Deep-Sea Research, 1963. Vol. 10, pp. 735 to 747. Pergamon Press Ltd. Printed in Great Britain.

Rossby waves in ocean circulation*

D. W. MOORE Department of Mathematics, Bristol University

(Received 12 August 1963)

Abstract--It is shown that damped, stat ionary Rossby waves can occur in the ocean superimposed on a steady west to east flow. A model o f wind-driven ocean circulation in a two dimensional homo- geneous ocean is constructed in which Rossby waves occur in the northern port ion of the basin.

I N T R O D U C T I O N

SINCE the work of SmMMEL (1948) it has been recognized that any dynamic theory of wind-driven ocean circulation must include the effect of the variation with latitude of the normal component of the earth's rotation if it is to have any chance of explain- ing the most striking of the observed features--the western boundary current. It is not difficult to modify the equations of motion to include this effect, but even if one makes the customary approximation of representing the turbulent transfer of momen- tum by an eddy viscosity, one is still faced with an intractable problem. Theories of ocean circulation can, in fact, be characterized by their method of overcoming the intractability of the full Navier-Stokes equations and two types may be distinguished. In the first place one has the frictional theories of STOMMEL (1948) and MUNK (1950) in which the non-linear inertia terms are discarded completely and the/3 term balanced against the viscous stresses. In the second place one has the inertial theories (FomNOFF 1954; MORGAN, 1956; CHARNEY, 1955; CARRIER and ROBINSON, 1962) in which the fi'ictional terms are dropped from the equations. Interesting critical accounts of these theories have been given by STOMMEL (1958), FOmNOFV (1962) and CARRIER and ROBINSON (1962).

In view of these contrasting theories it seems worthwhile to examine a situation in which both the inertia terms and the frictional terms can be included exactly in the solution. In Section 2, possible flows in the neighbourhood of a latitude of vanish- ing wind stress curl are examined on the basis of the exact equations. It is shown that the problem can be reduced to that of solving an ordinary third order, non-linear equation. In the sense that such an equation can be integrated numerically in a straightforward fashion an exact solution has been found but, rather than examine this, attention has been concentrated on determining the relative imlSortance of viscous and inertia forces in this restricted situation. The results are described in terms of a Reynolds number R Ua/2/fl ~ v, where U is the velocity scale of east- west flow and v an eddy viscosity. If R < 1 the results are shown to coincide with Munk's theory, whilst if R > 1 the inertial theories are valid, except in viscous sub- layers on the continental walls. The nature of these sub-layers is examined and it is

*Contribution No. 1417 from the Woods Hole Oceanographic Institution.

735

736 D . W . MOORE

shown that if the external flow is that of FOFONOFF (1954), no viscous sub-layer can exist on the eastern boundary.

In Section 3 the exact equations are examined in more detail in the case of no wind stress and some exact solutions derived. It is shown that if the flow far from the continental boundaries is west to east solutions representing damped Rossby waves exist. FOFONOFF (1961), who has found similar waves in a slightly different situation, has suggested that these Rossby waves may play a role in ocean circulation and in Section 4 of this report a model of ocean circulation involving these waves is con- structed. The flow found is east to west in the southern portion with a boundary current on the southern half of the western boundary whilst in the northern portion of the basin there is no boundary current but instead a system of damped Rossby waves which decay on a basic west to east flow.

The model is also novel in that it seems most appropriate to an overall ocean Reynolds number of order ten or less and so possesses some features of earlier models which were based on either zero or infinite overall Reynolds numbers. In particular, if the Reynolds number is of order ten, the Gulf Stream thickness is independent of the assumed eddy viscosity so that the model retains one of the most attractive features of the inertial theories.

