The balance and redistribution of potential vorticity within the ocean

Dynamics of Atmospheres and Oceans, 1 (1977) 299--321

299

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

THE BALANCE AND REDISTRIBUTION OF POTENTIAL VORTICITY WITHIN THE OCEAN

RI C HAR D E. THOMSON and R.W. STEWART

Environment Canada, Ocean and Aquatic Affairs, Pacific Region, Victoria, B.C. (Canada) (Received February 5, 1976; revised and accepted November 5, 1976)

ABSTRACT Thomson, R.E. and Stewart, R.W., 1977. The balance and redistribution of potential vorticity within the ocean. Dyn. Atmos. Oceans, 1: 299--321. The balance and redistribution of potential vorticity in a turbulent, inhomogeneous ocean is examined analytically. In the first part of the paper, we derive a vorticity conservation law describing the way in which vorticity is lost and gained in a closed ocean basin. We show that the potential vorticity within the ocean can o n l y be altered by molecular diffusion at the bounding surfaces, or by the advective input of vorticity via the turbulent velocity at the free upper surface. An imbalance between these two effects within the body of the ocean leads to a net temporal change in the total potential vorticity, which we are able to express in terms o f the temporal changes in the magnitude and orientation of a vector formed by the surface velocity and density and the earth's rotation. Finally, the steady-state balance of integrated vorticity is found to be between the surface wind-stress curl and the molecular diffusion at the solid boundaries. We then consider the localized redistribution o f potential vorticity in detail. In recent papers, Green (1970) and Welander (1973) have purported to show that an internal redistribution of potential vorticity can give rise to important forces in the atmosphere (Green) and the ocean (Welander). Using their recipe leads in some cases to such unphysical results as a breakdown o f the laws o f conservation of angular momentum. Such consequences cause us to question the validity of the argument, and lead us to a close examination of Welander's mechanism. (Green's argument is more general and does not lend itself to such close examination.) Our investigation reveals important flaws not only in Welander's argument, but also in the classical mixing-length theory of Prandtl (1925) and of Taylor (1915) on which it is modelled. More specifically, the internal redistribution process presented in these theories is found to be incomplete since it fails to take into account the lift force exerted on individual parcels o f fluid as they are transported by the turbulent motions. Inclusion of these lift forces not only permits the unification of the classical mixing-length theories, but also shows that the forces claimed by Welander do not in fact arise.

INTRODUCTION

Although the distribution of incoming solar energy over the earth's surface is well known, it is the manner in which this energy is redistributed t h a t must be understood in order to explain the ocean--atmosphere circula-

300 tion. Since both the turbulent and mean fluxes of quantities such as heat, m o m e n t u m and vorticity are involved in the redistribution processes, the problem is rather formidable. Nevertheless, it is Oossible in the case of m o m e n t u m fluxes to obtain some very general results using the concept of potential vorticity. This particular property of parcels of stratified fluid on the surface of the rotating earth has the advantage that it obeys a rather simple conservation law. In the present paper, we deal primarily with the consequences of this conservation law upon the circulation of the ocean, although extension of our results to the whole ocean--atmosphere system should be relatively straightforward. The use of vorticity arguments in theories of ocean circulation began more than a quarter of a century ago when Sverdrup (1947) introduced the idea that wind~lriven ocean currents could be understood in terms of a balance between vorticity introduced by the curl of the wind stress and the change in planetary vorticity of water transported meridionally. Further development of this concept began early, most notably through the work of Stommel and a series of collaborators who applied it to the discussion of both wind
301 1932 and 1970. In the atmosphere, the formation of such isolated regions of large relative vorticity is a c o m m o n occurrence. Most important, though, is the fact that any meridionally displaced fluid parcel receives relative vorticity as a consequence of the variation of the vertical c o m p o n e n t of the earth's rotation. Thus, even in the absence of external vorticity sources, this internal vorticity gradient is capable of generating relative vorticity. In the past, meteorologists such as Eady (1949) have attempted to invoke horizontal diffusion on planetary scales as a mechanism for poleward energy transport, while oceanographers have, on dimensional arguments, tended to rule o u t its importance except near boundaries. Recently, Green (1970), in attempting to account for the energy required to maintain the zonal components of surface winds in the presence of frictional braking, has proposed a refined mechanism of transport based u p o n the horizontal transfer of largescale eddies. Similarly, Welander (1973) has derived an eddy mixing theory for the oceans which appears capable of complementing the wind-induced Sverdrup transport. In the region of the North Atlandtic, this leads to a more realistic transport for the Gulf Stream b y a doubling of the transport induced b y the winds alone. Basically, these theories propose that quasigeostrophic eddies can generate a net meridional flux of vorticity because of the nature of the variation of the Coriolis parameter with latitude (the fi-effect). In spherical coordinates, an additional term arises as a consequence of the meridional variation of ft. A mean meridional circulation is then produced through gradients in the Reynolds stress corresponding to this flux of vorticity. According to these studies therefore, it would seem that horizontal mixing on the geostrophic scale does play an essential role in maintaining oceanic and atmospheric circulations. However, there arises a number of important difficulties which cast d o u b t on their validity. To begin with, Welander finds that use of Taylor's (1915) mixing-length concept for vorticity leads to a force which is everywhere to the west. If this notion were applied to a layer of fluid entirely covering a rotating sphere, the layer would experience a decrease in its angular m o m e n t u m w i t h o u t there being a corresponding increase in this quantity elsewhere in the system, contrary to the law of conservation of angular m o m e n t u m . A similar conflict arises in Green's analysis if the vertical c o m p o n e n t of eddy diffusivity is assumed uniform. In addition, it is impossible to construct a stress field corresponding to this westward force, since the stress gradient would necessarily be monotonic over the surface of the globe. Such a monotonic stress gradient is not consistent with the fact that the zonally averaged Reynolds stress must vanish at both poles. Although Welander uses his arguments for zonally b o u n d e d oceanic regions in which the stress need not vanish and Green considers the atmosphere to be baroclinic enough to justify the use of a spatially varying eddying coefficient, the fact that such fundamental conflicts arise leads us to suspect that the authors may be avoiding rather than accounting for these difficulties. The work presented here is the result of an a t t e m p t at reconciliation of the apparently correct analysis of the above papers and the physical discrep-

