Chapter VII Wind Driven Circulation

CHAPTER V I I

Wind Driven Circulation

7.1

The Turbulent Ekman Layer

The solution (5.2.2) for the laminar Ekman layer has two components of velocity, these being given by linear combinations of the real and imaginary parts of exp ki”2(f/v)1’2z,where z denotes the vertical coordinate, f the Coriolis parameter, and v the viscosity. The particular solution (5.2.4) is maintained by the pressure gradient of a geostrophic flow, and another particular solution can readily be obtained for the case in which a horizontally uniform wind stress of p7 (dynes/cm) acts on the free surface of a semi-infinite liquid of density p. Since there is now no horizontal pressure gradient, we set G = 0 in Eqs. (5.2.11, and since the velocities must be finite at z = -00, we set A = 0 in (5.2.2). The remaining integration constant B is then determined by setting T / V equal to the derivative of (5.2.2) at z = 0. The resulting solution is called an Ekman spiral, because of the way in which the vector formed from u, u rotates with depth. Both of the laminar Ekman layers mentioned above become unstable and turbulent as the boundary layer Reynolds number increases. This number is based on the Ekman depth (v/f)”’ and for the wind driven problem, the velocity scale is ~ ( v f ) - ” ’ , where 7 = 171. Thus, the Reynolds number is (7.1.1) R = ~(v/f)~’~/(vf)% = (7/Vf) 102

7.1.

103

T H E TURBULENT EKMAN LAYER

The minimum critical value of this boundary layer Reynolds number varies between 10 and 100 (Tatro and Mollo-Christensen, 1967; Lilly, 1966; Faller and Kaylor, 1967), depending on whether the Ekman layer is driven by a wind or by a geostrophic pressure gradient. Most experiments have been performed under the latter conditions, and at R = 500, the flow is rather turbulent. The value of (7.1.1) is lo6 for a typical wind stress of 1 dyne/cm2, so that the oceanic boundary layer is turbulent, even in the absence of other mechanisms (Assaf et al., 1971). Since so little is known about turbulence in nonrotating shear flows, it is both fortunate and remarkable that the integral property discussed below is independent of the detailed mechanics of the eddies. Consider a semi-infinite ocean subjected to a uniform wind stress 7 , and let V(z) denote the horizontally averaged velocity in the water. The fluctuating component V‘(x,y, z, t ) is associated with the turbulent eddies, and is schematically indicated by the loops in Fig. 7.1. Although a detailed description of the eddies is not necessary for the derivation of Eq. (7.1.6) the nomenclature introduced below will be useful later on. We shall use the terms “small scale” and “momentum transporting eddies” synonymously, to denote those horizontal wavelength components of the turbulence that are predominant in the downward transport of the momentum supplied by the wind. The space-time scale of these eddies is such that they are neither in geostrophic nor hydrostatic balance. The turbulent velocities decrease with depth, and we denote the characteristic depth of the turbulent Ekman layer by z = -he. The value of he is much less than the total depth of the ocean, and is estimated by the similarity theory given in Section 8.1. When 7 is uniform, V(z) is also confined to a depth h e , and the term “pure boundary layer” will be used to refer to the case in which all of the wind induced velocities vanish at great depths. For the case (Section 7.3) in which T varies horizontally, we will find, on the other hand, that wind driven motions are induced at great depths.

tk 2.0

z--h, -

I/’

I

FIG. 7.1 Perspective diagram of a uniform x-directed wind stress T acting on the free surface (z = 0) of a semi-infinite ocean. k is the unit vertical direction and M is the vertically integrated value of the mean current. The small scale and turbulent eddies are schematically indicated by a loop.

104

VII. WIND DRIVEN CIRCULATION

We now consider the statistically steady field in the pure boundary layer, and form a momentum budget for the infinite horizontal strip between z < 0 and z - dz. Let pe(z) denote the average downward transport of horizontal momentum, or the stress, acting on the top of this strip. The stress on the bottom of the strip is then given by -p8(z - dz), and therefore p aO/az dz is the average horizontal force due to the action of the turbulence. The average Coriolis force is p dz f k x V, and since there is no horizontal pressure gradient in this case, the relation

f k x V(z) = dQ/dz

(7.1.2)

expresses an exact momentum balance for the turbulent boundary layer. Since the Reynolds stress 8 must vanish at great depths, and since p 8 must equal the given wind stress at z = 0, we have the boundary conditions and

~(O)=T

8(--)=0

(7.1.3)

Therefore, the vertical integral of (7.1.2) is

fkxM=r

(7.1.4)

where 0

M= jVdz

(7.1.5)

-m

By sketching the orthogonal triad (M, k, r ) , it is readily seen that (7.1.4) is equivalent to (7.1.6) M = ( r x k)/f This beautifully simple relation, when extended to the case of variable 7 ,f in Section 7.3, provides the basis for the theory of the wind driven ocean circulation. 7.2 Where Does the Momentum Go?

