Thermoacoustic internal gravity wave turbulence in the Earth's lower atmosphere

TL;DR

The paper develops a two‑dimensional Boussinesq framework to study the nonlinear coupling between internal gravity waves and thermal waves arising from temperature‑dependent density inhomogeneity in the Earth's lower atmosphere. It analyzes the linear thermoacoustic IGW mode and performs nonlinear spectral simulations on a 512×512 grid, revealing an inverse cascade of large-scale velocity structures and forward cascades for density and temperature fluctuations. In the troposphere, turbulence intensifies with horizontal and vertical spectra E(k_x) ∼ k_x^{-1.67} and E(k_z) ∼ k_z^{-2.89}, accompanied by a rising diffusion coefficient, indicating strong vertical mixing; in the stratosphere, energy transfer saturates, yielding weaker turbulence with E(k_x) ∼ k_x^{-1.83} and E(k_z) ∼ k_z^{-1.03}. These results highlight thermoacoustic IGWs as a mechanism for scale‐dependent energy transfer and vertical mixing, with significant implications for atmospheric dynamics and weather patterns.

Abstract

We propose, for the first time, a two-dimensional model for the nonlinear coupling of internal gravity and thermal waves in the presence of temperature-dependent density inhomogeneity due to thermal expansion and thermal feedback in stratified fluids of the Earth's lower atmosphere ($0-50$ km). Such a coupling gives rise to the evolution of thermoacoustic internal gravity waves (IGWs), which are distinctive from the known IGWs in the literature. We perform numerical simulations to study the nonlinear interactions of velocity and density perturbations associated with the IGWs and thermal fluctuations corresponding to the thermal mode. We show that solitary vortices of IGWs coupled to the thermal wave can lead to thermoacoustic turbulence. We observe the formation of large-scale velocity potential flows and small-scale structures in the density and temperature profiles. Interestingly, while the wave energy spectra exhibit power laws: $ k_x^{-1.67}$ and $ k_z^{-2.89}$, respectively, for horizontal and vertical wave numbers, in the troposphere ($0-15$ km) with negative temperature gradient, the same in the stratosphere ($15-50$ km) with positive temperature gradient tend to relax toward $k_x^{-1.83}$-horizontal and $k_z^{-1.03}$-vertical spectra. We find that while the energy spectra in the tropospheric turbulence are consistent with the observed phenomena without temperature gradients, those in the stratosphere differ.

Figures (10)

• Figure 1: Dispersion curves for the real wave frequency $(\omega_r)$ and the instability growth rate $(\gamma)$ of thermoacoustic internal gravity waves in the case of $C_0\equiv d\overline{T}/dz<0$ are shown for different parameter values. The solid, dashed, dotted, and dash-dotted lines, respectively, correspond to $k_z=0.1,~\kappa^\prime=0.00085,~q_{T_0}=0.03$; $k_z=0.3,~\kappa^\prime=0.00085,~q_{T_0}=0.03$; $k_z=0.1,~\kappa^\prime=0.0013,~q_{T_0}=0.03$; and $k_z=0.1,~\kappa^\prime=0.00085,~q_{T_0}=0.035$. The other fixed parameter values are as in Table \ref{['tab-parameter']} for $C_0<0$.
• Figure 2: Dispersion curves for the real wave frequency $(\omega_r)$ and the instability growth rate $(\gamma)$ of thermoacoustic internal gravity waves in the case of $C_0\equiv d\overline{T}/dz>0$ are shown for different parameter values. The solid, dashed, dotted, and dash-dotted lines, respectively, correspond to $k_z=0.1,~\kappa^\prime=0.000279,~q_{T_0}=0.0355$; $k_z=0.3,~\kappa^\prime=0.000279,~q_{T_0}=0.0355$; $k_z=0.1,~\kappa^\prime=0.001,~q_{T_0}=0.0355$, and $k_z=0.1,~\kappa^\prime=0.000279,~q_{T_0}=0.04$. The other fixed parameter values are as in Table \ref{['tab-parameter']} for $C_0>0$.
• Figure 3: Evolution of the potential function ($\widetilde{\psi}$), density fluctuation ($\widetilde{\chi}$), and temperature ($\widetilde{T}$) at different times (a) $t = 0$, (b) $t = 5$, (c) $t = 10$, and (d) $t = 20$ is shown [Simulation of Eqs. \ref{['eq-mom5']}-\ref{['eq-temp5']}]. The subplots for $\widetilde{\psi}$ (left panel) illustrate the emergence of large-scale structures in the potential field due to an inverse cascade process. The middle and right panels highlight the development of small-scale eddies, resulting from the forward cascades of density $(\widetilde{\chi})$ and temperature $(\widetilde{T})$ fluctuations. The parameter values are as in Table \ref{['tab-parameter']} for the troposphere.
• Figure 4: Evolution of the total energy $E$ with time $t$ [Eq. \ref{['eq-energy']}] is shown. A rapid transfer of energy takes place in the evolution at a later stage of the time period. Such a rapid transfer of energy is due to the instability of IGWs by the effects of the thermal expansion and thermal feedback of the medium to the temperature and density perturbations. The parameter values are as in Table \ref{['tab-parameter']} for the troposphere.
• Figure 5: The wave energy spectra $E(k_x)$ and $E(k_z)$ corresponding to the horizontal and vertical wavenumbers are shown at different times $t = 5, 10, 15,$ and $20$. The spectra show a power-law behavior in the inertial range, following $k_x^{-1.67}$ and $k_z^{-2.89}$ for the horizontal and vertical spectra, respectively. The parameter values are the same as given in Table \ref{['tab-parameter']} for the troposphere.