### Seven Ways To Model Fusion Plasmas

__Seven Ways To Model Fusion Plasmas__

__Introduction__
A few years ago a man named John Hedditch quit his coding
job at Google to follow his dream. He
wanted to work in fusion research. This
was not a sound financial choice. Fusion
is under resourced and it certainly does not pay as well as Google. However,
Dr. Hedditch had a vision that most people in this space share: a green energy
future for the human race. John is an Austrian, so he went to work at the
University of Sydney under Dr. Joe Khachan.
The team worked hard - and on August 2

^{nd}2018, Dr. Hedditch submitted a paper laying out some new ideas on how to model plasma [2].
John Hedditch (Left)
and his boss Joe Khachan (right).

Hedditch wants to use field theory to model plasma
equilibrium. This is stealing an idea
that is commonly used in the world of quantum physics. Field theory asks how charged particles
behave in a field. This is like modeling
a spider web. The charges on the
particles create a web of electromagnetic forces – that push and pull on one
another. The math used, is based on this
concept. It is a complex idea that would
be tough to code in a computer. However,
Hedditch is just using the math to uncover some new insights about plasmas in
places where there is no magnetic field.

__Current Problems:__

You might think we have already mastered plasma modelling –
but that is wrong. There is a body of
evidence that our models consistently fall short of reality. For example, tokamaks experience

*anomalous transport*. This is a catchall term for anytime a plasma leaks out for seemingly no reason. Whenever trapping fails to live up to the classical predictions, anomalous transport is the reason. This one example of a gap between reality and our predictions. One flaw in our plasma models is the concept of the super-particle. One super-particle will represent about 1,000 real ions or electrons. The computer tracks these – because no simulation has been able to track every electron and every ion inside plasma. The use of super particles becomes a problem when you need the plasma to be mostly negative or positive. If you need more (+) ions in there, you might be out of luck. The difference populations can be so small, the computer will never see it. Bottom line: super particles work – until they fail. These are just some examples of flaws with our plasma model. Well, models. This post will present seven ways in which we can model fusion plasma. The fact that there are so many methods, should tip you off that this research is still evolving.

__Executive Summary:__
This post covers seven ways to model plasma: charged ideal
gases, charged single fluid, two charged fluids, gyrokinetic, kinetic,
everything and quantum. The post was
inspired by a recent paper trying to model plasma using Quantum Field
Theory. It opens with short histories of
the charged ideal gas model and magnetohydrodynamics. An example from magnetic mirrors is used to
show flaws in the MHD model. Next, the
two-fluid model is explained based on differences between the ion and the
electron. After this, particle-in-cell
is covered and how it fits with gyrokinetics and kinetics. Gyrokinetics is shown to simplify ions’
corkscrewing motion. Kinetics is shown
as (BIC) blob-in-cell modeling. Quantum
and klimontovich (everything) models are covered at the end.

__Part 1: Historical Methods__

__Charged Ideal Gas:__

Plasma modelling got started after World War II. Back then, all physicists could do was base
their approaches off older methods. The
first theory applied to plasma was the ideal gas. The first person to apply these concepts was
Lyman J Spitzer. In the 1950’s, Spitzer
was a professor at Princeton. What he did
was relatively simple. He combined the
physics of ideal gases, with electrostatics; an out came a model for plasma
behavior.

The concept of an ideal gas is widely used to estimate how
gas behaves. Most people know it as the
law (PV=nRT) but few understand there are real assumptions behind it. For example, ideal gases can bounce off one
another with no lasting problems (aka: perfectly elastic collisions). Real life is not like that. Spitzer stole these assumptions and modified
them to apply to a plasma. But unlike
gas a plasma is charged. He added in electrostatics
to deal with these charge effects. The
Spitzer model treats plasma like an ideal gas, with a charge on it.

This plasma model was covered in Spritzers’’ 1956 book

*The Physics of Fully Ionized Gases*. Anyone interested in fusion should get a copy.

