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NUMERICAL AND ANALYTICAL ANALYSES
OF A TORNADO MODEL
by
PATRICK ALAN SCHMITT, B.S.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
/ ^ ^ - i »
May, 1999
v
' ',.^.j ACKNOWLEDGEMENTS
> I would like to first thank the academy, I mean my parents, whose support has
been the driving force behind all my work. I wish also to sincerely thank Dr. Gilliam
and Dr. Shubov for providing so much of their time to aid me in this endeavor and
for the push to stay on track. I also acknowledge the Faculty and Staff for putting
up with my antics.
I feel I must thank some of my fellow teaching assistants individually. My room
mate Chris, for putting up with me these last few months. April for her ever sunny
and glowing opinions on graduate school. Leah for being herself, and being a good
sport. Clint for keeping the humor low and stupid. Scott for his decisive nature (pick
a degree!). Dan for being about as good a friend as a guy can have. Finally, I would
like to thank all the little people who I have crushed along the way.
11
CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT iv
LIST OF FIGURES vi
I INTRODUCTION 1
II NAVIER-STOKES EQUATIONS 2
2.1 General 2
2.2 Cylindrical Coordinates 3
III BURGERS-ROTT MODEL 7
IV STREAMLINES OF THE FLOW 12
4.1 Method 1: An Analytic Approach 13
4.2 Method 2: A Numerical Approach 16
4.3 Pure Burgers-Rott Model 17
4.4 Sources and Sinks 21
V CONCLUDING REMARKS AND FUTURE RESEARCH 31
BIBLIOGRAPHY 32
111
ABSTRACT
During the course of this thesis work, we will be studying and analyzing one of
the first tornado models derived from the Navier-Stokes equations for incompressible
fluid called the Burgers-Rott model. We will completely present the model and its
derivation, but will present this derivation autonomous from the original work. Our
goal is to consider all cases of the model, including those which due to physical
restrictions do not produce a tornado. In doing so, we will go beyond the original
work by using computer simulation to graphically interpret our results.
IV
LIST OF FIGURES
2.1 Cylindrical Coordinates 3
2.2 Translated Curvilinear System 4
3.1 The cylinder Q 8
3.2 The disk CR 10
4.1 Error Analysis of r versus z 16
4.2 Plot of the two methods together 17
4.3 Plot with Too = 2000, a = .02, i^ = 4 18
4.4 Plot with Too = 2000, a = .02, u = l 18
4.5 Phase plot with u = 6-solid, v = 4-dash, v = 1-dot 19
4.6 Plot with Too = 2000, a = .06, z/ = 6 19
4.7 Phase plot with a = .02-solid, a = .06-dash, a = .08 -dot 20
4.8 Plot of Too = 5000, a = .02, ly = 1 20
4.9 Phase plot with Too = 2000-solid, Poo = 5000 -dash, T^ = 7000-dot . . 21
4.10 Plot with Too = 6000, a = .009,6=-.005,1/= 3 23
4.11 Plot with Too = 6000, a = .02,6=- .005, i/ = 3 24
4.12 Plot with Too = 6000, a = .06,6=-.005,1/= 3 24
4.13 Plot with Too = 6000, a = .009,6=-.005, z/ = 1 25
4.14 Plot with Too = 9000, a = .009,6=-.005, z/ = 1 25
4.15 Phase plot with Too = 6000-solid, Too = 9000 -dash. Too = 11000-dot . 26
4.16 Plot with Too = 3000, a = .01,6= .01,1/= 5 26
4.17 Phase plot with ^ 27
4.18 Plot with Too = 3000, a = .01,6= .01,1/= 3 27
4.19 Phaseplot witha = .01,0 = .02 28
4.20 Plot with Poo = 3000, a = .01,6 =.01,1/= 3 28
4.21 Plot with Too = 6000, a = .01,6 =.01,1/= 3 29
4.22 Plot with Poo = 3000, a = . 0 1 , 6 = .005,1/= 3 29
4.23 Phase plot with 6 = .005,6 = .01 30
VI
CHAPTER I
INTRODUCTION
In 1948, J. Burgers presented a model for tornadic motion based upon equations
developed by Navier and Stokes that describe particle motion through an incompress
ible fluid. This work was later extended by Rott in 1958, and took the name of the
Burgers-Rott model for tornados. The Burgers-Rott model was one of the very first
models of this kind developed. At the time of its incarnation, very little was really
known about these violent storms, as is apparent in the assumptions made by the two
mathematicians. The most glaring drawback of this model is the fact that boundary
conditions have not been considered. In effect, making the tornado engulf the entire
planet. Temperature is another physical phenomenon that is not taken into account,
but is known to play a roll in the occurrences of tornados. Even though, there are
limitations to the reality of this model, it is important to understand as much about it
as possible before discussion of more detailed models should take place. In completely
analyzing the Burgers-Rott model, insight may be gained that will aid in the study
of more complicated models.
