wave-driven rotation and mass separation in rotating magnetic mirrors

8/3/2019 Wave-driven rotation and mass separation in rotating magnetic mirrors

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Wave-driven rotation and mass separation in

rotating magnetic mirrors

Abraham J. Fetterman

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Astrophysical Sciences Program in Plasma Physics

Adviser: Nathaniel J. Fisch

January 2012


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c Copyright by Abraham J. Fetterman, 2011.All Rights Reserved


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Abstract

Axisymmetric mirrors are attractive for fusion because of their simplicity, high plasma pressure

at a given magnetic pressure, and steady state operation. Their subclass, rotating mirrors, are

particularly interesting because they have increased parallel confinement, magnetohydrodynamicstability, and a natural heating mechanism.

This thesis finds and explores an unusual effect in supersonically rotating plasmas: particles

are diffused by waves in both potential energy and kinetic energy. Extending the alpha channeling

concept to rotating plasmas, the alpha particles may be removed at low energy through the loss

cone, and the energy lost may be transferred to the radial electric field. This eliminates the need

for electrodes in the mirror throat, which have presented serious technical issues in past rotating

plasma devices.

A high azimuthal mode number perturbation on the magnetic field is a particularly simple way

to achieve the latter effect. In the rotating frame, this perturbation is seen as a wave near the

alpha particle cyclotron harmonic, and can break the azimuthal symmetry and magnetic moment

conservation without changing the particles total energy. The particle may exit if it reduces its

kinetic energy and becomes more trapped if it gains kinetic energy, leading to a steady state current

that maintains the field. Simulations of single particles in rotating mirrors show that a stationary

wave can extract enough energy from alpha particles for a reactor to be self-sustaining.

In the same way, rotation can be produced in non-fusion plasmas. Waves are identified to produce

rotation in plasma centrifuges, which separate isotopes based on their mass difference.

Finally, a new high throughput mass filter which is well suited to separating nuclear waste is

presented. The new filter, the magnetic centrifugal mass filter (MCMF), has well confined output

streams and less potential for nuclear proliferation than competing technologies. To assess the

usefulness of the MCMF, a metric for comparing mass filters is developed. With this metric, the

MCMF is compared with other mass filters such as the Ohkawa filter and the conventional plasma

centrifuge.

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Acknowledgements

This thesis exists only due to the extraordinary support of many individuals. Foremost among them

is my adviser Nat Fisch, who has patiently walked me through basic problem analysis enough times

that I can hear him in my head. His moral support and encouragement have never flagged, and hehas been remarkably understanding of odd hours and locations. I look up to his ability to be both

critical and constructive, and I feel I will never be able to learn enough from him. I sincerely hope

our collaborations will continue in the not-distant future.

This interaction, and my entire experience at Princeton, has only been possible through the

super-human patience of Barbara Sarfaty, the graduate program administrator, but truly the glue

that holds everyone together.

I could not continue if I did not thank my fiancee Lisa Qiu. She has loved and supported me

under extreme conditions of thesis-writing, research, and worrying about research. The happiness

of time spent with her provides light in darkness, and has inspired a flurry of productivity this past

year. I eagerly look forward to our lifetime together. I love you KINICHI!!

I am grateful for the loving support of my parents, Bob and Deb Fetterman, as well as my brother

Aaron. I have been blessed by their constant support and encouragement, and would not be who I

am today without them.

I am also very grateful for my many co-conspirators at Princeton. Mike Sekora has inspired me

to strive for awesomeness. Seth Dorfman has helped me produce an epic video, eat many foods, and

hear many enigmatic jokes. Danielle Fong has supported my preposterous ideas and eventually hired

me. Josh Kallman, Laura Berzak, Adam Hopkins, and Jess Baumgartel kept me entertained, while

my fellow party officers, Luc Peterson, Nate Ferraro and Lee Ellison kept me happily distracted

while at lab.

I appreciate the support of Andrey Zhmoginov, who helped me get my bearings on mirror

plasmas, simulation, and diffusion paths. I would like to thank Yevgeny Raitses for a year of

intensive experimentalist boot camp, and for ongoing useful discussions in various realms.

Thanks to my readers Hong Qin and Phil Efthimion, who dealt graciously with very shortdeadlines.

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For Lisa, my love, the light of my life.

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Mass separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theory of rotating plasmas 7

2.1 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Wave-Driven Rotation in Mirrors 12

3.1 Branching ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Producing rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Alpha Channeling 19

4.1 Diffusion paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Plasma response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Reactor implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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5 Contained modes 30

5.1 Eigenmode equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Contained mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Monotonic rotation profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4 Cyclotron damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.5 Application of modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Countercurrent plasma centrifuge 42

6.1 Comparison with gas centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.2 Overview of design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.4 Countercurrent production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 The magnetic centrifugal mass filter 54

8 Metrics for comparing plasma mass filters 59

8.1 Nuclear waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.2 Plasma mass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8.3 Plasma centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.4 Ohkawa filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.4.1 Light Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.4.2 Heavy Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.4.3 Separative Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.5 Magnetic centrifugal mass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.5.1 Light boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.5.2 Heavy boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.5.3 Separative Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8.6 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.6.1 Separative power per volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.6.2 Energy use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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9 The need for a high throughput mass filter 76

9.1 Plasma mass separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

9.2 Comparative utility of a separation method . . . . . . . . . . . . . . . . . . . . . . . 79

9.3 Utility of mass separation in waste management . . . . . . . . . . . . . . . . . . . . . 81

9.3.1 Hanford Nuclear Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.3.2 Spent nuclear fuel reprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

10 Conclusions and Future Work 86

10.1 K ey Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

10.2 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.3 Mass filter concept development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.4 Fusion concept development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A Diffusion relations 92

B Derivation of effective potential 94

C GDT Proposal 96

C.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C.3 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

C.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Chapter 1

Introduction

Devices with rotating plasmas provide unique opportunities for particle manipulation [Lehnert, 1964].One reason is that the radial electric field which used to produce the rotation provides an energy

source that can readily be tapped into. Another reason is that the centrifugal force produced by

rotation naturally discriminates between heavy and light particles.

The goal of this thesis is to explore how rotating mirror plasmas may benefit from new physics

ideas (alpha channeling) for the purpose of controlled nuclear fusion, and to examine how techniques

introduced here could be used for new applications (nuclear waste separation).

First, the alpha channeling concept is extended to super-thermal rotating mirrors. It is shown

that energy can be transferred from waves or particles into rotation. Because of past difficulty

in exceeding the Alfven critical ionization velocity (CIV) in rotating mirrors, the use of methods

described here to drive rotation could transform rotating mirrors into practical devices. Examples

for driving rotation are explained for both fusion and isotope separation problems.

Secondly, the magnetic centrifugal mass filter (MCMF), a new concept for separating nuclear

waste, is introduced. The MCMF is compared to other plasma mass separation methods, and its

use as a high throughput mass filter is proposed.

1.1 Background

A rotating magnetic mirror is a simple mirror with a radial electric field. This electric field creates

an azimuthal drift that is the same for all charged particles. We can imagine that the field lines

or flux tubes spin around the device axis. The ions are confined to the midplane like a bead being

spun on a string. Although electrons initially escape, as they leave, the device becomes positively

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Figure 1.1:

charged. This causes it to hold in electrons and expel ions. If the device spins rapidly compared to

the ion thermal speed, the confinement along field lines can be improved dramatically.

A charged central electrode can produce the radial electric field [Lehnert, 1971]. Alternatively,

this field can be generated and better controlled with segmented electrodes in or past the mir-

ror throat. In either case, good conductivity between the plasma and the electrode is essen-

tial [Bekhtenev and Volosov, 1978]. Unfortunately, good conductivity to electrodes frequently leads

to unacceptable particle losses.

A wide variety of rotating devices have been constructed in the past 50 years [Lehnert, 1971].

Early devices had low levels of ionization and were poorly diagnosed, which made rigorous analysis

difficult. However, the devices were comparable to current machines, with radii on the order of 30

cm and estimated densities on the order of 1015/cm3. An important property of these devices is that

none achieved a rotation speed exceeding the Alfven critical ionization velocity (CIV) [Alfven, 1960,

Lai, 2001].

