Simple analytical model for fluid flow in "Mushroom cloud"

In potential flow theory there are simple analytical models (formulas) for velocity-field of elementary features (like source, sink, dipole, vortex etc.)

• Is it possible to write simple analytical expression for flow (i.e. velocity field $vec v(vec r)$) inside ascending mushroom cloud (i.e. bouble of hot air creating single toroidal vortex)?
• Is it possible to analytically integrate motion in such velocity field, in order to obtain explicit formula for time dependent tranjectory $vec r(vec r_0,t)$ of each test particle (each fluid element) in the flow depending on initial position $vec r_o$ and time $t$ ?

Note: In reality this will involve turbulences making the shape more complicated (more vorticies). I don’t want to model these turbulences,(see. Rayleigh–Taylor instability), just the single primary toroidal vortex on top of column of up-drafting air.