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Effective pressure is ice-overburden pressure minus water pressure beneath the ice. It controls creep closure of channels and cavities; smaller positive $N$ weakens that closure and favors separation of ice from bed obstacles. It is therefore central to both basal sliding and subglacial drainage.

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.

Glaciology studies glaciers, ice sheets and related snow and ice through their mass balance, deformation and interaction with the environment. A glacier gains material through ice accumulation and loses it through ice ablation; its motion also transfers material between locations. Changes in surface height therefore combine local mass balance with flow convergence or divergence.

Velocity measurements, thickness profiles and repeated surface surveys constrain that budget. Basal sliding and internal deformation can produce similar surface motion with different stress and water conditions at the bed. Combining these observations is necessary to infer how an ice mass responds to changing atmospheric or oceanic forcing.

A constitutive law for ice creep relates deviatoric stress ${\tau }_{ij}$ and strain rate ${\dot{ϵ}}_{ij}$ through an effective stress ${\tau }_e$ and a temperature-dependent rate factor $\mathcal{A}$. The commonly used exponent is near three. Homogeneity of degree $n$ explains the combination $u/N^n$ in pressure-dependent basal sliding laws when geometry and rate-factor scales are fixed.

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.

Basal sliding allows ice to pass over bed obstacles by enhanced deformation and regelation: pressure-induced melting on the upstream side and refreezing downstream transfer ice around small obstacles. Cavitation opens water-filled gaps on the downstream side. Their size and the remaining ice-bed contact depend on effective pressure in subglacial drainage $N=p_{\mathrm{ice}}-p_{\mathrm{water}}$ as well as basal shear stress $\tau$. At fixed $\tau$, reducing $N$ generally enlarges cavities and reduces resistance, giving a sliding law $u(\tau ,N)$ with $u_N<0$ in this regime.

A linked cavity drainage system connects these cavities through passages along the bed. Under the prescribed decreasing relation $N(Q)$, the chain rule gives $\boxed{u_Q=u_N{N}^{\prime }(Q)>0}$. This conclusion uses that hydraulic relation; it need not apply to a different drainage regime.

With $t=\epsilon \tau$, the ice continuity equation becomes $H_{\tau }+\epsilon (uH{)}_x=\epsilon s^{\prime }(x)$. For bounded spatial gradients over $\tau =O(1)$, $H$ is independent of the fast seasonal time to leading order. Meanwhile $S^{\prime }(Q)=\epsilon \kappa$ gives $S_t=\kappa Q_{\tau }$, soLet $J(\tau )={\int }_0^{\tau }M(r)dr$. Direct differentiation proves thatsatisfies the equation and $Q(0,\tau )=0$. This is the boundary-generated solution, or a periodic-forcing solution with the forcing continued into its past. An independently prescribed initial profile would also need matching in the region whose characteristic curves have not yet reached the upstream boundary.

If $M>0$, $J$ is invertible and rearranging gives $x=[\tau -J^{-1}(J(\tau )-\kappa Q)]/\kappa$. For $M=1-\delta \cos \tau$ with $|\delta |<1$,Substitution, followed by the sine-difference identity, givesFor fixed $Q$ and nonzero first-order amplitude, the extrema occur at $\tau =\kappa Q/2+n\pi +O(\delta )$. Since $x=Q+O(\delta )$, the seasonal phase lag in linked cavity drainage gives the stated phase relation $\boxed{\tau \approx \kappa x/2(\mathrm{mod}\pi )}$. Maxima and minima alternate; when $\sin (\kappa Q/2)=0$, the first-order approximation is flat and does not locate individual extrema.

Glacier sliding is motion of the ice base relative to its bed, distinct from internal ice deformation. Near a melting bed, basal water, pressure-melting/refreezing around obstacles and creep past bed roughness can permit motion; a cold frozen bed can suppress it. A simple sliding law is increasing in basal traction, for example $u_b\propto {\tau }_b^m$ on the positive-traction branch, with resistance also affected in richer models by roughness and basal water pressure.

Slow ice motion has negligible inertial acceleration. In a long shallow glacier, leading vertical stress is hydrostatic and longitudinal stress gradients are neglected relative to transverse shear gradients. Resolving gravity downslope and normally yields the two stated balances. This approximates a two-dimensional slab/valley centreline; lateral drag, rapidly varying geometry and strong longitudinal-stress regions need additional terms.

With a stress-free surface,Put $K=\rho g\sin \alpha$ and $D=1-\cot \alpha h_x$, assuming positive driving shear for the stated power law. Integrating from the bed givesThe surface speed is $u_s=u_b+A(KD{)}^n{h}^{n+1}/(n+1)$. Mass conservation, with $a$ measured as ice-equivalent thickness accumulation, is $\boxed{h_t+q_x=a}$.

