Quadratics on the Sun

Thursday, I read (via @wired) about a really interesting tornado on the surface of the sun.

This twister was reported at nearly 186,000 mph with temperatures “between 90,000 and 3.6 million degrees Fahrenheit.”  Pretty stunning.

I thought the twisting winds might make a really interesting multivariable calculus problem, but a physics colleague, John Burk (@occam98) asked, “I wonder what the F-number is on a tornado with 186,000 mi/hr winds?”

OK, a new direction.  I started by finding the Fujita Scale for tornado categories:

F0

F1

F2

F3

F4

F5

F6

40-72 mph

73-112 mph

113-157 mph

158-206 mph

207-260 mph

261-318 mph

319-379 mph

Reminding me that the damage caused by a tornado is connected to the square of the velocity of the winds.  John found a “phenomenal quadratic fit” to the Fujita Scale for tornadoes by using the midpoint of each range.  I repeated John’s analysis and found basically the same results.  I also added a residual analysis.

There are very few data points here, so I shouldn’t have been all that surprised by a residual pattern.  As John’s analysis suggested, the quadratic is very close to the data points–residuals\in{[-0.607,0.476]}–a very small interval relative to the dependent range.  But as a math teacher, I began to wonder.

  1. How you would set the range of each category if you use midpoints?
  2. Can the fit be any better?
  3. Once you get an equation, do the coefficients tell you anything?

I’m not sure how to answer the first question in any sort of non-arbitrary way, so I turned my focus to two other approaches:  using the minimum and maximum wind speeds for each F-category.  Those graphs and their residuals follow.

So all three appear to fit the F-scale data very well, with maybe a slightly less obvious residual pattern in the min and max curves (although I wouldn’t stake any deep claims on that assertion based on so few data points). The residuals range for the minimum speeds is residuals_{min}\in{[-0.929,0.667]}, and is residuals_{max}\in{[-0.429,0.571]} for the maximum speeds.  If forced to make a call, perhaps the maximum speeds are better for their smaller overall residual range and possibly less defined pattern.  Also, using the equation for the maximum (or minimum) wind speeds avoids the vague endpoints issue for the Fujita scalings the mean wind speed approach encountered.

I don’t recognize anything about the coefficient values, so I tried converting the three equations into factored and vertex forms with nothing really enlightening there either.  I don’t know what I was hoping to find, but none of the coefficients in any of the forms seem to be sharing any secrets.

For now, I think I’ve run to the end of my reverse-engineering of the Fujita Scale.

Returning to the solar tornado that inspired all of this, I used my CAS to solve for the minimum and maximum Fujita scales based on these data, getting that the solar tornado would be rated somewhere between an F-270 and an F-285 tornado if it happened on Earth.  Wow.

In the end, perhaps it’s time for me to study the development of the Fujita Scale, but I’m pretty convinced from these tight fits that it was not a purely random equation.

In the meantime, I hope you find here a (not so surprising) connection to quadratic functions and possibly something to provide a deeper connection between mathematics and science–something woefully underrepresented for too many students.

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