Download Coil Design and Related Studies for the Fusion-Fission

To my wonderful daughter Hilda
and my lovely wife Frida
List of Papers
This doctoral thesis is based on the following papers, which are referred to in
the text by their Roman numerals.
Reviewed journal papers and manuscripts:
I
Hagnestål, A., Ågren, O., Moiseenko, V.E., Field and Coil
Design for a Quadrupolar Mirror Hybrid Reactor, Journal of
Fusion Energy 30, 144 (2011).
II
Hagnestål, A., Ågren, O., Moiseenko, V.E., A Compact NonPlanar Coil Design for the SFLM Hybrid, Journal of Fusion
Energy 31, 379 (2012).
III
Hagnestål, A., Ågren, O., Vacuum Field Ellipticity
Dependence on Radius in Quadrupolar Mirror Machines,
Journal of Fusion Energy 31, 448 (2012).
IV
Hagnestål, A., Ågren, O., Moiseenko, V.E., Radial
Confinement in Non-Symmetric Quadrupolar Mirrors, Journal
of fusion energy: DOI: 10.1007/s10894-012-9573-x (2012).
V
Hagnestål, A., Ågren, O., Moiseenko, V.E., Finite
corrections to the magnetic field in the SFLM Hybrid,
Manuscript (2012).
VI
Ågren, O., Moiseenko, V.E., Noack, K., Hagnestål, A. Studies
of a Straight Field Line Mirror with Emphasis on FusionFission Hybrids, Fusion Science and Technology 57, 326
(2010).
VII Ågren, O., Moiseenko, V.E., Noack, K., Hagnestål, A., Radial
Drift Invariant in Long-Thin Mirrors, The European Physical
Journal D 66, 28 (2012).
VIII Noack, K., Moiseenko, V.E., Ågren, O., Hagnestål, A.,
Neutronic model of a mirror based fusion-fission hybrid for the
incineration of the transuranic elements from spent nuclear fuel
and energy amplification, Annals of Nuclear Energy 38, 578
(2010).
Conference papers:
IX
X
XI
XII
Hagnestål, A., Ågren, O., Moiseenko, V.E., Coil design for the
Straight Field Line Mirror, Presented as a poster presentation
at the OS-2008 conference at Daejon, Korea, published in the
conference proceedings in Transactions of Fusion Science and
Technology, 55 (2T), 127 (2009).
Hagnestål, A., Ågren, O., Moiseenko, V.E., Theoretical field
and coil design for a single cell minimum-B mirror hybrid
reactor, Presented as a poster presentation at the OS-2010
conference at Novosibirsk, Russia in July 2010, published in the
peer-previewed conference proceedings in Transactions of
Fusion Science and Technology 59 (2T), 217 (2011).
Hagnestål, A., Ågren, O., Moiseenko, V.E., Coil System for a
Mirror-Based Hybrid Reactor, Presented at the FUNFI
conference at Varenna 2011 as a poster presentation and
published in the conference proceedings “Fusion for Neutrons
and Subcritical Nuclear Fission”, AIP Conference Proceedings
1442, 217 (2012).
Noack, K., Ågren, O., Källne, J., Hagnestål, A., Moiseenko,
V.E., Safety and Power Multiplication Aspects of Mirror
Fusion-Fission Hybrids, Presented at the FUNFI conference at
Varenna 2011 by Klaus Noack and published in the conference
proceedings “Fusion for Neutrons and Subcritical Nuclear
Fission”, AIP Conference Proceedings 1442, 186 (2012).
The author has also contributed to the following work, not included in the
thesis.
A.
B.
C.
Hagnestål, A., Ågren, O., Moiseenko V. E., Coil design for the
SFLM Hybrid, Proceedings of the EPS conference 2012.
Noack, K., Ågren, O., Moiseenko, V. E., Hagnestål, A.,
Comments on the power amplification factor of a driven
subcritical system, Annals of Nuclear Energy: DOI: 10.1016/
j.anucene.2012.06.020 (2012).
Ekergård, B., Boström, C., Hagnestål, A., Rafael Waters and
Mats Leijon, Experimental results from a linear wave power
generator connected to a resonance circuit, Wiley
Interdisciplinary Reviews: Energy and Environment: DOI
10.1002/wene.19 (2012).
There are also a number of conference papers from the Open Systems,
Alushta, FUNFI and EPS conferences to which the Author’s contribution is
small.
Contents
1.
2.
2.1.
2.1.1.
2.1.2.
2.2.
2.3.
2.4.
2.5.
2.6.
2.6.1.
2.6.2.
2.7.
2.8.
2.9.
2.9.1.
2.9.2.
3.
3.1.
3.1.1.
3.1.2.
3.1.3.
3.1.4.
3.2.
3.2.1.
3.2.2.
3.2.3.
3.2.4.
3.2.5.
3.2.6.
3.2.7.
3.2.8.
3.3.
3.3.1.
3.3.2.
3.3.3.
Introduction ...................................................................................13
Fission power ................................................................................15
Fission power today.......................................................................15
Neutron multiplication...................................................................16
Negative feedback factors and delayed neutrons ..........................16
Resources.......................................................................................17
Front end........................................................................................19
Back end ........................................................................................19
Nuclear non-proliferation ..............................................................20
Transmutation................................................................................21
Transuranics ..................................................................................21
Long-Lived Fission Products ........................................................21
Breeding ........................................................................................22
Critical fast reactors.......................................................................23
Driven systems ..............................................................................23
Accelerator-driven systems ...........................................................25
Fusion-driven systems ...................................................................25
Fusion energy and plasma physics ................................................29
Basics in fusion .............................................................................29
Plasmas..........................................................................................29
Fusion reactions.............................................................................29
Fusion and confinement ................................................................30
Resources.......................................................................................31
Plasma physics ..............................................................................31
Magnetic confinement ...................................................................31
Particle drifts .................................................................................32
Collisions in fusion plasmas..........................................................34
Radial transport .............................................................................35
Kinetic theory: the Vlasov equation ..............................................35
MHD Equations.............................................................................36
Diamagnetism, MHD equilibrium and the concept of ................38
Plasma instabilites .........................................................................39
Mirror machines ............................................................................39
End confinement and electron temperature ...................................39
Mirror geometries..........................................................................42
The flute instability in magnetic mirrors .......................................44
3.4.
3.4.1.
3.4.2.
3.4.3.
4.
4.1.
4.2.
4.3.
4.4.
4.5.
4.5.1.
4.5.2.
4.6.
5.
5.1.
5.2.
5.3.
5.3.1.
5.3.2.
6.
6.1.
6.1.1.
6.1.2.
6.1.3.
6.1.4.
6.1.5.
6.1.6.
6.2.
6.3.
7.
7.1.
7.2.
7.3.
7.4.
7.4.1.
7.4.2.
7.4.3.
7.5.
7.6.
8.
8.1.
8.1.1.
8.1.2.
8.1.3.
Other devices for magnetic confinement.......................................46
Tokamaks ......................................................................................46
Stellarators.....................................................................................47
Other schemes for magnetic confinements....................................47
Theory for the magnetic mirror vacuum field ...............................49
The long-thin approximation.........................................................49
Flux coordinates ............................................................................50
Flux tube ellipticity........................................................................53
Flute stability.................................................................................55
Drifts and neoclassical transport in mirror machines ....................56
Particle drifts .................................................................................56
Neoclassical transport....................................................................58
The Straight Field Line Mirror field..............................................60
Finite ß effects on the mirror magnetic field .................................63
The equilibrium .............................................................................63
The plasma currents.......................................................................65
Magnetic field from plasma currents.............................................66
The method from Paper V .............................................................66
The method from Paper VII ..........................................................67
Superconducting coils in fusion ....................................................69
Properties of superconducting coils...............................................69
Superconductivity..........................................................................69
Superconducing magnets...............................................................70
Cooling and shielding....................................................................71
Power supply and stability.............................................................72
Internal structure............................................................................72
Use of copper coils in reactor scenarios ........................................73
Mirror machine coil types .............................................................73
Existing or earlier mirror machine coil systems ............................74
The SFLM Hybrid project .............................................................75
Geometry .......................................................................................75
Plasma parameters and magnetic field properties .........................76
Radio frequency heating................................................................78
Fission mantle and shielding .........................................................79
Geometric design...........................................................................79
Reactor safety ................................................................................81
Cooling ..........................................................................................83
Electron temperature .....................................................................84
“Divertor plates” and heat load .....................................................85
Summary of results and discussion ...............................................87
Design of coils and magnetic fields...............................................87
Optimization methods ...................................................................87
Optimization of vacuum magnetic fields.......................................89
Function modelling with splines ...................................................90
8.1.4.
8.1.5.
8.1.6.
8.1.7.
8.1.8.
8.1.9.
8.1.10.
8.2.
8.2.1.
8.2.2.
8.3.
8.3.1.
8.3.2.
8.3.3.
8.3.4.
8.3.5.
9.
10.
11.
12.
13.
14.
Superconducting coil modelling....................................................91
Results from Paper IX ...................................................................98
Results from Paper X.....................................................................98
Results from Paper I ......................................................................99
Results from Paper II (and XI) ....................................................104
Results from Paper III .................................................................107
Results from Paper V...................................................................109
Radial transport and radial invariant ...........................................111
Results from Paper VII................................................................111
Results from Paper IV .................................................................113
Discussion ...................................................................................116
Discussion on coil accuracy requirements...................................116
Discussion on coil calculation limitations ...................................117
Discussion on finite ß issues........................................................117
Discussion on radial invariant, E and low ß limit .......................117
Comparing axisymmetric and quadrupolar drivers .....................119
Conclusions .................................................................................121
Future studies ..............................................................................123
Summary of papers......................................................................125
Sammanfattning...........................................................................129
Acknowledgements .....................................................................131
References ...................................................................................133
Abbreviations and nomenclature
ADS
BOC
BWR
CTD
D
ELM
EOC
FDS
FTWR
GDT
IAEA
ICRH
ITER
LBE
LLFP
LLNL
LOCA
LWR
MHD
NEA
PWR
RW
SABR
SFLM
SKB
T
TRU
WR
Accelerator-Driven System
Beginning Of fuel Cycle
Boiling Water Reactor
Coolant Temperature Density (effect)
Deuterium
Edge-Localized Mode
End Of fuel Cycle
Fusion-Driven System
Fusion Transmutation of Waste Reactor
Gas Dynamic Trap
International Atomic Energy Agency
Ion Cyclotron Radio frequency Heating
International Thermonuclear Experimental Reactor
Lead-Bismuth Eutectic
Long-Lived Fission Products
Lawrence Livermore National Laboratory
Loss Of Coolant Accident
Light Water Reactor
MagnetoHydroDynamic
Nuclear Energy Agency
Pressurized Water Reactor
React & Wind
Subcritical Advanced Burner Reactor
Straight Field Line Mirror
Svensk KärnBränslehantering
Tritium
TRansUranics
Wind & React
B
B̂
E
F
j
R
T
V/m
N
A/m2
m
Magnetic field
Unit vector parallel to B
Electric field
Force
Current density
Position vector
u
v
xc = (xc,yc,zc)
A
a
B
c
D
f
g
I
keff
M
m
m
n
n
p
p
p
P
Pfiss
Pfus
Q
Q
Qr
q
Rm
rg
r0
s
s
T
kBT/e
v
v
W
x0
y0
Z
0
m/s
m/s
m
m-1
m
T
m
T/m
A
kg
kg
kg
1/m3
N/m2
N/m2
N/m2
N/m2
W
W
N/m2
C
m
m
m
m
K
eV
m/s
m/s
J
m
m
-
Mass velocity in MHD
Velocity of a particle
Gyro center position
Curvature vector
Number of proton masses in a nucleus
Plasma radius in a mirror machine
Magnetic field modulus on the z axis
Axial scale length of a mirror machine
Diffusion constant
Distribution function
Quadrupolar field contribution
Current
Effective neutron multiplication
Ion mass
Electron mass
Particle mass
Neutron or number of neutrons
Particle density
Scalar pressure
Parallel pressure (vs magnetic field)
Perpendicular pressure (vs magnetic field)
Total perpendicular pressure (incl. B)
Fission power
Fusion power
Total parallel pressure (incl. B)
Fusion Q, produced power/input power
Fission to fusion power ratio
Electric charge
Mirror ratio
Gyro radius of a particle
Radial Clebsch coordinate
Arc length coordinate along B
Arc length-like coordinate
Temperature
Temperature (thermal energy)
Velocity component perpendicular to B
Velocity component parallel to B
Energy
x-like Clebsch coordinate
y-like Clebsch coordinate
Atomic charge
Fraction of delayed neutrons
Plasma pressure/magnetic pressure
at the midplane
0
m
m
0
c
mfp
eV
Rad/s
Rad/s
J/T
V
Tm
C/m3
kg/m3
m2
rad
m2
s
1/s
m
Energy of a particle
Gyro angular frequency at the midplane
Gyro angular frequency
Magnetic moment
Electric scalar potential
Magnetic scalar potential
Charge density
Mass density
Cross section
Angular-like flux coordinate
Normal (Radial-like) flux coordinate B0r02/2
Collision time (time between collisions)
Average number of neutrons per fission
Collision frequency
a/c, used for ordering in paraxial approx.
Mean free path
1. Introduction
The world’s demand for energy is increasing and the increase is likely to
continue for many years to come. The main resource for energy production
today is fossil fuels such as coal, oil and gas. Since these resources are
limited and prices are rising, other energy sources ought to replace them.
The environmental impact of the fossil fuels is also a large concern targeted
by many governments all over the world and the need for a new clean energy
source is rather urgent. A few alternative energy sources are available.
Renewables can give a significant contribution to the energy production.
There is definitely enough renewable energy sources to fulfill today’s energy
needs, but a problem is how to harvest the often intermittent renewable
energy resources in an economically and an environmentally (often referring
to acceptable by the public) viable way. It is far from obvious and perhaps
not even likely that renewables can provide all energy demanded by the
worlds growing population within the next 100 years. The remaining
candidates are few, and fission energy is likely to play a significant role in
the future energy production. Fission energy is a stable base energy supplier
in many countries including Sweden. Fission energy faces a rather massive
resistance from a large part of the world population due to the accident risk,
the waste management problem, problems with past and present uranium
mining and the risk of nuclear proliferation. Recent events in Fukushima
have further diminished the public trust in fission power, and Germany has
now decided to decommission all their fission plants. Advantages of fission
are that it is almost free of CO2 emissions, that the energy production cost is
quite low and currently lower than for most renewables (except hydropower)
and that the energy production is independent of weather and time of day. In
addition, the technology to burn U-235 in light water reactors (LWRs) and
some other reactor types is known and has been well tested during the last 50
years of commercial operation. As will be pointed out in section 2, the
available resources for fission are enormous, taking into account future
breeding technologies not existing commercially today.
Another possible future source of energy is fusion energy. The available fuel
sources for fusion are enormous. Just to illustrate how energy dense fusion
fuel is, a comparison can be made with a coal power plant. A 1 GW coalfired power plant consumes about 2.7 million metric tons of coal a year,
while a 1 GW fusion device would consume about 250 kilograms of
13
deuterium and tritium [1]. The nuclear waste problem from fusion would be
almost negligible (there would be some low-active waste from activated
reactor parts) and the overall environmental impact would be very low.
Fusion power would therefore be an excellent solution for energy supply, but
a serious problem with fusion energy is that it is hard to find a reactor
configuration in which the energy gain, i.e. produced energy/consumed
energy (the fusion Q factor), is sufficiently large. Despite the worldwide
fusion research efforts since the 1950ies, commercialization of fusion power
is still very far from realization. The complexity of the problem has proved
to be greater than first anticipated, and there is still no commercial reactor
scenario identified today. This implies that there is still at least 50 years
before commercialization may become possible, in any case for magnetic
fusion. There may however be another way for fusion research to contribute
to society, which also probably could be realized in a shorter time scale. To
increase fast fission reactor safety, subcritical reactors have been proposed
(see for example Ref. [2]). Subcritical reactors are not self-sufficient in
neutrons, and are driven by a neutron source. Fusion devices can be
developed to excellent neutron souces, and it seems possible to combine a
fusion reactor with a subcritical fission reactor into a fusion-fission (or
hybrid) reactor. The fusion device in a hybrid reactor would be much less
complicated to accomplish than a pure fusion device. Several fusion devices
available today can with moderate extrapolation reach a sufficient fusion Q
for becoming drivers for hybrid reactors. The immediate aim of such a
device is transmutation of transuranics in combination with energy
production. Another aim for long term sustainability is also breeding of
fissile material. This doctoral thesis is about a fusion-fission reactor concept
called the SFLM Hybrid which is based on a single cell magnetic mirror
fusion device. The main work in the thesis is about the magnetic coil system
and the magnetic field for that concept. Work has also been done on a radial
invariant and effects of asymmetry in quadrupolar mirrors.
The outline of this doctoral thesis is as follows. Section 2-6 contains theory
and background information. In section 2, basics of fission power is
described as well as the possible role for fusion-fission devices. Section 3
introduces the reader to fusion and plasma physics. Section 4 gives some
theory for the magnetic mirror vacuum field and section 5 incorporates the
modification of the magnetic field from the plasma currents. Section 6 gives
a brief introduction to superconducting coils. Section 7 describes the SFLM
Hybrid project, and section 8 gives a summary of the results in the thesis.
Section 9 summarizes the conclusions made, and section 10 gives
suggestions for future work. Section 11 gives a short summary of the papers
in this doctoral thesis, section 12 gives a short summary in Swedish, section
13 contains acknowledgements and section 14 contains the references. The
papers included in this doctoral thesis can be found after section 14.
14
2. Fission power
A fusion-fission reactor gains the vast majority of its energy from fission.
However, fission power is already a widely used source of electric power in
the world. Thus, a question naturally arises: what is the point with a fusionfission reactor? A fusion-fission device is a more complex and expensive
reactor than an ordinary fission Light Water Reactor (LWR). Also, the LWR
technology is well tested during many years of operation (50 years or so),
although the risk of nuclear accidents can never be completely eliminated.
The point is that a fusion-fission reactor has the possibility to operate with a
fast neutron spectrum with better safety margins than critical fast reactors
have. A fast spectrum is required for transmutation of minor actinides and is
desirable for breeding of fertile nuclear fuel. In order to understand the
possibilities with fusion-fission reactors, some basic knowledge of fission
power, resources and the problems associated with fission power is needed.
In this section, background information about fission is provided and the
possible future role of fusion-fission devices is described in the end of the
section.
2.1. Fission power today
Fission power is a well-known subject that can be learned from standard text
books; see for example Ref. [2]. A brief overview is given here. Commercial
reactors today are thermal reactors, which means that the neutrons are
slowed down (moderated) to thermal energies. With thermal neutrons,
reactor safety is improved and the probability of fission in U-235 and
plutonium is increased. Most thermal fission reactors are LWRs, where the
water is an efficient moderator.
LWRs work in the following way: The fuel rods that form the core are
arranged in a water tank. To control power, control rods that absorb neutrons
are used as regulators. Water is either boiled at the fuel rods (Boiling Water
Reactor, BWR) or heated by the fuel rods and boiled in a separate system
(Pressurized Water Reactor, PWR). The steam goes through a turbine
generating electricity, condensates and is returned to the system.
15
2.1.1.
Neutron multiplication
The effective neutron multiplication constant keff is a key parameter in
fission. The parameter keff can be interpreted as the ratio of neutrons in any
generation to the number of neutrons in the next generation. Todays fission
reactors are critical reactors, which means that they operate with keff = 1. The
parameter keff depends on many factors and can for a LWR be expressed in
the so-called 4, 5, or 6-factor formulas where different effects are extracted
into a number of factors. For example, the 6-factor formula is
keff
fp PFNL PTNL
(2.1.1)
where is number of fission neutrons produced per absorption in the fuel, f
is the absorption in fuel probability, p is slowing down without being
absorbed probability,
is the fast fission factor and the last two are
correction factors for neutron leakage of thermal and fast neutrons [2]. By
following a neutron from its creation, the cascade of neutrons that will
follow from fission reactions triggered by this neutron or later generations of
neutrons in the cascade can be viewed. For critical reactors, the average ratio
of the number of neutrons in the next generation to the number of neutrons
in the present generation in this cascade remains at a nearly constant value
keff for many generations, since all neutrons have the same source (fission) as
the first neutron and thereby the same energy spectrum. For driven systems
which have an external neutron source, this is not the case. The source
neutrons in driven systems have a different source than the neutrons in
subsequent generations which are fission neutrons, and thus have a different
energy spectrum. This means that the first generation (and to some extent
even the second) will have a different neutron multiplication than the
subsequent ones due to the energy dependence in different cross sections and
since the number of neutrons produced per fission reaction increases with
neutron energy [3]. Also, the location of the neutrons has a different
distribution.
2.1.2.
Negative feedback factors and delayed neutrons
Since LWRs operate at keff = 1 it may be questioned how the reactor can
remain stable. The time between two generations of neutrons is in the order
of 10 5 seconds. For a slightly supercritical keff = 1.0001 this corresponds to a
neutron multiplication (and hence power increase) of 1.0001100 000 after one
second. This is a large number (about 20 000). What saves the situation is a
combination of two effects. The first effect is that keff is lowered by an
increased temperature. The negative feedback factors on keff however need
some time to become effective due to the need for heat conduction and
boiling gas expansion (in the order of a second), and this effect alone is not
sufficient for reactor stability. The second effect is that about 0.65% of the
16
neutrons are delayed in LWRs, and come from decay processes in the fuel
on average 15 seconds or so after the fission reaction. This makes the
changes in the neutron flux much slower as long as keff 1.0065 1
,
where is the fraction of delayed neutrons. Due to the fraction of delayed
neutrons, a moderate increase i keff only causes a slow increase in power and
neutron flux. The negative feedback factors are effective at this time scale
and respond to such a moderate increase in keff by reducing keff to unity at a
new power balance point. Thereby, the reactor stays critical (or subcritical) if
changes in keff are sufficiently slow and within some range. In practice,
operation at LWRs follow predefined schemes for moving control rods etc.
to ensure that supercriticality accidents should not happen as long as those
rules are followed.
There are two main negative feedback effects on keff that arise from higher
temperature. One comes from the density decrease in water (or in a BWR
from a higher steam percentage) which deteriorates the moderation of
neutrons. The other effect is that resonance cross sections for neutron
capture for epithermal neutrons in U-238 are broadened by the Doppler
effect. This causes a larger fraction of the neutrons to be captured in U-238
when the temperature is increased.
2.2. Resources
The fuel used today in fission comes from uranium ore. The natural uranium
extracted from the ore consists of 99.3 % U-238, 0.7 % U-235 and negligible
amounts of other isotopes. Most reactor types do not use natural uranium
directly, but instead uranium enriched in U-235 up to a level of typically 4 %
for LWRs. This enrichment process is similar to that which is used for
producing nuclear weapons, where an enrichment level of 20 % is enough to
build a bomb but 90 % or more would be desirable [4].
The reported amounts of available uranium for power production varies a lot
depending on information source, partly since the amounts of uranium
considered available varies with the uranium price and different prices are
used in different information sources. The price depends on the percentage
of uranium in the ore. A rough scaling is given in Ref. [5], where uranium
ore is assumed to have a log-normal distribution over the world. This means
that a tenfold decrease in ore grade would correspond to a 300-fold increase
in amount of recoverable uranium from that ore in the Earth’s crust [5]. The
Red Book [6] by IAEA has been produced in 24 editions since 1965 and is
here regarded as a reasonably reliable source. The avaliable resources are
sufficient for at least 100 years with the current uranium consumption at a
fuel price of 260 USD/kg [6]. If the price is increased, there will be a lot
more uranium available. Since the price of fission fuel is in the order of
17
0.005 EUR/kWh and the uranium cost is about 50% of the fuel cost [7]
(0.0025 EUR/kWh) there is a large margin to increase the uranium price
without having a dramatic increase in electricity price.
Another possibility is to extract uranium from seawater. Seawater contains
3-4 ppb uranium, and the total amount of uranium in the seas is estimated to
4 gigatons, corresponding to about 100 000 years with the consumption rate
of the world today [6]. There is no technique available today for extracting
uranium from the sea at a competitive cost, but research is going on in Japan
[6]. Currently the extraction cost is about 700 USD/kg [6] compared to
todays market price of 100 USD/kg, which does not seem to be an
unreasonable price in the future.
When discussing resources, the possibility of breeding should also be taken
into account. Today, only about 1% of the energy in the original uranium ore
is consumed (converted) in the fission plants (see breeding, section 2.7). If
future technology could solve the safety problems with breeding and TRU
burning, there is a factor of 100 times more energy resources available from
uranium. Also, then the mining costs per kWh will be much lower, enabling
mining of lower-grade ore to be commercially feasible and thus increasing
the amount of available uranium with a factor of 90 000 according to the
very rough log-normal distribution [5], giving in total 9 000 000 times more
fuel. There are however other problems associated with mining (such as
difficulties of extracting uranium from very low-grade ore), and it is
probably overoptimistic to believe that such a large portion of the earths
crust will be available for mining purposes. This is however a rough
indicator of how much fuel there could be.
If the safety problems with breeding are solved, also thorium can be used for
energy production. Thorium is about 3 times as abundant as uranium [8].
There are no facilities today that use thorium for commercial energy
production, but a few test reactors have been built and some are under
construction [9]. Specifically, India is aiming for a thorium fast breeder
reactor that is supposed to be operational in 2013-2014 [10]. China has also
started a program for thorium reactors [9]. For comparison with fusion fuel,
C. Rubbia has claimed that the availability of fission fuel is about the same
as the availability for D-T fusion fuel [11]. Lithium is 7 times as abundant as
thorium in the earths crust, but only 7.5% of the lithium is Li-6 which is the
primary isotope used. With this in regard, thorium is 4 times more energy
dense than lithium (per unit mass) and the available energy resources for
fission and fusion is about the same [11].
To summarize, the resources for fission power are vast. If breeding
technology becomes commercial, the available resources are likely to be
18
immense. Although breeding is not the main target in this project, fusionfission hybrids seem well suited also for breeding, although critical fast
breeder reactors are likely to be considerably cheaper since they do not need
a fusion driver.
2.3. Front end
The uranium used today originates from uranium mines. Uranium mining
has caused environmental problems during the early stages of nuclear power
and continues to do so today in some development countries, where
environmental regulations are weak and risks of prosecution are small.
Numerous examples are presented by Greenpeace [12] and are used as an
argument against fission power. It is however evident that uranium mining
can be done in ways that are safe for the personnel and has a similar
environmental impact as other metalliferous mining activities [13]. Such
mining has taken place in Canada and Australia where most mines have ISO
14001 certification [13]. Thus, this is not an argument against uranium
mining in general but against uranium mining using environmentally benign
techniques with lack of control.
The mining capacity in the world today is lower than the consumption. The
reason for this is that 25-50% of the uranium supply the last years has come
from stockpiles of uranium and from downblending of weapons-grade
uranium from nuclear weapons [6]. Since this source soon will diminish, the
mining capacity needs to increase.
2.4. Back end
The fission power industry must be able to handle the whole life cycle of the
fission fuel which implies that the spent nuclear waste must be taken care of.
