Mean-Flow Effects of Thermal Tides in the Mesosphere and Lower Thermosphere

JUNE 2017 BECKER 2043

ERICH BECKER Leibniz Institute of Atmospheric Physics, Kuhlungsborn,€ Germany

(Manuscript received 27 June 2016, in ﬁnal form 24 March 2017)

ABSTRACT

This study addresses the heat budget of the mesosphere and lower thermosphere with regard to the energy deposition of upward-propagating waves. To this end, the energetics of gravity waves are recapitulated using an anelastic version of the primitive equations. This leads to an expression for the energy deposition of waves that is usually resolved in general circulation models. The energy deposition is shown to be mainly due to the frictional heating and, additionally, due to the negative buoyancy production of wave kinetic energy. The frictional heating includes contributions from horizontal and vertical momentum diffusion, as well as from ion drag. This formalism is applied to analyze results from a mechanistic middle-atmosphere general circulation model that includes energetically consistent parameterizations of diffusion, gravity waves, and ion drag. This paper estimates 1) the wave driving and energy deposition of thermal tides, 2) the model response to the excitation of thermal tides, and 3) the model response to the combined energy deposition by parameterized gravity waves and resolved waves. It is found that thermal tides give rise to a signiﬁcant energy deposition in the lower thermosphere. The temperature response to thermal tides is positive. It maximizes at polar latitudes in the lower thermosphere as a result of poleward circulation branches that are driven by the predominantly westward Eliassen–Palm ﬂux divergence of the tides. In addition, thermal tides give rise to a downward shift and reduction of the gravity wave drag in the upper mesosphere. Including the energy deposition in the model causes a substantial warming in the upper mesosphere and lower thermosphere.

1. Introduction The energy deposition is related to thermodynamic irreversibility. In the basic hydrodynamic equations of mo- The dynamical control of the middle atmosphere is tion, this irreversibility is expressed in terms of mechanical mainly due to waves that are generated in the tropo- and thermal dissipation due to molecular viscosity and sphere and then propagate to higher altitudes where heat conduction. In the troposphere, the dissipation takes they become dynamically unstable, break, and cause place at the end of macroturbulent cascades of kinetic wave–mean-flow interactions. Nonorographic gravity energy and available potential energy (e.g., Augier and waves and thermal tides (herein ‘‘tides’’) can propa- Lindborg 2013; Brune and Becker 2013). These cascades gate over many scale density heights before they are maintained by the generation of available potential break or are attenuated by other dissipative processes. energy due to differential heating (radiative, latent, and Therefore, these waves can generate quite strong sensible heating). The mechanical dissipation (i.e., the mean-flow effects per unit mass in the mesosphere and frictional heating due to molecular viscosity) is of funda- lower thermosphere (MLT). The mean-flow effects mental importance in the Lorenz energy cycle. As stated are usually described in terms of the divergence of the by Lorenz (1967), the frictional heating is the net diabatic Eliassen–Palm flux (EPF) that drives a residual cir- heating of the atmosphere in climatological equilibrium. culation [see Andrews et al. (1987)]. The underlying In the middle atmosphere, turbulent energy cascades theoretical framework is known as the transformed are induced by the breakdown of gravity waves (GWs) Eulerian mean (TEM) equations. However, the TEM that propagate upward from the troposphere, leading to equations do not include the energy deposition that mechanical and thermal dissipation (e.g., Lindzen 1981; accompanies the EPF divergence. Hines 1997; Lübken 1997; Fritts and Alexander 2003). On the other hand, the convergence of the vertical potential Corresponding author: Erich Becker, [email protected] energy flux of GWs gives rise to the energy deposition that

DOI: 10.1175/JAS-D-16-0194.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/27/21 04:49 AM UTC 2044 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 contributes to the large-scale heat budget. The relation to approximately the lower mesosphere in winter and drive between the dissipation rates and the energy deposition is the residual circulation in the stratosphere [see the review not straightforward. This holds not only for GWs but also of Alexander et al. (2010)]. Traveling planetary waves in for large-scale waves that may account for a significant the MLT, on the other hand, are generated in situ by dy- vertical energy transport. The main purpose of the present namical instabilities and generate an EPF divergence that paper is to derive an expression for the energy deposition opposes the GW drag (Plumb 1983; Norton and Thuburn of waves resolved in middle-atmosphere climate models 1996; McLandress et al. 2006; Pendlebury 2012). There are and to estimate the importance of the energy deposition of also stationary wave structures in the MLT, especially thermal tides. in the northern winter hemisphere. These are known to It is well known that the energy deposition of be caused at least partially by regional differences in GWs yields a substantial contribution to the large-scale refraction/filtering of GWs at lower altitudes and the heat budget of the MLT (e.g., Fomichev et al. 2002). This resulting longitude-dependent GW drag in the MLT contribution is said to be direct since it is not related to (Smith 2003; Lieberman et al. 2013). As will be discussed adiabatic heating or cooling by the residual circulation nor in section 2, quasigeostrophic waves are not expected to to the downward heat flux associated with GW damping. yield a significant energy deposition in the MLT. The correct formulation of the energy deposition was first The morphology of thermal tides has frequently been mentioned by Hines and Reddy (1967). Estimates for di- studied via observation and modeling. Reviews can be found rect GW heating were first given by Liu (2000).Further in Forbes (2007) or Smith (2012), for example. The linear accounts of the direct heating of GWs can be found in theory of tides (Chapman and Lindzen 1969) and linear several studies (e.g., Becker 2004; Akmaev 2007; Becker numerical models that include specific damping mechanism and McLandress 2009; Yigitetal.2008 ; Shaw and Becker for tides (e.g., Hong and Lindzen 1976) are usual tools to 2011). Using geometric height as the vertical coordinate, a interpret tidal observations or the results from comprehen- main result is that the energy deposition is given by the sive GCMs. Linear numerical models of tides usually also convergence of the vertical pressure flux, which is positive include nontrivial mean flows (e.g., Grieger et al. 2004). in the case of wave breaking, plus an additional term given Overall, the refraction, breakdown, and damping of by the negative momentum flux times the wind shear. GWs in the MLT are subject to a regular daily cycle that is Hence, the energy deposition can be expressed in terms of induced by the tidal variations of the large-scale winds. wave fluxes and the mean flow. It is thus tempting to as- Such GW effects then can give rise to a modulation of the sume that the energy deposition is automatically simulated tides. The role of GWs in modulating the phases and am- in a general circulation model (GCM) with regard to waves plitudes of tides has been the subject of many model studies that are explicitly resolved. In the present study we show (e.g., Lu and Fritts 1993; Ortland and Alexander 2006; Liu that this assumption is not valid and that the energy de- et al. 2008, 2014). It is typically found that the GW drag position of resolved waves is mainly due to the frictional alters the phases and vertical wavelengths of the tides, heating associated with the momentum diffusion that whereas the GW-induced turbulent vertical diffusion gives damps the waves. It is therefore essential to include the rise to the damping of the tides. As pointed out by Ortland complete frictional heating in a GCM in order to simulate and Alexander (2006), the simulated effects of GWs on the the heat budget of the MLT correctly. This frictional tides are sensitive to the details of how the GWs are rep- heating can be represented in a GCM by means of resented. Therefore, a general consensus of how the mor- the shear production associated with the employed mac- phology of tides is modulated by GWs has not yet been roturbulent diffusion scheme. Assuming a quasi-stationary reached. In particular, model estimates usually involve GW turbulent kinetic energy equation, the shear production parameterizations that are based on the conventional equals the molecular frictional heating minus the adiabatic framework of quasi-stationary and single-column dynamics conversion that takes place at the turbulent scale (e.g., van for the GWs. As noted by Senf and Achatz (2011),this Mieghem 1973). In addition, molecular diffusion and ion approach may lead to significant deviations for the tidal drag become important for wave damping in the thermo- amplitudes and phases when compared to the corre- sphere (Hong and Lindzen 1976; Vadas and Fritts 2005; sponding results from a three-dimensional and transient Vadas and Liu 2009), and corresponding frictional heating ray-tracing model. Further complications arise from wave– rates have to be considered as well. wave interactions among different tidal modes, as well as Atmospheric waves contributing to the wave spectrum in among tides and traveling planetary waves (e.g., Norton the MLT other than nonorographic GWs and tides usually and Thuburn 1999; Moudden and Forbes 2014). do not propagate over the entire altitude range. For ex- The question of how tides modulate the temporally ample, quasi-stationary planetary Rossby waves and oro- averaged zonal-mean GW drag was investigated by graphic GWs forced in the troposphere propagate only up Miyahara and Wu (1989). They employed an idealized