The Alfven CIV is the speed at which the ion rotation energy equals the ionization potential of

neutrals. Adding energy to increase the rotation speed above the CIV limit only results in increased

ionization of neutrals. The CIV limit persists even in fully ionized plasmas due to creation of neutrals

through sputtering of end electrodes or end insulators [Lehnert et al., 1966]. The CIV limit has been

observed in experiments over many orders of magnitude in density [Lehnert, 1971, Lehnert, 1974b].

The Alfven CIV limit was finally broken around 1991 at the PSP-II device in Novosibirsk

[Abdrashitov et al., 1991]. In this device, extensive electrode conditioning and shaping was used

to reduce sputtering and the plasma density was extremely low [Volosov, 1984]. The result was a

rotation speed of 2 106 m/s, 40 times larger than the Alfven CIV, unambiguously showing thatit is possible to exceed the CIV limit. Another major accomplishment on PSP-II was the demon-

stration of a magnetohydrodynamically (MHD) stable mirror plasma using only sheared rotation for

stabilization [Volosov, 2009].

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1.2 Fusion

Axisymmetric mirrors might be attractive fusion reactors because they are simple, naturally steady

state, and able to achieve high beta values. Compared to closed devices, the open geometry allows

easier fueling and ash removal along field lines. They also may require less neutron shielding thantoroidal devices because the plasma is simply connected.

Mirrors with supersonic rotation have particular advantages. However, there is a principal issue

with simple mirrors, the lack of parallel confinement. In rotating mirrors this issue is reduced by

the strong centrifugal potential [Lehnert, 1974b]. While simple mirrors have an energy confinement

time that is proportional to ln Rm, where Rm = Bm/B0 is the magnetic mirror ratio, supersonic

mirrors have confinement times that scale like eRm (see Chapter 2 and [Bekhtenev et al., 1980]).

In addition to poor parallel confinement, simple axisymmetric mirrors suffer from MHD and

phase-space instabilities [Ryutov, 1988]. While non-axisymmetric mirrors can benefit from MHD

stable anchors, they are subject to much faster radial transport. Sheared rotation seems to be

the most natural way to produce stabilization while keeping axisymmetry, and the power of sheared

rotation to produce stability has been emphasized by the PSP-II and more recent gas dynamic trap

(GDT) experiments in Novosibirsk, Russia [Beklemishev et al., 2010, Volosov, 2009]. In terms of

phase-space instabilities, mirrors with supersonic rotation have large enough confinement potentials

that ions are Maxwellian, so that loss cone instabilities are not an issue.

In addition to having better parallel energy confinement and stability, the rotating mirror nat-

urally addresses the issue of plasma heating. As soon as a neutral is ionized, the ion and electron

drift apart in the magnetic field, each moving radially toward the oppositely charged electrode. Each

particle gains a kinetic energy 2WE0 from the field, where WE0 = mv2E/2 is the rotation energy and

vE = Er/Bz is the EB drift velocity. Half of the kinetic energy is in directed motion (rotation),and half is in perpendicular kinetic energy that may thermalize with the plasma. The rotation

clearly has a bias toward ion heating because of the much larger ion mass.

The heating is extremely efficient because most of the energy can be recovered when particles

leave the trap. This is because ions leave at a small radius (reducing directed kinetic energy) andmust overcome the centrifugal potential (reducing thermal kinetic energy). For a mirror with radial

mirror ratio Rr = r20/r

2m, the energy 2(1 R1r )WE0 will naturally be recovered if particles are

removed at the mirror throat. This means that only 2WE0/Rr must be spent for each particle

passing through the system.

The efficiency can actually be improved further by using end electrodes with a specific shape

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[Bekhtenev et al., 1980, Volosov, 2005]. These electrodes remove particles where they are at peak

potential energy (i.e., where the thermal velocity is in the opposite direction as the rotation velocity).

In the usual orientation (outer wall charged positive), this is at the outer radius of the gyro motion

for ions and the inner radius for electrons. In theory, it is possible to recover over 99% of the ion

energy using continuous shaped end electrodes or insulators [Volosov, 2005].

The high recovery efficiency has led to the suggestion of using centrifugal traps for non-equilibrium

fusion of p 11 B [Volosov, 2006]. The asymmetrical centrifugal trap (ACT) proposed for this pur-pose is actually a reverse mirror, with lower magnetic field at the ends than the midplane, which

allows more efficient direct conversion in the end regions. The unique confinement properties of

this device inspired us to suggest a plasma mass filter for nuclear waste separation, the magnetic

centrifugal mass filter (MCMF).

1.3 Mass separation

Early in the study of rotating plasmas it was realized that ions of different mass would be sepa-

rated radially by collisions [Bonnevier, 1967]. This was demonstrated in early devices, but sepa-

ration remained low because of the high neutral densities [Bonnevier, 1971]. The Alfven critical

ionization velocity also limited the achieved rotation speeds. It was eventually shown that be-

cause of heating, partially ionized plasmas were fundamentally limited in possible separation fac-

tor [Wijnakker and Granneman, 1980].This was partly overcome by the vacuum-arc plasma centrifuge [Krishnan et al., 1981]. These

used fully ionized plasmas produced by a pulsed arc, and demonstrated enrichment of zirconium

isotopes by 1700% [Krishnan, 1983]. Large enrichment ratios were produced by maintaining a low

temperature, on the order of 1 eV, achievable because of the very low neutral density. The Alfven

CIV was not exceeded in these experiments by a large factor.

In more recent years, interest has grown in using rotating plasmas to separate nuclear waste

[Ohkawa and Miller, 2002]. Plasma mass filters can be superior to chemical filters because they can

replace several separation steps with one, they are not sensitive to the chemical composition of the

input feed, and they do not need recycling or recapturing of water or salts.

Nuclear waste is seen as a simpler separation problem than isotope enrichment. To reduce the

total mass of high level waste, the mass filter is used to separate bulk elements like aluminum from

strontium-90 and other radioactive fission products. This could result in a four-fold reduction in

the high level waste mass [Gilleland et al., 2002]. It is possible that reducing the mass of waste will

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significantly reduce the cost of processing and disposing of the waste.

The Archimedes Technology Group pursued rotating mirrors for this purpose, using a plasma

mass filter design first suggested by Ohkawa [Ohkawa and Miller, 2002]. In this filter, the rotation

is such that the outside wall is the negative electrode and serves as the collection point for heavy

particles. The rotation speed is chosen so that heavy ions, which experience a larger centrifugal

force, cannot be confined radially by the magnetic field, while light particles are magnetized and

removed axially from the device.

1.4 Opportunities

This thesis focuses on a few ways rotating plasmas might be more effective for the above purposes. In

tokamaks, rapid removal of alpha particles might double the fusion reactivity [Fisch and Rax, 1992].

The rapid removal of alpha particles with waves, or alpha channeling, was recently extended to

mirrors [Fisch, 2006]. Here we extend the concept further to rotating mirrors. A unique aspect is

that alpha particle energy can be transferred to the radial electric field rather than a wave. This

allows a direct conversion of the alpha particle energy back to ion energy.

This concept led us to a more general idea, driving rotation in rotating mirrors with waves.

The end electrodes have always caused difficulty because the need for conductivity with the central

plasma conflicts with the low density needed to avoid the Alfven CIV limit. Driving the rotation

volumetrically with waves would allow us to avoid the CIV limit by design. This is therefore not just an improvement, but a fundamental change in the potential of rotating mirrors.

Because of the usefulness of driving rotation with waves, we describe how rotation can be driven

in the absence of a kinetic energy source, for example in a plasma centrifuge. We found that the

radial potential energy in rotating mirrors can be exchanged for particle kinetic energy or wave

energy. To describe possible ion-wave-potential interactions, we define the branching ratio fE , the

increase in a particles potential energy over the decrease in kinetic energy.

A convenient way to convert alpha particle energy to rotation energy is a fixed magnetic ripple,

which has a branching ratio fE = 1. The fixed ripple appears as a wave at the alpha particle

cyclotron frequency in the rotating frame, so it can cause radial diffusion of the ions. Because the

ripple has no energy in the rest frame, it consumes little or no power, resulting in efficient conversion

of alpha particle energy.

