On the energetics of mean-flow interactions with thermally dissipating gravity waves

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jd007908, 2007 On the energetics of mean-flow interactions with thermally dissipating gravity waves R. A. Akmaev 1 Received 10 August 2006; revised 26 January 2007; accepted 29 January 2007; published 15 June [1] Previous studies have demonstrated the importance of downgradient transport by dissipating waves and particularly of downward heat fluxes by gravity waves undergoing thermal dissipation. With a few exceptions, however, this effect has not been represented in gravity-wave parameterizations commonly employed in global numerical models. A general expression relating the heat flux to the wave energy deposition rate caused by thermal dissipation is obtained within the standard linear-theory approach. Although the flux is directed down the gradient of potential temperature, it is not proportional to its magnitude, i.e., is not formally diffusive as commonly represented. With necessary assumptions regarding the partitioning of the total wave energy deposition rate between the thermal and frictional channels, the heat flux may be calculated within any suitable parameterization of gravity-wave drag. The general relation may also be used to estimate net heating rates from observations of wave heat transport. In a more general thermodynamical context, it is noted that gravity-wave dissipation results in atmospheric entropy production as expected for a dissipative process. Without friction, entropy is produced under the conservation of the column potential enthalpy. Thermally dissipating waves thus represent an example of an entropy-generating process hypothesized in the literature but not identified before. Although the downward heat transport results in a local cooling of upper levels, the integrated net effect of the wave energy deposition and heat transport combined is always heating of the whole atmospheric layer in which the dissipation occurs. Citation: Akmaev, R. A. (2007), On the energetics of mean-flow interactions with thermally dissipating gravity waves, J. Geophys. Res., 112,, doi: /2006jd Introduction and Overview [2] Gavrilov and Shved [1975a] were first to show that the net thermal effect of plane gravity waves in a horizontally uniform windless atmosphere (more precisely, a background state with no vertical wind shear) is determined by the convergence (negative divergence) of the wave enthalpy flux F T ¼ rc p T0w0: Standard notation is used: the overbar and the prime denote mean quantities and deviations from the mean, respectively; the mean density r(z) is assumed to depend on height z only, as any other mean variable; the specific heat at constant pressure c p is assumed constant, along with the gas constant for dry air R, specific heat at constant volume c v, and the ratios k = R/c p and g = c p /c v ; T 0 and w 0 are wave fluctuations of temperature and vertical velocity, respectively. The enthalpy flux F T in equation (1) clearly differs from the 1 Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado, USA. Copyright 2007 by the American Geophysical Union /07/2006JD ð1þ wave energy flux F E as commonly defined [for example, equation (A13) in Appendix A]. This means that the net heating rate of the mean state does not equal the convergence of F E, as has been sometimes postulated [e.g., Hines, 1965; Lindzen, 1990]. [3] It is convenient to introduce potential temperature q = T/P, where P =(p/p 0 ) k is the dimensionless Exner function, p is pressure, and p 0 is some reference pressure. Linearizing this definition, q0 ¼ T 0 q T k p 0 p ; expression (1) may be rewritten as F T ¼ p0w0 þ F q : The first term in equation (3) is recognized as the vertical wave energy flux in a windless atmosphere, and F q ¼ rc p P q0w0 will be loosely referred to as the vertical wave heat flux. F q is closely related to the flux of potential enthalpy F q / P [see equation (27) below]. In layers of thermal dissipation, ð2þ ð3þ ð4þ 1of12

2 q 0 and w 0 are no longer in quadrature as in conservative waves, and F q is nonzero and generally nonconstant. In addition to the wave energy flux then there is a mean waveinduced heat transport and local cooling and heating associated with it. [4] Plumb [1979] elucidated the role of dissipation in downgradient mean fluxes of conserved quantities following earlier work on wave tracer transport [Clark and Rogers, 1978; Holton, 1980]. Although he did not specifically address the heat transport, F q is clearly an example of such a downgradient flux. It is always directed downward irrespective of the direction of wave propagation because gravity waves can only exist in a stable background stratification, i.e., if the vertical gradient of mean potential temperature is positive (@ q/@z > 0). It should be emphasized that the downgradient transport does not necessarily require the wave to dissipate, but it rather requires the dissipation of wave-induced fluctuations of the conserved quantity [Plumb, 1979]. Of course, thermal dissipation causing the heat flux would imply wave dissipation, but if, for instance, a wave encounters only mechanical (frictional) dissipation, there would be no heat flux, at least to second order in wave perturbations [Plumb, 1979]. [5] Conversely, a chemical tracer would be transported down its mean gradient in the presence of a sink regardless of the wave dissipation. Numerical simulations have revealed, for example, a strong downward transport of atomic oxygen by tidal waves in the mesosphere and lower thermosphere, driven by its chemical losses in recombination processes [Akmaev and Shved, 1980; Forbes et al., 1993]. Within a purely kinematic approach, i.e., with no specific assumptions on wave dynamics and dissipation mechanisms, Plumb [1979] was able to represent the downgradient tracer flux in a diffusive form. The applicability of this representation to the heat transport will be discussed in more detail below. [6] In a well-known study, Walterscheid [1981] considered a particular case of heat transport by gravity waves dissipated by molecular heat conductivity in the upper atmosphere. According to his simple physical interpretation, thermal dissipation induces a downward heat flux by adding heat to cold air parcels displaced upward by the wave motion and removing heat from warm parcels displaced downward. This interpretation is fully consistent with the mechanism of downgradient transport of conserved quantities by Plumb [1979] as the addition and removal of heat from the