Comparisons of amplitudes of wave variations of atmospheric characteristics
obtained using direct numerical simulation models with polarization relations
given by conventional theories of linear acoustic-gravity waves (AGWs) could
be helpful for testing these numerical models. In this study, we performed
high-resolution numerical simulations of nonlinear AGW propagation at
altitudes 0–500 km from a plane wave forcing at the Earth's surface and
compared them with analytical polarization relations of linear AGW theory.
After some transition time

Observations show the frequent presence of acoustic-gravity waves (AGWs) generating at tropospheric heights and propagating to the middle and upper atmosphere (e.g., Fritts and Alexander, 2003). These AGWs can break and produce turbulence and perturbations in the atmosphere (Gavrilov and Yudin, 1992; Gavrilov and Fukao, 1999). For example, sources of AGWs could be mesoscale turbulence and convection in the troposphere (e.g., Fritts and Alexander, 2003; Fritts et al., 2006). Turbulent AGW generation may have maxima at altitudes 9–12 km in the regions of tropospheric jet streams (Medvedev and Gavrilov, 1995).

Non-hydrostatic models are useful for direct numerical simulations of waves and turbulence in the atmosphere. For example, Baker and Schubert (2000) simulated nonlinear AGWs in the atmosphere of Venus. They modeled waves in the atmospheric region with horizontal and vertical dimensions of 120 and 48 km, respectively. Fritts and Garten (1996), also Andreassen et al. (1998) and Fritts et al. (2009, 2011), simulated the instabilities of Kelvin and Helmholtz and turbulence produced by breaking atmospheric waves. These models simulate turbulence and waves in atmospheric regions with limited vertical and horizontal dimensions. The models exploited spectral methods and Galerkin-type series for converting partial differential equations (versus time) into the ordinary differential equations for the spectral series components. Yu and Hickey (2007) and Liu et al. (2008) developed two-dimensional numerical models of atmospheric AGWs.

Gavrilov and Kshevetskii (2013) described a two-dimensional model for high-resolution numerical simulating nonlinear AGWs using a finite-difference scheme taking into account hydrodynamic conservation laws as described by Kshevetskii and Gavrilov (2005). This approach increases the stability of the numerical scheme and allows us to obtain non-smooth solutions of nonlinear wave equations. This permitted us to get generalized physically acceptable solutions to the equations (Lax, 1957; Richtmayer and Morton, 1967). Gavrilov and Kshevetskii (2014a) created a three-dimensional version of this algorithm for simulating nonlinear AGWs in the atmosphere. They modeled waves produced by sinusoidal horizontally homogeneous wave forcing at the Earth's surface.

Karpov and Kshevetskii (2014) used a similar numerical three-dimensional model to study AGW propagation from local non-stationary wave excitation at the Earth's surface. They showed that infrasound going from tropospheric sources could provide substantial mean heating in the upper atmosphere. Dissipating nonlinear AGWs can also create accelerations of the mean flows in the middle atmosphere (e.g., Fritts and Alexander, 2003). However, details of the mean heating and mean flows created by non-stationary nonlinear AGWs in the atmosphere need further studies.

Numerical models of atmospheric AGWs require verifications. For plane stationary wave components with small amplitudes, conventional linear theories (e.g., Gossard and Hooke, 1975) give the dispersion equation and polarization relations, which connect wave frequency, vertical and horizontal wave numbers and ratios of amplitudes of different wave field variations. One can expect that such relations could exist between corresponding parameters of the numerical model solutions. Therefore, theoretical polarization relations could be useful for verifying direct simulation models of atmospheric AGWs.

In this paper, using the high-resolution numerical three-dimensional model by Gavrilov and Kshevetskii (2014a, b), we made comparisons of calculated ratios of amplitudes of different wave fields with polarization relations given by the conventional linear AGW theory. We considered simple AGW forcing by plane wave oscillations of vertical velocity at the surface, which is similar to the assumptions made in analytical wave theory. We found height regions of the atmosphere where numerical results agree with analytical ones, and regions of their substantial disagreement.

Theoretical dispersion equation and polarization relations are widely used for developing simplified parameterizations of AGW dynamical and thermal effects in the general circulation models of the middle atmosphere. Therefore, comparisons of numerically modeled and analytical polarization relations are useful for both verifications of numerical models, and obtaining limits of analytical relation applicability and for verifying AGW parameterizations.

The three-dimensional numerical AGW model calculates velocity components

The used numerical scheme is analogous to the two-dimensional algorithm described by Kshevetskii and Gavrilov (2005). It is a modification of the method by Lax and Wendroff (1960). This algorithm involves the conservation laws of momentum, mass and energy. The main differences of our scheme from the classical Lax and Wendroff (1960) algorithm are the implicit approximating equations of hydrodynamics at the first half step in time, which diminish errors of description of acoustic waves (Kshevetskii, 2001a, b, c).

