1. Introduction

At present, the dynamical basis for global numerical weather prediction and climate models is the set of quasi-static primitive equations. Since the nonhydrostatic motions typical of dry and moist atmospheric convection have such small horizontal scales, and since they cannot be accurately simulated with the quasi-static primitive equations, the collective effects of these motions must be parameterized in quasi-static primitive equation models. However, in the not-too-distant future, it will be possible to construct and run global numerical weather prediction and climate models based on the exact primitive equations, with much more accurate treatments of the moist thermodynamics, and with cloud-resolving spatial discretization. In such nonhydrostatic models, cumulus cloud fields will be explicitly simulated, eliminating the need for cumulus parameterization. This pushes the frontier of empiricism back to the parameterization of the microphysics of the precipitation process.

The purpose of the present paper is to extend the potential vorticity conservation principle to nonhydrostatic models with Ooyama's (1990, 2001) form of moist dynamics and thermodynamics. We begin by reviewing the exact, nonhydrostatic primitive equations for a moist atmosphere in section 2. In section 3 we derive the generalized potential vorticity principle. Equations (20) and (21) are our main results, the latter form being useful for physical interpretation. In section 4 we discuss one of the many possible invertibility principles (depending on the particular balance conditions) associated with the generalized potential vorticity. Section 5 contains a derivation of the “equivalent potential vorticity” and a discussion of why this form is unacceptable from the standpoint of possessing an invertibility principle. Conclusions are given in section 6.

2. Nonhydrostatic primitive equations for a moist atmosphere

Consider atmospheric matter to consist of dry air, airborne moisture (vapor and cloud condensate), and precipitation. Let ρ_a denote the mass density of dry air,¹ ρ_m = ρ_υ + ρ_c the mass density of airborne moisture (consisting of the sum of the mass densities of water vapor ρ_υ and airborne condensed water ρ_c), and ρ_r the mass density of precipitating water substance. The total mass density ρ is given by ρ = ρ_a + ρ_υ + ρ_c + ρ_r = ρ_a + ρ_m + ρ_r. The flux forms of the prognostic equations for ρ, ρ_m, and ρ_r are given in (1)–(3), in which u denotes the velocity (relative to the rotating earth) of dry air and airborne moisture, and u + U denotes the velocity (relative to the rotating earth) of the precipitating water substance, so that U is the velocity of the precipitating water substance relative to the dry air and airborne moisture. The term Q_r, on the right-hand sides of (2) and (3), is the rate of conversion from airborne moisture to precipitation; this term can be positive (e.g., the collection of cloud droplets by rain) or negative (e.g., the evaporation of precipitation falling through unsaturated air).

The total entropy density is σ = σ_a + σ_m + σ_r, consisting of the sum of the entropy densities of dry air, airborne moisture, and precipitation. Since the entropy flux is given by σ_au + σ_mu + σ_r(u + U) = σu + σ_rU, we can write the flux form of the entropy conservation principle as (4), where Q_σ denotes diabatic processes such as radiation.

Next, we can write the equation of motion as (5), where ζ = 2Ω + ∇ × u is the absolute vorticity; Φ the potential for the sum of the Newtonian gravitational force and the centrifugal force; ρ⁻¹∇p the pressure gradient force, with p = p_a + p_υ the sum of the partial pressures of dry air p_a and water vapor p_υ; and F the frictional force per unit mass. The derivation of (5) is given in Ooyama (2001) and is reviewed in appendix B.

