On the bottom relief effect on wind-driven currents in a homogeneous ocean

Dynamics of Atmospheres and Oceans, 2 (1978) 293--320 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

293

ON THE BOTTOM R E L I E F E F F E C T ON WIND-DRIVEN C U R R E N T S IN A HOMOGENEOUS OCEAN

V.A. MITROFANOV

Yaroslavl State University, Yaroslavl (U.S.S.R.) (Received July 18, 1977; Accepted February 8, 1978) ABSTRACT Mitrofanov, V.A., 1978. On the bottom relief effect on wind-driven currents in a homogeneous ocean. Dyn. Atmos. Oceans, 2: 293--320. The three-dimensional model of stationary wind-driven currents in a homogeneous ocean of a variable depth is investigated. The model is linear but includes horizontal and vertical turbulent mixings. Two cases of the behaviour of the isolines of the function f/H are considered, namely: (1) all isolines f/H start at one part of the coastline and end in another part of it, and (2) a certain isoline f/H exists which is tangential to the coastline. Here f is the Coriolis parameter, and H is the depth of the ocean. The first case is the simplest one; it arises in particular if H = constant and the coasts are meridional. The second case is marked by the boundary current separation from the coast. The paper deals with the boundary layers which arise at the surface, bottom, side boundary and inside the ocean. INTRODUCTION T o u n d e r s t a n d t h e n a t u r e o f t h e o c e a n i c c i r c u l a t i o n it is n a t u r a l t o o u t l i n e s o m e leading f a c t o r s a n d t o investigate t h e m with simple models. T h e simplest m o d e l is t h a t o f a h o m o g e n e o u s o c e a n . As is w e l l - k n o w n , m a n y e f f e c t s such as w e s t e r n intensification, coastal upweliing, etc., o b t a i n e d in t h e f r a m e w o r k o f t h e m o d e l are also characteristic o f a n o n - h o m o g e n e o u s o c e a n . Besides, t h e effects arising in a h o m o g e n e o u s o c e a n m a y be m o d e l l e d in a l a b o r a t o r y in a fairly simple w a y t h u s p r o v i d i n g an e x p e r i m e n t a l test o f t h e t h e o r y . T h e latter is n o t u n i m p o r t a n t in view o f t h e great d i f f i c u l t y o f o b s e r v a t i o n s carried o u t in t h e real o c e a n . T h e p a p e r deals w i t h t h e p r o b l e m o f t h e b o t t o m relief e f f e c t o n winddriven c u r r e n t s in a h o m o g e n e o u s o c e a n . A c c o r d i n g t o t h e E k m a n t h e o r y , a p a t t e r n o f c u r r e n t s in a h o m o g e n e o u s o c e a n d e p e n d s u p o n t h e peculiarities o f t h e f u n c t i o n f / H w h e r e f is t h e Coriolis p a r a m e t e r a n d H is t h e d e p t h o f t h e ocean. In particular, t h e coastal i n t e n s i f i c a t i o n m a y exist o n l y near t h o s e parts o f t h e coastline at w h i c h a(f/H)/at > 0. Here t is a c o o r d i n a t e a l o n g t h e coastline w h i c h grows t o t h e positive direction. By positive we m e a n t h e d i r e c t i o n o f t h e m o v e m e n t a l o n g t h e coastline w i t h t h e o c e a n lying t o t h e

right.

U, V, W, p

The interior region:

1

e

Elh !2

1

1

~ 1/2 --v

e

E 1/2

E 112

E 1/2

1

1

5

Ely/2

E,1/2

1

5

1

horizontal

horizontal

vertical

Velocities

Scales

~lt2,rlt2

Elv/2/E 1/2

w rl/2 ~ vl / 2 /, ~:'h

Ev1 /2

vertical

The region fulfils the condition for the vertical velocity at the surface and the bott o m of the ocean and the condition for the horizontal velocity at the side boundary.

The region eliminates the discrepancies in the side and b o t t o m layers at the b o t t o m and the side b o u n d a r y of the ocean, respectively.

The layer eliminates the discrepancy in the corner region 3 at the ocean surface; it provides a balance of the vertical fluxes in the ocean and is characterized by a strong vertical c u r r e n t ; i t causes the condition for the horizontal velocity at the b o t t o m .

The region eliminates the discrepancy in the surface layer at the side b o u n d a r y ; it is characterized by strong horizontal and vertical currents; it causes the discrepancy in fulfilling the condition for the vertical velocity at the ocean surface.

The layer fulfils the conditions for the horizontal velocity at the ocean b o t t o m ; it causes the discrepancy in fulfilling the conditions at the side boundary.

The layer fulfils the conditions for the horizontal velocity at the ocean surface: it is characterized by a strong horizontal current; it causes the discrepancy in fulfilling the conditions at the side boundary.

F u n c t i o n and concise characteristics of the region

The thickness e of the b o t t o m layer depends upon the b o t t o m slope and the relation between E h and E v (see Section 2). The value 5 of the characteristic velocity will be of different order in various subregions of the interior region (see Table II).

6

The corner region:

5

Um ~ VFrl ~ Wm

The side Cupwelling) layer: Ul, Vl, Wl, Pl

Uk, Ok, Wk

region:

The corner

Ub, Ub, Wb

The b o t t o m layer:

4

3

2

The surface (Ekman) layer:

1

UE, OE, WE

Denotation and associated variables

Layer No.

Division of the ocean into regions

TABLE I b~

¢D

295 In addition, the b o t t o m relief effect may cause the appearance of strong internal currents in a homogeneous ocean. For example, if H depends upon the meridional coordinate y only and varies so that a(f/H)/ay = 0 at a certain latitude line, then an intensive horizontal gyre is formed near this line (see the analytical solution b y Kamenkovich and Mitrofanov, 1971 and the laboratory experiment b y Brink et al., 1973). Another interesting effect arises in a homogeneous ocean when the isoline of the function f/H is tangential to the coastline. Kamenkovich and Reznik (1972) showed, in the framework of the Ekman theory, that the b o u n d a r y current separated from the coast in the vicinity of the point of tangency and followed along one of the branches of the isoline in question. The authors" suggested that the effect obtained should be used as an explanation of the Gulf-Stream separation from the western boundary. Thus it was ascertained, in the framework of the Ekman model, that the b o t t o m relief effect may be important. The consideration of the problem of the b o t t o m relief effect is most sensible in the framework of the three-dimensional linear model of a homogeneous ocean. Such a model is more complete in comparison with the two~limensinnal one, as it takes into account horizontal turbulent mixing. Moreover, it is then possible to describe the essential details of vertical circulation in the ocean. The three-dimensional model of a wind-driven circulation in a homogeneous ocean with a constant depth has already been considered in the literature. Pedlosky (1968) investigated the model in a particular case where the coefficients of horizontal and vertical turbulent mixing satisfied a certain relation. Reznik (1970) analysed the same problem with different relations b e t w e e n these coefficients. The generalization of the Pedlosky (1968) and Reznik (1970) problems in the case of a variable depth of the ocean is n o t trivial. The analysis of the problem is complex even in the quasi-static approximation. Therefore we shall first give a short summary of the regions into which the ocean is divided (Tables I and II). The schemes of the division of the ocean into the regions are represented in Figs. 1, 4. In section 1 the initial problem is reduced to the non-dimensional form. The small parameters Eh and Ev appear as horizontal and vertical Ekman numbers, respectively. In section 2 the interior region is distinguished within which horizontal currents do n o t vary along a vertical. The b o u n d a r y layers at the surface and b o t t o m of the ocean are taken into consideration. The equations describing the motion in the interior region are also deduced. In section 3 the side (upweUing) layer and the corner regions 3 and 5 (Fig. 1) are investigated. The boundary conditions are deduced for the interior region. The character of the currents in the interior region depends crucially u p o n the behaviour of the isolines f/H. Thus t w o problems are considered further. Section 4 deals with the problem of circulation in the basin where all iso-

T h e t r a n s i t i o n region:

T h e t r a n s i t i o n region: s t Va, Va, Pa

11

12

T h e coastal layer: s Otn,Pn On,,

9

T h e i n t e r n a l b o u n d a r y layer: 0~ ~ r vi, Pi

T h e coastal layer: ~, t On, Pn

8

10

T h e q u a s i - g e o s t r o p h i c region: r v~, Üg, pg

D e n o t a t i o n a n d associated variables

7

L a y e r No.

