Frequency-vertical-wavenumber and horizontal-vertical-wavenumber spectrum

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 $n(\bm{k},m)$ is given by Eq. (2).

The corresponding vertical wavenumber-frequency spectrum of energy is obtained by transforming $n_{\bm{k},m}$ from wavenumber space $(\bm{k},m)$ to the vertical wavenumber-frequency space $(\omega,m)$ and multiplying by frequency. In the high-frequency large-wavenumber limit,

The total energy density of the wave field is then

Thus, we also it convenient to work with the wave action spectrum expressed as a function of $\omega$ and $m$ Therefore we also introduced (38). The relation between $a$, $b$ and $\widetilde{a}$, $\widetilde{b}$ reads: