A vertically integrated treatment of ice stream and ice shelf thermodynamics

The extremely small vertical shear in ice stream and ice shelf flow simplifies the equations, which govern their thermodynamic evolution. Complemented by the widely used shallow shelf approximation used to simplify the ice flow momentum balance, a vertically integrated formulation of heat transfer presented here reduces the dimensionality of the thermodynamic problem from three to two (plan view) dimensions and thus significantly reduces the computational cost of treating ice stream and ice shelf thermodynamics in models. For realistic conditions, errors in ice stiffness parameter, ice thickness, and speed caused by the vertically integrated treatment of heat transfer are less than 5% of magnitudes of these values compared to the standard three-dimensional thermomechanical computations. In addition, for the specific case of ice shelves with strong bottom melting, the governing equation describing evolution of the vertically integrated ice stiffness parameter is derived, which further reduces computational cost. The presented error analysis and formulations of ice stream and ice shelf thermodynamics in terms of the vertically integrated temperature allow the thermodynamic effects on ice deformation to be easily incorporated into studies that traditionally disregard them.