parcels is precisely a manifestation of the damping of temperature fluctuations. A similar interpretation was proposed to explain the downward transport of atomic oxygen by its enhanced chemical losses in air parcels periodically displaced into denser layers by tidal motions [Akmaev and Shved, 1980]. The downward heat transport inevitably results in cooling of upper levels, and the terms dynamical cooling or wave cooling [Walterscheid, 1981] have been commonly used in the literature. Their exact meaning will be discussed in more detail below as well. [7] At about the same time, Lindzen [1981] developed the first practical parameterization of momentum deposition by breaking monochromatic gravity waves [see also Holton, 1982]. The deposition of horizontal momentum or wave drag associated with the wave breaking or, more generally, dissipation is crucially important in maintaining the dynamics and structure of the middle and upper atmosphere [Fritts and Alexander, 2003]. Amplitudes of upward propagating waves grow exponentially with altitude in response to the decrease of r(z). The celebrated parameterization by Lindzen [1981] is based on the closure hypothesis by Hodges [1969] stating that it is the turbulence generated by a breaking wave that prevents its amplitude from exceeding the static stability threshold, so that the wave becomes saturated. In its original formulation, this parameterization explicitly prescribes both frictional and thermal wave dissipation via the same eddy mixing coefficients or, equivalently, with a turbulent Prandtl number Pr = 1 [Lindzen, 1981; Holton, 1982]. Until recently [Becker, 2004], the downward heat transport by dissipating waves had not been accounted for in model implementations of Lindzen [1981] and other similar parameterizations [e.g., Gavrilov, 1990]. [8] Generation of turbulence by a breaking wave would also result in a mean turbulent downward heat transport if the eddy-mixing coefficients damping the wave are assumed to act on the background stratification as well. It is readily shown that the mean turbulent heat flux is then directly proportional to the wave heat flux [Schoeberl et al., 1983; Liu, 2000; Becker, 2004]. This means that in the original scheme by Lindzen [1981], the supposed thermal eddy mixing enters the mean heat transport twice: once directly, by acting on the mean stratification, and also indirectly, by dampening the wave and inducing the wave heat flux [Schoeberl et al., 1983]. This may result in excessive cooling of upper layers since considerable eddy-mixing coefficients are required to dampen the breaking waves and generate a wave drag thought to be needed in the middle and upper atmosphere. [9] One solution to this problem would be to assume large Prandtl numbers or low thermal eddy-mixing coefficients compared to momentum mixing coefficients either for the turbulent mean transport or for the turbulent wave dissipation. It was suggested by Chao and Schoeberl [1984] that the Prandtl number for the wave dissipation Pr W 1 because turbulence generated by breaking waves is localized in narrow unstable layers with nearly adiabatic stratification and so cannot contribute effectively to wave dampening. This would imply that breaking waves primarily encounter frictional dissipation and substantially reduce the wave heat flux. If correct, this assumption would justify the absence of wave heat transport in model implementations of the parameterization by Lindzen [1981]. [10] On the other hand, Fritts and Dunkerton [1985] argued, based on the same turbulence localization hypothesis, that the Prandtl number for the mean transport Pr M 1, whereas Pr W 1. In this case, turbulent dissipation of both temperature and momentum fluctuations is sufficient to saturate the wave amplitude, but the mean eddy heat flux becomes negligibly small. According to this approach, there still remains the downward wave heat transport driven by the thermal dissipation to be accounted for in large-scale models [Liu, 2000; Becker, 2004]. [11] Recently, Medvedev and Klaassen [2003] also incorporated the wave heat transport into their spectral gravity wave parameterization. In their scheme, the properties of the incident wave spectrum define a damping coefficient, which acts equally on wave temperature and velocity perturbations. 2of12

3 Unlike in the parameterization by Lindzen [1981], however, the waves are not assumed to dissipate by turbulence but rather by nonlinear interactions across the wave spectrum. Consequently, there is no mean turbulent heat or momentum transport. This approach is consistent with some numerical simulations of wave breaking suggesting that it is primarily wave overturning that limits the wave amplitude [Walterscheid and Schubert, 1990]. Free convection resulting from the overturning may even transport heat upward in unstable layers, while turbulence is just an end product of the convection and does not have to be very strong to limit the wave amplitude. This is equivalent to the assumption of Pr M 1 for the mean turbulent transport while the wave thermal dissipation and downward heat flux remain substantial. [12] It is clear from the preceding brief overview that different approaches have been employed in the past to account for the wave heat transport, if any. The effect has been incorporated into two particular gravity-wave parameterizations [Lindzen, 1981; Medvedev and Klaassen, 2003]; to date, only two modeling studies have examined its possible role in the middle atmosphere [Medvedev and Klaassen, 2003; Becker, 2004]. [13] This work was initiated by a specific need to account for the heat transport within the Doppler-spread parameterization (DSP) [Hines, 1997] presently employed in many global numerical models. The task is not entirely trivial because, unlike in the parameterizations by Lindzen [1981] and Medvedev and Klaassen [2003], the DSP does not explicitly prescribe a damping or an eddy-mixing coefficient that could be used to calculate the covariances of T 0 or q 0 with w 0 in equations (1) or (4). Eddy mixing coefficients acting on the background state are estimated, but it is important to realize that turbulence is not considered the cause of wave damping but rather the final product of wave obliteration [cf. Walterscheid and Schubert, 1990]. The DSP explicitly calculates only a limiting vertical wave number as determined by the wave spectrum at each altitude. The wave momentum deposition rate is then calculated by integrating over the spectrum and the wave energy deposition rate from the convergence of