Noise rectification in quasigeostrophic forced turbulence

Abstract

We study the appearance of large-scale mean motion sustained by stochastic forcing on a rotating fluid ~in the quasigeostrophic approximation! flowing over topography. We show that the effect is a kind of noiserectification phenomenon, occurring here in a spatially extended system, and requiring nonlinearity, absence of detailed balance, and symmetry breaking to occur. By application of an analytical coarse-graining procedure, we identify the physical mechanism producing such an effect: It is a forcing coming from the small scales that manifests itself in a change in the effective viscosity operator and in the effective noise statistical properties. Numerical simulations confirm our findings.@S1063-651X~98!14211-3# Nonlinear interactions can organize random inputs of energy into coherent motion. This noise-rectification phenomenon has been discussed in several contexts, including physics and biology @1#. Three ingredients are needed to obtain this kind of noise-sustained directed motion: nonlinearity, random noise lacking the property of detailed balance, and some symmetry-breaking feature establishing a preferred direction of motion. It has recently been shown numerically @2# that directed motion sustained by noise appears in quasigeostrophic twodimensional fluid flow over topography. The average flow at large scales approaches a state highly correlated with topography that disappears if noise or nonlinearity are switched off. The small scales of the flow follow a more irregular behavior. In this paper we establish that these topographic currents arise from a form of noise rectification, occurring here in a spatially extended system. To this end we analytically calculate a closed effective equation of motion for the large scales of the flow, by coarse graining the small scales. From this effective equation, the forcing of the small scales on the large ones ~sustained by noise and mediated by topography! is identified as the mechanism responsible for the directed currents. It appears as a renormalization of the viscosity operator in such a way that it favors relaxation toward a state correlated with topography, instead of toward rest. The effect becomes more important for increasing nonlinearity. More interestingly, this forcing vanishes when noise satisfies the detailed balance property revealing that the effect shares the same nonequilibrium origin of other noise-rectification phenomena. The presence of topography provides the symmetry-breaking ingredient needed to fix a preferred direction.