E 112

E 1/8

E 7/24

E 3/14

E 1/7

E 2/7

E 1/4

E 112

E 1/6

E 1/2

1

1

1

1

length

Velocity 5

E1/4

E 1/3

E 1/3

1

witdh

Scales

T h e region is l o c a t e d near t h e p o i n t 0 a n d fulfils t h e cond i t i o n s at t h e s e g m e n t o f t h e b o u n d a r y l" w h i c h adjoins t h e i n t e r n a l b o u n d a r y layer.

T h e region is l o c a t e d in t h e vicinity o f t h e p o i n t 0 a n d fulfils t h e b o u n d a r y c o n d i t i o n s near t h e p o i n t ; it is chara c t e r i z e d b y t h e s e p a r a t i o n o f t h e s t r o n g coastal c u r r e n t f r o m t h e b o u n d a r y F.

T h e layer p r o v i d e s a b a l a n c e of t h e h o r i z o n t a l fluxes in t h e o c e a n a n d e l i m i n a t e s t h e d i s c o n t i n u i t y o f t h e quasig e o s t r o p h i c s o l u t i o n at t h e isoline f / H t a n g e n t i a l t o t h e b o u n d a r y F; it is c h a r a c t e r i z e d b y a s t r o n g h o r i z o n t a l curr e n t ; it is n o t valid in t h e vicinity of t h e p o i n t 0.

T h e layer fulfils t h e c o n d i t i o n s at p a r t IF"3 o f t h e b o u n d ary; it is n o t valid in t h e vicinity of t h e p o i n t 0.

T h e layer fulfils t h e c o n d i t i o n s at p a r t F 2 o f t h e b o u n d ary; it is c h a r a c t e r i z e d b y a s t r o n g h o r i z o n t a l c u r r e n t ; it is n o t valid in t h e vicinity o f t h e p o i n t 0.

T h e region is c h a r a c t e r i z e d b y t h e small f r i c t i o n value; t h e s o l u t i o n for it d o e s n o t satisfy t h e b o u n d a r y condit i o n a n d has a d i s c o n t i n u i t y at t h e isoline f / H t a n g e n t t o t h e b o u n d a r y P.

F u n c t i o n a n d consise c h a r a c t e r i s t i c s o f t h e r e g i o n

Division of t h e i n t e r i o r region o f t h e o c e a n near t h e p o i n t 0 w h e r e t h e isoline f / H is t a n g e n t t o t h e side b o u n d a r y ( t h e case E h = E v = E )

T A B L E II

b~

297 lines f / H start at one part of the side boundary and end in another part (Fig. 2a). The coastal b o u n d a r y layers are taken into consideration and the solution in the quasi-geostrophic region is constructed for the case Eh = E~ = E. Section 5 deals with the problem of circulation in the basin where a certain isoline f / H is tangential to the side boundary (Fig. 3a). The internal bound. ary layer forming in the vicinity of this isoline is investigated in the case Eh = E~ = E. The transition regions 11 and 12 (Fig. 4) arising near the point of tangency are also considered. In section 6 the results of the analysis of the problems in question with different relations between Eh and E~ are shown. 1. OUTLINE OF THE PROBLEM Let us consider a linear problem of stationary wind~iriven circulation in a homogeneous ocean of a variable depth, which is situated in temperate latitudes. Accepting the approximations of the hydrostatic ~-plane, we shall write the initial equations and b o u n d a r y conditions in the form: --fi) = - - P x / P + A h A h u + Avu~z fu = --py/p

+ Ah/khi) + AvVzz

(1.1)

Ux + i)y + Wz = 0 p A v u z = " - ' f , PAvVz = "-r y, w = 0

at z = 0

u =v=w=0

at z = H ( x , y )

u=v=0

at F ( x , y ) = 0

(1.2)

Here u, v, w are components of the velocity along the axes o f the Cartesian coordinates x, y, z directed eastward, northward and vertically down, respectively; p ( x , y) is the deviation from the hydrostatic pressure pg = g p z ; g = constant is the gravitational acceleration; p = constant is the density of water; f = f* + ~*y is the Coriolis parameter in which f* and ~* are constants; Ah, Av are coefficients of horizontal and vertical turbulent mixing, respectively; Ah = ¢}2/~}X2 + ~2/ay2 is the two-dimensional Laplace operator; Tx, r y are components of the vector ~ of the tangential stress of the wind on the ocean surface; H ( x , y ) is the depth of the ocean; F(x, y) =0 is the equation of the side boundary *. If it is n o t specified otherwise, we shall denote a partial derivative by a corresponding b o t t o m index and a vector c o m p o n e n t by a top index. Let us introduce characteristic scales L* of the width and H* of the depth of the ocean, T* of the tangential stress of the wind, V* of the horizontal * The region of the ocean is assumed to be simply connected and the side boundary vertical.

298

v e l o c i t y , W* o f the vertical v e l o c i t y , P* o f pressure so t h a t V* = T * / p ( f * A v ) 1/2, W* = V * H * / L * , P* = p V * L * f * . T h e c h a r a c t e r i s t i c value o f t h e v e l o c i t y on t h e surface o f t h e o c e a n is t a k e n as t h e scale V*. T h e scale W* is o b t a i n e d f r o m the e q u a t i o n o f c o n t i n u i t y and P* f r o m t h e a s s u m p t i o n t h a t t h e c u r r e n t s are q u a s i - g e o s t r o p h i c in t h e m a i n region o f t h e ocean. We shall i n t r o d u c e n o n d i m e n s i o n a l ( p r i m e d ) variables b y t h e f o l l o w i n g f o r m u l a e : (x,y)=L

* • (x',y'),

( u , v ) = V* " ( u ' , v ' ) , T=T*.~',

z=H

* .z',

w = W* " w ' ,

f=f*'f'=f*(l

H=H

p=P*

* . H'

"p'

+~y')

w h e r e fl = fi* L * / f * . Passing o v e r t o t h e n o n - d i m e n s i o n a l variables in t h e e q u a t i o n s (1.1) a n d t h e b o u n d a r y c o n d i t i o n s (1.2) a n d o m i t t i n g t h e p r i m e s we o b t a i n : (1.3)

- - f v = - - p ~ + E h A h U + E~u~ z f u = - - B y + E h A h V + EvVzz

(1.4)

Ux + Vy + W z = 0

E~/2Uz =

1/2 v~ = - - ' r ~, w = 0

atz = 0

(1.5)

u = v= w = 0

at z = H ( x , y )

(1.6)

u = v= 0

at F ( x , y) = 0

(1.7)

- - ' r ~, E ~

w h e r e E h = A h / f * L * 2, E v = A v / f * H * 2 are t h e h o r i z o n t a l a n d vertical E k m a n n u m b e r s , r e s p e c t i v e l y , F(x, y ) = 0 is t h e e q u a t i o n o f t h e side b o u n d a r y in its non-dimensional form. A c c e p t i n g L* = 5 • 10 8 c m , H* = 5 • 10 s c m , f* = 10 - 4 sec - 1 , Ah = 10 8 c m 2 sec - 1 , Av = 10 2 c m 2 sec -1 we o b t a i n Eh = Ev = 4 • 10 - 6 . B u t t h e values o f t h e c o e f f i c i e n t s Ah, A~ are n o t k n o w n f o r a real o c e a n a n d t h e values Eh, Ev m a y b e o f q u i t e d i f f e r e n t orders. T h u s we shall m a k e an a s y m p t o t i c a l analysis o f t h e general p r o b l e m ( 1 . 3 ) - - ( 1 . 7 ) w i t h t w o i n d e p e n d e n t small p a r a m e t e r s Eh, Ev. 2. S UR F AC E AND BOTTOM LAYERS Equations

for the interior region

L e t us i n t r o d u c e t h e c o m p l e x f u n c t i o n s U = u + iv, T = T ~ + i~ y. As p d o e s n o t d e p e n d o n z in t h e quasi-static a p p r o x i m a t i o n , t h e f o l l o w i n g e q u a t i o n f o r t h e f u n c t i o n U~ is o b t a i n e d f r o m (1.3): EhAhUz

+ EvUzzz -- ifUz = 0

(2.1)

299

From (2.1) one can see t h a t Uz - 0 everywhere in the ocean except in the boundary layers of the thickness E~/2 or E~/2. Considering the smooth functions T, H it is natural to assume t h a t such layers would form only at the boundaries z = 0, z = H, F(x, y) = 0. The function U does n o t depend on z and according to (1.4) w depends linearly on z in the rest (interior region) of the ocean. Let the m o t i o n in the interior region of the ocean be described with the help of the functions ~r(x, y), u)(x, y, z),/~(x, y) *. Then the function U does n o t satisfy the conditions (1.5). To eliminate the discrepancy we shall introduce a stretched variable [ = z/Ely 12 and correction functions UE(X, y, ~), WE(X, y, ~') describing the m o t i o n in the ocean surface layer of thickness Ev1/2 (the Ekman layer). At the first approximation the determination of the functions UE, WE is reduced to the ordinary Ekman problem and therefore is n o t considered here. We shall deduce only the expressions for the full flux S~ = E~/2 • f ~ U~ d~ of the surface current and velocity WE at the surface of the ocean which are necessary for the further analysis. We have (see Pedlosky, 1968): SE = E~/2SEo + ... , SEO = T / i f

(2.2)

WE[~=O = diVhSE = __p.1/2_vrotz(¢/f) + ...