the wave energy flux [Hines, 1999; Akmaev, 2001]. [14] Further analysis presented here suggests that certain assumptions have to be made to estimate the downward wave heat flux and the resulting net heating rate from the quantities explicitly defined in the DSP and other parameterizations primarily focusing on representing the wave drag. These estimates depend, in particular, on the assumed partitioning of the total wave energy deposition rate between the thermal and frictional channels. Some additional results of this analysis appear to be of more general interest and are presented here as well: [15] (1) A general expression is obtained relating the wave heat flux with the wave energy deposition rate due to thermal dissipation. With the necessary assumptions just described, the wave heat flux may also be related to the total wave energy deposition rate. These expressions essentially establish a relationship between the wave heat and momentum fluxes and should be applicable to any parameterization of the gravity-wave drag. [16] (2) It is noted that although the heat flux is directed down the gradient of potential temperature, it is not proportional to the magnitude of the gradient and so is not formally diffusive, as commonly parameterized [e.g., Medvedev and Klaassen, 2003; Becker, 2004]. Interestingly, the same conclusion applies to the eddy heat flux as represented in the parameterization by Lindzen [1981] because, as already noted, the two fluxes are simply proportional to each other. This is contrary to the common belief that the parameterization by Lindzen [1981] produces diffusive damping of background temperature perturbations with possible implications for interactions of parameterized gravity waves with large-scale waves such as tides. [17] (3) It has been suggested in the past that dissipating gravity waves deposit their energy and so should heat the mean state everywhere in the layer of dissipation [e.g., Hines, 1965; Lindzen, 1990]. The very notion of wave cooling [Walterscheid, 1981] may therefore seem counterintuitive, and properties of dissipating waves are briefly discussed from a more general thermodynamical perspective. It is shown that although the downward heat transport results in a local cooling of upper layers, the net effect of wave dissipation is always to increase not only the column enthalpy (or potential and internal energy), as has been mentioned in previous studies [e.g., Walterscheid, 1981; Hickey et al., 2000], but also the column entropy in the whole layer of dissipation, which is to be expected for a dissipative process. [18] (4) Considerable wave heat fluxes have been observed in simultaneous measurements of temperature and vertical velocity perturbations [e.g., Tao and Gardner, 1995; Gardner and Yang, 1998; Liu and Gardner, 2005]. These observations suggest that gravity waves indeed encounter substantial thermal dissipation in the upper atmosphere. These results have been criticized in the literature [Fritts, 2000], and the analysis presented here may facilitate their correct interpretation [cf. Medvedev and Klaassen, 2003]. 2. Net Wave Heating Rate [19] In this section, the net wave heating rate is restated following the common procedure [Gavrilov and Shved, 1975a; Plumb, 1979, 1983; Gavrilov, 1990]. Some further details of the derivation are presented in Appendix A for completeness. [20] The standard set of dynamical equations in a twodimensional, plain, compressible, and nonrotating atmosphere may be written in the form: r du ¼ X ; r dw dt þ rg ¼ ð6þ dt dp rc p dt dt ¼ Q þ þ þ ¼ 0; p ¼ rrt: ð5þ ð7þ ð8þ ð9þ 3of12

4 Here x and z are Cartesian geometric coordinates along the horizontal and vertical axes, respectively; p, r, and T are the air pressure, mass density, and temperature, respectively; u and w are the horizontal and vertical velocities, respectively; and d/dt + u@/@x + w@/@z is the material time derivative following the air motion. [21] In the momentum equations (5) and (6), X and Z represent forces other than gravity g. Frictional dissipative forces are of particular interest, in which case X and Z are linear in velocity components. In the thermodynamic energy equation (7), Q represents heating rates by such processes as radiation and molecular or eddy heat conduction. Frictional dissipative heating D in equation (7) is a quadratic positive-definite function of the velocity components representing the conversion of kinetic energy into heat [e.g., Van Mieghem, 1973]. The set of equations is completed with the continuity equation (8) and equation of state (9). [22] All variables are separated into the mean and the wave perturbation: ðþ ¼ ðþ þ ðþ 0 ðþ 0 ¼ 0 g: ð10þ The mean properties are usually averages over the horizontal wavelength and are assumed to depend on z only. Index z is also used to denote the vertical derivative of a mean quantity: ðþ As usual, stationarity will be assumed for the means, but in some instances, their time derivatives are retained to formally identify the variables whose tendencies are represented in the corresponding equations. Using equation (10), full equations (5) (9) may be separated into a set of linear wave equations (A1) (A5) (see Appendix A) and mean equations, in which only terms up to second order are retained [Plumb, 1979, 1983; Gavrilov, 1990]. [23] As is well known, linear equations (A1) (A5) describe two types of waves: gravity waves and higher frequency acoustic waves [Hines, 1960; Gossard and Hooke, 1975; Lighthill, 1978]. Some previous studies have used hydrostatic equations in pressure coordinates from the outset [e.g., Walterscheid, 1981; Medvedev and Klaassen, 2003; Becker, 2004]. This approximation excludes acoustic waves, as do the Boussinesq and inelastic approximations in geometric coordinates [Gossard and Hooke, 1975; Lighthill, 1978; Fritts and Alexander, 2003]. Although this study focuses primarily on internal gravity waves, a more general case is considered first in this section with acoustic waves retained, keeping in mind that they may represent a substantial additional source of energy in upper atmospheres of planets [e.g., Schubert et al., 2005]. Following the approach of Gavrilov [1990], further approximations are introduced consecutively in the next section to identify the applicability of specific statements and relations obtained. [24] Balance relations for different forms (potential, kinetic, and total) of wave and mean energy may be derived from the two sets of equations (see Appendix A). In particular, the mean of the thermodynamic energy equation (7) describes