1. Introduction
Three-dimensional (3D) numerical models are important tools used in the quantitative study of the middle atmosphere and interpretation of satellite measurements. Well-designed numerical models in the middle atmosphere often consist of several types of comprehensive modules representing radiation, dynamics, photochemistry, and transport. Among these four types of modules, the dynamics or the dynamical core of a numerical model plays a critical role in organizing and coupling different physical processes in a consistent manner. This is mainly due to the dynamics being controlled by fluid mechanics that is a continuum, behaving highly nonlinearly, and evolving with time on many different scales. The other modules play complementary roles. For example, the physics module for calculating radiative heating and photolysis rates provides the parameterized force to drive the dynamics and photochemistry. This partially affects the dynamics with bounded or controllable uncertainty because the time-independent problem of radiative transfer is well defined and solvable for a given input of atmospheric parameters, such as temperature and species, solar irradiance spectrum, and kinetics database (e.g., Zhu 2004; Mlynczak and Zhou 1998; Zhu et al. 2007). From the perspective of satellite measurements and the global modeling of atmospheric structure, it is often the large-scale dynamics and transport that can be observed directly with instruments and simulated explicitly with models. For example, tidal winds were directly observed from the Upper Atmosphere Research Satellite High Resolution Doppler Imager (UARS/HRDI) and were used to estimate the momentum deposition based on the derived velocity correlation terms that characterize the effect of tides on the zonal mean flow (Lieberman and Hays 1994). The temperature measured from the Sounding of the Atmosphere using Broadband Emission Radiometer (SABER) onboard the Thermosphere, Ionosphere, Mesosphere, Energetics and Dynamics (TIMED) satellite has been used to derive planetary-scale waves for both temperature and winds and the associated wave force terms such as the Eliassen–Palm flux divergence of tides in the middle atmosphere (Zhu et al. 2005, 2008). These large-scale fields derived from satellite observations can be directly assimilated into numerical models in a diagnostic analysis to gain additional physical insights into middle atmosphere dynamics (e.g., Akmaev 1997; Zhu et al. 2005). On the other hand, the momentum and energy sources produced by subgrid-scale motions have to be parameterized.
For middle atmospheric modeling studies, the subgrid-scale motions can be most effectively represented by gravity waves because of their ability to transport momentum and energy over a large spatial distance. Generated mainly in the troposphere by mechanisms such as flow over topography (e.g., Nappo 2002; Teixeira et al. 2004), convection (e.g., Alexander et al. 1995), shear instability (e.g., Lindzen 1974; Scinocca and Ford 2000), and geostrophic adjustment (e.g., Zhu and Holton 1987), gravity waves can propagate upward into the mesosphere and lower thermosphere (MLT) (e.g., Hines 1960). It has been well established that zonal forces generated by breaking gravity waves are crucial to maintain large-scale dynamics and transport in the middle atmosphere, especially in the MLT region (e.g., Lindzen 1981; Holton 1983; Fritts 1984; Holton and Zhu 1984; McIntyre 2000; Holton and Alexander 2000). It is also known, both observationally and theoretically, that wave generation and propagation are 3D in space (e.g., Fritts 1984, 1995; Zhu 1987; Sato 1994; Marks and Eckermann 1995; Broutman et al. 2001, 2002, 2004). On a slowly varying time scale or averaged over a wave period, a propagating gravity wave will not have any effect on the background wind or temperature unless the wave is subject to dissipation in wave action (e.g., Lighthill 1978; Zhu 1987; Eckermann 1992). On the other hand, a dissipative wave could (i) induce a momentum drag on the background flow, (ii) produce eddy mixing in the background dynamical and tracer fields, and (iii) generate a heating–cooling dipole on the background temperature field. The mechanisms that contribute to the dissipation of a gravity wave disturbance in the middle atmosphere are convective and dynamical instabilities, radiative damping, eddy diffusion, and nonlinear wave–wave interactions. In the lower thermosphere, molecular diffusion makes an additional contribution. Among these dissipative processes, the convective and dynamical instabilities associated with wave breaking are believed to be dominant in producing momentum drag and eddy diffusion in the background flow (Fritts and Rastogi 1985). This is mainly because the characteristic gravity waves in the middle atmosphere are of high frequency and long vertical wavelength (e.g., Hirota and Niki 1985; Zhu et al. 1997), such that they can propagate over a large distance without significant damping before wave breaking. In addition to the drag and eddy diffusion that have been extensively studied and have been parameterized in many middle atmosphere models, it has been recognized recently that the dynamical heating associated with the downward heat flux by dissipative gravity waves also needs to be included in middle atmosphere models (Becker 2004; Akmaev 2007). Based on the rocket measurements of neutral density fluctuations, Lübken (1997) found that the dynamical heating rate near the mesopause estimated from the turbulence energy dissipation rates amounts to 10–20 K day−1, which is comparable to the magnitudes of radiative and chemical heating rates in the same region. In summary, the most effective dissipation of gravity waves—wave breaking—produces three important effects on the background state: wave drag, eddy diffusion, and wave heating. Here, we call these three dynamical and energetic consequences resulting from gravity wave breaking in the middle atmosphere the “breaking trinity.”
