Thin-film icicle-ripple instability
= 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 relation>
$$
\frac{h\sigma}{V}
=\frac{\alpha h}{1-i\Gamma\alpha^3h}.
$$
The fastest-growing mode has $\alpha_m=5^{-1/6}(\Gamma h)^{-1/3}$ and travels upward at speed $\sqrt5V/6$ when the vertical coordinate points downward.