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GCM where the zonal-mean flow was considered as the thermodynamic variable. This holds also for the basic flow in the parameterization of GWs. They found KMCM. Therefore, in order to interpret model results that tides caused a westward zonal-wind anomaly in the with regard to the energy deposition, we must analyze the mesopause region and in the lower thermosphere that was enthalpy budget. For this purpose we shall use a formalism related to the westward EPF divergence of the tides. This that is analogous to the derivation of the energy deposition wind anomaly then shifted the eastward GW drag in the owing to GWs (e.g., Becker 2004). summer hemisphere to higher altitudes, and the drag per When using the primitive equations with pressure as unit mass became significantly stronger. Becker (2012) vertical coordinate for this analysis, the subgrid-scale used a high-resolution mechanistic GCM with simplistic terms look cumbersome, and the notation cannot easily tidal forcing and resolved GWs. He found that, contrary to be reconciled with the notation commonly used in GW the result of Miyahara and Wu (1989), tides gave rise to a theory. Alternatively, when using z as vertical co- downward shift and weakening (per unit mass) of the GW ordinate, the averaging over scales is much more drag in the summer MLT. Becker (2012) interpreted the complicated; it requires density-weighted averages and downward shift as a result of the nonlinear dependence of an incompressibility approximation to exclude sound GW instability on the tidal wind variations. waves. Therefore, in order to keep the mathematical In this paper we use a mechanistic GCM to investigate procedure as simple as possible, we utilize the primitive the role of tides on the general circulation of the MLT. As equations in the anelastic approximation. usual, GWs are parameterized and evaluated at each model The anelastic approximation invokes a hydrostatic ref- gridpoint and time step. We focus on the energy deposition erence state that depends on height z only. Thermody- ~ of the tides and its importance for the zonal-mean sensible namic variables are then expanded as X 5 Xr(z) 1 X(l, f, heat budget of the MLT, and we provide estimates of the z, t), where X represents either density r,pressurep, generalized EPF divergence of the tides. In addition, we temperature T, or potential temperature Q.Furthermore, perform sensitivity experiments in order to estimate the l is longitude, f is latitude, and t is time. Since there are zonal-mean model response to 1) the thermal excitation different kinds of anelastic and pseudo-incompressible of tides and 2) to the energy deposition owing to param- equations in the literature (e.g., Durran 1989), and since eterized GWs and resolved waves. Results are based on these equations use the Exner pressure as prognostic simulations using an updated version of the Kühlungsborn thermodynamic variable, we derive in the appendix a Mechanistic General Circulation Model (KMCM) as de- version of the anelastic primitive equations that includes scribed in Becker et al. (2015). This model includes explicit the enthalpy equation [see Eqs. (A19)–(A22)]. In this computations of radiation and the tropospheric moisture section, we extend these equations by external diabatic cycle, as well as energetically consistent parameterizations heating, as well as by subgrid-scale closures for ion drag of GWs, turbulent diffusion, and ion drag. and diffusion of momentum and sensible heat, and we In section 2, we revisit the energetics of GWs using an average over the scale of GWs in the single-column ap- anelastic version of the primitive equations, which are proximation. This yields the following forced-dissipative derived in the appendix, and we derive an expression for anelastic primitive equations: the energy deposition owing to resolved waves. A description of the KMCM and our simulations is given in v2 =p › v 5 v 3 (f 1 j)e 2 w› v 2 = 2 section 3.Insection 4, we present KMCM results regarding t z z r 2 r the contributions of the tides to the zonal-mean momen- 1 1 › [r (K 1 n)› v 2 r v0w0] 1 =(K S ) 2 Dv, tum and heat budgets of the MLT, and we discuss the r z r z z r h h model response in the aforementioned sensitivity experi- r ments. Our conclusions are summarized in section 5. (1) p~ r~ › 52g , (2) zr r 2. Energy deposition: Parameterized gravity waves r r versus resolved waves 5 = 1 r21› r 0 v r z( rw), and (3) GCMs are usually based on the primitive equations with enthalpy (sensible heat) as the prognostic

c n ~ p Kz T K c d T 52wg(1 1 T/T ) 1 Q 1 Q 1 › r r › Q1 › T 2 T0w0 1 c = h=T p t r rad lat r z r Q z z p r Pr r Pr Pr 1 1 n › 2 1 2 1 1 1 n › 0 2 1 0 2 1 0 0 2 21 0 0 (Kz )( zv) KhjShj v (Dv) (Kz )( zv ) KhjShj v (Dv ) gTr T w . (4)

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The nomenclature is defined in the appendix; v and Note that ion drag does not represent an internal force = denote the horizontal velocity field and nabla oper- like stress divergences due to pressure and momentum ator, j is the relative horizontal vorticity, f is the fluxes. Additionally, ion drag is linear in the flow velocity,

Coriolis parameter, and w and ez denote the vertical and the associated rate of change of kinetic energy is al- velocity and unit vector in the vertical direction. The ways negative at each point in the flow. The same holds GW perturbations are denoted by primes, and the when the Lorentz deflection is included in the tensor D. average over the GW scale is indicated by an overline Following Schmidt et al. (2006), we assume that the ki- for the vertical fluxes of horizontal momentum and netic energy transferred to the plasma is thermalized lo- temperature in Eqs. (1) and (4), v0w0 and T0w0,aswell cally so that the frictional heating due to ion drag is given as for the frictional heating rates in the last row of by the negative rate of change of the kinetic energy.2 Eq. (4). All other flow variables are assumed to be As mentioned in section 1, the turbulent frictional averaged over the GW scale. heating mainly represents the molecular frictional heat- The vertical momentum diffusion in Eq. (1) is ing in the real atmosphere, which takes place at the standard and involves the turbulent and molecular microscale and results from wave breaking and the so- diffusion coefficients Kz and n.Thistermiswritten induced forward kinetic energy cascades. In the ther- along with the momentum deposition due to GWs. The mosphere, the molecular kinematic viscosity becomes horizontal momentum diffusion in Eq. (1) is formu- increasingly important with altitude and gives rise to lated with a stress tensor; that is, Sh is the horizontal damping of GWs (Vadas and Fritts 2005; Vadas and Liu strain tensor and Kh the turbulent horizontal diffusion 2009). The turbulent vertical diffusion coefficient is coefficient [see Becker and Burkhardt (2007),their therefore completed by the kinematic molecular viscos- Eqs. (7) and (8)]. The tensor D describes the damping ity. The corresponding molecular heat conduction is in- rate coefficients for ion drag [see Liu and Yeh (1969)]. cluded in terms of diffusion of temperature instead of The diabatic heating rates in Eq. (4) include radiative potential temperature.3 and latent heating, Qrad and Qlat, respectively, as well The system of Eqs. (1)–(4) conserves energy regarding as the vertical and horizontal diffusion of heat (last two the dynamics of the mean flow (see appendix). Energy terms in the first row), where the former is written conservation also holds regarding diffusion of the mean along with the convergence of the vertical GW heat flow. Here, we have assumed a diagnostic turbulence flux. The turbulent diffusion coefficients are scaled model—that is, a quasi-stationary turbulent kinetic energy byaPrandtlnumberPr. The second row in Eq. (4) equation [see section 3 in Becker (2004)]. The system is, represents the frictional heating due to momentum however, incomplete regarding the nonlinear GW terms. diffusion and ion drag, as well as the negative buoy- The reason is that the GW kinetic energy is another ancy production of GW kinetic energy. prognostic variable that must be evaluated. To formulate In accordance with the usual formulation of subgrid- the GW kinetic energy equation, we add momentum dif- scale closures for diffusion and GWs in GCMs, we have fusion and ion drag to the momentum equation [Eq. (A19)] assumed that the diffusion coefficients are mean-flow and linearize about a mean flow. We then multiply by v0 quantities. Furthermore, we have utilized the incompres- and average over the GW scale. Since the diffusion ten- r~ r 52Q~ Q 52~ sibility assumption—that is, / r / r T/Tr [see dencies are assumed to be small, the work performed by Eqs. (A5) and (A8)]—to rewrite the adiabatic heating friction on a control volume of the size of the GW scale is term. Since the frictional heating is quadratic in the hori- negligible, and we obtain the following approximations: zontal velocity field, it includes contributions from the resolved flow and from the GW perturbations. All r21 0 › r 1 n › 0 52 1 n › 0 2 r v z[ r(Kz ) zv ] (Kz )( zv ) and (5) frictional heating rates specified in Eq. (4) are posi- 0 = 0 52 0 2 tive definite, as is required by the second law of thermo- v [ (KhSh)] KhjShj . (6) dynamics.1

