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You are here: Experts > Science > Weather > Meteorology (Weather) > Tornado outbreaks as patterned mathematical processes
Meteorology (Weather) - Tornado outbreaks as patterned mathematical processes
Expert: Ryan Hastings - 11/2/2009
Question
QUESTION: Would you care to verify that
//sin x cos x-sin^2x =the cardioid
You see, this is a derivation of the balancing terms of the pressure field and horizontal velocity equations found in first year dynamic meteorology from
du/dt=-1/rhopartial pressure/partial x+fv
dv/dt=-1/rhopartial pressure/partial y-fu
The point is that past tornado outbreaks can be resolved to show that the cardioid is involved in placing storm cells in a cardioid series/sequenced process, hence, tornado outbreaks are not random but deterministic!
ANSWER: Well, first off tornado outbreaks are not random events at all. They are caused by a number of well-known atmospheric processes, and it's not uncommon for the possibility of a major outbreak to be recognized several days before the event. They are not random, but there is a limit to our ability to predict them more than a few days in advance because the atmosphere is a chaotic system.
The equations you mention are the result of rewriting the Navier-Stokes equations for an observer in a rotating frame and scaling them for the synoptic scale. As such, they're okay for getting a first approximation for large scale behavior; but they're mostly useful as a starting place for deriving quasigeostrophic theory. Certainly the mathematics of the occurrence of a tornado itself would require including the stress tensor as well as finding a way to treat turbulence. And for predicting storm cells you'll definitely need some form of the conservation equations for energy and mass and an equation relating vertical velocity to buoyancy.
I do not understand what you mean when you say that "the cardioid is involved in placing storm cells in a cardioid series/sequenced process."
---------- FOLLOW-UP ----------
QUESTION: You say that you do not understand that cardioid, being the solution to the pressure balancing terms, shows up as the method by which storm families are related to one another in an unfolding tornado outbreak. These storm families are related to one another by means of a series/sequencing process. The Browning Study of the May 26, 1963 Oklahoma process can now be understood as storms fitting the expected locations on the curve of the cardioid with that curve's known critical points. Therefore, the area involved in tornado outbreaks appears to be associated with the nature of the areal impulse inducing it in a 1:3 ratio, just as the circle(pressure field) is related to the cardioid half(tornado strike area). That is consistent with the Browning Study findings although not known to be at the time! Incidentally, the Navier-Stokes relationship you brought up concerns the relationship of atmospheric drag with bulk viscosity, a critical factor toward the limiting factor in inducing tornadoes!
Answer
We can try taking this a piece at a time.
1. I'm not sure what you mean by "solution to the pressure balancing terms." There are an infinite number of solutions to those equations (I don't know what you mean by the "pressure balancing terms" though), and they can take many forms. That's what weather--on the synoptic scale--is. Supercell thunderstorms and tornadoes are not, however, possible solutions to those two equations, because (a) there's no vertical velocity, (b) there's no continuity equation, and (c) there's no energy equation. Those two equations are simply statements of the conservation of horizontal momentum, and a lot of other processes are involved in causing storms.
2. The cardioid, as I understand it, is the trajectory of a point on a circle as that circle is rotated around another, fixed, circle. This is sort of like the Fujiwara effect, I guess, but I don't think that's one of the important effects of storm interaction. Cold pool-updraft interactions are going to generally be more important. To be fair, storm interaction is rather poorly understood; in fact I plan to spend the next several years of my life studying them intensely. I will not be using those two synoptic scaled equations to study them, though, because they are incomplete and irrelevant for storm-scale interactions. (For example, I'm going to throw out the Coriolis term, because it's completely irrelevant for the scales of storm interactions, which happen over twenty minutes or so.) Furthermore, on days of severe storm outbreaks, there's almost never anything happening on the synoptic scale than resembles the cardioid, if I understand it correctly.
3. I am not sure what the Browning study is. I looked up a few papers by Browning, but they were all from the 1960s and have either been updated--we've learned a great deal about supercells and tornadoes since his pioneering work--or become part of common knowledge (which is why I have never read his papers). I would appreciate a citation, though.
4. I'm not even really sure what that sentence that begins with "Therefore, the area involved..." even means. I think you're trying to pack it too full.
5. The equations you cite are ultimately derived from the Navier-Stokes equations, only ignoring viscosity, assuming constant density, rejecting every part of the stress tensor that is not from pressure, and rewriting it for an observer in a rotating frame of reference.
I'm sorry to sound so discouraging or antagonistic, but I think there's a combination of you misunderstanding some meteorological concepts and a whole lot of me not understanding what you are talking about. Have you written this up in a more complete way, starting from more basic explanations? Really, though, what it boils down to, is that if you can use this to actually predict cell or tornado position in real storms, then you'll have something of interest. Even if the curve happens to fit a few outbreaks that have happened before, that doesn't mean it has physical relevance.
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