302

ancies. Our effort will be mainly directed toward Welander's theory, since unlike Green he provides a recipe for determining the force induced by potential vorticity redistribution. First, however, we derive a generalized theorem for the vorticity balance in a closed ocean basin without heat sources, where it is shown that the input of potential vorticity must take place at the free upper surface through direct turbulent transport and b y molecular diffusion of m o m e n t u m across isopycnals. At solid boundaries only molecular diffusion can alter the net amount of potential vorticity while away from any boundaries such vorticity can neither be created nor destroyed. We then limit our discussion to the oceanic interior in which molecular friction is assumed unimportant, and consider in detail the recipe proposed b y Welander for the mixing of potential vorticity. The fundamental conflicts in this theory lead us to re-examine the whole mixing-length concept including Taylor's ideas on vorticity transfer, as well as Prandtl's ideas on momentum transfer. Accordingly, we find that in previous literature authors have failed to account for the presence of a lift force exerted on fluid parcels moving into a region with different vorticity than at the origin. This force, generated b y the interaction between the translational velocity of the vortex core of fluid and the irrotational circulation which necessarily surrounds it, is in Welander's theory equal b u t opposite to the vortex force producing his westward acceleration. As a consequence, the mixing of potential vorticity is incapable of generating a net force on the fluid. It appears that only in the vicinity of strong 'boundary' currents where the horizontal gradient of mean relative vorticity is comparable to fl is it possible for vorticity mixing to significantly alter the flow. EQUATIONS OF MOTION

L e t u = (Ur, Uo,Ue) be the velocity of a fluid in a spherical coordinate system (r,0,¢) which is fixed to the rotating earth, where r is the radial distance measured positively upwards, 0 is the co-latitude, and ¢ is the azimuthal angle about the axis 0 = 0 (¢ positive eastward). With ~ the angular velocity of the earth, t the time, p the density, ~ the pressure (including the centripetal acceleration terms) and v the kinematic viscosity, the conservation of momentum requires that:

8u + (~" ~t

V)u + 2 ~ X g = - - ~ I P + ~ + vV2~

(1)

where ~ = VcP is a conservative b o d y force ( ~ being a potential). Taking the curl (•x) of eq. 1 yields the vorticity equation: D( ! +

: f(! +

-- w × w +

(2)

303

in which D a D--t = a~'t + ( ~ " XT)

(3)

is the material derivative, and: ~ = Vx]~

(4)

is the relative vorticity and a = 1/p is the specific volume of the fluid. The system of equations is closed through use of the conservation of mass equation which takes the form: V'~ =0

(5)

for an incompressible fluid without heat sources for which: Dp Dt - 0

(6)

The inherent difficulties associated with the third term in eq. 2, which expresses the generation of vorticity by the pressure--density field, are elimin a t e d by taking the scalar product of the equation with the density gradient. This operation, together with eq. 6, yields an equation for the potential vorticity, P: DP_ Dt

(7)

vp

where: P =

+

vp

= V" [ ( ~ + 2 ~ ) p ]

(S)

and in which ~ + 2 ~ is the absolute vorticity referred to an absolute or Newtonian frame of reference. A generalization of eq. 7 has been given b y Ertel (1942). In regions of negligible molecular friction, eq. 7 expresses conservation of potential vorticity following the fluid motion, viz., DP - 0 Dt

(9)

T h e averaged e q u a t i o n s

It is n o w assumed that the transporting and transported quantities may be separated into a mean value (overbar) plus a fluctuating value (prime), with: ~'=0 for any quantity ~ . The overbar will be understood to refer the ensemble average or to the zonal average.

304 Using eq. 3, the averaged form of eq. 7 becomes: a P + g . VP + u' • VP'

=/,'V2~ "

•

Vp

which, though use of the continuity equation (eq. 5) and two vector identities, can be written: -

aP + v - (~P-) + v • C~'/*) = v • ( v v 2 u x v p ) at

(10)

-

where the advective terms are now expressed through the divergence of potential vorticity transport, and the molecular diffusion of potential vorticity through the divergence of the vorticity generated by the diffusion of m o m e n t u m along isopycnals. Eq. 10, with molecular friction omitted, will be considered in detail in a later section (p. 308) with regard to the mixinglength concept for potential vorticity redistribution. Before that, however, we will determine the implications of the full equation for a closed ocean basin by using the non-averaged form of eq. 10. In doing so, we derive an overall vorticity conservation law for the ocean. THE POTENTIAL VORTICITY BALANCE IN A CLOSED OCEAN BASIN Consider an enclosed oceanic region of volume V, having solid sides and b o t t o m , and a free upper surface. If S is the area of the boundaries, then integration of eq. 7 over the volume followed by application of the divergence theorem yields:

d fpdy=f(vV2u)×Vp.nds_f~.nds ~ ~

dt

V

S

(11)

S

where d/dt is the total derivative with respect to time, ~ is an outward unit normal to S and where the frictional term has been rewritten in terms of the velocity. The requirement that there be no mass transport throught the solid boundaries gives the condition: u.n

