Geological Problems

A number of important geological problems can be modeled by the instantaneous heating or cooling of a semi-infinite half-space. In the middle of the nineteenth century Lord Kelvin used this solution to estimate the age of the Earth. He assumed that the surface heat flow resulted from the cooling of an initially hot Earth and concluded that the age of the Earth was about 65 million years. We now know that this estimate was in error for two reasons – the presence of radioactive isotopes in the mantle and solid-state thermal convection in the mantle.  Instantaneous Heating or Cooling of a Semi-Infinite Half-Space 277 . Heating of a semi-infinite half-space by a sudden increase in surface temperature. In many cases magma flows through preexisting joints or cracks. When the flow commences, the wall rock is subjected to a sudden increase in temperature. Heat flows from the hot magma into the cold country rock, thus increasing its temperature. The temperature of the wall rock as a function of time can be obtained by solving the one-dimensional, time-dependent heat conduction equation for a semi-infinite half-space, initially at a uniform temperature, whose surface is suddenly brought to a different temperature at time t = 0 and maintained at this new temperature for later times. This solution can also be used to determine the thermal structure of the oceanic lithosphere. At the crest of an ocean ridge, hot mantle rock is subjected to a cold surface temperature. As the seafloor spreads away from the ridge crest, the near-surface rocks lose heat to the cold seawater. The cooling near-surface rocks form the rigid oceanic lithosphere. We now obtain the solution to Equation (4–68) in a semi-infinite half space defined by y > 0 whose surface is given an instantaneous change in temperature. Initially at t = 0, the half-space has a temperature T1; for t > 0, the surface y = 0 is maintained at a constant temperature T0. As a result, heat is transferred into the half-space if T0 > T1, and the temperature increases. If T1 > T0, the half-space cools, and its temperature decreases. The situation is sketched in Figure 4–20 for the case T0 > T1.