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 o f Mathematics, Bristol University
(Received 12 August 1963)
Abstract--It is shown that damped, stationary 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 h o m o geneous ocean is constructed in which Rossby waves occur in the northern portion o f the basin.
INTRODUCTION
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 explaining 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 momentum 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 vanishing 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 eastwest 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 sublayers on the continental walls. The nature of these sub-layers is examined and it is
*Contribution No. 1417 from the W o o d s 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 constructed. 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.
FLOW
NEAR
A LATITUDE
OF
VANISHING
WIND
STRESS
CURL
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 bottom 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
u b--X
bu
+ v-- --fv
oy
1 bp
--
p 3x
ubV + vbv
~x
+
1
~y + f u --
W+
- - - + vV ~ v,
p by
bu + by
3x
by
0
vV~u
(2.1)
(2.2)
(2.3)
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. Now (2.3) can be integrated by means of a stream function
~b(x, y) such that :
u-and using the relations :
~y
,
v=----,
bx
(2.5)
Rossby waves in ocean circulation
~(f~) : f ~ -~~ - -
737
(2.6)
iv,
~y (f~b) =-f ~ + ~b~Y
3f = fu ÷ fl~b,
(2.7)
one has finally:
u -3u- + v - - 3u
-3x
by
p bx
I 3p' + W+~, V2 u
(2.8)
u -by
- + v - - - 4by, ~
. . . .I
(2.9)
~x
by
bp'
p 3y
+~
V2 v,
where :
p' = p + p f ~b.
(2.10)
The equations in this form show clearly that it is the variation of the Coriolis parameter 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 m o m e n t u m is to be transferred across
y = 0 one must have
v:0,
--bu=0
by
on y = 0
(2.11)
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 NavierStokes 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)
-? y g I (x)
+ y~ K s (x) + . . .
(2.13
then substitution into (2.8) leads to the equation •
ff'=--lbP--'+Ko(x)+yKl(X)+Y~K2(x)~
...+~,f"
(2.14)
P = c(y) + x(x) + y f Kl (x)dx + y2 f K~(x)dx + . . .
(2.15)
thus :
p bx
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
dy
_ yff,, + yf,2 _ f l y f _
f Kx(x) dx--2yf
K~(x)+...--vYf".
(2.16)
Thus by inspection of the powers of y in (2.16) :
dC
--
dy
-- Cy, say, and K1 (x) = 0
(2.17)
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 :
vf"' -- 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
~
f ( L ) --=f ' (L) = 0
J
(2 20)
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) transport 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,
\
J"~ /
EV=--
C,
R(o2) -- K (U
x)
(2.22)
whence :
*Note that the constant of integration can be absorbed in the unknown constant, C in (2.19).
Rossby waves in ocean circulation
F (0) = F'(0) = 0
739
(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'"
R
- -
=
F + E -- 1{ (~").
(2.26)
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-~)
(2.28)
near ~" = 0 and :
F=
(1 -- e~-L ( ~ ) ½)
(2.29)
near ~" = L (~/U) t. These solutions fail to satisfy the conditions F ' ( 0 ) = 0 and
F ' (fli/U t L) = 0, which state that the tangential velocity is 0, so that viscous sublayers 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. Reference 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. Furthermore 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)
(x*) -+ 1 as x* -~ ~
F * (0) - - F * ' (0) = 0, f * '
(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 decelerating, 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.
EXACT
SOLUTIONS
OF
THE
FREE
EQUATIONS
If one takes K (x) ~ 0, so that there is no wind stress (2.19) takes the form :
vf'"
-
C=
ff"
_ f , 2 + /3f.
(3.1)
One can easily verify that :
f=
-- U(1 - - e - ~ x )
(3.2)
is a solution of this equation provided that :
C = flU and
vz3 -- 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
If 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 :
~x + ~2 + % = +
-
U
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. N o w suppose that
the roots are not all real, so % = p ÷ iq, a s = p - - iq.