Consolidating our results so far, the prognostic equations for the total mass density ρ, the mass density of airborne moisture ρ_m, the mass density of precipitation content ρ_r, the entropy density σ, and the three-dimensional velocity vector u are

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The diagnostic variables appearing explicitly in (1)–(5) are the terminal fall velocity U, the entropy density of precipitation σ_r, the source terms Q_r and Q_σ, and the total pressure p. In the determination of U, σ_r, Q_r, Q_σ, and p, several other diagnostic variables are introduced. The additional diagnostic variables introduced in (6)–(13) are the mass density of dry air ρ_a, the thermodynamically possible temperatures T₁ and T₂, the actual temperature T, the partial pressure of dry air p_a, and the partial pressure of water vapor p_υ, the formulas for which are given below. To summarize, the diagnostic relations are

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Note that, although there are four types of matter (with densities ρ_a, ρ_υ, ρ_c, ρ_r), there are only three prognostic mass continuity equations [(1)–(3)]. The separation of the predicted total airborne moisture density ρ_m into the vapor density ρ_υ and the cloud condensate density ρ_c is accomplished diagnostically in the two alternatives of (12). In the absence of condensate, all the predicted airborne moisture ρ_m is in vapor form so that ρ_υ = ρ_m and ρ_c = 0, while if the vapor is saturated, ρ_υ = ρ^*_υ(T) and ρ_c = ρ_m − ρ_υ. The formulas for the entropy density functions S₁ and S₂, from which the temperatures T₁ and T₂ are diagnosed, are given in appendix A.

In the context of numerical modeling, the procedure for advancing from one time level to the next consists of computing new values of the prognostic variables ρ, ρ_m, ρ_r, σ, and u from (1)–(5). The diagnostic variables required for the prognostic stage are determined by sequential evaluation of (6)–(13), namely, the diagnosis of the dry air density ρ_a from (6), the thermodynamically possible (wet bulb) temperature T₂ from (7), the entropy density of precipitation from (8), the thermodynamically possible temperature T₁ from (9), the choice of the actual temperature from (10), the dry air partial pressure p_a from (11), the water vapor mass density ρ_υ, the airborne condensate mass density ρ_c, and the water vapor partial pressure p_υ from the appropriate condition in (12), and the total pressure from (13). Since they are not essential to our discussion here, we have omitted the parameterization formulas for the terminal fall velocity U and the source terms Q_r and Q_σ. See Ooyama (2001) for further discussion.

To obtain a better feel for the somewhat unfamiliar system (1)–(13), it is interesting to note the limiting forms of these equations for the case of a perfectly dry atmosphere (i.e., ρ_υ = ρ_c = ρ_r = 0). In that case the ρ_rU term in (1) vanishes, Eqs. (2) and (3) are dropped, the σ_rU term in (4) vanishes, and the precipitation contribution to F (see appendix B) in (5) vanishes. In addition, the diagnostic equations (7), (8), (10), and (12) are dropped; (6) and (13) reduce to ρ = ρ_a and p = p_a; (9) reduces to c_Valn(T/T₀) − R_aln(ρ/ρ_a0) = σ/ρ, from which T is diagnosed; and (11) reduces to p = ρR_aT, from which p is computed.

At present it is not feasible to numerically integrate moist, nonhydrostatic, “full physics” models over the whole globe with 1–2-km resolution. However, it is possible to perform such 1–2-km-resolution integrations over a single hurricane, for example. Such integrations advance the art of hurricane modeling to a new level that involves much less physical parameterization. In order that such full physics models can be interpreted in terms of well-established principles of geophysical fluid dynamics, we now derive the potential vorticity principle associated with the system (1)–(13). As we shall see, the only equations in the set (1)–(13) that are needed for the derivation of the potential vorticity principle are (1) and (5).

3. The potential vorticity (PV) equation

Consider the fundamental identities ∇·(a × b) = b · (∇ × a) − a · (∇ × b) and ∇·(Aa) = A∇·a + a · ∇A, for the arbitrary vector fields a and b, and the arbitrary scalar field A. Choosing a = ∇ψ and b = ∂u/∂t in the first identity, we obtain ∇·[∇ψ × (∂u/∂t)] = −∇ψ · (∂ζ/∂t), since ∇ × ∇ψ = 0 and ζ = 2Ω + ∇ × u. Choosing A = ∂ψ/∂t and a = ζ in the second identity, we obtain ∇·[ζ(∂ψ/∂t)] = ζ · ∇(∂ψ/∂t), since ∇·ζ = 0. Taking the difference of these two results we obtain

