Ice sheet dynamics

Ice sheet dynamics combines mass conservation with the forces driving and resisting ice flow. For thickness $H$ and depth-integrated horizontal flux $\mathbf{q}$, a simple fixed-bed budget is ${\partial }_{t}H+\nabla \cdot \mathbf{q}=a$, where $a$ is net accumulation in ice-equivalent units. A local increase in thickness can therefore come from accumulation or convergent transport.

Surface slopes supply gravitational driving stress, resisted by internal deformation, basal sliding laws and lateral or longitudinal stresses. The shallow-ice approximation and shallow-shelf approximation retain different balances. Near a grounding line, flotation and contact with the bed must be included when connecting an inland ice sheet to an ice shelf.

An ice cap is a dome-shaped terrestrial ice mass that flows outward from one or more ice divides and is smaller than an ice sheet.

In the idealized normalized model $\dot{M}=1-\beta z-\alpha /(\Gamma z-s)$ on the cold branch $\Gamma z>s$, snow gain and melt loss balance at two roots when $\Gamma -\beta s>2\sqrt{\alpha \beta \Gamma }$. Assume ice height increases with mass. Differentiation gives $d\dot{M}/dz=-\beta +\alpha \Gamma /(\Gamma z-s{)}^2$, so the higher root is stable and the lower root unstable. The roots merge at the displayed saddle-node bifurcation. Here $s$ is a sea-level Celsius temperature, so absolute kelvin values cannot be substituted into the reciprocal law. The singularity at the melting threshold limits this model's physical scope.

The shallow-ice approximation uses the small ratio of ice thickness to horizontal extent to make pressure hydrostatic and balance its horizontal gradient primarily by vertical shear stress.

For a two-dimensional Newtonian ice sheet of thickness $h$ on a flat no-slip bed, hydrostatic pressure and vertical shear give the depth-integrated fluxwhere $\nu$ is the kinematic viscosity. Mass conservation is $h_t+q_x=a$ for accumulation rate $a$.

On a horizontal bed, the shallow-ice approximation with a linear basal drag law gives $u=-(\rho g/\mu )h_x(h/\beta +hz-z^2/2)$ and $q=-(\rho g/\mu )(h^3/3+h^2/\beta )h_x$. With no ice accumulation or ice ablation, mass conservation is $h_t+q_x=0$. Internal shear supplies the cubic mobility and basal sliding the quadratic mobility.

A locally steady glacier nose moving at speed $v>0$ has $q=vh$ by mass conservation. At distance $s$ behind its edge, the linear basal drag law then gives $s=(\rho g/\mu v)(h^3/9+h^2/(2\beta ))$. The thin nose has $h\propto s^{1/2}$ from basal sliding; a thicker outer nose has $h\propto s^{1/3}$ from internal deformation. A thick bulk therefore still has a sliding-controlled inner tip. The divergent idealized edge slope limits the immediate validity of the shallow-ice approximation.

The two mobilities in the shallow-ice flux with linear basal drag are equal at thickness $H_{\ast }=3/\beta$. A conserved half-volume $V_g$ gives length $L_{\ast }=V_g/H_{\ast }$ and time $T_{\ast }=3\mu L_{\ast }^2/(\rho g{H}_{\ast }^3)$. Nondimensional thick profiles spread as ${\tau }^{1/5}$; thin sliding profiles spread as ${\tau }^{1/4}$. Matching the centre of the mass-normalized early planar volume-conserving nonlinear-diffusion similarity to unit depth estimates ${\tau }_{\mathrm{tr}}=3/(10I_3^2)\simeq 0.42385$. The numerical estimate depends on the matching convention and the initial-profile relaxation.

For a thin Newtonian fluid film on a conical slope at angle $\alpha$, neglecting the thickness-gradient contribution to the driving pressure gives volume flux per unit width $q=D{h}^3$, where $D=g\sin \alpha /(3\nu )$. Radial mass conservation isThe distance $x$ is measured along the slope; the horizontal radius is $x\cos \alpha$, so total volume is $2\pi \cos \alpha \int xhdx$.

For ice accumulation $a=A(1-x/\ell )$, regular zero flux at the apex givesThe steady zero-thickness terminus is at $3\ell /2$, below the snowline $x=\ell$. Accumulation above the snowline exactly balances ice ablation below it.

After ice accumulation and ice ablation cease, the similarity solution conserving $M=\int xhdx$ isIt is dry outside this interval. At its terminus the thickness $h_N$ is nonzero: the Rankine-Hugoniot condition gives ${\dot{x}}_N=q_N/h_N=D{h}_N^2=x_N/(5\tau )$. This is an entropy solution of the gravity-only kinematic wave model and a long-time approximation for general finite-volume initial profiles. Local pressure-gradient effects smooth the idealized discontinuous front.

Putting $y=x^{5/3}$ and $w=x^{1/3}h$ transforms $h_t+x^{-1}(xD{h}^3{)}_x=0$ into the scalar conservation lawAlong smooth characteristic curves, $w$ is constant and $y$ grows at speed $5D{w}^2$. Starting from $h_0(s)$ givesAfter characteristic crossing, the Rankine-Hugoniot condition and entropy solution select the physical front.

