The Stefan Problem

A number of important geological problems involve the solidification of magmas. We assume that the magma has a well-defined melt temperature at which the phase change from liquid to solid occurs. Associated with this phase change is a latent heat of fusion L. This is the amount of heat that is liberated upon the solidification of 1 kg of magma. Heat conduction problems involving phase changes differ from problems we have already solved in two major ways. First, we have to determine as part of the solution where the phase change boundary, that is, the interface between solid and liquid, is located. The position of this boundary obviously changes as solidification proceeds. Second, we have to account for the latent heat of fusion, which is liberated at the solid–liquid interface as solidification takes place; this additional heat must be conducted away from the phase change boundary. The first problem we consider is that of a horizontal layer of magma that is solidifying from its upper surface downward as a result of being cooled from above. We assume that the upper surface is maintained at a constant temperature T0. An example of this would be the solidification of a lava flow. Because of heat loss to the surface the solid layer grows thicker with time. A lava flow also solidifies at its base. However, if we assume that the magma is extruded at its melt temperature, then as long as there is still a liquid region, the solidification from the top and bottom can be treated independently. This also means that the overall flow thickness is unimportant in describing the solidification process as long as a molten region is present.

In this section, we will consider the solidification from above; in the next section, we will treat the solidification from below. The solidification of a lava flow from above is essentially identical with the freezing of a lake. This is the problem for which Stefan (1891) first obtained the solution developed below.