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Defining ψ̇ ≡ Dψ/Dt, where D/Dt = ∂/∂t + u · ∇ is the material derivative, and using the triple vector product ζ(u · ∇ψ) = u(ζ · ∇ψ) + ∇ψ × (ζ × u), we can write (14) as

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Using the continuity equation (1), in the form Dρ/Dt + ρ∇·u + ∇·(ρ_rU) = 0, we can put (15) in advective form and eliminate ∇·u to obtain

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We now combine (16) with the momentum equation (5). Substituting from (5) for ∂u/∂t + ζ × u, noting that ∇·(∇A × ∇B) = 0 for any scalar functions A and B, (16) reduces to

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For two-dimensional flows with line symmetry or circular symmetry, the vectors ∇ψ, ∇ρ, and ∇p all lie in the same plane so that ∇ρ × ∇p is perpendicular to ∇ψ, and the solenoidal term ∇ψ · (∇ρ × ∇p) vanishes no matter how ψ is chosen. However, for general three-dimensional flows, how do we choose the scalar function ψ in such a way that the solenoidal term ∇ψ · (∇ρ × ∇p) vanishes? To accomplish this, we follow Ooyama (1990) and first define the virtual temperature T_ρ by

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so that T_ρ is the temperature that dry air would have if its pressure and density were equal to those of the given sample of moist air. Note that the virtual temperature T_ρ is equal to the actual temperature T when p_υ = 0 and ρ_m = ρ_r = 0. Next let us define the virtual potential temperature θ_ρ by

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where p₀ is a constant reference pressure and κ = R_a/c_Pa. Since θ_ρ can be written as a function of p and ρ only, we have ∇θ_ρ = (∂θ_ρ/∂p)_ρ∇p + (∂θ_ρ/∂ρ)_p∇ρ. Since ∇p · (∇ρ × ∇p) = 0 and ∇ρ · (∇ρ × ∇p) = 0, this implies that ∇θ_ρ · (∇ρ × ∇p) = 0, so the choice² ψ = θ_ρ annihilates the first term on the right-hand side of (17). Thus, (17) reduces to

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The generalized potential vorticity principle (20) is our main result. It should be noted that (20) is a generalization of the well-known (dry) Ertel (1942) potential vorticity principle in three respects: (i) the total density ρ consists of the sum of the densities of dry air, airborne moisture, and precipitation; (ii) θ_ρ is the chosen scalar field; and (iii) precipitation effects are included in F and in the last term of (20a). In a completely dry atmosphere, the total density ρ reduces to the dry air density, the virtual potential temperature θ_ρ reduces to the ordinary dry potential temperature, and precipitation effects disappear, so that (20) then reduces to the ordinary (dry) Ertel PV principle.

It is worth noting that (20a) can be written in a form that is more physically revealing for thermally forced, frictionally controlled, balanced atmospheric motions. Defining j = ∇θ_ρ/|∇θ_ρ| as the unit vector normal to the θ_ρ surface and k = ζ/|ζ| as the unit vector pointing along the absolute vorticity vector becomes (20a)

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This form emphasizes the exponential nature of the time behavior of P for material parcels moving through a region with approximately constant rate processes associated with F, θ̇_ρ, and ρ_rU. For example, in the intense convective region of a hurricane, k tends to point upward and radially outward. Since θ̇_ρ tends to be a maximum at midtropospheric levels, air parcels flowing inward at low levels and spiraling upward in the convective eyewall experience a material increase in P due to the P(k · ∇θ̇_ρ)/(k · ∇θ_ρ) term. This material increase of P can be especially rapid in lower tropospheric regions near the eyewall, where both P and (k · ∇θ̇_ρ)/(k · ∇θ_ρ) are large. Although the (k · ∇θ̇_ρ)/(k · ∇θ_ρ) term reverses sign in the upper troposphere, large values of P are often found there because the large lower tropospheric values are carried upward into the upper troposphere.

