Wind-drift near the equator

Wind-drift near the Equator HENRY STOMMEL

(Received 20 July 1959) Abstract--The principal result of the model is to show that a linear theory constructed from Ekman spiral and geostrophic current does not suffer singularities at the equator and even exhibits some qualitative similarity to the Equatorial Undercurrent, but that in order to account for the observed amplitude of the phenomenon, non-linearities are not altogether negligible. (1)

INTRODUCTION

IN EKMAN'S well-known solution of drift of water in a rotating homogeneous ocean of infinite depth when acted upon by a steady stress applied at the surface, the depth of the spiral, and amplitude of the currents increase without limit as the vertical component of rotation (or latitude) approaches zero. It is possible to add an arbitrary infinitesimal deformation of the free surface to Ekman's homogeneous model and an associated pattern of geostrophic f l o w - i n d e p e n d e n t of d e p t h - w h o s e Coriolis forcejustbalances the arbitrary pressure gradient. Infact, Ekman uses suchadditionai geostrophic velocities to satisfy conditions of no flow through coastal barriers. The one-dimensional nature of the simplified equations used by Ekman does not permit satisfying the true boundary condition at a vertical coastal barrier : namely, that the velocity normal to the barrier shall vanish at all depths. Instead of trying to solve the details of this more complicated problem, which evidently would involve a higher-order set of dynamical equations, Ekman simply adds a pressure field sufficient to cause the total vertically integrated component of velocity normal to the coast to vanish. When the ocean is of finite depth, a finite geostrophic current is required, and when the bottom is rigid it is necessary to allow for a second frictional boundary layer, or current spiral at the bottom. This latter complication can be avoided by asserting that the bottom boundary condition is one of no s t r e s s - such as can be achieved by letting the viscosity become very small near the bottom. In these ways Ekman is able to avoid the indeterminacy of additive geostrophic motion in his simple one-dimensional (vertical) dynamical system, to make an approximate allowance for the influence of lateral boundaries, and also to compute patterns of flow in homogeneous ocean models under various distributions of wind and coasts. The purpose of the present paper is to show that this simple type of analysis can be extended across the equator, and that there is indeed no singularity at the equator. Thus, in principle, there is no reason to suppose that Ekman's solution ' blows up ' at the equator, as is sometimes asserted in qualitative discussions. However, it will be shown that the model is consistent only when the amplitude of the velocity is kept at an unrealistically low value by imposing a very small value of wind stress, or a large eddy viscosity. When realistic values are introduced and the solution for the velocity field is used to compute the magnitude of the neglected inertia terms it is found that they are not truly negligible and that the model presented here becomes 298

Wind-drift near the Equator

299

inconsistent. Therefore it seems that the model does not apply as it stands to the Pacific Equatorial Undercurrent (Cromwell Current). From a pedagogical point of view however the model may be of some value because it exhibits qualitative features similar to the Cromwell Current, because in an ocean where mean wind stress along the equator is less (Indian Ocean) it might actually apply, and because it embraces the very simplest linear type of dynamical scheme which one is bound to investigate before pursuing more complex models including lateral friction, inertial terms, etc. At one time Dr. VERONIS and I considered introducing the heat-transport equation in a linearized form such as used by LINEYKIN(1955, 1957) and STOMMELand VERONIS (1957) and developing the temperature, pressure and velocity fields about the equator in a power series in latitude (y), but such an analysis is also inapplicable to the actual Cromwell Current because of the neglect of inertial terms, and we decided that it did not reveal enough additional information to publish. In this same issue Dr. CHARNEY shows how the inertial terms can be included in the simple homogeneous model, and in the paper after that Dr. VERONIS shows how the thermal field can also be included. (2)

THE

SOLUTION

OF

THE

LINEAR

PROBLEM

The model consists of a homogeneous layer of water of uniform depth h straddling the equator. Acting upon the upper surface there is a westward wind stress r. It does not appear likely that a current like the Cromwell Current is caused by a detailed structure of the wind stress as a function of latitude, so the wind stress is assumed to be uniform. The x-axis is direct toward the east, the y-axis northward, the z-axis upward and z = 0 at the bottom of the layer. The eddy viscosity is a constant A. The linearized equations of motion, in which latitude enters only parametrically through the vertical component of the earth's angular velocity o~ are of the form : -- 2a, v ---- A 2~ou = A

~u

1 bp

bz ~

p bx

b2 v

1 bp

bz 2

p by

where p is pressure, p density, and u, v the x and y components of velocity. We assume the water to be homogeneous in density so that a p / b x and bp/by are independent of depth. The solutions are of the form 4

u q- ir :

Z c i e ~iz --~ G 1 + i G z 1

where ~l----r(lq-i),

~2-----r(I--i),

%=r(--lq-i), r :

Gx and G~ are real constants.

%=

r ( - - l -- i)

/oga

We take z = 0 at the bottom where we assume

A b_._uu= A --bY= 0. Making the bottom a plane of no stress has the virtue of avoiding bz

bz

the necessity of calculating an additional functional layer at the bottom, thus simplifying the analysis.

