the mass of black hole, $W$ $(=\int\alpha Pdz)$ is the integrated viscous stress (with $\alpha$ and $P$ being the viscosity parameter and pressure, respectively), $T_{\rm c}$ is the central temperature of the disk, $v_R={-{\dot M}}/{2{\pi}R{\Sigma}}$ is the radial velocity of the matter, $C_{\rm p}$ is the specific heat with constant pressure, and ${\Omega}_{\rm K}$ is the Keplerian angular frequency.

There are several terms on the right-hand side of equation (\[energy\]). The disk heating in the non-irradiated disk is given by the viscous heating; $$\label{qvis}
     Q^{+}_{\rm vis}~=~{3\over 4}W{\Omega}.$$ The external-heating terms due to indirect and direct irradiation of the disk, $F_{\rm i}$ and $F_{\rm d}$, are given by equations (\[fi\]) and (\[fd\]). In the present time-dependent study, we included irradiation as heating in the vertically averaged structure, and did not solve the vertical structure of the disk with irradiation input from above the surface (cf. Tuchman et al. 1990). The radiative cooling is given by $$\label{qrad}
    Q^{-}_{\rm rad} = {\sigma}T_{\rm eff}^4,$$ where $\sigma$ is the Stefan–Boltzmann constant. Finally, $Q_{\rm dif}$ is the heat-diffusion term, defined by $$Q_{\rm dif}
   ={{\nu}_{\rm th}}{{C_{\rm p}\Sigma}\over {2R}}{\partial\over{\partial R}}
     \left(R{{{\partial}{T_{\rm c}}}\over{{\partial}R}}\right),$$ and ${\nu}_{\rm th}(=~2W/{\Omega}{\Sigma})$ is the thermal diffusivity. Here, we use the $\alpha$ formula with the same 