Sasaki’s Entropic Balance Tornadogenesis:
Where We Are, How We Got Here, and Where
we are Going
William Frost
Dr. Douglas Dokken, Advisor
September 3, 2016
Contents
1 Introduction
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
2 Introduction of Concepts
2.1 Mathematical Concepts . . . . . . . . . . . . . . . . . . . . . . .
2.2 Meteorological Concepts . . . . . . . . . . . . . . . . . . . . . . .
2.3 Parts of a Storm . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
4
5
3 The
3.1
3.2
3.3
Tornadogenesis Attractor
Sasaki’s Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . .
How to Proceed . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Smale-Williams Attractor . . . . . . . . . . . . . . . . . . .
7
8
9
12
4 Future Work
13
5 References
15
1
Introduction
Dr. Y.K. Sasaki was an eminent meteorologist and researcher. For the field
of tornadogenesis, he proposed his Entropic Balance Theory, provided a new
physical formalism for tornadogenesis, and suggested an interesting and unusual
connection with quantum field theory. The most relevant to our investigations
was his Entropic Balance Theory, which restructured the classical equations of
velocity, vorticity, and helicity in terms of entropy, allowing the developement
of tornadoes to be understood in terms of potential temperature gradients. His
work was the basis for the majority of this research.
1
1.1
Problem Statement
We want to examine the mathematical consequences of Sasaki’s Diagnostic
Euler-Lagrange equation as derived in his paper Entropic Balance Theory and
Variational Field Lagrangian Formalism: Tornadogenesis and by Swenson and
Theisen in Tornadogenesis: The Birth of a Tornadic Supercell and elaborate
the idea of the tornadogenesis attractor by explaining the ways it appears and
proposing a new part of it. Then, we want to establish a thorough framework for
future research by clearly explaining Sasaki’s ideas involving Entropic Balance
and Quantum Field Theory.
2
Introduction of Concepts
Tornadogenesis has a lot of math and science behind it and a corresponding
amount of jargon. The mathematical concepts behind tornadogenesis are fairly
basic for anyone familiar with vector calculus. For those not familiar with vector
calculus, the trick is that its concepts are a lot easier than they will ever seem
from a textbook description. For the meteorological side, all that is required is
that one understand the small list of basic terms that all the others are based
on. If those endorsements makes you tremble, then fear not, because all effort
spent here is to understand tornadoes, and is thus worthy of your time. The
purpose of this section is to create an easy to understand index of necessary
terms, organized by category.
2.1
Mathematical Concepts
This section introduces the mathematical concepts as well as their use in this
topic.
Vector The most mathematical concept we have here is the vector. A vector
is a directed quantity. These are often represented as arrows, where the bottom
of the arrow is the point the vectors at, the length of the body is proportional to
the magnitude of the vector, and the direction of the vector is the direction of
the arrow. Vectors are a very convenient way to represent quantities like force
and velocity, which by definition, have both magnitude and direction.
Vector Field If we associate each point in a space a vector, we have created
a vector field. In meteorology, wind is a vector field. After all, every point in a
storm is blowing somewhere. Naturally, these are much more complicated than
vectors because we have to talk about the behavior of the entire set of vectors,
but that also allows us to do some interesting things. One important thing we
do is track how a thing would move through the vector field. When the vector
field does not change, this is fairly simple: just pick a point, see where the
vector points, follow it, see where the vector at that new point is going, and
repeat until you have as much information as you need. The paths formed in
2
this manner are called trajectories, because it is the path that an object would
actually follow.
Vorticity The next important mathematical concept is vorticity. Vorticity is
defined as the curl of a vector field (∇ ×V ), and is represented by ω. Intuitively,
vorticity is the amount the vector field is rotating at each point. It is calculated
with a cross product, so you can use the right-hand rule to get a rough idea is
going on. Curl the fingers on your right hand in the direction of the vectors
in the field around the point you are concerned with, and your thumb will
be pointing in the direction of the vorticity. The magnitude of a cross product
increases proportionally to the magnitudes of the components, and to the sine of
the angle between them. When meteorologists talk about vertical vorticity, that
means the air in that area is rotating counterclockwise parallel to the ground
when viewed from above.
