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Fusion energy : burning questions
Jakobs, M.A.
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Fusion Energy - Burning
Questions
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. F.P.T. Baaijens, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op maandag 14 november 2016 om 16.00 uur
door
Merlinus Ambrosius Jakobs
geboren te Eindhoven
Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de
promotiecommissie is als volgt:
voorzitter:
1e promotor:
copromotoren:
leden:
prof.dr. K.A.H. van Leeuwen
prof.dr. N.J. Lopes Cardozo
dr. R.J.E. Jaspers
dr.ir. L.P.J. Kamp
prof.dr.ir. D.M.J. Smeulders
Prof.Dr. D. Reiter (Heinrich Heine Universität Düsseldorf)
dr. D.J. Ward (Culham Centre for Fusion Energy)
prof.dr.ir. B. Koren
Het onderzoek dat in dit proefschrift wordt beschreven is uitgevoerd in
overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening.
The witches’ ride to the devil’s castle,
where we meet only ourselves, ourselves, ourselves. . .
Dag Hammarskjöld
Waymarks
A catalogue record is available from the Eindhoven University of Technology
Library.
Jakobs, Merlijn
Fusion Energy - Burning Questions
Eindhoven: Technische Universiteit Eindhoven, 2016.
ISBN: 978-90-386-41-63-8
NUR 926
Cover:
Original image ‘The Wizard’ CC BY 2.0 by Sean McGrath
Solar image by ESA/NASA/SOHO
Photo montage by SuperNova Studios
Typeset by the author using LATEX 2ε .
c 2016 Merlijn Jakobs
i
Summary
Nuclear fusion is the process in which two atomic nuclei are joined together to
form a heavier one, thereby releasing a large amount of energy. It is the energy
source of all stars in our universe. Its application as an energy source on earth
would have several appealing properties like a virtually inexhaustible fuel, inherent
safety and the absence of long-lived radioactive waste. It is therefore an attractive
candidate to contribute to the world energy supply.
Currently the first power producing fusion reactor ITER is under construction
in southern France, and, if successful, a first generation of electricity producing
demonstration reactors is foreseen to follow in the 2040-2050 time frame. Present
day fusion reactors require external heating power to achieve the high temperatures
needed for fusion, but energy-producing reactors will have to rely (to a large extent)
on self-heating by the alpha particles that are produced in the deuterium-tritium
fusion reaction.
A fusion reactor will therefore ’burn’, much like an ordinary wood-burning
stove. You fill it with fuel, kindle it (i.e. inject heat until the ignition temperature
is reached) and once ignited the system will find an equilibrium ’burn’ temperature.
The only thing the operator has to do is to regularly add new fuel and remove
the ash (i.e. the helium that is produced in the fusion reaction). This thesis deals
with the properties of these burn equilibria, what determines their fusion power
and position in the operational space of the reactor, and how the system reacts to
a perturbation of its equilibrium state.
There are several parameters that govern the burn equilibria in a burning
plasma. One of the most important is the energy confinement time τE , a measure
for how fast energy is lost from the plasma. Because it is difficult to calculate
the energy transport in a fusion plasma from first principles, often scaling laws
are used which express the energy confinement time in engineering or physics
parameters. We have found an expression relating the electron density ne at the
operating points to the temperature, by eliminating τE from the equations using
such a scaling law.
We showed that the so-called burn contours, i.e. the contours in the operational
space of the reactor spanned by the plasma density and temperature, are exactly
ii
Summary
the same for all reactors, apart from a normalisation factor of the density which
contains the design values of the reactor, such as its dimensions and magnetic field
strength. This finding implies that the results of the analyses of the burn equilibria
are generic, i.e. are of application to any reactor design that follows the same τE
scaling.
One of the salient results of the analysis is that, for a given reactor, the power
output will generally not increase if the energy confinement is improved. Good
confinement - one of the central goals of fusion research - is still a highly desirable property as it allows smaller reactors to ignite and burn, but in the existing
conceptual power plant designs an improvement of confinement does not bring
any benefit. This also means that the fusion output power of such a reactor will
respond only weakly to (small) changes in τE , disqualifying it as a useful control
parameter. However, reducing τE too much, say by 30% or so, will quench the
reactor.
This result is directly connected to a second parameter that has a big influence on the operating points of a reactor, the ratio between energy and particle
confinement time ρ = τp /τE . Generally, energy and particle transport are linked,
which would result in ρ ≈ 1. However, particles that hit the wall can return to the
plasma, but they lose their energy in the process. This is called (edge) recycling
and is the main reason that ρ is expected to be between 5 and 10 in a reactor.
The value of ρ determines the accumulation of helium ash in the plasma, and the
fusion power output reacts strongly to variations in ρ. This makes it a candidate to
control the fusion power of a burning reactor, if a means can be found to effectively
change the value of ρ, for instance by changing the rate at which particles are
pumped from the reactor exhaust. It should be kept in mind, however, that the
efficiency of the reactor is highest at low values of ρ (say < 5), while the burn
can become unstable when ρ nears 10 (as we shall see) and no burn is possible for
ρ > 15.
This would suggest aiming for a high value of ρ, but it is not that simple unfortunately. A fusion reactor needs to breed the tritium it consumes from lithium, as
tritium does not occur naturally on earth. The tritium breeding ratio, the amount
of tritium bred divided by the consumed amount, just exceeds one, requiring tritium losses to be minimised. One of the ways of doing this is reducing the number
of cycles tritium needs to make through the reactor before it fuses. The tritium
burn-up fraction, the amount of tritium that fuses before being exhausted from
the plasma, therefore needs to be as high as possible, which requires a long particle
confinement time, or high value of ρ.
The first demonstration reactors will most likely still require some amount of
external power (to drive the plasma current, with heating only a side effect), and
this changes the shape and position of the burn contours in the reactor operating
space. Most importantly, it increases the fusion power output, but in most cases
not enough to compensate for the conversion losses associated with the generation
Summary
iii
of the heating power.
The plasma in a reactor will always contain some impurities and the inclusion
of those in the analysis shows that especially impurities with a low atomic number
Z have a big impact on the fusion power, because they are very effective at diluting
the fuel. The amount of impurities can increase through a change in the source,
or by being better confined because of an increase in ρ. The latter would have a
double effect: both the helium and the impurity concentration will increase, which
has an even stronger impact on the fusion power. The upside to this effect is that
the fusion power becomes more sensitive to the external heating power for higher
values of ρ and impurity content.
We have analysed the stability of the operating points and, although the system
possesses many interesting properties (including saddle points, several different
bifurcation points, limit cycles, and damped or growing oscillations), the upshot
is that (virtually) all reactor relevant operating points are stable except for ρ >
10. However, the addition of external heating also stabilises these equilibria, so
stability considerations will most likely only have implications for the case of a
reactor design with little or no external heating.
Finally, we show that the current form of the τE scaling law can result in
bizarre predictions when applied to burning plasmas. First of all, ignition should
be possible at arbitrarily low densities, arbitrarily low power and arbitrarily small
reactor size. Secondly, a small change in the density or power dependence of the
scaling law, which has a negligible effect on the predicted value of τE , results in
wildly different operating points and fusion power.
These unphysical results are the consequence of the coupling between the density and the heating power in a burning plasma, which leads to a singularity in
the burn condition for a particular combination of the n- and P -dependence in the
τE scaling. This might be a point of academic interest only, were it not for the
strange coincidence that the family of 5 scaling laws that are used in the ITER
physics basis, all happen to exhibit precisely this pathology. Put very succinctly,
these scalings laws approximately have τE ∝ n0.4 and P −0.7 , and this means that
if for P the fusion power Pfus ∝ n2 is substituted, the well-known triple product
nτE T becomes independent of density and confinement time, i.e. it reduces to T .
We have no explanation for the fact that the ITER scaling laws all happen to have
this peculiar behaviour, the data base on which they are based does not contain
burning plasmas at all.
Summarising, this thesis shows that the particle confinement is an attractive
candidate for burn control, whereas the energy confinement is not. The operating
points for future reactors are stable and their stability is increased by the addition
of external heating power. The stability properties of the burn point are, however,
complex and might need to be considered in the design of a fusion reactor. The
applicability of current τE scaling laws to burning plasmas is questionable at best,
and an effort should be undertaken to obtain data points for burning plasmas.
iv
v
Contents
Summary
i
1 Introduction
1.1 Ignition and burn . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Burn stability and sensitivity . . . . . . . . . . . . . . . . . . . . .
1.3 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
4
5
2 Theory
2.1 The fusion reaction . . . . . . . . . . . . .
2.2 The tokamak . . . . . . . . . . . . . . . .
2.2.1 Operational limits . . . . . . . . .
2.3 Transport and confinement . . . . . . . .
2.3.1 Classical transport . . . . . . . . .
2.3.2 Neo-classical transport . . . . . . .
2.3.3 Anomalous or turbulent transport
2.3.4 L and H mode . . . . . . . . . . .
2.3.5 Sawtooth crashes . . . . . . . . . .
2.3.6 Energy confinement time . . . . .
2.3.7 Scaling laws . . . . . . . . . . . . .
2.3.8 Particle transport and confinement
2.4 Helium transport . . . . . . . . . . . . . .
2.4.1 Helium profile . . . . . . . . . . . .
2.5 Tritium breeding and burn-up fraction . .
2.6 Power balance . . . . . . . . . . . . . . . .
2.7 Burn equilibria . . . . . . . . . . . . . . .
2.8 Reactor studies . . . . . . . . . . . . . . .
2.9 Stellarators . . . . . . . . . . . . . . . . .
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7
7
9
11
12
12
13
14
15
17
17
18
19
23
23
25
27
31
33
35
vi
3 Burn equilibria
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Burning plasmas . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . .
3.3 Burn equilibria with impurities and Pext . . . . . . . . . .
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.3.2 Temperature domain of a burning plasma . . . . .
3.3.3 Helium fraction with external heating . . . . . . .
3.3.4 Burn equilibria with external heating . . . . . . . .
3.3.5 Impurities . . . . . . . . . . . . . . . . . . . . . . .
3.3.6 Power output with external heating and impurities
3.3.7 The effect of Pext on net electric output . . . . . .
3.3.8 Uncertainties in scaling laws . . . . . . . . . . . . .
3.4 Discussion and conclusions . . . . . . . . . . . . . . . . . .
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37
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48
49
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60
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4 Burn stability
4.1 Introduction . . . . . . . . . . . . . . . . . . .
4.2 Theory . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Burn equations . . . . . . . . . . . . .
4.2.2 Stability of a two-dimensional system
4.2.3 Bifurcation theory . . . . . . . . . . .
4.3 Reduced system . . . . . . . . . . . . . . . . .
4.3.1 Derivation . . . . . . . . . . . . . . . .
4.3.2 Jacobian matrix of the reduced system
4.3.3 Normalisation . . . . . . . . . . . . . .
4.3.4 Reduced system stability . . . . . . .
4.3.5 Physical interpretation . . . . . . . . .
4.3.6 Low temperature stability . . . . . . .
4.3.7 High temperature stability . . . . . .
4.3.8 Phase portrait . . . . . . . . . . . . .
4.3.9 Stability for different scaling laws . . .
4.3.10 Stability with external heating . . . .
4.3.11 Reactor comparison . . . . . . . . . .
4.4 Full system . . . . . . . . . . . . . . . . . . .
4.4.1 Jacobian matrix of the full system . .
4.4.2 Full system stability . . . . . . . . . .
4.4.3 Eigenvectors and eigenvalues . . . . .
4.4.4 Low temperature stability . . . . . . .
4.4.5 High temperature stability . . . . . .
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vii
4.5
4.4.6 Stability for different scaling laws . . . . . . . . . . . . . . . 102
4.4.7 Reactor stability comparison with external heating . . . . . 102
Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 106
5 Sensitivity of burn contours to form of scaling
5.1 Introduction . . . . . . . . . . . . . . . . . . . .
5.2 Theory . . . . . . . . . . . . . . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Operating contours . . . . . . . . . . . .
5.3.2 Density and power coupling . . . . . . .
5.4 Discussion and conclusions . . . . . . . . . . . .
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109
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6 Discussion and conclusions
121
7 Outlook and recommendations
125
Appendix A Partial derivatives for the Jacobian
127
A.1 Reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2 Full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Appendix B Derivation of ne as function of T
131
Appendix C Neoclassical confinement time
133
Appendix D Alternative scaling for the confinement time
135
Bibliography
137
Acknowledgements
147
Curriculum Vitae
149
1
Chapter 1
Introduction
Fusion is a fascinating phenomenon. The ’simple’ and elegant process of joining
two elements into a heavier one has lit up our universe since the birth of the first
stars and, in the case of our sun, enabled life to evolve on earth. Ever since we
understood this seemingly unlimited source of energy, the harnessing of its power
has stood as one of the great challenges of physics. And we are fortunate to live in a
time when our collective efforts are about to culminate in the first demonstration
of controlled fusion as an energy source. Successful operation of the large test
reactor ITER will hopefully lead to the construction of one or more demonstration
reactors, which for the first time will provide fusion electricity to the grid.
To fuse two nuclei the Coulomb repulsion, due to their respective charges,
needs to be overcome. This can be achieved by heating up the fuel to, typically,
150 million degrees centigrade, or 15 keV1 . The success of future fusion reactors
depends on the ability of the fusion process to maintain this temperature with little
or no external power, so the basic question is: can we create a fusion reactor that
works like a stove? You put in fuel, heat it until it reaches the ignition temperature
and after that it will burn indefinitely, as long as you refuel on time and remove
the ash.
When thinking about the design of such a reactor, several questions arise. How
high are the ignition and burn temperatures (which are generally not the same)?
What is the power output? How much ash can be tolerated in the machine? Is
the system stable? What happens in case of a disturbance? Do we need to control
it? And if this is the case, can we?
1 The electronvolt (eV) is a unit that is often used in plasma physics and corresponds to
approximately 11000 degrees Kelvin.
2
1.1
Chapter 1 Introduction
Ignition and burn
At the temperatures required for fusion, the fuel has become a plasma, the fourth
state of matter. In a plasma, the nuclei are stripped of their accompanying electrons and form a soup of charged particles. This has the advantage that it can be
contained in a magnetic field that reduces the heat loss from the plasma by several
orders or magnitude. Also, it prevents the plasma from touching the walls of the
reactor.
Charged particles can travel freely along the field lines, like beads on a string,
but perpendicularly to them they are restricted to a gyrating motion. If the field
lines were to touch the reactor wall, there would be excessive heat and particle
losses, so to avoid these the field lines in a fusion reactor are bent such that they
close on themselves, resulting in a toroidally shaped magnetic field.
The sun creates energy by fusing hydrogen atoms into helium [1], but the fusion
reaction used in reactors on earth is between the hydrogen isotopes deuterium (D)
and tritium (T) because this reaction has a higher chance of occurring for a given
temperature. The products of the reaction are an alpha particle (helium nucleus),
a neutron and an amount of energy:
D+ + T+ −→ He2+ + n + 17.6 MeV.
(1.1)
The energy is released in the form of kinetic energy of the alpha particle (3.52 MeV)
and the neutron (14.1 MeV), with the lions share going to the neutron because of
conservation of momentum.
The neutron escapes the magnetic field unhindered and is absorbed in the
wall, where its energy is converted into heat. This heat is then extracted and used
to power a generator. The alpha particle on the other hand, is confined by the
magnetic field and will heat the plasma by transferring its kinetic energy through
collisions with plasma particles. It is this process that will have to provide most
of the heating power in a fusion reactor.
The plasma loses energy through radiation and conduction, and at low temperatures these losses outweigh the alpha heating power from the fusion reactions.
This means that external heating is required to make up the deficit, which is undesirable from an economic point of view because it lowers the efficiency of the
plant.
Fortunately, it turns out that for a large enough reactor, the fusion power
increases faster with temperature than the radiation and conduction losses. So
at a certain temperature the external heating can be switched off and the plasma
heats itself.
This is a precarious balance, because the fusion power has a stronger response
to variations in temperature than the radiation and conduction losses. This means
that a small temperature perturbation will grow, making this an unstable equilibrium. Intuitively, it can therefore be thought of as the ignition point. Bear in
1.1 Ignition and burn
3
mind that the temperature at this point is also determined by the reactor, not
only the fuel.
A positive temperature perturbation will be kept in check by the conduction
losses, that will ultimately outweigh the fusion power. There is therefore a second, stable, equilibrium at higher temperature, which corresponds to the natural
understanding of a ‘burn point’.
This dynamic behaviour is represented in figure 1.1, which shows the time
derivative of the temperature (Ṫ ) as a function of temperature (T ) for a hypothetical reactor. For low temperatures Ṫ is negative, indicating the need for external
heating, until at the ignition temperature it crosses zero. Beyond that Ṫ is positive, which means that the temperature in the reactor will increase on its own
accord until the second zero crossing at the burn temperature.
Ṫ (keV/s)
0.5
ignition
0
burn
−0.5
0
5
10
T (keV)
15
Figure 1.1: The time derivative of the temperature (Ṫ ) plotted against the temperature for a hypothetical fusion reactor. For low temperatures, Ṫ is negative
and external heating is required. The curve then crosses the horizontal axis and
Ṫ becomes positive, so the temperature of the plasma will increase by itself, until
the stable temperature is reached at the second zero crossing.
There is a limit to the amount of fuel (deuterium and tritium) and ash (helium)
that a reactor can contain, as is the case in a normal stove. Because the fusion
power scales with the fuel density squared, one wants to operate close to this limit.
Every helium particle takes the place of two fuel particles, thus lowering the power
output, and therefore needs to be removed from the plasma after it has had time
to transfer its energy.
Because the helium and fuel are mixed, selective removal of one particle species
is complicated. Consequently, rapid removal of helium results in a low burn-up
fraction of the fuel, because it is exhausted from the plasma before it has had time
to fuse. The fuel can of course be separated from the helium and be recirculated,
4
Chapter 1 Introduction
but this process is inevitably accompanied by some losses. As tritium does not
occur naturally (it has to be made, or ’bred’, in the reactor) and the maximum
tritium breeding ratio (defined as the average number of tritium atoms bred per
fusion reaction) is only slightly larger than one, these losses can be ill afforded.
Reducing the fuel recirculation on the other hand, by keeping the particles in
the plasma longer, will result in a higher burn-up fraction. But this comes at the
cost of a lower power output because the helium concentration will also increase.
Furthermore, there is also a limit on the plasma pressure, often referred to as
the Troyon or β-limit [2]. The exact value of this limit depends on the shape of
the plasma, but exceeding it inevitably leads to the development of a magnetohydrodynamic (MHD) instability that changes the geometry of the magnetic field
and causes the plasma to disrupt, potentially damaging the reactor.
1.2
Burn stability and sensitivity
A fusion plasma is a very dynamical system. There is regular redistribution of particles, energy and current by the sawtooth instability, possible changes in transport
due to the interaction of fast alpha particles with the magnetic field or turbulence,
or changes in power output and confinement due to the gradual build-up of helium
ash in the plasma core.
Such phenomena will nudge the plasma out of its burn point and the question
is: where will it go from there? Will the plasma drift away from its burn point?
Will it return to the previous equilibrium? In either case, will it cross operational
limits on these excursions, such as the β-limit? Can we control these excursions?
What happens to the fusion power? In short: will a fusion reactor burn like a
candle or will it make uncontrollable excursions in temperature and power? The
latter is of course highly undesirable behaviour, since not only would the utility
companies not like it, it would also put harder requirements on the plasma facing
components and structural materials in the reactor.
Furthermore, the energy transfer from alpha particles to the plasma is not
instantaneous but happens gradually. The time scale of this transfer depends on
plasma parameters such as density, temperature and composition and introduces a
time delay between variations in density and temperature, and the heating power
delivered to the plasma. This could introduce oscillatory behaviour or change the
stability properties of the burn equilibria.
To model the performance of a fusion reactor, descriptions of the energy and
particle losses are needed. The common approach is to use scaling laws that
predict the energy confinement time τE and particle confinement time τp (measures
of how fast the plasma loses its thermal energy and its particles, respectively),
taking machine and plasma parameters as input. This allows the study of burn
equilibria as a function of density, temperature, energy and particle transport,
and investigation of the sensitivity of, for instance, the fusion power to these
1.3 Research questions
5
parameters. However, these scaling laws have been developed using fusion reactors
in which alpha heating of the plasma was (almost) completely absent, and caution
needs to be exercised when applying them to burning plasmas.
1.3
Research questions
The main question this thesis tries to answer is the following:
What are the properties of burn equilibria in fusion reactors?
Whereby with properties we mean:
• the contours in the operational space of the reactor spanned by plasma temperature and density where stable burn is possible;
• the dependencies of these contours on parameters that are under operator
control, such as the density, and those that are much less so, such as the
particle and energy confinement time and plasma purity;
• the stability of the burn under perturbation of these parameters, and the
level of perturbation that can be tolerated before the burn quenches.
We’ll articulate these aspects in four sub-questions below.
What parameters determine the temperature and composition of
the plasma at the burn equilibria and how sensitive is the system with
respect to these parameters?
To keep the cost of electricity down, we want to maximise the power output
of the reactor which requires operation close to the density limit, limiting its
effectiveness as an actuator for control of the power. In a burning plasma, the
only other parameters at the disposal of the operator are the energy and helium
removal, leading to the question
How does the power output of a burning plasma respond to changes
in energy confinement or particle transport?
Not only the position, but also the stability of the equilibrium is of importance,
because it determines the level of control that is needed. And while a burn equilibrium might be stable, the evolution of the system in phase space in response
to a perturbation might still lead to a violation of an operational limit, be it a
fundamental physics limit for the plasma, or a material limit for the reactor. It
is therefore of importance to know how the system responds to perturbations of
the equilibrium and whether this leads to a reduction in the accessible operating
space:
What are the stability properties of the operating points?
The last point concerns the use of scaling laws for the energy confinement
time. While it is common practice to use them to predict the performance of
future experiments, they are based on databases without burning plasma entries.
6
Chapter 1 Introduction
Applying them to burning plasmas might uncover sensitivities that are not present
in externally heated plasmas.
How sensitive are the burn equilibria to errors in the scaling laws
for the energy confinement time?
This thesis is organised as follows. Chapter 2 provides the theoretical framework of burning plasmas based on existing literature, followed by an analysis of
burn equilibria - and the influence of density, particle and energy confinement on
these equilibria - and the effect of external heating and impurities in chapter 3.
Subsequently, chapter 4 provides a linear stability analysis of burn equilibria, for
both a two dimensional and a four dimensional system. The sensitivity of burn
equilibria with respect to scaling laws is investigated in chapter 5. The final chapters, 6 and 7, provide the conclusions and outlook towards possible future research.
7
Chapter 2
Theory
2.1
The fusion reaction
Fusion is merging of two atomic nuclei into a heavier particle. For reaction products
up to iron, the mass of the resulting nucleus is slightly smaller than the sum of
the masses of the fusing particles. This mass difference m is converted into energy
(E), described by Einstein’s famous E = mc2 with c the velocity of light [3]. So in
principle a lot of reactions could be used as an energy source, but there are some
factors that limit the choice to only one realistic candidate.
Firstly, there is a tradeoff between overcoming the Coulomb barrier and the
time the particles are close enough to interact. Atomic nuclei carry a positive
charge and repel each other. To overcome this repulsion, the particles need to have
enough kinetic energy1 . Although a higher initial velocity will bring the particles
closer together, thereby increasing the chance that they will fuse, it also reduces
the time they spend in each others vicinity which reduces the fusion probability.
It turns out that the fusion probability, or cross-section σ, has a maximum and
the particle energy at which this optimum occurs is reaction specific.
The repulsive force between two particles with charge Z scales with Z 2 , while
the kinetic energy scales only with the mass of the nucleus, which ∝ Z. Particles
with higher charge need a higher velocity to overcome the Coulomb barrier, thus
making it harder to fuse them. And indeed, low Z particles generally have higher
cross-sections. For a given element however, reactions with heavier isotopes are
favoured because for equal energies they have a lower velocity.
It is no surprise therefore that fusion reactions involving light elements like
hydrogen and helium have the highest cross-sections, or reactivity. The reactivity is the integral of the product of velocity and cross-section of the reaction
1 Another way of overcoming the Coulomb barrier is to create a very high pressure, which is
the case in the core of stars and for inertial confinement fusion. Because this thesis deals with
magnetic confinement fusion, we will not discuss this further.
Chapter 2 Theory
8
over a Maxwellian temperature distribution. This is relevant in case the reactions take place in a plasma where the energy of the individual particles follows
a (Maxwellian) distribution function. Figure 2.1 displays the reactivity, denoted
hσvi, for the deuterium-tritium (DT), the DD and the 3 HeD reactions based on
the fitting formulas provided by Bosch and Hale [4].
10−20
DT
D
D
10−29
He
D
10−26
3
hσvi (m3 /s)
10−23
10−32 0
10
101
102
103
T (keV)
Figure 2.1: The reactivity of three fusion reactions involving hydrogen isotopes.
The DT reaction has the highest reactivity for temperatures up to several hundred
keV. Please note that, although plotted here up to 1 keV, the parametrisation of
the reactivities from [4] is only valid below 100 keV for the DT and DD reactions,
and below 190 keV for the 3 HeD reaction.
A second consideration when picking a fuel is availability. The 3 HeD reaction
has the advantage that it is (mainly) aneutronic, which reduces the radioactive
activation of the machines, increases the lifetime of components, enables more
neutron susceptible technologies and diminishes the need for neutron shielding.
Unfortunately, 3 He is exceedingly rare on earth and thus seems unlikely to be
used for fusion on a commercial scale2 . Moreover, the 3 HeD reaction requires
temperatures that are an order of magnitude higher than the DT reaction, which
is problematic because of the β-limit (see section 2.2.1).
2 There are significant resources of 3 He on the moon though, which might become accessible
in the future [5].
2.2 The tokamak
9
The DD reaction has neither the advantage of being aneutronic, nor of having
the highest reactivity. On top of that, the energy released per reaction is, at 3.70
MeV, much lower than for DT (17.6 MeV) or 3 HeD (18.3 MeV). This leaves the
DT reaction as the only realistic candidate at this moment.
While deuterium is a naturally occurring isotope of water and there is plenty
available on earth, this is not the case for tritium. Tritium has a half life of 12.3
years and does therefore not occur naturally, so it has to be produced artificially.
This can be done by irradiating lithium with the neutrons released in the DT
reactions, and will be covered in more detail in section 2.5.
2.2
The tokamak
The high temperatures needed for fusion require the fuel to be kept away from
the walls of the reactor and while there are several ways in which this can be
achieved, the most promising approach for reactor development relies on magnetic
fields to confine and position the plasma. Charged particles can move freely along
magnetic field lines, but are restricted in their perpendicular motion due to the
Lorentz force. To avoid end losses, the magnetic field is usually bent into a toroidal
shape.
The most successful reactor concept to date is the tokamak, invented in Russia
in the 1950s by Sacharov and Tamm. It derives its name from the Russion acronym
for ’toroidal chamber with magnetic coils’: тороидальная камера с магнитными
катушками (toroidal’naya kamera s magnitnymi katushkami). The results from
experiments on the first tokamak, T1, were presented to the world at the second
Geneva Conference on the Peaceful Uses of Atomic Energy in 1958 [6], although the
device was at that time still unnamed. A schematic representation of a tokamak
can be found in figure 2.2.
A tokamak consists of a toroidally shaped vacuum vessel, which is surrounded
by coils that generate a toroidal magnetic field (Bφ , see figure 2.2). The curved
nature of the field causes the particles to drift, necessitating a helical transform of
the field lines. This is achieved by running a current through the plasma, which
induces a poloidal magnetic field (Bθ ). The resulting field lines have a helical shape
and form a set of nested flux surfaces which are isothermals and isobars (a detailed
derivation of the magnetic equilibrium in a tokamak can be found in [7], but the
intuitive picture is that particles are free to travel along the field lines, smoothing
out variations in pressure and temperature). The safety factor q is defined as the
number of toroidal turns a field line has to make to complete one poloidal turn.
In a cylindrical approximation this is given by q = rBφ /RBθ .
The plasma current is driven by operating the plasma as the secondary winding
of a transformer, the primary of which is the central solenoid placed in the central
opening of the vacuum vessel. A final set of coils generates a vertical field that
prevents the plasma from expanding, shapes it, and positions it in the vacuum
10
Chapter 2 Theory
Figure 2.2: A schematic representation of a tokamak, with the vacuum vessel
omitted for clarity. The toroidal field coils are shown in light blue, the poloidal
field coils in silver, the central solenoid in green and the plasma in purple. Image
courtesy of EFDA-JET.
2.2 The tokamak
11
vessel.