(2.3) y where diVhSE = S~,, + SEy. The function U does n o t generally satisfy the conditions (1.6). To eliminate the discrepancy we introduce a stretched variable ~ = (H -- z ) / e and correction functions Ub(X, y, 7), Wb(X, y, ~7) describing the motion in the ocean b o t t o m layer of thickness e. We then have the problem: E h A h U b + E h e - - l [ A h H . Ub~ + 2(Vh H - V h V b ~ ) ] -I"

+ E h e - - 2 ( V h H ) 2 U b ~ +Eve - i Ub,, - - ifUb = 0

(2.4)

e(Ubx + Vby) + H,,Ub,~ + HyVb, -- Wb,~ = 0

(2.5)

0 + Ubl¢7=o = 0

(2.6)

from (1.3), (1.4), (1.6), for the functions Ub, Wb, where Vh = ( a l e x , ~/~y) is an operator of the horizontal gradient. It follows from (2.4) t h a t the thickness of the b o t t o m layer depends u p o n the relation between Eh and Ev: I E /2 at Vh H = 0 and atV h H : f : 0, E v > > E h e = ~E 1/2 a t V h H ~ 0, Ev ~ Eh orEv <<~ Eh

* Here we m e a n that the f u n c t i o n s [/, t~,/~ and the correction f u n c t i o n s i n t r o d u c e d below depend u p o n the parameters E h and E v and are expanded into asymptotical series by powers of Eh and E v.

300

Thus, the horizontal mixing plays a substantial role in forming the b o t t o m layer if VhH ¢ 0 and Eh ~ Ev or Eh > > Ev. Let /) = 6Do + ... in a certain region of the ocean. Then we have: Vb = 5Ubo + .... Ubo = - U o exp[--(1 + i)f ~

"7]

(2.7)

from (2.4), (2.6) for the same region where: 1 at VhH = 0 and at V h H =/: 0, E v ~ > E h

X = (Ev/Eh) + (VhH) 2 at V h H ¢ 0, Ev ~ Eh . (VhH) 2 at vhH ¢ 0, Ev < < Eh From (2.7) we find the full flux of the b o t t o m current to be: Sb = e • :

Vb dv =--eS(]0(1 - - i ~ } x / ~ + ...

(2.8)

o

From (2.5), (2.6) we obtain the expression for Wb at the b o t t o m of the ocean: --WD In=0 = Hx~ + Hy~ + diVaSb

(2.9)

Considering the interior region we have: --ffi = --/~x + E h A h t~ (2.10)

ft~ -----~y + E h A h 0

ux + Vy + Wz = 0

(2.11)

Cvtz=o + w~l~:o = 0 , ~vl;=H + Wbln=o = 0

(2.12)

from (1.3)--(1.6) for the functions (], &,/~. Since & is a linear function of z, we find: Cgz = H - I [ H x f + Hyf) + d i v h ( S E + S b ) ]

(2.13)

from (2.3), (2.9), (2.12). By cross differentiating (2.6) and using (2.11), (2.13) we obtain the vorticity equation for the interior region:

(f / H ) x f + (f /H)yO + (Eh/H) Aa (~ = (f /H2 )diva (SE

+

Sb)

(2.14)

The first two terms in the left part of (2.14) depict the horizontal advection of the potential vorticity f/H; the third term the diffusion of the relative vorticity & (note t h a t & = fly -- fix is in the leg-handed system of coordinates). The right part of (2.14) represents the sources of vorticity due to the effect of the surface and b o t t o m layers. The equations (2.10) and (2.14) form a closed (two-dimensional) set which describes the motion in the interior region of the ocean. Boundary conditions for this set will be found from the analysis of boundary layers of thickness E 112 at the side boundary F.

301

3. SIDE L A Y E R S

Boundary conditions for the interior region For the sake of simplicity, let us assume that the ocean has a side boundary at x = 0 and lies at x > 0. Then it follows from (2.2) that SE and thus UE do n o t turn generally to zero at x = 0. To eliminate the discrepancy we shall introduce a stretched variable ~ = x / E 1/2 and correction functions Uk (~, y, ~), wk(~, y, [) describing the m o t i o n in the corner region 3 (Fig. 1). In the general case the corner region is situated in the vicinity of the line F(x, y) = 0, z = 0 and has the scales E 1/2 and E~/2 along the normal to the side b o u n d a r y and the vertical, respectively. Due to its complexity the determination of the functions Uk, wk is n o t considered here. We shall find only the full flux Sk = E~ 12 • f ~ Uk d~" of the c o m e r current and the expression for wk at the surface of the ocean. From {1.3), (1.4), (1.7) we have (see Pedlosky, 1968):

EhSk~y + S k ~ ~ ifSk = 0 SEIx=O + Sklt=0 = 0

which gives the following: Sk = ~vvlt2~o + ..., Sko = --S~o wk I1"= o

=

(EvlEh)I/2S~o~

'+

...

I~ =o exp[--(1 + i ) x / ~ • ~]

(3.1) (3.2)

It follows from (3.1), (3.2) t h a t wk does n o t turn to zero at the surface of the ocean. To eliminate the discrepancy we shall introduce correction functions Ul(~, Y, z), wl(~, Y, z), Pi(~, Y, z) describing the m o t i o n in the side layer of the E h1/2 thickness (the upwelling layer). The function UI, together with the function U, should satisfy the condition (1.7): 3-h-~-~ ~_ ~

4 .1 <--

--" --~ 1

J

6

Fig. 1. T h e s c h e m e o f t h e o c e a n division i n t o regions (a vertical cross-section). T h e conventional b o u n d a r i e s b e t w e e n t h e regions are p l o t t e d b y t h i n lines. T h e regions are n u m b e r e d a c c o r d i n g t o Table I. T h e a r r o w s indicate s c h e m a t i c a l l y t h e w a t e r m o t i o n arisen near the side b o u n d a r y I~ if Tt > 0 at l'.

302 g], should satisfy the condition (1.7): UIx=o + U/l~=o = 0

(3.3)

So the determination of the functions for the side layer should give the boundary conditions for the interior region equations (2.10), (2.14). From (2.1), (3.3) we have: EhCyy + E,,¢zz + dp~ - - i f ¢ = 0

(3.4)

¢=0at~=0 for the function ~b = Urz. As ¢ must vanish at ~ -~ ¢~ it follows from (3.4) that ¢ -= 0. Thus, Uz does not depend on z and according to (1.4) wr depends linearly on z. Since Ut does not turn to zero at the b o t t o m of the ocean, it is necessary to introduce correction functions Um(~, y, r/), wm(~, y, r/) which describe the motion in the comer region 5 (Fig. 1). In the general case the corner region is situated in the vicinity of the line F(x, y ) = O, z = H ( x , y ) and has the scales E 1/2 and e along the normal to the side boundary and the vertical, respectively. The functions Urn, Wm should also eliminate the discrepancy owing to the fact that the function Ub does not turn to zero at the side boundary. According to (2.7), (3.3) Ub Ix =o is of the same order as that of Uz. Therefore the function Um will be of the order O ( U r ) . Due to its complexity the determination of the functions Urn, wm is not considered here. We shall find only the expression for Wm at the b o t t o m of the ocean. From (1.4), (1.6) we have: --Wm]~= 0 = H x u I + HyU r + E h l / 2 Z X ~ + SYmy

(3.5)

where S m = e f o Um dr/is the full flux of the corner current. Coming back to the side layer, let us write the equations and the boundary conditions for the functions uz, v~, wt, Pr- From (1.3)--(1.6) we have: - - f v z = - - E h l / 2 p r ~ + u r ~ + EhUzyy

fur = --Pry + v t ~ + Ehvty~

(3.6)

E h l / 2 u l ~ + Vly + Wlz "= 0

Wl Iz=O + W~ ]~=o = 0, W r Iz=H + Wm I,=O = 0 By using (3.2), (3.5), (3.6) one can show that the correction functions for the side layer should be found in the form b~l/2 Vr = --v

V r o ( } , 3') +

- . -

W r = ( E v / E h ) l / 2 W r o ( ~ , y , z ) + ... Pt = (Ev "Er~)l/2 p r o ( ~ , Y ) + ...

where Uzo, win, Pro do not depend on Eh, Ev.

