On convective instabilities and differential rotation in rapidly rotating stars: A new quasi-geostrophic approximation

In rapidly rotating convective stars, the effect of Coriolis forces is predominant and the dynamics of convective instabilities is fundamentally different from that in slowly rotating stars. Coriolis forces not only enforce two-dimensionality of the fluid motions driven by convective instabilities but also generate strong differential rotation even in the vicinity of the onset of the instabilities. This characteristic suggests a quasi-geostrophic approximation taking advantage of quasi two-dimensionality of the fluid motions. We investigate convective instabilities and differential rotation generated by the instabilities in rapidly rotating stars using a new quasi-geostrophic approximation incorporating full spherical geometry and the equation of mass conservation. We model a rapidly rotating stellar convection zone of inner radius r(i) and outer radius r(o) by a large-gap spherical annulus defined by s(i) <= s <= s(o) and -(r(o)(2) - s(2))(1/2) <= z <= (r(o)(2)-s(2))(1/2), where ( s, phi, z) are cylindrical polar coordinates with the axis of rotation at s = 0 and r(i) = s(i) < s(o) approximate to r(o). In the limits si ! 0 and so ! ro, the corresponding spherical annulus represents a fully convective rapidly rotating star in the full sphere. For the purpose of comparison, fully three-dimensional solutions in spherical geometry are also calculated. It is demonstrated that the primary features of the convective instabilities in rapidly rotating spheres or spherical shells are quantitatively captured by large-gap spherical annuli using the new quasi-geostrophic approximation, providing an exciting opportunity to analyze and compute convective flows in rapidly rotating stars ( very small Ekman numbers), which cannot be achieved by fully three-dimensional numerical simulations. Implications of the results for several rapidly rotating stars are also discussed