There are two main ways to handle the problem. One is to get rid of some of
the environmentally benign isotopes in the fuel by transmutation and fission,
and store the remaining waste. This is described in section 2.6. The other
method is to create a geological repository for the fuel such that the fuel
remains safely stored for 100 000 years. After 100 000 years, the strongest
radiating radioactive isotopes has decayed and the rest is regarded as fairly
safe to leave in the ground. Sweden is in the front line of the development of
geological repositories, and a repository large enough to store the Swedish
nuclear waste is planned to be built near the fission plant Forsmark about 75
km outside Uppsala by Svensk KärnBränslehantering AB (SKB) [14]. The
repository is to be built 500 m below the surface where it is assumed to be
below the permafrost during an ice age (assumed to maximally reach
somewhere around 400 m). The fuel rods are placed in thick copper canisters
with cast iron inserts. The copper canisters are put in bentonite clay in
19
prepared caverns in the bedrock [14]. The Forsmark bedrock is considered to
be very stable.
A lot of research has already been made on this SKB model called KBS-3,
but still some issues are debated, for instance the corrosion rate of the copper
canisters. Natural questions also arises concerning nuclear non-proliferation,
since the plutonium weapon-grade quality will increase in time due to the
shorter half-life of Pu-240 (6 500 years) compared to Pu-239 (24 100 years).
Although such patient terrorist organizations are unlikely to exist, the
political situation 20000 years from now is hard to predict and it is
inconvenient that high-quality weapons-grade plutonium will lie buried in
the ground for tens of thousands of years. Also, there is always a possibility
to use the spent fuel as a dirty bomb, where the explosion is created with
conventional explosives. Another risk is treasure hunting, since some of the
fission products are rare and valuable.
The cost of the geological deposit in Sweden is estimated to about 12 billion
euros in current monetary value [14]. This is financed by a nuclear waste
fund, which receives 0.001 euro for every kWh fission energy sold.
2.5. Nuclear non-proliferation
Nuclear non-proliferation, to prevent nations and groups to develop nuclear
weapons, is probably the largest concern for nuclear power. The strategy is
to have a sophisticated control of weapons-grade fissile material such as
enriched uranium and weapons-grade plutonium, and to prevent the spread
of knowledge and technology required to produce such materials, in
particular enrichment facilities for uranium. Uranium with more than 20 %
U-235 is considered weapons-grade uranium. Plutonium is considered
weapons-grade if it contains less than 7 % Pu-240. This is due to a
considerable rate of spontaneous fission in Pu-240, which could cause
predetonation in a bomb before sufficient plutonium mass is assembled.
Even with weapons-grade plutonium, a plutonium bomb must be of
implosion type to give a massive explosion, which is considered more
complicated to build than the simpler uranium bomb. LWRs can be used to
produce weapons-grade plutonium if the fuel is removed from the reactor
shortly after refueling (1-3 months). Pu-240 is formed from neutron capture
in Pu-239, and therefore the plutonium is weapon-grade in the beginning of
the fuel cycle and the amount of Pu-240 increases continuously towards the
end of the fuel cycle.
20
2.6. Transmutation
Transmutation is the process of changing one nucleus to another (or two
others) by neutron capture or fission. In fission power, transmutation can be
used to get rid of undesired radiating species in the nuclear waste in order to
reduce the geological storage time [15]. There are two groups of elements
that are targets of transmutation: transuranics (TRU) and long-lived fission
products (LLFP). It should be pointed out that transmutation is not
recommended by all sources, see Ref. [16]. A list of the most important
targets of transmutation is given in Ref. [17]. Complications with
transmutation are the reprocessing of the waste necessary to separate the
different species [16] and to find safe reactors that can handle sufficient
densities of the transmutation targets. Another concern is that industrial
transmutation may be connected with nuclear proliferation.
2.6.1.
Transuranics
Transuranics are created from neutron capture in U-238 followed by
subsequent neutron captures and decays. The main component of the
transuranics is Pu-239, which is the main component responsible for the long
geological storage times for the nuclear waste with a halflife decay time of
24 100 years. Radiotoxicity is a measure of how dangerous a radionuclide is
for the human body. After 200 years of storage, TRU contributes with the
major part of the radiotoxicity of the nuclear waste [16]. Transuranics are
much more radiotoxic than fission products since they typically are emitters (emits helium nuclei) and fission products typically are -emitters
(emits electrons) [17]. Pu-239 is produced in LWRs but also to some extent
fissioned. To transmute minor actinides, the transuranics except plutonium, a
fast neutron spectrum is required [18]. The options to produce this fast
neutron spectrum are fast reactors and driven systems, but there seems to be
a consensus that driven systems are required for the transmutation of minor
actinides for reactor safety reasons [18]. Fission is always the goal of
transmutation of transuranics, since the decay chain to stable isotopes is long
for the elements in this group and since neutron capture will only result in
another transuranic isotope [16]. This implies that transmutation of
transuranics produces a lot of thermal energy which could be possible to
utilize in a power plant.
2.6.2.
Long-Lived Fission Products
Fission products are the rest products from fission and elements created by
subsequent neutron capture and decay from these. Fission of one isotope (for
example U-235) can result in a lot of different combinations of fission
products. Some are stable, some have short half-times and 7 are long-lived.
21
Fission products with a half-life of less than 90 years are classified as
medium-lived or short-lived. The largest radiation emitters are Sr-90 and Cs137, with a half-life of about 30 years [16]. These cannot be transmuted due
to a low cross section for neutron capture and must be stored until they
decay, which takes up to 500 years [16]. Since Sr-90 and Cs-137 anyway
need to decay, there is no point in transmuting any medium-lived or shortlived isotopes. Targets for transmutation have therefore been the 7 long-lived
fission products (LLFP). Of those, Tc-99 and I-129 are the isotopes that have
been targeted in most studies [16], since they give the largest contribution to
radiation [17], are possible to (slowly) transmute with thermal neutrons and
since they are mobile enough in the environment (soluble in water) [17] to
pose a large threat for geological storage [15]. Cs-135 is hard to transmute
without isotope separation [16]. The other 4 isotopes are either in smaller
amounts (at least for LWR waste from fissioning U-235), or considered
immobile in the environment (like the noble metal isotope Pa-107).
Transmutation of Tc-99 and I-129 can be done in LWRs [16] or possibly
with faster neutron spectras [19]. Transmutation of fission products does not
produce significant amounts of energy, and consumes neutrons.
Although it may be possible to transmute some of the LLFP, there is now a
general consensus in the fission community that transmutation of LLFP is
not necessary. Studies have shown that the radiation doses given from LLFP
leaking from a geological repository to the most exposed groups of humans
in any of the investigated scenarios are several orders of magnitude smaller
than the background radiation [20]. Thereby, transmutation efforts are
nowadays focused on transuranics, in particular plutonium and americium.
2.7. Breeding
Fertile nuclei are nuclei which are not fissile in a self-sustainable way, but
may be transformed into fissile nuclei by neutron capture. Breeding is the
process of turning fertile nuclei into fissile ones. The two species mainly
considered for breeding are U-238 and Th-232. U-238 is transmuted into Pu239 and Th-232 into U-233 by neutron capture and subsequent decay
processes. The point with breeding is to produce more fuel, where there is a
factor of 100 to gain only on the uranium. At a first glance, this may seem to
be in contrast to transmutation, where the aim is to get rid of plutonium.
However, both can be used in combination, and the final waste may still be
nearly cleansed on plutonium if a working transmutation scheme can be
implemented. Breeding is already present to some extent in todays LWRs,
but the gain of fissile fuel is less than the consumption. To have a net gain in
fissile fuel, which is the goal of a breeder reactor, a fast neutron spectrum is
preferred since a fast neutron spectrum produces more neutrons per fission
22
on average [16] and the capture-to-fission ratio is smaller, giving a
significantly larger in Eq. (2.1.1) [2].
2.8. Critical fast reactors
Critical fast reactors are critical reactors with a fast neutron spectrum, which
differ from LWRs that use thermal neutrons for fission. The goals of fast
reactors are transmutation, breeding and also hydrogen production enabled
by the high temperature of the coolant [2]. Fast reactor programs have
however been plagued by safety problems etc. [16] and have not yet been
commercialized. The main two stabilizing negative feedback factors in
LWRs on keff are Doppler broadening in U-238 and effects on moderation
due to water density changes (in BWR, void percentage). In fast reactors,
water is not present since the water would moderate the neutrons. Instead,
liquid metal (lead, sodium, lead-bismuth), molten salt or gas is used as
coolant [2]. The Doppler broadening effect is much less effective since a
smaller fraction of the neutrons that may undergo fission pass the neutron
capture resonances and since there often is less U-238 in the fuel. Another
feedback factor on keff , fuel/coolant heat expansion, plays a role in fast
reactors. A safety concern is also that the fraction of delayed neutrons is less
in plutonium ( Pu-239 = 0.26%) and even less in americium ( Am-241 = 0.12%)
and curium ( Cm-244 = 0.13%) [21]. Loss of coolant, in particular for sodiumcooled fast reactors, is a safety concern. Replacement of the metal coolant by
water around the fuel could lead to a catastrophy. Together, this makes the
fast reactor concepts much less safe than thermal reactors, and breeder
reactors have today not reached public acceptance. Critical fast reactors are a
key area for the generation IV studies.
2.9. Driven systems
Driven systems are non-critical reactors with keff less than unity. Thereby, an
external source of neutrons is required to maintain the neutron flux. The
reactor safety in a driven system is not dependent on negative feedback
factors and delayed neutrons, although they may still add somewhat to
safety. Instead, a driven system relies on the keff that keeps the reactor
subcritical. In all possible scenarios, keff must be kept below unity (plus the
fraction of delayed neutrons) [21]. For Accelerator Driven Systems (ADS) a
typical value is keff = 0.97. This is a larger margin than the delayed neutron
fraction give in LWRs (0.0065) and it may seem that driven systems are
safer. However, a driven system is more complex, has much smaller
negative feedback factors (if any) on temperature and is in several aspects
regarded as less safe than LWRs. The goal of driven systems is primarily
transmutation in combination with energy production [18]. Driven fast
reactors could also be used to accomplish breeding, although competition
23
from fast critical reactors may be too strong. As mentioned earlier, Stacey
claims that driven systems are necessary to accomplish transmutation [18],
see also Ref. [2]. The very point of driven systems is that they can perform
the tasks that require a fast neutron spectrum with a much larger safety
margin than critical fast reactors have, and that they are almost independent
of the fraction in the fuel.
The fission core provides a rather large neutron multiplication which gives a
large power multiplication. If the neutron multiplication keff is assumed to
be constant for each successive neutron generation, which is not exactly true,
in particular for the first generation, the total number of neutrons n produced
for each source neutron on average can be calculated as a geometric series.
n
keff2
keff
keffi
3
...
keff
i 1
keff
1 keff
(2.9.1)
For keff = 0.96, this would increase the total number of neutrons produced by
a factor of 25, and for keff = 0.98 this gives n 50. It is obvious that keff
should be as high as possible constrained by the safety issues. An
approximation of the total energy gain is
Qr
rh nc next
W fis
keff
Wsn 1 keff
(2.9.2)
where W fis 195 MeV is the average energy produced in a fission reaction,
Wsn is the average energy cost for each source neutron, v 2.9 is the
average number of neutrons produced in each fission reaction, nc is the
fraction of source neutrons that enter the fission core, next is a correction
factor that takes effects of different source neutron energy into account and
rh is the fraction of heat contribution that does not come directly from fission
reactions (primarily decay of fission products).
There are two types of neutron sources considered today. One is ADS, and
the other is fusion. For fusion devices to be useful as neutron sources, they
do not only need to fulfill the requirements for transmutation (concerning
energy consumption, environmental impact, overall cost etc.). They must
also be competitive against ADS systems. It is today not obvious which
system will be superior, and at the end the overall cost and reliability are
likely to be important factors. Also, the safety restrictions on keff for the
different systems will strongly affect the energy efficiency, and this depends
on the layout of the fission core.
24
2.9.1.
Accelerator-driven systems
ADS is the most developed concept for driven systems today. An extensive
comparison with fast reactors can be found in Ref. [15]. The system consists
of a proton accelerator that produces a proton beam. The beam is injected
into a spallation target of heavy metal, typically lead, which is aimed to
produce about 20-30 source neutrons for each injected proton. The source
neutrons enter the fission core which surrounds the spallation target [15].
The energy of the injected neutrons is in the order of 1 GeV, and the electric
energy efficiency of the accelerator is up to 50% [15]. The average energy
for the spallation neutrons is about 1.6 MeV, and the average electric energy
cost to produce one neutron is about 100-150 MeV. The ADS technology
faces some challenges [22]. One challenge is to increase the average
accelerator beam current. Another challenge is to maintain the neutron
production in the spallation target, i.e. to prevent vaporization at the beam
target point. Another concern is the utility, which presently is very low.
2.9.2.
Fusion-driven systems
Fusion-driven systems (fusion hybrids) use a fusion device as neutron
source. The concept was proposed already in the early days of fusion, and
was persued by H. Bethe [23] and others (see for example Ref. [24]) in the
1970ies. The Three-Mile-Island accident in 1979 led to a decline in the
fusion-fission research, and especially after the Chernobyl accident in 1986
it seems that the fusion community wanted to keep a distance from fission to
avoid negative publicity. It even seems that some researchers in fusionfission had problems both to publish their work and to get financed due to
this policy (see the acknowledgement in [25]). Also, after the Fukushima
accident, several countries have an ambition to avoid fission energy if
possible. Fusion is however in several ways naturally linked to fission. A
fusion neutron source (like any sufficiently intense neutron source) can be
used for breeding of weapons-grade plutonium or U-233 (which also
probably can be used for production of nuclear weapons) from thorium.
Fusion therefore has a link to the nuclear non-proliferation problems, which
was pointed out already by L. Lidsky in his criticism against fusion [26].
Also, it has been questioned if a tokamak fusion reactor can breed sufficient
amounts of tritium be self-sufficient. If fusion reactors cannot, fission
reactors or fissile inserts in fusion reactors are probably required for
producing tritium. Fission reactors are the source of tritium today.
In the new millennium there has been a renewed interest in fusion-fission,
and the subject is being pursued by several groups. Some of them are (in no
specific order):
25
1. The SFLM Hybrid project. This is what this thesis is about.
2. Researchers at Budker Institute who studies a mirror-based hybrid
scenario using the axisymmetric Gas Dynamic Trap (GDT) and
the new modified GDT as a driver [27].
3. R.W. Moir et al. at Lawrence Livermore who recently presented
an axisymmetric mirror-based concept [28].
4. S. Taczanowski et al. has some activities in mirror-based fusionfission [3].
5. W.M. Stacey et al. at Georgia Tech who have studied several
tokamak-based concepts with downscaled ITER parameters [29].
6. Y. Wu et al. in China who studies tokamak-based hybrids (several
FDS concepts) and are putting large resources into hybrid studies
[30].
7. M. Kotschenreuther et al. of Institute for Fusion Studies who
studies tokamak-based hybrids [31].
8. M. Gryaznewich et al. at Culham Laboratory is examining the
possibilities to use spherical tokamaks as neutron sources [32].
9. A Russian program has recently been initiated to build a sperical
tokamak neutron source.
10. V. E. Moiseenko et al. in Kharkiv, Ukraine have an experimental
stellarator-mirror facility aimed for fusion-fission and to become a
neutron source [33].
11. H. Yapici et al. are working with fusion-fission using catalyzed
fusion as a driver [34].
12. M. Ragheb, A. N. Eldin et al. at the University of Illinois who are
considering thorium breeding using hybrid reactors [35].
13. F. Winterberg has presented ideas concerning fusion-fission
reactors [36].
14. W. Manheimer is advocating fusion-fission [25].
Several types of fusion devices may be used as neutron sources, where the
tokamak is the most studied source so far. The two largest theoretical
tokamak hybrid projects that exist today are the FTWR in USA [29] and the
FDS [30] in China. Both are based on tokamaks with downscaled ITER
parameters. The strength with the tokamak concept is the reasonably good
plasma confinement, which allows for large safety margins on keff. However,
since the power multiplication in the fission mantle probably can be large
( 100), it is not obvious that such a confinement is crucial for hybrid
reactors. This enables the use of other concepts. Tokamaks have some major
drawbacks that make them less suited for hybrid reactors if a better suited
fusion device can have sufficient plasma confinement for hybrid reactor
configuration. The tokamak cannot (at least presently) be run in steady-state
due to the need for inductive toroidal current drive. The pulses will be in the
order of 20 minutes for ITER, which would correspond to a pulsed gigawatt
26
power source for the grid and could cause material problems. There is also a
lack of space in a tokamak for the fission mantle due to all instrumentation
and plasma heating, which typically causes more than half of the source
neutrons to escape from the reactor without having the chance to produce
fission. Specifically, in SABR only 39 % of the fusion neutrons enter the
fission core [37]. Large scale plasma activities (disruptions and instabilities)
are also concerns for tokamak hybrid reactor scenarios.
Mirror hybrid reactors have been considered by Bethe [23], Taczanowski
[3], Moir [28][38] and Noack et al [27] at the Budker Institute in
Novosibirsk, Russia. Mirror machines are relatively simple, can operate in
steady-state and can conveniently decrease the plasma heat load on the walls
by using magnetic expanders and thereby increasing the wall area. The
fraction of source neutrons that escape without entering the fission mantle
can be less than 10% and the first wall neutron load can be made acceptable.
The major concern is the electron temperature in the plasma. For the SFLM
hybrid case, the approximate formula for the energy amplification
Qr Pfiss / Pfus , the produced fission power divided by the produced fusion
power, is
Qr
where keff
case.
1.2
195
0.97
150
17.6 2.9 0.03
0.97 has been selected and the product rh nc next
(2.9.3)
1.2 in this
The 14 MeV fusion neutrons are about ten times more energetic than the
average spallation source neutrons and the fission neutrons. Source neutrons
with energy larger than 6.5 MeV produce essentially more fission neutrons
than strongly moderated source neutrons. In Ref [3], the high energy of the
fusion neutrons gives about 50 % extra fission neutrons in the first neutron
generation. A too strong moderation of the source neutrons before they reach
the fission core should therefore be avoided.
27
28
3. Fusion energy and plasma physics
A fusion-fission reactor needs a fusion driver, and the main work in this
doctoral thesis is about the mirror fusion driver. In this section, some basics
about fusion and plasma physics are given, as well as some information
specific to mirror machines.
3.1. Basics in fusion
3.1.1.
Plasmas
When a gas is heated to a high temperature, the molecules start to break up
and the atoms are ionized. An ionized gas is called a plasma. Plasmas exist
in many applications, such as for example low ionized plasmas in high
voltage circuit breakers. In nature, plasmas exist for example in a lightning
bolt. Outside the Earth, most matter is in plasma state, including all stars and
the Sun.
3.1.2.
Fusion reactions
Fusion energy is produced by joining two lighter nuclei into a heavier one,
plus other particles in some cases, where the resulting particles have a lower
total rest mass than the original ones. The rest mass difference is released as
energy. The most commonly targeted reaction is
D + T = He + n + 17.6 MeV
(3.1.1)
where D is deuterium (hydrogen with 1 neutron), T is tritium (hydrogen with
2 neutrons) and n is a neutron. The neutron receives 14.1 MeV and the alpha
particle (He) 3.5 MeV [39]. There are also a number of other reactions, for
example D + D reactions giving either He-3 + n + 3.2 MeV or T + p + 4.0
MeV [39]. D-D fusion requires higher colliding energies than D-T fusion
and gives less energy in each reaction, but uses on the other hand only
deuterium.
To initiate a fusion reaction, the electric force repelling the two nuclei that
should react must be overcome. When the nuclei get close enough, the
attracting nuclear force become greater than the repelling electric force and
the nuclei will undergo a fusion reaction. This implies that the particles must
29
have a high velocity, and also a lot of “target” particles to collide with. For
thermonuclear D-T fusion, an ion temperature of about 10 keV is required
[39], i.e. about 100 million °C. The densities in magnetically confined fusion
plasmas are typically in the order of 1019-1020 particles per cubic meter. This
is far less than in air at atmospheric pressure, having about 2 1025 molecules
per cubic meter. The reason for the “low” density compared to air is that
since such a high temperature is required, the pressure would be huge and
the magnetic field required (> 100 T) to contain such a pressure would be
too high. Fusion plasmas that are considered here are fully ionized, at least if
impurities are neglected.
3.1.3.
Fusion and confinement
One simple idea to create fusion would be to accelerate deuterion ions and
let them burst into a solid target containing tritium. This approach will
however not work for energy production, since the ions will lose their energy
through collisions too fast and too few fusion reactions would result per
incoming ion to produce a net gain in energy [39]. Another idea is to use
muons to catalyze fusion reactions, sometimes referred to as cold fusion but
more often as muon catalyzed fusion [40]. Muons are subatomic particles
with the same charge as an electron and a mass about 200 times larger, and a
“muon hydrogen atom” (where a muon replaces the electron) is compact and
has a much smaller Coulomb barrier for fusion. Muon catalyzed fusion can
be accomplished at room temperature. To the author’s knowledge, there is
no reactor scenario for muon catalyzed fusion yet, but studies aimed at
catalyzed fusion-fission are carried out [41].
The main path of fusion research is to heat a D-T mix to a hot plasma having
fusion temperature. The difficult task is to confine the plasma. A well
working fusion reactor is the Sun, which confines the plasma with strong
gravity forces. That option is however not available on the Earth. The fusion
plasma is also far too hot for any container material to withstand, so it cannot
be kept in a container without other means of confinement. Today, there are
two main paths pursued in fusion research; inertial confinement and
magnetic confinement. Inertial confinement is pulsed, and aims to find a way
to heat fusion fuel (typically a frozen pellet) into a hot plasma very fast and
keep it together as long as possible with momentum transfer from lasers or
similar radiation sources. The H-bomb is an example of inertial fusion,
where a fission bomb supplies both heating and compression with radiation
targeting the fusion material. The National Ignition Facility in USA is a large
inertial fusion experiment where lasers are used to target a frozen D-T pellet
[42]. In this doctoral thesis, magnetic confinement is addressed, and
henceforth only magnetic confinement will be discussed.
30
3.1.4.
Resources
In fusion, several fuels can be used. This doctoral thesis is focused on D-T
fusion. Deuterium is widely abundant in the world. About 1/7000 of the
hydrogen atoms are deuterium. There is a lot of hydrogen in the oceans, and
the available amount of deuterium is huge. For D-T fusion, the deuterium
resource will not be limiting. Tritium is produced from neutron radiation of
lithium, mainly from Li-6, and has a lifetime of about 12 years. The
resources of lithium are large, and would probably last for millions of years
of intensive utilization [11].
3.2. Plasma physics
In plasma physics, three different detail levels can be used whilst performing
calculations. The first is to look at single particle motion in the magnetic
field. From this approach, particle drifts can for example be derived.
However, in these calculations effects of the surrounding particles are not
taken into account, and all the important effects related to multi particle
interaction are thereby lost. It is impossible to take into account all the
plasma particles as single particles, since there are typically 1019-1020
particles per m3 in the plasma and each particle gives a force on all other
particles. The second approach is to look at the plasma from the statistical
mechanics viewpoint and use the kinetic theory. Kinetic theory takes most
effects into account but it is often a tremendous task to carry out the
calculations. The third approach is to treat the plasma as an electromagnetic
fluid and use the MHD (MagnetoHydroDynamic) equations. MHD theory is
easier (although not necessarily easy) than the kinetic theory, but some
information is lost and not all effects can be found. In this section, some
basics of plasma physics are explained.
3.2.1.
Magnetic confinement
In a fully ionized plasma, all particles are charged. Magnetic confinement is
based on the Lorentz force,
F
q (E v B )
(3.2.1)
where v is the particle velocity. The magnetic force is perpendicular to both
B and v, forcing each particle to gyrate around a certain magnetic field line
(to leading order). Thereby the particles are confined, at least to some extent,
in the directions perpendicular to B if the magnetic field is strong enough to
give a sufficiently small gyro radius (Larmor radius). The gyro radius rg is
given by
31
rg
v
qB
mv
,
qB
(3.2.2)
m
where
is the gyro angular frequency. For comparable temperatures, the
ions have much larger gyro radii than the electrons due to the larger ion
mass. The ratio is proportional to ( M / m)1/ 2 , giving the deuterium ions a
radius that is about ( M D / me )1/ 2 60 times larger than the electron radius
[43]. A typical ion gyro radius in a fusion plasma is in the order of 1 cm (B
=2 T, Ti = 10 keV gives for deuterium ions rL,i = 0.7 cm) while the electrons
typically have a gyro radius less than 1 mm.
In the direction parallel to B the particles movement is not as restricted, and
the particle trajectory is locally helix-like. The problem is to find a way to
confine the particles along B. There are several solutions to this problem.
The most obvious choice is to make a toroidal (doughnut-shaped) magnetic
surface where the flux surfaces are closed, although most of the field lines
are not closed (construction of “closed”, or nested, magnetic surfaces is
straightforward for axisymmetric tokamaks but a difficult task for
stellarators). Tokamaks and stellarators are toroidal devices. Mirror
machines, which will be considered here, have an open magnetic
configuration and a straight pipe-like vacuum chamber for the plasma. The
ends of the pipe are magnetically plugged for the plasma by the reflecting
magnetic mirror force, which arises when particles move from weaker
magnetic field to stronger, see section 3.3.1.
3.2.2.
Particle drifts
Particles are restricted in the perpendicular direction by the magnetic force.
However, if a force perpendicular to B is applied on the particle, the gyro
radius varies during the gyro period which results in a particle drift
perpendicular to both the magnetic field and the force. With B B0 zˆ is
constant, E = 0 and F qB0 v B F0 xˆ , the equations of motion give
vx
vy
F0
, vy
m
vx , vz
v
0
(3.2.3)
where
vx
2
vx
0
vy
2
vy
qB
F0
m2
(3.2.4)
If F0 = 0, the solution to these equation is a helical trajectory with a circular
movement in the projection on the xy-plane as described earlier with a gyro
frequency given in Eq. (3.2.2).The guiding center variables
32
xc
x
vy
,
yc
y
vx
(3.2.5)
are constant (no perpendicular drift of the guiding centers). With a finite F0,
a guiding center drift appears. With F0 constant, the solutions are
vx
v 0 cos t
vy
v 0 sin t
(3.2.6)
F0
qB0
with the chosen initial condition v (t 0) v 0 xˆ . For a more general case the
velocity in vector notation can be written (apart from higher order gyro
oscillations)
v
v Bˆ v g
vd
(3.2.7)
where vg is the gyrating part and vd is the drift velocity averaged over the
gyro motion. There are a number of forces that can cause drifts. The gravity
drift is one example. Since electric fields give a force that is charge
dependent, the electric drift is charge independant. Magnetic field gradients
and curvature gives rise to charge dependent so-called B and centrifugal
or curvature drifts. The leading order gyro-averaged total drift velocity for
time-independent fields can be written
vd
BB
B
3
q
B
E B
B2
mv 2 B (Bˆ Bˆ )
q
B2
(3.2.8)
where the magnetic moment
mv 2 / 2 B
(3.2.9)
is an approximate constant of motion and the gravitational drift has been
omitted as well as the drift from a nonuniform E. The first term is the E B
drift, the second is the drift from a gradient in B and the third term is the
curvature drift, where (Bˆ Bˆ )
1/ rc rˆc is the curvature [44] and rc is
B 0 ) this simplifies to
the curvature radius. For a vacuum field (
vd
E B
B2
2
1 B mv
(B
q
B2
)
(3.2.10)
which would be zero for E = 0 and
0 . The guiding center velocity
v gc v and the energy conservation for the particle can be rewritten as a
constant energy for the guiding center motion,
m 2
q (x gc )
B (x gc )
(3.2.11)
v gc const.
