Turbulence closure for the weakly nonlinear stochastic waves requires,
besides weak nonlinearity, randomness in both the phases and the
amplitudes of the Fourier modes. This randomness, once present
initially, must remain over the nonlinear evolution time.
Finding out to what extent is this true is the main goal of the
present Letter. For this analysis we derive an evolution equation for
the full probability density function (PDF) of the wave field.
We will show that, for any statistics of the amplitudes, phases tend
to stay random if they were random initially. If in
addition the initial amplitudes are independent variables they will
remain independent in a coarse-grained sense, i.e. when considered in
small subsets which are much less than the total set of modes.
, Yuri V. Lvov
and Sergey Nazarenko