2. F L O W N E A R A L A T I T U D E OF V A N I S H I N G W I N D STRESS C U R L

The ocean is taken to be of uniform depth and the motion is two-dimensional and parallel to the surface. Thus the wind stress must be regarded as a body force distributed uniformly through the depth of the ocean and bot tom friction must be ignored. Let O X , 0 Y be rectangular co-ordinate axes and O X is in the west to east direction and O Y in the south to north direction. Then the Navier-Stokes equations assume the form

bu bu 1 bp + v - - - - f v - - + W + v V ~ u (2.1)

u b-- X oy p 3x

ubV + vbv 1 ~x ~y + f u - - - - - + vV ~ v, (2.2)

p by

bu + by 0 (2.3) 3x by

In these equations (u, v) is the fluid velocity at (x, y), f the Coriolis parameter, p the pressure and v a coefficient of kinematic viscosity. W is the wind force which, as is customary, has been taken to be in the east-west direction. It is now further assumed that the latitudinal variation of the Coriolis parameter is given by the linear approximation

f = fo ÷ / 3 Y , (2.4)

where/3 is a constant. ~b (x, y) such that :

Now (2.3) can be integrated by means of a stream function

u - - , v = - - - - , (2.5) ~y bx

and using the relations :

Rossby waves in ocean circulation 737

one has finally:

where :

~(f~) ~ : f ~ - ~ - - iv,

~y (f~b) =- f ~ + ~b 3 f = fu ~Y ÷ fl~b,

(2.6)

(2.7)

3u 3u I 3p' V2 (2.8) u - - + v - - - - + W+~, u 3x by p bx

by by I bp' V2 (2.9) u - - + v - - - 4 , ~ . . . . +~ v, ~x by p 3y

p' = p + p f ~b. (2.10)

The equations in this form show clearly that it is the variation of the Coriolis para- meter which is dynamically important*.

Suppose that y = 0 is the boundary between two regions of disconnected motion of the ocean. Then since neither mass nor momentum is to be transferred across y = 0 one must have

v : 0 , - -bu=0 on y = 0 (2.11) by

Thus, under mild restrictions of analyticity, the stream function 4, (x, y) must possess a power series expansion near y = 0 of the form

(x,y) == yf (x) + Y~A (x) + . . . (2.12)

Sufficiently close to y = 0 the flow will be described by the stream function yf(x) and the higher terms will not be considered in this analysis. In general, contributions from these higher terms will enter the equation f o r f ( x ) since the situation far from the line y : 0 will affect the local flow through the ellipticity of the governing Navier- Stokes equation. These higher terms satisfy differential equations involving f (x) so that one is faced by a system of difference-differential equations, and the expansion (2.12) will only furnish a solution when the boundary-layer approximation is made in the basic equations. The basic equations are then parabolic with characteristics parallel to the y axis and one can proceed recursively.

I f it is supposed that the wind stress term can be expanded in the form :

W (x, y ) : K o (x) -? yg I (x) + y~ K s (x) + . . . (2 .13

then substitution into (2.8) leads to the equation •

f f ' = - - l b P - - ' + K o ( x ) + y K l ( X ) + Y ~ K 2 ( x ) ~ . . . + ~ , f " (2.14) thus : p bx

P = c(y) + x(x) + y f Kl (x)dx + y2 f K~(x)dx + . . . (2.15) P

where C (y) is an unknown function of integration and x (x) a known function of x. On substituting this expression into (2.9) one has :

*If ~ = 0 one recovers the result (TAYLOR, 1917) that rotating a two-dimensional viscous flow alters only the pressure field.

738 D . W . MOORE

dC f Kx (x ) d x - - 2 y f K ~ ( x ) + . . . - - v Y f " . (2.16) _ yff , , + yf ,2 _ f l y f _ dy

Thus by inspection of the powers of y in (2.16) :

dC -- -- Cy, say, and K1 (x) = 0 (2.17)

dy

The last condition implies that ~ W/3y = 0 on y = 0, that is to say the wind stress curl must vanish on the ocean boundary as defined by (2.11). Thus one has, defining:*

K (x) = -- ½ f 1(2 (x) dx, (2.18) L /

the third order equation :

v f" ' -- C -- K ( x ) = i f " _ f , 2 + fir. (2.19)

So far nothing has been said about boundary conditions. If it is supposed that x = 0 and x = L are fixed continental boundaries then the boundary conditions on the function f ( x ) are :

f ( 0 ) = f ' (0) = 0 ~ (2 20)

f ( L ) --= f ' (L) = 0 J The solution of (2.19), which is a fourth order equation when the unknown constant

C is eliminated by differentiation, is uniquely determined by (2.20) once the function K (x) is prescribed. When K (x) is a constant, so that the wind stress curl is zero, the system has the unique solution f = 0. This simply means that an external force is needed to maintain a steady flow in a dissipative ocean with boundaries.

