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

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