Plume rise and spread in a linearly stratified environment

We present analyses of plume rise into a linearly stratified environment, either with or without a uniform horizontal flow. In the case of a still ambient, we collate results on plume spreading height and volume flux, enabling the speed of the spreading intrusion in the buoyancy-inertia regime to be expressed in terms of the fundamental parameters of plume buoyancy flux and ambient buoyancy frequency. A theoretical expression for the final volume flux emanating from the plume-rise region, in terms of maximum rise height, is also derived. Hence it is shown that the ratio of the intrusion radius to the maximum rise height is a simple function of ambient buoyancy frequency and time. In the case of a wind, we analyse the theoretical model for a rising plume to obtain predictions for the downwind volume flux, and subsequent lateral spread, in the limits of strong and weak wind. We identify a regime of very weak wind, which may be modelled as a passive advection of plume flow in the still case as a first approximation. Numerical solutions of a general model are presented which show that it predicts a peak in entrainment, and hence volume-flux growth, in the case of intermediate wind strength. We verify the crosswind model predictions of lateral spread, upwind penetration and entrainment by comparison with large-eddy simulations.