I f the viscous and non-linear terms are neglected one has simply :

- - C - - K (x) = flf (2 .21 )

These approximations are applicable to the mid-oceanic regions and it is clear that (2.21) is the analogue in the present simplified system of the SWRDRUP (1947) trans- port equation. Clearly, even when a choice of C is made the boundary conditions (2.20) cannot be satisfied and the best one can achieve is f ( 0 ) = 0 o r f ( L ) = 0. Thus the neglected terms will intervene in boundary layers on the continental boundaries.

The subsequent analysis of these boundary layers is clarified if dimensionless co-ordinates and a dimensionless stream function are introduced. The choice of the scales involved is quite arbitrary and will not affect the results, hut as it is the object of the analysis of this section to compare viscous and inertial boundary layers it is convenient to choose scales natural to one of them. The mid-oceanic velocity U = 0 (K/B) is a suitable velocity scale and the inertial boundary layer thickness L - (U/~)½ is taken as the length scale. Thus one defines :

- - U F = f , ( V ) ½ ~ _ x , E V = - - C, R(o2) K ( x ) -- U (2.22) \ J"~ /

whence :

*Note that the constant of integration can be absorbed in the unknown constant, C in (2.19).

Rossby waves in ocean circulation 739

F (0) = F ' (0) = 0 (2.24)

where the Reynolds number, R is defined by :

R = U312/v ~i. (2.25)

The solution is thus determined essentially by two parameters, R and ( B / U ) t L and in general nine possible cases can arise. However, (fl/U)~ L, which is the ratio of the ocean breadth to the inertial boundary layer thickness, is large and one need only consider the variation of R. There are two cases to consider.

Case I : R < 1

In this case the left-hand side of (2.23) will be very large and viscous and inertia forces will not balance unless the third derivative of F is small, say 0 (1/8 a) where 8 is the length scale of x variation in the dimensionless system. Then l/R83 = 0 (1) so that 8 = 0 (R-~); furthermore the non-linear terms on the right-hand side of (2.23) are 0 (R ~) and can be neglected. Thus (2.23) takes the form :

1 F ' " F + E -- 1{ (~"). (2.26) - - =

R

If one has 8 < (fl/u)½ L the solution represents a slowly varying interior flow with thin boundary layers on the continental boundaries and, indeed, if one takes R (~-) = W0 ~', so that the wind force is independent of longitude, one recovers the boundary layer form of MUNK'S (1950) equations.

Case H R >> I

Inspection of (2.23) shows that in any region where x derivatives are of 0 (1), the left-hand side is negligible, so that in the interior of the ocean one has :

0 = F '~ -- F F " + F + E -- 1{ (~) (2.27)

However, the solutions of this inertial equation will not in general satisfy all the boundary conditions (2.27) and regions where the viscous term becomes important will exist near the boundaries. Since the solutions of (2.27) are themselves of inertial boundary layer character, owing to the assumption (t/U)½ L < I, this thin viscous region will be called the viscous sub-layer. The nature of these viscous sub-layers will now be examined for the case of FovONOVV's (1954) free inertial solutions. Thus K (~,) = 0 and one may verify that a solution of (2.27) is :

F = (1 -- e -~) near ~" = 0 and :

(2.28)

F = (1 -- e~-L (~ ) ½) (2.29)

These solutions fail to satisfy the conditions F' ( 0 ) = 0 and near ~" = L (~/U) t. F' (f l i /U t L) = 0, which state that the tangential velocity is 0, so that viscous sub- layers will arise. Considering these layers, let F = AF* and let ~" = 8x* near the western boundary and let 8 x * = L f l t / U t - ,~ near the eastern boundary. Then (2.23) becomes

