  Differentially rotating, "advection-dominated" accretion flows are considered
in which the heat generated by viscous dissipation is retained in the fluid.
The equations of time-dependent quasi-spherical accretion are solved in a
simplified one-dimensional model that neglects the latitudinal dependence of
the flow. A self-similar solution is presented that has finite size, mass,
angular momentum and energy. This may be expected to be an attractor for the
initial-value problem in which a cool and narrow ring of fluid orbiting around
a central mass heats up, spreads radially and is accreted. The solution
provides some insight into the dynamics of quasi-spherical accretion and avoids
many of the strictures of the steady self-similar solution of Narayan & Yi.
Special attention is given to the astrophysically important case in which the
adiabatic exponent gamma=5/3; even in this case, the flow is found to be
differentially rotating and bound to the central object, and accretion can
occur without the need for powerful outflows.
