## 1. Introduction

The equations governing fluid dynamics are fundamentally conservative, representing conservation of momentum, energy, and mass (Batchelor 1967). The equations of climate modeling, which are based on dynamical approximations to the governing equations of fluid dynamics, preserve the conservation properties to avoid introducing spurious sources or sinks of the conserved quantities (Lorenz 1967). The numerical solution to the equations of climate modeling requires the parameterization of processes that occur on scales smaller than can be represented by the discrete model grid. Physical processes occurring on these subgrid scales are important to the resolved energy and momentum budgets and necessarily respect the same conservation principles.

^{−2}in the global mean. Many models treat this energy transfer locally; that is, any resolved-scale kinetic energy tendency due to subgrid-scale momentum flux convergence or vertical diffusion is assumed to be balanced locally by a thermodynamic energy tendency of the opposite sign; that is,

*N*is the buoyancy frequency. Dynamics occurring on subgrid scales include convection and gravity wave propagation and are nonhydrostatic. Therefore, the framework must account for the interaction between two flows: a hydrostatically balanced resolved flow operating on long and slow spatiotemporal scales, and a nonhydrostatic subgrid-scale flow operating on short and fast spatiotemporal scales.

The tools of systematic multiple-scale perturbation theory from applied mathematics (Kevorkian and Cole 1981) are naturally suited for developing a framework to study the interaction of physical phenomena occurring on multiple spatial and temporal scales (i.e., the resolved and subgrid scales). Klein (2000) and Majda and Klein (2003) have shown how this theory can be used to systematically derive balanced models in the midlatitudes and tropics. Here, the goal is not to derive an asymptotic framework from first principles. Rather, the goal is to find an asymptotic framework that leads to the equations of interest and to use that framework to define a self-consistent treatment of energy and momentum in the context in which there is an imposed separation of horizontal length and time scales but vertical coupling within each column. This is the case of relevance to climate models. Note that the assumed time scale separation imposes statistical stationarity on the subgrid-scale processes.

Section 2 introduces the equations and relevant nondimensionalization. We introduce the multiple-scale framework in section 3, including the derivation of momentum, thermodynamic, and continuity equations for the resolved and subgrid-scale flow including the resolved-scale total energy budget. The horizontal space and time scale separation between the resolved and subgrid scales is used to define wave-activity conservation laws on the subgrid scale in section 4, which are used to close the interaction terms. In section 5 the subgrid-scale dynamics are reduced to a subset satisfying the anelastic constraint, which is an important regime for applications (most subgrid-scale parameterizations are formulated using the anelastic equations), and implications for the wave-activity conservation law closures are discussed. We show how the dissipation of subgrid-scale kinetic energy can be assured to lead to an increase in thermodynamic energy and to entropy production. The paper concludes with a summary and discussion in section 6.

## 2. Preliminaries

*ρ*,

*p*,

**v**,

*θ*, and

*π*are the density, pressure, velocity, potential temperature, and Exner function (

*π*=

*p*, where

^{κ}*κ*=

*R*/

*c*with

_{p}*R*being the dry gas constant and

*c*the specific heat at constant pressure), respectively;

_{p}**ẑ**is the unit vector in the vertical direction; and

**S̃**and

_{v}*S̃*are (dimensional) source–sink terms meant to represent the interaction with a turbulent microscale in which resolved-scale kinetic energy is ultimately dissipated and converted into thermodynamic energy (see section 4). Although a complete treatment of the problem will require treatment of the effects of moisture, as a first step we treat only dry dynamics.

_{θ}*ϕ*is the latitude on the tangent plane centered at

*ϕ*

_{0}and

**ŷ**is the unit vector in the meridional direction. In (5),

*M*is the Mach number, Fr is the Froude number, Ro is the Rossby number,

**S**and

_{v}*S*are the nondimensional versions of

_{θ}**S̃**and

_{v}*S̃*, and time is scaled advectively. The three nondimensional numbers—Rossby, Mach, and Froude—are defined as

_{θ}*H*, a length scale relevant for the mesoscale, rather than a planetary length scale. For the chosen reference length scale, the Mach and Froude numbers are equivalent (i.e., for

*H*=

*p*

_{ref}/

*gρ*

_{ref}then

*γ*=

*c*/

_{p}*c*so that

_{υ}*κ*− 1 = −1/

*γ*.

