Energy and dissipation spectra of waves propagating in the inner surf zone

The spectral behavior of  random sawtooth waves propagating in the inner surf zone is investigated. 
We show that the elevation energy spectrum follows a universal shape with a $\omega^{-2}$ tendency in the inertial subrange and an exponential decay 
in the diffusive subrange ($\omega$ being the angular frequency). A theoretical spectrum is derived based on the similarities between sawtooth waves 
in the inner surf zone and  Burgers wave solutions. A very good agreement is shown between this theoretical spectrum and laboratory experiments covering 
a large range of incident random wave conditions. An equation describing the universal shape of the dissipation spectrum is also derived. 
It highlights that the dissipation spectrum is nearly constant in the inertial subrange, in agreement with previous laboratory observations. 
Our results can be useful to improve broken wave dissipation parametrizations in stochastic spectral wave models.