303

By substituting these expansions in (3.6) and using the expressions (3.2), (3.5} we have: --fVto = --Pm~ + Uto~ (3.7)

r u m = U~o~ Ulo ~ + W l o z = 0

wmlz=0 = --S~o~, WloIz=H0 = 0

(3.8)

where Ho = H I x = o . As wz is a linear function o f z it is found from (3.8) that: wl0-

z -- Ho Ho S~o~

Then by integrating (3.7) with respect to ~ and using (3.1) we obtain: Um =--t-I~olSko , Plo - 0

(3.9)

Thus, at the first approximation the full f l u x S~ = U~H of the side current is equal in magnitude to the full flux S~ of the c o m e r current at the same vertical and opposite to it in direction. The b o u n d a r y conditions for the equations (2.10), (2.14) are obtained from (2.2), (3.1), .(3.3), (3.9): 0 = - - E } / 2 T / i f H + ... at x = 0

(3.10)

One can easily check on the validity of the conditions (3.10) for any smooth side b o u n d a r y F(x, y) = 0. Finally let us consider the water exchange between the interior region and the remaining regions of the ocean. From (2.2), (3.1), (3.8), (3.10) we have the equality of the fluxes of the water entering the side layer from the interior region, region 3 from the side layer, and the surface from region 3: --(GoH)x=o = - - / W z o

I~=o d~ = S~.o Ix=o

o

As Wto = 0 at z = H o region 5 and the b o t t o m layer do n o t take part in the water exchange with the others at the first approximation. Thus all water entering the side layer from the interior region will have to come to the surface layer through region 3. If r y > 0 at x = 0 the upweUing will take place on the whole in the side layer (Fig. 1). If Ty < 0 at x = 0 then the water m o t i o n will be reversed there. In the general case the coastal upwelling will occur near those parts of the b o u n d a r y F where Tt > 0 and the downweUing near those parts of F where r t < 0. Here Tt is the projection o f the vector ~ on the positive direction along the coastline (see Introduction). 4. QUASI-GEOSTROPHIC REGION AND COASTAL BOUNDARY LAYERS

Using (2.10), (2.14), (3.10), (2.2), (2.8) let us write the problem for the interior region of the ocean in the general form of:

304

- - [ ( k X Vh) = - - V h . P + EhAWh Vh

vh (Hf__) • v H + ~En A h ~ =

[ divh(SE + Sb)

Vh lr = _ ~--v/ 2 (kX'r) r + ... fH

(4.1) (4.2)

Here: Co=rotzvh Sb = - e

SE

E~/2 (k ×f T)

[vh + ( k x v h ) ] + ...

.oo

(4.3)

vh is a vector of the horizontal veloeiW and k is a unit vector along the z-axis. We shall omit further the marks . . . . over the functions vn, t5 describing the motion in the interior region: The analysis of the problem (4.1), (4.2) will be made with the help of the method of matched asymptotical expansions (Van Dyke, 1964; Cole, 1968). At first we shall consider the case where E h = Ev -- E. The necessary modifications for the general ease will be depicted in Section 6. For convenience of the analysis, let us set positive directions at the isolines f/H and boundary F. We shall assume the direction of the movement along the isoline f/H as being positive if greater values of the function f/H lie at the right side. The direction of the movement along the boundary F will be assumed as positive if the region of the ocean lies at the right side. The analysis of the problem (4.1), (4.2) may be made in eases with differ-" ent behaviour of the isolines f/H. In the present section we shall consider the simplest ease in which all isolines f/H will start at part F1 of the boundary and end in part F2 (Fig. 2a). To construct the solution far from the boundary F it is convenient to pass

t

C

(a)

(b)

Fig. 2. a. T h e s c h e m e o f t h e isolines f/H in t h e simplest case ( S e c t i o n 4). T h e a r r o w s at t h e isolines f/H a n d b o u n d a r y F i n d i c a t e t h e positive directions, b. T h e s c h e m e o f t h e integral c i r c u l a t i o n arisen in t h e same basin if rotz(t/H) > O.

305

over to the following curvilinear orthogonal coordinates: q = q ( x , y), r = r ( x , y)

such that the coordinate lines q = constant would coincide with the isolines f / H . The coordinates q and r increase while moving to the side of the growth of the function f / H and along the isolines f / H to the positive direction, respectively. (See the directions of the basis vectors q, r in Fig. 2a.) Then the equations (4.1) may be written in the form: --fvr =

_

_

Pq + + - -o~r + ... ! mq E{oL ~m~ m~ t

)

= - - P_L + E [ D ~ q + or r

mr f

o~ o mq

~m~ f

m-~ +''" /

(4.4)

or + E oLr r ~ r H m~mr

_ r

v~+ m~

+ s .o+ s r+ L

.fl

mq

...]

+ ~

m~mr~

mr

Here v q , v r, S~,, S~ are physical components of the vectors Ph, Sb and m q , m r are Lame coefficients in the q, r-coordinate system *. We do n o t write d o w n those terms of friction in (4.4) which contain derivatives of the functions v q, v r, S ~ , S~ as these terms do n o t play a substantial role in the following analysis. We shall also notice that the term ( f / H ) r ( v r / m r ) is zero since the function f / H does not depend on r: The friction is assumed to be small in the region far from the b o u n d a r y V and the motion is supposed to be quasi-geostrophic there. Representing the solution in the form: Vq

=

v~(q, r;E)

=

Etl2v~o(q, r)

+

...

(4.5)

Vr = v ~ ( q , r ; E ) = E1/2Vreo(q, r) + ... P = P g ( q , r; E ) = E Z / 2 p e o ( q , r) + ...

and using (4.4) we have: fVeo

1 apeo mq

~q

* T h e c o n n e c t i o n b e t w e e n t h e c o m p o n e n t s v q, v r a n d u, v is e x p r e s s e d b y t h e f o r m u l a e ;

v q = m q ~~)q xu+mq

~q V, I f = m r ~aur + m r ~-~

ar ~v.

306 fv o =

1 0 mq aq

1

~Pgo

m r

~r

v~0 = - H - ~ rotz

From the equations the expressions: v~°=

f H~ r°tz ( f ) / ~

vg° = fm]

--

a f ~q(H) (fv~°mr)

(4.6) dr + h(q)

rl(q)

are found where r = rl(q) is an equation of part F1 of the boundary and the function h ( q ) should be determined. (For brevity we do not write the expressions for p. ) One can easily see that the functions v~,, v~ do not satisfy all the conditions (4.2) at the boundary F. To eliminate the discrepancy it is necessary to construct the boundary layer at F. In order to construct the solution near the boundary F it is convenient to pass over to such curvilinear orthogonal coordinates: s = s(x, y), t = t(x, y)

that the coordinate line s = 0 would coincide with F, and the coordinates s, t would increase while moving into the region of the ocean and along F to the positive direction, respectively. (See the directions of the basis vectors s, t in Fig. 2a.) Then the equations (4.1) may be written in terms of the coordinates s, t by analogy with the equations (4.4}. It suffices to substitute s and t for the indexes q, r, respectively in {4.4). (For brevity we do not write these s equations.) Then v s, v t, Sb, Sbt will be the physical components of the vectors Vh, SD and ms, m t the Lame coefficients in the s, t coordinate system. The connection between the components v s, v t and vq, v r will be expressed by the formulae: vS _ m s mq

~ s va + _m_ s ~ s r ~q m r ~rr v '

v t-

rn t ~ t rnq ~ q

rrt t ~ t or

vq + - - -

(4.7)

rn r Dr

Let us represent the solution near the boundary F in the form: v s = v,~(o, t; E ) = E ~ v ~ o ( O ,

t ) + ...

