Iceberg Buoyancy and Capsize: Metacentric Height and Righting Arm (2D)

Presets

Live readouts

Graphs

Heel angle and swell slope against time

Righting arm GZ against heel angle (dots: equilibria)

Potential energy against heel angle (minima are stable)

Numerical checks

running...

How to use

Pick a preset, then press Calve new berg to drop a fresh block with the tilt you chose. The yellow M is the metacentre: when it sits above G (the red dot) the berg rights itself, when it falls below G the berg rolls over. Try: (1) Tall and thin and watch it roll to lie on its side, then read the stable heel angles in the readout; (2) Marginal and nudge the width slider by a few metres either way; (3) Waterline erosion: nothing is wrong with the berg until w/h crosses 0.75, then it flips; (4) Swell resonance with the swell period equal to the natural roll period; (5) lower the sea-water density to see the berg sit deeper. Space pauses, R resets.

The science

Archimedes. A floating body displaces its own weight of water. For uniform ice of density ρ_i in sea water of density ρ_w the submerged fraction is

f = ρ_i / ρ_w = 917 / 1025 = 0.895, draft d = f h, freeboard = (1 - f) h

so about nine tenths of a block berg is below the surface. Fresher water (lower ρ_w) makes the berg sit deeper; lower-density firn-rich ice sits higher.

Stability of a rectangular cross-section. Let w be the width at the waterline and h the height. With the centre of gravity G at mid-height, the centre of buoyancy B at mid-draft, and the second moment of the waterplane I = L w³/12 over displaced volume V = L w d, the metacentric radius is BM = I/V = w²/(12 d). The metacentric height is

GM = KB + BM - KG = d/2 + w²/(12 d) - h/2

The upright berg is stable when GM > 0. Substituting d = f h gives the critical aspect ratio

w / h > √( 6 f (1 - f) ) ≈ 0.75 (for f = 0.895)

This is the iceberg capsize criterion used in glaciology (Burton et al. 2012; MacAyeal et al.): a tall, thin berg is unstable, a wide, thin tabular berg is very stable. A berg that is eroded at the waterline gets narrower relative to its height and can cross the threshold.

Large angles. This page does not rely on the small-angle formula. For every heel angle θ it rotates the rectangle, finds the waterline that displaces exactly f w h of area (bisection), clips the polygon to get the exact centre of buoyancy B, and uses the righting arm GZ = -B_x measured from the vertical through G. The roll equation is

I_eff θ'' = ρ_w g A_sub B_x(θ - φ(t)) - c θ', I_eff = (1 + k_add) ρ_i w h (w² + h²)/12

with swell slope φ(t) = φ₀ sin(2πt/T). The potential energy is U(θ) = -ρ_w g A_sub B_y (G above B). Its minima are the stable floating attitudes: upright when GM > 0, otherwise a heeled angle or lying on its side, where the roles of w and h swap.

Natural period. For small rolls T = 2π √( I_eff / (ρ_w g A_sub GM) ). A 100 m berg rolls in about two minutes, which is why the page offers a time-speed slider.

Checks (run in the page). (1) displaced area equals f w h at four heel angles; (2) numerical GZ/sinθ at 0.06 degrees against the analytic GM; (3) a bisection on the sign of the numerical righting arm recovers √(6f(1-f)); (4) dU/dθ from finite differences equals the roll torque; (5) energy is conserved in an undamped roll; (6) the simulated small-oscillation period matches 2π√(I/(ρ_w g A GM)).

Simplifications. Two-dimensional cross-section (long berg, no end effects); quasi-static heave (the berg always floats at its equilibrium draft); hydrodynamic added inertia and damping are single numbers (k_add, damping ratio), not a wave-radiation solution; swell enters only as a tilt of the local sea surface; melt is accelerated (days per second) and reshapes the rectangle without refreezing; real bergs are irregular and rarely rectangular. Fracture mechanics of the calving itself is not modelled: the Calve button creates a new block of the chosen size with an initial tilt.