The equatorial current in a homogenous ocean

Deep-Sea Research, 1971, Vol. 18, pp. 421 to 431. Pergamon Preu. Printed in Great Britain.

The equatorial current in a homogeneous ocean A. E. GILL* (Received 28 January 1970; accepted 20 March 1970) Al~tract--The throe-dimensional circulation in a homogeneous ocean is examined in the equatorial region. It is found that, for usual values of the parameters, the currents there are controlled by lateral friction rather than vertical friction. F o r a westward stress at the surface, the current, relative to its vortical average, consists of a westward flux in the surface layer and a n eastward equatorial undercurrent. The transport of the current is independent o f the value o f the friction parameter and is equal to 14 x 10a ms/see for each dyne cm -s (10 -x newton m -~) of wind stress. The current ©xistl oven if there is no east-west pressure gradient. A picture o f the meridional c ~ u l a t i o n associated with the current is obtained, and the dynamical reasons for the existence of the current are Oven. INTRODUCTION

I~ TWO recent studies (P~DLOSKY, 1968; JOI-I~SON, 1968), the three-dimensional circulation in a homogeneous ocean was considered. As far as the verticaUy-integrated currents are concerned, these models give the same results as the well-known twodimensional models (see ROBrNSON, 1963). Away from the boundaries at the sides, the currents also have a weU-understood structure. The currents are independent of depth except in thin Ekman layers at the surface (and possibly the bottom). Relative to the uniform currents below, the surface Ekman layer carries a flux equal in magnitude to the wind stress divided by the Coriolis parameter, and directed at right angles to the surface stress (to the right in the northern hemisphere and to the left in the southern hemisphere). The most interesting new feature of the three-dimensional models occurs when this flux does not vanish at the side walls (which were taken to be vertical in the models). In order to preserve continuity, the flux is absorbed by a boundary-layer on the side walls. This layer was called an "upwelling" boundary-layer by Pedlosky and has an e-folding thickness equal to (A/a[sin ~])J (1) where A is the lateral eddy-viscosity coefficient, ~ is the angular velocity of the earth and ~ is the latitude (positive northwards). Both the three-dimensional studies referred to made use of a beta-plane approximation and excluded the equatorial region. The purpose of this paper is to describe the new feature that arises when the equatorial region is included in a linear homogeneous ocean model. Inertial effects are ignored and the vertical eddy viscosity coefficient, v, is assumed to be small in relation to the lateral eddy viscosity coefficient, A, in a way which will be precisely defined later. As far as the vertically-integrated currents are concerned, the equations are the same as for a two-dimensional lateral-friction model like that studied by M ~ K (1950). *Department o f Applied Mathematics and Thoaretical Physics, Cambridge

421

422

A . E . GILL

The equations are no different in the equatorial region from other regions, so no special features are to be expected in the field of vertically-integrated currents near the equator. The new feature arises when the westward (or eastward) stress does not vanish at the equator. This drives an Ekman flux away from (or towards) the equator on both sides. In order to preserve continuity, upwelling (or downwelling) occurs in a boundary layer at the equator, and poleward (or equatorward) motion occurs below the Ekman layer. Thus the surface stress drives a meridional circulation (depicted in Fig. 2 below). The upwelling (or downwelling) region could conveniently be called the ' equatorial upwelling layer' (or downwelling layer). The equatorial upwelling layer exists because of the vanishing at the equator of the components ( - - f v ' , f u ' ) of the Coriolis force per unit mass, where f--

2 0 sin

is the Coriolis parameter and (u', v') are the horizontal components of velocity. An analogy may be drawn with the side-walI upwelling layer which exists because of the vanishing of ( - - f v ' , f u ' ) at the side wall, though in the latter case because of the vanishing of velocity rather than because of the vanishing o f f . The forces which are normally balanced by the Coriolis forces, such as the wind force at the surface and the pressure gradients within the fluid, tend to drive the fluid at the equator in the direction in which they act until a balance is achieved with some opposing force. In the present model the opposing force is due to lateral friction. In earlier models of the equatorial undercurrent (STOMMEL, 1960; CHARNEY, 1960; VERONXS, 1960; ROBINSON, 1966), the opposing force was due to vertical friction with, in some cases, inertial effects as well. THE

