The shallow-ice approximation (SIA) treats ice as slowly sliding material where vertically integrated flux follows Glen’s (power) law: effective diffusivity grows strongly with thickness and surface slope. In 1D, conservation ∂H/∂t + ∂q/∂x = M with q ∝ −H^{n+2}|∂s/∂x|^{n−1}∂s/∂x and surface s = b + H produces characteristic thickening near divides and drawdown toward a calving front. This page uses explicit finite volumes on x ∈ [0,1] with a planar bed tilt α, zero flux at the divide, H = 0 at the terminus, and uniform accumulation M for intuition only — no thermomechanical coupling, sliding, or hydrostatic bed.
Who it's for: Cryosphere introductions after ODEs; companion reading to SIA derivations in glaciometry texts.
Key terms
Shallow-ice approximation
Glen’s law
Ice flux
Basal shear stress
Accumulation
Real glaciers need 2D/3D thermomechanics and sliding; this page isolates how a nonlinear diffusive flux redistributes thickness once surface slopes drive q in Glen’s n-law form.
Live graphs
How it works
Cross-section evolution of ice thickness on a planar bed: nonlinear diffusion from Glen’s flow law in the shallow-ice approximation, with a textbook basal shear τ_b ≈ ρgH sinα readout at a fixed probe.
Key equations
∂H/∂t + ∂q/∂x = M; q ∝ −H^{n+2}|∂s/∂x|^{n−1}∂s/∂x; s = b + H; τ_b ≈ ρ g H sin α
Frequently asked questions
Are ρ, g, and the τ_b number field-calibrated?
No. ρ and g are standard constants for the τ_b ≈ ρgH sinα panel; model thickness is rescaled by a fixed factor for display in kPa. The evolution uses a lumped mobility control, not a measured rate factor A(T).
Why can the profile look noisy at high mobility?
Explicit integration of a strongly nonlinear diffusion needs small effective Courant numbers; raise mobility slowly or pause if short-wavelength ripples appear.