*p*and a mesoscale

*m*; that is,

**x**

*= (*

_{p}*H*/

*L*)

_{p}**x**,

*z*= (

_{p}*H*/

*H*)

_{p}*z*,

*t*= (

_{p}*H*/

*L*)

_{p}*t*,

**x**

*= (*

_{m}*H*/

*L*)

_{m}**x**,

*z*= (

_{m}*H*/

*H*)

_{m}*z*, and

*t*= (

_{m}*H*/

*L*)

_{m}*t*. We choose our scalings bearing in mind the dynamical models required on each scale and according to (2) and (3) we set

*H*/

*L*=

_{p}*t*/

_{p}*t*=

*ϵ*

^{2}, with

*ϵ*≈ 0.1,

*H*/

*H*= 1,

_{p}*H*/

*L*=

_{m}*t*/

_{m}*t*= 1, and

*H*/

*H*= 1, whence

_{m}*z*=

_{p}*z*=

_{m}*z*and we refer only to

*z*.

^{1}Given that time is scaled advectively and that

*t*/

_{m}*t*= 1, with

*t*∼

_{m}*O*(

*N*

^{−1}) according to (3), implies a reference velocity scale

*U*∼

*NH*and thus Fr

_{int}=

*U*/

*NH*∼ 1, where Fr

_{int}is the internal Froude number. According to the choice of the reference velocity scale, the Mach and the (external) Froude numbers in (7) are near unity. Given the multiple space and time-scale dependence of a field

*f*, the time derivative and total horizontal gradient are then

**v**= (

**u**,

*w*) and each field is expanded in

*ϵ*for example,

*t*and

_{m}**x**

*(but not*

_{m}*z*) with the property that

*f*, which is an ansatz to eliminate secular growth. Note that this corresponds to an area-weighted average. For the velocity, we adopt a mass-weighted average decomposition such that

_{p}*ρ*

_{p}**u**

_{p}=

*ρ*=

_{p}w_{p}*M*

^{2}; limits of this parameter are considered in section 5. In terms of the thermodynamics, the ansatz (11) implies a leading-order thermodynamic state that varies only on the slow and long planetary scales and acts as a background state for the mesoscale. The source–sink terms

**S**and

_{v}*S*are assumed to take the following forms:

_{θ}*=*

_{p}*U*/Ω

*L*∼ 1 is the usual planetary-scale Rossby number. Thus, it is clear that on the planetary scale, horizontal advection, vertical advection, and rotation are of the same order. The limit of small

_{p}*ϵ*

^{2}=

*H*/

*L*as represented in (12) is taken with

_{p}*M*, Fr, and Ro

*held fixed.*

_{p}## 3. Equations for the mesoscale planetary-scale interaction

*O*(1); that is,

*M*for Fr. Here and henceforth all velocities are leading order and superscripts are dropped unless otherwise indicated. The mesoscale dynamical equations are obtained at

*O*(

*ϵ*). The mesoscale horizontal and vertical momentum equations are

**∇**

*is the three-dimensional gradient on the mesoscale. Note that the ansatz (11) together with the assumed scaling has not eliminated compressibility effects on the mesoscale [cf. the first term on the left-hand side of (18)].*

_{m}*O*(

*ϵ*

^{2}). To obtain them, we apply the mesoscale average to the

*O*(

*ϵ*

^{2}) equations and account for the mesoscale average applied to the

*O*(

*ϵ*) equations.