2
33
There are drifts for time-dependant fields as well, but those are not addressed
in this text.
3.2.3.
Collisions in fusion plasmas
In a plasma, particles collide. To describe collisions, a few concepts are
useful. A cross section is used to express the likelihood of interaction
between particles. Cross sections are dependent on the particle velocity, and
collision cross sections are typically denoted pq where p and q are the two
colliding particles. An example of a cross section is the 90° electron
scattering cross section of ion-electron collision
ei
Z 2 e 4 ln
4 02 me2 v 4
(3.2.12)
where ln is about 20 and Z is the atomic charge [43]. The mean free path
mfp is the distance particles on average can move in between collisions. The
collision frequency pk describes how often particles will collide on average,
and the collision time (or drag time) pk =1/ pk is the average time between
collisions. We have for a nearly Maxwellian distribution
v th
pk
mfp
nk v th
1
pk ,
mfp
nk
(3.2.13)
pk
Only a small fraction of the collisions lead to fusion reactions. The
remaining collision processes are more or less distant Coloumb collisions
where the momenta of the colliding particles are changed by the localized
electric forces around the charges. The main contribution of the average
momentum change for a particle comes from cumulative weak distant
collisions. The difference in mean square momentum impact of the collisions
that change the particle velocity less than 90° compared to those that change
the velocity more than 90° are about a factor of 70 [43]. This means that
velocity changes of particles in a fusion plasma mainly happens in small
steps distributed in time which is an important property of hot plasmas. The
collision frequency for low angle scattering goes down with increasing
temperature, as can be seen in Eq. (3.2.12), but does not become zero. The
properties described above are for electron-ion collisions, but similar
properties yields for electron-electron scattering and ion-ion scattering. An
important difference is that the diffusion impact of electron-electron
collisions are on a much faster time scale than for ion-ion collisions due to
the higher charge to mass ratio for electrons. Another important difference is
that through ion-ion collisions and electron-electron collisions, energy
transfer is efficient, but for electron-ion collisions the energy transfer is
slower due to the large mass difference between the colliding particles.
Thereby, the electron temperature and the ion temperature may be different.
34
3.2.4.
Radial transport
Radial transport, i.e. particle transport perpendicular to the magnetic surfaces
in the radial direction, occurs through different processes. One is through
collisions. If particles collide and their momentum is changed, they move
from the original encircled flux line to a new position. The step-size is in the
order of the gyro radius. This collisional transport is called classical
transport. However, there are other mechanisms through which radial
transport occurs, and those typically deteriorate the plasma confinement
considerably more than the classical collisions. The problem of constructing
a working fusion energy producing test facility would probably have been
much less challanging if the classical transport was the only transport
mechanism present.
Another mechanism for radial transport is the neoclassical transport. This
transport is also based on collisions, but the impact is in many cases far
worse. In a magnetic field, the guiding centers do not follow a single flux
line in any fusion devices except for the SFLM vacuum field, since there is
magnetic field curvature and gradients in B. These cause particle drifts that
deform the particle orbits. In tokamaks, some projected guiding center orbits
of mirror-trapped particles are banana-shaped and are called “banana” orbits.
Those large-scale orbits have typically much larger widths than the gyro
radius, and a collision for such a particle will lead to a much larger radial
step than the gyro radius on average. The diffusion coefficient in “random
walk” processes can be estimated from
D
coll
where coll is the collision frequency and
collision process.
r2
(3.2.14)
r is the typical step length in the
There are also other radial transport mechanisms caused by fluctuations or
formation of nonlinear structures, which are referred to as anomalous
transport. In many cases, for example in tokamaks, anomalous transport
gives the major contribution to radial transport. The anomalous transport is
important but complicated and is not discussed any further in this text.
3.2.5.
Kinetic theory: the Vlasov equation
A very brief overview of kinetic theory is given in this section. In kinetic
theory, the plasma is treated from a statistical mechanics viewpoint. The
plasma particle densities are described by the distribution function, or
which is a function
probability density, f (x, v, t ) for the particle species
of 6 dimensions plus time. The 6-dimensional (x,v) space is called the phase35
space. If collisions are neglected, which is a good approximation for a hot
plasma, the time dependence of f (x, v, t ) in phase-space is determined by
the Lorentz force and E and B fields. The Boltzmann equation describes the
evolution of f (x, v, t ) with a force F and a collision term at the r.h.s.
df
dt
f
t
v
f
F
m
f
v
f
t
(3.2.15)
c
Replacing the force F with the Lorentz force and setting the collision term to
0 gives the Vlasov equation
f
t
v
f
q
(E v B )
m
f
v
0
(3.2.16)
The E and B fields should be interpreted as the smoothened fields where the
fluctuating fields from the nearby individual particles do not show, and the
collision term ( f / t )c represents the forces from the near fields from
individual particles and plasma heating. Collision terms can be added, such
as in the Fokker-Planck equation [43], but this will not be discussed in this
text. The Vlasov equation takes most effects into account that can not be
handled with MHD theory, but is quite difficult to use. An example of an
effect where kinetic theory is required is Landau damping.
For the equilibrium, an arbitrary function of constants of motion ci (x, v ) ,
f f (c1, ..., cn ) is a solution for the Vlasov equation. A question is how
large n is.
3.2.6.
MHD Equations
A brief overview of the Ideal MHD equations is given in this section. The
Ideal MHD equations are basically fluid equations combined with Maxwells
equations. They are derived from the Boltzmann equation [45][46],
f
t
v
f
q
m
E v B
v
f
f
t
0
(3.2.17)
c
and the
where f is the distribution function for the particle type
gravitational force has been neglected. By taking moments of the Boltzmann
equation (integrating over velocity space),
Qi
V
f
t
v
f
where the different Qi are
36
q
m
E v B
v
f
f
t
d 3v 0 (3.2.18)
c
Q1
m
Q2
m v
Q3
( mass )
(momentum)
(3.2.19 a-c)
2
m v / 2 (energy )
the two-fluid equations can be derived for the two species ions and electrons
(we assume here that there is only one species of ions). These equations can
then be joined to form the single fluid equations. By invoking a number of
approximations and assumptions, one arrives at the Ideal MHD Equations.
The approximations are [45]
1. Quasineutrality. It is assumed that 0 E / en 1 , ne ni n .
2. The displacement current is neglected. Thereby, it is assumed that
thermal velocities and phase velocity of waves is considerably lower
than the speed of light.
3. The electron inertia is neglected.
4. The gyro radius is small.
5. The plasma resistivity is negligible.
6. High collisionality, which means that the pressure can be modelled
as a scalar pressure.
The Ideal MHD fluid equations then becomes
u
t
j B
m
t
(
m
E u B 0
d p
0
dt m
p
Momentum balance equation
(3.2.20)
u) 0
Continuity equation
(3.2.21)
Ohm’s law
(3.2.22)
Adiabatic equation of state
(3.2.23)
and should be combined with the Maxwell equations
B
E
B 0
0
j
B
t
Amperes law
(3.2.24)
Faradays law
(3.2.25)
Gauss law for magnetic fields
(3.2.26)
where
is the ratio of specific heats (
would correspond to
incompressible flow,
5 / 3 corresponds to the adiabatic equation in Ideal
MHD), m is the mass density and u is the local mass velocity. The
adiabatic equation results when it is assumed that there is no heat conduction
[46]. Although the high collisionality assumtion is virtually never satisfied in
fusion plasmas, the Ideal MHD equations can predict many phenomena that
37
have been verified experimentally. Examples are MHD instabilities, causing
dramatic plasma displacements, and the frozen in flux theorem predicting
that plasma tends to be tied to the magnetic flux lines for time dependent
magnetic fields if the plasma resistivity can be neglected.
Stability within Ideal MHD can be analyzed by linearizing perturbations
around a static (u 0) equilibrium. A selfadjoint boundary value problem
arises by considering Fourier components of perturbed quantities, i.e.
B
B (x)e i t etc. resulting in an equation of the form
2
F(
m
)
(3.2.27)
is a small plasma displacement and F ( ) is linear in
where
first derivatives. Selfadjointness leads to the energy principle
W
2
2
m
d 3x
,
Fd 3 x
W
and its
(3.2.28)
(x) , the system is stable within Ideal
If 2 0 for all test functions
MHD. Using simplified analythical test functions is the most common way
to identify MHD instabilities.
3.2.7.
Diamagnetism, MHD equilibrium and the concept of
In a magnetically confined plasma, each gyrating particle creates a magnetic
field in the opposite direction to the external field B, and will thus partly
cancel the magnetic field. The plasma is thereby diamagnetic. The force
balance for equilibrium becomes (from Eq. 3.2.20)
j B
p
(3.2.29)
and combined with the other two Ideal MHD equilibrium equations Eq.
(3.2.24) and Eq. (3.2.26) this gives
p
j B (
B) B /
0
(B
)B
(B
)B
( B 2 / 2) /
0
(3.2.30)
This can be reorganized as
p
B2
2 0
(3.2.31)
0
The second term on the l.h.s. acts as a “magnetic pressure” and is called the
magnetic pressure. The term on the r.h.s. is called the magnetic tension and
has to do with magnetic field line curvature. If the curvature can be
neglected, which is reasonable in some cases, the sum of the magnetic
pressure and the plasma pressure is constant. To give a measure of how
much pressure there is in the magnetic field,
38
p
B /2
(3.2.32)
2
0
is defined. For the most common definition of , B is the vacuum field. A
device has a certain limit for MHD stability, where the limit depends on
geometry and magnetic field in particular. Mirror machines have higher
limits than tokamaks and stellarators, where for mirror machines normally
is defined at the midplane (x = y = z = 0) where B has a local minimum. In
this thesis 0 will be the value at (x = y = z = 0) and will refer to the
local . Also, for mirrors the perpendicular pressure component p should
be used instead of p in the definition of .
3.2.8.
Plasma instabilites
To confine a plasma, it is not sufficient to find a device that gives a
satisfactory equilibrium. The plasma must also be stable, at least against
MHD stabilities. The consequences of MHD instabilities are normally a
rapid loss of confinement, and there is a general consensus in the fusion
community that MHD stabilities must be avoided in fusion devices [45]. If
the plasma is slightly displaced from its equilibrium position, the system is
stable if the plasma returns to the equilibrium position and unstable if the
displacement continues to grow. As stated by the Energy Principle for the
Ideal MHD theory [45], instabilities occur whenever the plasma can displace
in such a way that it reduces the total potential energy of the system.
There are many types of instabilities that can occur in a plasma, and some of
them are briefly described here. Usually, the most serious instabilities are the
MHD instabilities, which can be derived from the MHD equations. Other
types of instabilities are tearing modes, which feed upon magnetic
reconnection that decreases the magnetic field energy. There are also many
small scale instabilities called microinstabilities or velocity space
instabilities that have many different causes. These instabilities typically
increases transport, but on a less dramatic scale than MHD instabilities.
3.3. Mirror machines
3.3.1.
End confinement and electron temperature
The end confinement is the weakest point of a mirror machine, and it is here
for simplicity described for a single cell magnetic mirror. Both ions and
electrons are confined through the magnetic mirror effect, where the
confinement for each particle depends on the particles pitch angle in the
midplane. If we first assume that E can be neglected, particles having high
perpendicular (compared to the magnetic field lines) velocity compared to
39
parallel velocity will be confined, and particles that do not will escape the
magnetic trap. Particles that are lost longitudinally are said to be in the loss
cone, where the midplane pitch angle
arctan(v / v ) is lower than the
loss cone angle
c
arcsin 1/ Rm
(3.3.1)
where Rm Bmax / B0 is the mirror ratio, the ratio between the magnetic field
modulus at the mirror end and at the midplane [47]. The reflection
mechanism arise as a consequence of the conservation of the magnetic
moment,
mv 2 / 2 B and the energy
mv 2 / 2
B q . From Eq.
(3.3.1), it can be seen that some plasma particles are immediately lost upon
creation (ionization). The longitudinal confinements of the remaining
particles depend on how long they can stay out of the loss cone. This is
determined by collisions and high frequency instabilities [47], which will
isotropize the plasma. The timescale of angular scattering of particle species
is much shorter for electrons than for ions due to the electron’s lower mass,
where the ratio for deuterions to electrons is ( M / m)1/ 2 60 . Thus, the
electrons will leak out to the loss cone much faster than the ions. However,
since the plasma then will get a surplus of ions, a positive plasma electric
potential called the ambipolar potential will be built up in the plasma after
a very short period of time. This potential will confine all but the most
energetic electrons, so that the ion loss rate is the same as the electron loss
rate which ensures quasineutrality of the plasma (i.e. that there are almost
equal numbers of positive and negative charges). The loss cones are thereby
converted to hyperboloids of rotation, nicely illustrated in Ref. [47] p. 1634.
A sketch of the loss cones for ions and electrons including the ambipolar
potential is shown in figure 3.1.
Figure 3.1. A sketch of the loss cones in a magnetic mirror.
Typically, e , where
is the ambipolar potential difference, is about 46 times kbTe, confining all electrons with lower energies and expelling all
ions with lower energies (partly dependent on where E is located) [47]. This
40
implies that the electron distribution is close to a Maxwellian distribution.
For high energy particles, the hyperboloids asymptotically approach the loss
cone shape. A measure of the time scale on which electrons are scattered 90°
on average due to Coloumb collisions with other electrons is approximately
ee
5.5 108 Te3 / 2 / ne
(3.3.2)
for fusion plasmas where ne (cm-3) is the electron density and Te is in keV
[47]. For a fusion plasma with Te = 10 keV and ne = 1014 cm-3 this
corresponds to ee 0.17 ms [47], which is a short time scale. For ion-ion
scattering, the time scale is approximately
1.25 1010 Ei3 / 2 A / ni
ii
(3.3.3)
where A is the atomic number (number of proton masses) [47]. The
scattering time is about 50-60 times larger for deuterium ions than for
electrons, and estimates the time for pitch angle scattering loss through the
mirror ends. The ion-ion scattering time is too high to allow for efficient
fusion energy net production in a conventional mirror, and the theoretical
limit of fusion Q is only slightly above unity in such machines [47]. In
addition, high ion temperatures (50-220 keV) are required.
Since the electrons are confined electrostatically, the electron temperature
will be much lower than the ion temperature. This is because whenever an
either cold or hot ion is lost, one of the hottest electrons will be lost together
with it. Also, if neutrals (for example impurities) enter the plasma, cold
electrons will replace the hot plasma electrons when the inserted ion leaks
out. It is, however, primarily the ion temperature that is important for fusion
reactions, and plasma heating is often designed to heat ions. The problem
with a low electron temperature is that the electrons will cool the ions by
Coloumb collisions, the so-called electron drag. This effect will increase the
power loss but only to a minor extent increase the particle loss. The power
loss from ions to electrons is for energetic ions approximately described by
the ion energy density formula
W
where the electron drag time
d
d
W0 e
t/
d
(3.3.4)
is
5.5 1011
A 3/ 2
Te / ne
Z2
(3.3.5)
and Z is the ion charge [47]. For deuterions, this corresponds to d 11 ms at
ne = 1014 cm-3 and Te = 1 keV [47]. The electron drag is today and has indeed
earlier been the dominant energy loss mechanism in mirror machines, and
the primary limiting mechanism for the fusion Q. There have however been
successful experiments rather recently indicating that higher electron
41
temperatures can be reached. In the axisymmetric Gas Dynamic Trap (GDT)
experiment, an electron temperature of 250 eV has been reached (by forcing
a rotational shear by potential plates) [48], and in the multimirror GOL-3 an
electron temperature of as high as 2500 eV has been measured by Thompson
scattering (which is considered a reliable method) [49]. According to D.D.
Ryutov et al. [50], the lower electron temperature of earlier experiments was
“not due to an intrinsic failure of mirrors, but rather a result of a particular
operational mode”. With proper reduction of the neutral gas in the expander
tanks, the electron temperature may approach classical values as in the GDT
experiment [50]. In the theoretical design for an axisymmetric fusion-fission
reactor by Moir et al. [28], there are even measures taken (gas injection) to
lower the electron temperature to 3000 eV to avoid microinstabilities caused
by a “hole” in velocity space that would increase the ion transport to
unacceptable levels. The reason is that the sloshing ion peaks are predicted
to be reduced by a higher electron temperature, which would deteriorate the
warm plasma stabilization of microinstabilities. The mechanisms responsible
for recent increases in electron temperature vary between experiments, but
rotational shear, multimirror confinement and enhanced power for plasma
heating have been shown to be favorable. Further enhancement in Te may
result from increased power for heating and control of neutral gas and
background plasma density.
3.3.2.
Mirror geometries
Different mirror geometries have been explored by different groups over
time. The base geometry is the axisymmetric single cell mirror. The
magnetic field lines for such a field are shown in figure 3.2.
Figure 3.2. Magnetic field lines in the mirror cell for a single cell axisymmetric
magnetic mirror.
This field is however grossly unstable to flute modes, and has a poor plasma
confinement. There are a couple of ways to stabilize such a field. One is to
stabilize the field by adding a quadrupolar (or other multipolar) field
component. This was shown in an experiment by Ioffe in the 1960ies [51].
42
The successful XIIB experiment in the late 1970ies used quadrupolar
values approaching unity [52]. A quadrupolar
stabilization to demonstrate
field flattens the flux surfaces at the ends as shown in figure 3.3 where field
lines for the SFLM field are shown.
Figure 3.3. Magnetic field lines on a flux surface in the Straight Field Line Mirror.
Another method is to have strongly stabilizing regions at some locations,
where a possibly smaller but finite plasma pressure in these regions will add
so much to stability that overall stability to flutes is achieved. In tandem
mirrors, stabilizing anchor cells have been used. In axisymmetric mirrors,
magnetic expanders (or cusps) outside the confinement region provide
stability, in combination with other effects [50]. Stabilization of
axisymmetric mirrors at high
has been demonstrated with the Gas
Dynamic Trap by the Budker Institute in Novosibirsk, Russia [48].
Since the single cell mirror concepts are not likely to have an end
confinement that is good enough for pure fusion, at least not as compact
devices, some improvements have been made. The main path is the tandem
mirror, described in for example Ref. [47]. The basic idea with the tandem
mirror is to achieve electrostatic confinement of both ions and electrons of
the bulk plasma. To accomplish this, the large main mirror cell is surrounded
by two smaller plug cells in which a positive hump in the electric potential is
created. A first idea of the tandem mirror “plugging” was proposed by
independently by Dimov et al. [53] in the Soviet Union and by Fowler and
Logan [54] in the USA. The idea was to increase the plasma pressure in the
plug cells with neutral beam injection. This however required high neutral
beam injection particle energies of 0.5-1 MeV and a high magnetic field in
the plug cells, which would be expensive but perhaps possible to obtain with
a fusion plasma [47]. A modified concept with thermal barrier was then
proposed by Baldwin and Logan [55]. The electric potential in this concept
has a negative hump in potential between the plug cell and the central cell,
separating the electrons in the central cell from those in the plug cells. The
size of the positive potential hump depends on the electron temperature and
by introducing this negative “thermal” potential barrier, the electrons in the
43
plug cell can be heated separately if the system is stable. This would make
the cost in energy for creating the positive potential hump much lower, since
one may choose to only heat the electrons in the smaller plug cells [47].
There are also other concepts. The multimirror is one idea which is tested
experimentally in the GOL-3 experiment at the Budker Institute (see for
example Refs. [49][56]). The basic idea is to have many aligned mirrors and
a strongly collisional and high density plasma having a mean-free path for
ion 90° scattering of the same order as the mirror cell length. In this way, a
leaking ion only leaks one or a few mirror cells before being trapped again.
An electron temperature of around 2.5 keV has been reached with a heating
based on an injected relativistic electron beam [49]. At the Budker Institute,
there is now a new design project, aimed to combine favorable results from
GDT and GOL-3. The new device will be axisymmetric where the long
central cell is followed by shorter mirror cells on each side. Electron beams
in combination with Neutral Beam Injection (NBI) and Ion Cyclotron
Resonance Heating (ICRH) will be used for plasma heating.
3.3.3.
The flute instability in magnetic mirrors
The flute instability in magnetic mirrors radically deteriorates the
confinement, and was observed already in the first mirror experiments. The
effect is that the plasma edge rapidly will change its shape, moving parts of
or the entire plasma radially outward. The instability can look very different,
depending on the mode number m that is dominant, but the instability has a
long longitudinal wavelength. The perturbation can be viewed as an
alternating radial disturbance of the plasma edge depending on the angle
where m is the number of oscillations around the plasma edge, roughly
looking like
r
rb
ri cos(m
0
0
)
(3.3.6)
where rb is the original plasma radius and ri is the instantaneous amplitude of
the growing perturbation. In figure 3.4, two plasma edges are shown with
growing flute instabilities having m = 6 and m = 1. The flute instability
corresponds to a perturbation (or test function) that is not varying in z [57].
The flute-like instabilities that are dependent on z, which may be triggered
by a finite ß, are called MHD ballooning modes. The flute instability is a
Rayleigh-Taylor instability that is driven by the magnetic field curvature. A
physical picture of the flute instability can be made from particle drifts. The
field line curvature in the radial direction will cause a charge dependent
curvature drift which will separate the charges, making positive charges drift
in one azimuthal direction and negative charges in the opposite direction. If
there is a small perturbation, which will always be present, the charges will
be separated on the peaks of the perturbation, as shown in figure 3.5. This
44
charge separation will give rise to an electric field, which will in turn cause a
radial E B drift in that peak. If the magnetic field lines bend outward
(concave) from the center when moving from the midplane, the E B drift
will flat out the perturbation. This is stabilizing and is referred to as “good
curvature”. If the field lines instead bend inwards (convex) like in an
axisymmetric mirror, the peaks will drift outwards. This causes the
perturbation to grow and is referred to as “bad curvature” [43][47].
Figure 3.4. The plasma surface in an axisymmetric mirror cross section showing
flute instability for m = 6 and m = 1. The dashed line shows the plasma deformation
after a short time.
Figure 3.5. Perturbed plasma edge showing the charge separation from magnetic
field radial curvature. The view is perpendicular to the magnetic field.
However, as mentioned earlier, there are ways to design a magnetic mirror to
avoid flute instabilities. The first theoretical prediction was made by
Rosenbluth and Longmire [58] and is called the average minimum B
criterion. It is
zend
r0
zend
dl
B
zend
zend
B
dz
0
(3.3.7)
is the
where the integration is to be taken along a magnetic field line and
normal (radial-like) component of the field line curvature. It states that on
average along a magnetic field line the magnetic field strength should grow
stronger outwards in the r0 direction to grant stability. This is equivalent to
having more good curvature than bad on average in the low
limit.
However, this criterion does not take into account the varying pressure
profiles along the field lines. A more genuine flute stability criterion
including the effect of pressure weighting is derived by Kaiser and Pearlstein
[57]. It is
45
zend
pˆ
zend
dz
B
0,
pˆ ( pˆ
pˆ ) / 2
(3.3.8)
where p̂ is a radially independent pressure function.
3.4. Other devices for magnetic confinement
3.4.1.
Tokamaks
Most fusion research today is directed towards the tokamak concept. The
tokamak was invented in the Soviet Union during the early days of fusion
research by A. D. Sacharov and I. E. Tamm. The concept is a toroidal
(“doughnut-shaped”) device. If a simple solenoid would be used to form a
toroid to confine plasma, no plasma confinement would be achieved. The
reason for this is that the field curvature would cause a charge-separating
drift, where the positive ions drift up/down and the electrons drift in the
opposite direction. This causes a vertical electric field that will give a charge
independent radial drift outwards of the entire plasma which will hit the
outer wall. Therefore, equilibrium is never established. This problem is
solved in the tokamak by inducing a toroidal plasma current. This current
gives a helical-like toroidal magnetic field which does not suffer from an
average charge separation by the drifts. Examples of field lines can be seen
in figure 3.6, with a normal plasma current in (a) and a heavily exaggerated
plasma current in (b) to give a good view of the helical field lines.
a
b
Figure 3.6. Tokamak orbits with a normal plasma current in (a) and a heavily
exaggerated plasma current to show the helix shape in (b). Note that the magnetic
field lines, and thereby the orbits, are generally not closed with the exception of
rational surfaces.
The ITER project is a huge international tokamak fusion project with the aim
“to demonstrate that fusion is an energy source of the future” (see Ref. [59]).
The strongest argument for tokamaks is the reasonably good plasma
46
confinement compared to other devices [60]. Tokamaks have however also,
as all fusion devices, some drawbacks. The largest drawback is probably that
there currently is no way to drive the plasma current in steady-state with
fusion parameters, although some progress with steady-state tokamaks
(advanced tokamaks) has been made where the bootstrap current is an
important parameter [60]. Toroidal current drive is today arranged by
transformer action with an increased magnetic flux generated by a large
central solenoid. This magnetic flux has to be increased to induce a toroidal
electric field in the plasma, and when the maximum magnetic field strength
in the central solenoid is reached, the reactor must be restarted. This is a
major drawback, both for cost efficiency, the electric grid and material
durability reasons, and for ITER this limits the full power pulse length to
500-800 s [60]. This is also a drawback for tokamak hybrids. There are also
severe material problems, where the divertor plates in ITER is predicted to
have a lifetime corresponding to about 7000 pulses [61]. The high heat and
neutron material load in the divertor is a severe problem for reactor scenarios
in all compact fusion devices. The problem is however even larger for
tokamaks unless the ELM (Edge Localized Mode) instabilities, which are
present in H-mode, associated with large intermittent plasma heat loads
deposited on the divertor wall, can be suppressed. The material problems (to
withstand both a massive heat load from the plasma and the heavy neutron
bombardment from the fusion neutrons) are a cumbersome argument against
many pure fusion devices that needs a solution, but not an argument against
fusion-fission where the heat and neutron loads can be made substantially
lower.
3.4.2.
Stellarators
The stellarator concept is also a toroidal device. In a stellarator, there is no
plasma current. Instead, the rotational transform is created with external
windings. This eliminates the strong drawback tokamaks have with the
plasma current, and allows for steady-state operation. However, as all fusion
devices, the stellarator has some problems. Currently, the limit is low, 3%
or so. Also, there could be some problems with stochastic field lines, and to
assure that the drifting gyro centers will remain within a flux tube. However,
the stellarator may prove to be the best alternative for pure fusion in the
future. The Wendelstein 7x is a stellarator project in Germany that is planned
to start operation in 2014 (see for example Ref. [62]). This device has
complex 3D superconducting coils.
3.4.3.
Other schemes for magnetic confinements
During the more than 50 years of fusion research many devices have been
proposed, each with its own benefits and problems. To mention some of
47
them, there are reversed field pinches (RFP), z-pinches, -pinches, FRCs,
multipoles, ion rings, bumpy tori mirrors etc. Those will not be described in
this text.
