Internal temperature boundary condition
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Currently the model uses a LW boundary condition set by the surface temperature. That is, it sets the upward stream longwave flux at the surface according to some surface temperature (Tstar).
Most other models (e.g. HELIOS) instead set the net upward-directed longwave flux at the bottom-most level. They then express this as an internal temperature Tint (through σT^4).
For comparison with other models, it would be useful to include the latter boundary condition as an option. This can be done by simply adjusting the current configuration (Tstar) according to:
σTint^4 = σTstar^4 - S - F
where S and F are the downward-directed solar and thermal streams respectively.
Note that the implementation of a surface boundary condition (from the opposite perspective) is discussed in the Appendix of Malik+19 for HELIOS: https://iopscience.iop.org/article/10.3847/1538-4357/ab4a05
With the nonlinear method, this should be relatively easy to implement by requiring that all levels maintain a total flux which is equal to F_int = σTint^4. The flux residual (for the cost function) is then simply F_current - F_int.
In addition to the above boundary condition T_int, it might also be helpful to specify T_eff, as the effective internal temperature. With this case, each level has to maintain a flux of F_eff = σTeff^4 - F_asf where F_asf is the absorbed stellar flux. Having both is helpful - in the absence of irradiation, F_eff = F_int.