Convection in rotating spherical fluid shells with inhomogeneous heat flux at the outer boundary

摘要

The Earth's core is subject to a laterally varying heat flux at the outer boundary, which may account for correlations between the geomagnetic field and lower mantle structure. Studies of nonmagnetic, rotating convection in a spherical shell with fixed temperature boundary conditions have revealed flows resonating with, or locked to, the boundary anomalies when the length scales of the convection are close to those of the boundary condition. Here we study a similar system but for fixed heat flux upper boundary conditions, as in the Earth's core. We first map out the onset of thermal instability in a rotating shell of aspect ratio 0.4 for uniform outer boundary cooling with both rigid and stress-free boundaries. A preference for large scale (azimuthal wavenumber m=1) flows, not observed for the uniform temperature case, persists to Ekman numbers down to almost 10(-4). The preference for large scales is greatest for rigid boundaries and high (>= 1) Prandtl numbers. Hemispheric asymmetry appears in the weakly nonlinear regime, with small scale columnar convection coexisting with larger scale flows. We next study the effect of heterogeneous cooling of strength E proportional to the Y-2(2) spherical harmonic, which resembles the assumed heat flow at the core-mantle boundary. We illustrate the results with two sets of parameters, (I) when the most unstable mode in the uniform case is m=2, the same as the heterogeneous boundary condition, and (2) when it is m=1. We follow the numerical solutions from steady flows dominated by boundary heating, through periodic flows drifting at non-uniform rates, to chaotic flows. In case (1) both thermal convection and boundary-driven flow are dominated by the same azimuthal wavenumber m=2 as the boundary condition; they give way to periodic flows of the same symmetry (even m) at low boundary heterogeneity. At higher epsilon the symmetry is broken as modes with odd m are excited. In case (2) at low epsilon the m=1 and m=2 modes compete, while higher epsilon imposes an m=2 symmetry. Boundary effects depend strongly on the most unstable wavenumber at onset of convection with uniform boundary cooling: these simple linear results are a good guide to the probable behaviour of more complex, nonlinear regimes and have already been used to find suitable parameter ranges in a geodynamo calculation.