The theoretical work presented below addresses the asymptotic power laws of a three-dimensional action spectrum. In order to connect with that work, note that a horizontally isotropic power-law form of the three-dimensional wave action is given by Eq. (2).
The corresponding vertical wavenumber-frequency spectrum of energy
is obtained by transforming from wavenumber space
to the vertical wavenumber-frequency space and
multiplying by frequency. In the high-frequency large-wavenumber
limit,
The total energy density of the wave field is then