= Newtonian grounding-line flux
{title2=$q_G$}
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 $g'=g(1-\rho/\rho_w)$ 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>, $q=-gh^3h_x/(3\nu)$. In a locally steady region with negligible accumulation, <mass conservation> gives $u=q/h$ and $u_x=-qh_x/h^2$. Imposing the <unbuttressed Newtonian grounding-line stress> condition $u_x=g'h/(8\nu)$ at the same matching section then yields
$$
q^2=\frac{gg'}{24\nu^2}h_G^6,\qquad
q=\frac{\sqrt{gg'}}{\sqrt{24}\,\nu}h_G^3.
$$
This 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.<Marine ice sheet dynamics. Part 2. A Stokes flow contact problem by Christian Schoof>[Sections 2 and 5–6, constant-viscosity linear-slip model and matched contact-line conditions]{ref} The patch above should therefore be used as a transparent closure example, rather than as a universal predictive grounding-line law.
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