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A Lagrangian View of Wave-Mean Flow Interaction
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
83
A LAGRANGIAN VIEW OF WAVE-MEAN FLOW INTERACTION R. Grimshaw
Department of Mathematics U n i v e r s i t y of Melbourne P a r k v i l l e , V i c t o r i a 3052 Australia
It i s shown how t h e wave a c t i o n e q u a t i o n i s r e a d i l y o b t a i n e d from a Lagrangian Formulation of t h e e q u a t i o n s of motion. The c o u n t e r p a r t t o t h e wave a c t i o n e q u a t i o n i s t h e mean-flow e q u a t i o n which c a n a l s o be simply d e r i v e d i n a Lagrangian Eormulation; i n t h i s e q u a t i o n t h e wave f o r c i n g c a n b e expressed e i t h e r by a r a d i a t i o n s t r e s s t e n s o r o r by t h e wave pseudomomentum. A f t e r d e v e l o p i n g t h e t h e o r y i n a g e n e r a l framework t h e r e s u l t s a r e a p p l i e d t o a compressLble f l u i d . In p a r t i c u l a r i t is shown t h a t , u s i n g s p h e r i c a l p o l a r coo r d i n a t e s , t h e wave a c t i o n e q u a t i o n and t h e mean-flow e q u a t i o n t a k e Eorms u s e f u l f o r a p p l i c a t i o n s f n s t r a t o s p h e r i c meteorology. F i n a l l y t h e t h e o r y i s a p p l i e d t o a s e t of e q u a t i o n s which d e s c r i b e t h e i n t e r a c t i o n of b a r o t r o p i c t o p o g r a p h i c , @-plane Rossby waves w i t h a mean Elow.
91.
INTRODUCTION
The i n t e r a c t i o n between waves and mean flows is a t o p i c w i t h r e l e v a n c e t o many d i E f e r e n t b r a n c h e s of p h y s i c s . I n r e c e n t y e a r s c o n s i d e r a b l e i n t e r e s t in t h i s s u b j e c t h a s developed w i t h i n t h e g e o p h y s i c a l f l u i d dynamics community. Here, as e l s e w h e r e , t h e b e n e f i t s of d e s c r i b i n g t h e L n t e r a c t i o n w i t h i n a L a g r a n g i a n framework a r e now g e t t i n g i n c r e a s i n g r e c o g n i t i o n . Thus, on t h e one hand, t h e r e s p o n s e of t h e waves t o t h e mean flow i s b e s t d e s c r i b e d by t h e wave a c t i o n e q u a t i o n , which i s a n out-growth from t h e a c t i o n p r i n c i p l e s of c l a s s i c a l mechanics. In t h e f l u i d dynamics c o n t e x t t h e c e n t r a l r o l e of t h e wave a c t i o n e q u a t i o n w a s f i r s t recognized i n t h e p i o n e e r i n g work of Whitham (1965, 1970) and B r e t h e r t o n and G a r r e t t (1968). On t h e o t h e r hand, t h e r e s p o n s e of t h e mean f l o w t o t h e waves i s d e s c r i b e d w i t h i n t h e c o n t e x t of a g e n e r a l i z e d Lagrangian-mean Eormulation, i n which t h e a c c e l e r a t i o n of t h e Lagrangian-mean f l o w due t o t h e waves i s d e s c r i b e d e i t h e r by t h e r a d i a t i o n s t r e s s t e n s o r , O K by t h e pseudomomentum. The p r e l i m i n a r y i d e a s were developed by D e w a r (1970) and B r e t h e r t o n (1971) and r e c e i v e d t h e i r f u l f i l m e n t i n t h e g e n e r a l i z e d Lagrangian-mean t h e o r y of A n d r e w s and McIntyre (1978a,b). A g e n e r a l review of b o t h wave a c t i o n and wavemean Elow i n t e r a c t i o n i s c o n t a i n e d in a r e c e n t review by Grimshaw (1984).
In t h i s a r t i c l e w e s h a l l f i r s t g i v e a b r i e f a c c o u n t of t h e g e n e r a l t h e o r y based o n a L a g r a n g i a n f o r m u l a t i o n of t h e e q u a t i o n s of motion in o r d e r t o s e t t h e s c e n e f o r t h e developments of subsequent s e c t i o n s . A more comprehensive a c c o u n t of t h e g e n e r a l t h e o r y i s d e s c r i b e d by Grimshaw (1984). Then i n 92 we s h a l l d e s c r l b e t h e i m p l e m e n t a t i o n of t h e g e n e r a l t h e o r y f o r a c o m p r e s s i b l e f l u i d u s i n g t h e In 93 t h i s g e n e r a l i z e d Lagrangian-mean t h e o r y of Andrews and P k I n t y r e (1978a,b). t h e o i y i s r e c a s t i n s p h e r i c a l p o l a r c o - o r d i n a t e s and we d e m o n s t r a t e how t h e wave a c t i o n e q u a t i o n and t h e Lagrangian-mean flow e q u a t i o n s a r e r e l a t e d t o t h e Eliassen-Palm r e l a t i o n s and t h e Charney-Drazin theorems of s t r a t o s p h e r i c meteorology. Then i n 94 w e d e s c r i b e how t h e g e n e r a l t h e o r y c a n b e used t o s t u d y b a r o t r o p i c t o p o g r a p h i c Rossby waves on a p-plane.
84
R. Grimshaw
To d e s c r i b e t h e g e n e r a l t h e o r y we s h a l l suppose t h a t t h e p h y s i c a l system is s p e c i f i e d by t h e vector-valued f i e l d Q ( t , x ) where t i s t h e time and xi ( i = 1, 2 , 3) a r e s p a t i a l c o - o r d i n a t e s . i I n t h e a b s e n c e of d i s s i p a t i o n t h e p h y s i c a l system obeys a v a r i a t t o n a l p r i n c i p l e w i t h Lagrangian d e n s i t y Then t h e e q u a t i o n s of motion a r e L(c$~, Qxi, Q; t , x i ) .
where t h e g e n e r a l i z e d f o r c e Q r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . Our d e r i v a t i o n of t h e wave a c t i o n e q u a t i o n w i l l b e based o n t h e i d e a s of Whitham (1965, 1970) b u t w i l l f o l l o w t h e development by Hayes (1970) more c l o s e l y . Thus, w e suppose t h a t @ ( t , x i ; 8 ) depends o n t h e ensemble parameter 8 such t h a t
e+
Q ( t , xi;
2n)
Q ( t , xi;
=
el,
(1.2)
and t h e n d e f i n e t h e a v e r a g i n g o p e r a t o r
For s i m p l i c i t y w e s h a l l d e n o t e t h e mean f i e l d <$> by 3 . The a v e r a g i n g o p e r a t o r commutes w i t h d i f f e r e n t i a t i o n and h a s o t h e r s i m p l e and obvious p r o p e r t i e s (Andrews e n e x t d e f i n e t h e wave p e r t u r b a t i o n o r d i s t u r b a n c e f i e l d andAMcIntyre, 1978a,b). W by Q where
(1.4)
C$=?+C$
We s h a l l assume t h a t Q i s O(a) where a i s a measure of wave a m p l i t u d e , a l t h o u g h t h e r e i s a t p r e s e n t no r e s t r i c t i o n on t h e magnitude of a vis-a-vis t h a t oE I$,nor I f (1.1) is m u l t i p l i e d on t h e i r r e l a t i v e s c a l e s w i t h r e s p e c t t o t and xi. by $ e ( i . e . a$/ae) and t h e r e s u l t a v e r a g e d , i t may b e shown t h a t
(1.5a) (1.5b)
where =
@i
aL < Q e -->, wx
(1.5~)
i
a =< ieQ >.
(1.5d)
ai
Equation (1.5a) is t h e wave a c t i o n e q u a t i o n , * i s t h e wave a c t i o n d e n s i t y , is t h e wave a c t i o n f l u x I n d 8 is a t e r m r e p r e s e n t i n g t h e e f f e c t s of d i s s i p a t i o n . B o . t h k and 6, are O(a ) wave q u a n t i t i e s , and a r e t h e a p p r o p r i a t e g e n e r a l measures of wave a c t i v i t y . A p p l i c a t i o n s of ( 1 . 5 a ) depend on d e l i n e a t i o n of t h e f a m i l y C$(t,x i ; 0 ) . One c h o i c e which fs commonly used i s t o i d e n t i f y 8 a s a phase s h i f t , s o t h a t t h e a v e r a g i n g o p e r a t o r c a n b e i n t e r p r e t e d as a phase a v e r a g e . Thus w e p u t
Q ,
a nd
.
A
=
$ ( t , xi;
s ( t , xi)
w = -s
- el,
(1.6a) (1.6b)
t'
Ki
= sxi'
85
A Lagrangian View of Wave-Mean Flow Interaction Here s ( t , x i ) is t h e phase, 4 i s p e r i o d i c i n s , w i s t h e wave frequency and i s t h e wavenumber. It f o l l o w s t h a t
a
Ki
(1.7)
i
and t h e wave a c t i o n e q u a t i o n (1.5a) t a k e s t h e form o b t a i n e d by Whitham (1970) i n t h e absence o f d i s s i p a t i o n . Another c h o i c e i s t o i d e n t i f y 9 w i t h a c o - o r d i n a t e so t h a t t h e a v e r a g i n g o p e r a t o r i s a c o - o r d i n a t e a v e r a g e . It i s remarkable t h a t t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) is f o r m a l l y e x a c t , v a l i d w i t h o u t any a p p a r e n t r e s t r i c t i o n o n wave a m p l i t u d e o r w i t h o u t any assumption t h a t t h e mean f i e l d i s s l o w l y v a r y i n g w i t h r e s p e c t t o t h e waves. There i s , of c o u r s e , a n a s y m p t o t i c e r r o r i n c u r r e d in t h e i d e n t i f i c a t i o n of t h e f a m i l y ( 1 . 2 ) (Hayes, 1970). Turning next t o t h e mean flow e q u a t i o n w e f i r s t d e f i n e t h e " u n d i s t u r b e d " Lagr a n g i a n by
Note t h g t a l t h o u g h Lo does n o t depend e x p l i c i t l y o n t h e wave f i e l d $, t h e mean f i e l d s 4 g e n e r a l l y c o n t a i n O(a2) wave-induced components. Next w e p u t
L~ i s a n o ( a 2 ) wave p r o p e r t y , and i t s e x p l i c i t dependence o n t , x i l i n c l u d e s i t s dependence on t h e mean f i e l d s 0. I f we now m u l t i p l y (1.1) by $. and a v e r a g e J t h e r e s u l t , we c a n show t h a t
aT. aT . -&! +&
=
-<
i
ax1 > + , j
(1.10a)
j
where
(1.10b)
>
T.. = J1
- n,>bji'
(1.1Oc)
is t h e pseudomomentum, and -T.i t h e c o r r e s p o n d i n g f l u x . The s i g n s are H e r e -T a g r e e w i t h h i s t o r i c a l conventl!on (Andrews and M c l n t y r e , 1978b). A chosen similar e q u a t i o n can b e o b t a i n e d f o r t h e pseudoenergy by m u l t i p l y i n g ( 1 . 1 ) by 0, and a v e r a g i n g t h e r e s u l t (Grimshaw, 1984). Note t h a t when 9 is i d e n t i f i e d w i t h t h e wave a c t i o n d e n s i t y A (1.5b) r e d u c e s t o -Tjo, and t h e t h e c o - o r d i n a t e -x j’ wave a c t i o n f l u x 6 ( 1 . 5 ~ ) r e d u c e s t o -Tji; a l s o e q u a t i o n (1.10a) t h e n r e d u c e s t o t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) .
