One of the central discoveries in Wave Turbulence was the power-law
Kolmogorov-Zakharov (KZ)
spectrum,
, which realise themselves in presence of the energy sources and sinks
separated by a large inertial range of scales. The exponent depends on
the
scaling properties of the interaction coefficient and the frequency. Most of the previous WT
literature
is devoted to study of KZ spectra and a good review of these works can be
found in [1]. We are not going review these studies here, but instead
we
are going to find out how adding such energy sources and sinks will this modify
the evolution equations for the statistics.
Instead of the Hamiltonian equation (1) let
us consider

where describes sources and sinks of the energy, e.g. due to instability and viscosity respectively. Easy to see that this linear term will not change the structure of the -mode PDF equation (65) but it will lead to re-definition of the flux:

(93) |

In the one-mode equations, this simply means renormalisation

(94) |

Thus we arrive at a simple message that the energy sources and sinks do not produce any ``sources'' or ``sinks'' for the flux of probability.

In the inertial range, there is no flux modification and one can easily find solution of (90) for the one-mode PDF,

(95) |

where is the KZ spectrum (solving the kinetic equation in the inertial range). However, it is easy to check by substitution that the product of such one-mode PDF's, , is not an exact solution to the multi-mode equation (65). Thus, there have to be corrections to this expression related either to a finite flux, an amplitude correlation or both.