Ice Sheet Isostasy (3D)

The 3D block

The ground, ice and mantle surfaces use Blender-baked textures; the plate surface, ice dome and flow arrows are deformed live from the same spectral model as the 2D page. The research station and cairns are Blender models that ride on the moving surface, so you can watch the land sink under the ice and rebound after melting. Vertical motion is exaggerated.

What you are looking at

An ice sheet grows, holds, melts and disappears on a repeating schedule. Its weight pushes the elastic lithosphere down into the viscous mantle. The mantle must flow sideways to make room, which takes thousands of years, so the crust is always a little behind the ice. After the ice is gone the land keeps rising for many thousands of years: this is post-glacial rebound, still measured today in Scandinavia, Canada and the Baltic.

Isostasy: Airy equilibrium

If the crust could float freely, a column of ice of thickness h would sink until the mantle it displaces weighs the same: w = rho_ice h / rho_mantle. With rho_ice = 917 and rho_mantle = 3300 kg/m3 that is about 28 % of the ice thickness: a 3 km ice sheet presses the surface down about 830 m. This "Airy" depression is the yellow dashed line.

Lithosphere flexure

The real lithosphere is stiff, so it spreads the load over a wider area. For an elastic plate of flexural rigidity D = E Te^3 / (12 (1 - nu^2)) on a fluid foundation the deflection obeys D w'''' + rho_m g w = q. In Fourier space each wavenumber k responds as w_k = q_k / (rho_m g + D k^4). The natural length is the flexural parameter alpha = (4 D / (rho_m g))^(1/4), about 110 km for Te = 60 km. A wide ice sheet (much wider than alpha) behaves like Airy; a narrow one sinks less and is surrounded by a bulge.

Viscous relaxation

The mantle is a very viscous fluid. For a half-space with viscosity eta each Fourier mode relaxes exponentially, dw_k/dt = (w_eq,k - w_k) / tau_k with tau_k = 2 eta k / (rho_m g + D k^4). Without the plate this is the classic tau = 4 pi eta / (rho g lambda): for eta = 10^21 Pa s and a 3000 km wavelength it is about 4 thousand years. Long wavelengths relax faster than short ones until the plate stiffness takes over. The page integrates the 128 cosine modes with the exact exponential update, so the ice load can change smoothly in time.

Numerical checks on this page

• With the plate rigidity set to zero the series must give exactly the Airy depression, and the displaced mantle volume must equal the ice mass divided by the mantle density.
• A line load on a plate of the same rigidity is solved by finite differences and compared with Hetenyi's analytic result: central deflection q alpha^3 / (8 D) and the first zero crossing at 3 pi alpha / 4.
• The relaxation of the dominant mode after one relaxation time must reach 1 - 1/e, compared with an independent RK4 integration of the same equation.
• The equilibrium shape from the cosine series is compared with a direct finite-difference solution of the fourth-order plate equation on the same box.

What is simplified

The model is a flat, one-dimensional (infinite ridge) cross-section on a periodic box, not a sphere: the depression is measured relative to the far field, the mean level of the box is held fixed, and a ridge-shaped load is more efficient than a round ice sheet. The mantle is a single uniform Newtonian half-space (real mantle viscosity varies with depth and the asthenosphere is weaker than the deep mantle), the lithosphere is a thin elastic plate, and there is no gravitational self-attraction, no change of sea level and no ice-sheet dynamics: the ice schedule is simply prescribed. The result is a teaching model of the physics of rebound, not a prediction for any particular site.

What to try

• Compare Te = 15 km and Te = 150 km: the stiff plate sinks less and spreads the depression.
• Raise the viscosity by a factor of 20 (the slow-mantle preset) and watch the crust lag far behind the ice.
• Shrink the ice sheet to 100 km: it does not reach the Airy depression because the plate carries part of the load.
• Press "Jump to full ice" and then follow the uplift rate curve after melting.