Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν₀ + ν₁ exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10⁷ and Rayleigh numbers of up to 10⁸ are studied.For intermediate viscosity contrasts (around 10³), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z_r (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10¹⁵?5 × 10¹⁶ m² s^?1 for viscosity in the thermal boundary layer below the plate.