Observed atmospheric and oceanic internal wave spectra, when analysed in an Eulerian frame of reference, exhibit a large-wavenumber ‘tail’. In one-dimensional vertical-wavenumber (k3) spectra, it is typically proportional to |k3|−3.

In 1989, K. R. Allen and R. I. Joseph showed that a large-wavenumber tail was to be anticipated as a consequence of Eulerian nonlinearity, and they derived relations for the coefficients of both horizontal and vertical spectra of the form |k|−3. The coefficients were obtained only for the wave-induced vertical-displacement spectra, and only for an input spectrum having a certain ‘canonical’ frequency variation derived on other grounds.

The present work builds on that of Allen & Joseph. It is more general in some respects, more limited in others. It provides a more transparent form of analysis, it treats a broad class of wave variables, and it does so for input (Lagrangian) spectra that can be chosen by the user, free from any constraint to canonical or other restricted forms. It provides relations whereby the full Eulerian spectrum may be determined numerically, once the input spectrum has been chosen, and it provides analytic forms applicable at large wavenumbers for horizontally isotropic spectra. The derived one-dimensional vertical-wavenumber spectra are discussed in relation to observations.

Certain shortcomings in the development, both as given by Allen & Joseph and as found here, are identified and discussed.