Our goal in writing this thesis, is to provide some graphical representation to the
streamlines of the Burgers-Rott model, especially the cases not solvable analytically.
We plan to use these graphs to visually explain the effects of individual parameters
on the system as a whole.
CHAPTER II
NAVIER-STOKES EQUATIONS
2.1 General
In order to study any type of tornado model, we must first begin with a discussion
of some basic fluid dynamics. We will be mainly interested in viscous fluid. When we
deal with these flows, we also need to take into account parameters such as pressure
{P{^^y,z,t)), viscosity (dynamical, /z ), density {p = constant for incompressible
flow), velocity of the flow (v), and external force (F) [2]. Using these parameters,
Euler argued that the velocity field of an "ideal" fluid would satisfy.
^ + ( v - v ) 7 ; = - v p - f /
where
p = —J = -• p p
Both Navier and Stokes later added a viscosity term to the flow to obtain the equations
dv
where
^ -i-(v-V)v-iyAv = -Vp + f at
ly = — (kinematic viscosity).
Together with the continuity equation, the Navier-Stokes system becomes
Tr + (v-V)v = -Vp + f + i/Av at ^ '
V 'V = 0.
If we assume the external body force is potential (i.e., / = — Vf/), the system
becomes
dv — + {v • V)v = uAv - V (p -\- U) ^ (2.1) V-v = 0
In this system, {v • V)v represents convection, and describes energy transport as
a result of particle motion, uAv represents diffusion in the system and describes
dissipation of energy [2]. System (2.1) will form the basis for the rest of the discussion
as we turn to a specific tornado model.
2.2 Cylindrical Coordinates
In this section, we turn to a study of models. In this work, we are interested
in studying solutions of (2.1) which exhibit rotation about a vertical axis. If we
think about the path a particle would take once caught in the tornado, we expect
to see a circular motion about a vertical axis. We naturally then want to be able
to describe this type of motion. For this reason it is useful to express the system
(2.1) in cylindrical coordinates, which better describe the circular motion about the
z-axis by combining the usual polar coordinates for x and y, and the usual z-axis.
The coordinates (x, y, z) translate to cylindrical coordinates using the substitutions
X — r cos 9, y = r sin 9, and z = z
r - lines, z,9 = constant
9 - lines z, r = constant
z - lines r.,9 — constant
z -Une
r - Une 0 -Une (circle)
Figure 2.1: Cylindrical Coordinates
At every point of R^ space we have three coordinate unit vectors, e'r, e*e, and e*
—* -*
Cr = cos i -I- sin ^ j —• —*
60 = — sin ^ i + cos 9 j
6r K.
We also have dcr ^ dee
= ee, = —e*r, and all other derivatives are equal to zero. The 09 ' d9
coordinate vectors e'r, e , and e* form an orthonormal system as seen in (2.2).
^0 AB,
zk = zez(R)
*• y
Figure 2.2: Translated Curvilinear System
With this and a lengthy calculation, the Navier-Stokes system (2.1) written in
terms In cylindrical coordinates, the Navier-Stokes system becomes
dVr ^ ^ ^ VQ -\- V • VVr =
r
V0Vr
dt dve dt
+ V ' VVQ +
-\- V • VVz =
dp^ f. Vr 2 dve dr \ ^ r^ 7-2 QQ
I dp ( . 2 dvr V0
r r d9 I dp
r dz -\-i/Av,
dt ^ d ldv0 dv, r dr r dr oz
(2.2)
(2.3)
(2.4)
(2.5)
Which is
dvr ~dt
d d d VI + Vr^ + Ve^^ + V,^- ]Vr- -^
dr
dp
dv0
dt
dr
d_
dr
dp
+ z/
dz dVr
dr
dr + z/
d9
ld_
r dr d_ d9
1 d
+
V0 -
1 d^Vr
VrV0
+ d\ dz^
Vr
dVr
r' 12,
r dr \ dr
1 d'^V0 d'^V0
'^^W'^'d^ V0_
dvz dt
d d d + VrTT + VOT^T: + Vzl^ Vz "
VI
dr dp dz
+ 1/
d9
1 d
dz dv.
r dr \ dr + 1 d^v, + d^v. V,
r2 d9'^ dz^
The continuity equation is of the form
I d . , dvz (rvr) + r dr dz
0.
