Reynolds Stress and Eddy Diffusivity of β-Plane Shear Flows

Abstract

Abstract The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear γ, on a β plane, is calculated in conjunction with the eddy diffusivity of a coevolving passive tracer. The flow is damped by linear drag on a time scale μ−1. The stochastic forcing is white noise in time and its spatial anisotropy is controlled by a parameter α that characterizes whether eddies are elongated along the zonal direction (α < 0), are elongated along the meridional direction (α > 0), or are isotropic (α = 0). The Reynolds stress varies linearly with α and nonlinearly and nonmonotonically with γ, but the Reynolds stress is independent of β. For positive values of α, the Reynolds stress displays an “antifrictional” effect (energy is transferred from the eddies to the mean flow); for negative values of α, it displays a frictional effect. When γ/μ ≪ 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With γ = β = 0, the meridional tracer eddy diffusivity is , where υ′ is the meridional eddy velocity. In general, nonzero β and γ suppress the eddy diffusivity below . When the shear is strong, the suppression due to γ varies as γ−1 while the suppression due to β varies between β−1 and β−2 depending on whether the shear is strong or weak, respectively.