To exhaust the helium produced in the fusion reaction and to create a well
defined plasma-wall interaction region, most tokamaks are equipped with a socalled divertor. Usually located at the bottom of the vacuum vessel, this is a
region where the field lines intersect the wall. The transition surface between
closed flux surfaces and open field lines is called the separatrix and the plasma
outside this surface is referred to as the scrape-off layer.
Current tokamaks rely on external heating to create the necessary condition
for fusion. The ratio between the fusion power Pfus and external heating power
Pext
Q=
Pfus
,
Pext
(2.1)
is often used to gauge reactor performance. In case of a burning plasma in which
the alpha particles provide the required heating power, Q is infinite.
2.2.1
Operational limits
In equilibrium, the pressure gradient ∇p in the plasma has to be balanced by the
Lorentz forces arising from the plasma currents and the magnetic field
∇p = J × B,
(2.2)
with J the current density and B the magnetic field. Because the magnetic field
coils constitute a large fraction of the cost of a fusion reactor, ideally the ratio
between plasma and magnetic pressure
β=
p
B 2 /2µ0
(2.3)
would be one, so there is no ’wasted’ magnetic pressure. Unfortunately this value
is unattainable due to the existence of MHD instabilities, or modes as they are
often referred to.
The maximum value of β that can be achieved in a tokamak, before large scale
MHD modes become unstable, can be expressed as
βmax = g
IM
,
aB
(2.4)
and is generally referred to as the Troyon limit. Here IM is the plasma current
in megaAmperes, a the minor radius in meters and B the magnetic field on axis
in Tesla. Extensive stability calculations for a wide range of pressure and current
profiles by Troyon et. al [2] found the value of g to be 0.28 N/A2 , in fusion
literature often used without units.
12
Chapter 2 Theory
Another limit that needs to be respected when operating a tokamak concerns
the electron density ne . Empirically it was established that above the Greenwald
density (in units of 1020 m−3 ) [8, 9]
IM
,
(2.5)
πa2
a disruption, an event in which control over the plasma is lost and which can damage the reactor, becomes very hard to avoid. Advanced regimes allow operation
up to ne ≈ 1.5nG [10] which results in approximately double fusion power output
compared to operation at ne = nG .
nG =
2.3
Transport and confinement
The study of energy transport, and to a lesser extent, particle transport in magnetically confined plasmas has for a long time been a major part of fusion research.
The first calculations in the 1950s only took classical transport (i.e. collisional diffusion across a straight B-field) into account. However, it was quickly realised
that with the introduction of curved magnetic geometries, classical transport was
greatly enhanced because of the drift motions and the trapping of particles (due
to the variation in field strength along a field line), and this realisation led to the
study of neoclassical transport.
When the reactors became bigger en more advanced and temperatures increased, the predictions again turned out to be far off the mark and this time
there was no easy explanation, hence the name ’anomalous transport’. Increasing diagnostic capabilities and physical understanding led to the insight that this
was in fact turbulent transport, which to this day is not completely understood,
although advanced numerical models are reaching the point where experimental
results can be reproduced and predicted. This section will briefly introduce the
three forms of transport and the way the resulting confinement is modelled at
reactor level.
2.3.1
Classical transport
Previously it was stated that plasma particles stick to the field lines like ’beads
on a string’. This picture is not entirely accurate. In a homogeneous, straight
magnetic field the particles gyrate around the field lines with the Lorentz force
acting as the centripetal force, with the cyclotron frequency ωc and gyroradius ρg
(also referred to as Larmor radius) given by
qB
m
mv⊥
ρg =
,
qB
ωc =
(2.6)
(2.7)
2.3 Transport and confinement
13
with q denoting the charge of the particle (e for an electron or hydrogen nucleus),
v⊥ the velocity perpendicular to the magnetic field, m the mass of the particle and
B the magnetic field strength.
In case of a collision, particles can ’hop’ onto a different field line which is
typically one gyroradius away. This results in classical diffusion with the diffusion
coefficient of the order of χCL = νei ρg , where νei is the electron-ion collision rate.
Assuming a Maxwellian velocity distribution it can be shown to be [7]
νei =
with ln Λ ≈ ln
3/2
12π0
Te3/2
1/2 3
e
ne
√
2
e4
3/2
12π 3/2 20 m1/2
e T
ln Λ,
≈ 15−20 the Coulomb logarithm. Classical transport
1/2
−3
n20 /B 2 Tk
coefficients amount to χCL
e = 4.8 × 10
1/2
0.10n20 /B 2 Tk ≈ 7.2 × 10−4 m2 /s.
2.3.2
(2.8)
≈ 3.4 × 10−5 m2 /s and χCL
=
i
Neo-classical transport
Because the magnetic field is curved, the field lines are ’compressed’ on the inside,
and ’rarified’ on the outside of the torus. This creates a 1/R2 gradient in the field
strength that points towards the center of the torus. The curvature and gradient
give rise to the so-called curvature B drift and gradient B drift, respectively.
Furthermore, if there is a (radial) electric field the particles will experience an
E × B drift. Together these drifts change the trajectories of the particles in the
plasma; the radius of their gyration changes periodically over an orbit and they no
longer follow the field lines, resulting in a drift motion perpendicular to the field.
A second effect originating from the gradient in the field, is that particles can
2
become ’trapped’. The magnetic moment µ = mv⊥
/2B of a particle, with v⊥ its
velocity perpendicular to the magnetic field, is conserved. Because the magnetic
field is stronger on the inside of the torus than on the outside and the magnetic
field lines are twisted, a particle starting at the outside and following a field line
will initially move along the gradient. To keep its magnetic moment constant,
v⊥ has to increase and conservation of energy dictates that its parallel velocity
vk decreases. If its initial parallel velocity was too small, at some point it will
decrease to zero and it will reverse direction. It now follows a field line in the
opposite direction, until it again runs up against the magnetic ’hill’ and reverses
direction again. These particles are ’trapped’ and will bounce back and forth.
Looking in the poloidal plane, the centre of mass of particles traveling around
the torus in the direction of the magnetic field, describes a circle that is shifted
inwards and is slightly larger than the flux surface its associated with. For particles
traveling agains the magnetic field the opposite holds true: they are on a trajectory
that is slightly smaller and shifted outwards. This means that a trapped particle
Chapter 2 Theory
14
that reverses direction does not retrace its original path exactly, but follows a more
or less parallel trajectory. The resulting orbit looks like a banana in the poloidal
projection, and the step size for collisional transport of these particles is not their
gyroradius, but the width of their so-called banana orbit.
An approximate scaling for the resulting neoclassical diffusion coefficient is
χNC ∝ q 2 ε−3/2 χCL , with ε = a/R0 the inverse aspect ratio. Neoclassical transport is rock bottom: a tokamak cannot do better than this and although the
proportionality factor to classical transport looks inconspicuous, it turns out to be
a factor of 100 larger for tokamaks with a large aspect ratio.
2.3.3
Anomalous or turbulent transport
In most cases the transport is orders of magnitude larger still than neoclassical
transport and this phenomenon is referred to as anomalous transport. It is caused
by turbulence and a complete description is extremely complicated due to the nonlinear nature of the turbulence. Turbulent transport is convective, which sets it
apart from classical and neo-classical transport, which are both diffusive. However,
it turns out that for most purposes it works quite well to describe turbulent transport with an effective diffusion coefficient χT ∝ γmax L2c , where γmax is the growth
rate of the fastest growing mode and Lc is the turbulence correlation length [11].
Although there are many forms of turbulence in a tokamak plasma, two electrostatic drift wave instabilities are the major drivers of turbulent transport under
fusion conditions. For ion thermal transport this is believed to be the ion temperature gradient (ITG) instability [12, 13, 14], and the electron transport is dominated
by trapped electron modes (TEM) [12, 13, 15].
For drift wave turbulence, the value of Lc scales with the gyroradius ρg in
the limit of small ρ∗ = ρg /a, which results in so-called gyroBohm scaling with
a diffusion coefficient χT ∝ ρ∗ T /eB. This in contrast to the Bohm scaling that
applies to modes with a size comparable to the plasma dimensions (or minor
radius), which follows χT ∝ T /eB [16, 11].
Often there is a threshold gradient above which the turbulence growth rate
increases sharply. The result is a corresponding sharp increase in diffusion coefficient, an effect which is referred to as profile stiffness, because above the threshold
the gradient responds much less to changes in the heat flux. This phenomenon
is illustrated in figure 2.3, where the heat flux is plotted as a function of the
dimensionless temperature gradient length
R dTi
R
=
.
LTi
Ti dr
The diffusion coefficient in the stiff region of the plasma is often approximated
2.3 Transport and confinement
15
χ1
χ0
Threshold (κ)
Figure 2.3: Schematic representation of energy transport in a tokamak as a function of the normalised temperature gradient. For low temperature gradients, the
heat flux increases linearly with R/LT , until a critical value is reached. Beyond
this value, the gradient becomes stiff, i.e. it hardly responds to changes in heat
flux anymore. Figure adapted from [18].
with a critical gradient model [13]
R
R
χT = χgB χs
− κc H
− κc + χ0 ,
LTi
LTi
(2.9)
with χgB = q 3/2 T ρg /eBR the gyroBohm normalisation, χs the stiffness level, H
the Heaviside step function, κc the threshold (with a value around 5 often found
for reactor relevant tokamaks, although there are also parametrisations based on
gyrokinetic simulations [17]) and χ0 the level of residual transport in the absence
of turbulence.
2.3.4
L and H mode
In the eighties a new regime of operation was discovered in the ASDEX tokamak [19]. ASDEX was one of the first tokamaks equipped with a divertor, and
when enough heating power was supplied, the plasma would ’jump’ to a state in
which the confinement was roughly a factor two better than before. The new
Chapter 2 Theory
16
1
8
Experiment (Scans)
Core
Plasma Temperature
6
Tiexp (0.4) (keV)
Internal transport barrier
(ITB)
Sawteeth
Edge localized modes
(ELMs)
L—mode
Edge transport
barrier
in H-mode
JG98.483/35c
2
Neutrals
0
4
0
1
Normalised radius r/a
Tiexp (0.8) (keV)
Figure 1: Schematic view showing regions with different Figure 2: Link between central and edge ion tempera
Figure 2.4: Typical radial
temperature profiles in a tokamak for for
different
operating
a series of ASDEX-Upgrade discharges [30].
transport characteristics in tokamak.
regimes. When going from L to H mode, an edge transport barrier is created which
results in very steep temperature and pressure gradients at the plasma edge, and
elevates the core temperature. Figure courtesy of EFDA-JET.
JG99.238/4bw
Ti(0) and Te(0) (keV)
Ion temperature (keV)
regime was dubbed H-mode (high confinement) and the ’normal’ regime retroactively received the name L-mode (for low confinement).
The improved confinement originates from a transport barrier at the plasma
edge, where the pressure Pulse
gradient
creates
a radial electric field that
drives E × B
No: 47543,
47545, 47546
15
= 8-18MW
shear flows that locally10.0reduce the turbulent transport [20, 21]. ThisPNBIcan
be seen
8.0
Type-I ELMs
in a strong reduction in the balmer α radiation around the plasma [22],
indicating
6.0
a reduction in outward particle flux. The results are steep temperature en density
gradients at the plasma4.0edge, and because the core transport remains unaffected,
10
it looks like their respective profiles are placed on a pedestal, which is illustrated
in figure 2.4. Because the pedestal raises the temperature and density over the
2.0
whole cross section of the
plasma, it has a large contribution to the total stored
Ions
energy W and therefore the confinement time.
5
The H-mode comes1.0at a price though. The transport barrier at the plasma Electrons
0.8 that the pressure gradient keeps increasing until it hits a
edge is usually so strong,
0.6
(local) stability limit, which
triggers an edge localized mode (ELM) that ejects up
to 10% of the stored energy from the plasma [23]. This energy (and the particles
IP = 1.8MA / Bt =3.
0
0.4
0
1
3
that carry it) travel through
layer
the 3.7
divertor,
where
they hit
the 2
3.5 to3.6
3.8
3.0 the
3.1 scrape-off
3.2
3.3
3.4
Tiped and Teped (keV)
radius (m)
wall. The short timespan (≈1 ms) ofMajor
these
events results in transient heat loads
of up to 1 GW/m2 on
the divertor surface [24], which may damage the divertor.
Figure 3: Ion temperature profile for a series of JET shots Figure 4: Link between central and edge temperatur
For this reason, a reactor
will edge
require
form
of ELM[32].
mitigation
toJT60-U
protect
it. [33].
a series of
plasmas
with varying
densitysome
and edge
ion temperature
18
2.3 Transport and confinement
17
Analogous to the pedestal, the plasma can also develop transport barriers in
the core, often associated with a region of strong flow shear and the presence of
flux surface with rational values of q [25]. These regions are referred to as internal
transport barriers (ITBs) and might be used in advanced reactor scenarios.
2.3.5
Sawtooth crashes
An important mechanism in (particle) and energy transport in the center of the
plasma is the sawtooth mechanism. This takes its name from a sudden drop in
temperature in the center of the plasma, followed by a gradual recovery, until the
process repeats itself. When the central temperature is plotted as a function of
time, the resulting graph has a distinct sawtooth shape.
The sawtooth crash is caused by the central value of the safety factor q dropping
below one. This triggers an MHD instability, in which the hottest, central part
of the plasma is pushed outwards and replaced by cooler plasma. The result is
an outward propagating heat flux and a flattening of the temperature, and to a
lesser extend, density profiles. Because the fusion power Pfus ∝ p2 , the effect on
the fusion power can be significant.
Generally, sawteeth will not cause a disruption, but if they become too big
they might destabilise other, more harmful, MHD modes, like neoclassical tearing
modes. They can also play a role in flushing accumulating impurities from the
plasma core, but this is a double-edged sword as they can also help impurities
penetrate into the plasma center [26].
2.3.6
Energy confinement time
From a reactor point of view, the overall transport properties of the plasma are
more relevant than the precise values of the transport coefficients at each radial
(and poloidal) position. These global properties are reflected in the energy confinement time τE which is a measure of how long the plasma retains its energy
τE =
W
Pcond −
dW
dt
.
(2.10)
Here, W = 3/2ntot T is the total internal energy of the plasma, with ntot = ne +
Σi ni Zi , and Pcond is the conducted power. In equilibrium dW/dt = 0, so the
definition simply becomes
τE =
W
.
Pcond
(2.11)
Because Pcond is hard to measure, often a different version (with a slightly different
notation) of the confinement time is used
τ̃E =
W
,
Ploss
(2.12)
Chapter 2 Theory
18
in which case Ploss = Prad + Pcond , the total loss power. In experiments with only
external heating, this makes determining τ̃E simply a matter of measuring the
temperature and density, because in equilibrium Ploss = Pext .
2.3.7
Scaling laws
Because of the complicated nature of the transport processes the expected confinement time for a new experiment is often calculated using scaling laws, which
provide τE as a function of engineering or physics
The most common
Q parameters.
i
approach is to fit a function of the form τE ∝ i pα
,
known
as
a power law, to
i
the data.
This can be done either in engineering variables, like major and minor radius,
plasma current, magnetic field, density, power, etc, or in physics variables like the
Bohm time
τB =
a2 B
∝ ε2 R2 BT −1 ,
T
(2.13)
the normalised ion gyroradius
∗
ρ =
2eT
Mi
1/2
√
Mi T
Mi
∝
,
eBa
εRB
(2.14)
the ratio of plasma and magnetic pressure
β ∝ nT B −2 ,
(2.15)
the normalised collision frequency (collision frequency divided by the bounce frequency of trapped particles)
ν ∗ = νii
Mi
eT
1/2 R
a
3/2
qR ∝ nRT −2 qε−3/2
(2.16)
and the cylindrical safety number
qcyl =
RB
f (κ, δ) ∝ BRI −1 ε2 κ,
ε2 I
(2.17)
with f (κ, δ) a function of the plasma triangularity δ, the elongation κ = b/a, (with
b and a the diameter of the poloidal plasma cross section along the principal axes),
and T the ion temperature in eV [26]. The values obviously vary over the profile,
but can be approximated by their volume average for a global analysis. Using the
above definitions, a linear transformation can be made between the engineering
and physics variables and their respective exponents.
The number of free parameters in the fit can be reduced by placing constraints
on the exponents using the method developed by Kadomtsev [27] and Connor and
2.3 Transport and confinement
19
Taylor [28]. This relies on finding linear transformations of the physics variables
under which the governing equations are invariant. Applying these transformations
to the general form of the scaling law then constrains the exponents.
For instance, the Kadomtsev, or high β, constraint demands that the exponents
satisfy 4αR − 8αn − αI − 3αP − 5αB = 5. The Bohm and gyro Bohm constrained
scalings add αR −7αn −4αI −7αP −5αB = 0 and 6αR −22αn −9αI −12αP −15αB =
0, respectively, to the high β constraint.
One of the first concerted efforts in compiling a database with results from
different tokamaks was made in the eighties, which resulted in the ITER89P Lmode scaling [29]
0.85 1.2 0.3 0.5 0.2 0.5 0.1 −0.5
τ̃E = 0.048IM
R a κ B A n20 P
,
(2.18)
where R is the reactor major radius, A the average ion mass in amu, n20 the
electron density in 1020 m−3 and P the heating power in MW. Until the first Hmode scalings were published in the nineties, the L-mode scaling was also used for
H-mode discharges by multiplying the predicted confinement time with an H-factor
fH = 2.
In reference [26] a comprehensive review of confinement data was made, resulting in a set of closely related forms of the τE -scaling. The IPB98(y,2) is the most
commonly used scaling law for H-mode plasmas and also the recommended scaling
law for reactor extrapolations [31]. The value of τE that it predicts, is plotted in
figure 2.5 against the measured value of τE , for experiments in the confinement
scaling database [32, 33]. Its parameters are given in table 2.1, together with the
other H-mode scalings presented in the ITER physics basis [26]. These scalings
differ in the definition of κ (κ = b/a vs κ = πa2 /area and the database that
they are based on (the differences between the databases lie mainly in the type of
external heating for the plasma discharges that they contain).
2.3.8
Particle transport and confinement
For a burning plasma it is of importance how fast the ash is removed, relative to the
burn rate, as this determines the burn-up fraction. This ratio is governed by the
relation between particle and energy transport. Particle transport is somewhat
different process from energy transport, but it works by the same mechanisms:
diffusion and turbulence (convection).
Experimental observations put the particle diffusion coefficient at the same
order as the energy diffusion coefficient [34, 35, 36, 37, 38].
Dp ≈ χE ,
(2.19)
which agrees with the transport being dominated by turbulence. This finding is
confirmed by gyro-kinetic simulations, that also reveal a convection term, the sign
of which depends on the ratio between electron and ion heat flux [39, 40].
Chapter 2 Theory
20
Plasma Phys. Control. Fusion 50 (2008) 043001
10
10.0
τEexp
1.01
0.1
0.1
Review Article
COMPASS
JT-60U
PDX
TFTR
ASDEX
DIII–D
MAST
START
ITER
AUG
JET
NSTX
TCV
C-Mod
JFT-2M
PBX-M
TdeV
0.001
0.001
0.001
0.001
JG06.455-1c
0.01
0.01
0.01
0.01
0.1
0.1
98(y,2) (s)
τ th,IPB98(y,2)
τ
11
10
10
E
Figure
34. Plot value
of the measured
H-mode
thermal energy
confinement
versus that predicted by
Figure 2.5: The
predicted
of τE from
the IPB98(y,2)
scaling
plottedtime
against
the
IPB98(y,
2)
scaling
expression.
The
symbols
indicate
data
from
different
tokamaks as noted
the measured values [30]. Figure courtesy of EFDA-JET.
in the legend.
where ε is the inverse aspect ratio (ε ≡ a/R), B is the toroidal magnetic field in T, and n is the
line-averaged density in 1019 m3 . However, the anticipated operating regime for ITER is not
L-mode, but ELMing H-mode. Therefore, an H-mode database was formed [132] containing
both ELM-free and ELMing H-mode data. A full description of the H-mode database and its
variables can be found in [133, 134] and a description of the L-mode database in [21].
Both the L-mode and H-mode databases have been routinely updated. The current
recommended expressions to be used for the scaling of energy confinement time with selected
dimensional variables are ITER97-L [21] for the Lmode and IPB98(y) and IPB98(y, 2) [20]
for the H-mode. Unlike the Goldston and ITER89-P expressions, both of these scaling
expressions pertain to the thermal energy confinement time (τth ) rather than the global energy
confinement time (τE ), which includes the energy content in fast ions from auxiliary heating.
Scaling
IPB98(y)
IPB98(y,1)
IPB98(y,2)
IPB98(y,3)
IPB98(y,4)
C (10−2 )
3.65
5.03
5.62
5.64
5.87
I
0.97
0.91
0.93
0.88
0.85
B
0.08
0.15
0.15
0.07
0.29
n
0.41
0.44
0.41
0.40
0.39
P
-0.63
-0.65
-0.69
-0.69
-0.70
R
1.93
2.05
1.97
2.15
2.08
κ
0.67
0.72
0.78
0.78
0.76
ε
0.23
0.57
0.58
0.64
0.69
A
0.20
0.13
0.19
0.20
0.17
N
1398
1398
1310
1273
714
RMSE
(%)
15.8
15.3
14.5
14.2
14.1
ITER
τE (s)
6.0
5.9
4.9
5.0
5.1
Table 2.1: The exponents of the different parameters in the IPB98(y) and IPB98(y,1–4) scaling laws for τE in H-mode
plasmas [26]. The IPB98(y) and (y,1) scalings are based on the ITERH.DB3 dataset for ELMy H-modes, the first
using κ = b/a and the latter using κ = area/πa2 . Scalings IPB98(y,2–4) also use κ = area/πa2 , but are based on a
restricted dataset: (y,2) on ITERH.DB3 restricted to NBI heated discharges, (y,3) on the same as (y,2) but without
the Alcator C-MOD data and (y,4) on the same as (y,2) but using only data from the five ITER similar devices. All
scalings meet the Kadomtsev constraint, except for (y,3) which is just a free fit to the data.
2.3 Transport and confinement
21
22
Chapter 2 Theory
For our purposes a detailed treatment of particle transport is not practical.
Because we are most interested in the global particle transport, it seems reasonable
to introduce a particle confinement time, similar to the energy confinement time.
The definition of τp is then, analogous to the definition of τE , the total particle
content N divided by the net flux of particles Γp , leaving the plasma through the
separatrix:
τp,core =
N
.
Γp
(2.20)
Although the definition looks innocuous, there is a caveat which complicates matters somewhat. When a particle exits the plasma and hits the wall, it loses its
energy and electric charge. The, now cold and neutral, particle either enters the
pumping duct that leads to the pumps that maintain the vacuum, or it reenters
the plasma and is ionised again. The latter process is often referred to as recycling.
Depending on the value of the recycling coefficient Rcyc , τp can be a lot longer
than the primary particle confinement time τp,core . The value of Rcyc , defined
as the ratio of the recycled particle flux and the total particle flux, depends on
the wall and plasma conditions, but can reach values up to 0.95 or even higher.
Hence, a distinction needs to be made between core and edge particle transport.
A detailed discussion and two alternative descriptions can be found in [41], but
for simplicity we will stick to the more general definition of the global particle
confinement time τp = τp,core /(1 − Rcyc ).
Experimental observations indicate that the ratio ρ between τE and τp is approximately the same for different species, which makes it a suitable figure of merit
for helium transport [41, 24]. The energy and particle confinement times are then
related;
τp = ρτE
(2.21)
and, depending on the plasma conditions, ρ varies from ± 5 up to around 30,
with higher values for L-mode and ITB plasmas and lower values for elmy H-mode
plasmas [42, 37, 43, 44]. For the remainder of this thesis, we will write τp for the
particle confinement time as defined in equation (2.21), unless explicitly stated
otherwise.
Although we assume the same confinement time for all particle species, there
can be a difference due to the different transport and edge recycling coefficients,
and different pumping efficiencies. All of these depend on the atomic mass and
the effective ion charge
Zeff =
Σj nj Zj2
,
nj Zj
with nj the density of particle species j, and Zj its atomic number.
(2.22)
2.4 Helium transport
23
The figure of merit for the relative efficiency of the helium exhaust, is the
enrichment ratio η, defined as the ratio of the helium concentration at the separatrix to the helium concentration in the exhaust gases (that are removed from the
system)
ηHe =
ΓHe ncore
DT
.
2ΓDT ncore
He
(2.23)
Values of ηHe = 0.3 have been found in AUG with the original divertor [45], while
after the upgrade to the ITER like divertor values up to 1 were reported [46].
Similar results (ηHe between 0.1 and 1) were obtained at DIII-D [37], JT60-U [43]
and JET [47]. The ITER physics basis states that ηHe ≥ 0.2 is required for ITER
to be successful [48], but future reactors might require higher values to meet the
tritium breeding and recycling targets.
2.4
Helium transport
Helium ash removal is of critical importance to the success of a fusion reactor.
Remove the ash too quickly and it will not be able to transfer its energy to the
plasma to heat it. Remove it too slowly, and it will dilute the fuel too much,
resulting a lower fusion power and possibly extinguishing of the reactor.
Besides the volume averaged helium content, it is also the spatial distribution
that matters. Helium in the plasma center will have a much larger effect on reactor
performance than helium in the plasma edge, because of the peaked fusion power
profile that is expected.
The issue of ash removal from future fusion machines is a rather complicated
problem that is difficult to investigate in present machines, due to the fact that
there is no significant production of helium in the core. Furthermore, in contrast to
present day machines, the sawtooth period in ITER is expected to be significantly
longer than the energy confinement time, resulting in a different effect on the
transport of impurities from the core towards the edge of the plasma. Predictions
for the helium transport and profile in future reactors therefore rely heavily on
numerical simulations.
2.4.1
Helium profile
The helium profile in the plasma is determined by three things: the source profile,
the transport of the helium produced in the plasma core towards the edge and the
boundary condition at the edge, set by the recycling (which in itself is determined
largely by the pumping efficiency).
The source profile can easily be determined if the fuel density and temperature profiles are known. The alpha particles are created with an energy of 3.52
MeV and it takes some time before they have thermalised, during which they do
24
Chapter 2 Theory
not necessarily stay on the same flux surface. The thermal helium source profile
therefore is not exactly the same as the fusion power profile. Since this thesis only
deals with volume averaged values for the temperature and particle densities, this
is of little interest at this point, but needs to be considered for a detailed reactor
design.
Already in the 1960s the issue of helium transport was investigated, but only
in last decade of the century did people start to self-consistently study the effect
of helium in burning plasmas [49, 50, 51, 52, 53]. The most popular approach was
to use a zero-dimensional (0D) model that only included the power balance and
the alpha particle balance, resulting in a cubic equation for the helium fraction in
the plasma, with the ratio between particle and energy confinement time ρ as a
free parameter, which will be discussed in more detail in section 2.7.
Currently, a full understanding of helium (or impurity) transport in fusion
plasmas is still lacking. It can either be described as a fully diffusive process,
or as a combination of an effective diffusion and an inward pinch, resulting in a
transport equation of the form
ΓHe = −DHe ∇nHe + VHe nHe ,
(2.24)
with DHe and VHe the flux surface averaged diffusion coefficient and pinch velocity
respectively.
Both approaches have been used for ITER modeling. In [54], only the diffusive term is taken into account, using the 1.5D BALDUR code with both the
MMM95 and the Mixed Bohm/gyro-Bohm transport models to obtain the diffusion coefficients for the core transport. Three different pedestal models provided
the boundary conditions at the top of the pedestal, while the helium density at
the edge was calculated using Zeff and a specified impurity (Beryllium) content.
The resulting helium profiles are only slightly peaked, and the effect of sawtooth
oscillations on the central helium density is rather limited. The sensitivity study
shows that the central helium density is strongly dependent on the helium fraction
at the edge.
The approach of combining diffusion with a pinch has been taken in [55]. Using PTRANSP (predictive TRANSP [56]) and calculating the helium profile for
different values of DHe and VHe , it was found that the dependance of the central
helium density on the recycling increases with the inward pinch velocity, as would
be expected for a boundary value problem. The opposite holds for an outward
pinch velocity of course, in which case hollow helium profiles are expected. The
resulting predicted power for ITER ranges from 320 MW for the outward pinch
(for both large and small diffusivity, independent of the recycling) and 170-240
MW for the case of an inward pinch. This would give values of Q ranging from
4.6 to 6.5.
The most advanced predictions of the helium transport in ITER come from
(non-) linear gyrokinetic simulations. The results presented in [39] show that the
2.5 Tritium breeding and burn-up fraction
25
prospects for ITER look rather promising. In this case, the helium transport
equation reads
∂nα
nα
+ ∇ (−DHe ∇nHe + VHe nHe ) = ∗ ,
∂t
τsd
(2.25)
∗
with τsd
the thermalization time for fast alpha particles. The helium concentration
in the plasma center was found to be of 5% for a recycling factor Rcyc = 0.97, and
for the ITER specification Rcyc = 0.9 this decreased to 1.7%.
