It has been suggested that rapid thinning of the West Antarctic ice sheet is due to increased melting under ice shelves caused by a gradual ocean warming (Shepherd et al., 2004). Payne et al. (2004) showed that such melting could lead to an acceleration of grounded ice flow. In this paper, we analyze the response of a marine ice sheet to different perturbations near the grounding line using a numerical ice sheet model that takes into account longitudinal stress coupling and grounding line migration at subgrid precision, based on a novel technique. Results show that stress transmission or longitudinal coupling across the grounding line plays a decisive role. The grounding line migration is a function of the length scale over which the basal conditions change from frozen to the bed to floating, the "transition zone." We demonstrate that thinning of the ice shelf due to bottom melting has a negligible effect on the grounded ice mass. Only perturbations at the grounding line or reduction in buttressing of the ice shelf substantially thins the grounded ice sheet. Inclusion of lateral drag does not alter these results qualitatively. Marine ice sheets with large transition zones, such as ice streams, seem highly sensitive to such perturbations compared to ice sheets with small transition zones, such as an abrupt ice sheet/ice shelf junction.