Thin-film icicle-ripple instability

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.