The nonlinear turbulent interactions between acoustic gravity waves are investigated using two-dimensional nonlinear fluid simulations. The acoustic gravity waves consist of velocity and density perturbations and propagate across the density gradients in the vertical direction in the Earth's atmosphere. We find that the coupled two component model exhibits generation of large-scale velocity potential flows along the vertical direction, while the density perturbations relax toward an isotropic random distribution. The characteristic turbulent spectrum associated with the system has a Kolmogorov-like feature and tends to relax toward a k−5/3 spectrum, where k is a typical wave number. The cross field diffusion associated with the velocity potential grows linearly and saturates in the nonlinear phase.
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 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 kx and kz 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 ρ0 (z) and p0 (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 dp0/dz = −ρ0g, where g = −g is the acceleration force on the atmospheric fluid due to gravity, and is the unit vector in the vertical direction. If ωg2 < 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]
 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 kxkz. In such cases the fluctuations merely oscillate with the Brunt-Väisälä frequency. In the other limit kxkz, the AGWs are dispersive and have anisotropic propagation. Equations (3) and (4) possess the nonlinear terms y × ∇∇2ψ · ∇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.
 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  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.
 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.
2. Nonlinear Simulations
 To investigate the nonlinear mode coupling interaction and turbulence aspects of acoustic gravity waves, we have developed a two-dimensional spectral code to numerically integrate equations (3) and (4). The 2-D simulations are not only computationally simpler and less expensive compared with the full 3-D simulations, but they offer significantly higher resolution even on moderately sized small-cluster machines like the IGPP (Institute of Geophysics and Planetary Physics, at University of California Riverside) Beowulf. The spatial discretization in our code uses a discrete Fourier representation of the turbulent fluctuations. The evolution variables use periodic boundary conditions. The initial isotropic turbulent spectrum is chosen close to a k−2 spectrum with random phases in all the directions. The choice of such (or even a flatter than −2) spectrum treats the turbulent fluctuations on an equal footing and avoids any influence on the dynamical evolution that may be due to initial spectral nonsymmetry. The equations are advanced in time using a Runge-Kutta fourth-order scheme. The code is made stable by a proper dealiasing of spurious Fourier modes and by choosing a relatively small time step in the simulations. Our code is massively parallelized using Message Passing Interface (MPI) libraries to facilitate higher resolution in a 2-D computational box. The computational domain comprises 5122 modes in two dimensions. Other simulation parameters in the normalized units, as described above, are H = 10, ωg = 0.02, and L = 4 × 4 (two-dimensional box size).
 All the fluctuations in our simulations are initialized isotropically with random phases and amplitudes in Fourier space. The initial fluctuations do not possess any flows or mean fields. The latter can be generated in our simulations by means of nonlinear interactions. Fourier spectral methods are numerically almost non dissipative compared to the existing finite difference methods and are thus remarkably successful in describing turbulent flows in a variety of plasma and hydrodynamic fluids. They also provide an accurate representation of the fluid fluctuations in Fourier space. Because of the latter, nonlinear mode coupling interactions preserve ideal rugged invariants of fluid flows, unlike finite difference or finite volume methods. The conservation of the ideal invariants such as energy, enstrophy, magnetic potential, helicity etc. in the turbulence is an extremely important feature because these quantities describe the cascade of energy in the inertial regime, where turbulence is, in principle, free from large-scale forcing as well as small-scale dissipation. We include small-scale dissipation, however, to distinctly extend the inertial range that helps push the spectral cascades further down to the smallest scales. The latter leads to an improved statistical average of spectral indices. The numerical validity in the simulations is checked by monitoring the integrable form of the energy of the system.
