ acceleration
(the
cI, 8)
m / s 2, s i n c e
,Fr,
is
force
of gravity).
a measure
Fr = % / / g L
flo2aoCOS2~O = 0 . 0 1 6 9
of
the
It
is defined
significance
m/s 2 at of
the
$=45 °.
gravitational
by
o"
where I01 = %,and 0 is a ch~acteristlc velocity whereas Lois a ch~acteric I engt h. t
We m a y two
point called
forgo
£rames
the
a denote t=O.
subscipt
coincide the
at
on ~,
since
position
vector
we a r e
at
liberty
time t under
the particular at
to
assume
consideration.
some c h o s e n t i m e ,
that
the
At t h i s
w h i c h may t h e n
be
2 THE CORIOLIS A C C E L E R A T I O N
In the equation (1,5) the terme 2~ A ~ , where ~ denotes the relative velocity , is an apparent acceleration known as the ~anlo2/4 ~ r exist only if there is motion with reference to a movin E frame earth.
which such as
the
For the Coriolis acceleration we have :
(2,1)
2~ ^ ~
=
20 s i n ~ o
(~ ^ ~) u + 2~ cos ~ o
( ] ^ u~l
If u, v and w are the components of the relative velocity u : ~ = u ~ + v ~ + w ~ , then (2,1) becomes u
2~ ^ ~ u =
(2,21
(2~] w c o s
:
~ -
o
20 v sin o
~)
+ 20 u sin ~ j o - 20 u cos ~ ~ . o
If we let the symbol
(2,3)
be
f = 2 0 sin o
called
the
b~ca~ ~aa/m2/~ paname/e~,
the
a c c e l e r a t i o n in terms of its components
(2,41
^
=
fcu
-
+
final
expression
for
the
Coriolis
is
df
(wt-
u~)
The importance of the Coriolls a c c e l e r a t i o n in relation to the inertial forces is given by the R ~
(2,5)
P~ -
aumI~_a, Ro, which is defined as U /L oo f
,
o
where fo m 2OoSin ~o' with ~o = Constant When
~
>>
It
modification
of
Coriolis
force
, the are
in-between situation
t tt ttt
That That That
is is is
£or £or for
Coriolis flow likely
forces
pattern, to
be
. are
but
likely when
dominant
. F o r the a g n ~ t c
Ro
to <<
When
cause Itt Ro
only
the ~
slight
effects
I tit
we
- ~ce~e atmospheric motions,
the ~ Rossby numbers. the ~nl~l~ Rossby numbers. RO = 0 ( 1 1 .
a
of
the
have
the
we have :
L s
lOem and
o
motions,
Ro ~
we have
atmospheric
10 -I , but
the m~aa
(or n 2 4 / x ~ )
: L s 10Sm and Ro s I. Finally,
for
o
process,
Here we p r e f e r
for
we have rather
to use,
instead
- a~x~e atmospheric
the case of b~ca£-a~x~e
: Lo~ 104m and Ro ~ 10
of the Rossby number I~,
the parameter
1/f o t - Ki , o
~'~,o, '" which
is called
the X/~e~ ~ .
and if t o is the advective
BJERKNES"
In (2,8),
is a characteristic o Lo/U o, then : Ki m Ro .
time scale,
t
time scale
THEOREM
It is natural
for the rotating observer to define the circulation of a circuit
as ~ u~.d~ = r (~) r
and the absolute circulation
:
is then
.jj
-,-
^
But (~ A ~).d~ = (~ A ~±).d~± , and therefore
(z,8) where
r.- r~= ~.~(~±Ad~±) = 2~oZ(~±) , ~(~±)
equatorial
is the area plane,
enclosed
by the normal
~± of ~ onto
the
provided that the orientation of ~ ± correspondin E to that of
is related to ~ by the right-hand This is the part of Bjerknes' illustration,
consider
enclosed
Ao,
area
projection
screw rule.
theorem concernin E the effect of rotation.
a circuit
~
o
and the greatest
which
is horizontal
diameter
of which
at
latitude
is small
As an
~>0,
compared
has with
the earth's radius a . o Then ~(~±] = Aosin approximately, of
the
and hence,
atmosphere
circulation
F . P
mass
a meridional containinE
~
translation, o
is
without
sufficient
to
any deformation, create
relative
3
THE ATMOSPHERE AS A CONTINUUM
The Eeneral laws of mechanics and thermodynamics establish the basic working scientific principles of the atmosphere.
The meaninE that properties have in
this atmosphere depends on an understandinEs of a continuum. For instance the meaning of pressure in the atmosphere is often defined as the total force per unlt area
imposed on each of the unlt
areas of the atmosphere
and can be
thouEht of as the force per unit area on a solid unit surface immersed at any point In the atmosphere owing to the of molecules at the surface.
continuous impinEement and bouncing off
A Eiven mass of atmospheric air
in a constant
volume and at a constant temperature is always undeF the same pressure.
This
is
this
true
for
statement
the
may
be
thermodynamically verified
from
pure
Boyle's
substances. and
In
Charles's
particular
laws
peFtaining
to
perfect gases. Boyle's law states that during an isothermal process the ratio of
the pFessure
to
the density
isobaric
(constant
absolute
temperature
continuum
let
us
pressure) is
is constant.
process
constant.
postulate
that
the
From the
Charles's
pFoduct a
of
law states the
mathematical
properties
at
density
point
any
one
that and
an the
of
vlew,
a
point
can
be
expressed in terms of the pFoperties at a neiEhboring point. This is because the
property
and
its
de~-ivatives
are
continuous
in their
variations
wlth
space. From the mechanics point of view,
the atmospheFe is a thin layer of gaseous
mixture surroundlng the surface of the earth which remains attached to the earth by the pull of the gravity. The
atmosphere
is made
up of a numbeF
of
layeFs,
each characterized by a
distlncly diffeFent temperature distribution. The layer nearest the surface of the earth, characterlzed by a linear decrease of
temperature with altitude,
is called
the
~
or
the
~
layer
wheFe the average rate of temperature decrease with altitude is approximately 6.5°K/km . We shall see that the temperature gradlent the troposphere varies a gFeat deal,
in the lowest part of
whereas in the upper
layers It remains
essentially unchanEed. The troposphere contains about 80 percent of the total atmospheric mass.
It
therefore
most
it
is
is the
layeF In contact
influenced
by
energy
wlth the earth's transfer
throuEh
surface and radlation,
evaporation, condensation and convection. Thls layer is approximately IS km in thickness and represents the limit within which conventional air flights take place.
The
troposphere
is also the
layer
In which man-made
pollution fFom
industrial wastes in principally confined, and where most cloud formations are found.
Dynamically
speaking,
layer nearest The
atmosphere
continua
the
troposphere
is stable,
constitute,
called
fluids-
from
the
a
mechanical
atmosphere
we assume that the atmosphere
law of perfect
gases with constant
(3,1)
p = R p T
R = Cv(~
-
I)
internal
and
, ~ = Cp/ C v,
density,
those
portions
e = C T
of
the
p
view,
a
p
and C
v
category
Concerning
of the
of da@ oL~ fulled by the heats,
namely
:
,
v
where
of
Nea~x~W~z~ ~ / ~ .
consists
speciflc C
T is the absolute
point
is a
thermodynamics
atmosphere
but
the surface of the earth are often unstable.
is the atmosphere temperature
pressure
, p
of the atmosphere
is the
and e its
energy per unit mass.
For dry air
: C = I.OIS J°K/g p
and g = 9.8066
m/s 2,
then the adiabatic
lapse
rate
(3,2)
g__ - ~ - 1 C = 7
g _ 9.66OK/k R
P
If
the
relative
disturbed
from
velocities the
value
defined by the relations
are it
m
small
would
the
have
pressure
in
the
the
and
) , pm(z
"at~u~Lan~" altitude
is
principles
assumed requires
known,
p (z)
) and T ( z ) ,
z , as defining
basic state upon which fluctuations state
only
slightly
motion,
p (z),
p(z)
dzm + g p~ = 0
and we can think of pm(z namely
be of
:
dpm (3,3)
will
absence
= RT ( z )
functions
'
of but one variable,
a standard
atmosphere,
i.e
,a
due to the motion occur. The basic standard
although
in
the consideration
fact
its
of mechanism
determination
from
such a radiative
first
transfer
in the atmosphere. In simple temperature
case
we
T (z),
have
from
dT~ (3,4)
the
first
law thermodynamics,
for
the following equation
k(Tm)
A + Rm(T m) = 0
dz
t Note that for d r y air ~ = 1 . 4
and
dR(T) m
with - dz
p Q (T),_
the gas constant
R=287 J'K/kg.
the
standard
A
where k(Tm)
is the coefficient
of thermal conductivity and Q ( T )
of heat supply per unit mass by radiative this Course,
it is sufficient
heat transfer.
is the rate
For our purpose,
in
to assume that Qm is a known f u n c t i o n of T ( z )
and we suppose also that the influence of the rate of heat by radiative heat transfer
on
atmosphere. variation transfer quantities
the
atmospheric
motions
is
essential
even
with
the
standard
Doing this we consider only a mean standard heat source and ignore therefrom.
will
be
For
our
purpose
sufficient
are p (0),
and
as
this a
p (0) and T (0)
modelling
result (e.g.
the
of
the
radiative
thermodynamics
the values
heat
reference
of standard
state
at
the ground level). In
this
case
the
hydrostatic
equation
(3,3)
dimensional ratio of reference quantities
(3,5)
gHo p(O)
Bo =
involve
the
following
non
:
Ho R T(0) g
and
this
ratio
characteristic
Bo
is
called
the
Z
o
~
length scale for the vertical
n~T~e~.
In
(3,5),
motion and it follows
H°
is
a
from (3,5)
that
(3,6)
H
m R T (0) / g
is a characteristic
length scale for the standard altitude z . It follows also
f r o m (1,8) and (3,5) that
M
(3,7)
2 0
Fr z = ~ - -
e
0
BO
where
(3,8)
H° = u ° / 4~ R T®(0)"
is called the characteristic ~ a ~ A ~
(3,9) is called The N ( z )
e = H 0
the ~
0
~
d e f i n e d by
/ L .
0
and
Nf(zm)
(3,10)
=
is called the Z a ~ t ~ - ~
g
[W-
~
1 g +
dT
m ]
or the natural frequency of oscillations
of a vertical column of "standard" atmospheric mass given a small displacement from
its
a~e~N
equilibrium
position,
ffAe a~ndaad
~
~
~
~aaeal
The existence
of characteristic
scales
is exploited
by the
introduction of
nondimensional quantities denoted by primes, i.e, z - H z'
,
z
= H
z'
,
T
= T (0) T'
and then ecO N~2Cz~) -= BO
(3,11} where
o
is a dimensionless
dT' }
{ ~
-
i
+
ntexzaaae a~ #Ae atxutdand ~
and
N~Cz~)
is
the dimensionless V~is~l~ internal frequency. Finally, we obtain the following relation : (3,12)
z'
= Bo z '
,
between the dimensionless z' and z' (local altitude). MODELLING OF THE TURBULENCE
If F(~)
is the frictional
force
in the atmosphere,
then for Newtonian fluid
like dry air
c3,13)
(~.~),
=.
where ~ is the gradient operator,
~ is the coefficient of viscosity,
but the
representation (3,13) based on laminar friction is valid only if ~ is constant (~ m ~o ) and if the so-called Stokes hypothesis (which amounts the neglecting the bulk viscosity)
is adopted.
These approximations will be adequate for our
purpose. According
to our view,
in this Course,
the problem of modelling
turbulence
should not be considered at the same level as the one of building theoretical models for atmospheric motions.
The coefficient value is u Strictly
u° - ~ o / P ( O )
is the kinematic
viscosity and for dry air the
= 5 m2/s.
o speaking
is the kinematic eddy (turbulent) coefficient of o viscosity and it is only from this eddy coefficient of viscosity that we can take
into account
u
the effect
the earth's surface.
of "turbulence"
Of course,
in the boundary
such a procedure
layer close
is much over-simplified
to but
it will be sufficient for our purpose. In the lowest few kilometres of the atmosphere,
the eddy viscosity depends on
both the topography and the background winds. The simplified form (3,13) for the frictional force (viscous force density) a
bastardized
representation,
valid
only
in
Cartesian
coordinates,
is
invariant
under
is
which
represents the three-component relations. A
form
of
the
viscous
force
dencity
that
a
change
of
coordinate system is:
(3,14)
=.
But if u
is constant, as it is in the lowest 40 km of the atmosphere, then it o is not possible to use (3,14), because if u is constant in the troposphere, 0 then ~ varies, so that a more complicated form must be used for the viscous
force density
(3,1s)
=
1 ~(~. u~) + { ( ~ A
+
~) A u ~}
The c o m p l e t e h y d r o d y n a m i c e q u a t i o n s f o r a v i s c o u s a t m o s p h e r e
i n v o l v e not only
the addition
To be c o m p l e t e l y
accurate,
of a term like
viscosity
must
(3,13)
be
t o t h e momentum e q u a t i o n .
included
in
the
energy
equation
as
well.
That
A
involve the addition it
is
the
of a term ~(~),
production
of
thermal
which is a q u a d r a t i c energy
by
the
mechanical energy,
in the first law of thermodynamics.
The 9 R e ~ r u ~ n A e ~
is
(3, 16)
functional
viscous
o f ~ and
dissipation
of
Re = U L /p o o o
and this show relative importance of the inertia to the viscosity. The
measure
of the ratio
between
the frictional
and Coriolis
forces
ls
the
10
(3, 17) An important both
the
earth's
Ro
Ek -
Re
l)
o
-
f L2
o o
feature of large synoptic scale motions
Kibel
and
troposphere
in the immediate
Ekman
numbers
vicinity of the equator, (for
In
situation,
o
realistic
meteorological
synoptic
frequently,
for
scales,
may assume
one
small.
the
A
typical
value
the
synoptic
in
situation
prediction that
we have Ki<
motions
the
of
atmospheric (but
c
o
to
~ 1).
to
•
c
o phenomena
Ek
in
the
Except
0
if the characteristic t
o
atmosphere, Mo<< I
corresponds
1~<<1
we h a v e
of
is that
u = 5 m2/s.
is I0 -3, using value of eddy viscosity
time scale t > > 1 0 4 s any
are
in the atmosphere
~ 10 s s )
and
10 -2 a n d at
.
usually Bo ~
regional
1.
and
a
But, local
11
BACKGROUND READING For
further
details
concerning
the
physical
nature
of
the
atmosphere
reader is referred to : ESKINASI, S. (1975)
_ ~b~b/~ec.Aan/2~ and U
~
Academic Press Inc., New York.
HOUGHTON, J.T.
(1977)
_ ~/~e ~
o~ ~
,
Cambridge University Press.
SCORER, R.S.
(1978) Ellis Horwood limited, Publisher, Chichester (England).
a~ aa~ ~
,
the
CHAPTER II DYNANICAL AND THERMODYNANICALEQUATIONS FOR ATMOSPHERIC MOTIONS 4
. THE
BASIC
EQUATIONS
In a coordinate frame rotating with the earth the momentum equations is
(4,1)
P DED~= _ ~ p + p g _ O (2 ~ A u->] + ~o ~
where,
u~ i s
the
velocity
D
(4,2)
Dt
vector
observed
in
the
earth
frame
~3 (~.u->)
and
+ ~.i~ ,
8 8t
-
as
Do
~ + _ ,_
is the material (or convective) derivative . The conservation law of mass is expressed by the equation of continuity : D
(4,3)
~(Log
and the f i r s t
law o f thermodynamics by the energy e q u a t i o n :
p ~
(4,4)
p) + ~ . u~ = 0
1/p) = ko~2
+ p p ~DC
A
T + zCu ~) + p= Q=(T W)
A
where k is the constant coefficient of thermal eddy conductivity and X(u) is o expressible as : A xc
A )
=
,
where t
The t r a n s p o s e tensor I=(~lj), The
gradient
|S 8 v e c t o r .
"= CaL)
o£ a t e n s o r
where ~ijls oF t h e v e l o c i t y :
the ~
LCalj)
Kronecker
i s d e £ i n e d by a ~ i = a i j .
delta,
is a tensor,
is the unit
but ~ . ~
I.....
The
tensor. l a r and ~ / ~
13
-
(4.5)
[ ~u~
.--,
]_2~ , ( ~ . u~)
+ (~u~)"
is called the viscous stress tensor. Thus,
as I/ -- ~*o ,
zCu)
2 (~.u~)2
~o
(4,6) = ~o
- S
+ ~
(def u ~) . (def u ~)
,
where def ~ - ~ ~ + C~ ~)" is the velocity deformation symmetric For the acceleration
D~ / Dt
a small calculation
tensor
.
shows that
D~ _ a~ + c~.~) Dt
8t
2
(4,7) at
where
q =
]~[
and
~ = ~(~ A ~),
and
from
this
form,
D~ / Dt are easily found for any system of curvilinear
the
components
orthogonal
of
coordinates.
With the equation of state
(4,8)
p = R p T
and the thermodynamic
(4,9)
relation
e = CW T ,
the equations
(4,1)
, (4,3) and (4,4) provide a closed system for u, p and T .
SPHERICAL COORDINATES
It is helpful denote azimuth
the (A),
to employ
corresponding latitude
now spherical velocity
(4,10)
components
(~) and radius
in terms of these coordinates
coordinates
: A, ~,
in the
(r) respectively
the llne elements
r and
direction .
is given by :
dcr2 = r 2 cosZ~ dA 2 + r 2 d~ 2 + dr z ;
of
let u, v, w increasing
14
hence
the metric
coefficient
h
are
:
h
= r cos
!
2
=
r
h
and
3
= I
.
Thus
(4,11)
~ _
18
i 8 r cos 9 8 A
3+8
t+F~
~F
and
(4,12)
~ . u~ _
r
_~_@
Finally,
~2 _
8
v8
operator
1
in unit v e c t o r s
(4,1sa)
at_
o
(4,1Sb)
at ar
8r
(4,1Sc)
BA
From
(4,13)
and
-
aj:
au
-
+
+
(4,14),
with
:
0
~+
(4,1S),
r
2
tan ~
_ 3
are
8
E~
+
28
---A
F~F
:
;
;
a~ _ -t sin ~
= I cos
that
u au v au au uw uv + r cos ~ ak + r ~ + w ~-{ + - - - r - - t a n r
av ~ +
aw
1
a2 ar 2
the d i f f e r e n t i a t i o n
_~
it f o l l o w s
Dt
and from
we have
-k c o s ~ + j s i n ~ ,
(4,16)
2w
8
r 2 a~ 2
during
a~ - a~ arm
(4,15)
Du ~
8w
(32 + __I __a2 +
r2cos2~ ak 2
The c h a n g e s
8
+ r cos ~ 8 ~ (cos~ v) + ~-~+ - 6 '
r cos ~ 8-X + F a - 6 + w 8 - 6 "
f o r the L a p l a c i a n
(4,14}
1
8u ~p 8 A
u
u.v=
(4,13)
1
cos
~ ~ ~ 1
u av v av av vw u2 ] + + w + + tan ~ ~ r cos ~ ak F S-~ ~ --r r-J
u
aw
r cos ~ aX
vaw
+ r a-~ + w
the Laplacian
operator
a.
u2 + v
ar
r
of ~ is :
=] ~
'
15
~u~=
Au r
{
+
(4,17)
2
Av +
u
2
r
COS
r
In the equation of energy
z
aw 8A
~o
V
+ 2 tan ~
2
P
cos
r cos
2 8~ + 2
r
the expression
2
~0
av } @A
OU } e 8A. J
~
r cos ~
r
tan ~ 2-~--
r" c o s
0v
z
(4,4),
2
2
~O
aW 2 8~
,..
t
+
2
P
2
COS
aA j
fop the dissipation
"
function
becomes :
A~
~o{
X(U) -
-6
(4,18)
+.old2
where d A
(d11 - d22 )2
12
+
(d22
-
d33)
.+
(d33
- d11)
.}
+d2 +d 2 ] 23 31 '
( i, J = 1,2 and 3 ) are the components of the rate-of-strain tensor =
u) , and for the compenents of the viscous stress tensor
x we have the expression :
Tij = //0 { dlj - 2 (~.u~) ~lj } •
The expression
(4, 18) for X(u) in terms of d lj is valid in general oPthogonal coordinates and in spherical coordinates,
dll
=2
[
d22 = 2 d33
1 au r cos ~ @A F ~
+
+
w
-- r
tan~v
] ;
r
r
2 aw ar '
(4,19) d12
_ cos ~ a (co__~) + I av ~ a~ r cos ~ aA'
a d23 : r ~ dsz
_
rv_ I 8w . ( ) + g a--~ ' 1
aw
a
u
r cos ~0 a~. + r 5-r ( ) "
16 NON DIMENSIONAL FORM
We i n t r o d u c e
the f o l l o w i n g
(4,20)
x = aoCOS~o A
where ~o is a r e f e r e n c e
origin
of
earth surface k=O We
latitude.
this
right
:
y = ao(~ - ~o )
It f o l l o w s '
handed
,
z = r - a° ,
immediately
a a a-~ = ao a-y
curvlllnear
that
a a a r - eqz "
'
coordinates
(for a flat g r o u n d we have r = a o) at
system
latitude
lles
on
the
~o and l o n g i t u d e
. suppose
region,
therefore
distant
therefore, are
,
a a 8A - aoC°S~o a-x
(4,21)
The
transformation
from
equations
introduced
the
atmospheric
equator,
new longitude may
be
motion
around
occurs
some
~
between
o
= L
o
/ a
o
so-called
E-plane approximation
At first
it
put the b a s i c e q u a t i o n s
to
t' =
(see,
x and y
and
are
the
obviously
length
of a t m o s p h e r i c
the
nondimenslonal
x'=x/L,
o
v' = v / U ° ,
p' = p / pro(O)
,
are
y'=
quantities
and
i n t r o d u c e d as f o l l o w s
Y / L° '
w' = w / W o ,
(4,23)
Q=(1),
of the
to
form .
(marked by primes)
T'
coordinates
motion
the s e c t i o n 5).
introduce
u' = u / U 0 ,
/
~o
Although
they
they will be the C a r t e s i a n
t / to ,
Q~ = Q=
latitude
in terms of w h i c h
approximation,
horizontal
in n o n d i m e n s i o n a l
variables
mld-latitude
,
(ao),
is c o n v e n i e n t
unity.
a
that for small
the c h a r a c t e r i s t i c
(L o) a n d the e a r t h radius
The n o n d i m e n s i o n a l
without
In
central
a n d latitude c o o r d i n a t e s
rewritten
in the e x p e c t a t i o n
(4,22)
the r a t i o
the
s i n ~o ' cos ~0° a n d t a n 10° are all of o r d e r
in p r i n c i p l e
basic
that
= T / TIn(O) ,
:
z' = z / Ho '
p' = P / p=(O) ,
17 where
Wo
is
a
characteristic
characteristic
value
[}sine (4,23),
(4,20)
some r e a r r a n g e m e n t
vertical
o f mean r a d i a t i v e and
of
(4,21)
the
in
terms,
velocity
transfer
(4,12) the
scale
A
and
Q(ml)
is
a
.
and
(4,13)
~
we r e a d i l y
e q ~
in
obtain,
the
after
nondimensional
form : S DD~_~p'+ p, { Wo aw ' w' e~o ~-~'+ 2 Wo~ ° I + ~ o
z'
1 / cos ~ a (cos ~ v') + 1 + e ~ z' By' o o + (cos ~o)/(cos ~) u^~="au'} = 0
(4,24)
#
I +~z' 0 0 cos ~o°
where
D' 0 cos ~ ,a I ,0 ~o S S ,v ~--,+ -oy eo ~,-~-[,+ I + c ~ z' u ~-~,+ 1 + e o~ oz o o
(4,25)
L no=- W ° / U °
and S is the ~ b u ~ /
~
,
S - U
o
o
t
o
w '@
O--z' '
'
(SRo m Ki).
Then, a convenient scale for the vertical velocity is:
(4,26) Using (4,11),
~o / eo this
choice
(4,16),
of
m
1
scale
(4,17),
~
W
o
m
together,
the ~
e U
o o
with
the
cam4~
relations
o~ t2~e m
(4,20)-(4,23)
~
e q ~
and (4,1)
now becomes :
D' u' eo o u' w' tar* ~0 v' S D-~ + I +" e'o~° z' - ~ o 1 Y e ~ z''u ' o o
p'
e ] cos ~o/COS ~ 1 sin ~ v' + o cos ~ w' + Ro sin ~o Ro sin ~o ~ 1 + e ~ z' o o
(4,27)
1 { A'u' + ! cos ~o/COS ~ L Re
+ 2 ~o e
3
.i + eo~oZ'
(~'.~') _ ~2
ax'
cos ~o/COS ~ aw' )2 a--x' - 2 ~otS-n °(l+e~z' o o
1
ap' ~Mo2 ax'
t/cos2~
o (1 + eo~oZ')2
co__ o COS
a±, I
(I + e ~ z')2 ax' J ' o o
u'
18
where 2 2 cos ~ o / C O S
]%, =
82
1
(I + c 8 z')2 8x ' 2 oo (4,28)
1
82
(1 + e 8 z , ) 2 oo
tan
E: aZ' 0
8y, 2
c3 2 8o 8 + ay' 1 + eo6oZ'e ° 8 z '
°(1 + e 8 z')2 o o
2
82
+
'
and 8W' W' ~-~, + 2 Co8 ° 1 + e a z ' o o cos ~olCOS au' + 8x' 1 + ~ : 8 Z' 0 0
~ ' . u~ ' (4,29)
Analogous
m
~
e q ~
@
1{
(4, 3 0 )
can be written down ~
~8 D'v' o o - v'w' S D--~- + 1 + e 8 z ' o o
p'
Re
p'
1 sin Ro sin A ~v ~
u'
~Po
Se
I + J
l
+
°°(1+
v ' a n d w':
tan ~ u'2 + 8o I ~ c ' ~ z ' o o I 1 ap' 1 + eO($oZ' ~.M02 Oy'
z
3
+2ae
llcos ~ 8 (cos ~ v') + I + ~ 8 z' By' o o
~
1 + c 8 z' o o
(~'.u ~')
-
I/c°s2~
82
02 D'w' u'2+ v'2 ~ + eo~o I + eoSoZ'
~o Ro
cos W u' c o s ~o
I
+ (4,31)
~-,
+Bo
~.Mo2 ( Op' e A'w'
-
- 2 8 c 0 °(1
p,
)
2 82e 2 o o(1 + e 8 z')~ o o
-
1 + c 8 z')2 o o
The no nd im en si on a l
(4,32)
2 8oC ° ( I
+ c 8 z')2 o o
form of the e q u a t i o n
p'
= p'
T'.
w' I
o_~v' + 282e tan --~ ay' o o
c o s ~o/COS ~0 8 u ' -
v'
o ,¢1 + c ~ z , ~,2 o o cos ~o/eOs ~ Ou' } a~w' _ 2 8 tan ~ . . . . . , Oy' o (I + eoSoZ') 20x'
eo~oZ, )
By'
a--~'
v'
(1 + c S z ' ) o o
} "
of state
(4,8)
is
2
19
Finally,
using
equation
of energy
(4,20)-(4,23) (4,4)
and
(4,9),
(4,18),
in the nondimensional
(4,19)
we r e a d i l y
~ - 1 S D - -' p ' - 1 I A ' T ' + - ~-1 ~ Dt' P r Re 2
D'T'
p'S Dt'
obtain
the
form :
c
(4,33)
o
M 2
Z'( u~';co,~o) *
o
Re
A
1 Bo2(ro dR'm(T~o) Pr 2 c Re dz' 0
where t
_( 4 , 3 4 )
R R=(1) and (tO = gk%
,
z'= = Bo z'
Pr
~o C P k
- -
0
i s the ~nxzad/~ nam&e~ .
FoF z'we have the following expression : 2
;, :
eo {{ 2 { [ c°s ~°o @u' ( i + e ~ z')2 3 cos ~ Ox'
~o
00
+
[av' w' - - + a 0c 0 - (l+CoaoZ') ay'
cos ~0 au' +
ax'
cos
(4,35)
+
[ cos
tan IP v' - --av' ] 2 Oy'
aw' ]2 [ ~-, + (l+CoaoZ')
aw' ~-~-,
tan ~ v' - a e w' 12 ~0
0
J
)
aW'
]
0
~j@ cosU--P~ ~V' ] ) + COS ~002(9X'
~ ~-;,,, (
cos 9
[ (l+e°6°z')2
+
- - )~ +z' c az' ( 1+c 0 0
~0
[ +
V'
a
c°s%a.'__ Co c o s ~
ax'
(1+%% +
CO
o a-y'
2
a az'
]}} 0
~0 z' )
"
A
d%CT'2
,
L )=-_ p'==p=/p=(O)
dz'
, and ~ ( 1 )
@o
and T ' = T / T ( 0 ) .
is the characteristic
value
o f R=(T=);
20 5
.
f -PLANE AND B-PLANE APPROXIMATIONS o
THE
If the horizontal scale of the atmospheric motion considered is "small" with respect
to
the earth radius,
nondimenslonal
equations
i.e.
(4,24),
if ~ <
performinE the following limiting process :
(5, i)
--) 0 , with t', x' ,y' and z' fixed. o
When ~o~
, in Cartesian metric centered at some reference
latitude ~o and
longitude A=O, that locally approximates the spherical metric in this chosen neighborhood of the earth's surface, the effects of spherlcity are retained by approxlmating f=2fl sin ~ ,the local vertical (or radial) component of 2~, wlth o
a linear function of y, a latltudlnal coordinate which is measured positive northward
from the reference
latitude.
B-plane approximation and was first
This
approximatlon
is known
introduced by Rossby.
as
The symbol
the
~ has
been traditionally used to denote df/dy = (1/ao)/(df/d~0). If is important to mention here that the B-plane approximation consists solely of a set of geometric approximations and it does not involve any assumptions regardin E the relative magnitudes of the various dynamic terms in the basic equations. Since this point has not been emphasized in the standard references on the B-plane approximation and because of the fundamental so-called
B-plane
equations,
we
feel
that
it
is
importance of the
important
to
present
a
thorough account of this approximation. Let
us
now
assume
normalization (4,31)
and
(4,23),
(4,33)
that and
are all
we
have
that
the
of order
chosen
our
quantities unity.
scales in
Since
correctly
(4,24), we shall
(4,27),
in
the
(4,30),
be dealing
With
motions whose lateral scale L ° will generally be smaller than ao, the quantity ~oy', in : ~=~0o+~oy', will be small compared to unity and cos~0, sin~0, tan~ can be expanded
(y' =0)
in a convergent
Taylor
series
about
the
reference
:
(5,2)
tan
tan ~ = tan ~o{ 1+~
~o
1 ~o y 0(602 ) } o cos ~o sin ' +
latitude
~0°
21 :
From (4,26) and (6,2) it is clear that
D'
a
S 5[=
,a
S ~,+
,8
,0
u ~-~,+ v --,+
_ ~ o { t a n 9o y,u, a__ ax' + and to zeroth order in 8 S DD' y,-
(S,3)
eoZ'(U ,0~-~,+ v ,0--,) ay } + O(ao2)
we have, obviously,
o
S ~a6 , + u ~ ,,0
+ v,OF g , + w ~,,a.
Thus, although accordin E to cos 90 X' - - A 0 x'
and
y'
surface
are
z'=O,
orthogonal under
the
Cartesian
coordinates
so-called
E-plane)
y' -
'
9 - 90 ~ ' 0
curvilinear
coordinates
lying
on
the
spherical
approximation
that
that
can
is
be
tangent
~ <
the
surface
z'=O
at
the
reference
latitude 90 and longitude A=O. For scales L <108m, we have 6 <10 -I and hence (S,3) will then be valid provide 0 0 90 corresponds to mid-or low latitudes, so that tan 90 s 1. Indeed approximation (5,4)
~otan 9o << I
rules
out
the
application
of
the
resulting
fo-Plane
equations
(see,eqs.(S,G)-(S, lO) below) to be study of motions at high latitudes or in polar
area.
For
these
cases,
a
different
approach
involving
cylindrical
coordinates must be used. If now we expand the metric factors (4,27), 4,30), 6o
is
set
Dropping
(s,S)
equations
(4,24),
(4,31) and (4,33), according to (S,2) and if in these equations
equal
the
in the nondimensional
to
primes,
zero,
the
we
readily
so-called obtain
fo-PlBne the
approximation
following
Dp + p~.~ = o ; S ~5 P
SB_[ +
A v~) +
Ro
(S,6)
o
Bo
1
tan 90 Ro wl
a2v~ Re
+ - - 2 -Oz - 2 + 3 ~(~" u~)
o
;
+ -~Mo2
ae£
~
emerges. ~-pIxIne
22 BO
t a n ~o° Ro
Dt (6,7)
0
-,
~2 w s
(5,8)
~M2 0 z
~M2 0
}
.,o
Oz s
0
p = pT ;
DT
~-1 S D p _ 1 1 {~ST +
p S Dt
¥
Dt
PP Re
1
aST
}
c 20z 2 0
+
-
M2 f~(au
~'-1 2
o
s + c°
(o~u +
(6,9)
c ° Re
c2 aw )2 oF~
~Oz
+
ov
)~ + 2
~
(Or az
+
c2 aw )2 o~
+
fcOU)~ + (ov)~ L ax
(~)~l
+
ay
az
j
BoZcro dR (T) o o -
32_ (~.U~)2
+ P r (2Re
dz
0
In t h e s e f o - p l a n e n o n d i m e n s i o n a l e q u a t i o n s
~=v~+%w~,
oo
(5,5)-(5,9)
~=~+~-~
we have
,
0
(s, 10)
~j,
s ~ n = s ~ a+
v=u1+vj,
~.6+Waz.
In many studies of atmospheric motions on synoptic scale (~o~10-I) the system of "exact" nondimensional equations is
reduced
to
approximation retained intuitive
describe
the
the
variability
of
while the earth curvature and
nondimensional
(4,24),
so-called the
(4,27),
Coriolis
is neglected.
heuristic
argument)
equations
in spherical
by
parameter f
0
and ( 4 , 3 3 ) ,
with
In
latitude
this is
T h i s i s a c c o m p l i s h e d ( f r o m an
neglecting
all ~
O(~o)-terms for
the
in
the
second
term
parameter.
The r e s u l t i n E e q u a t i o n s a r e o f t h e same form as case the constant Coriolis
(4,31)
approximation.
parameter
coordinates
in the Taylor expansion of the Coriolis
(4,30),
E-plane
(6,6)-(6,9).
is substituted
by :
However,
in this
23 (S, lla)
f s fo + BoY
with
(S, ilb)
~o= (I__ dr)
a o d9 9=~ o "
In this case in the equation (5,6), for ~, we s u b s t i t u t e the term ~-~ by the term :
1
(S,12)
(~-~ + ~y)(~ A v~),
where ~ is the following nondimensional parameter (5,13)
~ _
Synoptic-scale
o t a n ~o Ro
atmospheric
m
0(1).
motions
are
c h a r a c t e r i z e d by
U ~10 o
m/s
and
L ~I06m. o But in mid-latitude : f ~lO-41/s,so that o o
~ 1 0 -1
and
R o ~ l O -1
and if these scales are used we have : ~~1 . For these synoptic-scale atmospheric motions, we have likewise Go
H = ~
<
~
c ~Ro2~10 -2,
o o since the characteristic vertical scale of motions, Ho, iS of the order of I04m in all cases of interest, Thus,
it
(4,24),
is
necessary
(4,27),
(4,30),
in the troposphere. to
examine,
in
the
nondimensional
basic
equations
(4,31) and (4,33) the following limiting process (for
the case of Re~m) : I 6o--)0 , Ro --)0 , Co--)O c° t' x' y' with ~, ~ - Ro 2 and , , , z' fixed .
(5,14)
The
limiting
special
process
significance
(5,14),
with ~=0(-°) chosen Ro because it examines ~
to be
of
order
d ~ ,
one,
has
when
the
planetayy vorticity gradient contributes equally with the relative vorticity gradient to the overall vorticity balance. meaning
of this ordering
relation
then
This recognition of the particular gives
confidence
that
the
numerical relations between the parameters are not merely fortuitous.
observed
24
The detailed derivation and justification of the geostrophic approximation and the quasi-geostrophic
main equation is deferred to section 9 and Chapter V .
6 . THE EQUATIONS F O R T HE LARGE-SYNOPTIC SCALE ATMOSPHERIC PROCESSES
In any reallstlc
situation,
in the atmosphere,
Re>>l and now we consider
the
following hydrostatic main limiting process: lw
(6,1)
Co--) 0 , and Re -9 m , ith
e2Re m Re± = 0(i) 0 and t', x', y' , z' fixed
~
~
~ap4~e
(4,27),
~ If
~ we
~
(4,30),
~o=0(1), (4,31) and
then
we obtain,
(4,33),
from
the equations
the full
equations
(4,24),
fop the ~zage- ~
aco/e
. drop
the
primes
e4ao2Lon~
we
for
can the
write
the
following
n~xt #xzaqem/,
set
viscous
of
and
nondimensional non
adiabatic
atmospheric motions :
(6,2)
Dp (aw 1 a cos ~0 Bu) S b-[ + p a-z + cos ~ ~y(COS~ v) + ax = 0 ; cos
[6,3]
p
(6,4)
S Dv tan~ u 2 + I sin~ u~ P ~ + ~o Ro sin~o J +
(6,6)
a-~ ap +BoP = 0
(6,6)
p S
S Du ~-
DT Dt
a°
tan~ uv
~-1S ~
;
I - ~
sin~ v~ cOS~o I ap sin~o J + .c o s ~. .~Mo2 . 8x. 1 ~o
ap ay
a2v 1 Re± 8z 2
p = pT; 02T Dp _ 1 1 + Dt P r Re± Oz 2 .e±o
~1_!_Bo2 ~ ® Pr Re£ o dz ' t Where ~=~0+~0 y , but ~0~i.
i a~u . Re± Oz 2 ;
[ az
+ (av)21 az
J
25 where D a S E6 = S ~
(6,7)
These h y d r o s t a t i c
c°S~o a a + - u ~-~ + v ~ cos9
model e q u a t i o n s
a + w O-~ .
(6,2)-(6,6) c o n s t i t u t e
a
very
significant
system for the large-synoptic scale atmospheric motions in a thin layer as the troposphere around the earth sphere.
7
.
THE CLASSICAL P R I M I T I V E EQUATIONS
When non
Re± m = we tansent,
motions,
the
obtain
viscous set
instead,
and
of
non
~
of
hydrostatic
adiabatic
equations
larse-synoptic
@n/m/Zi~e eqaa2/an~ for
(6,2)-(6,6)
scale
the
nan
for
atmospheric ~
but
non viscous and adiabatic atmospheric motions. In particular case, last
if ~ --~ O, with the relation (5,13) (E-effect), in these o primitive equations, then we obtain the classical, X ~ ,
generalized
Dp
(8u av aw) = 0 ~ + ~-~+ az
S~+
p Du
p
S b-t-
p
S
(1
]
1
Ro + ~ y ) v
D6- Ro +
~y)u
+
+
(7,1) I 8p
~--~-
+ Bo = 0
;
ap
7M 2 8x - 0 ; o
7 M 2 ay o
-
0
•
;
p=pT; DT S p Dt
7-i S Dp 7 ~[=
0 ;
where S D
b - F : Sa~ + u K8 ~ + v ~a+
a wa-~.
These equations (7,1) are usually rewritten in pressure coordinates developed by Eliassen .
26 THE PRIMITIVE EQS. IN PRESSURE COORDINATES In the
system
point's its
position
location
variable
pressure
its
(p=constant)
coordinates,
projected
along
are
projected its
of
the
axis.
with
respect plane.
with
respect
to
to
(x,y)
plane,
The
but the pressure
to
another
corresponding
The " v e r t i c a l "
pressure,
coordinates
"horizontal"
f r o m one p o i n t
onto a horizontal
derivative
usual
onto a horizontal vertical
differences
surface
the
but
in
the
some
differences
directed
of
a
isobaric
of
position
of a variable along
a
p denotes
derivatives
derivative
is
denote
the
is
vertical
axis. The d e p e n d e n t that
isobaric air
variables
are
p becomes one of the surface
speed dz/dt
(7,2)
A useful
becomes a dependent is
~
form
unaffected independent
of
t a k e n o v e r b y ~,
by t h e c o o r d i n a t e variables, variable,
the total
the
transformation
height
and
the
derivative
except
z of a particular
role
of
the
vertical
of pressure
Dp
-
Dt
"
the continuity equation
in which
the
pressure
p
is
the
vertical coordinate is found by considering an elemental column of atmosphere confined between the isobaric surfaces p and p-ap. The mass of
the column am is equal
to pax.ay.~z which by the hydrostatic
ap equation ( ~-~ + pBo = O) gives : I
~
Bo D (ax.ay.ap) (am) m ax.~y.ap Dt Bo = 0 ,
and carrying out the differentiation and taking the limite we find :
Note
~ + ~8~: o .
au + av
a-~
(7,3)
that,
as
with
many
other
primitive
equations
of
tangent
atmospheric
motions written in the isobaric system, it does not involve density
(7,4)
p =
Naturally, that
the main advantage of the pressure coordinates steem to the fact
according
synoptic
T "
scale
to
the
tangent
hydrostatic motions,
approximation is
to
the
leading
atmosphere,
order
in
for
the
hydrostatic
27 equilibrium. altitude
This
guarantee
that
the
pressure
z at x and y fixed and consequently
is a monotonic the change
function
of
is mathematically
sound. The procedure for transforming derivatives
[
in the (x, y, z) coordinates
derivatives in the (x, y, p) coordinates is very simple. we have :
8
ax
(7,5)
8
_ 8
8~ 8
8
ax + ~ ° p ~ ; 8
B
_ 8
ay
8M 8
+ Bop
B
opt:--; S ~
= -Bop
ay
;
~
8J{ 8
= S ~-[ + S B o p ~
~-
into
In dimensionless form
;
where (7,6)
z = M(t,x,y,p)
is a dependent variable with M as the local height above the flat ground of an isobaric surface. As consequence of (7,5) we have : 8
D
S~-6=
S~-[+
8
8
8M
8M
8H
8
U~-~+
V~+
Bop(S~I-+
u~-~+
v~-~-
w) ap-- ;
ap
cap
8p
8~
8~
8~ ~-~
but in nondlmenslonal form
ap
= S ~-+
(7,7)
u b - ~ + v ~--~+ w ~ - ~ = Bop(S b-E + u ~-~+ v
-
w)
.
Therefore D
(7,8)
8
8
8
s ~ - 6 = S b-6 + u ~ = s ~ +8
8
+ v b-~ + ~ a-~
~" v ~+
~ a8- ~ "
Summarize the simplifications attached to the use of pressuce coordinates, rewPlte the complete set of resulting tangent primitive equations
in the
following form :
a~ ~
v ~v + ~ a~ + ~. ~+
( 1Ro _ + 8y)(~A
Bo
v~) + 1
~=0 0
~.~+ ~ = o ; (7,9)
a~
T = - Bop ~-~ ; S
8T
+
v~.~T
aT ~P
+ ~(----
~'-1 T_ p
-)
= o .
;
we
28 For the primitive equations ~
(7,9) we have only the possibility to impose b~a
conditlons : t=O : -> v=v->o and T=T °
(7,10)
as the system of primitive equations
(7,9) is of the tA/nd order relative
to
we have
on
time . For
these
the ~
primitive
(7,9)
8R = pBO(S a-E
b)
according
. THE
to
following
~
~
the f0-Plane
v.D:N),
+
MODEL
EQUATIONS
a nonvlscous
adiabatic
equations
atmospheric
(5,5)-(5,9)
it
is
described
Bousslnesq
convenient
(Remm) a n d we s u p p o s e
in
.
0
the
motion
that
c ml , but Rom~ a n d ~ 0 To d i s c u s s
R=0
on
(7,7).
BOUSSINESQ
We c o n s i d e r
1903,
the
ground :
(7,11)
8
equations
approxlmatlon,
to introduce
first
introduced
the hydrostatic
by
equilibrlum
Bousslnesq standard
in state
by : , p=pm(z
~=0 , w=O , p = p ( Z )
(S,1)
where z =Boz ( s e e , ( 3 , 1 2 ) ) (5,5),
Equations indentically
.
(5,6),
(5,9),
with
satisfied and (5,7), dp~ -dz + p~
(8,2)
) , T=Tm(z ),
=
0
,
Rem~
, c El, O
p®=
p
T
,
and
~m0,
are
then
since z~= Bo z.
Thus for any given temperature distribution T ( z ) , find p ( z )
Ro~
(5,8) reduces to
(8,2) can be integrated to
and also pm(z ).
To describe the atmospheric motions which represent departures from the static standard
state
perturbation relations :
(8,1),
density
(8,2), ~
and
we the
introduce
the perturbation
perturbation
temperature
e
pressure
~,
the
defined
by
the
29
(8,3)
Substituting
p = p (z)(1+~)
;
P = p (z)(l+e)
;
T = T (z)(l+e)
.
(8,3)
into
(5,5)-(5,8),
where Remm,
c0ml, Romm and ~ 0 ,
we find
the following emnc/ set of equations, for ~ = ~ +w~, ~, e and 8 :
T®(z) CI+~)S D--~u~+ Dt
~M 2
BO
~
= (I+~)
~e
Be
[
o
~
~M °
dTm(zm) ] ~.~
(8,4)
(I+~)S De Dt
=
~-i S D~ ~ ~-+
W+e+W8
(I+=) ~
Bo
~-I + ~ dz
=0;
.
Let us consider the " Z a a ~
cuae", first introduced by Zeytounian in 1974,
when : (8,S)
Bo--) 0
with the ~
and Mo--) O, for t, x, y, z fixed,
~ Bo
(8,6)
Id o
In thls case : z If we r e p r e s e n t
__ ~ = 0 ( I )
.
= Boz - BM o z --~ O, as M---> 0 . o the
solution
of
equations
(8,4)
by asymptotic
expansions
the formt: v~=v~+ .... B
W = W +
(8,7)
t
B
...
,
~ = b~o~B+
..°
~=M~+ o B
... ,
8=M8+ 0 B
....
,
I n t h e Appendix I we g i v e
a brief
as
of
they
are
used
to
study
introduction
singular
t o boundary l a y e r t e c h n i q u e s
perturbation
problems
.
of
30 we can easily show that the limiting functions, the ~
,
~
B
S
Dt
= O; s
1 8rib +
Dt
8w
(8,8)
B
-
~ 8z
Os;
3=0;
8z
s
3+
~+
__ wB=o ; [dzDz --o
Dt (~B =
vs, ws' ~B' (~S and 03, satisfy
equet6on~, namely
B+I~_ ~
Dws S
~
-)
GO
-O B
,
where 0
S ~D= S ~ + We note
that
in the
v~B"~ + W
a___
.Oz"
limiting Boussinesq
process
(8,5),
(8,6),
we have
(in
dimensionless form) that : Tm(zm) --) T (O)ml. The asymptotic theory, allows us to obtain not only the classical Boussinesq equations
but also to define
these equations are obtained.
the
limits of the approximation
Namely,
for the characteristic
through which
vertical
height
H ° we have, from the (8,6), the following limitation : Bo B=--~
(s,s)
U 1
~Ho
M
0
.
g
0
T h e n H ~103m, f o r
[ RT (0) ]I/2
~ o
the usual
values
of U
and T (0).
0
Otherwise if we consider (3,7), with (8,6), we see that: Fr2m
U
2
o
~ ~M--> O, as M--> O,
gHo Bo
when B - M
1 and c om 1 (~ i s f i x e d ) . 0
9 • THE QUASI-GEOSTROPHIC MODEL EQUATION We
start
from
the
tangent
prlmltlve
equations
adiabatic atmospheric motions and we introduce, number KI=SRo.
(7,9),
for
the
nonvlscous,
in the place of Ro, the Klbel
In this case we rewrite the equations
(7,9)
in the following
31 form :
fa~
av~.l
1 c~.~+,.,,
k Bo
+
0
- - ~ = Ki
O;
a~
(9,1) aM T = - Bop ~-~ ;
~T _ a T + .~ 1 (v~.~T + ~ a[~__l~ i-I 0 TP1 ]) = where 1 .Ki)2
(9,2)
Aom ~-~L~-
•
o
We c o n s i d e r
:
now the q u a z ~ - q ~ / / m / ~ p n a c e x ~
(9,3)
Ki --~ 0 and M--~ O, for t, x, y, p fixed, o
with the ~ t m / / a ~ ~ C9,4)
1 Ki.2 -~S - (~--J,, ~ ~o = 0 ( i ) o
We r e p r e s e n t
the
solution
of
equations
(9,1)
by asymptotic
expansions
of
the
form : .-) v~ = v
qg
w=Kiw
(9,5)
+ Ki ~ v
ag
+ ...
+ ...
qg
= MoCP) + Ki R qg + K i 2 R a g + T = To(P) where
the
and Ro(p)
standard is
+ Ki T qq + . . .
temperature
deduced
To( p ) i s
from the
hydrostatic P To(q)
c9,
)
= 1
if
is
made o f
the
c•.o n d i t i o n
When Ki->O, J t = ~ o ( P ) + K i ~ t
a
D
, related
to
equilibrium,
the
heat
balance
namely
dq
R (I) = 0, on the o From the first equation of (9, I) we find : t
use
q
@
ground t
+ . . . =0 ( s e e , ( 7 , 1 * ) ) leads : R o ( P ) = O and we s u p p o s e qg t h a t p=O ( i n d i m e n s i o n l e s s f o r m ) i s the s o l u t i o n o£ t h e e q u a t i o n ~ o ( P ) = O .
equation
32
~A~qg +;~Bo~R =0, O qg and that is the well known qema/nopAtc aebz//mn : v -~ qg
(9,7)
XoBO(~^ i~t qg ).
=
From the second equation of (9,8)
~ . v~
= 0
qg
and
we get :
(9,1)
,
8~ ~ . v~
(9,9)
+
qg = 0
ag From t h e
first
relation
:
(9,10)
.
8p
equation
of
(9,1)
we f i n d
..>
for v ag , the following
afterwards,
a~ = ~ A { BOAo~Rag +
v~ ag
/3-->
q~, + ~1 v~qg 8t
From the h y d r o s t a t i c e q u s t l o n we have (9,11)
T = - Bop
qg
8M qg ap
and equation for T leads : aT
oqg=
Is 0tq°+ q0q0] 2 [ a~
: - B O ~%,p,
s
+~
.~
]a~
qg
qg
8p
where d Log To ] (9,13) Using
(9,14)
Ko(P ) _ To(P ) (9,9),
with
(9,10)
the
a
~'~,___~1_ P
+
and
(9,12),
Bo~o a~t ( qq a S
8x
dp
8y
ax
we derive
qg
8y
where
(9,15)
= ~o~ + 82
with
8X 2
a [ p~ ~ a ]
s~ 82 8y 2
, 0 .
8___) } ax
the mo/n eqao2/o~ o~ tAe
~qg+
#
S
a~
qg- o ,
8x
33 We observe that (9,14) contains only one derivation with respect to t and, as consequence,
only one initial condition must be supplied ?
We observe that this loss of initial conditions from the primitive equations corresponds ~u~ ~
to the fact
~
q ~
The boundary condition, (In
the
~
that the limiting process t.
~
(9,3),
wlth
(9,4),
~Ate~a
that must be suplied on the ground p=l, can be derived
~
co~)
from
(7,11),
if
we
take
into
account
(9, 12),
aJ{ 1 To(1) [ 0 k___~o(O~ q~ + ~l- + q~ 0 8t Bo K ( I ) S cgx ON
(9, 16)
0
~qg ~ )l~qg: 0 , By ~ ]
ap
on p=l. We don't specify the boundary conditions that must be applied at the upper end of the atmosphere, p=O, and at infinity in the horizontal plane. These may be, for example, that
p
2
a~
c
qg)2
ap
~
t
~
~
In the Chapter the
Chapter
~
III
IV t h e
~
we c o n s i d e r
k
S
o
)2 qg
.
t h e wave phenomena
problem of filtering
internal
in the atmosphere acoustic
and g r a v i t y
and i n waves.
34 BACKGROUND READING
For a extensive treatment of concepts of the Mechanics of Fluids the reader is referred to : GOLDSTEIN ,S. (1960) _ ~ e a Y & g d ~ ~
.
Intersclence Publishers, LTD, London (Chapters 1 and 2), and MEYER, R.E.
~,m~_.~
(1971)
~ ~
~
~f~t 9~.~.
Wiley-Interscience, New York (Chapters 3 and 6). The equations of motions
in several useful curvilinear coordinate frames can
be in : BATCHELOR, G.K.
(1967)
~J~m~d~o~ ~ ~
~
.
Cambridge University Press (Appendix 2). Concerning the E-plane approximation the reader is referred to :
LEBLOND, P.H. and MYSAK, L.A.
(1978) _ W o > ~ e ~ A e O c 2 a n . Elsevier Scientific Pulishing Company, Amsterdam (Chapter i).
For the Boussinesq and quasi-geostrophic model equations see, for instance : ZEYTOUNIAN, R. Kh.
(198S)
_
~
~ec22~
~
~
.
~%
~agm ~
Int. J. Engng. Sci.,Vol.23,
~
~
o~
n ° II, pp. 1239-1288,
and
PEDLOSKY, J.
(1979)
_
m
~
~
m ~ .
Springer-Verlag, New York (Chapter 6). Finally, concerning the various model equations for Weather Forecasting see :
35 MONIN, A.S.
(1972 ° _ W
~
~
a~ a P ~
The MIT P r e s s . this
last
book
Mechanicians
is
an
(Students
Ca~bridge
indispensable
one-volume
and Researchers)
/~g~Lcs.
M~ss.,
involved
U.S.A,
reference
in Meteorology,
and KIBEL, I . A .
(1963) W~e~ T h e Mac M i l l a n
~
~ Company.
and
(Translation).
text
for
Fluid
CHAPTER III WAVE PHENOMENAIN THE ATMOSPHERE 10 . N A V E E Q U A T I O N
In order
FOR INTERNAL
WAVES
to construct a fluld dynamics theory for atmospheric flows
important to clarify first o f
it is
all what are the possible types of atmospheric
motions In adiabatic, nonvlscous processes. All
these
atmospheric
classlflcatlon waves, rest.
l.e.
it
is
small
and
variable,
by
sufficient
the
to
character
consider
of
the
of the atmosphere
the
density
"standard
and
z =Boz
and
of
relative
for
~ - a # 4 o ~ to the
state of
functions
of
but
(in dimensionless
~o
their
is specified by a zero
temperature
altitude",
a~ea,
case
called the standard atmosphere,
pressure,
namely
have
oscillations
Such a state,
veloclty
motions
one
form),
where Bo is the Bousslnesq number. This state of rest waves
are
is assumed
studied.
equatlons(5,5)-(5,9),
to prevail
We
start,
wlth
(5,10),
in
far from this
the domain,
section
where Re=re. If we
I0,
where of
the
fo-plane
look for solutions of
these equations by expansions of the form:
V • _~ ,-+ W
(10,1)
=
W'
...
;
p = p ( z o) (1 + c~'+ . . . )
;
T = T (z)
;
performing
perturbatlon
+
;
p = p (z®) (1 + ~'+ ...) ;
CO
and
...
the
~o
(1+
e'+
classical
quantities
...)
11nearlzat lon
-~,w' ,=' ,~' v
and
e'
process
with
(products
themselves
or
of
with
the their
derivatives are neglected) we can easily show that the functlons v-9,w',x',~ '
37 and e' satisfy the following linear system of equations:
S
Ov v ) + co ~~' + ~ 1 (~ A ~, 1 w'~+ Ro tan ~o
c ~ S aw'
at
Co
1
~,.~
Ro
tan ~o
+
T ® ( z®)
- - ~ ' ~H 2
a~'
~M 2
az
m
_
Bo ~M 2
0
a
~V ,
aW'_
;
o
T (z) ®
=0
0
Bo
[
dTm(zm) 1
(lO,2)
o0
C~' = ~' - e';
where
S ~-~ @ (e' - ~-1 ,)~
+ om N~(z
~°NZ(z = mm m )
[7~1 '
~®tz JB°
+
) w' = 0 ,
dT dz
]
,
Z
---- S O Z ,
o0
according to (3,11) and (3,12). Now,
if
we
take
into
consideration
that
we
have
tara main
atmospheric
situations: (I) Co<
(2) e=O(1), but Romm, C
o
we can neglect the terms proportional
tO:Ro tan ~o in the linearized equations
(I0,2),
Otherwise,
corresponding to (5,5)-(5,9). dT(z) ~ dz
(10,3a) considering
the
case
we assume that
m 0 , of
an
dx~e~aa~
standard
atmosphere.
In
consistently with the fact that we are dealing with nondimensional we may take (10,3b)
T (z)
in the equations
(10,2).
e 1
N~(z ) m 1
and
o
m 13o ~-1
this
case,
quantities,
38
It is possible to write the equations (10,2),
taking into account
(10,3),
in
matrix form in the case where c0/Ro m 0 and for a non-trivial solution of this matrix equation the determinant of the coefficient matrix must vanish. Then we may obtain,
for the relative pressure perturbation ~', a wave equation which
reads:
(10,4)
Ot 2 L o o
Ot z
-
:~
= 0
where
( l O , Sa)
= ~+0
az~ - ~
~
-
L--~-oJ
and (10,5b)
Bo (x° ~2 + _ _
~ =
~M 2 o
If we set,
.
-
Ro 2
~
in (10,4), BomO, Romm and Coml, the wave equation (10,4) reduces to
the classical one for acoustics. For
the
~uU
wave
equation
(10,4)
it
is necessary
to
specify
~ou,t initial
conditions for O~'/Ot, when t=O. Namely:
(i0,6)
t=O :
0~'
02~ ' 03~ '
Ot'
Ot 2'
Ot 3'
04~ ' andOt 4
are given functions of ~=x~+y~ and z.
Otherwise
this wave equation
(10,4)
solution it is necessary to specify ~ the ~
(i0,7)
is of second
order
in z,
boundary conditions.
In particular, on
z=O: w'=O ==# [~z - ~o~~-l~Ox'~ J~=
O.
(10,4) may be expressed as a superposltion of
elementary waves as follows:
~-ea"' Boz/2 "®
~.(~;~)~(z;~)d~ o
where
its
ground we obtain the following condition:
A fairly general solution of
(lO,8)
and for
I~lS= m 2 + n S s k 2
and
i~.~=l(mx+ny),
e i~'~ dmdn,
39 The eigenvalue
k and eigenfunction
E are determined
from the Y b ~ u ~ - E ~
type equation~ d2~
(10,9)
+ ( k - .~--~)~= 0 ".t
dz 2
with the boundary condition (see (10,7)):
(lO,10) But,
dz--d~+ ~ - B o ~
the differential
= O, on z=O.
equation
(10,9)
is of second order
in z, and for its
solution it is necessary to specify a second boundary condition on the upper limit of the troposphere when z--) +m. For large z every solution of (10,9)
is
bounded if [B--°]2
(10,11)
and
the
spectrum
< A. <
is
a
continuous
one
with
just
one
discrete
eingenvalue
embedded within it, namely (10,12)
k°
=
[B--°] 2
(~'-I)
,
and the corresponding eigenfunction is : ~o(Z,ko)me-(2-~)B°z/2~"
(10, 13)
The functions ]] (T,A), mn
where T=t/S,
must satisfy the following
equation:
(10, 14) where
(lO,15)
+ 2~o
I
2Mo
-
with o
_ Bo ~----!1.
2
-
1
2 ojj
2° o {i ~
k2 ~
Bo 1
differential
40 If
the
system
of
eigenfunctions
~(z,A)
is
~
(corresponding
at
the
eigenvalues Bo2/4
can be represented in the form (10,8)
Z(z,A).
Therefore
we
obtain,
for
the
in terms of the eigenfunctions
equation
(i0,14),
well-posed
initial
values problem and frequencies of the internal waves (for isothermal standard atmosphere) are determined from the reiation (10, 18)
a,g=~o{ lr~ (I -;%o)I/2} .
We note that ~ a
(sign +) corresponds to internal ~
waves (modified by
gravity) and v
(sign -) to ~ waves (modified by compressibility). g If specially we consider the limit: Bo--) 0
we
obtain
and
two-dimensional
k---)O,
acoustic
waves,
in
which
there
are
no
vertical
osciliations of the air particles, with the frequencies 2
(10,17)
1
= llm
BO-) 0
p,a
m --+ Ro 2
A-)o and in this case :llm ~-0.
These
k2 -M2
o
are
the
only
pa4<~&le
waves
when
the
Bo-)o g
atmosphere is a ~
11
. THE WIND
one (in this case, of course, Ao-O).
DIVERGENCE
EQUATION
FOR
TWO-DIMENSIONAL
INTERNAL
WAVES
It is occasionally advantageous, especially in Meteorological applications, to find the differential equation that describes the wind divergence, rather than the one that describes pressure perturbation ~'. ~-isothermal
We consider in this case a
standard atmosphere dT ( z )
(11,1)
-
oo
-
GO
m 0
,
dz 8/ld a
two-dtmenslonal motion (~ m ~--~) w i t h Rom~. I n t h i s case,
instead of
(10,2) we have the f o l l o w i n g l i n e a r system o f equations, f o r u' ,w' ,~' and e ' ,
41
au' T (Boz) ax'_ S ~ + ~M 2 ax 0 ;
(ll,2a)
0
(11,2b)
c:S ~-~Sw'+ T~(BoZ) aza~' Bo__ e' = 0 ,~M 2
~M ~
0
(11,2c)
~ S a~'+ at
(11,2d)
ae'+ S~
-
0
~
dz
W'
= 0
and the wind divergence
,
Xz is given by
aU' aW'
,
Z2 - ~
+~- .
I n s t e a d of (10,4) we are a b l e to o b t a i n , s i n g l e wave e q u a t i o n f o r ~2" a' l o o
0 ;
dT (Boz)
1
(~-l)x~ +
since Bo dT~(z~)/dz~ = dT~(Boz)/dz
(11,3)
W' =
]3o
az
a2_
from ( 1 1 , 2 a ) - ( 1 1 , 2 d ) ,
S za 2
dT (Boz) ]a fS2 a21
0×2 +
at 4
dz
J~-£t
o--CJ
82
Bo
(11,4)
the f o l l o w i n g
~M z
O~ N~(Boz)T(Boz)~x21 z~=O"
0
For the equation (11,4) we may consider as a tentative solution a plane wave, i.e., , , Ikx-l~ z~(t,x,z) = ~2tzje ,
(11,5)
with T= t/S , and we set Z(~) = e -~/2
(11,6)
Z2(~-1(~))
,
where t (11,7) t
In
= Bo
particular
case,
~o
for
T~(Boz) = 1,
dz' - ~(z) dToo(Boz, ) the
isothermal
~ = BOZ = z®,
standard
atmosphere
we h a v e
and z = - 1 ( ~ ) = ~/Bo.
42
Finally,
for Z(E) we obtain the following equation: fe°k~z
1
d2Z dE2 +
(11,8)
[---~] Tm(Boz) + ~(~2;Boz)
= 0
- ~ -
with
o
(11,9)
To
}Z
2~2+
~(~02;Boz) = T=(Boz)
the equation
conditions
(11,8),
for
Z(E],
we must
~1~o Bo N2m(BOZ )T m (BOZ) "
add
the
following
two
boundary
: d__ZZ- 11 dE
(11,10a)
1 ~o
k.~ Z = O, on E = O ;
JJ
and lim [Z(E) I<= .
(11, lOb)
The
E~+m
homogeneous
(ll,lOa), the
set
(II,8),
with
the
homogeneous
have in general only the zero solution, of
eigenvalue present,
equation
eigenfrequencies
appearing
case,
c O ~ 0 , assuming that,
instead
of
(11,8)
and
d2Z
dZ
o
4 Zo+ P'z#(E)Zo = 0 ;
-
1
2
( 5 - /~ ) Z o =
0
, on
E = O,
dE IZo(=)l<=
Here,
Mo, Bo and ~ are 0(i).
(Ii,I0)
1
o
dE 2
(1969).
"non-standard" for the
is substantially simplified by
hydrostatic eigenvalue problem:
(II,II)
difficult
problem for which we refer to Dikij's book
the limiting process:
condition
unless ~ is equal to one of
in a quite
we merely point out that this problem
boundary
,
where Zo(E;/~) -= lira Z(E;Co,Mo, BO,~,e,k), 8 -)o EOflxed
we
obtain
the
In this following
43 with 2
(11,121
/~ =
k2
I
-2
7M2 o
and o
(11,13) Besides
@(g) = ~-~ N (Bo~p-l(g))T (Bo~o-l(g)). in the case
of
~en4~ ~
l~a/qxu%~x~ ~ ,
following singular perturbation problem,
----~= + {~z@(~)
d2Z
_
when k-)~, we obtain the
instead of (11,8),
(11,10):
e°2 Tm(Bo- I ( ~ ) ) B~
d~ 2
+ ]2[
2
c
o __
1
__ Tm(Bo~ I (~)) - 41-]}Zm= 0 , ~rB2 A 2
(11,14) v
2
dZ m
+
2 v 2 )Z = 0 , o n ~ = O,
(A2_
d~
iz®(®)l< ® , ~2
1 1 ~M2 2 o
when
u2m I-----~ 0 . k2 t This problem (11,14) is amenable to a treatment by the double-scale technique, but presently it is not resolved. At
this
process.
(11,15)
point
we
must
add
that
it
is possible
to
~ --) m and M ~ o
In this case we obtain,
C11,15)
o
model.
instead (Ii,9),
~'((a2;I~::)(p-lc(~)) --- A21~(~), A.2E k2 1 (8,5), with
the
(8,6), but
it is necessary to start of generalized Boussinesq equations
equations (12,3) below). See
limiting
O, with ~Me = M = 0(1),
On the other hand we may consider the Boussinesq case
t
other
For instance:
which lead to the so-called ~
for this
consider
Appendix
2
for
a
brief
discussion
of
this
technique
.
(see,
44 12
•
BOUSSINESQ
When
GRAVITY W A V E S
Bo<
Z
(12,1)
E - ~ - M z B
in b u i l t
0
into the f u l l exact equations (8,4) and we can now d i r e c t l y set in
(8,4): (12,2)
= = M2~
,
0
~ = S~
, 8 = MS
0
0
Then (8,4) takes the form:
(12,3a)
D~ T- (~E) S ~- + ~ ~ = _Mo;S D~ D~" ;
(12,3B)
S ~+Dw T. (~E) aza~ wB~ = _Mo;[S ~Dw - ~Be];
(12,3c)
~.~ +
~W
8w)
= -Ho
+ ~(~'v~ + a z
dTQo(BE)
_
-
-
dE
,
W
TGO(B
E)
]
+ M02~[~ + dTm(BE)~ ]T--~-Bo~ ; w
(12,3d)
~ + 8 = M (~ - ~.~) ;
(12,3e)
S
o
+ ~rm(BE)w = -M° S nt
where
(12,4)
r(k)
=
dE
The Bousslnesq system (8,8) is nothing but the full exact system (12,3) written at ~ as
q2na with respect
B=O(1) (in this case
E--) O,
o
The ~ ~ a ~ e ~ method of double height scales t. t
Mo---)O, when
M--> 0).