v t = v~,(o, t; E ) = E ~ v t , o(O, t ) + ... p =p,(o,

t;E)

= EVp,o(O,

(4.8)

t ) + ...

where o = s / E ~, ~ > O. We shall consider how the discrepancy in fulfilling the condition (4.2) for the c o m p o n e n t vs, normal to F, is eliminated. One can

307

easily see t h a t this discrepancy may by eliminated only by the b o u n d a r y layer in which the c o m p o n e n t v~ of the velocity and the water transport through the cross-section are of the order O(EZ/2). Therefore a = 1/2, ~ + ~ = 1/2. Then using (4.1) and (4.3) gives ~ = 1/3,/~ = 1/6, 3, = 1/2. Substituting (4.8) in the equations (4.1) written in terms of the coordinates s, t we have: --fo

t On

fo~o

0

_ --

=

1

" m8o

Op.o 00

1 0p~o --+ into 0 t

1

02vt~0

m s2o

002

(4.9)

1

0 (~) v~0_0 0 3t VnO _____ Hom~o 003 Ot o into where coefficients marked by the index " 0 " are taken at s = 0. The b o u n d a r y conditions (4.2) and the matching conditions between the expansions (4.5) and (4.8) give:

V~o = O at o = O and o -~ +oo (4.10)

V,o8 = --(¢t/fH)s=o at o = 0 s ~ o ~ Vgo Is= o a t o - ~ +oo

(4.11)

The problem (4.9), (4.10) is solved as follows: t

V,o = O, V,oS = __(¢t/fH)o for part F1 of the boundary where O(f/H)/Ot < O. Thus we obtain: (4.12)

VSgO[rl = --(T t / f H ) r l

from (4.11). Thus the required boundary layer does n o t form at part F1 and in the first approximation the quasi-geostrophic solution satisfies the condition (4.2) for vs. The solution of the problem (4.9), (4.10) is as follows:

t =c(t) exp(--ok/2) s i n ( o k , ~ 2 ) UnO V~o = ---f~ o

Homto

x/gHornto exp --

× IQt s i n ( O 2 x / 3 + 3 ) - - o Q k t

sin(~-----)]

for part F2 of the boundary where a(f/H)/at > 0. Here:

CHrnS~ 0 [ ~7 1'3 ~/-3 c . k(t) = ~ UH]J , Q(t) = - ~ - . o r n s o L rnt ~t s=o

(4.13)

308

and c(t) is an arbitrary function. Let us choose the function c(t) in order to satisfy the condition (4.11). Substituting (4.13) in (4.11) gives: --(Tt/fH)F2 -- Q t / ( H m t )F 2 = V~01r 2

(4.14)

from which one can easily find the function Q(t) and hence c(t) accurate to a constant. Thus, the boundary layer eliminating the discrepancy in fulfilling the condition (4.2) for v s is constructed at part F2. It is analogous to the Munk layer near the western coast in the Pedlosky (1968) problem. We note that the water transport through the cross-section of this layer is equal to E1/2Q in the first approximation. Using (4.7) it is easy to express V~o through v~0 and v~0. By substituting the obtained expression in {4.12) one finds:

h(q)=(m~

q

f drl Rmr) H dq

(4.15) r=rl(,)

where:

Using (4.15) one can transform the formulae (4.6) to:

V~o =

R

"rr

H

fH (4.16)

r 1 ~ V~o = Hmq aq

(q)

Rm~ dr + - fH

It follows from (4.16) that the velocity of the quasi-geostrophic current depends n o t only upon rotz (T/H) but also upon the components of ~. Let us now find the components of the full flux S = SE + Sb + VhgH of currents in the open ocean. Using (4.3), {4.16) we have S q =--El~2 R +...

a

'

S r = E l l 2 mqa---q ( [~ Rmrdr) + ... rl(q)

(4.17)

Consequently the full flux S is determined by the value of rotz(,/H) only. We substitute now the expression for v~0 in (4.14) and make use of the formulae (4.16). After certain transformations we obtain: r2(q)

Q = (f

Rmr dr)¢=~2(t ) + A

(4.18)

rl(q)

where r = r2(q) is an equation of part F2 of the boundary and q = q2(t) is a connection between q and t at F2. The constant A may be determined from the obvious condition that the full water transport through the vertical cut

309

passing along a line q = constant is equal to zero. In the first approximation we have: r2(q )

Vt,oHomso do + (f 0

S~rnrdr)q=q2ct ) = 0

rl(q)

from which it is easy to find that A = 0 *. We shall n o t consider in detail how the discrepancy in fulfilling the condition (4.2) for the c o m p o n e n t v t , tangential to F, is eliminated. Let us note in passing t h a t the discrepancy is to be eliminated at part F2 by the following terms of the expansions (4.8) and at part I"1 by the b o u n d a r y layer which is described with the help of the expansions having the form of (4.8) with ~ = 1/3, ~ = 7 = 5/6, ~ = 1/2. 5. I N T E R N A L B O U N D A R Y

LAYER

We shall now consider the more complex case of the behaviour of the isolines f / H in which a certain isoline f / H is tangential to the side b o u n d a r y F at point 0 (Fig. 3a). In this case the isolines f / H will start at parts P l , F3 of the boundary and end in parts F2, F4. Let us introduce systems of coordinates q, r and s, t as it was done in Section 4 placing for the sake of convenience their origins at point 0. Then the branch of the isoline f / H tangent to F which starts at point 0 (the line q = 0, r > 0) will divide the ocean into regions D1, D2 (Fig. 3a) to which the results obtained in section 4 are applied. In particular the quasi-geostrophic solution will be expressed in region D1 by the formulae (4.16) and in region D2 by the formulae (4.16) with the substitution of rl(q) by r3(q) where r = r3(q) is an equation of part F3 of the boundary. It is easy to find the form taken by the function r3(q) for small q. Using the conditions of tangency ** of the lines q = 0 and s = 0 at point 0: I

~2q

aqato=0,_~o<0 we have: r3(q) = ao(--q) 1/2 + al(--q) + ...

(5.1)

where a0, a l , ... are certain constants. It follows from (5.1) that the function v~0 has a discontinuity of the second kind at the line q = 0, r > 0. To eliminate

* We note that the constant A may be also determined from the matching conditions between the boundary layer at part ['2 and the transition regions which are formed near the points of extremum of the function f / H (the points P1, P2 in Fig. 2a). The scales of the regions along the s, t-axes are equal to E 2/7 , E 1/7 , respectively. ** We assume that the first-order tangency takes place.

310

0 (a)

(b)

Fig. 3. a. T h e s c h e m e o f t h e isolines f~H in t h e case o f t h e isoline f/H t a n g e n t t o t h e b o u n d ary F (Sec. 5). T h e a r r o w s at t h e isolines f/H a n d b o u n d a r y I~ i n d i c a t e t h e positive directions, b. T h e s c h e m e o f t h e integral c i r c u l a t i o n arisen in t h e same basin if r o t z ( ~ / H ) > O.

the d i s c o n t i n u i t y it is necessary t o c o n s t r u c t the b o u n d a r y layer n e a r this line. L e t us r e p r e s e n t the s o l u t i o n near the line q = 0, r > 0 in the f o r m of: v q = v~/(O, r; E ) = EavTo(O, r) + ... v r = or(O,

r;

E ) = E~v~o(O, r) + ...

