To assess the influence of fault heating on the descending lithosphere, let us consider the simplified geometry illustrated in Figure 4–54. The surface plate approaches the trench with a velocity u at an angle φ to the normal to the trench and descends into the mantle at an angle θ to the horizontal.
The linear chain of active volcanoes lies at a distance dv above the slip zone. An x, y coordinate system is set up in the descending plate as shown. The solution for constant heat addition to a uniform half-space can be used for this problem if several assumptions are made: a. A substantial fraction of the heat produced on the fault zone is lost to the underlying descending lithosphere. This is a good approximation because the cold descending plate is the dominant heat sink.
b. The initial thermal structure of the lithosphere can be neglected. Because thermal conduction problems are linear in temperature, their solutions can be superimposed. The heat addition problem can be treated independently of the ambient conduction problem as long as the required boundary conditions are not violated.