## 1. Introduction

**u**is the horizontal velocity,

*z*

_{ρ}is the isopycnal thickness, and the tilde represents an average along an isopycnal surface, plays an important role in turbulent tracer transport. It is important because average tracer quantities are advected not just by the

*Eulerian mean velocity*

**ũ**but by the total transport velocity given by

**û**

**ũ**

**u**

**u*** associated with the divergent component of thickness flux is effective in tracer transport—see Greatbatch (1998) and the comments below regarding gauge fields.] This was originally realized in the atmospheric community [for a review, see Andrews et al. (1987)] but not highlighted in the ocean case until Gent et al. (1995), when the idea was introduced in connection with the mesoscale turbulence parameterization of Gent and McWilliams (1990, henceforth referred to as GM90).

**∇**

_{ρ}is the horizontal gradient in isopycnal coordinates and

*κ*is a scalar diffusivity coefficient. In the case that

*κ*is constant, or at least independent of

*ρ,*this amounts to simple Fickian diffusion of thickness along isopycnal surfaces, which is a plausible choice from the point of view of baroclinic instability. To go beyond this one needs some sort of turbulence theory; the simplest type is embodied in a stochastic model of turbulence (Monin and Yaglom 1971). Such a model is behind simple one-point or mixing-length theories, which are still important in practice and which form the foundation for more elaborate turbulence closures. A stochastic model is of interest because it makes minimal assumptions about the nature of the turbulence: It only assumes that turbulence exists and that it is random and Markovian, that is, that turbulence statistics are independent of its past history. These assumptions are expected to apply in fluid turbulence provided time averages are taken over timescales much longer than the Lagrangian integral timescale and the turbulence is not strongly inhomogeneous. Such minimal dependence on the details of the turbulence is important in view of the large uncertainties prevailing in geostrophic turbulence models. However, until recently it was not possible to apply a stochastic theory to the case of ocean mesoscale turbulence because the theory assumed incompressible flow or a divergence-free velocity field. Because the ocean is strongly stratified, it is believed that mesoscale turbulence is highly nonisotropic, with mixing confined to isopycnal layers. It is therefore most natural to deal with the turbulence in isopycnal coordinates, in which case the two-dimensional velocity field is generally divergent and this requires a compressible stochastic theory of turbulence.

Such a theory, applicable to tracer transport, was recently developed by Dukowicz and Smith (1997, henceforth referred to as DS) and it validated the parameterization of GM90 in general terms. The weak point of the theory is a postulate that relates Lagrangian to Eulerian mean velocities. The present paper presents a further development of the theory for the ocean case, which avoids the need for the aforementioned postulate under certain conditions, and this allows us to go beyond the conceptual basis of the bolus velocity parameterization of GM90 when these conditions prevail.

## 2. Stochastic theory

*p*(

**x**,

*t*|

**y**,

*t*

_{0}) such that the probability of finding a particle in volume

*d*

**x**centered around point

**x**at time

*t,*given that it was in volume

*d*

**y**centered around point

**y**at time

*t*

_{0},

*t*≥

*t*

_{0}, is

*p*(

**x**,

*t*|

**y**,

*t*

_{0})

*d*

**x**. The assumption of Markovian statistics allows one to write down a Fokker–Planck equation for the time evolution of the probability density function. Specifying the appropriate choice for the probability density function (DS) leads to a theory of turbulent transport in a compressible medium. We apply the theory to the continuity and tracer equations of interest for the ocean, in which case the equations in isopycnal coordinates take the form

*h*(≡

*z*

_{ρ}) for convenience and

*ϕ*represents any tracer concentration per unit volume satisfying the equation

*D*

_{t}

*ϕ*

_{t}

*ϕ*

**u**

**∇**

_{ρ}

*ϕ*

**u**is the two-dimensional horizontal velocity on the isopycnal surface. These equations manifestly have the form of two-dimensional compressible equations to which the stochastic theory is applicable. The theory gives equations for the evolution of averaged quantities, which are defined as

*ϕ̂*

**v**, which we will refer to as the

*Lagrangian mean velocity,*is defined by

**K**

**x**,

**y**,

**z**are to be interpreted as two-dimensional position vectors in the horizontal projection of an isopycnal surface. The limit as Δ

*t*→ 0 makes sense only for timescales much longer than the Lagrangian integral timescale, in which case

