Gravity-driven viscous flow over partially lubricated bed

TL;DR

This paper analyzes gravity-driven viscous flow over a sloped bed with a finite partially lubricated patch, developing a simple SSA-based analytical model that captures how the patch induces stress and velocity perturbations in both Newtonian and power-law fluids. The model shows that perturbed stresses decay exponentially away from the patch, with the coupling length scaling as approximately $CL \,\sim\, h/\sqrt{\gamma}$ and the perturbation amplitude scaling linearly with slope and patch length, while the patch-scale velocity within the patch follows a distinct length scale. Numerical simulations using Elmer/Ice validate the analytical predictions for the Newtonian case and extend the results to nonlinear rheologies and sliding outside the patch, providing generalized scaling laws and numerical prefactors. The findings bear on ice-sheet dynamics, suggesting how rapid lake drainage could trigger hydrologically driven stress perturbations and cascades, with the coupling length and magnitude governed by patch geometry, thickness, slope, and boundary conditions outside the slippery region, and with inland regions potentially experiencing weaker coupling due to gentler slopes.

Abstract

We present an investigation into the response of a viscous fluid flowing over a sloped bed across a spatially finite patch of basal lubrication. We present a simple analytical model that captures the fundamental structure of such lubrication-induced stress and velocity perturbations in Newtonian fluids, as well as scaling arguments and numerical experiments that extend our analysis to power-law fluids. These analyses concisely reveal the underlying relationships between the system parameters (fluid thickness, $h$, slope, $α$, slippery patch length, $\ell$, and sliding condition outside of the slippery patch, $γ$) and the magnitude and spatial extent of the resulting perturbed stresses, $τ_{xx}$, and velocities, $u_p$. From these results, we conclude that the induced stresses are exponentially decaying functions of distance away from the patch location, and show that the amplitude of the perturbations scales linearly with surface slope and patch length while the decay length scales with thickness and patch length, and is critically dependent on the basal boundary condition outside of the slippery patch. These fundamental relationships can be incorporated into more complex models to investigate whether rapid lake drainages on ice sheets, which create a partially lubricated bed, can generate sufficient stress and velocity perturbations in the overlying ice flow to trigger lake drainage cascades.

Figures (10)

• Figure 1: Example numerical domain displaying the horizontal velocity (top) and normal deviatoric stress (bottom) result of an numerical experiment run with Elmer/Ice. Boundary conditions are set sufficiently far from the patch location such that the velocity in the far field is consistent with the sliding-accommodated shallow ice approximation (right) and $\tau_{xx}$ returns to zero.
• Figure 2: Depth-averaged numerical results for a selection of the numerical experiments with Newtonian ice and linear sliding law outside the patch (e.g., $n=1, m=\frac{1}{n}$) for direct comparison to analytical model, showing (a) the perturbed stress across the domain, (b) non-dimensionalized stresses across the domain in the upstream, patch, and downstream regions, compared to the analytical model (green), and scaling relationships and regression fits (green) for (c) coupling length and (d-f) stress. The solid green lines in (c-f) are the best fits, with the shaded areas representing the $\pm1\sigma$ envelope of the data scatter.
• Figure 3: Depth-averaged numerical results for a selection of the numerical experiments with Newtonian ice and linear sliding law outside the patch (e.g., $n=1, m=\frac{1}{n}$) for direct comparison to analytical model, showing (a) the perturbed velocity across the domain, (b) non-dimensionalized velocity across the domain in the upstream, patch, and downstream regions, compared to the analytical model (green), and (c) scaling relationship and regression fit (green) for the maximum velocity magnitude. The solid green line in (c) is the best fit, with the shaded area representing the $\pm1\sigma$ envelope of the data scatter.
• Figure 4: Depth-averaged numerical results for a selection of the numerical experiments with Newtonian ice extended to include nonlinear sliding law outside the patch (e.g., $n=1, m=\frac{1}{n},\frac{1}{2n}$) showing the perturbed stress (a) and velocity (b) across the domain. Scaling relationships for stress magnitude (c), coupling length (d), and velocity magnitude (e) are consistent with the scaling arguments in equations \ref{['eqn:xbl_ext']} and \ref{['eqn:veloscale_ext']}. The solid green lines are the best fits, with the shaded areas representing the $\pm1\sigma$ envelope of the data scatter.
• Figure 5: Ice-surface numerical results for a selection of the numerical experiments with Newtonian ice extended to include nonlinear sliding law outside the patch (e.g., $n=1, m=\frac{1}{n},\frac{1}{2n}$) showing the perturbed stress (a) and velocity (b) across the domain. Scaling relationships for stress magnitude (c), coupling length (d), and velocity magnitude (e) are consistent with the scaling arguments in equations \ref{['eqn:xbl_ext']} and \ref{['eqn:veloscale_ext']}. The solid green lines are the best fits, with the shaded areas representing the $\pm1\sigma$ envelope of the data scatter.