2 Ion drag does not perform any net work on a control volume of 1 Once the (symmetric) stress tensor for momentum diffusion is the flow. This is different for internal forces, where the work per- given, the associated frictional heating (also known as the shear formed is equal to the surface integral of the stress tensor times the production) follows from the energy conservation law for arbitrary velocity field. fluid volumes [see text of Lindzen (1990)]. Other forms of the 3 Gassmann and Herzog (2015) recently discussed that the tur- frictional heating are sometimes used in climate models. But these bulent vertical diffusion of Q may lead to negative thermal dissi- are neither consistent with the energy conservation law nor with pation. This issue is potentially important for the heat budget of the the second law [see discussions in Becker (2003) and Becker and MLT. However, it is beyond the scope of the present study to Burkhardt (2007)]. further address this problem.

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Using the single-column approximation and the hydro- GW potential energy equation. To derive this equation, static and continuity equations for the GWs, the GW we linearize Eq. (A22) about a mean flow and add kinetic energy equation can then be written as vertical and horizontal diffusion of heat (in terms of the temperature perturbation T0). We then multiply by › 1 › 0 2 0 2 2 ( t w z)v /2 g T /(Nr Tr ), where Nr is the buoyancy frequency of the 52 0 0 › 2 r21› 0 0 1 21 0 0 reference state, and we average over the GW scale (v w ) zv r zp w gTr T w while noting that Tr and Nr vary much more slowly with 2 1 n › 0 2 2 0 2 2 0 0 height than r or Q . The assumption that the diffusive (Kz )( zv ) KhjShj v (Dv ). (7) r r GW heat fluxes are negligible across the GW scale An assumption implicitly made in GW parameteriza- leads to approximations that are analogous to Eqs. (5) tions is that the left-hand side (lhs) of Eq. (7) equals and (6): zero. In this quasi-stationary case, the frictional heat- 1 K 1 n K 1 n ing and buoyancy terms in Eq. (4) can be eliminated T0› r z › T0 52 z (› T0)2 and (10) r z r z z via r Pr Pr K K h 0 h 2 2 2 T0= =T 52 (=T0) . (11) E 52r 1› p0w0 2 ( 0w0) › 52gT 1T0w0 r z v zv r Pr Pr 1 1 n › 0 2 1 0 2 1 0 0 (Kz )( zv ) KhjShj v (Dv ). (8) Using these results, the GW potential energy equation in the single-column approximation can be written as Here, E represents the well-known energy deposition of GWs (e.g., Hines 1997). Equation (8) shows that the g2T02 (› 1 w› ) energy deposition can be expressed in terms of the GW t z 2 2 2Tr Nr fluxes and the mean wind shear. The right-hand side g2 K 1 n K (rhs) of Eq. (8) represents the dissipation of GW kinetic 52gT21T0w0 2 z (› T0)2 1 h (=T0)2 . r N2T2 P z P energy minus the buoyancy production. The buoyancy r r r r 2 21 0 0 5 Q 1Q0 0 (12) production in Eq. (7) (i.e., gTr T w g r w )isusu- ally negative in regions of nonconservative wave propagation. As a result, each term on the rhs of Eq. (8) is Hence, in the quasi-stationary case, the buoyancy positive. Note that we retain the horizontal diffusion term on the lhs of Eq. (8) corresponds to the thermal terms for the GW dynamics. These terms are usually ig- dissipation—that is, the dissipation of the GW potential nored in GW parameterizations but are important in energy via the turbulent and molecular diffusion of heat. GW-resolving simulations (Becker 2009). For the sake of Since we have neglected any baroclinic–barotropic in- completeness we also consider the effect of ion drag on stability in the derivation of (12), the buoyancy term the GWs. Inserting Eq. (8) into Eq. (4) leads to can only convert wave kinetic energy into wave potential energy. In fact, the buoyancy production of GW c d T 52wg(1 1 T~/T ) 1 Q 1 Q kinetic energy is negative in the process of energy p t r rad lat deposition. This may be contrasted to the Lorenz en- c K T n 1 p› r z r › Q1 › T 2 T0w0 ergy cycle in the troposphere, where wave potential r z r P Q z P z energy is generated via baroclinic instability (main- r r r r K tained by differential heating) and converted into 1 c = h=T 1 E 1 (K 1 n)(› v)2 p P z z wave kinetic energy. r From the foregoing analysis of the energetics of GWs 1 2 1 KhjShj v (Dv), (9) we can conclude that the energy deposition is not automatically captured in a GCM. The combination of which corresponds to the thermodynamic equation Eqs. (4) and (8) rather suggests that the energy de- for a GCM with energetically consistent parameteri- position of resolved waves is only fully accounted for zations of GWs, turbulent and molecular diffusion, and when the frictional heating is included in the thermo- ion drag. dynamic equation for the resolved flow. According to Eq. (8), the energy deposition primarily The energy deposition is maintained by the pressure equals the dissipation of kinetic energy by momentum flux (or the potential energy flux in p coordinates). In diffusion and ion drag. The remaining buoyancy flux turn, the pressure flux is proportional to the vertical term, on the other hand, can be related to the dissipation momentum flux. Since the generalized EPF divergence of GW potential energy. This is readily seen from the of the tides includes a strong contribution from the

Unauthenticated | Downloaded 09/27/21 04:49 AM UTC 2048 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 vertical momentum flux (Miyahara and Wu 1989), we › 5 y 1 j 2 › 1 r f 21 t[u] res(f [ ]) wres z[u] ( ra cos ) div(EPF) anticipate that the convergence of the vertical pressure 1 flux owing to tides is significant and will produce a sig- 1 › fr [(K 1 n)› u] 2 r [u0w0]g r z r z z r nificant energy deposition at high altitudes. On the other r 1 = 2 hand, waves that can approximately be described by ex [ (KhSh) Dv] quasigeostrophic theory possess no significant vertical (15) momentum flux and will therefore not produce a relevant energy deposition in the MLT. and From Eqs. (1)–(3) and (9) it is straightforward to write ( ! down the zonally averaged primitive equations in the [T*y*] EPF 5 r a cosf e 2[u*y*] 1 › [u] TEM framework with the complete enthalpy equation. r y z 1 › g/cp z[T] We use the notation of Lorenz (1967) and denote zonal !) means by brackets and deviations from the zonal mean [T*y*] 1 e (f 1 [j]) 2 [u*w*] , z 1 › by an asterisk. The residual circulation in the anelastic g/cp z[T] case is defined as4 ! (16) y y 5 y 2 1 › r [T* *] res [ ] z r and (13) r 1 › where ex and ey denote the unit vectors in zonal and r g/cp z[T] ! meridional direction, respectively, and the symbol a de- › f [T*y*] notes Earth’s radius. Assuming that only waves con- w 5 [w] 1 cosf . (14) res f 1 › a cos g/cp z[T] tribute to the buoyancy production—that is, that 2 21 ~ gTr [w][T] is negligible in the zonally averaged form The transformed zonal-mean zonal momentum equa- of Eq. (9)—the TEM thermodynamic equation can be tion and EPF components then are5 written as