740 D . W . MOORE

~2

:k R 3 ~ F* ' " ---- 32- (F *'z - - F* F*") + hF* -- 1 (2.30)

where the upper sign refers to the western boundary whilst the boundary conditions on the boundaries are in both cases F* ( 0 ) = F* ' ( 0 ) = 0. Following the usual boundary layer procedure, one imposes on F* the condition that the y component of velocity should tend as x* -+ oo to the value given by the inertial solution. Re- ference to (2.28) and (2.29) shows that in both cases this yields the boundary condition:

AF*' ~ 1 as x * - + o o (2.31)

and since the point of the scaling is to have F* of 0 (1) one chooses A = 3. Further- more if viscous and inertia terms are to balance A / R 3 z ~ 0 (1) so that 3 = R-~ say. The linear/3 term in (2.30) is now seen to be only R-'-' and can be neglected, so that finally one has :

± F* ' " = F *'2 -- F* F*" -- 1 (2.32)

F * (0) - - F * ' (0) = 0, f * ' (x*) -+ 1 as x* -~ ~ (2.33)

The equations when the upper sign is taken are identical with those for the boundary layer near the forward stagnation point of a cylinder (GOLDSTEIN, 1938) but when the lower sign is taken the equations can be reduced to those for the boundary layer near the rear stagnation point and it is known that, since the flow is rapidly decelerat- ing, the boundary layer equations have no solution in this case. Thus no viscous sub-layer can exist on the southern part of the eastern boundary in the present case-- presumably if a viscous sub-layer were to be established farther north, where the flow is less rapidly decelerating, it would separate before the southern boundary was reached. (CARRIER and ROBINSON (1962) have pointed out that this deduction depends on the assumption of a constant, lateral eddy viscosity).

A mathematical argument to demonstrate that no solutions exist when the lower sign is taken is given in the Appendix.

3.

I f one takes K (x) ~ 0, so that there is

v f ' " - C = f f " One can easily verify that :

f = - - U(1

is a solution of this equation provided that :

C = flU and

EXACT S O L U T I O N S OF THE FREE E Q U A T I O N S

no wind stress (2.19) takes the form :

_ f , 2 + /3f. (3.1)

- - e - ~ x ) (3 .2)

vz 3 - - U~ 2 + / 3 ~ 0. (3.3)

Thus, since (3.3) has, in general, three distinct roots, three distinct exact solutions have been found. However, since equation (3.1) is non-linear, they cannot be added to construct more general solutions. Furthermore, no choice of U, save the trivial one U = 0, will a l l o w f ' (0) also to be zero, so that the exact solutions cannot satisfy the required boundary conditions at a continental boundary, i.e. they represent states of motion in the interior of the ocean.

Rossby waves in ocean circulation 741

I f one puts v ----- 0 one has : - - U ~ ~ q- fl = 0 (3 .4)

so that e = 5: ( f l / U ) t , in agreement with FOFONOFF'S (1954) inertial theory. I f U > 0 the roots are real and taking the positive root one has a flow which decays to a uniform east to west flow as x ~ oo. I f U < 0 the roots are pure imaginary and no uniform state is achieved as x -7 oo - - this is, of course, just FOFONOFF'S result that a uniform inertial flow can only be from east to west. It is of interest to generalize these results to the full equation (3.3). It is easily shown that i f :

U ~> /~/3 v+~J3 (3.5)

(3.3) has real roots whilst in the contrary case it has a pair of conjugate, complex roots. I f ~1, az, % are the roots one has :

U ~x + ~2 + % = + - v

% ~2 + ~2 % + ~2 % = 0 (3.6)

I f the roots are all real, then by (3.5) U > 0 and it follows from the first and last equations of (3.6) that one root is negative and two are positive. Now suppose that the roots are not all real, so % = p ÷ iq, a s = p - - iq.

