# Character of the temperature profile

Several temperature profiles calculated from Equation (6.24) are shown in Figure 6.6a. For the conditions assumed, the ice is nearly isothermal in the upper few hundred meters and then warms rapidly near the bed. Higher vertical velocities, resulting from higher accumulation rates at the surface, increase the thickness of the isothermal zone and decrease the basal temperature.

In essence, cold ice is advected downward from the surface, and the upward-moving geothermal heat warms this descending ice. With higher rates of advection (higher vertical velocities), the heat supplied can warm a smaller fraction of the descending ice, so the ice column as a whole is colder.

The shape of the temperature profile can be understood qualitatively in the following way. Consider the three elements of ice labeled A, B, and C in Figure 6.7. All three are moving downward, but because w decreases with depth (Equation (6.15)), element A will be moving fastest and element C slowest. As element C moves down, it must warm up and this requires heat. Thus, the heat flux out of the top of this element will be less than that into the bottom, and the temperature gradient required to conduct this heat will be less at the top of the element than at the bottom. However, w is small in this part of the glacier, so despite the

Figure 6.6. Calculated temperature profiles in polar ice sheets. (a) Accumulation zone. Vertical velocity is negative or downward. Parameters, other than w, are the same for all curves. Dashed profile labeled "R: w = -0.50" is calculated from Equation (5.25) for w. (b) Ablation zone. Vertical component of velocity is positive or upward. (Modified from Hooke (1977), Figure 3. Reproduced with permission of The University of Washington.)

Temperature, °C

Figure 6.6. Calculated temperature profiles in polar ice sheets. (a) Accumulation zone. Vertical velocity is negative or downward. Parameters, other than w, are the same for all curves. Dashed profile labeled "R: w = -0.50" is calculated from Equation (5.25) for w. (b) Ablation zone. Vertical component of velocity is positive or upward. (Modified from Hooke (1977), Figure 3. Reproduced with permission of The University of Washington.)

comparatively high temperature gradient here, this element does not have to warm up very much and the change in temperature gradient through it is small, as shown. Element B has a higher velocity, and the temperature gradient is still comparatively high here, at mid-depth in the glacier, so this element must warm up a lot. Thus, here the change in temperature gradient through the element is rather large. Element A has the highest vertical velocity, but at this level in the glacier nearly all of the heat introduced at the base has been consumed in warming deeper ice. Thus, the temperature gradient here is quite low, and despite

Figure 6.7. Qualitative illustration of effect of downward vertical velocity on a temperature profile.

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Bed its high velocity, element A does not have to warm up very much. Thus, again, the change in temperature gradient through the element is small, as shown.

Later we will examine temperature profiles in the ablation area, where the vertical velocity is upward. However, the reader may find it both challenging and instructive to try to deduce the character of the profile there, using the logic just presented.

Continue reading here: Error introduced by the assumed vertical velocity distribution