MODEL

Consider a homogeneous ocean of uniform depth H. The ocean has a vertical eddy-viscosity coefficient v and a lateral eddy-viscosity coefficient, A. Currents are driven by a wind stress applied at the surface. These currents are assumed to be so small that inertial effects may be ignored. The (dimensional) equations are --

212 sin ~ v. . . .

a -1 sec (~ g 71;~' q- F '~ -k- v u'z,~,

(2)

2-Q sin ~ u' :- -- a -1 g ~ ' -~- F'~ + v v'z,~.

(3)

sec $ [ua' + (v cos $),] + w'~, = 0

(4)

where u', v', w' are the velocity components corresponding to the co-ordinates A, 4', z' respectively, and where A is longitude, $ latitude and z' distance upwards from the bottom. F 'a and F'* are the lateral friction terms, g is the acceleration due to gravity and a is the radius of the earth. The hydrostatic approximation has been made so that the pressure gradient is provided by a surface elevation 7' relative to a geopotential surface. As in earlier models of the undercurrent, the Coriolis acceleration associated with the horizontal component of the earth's rotation has been neglected. This can be justified a posteriori provided a certain condition is fulfilled. This condition will be discussed later. Attention will be confined to the equatorial region, and the equatorial upwelling layer in particular. Thus the lateral friction terms are approximated by

The equatorial current in a homogeneous ocean F'~ =

a-~ A u ' ~

423

(5)

F'~ = a-2 A v ' ~

The upwelling is driven by the westward (or eastward) component of stress which is assumed to be constant. The northward component of stress is taken to be zero. The boundary conditions at the surface are therefore PUz"

t We,

~---

--

7"

=---0

)

at

z' = H,

(6)

where z is a constant. It is assumed that bottom friction is not important and the boundary conditions at the bottom are taken to be u'z,=v'z,:O

at

z':0.

(7)

(It is perhaps appropriate to think of this lower boundary as a surface of discontinuity of density rather than as a solid boundary.) NON-DIMENSIONAL

Let

z' = Hz,

FORM

u' = • u/212H,

w' = T w/212a,

v' : 7" v/2~2H

(8)

~1' : aT"~7/gH.

Then the above equations become --

sin ~ v = -- see ~ ~/a + E n u ~ + E v uzz

(9)

sin de u = -- '1, + E n v~, + Ev vzz

(10)

sec ~ [u~ + (v cos ~b)~] + wz = 0,

(11)

where the two parameters EH and Ev are given by (12)

EH : A/2g2a ~ "~ . Ev : v/2f2H 2

3

EH is called the horizontal Ekman number and E v the vertical Ekman number. The boundary conditions are Evuz=

1,

vz=O

Uz=Vz=O

at

z=

at

z=O

1"~.

(13)

J

VERTICALLY-INTEGRATED EQUATIONS Let

1

1

f~= f u d z ,

~= f vdz.

0

(14)

0

Then the vertical integrals of (9), (10) and (11) give --

sin ~b ~ = -- sec ~ ~7~ + EH ~ sin ~ fi = - - ~7c,+ EH ~ , ~a + (~ cos 4)* = 0.

-- 1,

(15)

424

A.E.

GiLL

These equations involve only the vertically-integrated velocities. The relevant solution depends on the geometry of the ocean basin involved. For instance, if the ocean basin is simply connected (and does not cover the whole globe), the relevant solution may be seen by inspection to be ~=~=0, (16) ~,=0, s e c $ , / a = ~-1. For such a basin, the horizontal transport (~, fi) is zero. The wind stress is balanced by an east-west pressure gradient (high to the west). For the east-west pressure gradient to exist, it is necessary for a side wall to cross the equator. If no such side wall exists, different balances result. For example, consider the case of a zonal channel with side walls at latitudes ~ = ± ~o. In this case, the relevant solution of (15) is given by = O,

~:0,

(t = ½ E n -1 (~2 _ $0z),

(17)

~7,:--sin6u.

