We present one of the first physically-motivated two-dimensional general relativistic magnetohydrodynamic ( GRMHD ) numerical simulations of a radiatively-cooled black-hole accretion disk .
The fiducial simulation combines a total-energy-conserving formulation with a radiative cooling function , which includes bremsstrahlung , synchrotron , and Compton effects .
By comparison with other simulations we show that in optically thin advection-dominated accretion flows , radiative cooling can significantly affect the structure , without necessarily leading to an optically thick , geometrically thin accretion disk .
We further compare the results of our radiatively-cooled simulation to the predictions of a previously developed analytic model for such flows .
For the very low stress parameter and accretion rate found in our simulated disk ( \alpha \approx 0.003 , \dot { M } / \dot { M } _ { Edd } \approx 5 \times 10 ^ { -6 } ) , we closely match a state called the “ transition ” solution between an outer advection-dominated accretion flow and what would be a magnetically-dominated accretion flow ( MDAF ) in the interior .
The qualitative and quantitative agreement between the numerical and analytic models is quite good , with only a few well-understood exceptions .
According to the analytic model then , at significantly higher \alpha or \dot { M } , we would expect a full MDAF to form .

The collection of simulations in this work also provide important data for interpreting other numerical results in the literature , as they span the most common treatments of thermodynamics , including simulations evolving : 1 ) the internal energy only ; 2 ) the internal energy plus an explicit cooling function ; 3 ) the total energy without cooling ; and 4 ) total energy including cooling .
We find that the total energy formulation is a necessary prerequisite for proper treatment of radiative cooling in MRI accretion flows , as the internal energy formulation produces a large unphysical numerical cooling of its own .
We also find that the relativistic cooling functions must be handled carefully numerically in order to avoid equally unphysical heating or cooling runaways .