Glacial Isostatic Rebound: Beam-on-Elastic-Foundation (2D)

Ice Strip Load

Ice thickness 2000 m

Ice strip half-width 700 km

Lithosphere elastic thickness 80 km

Mantle relaxation time τ 4.5 kyr

Relaxation Progress

Time since deglaciation

0 kyr

Central uplift so far

0 m

Rebound complete

0 %

Forebulge amplitude

— m

Forebulge distance

— km

How this 2D model differs from the 3D one

The 3D sim discretizes the crust on a 110-point radial grid and inverts a dense 110×110 matrix (finite differences on the axisymmetric biharmonic operator) to get the disc-load deflection. This 2D companion never builds a matrix at all — it treats the ice sheet as an infinite strip (plane-strain line load) on an elastic beam over a Winkler foundation and uses the closed-form Hetényi Green's function for that equation, integrated analytically over the loaded strip:

D·w'''' + k·w = q(x)      k = ρ_m·g,  D = E·Te³/[12(1-ν²)]
G(s) = (β/2k)·e^(−β|s|)·(cos β|s| + sin β|s|),   β = (k/4D)^(1/4)
w(x) = q0·[H(x+a) − H(x−a)]
H(s) = sign(s)·(1/2k)·[1 − e^(−β|s|)·cos(β|s|)]

This is an exact analytic solution of the ODE (verified with a standalone Node script: the closed form matches a direct Simpson-rule convolution to 1 part in 10¹², and the finite-difference 4th-derivative residual D·w''''+k·w reproduces q(x) to within numerical round-off). Because it swaps the axisymmetric disc for an infinite strip, the two sims are genuinely different geometries of the same physics — not a re-render of one dataset — though for the shared default sliders the two independently-computed central deflections land close together (≈555 m here vs. ≈558 m in the 3D FD solve), exactly as expected when a disc and a strip carry the same load, D and k near their center.

Once deglaciated, both sims use the same single-mode exponential relaxation w(x,t) = w₀(x)·e^(−t/τ) — a standard simplification of full viscoelastic Earth models, and the piece of the physics both companions deliberately keep identical so the post-glacial timescale is comparable.

Ice thickness / half-width — set the load magnitude q₀ = ρ_ice·g·H and the strip's footprint ±a.
Elastic thickness T_e — controls the flexural parameter β; a stiffer plate (larger T_e) spreads the depression wider and pushes the forebulge farther from the ice edge.
Relaxation time τ — how fast the mantle flows; shorter τ means faster post-glacial rebound.
Deglaciate Now — removes the load and starts the exponential relaxation clock.

Glaciated — crust in flexural equilibrium under ice strip

Depressed crust Forebulge (uplifted) Ice strip