2:
To make f u r t h e r p r o g r e s s w e now assume t h a t - t h e mean f i e l d s v e l o c i t y ii and a vector-valued mean f i e l d h , where dh - + A dt
a; ij
i
h--=O,
ax
j
7
c o n s i s t oE a mean
(1.11)
86
R. Grimshaw
x
where u r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . In a p p l i c a t i o n s w i l l incorporate t h e e f f e c t s of mean d e n s i t y , mean f l u i d d e p t h e t c . (compare Garrett (1968) and Note t h a t i n t h e absence of d i s s i p a t i o n A i s a mean q u a n t i t y Dewar-(1970)). ( A = 1); however i t i s u s e f u l i n a p p l i c a t i o n s t o l e t h have a f l u c t u a t i n g , d i s s i p a t i v e component. We n e x t d e f i n e
which i s thg: time d e r i v a t i v e f o l l o w i n g t h e meanlmotion, and assume t h a t L1 only t h r o u g h i t s dependence on d4ddt; f u r t h e r a p a r t from t h e depends on dependence L1 o! Ci through d $ / d t any o t h e r e x p l i c i t denpendence o f L1 o n These h y p o t h e s e s ii i s b i l i n e a r i n ui and t h e d i s t u r b a n c e v a r i a b l e s Q and $x
+
OF
i
.
a r i s e from t h e f a c t t h a t t h e f u l l Lagrangtan i s u s u a l l y a t most q u a d r a t i c i n t h e v e l o c i t y f i e l d . The mean flow e q u a t i o n c a n now b e o b t a i n e d by a v e r a g i n g (1.1). The d e t a i l s of t h e d e r i v a t i o n a r e d e s c r i b e d by Grimshaw (1984), and t h e r e s u l t i s
a aLO aLoE ( Y ) +a F- b_ aui j j a;,
+ R
where
ij
=
~
a R i jax j
-T
ij
<
~
aL axi
+;.T
10
J
.? aLO iLOsij)
A
>e + -
ah
iJ
(1.13a)
,
i j
aL 1 h-->
(1.13b)
?A
Here R i j is t h e r a d i a t i o n stress t e n s o r , a l t h o u g h n o t e t h a t t h e terminology i s not u n i v e r s a l . A l s oe d e n o t e s t h e e x p l i c i t d e r i v a t i v e of L w i t h r e s p e c t t o xi when t h e d i s t u r b a n c e f i e l d s Q and t h e mean v a r i a b l e s Ci, A a r e a l l held c o n s t a n t . Equation (1.13a) c a n b e recognized as t h e mean momentum e q u a t on and d e m o n s t r a t e s t h e manner in which t h e r a d i a t i o n s t r e s s t e n s o r , a n O(a ) wave q u a n t i t y , i s r e s p o n s i b l e f o r t h e a c c e l e r a t i o n of t h e mean flow. An a l t e r n a t i v e t o (1.13a) i n v o l v i n g t h e pseudomomentum i n s t e a d of t h e r a d i a t i o n stress t e n s o r c a n b e The r e s u l t is o b t a i n e d from ( 1 . 1 3 a , b ) by u s i n g (1.lOa).
!
a (-=aLO
at
i
)
+ aa x( -u j j
aLo ) hi
>e
+
+ - < -aA ax. J
+
. h aL a;> iJ
$x Q>. i
T h i s i s t h e form p r e f e r r e d by Andrews and McIntyre (1978a). s e c t i o n by n o t i n g t h e a l t e r n a t i v e e x p r e s s i o n f o r Rij,
+ax a no> i
(1.14) We c o n c l u d e t h i s
(1.15)
a7
A Lagrangian View of Wave-Mean Flow Interaction Here (i3Ll/a$x2)d d e n o t e s t h e d e r i v a t i v e w i t h r e s p e c t t o constant.
J
S i m l l a r l y I t c a n b e shown t h a t
8,
=
ix
4 J
when d $ / d t i s h e l d
-
(1.16a)
Uik+4+
(1.16b)
where
92.
GENERALIZED LAGRANGIAN-MEAN
FORMULATION
The equati-ons of motion f o r a c o m p r e s s i b l e f l u i d a r e (2.la) (2.lb) (2.lc) (2.ld)
where
Here x: i s t h e Eul r i a n c o - o r d i n a t e such t h a t a f l u i d p r t i c l e a t x i h a s The n o t a t i o n i s s t a n d a r d ; t h u s Qi i s t h e c o n s t a n t a n g u l a r v e l o c i t y v e l o c i t y ui. of t h e frame of r e f e r e n c e , @ ( x i ) i s t h e p o t e n t i a l f o r g r a v i t a t i o n a l and c e n t r i f u g a l f o r c e s , p(p, S ) i s t h e thermodynamic p r e s s u r e , p i s t h e d e n s i t y and S i s t h e e n t r o p y . The terms Xi and h r e p r e s e n t t h e e f f e c t s of d i s s i p a t i v e and d i a b a t ic e f f e c t s r e s p e c t i v e l y
.
The a p p r o p r i a t e L a g r a n g i a n f o r m u l a t i o n of t h e s e e q u a t i o n s i s t h e g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n of Andrews and McIntyre (1978a), which c o n t a i n s a comprehensive a c c o u n t of t h i s t h e o r y . Here we s h a l l g i v e only a b r i e f o u t l i n e ( s e e a l s o McIntyre (1977, 1980), Grimshaw (1979) o r Dunkerton (1980)). L e t xi b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s and l e t c i ( t , xo) b e t h e p a r t i c l e displacements defined s o t h a t
Then, f o r any g i v e n ui t h e r e i s a u n i q u e " r e f e r e n c e " v e l o c i t y i l ( t , x . ) s u c h J t h e p o i n t x i moves w i t h t h a t when t h e p o i n t xi moves w i t h v e l o c i t y The material time d e r i v a t i v e ( $ . l a ) i s t h e n g i v e n by v e l o c i t y ui.
The g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n i s now o b t a i n e d by l e t t i n g mean v e l o c i t y , and r e q u i r i n g t h a t
= 0.
ui
be t h e
(2.4)
Thus (2.3) a g r e e s w i t h ( 1 . 1 2 ) . The r e a d e r should n o t e t h a t o u r n o t a t i o n d i f f e r s i n one s i g n i f i c a n t r e s p e c t from Andrews and McIntyre (1978a,b); h e r e ii d e n o t e s
88
R. Grimshaw
zt
t h e Lagrangian-mean v e l o c i t y , r a t h e r t h a n used by Andrews and McIntyre (1978a,b), who u s e t h e s i n g l e o v e r b a r t o d e n o t e E u l e r i a n means. No c o n f u s i o n should arise as E u l e r i a n means w i l l n o t b e d i s c u s s e d i n t h i s a r t i c l e .
-
Next w e d e f i n e a mean d e n s i t y p s o t h a t
au g+p - = i ax
0.
(2.5)
i
It i s a n immediate consequence of ( 2 . l b ) and ( 2 . 5 ) t h a t
-
PJ where
(2.6b)
= P,
J = det[
axi ~
ax.
1.
(2.6b)
J
We l e t K i j b e t h e i , j
-
c o - f a c t o r of J , and s o
Note t h a t U i j is t h e d e r i v a t i v e of 3 w i t h r e s p e c t t o B x i / a x . and aU /axj = 0 . J ij The momentum e q u a t i o n ( 2 . l a ) now becomes
-
dui p-+ dt
-
2 p ~ Q u ijk j k
ui = ui
(2.8a)
.) = pXI,
iJ
j
-
where
.
@ a + p- -a-axi ;+-(pK ax d5,
+dt
(2.8b)
The g e n e r a l i z e d L a g r a n g i a n m e a n e q u a t i o n s a r e t h u s ( 2 . 8 a ) , (2.5) mean f i e l d s a r e Gi, p and S and t h e wave p e r t u r b a t i o n f i e l d i s Lagrangian is
and ( 2 . 1 ~ ) . The A suitable
ci.
- 1 p{.j uiui
+
E
B
X’U
ijk i j k
-
@(xi)
-
-
E(p, S)}.
(2.9)
Here w e r e c a l l t h a t p i s d e f i n e d in terms of p by ( 2 . b a ) , and E ( p , S) is t h e i n t e r n a l energy p e r u n i t mass, so t h a t
(2.10)
where T is t h e temperature. V a r i a t i o n s i n x; ( o r e q u i v a l e n t l y i n ti) g i v e (2.8a) w i t h Q i = pXi. I n o r d e r t o k e e p the-correspondence w i t h t h e g e n e r a l t h e o r y o f $1 a s c l o s e as p o s s i b l e w e d e f i n e Qs = pT s o t h a t v a r i a t i o n s w i t h r e s p e c t t o S-can b e d e f i n e d . The v e c t o r A of $1 i s i d e n t i f i e d as t h e 2-vector w i t h components p and S , and w e n o t e t h a t ( 2 . 1 ~ ) and (2.5) h a v e t h e r e q u i r e d form (1.11).
89
A Lagrangian View of WaveMean Flow interaction The wave a c t i o n e u a t i o n c a n now b e o b t a i n e d from t h e g e n e r a l t h e o r y of 51, u s i n g ( 1 . 5 a , b , c , d ) and q l . l 6 a , b ) .
where
,. ac.,
.4= < p -
ae
(2.11c)
8 =< -. p ( ac.,
ae Xi
as T>. +-
(2.11d)
ae
ThLs a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978b) a l t h o u g h t h e d i s s i p a t i v e term& h a s been w r i t t e n i n a d i f f e r e n t form h e r e . F o r l i n e a r i z e d wave motion, i t i s u s e f u l t o i n t r o d u c e t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n
%+
=
2 O(a ) .
(2.12)
It may t h e n b e shown t h a t
(2.13)
Note t h a t t h e second term h e r e i s i d e n t i c a l l y non-divergent and c a n b e o m i t t e d from (2.11a). To d e r i v e t h e mean flow e q u a t i o n s we f i r s t i d e n t i f y LO from (1.8).
Thus
The mean f l o w e q u a t i o n i s t h e n o b t a i n e d from where L i s g i v e n by ( 2 . 9 ) . ( 1 . 1 3 a , b ) , o r more d i r e c t l y by a v e r a g i n g ( 2 . 8 a ) . The r e s u l t i s (2.15a) where and
-
-
P = P(P,
S),
(2.15b) (2.15~)
It c a n b e v e r i f i e d t h a t Rij i s t h e same r a d i a t i o n stress t e n s o r d e f i n e d by An a l t e r n a t i v e form of (2.15a) i n v o l v i n g t h e pseudomomentum -Ti0 i n s t e a d (1.14). of t h e r a d i a t i o n stress t e n s o r c a n b e o b t a i n e d u s i n g t h e p r o c e d u r e d e s c r i b e d i n $1 and a v e r a g i n g ( s e e ( s e e (1.14), o r more d i r e c t l y by m u l t i p l y i n g ( 2 . l a ) by ax;/ax j Andrews and McIntyre (1978a)). The r e s u l t i s
90
R. Grimshaw
;a = + aTio - a t + - aa ~ (. U- j T i o ) Tjo 2 ax, J +
lt =
where
a + E> +m- < P
-
P<
'j
2 (; 2 + d5
ax' axi
> + <; T^ as >, ax,
d5
EjklQ$,)>'
and
(2.16a)
(2.16b)
(2.16~)
T h i s a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978a) a l t h o u g h t h e d i s s i p a t t v e terms on t h e right-hand s i d e of (2.16a) have been w r i t t e n i n a d i f f e r e n t form h e r e . s3.