Then : ~2 ( f + q~) = - - B/v (3 .7 )

2% p q- (p2 q_ q2) = 0 (3.8)

Equation (3.7) shows that ~1 < 0 and then (3.8) shows that p > 0. Thus in either case, two of the roots represent solutions which decay as x increases and the other root represents a solution which decays as x decreases. Thus in contrast to the inertial case one has solutions which decay to a uniform flow as x increases whatever the sign

of U, but if U > ¢31/3 v+2/3 the decay is oscillatory. I fp < q the solutions represent

slowly damped stationary waves on a basically west to east flow. In general it is neces- sary to solve the cubic numerically to determine p and q, but if Bv2/U z < 1 one can easily show t h a t :

q = ' P - - 2 U 2 (3.9)

I t must be borne in mind that, since (3.1) is non-linear, the conjugate complex solutions obtained in this case cannot be combined to form a pair of real decaying oscillatory solutions. Thus further numerical work seems desirable to determine the form of these damped solutions.

FOFONOFF (1961) has suggested that a model of ocean circulation in a rectangular basin might be constructed which had a basic east to west flow in its southern half and basic west to east flow with superimposed, damped Rossby waves in its northern half. The existence of exact solutions of damped Rossby wave type is encouraging and in the next section a model with this idea as its basis is constructed.

742 D . W . MOORE

4. A M O D E L OF O C E A N C I R C U L A T I O N

In this section the wind-driven circulation in a rectangular basin of uniform depth is considered. The wind W (y) is independent of x and dW/dy = 0 when y = 0 and when y = L' (the axes are orientated as in Section 2).

On eliminating the pressure terms from (2.8) and (2.9) one finds that one must solve the equation :

~ dW b~b b (V 2~b) b~b b (V 2 ~ b ) + f l _ _ = + v V a~h (4.1) by 3x bx 33' 3x dy

with the boundary conditions "

4',--b~b == 0 on x = 0, x = L; (4.2) bx

~b, 32 4, 0 on y = 0, y = L'. (4.3)

As a first step towards producing a tractable problem we may assume that the stream function has a boundary layer character on the continental walls x = 0 and x = L so that, symbolically b/bx >> b/by. Then (4.1) becomes :

34 33 4' 34 33 ~ b4J _ Wo ~ ~ry 3 4 ~b (4.4) by bx 3 -- ~ x b y b x ~ + /bbx L' sin L" + v bx 4_,

where, in addition, we have specialized to the wind distribution'

,ry. w ( y ) = - w 0 c o s - - ( 4 . 5 )

L' The boundary conditions are unaltered.

If one expands ~b about the lines y = 0 and y = L' one can apply the arguments of Sections 2 and 3 to the leading terms of the expansions. Far from the boundaries, the assumed wind will produce a west to east motion in the northern portion of the basin and an east to west flow in the southern portion. Thus near the northern boundary one anticipates that the solution will be damped Rossby waves since the free solutions have this form when U < 0. Near the southern boundary U > 0 and the free solutions have a boundary layer character.

How can these solutions join to form a closed flow pattern ? We shall try to answer this question by constructing a linear model of the non-linear boundary-layer equation (4.4). It is worth stressing that we are seeking a model of, rather than an approxima- tion to, the non-linear equations. We do not assert that our solutions approximate to actual ocean circulations for limiting values of the physical parameters - - rather, we aim at reproducing the general features of the actual situation qualitatively. If the model predicts the general features correctly we can assert that the forces it neglects affect only details of the ocean circulation pattern. But how is such a model to be constructed ?

The criterion is a realistic representation of the advection of vorticity in the bound- ary layer; if this criterion is not satisfied the model will retain no features of the exact solution. Thus, for example, in linearising the Blasius boundary layer equations, one replaces the term,

Rossby waves in ocean circulation 743

3u 3u / / - - % - / ) - -

3x by

by the linear term U bu/bx, where U is the mainstream velocity. One regards this as realistic, since the Blasius boundary layer arises from a balance between tangential advection of vorticity and normal diffusion of vorticity. In the present case, both normal and tangential advection are present and a realistic model would include both. Unfortunately, a constant tangential advection coefficient will not do, since, at the outer edge of the boundary layer the advection is purely normal. To include such a variable term in the model would greatly complicate it and, it will simply be omitted.