^{2}The horizontal momentum equation on the planetary scale is

**e**

_{z}= (

**e**·

**ẑ**)

**ẑ**, and is derived using (18). The planetary-scale potential temperature equation is

*ρ*and

_{m}*π*, respectively, and using (18). Note that (27) has been divided by

_{m}*M*

^{2}so that when calculating the total energy budget the pressure-work term on the planetary scale can be associated with the corresponding term in the planetary-scale kinetic energy budget. We choose to write the planetary-scale potential temperature equation in the above form so that mesoscale kinetic energy conversion terms can be identified. We note that the first two terms on the right-hand side of (27) can be associated with diabatic effects on the mesoscale because

*θ*[analogous to (28) and (29)]. Finally, the planetary-scale continuity equation is

_{m}It is clear from (26) and (32) that mesoscale fluxes of momentum, potential temperature, and pressure drive the planetary-scale flow, which obeys hydrostatic dynamics (15). The quasi-linear mesoscale [(17), (18), and (22)] is coupled nonlinearly through these eddy fluxes to the planetary scale. This interaction has implications for the planetary-scale energy and momentum budgets.

**u**

*times (31), is*

_{p}*z*.) It is clear from (33) that integrating over

*z*leads to conservation of zonal momentum.

**v**

*and taking a mesoscale average, we obtain the mesoscale kinetic energy equation on the planetary scale:*

_{m}**u**

*and adding it to*

_{p}*K*multiplied by (31):

_{p}*K*= |

_{p}**u**

*|*

_{p}^{2}/2. As we expect for hydrostatic dynamics,

*K*does not include

_{p}*w*. Planetary-scale kinetic energy is changed locally by pressure work and mesoscale momentum fluxes. Upon adding

_{p}*T*/

_{p}*κM*

^{2}times (31) to (32), we obtain the planetary-scale internal energy equation:

*=*

_{p}*z*/

*M*

^{2}, the planetary-scale potential energy equation can be calculated using (31) as

## 4. Understanding the interaction across scales: Wave-activity conservation laws

*A*is the wave-activity density and

**F**its flux, both being quadratic for linear dynamics, and

*D*is the wave-activity source–sink term. In the conservative case

*D*= 0. Note that after performing a mesoscale average, (39) becomes

Wave-activity conservation laws play a central role in the study of fluid dynamical disturbances to a specified background state. In the case of the large-scale circulation of the atmosphere, the Eliassen–Palm wave activity has been crucial to theoretical analysis (Andrews et al. 1987; Shepherd 2003; Vallis 2006). Wave-activity conservation laws can be generally derived using the Hamiltonian structure of geophysical fluid dynamics (Shepherd 1990). This framework should, in principle, allow one to consider general disturbances to a background flow, without making any Wentzel–Kramers–Brillouin (WKB)-type assumptions, and be extendable to finite amplitude. In the case of the dynamics derived in the previous section, these conservation laws can be derived in the usual way because the planetary scale acts as a horizontally and temporally homogeneous background flow for the mesoscale dynamics according to the ansatz (11). (The scales are, however, coupled in the vertical, as is assumed in the parameterization of subgrid scales in climate models.) In particular, the *x _{m}*,

*y*, and

_{m}*t*symmetries in the background flow lead to pseudomomentum and pseudoenergy conservation laws (Shepherd 1990).

_{m}Shaw and Shepherd (2008, hereafter SS08) derived wave-activity conservation laws for three-dimensional disturbances to a horizontally homogeneous background flow with disturbances governed by the anelastic or Boussinesq equations. In the current nomenclature, the mesoscale is considered the disturbance and the planetary scale is the background flow. The mesoscale dynamics derived in section 3 are not explicitly anelastic; however, the results of SS08 can be extended to the situation considered here. A detailed derivation can be found in appendix A. The anelastic form of the mesoscale fluxes and multiscale interactions, and their connection to the wave-activity conservation laws, is presented in the next section.

*x*and

*y*pseudomomentum fluxes with corresponding source–sink terms

The vertical wave-activity fluxes in (41) are important in driving the planetary scale. In particular, the pseudomomentum (41b) forces the planetary-scale momentum via (33), whereas the pseudoenergy (41a) contributes to forcing the total energy on the planetary scale via (38). The wave-activity conservation laws provide a means of relating the mesoscale fluxes in (33) and (38) to source–sink terms on the mesoscale. The other mesoscale terms on the right-hand side of (38) have already been related to source–sink terms on the mesoscale. In the absence of wave-activity sources–sinks, the subgrid-scales do not contribute to the planetary-scale budgets and thus (41a) and (41b) can be thought of as “non-acceleration” theorems (Charney and Drazin 1961; Eliassen and Palm 1961) for the effects of subgrid-scale disturbances.

