eat is actually generated throughout the fluid probably alters the quantitative criteria for convective instability but should not change anything at a qualitative level. On the other hand, certain contrasts are of greater importance.

One that requires mention is that the pressure in these circumstances, unlike in classical fluids, depends only on the radiation, and is therefore independent of density. Consequently, thermal diffusion can alter the pressure with no change in density. In the classical analog, thermal diffusion can quench convective instability by adding heat to falling low-entropy fluid elements, thereby forcing them to expand until their density no longer exceeds the surrounding density. This occurs when $\kappa_{th}$ is large enough that ${\cal R}$ falls below the critical value. By contrast, in the radiation-dominated case, the density remains unchanged in the face of thermal diffusion. This is why convection persists even when $k_* > \sqrt{G\tau}$; i.e., the radiation analog of the Rayleigh number does not accurately predict the effect of thermal diffusion on the instability.

Perhaps the most important contrast with classical fluids is that there is no easy definition of the kinematic viscosity. Bulk orbital shear leads to angular momentum transfer via magnetic forces (Balbus & Hawley 1998); these are, of course, entirely absent in our treatment. However, stresses arising from other shears may be quite different, whether they are caused by magnetic forces or other mechanisms (for example, photon diffusion: Agol & Krolik 1998). For this reason, it is unclear whether the conventional equation of the $r-\phi$ shear stress with a nominal viscosity (as in the Shakura-Sunyaev $\alpha$ formalism) is appropriate here. In any event, none of these possible viscosity mechanisms is present in our equations; not even radiation viscosity is considered, for no radiation shear stress exists in the pure diffusion approximation. Thus, in a formal sense, neither ${\cal R}$ nor ${\cal T}$ exists in our treatment