A vector field exhibiting large vorticity out of the page
Helicity Helicity is the next step beyond vorticity. Defined as the dot product
of vorticity with the vector field it came from (ω · V ), helicity can be thought
of as the entanglement of the velocity and vorticity vectors at a given point, or
in a storm, the degree to which the air is moving in a helical fashion. If you are
unfamiliar with the vector dot product operation, it’s value increases directly
with the values of the factors, just like an ordinary product, but also with the
cosine of the angle between them, meaning that when both vector quantities are
pointing in the same direction, the value is simple the product of the factors,
and also points in that direction.
Attractor This concept is the trickiest, and also the most important. An
attractor is defined in the study of dynamical systems as the state towards
which the system evolves from a wide variety of starting conditions. Points of
equilibrium are attractor. In the case of a marble in a bowl, the bottom of
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the bowl is an attractor. However, not all attractors are that simple. Chaotic
systems produce complex, fractal attractos called ”strange attractors.” In the
context of physical systems, attractors represent the ideal system. Attractors
appear in phase space, a space which has all the parameters of the system as
axes, not just the position. By understanding the properties of attractors, we
can understand what a system ”wants” to do.
2.2
Meteorological Concepts
Meteorologists often use the following terms to describe the concepts and behavior of the weather (AMS).
Process Any of the things we talk about going on in the atmosphere that
cause weather are properly called a ”process.” Any time an amount of air moves
up or down, side-to-side, gets hot, gets cold, or spins in figure eights, some sort of
process is going on, and meteorologists being scientists, every one has a name1 .
Parcel The most basic unit of air that can undergo a process is called a parcel.
Parcels are without a definite size or shape. In fact, they are totally abstract,
and exist primarily to illustrate meteorological concepts in simple thought experiments, making parcels a powerful tool.
Barotropic Process Under all conditions we regularly experience in the atmosphere, air behaves like an idea gas, following the Ideal Gas Law P Vn = RT .
In ideal atmospheric conditions, there are not disturbances, and pressure decreases smoothly with altitude. Any process done with such a smooth pressure
gradient is called barotropic. This is espcially significant when calculating vorticity, as in barotropic conditions, horizontal vorticity will be zero.
Baroclinic Process Such smooth conditions only occur in very strange circumstances in the real atmosphere. Normally, the pressure gradient is sloped,
so that if you drew lines of equal pressure, called isobars, they would not be parallel to the ground. This condition creates horizontal vorticity, called Baroclinic
Vorticity, which can feed storms.
Adiabatic Process The simplest process that can occur is one where no
energy is added. The physical properties of a parcel can still change, because
pressure, volume, and temperature all balance each other out, so any variable
can change if the others balance it out. Such a process is called adiabatic. The
key sign of an adiabatic process is that it is reversible, meaning that if you
move a parcel of air adiabatically, then bring it back to its original position, its
temperature, pressure, and volume will all be the same as they were originally.
1 air moving in figure eights has not yet been observed, and as such, the author stresses
this is purely for example purposes, and should not be taken to in any way reflect current
meteorological thought
4
Potential Temperature This concept addresses the main problem brought
on by the Ideal Gas Law: with three variables in equilibrium with each other,
it is very hard to compare any parcels of air to each other. Because the ability
to compare things is very useful, meteorologist created Potential Temperature.
The potential temperature of an air parcel is the temperature it would have if
it was brought adiabatically down to a constant pressure. This means that if
we only look at constant molar quantities of air, we also eliminate the pressure
variable, leaving only temperature to vary, allowing quantitative comparison.
Potential temperature is calculated with the following formula:
cR
P0 P
θ=T
P
Where θ is potential temperature, P is pressure, P0 is the pressure at our
constant pressure (traditionally sea level is used where the is pressure 1 000
kPa), R is the Ideal Gas Constant, and cP is the specific heat for constant
pressure. Potential temperature can also be directly related to entropy by this
equations:
S = cP ln (θ) + S1
Where S is entropy, and S1 is the entropy of the water vapor mixed in with
with air.
