Straight-sided solutions to classical and modified plume flux equations

The classical plume model due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1-23) is re-cast in terms of the non-dimensional plume radius, the plume 'laziness' defined as the squared ratio of the source radius and the jet length, and the buoyancy flux. It is shown that many of the key results of this classical model can then be read straight from the equations without recourse to solving them. Based on this observation, derivative models that consider plumes propagating through stratified environments or undergoing chemical reactions are similarly re-cast. We show again that key results can be read straight from the governing equations and results that have previously only been demonstrated numerically can be found analytically. In particular, we unify two previously distinct models that consider plumes propagating through stable and unstable stratified environments whose stratification has a power-law dependence on height. We present analytical solutions for the range of stratification power-law decay rates for which straight-sided plumes are possible. This result unifies the sets of solutions by Batchelor (Q. J. R. Meteorol. Soc., vol. 80, 1954, pp. 339-358) and Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229-248). We are able to explain the unstable behaviour previously found when the power lies in the range (-4,-8/3). Finally we show that this method also has limited advantages when applied to plumes with unsteady source conditions.