=0

(12)

at solid boundaries. When eq. 12 is applied to eq. 11 we then obtain:

a__.fedv=

dt

V

vp. S

a s - fP

-aA dA

(13)

A

where n A is the outward unit normal at the free surface having area A and where: f P u . ~ dS S

=

f P ~ - n A dA A

(14)

305 The latter relation shows t h a t the only way the velocity can transport potential vorticity into or out of an enclosed ocean is via the free surface. When this term vanishes, convergence of potential vorticity transport at one location within the ocean requires t h a t there be a divergence at another. This does not prevent a diffusive exchange at any of the bounding surfaces. In fact, this is precisely the role of the molecular friction term in eq. 13. If we separate the integration of the diffusive term in eq. 13 into an integral over the free surface plus an integral over the b o t t o m , the integrated potential vorticity equation may be written as:

dfPdV= f(vV2~)'(VpXI!~)dB-- f[Pu V

B

- - v V ~ X Vp] "P.A dA

(15)

A

where B (>A) is the area of the solid boundaries of the ocean basin. In this form, eq. 15 expresses a balance between the temporal variation in the total potential vorticity within the ocean and the sum of molecular vorticity diffusion through the bounding surfaces and the flux of potential vorticity through the free surface. It further shows that potential vorticity is completely conserved within the b o d y of the ocean, and can only be modified by effects at the boundaries. Moreover, as nonzero values of the diffusive term can only occur where isopycnals intersect the boundaries, the integrated contribution from large regions of flat b a t h y m e t r y will be negligible. Therefore, the greatest contributions to eq. 15 will come from the free surface and from the sloping areas of the ocean, particularly those associated with relatively large values of the density gradient such as in western b o u n d a r y currents and upwelling regions. As the implications of the latter statement may appear to be counterintuitive, however, a few clarifying words are in order. To begin with, we have shown that if Vp is small and parallel to ~B, the normal to the b o t t o m -- and surely over much of the flatter parts o f the ocean it must be small and parallel to n~ -- this term becomes unimportant. This seems curious in light o f the fact that in Stommel's (1948) classic paper on westward intensification of wind
nearly

306 which concentrate on the portions of the ocean where the direction of Vp is significantly different from that of n~ even though intuition and other calculations which do n o t make use of this integration may concentrate on portions of the surface where Vp and n8 are nearly parallel. If for now we just consider the left side of eq. 11 and use the expression for the potential vorticity (eq. 8), we can again apply the divergence theorem to obtain:

V

8

Since we are dealing with a real fluid, the "no-slip" condition must hold at all solid boundaries. This, and use of an orthogonal curvilinear coordinate system having one axis always normal to the boundary (see e.g. Batchelor, 1967, p. 599), yields the result: ~.n=0 at solid boundaries, while application of the no-slip requirement to the incompressibility conditions (eq. 6), shows that in the absence of heat and salt diffusion:

~}P- 0 at solid boundaries. Subject to these two conditions, eq. 16 reduces to:

d j'VdV= d f p ( ~ . 2~).n~dA V

(17)

A

so that considering eq. 17 as the dependent quantity in eq. 15, we see that the time rate of change of total potential vorticity resulting from the imbalance of vorticity input at the bounding surfaces manifests itself as a time rate of change of the surface integrated normal component of absolute vorticity. We may further simplify eq. 17 by using the vector identity,: p~ = vx(pu)

-

vp x u

and Stoke's theorem relating the integral of the curl of a vector over a surface to the line integral around its bounding curve, C, to obtain:

d_ddtf P d v = d V

f (p28 +~ x vp) ' nA dA

(18)

A

where the no-slip condition has been applied along C. For an ocean of constant surface area, then, temporal changes in the total potential vorticity are measurable through the temporal changes in the magnitude of the surface

307 vector p 2 ~ + ~ X ~Tp as well as changes in the angle it makes with the surface normal. If we n o w substitute eq. 18 into eq. 15, we arrive at an equation for the integrated vorticity balance:

f (,o28 +

x vp).

A

=

f

• (vp x

B

- f(P

-

x vp)-

dA

(19)

A

which involves integrals over the bounding surfaces of the ocean only. Assuming that the left side of eq. 19 vanishes when averaged over a long enough period of time, the time averaged version of this equation then expresses a balance between vorticity transported through the ocean surface b y the velocity fluctuations and that diffused through the boundaries by molecular friction. Using the requirement t h a t ~ = 0 (where t~A is the velocity normal to the ocean surface and --t is an average over any reasonable length of time) together with the fact that ~A P't can be easily related to the mean surface wind-stress curl (see the following section), eq. 19 simply becomes a restatement of the usually accepted notion that the vorticity added through the upper oceanic layer by the winds must be diffused through the solid boundaries. Away from these boundaries there can be no net generation or dissipation of potential vorticity, only a redistribution. The latter conclusions may have important implications to mechanisms claiming to be capable of generating net vorticity within a closed ocean basin b y internal readjustment alone. For example, in a recent numerical experiment b y Holland (1973), so called "pressure t o r q u e s " associated with b o t t o m topography are suggested as a means of generating net vorticity within the ocean. According to our work, however, such an effect must cancel over the volume of the ocean whenever the flow is governed b y mass conservation following the motion (Dp/Dt = 0). This follows from the requirement that compression of the vortex lines in topographic regions where the vertical velocity, w, is upward must be cancelled elsewhere b y vortex line stretching where w is downward. If there is to be a net torque on the ocean through the mechanism proposed b y Holland, it is necessarily related to the removal of this constraint so that Dp/Dt ¢ 0 along streamlines, making it difficult to relate our results to his. Further work would require determination of the effect, if any, of variations in density following the motion on the overall vorticity balance. It should be noted that the considerations revealed here in no way conflict with the notion that the interaction of b o t t o m topography and sloping isopycnals might influence ocean transport locally. It is well established that sloping topography affects transports. This work does, however, cast d o u b t upon the interpretation implicit in the expression 'pressure torque', and