__Charged Fluid:__
About the same time that Spitzer was developing his models,
another approach to modelling plasma was taking shape. When it is dense, bulk plasma can be treated like
a fluid that conducts electricity. A
conducting fluid. The approach is to combine
the math that governs fluids, with the math that governs electromagnetism. This method was taking shape in the fifties
and sixties at places like the Courant Institute at NYU. The Navier-Stokes equations are the
mathematics used to predict fluids. Correctly
formulated, these equations can predict how fast water flushes down a toilet;
or the shape water takes as it exits a faucet.
The equations are a set of coupled partial differential equations. In many situations, these cannot solve it
directly. Today, computational fluid
dynamics codes are used to model fluids.
Plasma is also charged, are so it conducts electromagnetic forces in
much the same way any wire would. The
math predicting this is known as Maxwells’ equations. In the 19

^{th}century, James Clerk Maxwell outlined the set of equations that modeled how electric and magnetic fields moved through charge material. The idea was to combine these two sets of equations to model plasma. This is magnetohydrodynamics. Magnetohydrodynamics treats the plasma like a single, continuous, conducting fluid.
MHD can easily model this system. You can write the boundary conditions. You can solve the equations. The math will tell you that everything will
work out fine. Except it doesn’t. In real life, mirrors have problems. The plasma leaks out the ends. Recently, a strong argument has been made
that we can still make this idea work for fusion power [4]. But, like so many other fusion technologies,
America is failing to invest in this and leaving the door open to other
nations. Today, we have no large mirror
machine anywhere in the United States. MHD
mathematics will not capture the flaws in a mirror machine. On paper, everything will look fine - in
reality, there will be problems. This highlights
an important concept in plasma modelling.
The simpler the math, the more it deviates from real life. This becomes more apparent as we move through
other modelling approaches.

__Part 2: No Computer Needed__

__Introduction:__
As you might expect, things have changed significantly since
1956. Back then, physicists could only
draw upon models that had existed before.
Plasma? Lets’ just pull out the concepts
of ideal gases and electrostatics to address it. Today things have changed. New models have evolved to meet specific
physics problems. They are

*evolutions*and*extensions*of Spitzers’ and Alfvéns’ work. Some of these approaches are identified and laid out below.
This picture has a lot of information. Five models are presented. At one extreme is magnetohydrodynamics; that
method treats plasma like a charged fluid.
The MHD approach has very simple mathematics. But, it captures the least amount of real plasma
phenomena. At the other extreme is the
Klimontovich model [8]. This tracks everything - all the time. That math can be impossible to solve. Along the spectrum, we transition from models
that can be solved using pencil and paper, to approaches that require computers. As the transition occurs, the math gets more
complicated but the detail increases.
Each method has its’ own strengthens and weaknesses. Ideally, you would pick the method that is
perfect for the problem at hand. The
first method, MHD was already laid out above.

__Two Fluids:__
The only other pencil and paper approach is the two-fluid
model. The two-fluid approach is
inspired by the difference between the ion and the electron. This difference is highlighted by this table
below [11-14].

An ion is thousands of times bigger and heavier
than an electron. If an ion was a
marble, an electron would be a speck of dust.
This fact drove researchers to consider modelling these plasma
populations differently. To basically treat
plasma it like two over-lapping fluids.
In this approach, the electrons have no mass. This is reasonable when you consider how
lightweight a real electron is when compared to an ion. But in this approach, the electrons’ charge,
matters. The electron exerts itself,
through the electromagnetic force it generates.
Therefore when these particles pass one another in a simulation – the EM
force is tracked – but the masses can flow right through one another, like
ghosts passing each other [7]. The plasma
is treated like two overlapping fluids.
They only interact through their electric and magnetic fields. This is depicted below.

Obviously, a two fluid model will capture more effects than
a single fluid model. Hence, MHD does
not model as much physics as the two fluid model will. Unfortunately, two-fluid methods take longer
to solve. Many multiples of computing
time longer [9]. Also - as you might expect – the two-fluid
approach does a good job mimicking effects that need high concentrations of one
kind of plasma. Like when just the ions
bunch together. One example of this is a
drift wave. These form inside
tokamaks. The ions gather together. They form a swell and wave of plasma with electron
clumping together between the ion waves.
Drift waves can only happen under certain conditions. They move around ringed systems like
stellorators, spherical tokamaks and normal tokamaks. Below is a list of plasma phenomena, compared
by which model handles which effects better [9].