! ! › › f g 1 [T*y*] f c › [T] 52c y [T] 2 c w 1 › [T] 1 [Q ] 1 [Q ] 2 › r [T] 1 r [T*w*] p t p res a p res c z rad lat r z rg/c 1 › [T] a r p r p z 1 T [K › Q] [y› T] c 1 › c r r z z 1 z 2 [T0w0] 1 p [= (K =T)] r z p r Q h r r Pr Pr Pr 1 2 g 1 1 n › 2 1 2 1 [E] [T*w*] [(Kz )( zv) ] [KhjShj ] [v (Dv)]. (17) Tr

Equations (13)–(17) are equivalent to the corresponding In practice, we use pressure as vertical coordinate TEM equations given in [Andrews et al. (1987),theirEqs. when computing the energy deposition due to the re- (3.5.1), (3.5.3), (3.5.2a), and (3.5.2e)],6 except for the in- solved flow. Assuming that resolved waves rather than clusion of GW effects, heat diffusion, and frictional the zonal-mean flow contribute to the frictional heating heating. Applying the interpretation of the energetics of in the middle atmosphere, the energy deposition due to GWs to the TEM equations, we can conclude that the last resolved waves can be computed using row of Eq. (17) represents the complete zonal-mean energy deposition that is due to parameterized GWs and the 5 R v 1 1 n r› 2 Erw [T* *] [(Kz )(g pv) ] resolved flow. While the buoyancy flux term [second term p in the last row of Eq. (17)] is implicitly included in Eq. 1 2 1 [Kh j Shj ] [v (Dv)], (18) (3.5.2e) of Andrews et al. (1987), who used potential temperature instead of enthalpy as prognostic variable, where the subscript rw indicates resolved waves. The the energy deposition was not considered. negative buoyancy production term has been reformulated according to Shaw and Becker (2011),wherev represents the pressure velocity. Furthermore, the vertical wind shear 4 The quasigeostrophic version was given in Becker (2012). has been rewritten according to the hydrostatic formula 5 The zonally averaged hydrostatic, continuity, and gradient- wind equations are straight forward and not further specified here. and the horizontal diffusion and dissipation is evaluated at 6 It is readily shown that [T*y*]/(g/cp 1 ›z[T]) 5 [Qy*]/›z[Q] surfaces of constant pressure [see Becker and Burkhardt holds in the anelastic approximation. (2007)]. The energy deposition given by Eq. (18) relates to

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FIG. 1. Annually and horizontally averaged diffusion parameters in the control simulation. (a) Mixing-length- based vertical diffusion coefficient (black) and the corresponding vertical mixing length (gray). (b) Vertical diffusion coefficients due to GWs (black) and molecular viscosity (gray). (c) Nonlinear, Smagorinsky-type horizontal diffusion coefficient (black solid) and the corresponding horizontal mixing length (gray solid). The back dashed line gives the additional linear horizontal diffusion coefficient. Note the logarithmic scaling for the molecular diffusion coefficient in (b), as well as for the horizontal mixing length and diffusion coefficients in (c). The displayed altitude regime reaches from about 15 to 200 km. the tides if we insert just the tidal wind field and tidal equidistant azimuthal directions at about 600 hPa. The temperature perturbation. following set of tunable parameters are used [for defini- tion of symbols, see Becker and McLandress (2009)]: the 2 2 minimum vertical wavenumber is m 5 5 3 10 4 m 1;the 3. Model description 0 maximum of the Desaubies spectrum at the launch level 5 3 23 21 We use an updated version of KMCM as described in is set to a vertical wavenumber of m*j 4.2 10 m ; Becker et al. (2015, hereafter BKL15). In the following the total squared horizontal wind variance at the launch 2 22 we describe the main features of this model. level is sh 5 0.87 m s between 358S and 358N, and sh 2 The KMCM has a standard spectral dynamical core decreases to 0.76 m2 s 2 toward the poles; the horizontal combined with a hybrid vertical coordinate (Simmons and wavelength is 380 km; and the fudge factors are f1 5 1.5, Burridge 1981). We apply the same horizontal resolution f2 5 0.3, and f6 5 0.35. as in BKL15—that is, triangular spectral truncation at The KMCM employs a standard local vertical diffu- wavenumber 32, but an extended model top, using 80 full sion scheme (Holtslag and Boville 1993). Additional 2 modellayersupto83 10 7 hPa (;200 km). The model is diffusion coefficients result from the GW schemes and equipped with an idealized radiation scheme, as well as from molecular diffusion in the thermosphere. To en- with an explicit tropospheric moisture cycle, where the sure a consistent scale interaction between parame- method of Schlutow et al. (2014) is used to compute the terized GWs and the resolved flow, the complete transport of water vapor. Land–sea contrasts are taken vertical diffusion coefficient is used to damp parame- into account by including orography, land–sea masks for terized GWs below their critical levels. In particular, several surface parameters (albedo, relative humidity, the nonorographic GWs are strongly damped in the ther- heat capacity, and roughness length), and a simple slab mosphere by molecular diffusion; they have no notable 2 ocean in order to close the surface energy budget and the effect on the circulation above about 10 4 hPa (;110 km). radiation budget. This method is also applied in the Whole Atmosphere Parameterizations of GWs consist of the McFarlane Community Climate Model (WACCM) (Garcia et al. scheme for orographic GWs (McFarlane 1987)andan 2007, their appendix A) in the context of a Lindzen-type extended Doppler-spread parameterization (DSP) for parameterization. The solid gray line in Fig. 1a shows the nonorographic GWs (Becker and McLandress 2009) prescribed asymptotic mixing length above 15 km for local thatisbasedontheideasofHines (1997). The non- vertical diffusion scheme. The solid black curve in the orographic GWs are launched isotropically in eight same panel gives the corresponding vertical diffusion

