On Elementary Plasma Physics and its Magnetic Confinement in Toroids

Submitted in Requirements for the Completion of PHYS 492

ELEMENTARY PLASMA PHYSICS

AND ITS MAGNETIC

CONFINEMENT IN TOROIDS

CREATED BY BRIAN HALLEE

FROSTBURG STATE UNIVERSITY

COMPLETED ON MONDAY, MAY 10, 2011

ELEMENTARY PLASMA PHYSICS AND ITS MAGNETIC CONFINEMENT IN TOROIDS

1

ABSTRACT

The continuing of investment into the engineering of nuclear fusion is of utmost

importance, as its application remains to be one of the only emission-free commercially viable

energy sources in the long-term future. The chief impetus in the advancement of fusion to the

commercial stage is the mathematical challenges barring the plasma from significant

confinement time. This report will aim at briefly describing the progress of such research over

the previous decades, and shedding light on properties such as the definition of a plasma, the

modern-day tokamak, plasma control issues, elementary plasma physics, the

magnetohydrodynamics of idealized plasma, and basic electromagnetic theory of fusion reactor

coils which is so critical to the proper confinement of the plasma. The quantitative theory

behind the engineering of such reactors and the significant facts and figures announced

through this report are derived from both current and historic experimental research. It is

hoped that the relatively deep mathematical content presented in this report is self-contained

through the use of relative diagrams and appendices to supplement the reader on background

issues.


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TABLE OF CONTENTS:

Chapter 1: Elementary Plasma Physics: …………………………………………………………...……. (3)

A Brief History …………………………………………………………………………..….…...……….… (3)

Forming A Plasma ………………………………………………………………………………………… (4)

Defining a Plasma ………………………………………………………………………………………… (5)

Chapter 2: Problems in Practical Plasma Physics: ……………………………………….………… (8)

Fusion Energy and the Tokamak Concept ……………………………………………….……… (8)

Feedback and Other Electromagnetic Control Issues ………………………..………….…. (11)

Chapter 3: Magnetohydrodynamic Theory and the Grad-Shafranov Equation: …..…. (13)

Determining the Poloidal Flux Function ……………………………………………………...… (14)

Electric Fields Inside a Tokamak ………………………………………………………………...… (19)

Equilibrium of the Plasma and the Grad-Shafranov Equation …………………….…… (20)

Magnetic Forces on the Tokamak Walls ……………………………………….……………… (26)

▫ Magnetic Forces in the Radial Direction ……………….…………………………… (28)

▫ Magnetic Forces in the Axial Direction ……………………………………………… (31)

Chapter 4: Green’s Function of Axisymmetric Magnetostatics and Convolution: ….. (33)

Distributions and the Dirac-Delta Function ………………………………………………...… (33)

The Green’s Function and its Properties …………………………………………………….… (35)

The Green’s Function of Magnetohydrodynamics ………………………….……………… (37)

Appendix: ………………………………………………………………………………………………….……… (43)

A.1: The Divergence of the Cross Product ………………………………..…………………… (43)

A.2: Formulating an Expression for Current Density ……………….……………………… (44)

Works Cited …………………………………………………………………………………………...………… (47)

Endnotes ………………………………………………………………………………………...………………… (48)


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Figure 1.1: A 3-D schematic for a basic stellarator

fusion reactor.

ELEMENTARY PLASMA PHYSICS

A BRIEF HISTORY

While the rigorous theory of plasma physics has only been developed over the

previous half-century, the long-term implications of this medium may prove to make it one

of the most important regions of physics. The history of plasma theory dates back to late

World War II when the nuclear fission program was in full force. Following the war, and

more specifically the unnervingly rapid development of Soviet fission technology, adept

scientists realized that even more destructive bombs would be needed to stave off the

Russian threat. Such a bomb was deemed “thermonuclear”, as its required temperatures

needed to reach a mind-boggling hundred million degrees Kelvin for rapid fusion (as

opposed to nuclear fission) to take place. These fusion bombs had theoretical yields in the

millions of tons of TNT which would, in turn, cause the fission bombs over Hiroshima and

Nagasaki to appear as children’s toys. It was at this point that the science of plasma physics

can be said to have begun. While there were certain outspoken individuals opposed to the

calamity expected to be wrought from fission weaponry, the rush and uncertainty of war-

time silenced most of the would-be opponents of atom bombs. However, when the fusion

bomb began to hit the drawing boards, many scientists felt the ethical implications of such

a massive explosion were simply too much to bear.1 The civilian casualties collected from a

single well-placed fusion-bomb would easily reach tens (if not hundreds) of millions.

Nonetheless, the realization that fusion could so easily lead to energy release in weaponry

lead these ethical scientists to search for peaceful, commercial uses for fusion energy.

Even so, fusion still requires an ambient temperature in the millions of degrees Kelvin

(temperatures at which only

plasma is the predominant state of

matter). Thus, the engineering of a

nuclear fusion energy plant would

require some serious background

knowledge in plasma and its

interactions with other materials,

at the very least. Perhaps the

father of what could be deemed a

“fusion reactor” is Lyman Spitzer

whose own invention, the

stellarator (Figure 1.1), is still

considered a relatively viable


4

concept for energy production.2 In short, the stellarator is a twisted toroidal figure

(sometimes bent in a figure-8) that confines the plasma in an “infinite cylinder” via

magnetic coils gently wound around the shape of the chamber. Such a device is considered

a “magnetic confinement” approach to fusion energy, and this type of confinement is the

one of interest for the duration of this report. Just how the fusion energy is gathered from

such a device is not of interest here as such a topic is one riddled with engineering (as

opposed to physical) concerns. Many problems have plagued the stellarator concept, (in

which most are also familiar to other fusion reactor devices), such as energy confinement

time, plasma boundary control, heating to thermonuclear temperatures, etc.

Subsequently, it was only just over a decade after Spitzer’s development that Soviet

physicists developed the tokamak reactor (See Chapter 2) that showed great promise in

mitigating these issues. The international fusion and plasma physics community were so

impressed by the potential of these devices that they were immediately catapulted to the

frontlines of plasma research. Fast-forwarding to present day, this concept still remains the

top-candidate as a commercial fusion reactor, and most of its physical barriers remain in

the realm of plasma physics. While this report is chiefly concerned with the ideal toroid in

plasma modeling, frequent mention will be made to the tokamak due to its toroidal nature

and present-day significance.

FORMING A PLASMA

While the preceding argument may give the reader the sense that plasma is a rather

exclusive medium, in reality this state of matter makes up most of the universe in its

present state. All continuously-burning stars including our Sun are simply massive spherical

entities of hot plasma undergoing fusion reactions at the thermonuclear core. In order to

gain an appreciation of the physical nature of the exceedingly mathematical arguments in

the chapters that follow, it is important to give a semi-detailed account as to how plasma is

formed and interacts with its self and other mediums.

We can think of plasma as the state that immediately follows the gaseous state. As such,

even the layperson realizes that given enough heat, a solid will slowly melt to a liquid.

Similarly, adding further heat will cause the liquid to boil and evaporate to the gaseous

state. What many do not realize is what happens when a volume of gas is granted so much

heat energy that its own atomic foundations begin to fall apart. Specifical ly, as a gas begins

to reach plasma-producing temperatures, the individual electrons and positively charged

nuclei begin to dissociate, or simply separate into free-flowing ions. This notion is what

separates plasma from the other three states of matter. A simple molecule such as


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Figure 1.2: Graph depicting the

quality of the plasma in pure Hydrogen

over a span of temperatures.

Figure 1.3: Electric fields present inside a

free plasma.

hydrogen (H2),(which is not associated with electrical properties in any way at normal

temperatures), may accept and produce electric currents and allow for shaping via

magnetic fields in its plasma form. This susceptibility to electromagnetism is what makes

plasma so special and useful in the quest for fusion energy. 3

Nevertheless, as with nearly all natural

phenomena, plasma tends to resist unnatural

phenomena that attempt to control it in tight,

organized fashion. At degrees of high

ionization (See Figure 1.2), the plasma will

naturally exhibit trillions upon trillions of

infinitesimal electric fields caused by the free

electrons and ions present. The constant

attractive and repulsive forces present in hot

plasmas cause for some major headaches in

fusion reactor designs due to the

considerable amount of instabilities caused by

such effects. Figure 1.3 below depicts the wild

variance of electric field through a differential

distance within the plasma. It should be noted,

however, that all other things being equal, a

plasma as a whole contains no net charge.

Taking into account each individual charge will add up to the original charge of the gas that

dissociated into the plasma which was, of course, electrically neutral.