The two fluid approach is convenient because its’ math is
complex - but solvable. These equations
represent about the edge of what can solved using paper and pencil. Any more complexity - and a computer must be
used. This solvability is why the plasma
physics bible:

*The NRL Plasma Formulary*relies so heavily on the two fluid model. Though it has other information, the two-fluid model make up the core of the reference. Below are a selection of some of these equations. Understand that this method cannot capture everything that happens inside a fusion reactor. The math is not wrong; but it is limited in what it can do.

__Part 3: Computers Required__

Beyond two-fluid, we move into approaches that are designed
to be solved by computers. These methods
have more details because they try to track single particles; or the fields
they generate. That is why the next
three systems evolved from modelling plasma like a charged gas. It is granular in detail. A computer is complex enough to deal this
level of detail inside a plasma. It can
tracking what individual particles or blobs are doing inside the machine. It does this by fracturing the reactor into
small cells. Little volumes, in which
particles or blobs move through. Today,
this particle/blob in cell method is the most common way we model plasma. These are our workhorse approaches.

__Particles in little cells:__
In
the 1960’s, Los Alamos National Labs was the place to be if you wanted to do
computing [5].
Workers there had access to supercomputers way before the rest of
us. It was no surprise then that Los
Alamos was the birthplace of computational modelling. One group, the T3 group, led the way. The team was led by Frank Harlow. Frank and his group developed what is now a
common approach to modeling plasma – particles in cells. Today, particle-in-cell (PIC) is a common
method for dealing with plasma. If you
use it, you owe a debt of gratitude to and his team. Interestingly, the T3 group was ignored and
even criticized by their peers [6]. They published when nobody else seemed to
care - and they were ultimately vindicated. Below is a visual homage to Franks’ team [6].

Particle in cell fractures the reactor into small cells and
tracks the plasma as it moves through these cells. The computer tracks super particles. One super particle might represent 1,000 real
ions. The equations controlling the
motion of these objects can be simple, or they can be complex. Different models use different
mathematics. The simplest PIC codes
combine Newtons’ equations of motion with the Lorentz force.

__How PIC works__
Here is how it works.
Within each cell, the computer figures out the force acting on that
particle. This is based on the local
magnetic and electric environment. The
fields act on the charged particles - creating a Lorentz force. That force is fed into Newtons’ equations of
motion. These are the simple equations
that figure out how objects move under any force. Normally when we think about this, we think
about big objects like falling apples. But,
the same equations work for very small things, like little ion. Once the motion for every particle is
calculated – the computer takes a step in time.
Everything moves. Now the
computer has to re-evaluate everything.
It figures out the new forces and then moves everything again. Overtime the motion of the plasma is
uncovered.

I might add here, that you can do this math, on your own,
using excel. This becomes useful when
you want to understand how a single ion will move through a fusion
reactor. It gives you an intuitive feel
for a machine and the electromagnetic environment inside it. It can also answer basic questions, like,
will this ion injection beam be trapped by the surrounding magnetic field? Simple excel models have helped me model
machines like fusors, the levitating dipole (LDX), magnetic mirrors and
experimental fusion reactor approaches.

__PIC & Gyrokinetics__
Inside these little cells, the most common motion of a
particle is a corkscrewing one. This is
because of the nature of the Lorentz force.
It is wiley force. It introduces
this twisting motion as a new wrinkle in the motion of plasma. For example, if you had an electron sitting
still, and you directed a magnetic field towards it, the electron would start
to spin. It would spin
counterclockwise. Meanwhile the ion
would move in the opposite direction.
These particles move around the magnetic field, with some radius. While they do so, they are often venting off energy,
as light. This motion, and the
mathematics governing it, is shown below.

