## 1. Introduction

[2] Studies of acoustic gravity waves (AGWs) [*Yeh and Liu*, 1974; *Fritts and Alexander*, 2003; *McKenzie and Axford*, 2000; *Stenflo*, 1987; *Jovanovic et al.*, 2002] in the Earth's middle atmosphere, in the solar atmosphere, as well as in planetary magnetospheres have been motivated by the need to obtain accurate predictions of the dynamics of the atmosphere under various meteorological conditions, and include the profiles of the density, pressure, and the presence of shear flows in the winds. The AGWs are low-frequency disturbances associated with the density and velocity perturbations of the atmospheric fluid in the presence of the equilibrium pressure gradient that is maintained by the gravity force. The frequency *ω* of the AGWs is given by the dispersion relation [*Stenflo*, 1987, 1996; *Jovanovic et al.*, 2002]

where *k*_{x} and *k*_{z} are the components of the wave vector along the *x* and the *z* axis in a Cartesian coordinated system, *H* is the density scale height, and the squared Brunt-Väisälä frequency is

where *ρ*_{0} (*z*) and *p*_{0} (*z*) are the equilibrium density and pressure that are inhomogeneous in the vertical direction (denoted by the coordinate *z*), and *γ* ≈ 1.4 is the adiabatic index. At equilibrium, we have *dp*_{0}/*dz* = −*ρ*_{0}*g*, where **g** = −*g* is the acceleration force on the atmospheric fluid due to gravity, and is the unit vector in the vertical direction. If *ω*_{g}^{2} < 0, it turns out from (1) that the AGWs are unstable. When the amplitudes of the AGWs become large, the modes start interacting among themselves. The mode couplings are governed by [*Stenflo*, 1996]

and

where *ψ* (*x*, *z*) is the stream function, *χ* (*x*, *z*) is the normalized density perturbation, *D*/*Dt* = (∂/∂ *t*) + **v** · , **v** = (∂*ψ*/∂*x*) − (∂*ψ*/∂*z*) is the fluid velocity, and ^{2} = (∂^{2}/∂*x*^{2}) + (∂^{2}/∂*z*^{2}). Time and space coordinates in equations (3) and (4) are normalized respectively by their typical values *t*_{0} and _{0}. The other variables are normalized as follows; *ψ*/*ψ*_{0} = *ψ*′, (*t*_{0}_{0}/*ψ*_{0})*χ* = *χ*′, *t*_{0}^{2}*ω*_{g} = *ω*_{g}′, *H*/_{0} = *H*′, _{0}∇ = ∇′, *t*_{0}*D*/*Dt* = *D*/*Dt*′. The primes have been removed from equations (3) and (4). Stationary nonlinear solutions of (3) and (4) in the form of a double vortex and a vortex chain have been presented by many authors [*Stenflo*, 1987, 1996; *Jovanovic et al.*, 2002; *van Heijst and Kloosterziel*, 1989; *Aburdzhaniya*, 1996; *Pokhotelov et al.*, 2001; *Yeh and Liu*, 1972].

[3] The coupled nonlinear equations conserve total energy *E* in the absence of sources and sinks. In the normalized units, it is

It is noteworthy from the linear dispersion relation that the group and phase speeds of the AGWs are negligibly small when *k*_{x}*k*_{z}. In such cases the fluctuations merely oscillate with the Brunt-Väisälä frequency. In the other limit *k*_{x}*k*_{z}, the AGWs are dispersive and have anisotropic propagation. Equations (3) and (4) possess the nonlinear terms _{y} × ∇∇^{2}*ψ* · ∇*χ* and _{y} × ∇*ψ* · ∇*χ* that are analogous to the polarization and diamagnetic nonlinearities in inhomogeneous drift wave turbulence. The former cascades energy toward larger scales, while the latter leads to the formation of smaller scales due to forward cascades. In the next section, we shall explore the nonlinear interactions of acoustic gravity waves.

[4] The nonlinear interaction between finite amplitude AGWs in the Earth's atmosphere [*Gill*, 1982] leads to energy transfer, resonantly and nonresonantly between the high- and low-frequency parts of the spectrum. Resonant interactions between the AGWs play also a key role in a rotating atmosphere [*Axelsson et al.*, 1996]. Many analytical theories [*Stenflo*, 1994] describing nonlinear couplings between different wavelength acoustic gravity modes predict the formation of vortices, which can be important for weather predictions. *Gardner* [1994] explains vertical wave number spectrum by means of a scale-independent diffusivity by assuming the damping effects of molecular viscosity, turbulence, and off-resonance wave-wave interactions. Similarly, [*Beres et al.*, 2005] presented an implementation of a physically based gravity wave source spectrum parameterization over convection in Global Circulation Model.

[5] With the objective of developing a self-consistent turbulent spectrum of AGWs we, in this paper, present the turbulent properties of nonlinearly interacting AGWs that are governed by equations (3) and (4). Specifically, we investigate by means of computer simulations the statistical properties of the turbulent spectrum arising from dual cascading. The fluid diffusion across the density and pressure inhomogeneity directions in the presence of coherent structures is also deduced. Furthermore, we consider the statistical turbulence properties of the nonlinearly interacting AGWs. Specifically, we perform computer simulation studies of the acoustic gravity mode cascading as well as the resulting structures and turbulent spectra. The diffusion of the fluid across the density and pressure gradients in the presence of large-scale acoustic gravity mode structures is also investigated.