Unauthenticated | Downloaded 09/27/21 04:49 AM UTC 2050 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 coefficient in the climatological horizontal mean from the molecular composition above the mesopause, the the control simulation that is further described below. current version of the KMCM applies altitudinal profiles Figure 1b displays the corresponding vertical diffusion for the heat capacity, gas constant, and molecular vis- coefficients due to molecular viscosity (gray curve) and cosity according to Vadas (2007). to turbulence generated by the breakdown of param- Model results are computed from a 15-yr control eterized GWs (black curve). The vertical momentum simulation (after equilibration of climatology). Snap- diffusion of the resolved flow is completed by the as- shots of the spectral state vector are archived every sociated frictional heating according to an energy 90 min. All model quantities are reconstructed from preserving vertical discretization (Becker 2003; Boville these data. The tidal variations for a particular month and Bretherton 2003). are computed in the following way: All saved spectral GCMs generally require some scale-selective damping state vectors of the 15 realizations of that month are of the horizontal wind in order to balance the horizontal rearranged such as to combine them into a single daily energy and enstrophy cascades associated with the re- cycle; this model day then consists of 16 universal times solved flow. The KMCM employs a harmonic horizontal (i.e., 1.30 a.m., 3.0 a.m., ... , 24.0 p.m.), where each momentum diffusion for this purpose. The scheme is daytime contains 15 (number of years) times 30 (number based on a symmetric stress tensor formulation and of days for each month) snapshots. Taking the average includes the frictional heating. The horizontal diffusion over these 450 snapshots for each daytime gives a mean coefficient consists of a Smagorinsky-type (nonlinear) daily cycle representative for the chosen month. The part and an additional linear part. Detailed information mean daily cycle should include only the zonal-mean about the computation of the horizontal diffusion state, stationary waves, and tides. Because of the final terms is given in Becker and Burkhardt [2007,their number of snapshots that are averaged for each daytime, Eqs. (13), (15), (17)–(23), and (31)]. The free parame- other wave components may also contribute to the di- ter of the nonlinear part [Eq. (13) in Becker and agnosed tides—for example, higher harmonics of the Burkhardt (2007)] is the horizontal mixing length as a quasi-2-day wave. For the purpose of the present study, function of height. The profile applied in the present however, our simplified diagnostics of the total tidal model is given by the gray curve in Fig. 1c. The resulting wave field is considered to be sufficient. Smagorinsky-type horizontal diffusion coefficient is dis- Figure 2 illustrates the zonal-mean climatology for July played by the solid black curve (climatological horizontal- and January of the control simulation in terms of the mean result from the control simulation). The additional temperature along with the residual mass streamfunction prescribed horizontal diffusion coefficient (dashed curve and the ion drag (ID) and the zonal wind along with the 2 in Fig. 1c) is significant above about 10 5 hPa and provides sum of the resolved EPF divergence and the parameter- a sponge layer in the upper model domain. ized GW drag (GWD). The simulated wind and temper- The complete momentum diffusion applied in the ature fields are very reasonable, and the wave driving in the KMCM preserves angular momentum and energy for MLT compares well with other models (e.g., Richter et al. arbitrary fluid volumes. In particular, the vertical energy 2008). In particular, the model simulates a cold summer flux and EPF by resolved waves are precisely deposited mesopause and a warm winter polar stratopause as a when these waves are damped by diffusion. This holds consequence of the GW-driven residual circulation in the also for the sponge layer as formulated in the KMCM. In upper mesosphere. Figure 2 furthermore confirms that the contrast, angular momentum and energy conservation hemispheric differences due to stronger quasi-stationary are strongly violated when Rayleigh friction is used as a Rossby waves in the northern winter stratosphere are sponge layer. well captured. This becomes evident when comparing the To properly describe the damping of tides in the ther- polar night jets in Figs. 2b and 2d. As a result of this mosphere, we also include ion drag and the associated asymmetry, the winter westward GW drag in the MLT is frictional heating. Ion drag is particularly efficient in clearly stronger in July than in January. As shown by damping large-scale waves that have periods larger than BKL15 and Karlsson and Becker (2016), this leads to about 1 h and propagate perpendicular to the direction of a colder summer polar mesopause during July than dur- Earth’s magnetic field. Here we use the same formulation ing January via the interhemispheric coupling mecha- as in Fomichev et al. (2002, their section 3.3); that is, we nism. The zonal-wind structure in the MLT exhibits not consider only the damping of the horizontal wind and only the wind reversals in the mesopause region but also 2 neglect the Lorentz deflection, we take the vertical profile additional reversals around 10 5 hPa (;150 km) in sum- 2 for the damping coefficient from Hong and Lindzen mer and 3 3 10 5 hPa (;130 km) in winter, which com- (1976) for solar minimum conditions, and we use a simple pares well with results from comprehensive models formula for the dip angle. To account for the changes in (Fomichev et al. 2002; Schmidt et al. 2006).

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FIG. 2. Zonal-mean model climatology in the control run. (a) Temperature (colors), residual mass streamfunction 2 2 (black contours drawn for 61, 10, 100 3 109 kg s 1 below 1 hPa, and for 60.0001, 0.001, 0.01, 0.1 3 109 kg s 1 farther 2 2 above), and ion drag (white contours drawn for 65, 10, 20, 40, 80 m s 1 day 1) for July. (b) Zonal wind (colors) and the sum of the resolved EPF divergence plus the parameterized GW drag (contours drawn for 65, 10, 20, 40, 80, 2 2 120 m s 1day 1 above 100 hPa) for July. (c),(d) As in (a) and (b), respectively, but for January.

It is important to validate the simulated tides. Figures 3a To estimate the response of the MLT to tides, it is and 3b show the mean zonal-wind tides at 528NforJuly necessary to perform a sensitivity experiment where the and January after averaging the tides at the same local excitation of tides is excluded. For this purpose, the ra- time over all longitudes. There is a predominant diurnal diative transfer calculations were performed as in the tide in the stratosphere and lower mesosphere, whereas a control run, but only the zonal means of the shortwave semidiurnal tide develops above about 0.03 hPa (;70 km). heating rates were fed into the thermodynamic equation This behavior is also reflected in Figs. 3c and 3d,which of the model. In this way the excitation of thermal tides show the meridional-wind and temperature tides corre- due to absorption of solar radiation was excluded while sponding to Fig. 3a. The transition from a diurnal to a the zonal-mean radiative forcing of the model remained semidiurnal tide with altitude in the extratropics has been unchanged. The corresponding model experiment will known also from other studies [see section 4.3 in Smith be referred to as the no-tides simulation. A second (2012)]. Figures 3a,c,d maybecompareddirectlyto sensitivity experiment compares the control simulation ground-based observational results of Gerding et al. (2013, to a perturbation run where the complete energy de- their Figs. 4 and 5) obtained at a single station in northern position due to parameterized GWs and resolved waves middle latitudes. Though the tidal amplitudes are some- was removed from the thermodynamic equation (no- what smaller in the model, the phases fit reasonably well to energy-deposition run). the observational estimates. In particular, we expect that the simulated tidal variations of the zonal wind give rise 4. Results and discussion to a modulation of the instantaneous GW drag. A more detailed analysis of the morphology of the tides simulated In this section, we shall first discuss the mean-flow with the KMCM is beyond the scope of the present study. effects of tides as deduced from the control simulation.

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FIG. 3. (a) Mean zonal-wind tide at 528N in July. (b) As in (a), but for January. (c) As in (a), but for the mean temperature tide. (d) As in (a), but for the mean meridional-wind tide. The mean tides are obtained by averaging the daily cycle with regard to local time over all longitudes. The displayed altitude regime extends from about 20 to 120 km.

This provides the basis for the interpretation of the is maintained by the strong westward GW drag aloft model response to tides and energy deposition. (cf. Fig. 2b). The colors in Figs. 4b and 4d confirm former results of a. Mean-flow effects of thermal tides Miyahara et al. (1993) that the tides exhibit a negative EPF Figures 4a and 4c show the EPF divergence of the divergence in the lower thermosphere. In the present resolved waves during July and January in the control model simulation, the westward tidal wave drag maxi- simulation above 50hPa (;20 km). The contribution mizes at subtropical and middle latitudes around the lower from planetary waves (zonal wavenumbers 1–6 only) to edge of the sponge layer, reaching values of about 220 to 2 2 the resolved EPF divergence is indicated by the black 230 m s 1 day 1 during winter. There is an eastward tidal contours in Figs. 4b and 4d below 0.001 hPa (;90 km). EPF divergence at polar latitudes in either hemisphere While the upper troposphere (not shown) and the north- that indicates nonmigrating components with eastward ern winter middle atmosphere exhibit the expected phase propagation. The predominantly westward wave westward wave driving due to planetary Rossby waves, driving induces poleward circulation branches in either the summer mesopause region (i.e., around 0.003 hPa) hemisphere above about 0.001 hPa (;90 km) as is visible in confirms the well-known westward EPF divergence of Figs. 2a and 2c. Figure 5 provides a more detailed picture in situ generated, westward traveling baroclinic waves of the residual circulation in the thermosphere, showing (e.g., Norton and Thuburn 1996; Pendlebury 2012). that the poleward circulation branches reach up to about 2 Below that region, there is a significant positive EPF di- 10 5 hPa (;150 km) in summer and up to the model top in vergence that mainly reflects the resolved GW activity that winter. They are accompanied by upwelling in the meso- is simulated even in a T32 model (McLandress et al. 2006). pause region at low latitudes above 0.003 hPa (;85 km) and The model produces an eastward EPF divergence downwelling at high latitudes. The poleward circulation in during July in the southern winter polar mesosphere. the lower thermosphere in summer largely contributes to As shown by McLandress et al. (2006), this feature is the rapid increase of the temperature with height above the due to eastward-traveling planetary waves that de- mesopause. The poleward circulation in the winter lower velop as a consequence of the reversed baroclinic in- thermosphere deviates from results obtained with the stability in the southern winter lower mesosphere WACCM, which show a winter-to-summer pole residual (strong temperature increase toward the pole), which circulation above the mesopause (Smith et al. 2011; Tan

