We consider the long-term evolution of gaseous disks fed by the vaporization of small particles produced in a collisional cascade inside the Roche limit of a 0.6 M _ { \odot } white dwarf .
Adding solids with radius r _ { 0 } at a constant rate \dot { M } _ { 0 } into a narrow annulus leads to two distinct types of evolution .
When \dot { M } _ { 0 } \gtrsim \dot { M } _ { 0 ,crit } \approx 3 \times 10 ^ { 4 } ~ { } ( r _ { 0 } / { 1 ~ { } km } ) ^ { 3.92 } { g~ { } s ^ { -1 } } , the cascade generates a fairly steady accretion disk where the mass transfer rate of gas onto the white dwarf is roughly \dot { M } _ { 0 } and the mass in gas is M _ { g } \approx 2.3 \times 10 ^ { 22 } ~ { } ( \dot { M } _ { 0 } / 10 ^ { 10 } ~ { } { { g~ { } s ^ { -1 } } } ) ~ { } % ( { 1500 ~ { } K } / T _ { 0 } ) ~ { } ( 10 ^ { -3 } / \alpha ) g , where T _ { 0 } is the temperature of the gas near the Roche limit and \alpha is the dimensionless viscosity parameter .
If \dot { M } _ { 0 } \lesssim \dot { M } _ { 0 ,crit } , the system alternates between high states with large mass transfer rates and low states with negligible accretion .
Although either mode of evolution adds significant amounts of metals to the white dwarf photosphere , none of our calculations yield a vertically thin ensemble of solids inside the Roche limit .
X-ray observations can place limits on the mass transfer rate and test this model for metallic line white dwarfs .