We have also identified opportunities to improve the Ohkawa type mass filter. A number of

issues are apparent in the Ohkawa design: heavy particles are broadly dispersed in the device, they

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consume a significant amount of energy, multiply charged ions may be mis-identified, and the device

must be collisionless. We developed a scheme to separate particles by mass that does not have these

issues, called the magnetic centrifugal mass filter (MCMF). We also developed a metric to compare

our new device to the Ohkawa filter and to the plasma centrifuge.

1.5 Organization of the thesis

In Chapter 2 we will introduce some basic theory on rotating plasmas. We will describe the cen-

trifugal confinement produced by rotation, and briefly cover MHD stability.

Chapter 3 will introduce wave-particle interactions and the branching ratio. Creation of diffusion

paths will be discussed, but applications will not be described until later.

Chapter 4 will discuss alpha channeling, stationary waves, and some simulation results.

Chapter 5 will describe a contained Alfven eigenmode that is unique to plasmas with strong

sheared rotation. This could potentially increase the efficiency of exciting stationary waves by an

order of magnitude.

Chapter 6 will discuss using waves to drive rotation in a plasma centrifuge. It will also suggest

ways to drive a countercurrent flow pattern using waves, which could multiply the separative power

of the plasma centrifuge.

Chapter 7 will then discuss a new mass separation device, the magnetic centrifugal mass filter

(MCMF).Chapter 8 will suggest a metric for comparing mass filters, and derive the value for the plasma

centrifuge, Ohkawa filter and MCMF.

Chapter 9 will review the need for mass filters in nuclear waste disposal.

Finally, Chapter 10 will provide some concluding thoughts.

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Chapter 2

Theory of rotating plasmas

Rotating mirrors have the magnetic geometry of a simple mirror, in which two high field mirrorcoils are attached to the two ends of a solenoid. Electrodes in or beyond the mirror throat produce

a radial electric field, and the resulting E B drift produces azimuthal rotation. With sufficientelectron mobility, the field lines are equipotentials and so the electric field is always perpendicular

to the magnetic field.

A consequence of equipotential field lines is that the E B rotation about the axis at a constantfrequency . This property is known as isorotation [Lehnert, 1971]. We show this by considering

a tube of flux with width dL (in the r z plane). The electric field E is related to the electric

potential by E = d/dL, and the magnetic flux is related to the field B by d = 2rBdL.The rotation frequency may then be written as,

= E/ (rB) = 2d/d. (2.1)

If is just a function of , the rotation frequency is also a function of onlyit may vary across

field lines but not along the same field line.

Because particles in the crossed E and B fields rotate about the axis at constant frequency, there

is a component of the centrifugal force along the magnetic field line pushing from the mirror throat

(small radius) to the midplane (larger radius). This results in a centrifugal potential that confines

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heavy particles to the midplane. The magnitude of the centrifugal potential is,

Uc =1

2m2r20

1

2m2r2m,

=1

2

m2r20 1 r2m/r

20 ,

= WE0

1 R1m

. (2.2)

Here r0 is the field line radius at the midplane, rm is the field line radius at the mirror throat,

Rm = Bm/B0 = r20/r

2m is the mirror ratio, and WE0 =

12

m2r20 is the rotation energy of a particle

at the midplane. The centrifugal potential for ions Uci can be several times the ion thermal energy

Ti, which results in significant improvement in the ion confinement time.

2.1 Confinement

We can get an estimate for Uci/Ti using the equations for ion and electron energy balance,

WE0i

1 R1m e0 + Wmi + Qie = WE0i, (2.3)

WE0e

1 R1m

+ e0 + Wme Qie = WE0e. (2.4)

Here 0 is the ambipolar potential of the mirror, Wm is the energy of a particle when it leaves through

the mirror throat, and Qie

is the energy transferred from ions to electrons. Because mi

me

, the

electron rotation energy WE0e is negligible,

WE0i

1 R1m e0 + Wmi + Qie = WE0i, (2.5)

e0 + Wme Qie = 0. (2.6)

If the confining potential is much larger than the temperature of a species, we can use the relation

for the energy of a particle exiting a mirror Wm T [Pastukhov, 1974]. Then, adding Eqs. 2.5 and

2.6,

Te + Ti = WE0i/Rm. (2.7)

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We can now find 0 by using the ambipolar condition,

ne/e = ni/i, (2.8)

n2eeUe/Te/UemeTe = n

2i eUi/Ti/UimiTi . (2.9)

Use has been made of the confinement time for particles in a mirror with a potential, eU/TU/T[Pastukhov, 1974]. Assuming the centrifugal confinement of electrons is negligible and ne = ni,

e(Ucie0)/Tie0/Te =e0

meTe

(Uci e0)

miTi. (2.10)

Taking the log and writing Uci in terms of Te and Ti with Eqs. 2.2 and 2.7,

e0 = Te (Rm 1) +TeTi

2 (Te + Ti) log U2

i

miTie220meTe

. (2.11)

For sufficient mirror ratios we can neglect the last term, so that,

e0 Te (Rm 1) , (2.12)

Ui Ti (Rm 1) . (2.13)

Using these with Eq. 2.11 to find better approximations,

e0 Te (Rm 1) + TeTi2 (Te + Ti)

log

miT

3i

meT3e

, (2.14)

Ui Ti (Rm 1) TeTi2 (Te + Ti)

log

miT

3i

meT3e

. (2.15)

These confining potentials produce a change in the ion and electron loss rate, which in turn

reduces the parallel energy loss rate. This loss rate is modified in the same way as the particle loss

rate eU/TT /U (since each leaving particle has energy = T [Pastukhov, 1974]).In addition to changing the energy loss rate, the potential modifies the density profile along

the field line. We expect the density along the field line to follow a Boltzmann distribution,

n n0e(UU0)/T. To better understand the equilibrium distribution, we will develop an MHD

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description of the rotating plasma. We start with the equilibrium MHD equations [Lehnert, 1971],

(nv) = 0 (2.16)

v v = p + 1c

J B, (2.17)

0 = E +1

cv B 1

cneJ B. (2.18)

If we consider v = , assume azimuthal symmetry, and ignore the parallel electric field (because

this is ideal MHD),

2rr = p + 1c

J B, (2.19)

0 = E +1

cv B 1

cneJ B. (2.20)

We will assume p/r = 0 to make the effects of rotation more clear. Rearranging Eqs. (2.19)

and (2.20), and using Amperes law B = 4J/c,

r =cE B

B2 mc

eBz2r, (2.21)

J = 2rc/Bz, (2.22)p

z= 2rcBr/Bz, (2.23)

B2z

r

= 82r. (2.24)

Here we can see the equilibrium of the system: The centrifugal force produces an ion drift which is

the origin of an azimuthal current. The azimuthal current with the axial field produces a radial force

to balance the centrifugal force. The current also produces axial confinement of plasma to the extent

that the magnetic field has a radial component (ie, between the solenoid and mirror throat). The

outward pressure of the rotating plasma is balanced by a reduction in the magnetic field pressure.

2.2 Stability

A major consequence of Eq. (2.24) is that the effective betatypically plasma pressure over magnetic

pressureneeds to be modified to include the centrifugal pressure as well. We find the effective beta

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(hereafter referred to as beta),

(r) =n (Ti + Te) +

r0

nm2rdr

B2/8. (2.25)

Unlike the thermal pressure, the centrifugal pressure increases with plasma thickness. A large volume

of plasma rotating faster than the thermal speed thus comes at the expense of reduced density or

increased beta.

Rotating mirrors share with other axisymmetric open traps relatively high limits on beta ( 1). Early studies found the maximum equilibrium beta to be about 2/3 [Lehnert, 1974a]. Recent

experiments on the gas dynamic trap (GDT) in Novosibirsk have achieved beta values of 0.6 with

small amounts of sheared rotation [Bagryansky et al., 2007]. The Maryland Centrifugal Experiment

has also regularly achieved c 1 with supersonic rotating plasmas [Teodorescu et al., 2008]. Inpractice, the beta limit may depend strongly on the ability to produce stable rotation and density

profiles [Panasyuk and Tselnik, 1975, Lehnert, 1976].

