Rain on Snow

What is modelled

A cold snowpack has a cold content: the heat needed to bring it to 0 C. Rain that falls on it first refreezes and warms the pack. Once the pack is at 0 C, further heat from rain and wind melts snow, the pack holds a little liquid water like a sponge, and everything beyond that runs out and is routed down the river.

Equations

• Cold content CC = SWE ci |Ts|, with c_i = 2100 J/(kg K).
• Refreezing mf = min(rain, CC / Lf), L_f = 334 kJ/kg; melt M = (q − CC) / Lf after ripening.
• Heat input q = ρw cw i Tr + (1.5 + 1.6 u) Ta.
• Holding capacity Wc = θ SWE; excess runs off into a linear reservoir Q ← Q e−Δt/K + R (1 − e−Δt/K).

Checks run in the page

• The water balance closes to rounding error.
• The ripening time equals the cold content divided by the heat flux.
• The reservoir follows I (1 - e^(-t/K)).
• The rain needed before the first outflow equals the refrozen water plus the holding capacity.

Honest simplifications

The pack is one uniform layer at one temperature, the turbulent heat flux is a simple linear bulk formula, rain is steady while it lasts, there is no snow settling or channelised flow inside the pack, and the river is a single linear reservoir. It shows why rain on a cold pack is quiet and rain on a ripe pack is dangerous, not a flood forecast.

• Press Cold dry pack and see the long delay before the first outflow.
• Press Ripe pack, warm storm and compare the flood peak with bare ground.
• Press Thin snow and watch the pack hardly matter.