GENERALIZED ELIASSEN-PALM AND CHARNEY-DRAZIN THEOREMS
One of t h e most promising a r e a s f o r a p p l i c a t i o n of t h e g e n e r a l r e s u l t s of $2 i s s t r a t o s p h e r i c meteorology. With t h e a v e r a g i n g o p e r a t o r i n t e r p r e t e d as a z o n a l a v e r a g e , t h e c o u n t e r p a r t s of t h e wave a c t i o n e q u a t i o n (2.11a) and t h e mean f l o w e q u a t i o n (2.15a) ( o r (2.16a)) are u s u a l l y c a l l e d Eliassen-Palm and Charney-Drazin theorems r e s p e c t i v e l y , a f t e r t h e p i o n e e r i n g work of E l i a s s e n and Palm (1961) and Charney and D r a z i n (1981) ( f o r r e c e n t reviews see Andrews and & I n t y r e (1978c), McIntyre (1980), Dunkerton (1980) and Uryu (1980)). To o b t a i n t h e s e theorems i n t h e i r most g e n e r a l form w e must f i r s t e x p r e s s t h e e q u a t i o n s of motion i n s p h e r i c a l A s i m p l e and c o n v e n i e n t way of d o i n g t h i s i s t o i n t r o d u c e polar co-ordinates. s p h e r i c a l p o l a r c o - o r d i n a t e s i n t o t h e Lagrangian (2.9); t h i s i s t h e approach used by R r e t h e r t o n (1982) f o r small-amplitude waves. The c o r r e s p o n d i n g r e s u l t s i n c y l i n d r i c a l p o l a r c o - o r d i n a t e s are d e s c r i b e d by Grimshaw (1984). Thus w e l e t t h e c o - o r d i n a t e s x i of 51 b e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , h a n d v, where r i s t h e r a d i u s , h t h e c o - l a t i t u d e and v t h e l o n g i t u d e . These a r e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , whose E u l e r i a n c o u n t e r p a r t s x a r e I. r ' , A' and v ' . The p a r t i c l e d i s p l a c e m e n t s are t h e n d e f i n e d by
The v e l o c i t y components Ln t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s a r e u
where
= _d _r ' dt
,
d h' dv' u2 = r' , u3 = r' sin h' dt ' dt
(3.2a)
(3.2b)
Here Ui are t h e mean v e l o c i t y components in t h e L a g r a n g i a n c o - o r d i n a t e d i r e c t i o n s , which must b e c a r e f u l l y d i s t i n g u i s h e d from t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s . The Lagrangian i s a g a i n g i v e n by ( Z a g ) , where, f o r s i m p l i c i t y , we s h a l l suppose t h a t t h e a x i s of r o t a t i o n i s i n t h e N-S d i r e c t i o n . Thus t h e L a g r a n g i a n i s
91
A Lagrangian View of Wave-Mean Flow Interaction
- 1
u3 - ~(x;) - ~ ( p , s)},
r2 s i n A PI- u u + 9 r ' s i n 2 i i
r f 2 s i n A ' PJ
where
=
i.
r2 s i n A
(3.3a) (3.3b)
and J i s a g a i n d e f i n e d by (2.6b) b u t now xi and xi a r e t h e s p h e r i c a l p o l a r coo r d i n a t e s . V a r i a t i o n s i n Si now g i v e t h e e q u a t i o n s o € motion.
-p { dul ~
dt
L
L
u2 - I u 3 - 2Q s i n h' u3 - -T K r
+ aQ ar
}
(3.4a)
(3.4b)
(3.4c) The c o u n t e r p a r t of ( 2 . 5 ) i s
w h i l e S a g a i n s a t i s f i e s (2.1.2). The wave a c t i o n e q u a t i o n c a n now b e o b t a i n e d from ( 1 . 5 a , b , c , d ) .
-a + - ( -a where
=
-
at
art r2 s i n ~ p < u
+ r2
ae
1
sin h
;=
ui
axi
hi
+
rf
*i )
=
$ u2 + r'
l p ~
<,I2
+
s i n h'
The r e s u l t i s (3.6a)
I
s i n A’
a v r >, s i n 2 A’ -
ae
ac ae pKji>,
ae
u
3
> (3.6b)
(3.6~)
92
R. Grimshaw
b a nd
h
1
=
=
-
r 2 sin Ap
ae
i
as +-T>.
(3.6d)
ae
h ' = 1; h = r, h ' = r ' ; h = r s i n A, h ' = r ' s i n 1 2 2 3 3
A’.
(3.6e)
Here w e remind t h e r e a d e r t h a t xi a r e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , A and v. For l i n e a r i z e d wave motion w e i n t r o d u c e t h e E u l e r i a n p r e s s u r e - p e r t u r b a t i o n p' by ( 2 . 1 2 ) . It may t h e n b e shown t h a t
The second term i s i d e n t i c a l l y t h e b a s i c f l o w is z o n a l ( i - e . i n t e r p r e t e d as a z o n a l a v e r a g e e q u a t i o n (3.6a) r e d u c e s t o t h e Andrews and McIntyre ( 1 9 7 8 ~ ) .
non-divergent and c a n be o m i t t e d from (3.6a). When and z2 a r e O(a2)) and t h e a v e r a g i n g o p e r a t o r is ( i . e . 8 i s i d e n t i f i e d w i t h -v) t h e wave a c t i o n g e n e r a l i z e d Eliassen-Palm r e l a t i o n d e r i v e d by Also -(r2 s i n ?-,)-!& i s t h e a n g u l a r pseudomomentum.
The mean flow e q u a t i o n s c a n now b e o b t a i n e d from (1.13a) ( o r ( 1 . 1 4 ) ) ( a f t e r a l l o w i n g f o r t h e p r e s e n c e of t h e g e o m e t r i c a l f a c t o r s h i i n L ) , o r more d i r e c t l y by a v e r a g i n g ( 3 . 4 a , b , c ) . The e s s e n t i a l s t e p in t h i s l a t t e r approach is
(3.8a)
where
R
ij
=
6
.<,I2
iJ
sin A' PJ
-
r 2 sin A
p"> (3.8b)
*
-
Here p is p ( p , 9) and R i j i s t h e r a d i a t i o n stress t e n s o r ( i . e . t h e a n a l o g u e o f ( 2 . 1 5 ~ ) in s p h e r i c a l p o l a r c o - o r d i n a t e s ) . For i n s t a n c e , t h e z o n a l mean flow e q u a t i o n is
-
1
aR
A= p
r2 s i n A a x j where
M =
A’ u3
+ GrI2
s i n 2 XI>.
A' X3>,
(3.9c)
(3.9b)
Here M is t h e z o n a l mean s p e c i f i c a n g u l a r momentum; € o r t h e r e l a t i o n s h i p between M and t h e z o n a l mean f l o w , see Dunkerton (1980). When t h e a v e r a g i n g o p e r a t o r is i n t e r p r e t e d as a z o n a l a v e r a g e ( i . e . 0 i s i d e n t i f i e d w i t h -v) t h e o f f - d i a g o n a l components of R v j are i d e n t i c a l t o Bq; simply compare (3.8b) w i t h ( 3 . 6 ~ ) . Note t h a t t h e d i a g o n a l components of R v j do not now a p p e a r in ( 3 . 9 ~ ) . With t h e s e i n t e r p r e t a t i o n s , and u s i n g ( 3 . 6 a ) , e q u a t i o n ( 3 . 9 ~ )becomes
A Lagrangian View of Wave-Mean Flow Interaction
dM dt
--
dt
(
r2L -?*--+ pr
pr2 s i n A
sin A
x3>.
93 (3.10)
With t h e f u r t h e r r e s t r i c t i o n t o l i n e a r i z e d waves on a z o n a l b a s i c flow (3.10) r e d u c e s t o a g e n e r a l i z e d Charney-Drazin theorem ( s e e t h e s i m i l a r r e s u l t s o b t a i n e d by Andrews and McIntyre ( 1 9 7 8 ~ )and B r e t h e r t o n ( 1 9 8 2 ) ) . It f o l l o w s t h a t M w i l l change o n l y i n r e s p o n s e t o wave t r a n s i e n c e r e p r e s e n t e d by t h e term i n v o l v i n g k i n (3.10), o r due t o d i s s i p a t i v e e f f e c t s r e p r e s e n t e d by t h e terms i n v o l v i n g 3 and X3.
S4.