the balance of combined (mean plus perturbation) total potential energy [Plumb, 1983]. Subtracting the wave potential energy equations (A8) and (A9) from the mean of equation (7) and using equations (2) and (A6), the total mean-enthalpy tendency may be written in the following p T ¼ Q þ DðuÞþh: ð11þ Of course, the left-hand side formally vanishes under the stationarity assumption and is retained here only to indicate that the right-hand side has the meaning of the total tendency of mean enthalpy. Other tendency terms such as the mean pressure tendency are not shown. [25] Because the frictional heating rate D in equation (7) is a quadratic function of velocity components, its average D is a sum of two terms representing the dissipation of the mean and wave kinetic energy, respectively. The former term is written explicitly in equation (11) as D(u) because the heating rate by frictional dissipation of the mean vertical motion is formally of fourth order [equation (A6)] and may be neglected. The wave frictional heating is incorporated into the last term in equation (11) (see below). (It may be worth noting that if Q is a nonlinear function of temperature, as in the case of infrared radiation, its mean Q will also consist of two terms, including a wave contribution of second order, but this well-known effect is outside the scope of this study.) [26] The total net wave contribution to the mean-enthalpy tendency in equation (11) is pt ¼ F ð q Þ z þre; ð12þ W where the first term represents the effect of the downgradient heat transport. The total wave energy deposition rate per unit mass e in equation (12) is the sum of the deposition rate of the wave potential energy due to thermal dissipation e T [equation (A17)] and the frictional deposition rate of the wave kinetic energy e F [equation (A14)]: e ¼ e T þ e F 0: ð13þ As already mentioned, e is explicitly calculated in the DSP and is usually available in other parameterizations as well. [27] Because the DSP does not specify the mechanism of wave dissipation in detail, to estimate h in equation (12), it would be desirable to find a suitable general expression for F q, independent of specific assumptions about these mechanisms. This may be accomplished by introducing further assumptions as described in the next section. Other forms of h have also been derived [e.g., Gavrilov and Shved, 1975a; Gavrilov, 1990] and are presented in Appendix A [equations (A18) and (A19)] for completeness. 3. Heat Flux by Gravity Waves [28] As is well known, in stationary conditions, F q is proportional to the dissipative loss rate of gravity-wave potential energy (A8), which is proportional to the covariance q 0 Q 0 [e.g., Plumb, 1979]. In turn, the thermal loss rate with the opposite sign is exactly the contribution by gravity waves to the total rate of deposition of wave potential energy e T [equation (A17)]. There should be a direct relation between F q and e T for gravity waves. 4of12

5 [29] This also implies that the downward transport of potential enthalpy is a property of gravity waves and not of acoustic waves [e.g., Schubert et al., 2005]. To obtain a relation between F q and e T, the derivation will now be restricted to gravity waves alone. This essentially means that the terms containing pressure oscillations are neglected with one notable exception being the wave energy flux F E (equation (A13) [Hines, 1960; Gossard and Hooke, 1975; Lighthill, 1978; Gavrilov, 1990]). Neglecting the acoustic potential energy (A9) and its dissipative contribution to e T [equation (A17)], the gravity-wave energy deposition rate due to thermal dissipation becomes e T k r! q 0 Q 0 qz H : ð14þ [30] Comparing equation (4) with equation (14), it immediately follows from equation (A8) that in stationary conditions, F q ¼ re T H P ; ð15þ where H P = H/k is the scale height for the Exner function H P ¼ 1 : ð16þ As already mentioned, the general relation (15) expresses a well-known fact that the wave flux of a conserved quantity is proportional to its dissipative loss rate [Plumb, 1979]. However, the particular form (15) has not been presented before and is one of the main results of this study. One important conclusion immediately following from equation (15) is that because the downward heat flux F q is directly proportional to e T, there can be no heat transport with no wave energy deposition. More specifically, gravity waves undergoing only frictional dissipation (e T = 0) do not transport heat. Since both F q 0 and e T 0 are proportional to q0q0, their signs are determined by the negative definiteness of this covariance as appropriate for a truly dissipative process [Plumb, 1979] (see also Appendix A). The proportionality coefficient in equation (15) does not depend on such wave properties as frequency, phase speed, horizontal or vertical wavelength, etc. This relation is therefore applicable to any monochromatic gravity wave or a superposition (spectrum) of waves. [31] Following Plumb [1979], no specific assumptions have been made up to this point on details of the dissipation mechanisms except for their linearity and the positive definiteness of both terms in equation (13). The general relation (15) should therefore be applicable to any gravitywave parameterization scheme. However, because e T represents only part of the total wave energy deposition rate e, a relation between F q and e would also be desirable. To obtain such a relation, the partitioning of e between the thermal and frictional channels (13) has to be explicitly specified. It is also important to clearly state what additional assumptions have to be invoked. [32] Further discussion will be limited to gravity waves with short vertical wavelengths, an approximation commonly used in parameterization schemes [e.g., Lindzen, 1981; Holton, 1982; Gavrilov, 1990; Hines, 1997; Liu, 2000; Medvedev and Klaassen, 2003; Becker, 2004], which substantially simplifies the wave polarization relations [Fritts and Alexander, 2003]. This assumption allows, in particular, to neglect the contribution of the vertical motion to the wave kinetic energy (A11) and to the frictional energy deposition rate e F. It will also be assumed that dissipation is sufficiently weak [Plumb, 1979], so that to a good approximation, essentially nondissipative polarization relations may be used [e.g., Gavrilov and Shved, 1975b; Vadas and Fritts, 2005]. [33] The linear dissipative terms are assumed to take the simple form [Plumb, 1979]: Q 0 ¼ a T rc p T 0 ¼ a T rc p Pq 0 ; X 0 ¼ a F ru 0 ; ð17þ ð18þ where a T 0 and a F 0 are the