A tornado-like vortex equation...

This is a project I started this past Summer and here's what I got: a tornado-like vortex model, in particular, a steady-state Beltrami-flow cyclone in cylindrical coordinates that satisfies Dirichlet boundary conditions. A sketch of a similar derivation is in my last post.

The second image is my contour plot renderings showing each velocity component in the meridional r-z axis for arbitrary shear and circulation values. The two subsequent images (not by me) compare these to simulation results.

Seeing that the Beltrami condition seems to match the simulation results in Giovea, et al. (2025) [1] (pg. 19 and 25) and Liu, et al. (2020) [2] (pg. 9, 11, 13) given a no-slip condition at z=0, a laminar tornado may be a Beltrami flow type (though this is pure speculation).

However interesting though, a small decrease in the ground friction, Cd (drag coefficient), greatly increases a vortex's potential to break down into a two-cell vortex (see Sullivan's vortex (1959) and Bellamy-Knights (1970)). Relating ground friction (in conjunction with swirl, Sr, and Re) to the flow geometry has been explored by Serrin (1972), but required discretized FEM.