Shear heating instability in the earth's upper mantle

We have studied the effects of radiogenic heat sources Q_m and heat flux from the deep mantle q∞ on the multivalued relation between shear stress at the base of the lithosphere τ and plate velocity u₀. If there are sufficient heat sources in the asthenosphere or if considerable heat flows up from the deep mantle, τ increases with u₀; such states are called subcritical. However, if neither of these conditions exist then τ decreases with u₀; such states are called supercritical. The uncertainties in both Q_m and q∞ preclude any inference about the τ vs. u₀ relation for the earth's lithosphere. We have also examined whether shear heating instability, associated with the strong temperature dependence of mantle viscosity, can occur in these asthenospheric shear flows. Based on a linear, one-dimensional analysis with coupled temperature and horizontal velocity perturbations, we find that all flows are stable for a constant surface velocity boundary condition while supercritical flows are unstable if basal shear stress is fixed.