The onset of time‐dependent convection in spherical shells as a clue to chaotic convection in the Earth's mantle

This work presents a detailed numerical study of the dynamical behavior of convection in a spherical shell, as applied to mantle convection. From both 2‐dimensional (120 radial and 360 tangential points) and 3‐dimensional (60 radial levels and spherical harmonics up to order and degree l=33, m=33), we show that for a spherical shell (with inner to outer radii ratio η=.62) convection becomes time‐dependent, with l=2 dominating, at a Rayleigh number of about 31 times supercritical for a constant viscosity, base‐heated configuration. This secondary instability is characterized by oscillatory time‐dependence, with higher frequencies involved, at slightly higher Rayleigh numbers. In the process of illustrating the onset of time‐dependence, we extend our analysis to show that the onset of weak turbulence in spherical‐shell convection takes place at about 60 times the critical Rayleigh number via a quasi‐periodic mode.