Grounding line

The grounding line is the transition where an ice sheet detaches from its bed and begins to float as an ice shelf. Ice thickness there obeys the hydrostatic flotation condition.

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.

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.

With constant volume flux per unit width $Hu=q_0>0$, define $\beta =l/\lambda$, $\mathcal{H}=H/\beta$, and $\mathcal{Q}=\nu q_0/(g{\beta }^3)$. The friction closure gives $H_x=\alpha -q_0\nu /(\beta g{H}^2)$. Differentiating constant flux and applying the unbuttressed Newtonian grounding-line stress condition gives $q_0{H}_x/H+g^{\prime }{H}^2/(8\nu )=0$. Substitution proves the displayed relation. For $\alpha \ge 0$, $g^{\prime }>0$ and $\mathcal{Q}>0$, the polynomial $(g^{\prime }/8g){\mathcal{H}}^5+\alpha \mathcal{Q}{\mathcal{H}}^2-{\mathcal{Q}}^2$ increases strictly on the positive half-line, begins negative and tends to infinity; hence it has exactly one positive root. Its root satisfies $\mathcal{Q}>\alpha {\mathcal{H}}^2$, so the thickness decreases seaward at that steady grounding line.

For a two-dimensional Newtonian fluid ice sheet attached to an unbuttressed ice shelf, ice-sheet flotation gives $\rho H={\rho }_{w}b$. The difference between the depth-integrated ice and ocean hydrostatic pressures is $\rho g{H}^2/2-{\rho }_{w}g{b}^2/2=\rho g^{\prime }{H}^2/2$, where $g^{\prime }=g(1-\rho /{\rho }_w)$. Shelf force balance integrates $4\mu H{u}_x-\rho g^{\prime }{H}^2/2$ to a constant; absence of a buttressing force makes that constant zero. The shallow-shelf approximation therefore gives the displayed boundary condition, or $u_x=g^{\prime }H/(8\nu )$ with kinematic viscosity $\nu =\mu /\rho$.

Matching a slowly varying grounded Newtonian sheet to a freely extending Newtonian shelf gives, in a standard two-dimensional model,where $h_G$ is the flotation thickness and $g^{\prime }=g({\rho }_w-\rho )/{\rho }_w$ is the reduced gravity.

A shelfy stream is a grounded, rapidly sliding ice stream whose internal resistance is dominated by longitudinal extension, as in an ice shelf, but which retains a basal drag. It is described by the shallow-shelf approximation.