The
two-varlable
expansion
procedure
(12,3) are amenable to a treatment by the
is
briefly
discussed
in
the
Appendix
2.
45
FollowlnE Bois (1976) we consider the linearization of (12,3) around the state of rest:
S~-~+
~'
=0 ;
^
8w'
s ~
(12,S)
+
B e' = 0 ;
T (B~) a~'
aq-
~.#,+ aw' ~-~ :
-MoS am' ~-[ +
Mo(~+
d
--(T (B ~)) - -w'
d~ "
T ( B ~)
S ae' ~-~ + ~r (B~)w'= MOP-1S a~' 8-t ; e' + {a' = M ~'.
o
We consider a plane progressive wave solution,
in an unbounded atmosphere,
for
the linear system (12,S). Writing -) V W'
W
(Z;Mo)ei~
(12,6)
,
(~
e
e'
where @ = ~ot/S - koX, and substituting linear system of ordinary differential
this form equations
(12,6)
into
(12,5) gives a
for ~,w,~,~ and e which can
be reduced to one equation for w(z;Mo), namely:
c12,7)
dz - -
+
-
w
:
)
+
C)
Jdz
'
where ~=MoZ and # (B~)mBF (B~). The structure of solutions of equation (12,7) can be examined by considering the case
46
m 0
dC and in this appear: if z >
VO
In
~T=CB~)
~ 1
c a s e @=(B~) ~ ~2 g - 1 = c o n s t a n t .
At o r d e r
2
2
<
0
(~-1)
the
solutions
of
(12,7)
z e r o i n M two c a s e s 0
are
oscillatory
and
if
(~-I) they are e x p o n e n t i a l .
the
where
case
determined
@®(Be)
by a ~
p
is e
ao2
~
~ m~.For
most adequate method is that of b~e ~ .
(12,8) and write
~ _ ~(C) M that
(12,9) satisfies,
a
solution the
of
problem
(12,7)
can
be
under study
the
We introduce the two variables:
and ~ = M z o
the function
w(Z,Mo)=W'(~,~;M o) with respect to the two variables ~ and ~, an equation which must
be identified to (12,7) if ~ and ~ are related to z by (12,8). Thus: d _d~a dz dE ~
(12,10)
d2
dz2 -
-
a + MO 0-~
- fd~]2 a2 + M [ dz'P a
l~J a~~
d~
f12
°La-~ , 2 ~ a-~-~]'
so that (12,7) reads
a~ 2
(12,11) d~ (
Let us seek w
o
in the form:
82w"
dZ~/d~ 2 aw I
1
+
)] dw
m
47
w':Wo<~,E)+.o<(~,E)+ ....
(12,12)
e
and at order 0 in Mo, W 0 satisfies the equation O
f_~?2a.o( °2
l~J
( 1 2 , 13)
This
equation
is
a
k°
+
.(~E) ~o "~
differential
(concerning the dependance
equation
in ~) and without
= 0
with
.
constant
coefficients
less in generality we can search
the elementary solution in the form:
~(~,E)
= Ao(E)ei~
d~ which gives for ~ the equation: fd~l 2_ rko~ 2 r
l~J
(12,9)
-
^
2
lCJ[~.(.E)-%]
Thus i
(12,10)
WO(~'E) = A+o ei~
+A-e-i~o
where ~ has the value:
C I ~ d~(C')
(12, 11)
~ = Moo o
dE'
dE'
d~ and a~ i s one o f the two r o o t s o f (12,9). From
(12,9),
if
~ (BE)<@3,
the
solutions
(12,10)
are
real
exponentials. If
~ (BE)>cr3, these solutions are oscillatory. At this stage the two functions A ~ are undetermined and in order to obtain o functions we must write down the equations satisfied by w*. This I equation reads: these
e
(12,12)
d~ 2 0w i ag 2
+
k°
= -.~ .[2~+
~(BE) - ~o
t
~
7[ <,Ao +c¢,,~oc~,~)
1
+ ~J"
/e
d~,~.'<,~1_-1e-i~
}.
48
In order that w[(~,~) be na ~
singular than w~(~,~) it is necessary
thatt:
=0,
2 ._~° +
(12,13) 2 d~ o +
a(~)
~
Hence
d•
Ao=O
.
c-ToO
(12,14)
A~(() -
and _
I
{C~oei~
Cooe_i~
where C r~ are two arbitrary constants. oo In the case of a ~ ~ solution (12, 15) must be replaced by i
( 12, 10)
where D ~ are also two arbitrary constants. oo In both formulae (12,15) and (12,16), ~ is given by the relation
(12,17)
~ =k ~ 0
1 I'" dE'.
~ 0
0
First the formulae (12,15) and (12,16) allow one to examine the structure of the solution of (12,7). t
We e l i m i n a t e
the
"secular"
1 T (B~)
terms;
see
the
Appendix
(~d dTm(B~)) _= d&%°m(B~) d~
dE
2.
We n o t e ,
also,
that:
49
d$ does notr v a n^i s h ]-I/2 ) the If 4)m({3(~) has a moderate variation (in fact if ~-~ behaviour of oscillatory solutions when
(12, 18)
p (Bg) =
exp
{_I
[
z--> +m, is that of [p (B~)]
+
dE'
. But
i.
JT (Bg'))
o
It
can
be easily
verified
that
if
T (B~)~(~:)~,
where
cr.zO,
w h e n g----* +~,
then
p (B~)--> O, when g--~ +~. Hence the only oscillations of the form (12, 15) which can satisfy a damping condition at infinity,
are those such CO+0 =C-=0, O0
these
or
is
no motion
.For
motions
such
that
C÷
C- i s
not
zero,
the
i.e.
damping
proceeds only from dissipation (non adiabatic effects, with Remm). TURNING POINT LOCAL PROBLEM
Now consider the case where d$/d~ vanisches at a point ~=~o" This point
is
defined by the equation
(12,19)
Oo ' 2:@ (B~o).
In the neighborhood of ~othe solutions of the form (12,15) or (12,16) are no more valid:
G0 is called a b/nnin~ @a/n~ for equation
(12,7).
In order to
study the solution of (12,7) in the neighborhood of Go, it is necessary to make a ~
expansion in the form:
~CBC) (12,20)
=J+ o
cc-co) d~®
+... ~=~o
dE
2 + Mo(Z_Zo) dSm
cr°
=
since
~=MoZ
dE
+ .... C=C 0
(Zo~ ~o). M
o
In the neighborhood of Go the function w(z,M o) satisfy the followin E equation, instead of (12,7),
(12,21)
d2w k2 -+ Moo 2 % (d:)(Z_Zo)W
dz 2
o-
÷ Mo~ (~:o)dd~ = O(Mo )
o
where (i2,22)
X°°(~O) - - - d ~
~=~o' ~°°(~°) -
d~
~=~o"
50 The equation
(12,21),
Asymptotic Expansion.
when M--~ O, can be studied by the Method of Matched o The expansion (12,12) is an outer asymptotic expansion
and it is necessary to consider an inner asymptotic expansion in neighborhood of Go. We introduce an Inner variable $
(-(o
~
(12,23)
associated with the inner (local) asymptotic expansion (12,24)
w = ¢ + ~
o
o
~ + ....
6>0.
1
Substituting (12,23) and (12,24) into (12,21) gives:
d~ (12,25)
k ~
d~2O + Mi-3~F o] o
-
-^A +
We wish to choose ~ so that first term, differentiated leading term,
<- ~
d~ (~0)J
in (12,25),
d~ +
I
÷ . . . .
o.
which is the most highly
is a dominant teFm in balance with at least one
of the other two leading terms (index "o"). Obviously
the only good choice
is the balance between
terms in (12,25) and that requiFes
(12,26)
But
1-3~
this case it
in
(12,27)
= 0
~
~
-
1 3
"
necessary that
is
l-a = # ~
2 ~ - 3 "
Finally, for ~ (~) we have the following equation 0 d2~ d~20 [k°]2 .. , A ^ (12,26)
--
+ [~oo~ ~'tq°Jz%=
0,
of which the general solution is (12,29)
t
See
) AA w0(z
the
Appendix
1
= a*ooAi(Z)
t'or
the
A•
+ a-ooBi(Z) =- Wo(Z) ,
application
of
the
MMAE.
the first and second
51 where ,-k ~ 2/3
(12,30)
[oj
z=
1/3
Ix=(%)l
and Ai and Bi are the Airy functions of first and second kinds. aoo; can be related
to the constants
the means of an asymptotic Z ((o)<0. (12,16))
(12,31a)
The
two
outer
C ~oo and D;oo of
(12,15)
The constants
and
(12,16)
matchin E.
In order to fix ideas, we assume that m Wo(~, ~) has the form (see, (12,15) and
solutions
:
. wo(~, ~) -
i
fC;oeiE
W~sc(E,E)
, if E<Eo'
_ ) , + Cooe-i~
and
(12,31b)
we(~,~) -
o
1
/
" p 3 Wex
^
d~
(E,E)
,
e -~ oo
D+
if
E>Eo,
since we assume that we--~ at infinity. o If we note that
-
~-~o
and z
=
[b~ 2/3
Iz=C%)l
0
then ¢ (12,32)
ltm
^"
w (Z) = llm o
Z-~-=
M2/3 0
'~<'~o
and (12,33)
lZ3 ~-~o
sc(~'~)'
~-~o
Ae
l i r a Wo(Z) = l i r a
Z->+®
•
Wexp(~,~),
~-~o E>Eo
t
by
The Van Dy ke m a t c h i n g c r i t e r i o n is (see, Appendix 1): Outer expansion [inner e x p a n s i o n ] - - ~ I n n e r e x p a n s i o n [ o u t e r e x p a n s i o n ] . For t h e ~]Jt~ order terms the relations ( 1 2 , 3 2 ) and ( 1 2 , 3 3 ) a r e o£ good matching conditions .
52
Z:
By using the asymptotic expansion of the Airy functions for range
AiCZ)~'l~ (-Z)-I/4sin(@o + ~1 ; Bi(Z)~-!~
(12,34)
(-z)-l/acoS(@O+~) ~
;
2
~0 ~ 3 ( - Z ) 3'2, we obtain for the ~<~o t h e f o l l o w i n g
(12,35)
outer
expansion
116
~,o(Z)-:%
Ho I~:-%l-~'%~osin(V'o +
:
lI
~),
if we take into accont that the only function which can be matched to (12,31a) then is the function Ai(Z). In (12,38) we have :
o
&2%-2,3 ~°= L o
(12,36)
i~®(~o) 11,a~--~",
2 ko~;~ix(%)lW21C-%l
@0= 3
MO
3,2
'
~<~o"
On the other hand, we have in the neighborhood of Co, for ~<~o:
e~O
2 1/2
where ~o= koJ° 11-* Cu1~; I du, with *=Cul - ~+ Cu-%1
de=
u=%
Therefore : (12,381
~oo
~ = M- - ~o ' (~<~o)" o
So that we deduce the following inner expansion for w" • o sc"
(12,39)
W:sc(~,~l - [~ O.(%'J^i-I'2 _ -i(~o/Mo )+ i~ 0 ] + Cooe ~
since
[~ (~)~] w2~ r
1112f(rO]-112
[~ (%1] ^
[~]
114
111
I~ (%1 I I~-%1
53 Identification ~<~o ) y i e l d s
of
(12,35)
(valid
f o r Z->-=) and
(12,39)
(valid
for
~-~o'
with
: -i~o/M0 + i~/4 C÷ = A e oo o C~O= A e +i~O/MO - i~/4
(12,40)
'
o
with
k°
A=
(12,41)
o
I-----
Iz®(~o)
2 ~4"~-
o
oo"
The writing of the expansions for ~>~o would give the relation between a ~ and D- in the same manner (here we have D ÷ mO a priori). oo oo By eliminating a ÷ between C* and C- in (12,40) we obtain oo oo oo (12,42)
Coo= C*ooe
21XO
~0 ' '
Zo=
M
4
o
"
It is possible to place the meaning of the turning point Go and of the formula (12,40) in evidence : the existence of ~o involves the disappearing of the wave which propagates if ~<~o'
the amplitude of the wave in the neighborhood of Go becoming O(Mol/e),
from
(12,40)
and
(12,39),
(12,23) and (12,26),
in an
interval
of
length
[~-~o I =O(K/3),
from
(12,27) .
For <>~o the waves decreases as exp(-~),
i.e. as exp(-~/M O) with respect to <.
Hence beyond Go there is no more oscillation, level of the oscillation.
and
The relation (12,40) then is a relation giving the
reflection coefficient between the incident wave C ÷ e i~ and the reflected wave oo C- e -1~.
oo
(12,42)
from
the reflection Note
that
equations
the r a t i o
of amplitudes
in t h e r e f l e c t i o n
is
[Coo/C~o [
=
1.
But
i s a c c o m p a n i e d o f a c h a r g e o f phase.
by t a k i n g (12,5),
it
the is
dissipative possible
effects
to solve
the
into
account
in
the
problem of
the
~
linearized
in a simplified manner. 13
. THE ROSSBY WAVES
F or the derivation of the equation satisfied by Rossby waves it is necessary to take into account the B-effect (see (5,12) and (5,13}),
according to which
variation of the Coriolis parameter is taken into account through the term: Ki ~-- y (~ A ~) in the equation for ~ (see, the system (9,1)).
54 After
llnearizatlon
(9,14),
of
relative
to the state
plane wave solutions,
namely:
(13,1)
Real(~qo(p)exp[i~'~-st]}
~0 =
and this l e a d s
of rest,
we may
look a
to the dispersion relation
O" =
(13,2)
BO~.o(mZ+n2)+~ - o-R , k
In (13,2)
the scalar ~ is the eigenvalue
corresponding
to the Sturn-Liouville
equat ion
(13,3)
d
[
p2
d~q°l
d-PlKo-~"
with suitable homogeneous
The phase
v
'
is always
R
less than 6,
thus existence
of Rossby
that both m and ~ be non zero.
speed of a Rossby
x-direction
qo
boundary conditions.
The Rossby waves frequency waves requires
+ ~¢~ = 0
dp J
wave
(in the absence
of a basic
current)
in the
is R
_
(13,4)
c x m
and retrogresses
toward negative x, i.e.,
BoA k2+ o
the West.
But the phase speed in the
y-direct ion
% (13,5)
c Y n
can be positive
i.e.,
on the
sign
-6 -
m
BoA k2+ o
or negative
depending
on the orientation
of the wave
vector,
o f ~. n
SLOW VARIATION OF THE WAVE AMPLITUDE
If A is a small and spatial
parameter
variations
that
is a measure
of Rossby waves,
of the slowness
}f can be written q0
of the temporal
(instead of (13, I)):
55 where
~o= ~ea~{Ao(T,x,Y)exp[i(mx + n y and
T=At,
X=Ax,
Y=Ay
are
the
st]}
a~o~m variables,
while
t,x,y,
are
the
variables. The function xl(t,x,y;T,X,Y)
is solution of the order -A equation and the
calculation of X1 presents us with an apparent dilemma. The right-hand side of this order -A equation is, as far as its dependence on t, x and y is concerned,
oscillating with the frequency of homogeneous
solution of the equation for Xl. That is, for a fixed m and n the forcing term on
the right-hand side
of the
order
-A equation for Xl oscillates
at
the
natural frequency of oscillation of the system. In
a
manner
precisely
analogous
to
the
resonant
forcing
of
an
undamped
oscillator (see Appendix 2), the solution for X i would then grow linearly with t, i.e., would contain a secular growth with t, in which case X1 A --= Xo
O(At) ~ T
so that in a time t=O(~), for which T=O(1),
the second term in the expansion
(13,6) would become as large as the first. Of course it is precisely for times t=O(~) that we wish to discribes the evolution of the
w~ve
amplitude,
so we
must insist that the expansion (13,6) remain valid for this length of time. To
accomplish
right-hand
side
this
we
of
the
must
remove
equation
the
for
resonant
X1 by
forcing
insisting
that
term
from
(elemination
the of
secular terms):
8A (13,7)
S.
aT
o
2~ m + E/BoA R
.
.
o
m2+n2+ ~/BoA °
8A
o
aX
2~ n + E/BoA .
R
m2+n2+ ~*/BoA°
and that equation (13,7) rules the dependency of A and Y. The vector form of (13,7) is simply
(13,8)
where
aA S° aT
+ (~.~)Ao= O,
0
o
8A
0
0
,
aY
on the ~
variables T,X
56
g
g
g
~=89 ~i
@9 + ~-qj ;
(13,9)
2
2
m - n - --ILUBoA0
ug- BOAo (m2+ n2+ ~/BOAo)2; 2mn v g - BOAo (m 2+ n2+ g / B ° A ° ) ~.
Therefore A
is constant for an observer moving with velocity 9 , or :
0
g
(13, 1o)
Ao: Ao(~ _ Tt ) Sg
where ~ = X~ + Y2 is the position vector in the (X,Y) plane*. To the first approximation the envelope of the Rossby wave packet moves with g
, which
is called the gnaa~ a~_/o~/~, the X and Y components of which are
given by (13,9). In contrast to the phase velocity, vector rules of projection.
the group velocity does satisfy the usual
It can be verified directly from (13,9) and (13,2)
that: a~ R U = -g Om
a~ and
v g
R
On
(13,11)
~ =
a~ a~ a~ n ~ + R~ m am 8n a~
w h e r e a__ d e n o t e s
the
vector
g
gradient
with
respect
to
wave number
.
Since (13,11)
~
where v *
The
R
is
the
chief
of
the
the slow
enriches
our
however
give
tions
i£
ciently for
speed
virtue
separates
o£
larger
,
R
of
the
the
problem
a
applied the
What to
of
9
o£
for
the
is
much of
the
In
Rossby
scales in
is
space
the
more
present
whlch
nonlinear
the
is
of
crests
it
time to
the
the (13,7).
method
methods Rossby
are waves.
problem
method
higher-order
the
the
in
systematically
from
problem to
that
heuristic
interaction
that
leading
derivation
important
wave
and
approximations
systematic for
the
multiple
dynamics
variations.
problems
problem
o£
local
-scale
device
advance
method for
understandln
desired.
example
= kv
R
It
only does,
correccan
be
effi-
Inadequate,
57 direction of the wave vector,
(13, 13)
~ + k (gVR VR a~
(9 ~ (gVR ~-~(kv R) : ~ VR+ k (9~
:
Hence unless the phase speed is independent of ~, the group velocity and the phase speed will be different in both magnitude and direction. Waves
for
which
~ ~
v are
g
called
R
" ~ " ,
and
the
~ox~a/~
u~u~e
is
BAROTPOPIC CASE
The l i n e a r i z e d
(13,14)
form of (9,14)
is
@•t_oAom"'qo _• B
(9.,qg -- 0 -~-~l}J. + s(9×
+ (9 f p2
(9~'qg
But on the flat ground we have, from (9,16),
•
the following linearized bondary
condition : (13,1S)
(gR' 1 To(l) (9 q----~g+ - at Bo Ko(1 ) (gt
qg
= 0, on p=l.
Integrating (13,14) fpom the level p=0 (infinity) at the gPound p=l yields
(13,16)
BoA°
(~2R~g)dp +
qgdp +
(gx
-0
~ LKo-75T-~-pJ
If we suppose that on the upper limite (p---W3] of the atmosphere,
= when p---)0, we
have :
(9M'qIg ---)0
(13,17) then we obtain,
(13,18)
'
in place of (13,16) and with (13,1S), the following equation : BOAo
J'~ (~e}{,o)dp+ ~i'° 8.,(gxqgdp = o
In the ~
case, ~me~
(13,19)
R~gm ~b(t,x,y)
tAo/ :
( - - ~o T -~
o
qg ~ - . -, (gt
58 an~,
and
we obtain
from
(13,18)
the
following
~
~
e q ~
for }{b(t,X, y)
(13,20)
AoTo(1) }{b + B°AoS 8x
If the linear wave equation (13,20) is multiplied by ~b' a little manipulation yields
(13,21)
~
+
XoTo(1 )
+ ~.~-H l b~ [ a t ]
~ 2BOAoS
=
in
is
0
which has a direct interpretation in terms of energy. It
is apparent
that
the
term
in the
square
bracket
[]
(13,21)
the
nondimensional form of sum of the kinetic plus potential energy, whose rate of increase with time balances the divergence of the flux vector
(13,22) so
that
b the
energy
[atJ
2BOAoS
@b(t,x,y)
for
the
(linearized) satisfies the ~ a~
( 13, 23)
at
' barotropic
quasigeostrophic
flow
~u~ :
b+~.9=o b
For a Rossby wave packet
(13,24)
~b = ¢bCOS 8
, 8 = mx + ny - o't
where ¢b can be assumed as a constant
during the derivation relative
to t,x
and y . In this case the energy to the lowest order is:
(13,25)
gb=
and as written,
¢~k2~in20
~T__~(1) ~cos2e
+
2
0 0
at any fixed point,
the wave about the oA~eaaq~ ~w~Ix~e 2 (13,26)
so that
<~> b
= [kZ+
1 ]'~ AoT ° ( I ) '
b
varies rapidly with half the period of
59
2 (13,27)
=
2 e
It is the average over the period, the
<~ >, which gives a stable definition of
b
local wave energy and is the appropriate
<~b >" Note that <~b > varies ~ The energy flux vector ~ ,
2
over the packet with #b"
from (13,22) and (13,24) is :
Jb = -#b2~ ~ COS28 - 2BoA S
(13,28)
definition of the ~
,~cos%
o
whose average over a period is
(13,29)
~[_~.~
1
¢b
= g
2BOkoS
•
But for the barotroplc Rossby waves, we have,
-m~ BOAo S
2+ n2+
and z
1
n
2
1
1
U --
gb BOAoS [k2+~ ] 2
V
gb
l
and therefore ( 1 3 , 2 9 ) (13,30)
2~
ms
BoA S 0
k2+
1
can be written as
<~ > = b
b
2
:
gb
Hence the average energy flux vector is equal to the wave energy multiplied by the group velocity. Or since ~ averaged over a wave period, (13,31) ff~
~
gb is
independent of space (13,23) becomes, when
- ÷ ~gb" ~ --~ ,
defined
by
O.
(13,26),
~
c o n ~
~
~
~tLtA
6O 14 . THE ISOCHORIC t NONLINEAR WAVE EQUATION (LONG' S EQUATION)
We consider the equations (5,5)-(5,9), wlth : ~8-=0, (:0-1,
Ro-=00, and Re-m ;
the corresponding equations are the usual Euler's equations and we can write the following system, with the dimensions : Du~
p ~ - + ~p + pg~ = 0 ;
•
Dp N +p~.u*=o
;
(14,1) p=RpT D~
where s=c v Log The e q u a t i o n s the
limit
=
p/p~
0
; ,
, ~= c p /c v , for a perfect gas .
of ~
m o t i o n may be o b t a i n e d ,
: ~
(c =0(1)
&u/
p
c --~).
The
v
formaily,
formal
~
from
limit
(14,1),
in
provides
an
a/iswer: S
(14,2)
M~ls
--c Logp , T--~T --P
is
i Ds Dp ~=O--~D-E=
C
p/p ;
1P
~. u ~ = 0 .
0
The resulting equations are the equations for the isochorlc motions :
v~
PD~ + (14,3)
+ pg~= 0 ;
~p
~..*=o
;
Dp b-E- = 0 . For the a/ead~ flows the equations (14,3) are written in the following form :
{~
2
^~^~÷!~+g~=o p
,
(14,4)
~.~=o
,
~.~=o,
where
q = I~1. t
For
the
full
discussion
of
isochortc
motions,
see
the
book of
Yih
(1980)
.
61 FUNCTIONS
OF SPAGEGURRENT
REPRESENTATION
In
the
~ AND X •
OF T H E M O T I O N B Y TWO E Q U A T I O N S
general
case
of
motion
which
is
IN @ A N D X •
steady,
three-dimenslonal
rotational t ,we are led to introduce two stream functions.
and
It is evident that
the equations of continuity in the system (14,4) is satisfied by putting:
(14,5) The
~ = ~¢ A ~X ;
two
stream
functions
¢
and
~
constituting
the
generallsatlon of the notion of function of plane stream.
three-dimensional The t h i r d
equation
of the system (14,4) gives us then: (14,6)
p = p(¢,x),
that is to say that the specific mass is ~
throughout the lenght of
each stream line. It is well known also that one of the integrals of the system (14,4) is the
equo/zon o~ $ ~ : 2 q + ~ p + gz = ~(@,z)
(14,7)
the function ~(@,X) b e i n g
,
itself also conserved throughout the lenght of each
stream llne. Using relations (14,6) and (14,7) the first equation of (14,4) is written in the form P
~A ~ A ~ :
(14,8) or better
~$ + --2~o , P
still
u ~ ^ ~ A u ~ = ( a e + p ap
(14,9)
= (ae~+~2Pas-~P~ )~" t
See (~
the
book
o£
~ ~ / t ~ ( ~
Heidelberg.
Zeytounlan .
(1974)
Lecture
Notes
: ~Q~g& in
~tt/t ~ g ~ @ C O J ~ t ~
Physics,
vol.27.
~LOJ~t4~
Springer-Verlag,
62 If relation (14,5) is taken into account we obtain on the left hand side of the equation (14,9) a double vectorial product:
(~@ A ~X) A (~ A ~) which can
be put in the form
The
last
two
relatlons,(14,9)
and
(14,10),allow
the
two
following
scalar
equations to be written:
8~
P
8p
(14,11)
These two expressions ( 1 4 , 1 1 )
are the two other first integrals of the system
of equations of the isochoric steady motion (14,4). We state,at once, that the arbitrary functions ~ and p must be determined from boundary
conditions
at
infinity
upstream
far
flow
non-disturbed
by
the
obstacle and supposed known. Here
we
suppose
b ~ o - ~
that
(in
the
altitude, denoted by z (14,12)
w
at
planes
upstream
(x,z))
and
the
non-disturbed
uniquely
flow
is
a
function
of
for
u~ e n a b l e s
it
at infinity upstream. Letting
W
= u
infinity
co
w
be the speed far upstream of the obstacle,we will write that: (14,13)
for x--> -m, u=U en (z),
v=w=o .
DETERI~INATION OF THE FUNCTIONS ~ AND p In the
framework
of
hypothesis
(14,13),
the
to be written that at infinity upstream:
(14,14)
O = -
(~)d~ -= "0
,
relation
(14,5)
63 z
being, therefore,
fiow.
In
this
the altitude of a stream line in the basic non-disturbed
particular
case
the
second
stream
function
X
at
infinity
upstream is a plane X = y = constant.
(14,15)
We will suppose implicitly that the solution of the problem considered ought to be uniformly bounded at all points of the infinite plane (x,y). We mention that @ is uniform function of z ,but that z (@) is only a uniform function of while U ( z ) If,
is strictly positive.
now ~r=O determine
the wall
of
the obstacle,we
will
have,
consequence of the nature of the flow at infinity upstream,
as
that
a direct
in all the
region occupied by the moving fluid = ~ (¢)
(14,16)
and p = p ( ¢ )
In this case the first equation of the system (14,11) becomes homogeneous: the stream surface @=constant being then also a vortex surface,
and the second
equation of the system (14,11) has a second member,a unique function of @ :
d~ (14,17)
-(
p dp®
~ + -2 d@ pm d@
).
If we note that from relation (14,7) for ~®(@) we can write : p --m P®(~)
2 $ (@) - (~ + gz) ,we will obtain, in place of expression (14,17),
(14,18)
1
dp~ q2
p (@) d¢
(2 + gz)
1 dd@ (P~$~)" p~
But the second equation of the system (14,11) in taking into account (14,18) as also (14,14) and (14,15) gives at infinity upstream
1
d
p--~-~ (p~)
1 -
U2
dp~ (oo
P~ d~
2
+
gz®)
dU -
dz
Finally, we will obtain in place of (14,11) the following system of equations for @ and X:
64
P~
(14,19)
d~
2 1
~
d~
P ~ d@
2
- U ~
The two equations
+ g(z-z (@))
.
(14,18) are the generallsatlon of the tbree-dlmensional
of the equation
of Long
(19S5)
two-dimenslonal,
stratified,
obtained
for a plane
stream
function
case in a
incompressible flow.
PLANE FLOW:LONG'S EQUATION
If we suppose in system(14,19) @=@ (X,Z) p
and
that
Zmy
we will obtain for the plane stream function @p(x,z) the classical equation of Long :
a2@p
+
@X 2
8s~
p
1
[ (81~.___[2+p)( al~._~p )2 ]
1
----~ + 2 ~ aZ
aX
aZ
(14,20) - U
where Let
dU
cogo d@p ~
U2
1
__dP~ { --~ + g(z-z (~p))}, d@p
2
U Cz (@p))mU C%) z
+ ~(x,z) represent the altitude of a stream llne in the disturbed flow
in such a way that the expression z - 8(×,z} = z®(~p) remains constant ~long the length of the stream llne (see the Flg. l below). t
This equation
has been d e r i v e d a t
first
by D u b r e l l - J a c o t i n
i n 1935.
65
Z
H=o
"upstn:am infinity"
- ~(~x-,I") / downstream
|
-LP..
:~,,.
+L~
0
~: .'eee ~
,
z
p/~7,.
We obtain, in place of the equation (14,20), for the function 6(x,z):
026 026 - -
Ox 2
+
,
+
~
2~-a6
Oz 2
[(06)2+5_~(o6)2]~az JJ dzd-~-(L°g(U~P= ) ) , : o
(14,21) g
dp=
--6, U pm d z 2
where zm
z ($p) = z - ~ ( x , z )
Long(1955) has remarked that the nonlinear terms in equation (14,21) disappear
66 if: (14,22)
U~pco=constant, dpco _ constant . dz co
The equation which results from this reduces to the equation of Helmoltz: a2~ (14,24)
a2~
-- + + cr2~ = 0 , ax s az s o
with 2
(14,24)
Cro
g
=
D epm
-
dZ
constant .
The dominant feature from the mathematical point of view is that the linearity of
equation
(14,23)
perturbations.
is
not
related
But an important
to
any
one
difficulty remains:
hypothesis
it
is that
of
small
the boundary
condition on the wall of the obstacle*, (14,25)
z = h (x) ~
~(x,h (x)) = h (x) ,
P
is n~xt linear
and
cannot
P
be
P
linearized
without
invoking
the
hypothesis
of
small disturbances.
15 . B O U S S I N E S Q ' S
TIIREE-DIMENSIONAL
LINEARIZED
NAVE EQUATION AND
RESULTS OF THE CALCULATIONS
The full system of equations (14,4) is nonlinear; small
perturbations
is
to
linearise
this
the purpose of the theory of
system
disturbed flow does not contain strong perturbations
by
supposing
that
in relation to the basic
non disturbed flow. We introduce the perturbations p', p', u', v'
and
w'
the corresponding hydrodynamical element and we note**: *
t*
z=h
P plane
(X)
is
of
the
(X,Z)
assume,
We
recall
trace
that
a
for
co
+ P~g = O.
the
obstacle,
z=h(×,y),
three-dimensional
in the
y=constant.
priori, at
dP m
dz
of
that
infinity,
all
the
upstream,
perturbations Z --Z
00
and
are
the
of the same order.
We
of
67
p = pm(z ) + p'(x,y,z) ;
(15,1}
p = p (z®) + p'(x,y,z)
;
u = U(z®)
;
+ u'(x,y,z)
v = V (z) + v'(x,y,z) ; w = 0 + w' ( x , y , z )
,
and the basic motion at infinity, upstream,
having one speed
We obtain, from equation (14,4), taking into account (1S,1) and neglecting the higher order terms,the following linear system:
p®(U® aU' ~-~ + V® aU' ~ + p®CU a v ' + v® a v' + ~--~ ~--~
dU
+ ~--~ - 0
dV
ap'_ +~-~-0
dz (1S,2}
aw' zz..+~xV
p (U
U
axaP'+ v
®w')
aw' ap' -~..ay) + - ~ +
aV' aW' ay + ~ =
gP'= 0 ;
d Log Pm
+aP' ~
dz aU' ~÷
8p' -
®w') dz
0
prow' = 0 ;
'
dz since ~-~m I. We note that:pu'=u,
pmv'=J, and p®w'=w; from the first two equations and the
last equation of system (15,2) we obtain an equation in w and p':
(u a
~+
a__) d Log p.
V®
8y
(1S,3) =
-
(
~-
dz a2p , a2p , + --~) . (ax 2 ay
a~) 8z
d U @~ + ---ax dz
÷
dVco aw ay dz - -
68 We
obtain
a second
equation
relating
w and
p'
from
the
third
and
fourth
equations of system (15,2), which gives us:
p~ _
d Log
(15,4)
g
w - ( u ~-~-+ a V
dz
= ( u a~-~+ V L a y) ap' a-z "
~)2 ay
By eliminating p' from (15,3) and (15,4) we obtain the following equation in
(% a
~)2
p2~ + a2~ + a2~
+ V ay
Lax2
ay2
] _ g d Logp~ p~ + a~
az 2
Lax2
dz
ay2J
- ( U a~-~-+V a---)[d2U~a~ + ----d2V' aw ~] ay I~z 2
(15,5)
=- (U a
~-~
dz 2
a_..) d { ay
+ V
d-z
ay
a (U
~-~ + V
8__) d L°g P,~ } ay
dz
w
We
obtain
.
BOUSSINESQ'S APPROXIHATION We see the case where
(15,6)
p (z) = P0exp(-~z)
,
and we pass to the non-dimensional variables:
~ = Rx , n
= ~ ,C _ z
0
0
0
and to the non-dimensional speeds U U _ m
where
U°
is
a
equation ( 15, 5)
V ®
U0
~o
constant
'
V_ ~
speed
®
V0
oo
'
~-
0
PoU~
characteristic
of
the
flow.
from
69
~_)2 p ~ + a ~ + ~ 2 ] + ~o P~ + ~2~1 + V= a~ ta~2 a 2 a~:2 La~ a2j
c% ~
- c% a
(1s,7)
a_)
= 2~.0(% a
a=
d~% a=
C% a~
am) d
a~)
where
~) =
(15,8)
~H z
g
_
_
o (uO)2
o
;
/3H o
~.-
o
2
o0
We observe that relation
2k
(15,9)
o _ Fr 2
U° =
, where Fr -
~o is the Froude
number
troposphere
Fr 2 will
terms
2k0, in
with
relation fact, the
(vertical)
be of the order (15,7),
to the other
calculation Hence,
be
pm(z=)
are
0
10 -2 -
neglected
of Boussinesq
If H is the 10 -3
as
neglected
first
(15,7).This
ecce4~
of
the
that
the
approximation
in
, which
a
(for isoehoric
thickness
approximation
motion)
when
shows
they
and
is,
in
in this case
intervene
in
the
of
an
of the force of Archimedes.
if we
asymptotic
of
can
o£
flow.
terms of the equation
the approximation derivatives
for our
again
seek
development (~ =
we will obtain for
O~ 0
a
solution
of
equation
+
~
0
OJ + 1
in
the
form
...
~o' as a first approximation,
(0= a
(15,5)
of the type
a_._)2
the following equation:
a2¢°0+ a2~o+ a2t~o
a2Wo+ a2Wo
(15,10) =0
.
70 PARTICULAR CASE
When U m U ° = constant and V°m 0 we obtain,
in place of equation
(15,10),
the
equation
[0. 0. a. ].Oo
(lS, 11)
_
+ __
8~ 2
which
has
been
+
+ 9o
an 2
8~ 2
investigated
by
[o.o ]
8~ 2
Kibel
o+
8~ 2
(1955),
o
= 0
an 2
Wurtele
(1957)
and
from
the
Crapper
(1959). We can again seek ~ in the form: 0
~o = ~oo(~)exp{ i (k~+~n)} which gives for ~oo(~) the following equation: d2~ (15,12)
oo d~2
+ Ao(C)~oo = 0
where k2+~ 2 Ao(~) = 9 ° (15,13)
(%k+L~)
(k2+~ 2)
2
(d2OJdC2)k + (d%TJdC2)~
-
Uk+V~ this
equation
equation different
(15,12),
wlth
(15,13),
which
originates
general
(15,10) has been obtained and analysed by Sawyer (1962) and also by a method by Veltichev
(1985}.
CONSIDERATION
OF THE GENERAL EQUATION
We introduce,
in (15,10), , Goo(C) = U .° v / U ~
(15,14)
+ V ®2
V tILn(x
(C)
-
m
O
,
(15,10)
71 and let da
icE)
( dE
=
2
co
dmG
1
dE
)
co
(15,15)
d2~
dG
dE2
The equation
2~
d~
dE dE
(15,10) may be written in the following form: a2~
(cos ~
B
a
~
+ sin am ~ a
(16,i6)
82
The
solution
8~ z
a2~ o+
o + A(~)~o}
an 2
@E2
8~
8~
__o)
o
+ sin ~co ~-~ )(sin cz -- - cos a~
co a T
82 o+
of equation
non-dimensional
8~2
a2~ o+
a
+ B(E)(COS (xco ~
+ D(
)2 {
8n z
o) = 0 .
(15,16)
parameters
will
be a function,
therefore,
of three
A(~), B(~) and
g D = ~ H2 - m DCE) • o G~(~)
C15,17)
SOLUTION OF THE EQUATION (I5,11).ZEYTOUNIAN'S WORK
We will write the boundary
conditions
(15,11);
considering
the
altitude
quantity
of the same
order
a~' I for C=0 : ~o=a-~; (16,18)
with h' (~,n) - H1 h' (Ho~,Hon) o
with equation
the
also
relief
as the hydrodynamic
conditions:
for E:I :
of
which must be associated
~o
,
0 ,
h'
as
being
perturbations,
a small
we obtain
the
72 and if suppose
that the tropopause
can be represented
by a rigid horizontal
plane found at an altitude H . o
Now the new non-dimensional
variables
(15,19)
Y = aoW
X = aO~ .
with a =H /L 0
0
0
and L
funct ion
o
an horizontal
lenght,
are introduce and the new unknown
a~
(15,2o)
.
= ~o-(l-C)ao_ __
~(X,Y) = ~I h' (LoX,LoY).
where
0
We obtain
for .(X, Y, ~)the following non-homogenous
(15,21)
{ aS(o0--~2 +
8X2
0--~2 a~ } 02, + ~)(as, + O s , ) = - ( I - ~ ) , ( X . Y ) ayS )+ 8~s 8X s o 8X s ayS
as a2 [as a3(~ + :D 8(~ ] with ~(X,Y) - ao(ax - - s + a-~) o aX 3 o a-X The
solution
.(X,Y,¢)
equation:
having
to
satisfy
" also
the
homogeneous
conditions:
(15,22)
Q=O ,
f o r ~=0 and ~=1 .
First we develop (1-~) in the interval (0,1] In serles of sln(n~): 2 1-~ = ~
m 1 ~. ~ sin(n~)
, ~e(0,1]
n=1
and seek the solution . in the form :
(15,23)
n =
nn(X.Y)sln(n=~) n=l
.
.
,
bondary
73 which satisfies conditions(iS,22). a2 (15,24)
Assuming
a2
equation
(15,24)
n
82Q
a~(ax2U + ayZ----r)
that ~(X,Y)
We obtain for Q the equation: 82Q
n + (~o_n22)
in
the
n + ~
aX 2
is a symmetric
aX 2
function
semi-plane
a2Q 0
n -
8y2
2
n~
~(X,Y)
in Y we seek the solution of
{-~<X<+~
,
O
by
imposing
a
condition of the type : 8Q (15,25)
n = 0 , for Y=I .
aY We construct
solution of equation (15,24) satisfying also the conditions in X 8Q
(15,26)
Q
_
n -
n
For this reason,
8X
82Q
83Q
n 8X 2
_ n #X 3
> 0
,
for
in the (X,Y) plane we construct,
X -->-m.
between the straight
lines
Y=O and Y=I, q-I equidistant straight lines : t
Y = ~- , j = l , 2 , . . . , q - 1 , J q and we develop the function ~(X,Y) as a limited Fourier series in relation to a~ Y j , by s u p p o s i n g t h a t ~-----~ O, f o r Y ---* Y ml : q (15,27)
J
(X, Yj ) = ~j(X) =
~
^
• (X)cos(~j)
m=O
with 1 ~(x,r_)cos(mn-~) + _1 ~ + (-I)" am(x ) = 2_ q -~. q r:1 q q q o q
We note that : g (X, Yj) = Qn(X, j)~ -= [~ j(X)
and we seek the solution Qnj(X) in
the form : q
(15,28)
hi(X) =
~g m=o
(X)cos(~j)
q
;
74
then,
after"
82fl ~ ay 2
expressing
a2"n] ay2 [ j we o b t a i n
(15,29)
for
Q
nm
J
b y a sum o f t h e
type
[15,28)
:
q~ ~ sin2(m~). (X) cos(~j) q
m=O
(X) t h e
d4Q a 20 - -dX - 4
I
[
following
n.
differential
equation
:
d2~
L°- n2~2 - a20q2sinR(m~']q'-)j dX 2nm
+
_ q2sin2(.....jg_m~_ n q
0
= _ 2__ t~ . xn m
nm
We will consider the following two cases : (I) 9 >0 : this Is the case of a ~xza~e stratification of the atmosphere; 0
then
obtain
for" f~ (X) t h e
n
solution
:
(X) rim ~na2Q__nm onm A
(15,30)
exp[-Xnm(X-X' ) ] ~ (X')dX' m
+ ~X
exp[Anm(X-X')] ~m(X')dX'
+ 2 nm ~nm
with A
(is,31)
=
/
{-
1 2
~
~
am
(XX')
~o -n ~ -aoq sin
nm
I ~nm =
sin
m
(X')dX'
L~-J
aR 0
'
~o -n x -aoq sin L~--I ,. +
2
ao
Ohm
=
a o2
q
+ -qZslnZ(m~)~) q 0 a o2
'
we
75 (II)
~ <0 o
stratification;
:
this
is
the
we o b t a i n
(x)
-
for
case ~
of
(X) t h e
an
atmosphere
solution
having
an
:
_ exp[-pn (X-X')] ~(X' )dX'
~na~RnmP.-
÷nn
+ ~Xexp[Pnm(X-X')1 ~'(x')dx' (15,32)
÷W
- Pnmun~X . exp[unm(X-X')]~m(X')dX'
I
PnmV~X n. exp[-Vnm(X-X')]~.(X')dX' ,
-
with
I
~)
2 2
2 2
2,m~,
=
+ Rnm
;
o
~-- /
(15,33)
i I~~o"~'n2~2~a2q2s o2 in~(~) q m ~nm1 2-
])am
ao
/ [ - 2)0+n2II~a~q2sin2(q) ] 2 R
nm
.
.
.
.
4 +
a2
0
The two solutions obtained
q2sin2(m~)2) o " g
o
: (15,30) for 2)o>0 and
(15,32) for 90<0 evidently
satisfy the condition at infinity upstream : d~ _
nm
d2~ nm
dX
--
rim_
dX 2
d3n
nm
~
O,
dX 3
for X --~ -m , whichfollows from (15,26) . ANALYSIS OF THE SOLUTIONS
In the first
case, for
which are well defined:
(15,30) AND(15,32)
9 >0 we have systems of stationary waves of two kinds o
76 (i) waves whose amplitudes decrease exponentially as they move away from the obstacle; (2)
sinusoidal
periodic
waves
which
are
uniquely downstream of an above the obstacle;
caused
by
the
obstacle
and
these periodic waves, which do
not decay give rise to zones, downstream of the obstacle, with vertical speeds alternatively positive or negative. These zones being,
in general, perpendicular to the basic non disturbed flow,
come from infinity upstream (in the planes ~=constant parallel to the ground). In the second case, when D <0, the solution (IS,32) o
is uniquely comprised of
waves of the first type. The solution for the perturbations,
w', of the vertical speed will be obtained
by the formula o
(15,34)
w' J
In a pratical but
it
is
P°U°° (1-~)a ° ~-~ + ~ Poo n=l
~.f~nm(X)sin(n~)cos(m~ ) m=O
way the obtained solutions may be realised often
preferable
to
apply
these
.
in different
solutions
for
a
wuys, simple
obstacle,deflned in such a way that we can easily calculate(analytlcally) integrals which comprise the solutions obtained in (15,30) and (IS,32). are treating the linear problem here,
the
As we
the sum of these simple solutions will
give the more general solution for a more complex relief. We will therefore write that
~(x,Y) = Z s(x,Y) !
and we will take ~i(X,Y) in the form of a paraboloid of revolution having its summit at the point ~=~0and a base radius(on the plane ~=0) equal to l/q, that is to say that I ~I(X,Y) = Go l-qe(X2+Y 2)
i for X2+Y2< -
(lS,3S)
;
q ~I(X,Y) = O, for all the other values X,Y .
We give here the formulae for the caculatlons in the case of a typical model with only a single intermediary level in altitude (~=1/2). In this case we can write,
in place of solutions
(lS,30)
(for ~ >0), o
the
77 formula: dF Xm
---
exp[-k
_ 2_
aoQm
_
(X-X')] m
(15,36)
f
-2
coS[~m(X_X, )]
-co
+ co
dF
m dX' - [ exp[+Am(X-X')] ~ dX' dX' JX dX'
}
m dX' dX'
,
with q
g=~
'
J= ®=o a~
F(X,Y) m a ° aX where Fjm F(X,q~) If for
q ^ F]= m=o EFm(X)c°s(~J),
,
.
j=l,2 .... q
, F m O, J
then dF
there
will
remain
(X). It is also evident that
m 9 O, for X--~ dX Using formulae (15,35) for ~I(X,Y) we obtain:
only
co
m
F(X,Y) = -2aoq2~o X, and F
~_ m
Therfore - dX
m
o
q-- = -2aoq~oX"
= -2aoq(o m K ° and we denote by M the quantity
2 --
-
4q ~o
-
K
2
0
E
a
aoQ m
- constant.
o Qm
We have that
I '[
2
Qm= A2m + ~
m51. j.
Qm + ~Do_8_ a q 2 s t n 2 mn q
"
'
the
term
Fo/q for
78 We n o w p e r f o r m the i n t e g r a t i o n obtain respectively
(15,37&)
in r e l a t i o n to X (see the s k e t c h below),
for Zm(X) the f o l l o w i n g f o r m u l a e
~o
8q
Xm=
a,
A X
--
A
e m sh(-~)
a n d we
: I
X ~= - -q "'
'
0 qmXm
Zm:-
(15,37b)
8q
ao
~o
{-~lul/qSh(;llln x)
QmAm
e
Xm" ao Qmkm
(15,37c)
0
x •-
These formulae
- I/q
-
2
~X
I*.
In[~* (X+ !)
[ m
mcin
0
÷ I/q
x•÷
we o b t a i n for zm(X) the f o l l o w i n g
Zm=
2 ^
sin[ ~(X-X')]
11,-_ jj'
I :; X _~ + _'
q
cos(#X)
P'm
m
~
q
(15,37) are u n i q u e l y v a l i d w h e n m~O a/nd m~q.
case f o r m=O and m=q;
(15,38)
~
M ~.
r
I/q
-
q
'
X a
x
.'-
Let us look at the
formula
(for 9o>0):
Fm(X')dX''
t8., / : D 0- 8 0
w h i c h g i v e s us the f o r m u l a e ~(m- 0 ,
(for m=O and m=q): 1
w h e n X < - q- ,"
^
(15,39)
,6
Zm-- - - - q ao
A3 sln
#/m
^ cos
q •
8q ~;o
8q ~o
0 ~m
0 ~m
Zm= a - - ^2 -- X - E - A 3- - s i n
-- X
,
,
when X z + - •
q
I~( X + ~1)
q ' ,
when
-. q
I/q
ao ~ m with
+
'.
- X-< + qi . - q! <
79 RESULTS OF THE CALCULATIONS fl
In Flg. 2 we have represented the range of vertical speeds p w'/PoU~(oOn the plane (=I/2 The values of the parameters are the following:
I Ho= 8.103 m, Lo= 96.103 m, ao: 1~; ~o=0, I
and q = 24 ,
which gives us a parabolic of 800m in height and 8.103m in diameter. U°=l?m/s,
~ mO
, the
basic
flow being
in the
direction
x>O
has
Finally a stable
stratification, with g ~ l O -3 I/s 2 ,which gives ~o=25. By superimposing typical solutions corresponding to an obstacle in the form of a paraboloid of revolution we can obtain ranges of vertical planes z=constant above varying sites; made above the region of Cantal
in particular,
speeds
in the
calculations have been
in the Massif Central of France and for the
region near the Basin of Arcachon, which, according to Trochu (1967), predict the distributions of rain distributions given
in these regions very well.
by Trochu are very
fascinating
The
calculated rain
indeed
(see,
Fig. 3 and
Fig. 4). We
note
that,
stratified (1969)
although
flow
is
over
linear
barriers
more s y s t e m a t i c
and,
of revolution
For
of
calculation
represented reliefs the
the
coast
sea from the of
represented each
land,
the
of
has
represented
For the to
calculations
above
form
of
the an
algebraic
of an obstacle
the
role
to
case
region of
investigated
the
by means
that
that
barrier
due t o
and have
region
( F i g . 4)
of
simple
relief
obtained the of
we h a v e
typical
simple
which separates the
sea.
influence
( F i g . 3)
of
coast,
the
Cantal a
sum
the
of the
the
work
results.
plays
relation
for
Arcachon Basin
we n o t e
in
three-dimensional Zeytounian's
difference
With
in
coast
of each paraboloid the
sum o f
relief the
the
of
type
the
these region
(15,35)
with
o f Go a n d q v a r i a b l e s . calculations
conclude
region.
the
the
in
been
his
give fascinating
zones
the
steady before,
b y s u c h a sum, we h a v e c a l c u l a t e d
point
for
solved
of revolution;
land
influence,
values
in
as paraboloids
roughness
at
rain
been
in any case,
form of a paraboloid the
equations
have
that
they
on the Cantal reflect
quite
( F i g . 4) well
a detailed the
local
analysis structure
has enabled of
rain
us
in this
80
v fO g 8 7 6 5 3 2 f o -f -2 -3 -¢ -5 -6 -7 -8
-9 -~0
81 N
f
o~
0
o
,.~
~.
o,s
o
3 : We~ ~
sta2~£e a
~
•
~#
~ ;
~ :
~"
""
"
•
0
.~w.o,cAon Sa,~.n ~ ~ = 240 ° and (e
naZcAed.
~ = 50. 0
a
82
.'.'.'." •° ' . ' , ' . °
/
i
::.:':':':'I /
,.......:.:.:, ..o.o.-.-o%-, ::.:.:.:.',:,:: .:...-.-......
i:i:!:i:i~
iiiii!iiiii !
~\(-
±::
:i:i:i:i: ::::::::: i:i:!:i:i
/~JPJU..~ I..
/ ',
::. ::~:
:L
T
".','.'.'.'.
"':':',:.:-
/
~
:
I
l
f
'
p#F.N~r~AT
',
:::::
~
,/
!
I
/
.'" : : :.
l
," ~
/
I
p
~
,;
l
e
atAs~-sE
I
,"
~,
"...... "'""""
~
~
:i
~ ~
"~/'~--
IIUL,~
..~.~
~._~__~
..':':':':'"'.':..~
:':'."'."':':':':'p
..:::~:::i':" • i.:,,'~:.."...-."
I
i ~/~
/ I *
.:
~)l ."::'" " ". I i """'":'""" • :.:::.'...: :.'~::'.. :'/"'::::::::" . : . :: ',/ / ]e ":':'::,: I
,
"('~[.~. / "7..:.> , .'.
.'.'~:'/I
I
~
I
~
,::..~..'.?.'.:.: :i~."~::: :.'.',;'...'...-. ,'.', ..".'"
."
"~""'/"-':""1:':':'
I
~.
4 • w~
" "
---.e
,y~
OA~
a ~
~e.r~t,'~.o.~(~o.n.to.~); ,~ =270 ° ~ oo
~
~
*
~o.a~.~.
~ =BO. 0
83 BACKGROUND READING
For
a extensive
treatement
of
the concept
of waves
in fluids
the
reader
is
referred to:
LIGHTHILL, J.
(1978) Cambridge
Concerning
BEER, T.
the waves
University Press.
in the atmosphere,
see:
(1974) ~ m ~ u ~ v ~ e ~ . Adam H i l g e r ,
London.
and DIKIJ,L.A.
(1969) _ ~&e U~eon~ o~ ~ (in Russian).
o~ //~e ~ a a / A ' ~ ~ Guldrometeo-Izdat,
For the nonlinear aspects of waves,
WHITHAM, G.B.
(1974)
Moscow.
see:
~ w ~ o u t a a d ~ ~ . J. Niley et sons.
REFERENCES TO WORKS CITED IN THE TEXT
BOIS, P.A.
(1976) - Journal
CRAPPER, G.D.
de M~canlque,
15, 781.
(1959) - J. Fluid Mech.,vol.6,pe_vtl,51.
DUBREIL-JACOTIN, M.L.
(1935) - Atti Accad.
Lincei Rend. Ci. Sci. Fis. Mat. Nat(6)
21,344-346. KIBEL, I.A. LONG, R.R.
(1955) - Doklady Akad. (1955) - Tellus,
SAWYER, J.S.
lO0,n°2,
247-250.
7, n°3, 342-357.
(1962) - Quart.
TROCHU, M. (1967) - C a l c u l
Nauk,
J. Roy. Met.
Soc.
voi.88
, n°378,412.
d'un champ de vltesse verticale en m~som~t~rologie
Application.
"Etude de Stage",
Ecole de la m~terologie,
Paris. VELTICHEV, I.
(1965) - T r a v a u x du C e n t r e Mondial M ~ t ~ r o l o g l q u e de Moscou, n°8,
p.45
(in Russian).
:
84 WURTELE, M.G.
(1957) - Aero-revue.
32, n°12; see also:
Beitr. Phys. Atmos.
29,242-252. YIH, C.S (1980) - W b ~ Z 6 ~ / ~ . ZEYT0~NIAN,R. Kh.
(1969)
Academic Presss,
-
~b~
~
~
o~ ~
~
London.
~Aen~u~en~
~
U~e ~
~b~
~
cu~
.
Royal Aircraft Establishment. Library translation n ° 1404, December ZEYTO~/NIAN,R. Kh.
(1974)
-
~o/e~
aa~
1969.
~es ~
~
~
~
de
~bdx~es
~ . Lecture Notes in Physics,vol.27. Heidelberg.
Springer-Verlag,
CHAPTER IV FILTERING OF INTERNAL WAVES We have already noted that the basic model equations (see, the Chapter II) are formulated a view to ~ equations
eco~
for atmospheric
16 . HYDROSTATIC
~
motions,
out of the solutions of "exact"
because
such waves are
of
na ~ % o o n ~
FILTERING
In the hydrostatic approximation, whan e
--~ O, we see that the general wave o equation (10,4) is highly deEenerate, and the consequences of this will be: lira (J) m ~ C
0
--->0
a
and
lira ( J ) m J E
0
g
--->0
ghs
with
i.e., are
_
2
(16,1)
+ k 2 7-1
1
%hs RJ all
internal
severely
slightly
wtll
'
waves a r e f l l t r e d ,
The f r e q u e n c i e s
over-estimated,
the closer
X--~-
acoustic
distorted.
[~_oo~2
but
of
the smaller
be t h e e s t i m a t e ;
nemely,
and t h e
the gravity
is
k (l.e. the
those
internal
internal
gravity
waves a r e longer
in
are
gravity
waves
this
case
the waves),
waves,
which
correspond to k2<>l, remain. The
frequencies
of
two-dimensional
acoustic
waves
(see,
(10,17))
remain
unchanged and this justlfies the use of the primitive equations (7,9) for the descrlptlon of synoptic processes. The reduction of the order of (10,4) from four to two with respect to time t also indicates the non-unlqueness of the double limiting process c ~ O, t 9 0 . o Near t=O we must formulate the problem of adjustment to hydrostatic balance (see,
the
section
19
in
the
Chapter
V).
This
problem
of
adjustment
to
hydrostatic balance makes it possible to solve the fundamental problem of the relationship between the
initial conditions for primitive equations and the
true initial conditions for full adiabatic, nonviscous atmospheric equations. We note that the hydrostatic filtering
in these full equations altered the
86
h y p e r b o l i c character of these equations; and Sundstrom Let
us
see for
note
that
even
in
the
steady-state
case
(Strouhal
limiting process eo-)O leads to the singular perturbation with the fact
that
the horizontal
short
lee waves downstream of the barrier) how these horizontal must obviously the question process
short
internal
long gravity
that
internal
this quetlon
scales.
when
Is very
Co~ 0,
important
the theory of long waves in the atmospheric Finally, about
we
by
note
the
that
the
generation
duration of this process front
of
internal
a~=20km/min) requiring adapt
the
.
and
scattering
acoustic
main
waves
thickness
of
of
remains
unanswered
for a complete
in spite
of
understanding
of
(regional)
hydrostatic internal
to the
motions.
balance
is
acoustic
brought
waves.
the
(with
atmosphere
After
this,
the
sound
(troposphere)-a
the atmosphere
(9,7).
This
The for a speed
process
continues
statement
will
£o be
19 and 21.
FILTERING
, B=O(1)
o
the wave
out as e 9 0 we o unfortunately,
with short gravity waves
It is clear that in the Boussinesq approximation,
Bo=~M
But for now,
traverse
equilibrium
in the sections
in connection (for exemple,
the same as the time required
to
in all.
to the state of geostrophic
BOUSSINESQ
the
0)the
In order to investigate
meso-scale
to
is approximately
only a few minutes
made more precise
17
adjustment
~
waves
gravity waves are filtered
of the process
waves,
number
problem
gravity
are filtered out.
use the method of multiple
of the conversion
wlth
the fact
the work of Oliger
instance
(1978).
equation
when
but M -~ 0,
o is again
(10,4)
highly
degenerate
and as a consequence
of
this: 2 a
@2__, g
(17,1)
Thus,
all
the
~iR~ + C~-l)~2i~ 2
2 = gB
internal
2k2 o
acoustic
+
waves
are
filtered
again,
and
the
gravity waves are severely distorted. The order of the wave equation terms of t is again reduced from four to two, the double
(10,4)
in
the nonuniqueness
of
limiting process: ^
(17,2)
indicating
Bo = BM
^
o
,
B = 0(1)
,
M--~ 0 o
and
internal
t--~ 0 .
87
The problem to adjustement to a Bousslnesq state must be formulated near t=0; this problem will make it possible to formulate the initial conditions for the Boussinesq equations correctly (see,the section 20 in the Chapter V). On the other hand, we see that the Boussinesq approximation is correct only if the characteristic height H 0 of the atmospheric motions being considered
is
such that : a2(O) -
(17,3)
-
~'g
[
ground level. When
e --i
capable
.
.
(0)
=
HB ,
is the speed of sound in the standard atmosphere at the
O
of
_~o a
g~"
i
i.e. , L -H -H
0
B
"I 112
where a=(0)=I~RT=(0) I &
^ u H° ~
>>
0
describing
then
B
the
atmospheric
Boussinesq processes
model
only
equations
locally;
(8,8)
are
therefore
the
behavior of the solutions of these equations at infinity must be determined. But,
in the general case,
a question still remains unresolved:
what outside
equations supplement to the Boussinesq equations and are joined to them via the radiation conditions in the steady-state case? This was done in the work of
Guiraud
and
Zeytounian
(1878)
in
the
steady-state
plane
adiabatic
nonviscous case (see, the Chapter VI of the present Course). Hence
it
is
necessary
to
elucidate
Boussinesq model equations (8,8) infinity; was
the
behavior
of
the
solution
condition
by
Gulraud
(1878)
is satisfied at
who
infinity.
showed
that
the
In section 20
analysis of the problem of adjustement being we not only that
the
(obtained initially by Zeytounian (1974)) at
for the three-dimensional model of steady lee waves,
resolved
of
we find
are not
of
radiation
a comprehensive
to a Bousinesq state,
the Boussinesq equations
this problem
classical for the
the
time
hyperbolic
type. Finally, ability
when the Boussinesq approximation is used to
take
account
of
the effect
of
"correctly",
a change
altitude on the tropospheric process being considering; follows from the fact
we lose the
in stratification in particular,
that for this approximation the full
with this
Eulerian energy
equation (the third eq. of (8,4)) is replaced by the conservation equation:
(17,4)
S
~-~-
B+
--
+
z z
-=0
=
0
.
88 18
. GEOSTROPHIC
FILTERING
Superposed to the hydrostatic
limiting process,
¢--~ 0 (see, the section 16), 0
we can consider the following quasi-geostrophic
limiting process
(see,
(9,3)
and (9,4)): Ki---> 0 and Mo--~ O, with A =l---rKi]2 o ~S[NoJ = 0(I),
(18,1)
the so-called geostrophic filtering.
(18,2)
2
and amongst filtered
ghs
--~ ~
,
the fast waves,
out
nonviscous
after
for the ~
two of them,
Co---) 0
atmospheric
and
(18,1)
the gravity and acoustic
(18,1).
processes
geostrophic approximation The
In this case we have
at
Thus
for
synoptic
forecasting
scales
it is sufficient
ones,
adiabatic
according
to prescribe
are
to
the
initial
value
, only, which satisfies to equation (9,14).
qg
initial
values
limiting process
of
the
other
meteorological
(18,1) and a new initial
fields
are
lost
during
layer result from the double
the
limit
process: Ki---)O and t--->O, and this initial layer describes a process of adjustement the
section
21),
i.e.,
the
adaptation
of
the
to geostrophy
hydrostatic
fields
(see,
to
the
geostrophic equilibrium state -) qg
0
qg
We must add that in the filtered out quasi-geostrophic model we have only one equation system
of
atmospheric adaptation (inner)
(see,
full
hyperbolic
motions. arises
(9,14))
In
for Euler
connection
the
evolution
equations with
the
and for each of the model
of
H qg '
for loss
instead
adiabatic, of
(outer,
the
nonviscous
meteorological
in time)
of
field
equations
the
initial problem of adjustement must be examined near t=O in order to
obtain the "correct" initial values, which are needed for its unique solution. Finally,
we not that adjustement of the meteorological
the generation,
dispersion,
and damping of the ~
fields as a result of
internal waves.
89 BACKGROUND READING
Concerning the problem of filtering of internal w~ves, see: MONIN,
S.A
(1972) _ Weo/Ae~
~
~
o~
MIT Press,Cambridge, ZEYTOUNIAN,
a
~a~m
~
~A~.The
Masschusetts (see, the sections 4-9),
R. Kh.(1982) ~
eu~
a~
~z~e~.
~
Izvestiya,
Atmospheric and Oceanics Physics. Vol. 18, n°6, 583-601 (Russian Edition), and
ZEYTOUNIAN, R. Kh. (1990) _ ~
~
Springer-Verl~g,
o~ ~
~ .
Heidelberg (see, the Chapter III).
REFERENCES TO WORKS CITED IN THE TEXT
GUIRAUD, J.P. (1979) _ Comptes Rendus Acad. Sci. Paris (A), 288, 43S. GUIRAUD,
J.P.
and ZEYTOUNIAN,
R. Kh.(1979)
_ Geophys.
Astrophys.
Dynamics ,12,61. OLIGER, J. and SUNDSTROM, S.(1978) _SIAM. J. Appl. Math., 35, 419. ZEYTOUNIAN, R. Kh.(1974) _ Arch. Mech. Stosowanej, 26, 499.
Fluid
CHAPTER V UNSTEADY ADJUSTMENT PROBLEMS The
basic
approximations
are
formulated with
a
view
to
filtering
acoustic
waves out of the solutions of equations for atmospheric motions, because such waves are of no importance concerning weather prediction. when
considering
equations,
the
approximate,
simplified
set
of
On the other hand,
equations
(primitive
Boussinesq equations or quasi-geostrophic model equation),
allowed to specify a set of "exact" equations.
This
initial conditions ~
is due to the fact that
one is
in number than for the the
limiting process which
leads to the approximate model, filters out some time derivatives. Due to this one
encounters
o x ~
the
problem
and
~n
~
ZAe ~ ,
consistent
~
of
~
deciding
Uw~e ~,
o~e
a~Ao/ ~ ~
e q ~ ?
c o ~
~ The
latter
with the estimates of basic orders
~Ae
one
~
are
mo4t
cond/2/an~ not
of magnitude
in
implied
general by the
asymptotic model. A physical process of time evolution is necessary to bring the initial set to a consistent concerned.
level as far as the orders of magnitude
is
Such a process is called one of ADJUSTMENT of the initlal set of
data to the asymptotic structure of the model under consideration. The process of
adjustment,
Meteorology,
which
occurs
in
many
fields
of
Fluid
Mechanics
besides
is short on the time scale of the asymptotic model considered,
and o/ Ute end o~ ~/, in an asymptotic sence,
u~e o ~
~
~
ZAe ae/ o~
UuUUaZ~~u/2a~e~Aemade& If we consider our basic model equations (see, Chapter II) in such a case it is
necessary
~n,ee,
to
tc a B
elucidate
~
~
the
problems
of
the
~
~a
A q ~
and ~ q ~ t a t c o p ~ .
A number of adjustement problems occur
in Fluid Mechanics being related to
loss of initial conditions as a consequence of loss of time derivatives during some limiting processes leading to a simplified set of equations. We refer to Just
one
of
Dynamicists. function
them
which
is
most
celebrated
and
has
intrigued
many
Fluid
It is the loss of initial conditions for the full distribution
when
one
goes
from
the
Boltzmann
equation
to
the
Navier-Stokes
equations by letting the ratio of the mean free path to macroscopic
length
scale
(1975;
(Knudsen
Chap. V, §.5).
number)
go
to
zero;
see,
for
example,
Cercignani
91 To the best of our knowledge these problems are solved by rescaling the time and
possibly
some
dependent
variables
leading
to
a
so-called
/2d/to/
problem. Depending on the kind of problems, when
the
rescaled
time
goes
to
we may have mainly two kinds of behavior
infinity.
Either
one
may
have
a
tendency
towards a limiting steady state or an undamped set of oscillations (think, for example,
of
the
inertial
waves
in the
inviscid
problem
of
spin-up
for
a
rotating fluid; see for that Greenspan (1968; §24)). For the terminology of the initial layer as adapted to this kind of singular perturbation problems we refer to Nayfeh (1973, p.23). 19
It
ADJUSTENENT TO HYDROSTATIC BALANCE
.
is
obvious
that
the
classical,
Kibel,
primitive
equations
(7,1)
are
obtained through the following limiting process:
f~
---~O, keeping t, z and the horizontal ~sition fixed during the process,
(19,1)
applied to the full equations for the tangent non-hydrostatic,
adiabatic and
nonviscous atmos)heric motions:
S ~ + p
p
N
.