In the steady state model considered above, there is a downward flux of horizontal momentum at the free surface, but no flux at great depths. Since the model is also horizontally homogeneous, it may then appear that the momentum supplied by the wind is disappearing in the boundary layer. The paradox is resolved in the following instructive generalization. Let E@(r) denote the absolute azimuthal velocity of an undisturbed axisymmetric vortex in a semi-infinite liquid, where r denotes the radial distance from the vertical axis of the vortex. The vortex is in equilibrium because of the balance of the centrifugal force with the radial pressure gradient, and u i / g r is

105

7.2. WHERE DOES THE MOMENTUM GO?

the slope of the free surface. We suppose, however, that g is sufficiently large so that this slope can be neglected in that which follows, and thus the free surface is taken at z = 0. The problem then is to determine the velocities induced by the application of an axisymmetric wind stress 7 ( r ) on z = 0. In particular, we want to construct the circular analog of Section 7.1, and to consider the budget of absolute angular momentum, this being the appropriate dynamical invariant. We will take into account the vertical transport of angular momentum by the turbulence and the lateral transport of angular momentum by the wind induced mean motion, but we will neglect the relatively small viscous stress that arises from the lateral variation in a,(r), should the vortex not be in a state of solid body rotation. The radial ( ~ ~ ( rand ) ) the azimuthal ( ~ ~ ( rcomponents )) of 7 will induce a mean radial velocity iC,(r, z) in the water, and the mean azimuthal component is denoted by a,(r) t “(r, z). We now ask for that distribution of 7 which will produce a “pure” boundary layer, in the sense that a,, a+ vanish at great depths. A necessary condition for a pure boundary layer is that the radial volume flux m = 217

/

(7.2.1 )

dz riir(r, z)

-m

be independent of r, for otherwise the divergent flow will induce motion at great depths (cf. Sections 5.3 and 7.3). Should the undisturbed vortex be contained in a circular annulus offinite radial width, then a constant value of (7.2.1) could be realized by having the same kind of clearance space at the vertical boundaries of the annulus as in the boundary layer flow of Fig. 5.la. Another necessary condition for a pure boundary layer is provided by the conservation of absolute angular momentum. Consider a control volume located between radii r and r +dr, and having semi-infinite vertical boundaries. The torque (in consistent units) of the wind on the top of this ring is (2nr dr)rpTQ(r)

(7.2.2)

In the steady state, this torque is balanced for by the difference between the fluxes of angular momentum of the fluid at the vertical boundaries of the control volume. The absolute angular momentum (per unit volume) at any radius is given by rp(UG tic+), and therefore the radial flux of angular momentum is 27rp

/

dz (r&)r(G~t i,)

= pYii,m

+ O(ii,C+)

(7.2.3)

-m

where the last term indicates the part of the integral that is quadratic in the amplitude of the velocity components induced by the wind. The prii,m term, on the other hand, is linear in this amplitude, and therefore dominant for sufficiently small m (or 7). The value of (7.2.3) at r t d r minus its value at r

106

VII. WIND DRIVEN CIRCULATION

must equal (7.2.2), according to the angular momentum principle, and therefore we have 27rr2r0 = (a/ar)[%,m +O(iiP@)l

As mentioned above, the quadratic (ii,.G@) term can be neglected for sufficiently small 7 , and since m is independent of r, we obtain

From (7.2.4), we conclude that a necessary condition for a pure boundary layer is that the torque of the wind (rro) must be proportional to the absolute vorticity

(aa@/ar)+(a&>

(7.2.5)

of the undisturbed vortex at all r. (If r does not satisfy this proportionality, then the divergences in the Ekman layer will generate motions at great depths, as indicated by the theory of the following section.) The coefficient of proportionality between wind torque and (7.2.5) is the mass transport function rn The correspondence between (7.2.4) and (7.1.6) can readily be made by considering the special case in which the undisturbed vortex is in solid body rotation with angular velocity f/2, so that = fr/2. For simplicity, we also take r,.(r) = 0, and Eq. (7.2.4) then reduces to m/2m = r@/f

(7.2.6)

