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A Newtonian fluid model of an ice sheet does not determine its grounding line volume flux per unit width from thickness alone without a basal law and matching conditions. Let $h_G$ be flotation thickness, $\nu$ the ice kinematic viscosity, $g$ gravity and $g^{\prime }=g(1-\rho /{\rho }_w)$ the reduced gravity. A local cubic scaling can be illustrated by patching two explicit approximations on a horizontal bed.

For no-slip Newtonian shallow-ice flux, $q=-g{h}^3{h}_x/(3\nu )$. In a locally steady region with negligible accumulation, mass conservation gives $u=q/h$ and $u_x=-q{h}_x/h^2$. Imposing the unbuttressed Newtonian grounding-line stress condition $u_x=g^{\prime }h/(8\nu )$ at the same matching section then yieldsThis is the result of the stated elementary patch, not a resolved grounding-zone solution. It shows why a bare numerical coefficient cannot be inferred from Newtonian rheology alone.

Basal sliding, longitudinal stresses and contact constraints require a transition calculation; they can change the flux-thickness law as well as its coefficient. A systematic Stokes flow treatment matches the outer shearing/sliding sheet to an extensional boundary layer and supplies effective flotation and flux conditions.^[ref] The patch above should therefore be used as a transparent closure example, rather than as a universal predictive grounding-line law.

The Newtonian shallow-ice flux on the flat bed isso mass conservation givesAt the ice divide, symmetry gives $h_x(0,t)=q(0,t)=0$. At the terminus, $h(x_N,t)=0$ and the moving-front mass balance applies; in a steady state $q(x_N)=0$. At the snowline, $h(x_s,t)=h_s$ and both $h$ and $q$ are continuous, hence $h_x$ is continuous because $h_s>0$.

In a steady state, $q_x=A$ in the accumulation region and $q_x=-A$ in the ablation region. ThereforeFlux continuity gives $x_N=2x_s$. Integrating the flux law gives the piecewise shapeMatching $h(x_s)=h_s$ yieldsThe largest surface slope is of order $\sqrt{\nu A/g}/h_s$, so the shallow-ice approximation requires