Treatment of Density Variations in Ice-Flow Models using the Shallow Ice Stream Approximation

In most models of large-scale ice-sheet dynamics, horizontal density variations within the ice are largely ignored. Ice-sheets typically comprise a core of meteoric ice, and an overlying layer of lower-density firn of variable thickness. This gives rise to spatial variation in the average density of the ice at each point on the surface, which in principle will modify the glacial dynamics.  A common approach to handle density-variation in the ice is to adjust the thickness of the glacier to the equivalent height of constant-density meteoric ice. We refer to this as the density-to-thickness (D2T) adjustment method. While this approximation preserves the total mass of the ice-column at each spatial coordinate, it introduces additional unwanted terms in the momentum equations, and misses other correction terms. 

In this study, we investigate the D2T adjustment approximation in detail, and consider a number of alternative formulations to handle the density variation in the ice-sheet, based around the modified field equations that we derive in the presence of a variable density field. The alternative formulations include: a static density distribution in which accumulation and compactfication of the firn layer counteracts the advection of the density field so that the time-evolution of the density field can be ignored; or alternatively a time-evolving density distribution with advects with the ice-flow, such that the material derivative of the density field is zero. 

These different formulations are studied in detail within the framework of perturbation analysis. We derive transfer functions for the induced perturbations in both the glacial thickness and velocity, in response to a small perturbation in the density field. We study the frequency profile of the response and its temporal evolution. This helps us gain a deeper understanding of the different formulations, and their impact on glacial dynamics. Within the numerical ice-flow model Úa, we compare the D2T adjustment method to an approach which explicitly includes the density variations, applied to numerical simulations of the Western Antarctic region containing Pine Island and Thwaites Glaciers.