Stefan Frost Penetration Depth

Josef Stefan published this in 1891, working on the thickness of polar ice, and the argument is beautifully simple. Assume all the heat removed at the surface goes into freezing pore water — none into cooling the soil, none arriving from below. The frozen crust conducts that latent heat out through its own thickness, so the thicker it gets the slower it grows, and integrating gives $X = \sqrt{2 k_f n \Delta T t / L}$. The product $n\Delta T t$ is the surface freezing index, usually quoted in degree-days; this page asks for its two factors separately because there is no degree-day unit on this site and inventing one would be worse than spelling out the definition. A reader holding an index of 720 °C·days directly can enter it as 720 C° over 1 day and the arithmetic is identical.

The square root is the practical content. Depth grows as the square root of the index, so time grows as depth squared: reaching twice as deep takes four times as long. Two consequences follow. Burying a service another half-metre buys protection far faster than it costs excavation. And a single brutal fortnight rarely does the damage a long, moderately cold, snowless winter does, because the index is an integral and the front does not care how the cold was distributed.

Now the honesty, and it is substantial: Stefan's equation over-predicts, typically by 20 to 40% in temperate ground. It ignores the sensible heat of the soil — the energy that has to be removed just to cool the ground from its starting temperature down to freezing before any water can change phase — and it ignores the geothermal heat arriving from below. Both oppose the front and neither is in the equation. The standard correction is the modified Berggren coefficient $\lambda$, applied as $X_{\mathrm{corrected}} = \lambda X$, where $\lambda$ is read off a chart against two dimensionless ratios: the thermal ratio, comparing the ground's initial temperature above freezing with the mean surface depression, and the fusion parameter, comparing the sensible heat to be removed with the latent heat. It comes off a chart. There is no closed form for it, it is not on this site, and there is no honest way to fake one. Treat a Stefan depth as an upper bound rather than a design figure, and use your local code's frost line for anything that gets built.

Two inputs deserve care. The surface factor $n$ converts an air freezing index into a surface one, and it is doing enormous work: roughly 0.9 for bare gravel or pavement, 0.7 for turf, and as low as 0.2 to 0.4 where snow lies deep all winter. Snow is close to a blanket of insulation, and it is the single largest control on frost depth in most climates — two sites with identical air temperatures can differ by half a metre because one of them keeps its snow. The volumetric latent heat $L$ is 334 kJ per kilogram of freezable water times the kilograms of water in a cubic metre, which runs from about 5 × 10⁷ J/m³ in a dry sand to over 1.5 × 10⁸ in a saturated clay. Note freezable: fine-grained soils hold a film of unfrozen water around every particle well below 0 °C, so a clay changes less water to ice than it contains.

Fitting $n$ to a measured frost depth is legitimate and often the best thing to do, because it folds in everything the derivation left out — but the resulting $n$ then belongs to that soil, that cover and this particular uncorrected equation, and should not be carried anywhere else.