Abstract

The equations describing the dynamics and thermodynamics of cloudy air are derived using the theories of multicomponent fluids and multiphase flows. The formulation is completely general and allows the hydrometeors to have temperatures and velocities that differ from those of the dry air and water vapor. The equations conserve mass, momentum, and total thermodynamic energy. They form a complete set once terms describing the radiative processes and the microphysical processes of condensation, sublimation, and freezing are provided.

An equation for the total entropy documents the entropy sources for multitemperature flows that include the exchange of mass, momentum, and energy between the hydrometeors and the moist air. It is shown, for example, that the evaporation of raindrops in unsaturated air need not produce an increase in entropy when the drops are cooler than the air.

An expression for the potential vorticity in terms of the density of the moist air and the virtual potential temperature is shown to be the correct extension of Ertel's potential vorticity to moist flows. This virtual potential vorticity, along with the density field of the hydrometeors, can be inverted to obtain the other flow variables for a balanced flow.

In their most general form the equations include prognostic equations for the hydrometeors' temperature and velocity. Diagnostic equations for these fields are shown to be valid provided the diffusive timescales of heat and momentum are small compared to the dynamic timescales of interest. As a consequence of this approximation, the forces and heating acting on the hydrometeors are added to those acting on the moist air. Then the momentum equation for the moist air contains a drag force proportional to the weight of the hydrometeors, a hydrometeor loading. Similarly, the thermal energy equation for the moist air contains the heating of the hydrometeors. This additional heating of the moist air implies a diabatic loading for which the heating of the hydrometeors is realized by the moist air.

The validity of the diagnostic equations fails for large raindrops, hail, and graupel. In these cases the thermal diffusive timescales of the hydrometeors can be several minutes, and prognostic rather than diagnostic equations for their temperatures must be solved. However, their diagnostic momentum equations remain valid.

Anelastic and Boussinesq versions of the equations are also described.

1. Introduction

A fundamental issue in the modeling of cloud systems is the choice of the governing equations. Studies of deep, nonhydrostatic, precipitating clouds require the use of the fully compressible equations or an anelastic set that filters sound waves. Cloudy air contains water vapor, liquid drops, and ice particles. The drops and particles may have different temperatures from that of the air. They also move relative to the air. Numerical models require that the physics be expressed as continuum equations.

The purpose of this paper is to present a general set of continuum equations for cloudy air that accurately describes this multiphase, multivelocity, and multitemperature system. Despite the fundamental importance of such a set of equations, the meteorological literature on the subject does not appear to have addressed this issue in a consistent manner. Two issues in particular require our attention here. Recent textbooks (e.g., Cotton and Anthes 1989; Emanuel 1994; Houze 1993) note that the momentum equation for moist air should contain a force term associated with the weight of the hydrometeors. However, this so-called liquid water loading term (henceforth called hydrometeor drag) is not derived in a systematic fashion and the assumptions inherent in the approximation are not enumerated. It is not clear whether the approximation is valid equally for the smallest of cloud droplets whose terminal velocity is vanishingly small and for the heaviest of hailstones whose fall speeds reach tens of meters per second. The second issue is the traditional meteorological assumption that the thermodynamics of the cloudy air can be considered to be isothermal in the sense that the air and hydrometeors have the same temperature. This single-temperature assumption enables the thermodynamics of the system to be treated in terms of a conservative moist potential temperature or moist entropy. There are several formulations of the cloud entropy equation. Hauf and Holler (1987) provide a review and synthesis of this approach. However, their development is based on the validity of the Gibbs equation that only holds for a single temperature system. Dutton (1976) notes that the Gibbs equation holds only if the system is losing mass or if the mass gain is from a system with the same pressure and temperature. Its extension to multitemperature flows is not clear.

This paper has several specific objectives. The first is to derive the compressible continuum equations for cloudy air. The derivation builds upon the extant literature of multiphase, multicomponent flows and tailors the presentation for the meteorological community. Here, the air is assumed to be a compressible, viscous, conducting fluid obeying the ideal gas law. For simplicity of the presentation we consider only two general classes of hydrometeors: liquid drops and ice particles. This simplicity enables the interaction between hydrometeors to be described. The generalization to a spectrum of hydrometeors is straightforward and not described here. The second objective is to provide a clear derivation of the concept of hydrometeor drag and to delineate the assumptions inherent in the approximation. We show that the concept of the drag holds for both droplets and hailstones. For both, a diagnostic relation (the terminal velocity) may be used to determine the drag. The third objective is to relax the traditional meteorological assumption that the temperatures of the air and hydrometeors are the same. A single prognostic equation for the total entropy is found for multitemperature flows. Having laid this foundation, the manuscript then derives the form of the potential vorticity equation that is consistent with the preceding development. Our result disagrees with that of Schubert et al. (2001) whose momentum equation does not properly consider the hydrometeor drag or the redistribution of momentum during a phase change. Lastly the fifth objective is to describe the impact of making the anelastic approximation on the compressible set of equations. The equations represent an improvement and refinement of those developed in Bannon (1996).

Section 2 describes the general formalism used to derive the conservation equations for mass, momentum, and energy. The formalism for the moist air uses standard fluid theory for multicomponent flows (e.g., Bird et al. 1960; Aris 1962) and the air–hydrometeor interactions are treated using the theory of Crowe et al. (1998) for multiphase flows. The restrictions implicit in this formalism are discussed. The first application of this formalism is to the conservation of the dry air. Section 3 treats the conservation equations for the water substance. Section 4 presents the ideal gas law for moist air. Section 5 derives the momentum equation that includes the effects of momentum exchange during phase changes. The assumption that the hydrometeor's momentum equation is dominated by the viscous diffusion of its momentum to the moist air yields a force balance that can be used to diagnose the hydrometeor's velocity relative to that of the air. This equilibrium or terminal velocity is generalized here to include the dynamic pressure gradient force. The momentum equation for the moist air includes the drag exerted on the air by the hydrometeors.