thermal and frictional damping coefficients, respectively. The second expression in equation (17) follows from equation (2) neglecting pressure perturbations. In the case of viscous or diffusive dissipation, the damping coefficients simply equal the corresponding mixing coefficient times the vertical wave number squared under the assumption of short vertical wavelength. The simple form of equations (17) and (18) is thus sufficiently general to encompass such common mechanisms of wave dissipation and saturation as the turbulent mixing [Lindzen, 1981], molecular heat conduction and viscosity [Walterscheid, 1981], nonlinear spectral [Medvedev and Klaassen, 2003], or radiative damping [Zhu, 1994]. [34] Substituting equation (17) into equation (14), the wave thermal energy deposition rate may be written as g e T ¼ a T qz q 0 2 0: q ð19þ With equation (18), the frictional energy deposition rate in equation (A14) is simply e F ¼ a F u 0 2 0: ð20þ From the simplified polarization relations for short gravity waves, q 0 2 ð q z q=gþu 0 2 [Fritts and Alexander, 2003], and the total wave energy deposition rate (13) may now be represented as e ¼ ð1 þ PÞe T ; ð21þ where the ratio P = a F /a T characterizes the partitioning of e between the frictional and thermal wave dissipation. It may be considered a generalized Prandtl number as it reduces to Pr W if the waves are assumed to dissipate via turbulent or molecular viscosity and heat conduction [Lindzen, 1981; Vadas and Fritts, 2005]. With equation (21), equation (15) may now be written as F q ¼ 1 1 þ P re H P ; ð22þ 5of12

6 relating the downward heat flux with the total wave energy deposition rate. Of course, equation (21) is formally inapplicable if a T =0ore T = 0, in which case equation (22) simply reduces to F q = 0 according to equation (15). [35] Formula (22) also implicitly relates F q to the wave momentum deposition rate, which is closely linked to e [e.g., Akmaev et al., 1997; Hines, 1999; Akmaev, 2001], and so may, in principle, be used in any gravity-wave parameterization. It is important to emphasize, however, that in order to calculate the net wave heating rate h in equation (12), the partitioning parameter P has to be known. [36] As indicated earlier, the DSP does not specify a detailed mechanism by which individual waves, spread by the action of the total spectrum beyond the limiting vertical wave number, are obliterated in the middle atmosphere. The simplest approach would be to assume P 1, which is conceptually similar to the assumption of Pr W 1 for wave dissipation [Lindzen, 1981; Holton, 1982; Fritts and Dunkerton, 1985]. Formulae similar to equation (22) have been derived under this assumption, in which case the coefficient depending on P in equation (22) equals 1/2 [Medvedev and Klaassen, 2003; Becker, 2004]. This corresponds to an implicit assumption that e T = e F = e/2. If, on the other hand, P 1 can be justified, then F q 0 and may be neglected [cf. Chao and Schoeberl, 1984]. Alternatively, P may be considered a tunable parameter of the parameterization. [37] The DSP also allows for a transition to wave dissipation by molecular viscosity and heat conduction in the thermosphere. A viscous limiting vertical wave number is calculated and compared to that defined by the wave spectrum [Hines, 1997]. When the former becomes smaller than the latter, it is assumed that the molecular dissipative processes begin to dominate wave dissipation. Above the transition altitude, P may then be formally set to the molecular Prandtl number Pr W 0.7 [Vadas and Fritts, 2005]. However, detailed numerical simulations show that e T and e F remain quite close in planetary thermospheres even without the approximations adopted in this section [e.g., Hickey et al., 2000]. It therefore remains unclear if assuming P 6¼ 1 in the thermosphere would be warranted, especially in view of the approximate manner in which the transition into the thermosphere is represented in the DSP. To summarize, P 1 appears to be a reasonable first choice for the DSP or other suitable parameterizations. With any choice of P, the addition of the wave heat transport may require retuning of other related parameters in large-scale models such as the mean eddy-mixing coefficients. [38] Using only kinematic arguments, Plumb [1979] was able to represent the downgradient flux of a tracer in a diffusive form. Consistent with the approximations adopted here, q 0 q z z 0 [Plumb, 1979], where z 0 is the perturbation vertical displacement. Using equations (14), (15), and (17), the heat flux may formally be written as where [Plumb, 1979] F q ¼ rc p ; K ¼ a T z 0 2 : ð23þ ð24þ Formula (23) is recognized as a common expression for a diffusive turbulent heat flux with an eddy-mixing coefficient K and similar expressions hold for any conserved quantity. According to equation (24), the diffusion coefficient K is proportional to the sink strength a T, so that different conserved quantities would diffuse at different rates [Plumb, 1979]. Mean wave transport is a second-order wave phenomenon, and equations (23) and (24) also reflect the obvious fact that the diffusion coefficient and the downgradient flux scale as the wave amplitude squared. [39] The wave heat flux F q has been represented in a similar diffusive form in previous studies [e.g., Schoeberl et al., 1983; Medvedev and Klaassen, 2003; Becker, 2004]. It should be noted, however, that although potential temperature is a conserved quantity [Plumb, 1979], it is of course not a passive tracer in the sense that wave properties depend on its mean distribution (background stratification). It follows then that although F q may formally be written in the form (23), it is not really proportional to the negative gradient of potential temperature. Using the same polarization relations as before and equations (19) and (24), it is readily shown that e T = N 2 K or, using equation (21), that equation (24) is equivalent to K ¼ 1 e 1 þ P N 2 ; ð25þ where the buoyancy frequency squared N 2 = g(q z =q). [40] Expressions similar to equation (25) have been obtained in previous studies [Schoeberl et al., 1983; Medvedev and Klaassen, 2003; Becker, 2004]. The presence of N 2 in the denominator of equation (25) reflects the fact that the amplitude of z 0 depends on the background stability (is inversely proportional to N). Substituting equation (25) into equation (23), the vertical gradient of potential temperature will cancel in the numerator and denominator, and F q will revert to the nondiffusive form (22). In the case of a chemical tracer, there