308

sho~vs that at least within the framework of our assumptions there is no net introduction of potential vorticity b y pressure. THE REDISTRIBUTION OF POTENTIAL VORTICITY

The previous discussion, although appropriate to a closed ocean basin, tells us nothing about the local effect of a vorticity redistribution. We therefore return to eq. 10 and make the simplifying assumption that molecular friction is only important near solid boundaries, thereby obtaining the averaged version of the conservation equation (eq. 9) in the form: aP ~ . at

~TP+ ~7 • (~'P') = 0

(20)

in which the turbulence appears as forcing term for the mean quantities. The averaged value of the potential vorticity can be obtained from eq. 8 and is given by: P = (~ + 2 ~ ) . Vp + ~ " Vp'

(21)

To calculate/~'P' we apply the concept of a 'mixing length' as first introduced b y Prandtl {1925) as a turbulent analogy to the mean free path in the kinetic theory of gases. (We shall follow the procedure of Welander as closely as our purpose permits.) In this concept, the value of a certain fluid property is conserved as a parcel of fluid is transported over a displacement / b y the fluctuating velocity. The parcel is then allowed to mix with its surroundings. Immediately prior to this mixing, however, the parcel produces a fluctuation in the fluid property because of the difference in the mean value of the transported property at the beginning of its 'free' trajectory from the mean value near its location at the time of mixing. This fluctuation is thus proportional to l and to the mean gradient of the property: ~."~- - - l • ~7~

(22)

for any quantity ~P = ~P' + ~ . In Prandtl's (1925) theory the transported property was taken to be momentum. On the other hand, Taylor (1915) t o o k the property to be vorticity, and it is in fact Taylor's theory which is the p r o t o t y p e of Welander's work. Ostensibly, vorticity transport theory would seem to have a better physical basis than m o m e n t u m transport theory, since the momentum of a parcel can be exchanged with the surroundings b y pressure forces, while vorticity can only be exchanged b y some kind of diffusion or mixing. As we shall see, however, the question is more complex than that and in fact neither theory adequately describes the situation, even in terms of its own simple assumptions. Every correct fluid
309 tions. It is thus perhaps surprising that no serious attempt seems to have been made to reconcile m o m e n t u m transport and vorticity transport mixinglength theories. Indeed, closer examination shows that the two cannot be correctly separated. In particular, the simple theories omit t w o exceedingly important facts which are needed to reconcile one with the other and which, when included, importantly modify the results obtained. The first of these arises from the fundamental difference between the motion of a molecule, in the c o n t e x t o f the kinetic theory of gases, and the motion of a fluid parcel. With the former, the motion between collisions is 'free'. With the latter, on the other hand, the motion must remain part of that of the fluid continuum, and the parcel must interact with the rest of the continuum during its trajectory. N o w if there exists a vorticity gradient, the displaced parcel will find itself with vorticity differing from the surrounding fluid. As the parcel continues to move, this vorticity difference will give rise to a 'lift' force -- which will then modify the m o m e n t u m o f the parcel. We shall see that this lift force just cancels the force calculated b y Welander, so it is far from negligible. The second omission in the classical theories derives from the fact that a parcel moving relative to its surroundings must be bordered b y regions of vorticity -- in the idealization b y vortex sheets -- which have opposite signs on opposite sides of the parcel. Therefore, if a parcel moves in a velocity gradient (or is accelerated b y the lift force) so that it has m o m e n t u m transverse to its trajectory different from that of its surroundings, it will have vorticity of one sign in front of it, and of the opposite sign behind it. Vorticity of one sign is thus effectively transported farther than is vorticity of the opposite sign, with a resulting important modification in the vorticity transport expression. Let us take a simple case to illustrate these points. As is customary in discussion of this kind, we shall specify the motion to be two dimensional with mean velocity (U, V) in Cartesian coordinates (x,y). Consider a single parcel of fluid in a fluid b o d y whose motion, except for that of the parcel, is directed in the x direction such that: V=O except in the parcel and its vicinity, and: U = U o(y) for t < 0 U = U(y) for t t> 0

The parcel, with area A, is started impulsively from y = 0 at time t = 0, and moves with an initial velocity: I

t

u =(O,v),

Vt

¢0

relative to the surrounding fluid. Its initial vorticity, which is conserved during its trajectory, originates from the mean shear at its original position,

310

thus: OY /y=0 The parcel then moves a distance l in the y direction, and mixes. Let us assume that the mixing occurs over a range: y=l + - 5/2

and that a length L measures the lateral separation between such mixing parcels. Continuity demands that the surrounding fluid move in the negative y direction by a distance A / L , over the range 0 < y < l. Moreover, as part of the mixing range 5 includes ambient fluids, in addition to that advected-in with the parcel, we have ~ > A / L . According to pure vorticity transport mixing length theory the final result can be schematized as shown in Fig. 1. The value of the vorticity near y = l after mixing has taken place will be ~'(l) (1 - - A / S L ) + ~(O)A/SL, which to the approximation of a linear vorticity gradient equals: ~U° ~y

~2U° 1(1 --5-~)

~y2

and the total vorticity is conserved. The velocity distributions corresponding to the above vorticity distributions are shown in Fig. 2. It should be noted that m o m e n t u m is n o t conserved in this process. Thus, the concern which caused us to examine closely the work of Green and in particular, of Welander arises in this very primitive example of the use of the vorticity transport idea. This idea clearly fails the test of consistency with one of the most basic of conservation laws. Now let us examine the case of pure m o m e n t u m transfer. Once more we consider a single parcel of fluid, but this time we take m o m e n t u m to be the

before

'

after

/

'

8

/

y--~

y--ol Fig. 1. The d i s t r i b u t i o n o f relative vorticity, ~', b e f o r e and after mixing. A parcel o f fluid o f area A has b e e n t r a n s p o r t e d f r o m y = 0 to y = I t h e n m i x e d with the s u r r o u n d i n g fluid over a length 5. C o n t i n u i t y requires t h a t t h e m a i n b o d y o f fluid shift a distance A/L t o w a r d y = 0, w h e r e L is t h e lateral s e p a r a t i o n b e t w e e n fluid parcels. The dashed line s h o w s t h e vorticity d i s t r i b u t i o n prior to advection.