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
Here Vr, V0, and v,, are the components of the vector v with respect to the basis e ,
60, 6,, i .e . ,
v{r, z) = Vr(r)6r + V0(r)60 -I- Vz(z)ez.
We also need the vorticity form of these equations. Vorticity is defined as w = V x -y,
and the equation for vorticity is
dw 'dt
-\- (v • V) w = (w • V) V.
The vorticity equations in cylindrical coordinates take the form
dWr d d d
dt + r a ^ + a + ^a '"^ d d
= I Wr-^-\-Wz— ]Vr-\-U Id f dw
dwf d d d r dr \ dr
VeWr
r I 1 d Wr d Wr Wr r^r- 1 + -r^rr^T- + r2 d9^ dz"^
dt + r a ^ + ^ ^ ^ + ^ ^ ^ r ^ + r
d d\ weVr = Wr-^ + Wz— V0 H h 1/
or oz J r
1 5 / dw0\ 1 d'^W0 d'^W0 r dr \ dr I r^ d9'^ dz"^
W0
dwz dt
d_ dr
d
d_ d
o' d
+ |^.Tr- + - . ^ + - . ^ l ^ .
= I Wr-^-\-Wz— ]Vz-\-iy 1 5 / dwz\ 1 d'^Wz d'^Wz r dr \ dr I r^ dz^ dz"^
We end this preliminary discussion by giving some relationships between the compo
nents of V and w that will be used later.
1 dv^ dvf, Wr =
W0 =
r d9 dz dVr dvz dz dr ' 1
w^. = -d . . dvr
d-r^'"'^ - -de
6
CHAPTER III
BURGERS-ROTT MODEL
In this chapter, we turn to the main object of interest in this thesis, the Burgers-
Rott tornado model. This model was first studied by Burgers (1948) and was later dv
extended by Rott (1959). We seek an axially symmetric (i.e., ^ = 0) stationary (i.e.
dv
'di = 0) solution to Navier-Stokes equations. Which implies that v(r, 9, z, t) = v{r, z).
Burgers had the idea to seek solutions to Navier-Stokes equations in the form
v{r, z) = Vr{r)er + V0{r)60 + Vz{z)ez (3.1)
Theorem: Solutions to (3.1) must be of the form
Vr{r)
veir)
Vz{z)
—ar +
r CXD
27rr2
2a{z — ZQ)
b
r
1 — exp ar 2u
c + -r
(3.2)
(3.3)
(3.4)
where a > 0, Too > 0, 6 and c are constants. The Burgers-Rott solution corresponds
to the case 6 = 0, c = 0. We present a proof given in [6].
proof: We first note the under the assumptions in (3.1) the Navier-Stokes system
becomes
Vr dVr dr
Vc dP
dr + 1/
dV0 VrV0 Vr— \-
dr = —z/
V
r dVz
^ dz dv,.
1 d
r dr
dVr
r
dP
r dr \ dr
d'^v.
dVr\ dr J
V0' fpZ
Vr
IY*£i
^y-
\ d , ,
r dr dz
dz ' ' dz^ '
0, (continuity equation).
We will also need the vorticity form of these equations where
dw
'dt -\- {v-V)w = (w •S/)v-\- i/Aw.
Writing this equation out we find that only one term provides new information:
dw, V ^ dr ^ dz
dVz V d ( dwz H r
rdr\ dr
where
w, Wz{r)
M'r) = --^{rv0{r))
Wr = W0 = 0.
The continuity equation (3.5) can be written as
1 d dv. r dr dz
= constant = 2a
which implies two things
Vz{z) = 2a{z- zo),
r
There are several possibilities for the Vr term dependent on 6.