Section 6 considers the thermodynamics of the cloudy air system and derives an internal energy equation that includes the effects of the exchange of energy during phase changes and the dissipation associated with the air–hydrometeor drag. Similar to the treatment of momentum, the assumption that the hydrometeor's thermal energy equation is dominated by the diffusion of its internal energy to the moist air yields a thermal energy balance that can be used to diagnose an equilibrium or terminal temperature for the hydrometeors. Then the internal energy equation for the moist air includes the heating of the hydrometeors. Sections 7a and 7b describe prognostic equations for the total pressure field and for the equivalent potential temperature, respectively. Section 7c derives an equation for the total entropy of the moist air if the flow is multitemperature and describes the irreversible sources of entropy. Section 8 deals with vorticity and derives an equation for the virtual potential vorticity. Section 9 concludes with a discussion of the assumptions made. Though the symbols used are described in the text as they are introduced, appendix A presents a symbol list for the convenience of the reader. Appendix B develops an anelastic set of equations that eliminates acoustic waves. The Boussinesq set is also described.

2. General approach and the conservation of dry air

We consider a fixed volume V in space with surface S. Let χ describe the amount of a flow property per unit volume. The change in the property in the volume V is given by the equation

 
formula

where uχ is the velocity with which χ is being transported, and χ̇ describes the rate at which property χ is being created. Using the divergence theorem and noting that the volume V is arbitrary, we have

 
formula

For example, letting χ be the density of dry air ρa with velocity ua yields the traditional continuity equation

 
formula

where the material derivative is defined following a dry air parcel

 
formula

Property χ may move with a velocity uχ that differs from the dry air velocity ua. Let uχua + vλ where vχ is the velocity of χ relative to that of the dry air. Then our conservation statement for χ may be written as

 
formula

Our equation states that the amount of flow property per unit mass of dry air χ/ρa will change following the dry air if there is a source of χ or if there is a convergence of χ moving relative to the air parcel. The latter process we call fallout because we anticipate that there will typically be divergence of χ relative to the dry air. In the sections to follow we apply our conservation relation (2.6) to the water substance, momentum, and energy of the flow, noting that if χ is the sum of quantities with different velocities, then the fallout term is the sum of the fallout for each quantity.

Despite the mathematical rigor in the derivation of the conservation statement (2.6), the assumption of a continuum hypothesis involves some physical simplifications. Three simplifications (e.g., Crowe et al. 1998) are noted here. The first is that the flow variables represent volume (Crowe et al. 1998) or ensemble (Drew and Passman 1998) averages. For example, the air velocity ua contains not only macroscale variations but also microscale variations around each droplet and particle. If included these variations would add microscale eddy flux divergences to the prognostic equations and necessitate closure assumptions. In practice, such fluxes could be absorbed into the formulation of the subgrid-scale processes parameterized in numerical solutions of the equations.

The second simplification of the continuum hypothesis is the error made in the definition of the density variables. For example, the moist air density ρm is underestimated because it neglects the fact that the hydrometeors comprise a certain fraction αhρl/ρdrop + ρi/ρpart of the volume of space. Here, ρl and ρi are the continuum density of the distribution of the liquid water and ice, and ρdrop and ρpart are the densities of a droplet and particle, respectively. Thus the true air density is ρm/(1 − αh). Because the hydrometeor volume fraction αh is typically O(10−5) or less in the atmosphere, this error is an acceptable one.

The third simplification of the continuum hypothesis is the neglect in (2.1) of the presence of drops and particles (see Fig. 1) that lie on the boundary of any given control volume V. Crowe et al. (1998) justify this neglect by invoking the principle of complementarity for which each boundary drop has its complement and that their net effect is equivalent to one interior drop. Here, we note that the area fraction σh of any such boundary drops is small [σh ∼ (αh)2/3 = O(10−10/3) ≪ 1] and the neglect of such drops is reasonable.

Fig. 1.

Schematic diagram of volume V with boundary and complementary drops on its bounding surface S.

Fig. 1.

Schematic diagram of volume V with boundary and complementary drops on its bounding surface S.

Lastly we note that we make the traditional assumption (e.g., Hauf and Holler 1987) that the thermodynamic systems of interest are infinite in the sense that the ratio of surface to volume contributions is small. Then, say, the entropy of the liquid water is that of a single large drop and any contribution to the entropy associated with a set of smaller drops (with the same net mass) is ignored.

3. Continuity equations for water

We treat the moist air as a binary fluid (e.g., Bird et al. 1960) composed of dry air and water vapor, and allow for the diffusion of one component relative to the other. The equation for the conservation of the water vapor with density ρυ and velocity uυua + vυ may be found by letting χ = ρυ in (2.6). Then the equation for the vapor mixing ratio rυρυ/ρa is

 
formula

where the first two terms on the right side represent the loss of water vapor due to condensation and deposition, respectively. The third term is the diffusion of the vapor relative to the dry air:

 
formula

Following Pruppacher and Klett (1997, 502–503). we ignore the effects of thermal and pressure diffusion but allow for that due to a gradient in concentration. We have therefore ignored the effects of diffusiophoresis. Following Bird et al. (1960, section 16.1), the diffusion velocity of water vapor relative to air is vυ = −Dυ∇ ln rυ, where the vapor diffusivity Dυ has a value of 0.211 cm2 s−1 at STP and increases with temperature and inverse pressure. Thus the diffusive effects are significant only on the microscale. For example, if the e-folding length scale of the vapor is 1 cm or larger, the diffusive velocity is 0.2 cm s−1 or less.

In contrast, fallout may be significant for the liquid and ice phases because their velocities can differ substantially from that of the dry air. Letting χ = ρl, the density of the liquid water phase with velocity ul, in (2.6), we find an equation for the liquid water mixing ratio, rlρl/ρa,

 
formula

where the liquid water fallout is defined as the divergence of the flux of liquid water relative to the dry air parcel:

 
formula

and vl is the liquid water velocity relative to the dry air such that ulua + vl. Similar relations hold for the ice phase

 
formula

where deposition and freezing are sources of ice and the ice fallout is defined analogously to (3.4) using the ice velocity uiua + vi. The sum of the three continuity equations [(3.1), (3.3), and (3.5)] for water substance states that, relative to the dry air parcel, the loss of water is due to the net fallout of the liquid and solid water and to the diffusion of water vapor.

