Modelling Transient Ice Flow With a Limited Domain

To develop an ice sheet model that can run fast and assimilate available data it is advantageous to limit the model domain to include only the relevant portions of the ice sheet. For a model where the domain is limited there is no terminus and the ice sheet span is unknown; the only ice thickness information comes from the initial condition. Therefore, calculations of flux crossing this downstream boundary must be carefully considered. We compare different options for calculating flux at the downstream boundary. A simple approach would be to linearly extrapolate the flux downstream, however this is a poor assumption for a transient problem. A more reasonable approach would be to extrapolate the ice thickness and surface slope and calculate the flux crossing this point dynamically. However, this also is not good enough. This improper accounting of the flux makes the model incapable of holding steady state. Calculating the flux kinematically is preferable to the dynamic calculation but forces instantaneous export of any additional mass. The kinematic calculation allows the model to return to the inital steady state but does not necessarily represent what is glaciologically realistic. The correct treatment of this problem involves embedding the limited domain model in a external ice sheet with an extended domain of full span. The same ice divide thickness is used to calculate the extended domain with a Vialov model. The flux crossing the end of the limited domain model is the same as the flux crossing that position in the extended domain model. Using the extended domain model we calculate impulse response functions for the ice sheet in response to spatially uniform, impulsive accumulation perturbations. The convolution of the impulse response function with the time series of spatially averaged volume input perturbations will approximate how ice moves through the system and when it arrives at the downstream boundary. This ensures that flux moves through the system at a physically meaningful rate. We present the result of including this boundary condition in our limited-domain transient ice flow model. This type of model can be used in a wide range of applications, particularly for looking at transient flow near an ice divide (near ice core sites).