DEFINING A PLASMA

We may take a brief conceptual

hiatus by applying firm mathematics to

the definition of a plasma via a factor

called the Debye length. We start with

the idea that at small enough scales the

plasma will effectively act like a point

charge or something similar. To define a

plasma, we wish to make use of a

distribution of charge that acts to attenuate

or shield electric potentials at certain


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points in space. The Poisson equation from elementary E&M states the following:

(Eqn. 1.1)

In Eqn. 1.1, represents the charge density, the electrostatic potential, and the

electric permeability constant. Now, we can alter Eqn. 1.1 to work with the multitude of

charges in our plasma by defining ne equal to the density of electrons in the local vicinity of

our point of reference (say, electrons/meter3), and n0 to be the average ion or electron

density throughout our plasma. Letting e equal the absolute electrostatic charge per

electron as usual, we may write the following:

( ) ( )

(Eqn. 1.2)

In order to work with Eqn. 1.2, we need some way of relating the local electron density to

the average charge density. Making use of the thermodynamic tool called the Maxwell -

Boltzmann Distribution we can do just that. Doing so, the electron density can be written

as follows:

( )

However, (as we will see in a moment), it will be convienent to rewrite the local density

using the first-order approximation of the Taylor series for ex:

( ) (

)

Substituting this value into Eqn. 1.2 and factoring out the n0, we see that the 1’s cancel and

we are left with the following:

( )

(

)

( )

( )

The above equation is an easily separable, second-order differential equation. The

substitution is currently a way of simplifying the expression. However, it should be

noted that this value has units of length, and it will play an important role in our definition

shortly. We can evaluate the above expression either quickly by hand or recalling the

formula for such a separable equation. Solving for ( ) we achieve the following:

( ) (

)

(Eqn. 1.3)


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The term in parentheses arises from the initial condition that if r approaches zero, we

achieve an electrostatic potential proportional to a single charge. On the other hand, if we

consider r to be an appreciable distance from the plasma, we have the exponential term

that includes the shielding of potentials due to other charges in the plasma. With this

duality duly mentioned, it is now appropriate to isolate and describe its significance to

plasma definition. Undoing the substitution we made earlier with a , we have the

following expression for :

(

)

(Eqn. 1.4)

Eqn. 1.4 represents the Debye length mentioned previously, and it places quantitative

restrictions on the charges for the aggregation to be considered a plasma. Specifically, the

distribution is considered a plasma when the distance of the point of reference is much

greater than a Debye length. The “much” is a rather loose restriction, and no text is so

brazen to place a firm restriction on exactly when an electrostatic charge distribution ends

and a plasma begins. To conclude the argument, we can apply Eqn. 1.4 to place a

restriction on the electron density to constitute a plasma. If we follow along with the radial

coordinate expressed in Eqn.s 1.2 and 1.3, we may depict a spherical Gaussian surface

filled with our plasma with a radius of one Debye length. Making further use of the average

ion density n0, we can express the number of ions inside of our Gaussian surface using the

following formula:

If we wish for the Gaussian surface to contain what we consider a plasma, then we must

place the following restriction on our ion-count inside the sphere:

(

)

(Eqn. 1.5)

Thus, we require the ion count to greatly exceed a handful of charges producing the typical

elementary effects encountered in E&M. In this case, the “much” in “much greater than

one” in Eqn. 1.5 is not a loose term. To gain an idea of the orders of magnitude typically

dealt with in applied plasma physics the typical fusion plasma will exhibit an ion-count of

with a Debye length of .4


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PROBLEMS IN PRACTICAL PLASMA PHYSICS

While the “jist” of this report is to inform the reader on plasma kinetics and how to

control it using electromagnetic fields, these issues are almost always of practical

consideration in the form of fusion energy. More specifically, the next chapter’s

derivations on plasma control make assumptions based on the tokamac design, as this

concept has (by far) seen the most success in recent decades. In the perspective of a

plasma scientist, success is seen as the probability or proximity of a certain design to

“break-even” in terms of energy expended and gained. As such, the goal of this chapter is

to familiarize the reader with the tokamak scheme, and present some of the major

contemporary issues concerning magnetic control of plasma, and how scientists are

approaching the problem. The terminology gained in this chapter will be of aid in the

following chapter when an attempt is made to mathematically describe and solve some of

these confinement problems.

FUSION ENERGY AND THE TOKAMAK CONCEPT

In the preceding chapter, a formal analysis of the fusion reaction inside the plasma

was given. However, the actual device in which these reactions are allowed to act

continuously was never mentioned. While fusion energy has been a hot topic since Lyman

Spitzer’s “Stellarator” of the 1950’s, the concept began to take a practical turn when the

Russian-developed tokamak design hit the scene in the mid 1960’s. The quirky word itself

stands for Toroidalnaya Kamera ee Magnitaya Katushka (toroidal chamber with magnetic

coils). A score or so tokamaks have been erected in various parts of the world since the

development; many of which have seen great success. Currently, all eyes in the plasma-

physics community are focused on the International Thermonuclear Experimental Reactor

(ITER) in France, for which the groundbreaking has just begun. ITER represents a

cooperation between the entire European Union and the world’s richest countries, and has

been on the drawing board since the late 1980’s. The program is slated to conclude in 2018

with ignition being reached sometime in 2020. The significance of ITER is the fact that

plasma physicists are convinced it will not only reach break-even, but expel 500MW out for

every 50MW bestowed. This very fact was one the initial impetuses for the research

behind this report, and I hope that it places great significance and value on the deep and

rigorous mathematics inherent in the following chapters.

The tokamak design can be broken up into four major components relevant to the

magnetic control of plasma. Figures 2.1 and 2.2 below depict the tokamak chamber and

magnetic components respectively for your reference. First, the plasma itself is housed


9

Figure 2.1: The schematic for ITER currently being

constructed in France.

within a large vacuum vessel lined with specialized metals to resists the high temperatures

inherent inside. As we will see later, this vessel has important conductive properties so

that residual currents may be induced for the gain of plasma control. Next, behind the

vacuum vessel (encasing it, if you

will) is the Lithium blanket which

absorbs the neutrons produced by

the Deuterium-tritium reaction and

allows them to interact with Lithium

ions. As previously mentioned, the

chemical-reaction resulting from

Lithium and high-energy neutrons is

a production of Tritium. So, this

ingenious device allows for passive

fueling of the tokamak. Lastly, and

most importantly, are the Toroidal

and Poloidal coils. The toroidal coils

(surrounding the vacuum vessel no

unlike concentric circles) are

responsible for producing the

poloidal field. For this reason, the

toroidal coils are typically referred to

as the poloidal-field (or PF) coils. The

poloidal coils (which “ring” around the vacuum vessel) are responsible for, to no surprise,

producing the toroidal magnetic fields. As such, they are referred to as the TF coils. A

third and final set of coils which are usually lumped in with the PF coils are the central

solenoid (CF) coils. These are located at the very center of the tokamak, and they act to

induce a current into the plasma which enables it to reach thermonuclear temperatures

and aid the fusion process. It will be demonstrated later that while inducing plasma

currents are enviable for nuclear fusion, the magnetic effects of this cannot be ignored

which adds a great deal of complexity to the modeling.

It should be apparent from E&M and the discussion of the previous chapter that the

plasma (a group of charged particles) will exhibit motion according to setting Newton’s

second law to Lorentz’s law:

( ) (Eqn. 2.1)


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Figure 2.2: Demonstrating the TF coils, blanket, and CS

coils in the center. The plasma roughly exhibits the helical

motion as shown.

Figure 2.3: Demonstrating the pulsed-nature of

tokamak operation.

This demonstrates that a

particle will exhibit a

cyclotron motion in the

presence of an orthogonal

magnetic field. So, in crudely

combining the PF and TF

coils’ via superposition we

expect that the particles will

move in helical paths

throughout the vacuum

chamber. Due to the special

properties of a torus,

Maxwell’s equations hold in

general form, which allows

the helical path to reach no

end. This is experimentally verified, and while there are many properties that do instill

volatilities in the plasma, it will generally follow this helical path while in operation.

What many people outside of fusion-related work do not know, is that tokamaks are

inherently pulsed devices. That is, the plasma control operation is broken up into three

steps: ramp-up, constant operation, and ramp-down (See Figure 2.3 below). At the current

state of tokamak reactors, the flat-top phase only exists on the order of seconds. This is

one of the issues to be addressed

with the ITER experiment. While

this pulsed-nature is not of major

importance in our plasma control

modeling, it has much significance

on the forces generated against

the vacuum chamber’s walls. Eqn.

2.1 also applies to the particles in

the toroidal walls. Thus, as the

velocity of the plasma particles

quickly decreases to zero, large

forces will be exerted on the

vacuum walls, and this must be

compensated properly. These

forces will briefly be discussed in


11

Chapter 3.

FEEDBACK AND OTHER ELECTROMAGNETIC PLASMA-CONTROL ISSUES

As mentioned, there are a multitude of issues currently plaguing plasma physicists

that need to be solved in order for useful, continuous energy to be extracted from a

tokamak device. Chiefly, these scientists must decide where to draw the line between

accuracy and simplicity in their modeling techniques, as is often the case in applied

physics. The Wikipedia article on plasma stability lists a grand total of 57 known

instabilities that occur when one attempts to confine plasma with magnetic fields.

Naturally, it must be decided which of these are “showstoppers” in the tokamak concept,

and how to model the magnetic fields in a way that diminishes them using the simplest

mathematics possible.

Plasma control issues are typically separated into two camps: electromagnetic control and

plasma kinetic control. The latter focuses more on plasma quality, heat energy, pressure

and the like. Thus, we are focused solely on the former. Of the predominant texts used to

carry out the research of this report5678, the following are the general assumptions in order

to carry out meaningful control-modeling of the plasma:

1. Allow the kinetics of the plasma (as a whole) to be described by a finite amount of

parameters

2. Ignore the mass of the plasma. This is justified due to the massive magnetic fields

present in the vacuum chamber. The masses of the individual ions simply have no

effect on their inertia.

3. Allow for uniformity of the plasma in the toroidal angle. This is described by the

term axisymmetric. Of all the assumptions listed here, this is perhaps the most

important. It allows for us to develop the theory using only two dimensions (radial

and vertical).

4. Allow for a constant resistivity in the plasma. It is understood that, while resistivity

typically inversely varies with temperature, uncertainties in resistance models are

usually large.9 This will be of great help in our development of the

magnetohydrodynamic (MHD) theory in the following chapter.

The first modeling issue concerned with plasma confinement is the problem in estimating

the boundaries of the plasma inside the vacuum vessel. Ironically, the goal of fusion

science is to maintain the plasma as close to the walls as possible without touching them.

This ensures that the maximum amount of power is being achieved from the fusion

reactions taking place inside. Although the walls are constructed using some of the most


12

heat-resistive compounds in the world, a significant encounter with thermonuclear plasma

would be catastrophic. Mind you, while the safety of such systems is greatly assured

(fusion reactors pose practically no harm to those operation them), the cost of such an

event would amount to what could be considered a catastrophe. This is greatly

compounded in the scenario of a fusion power plant, where any downtime is a serious

issue.

