Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere

The anisotropic characteristics of small-scale forced 2D turbulence on the surface of a rotating sphere are investigated. In the absence of rotation, the Kolmogorov $k^{\text{−5/3}}$ spectrum is recovered with the Kolmogorov constant $C_K\text{≈6,}$ close to previous estimates in plane geometry. Under strong rotation, in long-term simulations without a large-scale drag, a −5 slope emerges in the vicinity of the zonal axis ${\mathrm{(k}}_x\text{→0),}$ while a −5/3 slope prevails in other sectors far away from the zonal axis in the wave number plane. This picture is consistent with the new flow regime recently simulated by Chekhlov et al. [Physica D 98, 321–334 (1995)] and Smith and Waleffe [Phys. Fluids 11, 1608–1622 (1999)] on the beta plane. The concentration of energy in the zonal components and breaking of isotropy are caused by the strongly anisotropic spectral energy transfer and the stabilization of zonal mean flow by the meridional gradient of the planetary vorticity. The sharp tilt-up of the spectrum along the zonal axis was qualitatively understood through the scale-dependent stability property of the zonal flow. Under planetary rotation, the capacity for the zonal jets to hold energy and remain stable sharply increases with an increase of the meridional scale of the jets. In our simulations that were virtually inviscid at the large scales, the energy spectrum along the zonal axis tilts up all the way to the largest possible scale, indicating an apparent up-scale energy “cascade” along the zonal axis. This apparent up-scale cascade corresponds to a process of continuous mergers of zonal jets that does not cease until reaching the largest scale. This picture is consistent with the inviscid scenario for jet merging discussed by Manfroi and Young [J. Atmos. Sci. 56, 784–800 (1999)]. It contrasts the viscous scenario (for flows under the influence of a constant bottom drag) simulated in several previous studies, in which a distinct and finite jet scale emerges asymptotically.