A canonical statistical theory of oceanic internal waves

Abstract

We use the methods of statistical mechanics to develop a theoretical relationship between the observed oceanic spectra and the probability distributions usually studied in statistical mechanics. We also find that the assumption that in terms of Lagrangian variables the oceanic internal wave field is near canonical equilibrium (i.e. the internal wave modes are populated in accordance with a Maxwell–Boltzmann-type distribution) yields expressions for the various marginal or reduced Eulerian spectra associated with both moored and towed measurements which are in striking qualitative agreement with experiment. In developing this theory it is important to distinguish carefully between Lagrangian and Eulerian variables. The important difference between the two sets of variables is due to the advective nonlinearity (i.e. (v∇) v where v is the Eulerian velocity) which is present only in the Eulerian frame. Our method treats the dynamics within the Lagrangian frame, where because of the absence of the advective nonlinearity it is fundamentally simpler, and then transforms to the Eulerian or measurement frame. We find that at small wavenumbers the four-dimensional Eulerian frequency wavenumber spectrum is approximately equal to the corresponding Lagrangian frequency wavenumber spectrum. At large wavenumbers, however, advective contributions become important and the two types of spectra are significantly different. While from a Lagrangian frame point of view the system is entirely wavelike, at large wavenumbers the Eulerian spectrum is not confined to the dispersion surface and the system, from an Eulerian frame point of view, is not wavelike. Further, the three-dimensional Eulerian wavenumber spectrum exhibits a large-wavenumber advective tail which decays as a power law and results in one-dimensional marginal spectra which are in excellent qualitative agreement with experiment. The above features are exhibited independent of the detailed nature of the underlying Lagrangian frequency wavenumber spectrum.