6 < 0 =^ a sink of fluid on the z-axis
6 = 0 ^ a tornado
6 > 0 =^ a source of fluid on the z-axis
(3.5)
(3.6)
(3.7)
Recall, that the Burgers-Rott model corresponds to the case 6 = 0. We will focus on
this case and leave the other two cases of 6 for later discussion.
Consider Q, a cylinder with dO, its surface shown in the following figure.
Q
>• z.
Z2
Figure 3.1: The cylinder Q,
8
Recall, we are dealing with an incompressible fluid. This implies that the flux of
V through the surface is zero, if there are no sources or sinks inside Q.. The flux of v
through do. is given by
F = -'KB?2a{zi - ZQ) + 'KB?2a{z2 - ZQ) -\- 27rR{z2 - Zi)(-aR 4- —)
which are the flux of the bottom, the flux of the top, and area of the lateral surface
of our cylinder, respectively. So,
F = 27rR'^a{z2 - zi) - 27rR^a{z2 - Zi) + 27r6( 2 - Zi)
= 27rb{z2 - Zi).
Since Z2 ^ Zi, we conclude 6 = 0.
Let us now return to our proof. After substituting (3.6)-(3.7) into (3.5), we get
the following equation
dr^ +
v J r v dWz 2a —± + —wz = 0. dr V
Now, the fact that 6 = 0 gives us
rw"^ + 1 + 1/
I 2a w. H rWz = 0.
We rewrite this as
which is
or
rw'[ -h w^ + - {fw'^ -h 2rw^ = 0
(^^;)'+J(^v)' = o,
(rw'^) + - (r'^Wz) = Co.
Dividing both sides by r and using an integrating factor of exp ( ^ ) , we obtain
w.ir)exp(^)) =jexp(^
9
Therefore
vJz(f) = exp I ^ I I ci + Co / - exp f -r— ] ds ] . I\ 2v J \ J s "^ \2u
To obtain the Burgers-Rott solution and to avoid a strong singularity, we set co = 0.
This gives us
,2
Wz{r) = c i e x p ( ^ ^ ) .
and we compute the circulation v over the disk CR
R
(3.8)
Figure 3.2: The disk CR
(p V JCR
•27r rR
• dr = I I Wz{r)rdrd9 Jo Jo
• • / " = 27^C^ I exp ( —-— I rdr
'o \ 2i/
_ _ C J l - e x p ( —
Passing to the limit R oo. We obtain
27ny
a Ci = Foo < 00
where Too is called the circulation at infinity. So,
aTr Ci =
CXD
27n/
Substituting back into (3.8), we get
aV ^^(")^2 ' P
—ar
~2v'
10
Let us now show why CQ = 0. Consider the case when CQ ^ 0. We have
1 fas'\ = hir + - I — H .
2\2y J
So, we obtain a logarithmic term and a convergent power series (which can't be
summed in a closed setting). Hence, with Co / 0 the circulation over Cij -^ oo
as i? —> oo, which is physically meaningless since the whole vortex has an infinite
energy.
To finish the proof, we now compute V0{r) by first substituting (3.8) into the
equation
r ar
This gives us the following ordinary differential equation for V0(r)
I d . , ,. Food f-ar - — \rv0{r)) = - — exp
2
r dr 27rz/ V 2v
Multiply by r and integrate to obtain
Fonfl f (—as^ ''" = '-^l"'''^\-^^^'^^-Dividing by r, we have
Poo / . (—ar^W c oo
'" = ^rV-'''n-2^ )^r-Finally, when c = 0, we obtain the desired Burgers-Rott solution.
11
CHAPTER IV
STREAMLINES OF THE FLOW
This chapter is devoted to a numerical and analytical analysis of streamlines for
the Burgers-Rott model. A streamline is a curve in R space defined by a parametric
equation
R{t) = x{t)i-\- y{t)j-^ z{t)k
satisfying
where
v(x, y, z, t) = 'y^(x, y, z, t)i-\- Vy{x, y, 2, t)] -\- Vz{x, y, z, t)k (4.1)
is a vector field. Thus, we obtain the nonlinear system of ordinary differential equa
tions
x = v, (4.2)
y = vy (4.3)
z = Vz. (4.4)
In other words, we can think of a streamline as the trajectory of motion that a phys
ically infinitely small fluid particle takes. We again switch to cylindrical coordinates
and in doing so obtain a stream function which for the Burgers-Rott model has the
form
R{t)=^r(t)er-\-z(t)ez.
