Generated using V3.0 of the official AMS LTEX template–journal page layout FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! Reynolds stress and eddy diffusivity of β-plane shear flows

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 co-evolving passive tracer. The flow is damped by linear drag on a time scale μ. 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), along the meridional direction (α > 0) or are isotropic (α = 0). The Reynolds stress varies linearly with α and non-linearly and non-monotonically with γ; but the Reynolds stress is independent of β. For positive values of α, the Reynolds stress displays a “anti-frictional” effect (energy is transferred from the eddies to the mean flow) and a frictional effect for negative values of α. 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 v/(2μ), where v is the meridional eddy velocity. In general, non-zero β and γ suppress the eddy diffusivity below v/(2μ). When the shear is strong, the suppression due to γ varies as γ while the suppression due to β varies between β and β depending on whether the shear is strong or weak, respectively.