TOPOGRAPHIC, @-PLANE ROSSBY WAVES
The r e s u l t s of $2 and 53 have been o b t a i n e d w i t h o u t any r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e s c a l e of t h e waves, o r w i t h o u t making any of t h e a p p r o x i m a t l o n s such a s t h e h y d r o s t a t i c a p p r o x i m a t i o n , o r t h e q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n , o f t e n used i n g e o p h y s i c a l f l u i d dynamics. These v a r i o u s a p p r o x i m a t i o n s c a n b e used i n p o s t e r i o r i i n t h e e v a l u a t t o n of t h e wave a c t i o n d e n s i t y Jk ((2.11b) o r ( 3 . 6 b ) ) , t h e wave a c t i o n f l u x B 2 ( ( 2 . 1 1 ~ ) o r ( 3 . 6 c ) ) , e t c . However, i t i s i n s t r u c t i v e and sometimes more u s e f u l t o d e r i v e a wave a c t i o n e q u a t i o n and a mean flow e q u a t i o n d i r e c t l y from a n approximate s e t of e q u a t i o n s . In t h i s r e s p e c t t h e g e n e r a l t h e o r y of § l p l a y s a c o - o r d i n a t i n g r o l e . To i l l u s t r a t e t h i s a s p e c t w e s h a l l d e r i v e t h e wave a c t i o n e q u a t i o n and t h e mean f l o w e q u a t i o n s € o r a homogeneous f l u i d u s i n g t h e h y d r o s t a t i c and @-plane a p p r o x i m a t i o n s . In thLs c o n t e x t t h e wave p e r t u r b a t i o n s a r e g e n e r i c a l l y c a l l e d Rossby waves, and d e r i v e t h e i r wave-like c h a r a c t e r from a combination of t h e t o p o g r a p h i c s l o p e , t h e @ - e f f e c t and t h e s h e a r of t h e mean flow. Many a u t h o r s have s t u d i e d t h e i n t e r a c t i o n of Rossby waves w i t h mean flows ( s e e Dickinson (1978) f o r a review). From t h e p o i n t of view adopted i n t h l s a r t i c l e t h e p a p e r s by Grimshaw ( 1 9 7 7 ) , Rhines and Holland (1979), Young and Rhines (1980) and Huthnance (1981) a r e p a r t i c u l a r l y r e l e v a n t , a l t h o u g h n e a r l y a l l t h e r e s u l t s o b t a i n e d by t h e s e a u t h o r s a r e r e s t r i c t e d t o s m a l l a m p l i t u d e waves, o r s u b j e c t t o some o t h e r r e s t r i c t i o n c o n c e r n i n g t h e s t r u c t u r e of t h e mean flow. The r e s u l t s t o b e o b t a i n e d below c o n t a i n no r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e scale of t h e waves v i s a - v i s t h e mean f l o w , and hence c a n b e r e g a r d e d as a g e n e r a l i z a t i o n of t h e c o r r e s p o n d i n g r e s u l t s of t h e s e a u t h o r s . The E u l e r i a n e q u a t i o n s of motion f o r a homogeneous f l u i d i n t h e h y d r o s t a t i c a p p r o x i m a t i o n are (4. l a ) (4.lb) (4.1~) where
H = h(x', y ' )
f = f and
0
+
@1x '
-d= = + au , , + v a Ya ' dt
+
+
5,
(4.ld)
p2y1,
(4.le)
a
(4.1f)
94
R. Grimshaw
Here ( x ' , y ' ) a r e E u l e r i a n h o r i z o n t a l c o - o r d i n a t e s such t h a t a f l u i d p a r t i c l e a t (x’ y ' ) h a s h o r i z o n t a l v e l o c i t y components (u, v ) , w h i l e is t h e f r e e s u r € a c e d i s p l a c e m e n t , and h ( x ' , y ' ) i s t h e e q u i l i b r i u m d e p t h of t h e f l u i d . The terms F and G r e p r e s e n t t h e combined e f f e c t s of wind s t r e s s f o r c i n g and d i s s i p a t i o n . One immediate and useful consequence of ( 4 . 1 a , b and c ) i s t h e p o t e n t i a l v o r t l c i t y equation
<
where
dd tx = E ,
(4.2a)
HX = q + f ,
(4.2b) (4.2~)
and
(4.2d)
To u s e t h e g e n e r a l t h e o r y of $1 w e mst f i r s t r e c a s t t h e s e e q u a t i o n s u s i n g t h e g e n e r a l i z e d Lagrangian-mean Eormulation. The development i s similar t o t h a t d e s c r i b e d i n $2 f o r a g e n e r a l c o m p r e s s i b l e f l u i d . Thus, w e l e t x and y b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , and l e t 5 and 7) b e t h e p a r t i c l e d i s p l a c e m e n t s d e f i n e d so t h a t x ' = x + E .
=
y ' = y + q ,
= 0,
and
(4.3a) (4.3b) (4.3c)
Here i, a r e t h e LagrangLan-mean v e l o c i t i e s . We l e t J b e t h e J a c o b i a n of t h e t r a n s f o r m a t t o n from (x, y ) t o (x’, y ' ) w i t h c o - f a c t o r s K i j ( i . e . (2.6b) and ( 2 . 7 ) r e s t r i c t e d t o h o r i z o n t a l co-ordinates). Then ( 4 . 1 ~ ) becomes (4.4a) where
-
-
Note t h a t H i s mean q u a n t i t y .
H = JH.
(4.4b)
The e q u a t i o n s o f motion ( 4 . l a , b ) become
Here we are t e m p o r a r i l y u s i n g i n d e x n o t i o n where u1 = u, u2 = v e t c . a b l e Lagrangian i s
A suit-
95
A Lagrangian View of Wave-Mean Flow Interaction
2
where
u u
i i
+
f(k
-
+
e) u }
x
, i i
+
e = f x’ 0-
p,x'
2
g(hC
+
i
,
I
21
f
< 2 )J -
gH6,
2
(4.6a)
(4.6b)
p2y' j . I
Here i,, j and k, a r e u n i t v e c t o r s i n the_ x , y and z - d i r e c t i o n s r e s p e c t i v e l y . g i v e (4.4b). W e are V a r i a t i o n s i n Ei g i v e (4.5) w i t h Q i = HFi; v a r i a t i o n s in now i n a p o s i t i o n t o a p p l y t h e g e n e r a l t h e o r y of $1, n o t i n g t h a t (4.4a) h a s t h e r e q u i r e d Eorm ( 1 . 1 1 ) .
<
The wave a c t i o n e q u a t i o n i s g i v e n by ( 1 . 5 a , b , c , d ) , (4.7a)
- xi
+
ae
1
(k
x
e)i>,
(4.7b)
I
(4.7c)
8
=
- ac,
~
ae
F
i
>.
(4.7d)
An i n t e r e s t i n g f e a t u r e of (4.7b) i s t h e manner t n which t h e p-plane e f f e c t , h e r e r e p r e s e n t e d by t h e p a r a m e t e r s pi, m o d i f i e s t h e e x p r e s s i o n f o r t h e wave a c t i o n d e n s i t y (2.11b) o b t a i n e d i n $2. T h i s is more a p p a r e n t from t h e a l t e r n a t i v e expression t o (4.7b).
(4.8a) where
-
€ = €
0
+ p x + p y . 2 1
(4.8b)
F u r t h e r , i t c a n b e shown t h a t ( 4 . 8 a ) a g r e e s w i t h (3.6b) when t h e a p p r o p r i a t e p-plane, h y d r o s t a t i c and " t r a d i t i o n a l " a p p r o x i m a t i o n s ( i . e . n e g l e c t of t h e h o r i z o n t a l component of t h e e a r t h ' s r o t a t i o n ) a r e made i n ( 3 . 6 b ) . For l i n e a r i z e d wave motion t h e l a s t term i n k ( 4 . 8 a ) c a n b e n e g l e c t e d a n d m ? i s g i v e n by
where C ' i s t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n (compare ( 2 . 1 2 ) ) and t h e second term i n (4.9) i s i d e n t i c a l l y non-divergent, and hence c a n b e o m i t t e d from ( 4 . 7 a ) .
96
R. Grimshaw
The mean flow e q u a t i o n s a r e o b t a i n e d from (1.13a) o r (1.14) a f t e r a l l o w i n g f o r t h e e x p l i c i t dependence o n xi c o n t a i n e d i n t h e term i n v o l v i n g e i n ( 4 . 6 a ) , as a consequence of whtch t h e Lagrangian L i s n o t b i l i n e a r i n ui and The f o r m e r i n v o l v e s t h e r a d i a t i o n stress t e n s o r and i s r e a d i l y o b t a i n e d by a v e r a g i n g ( 4 . 5 ) . However, i n t h i s c o n t e x t , i t i s not so u s e f u l as s i g n i f i c a n t wave-forcing terms a r e c o n t a i n e d i ne ( s e e (1.13a)) which a r i s e from t h e e x p l i c i t dependence of f and h on xi. The a l t e r n a t i v e form (1.14) i s p r e f e r r e d , and i s most simply o b t a i n e d by m u l t t p l y i n g ( 4 . 5 ) w i t h a x i / a x j and a v e r a g i n g . The r e s u l t i s
ci.
where
(4.10b)
(4.10d)
and
A comparison of (4.10a) w i t h (1.14) o r (2.16a) s u g g e s t s Lhat Pi(4.10b) i n (4.10a) p l a y s t h e r o l e of pseudomomentum, a l t h o u g h w e n o t e t h a t HPi d i f f e r s from t h e pseudomomentum - T i o ( l . l O b ) , where h e r e (4.11) The d i f f e r e n c e arises due t o t h e afore-mentioned p r e s e n c e of L.
e,
i n t h e Lagrangian
In many a p p l i c a t i o n s t h e mean-flow e q u a t i o n (4.10a) i s dominated by t h e nearg e o s t r o p h i c b a l a n c e between t h e mean C o r i o l i s f o r c e and t h e mean p r e s s u r e g r a d i e n t . In t h i s case a more u s e f u l form of t h e mean flow e q u a t i o n i s o b t a i n e d W e find that by a v e r a g i n g ( 4 . 2 a ) .
g =E, -
(4.12a)
where
(4.12b)
and
(4.12~)
Note t h a t (4.12b and c ) d e m o n s t r a t e t h a t (4.12a) i s a l s o a n immediate consequence of (4.10a). The s i m p l i c i t y of (4.12a) i s s e l f - e v i d e n t ; t h i s e q u a t i o n i s a
97
A Lagrangian View of Wave-Mean Flow Interaction
g e n e r a l i z a t i o n of similar r e s u l t s which h a v e a p p e a r e d i n t h e l i t e r a t u r e , u s u a l l y f o r l i n e a r i z e d wave m o t i o n ( s e e Rhines a n d Holland ( 1 9 7 9 ) , o r Huthnance ( 1 9 C l ) ) . In p a r t i c u l a r , f o r t h g s e s i t u a t i o n s where t h e dominant component o f is T/n and t h e d i s s i p a t i v e term E i s small ( e . g . where t h e waves a r e s l o w l y v a r y i n g r e l a t i v e t o t h e mean f l o w ) , e q u a t i o n (4.12a) s u c c i n c t l y d e m o n s t r a t e s t h a t t h e mean f l o w c l o s e l y follows t h e geostrophic contours.
X
REFERENCES Andrews, D.G. and M c I n t y r e , M.E., 1978a: "An e x a c t t h e o r y of n o n l i n e a r waves on a Lagrangian-mean flow", J . F l u i d Mech., 8v , 609-646. Andrews, D.G. a n c M c I n t y r e , M.E., J . F l u i d Mech., 8 9 , 647-664.
1978b:
"On wave-action and i t s r e l a t i v e s " ,
Andrews, D.G. and M c I n t y r e , M.E., 1978c: " G e n e r a l i z e d E l t a s s e n - P a l m and Charney-Drazin theorems f o r waves o n a x i s y m m e t r i c mean f l o w s i n c o m p r e s s i b l e 175-185. a t m o s p h e r e s " , J. Atmos. S c i . , 3, B r e t h e r t o n , F.P., 1982: " C o n s e r v a t i o n of wave a c t i o n and a n g u l a r momentum i n a s p h e r i c a l atmosphere". P r e p r i n t . 1968: "Wavetrains i n inhomogeneous B r e t h e r t o n , F.P. and Garrett C . J . R . , moving media", Proc. R. SOC. London, S e r . A . TET, 539-554. Charney, J . G . and D r a z i n , P.G., 1961: " P r o p a g a t i o n o f p l a n e t a r y - s c a l e d i s t u r b a n c e s from t h e lower i n t o t h e u p p e r atmosphere", J . Geophys. R e s . , ii6, 83-109. D e w a r , R.L., 1970: " I n t e r a c t L o n between hydromagnetic waves and a timed e p e n d e n t , inhomogenous medium", Phys. F l u i d s , 3 3 , 2710-2720.
D i c k i n s o n , R . E . , 1978: "Rossby waves - l o n g p e r i o d o s c i l l a t i o n s o f a t m o s p h e r e s and o c e a n s " , Ann. Rev. F l u i d Mech. 159-198.
m,
Dunkerton, T . , 1980: "A L a g r a n g i a n mean t h e o r y of wave, mean-flow i n t e r a c t i o n w i t h a p p l i c a t i o n s t o n o n a c c e l e r a t i o n and i t s breakdown", Rev. 387-400. Geophys. Space Phys.,
m,
[LO] E l i a s s e n , A. and Palm, E., 1961: "On t h e t r a n s f e r of e n e r g y i n s t a t i o n a r y 1-23. m o u n t a i n waves", Geofys. Publ. = ( 3 ) ,
[ll] Garrett, C . J . R . ,
1968: "On t h e i n t e r a c t i o n between i n t e r n a l g r a v i t y waves and a s h e a r flow", J. F l u i d Mech., 71, 497-512.