Recall equation (3.1),
V = Vr{r)er + V0{r)e0 -\- Vz{z)ez.
Combining the above formula for v(R{t)) with this expression, we get
dr _ dcr dz dt ^ dt dt
12
By the chain rule
Therefore,
Hence
dCr d9 = 17^9
dr
dt dt
d9 dz
' = Tt''+'irt"'^Tt'-
Vr = r
< V0 = r9
Vz = Z
(4.5)
We consider the system (4.5) for the Burgers-Rott model (3.2) - (3.4), and the
case 6 = 0. This system of ordinary differential equations will be solved in two ways:
analytically and numerically. We begin by looking for an analytic solution. We have
the Burgers-Rott solution
—- = — ar, a > 0 constant dr ~di (m_ 'di dz 'di
oo
27rr2 1 — exp ar
2y
— = 2a(2; — ZQ).
(4.6)
(4.7)
(4.8)
4.1 Method 1: An Analytic Approach
We note that because of the special structure of the equations, we can solve the
system of equations explicitly in terms of known functions. Namely, using the chain
rule, we can eliminate the t variable in the second and third equation to obtain
d9 ^ -Foe dr 2a7rr^
1 — exp ar' 2v
dz -2 — = [Z-ZQ).
dr r
Once we solve these equations for 9{r) and z(r), we note that from (4.7), r(t) = r(0)exp(-at)
(4.9)
(4.10)
(4.11)
13
so that
9{t) = 9{r{t)) and z{t) = z{r{t)).
Let us begin with the computation of the solution for 9. We have
d9 - r oo
dr 2a7rr^ exp ar 2z/
Which we rewrite as
d9= ^°° 2a7rr^
1 — exp ar 2u
dr.
Integrating both sides
TO _ p
de= ' h'=l 2a7rs^ 1 — exp as
2v ds.
Here we integrate from r to ro based on equation where we see that for a fixed initial
value ro the function r is a decreasing function of t. Thus for 9, we obtain
9 = -Foe r i - ^ ^ p ( - i ^ ) 2a7r /
ds. (4.12)
as In order to simplify this expression, we make the substitution w = ——, which converts
to
oo
ar r 0
2" 1 — exp{—w) 9 = — ^ / -—~"\' ~'dw. W'
(4.13)
This integral is not solvable in closed form, due to the fact that 9 cannot be
expressed in terms of elementary functions. So, we turn to special functions for
help. Our integral looks very much like the special function known as the exponential
integral. This special function is of the form
/•oo
Ei(x) = / t~^ exp{-t)dt. J X
The exponential integral is, by analytic continuation, a single-valued function in
the complex plane cut along the negative real axis [3].
14
Now, the main difference between the exponential integral and the integral we
wish to solve, are the bounds of integration. We can rewrite the formula in the form
9 = ^ ^ STTI/
•'- oo
/•oo
/ ar~
1 — exp(—w) dw
W'
STTU
r i- oo
- 1 lOO
SITU
which finally gives
w
1
(1 — exp{—w)) ar" 2u
W (1 — exp(—w))
/•OO
2v
-L /•OO
2v
1 — exp(—w) w"^
^ —exp(—w)
dw
dw 21/
OO /.OO
2v
W
— exp(—u;) w
dw
9 -r oo
87ri/
- 1
w (1 — exp(—w))
2v
2v
-Ei ar^ ~2iJ
Ei ar^ 2y
(4.14)
Since we will use the computer to calculate these exponential integrals, this will be
the stopping point in our calculation of 9.
We now turn to finding z(r). We have
dz -2 -7 - = {Z-ZQ)
dr r
and since this equation is separable we have dz = —dr,
Z- ZQ r
integrating both sides of this equation yields
ro + C ln(2; — ZQ) = —21ns
^ - ^0 = c ( ^ )
We assume that a tornado will be, for our purposes, starting on the ground. So we
choose 20 = 0 and go about determining c. Recalling (4.8), we have
z = cexp(2at)
15
but at time t = 0, z should equal z(0). Therefore,
z = z(0) exp{2at). (4.15)
4.2 Method 2: A Numerical Approach
To solve these equations using a numerical approach, we use the built in Matlab
Runge-Kutta solver, ode45. We carry out this numerical method of solutions in
order to compare the numerically generated answer with the exact analytic answer.
