Introduction

Oceanic internal waves, whose restoring force is buoyancy in stratified fluid, exist widely in oceans. Internal waves are excited by topographies, tides and atmospheric disturbances. Their energy is transferred by nonlinear interactions among wavenumbers from large scales to small scales, and is dissipated in the small spatial scales of the Navier-Stokes turbulence generated by breaking of internal waves. Internal waves play a significant role in the general circulation of oceans and hence the climate of the Earth.

Energy spectra of internal waves are very broad with horizontal wavelengths varying from $10$ meters to $10^5$ meters, vertical wavelengths from $10$ meters to $10^3$ meters, and time periods from $10^3$ seconds to $10^5$ seconds. Surprisingly and despite all complexities, energy spectra of internal waves in the oceans appear to be somewhat universal, and are given by the Garrett-Munk (GM) spectrum. The GM spectrum was initially observed in the North Atlantic Ocean (Garrett & Munk, 1972,1975,1979).

In this paper we address the question of whether such a universal spectrum can be reproduced in direct numerical simulations. Similar universal spectra have been observed in direct numerical simulations for other wave-interacting systems on water surfaces, in particular, for gravity-wave and capillary-wave systems (Lvov, Nazarenko & Pokorni, 2006; Yokoyama, 2004; Onorato et al., 2002; Pushkarev & Zakharov, 2000; Dyachenko, Korotkevich & Zakharov, 2004). Unlike these systems, it appears from our direct numerical simulations that energy spectra of internal waves are not universal. Rather, accumulation of energy happens around the horizontally longest waves. We demonstrate that the energy transfer in inertial wavenumbers is dominated by scale-separated interactions with this accumulation. Such an accumulation around the longest waves has been observed also in the oceans. Furthermore, deviations from the Garrett-Munk spectrum were recently categorized (Lvov, Polzin & Tabak, 2004) thus mounting additional doubts to the possibility of any universality of the oceanographic spectrum for internal waves.