+
=0;
+~y
Co ( ~ A v ~) + _ _
1
9}
taneo ~wl
(19,2)
I + --
~ p = 0 ;
I
o Dt
tan
Ro
p = pT;
S~6
S D~
where
We
observe
that
the
=0,
--- S a~
v + w~-~ v : ul~ + vj. + ~.]~ a and ~
initial
data,
for
equations
(19,2),
need
not
fit
the
92
hydrostatic
b a l a n c e and,
with respect general
in p a r t i c u l a r ,
to the horizontal one,
case,
we
assume
that
at
the vertical so that
the
velocity
n e e d n o t be O(e)
in order to consider
initial
time
c w
is
o
of
the most
order
0(I).
Accordingly we get as initial conditions, for the full equations (19,2),
t=O : v~=~~°, cw=W ° , p=pO and p=R °,
(19,3)
where ~°,W°,P° and R ° are given functions of z and of the horizontal position. On the other hand when considering the primitive equations (7,1) we must give only the initial values of ~ and p, since the initial value of p yield, from the initial value of p and the relation T=p/p yield
the hydrostatic balance,
the initial value of T. The initial values of ~ and p have nothing to do with the
corresponding
(19,3)).
initial
Consequently,
conditions
we
get
as
for
the
initial
full
equations
conditions
for
(19,2) the
(see,
primitive
equations (7,1): -> -)o
(19,4)
t=O : v=v
and
p=p
o
,
-)o v is different from ~o and pO is different from R °.
where
Two of the initial conditions (19,3) have been lost during the process and two questions arises: I) How have these initial conditions been lost? -)o pO 2) How are v and related to ~°,W°,P° and R°? Regarding the first question, the answer is simple. According to the primitive equations
model
(7,1),
p
is related
to
p
by
the
equation
of
hydrostatic
balance, while w, as noticed for the first time by Richardson (1922;
see the
Chapter
of
V
of
his
book),
is
computed
by
the
process
of
solution
the
primitive equations (7,1). All this hold true at the initial time as well. As a matter of fact
if we
consider
the primitive
equations
(7,9),
use of
pressure coordinates, we observe that the two main variables of the primitive model
(7,9) are ~ and T.
The situation
is slightly reminiscent
of
which occurs
in classical boundary layer theory if one considers
deduced
~ by a kind
from
of
divergence-free
condition,
namely
the one
that ~ is the
second
equation of (7,9). We should keep in mind that like ~, H is known, apart from an integration constant, when T is known. Hence, w, the vertical component of velocity
is not
afterwards,
a primary
variable
in the
sense
that
it may
be
computed
through use of (7,7), when ~ and ~ have been computed themselves
from the knowledge of ~, T and p.
93 Now we intend here to adress ourselves to the second question. In the present case
it is fairly obvious that the proper rescaling,
and dependent variables,
For
ist: A
,
(19,S) the
adjustement
the time
, VmV , CW=-~ , pm~ and
to
hydrostatic
balance
pm~ .
problem
we
use
the
following
limiting process: (19,6)
e --9 0 , with ~ and ~ fixed, o
and the horizontal position is also fixed during the limiting process
We n o t e t h a t
through the process
A A A A Vo, wO, Po a n d So f o r t h e l i m i t i n g
Let us set
(19,6).
~=-0(i).
values
(functions
o f ~, ~ a n d o f
the horizontal position),
it is straightforward to derive the set of limiting
bl///x~&z~e~~,
,
~
namely:
[8~
OPo ^
ado
2A 0 -----] + - ~M°P° B~ + o a~J a~ A --BP°+ B_ %~WAo ,] = o , . 8~ a~L J
(19,7)
A
+ SO= 0 "
A
oil ]
=0,
A
(19,8)
A
a~
a~
a~
o + ~ __°
o a~
= 0 ,
if Sml (tomLo/U o) and Boml (HomRT (0)/g) . The system dimensional obtained
(18,7) ' fop A PO' A P0 a/Id ~0' ~
vertical
through
the
solution
motion of
is identical in
(19,7)
the with
to the equations for On~A atmosphere. 0nece w ° has been the
initial
(19,3)): (19,9)
~----0 :
~=-W°, ^ o Po=R ^ 0 po=P,
t We note that the horizontal posltlons not play of rSle in the problem of adjustment to hydrostatic balance.
conditions
(see,
94 and proper boundary conditlons on the ground and at infinity, we may use the A
transport equation (19,8) in order to compute v~° using the inltlal condition:
(19,10) The
A
~:0
: ~
0
:~)o, A
(19,9)
equation
merely
says
that
v
A
0
is
convected
without
change,
vertically, with the velocity wo.
EQS.(19,7)
NUHERICAL SOLUTION OF
The set of equations (19,7) have been solved numerically by 0utrebon (1981) using the slip condition:
(19,11)
~0 :0
at
~
0
:0
and enforcing A
0~°- O, --OP° a~ a~ + ~o= 0
(19,12)
at the maximum altitude z=20km, used in the computatlonal grid t. The initial conditions were chosen to be the standard equilibrium atmosphere, ^ concerning the thermodynamic state, while for w a triangular profile was 0
used,
with ~
being zero at the ground and above z=10km with a maximum at
0
z=3km. Two numerical computations were run; the first one, with results shown on the Fig. S,
corresponds
while
for
the
to a maximum
second
significantly higher,
(with
in the
results
namely 5m/s.
computations was 26km/a®(0),
not
^
initial
value of w
reproduced
equal to Im/s, o here) the maximum was
The unit of tlme used for the numerical
where a {0)
is the speed of the sound at
the
ground level for the standard atmosphere. The process of the adjustement,
to hydrostatic balance is composed of three
main phases.
During the first phase of adjustment typical profiles of which
being
at
shown
direction t
The
numerical
(1973) a
of
which
method
pratical dispersion.
for
t=0.058,
the
vertical
code is
are
the
reducing
investigation
used
by
characterlsed
velocity Outrebon
generalisation dispersion of
and is
of in
constructive
by
several
rather
strong
inspired the
work
convective difference
of of
one Fromm
difference
of
inversion
of
perturbations Lerat
(1969)
and
schemes
approximation
Peyret
concerning and of
the
reduced
the in
95 temperatures and pressures. FoF the second run, here,
a
shock
wave
is
formed.
There
is
a
with results not
second
phase
reproduced
dul"ing which
the
vertical velocity decays to zero, while the temperature end pressuFe pFofiles approach to the equilibrium ones. Typical such pFofiles ere shown for t=0.180. The third phase Is the ultimate phase of adjustement during which convergence ^
to aZeox~ state
is achieved;
we have shown typical
profiles at
t=0.998.
We
observe that, thanks to the unit of time used for numerical computations, the dimensionless
time
t used for the presenting the
numerical
results
may be
identifyed with the one used in (19,7). We note now that the basic enquiry about adjustement to hydrostatic balance is one
of
assessment
whether
asymptotically stable.
or
not
the
model
of
hydrostatic
balance
is
As an indication for this let us consider what may be ^
^
called the ~ a Z L c ~ aAi~ Ao(t) which is a solution to:
d~
d~ o _ ~o(~,~O+~o(~))
(19,13)
The m e a n i n g o f ~ o ( ~ )
is that
in (19,7) and the equation
it
' ~ o (o) ~ o.
allows to integrate
(19,8),
at once the
last
equation
namely:
A
Po ^-~ -= ~o(t,~ ) = zo(~ _ Xo(t)) '
( 19, 14)
Po
where
zo
- -po
, and
(R°) ~
A
(19,15) The
~o(~,~ ) = ~o(~ _ ~o(~))"
Fig. 6,
shows
according
to
0utrebon
(1981)
the
aspect
function of ~ for various values of ~0 and for the same Fig. 5. We definitely see that ~ More
pFecisely
than
that
we
of
~o(t)
as
a
initial data as in
tends to a limit when ~ --) m . o may even say that most of the
adjustement
isaccomplished when ~>0.5 and that the final approach to equilibrium is rather slow. If we let ~ --9 m we find, through matching and adjustment to hydrostatic balance
t~a ~
model,
that
tAe ~
(19,2), ~ z ~ ~
c o n ~
~
W o~o~~~
~
~
®. o
ead
96 We give on Fig. 7, of
~
0
according to Outrebon(1981)
a graph of ~
as a function
0
.
This shows that the vertical shift is a quite slgnlficant phenomenon. other hand it is difficult to maintain that
the
phenomenon
is
On
of
the
pratical
importance for weather prediction according to the primitive equations
(7,1).
One reason, amongst a number of good ones, being that it is doubtful that initlal
condltlons
might
be
sufflciently
precise
for
rendering
the
worthwhile
a correction based on the vertical shift! The best argument for considering adjustment to hydrostatic balance
rests
on
the investigation of stabillty of the hydrostatic model. What the computations by Outrebon tell us is that there is built into the equations a mechanism which drives back the atmosphere to a state of and that the transient tlme tied to this
mechanism
is
quite
hydrostatic less
than
active balance
the
necessary for a sound wave, starting from the ground, to cross back and
time forth
the whole of the troposphere. Finally, we are able to write the following relations for the of primitive equations (7,1), namely:
(19, 16)
~o
o
~"
I o(
A
lim
=
o)
.
The final result may be stated very simply by saying that:
c ~ e - ~ ,
~
~
.
initial
values
97 iI
I: 4~. 4
Z
0~%
a
t
I
I
;
F"
i'
't
I
J
415!' 4~q 4,ii
0~. t
"%
4~c~
o(o oz 2.
o o,~ ~=0
.....
I,~ ,
~=0,058
~.jc,
~.,$
~
,
3j6 £=0,180
qj~ .......
S/L £=0,998
-
98 0"')
~ !
/o~ /o~ t,/
'
0
I
'
.~
I
.~
• ~.. 6; v ~ u . ¢ ~ ~ , . ~ ~
o
!
'
a ~
•
L
.0
'
• ~ . 7- ~a~:~ ~
I
.5
I
I
1
I i
I
'
I
'
t.0 a t the ~
'
.~ ~ ta~
I i I
I
f.~ t . ~
.8
I
~
I
"1.5 ~ ~
I
to
2.O
99
THE ULTIMATE PHASE OF ADJUSTMENT TO HYDROSTATIC BALANCE.
Let
us
examine
now a n a l y t i c a l l y
balance.
We a s s u m e d e c a y
respect
to
according limiting Setting
the to
a
state
the
and study
ultime
the
way i n t h e
assumed
limiting
state
~_aaiq2xt
theory
obtained
and retaining
only
decay
the
towards remaining
behaves by
linear
when
t
the
perturbation goes
perturbing
we o b t a i n ,
^ = po
for
7,
~(~){I+.},
(19,7)
aw
+ #m(~)a._~.~
o a~
o
0_~ + Ow a~ aA
_ 8
a~
[
1 I + ~co(~) o
(19,18)
=
linear
0
version
of
(19,7):
:
<] d~
w = 0 ;
=0; o T[ =
~0 + 8.
From (19, 18) we get the following equation for w: a2w S2--+ o a~2
( 19, 19)
d# ~° a2w Ow ° - - + - - = dz~ a~ 2 a~
tt
0 .
Let us change from w to W defined by W = w exp
( 19, 20)
~
~:(~)
,
we get the equation:
(19,21)
o #oo(~) a~2
a~2
1
4~o0(~)[1
o I" We consider the limit matching
of
2d 'ol W = O d~ j
,
o
0, AP0 and AP0
with the hydrostatlc,
are equal to the initial
+
when
~--->m, with A fixed.
primitive e q u a t i o n s ,
values
of T, p
model, these
Through
~o00, p0A°°and
a n d p according to the p r i m i t i v e
equations (7,1). Accordin 9 to the work of G u i r a u d and Z e y t o u n l a n
(1982).
with
infinity
around
terms.
^ ^~ ^ 00= PoCZ){l+~}, ~o= ~(~){l+e}, ~o~
w, 8 a n d w t h e f o l l o w i n g ,~M 2
St
to
t
(19,17)
Am P0
hydrostatic
w,
the
100 which is
nothing
else
but
the
well-known equation
of telegraphy
provided
that
#m d o e s n o t d e p e n d on ~ . 0
Hence,
#o--1 ( w i t h t h e n o n - d i m e n s i o n a l A
z
and we s e t
1_~
~=E' and
variables),
~=2~
'
o
W(2MoT, 2~) --- 2~(T,~).
The solution of the equation 82~ (19,22)
a2~
- - 2 - X = 0, a~2 - 8~
corresponding t o c o n d i t i o n s :
(19,23)
Z(T,0)
=
Z(T,m)
z(o,~)
= z
0
= 0
C~)
;
;
eX E~-~=o = Zl(~)
is
readily
obtained
through
,
application
of
the
sine-Fourier
transform
(refer
to Titchmarsch (1948)). The solution of (19,22), with (19,23) is the following:
(19,24)
X(T,~) = IoF(~,~,~)~l(~)d~ + ~ I ° F(T,},~)X°(~)d~ ,
with
F(T,},<) = ~ F s i n [ ~
Jo
/ 2÷I
T]
sin(u})sin(u~)du
.
What we need now is an estimation of F(T,},~) when T--+ ~. This is most easily got by using complex integration in order to transform the integral along the positive real axis 0
(19,2S)
F(T,},<) s
3/2 cos(T÷~) T
when
r---)+m .
101 This calculation shows that, when ~--~ +m, w behaves in the following way (the reader must keep in mind that #m~l): o
(19,26)
w s M~o/2COS
We see from aemxz/~% ~
this
last
but
there
+ 4J~3/2 e
formula is a
by
that
w decays
tton ~
(19,26)
to
o/ ~ is
not
an
zero
like
.
We
artefact
(~)-3/2 p/~oAg/x/e.d note
of
that
uniformity
reveled
the
estimation;
it is a c t u a l l y related to the double limiting process:
the
non
asymptotic
lim applied first and then lim .
The
combination
asymptotic is
Outrebon's
analysis
asymptotically
ef///ade an
of
gives
/mnea one
context
the
Chapter
VII).
to
a strong
stable
. As a m a t t e r be
reader
but
of
fact
matched is
numerical evidence
that ,
the
with
an
referred
calculations
to
that
this
outer
model
one.
recent
For
book of
of
hydrostatic
approximation
hydrostatic
the
the
and
a
be
~
at
considered
discussion Zeytounian
above
approximation
should should
the
in
ALgA as
a related
(1990;
see
the
20 • ADJUSTMENT TO A BOUSSINESQ STATE
As explained
in the section 8 (see the Chapter
non viscous,
state
(20,1)
Bo = ~ M
II) to a Boussinesq,
adiabatic
(8,8) results from the limiting process
o
, M--~ 0 o
and
The so-called Boussinesq equations
~,t,x,y,z fixed.
(8,8) imply a divergence free f l o w
~w
(20,2)
~ . v~ + ~ B
= 0 8z
and the constraint (20,3)
e +8 =0 , B B
where the Boussinesq state is characterised by
(20,4)
-> -) +.. V=VB
,
W=W +.. B
,
Ir=M 2
0
~
+..
B
, ~=M
~
0 B
+..
, e=M
8
0 B
+..
102 This constraint (20,3) is tied to the loss of the a~/at derivative (in (8,4)) by the Boussinesq limiting process (20,1) and we should expect an adjustment process to the Boussinesq state. We discuss this adjustment, unsteady process, below by relying on a work to Zeytounian (1984). Let us set
~=t/M0
(20,5)
and apply,
in place of (20,1),
t h e new, i n i t i a l ,
Bo = ~M ~ M--> 0 , a n d
(20,6) We
,
rewrite
the
exact
full
fl, t , x , y , z
equations
limiting
process:
fixed.
(8,4)
setting
@
for
any
quantity
f
considered as a function of ~ instead of t, and we get the following system of equationst:
A
(2o,7a)
T (~Mz)
(1+~) s au~+ 8~
~(bto)
=
0 ;
~M °
(20,7b)
S --a~ + M ( 1 + ~ ) ~ . ~
(20,7c)
(1+~) S a8
(20,7d)
~=~+8+~@.
a~
~
~ ~ - - (1+~) O~ +
+ 0(M o) = 0 ;
o
~-1 sa~--+O(Mo ) = o
,
For the equations (20,7) we assume as initial conditions: A -) -)0
(20,8)
~=0 : u=u ,
where u 9°, o ,
o
A
0
A
~=~ ,
w=w
0
,
~=80
and 8 ° are given functions of x, y and z, and we set up an
asymptotic (initial) expansion
(20, 9)
•~ ,
~
U
U
A
A
^
=
A
. A-) U
0 0 +
M0
A (B
0
We o b s e r v e O(M)
that
so that
equations
the
(8,4)
all
the
result or
the
ll-
. . . .
*l IJ
dissipative below full
I
@
O.
t
11
A 111
is
terms the
viscous,
same non
are
comrised
whether adiabatic
we
within
the
start
from
equations
ones
terms the (Re,co).
103
We get at first:
{
(2o, l o )
~o
=
~o
(2o,11)
0
~°ml/(~-l)
But,
according
Van
Dyke
~ o
o
if:
(20,8),
~ =A =~ = o o o that
(20,12)
O=
for ~=0, where o
n°-constant o of matched
In
this
A -> A ^ UO'III' (~1
case,
and
MO~
, o
1
going
1
o
- 0 ,
@o" e°'
0
is necessary
0
--
(20,8) we can prove that:
match with the outer expansion only
a~
o o
~. o,
O'
to method it
~-1
a~ + @+ ~ I.
and where
(1984))
o
o _ 0 ,
Using the initial conditions
if
a@
a~
~o = 0 ,
. asymptotic
that
(20,4),
the
expansions
expansion
(see,
(20,S),
when
when t --) 0. This matching
consequently
it
is necessary
for example,
that
we
~--) =,
to
is possible suppose,
in
MO 0 1 and 8 O= M 0 8 0 1
' O=
and O ° are given functions of x,y, and z. I
to
the
next
order,
we f i n d
the
following
system,
for
conditions,
for
and 01: A au ~
o
o + ~(~ I a~ i+ - - 9"
S
(2)
= 0 ;
~+~.~=o; o
S
a~
(20,13)
o
11[
1= ~(~
1+
--
1
-
~)
;
@1---- Q1- ~1'
and,
from
(20,12)
and
(20,8)
we
have
(20, 13),
(2o,14)
e:o:
A
=U
,
C,:,o 0 o 1 1 ' 1=e .
the
following
initial
104
If the ground is flat, then we have the following boundary condition: A
z:O :
(2o, l s )
H2=o. 0
THE UNSTEADY ADJUSTMENT PROBLEM
Let us set:
u-~ = ~ o + ~ A@,O ,
(2O, lea) and
A
A
H 0: ~ ~0 + ~ A ~0
(20,16b)
In this case for ~0 we obtain the following relation *: ^
0
8~# o
(20,17)
S
-
-
+
A
(~
+
--
I
-
(~
0
=
O,
OL
a
A
and for ~0 we have: ~o-= ~ °. Therefore we get ' for ~o' ^ the following initial value problem: 2A
S 2 a ~o 8~ 2
(20,18)
=
O:
A A~o= O, ^
~o = o
S
o
O~°
~I
The solution of this problem is straightforward
(20, 19)
and , from it we get:
£---> oo : I~QO I--> 0 , laQo/a£ I---> o .
Coming back to (20,16b) and (20,17) we obtain: A
A
o
8~o t
S
^
could
B~
1
~"
11[° A 1 0 + 6) + - - - (dI is a n h a r m o n i c f u n c t i o n in the w h o l e of s p a c e a n d we
at assume
1
0
~ 9" that be
horizontal
this
h a r m o n i c f u n c t i o n is a c t u a l l y zero, a n
Justified scale.
assumption
that
o n l y t h r o u g h m a t c h i n g w i t h a s o l u t i o n v a l i d o n a large
105
and, from this we get (20, 20)
o
A
+m :
~
~ --~ ~ A ~o
^
o
o 1
1
1
This solves the m a i n issue by showing that adjustment at least for the initial conditions type.
Furthermore,
for
the
to a Boussinesq state,
(20,14), with (20,12),
Boussinesq
equations
(8,8)
is of the decaying we
get,
as
initial
conditions: o (20,21)
The reader and
~ .More
t=0 :
should
keep
precisely,
~ = ~ A ~o , 8 = _~I _ o u B
B
in mind that from
the
B
of ~B may be computed once the
~"
no i n i t i a l
Boussinesq
initial
1
.
conditions
equations
value
(8,8)
_~
are the
required
on T[
B
initial
value 0
0
of UB is known and £0B={01-7£I/~,
for t=O. What happens if the initial values no, o
and 8 o not satisfying the conditions
(20,12) is not known? 21
. ADJUSTMENT TO GEOSTROPHY
It is not difficult to find, by trial, that the adjustment
(21,1)
~tKi
and a p p l y i n g the l i m i t i n g
(21,2) where
is good by setting:
process
Ki--~ 0 , Mo--~ 0 , Ao,~,x,y, p fixed,
0----V~t~J
Let us set ~ for any quantity f, considered as a function of ~ instead of t. We expand the various quantities according to:
01
V
106
!o + Ki
(21,3)
+
...
0
1
0
and we substitue into (see, ( 9 , 1 ) ) : ^
XoBO ~ ~
^
+ Ki
+ (1+~Ki
y)(~ ^ ~)
r
tv~v
+ ~
=
O;
~.-~+~:o; a~
(21,4)
+ Bo p ~
= 0 ;
at
lap
~"
One finds, at first:
a~ (21,S)
°
=
0
,
a~
a~ o _ O,
Bop
___oo + ~ = 0
ap
o
and we get, at once, that ~o' ~oand ~omPAo are functions of p alone, namely (see, the section 9):
(21,6)
~o: %(P)' go: To(P) : -Bop
OR
o ~o:
ap'
P
Toy
"
^
In order to find equations for v~o, ~o' ~I and 41, we have to go to higher order: ^
(21,7a)
(21,7b)
o~
^
^ o + ~ ^ ~ + X0B o ~0= o at
~.~+ 0
ap
o = 0
1
;
107
O~l
Ko(P)
(21,7c}
S
(21,7d)
~' = - B o p
8t
A ~:
P
0 ;
o
a~ t ap
1
dT
where
K(p)
To( p ) - p - dp of pressure alone.
function
=
The s y s t e m
~
~
of
equations
~
,
0
(21,7)
(see,
is
(9,13)),
the
must be considered
system
governing
restricting
the
analysis,
set:
~o= ~'~o+~ ^ ~o'
a n d we d e r i v e ,
at once,
from the
equation
(21,7a)
cat last
relation
at J
(21,9) A
o
as functions
related
zero
by the be
implies
that
the
ot
of the
two h o r i z o n t a l
.
two e x p r e s s i o n s
coordinates,
Cauchy-Riemann relations.
polynomials
= 0
aSo + ~'oB°~,- (~o ~d ~o--"~o+ 0-)-- '
a~oo
(21,10)
should t
o~
A
(21,8)
are,
uacAead~ ~
.
We may, w i t h o u t
This
the
as a given
and physical
x,
Liouville
evidence
y,
harmonic
theorem
suggests
tells
that
they
ones which
us that are,
: A
--°% + ~oB°~: ~o '
(21,11) From t h e
last
two e q u a t i o n s
~
(21,12)
t
As
a
matter
solution the carry
~&° + ~o= o
at
of
val Id
sphere over
result necessary
call
to for
the
we d e r i v e
ae~ 1 Ko(P) atop 2
p
whole
analysis
at
(21,7)
physical
the
example,
analogous to
-S Bo
=
fact, on
(for, the
0
of
evidence sphere.
see which one
physical
the
should If
equations
follows
concerning evidence
be r e p l a c e d
we s t a r t
(6,2)-(6,6)
from (21,1) (21,10)
by m a t c h i n g
from primitive , where
and (21,2)
in order
to
Justify
it
on
ae.L---~)
we s h a l l
above and then
with a
equations
will
(21,11).
and
find
they
indeed,
a
n o t be
108 and
going
back
to
the
first
(21,7)
we
give initial values A of v 0 we may use the
for
two
equations
of
find
a
couple
of
A
equations
for v~0 and ~t' namely:
A
a~
(21,13)
o
o
a [
p2
i
a2~1~ = 0 .
A
It
is
obvious
that
Concerning
the
horizontal
velocity
section 19).
we
have
initial
to
value
according
to
the
primitive
o
initial
equations
Concerning the i n i t i a l value of ~
v
and
for
value
model
~ . 1
of
the
(see,
the
we may t e n t a t i v e l y use the
i n i t i a l value of ~ f o r the p r i m i t i v e equations but t h i s works only i f
this
i n i t i a l value may be s e t under the form: ~o (p)+Ki~°'I then we get A
(21, 14)
(~=0 : v~ --~V-)o
J} =~o
,
o
1
1
°
Whenever the i n i t i a l value appropriate to the p r i m i t i v e equations cannot be put under holds.
the form ~o(p)+Ki~°
we must
The second of equations
o-sBo
(21,1s)
expect
(21,13),
that
according
Ko)aCa
another to (21,8),
adjustment
:o,
but from (21,11) we o b t a i n
[a O+ o) 2A
(21, 16) Finally,
(21,17)
where
x°B°
m
a~
=
-
m
at~2
from (21,15) and (21,16),
we obtain a single equation for Do:
+
a~ 2
a-P
8p JJ
a2
a2
ax 2
8y 2
__
S
~2
A
~P°+ ~
process
can be written:
Ko(P) 0p J
0
109 When K o ( P ) = l , able
to
the equation
settle
the
( 2 1 , 17) i s t h e o n e o b t a i n e d
main
issue
of
the
adjustment
by Kibel
problem
(1957).
which
is
He was to
know
A
whether or not v~° and ~leVolve towards the geostrophic balance (9,7) when 6-)~o. As a matter of fact one has A
(21, 18)
-) 6---) +m : v ---> v 0
> . , ~---9 . , ~ , qg 1 qg
with the geostrophic relation:
(21,19)
~A~
qg
+ X OB ~0
qg
=0.
There is an important observation,
which was known to Kibel and which concerns
the way in which Lim ~ m~"
6 ~+~
I
1
i s r e l a t e d t o the i n i t i a l
values (21,14).
One s t a r t s from A
(21,20)
~.
A
o ÷ ~ A v°
= 0,
o6 which follows the first of (21,13) and we transform it, thanks to the second of (21,7), to
^ 06
a~ 0p
then, using the last two equations in (21,7), we get
(21,21)
~-~
KoCh) a p A
=0
.
A
Now if we integrate this last equation between ~=0 and [=m, and if we use the A
geostrophic balance for limiting values of v~0 and ~I' when 6--9+m, we get:
(21,22)
ar a 11 7 + sBo iK0-j EFJ r
110 This is an equation from which, with suitable boundary conditions on p and x, y, we may deduce the value of ~m. I From (21,22) we obtain the initial equation (9,14),
namelyt:
(21,23)
Bo A Rqg
_ t-o=
~.
condition
that
A -)° v + SBo
must
be
supplied
for
Ko,P ,
where A is the operator (9,15). We observe that from the solution derived by Kibel, it
Ko(P) =constant,
appears
that
the
limiting values tends to zero like i
which is restricted to A between (~'u~i ) and their
differences
61/2
osc(~), where osc(~) stands for some
bounded functions which oscillate like a cosine function. We mention for further study that the geostrophic balance occurs in a number of other situations, with various processes of adjustment discussed in Blumen
(1972). The Figure 8 below gives an example of the adjustment of meteorological fieds (after Monin (1958)) in the baroclinic atmosphere. In this case, there were no pressure perturbations at the initial
instant
of
time, and the velocity field corresponded to a plane-parallel flow of the type of tangential discontinuity along the ordinate axis (the initial of the horizontal velocity v -)° is given by the velocity field changed only slightly
as
limiting distribution of the horizontal
a
dotted line result
velocity
of
in
Figure 8). The
adjustment;
~(x)
(the
decreased by 3Z from losses due to the generation of fast the formation of inhomogeneities in the pressure field). " ~ "
distribution
kinetic
gravity The
to the velocity field: a distinct dip was
see
energy
waves
pressure
produced
(see the limiting distribution of the altitudes of the isobaric
the and field
in
surface
it M(x)
at ground level; it dropped by 4 dkm along the ordinate axis). The problem of the adjustment to geostrophy
in
the
case
of
barotropic atmosphere was first formulated by Rossby (1938)
a
and
hydrostatic Cahn
(1945)
and solved by Obukhov (1949). For the baroclinic atmosphere, this problem
was
treated by Bolin (1953}, Kibel (19aS; without taking the two-dimensional waves into-account), Veronis (19S6), Fjelstad (1958) and Monin (1968). See also excelent review by Phillips (1963) dealing with geostrophic motions. t
From ~ m1 This
matching
must
with
coincide
result
has
the with
been
main the
at
outer initial
first
quasi-geostrophtc value
obtained
~ by
qg
t=O
Guiraud
region for
the and
this
limit
value
equation
(9,141.
Zeytounlan
(1980).
the
111
~/s u~,~ J~,.,
a"
---~(~)
I
"-8
(~/~J~ .~arP.r~ (1968)).
112 BACKGROUND READING
Concerning
the
problem
lectures
I and II of
GUIRAUD,
J.P.
of
adjustement
of
meteorological
fields,
see
the
(1983) Mecanique Theorique, (Unpublished
Universit~
manuscript;
de Paris
C.I.S.M,
6.
Udine,
Italy).
REFERENCES TO WORKS CITED I N THE TEXT
CAHN, A.(194S) _ J. Meteorol.
2, 113-119.
C. (1976) _ ~
CERCIGNANI,
and ~ Scottish
BLUMEN, W. ( 1 9 5 3 ) BOLIN, B . ( 1 9 5 3 ) FJELSTAD, FROI~,
J.E.
(1958)
(1969)
GREENSPAN,
H.P.
5,
(1968)
of Fluids,
_ ~
~n4~
~
a~un$~.
and Space Physics,
10,
485-528.
Norske Videnskaps-Akad. Oslo,20,1.
vol.12,
suppl. II,
o~ ~
GUIRAUD, J . P .
a n d ZEYTOUNIAN,
R. Kh.
GUIRAUD,
a n d ZEYTOUNIAN,
R. Kh.
J.P.
~
373-385.
_ Geofys. Publikasjoner
_ Phys.
~
Academic Press.
_ Reviews of Geophysics
_ Tellus,
J.E.
•
~ .
(1982)
3-1I,
12.
Cambridge Univ. Press
_ Tellus,
(1980)
pp. I I ,
3_44, 5 0 - 5 4 .
_ Geophys. Astrophys.fluid
Dynamics
I__SS, 2 8 3 . KIBEL, KIBEL,
I.A.
(1955)
I.A.
_ DAN SSSR,
(1967)
_
~
104,
60-63
(in
Sa2.nm.da~_2/.on Ze
Russian).
~
A g ~ u : ~ ~
p e a / ~ W e x ~ ~ , LERAT,
A.
a n d PEYRET,
R.
(1973)
MONIN, A . S . MONIN, A.S.
(1958) (1969)
_ I z v . Akad. Nauk SSSR, _ ~ a ~
paqad~
o~
Moscow ( i n R u s s i a n ) .
_ C.R. Acad. t.277
~e.tAad
Sci.
Paris,
t.276
A, 7 6 9 - 7 6 2
and
A, 3 6 3 - 3 8 6 . set.
fco/c q
Geofiz. ~
497.
~.Izd.
Nauka,
Moscow
(in
Russian). NAYFEH, A.H.
(1973)
_
OBUKHOV, A.M.
(1949)
_ I z v . Akad. Nauk SSSR, s e t .
OUTREBON,
(1981)
_ Correction
P.
~ e a ~
me/~.
John Wiley and Sons.
d e Fromm p o u r
Geogr. i Geofiz., les
13,
sch&mas Y ; e t
281.
applications
g
a u ph&nom&ne d ' a d a p t a t i o n M~t&orologie.
au quasi-statisme
These de 3e cycle. Universit~
M&canique Th&orique. PHILLIPS,
N.A.
(1963)
_ Rev. G e o p h y s . ,
l,
2,
123-176.
en de Paris
6,
113 RICHARDSON,
L.F.(1922)
_
WeaZhe~ P
~
&9 N
~
~aacea~.
Cambridge-reprinted by Dover Publications in 1966. ROSSBY, C.G.
(1938) _ J. Marlne Res. l, 239-263.
TITCHMARSH,
E.C.
(1948)
_
8
~
Za
tAe
~Aeon~
o~
~
O~4n~/a
Oxford, Clarendon Press. VAN
DYKE,
M.
(1962)
_
~
~
methad~
/~
~&zid
~
~
.
Academic
Press, N-Y. VERONIS, G. (1956) _ D e e p -
Sea Res., 5, n°3,
ZEYTOUNIAN,
R. Kh.
ZEYTOUNIAN,
R. Kh. (1990) _ Magm ~
157.
(1984) _ C.R. Acad. Sci., Paris, t.299, mad2M~
Springer-Verlag,
o~ ~ Heidelberg.
I, n°20, ~ .
1033-36.
CHAPTER Vl LEE WAVE LOCAL DYNN,,IIC PROBLEMS
When Remm we o b t a i n , equations
for
possibility
the
instead
of
adiabatic,
fo-plane
equations
non-viscous,
t o impose, on t h e ~
(5,5)-(5,9),
atmosphere
and
q]taund, t h e f o l l o w i n g s l i p
we
the have
Euler only
condition:
z=O : 14=0. For adiabatic, we set,
non-viscous,
atmospheric phenomena at local scales
instead of Wlz=o= 0 rX-Xo
on z :
~hp-~--o
where
h
(when c =1) 0
, Y-Yo]
w
. ,-~-o j:
H
=
~v~.
H
r.X-Xo
Y-Yo]
~h[^-E~° . ,-E~-o j .
H
°'=H~ ' ~=T~ ' P=--9°m and c -o L° 0
0
0
0
The local ground h is characterized by the length scales ho, ~
and mo; X=Xo,
y=y ° is a local origin and h(O,O)ml, but h(m,m)mO. Here
we
start
with
equations ( 5 , 5 ) - ( 5 , 9 ) , 22
. EULER'S
the
Euler
equations
in
dimensionless
form,
i . e . the
where Remm.
LOCAL DYNAMIC
MODEL EQUATIONS
In connection with the Euler equations
(equations
(6,6)-(5,9),
where Rem~) we
can formally consider the following b~e local limiting processes:
(22,1)
e--) 0 , with t,x,y and z fixed, o
and (22,2) where
c --~ 0 , with t,x,y and z fixed, o
115
t
(22,3)
{ = e '
~ _ x-x o c
o
~ _ Y-Yo
'
c
o
Considering the Euler equations with the boundary condition:
we are model
on z = ~ h ( X x , p y )
w = o"v~.~h(kx,~y),
(22,4)
led to the of
local
limiting
steady
process
dynamic
which
(22,1),
For
prediction.
is closely this
model
related we
obtain
following set of limiting equations: N
a v~
a~°
(22,5b)
-~o~ ~o÷ % aT
(22,Sc)
~o= ~o~o ; ~.
(22,5d)
1
_
Po ~ o
~o ~o~
a~
--J
a~
+ Bo
]
= 0;
N
~ ~
aP°W° -
(PoVo) + az
P0Vo" --~
= __19 +
~po = 0 ;
I ~i a~°
+
0
;
~
(22, Se)
where
1
(V~o.~)~. ~ oaz° , -
(22,5a)
f
aTo
~ ~o" ~o ÷-w° LP° ~o_ ~-1 -
@z
=0,
zJ
= C0
and
(22,6)
. .Po' To ) m clim->o(~,CoW, p,p,T) (Vo' .Wo'. Po' o t,x,y,z fixed N
At this point,
formal matching with the primitive model equations
to the
(7,1):
the
116
~(t,Xo,Yo,Z) lim
(22,7)
~o
0
Po
P(t,Xo, Yo, Z)
~
P(t,Xo, Yo, Z)
Po ~
T
T(t,Xo, Yo, Z)
o
and
a[z.Bop]jXo, Yo:O
(22,8)
may be interpreted as providing
, (p-pT)xo,
0 ,
lateral boundary conditions at infinity for
the locai steady dynamic model (lee waves model), which take into account the prediction at X=Xo, Y=Yo according to the primitive equations (7,1). Of course, internal
it is necessary to resolve the vertical structure problem for the
lee waves of same type as the one considered
in section
11 of the
Chapter III. For the model equations (22,5) we get as condition on the ground:
(22,9)
Wo:
However,
on z :
~h(Ax,.y).
if the initial conditions for the full Euler equations contain x and
(22,10) where
o~.~h(Ax,.y),
t=O: v~=~0, ew=W °, p=pO ~o,
W o,
pOa_nd R 0 are
given
positions:
~ ~ ~ ~~ ~~ x=x1+yj and x=x1+yj,
equations,
limiting process
p=R o,
functions
of
z
and
of
the
horizontal
it is necessary to consider also, in the Euler
(22,2),
which lead to the local unsteady dynamic
evolution model in lieu of the equations of adjustment to hydrostatic balance (19,7),
(19,8).
This last model (local nonlinear adjustment equations)
is the
most complete one, but it is coupled to the primitive equations model! Therefore, variables, setting:
if we
introduce
X-Xo ~ Y-Yo x= e: and y= e 0 o
a
fast
time,
t= t-tO,
and
the
fast
horizontal
o
, and
if we use
the
limiting
process
(22,2),
117
~. ~. N" ~" T*)O m elim-.->o(~,eoW, P,p,T)v (Vo'Wo'Po'Po'
(22,11)
o
N
D
N
t,x,y,z fixed it is straightforward
to derive,
from the full Euler atmospheric ~O
.O
Ne
equations,
Ne
NO
the following set of limiting unsteady equations fop Vo, Wo, Po,p o and To:
S
~-
o
o
Oz
8{ NO
.~)Wo+
~e
;
~e
o +
.....
az ~
o
0
PO ~Mo
N0
a{
(22,12)
=
~
+ Bo
= O;
az
No~e
Po= PoTo ; Ne
NeNO
Sap° ~. N'~O, aPoWo +
o{
(PoVo / +
~e
NO
Po
We note that,
S OT _____0_0
÷
•
a{
b:]
if at the initial
- 0 ;
Oz
~e
--
Ne
~e
N'rN'O'Oo'o --
+ v .Dp^l+w^ -~-I[s0Po -t a{ °J L °
~r az J
time t=t 0 we have a set of initial
values,
functions of z and x, y, then in limiting process (22,2) we pose {=te~0- ; then {--)~, i.e., (22,5),
when t--It°, we obtain
the
local
steady
corresponding to limiting process (22,1),
For the unsteady equations and corresponding
(22,12)
dynamic
model
equations
where t=t 0 is a parameter.
we get as condition on the ground ~e
N0
Ne
_0
initial conditions for v o, w o, Po' and Po at {=0,
(22,9)
where the
initial values (see, (22,10)) are given functions of z and x, y. The
equations
conditions,
describe
local situation situation conditions!
(22,12),
with the
~
(corresponding
(with another
time
the condition n
~
(22,9) ~
and corresponding @aa~
to a fixed time t o ) changes to), under
the
influence
of
and
initial
show
how
a
into another local the
initial
b~
118 CONSISTENCY OF THE MODEL EQUATIONS (22,5) AND (22,12) If we are sure that our singular perturbation problem (related to the limiting process So-)0) can be resolved by the method of matched asymptotic expansions (see Van Dyke (1975)),
then we have the following matching conditions, between
the limiting processes:
(22, 13)
lim
lim o
C -)0
t-)o
I~[-~ t,~,9, z
= lim {-~
lim lim
,t=-- ;
~,x,y,z
]y [-)m fixed
(22,14)
~ t C
lim o
C -)0
fixed
- lim
~-~o Co->°
, t= t-t°
So->°
L~,~,z
I
t,~,~,z
fixed
fixed
c
for t=t °
Naturally the nature of matching conditions depends vitally of the behaviour of solutions to the
local problems when either one of t, ]xl,
[~[ tends to
infinity. If,
in
particular,
equations
(22,5),
we when
consider
the
behaviour
of
the
steady
solutions
Ix2+y2[--)m, we can suspect
that
the variations
of
with
respect to x and y occurs through two different scales. One of them grows with ~
'
but the other one corresponds to internal waves,
as discussed in
~2~2 and its scale remains of order one, when Ix +y [--)~. The reason ~2N2 for the existence of such waves is that, when Ix +y I--~, the relief is flat
Chapter I I I ,
and the perturbations obey the linear equations with slip on a flat ground; this is precisely the situation discussed in Chapter III. When ]x2+y2[--gm the horizontal wavelength of these waves becomes very small in comparaison to the distance to the relief and they appear, waves radiated away. excited,
trapped
Finally,
lee waves
locally, as plane
we note that it is possible to show that once
travel
along
the rays
variation may, at least in principle, be computed;
and
that
their amplitude
it is almost evident that
this amplitude decays when travelling awry from the relief but a formal proof is difficult.
119
23
.
MODEL EQUATIONS FOR THE TWO-DIMENSIONAL STEADY LEE WAVES
Here we start from the local steady Euler's dynamic model equations which
has
been
obtained
in
section
22.
But
we
consider
(22,5),
only
the
two-dimensional case : (23,1)
a__ ~ 0 ~
~--~0, with x, y and z fixed.
a~
Consequently we can write (the marks have been dropped from the nondimensional quantities): p
1
[uaW a~] ~--~ + w +
aP-
0 •
ax
~M 2
o
p (23,2)
F ~ u + w04
,
)
+ --~[~ o
OLu
+ Boo = 0 ;
-
apu apw 8--~--+ a-~ -- = 0 ; p=pT ; [uaT
~-lr ap --y-Lu~ + wa~p l
a~]
~+w
=o,
with the following boundary slip condition: w = o'u
(23,3)
dh (x) P-
-
,
dx
on
z
=
~h
P
(kx).
The function h (Ax) is of order unity being identically zero for p (23,2) and (23,3) four length scales ratios enter, namely: h { ~ (23,4)
=
H o
,
H
÷
k
=
0
,
Bo
=
[x[>l. In
o
RT
(0~
'
g and M z=
U~/~g
o
RT(0) g
The l e e wave t w o - d i m e n s i o n a l of the aoa/~
equations
Perturbations
are
assumed
steady (23,2), to
vertical
velocity
w on
the
mountain
z = v h (Ax)
there
is
P
form):
with
be top. a
problem
is considered
the
framework
(23,3).
confined
to
We a s s u m e
uniform
within
flow
the also with
troposphere that velocity
far
with ahead (in
vanishing from
the
dimensionless
120
u=l, w=O, when x--x-m,
(23,5)
and we set z The equations
for the altitude far a head (where xm-m). (23,2) are reduced as usual by introducing the stream function
@(x,z), such that: (23,8)
p u = - a_..~ a n d
p w = + a._~
ax '
az
but for convenience we follow the common technique in the theory of lee waves (see, Long (1953) and Zeytounian (1979)) which amounts to replacing the stream function
by
the
~
~
a~
t]~e
ab~zun//ne
~(x,z)
(in
dimensionless form), in such a way that: (23,7)
z = Bo(z-v3). m
For the system of equations (23,2) we have, first, the Bernoulli's equation: 2 2 1 u +____~w+ _ _ 2 ~o(~,_1)
(23,8)
p~'-1 ~(~) +
Bo
z = I(¢),
lr<
where plr
~--~ + w
= O,
and, secondly, the vorticity theorem (see Zeytounian (1974)): au a-z
(23,10)
aw -P d{~ a-x=
Equations (23,8)-(23,10)
1 P dLog~1 M~o(~_I)~ p d~ J "
c o n t a i n the a r b i t r a r y
functions
s t r e a m f u n c t i o n @. The r i g h t - h a n d s i d e o f e q u a t i o n from t h e c o n d i t i o n s distributions (23,11)
in the unperturbed flow (see,
I(@) and ~(@) o f the
(23,10)
(23,5)),
c a n be d e t e r m i n e d where t h e v e r t i c a l
of a l l e l e m e n t s a r e known; i n p a r t i c u l a r : p=p (zm), p=pm(zm) and T=T (z), when x--+-m.
Through some lengthy but quite straightforward computations one may derive a quasi-linear (23,6),
elliptic
differential
(23,7) and (23,8).
out the nondimensional
equation
for
~(x,z),
from
(23,10)
with
It is convenient to write this equation by singling
density perturbation ~=
P-Pm Pm
as an (implicit)
function
121
of 8 and its first order derivatives, namely: (23,12)
(l+ao) ~ - l =
1-
0.2
88
+
+ lazJ
(l+e) 2
- ~
1
where
Tm(zm) m Tm(Boz - Bov~).
Then the equation for ~(x,z) is rather awkward-looking one (Zeytounian(1979)): (23,13)
+ --+
+ o
_ 1 a[~_ a~ I+~
a~ a~
~
+ az az
w-~. L-~ As
is
well
known
(Long
I a_~]
t
(1953),
,
Zeytounian
LtaxJ
tazj
(1969,1979))
the
~ appropriate
bounadary conditions are: ~(x,vh (Ax)) = h (Ax), where ]x]
(23,14)
~(x,~-~) = O; lim r[8~[ +
x~+®tlaxl Concerning
the
second
a~ ]
~
condition
<+-.
of
(23,14)
(for
x=-m)
it
consequence of Long's hypothesis of n~ upstream influence;
is
merely
the
the role of this
hypothesis was clarified by Mc Intyre (1972) who showed (on a linear version) that third
it emerges from a careful condition
of
(23,14)
is
examination of the transient beheviour. the
consequence
velocity w on the top of the troposphere.
of
the
vanishing
The
vertical
The proper way to formulate
the
upper boundary condition relating to the top of the troposphere would be to match the previous model of lee waves in the troposphere with another model for waves in the upper atmosphere, taking into account the dissipation of such waves by viscosity at very high altitudes.
The works of Yanowitch
(1967),
122 Thomas and Stevenson (1972) and Bois
(1984) would be helpful
in formulating
this matching process. Needless to say, the problem (23,12)-(23,14) is ~ reducing the system of equations
and the benifit of
(23,2) to a single equation for 6(x,z)
is
lost through the awkward nonlinearity. The interest of the present formulation lies in the fact that it allows to work with asymptotic techniques in a very convenient way and we shall proceed with this now.
24
. BOUSSINESQ'S
INNER
SOLUTION
If we assume that: (24,1)
Bo = ~M o, Mo--eO, with ~,x,z,@ and A fixed,
a n d we s u p p o s e
that,
(24,2)
according
to
(23,12),
~ = O(Mo),
we find that lim
6o(X,Z) is a solution of the Helmholtz equation:
6
M 9o
o
fixed
826 826 ~ + ---2o + K2 6 = O, 8x 2 Oz 2 oo o
(24,3)
where the constant K 2 is given by: oo (24,4)
oo
z=O
~" L ~"
and is assumed to be positive. ¢°=0(I),
where o
In the limiting process (24,1) we assume that
is a dimensionless measure of
atmosphere; we recall that: Bo "°N2(zoo oo co] = T--~-~) On t h e o t h e r
hand,
(23,14)
leads
to:
1
+
~
]"
stability of
the standard
123
~o(X,¢h (Ax)) = h (Ax), when Ixl
(24,5)
0
=
llm :-I X_~=[ OXl
a~
+
b--~]<+~'
but the third condition of (23,14) is no longer valid (in the framework of the Boussinesq inner approximation ) and a matching condition must be used instead (see, the following section 25). The
problem
(24,3)-(24,S)
for
a semi-infinite
half
plane
zZCh (Ax) must be p complemented by a condition relating to r---)m, in order to achieve uniqueness (here we use polar coordinate r and @ defined by: x=rcos~, From
the
(1959)),
general
results
relating
to
the
we known that such a condition,
Helmholtz
z=rsin~,
equation
r2=x2+z2).
(see,
assuming that n~ ~
Wilcox
are radiated
in%rards, reads:
(24,6)
~o ~
sine ~eal ~r
where
the function G(cos~)
mountain
through
is arbitrary and must
25 our
goal
,
depend on the form of the
h (Ax). As a reminiscence p condition of (24,5) we impose : G(cos@) m O, for COS~ < O. In section
the
(cos~) exp[i(K00r-~/4)]
~
function was
to show
that
Long's
model t
is
of the
the first
second inner
approximation of an outer-inner approximation scheme. SEMICIRCULAR MOUNTAIN
We consider now the case of a mountain ridge with a semicircular cross section of radius r
o
(24.7 t
Long's
h(x=[o
classical
4Z~2~ condition We
consider
this
model :
is
IxI>ro'IXir°' obtained
i£
we
~o(X,l)=O. model
in
the
section
28.
add
to
(24,3)-(24,5)
the
Following
124 and stream function @o(X,Z) satisfies the equation t a2@o a2@ ° 2 -+ -+ ko(@ o- UoZ) = O. 8x 2 8z 2
(24,8)
For (24,8) the boundary conditions are: (24,9)
@o = 0 ,
on z = h(x);
(24,10)
@o = UoZ ,
for x --~ =
In order to satisfy the condition
(24,9) for the selected shape of the ridge
we transform to polar coordinates X = rcos~, z=rsinO, rZr o, O s O ~ and set
Ao= ~o-
uz
Equation (24,8) for A O, in the new coordinates,
is:
aA 1 a2A o + _1 o + _ _ _ ° + k2A = O. 8r 2 r Or r 2 8~2 o o
82A (24,11)
conditions
(24,9) then takes the form of two conditions:
(24,12)
ao(r,O) = ao(r,=) = O;
(24,13)
Ao(ro,O)
= UoroSinO.
For a unique solution of the problem (24,11)-(24,13)
it is sufficient that the
solution of equation (24,11) should satisfy two conditions: (1) for some constant Co, which is independent of position,
rI12~0 I s Co;
(24,14)
(2) for all angles within some range, smaller than It,
(24,15)
lim (rt/2Ao) = O.
r-)~ dT
0
~U 3
°{ 1
dZ m zm=O
125 The
first
of
these
requirements
is actually
the
condition
of
generalized
boundedness of the solution of equation (24,11). The
second
amounts
to
requirements
of
particulary
rapid
damping
of
the
solution along some selected directions, with the damping occuring faster than -112 the reduction in r with increasing r. In the problem at hand this direction is apparently toward the incoming flow. consideration that solution of
It is clear from physical
(24,11) should damp out most rapidly in the
direction toward the oncoming flow. Hence in order to obtain a unique solution of the problem (24,11)-(24,13), the second condition
(24,15) must be satisfed in the direction @=~
study carried out by Magnus (1949),
(before the
such a condition was used by Lyra (1943)
to single out the unique solution of the Helmholtz equation). Since
A0[~= =0 for all
rZro,
and solution of
(24,11)
is continuous
every-
where, following Merbt (1959), the second condition (24,15) can be used in the form: (24,16)
lim (r 1/2 r-~
Separation
of
variables
8A
o) = O.
a@ is
possible
by
seeking
the
solution
of
equation
(24,11) in the form: A
= FCr)GC@)
o
and the equations governing F(r) and G(@) are then:
(24,17)
dr 2
+5~+
-
r ~
F=O;
d2G (24,18)
- -
d@ 2
+ n2G = 0 ,
where n is a constant.
By virtue of boundary conditions (24,12) the solution
of equation (24,18) has the form: G(@) = sin(n@), with n assuming integral values (n=1,2,3 .... ). Equation
(24,17)
stratification)
is its
an
n-th
solution
order are
Bessel
Bessel
equation.
functions
For
with
real
real
k
(stable o argument; for
126 imaginary k ° (unstable stratification)
they are Bessel functions of imaginary
argument. The former are oscillating and the latter exponential
functions.
follows
in
directly
that
wave-type
disturbances
may
arise
only
those
It air
masses possessing stable stratification in the undisturbed state. In general, stratified
the assumption that a large vertical extent of the atmosphere unstably
is doubtful
in the
majority
of
problems.
Hence
it
is is
assumed here and in all subsequent problems that the incoming flow as stable stratification: -
dT dz
<
~'-I
z =0 co
The solution of equation (24, 11) can be expressed in the form: (24,19)
Ao(r,~) = ~ A J n=1%
where Jn(kor)
function,
n
n
(kr)+
BnYn(kor)}sin(nO)
0
is a Bessel function of the first kind and Yn(kor)
w h i l e A and B a r e c o n s t a n t s n
n
t o be d e f i n e d .
Boundary condition (24, 13) can be used to relate A A - o o I Jl(koro )
(24,21)
An= -XBn,
where x =Y (k r )/J (k r ) n
0
Coefficients damp out
and B :
n
n
Ur
(24, 20 )
n
is a Weber
0
B
n
n
0
0
'
x,B1;
nz2, n=l 2, '
....
remain undefined and requirement
at high z exuulx~ be used to eliminate
Jn(kor)--90,
that
the disturbances
this
indeterminacy,
of ¢
because
and Yn(kor) --~0 as r--~!
The condition (24,18) is equivalent to the requirement that:
z=OJ= 0
(24,22) X-)'-~
Here u'
denotes
the disturbance
of the horizontal
velocity.
Now substitute
solution (24,19), with (24,20) and (24,21) into condition (24,18). condition
is
stated
asymptotic expansions,
at
r--)~, functions
J
n
and
Y
are
in which
=n = ~n sin(kor + ~-- - ~) and cos(kor + ~- - ~)
replaced
Since this by
their
127 are first transformed
into functions
the equation thus obtained,
having
(kor + ~) as their argument.
In
the coefficients of terms such as
r-nsin(kor + ~) and r-ncos(kor + ~) are
equaled
coefficients
to
zero.
This
B . Without
detail, we note that Merbt, Following
this
considered, The
to
analyzing
n
pratical calculations,
leads
a
this
system
of
system
algebraic
of
algebraic
for
equations
in
in the previously cited work, points out that, for
only a few of the first coefficients
approach,
equations
only
two
harmonics
in
need be retained.
solution
(24,19)
will
be
assuming all B m0 for nz3. n
final
form
of
the
function is (Kozhevnikov
approximate
expression
for
calculating
1 R = r__ , Co= J (m)(l+xlx2) ro 1
- x2J2(mR)]sin~ cosO,
, m = kor o.
The stream function field in the vicinity of a mountain was the solution (24,23) with:m=2~ro=ikm,
calculated
6 ~"-6,12"--" , , ,
0
-4
-3
-2
-1
I
0
using
Uo=6 m/s and B ~___[I+ _ _ l=4OC/km dz= z==OJ
The results are plotted in dimensionless form in Fig. 9 below.
-I,0 0,5 j.
stream
(1963)):
(24,23) Ao= Uoro{Co[JI(mR) + x2Y1(mR)]-R}sinO-Co[Y2(mR)
I
the
I
• ~. 9: Ytn,eam,£La~a~Lt.Ar~=2 (
2
3
~
~
~
5
(1963))
6.Z/to
128 It is clear
that
substantial.
the disturbances
Very
strong
waves
produced
appear
in
by the the
mountain
~,e part
of
ridge the
are
quite
flow.
Their
amplitude
is damped as ones moves away horizontally from the mountain and,
addition,
depends markedly on altitude.
the flow are practically of 4-5. z-9~
imperceptible
It follows from solution
and
axe)
x--¢m. The
above
with a horizontal
Fig. axis
rotors are shifted somewhat to an aircraft
wing.
Disturbances at distances
(24,23) 9
in the windward part of exceeding
that disturbances
shows
that
form above
closed
r ° by a factor
disappear
streamlines
the ridge.
First
that of the outside airflow,
also at
(so-called
we note that
downstream and have a streamlined
The air within the rotor moves
in
shape,
the
similar
in the same direction as
with the exception only of a small leading part.
The air in the center of the rotor is at rest,
while
in the peripheral
it moves at velocity close to that of the free flow. rotors form above the mountain,
layers
In the case at hand bsa
with the center of the lower rotor situated at
an altitude of about 2.5 km and that of the upper located at 5.7 km. The lower rotor
is inclined noticeably
entire
mass
altitude
2.8
between
the
of km,
enclosed
to pass
mountain
approximately particles
air
to the horizontal.
fourfold,
in
the
through
and
free
flow
a narrow
the
rotor.
attaining
It,
between
(only Here
(in
the
so to speak, the
about the
case
800
wind
under
forces
ground
and
m wide)
24
the
passage
speeds
study)
the
increase m/s.
Air
located sufficiently far ahead of the mountain ridge at altitudes of
from I to 2.8 km, rise upward after passing around the rotor from the bottom. Velocity reversal regions also exit here. is similar to the one Just examined, pair played by the streamline and
trailing
part
with the role of the mountain ridge-rotor
moving
of the rotor.
The flow pattern between the rotors
in the immediate vicinity of the bottom
Above
it,
speed increases three-or fourfold downstream. aircraft.
as
above
the mountain,
the
wind
This point is very dangerous for
As a whole the generation of rotors above a mountain ridge results,
on the one hand,
in increasing wind speeds between them and in the vinicity of
the
on
ridge,
and
the
other,
in
the
formation
of
high-amplitude
waves
downstream from them. We note
that,
the smaller
number of rotors.
m
(smaller
For example,
r ° and
larger
%),
the smaller
is the
not more than one rotor can form when m=l.8,
while when m=l.S na rotors form at all. Finally
the reader
original sources
interested
(Kozhevnikov
in other computed examples (1968) and Miles (1968)).
is reffered
to the
129 25
. OUTER,
The
GUIRAUD'S AND ZEYTOUNIAN'S SOLUTION
condition
1
6(x,~-o)=0
(see,(23,14)),
at
the
top
of
the
troposphere
is
outside the domain of validity 6
as an approximation of 6 which is obtained o through the inner limiting process (24,1). 1 In order to obtain an outer approximation to which the condition 3 ( x , ~ ) = 0
may be applied, we set: (25,1)
w(~' ~; MO) ~=MoX, ~=MoZ, 6=M 0 1/2 ~(~,~;Mo) , e= M3/2 0
The choise of scaling in (25,1) is dictated for 6, by matching with (24,6) and for w, by equation (23,12). 1 In this case, instead of 6(x,~-r)=0, we have DO
(25, 2)
~(~,
~;
Mo)=0
and otherwise hp(A~/Mo)--)0, with Mo---)0, except a t From the
inner approximation we except
~m0.
that the perturbations
in the outer
region consist of lee waves with a wavelength of the order of M relation (24,6)), a ~ particular
~
(see, the o at the scale of the outer region. As a consequence, there is
built
into the solution and we must take care of it. FoF the
case dT - -dz
m r
o
= constant
we h a v e
(in dimensionless form):
(26,3)
T®(z®)
= 1 -
~ r ° z g
m
= 1 - R_r o ~ g
~
+ M3/2R_r o ~ , o
g
if we take into account (23,7) and (25,1). As consequence of (25,1),
(25,4)
z = §({
(25,3) and
- 0-M3/2~) 0
we find, from (23,13), the following eu22~ equation fop ~(~,~;Mo):
130
a2~
(2s,s)
-
-
÷
a( 2
-
+
-
¢=(C) ¢®(C) ~ ~
+
=
2 '
o
with, from ( 2 3 , 1 2 ) ,
¢=(C)~
o
1
(25,6)
= _o" ~¢ (C)K2__~_, ~ + oo
In
(25,5)
and (25,6)
(25,7)
we h a v e
0<¢ (()
m
K2 oo
I
-
~
g
(25,6)
If we take into account
; ¢ (0)
r~
~ K2 oo
in (25,S) we see that the term
1
¢~(C) ~ oo
is ~o~ dominant,
and strictly
the followin E ~
a2~
(2s,8)
-
-
ones
equation.
and
for
the oa/mF% approximation
we
have
¢=(C)
a2~ ÷
Og 2
It is possible outer
speaking,
e ~ :
+
OC2
-
-
M2 o
to consider the
Therefore,
(in an asymptotic
Helmholtz
model
equation
sense
) the equation
(24,3),
for
~o(X,Z),
(25,8) as
as
inner
it is shown that the upper condition
w=O , on z=l/Bo belongs to an e u 2 2 2 % ~ n / / ~ p r o c e s s :
(25,9)
Bo=~H o , Mo-~ 0 , w i t h
~,
~,
~ fixed.
Such an outer asymptotic approximation is worked out by Guiraud and Zeytounian (1979) and it is shown that the upper and lower boundaries of the troposphere alternately reflect the
inner
internal
(Boussinesg)
short gravity waves excited
approximation,
to the scale of the outer region. built
with a wavelength
As a consequence,
into the solution and we must take care of it.
by the lee waves of of the order
there The
is a double important
of M0, scale
point
is
131 that
these
~eed~
that
n~
occurs on the inner flow close to the mountain (to lowest order).
short
gravity
excited
waves
propagate
downstream
and
We
should understand the upper boundary condition as an artificial asymptotically an e ~
on
the
inner flow
(Boussinesq flow)
one,
which
having is
the
with (24,6),
was
only really interesting one. The equation (25,8) with the boundary conditions:
(2s, lo)
~(~,o)=o
,
~((,1/~)=o
,
as well as the matching condition, at the origin, ~=0, ~=0,
solved asymptotically for Mo--> 0 and a double scale structure emerged from the solution (on account of the term (¢m(~)/~o)~ in equation (25,8)). As a matter of fact, according to the technique of multiple scales (see, the Appendix 2), we set:
(25, Ii)
~ = ~o(~,(;X) + Mo~IC~,(;Z)+ . . . .
where
X -
e(~,~) M
'
o and
from (25,8), we find for ~
o
the following equation:
We choose 8(~,~) to be a solution of
fae32 [_ae]2
(26,13)
l J+l j-
¢C~) oo
and an obvious solution to
a2~ (25,14)
a2
0 + K2 ~ = 0
oo o
is then
(2s,ls) where (24,6).
~o(~,~;X) = Reu~ {A(~,~)exp[i(Ko0Z - ~/4)] } the phase difference ~/4
is used for effecting
latter
matching with
132 In order to obtain an equation for the amplitude order and we find,
from
(25,8),
for ~I(~,~,X)
A(~,~)
we go to the next
the following
nonhomogeneous
equat ion:
Oz~, + K 2 ~ = _ K ~ oo )
(25, 16)
(:92
~m((~
O01
+
21 ~(90 82~ o
+
8,~8"~+
.
80 82~o] aC --
8--~ j
_ g=c~)Se 880
8Cz where (25,17) and aecuIx~ terms appears
in the solution of equation
(25,16) for ~l(~,~;X)
which we must remove. If we take into accont the solution (2S,15), for ~o(~,~;X), we obtain,
instead
of (25, 16): (25, 18)
1 °2g, OZ2 +~
¢=(g)
+ iKoo~: Oe~+
~O0
+
where
+ ac2
~'(~)
A : 0
= exp[i(Koo z - ~/4)].
Accordingly,
by elimination
of the secular
terms,
we find
that
A(~,~)
is
determined by: (2S,19)
2 0[~__ 8A
We observe (~u~x} c
~
that
O e 0~]
+ E~
the equation ~
+
+
(25,19)
a~ 2
- ~
(~)
08
is an ordinary
of the equation (25, 13), namely:
O0 dLogA
00
--
[ 028
020 1
+ --
.
A
=
0
differential
equation
133 We
must
consider
point F(~,~),
those
characteristic
rays
which
eventually
reach
a given
having come from the origin (~=0,(=0) and having been reflected
many times on ~=0 and ~=I/B . Let s be the distance along the ray from the origin to the point P; through a straightforward computation t, we find as a parametrization of the ray:
A
.I/B
(25,21)
dt
~=sA, with s = 2~(n)J
m~. t)
0
A2]112
O0 + (-1)n
~o~
dt ~.oo(t) ] 1 / 2 . _ _ A2 [K 2
O0
where n is the number of reflections
experienced
by the ray before
reaching
the point ~, while ~(n) is the smallest integer greather than or equal to n/2. We observe reflections,
that k is constant and
that,
near
all along the
the ray,
origin,
even after
(25,21)
reduces
any number to:
of
x=rcosO,
z=rsinO, s=r, which shows that A is related to the angle between the ray and the horizontal at the origin (k=cosO). Let us denote by 8 (~,~) the value of 8 for the solution which corresponds
n
the ray which reaches the point ~ after n reflections,
A 1/B (25, 22)
8 . 8 . ~k .+ 2~(n)[ .
k 2 ]1/2 dt
.~ (t)
Jo LK o
n
+
(-1)
n
J ~
0
and near the origin we have: 8=p , where
we obtain:
.~m(t) .__
A 2] 1/2dt '
LK e
O0
P=Mor=(~2+~2)1/2. Hence
and @@n
= 0, which is the definition of A .
~--~IA=An t
Seep
Zeytounian
(1990;
Sect.13.2.4),
n
for
the
further
details.
to
134
Finally,
we obtain the following relation for the function An(~,~)$:
d r (25,23)
r2r~= (¢~
~21"2A1r
d~-~--~ L O g [An ['K-2 oo
nj
nJA=
const
=
~®(~)
,
n
where ^
(25,24)
An= 2~(n)
i,I,..[~oa(#t()t
.1/2 dt + (-l)n
.[~m(t )
O0
~1/2 d t .
O0
Then, writing ~(E)
(25,25)
=
exp{lllVco(t)dtt
we obtain the following relation for the coefficient An(~,~):
t/2 (25,26)
where
the same function that
0 is the
A1/2[#m(~)/K~o - A2] 1/4 n
n
easly checked
proper
(2S,26) and 8
n
(1-A2) 1/4~oo(~)G(A)
G(A)
with A=cosO as appears
(25,26)
matches
(~,~,X)
= ~8_X~ ~"
eu/e~ asymptotic
with
(24,6)
in (24,6).
and
Z A (~,~)exp{i[KooMolSn(~,~) tnE~ n
approximation,
It is then
where
the
+ n/[ - 4]}}
A
n
are
defined
, by
by (25,22).
$ From t h e solution (25,15) of (25,14) we can c o n s t r u c t another which is able A to s a t i s f y the b o u n d a r y c o n d i t i o n ~ = 0 and ~=I/B. That solution is
~o(~,~;Z) = ~ea~ ~ ~ A (~,~)exp{i[KooMolen(~,~) tnE~
n
* n~
where • stands for the set of n u m b e r s of all p o s s i b l e reflections which r e a c h the given point. At the origin the m a t c h i n g w i t h (24,8) is to be used, namely: r2K
-,,112
°° I
when p---Mor~ , for each n.
(coso .
-~} ~]}
for rays
,
135 The
main
purpose
understand
of
two-dimensional between
the
present
section
2S was
the r61e of Kozhevnikov-Miles's
the
describing
atmosphere
linear
(see
Helmholtz
lee waves
the
section
24)
(24,3)
and
equation
in compressible
twofold.