Latent Heat Air gets colder as altitude increases. This gives rise to a very
interesting result. Air in the troposphere, the part of the atmosphere where
weather occurs, has some level of moisture in it. As that moisture gets colder,
it condenses, and eventually freezes. However, because energy is conserved in
a closed system like a parcel of air, the air must have taken on the energy the
water lost as it went to a lower energy state. The air may be getting colder,
but water condensing and freezing more than makes up for it. When the air
becomes more energetic, its pressure decreases, and less dense parcels rise. This
process, called Latent Heat Release, gives updrafts a powerful kick if they can
lift air high enough its water condenses, and another when it freezes. This
means strong storms have tools to get even stronger.
Streamline When a vector field evolves with time, as the wind in a supercell
does, it becomes a lot more complicated to understand how things move through
it. One thing we can do is pretend the field does not change with time, and take
the trajectories at a single instant in time. We call these paths streamlines. In
meteorology, streamlines are useful because they give us an idea of what is going
on in a storm or a simulation, and we can more easily perform calculations on
a single instance of time.
2.3
Parts of a Storm
Most tornadoes are caused by a particular type of thunderstorm called a Supercell. Supercells have broadly regular structure. Theses structures are identified
5
below, as is their importance in tornadogenesis (AMS).
Mesocyclone The mesocyclone is the most characteristic part of the tornado.
Also called the ”Rotating Updraft,” the mesocyclone is the core of the supercell.
It is medium scale because it is smaller than the supercell overall, but is much
bigger than features like tornadoes. It is fed by low-level warm, moist, air and
the rotation it carries with it.
Forward Flank An important source of that warm air and vorticity is the
forward flank. On the forward flank, wind shear before the storm forms causes
coils to form. When the supercell travels onto these coils, the low pressure
mesocyclone sucks these coils up. Sometimes a coil will appear as a tail cloud,
a horizontal rotating cloud at the base of the storm. These tilt upwards, contributing their moisture and rotation to the storm.
Overshooting Top Air carried up by the mesocyclone gains a large amount
of vertical velocity as it rises up to the top of the troposphere, the lowest part
of the atmosphere where all weather takes place. This maximum height, called
the tropopause, varies depending on latitude and local temperature from about
30 000 to 56 000 feet and is responsible for the flat top of the anvil cloud.
However, the momentum imparted by the mesocyclone carries air directly over it
beyond the tropopause creating the Overshooting Top. This feature sometimes
carries air up to 70 000 feet high. Like anything thrown against gravity, this air
has to eventually come crashing down.
Rear Flank Downdraft The most important supercell feature in tornadogenesis is the Rear Flank Downdraft. The Rear Flank Downdraft, is exactly
what its name suggests: a downdraft on the rear flank of the supercell (the
side opposite the direction of the motion). Also called the Hook Echo, for its
apperance on reflectivity radar, it is fed by air falling out of the mesocyclone,
and rotates. Theodore Fujita, the meteorologist behind the Fujita (F) Scale of
Tornado intensity, first formulated the Recycling Hypothesis in 1975 (Fujita).
He proposed that the RFD caused tornadoes to form through three steps: first,
when the RFD reaches the ground, some of the air that splashes on the ground
is sucked back into the updraft, creating a convergence zone near the updraft,
which will feeds it further; second, cold air and rain descend into the convergence zone, carrying with them angular momentum rom aloft; finally, as that
moisture and angular momentum are recycled into the updraft, they add their
energy and tangential acceleration, respectively, building the tornado (Kis, et
al.). The original formulation has been criticized over the years and alternatives suggested, but the RFD’s critical role in tornadogenesis has been widely
reaffirmed.
Hierarchy of Vorticies Supercells are characterized by a Hierarchy of Vortices. At the highest level, the Hook Echo wraps through most of the storm.