We assume that the rates of condensation, deposition, and freezing are known from cloud microphysical considerations. Expressions for the hydrometeor velocities relative to that of the air are given in section 5.

4. Ideal gas law for moist air

The total pressure p of the air parcel of temperature T is

 
formula

where R is the ideal gas constant for dry air. The factor of ɛ = 0.6220 is the ratio of the mean molecular weight of water vapor to that of dry air. Equation (4.1) is a consequence of the assumption of ideal gas behavior for both the air and the vapor, and of Dalton's law. The equations of state for the dry air and water vapor are

 
formula

respectively. Here pa and e are the partial pressure due to dry air and water vapor, respectively.

5. Momentum equation

The theory of multicomponent fluids (e.g., Bird et al. 1960; Landau and Lifshitz 1959) indicates that the momentum of the moist air ρaua + ρυuυ is transported by the mass-averaged velocity u,

 
u
ρauaρυuυρm
(5.1)

where ρmρa + ρυ, is the density of the moist air. Then the momentum per unit volume is taken as χ = ρmu + ρlul + ρiui. The external forces acting on any volume of cloudy air are the total ambient pressure gradient force, frictional forces, and the force due to gravity. Then the statement for the conservation of momentum is, using (2.6),

 
formula

where u is the total momentum per unit mass of dry air. The left side is the time rate of change of u following an air parcel. Changes in u following an air parcel are due to the sum of the pressure gradient force, divergence of the viscous stress tensor of moist air σ, gravity, and momentum fallout terms on the right side. The latter is

 
formula

It is convenient to rewrite the fallout terms. For example the liquid momentum fallout is

 
formula

and similar expressions hold for the ice fallout and diffusion fallout terms. The air-hydrometeor drag forces are not present in (5.2) because they are internal forces. For example, by Newton's third law, the sum of the drag force exerted on the drops by the air plus that exerted on the air by the drops is zero.

In general, the rate of change of momentum may be written as the sum of a contribution due to the rate of change of velocity and one due to the rate of change of mass. Then (5.2) becomes

 
formula

Note the change in the material derivatives on the left side. We have introduced material derivatives moving with the moist air, liquid water, and ice velocities. For example,

 
formula

and similar expressions hold for the liquid drops and ice particles. We now use the fact that the velocity of the moist air is, to a very good approximation, the velocity of the dry air

 
formula

where the approximation is justified based on the smallness of both the vapor mixing ratio rυ ≪ 1 and the diffusion velocity |vυ| ≪ |ua|.

We use (5.6) and simplify the term in braces in (5.5) using the continuity equations to yield

 
formula

where

 
formula

represents the momentum exchange associated with changes in phase. The physics of this exchange is discussed later in this section.

In order to write a momentum equation for the air we first introduce the equations of motion for the liquid drops

 
formula

and ice particles

 
formula

where the density is the mass of the particle divided by its volume. The pressure gradient force, gravity, and the drag force per unit mass exerted by the air drive the hydrometeors' motion. Here the viscous decay times τυl and τυi are functions of the shape of the hydrometeor and the relative Reynolds number (e.g., Pruppacher and Klett 1997, chapter 10) that is based on the velocities of the hydrometeors relative to the air. Because this relative velocity is the difference between the air velocity and that of the hydrometeor, it is Galilean invariant. Note that the drag contains a contribution by the microscale pressure field that surrounds the hydrometeors. [Rigorously, the drag forces in (5.8) are proportional to the hydrometeor velocity relative to the moist air velocity u. Here we use (5.6) and write them relative to the velocity of the dry air.]

Substituting the relations (5.8) into the total momentum equation (5.7a) yields a momentum equation for the moist air

 
formula

where

 
formula

represents the net microphysical momentum forcing of the moist air. The air parcel is subject to a pressure gradient force that is the ambient pressure gradient force (reduced by the fraction αh), gravity, and a force equal but opposite to the drag force exerted on the hydrometeors by the air. The latter is a consequence of Newton's third law.

The traditional approximation is to assume that the hydrometeors' motion is that of the air plus a terminal velocity. The rationale for this assumption follows. If the viscous decay time is much less than the dynamic timescale τD, the accelerations on the left side of the hydrometeor momentum equation (5.8) may be ignored yielding a force balance for the liquid drops:

 
formula

and one for the ice particles:

 
formula

Figure 2 presents a schematic diagram of this balance. Because the density of the droplet is much greater than that of air, the pressure gradient force per unit mass is much smaller in magnitude than gravity and the force balance is primarily between the drag and gravity. The hydrometeor's velocity relative to the air may be diagnosed from (5.11). Using these drags in (5.9) yields

 
formula

where the fourth term on the right is the hydrometeor drag. Our derivation demonstrates that this loading represents only part of the drag force exerted by the hydrometeors on the air. Note that (5.12) follows directly from (5.7) if one assumes that the inertia of the hydrometeors is negligible.

Fig. 2.

Schematic force balance on a hydrometeor. The viscous drag per unit mass D balances the force per unit mass due to the pressure gradient −(1/ρdrop)∇p and gravity g. The drag is in the opposite direction to the hydrometeor velocity relative to that of air vl

Fig. 2.

Schematic force balance on a hydrometeor. The viscous drag per unit mass D balances the force per unit mass due to the pressure gradient −(1/ρdrop)∇p and gravity g. The drag is in the opposite direction to the hydrometeor velocity relative to that of air vl

If one ignores the dynamic pressure gradient forces in the expression for the drag force, then υl is the vertical terminal velocity −VTlk that is observed in the laboratory:

 
formula

where ps is the static pressure. Substituting into the momentum equation yields

 
formula

where the pressure is the sum of a static and dynamic contribution p = ps + pd. But the hydrometeor volume fraction αh is O(10−5) or less and may be ignored in (5.14). Then the momentum equation is identical to (5.12) but with the velocity departures given by the terminal fall velocities:

 
formula

where VTl and VTi are the (positive) terminal fall speeds of the liquid and ice particles, respectively.

