preset number of clumps are placed on the grid. The clump positions are sampled from a probability distribution derived from a smooth density distribution as described in section \[DiskGeometry\]. If two clumps overlap in position, their densities are added. Each clump’s radius is set to a fixed number of grid zones in a given simulation, and the logarithmic radial spacing of the grid causes the size and optical depth of the clumps to grow with increasing distance from the SMBH. Overlaid on the clumps is a diffuse, smooth background gas distribution that is generated by multiplying the density distribution from section \[DiskGeometry\] by 10$^{-2}$. An example is pictured in Figure \[BuildingUpSnapshot\].

![A clumpy gas distribution corresponding to the fiducial parameters in Table \[FiducialParameters\] and the clumping parameters for model 3 in Table \[ClumpingParameters\]. Not pictured is the diffuse background gas. The white cube drawn at the center has a side length of 2 parsecs.[]{data-label="BuildingUpSnapshot"}](f2.png){width="50.00000%"}

For each clumpy gas distribution, the total mass in the computational domain was set equal to that of our fiducial smooth density model. The parameters varied between each clumpy gas distribution are the ratio of the clump diameter to the radial distance of the clump from 