We investigate the nature of subcritical, finite-amplitude, double-diffusive convection in the infinite Prandtl number regime, applicable to magma chambers and the D-layer at the core-mantle boundary, by a twodimensional, finite-element method based on streamfunction, compositional and temperature fields. Grid refinement is used for resolving the disparately-scaled thermal and chemical boundary layers present for the large ratios of the thermal to mass diffusivity (Lewis number) characteristic of magmas. In the diffusive regime a large enough Le is required for the establishment of steady double-diffusive convection under subcritical conditions. This critical Le varies nonlinearly with the buoyancy ratio Rp, the ratio of the chemical to thermal buoyancy. For large Le, the steady-state heat transport in the diffusive regime depends weakly on Le and approaches that found for pure thermal convection. In accordance with steady-state boundary-layer scaling, the chemical Nusselt number Nuc is found numerically to vary as Nuc= 1.02 Le0.49Nut for stress-free boundaries and Nuc. = 0.96 Le0.34Nut for rigid boundaries, with Nut the thermal Nusselt number. For larger aspect-ratios a more complicated bifurcation pattern with Le is found, with the sequence ranging from no steady states, to three steady states and then to a single elongated cell, as Le is increased. Subcritical steady-state solutions can be attained by integrating the set of timedependent double-diffusive equations. Applications of these results to the chemical boundary layers at the core-mantle boundary would suggest the D-layer, if it is chemically stratified there, must be a timedependent feature. Time-dependent calculations show a strong sensitivity to the initial conditions. Subcritical convective solutions in the finger regime exhibit transitions, leading to complex timedependent flows. The tendency to form narrow cells in the subcritical, finite-amplitude, finger regimes may account for laterally variable composition in a nearly conductive thermal state. Subcritical finger instabilities are found to be able to penetrate through the entire layer in a narrow slot, as in finite Prandtl number calculations.
Bibliographical noteFunding Information:
We thank Frank Spera and Curt Oldenburg for stimulating discussions. This research has been supported by German grant D.F.G. Eb-56/11-2 and the American grant NSF EAR-86-08479. We are grateful to Kari L. Rabie and Anne Boyd for their assistance in preparing this manuscript.
- magma chambers
- subcritical bifurcation