This simulation models seasonal freeze-thaw dynamics in permafrost using the Stefan equation, Z = √(2k·DDT / (λ·ρ)), where the thaw depth Z grows with the accumulated degree-day thawing index (DDT) through the soil's thermal conductivity k and shrinks with its latent heat of fusion λ. A sinusoidal annual air-temperature cycle drives the seasonal freeze-thaw front, and the soil column plot shows the resulting active layer — the band of ground that thaws each summer and refreezes each winter above the permanently frozen permafrost table.
A soil column split into a thawed active layer (brown) sitting above frozen permafrost (blue), plus a live temperature-vs-depth profile line. A second chart plots active layer depth across all 365 days of the year, so you can see the seasonal bulge as summer heat drives the thaw front deeper before winter refreezes it back toward the surface.
Set the mean annual air temperature and summer peak temperature to define the seasonal swing, then tune the soil's thermal conductivity k and latent heat λ, which control how fast heat penetrates and how much energy is needed to thaw a given volume of ground. Press Animate Annual Cycle to sweep through all 365 days and watch the active layer depth, thaw front, and day counter update live.
The Stefan equation was originally derived in the 1890s to describe the thickness of sea ice, and it remains the standard first-order model geologists use to estimate permafrost active-layer depth. Because thaw depth scales with the square root of accumulated warmth, doubling the degree-days of summer heat only increases thaw depth by about 41%, not 100%.
Permafrost is ground that stays at or below 0°C for at least two consecutive years, common across Arctic and high-altitude regions. The active layer is the shallow band directly above the permafrost table that thaws every summer and refreezes every winter; its thickness — the thaw depth Z in this simulation — is what most affects plant roots, building foundations, and buried infrastructure.
The Stefan equation, Z = √(2k·DDT / (λ·ρ)), estimates how deep the thaw front has penetrated given the accumulated warmth above freezing. DDT is the degree-day thawing index, essentially a running total of daily air temperatures above 0°C; k is thermal conductivity (how easily heat moves through the soil), and λ is the latent heat needed to melt the ice in a given volume of ground. Because Z depends on the square root of DDT, thaw depth grows quickly at first each summer, then slows down.
Mean annual air temperature and summer peak temperature set the sinusoidal seasonal cycle the whole simulation is driven by. Thermal conductivity k represents how efficiently heat conducts through the particular soil or rock type — wetter, denser soils generally conduct heat faster. Latent heat λ reflects how much energy the ground's ice content absorbs before it melts, so higher λ (more ice-rich soil) means slower, shallower thaw for the same amount of summer warmth.
Warmer summers raise the degree-day thawing index DDT, and since thaw depth scales with the square root of DDT, any sustained rise in air temperature pushes the thaw front deeper before winter can refreeze it. Over years this can permanently thicken the active layer, degrade the underlying permafrost table, and destabilise structures built on ground that used to stay frozen year-round.
Permafrost stores vast amounts of organic carbon that has been locked in frozen soil for thousands of years. As the active layer deepens and permafrost thaws, microbes can decompose that organic matter and release carbon dioxide and methane, both greenhouse gases, creating a feedback loop where thawing permafrost contributes to further warming that deepens the active layer even more.