is no cancellation of the gradients, and the tracer flux remains diffusive [Plumb, 1979]. [41] It is also of interest that the turbulent diffusion coefficient in the parameterization by Lindzen [1981] [see also Holton, 1982] may be represented in the same form as equation (25) but with P =0[Akmaev et al., 1997; Becker, 2004]. This means that the eddy heat flux postulated in this parameterization is not actually diffusive because it is not proportional to the negative vertical gradient of potential temperature. This in turn means, for example, that the vertical wave heat transport as represented here [equation (22)] or in the Lindzen parameterization of wave-induced turbulence would not dampen large-scale waves such as tides in the way expected for a strictly diffusive damping. [42] On the other hand, F q may not be entirely independent of the background stratification. It follows from the conservation of vertical momentum or wave-action fluxes that the total wave energy density E [equation (A16)] is proportional to N in conservative gravity waves [Bretherton and Garrett, 1969; VanZandt and Fritts, 1989]. Higher energy deposition rates may then result in layers of enhanced atmospheric stability with the same strength of thermal damping a T. It should also be kept in mind that gravity waves are more prone to reflection in layers with lower N 6of12

7 [Fritts and Alexander, 2003] resulting in weaker wave activity, energy deposition, and heat fluxes. This implies that stronger heat fluxes may generally be expected in layers of enhanced atmospheric stability and vice versa. The exact form of this dependence is generally unknown but is likely weaker than the linear proportionality of F q to q z or N 2 as in equation (23). [43] Expression (23) is still formally valid if used along with equation (25). This may facilitate the incorporation of the wave heat transport into those model implementations of gravity-wave parameterizations that already employ eddymixing coefficients related to wave dissipation, saturation, and breaking [Lindzen, 1981; Holton, 1982; Hines, 1997]. 4. Thermodynamics of Wave Dissipation [44] Relations derived here allow to formulate some more general thermodynamical properties of dissipating gravity waves. It is well known in atmospheric thermodynamics that an isothermal state T(z) = const is the state of maximum entropy or the ultimate state of thermodynamical equilibrium of an isolated atmospheric column, i.e., under the conditions of conservation of the total potential energy (column enthalpy) and mass [Bohren and Albrecht, 1998; Verkley and Gerkema, 2004]. (A horizontally uniform, dry, hydrostatic mean state with constant composition is assumed, consistent with the approximations adopted in previous sections). [45] Entropy maximization under other constraints would generally result in different temperature distributions. It has been hypothesized, for example, that some dynamical processes such as convection or turbulence may generate entropy while conserving the column potential enthalpy instead [Ball, 1956; Bohren and Albrecht, 1998]. This formally results in an isentropic state q(z) = const as the state of maximum entropy [Bohren and Albrecht, 1998; Verkley and Gerkema, 2004]. However, no specific entropygenerating process has been identified before as exactly conserving the column potential enthalpy [Verkley and Gerkema, 2004]. [46] It is readily shown that a gravity wave undergoing purely thermal dissipation is a process exactly conserving column potential enthalpy. Assuming that there is no friction (e F = 0) and using equations (12) and (15), the wave-induced tendency of the mean enthalpy may be written in the following p T ð re TH P Þ þ re T : ð26þ Dividing by P and using relation (16), this equation may be rewritten in the form of a potential-enthalpy tendency where the two terms on the right combine exactly into one fluxdivergence p q re T H P : P Note that the flux term in equation (27) has the meaning of the flux of potential enthalpy and, as mentioned in section 1, differs from the wave heat flux F q [equation (4)] by the factor P. The right-hand side of equation (27) vanishes upon vertical integration over the whole layer of dissipation, i.e., over the layer where e T 0. This means that a thermally dissipating gravity wave transports heat downward and simultaneously deposits energy, as represented by the two terms in equation (26), at the rates exactly conserving the column potential enthalpy. [47] Following Goody [2000], it is straightforward to show that equations (26) or (27) formally imply production of the mean column entropy even in the absence of friction. Dividing equation (27) by q and integrating vertically by parts over the whole layer of dissipation, the resulting column-entropy tendency may be written in the form Z ds re T H P ¼ dt WT fe T0g P qz q 2 dz 0; ð28þ where S is the vertically integrated mass-weighted specific entropy c p ln q, and the inequality holds in a stable stratification. [48] Although the wave heat flux F q is generally not proportional to the negative gradient of potential temperature, as discussed in the previous section, its magnitude may be expected to increase with static stability. The downward heat transport would thus tend to drive the mean state to adiabatic stratification, which is the state of maximum entropy under the conservation of column potential enthalpy. A thermally dissipating wave clearly represents an example of the hypothetical dissipative process that produces the column entropy while conserving the column potential enthalpy. [49] It is worth to emphasize again the role of the second, energy deposition, term in equation (26). If this term is neglected and only the flux-divergence term is retained, as is often done in model parameterizations of turbulent heat transport, the column entropy production rate would be negative in a stable stratification, contrary to what is expected for such dissipative processes as wave dissipation or turbulence [Goody, 2000]. The physical resolution of this paradox is straightforward: To transport heat vertically in a stable stratification, work has to be done against buoyancy [Ball, 1956]. This work is expressed by the second term in equation (26), and it is deposited into the total potential energy or column enthalpy. As just discussed, the two terms in equation (26) combine to produce column entropy even in the absence of friction. Walterscheid [1981] also demonstrated the mean entropy production in the case of gravity waves dissipated by molecular heat conduction. However, he only considered the second, energy deposition, term in equation (26) and neglected the offending flux term. It is the flux term in equation (26) that results in the negative entropy production [Goody, 2000] if the wave energy deposition is neglected. [50] To some extent, pure thermal dissipation is an idealization and perhaps only radiative damping [Zhu, 1994] would qualify as such a process among the possible wave dissipation mechanisms discussed in previous sections. In a more general case with frictional dissipation present (e F 0) as well, there would be additional positive terms proportional to e F in equations (26) (28). Vertically integrating as before, it then follows that, in general, 7of12

