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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 .
H. A. Weidenmiiller Max-Planck-Institut ftir Kemphysik Postfach 10 39 80, W-6900 Heidelberg, FRG J. Wess Lehrstuhl fiir Theoretische Physik Theresienstrasse 37, W-8000 Miinchen 2, FRG J. Zittartz Institut fiir Theoretische Physik, Universit~it K61n Ziilpicher Strasse 77, W-5000 K61n 41, FRG
Managing Editor W. Beiglb6ck Assisted by Mrs. Sabine Landgraf c/o Springer-Verlag, Physics Editorial Department V Tiergartenstrasse 17, W-6900 Heidelberg, FRG
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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
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
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 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 ~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):
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 , .'.
.'.'~:'/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 k2l, 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):
:
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
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-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(~,~)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
_~
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 __~oO,
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 -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/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 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
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~lO0 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,
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=(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"
~
>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 -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 0O) 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 eO) 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
rr
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
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
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((~=0, 33 un.d. ~=0, I 0 ) .
I
4
4
•
•
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II
&
12
28g
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I
II
I UI
I
ii
J
0.$ 0.6
"
O.q.
-
I,.
0.I
0
-I00
-eO
~
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I
II
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I
f-
-f~)
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I
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I
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-20
II
I
I
0
l
it
I
ii
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20
40
• ~.37: ~ 0 ~ / ~
60
~
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(a=O=1? und 8=0, 10).
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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
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)