The last terms in (5.15) [and (5.12)] require some discussion. They accurately describe the exchange of momentum during a phase change. Consider the case of the condensation of water vapor. The Lagrangian vertical momentum equation in the absence of forces is

 
formula

where we have used the vapor continuity equation to write the last equality. To understand this equation consider the one-dimensional motion of a body of mass m subjected to no forces. Its equation of motion, representing the conservation of momentum, is d(mw)/dt = 0, where w is velocity. Alternatively we have an equation dw/dt = −(w/m) dm/dt that is formally equivalent to (5.16). As an example of (5.16), the change in the air velocity Δwa due to the condensation of ρa Δrl kilograms of liquid drops per unit volume is

 
formula

where rυ0 is the parcel's initial mixing ratio. The condensation produces an upward motion of the air as the drops fall at their terminal velocity. As an example, let rυ0 be 10−2 and assume all the vapor condenses to form 500-μm drops with a terminal speed of 2 m s−1, then the upward velocity of the air would change only 2 cm s−1. Conversely, for the same mixing ratio, the change in air velocity would be 30 cm s−1 downward as hail with a terminal speed of 30 m s−1 sublimates.

The final expression for the change in velocity of the moist air with density ρm = (1 + rυ) ρa is, using (5.12),

 
formula

where

 
rlrigphase
(5.19)

is the approximate momentum forcing of the moist air associated with the hydrometeors. We have written the equation relative to a rotating frame of reference with gravity redefined to include the centrifugal effects associated with the rotation.

In contrast, the momentum equation advocated by Ooyama (2001) and Schubert et al. (2001) for cloudy air with hydrometeors moving at their terminal velocity is, in the present notation,

 
formula

where ρt = ρa (1 + r) is the total density. Comparison of (5.20) and (5.18) indicates that (5.20) overestimates the inertia on the left side of the equation, neglects the velocity change due to a phase change, and includes a spurious advection term. As shown in section 8, this overestimate of the inertia leads to an incorrect solenoidal term in the vorticity equation that affects their choice of potential vorticity. Lastly the use of (5.18) and (5.11) produces no contradiction (cf. Ooyama 2001, p. 2077) regarding the specification of the hydrometeor velocities and the lower boundary condition.

6. Thermodynamic energy and internal energy equations

The total thermodynamic energy per unit volume is χ = ρaea + ρυeυ + ρlel + ρiei. Here, the total energy is the sum of the kinetic and internal energies. For example, ea = ka + ia. The sources of energy are that due to work and that due to radiative heating and the diffusion of heat. Then the energy equation is

 
formula

The left side is the time rate of change of the total thermodynamic energy per unit mass of dry air following an air parcel. The right side describes the sources of thermodynamic energy due to work, radiative heating, diffusion, and fallout. The rate of work done by the external forces is

 
formula

The work associated with the air-hydrometeor drag forces is not present here because they are internal forces. The first term is the work done by the pressure field associated with expansion of the moist air. We take the hydrometeors to be incompressible and ignore the work done associated with a change in hydrometeor size or shape. The next two terms are the rates of work done by the pressure gradient force and by gravity. The last term is the rate of total work done by the stress tensor. Note that a change of phase does not produce a change in the total thermodynamic energy. Instead, the phase change will produce an exchange of the energy among the constituents.

Here rad is the total radiative heating rate per unit mass of dry air and

 
formula

is the heating rate due to transport of energy by thermal conduction of the moist air with conductivity kT and of that due to enthalpy transport associated with the interdiffusion of mass (e.g., Bird et al. 1960). The latter is typically ignored in cloud microphysics (Pruppacher and Klett 1997, p. 507). Here we assume that the enthalpy of the water vapor is cT. The fallout of energy is

 
formula

where the mean energy for the moist air is e ≡ (ea + rυeυ)/(1 + rυ). Equation (6.4) describes the convergence of liquid and solid water energies into the dry air parcel as well as that due to the diffusion of water vapor. The change in energy relative to the dry air parcel is the sum of the energy lost due to mass fallout from the parcel plus that due to advection of energy at the liquid and ice velocities.

An alternative representation of the energy equation collects the terms associated with phase changes onto the right side of the equation and uses the expressions for the energy fallout (6.4) along with the continuity equations. The equation becomes

 
formula

where

 
formula

represents a modification of the transport heating due to the diffusion of water vapor, and

 
formula

represents the total energy change associated with a change in phase. Note the change in the material derivatives on the left side of (6.5).

An equation for the internal energy may be found by dividing the total thermodynamic energy into contributions due to kinetic and internal energy. Equations for the kinetic energy may be derived from the momentum equations (5.8) and (5.9) [but without the approximation (5.6)] in the usual manner. Substituting these results into (6.5) we have

 
formula

where

 
formula

is the rate of dissipation of kinetic energy by the frictional and air-hydrometeor drag forces. Note that this term is positive definite and always acts to increase the internal energy. The presence of hydrometeors with a mixing ratio of 10−2 and a terminal fall speed of 10 m s−1 (implying a viscous time τυVT/g ∼ 1 s) leads to a warming rate of 3.6 K h−1 for the air. Pauluis et al. (2000) note the importance of this dissipation in the tropical atmosphere and estimate it to be 2–4 W m−2. In contrast the frictional dissipation of the kinetic energy is only 0.15 W m−2 for large shears of magnitude 1 s−1.

The change in internal energy due to a phase change is

 
formula

This result indicates that the heating due to a change in phase includes the change in internal and kinetic energy. Because the terminal fall speeds are no greater than 50 m s−1, the kinetic energy terms in (6.10) may be ignored. Note that (6.10) contains the rate of phase change multiplied by the difference in the internal energy between the two phases. Internal energy appears here rather than enthalpy because (6.8) is an internal energy equation and not an enthalpy equation.

The change in internal energy per unit mass is defined as the product of the specific heat capacity and the temperature change. The internal energies of the water substance are related to each other by

 
formula

where cl and ci are the specific heats for the liquid drops and ice particles, respectively. These relations follow from integration of Kirchhoff's relations,

 
formula

from absolute zero to the air temperature T. Here the enthalpy of vaporization lυ, sublimation lS, and fusion lf, are functions of temperature and c = cυυ + R/ɛ is the specific heat at constant pressure for water vapor. The integral terms in (6.11) take into account the fact that the hydrometeors need not be at the temperature of the air. Here, cυa and cυυ are the specific heats at constant volume for the dry air and water vapor, respectively.

