Icicles on an Eave: Growth and Drips

What you see. The eave of a house with a snow-covered roof. Meltwater from the roof runs down seven icicles; in the cold it freezes onto them and they lengthen (time-lapse), while the part that does not freeze collects at the tip and falls as drops (real-time drops, drawn 2.5 times larger). Air temperature, wind and roof meltwater set the growth and the drip rate.

Growth. The water film stays at 0 °C and loses heat to the air by convection and to the cold sky by radiation: q = h (Tf − Ta) + εσ(Tf⁴ − Tsky⁴) with h = 5.7 + 3.8 v and a sky 18 K colder than the air. A cone of length L and radius εL has area A = π R √(R² + L²) and volume V = π R² L/3, so ρi V' = q A / Lf gives the constant lengthening rate L' = q √(1+ε²) / (ε ρi Lf), a few centimetres per hour in a hard frost, in line with field observations. Longer cones need more water: if the supply is smaller than q A / L_f the icicle becomes water-limited and its growth slows as 1/L².

Dripping. The unfrozen flow m_d = m_w − m_f reaches the tip, where a drop detaches when its weight equals the surface-tension force, m g = 2 π rt σ F (Tate's law, F = 0.7), so the period is m / m_d. As the icicle grows the flow that drips decreases and finally stops.

Honest simplifications. The icicle is a smooth cone of fixed taper; real icicles have ripples about 1 cm apart (not modelled), a hollow core and a tip shape that depends on impurities and wind. The film temperature is fixed at 0 °C, evaporation, solar heating and heat conducted through the roof are ignored, the sky temperature offset is a rough value, the Harkins-Brown factor and the 1.5 mm tip ring are assumed, and the other six icicles are the main one scaled by fixed factors.

Checks. (1) the closed-form linear growth against integration of the full volume balance; (2) exact T⁴ radiation against its linearisation; (3) water ledger and drop count over 20 minutes; (4) Tate drop size against the capillary length (Bond number of order one).

• Run Cold clear night: the main icicle lengthens at a constant rate while the tip keeps dripping (heat-limited growth).
• Pick Dry roof: all the meltwater freezes on the way down, there are no drips and the growth rate falls as the icicle lengthens (water-limited).
• Pick Windy and cold and Marginal thaw: convection sets the heat loss, so wind speeds growth and a thaw nearly stops it.
• Raise the taper (Blunt): a fatter cone has more surface per length, so it grows more slowly in length.