**K**

*φ*

**k**is the unit vector in the vertical direction and

*ψ, χ*are arbitrary scalar gauge fields, which are not specified by the stochastic theory. We will not focus on these fields henceforth since they play no role in tracer transport, except in section 7 where we will indicate a possible relationship among them. It is important, however, to keep the gauge fields in mind when comparing parameterizations to corresponding experimentally or computationally derived turbulence correlations. Note also that we have implicitly assumed that it is permissible to directly compare Eqs. (12)–(13) to (7)–(8). This may not be the case in general since the averages in (12) and (13) cannot all be the same as those specified in (6) and (9), namely, probability averages on an isopycnal surface, since such averages apply only to quantities satisfying the advection equation (5) [e.g.,

**ũ**in (12) and (13) cannot be a probability average since

**u**does not satisfy (5)]. We have therefore implicitly invoked a form of the “ergodic hypothesis”; that is, we have made the assumption that isopycnal probability averages are approximately equivalent to isopycnal Eulerian averages.

**v**. In the case of divergence-free flow,

**∇**

_{ρ}·

**û**= 0, Monin and Yaglom (1971) introduced the postulate

**v**

**ũ**

**∇**

_{ρ}

**K**

**∇**

_{ρ}·

**û**≠ 0. Therefore, substituting (16) into (14), one obtains

## 3. Potential vorticity

*f*is the Coriolis parameter (planetary vorticity) and

*ζ*=

**k**· (

**∇**

_{ρ}×

**u**) is the relative vorticity, obeys (5) and therefore may be viewed as a tracer for the purpose of the stochastic theory (see DS for a comment on

*q*as an active tracer). Therefore, according to (15), we have

*q̂*=

*h̃*

^{−1}

*f*+

*ζ̃*

## 4. Planetary geostrophic regime

*q*

_{∞}=

*f*/

*h.*An illuminating discussion of this regime is given by Pedlosky (1984). Furthermore, de Verdière (1986) demonstrates that this regime is subject to baroclinic instability, and therefore contains turbulence. Given that we have exact geostrophy:

*π*is the Montgomery potential and

*ρ*

_{0}is a constant reference density, we may follow an argument in Greatbatch (1998) comparing thickness-weighted and unweighted averages of (23), to obtain

*q̂*

_{∞}=

*f*/

*h̃.*Alternatively, it is easy to show that (20) reduces to (24) when

*q*=

*q*

_{∞}. Comparing (24) to (20), we see that in the PG limit we can neglect

*β*= ∂

_{y}

*f.*We observe that this is the same as (17) except that the bolus velocity is augmented by a velocity contribution proportional to

*β.*In the case of isotropic mixing,

**K**

*κ*

**I**

*κ*> 0, this term corresponds to a poleward component of bolus velocity in each hemisphere. We also have the following result for the Lagrangian mean velocity:

## 5. Mesoscale regime

*f*

_{0}is the “local” Coriolis parameter, taken to be constant. Gill (1982) shows that quasigeostrophy is valid provided (i) ε

_{L}=

*βL*/

*f*

_{0}≪ 1, (ii)

*τf*

_{0}≫ 1, and (iii)

*R*

_{o}=

*U*/

*f*

_{0}

*L*≪ 1, where

*L*is a horizontal length scale such as the Rossby radius, and

*τ*and

*U*are characteristic time and velocity scales, respectively.

*π*′ = Π

*e*

^{i(kx+ly−ωt)}, as in Killworth (1997), where

*i*=

*k, l*are the horizontal wavenumbers,

*ω*is the frequency, in general complex to allow for growth of instabilities, and Π is a complex amplitude. The spatial average, in the limit of averaging over very many wavelengths, is given by

*A*is some arbitrary convex averaging area on an isopycnal surface. Noting the identity

**∇**

_{ρ}

**∇**

_{ρ}

*π*

**∇**

_{ρ}

*π*

**∇**

_{ρ}

**T**

**T**

**I**

*A.*Since there is no contribution from the interior, the correlation will scale with the ratio of the periphery length to the area, which may always be made small by choosing a suitably large averaging area

*A.*That is, in quasigeostrophic flow,

The question becomes, is this result useful for QG turbulence? Typically, turbulence correlations are most meaningful in an ensemble average. However, ensemble averages are mainly a theoretical concept, and typically space or time averages are substituted for practical reasons. This is justified by the “ergodic hypothesis” (Monin and Yaglom 1971), which allows substitution of time averages for ensemble averages in stationary turbulence and spatial averages for ensemble averages in homogeneous turbulence. Therefore, the result (31) may be taken as an indication that in homogeneous quasigeostrophic turbulence the correlation

*g*→ ∞) or a constant-depth barotropic flow with a rigid-lid boundary condition, described by the barotropic vorticity equation. For such a flow (i.e., where the rigid-lid approximation is valid), the thickness

*h*is nearly constant and

**K**

*υ*′ = 0 on the north and south boundaries. This, therefore, implies the existence of a global constraint on any parameterization, whereas the present parameterization, like most others, is local when

**K**

*f*+

*ζ̃*

*f*in the PG case. This is our principal result. It is derived using only (15), a robust prediction of stochastic theory, the identity (20), and the fact that

*z*coordinates is useful for practical applications and is given in the appendix.

