Rossby wave nonlinear interactions and large-scale zonal flow formation in two-dimensional turbulence on a rotating sphere

Two-dimensional Navier-Stokes turbulence on a rotating sphere is one of the most fundamental mathematical models for describing the dynamics of planetary atmospheres and oceans. Despite its great simplicity, this system is known to have a solution with an anisotropic non-uniform large-scale structure similar to the zonal flows similar to those on Jupiter and other giant gas planets. Although westward circumpolar zonal flows are formed in the free-decay problem [1, 2] and multiple zonal band structure is formed in the forced problem [3, 4], the formation mechanism of the large-scale zonal flows has not yet been fully clarified. From the mean flow non-acceleration theorem based on the weakly non-linear theory [5, 6, 7], the effect of viscosity is sometimes considered to be its essential factor. However, there is no guarantee that the suggestion from the weakly nonlinear theory hold for fully nonlinear systems. In fact, Obuse and Yamada [8] reported the formation of large-scale westward circumpolar zonal flows in an unforced two-dimensional turbulence on a rotating sphere even when considering inviscid flows, which strongly suggests that the main factor in the formation mechanism of large-scale zonal flows is the nonlinearity of the Navier-Stokes or Euler equations, not the dissipation by viscosity.In this study, we consider two-dimensional Navier-Stokes equations on a rotating sphere, and focus on three-wave nonlinear interactions of Rossby waves, which are linear solutions of this system, to investigate the factors directly involved in the mechanism of large-scale zonal flow formation. The three-wave non-resonant nonlinear interactions of Rossby waves are investigated in detail, by calculating the time derivative of energy of zonal Rossby modes. The obtained results suggest that the formation of the westward circumpolar large-scale zonal flows is directly caused by non-local energy transfer due to near-resonant interactions.[1] S. Yoden and M. Yamada, “A numerical experiment on two-dimensional decaying turbulence on a rotating sphere,"  J. Atomos. Sci., 50, 631-643 (1993)[2] S. Takehiro, M. Yamada, Y.-Y. Hayashi, "Energy accumulation in easterly circumpolar jets generated by two-dimensional barotropic decaying turbulence on a rapidly rotating sphere", J. Atmos. Sci., 64, 4084-4097 (2006)[3] T. Nozawa and S. Yoden, "Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere," Phys. Fluids, 9, 2081-2093 (1997)[4] K. Obuse, S. Takehiro, M. Yamada, "Long-time asymptotic states of forced two-dimensional barotropic incompressible flows on a rotating sphere", Phys. Fluids., 22, 156601 (2010)[5] G. Charney and P.G. Drazin，"Propagation of planetary-scale disturbances from the lower into the upper atmosphere",  J. Geophys. Res., 66, 83-110 (1961)[6] A. Eliassen and E. Palm, "On the transfer of energy in stationary mountain waves",  Geofys Publ., 22(3), 1-23 (1961)[7] D.G. Andrews and M.E. McIntyre, "Planetary waves in horizontal and vertical shear: The generalized Eliassen-Palm relation and the mean zonal acceleration", J. Atoms. Sci., 30(11), 2031-2048 (1976)[8] K. Obuse and M. Yamada, in preparation