Then :
~2 ( f
+ q~) = - - B/v
2% p q- (p2 q_ q2)
=
0
(3.7)
(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 >
¢31/3v+2/3the decay is oscillatory. I f p < q the solutions represent
slowly damped stationary waves on a basically west to east flow. In general it is necessary to solve the cubic numerically to determine p and q, but if B v 2 / U z < 1 one can
easily show t h a t :
q=
'
P-- 2 U2
(3.9)
It 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.
4.
A
MODEL
OF
MOORE
OCEAN
CIRCULATION
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 :
b~b b (V 2~b)
by 3x
~
b~b b (V 2 ~ b ) + f l _ _ =
bx 33'
3x
dW + v V a~h
dy
(4.1)
with the boundary conditions "
4',--b~b == 0 on x = 0, x = L;
bx
(4.2)
~b, 32 4,
(4.3)
0 on y = 0, y = L'.
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
34 ~b
by bx 3 -- ~ x b y b x ~ + /bbx
L' sin L" + v bx4_,
(4.4)
where, in addition, we have specialized to the wind distribution'
w(y)
=
-
,ry.
L'
w0 cos --
(4.5)
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 approximation 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 boundary 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 advection. 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
b4 ~b
-- U ( Y ) ~ x ~ %- /33x
L' s i n L , %-v bx4
(4.6)
If one writes ~b = ~b* sin zry/L', one has :
33 ~b*
b~b* _ Wo ¢r
_____34
~b*
-- U(y) 3x 3 + /3 3x
L' + v bx 4 ,
(4.7)
with boundary conditions
~b*, 3~b* _ 0 on x = 0 and x = L
bx
(4.8)
and
b~b* _ 0 on y
3y
0 and y = L'.
(4.9)
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
~,
=
Wo zrx
/3]L- %- ~V ~ Ai e ~ix %- Ao
(4.10)
744
D . W . MOORE
where A~ are constants A~ are the roots of the cubic,
vAa -- U ( y ) A2 + / 3 = 0.
(4.11)
It 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 equation are similar to the
exact solutions of Section 3. Moreover, the discussion of the nature of the roots
given there m a y 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 A1, A2 say and one negative root, A3. The
b o u n d a r y conditions (4.8) lead to the equations :
Ao + A1 + A2 ÷ As = 0
W0w
/3L'
A0@
AIA1--AzA2--AsAs=0
Wo zr L
~ - L 7 - @ A 1 e -a,
Wo ~r
fl L '
Al Ax
e-AlL
L
+ A2
- - A2 A 2
e-;%L
e-A~L
@ Aa
- - A3 A s
e-a.~L
.
e-A3L
(4.12)
= 0
----- 0.
T h e solution of this system is greatly simplified if, A1 L, A2 L, -- haL >> 1, (so that
the b o u n d a r y layer thicknesses are small c o m p a r e d to the breadth of the basin) and
in this case one finds that :
~b*
Wo ~" (
= ~Z;
4~x -~
(A2 As L + A~ -- As)
a 3 (A 2 - - A1)
(A~ As L + Ax + As) e_a~x
Az (Ax -- A2)
e-a' * +
1
-~- ~
e As(L
_ ,,)
--
1}
L -- A3_"
(4.1 3)
It should be noticed that A~ and hence the constants A~ are functions of y, since U (y)
is involved in the fundamental cubic (4.11). The y dependence of the solution is thus
parametric.
As in the MUSK (1954) solution the eastern b o u n d a r y layer is ' invisible ' since the
term giving rise to it is 0 (1/As) c o m p a r e d to other contributions to ~b* of order L.
Case 2 :
U(y) <
1'+2/3
In this case the roots are A1 = p + iq A2 = p -- iq and As < 0, where p > 0.
Thus the general solution (4.8) m a y 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, -- A3 L >> 1 the solution is :
= ~L' \x+
( 1)
L+~
e- p * c o s q x +
(l
- - - q + - - q + q ' ~ 3 e -p*sinqx
1
1 e%(L_x) l
- L - - X-~+ z~
3"
(4.15)
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[/3cos ( ~ 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 > ½L', 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 - L'
(4.16)
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
1o6
lOO
107
10 s
109
10
1
10-1
ocean and for such values the crudeness of the model inertia terms would be serious.