(5.2)

p = P i ( O , r; E ) = E * p i o ( O , r) + ...

w h e r e 0 = q / E ~, K > 0. T h e w a t e r t r a n s p o r t t h r o u g h t h e cross section o f the b o u n d a r y layer m a y be d e t e r m i n e d f r o m t h e c o n d i t i o n t h a t the full w a t e r t r a n s p o r t t h r o u g h t h e vertical c u t C passing along t h e lines q = +E ~ and r = r0 is equal t o zero. ( T h e c u t C is p l o t t e d b y a d o t t e d line in Fig. 3a. T h e parameters u and r0 are c h o s e n so t h a t 0 < v < K, 0 < r0 < r4(0), w h e r e r = ra(q) is an e q u a t i o n o f p a r t Fa o f the b o u n d a r y . ) In t h e first a p p r o x i m a t i o n we have: +~

r2(q)

v[oHomqodO = E1/2(f

EK+~ f --o~

rl(

Rmrdr)o

(5.3)

q )

(In the present section values m a r k e d b y t h e index " 0 " are t a k e n at q = 0.) Besides, the c o m p o n e n t v~ o f the velocity m u s t be o f the o r d e r O(El/2). There. fore using (4.4) we shall find t h a t ~ = fl = 1/4, a = 7 = 1/2. S u b s t i t u t i o n o f (5.2) in t h e e q u a t i o n s (4.4) gives:

-forgo

10Pio =

foV~/o = - -

1

mqo

~)0

(5.4)

1 aPio mro 3 r

V?o-

Iv m.H !o aO

o (5.5)

311

7

Fig. 4. The scheme of the division of the interior region of the ocean near the point of tangency of the isoline f/H to the side boundary I~ (a horizontal cross section). The conventional boundaries between the regions are plotted by thin lines. The regions are numbered according to Table II. The arrows indicate schematically the water motion arisen near point 0 if rotz(T/H ) > 0 in region D 1 . w h e r e k = 1 + (VhH) 2 . B y d i f f e r e n t i a t i n g (5.5) with r e s p e c t t o t9 a n d using (5.4) it is easy t o r e d u c e t h e set (5.4), (5.5) t o t h e e q u a t i o n : ~r + k l ( r ) ~

- - k2 (r)~oo = 0

(5.6)

f o r t h e f u n c t i o n ~ = (fmq)oV~o, w h e r e : k l (r) = [ m ~ / H m 3 ( f / H ) q ] O , k 2 ( r ) = [ m ~ / m q H 2 ( f / H ) q

]O

In o r d e r t o o b t a i n t h e b o u n d a r y c o n d i t i o n s f o r t h e f u n c t i o n ~b we shall consider t h e t r a n s i t i o n regions in the vicinity o f p o i n t 0. T h e y are t w o (fig. 4). R e g i o n 11 is o f t h e scales E 2/7, E x/7 along t h e s-, t-axes (or q-, r-axes) respectively. T h e s o l u t i o n in region 11 m a y be r e p r e s e n t e d e i t h e r in the f o r m of: t)s = USc(Sc, tc; E ) = ES/14vSco(Sc, tc) + ... v t = vtc(Sc, tc; E )

= E3/14Utco(Sc,

t.c) +

...

P =Pc(So, t o ; E ) = E l l 2 P c o ( S c , to) + ...

w h e r e s~ = s / E 2/7, tc = t / E 1 / ~ , . o r in t h e f o r m of: vq = vqc(qc, rc; E ) = ES/14~qco(qc, rc) + ... or = ~c(qc, rc; E ) = Ea/14~co(qc, rc) +

"'"

(5.7)

P = Pc(qc, rc; E) = E l l 2 ~ c o ( q c , re) + ... w h e r e qc = q~ E217, rc = r / E 1/7. Expressing t h e variables sc, tc t h r o u g h qc, rc a n d using t h e f o r m u l a e (4.7), it is n o t d i f f i c u l t t o d e t e r m i n e t h e c o n n e c t i o n b e t w e e n t h e f u n c t i o n s ~ 0 , vt0, Pc0 and vg0, ffco, Pc0. H o w e v e r , t h e d e t e r m i n a t i o n o f these f u n c t i o n s is r e d u c e d t o t h e s o l u t i o n o f t h e c o m p l e x m a t h e m a t i c a l p r o b l e m and is n o t c o n s i d e r e d here. A similar p r o b l e m is c o n s i d e r e d b y Creegan e t al. (1975).

312

Region 12 is of the scales E v/24, E 1/s along the s-, t- axes respectively. The solution in region 12 is given in the f or m of: v ~ = v~(s~, t~; E ) = E1/2VSao(Sa, t , ) + ... ut

t t = va(s,, ta", E ) = E 1/3 v~0(s~, t~) + ...

(5.8)

p = p~{sa, t,; E ) = E1/2pao(Sa, ta) + ... s t where sa = s / E v/24, t, = t i E 1/s. The functions va0, V~o, Pa0 may be f o u n d if the matching conditions between the expansions (5.2) and (5.8) are used. From these conditions one can also obtain the b o u n d a r y condition for the function Oatr=O,O< O. The b o u n d a r y condition for the f unc t i on ¢ at r = 0, 0 > 0 may be obtained from the equality of the expansions v~(q, r; E ) a n d v[(O, r; E ) at the line r = 0. Actually, if E is sufficiently small, the internal b o u n d a r y layer will be much wider in comparison with the transition region 11 (E 1/t > > E 2/7 for E -* 0). Therefore region 11 is unable to determine the behaviour o f the expansions (5.2) at the line r = 0, 0 > 0; thus the equality o f the expansions (4.5) and (5.2) together with their r-derivatives must take place at the line * Finally, we shall have the following condition:

¢=0atr=0,0¢0

{5.9)

The condition (5.3) allows us to obtain also: f

d~dO =

_~

at 0 < r < r4(0)

Rrn~dr rl(q)

(5.10)

o

from which it follows t hat the f unc t i on ¢ is locally integrable in the region --co < 0 < +~, 0 ~< r ~< r4(0). Thus, we come to the problem of constructing a non-trivial solution of the equation {5.6) in the region - - ~ < 0 < +~, 0 < r < ra(0) which satisfies the condition (5.9) and is locally integrable in the region - - ~ < 0 < +~, 0 ~< r ~< r4(0). Such a solution is constructed in the Appendix and is of the form ~b =

~

c ~ (~)

(5.11)

~=0

where c~ are u n k n o w n constants, ¢(5) 1 =-exp[ - - $a ~ l ( r ) - - ~ 2 / ~ 2 ( r ) ] ~ " ~ a ( ~ 0 ) d ~

foo

7r

(5.12)

0

r,. E) and vr(o, r; E) at the line r = 0, * W e d o not use the equality of the derivatives of Vg(q, r since the equation (5.6) is of the firstorder with respect to r.

313

(--I)"

at a = 2n, n = 0, 1, 2, ...

cos

~ ( / ~ ) = ~ (--1)" sin p ~l(r) =

a t a = 2 n - - 1, n = 1, 2, ...

/ kl(r)dr,~2(r) = / k2(r)dr 0

0

It is obvious t h a t the problem (5.6), (5.9), (5.10) does n o t determine uniquely the function ~. To determine the constants c~ it is necessary to use the matching conditions between the internal b o u n d a r y layer and the transition region 11, those which require the equivalence of the expansions (5.2) at O -~ 0, r ~ +0 and the expansions (5.7) at qc -* o% r~ -* +oo. At first let us find the asymptotic expansion for the functions ~(~) at r -, +0. It may be obtained by a formal expansion of the function e x p [ ~ 4 k l ( 0 ) r ~4~1(r) -- ~2~2(r)] under the sign of the integral (5.12) into a Taylor series with respect to r in the vicinity of r = 0 and by a rearrangement of the order of the integration and summation operations *. Limiting ourselves by the first term of the expansion we have: l+a ~)(CO _-- r

4

g~

t~

"[" ... at r

(5.13)

-~ + 0

where J~'(g) = lr

exp[--kl(0)~'4]~qJ~(g~')d~" " 0

Then using (5.11), (5.13) let us write the first term of the ~ y m p t o t i c a l expansion of v[(O, r; E ) at 0 -* 0, r -* +0: Dr(0, r;

E) = E 114 ~

~=0

dar

4 j~

\r 114/

+

"'"

(5.14)

where ~ = c J ( f m q ) o o . Passing over to the variables qc, rc in (5.14) we obtain the series: m

v~(qcEll2S, rcEllT;E) = ~ affiO

6--~

E

28

l+a

~rc

4 j~(q_~l +... \ r~, -!

According to the matching conditions the series obtained must give the asymptotical expansion of ~(qc, re; E) at qc "* ¢¢, rc -* +~. But according to (5.7) such an expansion must n o t contain the terms of the order which * One can easily check upon the validity of the procedures by dividing the integral (5.12) into the sum f or p + f:r~', where 1/2 < v < 1/4. T h e n fr+®will be transcendentally small and the value ~ 4 k l ( 0 ) r - - ~4Kl(r ) - - ~2K2(r ) will be small in f ~ .

