subsection 3.1). Equation (\[fd\]) shows that direct irradiation is a function of the gradient of the disk height, and is hence very sensitive to small variations in the disk profile. Analytic models assuming $d\ln H/d\ln R$ = constant may exaggerate the irradiation by minimizing the shadowing that may occur even in the steady state. They are certainly not adequate for time-dependent disks. The height of the region from which the irradiation arises varies with time and the disk height profile varies with both the irradiation and time. This makes the shadowing of the outer disk a complex, time-dependent phenomenon. We assume that no direct irradiation flux is added to the shadowed portions of the disk, while the indirect irradiation heats the whole disk in accord with equation (\[fd\]).

Basic Equations and Boundary Conditions
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Once ${\dot M}_{\rm in}$, and hence $F_{\rm i}(R,t)$ and $F_{\rm d}(R,t)$, are determined, the irradiated flux is added to the flux generated internally in the disk by viscous heating through an implicit numerical method, and other physical variables are computed in each time step (see Tuchman et al. 1990 for details). A modification of the outer boundary condition by irradiation affects the whole vertical structure of 