the disk, including the mid-plane values (Tuchman et. al. 1990). This is a reasonable first approximation, but could be improved by including more sophisticated radiative transfer effects. In this formulation, the irradiation fluxes are parameterized by two constants, $C_{\rm X}$ and $A$, which are, by assumption, independent of the radius and time.

The basic equations of the time-dependent thermally unstable accretion-disk model (for details, see Kim et al. 1992 and references therein) are the mass conservation, $$\label{mass}
  2{\pi}R{{{\partial}{\Sigma}}\over{{\partial}t}}
       ={{{\partial}{\dot M}}\over{{\partial}R}} +2{\pi}R S_0(R,t),$$ the angular momentum conservation; $$\label{angmom}
  2\pi R{\partial(\Sigma\ell)\over\partial t}
       ={\partial(\dot M\ell)\over\partial R}
        +2{\pi}R{S_0(R,t){\ell_{\rm K}}}
        -{{\partial}\over{{\partial}R}}({2{\pi}R^2}W),$$ and the vertically averaged energy equation; $$\label{energy}
  {{{C_{\rm p}}{\Sigma}}\over 2}\left({{\partial}\over{\partial t}}
       +{v_R}{{{\partial}}\over{{\partial}R}}\right){T_{\rm c}}
       =Q^+_{\rm vis}+F_{\rm d}+F_{\rm i}
        - Q^-_{\rm rad}+Q_{\rm dif},$$ where we use cylindrical coordinates ($R, \phi, z$), with the origin at the black hole and the $z$-axis perpendicular to the plane of the disk ($z=0$). Also, $\Sigma$ is the surface density, $S_0(R,t)$ is the mass source function defined by $${\dot M}_{\rm T}(t)={\int}2{\pi}RS_0(R,t)dR,$$ with ${\dot M}_{\rm T}$ being the mass transfer rate from the companion star, $\ell=(GMr)^{1/2}$ is the specific angular momentum of matter in the disk, $\ell_{\rm K}$ is the specific angular momentum of the stream from a companion star, $M$ is 