In case of the ITER reference scenario, a rather flat density profile (i.e. fast
core particle transport compared to particle removal from the edge) is expected
for global helium concentrations of 2% and higher. In that case the shape of the
helium profile is expected to almost exactly match the electron density profile [39].
The cause of the relatively flat helium profile is the fast core transport of helium
compared to the time it takes a helium particle to enter the pumping system from
the plasma edge. Hence, the total helium concentration is mostly determined by
the efficiency of the pumping system [37, 45].
The precise value of ρ that still allows ignition is usually found to range from
5 - 10 [53, 57], depending on the precise model used. Early simulation of ITER
performance with the BALDUR 1.5D code give similar results, but also put requirements on the recycling coefficient to obtain the desired values of ρ [50, 52].
Although the precise knowledge of the transport processes involved has greatly increased over the past two decades, the fundamental requirements have not changed
much. The ITER physics database still lists a value of ρ between 5 and 10 [24].
2.5
Tritium breeding and burn-up fraction
Tritium is an unstable isotope of hydrogen, with a half life of 12.3 years. Consequently, tritium does not occur naturally on earth, but it can be made by irradiating lithium with the neutrons from the DT fusion reaction. The possible reactions
with both lithium isotopes, 6 Li and 7 Li, are given by:
6
7
Li + n → 4 He + T + 4.8 MeV,
4
0
Li + n + 2.5 MeV → He + T + n ,
(2.26)
(2.27)
where the neutron n0 released by the 7 Li reaction has less energy than the original
neutron.
The 6 Li reaction has a large cross section (940 barn) for thermal neutrons,
resulting in nearly all of them being captured and breeding a tritium atom once
they have slowed down enough. Because of the 4.8 MeV released in this reaction,
this can have a substantial positive contribution to the overall power balance of a
fusion reactor. The 7 Li reaction is endotherm and has a much lower cross section
26
Chapter 2 Theory
(order 0.3 barn for neutron energies above 5 MeV), but does offer the possibility
of breeding more than one tritium atom per neutron.
Because of the inevitable losses associated with tritium handling and the natural decay, the tritium breeding ratio
TBR =
tritium bred
tritium burnt
(2.28)
needs to be above one to compensate for this. On top of that, a small surplus
is required to obtain a starting inventory for new reactors when they are under
construction. The current world production of tritium (mostly from CANDU
fission reactors) amounts to several kilograms per year [58], which is at best of the
same order as the start-up inventory off a single fusion power plant, depending on
the time it takes to extract the tritium from the breeding blanket a
Figure 2.6: The cross sections of the relevant tritium breeding and neutron multiplication reactions [59].
The tritium breeding in a fusion reactor is foreseen to take place in a blanket
surrounding the plasma, between the plasma facing first wall and the vacuum
vessel. Even so, not all neutrons released in the fusion reaction will end up reacting
with lithium, due to absorption by structural materials and openings in the blanket
for the divertor, heating, diagnostics and fuelling systems. To obtain a TBR > 1,
2.6 Power balance
27
a neutron multiplier is integrated in the blanket, most likely beryllium or lead [59]
n + 9 Be + 3 MeV → 2He + n0 + n00 ,
0
00
n + Pb + 10 MeV → Pb + n + n .
(2.29)
(2.30)
The cross sections for the lithium, beryllium and lead reactions are plotted in
figure 2.6.
The achievable TBR depends on the exact design of the reactor and blanket,
and on technology that is still under development (ITER will be the first reactor
to test tritium breeding blanket modules), but is expected to lie between 1.08 and
1.15 [60, 61, 62, 63].
Whether the achievable TBR values are sufficient depends strongly on the
tritium burn-up fraction fb , which is the probability that a tritium atom injected
into the plasma will actually fuse. This is defined as the fusion rate (since every
fusion reaction consumes one tritium atom) divided by the rate at which tritium
is lost from the plasma, which is the sum of the fusion and the loss rates:
fb =
nD nT hσvi
,
nD nT hσvi + nT /τp∗
(2.31)
with nD and nT the deuterium and tritium densities.
The burn-up fraction has implications for the required tritium breeding ratio,
because 1/fb is the average number of cycles a tritium atom needs to make through
the system before it fuses. The time it takes for a given tritium atom to fuse is
the product of cycle time and number of cycles, and if one cycle takes one day,
decay losses cannot be neglected because they amount to about 1% over 50 cycles.
Of course the 3 He produced in by the decay of tritium can be converted back into
tritium by letting it absorb a neutron in the blanket, but this is still a net loss
because that neutron can no longer breed tritium from lithium (in other words: it
takes two neutrons to make one tritium atom in that case).
Part of the tritium might adsorb onto the surfaces of tritium handling systems
in places where it would be difficult to recover. This needs to be taken into account
in a reactor design to prevent the buildup of a large tritium inventory, which would
be unacceptable both from a safety and a TBR point of view. Modelling of the
tritium circulation puts the minimum at fb = 0.02 and above 0.05 if the reactor
also needs to breed the startup tritium inventory for a new reactor every 3 to 5
years [64].
2.6
Power balance
The following section provides a brief overview of the energy balance of a fusion
plasma, the derivation of the burn criterion and stable and unstable operating
points, where we will follow the approach taken by Freidberg [7]. Subsequently
28
Chapter 2 Theory
the energy balance will be complemented with the helium particle balance, which
will lead to a modified version of the burn criterion and closed ignition contours,
which is based on work presented in refs. [41, 49, 53, 65].
In order to keep the plasma in a fusion reactor at the required temperature, the
heat losses from the plasma need to be balanced by a heat source, a requirement
which can be written down in the form of an energy balance
Sα + SΩ + Sext = Srad + Sκ ,
(2.32)
with Sα the heating power density from alpha particles, Sext the external heating
power density, SΩ the Ohmic heating power density, Srad the power density of
radiation losses and Sκ the power density of the conduction losses.
Assuming no fast particle losses3 , the alpha power density delivered to the
plasma is given by
Sα = nD nT hσviEα ,
(2.33)
for a plasma with nD and nT the deuterium and tritium densities, hσvi the reactivity of the plasma and Eα = 3.52MeV the alpha particle energy. In case nD = nT
and no impurities, this reduces to
Sα =
1 2
n hσviEα .
4 e
(2.34)
The Ohmic heating power plays a role in case there runs a (large) current in
the plasma, for instance in a tokamak. The power density is given by
SΩ = ηJ 2 ,
(2.35)
with η the electrical resistivity, which scales with T −3/2 , and J the current density.
Since the Ohmic power density is comparatively low at fusion relevant temperatures, we will neglect it from now on. Note however, that Ohmic heating plays an
important role during the first phase of the discharge when the plasma temperature
is still quite low.
The first term on the right hand side of the energy balance is the radiation
losses. In the core of the plasma these consist of two parts: Bremsstrahlung and
synchrotron radiation. The Bremsstrahlung radiation power density is given by
!
X 21/2 e6
Z 2 nj ne Te1/2 ,
(2.36)
SB =
5/2
3/2
3
3
3π
ε0 c hme
j
3 Some fast particle losses are expected due to imperfections in the magnetic field and interactions with MHD instabilities (mostly Alfvénic modes), but the magnitude of these losses is
expected to be a few percent at most [66].
2.6 Power balance
29
with Z the ion charge, nj the density of ion species j and Te the electron temperature. For future references it is convenient to define
!
1/2 e6
2
CB =
.
3/2
3π 5/2
ε30 c3 hme
Synchrotron radiation is emitted by charged particles because of the acceleration associated with their gyration around the magnetic field lines. Because of
re-absorption and reflection on metallic surfaces, quantification of net radiated
synchrotron power requires an involved calculation, although simpler fitting formulas have been derived [67]. The emitted synchrotron power in ITER is negligible compared to the Bremsstrahlung everywhere except in the core of the plasma,
where they are approximately equal [68]. However, synchrotron radiation losses increase rapidly with temperature and at higher plasma temperatures can increase
to about 20% of the total energy losses from the plasma (or roughly twice the
Bremsstrahlung losses) [67]. Nevertheless, because synchrotron losses also depend
strongly on the temperature profile, first wall reflectivity and reactor geometry, we
have choses to neglect them in our analysis.
At the plasma edge (or in case high Z impurities are present) line radiation,
either from charge exchange or non-fully ionised atoms, also forms a loss mechanism. However, line radiation from the plasma core is only important during the
so-called burn through phase at the beginning of the discharge. Once the plasma
reaches fusion relevant temperatures line radiation from the core can be safely
neglected for our purposes.
Combining equations (2.34), (2.36) and (2.11), and setting Pext = 0, a criterion
for ignition can be derived for a 50:50 DT plasma:
ne τE =
3T
,
− CB T 1/2
1
4 hσviEα
(2.37)
which shows that the product of density and confinement time required for ignition
can be expressed as a function of T . In fusion research it is common to multiply
both sides by T , to obtain the triple product ne τE T on the left. The triple product
is a convenient figure of merit for a reactor because generally speaking the pressure
(product of density and temperature) scales with the magnetic field and the energy
confinement time scales with machine size, both of which have a big share in the
cost of the reactor. Increasing this triple product for a given reactor is therefore
a good measure of progress. Figure 2.7 plots the minimum value of the triple
product as a function of T , where it can be seen that the Bremsstrahlung has the
effect of increasing the minimal ignition temperature, but doesn’t affect the burn
equilibria at higher temperature.
The curves plotted in figure 2.7 are contours of Ṫ = 0 with the dot denoting
differentiation with respect to time (t), but the graph provides no information on
the stability of these equilibria. In this case an equilibrium is stable if the second
Chapter 2 Theory
30
60
ne τE T (atm s)
With Bremsstrahlung
ahlung
30
Bremsstr
40
Without
50
20
10
0 0
10
101
T (keV)
102
Figure 2.7: The triple product at ignition plotted against T . The dashed line is
the ignition curve without radiation losses, which only deviates significantly at
temperatures below 10 keV.
2.7 Burn equilibria
31
derivate of T is negative, so T̈ < 0, and unstable if T̈ > 0, which can be seen in a
graph that plots Ṫ as a function of T . To do this, the unknown confinement time
needs to be eliminated from the left hand side of the burn equation, to which end
a scaling law, such as the IPB98(y,2) scaling [26], introduced in section 2.3.7, can
be used.
Figure 2.7 plots Ṫ as a function of T for a hypothetical fusion reactor (the
ignition experiment presented by Freidberg in table 14.3 on page 520 in ref. [7]),
at a density of 1.1 × 1020 m−3 . While it is necessary to choose specific values for
the engineering parameters, the graph would look similar for any reactor capable
of reaching ignition.
For temperatures between 0 and roughly 5 keV Ṫ < 0, meaning that external
heating is required to keep the plasma stable at a temperature in that range.
Between 5 and 17 keV Ṫ > 0, so once the plasma enters this range the temperature
will increase by itself until it reaches 17 keV, since above 17 keV Ṫ < 0 again
(without external heating). This also means that the equilibrium at 5 keV is
unstable and the one at 17 keV is stable (albeit at a higher fusion power than at
5 keV.
Given the characteristics of the equilibria it seems logical to refer to the unstable equilibrium at lower temperature as the ignition point, since from that point
onward the plasma will sustain itself and the external heating can be switched off.
The second, stable, equilibrium at higher temperature can be seen as the burn
point, since this is the operating point the plasma will converge to in the absence
of external control.
2.7
Burn equilibria
The assumption of a pure DT plasma is useful to understand the power balance
in a fusion plasma and how this translates to ignition and stable burn, but is
not self-consistent because it neglects the alpha particles produced by the fusion
reaction. Since these form the energy source that keeps the plasma at the required
temperature, a proper treatment of fusion plasmas needs to include the helium
ash. In this section we will introduce the model used by Reiter et al. and Rebhan
et al. [41, 49, 53, 65], which includes the helium concentration self-consistently and
also allows a (fixed) concentration of impurity ions.
Using the particle confinement time as defined above, the helium balance in
the plasma reads
nD nT hσvi =
nα
,
τp
(2.38)
with nα the helium density. Given the fact that the densities of the different
ion species are coupled through the electron density, it is convenient to define a
Chapter 2 Theory
32
dilution parameter
fi =
ni
= 1 − ZfZ − 2fα ,
ne
(2.39)
ni = nD + nT being the fuel density, fα = nα /ne and fZ = nZ /ne where nZ is the
impurity density. The total particle density ntot then is
ntot = ne + ni + nα + nZ = ne ftot ,
(2.40)
with ftot = 1 + fi + fα + fZ = 2 − (Z − 1)fZ − fα being the total particle fraction.
Unless stated otherwise, we will assume nD = nT from this point onwards, so
ne = ni + 2nα + ZnZ .
(2.41)
Using these definitions, equation (2.38) can be written as
ne τE =
4fα
ρfi2 hσvi
(2.42)
6ftot T
,
− 4Rrad (T )
(2.43)
and equation (2.37) changes to
ne τE =
fi2 hσviEα
with
Rrad = fi RB,1 + fα RB,2 + fZ RB,Z
2 √
1
4
Z
= CB T fi gff
+ 4fα gff
+ Z 2 fZ
.
T
T
T
(2.44)
(2.45)
These are the Bremsstrahlung losses written in a slightly different form, with
gff (Z 2 /T ) the Gaunt factor.
√ With respect to equation 2.4 in [41], we have used
the approximation gff = 2 3/π from Wesson [69] and neglected any line radiation
losses from impurities (which make only a minor contribution for low Z impurities
with temperatures between 5 and 100 keV).
We have now two expressions for ne τE , so combining equations (2.42) and (2.43)
and substituting fi = 1 − ZfZ − 2fα results in a cubic expression for fα :
a0 + a1 fα + a2 fα2 + a3 fα3 = 0,
(2.46)
2.8 Reactor studies
33
with
3 a0 = − T fZ3 (Z 2 − Z 3 ) + fZ2 (4Z 2 − 2Z) + fZ (1 − 5Z) + 2 ,
2
3 2
a1 = − T fZ (4Z − 5Z 2 ) + fZ (14Z − 4) − 9
2
Eα 2 2
+
fZ Z − 2fZ Z + 1
ρ
4
+
[(fZ Z − 1)RB,1 (T ) − fZ RB,Z (T )] ,
ρhσvi
4Eα
3
[fZ Z − 1]
a2 = − T [fZ (4 − 8Z) + 12] +
2
ρ
4
+
[2RB,1 (T ) − RB,2 (T )]
ρhσvi
(2.47a)
(2.47b)
(2.47c)
and
a3 = 6T +
4Eα
.
ρ
(2.47d)
Equation 2.46 can be solved to obtain the helium concentration as a function
of T. This result can then be inserted into equation 2.42 to plot the burn contours
in the ne τE , T -plane, something that we will come back to in section 3.2.3.
2.8
Reactor studies
Even though construction of ITER, the first fusion reactor with Q > 10, has not
yet been finished, most ITER partners are already developing preliminary reactor
designs for a commercial fusion power plant. For the European countries this is
the power plant conceptual study (PPCS) [70, 71].
The PPCS includes five designs for a demonstration reactor, labelled PPCS A,
AB, B, C and D. They differ from each other in the maturity of their technology,
and are anticipated to be representative of the first three or four generation power
plants. Their main design parameters are summarised in table 2.2
The PPCS A, AB and B designs rely on materials and technology that is
currently being developed, and expect an improvement in plasma parameters,
mainly density and pressure, of about 20% over the ITER values. For PPCS
models C and D, advanced materials need to be developed and another gain of
about 20% in plasma performance is required. In the remainder of this thesis the
reactor designs from the PPCS will be used to investigate the burn equilibria and
their stability.
34
Chapter 2 Theory
Table 2.2: Proposed parameters for the PPCS reactors. The plant efficiency is
defined as the ratio between net power and fusion power, with net power the
electric power delivered to the grid. Adapted from [70].
Parameter
Major radius (m)
Minor radius (m)
Aspect ratio
Bφ (T)
Ip (MA)
Avg. ne (1020 m−3 )
ne /nG
βN (thermal/total)
H98
Zeff
Pfus (GW)
Blanket gain
Pnet (GW)
Padd (MW)
Plant efficiency
Model A
9.55
3.18
3
7.0
30.5
1.1
1.2
2.8/3.5
1.2
2.5
5.00
1.15
1.55
246
0.31
Model AB
9.56
3.19
3
6.7
30.0
1.05
1.2
2.7/3.5
1.2
2.6
4.29
1.18
1.50
257
0.35
Model B
8.6
2.87
3
6.9
28.0
1.2
1.2
2.7/3.4
1.2
2.7
3.60
1.39
1.33
270
0.37
Model C
7.5
2.5
3
6.0
20.1
1.2
1.5
3.4/4.0
1.3
2.2
3.41
1.17
1.45
112
0.42
Model D
6.1
2.03
3
5.6
14.1
1.4
1.5
3.7/4.5
1.2
1.6
2.53
1.17
1.53
71
0.6
2.9 Stellarators
2.9
35
Stellarators
The stellarator is an alternative type of fusion reactor, which, like the tokamak, relies on magnetic confinement in a toroidal geometry. Contrary to tokamaks, which
are (quasi-) axisymmetric, stellarators generally have a complicated magnetic geometry. As a measure of the helicity, or pitch, of the field lines, the rotational
transform ι is used for stellarators, which is related to the safety factor q in a
tokamak in the following way:
ι=
q
.
2π
(2.48)
The magnetic field in a stellarator is completely generated by external coils
and it therefore has no plasma current. This makes it an inherently steady state
design, with the added benefit that it is immune to disruptions. The absence
of disruptions also means that there are no hard operational limits like the β and
density limits in a tokamak. Although there are (MHD) instabilities in stellarators,
they result in a degrading of the confinement, which brings the plasma back to a
stable regime.
Due to the complicated structure of the magnetic field, stellarators always suffered from worse confinement than tokamaks. However, recently the construction
of the first optimised, large scale, stellarator has been completed in Greifswald,
Germany [72]. If successful, this might lead to a stellarator reactor design, dubbed
HELIAS [73].
The transport processes in stellarators are the same as in tokamaks and consequently, the same approach using a scaling law for the energy confinement time
is used [74]:
τE = 0.134a2.28 R0.64 B 0.84 ι0.41 ne0.54 P −0.61 ,
(2.49)
where ι takes the place of the plasma current that is present in the scaling laws
for tokamaks.
Apart from being inherently stable, stellarators have the benefit of requiring no
external heating for current drive, thus having a lower recirculating power fraction
than a tokamak of comparable size. Although stellarators will not be treated
explicitly in the remainder of this thesis, the analyses performed are also valid for
stellarators since the scaling law for τE has the same form.
36
37
Chapter 3
Burn Equilibria
3.1
Introduction
When designing a power-producing fusion reactor, one needs to know how it will
operate exactly. Necessarily, future fusion reactors are thought to operate at, or
very close to, ignition, where the alpha particles provide essentially all heating
power, otherwise they would never succeed in generating electricity at a competitive cost. However obvious this may be, it doesn’t say anything about the
composition and temperature of the plasma in operating point. Yet this is crucial information for maximising performance within the operational and material
limits of the reactor. Furthermore, knowledge of the stability and sensitivity to
changes in plasma and machine parameters of the operating point is important for
burn control purposes.
A good starting point for the analysis of the operating space in a reactor is
a global 0D model based on the energy balance of the plasma. The model can
then be expanded by including the particle balances of deuterium, tritium and
helium to investigate the effects of fuel burn up and ash accumulation. This has
been done ad hoc, by assuming the helium density is a certain fraction of the fuel
density, and self-consistently where the helium density is calculated using a particle
confinement time which is proportional to the energy confinement time, a method
which was first introduced by Reiter et al. [41]. Furthermore the effect of profile
shaping was investigated in a follow-up paper by Reiter et al. [49], in which the
shaping factors were defined by the volume averaged value of the corresponding
variables, and it was shown that a change in one of the profiles is equivalent to a
translation of the system in the ne , T plane.
Later, Rebhan et al. resolved the discrepancy introduced by the different definitions of the confinement time (including or excluding radiation losses) [53], and
investigated the burn stability of the old ITER design using the ITER89P L-mode
scaling to eliminate the confinement time from the equations [65].
38
Chapter 3 Burn equilibria
Since an analysis based on the ITER98 H-mode scaling is lacking and this scaling forms the basis for most reactor studies, it seems prudent to carry one out.
Building on the work of Rebhan et al. [65] we will derive an analytical relationship between the density and temperature in a burning plasma and compare the
obtained burn contours to those based on the ITER89P L-mode scaling. Furthermore, we will use this expression to investigate and comment on the sensitivity of
the system with respect to changes in energy and particle confinement.
Due to the nature of the scaling laws, the solutions that we find for the operating contours extend over many orders of magnitude in electron density. Obviously,
solutions outside the density range on which the scaling law is based should be
treated with extreme caution, and solutions far outside this range have no physical
meaning whatsoever.
In a real tokamak there are several mechanisms that will limit the density of
the operating contours to more reasonable levels. The high density regime is not
accessible because of the Greenwald and Troyon limits. On the low density side,
there are at least three things to consider. At high temperatures and low densities,
synchrotron radiation will become the dominant loss mechanism, because it scales
linearly with density, as opposed to the Bremsstrahlung and fusion power which
have a quadratic density dependence. We have neglected synchrotron radiation
losses because at power plant relevant densities they have only a minor effect on
the energy balance.
Then there is the LH-transition, which has a power threshold which scales
roughly linearly in ne [75], meaning that for very low fusion power the plasma will
not enter H-mode.
And finally, the alpha slowing down time depends on the density, and cannot
become too high, otherwise the alpha particles will be lost before they can transfer
their energy. The relevant time scale here is τE and not τp , because alpha particles
cannot be recycled at the edge while maintaining their energy.
For consistency with earlier work and because we are interested in the shape
of the solutions, we have chosen to plot the full operating contours in many cases.
However, when discussing the effect of different parameters on fusion power, we
have chosen an electron density ne = 1 × 1020 m−3 , or selected a reactor relevant
density range.
Because all current reactor designs feature some amount of external heating, we
will investigate the effect this has on the operating contours. We will then extend
this analyses to include the effect of impurities and look at the sensitivity of the
net electric power to the external heating power. The chapter will conclude with
the effect of uncertainties in the confinement scaling on the operating contours and
fusion power.
This chapter is a synthesis of two papers, which are complementary to each
other. Section 3.2 is included here as published in Nuclear Fusion, while an adapted
version of section 3.3 will be submitted to Nuclear Fusion.
3.2 Burning plasmas
3.2
Burning plasmas1 ;
3.2.1
Introduction
39
In a fusion reactor of the type tokamak, a plasma of the hydrogen isotopes deuterium and tritium is kept at a temperature of hundreds of millions Kelvin, confined
in a toroidal geometry by means of magnetic fields. To start the reactor, external
heating is applied to bring the plasma to the ‘ignition’ point: a combination of sufficiently high temperature and density at which the heating power delivered by the
fusion reactions balances the heat loss. Once the plasma is ignited its temperature
– and thereby the fusion power – increases autonomously until the stable ‘burn
temperature’ (Tburn ) is reached. Above this temperature the heat losses increases
faster than the fusion power. For a given Tburn , the electron density ne is therefore
determined by the reaction rate [4] and heat loss rate, which is the sum of the radiation and conduction losses. The latter are conveniently expressed by the energy
W
, defined as the ratio of the kinetic energy content W
confinement time τE = Pcond
of the hot plasma and the conductive power losses Pcond . As the heat loss is the result of complex turbulent processes, empirical scaling laws are used which express
τE as a function of operational parameters such as the geometry of the reactor, ne
and heating power. There are but a few global parameters under operator control
that influence Tburn and might be used for burn control. Important ones are ne ,
the mixing ratio of the two fuel components deuterium (D) and tritium(T) and
the quality of confinement, expressed by the H-factor H98 = τE /τIPB98(y,2) , i.e.
the value of τE compared to the scaling law prediction. A fourth and less obvious
burn control parameter is the ratio of particle and energy transport
τp
ρ=
(3.1)
τE
In tokamak reactors, particle confinement is much better than energy confinement, with ρ typically between 5 and 10 [34], with values of 10-30 also reported [76].
The paradox of the fusion reactor is that whereas good energy confinement is essential to reduce power losses, good particle confinement makes the reaction choke
on its own ash. The effect of particle confinement on the burn equilibrium is evident from the contours in the ne τE , T -plane (assuming T = Te = Ti ) for which
the fusion power heating balances the losses, an analysis already performed by
Reiter et al. [41]. Note that whereas the contours are open towards high energy
confinement when the choking effect of particle confinement is neglected (ρ = 0),
taking this effect into account closes and constricts the contours. For ρ > 14.7 no
sustained burn is possible. To complicate matters, a further constraint comes from
the fuel cycle, which requires ρ to be sufficiently high as was shown by Sawan et
al. [64].
1 Published as: Jakobs, M.A., Lopes Cardozo, N.L.C. and Jaspers, R.J.E., Fusion burn equilibria sensitive to the ratio between energy and helium transport, Nucl. Fusion 54 (2014) 122005
Chapter 3 Burn equilibria
40
1017
1016
ρ=
3
ne τE (s/cm )
ρ=0
ρ=1
ρ=
ρ=
5
9
13
1015
1014
10
100
T (keV)
Figure 3.1: The plasma operating contours (POPCON [77] in the ne τE , T -plane
for different values of ρ. For ρ = 0 the curve is open because there is no helium to
choke the reaction. For increasing values of ρ the operating range in T and ne τE
becomes more and more limited, until it vanishes for ρ = 14.71.
3.2 Burning plasmas
41
For these reasons, we address the question how the Tburn and Pburn change
under variation of H98 and ρ, as well as ne , while assuming that the fuel mix is
50-50. We first introduce the basic equations and definitions of the 0D-model,
following the work of Freidberg [7], Reiter et al. [41] and Rebhan et al. [53, 65].
Although inclusion of profile effects will quantitatively change the analysis, Reiter
et al. [49] showed that the qualitative properties of the system remain the same.
We then present an expression for the ne (T ), construct universal burn contours
and derive two new results for the influence of energy and particle confinement on
the burn equilibrium.
3.2.2
Theory
The energy balance of a burning fusion plasma is approximated by
Sα = Srad + Sκ ,
(3.2)
with Sα the alpha particle heating and Srad and Sκ the losses due to radiation
and conduction, respectively. External and Ohmic heating are neglected as they
have a minor influence on the burn equilibrium. The alpha power density is given
by Sα = nD nT hσviEα , where hσvi is the DT-reactivity [4], Eα = 3.52 MeV the
energy of the alpha particle that is produced in the DT-reation, while nj denotes
the number density of species j in units of m−3 . The dominant radiation loss
P
1/2
is due to the Bremsstrahlung, given by SB = j CB Z 2 nj ne gff Te,keV with CB =
5.35×10−37 Wm3 , Te,keV the electron temperature√in keV, Z the ion charge, gff the
Gaunt factor which we have approximated with 2 3/π and the summation is over
all ion species. To account for the helium density (nα ) resulting from the fusion
reactions we write nD nT hσvi = nα /τp , thereby assuming that the confinement of
alpha particles is the same as that of other species. We further introduce the fuel
dilution parameter fi = (nD + nT )/ne and the alpha fraction fα = nα /ne . Using
these notations, ref [41] finds
ne τE
ne τE
=
=
4fα
ρfi2 hσvi
6ftot T
,
fi2 hσviEα −4Rrad (T )
(3.3)
(3.4)
and by solving these equations arrives at burn contours, i.e. the contours in
the ne τE , T -plane for which the alpha heating balances the losses.
3.2.3
Results
We have used the same formalism to produce the burn contours in shown in figure 3.1. Note that ne τE must exceed a critical value for burn to occur. For given
ne τE there are two solutions: the unstable ignition temperature (left hand branch)
Chapter 3 Burn equilibria
42
and the stable burn temperature (right hand branch), in agreement with the intuitive picture of ignition and burn.
In this calculation τE is an independent parameter, whereas in fact it depends
on plasma parameters. Rebhan et. al [65] proposed a self-consistent analysis
by using a scaling law which expresses τ̃E = W/(Pcond + Prad ) as a function of
plasma parameters. They used the ITER89 L-mode scaling [29] to find burn
contours in the ne , T -plane for this specific scaling, for a specific choice of reactor
parameters. We follow this approach, using the now more relevant scaling for Hmode confinement, the IPB98(y,2) -scaling [26], which is commonly used to predict
the performance of future fusion devices. Since the radiation losses are not included
in IPB98(y,2) , the method applied in ref [65] cannot be used. Instead, we inserted
the expression for the alpha heating power to eliminate the confinement time and
derived an expression for ne as a function of T which is valid for all scaling laws
of the form τE = KAk nle P −m :
ne =
4fα Ak
ρK
1
1−2m+l
1 2
f hσvi
4 i
m−1
1−2m+l
m
(Eα V ) 1−2m+l .
