A 20 km alpine valley glacier modelled with the shallow-ice approximation: snow turns into ice above the equilibrium line, flows under its own weight by Glen's law, melts below it. Change the climate and watch the terminus respond over decades to centuries.
Starts from the equilibrium glacier of the current settings, steps the ELA up 100 m and times the volume response against the simple estimate H/|bt|.
The science
Mass balance. Net balance is b(z) = min(β (z − ELA), bmax) in metres of ice per year, with β a few thousandths per year and bmax = 2 m/a (typical of mid-latitude glaciers, here a modelling choice). Above the ELA snow accumulates, below it melt exceeds snowfall.
Ice flow. In the shallow-ice approximation the basal shear stress is τ = ρ g H |∂s/∂x| (ρ = 910 kg/m³) and Glen's law ε̇ = A τn with n = 3 gives the surface speed us = 2A/(n+1) τn H. The ice flux is q = −Γ Hn+2 |sx|n−1 sx, Γ = 2A(ρg)n/(n+2), and the thickness obeys ∂H/∂t = −∂q/∂x + b(z) on a 125 m grid, solved with backward Euler and two Picard iterations (stable at 0.5 a steps).
Rate factor. A(T) follows an Arrhenius law with the typical temperate-ice value 2.4×10-24 s-1 Pa-3 at 0 °C and activation energies 115 / 60 kJ/mol (warmer / colder than −10 °C). These are typical literature values used as round numbers, not a fit to one glacier.
Sliding. An optional Weertman-type law ub = C τ3 with C = s × 10-14 m a-1 Pa-3 (the slider s) adds speed with no change to the equations of mass conservation. The coefficient is illustrative.
Checks. The Halfar dome is an exact similarity solution of the same equation for a flat bed without mass balance (H ∝ t-1/11, L ∝ t1/11); the numerical model reproduces it. The velocity law is checked by integrating Glen's law through the depth, the steady state by the zero net balance, and the response time by the Jóhannesson et al. (1989) estimate τ ≈ H/|bt|, which is only an order-of-magnitude rule (factor 2 band).
Simplifications. One flowline per unit width: no valley-width changes (the 3D width is drawn from the real valley shape but does not feed back), no longitudinal stress coupling, no ice temperature evolution, no calving, fixed bmax. The valley is a fictional alpine valley, not a real glacier, and the numbers are illustrative of mid-latitude glaciers. Surface speeds across the valley use a parabolic factor for display only. Models built with Blender, scene by Three.js.