In order to write an internal energy equation for the air we introduce the thermal energy equations

 
formula

for the drops and particles, respectively. The temperature of a hydrometeor can change due to changes in phase, radiative processes, and conductive exchange of heat with the air. The changes due to a phase change are proportional to the rate at which the mass of water condenses (cond), freezes (reez), and deposits (dep), onto the drop or particle. We assume that these rates are known from microphysical considerations. Here ndrop and npart are the number density of the particles and droplets, respectively. The thermal conductive timescales τci and τcl for the ice and liquid are functions of the relative Reynolds number and hydrometeor shape (Pruppacher and Klett 1997, chapter 13).

An equation for the air temperature follows from (6.8) and (6.13). We find

 
formula

where cυm is the specific heat of moist air at constant volume, and the total heating of the moist air m is the total heating q̇, less that of the hydrometeors:

 
mrllrii
(6.15)

In a manner similar to the derivation of the momentum equation for moist air, we assume the conductive/ventilative decay time is much less than the dynamical timescale. Then the left side of the hydrometeor thermal energy equations (6.13) may be set to zero yielding thermal energy balances of the form

 
formula

As a consequence the approximate form of the internal energy equation for moist air is

 
formula

where the total heating rate

 
raddifphasedis
(6.18)

is the sum of the heating associated with radiative processes, with conduction, with phase changes, and with viscous dissipation. Comparison of the temperature equation (6.14) and its approximation (6.17) indicates that the approximation (6.16) has “loaded” the hydrometeor heating onto the moist air in analogy with the loading of the hydrometeor forces in the momentum equation. [Note that (6.17) follows directly from (6.8) if one assumes that the thermal inertia of the hydrometeors is negligible.] One consequence of this thermal loading is the neglect of the heat transport associated with precipitation. This neglect implies a spurious heating of the atmosphere given by ρwclPΔT, where P is the precipitation rate and ΔT is the temperature difference between the surface and the location of the condensation. For a precipitation rate of 4 mm d−1, the columnar heat source is 4 W m−2 for a temperature difference of 20 K.

7. Pressure, virtual potential temperature, and entropy equations

a. Pressure equation

An alternative to the thermal energy equation that describes the rate of change of the temperature field is an equation for the total pressure. We start from (6.14) divided by T and use the equation of state (4.1) to eliminate the temperature field, and then use the continuity equation for water vapor. The result,

 
formula

is a prognostic equation for the total pressure field. Here, cυm = cpmRm is the specific heat of moist air at constant volume, Rm = R(1 + rυ/ɛ) is the ideal gas constant for moist air and cpm = cpa + rυc is the specific heat of moist air at constant pressure. We note that (7.1) is based on relation (6.14); the approximate equation for the pressure is found, using the approximation (6.17), to be identical to (7.1) but with the total heating (6.18) in place of the heating of the moist air (6.15).

b. Virtual potential temperature equation

The virtual potential temperature is

 
formula

where Tυ is the virtual temperature, p00 = 1000 mb, and ρm = ρa(1 + rυ) is the density of moist air. Because

 
formula

we may use the pressure equation (7.1) and the continuity equations to find an equation for the virtual potential temperature. The result is

 
formula

where

 
formula

is the source term for the virtual potential temperature. We note that the two quantities in the braces are always nonnegative. The first quantity is O(10−3) and vanishes exactly if one ignores the effect of moisture on the specific heats. The second is O(10−1). Heating and flow divergence will increase the virtual potential temperature while a loss of water vapor will decrease it.

c. Entropy equation

The rates of change of the entropy of dry air and of water vapor are given by

 
formula

respectively, where we have used the relation (4.2b) for the vapor pressure in deriving (7.7). Using the equation of state for dry air (4.2a), the dry air pressure term may be rewritten in terms of the temperature and density of the dry air. The result is

 
formula

where Rm = R(1 + rυ/ɛ) is the ideal gas constant for moist air. Then (2.4) and (6.14) imply

 
formula

or, after some rearranging, we obtain an entropy equation for the moist air:

 
formula

where the first term on the right side represents the entropy source due to heating of the moist air and the second term is the source due to a loss in water vapor. The last three terms are sources of entropy related to the diffusion of the water vapor relative to the dry air.

We next obtain an equation for the total entropy. The entropy equations for the hydrometeors are given by (6.13) divided by their respective temperatures. The specific entropies are defined by

 
formula

for the hydrometeors, and by

 
formula

for the dry air and water vapor. Here, the common constant s0si(Ttp) is the entropy of ice at the triple point (denoted by a subscript tp) where etp ≡ 6.11 mb and Ttp ≡ 273.16 K. A total entropy equation follows from (7.10), (6.13), and (6.10). One finds

 
formula

where we have ignored the kinetic energy terms in (6.10). The four terms on the right side of (7.13) describe the sources of entropy. The first is that due to radiation,

 
formula

while the last is that due to fallout,

 
formula

Both (7.14) and (7.15) have a straightforward physical interpretation.

The entropy change due to diffusion is

 
formula

The first three terms describe a conductive and diffusive transport of entropy. The last four terms are positive definite and describe irreversible increases in entropy due to conduction and mechanical dissipation of kinetic energy in the moist air and conduction between the air and the hydrometeors.

It is of interest to compare the relative magnitudes of these entropy sources. Consider droplets with a mixing ratio of 10−2, a terminal fall speed of 10 m s−1 (implying a viscous time τυVT/g ∼ 1 s), and a conductive timescale of 100 s that are 10°C cooler than the air at 290 K. Then the rate of entropy production due to the dissipation by the air-hydrometeor drag forces is 3.4 × 10−3 W kg−1 K−1 and that, due to air-hydrometeor conduction, 2.1 × 10−3 W kg−1 K−1 is comparable in magnitude. In contrast, the rate due to the conduction of heat in the moist air with large thermal gradients of 1 K m−1 is 2.4 × 10−7 W kg−1 K−1, and that due to a radiative warming rate of 1 K day−1 is 4.0 × 10−5 W kg−1 K−1. Both are negligible compared to the air-hydrometeor contributions.