(3.5)
Here K is a constant that depends on the engineering parameters of the reactor,
A is the average ion mass in amu and P the power deposited in the plasma (by
the alpha particle or external sources).
Since K and the plasma volume V are the only reactor specific parameters in
this equation, ne (V −m K)1/(1−2m+l) represents a normalised density that is the
same for all fusion reactors that follow the same scaling law.
Figure 3.2 shows the POCPONs for ITER ID [78] and 3 conceptual reactor
designs PPCS models A - C as described in the conceptual power plant study [70].
There is a large difference between the IPB98(y,2) and ITER89P scaling for the
ITER ID reactor. The solid curves for ITER ID and PPCS models A - C are
isomorphic, which can be shown by applying the normalisation described above.
It is important to note that while the formalism using the scaling laws leads
to burn equilibria at values of ne and T that are far from the normal operating
conditions of a fusion reactor, these are probably artefacts due to the mathematical
form of the scaling laws. Reliable extrapolations can only be made in the parameter
range where the database on which the scaling laws are based is well populated,
i.e. with ne in the range 1019 to 1021 m−3 .
To elucidate the role of particle confinement, while zooming in on the reactor
relevant ne range, figure 3.3 shows the burn contours of PPCS model A in the ne , T plane. The plot shows clearly how, at constant density, the reactor will move from
ignition at temperature of 5 to 8 keV to burn at a temperature around 30 keV,
while the fusion power at the same time increases by an order of magnitude. The
fusion power at ignition and burn depends quadratically on ne , and therefore the
Greenwald density limit nG = Ip /πa2 is of fundamental importance. For all but
the lowest ρ-values this limit is more restrictive than the Troyon pressure limit
3.2 Burning plasmas
43
1020
Density limit
ne (m
−3
)
1021
PPCS A
PPCS B
PPCS C
ITE
R
ID
1022
β li
mit
ITER ID 89L
1019
4
5
6 7 8 9 10
20
30
40
50
T (keV)
Figure 3.2: The POPCONS for ρ = 5 for the ITER ID [78] and the PPCS A,
B and C designs [70]. The contours for the PPCS reactors are made with the
IPB98(y,2) scaling, as is the solid blue ITER ID curve, whereas the dashed ITER
ID contour is created using the 89L-scaling following the procedure developed
in [65]. The Greenwald density limit and the Troyon β limit for the ITER ID
design are plotted. They also give a good indication of these limits for the PPCS
designs, although the precise position is reactor specific.
2G
W
10
ρ=0
nG
1G
20
W
-3
ne (m )
500
200
100
1019
ρ=5
W
ρ = 10
5G
ρ = 14.71
Chapter 3 Burn equilibria
44
4
5
β
lim
it
MW
MW
MW
6 7 8 9 10
20
30
40
50
T (keV)
Figure 3.3: The operating contours for ρ = 0, 5, 10 and 14.71 for the PPCS A
design made with the IPB98(y,2) scaling. The colour indicates the fusion power in
each operating point and the dashed dotted lines indicate lines of constant fusion
power. The Greenwald density limit and the Troyon β limit are also indicated.
3.2 Burning plasmas
I
45
p
given by βmax = gT aB
with Ip in MA and the Troyon factor gT = 0.03.
Calculating the β limit requires knowledge about the composition of the plasma
and that is only available on the equilibrium contours, so plotting it is not straightforward. We have taken the following approach: for figs. 3.2 and 3.3 we have
calculated the β limit for a pure DT plasma, which results in a underestimation
of ne of at most 20%. For figs. 3.4 and 3.5 we have taken the value of fα and T
at the equilibrium and used that to calculate the value of ne and subsequently the
fusion power at the β limit. This results in a β limit that has two values a one
value of ρ and H98 (because every point on an equilibrium contour is associated
with a point on a β limit contour).
Figure 3.4 shows that the Tburn , and therefore Pburn , will change under variation
of ρ. In other words, if by some process in the plasma or in the exhaust the ratio
between particle and energy confinement changes, this will significantly affect the
output power of the reactor. This may be a point of concern as it may lead to
unwanted excursions of Pburn , but may also have potential as actuator for burn
control. The dependence of fusion power (at burn equilibrium) on ρ is depicted
in figure 3.4 for PPCS models A, B and C, for ne = 1 × 1020 m−3 . These curves
show a lower (ignition) and upper (burn) branch that meet at ρmax , the highest
value of ρ that can be tolerated at this particular density. For PPCS model A
ρmax ≈ ρcrit , the fundamental limit on ρ set by the Bremsstrahlung and fusion
cross section for the DT-reaction. For PPCS models B and C ρmax < ρcrit . Along
most of the upper branches, i.e. in the burn equilibria for 5 < ρ < 10, Pburn
is approximately inversely proportional to ρ. Note that part of the high power
branch is not accessible because it exceeds the β limit, but the expected impact
for future reactor designs is minimal because constraints on the achievable tritium
breeding ratio will most likely set a lower limit of ρ = 5 [64].
In another projection of the parameter space, the influence of H98 on Pburn
can be analyzed. Figure 3.5 displays Pburn as a function of H98 for ρ = 5, 10
and ne = 1 × 1020 m−3 , for PPCS model A. We see that below H98 = 0.73 and
0.83 for ρ = 5 and 10 , respectively, there is no ignition because the confinement
is too low. For the stable burn branch (top half of the contour), increasing H98
first results in a steeply increasing Pburn until a maximum is reached at H98 =
1.1 to 1.3, depending on ρ. A further increase in H98 will lower Pburn because the
increased helium content due to the better confinement chokes the fusion reaction.
For ρ . 9, some parts of the high power branch exceed the β limit, which needs
to be taken into account when choosing the operating point for a reactor.
On the unstable ignition branch (bottom half of the contour), an increase in
H98 beyond its minimum value initially results in ignition at lower temperature
and power. Both branches meet again at the maximum H98 = 6.38 and 2.79
(ρ = 5 and 10) and beyond that there are no more burn equilibria, i.e. the fuel
has become so diluted that the fusion power can no longer balance the conduction
and radiation losses.
Chapter 3 Burn equilibria
46
104
Fusion power (MW)
β limit
β limit
PP
C
β limit
PP
CS
PP
CS
C
SA
B
103
5
10
15
ρ
Figure 3.4: The fusion power at a constant electron density ne = 1 × 1020 m−3 as
a function of ρ for PPCS models A, B and C. It can be clearly seen that there
above the critical value ρ = 14.7 there are no burn equilibria for PPCS model A.
For PPCS models B and C, the maximum value of ρ is lower because the burn
contours are shifted towards higher densities with respect to PPCS model A. Below
the maximum value of ρ there are two equilibria: the unstable ignition branch at
lower fusion power and the stable burn branch at higher power. The latter is
especially sensitive to changes in ρ, but can exceed the β limit (dash dotted lines)
for low value of ρ (the dotted part of the curves). The PPCS D design does not
ignite in the current model below ne ≈ 1 × 1020 m−3 .
3.2 Burning plasmas
47
Fusion power (MW)
104
it
im
βl
103
β lim
it
ρ = 10
ρ=5
0.5
1
1.5
2
2.5
3
3.5
4
H98
Figure 3.5: The fusion power as a function of the H-factor H98 = τE /τE,98 for
PPCS model A at a density of ne = 1 × 1020 m−3 . While it is not surprising that
a too large reduction in the H-factor will lead to a loss of burn, the result when
H98 increases deserves more attention. The fusion power initially increases but
reaches a maximum around H98 = 1.3 for ρ = 5 and at H98 = 1.14 for ρ = 10,
beyond which it drops until at some point the burn equilibria vanish (H98 = 6.4
and H98 = 2.8 respectively. The dash-dotted lines indicate the β limit for the
two contours, and at low values of ρ this is the limiting factor of the fusion power
(dotted part of the contour).
Chapter 3 Burn equilibria
48
In short, for a given reactor there is no gain to be expected from improvement
of energy confinement. Either the plasma exceeds the β limit, or the power output
decreases. Rather, the reactor should be designed in such a way that its operating
point is at Hmax , provided it does not conflict with the β limit. Of course, better
confinement does allow one to reach ignition in a reactor with smaller dimensions
and lower Pburn .
3.2.4
Conclusions
We have derived an analytical expression (3.5) relating T and ne in a fusion reactor
with self-consistent treatment of fuel burn up and helium accumulation, using the
IPB98(y,2) scaling law for confinement time. This expression is valid for all fusion
reactors that obey the same energy confinement scaling if one takes into account a
scale factor that depends on the reactor parameters. Using these results we have
plotted the burn contours of the PPCS A design in the ne , T plane, including the
curves of constant fusion power and the Greenwald and Troyon limits.
The fusion power at these equilibria was found to be very sensitive to changes
in ne , ρ and H98 . The fusion power scales quadratically with the density around
the Greenwald density, although this will be different for reactors that have a
minimum density for ignition that is close to this limit. The dependence on ρ is
especially strong for intermediate to high values of ρ, and since the value of ρ is
to a large extent determined by the helium exhaust at the plasma edge, this offers
possibilities for burn control using helium pumping [79, 80]. The effect of H98 on
the fusion power could have implications for advanced tokamak scenarios where
values of H98 well above 1 are expected.
3.3 Burn equilibria with impurities and Pext
3.3
Burn equilibria with impurities and Pext
3.3.1
Introduction
49
The analysis in section 3.2 deals exclusively with equilibria in burning plasmas
without external heating, which is unlikely to be achieved in a tokamak because
of the need for non-inductive current drive. Furthermore, we have only looked
at ’pure’ plasmas so far, containing only deuterium, tritium and helium. A real
fusion plasma always contains a non-zero amount of impurities, be it beryllium or
tungsten from the reactor wall, or for example neon or nitrogen to increase the
radiated power in the divertor.
This section therefore investigates the burn equilibria in PPCS model A with
external heating. First we derive a way to determine the minimum and maximum
temperature on a POPCON, after which we present a procedure to determine the
helium fraction in a plasma with external heating. Then we analyse the operating
contours with external heating, and describe the effect of impurities on the operating contours and the power output at a given operating point. Finally, we study
the effect that external heating has on the net electrical power delivered to the
grid.
3.3.2
Temperature domain of a burning plasma
It turns out that we do not need to solve equation (2.46) completely to be able to
say something about the solution. In fact, we can already determine the allowed
temperature domain of a burning plasma by looking only at the determinant
∆ = 18a3 a2 a1 a0 − 4a32 a0 + a22 a21 − 4a3 a31 − 27a23 a20 ,
(3.6)
which is only a function of ρ and T .
Equation (2.46) has three real solutions if ∆ > 0, two real solutions of which
one is a multiple root if ∆ = 0 and one real and two imaginary roots for ∆ < 0.
The solutions we are looking for have to satisfy 0 < fα < 0.5, because fα = 0.5
corresponds to a pure helium plasma and fα = 0 is only possible in case τp =
ρτE = 0, which for finite ρ would mean that ne τE = 0 which has no physical
relevance.
It turns out that there is a real root fα > 0.5 for 1 keV < T < 1000 keV that we
can discard. Because we are looking for physically meaningful solutions (fα has to
be real), this restricts us to the temperature domain where ∆ ≥ 0. In figure 3.6 the
discriminant is plotted for several values of ρ in the domain 1 keV < T < 1000 keV.
Below 4 keV ∆ > 0, but the two real roots here are both negative and thus have
no physical meaning. This leaves us with ∆ > 0 in a temperature window ranging
from 5 to several hundred keV, depending on the value of ρ, where the solutions
Chapter 3 Burn equilibria
50
2
∆/∆peak
1
0
102
ρ = 0.1
101
ρ=1
−2
100
ρ=5
9
ρ=
3
ρ=1
−1
103
T (keV)
Figure 3.6: The discriminant ∆ of equation (2.46) plotted as a function of T for
different values of ρ. The values are normalised to the peaks around 10 keV.
3.3 Burn equilibria with impurities and Pext
51
satisfy 0 < fα < 0.5. Thus, the second and third zero crossing in figure 3.6 are the
lower limit Tmin and upper limit Tmax of the accessible temperature window, and
by definition the two roots coincide at these points.
3.3.3
Helium fraction with external heating
The derivation of a cubic equation for the helium fraction by means of equations (2.42) and (2.43) no longer works for a plasma with external heating because
the product of ne and Pext shows up in the resulting equation. Instead we can
divide both equation (2.42) and (2.43) by ne and equate the two expressions for
τE that we obtain that way.
Solving for fα again yields a cubic equation
0 = a0 + a1 fα + a2 fα2 + a3 fα3 ,
(3.7)
with coefficients
3 3 2
3
2
2
a0 = − T fZ (Z − Z ) + fZ (4Z − 2Z) + fZ (1 − 5Z) + 2 n2e ,
2
3 a1 = (− T fZ2 (4Z − 5Z 2 ) + fZ (14Z − 4) − 9
2
Eα 2 2
+
fZ Z − 2fZ Z + 1
ρ
4
Sext
+
(fZ Z − 1)RB,1 (T ) − fZ RB,Z (T ) + 2 )n2e ,
ρhσvi
ne
4Eα
8RB,1 (T )
1
3
(fZ Z − 1) −
,
a2 = 2 − T (fZ (4 − 8Z) + 12) +
ne
2
ρ
ρhσvi
Eα
n2e ,
a3 = 6T + 4
ρ
(3.8a)
(3.8b)
(3.8c)
(3.8d)
with Sext the external power density. These differ from the coefficients in equation (2.47) by a factor of n2e and have an extra term 4Sext /ρhσvi in a1 . In this
case only one of the roots satisfies 0 ≤ fα ≤ 0.5, and the solution is
fα = −
a2 + C +
3a3
∆0
C
(3.9)
,
with the discriminant ∆ defined by equation (3.6) and
∆0 = a22 − 3a1 a3 ,
∆1 =
C=
2a32
(3.10)
27a0 a23 ,
!(1/3)
− 9a1 a2 a3 +
p
∆1 + −27a23 ∆
2
(3.11)
.
(3.12)
52
Chapter 3 Burn equilibria
All terms on the right hand side of equation (3.9) are either a free parameter or
a function of T , so we have the desired expression for fα . The catch is that from
equation (3.9) alone we cannot tell what the equilibrium temperature is.
One possible solution is to use the obtained value for fα to compute Pfus and
insert this in the scaling law to compute τE . This value of τE can then be equated
to the value of τE obtained from equation (2.42). Solving for T then yields the
desired equilibrium temperature. This cannot be done analytically, but is easily
achieved numerically and this procedure is much faster than finding the equilibrium
by solving the energy and particle balance simultaneously for fα and T .
3.3.4
Burn equilibria with external heating
The operating contours presented so far have been obtained for a plasma without external heating. This provides insight in the operational space of an ignited
plasma, but future reactors are designed to operate with some external heating for
current drive, and possibly control purposes. This leaves us with density, temperature, ρ and Pext as parameters, of which three can be chosen ’freely’ (respecting
machine and plasma limits of course).
First, we want to know how the addition of external heating changes the operating contours that were obtained previously. Figure 3.7 shows this for the PPCS
A reactor design with different levels of external heating. The solid lines represent
the operating contours for ρ = 5 (blue) and ρ = 10 (red), and the dashed lines
represent operating contours with different levels of external heating.
As expected, the equilibria with external heating almost coincide with the
contours obtained previously at high densities, because in this region the external
heating is insignificant compared to the fusion power. These equilibria are well
above the Greenwald density or the β limit and therefore of little meaning for
reactor design, except for a reactor so large that the operational contours are
shifted towards lower densities.
The equilibria at low density and intermediate temperature (between 5 and 20
keV) have disappeared for reasonable amounts of external heating for PPCS model
A, because there the external heating outweighs the power losses from the plasma.
This is of little consequence, since these equilibria are mainly a mathematical
artefact and are not relevant for realistic reactor designs because of the (extremely)
low densities.
When following a burn contour toward lower temperatures, the curves with
external heating start to deviate and they converge on a minimum density that is
only a function of external heating and independent of ρ. This density is determined by the balance between radiation losses and external heating because there
is little alpha heating at these low temperatures.
After reaching this (local) minimum in the density, the curve will turn upwards
again when the temperature decreases even further. Since our model does not
53
ρ=5
ρ=5
ρ = 10
1022
ρ = 10
3.3 Burn equilibria with impurities and Pext
1021
ne (m−3 )
200
100
MW
MW
nG
50 MW
1020
β lim
it
10 MW
20
0M
10
1019
W
0M
50
M
W
W
10
1018
4
5
6
7 8 9 10
20
T (keV)
30
40
M
W
50
Figure 3.7: POPCON plot for PPCS model A with fixed external heating power
for ρ = 5 (
) and ρ = 10 (
). The solid lines represent contours without
external heating. For high densities, the curves almost coincide with the burn
curves because the external heating is only an insignificant fraction of the fusion
power. The equilibria at low densities have disappeared because in that case both
the conduction and radiation losses are very small and outweighed by the external
heating. On the low temperature side, the equilibrium is mainly determined by
the radiation losses because there is hardly any fusion power due to the low cross
section at these temperatures. On the high temperature side, the equilibrium is
also set by the balance between radiation losses and external heating, but the curve
has a downward slope because an increase in temperature has to be compensated
by a reduction in density to maintain equilibrium.
Chapter 3 Burn equilibria
54
include (impurity) line radiation and Ohmic heating, the results for temperatures
below a few keV should be treated with extreme caution.
On the high temperature (stable) branch, the external heating curves also
deviate from the burn contours for decreasing temperature. In this case the curves
trend towards an asymptote determined by the degradation
√ of confinement with
increasing power and the increase in radiation losses with T . In this temperature
range, around 70 keV, the reactivity is almost temperature independent and the
fusion power is only sensitive to fuel density.
From a reactor perspective, the interesting area are the stable operating points
around the Greenwald density and just below the β-limit. For a given electron
density, adding external heating shifts the operating points to a higher temperature
(and consequently higher fusion power). The shift becomes larger for increasing
values of ρ, and causes the curves of different values of ρ to converge on the same
asymptote.
This means that depending on the precise design of the reactor and the corresponding positioning of the operating contours in the ne , T -plane, either ρ, the
external heating power Pext , or a combination of both could serve as an actuator
for burn control.
Finally, adding external heating changes the sign of dne /dT : when the curve
starts to deviate from the operating contour without external heating, it enters
a regime where the temperature increases with decreasing density on the stable
branch (and vice versa on the unstable branch). This needs to be taken into
account when designing a controller for the reactor.
In contrast to the PPCS A, B and C designs, the ITER reactor is not anticipated to ignited. Hence there is always a non-zero amount of external heating
required, and the operating contours look very different from those for the PPCS
A reactor, which is illustrated in figure 3.8.
3.3.5
Impurities
Although all expressions presented so far allow for the presence of impurities in the
plasma, the previous sections have neglected their effect on the operating contours.
This is of course not realistic and therefore this section presents operating contours
for several impurities at different concentrations in the plasma. It turns out that
the main effect of impurities is a contraction of the operating contours for the
reactor, and therefore a reduction in fusion power on the stable burn branch,
and an increase in fusion power on the unstable ignition branch. The maximum
allowable value of ρ is lowered.
Every impurity has a critical concentration above which the burn can no longer
sustain itself. The fusion power at this critical concentration is not zero however,
because at a given temperature and density, the radiation losses are not zero.
Low and high Z impurities have a different effect, because the Bremsstrahlung
losses scale with Zeff which is a quadratic function of Z, whereas the fuel dilution
3.3 Burn equilibria with impurities and Pext
2
55
·1020
nG
100
70
50
30
ne (m−3 )
1.5
β -l
im
it
1
20
0.5
10
30
5
5
50 70 100
20
10
15
T (keV)
20
25
30
100
Figure 3.8: POPCON plot for ITER at ρ = 5 with constant levels of external
heating (labels are in MW). There are no equilibria for Pext = 0, i.e. ITER does
not ignite in our model.
Chapter 3 Burn equilibria
56
scales linearly in Z. However, this is only a minor effect compared to the overal
contraction of the burn contours.
1045
ρ=5
1% Be
1040
W
0.5% Ne
ρ = 10
1% Be
1030
0.
0
1%
2% Be
1025
W
1% N
ne (m−3 )
2% Be
%
1035
N
01
0.
1%
0.5% Ne
nG
1020
β limit
1015
1010
4
5
6
7 8 9 10
20
T (keV)
30
40
50
Figure 3.9: PPCS model A POPCON plots for plasmas containing different impurities (beryllium, nitrogen, neon and tungsten) at different concentrations. The
presence of impurities dilutes the fuel and enhances the radiation losses by increasing Zeff . The result is a large reduction in the accessible density and temperature
range for a given value of ρ, and consequently the maximum value of ρ is lowered.
Note that although low Z and high Z impurities affect the plasma in a different
way (because the dilution is linear in Z whereas Zeff scales with Z 2 ), this is only
a minor effect compared to the reduction in operating range.
3.3 Burn equilibria with impurities and Pext
57
Of more importance to a particular reactor design is the effect of impurities in
the plasma on the fusion power and energy gain factor Q. Figure 3.10 shows the
fusion power and helium fraction (left plot), and Q and Zeff as a function of the
beryllium fraction fBe = nBe /ne in the plasma, for ρ = 5 and 10 in PPCS model
A at a density of ne = 1 × 1020 m−3 with 246 MW of external heating.
The fusion power drops rapidly with increasing values of fBe , as do fα and Q.
Although the curves for different ρ values of converge on each other, the relative
effect is fairly similar, with a decrease 22% and 23% at Zeff = 1.5 respectively. It
is clear that the PPCS estimate of Zeff ≈ 2.5 yields unacceptable results for the
fusion power if helium and beryllium are the only impurities.
10
ρ
ff,
Ze
10
=
5
ρ=
ff,
Ze
Qρ
30
2.5
10
us
,ρ
20
2
=
,ρ
2.5
5
fα
Pf
0.1
,ρ=
Zeff
fα
Q
5
fα
5
=
5
=
s,ρ
Pfus (GW)
3
40
0.15
P fu
7.5
0.2
=
10
0.05
1.5
10
Q
ρ=
10
0
0
0.05
0.1
fBe
0.15
0
0
0.05
0.1
0.15
fBe
Figure 3.10: The fusion power and helium concentration (left plot), Q and Zeff
(right plot) for PPCS model A, with 246 MW of external heating, as a function of
) and 10 (
) for the high temperature
beryllium concentration fBe for ρ = 5 (
equilibrium at ne = 1 × 1020 m−3 . The relative decrease of fusion power is more
or less independent of the value of ρ. Even a moderate value of Zeff = 1.5 already
leads to a decrease in fusion power of 22% (ρ = 5) and 23% (ρ = 10)
.
The impurity content in ITER or future reactor plasmas cannot be know precisely of course, but JET reported Zeff = 1.2 in the core plasma during divertor
operation with the ITER like wall [81]. This was in a divertor plasma without
significant fusion power and therefore no helium content. Neglecting the tungsten
contribution (so assuming a ’pure’ D-Be plasma), this amounts to a beryllium concentration fBe = 0.017, which corresponds to a 17% decrease in fusion power for
PPCS model A relative to a plasma without beryllium pollution. The erosion rates
reported in [81] are probably on the low side for a fusion power plant, because of
the presence of fast He particles impacting on the first wall.
Chapter 3 Burn equilibria
58
The effect of tungsten in the plasma on the fusion power is much weaker than
that of beryllium for similar values of Zeff , as can be seen in figure 3.11, which
displays the fusion power and helium fraction fα on the left, and Q and Zeff
on the right, as a function of the tungsten concentration in the plasma. Both
JET and ASDEX report tungsten concentrations up to 10−4 , but we have taken
fW = 3 × 10−4 as the upper limit to obtain Zeff values that are foreseen in the
PPCS.
0.2
10
40
3
Pfus,ρ=
5
30
Pfus,ρ=10
5
0.15
2.5
20
10
2.5
10
fα,ρ=5
0
1
2
fW
3
−4
·10
0.1
0
2
Qρ=10
ρ=
ff,
Ze
5
1.5
ρ=
ff,
Ze
0
Zeff
10
fα
Q
Pfus (GW)
7.5
0
Qρ=5
fα,ρ=
1
2
fW
3
−4
·10
Figure 3.11: The fusion power and helium concentration (left plot), Q and Zeff
(right plot) for PPCS model A, with 246 MW of external heating, as a function of
tungsten concentration fW for ρ = 5 (
) and 10 (
). Also for tungsten, the
relative decrease of fusion power is more or less independent of the value of ρ, but
the effect is much weaker for the same value of Zeff than for beryllium.
Whereas for beryllium the relative decrease in fusion power is 22% at Zeff = 1.5,
the corresponding reduction in power for tungsten is only 2%. This is caused by
the quadratic dependence of Zeff on Z, and the quadratic dependence of the fusion
power on the fuel density, an effect which was already reported in [82]. The atomic
number of beryllium is 4, and it takes quite a lot of it to obtain the same value
of Zeff compared to tungsten with an atomic number of 74, for which a small
concentration already leads to a rather large change in Zeff .
Please note that the effect of high Z impurities is not limited to fuel dilution
and an increase in Zeff which results in higher Bremsstrahlung losses. They will
most likely not ionise completely and therefore emit line radiation, which can have
a non-negligible effect on the (local) energy balance in the plasma. The resulting
temperature decrease might very well result in a significant loss in fusion power,
but a complete treatment is outside the scope of this thesis.
3.3 Burn equilibria with impurities and Pext
59
So far we have looked at the effect of impurities on the fusion power in isolation,
but most likely a change in particle confinement time because of a change in ρ will
be accompanied by a change in impurity content (assuming the impurity source
stays the same of course). Figure 3.12 shows contours of constant Pfus in the
fBe , ρ-plane, as well as contours of constant beryllium influx SBe (dark blue lines).
10
5,0
00
4,0
00
6,0
8
00
5,0
00
7,0
00
02
6
6,0
0.0
ρ
8,0
0.001
00
4
9,0
0.0
00
00
05
7,0
00
8,0
00
11,
000
12,
000
1
10, 0.0
000
9,0
00
2
0.02
0
0.5
1
1.5
2
2.5
fBe
3
3.5
0.05
4
4.5
5
−2
·10
Figure 3.12: Contour plot of the fusion power (coloured lines) in MW for PPCS
model A with Pext = 246 MW as a function of the beryllium concentration fBe
and ρ, at a density of ne = 1 × 1020 . The impurity concentration is of course also
a function of ρ, and to indicate the effect of this coupling the contours of constant
beryllium influx are also plotted (dark blue lines). The labels denote particles per
second per m3 in units of 1020 .
If the impurity source stays constant during a change in ρ, the plasma will
move parallel to one the SBe = constant contours. This would have a significant
effect on the fusion power: for instance, if ρ increases from 8 to 9, and the original
beryllium concentration was 2%, the fusion power decreases with roughly 1 GW,
or 18%.
Of course it is unlikely that the impurity influx will stay constant when the
fusion power changes by such a large fraction, but the example nevertheless shows
Chapter 3 Burn equilibria
60
just how sensitive the fusion power output is to small changes in ρ.
3.3.6
Power output with external heating and impurities
In section 3.3.4 we analysed the effect of external heating power on the position of
the burn equilibria in operating space. Obviously, when the equilibrium changes, so
does the fusion power, and figure 3.13 displays Pfus (left y-axis) and the derivative
dPfus /dPext (right y-axis) as a function of Pext for ρ = 5 and 10 (blue and red curves
respectively) for fBe = 0 and fBe = 0.025 for the PPCS A design at ne = 1020 m−3 .
·104
1
20
Pfus,fBe =0
s f
dP fu t,
dP ex
15
0. 0
25
0.5
dP
dP fus ,
ext f
Be=
0.25
dPfus
dPext , fBe=0
0
0
50
10
Pfus,fBe =0
dPfus
dPext
Pfus,fBe =0.025
Be=
Pfus (MW)
0.75
Pfus,fBe =0.025
0
5
dPfus
dPext , fBe=0.025
100
150
Pext (MW)
200
0
Figure 3.13: The fusion power Pfus (left axis) and its derivative with respect to
the external heating power dPfus /dPext (right axis) for PPCS model A at ne =
) and ρ = 10 (
), with fBe = 0 and 0.025. The fusion
1×1020 m−3 for ρ = 5 (
power increases with Pext , but this effect decreases for increasing Pext . Equilibria
at higher ρ and fZ are more sensitive to Pext than those are lower ρ and fZ values,
but this difference also decreases for higher values of Pext , both in absolute and in
relative terms.