*c*is the phase velocity in the direction of symmetry associated with

*A*

^{P}), which is derivable from Noether’s theorem (see SS08). This relationship holds for a monochromatic wave. However, because

*x*,

_{m}*y*, and

_{m}*t*, the quantities can be decomposed into spectra and the relationship holds for each wavenumber–frequency pair (with its own

_{m}*c*). In fact, the wave activities need not correspond to waves at all; we may just define

*c*as

*c*is the streamwise phase velocity. As noted by SS08, (43) is the generalization of the first Eliassen–Palm theorem (Lindzen 1990) to three dimensions with a veering background flow.

_{s}^{3}Relations (42), (41), and (43) are key to ensuring the consistency of energy and momentum conservation on the mesoscale. In particular, the relations (41) and (43) imply

Relation (45) represents the generalization of the second Eliassen–Palm theorem to three dimensions with a veering background flow and to dissipative dynamics. Lindzen (1973) extended the second Eliassen–Palm theorem to include diabatic forcing but assumed WKB conditions in the vertical. The plane-parallel version of (45) with **S _{v}**

^{m}= 0, in particular the first line and last equality, corresponds exactly to (8) in Lindzen (1973) and to (8.24) in Lindzen (1990). Here we have shown that the second Eliassen–Palm theorem can be derived systematically using the Hamiltonian structure of geophysical fluid dynamics in the general context of the interaction between a mesoscale (subgrid-scale) flow and a planetary-scale (resolved) flow. Furthermore, we have generalized the relation to three dimensions with a veering background flow, and the wave-activity source–sink terms take full account of the background vertical shear (there is no WKB-type requirement on the background flow).

*dV*=

*dx*vanishes under appropriate boundary conditions. Before applying the integral, we must partition the planetary-scale thermal source/sink term

_{p}dy_{p}dz*S*into the rate of work done by the mesoscale source/sink term

_{θ}^{p}**S**

_{v}^{m}and the contribution from planetary-scale external sources/sinks. In particular, we set

**d**

*is the symmetric shear stress tensor on the mesoscale with (*

_{m}**d**

*)*

_{m}*= [(*

_{z}*d*)

_{m}*, (*

_{xz}*d*)

_{m}*, (*

_{yz}*d*)

_{m}*] and*

_{zz}**S**

_{v}^{m}=

**∇**

*·*

_{m}**d**

*;*

_{m}*Q*is a mass-weighted diabatic source/sink term. The second term on the right-hand side involving

_{θ}^{p}**d**

*is recognized as the transfer of the turbulent shear stress by the mesoscale velocity. This partitioning isolates the external (radiative) heating from the internal heating due to kinetic energy dissipation. (To incorporate latent heat release would require a full treatment of moist processes to ensure total energy conservation). We choose not to introduce an explicit stress tensor formulation of*

_{m}**S**

_{v}^{m}so as to keep the framework general enough to admit different closures. (The closure will depend on the particular physical processes being parameterized.)

*dV*(assuming vanishing boundary conditions), and assuming

*Q*= 0 we obtain

_{θ}^{p}*Q*= 0.

_{θ}^{p}*κ*/

*π*, adding ln

_{p}θ_{p}*θ*multiplied by (31), and using (46). To ensure that the planetary-scale entropy increases as a result of exchanges of energy and momentum with the mesoscale, we require that the vertical integral of (48) be positive definite when

_{p}*Q*= 0—that is, when the atmosphere is a closed system (a condition that is required for the second law to hold). (In practice, ∫

_{θ}^{p}*κQ*/

_{θ}^{p}*T*< 0 balances the entropy production.) Under such conditions, ensuring the positivity of the right-hand side of (48) (with

