General weak turbulence theory of resonant four-wave processes.
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In this article, the time evolution of wave correlations (cumulants) in a uniformly turbulent ensemble of weakly nonlinear, dispersive systems is discussed and a kinetic equation for the spectral energy density of the waves is derived in situations where the resonant three-wave decay condition cannot be satisfied.Abstract:
The time evolution of wave correlations (cumulants) in a uniformly turbulent ensemble of weakly nonlinear, dispersive systems is discussed. With closure of the hierarchy for wave correlations appropriate to the inclusion of resonant four‐wave processes, a kinetic equation for the spectral energy density of the waves is derived in situations where the resonant three‐wave decay condition cannot be satisfied. The resulting kinetic equation is a nonlinear integrodifferential equation with driving terms trilinear in the energy density. Some general properties of this equation are discussed including associated conservation relations and the law of increase of entropy.read more
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References
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Book
The theory of homogeneous turbulence
TL;DR: In this article, the kinematics of the field of homogeneous turbulence and the universal equilibrium theory of decay of the energy-containing eddies are discussed. But the authors focus on the dynamics of decay and not on the probability distribution of u(x).