The entropy production due to phase changes is

 
formula

where

 
formula

describe the irreversible entropy changes associated with a phase change. In deriving (7.18) we have taken the specific heats for the hydrometeors to be constants in (6.11). We assume that the condensation and deposition occurs under saturated conditions in thermal equilibrium. Then the entropy production is zero in these cases. In contrast, evaporation, sublimation, and melting/freezing can occur in unsaturated conditions with a temperature difference between the hydrometeor and the air. Then an entropy change is possible.

Consider the case of the evaporation of droplets. Noting that lυ(T)/T = s*υ(T) − sl(T), where the asterisk denotes saturated conditions and assuming small temperature differences, we find that the first and fourth terms in (7.17) may be written

 
formula

where Hl is the relative humidity relative to liquid water. In the absence of temperature differences, (7.19) indicates there is a nonnegative source of entropy (Emanuel 1994) when there is evaporation (cond < 0) of droplets in unsaturated air (0 < Hl < 1). The second term in braces increases this entropy source in the nonisothermal case. However the third term, representing the difference in the decrease in the entropy of the drops during evaporation and the increase in the entropy of the air, can alter this conclusion. For example, for drops 20°C cooler than air at 290 K with a relative humidity of 75%, the first term is −132 J kg−1 K−1, the second is −10 J kg−1 K−1, but the third is 638 J kg−1 K−1, and the net effect is a decrease in entropy during evaporation. A similar argument may be made for the terms in (7.17) associated with deposition and freezing.

It is important to note that the derivation of the entropy equation (7.13) has been accomplished starting with the thermal energy equations (6.13) and (6.14). It is incorrect to derive an entropy equation by employing the formalism of section 2 to the total entropy because there is no a priori statement of entropy conservation. Such an approach does not yield the correct result nor does it give guidance on the correct formulation of entropy sources, such as the diffusion term (7.16).

8. Vorticity and potential vorticity equations

The vorticity equation is the curl of the velocity equation (5.9)

 
formula

where we have included rotation and neglected friction and the fractional volume of the hydrometeors. Here ωa = 2Ω + ∇ × ua is the absolute vorticity. Using the continuity equation for moist air in the form

 
formula

to eliminate the velocity divergence yields

 
formula

Introducing the virtual potential temperature equation (7.4) with the approximation (5.6) yields a potential vorticity equation of the form

 
formula

where the virtual potential temperature is

 
formula

Because the virtual potential temperature is only a function of the total pressure and the density of moist air, the baroclinic term vanishes identically. The sources of potential vorticity in (8.4) are associated with sources of virtual potential temperature, momentum, and moist air. In the absence of radiative heating and phase changes, the potential vorticity equation reduces to

 
formula

for nondivergent flow and indicates that there is a possible source of virtual potential vorticity associated with the drag of the hydrometeors with density ρh = ρl + ρI. We note that the virtual potential temperature reduces to Ertel's potential vorticity in the absence of water vapor and hydrometeors.

The crucial difference of the present analysis with that of Schubert et al. (2001) is that the solenoidal term in (8.1) is correctly given as the cross product of the gradients of the density of the moist air and the total pressure. Schubert et al. ignore the velocity difference between the hydrometeors and the air and obtain a solenoidal term involving the total density (i.e., the density of the moist air plus that of the hydrometeors). In a manner similar to that in Schubert et al., it can be shown that an invertibility principle holds. Specifically, knowledge of the virtual potential vorticity (8.5) along with the hydrometeor density field ρh enables one to solve for p, ρm, θυ, Tυ, and the horizontal winds when the flow is in hydrostatic and geostrophic or gradient wind balance.

9. Conclusions

We have formally derived the equations for cloudy air using the theories of multicomponent fluids and for multiphase flows. The equations require expressions for the radiative processes and for the microphysical processes such as the condensation rate and equations describing the change in hydrometeor temperature and momentum. The equations may be readily extended to include a distribution of water droplets and ice particles. The general set of equations consists of the equations of state (4.2), the continuity equations [(2.4), (3.1), (3.3), (3.5)], the momentum equations [(5.8a), (5.8b), (5.9)], and the internal energy equations [(6.13a), (6.13b), (6.14)]. If one ignores the inertia of the hydrometeors, then the three prognostic equations for momentum are replaced by a prognostic equation for the moist air velocity (5.18) and diagnostic ones for the hydrometeors (5.11a) and (5.11b). If one ignores the thermal inertia of the hydrometeors, then the three prognostic internal energy equations are replaced by the prognostic equation for the moist air temperature (6.17) and diagnostic ones for the hydrometeors (6.16a) and (6.16b). We note that the prognostic equation for the air temperature (6.14) may be replaced with that for another thermodynamic state variable, for example, the total pressure equation (7.1), that for the virtual potential temperature (7.4), or that for entropy (7.10) or (7.13). The equations lead naturally to (8.4) as the correct extension of Ertel's potential vorticity for a cloudy atmosphere.

In addition to the simplifications implicit in the continuum hypothesis we made two explicit assumptions, namely, that the viscous and conductive timescales for the transport of momentum and heat from a hydrometeor is small compared to any dynamical timescale of interest. We now discuss these timescales. The dynamical timescale τD is estimated to be the vertical cloud scale H divided by the updraft speed W. Because H is about 2 km or greater and W is about 50 m s−1 or less, the dynamical timescale is 40 s or greater.

The viscous timescale may be expressed in terms of the terminal fall speed using (5.13). Substituting the hydrostatic relation for the pressure gradient force indicates that the viscous timescale is approximately the terminal fall speed divided by the acceleration due to gravity. The largest fall speeds (Pruppacher and Klett 1997, p. 420) are about 10 m s−1 for large raindrops (radius ∼0.4 cm) and about 10 to 50 m s−1 for large hailstones (radius ∼5 cm) (Pruppacher and Klett 1997, p. 444). Thus the viscous timescale is the order of 5 s or less and is considerably smaller than any dynamic timescale of interest in the evolution of a cloud. Thus the approximation (5.11) is justifiable.