_{p}dV*Q*= 0) ensures that there is an irreversible increase in planetary-scale entropy, consistent with the second law. The right-hand side of (48) can be related to the wave-activity conservation law source/sink terms using (45) such that entropy production requires

_{θ}^{p}## 5. Anelastic dynamics

**∇**

*· (*

_{m}*ρ*

_{p}**v**

*) = 0. Understanding how the planetary-scale momentum and total energy budgets are affected by an anelastic constraint on the mesoscale is relevant for climate models because the theoretical formulations of many state-of-the-art subgrid-scale parameterizations are based on the anelastic equations. A reduction of the planetary-scale momentum and energy budgets derived in section 3 can be made by neglecting those terms affected by the anelastic constraint. In particular, a reduced set of planetary-scale equations with mesoscale fluxes satisfying the anelastic constraint can be obtained by neglecting terms in (27) proportional to*

_{m}The neglect of these terms to obtain the reduced planetary-scale dynamics can be justified by considering the small Mach number limit of the dynamics presented in section 3. [Note that section 3 considered an *O*(1) Mach number.] It is clear from (18) that in the limit of small Mach number, the anelastic constraint **∇*** _{m}* · (

*ρ*

_{p}**v**

*) = 0 is recovered. In this limit, the integrity of the mesoscale thermodynamic equation (17) can be preserved if the stratification on the planetary scale is assumed to be*

_{m}*O*(

*M*

^{2}). A weak stratification is a well-known assumption of the anelastic equations (Lipps and Hemler 1982; Klein 2000). It is clear that these assumptions result in the neglect of terms proportional to

*θ*/∂

_{p}*z*=

*O*(

*M*

^{2}). As in the previous sections, its source is diabatic effects on the mesoscale.

Zeytounian (1990) derived the small-Mach number limit of the hydrostatic primitive equations, which he called the reduced hydrostatic primitive equations, by first applying the shallow atmosphere approximation (equivalent to *ϵ* ≪ 1 in section 3) and then applying the small Mach number limit (see his section 7.4). The small Mach number limit was applied to the hydrostatic primitive equations in pressure coordinates, and the leading-order pressure and enthalpy contributions were *O*(*M*^{2}). Our reduced planetary-scale dynamics are in agreement with (7.57) of Zeytounian (1990).

Interestingly, the anelastic constraint only affects the horizontal wave-activity fluxes; the vertical wave-activity fluxes are unchanged. A plausible reason for this is that the hydrostatic primitive equations admit only horizontally propagating sound waves, so any coupling with the mesoscale through compressible dynamics can only occur through the horizontal fluxes. Thus, the anelastic limit, which filters out sound waves, has no effect on the vertical fluxes, which are responsible for forcing the planetary-scale momentum and energy.

As in section 4, the wave-activity conservation laws provide a means of relating the mesoscale flux terms in (50a) and (50b) to mesoscale source–sink terms. [The other terms on the right-hand side of (50b) have already been related to source–sink terms on the mesoscale.] As discussed in the previous section, (53) is the generalization of the second Eliassen–Palm theorem for the case of anelastic dynamics.

*Q*= 0 is preserved under the anelastic constraint. Proceeding to the second law of thermodynamics, under the anelastic approximation (48) becomes

_{θ}^{p}*Q*= 0) requires

_{θ}^{p}**d**

*)*

_{m}*, the right-hand side of (55) can be rewritten according to*

_{z}*K*being a vertical diffusion coefficient. For

_{z}*K*> 0, the entropy production constraint is clearly satisfied in this case. The terms in the entropy production constraint can be associated with different subgrid-scale energy transfers. The first term on the left-hand side in the first line of (55), proportional to the wave-activity source–sink terms, is associated with nonlocal mesoscale energy transfer, the second with local transfer of mesoscale internal–potential energy to planetary-scale internal energy, and the final term with local kinetic energy transfer from the mesoscale to turbulence.

_{z}According to the first Eliassen–Palm theorem (43), upward propagation implies that momentum flux deposition must drag the background flow toward the phase velocity of the waves, such that the nonlocal transfer term in (55) is positive in the wave dissipation region. We obtain the same result from the requirement of entropy production, provided the local transfers are not positive. Our framework, however, shows that a complete statement requires consideration of the local transfers as well as of the wave source region (which is not generally taken into account in gravity wave parameterizations).