In real plasma, no electron starts by sitting still. No ion does either. Everything is moving. Everything is constantly moving. Material is swirling and mixing. The fields are pulling mass in all kinds of
directions. Plasma is flowing into and
out of the cloud with some velocity.
Inside this melee of motion, a corkscrewing path is the common thing you
would see. If an electron or an ion
starts with some tangential velocity, it will corkscrew around the magnetic or
electric field. It will corkscrew back
and forth these field lines like cars on the highway. Moreover, if the plasma enters a dense field,
it can flip directions and corkscrew backwards.
Even more amazing, if enough plasma gets together, its’ motion can make
its’ own field – and reject the outside field.
This is shown below.

Gyrokinetics tries to simplify this crazy motion by ignoring the corkscrewing path. While the real ion might move in a twisting path, the simulation models a ring that flies straight. The ring represents the corkscrew. Gyrokinetics tracks this ring as it moves through the different cells of the simulation. This simplification allows the gyrokinetics model to do all the work with only five numbers. These are: X, Y, Z, Vz, VTheta. The next model – Kinetics - adds an additional variable. Its’ six numbers are: X, Y, Z, Vx, Vy and Vz. This will make the kinetic model more comprehensive. Realize: this motion is integrated with particles in cell (PIC) code. Gyrokinetics works and it is simple enough to run on desktop computers – but it runs into problems over long periods of time. As you run the simulation longer, the code will stray further and further from reality.

__Kinetics__

Kinetic theory represents the state-of-the art plasma
simulations taking place today. Quantum
approaches may be more advanced – but they are just emerging in this
field. Kinetic codes are commercially
available and can be run on platforms like supercomputers. They have been used pretty regularly since
the late 90’s. A leading place for this
kind of work is the National Energy Research Scientific Computing Center in
California. The image below was
generated using a kinetics code.

The term kinetic plasma modelling is not strictly
defined. Some people consider
gyrokinetics just a subset of the bigger kinetic approach. Other people view kinetics as encompassing
PIC, BIC and anything in between [15]. They refer to these approaches as: kinetics,
fluids and anything that is a hybrid approach.
I have decided to separate out these methods because it is easier to
explain.

__BIC Blobs-In-Cells:__
What sets kinetics apart is that it models a blob instead of
a particle. The code could be called
blob-in-cell. A blob is a simulated
chunk of the material. This simulated
blob has all the gross properties of that collection of plasma. It tracks this blob, as it moves through
cells. Whereas before we had
representative or super particles, now we have volumes of plasma, blobs, to
track. These are little volumes of
fluids – like thousands of charged droplets floating inside the reactor.

__Each Blob:__
The blob has all the properties of that chunk of
plasma. That means it has some
distribution of energy. Much like a
crowd of people moving through the airport – plasma moves at different
speeds. Certain people will be walking,
running or on a moving walkway. Certain
ions will be speeding, fast and slow.
These speeds are tracked using a velocity distribution. The kinetics
code tracks the density and velocity distribution inside each blob of
plasma. That can be assigned to be some
specific curve, like a Maxwellian distribution.

__Many Blobs:__

But how do these blob move and interact? The code handles this motion in much the same
way it deals with the particles. It
breaks the calculation into a series of steps. It starts with some arrangement of blobs in
cells. For example, a dense blob might
be next to small blob. Next, the code
figures out the density in each cell.
This like an accountant tallying up how many (-) electrons and (+) ions
there are in each place. The key amount is
the net charge. If there are more
electrons, that blob is marked as negative.
Following this, the code figures out how this charge is moving. This includes the direction, speed and amount
that is leaving each cell. Since all
moving charge is a current – this motion creates a current density. The fourth step is solving for the magnetic
fields. Each blob creates and reacts to
the little fields surrounding it. Based
on this - a Lorentz force is calculated.
That force, moves the plasma. Finally,
the code steps forward in time and everything changes. Material is re-arranged. From there, the cycle repeats itself many
times. As this continues, we can capture
many of the physical events that happen inside a fusion reactor.

