Compactive Viscosity of Snow (Kojima)

Snow settles fast the first night and then appears to stop. This equation is why. The compactive viscosity climbs exponentially with density, so the act of compacting is precisely what makes further compaction harder. Settlement is self-limiting, and the limiting is severe: at the usual coefficient of 0.021 m³/kg the viscosity doubles for every 33 kg/m³ gained. Snow at 300 kg/m³ is sixty-seven times harder to compact than the same snow at 100.

Kojima published the relation in 1967 from measurements on seasonal snow; Anderson wrote it into the SNTHERM model in 1976 with $\eta_0 = 3.6\times10^{6}$ Pa·s and $f = 0.021$ m³/kg, and those two numbers have been carried into a great many land-surface schemes since. Both are fitted constants, and the pair only means something together: an $\eta_0$ from one study used with an $f$ from another produces an internally consistent number that is comparable to nothing.

Note what $f$ actually is, because it is the kind of quantity that quietly breaks calculations. It multiplies a density and the product has to be dimensionless, so $f$ carries the reciprocal of a density — cubic metres per kilogram. It looks like a bare number in the equation and it is not one. If it is ever refitted against densities in g/cm³ instead of kg/m³ it changes by a factor of a thousand, and the resulting viscosity is wrong by $e^{21}$, which is about $10^9$. That failure is silent: nothing in the arithmetic complains, and the answer is merely absurd rather than undefined.

$\eta_0$ is an extrapolated intercept, not a property of anything. It is where the fit would land at zero density, and there is no snow at zero density. Treat it exactly as the ceramics pages treat a fitted $\sigma_0$: useful for comparing one dataset to another, meaningless on its own, and quoted honestly only with the density range it was fitted over.

And temperature is the term this page does not carry, which is a real omission rather than a rounding. The full parameterisation multiplies through by $e^{c_5(T_0 - T)}$ with $c_5$ around 0.08 per kelvin, so the viscosity roughly doubles for every 8 K of cooling — snow at −20 °C is about five times stiffer than the same snow at −1 °C. Across a real winter that factor is often larger than anything density does. The values on this page should be read as applying to snow near its melting point, and a genuinely cold pack scaled up accordingly.

The practical use is calibration. Measure two densities on a layer with a known overburden and a known interval, back out a viscosity from the settlement page, and repeat at three or four densities. Plot $\ln\eta$ against $\rho$: the relation says that plot is a straight line of slope $f$ and intercept $\ln\eta_0$. If it curves, something else is changing as the density changes — grain shape, temperature history, liquid water content — and the curvature is real information about your snow rather than an inconvenience to be fitted away. Two points give a slope and nothing more.