Decided to go a bit more into what Fujita's ground marking method was and what it was trying to do.
One feature of tornadoes is a transition between air flow in the boundary layer near the ground and that in the vortex above, which WS Lewellen termed the 'corner flow' region. Fluid dynamics ain't my field, but there are plenty of articles out there for those who do understand it. The upshot is that the near ground air flow in tornadoes changes direction rapidly from inwards to upwards. This seems to be part of why they're so destructive. It can generate high vertical velocities, so tornadoes easily lift objects in a manner that tropical cyclones and other extreme winds don't.
This abrupt change means that objects being carried in the flow near the ground can be left behind. The region of strongest corner flow isn't uniform due to turbulence etc. This means tornadoes have areas where debris are swept up and deposited. Some tornadoes have the right flow structure and encounter the right ground conditions (such as loose crop stubble) to leave behind debris piles that are more or less cycloidal, and it is those marks that various authors attempted to use to measure the wind speeds of tornadoes.
Regardless of the exact method, they involve assuming the region where a given mark is formed is compact enough that is can effectively conceived as 'drawing' it and that this is representative of the actual windspeed. Unfortunately the thesis I mentioned in my
previous post doesn't clearly answer these questions. But one interesting thing it does conclude is that unlike what Fujita thought, you don't need subvortices to produce cycloidal marks, nor do the marks necessarily coincide with them when they are present. The methods also involve assuming that the marks are regular i.e. maintain the same relative distance from the tornado centre (though Fujita seems to have a way around this) which like some other proposed non-standard indictors assumes tornadoes are axisymmetric (they're not).
Fujita defined the translational speed as
U and the tangential speed (speed of rotation) as
V, with the radius between the mark and tornado centre as
R. He defined the cycloid with the
x axis as the tornado path for
y = R sin ωt and
x = Ut+R cos ωt (
ω - angular velocity,
t - time from initial
x-axis crossing). Taking the rotation (turning) number for a mark
c = ωt/2r, substituting it into the cycloid equation and differentiating gets
dy/dx = (dy/dt)/(dx/dt) = (V cos 2πc)/(U–V cos 2πc) =
(n cos 2πc)/(1–n sin 2πc) where
n = V/U. A loop is created when
n exceeds 1 (which I think would technically make it a prolate trochoid) whose width can be defined as the difference in
x when the slope changes from ∞ to -∞, when
1–n sin 2πc = 0 or 2πc = sin^-1(1/n) . The time for this rotation angle is
t = 2πc/ω = 2πcR/V and using the first equation obtains

. The width of the loop (
w) is therefore

. Finally, the relative loop width (
W) can be defined

which varies only with
n.
Here calculating the wind speed seems simple provided the translational speed is known with reasonable accuracy, it's simply the tangential speed plus the translational speed,
U+V or
U+U*n.
However, Fujita continues. He describes a case "where a large number of suction marks appear, [so] it is very difficult to identify the loops produced by a single suction spot that has rotated more than once around the tornado center". It then moves to talking about loop shift (
s) and calculating relative loop shift
S = s/2R = 2πR/n = π/n, and that as a result s can be found from w. But I'm not sure why it's necessary to calculate
S considering wind speed can be found from
w only.
He then applies it to the Greentown tornado (L2). I had to read it a couple of times to understand it. By graphing out the estimated start and end times vs distance he shows that the speed of tornado family L was fairly steady, so takes the 62.5 mph as the tornado's translational speed (
U). By examining the track he concludes the marks made seven revolutions around the centre in the period shown in the photo (presumably this is where
s or
S comes in?). He measures the distance between the loop tops and the estimated centre of the track. Then he groups the marks by the seven locations and uses the calculated speeds from
w (I think to calculate and average time of revolution and from that the average tangential speed. This is the weakest part of the procedure for me, it seems to be for calculating an average. I'm not sure why it's necessary and it's not well explained, he talks about the variation in core size and the distance of the marks from the centre, but never explicitly says the averaging is to account for this. Finally he adds the translational speed on to arrive at a windspeed for the seven locations: 172, 176, 173, 180, 180, 173 and 166 mph.
(TBC)