First
model of lee waves
troposphere.
and
we
examine
the
full
wanted
to
in an unbounded the
set
The answer
of
relation equations
to this question
has been given in section 24 and it may be stated that, Long's equation
(24,1)
and
to
Kozhevnikov-Miles's
inner approximation of the mountain
model
(for
the
semicircular
mountain)
for small mach numbers M 0 of incoming flow,
being of the order of M 0 in comparaison
belong
an
a length scale
to the height
of the
troposphere. The
second
boundary
point
that
condition,
answer
to
this
shown
that
question
the
The
(1959),
internal
scaling.
are
feedback the
going
investigate in
short
found
the
by
waves
according internal
are
point
to
lost
is provided
excite
alternately
The important
is
model,
waves
They
reflected
wanted
outward
Kozhevnikov-Miles's and Merbt
we
which
was
the
the
present
which
are
section
boundary
waves
generated
waves
to propagate
in the main part
and
may
along
from the upper and
be
the
upper
model.
2S
where
at
The it
is
infinity
rays
of the
in
of
equation
lower boundaries
as an artificial
troposphere.
approximated
is that these short waves propagate
condition
of
to the radiation condition of Lyra (1943)
one,
effects on the inner flow which is the only really
by
multiple
(25,13),
being
of the troposphere.
downstream
occurs on the inner flow close to the mountain.
upper
r61e
Kozhevnikov-Miles's
and that no
We should
having
understand
asymptotically
no
interesting one.
26 . LONG'S CLASSICAL PROBLEM
In
the
section
particular 14)
in
one obtains, streamline
case
a duct
with
curvilinear
8=z-z
+ __ + 2~ 8x 2 az 2 o being
the fluid
bottom
z=h
isochoric p
(x),
and
flow a
(see,
rigid
wall
nonperturbed density,
streamline
altitude
flow t h e we f o l l o w
density p t h e work
(@p), according
to the section
flow
supposed at
is
= 0
velocity to
infinity only
upstream.
infinity We
upstream
function of
Z
for X---)-00, and ; Z
O0
is
the
for °°the I s o c h o r l c t h e r e f o r e p--p00(Z00).Here
n o t e that
of Zco------Zm(~p)and (1969).
function
of Zeytounlan
at
be a l i n e a r
the Z=Ho,
for the 14:
02~
__
p-- t h e
two-dimensional
in the case U2 p = constant t, an equation of Helmholtz
02~
¢0
the
altitude variation,
(26,1)
t U
of
136
2=
with
__g
o
dpm
2 dz U p~
and the boundary conditions: Z = h (x) : 8 = h (x) , p P z--H
(26,2)
:
o
x --~-m
8--0,
: ~ ---) 0 ,
181 uniformly In order to build a numerical problem
(26,1),
conditions;
(26,2)
it
bounded for x --~ +m .
algorithm leading to the solution of the Long's
is
more
convenient
to
use
homogeneous
boundary
therefore new variables are introduced:
z/H (26,3)
with
~ = x/H
,
o
~ -
o
.(~)
-
i - ~(~)
~(~) - h p ( H o ~ ) H
o
Beginning with equation (26,1) one obtains,
(26,4)
for the function
8
~(~,C) = - H- + 71 + (1-~)C o
a partial derivatives equation with variable coefficients of the form :
a2@ (26,5)
+ 21_--~(~-1)-a~ 2
where
K2=H2~ 2 o o o conditions
(26,6)
a2~
and
-
~sdn/d~
i+~2(~-z) 2 a2@ +
a~aK
,
with
~((,0)= O, @(~,I)=i,
(1-~)
which
2
must
~(-~,~)=~;
aK 2
be
associated
the
boundary
137 the function @(~,~) being uniformly bounded in the whole infinite domain
R : {-~ s ~ s ~, 0 s ~ ~1}. It is obvious that the problem and the main difficulty
(26,5)-(2B,6)
can only be solved numerically
is that we cannot see how to take
into account
the
conditions: @(-m,~)m~ and @(~,~) uniformly bounded in the R? First,
we can try to bring the point ~=-m back to a point ~=~o'
upstream
of the obstacle;
but
the computations
carried
far enough
out t show
solutions obtained in this manner are only valid if the problem
that
the
(relative to
~), for equation (26,5) does not admit any increasing exponential components. Let us remark that the exact solution of problem (26,5)-(26,6) does not admit such components.
This is obvious if the linearized problem
examined with the assumption that the condition ~=h
(26,1),
is written,
p
(26,2)
not on the
profile z=h , but approximately for z=O. p In this case the solution of problem, uniformly bounded in the domain R, easily written in the following form tt
(26,7)
~(~,~)
= Ho ( ~ - I ) w ( ~ )
+
~ ~)sin(n~) ~ ~n ( n=l
}
,
where:
(26,8a)
~n(~) =
2
l
n~[Ko2-n2~2]1/2 _msin[[K2-n2~2]l/2(~-~')][ K~(~') K
+ W(g' )]dg' ,
1
(26,8b)
when n< --~0~[;
[In + ~(~')]d~'
:] '
K when n> 0
7[
"
We note that the function ~n(~) is the solution of the following equation 2
t
According
tt
See,
£or
to
Zeytounian
instance,
(1963
Kotschin,
2
and
1964).
Kibel
and
..
Ros6
(1963).
is
is
138 with the boundary conditions
~n(-m)=O and I~nl uniformly bounded for -m<~<+m .
(26,10)
PARASITE SOLUTIONS
Let
us
again
consider
the
equation
(26,9)
for ~n(~); the solutions of the
homogeneous equation corresponding to this equation are of the form:
20, for a l l
,26
+ BoCOS[[:-n2.2)"2] n
CoeXp[,[n2 2K:]"2]+ 0oeXp[[n2 2
for all n>K /~. o When we wrote the solutions
(26,8),
we have
, ~
the
solutions
(26,11),
for the nK /~; similarly when ~--~-~, the first of o 0 these solutions will not tend toward zero, and the second will not be bounded on the whole infinite straight line -~ ~ ~ ~ +~. Let us now suppose that we want to numerically solve this same problem and, for this, we transform it (relative to ~),
into a Cauchy problem;
that is, we
associate with the equation for ~ (~)i the initial conditions n
(26,12)
~n(~o)
: ~n(~o)
= O,
~0 being a point far enough upstream of the obstacle. However,
from solution
(26,8)
it can be seen that
the exact solution of our
problem at the point ~=~0 is different from zero and hence (the equation being linear) the error @(~,~) the boundary conditions
introduced into the solution through the transfer of (relative to ~) will be given by the solution of the
following problem: a2~
a2~
--+
Og2 Og2 (26,13)
@=0 f o r a¢
¢=e I ,
+ K2@ = 0, o
C=0 a n d < : i ,
a-~=e2 for
g=go'
139
where ~l(~o,O)~i(~o,l):O (26,14)
~2(~0,0)~2(~0,I)=0.
and
~i(~0,~) =
~ ~tn(~o)Sin(n~),
Hence one can
write
i=1,2,
n=1
and we look for the solution of the problem (26,13) a form similar to (26,7):
(26,1S)
~(~,~)
=
~ ~n(~)sin(n~). n=l
The ~n(~) satisfying the homogeneous equation
with the boundary conditions:
~n(~O)=~ln(~O) and ~n(~O)=~2n(~o). One c a n s e e t h a t increasing flow
for
n>K /H t h e s o l u t i o n
of this
0
compenents of the exponential
downstream
of
the
obstacle,
problem will
type which will
and
which
can
include
completely
even,
in
parasite
modify the
certain
cases,
as well
as those
nonlinear
problem
t h e c o m p u t e r memory. The c a l c u l a t i o n s of
Pekelis
(26,S), linear
(26,8)
relative
show
if
the
the
~)
if
that
equation
the
the
valid
out
(Zeytounian
numerical
situation
solution
that
has
in the nonlinear
(1964)) of
been
the
pointed
out
for
the
case.
p r o b l e m c a n be t r a n s f o r m e d
into
a boundary-value
problem
we a p p l y t h e c o n d i t i o n s :
~n(~O ) =
(26,16)
to
we h a v e c a r r i e d
concerning
problem is still
L e t u s now s e e (still
that
(1966)
(26,9);
~n(~N)=O,
0 ~
~N
being
a
point
for
enough
downstream
of
the
obstacle. Reasoning in a manner similar to that used before, one can see that, in this case,
and for n
i.e.,
we
shall artificially add to the solution of the equation the solution of the corresponding homogeneous equation with the boundary conditions:
5n(~o)=O but
~n(~N)=~2n which is in fact, a linear combination of sines and cosines of type (26, iia), whose amplitude is of the same order as that of the exact solution. The solution thus obtained, points,
will
fto~ ~
even though it stays uniformly bounded at all
toward
zero
perturbed the flow being studied.
for
~--->-m and
therefore
will
completely
140 THE N O N L I N E A R PROBLEI~
In
a
numerical
computation
conditions
(26,6),
to
variable,
the
C
differential
of
equation
it is convenient which
equations
to effect
gives,
in ~.
(26,5)
instead
For
that
we
satisfyin E
first
of
the
boundary
a discretization
relative
(26,5)
rewrite
a
the
system
of
equation
ordinary
(26,5)
in a
d/~form:
(26,17)
+ (C-l) ~
= (1-~)~[(1-C)~ To r e s o l v e
equation
(26,17)
ilim
(I-~)~
+ (C-l)
- ¢ + C].
w i t h t h e boundary c o n d i t i o n s :
~(~,0)=0, (26,18)
+
0(~,1)=1,
~(~,~)=0,
llim O(~,~)<~,
we are
going
the v e r t i c a l applied
to
make use o f
coordinate
the
method o f
"
~
, ~ "
according
C. T h i s method, w h i c h i s due t o D o r o d n i t s y n
t o the r e s o l u t i o n
of dlfferential
to
(1958),
is
k Ck=~ ( k = 1 , 2 . . . . .
n)
e q u a t i o n s o f the t y p e :
a~ aC - g(C)-
(26,19) Integrating
equation
(26,19)
with
respect
to
C from 0 to
gives
~kWe r e p r e s e n t
Ck
=
~o= ~o gCC)dC m
gk'
(k=l,2 .....
g(C) i n t h e f o r m o f a p o l y n o m i a l
n).
taking
the v a l u e s gk = g(Ck):
n
(26,20)
E AsCS;
g(C) = go +
9=1
in this
case we may w r i t e
(26,21)
that
~k = B go +
~
,
k = 1,2 . . . . .
n.
S=I
In writing equation
the relation
to determine
(26,21)
for
the n parameters
A i , A 2 .....
An
all
the
k values
we obtain
a system
of
141 as
a function
into
(26,20)
of ~o and ~k" Finally
we w i l l
obtain
by putting
the expressions
for these A s
the relation
n
~'k= pok/,tO + ~ pks ~,, , k=1,2 . . . . . n.
(26,22)
8=1
The c o e f f i c i e n t s
Pko a n d P~, b e i n g f u n c t i o n
o f n,
are determined
once and for
a l l . For example, the calculations give (see, Zeytounian (1968)): n=l : pl= -I and pl= 2 ; o 1 p1= 1 p1= 2, pi= 1 n=2 : o -2' I 2 2 ; p2= i n=3 :
p2= -8, p2= 4 ;
0
'
pl=
1
o
-3'
p2= o
1
2
pl= 3
1
~'
1
p',= 3
2'
2
pl=
1
3
-6 ;
2'
p2= - 6 p2= 3, p2= 1
2
p3= -1, pS= 27 o
"2'
1
3
pS=
27
2
--'3'
2
~;
pS= 13 :3
"~
;
For n=m we will have m(m+l) coefficients pk with k=l,2 ..... m and s=1,2 ..... m. s
Using
(26,22)
equation
and
from
the
fact
that
~
k - ~0 ~k = Polo
Let us @k = @(~,~)
~
will
obtain,
in place
be the value form
~ P ~s'
with k=1,2, "" . 'n.
of the stream
(26,17),
function
at
in place of the partial a ~
~
ond/non~
the
level
derivative
~
~k=k/n, equation
~
te
@k' U~tA k=l,2 ..... n.
To do this we apply the relation
(26,23),
by writing equation
(26,17)
form of a system of two equations:
(26,24)
of
s=l
with the purpose of constructing, divergent
n
pks +
s=l
the
we
(26,19), a system of algebraic equations of the form:
(26,23)
in
~k=Zk-EO
a@ _ (1-n)~ ;
at
a2~o
(26,25)
= -(1-n)--
+ 2~ ~
+ n ~b +(
a~ 2
in the
142
we o b t a i n then,
(26,26)
t a k i n g i n t o a c c o u n t t h a t ~omO and ~nml: n-1
<1-.,,)<,><:1=o'<1--,,)<.,o+ ~:,%+ ,"; 6----1
d2~ k
-<,--,,>
+,.:'"<+-~,,,,<+ <,--,,><:[
d~ 2
d~
: ,=<~<1-.,,><- <1+-,:')<,>o, ,~:= l ,,'<+ ~ ,~' " ,--'l " L L nJ J n
2
s
(26,27) n-1
k
S
L d~
~=I
k=1,2 ..... n.
With t h e a i d of e q u a t i o n
(26,26)
we can e l i m i n a t e
from e q u a t i o n
(26,27)
all
t h e ~k (k=1,2 . . . . . n); we t h e n o b t a i n one s i n g l e e q u a t i o n c o m p r i s i n g Ck and Uo ) namely:
(26,28)
[
d2~k
(1-~)--
- 2~ d@k - ~¢k - (1-~)K~ (1-~)~-~
d~ 2
d~
n-if k
+ F~¢s
+ ~, t E" d ~ s=l
n
1-~
d~
~
n
}
= Gk~
o o
s n 1+ 1 -
+ K P (1-~)~;
S=1
E"=,2,~(s-1), (26,29)
Fkl m~l s
.-.
pkpm 1+ 1m=l
m
s
[In'-) ~~2 ]
Gk= (1+~2) ~ pk _ 0
k=l,2,...
s=l
,n.
S
P P s=l
+
1+ 1-
s~ ~
1)
~<~
'
+~l
k nJ
+ Dk
n
143 To e l i m i n a t e
w ° from
(26,28),
we r e m a r k that for k = n w h e n ~k=l we o b t a i n Fn
(26,30)
_
m°=
From ( 2 6 , 2 8 )
_°: . °i 1
Gn + Gn o o
and (26,30)
for
@k w i t h k = l , 2 . . . . . t the following type :
strictly
2~
d~2
In this
last
d~ + k
(26,31)
we o b t a i n
us a single
a differential
equation system of
~-~o ~~_~ [~-~-~+~]
I-D d~
equation
mo' w h i c h g i v e s
speaking
d~k
+~k
.
Gn o
we c a n e l i m i n a t e
n,
d2~k
(26,31)
d~ o
the
)
~k..
=
coefficients
~k s,n
k spn
and
xk n
are
of
the
form:
s,n
(26,32)
s,n
=
=
Nk =
1-71L s
1-~[ s
I
-_9°E
;
Gn O
-~°
Gn o
F
;
O Dn _ Dk _ ~ n
O
.
n
o
o
NONLINEAR MODEL WITH ONE
INTERHEDIARY LEVEL If we c o n s i d e r level k=l,
~i=I/2,
the more s i m p l e case of a model between
in the s y s t e m
~o=0
(26,31)),
and
~2 =l
for
e q u a t i o n for the f u n c t i o n
According
to
Zeytounian
(1964,1969).
(then
the s y s t e m
nm2
with only a ~ and
(26,31)
k admits we shall
intermediary only
one
obtain
value,
only
one
144 This equation may be written in the form: d2¢ 1 (26,33)
de 1 + p(~)
- -
+ q(~)@1
d~ 2
r(~),
=
d~
with:
2~ rl . 2 ] = --c~-[~.~, - 1 ;
p(~)
(26,34)
r1+3 +s
+
where
~(~) = ( 1 - ~ )
1 + i-~ ~ 2
. 2+3
.
211 jI.
g
Equation (26,33) must be solved with the boundary conditions: for ~---)--~: ~I--~; @I- must be uniformly bounded in the
(26,36)
Lwhole
The c a l c u l a t i o n s negative), process
at
internal
carried least
by which
the
for
~[-~,+~].
out
have shown that,
when the
one
value
the
conditions
of
~,
(26,36)
in are
transformed
q(~)~O
(is
then
the
a condition
of
coefficient
internal
[-~,+~], into
Cauchy: (26,36)
1 ~1(~o ) - 2 '
d~l 0 , d~ ~=~o =
with ~=~o being a point sufficiently distant no/ ~ .
This procedure
upstream from the obstacle,
(for a model with a single
if the parameter K 2 satisfies the relation o (26,37)
o
s
.2
Ls
level) remains coccect
L12-
jjj
"
145
Once ¢1(~) calculated
is
calculated
by f i n i t e
approximately at
differences
any level
method,
Cm=-~(m=l,2 ....
then n-l)
Cm(~) w i l l
by means o f
be the
f o l l o w i n g p o l y n o m i a l o£ t h e s e c o n d d e g r e e
(26,38)
@m(~) = [ 4@i(~) - 1 ]Cm- [ 4@i(~) - 2 ]~: .
F i g u r e 10-14 i l l u s t r a t e
the method p r e s e n t e d above.
F i g u r e 10 shows t h e f l o w above and downstream o f a t y p i c a l calculated
isolated
obstacle,
f o r K2=25 . 0
8 6
0
f
2
~
~a;. 1o: ~
6
~
#
and ~
tO
/2
~
o~ a~ ~
I
~
I
~S
I
~
~8
i
a~tazAe ~on K2=25. 0
I
2O
l
146
We o b t a i n
as seen downstream of the obstacle,
the eddy vortices
(closed
of
leading
the
obstacle,
streamlines) to
a
a regime of rotors
are periodically
configuration
liberated
analogous
to
during
which
by t h e c r e s t
that
of
the
Von
Karman v o r t e x . Figure put
11 r e p r e s e n t s
the
in our numerical
analytically single exact
in
result
curvature
of
level
of
Long.
an o b s t a c l e
downstream of the obstacle
1
A
We s e e
already This
an asymmetric
on t h e
example
/r--/,l~
I~
that
the
model
with a
approximation
shows
formation
We h a v e
o f Long ( c a s e a) o b t a i n e d
a satisfactory clearly
obstacle.
the
of
the
influence regime
to
the
of
the
of
rotors
( s e e F i g . 10 f o r c o m p a r a i s o n ) .
l
16
of
in this
gives
example
upstream
-.....~£a..) ~
18
downstream
( c a s e b) a l o n g w i t h t h a t
investigation.
intermediary solution
flow
12
I0
0,~.,,_,~ t018
,9
6
~
Z
#
Z K~
~;/x~. 11: ~Oag~ ~
a~ an. o/~aZav/e:
(a)-
~
2an~'a ~
(K~=24);
(b)-
~nam ~%e m a d ~ ~aa ~%6a / z v e e a t i q ~
(K:=24).
147 I n F i g u r e . 12 we h a v e r e p r e s e n t e d waves above Sierra-Nevada and
(e),
constructed
with real are
flows.
different;
infinity
0
0.9Oj~ 0.3~ 0.1
o
l
#
~
and also
solid
~
to
that due the
the values to fact
(a),(b)
values
and (d) o f t h e r e l i e f
of
K2 a n d a l s o
(Long
(1959)),
the
0
for
flows
(c)
comparaison
o f K2 i n t h e c o r r e s p o n d i n g f l o w s o in the speed profile at
differences that
our theoretical
models define
the
surface.
Z
'
,~
out
partly
!
_~
a
is
upstream
the calculation different
f r o m known o b s e r v a t i o n s
We p o i n t it
flow by a plane
for
5
3~
~<; =so
~
~
5
7~
6
~/'/k72~////H.,'/7,<,
~
~
~s
z~
k,,, az o
a
~L~. 12: ~uJ.~ a&ox~ YLen.m~-XepBad~: (a)- ~ K2=60 (b)- K2=90 0 ' 0 ' (c)- ~ ~ ~ K2=56 , 0 (d)- ~ . K2=30 (e)~ 0 K2=19. 0
~
2~
148 Figures
13
and
14
illustrate
the
method
of
Long
(see
below
the
remarks
concerning this method). Long could have obtained the flow of Fig. 14 if he had been
able
to
resolve
the
problem
condition ~=0 to the profile:
correctly
z=O.175(l+cos~x);
by
applying
the
since he did not
em:ec~ slip take this
condition into account but wrote a linearized condition he obtained the flow of Fig. 13 which is the flow above an asymmetric obstacle and has nothing in common with the flow sought in Fig. 14
~ 4 . 13: ~&~a~ ~
~.
aa ~
14: ~&2a~ ~
~
e ~
~
~
(
~
K2=26. ~A~ ao/m2/aa o~ .~a~ o
z=O, 17S(I+COS~X)
~ ) .
~o~ K2= 2S o
149 Long resolves the linear equation (26,1) with a linear slip condition: ~(x,O)=h (x) instead of the nonlinear condition (26,2). We then obtain an p analytical solution of the problem; but this solution is, in fact, only valid for values of h (x) which are P
intrinsically small.
However,
Long
takes for
h (x) a profile with a height of relative importance: h (x)=O,175(l+cos~x) and p p calculates the flow for K2=25, which gives him the flow in Fig. 13, where the 0 obstacle is represented by the stream line @p=O. The real flow that should be obtained for the obstacle h (x)=O,175(l+cos~x) with K2=25 by taking account of p
0
condition (26,2): ~(x,h (x))=h (x) is represented in Fig. 14. p p The method of Long enables the nonlinear flows to he uniquely unknown obstacles apriori-for example, flow
(nonlinear)
for
an
asymmetric
found
for
in the case of Fig. 13, Long obtain a
profil
whose
equation
is
o
h (x)=?;
the
condition @ = 0 being uniquely fulfilled on this unknown profile.
As for the
flow for the symmetric profile of Fig. 14, Long resolved nothing.
It would be
P
necessary
for
him
to
choose
a
certain
profile
such
that,
using
his
calculation, he had @p=O for z=0.175 (l+cos~x) which appeared troublesome!!
27
. MODELS FOR LEE WAVES THROUGHOUT THE TROPOSPHERE
Let us consider the case of lee waves involving the entire thickness of the troposphere: H0=RT (O)/g, and in this case Bo = 1. dT It can be assumed for the troposphere that: - ---~--F0 dz T
(z)
=
1 -
{R
with F~=constant,
F0 m z~ (with the dimensionless variables )
In this case of constant velocity upstream linear temperature profile,
we have that,
(for unperturbed mean flow)
in the equation
(23,13),
Bo2 1
r~r-I
dT ],
L(z-zLJlT*dz--J
-
,
o
where Bo ~'- 1 2 --[~'-IBo-/~ l 2:oo: ~,M2L ~" oj ' o
As
Bo = 1 a~e ~
that:
.o-
Ho
T(o)/r o
and
the term
proportional to
(27,1)
i.e.
150
B°-~o
(27,2)
2
~'Mo
2 O(I) , Zoo=
- 0(I) ~
when M --90. 0
In this case we obtain that
(27,3)
_~0 ~
7-I
Bo , when Mo---~ O.
NONLINEAR MODEL
We assume here that o=0(I) and in this case with (27,2) and (27,3), the asymptotic solution of problem (23,12)-(23,14) (27,4)
we find
in the form:
6=~o+ o(H o) , W=~o+O(Ho).
We then obtain, from ( 2 3 , 1 2 ) ,
(1+~o)~-1 = 1 -
Or
(27,5)
~o
:{
~'- 1Boo.~ ~" o 1-~-lBo(z-~
l_~'-lBoz
},_1
o
)
1
l~'-lBo(z_Cr~o)
'
where 6 satisfies the following limiting equation S 0
(27,6)
[1-~'-1Bo ( z - c r 6 ) ] , / B 2 6 ° L
•
o JLax 2
+
O26o
1
az 2
60 OWo +
l+~oLax
ax
o
o
az
az
1 8~olI ~zjj + (l+~o)2BOAoo~o = 0 ,
Aoom (~-I Bo - ~ o ] / ~ ° .
where Combining
(27,5)
and
(27,6),
we
obtain
a
single
function 6 o , which is very complex and nonlinear. 6
0
equation
for
the
The boundary conditions for
have the form (see (23,14)):
t From (27,3)
the .
exact
equation
(23,13),
take
into account
single
the
relations
(27,2)
and
151 ~o(X, Crh (Ax)) = h (Ax), when •p
(27,7)
P
~O(-~,Z00) = O, ~o(X,I/Bo)
Ixl
= O;
llm [ a~° + a~° ]<~.~-x_)+~ t t~x I o z • The
nonlinear
problem
two-dimensional boundary
(27, 5)-(27,7)
perturbations
layers
above
a
describes
throughout
locally
cuFved
the
the
mesoscale
troposphere
surface
of
outside
arbitrary
steady of
the
form
and
e Ievat ion. LINEAR MODEL
Equation
(27,6)
is linearized
on the assumption
that
the obstacle
height
small: ^
(27,8)
v<<
1
=~v=VMo
2
~
ho
o
since Bo ~ I. When v--~, we obtain the following linear problem, 2
in place of (27,5)-(27,7),
2
a__~o + a ~o + D(Boz)~o= O, ax 2 az 2
~o(X,O) : h (Ax), P
(27,9)
~o(X, 1/Bo) = O; ~o(-m,z
) = O;
x_~+co~ lOx i where
Ixl
@z
•
I~o:{ 1~iBoz 2(7-I) } ao; 1
(27, io)
=
2~,-I 4~ 2
1-
oz
.
is
152
Under ordinary meteorological
conditions,
D(Boz)
problem (27,9) duplicates that of Dorodnitsyn
>0. We note that the linear
(19SO).
The solution of (27,9) is of the following form: (27,11)
~o(X,Z) = Z
n=1
".CIZ " z = O
@ (x)z (z) n
n
where (27,12a)
@(x) = - ~ n l
exp[-~nlX-X'l] hp(AX')dx',
when~>0
and
~nCX) = 2~--~sin[~n(X-X')]hp(AX')dx',
(27,12b)
when 2 = n
n J --~
The
eigenfunctions
Sturm-Liouville
Z(z)
and
eiEenvalues
2
Mn
satisfy
_~<
the
O.
following
problem: dz 2
n+
~(BOZ) Xn= O,
(27,13) Zn(O) = Xn[ 1 ]
= O.
When Bo--)O, both the height z and the slow height ~=Boz appear in this problem (27,9).
Therefore
the perturbed solution is in fact singular
(when Bo --) 0),
since it incorporates two different length scales for height. It
would
be
(27,S)-(27,7)
interesting
analyze
the
behaviour
of
nonlinear
problem
in detail as Bo -~ 0 and A --->m so that O0
A Bo = K2= 0 ( I ) .
O0 O0 This would enable us better to understand the essential of the classical Boussinesq approximation. The numerical solution of the nonlinear problem (27,S)-(27,7) interesting
nonlinear
effects
for
the
lee
developed only partially by Pekelis (1976).
waves
that
have
should bring out thus
far
been
153
REFERENCES TO WORKS CITED I N THE TEXT BOIS, P.A.(1984) _ Geophys. Astrophys. Fluld Dynamics,
29, 267-303.
DORODNITSYN,
A,A.(19S8) _ in Proceed. of the Third All Union Math. Congr.,Vol.3,
DORODNITSYN,
A.A.
447,
(Akad. Nauk SSSR, Moscow).
(1950) _ Trudy Ts.I.P, 21 (48), 3-25 (in Russian).
GUIRAUD, J.P. and ZEYTOUNIAN,
R. Kh.
(1979) _ G e o p h y s . Astrophys.
Fluid
Dynamics, 12, 61. KOTSCHIN,
N.E., KIBEL,
I.A. and ROSE, N.V.
(1963) _ Theoretical Hydromechanics Vol.l, Moscow (in Russian).
KOZHEVNIKOV,
V.N.
(1963)
_ Izvestlya AN SSSR,
Seriya Geofizlcheskaya,
n°7,
1108-1116. KOZHEVNIKOV, LONG, R.R. LONG,
V.N.
(1968) _ Atmospheric and Oceanic physics,
Vol.4, n°l, 16-27.
(1953) _ Tellus, 5, 42-57.
R.R.
(1959)
_
in the
" R ~
~ea~
VOW";
Rockefeller
Institute
Press, New-York. LYRA, G. (1943) _ Z. Angew. Math. Mech., 23,1-28. MAGNUS, W.(1949) _ Seminar der Unlversit~t Hamburg. MC. INTYRE, M.E.
(1972) _ J. Fluld Mech.,S2,
Vol.16, n°6,
I/2 may.
209-43.
MERBT, H. (1959) _ Beltr. Phys. Atmos., 31, 152-161. t MILES, J.W. (1968) _ J.Fluld Mech., 33, part4, 803-814. PEKELIS, E. (1966) _ B u l l . Acad. Sci. USSR. Atmos. Ocean Phys., 2, 689. PEKELIS, E. (1976) _ BulI.Acad. Sci. USSR, Atmosph.
and Oceanic Physics, 12,
n°6, 470-477. THOMAS, N.H., and STEVENSON, WILCOX, C.H. YANOWITCH,
T.N.
(1972) _ J.Fluld Mech. 54, 495-506.
(1959) _ Arch. Rat. Mech. 3, 133.
N. (1967) _ J. Fluid Mech. 29, 203.
ZEYTOUNIAN,
R. Kh.(1963) _ Trudy of the Centre of Meteorological Calculations, Moscow, Note i, 72.
ZEYTOUNIAN,
R. Kh.
(1964)_ Bull. Acad. of. Sci. USSR, geophysics series,
ZEYTOUNIAN,
R. Kh.
(1968) _ La Recherche Aerospatlale,
ZEYTOUNIAN,
R. Kh.
(1969) _ The Physics of Fluids, Supplement
N°127, 59-61. II, Vol. 12, n°12,
part If, 46-60.
t
see
also:
BILES,J.W.Ctgsg) _ W a ~ e a and ~
~
~
Proced. twelfth Int. Congr. A p p l . Mech., S t a n f o r d . W.G.Vincentt, Berlin, Sprlnger-Verlag, pp. SO-76.
sept. 865.
at,w.~tl.~ed C ~ . Eds. M.lleteny[
and
154
ZEYTOUNIAN, R. Kh.
(1974) ~o~.
Lecture
Springer-Verlag, ZEYTOUNIAN,
R. Kh.
(1979)
_
Izvestiya
of
Notes
in
Physics,
Voi.27,
Heidelberg. Acad. of
Sci. USSR,
Atmospheric
and
Oceanic physics, Voi. 15, n°5, 498-507. ZEYTOUNI AN,
R. Kh.
(1990)
_
+g~/,t~at,e,t/~
m,ea~_,gLr~
~
~
S p r t n g e r - V e r l a g . H e i d e l b e r g ; C h a p t e r 13.
~o~z~.
CHAPTER VII BOUNDARY LAYER PROBLEMS
28
. THE
We s t a r t
EKMAN
LAYER
from t h e s o - c a l l e d
non-adiabatic viscous, (x,y,p)
atmospheric
equivalent
hydrostatic motions t.
These
of the equations
independent variables,
.ra~t -~ $1[~. ~ Ki~-V+ (28,1)
equations
(9,1)
for
the ~
equations written
are
v i s c o u s and
the
non
at the section
adiabatic,
9, u s i n g t h e
namely:
+
8~pp~}+ [ 1 + ~
Kly ]I
~
A
~] + ~l° Bo ~ Ki
BO2XoKi2 ~a[pav~ ;
:
(28,2)
~ ~.~ + 8~--~ = O;
(28,3)
T = -Bop
Kt {~8T
+
8_ _p°'
1Iv ~T+~ 8I~-l ~
S
.
A
(28,4)
The s i m i l a r i t y
P r ~-1 Ki 2
8v~ 2
er dR ~
S-~oP~- OdpY
relation Ki m Ek± = xoKi 2 ' with Xo=O(1) ' SRe----~
(28,5)
is
B°Zx°KiZ~'8- F 0T
-
motivated
by
the
fact
hydrostatlc-Navier-stokes
that
it
equations;
corresponds this
to
the
similarity
least
degeneracy
relation
o b t a i n e d by Z e y t o u n l a n (1976). t
See
the
(6,2)-(8,7)
section
8
(Chapter I I ) . For ~ / t ~ e r t ~ a t m o s p h e r i c motions, out the limiting process: ~0--)0 (~0---~0).
we c a r r y
in
has
of been
156
We note that (28,6)
~o/P~(O)
Ek±=
,
f
H2 o o
is the vertical Ekmannumber. The complete
derivation
Guiraud and Zeytounian principal first
expansion
of these
of the quasi-geostrophic (1980) asymptotic
(see,
theory,
the section 9,
model
(when Ki--~)
uses concurrently
Chapter
leads to the problem of adjustment
If), ~
in the
with the
local
ones:
the
to geostrophy
(see,
the
section 21, Chapter V) and gives the initlai condition that must be supplied for
the equation (9,14),
the second one leads to the Ekman steady layer and
classical Ackerblom's problem which gives the boundary condition at the ground that must be supplied for the main equation (9,14). The second inner (local) region corresponds to IP-lJ:O(Ki) and we call it the gkman (a22.ed~) region (see, the sketch below).
!
I
Within the inner steady ^ l-p p = - ~ and we have: (28,7)
Ekman
region
the
independent
variables
Ki--90, with t, x, y and ~ fixed.
a If we take into account that: -- =
ap
1
1
, and we assume that:
>h are:t,x,y,
157 .
•
A
v -.)
'
-1%
v I II ,
v-) 0
J ~
^
(~
(28,8)
b)o
J{
I
.
o
+
Ki
. . .
4o
T
A
P then,
from
Po
(28,1)-(28,4),
we
obtain,
as
first
approximation,
the
following
relations: 8~° - 0 ,
~ o = o , a~
o _ 0,
a~
(28,9)
Bo2
a
~o~o = I , F~ Xo But
for
the
full
equations
.^ ago. o
(28,1)-('28,4)
^ ag o
o a~
we
have
on
the
flat
ground
following boundary conditions:
(28,10)
As consequence
aT = ~ro~=(p), on R = 0. p ~-~
8 R- , ~ = Bop S ~
~ = 0 ,
of (28,10),
we have
a# (28,11)
a~
o _ 0 , e~ ~ = 0.
o
From matching with the main region,
(28,12)
A
when p--9 m, we get:
(~= ALim 40= TO(1), ALim ~0= ~ 0 ( I ) , 0 0, p-~o p-~o
but ~ - 0. 0
Hence, f o r the next o r d e r we f i n d the f o l l o w i n g s e t o f e q u a t i o n s : A
^
0
Ov~o
XoB°~I + C~Av~) o - Bo2x o ~p[po(I) ~p--p] = (28, 13a) Bo 2
Xo
0 f
0~1- I =
°;
O;
the
158
a~
^ O'
(28,13b)
a~ Bo a p
1
= T°(1)'
and from (28,10) we have also
^
a~
^ ~ Vo= O, ~1 = ~ S o
(28,14)
if we suppose
that
at
1
1 , To(1 )
the main radiative
a~ a~
transfer
1
- ~
R (1) o =
not
have
o~
= I
O,
of Ekman
boundary
layer structure. We note that the flat ground in the Ekman layer is characterized by (28,15)
= ^Pgo + Kl ^Po1 + . - .
From (28,13),
(28,14) we obtain, after the matching with the main region,
(28,16)
~l=Z
where T
qg,l
The l a s t
eT
qg
equation
a~
gives:
T (1) o Bo - Const
imply ^ Bo . pgo=-T-~).qg, x,
(28, 18) where
dT° 1 ~ , ~ p=l
of (28,13)
__L_1 _ a~
~ =0 1
+
(t,x,y,l)=-B ~ q g l (see, the relation (9,11)). o[ap Jp=l
(28, 17)
and
qg,1
~
qg, 1
= ~
qg
(t,x,y, 1).
~
~i
=R
+
qg, 1
T (i} o ^ ~ P
159 THE ACKERBLOM'S PROBLEM
A We consider now the first equation, for V~o, of ( 2 8 , 1 3 ) . We set A ~= v 0
A + ~'
qg,l
, with ~
0
m ~ ( t , x , y , 1). qg,1 qg
From matching with the main region, when ~--~o, we have A Lim v~'= 0 Ap-~
and
XBo ~ 0
+ ~ A ~ v = 0, qg,1 qg,1
and we obtain, from (28,17),
X
0
B ~ 0
1
A A + ~ A v~- ~ A v~' 0
0
A As consequence,
for ~' we get the Ackerblom's problem: o A a2~ '
BJx O _ ~ A ~ , o a~2
A
(28,19)
~' = -~ 0
qg,1
A
~'o~ o, , ~
A
o
=0,
on A Bo ~qg,1; P = -To(1--~)
~
--> ~ .
The solution to (28,19) is obtained in a standard way:
(28,20)
A
A
~'o - i~ ^ ~'o = -(v, 1 - ~ i~ ^ ~,I)~.,
where i m (-1) I/2 and
(28,21)
~- = exp{-
1+i
2Bo~x [~+
Bo
~
q0,1H" ]l
o
We consider find for ~
now the equation of continuity, the following relation:
for the system
(28,13),
and we
160
A A
A ~=
~_~1]
Bo
(~. i ) d ~
1
+ s To-j~L--~-Ctj~:~ , go
Pgo
~0: ~eXl~{[V~qg,1-i~A ~g,1](1-E) }"
where
A
^ = tp-PgoJVqg,l "^ ^ "~ VodP + /Bo2xo ( ~ A ~ v o),
But
Pgo and for ~ ,
we get:
A
(~.L)d~= ~.
~
A V0),
^
-> A
VodP :
o
PgO
Pgo since ~. v~
qg,1
:o,
.~A -- Bo qg,1 U P g o = - ~ V q g , l " ~ ' f q g , l = O " Therefore
^= ~I
(28,22)
since
=
L1m ~
p-~o
8~ 8~ at * _ at qg,1
I
=
S
~
Bo
is independent
8R
8t
qg,1
0
0
we have:
s0~
^= - w qg,1 = ~ qg (t,x,y, 1) : -Bo ~1
Finally,
we
derive
the
boundary
supplied
to the main equation
qg,l'
of 8.
But, from matching with the main region,
(28,23)
0
condition
(9, 14), namely
at
the
flat
~ a p Jp=l. ground
which
(with Xo---~oto/P~0(O)H~):o
must
be
161
(28,24)
a onp
where v ~
qg
a
= 0,
= I,
-=A B (2 A DR O O
1
qg
). This
last boundary condition
different of the slip condition (9,16)
(28,24)
is natucally
and take into account of the
of the Ekman boundary layer on the main quasi-geostrophic flow which is ruled by the equation (9,14). 29
. MODEL
EQUATIONS
FOR
BREEZES t
Here we consider only the influence of a localized thermal non-homogeneity of the flat ground surface. We assume on the ground, z=O, with the dimensional variables, that: (29,1)
T = Tm(0 ) + AToE[t_t ° x _~] 60'
where E is a known function describing the temperature field on the ground in a
localized
region
characterized
by
the
length
scales
~and
m °,
at
the
proximity of some origin (0,0). For the breeze phenomenon it is necessary of take
into account
the Coriolis
terms in the dynamical equations and for that we must assume: (29,2)
6°= m°z lOam.
On the other hand we have,
in our breeze phenomenon,
value of the vertical scale:
^% RAT __~o<< RT g
g
and
AT
0
T o- ~
(29,3)
<<
oo
In this case: t
Accordin9
to
Zeytounian
(1977).
1.
(0)
the following particular
162 ~o (29,4)
c = -- << i, ~o
since ATo<<~° ~, and we may consider the main limiting process (see,(6,1)): (29,S)
e --9 0,
in the
full
equations
Re ---)m, (4,24),
section 4, Chapter If), o
with e2Re E Re±= 0(I),
(4,27),
~
(4,30),
(4,31)
and
(4,33)
(see
the
~%o/ ~ ~ 0. o
We note, that from (29,6),
(29,6)
ITJ
' with °°
and (29,7)
Uo
~
As a m a t t e r
1 (RATo) 2 Re± g2 v 0
of fact
corresponding
to the typical
U ~ 5 m/s , u ~ 5 m2/s o o
and
values:
AT - 10°C it, o
we have 60~ 10Sm.
Finally,
for
t o we u s e ~ - l s and i n t h i s o S e
if
~o
is
a
case:
1 ~ 1 2Ro s i n ~ o
mid-latitude
and
R°=U/2floSin~o~v
is
the
Rossby
number
corresponding to ~. But we need also to specify the value of main characteristic
velocity Uo; we
can suppose that, for the breeze phenomenon,
(29,8)
Uo=~
, 'Z:o-A T o ~
and we introduce a Grashof number Gr±,
related to the Reynolds number Re± by
the relation: t
In
this
case
our
dimensionless
variables
Naturally ~ m~°/a <<1. 0
tt We note that ATo~
0
-U
if Rel~l.__ o
are
:t/t °, X/i °, y/m ° and Z/~ O.
163 G r l m Re~ : 2 ~RAT~(~0)20
(29,9)
Tm(O)u 2 0
Finally, our Boussinesq number Bo, in this case,
is given
by
the relation:
AOg
(29,10)
To~ R--TTOT= Bo << 1,
and we can consider the Boussinesq-Zeytounis.n's, T
(29,11)
To---> 0,
Mo---->0;
0
main limiting process:
A
~-- = 'r = 0(1), 0
with' Now,
-= I. if we repPesent
(4,27),
(4,30),
the solution of the full Navier-Stokes
(4,31) and (4,33),
equations
(4,24),
where ~ m0, by asymptotic expansions of the o
form: U' =U
0
+...
,
V' =V
0
+..
•,
W' =W
0
+...
,
(29,12)
A
with Z =~MoZ,
we can easily show that the function Uo, Vo, Wo, ~2' ~I and 81 (as functions of dimensionless vBriables t,x,y and z), under the hydrostatic
main limiting process
(29,5) satisfy the ~
I sa~o+ C~o.~)5+Woa~o+ [~o~yi(~ 8t
(29, 13a)
0•2 az
,
s,.oo
uj
8z
=
A
(91;
Z
~
~
:
a2v~
A V~o)+ 1~= Cr~W2az2°''
164
I ~'~0 o~° CqW
+
(29,13b)
: O;
' + ~ ~+
s
t
o
o
and oJl=-e I ' V~o=Uo~ . VoW, ~=a._. 1 . aT oq "> ax ~ J' These
equations
(29,13)
can
be
Navie-Stokes equations
(4,24),
under
(29,11),
the
instance,
conditions Zeytounian
fields) (29,13)
(4,30),
the
~oI~ ~ o ~ ~ e we
~
an
inner
(4,31)
The
outer
trivial
~
which d e t e r m i n e s t h e ~
Therefore,
1
- -
[dz/z==O j o
as
(29,12).
gives
w=
c~z2
~.~'=0.
considered (4,27),
(1977))
829
+~--+
degeneracy
and
(4,33),
(see,
(there
of the solutions
full
when 3o~0,
degeneracy ~
of
for is
of the equations
z=O.
must
consider,
for
model
equations
(29,13),
the
following
boundary c o n d i t i o n s :
IZ =0: ~Vo= Wo= O, el= A T = (t,x,y); (29,14)
~
+~: Vo= Wo= ~2 = Or---> O; 2
]x+y It
is clear
impossible
that
the
to specify
2
]---> ®: v = w =
0
order
of
o
may not
breeze
become
the condition
w ~0 0
at
zero at z=m! z=~
for
all
~=
2
e -->o.
1
equations
analogy with problems of Prandtl's w
0
(29,13)with
for w ° at z==.
boundary
If
It would
layer theory,
In many published t>O.
respect
one
a
z
makes
it
appear
that,
by
in the given problem
solutions
assumes
to
n
of the problem of ~
stratified
atmosphere, i.e.,
dT
(29,15) the above However, condition
(29,16)
~ ~-1
- dz= z =0 cicumstance
~ '
does not result
in any contradictions
T z =0 ~ ~-I if - d d-~ ~ ' then by virtue of equations e--, 0 for z--~ = we should have I
w=O 0
for z==.
in the equations.
(29,13),
for el, and
165
Note
that
condition
satisfied
~
(29,16),
.
not
Integration
(29,13) with respect
c 9,17
being of
a
boundary
continuity
condition,
equation
should
of
the
to z from 0 to m, while making use of (29,14),
"I
o
be
system
yields
0
o
We
notice
that
"boundaPy-layer" of: - d T d-~mz
~0
equations
(29,17)
"antibreeze"
(29,13)-(29,17), where
condition
(29,17)
(29,13)
but
is
it must
not be
consistent
enforced
as
with
the
a consequence
< ~-i since 8 ---) 0, with z--) m. ~ ' 1
The constraint so-called
the
gives
the
possibility
over the main breeze.
over
the
thermal
spot
to obtain The model
simulated
by
formation
problem the
(x,y)<~ o, has zero as a solution when E(O,x,y)~O
the r61e of initial conditions
the
of
the
of the breeze
function
E(t,x,y),
and this solution plays
for t=O.
The fact that the lower atmospheric
layer is generally stably stratified
leads
to a new solution as compared with the case of the neutrally stable stratified atmosphere
(see,
for this case,
the origin of the antibreeze The
breeze
model
constraint
problem
(29,17),
problem,
since
velocity,
w o, at z=m.
The presence in general fact
that
is not small
main breeze.
the
in equation
well
(29,13)
different
have
in a stably
(antibreeze),
have
we
is
from
the
the
the
by revealing
~
Prandtl
vertical
^[~-1
with the other terms, atmosphere
of
t=O
boundary
component
and layer
of
the
]
aT- I
a very
weak
is responsible
a perceptible in nature,
all published
solutions
backflow,
(such as the Coriolis force or unsteadiness compensate
with
classical
for
(1964))
§7,3)).
(29,14),
observations
At the same time,
Zeytounian
(1972;
for 8 i of the term T[~-~-- + dZmlz-- =0 w0 which,
compared
show
and from
stratified
or
instance,
constraint
(29,13)
known
no sntibreeze,
for
(see, Gutman
for the
compensating
should
exist
over
of the problem
induced
of the process)
by
other
flow the
either factors
and not serving to
the breeze.
This can be attributed t o be u n s u i t a b l e a t
_
to the fact that the methods of solution used are found dT m
<~-I z ~O
A11
these
methods
reduce
,
in
one
manner
or
another,
to
successive
166
]
-second approximation.
As a result
+ dz---~{z=0 wO'
is always found
the solution departs
from
in the
the n~ proper
branch and no matter how many approximations are sought, relations (29,16) and (29,17) cannot be satisfied. The
difficulties
in solving
conditions (29,14),
the
problem
defined
by
equations
with (29,17), are due to the fact that,
(29,13)
and
as result of the
^~-I dTm ] presence of the term XL:-~-- + d-~m z =0 wO' which must be taken into account already in the first approximation,
the system is found to be coupled
in a
very complex manner and reduces to a six-order nonlinear equation with respect to z. Several solutions are known in which damping of w with altitude z is 0 specified as a boundary condition. Such an approach was taken, for example, by Estoque
(1961)
and
differentiating It
is
thus
clear relations
figures
the continuity that
here
(29,16)
presented
conservation
Magata
in
o f mass ~
the and
(1965),
who
equation
with respect
continuity (29,17)
articles
equation will
by
na~ s a t i s f i e d
use
an
equation
proper
in their
and
by
t o z. may n o t
n o t be s a t i s f i e d .
Estoque
obtained
Magata
It
be s a t i s f i e d is
and
seen from the
that
the
law
of
solutions.
A SIMPLE SOLUTION
Consider now the solution of the m a d ~ attempt to galn some insight
simple problem,
with which we shall
into the complex interaction of meteorological
fields during breeze. The statement of the problem will be simplified as much as
possible.
Noting
that
the
most
characteristic
factor
in
the
breeze
mechanism is the existence of a temperature difference between the land and sea surface varying periodically dtu-ing the day, we consider the simplest case of linear horizontal variation of surface temperature with a periodic cycle in time. The coordinate origin is taken at the shore line on the assumption of a straight and infinite shore. The x-axis is directed normal to the shore and the y-axis along the shore. Then the entire process does not depend on y. FOr simplicity, no initial allowance is made for the Coriolis force, settin E Romm and ~mO. There is very little information on the turbulence coefficient during breezes, and it will hence be assumed that u = u =Const. By virtue of the o
above assumptions system of equations (29,13) assumes the form:
167
au S
+ Uo
at
an
2
au
o
+ Wo
ax
an
0
= - -az ax
au o
aw
+
__°
8x S
=
a2u
0
az 2
0;
a8 as + u -- 1 + w-- 1 + S w
1
o ax
at A
o az
o o
__2 m n2, Ao m where ~
AIr-1 dT and So= ~L ~ + d z
Also,
with
in accordance
where
-1/2
+ Grl
az
a8
the
E(t,x,y)
(29,19)
2
= A081 ;
aZ
(29,18)
au
o
a28 = Gr---1/2 - - 1 ~
o- z 2 '
] z =0 "
assumptions
m a d e s we s e t
= (ao+ alx)sint t,
a
and a are specified constants. Quantity a can be interpreted o I I some characteristic gradient of the underlying-surface temperature, i.e.,a the maximum difference characteristic
in some sufficiently of several
between the temperature
length of the phenomenon. small
kilometers
region
I
is
of land and sea, divided by the
Condition
in the vicinity
in both directions.
as
(29,19)
is satisfied
of the shore,
It can hence
also
best
of the order
be expected
that
the results of the solution will be most valid in this region. The picture of the phenomenon should be somewhat is made for the effects no
longer
problem
varies
takes
mechanism,
in
into
of regions
the
x
one may except
far from the shore,
direction.
account
factors
that
distorted,
However, most
since
since no allowance
where the
statement
characteristic
the main features
the temperature
for
the
of the phenomenon
of
the
breeze
have been
correctly obtained. Thus
a periodic
(29,18), (29,20)
solution
of the
problem
is required,
described
by equations
the conditions: u
o
= w
o
= 0, for z=0,
and (29,21)
A 81= T(a0+alx)sint,
Initial conditions t
With the angular
are not needed
dimensions
we
(constant)
speed
have o£
for z=0,
t>0.
in this case.
£or the rotation
time of
variable, Earth.
the
t/~o,
where
~0 is the
168 UsinE (29,21), the solution of system (29,18) is souEht in the form Uo= u(t,z), el= O(t,z) + x~(t,z),
(29,22a)
L
n = ~(t,z) + x~(t,z). 2
This s o l u t i o n can have p h y s i c a l meaning o n l y a t
moderate x.
Hence
boundary
and
(29,20),
c o n d i t i o n s w i t h respect t o x a r e disregarded. By
virtue
of
equation of
continuity
of
the
system
(29,18)
(29,22), (29,22b)
w m O. o
Substitution of the solution (29,22) into equations
(29,18) yields a system in
which the variables are not a function of x: a2u @U
-112
S~-[ = -~ + Gr±
az 2 a20
9 0 + u~ = G r i l / 2 S~[ (29,23)
az 2;
a2~ a~ S~ a~
=
Gril/2
Oz2
a~ = koO, ~ =
ko~.
This system of equations should be solved subject
to the followln E boundary
conditions: A
u = O, 0 =~ aoSint, ~ =T alsint , for z=O;
(29,24) which
Lu follow
0 = ~ = ~=
from conditions
~ = O, for z = ~, (29,20),
(29,21)
and
of
the
behaviour
of
the
solutions far from the wall z=O. To simplify mathematical
manipulations
we convert
to news variables
(symbols
with bars) usin E the expressions: Z (29,25)
=
=
~/~ G r i 114
~,
g
= T ' ^ a l g --
OalCri 1 ' '
= 2Cr ~I/2(alkO)2 ~2~,
=
u
= v~
oa
u, AoalGF1114 T^--
Cri 1"'
ao = ~ a 2 A ~ / ~ G r - 1 1 4 K . 1
o
o
As a result equations (29,23) take the form (the bars are dropped) if S-1:
169
82~ I 2 az ~'
( 2 9 , 2Ba)
a~ 8t
(29,26b)
a~ _ az ~;
(29,26c)
a2u au I -~ = -~ + ~ az2;
(29,26d)
a20 80 1 --~ + u v - 2 8z 2;
(29,26e)
am 8z - 0.
Of c o n d i t i o n s
(29,24)
(29,27)
only
~ = sint,
Solvin E equations
those for
the temperature
0 = a sint
for
o
(29,26a)-(29,26e)
it
is
a t z=O c h a n g e t h e i r
form:
z=O. possible
to find
successively
v,
~,
U, 0 and ~. It is clear that system of equations (29,28a)-(29,28e) represents the chain of interactions between physical factors in the breeze mechanism.
It follows from
equations (29,2Ba) that the horizontal temperature Eradient is produced in the atmosphere due to heatinE of air by conduction of heat from the underlying surface.
Equation
(29,26b)
shows
temperature Eradient should result horizontal
pressure Eradient.
that
the
appearance
of
the
horizontal
in the appearance in the atmosphere of a
Equation
Eradient induces the onset of wind:
(29,28c)
indicates that
the pressure
here an important r61e is played by eddy
diffusion. Equation (29,28d) demonstrates the opposite effect, exerted by the wind on the temperature field. source
in
the
heat
conduction
Nonlinear term uv represents a negative heat equation.
It
is
precisely
this
term
which
describes the wind transport of heat. Finally,
the
above
scheme
simplified to the utmost. field of meteoroloEical
is All
quite
rough,
since
it
the reEions of breeze
variables
interact,
and
here
is
on
and
the
an
important
played by nonlinear terms (in the equations of motion), are found to be identically equal to zero.
based
a
model
correspondin E r61e
is
which in this model
A particularly important r61e in
breeze is played by the vertical velocity field. The solution of equations
(29,26a)-(29,2Be) with boundary conditions
and (29,24) is elementary: (29,28a)
~ = exp(-z) sin(t-z);
(29,27)
170 +~ = exp(-z)cos(t-z ~);
(29,28b)
(29,28c)
z u =- --
(29,28d)
e = aoeXp(-z)sin(t-z) +
exp(-z)
v~
cos(t-z);
z
4~
exp(-2z)cos[2(t-z)]
+ ~1 exp( _2z)cos[2(t_z_8)] _ exp(zl/~)cos[2(t The expression for- ~
is not
given,
since
it
z
~)].
V~
is too cumbersome.
The
daily
pressure variation at the underlying-surface level is: (29,29)
=lz=o = aoCOS(t+ ~) + 0.02 sin2t.
Solutions (29,18) show that the structure of this breeze model in vinicity of the shore is similar to the wind and temperature progressive wave damping out with
altitude.
These
waves
move
from
the
ground
upward,
as
confirmed
by
observations of breezes. We now establish the instant when the wind appears at the ground on the of breeze.
8u instant,~-~[z=0=O.
At this
From this
it is found that
onset
the breeze
lags behind the variation in the soil temperature by 6 hours; this is a rough result, since observations yield from 2 to 5 hours.
It is important that the
above solution should point to the cause of this lag, which is inertia moving air.
30
• MODEL E Q U A T I O N S
OF THE SLOPE WIND
Consider a local wind arising above a slope, less than several which
from
exceeds,
the
degrees
temperature
in absolute
along the slope.
and
value,
the steepness of which is not
the deviation of
of
the
several
This wind will
free
the surface
atmosphere
degrees
be called ~ e
temperature
at
the
same
centigrades
and
change
~
~
of
altitude little
on the assumption
that it develops in an atmosphere at rest. The slope wind arises as a result of the difference in air temperature in the vicinity of the slope in the free atmosphere at the same altitude. in the case of breeze,
a major role
played by eddy heat coduction.
Hence, as
in the slope wind mechanism should be
171 As
staring
topography
of
system of
equations
slope
the
is
we
consider
assumed
to
the
vary
(29,13).
equations
according
to
the
The
following
equation:
(30,1)
z = ~oZ(X,M),
where =o = Ao with Zo = Max x,yeD
Iz(x,y) l and Ao= 2
RAT o (see the section 29). g
We now transform from variables x,y and z to new independent
[30,2)
z
We have the f o l l o w i n g
eo~(~.~). relations:
8 -8z
variables:
=
8
8
-8C'
-8x
8 -8~
=
~0
-
8Z 8
8
~
8y
8C'
8
an
--
8Z 8
a~ 8C'
~0
and 8w
~-~
Dv. + 0
o
8u -
az
8v 8~ o o + -- + --,
o
a~
a~
a~;
where
(30,3) Then
(~0= W O- (X0 elementary
UO+ ~
transformations
VO .
yield,
in
place
of
(29,13),
the
following
equations:
8~
at
o . (V~o.t~)~. ~o
8~ o
V0 )
a~ = Gri1/2
a2v~
1
+
--
=
Gril/2
o
o
8E 2 (30,4)
2
^
--=
T 01;
aE
t~.# + --S° = o ; 0
s
88 at
1 *~v.~*o, 0
8e 1
0
8~
828 1
+ ~po(%+ =o~o.~X)
1
8C 2'
172
where
~=
O
.
For equations
o
. %:Wo-%Vo.~:. ~:(Uo.V o) and ,o =
~-I~+ dZ~lzdT'l:0
(30,4) we must consider the following boundary conditions: -)
(30,6)
Vo= O, ~o= O, 8,= ~E(t,~,n) for ~=0, t>O and ~,n e D2;
(30,6)
v"~O'--'~O,
~0---)0, .2--)0, 81--)0 when I ~ + ~ 1 - ~ and ~ - ~ . .
But for the slope wind we have that: GFil/2<< 1, and we consider the limiting process: Grll/2--)O. When Grll/2--)O, with t,~,n,~ fixed we have only possibility to impose the conditions
(30,8) and these conditions
yield a trivial outer limiting solution: v= 0
~= 0
.= 2
e m O. 1
Now we introduce the following inner variables: (30,7)
~ :
~/Grl
1/4 a n d
~ o
0 Gril/4
and we get the inner limiting process:
(30,8)
Crll/2--~O,
with
t,~,~
and
~ fixed.
If we associate to (30,8) the following asymptotic expansions: A
~ V = V +
(30,9)
we obtain,
0
....
A A e = e + 0
....
.
2
A = . +
A for v, ~, ~ and ~, as functions
....
8 = v I
+
....
of t, ~, ~ and ~, the following
Saandun~bzqeaequations: A
(30,10)
(3o,11)
S
av~
a~
A
~
A
av~
+ ( v , ~ ) v + ~ =a~:+ -
O;
~1__~ ]
lRo,j~ ( ~ ^ ~ )
~0
+ !~~ , _ _ ~~ : ~ = _ _ ;
A a2~
a~ 2
173
(30,12)
i~.~
(30,13)
a$ 3 a$ ^ a2$ s R * v . ~ $ . ~ a~--* ~ "o % ~.~z - a~ ..
+ _a~ = 0;
But the equation (30,11) yield, with the matching,
m ~(t,~,~) : ~2(t,~,~,~:O) m O. On the other hand, the condition Gril/2<
(3o,14
e<<
(AO)2ATo/--~--~ 0
and as consequence we have that: Ro>>l and 8<
A
-~
a~
--= ~. V-~ + a~
(30, Is)
s
e q ~
u~:
A
a2v e
A
-~
o~ Ute ~
a~
°~6z$=__.
O;
~-~ + v ,
a~
a25
"C~o=oV.i~:z -
a~ 2 ^
Specific features of this system are that, first ~ = ~
A~
+ vw and ~ are wind
velocity components along "curvilinear coordinates" ~, W and ~*; second,
the
equations
the
containing
new
terms
makes
a11owance
in
explicit
form
for
buoyancy forces acting along the ~ and W axes; and third, new terms appear in the equation of heat conduction, and describe the wlnd transport of the heat flux component associated with stratification of the undisturbed atmosphere. The boundary conditions,
as in the problem of a breeze,
are taken to ensure
contact of air at the surface level: ^
(30, 16)
~ = ~ = 0 for ~ = O, t>O,
t According 7,12))
.
to the Kaplun's
correlation
theorem
(see,
Van Dyke (1975; section
174
specification of the slope temperature: (30,171
~ = ~ ECt,~,~) for ~ : O, t>O,
where E is assumed to be known function of time, and damping of disturbances of meteorological variables with increasing distance from the slope surface: A
(30,18)
= ~ = S~ o
{ v~
S--)O
for
I~.~1---~,
for ~__~.t
The atmosphere is again assumed to be initially at rest: A
(30,19)
~ = 0 , S = 0
f o r t=O.
LINEAR SOLUTION OF PRANDTL ( 1 9 4 4 )
Note that when the slope can be treated as an infinite plane and conditions (30,16)-(30,19)
are satisfied,
system
(30,16) becomes
linear,
since all the
unknown quantities cease to depend on ~:
S
(30,20) S
aft -
-
at
-
% -~ S
m
~
sin~o-
a2fi
a~2 '
a~S ~+aS ~.o=o fisi~o=a~ 2
and OmO, which does not detract from Here it is assumed that a--~Z-sin(~ a~ o ) ' a--~Z-O an the
generality
of
the
problem
and
the
continuity
equation
together
with
in mesometeorology,
when
condition (30,18) yield ~mO. System of equations
(30,20)
is the case,
infrequent
the interaction between the velocity and temperature
fields
is described by
linear equations. The
steady-state
(30,16)-(30,18)
solution
{^-
at ~o>0 and Eml i s A
u = --
(30,21)
of
e-@sin@;
S = ~e-~cos@,
equation
(30,20)
satisfying
conditions
175
where Thus
01" for
neutral
or
unstable
/3 0
sin
.
stratification
(~0sO)
of
the
undisturbed
atmosphere, equations (30,20) do not have steady solutions which would satisfy conditions
(30,16)-(30,18).
As expected,
the diurnal wind
(4>0)
is directed
upslope (~>0), while the nocturnal wlnd (4<0) Is directed downslope (4<0). It is interesting that,
according to solution (30,21), the maximum of ~ does
not
physical
depend on
6 o.
The
cause
for
this
is as
difficult for air to rise along a steeper slope,
follows:
it
is more
but then the buoyancy force
component is larger. It is seen from relation for ¢ that the boundary layer becomes narrower with increasing slope
stepness.
Conversely,
for
a shallower
slope
the
boundary
layer thickness becomes greater. Finally we
note
that
as a result
of
eddy
friction,
the
rising
heated-air
particles set into motion also air layers, which themselves are not heated. As a
result
of
their
ascent
these
layers
layers which are not set into motion. removal
of
heat,
start
to
descend
become
cooler
The latter, and
form
than
the
neighboring
upon being cooled by eddy
a
weak
consequence it is clear that the temperature deviations,
descending
~
ox:cond/~u~ b~ ~nxu~//~ a~Izd~z~ (30,21).
As
as well as the wlnd
speed at some distance from the slope, take on small negative values.
• ~ . 15: Pn.o~i.,f~_~o~. ~ and ~ d ~
flow.
176 31 . M O D E L P R O B L E M F O R T H E L O C A L THERMAL P R E D I C T I O N (THE TRIPLE DECK VIEWPOINT)
Here
we
rewrite
consider
only
problems
which
are
the thermal boundary condition
(31,1)
T (0-----~= 1 + To-" co
where E.O,
two-dimensional
and
steady
and
we
(29, I) in the following form:
, o~t z=0,
if I~o-<1.
Far upstream,
when x--~-= and E-O,
flow which is characterized
we have a ~
undisturbed
by an Ekman layer profile:
U EkIL x0 ~ I
(31,2)
we assume t h a t
----
'o.
X
z/~ 0
z/~ 0
'
with
t (31,3)
U0
%:
J
~ ~e] -1/2 -
~o
L2-
j
,
where:
~°U (31,4)
Re -
o
U /~o
uo are ~
2QoSin~ o
Reynolds and Rossby numbers,
If we nondimensionalize the dimensionless
(31,5)
So = ~
I f ~~103m,
t h e n Bo<
case
x and z with ~,
the following Boussinesq
length ~o.
then we introduce,
number:
~Og .
but
if
~~103m t h e n we h a v e 2~o>>1 a n d a l s o
(when ~~103m),
2~o = ~e -1/a ~
with (31,7)
based on local scale
the coordinates
problem,
Therefore, (31,6)
o
and ~o -
2a m = a-zT > 2.
we c a n a s s u m e t h a t
~c = ~e -l/m, o
2~o<<~e.
in
177
For
Uo~10m/sec,
Po=Sm2/sec
and
fom2~oSinWoSl0-11/sec,
case
the
of
~~103m
leads to m=5. For this case we have the possibility of using: U
I
RT (0)
~o _£o kJ
(31,8)
=
g
~°/Mo~ ~ ~ 1,
~
and the Boussinesq approximation is correct. The value m=5 is the same as the one used by Smith (1973) and Smith, Sykes and Brighton
(1977) for the flow over an isolated two-dimensional
boundary
layers.
For
the Boussinesq
stratified
fluid
see
short
hump
in
the work of Sykes
(1978). When m=5 we have t h a t : [~o - ~ Re -3/8,
(31,9)
where Re -
UL
uo o
o
o
According to the Boussinesq approximation
(see the section 8) and taking into
account the relation (31,9) we have the possibility to formulate the following d
~
~
problem,
if we impose that
To---) O, with Mo---) 0 and To/Mo=Ao~l$:
2u
au
us~
aw
u~-~
1 a~ ax
au
+
ws~+
+
wsE
aw
+
-
s o
I a.
B
~" az
~
a2u
C e
=
1 __ az 2] ;
+
x2
s o
+
- -
u~-~ + w~--~" +
dT
=0
= P r laX 2
(~ = - 8, z = 0: u = w = 0, 8 = AoE(X),
this
'
au aw ~-~+ ~--~=0;
(31,10a)
In
]
c3z2] '
case
we
obtain
ATo~
since MO=U O / / ~ ? ~"
For
AT
0
the
£ollowing
0
.od TO= ATo~=(O).
0sxsl;
valuation:
2-----j; 0Z
178
.-~-90.1-expC-z/m°)c°s(z/a:°) = 8 =
x --> -co [ u - 9
m
Uco(Z/~o ) t ,
w --9 0,
(31,10b)
We n o t e
that: z if -- --~ co , t h e n u - 9 o
i,
f o r x - 9 -co,
II) If --z --9 0 , t h e n u ~ _z o o
f o r x - 9 -co.
I)
Now z=0,
if we it
require
to
is n e c e s s a r y
to
Into
account
introduce
the
the
boundary
conditions
on
the
ground
inner v a r i a b l e
z
A z -
(31,11)
take
- -~,
~>1
o and
in t h i s c a s e
(31,
12)
From
(%-1
u ~ 0c o
the
~ = z / ~ O,
first
equation
is
it
of
~-1 A 8 4
u ~
necessary
(31,13) Finally
we
z,
+
i
we
establish analysis
that
%
verify
that,
if
(%-1A, u~0c utx,~), o
with
...
