The instability theory of drumlin formation applied to Newtonian viscous ice of finite depth

The Hindmarsh instability theory of drumlin formation is applied to the study of interfacial instabilities, which may arise when ice flows viscously over deformable sediments. Here, the analytic form of this theory is extended to the case where the ice is Newtonian viscous and of finite depth, and where the basal till can be both sheared by the ice and squeezed by basal effective pressure gradients: previous authors assumed infinitely deep ice, based on the assumption that the developing waveforms had wavelength much less than ice depth. The previous infinite depth theory only allowed transverse instabilities to occur, and these have been associated with the formation of ribbed moraine; one of the purposes of extending the analysis to finite depth is to see whether three-dimensional instabilities, which might be associated with the formation of drumlins or mega-scale glacial lineations, can occur: we find that they do not. A second purpose is to calculate under what circumstances the infinite depth theory provides accurate prediction of bedform development in ice of finite depth d_i.We find that this is the case if the waveforms have a wavelength less than approximately 1.2d_i. Finally, the finite depth theory allows us to compute, for the first time, the response of the ice surface to the developing unstable bedforms. We find that this response is rapid, and we give explicit recipes for the surface perturbation transfer functions in terms of the perturbations to the basal stress and the basal topography.