However, the model with purely normal advection will prove to have the following features :

(A) It agrees qualitatively with the expansions of the exact solution near y = 0 and y = L'.

(B) It is asymptotically correct at the outer edge of the boundary layer. (C) It agrees with MUNR'S (1950) solution when the Reynolds number tends to

zero.

Its main weakness, and a serious one, is that it alters the character of the basic differential equations, replacing parabolic partial differential equations by ordinary differential equations. This deficiency is a consequence of omitting tangential advec- tion. In defense of the model one may note that the tangential flux of fluid in the boundary layer is determined at every section directly in terms of the normal velocity of the external flow so that, in effect, the omitted tangential advection is determined by the normal advection. Thus we may hope that only the details of the velocity distribution in the boundary layer will be wrongly predicted.

The model equation will be taken to be :

33 ~b b~ _ Wo 7r Try b 4 ~b (4.6) -- U ( Y ) ~ x ~ %- /33x L ' s inL, %-v bx 4

I f one writes ~b = ~b* sin zry/L', one has :

33 ~b* b~b* _ W o ¢r _____34 ~b* (4.7) - - U(y) 3x 3 + /3 3x L' + v bx 4 ,

with boundary conditions

and

~b*, 3~b* _ 0 on x = 0 and x = L (4.8) bx

b~b* _ 0 on y 0 and y = L'. (4.9) 3y

This last condition is a consequence of (4.3) and it is automatically satisfied by any solution of (4.7) so long as dU/dy = 0 at y = 0 and y = L'. This condition on dU/dy will be assumed to be satisfied in future.

The general solution of (4.7) is : 3

~ , W o zrx V ~ = /3]L- %- ~ Ai e ~ix %- Ao (4.10)

744 D .W. MOORE

where A~ are constants A~ are the roots o f the cubic,

vA a - - U(y ) A 2 + / 3 = 0. (4.11)

I t will be noted that (4.11) is identical with the cubic (3.3) obtained for the exact solutions of Section 3, so that the solutions of the model equat ion are similar to the exact solutions of Section 3. Moreover , the discussion of the nature of the roots given there may be carried over. There are thus two cases to consider.

Case 1 : U (y) ~ 8113 V +2/3

In this case there are two positive roots A 1, A 2 say and one negative root, A3. The boundary condit ions (4.8) lead to the equations :

Ao + A1 + A2 ÷ As = 0

W0w A I A 1 - - A z A 2 - - A s A s = 0 /3L'

Wo zr L L e-;%L e-a.~L (4.12) A 0 @ ~ - L 7 - @ A 1 e -a, + A 2 @ A a . = 0

Wo ~r e-AlL e-A~L e-A3L fl L ' Al Ax - - A2 A 2 - - A 3 A s ----- 0.

The solution of this system is greatly simplified if, A1 L, A2 L, - - haL >> 1, (so that the boundary layer thicknesses are small compared to the breadth of the basin) and in this case one finds that :

~b* Wo ~" ( (A 2 A s L + A~ - - As) (A~ A s L + Ax + As) e_a~ x = ~ Z ; 4~ x -~ a 3 (A 2 - - A 1) e-a' * + A z (A x - - A 2)

1 _ ,,) 1 } (4.1 -~- ~ e As(L - - L - - A3_" 3)

I t should be noticed that A~ and hence the constants A~ are functions of y, since U (y) is involved in the fundamenta l cubic (4.11). The y dependence of the solution is thus parametric.

As in the MUSK (1954) solution the eastern boundary layer is ' invisible ' since the term giving rise to it is 0 (1/As) compared to other contr ibut ions to ~b* of order L.

