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 hot ter buoyant asthenosphere beneath it, leading to the possibility that the l ower part of the mantle lithosphere could sink into the mantle in convectiv e downwelling. Such instabilities are driven by the negative thermal buoyan cy of the cold lithosphere and retarded largely by viscous stress in the li thosphere. Because of the temperature dependence of viscosity, the coldest, and therefore densest, parts of the lithosphere are unavailable for drivin g the instability because of their strength. By comparing theory and the re sults of a finite element representation of a cooling lithosphere, we show that for a Newtonian fluid, the rate of exponential growth of an instabilit y should be approximately proportional to the integral over the depth of th e lithosphere of the ratio of thermal buoyancy to viscosity, both of which are functions of temperature, and thus depth. We term this quantity 'availa ble buoyancy' because it quantifies the buoyancy of material sufficiently w eak to flow, and therefore available for driving convective downwelling. Fo r non-Newtonian viscosity with power law exponent n and temperature-depende nt pre-exponential factor B, the instabilities grow superexponentially, as described by Houseman & Molnar (1997), and the appropriate timescale is giv en by the integral of the nth power of the ratio of the thermal buoyancy to B. The scaling by the 'available buoyancy' thus offers a method of determi ning the timescale for the growth of perturbations to an arbitrary temperat ure profile, and a given dependence of viscosity on both temperature and st rain rate. This timescale can be compared to the one relevant for the smoot hing of temperature perturbations by the diffusion of heat, allowing us to define a parameter, similar to a Rayleigh number, that describes a given te mperature profile's tendency toward convective instability. Like the Raylei gh number, this parameter depends on the cube of the thickness of a potenti ally unstable layer; therefore, mechanical thickening of a layer should sub stantially increase its degree of convective instability, and could cause s table lithosphere to become convectively unstable on short timescales. We e stimate that convective erosion will, in 10 Myr, reduce a layer thickened b y a factor of two to a thickness only 20 to 50 per cent greater than its pr e-thickened value. Thickening followed by convective instability may lead t o a net thinning of a layer if thickening also enhances the amplitude of pe rturbations to the layer's lateral temperature structure. For the mantle li thosphere, the resulting influx of hot asthenosphere could result in rapid surface uplift and volcanism.