Choose $h=H\hat{h}$, $x=l\hat{x}$, $u_b=U{\hat{u}}_b$ and $t=(l/U)\hat{t}$, with $U=(KH/R{)}^m$. Scale accumulation with $HU/l$. The dimensionless basal law and parameters areThe resulting flux is exactly $h{u}_b+\delta (1-\mu h_x{)}^n{h}^{n+2}/(n+2)$.

When internal deformation is small and slope corrections are uniformly small, $u_b\simeq h^m$ andFor a steady accumulation profile and prescribed upstream flux $q_{\mathrm{in}}$, $\boxed{h_{\ast }(x)=[q_{\mathrm{in}}+{\int }_0^{x}a(\xi )d\xi {]}^{1/(m+1)}}$. For constant $a$ and zero upstream flux this is $(ax{)}^{1/(m+1)}$ on the region of positive flux.

A small thickness disturbance $\zeta$ satisfies ${\zeta }_t+[c(x)\zeta {]}_x=0$ with $c=(m+1)h_{\ast }^m$. Hence it travels down-glacier along $dx/dt=c>0$, with amplitude changing as $d\zeta /dt=-c^{\prime }\zeta$. Since surface speed is $u_s\simeq u_b=h_{\ast }^m$ in this sliding-dominated limit,The thickness wave therefore outruns individual ice particles. This is the sliding-dominated glacier kinematic wave.

Sliding and cavities. Regelation is melting under increased pressure and refreezing when the pressure is released. For ice near its melting point, the Clapeyron equation gives a lower melting temperature at greater pressure. Against an obstacle, ice melts on the high-pressure upstream face, the water passes around it, and it refreezes on the lower-pressure downstream face. Transport of the released latent heat back towards the melting face sustains the process. Together with viscous deformation around obstacles, this permits basal sliding; regelation is especially effective for small obstacles, for which heat conduction distances are short.

Subglacial cavitation occurs where the contact pressure predicted behind an obstacle falls below the water pressure: the ice separates from the bed and a water-filled cavity opens. The effective pressure in subglacial drainage isthe ice overburden pressure minus water pressure. Smaller $N$ facilitates larger cavities and reduces the resisting ice–bed contact. The exponent $n$ comes from Glen's flow law, with strain rate proportional to the $n$th power of deviatoric stress, commonly $n\simeq 3$. Its homogeneity makes $N^n$ the natural velocity scale, yielding $\tau /N=f(u/N^n)$ after including the fixed bed and ice-rheology scales.

Without cavities, material creep gives a drag law with $f(x)\propto x^{1/n}$. Cavities lower the resistance relative to that extrapolation. On a bounded-slope bed the drag is bounded in proportion to $N$; depending on bed geometry, the curve may level off or pass through a maximum and weaken as contact diminishes. The illustrated weakening curve is a qualitative possibility, not a universal formula for all beds.

Channels versus distributed drainage. In Röthlisberger channel drainage, frictional dissipation of flowing water supplies heat to melt the channel walls, while material creep of ice closes them under effective pressure. An area balance has the schematic formwhere $\Psi$ is hydraulic-potential drop per length, $L_f$ is latent heat per mass, and the positive constant $\mathcal{A}$ incorporates ice material creep and geometry. Water-flow resistance relates the flux to channel size and potential gradient. Balancing melting against closure produces the stated increasing relation $N_R(Q)$.

A channel gaining flux can maintain lower water pressure, since larger $N$ means smaller $p_w$. It draws still more water from neighboring routes, melts wider, and causes competing small conduits to lose flux and close. This positive feedback concentrates flow into an arterial network.

A linked cavity drainage system consists of water-filled lee-side cavities connected by narrow passages. Sliding over bed obstacles opens cavities; ice material creep tends to close them. The given decreasing relation $N_K(Q)$ means that carrying more water requires higher water pressure. A route gaining flux then has a higher pressure and diverts water towards neighboring routes. This negative redistribution feedback supports many connected routes and a distributed system.

Coexistence instability. At a coexistence point let $N_R(Q_R)=N_K(Q_K)$ and hold $Q=Q_R+Q_K$ fixed. A loss $\Delta Q_R=-\epsilon$ from the channel gives $\Delta Q_K=\epsilon$. The changes in the two equilibrium effective pressures areConsequentlyIf the bracket is positive, the cavity system's common effective pressure exceeds what the reduced-flux channel can support against material creep closure. The channel shrinks, loses further flux, and closes down. Thus the channel-cavity drainage instability condition is

Take $z$ upward from the horizontal bed and measure $x$ away from an ice divide. In the shallow-ice approximation, the hydrostatic pressure is $p=p_a+\rho g(h-z)$ and the horizontal balance for a Newtonian fluid isThe upper stress-free boundary condition gives $u_z(h)=0$. The resisting basal traction is $-\mu \beta u_b$, so the fluid-side shear stress satisfies $\mu u_z(0)=\mu \beta u_b$. This is a Navier slip boundary condition with slip length ${\ell }_s=1/\beta$. Integrating twice givesThe first term is the uniform basal sliding contribution, while the quadratic term is internal viscous deformation. Both are positive where the thickness decreases downstream.