311 after

before

8

y--O U

U

Fig. 2. The distribution o f m o m e n t u m associated with the vorticity distributions of Fig. 1. The dashed line shows the m o m e n t u m profile prior to adveetion.

q u a n t i t y t h a t is conserved. The result is shown in Fig. 3, where for consist e n c y with the vorticity argument above, we show the profile of Uo (y) to be visibly curved. In the vicinity of y = l, a velocity increase amounting to: U0(0 ) A / S L - - Uo(l ) can be seen to occur, but this is compensated by the decreased velocity over the range y = 0 to y = l -- ~/2 which results from the 'downward' movement of lower speed fluid required by continuity when the parcel moves from y = 0 to y = l. Vorticity is also conserved in this process, as is readily observed by first noting that the difference in velocity between y = l + ~/2 and y = 0 is the vorticity integrated over this range. Then, since the velocity for y > l + 5/2 and for y < 0 is unchanged, the net vorticity must also be unchanged. However, the behaviour of the vorticity is n o t trivial. As illustrated by Fig. 4, three regions of concentrated vorticity, idealized as vortex sheets, are generated: at y = 0, and y = l + ~/2 (In Figs. 3 and 4 the vorticity around y = I is shown as being zero, since the velocity is taken to be mixed.) It should also be noted that vorticity changes like those shown in Fig. 4 would occur even if the original vorticity were uniform. That these changes are not neglibefore

after

\

8

y--~

y=O -

U

U

Fig. 3. The distribution o f m o m e n t u m before and after mixing. The m o m e n t u m associated with the adveeted parcel o f fluid is mixed over a length 5 ; the main body of fluid shifting by a distance A/L to satisfy continuity. Dashed line shows m o m e n t u m profile prior to advection.

312 before

after

•8 j

y=~

"

/

D

/ ///

/ y=O

i

"

" //

"

Fig. 4. The distribution of relative vorticity associated with the momentum distributions of Fig. 3. Arrows represent vortex sheets generated at the onset of the momentum transfer process. gible is evident from the fact that they fully determine the velocity changes of Fig. 3. As was previously pointed out, the m o m e n t u m of the parcel may be partially exchanged with the surrounding fluid by pressure forces even within the framework of the concept of a parcel moving from y = 0 to y = l before mixing. This would lead to a decrease in the magnitude of the velocity increase around y =/,,with a corresponding increase in the velocity over part or all o f the range 0 < y < I. Both m o m e n t u m and vorticity would continue to be conserved. Thus, no possible form of this m o m e n t u m exchange could lead to the configuration of Figs. 1 and 2, in which vorticity is conserved but where the m o m e n t u m is not. It seems then, that something is seriously wrong with the vorticity transport idea in its simplest form. Let us examine it more closely, bearing particularly in mind the requirement that any correct argument must be valid in both its vorticity related aspects and in its velocity related aspects. There appears to be no question about the possibility of a parcel transporting its vorticity as it moves. Since the vorticity equation in two dimensions is : D__~= 0 Dt except for diffusion terms, this property of vorticity transport seems assured. Thus the error in the theory, if there is one, must be one of omission, not commission. When our parcel moves from y to y + dy, 0 < y < l, the redistribution of vorticity clearly calls, in the context of Figs. 1 and 2, for a deceleration of the fluid between y and y + dy, corresponding to the decrease in positive vorticity at y + dy. Quantitatively the change in velocity is given by: [~'(0) --

~(y)]A/L

Looked at from the point of view of the m o m e n t u m equation, this change

313

o f velocity must be caused b y some force. What force? There is in fluid mechanics theory a force known as the 'vortex force', which is exerted on surrounding fluid b y a moving vortex. Its kinematic value (i.e. per unit density) is: ~' X ~ per unit area, or: /~'X~ for a vortex of circulation ~ . A moving parcel of area A and vorticity ~(0) will thus generate a kinematic impulse ~(0)A in moving unit distance, and if this impulse is then imparted to the surrounding fluid along a length L of unit width, the velocity change will be ~(O)A/L. Meanwhile fluid of vorticity ~(y) moving in the opposite direction, as required b y continuity, will generate a velocity change -- ~(y)A/L. The total change is therefore [[(0) -~(y)]A/L which is just that given b y the vorticity argument. The velocity change produced b y the vorticity transport can therefore be identified dynamically with that produced b y the vortex force. Dynamically, however, every action must be accompanied b y a reaction in order that m o m e n t u m remains conserved. The reaction to the vortex force clearly must be on the moving parcel by the surrounding fluid. Moreover its magnitude must be: ~X~' which we immediately recognize as the lift force o f aerodynamics. Here, then, we identify the omission of the vorticity transport theory: the acceleration b y lift of the transporting parcel as it moves through the surrounding fluid. This acceleration gives rise to the kind of vortex sheets noted in Fig. 4, and thus to both a vorticity transport and vorticity distribution, which d i f f e r from those given in simple vorticity mixing theory. It should be noted that the presence of this lift force is also omitted, incorrectly, in simple m o m e n t u m mixing theory. But, in this latter case it represents only another interaction b y which the parcel exchanges moment u m with the surrounding fluid, opposite in sign, and not fundamentally different from those already k n o w n to exist b u t also omitted b y the theory. Our discussion so far has some relevance to traditional mixing-length t h e o r y as applied for example to wakes, jets and b o u n d a r y layers. In such cases, however, the t h e o r y is n o t ordinarily used as more than a framework for the insertion of empirical data in semi-empirical theory -- and is not much used nowadays even for that -- so is of little consequence in these contexts. On the contrary, in the context of geophysical fluid dynamics, and the possible importance of the transport and mixing of potential vorticity, it is of the greatest consequence. Thus, we n o w re-examine in more detail Welander's application of potential vorticity transport theory to the ocean, b u t taking into account the presence of the lift force. To do so, we return to

