Visualisation of the effect of the under-relaxation scheme. Panel (a): Map of the temperature residuals between states calculated with (ι₀ = 1/5) and without (ι_T,κ = 1) under-relaxation for the upper hemisphere of the FULL model in quiescence. Radial temperature profiles for these two states have been extracted at a height of z/r = 0.125 (dashed black line) and are shown in panel (b). The blue curve represents an additional state resulting from a larger under-relaxation parameter ι₀ = 2/5. The profile of the equilibrium temperature evaluated at every radius with Eq. (G.3) is shown as the dashed green line. Panel (b₁): Magnification of the region where the more generous underrelaxation description still deviates from the equilibrium solution.