One reason for doing this is to test our numerical method, since for the more general
Burgers-Rott model no explicit solution is available and we must rely on our numerical
solution. In our figures, we have plotted both the numerical and analytical solutions
and with E0 and Ez denoting the absolute value of the difference of these solutions,
we have
E0 = 7.7047e-^
E. = 1.4380e-^
In order to obtain this picture in Figures 4.1 and 4.2, we have chosen the following
values for our parameters that work well numerically: Too = 1000, nu = 5,a = .009
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
•
• 1
•
•
1 \
\
\ \
10
Figure 4.1: Error Analysis of r versus z
16
This results in graphs that very nearly coincide when we look at a 3-dimensional
plot of these streamline calculations.
y-axis -10 -10 x-aids
Figure 4.2: Plot of the two methods together
4.3 Pure Burgers-Rott Model
We now return to the equations (3.2) — (3.4) which are
—- = — ar, a > 0 constant dr 'di d9 'di dz 'di
oo
27rr2 1 — exp ar
2P
— = 2a{z — ZQ).
This, as we recall, was a special case with 6 = 0 in the Vr{r) term. We wish to see
if changing the parameters causes any interesting occurrences.
Let us start with the viscosity term v. Viscosity is basically a determination of
how much friction occurs in our system, or in other words, how much energy is being
dissipated. We can see this by looking at some plots of our system. We refer to (4.2)
as a starting point. Let us now look at what happens as we decrease z/, the viscosity
term, and fix all other parameters. We reduce the value of i/ to 3 as seen in Figure
4.3.
17
y-axis -10 -10
Figure 4.3: Plot with Poo = 2000, a = .02, z/ = 4
As we can see, there are more revolutions about the vertical axis. This is due
to the fact that the fluid is providing less resistance as the particle is caught in the
tornado. We now look at what happens when we decrease i/ to 1.
y-axis -10 -10 x-axIs
Figure 4.4: Plot with Too = 2000, a = .02, z/ = 1
Again, notice the increase in revolutions about the vertical axis. We should also
look to how 1/ is effecting the other variables such as ^. To investigate this we can
look at the phase plots oir x 9
18
Figure 4.5: Phase plot with v = 6-solid, v = 4-dash, v = 1-dot
We see that as the viscosity is decreasing, 9 is increasing. Since i/ only enters the
equation (4.7), we expect that changing this parameter has no effect on r{t) or z{t).
We have learned that as z/ decreases, 9 increases more rapidly which accounts for the
more rapid rotation of the streamline.
We now turn to the parameter a, the upflow gradient [3]. Looking at our equations,
we see that as a grows larger, 9 decreases. So, let us again start with the parameters
corresponding to (4.2). We begin by increasing a to .06.
y-axis -10 -10 X-axis
Figure 4.6: Plot with Too = 2000, a = .06, z/ = 6
We see that the slope of our plot is increasing . It may be easier to see this from
a phase plot, so let us look to a phase plot of r x 9.
19
Figure 4.7: Phase plot with a = .02-solid, a = .06-dash, a = .08 -dot
We notice that as the upflow gradient increases, 9 is decreasing. We also see by
looking at (4.6), that the parameter z is increasing with a. If we look back to (4.2),
we notice that z has attained a value of 0.25. Where as with (4.6), we see that z
has surpassed this value, implying that increasing a, causes our system to shoot up
towards inflnity.
The last parameter we wish to discuss is Too, inflnite circulation. Logically, we
would expect an increase in Too to correspondingly cause an increase in the number of
revolutions about the vertical axis. We turn to some plots to analyze this. We again
choose (4.2) as our starting point. Now, let us increase the circulation to Foo = 5000.
y-axis -10 -10
X-axis
Figure 4.8: Plot of Foo = 5000, a = .02, z/ = 1
20
As we predicted the the number of revolutions has increased. We should also look
to the phase plots r x ^ to see what other effects this increase has caused. We have
a> 15
Figure 4.9: Phase plot with Foo = 2000-solid, Foo = 5000 -dash, Foo = 7000-dot
Looking at this plot oirx9, we see that 9 increases when Foo increases. Similar to
z/, this parameter only enters equation (4.7) and varying its value produces no change
to r{t) or z{t).