6
The mesocyclone is very large, but focuses in the single low-pressure area in
which tornadoes appear. Tornado vortices appear, possible several at once, rotating around the mesocyclone. Inside each tornado vortex, there can appear
even and more powerful vortices called suction vortices (Church, Snow, Agee).
These are what cause the stories of a tornado ripping apart one half of a house
and leaving the other intact. At each change in scale, the volume of the vortex
decreases, but the force increases.
Image from Church, Snow, and Agee
3
The Tornadogenesis Attractor
The process of tornadogenesis appears to have an attractor behind it when
viewed from different perspectives. The simple fact that tornadoes come from
very regularly-formed supercells implies the existence of some dynamics leading
to that shape from initial conditions as widely varying as the locales and weather
conditions that spawn supercells. If this attractor could be found and formally
defined, it would be greatly useful in the study and prediction of storms, as
it would allow observations to feed directly into a pure mathematical model
to return likelihoods and severity of tornado formation (”Balance Theory and
Tornadogensis”).
This section first introduces the work of Dr. Sasaki leading up to the conclusion that an attractor exists in the process of tornadogenesis. Then, the
topology of attractors is introduced to allow readers to appreciate the value of
the Smale-Williams Attractor in relation to tornadogenesis. The attractors evidenced by both arguments are not necessarily different, as the arguments both
have flaws preventing them from fully describing a tornadogenesis attractor.
7
3.1
Sasaki’s Helicity
In this section, I will explain how Sasaki’s work revealed an attractor. First,
he formulated an Euler-Lagrange density function, and added the constraints
of conservation of mass and conservation of entropy. This equation distrbutes
mass and energy across the domain of space and time the supercell will exist in,
called Ω. Because we know that both mass and energy are conserved in closed
system, we define that domain to be large enough that the storm will be closed
so the extra constraints make sense. Recall that potential temperature makes
relates entropy and temperature, and that temperature is the energy conserved.
The Euler-Lagrange density equation looks like
1 2
|v| − U (ρ, S) − Φ − α {∂t ρ + ∇ · (ρv)} − β {∂t (ρS) + ∇ · (ρvS)}
L=ρ
2
This equation contains the ordinary physical potentials of kinetic energy ( 21 |v|2
with mass provided by ρ), potential energy (U ), and gravitational potential
energy (Φ). Conservation of mass and entropy are represented by the two terms
with greek multipliers before them: the α-term is conservation of mass, and
the β-term is conservation of entropy (”Balance Theory and Tornadogenesis”).
The form of these terms is a direct result of the divergence theorem applied to
moving fluids; all conservation laws have similar forms in this context. This
equation is then used to find the Action Lagrangian by integrating it over Ω like
this
ZZZZ
L=
Ldx3 dt
Ω
The Action Lagrangian contains every possible configuration of mass and energy
across Ω, but that tell us nothing about what will happen. Luckily, the physical
principle of Least Action says that the system that appears will be the one with
the least action. However, as this is a multi-variable equation, we will actually
get five equations of minima, one for each variable: v, S, ρ, α, β. The calculus of
variations is the tool used to find our state of least action. The five equations
produced are:
δv L = −∇α − S∇β = v
(1)
δS L = −US (ρ, S) + ∂t β + (v · ∇β) = 0
1
P
δρ L = ∂t α + S∂t β − |v|2 − U (ρ, S) − − Φ = 0
2
ρ
δα L = ∂t (ρv) + ∇ · (ρv) = 0
(2)
(3)
δβ L = ∂t (ρv) + ∇ · (ρvS) = 0
(5)
(4)
The consistency of this approach with classical physics is proven if the five
equations produced can be worked in a form of Newton’s Second Law. Here, a
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form of said law called an Euler Equation is produced2 :
∂t v + (v · ∇)v +
∇P
+g =0
ρ
(6)
For a simple understanding, see that the first term is clearly acceleration (the
second one is as well, though less obviously), and that the third term represents
mass, and the fourth, force. However,for our purposes, all but the first equations
are ignored, as they are prognostic, or time-evolution, equations, whereas the
first is a much more useful diagnostic equation relating fundamental quantities
to each other regardless of previous state or time frame.
