  We summarize the physical equations and analytic solutions of three versions
of the box model equations, suitable for the integral formulation of
axisymmetric gravity-driven particle currents with constant volume. The first
model is based on a simple constant resisting stress, while the second and
third models assume flow dilution by particle deposition. The third model is
characterized by assuming an interstitial fluid lighter than the ambient fluid.
All the calculations are performed on a flat topography. Ambient fluid
entrainment and cooling effects are not considered. All particles are assumed
to deposit at the same velocity.
