Investigating the basal friction law of Alpine glaciers from multi-decadal up to centennial observations

Understanding the physical laws governing glacier and ice sheet basal sliding speed is crucial for accurately predicting their dynamics and contribution to sea-level rise. However, basal sliding is controlled by complex processes linked to subglacial hydrology, which remains difficult to constrain. Previous studies based on Argentière glacier (France) suggest a simplification of the law describing the long-term evolution of sliding velocity, by proposing that long-term subglacial water pressure of hard-bedded glaciers is determined by the basal shear stress condition. The applicability of these findings to other glaciers and over long timescales has not been fully explored yet. In this study, we analyze multidecadal timeseries of surface velocities and elevations from various locations on seven Alpine glaciers, spanning from the early 20th century to the present. Using the Elmer/Ice finite element model, we solve for the full-Stokes equations to derive realistic estimates of basal sliding velocities and shear stresses from observed surface velocity and topography. Our analysis of these datasets reveals distinct basal friction behaviors both among glaciers and within different profiles of the same glacier. We identify three types of friction laws governing glacier dynamics. Most sites follow a Lliboutry-type law, where significant variations in sliding velocity occur under minor changes in shear stress. This behavior can be explained by the formation of water-filled cavities that grow as a function of sliding velocity under constant effective pressure. Other sites exhibit a Weertman-type law, characterized by a power-law scaling between shear stress and sliding velocity, implying constant cavity size through time. Finally, only Gébroulaz glacier follows a Coulomb-type law typical of sedimentary beds, where basal velocities increase dramatically beyond a critical shear stress threshold. For each measurement site, we derive a value of maximum shear stress CN and friction coefficient As and find that hard-bedded glaciers in the Weertman and Lliboutry regimes align along a unified friction law with similar values for the friction exponent m and the bed-shape exponent q. Our results show that the basal friction at all sites can be explained by a single friction law where the effective pressure either remains constant through time or scales with the basal shear stress. Further exploration of correlation between these friction laws and glacier geometric parameters, such as surface slope and bedrock roughness, may provide insights into the underlying mechanisms regulating the long-term effective pressure.