On the other hand v ---- 10a would give a low Reynolds number ocean similar to that
discussed by M t m g (1950).
20
I0
0
I
I
I
I
I0
~0
30
40
$0
3L"
Fig. 1. Lines of constant ~-0£"~ for rectangular basin with L = 5000 km and L" = 2000 kin.
The Reynolds number U3,~/v31/2 = 5.
It was decided to consider in detail the intermediate value v = 2 × 107 corresponding 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.
This work was done at the 1961 S u m m e r P r o g r a m m e in Geophysical Fluid
D y n a m i c s o f the W o o d s Hole O c e a n o g r a p h i c Institution a n d the a u t h o r wishes to express his gratitude
to the Fellowship C o m m i t t e e for the a w a r d o f a Postdoctoral Fellowship to a t t e n d this p r o g r a m m e .
T h e a u t h o r benefited greatly from discussions o f the work with regular a n d visiting staff m e m b e r s o f
the Institution a n d w o u l d like in particular to t h a n k Dr. GEORGE VERONIS for introducing h i m to the
p r o b l e m s o f ocean circulation a n d for his c o n t i n u e d interest in the work. T h e numerical work was
carried o u t on the M e r c u r y c o m p u t e r at Oxford a n d the a u t h o r would like to express his gratitude
to Dr. M. H. RO~ERS for his assistance with the p r o g r a m m i n g .
Acknowledgements
APPENDIX
T h e e q u a t i o n to be considered is :
F * " " = F *'z -- F * F * "
--
1,
(1)
with the b o u n d a r y conditions :
F* (0) = F* 1 (0) = 0,
(2)
F .1 ~ 1 as x* -+ o~
(3)
where the u p p e r sign gives the equation for the western b o u n d a r y a n d the lower sign gives the equation for the eastern b o u n d a r y . It will be s h o w n that in the latter case (1) has no solution which satisfies
(3).
Let F* ~ x *
+ g as x* -+ co
(4)
so that, g * - + 0 as x* ~ co.
T h e n g m u s t satisfy the equation :
± g(3) __ 2g(~) _ x * gtZ),
where the second order terms in g have been omitted.
If h = g(a) one has finally :
h (zl + x h (t) - - 2 h ~ 0
First, consider (6).
(5)
o n western b o u n d a r y
(6)
h (z) - x* h (~) + 2h -- 0 on eastern b o u n d a r y
(7)
h (x*) -- K ( x * ) exp {-- ¼ x * z }
(8)
If:
Rossby waves in ocean circulation
747
then K satisfies :
dzK
~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 (WHITTAKERand WATSON 1958).
Now as y -+ co for la r gYl < ~ lr
Dn (y) ~ e_Y214yn,
(11)
and, on using this result one finds that
--X*Z
h ~e
2- x *-3 or x .2 as x* ~ oD
the first term being the unique acceptable solution at oo.
Now consider (7). If
h* (x) = K ( x * ) exp { + ¼x .2}
then K satisfies :
dz K
axe2 + { 2 + ½ - - ¼ x *z} K = O,
(12)
(13)
(14)
whose general solution is :
K = ADz (x*) + BD_ 3 (ix*).
05)
Hence, one has :
h* ~ x *z or e 2- x *-3,
neither of which is acceptable as x* ~ oc. Thus (1) has no solution which satisfies (3) when the
lower sign is taken.
REFERENCES
CHARNEY,J. G. (1955) The G u 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 communication.
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) M o d e r n developments in f l u i d dynamics, pp. 1390. Oxford Univ. Press,
Oxford.
MORGAN, G. W. (1956) On the wind-driven ocean circulation. Tellus 8, 301-320.
MUNK, W. (1950) O n the wind-driven ocean circulation. J. M e t . 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 G u 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) M o t i o n 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.