Sloppy Model of a Tornado

DRAFT

CONE is the set of points specified by cone(u,theta) 0<=u<=1. The parameter u will be the up metric (z assuming ground xy). CONE represents the core of the tornado. That is the highest speed wall of wind. TORNADO is the set of points/n-tuples specified in the form t(r,theta,u,V) where V is the velocity vector at the point. We will computer V from speed and a normal of wind direction. SPEED is the set of speeds for all 3D points. SPEED will be determined from speed(r,theta,u). When we are in the area of interest of the cone, we can calculate the wind speed based on the following. Let r_cone be the radius of the cone for a given u value.

distance = abs(r - r_cone)
if distance < max_wallthickness/2
   speed(r,theta,u) = max_speed * (max_wallthinckness/2) * abs(r - r_cone) 
else
   speed(r,theta,u) = 0
endif

We will determine wind direction by overlaying a spiral over the cone and using the direction of the spiral as the wind direction. The basic formula for the spiral is
P_spiral = spiral(P) where P is a point of the form (r,theta,u)
P_spiral =(r,theta,???? [based on the number of twists in the cone from top to bottom.
Pick some step size delta_theta and for point P=(r,theta,u) calculate P_next=spiral(r,theta + delta_theta,u). The normal for velocity then is
N(r,theta,u) = | P_next - P |
If you would like to have a curved cone, describe a polyline pl with a parametric form pl(u) where 0<=u<=1. pl(u) = (x,y,z) so PL the set of points of pl is a line through 3 space. Now add to cone, speed and spiral the parameter ORIGIN a point (x,y,z) in space. Each point generated by the above functions will then be calculated relative to that point. Now do the calculations in the form
speed(r,theta,u,pl(u))
where the pl in the speed function is the added origin value.

CAVEATS

A cone does not have the taper you expect for a tornado. The wind speed model is probably hopelessly simplistic. If you follow a polyline or spine of the polyline, my model does not orient the cone cross sections with the direction of the polyline/spline. You would need to add a normal of the line to get that reasonable. This model does not change over time. It should both move though space and change it's dynamics overtime. However, that would require more guesses or information about how a tornado works. The moving thorough space should be easy, just add time to the polyline formula and determine some way to specify the changes in the line. You could morph from line to line over time.