Glacier Flow 3D

The model: a shallow-ice flowline

Ice thickness H(x,t) on a bed b(x) obeys mass conservation, ∂H/∂t = a(s) − ∂q/∂x, where s = b + H is the surface, a(s) is the surface mass balance and q is the ice flux per unit width. For slow, thin ice the shallow-ice approximation (SIA) says the basal shear stress is the driving stress, τb = ρ g H |∂s/∂x|, and Glen's flow law ε̇ = A τn with n = 3 then gives a nonlinear diffusion equation: q = −Γ Hn+2 |∂s/∂x|n−1 ∂s/∂x with Γ = 2A (ρ g)n/(n+2). The surface speed from deformation is us = 2A/(n+1) (ρ g |s'|)n Hn+1, so doubling the thickness multiplies speed by 16.

Climate and sliding

The mass balance is linear in elevation, a = min(β (s − ELA), amax): snow accumulates above the equilibrium-line altitude and melts below it. The rate factor depends on temperature through an Arrhenius law (Paterson and Budd fit), so cold ice is far stiffer than temperate ice. Optional basal sliding adds ub = C τb2; warm-based glaciers with sliding respond faster and can advance in surges. Raising the ELA (warming) shrinks the accumulation area ratio and the terminus retreats with a response time of decades to centuries, set by H / |aterminus|.

Numerics

Explicit finite volumes on a staggered 100 m grid, flux evaluated at cell faces from the two neighbouring surfaces, with the time step limited by Δt ≤ 0.1 Δx2/Dmax (D is the effective diffusivity). The scheme is exactly conservative, which the live budget check verifies.

How the 3D scene is built

Blender made the terrain (a trough bed z = b(x) + 0.15 (y/1 km)2 with ridged walls), moraine boulders, pines, a hut, the flag and the stakes, baked rock and meadow textures and exported as a Draco GLB. The ice is a live mesh: at each station its half-width follows the same parabolic trough, w = 1 km · √(H / 0.15 km), so the surface is flat across the valley and meets the walls exactly. The snow line on the rock is tied to the ELA. The vertical-exaggeration slider scales the whole scene in height.

Honest simplifications

One-dimensional flowline of constant unit width: no lateral drag, no valley shape factor, no tributaries; no ice-temperature evolution, calving or debris; SIA is poor on steep headwalls and at the very margin; the sliding law and linear balance profile are textbook approximations. It is an educational model, not a forecast of any real glacier.

Model checks in this page

(1) a steady ice dome on a flat bed with uniform accumulation must match the analytic Vialov profile H(2n+2)/n = 2 (a/Γ)1/n (L(n+1)/n − x(n+1)/n); (2) at that steady state the flux at mid-span equals the integrated accumulation, q(x) = a x; (3) the running budget ΔV = ∑(balance) − outflow is conserved to round-off.