The next major issue is one that has been man-made over the previous four decades of

fusion research. It has been determined that elongating the vacuum chamber in the

vertical directions (see Figure 2.1 as opposed to Figure 2.2) significantly improves the

performance-to-cost ratio. However, this invokes another instability (add one to the 57

mentioned earlier!) that must be accounted for in the numerical modeling techniques. We

will not touch on this here as it is a rather advanced issue. But, it is a noteworthy instability

and one that still contains very active research centered on mitigating its effects.

Theoretically, vertical instabilities can be alleviated through the use of superconducting

walls10. As such, it should be noted that most tokamaks in operation today utilize

superconducting TF and PF coils made of either Niobium-Titanium or Niobium-Tin11.

The final magnetic control issues consist of more specific plasma modeling concerns.

Specifically, while instabilities play less of a role, it is important to develop parameters on

the radial position of the plasma and wield the ability to fine-tune it using currents. Finally,

while not a significant topic in this report, plasma physicists also require a way to model

the strike points of the plasma inside of the vacuum vessel. This comes in handy as most

modern tokamaks make use of divertors that funnel escaping plasma into two specific

points in the poloidal plane. Currently, one of the most robust software suites for plasma

shaping is the eXtreme Shape Controller (XSC) originally developed for the Joint European

Torus in England. This is the first multivariable controller, and it allows shaping to a very

high degree of accuracy.

As one might have guessed, much of the current research on plasma control is grounded in

numerical techniques and computational methods far beyond the scope of this research,

as the “easy” theoretical work has long been concluded. Nevertheless, the theory

presented in the next chapter serves as an important foundation for fusion plasma physics,

and many of the derivations are completed to the point that numerical modeling may pick

up from there.


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MAGNETOHYDRODYNAMIC THEORY AND THE GRAD-SHAFRANOV

EQUATION

Now that the elementary kinematics of plasma and the problems concerning its

confinement in tokamaks have been developed, it is time to apply that knowledge in

mathematically describing the electromagnetism inside a tokamak vessel. The ultimate

goal of this section (and the motivation for this research) is to carefully and cleanly derive

the celebrated Grad-Shafranov equation of magnetostatics. This equation is pivotal to

magnetic modeling, and it serves as a jumping-off point for many numerical codes and

modeling algorithms in use today.

The field of Magnetohydrodynamics (MHD) is one that treats the plasma as a continuous

fluid and makes no distinction among the individual ions that it truthfully contains. It

attempts to completely describe the plasma under magnetic confinement once certain

parameters have been defined. As a result of this “fluid-like” assumption, MDH can make

use of the following two equations:

( ) (Eqn. 3.1)

(Eqn. 3.2)

The variables follow the usual notation, as represents pressure and represents mass

density. Eqn. 3.1 represents the continuity equation of fluid dynamics. In short, it states

that the rate at which something enters a system is equal to the rate at which it leaves.

Eqn. 3.2 results from applying Newton’s second law to an individual element inside the

plasma. Next, MHD makes use of Ohm’s law in its continuum format as follows:

(Eqn. 3.3)

In this case, represents the “conductivity”, which is defined as the inverse of resistivity .

Eqn. 3.3 is a result of the Lorentz force viewed from the reference frame of the plasma.

Lastly, in deriving the Grad-Shafranov equation, we will make extensive use of Maxwell’s

equations. They can be found and justified in any standard text on E&M 12. However, the

three most useful equations are shown below:

(Eqn. 3.4)

(Eqn. 3.5)


14

Figure 3.1: The cylindrical coordinate system for

tokamak MHD

(Eqn. 3.6)

These equations (in order) describe the impossibility of a monopole, Ampere’s Law, and

Faraday’s Law, respectively.

DETERMINING THE POLOIDAL FLUX FUNCTION

Now that the MHD equations have been specified, the derivation may begin. Naturally,

the use of the standard Cartesian coordinate system would soon present some hellish math

when applied to a toroidal geometry. Thus, it has become standard to use cylindrical

coordinates when dealing with the MHD of a fusion reactor. Figure 4.1 below demonstrates

how one can view the tokamak

scheme in cylindrical coordinates.

The z-axis denotes the height of

the device and it sits directly in the

center of the toroid where no

plasma is present. will denote

the toroidal direction inside the

vacuum vessel in a counter-

clockwise manner. Lastly, r (bold r)

will denote the vector of the

location of an element in question

such that:

( )

(Eqn. 3.7)

Perhaps the most notable consequence of the coordinate system of Figure 4.1 is the fact that

the plasma will (ideally) be axisymmentric. In other words, as long as the quality and shape of

the plasma is considered to be uniform in the direction, then then generic vector of Eqn. 3.7

is no longer a function of . We can state this in the following way:

(Eqn. 3.8)

Finally, we are able to introduce the magnetic flux function which will guide us through the rest

of the derivation. Recalling from elementary electrodynamics, magnetic flux represents the

amount of magnetic field integrated around some surface area. The flux we are particularly


15

concerned with is that which flows through ( ) in Figure 4.1. This is deemed the poloidal flux,

as each individual field line wraps around the vacuum vessel entirely in the poloidal plane.

Thus, we will denote this value with a Psi in the following way:

( )

∫

( ) (Eqn. 3.9)

The reasoning for the division by 2 is solely a matter of taste. Due to the fact that, in

cylindrical coordinates, dS represents , the integration of a full turn in ϕ will give you an

extra 2 . Thus, the division in Eqn 3.9 will rid us of that extra pesky term that would otherwise

trail us throughout the entire derivation. Succinctly, when all is said and done, the “true”

poloidal flux will equal . Now, we are particularly concerned with the individual

components of B. Thus, we begin to work with Eqn. 3.9 by taking the derivative with respect to

r:

( )

0

∫ ∫ ( )

1

∫ ( )

Currently, rho is a dummy variable to signify the differential r-direction. Now, due to the fact

that Bz is a function of both r and z, we must apply a special trick in order to differentiate

properly. This trick has been given the unofficial name of the Leibniz Integral Rule, and it works

as follows:

∫ ( )

( )

( ( ) )

( )

( ( ) ) ∫

( )

( )

( )

( )

( ) (Eqn. 3.10)

This trick allows us to evaluate and simplify the equation above. Applying it, we achieve

the following:

∫ ( )

( ) ∫

(

) (Eqn. 3.11)

As a result of Leibniz’s rule, we now have the r-derivative operating on rho functions in the

last term of Eqn. 3.11. In this way, we can say that the term vanishes, and we are left with

the following:

( ) ( )

(Eqn. 3.12)

This gives us a nice way to represent the z-direction as a function of the poloidal flux which

is the centerpiece of this derivation. We can now differentiate with respect to z in order to

solve for Br:

( )

∫ ( )


16

This is different from the previous scenario, as the differential is not related to the

integrating variable. Thus, we can bring the derivative inside as follows:

( ) ∫

This is messy, as we have no elementary mathematical tricks to integrate this. However, we can

make use of physics in the way of Gauss’s law for magnetics (Eqn. 3.4) by translating it to

cylindrical coordinates as follows:

( )

(Eqn. 3.13)

So, rearranging the terms of Eqn. 3.13, we have the following relation:

( )

(

) (Eqn. 3.14)

However, due to the fact that Bz is still inside the integral, we want to substitute the r’s in

Eqn. 3.14 with rho’s. Thus, making the Gauss’s equation substitution we have the following

equation:

( ) ∫

( ( )

( )

)

∫ ( ) ∫

( )

Considering the physics of the situation at hand, the first term in the above equation will

total to zero since there will (ideally) be an equal amount of inward field intensity as

outward. Evaluating the final integral, we achieve the following:

( ) ∫

Thus, we are able to form a very similar equation for B r:

(Eqn. 3.15)

As we shall observe here shortly, it will come in handy to combine both the radial and axial

components of the magnetic field into one single vector; namely, the poloidal field. This is

written as follows:

(Eqn. 3.16)

This particular component of the magnetic field will play an important role, thus we want a

more descriptive and compact way to represent Eqn. 3.16. While it may come across as


17

trivial, one way of going about this is to take the gradient of the toroidal angle ϕ. In

cylindrical coordinates, this appears as follows:

(Eqn. 3.17)

Now the formulation of Eqn. 3.17 and the following steps are more the result of

“tweaking” around with math than any common principle. Bear this in mind as I now claim

that the poloidal field is related to the cross product of Eqn. 3.17 with . To prove that

this is the case, we formulate the gradient of the poloidal flux:

(Eqn. 3.18)

The second step follows from Eqn. 3.8 which states that the toroidal angle is cyclic. No w,

we can take the cross product between the two preceding equations and see what we get.

|

(

)

|

(

)

(

)

(Eqn. 3.18b)

The final term above is exactly the values of B r and Bz in Eqn. 3.12 and Eqn. 3.15. Thus, we

can conveniently re-write Eqn. 3.16 (poloidal magnetic field) as follows:

(Eqn. 3.19)

Now that we have Bp, we can progress toward finding the current density via Ampere’s law

which works for toroidal geometries. Using Eqn. 3.5 and taking its divergence we write the

following:

( )

It is a fact of mathematics that the divergence of any curl equates to zero. This fact is

rigorously proved in the appendix A.1 using Ampere’s law as the example. Therefore, if the

left-hand side of the above equation equals zero, then the following must be true:

(Eqn. 3.20)

This has identical form to Gauss’s law for magnetism (Eqn. 3.4). It was this very law (which

naturally accounts for the divergenceless of magnetic fields) that lead us to describe the

poloidal components of B (Br and Bz) as a function of the scalar 𝜓. Therefore, we can

expect a similar formulation for Jr and Jz using another scalar function. We will denote this


18

new scalar with an f, and using the same methods as we did with B, (namely,

∫ ∫ ( )

) we achieve the following equations for current density:

(Eqn. 3.21a)

(Eqn. 3.22a)

Generally speaking, divergence of a vector indicates the relative expansion rate of the

flow. Thus, both the magnetic field and current density exhibit a lack of outward expansion

at a single point. This fact will also come in handy later when we develop flux surfaces.

