Concluding remarks

We tried to reproduce the universal energy spectrum of oceanic internal waves by direct numerical simulations. Instead of expected universal behaviour in large wavenumbers, the simulations show a tendency for energy to concentrate around the horizontally longest waves that are determined by periodic boundary condition. It also appears that the power-law exponents of integrated spectra are different from those of cross-sectional spectra. This is apparent in density-wavenumber spectra, whose integrated spectrum is strongly affected by the accumulation of energy in small isopycnal wavenumbers. It is shown that the accumulation strongly depends on the local value of inertial frequencies. We also find that the nonlocal interactions in the wavenumber space seem to be dominant in large isopycnal and density wavenumbers. Energy spectra in these wavenumbers are greatly affected by small isopycnal wavenumbers, since nonlocal interactions in the wavenumber space are dominant in energy transfer in large isopycnal and density wavenumbers.

On the other hand, energy spectra in the small wavenumbers cannot be universal since they are box-size dependent in direct numerical simulations with periodic boundary conditions. In the real oceans, $\beta$-effect changes $f$ and is important for Rossby waves, which are the longest waves in oceans. The topographies of seabeds and the alignment of continents can also alter the longest waves. The temperature gives seasonal variability and the atmospheric disturbance also gives fluctuations in large scales. Indeed, it is known that the large-scale motions cannot be universal in the real oceans.

Therefore, the small-scale properties cannot be universal if the nonlocal interactions are dominant in the energy transfer in large wavenumbers. It is in contrast to the surface-wave systems, which have the universal energy spectra in small scales as a result of cascades even if large-scale behavior is not universal.

Acknowledgmets This research is supported by NSF CMG grant 0417724. Y. L. was also supported by NSF CAREER DMS 0134955. We are gratefull to Kurt Polzin and Esteban Tabak for helpfull discussions. We are also thank YITP in Kyoto University (Japan) for allowing us to use SX8, where numerical simulations were performed.