  Experimental observations of turbulent buoyant plumes, produced by a constant
source of buoyancy, have been described with great success using a horizontally
averaged model for the conservation of mass, momentum and buoyancy flux.
However, experimental observations of plumes with time-dependent buoyancy
fluxes has proved more challenging for quantitative models. At each level in
the plume, the horizontal variation in velocity leads to an along-axis shear
and hence dispersive transport relative to the mean. With a time-dependent
source, axial dispersion of the dynamic properties of the plume has a key role
in the evolution of the flow. Using ideas of mixing length theory, we introduce
a model for this axial dispersion, and test the model by comparison with
experimental observations of plumes in which the buoyancy flux is suddenly
decreased or suddenly increased. In both cases, we show the transition from one
buoyancy flux to another is self-similar, and using the data, we find the axial
dispersion may be expressed as \beta Ub where \beta\ lies in the range
0.70-0.88 and U and b are respectively the horizontally averaged vertical
velocity and radius of the plume at height z and time t. Our model also reduces
to that of a classical plume when the buoyancy flux is steady.