The most important effect of rotation in axisymmetric mirrors is not increased confinement, but

magnetohydrodynamic (MHD) stability. The classical test for interchange stability from Rosenbluth

and Longmire [Rosenbluth and Longmire, 1957],

dl

p +prcrB2

> 0, (2.26)

where rc is the curvature radius, is almost always violated in axisymmetric mirrors since near the

midplane rc < 0. Many open traps have avoided this by using MHD stable anchor cells outside

the central solenoid [Ryutov, 1988].

Rotating mirrors can be made MHD stable with sheared rotation [Panasyuk and Tselnik, 1975,

Lehnert, 1976, Bekhtenev et al., 1980, Ryutov, 1987, Beklemishev et al., 2010], and this effect has

been reliably produced in a number of experimental devices [Bocharov et al., 1985, Ellis et al., 2005,

Bagryansky et al., 2007, Volosov, 2009]. Stability may be produced using a combination of tricks:

positive density gradients (dn/dr > 0) are stabilized by the centrifugal force; the sheared rotation

strongly stabilizes high mode numbers and somewhat stabilizes low mode numbers; parallel electron

currents (line-tying) can reduce the instability growth rate; finite larmor radius effects also tend to

stabilize high mode numbers. The latter three effects produce the vortex confinement that has

enabled = 0.6 discharges on the GDT [Bagryansky et al., 2007, Beklemishev et al., 2010].

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Chapter 3

Wave-Driven Rotation in Mirrors1

The limitation imposed by the Alfven critical ionization velocity (CIV) has been the most difficultto overcome, appearing in different applications across many orders of magnitude in pressure and

temperature [Brenning, 1992]. The critical velocity is vc =

2ei/mn, where i is the ionization

potential of neutrals and mn is their mass. As the power supplied to the plasma is increased to raise

the rotation velocity above this threshold, the power only produces electron heating and increased

ionization. The mechanism responsible for instability has been the subject of some debate as a result

of conflicting data from space-based experiments [Lai, 2001].

Besides the limitation due to the Alfven CIV, the use of electrodes to drive rotation may re-

strict the achievable rotation through other considerations [Bekhtenev and Volosov, 1978]. If the

electrodes are placed in the mirror throat, there must be sufficient plasma there to provide parallel

conductivity to the bulk plasma. If the rotation profile is instead determined by the perpendic-

ular conductivity (for example by using an electrode along the center axis), ionization and MHD

instabilities may develop in the plasma.

One can see that it would be desirable if, instead of relying on current driven by electrodes,

the rotation could be maintained by current drive within the bulk plasma. Rotation speeds small

compared to the ion temperature have been observed in tokamaks, and one possible mechanism

for this is the presence of waves near the ion cyclotron frequency [Ida, 1998]. The rotation may

be produced by absorption of wave momentum or by neoclassical ion viscosity [Chang et al., 1999,

Perkins et al., 2001, Eriksson et al., 2004]. We will show in this chapter that rotation may be pro-

duced in a similar manner in centrifugal mirror traps, and later show that with appropriate waves

1This chapter is based on Wave-Driven Rotation in Supersonically Rotating Mirrors, published in Fusion Scienceand Technology [Fetterman and Fisch, 2010c] and Channeling in a Rotating Plasma, published in Physical ReviewLetters [Fetterman and Fisch, 2008].

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this can be done so efficiently.

The primary source of momentum drag in centrifugal mirrors is the formation of new ions

[Bekhtenev et al., 1980]. When a neutral is ionized, the heavy ion moves one gyroradius towards the

negative electrode while the electron is nearly stationary, producing a charge separation. Processes

of thermalization and transport are ambipolar, so that this charge separation is maintained despite

collisions [Braginskii, 1965]. The energy lost due to the charge separation must be provided by the

external power source.

With wave-induced ion diffusion, the ions interact with the wave and travel radially within the

plasma, but electrons do not. If ions that move toward the positive electrode are allowed to exit,

but ions that move toward the negative electrode are not, the average charge separation of particles

leaving the device is decreased. This means that less energy must be supplied by the electrodes

to maintain the plasma. If the average radial distance the ion travels before exiting is at least the

initial gyroradius, the electrodes may be eliminated entirely.

Resonant particles can only absorb energy and momentum in a ratio proportional to their velocity,

which is /k. This momentum is ultimately balanced by the reactive force of the wave on the

antenna. This will be the foundation for producing plasma rotation with waves.

In this chapter we will first discuss the branching ratio, a new term to describe wave-particle

interactions in rotating plasmas. Then the method for using waves to produce rotation will be

described.

3.1 Branching ratio

When a particle interacts with a wave, it must absorb or emit wave momentum and energy in the

same proportions that they exist in the wave. For example, it cannot reduce the wave energy but

increase the wave momentum. This leads to a strong constraint on the change in the constants of

motion of the particle. If a particle interacts repeatedly with the same wave at random phase, its

motion can be described by a diffusion path in phase space [Karney, 1979].

This path describes constrained diffusion not only in velocity, but also in position. To see this,

consider a slab geometry with a uniform magnetic field in the z-direction and a uniform electric field

in the y-direction. Shift to the frame moving with vE = Ey/Bz, denoting terms in this frame with

a tilde. The particle is moving in circular orbits in this frame, as there is no electric field.

A wave with wave number kx and moving frame frequency has phase velocity vph = /kx, and

can interact with a particle moving with vx = vph. If the velocity is changed by vx, this will lead

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Figure 3.1: Interaction of a wave and a particle. In this diagram, because of the positive phasevelocity, ions that gain energy from the wave move out radially, while cooling ions move in.

to a change in the gyrocenter position ygc = vx/i, where i is the gyrofrequency. The changein velocity leads to the change in energy W = mivxvx, so that using the resonance condition we

find ygc = W /vphimi. From this relationship, we see that if vph > 0, the particle must movedown if it gains energy, and up if it loses energy.

In the moving frame, the interaction appears to be only between the particle and the wave, since

there is no electric field [Fisch and Rax, 1992]. However, if we consider the interaction in the lab

frame, the Ey field can be seen to play a significant role. Since the particle gyrocenter has moved by

ygc, there is a change in the potential energy q = qEyygc. Substituting our previous result,we find q = vEW /vph. This relationship is the basis for the branching ratio, fE = vE/vph.

The branching ratio is the increase in the potential energy divided by the total change in the

particle energy. For |fE | 1, energy is primarily transferred between the wave and the particlethrough kinetic energy, as in the stationary mirror case [Fisch, 2006, Zhmoginov and Fisch, 2008].

However, if |fE | 1, the potential energy change is at least as large as the kinetic energy change.To apply the branching ratio to a rotating plasma, we must introduce the centrifual force and

Coriolis effect. The first leads to an additional drift, and the second leads to an effective change

in the cyclotron frequency [Lehnert, 1964]. We define the ratio = vE/ri, so that the rotation

frequency i

2 and the cyclotron frequency i i (1 + 2) for small . Definingn = k/r, we then find the branching ratio,

fE n(1 + 4) /i + n2

. (3.1)

A full derivation is given in Appendix A. One can see the full solution in terms of and ni/ in

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III

III

0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5

20

10

0

10

20

n

i

Figure 3.2: Classification of wave interaction versus = E/i and ni/. In region I energy

transfer is primarily between the kinetic and wave energy, in region II energy transfer is betweenkinetic and potential energy, and in region III it is between wave and potential energy. The lines inthe middle of the regions are fE = 0, fE = 1, and |fE| , respectively.

Fig. 3.2 (for perpendicular waves, the y-axis is just n). We note that for < 1/4 particles areuntrapped, as in the band gap ion mass filter [Ohkawa and Miller, 2002]. We have divided the space

into three regions based on the type of interaction. Let us consider these regions one at a time.

Region I The simplest case, fE 0, is familiar as particle heating in a nearly stationary plasma.

At fE = 0, there is no change in the potential energy. For 0 < fE < 0.5 (the darker part of RegionI), there is a small increase in potential energy as the kinetic energy decreases, while for 1 < fE < 0the opposite is true. The radial motion of the particle may be seen as the drift due to an average

accelerating force, or as absorption of wave momentum.