4.4 Sources and Sinks
Now, we consider a more general case of this solution with b 0. In this case, the
Vr{r) term in the Burger's-Rott solution is
Vrir) = —ar + -r
b > 0 => source
6 < 0 => sink
causing our system to take the form
dr ^ r X X — = —ar + - , a > 0 constant dt r '
ar^ 1 — exp
~di oo
27rr2 2z/
dz ^ . \ — = 2a(2;-2;o). at
(4.16)
(4.17)
(4.18)
21
This results in the ordinary differential equation for w
d^w dr"^
+ 1 - * 1 + ^ V I r V
dWz 2a — - H Wz dr V
= 0.
At this point, we introduce an ansatz
Wz (r) = r'/''^z(r)
with u being an unknown function. So, we now have
r''u{r) + I 1 - - i r r'^u{r) ' a + -
z/ r r^u{r)
1
+ 2 r''u{r)
Which reduces to
u H—ru V
= r:ir-(i+9. Cir
Now, solving for u we obtain
u = exp [-h')\h'^''AYv''y^'^'^^'^ Returning to Wz-, we have
Wz — C\r^ exp [-YA Looking back, we recall
-^(^^^W) = . W
We can solve for V0 (r) and obtain
VQ <"=?/ •S}^i) exp {-Y/) ds.
= 0.
In contrast with the earlier Burgers-Rott model, we are faced with an integral which
cannot be solved in closed form. Thus in the present example, a closed form solution
to the system (10), is not possible. Therefore we turn to a numeric approach. To
look at this numerically, we will first calculate the integrand as a power series, and
22
integrate term by term
Tnna V0(r) = oo^
27r z/r
Fona oo»
27r z/r
Fooa
27n/r
J I
HH) 1 - as' 1 as
2z/ 2!
2z/
1
+ 3! as ,n
2z/ + X ds
( a\^ s^^"> ( a\^ •ds
1 2-I-- a •s^+^ -F ;-f ^ n 2z//
,6+ + 2 + ^ 2z/(4-h^)" ' 2!(6 + ^)
We focus on the following cases: 6 > 0, 6 < 0.
When 6 < 0, it makes the term Vr less than zero. When 6 > 0, there are two
possibilities for the Vr term: r < W | or r > J \ . These result from the fact that we
are evaluating our function from ro to r. So, if we substitute ro for r in (4.4) and
set this equation equal to zero, we can solve for ro. This results in ro = A / - , and
therefore, there are the two possibilities for r.
When r > J \ , this makes the term Vr less than zero. We assume from this a
downward sloping graph or a negative slope. The other case (r < w ^) results in the
Vr term greater than zero. In other words, a positive slope. We now look at a graph
of these cases to see if this is true.
We start with the case 6 < 0. Again, we choose values for our parameters that
work together numerically. We have Foo = 6000,6 = —.005, a = .009, z/ = 3
<C!—:
. , . • • •
-10 -to
Figure 4.10: Plot with Foo = 6000, a = .009,6 = - .005 , z/ = 3
23
Looking at Figure (4.10), we see the graph is decreasing which agrees with the Vr
term being less than zero. This can also be seen by studying the a plot of r x z.
We again turn to a discussion of the parameters and their effect on the system.
Let us use Figure (4.10) as a reference point, and begin our analysis by varying the
uplfow gradient, increasing a to a value of .02
-10 -10
Figure 4.11: Plot with Foo = 6000, a = .02,6 = -.005, z/ = 3
Just as in the earlier example, the slope decreases, or 9 is getting smaller. To
illustrate this better, we again increase the parameter, letting a = .015
Figure 4.12: Plot with T^ = 6000, a = .06,6 = -.005, z/ = 3
It is now clear the effect of a on our equations when 6 < 0.
24
We shift our focus to the parameter z/, returning to a starting point of (4.10). We
will decrease z/, or cause the system to become more ideal. So, we set z/ = 1.
0.14^
0.12^
0.1^
0.08-
0.06 V
0.04-
0.02-
10
• " ^ ' ^ • _ : ^ -
\ (••-•••^i--~:__
. • " : . '
5 " ^ ^"^'^-illi:'
y-axis -10 -10
Figure 4.13: Plot with Foo = 6000, a = .009,6 = -.005, z/ = 1
We notice that the decrease in the viscosity once again results in an increase in
circulation about the vertical axis. We expect that another decrease in z/, will once
again increase circulation.
