Stress gradient damping of thermoviscous ice flow instabilities

Calculations of thermomechanically coupled ice sheet evolution using the shallow ice approximation exhibit the development of fingering instabilities in velocity, temperature, and thickness, which have been argued to resemble and therefore explain ice stream formation. However, workers have had difficulty in obtaining comparable results. It has been suggested that this is due to ill-posing of the evolution problem when the shallow ice approximation is used. The influence of additional mechanical terms are examined, using both the full system and the hybrid lubrication/membrane stress approximation (the three-dimensional equivalent of the longitudinal stress approximation). Linearization techniques are used to formulate an analytical problem and a numerical eigenvalue problem. The analysis shows that the shallow ice approximation gives highly erroneous stability maps for the combination of shorter transverse wavelengths and longer along-flow wavelengths (i.e., ice stream dimensions), while the lubrication/membrane stress approximation and full systems give different but consistent results which remove the ill-posing. Calculations in two horizontal dimensions of thermoviscous coupling using the shallow ice approximation, which comprise a large proportion of ice sheet modeling results, need reexamination.