## Temperature distribution in polar ice sheets

In this chapter, we will derive the energy balance equation for a polar ice sheet. Solutions to this equation yield the temperature distribution in an ice sheet and the rate of melting or refreezing at its base. We will study some analytical solutions of the equation for certain relatively simple situations. A solution of the full equation is possible, however, only with numerical models. This is because: (1) ice sheets have irregular top and bottom surfaces; (2) the boundary conditions - that is, the temperature or temperature gradient at every place along the boundaries - vary in space and time; (3) longitudinal transport (or advection) of heat by ice flow cannot be handled well with the analytical solutions; and (4) there may be extension or compression transverse to the flowline, which makes the problem three dimensional. Furthermore, because the temperature distribution is governed, in part, by ice flow, and conversely, because the flow rate is strongly temperature dependent, a full solution requires coupling of the energy and flow (momentum) equations.

The thermal conditions in and at the base of an ice sheet are of interest not only to the glacier modeler, concerned with flow rates and the possibility of sliding, but also to the glacial geologist with interest in the erosive potential of the ice and processes of subglacial deposition.

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