300

HENRY STOMMEL

At the top surface, z = h, the stress ~- is taken to be only in the x-direction. These boundary conditions are enough to fix the four constants c i. The additive constants G1, G2 corresponding to geostrophic flow, are evaluated by taking the vertically integrated transports equal to zero. This might at first seem to be a very arbitrary step. It can be perhaps justified by stating first that since we will eventually allow ¢o to vary parametrically (co = fly) the well-known ' c u r l ' equation

fl f l v d z = curl ('r/p) hence, since curl (-r/p) = O, the vertical integral of v must also vanish everywhere. The continuity equation

~

~ ,.0

reduces to the form bx /'h

If, therefore, [ o u d z is set equal to zero at an eastern meridional boundary it also must vanish everywhere. The solution of the problem can now be written in the form : _¢ [cosh 0¢1 Z 1 )

u q- iv = ~ 1 P \s-~nh ~1 h

o¢i h

In order to evaluate the velocity components u and v as functions of z and ¢o it is necessary to separate real and imaginary parts of the right hand side. Writing Z -~ rz and H--- rh we obtain the following expressions : u

j Ch

1

-.

(si h . . +

J

cogh

sin

. cos

÷ sinh Z sin Z cosh H sin H q- s i n h Z sin Z sinh H cos H -- cosh Z cos Z cosh H sin H}

v

1

1

H (sinh ~ H -k sin S H)/sinhZC s i n Z sinh H c o s H -- c o s h Z c o s Z c o s h H s i n H -- cosh Z cos Z sinh H cos H --

s i n h Z sin Z cosh H sin H )

We can regard the H as containing the essential parametric variation of the quanti~ies with latitude. The following table gives some computed values

Wind-drift

301

near the Equator

H ,&_.._

0 2Ap 1.0

0-666

0.584

0.285

0.000

-- 0.180

-- 0.186

-- 0.083

0.081

-- 0.063

0.000

0.013

0.038

2Ao • u ~'h

-- 0.333

-- 0.254

-- 0.027

2Ap --4-y. v

0.000

0-151

0.128

2Ap TII

0.5

7r

-9, o o

IlH

"rh

2Ap --~- . v

zlh

./2

2Ap

-

IfH

I/H 2

Ttl

0.0

I/H z

At H =- 0, which corresponds to the equator, the expressions can be reduced to the following form,

u + iv=

~-h (,~

Z--A-?

I) -- ~ + ; 0

a simple parabolic velocity profile, with no north-south component. The case H -+ co corresponds to the combination of a thin surface Ekman layer underlain by an equatorward meridional geostrophic component. The figure shows the velocity vectors on either side of the equator. Quantitatively, the mean stress r at the equator is about -- 0-2 dynes cm -~ in the Pacific, and the layer depth (core of undercurrent) is h = 104 cm. The observed amplitude of the undercurrent at H -----0 and z/h = O, is about 150 cm sec -1, so that eddy viscosity A is 2.2 cm 2 sec -1. According to the model the undercurrent vanishes at a latitude where H = rr, so that using these values the northern border of the undercurrent is at a latitude corresponding to ~o ~ A X 1 0 - 7 sec -i, or about 0.t5°N. The observed boundary is more nearly at 1-5° latitude, so this agreement is not good. If the eddy viscosity is increased to 22.0 cmz sec -x, the computed northern boundary of the undercurrent agrees with that observed, but the velocities in the core of the undercurrent are reduced to 15 cm sec -~, which is much too low. Therefore, it appears that this linearized model does not apply to the Pacific Equatorial Undercurrent. Furthermore, using the above values of the wind stress and the two choices of eddy viscosity, a calculation shows that the non-linear inertial terms are of the same order of magnitude as the other terms in the neighbourhood of the equator, and should not be neglected. A reviewer has indicated several obscurities in the presentation which need to be cleared up. First, strictly speaking, the homogeneous layer lies upon a deeper resting layer of greater density. If there are pressure gradients in the homogeneous upper layer, but none in the lower layer, there must be a slope in the interface between them.

302

Hm~v STOMMEL

Therefore the upper layer cannot be of precisely uniform depth. Also it is worthwhile to emphasize that there is a local balance of pressure-gradient force on the layer and wind stress, and the uniformity of one guarantees that of the other. And finally,

Z/h

. -2e"

2f H

fo

~

N

Fie. 1. Schematic three-dimensional diagram of the combined frictional and geostrophic flow field in the neighbourhood of the equator, not showing the vertical component of velocity (which can in principle be determined from continuity).

the current transport field obtained in this model is very similar to that pictured by CROMWELL(1953, Fig. 6), and we might regard the present model as a mathematical analysis of Cromwell's dynamical description of the currents produced by an east wind at the equator.

Woods Hole Oceanographic Institution, Woods Hole, Mass., U.S.A. Contribution No. 1084 from Woods Hole Institution. REFERENCES

CROMWELLTOWNSEND(1953) Circulation in a meridional plane in the central equatorial Pacific. J. Mar. Res. 12, 196-213.