With the addition of Pext the fusion power increases, and this effect is stronger
for higher values of ρ and for higher impurity concentrations. For ρ = 10 and
fBe = 0.025 the fusion power increases by 1.8 GW when Pext goes from 0 to 246
MW, while for ρ = 5 and FBe = 0 the change is only 878 MW.
The sensitivity of Pfus to Pext decreases for higher values of Pext , and this effect
3.3 Burn equilibria with impurities and Pext
61
is stronger, both in relative and absolute terms, for higher values of ρ and fZ : the
value of dPfus /dPext for ρ = 10 and fBe = 0.025 at Pext = 246 MW is only 16%
of what it is at Pext = 0 MW, whereas for ρ = 5 and FBe = 0 the corresponding
ratio is 70%.
So for higher values of ρ and fZ external heating becomes a more effective tool
to increase the fusion power output of the reactor. But the ultimate goal is not
the fusion power, but the power delivered to the grid.
3.3.7
The effect of Pext on net electric output
The net (electric) power delivered to the grid, Pnet , is determined by the (overal)
thermal efficiency η of the plant, and the conversion efficiency ξ of the external
heating power. (Obviously the plant also uses a significant amount of power for
the cryostat, coolant pumps, and a host of other auxiliary systems, but their power
consumption is relatively insensitive to the fusion power and we will neglect it for
now.)
For the net power we can write
Pnet = PE −
Pext
,
ξ
(3.13)
with the electrical power PE given by
PE = η (Pfus + Pext ) .
(3.14)
Here we have assumed that the external heating power is delivered to the plasma
with 100% efficiency.
Combining these two expressions, we get
Pnet = ηPfus −
(1 − ξ)
Pext .
ξ
(3.15)
On the face of it, any amount of external heating will reduce the power delivered
to the grid, but this is only true if Pfus is independent of Pext . This is not the case,
and figure 3.13 shows that dPfus /dPext > 0 for reactor relevant levels of Pext , so
an increase in Pext also results in an increase in Pfus .
To determine what happens to Pnet if Pext changes, we take the derivative of
equation (3.15) with respect to Pext
dPnet
dPfus
1−ξ
=η
−
.
dPext
dPext
ξ
(3.16)
If dPnet /dPext is greater than zero, an increase in Pext will result in an increase in
Pnet . Some algebra allows us to transform this into a condition on dPfus /dPext :
dPfus
1−ξ
>
.
dPext
ηξ
(3.17)
62
Chapter 3 Burn equilibria
Using η = 0.31 as anticipated for PPCS model A [71], and ξ = 0.35 (reference [71] gives a value of ξ = 0.6, but this is based on negative ion source neutral
beam injection, which seems highly unlikely for a commercial fusion reactor.) the
condition is dPfus /dPext > 6.0. In figure 3.13 it can be seen that for ρ = 5 this
means that the addition of any amount of external heating leads to a reduction in
Pnet , while for ρ = 10 the limit lies somewhat above 100 MW, depending on the
impurity concentration in the plasma.
In the analysis above we have neglected the power gain from tritium breeding
in the blanket, which changes the numbers but not the argument. If we assume
all tritium breeding is done by the 6 Li + n → 4 He + T + 4.8 MeV reaction, the
total thermal power Pth is given by
4.8
Pth = 1 + TBR
Pfus ,
(3.18)
17.6
with TBR the tritium breeding ratio, which has a maximum value of approximately
1.15 [60, 61, 62, 63]. We can absorb this power multiplication factor in η and see
that it relaxes the requirement on dPfus /dPext by roughly 30%.
So depending on the values of η and ξ that can be achieved, adding external
heating may be positive or detrimental to the overall performance of the reactor.
Having said that, it seems unlikely that the foreseen level of external heating for
the first generation of fusion power plants is optimal from a cost of electricity point
of view.
3.3.8
Uncertainties in scaling laws
The scaling laws for the energy confinement time are based on a database containing several thousand experiments, and a scaling law is just a fit through the
datapoints, not an exact representation. The coefficients of the different parameters in the scaling law come with an uncertainty, which is also apparent in the
differences between the respective scaling laws in the ITER physics basis. These
are all based on different subsets of the same data, sometimes with different physics
restrictions. The variation in predicted energy confinement time is of the order of
one second, or roughly 25%, and the 95% log-nonlinear confidence interval for the
IPB98(y,2) scaling law is 3.5 - 8.0 s for ITER [26].
It stands to reason therefore, that a small change in one of the exponents
should only have a small effect on the burn contours. This is indeed the case for
changes to the exponents of all parameters, except ne and P . The burn contours
are very sensitive to variations in the scaling of τE with ne and P , as is shown
in figure 3.14. Reducing the P exponent compresses the burn contours along the
density axis, and increasing the exponent of P results in a stretching. The reverse
is true for changes in the exponent of ne . Already changes as small as one percent
result in a change of several decades in the predicted density range for the burn.
The temperature range remains unaffected as explained before.
3.3 Burn equilibria with impurities and Pext
63
1050
9
05
0.4
ne
1045
P −0.6969
ρ=
1040
5
ne0.414
1
1
P
P −0.6969
ne (m−3 )
1035
1030
683
−0 .
40
0.
ne
59
ρ=
5
ne0.4141
P −0.6831
1025
nG
1020
β limit
1015
4
5
6
7 8 9 10
20
T (keV)
30
40
50
Figure 3.14: Operating contours for PPCS model A for ρ = 5 and ρ = 10 applying
the normal IPB98(y,2) scaling, and the IPB98(y,2) scaling with a ± 1% change
in the exponents of ne and P respectively. These relatively small changes to the
scaling law result in comparatively large changes in the operating contours. The
accessible temperature range stays the same, since this is only a function of the
fusion and radiation cross sections combined with the transport modelling using a
confinement time, but the density range is heavily affected.
64
Chapter 3 Burn equilibria
This strong sensitivity of the operating contours to minor variations in the
energy confinement time scaling are clearly undesirable, because the exact position
of the operating point and the corresponding fusion power output have a major
impact on reactor design. Further investigation of this phenomenon is outside of
the scope of this section and will be addressed in chapter 5.
3.4
Discussion and conclusions
Future fusion reactors will to a large extent rely on alpha particles to provide the
necessary heating. If a fixed ratio between the energy and particle confinement
time is assumed, it can be shown that the resulting equilibria form closed contours
in the ne τE , T -plane [41]. We have derived an analytical expression for the temperature range that is spanned by such contours, based on the discriminant of the
cubic equation for the helium fraction in the plasma. This result is valid for all
reactors (not only tokamaks), for which the energy and particle confinement time
have a fixed ratio ρ.
Following the approach presented in [65], we also derived an expression for the
electron density as a function of temperature for reactors that follow the IPB98(y,2)
scaling law. Compared to the results presented in [65], the operating contours
obtained with the IPB98(y,2) scaling are shifted towards higher densities in the
ne , T -plane. This is because the electron density of the equilibria is extremely
sensitive to changes in the exponents of ne and P in the scaling law, something that
we will come back to in chapter 5. Please note that to mimic H-mode confinement,
the confinement time predicted by the ITER89P scaling is modified by an H-mode
factor fH , which in this case is taken to be fH = 2.
The contours obtained using the IPB98(y,2) scaling also have a shape that
differs from the ones found using the ITER89P L-mode scaling. This is because
the radiation losses are not included in the IPB98(y,2) scaling, as opposed to the
ITER89P L-mode scaling that does include them. An explicit treatment of the
radiation losses is thus required, and this does away with the artificial ’radiation
limit’ [53], eliminating the ’bump’ that is present on contours obtained using a
scaling law for τ̃E .
The use of scaling laws to eliminate τE from the burn equations results in burn
contours that span many orders of magnitude in density. Care has to be taken
when interpreting equilibria outside of the density range on which the scaling
laws are based, and often the contours extend to densities that have no physical
meaning. This is purely a mathematical artefact, which originates in the form of
the equations.
Burn contours in the ne τE , T -plane are universal since they only depend on the
reactivity and the radiation cross sections, but this property appears to be lost
when they are transformed to the ne , T -plane. This can be resolved by dividing the
expression for ne by the reactor specific terms in the scaling law for τE , resulting in
3.4 Discussion and conclusions
65
isomorphic burn contours that are valid for all reactors. The normalisation factor
is arbitrary, and the resulting isomorphic burn contours still extend over several
orders of magnitude in normalised electron density, depending on the scaling law
for τE .
The main benefit of this discovery lies in the fact that a sensitivity analysis of
the burn equilibria is also valid for all reactors: the results can simply be scaled
by the ratio of the reactor specific part of τE . We performed a sensitivity analysis
of the fusion power with respect to the three parameters that are under operator
control in a burning plasma ne , ρ and H98 .
The fusion power scales quadratically with electron density, apart from the
region around the minimum and maximum operating density on a contour. This
strong dependence makes the density a powerful actuator for control of the fusion
power. However, due to the desired operation close to the density limit the actual
achievable variation in density might be too small for practical use.
The fusion power is also very sensitive to H98 around the minimum and maximum values of H98 . In between, the power output reaches a maximum (and
minimum) and the different PPCS designs project H98 right at the value where
these extrema lie, effectively disqualifying the energy confinement as a control tool.
Also the precise value of ρ has a strong influence on the fusion power on the
burn branch, the sensitivity on the ignition branch is rather weak. Although acting
on the slowest time scale of the three parameters that were investigated, ρ might
be the most suitable actuator for power control in a fusion reactor operating on the
burn branch since it can be varied over a wide range without the risk of crossing a
stability limit. One needs to take the tritium breeding requirements into account
when considering this approach however, as values of ρ < 5 might result in a too
low tritium burn up fraction [64].
When including external heating in the model, several things change. For
reactors capable of ignition and high (≈ 25 keV) or low (<5 keV) temperatures, the
operating contours converge to a single contour that depends only on the amount
of external heating and not on the value of ρ. For intermediate temperatures,
the low density solutions have disappeared, because they feature such low fusion
powers that for realistic values of Pext the system is completely dominated by the
external heating. Only the high density solutions remain, dominated by the alpha
heating and consequently the curves with external heating almost coincide with
the original operating contours. In between the intermediate and the low and high
temperature regimes there is a smooth transition. The fact that contours with
different values of ρ but the same level of Pext converge to the same curve means
that depending on the reactor design and choice of operating point, either ρ, Pext
or a combination of both could be used as actuators for control.
The inclusion of impurities in the system results in reduction of accessible
temperature and density range. On top of that, the maximum allowable value
of ρ decreases for increasing impurity concentrations. There is a small difference
66
Chapter 3 Burn equilibria
between low and high Z impurities, due to the different scaling of fuel dilution and
conduction and radiation losses with Z. This changes the shape of the operating
contours, but it is insignificant compared to the contraction of the contours.
When looking at the effect of impurities on fusion power, the difference between high and low Z impurities becomes more pronounced. Because the required
concentration of low Z impurities is much higher for a given Zeff , they dilute the
fuel much more than high Z impurities. Since Pfus scales with the fuel density
squared, the corresponding effect on Pfus is even larger. The effect of increasing fZ
is larger for lower values of ρ, although this difference is smaller for high Z than
it is for low Z impurities.
Most likely a change in ρ will be accompanied by a change in the impurity content of the plasma, and the fusion power is very sensitive to this. This sensitivity
increases for higher impurity concentrations.
The addition of external heating has a positive effect on the fusion power
output of the reactor, and Pfus is more sensitive to Pext for higher values of ρ and
higher impurity fractions. Whether the addition of Pext is beneficial for the net
electricity production of the plant depends on the thermal and heating efficiency
of the reactor, but a reduction in external heating power would most likely result
in a larger Pnet for the PPCS A design.
Finally, we looked at the sensitivity of the operating contours to small changes
in the scaling laws for the energy confinement time. The predicted value of τE at
the ITER operating points varies relatively little between the different scalings in
the ITER physics basis and the predictions are robust against small changes in
the exponents of the individual scaling laws [26], but this is not the case for the
operating contours when small changes are made to the exponents of ne and P .
Even changes of 1% already result in the contours being stretched or compressed
along the density axis by many orders of magnitude. Although the corresponding
shifts in operating points around the Greenwald density and just below the β limit
are far less severe, we consider this an unphysical and also unwanted effect. A
more detailed investigation of this phenomenon can be found in chapter 5.
67
Chapter 4
Burn stability
4.1
Introduction
When designing a fusion reactor that is capable of (assisted) ignition, it is essential
to know the nature of the operating point. What will the plasma do when left
alone. Will it wander off to some unknown destination in phase space? Or will it
remain happily where it is. And if shaken or rattled by some external event, how
will it respond? In other words: is the operating point stable and if so, what is
the stability radius?
In the current understanding of a burning plasma two equilibria are identified:
an unstable one at low temperature and a stable equillibrium at high temperature [7, 65]. For a simple DT plasma without conduction losses this follows from
the fact that the reactivity has a maximum, whereas the radiation losses increase
monotonically with temperature. There are therefore two temperatures at which
the radiation losses are equal to the alpha heating power from the fusion reaction.
Because the energy balance contains only cross sections that follow from atomic
physics, the temperatures at these equilibria are independent of engineering parameters and can be determined from
√
hσvi = CB T .
(4.1)
The addition of conduction losses to the energy balance introduces a reactor
dependence in the system, but in general two equilibria remain. Again the one
at low temperature is unstable and the other is stable. Irrespective of the reactor
design, some general observations can be made. For instance, the temperature
of the stable equilibrium decreases drastically because at high temperatures conduction is the dominant energy loss mechanism (because of turbulent and, to a
lesser extent, neoclassical transport), reducing the importance of radiation losses.
68
Chapter 4 Burn stability
The unstable (low) temperature equilibrium is far less affected because in that
temperature range the radiation losses dominate.
If the conduction losses are higher than the alpha heating power for all temperatures (for instance because the reactor is too small), there will be no equilibria.
For a reactor with critical size, the alpha heating power will balance the losses at
only one point, and this equilibrium necessarily will be unstable. More precisely:
it will be a saddle point, with dT /dt < 0 on both sides of the equilibrium.
In a pure DT plasma, assuming fD = fT , the temperature is the only variable
(with the density being the free parameter). It is trivial to determine the stability
of the system, because the only eigenvalue of such a system is −J, the Jacobian.
While easy to analyse, it does not accurately represent a burning plasma because
the fusion reaction produces helium which has a finite residence time in the plasma.
The helium particles change the equilibrium by diluting the fuel, increasing radiation losses and even affecting the conduction losses. So any realistic description
of a burning plasma needs to take this into account.
Linear stability of the burn point is not the only consideration for the design
of a future fusion reactor. Of even greater importance is the convergence radius
of the equilibrium, or the size of the basin of attraction. The burn equations are
highly non-linear and although the Poincare-Lyapunov theorem states that a nonlinear system is stable in a region around an equilibrium of the linearised system,
the theorem does not say anything about the size of this region.
The questions this chapter aims to answer are the following: how does the
plasma respond to a disturbance of the burn equilibrium? Does it return to the
original equilibrium, or does it find a new one (possibly at T = 0)? And in doing
so, does it cross any operational limits? And what happens to the fusion power
during these excursions? Does the system have bifurcations and associated limit
cycles? In short: can we rely on the plasma to regulate the burn by itself, or does
it need to be controlled by external means?
First we will present the system of differential equations that describe a burning
plasma, and subsequently introduce a reduced system (with only two variables)
which is able to reproduce the most important dynamics. We will study the
stability of this system as a function of the free parameters by deriving an analytical
expression for the Jacobian and applying planar bifurcation theory [83].
Using these results we will investigate the effect of changes in the τE scaling on
the stability and compare different PPCS designs, both with and without external
heating and reflect on the implications for operating point selection and control
requirements.
Then we will return to the full system, derive the Jacobian, present the stability
of the system, and compare the respective reactor designs.
4.2 Theory
4.2
Theory
4.2.1
Burn equations
69
The burn dynamics can be described by four coupled, non-linear ordinary differential equations (ODEs). Rebhan and Vieth [65] performed a first analysis of this
system and determined the linear stability by deriving the Jacobian and evaluating its eigenvalues for the equilibria that they found using the ITER98P L-mode
scaling.
In their analysis the effect of helium ash accumulation and fuel dilution on the
ion mass was omitted, resulting in a symmetry in the fD and fT terms in the
Jacobian matrix. This symmetry enables a reduction of the number of dimensions
by developing the Jacobian with respect to the first row or column, which yields
ñD = −ñT to be an eigenfunction of the system. However, since a complete
analysis should take the mass dependence into account we did include this term,
breaking the symmetry and requiring a numerical treatment of the system.
The 0D model of a burning plasma features the variables nD , nT , nα and T [65],
and their evolution in time is governed by 4 differential equations:
dnD
dt
dnT
dt
dnα
dt
dT
dt
nD
,
τp
nT
= sT − nD nT hσvi −
,
τp
nα
= nD nT hσvi −
,
τp
3
2(sD + sT )T
nD nT
T
T
Srad
hσvi Eα + T −
−
= 3
+
− 3
.
2
τ
τ
ntot
E
p
2 ntot
2 ntot
= sD − nD nT hσvi −
(4.2a)
(4.2b)
(4.2c)
(4.2d)
Here hσvi is the reactivity of the plasma as a function of T , for which we have
taken the Bosch and Hale parametrisation [4], Eα = 3.52 MeV the energy of an
alpha particle, Srad denotes the radiation losses due to Bremsstrahlung and the
particle and energy confinement time are related through τp = ρτE , with ρ a free
parameter with a typical value in the range 5-10, although values up to 30 have
been reported [42, 37, 43, 44].
Equation (4.2d) was first derived from the energy balance by Rebhan et.
al. [65] using dW/dt = d/dt(3/2ntot T ) and solving for dT /dt. Consequently, equation (4.2d) contains some terms with an interpretation that may not be immediately obvious. Their meaning is most easily understood when considering that a
change in internal energy can be brought about by a change of temperature (at
constant density) or a change of density (at constant temperature). A change in
temperature is then equal to the total change in internal energy minus the change
in internal energy due to a change in density, divided by 3/2 times the total density
(note that the Boltzmann constant is absorbed into the temperature definition).
Chapter 4 Burn stability
70
The nD nT hσviT /ntot term denotes the temperature change due to the change
in density when two particles fuse. Since a fusion reaction doesn’t exchange energy
with the environment, the process has to be adiabatic (apart from the energy
gained from the change in binding energy obviously), so the kinetic energy of the
reacting particles has to be retained in the plasma.
The T /τp term compensates for the difference between particle and energy
transport. In case τp = τE particles are lost at the same rate as their energy,
so there is no effect on temperature. In other cases the change in temperature
is equal to the difference between particle and energy transport, with τp > τE in
general. Of course the use of energy and particle confinement times neglects the
close relationship between particle and energy transport, but this is unavoidable
in such a simple model.
Finally the 2(sD + sT )T /ntot term represents the cooling effect of refuelling. In
this case the fuel is assumed to be at zero kelvin, which is a good approximation
in case of pellet fuelling and gas puffing. Only in the case of NBI heating does the
new fuel have significant energy, but we will not include this effect in our analysis.
The electron density in the plasma is given by
ne = nD + nT + 2nα + ZnZ ,
(4.3)
ntot = 2nD + 2nT + 3nα + (Z + 1)nZ ,
(4.4)
the total particle density
and the average ion mass
A=
2nD + 3nT + 4nα + mZ nZ
.
nD + nT + nα + nZ
(4.5)
This burning plasma model implicitly assumes that dnZ /dt = 0, which is of course
most likely not the case in a fusion reactor. For our purposes is suffices though,
since we are mainly interested in the burn dynamics of the system, and not in
plasma wall interaction or impurity seeding, which are expected to be the main
sources of impurities in future reactors. From here onwards we will assume nZ = 0
for simplicity unless explicitly stated otherwise. However, relaxing this assumption
does neither change the analysis nor the conclusions.
The above system is four-dimensional, and its properties are determined by
the three free parameters sD , sT and ρ. Although the stability of the system can
easily be determined by calculating the eigenvalues of the linearised system, it is
complicated to investigate the behaviour of a four-dimensional system around the
equilibria and the dynamics of the system in general.
In the next section we will therefore present a reduced system that has only
two degrees of freedom and is determined by two parameters. This will allow the
application of planar stability theory, and we can study the transition between
different stability regions using bifurcation theory.
4.2 Theory
4.2.2
71
Stability of a two-dimensional system
Determining the stability of a non-linear system is generally not trivial. However,
the Poincare-Lyapunov theorem says that if a linearised system is asymptotically
stable in a certain point, the non-linear system will be stable in that point too.
Hence we will focus on the stability of the linearised system.
Determining the stability of a linear system is most easily achieved by determining the Jacobian matrix J, and subsequently calculating its eigenvalues at the
equilibria. Positive eigenvalues correspond to an unstable equilibrium and negative
eigenvalues to a stable point.
The eigenvalues of any (linear) system of equations can be determined by solving
det(J − λI) = 0.
(4.6)
In the two-dimensional case this can be written as
(j11 − λ) (j22 − λ) − j12 j21 = 0.
(4.7)
Defining p = j11 + j22 and q = j11 j22 − j12 j21 transforms this to
λ2 − pλ + q = 0,
(4.8)
so that the solutions are
λ1,2 =
√ 1
p± ∆ ,
2
(4.9)
with ∆ = p2 − 4q [84].
Depending on the value of ∆ the following cases can be distinguished:
i ∆ > 0: λ1 and λ2 are real and distinct,
ii ∆ = 0: λ1 and λ2 are real and equal, and
iii ∆ < 0: λ1 and λ2 are complex conjugates.
We can use p and q to determine the signs of λi , i = 1, 2, and thus the stability of
the equilibrium. The different possibilities, unstable (US), stable (S) and asymptotically stable (AS) are summarised in table 4.1. In case both p and q are zero,
λ1 = λ2 = 0 and the stability of the system is unknown (indicated with UK in
table 4.1). However, by looking at the different elements of J we can still obtain
information about the stability. If J = ( 00 00 ) is the system stable, and if J 6= ( 00 00 )
it is unstable.
72
Chapter 4 Burn stability
Table 4.1: Stability properties of a two-dimensional system as a function of the
elements of the Jacobian matrix J, with p = j11 + j22 and q = j11 j22 − j12 j21 .
p>0
p=0
p<0
4.2.3
q>0
q=0
q<0
US
S
AS
US
UK
S
US
US
US
Bifurcation theory
In order to fully appreciate the changes in behaviour between different regions in
the stability diagram of a burning plasma, we will briefly introduce the accompanying bifurcations. We will encounter them again in section 4.3.4.
The definition of a bifurcation is ẗhe division of something into two branches
or parts,̈ or, in the study of dynamical systems, a sudden change in the qualitative
or topological structure of the system. This abrupt change is brought about by a
small, continuous change in the bifurcation parameter(s) of the system.
From bifurcation theory we know that there can be local bifurcations in the
system at points where one or both eigenvalues has a real part that is equal to
zero [83]. This property can be exploited when looking for bifurcations: we will
find a single zero eigenvalue at points where q = 0 (λ1 = p and λ2 = 0), and a
double zero eigenvalue when p = q = 0 (λ1,2 = 0). In case p = 0 and q > 0 there
are two completely imaginary eigenvalues, and if p = 0 and q < 0 both eigenvalues
are real.
The type of bifurcation depends on the nature of the eigenvalues at the bifurcation point. For a single zero eigenvalue, the result is a saddle-node bifurcation, also
known as a fold, or limit point bifurcation [83, 85]. On one side of the bifurcation
there are two equilibria: a saddle point (two real eigenvalues: one positive and
one negative), and either a source (an unstable equilibrium) or a sink (a stable
equilibrium). At the bifurcation, these two equilibria meet and annihilate each
other, and on the other side of the bifurcation they have disappeared.
At the point where both eigenvalues are completely imaginary, the system goes
through a (Poincaré–Andronov–) Hopf bifurcation [83, 86], which signifies the
birth of a limit cycle. The bifurcation can be either supercritical, in which case
it is an attracting (stable) limit cycle, or subcritical if the limit cycle is unstable
(repelling).
A Bogdanov-Takens bifurcation occurs at the point where λ1,2 = 0 [83, 87].
Near this point the system has two equilibria, a saddle and a non-saddle (a source
or a sink), which annihilate in a saddle-node bifurcation. An Andronov-Hopf
bifurcation generates a limit cycle at the non-saddle equilibrium and finally a
4.3 Reduced system
73
non-local bifurcation called a saddle-homoclinic bifurcation also originates at this
point.
The saddle-homoclinic bifurcation occurs when a limit cycle collides with a
saddle point and connects it with itself, which does not depend on the local value
of the eigenvalues, but instead is determined by the global properties of the system [83].
There are several other bifurcations that can occur in systems with two or more
dimensions, but they are of little relevance in our case.
4.3
Reduced system
4.3.1
Derivation
There are several ways of reducing the dimensionality of the system. The most
logical reduction follows from the observation that the fusion power output has
a maximum at, or very close to, nD = nT , so it is reasonable to take this as a
property of the system. Because the transport of deuterium and tritium in our
model is exactly the same, this also means that sD = sT , hence this assumption
reduces the number of parameters by one.
A further reduction in dimensionality is facilitated by a coupling between ni
and nα , which can be achieved by fixing the electron density at a constant value.
Although this is not possible in reality because it would require an instantaneous
feedback system on the particle sources sD and sT , it enables us to isolate the
effect of helium accumulation on the burn dynamics at a constant density and it
has the added effect of making the system a bit less complicated.
The other possible relation between ni and nα would be through Zeff , but this
would also imply a coupling with T because in equilibrium fα = f (T ).
Combining the condition nD = nT with equation (4.3), allows us to express
the fuel density ni = nD + nT as
ni = ne − 2nα ,
(4.10)
which means we can combine equations (4.2a) and (4.2b) into an expression that
contains only one unknown, nα :
dni
1
ni
= si − n2i hσvi −
dt
2
τp
1
ne − 2nα
dnα
2
= si − (ne − 2nα ) hσvi −
= −2
,
2
τp
dt
(4.11)
where si = 2sD = 2sT . We can derive an expression for the refuelling rate
si = −2
dnα
1
ne − 2nα
2
+ (ne − 2nα ) hσvi +
,
dt
2
τp
(4.12)
Chapter 4 Burn stability
74
using equation (4.2c), which results in
si =
ne
.
τp
(4.13)
This result could have been anticipated by realising that in order to have a constant electron density, every particle that leaves the plasma needs to be replaced.
Since the fusion process only affects the number of ions, and leaves the number of
electrons unchanged, the only way an electron can leave the plasma is by transport
to the wall. The refuelling only needs to compensate for these losses, hence the
intuitive form of expression (4.13).
Also the expressions for the total particle density
and the average ion mass
ntot = 2ne − nα ,
(4.14)
2.5ne − nα
ne − nα
(4.15)
A=
take on a simpler form, as do the radiation losses
Srad = ne [(ne − 2nα ) Rrad (T, 1) + nα Rrad (T, 2)]
= ne (ne + 2nα )Rrad (T, 1)
(4.16)
where we have made use of the fact that Rrad (T, 2) = 4Rrad (T, 1)
Using these results we end up with two equations for the variables nα and T
dnα
1
nα
2
= (ne − 2nα ) hσvi −
,
dt
4
τp
2
(ne − 2nα )
3
nα T
T
Srad
dT
=
hσvi Eα + T −
−
− 3
,
dt
6ntot
2
ntot τp
τE
2 ntot
(4.17)
(4.18)
where in the latter the refuelling term has been combined with the particle transport term. The properties of the system are determined by the choice of the free
parameters ne and ρ.
4.3.2
Jacobian matrix of the reduced system
Writing xi = {nα , T }, the burn equations (4.17) and (4.18) can be expressed as
dxi
= fi (x1 , x2 )
i = 1, 2
dt
We can linearise the system around an equilibrium point xj0 as
2
X
dxi
∂fi =
(xj − xj0 ) ,
i = 1, 2.
dt
∂xj xj0
j=1
(4.19)
(4.20)
4.3 Reduced system
75
Determining the linear stability of the system requires that we find the eigenvalues
(and eigenvectors) of the Jacobian matrix J, whose elements jij = ∂fi /∂xj are
evaluated at the equilibrium points xj0 . Determining J comes with some extensive
algebra, and the full derivation can be found in section A.1. Here we will restrict
ourselves to the final result:
nα ∂τE
1
+ 2
,
ρτE
ρτE ∂nα
nα ∂τE
(ne − 2nα )2 dhσvi
+ 2
,
=
4
dT
ρτE ∂T
3
(ne − 2nα )(2nα − 7ne )
hσvi Eα + T
=
6n2tot
2
T 2ne τE
nα
∂τE
5n2 Rrad (T, 1)
− 2
,
−
1
+
− e 3 2
2
τE ρntot
ρntot ∂nα
2 ntot
dhσvi 3
(ne − 2nα )2
3
=
+ hσvi
Eα + T
6ntot
2
dT
2
nα
Srad
1
T ∂τE
−
−
+1
− 2
.