In terms of momentum flux or Reynolds stresses, the parameterization of subgrid-scale motions by gravity waves is fundamentally different from those in so-called turbulent viscosity models, where the Reynolds stresses are essentially parameterized by various length theories, including Prandtl’s mixing-length theory and K-epsilon models (e.g., Tennekes and Lumley 1972; Pope 2000). In the latter case, the momentum is carried and transferred by moving fluid parcels for a localized exchange. For mechanistic general circulation models (GCMs) with a high resolution that can explicitly resolve medium-scale gravity waves, the localized mixing-length theory can also be adopted to parameterize the effects of eddies with much smaller subgrid scales (Becker 2009). On the other hand, gravity waves carry and transfer momentum by pressure and velocity fluctuations over a large spatial distance. Such a nonlocalized transport of momentum also provides one of the important physical bases for changes in the tropospheric circulation to affect the MLT dynamics and climatology. From the perspective of wave–mean flow interactions in 3D flow on how the momentum and energy are spatially redistributed within the entire domain, the breaking trinity can be understood as three effects of wave propagation and dissipation on the 3D mean flow: (i) nonlocalized transport of momentum through wave propagation in 3D that remotely redistributes atmospheric momentum in both zonal and meridional directions from wave generation to wave dissipation regions; (ii) localized diffusive transport of momentum, heat, and tracers due to 3D mixing induced by wave breaking; and (iii) localized transport of heat by perturbing wave structures that redistributes the thermal energy within a finite domain. To quantitatively and self-consistently examine these effects on the 3D background state, we need a parameterization scheme that can provide (i) wave drag to the background 3D flow consisting of both the zonal and meridional components in the momentum equations, (ii) eddy diffusion to the dynamical and tracer fields that represents the mixing or the field smoothness by parameterized subgrid-scale motions, and (iii) wave heating to the temperature field induced by the wave breaking in the energy equation, all resulting from the dissipation of wave action caused by the interaction between 3D subgrid-scale waves and 3D background flow. This paper introduces a new parameterization scheme of drag, eddy diffusion, and wave heating imposed on a 3D background state by the breaking of gravity waves. The output of the new parameterization scheme will self-consistently include all three distinct processes associated with the breaking trinity by which the upward-propagating gravity waves influence the 3D large-scale dynamics and energetics of the middle atmosphere.
In section 2, we introduce a spectral parameterization scheme of gravity wave breaking that simultaneously produces the breaking trinity for given 3D wind and temperature profiles. The central part of the derivation is built on previous work by Holton and Zhu (1984), Zhu (1987), and Alexander and Dunkerton (1999), which was based on the general relationship between the wave action flux and the wave momentum, and also on a simple mapping approximation between the wave source spectrum in wave parameters and momentum deposition distribution in altitude. Section 3 shows the typical magnitudes and patterns of the breaking trinity to a 3D background field of wind and temperature derived from an output field of a high-altitude version of the Goddard Earth Observing System atmospheric model (GEOS-5). Finally, section 4 summarizes the paper.