~=2.
three
region,
f o r the n ~
824 a~ 2 +
that
of the s y s t e m
f o r the a p ~
= z / ~ o,
S-2~
....
to i m p o s e
u ~ u -9
II)
(31,10)
~-1 = 5 - 2 ~ ~
asymptotic
I)
for x - 9 -co.
then:
% and
A z ,
vertical
variables
are
necessary
for
the
(31,10):
where
I,
w h e n x - 9 -m, region,
where
~
U ~ U --9 l-e-Zcosz m Uco(z),
III)
~ = z/<,
for the ~ A U ~ ~0 u,
$ we note
that
~O/L 0
~ 3
wall
viscous
A A a n d u --9 z,
O' if m=S and as
w h e n x --~ -~,
region,
where
w h e n x --# -co. consequence
we have that:
Ugeost I~O0] ~ 1+0(~03 ) ' when ~E0 --90, and it is sufficient to take in the boundary
condition
for
Into account,
x --)-co, only the first term Uco(Z/0CO).
179 For
the
other
different (1981)).
case,
asymptotic
when
£O<~oL o3
analysis
and
~6>~Lo, ~
(see, for
it
example,
is
necessary
to
apply
a
the work of Smith et al.
But the case m=6 and m=4 can be analysed from the problem (31,10).
For the case m=3
it is necessary to start
Boussinesq approximation does not emerge.
from another problem,
where the
For m=3, we have ~~104m and we may
neglect the Coriolis terms in the local na~ Boussinesq equations. On the below sketch (Figure 16) we have demonstrated the triple-deck structure for the analysis of boundary Ekman layer flow interaction with the termal nonhomogeneity on the ground z=O (with the dimensionless variables).
A~ UPFER < INVISCID) REGI 0/~
( U
I
MIDDLE ( INVISCID REGION
(M) /
WALL (VISCOUS) LOWER REGION (L)
-~
×~"- ¢0
~.16- ~
~
~/@/~e-deck ~
~
~
o
e
44
~
(ca~e ~ £°~
x
180 THE TRIPLE-DECK STRUCTURE
We
shall
give
the
analysis
for
the
three
~,
regions:
•
and
~
of
the
order
one
triple-deck theory (see the Figure 16 above), with little dlscussion t. Beginning
in
the
m/dd~
dec~
(~),
where
x
and
z=z/g
O
are
the
coordinates, we expand the flow variables as:
u'(~) + m ~ G + . . . ;
u: W
(31,14)
0
:
...;
~C* [/ + 0 2
~ 7{ +
7[ =
~
8
0" ~ ~C 8 0
=
0
and substitute in problem ( 3 1 , 1 0 )
...;
+
...,
to flnd for the first lowest order:
Um(z "aG dU--~mw : O; Ja-'x + dz
aG
aE
g-if+----O; Oz
(31,1s) aE a~
~=0,
if we assume that:
(31,16)
~p=l, g,=l+@ and o=1.
This choice (31,16) is necessary if we want to obtain a significant degeneracy of t h e p r o b l e m ( 3 1 , 1 0 )
i n t h e lower (~) v i s c o u s r e g i o n i n t h e v i c i n i t y
A
o f z=O,
near the wall.
Notice that
the effects
o f t h e e x p a n s i o n of t h e b o u n d a r y l a y e r a r e O(~co) i n u
and OCm~) in w. Furthermore in the boundary layer,
t
The
reader
Zeytounian
betn 9 (1997;
referred see
the
to
Stewartson
"LeQon
IX") .
and
if
Williams
we
take
(1959)
into
original
account
work
or
181 that
~=2 ( s e e
(31,13)), .
(31,17)
we h a v e n e c e s s a r i l y
= a:2~ + O
and, by continuity, we obtain the form of the expansion for ~ in (31,14). Solutions for u(x,z) and w(x,z)
satisfying the upstream boundary conditions
8/'e: dUm{~)
(31,18)
~ = ACx)
and
dz
~ =
dA(x) U=(~), •
dx
which represent simply a vertical displacement of the streamlines through a distance -~oA(X). The flow in the appa,~ (~) dec& is driven by an outflow from the middle deck. Far from (31,18) we have: (31,19)
Lim w ( x , z ) z-)co
=
dA(x) dx
We introduce a new vertical coordinate z and we have
z=%~; in this
case
the flow expansions
u=
(31,2o)
Substitution
for
1 + ~ +
W----
~
W
0 2
If----
~
e =
~: o
0 2
+
...
-II +
...
-e +
....
in the equations of the
motion is inviscid: (31,21a)
a~ --
ax
+
i a~
-
--
~ ax
=
O;
^
(31,21b)
(31,21c)
~-~ + ~ ~ -
a~
a~
upper deck is:
...
0 2
the
a-x + a-z = O;
;
local problem
(31,1o) shows
t hat
the
182
(31,21d)
+
--
--
From (31,21) we obtain,
a2
[
(31,22)
8x 2
+
for ~,
~ = O.
a Helmoltz's
a2 + , + K2 ) a~ az 2 o ~=
equation
in a half space:
O,
where
(31,23)
2.2.. = ~--I ~-1
K~ o
~ddzT'~-~3z=0 ] "
~ I.'Y
Note that
in (31, 14) we have
according
to the last of equations
(31,15).
But a2~
o ~
-
axa~
=
~(x)
+
Q(~)
and consequently
1 dQ
- ^ ~ x(~). S d~
(31,24) However
the solutions
for 8 and ~ satisfying
the upstream
boundary
conditions
(for x--x-m) are:
(31,2S) Now,
for the equation
middle deck,
zCz) ~ 0 ~
QC~) =-=*{
m O,
~ ~(x).
(31,22)
we have,
as a consequence
of matching
with the
the following condition: Lim ~ ~ ~(x,O) z--->o
= ~(x) m Lim ~. ~----~
But 5(x,O)
as a consequence
=
dA(x) - d--~--'
of (31,19) and matching with the upper deck,
the following relation between ~ and A(x):
(31,26)
~-~
5~ z=O
o d---x + dx 3]
and we can write
183
This
last relation
(31,26)
is a boundary condition for the equation
(31,22)
for ~. It is obvious that the middle deck solution (31,18) do not satisfy the no-slip conditions on A=0. A situation which is remedied by the analysis of the lower viscous deck (~), where the stretched variable is Z
Matching
with
~
z
^ =
the
--
--
--.
2 0 0 expansions
(31, 14),
when
z=0(m o)
implies
the
inner
e xpans ions U
W
=
~
Substitution
A
+
...
=
3A (~W 0
+
...
=
2A ~T[ 0
+
...
(31,27)
~CU 0
of (31,27)
yields the following,
into the full equations
naaI/neo~, ~
~
of the local problem ~m% ~, ~ and 8:
A
z
(31,28b)
aft a~, --+--=0;
(31,28c)
^ a8 ^ u ~-~ + w
ax
02~
aA
a~
I
a~
a2~
Pr a~'
with the boundary conditions: = o: ~ = G = A
A
u--,z,
(31,29)
zA--->+m: (
o, 8 = ~ E(x), o
O~x~l;
~,---~, ~)-----~
dA ~(x)--,,O, A(x)---~O, 8-~---~;
A X--9-o0: AU--)Z + A(X), A--9-ZA ~-~, dA ~--90,
after matching with the middle deck (~)
(31,10)
184
(31,30)
u.,,,.
Lim ~.--90
o dz . =, ~, d A - ~ o u [z)~-~
= Lira ~--->+co
[o0; ~cs~ o
and taking into account that (z=~0~): (31,31)
U~(z) ~ z, dUm ~ 1, when z---)O.
d~ We note that for ~ we have the following expression A
z
(31,32)
a~ _ ~ ~ a~
= ~ [ ~d~ ÷ ~(x).
J
The specification of the problem (31,28)-(31,32) (31,26)
between
~(x)
and
interpretation of (31,26)
A(x),
since
is completed by the relation
~(x,0)m~(x).
The
well-known
is that the pressure ~(x) driving the flow in the
lower deck (~) is itself induced in the main stream,
i.e. the upper deck (~),
by the displacement thickness of the lower deck transmitted through the middle deck by the passive effect of displacement of the streamlines. The
aZ%en~
~£n9~
(3i,28}-(31,29) ~(x)
as
boundary
data
ae~-~idaced
to be solved known
prior
layer problems).
to
ceap//n~
in the
arises
lower viscous
the resolution
On the contrary,
(as this
because layer
is the pressure
the
problem
(~) does nn/ oxzcepZ case
in classical
perturbation
~(x)
must be calculated at the same time as the velocity components ~ and ~, as well as the temperature perturbation ~. Nevertheless,
it must be emphasized that this function ~(x) is not completely
arbitrary and that it is connected to the function A(x) through a relation. This last relation is obtained via the analysis of perfect fluid flow in the upper layer (~) (see (31,26)). If it is assumed that the parameter A (31,33)
ko<< 1 ~
then the equations
0
in (31,29) satisfies the condition:
TO<< M ,
(31,28) may be linearlzed about
layer profile by making a further expansion
the undisturbed
boundary
185
A U
=
A W
=
A Z
A,U ' 0
+
...;
~W' 0
+
...;
AO'
+ ...;
~(x) =
A~'
+ ...;
A(x) =
A A' + .... o
+
(31,34)
Finally
0
we
simply
o
record
here
the
solutions
obtainable
by
usin E
Fourier
transform in x, defined for the function f(x) by: f(k) = I f(x)e-ikXdx" In particular, we find instead of (31,26), the relation: =
= (31,35)
A(k)
iN -
~(k)
o - - , K2-k 2 o
where ,, (31,36)
N = o
r 2
2] 1/2
LKo-k J
, if
Ikl
Here we have applied the standard radiation condition for z--~+m, choosing the sign of No, for Ikl
186 REFERENCES TO WORKS CITED IN THE TEXT
ESTOQUE, M.A.
(1961) _ Q.J. Roy. Met. Soc.,vol.87,
pp. 136-146.
GUIRAUD,
and
_
J.P.
ZEYTOUNIAN,
R, Kh.
(1980)
Geophys.
Astrophys.
Dynamics, GUTMAN,
L.M.
(1972)
_
~
n
~
Za
tAe
paacea~. izdale1'stvo, Israel MAGATA, M. ( 1 9 6 6 ) _ Pap. PRANDTL, L.
(1944)
a.
Ceophys.,
o~
m
~
(1969). Transl.
vol.
16,
Jerusalem.
n°l.
_ ~fiAae~ du~cA d i e ~ t n S ~ . Braunschweig,
Vieweg and Sohn.
SMITH, F . T .
(1973) _ J.F.M,
vol.
SMITH,
SYKES,
BRIGHTON,
F.T,
Sci.
I_SS, 2 8 3 .
t]~
Guldrometeorologlcheskoe
Leningrad
Program for
Met.
n a ~
Fluid
R.I
and
67,
pt.
4,
803-824.
P.W.M.(1977)
J.F.M,
vol.83,
part
I,
163-176. SMITH, F.T, BRIGHTON, P.W.M, JACKSON, P.S. and HUNT, J.C.R.
(1981)
J.F.M.
113, 123.
STEWARTSON,
K. and WILLIAMS, P.G.
SYKES, R . I .
( 1 9 7 8 ) - P r o c . Roy. S o c . ,
(1969) _ Proc. Roy. Soc., London A312, 181-206.
VAN DYKE,
M.
(1976)
_
~
e
Press, ZEYTOUNIAN,
R. Kh.
L o n d o n A361,
~
met~
Stanford,
(1964)
_
Trudy
226-243. ~¢~ ~b~Zd
~ec~.
Parabolic
USA. of
the
World
Meteorological
Center,
Moscow, URSS; vol.3, pp. 19-74 (in Russian). ZEYTOUNIAN,
R. Kh.
(1976)
La M~teorologie du point de vue du M~canicien des Fluides. Fluid Dynamics Transactions,
ZEYTOUNIAN,
R. Kh.
(1977)
ZEYTOUNIAN,
R. Kh.
(1987)
_
J.
of
Engineering
Mathematics,
8, 289-362. vol.ll,
n°3,
241-247. _
~ea
~ Lecture
~oaetea
~
de
ta
~ c a ~
dea
II. Notes
Heidelberg.
in Physics,
voi.276;
Springer-Verlag,
CHAPTER VIII METEODYNANIC STABILITY
32 . WHAT I S
STABILITY
?
We consider here the main equation of the quasi-geostrophic model (9,14), with
(9,15): (32,1)
S
qo +
qg qo 8(x,y)
8t
= O,
where
(32,2) a n d Bog1,
X E1 t . 0
In the adiabatic nonviscous atmosphere (see the formula (9,16)) the boundary condition at the Earth's surface (reduction of the vertical velocity to zero) reduces to the form: ] + 8 (:Rqg, a~qg//ap) t = o, on. p=l. s aR qg + T~ o ( l ) IS ~8 [ aRqg ~
(32,3)
at
Ko(1)
ap
~,~
J
Let us consider a &o~b: ~t~a~, Us(y,p), having a purely zonal velocity (i.e., directed
along
the
circles
of
latitude)
which
is
expressed
from
the
geostrophic stream function RB(y,p): aR (32,4)
U(y,p)
= -
s 8y
This basic current is naturally assumed to be a solution to equation (32,1), with (32,2), and we consider now the evolution of a perturbation h(t,x,y,p) of t
We n o t e
that:
a(a,b) _ aa ab #--~,y) ax ay
aa ab a y ax"
188 this basic flow;
i.e.,
(32,5)
Mqg (t,x,y,p) = Ms(y,p) + h(t,x,y,p).
If (32,5) is inserted into (32,1),
with (32,2),
then the following equation is
obtained for h:
(32,6)
IS a ~_~.) a N ah 8(h,q) at" + Us q + 8y ax + 8--(-~-[ = O,
where q(t,x,y,p)
is the potential vorticlty (quasl-geostrophic
and baroclinlc)
of perturbation defined by
(32,7)
q :
+
s
8 f p2 8_~1 l.Ko--
.
The term ar3/ay in (32,6) is the gradient along the meridian of the potential vorticity of basic flow:
~_~IKo_~ ~~.~p Bl '
~2~2 B + S FI = BY + ay
(32,8) and we note that:
8R ay
(32,9)
One of the fundamental
U(y,p) ~ Stated
a2U
-
behavior o f h ( t , x , y , p ) ,
p2
aUB]
S a-P[Ko--~-~ ~p j"
questions
tAe ~
more p r e c i s e l y ,
a ["
ay2 B
to be clarified
o~ ~
this
is: ~
tAe q/c~en atnactaae
tZetd h ( t , x , y , p ) ?
means t h a t
given the
stemming from (32,5),
basic
flow Us(y,p),
must be s t u d i e d
the
in o r d e r to
determine wheter it increases or decreases. If it increases,
then the ~
of U s with respect
to h is ascertained.
The basic flow US can be sald to be "truly" a&xAte only when it is stable with respect
to ~
h. On the contrary,
~
takes places
if U
for e~ea one h. The equation dealing with h(t,x,y,p) a(h,q)/a(x,y)),
and generally
is quasi-linear
speaking
it
is quite
behavior of its solution under the boundary condition:
B
is unstable
(on account of the term difficult
to study
the
189
(32,10)
~ - + Us
+
h
-
~ +
TO( 1 )
8 (h, ah/ap)
+
according to (32,3) and (32,4), Thus the ~ way,
sJOx
ap
8(x,y)
TO( 1 )
= 0
on p=l,
(32,5).
case is often adopted and it is assumed that
the following equation,
which governs
the ~
lhI<
~
In this
pnm/~m~,
can
replace (32,6):
I~ °~ - +
(32,11)
Us
~)[~.,
8
82U
and ~
8h'
O,
p=l, we have the following boundary condition,
~÷%
L~ ÷ ~o--~ - ~ - - h' 8UB
ap
In
p2
o r p~ °%llOh'
ay2
+
[
+
order
to
solve
(32,11),
~
under
1
Ko(P)
lah'
~ % j ~ = 0.
the
condition
(32,12),
we
can
set
the
following
~3213~
h ~t x y p~ = ~ea1{~y p~exp[ik(x ;t]]}
where the zonal wave number k must be real since h' must remain finite for all x-->±=. It can be assumed that k>O. On the other hand, the phase velocity c can be
written
in
the
form:
c=c +Ic r
i'
and
therefore,
the
following
replace
(32,13): (32,14)
h'(t,x,y,p) = Real
(W,p)exp
t
exp ik
-
t
.
190
From (32,11) and (32,14) the following equation r e s u l t s for h(y,p):
[UB(y,p) _ c]{S a r p2 aa-~-~l+ ay2 - k2Fl}
(32,16)
+[.The boundary condition
82UB
r p2
Oy2
8o]].__0.
(32,12) at the Earth's surface,
p=l,
reduces
to the
form:
IUB(y,p)- C ] [ ~ +
(32, 16)
o-rI -r Ko(P)
OUB 1 ~ + ~
-
~]
Ko(P) U ]~ ~ "J = 0 ,
~p=l.
As a general rule, boundary conditions in y and p-->O must be superimposed on (32, 16). It
turns
out
that
those
conditions
are
Aomm92.nexx~
and
hence,
the
corresponding linear stability problem usually has only the trivial solution which is identically zero. One exception
is when k and
c are
linked by
a
relation depending
profile of UB(Y,p) which can be called by d / ~
on the
aeIa//a~ of the stability
problem. For a fixed profile UB(y,p),
if k is fixed, the dispersion relation allows a
sequence of complex roots in c. If c <0 for all the roots (we remark that k is i
real
and
positive),
then
the
perturbations
(called nan2u~ ~ )
attenuate
exponentially as a function of time and corresponding Rossby waves are stable for the type of perturbations considered. If,
however
ci>O for
at ~eaat ane normal
mode,
then
the
Rossby
waves
are
unstable for the perturbations of wave number k fixed. In the ~
~,
when: e
UB~ UB(Y), the instability process
is related essentially to the existence of the term
d 2 < / d y 2 and the situation is then referred a.s a However, when
UB= UB(Y,P),
191 &
~
~,
the
vertical
shearing,
aUs/Sp,
is
an
important
cause
of
instability and the corresponding process gives us the & a n x ~ ~ . The Eady (1949) model with: U m p, ~mO, B
is a simple and very good example
of baroclinic
(see, the next
instability
section 33). We
note,
finally,
that
for
the
~
~
the
equation (32,1),
with (32,2), reduces to the form:
(32, 17)
S
a~'~"
a(~* , ~ ' ~
q9
+
model
+ ~y)
qg
at
quasi-geostrophic
qg
=
O,
O(x, y)
with
(32, 18) A
method
~.~'= ~2:~" S}f'. qg
of
deriving
made~ (32,17):
qg
the
qg
& a n ~
ma/n
e ~
o~
~
qaaaZ-q~
it consists of taking the limit K0(P)--)O, in which
limit the
derivatives a~qg/Os no longer depend on p; s=(x,y). Therefore, for
~e
h (y)
for the barotropic the
following
(32,19)
33
e
instability,
equation,
in place
-
"
when we have UsmUs(Y), of
we obtain
(32,15),
d2U:] ~.
(k2+
O.
. THE CLASSICAL EADY PROBLEN
According to Drazin (1978) we consider an inviscid,
non-conducting,
~2xLtd (see
an
the
section
8 at
the
Chapter
rectangular channel whose cross-section 0 s y s L o, The
channel
gravitational
rotates
with
II)
in
infinitely
rigid
is given by
0 s z s H oangular
velocity
flo~
and
acceleration -g~.
It follows that the governing Boussinesq equations are (33,1)
Z o ~ long
au~ ! ~p + ~g(e-eo)~ ' at + (~.~)~ + 2~ o~A~ : _ Po
there
is
downwards
192
(~.~)e
(33,2)
ae at
(33,3)
~ . ~ = O,
,
= o,
where ~ is the fluid velocity relative to the rotetlng frame, the d e n s i t y , note also
p the r e l a t i v e
p=po[1-=(e-eo)]
pressure and 8 the t e m p e r a t u r e o f the f l u i d .
that Po a density
e ° a temperature
scale,
scale,
We
a the constant
coefficient of cubical expansion and ~ the unlt vector in the direction of the upward vertical. For Invlscld fluid the boundary conditions are (with ~=u~+v2+w~):
(33,4)
v=O at y=O,L o,
(33,5)
w=O at z=O,Ho.
C o n s i d e r now a b a s i c f l o w ~
B
which w i l l be p e r t u r b e d t o t e s t
its stability.
We
take the zonal flow i n c r e a s i n g l i n e a r l y with height:
0 B=
(33.8)
U
°zt H
0
where
U
0
is
a
velocity
scale.
is
This
balanced
geostrophlcally
and
hydrostatically by the b a s i c temperature AT
(33,7)
e= B
e+ O
--z
2~ U
0
-
H
0
0
y
¢gH °
and basic pressure 1
(33,8)
2fioUo 0
where AT
0
]
0
is a constant scale of basic vertical temperature difference across
the channel. Finally the basic density is
(33.9)
Ho
~gAToyJ J
We next scale the variables, denoting dimensionless ones by tildes:
193
x
,
[o
=
o
u
~=v
U=U'
~oo'~, 10)
o
~_
w
{e
~gH°
~_
o
,
L~o
'
eURo
o
2fl U L o o o
t
~, ~=
U'
o
~-
z
,
o
AT° ]]
-e-
o
g zj, o { p _ g I ~gPoAToz2~,
1
2floUoLoPo
H
o
H where Co= -~ and the Rossby number is defined by o
(33, 11)
Ro
U -
o
2fl L o o
The reason for scaling the vertical velocity with the Rossby number baroclinic
is that
instability occurs with w of order one when the Rossby number
is
small, as will be seen later (see, also, the section 9). It is also convenient at this stage to definie the Baaga~ ~ ( 3 3 , 12)
~B -
by
~ZgHoATo 4f12L2 o o
It
is
an
important
number
for
large-scale
meteorological
problems,
representing the square of the ratio of the buoyancy frequency to the Coriolis parameter. Note also that (33,13)
B = R i . Ro 2
where the R i c h a n z l a a n ~
(33,14)
is defined by
~gHoATo Ri - - u~
o and p is evaluated with e=e . B We w i s h t o t a / ( e AT >0 s o t h a t o stability. For the Westerlies (33,15)
Ro<
2
gHo 1 do dz' o
hot in
fluid the
is
above
atmosphere
the it
is
cold found
and there
is
static
that
c <<1 a n d $ < < 1 , o 2fl U
and therefore that the slope of the surfaces of constant density ~=
o o<
~gATo
194 From now on we drop the tildes oven the dimensionless variables,
and then the
basic flow becomes:
(33,16)
{ ~
z~, e B = - y ,
pB= - y z ,
B= for -m < x < +m, 0 ~ y,z ~i.
We perturb this dimensionless basic flow (33,16), writing (33,17)
~=%+~'
and substitute
these expansions
to
(33,1)-(33,3)
to
e = e B+ e ' ,
,
obtain
p=pB+p
' ,
in the dimensionless
for
the
perturbations
equations the
corresponding
following
governing
equations: (33,18a)
_ f a u ' + a u ' + ,au' + v ,Ou' ~ ,Ou' Kot~-6-- + ~ow ~-~- + Row' j~ - v' z~-~ u~-~ ay
(33, 18b)
_ Fay' .o~.
(33,18c)
2~ ~aw' aw' u, Bw' ,Sw' aw'1 O' 0p' eoKOI~-~- + Z~-~- + ~-~ + v Oy-- + Row'~ j . . . . Oz'
(33,18d)
ae'
(33,18e)
aU' av' a~+ --+ 8y
,av' Ov' ~ , O v ' ] u' + u ~ + v ' - - . .ow ~- ~ + Oy
88' a--~-+ z~-~-+ u ,Be' ~ - ~ + v ,@e' - ay -+
The b o u n d a r y c o n d i t i o n s (33,18f)
za~v '
IROc9W' =
Oz
_ ,0e' K ow~--
.
.
.
Op' Oy' .
v' + ~ w' = 0;
0.
(33,4) and (33,5) g i v e
v ' = 0 at y=0,1;
and (33,18g)
w'=0 a t z=0,1.
Linearizlng the perturbation equations (33, 18a)-(33, 18e), we find
(33,19a)
_ o tFau' au' + Row'j] - v' + ~ ap'_ . ~ + z~-~ - ~ - 0,
(33, iSb)
_ Fay' av' ] . O L ~ + z ~ j + u,
(33, 19c)
eoKOL~-~ + z ~
(33, 19d)
ae' Be' -8t + z~-~ -
2~ raw'
aw' ]
V'
+ ~--~ ap'_- 0 ,
- e ' + ~Bp'= O,
+ Bw'= O.
= _ap~ -ax'
195
Elimination of p' from (33,19a) and (33,19b) gives the vorticity equation I~ a
(33,20)
+
a ][Ov'
Ou'l
ay
-
Ow'
~ aw'
i f we use o f (33,18e). But in the geostrophic limit as Ro--¢O (with t,x,y, and z fixed),
(33,19a)-(33,19c)
v ' = ap'
(33,21)
equations
become
ap' u'=-a-~'
a--~'
e ' - ap' az
and therefore equation (33,20) becomes, f o r Ro ---->0,
I ~ - + z~-~l ( -82P' -+
(33,22)
a2p _---~, I aw' ay J - ~-~ = O.
ax 2
Similarly equation (33,19d) becomes a + a~_---,=ISY ~ xlj aopz '
(33,23) Elimination
o£ w'
from
ap'+ •w'= ax
equations
O.
(33,22)
and
(33,23)
finally
gives,
the
(33,18f)
and
f o l l o w i n g equation f o r the p e r t u r b a t i o n o£ the p r e s s i o n p ' :
(33,24)
I~
+ z4{
02p' az 2
If
we use
(33,21)
and
+B
(33,23),
[ a2p' ax 2
then
02P'I}
+ - -
the
Oy2
=0.
boundary c o n d i t i o n s
(33,18g) g i v e (33,25a)
8p'_ 0 Ox
at y=0,1;
(33,25b)
B + z~J~-~ -- I5-5 a lap' - ap'_ ax
0 a t z=O,1
In summary, we must s o l v e (33,24) with boundary c o n d i t i o n s stability
problem we use the method o f normal modes, t a k i n g :
(33,26) Now ~=n~
(33,25a, b). In t h i s
p'= ~(z)exp[ik(x-ct)]sin(~y). for
some
positive
integer
n
condition (33,25a) on the vertical wails. Equation (33,24) gives
in order
to
satisfy
the
boundary
196
(33,27)
~(k2+ 2 ) ~
_
O.
=
dz 2
Therefore
(33,28)
~(z) = A cosh(2qz) o
+ B sinh(2qz), o
where
1/2
q : ~[Z(k2+ n2~ 2)
(33,29)
for n=1,2 ....
and A
and B are some constants. o o Boundary condition (33,25b) at z=O gives Ao= -2cqB o,
so (33,30)
8 = Bo[sinh(2qz)
- 2cq c o s h ( 2 q z ) ] ,
f o r an a r b i t r a r y Finally
constant B of normalization. o condition (33,25b) at z=l gives the following eigenvalue relation [4q2sinh(2q)]c 2 - [4q2sinh(2q)]c - sinh(2q) + 2qcosh(2q) = O,
i.e.,
(33,31)
1 [(qcoth(q) -1)(qts_nh(q) -1)] I/2 c = - + 2 2q
It follows that the mode is stable
(kctsO)
if and only
if qmqcSl.2,
where qc
is defined as the positive root of the transcendental equation qts/uhq = 1. Therefore the modes are stable if and only if (33,32)
,
Z z Z (k,n) -
for given real k and positive
2 4q c k2+ n2~ 2 integer n. Therefore the flow is stable
only if: (33,33)
Z z max ZI(k,n) s 0.58
the maximum occuPing for k=O and n=l.
if and
197
The curves of marginal stability for the modes with n=l and n=2 are shown in Figure 17 below. Note that the longest waves are unstable for the largest values of ~, but that their growth rates are not the largest because kcl--) 0 as k --~ O.
0.6t 0.4
i r / = j]
Stable
,¢ 4' f
0.2
n =.2 ¢Y
0
~=4qc2/(k2+ n2~23 ~m% U~e ~
5
~e
~k
m~dea (n=l and n=2) ~
I0
tAe @ad~ ~
a~
198 34
.
THE EADY PROBLEM FOR A S L I G H T L Y
VISCOUS
ATMOSPHERE
The effect
of a little viscosity on baroclinic instability may be represented t a p p r o x i m a t e l y by taking thin viscous layers on the horizontal walls. We see that the
in a rapidly rotating slightly viscous
velocity
of
the
inviscid
generates a thin Ekman layer. a normal
(i.e.
the
"outer")
a d i s c o n t i n u i t y of
solution
at
a
rigid
wall
The vorticity of the inviscid solution generates
flux at the edge of the Ekman
conditions:
fluid,
layer,
so
that
the
inviscid
boundary
w=0 are replaced by
(34,1)
w = ±
-- j
[~-
{o
, at z =
H
o respectively,
where u
u sS m2/s). o It
may
be
shown
(as
is the kinematic viscosity
o
at
the
section
28)
that
(for the atmosphere
this
leaves
unchanged e x c e p t f o r t h e r e p l a c e m e n t o f boundary c o n d i t i o n s
Row' = +[21-EkJ]I/2fav'[~ay
(34,2)
J'
the
Eady
(33,18g)
we have
problem
by
at z = 0 , 1 ,
where the Ekman number is d e f i n e d as p
(34,3)
Ek -
2~
0
H2 o o
To justify this we i n fact need: R02<< Ekil2<< 1,
With ( 3 3 , 2 1 )
and ( 3 3 , 2 3 ) ,
(34,4)
Now
we
8
see
classical, (34,5)
t
So-called,
that
+ z
we
inviscid,
can
conditions
(34,2) g i v e r e s p e c t i v e l y
oxjoza1°p' -
-
re-use
almost
Ek
all
)"':[ our
8x 2
+
_a:':"
] 0/ z=O, i. 8y 2 ]'
calculations
for
the
Eady
problem of the section 33. As before we find (see(33,28))
~ = A cosh(2qz) o
Ekman
Ro U but - --- c ~ - 0(1]. Ekll2 0 V/2~ L 0 0
layers
and
+ B sinh(2qz)
see,
o
for
instance
the
section
28
(Chapter
VII).
199
etc...
Substituting (34,5) into (34,4),
instead of (33,25b), we get
Ao+ 2cqBo= ikA o, and Ao- 2q(1-C)Bo+ [Bo- 2 q ( 1 - c ) A o] t a n h ( 2 q )
= -iA[Ao+ B o t a n h ( 2 q ) ]
respectively, where A = ~ ~ [~Ekl I/2(k2+ n2~2)k
(34,6) Eliminating A
0
and B
o
we deduce the eigenvalue relation
4q2c(c-1) tanh(2q)
+ 4iAqc + 2 q ( 1 - i X )
- ( l + k 2) t a n h ( 2 q )
= O.
Therefore we obtain for c the following relation: c =
2qtanh(2q)
2
2qtanh(2q)
(q2+l)tanh(2q)
-- A2[ l-tanh2(2q) ]
-2q tanh(2q) 1/2
or else
1 . ~ ( I + T 2) 1 c = ~ - z 4q--------T---+ - 4qT
(34,7)
4T ( q - T ) ( q T - 1 )
- A2(1-T 2) 1112
2T where T = tanhq and tanh(2q) It
can
be
seen
that
(I+T) 2"
marginal
stability
occurs
where
1 c=~,
there
being
instability of the mode if and only if k is less than the positive root of ~2= (q-T)(1-qT) T It is easier to plot the marginal curve by regarding k as a function of q than vice versa.
Finally we note that EkB 2 _
_
~
A 2.
Ro 2 For
further
(1964).
details
the
reader
may consult
the
original
work
of
Barcilon
200 35
. MORE ON BAROCLINIC I NSTABI LI T Y
We consider here the stability problem equation (32,15), and we will assume that U
B
depends only on p:
(35,1)
UB~ UB(P).
We seek ~rave solutions h(y,p) of equation (32,1S) in the form
h(y,p) = ~(p)exp(ily).
(3S,2) Then,
setting
K 2= k2+
12 , we
obtain
for
the
complex
amplitude
~(p)
the
following equation
Us(P)
(35,3)
d
p2
where (3S,4)
~.Cp)
--
~ - ~l~oC-?~T
dp
For the equation (35,3) the boundary condition for p=1 (see (32,16)) reduces to the form:
(35,5)
[%cI) - cl~d~
JLdp p=l +
K°Cl) ~ } - %Cl)~ = o, To(ll) p=l p=l
where (35,6)
NB(1) -
dU B (p) dp
Ko( I ) I + -U (1). p=l To(l) s
We note that the boundary condition (35,6), for p=l,
is complex:
it contains
the eigenvalue c. Replacing it by the condition (NB(1)=O):
C3S,Sa)
~ dp p=l
+
~Cl)~cl To(l)
)=0,
we can prove s t a b i l i t y for ~B(P)>O, or: if Zs(p) changes sign once,then the flow is stable for (U~p)-K~Z~p)
201 The condition (36,6),
however,
must give rise to instability,
but only of a
type in which there cannot be more than one growing wave solution for each K. We can prove this by a finite - difference approximation of equations (36,3), with (36,6),
in which the segment Osp~l is broken up by the points Pl ..... PN-I
into N equal parts of length ~=~, and the equation and condition are written in the equivalent form:
(35,7)
B n- c
-1/2
[[oj ]{o
(3s,8)
,
B 0-C
where r 2
~2
+
K2~n - r2
n÷1/2 n
o ~o
} { =
NB
~2 J
B n~n=
oho'
is some average value of (S p2/ Ko(P)) between the point Pn and
n-1/2
Pn-1' while s2o = KO(1) / T o ( l ) " Then the following theorem holds (Dikii (1973)):
If all
(~B)n>O or all
(~B)n changes
(~B)n
sign once,
[(UB) n- KO](~B)n
and
if there exists
(the Fjortoft
condition),
a constant
K ° for which
then equation
(35,7)
with
the boundary conditions
R
N-I'
K
(36,10)
i - ~o= K(C)~o, K(c) = a + B=-~, a~O,
has no more than one pair of non-real complex-conjugate eingenvalues c. The
idea of the proof
involves the fact
as
of
graphically
hyperbola K(c),
intersections while
M(c)
is
the
that
the eigenvalues
function
a rational
M(c)=(~ I-
fraction
whose
are obtained
~0)/~o
denominator
polynomial of degree N-1 without any roots that are not real.
t
An a n a l o g Drazin
o/" t h e
and Reid
Fjorto£t ( 1981
; §22).
(1950)
theorem;
see
/'or
instance
the
and
book
o/'
the is
a
202 CONDITIONS FOR INSTABILITY
The n e c e s s a r y
conditions
for
instability
c a n be d e r i v e d
directly
from (35,3).
I f c = c + i c a n d c ~ O, t h e n ( 3 5 , 3 ) c a n a l w a y s be d e r i d e d by [UB- c ] , s i n c e r ! ! is never zero. If the resulting equation is multiplied by ~® ( t h e
[UB- c]
complex
conjugate
of
~)
and
integrated
of
p=l
at
p=O,
we
obtain
after
integration by part:
s
IO
2 p
1~a-
~ 2 dh
~ dp
_
IO
_2¢0 ~
~S (p)
__lRl=dp
1 UB(p)-c
+ ~jl h
i=dP
+ S L ip-----~ m
But
if
we t a k e
into
account
the
d~l
boundary condition
(35,S),
with
=0.
(36,6),
it
immediately follows that:
s iO p 2 dh~
(3s, 11)
1Ko--~-~ ~-~
+
2 dp
+ Ke~°l~l=dP
=
1
I 0 EB(p) iRl=dp I UB(P)-C
_. 'rK - - hp2 =® d~] + s [To_~l)T~ h
s "~o P
L o Lp
[uB-
c[ 2
NB(I)
Ko(1)(UB(1)-c)
] I~ ip_l ~ " -
Since
u B-
c
-
the imaginary part of (35,11)
-
c
+
ic
r
|
,
may be w r i t t e n
(35,12)
L~ ~
~
that
{35,13)
Hence, if c
i
is not to equal to zero, i.e., if the mode is to be unstable,
then the bracket { }, i n ( 3 5 , 1 2 ) ,
multiplied
b y c I must v a n i s h .
Therefore
the
203 vanishing
of
the
bracket
{
},
in
(3S,12),
is
a
necessary
condition
for
instability:
(35,14)
[o
i~,s
•
__
N (1)
cl 2 ~B(P)d p
J`11uB -
=
B
= 2
_
Ko(1)lu(1)-cl
~lhl~-`1"
The real part of (35,11) yields, with (35,14) for c i
O, and c o any constant,
(35,1s)
.(1)(U-Co~
r°
I~1 ~
] ~2
J`11us- e :
r°s IdOl = j`1[Ko(P~ld--~l
K=
+ ~
I~1
p
>
o.
Therefore certain stringent conditions must be satisfied by the basic state in order that (35,14) and (35,15) may be satisfied.
36
.
BAROTROPIC
INSTABILITY
We consider the equation ( 3 2 , 1 9 ) : d2~"
2 *
o,
=
w h e r e g2= k2+ S. We assume
that
at
atmospheric
flow
effectively
isolates
an
instability
consideration.
the
the
walls
exist
perturbations.
the region its
Although
from its source
containing clearly
surroundings
must
the
lie
region
an
the
artifice,
and a s s u r e s
within
of
the
this
that
should
region
under
Under these conditions it follows thatt:
[3"= O, y = Z l . case
equation
division
and
rigid
arise,
(36,2) In this
y=+l
by
the
necessary
(36,1) o
is
[UB(Y)-C],
condition
multiplied
for
instability,
by t h e
and i n t e g r a t e d
of
is
easily
complex conjugate y=-I
at
y=+l.
of
NO
obtained
h , after
Hence we o b t a i n
the after
i n t e g r a t i o n by p a r t :
;
+~
(36,3)
~.
dh
s
-
t
If
we
take
Into
account
I
s
dy + ,.,
+'
+~
1~'I2dY -
-1
(9,7),
(32,5)
_
d % ,C -~--~J[U(y)
-'1
and
(32,13)
for
the
barotropic
C]
dy
case.
if
O,
204
If we take into account ( 3 6 , 2 ) .
The Imaginary part of ( 3 6 , 3 ) may be written in
the following form: 2 • (36,4) to yield
Kuo's
The r e a l
part
---;dy ] l u . ( y )
-,
'
- cl 2
(1949) t h e o r e m . of
(36,3)
yields,
+1
with
(36,4)
,
f dy
(ci~ O) to exist
(36,4) shows that for unstable disturbances
barotropic sign
is the gradient of the basic state absolute vorticity.
dy~
relation
=
-1
d2U~ (Y)
m ~
2
J
-t
d [7"
c I = 0 and c o any c o n s t a n t ,
for
.+1
system,
within
the
the gradient range
of the absolute
(-I,+I).
That
is
to
The
in the
vorticity,
d R" dy ' must change
say,
inviscid
for
unstable
barotropic disturbances to exist , the basic velocity disturbtion must be such that
"2U'" Q B t y J"/d /, y2 is able to
sign.
This
include positive
motions
is Kuo's
the
(1949)
Influence
(negative)
of
over-balance
/3 t o make /3 - d 2U"B ( y ) / d y 2 change i t s
extension
of
Rayleigh's
/3 on
stability
the
/3 has a s t a b i l i z i n g
of
theorem of
instability
the
flow.
zonal
(destabilizing)
in the westerlies and a destabilizing
influence
(stabilizing)
Thus,
to a
on t h e wave
influence on wave
motions in the easterlies. Equation
(U:(y) - c O ) and d[3"/dy must be positively
(36,5) also shows that
correlated
within
(-I,+I)
for
free
disturbance
to
exist,
whereas
(36,4)
together with (36,5) show that, for unstable disturbance to exist, the product bS'" "d[7" must ty)--~-
be positive
change of sign of dD'/dy. in the U~(F)~'. ¥
region
of
positive
at
least
Further,
of
(-I,+I)
in addition
this relation requires
U:(y)d~,
and
small
in
the
lh'l 2-
region
to the
to be large of
negative
This condition has often been stated incorrectly to imply that the
existence of unstable disturbances point.
In part
requires U ~ ( y ) %
to be positive at every
205 37
. THE T A Y L O R - G O L D S T E I N AND S T A B I L I T Y
OF STRATIFIED
S H E A R ISOCHORIC
We
consider
here
FLOW
two-dimenslonal
stratified,
nonrotating
mean
flow
shear
EQUATION
Um(z)
hydrostatic
pressure
[U (z),O,O]
exactly
isochorlc equations
inviscid in the
motions isochoric
x-direction.
and density satisfies
as the
usual,
(v
m
0
flow If
8 m ~-~
and
in which p (z)
the basic
two-dimensional
and
O)
there
stably
is a steady
p (z)
state
of
Pm'
(nonlinear)
denote
the
Pm and
~ =
adiabatic,
(see the eqs.(14,3)):
Pb't +
o;
(37,1) Dp ~-=
O,
wlth D a + u~-~ O + w~-~ 0 and ~ = Dt _ Ot
I~_~ , ~.~] .
We now suppose there are small perturbations of this basic state. Thus for the total field we let: u' , O , w ' ) ,
u = (u+ -~
(37,2)
p = p.+
p ' , p = p=+ p ' ,
where the perturbation (primed) quantities are functions of x, z and t and are assumed to be small compared to their basic-state (37,2)
into
neglecting
(37,1), the
products
dropping the primes, (37,3)
using
the hydrostatic
of
all
(37,6)
perturbations
D u dU - -o + = w + 1 Op _ dz p-~ 8 - - ~ - O; Dt
(37,6)
Dt~ + p~ 1 ap Oz _ _ t g
DoP -Dt
relation
for
do00 + -w = O; dz
au Ow ~ + ~ - = O,
;
p=
quantities,
the following linearized equations:
Dw (37,4)
counterparts. and we
Substituting p= and obtain,
then upon
206
where
D
o _ a0 t + U 0~-~ . Dt We now assume that each dependent function is of the plane-wave form
(37,7)
#(x,z,t) ~
uw I = #(z)eik(x-ct) , k>O. P P
under this assumption ( 3 7 , 3 ) - ( 3 7 , 6 )
reduce to:
(37,8)
dU p.[ik(U m- c)u + ~ dz
(37,9)
dp p i k ( U - c)w + a-{= -pg;
(37,10)
dpm ik(U m- c)p + - - w = O; dz
(37,11)
iku + a-~ = 0,
w] + ikp = O;
dw
where u, w, p, p, Um and pm are functions of z alone. We now use
(37,11)
to eliminate
expression for p into (37,9).
u in (37,8)
and substitute
the resulting
Then, upon eliminating p by means of (37, I0),
(37,9) yields the following equation for w(z): (37,12)
d~[p (U - c)dW]
lO=-Zz [do= gdz d f dU
w]
-
÷ p k2(U - c)
]
w = O.
This is known as the Taylor-Goldstein equation; it was first derived by Taylor (1931) and Goldstein (1931) shear
flow.
When pm
in their studies of the stability of stratified
is held
constant
in the
first
two
terms,
retrieve the Boussinesq form d2w - -
dz 2
with
d2U dz 2
N2~ = - go dpm, for static stability, and p°=constant. p~ dz
w = O ,
we
readly
207 We note here two fundamental
restrictions
associated
with
concerns the two-dimensional
nature of the disturbance,
only
flow
parallel
to
the
mean
its origin in Y q ~ ' ~
were
introduced.
~Ae~v~eal (Squire
(1933)),
(37,12).
The first
i.e., waves travelling
This
simplification
which for a homogeneous
has fluid
states that:
"For each unstable three-dimensional two-dimensional
one travelling parallel to the flow".
For a simple proof of this,
see Drazin and Howard
this theorem to the case of stratified for
our
wave there is always a more unstable
consideration
of
fluids,
two-dimensional
(1966).
which provides
disturbances
always the fastest growing waves. On the other hand, normal to the flow are unaffected by the current The
second
approach
restriction
a
priori
concerns
eliminates
the
the
be dominated
the motivation
since
these
are
waves travelling strictly
decomposition
tnanaZea/ solution
by the growing
only
(Yih (1965)).
plane-wave
that would arise in any initial value calculation. solution will
Yih (1955) extended
(a continuous
However,
unstable
(37,7);
this
spectrum)
for large time the
plane-wave
modes and the
latter thus deserve first consideration. d2U For
flows
with
a
smoothly
elementary functions or
discontinuous.
in terms (1969)),
of
as solutions,
While
special
much
effort
varying
some
shear,
dz 2 independently
analytical
functions
(e.g.,
has
expended
been
~
O,
of whether
sollutions
see
(37,12)
Drazin
of
and
on seeking
no
p=
Howard
has
is continuous
(37,12)
numerical
longer
are (1966);
possible Thorpe
solutions
(see
Turner (1973) for a discussion of some of these).
SYNGE'S GENERALIZED RAYLEIGH CRITERION
More a century ago Rayleigh (1880) showed that for flow of a homogeneous fluid with rigid boundaries
(or boundaries
instability
the
/~ec~.
is An
that
analogous
profil but
more
instability was obtained by Singe However,
at
U(z)
infinity), should
a necessary
have
complicated
o~
~
necessary
condition for o~ze p ~
o~
condition
for
(1933) for the case of a stratified fluid.
his paper was overlooked for several decades and the same result was
proved independently by Yih (1957) and Drazin (1958).
208
To
prove t Synge's
nece.a~an~ condition,
eguation
( 3 7 , 12) w r i t t e n
(37, 14)
d-z
we
in the following
m°)
-
d-z
Since
d__~ _= dz
m
w
start
with
the
Taylor-Goldstein
form:
+ pm
--
-
k2c0
w = O,
(z 1
2 '
dU where
~ m U -
c.
w the
first
two
terms,
in
the
equation
(37, 14)
dz
can be written as
d (37,14)
hence
dw
d f d% ]w;
becomes N2
c37,,~:~
~
.
- 1.~lOo~
~ - o= j
÷ ok
w : o.
Next we multiply (37, 15) by w" (the complex conjugate of w), and Its complex conjugate
by w substract
and then
~2 [w* d z
1
integrate
w
-j rz2 -
f r o m z I t o z2:
d {oo,w'll
dz jjdz
2[d [ d U
1
2
::])
1
1
z
=o,
1 i
where
""lwl 2
i
I
=
~ .
If the boundaries are rigid or at infinity, integral
first
c = c + ic r
in
(37,16)
drops
out
after
an
w and w" vanish integration
there
by parts.
and the Putting
in the second integral and simplifying yields
1
c37,,7~
c
f
~
(%-c,
r d r d~o] I~ ~+ c~1 ~, t~lO<:~ rcoo-c
Iwl ~
____
-I-
c i
1
- 2p N2CUo - C r ) } d z = O.
Hence
if c >0 i
(unstable
waves),
the expression
in the curly
brackets
change sign and hence must vanish for some z~[zl,z2]. Thus formally,
t
According
to
Leblond
and Nysak
(1978; §43).
must
we obtain
209 "A necessary condition for a stratified
c37,18)
Ic -
c
shear flow to be a n ~
+
is that
2oN c%- %)
=
for at least one value of ze[zl,za]".
When p = constant
(homogeneous
fluid,
N~mO),
(37,18) reduces
to
condition
instability.
d~U -- O ,
dz 2 which
is
condition
Rayleigh's
well-known
(37,18) has no simple
unknown elgenvalue
necessary
interpretation,
for
however,
since
The
it involves the
c = c + ic . r !
MILES" SUFFICIENCY CONDITION FOR STABILITY
Of the many stability flow,
properies
the most celebrated
Richardson number
(37, 19)
(see, also,
Ri
Miles
(1961)
is undoubtedly
established
the stability
the formula
for stratified
criteria
shear
involving
the
(33, 14)):
N2 -
dU On the assumption
of analyticity
cenaLd/Zo~for stability
of
dz
and Peo' Miles
showed
that
a
is that:
1
Here we shall
present
Howard's
(1961) proof of this theorem,
which
is simpler
and does not require the analyticity assumption. We make,
first,
two transformations
where X=X(z) represents (37,20)
~-~
~
The rigid-wall at
z
=
z I and
differentiation
~-~ + ~-~ Z
- ~-~
boundary conditions z 2,
in the equation
a new dependent
which
we
will
in (37,20) gives
variable.
~
(37,12).
First
let w=xu,
Then (37,12) becomes
+ pm(N
- k ~ )X = O, z I- z ~ z 2.
imply that Z(z I) = ~(z 2) = 0 provided w ~ 0 assume
to
be
the
case.
Carrying
out
the
210
(37,21)
~
Suppose
now that
(o = U -
c ~ O,
~
+ p (N
X(z) for
is
- kZ~Z)X = O.
an ~
any z,
which be as differentiable
solution.
and
we c a n
as
is
Then c = c + ic is r ! branch of ~ for
choose
one
complex all
and
( z l , z 2)
U . Now s e t
= ~,I~ in (37,21).
After a little algebra it follows that
(37, 22) with
-
@(z 1)
= @(z 2)
integrating
over
~ dzJ
=0.
(zl,z2),
+ p~k2~ + p ~
Multiplying (37,22)
by
yields
¢"
- N
(the
after
complex
conjugate
integrating
by parts
1 [Z d f 1
¢ = O, z 1 of
¢)
and
1
I
+
P~)
dT
1
Equating
the imaginary part of (37,23) to zero,
Z
Hence
if c i>0 (unstable waves),
range of z. Thus, that -
+ 1
-dz
as Howard
+
(37,24)
put
be somewhere
~%12+ On the other hand i f -l~d z~ J
Z
we obtain
oo
-1
1
implies that - 4td l ~ U zm l Je +
it, a necessary
condition
for
N oo 2 < 0 for some instability
is
negative.
N2~ z 0 everywhere, then (37,24) implies that
dU~0 cl= O. Thus we obtain (for ~ * O) Miles' theorem. Unlike Singe's necessary condition for i n s t a b i l i t y , Miles' sufficient condition for s t a b i l i t y has a simple
physical
interpretation.
ratio of buoyancy to inertia,
<# ~
~xu,~
Since
Miles'
the
Richardson
theorem effectively
i~.,,~, u~-~ ~,~.~ ~.to.~ ~ . ~ . ~ .
number
represents
state that:
the
211
ItOI4ARD'S SEMICIRCLE THEOREM
This theorem defines a semicircular region in the complex c-plane in which the eigenvaiue
c
for
an
unstable
wave
is
located.
This
result
was
first
established by Howard (1961). Our starting point
is (37,21),
the equation for Z, where w=~Z=(U m- c)z.
this equation is multiplied by •
and the result
integrated over
If
(zl,z2) , we
find (a~ain, for rigid boundaries or boundaries at infinity)
Iz z
Setting
] C
=p~o~ 2 rLI~I id=l=+
(37,251 the
real
k = Izl 2 d= -
1
= o.
1
and imaginary
parts
of
pm[(U®- %)=- c ]
(37, 26a)
p . 21xl=dz
(37,251
equal
to
zero
a~
gives
PmN~Izl=d" = O, z
1
1
z
(37,26b)
e
p®(~-
e r)
~-
dz = O.
1
The
second
(1933)):
relation
"~
immediately
ci>O , ~
gives
another
U~/~e eacLa/~ ~
of
Synge's
paia~ z r aura ~
is sometimes rephrased to read:
" ~ ~
~
t~
cr ~
~
~ t/~ ~
o# uco"
Now l e t (37,27) Then
Q = Om
+
IZI
(37,26b) becomes, f o r cl>O,
(37,28)
z z ~;2 U Qdz = CrUz2 Qdz, 1
1
and by v i r t u e of (37,28),
(37,26a1 reduces to
(37,291 We now s u p p o s e
P~N=Izl =dz"
1 z
Then
> O.
that
1
1
(U)
mln
m a ~ U (z) ~
~ h m (U)
1
~x"
results Cr=Um(Zr)".
(Synge This
212 Z
Z
(37,30)
z f Z z Q(U _a)(Um_b)dz = fz2 U2Qdz - (a+b)f~2 U Qdz 1 1 1
0
+ abf z2 Qdz = Z
~ 2+ C2r
I
(a+b)c
r
1
÷ blf2Qdz÷ a- Z
20o.:, i dz
1
1
using (37,28) and ( 3 7 , 2 8 ) . But fz 2 Qdz > 0
and
2 Pm N
1
1
that c2+ c 2- (a+b)c + ab ~ 0 r
zl2dz > O; hence the inequality (37,30) implies
i
or equivalently
r
1
2
1
2
Thus we have ~ o a t e a d ' a ~ U w . c , w . m : "E~
~
oe~e ~ 0%e
[(a+b)/2,0]
~
c ~a~ ~ U~
~b~
anx~x~ m~ upp~
hn2/
[ci>O) m ~
c-p2xu~e ~
~e on o~ ce~
o/
o.nd, d/.ameZe~equa~ZaU~,eaaa,gee~U "
38 .THECONVECTIVE INSTABILITY PROBLEMt
Here in nondlmensional variables we consider the following equations, for the atmospheric motion: ~ -[+ ap
~.(pu~) 1
p~+__~p+ •~
(38,1)
=
0; 1
-~
Fr-~2~
~ = ~
l) 1 DE_ 1 fl ^T * ~( z - @ *2 E Dt p tRePr~
p = pT.
t According to Bois (1984).
]
~ u
~(~.u~); +
%Q(z)} ;
213 Note that these equations
(38,1) are normalized with the same length (Lo) for
horizontal and vertical scales (c ml). o
The quantity E, which
is the ~
temperature
of the atmospheric medium,
is related to p, p and T by the formulae: (38,2)
E =
T _ p(~-l)/~
pl/~ p
The quantity ~ appearing in the energy equation is the viscous dissipation and the terme Q(z) is a heat source term. We shall see later on that the presence of this term is necessary. First let us write the equations of equilibrium (~mO): dp= dz (38,3)
M2 + --~p= Fr 2
0;
p = p T ; d2T PrRe
From t h e
(38,3)
heat
source.
that
T
longer
at
the
natural.
It
T (z)
equation
on z by the
~ m
analogous
function
The f i r s t
depends
(38,4)
is
the
d z 2 + Qo Q ( z ) = O.
M2 FF 2
-
is
of the
intermediate
U2J~RT= ( 0 ) U~/gL 0
Boussinesq appears
as
related
to
system
(38,3)
o f ~z,
Bo
necessary
intensity
show t h a t
Q(z) the
of
the
hypothesis
where
Bo m -~eO
gLo - - ~RT(0)
number
the
(see,
the
formula
in
order
that
(3,5)), the
is
Boussinesq
approximation be satisfied (see, the section 8). Thus we assume: Q(z) is a function which depends on z by the intermediate of (38,5)
~ = ~z;
moreover
(38,6) and with this
2-
qo = ~ Qo' hypothesis
we h a v e
no
214 d2T(~) - d~ 2
(38,7) The e q u a t i o n This
(38,7)
equation
+ PrReQoQ( ~] = O.
provides
(38,7)
also
T (~) for shows
a given
why t h e
Q(~).
existence
of
Q(~)
is
necessary:
if
Q(~] = O, t h e n (38,8) For T
T ( ~ ) = a ~ + b. defined
V ~>0,
and decreasing
if
~ increases,
then
there
exists
a point
Go w h e r e T ( ~ o ) = O. For a realistic that
O(~)=O
locally
distribution
in the
valid.
troposphere.
For
a
approximation
is
introduce
two s c a l e s
4.
the
o f T (~)
But t h e
tropospheric
useless
and,
(see
the
Figure
Boussinesq Flow
the
in particular,
it
z and ~ (see,
18 b e l o w )
approximation uniformly
valid to
useless 12,
for
I
MESOSPHERE
\
TROPO-: I
,
I
{
,I
|
}
. m
0
~.
18: ~ a n Tc o ( ~ ) = 1 t h ~ aex~ t e m 4 ~ ~ a a ~=1 t h e aeo~ a 2 ~
La 288°K.
La 1 1 , 8 km.
then
only
Boussinesq
simultaneously
instance).
!
SPHERE ~i
is
is
the section
,,.11. ~
we c a n a s s u m e
215 Now, we assume that ~=M ~
(38,9)
Fr2=M, ~=Mz,
and we set i n ( 3 8 , 1 )
~ ,
~.~ ~ ~;
P = P=(E) + M2p; (38,10) P = P=(E)
+ Mzp;
T = T=(E) + MT.
Then (38,1) takes the forms:
-> 'ypoo(q ~
-> * ~p * 'yp ~ = -'y M ~
+
o~ + ~.(p u) + - - ~ ~ dp=
p=CE)@.u = -M f ~ t
(38,11)
dE
u + 3~(~.u) }
;
= MI~I Dp
PQo(E)g + Poo(E)~=(E)w-pp-~AT
;
Dt
dT dE
pw
- ~ DDT Y + ~-I - ~ ~} + O(M2); p T + T p = M(p - p T ) , where (38,12)
1 - E
~(E)
dE~
p~/~ with E = = P=
,
dE
The system (38,11) is the uniformly valid Boussinesq system for a dissipative flow. We
examine
related which
now
to the
situation
function can
considerably ground.
the
"~a~ea~e"
~
instability
temperature R (E)
effectively
is
of
profiles,
aeq~.
arise
in
a atmospheric namely,
Usual the
vary from a day to another
measurements
troposphere,
because
medium,
temperature
which
profiles
show where
is for
that
this
~(E)
can
of the radiation
from
the
216 When ~f~(C) is negative, the propagation of periodic waves is impossible (see, the
section
12)
that
there
appear
instability
effects
due
to
the
wave
propagation for which the velocity is of the form
(38,13)
u = ~(z)exp{o't + i~.~},
with ~ v . The correponding flows are unstable flows of the ~ a g ~ / g A - ~ n a n d type, and the question
of
their existence
experimentally
placed
in
is a
B&nard
evidence
problem.
(see, for
This
example,
existence Warner
has
and
been
Telford
(1983)). The theoriticai justifications of this existence (Manton (1974)) were proposed, between either
in general, two
as
atmospheric
levels
walls,
Co and
or
medium
by assuming that
free can
be
CI
the
(Co being
surfaces.
eventually
If the
considered
atmospheric
as
interval
medium
the
ground)
[Co,C I]
incompressible
is confined considered
is small,
with
a
the
constant
temperature gradient. A Rayleigh number can be defined as in the case of the classical B~nard problem. Cellular flows appear from a critical value of this l~yleigh number. The aim of the present section 38 is, by a more detailed analysis, to place in evidence a variable Rayleigh number, positive in the zone of convective flow, negative in the stable region, and which vanishes at the boundary. Since the problem deals with the ?/nea~ stability of the atmospheric medium, we seek the critical Rayleigh number.
THE EIGENVALUE PROBLEM
[
We seek u in the form (38,13) and for the system (38,11) we search a solution
(38,14)
~ : R(z)e~÷i~'~
~ : p(z)e~÷i~'~
= T (c)e(z)e ~t+i~'~
We note that in the relation (38,13) we have
(38,15)
~(z) =
U(z) ] V(z) , W(z)
and after eliminating the other quantities we obtain the following system for
217 the vertical velocity W(z) and the pseudo-temperature
(38,16)
~pmD2W + K 2 p 8
= ~6DW
- 2
+ M
8(z):
+
1
(38,17)
PrRe
D28 - v p 8
= p®~CEIW
~-~ +
dE
-dz +
P~ dE
-
~-I p + M 1 - M T-- ~ ~
O(M2);
[I D2 P tpm
- [a(E)-c(E)]d~'~'} + O(M2),
d2 1 f dTm ] with D 2 = - - - K 2, a(E) = + 2~J, c(E) = 1 dp. dz 2 dE Pm dE In (38, 16) and (38, 17) the pressure P(z) has the value: -
(38,18)
P(z) =
The system {(38,16),
~-~- crpo°
(38,17)}
-
-
-
-
+ O(M).
can be reduced to one equation
for W(z),
which
is:
.DBW_~e[1 + ~1]D4W -~j + cr2PmD2W
1
(38,19)
- K2p~W
PrRe2p~
PrRe2p~ dW + l~r2c(E)p~ ~-~ + O(M2), where
In
fact,
it
will
{(38,16),(38,17)} conditions.
as
here
as
equation
much
(38,18)
convenient
to
because
the
of
These conditions are those of equilibrium, dW W = ~-~= 0
(38,20) condition
be
of
zero
velocity
consider
the
associated
system boundary
namely:
at z=O and z-->+~; at
the
wall
(bottom)
and
vanishing
of
the
218 perturbation at infinity, (38,21)
O = 0
prescribed
at z=O and z-->+=;
temperature
at
the
wall
and
vanishing
of
the
perturbation
at
infinity. We finally have six homogeneous look for couple
boundary conditions.
(K2,~) for which the solution of
The problem then
(38,19)
is to
is no/: identically
zero. These couple can be defined by relations of the form K = X(v,M). Hence we assume that K can be written in the form
(38,22)
K = KoCV) + MKI(v)
+ ....
and in the following we compute only K . 0 PRINCIPLE OF EXCHANGE OF S T A B I L I T I E S
The aim of the present section is to eastablish the assertion that:
However,
in order to establish this property,
precisely particular, We assume
the
behavior
of
unstable
solutions
it is necessary for
large
to study more
altitudes
z's.
In
the following theorem will be used. that T=(<)
differentiable,
and
is a positive having
for
function,
large
<'s
which
the
is n times
behavior
of
<~
continuously with
6mO.
We
suppose that its m-th derivative has, for large <, following the behavior: (~ - i)...(~ - m+l)~ 3-m, ¥m~n. Let us set v = p + iq. Then:
"go/z p>O,
o.~ M,
~
~
emL~ ~
t./ua/: o x ~
,ten~ ~
~
B az~ ~,((r)>O mhLc~ d~ not d ~
a~
eq,a,et./..o~
(38,19)
~
.~u't/xx/~
o/:
219
[wl < Be -~(e)z, dW I ~-~
<
Be -~(V)z,
(38,23) dnW
7
<
Be_~(~)z
aNS~ z----)+co. The proof of this theorem is very technical because of the particular cases which must be considered in detail. We refer the reader to Bois (1979) for a detailed proof. above
theorem,
stabilities. (38,17).
By
The fundamental allow
one
to
inequalities establish
(38,23) set the
in evidence by the
principle
of
exchange
of
In this view we form an energy equation deduced from equation o multiplying (38,17) by the complex conjugate 8 of O and by
integrating from 0 to +co, we obtain:
jF+coO'[ V ~Ie ~_ 2 o - ~pcoO~z = ~
(38,24,
0 where ~
÷co • o pco•coWdz +
0
÷co
M~ ~l(Z,M)dz, 0
is a quadratic form of (e,W) and their successive derivatives.
I Now estimate the integrals in the relation
(38,24).
By integrating by parts
.d28 the term 8
dz 2
and by taking the boundary conditions (38,21) into account, we
have: (38,25)
e ~V--~--~eo e
= O -L~--~LI~ I
0
lel
0
÷ ~p.lel2~z In order to estimate the integral ~ = ~
÷co • m 8 pcoMco(~)Wdz we replace pco8 with the
o
t
Indeed, it is necessary to examlne the particular case where ~
3
is real.
220 aid of (38,16). Hence by setting D2W = G:
=
where
E
2
is
also
s+: ,.<<> K2
+ >,r+-,
o
a
o
quadratic
form
of
(W, 8)
and
their
derivatives.
integrating twice by parts a n d taking into account that ~ ( ~ ) only
through
(38,26)
g),
By
(depends on z
we o b t a i n :
':J
.+®r ~ ( ~ ) o
[God2w
'1.-~%-~ [
dz 2 -
K2G.,,] - --~ ] -j K12 ip ~ G •~ ® ((]W~dz J
+ Mf o
~3(z,M)dz,
where, in (38,26), we have set Zs(z,M) m ~2(z'M) + d(d~[G'dWL aE
dzdC'w]'
and so that ~
is also a quadratic form of (8,W) and their derivatives. 3 Now by exhibiting G in the first term of (38,26) and by integrating the term @
i
G W by parts, we finally obtain:
e+mf 1
2
Jp-~rlLI~dW 2+ I¢2 lwl2]~z
o
+ M~
~4(z,M)dz.
o Relation (38,24) takes then form:
(38,27)
j'+~£I rldel2+ o
l~-~-6LI~I K21el2] + +
vp.
lel 2 + I
ReK2~I c
12
,,-'o:.= rLIl~<,w :+ <:i,,i~]}<,z : _,.,r+°,<,:. ~ o
Let us set E(z,M) = ~(z,M) + iQ(z,M) in (38,27] and let us separate the real and the imaginary part of (38,27). Then, setting e-=p+iq we obtain:
(38,28) o
o
221
)=
r+=S1 i j I F ~ L Iriddle+ ~I ~iol~] + R-2S=IGI
(38,29)
o
~®[idw 2
*®
K2
o
÷= + Mf
~(z,M)dz.
o
From
the
two
equalities
(38,28)
and
(38,29)
we
can
deduce
two
important
assertions: first trin
suppose
E n=6).
that
From t h i s
~m(~)>O
V~ and
p>O.
We can apply
the
theorem
(by
theorem the integrai:
Mf ~(z,M)dz o
is ~
bounded by a term O(M).
In agreement with (38,22) we replace K by
K0, and thus:
(38,30)
jr+®J Vr-~-~ ' rl I. I de ~ i~+ I 0
p = -
1012] + - If®(( -K2Re I ) G i 2}dz 0
<0.
K2
o
o
This a s s e r t i o n i s i n c o n t r a d i c t i o n
w i t h the assumption. Hence
" I f Rm(~)z0 V~, then at z e r o t h order i n M, p i s e i t h e r negative or zero". This theorem, ~
~
Now consider
in fact, establishes the well known property that the R®(~) ~
a ~
~ .
the case where Rm(~)
changes
Rm(~) is continuous sO if ~S~o' and M=(~)zO
its sign.
In fact we assume
that
if ~Z~o; Go being the or~V root of
the equation ~(~)=0. In this case we have the following result: "~
~=(¢)~o,
~o~ ¢~¢o' ~
R=(¢)zo,
~o~ ¢z¢ o, R=(~o)=O, ~
This theorem proves the principle of exchange of stabilities.
o,~
In particular is
establishes that the threshold between a stable state and an unstable state is a stationary state
(o'=0).
222
THE B£NARD PROBLEM
The B&nard problem i s t h e p r a c t i c a l namely t h e s e a r c h o f t h e r e l a t i o n
r e s e a r c h of the
thresholds
of s t a b i l i t y ,
K=K(~) when Real(~)=O.
We now i n t r o d u c e t h e l o c a l R a y l e i g h number o f t h e c o n v e c t i v e flow: (38,31)
Ra(~) = - P r R e 2 p ~ ( ~ ) ~ ( ~ ) .
