the formula
I've successfully found that $m = 2\ln(x^2+y^2) $ look like a really static and not moving tornado.
But in the same time with vector equation I've found how to twist a cylinder.
$ r = \sin(v+\pi*u) $
$f(x,y,z) = (r * \sin(\pi*u),r * \cos(\pi*u), v) $
adding time to the vector
if we add $\tau$ to $r$ the cylinder start moving like this
the goal
So my goal is to make $m$ rotate like a tornado-ish. maybe incorporating the vector to the equation is the way, but as of now i'm hitting a wall.
in short, the goal is the make the cylinder and $m$ look like a tornado, so it can spine like a bit like a tornado.
I do not know what you are after, neither do I know anything about tornado’s, but perhaps something like an unstable spiral might be easier to describe such a shape. To be specific, I used a system of the form \begin{align} \begin{pmatrix} \dot{x}_1\\\dot{x}_2\\ \dot{x}_3 \end{pmatrix}= \begin{pmatrix} a & b & 0\\ -b & a & 0 \\ 0 & 0 & c \end{pmatrix}\begin{pmatrix} {x}_1\\{x}_2\\ {x}_3 \end{pmatrix},\quad b>a,a>0,c>0 \end{align} to have a growing unstable spiral. In the image I used $a = 0.05,b = 2,c = 0.01$, with an added sine input on the first coordinate to create the distortion ($2*\sin(t/10)$).