The shallow-ice flux with linear basal drag isNo ice accumulation or ice ablation appears. For a symmetric glacier, work on one half, $0\le x\le x_N(t)$, with $q(0,t)=0$, $h(x_N,t)=0$ and no outgoing volume flux. Its conserved half-volume per unit width is $V_g={\int }_0^{x_N}hdx$; the full glacier has volume $2V_g$. This specifies the volume convention rather than silently supplying an unspecified value. For a general finite release on the whole line, the late similarity solution is centred on its conserved centre of mass and uses half its total volume for $V_g$.

The internal-shear and linear basal drag law mobilities are equal at $H_{\ast }=3/\beta$. Suitable vertical, horizontal and temporal scales areWrite $h=H_{\ast }\mathcal{H}$, $x=L_{\ast }X$, $t=T_{\ast }\tau$. ThenThe initial typical thickness is $\mathcal{H}\simeq 10$, the initial extent is of order $L_{\ast }/10$, and its shear-controlled spreading time is of order $T_{\ast }/10^5$. The shallow-ice approximation also requires small aspect ratio: in particular the initial thickness and extent must satisfy approximately $100H_{\ast }/L_{\ast }\ll 1$. The equations model the broad shallow bulk; the very tip can require physics beyond that approximation.

The two limiting equations are porous medium equations of the form ${\mathcal{H}}_{\tau }=({\mathcal{H}}^n{\mathcal{H}}_X{)}_X$, with $n=3$ for a thick shear-dominated glacier and $n=2$ for a thin sliding-dominated glacier. Volume conservation and balance of the time derivative give exponent $\alpha =1/(n+2)$. Set $\mathcal{H}={\tau }^{-\alpha }f(\eta )$, $\eta =X{\tau }^{-\alpha }$. One integration, using zero volume flux at the ice divide, gives $f^n{f}^{\prime }=-\alpha \eta f$, henceThe planar volume-conserving nonlinear-diffusion similarity is thereforeThe Beta function fixes its constants:Consequently the explicit early shear and late sliding limits areTheir centre depths are $A_3{\tau }^{-1/5}$ with $A_3\simeq 0.842252$, and $A_2{\tau }^{-1/4}$ with $A_2=\sqrt{2/\pi }$. Thus the early extent grows as $t^{1/5}$ and the late extent as $t^{1/4}$. These are asymptotic regime profiles; they solve the respective limiting equations, not the full sum of both mobilities. For arbitrary finite-width initial data they are appropriate after the corresponding spatial relaxation, with a virtual time origin depending on that data.

The shear-to-slip transition in a shallow ice current occurs at thickness of order $H_{\ast }$. Using equality at the centre of the early similarity solution givesIf the initial profile is already approximated by that early similarity solution and its centre depth is 10, the virtual age is ${\tau }_0=A_3^5/10^5\simeq 4.2385\times 10^{-6}$, and the elapsed estimate is ${\tau }_{\mathrm{tr}}-{\tau }_0$. Other reasonable definitions of typical thickness change the order-one coefficient; the initial thickness alone does not uniquely determine an exact transition time or an exact initial profile. The robust nondimensional estimate is a transition time of order one in the $T_{\ast }$ scale.

Finally, let a local nose translate steadily at positive speed $v$ and write $s=x_N-x\ge 0$. This is a local traveling nose of a glacier with linear basal drag; the speed of the globally spreading glacier changes slowly with time. The local mass conservation equation integrates to $q=vh$, since both $q$ and $h$ vanish at the front. Therefore

The right side is strictly increasing, giving a unique positive thickness at every $s>0$. With $S=X_N-X$ and $c=d{X}_N/d\tau$, the same relation reads $cS={\mathcal{H}}^3/3+{\mathcal{H}}^2/2$. Its limiting profiles areThe square-root tip reflects basal sliding; the cube-root thicker region reflects internal viscous deformation. Even during the early thick regime, the very nose is thin and has the square-root inner form. At late times that sliding balance governs nearly the whole profile. The dimensionless crossover thickness is $\mathcal{H}=1$ and its distance is $S=5/(6c)$; the early thick cube-root outer profile matches this smaller sliding region. The diverging geometric slope at the idealized tip also marks the local limit of the long-wave shallow-ice approximation.

The shallow-ice approximation makes the pressure hydrostatic pressure,Horizontal Stokes flow balance and the stress-free boundary condition at $z=h$ givesoThe basal shear stress and the stated basal sliding law are thereforeIntegrating once more and imposing $u(0)=u_b$ yieldsFor the right half of an ice cap, $h_x<0$, so both basal sliding and internal deformation carry ice away from the ice divide.

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.

Subglacial drainage transports water beneath a glacier. The water pressure affects basal sliding through the effective pressure in subglacial drainage. Röthlisberger channel drainage concentrates discharge in melt-enlarged conduits; a linked cavity drainage system distributes it through connected spaces maintained by sliding and creep.

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 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.