The bolus velocity in this form represents a flux of tracer up the potential vorticity gradient, which results from a modification of (17) given by the second term on the right-hand side of (32), the so-called *β* velocity. However, this modification is typically of very small magnitude (as will be seen in section 6) and will therefore have little practical impact, except for the fact that it importantly changes the conceptual basis for the parameterization of bolus velocity, from being based on the mixing of thickness as in GM90 to being based on the mixing of potential vorticity, as previously suggested by Treguier et al. (1997).

## 6. The *β* velocity

*β*velocity,

*t*along the trajectory we have

*q*

*t*

*q̂*

*t*

*q*

*t*

*q̂*

*q*

*q*"(0), for example, indicates the initial condition. For a sufficiently small time increment, this may be rewritten as follows:

*q*

*t*

*q̂*

*q̂*

*t*

*q*

**x**

**∇**

*q̂,*

**x**=

**x**(

*t*) −

**x**

^{0}(0) is the displacement of the particle, and

*q*"(0) has been absorbed into the definition of

**x**

^{0}(0). We now multiply both sides by

*h*

**u**" and take some appropriate average (we retain the tilde sign to indicate that this is still an isopycnal average) to obtain

*q*=

*f*/

*h,*and therefore

*q̂*=

*f*/

*h̃*(we have dropped the subscript ∞ for simplicity), Eq. (35) may be written out as

*q*" ≈ −

*β*Δ

*y*/

*h̃*+ (Δ

**x**·

**∇**

*h̃*)/

*h̃*

^{2}. Thus, there are two components to the potential vorticity fluctuation: the first one is due to the change in the planetary vorticity

*f,*and the second one is due to a change in mean thickness

*h̃*along the trajectory. We will now assume that the mean thickness is constant since we wish to focus on the first component. This implies that meridional fluid parcel displacements are always negatively correlated with potential vorticity fluctuations. Alternatively, since

*q*" = −(

*h*′/

*h*)(

*f*/

*h̃*), fluid parcels displaced poleward will be stretched while parcels displaced away from the poles will be flattened relative to their mean thickness. Now, because the right-hand side of (37) is positive-definite and the diffusivity tensor is likely to be diagonally dominant, velocities

*υ*" will be positively correlated with displacements Δ

*y.*Therefore, the meridional potential vorticity–velocity correlation will always be negative. Because of Eq. (24), this implies that the associated bolus velocity will always be poleward. This is the content of Eq. (33); it shows that the

*β*velocity is inherent in the PG regime provided that turbulence is present. Note that the apparent singularity at the equator is merely due to the fact that the

*β*velocity given in (33) is valid only in the PG regime, which excludes the equator.

The existence of the *β* velocity has been specifically confirmed by Lee et al. (1997) in an eddy-resolving channel experiment performed with a three-layer isopycnal model. Using an appropriate buoyancy forcing, they obtained essentially constant mean thickness in the middle layer so that the predominant contribution to bolus velocity was due to (33). Using the value *κ* ∼ 1000 m^{2} s^{−1}, which they measure in the top and bottom layers and the values *β* ∼ 2 × 10^{−11} m^{−1} s^{−1} and *f* ∼ 0.83 × 10^{−4} s^{−1} characteristic of the center of the channel, (33) predicts a northward velocity *υ*^{*}_{β}^{−1}, in reasonable agreement with the value *υ*^{*}_{β}^{−1}, which they find in the experiment. As pointed out by Killworth (1997), this is a rather small velocity [even compared to the total bolus velocity—we typically find that **u*** ∼ *O*(1 cm s^{−1}) in eddy-resolving calculations].

