We choose a fiducial temperature. I chose the top of the D’’ layer. Heat flows through the D’’ layer from the CMB by conduction. Let us compute the temperature as we move from the top of the D’’ layer to the CMB. Assuming that 9TW (9 x 1012 W) is the heat issuing from the CMB, find the flux in W/m2 as q(r) = Q(CMB)/Area of surface. If we start at the top of D’’ layer, the area of the spherical surface is simply calculated. The radius of the top of the D’’ layer is found in the ExCel spreadsheet in box B27; note that the radius is measured in km in column B. Find the temperature at the next lower level (B26) using the conduction formula above. Find the thermal conductivity in column N. I chose to average the conductivity through the D’’ layer (top+bottom)/2 but this is just a fussiness.

You are not obliged to do your calculations with the ExCel spreadsheet … I did. Starting with the 2500K temperature at the top of D’’ (3630km), calculate the temperature at the next deeper level (3600km). I had calculated the heat fluxes at three radius in the D’’ later and placed them in Column R for the moment. Using ExCel should you choose to, you can assemble your calculation for the 3600km level by entering the following calculation into the temperature calculate box for 3600km: This is what I entered in box P26:

Note that the formula for the temperature increment from box P26 can be copied into box P25 for the next step. Note that this takes us into the core which we might assume follows an adiabatic gradient according to the form: In the spreadsheet, P is found in column I in gigapascals (x109), ap in column M (scaled by 10-6), density is found in column E and the heat capacity in column L. Incrementing down through the core (and note assuming purely adiabatic temperature increase) from the 3480 to 3200km level, I inserted this calculation into box P22, copied it into box P20 taking care to make sure that the correct data are being taken from the spreadsheet --- etc..

You can follow this recipe above the D’’ layer as well to the base of the lithosphere. I obtain a temperature-depth profile that looks like this: