## 1. Introduction

[2] The role of vertically propagating gravity waves (GW) in maintaining the circulation of the middle atmosphere is now widely recognized. In recent years most studies have primarily been concerned with estimates of momentum deposition, or the so-called gravity wave drag (GWD) exerted by dissipating and/or breaking GW. Relatively few studies have rigorously addressed the effect of the associated divergence of wave energy fluxes on the thermal structure of the mean circulation.

[3] Waves may affect the thermal structure of the atmosphere through several mechanisms. One such process is associated with the divergence of the wave's vertical flux of sensible heat, which arises because damping associated with GW breaking and/or saturation alters the phase relationship between fluctuations of temperature and vertical velocity. Explicit expressions for the corresponding heating rates have been derived by *Walterscheid* [1981] for molecular diffusion, and approximately by *Weinstock* [1983] and *Gardner* [1994] for diffusion associated with nonlinear interactions between GW harmonics. They all found that typical mesospheric magnitudes of downward wave heat fluxes, , are several K m s^{−1}, and that the corresponding flux divergence produces strong cooling rates up to several tens of K day^{−1} in the mesosphere. Although these numbers are difficult to reconcile with the much lower estimates of the monthly mean heat budget of the mesosphere [*Schoeberl et al.*, 1983], recent measurements of wave heat fluxes and their gradients with a wind-temperature lidar [*Gardner and Yang*, 1998] confirm the larger magnitudes estimated for individual profiles. It is worth mentioning here that the cited papers emphasized a wave-induced cooling of the mesosphere produced by this mechanism. However, the divergence of implies this cooling must be balanced by heating somewhere in the vertical column, because the fluxes themselves vanish at the lower and upper atmospheric boundaries, where waves are either assumed to propagate conservatively or to be absent. Measurements by *Gardner and Yang* [1998] revealed both heating and cooling due to this mechanism. To our knowledge, none of the other existing practical GW parameterizations used in middle atmosphere models includes differential heating due to GW.

[4] Another mechanism through which GWs may affect the thermal structure of the mean flow is the turbulent diffusion of heat. Some aspects of the early development of the concept of induced turbulent diffusion and heat transfer have been reviewed by *Ebel* [1984]. The turbulence appears as stochastic wind fluctuations associated with breaking gravity waves. Early models [*Hodges*, 1967; *Lindzen*,1981] postulated that the damping rate associated with this turbulence was precisely that needed to prevent the exponential growth of wave amplitude with height, i.e., to induce saturation of the GW harmonic. The turbulence associated with wave damping also affects the background wind (momentum deposition) and temperature (irreversible conversion of kinetic to internal energy). Thus both the drag and the heating depend strongly on the diffusion parameters. Earlier research [*Strobel et al.*, 1985; *Garcia and Solomon*, 1985] implied that this wave-induced turbulence equally affects disturbances and the mean flow, and that the diffusive coefficients which appear in the respective equations for the mean and fluctuating quantities are equal. In other words, the diffusion coefficient *D*_{GW}^{(u)} which relates diffusive fluxes of GW momentum with gradients of the GW wind field, **F**_{GW} = ρ*D*_{GW}^{(u)} ∇**u**′, was considered to be equal to ^{(u)}, the latter relating diffusive fluxes to gradients of the mean wind, = ρ^{(u)}∇. A similar assumption was invoked for the potential temperature diffusion coefficients, namely *D*_{GW}^{(θ)} = ^{(θ)}. While this hypothesis may represent a reasonable first step, it is clearly questionable on physical grounds, since the effects of a given turbulent field on motions of disparate scales are generally different. This latter property is recognized in theories of scale-dependent diffusion [*Weinstock*, 1990; *Gardner*, 1994; *Medvedev and Klaassen*, 1995], in which the corresponding coefficients associated with saturating or breaking GWs are different for each harmonic in the broad wave spectrum. In line with these arguments, the assumption ^{(u,θ)} = *D*_{GW}^{(u,θ)} has little physical justification.

[5] A related problem in the parameterization of cooling rates due to eddy thermal conduction is the uncertainty regarding the eddy or turbulent Prandtl number, *Pr* = ^{(u)}/^{(θ)}. The assumption of Pr = 1, in conjunction with uniform Hodges-Lindzen values of *D*_{GW} required to offset exponential wave growth and the implicit assumption that ≡ *D*_{GW}, has yielded excessively large values of associated diffusive cooling in the mesosphere [*Schoeberl et al.*, 1983; *Apruzese et al.*, 1984]. *Chao and Schoeberl* [1984], *Fritts and Dunkerton* [1985], and *Coy and Fritts* [1988] suggested that turbulence localized to the convectively unstable region of the GW may increase the turbulent *Pr*, leading to a reduction of eddy heat conductivity. This approach is physically justifiable for the case of a single breaking wave, and has allowed the derivation of the dependence of the eddy Prandtl number on GW parameters.

[6] For the case of a broad nonlinear spectrum, the derivation of the dependence of the eddy Prandtl number on GW parameters is considerably more difficult, and to our knowledge has not yet been attempted: nonlinear interactions create localized regions of instability which are spread throughout the wave field, rather than being confined to a particular wave phase. Thus for practical applications in general circulation models, the turbulent Prandtl number is usually treated as a tunable parameter. In the absence of definitive measurements or calculations of the turbulent Prandtl number for a strongly interacting wave spectrum, we shall assume its value to be unity. This assumption is consistent with the widespread instabilities expected in a broad spectrum, and we shall demonstrate that this value produces reasonable heating rates when used in conjunction with our gravity wave drag parameterization.

[7] The third effect of breaking gravity waves on the mean flow is the irreversible conversion of eddy kinetic energy into heat. The corresponding heating rate, ε_{GW} ≡ (∂/∂*t*)_{GW}, can be related to the vertical component of the diffusion coefficient by *c*_{p}ε_{GW} ≈ [*D*_{GW}^{(u)}]_{ZZ}(∂*u*′/∂*z*)^{2}, where the capital subscript “*ZZ*” indicates the vertical diagonal component of the diffusion tensor [see *Ebel*, 1984]. This formula has been used to estimate GW-induced heating in combination with various approaches for defining [*D*_{GW}^{(u)}]_{ZZ}. For example, *Schoeberl et al.* [1983] and *Gavrilov* [1990] (e.g., the latter's equation (24b)) employed Hodges-Lindzen-style approximations for the diffusion coefficient, while *Gavrilov and Yudin* [1992] utilized a semiempirical turbulence closure hypothesis for this purpose. A somewhat different approach was taken by *Hines* [1997] who invoked scaling arguments from the theory of homogeneous turbulence to relate ε to [^{(u)}]_{ZZ}. It should be noted that *Hines* [1997] does not distinguish between [*D*_{GW}^{(u)}]_{ZZ} and [^{(u)}]_{ZZ}.

[8] Several studies have bypassed the direct calculation of turbulent diffusion coefficients by applying linear wave considerations to estimate the dissipated kinetic energy. Such estimates are based on the linearized relation between vertical fluxes of energy, *F*_{e}, and momentum, *F*_{m}, for steady conservative waves, namely *F*_{e} = *cF*_{m} [*Jones*, 1971]. Here *c* is the observed horizontal phase velocity of the wave, and the lower case subscripts *e* and *m* denote energy and momentum, respectively. Assuming that the mean vertical velocity = 0 (so there is no energy flux by mean motions), the equation for the conservation of the horizontally averaged total energy, (*z*, *t*) + (*z*, *t*), can be written as

where we have further subdivided the kinetic () and potential () energies into mean (*M*) and wave/eddy (*E*) components. Invoking the steady wave approximation ∂(*K*_{E} + *P*_{E})/∂*t* = 0, together with *F*_{e} = *cF*_{m}, this becomes

Here we have explicitly recognized that ρ_{0}*a* = − ∂*F*_{m}/∂*z* represents the wave-induced horizontal drag acting on the flow. The following relation for the mean kinetic energy tendency

can be established independently from the mean momentum equation. It has been argued that the difference between these last two equations, ρ_{0} (*c* − ) *a* (which is positive because *a* and *c* − have the same sign), represents ∂*P*_{M}/∂*t*, or the rate at which wave energy is converted to heat. Arguments of this kind have been used by *Lindzen* [1973], *Gavrilov* [1990], and *Hines* [1997]. This approach is utilized to infer the diffusion coefficients either from polarization relations for GW [e.g., *Gavrilov and Roble*, 1994], or by employing scaling arguments from turbulence theory [e.g., *Hines*, 1997]. However, it must be noted that the relation between *F*_{e} and *F*_{m} for nonlinear, dissipative waves is more complex (see (38)), and that arguments based on linear wave considerations should used with appropriate caution.

[9] The purpose of this paper is to consider, in a self-consistent analysis, the energy conversion between gravity waves and the mean flow, and to derive rigorous expressions describing the thermal effects of GWs on the background fields by considering the full nonlinear energy budget. We suggest a means of representing these interactions for saturating waves, which is sufficiently general to be adapted for use with many GWD parameterizations and general circulation models (GCMs). We also provide examples and results using our own GWD scheme [*Medvedev and Klaassen*, 1995, 2000] (hereafter MK95 and MK00, respectively), in order to demonstrate typical patterns of heating/cooling rates associated with GW dissipation. A detailed investigation of the effects of GW heating on the middle atmosphere climate is beyond the scope of this paper.

[10] The paper is organized as follows. In section 2 we present a budget for the total spatially averaged energy of the flow, and derive general expressions for the evolution of the GW component and the horizontally averaged background, subject only to the hydrostatic and nonrotating approximations. In section 3 we consider the conversion of energy between waves and the mean flow (i.e., the “energy cycle”). The steady-wave approximation is introduced in section 4. A suitably general closure for the heating/cooling rates associated with breaking/saturated GWs is proposed in section 5. In section 6 we apply the MK spectral GWD parameterization to the general formulae obtained in the previous section to calculate wave heating rates for typical profiles of mean wind and temperature in a column model of GWD. In section 7 we present estimates of temperature tendencies due to GWs using simulations with the Canadian Middle Atmosphere Model. Conclusions are given in section 8.