The thermal response time is related to the viscous response time in the low Reynolds number regime by τT = (3 c Pr/2 cpa) τυ, where Pr = 0.72 is the Prandtl number for dry air and c is the specific heat for either the liquid or ice water. Then τT is 4 τυ and 2 τυ for small drops and particles, respectively, and the terminal temperature approximation (6.16) is justifiable. Pruppacher and Klett (1997, p. 546) estimate thermal response times of about 10 s for drops of radius 2 mm. Thus (6.16) for the larger drops is a more questionable assumption. Caplan (1966) also makes this point.

The case of large ice particles is more problematical. Figure 3 presents viscous and thermal response times for hail as a function of radius. The results are based on the appendices in Rasmussen and Heymsfield (1987) that detail representative terminal fall speeds and mean thermal ventilation coefficients for spherical particles. The figure indicates that the thermal response time is comparable to or larger than the dynamical timescale for hail of radius 6 mm or larger. The parameter settings in Fig. 3 are for surface values at 20°C with an air density of 1.22 kg m−3 and an ice density of 103 kg m−3. Typically the timescales increase with height as the air density decreases. The timescales decrease with decreasing ice density. For a radius of 50 mm the thermal timescale is 466, 434, and 256 s for an ice density of 1, 0.91, and 0.45 × 103 kg m−3, respectively.

Fig. 3.

Thermal τc and viscous τυ timescales as a function of radius for spherical hail and graupel.

Fig. 3.

Thermal τc and viscous τυ timescales as a function of radius for spherical hail and graupel.

In closing we note that the analysis does not address the specification of the microscale eddy flux divergences that formally result from the averaging used in the multiphase theory. Such specification remains an outstanding problem in the theory of multiphase flows (e.g., Drew and Passman 1998).

Acknowledgments

The National Science Foundation under Grant ATM-9820233 provided partial support for this research. The author benefited from fruitful discussion with Craig Bohren, George Bryan, Jeffrey Chagnon, Jerry Harrington, Dennis Lamb, Wayne Schubert, and Hans Verlinde. I thank reviewer Olivier Pauluis for his constructive criticisms of the manuscript.

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APPENDIX A

List of Symbols

  • ci, cl  Specific heat of ice water and liquid water respectively

  • cpa, cpm, c  Specific heat at constant pressure of dry air, moist air, and water vapor

  • cυa, cυm, cυυ  Specific heat at constant volume of dry air, moist air and water vapor

  • D  Drag exerted on a hydrometeor

  • D  Depth scale of convection

  • Dυ  Vapor diffusivity

  • e  Water vapor pressure

  • ea, ei, el, eυ  Specific thermodynamic energy of dry air, ice water, liquid water, and water vapor

  •   Mass-weighted thermodynamic energy of moist air

  • etp  Water vapor pressure at the triple point

  • ėfallout, ėphase, ėwork  Specific energy loss due to fallout, phase changes, and work done by the external forces

  • g  Acceleration due to gravity

  • H  Characteristic cloud-height scale

  • Hl  Relative humidity

  • Hρa  Density scale height in the dry base state atmosphere

  • ia, ii, il, iυ  Specific internal energy of dry air, ice water, liquid water, and water vapor

  • ka  Specific kinetic energy of dry air

  • kT  Thermal conductivity of moist air

  • k  Vertical unit vector

  • lf, ls, lυ  Enthalpy of freezing, sublimation, and vaporization

  • cond, dep, freez  Mass rate of condensation, deposition, and freezing

  • ndrop, npart  Number density of liquid drops and ice particles

  • p, pa  Total pressure and dry air pressure

  • pas, pad  Base-state and dynamic pressure for dry air

  • pd, ps  Total dynamic and static pressure

  • p00  Reference pressure = 1000 mb

  • PVυ  Virtual potential temperature

  •   Total heating rate per unit mass of dry air

  • dis  Heating rate per unit mass of dry air due to dissipation

  • dif  Heating rate per unit mass of dry air due to the diffusion of heat, enthalpy, and water vapor

  • i  Heating rate of ice particles per unit mass of ice

  • l  Heating rate of liquid drops per unit mass of liquid water

  • m  Heating rate of moist air per unit mass of dry air

  • phase  Heating rate per unit mass of dry air due to phase changes

  • rad  Heating rate per unit mass of dry air due to radiation

  • radi  Heating rate of ice particles per unit mass of ice due to radiation

  • radl  Heating rate of liquid drops per unit mass of liquid due to radiation

  • radm  Heating rate of moist air per unit mass of dry air due to radiation

  • trans  Heating rate per unit mass of dry air due to thermal conduction and interdiffusion

  • r  Total mixing ratio of water substance

  • ri, rl, rυ  Mixing ratio of ice water, liquid water, and water vapor

  • cond, dep, freez  Rate per unit mass of dry air of condensation, deposition, and freezing

  • dif  Rate of diffusion of water vapor per unit mass of dry air

  • fallout  Fallout rate of water per unit mass of dry air

  • R, Rm  Ideal gas constant for dry air and moist air

  • s  Total entropy per unit mass of dry air

  • sa, si, sl, sυ  Specific entropy of dry air, ice water, liquid water, and water vapor

  • s0  Entropy of ice at the triple point

  • sdif  Rate of entropy increase per unit mass of dry air due to diffusion and transport

  • ·fallout  Rate of entropy loss per unit mass of dry air due to fallout and diffusion of water vapor

  • ·phase  Rate of entropy increase per unit mass of dry air due to phase changes

  • ·rad  Rate of entropy increase per unit mass of dry air due to radiation

  • S  Surface of volume V

  • T  Temperature of dry air and water vapor

  • Td, Ts  Dynamic and static base-state temperature

  • Ti, Tl  Temperature of ice water and liquid water

  • Ttp  Temperature at the triple point

  • Tυ  Virtual temperature

  • u  Total momentum per unit mass of dry air

  • ua, ui, ul, uυ  Velocity of dry air, ice particle, liquid drop, and water vapor

  • uχ  Velocity of constituent χ

  • u  Mass-weighted velocity of moist air

  •   Total momentum forcing due to microphysics

  • fallout  Momentum loss per unit mass of dry air due to fallout

  • m  Rate of momentum gain of moist air per unit mass of dry air due to hydrometeor drag and mass exchange due to phase changes