Since the mass transport per unit azimuthal distance (m/27rr) equals the vertical integral of the radial velocity, and since the wind is directed in the azimuth (7,. =O ) , we see that (7.2.6) is in exact correspondence with (7.1.6) when the radius of curvature r is large. The momentum paradox in the latter problem is thereby removed by the fact that the divergence of the absolute momentum flux balances the wind stress in the horizontally homogeneous model of Section 7.1. The preceding discussion, moreover, need not be restricted to the case in which the original vortex is in a state of solid body rotation. Equation (7.2.4) implies that a pure boundary layer occurs whenever is proportional to the absolute vorticity, or to the sum of the Coriolis parameter and the relative vorticity. In this case, we, see that the Ekman transport is inversely proportional to the absolute vorticity, and this provides an introduction to an effect discussed further in the next chapter.

7.3

Sverdrup Theory

We now want to consider the case in which the r , f of Section 7.1 varies slowly with latitude, in accord with the global distribution of the mean

107

7.3 SVERDRUP THEORY

atmospheric wind stress. For maximum simplicity, we will use the Cartesian fl-plane approximation (Section 2.5), in which the x axis points eastward and the y axis points northward. For reasons mentioned previously, we shall also neglect the slight tilt of the free surface ( z = 0) which is associated with the large scale pressure gradient computed below. The global scale variation of T can hardly influence the 8 of the small scale turbulence, and therefore the local balance of the eddy stress with the Coriolis force is still given by (7.1.6). But the slow variation of r , f with latitude will produce a divergence in the Ekman transport M , and the resulting "suction velocity" (Section 5.3) will induce an additional field underneath the turbulent boundary layer. We now give two derivations of this effect, and the reader may prefer the second [following (7.3.8)] because it is mathematically simpler. The first derivation, however, is more revealing and useful later on. Consider the two layer model of Fig. 7.2, wherein the depth he of the turbulent layer is a small fraction of the thickness h of the upper layer, and the local value of the vertically integrated boundary layer flow is given by (7.1.6). south

north "TRADES"

"WESTERLIES"

P+AP ( 0 )

FIG. 7.2 A north-south vertical section through a two layer ocean driven by a variable zonal wind stress 7Cy). y is positive in the north direction and the x direction is into the page. h(x, y ) is the steady state depth of the interface which separates light fluid (density p ) and heavy fluid (p + A p ) . The region of small scale turbulence is confined to the depth he << h, and -we is the Ekman suction velocity.

By integrating the continuity equation with the boundary condition w(x, y , 0) = 0, we then find that the upward vertical velocity w = we at the bottom of the Ekman layer is given by

we=V.M

orby

w,=V-(~xk/f)

(7.3.1)

when (7.1.6) is used. The wind stress assumed in the model (Fig. 7.3) acts in the zonal direction, and has magnitude 7 0 ) . Thus, the northward Ekman transport is J I y = --7(y)/f(y) where fb)is the local Coriolis parameter, and

(7.3.2)

B = df/ay in that which follows.

108

VII. W I N D DRIVEN CIRCULATION north

X.0

south

FIG. 7.3 Schematic diagram of total mass transport streamlines in a square ocean basin. The assumed wind stress distribution is shown o n t h e right. TX,T y are the components of the total transport vector. The anticyclonic Sverdrup gyre extends only up t o the western boundary layer.

The corresponding value of (7.3.1), or

we = - ( a/ aY) ( T( y) /f )

(7.3.3)

is sketched in Fig. 7.2, and thus we see that downward directed velocities are forced into the fluid beneath the turbulent boundary layer at latitude y = L/2. Consider now the effect of (7.3.3) on the marked material column (Fig. 7.2) whose top is located at z -he and whose bottom is located at the interface z = -h(x, y ) which separates the upper layer from a very deep bottom layer of density p + A p . The. downward directed Ekman suction velocity tends to “squash” the cylinder in the vertical direction, thereby increasing its crosssectional area. The product of this cross-sectional area and the vertical component of the earth’s vorticity f thereby tends to increase. But the motion in the region z < -he must satisfy the circulation theorem (conservation of potential vorticity), and consequently a relative velocity must develop which will conserve the product of cross-sectional area with the normal component of absolute vorticity. This can be accomplished by a southward movement of the column, since f(y) decreases southward, and the circulation around the “squashed” cylinder can thereby be conserved. Thus we see how geostrophic velocities are generated in the water beneath the Ekman layer, and the concomitant pressure gradient will then cause the interface h(x, y ) to adjust, as indicated in Fig. 7.2. These quantities will now be computed for the case in which ~ ( y ) a n dits gradient are small. The suction velocities and the induced geostrophic velocities are then small, and the nonlinear terms can be neglected in the equation for