2. Parameterization of gravity wave breaking in a three-dimensional background flow
a. Wave–mean flow interaction in a three-dimensional atmosphere
In Eqs. (1)–(3), the momentum (ρ0u, ρ0υ) and total potential energy (ρ0cpT) under the Boussinesq approximation are all linear variables, so the eddy forcing terms on the right-hand sides of the mean momentum and energy equations can all be expressed in flux form, as shown in Eq. (4) (Holton 1975). As a result, the mean momentum and potential energies can be spatially redistributed through the flux divergences, but the integrations over an enclosed domain will be solely determined by the boundary fluxes. Mathematically, this is also because the average of a perturbed linear quantity vanishes (e.g., 
b. Extension of Lindzen’s parameterization to a three-dimensional atmosphere
We have already indicated that the convective and dynamical instabilities associated with wave breaking are believed to be the dominant mechanism of wave dissipation for the upward-propagating gravity waves in the middle atmosphere. The wave breaking can produce both the momentum drag and eddy diffusion to the background flow (Lindzen 1981). On the other hand, for slowly propagating planetary waves where radiative damping is the main dissipation mechanism, the effect of eddy diffusion by wave dissipation on the background state is small and often neglected (e.g., Matsuno 1970; Holton and Lindzen 1972). Since ωi represents the wave dissipation effect by either radiative damping or eddy diffusion (Holton and Zhu 1984), the wave dynamical heating associated with Eq. (11c) remains potentially significant for a slowly propagating wave with radiative damping as its main dissipation mechanism. The instabilities of an upward-propagating gravity wave are mainly caused by the exponential growth of wave amplitude due to the atmospheric density effect or the increase of λ due to the critical level and are modified by the static stability of the background state. The most widely used parameterization schemes have been those based on Lindzen’s (1981) theory of wave breaking and saturation. The term “saturation” refers to the growth cessation of an upward-propagating unstable wave component when its Richardson number reaches its critical value of ¼ (shear instability) or approaches zero (convective instability). Lindzen’s parameterization scheme (1981) was originally proposed for deriving Su and the associated Kzz−m for a zonal mean flow (
Alexander and Dunkerton (1999) proposed a simplified version of Lindzen’s parameterization scheme by neglecting the complex details of wave dissipation either below or above the breaking level zb. They assumed all momentum of a wave component to be deposited at the breaking level zb. By doing so, they mathematically have directly mapped the source momentum spectrum in horizontal wavenumber and phase speed at the lower boundary to a momentum deposition and eddy diffusion coefficient in altitude. Such a simplified parameterization not only increases the efficiency of computation; more importantly, it also combines all the uncertainties in the above three aspects [(i)–(iii)] into one of specifying the source spectrum of the waves. In principle, this makes it much simpler to validate and improve the parameterization scheme as more observations of gravity wave variance and spectra become available. Furthermore, we will show below that such a simplification also makes the evaluation of the newly included sensible heat flux Eq. (11c) straightforward. The 3D parameterization scheme that includes the breaking trinity introduced below essentially combines the formulations of Holton and Zhu (1984), Zhu (1987), and Alexander and Dunkerton (1999) to give a set of output profiles of (Su, Sυ, ST) and (Kzz−m, Kzz−T) as functions of altitude for a set of given input wind and temperature profiles of (
In the above equations, ε is an intermittency factor introduced by Alexander and Dunkerton (1999) that represents the ratio of the observed momentum flux to the modeled one at the lower boundary. It is a specified parameter in the parameterization scheme that can vary with time, space, and wave parameters. The eddy diffusion coefficient derived from the Lindzen-type parameterization refers to the dissipation and momentum mixing to the wave field. To apply it to the background mean state, we introduce an additional parameter εm to characterize its efficiency for mixing the momentum of the background fields. Holton and Zhu (1984) showed that there was a partial cancellation between the direct wave drag and the drag induced by the eddy diffusion, which smoothed the total drag on the right-hand sides of Eqs. (1)–(3). The current parameterization scheme includes hundreds to thousands of wave components in the source spectrum, which yields smooth drag profiles for typical wind profiles. Therefore, we set a smaller value of εm = 0.3 in the current scheme. The eddy Prandtl number Pr is defined as the ratio of the eddy momentum diffusivity to the eddy heat and tracer diffusivity. It also reflects the fact that the wave eddy diffusion coefficient Dzz defined in Eq. (24) is different than the eddy diffusion coefficient for the background state Kzz−m defined in Eq. (33). Both the mesospheric tracer measurements and careful model analysis of the breaking process show that Pr is much greater than 1 (e.g., Strobel et al. 1987; Strobel 1989; Chao and Schoeberl 1984). This is consistent with Eq. (8) of a vanishing sensible heat flux to a first-order approximation. We set Pr = 5 in the current scheme. Again, it is noted from Eqs. (27) and (28) that the partition of the deposited momentum in (x, y) direction is based neither on (cosδ, sinδ) nor on (cosθ, sinθ) but rather on (cosθ0, sinθ0).