Region II In this region around fE = 1, kinetic and potential energy are interchanged and wave

energy is nearly constant. If we consider a particle in the rest frame, we find that its energy,

E = W + q, must be conserved since vph = 0. The particle may scatter off of the stationary wave

in the x-y plane (the wave breaks the adiabatic invariant ), which allows stochastic motion in they direction.

Region III In this case, |fE| 1, so the wave energy is primarily converted to potential energy.We see that |fE| if the wave is travelling with the same speed as the moving frame, vph = vE .Thus the wave does not change the particles kinetic energy in the rotating frame W, but breaks the

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azimuthal symmetry and provides a source of angular momentum, by which the particle can travel

radially.

3.2 Producing rotation

We will use the description above of the interaction of a single particle with a wave to define waves

that may produce rotation in a mirror. We first will complete the picture of diffusion paths in

a full mirror machine. Then we will discuss how to fulfill two criteria necessary for alpha chan-

neling [Fisch, 2006]. The diffusion paths must be favorable for driving rotationthere must be a

population inversion along the diffusion path in the desired direction. Also, it is necessary to limit

the energy a particle can gain from the wave to prevent losing significant rotation energy to few

particles.

When particles interact with a specific wave, the perpendicular and parallel energy changes are

related through k and k [Chen et al., 1988, Fisch and Rax, 1992]. The potential energy change

is related to the perpendicular energy change through the branching ratio. Knowing these changes

in the wave region, it is easy to then determine the perpendicular, parallel, and potential energy at

the midplane. These will uniquely determine particle orbits if we ignore gyroangle, azimuthal angle,

and bounce position. Repeatedly interacting with the wave at random phase thus produces diffusion

along a one dimensional path in W0-W0-0 space (from Appendix A),

W0 = W/Rrf, (3.2)

W0 =

kvnc

+ (Rrf 1) n2E

c+

1 R1rf

W, (3.3)

q0 = RrfncE0

2cW (3.4)

Three diffusion paths are apparent in Figure 3.3 that allow us to remove trapped particles,

through either the loss cone or the outer radius of the plasma. Path (a) to the loss cone reduces

both the kinetic and potential energy of the particle, path (b) to the outer radius increases potential

energy but decreases kinetic energy, and path (c) increases both kinetic and potential energy. In

this diagram we have assumed that > 0, or Er < 0. This polarity is more common in rotating

mirrors because it leads to better ion confinement [Lehnert, 1971]. > 0 is also preferred in this

application since path (b), which produces rotation, has no opposite path leading to a loss region.

We wish to create a population inversion along path (b) to drive rotation. The waves resonant

parallel energy must be greater than the energy required to overcome the centrifugal potential for

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Figure 3.3: The shaded region indicates the wave resonance, dependent on radius. The dashed linesdepict diffusion paths that could eject particles from the plasma.

the particle to exit the loss cone. On the other hand, the wave resonant parallel energy must be

less than the energy at which new particles are created for there to be any particles in resonance.

For perpendicular waves, the particle will remain in resonance with the same wave and these simple

conditions on the resonant energy are sufficient to ensure favorable diffusion toward the exit. For

non-perpendicular waves, the overall diffusion path must still intersect the particle source and loss

region, but the resonance region of a single wave has no such restriction.

The final requirement for driving rotation is that the particle energy gain is restricted. The

energy gain may be limited in three ways. The simplest way is by the limited radial extent of the

plasma. Particles move inward as they gain energy, and once particles reach the center axis they

cannot move in further. Although this is a strong restriction on energy gain, for most configurationsthis energy is much higher than that imposed by the second method [Zhmoginov and Fisch, 2008].

The second way to restrict the energy gain is to use non-overlapping waves with ks 1, wheres is the gyroradius of source particles. The diffusion coefficient for resonance with the first cyclotron

harmonic is proportional to J1(k)2, which will go to zero at k 3.83, 7.02, . . .. As the particleincreases in perpendicular energy, increases, as does k = n/r. The diffusion coefficient will thus

approach zero at some point as the particle gains energy from the wave.

Energy gain is further limited by the fixed resonance region if the wave is not perpendicular.

As the particle changes its perpendicular energy, it also changes its parallel energy until it is out of

resonance ( kv = i is not satisfied). If there is no wave overlapping on the high-energy sideof the diffusion path, the particle will cease to gain energy from the waves.

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3.3 Summary

We have shown that it is possible to maintain the angular momentum of supersonically rotating

mirror machines using radio frequency waves. In principle, the rotation energy may come from an

arbitrary combination of kinetic energy and wave energy. Because electrodes are not necessary, it ispossible that past limits on rotation speed will not affect devices driven this way.

Two specific applications of these waves will be discussed. In the next chapter, waves will be

found that convert alpha particle kinetic energy to rotation energy in a fusion reactor. This is an

extension of the alpha channeling effect. Simulations will show that waves produced by a fixed

magnetic ripple convert kinetic energy to potential energy as predicted.

In Chapter 6, we will consider using waves to drive rotation in plasmas designed for isotope

separation. In this case, there is no kinetic energy source, so rotation must be created with wave

energy. Eliminating electrodes in this case has the added benefit of reducing interaction with the

outward flow of separated material.

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Chapter 4

Alpha Channeling 1

Alpha channeling was originally developed in tokamaks to more efficiently take advantage of thefree energy available from alpha particles, as well as to rapidly remove fusion ash from the sys-

tem [Fisch and Rax, 1992]. In alpha channeling, alpha particles interact stochastically with ra-

dio frequency waves, causing average outward diffusion due to a population inversion [Fisch, 1995,

Herrmann and Fisch, 1997]. Wave energy would then be used to drive electron currents or to heat

fuel ions. By removing alpha particles quickly, the fusion reactivity might be doubled in a system

with the same plasma pressure [Fisch and Herrmann, 1994].

The alpha channeling concept was recently extended to stationary mirror machines [Fisch, 2006,

Zhmoginov and Fisch, 2008]. In this case, the loss cone produces a convenient phase space bound-

ary through which alpha particles can be removed. Many of the benefits of alpha channeling in

tokamaks, such as increased fusion reactivity, also apply to mirrors. Wave energy in this case

might be used for heating ions in the central cell, or electrons in the plug region of a tandem

mirror [Zhmoginov and Fisch, 2009].

An alternate use of alpha particle energy is possible in supersonically rotating mirrors, as dis-

cussed in the last chapter. By moving particles across the radial electric field, alpha particle kinetic

energy can be transferred into potential energy directly. This not only allows efficient removal of

alpha particles and direct use of their energy, but also eliminates the electrodes that have been a

major limitation in producing centrifugally confined plasmas. It was found that particle energy, wave

energy, and potential energy could be interchanged using these waves, and a parameter describing

the exchange of energy between these sources was called the branching ratio.

1This chapter is based on Alpha channeling in rotating plasmas with stationary waves, published in Physics ofPlasmas [Fetterman and Fisch, 2010a].

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To convert a significant amount of the alpha particle energy into rotation energy, waves with

large n will be shown to be necessary. We propose that a simple way to produce these waves is to

introduce a stationary perturbation on the outside of the plasma, which will appear in the rotating

frame as a wave. Because the wave is stationary, the particles total energy in the rest frame will be

conserved. Therefore, only the kinetic and potential energy will be interchanged. The momentum

in the system is balanced by forces holding the antenna in place. This is similar to the stationary

magnetic resonance used in autoresonant ion cyclotron isotope separation [Rax et al., 2007].

We show here that a stationary antenna can produce waves that penetrate the plasma layer and

may be used for alpha channeling. By simulating a specific set of waves, we demonstrate that these

waves efficiently convert alpha particle kinetic energy to potential energy. Finally, we consider a

reactor based on the simulated parameters and find that rotation may be maintained in the absence

of electrodes.

This chapter will be organized as follows. In Section 4.1, we will review how radio frequency

waves produce diffusion paths in phase space. Then in Section 4.2 we will examine a rotating

plasmas response to a stationary perturbation. Lastly, in Sections 4.3 and 4.4, we will describe our

simulation of alpha channeling and its results.

