HRT

Abstract

The energy spectrum of stably stratified turbulence is studied numerically by solving the 3D Navier-Stokes equations under the Boussinesq approximation pseudo-spectrally. The resolution is $1024^3$. Using toroidal-polodial decomposition (Craya-Herring decomposition), the velocity field is divided into the vortex mode and the wave mode. For randomly forced flows, we show that there is a sharp wave number transition in the energy spectra of the vortex and wave modes. With the initial kinetic energy zero, the vortex-mode first develop a $k_{¥perp}^{-3}$ spectrum for the whole wavenumber range studied. Here, $k_{¥perp}¥equiv ¥sqrt{k_x^2+k_y^2}$. Then at large $k_{¥perp}$, a $k_{¥perp}^{-5/3}$ part appears with a sharp transition in the wave number. Meanwhile the wave-mode spectra show a $k_{¥perp}^{-2}$ first, and then $k_{¥perp}^{-5/3}$ part appears at high $k_{¥perp}$. Spectra for different values of the Brunt-Vaisala frequency are investigated. We present evidence that the $k_{¥perp}^{-3}$ part at the large scale in the vortex-mode spectra may be characterized as 2d turbulence. The small scale part is a Kolmogorov spectrum $¥sim ¥epsilon^{2/3}k_{¥perp}^{-5/3}$, where $¥epsilon$ is the horizontal energy dissipation of the vortex-mode. Finally, we discuss decaying stratified turbulence. We note that our present high resolution simulations support earlier simulations (Kimura & Herring 1996) in which the decay of total energy was found to be $E(t)¥sim t^{-1}$. An explanation for such a decay is proposed in terms of the shape of the toroidal and polodial energy spectra.