314 the calculation of the forcing term ~ ' P in eq. 20. Suppose a portion of the fluid at some position (r,O,q~) is transported b y a velocity fluctuation ~' to another position (0,0,0) where it then mixes with its surroundings (Fig. 5). In the mixing-length theory it is assumed that the fluid parcel begins its journey with the potential vorticity P (r,O,¢), the mean value of P at that point. This permits us to write: P(r,O,(o) = Po + vP0 " / + ... as the Taylor expansion about the origin (0,0,0) where: P0 = P(0,0,0) and where higher-order terms may be neglected. Since P is materially conserved following the fluctuating motion, the ensemble (or zonal) average gives: --

m

~ ' P = u'Po + ]~' • Po" l

(23)

where, in accordance with eq. 22: - -

VPo " / :

t

--Po

The zeroth order term, /~'Po will vanish (since ~t' = 0 by definition), whereby:

= (u'r,u'o ,u',)Po. t = (ur'u°'u~)

~~

+

-~-~ rsinO

~¢/

(24)

with I = (lr,l O ,lc~). In eq. 24, terms like u'ol v and U'rlo will be assumed negligible. Nonzero values of terms of this sort are associated with anisotropic turbulence, usually generated b y shear flows. While the turbulence in the ocean is very probably anisotropic, this anisotropy is irrelevant to the present discussion. Thus, for the sake of clarity, and to focus attention on the transport of potential vorticity, we will strip expression 24 of all unnecessary complications no matter h o w important they may be in real situations. On the other hand, the terms u'rl~, u'o lo and u'~l v cannot be neglected in our discussion since the signs of the components of / are determined b y the correr

/

{r=e,~l Fig. 5. The path of fluid parcel in mixing length theory; its initial position is at (r,O,~)).

315 sponding components of the fluctuating velocity. The diagonal elements of the 'eddy diffusion coefficient', Kii (a second order tensor), may then be defined via these terms, viz.: u~ly = K1i

where j = r,O,¢. Making the assumption that the turbulence is horizontally isotropic, we m a y further set: Koo = K¢¢

= K H

and :

Kr~=Kv as the horizontal and vertical eddy coefficients, respectively, for which it is customary to take KH > > Kv because o f the suppression of vertical motions in the ocean. Within the limits of the above assumptions, eq. 24 reduces to the approximate form: u'P=

0,

gnaPo r ~0'

K n ~Po~ rsin0 ~] + •

(25)

where we have separated the horizontal c o m p o n e n t from the vertical component:

to expedite the discussion that follows. Moreover, there is nothing to be gained by allowing for spatial variations in the eddy coefficients, and we will follow most authors b y taking t h e m to be uniform. The determination of/d'P in eq. 25 now requires an expression for Po. To obtain this, we use the fact that the gradient density in the ocean is nearly aligned in the negative r direction so that p = p(r). We further assume that the relative vorticity and the horizontal gradients of relative vorticity in the region under consideration are small compared to the planetary vorticity and the planetary vorticity gradient. With these assumptions, eq. 21 becomes:

where 2Ct r = 2~20 cos 0 for Fto =- [ ~ ]. The horizontal diffusion of potential vorticity will t h e n be influenced by the meridional variation of ~2r. On the other hand, the vertical diffusion of potential vorticity will be determined by the vertical gradient of the relative vorticity since ~2~ = constant in the vertical. Eq. 25 thus has the form: u'P=

( 0'- - 2 Kr H a a~ ~r ,0 ) + F

(26)

316 with :

which can now be used to determine the effect of a local redistribution of potential vortieity. The divergence of the vector t~P, as is seen in eq. 20, is a term in the expression for aP/~ t and therefore has the character of a forcing term. In particular, we wish to investigate the i__mplieations of the vortieity redistribution given b y eq. 20 when the term aP/~ t is zero. We begin by relating the stress, r , to the vertical component, ~, of expression (26) using the relationship :

go

~:;