4.1 Diffusion paths

The branching ratio defined in Chapter 3 was be used to characterize three regimes of wave interac-tions. For fE 0, alpha channeling in the rotating plasma will be the same as in the non-rotatingmirror caseenergy transfer will be primarily between the wave and the particle kinetic energy. Near

fE = 1, energy is primarily exchanged between kinetic and potential energy. Finally for |fE | ,the kinetic energy is constant while energy is transferred between the wave and potential energy.

Several conditions are necessary for alpha channeling to be successful. First, there must be a

diffusion path that connects the alpha particle source to a region of phase space in which the alpha

particle will be lost. The alpha particles should have low energy when they are lost, so that their

energy can be used in the plasma. Also, there should also be a limitation on the heating of alpha

particles, so that all particles eventually exit by the desired path.

Particles that interact with a wave satisfy the resonance condition n kv = ni, where is the rest frame frequency, n is the azimuthal mode number, is the ion rotation frequency, k is

the parallel wave number, v is the particle velocity along the magnetic field, n is an integer and i

is the rotating frame cyclotron frequency. In rotating midplane coordinates, these resonant particles

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lie on a line in phase space with slope (Rrf 1)1, where Rrf = Brf/B0 is the mirror ratio of theRF region. By placing multiple wave regions at different positions in the mirror, we can thus access a

large fraction of phase space using few waves, as in Fig. 4.4 [Fisch, 2006, Zhmoginov and Fisch, 2008].

The region will also be offset from the origin along W0 by the parallel resonant energy in the

wave region Wres = 12mv2, plus the centrifugal confinement potential between the wave region and

the midplane 12m2r2

1 R1rf

. If we use perpendicular diffusion paths, we can ensure particles

will leave at low energy by choosing a resonant parallel energy just above the centrifugal confinement

potential. We will also guarantee that there is a population inversion along the diffusion path between

the empty region of phase space at the loss cone and the source of alpha particles at high energy.

The final requirement for alpha channeling is that the energy gained by particles is limited. Two

methods suggested for limiting heating in stationary mirror machines are by taking advantage of

the radial limitation of the plasma, and through a zero in the diffusion coefficient due to finite k

effects [Zhmoginov and Fisch, 2008]. It is also possible to limit heating by using a wave that is not

entirely perpendicular. In this case, as a particle gains energy from the wave, its parallel energy

changes to move it out of resonance. If there is no overlapping wave on the high energy side of the

diffusion path, the heating will be limited. This is the limitation that will be used in the following

simulations. Disadvantages to this method are that multiple waves must be used, and alpha particles

may leave at higher energies than ideally possible.

Once the requirements for alpha channeling are met, we consider the most effective way to use the

alpha particle energy. To convert a significant amount of the kinetic energy into rotation energy, the

waves must have fE 1. The equality fE = 1 holds for stationary waves with = 0. The resonance

condition then dictates that these waves have n i/ 20 for practical values of and i.In a vacuum such a mode would decay like r|n|. Fortunately, the plasma in a centrifugal trap is

localized near the outside of the cylinder (as in Fig. 4.1), so that even with this decay there may

be sufficient magnitude for alpha channeling throughout the plasma region. The rotating plasma

response to a stationary magnetic perturbation will be calculated in the next section.

4.2 Plasma response

We will estimate the plasma response to a magnetic perturbation by treating the local geometry

as an infinite cylinder. We assume that the plasma occupies a region of width a from radius r1

to r0, and that the plasma is surrounded by a wall at radius rw (see Fig. 4.1). The wall carries

an externally imposed current j that produces the stationary magnetic wave. We will first solve

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Figure 4.1: A cross section of the geometry used in determining the plasma response. The central(c), plasma (p), vacuum (v) and wall (w) regions are labeled.

for the normal mode solutions in each of the vacuum and plasma regions. We will then connect

the solutions using appropriate boundary conditions, including the current j on the wall boundary.

This will allow us to determine the fields produced by the antenna everywhere.

For a given magnetic field strength and rotation speed, there is an optimum value for the plasma

thickness a. The reason it is not advantageous to use a thicker plasma region is that the cen-

trifugal force produces an outward pressure everywhere that must be balanced by the magnetic

pressure. Recall from chapter 2 that the ratio of the plasma pressure to magnetic pressure is

c = 8

nT + nmar02

/B20 . Thus for fixed c, with supersonic rotation the achievable density n

is inversely proportional to the plasma thickness. To maximize the fusion power output Pfus

n2a,

we find a/r0 T /m2r20 1/2Rm, according to Eq.(2.13).We can simplify calculation of the plasma response by neglecting the difference in rotation fre-

quency between species due to the centrifugal drift. The difference in ion and electron drifts leads to

an azimuthal current and a radial variation in the zero order magnetic field. If k /vA, the axialvariation of the field will be much greater than the radial variation in first order terms, and so we

may ignore the contribution of the zero-order azimuthal current. For different ion species rotating

at different frequencies, the same wave will appear at different frequencies due to the Doppler shift.

Because this difference is a small fraction of the overall frequency, we may ignore this effect as well,

assuming that the wave is far from any resonance.

Thus we are assuming in calculating the plasma response that every component of the plasma

is rotating at the same frequency . In the frame rotating with , the problem is then sim-

ply the response of a stationary plasma to an oscillating field with = n. We can eas-ily calculate the response of a two species cold plasma to such a field using the MHD equations

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[Bernstein and Trehan, 1960, Stix, 1992]. Assuming ck and e, one finds,

Bpr = c3nr

Bz + c4Bzr

(4.1)

Bp = ic4n

r

Bz + ic3Bz

r

(4.2)

Bpz = BJn (krr) + CKn(krr). (4.3)

Here B and C are constants, J and K are Bessel functions, and we have omitted a factor of

exp i

kz + n t

. We define the constants xi = ni/ne, m = ximi + (1 xi) mj , v2A =B2/4nem, s = eB/mc, x = ij/s,

2H=

2xs/ (xii + (1 xi)j),

c1 = 1 +(x i) (x j)

2 2x, (4.4)

c2 = 1 +

2

/

2

H 12/2x 1 , (4.5)

c3 = ic1

sk

2

v2A, (4.6)

c4 = ik

c22k2

2

2sc21k

2 c2

2

v2A

, (4.7)

1 =

c2k2

2

v2A

2

2

2sc21k

4, (4.8)

k2r = 1

1

c22k2 c22/v2A c212k2/2s

. (4.9)

Eq. (4.9) may be seen to be a formulation of the two-ion cold plasma dispersion relation, so that

the wave is equivalent to the fast Alfven wave.[Stix, 1992]

The solution in the vacuum regions satisfies B = , and 2 = 0. The z-component of thesolution in each region may be written, using only bounded solutions,

Bcz = AIn (kr), (4.10)

Bvz = DIn (kr) + EKn(kr), (4.11)

Bwz = F Kn(k

r), (4.12)

where I and K are the modified Bessel functions. The other components of the field are simple

functions of the above equations.

We finally consider the boundary conditions. At all boundaries, the perpendicular field Br

must be continuous. At the free boundaries of the plasma, at r1 and r0, Bz must be constant

across the boundary for magnetic pressure balance, while at rw the current produces the jump

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Bz

Br

80 85 90 95 100 105r cm

5

5

10

15

B G

Figure 4.2: Axial and radial magnetic field versus radius, for r1 = 80 cm, r0 = 100 cm, rw = 110cm. The wave is stationary in the lab frame, with n = 19 and k = 0.0463 cm, and plasmaparameters described in the text. The two components are out of phase with each other, and theazimuthal field is not shown.

Bwz = Bvz +4c j .

Solving the system Eqs. (4.1), (4.3), (4.10)(4.12), with the above boundary conditions, we will

have a solution to the magnetic field everywhere. To find the electric field, we can simply use

Faradays law, E = (1/c)B/t. In this case Ez = 0, so we can calculate the rotating framefields Er = /(ck)B and E = /(ck)Br.

