Thermal equilibrium curves of accretion disks driven by magnetorotational instability

Analogous to the HR diagram for stars, the thermal equilibrium curve encodes the thermodynamics of accretion disks by expressing the local balance between heating -- primarily via viscous dissipation -- and cooling -- typically through radiative transfer. These curves are commonly plotted as surface density versus effective temperature. When an S-shaped locus appears, local annuli become bistable, and limit-cycle oscillations arise when the external mass-transfer rate falls within an unstable band. This behavior underpins the disk instability model for recurring outbursts in cataclysmic variables. This paper reviews first-principles thermal equilibrium curves for accretion disks driven by magnetorotational instability (MRI), with emphasis on dwarf novae. Unlike the parameterized $\alpha$-viscosity approach, the curves are obtained by solving the governing equations with radiation magnetohydrodynamics simulations, thereby reproducing S-shaped loci without prescribing $\alpha$. The disk instability in dwarf-nova systems and the physical origin of angular-momentum transport (shear stresses) are also briefly reviewed. Notes on the stability of radiation-dominated accretion flows are included in the Appendix.