  A growing sample of white dwarfs (WDs) with metal-enriched atmospheres are
accompanied by excess infrared emission, indicating that they are encircled by
a compact dusty disk of solid debris. Such `WD debris disks' are thought to
originate from the tidal disruption of asteroids or other minor bodies, but the
precise mechanism(s) responsible for transporting matter to the WD surface
remains unclear, especially in those systems with the highest inferred metal
accretion rates dM_Z/dt ~ 1e8-1e10 g/s. Here we present global time-dependent
calculations of the coupled evolution of the gaseous and solid components of WD
debris disks. Solids transported inwards (initially due to PR drag) sublimate
at tens of WD radii, producing a source of gas that accretes onto the WD
surface and viscously spreads outwards in radius, where it overlaps with the
solid disk. If the aerodynamic coupling between the solids and gaseous disks is
sufficiently strong (and/or the gas viscosity sufficiently weak), then gas
builds up near the sublimation radius faster than it can viscously spread away.
Since the rate of drag-induced solid accretion increases with gas density, this
results in a runaway accretion process, during which the WD accretion rate
reaches values orders of magnitude higher than can be achieved by PR drag
alone. We explore the evolution of WD debris disks across a wide range of
physical conditions and calculate the predicted distribution of observed
accretion rates dM_Z/dt, finding reasonable agreement with the current sample.
Although the conditions necessary for runaway accretion are at best marginally
satisfied given the minimal level of aerodynamic drag between circular gaseous
and solid disks, the presence of other stronger forms of solid-gas
coupling---such as would result if the gaseous disk is only mildly
eccentric---substantially increase the likelihood of runaway accretion.