8 dissipating gravity waves generate both the column entropy and potential enthalpy. [51] It is important to note that the column enthalpy always increases as well, irrespective of the presence of friction. The first term in equations (12) or (26) simply redistributes heat in the vertical and vanishes upon vertical integration, while the second term can only make a nonnegative contribution even in the absence of friction. This is at variance with the conclusion by Walterscheid [1981] that e T does not represent a direct addition of heat to the mean state, and that a net heating only occurs in the presence of viscous dissipation. It is clear from equation (26) that dissipating gravity waves always contribute positively to the total potential energy of an entire atmospheric layer in which the wave dissipation occurs. Since the total potential energy is proportional to the column internal energy [Lorenz, 1967], the widely used terms wave cooling or dynamical cooling [Walterscheid, 1981] may be somewhat misleading. Of course, the downward heat flux cools the upper portion of the dissipation layer, but the associated heating of the lower levels in combination with the required in situ wave energy deposition may only increase the column enthalpy or internal energy. Recall once again that F q can only be nonzero if e T >0. 5. On Interpretation of Some Observations [52] Lidar observations provide compelling evidence for the thermal dissipation and downward wave heat transport in the mesosphere [e.g., Tao and Gardner, 1995; Gardner and Yang, 1998; Liu and Gardner, 2005]. Simultaneous measurements of temperature and vertical wind perturbations T 0 and w 0 have been used in these studies to estimate the enthalpy flux F T [equation (1)]. Negative (downward) fluxes are often observed in extended layers, which is indicative of thermal dissipation and may result in substantial cooling rates near the mesopause. [53] These observational estimates of heat fluxes and cooling rates have been criticized in the literature [Fritts, 2000]. Without going into all aspects of the discussion, some additional comments may be offered based on the general results presented here. One of the criticisms is that positive values of F T have been observed in many individual observations as well as in average profiles, especially at lower altitudes. According to Fritts [2000], this may be indicative of instrumental or data processing problems since negative heat fluxes are normally expected for dissipating waves. [54] It should be noted first that it is the heat flux F q [equation (15)] that becomes negative for thermally dissipating waves in stationary conditions [Plumb, 1979]. According to equation (3), F T differs from F q by the covariance p 0 w 0 which is positive for upward propagating waves and equals the wave energy flux F E in a windless mean state. It would then be natural for F T to turn from positive to negative values if upward propagating waves encounter a layer of dissipation or break, just as shown for average results in Figures 6 and 8 of Gardner and Yang [1998], for example. Secondly, as pointed out by Plumb [1979], the direction of wave fluxes of conserved quantities also depends on other conditions such as nonstationarity or possible generation of waves. For example, F q may be expected to be positive (upgradient) for waves decaying in time or in layers of secondary wave generation [e.g., Vadas et al., 2003]. The positiveness of F T at some levels therefore does not appear sufficient to dismiss the analyses of Tao and Gardner [1995], Gardner and Yang [1998], and Liu and Gardner [2005]. [55] On the other hand, the net wave heating rate h equals the convergence of F T [equation (A18)] only in a windless mean state [Gavrilov and Shved, 1975a]. Mesospheric observations are often conducted in the presence of strong background wind shears possibly produced by large-scale waves [e.g., Tao and Gardner, 1995; Gardner and Yang, 1998]. Therefore the estimates of ( F T ) z presented in these studies may only be considered partial contributions to the net wave heating rate h [equation (12)]. If F q may be estimated from these measurements, then equation (15) may be inverted to find e T and, with appropriate assumptions regarding P, the total net wave energy deposition rate e as well. The net wave heating rate h may then be calculated from equation (12). [56] Another group of temperature measurements in the upper atmosphere also appears to be relevant to the present discussion. Some in situ rocket and ground-based lidar observations indicate the presence of long-lasting adiabatically stratified layers q z 0 in the mesosphere, often associated with a temperature inversion layer near the bottom [Hauchecorne et al., 1987; Whiteway et al., 1995; Meriwether and Gardner, 2000; Williams et al., 2002; Lehmacher et al., 2006]. Several mechanisms have been suggested to explain these layers including gravity-wave breaking and associated convection and turbulent mixing [e.g., Hauchecorne et al., 1987; Whiteway et al., 1995; Lehmacher et al., 2006]. [57] Sometimes these adiabatic layers exhibit a tendency for a slow downward propagation as would be expected for upward propagating large-scale waves such as atmospheric tides. One possible mechanism may be static instability of these large-scale waves [e.g., Liu et al., 2000; Meriwether and Gardner, 2000; Williams et al., 2002]. However, at other times, the adiabatic and inversion layers are observed to persist for several days with no tendency for the downward phase propagation [Hauchecorne et al., 1987]. It should also be noted that temperature amplitudes of several tens of degrees would be required for large-scale waves with relatively long vertical wavelengths to become convectively unstable, which may be excessive for such waves as tides in the midlatitude mesosphere. This suggests that alternative mechanisms may be worth considering as well. [58] Essentially the same behavior may be expected in layers where thermal dissipation of gravity waves takes place, irrespective of the presence of convection and turbulence or large-scale waves. As discussed above, although the downward wave heat transport is not strictly diffusive, it may still be expected to drive the background state toward the adiabatic stratification and create a temperature inversion near the bottom of the layer of dissipation by depositing the heat extracted from upper layers. If substantial wave sources persist in the lower atmosphere, the adiabatic and inversion layers may be supported for several days. When the wave sources become weak, there would be no adiabatic and inversion layers, thus explaining their sporadic nature. With sufficient gravity-wave activity, an adiabatic layer may even 8of12