It should be pointed out that, strictly speaking, the first and second terms on the right-hand sides of Eqs. (1)–(3) represent two different ways of parameterizing the effects of subgrid-scale motions: nonlocalized wave forcing due to wave propagation and breaking versus localized mixing-length theory of turbulence. For a single gravity wave component, the wave-breaking processes occurring on scales much smaller than its wavelength dissipate the wave action and induce the explicit momentum and heat flux divergences shown in Eqs. (4) and (11). When there is a clear scale separation among the background grid-resolved motions, the smaller subgrid wave component, and a much smaller-scale mixing by wave breaking, the effect of the subgrid-scale eddy terms can be entirely included by the first terms (Su, Sυ, ST) in Eqs. (1)–(3). Including the second eddy diffusion terms on the right-hand sides of Eqs. (1)–(3) implies that the vertical mixing by the much smaller-scale motions that directly dissipates the wave component also diffusively mixes the grid-resolved motions. The diffusion terms are expected to become more important as more gravity waves components with different wavelengths are included in the parameterization scheme because this will make the separation among different scales less obvious. This analysis also suggests that the additional parameter εm introduced in the current parameterization scheme is empirical and it may vary with the setting of the wave source spectrum.
From Eqs. (15), (19), (21), and (26), we see that, at zb, ĉ and thus ω̂, FP0, and Π are all positive. Therefore, the signs of the drag components are solely determined by the wavenumber vector direction (cosθ0, sinθ0), whereas the heating rate and eddy diffusion coefficients are always positive. Previous studies (e.g., Walterscheid 1981; Liu 2000; Talaat et al. 2001) suggested that the gravity waves could induce dynamical cooling in the wave-dissipating region. A positive heating at zb in the current parameterization scheme is mainly due to the assumption that each spectral wave component is completely dissipated at the breaking level where the wave heating occurs. On the other hand, if one assumes a finite and a broad region of wave dissipation between zb and a higher level such as a critical level zc where the wave amplitude completely vanishes, then cooling could occur in most of the region between zb and zc. To accommodate such a cooling effect, we add a cooling term of the same magnitude in Eq. (30) immediately above zb, which leads to a net effect of downward transport of heat near the wave breaking level. Physically, this means that there is no net heat being induced within the atmosphere when its vertical heat flux vanishes in both the lower and upper boundaries. Mathematically, the heat flux term of an upward-propagating gravity wave packet shown in Eq. (11c) experiences two major stages: (i) it changes from 0 to a finite value as the wave breaks because of a finite jump in ωi at zb and (ii) it changes back to 0 again as the wave approaches a critical level and completely dissipates so that ω̂A → 0. Previous studies mainly focused on (ii) and overlooked (i). The net effect of Eq. (30), where the breaking waves heat the background atmosphere at zb and cool it one level above zb, is the downward transport of heat. We point out that the physical mechanism of this downward heat transport caused by a discontinuous wave breaking is slightly different from the same conclusion in previous studies (e.g., Walterscheid 1981), where the downward heat flux and the associated cooling only correspond to the above stage (ii).
Alexander and Dunkerton (1999) emphasized the dynamical importance of wave dissipation at zb rather than at zc. The above analysis of the heat flux and transport shows the energetic importance of wave dissipation at zb. Wave dissipation at zb produces a net downward transport of thermal energy to the mean state. Realization of ωi = 0 and ω̂A = 0 at the lower and upper boundaries respectively allows the finite downward heat transport to be evaluated self-consistently. This also makes Eq. (3) energetically consistent. In summary, the current parameterization of the breaking trinity extends the previous Lindzen types of parameterizations such as those by Holton and Zhu (1984) and Alexander and Dunkerton (1999) in two aspects: extending it to a 3D background flow and including a wave breaking-induced heating term. For a given set of input wind and temperature profiles [u(z), υ(z), T(z)] as a basic state in a model grid, the parameterization scheme of the breaking trinity outputs the vertical profiles of (Su, Sυ, ST, Kzz−m, Kzz−T) that allow one to calculate all the force terms by the subgrid-scale motions such as those on the right-hand sides of Eqs. (1)–(3).
