  We develop a Lagrangian stochastic model (LSM) of a volcanic plume in which
the mean flow is provided by an integral plume model of the eruption column and
fluctuations in the vertical velocity are modelled by a suitably constructed
stochastic differential equation. The LSM is applied to the two eruptions
considered by Costa et al. (2016) for the volcanic-plume intercomparison study.
Vertical profiles of the mass concentration computed from the LSM are compared
with equivalent results from a large-eddy simulation (LES) for the case of no
ambient wind. The LSM captures the order of magnitude of the LES mass
concentrations and some aspects of their profiles. In contrast with a standard
integral plume model, i.e. without fluctuations, the mass concentration
computed from the LSM decays (to zero) towards the top of the plume which is
consistent with the LES plumes. In the lower part of the plume, we show that
the presence of ash leads to a peak in the mass concentration at the level at
which there is a transition from a negatively buoyant jet to a positively
buoyant plume. The model can also account for the ambient wind and moisture.