The next critical concept in Sasaki’s writing is relative helicity. Defined as
helicy divided by the product of the magnitudes of the component velocity and
vorticity vectors, relative helicy will increase as a tornado forms (”Variational
Field Formalism”). Because of the definitions of dot and cross product, the
following is effectively a re-phrased the Pythagorean Identity sin2 θ + cos2 θ = 1:
2 2
v·ω
|v × ω|
+
=1
|v||ω|
|v||ω|
Its significance becomes clear when vorticity and its time-derivative are rewritten using the diagnostic entropic definition of velocity produced by Sasaki’s
Action Lagrangian as ω = 21 ∇S × (−S∇β) and ∂t ω = ∇ × (v × ω), respectively.
As relative helicity is maximized, the second term goes to one, which implies
the first term goes to zero, which implies that the vorticity ceases to change
with time, which implies that a steady state is reached. This implies firstly that
tornadic supercells will be longer-lived, and secondly that a non-linear attractor
could be behind this process.
These equations do not describe an attractor. Rather, because they describe
circumstances where the attractor will appear, we only get some insight into the
phase space where the attractor exists.
3.2
How to Proceed
Sasaki used physics and the calculus of variations to support the existence of
an attractor in tornadogenesis. This paper seeks to support the existence of
a tornadogenesis attractor using physics and topology. To do so, this section
will explain the topological principles behind such an attractor using a simple
example.
The Logistic Map One of the simplest attractors in existence is called the
Quadratic Family, or the Logistic Map (Chorin). It has served widely to introduce the subject of dynamical systems and attractors, and it will serve thusly
here. The Logistic Map is based on this quadratic family of equations:
Fµ (x) = µx(1 − x) = µ(x − x2 )
2 Full
derivations of equations 1-6 are found in Swenson and Thiesen, 2015
9
It is called a family of equations because of the family parameter, µ, which will
be consistent for any given instance of the equation, the behavior of the equation
as it is varied is the prime focus, instead of the changes of the value along the
x-domain. Dynamics are produced by observing the behavior of a single point
on the real number line under discrete time iterations by mapping the chosen
point the point given by the function Fµ (x) so that xt=1 = F µ(xt=0 ).
While this map illustrates many important features such as bifurcation, and
is very interesting to study for smaller µ-values, this paper is most concerned
with the example provided when µ > 4. In this circumstance, a lot of boring
dynamics can be eliminated. First, any point where x < 0 maps to lower and
lower values, rushing off towards negative infinity. Any point where x > 1 maps
down to a point less than zero, where it again rushes off to oblivion. Further,
when x = 0, it stays forever, and when x = 1 it maps to zero, where it again
stays forever. Inside the range [0,1], dynamics can be a lot more interesting.
Points will bounce back and forth for some time before getting boring. However, first observe the middle section. Every point on the open interval where
Fµ (x) > 1 will map to a point greater than one, and then on to negative infinity. Furthermore, there are two more open intervals on [0,1] that map to that
open interval. These intervals map out of [0,1] after two iterations. A pattern
emerges: every iteration maps out 2n more, smaller, open intervals. If this is
repeated infintely, nothing will be left but a scattered dust of points, called the
Cantor Set, and represented by Λ. Expressed more formally:
!
∞
[
Λ = I \ lim
Ai
i→∞
i=1
Where I is the interval [0,1] and Ai is all of open intervals that will map out of
[0,1] after i iterations.
While the utility of the Logistic Map’s example may be hard to grasp, it
exists. Beyond illustrative purposes, even its simple one-dimensional dynamics
10
show similarities to parts of the supercell. Namely, each iteration maps out
points, and the supercell does something similar at the top and at the bottom,
where air mass is is lost during recycling. We do not mean to suggest that the
Logistic Map is the attractor Sasaki’s physics imply, but rather to provide a
concrete example of how attractors work in storms, and to introduce concepts
critical to attractors.
