## Melting and freezing at the bed

In our analysis so far, we have tacitly assumed that the temperature at the base of the glacier is below the melting point. However, this assumption has not been incorporated into Equation (6.24) or (6.25). To be specific, if the bed is at the pressure melting point and melting is occurring there, some of the geothermal heat is clearly being used for that purpose and is not being conducted upward into the ice. Thus, our estimate of po is likely to be too high. If we inadvertently insert such a value of po into Equation (6.24) or (6.25), the calculated temperature at the bed, 0o, will turn out to be greater than the pressure melting temperature which is clearly impossible.

To obtain a correct solution for the temperature profile in this case, po must be adjusted downward. The procedure is straightforward. Because erf (0) = 0 and 0(0) = 0pmp, then at the pressure melting point temperature, Equation (6.24) can be solved for po, thus:

The melting point is depressed approximately 0.098 K MPa-1 (if the water produced is saturated with air) so, for example, 0pmp under 500 m of ice would be ~ -0.4 °C. Inserting the value of po obtained from Equation (6.26) into Equation (6.24) and solving for temperatures at other depths in the glacier will give the desired temperature profile. (Note that this approach is equivalent to solving Equation (6.17) with a temperature boundary condition at the bed.)

The basal melt rate, dm/dt, can also be calculated. The heat available for melting is the difference between the geothermal heat flux and the heat flux into the ice, or K (ftG - fto), where ftg is the gradient that would be required to conduct the geothermal flux upward into the ice. Thus, we obtain:

dt L

where L is the latent heat of fusion, and the result is in mm a-1 if the gradients are in K m-1.

It is also possible that water formed by basal melting at some distant locality has moved along the bed to the site at which the temperature profile is to be calculated. Until all such water is refrozen, perhaps incorporating sediment into the ice in the process, it will keep the basal temperature at the pressure melting point. Again, Equation (6.24) does not know about this water, so the intelligent scientist must intervene. Presumably, he or she has calculated basal melt and freeze rates further upglacier, and has kept track of how much of the water produced has not refrozen. In any case, the procedure is similar to that above, except that now the value of fto calculated from Equation (6.26) will be greater than that necessary to conduct the geothermal heat upward into the ice, and dm/dt in Equation (6.27) will be negative, indicating freezing.

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