On icicle ripples

Abstract

Natural icicles have an overall conical shape modulated by surface ripples. It has been noted from many observations of icicles formed in nature and in the laboratory that the wavelength of the ripples has a very narrow spectrum between about 8 and 12 mm and that, as time evolves, the phase of the ripples migrates upwards. In this pedagogical review, I explore some of the physical mechanisms that can cause and mediate the formation and migration of ripples on icicles using simple mathematical models. To keep the mathematics more straightforward and transparent, I confine attention to two dimensions. A key physical parameter is the surface tension between the film of water that coats an icicle and the air that surrounds it, which causes a phase shift between the film–air interface and the ice–film interface. I show that the wavelength of ripples is dominantly proportional to the cube root of the square of the gravity-capillary length times the thickness of the water film. At high film-flow rates, advection-dominated heat transfer coupled with the interfacial phase shift becomes the dominant driver of instability. Gibbs–Thomson undercooling provides an unexpectedly large stabilisation of small wavelengths at these large flow rates, sufficient to maintain wavelength selection at millimetre scales.