Equation (38,19) takes the simplified (38,32)
form ( ~ e ~
¢=0):
DeW + K2Ra(()W = -Ma(()D4[d~l + O(M2),
and we look f o r K when e-=O. We assume t h a t t h e r e e x i s t s Go s u c h t h a t : (38,33)
~<~o ~
The s o l u t i o n results
I)
Ra.(~)>O, ~ o
o£ t h e e q u a t i o n ( 3 8 , 3 2 ) can be computed and we g i v e h e r e o n l y t h e
o£ t h e c o m p u t a t i o n .
if
===~P~a(~)~O.
~>~o: Ra(~)
is
One o b t a i n s :
negative
and
the
solution
which
is
infinity is the sum of three monochromatic waves:
(38,34)
WCz,M)
= F.A ( ( ) e x p
j=l J
-
-
-
'
M
with
(38,35)
dE
N(~)
=
IK2RaC~)I 1/3
,
and
(38,36)
Aj(~) =
B° e x p [ - g C ~ ) ] J
Ll~J
-
gC~)
K~]
~ a(~)
=
-
::-~-~-d~.
o
damped
at
223
The s q u a r e positive.
roots
in
(38,35)
and
(38,36)
are
those
of
which
real
part
is
The c o n s t a n t s B°J a r e u n d e t e r m i n e d .
I f EsE 0, b e c a u s e o f t h e change o f s i g n o f Ra(E) , t h e d ~ j ' s
in (38,35)
dE
must read:
now
(38,37)
dE Hence
there
exists
one
d~j__ (corresponding
root
to
j=l)
for
which
dE vanishes.
__d~J can dE
From (38,37) ds] vanishes at the point Et defined by the equation:
dE (38,38)
K 2= N(E1)
and Et is a turning point for the equation (38,32) t. In what follows we assume that N(E) is a monotonous function.
Two cases must be considered,
whether E is smaller or greater than the u x ~
2) if Ei<ESEo:
according to
root of the equation (38,38):
the solutions have the form ( 3 8 , 3 4 ) where --d~J is given by
dE
(38,37). The relation (38,36) remains valid, 3) if E<EI: the waves of phases $2 and ~3 are unchanged. Now W reads
2
(38,39)
w= E
~Ij
3
.
r ~oj ( E ) .
A,j(E)exp[-"~--~-~-J +j~2Aj ~E)exp ~ - ~ - J .
r
•
The d~°lj / d~ ' s and AIj s read:
t
Concerning
the
(12,19)
(12,30)).
to
turning
point
see,
for instance,
the section
12 (equations
224
dl12 ___
d111 = _ _
d~
dC (38,40)
B lj exp[-g(~)] Alj(~)
Note that
=
only the waves of phases
other phases
~11 and ~12 are really
(~I' ~2 or ~3 ) contain a term of fast damping.
are inexistant bottom
= I IN(() - K 2q1/2 J ,
for
except
~2
and
in the
~3'
immediate
and
the
neighborhood
turning
point
oscillatory;
Hence these waves
of singular
~x
for
phases ~2 and ~3 represent classical boundary layers,
the
points
~i). The
(the
waves
of
while the wave of phase
~I represents a free boundary layer. 4) matched provides
In the neighborhood to
of ~I' the waves of phases
the wave of phase
two
relations
~01 which exists
between
Bn,
these relations,
we obtain between B
(38,41)
B12= B i l e x p
i
BI2 and
11
and B
~
12
only
B I.
By
for
~011 and ~012 must be ~>~I"
The
eliminating
matching
B I between
the reflection relation
= -B11ex p
i
+ ~
,
M
with
¢I = I~Iv/N(~} - K~ dE = ~II(~i).
(38,42)
0
This reflection
formula
(38,41)
can be compared
to the formula
(12,42)
for
periodic flows (see the section 12). Finally W depends linearly on three constants. by writing
the boundary
conditions
at z=O.
conditions now are:
(38,43)
W =
dW = D2W = 0
at
z=O.
These constants are determined From
(38,20)
and
(38,21)
these
225
SOLUTION OF THE B~NAP~ PROBLEH
Writing
the
condition
(38,43)
we
obtain
for
and B an j zj algebraic linear system, of which the righthand side is zero. This system is a system for B z, B2, B 3 if ~1 does not exist, exists.
the
constants
B
and for B11 , Bt2 , B2, B 3 if ~1
In this last case the additional equation is the equation (38,41).
differentiation
of each function Aj(~)exp[-~j(~)/M]
the zeroth order in M, the same functions, allows one to determine
completely
the
d~2/d ~
complexe
e
conjugate
of
to z are at
but multiplied by: -d~j/d~.
the equations. d~2/d ~,
with respect
and
that
dsz2/d ~
=
-d~11/d ~.
1) if ~z does not exist A-1
0,
where 1
A
1
=
1
-4~Jd,~
"
-4~,:/d~:
fL--~ d.flJ -
~2
fL--~ J -
~.dC J
l-]~-~ J 2_ ~
~ ,-f ~____.~_~,1~_ d~;~
L-;~-~J - '~
2) if ~I exists (38,45) where
1
A = O, 2
"
This
Note again that d~3/d ~ is
cancelling the determinants we finally obtain the following equations:
(38,44)
The
-4~,./d,~
"
By
226 1
1 A 21
A 11 1
1
d:
A 12
A22
1
1
[[ d(
j
d~
-i~,,/d:
d~_~1112+ K2
.'1
LL d: J
[d~]
J
" J
2+ K 2
A
2
A 13
A23
1
1
0
us
[~_~_.1112+ K2
~d~1i/d:
~11/d:
J
0
and we note that Let
Idd~_~l1] 2+ K2
show
e x p I - i l M~-~l + 4 ] ]
AI
that
,
equation
, (38,44)
has
no
solution.
To
that
purpose
introduce the quantities:
=
,
g =
d:
(38,46)
d~2
2- K
2,
~
Note that ~ and ~ are real.
- -• = e
II
, the determinant
,l d~l
,
~=
~ d:
[] d~l
2-
By remarking that
A1 , a f t e r
AI= 2i--~sin(8) +
some computations,
1 /(A. 2 1.1=~ Ixi=~t -
where we have set: S = Arg
~=
([d,.]'I ~-
.
(x2)e_2i/[/3 _
reads:
(X 2- ¢2)e2'/[/3
},
K2.
we
227 From the relations
it can then be deduced that [
d¢2 •
By remarking that Arg
~-
<0 and that the roots chosen are such that
dCj 12>0 we deduce that
d~ 2
d~ I
dE
dE
=
n
+
it
,
n>O,
t>O,
so that (38,46) also reads: #sin(e) iA = - 2 - +
1
(t - nVf33).
We deduce from this expression that iA
i
is a real quantity of which the sign
is constant (this sign is that of ~48). In
particular,
this
shows
that
A
I
ae~ea
property is that in the (K, Ra(O)) plane, a ~
vanishes.
A
consequence
of
this
the curve of equation Ei=O bounds a
region. From (38,38) this curve has the equation:
(38,47)
Ra(O) = K 4,
and the only solutions of the problem are the roots of the equation (38,45). As for the classical B~nard problem (see, for instance, Chandrasekhar (1961)) these roots can be studied only by trials and errors: denote by K of
(38,45)
for a given Ra(E).
We can draw the curve
discontinuous at the points 0 and K
A2(K).
the roots o This curve is
(see figure 19 below) and for given Ra(E) o we can numerically determine its smallest zero K and its largest zero K . 1 2
228
0
K"
• g. z9: Ze.,~x~~ t ~ ~
A2(K) ~
o,,toe~Ra(~)
RELATION WITH THE CLASSICAL B~NARDPROBLEM
In the classical B~nard problem the stability of the flow can be determined by drawing a curve flow.
This
in the
curve
(K, Ra) plane,
separates
two
other where the flow is unstable. flow
is stable
regions:
v F,a < F,a ,
one
is the Rayleigh number of the where
the
flow
The absolute stability
for any K, so that
that,
where ~
we obtain a stability
is stable,
the
is obtained when the threshold
~
C
such
229
the flow is stable for any K. The number Ra is defined by:
(38,48)
~
=
%~oLoPrrRe] VrJ 2'
where Fr, Re and Pr are the Froude, Reynolds and Prandtl numbers of the flow and L
is the characteristic length. The coefficient ~ is related to the law o o of the fluid medium and 8 ° is related to the temperature gradient. We have for the variation of density p with respect to the altitude: (38,49)
Po~ = i ÷ ~o~0(z - Zo),
(see, for instance, Drazin and Reid (198i)). The drawing of a stability curve is possible here only if the number Ra(~) is defined with the help of a unique parameter. For example, if ~m(~) is negative in an
interval of small
length
(with respect
to
the atmospheric
one)
then
denoting by z" the altitude (with dimension) where M (~) vanishes, we have O
o
e = z~/Ho,
(38,50)
since e is small the Boussinesq approximation corresponds to the value M ~ , hence e
(38,61) and ~ ( ~ ) its
M = zo/H o, v a n i s h e s a t t h e p o i n t ~=M. On t h e o t h e r hand,
derivative
in this
(38,52)
interval,
~(~)
The Rayleigh number,
e o
f o r z
o
approximating ~(~)
by
we can s e t :
= aoM(Z-l) = ao(~-M). in this case,
is defined with the help of one parameter
only, which is ao, stratification parameter of the atmospheric medium. From (38,50) we have: (38,S3)
Ra(~) = MPPRe2aop~(~)(l-z).
For such a Ra(~)
it is possible to draw a stability curve
in the
(K, Ra(O))
plane. Such a curve has been drawn in Bois (1979) for numerical values which correspond to those of air at the ground level. The
results
(see,
the
figures
critical Rayleigh number R ac=
20
and
21
below)
show
that
there
exists
a
740, from which the cellular flows appear. The
cellular flows are really cellular if ~<~i and exponentially small if ~>~I"
230
I
~~O)u~e / .... y
,
I .85
K
I\
~ou~
f
I<._ X
~4.21: ~
fto~ ~ ~I~ (x,z) pZa~.
231 When the cellular flows appear, the only part of W which is not damped is the term including ~tl and ~,2" Hence the only solution which is distinguishable, except in the immediate neighborhood of the ground, is the solution: (38,54)
W =
2 ~ Btjexp(-~lj(~)/M). J=t
Let us introduce the relation (38,41), then: (38,55)
W = C ~(~)sin(~ - ~1- ~/4),
so that W = D2W = 0 (38,56)
~ + ~/4.
It
Observe
that
.
observed difference
stability
threshold
with
corresponding
manner which as been used to define the is At
same i n b o t h p r o b l e m s . not constant least,
note
in the that
fluid at the rest,
incompressible
that
surface, this
which (740
boundary
located
altitude appears
instead
of
is
Moreover the Rayleigh
number:
the altitude
v e r y n e a r G0.
numerically 1100 f o r
conditions)
the Rayleigh
at
the
a classical
proceeds the
in
from
the
definition
is not
number o f t h e p r e s e n t
problem
whole flow. the
B6nard
problem,
treated
would be more realistic
stationary shear flow. for
by a free
important
of the
problem
t~
were reflected
can be numerically
determination Benard
it
the
m ~], M J
at the altitude ~ such that
~ = ~l+ ~ / 4 ,
The f l o w h e n c e a s
~
here
for
convenience
Such problems have been treated by Tveitereid confined
flows,
and
in
a
if it were treated for a basic
have
shown
that
perturbations, which occur in the flow, are longitudinal rolls.
the
(1974) first
232 BACKGROUND R E A D I N G
For a extensive treatment of the concept of the stability in fluids the reader is referred to: CHANDRASEKHAR,
S. (1961) _ ~ a a d ~
and H
~
~
.
Clarendon Press, Oxford, England. and
DRAZIN,
P.
and
REID,
W.H.
(1981)
_
~qziaa44~
~
.
Cambridge
University Press. Concerning the baroclinic and barotropic instability,
see the Chapter 7 of the
book of:
PEDLOSKY,
J. (1982) _ ~ e a p A ~ & ~ i d D ~ . S p r i n g e r - V e r l a g ,
New York Inc
REFERENCES T O WORKS CITED IN T H E TEXT
BARCILON,
V. (1964) _ J. Atmos. Sci., 21, 291-299.
BOIS, P.A.
(1979) _ Journal de M@cm_nlque, 18, 633-860.
BOIS, P.A.
(1984) _ " ~ a g m ~ t ~ Publ.
DIKII, L.A.
o ~ $ e a ~
~am~e4 ~n t A e o / m ~ " .
IRMA, Lille, vol. VI, Fasc. 4, n°2, pages II,1-II,89.
(1973) _ Izv. Acad. Sci. SSSR. Physics of the atmosphere and ocean, 9, n°12,
1312-1316,
(in Russian).
DRAZIN, P.G.
(1968)
DRAZIN, P.G.
(1978) _ in " ~ a / ~ 2 / ~ g o / d ~ Z a ~ e e p A g ~ " .
J. Fluid Mech. 4, 214-224.
Academic Press Inc., London (see, pages 139-189). DRAZIN, P.G. and HOWARD, L.N.
(1966) _ In "
~
ia ~
~ e ~ " ;
Accademic Press, New York, 9, 1-89. EADY, E.A. FJORTOFT, GOLDSTEIN,
(1949) _ Tellus,
R. (1960) _ Geophys. Publ.,
17, n°6, Oslo,
1-62.
S. (1931) _ Proc. Roy. Soc. London, A, 132, 624-648.
HOWARD, L.N. KUO, H.L.
1, 33-$2.
(1961) _ J. Fluid Mech.,t0, S09-612.
(1949) _ J. Meteorol., 6; I06-122.
233 LEBLOND,
P.H.
and MYSAK,
L.A.
(1978) _ Wo~t~e~ ~ n ~ A e Elsevier
Ocean.
Scientific
Publ.
Company,
Amsterdam. MANTON, MILES,
M.J. J.W.
RAYLEIGH, SQUIRE, SYNGE,
(1974) _ Austr.
J.W.S.
H.B. J.L.
J. Phys.,
(1981) _ J. F l u i d Mech., (1880)
_ Proc.
27,
496-609.
10, 496-608.
L o n d o n Math.
Soc.,
(1933) _ Proc.
Roy.
Soc.
London,
(1933) _ Trans.
Roy.
Soc.
Can.,
TAYLOR,
G.I.
(1931) _ Proc.
Roy.
Soc.
London,
THORPE,
S.A.
(1969) _ J. F l u i d Mech.,
TURNER,
J.S.
TVEITEREID, WARNER,
38;
A,
(1974)
_ Z.A.M.M.,
J. and TELFORD,
J.W.
YIH,
C-S.
(19SS) _ Q. Appl.
YIH,
C-S.
(19S7) _ Tellus,
YIH,
C-S.
(1966) _ D g n ~
142; 621-828.
27 (III);
1-18.
A, 132; 499-$23.
673-883.
(1973) _ Z a a g o ~ ~ ~ o / d ~ . M.
9, S7-70.
Cambridge
University
Press
64, 633-640.
(1967) _ J. Arm. Math.,
Sci.,
24;
374-382.
12; 434-43S.
9; 220-227. o~ N
a
n
~
~o/d4.
Mac Millan,
N e w York.
CHAPTER IX
DETERMINISTIC CHAOTIC BEHAVIOUR OF ATMOSPHERIC MOTIONS
39.
ATMOSPHERIC EQUATIONS AS A FINITE-DIMENSIONAL DYNAMI CAl. SYSTEM
In a non-adiabatlc,viscous,
atmosphere, a finite number of lower (large-scale)
modes of motion determines all the remaining modes,since the scale)
modes
are
strongly
damped
due
to
frictional
higher
force
and
(small-
dissipation
function and only replicate with decreased amplitude the fundamental modes of oscilIation (in particular, they have the same kind of spectrum). In connection
with these
issues Hopf(IS48)has
advanced
the hypothesis
equations
that
every set of phase trajectories
of the Navier-Stokes
for t --> +m to ~ / / e - ~
set. Thus the equations of a non-adiabatlc,
is attracted
viscous, atmosphere can be written in the following form: dl1 d t - ~;('/1;!~')'
( 3 9 , 1)
with
~={~1(t) ..... ~N(t)}
and
where
~k(t),e.g.,aPe
the
coefficients
in
the
Gslerkin approximation t and ~ is the bifurcation parameter. Non adiabatlcity and viscosity of the atmospheric motions gives rise not only to finite-dimensionallty of the phase space but also to dissipation of phase flow, i.e.,an umennge c At different
~
phase point
~ ~Ae @Au4e ~<~xune "downstream" as t--~+~. ~/o the quantity
div~(~ O) can be either
positive
(expansion) or negative(compression). The phase flow is called d / ~ (39,2)
t
In
if for every ~
o
we have:
A(9/O) =- div ~(~/o ) < O.
the
Free obtain
section
41
convection the
Lorenz
we
and
consider as
system
example (section
the of
classical
B6nard
finite-dimensional 42).
problem
for
dynamical
the
internal
system
we
235 Because of dissipation, attractors have ~
phase volume and dimensionality
smaller than N. More
recently
critical
interest
points
nor
has
been
closed
arounsed
curves
(limit
in attractors
which
cycles),called
~
are
neither
~
.
They belong essentially to third-order and higher-order systems,and
include
and the ~onea@ at/Ttae.byt (see the section 42). They are aptly called strange attractors because their properties are much more subtle than those of critical
points
and
limits cycles
(closed curves
in the
phase
plane
which
represent periodic solutions which are approached by neighbouring solutions as t--*±m respectively).
The
solutions
are
not
periodic,
althouth
entirely
deterministic, they share some properties of random systems. For example, several solutions which are initially arbitrarily close together may develop in substantially different way as time increases .The trajectories originating
in the
phase space
in which they wander for ever
appropriate
donta/zt o~ ~
t
tend
in a way that
to
a
subset
of
may appear to be
random. The correlation functions of quantities describing strange attractors,
e.g.
the second-order correlation
(39,3) decay
I ~
~ijCt) = Lim T~ rapidly
as
t
~+T~i
(s)~J(s+t)ds,
-T
increase
from
zero,
as
to
similar
correlations
of
transition,
in
quantities describing turbulence itself. Ruelle
and
Takens
(1971)
used
strange
attractors
to
model
marked contrast to Landau's model of spectral evolution (see the section 40). In Landau's model there is a succession of bifurcations as the Reynolds number increases, so that at each stage of transition the solution has components of differents
period:
2~/~i,
o~-~aLodlcfunction
2~/~2,etc ....
or,
precisely
the
solution
is
a
of time.
Spectral analysis of the solution gives resonance peaks at the frequencies ~1,~2,..etc
;yet
spectral
analysis
of
the
apea2odic solution
for
a strange
attractor gives a broad band of "noise" rather than high peaks. So
a
strange
attractor
is
suggestive
of
turbulence
itself,and
Landau's
spectral evolution is suggestive of the more orderly transition seen in the cellular motion of Benard convection. t
The domain of a t t r a c t i o n o f t h e s o l u t i o n ~----0 is the set of the point ~ 0 of ~ N s u c h t h a t ~ ( 0 ) = ~ 0 t h e n ~(t)---)~ as t---)co. So if the solution ~----0 is asymptotically nelghbourhood
stable of
0
then
in ~N.
its
domai n e
of
attraction
includes
some
236 In all
dynamical
systems
where
attractor has been determined,
the
general
solution
is aperiodic
and
it has proven to be a strange attractor.
the That
is, it is not topologically the product of several one-dimensional continua, in the sense that a smooth surface in three-dimensional space is the product of two continua.
Instead it is the product of several continua and one or more
~en/a~ sets, and an arbitrary intersecting curve, such as a line parallel to a coordinate axis,
intersects it in a Cantor set.
This is an uncountable nowhere-dense set;an example is the set of all numbers between 0 and 1 whose decimal expansions contain only O's and l's. For N=3 a strange attractor would be an 6n~/n//ecomplex of surfaces. STOCHASTICITY
Of special
"
interest to us here are the strange
attractors
,on which phase
trajectories display the following properties of stochasticity. (A)-
An
extremely
~
d e ~
on
~
cand/2/an~,due
to
exponential divergence of trajectories which are initially close together and leading
to
their
unpredictability
or
non
reproductibility
for
initial
conditions which are given with arbitrarily high (but finite) precision. (B)-The
e.~J~-e~
b~_Zca/e~,i.e,their
~
a~
arbitrarily
close
t]~e
a / ~
approach
to
any
o~ of
the
o ~ attractor's
points-which implies that their return infinitely often to the attractor-and the property that any initial nonequilibrlum probality distribution (measure) over the phase space (or,more precisely, over the region of attraction of the strange
attractor)reduces
to
some
limiting equilibrium
distribution
at
the
attractor (an invarlant measure). (C)-The ~
@aepen~: for any (measurable) subset A and B of the attractor,
the probability after emerging from A of arrival at B is proportional after a lone time to the meastu~e of B. A consequence of the mixing property
is the
fact that the time-averaged value: <¢[~(t)]> of any function ¢[~(t)] defined on the strange attractor is independent of the initial conditions ~o (for almost all ~0), and that this average value coincides with the average: over the (39,4)
invariant <#> m Lim T~
J
measure ( ~ ) : Ti
i T@[~(t)]dt 0
237 One mark of the mixing
property
is a rather a a ~
4ece~ of the correlation
function as T--9+=:
(39,S)
BJ'(T) = < [ ~ J ( t )
which
is
to
say
continuity
- <~J>ll ~ l ( t + T )
of
their
- <~'>1 >
Fourier
transforms
expedient
to have
with
respect
to
T,i.e.,their spectral functions. According
to Monin
(1986)
it appears
refer to the stochastic(random)
evolution
the
term TURBULENCE
(in the sense of (A)-(C),
the flow of a non-adiabatic,vlscous,atmosphere
above)
of
which possesses VORTICITY.
We note also that the existence of a broad band spectrum implies that then an auto-correlation function which tends to zero when the time increases. the auto-correlation
function measures
the time similarity,
Because
then the presence
of turbulence implies a loss of the similarity, a loss of memory of the initial states.
In other words
not permit
to predict
, the knowledge of the time behaviour the behaviour
in the past does
in the future:TURBULENCE
CORRESPONDS
TO
UNPREDICTABILITY. A very
important
consequence
of this
loss of similarity
initial states is the divergence of the trajectories neighbouring similitude) This
(then very similar)
fundamental
trajectories
(the
aam~
@ n ~
it
in phase space
diverge
(and
then
of the :two very
loose
their
if the regime is turbulent. property
(in
attractor(turbulence)related
corresponds
trajectories
(or memory)
is to
called the
iteration
fact
that
if twice
to even
sense
the
"sensitivity
performed
is nat due
the
to
of
(A),above)
divergence to
initial
i~% @n/ne/4~ gives
of
any
the
system
the
errors
same
which
strange
neighbouring
conditions"
indeed
infinitesimal
of very
or is
S.I.C.
It
predictihle
final
always
state) affect
L~ the
initial conditions and are exponentially amplified when the time runs. Lorenz (1963),
who invented the first and most famous example of deterministic
chaos
the appearance
through
three equations
of a strange
(see the section 42),
attractor
in the
gave this striking
solution
of his
illustration of the
S.I.C. Referring to the general unpredictability for (long-term) weather forecasting, Lorenz said that even a very small change of initial conditions produced
by
consequence
the
motion
of butterfly
wings
for the next future behaviour
dynamical system).
would
have
such as that
(unpredictible)
of the atmosphere
large
(considered as a
238 BIFURCATIONS AND I N S T A B I L I T I E S
In
the
framework
of
the
atmosphere-dynamical
equations,
the
origin
of
irregularity of turbulence is attributed to the instability, or the S.I.C, of the solution of the equation (39,1) at supercritical of ~. The instability of such a dynamical system is characterized by the positive L ~ p u n o v number (see the definition in the section 40) at a point of the phase space and the irregularity of the solution is interpreted as a nature of the strange attractor. Thus the irregular nature of the atmospheric turbulence
(chaotic behaviour)
is now fully understood in terms of the theory of dynamical systems. It is not obvious
that
dynamical
a
turbulent
system of
manifold theorem t
state
finite
of
the
atmosphere
dimensions,
but
it
can
is
be
described
guaranted
by
the
by
a
center
at the stage of instability where only a finite number of
unstable modes are involved. Consider a dynamical
system
(39,1),
assume
that
there
is a periodic
self
oscillating solution;this happens, for instance, in Rayleigh-B@nard convention problem (see the section 41). Now consider cases when this periodic flow loses its stability for ~ crossing O
a critical value ~ . Elementary possible bifurcations which might take place are well known: appear
a
one("~
new
motion
bifurcation"),
frequencies motion
periodic
("~o~
~
disappears
or
with
a
"
a
quasi into
eventually
atmospheric problem could
periodic
an
leaving
("~u/Ze.nnde bifurcation").Eventual lead
frequency
half motion
invariant place
to
symmetries
and
to richer
of
with
torus), a
the
or
type
group
of
two
natural
the
periodic
intermittency
invariances
possibilities
(see,
it can
fundamental
for
of
the
example,
looss (1984)). To study the stability of a periodic orbit,
it is necessary to look at the
~qu2/
linearized
~
,
i.e.,
the
eigenvalues
of
9
~
mx~
(after
elimination of eigen-direction tangent at the origin of the orbit). The total multiplicity of the eigenvalues crossing the unit circle determines the relevant dimension where asymptotic dynamical phenomena lie, for the full system
(dimension
situation
(for
of
example,
&e~e~), at a definite For
a
complete
companion
the
and
amplitude
center in
the
threshold rigorous equations
manifold).More Rayleigh-B@nard ~"
the
description see
precisely,
the
steady of
paper
this by
in
instability
state method Coullet
is
in
typical
the
replaced
and and
the
by
obtention
Spiegel
(1983).
ama2/ an oF
239 oscillating regime at a well defined frequency f harmonics
nfl).
modulated
with
Fourier
This a
means
constant
spectrum
of
that period
the
(2fl,3f I .... ); finally
the
(periodic
velocity
a phase
space
(including eventually higher 1 in any point of the box is
velocity
regime).On
contains diagram
only
the
other
sharp
obtained
hand
peaks
consists
at
the f
1
in a closed e
loop and such a closed loop is called a limit cycle and we say that,at ~ , the system bifurcates from the steady state(a fixed point)to a stabie limit cycle. The
second
step
breaking
the
appearance at a new threshold ~ former f for
1
simplificity
t.
new
recall
regime
regime
that
let is
its
us
called
properties
assume
that
biperiodic are
corresponds
to
the
•
>~ of a new frequency f2 superimposed to the
and more generally the two frequencies f
•
This
(mono)-periodic eo
only or
f
and
1
more
intermediate
and f
1
2
f
are ~ are
2
precisely
between
a
.
But
present
q~-peaiadtc
periodic
regime
and to and
chaotic one (aperiodic regime) it. Incommensurate means that
one
cannot
find
integers
n
and m, for two
frquencies fl and f2' as and f
I
and f
2
mf =nf 1 2 are unrelated.
The equation (39,1) possesses a unique solution
tt
~=w(t), satisfying
the
~°E~(t°),
initial
condition•
Here
~
is assumed
to
be
continuously
differentiable with respect to t for tzO. If for any c>O and any To>O,
there exists a time interval T(e,TO)>T ° and a
time t1(c,t o) such that t 2>t implies 1
l ~ ( t +T) - ~(t2) I < c, then ~=~(t)
is called quasi-periodic•
w(t) is called quasi-periodic
This simply means that a trajectory
if for some arbitrarily large time interval T,
~(t+T) ultimately (i •e. large t or t 2>t i ) remains arbitrarily close to ~(t). If ~(t)
is periodic,
trajectories periods. w
2
Thus
include
or
multiple
then ~(t)
quasi-periodic
also
periodic
if ~(t)=~iCt)+~2(t)
incommensurable,
periodic
it is obviously
quasi-periodic.
trajectories
and ~(t+T1)=~1(t),
is quasi-periodic• is
non
periodic
with
Quasi-periodic incommensurable
~(t+T2)=~2(t),
T I and
A trajectory which is nat (aperiodic);
cul o4~.~oxILc
240 In order to draw the corresponding phase space trajectories a two-dimensional representation can no longer be used. The limit cycle representing f
has
I
to
wind with the frequency f
and thus the representation, in a three-dimensional 2 oo takes the form of a torus. At ~ we say that the stable limit
phase space,
cycle bifurcates to an invariant torus T 2. The fact that the phase space is no more representable
in two-dimensions
us to use a transformation which allows to
the
~c2/n~) of a thFee-dimensional section is not only to
phase space.
obtain
complex three-dimensional
represent
a
The
interest
good two-dimensional
with very simple mathematical
(here in 3 diemensions)
section
( Y ~
of
Poincar~
useful
models labelled generically as
by a given plane ~. Then,
the
representation
phase diagram, but also to make
Very simply the Poincar~ section is a cut, a section,
simple
transformation
representation.
lowers
A limit cycle
by
one
"iterated
of all the
an ensemble
(phase space in
the 2
into a single point (analogous to a fixed point).
of
map".
trajectories
of
points
is
are
Obviously
dimensionality dimensions)
a
comparaison
obtained which corresponds to the Poincar~ section of phase diagram. this
leads
of
the
transformed
Much more interesting is the
following: The Poincar6
section for the
T 2) is a simple harmonics
of
complicated
closed
the
loop
trajectories (analogous
fundamental
to a
frequencies
section than a simple ellipse,
quasi-periodic
in a quasi-periodic limit may
cycle) give
regime
The
rise
presence
to
some
but as far as the regime
(torus T2; phase space at 3 dimensions)
(torus of
more
remains
the Poincar~ section is
a closed loop. Reciprocally
if the Poincar~ section is a closed
loop one can affirm that the
dimensionality of the corresponding phase space is 3. The
existence
dissipation.
of
attractors
is
In the phase space,
the fixed point
(corresponding
closely
related
to
the
the trajectory is attracted, to the equilibrium
state)
existence
of
converges toward
or to an attractive
limit cycle (corresponding to the oscillating reEime). Note that dissipation caa/ane_Zs phase space, the
meastu-e
of
the
phase
space.
More
more precisely dissipation generally
if
we
quasi-periodic regime in a dissipative system (as the atmosphere) a quasi-periodic attractor, What happens by a further or chaos
appears;
the
we deal with
or an attractiv torus. increase of the ~ parameter? At ~
the spectrum
is no more composed
OOO
of sharp
> ~
Oe
turbulence
lines
broad band noise begin to appear especially near zero frequency. space,
lowers
consider
but
also
In the phase
the trajectories are attracted on a complicated structure of "strange"
241 aspect
(and
of route Takens
strange
properties)
to turbulence
is
which
is
called
ab~xu~e ~
in very good agreement
(see the s e c t i o n 40),
through
with the
.
This
ideas
it is not yet proved
kind
of Ruelle
that at ~
o@e
and
it is
appearance of a third frequency which produces chaos! We
note
also
that
the
from a periodic
orbit
torus.
Instead,
the
points
is
appearance does not
appearance
generic.
The
of
an
invariant
imply the of
appearance
finitely
appearance
of
torus
an
many
T 2 at
a bifurcation
of orbits
dextae on that
periodic
invariant
T3
orbits
torus
and
at
fixed
the
next
bifurcation depends on existence of an orbit dense on the T 2 torus and hence, the bifurcation to an invariant T 3 torus seems unlikely. Finally,
if a periodic orbit on the T 2 torus goes round the long way "n" times
before closing,
then the bifurcation
is ~
with a sudden n-folding of
the period at the bifurcation.
40
. SCENARIOS
A first
bifurcation
may be followed
by further
bifurcations,
and
we may ask
what happens when a certain sequence of bifurcations has been encountered. In principle there is an infinity of further possibilities, to be specified,
but,
not all of them are e q u a l l y probable.
The more likely ones will be called aceanai~,
and below we shall examine three
prominent scenarii which have had theoretical and experimental In general,
in some sense
success.
a scenario deals with the description of a few attractors.
other hand a given dynamical system may have many attractors the Lorenz system at the section 41).
On the
(see, for example
Therefore,
several
scenarli
c o n c u r r e n t l y in different regions of phase space.
Finally,
a scenario does not
describe
its
may evolve
domain of applicability.
THE LANDAU-HOPF "INADEQUATE" SCENARIO After it
the first
is generally
the
first
bifurcation
the motion is generally
qusi-periodic
bifurcation
w i t h two p r i o d ,
leads
to
closed
orbits,
the
attracting invariant torus in the phase space. such time,
that such
its orbits as one
with two periods.
of
covers
the torus
the coordinates
Specifically,
e and ~ on the torus such that
It
second
after
can
space,
lead
to
if an
the motion is
then a resulting
phase
the second
was s h o w n t h a t
If, furthermore,
densely, in the
periodic;
a n d s o on.
function
of
is quasi-periodic
one can define two intrinsic angle -coordinate
242
O = ~ t + Const i and
the
orbits
is
dense
on
,
~ = ~ t + Const, 2
the
torus
if
and
only
if
~
After the next bifurcation there may be motion o n a T a torus, idea
behind
the
Landau-Hopf
O%depeadaa/ frequencies,
scenario
was
that
as
the motion is so irregular
soon
as
and
i
~
are
2
and so on. there
in appearence
are
that
The many
it must
be r e g a r d e d for pratical purposes as chaotic. Obviously
the appearence
of
turbulence
number of degrees of f r e e d o m
is r e l a t e d
to a system
One
corresponding
of
the
bifurcation
critical
a
large
(N).
There are various way in w h i c h this scenario can be ~ (a)
with
value
may
be
~
then,
of the ~ is exceeded,
motion for the system to follow,
and there
:
;
there
as
soon
as
the
is no n e a r b y stable
is a so-called ~
tnana/2/aa
to a m o t i o n involving more or less remote parts of the phase space, (b)
Although
bifurcation,
an
invariant
torus
generally
the orbit need rugl ~e de2%~e on it;
appears
at
it may return to
point after winding finitely many times around _ then the orbit the m o t i o n is periodic.
In fact
it is now believed,
theorem that closed orbits on the torus are more may lead to the Feigenbaum (c)
A possibility
bifurcations, is not
a
but
second
its starting is closed and
on the basis of Peixoto's
likely than dense ones;
this
(1978) scenario,
discussed
there appears
torus
the
by Ruelle
an invariant
a so-called
the m o t i o n is not quasi-periodic,
and
point
aZoxzno4e ~
Takens set
;
is that,
after
a few
in the phase space,
which
then,
as
explained
below,
but operriod/~.
THE RUELLE-TAKENS "STRANGE ATTRACTOR" SCENARIO
In the scenario of the early onset of turbulence proposed by Ruelle and Takens in
1971,
the
scenario,
Concerning
Ruelle
four
bifurcations
to be supercritical
each of which
scenario
first
is attracting
the existence
are
assumed,
as
in
the
Landau-Hopf
and to lead to invariant tori T k, k=1,2,3 and 4, between
its appearance
of these tori,
and be next
bifurcation.
see the discussion of the Feigenbaum
in the next subsection.
and Takens
prove
attractor
contained
Cartesian
product
in of
a
that,
on T 4, motion on a particular
T4
rather
is
likely.
two-dimensional
The
~unbm%
~
attractor and
a
kind is
of
strange
locally
the
two-dimensional
243 surface.
The vector field that yield the strange attractor
as unlikely, arbitrary; stated.
however one
can
imagine
Apparently,
manifold
that
cannot be dismissed
their particular choice of strange attractor
no
many
variations
one has
found
leads to a strange
of
it,
a specific
attractor
each
having
vector
precisely
is somewhat
field
the on
according
property
a specific
to the Ruelle
and Takens scenario{ The
important
some sense
idea their
likely,
circumstances.
strange
attractor
~
on
~
A ~
~
Lyapunov
and
strangeness
While
it
is
~r~
~
hence
measure
scenario
that
The
of
of
appear,
simultaneously
~
aat
equations.
to
external
deterministic to noise
a
even generic
in
existence
on T4: an
are
~
of
a
so~
£ n ~
~
in
continuous
field
power
small
with
to
describe
measurements of
strange
system
will
T4
the
Aa~
sense
spectrum.
perturbations
strange
attractor
this
how t h e
and
to
attractor,
set
is
of The
of
the
is
open
large
appearance
show let
by t h e
The
by Kifer
attracting
third
will
the
us
of
in the
measurable
reformulate
sight
two,
small
the
if
there
to
noise
This
intuitive
globally there is at most a small probability
much
altered
to
the
insensitive
noise
by small noise not
to
is a strange
seem
to
be
insensitivity
and
the chaos of the scenario
locally
about
o f t h e s y s t e m . The
external
points.
possibly
is
may be t o t a l l y
sensitive
counter
and
frequency
evolution
systems
most
established is
of
then
appear
turbulent
(bifurcation)
In effect,
point
When t h e
of chaotic
be accidentally fixed
one,
addition
systems
and at first
(1974).
exhibit
noise
as chaotic,
systems near transition
that order cannot very
In order
The n a t u r e noise.
is surprising
discovered
once
d o e s n o t mean t h a t
some b r o a d - b a n d
destroyed
evolution small
field
~Ae/4,
the
in
i s no e x c e p t i o n a l .
frequencies.
T h i s we i n t e r p r e t
RT s c e n a r i o
a
such
independent b a s i c
attractor.
vector
under
this
presence
that
are
Eckmann ( 1 9 8 1 ) ) :
power spectrum
three
by
~
of vector
in
say
motions
set
itself
the
the
it
attractors
a a T4.
fields,
sense.
manifest
(see,
is
words,
the
vector
of
~
characterized
in other
not
~
~
attractor
theoretic
consequences scenario
-
on strange
and are possibly
does
~
one on w h i c h ape
the
true
motions
property
~ ~
is
the constant
the
theorem
is a generic a~nge
dynamical system;
near
Their
a
of
is that
or at least not unlikely
certain
mm2/a~
paper
it has been is so strong
terms, by
much like noise,
to change stochastically
and
from on
244 bassin
(domain)
of
attraction
dissipative systems)
to
another.
contracts volume,
Although
it need not
the
contract
flow
(for
lengths.
the
If we
take snapshots of the flow at t=O, 1 and 2, say, we may have (see, the FiEure 22) the picture shown in (a) but could also get that of (b) or even that of
(c).
(o)
V
TIV
T2V
i 1-o
-°
(b! v
TIV
v
TIV
(¢)
I I
TEV
TZV
I
~:~. 2 2 : ¢ o ~ o ~ u o ~ u T ~ p A o ~
(a) "r~z,'n~oz,~",
In particular,
even if all points
in 7 (a finite volume in state space ~N)
converge to a single attractor M, one still may find that points which are arbitrarily close initially may get macroscopically separated on the attractor after
sufficiently
large time
interval.
This property
is called
"sensitive
dependence on initials conditions". It is na£ excluded for area-contracting flows, in dissipative dynamical systems. be called a strange attractor.
i.e.,
it can, and will,
occur
An attractor exhibiting this property will
245 The
solution
of
dissipative
dynamical
systems
separate
exponentially
with
time, having a positive Lyapunov characteristic exponent and thus the motion is characterized as chaotic with the appearance of a strange attractor. The ~ g u ~
characteristic ~ ~
(40,1)
of the ~Zo~a is defined as:
{~ I n ~----~f, ~(t)]
~ = Lim t ~
~(0)-->0
where the ~(t) are values of distance between initially neighboring solutions. This gives us a measure of the mean exponential initially neighboring solutions, important
is that ¢ ~
paa~,
rate of divergence of two
or of the ~
of the turbulence.
indicatin E that the movement
The
is chaotic.
It
would be expected that as Reynolds number increases the movement would become more chaotic, with a consequence increase in ~. The next property, after the Lyapunov exponent, characterize the turbulent motion is the d
which might be calculated to
~
of the strange attractor on
which the resides. That dimension should be a measure of the number of a ~ d ~
a~ ~ n e e ~
or actives modes of the turbulence.
The problem of chosing
which modes are active may prove to be formidable. The
early
discussion
of
atmosphere-dynamic
chaos
geometrical reconstruction of strange attractors, b~
dimension
atmospheric
(i.e.,
systems,
at other
the
o~
tools
differential dynamical systems.
of
are
was
chaos).
available
largely
based
on
which is possible only For
from
the
To an erEodic measure p,
moderately ea9~
a in
excited
theory
of
various parameters
are associated: (a)
characteristic
exponents
A IzA 2z. "" (also
from the multiplicative ergodlc theorem
called
(Oseledec).
The A
~ u ~
e~4~)
give the rate of
i exponential divergence of nearby orbits of the dynamical system; (h) en/na~,
h(p);
this is mean rate of creation of information by the
system, or X o 2 m ~ - Y i a a [ (c) ~
d / ~ ,
invariant; dimsp _ smallest Hausdorff dimension of a set
such that p(~)=l (see, Eckmann and Ruelle (1985)). We consider, now, dynamical systems such as map (discrete time, n):
(4o,2
i÷o
where X is a p-dimensional vector. To define the Lyapunov numbers, let
Jn = EJ(in )J(£n - 1 )...J(i 1 ) ] ,
246 A
where J(X) e (aF/aX) is the Jacobian matrix of the map, and let Jl (n)->J2(n)->" " "->jp(n) be the magnitudes of the eingenvalues of J . The Lyapunov numbers are: n
(40,3)
61 = Lira [ j l ( n ) ] l/n,
1=1,2 . . . . .
p,
n------)~
where t h e
positive
real
simply the logarithms
nth root
ts taken,
a n d t h e Lyapunov e x p o n e n t s
~t a r e l
of the 6. l
If the system is ~
t h e Kolmogorov e n t r o p y N
(40,4)
h = Z X I=1
1
N
where t h e ~ i s e x t e n d e d o v e r a l l
~
~'l"
rl.=l
Dimension
is perhaps the most
basic property of an attractor.
The relevant
definitions of dimensions are of two general types, thoses that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visists different region of the attractor. We define here the dimension of chaotic attractor, d, by: N
(40,5)
I
d = N - ~. I=1 N+I
where
,
~L N÷I N
~. ;ti< 0
and
I=1
~ ;it= h -> O. t=l
we have that: N
(40,6)
0~-Z---L-~ i=1
For
the
Lorenz
< i.
N÷I
chaotic,
strange
attractor
(see
the
section
42),
we
have:
d=2,06. The dimension of an attractor provides a way of quantifying the number of relevant degrees of freedom present in dynamical motion. The dimension d of an attractor,
if it is small
and non
integral,
confirms
that
the
dynamics
admits a low-dlmensional deterministic mathematical description characterized by a strange attractor. The
key
to
understanding chaotic
and ~ Exponential
behaviour
lies
in understanding
a
simple
have
finite
operation, which takes place in the state space.
divergence
is
a
local
feature:
because
attractors
size, two orbits on a chaotic attractor cannot diverge exponentially forever. Consequently the attractor must fold over onto itself. Although orbits diverge
247 and follow
increasingly
different
paths,
they eventually
pass c l o s e
must
to
one another again. The orbits on a chaotic attractor are ~
by this process,
much as a deck
of cards is shuffled by a dealer.
The anndamxwx~s of the chaotic orbits
result
The
of the shuffling
happens
repeatedly,
attractor
process.
creating
is, on other words,
process
~ab~a ~
of
the stretching
~
and
a ~z~cta~t: an object
~
that
is the
and folding
.
A
reveals
chaotic
more detail
as it is increasingly magnified. The
stretching
removes
the
stretch
makes
separated
and
folding
initial
operation
information
small-scale
trajectories
and
of
a chaotic
replaces
uncertainties
together
an
it
attractor
with
larger,
erases
new
the
systematically
information:
fold
large-scale
brings
the future,
uncertainty
by the initial
and all predictive po~a~ o~t ~tan~. attraction
and
Indeed
is
it
power
can exist.
is lost:
Finally,
we
divergence impossible
fluctuations
of to
covers
initially,
trajectories a
time
strange
interval
the entire
is simply ~tc ~
that
the get
measurement
there
note
After a brief
up
might
two
attractor
conditions
appear
attractor
area the divergence of trajectories
the
~
the
if
of
uncompatible!
the
phase
trajectories are attracted on an finite object at two dimensions: two-dimensional
Thus
In this light it is clear that no exact solution,
no short cut to tell specified
widely
information.
chaotic attractors act as a kind of pump bringing microscopic to a macroscopic expression.
the
space
in a bounded
is not possible.
On the contrary if we want to keep the possibility to form a strange attractor we have to consider an attractive object with 3 dimensions: case of the attractor described by Rossler on a two-dimensional
spiral,
(1976)
escape by emerging
for example in the
the trajectories can diverge into space and return toward
the centrum diverges again etc... For the same raisons, attracting
region
trajectories
we can expect that, being
in
3
which can be attracted
perpendicular
other one
from a torus T 3 (3 frequencies),
dimensions
an
instability
along a direction
(/WP~).
Then,
but
the
may
lead
to
diverge
along
the
on the contrary
to what happens
on a torus T 2 (fl,f2) whose instability lead to synchronisation
(limit cycle),
an instability on a torus T3(fl,f ,f 3) may lead to a strange attractor. It is through topological the
~
an~t ~
frequencies t
See,
of this kind that one can understand
a deterministic
system
with
3-independent
(3 degrees of freedom) may lead to turbulent behaviour.
for
example,
Schertzer
(198B).
consider
consideration
idea that
briefly
Sreenivasan In
the
the
and
Heneveau
Hlscellanea
fractals
in
(1986)
( section
atmospheric
47
and in
turbulence.
also the
Lovejoy chapter
X)
and we
248 We do
not
chaos
in
want
according Another
to
the
kind
Manneville though
leave
the
reader
motions
chaos
has
subharmonics
(false)
later
case,
is deterministic
_ at
that
deterministic
strange
least
attractors
through
is
(phase space
at
(1978)
not
a
an
biperiodism).
been proposed
there
by Felgenbaum
idea
through
intermittencies-has
this
been proposed
the
occurs
mechanism (or
of route-through In
on
always
Ruelle-Takens
(1980).
the
of route
to
atmospheric
by Pomeau and
strange
3 dimensions).
attractor,
A third
and corresponds
kind
to cascade
of
bifurcations.
THE FEIGENBAUM "CASCADE OF PERIOD DOUBLINGS" SCENARIO* While the Lorenz (strange) attractor appears in connection with a subcritical Hopf bifurcation, the LBundau-Hopf scenario and the Ruelle and Takens scenario both require
a sequence of supercrltlcal
bifurcations
leading to
invaris_nt
tori of successively higher dimension, arbitrarily high in the former scenario a~nd of
dimension
at
least
4
in the
latter.
However,
such
a
sequence
is
on
a
sequence
of
unlikely according to Peixoto's theorem. Felgenbaum
(1978,1980)
has
developped
a
scensrlo
subharmonlc bifurcations with period doubling. occur
in many ex~unples of
Furthermore,
as
It turns out that such doubling
iterated mappings
the number
n
of
doublings
based
and
simple
increases,
dynamical
the
systems.
behavior
of
the
system is governed by certain asymptotic laws that involve universal constants and
functions,
independent
of
the
system
under
study.
In
addition,
the
asymptotic laws appear to hold quite accurately fop rather small values of n. In particular', the values ~n of the dimensionless parsJneter ~, in (39,1),
at
which
~m
the
bifurcations
(doublings)
take
place
converge
to
a
value
geometrically, with ~n+l-
(40,7)
~n
0,21416938...
~n- ~ n - 1 for
large
motion ~,
n.
An n--d=, a t
approaches the
evidence
motion for
an
least
a continuous
spectrum
studied,
with
See also the work cascade of period
of of
this the
behavior
in
(1984)
universal
strange the
dimensionless
of Coullet and Tresser doublin 0 bifurcations.
the power spectrum
certain
is presumably aperiodic on a example
considirably higher values t
in the cases
parameter for
an
features.
attractor.
Lorenz
system r
analysis
of the At
There
is
(42,9)
at
than of
values the
249 studied
by
Lorenz.
Namely,
the
strange
attractor
that
appears
at
r=24,74
persist up to a value r=r'(~250). For r considlrably greater than r', there is a periodic orbit,
and as r is decaeox~ed toward r',
doubling at values r
-
r
,+I n
-
n r
0,214 . . . .
n-1
After the cascade of period doublings, point ~® an ia~ea~e c a ~ In an experiment,
is a sequence of
of r that converge to r" from above, with:
n
r
r
there
one expects beyond the accumulation
of noisy periods.
if one observes subharmonic bifurcations at ~i and ~2' then,
according to the scenario,
it is very probable for a further bifurcation to
occur near )I ~3= ~2- (#I- ~2 ~ ' where Bs4,66920 . . . .
In addition, if one has seen three bifurcations, a fourth
bifurcation become more probable than a third after only two, etc.
We note
that B is a universal number such that
j--~ ~ogl.j-
Ltm
(40,8)
~ 1 = -Log8
and one even has (40,9) At
the
]~j- pm[ ~ Constant B -j, as j--~. accumulation
broad-band numerical
point,
spectrum. and
This
physical
forced
will
Feigenbaum
grounds.
observed in most current ~ equations,
one
The
observe scenario periods
aperiodic is
behavior,
extremely
doublings
well
have
but
not
tested on
by
now
been
dimensional dynamical systems (Henon map, Lorenz
oscillator
with
friction,
Rayleigh-B6nard
convection,
etc..). Now we recall
the main steps of the Renormalization Group
analysis of
the
cascade of period-doubling bifurcations according to Argoul and Arneodo (1984, p. 274). RENORMALIZATION GROUP ANALYSIS Dynamical systems that exhibit such a cascade of period-doubling bifurcations are
in practice well
modelled by one-dimensional
maps with a single smooth
250 maximum such as:
(40,10)
f (x) = Rx(1-x). R
As we incFease fR(x),
the parameter
we observe successive
to chaos
which presents
R which detePmines
the height of the maximum of
steps of the cascade and a continuous
a strong
analogy
with second-order
phase
(see, Ma (1976)).
4
i
{ 0
~.
23. ~
G
p"
~
<~ [-1,+1] ~ ~
p
l
~
~
~
~ ~
a nmm~uua e q u a l t~ o a e a ~ ~
U~e t e ~ .
t
~
~
~
~ t~
fR(×)=l-~2, ~
~a
transition transition
251 In
the
neighborhood
which diverges (40,11) where
of
the
trasnition
according
T(R) ~ ( R - R ) is the
period
of
the
which displays the universal (40,12) where
one
can
define
and
an
a
law:
-u,
C
T
(R=Rc)
to the scaling
bifurcation
cycles,
"order"
parameter
behavior:
L(R) ~ (R - R )u C
L is the envelope
of
the
Lyapunov
characteristic
exponent.
In (45,16)
and (4S,17) we have (40, 13)
in2 in~ '
u -
and u can be assimilated depend on the explicit maximum.
Like
technique
in
in order
~s4,66920 ....
to a critical
phenomena,
to understand
these
Figure 23 above the renormalization
from t h e s i m i l a r i t y
and i t s ~
f2(x)
~
asymptotic
scale
that
one
can
universal
use
it dee~ no/
nature
of
Renormalization
properties.
As
its
Group
sketched
in
operation
w i t h a=f-~-)-, r e s u l t s The
in the sense
R
critical
1
exponent
form of f (x) but only on the quadratic
i n t h e shape t h a t
characterizes
f(x)
i n n e i g h b o r h o o d o f i t s extremum.
invariance
displayed
that one look for a fixed point g(x)
by
the
dynamics
of renormalization
at
R=R c, suggests
operation,
which must
satisfy the equation: (40,15)
~-Ig(~x) = g(g(x));
The coefficient the quadratic
in the Taylor
case,
series
==I/g(1). of g(x)
have been found
numerically.
In
one gets
g(x) = 1 - 1,5278 x 2 + 0,10481 x 4 + . . . . with ~=-2,5029... To
handle
spectrum
the of
approach
the
to such
renormalization
a critical operator
eigenvalue
lies outside of the unit circle.
associated
with an eigenvectop
situation, linearized
we
need
around
to
study
the
g(x).
Only
one
This single relevant
eigenvalue
CA(X) which is an even function of x.
is
252 As illustrated in Figure 24 the unstable manifold W
U
of
the fixed point g(x) is of dimension stable
1
while
manifold
the
W is S
of
codimension I. Therefore any one parameter path obtained by varying R in
f (x)
will
R
intersect
transversally R=R
C
W
for
.
On the way to criticality, near
W,
one
essentially direction
thus
the
unstable
eA,which explains
why the critical in
(40,11)
depend
feels
exponents
and
only
(40,12)
one
constant,
universal
namely
unstable
the
eigenvalue
~=4,66920... An experimental estimate in
of
~
measuring
parameter
R
n
24: ~ A , ~ e - ~
aAele.A
the where
2n-periodic
a
S
cycle
bifurcates
into
2n+I-periodic cycle; values
• ~.
consists
are
a these
~
predicted to
~
~
~
R
-
-
~
C
scale according to (R - R ) ~ ~-n. C
At the accumulation point of the perlod-doubllng cascade, W . Wherever this intersection point lies on W S
the
renormalization
operation
will
converge
S
to
asymptotic orbit displays a Cantor set structure results
have
been
obtained
by
g(x)
and we are on
C
which
illuminate
the
where the adherence of the (see the figure 25 below).
several
Lanford III(1982), Epstein and Lascoux (1981)).
C
,the successive iterations of
universal scale invariance of the dynamics at R=R
Rigourous
R=R
authors
(Eckmann(1983),
253
O
A: c==A~ L_Jc;
al V-lci
0
flo0o and c
~
~ar~or set.
C
THE POMEAU-MANNEVILLE "INTERMITTENTLY TURBULENT" SCENARIO
While
the
two other
(Ruelle-Takens)
and
associated with a and
unstable
points).
The
scenarii
have
pitchfork t
been
associated
bifurcations
point
general
idea
which
then
is best
both
the
for
which
t
stable
point
as
parameter
In
this
periodic
case orbit
a
the
for
~[0,2]
loses
its is
maps
stability
changed.
the
[-I,+I] and
this
(into simple
one-parameter family of iterated maps on the unit interval, we take f (x) = 1-Hx 2,
bifurcations
collision
disappear
explained
Hopf
(Feigenbaum),
"aadd~e node ~ " , i . e . ,
fixed
with
gives
x
into rise
one
is
of a stable
complex
fixed
example n+l
of
= f (x) ~
n
itself. to
a
stable
a
and
254 The f u n c t i o n f3 = f 0 f o f can be s h o w n to have a saddle node
for
~
= 7/4.For
~>I,7S,
f3 has a stable periodic orbit of
period
unstable collid have
three,
and
an
nearby.
The
two
one at
~=I,7S,
then
and
eigenvalue
both
1
(see
Figure 26) opposite. For ~ s l i g h t l y below 1,7S, the local
picture
near
x=O
is
that
if
shown in Figure 27. It
can
be
shown
~-I,7S=0(c) orbit
then
will
iterations small As
x
long
this
to
as
observer
the
typical
will of orbit
O(e -I/2) a
fixed
around orbit
interval
impression periodic
cross
interval
small
a
need
x~O. is
,
in
have
the
seeing of
~Fi 4. 26: ~,u~A ~ f3
an
a
period
three.
/ Once
one
has
left
the
small
interval,
the
iterations
of
the
map
will
look
rather like those of a chaotic map. Thus this map can be called intermittently turbulent
(see Figure 28 below).
255 't.O
- ~
:•;•/."-'~ ...--'~" ~ . - - ~ . .--",e"r. : ~
".', , • •
• •
• .
0,5
.
•:"
:
":.:•
,
• :.
.
:~':
.
•
•
*
~
. ~°o°
•
*o
."
~
,.. "o'..
D
-~..-",."~..""
-
.
°°
".°*""
° •
•
•
•
•
°°
~
°°°°o ..
.
°°
oe • • o,
0,0
•
:~..
°~2
•
"°
•
• % .o
-0,5 oO
-t'.0
I
I
200
0
.%
I
400
I
600
I
8OO
I000
i
ae/.q,A&e,-,,~h.ee~ ~. / ~ 1 , 7 5
, and..Lad/e..a.t/aq,
The problem with this arguments comes in the splitting into two regions. true
that
the
x ~ small be
interated
intervals
lost whenever
map may have
around contact
one
passes
near
sensitivity
points•
to
~=1,75.
period, the
these
but not chaos.
proof
of Jakobson
sensitivity value
For
the contact
of
to
initial
positive
parameter
values,
On the hand, (1980)
point•
on
In fact, values
will
have
effect
for may
we conjecture
near
to,
(very
and just
long)
stable
we also conjecture that a modification of
would
conditions
Lebesgue
conditions
But this destabilizing
that this will happen for an infinity of parameter below
initial
It is
show
that
(S•I•C•)
measure
near
truly
occurs 1,75
aperiodic
for a set
(according
to
behavior
with
of a parameter Eckmann
(]981;
p.651)). We
can
now
formulate
a
reasonable
version
of
this
scenario
for
general
dynamical systems:
Assume
a
one-parameter
family
close to a one-parameter
of
dynamical
systems
has
family of maps of the interval,
Poincar@
maps
and that these
maps have a stable and unstable fixed point which collid as the parameter is
varied.
Then,
as
the
parameter
is
varied
further
to
~
from
the
256
critical
parameter
behavior
of
value
~c'
one
random duration, ~
{V -
~c
will
see
intermittently
turbulent
with laminar phases of mean duration
-1/2
in between.
The
difficulty
precursors, stable
with
this
scenario
is
that
it
does
not
have
fixed point
(respectively
periodic
orbit)
may not be visible.
The first
would be that
long transients can be observed before the two fixed points The
second
bifurcations, turbulence based
kind
and,
at
(Collet
their
to
of
the
precursor "end"
and Eckmann
work
transitions
on
of
observation
turbulence
degrees
of
freedom,
can
geometries.
Nevertheless,
turbulence
is
in
such
as
cascade the
the
seen
in
(periodic orbits)
of
inverse
intermittent
We note
for
be
a
that
many
and
system.
physical
pitchfork
transition
Pomeau
Lorenz
On can
increasingly
to
Manneville
Intermittent
experiments.
The
is now well understood for systems with a
that
describing
fluid
motions
in
bounded
the best known example of intermittent transition to
parallel
flows
(the
structure
of these transition flows
characte~
of the instabilities
Sommerfeld equations.
is
this,
(1980)).
intermittent trasition to turbulence few
clear-cut
because the unstable fixed point which is goin E to collid with the
think of two ways out of this problem.
collid.
any
However,
so-called
"transition
is usually attributed
described
in their
flows").
The
to the subcritical
linear version by the Orr-
the precise connection between intermittency at
the onset and subcriticality of the instability needs to be made more precise. Intermlttency
in
transition
stability
analysis.
localized,
steady
relative
velocity
Reynolds
number
In
and
pipe
where
fluctuation
along
the
reaches
a
trajectory x~.
the variable is
probably
flows
for
finite-amplitude flow well
velocity of the fluctuation. system
flows
little
to
do
instance,
it
seems
fluctuations
can
exist
lines
(say along
defined
value,
with likely with
the x-variable)
dependent
linear
on
a
that given
when
the
the
relative
If one considers the fluid equations as dynamical
x plays
represented
has
the role of
(according
starting from and returning
to
time.
Pomeau
This
sort
(1983))
by
to the undistributed
of
an
localized
exceptional
parallel
flow at
The fact that such trajectories exist in a manifold of codimension one
(which fixes the values of the Re for a given relative all flows
it still
true for
a dynamical
system'in
velocity)
function
consider function spaces as the local state of the system,
in space of
space
i.e.,
(one must
the value of
257 a11
functions for a given x
is an element
of an
infinite dimension space
spanned by all admissible functions of the space variables independent of x). This helps one to understand how a well defined nonlinear stability criterion may exist and why,
in transition flows, the ~
the threshold. This picture, provide,
for
instance,
concrete
examples
and
a
tend to widely separated at
if true, needs to be made more precise.
detailed
structure
the behaviour of
the
for
critical
It could
fluctuations
intermittency parameter at
in the
threshold. Finally,
for a renormalization group and universality point
of view of the
Pomeau and Manneville intermittency, see the book of Guckenheimer and Holmes (1986; section 6.8). Concerning these three main scenarii, the reader can consult the recent paper of Monin (1986).
41
. BENARD PROBLEM FOR THE INTERNAL FREE CONVECTION
Thermal
instability often arises when a fluid is heated from below and the
classical example of this, described in this section, is a horizontal layer of fluid
with
its
lower
side
hotter
experiments were made by B~nard
than
(1900).
its
upper.
The
first
quantitative
Stimulated by B~nard's experiments,
Rayleigh (1916) formulated the theory of convective instability of a layer of fluid between horizontal planes. Hence, we consider internal free convection between two horizontal planes with the separation d' and having temperatures T'=constant,for the upper plane, and o 0 T'+ AT' (AT'=constant>O) for the lower plane, The main body of the present o o 0 section is concerned, specifically with the convection in the fairly deep layer of expansible liquid with an equation of state of the following simple form
(41,1)
p'=p'(T')
where p' is the density and T' is the temperature. For our expansible liquid we have possibility to construct the Froude and Mach numbers:
258
Fr
(41,2)
c%/%) - - ,
-
g'd'
M -
c%/%) C'AT'
o
o
o
where
p' is the kinematic viscosity at the temperature T~, g' is the o acceleration due to gravity and C' is the specific heat at the temperature T'. o o In the e ~ , Navier-Stokes, equations for the expansible liquid we have a fundamental small parameter,
(41,3~
~AT~
%:
namely
<< 1
where
'
60 = -
(41,4)
[~ dWl
,dT,JT,=T, 0
The parameter o
(41,5)
Gr
-
Fr is
the
Grashof
number
and
if
~-
PoUoC o
is the Prandtl n u m b e r (k'^u k' o thermal conductivity at the temperature T'=T ° and po-Po'= '(T''),o j then:
is
the
O"
Ra = ~ Gr - ~-ero,
(41,6) is the R~yleigh
number.
Finally
(41,7)
Bo -
M
Fr
is the Boussinesq number.
THE EXACT NAVIER-STOKES EQUATIONS We present problems.
here the equations that govern a wide class of thermal convection A
primed
quantity
will
denote
a
dimensional
quantity,
while
a
non-primed quantity will denote a dimensionless quantity. Let
there
be
which gravity velocity u:,
established acts
an
{x~},
in the negative
the pressure p',
i=1,2,3,
Cartesian
co-ordinate
system
in
x' direction. Then, for x' component of 3 i the density p' and the temperature T', we can
write the following exact equations:
259
(41,8)
ou.
p,
i
Dt'
P ~la-
,
i
+
i
ax TM k au' k
D D-~,(Log p') +
(41,9)
_&[oual
,.
+ g
+ ~
have
heat
postulated
conducting,
(41,1)
,FuUx~Jl
the
viscous
expansible
fluid
and
liquid
as
a
is a newtonian
consequence
ax; ax;
2
of
):
compressible,
equation
of
state
we have De' - C'(T'I DT' Dt' "Dr'
(41,11) where
that
J ax' I
= O,
ax' k
ax~ We
+
ax' J
e'
is
viscosity
the
specific
coefficients
X '= X'(T'),
'
internal
energy,
H'
and k' is the thermal
~'--- /s' (T')
k '= k'(T'),
and
the
A'
are
conductivity.
two
Finally,
dynamic
C'(T')
is
the specific heat for the expansible postulated
the equation
liquid, only function of T' since we have D _ a u' a (41 I). ~ , = ~-6,+ where t' is the ' j BX' '
of state
J
time. F or t h e
temperature
we have,
fop the
B~nard problem,
the
following
boundary
conditions: (41,12) Let
T'= T'+ AT' 0
C'o, Ao"
~o"
k'o and Define
(41,13)
C - C' '
0
3
Po' be
p' (T') at T'=T'. o
C'
at x'= 0
0
k'
A - A' '
0
o ,
Let do,
*
,
Up/do,
temperature,
values
go/Po ,
and T'= T' at x'= d'. 0
of
C'(T'),
~'
~ - ~
,
0
A'(T'),
k'
k - k' '
0
~'(T'),
(41,14) Equations
x
u' __ _ _ i
,
respectively.
d'o u %/%
(41,8)-(41,10)
,
and
P - P" PO
units
for
length,
New dimensionless
t' t - - -
transform to:
d°~/~' o, T
T' - AT,, o p -
nr'
velocity,
variables
defined as follows: x' i = --,
k'(T')
"
AT'o, d'o2/U0 and pog d'o be scaling
time and pressure,
3
, ,do , Pog
are
260
D(Logp)
¢9u
+ ax
P --
+ F-{
k
O;
k
i" P~i3
pC(T)-~- + B o p
(41,15)
=
=
--Sxj + Oxl
C~Uk _ 1 O ~ Ox
-Ox k
aT g'ffi
Ox
p
=
o + --' MA - go [OxkJ
I au
2
Bu J
Ox
+ _~O~o,°xl [ OXk j;
l
p(T),
where A=A(T), ~=~(T) and k=k(T). In the dimensionless system (41,15) we have four non-dimensional parameters: Fr - - - , g'd'
M -
o -
-
,
Bo
=
C'AT'
0
0
0
g'd' o - - ,
~
C'AT' 0
0
-
g o'C o '
k' 0
The boundary conditions (41,12) lead [~ = To+ 1
(41,16)
TO
with T o-
at x = 0, 3
at
x3
=
I,
%/AT;.
THE DIMENSIONLESS DOMINANT EQUATIONS
When no motions are presents (ui~O) , the equations
(41,15) require only that
the pressure distribution is governed by the equation (41,17)
ap 8--~-_ -p(T) 3
and the temperature distribution is governed by the equation (41, 18)
ax i
But if we suppose that the solution of equation (41,18) is simply TmTo, then
261
the corresponding
distribution of the density
is given by pml and with the
expression for p, equation (41,17) can be integrated to give : p=l-x 3. Let the initial state: u im O, TmTo, pml a n d p m l - x 3
(41,19)
be slightly perturbed.
Let u i denote the velocity in the perturbed state and
let the altered temperature distribution bet: (41,20)
T = T + e. o
Finally,
let
(41,21)
p
=
1
-
x + AFr 3
o
denote the altered pressure distribution, (41,22)
A
o
1 -
where
Ap;
Fr
g'p;d;
with Ap~ the pressure fluctuation in slightly perturbed convective motion. We shall now consider the problem of approximate expressions for the density p and the coefficients A, ~, k and C. For density p we obtain the following relation: (41,23)
p=l-co(e+Toe2+...)
where (41,24)
To
2
O'
dt'
T'=T' o
If the expression dE, T,
I
xp-
(41,25)
- i]
is &aanded when u --)0, then: o
(41,26)
p : 1 - eo~,
~ : e + Xpeoe2+...