-

109

7.3 SVERDRUP THEORY

conservation of potential vorticity (3.7.3a). Thus, we have the linearized version (3.7.4), or up = f aw/az

(7.3.4)

where u is the northward component of geostrophic velocity. Since w = we at z = -he, and w = -dh/dt at z = -h, the linear variation of w with z implies

-

aw/az = (we + V Vh)/(h-he)

2

(7.3.5)

we/h

because dh/dt = V V h is quadratic in V, and also because he 4 h. The approximations made in obtaining (7.3.4) and (7.3.5) are

The result of combining ( 7 . 3 3 , (7.3.4), and (7.3.3) is

uh = -(f/~)(a/a~)(~/f) = -0-l Way + T/f

(7.3.7)

This transport beneath the Ekman layer is comparable with (7.3.2), but the typical value of u is much smaller than the typical boundary layer velocity My/he. Thus, we see that when the northward velocity is integrated from the interface t o the free surface, the total transport function T, is given by the sum of (7.3.7) and (7.3.2), or

T,

= -0-l

ar/ay

(7.3.8)

This is a particular case of the Sverdrup relation (Robinson, 1963; Stommel, 1965), which states that the total northward transport is equal to /3-' multiplied by the curl of the wind stress. In the second and mathematically simpler derivation of the Sverdrup relation, we do not separate the total transport T into components (e.g., M) but directly integrate the linearized momentum equation in the vertical. Thus, the integral of the Coriolis force from z = -h to z = 0 is written as f k x T,and the integral of the wind induced force is r . Since the lower layer (Fig. 7.2) is not in direct contact with the wind stress, we may assume the motion therein to be at rest in the steady state. Therefore, g* V h is the horizontal pressure gradient force in the upper layer where g* is the reduced value of gravity, and the vertically integrated pressure force then becomes g*h V h. Therefore, f k xT= -V(g*h2/2) + r , and the vertically integrated continuity equation is V T = 0. By taking the curl of the first equation and by utilizing V T = 0 , one readily obtains the Sverdrup relation for the northward component of T as a function of curl r . The reader will also find it instructive to perform this calculation in spherical coordinates, so as to examine the validity of the 0-plane approximation used in deriving (7.3.8). In addition to the northward component T y , there is also an eastward

-

-

110

VII. WIND DRIVEN CIRCULATION

component T, of the transport vector T. The conservation of mass requires that T(x,y ) be nondivergent, and therefore

aT,/ax + aTy/ay = o There can be no transport normal to the eastern boundary, or

T&, Y ) = 0 and consequently the integration of the previous equation gives (7.3.9)

T, = (x-L)(a/ay)b-’ aT/ay

In Fig. 7.3, we have ar/ay = 0 and a2r/ay2< 0 at the northern boundary, and therefore T, > 0 for y > L/2, as shown by the clockwise circulation of the streamlines of T. We note, however, that (7.3.9) does not satisfy the boundary condition T, (0 y ) = 0 at the western wall, and the deep significance of this is discussed subsequently. Since T, changes sign near y = L/2, the eastward component of geostrophic velocity T,/h must also change sign, and the associated north-south slope of the interface (Fig. 7.2) can be computed as follows. Since the lower layer is at rest, the slope of the interface is given by Margules’ equation (4.4.9, and by using (7.3.7), we then find that the east-west slope of the interface is (7.3.1 0) Therefore, the slope is upward to the east along the central latitude (r = 0). Let us now examine the reason why the solution given above is unable to satisfy the western boundary condition T, (0, y ) = 0. The approximate vorticity equation (7.3.4) was based on (7.3.6), and the first two of these approximations can be written as

-

1 P IV V.Cl/luPI

- IWf - IullfL- r / f 2 L h

(7.3.1 1)

wherein we assume the lateral scale L of the wind to be the same as the radius of the earth, and (7.3.7) has also been used in (7.3.1 1). When the typical oceanic values

r = 1 (crn/sec)2, f =

sec-’,

h = lo5 cm, L

=5 x

10’ cm

are inserted, we see that the right hand side of (7.3.1 1) is lo-’, so that there is little doubt about the validity of the asymptotic solution. But the V -02term contains the highest horizontal derivatives in the exact vorticity equation. Therefore, the neglect of that term lowers the order of the differential equation, and thereby removes some of the solutions which are necessary for the satisfaction of all the inviscid boundary conditions. Accordingly, we now look for one of these “lost” solutions whose horizontal scale is much less than L , and for which the relations (7.3.1 1) and (7.3.4) are not applicable.

