## 1. Introduction

*Q*

_{λ}is defined as the vector dot product of the absolute vorticity

*ω*and the gradient of a scalar function, say,

*λ,*divided by the total density

*ρ*of the flow:and its time evolution is governed by an equation of the formwhere

**J**

_{λ}is the flux of potential vorticity. In this context, the choice of

**J**

_{λ}is arbitrary to within the inclusion of the curl of an arbitrary vector function. In particular, the addition of the vector product of two arbitrary scalar functions (e.g., ∇

*f*× ∇

*g*) will not change (2). However, Haynes and McIntyre (1987) show that a particular choice of

**J**

_{λ}enables the existence of an impermeability conditionsuch that the flux of

*ρQ*

_{λ}across any constant

*λ*surface vanishes. If

**J**

_{λ}is required to satisfy (3) in addition to (2), then the arbitrariness of

**J**

_{λ}is further restricted. For example, an addition of the form ∇

*λ*× ∇

*g*to

**J**

_{λ}will not change (2) or (3). This arbitrariness allows some flexibility in the choice of

**J**

_{λ}that can be invoked in specific situations to provide convenient formulations. Other desirable properties of the flux vector are that it contains a convective (sometimes called advective) component, that it satisfies a generalization of the uniqueness theorem of Bretherton and Schär (1993), and that it vanishes for a fluid at rest.

Schär (1993) has shown that one choice of potential vorticity flux may be expressed in terms of a Bernoulli function for a single-component flow when *λ* is taken to be the potential temperature *θ.* This result has been successfully utilized in a dry atmosphere (e.g., Schär and Durran 1997) and in a Boussinesq ocean (e.g., Marshall et al. 2001). An extension to handle multicomponent fluids (e.g., a salty ocean or a moist atmosphere with hydrometeors) is desirable.

**J**

_{λ}

*ρQ*

_{λ}

**u**

*λ̇*

*ω**λ*

**F**

_{λ}

**u**is the three-dimensional velocity,

**F**

_{λ}is an arbitrary force field per unit mass, and the evolution equation for the scalar

*λ*iswhere

*λ̇*

*λ,*and

*D*/

*Dt*is the material derivative. We note that (4) has the same form as that of Haynes and McIntyre (1987) if

*λ*is the potential temperature and

**F**

_{λ}is the frictional force. The next section introduces an alternative definition of (4) with

**F**

_{λ}representing the net force field per unit mass:Section 3 shows that, with this definition,

**J**

_{λ}is related to the kinetic energy of the flow bywhere the quantity in parentheses is the specific kinetic energy. This result is closely related to the generalized Bernoulli theorem (Schär 1993) but has a broader range of validity. Section 4 compares the present analysis to that of Schär (1993). We show that the alternative flux vectoris equivalent to

**J**

_{λ}in terms of its divergence. Here,

*B*is the Bernoulli function. Sections 5 and 6 present the flux vectors for a salty ocean and a cloudy atmosphere that are consistent with (4) and (7).

## 2. Generalized potential vorticity and its flux vector

**Ω**is the rotation rate,

*p*is the pressure, Φ is the geopotential, and

**F**is the frictional force per unit mass. Following Pedlosky (1986), we take the curl of (9), use continuity in the formto eliminate the velocity divergence, and combine the result with the gradient of the conservation equation (5) for the arbitrary scalar function

*λ.*The result is the general potential vorticity equationwhere

**= 2**

*ω***Ω**+ ∇ ×

**u**is the absolute vorticity. We retain the term associated with the curl of the gradient of the geopotential for reasons to be explained below. The baroclinity vectoris the curl of the pressure gradient force per unit mass. Following Obukhov (1963), the sources and sinks of potential vorticity may be written in terms of the divergence of a vector

**O**:whereThis result follows from the vector identity for the divergence of the product of a scalar and a vector and the fact that the divergence of a curl is identically zero. It is noted that there is no need for

*λ*to be chosen so that its gradient annihilates the baroclinity vector

**B**when forming a generalized potential vorticity equation.

It is straightforward to show, following Haynes and McIntyre (1987), that (13) may be written in flux form as (2) with flux vector given by (4) such that the impermeability theorem (3) holds. In addition, it is readily shown that (4) shares the uniqueness property of Bretherton and Schär (1993) for a flux vector that is linear in the net force **F**_{net}. With this choice of **F**_{λ}, the flux vector **J**_{λ} has the convenient property of vanishing for an atmosphere at rest. This desirable property motivated the retention of the gradient of the geopotential in (11) and (14).

## 3. Kinetic energy and the potential vorticity flux vector

*λ*with this equation. Using the “bac-cab” rule for the vector triple product term yieldsThe first term on the right-hand side can be written in terms of the transport of potential vorticity and the second may be rewritten using the

*λ*equation (5). One findsThen, because

**J**

_{λ}is defined by (4), we have the result (7). In the steady state, (7) reduces towhere the quantity in parentheses is the specific kinetic energy. Because the divergence of (18) is identically zero, the divergences of the convective and nonconvective components of the flux vector must cancel. Unlike the choice of Davies-Jones (2003), the flux vector (18) does not, in general, vanish in the steady state. However, like the vector of Davies-Jones, our choice (4) with (6) for arbitrary

*λ*does not reduce to the pure convective potential vorticity flux vector in the absence of frictional forces, and thus it violates one of the stipulations of Bretherton and Schär (1993).

## 4. Comparison with Schär (1993)

*λ*=

*θ*and

*λ̇*

*θ̇*

**J**

_{S}

*ρQ*

_{θ}

**u**

*θ̇*

*ω**θ*

**F**

*h*is the enthalpy. In his derivation he invokes the thermodynamic identity for a homogeneous fluid to express the pressure gradient in terms of gradients of thermodynamic state variables. The general form of this identity for a multicomponent fluid iswhere

*s*is the entropy, and the summation is over the chemical potentials

*μ*

_{l}of the mixture with the concentrations

*χ*

_{l}. Thus, in contrast to (20), the present expression (7) for the flux vector is more general and is valid for multicomponent fluids. For example,

*λ*can be the water mixing ratio of a moist atmosphere or the salinity of the ocean.

## 5. Potential vorticity flux vector for a salty ocean

*λ*= −

*σ*with source

*λ̇*

*σ̇*

## 6. Potential vorticity flux vector for a cloudy atmosphere

*T*

_{υ}is the virtual temperature,

*p*

_{00}= 1000 mb, and

*ρ*

_{m}=

*ρ*

_{a}(1 +

*r*

_{υ}) is the density of moist air. Here,

*r*

_{υ}is the water vapor mixing ratio. The flux vector that is consistent with the general flux vector (4) iswhere the net force isNote that the impermeability condition (3) holds for the moist potential vorticity.

## 7. Conclusions

This analysis has explored the definition of the flux vector for a generalized potential vorticity function. The results indicate that the general flux vector contains a contribution due to gravity and the pressure gradient force but that this contribution does not prohibit the utility of the conservation of the potential vorticity or the existence of an impermeability theorem. In addition, the analysis presents convenient expressions for the flux vector in terms of the kinetic energy of the flow rather than a Bernoulli function. The two versions of the theorem are physically equivalent for single-component flows, but the revised version using the kinetic energy easily allows for multicomponent fluids such as cloudy air and salty water.

## Acknowledgments

The National Science Foundation, under Grants ATM-9820233 and ATM-0215358, provided partial support for this research.

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