Mathematical Sciences: Semi-Strong Turbulence

在60年代，科学家们开始研究所谓的弱湍流，它可以用所谓的波的动力学方程来描述，类似于粒子的玻尔兹曼动力学方程。这样的湍流表现了许多色散波之间弱相互作用的动力学（就像海面上的非线性波）。我们发现在弱湍流和强湍流之间有一种明确的中间情况，我们称之为半强湍流。在这种情况下，湍流代表了弱相互作用波系统的动力学，但波动力学方程是不适用的，无论波的振幅有多小（例如，当波是非色散的时候，比如声波）。我们将从第一性原理出发，发展半强湍流理论，并将结果应用于声学湍流、浅水重力波湍流、长波对短波湍流的抑制、海浪湍流的角分布、光纤湍流、磁流体湍流等实际问题。此外，我们将实施一些计算和实际的实验来检验理论的预测。我们还打算发展一种新的非线性波的形式，这将使我们能够简化半强湍流的描述。

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.