Angular distribution of energy spectrum in two-dimensional β-plane turbulence in the long-wave limit

Abstract

The time-evolution of two-dimensional decaying turbulence governed by the long-wave limit, in which LD/L → 0, of the quasi-geostrophic equation is investigated numerically. Here, LD is the Rossby radius of deformation, and L is the characteristic length scale of the flow. In this system, the ratio of the linear term that originates from the β-term to the nonlinear terms is estimated by a dimensionless number, \documentclass[12pt]{minimal}\begin{document}$\gamma =\beta L_{\rm D}^2/U$\end{document}γ=βLD2/U, where β is the latitudinal gradient of the Coriolis parameter, and U is the characteristic velocity scale. As the value of γ increases, the inverse energy cascade becomes more anisotropic. When γ ⩾ 1, the anisotropy becomes significant and energy accumulates in a wedge-shaped region where \documentclass[12pt]{minimal}\begin{document}$|l|>\sqrt{3}|k|$\end{document}|l|>3|k| in the two-dimensional wavenumber space. Here, k and l are the longitudinal and latitudinal wavenumbers, respectively. When γ is increased further, the energy concentration on the lines of \documentclass[12pt]{minimal}\begin{document}$l=\pm \sqrt{3}k$\end{document}l=±3k is clearly observed. These results are interpreted based on the conservation of zonostrophy, which is an extra invariant other than energy and enstrophy and was determined in a previous study. Considerations concerning the appropriate form of zonostrophy for the long-wave limit and a discussion of the possible relevance to Rossby waves in the ocean are also presented.