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Snow Albedo and the Ice-Albedo Feedback

Fresh snow reflects almost 90 percent of sunlight, but it greys with age, soot and thinning - so it absorbs more, melts faster, and greys even more. Run a whole spring on a hillside, change the weather and the soot, and compare it with snow whose albedo is held fixed.

Albedo feedback (ageing, soot, thin snow)

Day of the season

-surface albedo now

-sunlight on the snow (W/m2)

-net energy into the snow (W/m2)

-melt rate (mm w.e. per day)

-snow water equivalent / depth

-snow gone on day (feedback vs fixed albedo)

Independent checks

Each row compares a closed form or textbook value with a separate calculation: the daily sunlight integral, an Euler albedo-ageing ODE, the melt energy bookkeeping and the planetary temperature by bisection.

The science

Albedo. Albedo alpha is the fraction of sunlight reflected. The snow absorbs (1 - alpha) S, where S is the sunlight reaching the ground: S = S0 E0 cos(zenith) tau (1 - 0.72 cloud), with S0 = 1361 W/m2 and the zenith angle from latitude, date and hour (cos z = sin phi sin delta + cos phi cos delta cos h). Fresh snow is about 0.88; old, wet snow about 0.5.

Energy balance. A ripe snowpack at 0 C melts when Q = (1 - alpha) S + L_down - 0.98 sigma (273.15 K)^4 + h_c T_air is positive. L_down = eps_a sigma T_air^4 (eps_a = 0.70 + 0.27 cloud), and h_c = 5 + 3.5 wind (W/m2/K) lumps the sensible and latent heat of the wind. Melt rate = Q / (rho_w L_f) with L_f = 334 kJ/kg: one W/m2 melts about 0.26 mm of water per day.

Feedback. Surface age raises the grain size: alpha_age = 0.5 + (alpha_fresh - 0.5) exp(-age / 7 d), and the snow ages three times faster while it melts. Soot lowers albedo by 0.14 (1 - exp(-c / 100 ppm)), and melting concentrates it at the surface (c grows as the snow thins). Below about 5 cm the dark ground shows through. Each effect lowers alpha, raises Q and speeds the melt: that is the ice-albedo feedback in miniature. Fresh snow buries the soot and resets the age.

Planet. Averaged over the globe, S0/4 (1 - A) = sigma T^4 gives an effective temperature of 255 K for A = 0.30; if more ice raises A the planet cools, if ice melts it warms, which is the same feedback on a world scale.

Simplifications. One point on the slope, ripe snowpack at 0 C from the start (no cold content, meltwater leaves at once), constant density 350 kg/m3, one sinusoidal day, a straight warming trend, simple cloud and clear-sky terms, bulk turbulent exchange, no rain, no drifting, no snow-line shading by trees. The aging time, soot law and ground albedo are illustrative values in the range of field studies, not fits to a site. The soot ppm value is a mass fraction.