If we wish to relate our newfound values for current density to the components of

magnetic field, we can do so simply by using Ampere’s law and separating the coordinates.

Breaking up Eqn. 3.5 by using the cylindrical curl, we achieve the following:

*

+ *

+

*

( )

+ [ ]

The strikethroughs in the above equation are a consequence of Eqn. 3.8. Now, considering

that this is a vector equation, we can equate the individual components as shown below:

(Eqn. 3.23a)

(Eqn. 3.23b)

( ) (Eqn. 3.23c)

We already have a convenient way of expressing the poloidal components of the magnetic

field via Eqn. 3.19. However, we are now in search of a compact way to express the

toroidal component of B. This can be done by using Eqns. 3.23a and 3.21 as follows:

*

+

(Eqn. 3.24a)

If we are feeling particularly lazy, we can eliminate the pesky μ0 by making the

substitution ( ) . We can do one better by recalling that the gradient of the

toroidal angle is

(Eqn. 3.17). This allows the toroidal B-field to be written

(Eqn. 3.24b). Armed with the toroidal component, we can now couple it with the poloidal

components to generate this extremely compact equation for the vector B:


19

(Eqn. 3.25)

The next goal is to obtain a similarly compact equation for the general current density

vector J. The current density plays an equal (if not, more important) role in the Grad-

Shafranov equation than the poloidal flux. Thus, we wish to develop it carefully and in full.

As part of an effort to spare the casual reader from overbearing math, a multitude of steps

toward achieve J in compact form is emitted. However, for the courageous or Type-A, the

derivation is worked out and displayed in its entirety in the appendix A.2. At the conclusion

of that section, the current density J is written in the following form:

𝜓 (Eqn. 3.26)

Taking a quick look at Eqn. A.2.2 in appendix A.2, we see that the first term in the above

equation is solely of radial and axial components of forcing function derivatives. Thus, if we

were to concern ourselves only with the toroidal component of current density, (which is

naturally the most important component considering this is the direction we wish for

current to flow), we can write the following:

𝜓

Or, as we shall see momentarily, the above can be written in a more helpful notation:

𝜓

(Eqn. 3.27)

Eqn. 3.27 is paramount to the final form of the Grad-Shafranov equation. Physically, it

allows for the toroidal component of current density to be described via a radial distance

and poloidal flux value.

ELECTRIC FIELDS INSIDE A TOKAMAK

While not of direct importance to the research of this paper, it is noteworthy to show

how Eqn. 3.27 allows for easy access to the mean electric field inside the plasma of a tokamak.

To find this, we must make use of the mathematical law that is the Kelvin-Stokes Theorem, and

a Maxwell equation that is Faraday’s Law (Eqn. 3.6). The Kelvin-Stokes theorem is written as

such:

∫ ∮

So, if we take Faraday’s Law and integrate it over the entire surface of S(r) as shown in Figure

4.1, we see that the Kelvin-Stokes Law comes in great handy:


20

∫ ∫

∫

( )

( )

( )

Recapping, the first two steps were a product of the Kelvin-Stokes Theorem whereas in the final

step we were able to integrate the right-hand-side of Faraday’s Law and pull out the time-

differential. But, we know from earlier that in our MHD model that the poloidal flux is defined

to be the final integral in the preceding equation divided by 2 . Thus, we can write:

∫

𝜓

( )

From here, we can use the same tricks as we did for current density and only take into

account the toroidal component of the field. We can view this as the “driving” component

of the field that pushes the negatively charged ions (electrons) through the plasma. A

consequence of our idealized plasma stream allows for us to consider the plasma and its

parameters uniform throughout a specific toroidal route. Thus, Eϕ becomes a constant in

the above integral, and the rest is simple calculus and algebra:

∫ ∫

𝜓

( )

Therefore,

𝜓 (Eqn. 3.28)

EQUILIBRIUM OF THE PLASMA AND THE GRAD-SHAFRANOV EQUATION

It is at this point in the derivation where the physics concerning the plasma itself begins

to play a large role. The previous sections of this chapter have assumed and absence of plasma,

which is fine. They concern only the electromagnetism of the PF and TF coils. Now, we wish to

apply this knowledge to the plasma directly assuming it obeys the MHD theory and is in perfect

equilibrium. In other words, we wish for the confining forces of the PF coils to balance out with

the outward forces generated by the plasma kinetic pressure.

Recall from basic thermodynamics that kinetic pressure has the units of force per unit area. If

we attempt to write the gradient of the kinetic pressure, (which should decrease as we

approach the vacuum walls), we may equate it to the following:

(

)


21

Figure 3.2: Depiction of the isobaric surfaces of

the plasma inside an ideal toroid.

However, we just mentioned that for a plasma element to be in equilibrium, its conflicting

forces must equal. Therefore, we can set the F in the above equation to the Lorentz Force (Eqn.

2.1) that signifies the inward force of the PF coils:

(

[ ]) (

[ ])

The second step follows from the alternative way to represent the Lorentz Force.13 The Del

operator is a space-derivative. Thus, this operator goes on to only affect the length vector l.

Taking this into account:

( ) (

) (Eqn. 3.29)

This preceding equation has an important physical consequence. Namely, the kinetic pressure

gradient is perpendicular to both the current density vector and the magnetic field lines. Using

the clean, convenient properties of divergence, we can thus construct the following equations:

(Eqn. 3.30a)

(Eqn. 3.30b)

These equations give us a quick way to signify the fact that these variables are indeed

perpendicular. We know that the gradient gives an outward directional-derivative vector. So,

the vector extends outward from the poloidal plane in all directions. Thus, if the current

density and magnetic field are perpendicular to these vectors everywhere, they must lie on

what is called isobaric surfaces. This is

aptly demonstrated in Figure 3.2 below.

We mentioned earlier that the plasma

pressure has been experimentally

determined (and following common

intuition) to wield maximum pressure at

the center and decreases outward.

Therefore, for the plasma to exhibit

equilibrium at the center, the magnetic

field (and current density) must have a

maximum value at this (pressure

center). In reality, this has been empirically determined to be true, and the special term

magnetic axis has been given to the point where all of this takes place. This magnetic axis is not

the true center of the tokamaks these days considering the additions of vertical elongations,

diverters, and the like. The most interesting conclusion from all of this occurs when we attempt


22

Figure 3.3: Depiction of the poloidal surfaces of the

plasma with added vectors using the syntax of Stacey3.

to solve for the direction of the poloidal flux relative to these vectors. We will use the magnetic

field as the “dummy” vector, since we know its poloidal values relative to the flux via Eqns. 3.12

and 3.15. Taking the divergence of B with the gradient of the poloidal flux, we achieve:

𝜓 ( ) (

)

(

) (

)

Thus,

𝜓 (Eqn. 3.31)

The last step followed from the general law of calculus that the order of the differentials plays

no role in the end result. As such, via Eqn. 3.31 we have proved that the magnetic field is also

perpendicular to the gradient of the poloidal flux. We know through the preceding kinetic

pressure derivation that B lies on

isobaric surfaces. Thus, we now know

that B (and subsequently J and p) can

be written as functions solely of the

poloidal flux. In mathematical terms,

we have just proved the following:

(𝜓)

(𝜓)

(𝜓)

Subsequently, because (𝜓) , we can write:

(𝜓)

With the preceding four equations in hand, we have arrived at the home stretch in deriving the

standard Grad-Shafranov equation. All that is required at this point is to re-write Eqn. 3.29 via

substitution of previously determined values of the vectors. While somewhat intricate the

following derivation is important and summarizes everything we have worked so hard to

develop. Therefore, it will be kept in this section of the report as opposed to an appendix. Using

Eqn. 3.29, we can separate the vectors into components as follows:

( ) ( )


23

Taking the cylindrical cross product of these two terms as we have done before:

|

| ( )

( ) ( )

Re-writing Eqns. 3.21a and 3.22a to account for ( ), we have the following:

(

) (Eqn. 3.21b)

(Eqn. 3.22b)

Now, using Eqns. 3.12, 3.15, 3.24b, 3.27 as substitutions, we may (for better or worse!) write

the following:

(

𝜓 (

𝜓)

) (

(

)

)

( (

)

(

𝜓)(

)

)

The stray ’s and negative signs have been left un-canceled and un-grouped for ease of

referencing where the substitutions were made. If we begin to cancel and group like terms in

the preceding equation, we form a nicer representation:

(

𝜓

) (

) (

𝜓

)

We can really begin to simplify the preceding equation when we account for the fact that we

are still dealing with idealized plasma. Thus, the aforementioned isobaric argument equates

the toroidal term of the equation to zero. So, making note of this and simplifying further, we

get:

𝜓 (

)

(

)

In case it went unnoticed, the preceding equation effectively separated the function into terms

solely of poloidal flux and force, respectively. Even better, the terms inside the parenthesis are

exactly the values of 𝜓 and , respectively. Now, the general notion of this section has been

to explicitly state how to construct equations for MHD scalars and vectors in the most compact

form humanly possible. This equation is not an exception. The time spent developing the

argument that J, B, p, and F are functions of 𝜓 is well worth the simplification it will lead to in

the preceding equation. Starting with the second term, we use the fact that (𝜓) and the

chain rule:


24

(

) (

)

(

)

𝜓 (Eqn. 3.32)

So, quickly re-writing the pressure-gradient equation to account for Eqn. 3.32:

𝜓 𝜓

𝜓

Next, accounting for Eqn. 3.27:

𝜓

𝜓

But, we can perform the same trick used for Eqn. 3.32 to on the left-hand side to achieve:

𝜓

𝜓

𝜓

Allowing for the cancellation of 𝜓 on each side of the equation and solving for current density:

(

)

Finally, if substitute Eqn. 3.27 back in again to make this a function of the differential elliptic of

the poloidal flux function and isolate that term, we achieve the celebrated Grad-Shafranov

equation:

𝜓

(Grad-Shafranov Equation)

This is a very interesting equation indeed, as the poloidal flux function shows up both as a

dependent and independent variable in the equation. Likewise, as a result of our findings along

the way to deriving this equation, we know that in solving the Grad-Shafranov equation for 𝜓

will allow us to formulate values for current density, magnetic field, and even the electric field.

