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 not radioactive heat production. Our simplified model consist 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)}}