  • phase  Rate of loss of momentum per unit mass of dry air due to phase changes

  • vi, vl, vυ  Velocity relative to that of dry air for an ice particle, liquid drop, and water vapor

  • vχ  Velocity of constituent χ relative to the dry air

  • v  Mass-weighted velocity of moist air relative to that of dry air

  • V  Volume

  • VTi, VTl  Terminal fall speed of ice particle and liquid drop

  • wa  Upward air velocity

  • W  Characteristic updraft speed

  • αh  Volume fraction occupied by hydrometeors

  • δs, δs, δsil  Irreversible entropy change due to evaporation, sublimation, and melting

  • ɛ  Ratio of molecular weight of water to that of air

  • ρa, ρm, ρυ  Density of dry air, moist air, and water vapor

  • ρ̇υ  Rate of creation of water vapor per unit volume due to diffusion, evaporation, and sublimation

  • ρas, ρad  Base-state and dynamic density for dry air

  • ρdrop, ρpart  Density of liquid drop and ice particle

  • ρi, ρl  Density of ice phase and liquid phase

  • ρiw, ρlw  Density of ice and liquid water

  • ρh, ρt  Density of hydrometeors, total density

  • σ  Viscous stress tensor for moist air with components σij

  • σh  Surface fraction occupied by hydrometeors

  • θυ  Virtual potential temperature

  • θυs, θυd  Static and dynamic virtual potential temperature

  • Θ̇υ  Rate of production of virtual potential temperature

  • τci, τcl  Thermal conductive timescale for an ice particle and liquid drop

  • τD  Dynamical timescale

  • τυi, τυl  Viscous timescale for an ice particle and liquid drop

  • Ω  Rotation vector of the earth

  • ωa  Absolute vorticity of the air

  • χ, χ̇  Generic flow property per unit volume and its creation rate

APPENDIX B

The Anelastic Equations

This appendix develops an anelastic set of equations suitable for the description of deep moist convection relative to a reference state. The development modifies the formalism of section 2 and corrects the formulation of Bannon (1996) that mishandles the hydrometeor drag term when making the closure assumption.

The base state atmosphere, denoted with a subscript s, is a dry atmosphere in hydrostatic balance that satisfies

 
formula

where, for example, ρas is the density of the dry air in the base state atmosphere. Other multiply subscripted variables have similar interpretations. The total pressure field is then defined as the sum of the static pressure for dry air and a dynamic component pd:

 
ppaszpdx, y, z, t
(B.2)

and similar relations hold for the other thermodynamic state variables. Since the base state is dry, all moisture variables are dynamic components.

The moist anelastic equations developed here also require (Bannon 1996) that: (i) the buoyancy force is a major component of the vertical momentum equation, (ii) the characteristic vertical displacement D of an air parcel is comparable to the density scale height Hρa ≡ (−d ln ρas/dz)−1, and (iii) the horizontal variations of the thermodynamic state variables at any height are small compared to the static value at that height. For example, ρad(x, y, z, t)/ρas(z) ≪ 1. Similar inequalities hold for the other state variables. The important exception is that the densities of the water fields can undergo O(1) variations.

The continuity equation for the dry air (2.3) is approximated by

 
formula

The argument for this approximation is standard (e.g., Bannon 1996). Then the generic conservation equation (2.6) for compressible flow is replaced with its anelastic counterpart

 
formula

As a consequence of (B.4), the three continuity equations (3.1), (3.3), and (3.5) for water substance hold for anelastic flow provided we define the mixing ratio of the water constituents in terms of the reference air density. For example, rυρυ/ρas ≪ 1.

In the anelastic approximation we linearize the ideal gas law (4.1) to

 
formula

and the dynamic pressure pd depends on the total vapor content rυ. The equations of state are

 
formula

for the dry air and the vapor, respectively.

The momentum equation may be derived following the development in section 5 but with ρasua replacing ρaua in the definition of the total momentum when substituting for χ in (B.4). Then (5.18) becomes

 
formula

where r is the total water mixing ratio, and we used (B.1) to eliminate the static pressure gradient force and have ignored friction. We next neglect the inertia of the water vapor because rυ ≪ 1 and introduce the density scale height to rewrite the equation in final form as

 
formula

The kinetic energy equation associated with (B.8) is, using (B.3a),

 
formula

The hydrostatic relation for the base-state (B.1) can be written as cpaθasd(Ts/θas)/dz = −g, where the base-state virtual potential temperature equals the base-state potential temperature for dry air θυs = θas. This result holds because the base state is assumed to be dry. Then we can rewrite the virtual potential temperature equation in the form of a static energy equation

 
formula

where in the last term, we have assumed that the hydrometeor heating is loaded onto the air. Energy conservation between the kinetic energy equation (B.9) and (B.10) requires

 
formula

where we invoke the anelastic maxim (Bannon 1996) that the thermodynamics is slave to the dynamics. This assumption completes the derivation of the approximate equations. Note that the closure assumption (B.11) does not include the mixing ratios of the hydrometeors because their presence in (B.8) represents the hydrometeor drag on the air and it is inconsistent to include them in a definition of virtual potential temperature. Thus, the present closure corrects the error in (6.8) of Bannon (1996). The equation for the total energy of the air,

 
formula

indicates that the momentum exchange phase makes no net contribution to the total energy. In the absence of heating and phase changes, the only energy source of the air is the work done against the hydrometeor drag.

To form a potential vorticity equation we take the curl of the velocity equation (B.8) and use the anelastic continuity equation (B.3b) to eliminate the velocity divergence. Introducing the anelastic form of the virtual potential temperature equation (7.4) with approximation (5.6):

 
formula

where

 
formula

yields the anelastic counterpart to the potential vorticity equation (8.4) in the form

 
formula

It is straightforward to obtain a Boussinesq set of equations from the present anelastic equations by replacing the base-state density with a constant density and taking the limit as the density scale height tends to infinity.

Footnotes

Corresponding author address: Peter R. Bannon, Department of Meteorology, The Pennsylvania State University, University Park, PA 16802. Email: bannon@ems.psu.edu