Incorporating Tidal Forcing into Long-Term Ice-Sheet Dynamics via Temporal Averaging

Long-term models for ice-sheets and glaciers are in general modeled as very slow, viscous fluids, which allows for large time steps on the order of years. However, over the past few decades, observations and theoretical studies have shown that ice streams exhibit short-term responses when it comes to external forcing due to ocean tides, which induce for example non-linear velocity variations.

Short-term models typically apply viscous-elastic ice rheologies to capture these effects. However, resolving tidal dynamics requires very small time steps on the order of minutes, making long-term simulations computationally challenging.

To address this, we propose a temporal averaging approach to include short-term tidal effects into efficient, long-term simulations. We present initial results based on a two-component modeling framework. The long-term component models the ice sheet as a viscous fluid governed by the p-Stokes equations with free surfaces. The short-term component describes the elastic response of the ice to tidal forcing, modeled by an elasticity problem driven by variations in hydro-static pressure due to ocean tides. This leads to a variational inequality of Signorini type, reflecting intermittent contact between the ice and the bedrock. 

As the tidal cycle causes the ice–bedrock contact zone to evolve in time, the basal boundary condition alternates between frictional contact and floating due to ocean-pressure. By exploiting the periodic nature of the tidal forcing, we derive effective, tidally averaged basal traction coefficients based on the varying grounding line position. These effective coefficients can be incorporated into the basal friction law of the viscous, long-term ice-flow model. The averaged friction coefficients are updated after a time step in the long-term model to take the geometric deformation of the ice sheet into account. 

This approach allows for efficient simulations that capture the influence of short-term tidal dynamics on ice-streams effectively, and relies on a clear separation between the short tidal time scale and the long-term viscous dynamics of the ice sheet.