What it does
In permafrost country only the top of the ground thaws each summer. That thawed layer is the active layer, and how deep it goes decides where roots, water and buildings can sit. The Stefan equation is the classic first estimate of that depth:
Z = √(2 k Iₛ / L)
where k is the thermal conductivity of the thawed soil (W/m/K), Iₛ the thawing index at the ground surface (degree-days above 0 °C, turned into °C·seconds), and L the latent heat needed to melt the ice in a cubic metre of soil, L = ρd × w × 334 kJ/kg from the dry density ρd and the gravimetric water content w. The same equation, with the freezing index and the frozen soil’s conductivity, gives the frost depth of a winter.
This page works out the index from your daily air temperatures (or takes it typed), applies an n-factor to get from air to ground surface, solves one to three soil layers one after another, draws the thaw curve through the season, and shows a sensitivity table. It is for study, not design. To see how heat spreads through a slab step by step, try heat conduction solver; for glacier mass balance and ice flow, glacier mass balance; and to name the soil itself, soil classification.
How to use it
- Pick Thaw depth (active layer) or Frost depth.
- Under Temperatures, paste daily mean air temperatures in °C, one day per line. A date or day number before each value is fine, and a header line is skipped. For frost depth, paste one winter, from autumn to spring. If you already know the index, choose The degree-days already and type it.
- Set the n-factor: the ratio of the surface index to the air index. Moss, plants or snow keep the ground cooler in summer and warmer in winter (n below 1); bare gravel is near 1. Use 1 if your index was measured at the ground surface.
- Under Soil, choose 1, 2 or 3 layers and give each one’s thickness, thermal conductivity (thawed and frozen), water content in per cent of dry mass, and dry density. The last layer carries on downwards.
How the layers are solved
The front first crosses layer 1 just as in the one-soil equation. Once it is deeper, the heat has to pass through every layer above it, so their resistances (thickness ÷ conductivity) add up. Fully thawing layer i takes L h (R + h ÷ 2k) of index, where R is the resistance of the layers above, and the front part-way into a layer comes from the same balance solved as a quadratic. With a single layer this is exactly the plain Stefan equation. The layer table shows the surface index needed to thaw each layer and, for daily data, the day it happens.
Worked example
The page opens with a made-up year of daily air temperatures and one silty soil: k thawed 1.2 W/m/K, water content 30 %, dry density 1400 kg/m³. The year has 136 days above 0 °C and a thawing index of 888 °C·days. With the n-factor of 0.8 the ground surface gets 710 °C·days. The latent heat is 1400 × 0.30 × 334 000 = 140 MJ/m³ (42 % water by volume), so
Z = √(2 × 1.2 × 710.32 × 86 400 ÷ 140 280 000) = 1.02 m.
The sensitivity table shows why the soil matters as much as the weather: 20 % more index (or a 20 % higher conductivity) gives 1.12 m, while 20 % more water gives 0.94 m.
Now choose 2 layers. The silt becomes a 0.5 m top layer over a gravelly sand (k 2 W/m/K, 12 % water, 1700 kg/m³). Thawing the silt takes 169 °C·days at the surface, which the example year reaches on day 184; the drier sand below thaws faster, and the front ends the summer at 1.52 m.
Switch to Frost depth: the freezing index of the same year is 3440 °C·days, the freezing n-factor 0.6 (snow cover) leaves 2064 °C·days at the surface, and in one frozen silt (k 1.8 W/m/K) the frost goes 2.14 m deep.
Tips
- Water content counts twice as much as you might think: wet, peaty ground stays shallow because each cubic metre holds far more ice to melt.
- An organic top layer with a low conductivity insulates the ground below; try 2 layers to see it.
- Download the curve as CSV to compare with probe or thermistor readings through the season.
FAQ
Why is my measured active layer shallower? The Stefan equation ignores the heat used to warm the soil up to 0 °C, so it usually overestimates a little. Fitting the n-factor to one year of measurements is a common way to calibrate it.
Where do the soil numbers come from? The starting values are rounded, made-up examples. Use lab or field values for your own soil.
Is my data uploaded? No. Everything runs in your browser and keeps working offline.
Where the numbers come from
- Formula (written into code by fibub, no tables copied): Stefan's solution for a moving freezing or thawing front, Z = √(2 k I / L), as given in permafrost and frozen-ground textbooks, e.g. French, H. M., The Periglacial Environment, and Andersland, O. B. and Ladanyi, B., Frozen Ground Engineering. The layer-by-layer form treats the layers above the front as thermal resistances in series.
- Latent heat of fusion of water, 334 kJ/kg (a general physical constant).
- Checked 2026-11-09 by hand and against NumPy: the index of the example year, and the layered depth by numerical integration of the front's energy balance on 200 random one-, two- and three-layer soils (agreement better than 1 part in 10¹¹).
- The daily temperatures in the example are a made-up year, not a real station. The soil values and n-factors are rounded, made-up starting values, not taken from any table; replace them with your own.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
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Last updated 2026-11-09.