S. Takehiro $^{a}$ and J.R. Lister $^{b}$
$^{a}$ Department of Earth and Planetary Sciences, Kyushu University, Fukuoka, Japan. $^{b}$ Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
Penetration of finite-amplitude columnar convection and induced mean zonal flows into an outer stably stratified layer in a rapidly rotating spherical shell is examined numerically. The basic state considered has the inner part of the shell unstably stratified with a homogeneous internal heat source while the outer part is stably stratified with a constant temperature gradient. Calculations are performed for several cases of different stratification of the outer layer. The Taylor number and the Prandtl number are fixed to $10^8$ and 1, respectively. The radius ratio of the inner and outer radii is 0.4. The Rayleigh number is roughly four times the critical value for each case. The boundary conditions are free-slip and fixed-temperature. Time integrations are performed until the convective motion was fully developed and the kinetic energy saturated. In the case of weak stratification, both small-scale convection columns and induced mean zonal flow penetrate the stable layer completely. This zonal flow in the stable layer is generated directly in situ by the local columnar convection through the Reynolds stress associated with the tilting structure of the columns. When the stratification is increased, penetration of columnar convection is weakened and convection cells are trapped below the layer completely. However, mean zonal flows still penetrate the stable layer to the surface although their strength in the stable layer slightly decreases. These surface zonal flows are the result not of in situ generation by the columnar convection, but of remote generation from the deep region. In the inner unstable regions, the Reynolds stress of small-scale columnar convection produces deep zonal flows. These deep zonal flows penetrate the stratified layer to the surface but the columnar convection itself does not. It is suggested that penetration of mean zonal flows in these cases is governed by viscous diffusion, since the numerical results shows that the extent of penetration is almost independent of the stratification; it is similar to the horizontal scale of the mean zonal flow; the time scale of mean zonal flow development is similar to the viscous diffusion time of the stratified layer. Our results suggest that if the Lorentz force plays a subdominant role, fluid motion in the stably stratified layer near the outer surface of the Earth's fluid core may be affected by the underlying large-scale deep zonal flows, even when the stratification is so strong that the small-scale columnar motion cannot penetrate the layer.