Ice sheets evolve over a wide range of time scales. They grow by snowfall, spread gravitationally, and diminish through melting or iceberg calving at the ice-sheet margin. The evolution of ice-sheets can be substantially affected by the rate of ice transport from their interior to their margins, and ice streams are the dominant transport mechanism in present ice sheets. Ice streams are bands of fast flowing glacier ice whose flow pattern varies both temporally and spatially. In particular ice-streams can become stagnant, reactivate, and flow in varying paths. In this thesis I investigate the dynamics that leads to ice-stream formation and their spatiotemporal variability. The two major dynamical factors I study are the frictional stress at the base of the ice and the non-Newtonian ice rheology. Both of these components are poorly constrained from observations, and may affect the stability of ice flow: the shear-thinning rheology of ice through shear instability, and the frictional bottom stress through the generation of melt water in the basal porous sediments that can lubricate the motion of the overlying ice. While we do not find a flow instability or ice-stream formation caused by the shear-thinning rheology, we do find that a triple-valued bottom sliding law can lead to ice-stream formation in our model and can account for various observed spatiotemporal characteristics of ice-streams. In particular the flow under such a sliding law can generate both steady and oscillatory ice stream solutions, independently of the shear thinning ice rheology. We then analyze the motion of the ice-stream shear-margins by linking the leading order dynamics of ice-streams to the Landau-Ginzburg reaction-diffusion equation.

Next, we study the consequences of the non-Newtonian ice rheology on ice flow under a triple-valued sliding law, and investigate the dependence of the ice-stream shear-margin width on the rheology. Finally, we study the spatiotemporal variability due to the interaction of two ice streams. We demonstrate that a spatially symmetric two-stream pattern can be unstable under an asymmetric perturbation, which results in a flow with asymmetric patterns that are maintained through the competition of the two ice-streams over a shared mass source. The rich spatiotemporal variability is found to mostly be a result of the triple valued sliding law, but non Newtonian effects are found to play a significant role in setting a more realistic shear margin width and allowing for relevant time scales of the variability.