Decimated Amplitude Equations in Turbulence Dynamics

Abstract

A scheme of decimation, or statistical interpolation, is developed for stochastic systems which exhibit dynamical and statistical symmetries among groups of modes. The end product is generalized Langevin equations for a sample set of explicitly followed modes. All the other modes are represented by the random forcing amplitudes in the Langevin equations. The random forcing is constrained by realizability inequalities and by appeal to the statistical symmetries. The latter yield expressions for moments of the joint probability distribution of forcing amplitudes and sample-set amplitudes in terms of moments of the sample-set amplitudes alone. This permits integration of the dynamical equations for an ensemble of sample-set amplitudes. Converging approximation sequences may be generated by systematically enlarging the set of constraints.

The decimation scheme (DS) bridges among several kinds of attack on the turbulence problem: moment hierarchy equations, renormalized perturbation theory (RPT), renormalization group methods, and direct modeling of large systems by systems with fewer modes. The DS does not appeal to perturbation theory, but in the limit of strong decimation it can be analyzed perturbatively. The direct-interaction approximation and other RPT approximations may thereby be obtained from appropriate moment constraints. One feature of the DS is that invariance to random Galilean transformation, a severe problem in the RPT treatment of turbulence, can be directly assured by constraints which relate 3rd-and 4th-order moments.