On the initial evolution of the weak turbulence spectrum in a system with a decay dispersion relation

The rate of change ε(k,t)=∂E(k,t)∕∂t of the energy spectrum E(k,t) of a weak turbulence in a system with a decay dispersion relation is investigated with particular interests in the initial stage of evolution such as the first several tens of periods. The theoretical predictions given by the kinetic equation of Hasselmann and that of Janssen which have been derived by the weak turbulence theory are compared with the results of direct numerical simulation (DNS) with a great accuracy. It is shown that ε(k,t) predicted by Janssen's equation correctly reproduces the rapid variation of ε(k,t) with the linear time-scale which is observed in DNS, while that predicted by Hasselmann's equation does not. It is also shown that the transient behavior of the asymptotic approach of Janssen's ε(k,t) to Hasselmann's one strongly depends on the wavenumber. For wavenumbers smaller than the spectral peak, the approach is exponential, while for wavenumbers larger than the spectral peak it contains a damped oscillation whose amplitude decays in time like 1∕t. A reasonable explanation for the origin of this damped oscillation is given in terms of the frequency mismatch of the three-wave “sum” interactions.