It is apparent from (55) that the sign of the various contributions to the entropy budget depends crucially on the structure of the background temperature *T _{p}*. If we consider the case of stable stratification on the planetary scale and assume that mesoscale mixing is downgradient, then

*T*are constrained by (55).

_{p}*Q*= 0. If the remaining nonlocal transfer term is positive-definite, then it is clear that the vertical exchanges of energy and momentum lead to a warming of the planetary-scale temperature (wave absorption warms the background), as argued by Lindzen (1990). On the other hand, if the term proportional to

_{θ}^{p}*c*in (53) and (45) is neglected, then an acceleration of the planetary-scale flow by momentum flux convergence leads to a decrease in entropy and local cooling, according to (57), in violation of the second law. This was the example considered in the introduction. Interestingly, the form of the thermodynamic energy tendency is not directly affected by the anelastic constraint on the mesoscale, although the sources of the tendency are [compare the left-hand sides of (53) and (45)].

_{s}However, the second law of thermodynamics is a global constraint and cannot be ensured from local considerations as in (57). For complete consistency, it is necessary to consider the entropy constraint (55), as discussed above. In this case the local transfers do not integrate out in the vertical column, as they do in the enthalpy budget.

## 6. Summary and discussion

By combining the theories of multiple-scale asymptotics and Hamiltonian geophysical fluid dynamics, we have derived a self-consistent (in terms of energy and momentum) theoretical framework for physical parameterization in climate models. We have derived energy (38) and momentum (33) equations for a nonlinear, hydrostatic resolved (planetary) scale, which include interactions with a quasi-linear, compressible or anelastic subgrid-scale (mesoscale) described by (17), (18), and (22). The temporal and horizontal spatial symmetries in the planetary-scale background flow for the mesoscale allow the construction of wave-activity conservation laws (A10), (A21), (B3), and (B6) on the mesoscale. These conservation laws are used to understand the fluxing of energy and momentum between scales and the ultimate dissipation of mesoscale kinetic energy, conversion to planetary-scale internal energy, and increase in planetary-scale entropy. In particular, they are used to relate mesoscale flux terms in the planetary-scale momentum and energy budgets to mesoscale source–sink terms, thereby providing generalized nonacceleration theorems.

The relationships between the subgrid-scale fluxes (45) and (53) derived here, which result from self-consistency in terms of energy and momentum conservation, are generalizations of the second Eliassen–Palm theorem and place strong constraints on the contributions of subgrid-scale fluxes to the resolved energy and momentum budgets. Any parameterization of subgrid-scale momentum fluxes must satisfy them to ensure the conservation of both energy and momentum and conform to the second law of thermodynamics. This includes wave drag, cumulus, and boundary layer parameterizations. In particular, entropy production is only guaranteed if (49) and (55) are satisfied. These relations also place strong constraints on parameterized fluxes and may be especially useful in cases where observational constraints are lacking. [Note that (45), (53), (49), and (55) are valid whether or not a phase velocity can be usefully defined.] For example, mixing by gravity wave breaking is constrained not only by energy considerations but also by the requirement of entropy production. Current treatments of kinetic energy dissipation that assume local conservation of energy by balancing a kinetic energy tendency locally by a thermodynamic tendency, as in (1), are incorrect according to (57c) and may lead to spurious sources/sinks of energy and a violation of the second law. [The local conservation formulation (1) can also be seen to be in error according to (46) because it neglects the second term on the right-hand side of the second equality, which involves a nonlocal transfer by the turbulent microscale flow.] Ensuring that particular parameterizations satisfy the entropy production constraint in the vertical column, (49) and (55), is the subject of future investigation.

While the motivation of the framework was the parameterization of subgrid-scale processes in climate models, numerical weather prediction models, which are generally of higher spatial resolution, must also parameterize the transfer of energy and momentum between the resolved and subgrid scales. Thus, the relationships between the fluxes derived here are also relevant for such models.