314

exceeds O(E3/14). Therefore all d~ except C0 must be zero * The constant Co may be found in the easiest way from the condition (5.10). Using (5.11), (5.12) we have:

Co =

Rmrdr 0

rl(q)

Let us note that Co may be determined from the matching condition in question as well if the function ~0(qc, re) is known. The function v[0 may be written n o w in the final form: +oo

r _

rio

Co

f

~r(fmq)o o

exp[--~4~l(r)--~2(r)]

cos(~0)d~

(5.15)

Using (5.4) it is also easy to write o u t the formulae: +co

Co

Pio =-~ f

e--X(~'r)~

-1

sin(~0)d$ + a(r)

(5.16)

o

+~o V~[o= 7r(fmr)----~oC° f e-X(~'r)[~3kl(r ) + ~k2(r)] sin(~0)d~

--

a'(r) (fmr)o

(5.17)

where X(~, r) = ~4~1(r ) + ~2~2(r ). The function a(r) is determined, accurate to a constant, from the matching condition between the expansion of v~[(O, r; E) at 0-~ +~o and the expansion of v~(q, r; E) at q -~ -+0. Since the first addendum in the right side of (5.17) vanishes when 0 -~ +oo we find that a'(r) =

(fm~v~o)o. In this way the solution in the internal boundary layer is constructed in the first approximation. We note that the matching conditions between the internal boundary layer and the quasi-geostrophic region and region 12 were not sufficient to construct the solution. The matching conditions between the internal boundary layer and the transition region 11 played a substantial role. We succeeded in finding the first terms of the expansions (5.2) in the internal boundary layer without solving the problem for region 11. However, in order to find the following terms of the expansions {5.2) it is necessary to solve the problem in question. Finally let us consider the general Character of the water motion near point 0. It follows from (5.3) and (4.18) that the transport of the internal boundary current is the same at all cross sections of layer 10 in the first approximation, and is equal to the transport of the coastal boundary current in layer 8 near point 0. Since the transport of other currents approaching region 11 is of the order O(E 1!2), the strong coastal current of layer 8 must transit continuThe result corresponds to the so-called principle of least singularity (Van Dyke, 1964).

315

ously into the strong internal current of layer 10. In particular, if rot~(¢]H) > 0 in region D~ then the strong coastal current will separate in region 11 from the side b o u n d a r y F and follow along the isoline f / H (Fig. 4). 6. B O U N D A R Y L A Y E R S I N T H E I N T E R I O R R E G I O N W I T H V A R I O U S R E L A T I O N S BETWEEN E h AND E v

The problems investigated in Sections 4 and 5 may be analysed considering different relations b e t w e e n the Ekman numbers Eh and Ev. The quasi-geostrophic solution for the general case will be as follows: Vq

~:a/2,r I , . r) .+ . p g. = .E¢1/2 = E v1/2 Vqs o ( q , r ) + .... vsr = ~v vgox~ . P g.o ( q ., r) +

where the functions v~0, V~o are given b y the formulae (4.6). In this analysis of the coastal b o u n d a r y layers the following cases occur: (1) Ev < < E ~ Is. Here the coastal layers are formed due to horizontal mixing. The layer at part Fu of the b o u n d a r y ensures the fulfilment of the conditions (4.2) and is described by the expansions: v~ -- ~v~l/2'"~no~v,r~t) + ..., v,t = 5V~o(O, t ) + " " , P n = 7PnO(O, t) + ...

(6.1)

where o = s / g , g -- E a113 , 8 _- E~1/2 / E h1/3 , 3' = E,,1/2 . The layer at part F~ fulfils the condition (4.2) for v~ only. (2) E~ ~ Ehz/s . Both horizontal and vertical mixing are substantial in the coastal layers. The layer at part l~z is described b y the expansions (6.1) and fulfils the conditions (4.2). The layer at part F~ fulfils the conditions (4.2) for v t only. (3) E~ > > E~/3. T w o boundary layers appear at part Fz. The layer which is wider is formed b y vertical mixing and is described b y expansions of the form (6.1) in which g = 7 = E ~ / ~ , 6 = 1. The layer fulfils the condition (4.2) for v~. The layer which is narrower is formed b y horizontal mixing and is described by expansions of the form (6.1) in which g m 7 ___171/2 ~h ~¢~1/4 v , ~ = 1 . The layer fulfils the condition (4.2) for v t. Only the narrow layer is formed at part F1. The functions v n*o , V t, o , P,,o entering (6.1) in these cases are determined easily from the solution of the appropriate problems (see Reznik, 1970). The coastal b o u n d a r y layers are unable to fulfil the condition (4.2) for v~ at part Fa in all cases considered. Hence it follows that the formulae (4.12) and (4.16) are valid for the general case. Thus we come to the following conclusions: ( t ) The quasi-geostrophic flow does n o t depend u p o n the relation between En and E~ in the first approximation. (2) The coastal current intensification arises in the general case only near part P2 o f the boundary. Passing over to the problem considered in Section 5 we shall obtain the discontinuity of the quasi-geostrophic solution at the line q = 0, r > 0 in the gen-

316 eral case as well. The boundary layer with the transport of the order O(E~/2) will be formed near this line. Here the following cases arise *" (1) Ev < < Eh. The internal boundary layer is formed by horizontal mixing and described by the expansions: v7 = Evl/2Vio(tg, q r) + ..., v[ = 5V[o(0, r) + ..., Pi = E~l/2pio(O, r) + ...

(6.2)

where 0 = q / ~ , ~ = Elh 14, ~ = ,~v 12112~-h /i~,1/4 . The functions V~o,rio",pio satisfy the equations (5.4), (5.5) with k = (VhH) 2 and are expressed by the formulae (5.15)--(5.17). (2) E v ~Eh. Both horizontal and vertical mixing are substantial in the internal boundary layer. The layer is described by the expansions {6.2) where the functions V~o, rio, Pio satisfy the equations (5.4), (5.5} with ~ = E ~ / E h + (VhH) 2 and are expressed by the formulae (5.15)--(5.17) as well. (3) E v ~ > Eh. The internal boundary layer is formed by vertical mixing. The layer is described by the expansions of the form (6.2} in which ~ = 5 = E~/a and the functions v~0, v[0, Pio satisfy the equations (5.4), (5.5) with = 1 and with the term (1/maqo ) (~ a V[o/0 0 a) omitted. In this case the determination of the functions V~o, rio, P~0 is reduced to solving the problem considered by Kamenkovich and Reznik (1972). Summing up the results obtained we may conclude the following: (1) The balance between the forces in the boundary layers of the interior region and the structure of the layers depends crucially upon the relation between Eh and E~. (2) The disposition and transports of the strong boundary currents providing the balance of the horizontal fluxes in the ocean do not depend upon this relation. CONCLUSION The analysis made permits to plot simply enough the patterns of the fluid circulations in the basins considered. For example, using (4.17) it is not difficult to find the full flux function: r

= Ely 12 f

Rmrdr

+ ...

rl(q)

for the open ocean. Closing the streamlines in the boundary layers we obtain the schemes of the integral circulation in the basins (Figs. 2b, 3b). In order to make the character of the three-dimensional circulation in a homogeneous ocean clear let us notice that the velocity in the interior region may be decomposed in two sums: ~)h --- 7JhR + ~ h r ,

W = W R -b WT. SO

that:

* For the sake of simplicity we assume that V h ¢ 0 within the layer.