So, we turn to the parameter Foo,
-10 -10
Figure 4.14: Plot with Foo = 9000, a = .009,6 = -.005, z/ = 1
25
Figure 4.15: Phase plot with T^o = 6000-solid, Foo = 9000 -dash, Foo = 11000-dot
In order to look at the graphs with 6 > 0, we must change the values of our
parameters. We start with Foo = 3000,6 = .01, a = .009, z/ = 5, to obtain a starting
point for our graphs. These values for our parameters give us the following plot
y-axis
Figure 4.16: Plot with Foo = 3000, a = .01,6 = .01, z/ = 5
Looking at a phase plot, we can see that at the point y ^, we have a stationary
point.
26
0.7
0.6
0.5
0.4
0.3
0.2
0.1
•
•
: 1 • ' 1 1 1
;l ; 1 :l ; 1 i ^
•
•
1 1.5
Figure 4.17: Phase plot with ^
We also note from this plot that when r < J\, the graph is increasing between 0
and this stationary point. When r > w ^, the graph is decreasing from the stationery
point to oo. Hence, there are two separate cells.
We turn to the analysis of the effects of viscosity, circulation, and upflow gradient
on the system using (4.17) as our starting parameters.
Let us start by varying z/. We decrease the viscosity to a value of 3.
y-axis -2 -2 x-axis
Figure 4.18: Plot with Foo = 3000, a = .01,6 = .01, z/ = 3
Similar to the previous cases, decreasing viscosity has the effect of increasing cir
culation about the vertical axis.
27
We now investigate the effects of the upflow gradient, a. We increase a to a value
of .02. We begin by looking at the phase plot of r x z
0.14
0.12-
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Figure 4.19: Phase plot with a = .01, a = .02
Notice that the stationary point is moving towards zero as a increases. We
also point out that the slopes are becoming steeper. This can also be seen on a
3-dimensional plot with these values.
y-axis x-axis
Figure 4.20: Plot with Too = 3000, a = .01,6 = .01, z/ = 3
Once again, we state that an increase in Foo increase the circulation around the
vertical axis, and by looking at the equations, we know it increases the variable 9.
28
y-axis -2 -2 x-axis
Figure 4.21: Plot with F o = 6000, a = .01,6 = .01, z/ = 3
We finally, look to the effects of changing 6. We decrease 6 to .005
y-axis
Figure 4.22: Plot with Foo = 3000, a = .01,6 = .005, z/ = 3
We also look at a phase plot of r x 2;.
29
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Figure 4.23: Phase plot with 6 = .005,6 = .01
Notice that changing 6 causes a change in the stationary point. So, if we were to
increase 6, we would be in effect moving the stationary point to the right. Also, this
causes a change on 9
30
CHAPTER V
CONCLUDING REMARKS AND FUTURE RESEARCH
In this thesis, we took the pre-existing tornado model developed by Burgers and
Rott, and completely analyzed and studied its componenets. We also reproduce
derivation of this model independently of the original work. Our goal, to construct a
graphical representation of this model was succeeded through the use of Matlab. The
model we studied was shown to be flawed in some of its assumptions, which pushes for
the study of more developed models. More sophiscated models have been developed
over the past forty years which consider more physical occurrences than the Burgers-
Rott model. Of course, adding these extra parameters causes the models to become
much more difficult and maybe even unsolvable. The hope of the author is that the
work presented here will in some way aid in the solving of the more complicated
systems.
31
BIBLIOGRAPHY
[1] J.M. Burgers, "A mathematical model illustrating the theory of turbulence,"
Adv. Appl. Mech. 1, 197-199, 1948.
[2] A.J. Chorin and J.E Marsden, "A Mathematical Introduction to Fluid Mechan
ics,'' Springer-Verlag, New York, 1-31, 1979.
[3] V.I. Shubov and D.S. GilHam, "A Mathematical Analysis of Tornado Dynam
ics, " Preprint Texas Tech University.
[4] N.N. Lebedev, ''Special Functions and Their Applications " Prentice-Hall, Inc.,
Englewood Cliff, New Jersey, 30, 1965.
[5] N. Rott, "On the viscous core of a line vortex," Z. Angew Math. Mech., 9,
543-553, 1958.
32
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