In the context of gravity wave propagation, Lindzen (1973) showed that the thermodynamic energy tendency due to gravity wave dissipation has the form (57b); however, a constant background wind was assumed in the derivation. In the context of gravity wave drag parameterization, the Lindzen (1981) parameterization does not represent the thermodynamic energy tendency according to (57b), although Becker and Schmitz (2002) extended Lindzen’s parameterization to include the correct tendency [see their Eq. (15)]. In the Hines (1997) parameterization, an analogous thermodynamic energy tendency is used; however, it is scaled by a fudge factor Φ_{5} with a suggested range of 1 ≤ Φ_{5} ≤ 3. According to the current analysis, Φ_{5} must equal unity to ensure consistency between energy and momentum conservation. In all the above studies, the thermodynamic tendency is only applied in the dissipation (wave breaking) region; the present framework implies that it should be also applied in the generation region (and be associated with the wave sources).

Under the time scale separation assumption of our framework, which is that typically made in the parameterization of subgrid scales in climate models, the mesoscale is forced to be statistically stationary on the planetary scale. This leads to the wave-activity conservation laws being diagnostic relations [e.g., (41) and (51)]. Some climate models include turbulent kinetic energy equations that are prognostic. To include prognostic effects on the mesoscale, the theory would need to incorporate an intermittency factor to preserve the scalings on the planetary scale.

A weakness of our framework is that the mesoscale dynamics are assumed to be quasi-linear, in the sense that they are described by the small-amplitude form of the wave-activity conservation laws. This would seem to limit the applicability of the framework (e.g., convection and boundary layer turbulence are clearly nonlinear processes). The introduction of an intermittency factor should also make it possible to allow a nonlinear mesoscale without disturbing the balances on the planetary scale, and the Hamiltonian framework ensures that the mesoscale wave-activity conservation laws can be extended to finite amplitude. [Extension to moist dynamics should also be possible; see, e.g., Bannon (2003).] Unfortunately, in this case the direct connection between the vertical wave-activity fluxes and the planetary-scale equations is lost because there is an additional term in the vertical fluxes involving advection of the wave-activity density (Scinocca and Shepherd 1992). (This problem also arises for the classical theory of wave–mean-flow interaction involving the Eliassen–Palm wave activity; see Andrews et al. 1987.) Since the planetary-scale equations remain the same in the case of a nonlinear mesoscale, our framework still provides a general understanding of the transfers of energy and momentum between the resolved and subgrid scales in this case. In particular, every pseudomomentum flux must be accompanied by a pseudoenergy flux. Thus, the parameterizations of convective momentum transport of Schneider and Lindzen (1976) and Gregory et al. (1997), which neglect the energetics, are in error according to (57a). Similarly, parameterizations of the turbulent dissipation of resolved-scale kinetic energy in the boundary layer that neglect the associated dissipative heating are also in error. Computing the actual importance of these errors in realistic applications is the subject of future investigation.

## Acknowledgments

This research has been supported by the Natural Sciences and Engineering Research Council of Canada, in part through a Canada Graduate Scholarship to the first author, and by the Canadian Foundation for Climate and Atmospheric Sciences. The authors are grateful to Dr. N. A. McFarlane and Dr. E. Becker for many helpful discussions and to Dr. R. Klein and two anonymous reviewers for helping to improve the manuscript. The first author also acknowledges support from the Canadian Meteorological and Oceanographic Society and Zonta International.

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## APPENDIX A

### Compressible Wave-Activity Conservation Laws

*κγ*=

*R*/

*c*, with

_{υ}*c*being the specific heat at constant volume,

_{υ}*q*=

**·**

*ω***∇**

*θ*/

*ρ*is the potential vorticity with

**=**

*ω***∇**×

**v**and

*dV*=

*dx dy dz*. (Recall that rotation is negligible on the mesoscale by assumption.) The third term in the Hamiltonian is recognized as the internal energy of the system. Using these functionals, we proceed to derive the relevant wave-activity conservation laws according to the procedure outlined in section 5 of Shepherd (1990). According to the ansatz (11), we consider the planetary scale as a horizontally and temporally homogeneous but vertically varying background flow for the mesoscale dynamics. Accordingly, the

*x*,

_{m}*y*, and

_{m}*t*symmetries in the planetary-scale flow lead to pseudomomentum and pseudoenergy conservation laws on the mesoscale.