The results for a stationary wave in a plasma with ne = 1.5 1013 cm3, B = 2 T, = 0.046,a = 20 cm, r0 = 100 cm and 50/50 DT ions are shown in Fig. 4.2. One can see that while the axial

field dies off rapidly, the radial field is nearly constant over the plasma region because the derivative

of the axial field remains large. Since E =

/(ck

)Br, this will lead to little radial variation in

the phase-space diffusion coefficient, which is proportional to |E|2 [Zhmoginov and Fisch, 2008].Because we have assumed the plasma to be of uniform density and rotation frequency, we

may be excluding important resonances influencing the accessibility of the wave.[Buchsbaum, 1960,

Stix, 1975] Since the wave frequency in the rotating frame is proportional to the rotation frequency,

a single wave will appear at a broad range of frequencies across a plasma with strong rotation

shear. Should there be a resonance, the wave may be absorbed at that layer, or may undergo mode-

conversion. Details of the density, temperature and rotation profile will be critical in determining

whether the wave will reach the inner plasma boundary.

4.3 Simulation

To demonstrate the effectiveness of alpha channeling using these waves, we simulated alpha particles

by calculating their full equations of motion in a mirror machine with four separate wave regions

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Figure 4.3: A cross section of the plasma with the wave regions indicated. The black lines denotethe inner and outer plasma boundaries. The dash-dotted line indicates the axis of symmetry.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

W0 MeV

W0

MeV

Figure 4.4: The resonance regions for the waves in rotating, midplane phase space. The diagonalline at 3.5 MeV indicates the alpha particle birth energy. The upward line by the x-axis indicatesthe boundary of the alpha particle loss cone.

(see Fig. 4.3). We have chosen specific parameters for the simulation, but these are not optimized

and only provide an example for the present discussion. The midplane magnetic field is B0 = 2T,

the ratio of rotation to deuterium cyclotron frequency is = 0.046, the plasma thickness is 20 cm,

the outer radius is 100 cm, and the mirror ratio is 5. This B0, and r0 requires a radial electric field

E0 = 88kV/cm and leads to a rotation energy WE0D = 200 keV, which with Rm = 5 gives the ion

temperature Ti = 40 keV and electron temperature Te = 12 keV (according to relations in chapter

2). With a midplane density of n0 = 1.5 1013 cm3, the centrifugal beta is c = 0.20.The properties of the waves are shown in Table 4.1. The field everywhere is determined by

solving the system of equations described in the previous section, with the radii r1, r0, and rw

varying like R1/2rf to remain on the same field lines. The currents in the wall layer are calculated

based on a specific value of Bpz (r0) and the given wave parameters. Waves are localized axially

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Table 4.1: Properties of the simulated waves.Rrf n k (cm1) L (cm) Bpz (r0) (G)1.0 -19 -0.0463 125 10

1.13 -22 -0.0435 100 101.3 -26 -0.0401 100 5

1.5 -30 -0.0390 75 5

0.1

1

10

102

103

texit

r1

r0

rw

0 1 2 3 4

W0

80

90

100

110

r

Figure 4.5: The amount of time to exit (in milliseconds) versus initial midplane gyrocenter radius(cm) and parallel energy (MeV). Dots indicate individual particles in the simulation, and white dotshave not left the plasma after 1 s.

using a Gaussian envelope with scale length L.

Due to centrifugal confinement, we expect the plasma density to be reduced in regions with

higher magnetic fields. Both reduced density and increased field lead to an increase in the Alfven

velocity, v2A R2rf exp(WE0(1 R1rf )/Ti). With higher fields we also find that larger azimuthalmode numbers are required to satisfy the resonance condition n i. If we consider the lowfrequency limit of the dispersion relation k2r =

2/v2Ak2 k2 , we see that since k2 = n2/r2 R3rf,the wave will decay more rapidly in regions with higher magnetic fields. For this reason we have

located waves in regions with relatively low mirror ratios.

0

1

2

3

4

Wexit

r1

r0

rw

0 1 2 3 4W0

80

90

100

110

r

Figure 4.6: The same as Fig. 4.5, for the alpha particle energy at exit (in MeV).

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The results of the simulations are shown in Figs. 4.5 and 4.6. Since particles were removed at the

wall rw = 110 cm, alpha particles born at the very edge of the plasma have a sufficient gyroradius to

exit immediately. This is why the exit time is smaller for these particles and the exit energy higher.

Some energy is recovered from these particles, however, since the outer wall is the positive electrode.

For the remaining particles, one can see that alpha channeling efficiency is much better at small

initial W. Although the waves are mostly perpendicular, they do have W/W < 0, leading to

a slope outward toward larger W0 for decreasing W0. Also, particles with large W are nearer to

the loss cone, and more poorly confined by the magnetic mirror force.

4.4 Reactor implications

It is clear from the simulations that a significant fraction of the alpha particle energy can be trans-

ferred to the potential. To discuss the efficiency of alpha channeling, we note that alpha particles

in this system are born with significant rotation energy (0.4 MeV) and thermal energy (3.5 MeV).

Without alpha channeling or radial losses, alpha particles would have to overcome the centrifugal

potential and exit at small radius, giving .64 MeV to the potential.

In our simulations, 26% of the particles exit the system immediately, either radially or through

the loss cone. 32% of the particles were removed by alpha channeling after 1 s, and the remaining

42% remained in the system. The particles subject to alpha channeling converted 64% of their energy

(2.5 MeV) to potential energy on average. If we assume that the remaining particles eventually leavethe plasma axially, the average energy that alpha particles add to the potential is 1.38 MeV. Since

the energy used to create an alpha particle is 0.4 MeV for the deuteron and 0.6 MeV for the triton,

0.38 MeV remains to compensate other energy loss from the plasma.

Energy loss in centrifugal traps is primarily due to longitudinal ion loss [Bekhtenev et al., 1980].

Deuterons scatter into the loss cone after a time D ii exp

WED

1 R1m

/Ti

, where ii is the

ion-ion collision time and Rm is the mirror ratio, and tritons have a similar dependence on energy

[Pastukhov, 1974]. In the plasma described above, ii = 1s, so that D = 55s and T = 400s.

The fusion timescale is fus = 300 s. Combining these times with the simulated parameters and a

plasma length of 40 m, we expect the power lost to be approximately 110 kW. If each fusion event

produces 1.38 MeV of potential energy and requires 1 MeV, the net power from alpha particles left

in the potential will be 150 kW, exceeding the power loss. The total fusion power produced will be

9 MW. Note that this reactor is not optimized for power production. The power could be increased

by increasing the magnetic field, decreasing the rotation speed, or allowing for higher plasma beta.

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Before alpha channeling, the efficiency Q would have been about 30. The above discussion

demonstrates that alpha channeling can allow global energy balance without recirculating power,

implying Q . While this efficiency improvement is substantial, more critical is that power doesnot need to be supplied to the plasma through end electrodes. The electrodes have typically reduced

the rotation velocity to the Alfven CIV limit, or else required extensive conditioning to avoid this

limit [Abdrashitov et al., 1991, Lehnert, 1974b, Lehnert, 1971]. In addition, the electrodes could be

a source of instability and require sufficient density for conductivity [Bekhtenev and Volosov, 1978].

Because eliminating the electrodes entirely is a critical issue, we note that this principle can still

be used to produce rotation even if power loss is much larger than expected. One way to compensate

for a higher loss rate is to decrease the rotation energy of the plasma, reducing the temperature by

the same factor. This would reduce the energy used to create the plasma, so that the alpha particle

power produced is a larger fraction of the overall power consumed. Alternatively, recirculating power

could be used to drive rotation directly with waves, which will be discussed in chapter 6.

Two effects have been considered in other works that contribute to the power loss, charge ex-

change and perpendicular heat loss [Bekhtenev et al., 1980, Lehnert, 1974b]. While the cross section

for charge exchange is much smaller than the ionization cross section, the energy that is lost in each

charge exchange event is larger since neutrals return no energy to the potential on exiting the trap.

Regarding the perpendicular heat loss, it will be important to determine the plasma density between

the hot plasma region and wall. Because the parallel loss time is much faster than the perpendicular

loss time, this density may be negligible. It is still possible that energy will be lost due to heat

transfer from a region of fast rotation to slow rotation, where the confinement potential is smaller.

