Rayleigh-Taylor instability and convective thinning of mechanically thickened lithosphere: effects of non-linear viscosity decreasing exponentially with depth and of horizontal shortening of the layer

Localized mechanical thickening of cold, dense lithosphere should enhance its gravitational instability. Numerical experiments carried out with a layer in which viscosity decreases exponentially with depth, overlying either an inviscid or a viscous half-space, reveal exponential growth, as predicted by linear theory. As shown earlier for a layer with non-linear viscosity and with a constant theological parameter (Houseman & Molnar 1997), a perturbation to the thickness of the layer grows super-exponentially; for exponential variation of the theological parameter, the time dependence of growth obeys an equation of the form (Z/L)((1 - n) )= (n - 1)(C beta gL(2)/nB(0))(n )(T-b - t),( ) where Z is the magnitude of the perturbation to the thickness of the layer; L is the characteristic e-folding distance through the layer for the theological parameter B, which is proportional to viscosity and reaches a minimum of B, at the base of the layer; n is the power relating stress to strain rate; C (similar to 0.4, for the experiments considered here) is an empirical constant that depends on wavelength; beta is the vertical gradient in density (assumed to decrease linearly with depth in the layer); g is the gravitational acceleration; t is the time; and t(b) is the time at which a blob of material drawn from the basal part of the layer drops away from the layer. A simple application of this scaling relationship to the Earth, ignoring the retarding effect of diffusion of heat, suggests that somewhat more than half of the lithosphere should be removed in a period of similar to 20 Myr after the thickness of the layer has doubled. The imposition of horizontal shortening of the layer accelerates this process. In the presence of a constant background strain rate, growth will initially be exponential as the non-Newtonian viscosity is governed by the background strain rate. Only after the perturbation has grown to several tens of per cent of the thickness of the layer does growth become super-exponential and yet more rapid. An application of this scaling and its calibration by numerical experiments presented here suggests that super-exponential growth is likely to begin when the perturbation approaches similar to 100 per cent of the thickness of the layer, or roughly 100 km, when applied to the lithosphere. Thus, where the crust has doubled in thickness in a period of 10-30 Myr, we anticipate that roughly half, or more, of the thickened mantle lithosphere will be removed in a period of 10-20 Myr following the initiation of shortening.