The Cardioid: the basis for all tornado outbreaks???

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I have been a little slow and have, over the last few days, finally read the entire thread by Mr. Van Grun. While somewhat interesting to read, it is not viable or tangable in the aspect he is trying to promote. This has prompted me to start a new thread (as the old thread is now locked)
My hope is that we can keep this thread from going down hill by ligit. responses....NO DOUCHBAG NAME CALLING STUFF, PLEASE.

I believe what Mr. Van Grun is refering to is called or better yet in direct relationship with the Mandelbrot Fractal. Not necassarly the Cardioid, at least not in this situation...that is, tornado outbreaks, tornado life cycle and even tornadogenesis.
It seems somewhat easy, at least to someone who is in the mathmatical field, to apply the basic equation r = 2a(1 - cos θ), this equation is in reference to both polar coordinates however keeping in mind that both polars have to be thought of or included....it wouldnt/can not work any other way.

Here is the problem with his (Mr. Van Grun) thought process:

He has not calculated for both polars. Simply stated, the Cardioid is an envelope of all circles whos centers are based on a fixed circle. In motion, they pass through each other but, only one point on the fixed circle.

What he is trying to explain, I think, is the one full cusp of "that" circle.
That equation(s), which he had given would be:

x = (a + b) cos(t) - b cos[(a/b + 1)t] And x = (a + b) cos(t) - b cos[(a/b + 1)t]


The above equations he gave do not in any way fit the verbal schematic he gave. If memory serves me correct, (it has been a long time since my college days)....They represent the basic epicycloid as far as the circumference only. Not the "cyclic" movement he was trying to explain.

With his verbal schematic and layout, the Holly Colorado tornado 3/27/08
should have never happened. It would have been out of the next "phase" or cusps. This could also include the Bird City 3/28/07 tornado as well.

The outer circle has to be the same radius of the inner circle for his "thought process" to work correctly. It simply does not fit. No questions, no exceptions.

I guess one could make the argument that Greensburg and Hallam and the 5/3/99 event for that matter would and do follow the basic Cardioid pattern. But there is no way, at least that I know of that could allow such mass chaos to happen along a curve per say, that would allow us to know exactly where tornadoes/tornadogenesis will happen.

Just my thoughts on the matter.

Edit: I would be interested in hearing from Mr. Van Grun to see what he has to say about this. Of course, I do not have the apparent mathmatical knowledge he does, but I did have to study some of this in electronics while I was in college. I can only assume that he will respond (hopefully) and I ask anyone interested to do the same.
 
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Folks,

It is not that difficult to understand. All I have been saying is given most if not perhaps all tornado outbreaks, there seems to be the tendency for those storm families to fit a series/sequenced process whose pattern is based upon the cardioid. I showed this derivation from the simple pressure field/horizontal velocity balancing of first year dynamic meteorology. I am saying that from what I have seen there is a strong connection between storm families(call these, initial tornadoes of every new family of storms,) to be contained within a cardioid series/sequencing process. I cited for example the Depauw, Orleans, Fountaintown, and Xenia families of storms. Connect the dots and see a cardioid lobe. What I am suggesting though is profound in that those locations associated with the cardioid's critical points should have the preference of dominating other nearby storms. Keep in mind the phrase: THE REAL APPROACHES THE IDEAL. Just as the Ekman Spiral doesn't always look exactly like that found in nature, so, the cardioid half will not look perfect as well. Variations on a theme! In fact Holton has said that a perfect Ekman Spiral would be unstable!
 
One matter of clarification I would like on this theory is...what is following the Cardioid pattern? Is it the individual tornado tracks, mesocyclone tracks, supercell storm tracks...what? No one has yet specified what it is that is following those tracks.

Otherwise, I recall a quote from my mesoscale meteorology class from Richardson: "Big whirls have little whirls which feed on their velocity, and little whirls have lesser whirls and so on down to viscosity". I think this refers to a way of describing the different scales of motion (planetary -> synoptic -> mesoscale -> microscale -> ...). Any organized rotation has smaller turbulent eddies associated with it. It seems like this theory, and the discussion of the Mandelbrot set along with it, fits that description of a tornado consisting of bigger whirls with associated smaller whirls and smaller whirls etc.
 
It would be helpful to post or refer to a viewable diagram that illustrates your proposal; and a statistical best-fit analysis would be a good idea, too.

What I think you're asserting is that there is a fit between some sample and a non-linear distribution. This is hard to do with your eyeball.

The story I've always enjoyed is that the "father" of chaos theory and fractals, Benoit Mandelbrot, didn't develop the theories from mathematics but rather from his observations that many distributions in economics and demographics were not Gaussian, i.e. "Normal". Specifically as I recall, he was looking at the distribution of household income in the population while working at IBM.

Many natural phenomena don't follow a Normal distribution. In particular the Levy distribution family is often associated with Mandelbrot's work. The density function has a significant positive skew which could resemble to the eyeball a cardioid lobe for small samples.
 
