Steady State Geotherm with Radioactive Heat Production

Let us analyse the steady state temperature state of the lithosphere as a function of the thermal conductivity structure and the radioactive heat production. The governing equation is

${\partial \over {\partial z}}\left( {k{{\partial T} \over {\partial z}}} \right) + Q = 0$

We assume that the radioactive heat production, $Q$ is exponentially decreasing with depth.

$Q = {q_0}{e^{ - {z \over {{z_e}}}}}$

The analytical solution for the temperature is

$T = - z_e^2{{{q_0}{e^{ - {z \over {{z_e}}}}}} \over k} + {c_1}z + {c_2}$

The constants ${c_1}$ and ${c_2}$ must be determined through suitable boundary conditions.

Our simplified model consists only of two domains, the crust and the mantle. Within both the temperature solution has the above form and hence a total of four constants have to be determined through boundary conditions. The latter are: 1) temperature boundary condition at the surface, 2) temperature boundary condition at the bottom of the lithosphere, 3) temperature at the Moho must be continuous, and 4) the heat flux at the Moho must be continuous. Note that we assume that there is no radioactive heat production in the mantle. The resulting analytical solution is implemented below.