Transition in energy spectrum for forced stratified turbulence

Energy spectrum for forced stably stratified turbulence is investigated numerically. The 3D momentum equation under the Boussinesq approximation is solved pseudo- spectrally with stochastic forcing applied to the largest velocity scales. Following Lesieur \& Rogallo (1989) and Carnevale {\it et.~al.}(2001), spectral eddy viscosity, $ \nu_t(k)=(a_1+a_2 \exp(- a_3k_c/k))\sqrt{E(k_c)/k_c}$, is used for small scale dissipation. Using toroidal-poloidal decomposition (Craya-Herring decomposition), the velocity field is divided into the vortex mode ($\phi_1$) and the wave mode ($ \phi_2 $). With the initial kinetic energy being zero, the $\phi_1$ spectra as a function of horizontal wave numbers, $k_{\perp}$, first develops a $k_{\perp}^{-3}$ spectra for the whole $k_{\perp}$ range, and then $k_{\perp}^{-5/3}$ part appears with rather a sharp transition wave number. Meanwhile the $\phi_2$ spectra shows $k_{\perp}^{-2}$ first, and then $k_{\perp}^{-5/3}$ part appears with the same transition wave number. According to Carnevale {\it et.~al.}, the transition wave number is understood as the Ozmidov scale with a correction by the coefficients of the buoyancy spectrum, $E(k) =\alpha N^2k^{-3}$, and the Kolmogorov spectrum, $E(k)=C_K\epsilon^{2/3} k^{-5/3}$. By equating these spectra, we obtain $k_b \sim (\alpha/C_K)^{3/4}\sqrt {N^3/ \epsilon}$. This assessment will be discussed. \par\medskip\noindent Carnevale,G.F. {\it et.~al}: 2001 J.~Fluid Mech. {\bf 427} 205--239.\par\noindent Lesieur, M. \& Rogallo, R. 1989 Phys. Fluids A{\bf 1} 718--722.\par