111

7.4. INERTIAL WESTERN BOUNDARY LAYERS

7.4 Inertial Western Boundary Layers The procedure to be used in modifying the general circulation picture (Fig. 7.3), is suggested by the problem considered in Section 2.6. We showed that a westward current (Fig. 2.3) having small relative vorticity { tends to intensify into a thin jet as it is forced toward high latitudes by the western wall, and the nonlinear terms V *Or are essential in the dynamics. Therefore, we expect a similar effect to occur as the flow in the southwest corner of Fig. 7.3 approaches the western boundary. An expanded view of the southwest corner (x = 0, y = 0) is given in Fig. 7.4, in which the solid curves labeled h = constant are also the isobars or streamlines of the geostrophic flow above the interface. We now “stop” the linear solution of Section 7.3 at some small longitude x = 6 , and then proceed to compute the flow further downstream by including the previously neglected nonlinear vorticity terms (Charney, 1955; Morgan, 1956).

I,, , , ,

x =o

,I,,

X.6

, , ,, , , , , , , A

,

,I

x=L

FIG. 7.4 The western boundary current and its connection with the geostrophic interior solution. See Figs. 7.2 and 7 . 3 and the text.

The present problem only differs from Fig. 2.3, because the model now includes an interface and the associated buoyancy g A p / p between two layers. Since the geostrophic velocities (7.3.7) at x = 6 are fixed, the Margules equation, or (7.3.10), indicates that the gradients in h will be negligibly small for an asymptotic case in which g is very large. Therefore, if we start the discussion with this case, then the variations in h can be neglected, and the rigid bottom calculation (Fig. 2.3) can then be applied to the upper layer in Fig. 7.4. Thus, h is constant, and absolute vorticity f t { is conserved, as columns are deflected northward by the western boundary. Iff denotes the Coriolis parameter at some high latitude (’y L/2), then the fluid on the boundary (x = 0) will have a

-

112

VII. WIND DRIVEN CIRCULATION

relatively large vorticity

-x and the east-west width is then given by (2.6.3), or 6

-

(7.4.1)

(f/f)"2

where hT is total volume transport of the north-going boundary jet. Continuity requires this transport t o equal the total southbound Sverdrup transport, as given by the longitudinal integral of (7.3.8), from x = 6 to x = L , and thus we have

%

L

[ dx (-TJ

=p-l

L

(aT/ay)dx

-

T L / ~

- r / L and 0 - flL. The width (7.4.1) of the jet then becomes 0

because

1 0

6

-

(7L/f2h)'/'

(7.4.2)

Since this width approaches zero, and since the right hand side of (7.3.1 1) also approaches zero as r .+ 0, it is most appropriate to refer to the jet as an inertial boundary layer. Let us now estimate the modification of the width of the western boundary current caused by a finite value of g The variations in h ( x , y ) are now dynamically significant, and we let Ah denote the typical variation in h(x, y ) across the width of the jet. From the geostrophic relation f V ( x ) = g ( A p / p )ahlax, we obtain the order of magnitude relation fv- g ( A p / p )A h / & ,where V - TI&.For large but finite g , we can use (7.4.1) as a first approximation in these relations, and thus we obtain (7.4.3)

If the right hand side of (7.4.3) is small, or (7.4.4) then Ah is small, the variation of h along the western boundary is also small, and (7.4.2) will indeed determine the width of the jet. Equation (7.4.2) can also be used for order of magnitude purposes, even when the left hand side of (7.4.4) is of order unity. But for such values of the parameters, the buoyancy effect? must be considered in any quantitative theory, and the reader is referred to the literature cited previously. The significant nondimensional number (7.4.3) equals the square of the raiio of (7.4.2) to the Rossby radius of deformation. This number is nearly equal to unity for the Gulf Stream and Kuroshio boundary currents.

t See Section 4.1 for an example of the inertial boundary layer when no present.

p effect is

1 . 5 . COMPARISON WITH OBSERVATIONS

113

The considerations of this section show that the Sverdrup solution (Section 7.3) can be accepted for the “interior” of the ocean, even though it does not satisfy the proper boundary conditions at the southwest boundary. The situation at the northwest boundary is discussed below.