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FIG. 4. (a) EPF divergence due to resolved waves for July. (b) As in (a), but for the tides only. The black contours 2 2 (drawn for 62, 4, 8, 16, 32 m s 1 day 1) display the EPF divergence due to planetary waves (zonal wavenumbers 1–6 only) up to 0.001 hPa. (b),(c) As in (a) and (b), respectively, but for January. The displayed altitude regime extends from about 20 to 200 km. et al. 2012). A possible explanation for this difference is that In the summer hemisphere, the negative vertical wind the westward EPF divergence of the tides is stronger in the shear that is induced by the tidal EPF divergence and by KMCM than in the WACCM. Also differences in the GW the radiatively determined state results into westward 2 parameterization regarding GW components with high ﬂow above about 10 5 hPa (see Figs. 2a and 2c). In zonal phase speeds that can cause an eastward GW drag in winter, the westward tidal wave driving is not strong the lower winter thermosphere may play a role. enough to prevent a positive vertical shear above about

6 21 FIG. 5. Temperature (colors), residual mass streamfunction (back contours for 60.001, 0.01, 0.1, 1, 10 3 10 kg s ), and the sum of resolved EPF divergence, parameterized GWs, and ion drag (white contours for 65, 10, 20, 40, 80, 2 2 120 m s 1 day 1) for (a) July and (b) January. The displayed altitude regime extends from about 70 to 200 km.

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FIG. 6. (a) Energy deposition due to resolved waves and parameterized GWs for July. (b) As in (a), but for the tides only. (c),(d) As in (a) and (b), respectively, but for January. The displayed altitude regime extends from about 60 to 200 km.

2 3 3 10 4 hPa that is dominated by the radiatively de- obliterated around the mesopause are efficiently dam- termined state and leads to eastward flow above about ped by molecular diffusion further above. 2 3 3 10 5 hPa (;130 km). The resulting zonal-wind Figure 7a shows the horizontally and annually aver- structure in the upper model domain induces an ion aged energy deposition due to all waves and due to only drag that drives a summer-to-winter pole circulation the tides (black and gray solid curves), confirming that (Fig. 5). the latter give the main contribution above about Figures 6a and 6c show the complete energy de- 0.0003 hPa (;100 km). The contributions from the lin- position in the MLT due to parameterized GWs and ear horizontal momentum diffusion (dashed lines in resolved waves for July and January. At middle latitudes Fig. 7a) reveal that the sponge layer becomes significant 2 and up to about 0.001 hPa, the model result confirms the only above about 3 3 10 6 hPa (;170 km). Even within well-known behavior for GWs: there is a sudden onset of the sponge layer, the major contributions to the energy the energy deposition around 0.01 hPa (;80 km) in deposition are due to other processes. Figure 7b shows 2 summer, reaching about 7 K day 1 around 0.005 hPa for the individual contributions to the energy deposition of the present model tuning; on the other hand, the energy the resolved waves that result from ion drag, nonlinear deposition in winter is significant throughout the me- horizontal diffusion (i.e., without the effect of the sosphere but generally smaller than during summer. sponge layer), vertical diffusion, and the negative 2 Other examples for this summer–winter asymmetry can buoyancy production. Above about 3 3 10 5 hPa be found in, for example, Lübken (1997,hisFigs.7and8), (;130 km), the frictional heating due to ion drag is the Fomichev et al. (2002, their Fig. 8a), and Becker (2004, dominant contribution. The frictional heating due to his Fig. 5). The energy deposition in Figs. 6a and 6c nonlinear horizontal momentum diffusion dominates exhibits a significant heating in the thermosphere of below that level. The frictional heating owing to vertical 2 several 10 K day 1. Figures 6b and 6d reveal that this momentum diffusion (mainly due to molecular viscos- energy deposition is mainly due to the tides. Parame- ity) and the negative buoyancy production (mainly in- terized GWs do not notably contribute in the thermo- duced by molecular heat diffusion) increase with sphere for the present model; those GWs that are not yet altitude up to the onset of the sponge layer. These terms

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FIG. 7. Energy deposition in the annual horizontal mean owing to different processes and wave types. (a) Complete energy deposition due to resolved waves and parameterized GWs (black solid line) and energy deposition due to only tides (gray solid line). The corresponding contributions from the sponge layer are indicated by dashed lines. (b) Energy deposition of resolved waves according to different processes: ion drag (solid), nonlinear horizontal diffusion (long dashed), vertical diffusion (dotted), and negative buoyancy production (short–long dashed). The displayed altitude regime extends from about 60 to 200 km. are always less important than the contribution from response of the parameterized GW drag during July either nonlinear horizontal diffusion or ion drag. (with results from the control run being included by According to these model results, the energy de- black contours). From about 0.01 to 0.0003 hPa (;80 to position of the tides contributes significantly to the heat 100 km) we can infer a reduction of the maximum GW budget of the lower thermosphere. The estimate of a drag in each hemisphere, which is accompanied by a 2 several 10 K day 1 heating rate is quite significant, es- slide amplification of the GW drag around 0.03 hPa pecially when compared to the cooling by molecular (;70 km). The details of this response may vary among 2 heat conduction which reaches about 2130 K day 1 in models, depending on the morphology of the tides and the upper model domain (not shown). the representation of GWs. Nevertheless, a qualitative explanation for a reduction and downward shift of the b. Model response to the excitation of thermal tides GW drag by the tides is straightforward (Becker 2012). We now consider the first sensitivity experiment de- Let us consider the dispersion relation for midfrequency scribed in section 3—that is, the differences in the con- GWs in combination with the fact that mainly GWs with 2 trol run from the no-tides run where all shortwave phase speeds of more than 10 m s 1 contribute to the GW radiative heating rates are substituted by their zonal drag in the extratropical mesosphere (Lindzen 1981). means. Corresponding differences will henceforth be These waves propagate nearly conservatively up to their denoted as model response to tides. Figure 8 shows the breaking levels. When they are refracted to higher in- response in the MLT for the zonal-mean temperature trinsic frequencies during the westward (summer) or and zonal wind during July and in the annual mean. The eastward (winter) phase of the zonal-wind tide, they still climatology of the control run is shown as black contours propagate nearly conservatively. The reversed phase of for reference. The model response to tides is charac- the tide, however, gives rise to their refraction to lower terized by a substantial warming and westward accel- intrinsic frequencies and shorter vertical wavelengths eration in the lower thermosphere. While the overall such that the GWs encounter dynamic instability at 2 warming in the altitude regime above about 10 3 hPa somewhat lower altitudes. The net effect is a downward (;90 km) is related to the energy deposition of the tides, shift of the breaking levels and a reduction of the maxi- the westward acceleration results from the predominantly mum zonal-mean GW drag per unit mass. westward tidal EPF divergence. The temperature and wind responses shown in Figs. 8a Figure 9 shows more details of the zonal-mean zonal and 8b are consistent with this notion. They are, however, forces and the energy deposition. Figure 9a depicts the superposed with the direct effects of the tides in the lower

