Theory of Convection in a Deep Rotating Spherical Shell, and Its Application to the Sun

Abstract

The theory of convection in a rotating spherical shell, when applied to the Sun, should ultimately satisfy at least three broad constraints. The radial heat flux by the convection must not vary significantly with latitude; an equatorial acceleration must be produced; and the convection must give the right dynamo action to produce the gross features of a solar cycle.Important quantities to look for in the observations with which a convection theory can be compared include evidence of global velocities of giant convection cells, differences in motion features between low latitudes and high, persistence of motion features over successive rotations, evidence of excess brightness and variations in total solar luminosity, correlations of velocities between northern and southern hemispheres, and between north-south and east-west motions, and time periodic changes in motion fields.The general theory of convective motions in a rotating spherical shell such as the convection zone of the Sun has developed rapidly over the last several years, but is still a long way from providing quantitative models which agree in a satisfactory way with the main solar observations. Most work has (for mathematical reasons) employed the Boussinesq approximation, and most has been either linear or nearly so. Early work demonstrated the basic ability of global convection to transport momentum toward the equator, but whether an equatorial acceleration results depends on the effects of competing angular momentum transport processes, namely transport in the radial direction by convection, and transport by axisymmetric meridional circulations. The end result can be assessed only by nonlinear calculations.Some recent calculations directed at this question by the author indicate that, at least at Prandtl number of unity, convection growing from an initial state of solid rotation produces equatorial acceleration only when equatorial modes dominate and the rotational constraint is sufficiently strong, which results in a convective heat flux which is strongly dependent on latitude. When the rotational constraint is broken enough to give convection at all latitudes and therefore more nearly uniform heat flux, equatorial deceleration and angular momentum mixing in latitude results. A few calculations for Prandtl number substantially smaller than one give results which suggest it is possible to produce equatorial acceleration and nearly uniform heat flux, but these solutions appear not to be unique. In particular, the amount of differential rotation present in the initial conditions appears to be important in determining the final state.