In the same way we can obtain,
for A, /~, k and C,
approximate relations:
t
With
and
the
we
dimensions
we
have
e=(T'-T;VA%
suppose
that
e
is
of
the
order
of
unity.
the following analogous
262 A'
A = 1 - Co~0,
i0 = ~
(8 + XlCoe2+...),
%J
= 1 - Co8,
M' g = 6o--~(8 + Xgeo82+...),
k = 1 - Col,
K' f = --~ (8 + xkCo82+...),
(41,27)
%)
N' C = 1 - Co~,
where
A~,
MR ' K~,
(41,4)),
while
formula
(41,2S)),
quantities, Now,
~ =
and N~ are
xA,
if we take
x c are
pespectively
x A, x , x k and Xc, into account
and 8 the following
(e + XcCo82+...),
the quantities
x , x k and but
--~ % the
for
remains
to 60
quantities A,
~,
bounded
the preceding
set of dominant
p
similar
k
(see
similar and
C;
, the
to Xp these
formula
(see,
the
last
foup
w h e n eo---)O.
results,
d i m e n s i o nless
then,
we obtain
pepturbations
for
u i,
equations:
au
(41,28)
k _ CO ~ ax
+ O(C );
k
(41 , 29)
Du
- ' ~ + Ao ( l - c ° e . -Dt
O~
~l
- Au + c ° - Cr e a i 3 i
.,fr U~jax ±r°lLo<j I+
o ° [o[-au-
M'
r
6 ° Oxj N'
(41,30)
1- 1+
De + Co Co8 D-[
Ox
i [au au] ]2 i + ax
, o{
solution
(41,31)
of these
equations
ax
]
M' + g M ~w 0 6o The
au -Ox
(41,28)-(41,30)
conditions: e = l a t x =0 3
]
Ox
}]} +OCc :). i
o ( l _ x 3 ) + AoM ~ ~
and 8=0 a t x = I . 3
eo
i
au + ]
Ox must
[ ol aFsae ] K'
+gM
the b o u n d a r y
au +
'I]2
, ~ 6o
ax i L
+ O(c
:).
oxJ
i be sought
which
satisfy
263 THE "DEEP" CONVECTION LIMITING EQUATIONS
If we introduce, in place of 8, the new perturbation for the temperature (41,32)
E = ~ Gr(e
in such
a case
we o b t a i n
for
E=O a t
x =0 3
(41,33) in For
place
of
+ x3-1), E the
classical
homogeneous
boundary
and x =1, 3
(41,31).
simplicity
we c a n
assume
Aoml
(41,34)
in the
equation
and i n this c a s e
(41,29)
Ap~mp~
that: .
ld~J
Then, the corresponding expression f o r the pressure p e r t u r b a t i o n (41,32))
conditions:
is
given
(according to
by: x3 v
i and t into ~ • and we suppose that c o--~ (with Finally, we change u i into ~-x i and T fixed) in new equations, for v i, R and E, with$: (41,36)
e 0 Bo = ~ o = 0 ( 1 ) .
M Gr ~
Entering
the
formal
8v aX 1
(41,37)
O"
limiting
k
=
process
tt
,
we o b t a i n
the
following
set
of
dee~
O;
k
Dvi
+
DT
OR ~-~
-
i
E6
-
i3--
Av
I'
(i=1,2,3); av
av
2,
[l+&° (l-x3)] D[D-~- RAY31 = AE + ~ [ axt + 0x] ] J i with t h e b o u n d a r y (41,38)
conditions,
for
E:
E=O o n x =0 a n d x = 1 . 3 3
D 8 We note that: D-T = ~
+ vj ~
0
02
In
this
t$
Zeytounian
case
we
suppose
(1983,1989).
that
02
end A=--Ox 2 + Ox 2 - + Ox 2 - and 6 i 3 mO f o r i~3, 633m l . I
t
02
Bo>>I.
2
3
264 In
the
deep
convection
equations
(41,37)
we
k' u'
.
have
the
following
three
parameters:
[
(41,39)
-
'
k'
g,O
o
ira =
o o
C'
o
Hence if
then it
C'
d' =
(41,40)
o
o
g, ,"
~o
is necessary to consider a $4ana~ ~
~y~ ~ e
dge~ ~
and
in this problem we have a new parameter 8 o.
THE "SHALLOW"
CONVECTION
LIMITING BOUSSINESQ EQUATIONS
If 8---~ in the equations (41,37) we find again, instead of (41,37), o classical Boussinesq equations for B@nard shallow convection problem. In this
case,
when
c --~ (with x and o f remain bounded (of the order of unity) t. Entering
the formal
limiting process,
T fixed),
the
Boussinesq
number
0o--~0, we obtain the following
the
Bo
set of
classical equations for the Rayleigh-B~nard problem: Ov
Ox
(41,41)
k
=
O;
k
1 Dvl - -
o"
ON +
Dx
-
-
9×
--
~13 = AV 1 ;
DE ~-~- Rav3=AE.
The
Rayleigh-B~nard
problem
for
the
convective
instability
consists
investigation of the stability of the following basic convective flow:
in the
265
u~- O;
T' = To+ AT'o~(1-x'Jd'o);
(41,42)
P' = g' Po'd'o ( 1-%/d' o ) + APoPr ( 1 -x'./d'o ) ( x'3/2d'o )' starting with the equations (41,41) for the dimensionless perturbations v i, R and E. 42
.
THE LORENZ DYNAMICAL SYSTEM
A simplified two-dlmensional model: v2mO and the variables are a~/ function of x 2, permits the introduction of a stream function @ such as
8~
(42,1) the
8~
v l = ~-X--' v3= -~x- " 3 1
vorticity
~ is defined
(42,2)
as 82 82 with ~2= _ _ + _ _ 8x 2 8x 2
~ = ~2~
I
In the starting equations
(41,41),
the
3 pressure
R
may be eliminated and we
get:
1 8
2
~
I 8(@,~2@)
@ = ~8(xt,x 3)
8s
~2(~2@);
+
8x I
(42,3)
BE.
Let
us
OSXl~,
describe where
8(@,s) 8¢ - 8(xl,x3) + IRa
convective
~
may
be
movement
the
= ~2E.
in
a
rectangular
dimensionless
length
of
horizontal size of one "convective cell". In this case the adopted boundary conditions are: a2v (42,4a)
v = O,
3
8X 2 3
3 _ O,
E = 0 at
x =0 and x = I ;
3
3
domain the
O~-x -<1 3 "box", or
and the
266
~2 v
BE I _ O, ~--~= 0 at x I=0 and x 1=~o" ax 2 I I equations (42,3) satisfying all the boundary
(42,4b) The
Vl= O,
solution
of
(42,4a, b) may be expanded in a double Fourier series,
I
@(T,x,,x a) =
(42,5)
®
the
solution
of
= the
®
~
~
I=0
J=l
0 Fi~rxl]
B,,C )cosl--jcosCj-x3),
nonlinear
problem
(42,3)-(42,4a, b)
considered as the superposition of the eigenfunctions Concerning
this
linear
as:
~All(T)sin[~-!]sin(J~x3);
1=1 J=l
~(T,X1,X3)
since
E
conditions
convection
classical
problem
will
be
of the linear problem. see,
for
instance,
the
book of Drazin and Reid (1981; pp.37-62). The next step is to substitute expansions
(42,5)
use the Galerkin technique
(see, for instance,
(1984;
requiring
chapter
function (42,3)
of
VI,
§3)),
the set
(42,5).
is multiplied by
After
the
into equations
(42,3) and to
the book of Platten and Legros
residue
to
this substitution,
fP~X1~
sinl-7-:Isin(q=x 3) and
be
orthogonal
the first
to
each
equation
the second equation of
of
(42,3)
0 rPRXl~
by cos~--~Jsin(q~x3). 0 and %, evolution
Then these new equations are integrated over x I between
o and over x s between 0 and I. The orthogonality condition lead to the equations
for each
Fourier
coefficient
A
pq
and B
Pq
, with
p=O ....
and q=l .... ~. Thus we have replaced a system of nonlinear partial differential equations by an infinite set of nonlinear ordinary differential
equations
(n.o.d. eqs.) for
the coefficients A (T) and B (T). pq pq For numerical use, the expansion (42,5) has to be truncated and infinite set of n.o.d, eqs. reduces to a finite set of n.o.d, eqs. to be numerically integrated. Many types of truncations may be adobted such as isMand
jaN
or i+j~K, which lead to different truncation errors. Here
we
adopt
representation).
the
truncation
Moreover,
scheme:
i+jsK
with
k=2
("minimum"
it may be shown that the Fourier coefficients with
(p+q) odd do not contribute
(they tend to zero
when T---)~ even
they were different from zero). Therefore, for K=2 we have:
it
initially
267
¢ = A(t)sin(~roXl)sin(~x3)
(42,6)
E = B(t)cos(~roX,)sin(~x 3) + C(t)sin(2~x3) , u s i n g only three F o u r i e r
coefficients,
I ro= ~-,
where:
1 A = -Cr A11'
t = cT,
o
The e v o l u t i o n e q u a t i o n s f o r ( A ( t ) ,
1 B = ~
1 C = ~
Bl1'
Bo2.
B ( t ) and C ( t ) are:
dA ~ 2 ( r °2 + I ) = IRaB ~ r ° + ~ A ~4(r~+1)2; -~ ~ ~ dd BK = -~ A ~ r o- ~2(r °2 +I)B - ~ A C 2 r o ;
(42,7)
c~ Finally,
dC
AB
=
2
c~--~
if we introduce
T = ~---(l+r~)t" Cr
IRa Y = 2 the f o l l o w i n g
dY
(42,9)
,
0
(42,8)
dX ~-~
~
2
.
new variables:
2
we ob t ai n
r o- 4 C
r
X - 1
o
A,
V~( l + r o2) 2
~r 0 - B, ~ ( l + r o2)
Lorenz
~r Z = -Ra
2 0
C,
l+ro2
systemt:
= -cX + cY; = rX -Y - XZ;
dZ ~-~ = -bZ + XY, where r (42,10) It
r -
4(
2
4
0
1+r~) 3
IRa,
b -
l+r~
.
has been proved by Lorenz t h a t a n x ~ W ~
system (42,9) i s ~
t
t
and i t s
futur
( a p e r i o d i c ) t r a j e c t o r y o f the
i s e s s e n t i a l l y u n p r e d i c t i b l e evan i f
t h e system (42,9) i s d e t e r m i n i s t i c . This
simply
means t h a t
amount, t h e y w i l l for
if
~
initial
states
differ
by
an
~nV~lu:24~
i n e v i t a b l y e l v o v e towards two c o n s i d e r a b l y d i f f e r e n t s t a t e s
large t.
t
See
tt
A t r a j e c t o r y ~o(t) is stable at a point ~~~P(to'~ ~) I f any other t r a j e c t o r y
Lorenz
(1963)
passing sufficiently
t--~.
-) close to ~P(to, XO) a t t = t
o
remains close to ~o(t) as
268 In atmosphere
there are always errors
in observing an instantaneous state,
i.e., initial conditions are not known sufficiently accurately, and therefore an acceptable prediction of the distant future is impossible if the behaviour is nonperiodic,
even
in
deterministic
importance in the study of ~
~
systems.
This
statement
is
of
some
.
LANDAU HODEL EQUATION If we suppose that in the Lorenz system (42,9) we have for Y and Z the steady solutions: XY Z = ~-- and
Y = rX - XZ,
then we obtain for X the following evolution equation (42, ii)
X dX - -~ X + ~ r - dT 1 + X2/b"
Near
(near
threshold
the
critical
point
when r ~ l )
we o b t a i n ,
from
the
e v o l u t l o n equation (42,11), the Landau model e q u a t i o n (Landau (1944)): ( 4 2 , 12)
~-~
where c=r-1, This
if
explalns
using
a
we n e g l e c t the
double
perfect
Fourier
the
terms
Xs . . . . .
agreement
near
expansion,
the
truncated
critical to
the
point
between
lowest
order,
Landau model. Rewriting
the
Landau equation
( 4 2 , 12) a s a l i n e a r
+ --
b'
Ixl 2
in
1 / I X I 2, n a m e l y
t =o-~,
we find the explicit general solution (42,13)
IXI2
bc +
I ~ -I ~ if
c~O,
[ ~X~ o
~]
exp(-2ct)
w h e r e X is the initial value of 0
Ixl.
Therefore X2 (42,14)
Ix12= 1
o
1°
D-E Xo+ ( 1 - E~ Xo)eXp(-2
t)
We have that b>O; if e>O then the solution (42, 15) give
I Xl-Xoeet
as t-->-®
analysis and
the
269 and X--)0, just as in the linear theory, but
o
(4216)
IXi IXel
whatever
the
value
of
ast*®
X O.
This
is
called
~
a/x~/t~,
flow being linearly unstable for e>O hut settling down as a new eventually.
The new flow is, moreover,
independent of the
the
basic
linear flow
initial conditions
except through the phase of the complex amplitude X of the dominant mode;
it
has
to
period
equilibrate,
2~[/~ if ~ ~0 or is steady if ~ =0. The disturbance I I I because its amplitude tends to X after a long time: e
(42, 17)
X~(bc) 1/2
Thus, when 0
Ixl6,
e
as r ~ l
Is small and even if hlgher-order terms, for example one
were included on the right-hand side of the Landau equation,
(4218~
dlXl dt 2 =
eJX
they would remain small be unchanged.
12 ~ iX I4
t = 2o" "w,
and the qualitative
The typical developement of
29 below and the dependence ]X]:O,
is said
character of the solution would
IX] with time is sketched in Figure
of the amplitudes
of the equilibrium
solutions
]X]:X e upon r:l+e in Figure 30 b e l o w '
IXI
. . . . . . . . . .
.........
| . . . . .
.
.
.
.
.
.
.
.
.
.
.
Xe
Wo 0
t
According
to
D r a z i n and Reid
(1981;
p.373).
.
.
°
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
°.
270
IX{
Stable V"='I.
0 ~.
The
branching
called
a ~
3o: ~
of the curve .
The
~
~
m
!
equation
,,,D,.. r
Ixl=lx(r) I.
~
of the equilibrium
Landau
!
~
solutions
implies
that
at
the
r=l,
{X{=O
solution
is
{X{=O,
which represents the steady basic flow, is stable for rl (e>O) and that it exists, In more
{X{=Xe, which represents the new laminar flow,
i.e. for e>O.
complete
models
of
atmosphere-dynamic
stability
there may be further bifurcations from the solution least
stable
solution If e
normal
mode of
we shall
see
that
{X[=O, e.g. where the next
the basic flow becomes
unstable,
and
from the
{Xl=Xe.
(b>O) equation (42,12) confirms that the disturbance decays
with the linear theory,
case
the
term
in accord
i.e.
{X{~Xoeet In this
is stable where
as t--4+m and Xo--)O.
_b{X{4m of equation
(42,18)
remains small for all time if it is initially small.
due
to
the
nonlinearity
271 43
.
THE LORENZ (STRANGE) ATTRACTOR
Now we consider the Lorenz system (42,9).
system that is neither equations
(42,9)
It is hard to imagine a much simpler
linear nor two-dimensional,
nevertheless
do
very
discovered numerically a striking mathematical known as the Lorenz
(strange)
attractor
but the solutions to these
complicated
things.
Lorenz
(1963)
structure which has come to be
and which occurs for these equations
with 8 b=~, ~=I0,
The exact
r=28.
parameter values are
solutions definitely
not
crucial,
o t h e r r e g i o n s o f parameter space.
It
the
specific
the
behavior of
should a l s o be noted t h a t ,
overwhelming numerical evidence, t h e r e i s that
but
typical
does depend on t h e parameters and i s q u i t e d i f f e r e n t
structure
about
equations.
It
to
is
the
not
to
in spite of
t o my knowledge no complete p r o o f
described a c t u a l l y
hard
in
see
that
does
it
occur
does
for
occur
these
for
equations. The
phenomenology i s
as
follows:
The
equations
admit
s o l u t i o n s , one a t t h e o r i g i n and the o t h e r two (which we w i l l r e f e r t o as c e n t e r s ) a t X=Y=±V~b(r-1); Z=r-1. a r e unstable.
Orbits
that start
three
denote by C± and
A11 t h r e e s t a t i o n a r y s o l u t i o n s
near the o r i g i n escape m o n o t o n i c a l l y ; those
t h a t s t a r t near the c e n t e r s escape through growing o s c i l l a t i o n s . is
computed s t a r t i n g
what
is
from some more o r
found w i t h o u t
stationary
exception is
If
a solution
less randomly chosen i n i t i a l
that
the
orbit
will,
after
point,
an i n i t i a l
t r a n s i e n t regime o f v a r i a b l e length, s e t t l e down t o a motion i n which, most o f t h e time,
it
can be thought o f as p e r f o r m i n g o s c i l l a t i o n s
c e n t e r s . The o s c i l l a t i o n
grows i n a m p l i t u d e ; when i t
t h e o r b i t a b r u p t l y makes a t r a n s i t i o n t o o s c i l l a t i o n This o s c i l l a t i o n
a g a i n grows and the o r b i t
t o o s c i l l a t i n g about the f i r s t The
amplitude
of
size,
about the o t h e r c e n t e r .
e v e n t u a l l y makes a t r a n s i t i o n back
and i t
immediately
after
transition
varies
from
i n t u r n determines the number o f o s c i l l a t i o n s
b e f o r e the next t r a n s i t i o n .
The sequence o f
transitions
and the
appears random,
reaches a c r i t i c a l
the
c e n t e r , and so on.
oscillation
t r a n s i t i o n to t r a n s i t i o n ,
about one o f
continuous (see F i g u r e 31 below).
number o f
power s p e c t r a o f
Thus,
the motion hot
oscillations the
between
c o o r d i n a t e s are
appears c h a o t i c and
s a t i s f i e s the standard o p e r a t i o n a l t e s t f o r c h a o t i c b e h a v i o r .
272
~W.31: ~ae ~
~
tAe ~on~,~ ~
•~ The
mathematical
object
~
W ~
x ~
W
(42, 9) (o,~ a ~
~
~
responsible
o
~cx~e).
~5. for
this
behavior
schematically in Figure 32 below.
c;
--j o
1
~ic~.32:.4 acd~mu2ic ~Leu~ o~ tAe ~onR.n~ u22nncZo~.
is
sketched
273 This
sketch
(see,
three-dimensional
above)
state
approximately to scale; the
mathematical
structure
for
a
the
family
Lorenz
of
It
more
transparent.
To
a first
motion
constitutes
band,
flows
counter-clockwlse
in
is
the
not
even
approximation,
looks like two reasonably flap loops of ribbon,
The
orbits
system.
proportions have been distorted in the hope of making
structure
other along a central band.
represents
space
on
the
one lying above the
and the two glued together at the bottom of that arounds
the
right.
the
loops,
Going
clockwise
once
around
a single oscillation around C + . Orbits
on
the
the
left
and
right-hand
beginning
loop
either above or
below the ribbon are attracted quickly down to its immediate vicinity and then follow the flow on it. The double-loop the
solution
flow;
any
point
on
structure
it has
an
forward and backward for all tlme without The central band is divided
pattern
typical
of
growing
orbits.
Orbits
invariant
can
be
the
left
both
this
straight
these orbits are exceptions
followed of
under
traced
in half by orbits that flow essentially
oscillations to
that
leaving it.
down to the stationary solution at the origin; the
is strictly
orbit
by
transitions
boundary
will
displayed
make
their
to by
next
oscillation around C ; those to the righ will go next around C . The fact that +
oscillations example,
a
loop
started out. occurs
around
the
around
centers C+
brines
are
growing
in
orbit
back
the
amplitude to
the
means left
that
of
for
where
is
A transition from oscillation around C + to oscillation around C
when an orbit
making a loop around
C + comes back to the
left of the
dividing boundary. The central band divides in half laterally at the bottom of this boundary, each half,
after having made a loop around the appropriate center,
wide enough to cover almost the entire top of the band. apart
laterally
as
they
flow
around
the
loops,
and
and
has become
Thus orbits are pulled this
accounts
for
the
observed sensitive dependence on initial conditions. A typical orbit on this structure wanders over the surface, close
to
each
nontypical
point
orbits.
infinitely
often.
There
We have already mentioned
are,
however,
the orbits
orbits,
all
unstable,
as well
as orbits
a
great
many
in the middle of the
central band which simply converge down toward the origin. periodic
coming arbitrarily
There are also many
with more
subtle
kinds
of
atypical behavior. We next take a closer look at the ribbons and argue that they cannot be simple surfaces
but must
rather have ~
m~
b~.
Start
at
the top of the
central band where there are two approximately parallel ribbons, the other.
As we have drawn the picture,
one on top of
the upper ribbon is made up of orbits
274 returning
to the central
the orbits the bottom, in two
band after a loop around C ÷ ; the lower,
flow down the central
band,
they form a two-sheeted
with
half
going
left
around
C
up of orbits
whose
previous
and
half
right
has at least
circuit
was
by the flow but the separation of the central true for central
band actually
the
lower ribbon,
remains nonzero,
band actually has four
going
around
argue
in
sheets.
C÷
this
This
and way
C
are
we
object
see
therefore
C + , the
instance
the
the structure
ribbons
of what
the
the upper one
lower
than just one.
four-sheeted,
all
C ÷ . Thus, of
orbits
together
so the upper ribbon at the top
and
come
Thus,
so
must
has
The same
at the bottom
layers rather than just two.
actually that
is an
around
At
laterally
are carried closer
has two layers rather and
to split
two sheets,
around
whose previous circuit was around C . These sheets
around C . As
are drawn together.
surface which proceeds
ribbon of orbits going around C ÷ or C made
the two ribbons
on.
have
of the
the ribbons
Continuing
infinitely
to be
is
called
to many
a
(~aaea~) o22aacta~. Let
us
now
investigate
phase flow
(39,2)
the
Lorenz
is negative
equations
(42,9).
The
divergence
of
the
volume.
The
for these equations:
= -(v+b+l), so
that
all
trajectories
migrate
towards
a certain
set
of
zero
quant i t y: Rsatisfies
Ix2+ Y2+
(Z-r-~)2] 1/2
the condition d~ - - < -C ~f + C dt I 2
(43,1)
with positive C I and C2, so that all trajectories
enter the sphere
C 2 - ~2 . C
~
1
The s y s t e m
does not
change under
(X,Y,Z) ~ For
r
the
unique
origin.
For rzl
saddle
point
one-dimensional
fixed
(the onset with ones-
a
the
substitution
(-X,-Y,-Z). point
is
the
of convection)
two-dimensional the
separatrixes
stable
vertex
it loses stable
F ÷ and
appear C +, C-, towards which the separatrixes
0
move.
the
its stability
manifold r-),
at
and
two
and
coordinate (becoming
two
a
unstable
new fixed
points
For v
275 while
for
v>b+l
(following
Lorenz,
we
will
henceforth
investigate
the case
0-=10, b=8/3) they are stable for 1 < r < r e - v(v+b+3)
(43,2)
24, 74
o'-b-1
e
while for r>r , they lose their stability. For
r=r1~13,92
saddle
point.
we
find
For
a bifurcation
Cantor
periodic motions);
set of curves
structure,
including
of Q
a
the
separatrixes
return
to
time,
has
loops
separatrixes
saddle-point periodic motion L +, L- around the fool C +, C- (at the same which
the
the
appear
invariant
of
which
there
there appears an
r>r I out
for
which
1
is not
denumebrable
an attractor
set
of
and
saddle-point
the separatrixes r ÷, r- intersect and move towards the foci
C-, C+. For
r=r2s24,06
the
separatrixes
r +,
r-,
in place
of
the
foci,
are
curled
around the cycles L-, L +, and instead of ~I' there appears an infinte Lorenz attractor ~2 whose region of attraction is limited by the stable manifolds of the cycles L-, L + (so that excitation of randomness is stable, on
this
is " hard").
For r>r
including O, F +, F- and therefore there is no strucural
attractor
the
periodic
motions
are
everywhere
dence
it 2 stability;
(capable
of
undergoing a sequence of period-doubling bifurcations and of disappearing as r grows only by way of adhesion to the loops of the separatrixes). cycles
L +,
L-
contract
to
the
points
C +,
C-
and
the
latter
stability.
a
d
b
z
• ~. 33- B (a)
Jr
z
e ~ ~
~
c
~
f
~oz~
1 < r < r 1, (&) ~
(c)
~o~ r l <
r < r 2,
(d)
(e)
~r2<
r < r 3,
(e) e o ~ r
~
r = r 1, r = r 2, = r3.
z
z
O
For r=r , the lose
their
276 For r'
r decrease from r
oe
to r2,
the phase
point
M(t)
stays
within the
attractor , while for r
it loses stability and M(t) moves toward C ÷ or C-). 2 An example of a trajectory in the attractor (for r=28, intersectinE the plane
Z=27)
is
shown
in
FiEure
34
below,
which
is
the
work
of
Oscar
Lanford
(Berkeley Univercity).
z
X
~.
34: :~on~v~ ~
~
~
r = 28.
It start at the coordinate origin, circles C ÷, and the unwinds and is drawn to C-, leaves C- and spirals toward C ÷, etc., while the period of rotation around C ÷ or C- equals 0,62 and the radii of the spirals change by 6X per rotation. As
pointed
out
by
Lorenz
himself,
for
this
example
the
Zn÷l = [3(Zn ) of successive maxima has the trianEular form, everywhere; it is ergodic end has the mixin E property. Mori
and
Fujisaka
Lorenz attractor negative neEative to
zero;
However,
and
(1980)
the
and for for for
have
as a function dimension r=l
l
which grows for
2
r>r
(the it
of
the
appearance Is
greater 2
calculated of r
and for
again
than very
the
2
attractor of
the
negative
zero
Lyapunov
( t h e s e c o n d o n e ~ =0,
there
large
r
is
Poincar@ while
exponent the
third
d=2+A1/IA31). convective and
also
for
"rollers") r=r"
appears
multiply
For
a
it
mapping
In A
(Z) I>I for
1 3
r
reaches
new b r a n c h
interrupted
the
~ =A-~
1
it
is is
reduces zero. of
R
1
b y "b~=~i~e"
277 of zero values, corresponding to periodic motion (see the example in Figure 35 below, for b=4, ~=18; here for r=40 the dimension of the Lorenz attractor is
d=2,06).
I O
.... _ -~.m- - - -- Sy.- - - e . t 7"
/00 ~x~. 35: ~
2OO
JO0
r
o~ ~max= ~i a~ ~xtctinrt o~ r ( &=4 a~ct o"=16).
278 BACKGROUND READING
For a good introduction
to
SCHUSTER,
De/~~Aoz~;
H.G.
(1984)
"Deterministic
Chaos"
~
Physik-Verlag,
$
the
~
reader
is referred
to:
.
Weinheim (R.F.A.),
and also BARENBLATT, G . I . ] IOOSS, G.
(1983)
and
Pitman a d v .
JOSEPH, D.D.
Publ.
Program
Boston-London-Melbourne.
Concerning the theory of the Lorenz system, see the book of: SPARROW,
C. (1982) _ ff~e ~ o n ~
~ q ~ :
B
~
,
c ~
o ~ .
and a Z o x ~
Springer
REFERENCES TO WORKS CITED IN THE TEXT
ARGOUL, F. and ARNEODO, A. (1984) _ in J.M.T.A, n°special BENARD, H. (1900) _ Rev. C~n. Sci. Pures Appl. COLLET,
P.
and
ECKMANN,
J.P.
(1980)
_
1984; pp.241-288.
12, 1261-1271,
~ea~
,u%a~ an
d g n ~
~ .
P.H. and SPIEGEL, E.A.
(1983) _ S.I.A.M.J.
~Ae
~
cu~
Progress in Physic,
Vol I. Birkadser, COULLET,
1309-1328.
Basel.
Appl. Math., Voi.43,
n°4, 776-821. COULLET,
P.H. and TRESSER, C. (1984) _ in J.M.T.A,
DRAZIN, P.G. and REID, W.H.
(1981) _ H V d a n d g ~
n°special 1984, pp.217-240. ~
.
Cambridge
University Press. ECKMANN,
J.P.
(1981) _ R e v i e w s
of Modern Physics,
voi.53, n°4, part i,
pp.643-654. ECKMANN, J . P .
(1983) _ in "
~
& e A o ~ o ~ b~ d
L e s H o u c h e s Summer S c h o o l Eds. ECKMANN, J . P . EPSTEIN,
a n d RUELLE,
D.
H. a n d LASCOUX, J .
FEICENBAUH, M.J.
(1978) _ J .
~
~ (1981).
Iooss,
Helleman and Stora,
(1985)
_ Rev.
(1981) Stat.
Mod. P h y s i c s ,
_ Comm. Math. Phys.,
vo1.19,
~ " .
Phys., 25-52.
North-Holland. 57;
617-656.
vo1.81;
437.
279 FEIGENBAUM,
M.J.
(1880) _ Comm.
Math.
and also: GUCKENHEIMER,
J.
and
HOLMES,
Phys.,
77; 65
Los Alamos Science,
mh.
(1986).
N
vol.l;
~
4-27.
O~_d2/at/an~,Dqnamica~
~ec/an~-~ge/d~.
Springer-Verlag,
New-York. HOPF,
E. (1948) _ Comm.
IOOSS,
G.
(1984)
_
Pure and Appl.
in " f f ~ ed.,
JAKOBSON,
Math.,
vol.
I; pp. 303-322.
and ~AoxfAc ~Aenam2_n~ ~
p. 185. Elsevier Sci. Publ.
M. (1980) _ in "Proceed.
Intern.
Conf.
~u/d~";
T. Tatsumi
B.V.(North-Holland).
on Dynam.
Syst".
Northeastern University. KIFER,
J.I.
(1974) _ Math.
LANDAU,
L.D.
LANFORD
III (1982)
LORENZ,
E.N.
LOVEJOY,
U.S.S.R.
Izvestija,
(1944) _ C.R. Acad. Sci. U.R.S.S., Bull.
Amer.
Math.
(1963) _ J. Arm. Sci.,
S.
and
SCHERTZER,
D.
44, 311-314.
Soc.,
Vol.20;
(1986)
_
8, 1083.
Vol.6; 130-141.
Bull.
Vol.67, MA, S.K.
(1976).
~ a d e a n ~Aeaa~ Reading,
PLATTEN,
(1986)
J.K.
(1980)
_ Soviet
a n d LEGROS,
cn/Zico~
of
the
n°l,
American
(1986)
(1983)
_ Lect.
Physics J.C.
Notes in Physics,
Adv. P u b l . POMEAU, Y. a n d MANNEVILLE, P.
- USPEKHI, v o l . 2 9 ,
(1984)
_ ~
~
n°9,
~
ROSSLER, O.E.
(1976)
_ Phys.
SREENIVASAN, K.R.
132,
181.
pp.843-868.
~iqadx/~. Heidelberg. p.295.
Pitman
Program. (1980)
_ Philos.
RUELLE, D. a n d TAKENS, F.
pp. 2 1 - 3 2 .
vol.
_ I n "Naa2/aeo~ Dqnom/r_~ a n d ~ ~ " ;
RAYLEIGH, L o r d ( 1 9 1 6 )
Soc.,
pheanm2_aa. B e n j a m i n ,
Springer-Verlag, POMEAU, Y.
Meteo.
Mass.
MORI, H. a n d FUJISAKA, S. MONIN, A.S.
a~
427.
Mag.
Letter.,
(1971)
_ Comm. Math. (6)
32,
s e t . A, v o 1 . 6 7 ;
(1986)
77,
189.
629-646.
_ Comm. i n Math.
a n d MENEVEAU, C.
Phys.;
_ J.
p. 397.
Phys.,
Fluid.
20,
167-192.
Mech.,
vo1.173;
pp.367-386. ZEYTOUNIAN, R. Kh.
(1983)
_ CRASc., S e r i e
ZEYTOUNIAN, R. Kh.
(1989)
_ Int.
ZEYTOUNIAN,
R. Kh.
(1989)
J.
Engng.
I,
t.297;
Sci.
pp.271-274.
Vol.27,
N°ll,
_ ~ a @aao~ o n ~%e ~naaa£O.on Za ~ of Mechanics, Warszawa 1989.
vol.
41,
issue
2/3,
pp. ~ pp.
1361-1366. .
Archives 383-418;
CHAPTER X
MISCELLANEA 44 . INTERNAL SOLITARY WAVES IN AN ISOCHORIC FLOW We start here with the equation
(14,21) for the function ~(x,z),
which is the
displacement of a particle above its equilibrium height (in the steady case). If U =constant then we obtain,
instead of (14,21),
O~
o:',
the following equation:
rro< l:'+
0x-~ + az 2 + ~ 2 ~ - - Llax j
o~
<:">= dz
(44,1)
g
1
U02 P~
dP¢o
--
~ = O,
GO
U 0 ~ constant,
dz
w
where
z
in the undisturbed
is the height,
region,
of the fixed
streamline
through the point (x,z). The mathematics an exponential (44,2)
is a bit simpler if we direct attention to isochoric flow with basic density gradient p (zm) = p (O)exp[-XoZ=].
We now non-dlmenslonallze
according to the following scheme:
x W=hz ' A=~-~, ~ a ~=Xoh' ~=[~]z , ~=X' ~=~,
(44,3)
where we consider a flow in channel of height h; the characteristic wave-length
is A and the amplitude of the disturbance
in terms of the quantities
(44,4)
in (44,3),
a~
~
is of the order of a.
equation (44,1) with (44,2) becomes
a2A + --a2A + ~[~ a[~] 2+
~a~=
raA]2
=~[~J -
2 aA] + ~ A
a~j
where (44,5) In
non-dimenslonal
2 ~ -
gxoh2 Uo 2
form
the
boundary
horizontal
conditions
are
written
= o,
281
(44,6)
A=O
Let us now assume solutions
"0=0 and "0=1.
~
various
of (44,4),
for ~ in terms of ~,
expressions
/3 and search for
(44,6) in the form$:
A = AO0 + ~A 1 0 + ~AoI+ ~SAn+...
~=13
THE CASE OF
In this case we have:
A2=h/x . The first approximation 0
is
82A (44,8)
oo + ~ A
a"02
and since A
O0
= 0
oo oo
must vanish at the top and bottom of the channel,
the solution
is (44,9)
Aoo = ~o(~)sin(n~"0),
where Do(E) is arbitrary.
<=
n2~ 2,
A second approximation
a2A
01 + n2 2A = _ 2 A _ a"02 o1 o1 oo
aZA
oo + a~2
is: aA
0___00 a"0
or
d29
B2A
with solution
d2~ (44,10)
AOI = D1sln(n~"0) + 92cos(n~"0) + ~--n~"0cos(n~"0)[ 9 + --~°"0sin(n~"0), 2
where D
I
and ~) are arbitrary 2
at .0=0. Then t h e c o n d i t i o n d2~?
(44,11)
o +
d~2
This has ~a solution
no s t e a d y - s t a t e t
According
to
Long
of ~. But ~2mO,
since
+
dE 2
A01 must vanish
t h a t A =0 a t "0=1 y i e l d s O1
o2 ~)= oi o
O.
for ~o(~) that vanishes
disturbance R.
functions
0
f o r ~=/3 t h a t
(1965).
at elther ~=~ or ~=-~.
vanishes at infinity.
There i s
282 THE CASE OF ~=~
Again the first approximation is 2
Aoo= ~ o ( ~ ) s i n ( n ~ ) ,
(7 = oo
2 2
n~
and a second approximation is
(44,12)
a2A
lO + n 2 2 A = lo an 2
a2A
oo a~2
o"2 A 1o
oo
2 D 2° + (r:o~)o]" = -sin(n~n) _f d d~
t
A solution satisfying A =0 at n=0 is lO AlO(~,19) = ~ (~)sin(n=n) + 2 n ~ C o s ( n ~ w ) ~ 2~)° + o'12o~)o] I Ld~2 and requirement that A =0 at W=I yields lO d2~)
°+ ~ ) = 0 .
d~2
1o 0
Again there is an steady-state disturbance for ~=~ that vanishes at i n f i n i t y . THE CASE OF ~ = ~
We again have A = Do(~)sin(n~n) oo
and V~o= n2~ 2
and t h e d i f f e r e n t i a l equation for A is the same as eq.(44,12), IO
the first term on the right is missing. Therefore,
(44,13)
2 o" AIo(~, n) = ~1(~)sin(n~w) + IO~o(~)WCOS (n~W). 2n~
Since A =0 at W=I, we see that: lO (44, 14)
o"2 IO=0,
The equation for A
Ol
is
A IO= ~i (~)sin(n~n)"
except that
283
a2A 0____! + n2 2A
(44, 14)
an2
Ol
= _ 2 A
Ol oo
+
aA
oo
an
= -
AoI=O
+ n=9o(~)cos(n~n).
at n=O is 2
o" :D + °~4)o(~)ncos(n,~ n) + --~°nsin(n~n).
Ao1(~,n) = 9t(~)sin(n~n) Since A
oi
=0
2n~
2
at n=l, we see that
(44, 15)
o"2 = O,
Finally,
the equation for Air is
Ol
A
Ol
= ~ slnCn*rn)
+ --°nsin(n~n).
1
2
a2A oo
aeAll + n21z2A = an 2 11
21_[ _OAoo ] 2 0 +A _
a~c2
an
[o:]
1___.oo_ cr2 A
zz oo
an
d2~)
-- -
d~2
-
+ o"
°n2~2(I
190
sin(n~n)
+ n~
1
(~)cos(n~n)
+ cos(2n~n))
4 so that (44, 16)
A11 = 92(~)sin(n~n)
+
+ Since A
n
[ (-1)n-1] is
6
cosCn~n)
cosCn~n)[ + n- 2n~
( . ) sin(nt¢~) t ~ n -~
1:D2,- ) ~- o t ~ +
d2D o+ d~ 2
o.: 1Do1
~(~3cos (2nrtn). 12
at y = l ,
=0
11
(44, 17) If
2)~(~)
ei~etL
differential
the
first
with solution
+ term
d~ 2 vanishes
+ <1~)o completely,
equation has na solution vanishing at ~ - - ~ .
For odd n, however we get d2~) (44,18)
6
o+o.~ d~ 2
2 2 1~)o+ ~n~) o = O,
n=1,3,5 .....
and
the
remaining
284
(44,19)
9o(~)
vanishing Notice
at
that
l~l=.,
= -
if 2 1
if we regard
of the disturbance, (44,20)
_
To the present
9~ 2 l*aech2[~iv 4n~ L2 is ~
]
n=1,3,5 ....
nj'
.
a in (44,3)
as the n o n - d i m e n s i o n a l
maximum
amplitude
then 9~ 2 11 = I , 4n~
n=1,3,5 ....
order then, h = ~csech
(44,21)
sin(n=
,
n=l, 3,5 .....
2 2 2 4nff o- = n ~ - 9----~4N ,
n=l, 3, 5 . . . . .
THE CASE OF ~= z~ As in the preceding
case,
2
we have
2 2
~00 = n ~ ,
AO0 = Do(E)sin(n~n) ;
2
In addition,
the
0"10= 0
Ato = ~1(~)sin(nrr~);
°'012= 0 ,
Aol = ~)t(~)sin(n~n)
solution
for
Alt i n
d2D we o m i t Equation
the
terms
( 4 4 , 17)
involving then
reveals
cr2 = 11
0
- d~ 2
( 4 4 , 16)
+ --~°nsin(n~n)'2
and
( 4 4 , 17)
is
applicable
and
that
2)=0 o
if
An
=
n odd,
92 (~)sin(n~n)
+
and that
2
11
+
=0 i s
n if
~ (~) sin(n~n) i ~ n
cos(n~n)
-3 + cos(2n~n)
12 n=2,4,6 . . . . . Continuing,
we obtain "220=0,
and
if
o
Hence (44,22)
here
A2o = ~2(~)sin(n~n),
n=2,4,6 .....
,
e~en.
285
OZA - - 21 + n2~2A
(44,23)
On 2
OZA 8A 20 = -or2 A - ~ oo + 21 21 oo O~ 2 On d2~) o -+ ~1~o]sinCn~n) d6 2
- ~oHl(~)n2u2(l
The solution satisfying A
oo
OA
On
1o
On
+ n~H2(~)cos(n~n)
+ cos(2nun)).
=0 at n=O is 21
A21= ~)3(~)sin(n~n) + l~Do(~)Hl(~)cos(n~n)
+ H
OA
(') sin(nun) 2 ~ n ~
9 o (~)
2
Applying the condition that A d2~)
d2~) + nc°s(nun)~ [ - d~ - 20 +
9 o (~)
HI(~)+
6
ffl(~)cos(2n~n).
craig)o] n=2,4, 6 ....
=0 at n=l, we get 21
° + ~ =2100.
d~2
There is an steady-state disturbance for ~= 2~ that vanishes at infinity.
THE CASE OF
~=~2
As in the previous case (~=a2~) we have: A
= 2) sin(nlrn), 0 2
2 2
= n ~ , O0
~
2
A
= H sin(nun), 10 I
2
,2
= ~ = O, I0 20
v
= O, Ol
A
= H sin(nun); 2o 2
~I1= 0 ,
and A
(44,24)
9 = :I)sin(n~) + --~°~sin(n~), ol 1 2
All =
~2
2+ g H I
sin(nun) + --~°[2cos[n*rn) - 3 + cos(2n~n)]. 12
As in the case: ~= 2~, the derivation of (44,24) yields the r-equir'ement that n be even. Also
286
82/,
_ _ 02 + n2 2~ 8.02
02
8A
= _02 ~ + 02 oo
0__..__/_1
8.0
2 -0- D s i n ( n g . 0 )
+ ~ ngcos(ng.0) 1
02 0
0 + --n~cos(n~), 2
+ --2s°in(n~) 2
n=2,4, 6 .....
so that (44,2S)
2 ~o (~) + 0-02 ncosCn~n) 2n=
(~) o ncos(n~n) 4nx
Ao2= ~4(~)slnCn~n ) _
+ z~XDo (~)
+
4n2~ 2
cos(nxn)
.0sinCn~),
+
inCn~n)
n=2,4,6 .....
2 The condition
at .0=I yields: ~o(~)
=
2
1
4~2~)o(~) ~ ~o~ = ~,
and
(44,26)
/`02= D4C~)sinCn~.0)
~1 (~)
+
~0 (~)
.0sin(n~n) +
2
2
.0 sin(n~n).
8
Final ly, azA
a2A
12 + n 2 Z A = 8.02 12
oo
8~2
aA
01 8n
aA
O0
+
aA
8~
1____~1_ 0 2
8n
A
02 1o
d2~) - _
o + 0-;22)o]Sin(n~n)
dg 2
DoC~)n~cos(n~n)[Dl(~)n~cos(n~w)
2
+
1
cinCn~.0)] + D2(~)n~cos(n~.0)
(~) cinCn~n)
4
- --~sin(n~n) 6 T h e solution
is:
v C~)
+ - I-
2
n~.0cos(n~.0)
9~Cg) 6
_ 0.2 A 12 oo
287
(44.27)
d2~) o + o.~2~o(~ ) d~ 2
A12(~,I~) = -[¼~1 (~) + ~--- i ( ~ )
2cos(n~n) +
~)2 (~] + --wsin(n~n) 2
_
n,,~(~)1~_~_ os(n,,n ) in(nun)
S 2 + ~)o(~)sin(2n,rn)
- ~o(~)~t(~){2I- - ~cosC2n~n)}l - ~~:(~) ( n
+
n2~2)[ -
11 3n2~ 2
4 cos(2nlrn) + ---sin(2nr~n)]
9n3~[3
+ DS (~)sin(n~n) + !9 3 o (~)Dl(~)cos(n~) The condition at n=l yields
d2D (44,28)
o + o.~2~o- 6n ~ 2 o = O. d~ 2
This has a s o l u t i o n 9o-2 12cech2 [~io.12]. n/[ vanishing at {E{=-, if o.~2 is negative. (44.29)
90(~) =
Thus
9o.2 (44,30)
12
- -1 n~ and, t h e r e f o r e , to the present order. = -~sech 2
(44,31)
sin(n~ ).
n=2.4.6 .....
[ o.2= n2 2+ ~ 2 _ 9"- -n~~_2
Notice that if we seek other solutions by putting ~= 3~, ~=~/33, etc.., all subsequent cases will lead to equation (44,28) with the d2~)Jd~2-term missing. It follows immediately that ~)o(~)=-0, and that there are not other solutions of the kind we are looking for. The
solutions
(44,21)
and
(44,31)
represent
disturbances
with
maximum
amplitude at ~=0, dying off monotonically and symmetrically on both sides. The speeds of propagation are determined by values of o-2 in (44,21) and (44,31)
288
respectively.
They
are
~
~
disturbances
to n=1 and n=2,
Bee schown
similar
to
those
discussed by Keulegan (1953). Two examples, below.
corresponding
Since /3 is always positive,
The wave
we see from
in the lowest portion of the channel
in Figures
(44,21) and
36 and 37
(44,31)
that ~>0.
is one of elevation
if n odd,
and one of depression if n even. We recognize
that
if closed
streamllnes
appear
in the flow,
situation In which density increases with height. some of the cases considered above, be possible, amplitude,
_
.
-12
nevertheless,
Notice,
we have a local
finally,
too that in
in which there is no solitary wave,
it may
to find solutions for an infinite train of finite-
internal waves.
.
._
-8
.
. _
-4
j
|
0
((~=0, 33 un.d. ~=0, I 0 ) .
I
4
4
•
•
8
II
&
12
28g
~.0
J
I
I
II
I UI
I
ii
J
0.$ 0.6
"
O.q.
-
I,.
0.I
0
-I00
-eO
~
I
|
I
II
I
I
f-
-f~)
II
l
I
I
I
]
-40
I
-20
II
I
I
0
l
it
I
ii
•
e
0
20
40
• ~.37: ~ 0 ~ / ~
60
~
~
U ~
(a=O=1? und 8=0, 10).
eO
I00
290
FREE SURFACE SOLUTIONS
If the
upper boundary surface
is ~ee,
then we have
the following
condition
(instead of A=O on W=I)
aA ~ll'_aAl ~ roAl~l J A = ig~ - T L L ~ j + ~'L~J J'
(44,32)
on ~=~+~.
In the work of P.D. Weidman (1978) the mode shapes for internal solitary waves propagating the
limit
upper
in a ~
of
boundary
exponential
stratified
"~"
stratification.
and
a
free
fluid of finite The solutions
surface
stratification.
are
The profound
depth are determined
obtained for both
compared changes
with
which
known
results
depending
the upper surface
is fixed or free are derived
from the strong
the
when
weak.
free
surface
supporting
the
qualitative
the
stratification
conjecture
differences
that
in
the
the
is
free
modal
structure
can
when
the
no
is
of
presented
longer
fluid
for
whether
influence
Evidence
surface
on
in
a fixed
is
effect heavily
stratified. Now
we
consider
the
evaluated at ~=I+~A,
case
of
~=mB.
Since
the
Bernoulli
the variable A and its derivatives
equation
in (44,32)
expanded about the linearized position of the free surface, set of free surface conditions developements (44,33a)
~
A = 0 oo o0
at n=l; aA
(Aio+ Aoo
oo) + ¢2 A = 0 lo oo
an
2 A + cr2 A Ol
oo
oo
Ol
are first
The ordered
This gives:
at ~=I;
a2A oo at ~=1; a. 2
[.+,, %,+,, oo cqn
OAoo(l _ cr2 )
+
be
by inserting the asymptotic
(44,7) in the expanded form of (44,32).
(44,33b) (44,33c)
are then obtained
~=I.
must
n°° a~
ol
-
O,,oo1 _
o l an J
l~Aool2
2Lan
j
at
Oa'O,o~=i,
f o r the respective d i f f e r e n t i a l eqs. f o r Aoo, A i o , Aol and All.
If
one has
solved the f i x e d boundary problem, i t is a simple matter to make the necessary adjustements f o r the free surface. As an example, we take Long's results
( f o r exponential s t r a t i f i c a t i o n ) and
solve the free surface problem. The a p p l i c a t i o n of surface so Iut ions:
(44,33)
gives the free
291
hoo= I)o(~)sinCn~),
~oo= n2 =2;
A
2
lO
= ~1(~)sin(n~D),
A01 = D1C~)sin(nvrw) +
= 0 ;
:
sin(n~w) + °--°-----.WcosCn~w) /11~ '
2
01
2;
and the solution for &11 satisfying the lower boundary condition is
~)2 (~ )
d2~) 0
(~) sin(nun) + ~1 ~ W" ~
+
1
2
~9o(~) +
o
eos(2n~w)
12
~i(~)Wn-~Cos(n~w).
The free surface condition (44,33d) is satisfied if: d2D
7n~D2 d~2o +
(44,35)
(n add);
d2~
(44,36) showing
0 + cr2~ + 3 n ~ 2
d~2 that
the
11 0
even
as well
as
0
= 0 the
( n e~e~), odd
modes
are
determined
when
~=~.
Solutions of (44,35) and (44,38) vanishing at infinity agree with the results obtained by Benjamin (1966). We now compare the results for a linearly stratified fluid with those obtained by previous Long's paper for an exponentialy stratified fluid. For
~<
the
non-dimensional
streamline
deviations
for
both
cases
can
be
written (44,37)
A = sgnC~)l~Isin(n=n)sech2(~ ~)
and the associated non-linear phase speeds are then* c 2= c2(I+4~2), n We
first
observe
that
with
c = linear wave speed. n
a
rigid
upper
boundary,
both
density
produce qualitatively similar wave shapes for both even and odd $ For example, if surface, t h e n :
we 2
nx
consider ^
the
=._--~-aj~For the ~ I~"
case
o£
modes
modes.
Also,
stratification with the Free
linear
and
profiles
2
nx
^
~ ==--00~ For the e~U~ Z '
modes.
292 with
the
exception of
the
odd
modes
for
linear
stratification,
all
modes
shapes for a free surface are essentially inverted from what they were with a rigid top plate. about
It is remarkable that such profound changes can be brought
by simply removing
the rigid boundary.
Although
there
is a distinct
character change for the odd mode internal waves with a free surface in going from an exponential
to a
manifestation
basic
of
a
linear density profile, difference
between
this
constant
is apparently not a an
variable
Brunt
frequency fiuid. Rather, all the above sensitive changes in mode shape seem to be due to the strong influence of the free surface when the fluid is weakly stratified.
4S
•
D E E P CONVECTION
THE
EQUATIONS
The atmospheric flows are ~ Mach number flows U2 Mo2~ ~ << 1 ~RT(O) and
if
we
wish
to
avoid
the
constraint
(8,9)
imposed
in
the
Boussinesq
approximation (see, the section 8) it becomes necessary to analyze flows with very low Froude number since, according to (3,7), ~Mo 2 Bo = ¢ - ~
(45,1) if
it
Fr~Mo<
o Fr 2
is assumed
tha Bo=gHo/RT (0), ~ = c / c v and
eo=Ho/Lo are
of
the
order
unity. However,
the limiting process Fr--~O, i.e.,
in fact Mo--~O in the atmospheric
equations (for instance the equations (8,4) where the expression BoFr 2 must be substituted
Fr2mU2JgHo ) ¢ degeneracy of these equations to order zero . IT soon
for
become
~Mo 2,
clear
when
that
eoSl
it
and
should
be
assumed,
leads
to
according
a
very
to
strong
Zeytounian
(1974), that when Fr---~, the term (with the dimensionless quantities) t¢ +
t
This degeneracy is obtained in
ff
See
the
formula
is the
related section
(3,11)
or
~ ~°NZ(z ) -~ O, ~ m
dz to the 46. the
so-called
system
o£
<~O~-~O~e~
equations
(10,2).
~del
which
293 which means that we must consider the following double limiting process: 0
(45,2)
Fr --~ 0 ,
~°--> 0 ,
with -~-~ = 0(I). Fr 2
In order to confirm (45,2) it is sufficient to understand that when Fr--~O, the first equation of (8,4) implies that 8~Fr 2. Because of this the third equation of (8,4), not
in fact,
satisfied,
implies the constraint written in (45,2).
then
(when m°=O(1))
~.~mw
If the latter is
---W] with Fr--)O. This
is precisely
what bring on a very strong degeneracy of the basic exact equations
(8,4) at
order zero. Hence
it is necessary
that the characteristic
value of the V~is~l~
internal
frequency (with dimensions and for this we use a superscript asterisk) satisfy the relation: •
(45,3)
N (0) s
U
0
--=
10-31/s
H
0
s i n c e Bosl i m p l i e s t h a t Ho=RT (O)/g = 104m. The c o n s t r a i n t (1962)
(45,3)
is,
in fact,
the
i n o r d e r t o o b t a i n t h e so c a l l e d
o f e q u a t i o n s (which f o l l o w s of E u l e r ' s ~-~
+ e~ N
+
one
imposed by Ogura and P h i l l i p s
"anelastlc"
e q u a t i o n s v i a the system
to-plane equations):
~-lBo~
=
O;
b--6= O;
(45,4)
(~-1)~.~ + D L ° g A Dt
- O,
R
e =TR
where (45,5) with
= p-(~-l)/~ ,
'
D--tDm ~-[ +a
5.~,
S~I, e ml and Romm. 0
THE "ANELASTIC" EQUATIONS OF OGURA and P H I L L I P S
Let us return to (45,4) and replace M 2 by BoFr2/~. When Bo/~ = 0(I) and Fr---X),
the
l i m i t i n g form o f t h e s e e q u a t i o n s ( 4 5 , 4 )
following asymptotic representation:
can be sought by p o s t u l a t i n g
the
294
-) -) U = U0 R
(45,6)
= N + Fr2[q + ...; 0
2
i 8 = e + Fr2e ...; + .... o 2 To order zero, we have the following from the first equation of (45,4) (45,7)
e o ~ R o + ~-1~ Bo ~ = 0
~
No=
Ro(t,z),
Co= eo(t,z).
But, the second equation in this system (45,4) shows that ae ae _~o . ~ . ~ O=o ' o
at
az
and i f at the i n i t i a l instant Oo-Ooo(Z),_ thent: (45,8)
8 m e o
oo
(z) and
Again from the first equation D u~
(45,9)
R mR
oo
o
(z).
in (45,4), to order Fr 2, we obtain:
+ Coo(Z) ~n = ~-___! 1 Bo ~ 2 ~" e
~'-1 Bo ~
Dt
e
oo
(z)
~,
given the fact that dN
8
e --°° 2 dz
_-- _;r-1 B o ___2. ~"
e
oo To order zero, the last equation of systeme (45,4) yields:
~.u~:
(u~
o
since
o
~)d [Lo %0
]
oo
p=V]I/~-I/8.Therefore,
1
:-"OpooCZ
to order zero,
d~oo dz'
the following continuity
results: (45,10)
V.(Poo(Z)U" o) = O,
where
1/~'-1
Po0 = [7 O0 Let
us c o n s i d e r
the
/eO0"
second equation
t It might easily be thought that, W e~ 0
0
,~0.
similarity
It
will
relation
be
seen
(45,11),
of
In fact, e
further we
system
on,
(45,4).
oo
(Z)EI since
however,
have
eoo(Z) =
I + Fr2eo2(Z).
that
de
because
o____£o=O, if dz of
the
equation
295
It Elves: de De oo 2 w + Fr 2 - - + ... o dz Dt and
it is o b s e r v e d
that
the f o l l o w i n g
= O, similarity
relation
must
be
imposed
d8 (45, 1 1 ) where account
oo _ AoFo2(z)Fr2 ' dz A =constant and o the i n f l u e n c e
(45,11)
necessarily
it f o l l o w s
F (z) 02 of the
implies
is
function
reference
that
of
order
unity
stratification.
88oJSt=O.
On the
which
As
hypothesis
a
takes
matter (45,11),
that: De
(45, 12)
2 + AoFo2[Z)Wo = O. Dt
Let
us n o w r e t u r n
to the z e r o t h d r]
e
oo
order
~'-lBo
+
relation
(see
(45,7))
0
=
oo dz and derive
it w i t h r e s p e c t d 2 [7
e
to z. T h e r e
oo +
de
oo dz
oo dz 2 i.e. , to o r d e r
zero,
w h e n Fr-->O,
d r]
results:
oo _ e
dz
d 2 [3
d r] o______oo+ A r (z)Fr 2 oo _ O, oo dz 2 o 02 dz
we h a v e
d 2 [3 8
--°°
oo dz 2
_
0
~
N
oo
=
1
+
c
o
z
or e v e n c 8 + ~-lBo o oo ~ since
[
800(Z)
R
with
c = -~-lBo, o
Lime = O. oo Fr43
Therefore
(46,13)
= 0 ~
oo
(z)
= 1 + (AO~Fo2(Z)dz)
I
Fr 2
I eo2(Z)
= 1 - ~-lBo ~
z + P
02
( z ] F r 2,
into
of
fact
for
82
296
dP (45,14) Finally,
o2
dz we
representation
- 2F~lBoAo~Fo2 (z )dz.
arrive
at
conclusion
that
the
following
asymptotic
must be postulated: =
(45,15)
the
{ uFl~
+ --)br2U +
..;
2 " + Fr2{Po2(z) = U~o 1 - ~-lBoz
+
n 2}
+
I
l
I
8 = 1 + Fr2E + .... 2 to obtain
the following
for ~u , R and E o 2 2
Ogura
D~
(45,16)
~-I Bo ~
(45, 17)
~. (Poo(Z)U~ o)
(45,18)
D~(E2) = O;
Dt
+
and Phillips
~n
= ~-1Bo 2 ~
(1962)
type,
anelastic,
equations
E ~; 2
= O;
where
(46,19)
and
{ - -'2 _ = 82+ Ao~Foe(Z)dz
1 1/~'-1 Poo(Z) = [1 - ~ BozJ
a
D __ D--t a-t +
~Uo"~"
THE DEEP CONVECTION EQUATIONS ACCORDING TO ZEYTOUNIAN (Case of the adiabatic atmosphere)
Here,
our starting
point
is the system of Euler equations
and 8, wrltten without dimensions. with Bo fixed at the order unity.
We will consider The variables
(8,4) for 3, =,
the limiting process M--)O o t, x, y, z, as well as all the
297
other parameters
(S,~) remain fixed at the order unity when Mo--g3. When Mo--g3
[
we suppose satisfied the following asymptotic representation: u ~ : u ~ ÷ . . .
(45,20)
In the
this
case,
functions
;
a
(& =
K
O =
K O +a "'" ;
~
M2~ + 0 a
=
we h a v e u~a, wa,
the Oa
D~
s
~a+
Dt
"'"
;
....
following
adiabatic
deep convection
equations
for
and i ta:
T~(Z m) a+__~.:Bo e~; ~
~
a
a
a
Bo
a
(45,21) S
~T~(zm )
De
~-1 S
a
Dt
~
D~
a
+
Dt
Bo _ _
Z(z)wa=
O;
T~(z)
~ a = Wa+ Oa . where S D ~ - m S ~a
+ ~a .~,
and once the
following
hypothesis
is
made:
dT
(45,22) dz Z (z)
m _ -~-___!+ Z ( z ) K ~ '
being a function which takes into account a weak stratification with
the altitude
of the standard
unity in absolute values. (46,23)
T )- ( zm
to
and which
is assumed
of
the order
In the limiting system (4S,21) we have:
m 1 -
It is again pointed out that then according (4S,22),
atmosphere
Z ;
z
m
~Boz.
if (4S,21)
is to remain asymptotically
the temperature gradient - d T J d z
valid,
must be very close
(~-I)/~.
STEADY TWO-DIMENSIONAt CASE
We consider now the following steady two-dimensional adiabatic deep convection equations, accordinE to (45,21),
298
8u U
au
a
- -
+
W
-
aOx
1-~'- 1Boz a~ ~" a_
+
¥
aOz
@w a
U
a
-
+
1-~-lBoz 8~
@w a
W
- -
aox
O;
ax
aOZ
a_
Bo e
8z
~
a;
(4S,24) au - -
a
aw +
ax
a
az
an we note that
~
a
1-~-lBoz ¥
aea
ae a u -aax
w
Bo
_ _ - -
~-1 [u a=a
+ w-aaz
~ t aax
dLogp~(Boz)
Bo
1
dz
~
1-~-lBoz
Bo
awra]
+ w --. + l_~_lBozZ (Boz)wa = O; aaz )
The contlnuit F equation (thied equation of system (45,24)) is integrated if we
[~.
introduce the following generalized stream function Ca(X,Z): Ua= -exp
dz
]d,a
-. ; 1-~-IBozJdz
1%
(4s,2s
w = +exp{~--j. -a ~- -l-~-IBozJdx
The fouPth equation of (4S,24) then leads to the following first integral: (45,26)
Furthermore,
ea - ~ - I
. x®(Boz) + Bo{ . dz = E(@ ). a JI_~-IBoz a
the following vorticity equation
is obtained from the first two
equations of (45,24) when the third equation is also put to use: (45,27)
where
Ow ~ = a a 8x
u
1
Off an a + w___~a +
aax
asz
Bo 8 [e
I_~-IBoz
wa~a- ~
Ox
(45,25)
and
a"
~-1
~
]
aJ
'
'au a
By
taking
use
of
Oz
(45,27) , a second first integral: dz
j--TBo
~
(45,26),
we obtain
from
299 since
~_~Oa.~O@ac~x-~-'-z-]dd--~amO.
But
1 exp --
--
=
-
and hence
8x
a2@a
1
a_
1 - ~'~,IBoz
au
ax 2' 1
a2~
Oz Therefore,
a2~a (45,29)
az
Oz 2
(45,28),
from
_~
-ax 2
the following equation
a2@ + --
in @a is derived:
-I
]-{o 2
The
arbitrary
of @a'
functions
~(@a ) and
E(@a),
}
Bo dE
must
be
determined
from
the
boundary conditions.
4B
. MODEL MACH
EQUATIONS NUMBER
It soon become
FOR
LOW
ATMOSPHERIC
apparent
FLOWS
that
the quasi-geostrophic
in the section 9 was too approximate necessary
to
devise
a
new
approximation
limiting model,
but a more efficient
It thus seemed
reasonable
(atmospheric
motions
which
would
the
Mach number
low Mach number flows)
drop the idea of a low Kibel number.
equation
also
lead
Considered It was thus
to
a
simple
one.
to preserve
always
model
for many synoptic situations.
but,
M ° as
small
parameter
on the other hand,
to
The latter would be assumed of the order
unity. The
starting
point
once
again
consists
of
the
primitive
equations
which
we
300 write in the following form in accordance with the results of section 7 (see the equations (7,9)): (46,1)
s~-+ (~.~)~t+~+
+By c~^~) + - - ~ : o ; o
(46,2)
IS 8
(46,3)
~
+ ~
The s y s t e m o f e q u a t i o n s
~]~18;R
(46,4)
We I n t r o d u c e (46,S)
~÷
~'-I a_~} = 0;
['SJf]
= 0 (46,1)-(46,3)
and M, w h i c h f u n c t i o n s o f t , x , y F or s y s t e m ( 4 6 , 1 ) - ( 4 6 , 3 ) ,
w {8
forms a c l o s e d
~
in the e q u a t i o n s
~=uz+vj,
and p.
the following slip ~.~÷
system for:
= 0,
condition
is prescribed:
~r~o.
(46,1)-(46,3)
the horizontal
divergence
D m ~ . ~ = ~Bu + a - ~ Ov
and the vertical component of the eddy (46,6)
Q ~ ~.(~A~) =
av 8x
au By"
It now becomes possible, after a rather long but simple calculation to replace the vectorial equation (46,1) with two scalar equations for ~ and fl:
(46,7)
~" + ~'~
+ ~
+ _
lax a y
+ 1
÷ 8
ay
+ ~-~ ~-~ + ap ay
~ ÷ 8u ÷ --
~ 2 ~ _- O;
~,M2 o
(46,8)
~
+ ~'~ + ~
+ ~)~ + 8p 8x
ap ay +
+ 8Y ~) + ~v = O.
It will be remarked that the sllp condition (46,4) can be written for
(46,9)
p = Ps(t,x,y) provided that R(t,x,y,p )=0.
According to the definition of ~ (see (7,2)),
it is seen that ps(t,x,y) must
301 satisfy
the e q u a t i o n :
(46,10)
OPs
S
+ ~'~Ps = ~(t,x,y,p
8t
).
THE SO-CALLED "CLASSICAL" QUASI-SOLENOIDAL (QUASI-NONDIVERGENT) Here
we c o n s i d e r
MODELS EQUATIONS
the
classiacal
Monin-Charney
case
which
is
based
on
the
limiting process: (46,11)
M---~, o
In conjunction
with
.w/z~ a s y m p t o t i c
with t,x,y
the
limiting
expansion,
(46,12)
~
for
T
To
P
Pso
a~ --
o
OH o
-
Ox it
-
which
S means
~o=~o(t,p)
it
can
and from
l
let
us consider
of equations
the following
(46,1}-(46,3)
2 ~0 2
+ ....
2
T
2
Ps:
0
Oy is
noticed
02R
(46,13)
(46,11)
the system
To order zero (46,1} leads to
and from (46,2)
process
~o ~o
=
and p fixed.
that
02~
--
1
0~{ol
o + ~o [ 1 ° + j- ~- , pa-~atap ap 2 be
assumed
(46,3}
that
we f i n d
(46,10),
psomPso(t) satisfies
(46,14)
S
: O,
~o m ~ ( t ' p ) ' v
that
In
this
Dom~.~o=Do(t,p).
the following
case,
we also
have
according to
Finally,
relation:
OP.o
It
is
addition, form:
obvious the
that
Ot the
components
- ~o(t,Pso). order u
o
is
and v
o
consistent of ~
o
can
only
with
be represented
a
flat in
the
ground.
In
following
302 h 8~o
1 Uo= ~:DoX +
8¢0
8y h 1 O~Po 8¢o, Vo= 2~)0y + - - + - 8y 8x
(46,16)
8x
with +co
(46,16)
~o(t,x,y,p ) =
dx'dy',
~ o ( t , x , y , p ) L o g (x_x,)2+ (y_y,)2 --co
0v where
~o m
0u
o
8x
o
h and ~o(X,y)
is an arbitrary
function
which
is harmonic
8y
h ~ 2 = 0). with respect to x and y (amW ~/8x 2 + a~2$o/OY The problem which now arises is to make sure that to order zero, our functions are
independent
atmosphere
of
the
time
as a function only
t,
i.e.,
that
they
of the altitude
characterize
p.
To
adiabatic equation for the temperature T must be used, the expansion
this
the
end the
standard full
non
instead of (46,2), and
(46,12) must be extended to the term proportional
to Mo,
if we
take into account the followlng similarity relation
1//-~-Z--- ~ =
c46,17)
~>o
OCl),
o between R e ± and M ° (for the definition of Re± see that
secularities
do not
appear
during
(since the fluctuation of temperature T that the right hand member, t be zero :
(6,1)).
a sufficiently
In this
long
must remain bounded),
period
case, of
it is mandatory
dp
We t h e n r e c o v e r the s t a n d a r d atmosphere dPso
(46, 19)
Ro- Ro(P), To--- To(P), Wo= - -
mO,
dt
associated with the thermal balance
equation
(46,18).
