Viscous mountain spreading over an underthrust plate ($q=Uh-\rho g{h}^3{h}_x/(3\mu )$)

For a Newtonian layer with density $\rho$, dynamic viscosity $\mu$, thickness $h$ and a plate moving at $U$, a stress-free roof and no-slip base give $q=Uh-\rho g{h}^3{h}_x/(3\mu )$. Zero net mountain-material volume flux produces $h^3=9\mu Ux/(\rho g)$ when the contact height vanishes. Very near its idealized contact line the slope violates the lubrication approximation.

A lower-viscosity basal sediment layer changes the plate coupling in viscous mountain spreading over an underthrust plate. With mountain thickness $H$ and sediment depth $d$, equal densities give one hydrostatic pressure gradient $G=\rho g{\partial }_x(H+d)$. Fluxes are $q_s=Ud-G(d^{2}H/2+d^3/3)/{\mu }_s$ and $q_m=UH-G(d{H}^2+d^{2}H/2)/{\mu }_s-G{H}^3/(3\mu )$. Separate conservation of mass laws determine each layer. Steady zero mountain flux and sediment influx $U{d}_{\infty }$ give contact height $3d_{\infty }$ and surface slope $2{\mu }_{s}U/(9\rho g{d}_{\infty }^2)$.

In front of the contact line in the sediment-lubricated viscous mountain model, $h=d$ and $d_x=3{\mu }_{s}U(d-d_{\infty })/(\rho g{d}^3)$. For $\delta =d/d_{\infty }$ and $\ell =\rho g{d}_{\infty }^3/(3{\mu }_{s}U)$, integration gives $x/\ell =P(\delta )-P(3)$, with $P(\delta )={\delta }^3/3+{\delta }^2/2+\delta +\ln (\delta -1)$. The branch $1<\delta \le 3$ is unique and tends exponentially to the uniform far-field depth.