__Track Everything:__

Kinetics models are pretty detailed, but even they can miss
reality. This prompted a Russian
physicist to muse about the idea of modeling everything. The Klimontovich model is a fictitious approach
where we would track all the particles, all the time. This does not mean that every particle would
have a direct impact on every other particle.
Real plasma does not work like this.
Plasma has a Debye length. That
is the length over which a particle

*matters*to another particle. If it was one millimeter, then particles over a millimeter away have no direct impact on that ions motion. They cannot “see” one another. The Klimontovich approach is fictitious because no code attempts to model a full plasma reactor this way. It would be too complex. In 2018, the most advanced plasma code can do is a complete model of 1 billion (1E12) particles while keeping the total energy of the system constant (1E16 within accuracy) [16].

__Part 4: Quantum__

__Plasma__
Quantum methods represent the frontier of plasma
modelling. They’re new; like in the last
few years. Like all new things – they
are not fully understood, appreciated or implemented. Using quantum theory to model plasma is yet
another example of plasma physicists stealing ideas from other branches. That said, using quantum mechanics would be
much harder to code into a computer program.
But the method would allow us a much more granular view of what is going
on. There are two places we can steal
ideas from in the quantum world: field theory and quantum electrodynamics [2,
10]. This article only deals with field theory.

__Field Theory:__
Quantum
field theory is the idea that everything exists, moves and reacts inside
fields. Typically you think of gravity
when you think of field theory. Gravity
is a powerful field that we live in here on Earth. The fields inside a fusion reactor are far
stronger than gravity. An electric and
magnetic field can exert so much more force.
Moreover, charged, moving plasma can create its’ own EM fields. These overlap, interact and balance one
another, creating a web of forces. Field
theory treats all the particles as existing in this web of forces. This is like an invisible spider’s web that
connects all the particles. If one ion
is moved, it pulls and pushes all the material around it. That field is under tension. It is that tension that QFT models. The math reflects this. It treats each link in the field as a strand
under tension. Perturbations can be
modeled using this mathematics. That
treatment is known as Lagrangian mathematics. These expression contain a rich amount of
information: the interactions, tensions and behavior of the field over time.

Field theory feels suitable for times when we need the
plasma to do something - like make its' own field, or its' own structure. Examples of this are structures like the
Field Reversed Configuration or the Spheromak.
It would be interesting to apply Quantum Field Theory to fusion machines
like the magnetic mirror or the levitating dipole.

__Hedditch Paper__
John Hedditch wants to apply this unique mathematics to plasma
situations that cannot be modelled with other means. The situation he is interested in is environments
where there is no magnetic field. In his
paper, he uses field theory to show that such an environment is stable. That conclusion is interest. It lets us model a set of fusion machine
where the plasma self-magnetizes.
Environments like Lockheeds’ Compact Fusion Reactor, the edges of Tri
Alpha’s FRC or the inside of a polywell would all apply.

__Acknowledgments:__
The author would like to thank Dr. Thomas Dolan, Dr. Charles
Swanson and Dr. Tomas Linden and others for their assistance with this
article. For further reading on quantum
modelling of plasma, the reader is referred to Dr. Yuan Shis’ recent defense on
applying quantum electrodynamics to plasma modelling [10].

__References:__
1.
"Existence of Electromagnetic-Hydrodynamic
Waves" H. ALFVÉN, Nature volume 150, pages 405–406 (03 October 1942)

2. “A
different approach to the MHD equilibrium” Hedditch, John August 2nd, 2018,
4 Pages, submitted to Physics of Plasmas, ArXiv
.

8. "Yuri
L'vovich Klimontovich" Condensed Matter Physics, 2004, vol. 7, No. 3(39),
p. 439-442, English A.Zagorodny

12. Nave,
R. "Atomic Radii." Ionization Energy. Georgia State University, n.d.
Web. 15 Nov. 2012. .

13. Yoon,
Jung-Sik, Young-Woo Kim, Deuk-Chul Kwon, Mi-Young Song, Won-Seok Chang,
Chang-Geun Kim, Vijay Kumar, and BongJu Lee. "Electron-impact Cross
Sections for Deuterated Hydrogen and Deuterium Molecules." Reports on
Progress in Physics 73.11 (2010): 116401. Print.

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