48
4. Theory for the magnetic mirror vacuum
field
The magnetic vacuum field is the magnetic field when no plasma is present,
i.e. when = 0. When > 0, the magnetic field will be altered due to the
diamagnetic effect of the plasma. When plasma calculations are made in the
vacuum field, they are in general only valid for low plasmas. In this
chapter, some expressions used in several papers in this thesis are derived,
based on the magnetic vacuum field.
4.1. The long-thin approximation
In order to be able to achieve analytic expressions for calculations in a
mirror device, the common approach is to use the long-thin (or paraxial)
approximation. In the long-thin approximation, it is assumed that the
magnetic field lines make a small angle with the system axis near this axis
and are parallel to the axis at the axis [63]. This is true for points where the
distance to the axis is small compared to the characteristic scale of the field
variation along the axis [63]. For the vacuum field, there are no currents
within the confinement region. Thus, the magnetic field can be expressed as
B
(4.1.1)
m
where m is a scalar magnetic potential, and from
Laplaces equation,
2
m
B
0,
0
B ( z )dz
m
0
x2
y2
4
B
obeys
(4.1.2)
With the paraxial approximation and quadrupolar symmetry,
expressed as
z
m
x2
y2
4
g ( z ) o(
4
)
m
can be
(4.1.3)
where
a / c is the small parameter, i.e. in this case x or y divided by c,
for the expansion and prime denotes differentiation in z. Due to the
symmetry of a quadrupolar mirror, terms containing odd power of x and y
are omitted. Eq. (4.1.3) obeys to leading order Laplaces equation and has
two free funcions, B ( z ) and g ( z ) . B ( z ) is the magnetic field strength on
49
axis, and g ( z ) gives rise to the so-called quadrupolar field component. The
quadrupolar field component can be oriented to be directed towards the z
axis at the y axis and away from the z axis at the x axis. There are also other
components like octopolar components etc. but those are of higher order in .
The magnetic field components can be derived from Eq. (4.1.1). They are
Bx
Bz
x
g B , By
2
x2 d ( g B )
B
dz
4
y
g B ,
2
y2 d ( g B )
dz
4
(4.1.4 a, b, c)
neglecting third order terms in . As can be seen, both B and g contributes
to the Bx and By components.
4.2. Flux coordinates
In fusion, Clebsch coordinates are often used to hide the topology of the
magnetic field in the coordinate system. The curvilinear coordinate system
can be defined by
B
B0 x0
y0
(4.2.1)
where x0 and y0 are the two Clebsch coordinates and the constant B0 is the
magnetic field at x y z 0. The Clebsch coordinate x0 is an x-like
coordinate and y0 is an y-like coordinate, equal to x and y at the midplane
z 0 . Since B x0 0 and B y0 0, x0 and y0 are constant along B. A
natural choice for the third coordinate in the vacuum field is m . It is
however often more comfortable to replace m with the arc length s that is
equal to z at the z axis. The x0 and y0 coordinates can also be expressed in
cylinder-like Clebsch coordinates by introducing
x0
r0 cos 0 ,
y0
r0 sin
Often in literature, r0 is replaced by
divided by 2 . Then, B
often called flux coordinates.
0
, B
B0 r0 r0
0
(4.2.2)
B0 r02 / 2 which is the enclosed flux
0 , and these Clebsch coordinates are
The Clebsch coordinates and for example s form a curvilinear coordinate
system, where vectors can be represented in contravariant and covariant
form. The contravariant (“normal”) form for an arbitrary vector F is
F F i R / u i F i ei , where R is the position vector, the u i are the
curvilinear coordinates and Einstein’s summation convention is implied. The
covariant form is F Fi u i Fi ei , where the base vectors u i are dual
vectors to R / u i . The scalar product is A B Ai Bi Ai B i and the base
vectors can be converted between covariant and contravariant form using
50
e j ek
ei
ei e j e k
e j ek
ei e j e k
, ei
(4.2.3)
where the indexes can be cyclically shifted (see Ref [44]). From these, the
cross product can be easily found by solving for e j e k or e j e k . Another
useful identity is that ei e j e k 1/ ei e j ek is the determinant of the
Jacobian ( x, y, z ) / (u1 , u 2 , u 3 ) .
The point with the Clebsch coordinates is that x0 and y0 (or alternatively r0
and 0) are constant along a magnetic field line, and thus identify or label a
field line. The third coordinate s marks the position along the field line. To
find an expression for the Clebsch coordinates, magnetic field lines
x B [ x( z ), y ( z ), z ] can be traced by the field line equations
Bx
,
Bz
dx( z )
dz
dy ( z )
dz
By
(4.2.4)
Bz
where x and y are the coordinates for the selected field line at position z. The
field components in the vacuum field using the paraxial approximation are
inserted from Eq. (4.1.4) giving two first order differential equations
dx( z )
dz
g
B
2B
x( z ),
g
dy ( z )
dz
B
2B
y( z)
(4.2.5)
to first order in . The solutions are
z
x( z )
x0 e
0
g B
dz
2B
z
g
x0
B0 0 2 B dz
e
B
y0
B0
e
B
x0 ax ( z )
(4.2.6)
and
z
y( z )
y0 e
0
g B
dz
2B
z
0
g
dz
2B
y0 a y ( z )
(4.2.7)
where the integration constants are set to x0 and y0 which can be identified as
the Clebsch coordinates, thereby stating that x = x0 and y = y0 at z = 0. To get
a more intuitive understanding of the field line equations in Eq. (4.2.6) and
Eq. (4.2.7), the g / B integrand is the contribution from the quadrupolar field
component. The second term is just the axisymmetric resizing of a flux tube
due to different field strengths. To simplify the notation in the coming
expressions, we introduce
51
g
h1 ( z )
h2 ( z )
B ' 1 Bx
2B
B x
g B ' 1 By
2B
B y
(
2
),
(
2
(4.2.8 a, b)
)
Instead of using s, we introduce the arclength-like variable s (
inverse m ( s ) by
m
) and its
s
m (s )
(4.2.9)
B ( s )ds
0
and since the magnetic scalar potential along the z axis is given by
z
m ,0
( z)
(4.2.10)
B( z )dz
0
an inverse function s (x) m1,0 [ m (x)] can be used. By treating the off-axis
terms in Eq. (4.1.3) as a small parameter , a Taylor expansion gives
1
m ,0
s ( x, y , z )
z
1
d
[
1
m ,0
m ( x, y , z )]
z
m ,0
1
B
1
B
z
d
m ,0 ( z )]
[
x
2
y
4
1
m ,0
dz
x2
2
B
y2
4
g ( z)
(4.2.11)
dz
or more compactly
s ( x, y , z )
z
x2
h1 ( z )
2
y2
h2 ( z ) o(
2
4
)
(4.2.12)
The magnetic field B can now be expressed using s as
B
B( s ) s
m
1
x2
(h1
2
h12 )
B( s ) 1
x2 2
(h1
2
dh1
)
dz
y2
(h2
2
h22 ) zˆ
o(
3
)
dh2
)
dz
o(
4
)
In Clebsch coordinates ( s , r0 , 0 ) , this becomes apart from o(
4
B ( s ) xh1xˆ
yh2 yˆ
(4.2.13)
and the modulus is
B
B ( s , r0 , 0 )
52
B(s )
r02
[u1 ( s ) cos 2
2
0
y2 2
( h2
2
u2 ( s )sin 2
0
] o(
4
)
(4.2.14)
) corrections
(4.2.15)
where
u1,2 ( s )
B ( s )e
2
s
0
h1,2 ( s ') ds '
dh1,2 ( s )
2
[h1,2
(s )
ds
]
(4.2.16)
The expressions derived in this chapter can be used to examine some
properties of the magnetic field.
4.3. Flux tube ellipticity
A vacuum field flux surface for a mirror geometry is normally defined as an
extension of a circle centered on the z axis at z = 0 extended along the field
lines in both directions and thereby forming a tube. It is defined by
r0 const , i.e. [ x / ax ( z )]2 [ y / a y ( z )]2 const which for each z defines an
elliptical cross section of the flux tube. The flux enclosed by a flux surface is
constant. The flux tube ellipticity is a measure of how elliptic a flux surface
is and is a function of z. The ellipticity of a flux surface is to first order in
z
ell
a ( z) ay ( z)
max x
,
a y ( z ) ax ( z )
( z)
max e
0
g(z)
dz
B( z)
z
,e
0
g(z)
dz
B( z)
(4.3.1)
In Paper III, an expression for the ellipticity to 3rd order in
has been
derived. To derive such an expression, higher order terms of the solution to
Laplace’s equation (Eq. (4.1.2)) must be included. The solution is in
cylindrical coordinates
Al , n I l
m
l
n
n r i(l
e
c
n z / c)
(4.3.2)
where the I l are the modified Bessel functions. The odd terms in l (i.e.
l
1, 3,... are omitted due to the choice of symmetry. To order ( 5 )
z
B ( z )dz
m
0
4
r B ( z)
16
4
r2
B ( z ) g ( z ) cos 2
4
g ( z)
cos 2
3
(4.3.3)
h( z )
cos 4
4
where h( z ) represents the octupolar field. This gives the B components
r
r3
B ( z ) g ( z )cos 2
1 ( z , ),
2
4
r
r 3 g ( z)
h( z )
g ( z )sin 2
sin 2
sin 4
2
8
3
2
Br
B
(4.3.4)
,
(4.3.5)
53
Bz
B( z )
r2
B ( z ) g ( z )cos 2
4
r4
16
2
( z, )
(4.3.6)
where
1 d 3 B( z ) 1 d 2 g ( z )
cos 2
4 dz 3
3 dz 2
1 d 4 B( z ) 1 d 3 g ( z )
(
z
,
)
cos 2
2
4 dz 4
3 dz 3
1
h( z )
cos 4 ,
4
h( z )
cos 4 .
4
( z, )
(4.3.7)
(4.3.8)
The position vector along a field line can be parameterized using z, yielding
x B rB ( z )rˆ B ( z ) ˆ zzˆ . The field lines can now be traced with the field
line equation
drB ( z )
dz
Br
Bz
(4.3.9)
where only the radial part is needed for determining ellipticity. Inserting the
field components yields
drB ( z )
dz
rB ( z )
rB ( z )3
B ( z ) g ( z ) cos 2
1 ( z, )
2
4
r ( z )2
rB ( z ) 4
B( z ) B
B ( z ) g ( z ) cos 2
2 ( z, )
4
16
to fourth order in
(4.3.10)
which becomes
drB ( z )
dz
A3 ( z , )rB3 ( z )
A1 ( z , )rB ( z )
(
5
)
(4.3.11)
where
1
B
2B
A1
A3
1 B
4B 4
g
cos 2
3
h
cos 4
4
g cos 2
B
,
g cos 2
(4.3.12)
A1 ( z, ) .
(4.3.13)
By assuming that the term A3 ( z , )rB3 ( z ) is small, the equation can be
linearized. The field line radius can be written
rB
rB ,1
rB ,3
(
5
)
(4.3.14)
where rB ,1 is the first order solution and rB ,3 is the small contribution from
the third order terms. The differential equation then becomes
d
rB ,1
dz
rB ,3
A1 ( z , ) rB ,1
rB ,3
A3 ( z , ) rB3,1 3rB ,3 rB2,1
The first order solution is (compare Eqs. (4.2.6 – 4.2.7))
54
o(rB2,3 ) (4.3.15)
z
rB ,1
r0 e
0
B ( z ) g ( z ) cos 2
dz
2B( z )
(4.3.16)
and inserting this yields
drB ,3
f 2 rB ,3
dz
f1 ,
(4.3.17)
where
z
f1
A3 ( z , )r03e
z
3 A1 ( z , ) dz
0
,
3 A3 ( z , )r02 e
f2
2 A1 ( z , ) dz
A1 ( z , ). (4.3.18)
0
The solution to the differential equation is
z
rB ,3
e
z
z
f 2 ( z , ) dz
e0
0
f 2 ( z , ) dz
f1 ( z )dz
C
(4.3.19)
0
where the constant C is zero since rB ,3 ( z
z
rB ( z, ) rB ,1
rB ,3
r0 e 0
0) 0 . The total result yields
z
A1 ( z , ) dz
e
z
f 2 ( z , ) dz z
0
e0
f 2 ( z , ) dz
f1 ( z )dz
(4.3.20)
0
to fourth order in . The third order eccentricity or ellipticity (the cross
section may not be exactly elliptic to this order) becomes
ell ,3
max
ell
1
,
rB ,1 ( z )
,
ell
rB ,1 ( z )
ell
0
rB ,3 ( z )
/2
rB ,3 ( z )
0
. (4.3.21)
/2
4.4. Flute stability
The average minimum B criterion from Eq. (3.3.7) becomes in flux
coordinates using Eq. (4.2.14)
zend
r0
zend
zend
dl
B
zend
B / r0 dl
B2
0
zend
r0 (u1 ( z )cos
0
u2 ( z )sin
0 ) dz
B2
zend
0
(4.4.1)
0
W1RL ,2 RL
0
where
zend
W1RL ,2 RL
zend
dz 2
e
B( z )
z
0
h1,2 ( z ') dz '
2
[h1,2
( z)
dh1,2 ( z )
dz
]
0
(4.4.2)
55
It can be shown that the W1RL and W2RL expressions will give the same result
by symmetry. The flute stability criterion, Eq. (3.3.8), that takes the varying
pressure along a field line into account becomes
zend
pˆ ( z )
W1,2
zend
dz 2
e
B( z )
pˆ ( z ) ( pˆ ( z )
z
0
h1,2 ( z ') dz '
dh1,2 ( z )
2
[h1,2
( z)
dz
]
0
(4.4.3)
pˆ ( z )) / 2
This can be verified against Eq. (3.3.8) using the expressions given in Ref.
[57].
4.5. Drifts and neoclassical transport in mirror machines
4.5.1.
Particle drifts
In mirror machines, except for the exceptional case of the ideal SFLM field,
the magnetic field lines are curved which gives rise to curvature drifts. There
will also be gradients in the magnetic field which give rise to drifts. In the
vacuum field, these gradients can be expressed in terms of curvature since
0
B
BBˆ
ˆ B
0 j B
Bˆ
j
Bˆ Bˆ BBˆ
Bˆ
Bˆ
B Bˆ
B Bˆ
where 0 j BBˆ
Bˆ and
vacuum case when j 0 gives
Bˆ
B
Bˆ
ˆ
B
Bˆ
( B) .
(4.5.1)
which in the
(4.5.2)
The magnetic drifts in the vacuum field can then be divided into radial drifts
dr0 / dt that depend only on geodesic (angular-like) curvature and angular
drifts d 0 / dt that depend only on normal (radial) curvature. Since the
curvature components are important in this thesis, they are derived
explicitly. We now use the radial-like flux coordinate
r02 / 2 and derive
the curvature components as
Bˆ
Bˆ
s
Bˆ
0
0
(4.5.3)
If a field line is parameterized as R f ( x( z ), y ( z ), z ) where R f is the
position vector for a field line, Bˆ
R f / s which gives
ˆ
B
s
x
2
ds
x
2
Rf
y
2
Rf
dz 2
y
x
x
0
56
y
(4.5.4)
y
0
0
The curvature components on the r.h.s. can now be evaluated in the paraxial
approximation from
z
h1dz
x
x0 e
y
y0 e 0
z
h12
h1
h2 dz
z
z
x0 e 0
0
h2
z
2 sin 0 e 0
(4.5.6)
z
h1dz
1
cos 0 e 0 ,
2
(4.5.7)
z
h2 dz
1
sin 0 e 0 ,
2
h2 dz
z
(4.5.5)
h22 ,
h1dz
2 cos 0 e 0
0
z
h1dz
2 sin 0 e 0
h1dz
(4.5.8)
, (4.5.9)
0
z
y
y0 e 0
0
h2 dz
z
h1dz
h12 ,
h1
2 cos 0 e 0
h2 dz
y0 e 0
x
2 sin 0 e 0
h1dz
x0 e 0
y
h1dz
z
h22
h2
z
x
2 cos 0 e 0
z
h2 dz
2 sin 0 e 0
0
z
h2 dz
h2 dz
2 cos 0 e 0
(4.5.10)
0
which gives
cos 2
0e
2
z
0
h1dz
h12
h1
sin 2
0e
2
z
0
h2 dz
h22
h2
(4.5.11)
and
2 sin
0
0 cos
e
0
2
z
0
h1dz
h12
h1
e
2
z
0
h2 dz
h2
h22
. (4.5.12)
The magnetic drift can now with Eqs. (4.5.2, 4.5.3) be calculated as
2
1 B mv ˆ
(B
q
B2
v d ,m
2
1 B mv
q
B
2
1 B mv ˆ
(B
q
B
Bˆ )
Bˆ
0
Bˆ
0
)
(4.5.13)
.
Since Bˆ ( m ) / B and since the cross products can be evaluated from the
definition of contravariant base vectors [44],
R
R
m
0
0
0
R
m
m
m
0
(4.5.14)
2
B / B0
57
R
R
m
m
0
0
R
0
m
0
m
(4.5.15)
2
B / B0
0
where R is the position vector, the magnetic drift velocity becomes
2
1 B mv
q
B
B
12
2
q B
Bˆ
v d ,m
qB0
where
R/
written as
m
ˆ
B
m
0
R
B
2
0
0
0
(4.5.16)
R
0
0
2
mv / 2 . Since
(r0 ) r02 / 2,
R / r0 ( R /
)(d / dr0 )
( R / r0 ) / r0 the magnetic drift velocities for a particle can be
dr0
dt
d 0
dt
2
B
qB0
2
B
qB0
0
r0
,
(4.5.17)
(4.5.18)
where B B to first order in . It is from these equations clear that in the
vacuum field the radial magnetic drift is caused by a geodesic curvature, and
that the normal curvature gives rise to an azimuthal drift.
4.5.2.
Neoclassical transport
Neoclassical transport is in mirror machines caused by geodesic field line
curvature, which gives radial drifts away from the flux surface. If these
radial drifts are oscillatory and cancel during one bounce back and forth
across the mirror, the magnetic field is said to be omnigenous (in the average
sense), a concept introduced by Hall and McNamara [64]. For a quadrupolar
mirror, this requires that the device is symmetric in the sense that B ( z ) and
g ( z ) are even functions. Thereby, the radial drifts on opposite sides of the
mirror will be equally large but with opposite signs, and will almost exactly
cancel out. In such mirrors, a radial invariant I r r0 exists, which has been
derived in Paper VII. If this symmetry is not present, there will be a
collisionless radial transport that may be unacceptably large as pointed out in
Paper IV. The particle orbits in symmetric mirrors have a similarity with the
banana orbits in tokamaks and are here called banana orbits as well. These
58
gyro center orbits may have larger radial excursions than the Larmor radius,
and will lead to neoclassical transport.
Catto and Hazeltine [65] introduced the concept of local omnigenity as
referring to magnetic fields where the radial drift off a flux surface is zero
everywhere. The banana orbit widths in such fields are zero, and thereby
neoclassical transport of any kind is abscent. Axisymmetic fields are locally
omnigenious in the vacuum field since 0 vanishes everywhere, and even at
if fluctuations and non-ideal effects are ignored. The SFLM
arbitrary
vacuum field is an example of a locally omnigenious quadrupolar field, and
for well behaved even functions of B ( z ) there exists a non-trivial (i.e. the
axisymmetric solution g ( z ) 0 z is excluded) g ( z ) which gives a locally
omnigenious vacuum field in the paraxial approximation (see Paper VII).
0 everywhere. For locally
This can be accomplished by solving
0
omnigenious fields, the radial transport should be small unless there are
anomalous transport processes present or there for some non-ideal reasons
exists azimuthal electric fields.
In mirrors with finite ß, or that have a strong radial electric field, the
azimuthal drift may become an important factor for radial transport by
giving rise to the so-called resonant neoclassical transport. The resonant
neoclassical transport is a collisionless transport that occurs for particles
which have an azimuthal drift that allows them to come in “resonance” in the
sense that the number of bounces back and forth the mirror it takes before
the particle has travelled one lap around the z axis is a simple rational
number. For some particles, this drift can be very fast. In quadrupolar
mirrors that are not locally omnigenous, the two sides of the mirror causes
radial drifts that are in opposite directions. However, if one looks in the xyplane, if the drift is outward for positive z in the first and third quadrant, it is
opposite in the second and forth quadrant. This is illustrated in figure 4.1,
where the regions with outward ion drift are white and the regions with
inward ion drift are gray. The worst possible particle trajectory is the one
illustrated in figure 4.1. Here, the trajectory is in perfect phase and travels
half way around the z axis on one longitudinal bounce back and forth the
mirror. As can be seen in figure 4.1, the particle drifts outwards constantly
and will soon be lost, if d 0 / dt does not change rapidly with radius.
Particles with a ratio bounces/lap close to nbounce 2 4n will suffer from
neoclassical radial transport if the phase does not cancel the effect, where the
effect is larger for small n and largest for n 0 . Also faster spinning
particles like those with nbounce 2 /(1 2n) will experience resonant radial
transport. Resonant transport is predicted to be the most serious type of
neoclassical transport for at least some mirror fusion devices ([66], section
9.3).
59
Figure 4.1. This figure illustrates neoclassical resonant transport in a projection
where the rolled out circumferential surface with r0 = const is shown for a
quadrupolar mirror. The white regions have a negative geodesic field line curvature
and a corresponding positive radial drift, and the opposite yields for the gray
regions. As can be seen, the worst particle resonance has been selected for
demonstration where the drift is everywhere outward.
For particles that spin very fast around the z axis, a stochastic behavior is
expected [67] and this is referred to as stochastic neoclassical transport. To
summarize, there are three regimes of neoclassical transport that depend on
the angular drift per longitudinal bounce
0 of the ions:
1.
2.
3.
0
0
0
1 : Neoclassical transport.
1: Resonant neoclassical transport.
1 : Stochastic neoclassical transport.
4.6. The Straight Field Line Mirror field
The Straight Field Line Mirror (SFLM) field is a minimum B magnetic
mirror field derived by Ågren and Savenko [68]. It is defined by
B
B0
1 ( z / c) 2
(4.6.1)
2/c
[1 ( z / c) 2 ]2
(4.6.2)
and
g
where c is the axial scale length of the system. Alternatively, the SFLM field
can be written
B
60
B( s ) s
B0 x0
y0
(4.6.3)
where B ( s ) B0 /(1 s 2 / c 2 ) by Eq. (4.6.1) and the Clebsch coordinates are
to leading orders
x
o( 3 ),
1 z/c
y
y0
o( 3 ),
1 z /c
x 2 / 2c y 2 / 2c
s z
1 z/c 1 z/c
x0
(4.6.4 - 4.6.6)
o(
4
).
The field lines in the SFLM field are straight non-parallel lines, stretching
between two focal lines located at c with an angle of 90° between them
(see figure 3.3). This implies that in this vacuum field the gyro center drift is
abscent when there is no electric field, since no B -drift is present for this
particular vacuum field. For the vacuum field SFLM, B is parallel to B,
B
B(s) 0
(4.6.7)
Therefore, the both guiding center Clebsch coordinates are constants of
motion [69], and the field is locally omnigenius. Each guiding center
bounces back and forth on a single flux line. The arc length turning points in
the vacuum field are given by [69]
sturn
B0
c 1
.
(4.6.8)
Inserting this magnetic field into the stability criterion in Eq. (4.4.3) results
in W1,2 0 , since the second factor
2
h1,2
( z)
dh1,2 ( z )
dz
(4.6.9)
becomes zero for all z (or alternatively since
0 ). The SFLM field is
therefore marginally flute stable for all z in the low limit. Recalling that the
field lines are straight, this can intuitively be seen since no “bad curvature”,
being convex, or “good curvature”, being concave, exists; the field lines do
not have a curvature at all. This also implies that the flute stability will be
independent of the pressure profile in the low limit. However, the SFLM
field modulus is monotonically increasing with increasing z , having a
monotonically increasing strong gradient dB / d z . The mirror must be
ended somehow, and this cannot be done with a continuous B derivative in
z without concatenating the SFLM field with another field at some point.
These added parts of the field which will end the mirror also have to be
stable to flutes.
61
The flux tube ellipticity in SFLM is also low, even if it is possible to find
solutions that are stable to the flute criterion or average minimum B criterion
that have lower ellipticities, see for example Ref. [70]. For a mirror ratio of
four, the maximum flux tube ellipticity of SFLM is 13.9 [68]. To leading
order, the equation determining the SFLM flux surfaces is
K
x
1 z/c
2
2
y
1 z /c
2
(4.6.10)
where K is a constant. For the plasma surface, K = a, i.e. the midplane
radius. The SFLM flux tube ellipticity is
ell
( z)
1
z /c
1
z /c
Rm
Rm 1
2
(4.6.11)
For Rm >> 1, this becomes
ell
62
( z ) 4 Rm
2 o(
1
)
Rm
(4.6.12)
5. Finite ß effects on the mirror magnetic field
When there is a finite
present, the magnetic field will be significantly
modified by the associated plasma currents. In mirror machines, the
limit is typically high, and 0 1 has been reached in the
midplane
(short-fat) single cell quadrupolar mirror 2XIIB at LLNL [52]. In this
chapter, the equations required to calculate the magnetic field from the
plasma in equilibrium is derived. Also, a numerical method to calculate this
field, used in Paper V, is presented as well as a method described in Paper
VII.
5.1. The equilibrium
To find the magnetic field contribution from the plasma, the pressure
distribution must be determined. From the pressure distribution, the plasma
currents can be derived. To find the equilibrium, we follow Newcomb [71].
The plasma pressure tensor in flux coordinates is assumed to be of the form
P
p
ˆˆ
I BB
ˆˆ
p BB
(5.1.1)
where off-diagonal terms are neglected. By including the magnetic pressure,
the total stress tensor becomes
T
ˆˆ
P I QBB
(5.1.2)
P
B2
2 0
(5.1.3)
where
p
and
Q
B2
p
p.
(5.1.4)
0
The force balance can be found from
T 0 . The divergence of the first
ˆ ˆ . With use of the
term is simply P, and the second term is a dyadic QBB
vector formula
( AB) B( A ) ( A )B , the force balance then
becomes
63
ˆˆ
P I QBB
T
0
P Bˆ
QBˆ
ˆ
QB
Bˆ
(5.1.5)
is the magnetic field line curvature. This equation can now
where Bˆ Bˆ
be divided into perpendicular and parallel force balance, yielding
P Bˆ
( P)
Q
P
QBˆ
ˆ Q
PB
(5.1.6)
and
Bˆ
Q
B
B
ˆ
BB
Q
.
B
(5.1.7)
From the parallel force balance equation, Eq. (5.1.7), the relation between
the parallel and perpendicular pressure components on a single flux line can
be found. It becomes (see Paper V, Appendix A or [71])
p
p
B2
,
B/ s s B
s
p
B
L
p B/ s
ds
B2
(5.1.8 a, b)
where s is an arc length coordinate along B and s
L at the mirror throats.
Thereby, if either the parallel pressure or the perpendicular pressure is
specified, the other component can easily be determined.
