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Berg and sea
Thickness, width, ice density, salinity, water temperature and current set the freeboard, the stability limit and how fast the berg melts; the time-lapse sets days per second.
Waves
Model checks
The science
What you see. A tabular iceberg floating beside a 150 m ship for scale. In the time-lapse the berg melts: the sides and the waterline are eaten faster than the bottom, so it gets narrower relative to its thickness until it becomes unstable and rolls over. The end view shows the roll. Set the dimensions, ice density, salinity, water temperature, current and sea state; the Nudge button kicks the berg like a wave.
Floating. Archimedes' principle gives the submerged fraction f = ρice / ρwater: about 89 % in sea water, and a little more in fresh water, so roughly a tenth of the thickness shows above the sea.
Stability. For a rectangular cross-section of width W and thickness H the draft is d = fH and the metacentric height is GM = W²/(12 d) − (H − d)/2. The berg is stable only if GM > 0, that is W/H > √(6 f (1 − f)) (about 0.75). The dynamics integrate I θ'' = τ(θ) − c θ' with the exact righting torque τ = ρw g A xB, where the centre of buoyancy xB is the true centroid of the submerged part of the tilted rectangle (heave solved at every angle), so large angles and the capsized equilibrium at 90° come out of the same model. I includes 40 % added water mass.
Melt. Basal and side melt use the empirical Bigg-type form Mb = 0.58 |u|0.8 (Tw − Tice) / L0.2 in m/day with the ice at −10 °C, and wave erosion at the waterline Me = Ss (Tw + 2)/12 m/day. Water density is a simple linear fit in salinity and temperature.
Honest simplifications. The berg is a perfect box of uniform ice (real bergs have a keel, cracks, trapped air and irregular shape). Rolling is two-dimensional, per unit length, with no heave dynamics, drag on the roll beyond a damping ratio of 0.35, or wave-making; melt coefficients are empirical and the time-lapse speed is not real time. The roll period, damping and the sea-state torque are illustrative. Real capsizing bergs are tens of metres of rotating ice and are dangerous, so this model only shows the principle.
Checks. (1) closed-form GM against the numerical slope of the exact righting torque; (2) the critical width-to-thickness ratio by bisection against the formula; (3) the Archimedes area condition and the two equilibria; (4) the small-roll period against a time integration of the exact torque model.
What to try
- Pick Tall thin berg (unstable): it starts below the W/H limit, so any wave rolls it.
- Pick Stable giant and press Nudge: the roll period is minutes, and it rights itself.
- Run Melt until it rolls: the berg narrows faster than it thins until W/H crosses the limit in the fourth graph.
- Choose Porous glacier ice or Fresh lake water and watch the freeboard and the limit move.