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FIG. 8. Zonal-mean model response to thermal tides (colors) for temperature and zonal wind (a),(b) during July and (c),(d) in the annual mean. Results from the control run are inserted by additional black contours (for 120, 160, 2 200. ...K and 0, 610, 620, 640, 160, 180, 1100, 1120 m s 1). The displayed altitude regime extends from about 50 to 200 km. thermosphere. Figure 9b shows the annual-mean re- the no-tides run (not shown). This explains why the me- sponse of the EPF divergence owing to resolved waves. ridional structure of the temperature response in Figs. 8a 2 Above the mesopause the pattern largely reflects the and 8c reverses above about 5 3 10 5 hPa (;120 km) and EPF divergence in the control run, which in turn is is accompanied by a positive vertical shear in the zonal- mainly due to the tides (see Fig. 4). In the annual mean, wind response (Figs. 8b and 8d). the thermospheric EPF divergence is somewhat The response of the energy deposition in the lower stronger than during the solstices since the tidal am- thermosphere strongly resembles the energy deposi- plitudes maximize around equinox. The poleward cir- tion in the control run (Fig. 9d)—that is, a substantial culation branches in the lower thermosphere that are contribution to the large-scale enthalpy budget of the driven by the tidal EPF divergence lead to the structure of thermosphere is missing in the no-tides run. The me- the temperature response that is visible in Figs. 8a and 8c ridional distribution of the annual-mean energy de- 2 2 between about 10 3 and 10 4 hPa (;90 and 110 km); position response shows maximum heating rates at that is, the strongest warming is found at polar lati- middle and high latitudes in the thermosphere. This tudes. In turn, this temperature response is linked to indicates that those tidal components that propagate to the predominantly westward wind response visible in higher latitudes also propagate to higher altitudes as Figs. 8b and 8d, since the gradient-wind balance applies compared to other tidal components that are more to the zonal-mean flow at least qualitatively in the MLT confined to low latitudes. This is consistent with the (McLandress et al. 2006). The westward zonal-wind notion that the diurnal tide, which dominates in the response extends into the upper model domain and, MLT at low latitudes, has shorter vertical wavelength hence, induces a substantial eastward response of the and is thus more susceptible to damping by vertical zonal force due to ion drag (Fig. 9c). In turn, the ion drag diffusion than the semidiurnal tide, which dominates at response induces equatorward circulation branches above higher latitudes [Smith (2012),hersection4.3.2;see 2 about 10 5 hPa (;150 km) in the control run relative to also references therein].

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FIG. 9. Model response of wave–mean-ﬂow interactions when including thermal tides (colors). (a) Parameterized GW drag for July, (b) resolved EPF divergence in the annual mean, (c) ion drag in the annual mean, and (d) energy deposition in the annual mean. Results from the control run are inserted by additional black contours for (a) 620, 2 2 2 2 2 2 160, 1100, 1140 m s 1 day 1, (b) 65, 615, 225, 235 m s 1 day 1, (c) 65, 615, 125 m s 1 day 1, and (d) 5, 15, 25, 2 35, 45 K day 1. The displayed altitude regime extends from about 60 to 200 km. c. Model response to energy deposition altitudes relative to the no-energy-deposition run. Stated otherwise, the GW drag in the no-energy-deposition run As discussed in Zhu et al. (2010), Shaw and Becker is very unrealistic; it is much too strong (more than (2011),andZhu et al. (2011), there is no general consensus 2 2 450 m s 1 day 1 in northern summer; not shown) and also yet in the modeling community about the energy de- too high in altitude. This explains part of the very strong position of waves. Particularly the theory of the energetics and positive temperature response in the polar summer of GWs as outlined in section 2 was questioned by Zhu 2 MLT around 3 3 10 4 hPa (;100 km) in Fig. 10a.An- et al. (2010, 2011). The importance of the energy deposition other consequence of the energy deposition is a weaker in the MLT, however, can be demonstrated by comparing GW-induced vertical diffusion coefﬁcient. This gives the control run against the no-energy-deposition run in rise to stronger tides and stronger poleward circulation which the sum of energy deposition rates due to pa- branches in the lower thermosphere. The resulting rameterized GWs and resolved waves, E 1 E [see rw westward wind response (not shown) induces an east- Eqs. (8) and (18)], was subtracted from the rhs of the ward response of the ion drag like in the previous thermodynamic equation of the model. Results for the sensitivity experiment. Hence, also the structure of the temperature response during July and in the annual 2 temperature response above about 3 3 10 5 hPa mean are shown in Fig. 10. The heating effect of the (;130 km) is qualitatively similar in both cases. energy deposition is substantial in the MLT. The structure of the temperature response is similar to the re- 5. Conclusions sponse to tides (Fig. 8). A closer inspection of the model data showed that the energy deposition leads to lower We employed a mechanistic middle-atmosphere GCM 2 static stability in the control run from the upper meso- with a model top around 8 3 10 6 hPa (;200 km) in order sphere on. As a result, the parameterized GW drag in the to estimate the effects of thermal tides on the general cir- control run is strongly reduced and shifted to lower culation of the mesosphere–lower thermosphere (MLT).

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FIG. 10. As in Figs. 8a and 8c, respectively, but for the model response to energy deposition.

The explicitly simulated dynamics (including tides and example, the frictional heating in the troposphere is an planetary waves) is comparable to the dynamics simu- essential part of the Lorenz energy cycle; it is maintained lated with comprehensive middle-atmosphere GCMs. A by buoyancy production of wave kinetic energy through specific property of the present model is that subgrid- baroclinic instability and differential heating. In contrast, scale parameterizations for gravity waves (GWs), mo- the buoyancy production of wave kinetic energy is neg- mentum diffusion, and ion drag are formulated in an ative in the case of energy deposition, and wave kinetic energy conserving way. In particular, the energy de- energy is generated by the convergence of the vertical position related to the parameterized GW drag and the pressure flux (or potential energy flux). Hence, the energy frictional heating (mechanical dissipation) that arises deposition becomes relevant when vertical wave propa- from momentum diffusion and ion drag are included. The gation is accompanied by a strong vertical pressure flux. sponge layer of the model is due to a linear harmonic Since this flux is proportional to the absolute vertical flux horizontal diffusion that preserves angular momentum and of horizontal momentum, quasigeostrophic waves hardly energy. This sponger layer is only of secondary importance contribute to the energy transfer from low to high alti- since the ion drag is very efficient to dissipate most of the tudes. This is, however, different for GWs and tides. wave energy in the upper model domain. The climatology of the complete energy deposition as To derive an expression for the energy deposition of simulated with the present GCM confirms the well- resolved waves, we first recapitulated the energetics of known behavior for GW breakdown in the MLT, in- GWs in the single-column approximation for an anelastic cluding the summer–winter asymmetry. In addition, there version of the primitive equations. We emphasized that is substantial energy deposition in the lower thermosphere, the usual formulation of the energy deposition in GW where resolved waves are strongly damped by momentum parameterizations is based on the assumption of a quasi- diffusion and ion drag. This heating rate amounts to sev- stationary energy equation for the GWs. When this as- eral tens of kelvins per day and is mainly caused by sumption is not made, the energy deposition is given by the tides. the frictional heating minus the buoyancy production of We performed a sensitivity experiment with all radia- GW kinetic energy. Following this insight, which was also tive heating rates substituted by their zonal means such as mentioned in previous studies, zonal averaging of the to switch off the excitation of thermal tides. The differ- thermodynamic equation gives rise to a corresponding ence between the control simulation and the model run expression that can be used to diagnose the energy de- without tides reveals the tidal effects on the general cir- position of waves resolved in a GCM [see Eq. (18)]. culation. The difference in the lower thermosphere is The energy deposition in the MLT is the irreversible characterized by a warming that is very pronounced in the heating that occurs when vertically propagating waves polar regions. On the other hand, the energy deposition dissipate either as a result of wave breaking and turbu- of the tides is significant at all latitudes and has weak lence or directly as a result of molecular viscosity and ion maxima at middle and high latitudes. The redistribution drag. The energy available for this process is generated in of heat in the lower thermosphere is predominantly due the region of wave excitation at lower altitudes—for ex- to the westward Eliassen–Palm flux (EPF) divergence of ample, typically in the troposphere in the case of GWs. the tides, which drives poleward circulation branches in Though the energy deposition mainly corresponds to the both hemispheres of the lower thermosphere, with up- frictional heating, both terms are not equivalent. For welling in the tropics and downwelling at polar latitudes.