To determine the pressure distribution, the plasma feeding and heating must
also be taken into account. In the SFLM Hybrid, the intention is to use ICRH
(Ion Cyclotron Radiofrequency Heating) for heating, which would create a
sloshing ion pressure distribution where there is one pressure peak at each
side of the mirror. The plasma feeding has not yet been determined. The
effects of ICRH heating could in principle be calculated with complicated
ICRH-codes. However, in this thesis this is postponed to later work, and
prespecified simple pressure profiles have been used. We choose to follow
Newcomb [71] and Pearlstein et al. [72] and model the perpendicular
pressure as
p
p
,0
p
, r0
(r0 ) p
,B
( B)
(5.1.9)
where p ,0 is the perpendicular pressure at origo, r0 is a radial Clebsch
coordinate equal to x at the x axis and the functions p ,r0 (r0 ) and p , B ( B )
are normalized so that they are equal to unity at r0 0 and B Bmin . Here,
Bmin is the magnetic field modulus at origo. We have from Eq. (3.2.32) that
p
where
origo.
64
0
is the plasma
0
B02
(5.1.10)
2 0
at origo and B0 is the vacuum magnetic field at
,0
5.2. The plasma currents
The plasma currents are formed by gradients in the plasma pressure. Since
the aim of fusion devices with magnetic confinement is to confine plasma,
there must be pressure gradients since the plasma pressure (and density)
should vanish (or nearly vanish) at the boundary of the confinement region.
The most important contribution to the plasma currents in a magnetic mirror
is the diamagnetic current (however, in tokamaks there is a toroidal current
as well). Each charged particle in the plasma gyrates roughly around a
magnetic field line and forms a small current loop in the movement
projection on the plane locally perpendicular to B. The ions and electrons
gyrate in opposite directions so that the currents from the two species add.
These currents are diamagnetic, i.e. orientated so that the magnetic field is
reduced by the currents. When the pressure is constant in space, all currents
from the gyrations cancel. When there is a pressure gradient, the currents no
longer cancel but form the diamagnetic current. This is illustrated in figure
5.1, where the pressure is strongest at the center. The gyrating currents
clearly add to form the diamagnetic current (thick lines) since there will be
more circular currents on the high-pressure side than on the low-pressure
side.
Figure 5.1. This figure illustrates how the small currents from gyrating particles add
to form the diamagnetic current (thick line) when there is a negative pressure
gradient from the centre and outwards.
In quadrupolar mirrors, there is also a parallel current j Bˆ if the mirror field
is not locally omnigenous everywhere. This current is formed by the
neoclassical banana orbits, which add in a similar way as the gyro orbits
above since there normally is a negative pressure gradient radially outward.
This current is closely linked to the geodesic curvature of the field lines and
also has a sin 2 0 angular dependence if the azimuthal drift is small,
showing that the parallel current in adjacent quadrants have opposite
directions.
The plasma currents can be found from the equilibrium. The details of the
derivations can be found in Paper V, Appendix A or to a large extent in Ref.
65
[71] and are not repeated here. By applying the operator B̂
on Eq.
(5.1.6), the expression for the parallel current j becomes after some algebra
B3
0Q
j
Bˆ
s
p
p
B2
L
ds.
(5.2.1)
The perpendicular current j is most straight forward to derive from the
usual equilibrium relation
j B
and by taking B̂
(5.2.2)
P
on both sides of the equation, this yields
j
1
B
B2
p
p
p
.
(5.2.3)
5.3. Magnetic field from plasma currents
Due to the plasma currents, a finite ß will modify the vacuum magnetic field
and it is no longer possible to represent the magnetic field with a scalar
magnetic potential. The total magnetic field can be expressed as
B B v B pl
(5.3.1)
where B v is the vacuum field and B pl is the magnetic field from the plasma
currents. In Paper V, a numerical approach has been made. In Paper VII, a
different method is described.
5.3.1.
The method from Paper V
The magnetic field from the plasma, B pl
the currents by solving Poisson’s equation
2
A pl
A pl , can be determined from
0
j
(5.3.2)
in the plasma region and the surrounding vacuum. A Dirichlet boundary
condition for the magnetic vector potential A pl at any chosen boundary can
in the Coulomb gauge be calculated from the Coulomb integral
A pl
0
4
V
j
d 3x
x x
(5.3.3)
where the integration is to be performed over the plasma region and the
boundaries where A pl is calculated are well outside the plasma region. The
integral can be evaluated numerically with quite large integration steps, and
Eq. (5.3.2) is then solved numerically with a boundary condition determined
from Eq. (5.3.3).
66
To distribute the plasma pressure, the shape of the constant perpendicular (or
parallel) pressure contours in the z 0 plane must be determined. This can
be done by an iterative procedure, where those contours are traced
numerically. The pressure is distributed according to these profiles, and the
magnetic field is calculated. Then new contours can be calculated in the next
iteration step. The constant perpendicular pressure contours can be found
from From Eq. (5.1.6), where we have that
P
Q
p
B2
.
2 0
Q
(5.3.4)
By taking B̂ on both sides of the equation and evaluating the r.h.s., the
contours can be traced by a Runge-Kutta method. The field lines can also be
traced by a Runge-Kutta method, and then the pressure can be distributed.
5.3.2.
The method from Paper VII
In Paper VII, another method is described to calculate B pl . In the long-thin
approximation, the magnetic field can be expressed as
B
1
2
Bv
The gradient term can be determined from
integral (see also [69])
m , pl
1
8
(5.3.5)
m , pl
dV
B(x )
x x
B
0, yielding the Coulomb
(x )
s'
(5.3.6)
where the integration is to be performed over the plasma region. This is the
solution of
2
m , pl
1
B
s
2
(5.3.7)
with a proper boundary condition. In practice, Eq. (5.3.7) can be used to
solve for m, pl and the boundary condition can be calculated from Eq. (5.3.6)
in a similar way as in the method of Paper V described above.
67
68
6. Superconducting coils in fusion
In a fusion device based on magnetic confinement, the strong magnetic field
confining the plasma must be generated by a coil system. There are two
main options for coils: standard copper coils and superconducting coils. In
small pulsed experimental devices, copper coils are almost always used.
Typically, they however have too high resistive losses for fusion reactors,
and therefore superconducting coils – which have no resistance – are used.
The drawbacks with superconducting coils are the cost and the sensitivity to
nuclear radiation. For ITER, about 30 % of the total cost is associated with
the coil system and the cryostat [73]. In this section, basic characteristics of
superconducting coils and to some extent copper coils are described to give a
background to coil design.
6.1. Properties of superconducting coils
6.1.1.
Superconductivity
The electrical resistivity of metallic conductors (like silver or copper) is
reduced gradually with decreasing temperature. At a temperature
approaching 0 K, they have a low – but nonzero – resistivity (as long as they
do not become superconducting). In a superconducting material the
dependence on temperature is similar, but below a certain critical
temperature the electrical resistivity drops to zero abruptly. There are
basically three parameters that control if the superconductor is in
superconducting state [74]. They are:
1. Temperature. The critical temperature is dependant upon material,
but is below 140 K for known superconducting materials and often
as low as a few K.
2. Current density. There is a maximum allowed current density called
critical current density J c , and if the current density exceeds this
value the superconductor becomes resistive.
3. Magnetic field. The maximum allowed temperature for maintaining
superconductivity is lowered by magnetic fields, and the maximum
allowed magnetic field strength to maintain superconductivity for a
certain temperature and material is called the critical magnetic field.
69
These three limiting factors are not independent on each other. Generally, if
the temperature is lowered, the critical current density and the critical
magnetic field will become larger [74].
Another property of superconductivity is that magnetic fields can be
expelled from superconducting materials. This is known as the Meissner
effect.
6.1.2.
Superconducing magnets
There are many known superconducting materials, but very few of them,
only a few in 10 000, qualify for making superconducting magnets [74].
These are called magnet-grade superconductors if they are available
commercially. The desired properties are high critical current, very high
critical magnetic field and good strain properties.
The two types of superconductors that have been used in fusion coils due to
their high critical magnetic field are Nb3Sn and NbTi. The advantages with
NbTi are that it has good manufacturing properties and is not very sensitive
to strain. NbTi is a ductile material that can be wound on a coil [75]. The
drawback with NbTi is that the critical temperature is about 10 K, limiting
the maximum magnetic field to about 11-11.5 T with ITER parameters even
at 1.8 K operation [76]. NbTi is a commonly used superconductor and is
relatively cheap. Nb3Sn is a brittle material that cannot be wound easily. To
manufacture Nb3Sn coils, one can use Wind and React (WR) or React and
Wind (RW) techniques, where WR seems to be the better choice [75]. In the
WR manufacturing process the niobium and tin are wound separately and
then the Nb3Sn is formed by a heat treatment of typically 650-700°C for
about 180-200h [75][77]. After the heat treatment, the brittle conductors are
carefully taped. If a conductor is broken during this process, it cannot easily
be repaired. This makes the fabrication difficult and expensive, and the small
global production of these coils also add to the cost. In addition, Nb3Sn
strands are sensitive to strain. However, Nb3Sn has a critical temperature of
18 K, and with ITER parameters this corresponds to a maximum allowed
external magnetic field of more than 14.5 T [76]. The consequence of this is
that NbTi is used in low magnetic field regions while Nb3Sn is used in high
magnetic field regions.
Nb3Al has also been suggested as a superconducting material for fusion coil
systems and has been examined for ITER [78]. It is also considered for
DEMO [79]. Already in the 1970ies laboratory experiments showed that
Nb3Al had excellent properties concerning critical magnetic field ( B 20T
seems possible [80]), critical current density and strain. There has however
been problems to find an appropriate manufacturing process [80], but it
seems now that these problems have been overcome [80] and a large Nb3Al
70
coil was fabricated (with jelly-roll technique) and successfully tested in 2005
[79]. The critical magnetic field of this superconductor is even higher than
for Nb3Sn, and it seems likely that the brittle Nb3Sn conductors will be
obsolete in future if the manufacturing process of Nb3Al will be competitive
in price.
Also, HTS (High Temperature Superconductors) are under investigation for
future fusion plants and show promising properties [81]. HTS are already
used for the current lead to the coils, and the use of HTS here reduces the
power consumption for cooling the current lead significantly, by roughly a
factor of 4. This is since the heat conductivity of these materials is lower,
giving less heat transfer from the concatenation point to the resistive
conductor, and since there is no ohmic heating [81]. The energy
consumption of the coil system is however typically a small part of the total
cost, and reliability of operation may be more important [74].
6.1.3.
Cooling and shielding
To maintain superconductivity, the superconductors must be cooled. This is
done by using either liquid nitrogen or liquid helium. The choice of coolant
depends on the critical temperature. For type II superconductors with a
critical temperature above the boiling point of liquid nitrogen at 77 K at
atmospheric pressure, liquid nitrogen can be used. For type I
superconductors and type II superconductors with a critical temperature
below 77 K, liquid helium is used. A cooling system with liquid helium is
much more expensive, and could suffer from problems with solid air plugs.
In fusion, liquid (sometimes in combination with compressed) helium is used
to cool the coils. In the ITER cryogenic system, one system of 80 K
compressed helium precooled by liquid nitrogen is used to thermally shield
the coils and other systems, and the 4 K liquid helium coolant loops cools
the interior of the coils [82]. The 80 K system has a cooling capacity of 1300
kW and the liquid helium 4 K system has a cooling capacity of 65 kW [59].
The internal heating in the coils comes primarily from the neutron (and
gamma) radiation and for pulsed systems like tokamaks from AC heating.
The AC heating comes from hysteresis effects (in the superconducting
material), coupling losses (from currents between filaments) and induced
eddy currents [74]. During one pulse in ITER, 13 MJ is being deposited from
AC losses [82], which roughly would average about 17 kW for a 750 s pulse.
The nuclear heating is about 14 kW [82], making the two effects roughly
equal in magnitude for ITER. There is also external heating from heat
conduction and radiation.
71
In a fusion power plant, it is important that the superconducting coils are
sufficiently shielded from neutrons (and possibly some gamma radiation) to
reduce cooling power and radiation damage to the materials. Detailed
computations of these doses can be made with Monte Carlo simulations, as
has been done in Paper XII by Prof. Klaus Noack. Both the nuclear heating
and the radiation damage to the epoxy insulators can possibly set the
minimum required thickness for the shields [83], and without shielding both
these radiation limits would be violated by orders of magnitude. It is rather
easy to stop thermal neutrons with a few cm of boron shield or similar. The
14 MeV fusion neutrons are however much more difficult to stop, and
roughly 0.44-0.5 m thick shields are required in ITER on the inboard side
[84]. For shielding, a mixture of water (for moderation of neutrons and
coolant) and borated steel is typically used [84]. Note, however, that the
ITER coils do not seem to be dimensioned for steady-state operation [85],
and thicker shielding may be required for a reactor scenario. Also, note the
cost for coil heating: At 4 K, 500 kWh worked by the cryogenic system is
required to remove one kWh of heat [86]. The range 250 – 8000 kWh is
given in [74].
6.1.4.
Power supply and stability
The current feed to superconducting coils is supplied by a DC-source with
high current and low voltage, since only the feeder wires will cause a voltage
drop. It is of great importance to change the current slowly to prevent
mechanical stresses and heating from induced eddy currents. Heating from
eddy currents can cause temperatures over the critical temperature locally
which in turn gives rise to ohmic heating and overall heating with loss of
superconductivity in the entire coil. Loss of superconductivity in a coil is
called a quench. Quenches can arise from a variety of causes, for example
too high magnetic fields. The ability to avoid quenches is referred to as
stability for the superconducting coil and should also take into account
external events such as consequences of plasma instabilities [74].
6.1.5.
Internal structure
The internal structure of a superconducting coil is very complicated. A
superconducting coil consists of a large number of wound strands. Each
strand consists of a large number of superconducting filaments embedded in
a stabilizing material, for example copper. The current in each strand is
typically of the order of a few hundred amperes [75]. There are a few
different techniques that can be used in fusion coils for structuring the
strands, described in Ref. [75]. Cable in conduit is the chosen design for the
ITER coils. A number of strands – typically in the order of 1000 – are put
into a cable and the cable is wound many laps and embedded into structure
72
material. There is a cooling channel in the center of the cable. There is also a
cover of structure material outside the current-carrying region [76]. A
detailed description for the ITER central solenoid is found in Ref. [87].
A limiting factor in superconducting coil design is the stesses that arise from
the often very high magnetic fields. Stress can both hamper the operation
capabilities of the superconductor and damage the superconductor
permanently (especially Nb3Sn coils are sensitive). In general, coil curvature
should be kept low and the usual axisymmetric coils have good properties
for this. This is an advantage for axisymmetric mirrors compared to
quadrupolar mirrors.
6.1.6.
Use of copper coils in reactor scenarios
Copper coils are typically used in plasma experiments to reduce cost. These
experiments are almost always pulsed and stored energy from capacitor
banks or flywheels can then be used to create the magnetic field for a short
while. Even some large experiments have copper coils, like for example JET
(Joint European Torus) in Culham, England. However, the ohmic losses are
typically regarded too large for reactor scenarios. For spherical tokamaks,
however, superconducting coils cannot be used at the inboard side, since
there is not sufficient space for shielding. In the ARIES-ST study, a 3 GWth
spherical tokamak reactor is proposed [88]. The TF (Toroidal Field) coils are
made of copper, and the best option seems to be to cool them with water at
room temperature [88]. Using hot water or liquid lithium to increase the
power conversion efficiency does not pay off due to the increase in
resistivity [88]. Neither does it pay off to have the copper coils at cryogenic
temperatures even if the resistivity drops significantly, since the efficiency of
the cooling at such low temperatures is too low [88]. One problem in the
ARIES-ST project is that the inboard leg of the TF coils suffer from
radiation damage, and need to be replaced every third year or so [88].
Copper is rather sensitive to neutron damage, and has a low DPA
(Displacement Per Atom) limit. The ohmic heating in the TF coils for
ARIES-ST was calculated to 329 MW. This is reasonable (although quite
high) for a power plant that aims for a power production of 3 GWth.
6.2. Mirror machine coil types
For quadrupolar mirror machines, the first quadrupolar coils were the “Ioffe
bars”, see for example Ref [89]. These are similar to the quadrupolar coils
used in Paper I. In later designs, two types of coils have primarily been used
to produce the quadrupolar (and partly the axisymmetric) magnetic field
component. One is the baseball coil (with the shape of a baseball seam),
which is described in Ref. [90]. The other is the yin-yang coils [91], which
73
are “bent” elliptic coils ordered in pairs embracing each other. Both these
coil types are shown in figure 6.1. To add contributions to the axisymmetric
part of the field, circular coils are normally used, possibly combined with
elliptical-like racetrack coils (formed like a racetrack) at the elliptic regions.
A yin-yang pair is used to create an anchor cell of a tandem mirror, giving
rise to both the axisymmetric and the quadrupolar field components. A
baseball coil is typically used to produce the quadrupolar field and other
coils are normally needed to give a sufficient axisymmetric field.
Figure 6.1. Quadrupolar coil types for mirror machines, with a baseball coil in (a)
and a yin-yang coil pair in (b).
6.3. Existing or earlier mirror machine coil systems
A number of mirror machines have been built for fusion research. The coil
system of the Gamma 10 device can be viewed in Ref. [92]. It consists of
circular coils, racetrack coils and baseball coils. In the reactor design study
MARS (Mirror Advanced Reactor Study) at Lawrence Livermoore National
Laboratory (LLNL), yin-yang coils were used [93]. For the large MFTF-B
tandem mirror (40 m long) at LLNL, which was ready for operation in 1986
but never used due to budget cuts, both yin-yang coils and baseball coils
were used in combination [93].
74
7. The SFLM Hybrid project
Studies of a hybrid concept based on the SFLM started in 2007 at Uppsala
University. The aims are to examine the possibilities of making a mirrorbased fusion-fission device for transmutation of minor actinides in
combination with energy production, and to make a reactor design study.
The participants are Prof. Olov Ågren, Dr. Vladimir Moiseenko, Prof. Klaus
Noack, Anders Hagnestål (the author), Prof. Henryk Anglart and Prof. em.
Jan Källne. The concept is described in Paper VI, the coil system is
described in Papers I-II, IX-XI and the fission mantle is described in Papers
VIII, XII. The goal of the project at this stage is to show that it seems
probable that such a mirror-based fusion hybrid can be an alternative for
combined transmutation of TRU and energy production. In this section, most
parts describe results mainly produced by other members of the project
group, which is here regarded as background information.
7.1. Geometry
The SFLM Hybrid is an average minimum-B mirror machine with a mirror
ratio of 4. The orientation of the device can be either horizontal or vertical,
since both orientations allow for natural circulation of the coolant. Currently,
a vertical orientation is the selected option. The length of the confinement
region is 25 m. Beyond each mirror end a 6.25 m long magnetic expander
including a recirculation region is added, giving a vacuum chamber length of
37.5 m. The plasma radius is 40 cm at the midplane. The vacuum chamber
has a circular cross section with a 90 cm radius in the confinement region,
which expands to 100 cm at the beginning of the recirculation region. Near
the magnetic expanders, the vacuum chamber radius expands to about five
meters. The fission mantle is located outside the vacuum chamber
confinement region, having a total length of 26 m. The cross section of the
fission mantle is circular and has an outer radius of 2 m. In the first design in
Paper VIII, there was no neutron shield included in the fission mantle. In the
second design in Paper XII, a neutron shield which protects the coil system
is included. Outside the fission mantle the superconducting coil system is
located, where a radial 10 cm space separates the coils from the neutron
shield. The coils are described in section 8.1. Inside the vacuum vessel at the
recirculation region, antennas for radiofrequency heating are located. The
device stripped from coils is shown in figure 7.1.
75
Figure 7.1. The SFLM hybrid device (here horizontally orientated) where the coils
have been removed.
7.2. Plasma parameters and magnetic field properties
The vacuum magnetic field is described in chapter 8.1, and provides flute
stability with some margin in the vacuum field. The device is aimed for a
of about 0.4-0.6 with the latest coil design (Paper II), which may affect
stability. Therefore, a stability margin for maintaining flute stability has been
applied in Papers I-II. There are also other stabilizing effects such as line
tying (see for example Ref. [94]) and finite Larmor radius effects [95].
Furthermore, the magnetic expanders add to stability, although their main
purpose is to take care of plasma loss.
The device is in this version aimed for an initial fusion neutron production of
around 3.6·1018 neutrons per second. Each fusion reaction give one neutron
and in total 17.6 MeV 2.82 10 12 J. This corresponds to a fusion power of
about 10 MW and a fission power of about 1.5 GWth with a Qr 146 150.
These values are for the beginning of the fuel cycle (BOC) with keff = 0.97.
Control rods, suggested in Paper XII, or burning absorbers can be used to
keep a fixed value of keff = 0.97, and with this option the fusion power will
be held fixed. Without control of keff the fusion power needs to be gradually
increased to 20 MW in the end of the fuel cycle (EOC) to maintain the
power production of 1.5 GW, since keff will be reduced to 0.945 before
refuelling is performed due to burnout of fissile isotopes (see Paper VIII). A
76
fuel cycle is 311 days of steady-state operation (or nearly one year), which is
a standard fission reactor operation time. It is, from the fusion point of view,
strongly recommended to keep the fusion power fixed for several reasons:
1. The first reason is that the device must be dimensioned for the
maximum neutron (and fusion energy) production during the fuel
cycle. For the fixed keff case, only half the maximum neutron
production need to be achieved, which reduces the maximum
or the magnetic field strength. The
requirements of the plasma
neutron production is approximately given by
nr
nT nD
D T
v d 3x
(7.2.1)
V
where
is the averaged reaction rate for D-T fusion
D Tv
reactions, nr is the number of fusion reactions and nT and nD is the
number densities of tritium and deuterium respectively. If the
temperature is kept constant, the pressure is proportional to the
number densities nT and nD and thus the neutron production nr p 2 .
With only half the required maximum nr , the pressure can thus be
reduced by a factor 2 , corresponding to a reduction of the
with a factor 1/ 2 for the same magnetic
maximum plasma
is fixed, the magnetic field can be
field. If instead the plasma
reduced by 2 1/ 4 0.84. This factor can be very important for the
cost of the coil system, since by this perhaps all coils can be made of
the considerably cheaper material NbTi instead of Nb3Sn.
2. Following the first point, the heating system (ICRH antennas) and
the massive associated power feed can be dimensioned to half the
size, about 70 MW. Also, the power consumption for heating is
reduced to 2/3 on average, which would save in the order of
10 million euro/year.
is
3. The magnetic field can be optimized for the pressure profile. If
to be varied by a factor of 2 , it will probably not be possible to
avoid neoclassical transport, and other field properties could also be
suboptimal (perhaps higher ellipticity is needed). This is since the
curvature of the field lines is very sensitive to changes in the
magnetic field. The stability limits to ballooning modes also depend
, although we do not expect this to be problematic (see
on
introduction of Ref. [95]).
The drawback with operation at a fixed keff is perhaps that control rods may
complicate reactor safety.
The required fusion Q, the ratio of thermal power output to total electric
power consumed for the fusion device, is 0.15. This is a factor of 100 lower
77
compared to a pure fusion reactor (Q = 15), but it is still not obvious that it
can be achieved in a mirror device of this type. The critical parameter is the
electron temperature.
Note that an increase of the midplane magnetic field is presupposed when
for a certain neutron production is estimated, so that the
the required
mirror ratio would be about 4 on average with the diamagnetic effect
effects are included has
included. The final coil design where finite
however not been completed yet.
The plasma that leaks out at the mirror ends would be removed by vacuum
pumps at the expander region. The feeding of the plasma has not been fully
specified yet. New plasma could be injected using pellet injection from the
mirror ends, and gas feed can also be used. Another option is to use neutral
beam injection from the mirror ends, although the pitch angle may become
unfavourable. In the current design, there is no room for feeding with neutral
beams at the central cell since this would require holes in the fission mantle.
R. Moir et al. has dealt with this problem in their design by making the
fission mantle a bit shorter [28]. A disadvantage with pellet injection and gas
feed is that both methods may lower the electron temperature.
7.3. Radio frequency heating
For plasma heating, ion cyclotron radiofrequency heating (ICRH) is planned.
There are several reasons for this.
1. Calculations predict that the efficiency of the ICRH could be high,
theoretically more than 90% [96].
2. The antennas are small and can be fitted in at the elliptic regions just
outside the mirror ends, where the plasma density is low. There can
be two antennas at each end, giving a possibility for four antennas in
total. As mentioned above, the competing neutral beam technology
with midplane injection would require large holes in the fission
mantle to operate where fusion neutrons could escape, and thereby
such neutrons would not contribute to the fission power generation.
3. For a fusion-fission device, electric breakdown criteria for
sequenced antennas admit an ICRH power of 130 MW, which would
be required at EOC if no control rods are used (100 MW indicated in
[96]).
4. The ICRH may improve the end confinement since perpendicular
energy is added to the ions, and the ions thereby move further away
from the loss cone in velocity space.
78
7.4. Fission mantle and shielding
The fission mantle is thoroughly described in Papers VIII, XII. In Paper
VIII, two versions of the fission mantle are outlined, and in this doctoral
thesis the “near-term” option is described which is aimed for 1.5 GWth. In
Paper XII, the fission mantle is modified to include neutron shielding of the
coil system and to include small control rods. The calculations on the fission
mantle have been made with Monte Carlo simulations using the program
MCNP5. In the calculations, the geometry of the fission mantle is simplified
as axisymmetric layers of a certain thickness. Each layer constitutes of a
homogenous mixture of the present structure material, fuel, coolant etc. in
that layer. Several of the materials and concepts from the fission mantle
study are adopted from FTWR tokamak hybrid concepts (see for example
Ref. [29]), which have been studied since the 1990ies by W. M. Stacey et al.
at Georgia Tech in high technical detail. One important difference is that a
15 cm thick buffer region is added in the SFLM Hybrid to protect the first
wall from fission neutrons.
7.4.1.
Geometric design
The geometric design of the fission mantle is shown in figure 7.2. The 3 cm
thick first wall has an inner radius of 90 cm and is made of HT-9 steel.
Outside the first wall is a 15 cm thick reflecting buffer of Lead-Bismuth
Eutectic (LBE), which is the primary coolant material. It may seem strange
at first to have a reflector here. The primary reason for having the buffer is to
protect the first wall from fission neutron damage. Without such a buffer, the
lifetime of the first wall would be too short. The neutron damage primarily
comes from the fission neutrons since they constitute the majority of the
neutrons, and by reflecting them the lifetime of the first wall is extended
significantly to more than 30 years using a maybe somewhat high DPA-limit
(Displacement Per Atom) of 200. Another benefit of the buffer is that LBE is
a good neutron multiplier for fusion neutrons through (n,2n) reactions, and
the number of neutrons that enter the fission core is increased. On the other
hand, the buffer moderates the neutrons which gives lower average fission
neutron gain per source neutron than would otherwise be expected. The
combined effect is that Qr is slightly lowered by the buffer.
79
Figure 7.2. The radial structure of the fission mantle. The figure is produced by
Klaus Noack.
Outside the buffer is the fission core, which is about 22 cm thick in the socalled “near-term” option. The core is composed of fuel, structure/cladding
and coolant. The fuel is of dispersion type containing TRU-zirconium
elements enbedded in a zirconium matrix. The structure/cladding is made of
HT-9 steel.
Outside the fission core is a core expansion zone filled with LBE. This zone
will work as a neutron reflector, and will give a negative contribution to the
void effect since a loss of coolant accident (LOCA) will result in a poorer
neutron confinement. In this zone, two 2.5 m long and 1 cm wide boron
carbide annuli (control rods) are located which can be inserted axially all the
way to the midplane. They are used to keep keff constant during a fuel cycle
(see Paper XII), and can lower keff by about 0.04. Also, there is space for
special irradiation assemblies and extra fuel if needed in this region. The
core expansion zone is 15 cm thick.