ρntot
τE
τE ∂T
3T ntot
j11 = (2nα − ne )hσvi −
(4.21)
j12
(4.22)
j21
j22
(4.23)
(4.24)
Although consisting of long expressions, numerical evaluation of the Jacobian
matrix is straightforward and very quick, because of the analytical expressions for
all derivatives which makes computationally expensive numerical derivates unnecessary.
4.3.3
Normalisation
The density spans a range of several orders of magnitude, and consequently a
numerical treatment of the problem is prone to precision and rounding errors. This
can be solved by normalising the density, which we have done by expressing the
helium density as a fraction of the electron density, fα = nα /ne for the numerical
calculations. To a lesser extent the same holds true for the temperature, which
can be normalised to the equilibrium temperature Teq to obtain T ∗ .
So for the different terms of the Jacobian, we need to make the following
Chapter 4 Burn stability
76
substitutions:
n α = n e fα ,
(4.25)
∗
(4.26)
T
dnα
dt
∂
∂nα
dT
dt
∂
∂T
= Teq T ,
dfα
= ne
,
dt
1 ∂
=
,
ne ∂fα
dT ∗
= Teq
,
dt
1 ∂
=
.
Teq ∂T ∗
(4.27)
(4.28)
(4.29)
(4.30)
With these transformations, the relations between the elements of the Jacobian J
and the normalised Jacobian J ∗ take the form
∗
j11 = j11
,
ne ∗
j12 =
j ,
Teq 12
Teq ∗
j21 =
j ,
nα 21
∗
j22 = j22
.
4.3.4
(4.31)
(4.32)
(4.33)
(4.34)
Reduced system stability
For each combination of ρ and ne there are two equilibria (which possibly coincide),
and the elements of the Jacobian matrix depend on the local values of fα and T .
If we want to find the zero eigenvalues in the system, we have to solve q = 0 (or
p = q = 0), dfα /dt = 0 and dT ∗ /dt = 0 simultaneously. In reference [41] an
expression relating fα and T is presented, which can be used to express ne as a
function of T for different forms of the τE scaling law [65, 88]. Although an analytic
solution to the problem might exist if the inverse function of the relationship ne (T )
can be found, the resulting expression will probably not provide a lot of insight
into the physics behind the solution.
Alternatively, we can look for eigenvalues equal to zero on an equilibrium plane
[T ∗ , fα ](ne , ρ), which is the approach that we have taken. Looking at the shape
of the equilibrium contours presented in references [65] and [88], we see that the
equilibria form nested, closed contours in the ne , T -plane, with increasing values
of ρ resulting in smaller (contracted) curves.
The question now is what happens to the stability when we change either ne or
ρ. Changing ρ at constant ne means moving parallel to the T -axis, whereas varying
ne at constant ρ means moving along a burn contour (black curves in figure 4.1).
4.3 Reduced system
77
In doing so, the system will pass through regions of different stability, which we
will discuss in the following paragraphs.
Figure 4.1 shows the nature of the equilibria in the reduced system in the
ne , T -plane for the PPCS A reactor design [70]. On the low temperature side, the
equilibrium is a saddle point having both a stable (green in figure 4.1) and an unstable (red) eigenvector. Moving towards higher temperatures, both eigenvectors
first become unstable and subsequently acquire an imaginary part (blue), which
introduces oscillatory behaviour. After crossing into a stable oscillatory region
(magenta), the system becomes asymptotically stable (two stable eigenvectors),
until at the high temperature side a small region of oscillatory stable behaviour is
again encountered.
λ1
10
λ2
70
ne (m−3 )
1060
1050
1040
1030
1020
1010
5
10
20
50
T (keV)
100
200 5
10
20
50
T (keV)
100
200
Figure 4.1: The stability of the two eigenvectors of the reduced burn system for
PPCS model A [70]. The black lines indicate contours of constant ρ, starting with
ρ = 1 on the outside and ending with ρ = 14 for the innermost curve. Unstable
means oscillatory unstable
behaviour for a given eigenvector is denoted ,
behaviour,
oscillatory stable behaviour and
stable behaviour.
Starting at the lowest temperature on a contour of constant ρ, we find that the
equilibria are saddle points, with a stable and an unstable eigenvector. Following
the contour in the clockwise direction, the stable eigenvector also becomes unstable,
which happens exactly at the maximum density of the contour. This transition is
accompanied by a saddle-node bifurcation, which occurs when a system has one
eigenvalue equal to zero. In this case the source point meets the saddle point and
they annihilate each other.
For constant ρ, the bifurcation parameter is ne and we can understand the
78
Chapter 4 Burn stability
physical reason for the bifurcation as follows. When increasing the density, the
temperature difference between the stable an unstable equilibria is reduced, until
at the maximum density both equilibria coincide. A further increase in ne is not
possible without also changing the value of ρ.
Further along the contour, the eigenvalues acquire an imaginary part, resulting
in unstable oscillating behaviour of the equilibrium (blue points in figure 4.1). The
real parts of the eigenvalues now become increasingly smaller, until at some point
they become negative, corresponding to a stable oscillatory equilibrium (green
points). At the change from unstable to stable oscillations, both eigenvalues
are purely imaginary, which means that the system goes through a (Poincaré–
Andronov–) Hopf bifurcation. In this case the bifurcation is subcritical, resulting
in the birth of an unstable limit cycle.
Moving along, the limit cycle increases in size until it degenerates into a homoclinic orbit to the saddle equilibrium at the same density, but lower temperature,
where it disappears in a (saddle-)homoclinic bifurcation. This is a global bifurcation, meaning that it does not depend on the local parameters of the system, but
instead arises from the properties of the system at different points, contrary to the
fold and Hopf bifurcations, which are local bifurcations.
Traversing the contour further in the clockwise direction, for ρ ≤ 10.5, the
imaginary part of the eigenvalues can decrease to zero yielding a stable equilibrium.
Approaching the low density side of the contour, the equilibrium once more enters
the stable oscillatory part, before moving into the saddle region at the minimum
density on the contour by transitioning either through the unstable oscillatory
(ρ ≥ 8) or a stable region (ρ ≤ 8).
For 10.5 ≤ ρ ≤ 14.5, the system always has an oscillatory behaviour on the high
temperature side, but for ρ ≥ 14 there are no stable equilibria anymore, effectively
lowering the upper limit on the accessible value of ρ for a fusion reactor if stable
burn without external control is a requirement.
At the point where the fold, Hopf and saddle-homoclinic bifurcations meet
(close to the ρ = 8 contour at the minimum electron density), the system has a
Bogdanov-Takens bifurcation [87]. A Bogdanov-Takens bifurcation occurs when
the system has a zero eigenvalue of multiplicity two. There are two nearby equilibria: a saddle point and a node (sink or source), which annihilate via a saddle
node bifurcation. The non-saddle equilibrium undergoes a Hopf bifurcation that
generates a limit cycle that connects to the saddle point in a saddle-homoclinic
bifurcation (which we encountered a little while back).
4.3.5
Physical interpretation
Besides looking at the stability properties by themselves, we can also try to understand the physical mechanisms behind the stability or instability of the different
eigenvectors. This section will interpret the local stability properties in terms of
the physical mechanisms that drive them. Because the parameter space is too
4.3 Reduced system
79
large to be covered in detail, we will discuss the eigenvectors and eigenvalues at
different relevant points, all at a density of ne = 1 × 1020 m−3 for the PPCS A
design with the IPB98(y,2) scaling for τE .
The Jacobian matrix consists of the partial derivatives of the burn equations
with respect to the different variables. Evaluating it at an equilibrium point tells
us something about the sensitivity of the equilibrium to changes in both variables.
Looking at equation (4.7), it is clear that
∗
∗
j11
< −j22
(4.35)
∗ ∗
∗ ∗
j11
j22 ≥ j12
j21 .
(4.36)
is a necessary, but not sufficient condition for stability. The other requirement is
So the stability of the system depends on all four elements in J ∗ .
∗
Looking at the physical meaning, the top left term j11
represents the derivative
of dfα /dt with respect to fα . A negative value here means that an increase in fα
will result in a negative value for dfα /dt, driving the system back to equilibrium.
∗
The bottom right term j22
corresponds to the sensitivity of the rate of change
of temperature to changes in temperature, and again, a negative value has a stabilising effect.
When taking the top right and bottom left terms into account, the picture
becomes more complicated. They describe the change in dfα /dt and dT ∗ /dt, caused
by variations in T ∗ and fα respectively. Their effect is stabilising if they have
opposite signs, and destabilising in case they have the same sign, but the overall
stability is determined by the signs and values of all four elements of J ∗ .
Say an increase in helium content causes the temperature to rise. Even though
∗
j11
could be negative and drive the helium concentration back to the equilibrium
level, the temperature increase might promote an increase in helium content. If this
latter effect is stronger than the former, the system will be unstable. Notice that
∗
this effect can take place even when j22
< 0, it really depends on the magnitude
of the different terms.
We will take a closer look at the different terms that make up the different elements of the Jacobian matrix, and try to determine on physical grounds whether
they will be positive or negative. This will provide insight in the driving mechanisms behind the instabilities that are present in the system.
The first element, j11 , consists of three parts. The first
(2nα − ne ) hσvi
is always negative, because ne ≥ 2nα , and corresponds to the reduction in reaction
rate when the fuel dilution increases.
The second term
1
−
ρτE
Chapter 4 Burn stability
80
is always negative by definition. This leaves us with the last part
nα ∂τE
,
ρτE2 ∂nα
which is always positive. This can be seen by looking at equation (A.6) and
realising that nα < 0.5ne .
The second element, j12 , has only two terms. The first can be positive or
negative, depending on the sign of dhσvi/dT . Because the maximum of hσvi
lies at 67 keV, this term will be positive under reactor relevant conditions. This
represents the increase in helium production because of a higher reactivity if the
temperature rises.
The sign of second term of j12 is also determined by dhσvi/dT (equation (A.7)),
but has the opposite sign to the first term, because a higher temperature results
in a higher fusion power, which decreases the (particle) confinement time.
The third element j21 , turns out to be the term with the largest magnitude
for all temperatures at ne = 1 × 1020 m−3 . It also has a negative sign everywhere,
meaning that an increase in helium content will result in a decrease of temperature
(all else remaining constant). Equation (4.23) has three terms, the first of which
is
(ne − 2nα )(2nα − 7ne )
3
hσvi
E
+
T
.
α
6n2tot
2
This is the derivative of heating power per particle with respect to nα . An increase in helium content leads to lower fuel concentration and hence lower fusion
power. The ntot term in the denominator becomes slightly smaller for increasing
nα , but since nα < 0.5ne it can easily be seen that this term is always negative by
substituting ne − 2nα > ne − ne = 0 and realising that 2nα − 7ne < 0.
The second part reads
−
T ∂τE
2ne T ∂τE
2ne T
T ∂τE
+
−
−
,
ρτE2 ∂nα ntot ρτE2 ∂nα τp n2tot τE2 ∂nα
| {z } |
{z
} | {z } | {z }
1
2
3
4
where we have separated several terms to make their physical meaning clearer.
Here part 1 stems from the changes in particle losses due to a change in helium
content. This affects the particle losses because the average ion mass is present in
the IPB98(y,2) scaling, where τE ∝ A0.19 , so in this case this term is negative.
Parts 2 and 3 are the derivative of the cooling term from refuelling with respect
to the helium content. Part 2 represents the changes in refuelling that accompany a
change in particle confinement, which itself is caused by a change in helium content
(which is positive), and part 3 the changes due to a change in ntot when the helium
content changes. In the latter case the refuelling itself is not directly affected, but
the cooling effect is divided among more or less particles. This term has a minus
sign because a higher helium content means a lower number of particles.
4.3 Reduced system
81
Part 4 is due to a change in energy confinement caused by a change in helium content, which is negative because a larger helium content leads to a longer
confinement time and hence lower energy losses.
Finally, the third term in equation (4.23),
−
5n2e Rrad (T, 1)
,
3 2
2 ntot
represents temperature decrease due to the change in radiation losses, which in2
crease when nα increases due to the Zeff
term in the Bremsstrahlung.
Summarising j21 consists of three terms, of which the first and the last have a
cooling effect. Because the first term is the largest for all equilibria, the fact that
the sign of the second term depends on position of the equilibrium doesn’t matter.
Finally, j22 is made up of three terms. The first,
(ne − 2nα )2
dhσvi 3
3
+ hσvi ,
Eα + T
6ntot
2
dT
2
is positive and corresponds to the temperature increase due to the increased alpha
heating for a rising temperature.
The second term
nα
1
T ∂τE
−
+1
− 2
ρntot
τE
τE ∂T
looks complicated, but it represents the energy losses because of lower confinement,
and a cooling effect associated with the refuelling with cold particles. Determining
the sign of this term is not difficult, because equation (A.7) tells us that ∂τE /∂T
is negative for reactor relevant temperatures, so this term of j22 is negative.
The final term represents the increased radiation losses for higher temperatures,
and this obviously has a negative contribution to the derivative. The overall sign
of j22 thus depends on whether the increased fusion power outweighs the increased
losses and this changes with temperature.
4.3.6
Low temperature stability
We can apply the results from the previous section to different equilibria in the
phase plane. Starting with the low temperature equilibrium at ρ = 5, which has a
temperature of 6.24 keV, we can evaluate the normalised Jacobian matrix:
−0.011693 0.000094
∗
J =
.
−2.542608 0.158415
The (normalised) eigenvectors for the low temperature equilibrium are
v1 = (0.066202, 0.997806)
82
Chapter 4 Burn stability
and
v2 = (0.000556, 1.000000),
with v1 stable and v2 unstable. Although in both eigenvectors the alpha en temperature components have the same sign, the v2 has a much larger temperature
component than v1 1 . The temperature component in both v1 and v2 is destabilising, but it is the larger helium component that stabilises v1 .
4.3.7
High temperature stability
For the high temperature equilibrium at ρ = 5 and T = 29 keV, the Jacobian
matrix is
−0.086266
0.000082
J∗ =
.
−17.932381 −0.255725,
Compared to the low temperature equilibrium, we can observe that the only qualitative difference can be found in the bottom right term. This term has become
negative, reflecting the fact that at the high temperature equilibrium the losses
increase faster with temperature than the fusion power. The derivative of the
change in helium density with respect to the temperature is still positive, which
is caused by the fact that dhσvi/dT , which is always positive for reactor relevant
temperatures, is larger than ∂τE /∂T , which is negative because the increase of
power with increasing temperature results in a lower value of τE . The major quantitative difference is in the bottom left element, which shows that the sensitivity of
the temperature with respect to changes in the helium concentration at constant
ρ increases with temperature, which finds its origin in the strong temperature
dependence of hσvi.
In this case the eigenvectors are
and
v1 = (0.066202, −0.997806)
v2 = (0.000556, −1.000000),
which are both stable. Notice that again both vectors are nearly parallel.
From figure 4.1 we know that in between the unstable and stable equilibria
there is transition region where the eigenvalues acquire an imaginary component,
giving rise to oscillatory behaviour. Also, the values of the different elements of
the Jacobian change when going to higher or lower densities, or when changing
the value of ρ, but the overal picture stays the same.
1 Even with normalisation, J ∗ is highly asymmetrical and the eigenvectors are nearly parallel
(their inner product is 0.997843), so special caution is warranted when calculating the eigenvalues,
because the results are sensitive to small (rounding) errors in the elements of the Jacobian. Since
a 2x2 matrix is always in Hessenberg form, balancing does not help in this case (in fact balancing
a matrix in Hessenberg form changes the eigenvalues and vectors).
4.3 Reduced system
4.3.8
83
Phase portrait
We can divide the ne , T -plane into regions with different behaviour. Figure 4.2
shows the phase portraits for PPCS model A for different values of ρ at a density
of ne = 1020 m−3 . The red markers indicate the equilibria, whose position in
the ne , T -plane is indicated in the top middle graph. The red lines indicate the
separatrix between regions that converge towards a stable equilibrium and regions
that do not (i.e. that will lead to a distinguishing of the burn if no action is taken).
Starting from the top left image and moving anti-clockwise, the first image has
ρ = 5 and we can see that the unstable point on the left is indeed a saddle point,
with a stable eigenvector dominated by fα and an unstable one dominated by a the
temperature. The stable equilibrium at higher temperature is a stable improper
node, meaning that all orbits approach the node from opposite directions along
the same line, except for two orbits which come in from opposite directions with
a certain angle to the above mentioned line.
Both axis can be found by finding the matrix T ∈ R2×2 that satisfies T AT −1 =
J, with J the Jordan canonical form of A. Then define y(t) to satisfy the system
y 0 = Jy
(4.37)
and all orbits come in along the direction of the y1 vector, except for the two orbits
that come in along the y2 direction.
The separatrix between the stable and unstable region starts on the T -axis a
little above T = 5 keV in this case and increases more or less linearly until T = 10
keV and fα = 0.2, at which point the curve starts to flatten and disappears towards
T = ∞. Lowering the value of ρ will lift the asymptote of the separatrix towards
fα = 0.5, but it always stays below the physical limit fα = 0.5, which corresponds
to a complete helium plasma. In the scope of our model this would be a an
unrecoverable scenario without external heating, because a pure helium plasma
means no fusion power and consequently infinite energy and particle confinement
times.
If the value of ρ is increased to eleven, we see that the separatrix starts to
flatten a bit more strongly, but still approaches an asymptote that extends towards
T = ∞, albeit at a lower value of fα . The unstable equilibrium has moved up
a bit and the stable equilibrium is closer to the separatrix. At this point the
eigenvalues of the system at the stable equilibrium have acquired an imaginary
part and consequently this has become a stable spiral point. The pitch of the
spiral depends on the ratio of the real and imaginary parts of the eigenvalues.
Increasing the value of ρ until we hit the saddle-homoclinic bifurcation results
in the separatrix revolving around the stable equilibrium and connecting with itself
in the unstable equilibrium. This also means that the lower part of the separatrix
has disappeared from the system and the stable region of the phase space has been
reduced to the area within the homoclinc orbit.
Chapter 4 Burn stability
84
0.3
ρ=5
ρ = 11
ρ = 13.8375
ρ = 14.4
fα
0.2
0.1
0
0.3
fα
0.2
0.1
0
10
ρ = 14.7
0.3
20
30
40
ne
fα
0.2
0.1
0
10
20
30
T (keV)
40
5
10
20
T (keV)
50
Figure 4.2: Phase portraits for PPCS model A in the fα , T -plane of the reduced
burn system for ne = 1020 m−3 for different values of ρ. The equilibria are indicated
by corresponding red markers in the phase portrait and in the ne , T -plane (bottom
right), and the red line indicates the separatrix between stable and unstable regions
in the phase plane. The top left image at ρ = 5 shows a saddle point at low
temperature and a stable equilibrium (sink) at high temperature. When the value
of ρ is increased, an imaginary component is introduced in the system (top left).
A further increase in ρ introduces a limit cycle, that grows until it becomes a
homoclinic orbit that connects the saddle point with itself, orbiting the (high
temperature) stable equilibrium. For even higher values of ρ the stability of the
high temperature equilibrium changes: it becomes unstable with an imaginary
part, until for very high values of ρ the high temperature equilibrium also becomes
a source. Another effect of increasing ρ is that the two equilibria approach each
other, until they coincide at the maximum allowed value of ρ.
4.3 Reduced system
85
The homoclinic orbit becomes an unstable limit cycle which shrinks in size with
a further increase of ρ, until the system hits the Hopf bifurcation where the stable
equilibrium changes to an unstable one. Note that right at the Hopf bifurcation
there is an infinite number of limit cycles around the equilibrium, that at that
point is stable, but not asymptotically stable.
Towards the upper limit of ρ the imaginary part of the eigenvalues at the high
temperature equilibrium disappears again, changing this point into an improper
node, and the system is left with two unstable equilibria.
An interesting observation that can be made when comparing the 5 phase
portraits in figure is that the overall picture looks remarkably similar, with most
orbits converging towards a curve that resembles a skewed parabole. The equilibria
are located somewhere along this curve, and depending on the value of ρ, the
separatrix partially runs along this curve as well.
This observation could have implications for reactor start up or the design of
burn control systems, because some orbits might be highly undesirable as they will
cross the β limit, or put too much heat load on the first wall.
4.3.9
Stability for different scaling laws
The ITER physics basis contains five different scaling laws, (IPB98(y) and
IPB98(y,i), with i = 1, 2, 3, 4 [26]), and although their predictions for τE in ITER do
not differ much, it has already been shown that the operating contours they predict
for a burning plasma show large differences in density range [88]. In the following
section we will investigate the changes in stability for the equilibria between the
different scalings in the ITER physics basis.
Figure 4.3 shows the different stability regions in the ne , T -plane for the five
different energy confinement time scalings in the ITER physics basis. The global
picture looks similar for the first four scalings: y, and y(1,2,3), with an unstable
saddle point at low temperature and a second equilibrium at higher temperature,
the stability of which is determined by ρ and ne . All plots feature the same stability
regions, and they have a similar shape and position in the plot.
At first glance the IPB98(y,4) scaling shows a completely different picture,
but closer inspection learns that flipping the plot upside down makes it look very
similar to the other four. The reason for this is that the value of 1 − 2m + l, with
m and l being the exponents of the power and electron density in the scaling law,
has a different sign for the (y,4) scaling compared to the other four scalings. The
fact that this value is very close to zero for all scalings explains the large spread
in density range between the different scalings, as was explained in chapter 4.
Ignoring the fact that the plot for the last scaling is ’upside down’, the main
differences in stability can be found when looking at the intersections of the curves
of constant ρ with the stability boundaries. For the IPB98(y) scaling, the ρ = 14
contour extends to about two/thirds of the width of the unstable (dark blue) region
on the high temperature side. The same contour for subsequent scalings reaches
Chapter 4 Burn stability
86
98(y)
ne (m−3 )
1030
1025
1020
1015
1028
1055
1024
1040
1020
1025
1010
1016
98(y,4)
ne (m−3 )
98(y,3)
10100
1070
1075
1020
1050
10−30
1025
10−80
100
5
10
20
50
T (keV)
100
200
5
10
20
50
T (keV)
100
ne (m−3 )
ne (m−3 )
1070
ne (m−3 )
98(y,2)
98(y,1)
1032
10−130
200
Figure 4.3: Stability regions for the PPCS A design for the different τE scalings
in the ITER physics basis [26]. The
area is asymptotically stable,
asymptotically stable with an oscillation,
is unstable with a oscillation,
is unstable
and
is a saddle point (unstable).
4.3 Reduced system
87
even further, and for the (y,2), (y,3) and (y,4) scalings extends into the stable
(purple).
Similarly, the ρ = 10 contour doesn’t quite extend into the green area for the
(y) and (y,1) scalings, whereas for the other three scalings there is a significant
part of this curve that traverses the asymptotically stable (green) area of the plot.
The scaling laws for τE contain three parameters that change when moving
through the phase space of a burning plasma; the average ion mass A, the electron
density ne and the heating power P . Looking at the exponents k, l and m of
these parameters in the scaling law, we see that l decreases and m increases for
the subsequent scalings, whereas there is no clear pattern for k.
The expansion of the unstable and oscillatory region therefore correlate with
an increase in density and a decrease in power dependence, but we have not been
able to identify a physical mechanism for this.
For all scalings the point where both eigenvalues are zero lies roughly on the
ρ = 8 contour, but the density for this point varies greatly. Depending on the sign
of 1 − 2m + l, it can be found below or above the Greenwald density, but it is never
in the reactor relevant density range.
Figure 4.4 zooms in on the reactor relevant density and temperature range for
the PPCS A design, plotting the stability of the equilibria on curves of constant
ρ as well as the Greenwald density and β limit. In all subplots we see an unstable
saddle point on the low temperature side, and a, mostly stable, equilibrium at the
high temperature side. Only for ρ = 14 does this equilibrium become unstable,
but it already acquires an imaginary eigenvalue part above ρ ≈ 10, depending on
the scaling.
In the two plots on the top row, the ρ = 2 curve shows an oscillatory region (in
the bottom right of the plot). This is the intersection of this particular contour
with the long ’tail’ of the stable oscillating region, which could already be seen
in figure 4.1. The fact that it is not visible in the other plots is an artefact from
plotting only integer values of ρ.
Concluding we can say that for the scaling laws in the ITER physics basis there
are only minor differences in stability, both on a global level and in the reactor
relevant domain. Those differences manifest themselves at the high temperature
equilibrium, and correlate with the values of the power en density exponents in
the scaling law: a weaker power dependence and a stronger density dependence
result in an expansion of the unstable area to lower values of ρ. However, based
on the results of present day tokamaks [34], it seems unlikely that this will be an
issue for future reactors, although much higher values of ρ have also been reported
in limiter plasmas [76].
4.3.10
Stability with external heating
So far we have only considered ignited plasmas without external heating. This
approach allows for an analytical treatment of the system, but it is an unrealistic
Chapter 4 Burn stability
88
IPB98(y)
lim
it
ρ=
5
ρ=9
ρ = 13
1
ρ = 13
β
ρ=1
ne (1020 m−3 )
2
nG
IPB98(y,1)
IPB98(y,2)
IPB98(y,3)
IPB98(y,4)
ne (1020 m−3 )
2
1
ne (1020 m−3 )
2
1
5
10
15
20 25
T (keV)
30
35
40
5
10
15
20 25
T (keV)
30
35
40
Figure 4.4: Stability for the reduced system burn contours of PPCS model A
at different values of ρ for the different τE scalings in the ITER physics basis.
The asymptotically stable part of the contours is denoted
, whereas
is
asymptotically stable with an oscillation,
is unstable with an oscillation,
is unstable and
is a saddle point (unstable).
4.3 Reduced system
89
scenario for a tokamak reactor design, if only because some form of non-inductive
current drive will be needed. The external heating power required for this will also
affect the stability of the equilibria.
In the case of external heating, equation (4.18) changes to
2
(ne − 2nα )
nα T
T
Sext − Srad
dT
3
=
hσvi Eα + T −
−
+
,
(4.38)
3
dt
6ntot
2
ntot τp
τE
2 ntot
where Sext is the external heating power density in keV/m3 . Since Sext is independent of nα or T , it does not show up in the Jacobian, which consequently has
the same eigenvalues as before. Adding external heating does therefore not affect
the stability of the system directly.
MW
0M
W
15
0M
W
20
0M
W
25
0M
W
10
50
0.2
0M
50 W
10 MW
0
15 MW
20 0 MW
25 0 M
0 MW
W
fα (%)
0M
W
0.25
0.15
0.1
22
24
26
28
T (keV)
30
32
Figure 4.5: The position of the high temperature equilibrium in the fα , T -plane
as a function of Pext , at a density of ne = 1 × 1020 m−3 for the PPCS A design
for ρ = 5 (
) and ρ = 10 (
). For increased levels of external heating, the
equilibrium temperature increases and the helium fraction decreases. Whereas the
relative decrease in helium fraction is the same at roughly 5% at both ρ values, the
relative increase in temperature is significantly larger at ρ = 10 (9% versus 24%).
Indirectly there is an effect because the equilibria are shifted in phase space to
different values of fα and T , and the eigenvalues of the Jacobian evaluated at these
points are different. The shift in fα and T is illustrated in figure 4.5, which shows
the equilibrium position in the fα , T -plane as a function of Pext for the PPCS A
design at ρ = 5 and ρ = 10.
Increasing Pext results in a shift of the equilibrium to higher temperatures
and lower helium fraction, which can be understood as follows. Adding external
heating increases the temperature and consequently the reactivity of the plasma.
This increase, together with the resulting increase in fusion power, lowers the
energy confinement time, which results in a faster exhaust of helium ash.
90
Chapter 4 Burn stability
The relative shift in helium fraction is roughly 5%, and this value is more or
less independent of the value of ρ. The relative temperature shift, on the other
hand, increases for higher values of ρ. The reason this effect is stronger for higher
values of ρ lies in the ratio between alpha heating power and external heating. At
higher values of ρ the equilibrium temperature is lower and the helium fraction
higher, which implies a lower fusion power output. Adding a certain amount of
external heating will therefore have a bigger effect at high ρ values.