=p-lvxz- r

Substitution of eq. 26 into eq. 20 then shows that ~ represents the forcing that is usually associated with the wind stress in Wind-Driven Ocean Circulation models. The horizontal c o m p o n e n t of (26), on the other hand, appears to represent an additional forcing involving meridional turbulent transport of potential vorticity. It is this c o m p o n e n t of the mixing process, which appears capable of generating a mean transport comparable to that of the surface winds, that characterizes the theoretical findings of Welander. Physically, the origin of the horizontal mixing force described above can perhaps be best understood b y considering the motion of a fluid parcel having an absolute vorticity, ~ + 2 ~ , which is being advected b y the horizontal velocity ~'. The requirement that the potential vorticity of the parcel be conserved along its path and that the magnitude of t Vp [ be constant constrains the projection of ~ + 2 ~ along Vp to a constant value following the motion. Therefore, any meridional displacement of the fluid produces a change in 2 ~ • Vp which must be compensated b y the generation of relative vorticity, ~', within the fluid parcel. In particular, southward displacements generate positive relative vorticity while northward displacements generate negative relative vorticity. Since it is impossible for advective motions to generate vorticity in the inviscid fluid surrounding the parcel, the newly acquired relative vorticity is confined to the parcel itself. Thus, the fluid parcel behaves as a vortex core superimposed upon a background of relative vorticity,~, whose spatial gradients have been considered negligible away from the fluid boundaries. For the m o m e n t neglecting the lift force, this situation is exactly analogous to the one described earlier in our simple model of the vorticity transport process. Therefore after mixing the meridional distributions of potential vorticity and m o m e n t u m will have essentially the same forms as those shown in Figs. 1 and 2, the only difference being that the curvatures of both plots will be increased because of the nonuniformity of ~t~ with latitude. Consequently, the horizontal mixing force

317

derived in Welander's t h e o r y is actually the vortex force, ~' X~', whose magnitude u~ ~'r is measured by the horizontal c o m p o n e n t in eq. 26. It is this force which accounts for the consistently westward acceleration in the above theory. Moreover, as the vertical c o m p o n e n t of the curl of the ensembleaveraged value of this force is nonzero as a consequence of the nonuniform variation of the Coriolis parameter with latitude, it appears capable of exerting a torque on the fluid in a manner analogous to the wind stress curl on the ocean surface. Since it is easily shown that Xlx (~' X~')" Vpis identical to the divergence of the horizontal c o m p o n e n t o f ~ ' P obtained in eq. 26, the transport produced by this torque is derivable from eq. 20 as required. Up to this point in the present analysis, our presentation has n o t basically differed from that of Welander. Nor has there been any disagreement concerning the existence of a consistent westward force on the fluid generated by the turbulent horizontal mixing of potential vorticity carried meridionally by fluid parcels. What is missing from these theories, however is the reaction force induced o n the advected parcels b y the surrounding fluid. As indicated previously, this is manifested as a lift force ~ X ~' generated as a consequence o f the circulation Q round the moving vortex core of relative vorticity. We can show this by considering a parcel of fluid being advected meridionally over the earth's surface by turbulent motions with velocitiy u0. Without loss of generality, we may suppose that the parcel begins its journey with a zero zonal c o m p o n e n t (u~ = 0, initially) and zero relative vorticity. By the time it approaches the mixing region it will therefore have a net positive relative vorticity if the initial m o t i o n is southward or a net negative relative vorticity if the initial motion is northward. The rotation associated with this newly acquired relative vorticity will again be confined to the parcel itself. But the parcel cannot generate a vortex sheet around itself since there is no source for the vorticity in such a sheet, consistent with the concept of potential vorticity conservation. Thus, the parcel must be surrounded by an induced (relative) irrotational vortex of circulation equal to the total relative vorticity within the confines o f the parcel. The disturbance in the velocity field surrounding the parcel of fluid will therefore have the 1/r dependence characteristic of the irrotational flow it represents (see e.g. Batchelor, 1967, p. 539). Accordingly, there will be a total lift force, ~L, on the advected parcel which will always be to the east with magnitude: FL = pu'oC

(Batchelor, 1967, p. 406) where:

c-f

A

is the strength of the circulation associated with the parcel; positive for

318

southward advection and negative for northward advection. The ensemble averaged value of this eastward force over the region of homogeneous twodimensional turbulence under consideration is then: -F L =P

f ,J

u ot ~~ f d A

A

which is equal to the total westward force exerted on the surrounding fluid by the vortex force. The additional force on each advected vortex arising from the interaction of the local background vorticity and the parcel's motion (Batchelor, 1967, p. 539) will not affect this result since its ensemble average is zero. The resultant force on the fluid produced by potential vorticity redistribution therefore vanishes. We conclude, therefore, that with the assumption of homogeneous twodimensional turbulence and parcels starting with zero zonal velocity component, the effect on the mean flow is identically zero everywhere. The ensemble averages of the lift force and the vortex force balance at every point in the fluid. This makes the situation exactly analogous to what happens under the influence of the viscosity produced by random molecular motions; solid b o d y rotation is unchanged by viscosity. Turbulence, of course, can behave quite differently. It can be both inhomogeneous and significantly anisotropic to that Reynolds shear stress gradients may be generated, with resulting accelerations. However, in the case of such stress gradients, any m o m e n t u m or angular m o m e n t u m added to the fluid at one location must necessarily be compensated for at some other location. However, we have responded to an objection to a vorticity argument with a dynamic argument. Once more noting that every correct fluid mechanical theory must be valid with respect to the vorticity field as well as to the velocity field, we also need to support our results with a vorticity argument. To do this, we consider the configuration of Fig. 6a which takes into account the change in the velocity field brought about by both the vortex force and the lift force. (Reorientation of the velocity plot is allowable here because only the relative changes in velocity are relevant, n o t the absolute values.) The corresponding vorticity field is shown in Fig. 6b, where the arrows represent equal and opposite vortex sheets required to develop the velocity field of Fig. 6a. (Southward advection produces analogous results and will not be discussed.) It must be noted that these vortex sheets are not generated by the advection process, since that would be inconsistent with the conservation of potential vorticity, as we have already pointed out. Rather, the sheets are simply those which were produced when the fluid parcel was first set into motion, but which have now become reorientated in the latitudinal direction because of the curved path followed by the advected parcel. Since the lift force is normal to the direction of relative motion, it does not alter the relative speed, so the conservation of vorticity in the vortex sheets causes no difficulty. Now, for our results to be valid with