Outside the core expansion zone, which also acts as a neutron reflector, is
the radial reflector. The purpose of the radial reflector is to confine the
neutrons and thereby increasing keff per amount of fuel and reducing needs
for neutron shielding. Also, the tritium generation is located here. There are
axial reflectors (neutron end plugs) as well at the mirror ends which cover
the whole fission mantle endings. The material in these has been adopted
from the NEA benchmark for ADS, using 70% HT-9 steel and 30 % LBE
coolant. For the radial reflector, the LBE is replaced with the lithium-lead
coolant Li17Pb83 to generate tritium (T), where the lithium is 20% enriched
in Li-6. Tritium is primarily produced by the reaction Li-6 + n = He + T +
4.8 MeV [11]. It is vital that the device is self-sufficient in tritium.
Calculations show that tritium generation will be larger than the
consumption during the whole fuel cycle, averaging 3.60 times the
consumption in the first model in Paper VIII. This large tritium production
implies that the enrichment of Li-6 can be substantially reduced. The tritium
80
regeneration is sufficient in the newer fission mantle version in Paper XII as
well.
Outside the radial reflector, the neutron shield is located as specified in
Paper XII. Neutron shields for fusion is typically very thick, 0.5-1 m, since it
is hard to stop the 14 MeV fusion neutrons. For the SFLM Hybrid, the
shielding problem is however a fission shielding problem, since such a small
fraction of the neutrons that escape through the reflector is fusion neutrons.
Typical fission neutrons have energies of 1-2 MeV directly after fission. A
neutron shield of 25 cm thickness is therefore sufficient to reduce the nuclear
heating and damage in the coils to acceptable levels. The shield is made
from a mixture of stainless steel alloy S30467 type 304B7 (60 vol%), which
has 1.75 % wt% boron added, and water (40 vol%).
7.4.2.
Reactor safety
Reactor safety is a complicated subject, and much work is needed to analyze
various safety cases in the SFLM Hybrid project. The most severe accident
case to avoid is a supercriticality, preventing keff from exceeding 1 + ,
where is the fraction of delayed neutrons. The Chernobyl accident was
partly of this type, with known disastrous results. Also, a core meltdown
must be prevented. A core meltdown will probably destroy the facility and
cause a lot of sanitation work, and could also lead to radioactive leakage like
the (small) leakage in the TMI (Three Mile Island) accident.
In a reactor with a fast neutron spectrum, the presence of coolant in the core
effects keff directly in a few ways. First, the coolant acts as a reflector. How
much reflection a coolant causes depend on the coolant material. A change
of the coolant reflecting property will affect keff, and the amount depends on
geometry and location of the coolant. In most cases, presence of the
reflecting property will increase keff by confining the neutrons. Second, the
coolant will moderate the neutrons. This might affect keff in either direction,
depending on the neutron spectrum, the fission cross sections of the fuel and
the number of neutrons per fission, but will in the SFLM Hybrid decrease
keff. Third, (n, 2n) reactions in the coolant might increase keff and neutron
absorption in the coolant might decrease keff. In reactors with lead coolant,
the total impact of all these effects is that the presence of the coolant leads to
an increase in keff. It is however not so for all regions within a reactor,
especially not in the central core region where presence of coolant in leadcooled reactors cause a decrease in keff. The coolant can be voided in a
reactor, for example if the coolant loop has a leakage or if the coolant is
blocked by solid coolant etc. causing local boiling. If the coolant by leakage
or other effects is removed from the reactor, keff will be affected. This is
called the void effect. Lead-cooled cores have in most cases a negative void
81
coefficient if all coolant is lost (see for example Ref. [97]), implying that keff
will decrease when the coolant is removed. However, as stated above, this is
not always true for voiding subsections of the coolant by for example
coolant boiling. The core must remain subcritical in all possible coolant
distributions, and such safety case studies should be addressed.
In Papers VIII, XII, two safety cases concerning coolant void effects are
handled. The studies are made for a horizontal reactor orientation in Paper
VIII and for a vertical orientation in Paper XII. Since no detailed core layout
even has been done yet, the safety cases are simplified to a homogenized
model. The first case is loss of coolant accidents in the LBE loop, which is
investigated by calculating keff for a number of LBE coolant levels. The
result is a strong decrease in keff in all cases for both orientations, being more
pronounced for lower coolant levels. There is no indication that an increase
in keff would occur at any coolant level. The strong decrease in keff is caused
by the loss of the neutron confining coolant reflector in the core expansion
region. The second safety case examined is partial voiding in the core caused
by local boiling. This case is examined by voiding axisymmetric rings of
different thicknesses, and here the cases are the same for both reactor
orientations. The expected increase in keff is well below 0.02 and it seems
unlikely that this increase will exceed 0.02 for any local boiling. It should
however be noted that voiding of a small localized part, say a cooling
channel, has not been checked. The calculations suggest that keff = 0.97 is an
adequate reactor scenario value concerning these studied safety criteria. In
Paper XII, a continuation of a LOCA accident is examined where the core is
flooded with water to remove the residual heat. The case reveals that if only
the core is filled with water, the reactor becomes supercritical. If the whole
coolant region is flooded, the reactor remains deeply subcritical. Since this is
a possible case, there should be large and many connection channels
between the different coolant loops.
Safety cases concerning supercriticality where parts of the core are
moved/compacted due to core meltdown have not yet been adressed.
Although core meltdown should be avoided at all costs, a core meltdown
should not result in supercriticality. A drastic increase of keff (from 0.84 to
0.96) is pointed out in Ref. [98] in a horizontally oriented mirror machine
when the different materials (fuel, structure/cladding, coolant etc.) melts and
ends up in a sandwich structure on the ground. The reason for the large
increase in keff is that the fuel becomes more compactly distributed when the
large empty space in the vacuum chamber compacts. The SFLM Hybrid can
be oriented vertically, but this does not guarantee that such scenarios cannot
be found, for example if a part of the core falls into the vacuum chamber.
The described increase in keff may be a critical safety issue for magnetic
82
mirror hybrids, and should be studied for the SFLM hybrid concept.
However, there are most likely ways of handling such a scenario.
The criticality keff is also affected by the temperature. In a fast reactor, the
change in keff comes from the Doppler broadening effect and the CTD
(Coolant Temperature Density) effect. In the SFLM Hybrid case, the
Doppler broadening effect is small due to the fast neutron spectrum and the
very low content of U-238 in the fuel. For Doppler broadening, keff / T
was calculated to about 1.1 10 6 for the SFLM Hybrid where keff was
calculated at two different temperatures and the temperature dependence was
approximated as linear between those two temperatures. For the CTD effect
in the central core keff / T was calculated to about 4.27 10 5. for the
LBE. This reflects the void effects in the central core, and is not negligible.
If the entire buffer and expansion zone is added, the CTD effect reduces
somewhat to 3.37 10 5 , but there is still a positive increase in keff with
increasing temperature. Further analysis of the CTD effect has not yet been
made.
7.4.3.
Cooling
The cooling has been considered by Prof. Henryk Anglart at KTH for the
vertically orientated case, and some results are presented in Ref. [99]. The
LBE and Li17Pb83 coolants are pumped during normal operation, and the
pumping power can be well below 50 MW. The MHD forces have not been
included in this analysis, and they may add to the pumping power. However,
the cooling channels would be almost parallel to the magnetic field except at
the influx/outflux, which should make the MHD forces less significant than
would otherwise be the case. It may also be possible to reduce them with
dielectric pipe coating.
When the reactor is stopped, the residual heat (or decay heat) must be
removed which became evident to the broad public during the Fukushima
accident. The residual (or decay) heat comes from decay of fission products
and gradually fades away after shutdown. It however constitutes of more
than 5 % of the nominal power at shutdown, and cooling is needed for a
rather long time (at least weeks) afterwards. In the SFLM Hybrid, passive
cooling is planned for removal of residual heat. This removes dependence on
external pumps after shutdown. With vertical orientation of the SFLM
Hybrid, it is shown in Ref. [99] that it seems possible to have natural
circulation of the coolant that is sufficient for removal of the residual heat.
The circulation is driven by the heat differences in the core and the outer
loop. It may however probably also be possible to achieve this for the
horizontal case.
83
It seems sufficient to have only one point of influx and one point of outflux
of coolant along the z axis. Since the influx/outflux of coolant must be
regarded during coil design so that there is sufficient space available for the
pipes, this means that the coil system only needs to have gaps at the
recirculation regions.
A vertical orientation also simplifies the refuelling process, since this can be
done from the top. For a horizontal refuelling process, the coolant must be
prevented from leaking out.
7.5. Electron temperature
The weakest point in the fusion driver is probably the electron temperature,
which has been a major roadblock for fusion mirror machines since too
much input energy is lost due to electron drag. A typical requirement for
mirror fusion devices is Te = 10 keV [47]. In an experiment in the Gamma 10
tandem mirror device in Japan an electron temperature of about 650 eV has
been reported, indicated in Ref. [100]. For the axisymmetric Gas Dynamic
Trap (GDT) device an electron temperature of 250 eV [48][101] has been
achieved, measured with Thompson scattering. The multimirror device
GOL-3 has achieved an electron temperature of 2.5 keV. In Paper VI, an
estimate of the required electron temperature for the SFLM Hybrid has been
made. The requirement on Te is substantially relaxed due to the strong fission
to fusion power amplification Qr, and is estimated to 500 eV for a power
producing device. The estimate is based on two assumptions. First, the major
power loss in the mirror is assumed to be due to electron drag. Second, the
requirement for a fusion device with Q = 15 would be 10 keV or higher. The
strong Qr relaxes the requirement on Q to about 0.15 with Qr 100.
Assuming that the electron drag time increases as Te3/ 2 from Eq. (3.3.5), that
d is proportional to the power loss and that the densities ne are the same,
the result is
Te ,hybrid
Te, fusion
100
2/3
10 keV
1002 / 3
0.5 keV
(7.5.1)
Although this is a large relaxation of the required electron temperature, it is
not obvious that such a temperature can be reached in the SFLM hybrid.
Therefore, a scenario for increasing the electron temperature is outlined in
Paper VI. The scenario is based on reducing the plasma and neutral gas
density in the expander region by pumping. Thereby, the electrons would
lose less energy in that region from collisions (where line radiation,
secondary emission and other effects can be important).
84
7.6. Divertor plates and heat load
A critical point for fusion reactors is the divertor plates. The divertor plates
are the material surfaces that are designed to receive the majority of the
leaking plasma. The requirements on the divertor plates for toroidal systems
for pure fusion are very high, and the best material so far tested for ITER
would only last for about 7000 pulses (i.e. about 3 months of steady-state
operation) [61]. The divertor plates for ITER should be able to withstand
both a very high heat load from the plasma of about 5 MW/m2 (requiring
good heat conduction), the strong neutron load from fusion neutrons and the
pulsed mode of operation with varying heat loads which causes material
stresses. The maximum acceptable upper heat load limit of 10 MW/m2 is set
in Ref. [102], but the 5 MW/m2 is assumed to be more correct.
For the SFLM Hybrid, the situation is much better. Due to the strong Qr, the
fusion power is strongly reduced. The input power (at BOC) is about 70
MW. In such a fusion device, the radial losses are expected to be small
compared to the axial losses, not more than 10% of the total losses. It should
however be mentioned that this has not been checked for the SFLM Hybrid
yet, and some experimental work suggests otherwise [103], although this
may be due to electron cyclotron resonance heating. For the first wall, a
limiter would receive most of this load of about 7 MW. It is not expected to
be a problem to distribute this load on a surface large enough to withstand
the heat load, say 10 m2 or so, and it is expected that a larger radial heat flux
can be dealt with. The rest of the heat load comes from the axial losses and
will reach the expander. The expander can – and should – be made
sufficiently wide to distribute the heat load on a large area. For example, if
100 m2. This
the radius is 4 m, the total expander area will be 2 42
2
would give an average heat load of about 0.7 MW/m , which is a tolerable
heat load if it would be evenly distributed on the surface.
85
86
8. Summary of results and discussion
The results presented in this thesis can be grouped in the following way. In
Papers I-III, V, IX-XI, the results are associated with magnetic field and coil
design and those results will be summarized in section 8.1. In Papers VI,
VIII, XII the results are associated with the fission mantle and the overall
concept. These results are presented in section 7 and will not be repeated
here. In Papers IV, VII, the results are associated with radial transport and
constants of motion. These results are given in section 8.2. An overall
discussion of the results is given in section 8.3.
8.1. Design of coils and magnetic fields
To understand the results from these papers, it is necessary to present them
in cronological order, which is done in sections 8.1.5 - 8.1.10. The field and
coil design has been a process over 3-5 years, resulting in some intermediate
stages and a final vacuum design. The focus in this work has not been to
reproduce a magnetic field with as high accuracy as possible, but rather to
show that it is possible to find a coil set that reproduces the magnetic fields
within the given geometric constraints with a tolerable accuracy. First, the
optimization methods and coils used are described, and then the specific
results from the papers on coil and field designs are given.
8.1.1.
Optimization methods
The basic task for an optimization algorithm is to select the best element or
value from a number of available elements or values according to some
criteria. In this doctoral thesis the available values are function values of
some n-dimensional continuous cost function, where the cost is to be
minimized. It is here important to realize the difference between global and
local optimization. Local optimization, where a local minimum – not
necessarily the best minimum available – in the cost function is found, is
rather straightforward. To illustrate this, consider a 2-dimensional cost
function and interpret it as the altitude of some terrain. If an initial starting
point is given, it is simply just to walk downhill until you have uphill in all
directions to find a minimum. This can be done in numerous ways, where the
result will be the same but the processing time different. This is the principle
of local optimization. Global optimization, on the other hand, is much more
87
difficult. A global optimizer tries to find the best minimum of all available
minima. For a cost function with many parameters, say 30, it is often in
reality not possible to guarantee that all minima have been checked. Global
optimizers are therefore often heuristic and look for an as good minimum as
possible, but do not guarantee that it is the best minimum available.
There are numerous optimization methods available for both global and local
optimization. At this stage of the project, time has not been spent to identify
and implement/use a global optimizer. Instead, simple local optimizers have
been used. After trying some different local optimizers, an implementation
of the Nelder-Mead algorithm was selected. This method is heuristic and
does not always converge, but in practice this showed to be no problem in
the cases studied. The results were comparable to the results achieved with
the other methods, and the implementation worked well.
For coil optimization, there are two magnetic field components that are to be
reproduced, the quadrupolar field represented by g ( z ) and the axisymmetric
field B ( z ) . In Papers I, IX-X coils types that only give contribution to one of
the field components have been selected. Thereby, the coil optimization
problem is separated into two independent optimization problems, one for
g ( z ) and one for B ( z ) which simplifies the problem. The cost function (in
this case for B ( z ) ) typically punishes deviances from an ideal field and
power use and a typical cost function could be
zmax
fz
dz
0
B ( z ) Bc ( z )
2
B ( z)
2
ri I i 2
Ki
(8.1.1)
i
where Bc ( z ) is the B ( z ) contribution from the coil system and K i is a
weight factor for the power term. In some cases, more terms were added.
The power term is not at all correct for superconducting coils but serves the
purpose of restricting a waste of current to gain accuracy. It is important that
the power term includes all conductors to get an appropriate and practical
result.
Also, it proved useful to do the coil optimization by hand, which means that
the input parameters were changed manually and the result was observed
between each change. This was done in Paper II which at first may sound
reckless, but almost as accurate results were found with this method
compared to using an optimizer and it takes only an hour or so to do the
optimization. The numerical optimization process produces a lot of
impractical and strange solutions, and it is sometimes difficult to find a way
to avoid these and find a proper solution. For the coils, there are some
auxillary constraints that are not so easy to implement in a practical way in
the cost function. One is that the coils should not intersect, and should be
88
spaced with some margin (which though in principle can be implemented).
Another is engineering simplicity. Also, for the 3D-coils, no analythical
expression has been found for the contribution to B( z ) and g ( z ) , which
makes an optimization process very slow.
8.1.2.
Optimization of vacuum magnetic fields
The magnetic field can be optimized in many different ways. A list of
properties that should be addressed for coil design in tandem mirrors has
been given by Baldwin [93]. He listed 8 properties to be targeted. Many of
them refer to problems specific to tandem mirrors. In Papers I-II, IX-XI the
vacuum fields were optimized for field smoothness, flute stability with a
margin and low ellipticity. There are two of the properties relevant for
single-cell mirrors listed by Baldwin that are not fully targeted in the work in
Papers I-II, X-XI. The first is neoclassical radial transport, which is
associated with geodesic curvature causing radial particle drifts that creates
“banana” particle orbits [93]. This is not targeted in the field design in Paper
X and only to a certain extent targeted in Papers I-II, XI. It was argued in
those papers that in the case of a fusion hybrid, this property is not as
important as it would be for a pure fusion device, since the required particle
confinement time is considerably shorter and the majority of the losses still
will be end losses. The author has however after these papers were written
understood that they may be quite important, and that they should be
into account.
targeted in a future coil design that takes effects from finite
The other property is stability to MHD ballooning modes. The reason for not
targeting MHD ballooning modes (yet) is the later publication by Newins
and Pearlstein [95], where the authors expected that ballooning modes would
not be limiting in any tandem mirror device due to the strong stabilization
of finite larmor radius (FLR) effects. At this stage of the project, ballooning
mode analysis is therefore postponed to future studies.
The properties that are addressed are important. The field must be stable to
flute modes. This is the very point of having a quadrupolar field. Two
different methods have been used to determine flute stability. In Paper X, the
average minimum-B criterion from Eq. (3.3.7) has been used. However, this
criterion does not take into account the effects of a varying pressure along
the field line. In Papers I-II, this is corrected by instead using the flute
stability criterion given in Eq. (3.3.8). It is also important to obtain a fairly
low flux tube ellipticity. A too high ellipticity gives an impractically shaped
plasma which could hit the first wall in the strongly elliptic regions. In
practice, this reduces the plasma volume, and for the SFLM Hybrid it would
increase the required magnetic field strength to get a sufficient neutron
production. If the fans get too thin, plasma feeding may get complicated.
Also, a slight argument is that classical transport may increase somewhat in
89
the thin fans. It is also crucial to avoid too sharp gradients in the magnetic
field, since it will not be possible to create sharp gradients when the thick
fission mantle is present. It should be noted here that the stability
requirement and the low ellipticity requirement are contradicting, and the
task is to find a field that is suitable in both aspects.
8.1.3.
Function modelling with splines
One way to optimize the vacuum magnetic field is by modelling the B ( z )
and g ( z ) functions with equidistant cubic clamped B-splines, which has
been done in Paper X. Splines are chosen since they are fairly easy to
implement, have continuous derivatives up to second order and since the
control points only affects the function in a region close to itself. Clamped
splines are chosen, since with this representation the end derivatives can be
prescribed. Typically, between 10 and 20 control points are used for each
function. The control points are then used as variables in a cost function
which is optimized by a local optimization function (Nelder-Mead). Since
both B ( z ) and g ( z ) are even functions, it is sufficient to model half of
those functions. The boundary conditions for the spline representation of the
B function are
B (0)
B0 , B ( L )
Rm B0 , B (0) 0, B ( L) 0
(8.1.2)
where prime denotes differentiation in z and L is half the length of the
confinement region. For the g function, the only fixed boundary condition is
g (0) 0
(8.1.3)
due to symmetry and the rest of the boundary conditions g (0), g ( L) and
g ( L) can be chosen freely. The boundary conditions for B are at least
nearly fulfilled by locking the first and last spline control points to the
boundary value and placing them at z = 0 and z = L. The derivatives of B
can automatically be prescribed with the clamped spline representation. For
g, all control points are free and g ( L) is either prescribed to some value or
used as a variable in the optimization.
A cost functional to optimize has been chosen and the optimization
performed. The optimization has been made with the average minimum-B
criterion as stability criterion in Paper X. An attempt has also been made to
optimize the field using the flute stability criterion and a prespecified
sloshing ion distribution to model the pressure. It proved however hard to do
such an implementation, since the optimizer always selected a profile that
disqualified the pressure profile. It was obvious that such a simple method
was not sufficient, and that an equilibrium calculation was required to get
even close to appropriate results.
90
8.1.4.
Superconducting coil modelling
The superconducting coils need to be modelled somehow to determine their
size and to find the magnetic field that is generated from them. The size is
required both since there is an obvious auxillary condition that the coils are
not allowed to intersect and since the current distribution in a coil is affected
by its size. The widths of the coils are determined from several aspects. The
minimum width is obviously dependent on the coil current. The dependence
is three-fold. First, the magnetic field strength should not exceed the critical
magnetic field for the superconducting material anywhere. If so, a
superconductor quench will occur. Second, the coil must be strong enough to
withstand the huge magnetic forces. Third, the critical current density for the
superconductor must not be violated. Also, space is required for cooling
channels within the coils. To determine the dimensions for a coil set is
therefore a complicated task. For instance, the cooling requirements need to
be determined for the expected maximum heat load (which itself is not very
easy to determine, see section 6.1.3), and after this the cooling channels must
be dimensioned. However, what can be done conveniently and probably with
a fairly realistic result is to use an existing similar coil system as a reference.
The method employed is to calculate the average current density in the
winding pack for a coil system producing a similar magnetic field strength as
the device that should be modelled. Two existing or planned systems were
analyzed: ITER and the Japanese tokamak JT-60. In the (somewhat old)
specification for the ITER toroidal coils (Nb3Sn) they have a current
carrying region of about 0.85 m2, and carry a current of 11.5MA [76]. This
gives an approximate average current density of 1.35 kA/cm2. In the
remodelling of JT-60 for superconducting coils (NbTi), approximately the
double current density was used [104]. In Paper I, X, a current density of 1.5
kA/cm2 is assumed for the Nb3Sn coils, and in the lower field solution in
Paper II, where we hope that NbTi coils can be used, 2.6 kA/cm2 was
directly adopted from the JT-60SA NbTi coils [105]. The size of the
stabilizing structure material outside the current carrying region is
approximated by again looking at ITER and JT-60 toroidal field coils. Both
have a cover of structure material with a thickness of about 10 % of the total
coil width. Thereby, in total 20% extra width is added to the coils for
structure material. The cross sections of the coils are selected to be quadratic
for the circular coils, and quadratic-like for the quadrupolar and fishbone
coils. The magnetic field is then calculated from Biot-Savarts law, where the
winding pack has been divided into m m equidistant filamentary line
currents. With m 10 the solution becomes very accurate, but already m 1
gives a fairly good approximate solution in many cases.
91
Four different coil types have mainly been used in this thesis. The simplest
coil type is the circular coil, which has been used to produce the
axisymmetric field in Papers I, IX-XI. It only contributes to B ( z ) (in ideal
cases) and it is straight forward to calculate the contribution from one
filament as
B( z )
0
2
I
r
r
2
z
2
z
2
3/ 2
(8.1.4)
where r is the filament radius and z is the filament z position.
To produce the quadrupolar field component, the simplest coil geometry
consists of four parallel conductors with quadrupolar symmetry. Such a
configuration is shown in figure 8.1, where also the quadrupolar field is
illustrated with arrows.
Figure 8.1. Quadrupolar coil configuration where the arrows show the direction of
the quadrupolar component of the magnetic field.
Two different types of coils have been used that only contributes to the
quadrupolar field. One coil type that was evaluated in Paper IX was long
conductors like those shown in figure 8.1 but where the distance from the z
axis to the coil was a function of z. Since they proved not to be very useful,
they are not further commented here. Another type of quadrupolar coil that
has been used in Papers I, X look like symmetric Ioffe bars which are
interconnected at certain positions to modify the current. A series of
connected such coils are shown in figure 8.2. Note here that the current does
not form circles at the connection circles, since the direction of the current
shifts at each quarter-circle connection.
92
Figure 8.2. A 3D-image of three connected quadrupolar Ioffe bar-like coils. The
currents follow the black arrows.
The contributions to the quadrupolar field from such a coil can be grouped as
contributions from sets of four straight conductors g c , s and contributions
from sets of four quarter-circle segments g c ,c . The contributions to zeroth
order in r are
gc,s
3 0 I1
z2
z1
[( z
r '2 dz '
z )2 r '2 ]5 / 2
3 0 I1
K (z
z2 ) K ( z
z1 )
(8.1.5)
where
K (z )
[3r 2 2 z 2 ]z
3r 2 [r 2 z 2 ]3/ 2
(8.1.6)
r 2 (z
[( z z )2
(8.1.7)
and
g c ,c
3 0 I1 / 2
z)
.
r 2 ]5 / 2
In the equations, I1 is the current in the straight conductors, with a positive
sign for currents that give positive contributions to g. For gc,c in Eq. (8.1.7), a
positive sign shall be taken for coils with positive, inner z and negative,
outer z . The other two rings have a negative sign. The parameter r is the
coil radius. When making filamentary line currents of the straight segments,
the angular distance of the filamentary current from the center, q , changes
the contribution to gc,s in Eq. (8.1.5) with a factor of cos(2 q ) when the
symmetry is regarded. The g ( z ) field for such a coil can be seen in figure
8.3.
93
Figure 8.3. The g(z) field from one quadrupolar coil with end points at z
0.5 and
radius of 0.4, where the contributions from the different parts of the coil are shown.
These coils are good for optimization since the contribution to the g ( z )
function is well localized in z and since they only contributes to g ( z ) .
Compared to the baseball coil, they however waste current since baseball
coils contribute to B ( z ) as well.
All coil types described so far have only contributed to one of the functions
g ( z ) or B ( z ) . This is beneficial for the optimization process since it can be
divided into two separate optimization problems, one for B ( z ) and one for
g ( z ). However, this creates two layers of coils, and the size and cost of the
coils will be unneccesarily high. Also, the chain of quadrupolar coils has
some drawbacks. They are interconnected in a rather complicated way and
they have sharp corners that must be rounded off significantly. Even if they
are rounded off, the curvature of the conductors will be quite high. This is a
bad trait for superconducting coils since the strain in the condutors will get
high. To deal with these drawbacks, a new type of coil was invented. We
called it fishbone coil (not connected to fishbones in tokamaks) due to its
resemblence of a fish skeleton when several coils are put together. This type
of coil has a 3D-geometry and give contributions to both the g ( z ) and B ( z )
fields. A fishbone coil is shown in 3D in figure 8.4.
Figure 8.4. A fishbone coil shown from two different angles. The coil type is a 3D
coil with quadrupolar symmetry that give field contributions to both g(z) and B( z ) .
94
The coil geometry is best described by looking at the circumferential surface
that the coil is oriented on. The parameterized curve that describes a current
thread in the coil is at a constant distance from the z axis. Thus, the curve is
well described using cylindrical coordinates (r , , z ) with r kept constant for
each filament. The coil layout on that circumferential surface is shown in
figure 8.5.
Figure 8.5. The fishbone coil layout on the circumferential surface that wraps the
fission mantle. Only half of the circumference is shown.