It is also worth noting that the effects of precipitation contained in the last term of (20a) can be distributed over the other three terms to obtain an equation that is formally similar to the one usually given for the dry case. This alternative form is discussed in appendix C.

4. Invertibility principle

It is natural to ask if, under certain balance conditions, there exists an invertibility principle for P. The importance of the existence of an invertibility principle is hard to overemphasize. It is the existence of such a principle that makes P such a dynamically interesting quantity. To see that such a principle exists, consider an f-plane case in which a large-scale axisymmetric flow has a slowly evolving P(r, z, t) field. The slow evolution of the P field is due to radial and vertical advection of P, and the θ̇_ρ, F, and ρ_rU terms on the right-hand side of (20a). Since the evolution is slow, we can consider the tangential wind and mass fields as continuously changing from one hydrostatic and gradient balanced state to another. If we have a method of predicting P(r, z, t), the invertibility problem is to diagnostically determine, at each time, the tangential wind υ(r, z, t), the total density ρ(r, z, t), the total pressure p(r, z, t), the virtual temperature T_ρ(r, z, t), and the virtual potential temperature θ_ρ(r, z, t) from

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This constitutes a system of five equations for the five unknowns υ(r, z, t), ρ(r, z, t), p(r, z, t), T_ρ(r, z, t), and θ_ρ(r, z, t) with given P(r, z, t). Note that the solution of the invertibility problem gives us the total density ρ(r, z, t), the total pressure p(r, z, t), and the virtual temperature T_ρ(r, z, t). The determination of the actual temperature T, the partition of p between p_a and p_υ, and the partition of ρ between ρ_a, ρ_m, and ρ_r is not possible from knowledge of P only. In other words, solution of the invertibility problem gives only the parts of the mass field that are of direct dynamical significance.

In principle it is possible to use (22)–(25) to eliminate υ, ρ, and θ_ρ from (26) and thereby obtain a single partial differential equation relating the total pressure p(r, z, t) to the known PV distribution P(r, z, t). However, in (r, z, t) coordinates, the resulting partial differential equation is somewhat complicated. In the following two paragraphs we consider the transformation of the invertibility principle from the physical height coordinate to a total pressure type coordinate and to the virtual potential temperature coordinate. Both of these transformations result in simpler forms of the invertibility principle.

First consider the transformation to (r, ẑ, t) coordinates, where ẑ = ẑ_a[1 − (p/p₀)^κ] is the pseudoheight, with ẑ_a = c_PaT₀/g. With this vertical coordinate, the gradient and hydrostatic equations are (f + υ/r)υ = ∂ϕ/∂r and (g/T₀)θ_ρ = ∂ϕ/∂ẑ, where ϕ = gz is the geopotential and where the radial derivatives are now understood to be at fixed ẑ. In a similar fashion, the transformation of (26), along with the use of the gradient and hydrostatic relations, yields (27a), which can be regarded as a second-order nonlinear partial differential equation relating the geopotential ϕ to the PV. The boundary conditions for (27a) are that ϕ goes to a specified far-field geopotential ϕ̃(ẑ) as r → ∞ and that ∂ϕ/∂ẑ is given in terms of a known boundary θ_ρ at the top and bottom boundaries. Thus, the complete invertibility principle in ẑ coordinates is

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where ρ̂(ẑ) = ρ₀(1 − ẑ/ẑ_a)^(1−κ)/κ is the pseudodensity (a known function of ẑ). The solution of the nonlinear, second-order partial differential equation (27a), with the appropriate boundary conditions (27b) and (27c), yields the geopotential ϕ(r, ẑ, t), from which υ(r, ẑ, t) and θ_ρ(r, ẑ, t) can be calculated using the gradient and hydrostatic relations.

