Probabilistic Aspects of Equation of Motion of Forced Burgers and Navier-Stokes Turbulence

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Abstract
Physical requirements and limitations on the force terms of the equations of motion for forced Burgers turbulence and for a class of forced, incompressible Navier-Stokes turbulence are discussed from probabilistic point of view. A basic problem, to determine the appropriate normalization of equations of motion, is answered. The normalization and the physical requirements are shown to stipulate that the force terms must bear Gaussian and white character for their time dependence as an exclusive consequence of the central limit theorem of Rosenblatt. A range of physical phenomena is thus pointed out to substantialize Kraichnan-Wyld-Edwards type of equations of motion for turbulence. A problem is found in the definition, as stochastic partial differential equations, of such equations with Gaussian-white-noise forces in the inviscid limit, and a possible way to circumvent the difficulty is shown to be inherent in the central limit theorem itself.

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