Closed 1st-Order And 2nd-Order Moment Equations For Stochastic Nonlinear Problems with Applications To Model Hydrodynamic And Vlasov-Plasma Turbulence

Working along the lines of a procedure outlined by Keller, a technique is developed for deriving closed first_ and second_order moment equations for a general class of stochastic nonlinear equations by performing a renormalization at the level of the second moment. The work of Weinstock, as reformulated recently by Balescu and Misguich, is extended in order to obtain two equivalent representations for the second moment using an exact, nonperturbative, statistical approach. These general results, when specialized to the weak_coupling limit, lead to a complete set of closed equations for the first two moments within the framework of an approximation corresponding to Kraichnan's direct_interaction approximation. Additional restrictions result in a self_consistent set of equations for the first two moments in the stochastic quasilinear approximation. Finally, the technique is illustrated by considering its application to two specific physical problems: (1) modelhydrodynamicturbulence and (2) Vlasov_plasma turbulence in the presence of an external stochastic electric field.