[ 1 2 ] Grimshaw, R . , 1977: "Non-linear a s p e c t s o f s h e l f waves and mean c u r r e n t s " , in "Waves on Water o f V a r i a b l e Depth", e d . D.G. P r o v i s and R. Radok; pp 206211; L e c t u r e Notes i n P h y s i c s , bG, S p r i n g e r and A u s t r a l i a n Academy o f S c i e n c e , pp 235. [ 1 3 ] Grimshaw, R., 1979: "Mean f l o w s induced by i n t e r n a l g r a v i t y wave p a c k e t s 391p r o p a g a t i n g i n a s h e a r flow", P h i l . T r a n s . R . SOC. London, S e r A 417.
m,
[ 1 4 ] Grimshaw, R., 1984: "Wave a c t i o n and wave-mean f l o w i n t e r a c t i o n , w i t h 11-43. a p p l i c a t i o n t o s t r a t i f i e d s h e a r f l o w s " , Ann. Rev. F l u i d Mech.,
n,
98
R. Grimshaw Hayes, W.D., 1970: " C o n s e r v a t i o n of a c t i o n and modal wave a c t i o n , "Froc. R. SOC. London., S e r A m(T, 187-208.
(161 Huthnance, J.M.,
1981: waves", J. F l u i d Mech.,
"On mass t r a n s p o r t s g e n e r a t e d by t i d e s and long
m, 367-387.
[17] McIntyre, M . E . , 1977: "Wave t r a n s p o r t in s t r a t i f i e d r o t a t i n g f l u i d s " i n "Problems o f S t e l l a r Convection"; pp 290-314, ed. E . A . S p i e g e l and J.P. S p r i n g e r , pp 363. Zahn: L e c t u r e Notes in P h y s i c s [ 1 8 ] McIntyre, M . E . , 1980: "An i n t r o d u c t i o n t o t h e g e n e r a l i z e d Lagrangian-mean 152-176. d e s c r i p t i o n of wave, mean-flow a c c e l e r a t i o n " , Pageoph.,
m,
[19] Rhines, P.B. and H o l l a r d , W.R., 1979: "A t h e o r e t t c a l d i s c u s s i o n of eddyd r i v e n mean flows", Dyn. Amos. Oceans, 7 , 289-325. [20] U r y u , M . ,
Pageoph.,
1980:
" A c c e l e r a t i o n of mean z o n a l f l o w s by p l a n e t a r y waves",
m, 661-693.
[ 2 1 ] Whitham, G.B.,
1965: "A g e n e r a l approach to l i n e a r and n o n l i n e a r d i s p e r s i v e 273-283. waves u s i n g a Lagrangian", J. F l u i d Mech.,
[ 2 2 ] Whitham, G.B., F l u i d Mech., [ 2 3 ] Young, W.R.
m,
a,373-395. 1970:
"Two-timing,
v a r i a t i o n a l p r i n c i p l e s and waves", J .
and R h i n e s , P.B., 1980: "Bossby wave a c t l o n , e n s t r o p h y and energy i n f o r c e d mean f l o w s " , Geophys. Astrophys. F l u i d Dynamics, 'fs, 39-52.
83
A LAGRANGIAN VIEW OF WAVE-MEAN FLOW INTERACTION R. Grimshaw
Department of Mathematics U n i v e r s i t y of Melbourne P a r k v i l l e , V i c t o r i a 3052 Australia
It i s shown how t h e wave a c t i o n e q u a t i o n i s r e a d i l y o b t a i n e d from a Lagrangian Formulation of t h e e q u a t i o n s of motion. The c o u n t e r p a r t t o t h e wave a c t i o n e q u a t i o n i s t h e mean-flow e q u a t i o n which c a n a l s o be simply d e r i v e d i n a Lagrangian Eormulation; i n t h i s e q u a t i o n t h e wave f o r c i n g c a n b e expressed e i t h e r by a r a d i a t i o n s t r e s s t e n s o r o r by t h e wave pseudomomentum. A f t e r d e v e l o p i n g t h e t h e o r y i n a g e n e r a l framework t h e r e s u l t s a r e a p p l i e d t o a compressLble f l u i d . In p a r t i c u l a r i t is shown t h a t , u s i n g s p h e r i c a l p o l a r coo r d i n a t e s , t h e wave a c t i o n e q u a t i o n and t h e mean-flow e q u a t i o n t a k e Eorms u s e f u l f o r a p p l i c a t i o n s f n s t r a t o s p h e r i c meteorology. F i n a l l y t h e t h e o r y i s a p p l i e d t o a s e t of e q u a t i o n s which d e s c r i b e t h e i n t e r a c t i o n of b a r o t r o p i c t o p o g r a p h i c , @-plane Rossby waves w i t h a mean Elow.
91.
INTRODUCTION
The i n t e r a c t i o n between waves and mean flows is a t o p i c w i t h r e l e v a n c e t o many d i E f e r e n t b r a n c h e s of p h y s i c s . I n r e c e n t y e a r s c o n s i d e r a b l e i n t e r e s t in t h i s s u b j e c t h a s developed w i t h i n t h e g e o p h y s i c a l f l u i d dynamics community. Here, as e l s e w h e r e , t h e b e n e f i t s of d e s c r i b i n g t h e L n t e r a c t i o n w i t h i n a L a g r a n g i a n framework a r e now g e t t i n g i n c r e a s i n g r e c o g n i t i o n . Thus, on t h e one hand, t h e r e s p o n s e of t h e waves t o t h e mean flow i s b e s t d e s c r i b e d by t h e wave a c t i o n e q u a t i o n , which i s a n out-growth from t h e a c t i o n p r i n c i p l e s of c l a s s i c a l mechanics. In t h e f l u i d dynamics c o n t e x t t h e c e n t r a l r o l e of t h e wave a c t i o n e q u a t i o n w a s f i r s t recognized i n t h e p i o n e e r i n g work of Whitham (1965, 1970) and B r e t h e r t o n and G a r r e t t (1968). On t h e o t h e r hand, t h e r e s p o n s e of t h e mean f l o w t o t h e waves i s d e s c r i b e d w i t h i n t h e c o n t e x t of a g e n e r a l i z e d Lagrangian-mean Eormulation, i n which t h e a c c e l e r a t i o n of t h e Lagrangian-mean f l o w due t o t h e waves i s d e s c r i b e d e i t h e r by t h e r a d i a t i o n s t r e s s t e n s o r , O K by t h e pseudomomentum. The p r e l i m i n a r y i d e a s were developed by D e w a r (1970) and B r e t h e r t o n (1971) and r e c e i v e d t h e i r f u l f i l m e n t i n t h e g e n e r a l i z e d Lagrangian-mean t h e o r y of A n d r e w s and McIntyre (1978a,b). A g e n e r a l review of b o t h wave a c t i o n and wavemean Elow i n t e r a c t i o n i s c o n t a i n e d in a r e c e n t review by Grimshaw (1984).
In t h i s a r t i c l e w e s h a l l f i r s t g i v e a b r i e f a c c o u n t of t h e g e n e r a l t h e o r y based o n a L a g r a n g i a n f o r m u l a t i o n of t h e e q u a t i o n s of motion in o r d e r t o s e t t h e s c e n e f o r t h e developments of subsequent s e c t i o n s . A more comprehensive a c c o u n t of t h e g e n e r a l t h e o r y i s d e s c r i b e d by Grimshaw (1984). Then i n 92 we s h a l l d e s c r l b e t h e i m p l e m e n t a t i o n of t h e g e n e r a l t h e o r y f o r a c o m p r e s s i b l e f l u i d u s i n g t h e In 93 t h i s g e n e r a l i z e d Lagrangian-mean t h e o r y of Andrews and P k I n t y r e (1978a,b). t h e o i y i s r e c a s t i n s p h e r i c a l p o l a r c o - o r d i n a t e s and we d e m o n s t r a t e how t h e wave a c t i o n e q u a t i o n and t h e Lagrangian-mean flow e q u a t i o n s a r e r e l a t e d t o t h e Eliassen-Palm r e l a t i o n s and t h e Charney-Drazin theorems of s t r a t o s p h e r i c meteorology. Then i n 94 w e d e s c r i b e how t h e g e n e r a l t h e o r y c a n b e used t o s t u d y b a r o t r o p i c t o p o g r a p h i c Rossby waves on a p-plane.
84
R. Grimshaw
To d e s c r i b e t h e g e n e r a l t h e o r y we s h a l l suppose t h a t t h e p h y s i c a l system is s p e c i f i e d by t h e vector-valued f i e l d Q ( t , x ) where t i s t h e time and xi ( i = 1, 2 , 3) a r e s p a t i a l c o - o r d i n a t e s . i I n t h e a b s e n c e of d i s s i p a t i o n t h e p h y s i c a l system obeys a v a r i a t t o n a l p r i n c i p l e w i t h Lagrangian d e n s i t y Then t h e e q u a t i o n s of motion a r e L(c$~, Qxi, Q; t , x i ) .
where t h e g e n e r a l i z e d f o r c e Q r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . Our d e r i v a t i o n of t h e wave a c t i o n e q u a t i o n w i l l b e based o n t h e i d e a s of Whitham (1965, 1970) b u t w i l l f o l l o w t h e development by Hayes (1970) more c l o s e l y . Thus, w e suppose t h a t @ ( t , x i ; 8 ) depends o n t h e ensemble parameter 8 such t h a t
e+
Q ( t , xi;
2n)
Q ( t , xi;
=
el,
(1.2)
and t h e n d e f i n e t h e a v e r a g i n g o p e r a t o r
For s i m p l i c i t y w e s h a l l d e n o t e t h e mean f i e l d <$> by 3 . The a v e r a g i n g o p e r a t o r commutes w i t h d i f f e r e n t i a t i o n and h a s o t h e r s i m p l e and obvious p r o p e r t i e s (Andrews e n e x t d e f i n e t h e wave p e r t u r b a t i o n o r d i s t u r b a n c e f i e l d andAMcIntyre, 1978a,b). W by Q where
(1.4)
C$=?+C$
We s h a l l assume t h a t Q i s O(a) where a i s a measure of wave a m p l i t u d e , a l t h o u g h t h e r e i s a t p r e s e n t no r e s t r i c t i o n on t h e magnitude of a vis-a-vis t h a t oE I$,nor I f (1.1) is m u l t i p l i e d on t h e i r r e l a t i v e s c a l e s w i t h r e s p e c t t o t and xi. by $ e ( i . e . a$/ae) and t h e r e s u l t a v e r a g e d , i t may b e shown t h a t
(1.5a) (1.5b)
where =
@i
aL < Q e -->, wx
(1.5~)
i
a =< ieQ >.