All that is required is a forcing function F and its variance over r, and likewise for p. A multitude

of numerical codes such as the CREATE-L have been developed that work off the Grad-

Shafranov equation to determine analytic functions for quantities such as toroidal current

density using a certain number of constant parameters. Airiola and Pironti14 use the notation

shown in Figure 3.4 to denote the cross section of the tokamak including its magnet system. We

will use this to develop our final characterization of equilibrium-based plasmas. Succinctly, the

notation seen in this figure represents the following:

Individual poloidal-field coils (Including the Ohmic-heating coils)

∑ The entirety of the PF coil system

The conducting sheath (vacuum vessel) around the plasma


25

Figure 3.4: The tokamak cross section including the

PF and OH coils.

Plasma-containing area of the vacuum vessel

Vacuum space

It is at this point, the reader may

have concern as to how the

differential elliptic of 𝜓 can appear

in the Grad-Shafranov equation, and

Eqn. 3.27 the way that it does. As

previously mentioned, the second-

half of our derivation takes into

account the actual existence of the

plasma and the physics of its

kinetics. When we derived Eqn.

3.27, this was not the case. At that

point, we were solely concerned

with the electromagnetic effects of

the PF coils. Thus, in order to clean

up this issue in a compact form, we make use of the syntax of Figure 3.4 in the following way:

𝜓 {

(Eqn. 3.33)

The first equation of Eqn. 3.33 is something we have not mentioned called the homogenous

Grad-Shafranov Equation. Solving this equation is paramount to determining an analytic

function for 𝜓 alone, and that process is described in detail in Chapter 4. A final note worth

mentioning is the boundaries placed on the poloidal flux in order to solve for the partial

differential equations of Eqn. 3.33. Both are of the Dirichelet type, and only concern

themselves with the radial and axial location. Firstly, it is assumed that there is no residual

magnetic field as the vector r extends off to infinity.

𝜓( ) (Eqn. 3.34a)

The second condition is a bit more subtle. As depicted in Figure 3.3, the poloidal flux surfaces

(ideally) ring around the magnetic axis extending far beyond the walls of the vacuum vessel.

However, for a flux surface to exist exactly at r = 0, it would have to come in from in the

axial direction. This is not physically possible, and although there will likely be some residual

field lines in a real-world tokamak at the z-axis, we define the theory so that this is not the case:

𝜓( ) (Eqn. 3.34b)


26

Figure 3.5: Depiction of the tokamak from

top-view with a continuously-wound coil.

We will assume N turns in total around

the ring.

With the boundary conditions of Eqn. 3.34 and Eqns. 3.33 in check, we are at a point where

jumping into the numerical modeling or engineering of real world tokamaks is feasible.

Naturally, advanced empirical software is needed to come up with a reasonable expression for

(Something we just blindly wrote and never defined!). Nonetheless, the Grad-Shafranov

equation is very meaningful in both theoretical and experimental plasma physics and embodies

a significant milestone in the joint-effort of mid to late-century U.S., U.K., and Russian

physicists.

MAGNETIC FORCES ON THE TOKAMAK WALLS

One final topic worth mentioning is the practical issue of the forces exerted on the

tokamak walls in the event of a disruption of plasma current. As a consequence of the Lorentz

Force Law, fusion engineers must construct their tokamak vessels with a specific amount of

stress tolerance in order to combat these magnetic forces. In for us to model this issue, we

must take a quick deviation from the assumptions of magnetohydrodynamics and begin with

the most basic theories developed in

elementary electrodynamics. We will make

the assumption that the TF coils are placed

close enough together to approximate a

continuous solenoid bent into a toroid. Using

the terminology shown in Figure 3.5 gathered

from Weston Stacey’s text on the subject15,

we know from E&M that a solenoid exhibits

the following properties16:

( ) 2

(Eqn. 3.35)

From here, we can make use of a variant of

the Lorentz force law in differential form

where . Ideally, the only

appreciable force generated by the coils of

Figure 3.5 should reside in the toroidal (ϕ)

direction. Thus, the in the Lorentz law

becomes . Lastly, in order to make the following arguments more manageable, we will

assume that the average of the magnetic field throughout the poloidal plane is equal to half of

the field strength at the outer edge (R2). In equation form, this translates to:


27

Figure 3.6: Schematic of the

poloidal cross-section of an ideal

torus. The differential force is in

the direction of 𝑛𝑟 .

( ) (Eqn. 3.36)

Now, we make use of the preceding assumption by substituting it into the differential Lorentz

force law:

(

)

We can improve on the coordinate system of Figure 3.5 by viewing the radial distances as

relative to the midpoint R0. This makes sense considering that we are assuming a perfectly

circular poloidal cross section, and because all of the physics

that we are concerned with lie within the confines of the

toroid. This new coordinate system is depicted in Figure 3.6,

and it introduces a new variable (θ) which takes into account

the axial location of the point in question. Thus, by

introducing θ to the fray, we can subsequently eliminate any

use of (z) in the following argument. If we choose for θ to

equal zero when a point is located on the outermost edge of

the toroid, then we can write the radial distance of some

particular point using the following equation:

( ) (

( ))

If we denote the value as

, we can write the above

equation a bit more compactly:

( ( )) (Eqn. 3.37)

Plugging Eqn. 3.37 back into our current formulation for the Lorentz force:

( ( ))

The cross product in the numerator, in its current form, is somewhat ambiguous. We can clean

it up by pulling out the scalar , and taking the directional cross product using the right-hand

rule applied to figure 3.6:

( )

Whilst it is a given that we are accumulating a significant amount of r-terms (r, R, R0, etc…), it

should be remembered that represents the small vector extending from the center-point of


28

the toroid. Thus, we have shown that the differential force above is a force that extends in the

radial direction away from the toroidal vessel in all directions:

( ( )) (Eqn. 3.38)

At this point it makes sense to try to break this force into components R and Z, and attempt to

find the radial forces and axial forces exerted on the tokamak walls. Doing so is (relative)

straightforward integration considering we already have the force in differential form via Eqn.

3.38, and θ is the only variable we must take into consideration. To achieve this, we break up

into separate differentials for and , and make use of superposition in the end to find the

total force.

MAGNETIC FORCES IN THE RADIAL DIRECTION

Whilst the differential of Eqn. 3.38 may appear harmless to the untrained eye, in reality

integrating over θ will require multiple substitutions and trigonometric identities. The detailed

derivation that follows naturally beckons a separation of the R and Z integrals. Thus, this section

will solely be concerned with the radial forces. We can qualitatively deduce from Eqn. 3.38 that

as R becomes smaller, the force will become greater. Thus, we know before partaking in any

calculations that the inside toroidal wall will sustain greater forces. We begin the derivation by

taking a closed integral over Eqn. 3.38:

∮

( ( ))

To make this equation integrable, we separate the differential into ( ) to represent

the radial component. Making this change and pulling out the constants we have an equation

ready to be integrated:

∮

( )

( ( ))

(Eqn. 3.39)

The most straightforward way to begin tackling this integral is to make a u-substitution.

Choosing a substitution that actually leads to something meaningful takes some doing.

However, it has been determined that the most efficient way of transforming Eqn. 3.39 is to

make the somewhat ambiguous assumption that (

). Taking the differential, we

have

(

) . At this point, it is noteworthy to state the following three

trigonometric identities that will aid us in cleaning up the integrand with this new substitution:


29

(

) √

( )

(Eqn. 3.40a)

(

) √

( )

(Eqn. 3.40b)

Subsequently, we have the following equation for u:

(

) √

( )

( ) ( ( )) ( )

( ( )) ( ) ( ) ( )

From the final step, we can conclude that:

( )

Now, the reasoning for choosing such a u-substitution should be more apparent. We now have

a relatively compact way of constructing the cosines in our integrand. Likewise, due to the

secant present in our u-differential, we may now construct the following using Eqn. 3.40:

(

)

(

) (

( )

) (

)

( ) ( )

(

)

Substituting our newfound values for cos(θ) and dθ into Eqn. 3.39 we achieve the following:

∮

(

*

. (

*/

(

)

Throughout the substitution, I will omit the limits of the integral for two reasons. First, when

tan(θ/2) is re-substituted into the equation, the u-limits will become meaningless. Secondly,

taking the tangent of either 2π or 0 gives an upper and lower limit of zero. This tends to look

awkward and ambiguous. Cleaning up the integrand (and omitting the obvious ):

∮

( )

( ) . (

*/

∮

( ) ( )( )

∮

[ ]

∮

( )

( )( )


30

It may be realized from this final term that partial fractions (everybody’s favorite trick) may be

utilized to separate the integrand into two terms. The basic idea behind the partial fraction

separation is the following:

∮

( )

( )( )

*∫

∫

+

In other words, this mathematical trick allows for us to separate the fourth-order polynomial in

the denominator into a two-term, second order polynomial. We then solve for the terms A and

B in the following manner:

( ) ( ) ( )

Letting u = 1, we see that the above equation condenses to show that A = B. Therefore the

following equation holds true:

( ) ( )

Cancelling like terms and isolating A, we see that

. Substituting these values back into

our integrands and factoring some values:

*∫

( ) ∫

( ) ( ) +

In order to work with either of these integrals, we must make use of the following important

fact that will be treated as an axiom in this argument:

∫

(√

*

√ (Eqn. 3.41)

Putting Eqn. 3.41 to use on the first integral, we have . Thus ∫

( )

( ). Applying the fact to the second integral is relatively simple using the parenthetical

notation introduced in its last construction. Namely, ( ) ( ). Thus,

∫

( ) ( )

√( )( ) .√

/

Combining the two integrals back into one equation and cleaning up some terms we achieve

the following expression for the radial force:

0

( )

√ .√

/1


31

At this juncture, we are in a position to substitute the value for u (namely, u = tan(θ/2)) back

into the equation. Doing so gives us the following:

0

√ .√

(

)/1

We are now forced to evaluate the preceding anti-derivative as one normally would be

subtracting the function of 2π by the function of zero. However, we can save a great deal of

work by realizing that both 2π and zero will set the tangent function to zero. Subsequently, the

hyperbolic-arctangent of zero also equates to zero. Thus, the fairly complicated and messy

second term vanishes at both values. Naturally, theta also vanishes at zero so after all that work

we are left with and extremely simple value for the total radial force on the toroidal walls:

(Eqn. 3.42)

As such, if one knows the current passing through the coils and the number of coils present, the

radial force becomes a simple algebraic calculation. Of course, the preceding argument hinges

on the fact that the toroid is ideal and can thus be treated as a perfect solenoid. In reality, flux

leakage and non-uniform and non-linear geometry and plasma will lead to a very complex force

to accurately model.

MAGNETIC FORCES IN THE AXIAL DIRECTION

Fortunately, modeling the axial forces on the walls is a much simpler endeavor. This

stems back to the syntax introduced in Figure 3.6. In this case, the differential length

becomes , which leads to a more straightforward substitution. Likewise, due to the

uniformity and geometry of our ideal toroid, we must only integrate by half-a-turn and multiply

by 2 to find the total (as opposed to net) axial force. We start with the following equation:

∮

( )

( ( ))

(Eqn. 3.43)

If we simply take our u-substitution to be the denominator in Eqn. 3.43, then we may state the

following:

( )

( )

( )


32

Figure 3.7: The components of the radial and axial forces

and their superposition

Making the preceding substitutions, we have:

∮

( )

(

( ))

∮

After evaluating the integral we achieve a simple natural logarithm of u in the form | |. So,

changing the variable back to theta and evaluating at the limits we achieve:

| ( )|

|

| (Eqn. 3.44)

Interestingly, although the axial-derivation was simpler by milestones, the expression via Eqn.

3.44 is more complex in that it

requires knowledge of the inner

radii r of the toroid. To conclude

the argument with a meaningful

product, we can plug both Eqn.

3.44 and Eqn. 3.42 into

computational vector software

and come up with a depiction of

the radial and axial forces, and

their superposition in a uniform

toroid. Figure 3.7 depicts just

that, and demonstrates how adding the individual components of force via superposition leads

to a visual approximation of the real forces in relative-vector format on a hypothetical tokamak.


33

GREEN’S FUNCTION OF AXISYMMETRIC MAGNETOSTATICS AND

DISTRIBUTIONS

The motivation for constructing this section is chiefly due to the Ariola and

Pironti’s17 construction of the poloidal flux function written in terms of a Green’s Function:

𝜓 ( ) ∫ ( ) ( )

Being that I had never worked with (or been exposed to) a Green’s function, I quickly

became very intrigued as to how this equation could be derived. Even more grappling was

the task of isolating G(r,r’) and deriving its value into some concise form. As such, a

significant amount of my research became directed at the basic underpinnings of the

Green’s function and how it is defined. While Green’s Function is a longstanding method

for solving certain types of partial differential equations and is in no way directly tied to

the magnetostatics of plasma confinement, I have since learned that it plays a large and

useful role in ideal plasma physics, and I feel it deserves a semi-meaningful treatment in

this report. The following sections will attempt to build up the notion of a Green’s function

from properties of basic partial differential equations that lead to a distribution. In

conclusion, it will be stated just how scientists have allowed this special function to work in

plasma physics, and what the function itself signifies.

DISTRIBUTIONS AND THE DIRAC DELTA FUNCTION

To begin the derivation of a Green’s function, we must first construct the notion of

a generalized function or distribution. The best way to demonstrate this is to refer back to

a basic concept covered in introductory mechanics courses; the impulse. By nature, the

impulse deals with forces that act over very short periods of time. But in our case, we are

concerned with the mathematics of letting this small delta become even smaller (natural ly,

letting it approach zero). The fundamental way of writing the impulse is shown below:

J = ∫ ( )

(Eqn. 4.1)

Following through with the integration, this becomes:

∫

∫ ( )


34

3 2 1 0 1 2 3

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Figure 4.1: Graphing the Poisson kernel

over smaller values of y via Mathematica

software.

So, assuming non-relativistic reference frames, the impulse is solely dependent on some

quick change in velocity proportional to constant mass (momentum). We can visualize the

graph of this change as some relatively constant value suddenly shooting up to a higher

value. However, if we let t0 – t tend to zero, then we can reasonably say that the jump

from v0 to v1 is an instantaneous one. Moreover, in order to simplify the task at hand18 it is

convenient to let the area of the impulse in Eqn 4.1 to equal 1. Thus, with an

instantaneous jump from v0 to v1 we approximately have the unit step function. For this to

occur, the force f(t) (the forcing function) will have to act over an instantaneous delta, and

have an infinite amount of force. Worse, the very definition of Eqn. 1 says that f(t) must

take on the role of the slope of the impulse between t0 and t1, which is a discontinuity.

Fortunately, this example does not shake the very groundwork of what a function is.

Rather, f(t) sets us up to define something new that plays well with functions, but cannot

be expressed analytically like a function. Mathematicians have granted functions like f(t)

the name distribution, and the particular distribution we are interested in is the d irac-delta

function which is written as ( ). The notation can be misleading, as the function

doesn’t necessarily have to do with the difference between t and t0. Instead, it is a way of

conveying the fact that the distribution’s interesting effects occur when t is equal to t0.

Perhaps one of the best ways of defining the dirac-delta function is to use the Poisson

kernel and computational software to display the result based on different values of y. The

Poisson kernel is written as follows:

( )

( ) (Eqn. 4.2)

In this way, P(x,y) takes the vertical axis,

while x takes the horizontal. Consequently,

the area under P(x,y) will always equal 1. As

depicted in Figure 4.1, the differing values of

y will cause P(x,y) to develop very sharp

peaks centered on the vertical axis.

Specifically, these peaks are caused by

allowing y to take on smaller and smaller

values. Thus, a neat and rather elementary

way of writing the dirac-delta function can

be found by relating it to the Poisson kernel

as follows:

( ) *

( ) + (Eqn. 4.3)


35

Others19 typically attempt to illustrate the dirac-delta with increasingly elongated

rectangles of area 1. However, I feel the Poisson kernel is a more elegant representation.

There exists a rather crude way of obtaining a very useful formula concerning a continuous

function coupled to the dirac-delta function. If we take some general (continuous) function

( ) and multiply it by the dirac-delta and integrate we get the following expression:

∫ ( ) ( ) (Eqn. 4.4)

It is at this point where the rectangular approach to the dirac-delta becomes helpful. If we

re-write ( ) as ( ) such that fn takes on the form of area 1 rectangles

increasing in peak linearly with increasing n, then important properties begin to pop out.

First, if we take n to approach some ridiculously high number (but nowhere near infinity!)

then we can say that the time-delta is small enough over this range that ( )

( ). If this is the case, then gamma may be removed from the integral of

Eqn. 4.4, and we are left with the following expression:

( ) ∫ ( )

The integral is simply the area under the function fn over some very-small amount of time.

We know the area of these functions is always 1. Thus, we are left with only ( ).

Therefore, barring rigorous mathematical proofs outside the scope of this write -up, we

have determined the following property of the dirac-delta:

∫ ( ) ( ) { ( )

(Eqn. 4.5)

THE GREEN’S FUNCTION AND ITS PROPERTIES

With Eqn. 4.5 in hand, we are now in position to define the Green function. As you

may have gathered from the Green function-dependent equation resulting from the

differential elliptic operator in the chapter on magnetostatics, the Green function is a

mathematical trick (so to speak) for solving a partial differential equation. It works to turn

the P.D.E. into an easier, more manageable expression using the ideas of a distribution

formulated in the preceding section. The most efficient way in demonstrating its use is to

begin with an example. Specifically, we will begin with the non-homogenous simple

harmonic motion P.D.E. acted upon by some forcing function f(t).

( ) ( ) ( )


36

We can immediately make use of Eqn. 4.5 to re-write the function f(t) as follows:

( ) ∫ ( ) ( )

(Eqn. 4.6)

Since the function f(t) is integrated over all time consisting (possibly) of many impulsive

forces at different times, we can pick apart Eqn. 4.6 and choose to only concern ourselves

with a particular unit impulse at t’. As such, we are now solving the following P.D.E.:

( )

The fundamental principle here is that Mu isn’t necessarily equal to y due to our

manipulation of the forcing function. In fact, the first use of Mu was seen in the work of

mathematician George Green (1793-1841), so it this strange function μ is actually the

Green’s function, and it is denoted as G(t,t’). The notation signifies the particular impulse

at t’, while considering t as the independent variable. Therefore, the new P.D.E. is written

as follows:

( ) ( ) ( ) (Eqn. 4.7)

While it will be verified shortly, the solution for y based on this Green’s function is 20:

( ) ∫ ( ) ( )

(Eqn. 4.8)

Intuitively, the syntax of Eqn. 4.8 makes physical sense, as we formed G(t,t’) by doing away

with most of the force over time. Now, we attempt to “re-construct” y(t) by adding up all

those missing forces with via the Green function. Proving Eqn. 4.8 holds takes only a few

steps.