c. Specification of the source spectrum
A discrete source spectrum needs to be specified for the geopotential variance Φ02(θ0, c, K) as a function of three wave parameters: θ0, c and K. We have already indicated the reason for choosing these three independent wave parameters for prescribing the wave spectrum: θ0, c, and K do not vary with altitude for an upward-propagating wave component. In some parameterization schemes, such as the one by Warner and McIntyre (1996), the distribution and variation of the energy spectrum were prescribed as a function of λ and ω̂, both of which vary with the altitude. We assume a functional form that is separated in wavenumber vector direction: Φ02(θ0, c, K) = B1(θ0)B2(c, K). Following Matsuno (1982) and Holton and Zhu (1984), we assume an isotropic distribution of Φ02 in θ0 [i.e., B1(θ0) =
3. Numerical results of the parameterization scheme
A set of output fields near the Northern Hemisphere solstice (20050105) from version 5 of the Goddard Earth Observing System Data Assimilation System (GEOS-5 DAS) is used to test the parameterization scheme. The GEOS-5 atmospheric GCM is a weather–climate-capable model consisting of a finite-volume dynamical core and a physical package parameterizing four major groups of physical processes (Rienecker et al. 2008). The standard setting of the atmospheric model has 72 vertical layers extending from the surface to ∼70 km. The major physical processes contained in the model are (i) moist processes including cloud microphysics, (ii) shortwave and infrared radiation, (iii) drag and eddy diffusion parameterized by a 2D Lindzen-type of scheme for gravity wave breaking (McFarlane 1987; Garcia 1991), and (iv) surface processes in the atmospheric boundary layer. A high-altitude version of the GEOS-5 atmospheric model has 82 vertical levels extending from the surface to ∼100 km with a horizontal resolution of 2.5° longitude by 2° latitude. In Fig. 1, we show the zonal mean fields of the assimilated model winds and the parameterized breaking trinity calculated by the current scheme based on the output model winds: wave drag, eddy diffusion coefficient, and wave heating by gravity wave breaking. We do not show the temperature field in the figure because it only plays a minor role in the parameterization scheme by modifying the background static stability. Note that the magnitudes of the zonal mean zonal drag
Figure 2 shows the breaking trinity of
The wind distribution at a particular longitude sector could significantly deviate from its zonal average. This is similar to the case in which a measured single wind profile at a prescribed location could be completely different than its climatology. To see such an effect and further illustrate the 3D nature of the current parameterization scheme, we show in Fig. 3 the input wind field (u, υ) and the output force terms (Su, Sυ,
4. Conclusions
In this paper, we have developed a 3D spectral parameterization scheme to self-consistently include the “breaking trinity” of upward-propagating gravity waves for large-scale numerical models: (i) momentum drag that represents the nonlocalized transport of momentum through wave propagation in a 3D background flow, (ii) eddy diffusion coefficients that characterize the localized diffusive transport of momentum, heat, and tracers due to 3D mixing induced by wave breaking, and (iii) wave heating rate that captures localized transport of heat by perturbing wave structures to redistribute the thermal energy within a finite domain. For a set of given input wind and temperature profiles at each model grid (u, υ, T), the parameterization scheme returns five vertical profiles Su, Sυ,
The spectral parameterization has been developed by using a general relationship between the wave action flux and the wave momentum and heat fluxes developed by Zhu (1987) and a mapping approximation between the wave source spectrum and the vertical distribution of the momentum deposition developed by Alexander and Dunkerton (1999). When the parameterization algorithm is applied to a set of 3D wind fields output from a high-altitude version of the GEOS-5 atmospheric model, the derived zonal mean drag and eddy diffusion coefficient near the solstice mesopause are comparable to those derived in previous work and are required to produce the horizontal anomalous temperature gradient and the vertical zonal wind reversals near the solstice mesopause (e.g., Holton 1983). The derived wave heating and cooling rates near the mesopause are found to be 10–20 K day−1, which is comparable to the radiative and chemical heating rates. The filtering effect of the upward-propagating gravity waves is also well captured by the current parameterization scheme.
Acknowledgments
The authors wish to thank two anonymous reviewers for helpful comments on the original manuscript. Xun Zhu wishes to thank Hans G. Mayr for helpful discussions and suggestions in this study. This research was supported by the TIMED project sponsored by NASA under Contract NAS5-97179, NASA Grants NNG05GG57G and NNX09AH77G, and in part by NSF Grants ATM-0730158 and ATM-0640864 to The Johns Hopkins University Applied Physics Laboratory.
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Example of the winds from a high-altitude version of the GEOS-5 model (20050105) and the parameterized breaking trinity: wave drag, eddy diffusion coefficient, and wave heating by gravity wave breaking from the GEOS-5 model runs near the NH winter solstice. All fields have been zonally averaged over 144 longitudinal grids: (top left) zonal and (top right) meridional wind; (middle left) zonal and (middle right) meridional drag; and (bottom left) eddy diffusion coefficient and (bottom right) heating rate.
Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3302.1
Example of the parameterized breaking trinity: wave drag, eddy diffusion coefficient, and wave heating by gravity wave breaking derived from the zonally averaged wind and temperature fields near the NH winter solstice: (top left) zonal and (top right) meridional drag; and (bottom left) eddy diffusion coefficient and (bottom right) heating rate.
Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3302.1
As in Fig. 1, but on a longitudinal plane of 110°E, where the meridional wind is peaked.
Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3302.1