Fixed Points We have encountered fixed points already. They are simply
those points which map to themselves. Fixed points are important to the dynamics of attractors as they correspond to equilibrium points. We say a fixed
point is hyperbolic if f 0 (x) 6= 03 . Further, if f 0 (x) > 0, we say our fixed point is
repelling, and if f 0 (x) < 0, it is attracting. Graphically, any point that intersects
the line F (x) = x is a fixed point.
Periodic Points Periodic points are like fixed points in that they are found
in that they are easy to keep track of. Periodic points do not map to themselves
every iteration, but always return to their location after a fixed number of
iterations. Formally put:
P ern (F ) = x ∈ I|Fµn (x) = x
This set contains 2n points, easily proven by observing that every iteration n
creates an equation of order 2n. Fixed points can be thought of as periodic
points with a period of one. The set of periodic points with a given period will
also contain the periodic points with the period if its factors.
Eventually Periodic Points A further subset of all the elements of Λ is the
set of eventually periodic points. These points are all of those points which
begin by moving around Λ in a non-periodic manner, but eventually map to a
periodic point, where they fall into a regular orbit. In other words, every point
which maps to a periodic point but is not on its orbit is eventually periodic.
Sensitive Dependence on Initial Conditions This is the first important
condition of an attractor. Also called the ”butterfly effect,” from the work
of Edward Lorenz, the founder of Chaos Theory, it simple means that small
variations in initial conditions will lead to massive changes in state as time goes
on.
Density of Periodic Points This term is also simple in concept, though often
hard to prove. Periodic Points can be said to be dense if there is a periodic point
arbitrarily close to every other periodic point.
3 For dynamical systems of higher dimensions, fixed points are hyperbolic if the eigenvalues
all have a non-zero real part
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Topological Transitivity Like the last term, topological transitivity is more
difficult to define formally than informally. Informally, it means if you pick any
two points in a topologically transitive attractor, there will be a point arbitrarily
close to one of those points which will end up arbitrarily close to the other point
after enough iterations. There are two ways of expressing this formally:
f n (A) ∩ B 6= ∅
lim µ A ∩ T −n B = µ(A)µ(B)
n→∞
The first is the topological formulation, the second is the system of strong mixing
as it appears in the study of dynamical systems. What they both express is that
a topologically transitive attractor spreads each part of itself to all parts of itself,
even while the behavior of every element is deterministic.
3.3
The Smale-Williams Attractor
Then, the Smale-Williams Attractor is introduced and its similarity to supercellular processes is explained to support the claim that it represents an advancement towards the understanding of the attractor model of tornadogenesis.
The Smale-Williams Attractor, or the the Smale Solenoid, is an attractor
that is similar to the Logistic Map. It also has more concrete similarities to
supercells. We propose that the Smale-Williams attractor has enough similarities to the known properties of the Sasaki’s Tornadogenesis Attractor, that the
former can be used to understand the latter in more concrete terms (Chorin).
The Smale-Williams Attractor is created by mapping a torus to itself in
discrete steps, like the Logistic Map. First, the torus is created as a solid of
revolution by revolving a disc along a path defined by a circle, formally expressed
as T = S 1 × B 2 , where S 1 is the circular path, B 2 is the unit circle x2 + y 2 ≤ 1,
and T is the resultant torus. Points on this torus are indexed by the coordinates
θ, x, y, where θ gives the angle around S 1 , and both x and y are the ordinary
Cartesian coordinates on the unit circle. To iterate this torus, use the function
F (θ, p). This map transforms the x and y coordinates in the same way, so for
brevity p is used to represent both x and y.
1
1 2πiθ
F (θ, p) = 2θ, p + e
10
2
This does two things: along the θ axis, it stretches the entire torus by a factor
of two, making it wrap around itself. In the x and y axis, it first reduces the
area by a factor of 10, then displaces it (the + 21 e2πiθ ) term, so that the two
coils created by the extra θ length do not overlap each other. As this map is
iterated ad infinitum, the attractor begins to appear. Each iteration adds more
coils, which become finer and finer as it wraps around more and more times.