_{m}*δ*(*) represents the usual functional derivative and (20) has been used to derive the first variation of the Hamiltonian. It can be easily verified that

**is the state vector (a vector of the dependent variables) and**

*ξ***X**is the vertically dependent planetary-scale background state, subject to the condition that the first variation of

*ξ*=

**X**:

*q*is a higher-order term because it involves gradients on the long horizontal planetary scale and because Coriolis forces are higher-order terms.) In the more general case, we need to consider finite perturbation potential vorticity. In the general case the quadratic form of the pseudoenergy density is

_{p}*δπ*/

*κγπ*=

*δθ*/

*θ*+

*δρ*/

*ρ*according to (23). Upon making use of the Casimir coefficients given in SS08 and the ansatz (11), and equating the Mach and Froude numbers as done in section 3, the pseudoenergy density becomes

*W*=

*u*)

_{p}^{2}+ (

*υ*)

_{p}^{2}

*q*is the background potential vorticity gradient in the normal direction defined by

_{pn}**n̂**= sin

*α*

**x̂**− cos

*α*

**ŷ**, where

*α*is the angle between the background (planetary-scale) velocity and the

**x̂**direction. The effects of compressibility are associated with the second, sixth, and seventh terms. The pseudoenergy wave-activity conservation law, calculated by taking a time derivative of the pseudoenergy density and making use of the mesoscale dynamics (17), (18), and (22), is

*q*through the interaction with the background. In regions where the source–sink terms are zero, as is the case between the source and sink regions of the nonlocal transfer of energy and momentum, the mesoscale potential vorticity is zero. Thus, all terms in (A11) can be related to mesoscale source–sink terms.

_{m}**ŝ**= cos

*α*

**x̂**+ sin

*α*

**ŷ**. As for the pseudoenergy, the streamwise pseudomomentum Casimir is defined by the first two conditions and we can make use of the results of SS08. The general quadratic form of the streamwise pseudomomentum is

**ŝ**replaced by

**n̂**, such that the density is

*q*is the background potential-vorticity gradient in the streamwise direction. The streamwise pseudomomentum wave-activity conservation law, calculated by taking the time derivative of the streamwise pseudomomentum density and making use of the mesoscale dynamics, is

_{ps}## APPENDIX B

### Anelastic Wave-Activity Conservation Laws

*M*that are retained are those containing

*M*

^{2}/

*θ*, which is assumed to be

_{pz}*O*(1) under the weak stratification assumption discussed in section 5. The other terms involving

*M*are higher order under the anelastic approximation. In the anelastic case, the pseudoenergy and streamwise pseudomomentum densities satisfy the relation

**∇**· (

*ρ*

_{p}**v**

*) = 0, is*

_{m}*ρ*and

_{m}*π*.

_{m}Scaling parameters and their values.

^{1}

In keeping with our motivation, we consider only two time scales. We do not include the *O*(*ϵ*^{−1}) time scale in our framework, which is the mesoscale advective time scale. Its neglect does not fundamentally change what is derived in the subsequent sections.

^{2}

The mesoscale average of (22) contributes to *O*(*ϵ*^{2}) according to the mass-weighted average decomposition: *O*(*ϵ*^{2}) to satisfy

^{3}

Eliassen and Palm (1961) derived their first and second theorems for the case of steady, two-dimensional gravity waves, assuming conservative dynamics. However, they did not derive conservation laws that relate the vertical fluxes to conserved quantities. Subsequent authors (Bretherton 1966; Hines and Reddy 1967; Lindzen 1973) generalized the Eliassen–Palm theorems to nonsteady disturbances, but they appealed to WKB-type conditions in the vertical (large Richardson number) to define a wave packet with phase velocity *c* and the associated wave action.