Variational Methods in Ice Sheet Modelling

• Main Author: Brinkerhoff, Douglas John
• Format: Thesis
• Language: unknown
• Published: University of Montana 2013
• Subjects: cryosphere; modelling; variational techniques; Greenland; Ice Sheet
• Online Access: https://scholarworks.umt.edu/etd/1167; https://scholarworks.umt.edu/context/etd/article/2186/viewcontent/BRINKERHOFF_THESIS.pdf

A complete simulation of flowing ice requires knowledge of both the fundamental physical principles that govern the stress and energy balances and a framework for assimilating data into a model to help estimate unknown parameters. Modelling ice is complex, due to the large spatial extent of ice sheets, the multiple scales at which relevant physics operate, and the coupling between heat, stress, and ice rheology. As such, it is usually necessary to make approximations to the equations governing ice flow. At the same time, it is important to have an understanding of the specific assumptions that lead to these approximations. We develop a variational principle for Stokes flow, and neglect certain components in order to obtain the variational principle for the first-order approximation for ice flow. This result is fundamentally the result of assuming bed slopes to be much less than surface slopes, and that vertical resistive stresses are negligible. From a practical standpoint, using automatic differentiation tools on this functional yields a compact model of ice flow that automatically incorporates correct boundary condition. This model is compared to well known benchmark tests. We also present an improved model of ice thermodynamics that operates on enthalpy rather than temperature, avoiding many of the difficulties associated with phase change. We derive a method for inverting the Blatter-Pattyn ice sheet model in order to solve for the rate of basal sliding. This method uses the adjoint equations of the forward model to obtain the gradient of an error functional, and this is minimized using a quasi-Newton method. These methods are applied to an instrumented streamline of the Greenland ice sheet. We perform numerical experiments on this geometry in order to assess the sensitivity of thermal conditions at the ice sheet bed to perturbations in unknown parameters. The basal thermal regime is sensitive to changes in geothermal heat flux, with the location of the transition zone between cold and temperate ice being ...