Here there is no east-west pressure gradient. The wind-stress is balanced by lateral friction and the east-west transport is in geostrophic equilibrium with a north-south pressure gradient. MOTION

RELATIVE

TO

THE

VERTICAL

AVERAGE

Whatever the solution of (15), the motion relative to the vertically averaged motion is the same. This may be seen by subtracting the averaged equations (15) from their unaveraged counter parts (9), (10) and (11). The following equations result

--sin 4, z3 = E n ~ + E v ~ z z +

1,

sin $ ~ = E n f ~ + Ev fJzz,

(18)

sec $ [aa + (13cos 4)4,] + wz -: 0, where

~= u--tL

13= v - - 6,

(19)

are the velocities relative to the vertically averaged velocity. (~, ~) is sometimes called the baroclinic part of the velocity (BRYAN and Cox, 1968). The implications of this result are important, for it means that an equatorial layer may exist even if there are no cross-equatorial barriers to support a pressure gradient. (This idea may be applicable to the motion of planetary atmospheres even though the driving force may be different.) Since the baroclinic part of the motion is independent of the vertically-averaged part, there is no loss of generality in taking the special solution (16) for which the vertically averaged part vanishes. Then the carets ( ) may be dropped in equation (18). Selection of the special solution (16) does, however, affect the physical interpretation of the solution obtained, and this should be borne in mind. SOLUTION AT SOME DISTANCE FROM THE EQUATOR Outside the equatorial boundary layer, the lateral friction terms may be ignored. The horizontal components of the currents may be divided into two parts--a depth independent part (uz, v0 and a remainder (u*, v0, that is

The equatorial current in a h o m o g e n e o u s ocean

u = ~ ----- uX + u e,

v :

~ = d + v e.

425 (20)

The remainder is only important in the Ekman layer. For the depth-independent part, the vertical friction terms in (18) vanish, so at a distance from the equator, this part is given by sin~d---- --1, uz = 0 . (21) For the case where solution (16) is selected, this represents a current in geostrophic equilibrium with the east-west pressure gradient. As the equator is approached, this equatorward current increases in proportion with the distance from the equator. Clearly this solution cannot be valid at the equator, and so an equatorial boundary layer is required. The remainder (uL ve) of the horizontal component of the current satisfies -- sin ~ v • =

sin 9~ u e

Ev uzze ~ vzze

Ev

(22)

J

outside the equatorial layer. The solution to (22) is the well-known solution for an Ekman layer. Multiplying the second equation by the imaginary number i, and adding to the first equation gives an equation in the single dependent variable u* + iv e. The solution satisfying the boundary condition (13) at the surface is, for ~ positive, u ~ + i ve = (Ev

sin ~)-~ exp {(sin ~ / 2 E v ) t (1 + i) (z -- 1) -- ¼ ,ri}.

(23)

The Ekman layer has an e-folding depth (non-dimensional) of (2Ev/sin

(24)

$)t.

The transport (U e, Ve) associated with this flow is given by 1

l

[de = f u* d z = O, 0

Ve

= f ve dz = cosec 4,

(25)

0

and has the well-known property of being independent of the value of the friction coefficient. The current given by (21) can therefore be interpreted as the flow required to compensate the Ekman flux in order for the total transport (~, t~) to be zero.

THE EQUATORIAL UPWELLING

LAYER

In the equatorial upweUing layer, the horizontal components of the current can again be divided into a depth independent part (ux, vz) and a remainder (uL v0, as given by (20). For the depth independent part, the vertical friction term in 08) vanishes and so the equation is --~d

= E H ~ Z + I,

(2@

u I = E l l v ~ I,

where the approximation sin ~ ~ ~ has been made. Multiplying the second equation by i and adding to the first gives EH

(u1% i vg** -- i ~ (ux % i v9 = -- 1.

