Convective instability of a boundary layer with temperature- and strain-rate-dependent viscosity in terms of `available buoyancy'

Cold mantle lithosphere is gravitationally unstable with respect to the hotter buoyant asthenosphere beneath it, leading to the possibility that the lower part of the mantle lithosphere could sink into the mantle in convective downwelling. Such instabilities are driven by the negative thermal buoyancy of the cold lithosphere and retarded largely by viscous stress in the lithosphere. Because of the temperature dependence of viscosity, the coldest, and therefore densest, parts of the lithosphere are unavailable for driving the instability because of their strength. By comparing theory and the results of a ¢nite element representation of a cooling lithosphere, we show that for a Newtonian £uid, the rate of exponential growth of an instability should be approximately proportional to the integral over the depth of the lithosphere of the ratio of thermal buoyancy to viscosity, both of which are functions of temperature, and thus depth. We term this quantity `available buoyancy' because it quanti¢es the buoyancy of material su¤ciently weak to £ow, and therefore available for driving convective downwelling. For non-Newtonian viscosity with power law exponent n and temperature-dependent pre-exponential factor B, the instabilities grow superexponentially, as described by Houseman & Molnar (1997), and the appropriate timescale is given by the integral of the nth power of the ratio of the thermal buoyancy to B. The scaling by the `available buoyancy' thus o¡ers a method of determining the timescale for the growth of perturbations to an arbitrary temperature pro¢le, and a given dependence of viscosity on both temperature and strain rate. This timescale can be compared to the one relevant for the smoothing of temperature perturbations by the di¡usion of heat, allowing us to de¢ne a parameter, similar to a Rayleigh number, that describes a given temperature pro¢le's tendency toward convective instability. Like the Rayleigh number, this parameter depends on the cube of the thickness of a potentially unstable layer; therefore, mechanical thickening of a layer should substantially increase its degree of convective instability, and could cause stable lithosphere to become convectively unstable on short timescales. We estimate that convective erosion will, in 10 Myr, reduce a layer thickened by a factor of two to a thickness only 20 to 50 per cent greater than its pre-thickened value. Thickening followed by convective instability may lead to a net thinning of a layer if thickening also enhances the amplitude of perturbations to the layer's lateral temperature structure. For the mantle lithosphere, the resulting in£ux of hot asthenosphere could result in rapid surface uplift and volcanism.