Stochastic parametric resonance in shear flows

Abstract. Time-periodic shear flows can give rise to Parametric Instability (PI), as in the case of the Mathieu equation (Stoker, 1950; Nayfeh and Mook, 1995). This mechanism results from a resonance between the oscillatory basic state and waves that are superimposed on it. Farrell and Ioannou (1996a, b) explain that PI occurs because the snap-shots of the velocity profile are subject to transient growth. If the flows were purely steady the transient growth would subside and not have any long lasting effect. However, the coupling between transient growth and the time variation of the basic state create PI. Mathematically, transient growth, and therefore PI, are due to the nonorthogonal eigenspace in the linearized system.

Poulin et al. (2003) studied a time-periodic barotropic shear flow that exhibited PI, and thereby produced mixing at the interface between Potential Vorticity (PV) fronts. The instability led to the formation of vortices that were stretched. A later study of an oscillatory current in the Cape Cod Bay illustrated that PI can occur in realistic shear flows (Poulin and Flierl, 2005). These studies assumed that the basic state was periodic with a constant frequency and amplitude. In this work we study a shear flow similar to that found in Poulin et al. (2003), but now where the magnitude of vorticity is a stochastic variable. We determine that in the case of stochastic shear flows the transient growth of perturbations of the snapshots of the basic state still generate PI.