Icicle ripple

An icicle ripple is a corrugation of the ice-water interface produced while a thin flowing water film freezes. Coupling lubrication theory, surface tension, thermal conduction, and the Stefan condition determines its growth and migration.

For a water film of thickness $h$, lateral freezing speed $V$, ripple wavenumber $\alpha$, and $\Gamma =\gamma /(3\rho g)$, a long-wave model gives the dispersion relationThe fastest-growing mode has ${\alpha }_m=5^{-1/6}(\Gamma h{)}^{-1/3}$ and travels upward at speed $\sqrt{5}V/6$ when the vertical coordinate points downward.