An interesting observation is that the effect on the fusion power output of the
reactor, is almost linear in T , with the gradient dPfus /dT a function of ρ. For high
values of ρ it might therefore be interesting to consider external heating to increase
the power output of the reactor. As long as the gain in fusion power output is
larger than the additional external heating divided by the plant efficiency times
the heating efficiency, the net effect is positive.
To determine the stability of the system with Pext 6= 0, we first solve the burn
equations (4.17) and (4.38) to obtain the equilibrium values of nα and T . These
are used to obtain the Jacobian of the system at the equilibrium and subsequently
we determine the eigenvalues of the system.
Figure 4.6 plots the stability of the system in the ne , T plane for the PPCS A
reactor with different levels of external heating, at different values of ρ. The plot
on the top left is identical to the bottom left in figure 4.4.
It is immediately apparent that the green area at high temperature increases
for increasing levels of external heating and that the blue segment on the ρ = 14
curve disappears. At the same time the addition of Pext results in a change from
closed contours to open contours, which manifests itself in a change in the sign of
the slope of the high temperature equilibrium curves. Instead of having a positive
value of dne /dT , this now becomes negative.
On the low temperature side the equilibrium curves also change. Additional
heating first of all changes their slope from negative to positive, and secondly
causes them to curve upwards after hitting a local minimum in density, which
introduces a second stable equilibrium at low temperatures. Around the minimum density point, the system goes through a transition from unstable to stable,
passing through an oscillating area, with the same bifurcations as described in
section 4.3.4. The stability of the equilibrium to the right of the local density minimum is unaffected by the amount of external heating, it remains a saddle point
(which is always unstable).
Increasing the additional power has the effect of shifting the minimum density
on the low temperature side upwards, and at some point they disappear from the
plot (bottom middle plot). The relevance of these equilibria for reactor purposes
is minimal because of the very low fusion power output, but it basically rules out
a low power startup scenario at high density.
Concluding we can say that the addition of external heating has a beneficial
effect on the stability of the high temperature equilibrium, for which the stable
4.3 Reduced system
91
Pext = 0MW
Pext = 50MW
2
lim
it
ρ=1
ρ=5
ρ=9
ρ = 13
ρ=1
1
ρ = 13
ne (1020 m−3 )
β
nG
Pext = 100MW
Pext = 150MW
Pext = 200MW
Pext = 250MW
ne (1020 m−3 )
2
1
ne (1020 m−3 )
2
1
5
10
15
20 25
T (keV)
30
35
40
5
10
15
20 25
T (keV)
30
35
40
Figure 4.6: Stability for the reduced system burn contours of PPCS model A at
different values of ρ for different levels of external heating. The asymptotically
stable part of the contours is denoted
, whereas
is asymptotically stable
with an oscillation,
is unstable with an oscillation,
is unstable and
is a saddle point (unstable).
Chapter 4 Burn stability
92
area increases for increasing Pext . The stability of the low temperature equilibrium
is unaffected for higher densities, and a stable, third equilibrium is introduced
at an even lower temperature. The transition between the stable and unstable
equilibrium at low temperature occurs at the local density minimum.
4.3.11
Reactor comparison
There are currently four different prototype reactor designs under consideration in
Europe [71, 70], with PPCS models A and B being more conservative and PPCS
models C and D applying riskier technology extrapolations. This section analyses
and compares the stability for the four different designs at their designed level of
external heating.
Figure 4.7 plots the stability of the burn contours for the PPCS model A, B, C
and D designs for different values of ρ with the amount of external heating specified
in reference [71], which is 246 MW, 270 MW, 112 MW and 71 MW respectively.
We take the same approach as in section 4.3.10, first calculating the equilibrium
values of nα and T for the given values of ρ and Pext and subsequently determining the eigenvalues of the Jacobian. The resulting stability curves are plotted in
figure 4.7, in which the top left plot closely resembles the bottom right plot in
figure 4.6, since the level of external heating differs by only 2% between them.
All four reactor designs are stable in their complete operating range for their
designated level of external heating, but the type of stability differs between the
different reactors at different operating points. The operating space for the PPCS
A and B designs are almost completely asymptotically stable, the PPCS C and D
designs show considerable areas with oscillatory stable behaviour. For the PPCS
C design, this only occurs for ρ ≈ 2 or well above nG for high values of ρ.
4.4
Full system
Having studied the reduced system in detail, we will now return to the full system
of burn equations, as presented in section 4.2.1. First we will linearise the system
around the equilibria and derive the Jacobian matrix, and use this result to investigate the stability of the system in the ne , T -plane for different reactor designs
and energy confinement scaling laws. Finally, we will discuss the stability of the
system with external heating.
4.4.1
Jacobian matrix of the full system
Writing xi = {nD , nT , nα , T }, the burn equations (4.2) can be expressed as
dxi
= fi (x1 , x2 , x3 , x4 )
dt
i = 1, 2, 3, 4
(4.39)
4.4 Full system
PPCS A, Pext = 246MW
93
PPCS B, Pext = 270MW
2
lim
it
ρ=1
ρ=5
1
ρ=9
13
ρ=
ne (1020 m−3 )
β
nG
PPCS C, Pext = 112MW
PPCS D, Pext = 71MW
ne (1020 m−3 )
2
1
5
10
15
20 25
T (keV)
30
35
40
5
10
15
20 25
T (keV)
30
35
40
Figure 4.7: Stability for the reduced system burn contours of PPCS models A, B,
C and D at different values of ρ for the design value of external heating as specified
in [71]. The asymptotically stable part of the contours is denoted
, whereas
is asymptotically stable with an oscillation.
Chapter 4 Burn stability
94
We can linearise the system around an equilibrium point xj0 as
4
X
dxi
∂fi (xj − xj0 ) ,
=
dt
∂xj xj0
j=1
i = 1, 2, 3, 4.
(4.40)
To determine the linear stability of the system requires determining the eigenvalues (and eigenvectors) of the Jacobian matrix jij = ∂fi /∂xj evaluated at the
equilibrium points xj0 .
As with the reduced system, determining the derivatives used in the Jacobian is
not difficult, but involves some extensive algebra which can be found in section A.2.
Here we simply present the Jacobian of the full system:
1
nD ∂τE
+ 2
,
ρτE
ρτE ∂nD
nD ∂τE
= −nD hσvi + 2
,
ρτE ∂nT
nD ∂τE
= 2
,
ρτE ∂nα
dhσvi
nD ∂τE
= −nD nT
+ 2
,
dT
ρτE ∂T
nT ∂τE
= −nT hσvi + 2
,
ρτE ∂nD
nT ∂τE
1
= −nD hσvi −
+ 2
,
ρτE
ρτE ∂nT
nT ∂τE
= 2
,
ρτE ∂nα
dhσvi
nT ∂τE
= −nD nT
+ 2
,
dT
ρτE ∂T
nα ∂τE
= nT hσvi + 2
,
ρτE ∂nD
nα ∂τE
= nD hσvi + 2
,
ρτE ∂nT
1
nα ∂τE
+ 2
,
=−
ρτE
ρτE ∂nα
dhσvi
nα ∂τE
= nD nT
+ 2
,
dT
ρτE ∂T
j11 = −nT hσvi −
(4.41)
j12
(4.42)
j13
j14
j21
j22
j23
j24
j31
j32
j33
j34
(4.43)
(4.44)
(4.45)
(4.46)
(4.47)
(4.48)
(4.49)
(4.50)
(4.51)
(4.52)
4.4 Full system
j41
j42
ntot nT − 2nD nT
3
=
hσvi Eα + T
3 2
2
2 ntot
4(sD + sT )T
(ρ − 1)T ∂τE
1 ∂Srad
2Srad
+
− 3
+ 3 2 +
,
2
ρτE ∂nD
∂nD
n2tot
2 ntot
2 ntot
3
ntot nD − 2nD nT
hσvi
T
E
+
=
α
3 2
2
2 ntot
(ρ − 1)T ∂τE
4(sD + sT )T
1 ∂Srad
2Srad
,
− 3
+ 3 2 +
ρτE2 ∂nT
∂n
n2tot
T
2 ntot
2 ntot
2nD nT
3
= − 3 2 hσvi Eα + T
2
2 ntot
(ρ − 1)T ∂τE
6(sD + sT )T
1 ∂Srad
3Srad
+
,
− 3
+ 3 2 +
ρτE2 ∂nα
∂n
n2tot
n
n
α
tot
2
2 tot
3
∂hσvi
nD nT 3
= 3
hσvi + Eα + T
2
2
∂T
2 ntot
1 Srad
2(sD + sT )
T ∂τE
1−ρ 1
− 3
−
− 2
.
+
ρ
τE
τE ∂T
2T
ntot
2 ntot
+
j43
j44
95
(4.53)
(4.54)
(4.55)
(4.56)
This result differs from the Jacobian presented by Rebhan and Vieth [65] for two
reasons. Firstly, our result is valid for scaling laws for the energy confinement time
that do not include radiation losses, whereas ref [65] uses the ITER89P scaling [29],
which includes the radiation losses in the energy confinement time. This introduces
some extra terms in our Jacobian to account for this.
The second difference arises because ref [65] assumes a constant ion mass of
2.5 amu, which is only correct in case of a pure DT plasma. A burning plasma by
definition contains helium and this needs to be taken into account in the average
ion mass, which consequently will be ≥ 2.5 amu.
4.4.2
Full system stability
Using the expression for the Jacobian presented in section 4.4.1, we can determine
the stability of the different eigenvectors of the system for each equilibrium by
determining their respective eigenvalues. In figure 4.8 the stability of the different
eigenvectors of the system is plotted. Again, red is unstable, blue is oscillatory
unstable, purple is oscillatory stable and green is stable.
We can see that on the high temperature side the system is stable, whereas on
the low temperature side there is one eigenvector that turns the system unstable.
In the central region of the operating space their are several transitions between
stable and unstable behaviour for the different eigenvectors.
A transition between different stability regimes means that either the real or the
imaginary part of the eigenvalues has a zero crossing. There is only one point (near
Chapter 4 Burn stability
96
1070
ne (m
−3
)
1060
1050
1040
1030
1020
1010
1070
ne (m
−3
)
1060
1050
1040
1030
1020
1010
5
10
20
50
T (keV)
100
200
5
10
20
50
T (keV)
100
200
Figure 4.8: The stability of the different eigenvectors for a burning plasma in
the PPCS A reactor using the IPB98(y,2) scaling for τE . Unstable behaviour is
indicated by ,
signifies oscillatory unstable behaviour,
oscillatory stable
behaviour and
stable behaviour. The irregularities in the boundaries between
the different colours, for instance on the green-purple boundary in the two plots
on the right is caused by a an interchange of two or more eigenvectors.Please note
that the density and β limits are not taken into account in this plot.
4.4 Full system
97
the bottom of the plot where the blue parts end) where the absolute value of both
eigenvalues goes to zero, which corresponds to the Bogdanov-Takens bifurcation
described in section 4.2.3. Hence the fact that blue always borders red and purple
and never borders green. Similarly, purple is never adjacent to red. Note that
although it appears otherwise, there is actually a small slither of red between the
green and blue areas in the bottom right plot.
The irregularities on the border between purple and green that can be seen
in the two plots on the right of figure 4.8 are caused by a numerical difficulty
sorting the eigenvectors. When determining the eigenvectors, most algorithms will
determine the largest eigenvalue and corresponding eigenvector first, then reduce
the dimension of the matrix and repeat. If two (or more) eigenvalues happen to
approach each other closely enough, both in argument and absolute value, it can
be difficult to know which is which at the next point in parameter space. This can
of course be solved by looking closely at the trajectory of the different eigenvalues
and vectors, but developing a fail safe algorithm is rather involved and doesn’t
lead to new physical insight.
Although this plot allows us to investigate the stability of each eigenvector
in a particular equilibrium, it is hard to obtain the overal stability at a glance.
Figure 4.9 plots the stability areas of the complete system in a single figure, with
red the unstable and green the stable part. The system is unstable on the low, and
stable on the high temperature side. The transition between the two regions lies at
the minimum density of the burn contours below the Bogdanov-Takens bifurcation
(which occurs at ρ ≈ 8) for the low density part of a burn contour. For higher
values of ρ and at the high density part of the contour, the transition occurs at a
higher temperature and a higher respectively lower density.
Comparing this picture to figure 4.1 the overal stability of the four- and twodimensional systems looks very similar, lending greater credibility to the claim
that the reduced system captures the essential physics.
4.4.3
Eigenvectors and eigenvalues
Apart from looking at the stability of the individual eigenvectors and of the system
as a whole, we can also try and interpret the eigenvectors of the system in a physical
sense. Although new for the ITER physics basis scalings, a similar approach was
taken by Rebhan and Vieth for the ITER89P L-mode scaling [65].
They investigated three cases: one where they assumed that τE and τp kept
their equilibrium values during a perturbation, one where τE and τp followed the
ITER89P scaling law but the heating power was equal to the loss power, and one
where they equated the heating power to Sα = nD nT hσviEα 2π 2 κa2 R, again with
τE and τp following the scaling law.
We have only investigated the latter case, of course using the IPB98(y,2) scaling, since this is of most relevance for future reactors. Besides the fact that the
ITER89P scaling includes radiation losses and the IPB scalings do not, we have
Chapter 4 Burn stability
98
also included the presence of helium in the plasma on the average ion mass, something which was absent in the analysis in [65].
4.4.4
Low temperature stability
We will discuss the eigenvectors for the full system at the same equilibria as for
the reduced system, at a density of ne = 1020 m−3 and ρ = 5 for the PPCS A
reactor with IPB98(y,2) scaling. The normalised eigenvectors for the unstable,
low temperature equilibrium at T = 6.2 keV are displayed in table 4.2.
Table 4.2: The normalised eigenvectors and eigenvalues (λ) at the low temperature
equilibrium for a burning plasma in the PPCS A reactor using the IPB98(y,2)
scaling at a density of ne = 1020 m−3 and ρ = 5. The first eigenvector also has a
very weak fα dependence which doesn’t show up at this level of accuracy.
v1
v2
v3
v4
fD
0.01
0.21
−0.14
−0.48
fT
0.01
0.21
−0.14
0.29
fα
0
−0.03
0.05
0.05
T∗
−1.00
−40.96
0.98
0.83
λ
0.195
−0.018
−0.014
−0.012
For the low temperature equilibrium all four eigenvectors have four non-zero
elements, which agrees with our finding that the dimensionality of the system
cannot be reduced without discarding physics information, as was done in ref [65]
.
The first eigenvector v1 is unstable and is predominantly a temperature perturbation. This corresponds to the temperature instability that is present in the
simplest model for a burning plasma that contains only Bremsstrahlung losses
and alpha particle heating. At the low temperature equilibrium the reactivity
of the plasma has a stronger temperature dependence than the radiation losses.
Therefore a temperature perturbation will quickly grow until the plasma either
extinguishes (in case of a negative perturbation) or the temperature reaches the
stable equilibrium. Because a temperature perturbation also has an effect on the
plasma composition, v1 also has non-zero deuterium, tritium and helium components.
The second and third eigenvectors differ from the first in that they show much
larger deuterium, tritium and helium components. They both correspond to an
in-phase deuterium and tritium density fluctuation coupled to a temperature and
helium density fluctuation.
The fourth eigenvector v4 differs from the previous three in that the deuterium
and tritium fluctuation have an opposite sign. The temperature component is
4.4 Full system
99
the smallest for this eigenvector. This eigenvector probably corresponds to the
eigenvector (ñD , −ñT , 0, 0) that was found in reference [65]. The eigenvalue of v4
is indeed λ4 = −1/τp , and the reason the eigenvector looks different is caused by
the fact that we have included the ion mass dependence in the τE scaling law, which
causes a coupling between the temperature and particle densities that cannot be
removed.
Concluding we can say that it is the thermal part that causes the linear instability of the low temperature equilibrium in a burning plasma. The particle
dominated perturbations are all stable.
4.4.5
High temperature stability
The eigenvectors at the high temperature equilibrium (T = 21.5 keV) look a little
different and are listed in table 4.3. The most obvious difference is the fact that
the first two eigenvectors have acquired an imaginary part. Secondly the particle
components are a lot smaller (for almost the same normalisation: the particle
densities for both the high and low equilibrium are normalised to ne = 1020 m−3
and the temperature is normalised to the equilibrium temperature of 6.22 and 21.5
keV respectively).
Table 4.3: The normalised eigenvectors and eigenvalues (λ) at the high temperature equilibrium for a burning plasma in PPCS model A using the IPB98(y,2)
scaling at a density of ne = 1020 m−3 and ρ = 5. The first eigenvector also has a
very weak fα dependence which doesn’t show up at this level of accuracy.
v1
v2
v3
v4
fD (10−4 )
2 + 68i
2 − 68i
−1747
116
fT (10−4 )
2 + 68i
2 − 68i
1642
116
fα (10−4 )
2 − 9i
2 +9i
-102
328
T∗
1.0000
1.0000
0.9708
0.9993
λ
−0.210 − 0.120i
−0.210 + 0.120i
−0.074
−0.060
The unstable temperature dominated fluctuation at the low temperature equilibrium, is stable at the high temperature equilibrium, but has acquired an imaginary part which causes an oscillating behaviour. The magnitude of the imaginary
part relative to the real part of the eigenvalue increases with ρ, giving rise to higher
ratio of oscillation period to damping time.
The third eigenvector corresponds to the (ñD , −ñT , 0, 0) eigenvector found by
Rebhan and Vieth with eigenvalue λ3 = −1/τp , but in our case this is again
coupled to a temperature and (weak) helium fluctuation.
The fourth and final eigenvector is a combination of temperature en density
fluctuations, but it stabilises slower than the other three eigenmodes.
100
Chapter 4 Burn stability
Because of the lack of symmetry in the Jacobian, both at the high and low
temperature equilibria all eigenvectors are coupled density and temperature fluctuations. This makes a simple interpretation in terms of a pure temperature or
density perturbations difficult, but nevertheless we can identify the underlying
mechanisms of several eigenvectors.
One eigenvector is a density dominated fluctuation, with eigenvalue λ = −1/τp ,
which is expected since this is the particle transport timescale.
Another eigenvector corresponds to the thermal mode that is present in a pure
DT plasma. The eigenvalue of this mode changes with T : for low values of T it
is small compared to τE , for higher values of T the ratio increases to close to one.
This can be understood by realising that at low temperature the radiation losses
account for a large fraction of the total energy losses and these are not included
in τE , which is consequently much larger than the timescale of the fluctuation. At
higher temperature the importance of the radiation losses decreases, bringing the
energy confinement time and the eigenvalue of the mode closer together.
The other two eigenvectors are hybrid modes that consequently have timescales
somewhere in between −1/τE and −1/τp . Their eigenvalues also depend on the
precise location of the equilibrium.
We can plot the stability of the system in the ne , T -plane like in figures 4.8
and 4.1, or look at the composition of the eigenvectors in individual equilibria, but
this doesn’t tell us how the stability is affected by changing the parameters of the
system that are under control of the operator.
In the model that we used, the properties of the system are determined by
three parameters which, within limits, can be chosen freely: sD , sT and ρ. In our
calculations so far we have made the assumption that sD = sT , effectively giving
us two inputs that we can adjust.
The standard approach in this case would be to plot the stability boundary (or
the eigenvalues) as a function of sD and ρ-plane. However, because for each value
of ρ there are two equilibria that have the same refuelling rate and the boundary
of stability is near the maximum or minimum density on a ρ = const contour, such
a plot would not be very instructive.
Instead, we have plotted the lines of constant ρ and constant sD (or sT for that
matter) in the stability overview for PPCS model A with the IPB98(y,2) scaling
in figure 4.9. At low density the stability boundary intersects the contours of
constant ρ at the minimum density for ρ < 9. For higher values of ρ the stability
boundary deviates towards higher temperatures and consequently higher density
at the intersection with the ρ iso-contours. This trend continues at the high density
side of the contours, where the intersection lies below the maximum density.
The dashed lines that indicate constant fuelling rate are slightly curved downward and are just a bit lower at the high temperature side. The reason for this it
that in the center the value of ρ increases, which means a better particle confinement and consequently a less need for refuelling. At the low and high temperature
4.4 Full system
101
1070
1060
-3
ne (m )
1050
1040
1030
1020
1010
5
10
20
50
T (keV)
100
200
Figure 4.9: Stability plot for the PPCS A reactor with IPB98(y,2) scaling, including contours of constant ρ (solid lines) and constant refuelling rate (dashed lines).
Please note that the density and β limits are not taken into account in this plot.
102
Chapter 4 Burn stability
sides the particle confinement is lower, so for a given density there is a higher
particle transport. Finally the reason for the slope in the lines lies in the fact
that the fusion power increases with temperature, so the confinement time decreases towards higher temperatures. This effect weakens when the temperature
approaches the maximum in the reactivity around 70 keV, which can be seen in
the lines of constant refuelling rate at high density and temperature, which run
almost horizontally at high temperature.
Looking at the intersections between the lines of constant refuelling rate and
the stability boundary, it is obvious that it is only possible to cross the stability
boundary by changing the refuelling rate while keeping ρ constant for high densities
or high values of ρ. For ρ < 10 at low densities the stability boundary intersects
the constant ρ contours at the point where they are tangent to the lines of constant
refuelling rate.
4.4.6
Stability for different scaling laws
The ITER physics basis contains five different scaling laws, each based on a fit to
a different subset of the database, or using slightly different fit restrictions. The
effect of using a different scaling on the stability boundary in a burning plasma is
illustrated in figure 4.10 for the PPCS A reactor design.
Again green indicates stable behaviour and red unstable. The stability boundary hardly changes between the different scalings, except for the region around the
minimum density. There the position of the critical point, which corresponds to
the position of the Bogdanov-Takens bifurcation in the reduced system, varies.
Zooming in on the reactor relevant density area, as is done in figure 4.11, we
see that only at very high values of ρ there is a difference in stability between the
different scalings at the high temperature equilibrium. It is interesting to note
however, that although basically all high temperature equilibria are stable, they
all have at least two imaginary eigenvalues. A perturbation of these equilibria will
result in damped oscillations of the plasma parameters. From these plots there is
no way of telling how strong the damping is compared to the oscillatory part, but
in general higher values of ρ exhibit weaker damping (which makes sense because
these equilibria are closer to the unstable equilibria at low temperature).
4.4.7
Reactor stability comparison with external heating
Most reactor designs, even though capable of ignition, still employ some level of
external heating, mostly for current drive and control purposes [70]. Figure 4.12
shows the stability of the PPCS A, B, C and D reactor designs with their respective
levels of external heating, for ρ ranging from 1 to 14.
Obviously, the shape of the curves is exactly the same as in figure 4.7, but the
stability properties are somewhat different. The addition of two extra degrees of
freedom has resulted in the addition of an imaginary component to the eigenvalues
4.4 Full system
103
98(y)
ne (m−3 )
1030
1025
1020
1015
1028
1055
1024
1040
1020
1025
1010
1016
98(y,4)
ne (m−3 )
98(y,3)
10100
1070
1075
1020
1050
10−30
1025
10−80
100
5
10
20
50
T (keV)
100
200
5
10
20
50
T (keV)
100
ne (m−3 )
ne (m−3 )
1070
ne (m−3 )
98(y,2)
98(y,1)
1032
10−130
200
Figure 4.10: The stability regions for a burning plasma in the PPCS A reactor for
the five scalings in the ITER physics basis using the full system. Please note the
change in density range between the different plots, and be aware that the density
and β limits are not taken into account.
Chapter 4 Burn stability
104
IPB98(y)
it
ρ=
5
ρ = 13
ρ = 13
1
lim
ρ=9
β
ρ=1
ne (1020 m−3 )
2
nG
IPB98(y,1)
IPB98(y,2)
IPB98(y,3)
IPB98(y,4)
ne (1020 m−3 )
2
1
ne (1020 m−3 )
2
1
5
10
15
20 25
T (keV)
30
35
40
5
10
15
20 25
T (keV)
30
35
40
Figure 4.11: Stability for the full system burn contours of PPCS model A at different values of ρ for the different τE scalings in the ITER physics basis. The
asymptotically stable part of the contours is denoted
, whereas
is asymptotically stable with an oscillation,
is unstable with an oscillation and
is
unstable.
4.4 Full system
PPCS A, Pext = 246MW
105
PPCS B, Pext = 270MW
2
lim
it
ρ=1
ρ=5
1
ρ=9
13
ρ=
ne (1020 m−3 )
β
nG
PPCS C, Pext = 112MW
PPCS D, Pext = 71MW
ne (1020 m−3 )
2
1
5
10
15
20 25
T (keV)
30
35
40
5
10
15
20 25
T (keV)
30
35
40
Figure 4.12: Stability of the full system burn contours of the PPCS A, B, C and D
designs [70], for ρ ranging from 1 to 14. At low density and high temperatures, most
). At higher densities, they are stable, but the solutions in
equilibria are stable (
this region will oscillate (
), and the PPCS D design has a region with unstable
oscillating behaviour (
). Finally, at high density and low temperatures, there
are unstable equilibria (
).
106
Chapter 4 Burn stability
for most of the high temperature equilibria, compared to the stability of the reduced system. Only at lower densities do equilibria without oscillatory behaviour
still exist.
Again, the PPCS D design is the odd one out, and in this case the system
is unstable for higher densities at low temperatures, which were stable for the
reduced system with external heating. Unfortunately, the anticipated operating
point for PPCS model D (T = 12 keV and ne = 1.4 × 1020 m−3 ) lies within the
unstable density range, although the equilibrium temperature that our simulations
predict is somewhat lower than the value from the PPCS.
4.5
Discussion and conclusions
We have derived a simple two-dimensional system to study the operating point
stability of a burning plasma, by assuming a constant electron density and equal
deuterium and tritium concentrations. This allowed us to use planar bifurcation
theory to describe the transitions between regions with different stability in the
phase plane. Furthermore, we have analysed the physical mechanism behind the
stabilising or destabilising effect of the different elements of the Jacobian matrix.
In general, a burning plasma has two equilibria at a given density, one at
a lower, and one at a higher temperature. The low temperature equilibrium is
always unstable, and the stability of the high temperature equilibrium depends on
the density and ρ. For low values of ρ, the high temperature equilibrium is stable
for all but the highest densities (which are inaccessible anyway because they are far
above the Greenwald and Troyon limits). For high values of ρ the high temperature
equilibrium is stable for intermediate densities, and the stable range shrinks for
increasing values of ρ and it disappears completely when approaching ρcrit .
At the boundaries of the different stability regions, the system features different
bifurcations. In the two dimensional system these can be easily distinguished by
looking at the eigenvalues. Generally speaking, there is a local bifurcation in the
system when either one, or both, of the eigenvalues is zero or has a real part equal
to zero. These zeros occur at the boundary between different stability regions.
It turns out that the reduced system contains five bifurcations: two saddle node
bifurcations at the maximum and minimum density on a contour respectively, a
sub-critical Hopf bifurcation at the stable-unstable transition which results in the
birth of a limit cycle and a saddle homoclinic bifurcation where the homoclinic
orbit collides with the saddle point at the unstable equilibrium and disappears. The
final bifurcation is a Bogdanov-Takens bifurcation which occurs at the point where
both eigenvalues are equal to zero. This is a point where two fold bifurcations, a
Hopf bifurcation and a saddle-homoclinic bifurcation meet. The Hopf bifurcation
creates a limit cycle, which grows and collides with the low temperature saddle
point in the saddle-homoclinic bifurcation.
For low values of ρ both the reduced and the full system show linearly stable
4.5 Discussion and conclusions
107
behaviour at high temperature and linearly unstable behaviour at low temperatures, while the density has negligible influence on the stability properties. For
intermediate temperatures, which are only accessible at higher values of ρ for
intermediate densities, the system goes through a transitional phase, where the
eigenvalues acquire an imaginary part which gives rise to oscillatory behaviour.
The linear stability properties of a system only provide (very) limited information about its non-linear stability. For the reduced system we made streamline
plots that show the temporal evolution of the system in the ne , T -plane which can
be used to identify safe operating regimes, where a perturbation of the system will
not grow to violate the β-limit. The Greenwald limit cannot be exceeded in the
reduced system because we assume the density to be fixed.
For the four dimensional system it is not possible to make such streamline
plots. Also there is no guarantee that the orbits in phase space map out contiguous
volumes; there is a real possibility that some of the trajectories of the system pass
through a saddle point or even form interlocking loops.
While we haven’t found any trajectories starting at a stable, high temperature
equilibrium that cross the β or density limit, we cannot exclude that such orbits
exist. A perturbation of an unstable equilibrium at low temperature on the other
hand will almost certainly cross the β-limit in case the trajectory converges to
the high temperature equilibrium. In case the trajectory leads to an extinguishing
of the plasma there is the risk of crossing the density limit, since the particle
confinement increases and the refuelling rate stays the same.