9 be expected to expand downward, not unlike in the downward propagation of wind-shear layers in such phenomena as the quasi-biennial oscillation (QBO) driven by wave momentum deposition [e.g., Plumb and McEwan, 1978]. As indicated earlier, the enhanced stability in the inversion layer may stimulate an increase in gravity-wave activity, dissipation, and downward heat transport [Bretherton and Garrett, 1969; VanZandt and Fritts, 1989]. [59] Although this mechanism qualitatively explains some of the observed features of mesospheric adiabatic and inversion layers, large-scale waves may still play a role in preconditioning the atmosphere by producing alternating layers of lower and higher stability [Liu et al., 2000]. Such layers will in turn facilitate convective wave instability and dissipation or increased gravity-wave activity, respectively, both resulting in additional downward heat transport. 6. Conclusion [60] The initial purpose of this study was to examine various approaches to heat transport by gravity waves available in the literature and to generalize its representation for use in parameterizations that do not specify wave dissipation mechanisms in sufficient details. Plumb [1979] elucidated the role of sinks of conserved quantities in their downgradient wave transport, and the wave heat flux F q is an example of such a downgradient flux driven by thermal dissipation. Since the same dissipative process is responsible for the loss of the wave potential energy and for its deposition e T into the mean state, there should be a relation between F q and e T. Such a general relation (15) is obtained in this study and is one of its main results. [61] It is important to realize that in order to relate F q to the total wave energy deposition rate e, further assumptions are needed, in particular, regarding the partitioning P of e between the thermal and frictional channels (21); F q is then related to e and, implicitly, to the wave momentum deposition, by equation (22). These relations facilitate the incorporation of the heat transport into gravity-wave parameterizations that primarily deal with the wave drag and do not specify the wave dissipation mechanisms in details necessary for explicit calculations of the heat flux. The general results may also be used in interpretation of observations to obtain estimates of net wave heating rates in the middle and upper atmosphere. [62] It is emphasized that although the downgradient fluxes of conserved quantities may formally be expressed in a diffusive form [Plumb, 1979], F q is actually not proportional to ( q z ), as commonly represented. Although potential temperature is a conserved quantity according to the definition by Plumb [1979], it is not a passive tracer because wave properties depend on its background stratification. Still, generally larger heat fluxes may be expected in regions of enhanced atmospheric stability and vice versa in less stable layers. As a result, thermally dissipating gravity waves are expected to drive the background state to an adiabatic stratification. [63] General thermodynamic properties of dissipating gravity waves are briefly outlined. It is noted that dissipating waves always increase the column enthalpy, and thus the total potential and internal energy, as well as generate entropy in a layer where the dissipation occurs. Interestingly, gravity waves, undergoing pure thermal dissipation, generate the column entropy while exactly conserving the column potential enthalpy. This is well known in atmospheric thermodynamics to formally result in an adiabatic temperature profile. This result confirms the above suggestion that dissipating gravity waves drive the background stratification to adiabatic. Perhaps an even more interesting fundamental question to ponder would be why gravity waves tend to produce an adiabatic stratification that would preclude their vertical propagation. There is certain similarity of this behavior with the downward propagation of critical levels by wave momentum deposition in such phenomena as the QBO. Appendix A [64] Following the standard procedure, equations of wave energetics are derived from equations (5) (9) after their linearization with respect to wave perturbations. Substituting definition (10) into equations (5) (9) and retaining only linear terms, the wave equations may be written in the form: r Du0 Dt þ u ¼ X 0 ; ða1þ r Dw0 þ r0 g ¼ Z 0 ; rc p P Dq0 Dt þ q z w 0 ¼ Q 0 ; Dr þ r þ z w 0 ¼ 0; p 0 p ¼ r0 r þ T0 T ; ða2þ ða3þ ða4þ ða5þ where D=Dt + u@/@x is the material time derivative following the mean horizontal flow and sometimes called the Stokes operator [Gossard and Hooke, 1975]. Because of the stationarity and horizontal homogeneity to be assumed, the mean variables are interchangeable with the partial and with D=Dt. Following Plumb [1979], it is sufficient to assume initially that the dissipative terms X 0, Z 0, and Q 0 are linear in wave perturbations u 0, w 0, and q 0, respectively, without detailed specifications of the dissipation mechanisms. Their dissipative nature is guaranteed by the assumed negative definiteness of covariances q0q0 and p0q0 [Plumb, 1979] and the positive definiteness of the frictional heating term (see below). [65] Combining equation (A5) with equation (2), the perturbation thermodynamic equation is compactly presented in terms of q 0 in equation (A3), but it may also be written in terms of T 0 and p 0,orp 0 and r 0, for example. The various equivalent forms facilitate the derivation of equations for two different forms (gravity wave and acoustical) of wave potential energy. These may be combined with the 9of12