There is an alternative but equivalent way to arrive at the *β* velocity. Welander (1973), Rhines (1979), Rhines and Holland (1979), Tung (1986), and Greatbatch (1998) propose closely related parameterizations of the momentum equations based on the turbulent flux of potential vorticity. In the *u*-momentum equation there then appears a “friction force that is everywhere directed westward,” equal to −*βK*_{22} (Welander 1973). When balanced against the Coriolis term, this results in the *β* velocity of Eqs. (25), (33). From the atmospheric perspective, Tung (1986) points out that the presence of this term has important physical implications. Without this term, the winter stratospheric westerly jet would reach unrealistically large velocities and the temperature near the winter pole would be too low. It is of interest to note that the *β* velocity is associated with the presence of diabatic heating in the experiments of Lee et al. (1997) and in the atmospheric context of Tung (1986). Without diabatic heating it is likely that a statistical equilibrium consistent with the continuity and momentum equations would not be possible, and a poleward thickness flux would be prohibited.

From a theoretical perspective, a remarkable result by Killworth (1997) gives a bolus velocity that is equivalent to (25), and therefore also predicts the *β* velocity of (33), but from consideration of linear waves alone, in a perturbation theory about a slowly varying mean velocity. This in effect extends to turbulence a result derived from linear stability theory. However, a justification for the role of linear terms in turbulent mixing problems has been given by G. M. Lilley in Morris et al. (1990) for the case of incompressible flow. This theory, therefore, lends powerful independent support for the existence of the *β* velocity. It is not the object of this paper to discuss the pros and cons of the Killworth (1997) theory and the present theory. There is, however, an important difference between these two theories that is worth pointing out. The Killworth theory derives a form of the diffusivity *κ* that is proportional to the instability growth rate and therefore requires that the underlying mean flow be unstable for *κ* to exist; the present theory merely assumes that fully developed turbulence is present, in which case *κ* is formally given by (11). It should be noted that, when there is no turbulence, then *κ* is zero. In other words, the present theory, unlike the Killworth theory, makes no statement about the origin of the turbulence. It does require an independent specification of *κ,* in common with other parameterizations of its type, but then it leaves open the possibility that a future, more general type of turbulence theory will become available that will predict *κ* prognostically. Until that time, it is quite possible to have seemingly paradoxical situations arising from an ignorance of what the value of *κ* may be.

*d̃*/

*dt*≡ ∂

_{t}+

**ũ**·

**∇**

_{ρ}. This is the same as the equation in the Killworth theory except that the last term on the right-hand side was neglected. In the PG regime, geostrophic balance implies

*f*

**∇**

_{ρ}

**ũ**

*βυ̃*

*f*

**∇**

_{ρ}

**u**

*βυ*

*υ*′ is the meridional fluctuating velocity component. Multiplying the continuity equation by

*h*′, making use of (39), and taking an average we obtain

**K**

*β*-velocity correction, with the presence of baroclinic instability, consistent with Killworth (1997).

*β*velocity to Rossby waves because of its

*β*dependence. This is clearly incorrect, as is obvious from the above analysis, but it is possible to demonstrate this more directly. Let us consider the case of pure Rossby waves when the gradient of mean thickness vanishes. Equation (40) in the limit of slowly varying mean velocities then becomes

*β*velocity.

## 7. The gauge

*χ*

*q̂ψ*

*χ*and

*ψ,*as in (44) implies that the potential vorticity system contains only a single gauge.

## 8. Summary of results

*h̃*

**u*** retains an arbitrary rotational component specified by the gauge

*ψ*in addition to the largely irrotational potential vorticity term [see (32), where the dominant thickness-mixing term is irrotational for constant, isotropic

**K**

*ϕ̂*

*q̂,*we have

In summary, assuming the validity of the stochastic theory, we have deduced that under certain conditions the bolus velocity corresponds within a gauge to a flux directed up the potential vorticity gradient and is characterized by a general symmetric diffusivity tensor **K**

## Acknowledgments

We would like to thank Richard D. Smith and Peter D. Killworth for helpful discussions. This work was made possible by the support of the DOE CHAMMP program.

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## APPENDIX

### Transformation to *z* Coordinates

*z̃*to be the average depth of an isopycnal surface and

*ρ̃*

*z̃*[

*ρ̃*(

*x, y, z̃*(

*x, y, ρ, t*)

*ρ*], such that

*h̃*= ∂

_{ρ}

*z̃*=

_{z}

*ρ̃,*

*q̂*

_{z}

*ρ̃*(

*f*+ ∂

_{x}

*υ̃*

_{y}

*ũ*

_{x}

*ρ̃*

_{z}

*ũ*

_{y}

*ρ̃*

_{z}

*υ̃.*