 The initially randomly propagating acoustic gravity waves begin to interact linearly during the early phase of the simulations. As the waves acquire larger amplitudes, they begin to interact nonlinearly. During the nonlinear interaction phase, various eddies mutually interact and transfer energy between the modes. The evolution of the modes for density fluctuations and velocity potential fields are shown in Figures 1 and 2in the (x, z) plane. It appears that the density perturbations (i.e., χ) have a tendency to generate smaller length-scale structures, while the velocity potential cascades toward larger scales. This is consistent with the corresponding nonlinear terms in drift wave turbulence. The coexistence of the small- and larger-scale structures in the turbulence is an ubiquitous feature of various 2-D turbulence systems. For example, in 2-D drift wave or CHM turbulence, the plasma fluid admits two inviscid invariants, namely the energy and the mean squared vorticity (i.e., irrotational velocity fields). The two invariants, under the action of an external forcing, cascade simultaneously in turbulence, thereby leading to dual cascade phenomena.
 In these processes, the energy cascades toward longer length scales, while the fluid vorticity transfers spectral power toward shorter length scales. Usually a dual cascade is observed in a driven turbulence simulation, in which certain modes are excited externally through random turbulent forces in spectral space. The randomly excited Fourier modes transfer the spectral energy by conserving the constants of motion in k space. On the other hand, in freely decaying turbulence, the energy contained in the large-scale eddies is transferred to smaller scales leading to a statistically stationary inertial regime associated with the forward cascades of one of the invariants. Decaying turbulence often leads to the formation of coherent structures as the turbulence relaxes, thus making the nonlinear interactions rather inefficient when they are saturated. It seems that the acoustic gravity waves exhibit turbulence in a manner similar to two-dimensional hydrodynamic turbulence in which the density and velocity potential fluctuations are analogous to the hydrodynamic fluid energy and vorticity. The evolution of the turbulent energy associated with the modes (density and potential field) are shown in Figure 3 where the linear and nonlinear phases of the evolution are clearly marked by the curve. After the nonlinear interactions are saturated, the energy in the turbulence does not grow and remains nearly unchanged throughout the simulations. Correspondingly, the energy transfer rate shows a significant growth during the linear and initial nonlinear phases. However when the nonlinear interactions saturate, the nonlinear transfer of energy in the spectral space amongst various turbulent modes becomes inefficient and the energy transfer per unit time tends to become negligibly small as shown in Figure 4. The power spectrum associated with the acoustic gravity wave turbulence exhibits a spectral slope close to −5/3 as shown in Figure 5. This is indicative of the turbulent eddies being the dominant processes in the spectral transfer and is consistent with the Kolmogorov-like phenomenology.
 We finally estimate the turbulent transport coefficient associated with a self-consistent evolution of the small- and large-scale turbulent fluctuations. An effective electron diffusion coefficient caused by the momentum transfer can be calculated from Deff = 〈v(r, t) · v(r, t + t′)〉dt′, where v is the fluid velocity, the angular bracket represents spatial averages and the ensemble averages are normalized to unit mass. Since the 2-D structures are confined to a x-z plane, the effective diffusion coefficient, Deff, essentially relates the diffusion processes associated with the translational motion of the fluid elements of our nonlinearly coupled system. We compute Deff in our simulations from both the velocity field fluctuations to measure the turbulent transport that is associated with the large-scale flows. It is observed that the effective cross-field transport is lower initially, when the field perturbations are Gaussian. On the other hand, the diffusion increases rapidly with the eventual formation of longer length-scale structures. This is shown in Figure 6. The transport due to the electrostatic potential dominates substantially as depicted in Figure 6. Furthermore, in the steady state, the nonlinearly coupled modes form stationary structures and Deff saturates eventually. Thus, remarkably, an enhanced cross-field transport level results primarily because of the emergence of large-scale coherent structures in a turbulence dominated by the acoustic gravity waves.
3. Concluding Remarks
 In summary, we have investigated the turbulent properties of 2-D nonlinearly interacting AGWs by means of computer simulations. We found that the nonlinear equations governing the dynamics of the fluid vorticity and density fluctuations in a nonuniform fluid admit dual cascade, leading to a Kolmogorov-like energy spectrum. In such a cascade process, we observe the formation of the short and large-scale structures that are responsible for the cross-field (across the density gradient direction) diffusion of particles due to random walk processes. The present results are essential for understanding the turbulent properties of the atmospheres of the Earth and other planets.