7.5

Comparison with Observations

The interface in our model will be identified with some constant density surface at mid depth in the main thermocline of the ocean. Each of these density surfaces is observed to reach maximum depth near the latitude where the zonal wind stress changes sign, and thus we have qualitative agreement with the variation of the interface shown in Fig. 7.2 (Von Arx, 1962; Fuglister, 1960). The total clockwise circulation of the North Atlantic and North Pacific Oceans has been determined by a geostrophic calculation of the southbound flow, using observed east-west density gradients. The transports also have been determined from measurements of the compensating northbound flow in the Gulf Stream and Kuroshio boundary currents. By using realistic wind stresses, Munk (1950) obtained favorable comparison with the Sverdrup transport. Munk also gave a completely closed circulation theory by assuming a viscously dominated boundary layer, in contrast with the inertial western boundary current expounded above. The precise dynamics of the western boundary, however, has negligible effect on the Sverdrup interior solution, and the total amount of water circulating clockwise is obtained by multiplying the maximum value of (7.3.8) with the width of the ocean. Thus Munk obtained 36 x lo’* gm/sec for the North Atlantic and 39 x lo’* for the North Pacific. Subsequent direct measurements of the northbound transport in the Gulf Stream gave a value of 33 x 10” gm/sec at 30°N, but the transport continues to increase and reaches 147 x 10l2 gm/sec, 2000 km further downstream (Knauss, 1969). This large downstream increase in transport has not been satisfactorily explained. For this reason, and for others given below, we have not attempted to continue the western boundary flow northward in Fig. 7.4, and thus our circulation pattern is not yet closed. Reference is made to Hansen’s (1970) description of the synoptic variations in the position of the axis of the Gulf Stream jet. South of Cape Hatteras (35’N) the axis is relatively straight and steady, but meanders of increasing amplitude develop downstream. Typical wavelengths are 200-400 km, and phase speeds are about 8 cmlsec (as compared to maximum jet velocities of 100-200 cmlsec). Occasionally, the amplitude of a meander wave becomes so large that a ring of water detaches from the main stream (Fuglister, 1972) and thereby deposits a large volume of cold fresh coastal water on the seaward side of the mean stream or, inversely, a large volume of warm salty ocean water on the shoreward side of

114

VII. WIND DRIVEN CIRCULATION

the mean stream (Saunders, 1971). These meander waves may be related to the geostrophic instabilities discussed in Sections 4.3 and 4.4 and also to the large variations in the depth of the ocean bottom along the path of the Gulf Stream. The exchange of heat and salt that occufs when the meanders detach from the western boundary current may be important in the thermodynamical budgets of the tropical and polar oceans. Thus, the observations indicate the existence of important processes that are beyond the scope of the previous model, and therefore the dynamics of the northwest territory is left “open”. The circulation pattern in Fig. 7.3 is not only kinematically open in the upper left corner, but the overall ocean energetics is also open, as indicated below. The energy which is pumped downward at the bottom of the turbulent boundary layer is given by the product of the pressure @(x,y, -he) and the vertical velocity

-we = -V

*

(7.5.1)

M

when integrated over some constant level surface (z = -he) that lies beneath the Ekman layer. If dA denotes an area element on that surface, then the pressure work is (7.5.2) since the normal component of M at the vertical coastal boundaries must, in fact, vanish. Since the Sverdrup theory is linear, the kinetic plus potential energy which develops in the interior of the ocean must be entirely due to the pressure work (7.5.2). Since the geostrophic relation

f k x V = -Vz@ applies at z = -he, the value of (7.5.2) is #dA

T

xk

- V x k = 4 dA7

V

(7.5.3)

Thus, we see that the rate at which energy is supplied to the deep ocean is given by the product of wind stress and thegeostrophic velocity, or by TTJh. The geostrophic transport streamlines in the interior of the ocean (Fig. 7.3), imply a positive contribution to (7.5.3), and therefore energy is “still” being pumped downward into the steady Sverdrup interior! This energetic paradox is due to the fact that no mechanism has been provided for the transformation and dissipation of the work done by the wind, and a purely inertial western boundary current cannot be made to connect with the Sverdrup solution in the northwest corner of the basin. Munk (1950) solved this problem for the case of a homogeneous ocean by introducing lateral friction in the western boundary. But for a stratified ocean (Part 11), we will need to utilize some of the wind work to

REFERENCES

115

mix waters of different temperature (salinity) and to thereby satisfy the global .heat budget requirements of the ocean. Consequently, we will neither dispose of the prime energy source (wind work) nor close the circulation, until the thermodynamics is settled.

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