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2 At higher altitudes (above ;10 5 hPa corresponding to momentum and energy budgets in that region. Whereas ;150 km), the westward zonal-wind response induces an the EPF divergence of the tides and the tidal modulation eastward response of the ion drag. As a result, the lat- of the zonal-mean GW-drag have been analyzed in pre- itudinal structure of the temperature response reverses at vious studies, the energy deposition has not been in- these heights, with minimum heating or even cooling at vestigated until this present work. The mean-flow effects polar latitudes. caused by thermal tides require further investigation based The model response to the excitation of tides also on comprehensive GCMs that 1) extend into the ther- reveals a significant indirect effect on the general circula- mosphere, 2) employ consistent subgrid-scale diffusion tion in the upper mesosphere in terms of a downward shift schemes, and 3) include an improved representation of of the breaking levels of parameterized GWs. A plausible GW–tidal interactions. GW-resolving GCMs are expected interpretation is that the GWs become more dynamically to lead to promising new results in the future, since they unstable when the horizontal wind tide is in the direction account for GW intermittency and the generation of sec- of GW phase propagation, whereas the reduced instability ondary GWs. Such GW effects were not accounted for in in the opposite phase is less important. Indeed, the simu- the present study but are expected to be quite important lated zonal-wind tides in the summer MLT having for the dynamics and variability in the thermosphere. 2 amplitudes of 610 m s 1 are supposed to induce such a modulation of the mean GW drag. This interpreta- Acknowledgments. The author is grateful to S. L. Vadas tion suggests that events of strong activity of westward- and D. C. Fritts for inspiring discussions and comments on traveling waves in the summer mesosphere can also the manuscript. The comments by Jia Yue, two anony- cause a reduction and downward shift of the GW drag. mous reviewers, and the editor Anne K. Smith are grate- The energy deposition of resolved waves is incomplete fully acknowledged. This study was partly funded by the in GCMs when frictional heating is ignored or flawed. Leibniz Society via the SAW project WaTiLa. Though the contribution associated with molecular vertical momentum diffusion and ion drag is often included in GCMs extending into the thermosphere, it is the scale- APPENDIX selective damping of momentum by nonlinear horizontal diffusion that dissipates most of the energy of resolved An Anelastic Version of the Primitive Equations 2 waves in the MLT up to about 3 3 10 5 hPa (;130 km) in with Enthalpy as the Prognostic Variable the present model. Conventional measures to parameter- ize this damping such as hyperdiffusion, numerical filter- An anelastic version of the fluid dynamical equations of ing, or Rayleigh friction are not consistent with angular motion is advantageous in atmospheric applications when momentum, energy conservation, and the second law of it is necessary to exclude sound waves. For example, the thermodynamics; they can hardly be used to describe the linear solutions for gravity waves are more easily obtained energy deposition of resolved waves. At altitudes above when using the anelastic equations. Different versions of ;130 km, however, the ion drag dissipates most of the anelastic or pseudo-incompressible approximations have energy of resolved waves for the present model setup. been proposed in the literature. For an overview, the We demonstrated the importance of the energy de- reader is referred to Durran (1989) and Achatz et al. position by parameterized and resolved waves by per- (2010). Here, we follow the approach used by Vallice forming a second sensitivity experiment in which the (2006, his chapter 2.5). We first consider the full Eulerian energy deposition was neglected. We found that the en- equations in Cartesian coordinates, from which we will ergy deposition in the mesopause region causes the GWs then obtain the corresponding primitive equations. to break at lower altitudes. As a result of the damping As usual, we decompose density, pressure, tempera- effect that GWs have on the tides in the present model, ture, and potential temperature as the tides are significantly stronger from the upper me- r 5 r (z) 1 r~(x, y, z, t) sosphere on when the energy deposition is included. r 5 1 ~ Therefore, the model response to the energy deposition is p pr(z) p(x, y, z, t) comparable to the model response to tides, though the T 5 T (z) 1 T~(x, y, z, t) maximum temperature response of more than 100 K above r Q5Q 1 Q~ the summer mesopause is much stronger. r(z) (x, y, z, t), (A1) Summarizing, thermal tides have significant direct and indirect effects on the general circulation of the MLT that where the subscript r denotes a hydrostatic reference state are not negligible. Particularly the EPF and energy de- that depends only on z and where the deviations are de- ~ posited in the lower thermosphere are pivotal for the noted by r~, p~, T~,andQ, which depend on the spatial

Unauthenticated | Downloaded 09/27/21 04:49 AM UTC 2060 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 coordinates (x, y, z)andtime(t). The reference state ful- p r~ h 5 e 1 r 1 2 , (A9) fills the equation of state for an ideal gas with regard to r r r r the temperature and potential temperature; that is, RTr 5 R/cp pr/rr and Qr 5 Tr(p00/pr) with p00 5 1013 hPa. Here, R where again the approximation Eq. (A6) has been used. is the gas constant and cp the heat capacity per unit mass at When computing the total time derivative of Eq. (A9), constant pressure. we insert Eq. (A7) and differentiate the second term on The anelastic approximation consists of two parts. the rhs of Eq. (A9) only with regard to the reference First, the density in the continuity equation is replaced state. Furthermore, we make use of the hydrostatic re- r by r such that lation for the reference state, dzpr 52grr, where g is the acceleration due to gravity. The final result is 0 5 = (r v) 1 › (r w), (A2) r z r r~ = d h 5 c d T 52wg 1 2 . (A10) where and v denote the horizontal nabla operator and t p t r velocity field, respectively, and w is the vertical velocity. r Accordingly, the density velocity that performs work To derive the anelastic version of the horizontal Eulerian becomes d r 5 wd r , and an internal energy equation t r z r momentum equation, consistent with Eq. (A2) must be written as 21 ~ p 1 p~ d v 52(r 1 r~) =(p 1 p), (A11) 55 5 r r t r r d e cyd T wd , (A3) t t (r 1 r~)2 z r r we simply neglect r~, yielding where cy denotes the heat capacity per unity mass at con- d v 52r21=p~. (A12) stant density. The second part of the anelastic approxima- t r tion states that the perturbation quantities in Eq. (A1) are The anelastic approximation is less straightforward for small as compared to the reference values and that the the vertical momentum equation, density perturbation only depends on the potential temperature perturbation. Hence the exact thermodynamic 2 d w 52(r 1 r~) 1› ( p 1 p~) 2 g. (A13) relation, t r z r

dQ dr cy dp To preserve the buoyancy force, we ﬁrst multiply Eq. 52 1 , (A4) r 1 r~ r~ Q r (A13) by the density, r , and then neglect on the cp p lhs of Eq. (A13). After dividing by rr, we obtain leads to the anelastic version, 52r21› ~2 r21r~ dtw r zp r g (A14) Q~ r~ 52 Q r , (A5) as a preliminary result. Combining Eqs. (A12) and r r (A14) with Eq. (A2) allows to derive the kinetic energy since equation in ﬂux form: › r 1 = r 1 › r jr~j jp~j t( rk) (v rk) z(w rk) . (A6) r p 52 =~2 › ~2 r~ r r v p w zp wg , (A15)

These approximations allow the expansion of Eq. where k 5 (v2 1 w2)/2. The ﬂux form of the enthalpy (A3) as equation, Eq. (A10),is p r~ d e 5 r 1 2 2 wd r , (A7) r~ t r2 r z r › (r h) 1 = (vr h) 1 › (wr h) 52wgr 1 2 . r r t r r z r r r r and they imply that (A16) ~ Q T~ Adding Eqs. (A15) and (A16), the buoyancy terms 5 . (A8) Q proportional to r~ cancel. Furthermore, the ﬁrst term on r Tr the right-hand side (rhs) of Eq. (A16) can be written in a

The anelastic version of the enthalpy equation is ob- ﬂux form using dtF5wg. However, the pressure work tained by noting that on the rhs of Eq. (A15) cannot be written as a ﬂux

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