Now, assuming that θ_ρ is a monotonic function of the physical height z, consider the transformation to (r, θ_ρ, t) coordinates. With this vertical coordinate, the gradient and hydrostatic equations are (f + υ/r)υ = ∂M/∂r and Π = ∂M/∂θ_ρ, where M = c_PaT_ρ + gz is the Montgomery potential (based on virtual temperature), Π = c_Pa(p/p₀)^κ is the Exner function (based on total pressure), and where the radial derivatives are now understood to be at fixed θ_ρ. In a similar fashion, the transformation of (26), along with the use of the gradient and hydrostatic relations, yields (28a), which can be regarded as a second-order nonlinear partial differential equation relating the Montgomery potential M to the potential vorticity P. The boundary conditions for (28a) are that M goes to a specified far-field Montgomery potential M̃(θ_ρ) as r → ∞ and that ∂M/∂θ_ρ is given in terms of a known boundary Π at the top and bottom boundaries. Thus, the complete invertibility principle in θ_ρ coordinates is

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where Γ(p) = dΠ/dp = κΠ/p. The solution of the nonlinear, second-order partial differential equation (28a), with appropriate boundary conditions (28b) and (28c), yields the Montgomery potential M(r, θ_ρ, t), from which υ(r, θ_ρ, t) and Π(r, θ_ρ, t) can be calculated using the gradient and hydrostatic relations.

Of course, the ẑ-coordinate invertibility relation (27), the θ_ρ-coordinate invertibility relation (28), and the original relations (22)–(26) are simply different mathematical forms of the same physical principle. For the special case when all independent and dependent variables are interpreted in terms of their dry limits, the invertibility relation (27) is equivalent to the one solved by Hoskins et al. (1985) and Thorpe (1985, 1986), while (28) is equivalent to the one solved by Schubert and Alworth (1987) and Möller and Smith (1994). In their calculations these authors used a further transformation from the physical radius r to the potential radius R, which is defined by ½fR² = ½fr² + rυ. Regardless of this additional transformation, we can interpret the results of these previous “dry” invertibility studies in terms of our moist model, since the dry and moist invertibility problems are isomorphic under the interchanges θ ↔ θ_ρ, T ↔ T_ρ, etc.

In passing we note that the invertibility principle (27), expressed in the ẑ coordinate, or its equivalent, (28), expressed in the θ_ρ coordinate, represents only one member of a family of invertibility principles. Other members of the family are generated by replacing the gradient wind equation with different horizontal balance relations, for example, the geostrophic equation, the nonlinear balance equation, or the asymmetric balance equation (Shapiro and Montgomery 1993). In any event, the existence of such a family of invertibility principles indicates that the potential vorticity (20b), obtained through the choice ψ = θ_ρ, is the form that maintains a direct connection to the balanced dynamics and is hence the appropriate moist generalization of the well-known (dry) Ertel PV. In the next section we discuss a commonly suggested choice for ψ that turns out to be unacceptable from the standpoint of possessing an invertibility principle.

5. An alternative approach to the choice of ψ

Our approach in deriving (20) has been to choose ψ in such a way as to annihilate the ∇ψ · (∇ρ × ∇p) term on the right-hand side of (17). An alternative approach attempts to choose ψ in such a way as to annihilate the ζ · ∇ψ̇ term on the right-hand side of (17). One way to accomplish this, at least in an approximate sense, is to note that (4) can be written in the form

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where s = σ/ρ_a is the “dry-air-specific” entropy of moist air. Now consider a physical situation that warrants the neglect of the right-hand side of (29). For example, such a situation might occur when radiation and precipitation effects play a secondary role in the entropy budget. Then, defining the equivalent potential temperature by θ_e = T₀ exp(s/c_Pa), (29) reduces to θ̇_e ≡ Dθ_e/Dt ≈ 0 and the choice ψ = θ_e in (17) leads to

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which is sometimes referred to as the equivalent potential vorticity principle. The equivalent potential vorticity ρ⁻¹ζ · ∇θ_e has been used as a diagnostic in several studies, for example, in the analysis of model simulations of the rotation and propagation of supercell thunderstorms (Rotunno and Klemp 1985) and model simulations of extratropical cyclones with embedded latent heat release (Cao and Cho 1995; Persson 1995). A variant, using Hauf and Höller's (1987) entropy temperature for the choice of ψ, has been used by Rivas Soriano and García Díez (1997) to study the effects of ice.