317

so that: VhR = H - 1 S , Vh~. = - - H - I ( S E

+ S b ) , WR = z H - - I ( V h H • VhR ) ,

WT = zI't -1 [(VhH " VhT) + diVhSb] + (Z -- H)H-ldivhSE

'where S is the full flux of the currents in the ocean. Then VhR will be determined by the value rotz (~/H) in the first approximation and Vh~ b y the value T. The streamlines of the flow with the velocity VR will be closed in the interior region. The flow with the velocity v~ will transient into the currents of the surface, b o t t o m and side layers at the interior region "boundaries". Thus the three-dimensional circulation in a homogeneous ocean represents itself in the superposition of t w o fluid gyms, namely: (1) the "horizontal" gyre which is limited b y the interior region and caused b y the rotz(~/H) effect and (2) the "vertical" gyre which is spread to all regions of the ocean and caused b y the effect of ~ itself. The b o t t o m relief does n o t in fact exert its influence on the "vertical" gyre of the fluid. On the contrary the "horizontal" gyre peculiarities depend essentially u p o n the b o t t o m relief or more precisely upon the behaviour of the isolines f / H . We n o w consider certain questions of the practical application of the theory developed. First of all we note that the model constructed may be improved by introducing the non-linear factors. We also note the o p p o r t u n i t y of the application of the theory in laboratory modelling of ocean circulation and in modelling of the b o u n d a r y current separation from the coast in particular. At the same time it should be noted that the direct application of the theory given to the real ocean requires a careful approach. The fact is that it is n o t quite clear h o w to s m o o t h d o w n the depth of the ocean in the coastal zone and determine where to place a conventional side boundary. The behaviour of the isolines f / H in the coastal zone may essentially depend u p o n the m e t h o d of smoothing the depth. We shall notice in particular that there are no isolines f / H tangential to the coastline near, e.g., Cape Hatteras on the U.S. eastern coast on the maps of Welander (1968) and Ilyin et al. (1974). Therefore the explanation of the Gulf-Stream separation through the b o t t o m relief effect is problematic. On the other hand it is possible to explane the Alaskan Stream separation b y the existence of the isoline f / H tangent to the conventional side b o u n d a r y near the Aleutians (Thomson, 1972). Another difficulty consists in the fact that it is n o t enough to take only the first terms o f the asymptotical expansions for the practical calculations of the currents in the b o u n d a r y layers. The determination of the following terms of the expansions is often connected with great mathematical difficulties. These relate in particular to the internal b o u n d a r y layer for which the ratio of the second terms of the expansions to the first terms is of the order O(E l/s) while for the real ocean E = 4 • 10 --6 according to the estimates in Sec. 1.

318 ACKNOWLEDGEMENTS The author extends his deepest thanks to Dr. V.M. Kamenkovich for the initial suggestion of the problem, valuable advice and permanent attention. The author is also indebted to Dr. A.M. Ilyin for helpful discussions of mathematical problems. APPENDIX

Solution o f the problem (5.6) and (5.9) We shall construct the locally integrable solution of the problem (5.6) and (5.9) using the theory of distributions (for example, see Vladimirov, 1976). Let us continue smoothly the functions kl{r), k2(r) in the regions r ~< 0 and r ~> r4(0) and for the sake of simplicity assume them to be infinitely differentiable and positive at ---¢¢ < r < +¢¢. Let the function ~(0, r) be a locally integrable (in the region r ~> 0) solution of the problem (5.6) and (5.9). We shall continue it by zero in the region r < 0 and for simplicity assume that it will grow slower toward infinity compared will certain degrees of 101 and I rl. Then the function ¢(0, r) will define the regular functional (the tempered distribution ~):

(~, ~) = f ;

dp(O,r)~o(t~, r)dO dr

on the set of the infinitely differentiable {trial) functions ~0(0, r) decreasing toward infinity together with their derivatives faster than any degrees of 101- I and Ir 1-1 . Taking q~ as a basis we shall construct the distribution: Lq5 = Cr + kl(r) ¢oooo -- k2(r) ¢oo and then show that it equals zero (in the sense of distributions) everywhere except the point (0, 0). Let ~(0, r) be an arbitrary trial function equal to zero in the vicinity of the point (0, 0). Then such o > 0 exists that ~0 = 0 at 10l ~< o, lr[ ~< o. We shall represent the value of the functional LO at the function ~0 in the form of +~

(LO,~)=(O, L~)= f

= lim f

f

+~

f

CL~d0dr=

~bL~dO dr = lim I f

f

~L¢ dO d r + f

(O~)~=~ dO] (A.])

where L~ = --~,. + (k]~)oooo -- (k2~)oo. The first addendum in the square

319 brackets (A.1) is zero for any e > 0 as LO(O, r) = 0 at r > 0. The second a d d e n d u m for e < o is written in the form of: lim E "-'~ + 0

f

lim(f +f

~,:,o

6--' +0

--ae

0

+f ~

0

and is zero as well since ~b(0, 0) = 0 at 0 ¢ O. Thus the functional L~b is equal to zero at any trial function ¢ with the carrier which does not contain the point (0, 0). Hence L~ = 0 in the sense of distributions everywhere except the point (0, 0). According to the conclusion obtained above the carrier of the distribution L~b is the point (0,0). By the theorem about the structure of the distributions with the point carrier (Vladimirov, 1976) such distribution may be represented uniquely in the form of: m

L~= ~

c~,~=O

c~5(~)(0)6(~)(r)

(A.2)

where 6(~)(0), 5(~)(r) are derivatives of ~, ~ orders of 5-functions and c ~ are certain constants. The solution of the equation (A.2) will be found in the form of: tB

~b= ~ c~,~b(~) ~,~=0 where ~b(~°) are solutions of the equations: c~(r~ ) + k z ( r ) c ~ C~) o o o a - - k2(r)~b~~) = 6(a)(0)5(~)(r)

(A.3)

(A.4)

in the set of the distributions equal to zero at r < 0. At first let us find the distributions ~b(a°) for a = 0, 1, ..., m. Applying Fourier transformation Fo with respect to the variable 0 to (A.4) we obtain:

~ 0 ) + [}4kl(r) + }2/~2(r)]~(~o) = (--i})~5(r)

(A.5)

where ~b(~°) = Fo [~b(a°)] *. The solution of the equation (A.5) which is zero for r < 0 is of the form: (~(~o) = O(r) (--i})~ exp[--}4~l(r) --}2K2(r)]

(A.6)

where O(r) = 1 at r > 0, O(r) = 0 at r < 0, ~z(r) = f~ kx(r)dr, g2(r) = fro k2(r)dr. Applying the reversed Fourier transformation with respect to } to (A.6) we find: O(r)

f

exp[--~4~z(r)-- ~ 2 ~ 2 ( r ) ] ~ ( ~ 0 ) d ~

0

* If f(tg) is a regular function integrable at the interval --~ < 0 < +~ then Fo[f] = = I+__~f ( O ) e i~O dO.

(A.7)

320

where (-1)” *c&J) =

i (-1)”

cos /_4

at d = 2n, n = 0, 1,2, . . .

sin p

ata=2n-l,n=1,2,...

By analogy the distributions @(@) for p = 1, 2, . . . . m are determined which are of the form c$(~@)= {c#I’“~‘}f A(@), where {@‘“@‘}is a regular sum and A(@) is a functional concentrated in the point (0, 0). However the distribution q!~which is to be found must be regular. Therefore the irregular sum Zr= a,p= r c,~A(@) in the formula (A.3) must turn to zero. Hence it follows owing to the linear independence of the functionals Acap) that colp = 0 for fl= 1, 2, . . . . m. Thus the distribution @will be of the form

(A.81 The formulae (A.7), (AS) in the region < 6 < +m, 0 < r < r*(O) will give the solution found for the problem (5.6), (5.9). REFERENCES Brink, K.H. et al., 1973. The effect on ocean circulation of a change in the sign of /3.Tellus, 25 (5): 518-521. Cole, J.D., 1968. Perturbation Methods in Applied Mathematics. Blaisdell, London. Creegan, A. et al., 1975. A singularity in an oceanic boundary layer. Math. Proc. Cambridge Philos. Sot., 77 (Pt. 2): 439-446. Ilyin, A.M. et al., 1974. An experiment in the construction of a smoothed bottom relief of the world ocean. Oceanology, 14 (5): 765-769. Kamenkovich, V.M. and Mitrofanov, V.A., 1971. On the case of bottom relief influence on currents in the ocean. Dokl. Akad. Nauk S.S.S.R., 199 (1): 78-81. Kamenkovich, V.M. and Reznik, G.M., 1972. Onthe boundary current separation from the coast due to the bottom relief influence (linear barotropic model). Dokl. Akad. Nauk S.S.S.R., 202 (5): 1061-1064. Pedlosky, J., 1968. An overlooked aspect of the wind-driven oceanic circulation. J. Fluid Mech., 32 (4): 809-821. Reznik, G.M., 1970. On the problem of three-dimensional circulation in a homogeneous ocean. Izv. Akad. Nauk S.S.S.R., Fiz. Atmos. Okeana, 6 (11): 1163-1177. Thomson, R.E., 1972. On the Alaskan stream. J. Phys. Oceanogr., 2 (4): 363-371. Van Dyke, M., 1964. Perturbation Methods in Fluid Mechanics. Academic Press, New York, N.Y. Vladimirov, B.S., 1976. Uravneniya Matematicheskoi’ Fiziki. Nauka, Moscow. Welander, P., 1968. Wind-driven circulation in one- and two-layer oceans of variable depth. Tellus, 20 (No. 1): l-15.