319

° eo+ Ae

la s '

q

\# ,,

A

0o+ Ae

.I

i

m

t I I I I I

d

~eo

% Relative Velocity

Relative Vorticity

Fig. 6a. The meridional distribution of the zonal velocity component (reorientated for exemplary purposes) when the vortex sheets associated with the northward advected fluid parcel are taken into consideration. Area A and velocity Q lie to the right of the velocity distribution prior to advection (dashed line) while area B and velocity q lie to the left. Fig. 6b. The meridional distribution of potential vorticity (reorientated for exemplary purposes) corresponding to the velocity distribution of Fig. 6a; the arrows represent equal and opposite vortex sheets associated with the advected fluid parcel.

respect to the vorticity field, the net relative vorticity transported southward must equal the net relative vorticity transported northward. Using Figs. 6a and 6b, together with the fact that the advective velocities must be proportional to the meridional displacements for a given At, this relative vorticity balance is:

~qd + ½q~ = Q(d + ~) - - Q d

(28)

where q and Q measure the strength of the vortex sheets associated with southward and northward advection, respectively. The righthand side of eq. 28 simply expresses the fact that the vortex sheet, Q, on one side of the parcel is transported further to the north than the sheet, -- Q, on the opposite side of the parcel. An expression relating the strengths q and Q can be obtained from the condition that the total impulse imparted to the fluid parcel by the lift force (represented by the area A in Fig. 6a) is equal to the total impulse imparted to the surrounding fluid by the vortex force (represented by the area B), viz.: Q =2-~ (~ + d ) Substitution for Q in eq. 28 yields the required result. We may conclude with a brief outline of the results of this section. Firstly, we have re-examined the mixing length concept as used by Prandtl (1925) for momentum transport theory and by Taylor (1915) for vorticity transport theory. As both theories failed to take into account the interaction of advected fluid parcels with their surroundings, they have neglected the

320

presence of the lift force on these parcels. Omission of this force, and the omission of the influence of the vorticity bordering the parcels in Taylor's theory has been shown to render it invalid since it fails to comply with the conservation of m o m e n t u m law. Momentum transport theory, on the other hand, remains valid with respect to both m o m e n t u m and vorticity conservation although inclusion of the lift force is necessary for completeness of the theory. On the basis of these findings, we then re-examined the potential vorticity transport theory recently put forward b y Welander (1973). We have shown that the westward mixing force in his model, which appears to be capable of driving large-scale ocean circulation, is simply the vortex force exerted on the surrounding fluid by the meridionally advected parcels of fluid. As with previous mixing length theories however, the model omits the lift force. When included, the lift force just cancels the vortex force. This leads us to conclude that the resultant force on the ocean or atmosphere, arising through a turbulent redistribution of potential vorticity b y strictly internal means, must vanish. Consequently the inconsistencies in the theories of Green and Welander, with respect to the conservation of angular momentum and the form of the stress associated with the vortex force, are related to the fact that classical vorticity transport theory, which their arguments follow in detail (Welander) or in spirit (Green) omit the lift force. Finally, we have supported our findings, based on the mechanics of t h e turbulent mixing, with a more complete vorticity argument, including the effect of vorticity bordering moving parcels, which shows that the net relative vorticity transported meridionaUy during the advective process must also vanish. Thus, despite the appeal the mixing theories may have with regard to accounting for certain features of large-scale circulation, it does n o t appear that the horizontal redistribution of potential vorticity b y conserved processes is a mechanism capable of generating a mean flow. In fact, eq. 20 simply reduces to: --

--

1

~

J~ • VP = ~ r r ( r

-2

l~p Vx~ - r / I r l ) ~ _ ar

which yields the usual Sverdrup relation when integrated from a depth of no-motion to the surface. Our detailed discussion has been directed toward the work of Welander (1973), since his model is sufficiently fully described to permit close scrutiny. However we suggest that the general conclusion also probably applies to the model proposed b y Green (1970) for the generation of largescale atmospheric circulation. At the very least we would contend that the concept of transport of vorticity cannot be used uncritically. Before any ideas of vorticity redistribution are accepted, they should be couched in terms of some specific detailed model, which can be closely and critically examined in a manner similar to that we have applied to Welander's model. Our work shows that the notion of vorticity transport is attended by too many subtleties and complications to be dealt with less carefully.

321 ACKNOWLEDGEMENTS

Part o f this was done while one of us (R.E.T.) was a Visiting Lecturer in the Department of Mathematics, Monash University, Melbourne, and their support is gratefully acknowledged. REFERENCES Batchelor, G.K., 1967. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 615 pp. Eady, E.T., 1949. Long waves and cyclone waves. Tellus, 1 (3): 33--52. Ertel, H., 1942. Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z., 59: 277--281. Green, J.S.A., 1970. Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Q. J. R. Meteorol. Soc., 96: 157--185. Holland, W.R., 1973. Baroclinic and topographic influences on the transport in western boundary currents. Geophys. Fluid Dyn., 4 : 187--210. Parker, C.E., 1971. Gulf Stream rings in the Sargasso Sea. Deep-Sea Res., 18: 981--993. Prandtl, L., 1925. Bericht fiber Untersuchungen zur ausgebildeten Turhulenz. Z. angew. Math. M e c h . , 5 : 136--139. Stommel, H., 1948. The westward intensification of wind driven ocean currents. Trans. Am. Geophys. Union, 29 (2): 202--206. Sverdrup, H., 1947. Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Natl. Acad. Sci., 33: 318--326. Taylor, G.T., 1915. Eddy motion in the atmosphere. Philos. Trans. R. Soc., Set. A, 215: 1--26. Welander, P., 1973. Lateral friction in the oceans as an effect of potential vorticity mixing. Geophys. Fluid Dyn., 5 (2): 173--189.