Case 2 : U(y ) < 1 '+2/3

In this case the roots are A 1 = p + iq A 2 = p -- iq and A s < 0, where p > 0. Thus the general solution (4.8) may be written

~ , Wo ~x = ~-~w- + Ao' + A'I e -p~ cos qx + A' 2 e -px sin qx + A's e a3x (4.14)

and assuming that pL, - - A 3 L >> 1 the solution is :

( 1) ( l = ~ L ' \ x + L + ~ e - p * c o s q x + - - - q + - - q + q ' ~ 3 e -p*s inqx

1 1 e%(L_x) l (4.15) - L - - X-~ + z~ 3"

Rossby waves in ocean circulation 745

The flow near the eastern boundary is similar to the previous case, but the flow in the western half consists of damped oscillations.

So far little has been said about the choice of U(y ) in the model equation. ~qow the x-component of velocity in the mid-ocean is easily found to be

W°~r[/3 cos ( ~ r Y ) ] ( L - - x ) a n d t h i s i s f r o m e a s t t o w e s t i f y < ½ L ' a n d f r o m w e s t - f f

to east if y > ½ L', that is to say east to west in the southern half and west to east in the northern half. Thus we take U (y) to be :

~ry U (y) = q- Uo cos - - (4.16)

L'

which has the same y variation as the actual x-component of flow. The nature of the solution will depend very greatly on the values assigned to v

and U0. There is not much freedom with respect to the choice of U0, which must be of order 10 cm sec -x, but the values assigned to the eddy coefficient v have varied greatly in previous work. Given the values U0 = 10 and/3 : 10 -13 the relationship between the Reynolds number, R = Ual2/vfl 1/2 introduced in Section 2 and v is shown in TABLE 1. The choice v ---- l0 ~ would thus correspond to a high Reynolds number

Table 1

v R

1o 6 lOO 10 7 10 10 s 1 109 10-1

ocean and for such values the crudeness of the model inertia terms would be serious. On the other hand v ---- 10 a would give a low Reynolds number ocean similar to that discussed by M t m g (1950).

20

I 0

I I I I

0 I 0 ~0 30 40 $0

3L" Lines of constant ~-0£ " ~ for rectangular basin with L = 5000 km and L" = 2000 kin.

The Reynolds number U 3, ~/v31/2 = 5.

Fig. 1.

It was decided to consider in detail the intermediate value v = 2 × 107 correspond- ing to R = 5. The streamlines obtained are shown in FIG. I. It can be seen that

746 D . W . MOORE

damped Rossby waves carry departures from geostrophic conditions into the northern interior of the ocean. The existence of these waves suggests that the meanders observed in the Gulf Stream may not be wholly due to instability but may rather be a property of the steady solution. The streamlines are not very similar to actual ocean currents, but it should be remembered that in practice the line of vanishing wind stress curl and the coast line are not at right angles so that the flow turns through less than 90 ° in the northwest corner.

As R decreases from five, the waves disappear and the flow pattern agrees with MUNK'S in the limit R ~ 0. As R increases from five, the waves penetrate further eastward and eventually fill the entire northern half of the basin. There is a region of rapid transition at the latitude of vanishing wind-stress curl. However at such large Reynolds numbers the oscillatory flow pattern predicted would be highly unstable and would lead to intense turbulence in the northern portion of the basin if, indeed, it could ever be established. This turbulence would increase the overall eddy viscosity and hence the Reynolds number would decrease and a less oscillatory pattern at a lower Reynolds number would result. Thus the model suggests that the ocean has a built-in mechanism for keeping its overall Reynolds number based on eddy viscosity from becoming large.

A c k n o w l e d g e m e n t s This work was done at the 1961 S u m m e r P rog ramme in Geophysical Fluid Dynamics o f the W o o d s Hole Oceanographic Inst i tut ion and the au tho r wishes to express his grat i tude to the Fellowship C ommi t t e e for the award o f a Postdoctoral Fellowship to a t tend this p rogramme. The au tho r benefited greatly f rom discussions o f the work with regular and visiting staff members o f the Inst i tu t ion and would like in part icular to thank Dr. GEORGE VERONIS for in t roducing h im to the p rob lems of ocean circulation and for his cont inued interest in the work. The numerical work was carr ied out on the Mercury compute r at Oxford and the au thor would like to express his grati tude to Dr. M. H. RO~ERS for his assistance with the p rogramming .