We use a numerical scheme similar to the two-dimensional algorithm developed
by Kshevetskii and Gavrilov (2005). Used hydrodynamic equations (see Gavrilov
and Kshevetskii, 2013, 2014a) can be presented in the conservation law forms

In this study, we employ vertical profiles of background

The numerical model involves kinematic molecular heat conductivity and
viscosity, increasing versus altitude inversely proportional to the
background density. We also include background turbulent heat conductivity
and viscosity, taking their vertical profiles with the maxima of
10 m

The comparisons considered in this paper used relations obtained from a
theoretical model of monochromatic AGWs in the plain rotating atmosphere.
Conventional linear theories suppose that wave components

In this study, using the high-resolution nonlinear numerical model described
in Sect. 2, we simulated hydrodynamic fields produced by spectral AGW
components and compared ratios of their amplitudes with those predicted by
the analytical polarization relations Eqs. (7) and (8). To make simulations
matching the linear AGW theory (see Eq. 5), we used nonlinear AGWs with forms
of plane waves and suppose horizontally periodical distributions of vertical
velocity at the Earth's surface moving along axis

Gavrilov and Kshevetskii (2014a, b) demonstrated that, after triggering the
wave source at

To estimate AGW amplitudes in the numerical model solution, we calculated
standard deviations of the corresponding wave fields over all nodes of the
horizontal grid at the considered altitude. For the sinusoidal wave
component, this standard deviation is equal to a half AGW amplitude.
Therefore, ratios of amplitudes of horizontally homogeneous stationary
sinusoidal AGWs should be equal to the ratios of the corresponding standard
deviations. Simulated standard deviations of wave fields in horizontal planes
located at different heights grow in time throughout transition intervals
after activating the wave forcing and then tend to constant values different
at each height (see Gavrilov and Kshevetskii, 2014b). In the horizontally
periodical approximation of Eq. (1), these standard deviations are
approximately equal to half wave amplitudes at large

Standard deviations and their ratios for AGW with

Table 1 represents standard deviations at different altitudes calculated with
the numerical model and with analytical polarization relations and their
ratios for AGW with

For comparisons of numerically simulated values with analytical ones in
Table 1, we use a standard

Many numerically simulated AGW parameters do not match the respective
analytical values in Table 1. No matches are in the bottom part of Table 1,
which corresponds to the initial transition time interval. Gavrilov and
Kshevetskii (2014b) showed that vertical structures of transient waves are
different from those predicted by the linear AGW theory during the transition
interval after activating the surface wave source Eq. (11). The bottom part
of Table 1 shows that numerically simulated wave amplitude

Same as Table 1 but for AGW with

In the upper part of Table 1 for quasi-stationary AGWs at

Table 1 reveals the numerically simulated AGW momentum fluxes

Table 2 is the same as Table 1, but for AGW components with

Tables 1 and 2 contain comparisons of the numerical results and linear polarization relations at altitudes below 100 km, where considered AGW modes are quasi-linear and almost nondissipative. At higher altitudes, growing wave amplitudes and molecular viscosity and heat conduction lead to fast growing wave-induced mean flows, which violate assumptions of conventional AGW theories and change ratios of wave amplitudes of different hydrodynamic fields. Therefore, we found poor agreement between numerical and analytical wave results above altitude 100 km and do not include them in Tables 1 and 2. These disagreements become larger with increases in amplitudes of the lower boundary wave sources due to higher nonlinear effects and faster growths in the wave-induced jet streams above 100 km. To get better agreements, improved analytical AGW theories taking into account transient processes, high wave dissipation and fast changes in background fields are required.

In the areas of Tables 1 and 2, where numerical and analytical parameters are close, one can use analytical formulae for descriptions and estimations of the wave fields. Opposite to that, areas of substantial differences between numerical and analytical AGW parameters in Tables 1 and 2 reveal regions where numerical simulations are required.

Relations of linear AGW theory are frequently used for simplified parameterizations of AGW dynamical and thermal effects for their use in the numerical models of atmospheric general circulations (e.g., Lindzen, 1981; Holton, 1983; Gavrilov and Yudin, 1992, etc.). Similar parameterizations are also being developed for highly dissipative AGWs in the upper atmosphere (e.g., Vadas and Fritts, 2005; Yigit et al., 2008). Sometimes, different parameterizations give different results. Direct numerical simulation models of atmospheric AGWs may be useful tools for testing and verifying simplified parameterizations of wave effects.

In this study, we performed high-resolution numerical simulations of
nonlinear AGW propagation to the middle and upper atmosphere from a plane
wave forcing at the Earth's surface and compared them with analytical
polarization relations of linear AGW theory. Such comparisons may be used for
verifications of numerical models of atmospheric AGWs. Numerical simulations
show that, after triggering the wave source Eq. (11) at

Reasonable agreements between numerically simulated and analytical wave parameters in atmospheric regions, which correspond to the scope of the limitations of the AGW theory, may be considered as evidence of adequate descriptions of wave processes by the used nonlinear numerical model. Areas of substantial differences between numerical and analytical AGW parameters reveal atmospheric regions, where analytical theories give substantial errors and numerical simulation of wave fields is required. Direct numerical simulation models of atmospheric AGWs may be useful tools for testing and verifying simplified parameterizations of wave effects.

This work was partly supported by the Russian Basic Research Foundation, by the Russian Scientific Foundation (grant 14-17-00685), and by the Ministry of Education and Science of the Russian Federation (contract 3.1127.2014/K). Edited by: O. Marti