We, with our love of doughnuts, most often imagine tori with the θ axis parallel
to an imagined table, ready for some sugar or jelly. However, if we instead
imagine our Smale-Williams torus on an edge, it appears very similar to a system
12
of updrafts and downdrafts in a supercell. When downdrafts recycle into nascent
tornadic vortices, the structure that forms seems remarkable similar to that of
the nested loops of air in the Smale-Williams Solenoid. This similarity is most
noticible if one examines cross-sections of the solenoid, and of the hierarchy of
vortices.
The cross-sections of the Smale-Williams Attractor (Chorin)
4
Future Work
Modeling The work of this paper has been entirely theoretical. The first
thing that could be done to continue this work would be to begin simulations
and numerical models to examine the observations made in this paper for clarification, verification, or repudiation.
First and foremost, the author has found no attempt to model supercells by
Sasaki’s methods. Considering that easing modeling was one of Dr. Sasaki’s
prime concerns when developing the Entropic Balance Theory, developments
in this direction would be potentially fruitful, as well as honor his legacy. Researchers attempting to do so would want to become familiar with existing
supercell modeling techniques, and programs like AARPS and CAM1 which are
commonly used to create such models, and have a firm enough grasp of programming to be able to change how the programs would operate to account for
the techniques of entropic balance theory. Doing so would not only advance
Entropic Balance theory, but also meteorology as a whole.
To examine the relevance of the Smale-Williams attractor, models with
enough resolution to portray the hierarchy of vortices, ideally down to the suction vortex level, could be employed. There, snapshots at different instances of
time would be examined to find where vortex lines and velocity lines match up
most closely to determine where helicity is maximized. Doing so in stopped time
would allow streamlines to be used, simplifying the task, and easing visualization. The Smale-Williams seems to exhibit similar structure to a supercell, but
where is stil undetermined. For instance, this modelling could reveal that the
Smale-Williams attractor most readily represents the vorticity, and the borders
of its regions are where vorticity converges on zero. These experiments would
further elucidate the exact role of the Smale-Williams attractor and allow more
concrete statements about a tornadogenesis attractor to be made.
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Testing Sasaki’s Hypotheses Everything of meteorological significance in
this paper depends on the equations derived by Dr. Sasaki. These equations
are in turn based on his two hypotheses, and the associated meteorological process. The first hypothesis is that the time scale of microphyscial phase change
is strictly shorter than the time scale of supercells. The second hypothesis is
that variations of initial entropy levels are small enough to allow them to be
approximated by their ensemble means. Sasaki also defines a ”quasi-adiabatic
process” as a process where microphysical phase changes are instantaneous, with
adiabatic conditions before and after. Sasaki’s hypotheses seek to add thermodynamic legitimacy to quasi-adiabatic processes, which in turn legitimizes his
Action Lagrangian, from which the new type of helicity is derived (”Variational
Field Formalism”).
This author accepted the hypotheses as a basis for this work, but it must
be acknowledged that these hypotheses are not tested. Should the mathematics
described in this paper be involved in important conclusions one day, the validity
of the hypotheses will be of critical importance. Already, Sasaki’s work has
fascinating conclusions, which would make an investigation or an elucidation of
his hypotheses a worthwhile pursuit.
Quantum Field Theory Sasaki’s hypotheses are rooted in quantum theory.
He discussed this connection briefly, concluding that quantum variations should
be able to contribute significantly to storm-scale systems, compared to what
most meteorologists assume. While the author had insufficient knowledge and
experience to explore this path, it remains ripe for future exploration. In particular, Feynman’s Path integral, which connects classical and quantum mechanics,
was cited by Sasaki without elaboration in his explaination. Additionally, future
researches could explore a possible connection to the Aharanov-Bohm Solenoid
Effect, an observed quantum phenomenon wherein an electrically charged particle is affected by an electromagnetic potential, despite being in a region where
the magnetic and electric fields are zero, such as the inside of a solenoid. Notably, rotating air in a supercell is a solenoid, and if researcher can draw a
connection between quantum mechanics and fluid dynamics as Sasaki suggests,
the Aharanov-Bohm Solenoid Effect could be re-formatted to the new field.
14
5
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