Half the coil is shown, and the curve consists of straight lines and circle
segments in that plane. Such a curve can be parameterized using H, L and a
from figure 8.5, where a is the radius of the circle segments in that plane and
a negative L wraps the coil so that a positive coil contribution to B ( z ) gives
a negative coil contribution to g ( z ) . From the symmetry, we get H r / 2
since H is one quarter of the circumference. To parameterize the curve, it is
convenient to express the distances c and d in figure 8.5 in L, H and a and
the angle in d and a. Within reasonable values of H and a ( H 2a ),
d
aH 2
aL2
a2 H 2 (H 2
4a 2
H2
L2
L2
4a L )
4a L
2a 2 L
,
(8.1.8)
95
a2
c
(a d )2
(8.1.9)
a d
.
a
(8.1.10)
and
cos
1
Each coil is divided into (m m) current filaments. To parameterize the
filament curve it is divided into 8 subsections, 4 straight lines and 4 circle
segments (on the flattened out circumferential surface). For two filaments
that have different z positions the radii of the circle segments differ (see
figure 8.5). Also, for all filaments except those that are in the coil midline in
z, the radii of the circle segments varies along the curve where 2 circle
segments have radii a1 and 2 circle segments have radii a2 (see figure 8.5). In
the parameterization, n 0,1, 2,3 where each number represents one
quadrant of the coil and to simplify we define
1 for even n
n
(8.1.11)
1 for odd n
and
L
.
L
L
(8.1.12)
The filament circle segments are parameterized with the curve parameter t as
aeff
a
z,
n
x(t ) r cos
z (t )
n
(t ) n
(t ) ,
a
L eff
aeff
2
r
y (t ) r sin
1 cos
sin
t tmax / 2
tmax / 2
(t ) ,
t
t tmax / 2
tmax / 2
,
(8.1.13 - 8.1.14)
0, tmax , (8.1.15 - 8.1.17)
L
2
L
z
z0
(8.1.18)
and the filament line segments are parameterized as
z sin ,
(t ) n
2
1
c
r
x(t ) r cos
z (t )
96
n
zl
t
d
H
tmax
(t ) ,
L
z cos ,
L
2
2c
n
y (t ) r sin
t
tmax
L
(8.1.19 - 8.1.20)
L
2d
,
(t ) ,
(8.1.21)
t
L
zl
0, tmax , (8.1.22 - 8.1.24)
z0
(8.1.25)
where z a represents the filament displacement in z from the coil central
filament, z0 is the coil center z position and r is the radial distance to the
filament from the z axis.
An analytic expression for the magnetic field components for such a coil has
not been found. However, it is straight forward to calculate them numerically
using the parameterization above. The contribution to B( z ) is roughly
comparable to that for the circular coil, at least when the L parameter is
small. The contribution to the g ( z ) function is localized in roughly the same
way as the contribution for the quadrupolar coil shown in figure 8.3. In
figure 8.6, the contributions to B ( z ) and g ( z ) for a fishbone coil is shown.
Figure 8.6. The contribution to the functions B( z ) (a) and g ( z ) (b) for one
fishbone coil with parameters a 1 m, r 2.3 m, z 0 m, L 2 m and I 100 kA.
The field is calculated with a single filament at the center of the coil.
There are many advantages with these coils compared to other coil
combinations:
1. There will be only one layer of coils, which reduces the size of the
coil system significantly. Also, this most likely reduces the
maximum magnetic field in the coils. The cost of the coils is also
likely to be reduced, although the more complicated 3D geometry
may add to cost. However, 3D coils with more complicated
geometry have been manufactured for the Wendelstein 7-X
stellarator [106], and 3D coil manufacturing is here considered to be
a known technology.
2. Compared to baseball coils, these coils have softer curvature. Also,
they can be inserted into each other like a pile of drinking glasses,
which allows for more freedom during the design of a coil system.
Equally sized baseball coils cannot overlap. In the case of the SFLM
Hybrid, the ability to overlap is crucial.
3. The coils give contributions to both B ( z ) and g ( z ), and thus the
optimization problem is not separated as it was for earlier coil sets.
97
However, it is almost separated, since the current roughly controls
B ( z ) and the L parameter can afterwards be used to control g ( z )
when the current is determined. Changing L only has a very slight
effect on B ( z ) .
4. The coils are separate, and not connected to each other. This should
simplify engineering in the whole device.
8.1.5.
Results from Paper IX
This paper presents the first attempt in coil design, where the possibilities of
making coil systems for the SFLM vacuum field were explored. The main
result of this paper is the knowledge gained about the restrictions on the
fields imposed by the coils. Effort was in the beginning of this work put into
creating a coil set for the SFLM field up to a mirror ratio of four. It turned
out to be hard (or not even possible with some geometrical constraints) to
create the sharp gradients in z direction near the mirror ends in the
axisymmetric field (the B ( z ) function) and especially in the quadrupolar
field (the g(z) function). The difficulty to create such gradients is caused by
the minimum allowed radial distance to the coil system. Also, when trying to
produce those gradients, the field outside the planned confinement region
was not regarded. This was a mistake, which was not yet realized in Paper
IX. Since the gradients dB / dz and dg / dz are so strong, the field
components (the B ( z ) and the g(z) functions) form a huge overshoot outside
the SFLM region. Thus, the mirror ratio will be much larger than planned,
and the field will be grossly unstable to flute modes or have extremely
elliptical flux tubes. The actual results from these coils are rather
uninteresting, and will not be repeated here.
8.1.6.
Results from Paper X
At the time when this paper was produced, it became obvious that another
field than the SFLM field had to end the mirror, concatenated with the
SFLM field at some points before the mirror ends. The requirements selected
for such a field was that the whole field should be stable to flutes, that the
flux tube ellipticity should be minimized and that too sharp field gradients in
B ( z ) and g(z) should be avoided. It is however not obvious that such a
concatenated field would be optimal according to those critera, i.e. it is not
obvious that the best field for these criteria would partly consist of an SFLM
region. Therefore the problem of designing the whole field was addressed.
The field functions B ( z ) and g ( z ) was modelled using a spline
representation, and an optimization was made that targeted flute stability
(not taking the pressure distribution into account), low ellipticity and low
field gradients. A coil set was then created for this field. The coil set was
98
similar to the coils in Paper I (see below), and reproduced the field with
sufficient accuracy. These results are now obsolete, since the solution is
superseeded by the results in Paper I (which also is obsolete).
8.1.7.
Results from Paper I
In Paper I, the average minimum-B criterion was replaced by the flute
stability criterion given in Eq (3.3.8), which takes a varying pressure profile
into account. The SFLM field was used concatenated with an ending field at
z 8.75 . This ending field was finally modelled with splines and set up by
hand to avoid ripple, when some optimized solutions had been observed. A
margin was requested for the flute stability criterion to assure stability also
when the pressure profile would deviate somewhat from the pressure profile
selected for the computations. The resulting field is flute stable with some
margin, and is probably stable for all reasonable pressure profiles in the low
limit. The flux tube ellipticity of 20 is fairly low, and considered low
enough. The gradients of the field are acceptable. An advantage of using the
SFLM field at the center is also that there will be almost no neoclassical
transport from particles that stays within z 8.75 due to the omnigenuity
property of this field, and the overall neoclassical radial transport will
probably be quite low in the low
limit. The field characteristics are shown
in figure 8.7, where the midplane B is 2 T.
99
Figure 8.7. The selected field for the fusion-fission device. The figures show B in
(a), g in (b), W1 for both a constant pressure profile and a sloshing ion distribution in
(c), the flux tube ellipticity in (d) and the representative sloshing ion distribution in
(e).
A coil set was then optimized for this field. In this case, a current density of
1.5 kA/cm2 was used to dimension the coils. The resulting coil system
reproduces the magnetic field with satisfying accuracy for the SFLM Hybrid.
The field errors for the circular coils are shown in figure 8.8 and the field
errors for the quadrupolar field are shown in figure 8.9. In the confinement
region, the field errors are about 0.2% in B in the SFLM region. In the
spline region the errors in B are up to 1.5%, but this is expected to be of less
importance since it is not a ripple in the ordinary sense but rather a change of
the profile shape. For the g function, the relative errors are about 1-1.5%
with a maximum of up to 10% in g at the end of the confinement region,
100
cutting off the top of the pointed peak. In the expander region the errors are
larger, but this is expected to be of less importance. The coil set is visualized
in 3D in figure 8.10 and figure 8.11, where figure 8.10 shows the mirror
machine with the entire coil set and figure 8.11 shows the mirror machine
with the quadrupolar coils only. The axisymmetric coils are listed in table
8.1 and the quadrupolar coils are listed in table 8.2.
Figure 8.8. The relative error in the B( z ) function (a, b) and the B( z ) function (c).
Figure 8.9. The relative error in the g(z) function (a,b) and the g(z) function (c). The
reason for the sharp relative error peak at z = 13m in (b) is that g(z) changes sign.
101
Table 8.1. The circular coils on the positive z side for Paper I, defined by inner
radius, cross section center z coordinate, cross section width/height and current.
Inner radius (m)
2.45
2.45
2.45
2.50
2.50
2.57
2.66
2.89
3.09
3.09
3.09
2.87
2.77
2.55
z (m)
0.900
1.833
2.701
3.604
4.769
6.497
7.391
9.381
10.441
11.470
12.650
14.000
17.289
18.125
Coil width (m)
0.473
0.184
0.393
0.323
0.445
0.513
0.286
0.209
0.750
1.023
1.258
1.384
0.241
0.789
Current (kA)
2335
352
1609
1086
2061
2740
852
456
5852
10905
16497
19956
-606
-6482
Table 8.2. The quadrupolar coils on the positive z side for Paper I, defined by end z
coordinate, cross section width/height of the straight bars, cross section width/height
of the quarter-circle segments, current in the straight bars and current in the quartercircle segments. The inner radius for all segments is 2.1 m. The array of coils is cut in
two at the magnetic expander, indicated by zero current and zero width for the
straight segments.
End z (m)
2.000
3.781
5.416
7.000
8.700
9.602
10.957
12.500
13.800
16.250
16.900
17.600
19.375
21.005
102
Width
0.315
0.332
0.365
0.420
0.519
0.684
0.972
0.978
0.392
0.753
0.666
0.581
0
0.232
Width c. (m)
0.074
0.108
0.147
0.216
0.314
0.488
0.074
0.633
0.600
0.248
0.230
0.411
0.164
0.164
Current (kA)
1032
1147
1390
1840
2809
4869
9840
9954
1597
-5905
-4617
-3519
0
559
Current c. (kA)
58
121
225
484
1030
2485
57
-4178
-3751
644
549
1760
280
-280
Figure 8.10. The mirror machine with the entire coil set.
Figure 8.11. The mirror machine with the quadrupolar coils, where the
axisymmetric coils have been removed.
103
8.1.8.
Results from Paper II (and XI)
In Paper II, fishbone coils have been used to create the magnetic field. The
magnetic field has the same geometry as in Paper I, but is downscaled to
1.25 T at the midplane. A number of properties of the magnetic field that
would be produced by the coil set are shown in figure 8.12. The deviation
from the ideal field (see figure 8.7) in the confinement region is shown in
figure 8.12 (a) for B ( z ) and (b) for g ( z ). The maximum deviation in B ( z )
is about 2.5 %, which partly arises from a smoothening of a somewhat poor
concatenation point between the SFLM field and the ending field at
z 8.75 m. For g ( z ), the typical deviation is below 1 % except near the
mirror end where the quadrupolar field deliberately has been made weaker to
lower the flux tube ellipticity. This does not have a significant effect on the
flute stability function W1, since the plasma pressure is low in this region.
Figure 8.12 (c) shows the B ( z ) function and (d) shows g ( z ). In the
recirculation region and the magnetic expander, the ideal field profile is only
very roughly followed. The field that would be produced by the coils has
however similar properties in this region. In figure 8.12 (e), the pressure
weighted stability function W1 in the confinement region is shown and
compared with that of the ideal field. There is a stability margin to flutes in
the low limit, since W1 ( z 12.5) 0 . In figure 8.12 (f), the flux tube
ellipticity is shown. The maximum ellipticity is about 19.4, which is
somewhat lower than in Paper I since g ( z ) has been slightly modified. The
recirculation region reduces the ellipticity to about 1 at the magnetic
expanders and makes the plasma receiving “divertor plates” circular. With a
midplane plasma edge radius of a 40 cm, the outermost plasma edge
rvc ,min ( z ) is calculated for the coil set to first order. The result is shown in
figure 8.12 (g), and a magnification of the mirror end region for rvc ,min ( z ) is
shown in figure 8.12 (h). At the mirror end ( z 12.5 ), the outermost plasma
edge radius is 87 cm. At about z 13.5 in the recirculation region a
maximum of about 93 cm is reached, which illustrates the need to expand
the vacuum chamber radius to 1 m beyond the mirror ends. At the end of the
magnetic expander, rvc ,min (18.75) 4 m.
The resulting coil set consists of 30 coils. Since the coil set is symmetric,
only the 15 coils on the positive z side needs to be specified. Those coils are
described in table 8.3. The C-14 coils are the recirculation coils and have a
negative L which gives a wrapped coil that produces a negative contribution
to g ( z ) for a positive contribution in B ( z ) . The C-15 coils are the cusp
coils which have a negative current that gives a negative contribution to
B ( z ) . The C-15 coils are circular ( L 0 ). The coils do not intersect other
coils or other parts of the device, at least not significantly (only possibly
very slightly in the structure material, which can be handled in a detailed
design).
104
Figure 8.12. The components and some properties of the magnetic field that would
be generated by the coil set. The figures show the deviation from the ideal field of
B( z ) in the confinement region in (a) and g ( z ) in (b), B( z ) in (c), g ( z ) in (d), the
W1 ( z ) stability function in the confinement region in (e), the flux tube ellipticity in
(f) and the outermost plasma edge radius rvc ,min ( z ) in (g) and (h).
105
Table 8.3. The 3D coil parameters on the z > 0 side of the midplane.
Coil name
C-01
C-02
C-03
C-04
C-05
C-06
C-07
C-08
C-09
C-10
C-11
C-12
C-13
C-14 (recirculation)
C-15 (cusp)
z(m)
0.4
1.25
2
2.75
3.5
4.25
5
5.75
6.5
7.7
9.5
11
12.45
15.05
17.4
I(kA)
787
700
700
710
762
720
680
750
1100
1640
2750
5600
11150
11200
-5070
L(m)
0.915
1.085
0.905
1.05
1.035
1.06
1.20
1.68
0.88
2
2.33
2.32
0.71
-2.98
0
r(m)
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
3
a(m)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Coil width(m)
0.209
0.197
0.197
0.198
0.205
0.200
0.194
0.204
0.247
0.301
0.390
0.557
0.786
0.788
0.530
The resulting coil system is also shown in 3D in figure 8.13, where only the
coils are shown in (a) and the vacuum chamber, fission mantle and the
coolant influx/outflux has been added in (b).
Figure 8.13. The resulting coil system in 3D, where the coils are shown in (a) and
the coils with the vacuum chamber, fission mantle and coolant influx/outflux are
shown in (b).
The significant size reduction of this coil system compared to the coil system
in Paper I is interesting to note, and is shown in figure 8.14. It is due to the
following three causes:
106
1. The overall magnetic field has been lowered.
2. The goal is to use NbTi coils, at least for most of the coils, which in
combination with the weaker magnetic field implies that a higher
current density of 2.6 kA/cm2 can be used instead of 1.5 kA/cm2.
3. The fishbone coils produce both the B ( z ) and the g ( z )
components, which makes the overall coil size smaller and allows
the coils to be organized in one single layer.
Figure 8.14. A cross sectional comparison between the size of the old coil system
and the new coil system, where the cross sectional area of the largest coil of the new
coil system is about 27% of the combined cross sectional area (quadrupolar coil +
circular coil) of the largest old coils.
8.1.9.
Results from Paper III
In Paper III, the radial dependence of the vacuum field flux tube ellipticity
(or excentricity) is examined. To first order in the long-thin approximation
the flux tube ellipticity is independent of r0 . An expression to third order is
derived in Paper III, and is given in section 4.3. These solutions are
compared to a more or less exact solution where the field lines are followed
numerically from the midplane using the fourth order Runge-Kutta method.
Two devices are examined, the SFLM Hybrid and a hypothetical device
where the g ( z ) function is much more localized. The results for the SFLM
Hybrid can be seen in figure 8.15, where the traced outermost plasma field
line is shown in (a, b) along with the vacuum chamber wall, and the
ellipticity is shown in (c, d). In general, the 3rd order solution mitigates most
of the errors from the first order approximation. For the SFLM Hybrid, the
outermost field line is 3.5 cm further out in radius than would have been
107
expected from the first order approximation (97 cm compared to 93.5 cm).
The corresponding values for the ellipticity are 19.9 instead of 19.4.
Figure 8.15. The vacuum magnetic field properties for the SFLM Hybrid. In (a, b)
the radial dependence in z of the field line that intersects the midplane at
(r 0.4,
0) is shown and compared with the vacuum chamber first wall. In (c,
d) the ellipticity for r0 40 cm is shown. In (a–d), a first order paraxial
approximation, a third order paraxial approximation and an almost exact numerical
Runge-Kutta solution is compared.
The radial dependence of the maximum ellipticity is also examined for the
SFLM Hybrid, and is shown in figure 8.16. As expected, the ellipticity
increases with radius for the examined interval, and the 3rd order solution
seems to be fairly accurate up to about r0 0.4 m.
We suspected that the ellipticity dependence on radius would go up if the
g ( z ) function was more localized in z, like for example in a tandem mirror.
Therefore, such a case was examined. For this case, much to our surprize,
the ellipticity dependence on radius was weaker than in the SFLM Hybrid.
108
Figure 8.16. The maximum ellipticity as a function of the radial Clebsch coordinate
r0 in the SFLM Hybrid. A first order paraxial approximation, a third order paraxial
approximation and a more or less exact numerical Runge-Kutta solution is
compared.
8.1.10. Results from Paper V
In this paper, the finite
corrections to the magnetic field have been
calculated for a prespecified sloshing ion distribution function in the SFLM
Hybrid. The constant pressure profiles at the midplane have been
determined. It was found that they had a square-like octupolar distortion near
the plasma edge if no specific boundary conditions were applied for the
pressure, and that they were almost circular within most part of the plasma
region. These contours are shown in figure 8.17 for different
values, and
the shape of the contours is not sensitive to
although a slight increase in
the octupolar distortion can be observed with increasing .
Figure 8.17. Contours of constant pressure at the midplane cross section for
different ß values.
109
case, and
The flux tube ellipticity has also been calculated for the finite
the results are shown in figure 8.18. It was found that the ellipticity was
increased by the plasma currents. However, the increase in ellipticity was
more pronounced near the z axis than at the plasma edge, and the total
ellipticity decreases with radius in the plasma region. Also, although the
plasma presence increases ellipticity, the outermost field line radius is even
slightly decreased with finite , which is a beneficial result.
Figure 8.18. Flux tube ellipticities for different ß values and radii in (a-d), and the
corresponding outermost plasma radius in (e-f). The radial dependence of the
ellipticity is shown in (g).
110
8.2. Radial transport and radial invariant
In Paper VII, a radial invariant that is valid for fields with small drifts
(typically vacuum fields) has been found for particle orbits in quadrupolar
mirrors that exhibit a certain symmetry. The geodesic curvature in
quadrupolar mirrors give rise to gyro center radial drifts. It is therefore here
required that the geodesic curvature along a field line is an odd function of z,
so that the radial drifts on each side of the midplane cancel. This is fulfilled
if both functions B ( z ) and g ( z ) are even functions. If this is not the case, a
collisionless radial transport will be present, which is shown in Paper IV.
8.2.1.
Results from Paper VII
The main result in this paper is the derived radial invariant I r , which is valid
in fields where the drifts are small (typically vacuum fields). To sketch the
derivation (see Paper VII for details), we first define the field line equations
to first order as
xB ( z )
1
1
( z ) x0
(8.2.1)
yB ( z )
1
2
( z ) y0 .
(8.2.2)
and
The functions B ( z ) and g ( z ) are related to
B( z )
1
B0
z
(
)
1
1
1
( z ) and
2
2
( z ) by
(8.2.3)
( z)
and
g ( z)
1
1
( z)
1
1 ( z)
2
B0
1 ( z) 1
(z)
1
2 ( z)
2
( z)
(8.2.4)
where prime denotes differentiation with respect to z. We now specialize the
calculation to the case where B ( z ) and g ( z ) are even functions and B ( z )
has a minimum at z 0. Then 1 ( z ) 2 ( z )
( z ) and the function ( z )
can be divided into its even and odd components e ( z ) and o ( z ) . We write
( z) 1
where we choose e (0)
center can be written
o
e
( z)
o
( z)
(8.2.5)
(0) 0 . The equations of motion for the gyro
dx0
dt
y0 ( s ),
(8.2.6)
111
dy0
dt
x0 ( s ),
(8.2.7)
M ds
2 dt
B( s )
2
(8.2.8)
where
2
B
m
1
(8.2.9)
0
can also be divided
and bars denote guiding center values. The function
into even and odd components, yielding
2
B
e
m
1
e
e
(8.2.10)
o o
0
and
2
o
B
o e
m
1
e
o
.
(8.2.11)
0
With cylindrical Clebsch coordinates, the equations of motion become
dr0
dt
r0
o
( s )sin(2 0 )
(8.2.12)
and
d 0
dt
e
(s )
o
( s ) cos(2 0 ).
(8.2.13)
With these equations and the assumption that e ( s ) and o ( s ) are small
(which imply small drifts), the radial invariant can be defined as
I rN (x, v ) r0
2 x0 y0
r0
N
hn ( , ) cos n
n
cos
/4
n 1
(8.2.14)
where
hn ( , )
2
n
/4
d
0
o
s ( ) sin
n
/4
,
(8.2.15)
is a bounce time variable which satisfies ds / d
s,
is the time for one
bounce back and forth the mirror and N is the number of Fourier components
to include in the invariant. Eqs. (8.2.11 - 8.2.12) give a condition for locally
omnigenius equilibria ( r0 const ), yielding
112
o e
1
e
o
0
(8.2.16)
which can be satisfied for any choice of B ( z ) .
To illustrate the invariance of the radial invariant, it is shown in figure 8.19
for different N. As can be seen, as N become large, the invariant is almost
constant.
Figure 8.19. Variations of Nth order radial invariants versus the arc length s(t) over a
longitudinal bounce for N = 0, 1, 2 and 500. The curves are for a long-thin system
with some geodesic curvature. The plot for the guiding center radial coordinate r0 (t )
reveals a 2.5 cm banana width. The variations of the successive approximations for
the radial invariant decreases with N, approaching diminishingly small values for
large N.
An observation in Paper VII is also that the parallel invariant J to leading
and . Therefore, it is not sufficient to use J to
order is a function of
model a radial pressure profile. For this, the radial invariant can be used.
8.2.2.
Results from Paper IV
The purpose of Paper IV was to point out that quadrupolar mirror machines
must be symmetric in the sense that B( z ) and g ( z ) are even functions for a
radial invariant to exist. This is in some sense known for a restricted group
of mirror researchers, see for example Ref. [71] (Appendix D), but perhaps
not widely known. If this symmetry is not present, there will be a
collisionless radial transport (or leakage). This can intuitively be seen from
the geodesic field line curvature. If there is a geodesic field line curvature,
indicating that the magnetic field is not omnigenius in the local sense, there
will be radial drifts. If the symmetry is fulfilled, the radial drift on one side
of the mirror will be cancelled by an opposite radial drift on the other side of
the mirror, if the azimuthal drift in one bounce is small.
To calculate the orbits in this paper, two methods are applied. The main
method is to just follow the particle with a fourth order Runge-Kutta method
and calculate the Lorentz force. The magnetic field was calculated in a grid
of (100 100 300) points and quadratic interpolation was used to calculate
113
B at a specific point. The method turned out to be very accurate but fairly
slow. The second method was to use the equations of motion from Paper
VII, and this was used to check the first method. Due to the sensitivity at the
turning points, this method was actually (to our surprise) not much faster
than the first one if a good accuracy was required and equidistant steps were
used.
A simple coil configuration with 8 fishbone coils was varied to create fields
with fluctuating field line curvatures. Four different configurations were
tested:
1. A fully asymmetric mirror where one side was axisymmetric and the
other side quadrupolar.
2. A symmetric quadrupolar mirror.
3. A mirror with about 15% asymmetry in g ( z ) .
4. A mirror with about 1.5% asymmetry in g ( z ) .
For the asymmetric cases, there was a slight asymmetry in B( z ) as well. The
results for particle orbits can be seen in figure 8.20. The particle orbits can
be divided into quadrant locked orbits, that tend to stay in their quadrant, and
encirculating orbits, that encircle the z axis. This is determined by the
azimuthal drift. The encirculating orbits in asymmetric fields will be
oscillatory in r0 . For quadrant locked particles in asymmetric fields, the
radial drift will be either inwards or outwards dependent on which quadrant
they are in. For quadrant locked particles, there is a risk for a net radial drift
and thereby a collisionless radial transport.
114
Figure 8.20. In (a), drift trajectories in the non-symmetric mirror field for 16
deuterium ions that are launched at the midplane with different initial positions in
the x0 y0 plane. The small circles mark the initial positions for each ion in (a-c). The
ions are tracked for about 100 bounces back and forth the mirror in (a) and the two
particles that are still confined after 100 bounces are traced for up to 400 bounces in
(b). As can be seen, all of these particles are lost in less than 400 bounces for the
non-symmetric mirror. In (c), an ion with identical velocity and position as in (a)
was launched in the symmetric mirror and traced for about 200 bounces. As seen in
the magnification in (d), the trajectory is almost closed and the radial drift after one
lap around the z axis is only about 0-0.1 mm. In (e) and (f), the orbit is shown for the
error case where the coil parameter L is lowered by 20% on one side of the mirror,
and in (g) and (h) the corresponding orbit for the case where L is lowered by 2% on
one side is shown. In (e-h), the orbit appears to be drawn with a thick line, which is
due to the neoclassical “banana” orbits that the particle gyro center follows.
115
In general, one can see that for the fully assymetric mirror, the radial drifts
are totally unacceptable. For the 15% asymmetry, they seem to be too large
(unless a fast azimuthal particle drift is introduced, by for example finite
or a strong electric field). For the 1.5 % asymmetry case, the drift is small
and most likely acceptable. Mirrors with asymmetry smaller than this will
most likely not suffer from problems with radial transport of this type. For
the symmetric case, the radial drift is very small or totally abscent as
expected.
The calculations in this paper have been made on a vacuum field and the
electric field has been omitted. With a non-neglible finite
or a strong
radial electric field, they would no longer be valid, since both these effects
will give a fast azimuthal drift. A radial electric field is always present (if the
electron temperature is not zero) due to the ambipolar potential. This electric
field can be further increased with potential plates at the end plates.
8.3. Discussion
8.3.1.