(27)

426

A. E. GILL

The solution which gives (21) for both q5 -> ~ oo is given by It I AV i v 1

: : zr b [Ai (b(~) -- i Gi (b~b)],

(28)

where (29)

b : e ~'~ EH-~,

where A i is the Airy function (see JEFFREYS and JEFFREVS, 1946, §17.07) and Gi is a function defined by SCORER (1950). The thickness of the equatorial upwelling layer is EH~. Some properties of the solution (28) can now be given. The most interesting is that the transport of the equatorial undercurrent (which is eastward) is independent of the value of the friction coefficient. Using some formulae given by ROTHMAN (1954a, b) the result f u I d4, =~ ~r

(30)

is obtained. In dimensional terms Ha

f

(31)

ut dq~ ~ ": r:a/2£2.

--o0

In other words the transport of this equatorial undercurrent is proportional to the wind stress at the surface, but in the opposite direction. For each dyne cm -z (10 -1 newton m -z) of wind stress, the transport is 14 × 106 m3/sec. Now, a typical value of the wind stress at the equator is 0.3 dyne cm -z (HELLERMAN, 1967) corresponding to a transport of 4 × l06 ma/sec. For comparison the transport of the equatorial undercurrent relative to the earth, is 20-40 times 106 m3/sec (KNAuSS, 1960, 1966). A smaller value is obtained if the vertically averaged velocity is subtracted; for instance, if the vertical average is taken over the top 250 m, the relative eastward transport is about 15 × 106 ma/sec. This is clearly in excess of the value predicted for a homogeneous ocean, but it is interesting to note that the two values compared are o f the same order of magnitude, even though the real ocean is stratified and inertial effects are known to be important (KNAUSS, 1966). The Ekman fluxes in the equatorial region may also be calculated. The simplest way is to use the definitions (20) and (25) which imply U e ....... u t,

Ve . . . .

vL

(32)

In particular, there is a westward flux in the Ekman layer equal in magnitude to the flux given by (31). This flux is in the direction of the surface stress. PROFILES

Velocity profiles for the equatorial undercurrent of the model are shown in Fig. 1. They were calculated from the power series expansion of (28), namely E / ~ ( u z + i c I)==c(1 + i y a / 3 ! - - 4 y 6 / 6 ! -

7"4iy9/9 !+...)

+ c' ( i y -- 2y4/4 ! -- 5" 2iy7/7 ! -~- 8- 5.2y1°/10 ! - - . . . )

(33)

-- (yZ/2 ! 4- 3iy~/5 ! -- 6" 3yS/8 ! .... 9" 6' 3if11/l 1 ! -]- . . . )

where

y - E t t -~ ~,

(34)

The equatorial current in a homogeneous ocean

~C-- ~

45A

I

-05

-Io

Fig. 1.

--

427

2

3

/

--

Velocity profiles across the equatorial undercurrent, as given by (28).

where

c = 3 - t / " (]) ~ 1.2879,

and

c' = 3 -t P (]) ~ 0.9389.

(35)

The vertical velocity component is calculated from the continuity equation (last of (18)), which gives w = -- z v~r (36) below the Ekman layer. The behaviour of the functions for large [Eu-~ ~1 can be found from (28) and the relevant asymptotic expansions. The first terms may be deduced directly from (26), namely

MERIDIONAL

vI ~ -- 1/~

l

UI ~

J

-

2EH/~ 4

-

CIRCULATION

.

(37)

NEAR THE E Q U A T O R

In the neighbourhood of the equator, the continuity equation (last of (18)) is approximately ~ + w~ = O. Thus a stream function ~bmay be defined so that ~ =

-

~,

w =

4,~.

Outside the Ekman layer and the equatorial layer, the solution is given by (21), and so ~bis given approximately by 4~ = z / ¢

428

A.E.

GILL

Zffil

J

Zffi0 Fig. 2. The meridional circulation in the neighbourhood o f the equator in a homogeneous ocean. The solid lines show the streamlines as computed by W. D. McKee. The heavy broken lines mark the edges, based o n e-folding scales, o f the Ekman layer and o f the equatorial upwelling layer. The relative magnitude o f the two scales used for the drawing corresponds to a value o f Ev EH-t : 0"01. The arrows show the sense o f circulation for a westward surface stress. The contour interval is O. 1 EH-~

E~'"~, ~ Fig. 3.

Contours of E ~ t ~ drawn on the same scale as that used for Fig. 2 and corr~ponding to the same values of the parameters. They were computed by W. D. McKee.

