Simulations of nonlinear pore-water convection in spherical shells

Abstract

Hydrothermal convection of pore water of uniform viscosity within a permeable, internally heated spherical shell bounded by two concentric spherical surfaces of inner radius ri and outer radius ro is investigated by fully three-dimensional numerical simulations based on a domain decomposition method. We first determine the critical Rayleigh number for the onset of hydrothermal convection by expressing linear solutions in terms of spherical Bessel functions. It is found that the basic motionless state becomes unstable with respect to an infinitesimal disturbance characterized by a spherical harmonic of degree l, the size of which is strongly dependent upon the aspect ratio ri∕ro. However, the three-dimensional structure of convection cannot be determined by the stability analysis because of the mathematical degeneracy of the linear solution. A new numerical scheme using a finite difference method is then employed to simulate three-dimensional nonlinear convection near the onset of convection. When the aspect ratio (ro−ri)∕ro is moderately small, a large number of different stable stationary patterns with exactly the same Rayleigh number are found by using different initial conditions. The solutions are relevant to the convectively forced circulation of water in the interiors of early solar system bodies and outer planet icy satellites.