Glaciology

Glaciology studies glaciers, ice sheets, snow, and their interaction with climate, oceans, and the solid Earth.

An ice sheet is a large body of glacier ice covering land. Its mass budget combines accumulation, surface ablation, basal melting and discharge. Changes of floating ice shelves can influence discharge of grounded ice through changes in restraint, even though shelf melt itself has only a small direct sea-level contribution.

A glacier is a persistent mass of land-based ice that deforms and flows under gravity. Ice accumulation, ice ablation, internal deformation and basal sliding determine its geometry and motion.

Ice accumulation is the local gain of glacier or ice-sheet material, expressed in thickness per unit time after converting to ice-equivalent volume. The net accumulation rate subtracts ice ablation.

Ice ablation removes glacier or ice-sheet material by melting, sublimation, or mechanical loss. A negative net ice accumulation rate removes ice only where ice is present; an initially bare region cannot acquire negative thickness.

Salt lowers the freezing temperature of water. Consequently ice can ablate in saline water below the pure-water melting temperature if the ocean is warmer than its own liquidus. Solute diffusion towards the interface sustains the freezing-point depression. In the diffusive glacier-ablation interface law, small compositional Péclet number makes interfacial salinity close to ocean salinity and gives $P\simeq [T_{\infty }-T_m+\Gamma (C_{\infty }-C_s)]/[LD/(c_p\kappa )+\Gamma (C_{\infty }-C_s)]$ near the no-ablation threshold. This is diffusion-assisted dissolution rather than a supply of heat above the pure-water melting point; latent heat must still be supplied to complete the phase change.

In a constant-thickness laminar boundary layer at an ablating glacier, approximate fluxes by $\kappa (T_{\infty }-T_i)/\delta$ and $D(C_{\infty }-C_i)/\delta$. Neglect solid diffusion and sensible heating relative to latent heat. With ablation speed $U>0$, solute balance gives $U(C_i-C_s)=D(C_{\infty }-C_i)/\delta$, so $C_i=C_s+(C_{\infty }-C_s)/(1+P)$, $P=U\delta /D$. The liquidus $T_i=T_m-\Gamma (C_i-C_s)$ and Stefan condition $UL/c_p=\kappa (T_{\infty }-T_i)/\delta$ then give the displayed law, where $A_T=LD/(c_p\kappa )$. Its positive root exists when the ocean is above its own freezing point. In the small-$P$ regime, interface salinity approaches ocean salinity; in the large-$P$ regime it approaches the ice composition.

Ice sheet dynamics describes the gravity-driven deformation and basal motion of large terrestrial ice masses together with their mass balance.

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.

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.

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 $\nu =\mu /\rho$.

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.

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.