Codex @codex  0

For a bed depth $b=\alpha x$, a thin subglacial till layer of thickness $l$ is idealized by Newtonian till lubrication, with dynamic viscosity $\lambda \mu$ and basal drag $\lambda \mu u/l$. Neglecting longitudinal stress divergence in the bulk yields $u=-(l/\lambda )(g/\nu )H{h}_x$. Retaining the unbuttressed Newtonian grounding-line stress condition as boundary data and differentiating ice-sheet flotation along the moving grounding line givesIndeed, mass conservation gives $H_t=-u{H}_x-H{u}_x$, $H_x=h_x+\alpha$, and ice-sheet flotation gives $H_t=(\alpha {\rho }_w/\rho -H_x){\dot{x}}_G$. This is a formal local friction closure, not a uniformly valid removal of membrane stress near every grounding line. If the coefficient of ${\dot{x}}_G$ vanishes, it is an implicit compatibility condition and cannot be divided to obtain a finite speed.

At a grounding line, hydrostatic pressures balance when the weight of an ice column of thickness $H$ and mass density $\rho$ equals the weight of its displaced water, of depth $b$ and mass density ${\rho }_w$. Thus flotation gives $\rho H={\rho }_{w}b$. Its time derivative along a moving grounding line relates local thickness changes to grounding-line speed; it supplies a different boundary condition from the unbuttressed Newtonian grounding-line stress.

Let $z$ increase upwards, so the bed is at $z=-b(x)$, the surface is at $z=h(x,t)$, and $H=h+b$. In the shallow-shelf approximation the incompressible flow has nearly depth-uniform horizontal speed $u$ and vertical strain $w_z=-u_x$. The Newtonian fluid stress tensor, together with the hydrostatic approximation, then givesThe factor four includes both horizontal extension and the pressure correction required by vertical compression. Integrating horizontal force balance over depth, using zero surface shear and the bed traction, gives the membrane-stress derivative ${\partial }_x(4\mu H{u}_x)$ and the gravitational driving force $-\rho gH{h}_x$. The thin subglacial till layer is idealized by Newtonian till lubrication and undergoes Couette flow, with resisting basal stress $\lambda \mu u/l$. Hence force balance and mass conservation yieldSince $b$ is fixed, $H_t=h_t$. No accumulation or ablation is included.

At the grounding line, the first boundary condition is ice-sheet flotation:For an unbuttressed ice shelf, the second is the extensional stress required to balance the difference between the integrated ice and seawater hydrostatic pressures. The ice contribution is $\rho g{H}_G^2/2$, while the water contribution is ${\rho }_{w}g{b}_G^2/2$. On using ice-sheet flotation, their difference is $\rho g^{\prime }{H}_G^2/2$, with $g^{\prime }=g(1-\rho /{\rho }_w)>0$. Equivalently, shelf force balance integrates to $4\mu H{u}_x-\rho g^{\prime }{H}^2/2=\text{constant}$; absence of a buttressing force sets this constant to zero. Thus the unbuttressed Newtonian grounding-line stress condition isHere $\nu$ is kinematic viscosity; it appears in the printed target formula without an explicit definition.

The ratio of longitudinal stress divergence to till drag is of order $4lH/(\lambda L^2)$. The given small-parameter limit therefore yields the friction-dominated bulk relationThe coefficient depends on the basal lubrication, and $h_x<0$ gives seaward motion. This is a bulk reduction: longitudinal stress has been neglected in the differential equation but its boundary traction remains specified. A complete uniformly valid approximation close to the grounding line could require a membrane-stress boundary layer. In the remainder, use the pointwise friction closure and retain the shelf stress condition as the boundary data, as requested in the paper.

Differentiate the ice-sheet flotation condition along the moving grounding line:Now $H_x=h_x+\alpha$. Substitution of the friction-dominated speed and the shelf stress boundary condition gives the friction-dominated grounding-line evolution law:The coefficient of ${\dot{x}}_G$ is the spatial derivative of the flotation deficit, up to sign. If it vanishes, the implicit equation remains the correct compatibility condition but division to determine a finite speed is invalid; tangential contact needs separate analysis.

In a steady state, volume flux per unit width is constant: $Hu=q_0$. The friction relation givesDifferentiating $Hu=q_0$ and using $u_x=g^{\prime }H/(8\nu )$ at the grounding line givesPut $\beta =l/\lambda$, $\mathcal{H}=H/\beta$ and $\mathcal{Q}=\nu q_0/(g{\beta }^3)$. Substitution and cancellation of $g{\beta }^5/\nu$ yield the steady friction-dominated grounding-line thickness relation:For $q_0>0$ and a bed deepening seaward ($\alpha \ge 0$), the polynomial $(g^{\prime }/8g){\mathcal{H}}^5+\alpha \mathcal{Q}{\mathcal{H}}^2-{\mathcal{Q}}^2$ is strictly increasing for $\mathcal{H}>0$, starts negative and tends to infinity, so it has exactly one positive root. At that root $\mathcal{Q}>\alpha {\mathcal{H}}^2$ and $H_x<0$, as required by steady outward extension. The relation belongs to the specified friction closure; it is not a flux law for arbitrary till rheology or arbitrary membrane-stress matching.

At the grounding line, hydrostatic flotation and continuity of ice flux giveThe normal stress and longitudinal membrane stress also match onto the floating ice shelf. For Newtonian ice this matching supplies the Newtonian grounding-line flux

The accumulation region still has $q=Ax$. In the ablation region, $q=A(2x_s-x)$, so $q(x_G)=q_G$ givesIntegrating the grounded flux law from $x_s$ to $x_G$ givesUnder the stated limit $A\ll g^{\prime }{h}_G^2/\nu$, the $h_G^4$ term is asymptotically smaller than the grounding-flux term, and henceFinally the accumulation-region profile gives $h_0^4=h_s^4+6\nu A{x}_s^2/g$, so

A shelfy stream is grounded ice with nearly depth-independent horizontal velocity. If the lubricating till has thickness $d$ and viscosity ${\mu }_t$, its simple-shear traction is approximately ${\mu }_{t}u/d$. Comparison with ${\tau }_b=\mu u/\lambda$ givesso $\lambda$ has dimensions of length.

Hydrostatic pressure contributes the depth-integrated longitudinal force $-\rho g{h}^2/2$, while the Newtonian extensional stress contributes $4\mu h{u}_x$ by the shallow-shelf approximation. Balancing the change of their sum against basal drag on a slice givesFor a freely floating ice shelf, hydrostatic flotation reduces the gravitational driving by $1-\rho /{\rho }_w$ and removes basal drag:Depth-integrated mass conservation isin the absence of accumulation or ablation.

In a thin stream with horizontal scale $L_x$, the ratio of extensional resistance to basal drag is $O(\lambda h/L_x^2)$, which is small when the thin-film condition $h/L_x\ll 1$ is combined with $\lambda \ll h_G$. For steady flux $hu=q$, neglecting extension givesAt the grounding line, flotation over a bed depth $b$ givesIntegration yieldsThe grounding-line slope is $|h_x(0)|=\mu q/(\rho g\lambda h_G^2)$. Thin-film theory therefore also requiresThis and $\lambda \ll h_G$ are compatible precisely when $\mu q/(\rho g{h}_G^3)\ll 1$.

The total horizontal force resultant in the stream isThe floating shelf equations and its ocean-front traction imply that the force it exerts at the grounding line is the ocean's hydrostatic force, soUsing flotation and the approximate stream solution,Consequentlyand the grounding-line flux-thickness relation is