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 L_{D}IL → 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, γ = βL_{D}^{2}IU, 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 |I|>{√3}|k| in the two-dimensional wavenumber space. Here, k and I are the longitudinal and latitudinal wavenumbers, respectively. When γ is increased further, the energy concentration on the lines of I = ±{√3}K 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.