Steady State Geotherm

Let us analyse the steady state temperature state of the lithosphere as a function of the thermal conductivity structure. We assume that there is no radioactive heat production. Our simplified model consists of a sedimentary basin that overlies the basement, both of which have a constant conductivity throughout them. Fourier's law requires that the heat flux in the sediments and the heat flux in the basement is equal at the sediment-basement interface.

\[-k_{sed}\left( {T_{top} - T_{sed-bas} \over h_{sed}} \right) = -k_{bas}\left( {T_{sed-bas} - T_{bot} \over h_{bas}} \right)\]

The subscripts \(_{sed}\) and \(_{bas}\) refer to sediments and basement respectively. Basement in the current context stands for the entire lithosphere, excluding the sedimentary basin on top. \(k\) is the conductivity, \(h\) the thickness, \(T_{top}\) is the surface temperature, \(T_{bot}\) is the temperature at the bottom of the lithosphere, and \(T_{sed-bas}\) the temperature at the interface between sediments and basement. Solving for \(T_{sed-bas}\) yields

\[T_{sed-bas} = {h_{bas} k_{sed} T_{top} + h_{sed} k_{bas} T_{bot} \over \left( h_{sed} k_{bas} + h_{bas} k_{sed} \right)}\]