(1.5d)
ai
Equation (1.5a) is t h e wave a c t i o n e q u a t i o n , * i s t h e wave a c t i o n d e n s i t y , is t h e wave a c t i o n f l u x I n d 8 is a t e r m r e p r e s e n t i n g t h e e f f e c t s of d i s s i p a t i o n . B o . t h k and 6, are O(a ) wave q u a n t i t i e s , and a r e t h e a p p r o p r i a t e g e n e r a l measures of wave a c t i v i t y . A p p l i c a t i o n s of ( 1 . 5 a ) depend on d e l i n e a t i o n of t h e f a m i l y C$(t,x i ; 0 ) . One c h o i c e which fs commonly used i s t o i d e n t i f y 8 a s a phase s h i f t , s o t h a t t h e a v e r a g i n g o p e r a t o r c a n b e i n t e r p r e t e d as a phase a v e r a g e . Thus w e p u t
Q ,
a nd
.
A
=
$ ( t , xi;
s ( t , xi)
w = -s
- el,
(1.6a) (1.6b)
t'
Ki
= sxi'
85
A Lagrangian View of Wave-Mean Flow Interaction Here s ( t , x i ) is t h e phase, 4 i s p e r i o d i c i n s , w i s t h e wave frequency and i s t h e wavenumber. It f o l l o w s t h a t
a
Ki
(1.7)
i
and t h e wave a c t i o n e q u a t i o n (1.5a) t a k e s t h e form o b t a i n e d by Whitham (1970) i n t h e absence o f d i s s i p a t i o n . Another c h o i c e i s t o i d e n t i f y 9 w i t h a c o - o r d i n a t e so t h a t t h e a v e r a g i n g o p e r a t o r i s a c o - o r d i n a t e a v e r a g e . It i s remarkable t h a t t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) is f o r m a l l y e x a c t , v a l i d w i t h o u t any a p p a r e n t r e s t r i c t i o n o n wave a m p l i t u d e o r w i t h o u t any assumption t h a t t h e mean f i e l d i s s l o w l y v a r y i n g w i t h r e s p e c t t o t h e waves. There i s , of c o u r s e , a n a s y m p t o t i c e r r o r i n c u r r e d in t h e i d e n t i f i c a t i o n of t h e f a m i l y ( 1 . 2 ) (Hayes, 1970). Turning next t o t h e mean flow e q u a t i o n w e f i r s t d e f i n e t h e " u n d i s t u r b e d " Lagr a n g i a n by
Note t h g t a l t h o u g h Lo does n o t depend e x p l i c i t l y o n t h e wave f i e l d $, t h e mean f i e l d s 4 g e n e r a l l y c o n t a i n O(a2) wave-induced components. Next w e p u t
L~ i s a n o ( a 2 ) wave p r o p e r t y , and i t s e x p l i c i t dependence o n t , x i l i n c l u d e s i t s dependence on t h e mean f i e l d s 0. I f we now m u l t i p l y (1.1) by $. and a v e r a g e J t h e r e s u l t , we c a n show t h a t
aT. aT . -&! +&
=
-<
i
ax1 > +
(1.10a)
j
where
(1.10b)
>
T.. = J1
- n,>bji'
(1.1Oc)
is t h e pseudomomentum, and -T.i t h e c o r r e s p o n d i n g f l u x . The s i g n s are H e r e -T a g r e e w i t h h i s t o r i c a l conventl!on (Andrews and M c l n t y r e , 1978b). A chosen similar e q u a t i o n can b e o b t a i n e d f o r t h e pseudoenergy by m u l t i p l y i n g ( 1 . 1 ) by 0, and a v e r a g i n g t h e r e s u l t (Grimshaw, 1984). Note t h a t when 9 is i d e n t i f i e d w i t h t h e wave a c t i o n d e n s i t y A (1.5b) r e d u c e s t o -Tjo, and t h e t h e c o - o r d i n a t e -x j’ wave a c t i o n f l u x 6 ( 1 . 5 ~ ) r e d u c e s t o -Tji; a l s o e q u a t i o n (1.10a) t h e n r e d u c e s t o t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) .
2:
To make f u r t h e r p r o g r e s s w e now assume t h a t - t h e mean f i e l d s v e l o c i t y ii and a vector-valued mean f i e l d h , where dh - + A dt
a; ij
i
h--=O,
ax
j
7
c o n s i s t oE a mean
(1.11)
86
R. Grimshaw
x
where u r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . In a p p l i c a t i o n s w i l l incorporate t h e e f f e c t s of mean d e n s i t y , mean f l u i d d e p t h e t c . (compare Garrett (1968) and Note t h a t i n t h e absence of d i s s i p a t i o n A i s a mean q u a n t i t y Dewar-(1970)). ( A = 1); however i t i s u s e f u l i n a p p l i c a t i o n s t o l e t h have a f l u c t u a t i n g , d i s s i p a t i v e component. We n e x t d e f i n e
which i s thg: time d e r i v a t i v e f o l l o w i n g t h e meanlmotion, and assume t h a t L1 only t h r o u g h i t s dependence on d4ddt; f u r t h e r a p a r t from t h e depends on dependence L1 o! Ci through d $ / d t any o t h e r e x p l i c i t denpendence o f L1 o n These h y p o t h e s e s ii i s b i l i n e a r i n ui and t h e d i s t u r b a n c e v a r i a b l e s Q and $x
+
OF
i
.
a r i s e from t h e f a c t t h a t t h e f u l l Lagrangtan i s u s u a l l y a t most q u a d r a t i c i n t h e v e l o c i t y f i e l d . The mean flow e q u a t i o n c a n now b e o b t a i n e d by a v e r a g i n g (1.1). The d e t a i l s of t h e d e r i v a t i o n a r e d e s c r i b e d by Grimshaw (1984), and t h e r e s u l t i s
a aLO aLoE ( Y ) +a F- b_ aui j j a;,
+ R
where
ij
=
~
a R i jax j
-T
ij
<
~
aL axi
+;.T
10
J
.? aLO iLOsij)
A
>e + -
ah
iJ
(1.13a)
i j
aL 1 h-->
(1.13b)
?A
Here R i j is t h e r a d i a t i o n stress t e n s o r , a l t h o u g h n o t e t h a t t h e terminology i s not u n i v e r s a l . A l s o
!
a (-=aLO
at
i
)
+ aa x( -u j j
aLo ) hi
>e
+
+ - < -aA ax. J
+
. h aL a;> iJ
$x Q>. i
T h i s i s t h e form p r e f e r r e d by Andrews and McIntyre (1978a). s e c t i o n by n o t i n g t h e a l t e r n a t i v e e x p r e s s i o n f o r Rij,
+ax a no> i
(1.14) We c o n c l u d e t h i s
(1.15)
a7
A Lagrangian View of Wave-Mean Flow Interaction Here (i3Ll/a$x2)d d e n o t e s t h e d e r i v a t i v e w i t h r e s p e c t t o constant.
J
S i m l l a r l y I t c a n b e shown t h a t
8,
=
ix
4 J
when d $ / d t i s h e l d
-
(1.16a)
Uik+4+
(1.16b)
where
92.
GENERALIZED LAGRANGIAN-MEAN
FORMULATION
The equati-ons of motion f o r a c o m p r e s s i b l e f l u i d a r e (2.la) (2.lb) (2.lc) (2.ld)
where
Here x: i s t h e Eul r i a n c o - o r d i n a t e such t h a t a f l u i d p r t i c l e a t x i h a s The n o t a t i o n i s s t a n d a r d ; t h u s Qi i s t h e c o n s t a n t a n g u l a r v e l o c i t y v e l o c i t y ui. of t h e frame of r e f e r e n c e , @ ( x i ) i s t h e p o t e n t i a l f o r g r a v i t a t i o n a l and c e n t r i f u g a l f o r c e s , p(p, S ) i s t h e thermodynamic p r e s s u r e , p i s t h e d e n s i t y and S i s t h e e n t r o p y . The terms Xi and h r e p r e s e n t t h e e f f e c t s of d i s s i p a t i v e and d i a b a t ic e f f e c t s r e s p e c t i v e l y
.
The a p p r o p r i a t e L a g r a n g i a n f o r m u l a t i o n of t h e s e e q u a t i o n s i s t h e g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n of Andrews and McIntyre (1978a), which c o n t a i n s a comprehensive a c c o u n t of t h i s t h e o r y . Here we s h a l l g i v e only a b r i e f o u t l i n e ( s e e a l s o McIntyre (1977, 1980), Grimshaw (1979) o r Dunkerton (1980)). L e t xi b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s and l e t c i ( t , xo) b e t h e p a r t i c l e displacements defined s o t h a t
Then, f o r any g i v e n ui t h e r e i s a u n i q u e " r e f e r e n c e " v e l o c i t y i l ( t , x . ) s u c h J t h e p o i n t x i moves w i t h t h a t when t h e p o i n t xi moves w i t h v e l o c i t y The material time d e r i v a t i v e ( $ . l a ) i s t h e n g i v e n by v e l o c i t y ui.
The g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n i s now o b t a i n e d by l e t t i n g mean v e l o c i t y , and r e q u i r i n g t h a t
= 0.
ui
be t h e
(2.4)
Thus (2.3) a g r e e s w i t h ( 1 . 1 2 ) . The r e a d e r should n o t e t h a t o u r n o t a t i o n d i f f e r s i n one s i g n i f i c a n t r e s p e c t from Andrews and McIntyre (1978a,b); h e r e ii d e n o t e s
88
R. Grimshaw
zt
t h e Lagrangian-mean v e l o c i t y , r a t h e r t h a n used by Andrews and McIntyre (1978a,b), who u s e t h e s i n g l e o v e r b a r t o d e n o t e E u l e r i a n means. No c o n f u s i o n should arise as E u l e r i a n means w i l l n o t b e d i s c u s s e d i n t h i s a r t i c l e .
-
Next w e d e f i n e a mean d e n s i t y p s o t h a t
au g+p - = i ax
0.
(2.5)
i
It i s a n immediate consequence of ( 2 . l b ) and ( 2 . 5 ) t h a t
-
PJ where
(2.6b)
= P,
J = det[
axi ~
ax.
1.
(2.6b)
J
We l e t K i j b e t h e i , j
-
c o - f a c t o r of J , and s o
Note t h a t U i j is t h e d e r i v a t i v e of 3 w i t h r e s p e c t t o B x i / a x . and aU /axj = 0 . J ij The momentum e q u a t i o n ( 2 . l a ) now becomes
-
dui p-+ dt
-
2 p ~ Q u ijk j k
ui = ui
(2.8a)
.) = pXI,
iJ
j
-
where
.
@ a + p- -a-axi ;+-(pK ax d5,
+dt
(2.8b)
The g e n e r a l i z e d L a g r a n g i a n m e a n e q u a t i o n s a r e t h u s ( 2 . 8 a ) , (2.5) mean f i e l d s a r e Gi, p and S and t h e wave p e r t u r b a t i o n f i e l d i s Lagrangian is
and ( 2 . 1 ~ ) . The A suitable
ci.
- 1 p{.j uiui
+
E
B
X’U
ijk i j k
-
@(xi)
-
-
E(p, S)}.
