Depth-dependent rheology and the horizontal length scale of mantle convection

[ 1] Numerical simulations show that depth-dependent viscosity can increase the wavelength of mantle convection. The physical mechanism behind this phenomenon and its robustness with respect to model parameters remain to be fully elucidated. Toward this end, we develop theoretical heat flow scalings for a convecting fluid layer with depth-dependent viscosity. Bottom and internally heated end-members are considered. For the former, the viscosity structure consists of a high-viscosity central region bounded from above and below by horizontal low-viscosity channels. For internally heated cases, only a surface low-viscosity channel is present. Theoretical scalings derived from boundary layer theory show that depth-dependent rheology lowers the lateral dissipation associated with steady state convective rolls, allowing longer aspect ratio cells to form as the viscosity contrast between the channels and the central region is increased. The maximum cell aspect ratio is estimated from the condition that the pressure gradients that drive lateral flow in the channels do not become so large as to inhibit vertical flow into the channels. Scaling predictions compare favorably to results of numerical simulations for steady state cells. As the Rayleigh number driving convection is increased, small-scale boundary layer instabilities begin to form. This increases lateral dissipation within the channels and the preferred cell aspect ratio decreases as a result. Internally heated simulations show that a near-surface high-viscosity layer, an analog to tectonic plates, can suppress these small-scale instabilities. This allows a low-viscosity channel to maintain large aspect ratio cells for Rayleigh numbers approaching that of the present-day Earth.