Mr. Wolfson,

You are quite right when you suggest that this concept should be subjected to a critical analysis. My observations only point to what you suggest. As far as providing viewable insights and so forth, I can only suggest the events as many are on the web for people to review. I do not know about the Browning Study being readily available as I obtained that study from the US Technical Department many decades ago but if you are not familiar with it, you should come to know at least the history of this AFCRL study and its ramifications!
 
I do not know that this might belong on a different thread but in association of the cardioid matter, I bring up the mesocyclone. That unique cloud structure has caught my attention from a point of mathematical mimicking what might be called a hyperboloid of one sheet variation. My point concerns the volume of the mesocyclone cloud in a hyperboloid of one sheet nature because that volume is

piabH[1+1/3(H^2/c^2)]

The reason I bring this up is the 1/3piabH because that is the volume of a cone and don't tornadoes assume a shape of the cone? So, one starts with that cloud shape and volume of a mesocyclone and one has only to note that when the value H^2/c^2 goes to unity, one is left with the volume

4piabH/3

which could be interpreted as an updraft composed of right circular cylinder plus cone and note that it doesn't necessarily involve vorticity as a cause but might include it asn an effect. One can then take such a volume of a hyperboloid of one sheet and use it as a differential equation of the nature

y''+y'+piabH[1+1/3(H^2/c^2)]-4piabH/3=0

and solve! Any takers?
 
Oy! One at a time, Dave! (Short form: No. They are not always conical....)

On the first topic. If you would post a table of the variate data for the hypothesis you want to test, I'm pretty sure there are several people on ST that have access to a best-fit program and might be able to do that for you. My impression from what you've said is that the observations are "distance from... something", where the something is a track or centroid. But in any case you do need to get more precise to carry this much farther.
 
Alright, I am with ya!

Outside of the latitude and longitude plus time data, how could one set up such an analysis that would be satisfying? I had recommended the Palm Sunday outbreak of April 11, 1965. In fact, I would recommend repeating those isochrone profiles as found in the original Bradbury and Fujita, SMRP #86, University of Chicago study, of which the Monthly Weather Review story is based. Given the time plus earth grid coordinates, what would be satisfying?
 
Dave,

This might be very simple for you but you have to remember that you are talking to a bunch of people from diverse backgrounds. Some of us have had classes back in our college days that dealt with calculus and trig but have not used it in so long that we are only familiar with the terms but cannot remember what they mean. Some have no background. And some have the background but are having trouble figuring out where you are going with it. Heck, some of these guys finished college before Mandelbrot Fractals and Julia Sets were explored fully.

I must admit I'm kind of lost on what you are referring to. Are you referring to outbreaks that occur with extratropical cyclones? I guess I'm stuck with the visual representation of a cardioid pattern as tracing one point on a circle as it rotates around a fixed circle. Are you suggesting that there may be some relationship with helicity or vorticity in some predictable manner?
 
I appreciate that people on ST who are equipped to ask Mr. Van Grun questions that can help him communicate his ideas more precisely and productively are doing so. While I'm no mathematician, I have an open mind and I'm curious whether there might be something to Dave's cardioid theory. Since I have a personal interest in the 1965 Palm Sunday Outbreak, I'd be particularly interested to see that broken down from mathematical formulae into language and graphic illustrations that are accessible to non-mathematicians, if that is at all possible. I'm willing to step up the ladder as much as I can if the math-heads are willing to step down a few rungs. I understand that among mathematicians, equations are a form of language that might not be readily reducible to words and examples; just thought I'd put it out there. Remember, a lot of us, including me, don't have a clue what you're talking about! But speaking personally, I'm hoping to see it explicated enough on this thread that I can at least get a layman's gist of it.
 
Well, let's give it one more shot in small letters....

Dave, your hypothesis I think is that some spatial property of the tornadoes in outbreaks follows a cardioid distribution. So what one needs to test this is a table of the form instance, value, where value is an instance of the hypothesis you're testing.

If your hypothesis applies to individual tornado tracks, then you need to identify the coordinate axis and normalize the observations' distances from that axis as a function of the distance along the axis. You may be implying that the axis is a line connecting the initiation and dissipation points, but maybe not -- I still dunno. There are several other possibilities, or I'm off the track altogether :rolleyes: .
 
Bob,
Although I am well versed in the calculus (I just got a math degree and took many years of calculus courses) I myself am confused as to exactly what Mr. van Grun is trying to say. I haven't seen one picture yet. However, if it helps with the background material, check out the Wikipedia article on cardioids to get an idea for what the shape is and what parametric and polar equations can be used to graph them. If you have a graphing calculator like a TI-83 or 89, you can graph them there. They're pretty graphs.
 
Thanks everyone, I was about to crack some heads when I saw this thread re-appear tonight but I see a pretty good discussion at hand. Myself and the moderators will let the thread stand but we will be deleting noise or issuing warnings. Even though this is in fringe territory it needs to stay on track with a modicum of courtesy. If anyone feels it needs to be deleted for some reason, PM a moderator or myself (a mod would be best though) rather than derail. Thanks.

Tim
 
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