Discussion on coil accuracy requirements
When attempting to find an accurate solution for a coil design, the question
of how accurate the magnetic field needs to be arises. For a tokamak, the
ripple caused by the discrete toroidal coils give rise to increased
neoclassical-type radial ion losses. These are claimed to be (at that time) 1-2
orders of magnitude greater than the ordinary neoclassical losses in tokamak
fusion D-T plasmas in Ref. [107]. However, no such effects are expected in
a mirror. A heavy ripple of more than 1 % (top to bottom) in the
axisymmetric field is claimed to give rise to ballooning modes in tandem
mirrors in Ref. [108]. Later theoretical achievements by Newins and
Pearlstein [95], suggest that the FLR effects are so strong that theoretical
MHD ballooning instabilities would not be important for tandem mirror
experiments. In any case, ballooning modes are expected to be less important
for a single cell minimum-B mirror of the SFLM Hybrid type than for a
tandem mirror, since the regions of bad curvature are smaller. Also, the
ripple can be substantially reduced by using ferromagnetic inserts if such a
measure would be required [108]. Evidence of any serious problems with a
ripple of the size presented in this paper for a mirror machine has not been
found. Concerning the flute stability, the impact of the ripple is small and the
resulting field flute stability is checked in Paper II. Ripple effects on the flux
tube ellipticity are negligible. The ripple will cause particle drifts, but they
will be of oscillatory nature as long as the field is symmetric in the sense of
Papers IV, VII, and the banana widths from the ripple is expected to be
small.
116
8.3.2.
Discussion on coil calculation limitations
The coil calculations have been made using a rough scaling law for the coil
dimensions. This is not fully accurate. To produce real superconducting
coils, a considerably deeper analysis must be done. The coils must be
properly dimensioned to hold (both the whole coils and the internal
structure) for the very strong magnetic forces. This requires numerical
modelling of the coils. Also, the magnetic field must be kept below the
critical magnetic field for the superconductors everywhere within the
superconducting coils. This has not been checked. A detailed check would
require that the internal structure of the coils must be decided. Implications
of this are that there is an uncertainty of the position of the filamentary line
currents due to the uncertainty in coil sizes. The consequence will be a need
for recalculation of the currents, which is straightforward. There is also an
uncertainty if the geometric constraint that coils shall not intersect will be
fulfilled when the structure material is properly dimensioned. This can be
handled by for example changing the coil cross section shapes or moving
some coils. Also, the proper material of each coil has to be decided from the
maximum magnetic field strength. The aim in this study was to show that it
seems realistic that a coil set can be created which generates the magnetic
field with satisfying accuracy without violating the geometric constraints. If
problems should arise, there are ways to handle them. A somewhat lower
mirror ratio or alternatively a somewhat more long-thin configuration would
allow for coils for the mirror field to be constructed. In summary, coil
construction for the fusion-fission mirror system seems realistic.
8.3.3.
Discussion on finite ß issues
The constant pressure contours of the midplane become square-shaped with
finite ß in the SFLM Hybrid. This is in contrast to the result for tandem
mirrors calculated by Pearlstein et al. [72], where they found a diamondshaped distortion. Their distortion was however caused by the parallel
current. In the case with the SFLM Hybrid, the parallel current is small and
does not have a noticeable impact on the flux surface shapes. Other effects,
such as a small octupolar component in the vacuum magnetic field that come
from the coil system, contribute in the SFLM Hybrid case.
8.3.4.
Discussion on radial invariant, E and low ß limit
The derivation of the radial invariant is only valid when the azimuthal drift is
so small that the gyro center roughly stays at the same 0 value during one
bounce back and forth the mirror. With a finite , or strong electric field,
117
this is no longer the case. For finite , the drift from the gradient in B is
typically much larger than the curvature drift. The magnetic drift is
BB
B
3
q
B
vd ,m
mv 2 B
q B2
(8.3.1)
/(2 0.4)
which is
If B is assumed to be perpendicular to B, B
. Preliminary numerical calculations for a 7 keV
considerably larger than
deuterium ion in the SFLM Hybrid with 0 0.4 show that this drift
represents a change in 0 of the particles position in the order of 15° to 60°
in one single bounce back and forth the mirror, depending on r0 for the
particles position. For such high
values, the derivation of the radial
invariant is no longer accurate (although there may still exist a radial
invariant). The drift is almost proportional to , so typical drift velocities in
the SFLM Hybrid can be approximated from this value.
For a radial electric field, an approximation can be made. For the simulated
particle above, the time for one bounce back and forth the mirror is about
8 10 5 s. If we assume that the ambipolar potential is 5Te (where Te is in
eV) at the center and drops to zero just outside the plasma edge, the radial
electric field roughly becomes
Er
5Te
r
5Te
12.5Te
0.4
(8.3.2)
for the SFLM Hybrid where it is assumed for simplicity that Er is constant
in the plasma region (which is not true). The electric drift is
vd , E
E B
B2
E
B
12.5Te
2
6Te
(8.3.3)
where it is assumed that E B and B 2T. If we assume that Te 500 eV,
the drift during one bounce is 6 500 8 10 5 24 cm. At a radius of
10 cm, this represents an angular displacement of about 140° in one single
bounce. From this approximation, a reasonable validity limit for the
derivation of the radial invariant would be about Te 50 eV in the SFLM
Hybrid.
The drift from finite
and from the electric field are in different directions
for ions and will to a large extent cancel if they are in similar magnitude.
Then, the validity must be approximated from the sum of these drifts. The
E B drift is independent of a particle’s speed if E remains unchanged, and
the drift in one bounce will decrease with increasing energy of the particle.
For the magnetic drift, the situation is opposite and the drift during one
bounce will increase linearly with the velocity of the particle.
118
8.3.5.
Comparing axisymmetric and quadrupolar drivers
The other mirror-based hybrid systems that have been proposed recently use
axisymmetric mirrors as drivers [27][28], see also [109]. There are several
advantages with axisymmetric systems:
1. The coil system is considerably simpler. All coils are circular, which
also means that the magnetic field can be made higher since the
strain in the coils would be smaller. This would also reduce the cost
of the coil system.
2. The mirror ratio can be made higher, which is beneficial for axial
confinement.
3. The flux tube have a more convenient shape (axisymmetric), which
in the SFLM Hybrid would implicate that the plasma radius could be
about 80-85 centimeters (since there are no wide quadrupolar fans)
and the plasma volume would roughly be quadrupled. This means
that the magnetic field or the plasma
can be lowered while the
neutron production would remain the same, compared to the SFLM
case. This would also reduce radial transport, since each particle has
a longer radial distance to pass before reaching the limiters where
the particles are lost.
4. Axisymmetric devices are locally omnigenius, even to arbitrary ,
which eliminates all types of neoclassical transport. This may be
hard to accomplish with finite
in a quadrupolar device.
Stabilization of axisymmetric devices is however not as well explored as for
quadrupolar devices, and for the time being we avait more results from these
experiments. Explicity, there might be problems of getting a high electron
temperature since axisymmetric devices are partly stabilized by having
plasma in the magnetic expanders. In the SFLM Hybrid, the density in the
expander should be as low as possible to increase the electron temperature.
The electron temperature is probably the single most important parameter for
a mirror fusion driver, since it at least for the low electron temperatures
reached so far in mirrors directly controls the fusion Q and thereby the
amount of heating required.
119
120
9. Conclusions
Conclusions from the work in this doctoral thesis are:
1. A steady-state fusion-fission hybrid reactor with a single cell minimumB mirror as fusion driver has been proposed in Paper VI. The fusion
device is based on the SFLM magnetic field. There are currently no
major showstoppers known for this project, although a lot of issues have
not been examined in detail yet. A concern is the electron temperature,
and reactor safety needs to be addressed more in detail, especially the
consequences of a core meltdown.
2. The fission to fusion power multiplication in a mirror hybrid can be
made very high, about 150 with an assumed keff of 0.97 since the
geometry can be arranged so that almost all fusion neutrons enter the
fission mantle. The power multiplication for a mirror hybrid could be
substantially higher than what can be achieved for tokamak hybrids. A
mirror machine with a power multiplication of 150 have a strongly
relaxed requirement for the plasma confinement and even a thermal
fusion Q = 0.15 is expected to be good enough for energy production
compared to Q = 15 for a pure fusion reactor.
3. The SFLM magnetic field cannot constitute the entire mirror field in a
mirror machine, and has to be concatenated with some other field before
the end of the mirrors. This is necessary, since the z derivative of the
field on the z axis dB / d z must be zero at the mirror ends, and dB / d z
in SFLM is monotonically increasing with z . Such a concatenated field
has been selected in Papers I-II that has margins to flute stability in the
low limit, a maximum flux tube ellipticity of about 20 and sufficiently
soft gradients for a mirror ratio of four. The concatenation point is at
z 0.7c where c is the axial scale length of the device.
4. The further away the coil system is from the plasma, the harder it is to
produce large relative gradients ln B / z in the magnetic field. This
limits the fields that could be realized for a fusion-fission reactor, since
the fission mantle including shielding is in the order of 1.2 m thick (in
principle the fusion neutron shields in a fusion reactor would be of
almost the same thickness). This property also sets a minimum length of
the machine for a specific magnetic field.
5. A new type of coil has been invented, which we call the fishbone coil.
They have practical geometrical shapes and can be inserted into each
121
6.
7.
8.
9.
10.
11.
12.
13.
14.
122
other like a pile of glasses which offers great flexibility in coil design.
They contribute to both the B ( z ) and g ( z ) functions, which means that
only one layer of coils is required, and the contributions to B ( z ) and
g ( z ) can almost independently be controlled by the coil parameters.
They also have soft curvature which is beneficial for superconducting
coil design.
Two different coil systems have been found that reproduces the vacuum
magnetic field with satisfying accuracy and within the geometric
constraints for the SFLM Hybrid. The preferred version is the version
based on the fishbone coils.
There is sufficient space available inside the coil system to shield the
coils from neutrons.
The vacuum field ellipticity increases slightly with radius. For the SFLM
Hybrid, the ellipticity increases from 19.4 to 19.9, which increases the
outermost field line maximum radius by 3.5 cm.
A magnetic expander region with a recirculation region has been added
to the mirror machine at each end. By this, several advantages are
gained. The heat load from end losses can be distributed over an almost
arbitrarily large area, some extra stability to flutes is gained and the
electron temperature may rise due to density depletion in the ends.
The heat and neutron loads on both the first wall and the “divertor
plates” at the magnetic expanders are predicted to be tolerable. This is in
contrast to ITER (and supposedly most compact fusion devices), where
the divertor plates is only expected to last for less than 3 months of
steady-state operation or so.
A radial invariant in symmetric quadrupolar mirrors has been identified
that can be used to model radial pressure density variations.
It has been emphasized that assymetric quadrupolar mirrors have a
collisionless radial transport. For mirrors with a pronounced asymmetry,
this radial transport will be unacceptably large.
If a finite
is present in the SFLM Hybrid, the flux tube ellipticity is
increased. However, this increase in ellipticity is more pronounced near
the z axis, and at the plasma edge the increase in ellipticity is moderate.
A beneficial property is that even though the ellipticity is increased at
the plasma edge, the outermost field line on a flux surface have a smaller
maximum radial extension in the finite
case compared to the vacuum
field.
At the plasma lateral boundary (the plasma edge), the flux surface cross
sections have a square-shaped distortion at the midplane with finite .
Near the z axis, they are circular.
10.
Future studies
There are a lot of studies remaining in the SFLM Hybrid project and related
research areas, and some of them can be done by the author, others have to
be done by people with other competences. There is currently no long-term
project plan on which a decision has been made. However, the following
tasks could be done (in no specific priority order):
1. Find a proper financiation for the project. This is the single most
important point.
2. Specify how the plasma should be fed with deuterium and tritium, and
investigate the impact on the electron temperature etc. of such feeding.
3. Predict the electron temperature and the overall fusion Q.
4. Try to determine the proper equilibrium and magnetic field with ICRH
heating and optimize the coil system for such a realistic case. Also, here
the neutron production should be calculated more accurately and the
required 0 to produce the neutrons should be determined.
5. Investigate the need of and maybe design a (hopefully weak) correction
coil system that can modify the magnetic field profile. It is not expected
that the equilibrium can be determined with such accuracy that the
optimal field can be entirely determined in advance.
6. Write an overall project specification that includes all required systems.
Such papers can be published in for instance Journal of Fusion Energy.
7. Extend reactor safety studies.
8. Perform burnout studies for the fission core. So far, impact on burnout
has only been extrapolated from other systems (FTWR).
9. Examine the impact of the reactor poison Xe-135. This impact is
expected to be low for a fast neutron spectrum, but may still be finite.
10. Perform a detailed internal coil modelling, taking strain, cooling and
maximum magnetic field into account.
11. Continously evaluate the progress with axisymmetric mirrors. If
axisymmetric mirrors would prove to be well working with respect to
stability,
limits and electron temperature, they would most likely be
superior to quadrupolar devices.
12. Analyse ballooning stability properly.
13. Verify that fusion hybrids can be competitive against ADS systems and
critical fast fission reactors for burning actinides.
123
14. Examine the particle orbits for symmetric quadrupolar fields with a
finite , and investigate the impact of resonant and possibly stochastic
neoclassical transport.
15. Design of the magnetic field and overall geometry of the magnetic
expanders.
16. If the theoretical results seem promising, plans for building an
experiment should be made.
124
11.
Summary of papers
Paper I
Field and coil design for a quadrupolar mirror hybrid reactor
This paper describes a version of the magnetic field and coil design for the
SFLM Hybrid project, where the midplane magnetic field was 2 T. Magnetic
field properties are derived using the paraxial approximation. A magnetic
field, based on the SFLM field concatenated with another field to end the
mirrors, is constructed, optimized for flute stability (using the flute stability
criterion), ellipticity and field smoothness. A coil set is modelled
theoretically to produce the constructed magnetic field. The author has made
most of the work in this article. The paraxial formulas are derived by O.
Ågren.
The paper is published in Journal of Fusion Energy 2011.
Paper II
A Compact Non-Planar Coil Design for the SFLM Hybrid
This paper describes the version of the magnetic field and coil design for the
SFLM Hybrid project, where the midplane magnetic field was set to 1.25 T.
The magnetic field is in other aspects similar to that in Paper I. Magnetic
field properties are derived using the paraxial approximation. A new 3D coil
type, the fishbone coil, has been designed and the vacuum field can be
generated with satisfying accuracy with 28 fishbone coils and 2 circular cusp
coils at the magnetic expanders. The fish-bone coil has soft curvature and is
believed to be well suited for superconducting coils. The author has made
most of the work in this article.
The paper is published in Journal of Fusion Energy 2012.
Paper III
Vacuum Field Ellipticity Dependence on Radius in Quadrupolar Mirror
Machines
This paper investigates the radial dependence of the vacuum flux tube
excentricity (ellipticity). To first order in the long-thin approximation, the
ellipticity is independent of radius. A third order expression is derived and is
compared with numerical “exact” solutions. For the SFLM hybrid vacuum
field, the ellipticity increases slightly with radius and the outermost field line
125
is 3.5 cm further out than expected from the first order calculation. The
author has made most of the work in this article.
The paper is published in Journal of Fusion Energy 2012.
Paper IV
Radial Confinement in Non-Symmetric Quadrupolar Mirrors
This paper investigates the effects of non-symmetric quadrupolar and
axisymmetric fields in quadrupolar mirrors. The aim with the paper is to
clearly point out that if there is a non-symmetry of the geodesic curvature
with respect to the midplane, particles are not confined. The author has done
most of the work in this paper.
The paper is published in Journal of Fusion Energy 2012.
Paper V
Finite ß corrections to the magnetic field in the SFLM Hybrid
In this paper the finite
effects on the magnetic field is investigated for the
SFLM Hybrid. The flux tube cross sections at the midplane have been
calculated as well as the modification of the flux tube ellipticity from the
finite
. The parallel current has also been investigated. The author has
done most of the work in this paper.
The paper is a manuscript intended for journal publication.
Paper VI
Studies of a straight field line mirror with emphasis on fusion-fission
hybrids
This paper is a concept article, proposing a fusion-fission minimum B mirror
hybrid reactor using the Straight Field Line Mirror (SFLM) as a fusion
driver. In the paper, the strong fusion-fission power amplification is pointed
out. The paper reviews the SFLM concept, discusses radio frequency
heating, discusses Monte Carlo simulations and discusses results from coil
calculations. Also, a scenario for increased electron temperature is outlined
and advantages with mirror geometry compared to toroidal geometry are
emphasized. The article is written mainly by Olov Ågren with assistance
from Vladimir Moiseenko and Klaus Noack. The author has contributed with
the coil calculations. Since this is a concept article, the author has also been
involved in overall discussions of the project, especially regarding the size of
the device.
The paper is published in Fusion Science and Technology in 2010.
126
Paper VII
Radial Drift Invariant in Long-Thin Mirrors
In this paper, gyro center orbits for long-thin quadrupolar mirrors are
examined and a radial invariant is derived. It is also shown that the parallel
invariant J is insufficient to model radial variations in pressure. The radial
invariant can be used for this purpose. Most of the work in this article is
made by Prof. Olov Ågren. The author has contributed with numerical
calculations of particle orbits.
The paper is published in The European Physical Journal D in 2012.
Paper VIII
Neutronic model of a mirror based fusion-fission hybrid for the
incineration of the transuranic elements from spent nuclear fuel and
energy amplification
This paper presents the first fission mantle design for the SFLM Hybrid
project. Monte Carlo simulations suggest that the energy amplification of the
fission mantle would be about 150 at BOC. Reactor safety for LOCA and
local boiling of coolant is also addressed. The major contribution was done
by Klaus Noack (in particular all Monte Carlo simulations) with minor
contributions and proposals from Olov Ågren and Vladimir Moiseenko (on
some overall parameters and plasma questions). The author’s contribution to
this paper is small. The author has been involved in the axial sizing of the
device and has been working with the text.
The paper has been published in Annals of Nuclear Energy in 2010.
Paper IX
Coil design for the Straight Field Line Mirror
This paper presents early work in coil design, and is superseded by Paper II.
Two applications of mirror machines are addressed, a small material science
experiment and a short-fat fusion-fission device for transmutation. For the
material science experiment, a simple coil set is found, although with serious
drawbacks not realized when the article was written. For the transmutation
device, a coil set could not be found. The author has done most of the work
in this paper. Some of the formulas are derived by O. Ågren.
The paper was presented by the author as a poster presentation at the
conference Open Systems 2008 in Daejon, South Korea. The paper is
published in the (non-peer-reviewed) conference proceedings in Fusion
Science and Technology 2009.
127
Paper X
Theoretical field and coil design for a single cell minimum-B mirror
hybrid reactor
This paper describes a magnetic field and coil design, which is now obsolete
and superseded by the magnetic field and coil set given in Paper II. The
magnetic field is optimized for flute stability (using the average minimum-B
criterion), ellipticity and field smoothness using a spline representation for
the magnetic field components. A coil set is then produced theoretically that
reproduces that field. The author has made most of the work in this article.
The paraxial formulas are derived by O. Ågren.
The paper was presented by the author as a poster presentation at the
conference Open Systems 2010 in Novosibirsk, Russia. The conference
papers were peer-reviewed and this paper is published in the conference
proceedings in Fusion Science and Technology 2011.
Paper XI
Coil System for a Mirror-Based Hybrid Reactor
In this paper, the coil sets in Paper I and Paper II are compared and
discussed. The author has made most of the work in this article.
The paper was presented by the author as a poster presentation at the
FUNFI conference 2011 in Varenna, Italy. The conference paper was not
peer-reviewed and this paper is published in the conference proceedings in
AIP conference proceedings in 2012.
Paper XII
Safety and Power Multiplication Aspects of Mirror Fusion-Fission
Hybrids
In this paper, the fission mantle design is modified in two ways: shielding
for the coils is added and small annuli (control rods) to be able to keep keff at
constant level are added. Also, safety cases are examined for a vertical
orientation of the device and a safety case with water cooling is examined.
Klaus Noack has made most of the work in this article. The author has
assisted in specifying coil data and material. The author’s contribution to this
paper is small.
The paper was presented by Klaus Noack at the FUNFI conference 2011 in
Varenna, Italy. The conference paper was not peer-reviewed and this paper
is published in the conference proceedings in AIP conference proceedings in
2012.
128
12.
Sammanfattning
För att förhindra att en ny energikris uppstår och för att minska
skadeverkningarna på vår miljö, behöver de fossila bränslena – som idag står
för mer än 80 % av världens energiproduktion – ersättas med alternativ. De
alternativ som finns tillgängliga idag är förnybara energikällor och kärnkraft.
Kärnkraften ger en stabil basproduktion av el, vilken har stora möjligheter
att stå för en större del av den framtida elproduktionen än vad som sker idag.
Kärnkraften är dock – såsom andra energislag – behäftad med ett antal
problem av mer eller mindre allvarlig karaktär. Allvarligast är troligen
sambandet med kärnvapenspridning samt kanske risken för reaktorolyckor.
Dock utgör även förvaringen av uttjänt kärnbränsle ett problem. Det finns ett
antal radioaktiva ämnen i kärnbränslet som måste slutförvaras på ett säkert
sätt, och en förvaringstid på 100 000 år eller mer krävs för att
radioaktiviteten i bränslet skall sjunka till sådana nivåer att det kan betraktas
som säkert att lämna det i berggrunden. De ämnen som ger allvarligast
konsekvenser för levande varelser efter några hundra års förvaring,
transuranerna, går dock att bränna bort genom s.k. transmutation, vilket för
transuraner innebär fission. För att transmutation skall fungera krävs dock en
annan typ av reaktor än dagens lättvattenreaktorer. För att bränna americium
krävs med stor sannolikhet s.k. drivna system vilka måste drivas av en yttre
neutronkälla. Möjliga sådana neutronkällor är partikelacceleratorer och
fusionsreaktorer. Med drivna system finns också möjligheter att skapa fissilt
kärnbränsle ur fertila isotoper, s.k. breeding. Genom sådan teknik kan
bränsletillgången för kärnkraft bli mycket stor, och räcka för
världsproduktion av elektricitet i 10000-tals år genom att breeda U-238 och
thorium. Även s.k. snabba reaktorer eller breederreaktorer är tänkbara för
detta ändamål, och det är möjligt att drivna system får svårt att konkurrera
med dessa för breeding.
Fusionskraften har länge setts som ett möjligt alternativ för framtidens
energiproduktion. Tillgångarna på energiråvarorna är mycket stora. Dock har
problemet med att få fusionsreaktorer att leverera mer el än de konsumerar
visat sig vara mycket svårare än väntat, och det kommer inte att finnas någon
fusionskraft kommersiellt tillgänglig på åtminstone 30 år (förmodligen
betydligt längre). Det är dock möjligt att fusionsforskningen kan lösa ett
annat problem än det i förväg tänkta genom att fungera som neutronkälla till
drivna system och därmed möjliggöra transmutation och breeding med goda
129
reaktorsäkerhetsmarginaler. I denna doktorsavhandling presenteras ett
koncept på en fusion-fissionsreaktor baserat på en s.k. spegelmaskin. I den
rörformade spegelmaskinen innesluts fusionsplasmat m.h.a. den magnetiska
spegeleffekten, och neutroner skapas genom fusionsreaktioner.
Spegelmaskinen omges av en fissionsmantel vilken har en
neutronmultipliceringsfaktor som är mindre än ett, typiskt 0,97.
Fusionsneutronerna som skapas åker in i den omgivande fissionsmanteln och
genererar en kaskad av fissionsreaktioner. Energimultipliceringen som fås
med fissionsmanteln blir i storleksordningen 100-150 ggr. Därför blir
energiproduktionskravet för fusionsreaktorn bara 1/100 jämfört med en ren
fusionsreaktor. Detta krav kan flera olika typer av fusionsapparater komma
att kunna klara av. En av dessa är spegelmaskinen. Projektet SFLM hybrid
bedrivs för att teoretiskt påvisa att det verkar vara möjligt att bygga en
hybridreaktor baserad på en encellig spegelmaskin.
I denna doktorsavhandling har det huvudsakliga arbetet varit att beräkna
strömspolar för att skapa magnetfältet för SFLM hybrid. Olika
magnetspolsdesigner med supraledande magnetspolar har tagits fram. En ny
typ av spole har uppfunnits, vilken kallas fishbone coil och är lämplig att
använda för att producera magnetfältet till enkelcellsspegelmaskiner av
kvadrupoltyp. De beräknade spoluppsättningarna återskapar magnetfältet
med tillräckligt god noggrannhet och uppfyller de preliminära utrymmeskrav
som angetts. Den senaste designen är med stor sannolikhet möjlig att
realisera med befintlig teknik. Det radiella beroendet av ellipticiteten för
vakuumfluxrören i spegelfältet har också undersökts, och magnetfältet från
plasmat har beräknats. En viktig slutsats som kan dras beträffande
magnetfältet från plasmat är att även om plasmaströmmarna ökar
ellipticiteten för fluxrören i spegelmaskinen, så kommer fluxrören inte att få
större radiell utsträckning.
I övrigt har rörelsekonstanter och partikelbanor i spegelmaskiner behandlats,
där en radiell invariant har identifierats.
130
13.
Acknowledgements
First of all, I would like to thank my supervisor Prof. Olov Ågren for letting
me do PhD studies as Uppsala University, and for the support and guidance
in my work. Without him, this work would not have been possible. He is
also greatly acknowledged for his cheerful company during these years, and
for sharing his both broad and deep knowledge in physics and other various
subjects. I would also like to thank my assistant supervisor Dr. Vladimir
Moiseenko for the support, although the physical distance to Ukraine makes
the contacts less frequent. Vladimir is also greatly acknowledged for his
contributions to the project with his expertise in ICRH heating and fusion in
general. Prof. Klaus Noack is also acknowledged. He has a great expertise in
fission and hybrids, and the project would be less deep without his
contribution. Henryk Anglart is also acknowledged for providing knowledge
on liquid metal cooling. Jan Källne is acknowledged for arranging and
taking initiative to the successful FUNFI conference, the first fusion-fission
conference in a long time in Europe, in Varenna, Italy. I would also like to
thank my 3:rd supervisor Ladislav Bardos. The head of the department, Prof.
Mats Leijon, is acknowledged for support. The administrative staff at the
department is also acknowledged.
The Swedish institute is acknowledged for financing the two conference
visits to OS-2008 and OS-2010, and for financing my assistant supervisor
Dr. Vladimir Moiseenko with a grant. Liljewalchs resestipendium is
acknowledged for travel funds.
My office room mates Johan Abrahamsson and Emilia Lalander, as well as
my former office room mates Martin Ranlöf and Johan Lidenholm are
acknowledged for the great company. Johan Abrahamsson is especially
acknowledged for countless long discussions on electromagnetism,
pedagogic issues and a variety of other subjects. He is also acknowledged for
helping out with 3D images for this thesis. Mårten Grabbe, Linnea Sjökvist
and Kiran Kumar Kovi are acknowledged for the joyful teaching experiences
that we have had together. The rest of the staff on the division for electricity
is also acknowledged for their company during these years.
Last but definitely not least my family is greatly acknowledged. My
wonderful wife Frida should be greatly acknowledged for all love and
131
support and for taking the watch shift during the early mornings, late
evenings and weekends in which large portions of this doctoral thesis were
produced. I would also like to thank my soon 4 year old daughter Hilda for
letting her father sleep much more during the production of this thesis than
during the production of the licentiate thesis. My twin brother Johan should
be greatly acknowledged for the enthusiasm he induces, and my mother
Inger for taking a few child watches and providing support despite of the
relatively large distance to Gothenburg. Acknowledgements also go to the
rest of the family, my sisters Annica and Ulrika and my father Gary. My
parents-in-law, Lennart and Kerstin, are also greatly acknowledged for all
the support that they provide despite the long distance to Ömmesala city.
132
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