Mathematical Sciences: Semi-Strong Turbulence

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DMS-9423245 Saffman and Meiron In the 60s scientists began to study so called weak turbulence which can be described by a so called kinetic equation for waves, similar to the Boltzmann kinetic equation for particles. Such turbulence represents a dynamics of many dispersive waves weakly interacting with each other (like nonlinear waves on the sea surface). We have found that there is a well defined intermediate situation between weak and strong turbulence which we call {\it semi-strong turbulence}. In this situation the turbulence represents the dynamics of a system of weakly interacting waves, but the wave kinetic equation is inapplicable, no matter how small the wave amplitudes are (this happens, for example, when the waves are non-dispersive, like acoustic waves). We are going to develop the theory of semi-strong turbulence from first principles and apply the results to several practical problems, e.g. acoustic turbulence, the turbulence of shallow water gravity waves, the suppression of short wave turbulence by long waves, angular distribution of the sea wave turbulence, turbulence in optical fibers, magneto-hydrodynamic turbulence, etc. In addition we are going to implement some computational and real experiments to check the predictions of the theory. We also intend to develop a new formalism for nonlinear waves, which will allow us to simplify the description of semi-strong turbulence.