Deriving the Tornado Differential Equations

TORNADO DIFFERENTIAL EQUATIONS

Introduction

Many have asked me where the TDE, Tornado Differential Equations came from. In other words, they want to know how I derived them, so today I will do just that.

The entire purpose of the TDE's is to create a formula set that can describe a tornado or vortex without looking at its destruction. The Fukishima scale is nice, but it evaluates tornadoes based on their destruction, so it cannot be analyzed prior to impact, an ability which could save lives.

 Lets setup a few things first:

The X and Y equations must work in conjuction to create a circle, because if they are written incorrectly, they will create a Lorenz system.

You may wonder why not simply define x and y as:

(i.e. as a circle) and just make z=z and entirely eliminate the need for Diffs. Well the equation set written below will create an equation which will become unstable at extrema omnicron values( a system which I will describe towards the end of this entry). Tornadoes are also unstable when certain factors are strong enough:

Here is a .75 Omnicron core, which was built in a tornado simulator that I developed and may post about later on..75 is optimal, both from a physical and TDE solution standpoint.

Setup

We are going to have three coordinate variables and one for time. Time will simply step through numbers so we can say:

t=0...n

So this will be a 4 dimensional Diff but lets start with x and ẋ.

We need the x to rotate(curve) so:

ẋ=x+y

because the slope of the x axis is now t the y coordinate and the x coordinate combined we will have a curving bottom.

However now we need to fix the other direction so:

ẏ=x+y or ẏ=-x+y

but we need this to tornado to curve upwards as a function of either x or y. In order to do this we must add in a z term.

Thats easy enough:

ẏ=-x+y+z

 and to decrease the turning radius of the tornado we can add an xz term:

ẏ=-x+y+z+xz

so y turns tighter as you climb and turn, which competes with the ẋ equation.

Now it's z's turn, Tornadoes curve outwards in a parabolic fashion:

ż=-xy

but if we add a z term into the right hand side of numerator, we can keep all three equations as first order ODE's and acheive a faster curve because of the integration step used for solving ODE's:

ż=z-xy

Groovy, with that done we now have a framework for our ODE's, and we can begin solving for our coefficients of stability, which is a very important step.

To Summarize the previous step, and place all the space equations in a matrix we have:

Selecting the Input Variable

The goal of having a dynamic parameter within the TDE is to allow for simple manipulation of the equations by simply entering a number as a constant or coefficient. To determine the placement examine the equations and a vortex. Note that a vortex has a direct relationship between radius r at a height h, therefore we must simultaneously relate all three variables. However the equations are connected, each is solved by another, therefore we need modify only one equation. The ẋ equation contains only 2 variables(3 or 4 if you count the derivative and if you split dt and dx); therefore it is the natural choice for modification. All we need to do is simply place a coefficient, Ο, omnicron, in front of x because increasing this right hand expression, will increase dy which in turn decreases dz which causes the rate of height change to slow as the radius r becomes larger. Thus our tornado graph will be inverted but this is simply an aesthetic problem which is of no real concern.

Coefficients and Constants of Stability

The TDE Matrix

x=x-.5yy-(x^O)/O

Therefore as x increases it simultaneously decreases, as does y. The 3==>> x , and if 0<O<1, than:

is a graph of this( (x^O)/O) term as O is increased, therefore a user can drastically modify the tornado dynamics.

Our y and x term are related and create the vortex, so modifying one effects the other, therefore signs and magnitudes of coefficients are very important.

We need the Y to be heavy reliant on X and therefore:

ẏ=-2x+y+z+xz, because with Lorenz type system small coefficients lead to higher stability. The .01 y term is not irrelevant because of its, small magnitude, without it the X and XZ terms would be over powered, and the sinusoidal motion of the tornado would not occur. The xz term can be left with a 1 times multiplier, it is in the equation to assure that y is related to z and x, and by multiplying the two we achieve an exponential growth of the radius with respects to altitude. The Z term is kept small, but larger than the y term to place more emphasis on z in increasing y than on x, however x acts as a limiter of this exponentiation because its differential is not directly related to z.

Z is highly important, it allows for z to decrease exponentially as z is increased, and ensures that the increase in radius out paces the increase in height which achieves this effect:

Other Coefficients

There are coefficients within the equation that I have not yet shown a derivation for, these were discovered through experimentation with the Tornado simulator. In the following section I will demonstrate the Tornado Simulator and provide a synopsis of how these experiments lead to coefficients and constants.

Data and Analysis

The following data lists are explained in the presentation which follows them:

Core Diameters

19

15

56

23.3

23

118

91.4

30.8

30.4

42

80.1

75.1

52

48.3

102

28

Average= 52.15 pixels

Supersonic Blast destruction Method Data:

201.3 x of ellipse

78.1 pixels on y of ellipse

142 pixel blast length

35 pixel width of cone base

Supersonic Ellipse Evaluation Method:

Time(Minutes) and temperature(Celsius) in tornadoes center showing the cooling effect of the low pressure zone as well as the introduction of air.

1	21.06
2	21.06
3	21.06
4	19.81
5	19.88
6	19.88
7	19.88
8	19.88
9	19.88
10	19.81
11	19.81
12	19.75
13	19.75
14	19.75
15	19.75
16	19.75
17	19.69
18	19.75
19	19.69
20	19.69
21	19.69
22	19.69
23	19.69
24	19.69
25	19.69
26	19.69
27	19.69
28	19.69

Graphs of Each Function:(X,Y,Z)