We derive a new expression for the entrainment coefficient in a turbulent plume using
an equation for the squared mean buoyancy. Consistency of the resulting expression
with previous relations for the entrainment coefficient implies that the turbulent Prandtl
number in a pure plume is equal to 3
/
5 when the mean profiles of velocity and buoyancy
have a Gaussian form of equal width. Entrainment can be understood in terms of the
volume flux, the production of turbulence kinetic energy or the production of scalar
variance for either active or passive variables. The equivalence of these points of view
indicates how the entrainment coefficient and the turbulent Prandtl and Schmidt numbers
depend on the Richardson number of the flow, the ambient stratification and the relative
widths of the velocity and scalar profiles. The general framework is valid for self-similar
plumes, which are characterised by a power-law scaling. For jets and pure plumes it is
shown that the derived relations are in reasonably good agreement with results from
direct numerical simulations and experiments.