A scaling law for similar ice sheet flow

The time scales of the flow and retreat of the outlet glaciers draining Greenland and Antarctica and their potential instabilities are arguably the largest uncertainty in future sea-level projections. The associated stress and velocity fields are highly complex. Here we derive an exact scaling law from first principles that shows that the time scale of outlet-glacier flow is related to the inverse of 1) the fourth power of the width-to-length ratio of its topographic confinement, 2) the third power of the confinement depth and 3) the temperature-dependent ice softness. We show that idealized numerical simulations of marine ice-sheet instabilities (MISI) as found in Antarctica follow this theoretical prediction. In a further step we apply the scaling law to observations of different MISI-prone Antarctic outlets to compare their potential instability time scales. The simple scaling relation incorporates the full complexity of the ice stress field of a fast outlet glacier similar to the predictive power of the thermodynamic equations of an ideal gas. In quantifying the non-linear influence of glacier geometry and temperature on the ice dynamicsscaling law allows to investigate similar ice flow under future global warming.