Brightness of the Sun's small scale magnetic field: proximity effects
  The net effect of the small scale magnetic field on the Sun's (bolometric) brightness is studied with realistic 3D MHD simulations. The direct effect of brightening within the magnetic field itself is consistent with measurements in high-resolution observations. The high 'photometric accuracy' of the simulations, however, reveal compensating brightness effects that are hard to detect observationally. The influence of magnetic concentrations on the surrounding nonmagnetic convective flows (a 'proximity effect') reduces the brightness by an amount exceeding the brightening by the magnetic concentrations themselves. The net photospheric effect of the small scale field (~ -0.34% at a mean flux density of 50 G) is thus negative. We conclude that the main contribution to the observed positive correlation between the magnetic field and total solar irradiance must be magnetic dissipation in layers around the temperature minimum and above (not included in the simulations). This agrees with existing inferences from observations. 
I. Thaler, H.C. Spruit
astro-ph.SR