Conclusions

We have developed a quite general, natural Hamiltonian formalism for internal waves in a stratified, rotating environment. Our formulation gains much in simplicity, by restricting consideration to flows in hydrostatic balance, superimposed on a vertically arbitrary, but horizontally uniform shear and vorticity fields. The resulting Hamiltonian inherits much of the structure of the shallow-water equations, though with one extra vertical dimension. The use of isopycnal coordinates, whereby the depth $z$ is replaced by the density $\rho$ as the independent vertical coordinate, allows for a straightforward separation of the dynamics of waves and vorticity, by assuming the latter to be uniform on surfaces of constant density.

This Hamiltonian formulation allows us to derive a kinetic equation for the time evolution of the spectral energy density. In the limit of high frequencies, when the effects of the rotation of the Earth loose significance, exact steady solutions to this kinetic equation can be found, corresponding to the direct cascade of energy toward the short scales. This Kolmogorov-like family of spectra includes the empirically based prediction of Garrett and Munk, as well as much of the pattern of observed variability around it. It is conjectured that the selection principle yielding the GM spectrum from this bigger family may involve a solvability condition related to the small corrections due to the Coriolis effect. A full investigation of this issue, however, lies beyond the scope of the present article.