The equatorial current m a h o m o g e n e o u s ocean

429

in that region, so the stream lines are straight lines which pass through the origin. The stream function can also be calculated in the Ekman layer, using (23), and in the equatorial upweUing layer, using (28). Other methods are required to calculate the stream function in the remaining corner. Figure 2 shows streamlines computed numerically by W. D. McKee. Figure 3 shows the corresponding contours of eastward velocity. TIME

SCALE FOR ESTABLISHMENT OF THE EQUATORIAL UNDERCURRENT

The time-scale for the establishment of an Ekman layer i s f -1, where f = 212 sin is the Coriolis parameter. This time scale increases as the equator is approached and becomes of order (212)-1 En-t at the edge of the equatorial upwelling layer. If acceleration terms are added to the equations of motion, it is clear that this is also the time scale for establishment of the equatorial upwelling layer. This suggests a method of estimating a time scale for the establishment of the equatorial undercurrent in the ocean, as equal to the time scale for establishing an Ekman layer in the region of the undercurrent. Typical values of f i n the region of the undercurrent are about 10-6 see-1 which gives a time scale for the equatorial layer of the order of 10 days. NORTHWARD

WIND-STRESS

A northward (or southward) wind stress also produces a current in the equatorial upweUing layer. The solution is obtained from the above solution for a westward stress by the transformation ~ --> z3, ~ --> -- tL In this case ~ is antisymmetric and ~3symmetric about the equator. A small northward component of wind stress then makes the undercurrent asymmetric but does not alter the total transport. A CONDITION

ON T H E F R I C T I O N

PARAMETERS

What is the condition which determines when the equatorial layer is controlled by lateral friction rather than vertical friction? To answer this question, consider the way the thickness of the Ekman layer varies as the equator is approached. Equation (28) shows that the thickness increases like ~-J and at the edge of the equatorial upweUing layer, where ~ is of order E~* has a thickness of order Ev½E~-J. For lateral friction to be dominant in the equatorial layer, this value must be small compared with unity, that is

Ev .< Eu~.

(38)

430

A . E . GILL

Commonly used values of E v and E u (10 -8 < E v < 10-3, E u ½N 10-2) satisfy this condition. The corresponding configuration of the boundary layers is shown in Fig. 2, where the broken lines mark the boundaries of the Ekman layer and of the equatorial upwelling layer. They are based on the e-folding scale (24) for the Ekman layer, and an e-folding scale (4.5 En) t for the equatorial layer, deduced from the asymptotic expansion of the Airy function which appears in (28). THE C O N D I T I O N FOR N E G L E C T I N G EFFECTS OF THE H O R I Z O N T A L C O M P O N E N T OF THE E A R T H ' S R O T A T I O N

KNAUSS (1966) has shown that, for the observed undercurrent, the Coriolis acceleration associated with the horizontal component of the earth's rotation is small compared with other terms in the momentum equations. Under what conditions is the neglect of this effect consistent with the other assumptions of the present model ? An a posteriori examination of the neglected terms shows that they are most significant in the upwelling layer, where they are of order a -t H E u -~

relative to the terms retained. Therefore the condition for the neglect of these terms is that a -1 H E u -~ < 1. (39) Using the definitions of E u and Ev, conditions (38) and (39) may be combined to give (a/212)~ D A-~ ,< H < (A/2.Q)~ a -~, (40) the left-hand side of the inequality corresponding to (38) and the right-hand side to (39). Now, to obtain a width of the undercurrent about the same as the observed width, a value of A of about 104 m z sec -1 (10s cm z sec -1) is required. Using this value and a value of v of 10-a m 2 sec -1 (10 cm ~ sec-1), (40) becomes 2 4 m < H < 900m. Since the observed undercurrent has a depth of 100-200 m, a model based on (40) does not seem unreasonable. DISCUSSION