(

)

Now, placing Eqn. 4.8 in to substitute for y(t):

(

) ∫ ( ) ( )

∫ (

) ( ) ( )

∫ . ( )

( )/ ( )

∫ ( ) ( )

( )

The last followed from Eqn. 4.6. Thus, we have quickly proved that the Green’s function

is a solid way of solving the P.D.E., and that y(t) can be regained via the likes of Eqn. 4.8.

Solving for the actual function G(t,t’) itself, in this case, is as s imple as solving Eqn. 4.7 with


37

the usual tricks taught in a differential equations course, as the expression is no longer

partial!

The Green’s Function of Magnetohydrodynamics

With the ideas of the distribution and Green’s function fully deve loped, we are now

in a position to tackle the Green’s function of axisymmetric magnetostatics mentioned in

the preceding chapter. Truthfully, the methodology required to put the function G(r,r’) into

useful form (namely, one involving full elliptic integrals of the first and second kind) can be

said to be beyond the expertise of an undergraduate physicist. However, a significant

amount of time was spent in working different methodologies in gaining G(r,r’), so a quick-

and-dirty derivation will be presented excluding nearly all the details.

First, we want to recall the particular P.D.E. in question. The aforementioned formula

derived for the differential elliptic operator was stated to be the following:

𝜓 (Eqn. 4.9)

The differential elliptic (or Grad-Shafranov) operator is defined to be21:

( )

(

*

Or, following through with the differentiation:

(Eqn. 4.10)

Applying Eqn. 4.8 to this problem (and ensuring correct parameters for cylindrical

coordinates):

𝜓( ) ∫ ∫ ( ) ( )

(Eqn. 4.11)

The boldface-typed r signifies the vector-nature of the variable, as it concerns both the

and directions. (Remember that the toroidal direction ϕ is cyclic and uniform, so it is not

of concern.) It should be taken as a given22 that due to the definition of the differential in

cylindrical coordinates, the delta function is written as follows:

( ) ( ) ( )

Therefore, we use the methodology of Eqn. 4.7 on Eqn. 4.9 and achieve:


38

( ) ( ) ( ) (Eqn. 4.12)

We start on the quest to solve for G(r,r’) by taking the homogenous version of the above

equation into account; namely, 𝜓 Using perhaps the oldest trick in the physicist’s

book, we can allow 𝜓( ) ( ) ( ) via separation of variables. If we bestow the

“prime” notation to R-derivatives and “dot” notation to Z-derivatives, then the

homogeneous equation can be written as follows using Eqn. 4.10:

Dividing by Z and R,

(

)

(

)

Due to the fact that the R and Z terms equate as they do in the preceding equation despite

the different independent variables, we can say that both sides are constant. As is usually

the case, it is convenient to denote this constant as k2. As such, the Z(z) equation can be

written as follows:

This represents the familiar simple harmonic motion D.E., and has the following solution:

( ) ( ) ( ) (Eqn. 4.12)

The R(r) equation, however, is not so simple. Rearranging after the constant, we achieve

the following:

(Eqn. 4.13)

The form of Eqn. 4.13 resembles what is known as the modified Bessel Differential

Equation. Frank Bowman’s classical book on Bessel function gives the following formula 23:

( ) ( ) (Bessel D.E.)

* (

)

(

)+ (Solution)

where Jn and Yn are Bessel functions of the first and second kind. Also,q is a representative

variable that actually equals √ . Thus, referring back to Eqn. 4.13, we see that p = -

1, α2 = k2, β2 = 0, q = 1, and r = 1. So, using Bowman’s handy formula we arrive at a

representation for R(r):


39

( ) [ ( ) ( )]

While the imaginary parameters are somewhat normal for the modified Bessel’s, I (and

most others)24 prefer to eliminate complex variables from the scene. Thus, there is one

final Bessel Identity25 that will allow us to do just that:

( ) ( ) ( ) ( )

This fact transforms the complex-parameter Bessel functions into real modified Bessel

functions of the first and second kind. Although it is unnecessary to state the details of

these modified Bessels, it is crucial to note that Iβ tends to infinity as its independent-

variable approaches infinity. This is not physically helpful or meaningful, so we “assume”

(no hard proof will be shown here) that constant C equals zero, and R(r) can finally be

written as follows:

( ) ( ) (Eqn. 4.13)

Due to the fact that we have broken the poloidal flux (𝟁) up into different eigenvalues

based upon the constant k, we are left with the following function following separation of

variables:

𝜓( ) ( ) ( ) ( ( ) ( ))

So, in order to obtain the total poloidal flux, we simply integrate over all possible k-values.

Doing this and allowing the constants C3 and C2 to absorb C1:

𝜓( ) ∫ ( ) ( ( ) ( ))

(Eqn. 4.14)

Now, Arfken26 has a way of re-writing the dirac-delta function mentioned earlier in

cylindrical coordinates using some correlations of the delta to its Fourier Transform which

were unmentioned in the preceding sections. The new way of writing this is as follows:

( ) ( )

∫ ( )

However, we can avoid the issue of having negative values of k by realizing the even

integrand and making use of Euler’s equation as follows:

( )

( ) ∫ ( ( ))

(Eqn. 4.15)

Therefore, cleaning up the leading term with a unknown function ak and realizing the

G(r,r’) will assume a similar form due to the findings of the homogeneous solution, we

construct the Green’s function:


40

( ) ∫ ( ) [ ( )]

(Eqn. 4.16)

Now that we have the Green’s function, the rest is simply play in order to get a value for a k

and re-write the integral in some form useful to our applications. The remainder of the

work on Eqn. 4.16 will simply be glossed over as it contains math beyond the scope of this

report. 27 goes on to plug Eqn. 4.16 into Eqn. 4.12 and rearrange so that the following

equation is produced:

( )

( ) (Eqn. 4.17)

Pg. 914 of Arfken’s Mathematical Methods book28 re-writes this in the following interesting

way:

( )

( ) ( ) (Eqn. 4.18)

Where r< and r.> are the min and max of {r,r’}, respectively. Plugging this value of a k back

into Eqn. 4.16, we achieve a known function for G(r,r’):

( ) ∫ *

( ) ( )+ [ ( )]

∫ [ ( ) ( )] [ ( )]

(Eqn. 4.19)

Finally, two concluding identities will lead us to a function involving the elliptic integrals

we have worked so hard to achieve. Firstly, it may be apparent that Eqn. 4.19 gives G(r,r’)

in a form that can be dealt with using the Fourier Transform. Erdelyi 29 (pg. 49, transform

47) gives a formula particularly useful in dealing with Eqn. 4.19:

( ) ( ) ( ) ( ) ∫ ( ) ( )

( )

√

*

+ (Eqn. 4.20)

Therefore, it is obvious that the transform we will be conducting is a cosine transform.

After matching up the constants and variables in Eqn 4.19 with Eqn 4.20 (v = 1, a = r, b = r’,

= (z-z’)), we achieve the following:

( ) √

[ ( )

] (Eqn. 4.21)

It is now worth mentioning that Qn represents the Legendre function of the second kind.

This is not of too much importance, however, as Abramowitz’s classical book on


41

mathematical functions30 (8.13.12) gives the following direct relation to the elliptic

integral:

[ ] √

0 .√

/ ( ) .√

/1 (Eqn. 4.22)

K and E in the above equation represent the complete elliptic integrals of the first and

second kind, respectively. So, once again matching our variables in Eqn. 4.21 to those of

Eqn. 4.22, we achieve a Green’s function in terms of elliptic integrals:

( ) √

√

. ( )

/

[( ( )

) (√

. ( )

/

)

. ( ( )

*/ (

√

. ( )

/

)]

But, if we simplify the terms under the roots we see that r and r’ factor as follows:

( ) √

√

( ) ( )

0( ( )

* .√

( ) ( ) / (

( ) ( )

* .√

( ) ( ) /1

We can do much to simplify this messy function by defining a new constant 𝝹 (unrelated to

the earlier constant k) as follows:

( ) ( ) (Eqn. 4.23)

Substituting this into the Green’s function equation, the function condenses as follows:

( ) √

√ * (

) (√ )

(√ )+

Finally, if we simplify and factor out a

, we obtain our final expression for G(r,r’):

( ) √

* (

) ( ) ( )+ (Eqn. 4.24)


42

In the form of Eqn. 4.24, the Green function, and subsequently the poloidal flux, can be

determined to a large degree of accuracy using computational software such as

Mathematica or MatLab. The tedious work involved in finding the Green function in the

form of Eqn. 4.24 can be justified via the benefit in being able to solve the homogenous

Grad-Shafranov equation. This equation plays an important role in determining the plasma

shape so that numerical algorithms developed by plasma scientists may obtain an

approximation of the poloidal flux function fitting the available magnetic measurements.31

In summary, the Green function developed in Eqn. 4.24 satisfies the homogenous version

of Eqn. 4.9 everywhere with the exception of the point r’, as this is where the impulsive

source is located.


