Theoretical Estimation of the Superrotation Strength in an Idealized Quasi-Axisymmetric Model of Planetary Atmospheres

This paper presents a theoretical estimation of the strength of equatorial superrotation in planetary atmospheres by exploring a quasi-axisymmetric system that is zonally averaged primitive equations for a dry Boussinesq fluid on a rotating hemisphere with the effects of nonaxisymmetric eddies parameterized by eddy diffusion. The fluid is forced by Newtonian heating and cooling, and the horizontal eddy diffusion of momentum is assumed to be much stronger than the vertical one. In this system, the superrotation is maintained by the Gierasch mechanism, which possibly explains the superrotation of the Venus atmosphere by angular momentum transport due to the mean meridional circulation and horizontal diffusion. For the estimation, a quintic equation for a scalar measure of the superrotation strength is developed from the primitive equations. The quintic equation estimates the superrotation strength by its unique positive solution, which depends only on three nondimensional parameters: the external thermal Rossby number, the ratio of the radiative relaxation time to the timescale for the vertical diffusion, and the ratio of the planetary rotation period to the geometric mean of the timescales for the horizontal and vertical diffusion. The parameter dependence of the dominant dynamical balance is also investigated. The balance is a cyclostrophic, geostrophic, or horizontal diffusion balance, and in each balance, the equator-to-pole temperature difference is either nearly equal to that in the radiative-convective equilibrium state or significantly reduced by thermal advection. Steady-state or statistically steady-state solutions of the primitive equations are obtained by numerical time-integrations for a wide parameter range covering many orders of magnitude. The numerical solutions show that the theoretical estimates have a relative error of less than 50%, which is very small compared with the superrotation strength varying five orders depending on the external parameters, and show that the estimation is valid.