The physical reason for the existence of the equatorial currents in the present model is fairly clear from the structure of the solutions obtained. The wind stress at the surface produces an Ekman flux. At a distance from the equator, this flux is at right angles to the wind stress, the Coriolis force associated with the flux balancing the force of the wind on the surface. At the equator, the Coriolis force vanishes, so the force of the wind on the surface tends to accelerate the surface water in the downwind direction until a balance is achieved with friction forces. If the wind stress is westwards, the Ekman flux is away from the equator on both sides, so that a strong upwelling is forced in the equatorial region. In the case where (16) is the solution of the vertically-integrated equations, the equatorward flow, which is required below the Ekman layer by continuity, is in geostrophic balance with a pressure gradient. This balance is no longer possible at the equator where the

The equatorial current in a homogeneous ocean

431

Coriolis forces are zero, so the pressure gradient tends to accelerate the fluid in the direction of decreasing pressure. I f lateral friction is important, the down-gradient current, i.e. the undercurrent, is limited in magnitude by lateral friction. As a description of the observed undercurrent, a simple model such as this is far from complete, but it would seem to contain some of the essential elements. The important effects which are not included in the model are inertial effects and the effects of density stratification. For a westward wind stress, inertial effects would tend to increase the strength of the undercurrent because of the down-gradient acceleration experienced by a particle as it moves toward the equator. The sign o f this effect can be seen in the model of CrIARN~Y (1960). W. D. M c K ~ (private communication) has found, in fact, that if E r E H - t : 0.01, the transport o f the undercurrent is increased by a factor of four if the appropriate Rossby number, "ra/2DHA, is raised from zero to one. (If ~" = 0.3 dyne cm -g, H = 1 5 0 m and A = 104m~sec -t, then -ra/2DHA : 0"9). Stratification effects are more complicated. However, it can be seen that meridional motion like that shown in Fig. 2 implies overturning. Thus vigorous mixing near the equator would be expected. As KNAUSS (1960) has pointed out, such an effect produces horizontal density gradients which in turn produce currents along the equator. REFERENCES

BRYAN K. and M. D. Cox (1968) A non-linear model of an ocean driven by wind and differential heating: Part II. J. atmos. Sci., 25, 968-978. CI-tAm~Y J. G. (1960) Non-linear theory of a wind-driven homogeneous layer near the equator. Deep-Sea Res., 6, 303-310. I-I~Lea_V~N S. (1967) An updated estimate of the wind stress on the world ocean. Monthly Weather Rev., 95, 607-626. JEt~RBYS H. and B. S. JEFFREYS(1946) Methods of Mathematical Physics. Cambridge University Press. Jom~soN J. A. (1968) A three-dimensional model of the wind-driven ocean circulation. J. fluid Mech., 34, 721-734. KNAUSS J. A. (1960) Measurements of the Cromwell current. Deep-Sea Res., 6, 265-286. KNAUSS J. A. (1966) Further measurements and observations of the Cromwell current. J. marine Res., 24, 205-240. Mut,~ W. H. (1950) On the wind-driven ocean circulation. J. Meteor., 7, 79-93. PEDLOSKYJ. (1968) An overlooked aspect of the wind-driven ocean circulation. J. fluid Mech., 32, 809-821. Ronn~SON A. R., editor (1963) On the Wind-Driven Ocean Circulation. Blaisdell Press, New York. Rosn~SON A. R. (1966) An investigation into the wind as the cause of equatorial undercurrent. J. marine Res., 24, 179-204. ROTHMAN M. (1954) The problem of an infinite plate under inclined loading, with tables of the integrals of Ai (-4- x) and Bi (-4- x). Q. J. Mech. appl. Maths., 7, 1-7. RO~nV~AN M. (1954) Tables of integrals and differential coefficients of Gi (x) and Hi (-- x). Q. J. Mech. appl. Maths., 7, 379-384. SCORER R. S. (1951) Numerical evaluation of integrals of the form I = ~f(x)e ~°(~)dx and tabulation of the function Gi (x). Q.J. Mech. appl. Maths., 3, 107-112. STOMMELH. (1960) Wind-drift near the equator. Deep-Sea Res., 6, 287-297. VERO~_S G. (1960) An approximate theoretical analysis of the equatorial u n d e ~ n t ~ Deep-Sea Res., 6, 318-327.