43

APPENDIX

A.1 – The Divergence of the Cross Product

It is stated in Chapter three that the current density J is divergenceless due to its

relationship with magnetic field in Ampere’s Law. Specifically, this is due to the

mathematical truism of the divergence of the cross product being zero. The goal of this

section is to make quick proof of this fact using Ampere’s Law as the example. Thus, we

begin by taking the divergence of Eqn. 3.5 as follows:

( ) [ ] ( ) 0

( )

1

The final terms, of course, results from our usage of the divergence in cylindrical

coordinates. Now, if we work on the other side of the equation, we generate the following:

*

+ *

+

[

( )

]

Continuing along by taking the divergence of the above equation, we end up with the

following mess:

( )

*

+

*

+

,

*

( )

+-

This actually isn’t quite the mess that it appears to be. If we take a close look at each

individual term, we see that any particular component of B is being differentiated over an

orthogonal direction. Naturally, each component only changes in space over directions in

line with the component. Changes in any other direction equates to zero. Thus, each

derivative in the above equation equates to zero. Subsequently, the divergence of the curl

is zero, and we have the following expressions:

( ) (Eqn. A.1.1)

( ) (Eqn. A.1.2)

Thus, we can state with mathematical proof that the current density is indeed a

divergenceless vector.


44

A.2 – Formulating an Expression for Current Density

To begin our formulation of the current density in compact form, we start with

none other than Ampere’s law (Eqn. 3.5). Using what we learned about the vector B

through Eqn. 3.25, we can formulate the following expression:

( ) ( 𝜓 ) (Eqn. A.2.1)

It will be beneficial to separate the above equation into two terms and work them both

separately. As such, we will denote as Term 1, and ( 𝜓 ) as Term 2.

TERM 1

We can start to decompose Term 1 using the determinate method of the cylindrical

curl as follows:

𝜓 ||

||

[ ]

[ ]

*

( )+

*

( )+

*

+

*(

)

+

TERM 2

Term 2 can be decomposed using the same procedure as Term 1 by recalling the

value of 𝜓 (Eqn. 3.18b):

( 𝜓 ) ||

||

*

+

[

*

+

*

+]

*

+

The first and last terms in the above equation go to zero due to the cyclic nature of ϕ.

Therefore we are left with the following expression for Term 2:

( 𝜓 ) 0

𝜓

*

+1


45

So, combining terms 1 and 2 we have the following expression for the Ampere’s

Law:

*(

)

+ 0

𝜓

*

+1

If we once again make the substitution of ( ), we have the following:

*(

)

+ 0

𝜓

*

+1 (Eqn. A.2.2)

Now, if we had values for and 𝜓 we could (relatively) easily solve Eqn. A.2.2 for the

magnetic field. However, if we want a compact form similar to that of Eqn. 3.25 for B, we

still have a bit of work to do. We can start by dividing up the two terms in the above

equation into separate derivations as we did before. We will label the first term as Term A

and the second as Term B.

TERM A

We first note that the forcing function is solely a function of radial and axial

components. Thus, we can write the gradient of F as follows:

( )

Recalling that the gradient of the toroidal angle is

, if we work out the cross product

between it and , we get an interesting result:

||

||

*

+

Interestingly enough, this comes out to exactly equal Term A. Thus, we have developed an

extremely compact way of writing half of Eqn. A.2.2 without the mess of differentials.

TERM B

Unfortunately, Term B cannot be constructed using as elegant a method as Term A

did. The following steps are more the result of a guess-and-check approach, and substation

of new operators as needed. We see that the common operand in Term B is the poloidal

flux function 𝜓. Thus, we can begin by writing the gradient of this function:


46

𝜓

Multiplying this by

we get:

𝜓

Now, taking the divergence of this new equation:

( 𝜓)

(

)

(

)

*

+

𝜓

Next, if we multiply this equation by a r2 this time, we will achieve the value inside the

brackets of term B:

( 𝜓)

*

+

𝜓

Some peer-reviewed articles will leave the term in this form. However, we can do one

better by making use of what has been dubbed the differential elliptic operator. The

operator is defined as ( ) where 𝞆 is some general function represented in

cylindrical coordinates. Now, if we substitute the elliptic operator in the above equation

and multiply by

, we achieve:

( 𝜓)( )

*

*

+

𝜓+ 𝜓

This is exactly what defines Term B.

Piecing the two terms together we now have an amazingly compact way to write

Ampere’s law using MHD in cylindrical coordinates:

𝜓 (Eqn. A.2.3)

But, we know from Eqn. 3.5 that this also equals magnetic permittivity times the current

density. Thus we can finally construct a meaningful expression for the vector J:

𝜓 (Eqn. A.2.4)

Thus, we have proved that current density is simply a function of the forcing function F,

the poloidal flux 𝜓, and radial/axial distance. The compact nature of A.2.4 not only hides

nearly all of the minute details, but can also be evaluated in this form using modern

computational software. Demonstrating this, however, is beyond the scope of this report.


47

WORKS CITED Arfken, G. (1985). Mathematical Methods for Physicists (3rd Ed.). Academic Press.

Arzimovich, L. A. (1965). Elementary Plasma Physics. New York: Blaisdell Publishing Company.

Boas, M. (2006). Mathematical Methods in the Physical Sciences. Hoboken, NJ: John Wiley &

Sons.

Bowman, F. (1958). Introduction To Bessel Functions. Courier Dover Publications.

Bromberg, J. L. (1972). Fusion: Science, Politics, and the Invention of a New Energy Source.

London, England: The MIT Press.

Erdelyi, A. (1954). Tables of Integral Transforms. New York: McGraw-Hill.

F. Dini, S. K. (2004). Green Function of Axisymmetric Magnetostatics. Iranian Journal of Science

& Technology, Vol. 28, No. A2.

Freidberg, J. (2007). Plasma Physics & Fusion Energy. Cambridge: Cambridge University Press.

George Arfken, H. W. (2005). Mathematical Methods for Physicists. Burlington, MA: Elsevier.

Griffiths, D. J. (1999). Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice Hall.

J. Blum, J. L. (Vol. 24). The self-consistent equilibrium and diffusion code SCED. Computer

Physics Communications, 235-254.

M. Abramowitz, I. S. (1965). Handbook of Mathematical Functions. Dover.

Marco Ariola, A. P. (2008). Magnetic Control of Tokamak Plasmas. London: Springer-Verlag.

National Institute of Standards and Technology. (n.d.). Green's Function Tutorial: Introduction to

Green's Functions. Retrieved March 1, 2011, from Materials Reliability Division:

http://www.boulder.nist.gov/div853/greenfn/tutorial.html#example

Seife, C. (2008). Sun in a Bottle: The Strange History of Fusion and the Science of Wishful

Thinking. Viking Adult.

Stacey, W. (2005). Fusion Plasma Physics. Weinheim: Wiley-VCH.

Stacey, W. M. (2010). Fusion: An introduction to the Physics and Technology of Magnetic

Confinement Fusion. Weinheim: Wiley-VCH Verlag GmbH & Co.


48

ENDNOTES

1 Seife, C. (2008). Sun in a Bottle: The Strange History of Fusion and the Science of Wishful Thinking. Viking Adult. 2 Bromberg, J. L. (1972). Fusion: Science, Politics, and the Invention of a New Energy Source. London, England: The MIT Press. 3 Arzimovich, L. A. (1965). Elementary Plasma Physics. New York: Blaisdell Publishing Company. 4 Stacey, W. (2005). Fusion Plasma Physics. Weinheim: Wiley-VCH. 5 Marco Ariola, A. P. (2008). Magnetic Control of Tokamak Plasmas. London: Springer-Verlag.

6 Freidberg, J. (2007). Plasma Physics & Fusion Energy. Cambridge: Cambridge University Press.

7 Stacey, W. (2005). Fusion Plasma Physics. Weinheim: Wiley-VCH.

8 Stacey, W. M. (2010). Fusion: An introduction to the Physics and Technology of Magnetic Confinement Fusion.

Weinheim: Wiley-VCH Verlag GmbH & Co.

9 Marco Ariola, A. P. (2008). Magnetic Control of Tokamak Plasmas. London: Springer-Verlag.

10 J. Blum, J. L. (Vol. 24). The self-consistent equilibrium and diffusion code SCED. Computer Physics

Communications, 235-254.

11 Stacey, W. M. (2010). Fusion: An introduction to the Physics and Technology of Magnetic Confinement Fusion. Weinheim: Wiley-VCH Verlag GmbH & Co

12 Griffiths, D. J. (1999). Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice Hall.

13 Griffiths, D. J. (1999). Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice Hall


15 Stacey, W. M. (2010). Fusion: An introduction to the Physics and Technology of Magnetic Confinement Fusion. Weinheim: Wiley-VCH Verlag GmbH & Co 16 Griffiths, D. J. (1999). Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice Hall 17 Marco Ariola, A. P. (2008). Magnetic Control of Tokamak Plasmas. London: Springer-Verlag 18 Boas, M. (2006). Mathematical Methods in the Physical Sciences. Hoboken, NJ: John Wiley & Sons

19 George Arfken, H. W. (2005). Mathematical Methods for Physicists. Burlington, MA: Elsevier.

20 National Institute of Standards and Technology. (n.d.). Green's Function Tutorial: Introduction to Green's Functions. Retrieved March 1, 2011, from Materials Reliability Division: http://www.boulder.nist.gov/div853/greenfn/tutorial.html#example


22 Weisstein, Eric W. "Delta Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DeltaFunction.html

23 Bowman, F. (1958). Introduction To Bessel Functions. Courier Dover Publications.


49

24 F. Dini, S. K. (2004). Green Function of Axisymmetric Magnetostatics. Iranian Journal of Science & Technology,

Vol. 28, No. A2.

25 Weisstein, Eric W. "Bessel Function of the First Kind." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

26 Arfken, G. (1985). Mathematical Methods for Physicists (3rd Ed.). Academic Press.

27 F. Dini, S. K. (2004). Green Function of Axisymmetric Magnetostatics. Iranian Journal of Science & Technology, Vol. 28, No. A2.

28 Arfken, G. (1985). Mathematical Methods for Physicists (3rd Ed.). Academic Press.

29 Erdelyi, A. (1954). Tables of Integral Transforms. New York: McGraw-Hill

30 M. Abramowitz, I. S. (1965). Handbook of Mathematical Functions. Dover.


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