(2.9)
Here w e r e c a l l t h a t p i s d e f i n e d in terms of p by ( 2 . b a ) , and E ( p , S) is t h e i n t e r n a l energy p e r u n i t mass, so t h a t
(2.10)
where T is t h e temperature. V a r i a t i o n s i n x; ( o r e q u i v a l e n t l y i n ti) g i v e (2.8a) w i t h Q i = pXi. I n o r d e r t o k e e p the-correspondence w i t h t h e g e n e r a l t h e o r y o f $1 a s c l o s e as p o s s i b l e w e d e f i n e Qs = pT s o t h a t v a r i a t i o n s w i t h r e s p e c t t o S-can b e d e f i n e d . The v e c t o r A of $1 i s i d e n t i f i e d as t h e 2-vector w i t h components p and S , and w e n o t e t h a t ( 2 . 1 ~ ) and (2.5) h a v e t h e r e q u i r e d form (1.11).
89
A Lagrangian View of WaveMean Flow interaction The wave a c t i o n e u a t i o n c a n now b e o b t a i n e d from t h e g e n e r a l t h e o r y of 51, u s i n g ( 1 . 5 a , b , c , d ) and q l . l 6 a , b ) .
where
,. ac.,
.4= < p -
ae
(2.11c)
8 =< -. p ( ac.,
ae Xi
as T>. +-
(2.11d)
ae
ThLs a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978b) a l t h o u g h t h e d i s s i p a t i v e term& h a s been w r i t t e n i n a d i f f e r e n t form h e r e . F o r l i n e a r i z e d wave motion, i t i s u s e f u l t o i n t r o d u c e t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n
%+
=
2 O(a ) .
(2.12)
It may t h e n b e shown t h a t
(2.13)
Note t h a t t h e second term h e r e i s i d e n t i c a l l y non-divergent and c a n b e o m i t t e d from (2.11a). To d e r i v e t h e mean flow e q u a t i o n s we f i r s t i d e n t i f y LO from (1.8).
Thus
The mean f l o w e q u a t i o n i s t h e n o b t a i n e d from where L i s g i v e n by ( 2 . 9 ) . ( 1 . 1 3 a , b ) , o r more d i r e c t l y by a v e r a g i n g ( 2 . 8 a ) . The r e s u l t i s (2.15a) where and
-
-
P = P(P,
S),
(2.15b) (2.15~)
It c a n b e v e r i f i e d t h a t Rij i s t h e same r a d i a t i o n stress t e n s o r d e f i n e d by An a l t e r n a t i v e form of (2.15a) i n v o l v i n g t h e pseudomomentum -Ti0 i n s t e a d (1.14). of t h e r a d i a t i o n stress t e n s o r c a n b e o b t a i n e d u s i n g t h e p r o c e d u r e d e s c r i b e d i n $1 and a v e r a g i n g ( s e e ( s e e (1.14), o r more d i r e c t l y by m u l t i p l y i n g ( 2 . l a ) by ax;/ax j Andrews and McIntyre (1978a)). The r e s u l t i s
90
R. Grimshaw
;a = + aTio - a t + - aa ~ (. U- j T i o ) Tjo 2 ax, J +
lt =
where
a + E> +m- < P
-
P<
'j
2 (; 2 + d5
ax' axi
> + <; T^ as >, ax,
d5
EjklQ$,)>'
and
(2.16a)
(2.16b)
(2.16~)
T h i s a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978a) a l t h o u g h t h e d i s s i p a t t v e terms on t h e right-hand s i d e of (2.16a) have been w r i t t e n i n a d i f f e r e n t form h e r e . s3.
GENERALIZED ELIASSEN-PALM AND CHARNEY-DRAZIN THEOREMS
One of t h e most promising a r e a s f o r a p p l i c a t i o n of t h e g e n e r a l r e s u l t s of $2 i s s t r a t o s p h e r i c meteorology. With t h e a v e r a g i n g o p e r a t o r i n t e r p r e t e d as a z o n a l a v e r a g e , t h e c o u n t e r p a r t s of t h e wave a c t i o n e q u a t i o n (2.11a) and t h e mean f l o w e q u a t i o n (2.15a) ( o r (2.16a)) are u s u a l l y c a l l e d Eliassen-Palm and Charney-Drazin theorems r e s p e c t i v e l y , a f t e r t h e p i o n e e r i n g work of E l i a s s e n and Palm (1961) and Charney and D r a z i n (1981) ( f o r r e c e n t reviews see Andrews and & I n t y r e (1978c), McIntyre (1980), Dunkerton (1980) and Uryu (1980)). To o b t a i n t h e s e theorems i n t h e i r most g e n e r a l form w e must f i r s t e x p r e s s t h e e q u a t i o n s of motion i n s p h e r i c a l A s i m p l e and c o n v e n i e n t way of d o i n g t h i s i s t o i n t r o d u c e polar co-ordinates. s p h e r i c a l p o l a r c o - o r d i n a t e s i n t o t h e Lagrangian (2.9); t h i s i s t h e approach used by R r e t h e r t o n (1982) f o r small-amplitude waves. The c o r r e s p o n d i n g r e s u l t s i n c y l i n d r i c a l p o l a r c o - o r d i n a t e s are d e s c r i b e d by Grimshaw (1984). Thus w e l e t t h e c o - o r d i n a t e s x i of 51 b e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , h a n d v, where r i s t h e r a d i u s , h t h e c o - l a t i t u d e and v t h e l o n g i t u d e . These a r e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , whose E u l e r i a n c o u n t e r p a r t s x a r e I. r ' , A' and v ' . The p a r t i c l e d i s p l a c e m e n t s are t h e n d e f i n e d by
The v e l o c i t y components Ln t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s a r e u
where
= _d _r ' dt
,
d h' dv' u2 = r' , u3 = r' sin h' dt ' dt
(3.2a)
(3.2b)
Here Ui are t h e mean v e l o c i t y components in t h e L a g r a n g i a n c o - o r d i n a t e d i r e c t i o n s , which must b e c a r e f u l l y d i s t i n g u i s h e d from t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s . The Lagrangian i s a g a i n g i v e n by ( Z a g ) , where, f o r s i m p l i c i t y , we s h a l l suppose t h a t t h e a x i s of r o t a t i o n i s i n t h e N-S d i r e c t i o n . Thus t h e L a g r a n g i a n i s
91
A Lagrangian View of Wave-Mean Flow Interaction
- 1
u3 - ~(x;) - ~ ( p , s)},
r2 s i n A PI- u u + 9 r ' s i n 2 i i
r f 2 s i n A ' PJ
where
=
i.
r2 s i n A
(3.3a) (3.3b)
and J i s a g a i n d e f i n e d by (2.6b) b u t now xi and xi a r e t h e s p h e r i c a l p o l a r coo r d i n a t e s . V a r i a t i o n s i n Si now g i v e t h e e q u a t i o n s o € motion.
-p { dul ~
dt
L
L
u2 - I u 3 - 2Q s i n h' u3 - -T K r
+ aQ ar
}
(3.4a)
(3.4b)
(3.4c) The c o u n t e r p a r t of ( 2 . 5 ) i s
w h i l e S a g a i n s a t i s f i e s (2.1.2). The wave a c t i o n e q u a t i o n c a n now b e o b t a i n e d from ( 1 . 5 a , b , c , d ) .
-a + - ( -a where
=
-
at
art r2 s i n ~ p < u
+ r2
ae
1
sin h
;=
ui
axi
hi
+
rf
*i )
=
$ u2 + r'
l p ~
<,I2
+
s i n h'
The r e s u l t i s (3.6a)
I
s i n A’
a v r >, s i n 2 A’ -
ae
ac ae pKji>,
ae
u
3
> (3.6b)
(3.6~)
92
R. Grimshaw
b a nd
h
1
=
=
-
r 2 sin Ap
ae
i
as +-T>.
(3.6d)
ae
h ' = 1; h = r, h ' = r ' ; h = r s i n A, h ' = r ' s i n 1 2 2 3 3
A’.
(3.6e)
Here w e remind t h e r e a d e r t h a t xi a r e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , A and v. For l i n e a r i z e d wave motion w e i n t r o d u c e t h e E u l e r i a n p r e s s u r e - p e r t u r b a t i o n p' by ( 2 . 1 2 ) . It may t h e n b e shown t h a t
The second term i s i d e n t i c a l l y t h e b a s i c f l o w is z o n a l ( i - e . i n t e r p r e t e d as a z o n a l a v e r a g e e q u a t i o n (3.6a) r e d u c e s t o t h e Andrews and McIntyre ( 1 9 7 8 ~ ) .
non-divergent and c a n be o m i t t e d from (3.6a). When and z2 a r e O(a2)) and t h e a v e r a g i n g o p e r a t o r is ( i . e . 8 i s i d e n t i f i e d w i t h -v) t h e wave a c t i o n g e n e r a l i z e d Eliassen-Palm r e l a t i o n d e r i v e d by Also -(r2 s i n ?-,)-!& i s t h e a n g u l a r pseudomomentum.
The mean flow e q u a t i o n s c a n now b e o b t a i n e d from (1.13a) ( o r ( 1 . 1 4 ) ) ( a f t e r a l l o w i n g f o r t h e p r e s e n c e of t h e g e o m e t r i c a l f a c t o r s h i i n L ) , o r more d i r e c t l y by a v e r a g i n g ( 3 . 4 a , b , c ) . The e s s e n t i a l s t e p in t h i s l a t t e r approach is
(3.8a)
where
R
ij
=
6
.<,I2
iJ
sin A' PJ
-
r 2 sin A
p"> (3.8b)
*
-
Here p is p ( p , 9) and R i j i s t h e r a d i a t i o n stress t e n s o r ( i . e . t h e a n a l o g u e o f ( 2 . 1 5 ~ ) in s p h e r i c a l p o l a r c o - o r d i n a t e s ) . For i n s t a n c e , t h e z o n a l mean flow e q u a t i o n is
-
1
aR
A= p
r2 s i n A a x j where
M =
A’ u3
+ GrI2
s i n 2 XI>.
A' X3>,
(3.9c)
(3.9b)
Here M is t h e z o n a l mean s p e c i f i c a n g u l a r momentum; € o r t h e r e l a t i o n s h i p between M and t h e z o n a l mean f l o w , see Dunkerton (1980). When t h e a v e r a g i n g o p e r a t o r is i n t e r p r e t e d as a z o n a l a v e r a g e ( i . e . 0 i s i d e n t i f i e d w i t h -v) t h e o f f - d i a g o n a l components of R v j are i d e n t i c a l t o Bq; simply compare (3.8b) w i t h ( 3 . 6 ~ ) . Note t h a t t h e d i a g o n a l components of R v j do not now a p p e a r in ( 3 . 9 ~ ) . With t h e s e i n t e r p r e t a t i o n s , and u s i n g ( 3 . 6 a ) , e q u a t i o n ( 3 . 9 ~ )becomes
A Lagrangian View of Wave-Mean Flow Interaction
dM dt
--
dt
(
r2L -?*--+ pr
pr2 s i n A
sin A
x3>.
93 (3.10)
With t h e f u r t h e r r e s t r i c t i o n t o l i n e a r i z e d waves on a z o n a l b a s i c flow (3.10) r e d u c e s t o a g e n e r a l i z e d Charney-Drazin theorem ( s e e t h e s i m i l a r r e s u l t s o b t a i n e d by Andrews and McIntyre ( 1 9 7 8 ~ )and B r e t h e r t o n ( 1 9 8 2 ) ) . It f o l l o w s t h a t M w i l l change o n l y i n r e s p o n s e t o wave t r a n s i e n c e r e p r e s e n t e d by t h e term i n v o l v i n g k i n (3.10), o r due t o d i s s i p a t i v e e f f e c t s r e p r e s e n t e d by t h e terms i n v o l v i n g 3 and X3.
