Energy spectra of stably stratified turbulence

Energy spectra for forced stably stratified turbulence are investigated numerically using the Direct Numerical Simulations (DNS) with $1024^3$ grid points. The calculation is done by solving the 3D Navier-Stokes equations under the Boussinesq approximation pseudo-spectrally. Using toroidal-poloidal decomposition (Craya-Herring decomposition), the velocity field is divided into the vortex mode ($\phi_1$) and the wave mode ($\phi_2$). The $\phi_1$ and $\phi_2$ spectra as a function of horizontal wave numbers, $k_{\perp}$, has the form of \vspace{-0.6cm} \begin{eqnarray} E_{\perp\Phi_1}(k_{\perp}) =\left\{\begin{array}{l} \alpha \eta_{\perp\Phi_1}^{2/3}k_{\perp}^{-3} \quad(k_{\perp}< k_c) \\ C_K\varepsilon_{\perp \Phi_1}^{2/3}k_{\perp}^{-5/3}\quad (k_{\perp}>k_c)\end{array}\right. \;, \nonumber\\ E_{\perp\Phi_2}(k_{\perp}) =\left\{\begin{array}{l} \beta \sqrt{N\varepsilon_{\perp\Phi_2}} k_{\perp}^{-2} \quad(k_{\perp}< k_c) \\ C_K\varepsilon_{\perp\phi_2}^{2/3}k_{\perp}^{- 5/3}\quad(k_{\perp}>k_c)\end{array}\right. \;.\nonumber \end{eqnarray} where $\eta_{\perp\phi_1}$ and $\varepsilon_{\perp\phi_2}$ are the horizontal enstrophy dissipation based on the $\phi_1$ energy and the horizontal energy dissipation based on the $\phi_2$ energy, respectively. For both cases, $C_K\approx 1.2\sim2.0$ is obtained being close to the Kolmogorov constant. To understand the reason for the steeper spectra than the Kolmogorov -5/3 for large scales, inviscid calculations (truncated Euler's equation) without forcing are conducted. We verified that emergence of steeper spectra for large scales and thermalization spectra for small scales.