Another effect that should be included in overall power balance is energy recovered at shaped

endplates. It was assumed in the simulation that particles are removed at their gyrocenter after

exiting the trap axially. However, using shaped endplates the particles may be removed at the

high potential side of their gyro-orbit [Bekhtenev et al., 1980, Volosov, 2005]. In past proposals this

contribution has been significant, but it has been necessary to use many closely-spaced electrodes

to achieve this effect. Because end electrodes are not necessary in the present scenario, a continuous

surface may be used to absorb the outgoing plasma flux, improving the efficiency of energy recovery.

4.5 Summary

We have shown that stationary waves can sustain rotation in a supersonically rotating mirror. The

waves interact with alpha particles, converting alpha particle kinetic energy into potential energy

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and ejecting the alpha particles. Because the waves are stationary in the lab frame, they may be

produced by static magnetic coils and do not require significant energy input.

The most significant consequence of alpha channeling is that electrodes are not necessary to

maintain rotation in the plasma. Electrodes have been a source of numerous issues with rotating

plasmas, most prominently limiting rotation speeds to the Alfven critical ionization velocity. It is

expected that with the reduced design constraints, systems will be able to exceed the CIV without

the need for extensive surface conditioning or extremely low densities.

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Chapter 5

Contained modes 1

We have used the branching ratio fE in the last chapters to describe the wave-particle interactionin rotating plasmas. We showed that a fixed ripple in the lab frame, with vph = 0, will have fE = 1,

so no energy is transferred to or from the wave. This is a simple, passive way to drive rotation

and utilize alpha particle energy, in addition to removing the fusion ash. One challenge is that

high mode-number waves are required to satisfy the alpha particle cyclotron resonance condition,

m i/ 20, where is the rotation frequency and i is the cyclotron frequency. These modesare mostly evanescent in the plasma.

Although these waves are evanescent near the antenna, it turns out that they may be excited

efficiently as internal modes, much like in tokamaks. In the tokamak geometry, contained Alfven

modes exist that are destabilized by alpha particles and are responsible for turbulence in the ion

cyclotron range of frequencies [Coppi et al., 1997]. These modes may be useful in alpha channel-

ing in tokamaks, where there are advantages to using low frequency waves, like the ion Bernstein

wave [Fisch, 1995, Valeo and Fisch, 1994], although for somewhat different reasons than in mirror

machines [Fisch and Herrmann, 1995].

However, when ion Bernstein waves were excited in the Tokamak Fusion Test Reactor (TFTR)

to test alpha channeling, the surprising result was diffusion rates fifty times higher than those

predicted by quasilinear theory [Fisch and Herrmann, 1999, Herrmann, 1998]. Although it seemed

implausible, the only explanation for such a dramatic increase in diffusion was that a toroidal

cavity mode was excited. The existence of such high-Q cavity modes was later shown to in fact be

quite plausible [Clark and Fisch, 2000], and has recently been supported by measurements on the

1This chapter is based on Contained modes in mirrors with sheared rotation, published in Physics of Plas-mas [Fetterman and Fisch, 2010b].

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National Spherical Torus Experiment [Gorelenkov et al., 2010]. If the very large diffusion achieved

by these modes can also be accomplished in a tokamak reactor, alpha channeling waves might be

more efficiently excited. This would increase reactor efficiency and reduce the overall reactor cost

for the same power output.

What most interests us here is that there has been considerable experimental and theoretical

support for these kinds of modes and the effects they can play in alpha channeling in similar ge-

ometries. This lends support to the proposals outlined in this paper, where we describe a similar

contained mode for plasmas with sheared rotation. Consider a low frequency wave that is evanescent

outside the plasma. Inside the rotating plasma, this wave may appear at a higher frequency due to

the Doppler shift. As the rotating frame frequency increases, the evanescent wave may go through

a cutoff and become a propagating wave. As it propagates inward, it will reach another cutoff,

either due to convergence (since k2

increases as the radius decreases) or due to a decreasing rotation

frequency (for example if there is an inner wall). The Alfven wave is reflected at these cutoffs, so

the energy is contained and high amplitudes can be achieved.

Rotation shear has been found to stabilize high mode number, low frequency modes in mir-

ror plasmas.[Huang and Hassam, 2001, Cho et al., 2005, Volosov, 2009] It is therefore unusual that

sheared rotation should lead to waves with high mode numbers. However, in this case higher mode

numbers have higher frequencies in the plasma frame, and we will show that they are not effectively

damped by the velocity shear.

Because the modes can have high azimuthal mode numbers and zero frequency in the lab frame,

they may couple to the fixed ripple used for alpha channeling. This would increase the alpha

channeling efficiency by increasing the wave amplitude in the peak rotation region where most alpha

particles are produced. This mode may also provide an efficient method of plasma heating if the

rotating frame frequency coincides with a cyclotron harmonic.

Because the wave phase velocity is near the Alfven velocity, we expect zero frequency con-

tained modes to exist only if the rotation speed is near the Alfven speed. Recent experiments on

MCX have suggested that the Alfven mach number cannot exceed unity [Teodorescu et al., 2008,

Teodorescu et al., 2010]. These experiments observe a limit on the average Alfven mach number

defined by MA = Vp/aBvA, where Vp is the voltage across the plasma, a is the plasma width, and

vA is the Alfven velocity based on a line-averaged density. The theory supporting the Alfven mach

number limit requires a cylindrical plasma with uniform rotation. Because the rotation profile is

nonuniform, there is no conflict with the requirement that the peak Alfven mach number MA > 1.

In fact for values ofMA near unity it is very likely that MA > 1. We therefore think that the modes

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described here are of interest in MCX and other supersonically rotating plasmas.

This chapter will be organized as follows. In Section 5.1, we will derive the eigenmode equation

for the contained modes. We will then in Section 5.2 find the properties of contained modes assuming

a peaked rotation profile. Mode properties for plasmas without a strong peak will be addressed in

Section 5.3. We will then discuss ion cyclotron absorption of these modes in Section 5.4.

5.1 Eigenmode equation

The contained modes are localized eigenmode solutions to the MHD equations,

t+ v

v =

1

cJ B, (5.1)

0 = E +1

cv

B

1

cneJ

B. (5.2)

The first of these is the force balance equation, and the second is Ohms law. We will assume

the equilibrium is B0 = B0z, J0 = 0, v0 = r (r) , and E0 = r (r) B0r/c. We have defined(r) as the rotation frequency at radius r. This equilibrium neglects the diamagnetic effect of the

centrifugal force. The diamagnetic effect might be significant at desired rotation speeds if a/r is not

small. However, because the solutions are localized, the variation of the axial magnetic field is small

across the region of interest. Therefore, this is a good approximation even for finite beta plasmas if

the local magnetic field is used rather than the vacuum field.We seek solutions proportional to exp

im + ikz it

. The linearization of the second term

on the left of Eq. (5.1) is, to first order,

v v = imv1 2v1r + 2v1r + v1rr. (5.3)

Here we have omitted the dependence of on r and use primes to denote radial derivatives. The

terms on the right of Eq. (5.3) represent the Doppler, centrifugal, Coriolis, and convection terms

respectively. We will neglect the centrifugal and Coriolis effects, which are much smaller than theDoppler term assuming m 2. The linearized form of Eq. (5.1) is then,

(i + im) v1 = 1c

J1 B0 v1rr. (5.4)

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When we take the curl of Eq. (5.2), apply Faradays law and linearize, we find,

iB1 =

v1 B0 + v0 B1 1ne

J1 B0

. (5.5)

We finally get a single differential equation for B1 by substituting Amperes law, J1 = c4B1,and the velocity from Eq. (5.4) into Eq. (5.5). We define the plasma-frame frequency as = m,and find,

iB1 =

i1

4( B1) B0

B0

i v1rr

B0 + v0 B1

c4ne

( B1) B0

. (5.6)

Using this equation, one can find expressions for B1r and B1 in terms ofB1z and its derivative. A

differential equation for B1z can then be obtained. These calculations are performed in Appendix B.

A rough approximation of the result may be found by considering a plasma with solid body rota-

tion ( = 0) and waves with k = 0. In this case, we re

wave-driven rotation and mass separation in rotating magnetic mirrors

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