The choice ψ = θ_e is problematic in three respects: (i) in a precipitating atmosphere with radiative forcing, the term ρ⁻¹ζ · ∇θ̇_e is not exactly eliminated; (ii) because θ_e does not depend on ρ and p only, but also on ρ_m and ρ_r, the term ∇θ_e · (∇ρ × ∇p) is not eliminated from (30); and (iii) when attempting to set up an invertibility principle for ρ⁻¹ζ · ∇θ_e, analogous to the invertibility principle (22)–(26) for ρ⁻¹ζ · ∇θ_ρ, one finds that knowledge of the ρ⁻¹ζ · ∇θ_e field is not sufficient and that additional information on the moisture field is required for determination of the balanced wind and mass fields. In other words, ρ⁻¹ζ · ∇θ_e does not carry all the essential dynamical information on the balanced flow like ρ⁻¹ζ · ∇θ_ρ does. For balanced flow and slow manifold dynamics, the relevant part of the vector ζ is perpendicular to the θ_ρ surfaces, not to the θ_e surfaces. We conclude that, because ρ⁻¹ζ · ∇θ_e is not invertible, (20) is more useful than (30) for the study of moist, precipitating, balanced flows.

6. Conclusions

The physical model that serves as the basis for the derivation of the generalized potential vorticity principle (20) is the nonhydrostatic moist model [(1)–(13)]. In this model, pressure is not used as one of the prognostic variables, since it is not a conservative property and its use as a prognostic variable would lead to an approximate treatment of moist thermodynamics. Rather, the prognostic variable for the thermodynamic state is σ, the entropy of moist air per unit volume, with temperature and total pressure (the sum of the partial pressures of dry air and water vapor) determined diagnostically. There are several unique aspects of this model that are worth noting.

1. The model dynamics are exact in the sense that there is no hydrostatic approximation and no traditional approximation (i.e., selectively replacing the actual radius by the constant radius to mean sea level and neglecting Coriolis terms proportional to the cosine of latitude); this means that the associated angular momentum and energy principles are exact.

2. The connection between dynamics and thermodynamics is through the gradient of pressure, which includes the partial pressures of dry air and water vapor.

3. The first law of thermodynamics is expressed in terms of σ, the entropy density of moist air; all the usual approximations associated with moist thermodynamics are thereby avoided.

4. There is no cumulus parameterization; the frontier of empiricism is pushed back to the microphysical parameterization of the precipitation process through U and Q_r.

5. The model is modular in the sense that ice can be included by specifying E(T) to be synthesized from the saturation formulas over water and ice (Ooyama 1990).

Even with a nonhydrostatic moist model of this generality, it is possible to derive a PV principle that is a straightforward generalization of the well-known (dry) Ertel PV principle. Using (20b) it is possible to construct PV maps as diagnostics of the nonhydrostatic moist model. Such a PV diagnostic is a useful aid in understanding the relationship between nonhydrostatic moist convection and the large-scale balanced flow. Hausman (2001) has produced such PV cross sections for a nonhydrostatic hurricane model based on the axisymmetric, f plane versions of (1)–(13). The PV structure associated with the intense hurricane stage of the numerical simulation consists of an annular ring of very high PV (more than 200 PV units) extending through the whole troposphere between 10- and 15-km radius. In a fully three-dimensional model, such a PV structure would be subject to combined barotropic–baroclinic instability, the barotropic aspects of which were studied by Schubert et al. (1999). Hausman also compared cross sections of P, computed from (26), with cross sections of dry Ertel PV. The differences are quite small, indicating that dry Ertel PV and its invertibility principle can give an accurate description of the balanced aspects of hurricane dynamics if the frictional and moist-diabatic source/sink terms for the dry PV are accurately parameterized.

In closing we note that there is an impermeability principle associated with the flux form of (20), that is, the θ_ρ surfaces are impermeable to ρP, even though they are permeable to mass. This is discussed in appendix D.