For an initially bare cone with accumulation $a\simeq A$ near its apex, the early outer thickness is $At$. The radial gravity flux becomes comparable to accumulation over distance $x_a\sim D{A}^2{t}^3$. Setting $h=AtF(\eta )$ and $\eta =x/(D{A}^2{t}^3)$ givesRegular apex flux requires $F\sim (\eta /2{)}^{1/3}$ near zero. The boundary layer therefore approaches the local steady thickness $(Ax/(2D){)}^{1/3}$ while farther regions are still filling.

An ice divide is a line or point of maximum ice-surface elevation from which ice flows in opposite directions. Symmetry gives zero horizontal ice flux at the divide.

The snowline separates a region of net annual snow accumulation from a region of net ablation. A height-based idealization places it where the ice surface has a prescribed elevation $h^{\ast }$.

Basal sliding is motion of glacier ice relative to its bed. A sliding law relates the basal velocity to basal shear stress, effective pressure, roughness, or other bed properties.

For a shallow glacier with basal traction proportional to thickness and sliding law $u_b\propto {\tau }_b^m$, negligible internal-deformation flux gives $u_b\propto h^m$. After scaling, the thickness flux is $h^{m+1}$. Linearization around a steady thickness $h_{\ast }(x)$ gives ${\zeta }_t+[c(x)\zeta {]}_x=0$, $c=(m+1)h_{\ast }^m$. Thickness disturbances therefore propagate faster than the surface material speed $u_s\simeq h_{\ast }^m$ by factor $m+1$.

A steady accumulation profile and upstream discharge $q_{\mathrm{in}}$ give $h_{\ast }^{m+1}=q_{\mathrm{in}}+{\int }_0^{x}a(\xi )d\xi$. Along a disturbance ray, $dx/dt=c$ and $d\zeta /dt=-c^{\prime }\zeta$, so a spatially varying background also changes amplitude. The surface-speed ratio is specific to sliding-dominated flow; internal deformation and significant surface-slope corrections change the flux derivative.

A water-filled cavity opens where the normal contact pressure behind a bed obstacle would otherwise fall below subglacial water pressure. Reduced effective pressure in subglacial drainage enlarges cavities and changes the resisting contact geometry. For a fixed bounded-slope bed, the basal shear stress is bounded by a bed-dependent multiple of effective pressure; a sliding curve may saturate or weaken as cavities grow.

Pressure changes the melting temperature of ice according to the Clapeyron equation. Ice pushed against an obstacle can melt on its high-pressure side and refreeze downstream where pressure is lower. Conduction of the released latent heat back towards the melting region permits motion around the obstacle. Small obstacles favor this heat-transfer mechanism; larger ones can be bypassed mainly by ice deformation.

A linear basal drag law has resisting traction $-\mu \beta u_b$, with $\beta >0$ of dimension inverse length. It idealizes lubrication by subglacial till and is equivalent to a Navier slip boundary condition of slip length $1/\beta$ when the ice is a Newtonian fluid. It is a constitutive idealization, not a universal sliding law.

Subglacial till is a mixture of rock fragments and fine sediment beneath ice. A wet deformable layer can lubricate basal sliding. Its constitutive equation must be specified; Newtonian till lubrication is one idealization, rather than a claim that all subglacial till is a Newtonian fluid.

A thin layer of subglacial till idealized as a Newtonian fluid of dynamic viscosity $\lambda \mu$ and thickness $l$ undergoes Couette flow between stationary bedrock and ice moving at speed $u$. Its resisting basal shear stress has magnitude ${\tau }_b=\lambda \mu u/l$ and acts opposite to the sliding motion. Combining this drag with the hydrostatic driving force gives the bulk approximation used in the friction-dominated grounding-line evolution law.

An ice shelf is the floating extension of a grounded ice sheet. Hydrostatic flotation makes its surface slope and horizontal driving stress depend on the density contrast between ice and seawater.

Ice-shelf buttressing is the restraint that a floating ice shelf, through its contact with sidewalls or pinning points, can exert on upstream grounded ice flow. Thinning or loss of the shelf can reduce this restraint and increase discharge, but the response depends on geometry and cannot be inferred from floating-ice area alone.

Ice-shelf water has acquired characteristic properties through interaction with the underside of an ice shelf. Melting at depth can cool and freshen the water, including cooling below the surface freezing temperature because pressure lowers the freezing point. Its eventual density and contribution to deep-water formation depend on mixing with surrounding shelf and ocean waters.

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 $\nu =\mu /\rho$.

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.

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.

The shallow-shelf approximation treats horizontal velocity as nearly depth-independent and balances membrane stress, hydrostatic driving stress, and basal traction. For Newtonian ice of thickness $h$ and speed $u$, the extensional resultant is $4\mu h{u}_x$.

Short-wavelength corrugations on the underside of a highly viscous ice shelf relax through local Stokes flow. For a wavelength much smaller than the shelf thickness, the deformation occupies a depth of order the inverse wavenumber and the ice may be treated as a viscous half-space.