The Creation of 3D Zonal Flows, their Spacings, Rhines Scaling, Turbulence, and Inverse Cascades in Atmospheres and Oceans

The creation of large vortices and zonal flows by small-scale forcing on a β-plane has been examined numerically in 2D for over 30 years. Marcus (2000) showed in 2D that, at small length scales, the inverse cascade produces a k^-5/3 kinetic energy spectrum with vortices and, at larger scales, a k^-5 spectrum with east-west zonal flows. By mathematical tautology, zonal Fourier components (k) in the latter spectrum have RMS velocities proportional to βk^-2, meaning all components obey the Rhines relation. The largest size (zonal width) produced in our calculations was ~2πL_R (L_R is the Rossby deformation radius), which also obeys the Rhines relation (as k ≡ 2π/(2πL_R) is part of the k^-5 spectrum). Our 2D calculations were consistent with the interpretation that energy inverse cascades up to a length where potential energy dominates kinetic as, in the QG approximation, the ratio of the kinetic to potential energy is (kL_R)². Here, we use the 3D QG equation to ascertain if the 2D results for zonalization, Rhines relation, and inverse cascades are valid in 3D. Although the 3D QG equation does not formally have an L_R, we also determine whether the largest scale zonal flow (i.e., the terminus of the inverse cascade) occurs at the wave number where potential energy dominates kinetic.