The equa t ion to be considered is :

with the boundary condi t ions :

A P P E N D I X

F * " " = F * 'z - - F * F * " - - 1, (1)

F* (0) = F* 1 (0) = 0, (2)

F .1 ~ 1 as x* -+ o~ (3)

where the upper sign gives the equat ion for the western bounda ry and the lower sign gives the equa- t ion for the eastern boundary . It will be shown that in the latter case (1) has no solut ion which satisfies (3).

Let F* ~ x * + g as x* -+ co (4)

so that , g * - + 0 as x* ~ co. Then g mus t satisfy the equat ion :

± g(3) __ 2g(~) _ x * gtZ), (5)

where the second order terms in g have been omit ted. I f h = g(a) one has finally :

h (zl + x h (t) - - 2h ~ 0 on western boundary (6)

h (z) - x* h (~) + 2h -- 0 on eastern bounda ry (7) First, consider (6). I f :

h (x*) -- K ( x * ) exp {-- ¼ x *z} (8)

Rossby waves in ocean circulation 747

then K satisfies : d z K ~x.- 2 + {-- 3 + ½ -- i x*Z} K (x) = 0 (9)

whose general solution is : K = AD_3 (x*) + BDz (ix*), (10)

where A, B are constants and D n is the parabolic cylinder function (WHITTAKER and WATSON 1958). Now as y -+ co for la r gYl < ~ lr

Dn (y) ~ e_Y214 yn, (11)

and, on using this result one finds that

--X*Z

h ~ e 2- x *-3 or x .2 as x* ~ oD (12)

the first term being the unique acceptable solution at oo. Now consider (7). I f

h* (x) = K(x*) exp {+ ¼ x .2} then K satisfies :

d z K axe2 + { 2 + ½ - - ¼ x *z} K = O,

whose general solution is : K = ADz (x*) + BD_ 3 (ix*).

Hence, one has :

h* ~ x *z or e 2- x *-3,

neither of which is acceptable as x* ~ oc. lower sign is taken.

(13)

(14)

05)

Thus (1) has no solution which satisfies (3) when the

R E F E R E N C E S

CHARNEY, J. G. (1955) The Gu l f Stream as an inertial boundary layer. Proc. U.S. Nat. Acad. Sci. , Wash. 41, 731-740.

CARRIER, G. F. and ROSlNSON, A. R. (1962) On the theory of wind-driven ocean circulation. J. Fluid Mech. 12, 49-80.

FOFONOFE, N. P. (1954) Steady flow in a frictionless homogeneous ocean. J. Mar. Res. 13, 254-262.

FOFONOFF, N. P. (1961) Private communicat ion. FOFONOFF, N. P. (1962) Dynamics of ocean currents. In : The Sea : Ideas and Observations.

(Edited by HILL, M. N.) Vol. I, 323-395. lnterscience Publishers, New York. GOLDSTEIN, S. 0938) Modern developments in f lu id dynamics, pp. 1390. Oxford Univ. Press,

Oxford. MORGAN, G. W. (1956) On the wind-driven ocean circulation. Tellus 8, 301-320. MUNK, W. (1950) On the wind-driven ocean circulation. J. Met . 7, 79-93. STOMMEL, H. (1948) The westward intensification of wind-driven ocean currents. Trans.

Amer. Geophys. Union 29, 202-206. STOMMEL, H. (1958) The Gu l f Stream, pp. 202. University of California Press, Berkeley,

Calif., U.S.A. SVERDRUP, H. U. 0947) Wind-driven currents in a baroclinic ocean, with application to the

equatorial currents of the eastern Pacific. Proc. U.S. Nat. Acad. Sci., Wash. 318-326. TAYLOR, G. I. (1917) Mot ion of solids in fluids when the flow is not irrotational. Proc. Roy.

Soc. (A) 93, 99-113. WHITTAKER, E. Z. and WATSON, G. N. (1958) Modern analysis, pp. 347. Cambridge Univ.

Press, Cambridge.