A
t
See,
£or
the
de£inition
o£
time
of the approximate non adiabatic equation for T ,
d F dTol dRco lPoCp) l = %
c46,16)
so
0~ 0
and
R
co
the
section
3
and
the
equation
(4,33).
303 For this case,
the following system is derived from the system of equations
(46,1)-(46,3) for ~
and ~ _Bo~ :
0
2 ~
2
a~
o+
(46,20)
÷
^
%--o,
Ot
~.~o = o. System
(46,20)
describes
the
flow
of
an
incompressible
atmosphere
along
isobaric surfaces p=constant;this plane flow was projected onto the /3-plane. This flow is strongly uncoupled with respect to the altitude. The only means of obtaining a coupling with respect to p is to impose on this flow initial and lateral conditions (in x and y). From
the equations
(46,7) and
system for n ° and ~ ,
(46,8)
we
find
to
Oy (46,21)
order
zero
the
following
which is equivalent to (46,20):
Ox
Oy j-J
[s ~o+ J(¢o,)]~2@o+8°@°=0, .
ax
if we take into account that D mO and ~ mO. o o I n ( 4 6 , 2 1 ) we h a v e
Jt ( a , b ) :
System
(46,21)
Oa ab
Oa Ob
the
so-called
~
forms
quasi-nondivergent)
model.
ay ax"
The
first
classical
equation
in
quasi-solenoidal
(46,21)
is
the
(or
so-called
"~bzne~e" equation whereas the second one of (46,21) is an et~eb/Zion equation for ¢o" Certain remarks can now be made. Firstly,
(46,21) is of ~ order in t and necessitates o o condition : @o[t=o= ~o" How can ~o be found? Solution
system
resides
in
introducing a short gravity
waves
posing
the
time t=t/M
(which
exist
o at
problem
of
the
in order to take the
level
of
unsteady into account the
an
adjustment
in the quasi-solenoidal
model
(46,21),
by
the internal
primitive
(46, I)-(46,3)) which were filtered out during the limiting process Secondly
initial
equations (46,11).
why is there no derivation
304
with respect to the altitude p. As has already been pointed out, this quasisolenoldal
model
discribes
the
motion
atmosphere stratified
in horizontal
latter
independent
being
cancelling
totally
out
of
~0
leads
to
of
an
incompressible
(barotropic)
layers in the planes p=constant
of each other. the
Hence,
undesirable
to order
consequence
of
and the
zero,
the
forcing
the
motion into the horizontal planes p=constant. The problem might be expected to be remedied order.
if the expansion
(46,12)
is carried
out
to the next
following
If ~>I in (46,17) then from the (46,2) it follows that:
(48,22) where
K°Cp) -- - ~ t d - ~ and
T
is directly
2
related
~ -~,i T p'
to
~2'
satisfyinE
with
~0 to
(46,20),
by
the
relation: all (46,23)
T = -~p
2
2
ap
Next, we have
~_- ~ . 9 = _
(46,24)
2
and then
c46,2
2
a~ ap
'
a~
)
s
a~
5+
o+
+
at where x4
. Bo~ ~
4
(R=-H (p)+M2~ +M4H +. o
o,
ap o 4
o 2
""
).
It will be remarked that a couplin E with the altitude, p, exists thanks to the terms
a~Jap__ and
w2a~Ja p._
However
this
is
not
enouEh
and
this
main
description remains hlghly deEenerated. Let us now take a look at what happens
to the slip condition on the flat
ground. First of all, we have the following (46,26)
~o(Pso ) = 0
~
Pso--1.
We denote by fs-f(t,x,y, ps o) and we then have:
305
(46,27)
s Ps2 + ~ 2s=
dp 0
0
Ps2 =-
Ss The
above
relation
(46,27)
is
indeed
compatible
with
the
one
which
results
from (46,10): (46,28)
S 8ps2 + at
~o.~
p 2 = ~2s'
given the fact that IS ~-a~ + ~ . ~
+ ~ a~Ip=ps=
o.
THE GUIRAUD and ZEYTOUNIAN'S t RECENT RESULTS
The equations
(46,20)
corresponds
surfaces as discovered
to a kind of Froude blocking within isobaric
by Drazin (1961).
over any relief and must turn over
Such a blocked flow is unable to ride
it. This a serious drawback
of such a kind
of approximation
as is the fact that flows within two isobaric surfaces,
from each other,
are apparently disconnected.
higher
approximations
(46,25)).
but
only
in
A much stronger correction 1 A x = R x,
(46,29)
c46,3o~
parametric
way
is uncovered
(see,
is got in the far field.
for
at
instance
We put:
1 A y = ~ y
o
and we let: M ~ o
a
Some connection
close
o
while ~ and ~ remains 0(I). A
We find that:
f = ~cp~ + Mo~, ,.., = ~ ~, .,.
Again
~(p)
pressure,
is
but
the
L(p)
standard is
an
distribution
horizontal
wind
of with
pressure or, which is the same through ~o(p), be considered
separately.
The first
one deals
relief at all, even at finite horizontal See,
Zeytounian
and
Guiraud
(1984).
altitude, an
as
arbitrary
on altitude.
function
of
dependence
on
Two situations
with the case
distances.
a
when
there
must is no
We set up an expansion:
306
A
A
A
A
(46,31 )
~o ~'2+ " " " ;
7e and
the
following
vortex the
O
results
emerge.
which depends on the
field
90 .
vorticity
created
If
of
but
one assumes
A
The h o r i z o n t a l
distribution
As a m a t t e r
is
created.
...;
2
it
fact,
of vertical
Drazin's
may d e s c r i b e
that
this
velocity
model of
is
0
a
potential
vorticity
associated
is
to
how s u c h
distribution
9
unable
a
vorticity
vorticity
is
with
explain
how
evolves
one
localized
A
generates a
the
doublet
potential
of
information
vortex
potential
vortices
level
of
~
solution oft:
in
0
the
far
field.
A
and
which is not contained A
At t h e
9
both
9
One
finds
that
and
0
9
in the so-called
1
do
not
afford
and ~
2
plays
(48,1)-(46,3)
a
new f e a t u r e s
2
to the
aerodynamics
is
vertical
respect
analogous Euler
ones
concerned, flow,
quasi-solenoidal
any
quasi-solenoidal
occur.
One f i n d s
.
with
role
with respect incompressible
is
1
new
field
9.
to
be a
0
^
that
~
has
2
+
:
tx ÷ y ) ~
equation
9
A
+
This
it
A
the
(see,
(46,32)
the one
for
adiabatic played
instance,
rules
rules
the
the
We o b s e r v e
by
primitive the
acoustic
equations
Viviand
(1970)).
As f a r
phenomenon
phenomenon of that
equations
~o(p)
is
of
adaptation
adaptation
the
total
to amount
as to the of
contained in the isobaric surface p=constant and ^ which drives the potential vortex 9. o We h a v e b e e n d e a l i n g w i t h o n e o f two s i t u a t i o n s , the second corresponds to the ease
when there
which
to
acoustics
here
approximation.
vorticity
to
o.
~
is
i s some r e l i e f
(46,33)
A
A
9=
9+
at finite
distance.
Then one must put:
A 0
Ng+ 0
....
1
~ = M~+ 0
1
....
~ = H~+ 0
A
and
one
finds,
again,
that
-V°
...
1 A
is
a potential
vortex
but
91
is
no
longer
a
A
doublet
of
potential
vortices.
Rather
9
A
played
9
2
t We n o t e
and ~ that:
2
when t h e r e ~o(p)
was no r e l i e f .
= ~flo(t,x,y,p)dxdy.
1
and
~
1
play
now t h e
role
that
was
307 A
number
of
problems
arise
from
this
low
Mach
number
approximation.
blocking remains a mystery and we have only explained how blocking in the far field. generated
in
atmospheric
Related
the
lee
flows.
to that one would
of
mountains
Another point
low Mach number
approximation
(1980)
considerations
for some
understand p--~.
47
how the
the
concerns
near
the
about
low Mach number
Clearly further researchs
.
in
llke to understand context
of
how waves are
low
Mach
the three-dimensional
top of a relief
that
toplc).
approximation
number
nature of the
(see Hunt
Finally
First
is released
one
and Snyder
would
like
works at high altitude
to
when
are necessary.
FRACTALS I N ATMOSPHERIC TURBULENCE
Scaling
notions
scales
over
structures.
Meteorological
be
argued
In
the
notion
Richardson 1920s cascade been by
of
central ubiquity
fluctuations The
of
these
scaling
turbulent
of
from
"simple
scaling",
specify
the
the
scaling
said and
be
large
-
dimensions
scale the
themselves:
and
structures
are
ratio. can
The are
a
scale
regimes
equations
and
(or
scale
scaling
injection
characteristic
scaling
only
of
equations
of
fractal
to
involves
existence
energy
lack
of
atmospheric
of
are
of
orders in
of
of
of
often
only
a
small
the
to
small
k -s/3
order
scales
large
scale
viscous
scale
a fact
Since
Kolmogorov
be
Furthermore,
traced who
involving
that
of of
prediction,
dynamics
scales.
thousands possibility
scale.
can
weather
of the
in
atmosphere
atmospheric
turbulence,
the
allowing
magnitude
numerical
of
scaling
then, is
energy
to
in
the
a self-similar
scaling
most
back
ideas
notably
spectrum
have
expressed of
velocity
flows.
field
scaling
roughly
respectively,
9
father
large
studies
basic
-
dynamical
over
a model
velocity
the
is
that
the
scales
in atmospherical
develop
small
regimes
the
suggested
to
the
millimeters,
spanning
energy
system
scale
several
(1965),
also
the
spectra,
appearance
a
Dynamics,
law
occurs.
atmosphere,
regime
the
operation
largest
and
power
Navier-Stokes
dissipation
scaling
the
if
from the
the
kilometers a
range
Fluid
with by
the
a
directly
associated defined
and
a scale-changing
In
where
ranges,
with
precisely,
over by
associated
wide More
[nvarlant) related
are
~
affords
ideas.
since
it
of
all
The
occurs the
a
first when
prototypical scaling
only
statistical
one
of
example interest
parameter
properties.
with might
is
Assuming
which
to
be called
sufficient statistical
to
308 translation
invariance
and
isotropy
[including
reflection
symmetry),
the
fluctuations of the velocity depend only on the distance
ll{ between the points c47,1)
and
Avc ) =
19c + )
-
In this case, in dividing the scale ~ by A we reduce fluctuation by the factor AH: C47,2)
AVCUA) =d AVC~)/A";
H is the sense
(single) scaling parameter.
of
probability
distributions;
The equality hence
,=d,, is understood in the
the
scalin E
of
the
various
high-order statistical moments follows: (47,3)
= /A~(h)
where ~(h)=hH and <> indicates ensemble average. Since the energy spectrum is the Fourier transform of the covariance, we have a spectrum k -~
with ~=2H+I.
If one assumes that there exists a nonfluctuating density of energy flux to small scales (c) that is scale-invariant (the nonlinear terms in the N-S Eqs. conserve this flux,
while breaking up large eddies
into smaller and smaller
subeddies), then dimensional analysis gives (47,4)
AV
~
cl/3~1/3,
1 hence H=~,
5 ~=~.
@__~V~-2/3 (which diverges as ~--90), such behavior Note that since ax associated
with velocity fields
with singular
shears.
The
is already
problem
of
such
singular behavior was first discussed by Leray (1934). In the
1960s,
(1975)
for
Kolmogorov
summary
and
(1962)
and
development)
(1962)
Obukhov pointed
out
(see Monin that
and
scaling
Yaglom
generally
involves an infinite number of parameters: ~(h) is no longer linear in h. This is a richer behavior called "multiple scalling".The simplest way of expressing this
is to now consider e as a fluctuating quantity.
average
is independent of scale
(as before)
but
Its ensemble
spatial
is highly vriable
in each
realization of the cascade process. This variability (intermittency) is built
309 up
step
by
step;
large
eddies
~
modulate
the
flux
to
smaller
and smaller scales. Multiplicative
processes
phenomena,
including
just
as
one,
high-order
in
"outliers"
in
simple
the
data.
moments),
for
and
These
create fields
one
of
the
processes
the
are
easy
the
other
for
"multifractal" The
recent
to
simulate
Because a whole
of such multiplicative
averaged.
must
statistical
processes
depend
line,
plane,
or
therefore
speak
of
rather than "fractal" properties.
book
development
are
of
statistical
their
We
they
(not
(there are in fact two dual-dimension
fractal
which
orders
of
intermittent.
(for example,
over
interesting
divergence
phenomenon
not only on the scale but also on the dimension set)
of
of different
(which
that are extremely
properties
number
above),
related
singularities,
the statistical
a
described
is now involved
the
with
singularities
scaling
moments,
hierarchy of dimensions functions:
associated
a hierarchy
the
statistical
numerically),
are
by
Schertzer
of these topics.
Ecole Polytechnique,
and
Lovejoy
(1988)
on
NVAGI
gives
ample
The second NVAG workshop will be at the (former)
Paris, June 27 - July I, 1988.
DO DISSIPATIVE STRUCTURES IN FULLY DEVELOPED TURBULENCE FORM A FRACTAL SET?
What
is the
fractal
dimension
of this
set?
Answers
to
these
questions
are
interesting because they bring the theory of fractals closer to application to turbulence and shed new hight on some classical problems in -for
example,
the
growth
of material
lines
turbulence
in a turbulent
environment.
The
overwhelming conclusion of recent works t is that several aspects of turbulence can be described roughly by fractals and that their fractal dimensions can be measured. several
However, of
it is not clear how
its facets,
on can solve
problem of reconstructing self).
Speculations {
Although
actual
has yet
measurements
that
the
earlier
been
(up
the original
abound
made
given the dimension for
to a useful
set
several
suggests that these speculations, no effort
(or whether),
(that
~
leading
is,
of work
of
£or
example,
Sreenlvasan
turbulent
Mandelbrot
initiated largely by himself,
to put
in turbulent
them
shear
on firmer
flows.
The
and
Neneveau
(1986).
the
the turbulent
fully
ground paper
Meneveau (1986) is a first attempt at filling this gap. t See,
accurary)
of
inverse flow it-
flows
are
(1974,1975)
are plausible,
by ressorting Sreenivasan
to and
310 starting
with
turbulence mechanism
responsible
given
order
eddies
of
(196S),
of
(or size) the
stabllity energy.
Richardson
this
arise
preceding
and generate
has
It is well
- that
by Richardson,
in the 1920s,
in advancing
our undestandin8 description
of
turbulent of
that
theory of fractals t must be applicable
Mandelbrot,
for example,
facets"
claimed
turbulence-which turbulence-must are totally
has
It
all
a
that
appears
selfsimilar
with
a
basic
a
their
to which they transmit
their
to
at scales small
Reynolds size
number
enough
is unity.
Is of the order In a memorable
characteristic
measure
certain
of
it
(1941,1962),
Obukhov
has made remarkable
strides
that
to
they
the
are
"objects"
expectation
that
the
In the most basic sense,
" turbulence
along
involves
of
the
in
the
concepts
many fractal
geometric vast
aspects
literature
of on
from Euclidean geometry but,
with few hard results as yet".
dimension
objects, called
fragmentation
or
greater
than the object's D=LogN/Lo8(I/~),
each fractal
the ~
exponent
object which is made of N part,
of
rhyme
He has also led the way by his own investigations
it is strictly as
larger
-namely leads
involve fractals;
"they involve suggestions
a
flows.
investigation
ignored
of
of
to turbulence.
that
The
lose
verbalized
to the Euclidean dimension of classical
form
property
been
inadequate.
associated
which
a proper
necessarily
in his own words,
is
has remarked
that
of stability
(1948),
flows
scales-
developed
eddies
assumed
whose characteristic
of turbulent
that
are
is expected to terminate
(1945) and Weizs~cker
of a hierarchy
be
and cultivated by Kolmogorov
Onsager
fully
loss
turn
This theory of cascade,
(1941,1962),
and
of the in
that
to
lower bound on the scale
scale.
is this
these
thought
or scales of various orders.
of a smaller order
is, scale
the Kolmogorov
Analogous
been
is assumed
as a result
eddies
known that this
consisting
situation
order;
This reccurlng scheme
to be stable
It
it
consists of a hierarchy of eddies,
~
d
;
it
topological
object ~
has
the
dimension.
characteristic
each of which is obtained
of
a
from the
whole by a reduction of ratio e. Of course, other
a complete
quantities
how far the fractral scaling functions t
description
object
sets
See, M a n d e l b r o t ( 1 9 8 2 ) ,
to which Fractals.
of
demands
loosely
reference
must
be
made
For
a specification
speaking,
is from being dust-like-
only one of which is the fractal
and original
account
of fractal
such as &~caaan//~ -which,
of of
or the entire spectrum of
dimension. an
is a measure
enjoyable
311
It
is
known
turbulence helpful
(Batchelor is
~
is
the
turbulence, scale
of turbulence.
cubes
of LI=Lo/n
cubes
is
~/3
motion
simplest
Within
If we divide
of a
given of
this
according
ones
distribution
to
cube
a
unaltered.
field
certain
of
are
(fully
of smaller
in each of these smaller probabi i ity
similarity
or does not.
in this
developed)
(n>>l)
law. .
.
directly
of very
L ° is an integral
.
cubes of length L2=L1/n
This
sizes
rate
structure
arguments
into a number
extends
affected
is the binary one according
box either contains dissipation
small
arguments
length Lo, where
into second-order
reaches
the
scale-similarity
, the density of dissipation
distribution
until
that
scale-similarity
The essence
of these cubes
the probability
(1949))
that
a cube with sides
distributed
subdivision
of
it.
following.
consider
- -
Townsend and
in discribing
context
and
,
by
Further 1/3
leaves
to all scales viscosity.
The
to which a given high-order
It is this simple picture that we
shal I pursue.
AN UPDATE OF MANDELBROT'S WORK
Let
D
be
the
fractral
resolved the smallest cover
dimension
required
to
to its definition:
the
entire
~ = Log(L0/n) each cube has a volume
Since
~ =
is
this
meansVthat
global
(L^/~) 3-D
mean.
is contained
times
the
(du/dx) 2
Consequently,
have
D
can
be
calculated
( L J B ) 3, the total
volume
occupied
is
:
in these cubes,
global
in
the
the
average
dissipating
Kurtosis
--a
K
we
.
of the order
defined as
(47,7)
When
the number N of boxes of size regions,
or , N=
field.
-n
Since all dissipation them
dissipative
dissipation
by the cubes of active dissipation (47,6)
the
scales ~, and determined
according
(47,5)
of
pul' ~xj // f0ul ~xj 2
(or
the level of dissipation
value. cubes the
Assuming is
local
(Ln/~) 3-D
flatness
factor)
in
isotropy, times
the
of du/dx,
312
will
be
given
by
( L J ~ ) 2(3-D)
times
the volumes
occupied
by
the
dissipating
cubes. From t
:~'-2+D
C
we have:
(3/2) (3-~) Ke A
K = ( L / n ) 3-D o, "
(47,8)
where
u'A u ; A being
ReA=
streamwise flatness
velocity.
factor
of
the
If
we
(du/dx)
Taylor
mlcroscale
invoke
Taylor's
is the same
and
the
root-mean-square
frozen-field
as that
where now K is the Kurtosis of (du/dt),
u'
of
hypothesis,
(du/dt).
A plot
of
the LogK,
vs. LogRe A will yield the co-dimension
(3-~). M~ndelbrot from
Kuo
used and
available
this
argument
Corrsin
since then,
and Meneveau
(1986;
and,
(1971),
from an examination
estimated
and are plotted p.376).
D
to
This
dissipation
that
the slope
a D of 2,9.
Reynolds
More
have
to collapse
become
on a line
a version from Mandelbrot's
fractional
the
data
data
volume
~ = ( L / w ) 3-~
earlier
occupied
by
field is given by
For ReA<150, yields
means
2,6.
the Kurtosis
16 of the paper of Sreenivasan
The data may be considered
with a slope of 0.4, yielding a D of 2,73, estimate.
be
in Figure
of
in this
This
Figure
indicates
16 is decidely
either
numbers are less spotty or that
that
smaller
the dissipation
local
(=0,15), regions
which at
isotropy does not obtain.
low Both
are likely. As a final
remark
fractalike
behaviour
when
collection
of
perhaps
a
we note
that
it seems
viewed a
on
number
that
very of
turbulence
long
times
fractals
each
scales. of
different.
We think that this view can be reconciled
turbulence
now in vogue as an ensemble of semi-organized
t
Let
~
be an
dimension isolated ~asured.
~.
object
in three-dimensional
Let ~ a line
point-akin
element,
to the
Cantor
space
lntersectin
with
a £ractal
0 the
object,
discontinuum-
genuinely
its
Turbulence
is
is
slightly
roughly with the view of motions.
interface gives
whose dimension
which
loses
of
a set D
C
of
can be
313
4.
o~ ~
o~
~ ~
x o ~
~
.
314 REFERENCES TO WORKS CITED IN THE TEXT
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a n d TOWNSEND, A.A.
T.B.
(1966)
_ J.
Fluid
DRAZIN, P . G .
(1961)
HUNT, J . C . R .
a n d SNYDER, W.H.
KEULEGAN, G.H.
_ Tellus,
(1953)
(1941)
_ C.R.
(1962)
_ J.
LERAY, J .
a n d CORRSIN, S. (1934)
LONG, R.R.
(1971)
_ Tellus
17,
MANDELBROT, B . B .
(1974)
_ J.
MANDELBROT, B . B .
(1976)
_ J.
MANDELBROT, B.B.
(1982)
_
_ J.
Acad.
Fluid
_ Acta Math.,
(1986)
Fluid 51,
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London A199,239.
_ J.
13,
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96,
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671.
133-140.
82.
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1,
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Mech.
63,
Mech.
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Mech.
62,
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241-270.
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KOLMOGOROV, A.N.
Roy.
239-251.
(1980)
_ NBSJ.
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KUO, A . - Y .
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~ a ~
50,
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and
C o m p a n y , New-York. MONIN, A . S .
and
YAGLOM, A.M.
(1976)
_
a~ ~ a n / ~ , OBUKHOV, A.M.
(1941)
_ C. R.
OBUKHOV, A.M.
(1962)
_ J.
OGURA, Y. a n d PHILLIPS, ONSAGER, L.
(194S)
RICHARDSON,
L.F.
SCHERTZER,
D.
Fluid
N.A.
_ Phys.
(1966)
Acad.
Sci.
Mech.
(1962)
Rev.
68,
Y & ~
vol.
2.
~xt&/
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~ e c ~ :
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~ e c ~
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Mass.
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13,77.
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vol.
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173-179.
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Dover,
New-York. and
LOVEJOY,
S.
(1988)
~c~n~,~,
_
and
D. R e i d e l , SREENIVASAN,
K.R.
VIVIAND, H.
(1970)
WEIDMAN , P . D .
a n d MENEVEAU, C. _ J.
(1978)
_ Tellus,
WEIZSXCKER,
C . F . Von ( 1 9 4 8 )
ZEYTOUNIAN,
R. Kh.
(1974)
(1986)_J.
de M&canique,
_ Z.
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Non-~eaa
Hingham, Mass.
Mech.,vol.173,
pp.
357-386.
$73.
177-184.
Phys.
124,
614.
_ Archiwum Mechaniki Stosowanej,
26,
3,
499-$09
(Warszawa). ZEYTOUNIAN , R . Kh.
a n d GUIRAUD,
J.P.
(1984)
_
in:"Md~.
~ " .
~
~ e / n @ ~
Boole
J.J.H. Miller.
Dublin.
Press,
m
~
ed.
APPENDIX 1 BOUNDARY LAYER TECHNIQUES FOR THE STUDY OF SINGULAR PERTURBATION PROBLEMS
Throughout the discussion of Meteorological Fluid Dynamics we have made use of boundary layer techniques to approximate the solution of atmospheric equations with small (or high) parameters. Such methods are characteristic for the tools used to study certain singular perturbation problems
in Theoretical Fluid Mechanics.
For the reader who is
not familar with these methods we will give a short introduction to the topic of boundary-layer theory. A much more extensive treatement can be found in the book by Van Dyke (197S). The original idea, that a fluid of small viscosity could be approximated by an inviscid (perfect, for
non viscous) fluid in almost every spatial
narrow boundary
(190S)). outer
layers,
This treatement
expansions",
or
•
was first
presented
by
has been give the name, in
more
recent
Prandtl
region except (see,
"the method of
literature
"the
method
Prandtl inner and
of
matched
t
asymptotic expanslons ". Altghough these methods have not yet been shown to be completely rigorous in all situations, boundary layer theory have proved to be extremely important in the
analysis
of
the
Navier-Stokes
equations
(see, for
example,
Zeytounian
(1986 and 1987)). In general, it may be stated, & c a a d ~ - & z ~
~ A e o ~ S s ~
~ .
In this Appendix 1 we discuss perturbative method for solving a differential equation whose highest derivative is multiplied by the perturbing parameter s<
layer
is a narrow region where
equation changes rapidly.
By definition,
the
solution of
a
differential
the thickness of a boundary
layer
must approch 0 as c--->O. In this Appendix 1 we will concerned with differential equations
whose
solutions
regions of rapid variation.
exhibit
only
isolated
(well-separated)
It is possible for a solution to a perturbation
problem to undergo rapid variation over a ~
region (one whose thickeness
does not vanish with s--~). However, such region L~ r~at a boundary will consider such problems in Appendix 2. t
HMAE.
narrow
layer.
We
316 There are two standard approximations In the aaZe~ region varying,
so
multiplied but
it
is
by c.
the boundary
functions Thus
we
simpler
valid
to
neglect
Inside
a boundary
layer
is so narrow
of the differential can
that one makes
(away from a boundary
replace
a
approximate
layer)
any
solution
derivatives
of
layer the derivatives that
y(x) y(x)
theory.
is slowly which
of y(x)
we may approximate
are
are
large,
the coefficient
equation by constants.
single
differential
equations
equation
in each of several
every region the solution of the approximate unknown constants
in boundary-layer
the
of integration.
boundary or initial conditions
by
inner
a
sequence
and outer
of
much
regions.
In
equation will contain are or more
These constants
are then determined from the
using the technique
of asymptotic
m ~ .
LARGE 0 and SMALL o. ASYMPTOTIC SEQUENCES AND EXPANSIONS
Consider a family of boundary value problems ~ C depending c (<
Under
many conditions,
well-known
"method
of
a solution
perturbation"
ye(x)
i.e.,
of ~
as
a
on a small parameter can be constructed
power
series
in e
by
with
first term Yo being the solution of the problem ~o" When
such
an
expansion
perturbation
problem.
this regular
perturbation
converges
When ye(x)
as
e--~
~
b% x,
doe~ ao/ have a uniform
we limit
method will fail and we have a ~
have
a regular
in x as c-~/3, perturbation
problem. Below,
it will be convenient
to use the Landau order symbols 0 and o which are
defined as follows: Given two functions
f(e) and g(e),
f = O(g) (AI,1)
as c---~
f(e) ~
if
we write
is & o ~
as c--)O.
We write t f
(AI,2)
t
Often,
if
f<
is
= o(g)
f(e) ~
used
as
a s c--~O
--~ 0
an
a s c--)O.
equivalent
notation.
317 Consider
now a sequence
is called an ~
{~ (e)} n=1,2 ....
of e.
of functions
n
Such a sequence
aequ~nr.e if
(A1,3)
~n.i(e)
= O(~n(e))
a s e---~O
for each n=1,2 . . . . . A sum of terms of the form N
a (x)a (e)
n=l
is called
an e ~ m ~ m Z t c
infinite)
a s e----~ w i t h r e s p e c t
~
of the function
to the sequence
f(x;e)
{~ ( e ) }
to N terms
(N may be
if
n
M
(A1,4)
f(x;e)
-- ~, a n ( X ) ~ n ( e ) n=l
= 0 ( ~ M)
aS
(:---90
for each M=I,2 ..... N. if N=~,
the following
(A1,5)
notation
f(x;e)
~ ~a
n
n=l
Clearly,
an equivalent
is generally (x)~ (e)
used
as e--~.
n
definition for an asymptotic
expansion
is that
H-1
f(x;e)
- ~.an(Xl~nCe)
= 0(~ M)
as e--43
n=l
for each M=2
. . . . .
N.
I N T U I T I V E APPROACH OF THE HHAEt
In
any
perturbation
natural
0
x
ranges
asymptotic
example,
small
positive
a s e--~O,
over
bounded)
domain
sequence
{eJ})
some
(usually
(often
problem,
such
the
conditions
physical
power
an expansion
this solution may fail
in applications,
The
a
YeCX) ~ ~ a j C x ) ~ l ( e ) J=o sequence
perturbation
t
involving
parameter
e
it
is
is
an
to seek a solution of the form
(A1,6) where
problem
cannot
to satisfy all
considerations
will
~ as
be v a l i d
and e--~.
{~j(e)} In
uniformly
boundary conditions often
indicate
which
are so omitted).
following
discussion
is
taken
from
the
a
book
of
O'Nalley
(1974).
singular in
x;
for
(moreover, boundary
318 Instead, the expansion yo will be generally satisfactory in the "outer region" away from (part of) the boundary of ~). It will be called an eu/e~ asymptotic expansion
(or outer solution).
convergence,
In order to investigate regions of nonuniform
one introduces one or more stretching transformations
(AI,7)
~ = ~#(x;c)
which "blow up" a region of nonuniformity neglected boundary conditions,
(near a part of the boundary with
for example).
Thus x = -~,
~>0,
C might be used for nonuniform convergence at x=0. Then if ~ is fixed and c--~, x--K), while if x>0 is fixed and ~--~, ~ . Selection of correct stretching transformations by
physical
considerations,
or
is an art sometimes motivated
mathematically
(as
in
Kaplun's
concept
of
principal limits). In terms of the stretched variable ~, one might seek an asymptotic solution of the form CA1,8)
ylc(c) - ~ bjCc)~j(c) j=o
as c--)O
where the sequence
{~t (c)} is asymptotic as c--~, valid for values of ~ in J some "inner region". This will be called an inner asymtotic expansion and this inner expansion often accounts for boundary conditions neglected by the outer
expansion. expressed
The
inner region will
in terms of the outer
generally variable x.
shrink
completely
Hence,
the
as
e--)O when
inner expansion
is
In most problems, it is impossble to determine both the outer and the inner 0 i expansions ye and Ye completely by straightforward expansion procedures. Since both
expansions
original
problem
asymptotically in different regions, one might attempt to mo/cA them,
i.e., to
formally relate
should
represent
the
the outer expansion
yO
inner expansion yl in the outer region:
solution in the
of
the
inner region:
(yO)i and
the
(yl)O through use of the stretching
~=~Cx;c). The rules for even formally accomplishing this, in all generality, can be very complicated
(see,
through
of
use
an
Fraenkel
(1969)).
~seab~a domain
an
Justification ~
in ~
particular is
examples
difficult
Lagerstrom and Casten (1972) and also the book of Kevorkian and Cole
(see, (198i))
319 and,
in general, there is no apriori reason to believe that an overlap domain
(where matching is possible) exists. Once matching is accomplished, however, the asymptotic solution to well-posed problems becomes completely known in both the inner and outer regions. Frequently,
it
is
convenient
to
obtain
a
cem4~
~
y~
u~q~vun2~
in ~. One method of doing so is to lett c
(AI,9)
o
_( i)o
I
Y~ = Yc + Yc
making
the
appropriate
Ye
'
modification
if
several
regions
of
nonuniform
convergence (several inner regions) are necessary. We refer the reader to the cited references for more details of this important technique and many examples of its application.
ASYMPTOTIC HATCHING PRINCIPLE
One general asymptotic matching which involves the intermediate limits is that due
to Kaplun
(1967).
Another widely used asymptotic
that due to Van Dyke (1964). matching
principle
distinction
between
is
is
The amount of work involved in using Van Dyke's
considerable.
the
matching principle
idea
of
Fraenkel
(1969),
overlapping
principle ceano~ derive justification for
and
the
who
probed
asymptotic
into
the
matching
itself from Kaplun's principle of
intermediate limits. Further, there is a possibility, as Fraenkel demonstrated, that
Van
Dyke's
matching
principle
can
fail
when
the
asymptotic
sequence
contains logarithms. In order to relate the outer solution (AI,6) and the inner solution (AI,8) to each other,
one presupposes the existence of a region of overlapping,
where
the two expansions are valid. Kaplun and Lagerstrom (1957) admitted that there is no
apriori
reason for
the
regions
of
validity
of
the
inner
and
outer
expansions to overlap. Therefore, the results should be assumed to be apriori asymptotically correct. First we note that the a/mpZeat asymptotic matching principle due to Prandtl (1928) states x (AI, IO)
Lim y[ = Lim y~ , x-~O ~--~
where ~ -
$ Note that ( y )
Ey E so that ( y ) i = y C. Likewise, in the outer region,
----YE"
320
Another
widely used asymptotic
matching
principle
is that
due to Van Dyke
(1964), which states: I [The m-terms inner expansion of (the n-terms outer expansion)] = [The n-terms outer expansion of (the m-terms inner expansion)], where m and n are any two integers. on writes
n terms of the outer
Thus in order to find the left hand side,
expansion
in terms
of the
inner variable,
expands it for small e keeping the inner variable ~ fixed, and truncates the resulting expansion after m terms; and similarly for the right hand side. We proceed to give here t a variant form of asymptotic matching principle which is more expeditious precisely, side
in this context.
One still writes here (AI,IO). But more
the inner limit of the outer expansion,
in (AI,10)
series about
essentially
implies,
the inner boundary.
which is what the left hand
is represented
That
is,
by a formal
in the neighborhood
Laurent's
of x=O,
one
writes aj(x) = aj(O) + x d ~
x=O + O(x2)' j=1,2 .....
or x dao )] O(e2), y3(x) ~ ao(O) + c[~ ~ x=O + al(O + x with ~j(e)m£ j, and written in terms of the inner variable ~=~ (~I),
da° x=O +
(AI,11)
y3(x) ~ a O(0) + E[~ ~
It is clear that this principle basic principle due to Prandtl.
al(O
)]
+
O(e2).
is simply a rational
generalisation
of the
Note that this principle puts a less stringent
restriction on the domain of validity of the outer solution in that the latter is required to extend merely to the neighborhood of the inner boundary whereas the basic principle
due to Prandtl
requires
the domain of validity of the
outer solution to extend right up to the inner boundary -a probable
source
of the difficulties
higher order problems.
the
latter
method
develops
at
the
This may also be the reason why the present principle
succeeds where Prandtl's principle fails. Formally one may enunciate the present principle as follows: H[The n-terms formal Laurent's series expansion of the outer expansion about a
~the
inner boundary written in terms of the inner variable] = [ The
,formal outer limit of the inner expansion]. t
According t o Shivamoooi (1978).
n-terms
321 A SIMPLE BOUNDARY-LAYER PROBLEM
(AI,12)
[cy"+
[
y' + y = O,
y(O) = a,
Osxsl, c<
y(1) = b.
The exact solution of the two-point problem (At, 12) is S
(Al,13)
yc =
(a
e
2 _
b)e
S X 1
S +
(b
-
a
e
1
)e
S X 2
where -1
±
(1
S1,2 =
- 4 c ) */2 2~
One approximates
(A1,13) by
(A1,14)
ye = be l-x + (a-be) e x e -x/e + OCe)
-a form
which
is uniformly
valid
throughout
Osx~l
as
e-->O. Note
expansion cannot be obtained keeping either x or x/e fixed.
that
this
In the former case
one obtains yeo = bel-x + O(e)
for x>O
which is not valid in the boundary layer near x=O since y~(O) = be
a.
In the latter case one obtains y~i = be + (a_b)e-Xl~ + O(c), which is not valid as x--~l. This
suggests
that
we
represent
the
solution
by
expansion using the variables x and ~ m ~ -this E
different related matching
to
asymptotic each
other
principle
representations in a rational (that
besides
of
manner
-this the
Thus seek an outer expansion N-1
y°(x;e) ~ ~ an(X)E n÷ O(e N) n=o
different
ao2te function,
makes
determinate).
(AI, 15)
the
two
is the MMAE.
leads two
to
asymptotic
Since they are they the
different
should
be
asymptotic problems
322 where in accordance
with the outer
limit process one has m-1
o Ye - ~ a (x)e n
(AI,16)
n--O
a ( x ) = Lim e--)O
m
n
m
x fixed Substituting
equal
(AI,15)
powers of e,
(A1,17)
in equation
(AI,12),
and
equating
the
coefficients
of
one obtains
a 'o+
The o u t e r s o l u t i o n
a =o
O,
a ~ + a l -- - a "o '
is valid
etc.
everywhere except i n t h e r e g i o n x=O(c) t ,
so t h a t
one h a s :
(A1,18)
ao(1)
= b,
at(l)
= O,
etc.
Thus one obtains
(Al,19)
a (x) = be l-x, o
a (x) = b(1-x)e l-x, 1
etc.,
so that 0
Ye
(A1,20) For small except
c,
b[ I+c(l-x) ]el-X+ O(e2)"
this solution
in a small
rapidly
is close
interval
at x=O,
in order to retrieve
to the exact solution where
the exact
the boundary condition
(AI,14)
solution
everywhere,
(AI,14)
changes
there which is about
to be
lost. In order
to determine
magnifies
an expansion
the independent
valid
in the boundary
layer,
x=O(e),
variable as
-x E
~= so that equation
(AI, 12) becomes
(A1,21)
d y dy -+ + e y = O. d~ 2
2
Seek now an inner expansion
(A1,22)
t Since general,
yi(c~;c)
the
system and
one
(A1,17) of
these
N-I = ~. b n ( ~ ) e n + O(c N) n=O cannot boundary
take
on
both
conditions,
of
the y(O)--a,
boundary should
conditions, be dropped.
in
one
323
where in accordance with the inner limit process m-1
!
ye - ~ bn(E)en (AI,23)
bmC E) = Lim
n=o Em
E fixed Substituting
(AI,22)
in equation
(AI,12),
and
equating
the
coefficients
of
equal powers of e, one obtains d2b db ~ + o = O, dE 2 dE
(AI,24)
d2b db ----!I+ ----!I= -b o, dE 2 dE
etc.
Noting that the inner solution is valid only in the region x=O(e), (A1,2S)
bo(O) = a,
one has
bl(O) = O, etc.
Thus one finds
(AI,2S)
I bo(E) = a - Ao(l-e-E) ; b1(E) At(I-e-E) - [a - Ao(I+e-E)]E; etc.,
so that
(A1,27)
y~ = a - AO(1-e-E) + e{AI(1-e-E) - [a - Ao(l+e -E)]E} + O(e2).
Asymptotic matching p r i n c i p l e ,
according to (A1, 11), s t a t e s
be + c[be-be E] + O(c 2) = (a - A O) + c[A I- (a-Ao)E] + 0(c2), from which one has immediately (AI,28)
A = 0
a
-
be,
A = I
be
so that
Note that writing the outer expansion E=x/~,
(AI,20)
in terms of the
then, we have 0
Y~ = bee -CE + e(l-cE)bee -~E + O(e 2} which leads to the O(c 2) approximation
inner variable
324
{A1,30)
(y2)i ~ be[1 + c(l-~)].
Analogously, variable
writing
x for x>0,
the
inner
expansion
(AI,27)
in
the e -~ terms are asymptotically
terms
of
negllgible
the
outer
and we have
the O(c 2) approximation (AI,31)
(y~)O e (a_Ao)(l_x) + cAl.
Since ~=~, matching {to this order) wlll be accomplished by selecting a-A °= be ~
A °= a-be, AI= be,
which agree with (AI,28). Exressing
all approxlmatlons
in terms of the outer variable
x, then we have
the composite approximation y~(x) = [bel-x+
(A1,32)
(a_be)(l+x)e-X/C] + s[(l_x)bel-X_ bee-X/C],
which should be compared with the exact solution previously given. The inner and outer expansions are depicted in Fig. i. The asymptotic solution (see, Fig. 2) follows the inner solution near x=O and the outer solution x>O.
i
0
1
i
0
Yc ~
C
y "+ Y
'
4-
^ y=u, y(O)=O, y(1)=1;
c=0,1.
for
325
0I
.5
1.
ey"+y'+y=O, y(O)=O, y ( 1 ) = l /x~ the aagu2/.aa 1-x e ,~ tae ~ e ~ . e d ~ u ~ .
x
326 REFERENCES TO WORK CITED I N THE TEXT
FRAENKEL, L.E. KAPLUN,
S.
(1969) _ Prec. Cambridge Phil. Soc. 65, 209-284.
(1967)
_
In
~guLd
~
Lagerstrom,
~
L.
and
~O~qa/~2~
N. Howard,
and C.
S.
~
~
Liu,
(P. eds.)
A.
Academic
Press, New York. KAPLUN, S. and LAGERSTROM, P.A. KEVORKIAN,
J.
and
COLE,
(1957)
J.D.
J. Math. Mech. 6, 585.
(1981)
_
~ ~
~
~
~e2Aad
O'MALLEY,
P.A. and CASTEN, R.G. R.E.,Jr
(1974)
_
~
dp~
.
Springer-Verlag, LAGERSTROM,
in
New York.
(1972) _ S.I.A.M. Review 14, 63-120. ~
~
~
~
~
.
Academic
Press, New York. PRANDTL, L.
(1905) _ Verh. Int. Math. Kongr. 3rd, pp.484-491.
Tuebner, Leipzig.
PRANDTL, L. (1928) _ NACA TM-452. SHIVAMOC~I, VAN
DYKE,
B.K. M.D.
(1978) _ ZAMM, S_88, 354-3S6. (1964)
~
~
~e/hoz/~ 0%
~guZd
~echon/c~.
Accademic
~hdd
~ ~ .
Parabolic
Press, New york. VAN
DYKE,
M.D.
(1975)
_
~
Press,
ZEYTOUNIAN,
R. Kh.
(1986)
_
~
~etAod~
~
Stanford.
~e~
~ox~ge~
~b//xle~ I.
~
de
Springer-Verlag,
ga
~dcan~
Heidelberg.
de~
Lecture
Notes in PhFsics, vol. 24S. ZEYTOUNIAN,
R. Kh.
(1987)
_
~e~
~odege~
~taide~ II.
d
~
Sprlnger-Verlag,
Notes in Physics, vol. 276.
de
~
~dca~
Heidelberg.
de~ Lecture
APPENDIX 2 TWO-VARIABLE EXPANSIONS
Various
pb~vsical
disturbance
which,
non-negligible In solving construct
problems
are
because
of
an expansion
the main
which
active
over
feature of the method expansion
of
process expansions calculated
by
the
is to combine
is uniformly
central
used
in the
are not applicable, the
Pepeated
pPesence
of
long
time,
a
over
a
small has
a
techniques
to
application
of
long time
intervals.
for long times,
successive
illustrating
appropriate
previous
fop the calculation
elementary example
the effect of a small
valid
is the nn~% e ~ ,
type
rules must be established A typical
being
the
cunu~effect.
a given problem,
process
be
by
characterized
Appendix
i.
A
of a limit Since
limit
terms in the solution cannot
limits,
and
mope
importantly,
of these terms.
the kinds of problems
which arise is
linear damping on a linear oscillator.
We have the following problem: d2y dy -+ 2 e ~ - + y = O, dt 2
(A2,1)
d_~
y(O)=O,
=1, t=O
where c<
phenomena described by equation
as can be seen clearly
e
(A2,2)
(A2,1) occur over two-time
if the exact solution
scales
is written:
-et
y -
sin(V'l-c ~ "
t).
V/l_c 2" For
this
example
obtained
only
if
(A2,3) and
in
this
consider
the
expansion t ~ limit
it
evident
y(t;e)
= sint - etsint + O(c2t 2)
exact
of e -zt
O
nontrivlal
is expanted
approximation,
process
that
y(t;c)
and ~%t~e.
interval
is
the
solution to this Thus,
expansion
order.
this as
terms
can
be
in powers of c as follows
amplitude
(A2,2)
correction
decays
linearly
we see
that
the
Clearly,
this
expansion
regular discussed
perturbation in
Appendix
c can be chosen sufficiently
small
with
amplitude
is I.
(l-ct)
is only
identical Fop
time.
any
to
If we is the
valid
fop
an
inneP
finite
time
so that solution
(A2,3)
is
328 a good approximation Because in
~t
in this and
(uniformly
limit
e -et
the
results
e2t,
valid
to O(e) of the
and sln(v/1-g are
2" t )
are
not
uniformly
is associated
with the
exact
result
approximated valid
of
(A2,2).
by power series
over
the
entire
time
interval. Therefore,
this
expansion
is
only
example,
initially
valid
the
first
representation
for
large
secular
of
the
terms
representation It
is
wish
also to
that
the
process
tends
to
by (A2,2)
Any g e n e r a l
infinity.
the solution
is the
exact
will
occur
next
from
as a general
contradictory ~ t)
T = (A2,4)
e--dO,
Another
way o f
to
the
In
this
nonuniform
solution. the
Mixed
nonuniform
~=et
for
long
long
times
fixed.
saying
will
times.
as t h e
is
if
In
is e -et
However,
this
arise
that
fact,
itself;
this
limit
argument the
we
of
function
expansion.
must, of the
asymptotic
for
: s I n rq'l-J, I c E]
t)
is proposed
I n=2=
uniformly
~
1 + ~ ~ne
due
requirements
of e -ct
process
of the
in
term.
term in (A2,2).
The e s s e n c e
as a function
ctslnt
term
does not have an outer
method that
the
we e n c o u n t e r
sin(__
for
two d i f f i c u l t i e s .
expressed
2 t}
of
e -¢t
representation limit
cog e x i s t
defined these
need
presence
the
and stn(v/1-e
therefore
we
the
emtcost
mutually
e -et valid
sine
of
of the sin( l~-e
represent
to
term
times form
the only uniformly
the
due
mixed a e ~
evident
process
t fixed
c--)O,
and
limit
be able
to
two-variable
expansion
cope
simultaneously
method is
where each
with
to represent
term can be uniquely
two time variables N
}t
= "fast
time";
= ct = "slow time", where the ~
n
are unknown constants.
TWO-VARIABLE ANALYSIS
In
addition
("fast-tlme")
to
the
("slow-tlme")
variable
x.
variable ~ ranging over an unbounded
one
introduces
interval.
nonuniform convergence occurred at X=Xo, the fast variable
another
If, for example,
329
,f
(A2,5)
~ = E
X
f o r some p o s i t i v e Selection scales
g(s)ds
0
g m ig h t be a p p r o p r i a t e .
of a proper fast
occurring,
variable
may be b a s e d on d i f f e r e n t
physical
may be m o t i v a t e d by s i m p l e model e q u a t i o n s ,
times
o r may be l e f t
somewath arbitrary initially. One seeks a solution y(x,@;e) which is a function of ~ variables.
The original
differential
the slow and fast
equation becomes a pa/d/a~ differential
equation in the variables x and W and an asymptotic solution of it is sough having a power series solution (A2,6)
~ ~yj(X,~)C j
y(x,n;~)
a s £--)0,
J=O where t h e c o e f f i c i e n t s Substituting coefficients for not
this of
likes
the coefficients suffice
~uu/edaex~
a r e bounded f o r O-<x~l and f o r a11 0>-0.
expansion
to
into
the
powers o f e, yj(x,~).
determine
condition
we o b t a i n
equation
partial
Yj
t
term
yj
successively.
additional
S,
and
differential
Applying the boundary conditions the
on
differential
In
conditions
equating equations
will
generally
addition
must
be
to
the
imposed t .
Following Poincar~, one usually asks that certain "~ecx~a~ terms" (like, e.g., ~ke-~,
k>O)
be
~
.
These
somewhat
arbirary
requirements
can
be
motivated mathematically. We note that it was necessary to determine the form of the second-order coefficient Yl (by boundedness and ~
conditions)
in order to completely obtain the first coefficient Yo" This is typically the case of for two-time methods. It
is
important
coefficients
to
arise
also
in
a
observe
variety
of
that
equations
physical
with
slowly
applications
and
varying
engeneering
approximations. Two-time techniques are well suited to such problems. As an illustration of the two-variable method, consider the two-point problem e
(A2, 7)
dZy
+ a(x)~--~ y + b(x)y
dx2 y(0)
= ~,
where a(x) and b(x) are t
We
do
is
a
= O,
not
expect
generalized
arbitrariness.
to
y(1)
= 8,
infinitely
data r mine
asymptotic
the
differentiable y
expansion,
s
uniquely, but
functions since
we
need
the to
on
O-<x~l and
expansion
sought
eliminate
some
330 a(x)>O
there.
previously,
By
analogy
to
the
constant
coefficient
problem
the fast variable t (A2,8)
n = ~
a(s)ds 0
might be appropriate. Proceeding with this ~ requires
the solution y(x,~;c)
to satisfy the
differential equation a2(x)
a2Y
+ By] ~-~ + 2a(x)
[ a~ ~
(98x4~9+ ~da ~@y+
ay a(x)~-~-
(:32y + b(x)y + ~., = 0 Bx 2 since
d
dx
a
--
a(x) - -
+
ax
c
a --
O~
and d2 __
02 --
+
dx 2
$ The c o n d i t i o n are
easily
Under
the
the
@x 2
2a(x) ~
under which b o u n d a r y
deduced
from
a
transformation
equatLon
(A2,7)
B2 axaw
layer
canonical
form
theory of
a2(x)
da 1 a --- - dx c @W
+
@2
+
c2
(Appendix
a 2
can
be a p p l i e d
~2221dx2/t~ ~ r u ~
a s c--90
equation
1)
(A2,7)
.
1 x y(x) = e x p [ - ~ a ( s ) d s ] Y ( x )
takes
the
form
d2y /a2(x) + ida dx
2 - [4~
A s u f f i c i e n t c o n d i t i o n for t h e is merely
}
~-~
existence
of
b(x)
~#l~e
Y = O.
that
a(x)~O, Then i t a l w a y s i s p o s s i b l e inequality
to
find
da ~-~
<m, b(x)<m,Osxsl
sufficiently
small
a2(x) 1 da 4 ~ + 2 d--x- b(x) > O, is s a t i s f i e d
treated
one might expect nonuniform convergence as e--X) near x=O and that
and
in this
case
d2Y/dx 2 > 0
when
Y>O.
E
O~x~l,
so
that
the
331 Subst itut ing t he expansion y(x,n,e) ~ ~ Yj(X,n)e j j=o into this differential equation, we formally equate coefficients of each power of e separately to zero. Thus, from the coefficient of c' I_ we have aZ(x)
['.°__1 - -
÷ OYe
On 2
= 0
On
and, integrating, we obtain
(A2,9) where A
Yo(X,n) = Ao(X) + Co(x)e-n, o
and C
o
are undetermined.
Likewise,
from the coefficient
o of c , we
have 2
2
a2(x) [ a_yl + __aYl] = _2a(x) a_yo On 2 On OxOn
da aYo dx On
a(x)aYo _ b(x)Yo. Ox
Integrating with respect to n, then, a2(x)[ OYl .... + Yl] = -2a(x)OY° an
ax
da
.x,o fl L
ax
b(x)Yo]dn
and, substituting for Yo' we have
On +
for A
1
arbitrary.
L
" (x)
dCo dx
+
dx
da
~^~-cCo - b(X)Co]e-n = a2(x)A1(x)
We then integrate with respect to n. Since Yl must remain
bounded as n-->~ (i.e., as e--gO for x>O), we must have (A2,10)
dA a(x) --~0 + b(X)Ao(X) = O. dx
Likewise, to avoid a secular term in Yl which is a multiple of ne -n, we must have (A2,11)
dC a(x) ---~° + da a-~ Co - b(x)C o = O. dx
Thus, integration implies that Yl has the form (A2,12)
Yl(X,n) = A (x) + Cl(X)e-n
332 where A 1 and C 1 are so far arbitrary.
Applying the boundary conditions to
Y = Ao(X) + Co(x)e-n + e ( . . . ) , noting that
e -~ is a s y m p t o t i c a l l y n e g l i g i b l e at
x=0,
implies that
we must
select (A2,13)
Ao(1) = ~
Summarizing,
y ( x , ~ ; e ) = Ao(X) + Co(x)e-~ + e[AI(X) + C1(x)e -~] + 0(e 2)
where
Ao(x) = /3 exp[-j~ b(s) ds],
Fo- o O ]expEfo
(A2,15)
1
Co(0) = ~ - Ao(0).
then, we have begun to develop an asymptotic solution of the form
(A2,14)
and A
and
and C
1
,
are, as yet, undetermined.
LINEAR OSCILLATOR WITH SMALL DAMPING t
In finish oscillator.
discussion
of
two-variable
method
by
studying
(A2, t6)
y = Fo(T,~) + eFi(T,~) + e2F ( T , t ) + . . . .
involving the fast and slow times T = (i
+ e2~ + e3~ + . . . ) t , 2 3
We then use the chain r u l e to c a l c u l a t e According
damped
linear
We consider the problem (A2,1) and we assume that the solution has
a general asymptotic expansion of the form
t
the
to
Kevorkian
and Cole
(1981).
~=et.
333
aF 8F aF aF OF dy 0(1 + cZ~ ) + c 0 + c I + c2__~I + c 2 2 + O ( e 3 ) , dt aT 2 BT a'r oT o-c
(A2,17)
O2F
d2y -
-
+ 2C2~2
)
+ 2C
c3"r2
dt 2
02F
02F 0(1
0
(:3"rat
(A2,18a)
a2F ~(Fo ) = o
(A2, 18b)
a2F aF ~(F I) = -2 ____~o_ 2._~o,
C
02F
2
- -
2
(3.r2
a-r 2
02F
+ 2c2
1
aTa~
+ 0(c3).
ape
n
+ F = O, o
aw2
aTaT
aw
82F
02F
~(F2 ) = _2w2
(A2,18c)
1.
aT 2 +
Thus, the sequence of equation for the F
02F
+ C 2 .____.£0 + C
o
a2F 0
2
aT2
aw 2
OF
1
0
-2
oTaT
aT
OF -2--
1
o~
etc. The
first
of
these
is
the
equation
for
the
free
oscillations,
remainder have the appearance of forced
linear oscillations.
Fo=Fo(T,T),
which are the solutions
the free
linear oscillations
have the possibility of being slowly modulated.
(A2,19) According
while
Howevere, to
since
(A2,18a)
Thus, we have
FO(T,T) = Ao(T)cosT + Bo(T)sinT. to expansions
(A2,16)
and
(A2,17),
the
initial
conditions
of the
problem (A2,1) become
(A2,20)
Fo(O,O) = O,
aF °(0,0) = 1, aT
F (0,0) = O,
aF aF 2(0,0) = - ~°(0,0), aT at OF
F2(O,O) = O,
2(0,0) = -
aw Conditions for F
(A2,21)
the
0
aF
OF I(0,0) - ~ °(0,0). at 2 aT
yields initial conditions for A A(O) 0
= O,
B (0)= 0
0
and B : 0
i.
Nothing more can be found out about Ao(T) and Bo(T) without considering F I.
334 Substituting for F (A2,22)
into the right-hand side of equation (A2,18b) gives:
0
~(F i) = 2
+ A ° slnT - 2
dE
o + Bo cosT.
dE
The bracketed terms on the right-hand side of equation of [ on~.
Therefore
the particular
solutions
(A2,22) are functions
corresponding
to these
terms
would be functions of t multiplied by the mixed secular terms TsinT or TCOST. Such terms cannot be permitted to occur in the solution because they lead to unboundedness
with
respect
to
argument mentioned earlier:
T.
Alternately,
we
can
use
the
consistency
A term like eTsinT is inconsistent with a unique
F % since it could equally well be relabelled [sinT + 0(c 2) and would become 0(I) instead of being O(e) as assumed. Therefore,
we must eliminate
all
homogeneous
solutions
of ~(F%)=0 and
give the two first order ordinary differential equations for A dA --
o
dE dB
o
dE
+
A =
o
O,
A (0)
o
+ B = O,
=
this
and B : o o
0
Bo(O) = 1
o
and we find that
(A2.,23)
Ao(~)
= O,
Bo([)
= e -~
The uniformly valid expansion thus far is N
(A2,24) Comparing e-¢tsint
y(t,e) : e-tainT + e{AI([)cosT + BI([)sinT} + 0(E2). this is
with the exact
indeed
solution
the corFect
(A2,2),
uniformly
valid
we see
that
the first
approximation
of
term
the exact
result to 0(I). Now,
Ai(~),
Bl(t) and
the frequency
shift
w e are
to
be found from similaF
considerations applied to (A2,18c) for F . Thus far, we have 2
" Fo= e-tslnT'
FI = A i (~)cos~ + B i(t)sinT,
and (A2,25)
~(F2) =
dAi + A i dE
+ (2~2+ l)e -t
inT - 2
dt
i + BI
osT.
First, repeating the argument that homogeneous solutions of ~(F2)=0 cannot be permitted,
we must set the bracketed terms in (A2,25) equal to zero.
Solving
335 the
resulting
equations
for
A 1 and
B 1 subject
to
the
initial
conditions
AI(O)=BI(O)=O (which follow from (A2,20), for F 2) we find
(A2, 26)
= -~
Al(~)
~]te
,
Bi(~) = O. N
This means that cF
would be proportional
1
to cte-tcosT.
Again,
such a term
cannot be consistent because it can also be written as c2e-~TcosT + O(c s) and shift to O(c 2) in the expansion. One could also have required that
IF2/FII be
bounded to disallow such a term. Therefore, we must set
(A2,27)
1
2"
~z
All the necessary reasoning has now been explained to carry out the solution to
any
order
and,
in
fact,
to
solve
a
wide
variety
of
weakly
nonlinear
problems having the form ~+y+cf dt 2
=0.
For the problem (A2,1), the result (~2,28)
is
y(t,e)
= exp(-ct)
s e e n t o be t h e u n i f o r m I y v a i i d
sin
1 - g-- + O(c 2) t
general
asymptotic
+ O ( e 2)
expansion of
the
exact
solution to O(c) for times t of order i/~. For
a general
Chapter 6).
discussion of multipie scale
methods
see also
Nayfeh
(1973;
336 REFERENCES TO WORK CITED I N THE TEXT
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J.
and
COLE,
J.D.
(1981)
_
~
e
~
~ a ~ .
~ e ~
Springer-Verlag,
New York. NAYFEH,
A.H.
(1973) _ ~ e a ~
~e~.
Wiley,
/~
N e w York.
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Special
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issue J.M.T.A,
AUTHORINDEX*
Argoul, F. 278
E s k i n a s i , S. 11
Arneodo, A. 278
Estoque,
B a r c i l o n , V. 199,232
Feigenbaum, M.J. 2 4 2 , 2 4 8 , 2 7 8 ,
Barenblatt,
M.A. 166,186
G . I . 278
279 Fjelstad,
J.E.
Beer, T. 83
Fjortoft,
R. 201,232
B&nard, H. 257,278
F r a e n k e l , L.E. 318,326
Benjamin, T.B. 314
Fromm, J.E.
Bois, P.A. 45,83,153,212,
F u j i s a k a , S. 276,279
Batchelor,
G.K. 34
110,112
112
G o l s t e i n , S. 3 4 , 2 0 6 , 2 3 2
219,229,232 Blumen, W. 112
Greenspan, H.P. 9 1 , 1 1 2 , 1 5 6
B o l i n , B. 110,112
Guckenheimer, J.
B r i g h t o n , P.W.M. 177,186
Cuiraud, J.P.
257,279
87,89,99,112,130 153,186,306
Cahn, A. 110,112 C a s t e n , R.G. 318,32B
Gutman, L.N.
C e r c i g n a n i , C. 90,112
Holmes, Ph. 267,279
C h a n d r a s e k h a r , S. 227,232
Hopf, E. 2 3 4 , 2 4 1 , 2 7 9
Cole, J.D. 3 1 9 , 3 2 6 , 3 3 2 , 3 3 6
Houghton, J.T.
Collet,
Howard, L.N. 2 0 7 , 2 0 9 , 2 1 1 , 2 3 2
P. 256,278
C o r r s i n , S. 312,314
165,186
Hunt, J.C.R.
11
186,307,314
C o u l l e t , P.H. 238,278
I o o s s , G. 278,279
Crapper,
J a c k s o n , P.S.
Dikij,
G.D. 70,93
L.A. 4 2 , 8 3 , 2 0 1 , 2 3 2
186
Jakobson, M. 255,279
D o r o d n i t s y n , A.A. 140,152,153
Joseph,
D r a z t n , P.G. 1 9 1 , 2 0 7 , 2 2 9 , 2 3 2 ,
Kaplun, S. 173,319,326
266,269,278,296,
Keulegan, G.H. 288,314
305,314
K e v o r k i a n , J. 3 1 8 , 3 2 6 , 3 3 2 , 3 3 6
Eady, E.A. 191,232 Eckmann, J . P .
Kibel,
Eliassen,
*
See
also
A. 25
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SUBJECT
Absolute acceleration 2 --
INDEX
Burger number 193
Velocity 1,2
Cantor set 236,312
Acoustic waves 40,85
Chaos 234-279
Ackerblom's problem 159
Characteristic rays 132
Adjustment 90,91,97,101,I05,117
Conditions for instability 202
Airy functions 51,52
Conservation law 58
Anelastic equations 296
Consistency 118
Angular velocity 1
Contraction of volume 244
Antibreeze 165
Convective instability 212-231
Aperiodic regime 239
Coriolis acceleration 3
Asymptotic expansions 317 -
sequences 317
-
--
parameter 3
Correlation function 236,237
Average value 58
Curvilinear coordinates 16,21
Balance equation 303
Deep convection equations 263,296-299
Baroclinic 191
Depth parameter 264
Baroclinic instability 197,200
Dimension 246
Barotropic 40,67,58,190
Dispersion relation 190,218
Barotropic instability 203
Dispersive wave 57
Benard problem 222,225
Dissipation function 15
Bessel functions 126
Dissipative flow 234
E-parameter 23
Domain of attraction 235
E-plane approximation 20-22
Dominant equations 130,262
Bifurcations 238,275
Dorodnitsyn method 140
Bjerkne's theorem 4
Doublings 248-249
Boundary layer problem 155,
Dynamical system 234
172-173,315,321 Boussinesq
approximation
Eady problem 191 68,214
--
equations 3 0 , 2 6 4
--
filtering 86
--
gravity waves 44
--
number 7 , 2 1 3 , 2 5 8 , 2 9 2
Breezes 161 Brunt-V~is~l~ frequency 8
Eigenvalue problem 216 --
relation 196
Ekman number 9-10,198 - - r e g i o n 166 Energy equation 12 Entropy 245
345 Equation --
of Bernoulli
61
--continuity --state
-
--
--
12
13
Instabilities 238
telegraphy
100
Ergodlcity 236 Fast
variables
Intermittent --
116,332
Feigenbaum scenario Filtering
Inner variable 50,322 Instability 188
248
86,90
transition 256 turbulence 255
Isochoric model 43,60,280 Isothermal atmosphere 37 Kibel number 4
First integrals 62
Knudsen number 90
fo-plane approximations 20
Landau equation 268
-
-
equations 21,22
Laplacian operator 14
Folding 246
Large-synoptic scale 24
Fractals 247,307,309
Law of perfect gases 6
Free surface solution 290
Lee waves65,79-82,114,119,127,146-148
Frictional
Long
force 8
Froude number 2,268,292
equation 64
--problem
135,136
Functions of space current 61
Lorenz system 267
Galerkin method 266
Low Mach nomber flow 299
Generalized Boussinesq equations 44
Lower deck 178,180,183
Geostrophic filtering 88
Lyapunov exponent 245,277
--
relation 3 2 , 8 8
--
numbers 246
Grashof number 162,268
Mach number 258,292
Gravitational
Mandelbrot's work 311
acceleration 2
Gravity waves 40
Matching S 1 , 1 0 1 , 1 1 5 , 1 3 1 , 1 6 0 , 1 8 2 , 3 1 9
Group velocity 66
Metric
Howard's semicercle theorem 211,
Middle deck 178,180
212 Hydrostatic Boussinesq equations 163 -
-
-
-
coefficient
14
Miles condition
209
Mixing property
236
c a s e 42
MMAE 315
equations 24
MSM 6 6 , 8 6 , 1 3 1 , 3 2 7
--
filtering 86
Modelling of the turbulence
--
parameter 7
Momentum e q u a t i o n
Hyperbolicity 247
Navier-Stokes
Initial conditions 92
Newtonian fluid
-Inner
layer equations 93 Boussinesq's solution 122
--expansion
324
equations
258-260
6
Nondimensional
form 16-19
Non-isothermal
atmosphere
Nonlinear
8
12
problem 140,150
40
346 Normal modes 190
Stability 187
Outer approximation 134
Standard altitude 6
-
-
expansion 324
-
-
atmosphere 6-8
Stochasticlty 236
- - s o l u t i o n 128 Parasite solutions 138
Strange attractor 242-247,271
Poincar@ section 240
Stretching 246
Pomeau-Mannevile scenario 263-257
Strouhal number 17
Power spectrum 272
Sturm-Liouville problem 39
Prandtl number 19,260
Supercritical stability 269
Pressure coordinates 26
Synoptic scale 3
Primitive equations 25
Tangent motions 91
Principle of exchange
Taylor-Goldstein equation 206
of stabilities 218 Quasi-periodic solution 235 -
-
-
-
Thermal boundary condition 176 Triple deck 176-184
-geostrophic model 32
Troposphere 5
-solenoidal model 292,301
Turning point local problem 49
Radiation condition
123,124,135,185
R a y l e i g h number 2 1 6 , 2 2 2 , 2 2 9
Two scales method 46,$5,129,327
--variables
expansion 327-335
R e n o r m a l i z a t i o n 249-262
Unpredictability
R e y n o l d s number 193,209,268
U n s t e a d y a d j u s t m e n t p r o b l e m 104
Rossby number 3
Upper d e c k 178,181
-
-
Vertical
waves 5 3 - 6 4 , 5 9
R u e l l e - T a k e n s s c e n a r i o 242 Scenarii
241
Self-induced
c o u p l i n g 184
Self similarity
310
S h a l l o w c o n v e c t i o n 264
Singe's
c o n d i t i o n 29,31
criterion
207-209 26,66
S l o p e wind 173 Slow v a r i a b l e s Spherical
shift 9 5 ,9 8
-
-
-
-
equations 183 stress 13
- - e q u a t i o n 38,41
Slip conditions
Solitary
-
55
waves 2 8 0 , 2 8 8 , 2 8 9 coordinates
S q u i r e t h e o r e m 207
13-14
120
Viscosity effects 198
Wave energy 54
S . I . C 237 Similarity
displacement of the streamline
-
Secular terms 48,55,132,329
237
