The formation of cave scallops

We present a mathematical model for the formation of scallops in ice or limestone caves. The mechanism in each case is similar, and is opaquely related to the mechanism for the formation of dunes and drumlins. Turbulent fluid flow over a wavy bed induces a shift of the maximum bed shear stress upstream of the maximum elevation of the bed. The resulting enhanced shear rate causes enhanced melting or dissolution as appropriate upstream, and this causes an instabilty, which takes the form of a travelling wave. Nonlinear evolution of the waves requires a description of flow separation, and this is empirically accommodated in the model by allowing the melt or dissolution rate to depend on the slope angle of the interface. The resulting model predicts the formation of the cusps which are observed.