How long does it take a pond to freeze all the way through?
Given the air temperature and a pond's depth, roughly how long will it take for the pond to freeze solid from the surface down, and what physical law governs how the ice thickness grows with time?
History and references
Settled in 1891 by Josef Stefan (Stefan problem).
Kapitza's problem #83 asks for an order-of-magnitude estimate of how long a pond takes to freeze completely. The general mathematical framework for moving-boundary heat-conduction problems of exactly this type -- a phase-change front advancing into a medium as latent heat is conducted away -- was formalized by Josef Stefan in 1889-1891 while studying the thickness of polar sea ice, giving the now-standard square-root-of-time growth law (the 'Stefan problem'), which remains the basis for modern lake- and sea-ice forecasting models, refined further with weather-driven boundary conditions.
A backyard observation -- ice grows fast at first, then more slowly -- turns out to be governed by a named mathematical problem still used today to forecast sea ice in a warming Arctic.
Explanations that circulated
- Ice thickness grows linearly with time at constant outside temperature (the correct 'Stefan problem' solution shows ice thickness grows roughly as the square root of time, because the growing ice layer itself insulates the water beneath and slows further freezing)
References
- Über einige Probleme der Theorie der Wärmeleitung, Josef Stefan, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, 1891
Solutions 0
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