S4.
TOPOGRAPHIC, @-PLANE ROSSBY WAVES
The r e s u l t s of $2 and 53 have been o b t a i n e d w i t h o u t any r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e s c a l e of t h e waves, o r w i t h o u t making any of t h e a p p r o x i m a t l o n s such a s t h e h y d r o s t a t i c a p p r o x i m a t i o n , o r t h e q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n , o f t e n used i n g e o p h y s i c a l f l u i d dynamics. These v a r i o u s a p p r o x i m a t i o n s c a n b e used i n p o s t e r i o r i i n t h e e v a l u a t t o n of t h e wave a c t i o n d e n s i t y Jk ((2.11b) o r ( 3 . 6 b ) ) , t h e wave a c t i o n f l u x B 2 ( ( 2 . 1 1 ~ ) o r ( 3 . 6 c ) ) , e t c . However, i t i s i n s t r u c t i v e and sometimes more u s e f u l t o d e r i v e a wave a c t i o n e q u a t i o n and a mean flow e q u a t i o n d i r e c t l y from a n approximate s e t of e q u a t i o n s . In t h i s r e s p e c t t h e g e n e r a l t h e o r y of § l p l a y s a c o - o r d i n a t i n g r o l e . To i l l u s t r a t e t h i s a s p e c t w e s h a l l d e r i v e t h e wave a c t i o n e q u a t i o n and t h e mean f l o w e q u a t i o n s € o r a homogeneous f l u i d u s i n g t h e h y d r o s t a t i c and @-plane a p p r o x i m a t i o n s . In thLs c o n t e x t t h e wave p e r t u r b a t i o n s a r e g e n e r i c a l l y c a l l e d Rossby waves, and d e r i v e t h e i r wave-like c h a r a c t e r from a combination of t h e t o p o g r a p h i c s l o p e , t h e @ - e f f e c t and t h e s h e a r of t h e mean flow. Many a u t h o r s have s t u d i e d t h e i n t e r a c t i o n of Rossby waves w i t h mean flows ( s e e Dickinson (1978) f o r a review). From t h e p o i n t of view adopted i n t h l s a r t i c l e t h e p a p e r s by Grimshaw ( 1 9 7 7 ) , Rhines and Holland (1979), Young and Rhines (1980) and Huthnance (1981) a r e p a r t i c u l a r l y r e l e v a n t , a l t h o u g h n e a r l y a l l t h e r e s u l t s o b t a i n e d by t h e s e a u t h o r s a r e r e s t r i c t e d t o s m a l l a m p l i t u d e waves, o r s u b j e c t t o some o t h e r r e s t r i c t i o n c o n c e r n i n g t h e s t r u c t u r e of t h e mean flow. The r e s u l t s t o b e o b t a i n e d below c o n t a i n no r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e scale of t h e waves v i s a - v i s t h e mean f l o w , and hence c a n b e r e g a r d e d as a g e n e r a l i z a t i o n of t h e c o r r e s p o n d i n g r e s u l t s of t h e s e a u t h o r s . The E u l e r i a n e q u a t i o n s of motion f o r a homogeneous f l u i d i n t h e h y d r o s t a t i c a p p r o x i m a t i o n are (4. l a ) (4.lb) (4.1~) where
H = h(x', y ' )
f = f and
0
+
@1x '
-d= = + au , , + v a Ya ' dt
+
+
5,
(4.ld)
p2y1,
(4.le)
a
(4.1f)
94
R. Grimshaw
Here ( x ' , y ' ) a r e E u l e r i a n h o r i z o n t a l c o - o r d i n a t e s such t h a t a f l u i d p a r t i c l e a t (x’ y ' ) h a s h o r i z o n t a l v e l o c i t y components (u, v ) , w h i l e is t h e f r e e s u r € a c e d i s p l a c e m e n t , and h ( x ' , y ' ) i s t h e e q u i l i b r i u m d e p t h of t h e f l u i d . The terms F and G r e p r e s e n t t h e combined e f f e c t s of wind s t r e s s f o r c i n g and d i s s i p a t i o n . One immediate and useful consequence of ( 4 . 1 a , b and c ) i s t h e p o t e n t i a l v o r t l c i t y equation
<
where
dd tx = E ,
(4.2a)
HX = q + f ,
(4.2b) (4.2~)
and
(4.2d)
To u s e t h e g e n e r a l t h e o r y of $1 w e mst f i r s t r e c a s t t h e s e e q u a t i o n s u s i n g t h e g e n e r a l i z e d Lagrangian-mean Eormulation. The development i s similar t o t h a t d e s c r i b e d i n $2 f o r a g e n e r a l c o m p r e s s i b l e f l u i d . Thus, w e l e t x and y b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , and l e t 5 and 7) b e t h e p a r t i c l e d i s p l a c e m e n t s d e f i n e d so t h a t x ' = x + E .
=
y ' = y + q ,
= 0,
and
(4.3a) (4.3b) (4.3c)
Here i, a r e t h e LagrangLan-mean v e l o c i t i e s . We l e t J b e t h e J a c o b i a n of t h e t r a n s f o r m a t t o n from (x, y ) t o (x’, y ' ) w i t h c o - f a c t o r s K i j ( i . e . (2.6b) and ( 2 . 7 ) r e s t r i c t e d t o h o r i z o n t a l co-ordinates). Then ( 4 . 1 ~ ) becomes (4.4a) where
-
-
Note t h a t H i s mean q u a n t i t y .
H = JH.
(4.4b)
The e q u a t i o n s o f motion ( 4 . l a , b ) become
Here we are t e m p o r a r i l y u s i n g i n d e x n o t i o n where u1 = u, u2 = v e t c . a b l e Lagrangian i s
A suit-
95
A Lagrangian View of Wave-Mean Flow Interaction
2
where
u u
i i
+
f(k
-
+
e) u }
x
, i i
+
e = f x’ 0-
p,x'
2
g(hC
+
i
,
I
21
f
< 2 )J -
gH6,
2
(4.6a)
(4.6b)
p2y' j . I
Here i,, j and k, a r e u n i t v e c t o r s i n the_ x , y and z - d i r e c t i o n s r e s p e c t i v e l y . g i v e (4.4b). W e are V a r i a t i o n s i n Ei g i v e (4.5) w i t h Q i = HFi; v a r i a t i o n s in now i n a p o s i t i o n t o a p p l y t h e g e n e r a l t h e o r y of $1, n o t i n g t h a t (4.4a) h a s t h e r e q u i r e d Eorm ( 1 . 1 1 ) .
<
The wave a c t i o n e q u a t i o n i s g i v e n by ( 1 . 5 a , b , c , d ) , (4.7a)
- xi
+
ae
1
(k
x
e)i>,
(4.7b)
I
(4.7c)
8
=
- ac,
~
ae
F
i
>.
(4.7d)
An i n t e r e s t i n g f e a t u r e of (4.7b) i s t h e manner t n which t h e p-plane e f f e c t , h e r e r e p r e s e n t e d by t h e p a r a m e t e r s pi, m o d i f i e s t h e e x p r e s s i o n f o r t h e wave a c t i o n d e n s i t y (2.11b) o b t a i n e d i n $2. T h i s is more a p p a r e n t from t h e a l t e r n a t i v e expression t o (4.7b).
(4.8a) where
-
€ = €
0
+ p x + p y . 2 1
(4.8b)
F u r t h e r , i t c a n b e shown t h a t ( 4 . 8 a ) a g r e e s w i t h (3.6b) when t h e a p p r o p r i a t e p-plane, h y d r o s t a t i c and " t r a d i t i o n a l " a p p r o x i m a t i o n s ( i . e . n e g l e c t of t h e h o r i z o n t a l component of t h e e a r t h ' s r o t a t i o n ) a r e made i n ( 3 . 6 b ) . For l i n e a r i z e d wave motion t h e l a s t term i n k ( 4 . 8 a ) c a n b e n e g l e c t e d a n d m ? i s g i v e n by
where C ' i s t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n (compare ( 2 . 1 2 ) ) and t h e second term i n (4.9) i s i d e n t i c a l l y non-divergent, and hence c a n b e o m i t t e d from ( 4 . 7 a ) .
96
R. Grimshaw
The mean flow e q u a t i o n s a r e o b t a i n e d from (1.13a) o r (1.14) a f t e r a l l o w i n g f o r t h e e x p l i c i t dependence o n xi c o n t a i n e d i n t h e term i n v o l v i n g e i n ( 4 . 6 a ) , as a consequence of whtch t h e Lagrangian L i s n o t b i l i n e a r i n ui and The f o r m e r i n v o l v e s t h e r a d i a t i o n stress t e n s o r and i s r e a d i l y o b t a i n e d by a v e r a g i n g ( 4 . 5 ) . However, i n t h i s c o n t e x t , i t i s not so u s e f u l as s i g n i f i c a n t wave-forcing terms a r e c o n t a i n e d i n
ci.
where
(4.10b)
(4.10d)
and
A comparison of (4.10a) w i t h (1.14) o r (2.16a) s u g g e s t s Lhat Pi(4.10b) i n (4.10a) p l a y s t h e r o l e of pseudomomentum, a l t h o u g h w e n o t e t h a t HPi d i f f e r s from t h e pseudomomentum - T i o ( l . l O b ) , where h e r e (4.11) The d i f f e r e n c e arises due t o t h e afore-mentioned p r e s e n c e of L.
e,
i n t h e Lagrangian
In many a p p l i c a t i o n s t h e mean-flow e q u a t i o n (4.10a) i s dominated by t h e nearg e o s t r o p h i c b a l a n c e between t h e mean C o r i o l i s f o r c e and t h e mean p r e s s u r e g r a d i e n t . In t h i s case a more u s e f u l form of t h e mean flow e q u a t i o n i s o b t a i n e d W e find that by a v e r a g i n g ( 4 . 2 a ) .
g =E, -
(4.12a)
where
(4.12b)
and
(4.12~)
Note t h a t (4.12b and c ) d e m o n s t r a t e t h a t (4.12a) i s a l s o a n immediate consequence of (4.10a). The s i m p l i c i t y of (4.12a) i s s e l f - e v i d e n t ; t h i s e q u a t i o n i s a
97
A Lagrangian View of Wave-Mean Flow Interaction
g e n e r a l i z a t i o n of similar r e s u l t s which h a v e a p p e a r e d i n t h e l i t e r a t u r e , u s u a l l y f o r l i n e a r i z e d wave m o t i o n ( s e e Rhines a n d Holland ( 1 9 7 9 ) , o r Huthnance ( 1 9 C l ) ) . In p a r t i c u l a r , f o r t h g s e s i t u a t i o n s where t h e dominant component o f is T/n and t h e d i s s i p a t i v e term E i s small ( e . g . where t h e waves a r e s l o w l y v a r y i n g r e l a t i v e t o t h e mean f l o w ) , e q u a t i o n (4.12a) s u c c i n c t l y d e m o n s t r a t e s t h a t t h e mean f l o w c l o s e l y follows t h e geostrophic contours.
X
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