Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell III: Effects of velocity boundary conditions

Abstract

Abstract We perform numerical experiments for convection in a rotating spherical shell with either one or two nonslip boundaries, for the same Rayleigh and Taylor numbers (R = 104 to 105, T = 105) and the same temperature boundary conditions as used in Part II. We find that both convection and differential rotation energies are reduced compared with solutions with stress free boundaries, but the differential rotation is reduced by a much larger factor, demonstrating the constraining influence of rigidly rotating boundaries on the angular momentum profile. Differential rotation ceases to be a prominent, global feature of the flow, and may be difficult to observe in laboratory experiments. The convection spectrum with nonslip boundaries is also broader, and shifts much less toward low longitudinal wave numbers with increased R. What differential rotation remains is driven primarily by Coriolis torques from the axisymmetric meridional circulation, rather than by Reynolds stresses. For R≦5 × 104, the bouyancy driven axisymmetric meridional circulation is substantially larger than for stress free boundaries, due to Ekman boundary layers breaking the rotational constraints which otherwise suppress this inherently nongeostrophic flow. In the same Rayleigh number range, a greater fraction of the total heat flux is carried by convection than with stress free boundaries, also a result of the destabilizing influence of the Ekman layers. Solutions with stress free top and nonslip bottom behave similarly to the stress free top and bottom case at low Rayleigh number, because the convection occurs mostly outside the cylinder tangent to the inner boundary equator, so that the inner velocity boundary condition is not strongly felt. As R is increased, the convection and differential rotation feel this boundary much more strongly. Despite the many differences, the convection solutions for different boundary conditions have a number of similarities. These include location of peak convection amplitudes (the equator with a secondary peak at high latitudes), north-south roll orientation near the equator, Reynolds stress patterns, and the form of axisymmetric meridional circulation. Of particular interest is that the helicity profile of the convection is similar for all boundary conditions, but is of larger amplitude with nonslip boundaries. Our results suggest that differential rotation is likely to be much different, and of smaller amplitude relative to the convection which drives it, in a liquid planetary interior as compared to a stellar convection zone. This difference may result in favoring planetary dynamos of the α2 variety, and stellar dynamos of the α-ω type. These speculations need to be tested by model calculations with magnetic field included.