On the mathematical theory of plumes

 We reconsider the theory of turbulent plume formation provided by Schmidt (1941a,b) and its integral formulation, particularly that of Morton et al. (1956). A particular issue for the correct formulation of a mathematical theory is whether the plume is taken to have finite or infinite width, and whether the entrainment rate is prescribed or deduced. Fox (1970) showed that the entrainment rate for a plume can be deduced from the governing partial differential equations by the use of integral moment theory, providing one assumes expressions for the velocity and buoyancy profiles, but it is less clear if entrainment needs to be prescribed if the plume is taken to be of infinite width (as might be appropriate for a laminar plume). Here we choose an eddy viscosity model of a plume which differs from those previously used by allowing the eddy viscosity to vanish with the vertical velocity. We then show that for the ordinary differential equations describing the similarity solution for such a plume rising in an unstratified medium, their solution implies that the plume is finite, and that the entrainment rate at the plume edge is a consequence of the model formulation, and does not need to be hypothesised; and we also show that the entrainment coefficient which is thus determined is consistent with values obtained by experiment. We also show that the resulting velocity profiles differ from those found experimentally by their omission of the Gaussian tail, and we suggest that this discrepancy may be resolved in the model by the inclusion of the small molecular kinematic viscosity.