In the intermediate temperature range with imaginary eigenvalues there is also
the possibility that a perturbation of a stable equilibrium leads to a trajectory that
ends up in an unstable part of phase space, which can lead to either an extinguishing of the plasma and crossing the density limit, or an increase in temperature
which can lead to a violation of the β-limit.
The addition of external heating power lifts the low temperature equilibria to
inaccessible densities, and has a stabilising effect on the high temperature equilibria
at high values of ρ. Only for high levels of Pext , ρ close to ρcrit and ne ≈ 2nG does
external heating have a destabilising effect.
When comparing the stability of the four different reactor designs in the PPCS,
models A, B and C show similar stability characteristics. Because PPCS model D
is not capable of ignition and requires external heating to maintain the required
temperature, the position of the operating points deviates significantly from those
of the other three reactors. However, also for PPCS model D the operating points
are stable.
Although the dynamics of the four-dimensional system of burn equations are
much more complex than those of the two-dimensional system, the overal stability
looks very similar. The main difference being that all operating points for the
PPCS A, B and C designs lie in the stable region of phase space where at least
one of the eigenvalues has an imaginary component. In most cases the imaginary
108
Chapter 4 Burn stability
part is small compared to the real part, so possible oscillations will experience a
strong damping, but nevertheless this might need to be taken into account in a
detailed reactor design.
The inclusion of the ion mass dependence in our derivation breaks the symmetry between deuterium and tritium that was present in the results presented in
reference [65]. Consequently, the eigenvectors that describe a pure density and a
pure temperature perturbation that they identified do not exist anymore. Nevertheless, we were able to identify the corresponding thermal and particle dominated
eigenvectors in the full system. The other eigenvectors are hybrid modes for which
we have not been able to find a simple physical interpretation.
Concluding we can say that the stability properties of a two-dimensional burn
system with constant ne and D/T-ratio are in good agreement with those of the
full four-dimensional system. Operating points in the reactor relevant density and
temperature range are mostly stable, with the exception of those of the PPCS
D design. The use of different scaling laws yields only slightly different stability
properties at the operating points, and the addition of external heating has a
stabilising effect in all reactor relevant scenarios.
One point of concern that remains is the temperature evolution during the
start-up of the reactor. After heating the plasma to the ignition point, the temperature and helium content will evolve until they settle at their stable values.
However, when starting with a pure DT plasma, the temperature can overshoot
the equilibrium temperature by more than 10 keV, which will most likely violate
the β-limit.
To prevent this from happening the plasma could be started with a non-zero
helium concentration, but this would require a higher heating power. A better
solution might be to start with a pure DT plasma, but replace (part of) the
fuelling by helium injection once ignition has been reached. That way the reactor
could be started with a minimum amount of external heating while still preventing
the dangerous temperature overshoot.
109
Chapter 5
Sensitivity of burn contours
to form of scaling laws
5.1
Introduction
Unlike present day experiments, an economically viable fusion reactor cannot rely
on external heating to keep the plasma at the required temperature. While some
level of external heating might still be required, for example for non-inductive
current drive, the main source of power has to come from the alpha particles
produced by the fusion reaction. This constitutes a radical change to the dynamics
of the plasma, because it creates a strong link between the density n, temperature
T and fusion power Pfus .
The common method for evaluating reactor designs is to make use of a scaling
law for the energy confinement time τE , because a complete description of the
transport properties of the plasma is too complicated for this purpose. For lack
of a better alternative, this is also the common approach for reactors capable of
ignition (or in any case have predominant alpha heating). The implicit assumption
is that the plasma doesn’t know what form of heating is applied. From a transport
point of view, this seems reasonable, because the main drive for turbulent transport
is the temperature gradient which should be independent of the precise heating
method.
From the Lawson criterion [89] for ignition follows that to good approximation
the triple-product nτE T must exceed a critical value, (which is why the triple
product is commonly used as a metric of progress in fusion research). In a burning
plasma Pfus ∝ n2 , and the commonly used scaling law IPB98(y,2) [26] predicts an
energy confinement time τE ∝ n0.41 P −0.69 . Combining these two proportionalities,
we find that τE ∝ n−0.97 , according to which the triple product for an ignited
plasma is (almost) independent of density.
110
Chapter 5 Sensitivity of burn contours to form of scaling laws
This suggests that a fusion power plant could ignite at arbitrarily low density,
and consequently, low fusion power. Of course there are physical mechanisms
such as the alpha slowing down time, synchrotron radiation losses or the power
threshold for the LH-transition (all linearly dependent on density), which will put a
lower limit on the density. Nevertheless, these only become a factor below electron
densities ne < 1019 m−3 . So either ignition at such low densities is possible, or there
is a problem with the application of the current scaling laws to burning plasmas.
Scaling laws for τE have been used from the very beginning of fusion research
in an attempt to compare the results of different reactors and develop a basis on
which to design new (and better) experiments. In this regard they have been
highly successful, but care has to be taken when applying them outside the range
of plasma parameters present in the dataset on which they are based. This is
especially important in the case of burning plasmas, as they will explore parameter
regimes that are currently inaccessible.
A second problem arises from the assumption underlying all current τE scaling
laws: that the electron density ne and heating power P are independent of each
other. However, in a burning plasma density and power are strongly coupled and
consequently a scaling law should not treat them as such.
In this chapter we investigate the application of current scaling laws to burning
plasmas, and show that there is a singularity in the system. We will explore the
consequences of this singularity for the predicted operating contours and conclude
with some suggestions that might help resolve the issue.
5.2
Theory
In a burning plasma, the only parameters that are under direct control of the
operator are the electron density and fuel ratio. Given these two, the plasma
might find an equilibrium. The existence, stability and position in phase space
has been the topic of several studies. Early work on burning plasmas was done
by Kolesnichenko et al. [90], and Houlberg et al. [77] were the first to introduce
the plasma operation contour (POPCON) plot, that displays the operating contours in the ne , T -plane. Other studies looked at the required power for ignition,
for instance Mitarai et al. [91], and Reiter et al. [41] looked at the influence of
helium on the burn equilibria when it is taken into account self-consistently. Rebhan et al. studied the stability of the operating points for ITER ID ([78]) with
self-consistent helium treatment and the ITER89P L-mode scaling ([29] for the
energy confinement time τE (including an H-mode factor fH = 2 to mimic H-mode
behaviour).
All these studies apply a scaling law for τE in their calculation of the operating
points (dx/dt = 0, with x = [nj , T ] and j running over the different ion species),
either implicitly by solving the equations numerically, or explicitly by eliminating
τE from the equations by means of the scaling law.
5.2 Theory
111
We will follow the approach presented by Reiter et al. in [41], which was
expanded by Rebhan et al. in [53, 65], and largely adopt the notation introduced
therein.
The power balance in a burning plasma is
Pα = nD nT Eα hσvi =
W
+ n2e Rrad = Pcond + Prad ,
τE
(5.1)
where nD and nT are the deuterium and tritium density, Eα is the alpha particle
energy, hσvi the reactivity, ne the electron density, W = 3/2ne ftot T the internal
energy of the plasma and Rrad can be interpreted as the Bremsstrahlung reactivity
multiplied with the energy per interaction, defined by
2 1
4
Z
1
Rrad = CB T 2 fi gff
+ 4fα gff
+ ZfZ gff
.
(5.2)
T
T
T
Here fi = ni /ne = (nD + nT )/ne , fα = nα /ne , fZ = nZ /ne , T is the
√ temperature
and gff the Gaunt factor, which can be approximated by gff ≈ 2 3/π for fusion
plasmas [69]. The quantitative error introduced by this approximation is of the
order of 10%, but does not affect the stability or the dynamics of the system.
From equation (5.1) the burn criterion can be derived:
ne τE =
3
2 ftot T
1 2
4 fi Eα hσvi
− Rrad
,
(5.3)
with
ftot =
1
(ne + ni + nα + nZ ) = 1 + fi + fα + fZ .
ne
(5.4)
Note that ftot = 2 in case of a pure hydrogen plasma and limZ→∞ ftot = 1. A
similar condition can be derived for the alpha particle fraction fα . Since fi =
1 − ZfZ − 2fα , the alpha particle balance is
n e fα
1 2 2
ne fi hσvi =
,
4
ρτE
(5.5)
which translates to
ne τE =
4fα
,
ρfi2 hσvi
(5.6)
where we used the definition τp = ρτE . Equating (5.3) and (5.6) results in a cubic
equation for fα , which can be solved to obtain fα as a function of T . The result
is plotted in the left half of figure 5.1 for different values of ρ.
Substituting the result in equation (5.6) yields a self-consistent expression of
the burn criterion and the value of ne τE can now be plotted as a function of T
Chapter 5 Sensitivity of burn contours to form of scaling laws
ρ=
13
ρ
ρ=
= 9
13
5
=
9
0.2
ρ=
ρ=
fα
0.3
ρ
0
1
1016
1015
5
ρ
0.1
1017
3
ρ=0
0.4
ne τE (s/cm )
112
=
1
ρ=
10
T (keV)
100
10
T (keV)
1014
100
Figure 5.1: Helium fraction (left) and ne τE plotted as a function of T for different
values of ρ.
to obtain so-called plasma operating point contour (POPCON) plots, as shown in
the right half of figure 5.1.
For reactor design purposes, it is desirable to plot the operating points in the
ne , T -plane instead of the ne τE , T -plane. This requires elimination of τE from the
burn criterion and for want of a good description of the transport in a tokamak this
can only be done by means of a scaling law. Most scaling laws for the confinement
time in a tokamak take the general form
τE = Knle P −m ,
(5.7)
with P the external heating power delivered to the plasma and K a factor which
is obtained from several machine parameters. For the well-known ITER89P and
IPB98(y,2) scaling laws K is given by
0.85 1.2 0.3 0.5 0.2 0.5
K89 = 0.048IM
R a κ B A
K98 =
0.93 1.39 0.58 0.78 0.15 0.19
0.145IM
R a κ B
A .
(5.8)
(5.9)
Here R is the major radius, a the minor radius, IM the plasma current in MA, κ
the plasma elongation, B the applied toroidal field on axis and A the plasma ion
mass in amu. The exponents of density and power are l = 0.1 and m = 0.5 for the
ITER89 scaling and l = 0.41 and m = 0.69 for the IPB98(y,2) scaling. Note that
the average ion mass A depends on the plasma composition, and is therefore not
constant.
5.3 Results
113
There are two significant differences between the ITER89 and IPB98(y,2) scaling laws. Firstly, the former is based on L-mode plasmas, whereas latter is developed for H-mode. However, the most important difference for the problem at
hand is that the IPB98(y,2) scaling law does not include radiation losses in the
definition of the confinement time, where the ITER89 scaling does include these.
This has implications for the form the power balance takes, and consequently for
the calculation of the alpha particle content and the determination of the operating points. Henceforth we will use τ̃E to denote the confinement time including
radiation losses.
In [88] we presented an analytical expression for ne as a function of T using
the IPB98(y,2) scaling:
ne =
fα
ρK
1
1−2m+l
m−1
1−2m+l
m
1
2
(1 − 2fα − ZfZ ) hσvi
(Eα V ) 1−2m+l ,
4
(5.10)
but we did not include the derivation, which can be found in appendix B.
5.3
Results
5.3.1
Operating contours
Using equation 5.10 we can plot operating contours in the ne , T -plane, as is done in
figure 5.2 for the PPCS A design at ρ = 5 and 10. Simply following the math has
resulted in nicely closed operating contours, but they extend to either very high
or very low densities. The high density points can be discarded on the grounds of
being above the Greenwald density, the β-limit or both, but this is not the case
for the low density points.
Looking at equation (B.9), it is apparent that the confinement time scales
with nel−2m which is n−0.97
for the IPB98(y,2) scaling law. Of course there is
e
some effect from the variation in helium concentration over a burn contour, but
the main trend is determined by the density. Lower densities therefore result in
longer confinement times which, in combination with the alpha concentration at
intermediate temperatures, results in burn equilibria that extend to extremely low
densities.
The obvious thing to try to put a lower limit on the accessible density is to look
at the neo-classical confinement time, because this is the absolute upper limit on
confinement in a tokamak. So if with decreasing density the value of τE predicted
by the scaling laws at some point exceeds the value of τENC , this puts a lower limit
on the density of the operating contours. However, as shown in section C, τENC
also scales linearly with density and exceeds τE for all densities.
A second thing to look at is the LH-transition power threshold: when the
heating power becomes too low, the plasma will lose the H-mode confinement.
114
Chapter 5 Sensitivity of burn contours to form of scaling laws
1045
ρ
1040
=
5
ne (m−3 )
1035
10
ρ=
30
10
1025
nG
1020
β-limit
1015
4
5 6 7 8 910
20
T (keV)
30 40 50
70
Figure 5.2: The operating contours for the PPCS A design [70, 71], for ρ = 5
and 10. The contours extend to extremely high and low densities. The solution
might be trusted in the the density range on which the scaling law was based, i.e.
around the Greenwald limit and about a decade below. But also in this range the
near-degeneration of the solution leads to virtually vertical contours, and there
obvious way to tell where the solutions are no longer valid. In other words, from
the point of view of the scaling law, there is no good reason why burn could not
be achieved at densities of 1019 m−3 or even lower.
5.3 Results
115
This threshold has a roughly linear dependence on density [75], as opposed to the
alpha power which scales with density squared. Hence this puts a lower limit on
the accessible density range.
Another factor to take into account is the alpha slowing down time, which
also depends on density. At first glance, the relevant time scale appears to be
τp , but because τp is mainly determined by the edge recycling and alpha particles
cannot be recycled without losing their energy, it is actually τE that matters. In
other words: the alpha particles need to transfer their energy to the plasma before
hitting the wall.
5.3.2
Density and power coupling
The more fundamental problem stems from the coupling between power en density in a burning plasma. In present day fusion reactors, the heating power and
density can be chosen independently. Consequently, the confinement database is
populated with shots for which there is no coupling between density and power.
In a burning plasma, this is not the case. In the absence of external heating power
the temperature cannot be influenced directly, only the density is under control of
the operator.
When using the expression for the alpha power to eliminate the energy confinement time from the power balance (with the purpose of expressing ne as a function
of T ), the fraction 1/1 − m2 + l shows up on the temperature side of the equation.
Here m and l are respectively the exponents of the power and electron density in
the scaling law for τE .
Looking at equation (B.9), it is immediately obvious that there is a singularity
at z = 1 − 2m + l = 0. At this point equation (5.10) is no longer valid, and has to
be replaced by an expression from which the density dependence has disappeared:
K
1 2
f hσviEα V
4 i
−m
=
4fα
.
ρfi2 hσvi
(5.11)
In this expression, fi , fα and hσvi are all functions of T , so this is one equation
for one variable, T . It can be rearranged to
K
Ak
4
V Eα
−m
=
m−1
4fα 2
f hσvi
,
ρAk i
(5.12)
where the left hand side is constant (the factor A−k gets rid of the ion mass
dependence which was included in K) and determined by the reactor parameters.
The right hand side is reactor independent and only a function of T . Depending
on the reactor properties, this equation has two, one or no solutions, because
changing the reactor parameters changes the value of V and K, which determines
the intersections with the closed contours described by the expression on the right
116
Chapter 5 Sensitivity of burn contours to form of scaling laws
hand side. If the solutions exist, these are independent of density, meaning that
the burn contours have degenerated into vertical lines in the ne , T -plane.
A similar exercise can be performed for a scaling law including radiation losses.
First an expression for ne similar to equation (5.10) can be derived using the
ITER89P scaling law (see Appendix B), yielding:
ne =
Em
K
1
1−2m+l
4fα
ρfi2 hσvi
1−m
1−2m+l
(5.13)
,
with E = 32 ftot T 2π 2 κa2 R = 32 ftot T V .
In this case the expression at the singularity reads
K
Em
1
1−m
=
4fα
,
ρfi2 hσvi
(5.14)
which can also be rewritten to have only reactor dependent, constant term on the
left and reactor independent terms on the right hand side:
K
Vm
1
1−m
=
3
ftot T
2
m 4fα
ρfi2 hσvi
1−m
.
(5.15)
It turns out that the different scaling laws in the ITER physics basis are all
close to the singularity. As a matter of fact, the IPB98(y,4) scaling is located on
the other side of the singularity than the other four scaling laws, which results
in a burn contour that is ’mirrored’ along the density axis compared to the other
(flipped up-down around approximately the Greenwald density).
Table 5.1: Predicted values of τE for ITER [26]
Scaling
IPB98(y)
IPB98(y,1)
IPB98(y,2)
IPB98(y,3)
IPB98(y,4)
τE
6.0
5.9
4.9
5.0
5.1
The predicted confinement times from the different τE scalings in the ITER
physics basis are very similar (see table 5.1). Yet the fact that the scaling laws are
so close to this singularity means that small variations in the exponents l or m have
a major impact on the operating contours. Approaching the singularity results in
a stretching of the burn contours along the density axis, until at the singularity
the operating points no longer form contours but instead degenerate into two
5.3 Results
117
vertical lines, or isothermals. The positions of these lines on the temperature axis
is determined by the solutions to equation (5.12) or (5.15).
Because the right hand side of these equations is reactor independent, the
positions of the solutions is determined by the left hand side, i.e. the reactor
parameters. For large enough reactors there will be two solutions and for reactors
that are not capable of ignition there are none. In between there are reactors that
have a ’critical size’, where both solutions coincide and there is only one operating
temperature.
Ergo, for a large enough reactor, a hypothetical scaling law for τE with the
values of l and m such that z = 0 would provide reasonable predictions for τE .
Yet the burn contours would consist of two vertical lines in the ne , T -plane, and
ignition could be achieved at any power and density between the LH-transition
threshold, and the Troyon and Greenwald limits.
While the scaling laws yield similar values for the operating temperature around
the Greenwald density, which is to be expected because the data they are based on
contain mostly points in this region. However, the value of dne /dT varies greatly
between the different scalings and changes sign when crossing the singularity. This
is an issue because the required response of a control system depends on the value
of dne /dT : in the case of IPB98(y,4) an increase in density will result in a decrease
in temperature on the stable burn branch, which is the opposite of what is currently
expected.
A further illustration of the problem can be seen in figure 5.3, which displays
the burn contours of the ITER ID design for ρ = 3 using the IPB98(y,2) scaling law,
and
but for two different exponents l of the electron density: the original value n0.41
e
.
We
deliberately
chose
the
ITER
ID
design
because
a slight different value n0.35
e
the IPB98(y,2) scaling law predicts that it will not ignite. However, choosing a
different reactor design does not change the analysis below in any meaningful way.
The figure also shows the Greenwald density limit of nG = Ip /πa2 and the
Troyon pressure limit βmax ≡ 0.072ε(1 + κ2 )/2 for a pure hydrogen plasma.1
For the original value of l, the operating contour looks like we expect it to look.
The ITER ID design is slightly too small to achieve ignition, so the minimum
density on the contour lies above the Greenwald and Troyon limits. However, for
l = 0.35 the situation looks completely different: all of a sudden the ITER ID
design does ignite, and the maximum density at which it ignites lies well below the
Greenwald and Troyon limits. And the change in predicted confinement (for the
same density and power of course) is just a few percent, depending on the exact
value of the density.
To demonstrate the effect of variations in l or m, figure 5.4 shows the minimal
density at which the ITER ID design ignites for the IPB98(y,2) scaling, as a
1 The exact plasma β at a given electron density depends on the corresponding ion density,
which can only be determined on a burn contour. For all other points in the ne , T -plane this
depends on the trajectory in phase space taken by the plasma.
118
Chapter 5 Sensitivity of burn contours to form of scaling laws
1065
1050
1035
)
τE
τE
∝ n0
1020
.69
1P
0.4
e
∝n
−3
ne (m
−0
nG
β-limit
.
e 35
105
P −0.
69
10−10
10−25
5
10
20
T (keV)
50
100
Figure 5.3: Burn contours for the ITER ID design for ρ=3, using the IPB98(y,2)
scaling but with varying exponents l for the density (l = 0.41 and l = 0.35). This
small change in density dependence τE has a dramatic effect on the predicted burn
equilibria.
5.4 Discussion and conclusions
119
function of the density exponent l (left plot) and power exponent m (right plot).
m = 0.705
l = 0.38
ne
1025
1020
1015
0.3
0.35
0.4
l
0.45
0.5 0.6
0.65
0.7
m
0.75
0.8
Figure 5.4: The maximum (red curves) and minimum (blue curves) density at
ignition as function of l (left plot) and m (right plot). Small changes in the values of
l and m lead to large changes in the predicted density at ignition. The red and blue
curves represent the maximum and minimum of the dashed and solid contours in
figure 5.3 respectively. The left and right plot are not quite mirror images because
of the appearance of m in the numerator of the exponent in expression 5.10.
Since there is no obvious physical reason why 2m − l > 1 is not allowed, the
solutions on the ’other side’ of the asymptote cannot be disqualified at this point.
Yet it seems unlikely that such small changes to the scaling laws, which are well
within the error margins, can have such enormous effects on the operating points
in a burning plasma.
5.4
Discussion and conclusions
When applying the current τE scaling laws to burning plasmas, this leads to predictions for the operating points at extremely high and low densities. The high
density operating points can be discarded because they lie above the Greenwald
and Troyon limits, but this is not the case for the low density points. Of course,
the LH-transition power threshold, synchrotron radiation losses and alpha slowing down time will put a lower limit on the density, but this only happens below
ne ≈ 1019 m−3 . We suspected that including neoclassical transport explicitly might
solve this problem, but this is not the case as can be seen in section C.
While it can be argued that this a mathematical artefact that is of little consequence for real world applications, the predictions for the operating points are also
extremely sensitive to small variations in the exponents l and m of ne and P in
120
Chapter 5 Sensitivity of burn contours to form of scaling laws
the scaling law. We have used the expression for ne as a function of T in a burning
plasma to show that small changes in l and m lead to big changes in the predicted
minimum (or maximum) density on a burn contour. Moreover, for l + 1 = 2m,
the operating contours degenerate into two vertical lines in the ne , T -plane, which
means that the operating points have become independent of density. In other
words, ignition is possible at arbitrarily low densities and fusion power.
Unfortunately, we have not been able to identify a physical mechanism that
could be the reason for this problem. In a burning plasma, the density and temperature (and therefore the fusion power) are coupled, and one cannot be changed
without affecting the other. In fact, al else staying constant, the operator can
only change the temperature in a burning plasma by changing the density. This
coupling is of course absent in the confinement database.
In recent years considerable effort has been spent on the development of two
term scaling laws [92], combining a core and pedestal scaling. However, these still
treat density and power as independent parameters and will therefore fundamentally suffer from the same problems, although the precise value of the density and
power exponents might be further removed from the singularity than is the case
for the ITER scaling laws. For a more detailed treatment of these two term scaling
laws see section D.
We have therefore, unfortunately not been able to find a solution to this problem. It would be interesting to have shots in the confinement database that have
densities, temperatures and heating levels that are expected in a burning plasma.
Since these do not exist by definition, one could look for shots that would be operating points if a hypothetical fusion reaction was used that delivers more energy
to the plasma.
For instance, if the alpha particles in the DT reaction had an energy of 5 MeV,
or even 10 MeV, some present day devices would be capable of achieving ignition.
Looking at the energy confinement scaling of shots that would have burned if that
were the case might shed some light on the expected confinement scaling in burning
plasmas.
A second suggestion is to mimic a burning plasma in, for instance, JET, by
coupling the heating systems to the temperature and density in a feedback loop,
with a gain factor to compensate for the fact that JET does not ignite.
121
Chapter 6
Discussion and conclusions
At the beginning of this thesis several research questions were formulated. This
chapter will provide an answer to these questions using the results presented in the
previous chapters, and subsequently discuss these answers in the broader context of
developing electricity producing fusion reactors. The overarching research question
of the thesis was
What are the properties of burn equilibria in fusion reactors?
Because of the general nature of this question, several sub questions were raised
whose answers, when combined, provide a good overview of the properties of burn
equilibria.
What parameters determine the position of the burn equilibria in
operating space and how sensitive is the system with respect to these
parameters? The burn criterion for a pure DT plasma is a well known result in
literature, but little work has been done on burning plasmas with a self consistent
helium treatment. Taking helium accumulation into account leads to closed burn
contours in the ne τE , T -plane which are completely determined by the temperature [41], and these contours can be translated to the ne , T -plane [65].
The work presented in chapter 3 shows that burn contours in the ne , T -plane
are exactly the same for all reactors that obey the same τE scaling law, apart from
a scaling factor that is a function of the engineering parameters of the reactor.
Consequently, the burn equilibria in different reactors will coincide when plotted
on a normalised density scale. Only the position relative to the Greenwald en β
limits will differ because these depend on the engineering parameters.
Figure 6.1 plots two such universal operating contours in the normalised density
and T -plane. Note that we cannot indicate the Greenwald density and β-limit in
this plot, since these are reactor specific. Also, the contours extend over many
orders of magnitude in the density, which is an artefact of the mathematical form
of the scaling law.
122
Chapter 6 Discussion and conclusions
=
ρ
=
10
1030
1020
ρ
normalised density
5
1040
1010
1
1
10
T (keV)
100
Figure 6.1: The generic operating contours for any fusion reactor that follows the
IPB98(y,2) scaling for τE for ρ = 5 and 10.
How does the power output of a burning plasma respond to changes
in energy confinement or particle transport?
There are several parameters that affect the position of the equilibria and the
fusion power at these points. The most interesting parameters from a reactor
design perspective are the H-factor and ρ = τp /τE . The density also plays a role
of course, but since the fusion power scales quadratically with density over most
of the operating range, it is desirable to choose a point close to the density limit.
For most of the past sixty years the fusion community has focussed on increasing the energy confinement time, and considerable gains have been made. In fact,
for a given reactor design, the confinement time predicted by the IPB98(y,2) scaling is only 10 to 30 percent below the value at which maximum fusion power is
achieved. Increasing it beyond that value will result in lower power output for a
given reactor design, although further improvements in energy confinement will
allow the construction of smaller reactors.
The power output of a reactor also depends strongly on the helium accumulation in the plasma, which depends on both the temperature and ρ. For a given
density, the output power on the high temperature side of a burn contour scales
roughly inversely with ρ, creating an incentive to keep ρ as low as possible. Since
the value of ρ is mainly determined by the helium recycling at the plasma edge,
increasing the pumping capacity at the divertor might offer a possibility for burn
control through the particle confinement time τp .
Minimising τp might seem desirable from a power balance point of view, but it
comes at a price. The tritium burn up fraction is a critical parameter in reactor
123
design and this is closely linked to the tritium confinement time. When this
becomes too low, the tritium has to be recycled too often, resulting in unacceptably
high tritium losses which cannot be replaced because there is an upper limit on
the tritium breeding ratio that can be achieved.
In our model we have assumed that the confinement time for all particle species
is the same, which is not necessarily true. As already mentioned, a major factor
is the recycling at the plasma edge and this might well be different for different
particle species. Identifying or creating a mechanism that allows a controlled increase in tritium recycling (or decrease in helium recycling) would offer interesting
possibilities to increase the power production of a fusion reactor without impeding
its tritium breeding capacity.
What are the stability properties of the operating points?
In chapter 4 we have derived an analytical expression for the Jacobian of the
system of burn equations for a plasma that obeys a scaling law for the energy
confinement time of the form τE = KAk nle P m , which is the form of all scalings in
the ITER physics basis. We did this for both the full system of burn equations,
and a reduced system that contains only expressions for the helium density and
the temperature. This requires the assumption of a constant electron density and
nD = nT . The properties of the reduced system are governed by ne and ρ, whereas
for the full system it is sD , sT and ρ.
While the system exhibits a complex stability diagram and features interesting transitions and bifurcations, the reactor relevant operating points are stable,
except for very high values of ρ. The addition of a significant amount of external
heating, as foreseen for the reactors in the PPCS, stabilises also these operating
points. Only for PPCS model D does the stability of the operating point remain
a concern. Using a different scaling law has a negligible effect on the stability
properties of the operating points.
It seems reasonable to assume that some form of burn control will be necessary,
if only to maintain the fusion power at the desired level. In that case, most likely
a stable and robust operating point can be found for all values of ρ, where even a
sizeable disturbance of the equilibrium does not lead to a violation of an operational
limit on a timescale that cannot be dealt with by the control systems.
A point of consideration is the start-up of the plasma. To minimise the required
amount of external heating, a pure DT start is desirable, but this will result in an
overshoot of the temperature that could cause the plasma to exceed the Troyon
limit. This might be circumvented by injecting helium or changing the DT fuelling
ratio once the ignition temperature has been reached.
How sensitive are the burn equilibria to errors in the energy confinement time scaling laws?
As is shown in chapter 5, great care has to be exercised in using the current
τE scaling laws for burning plasmas. The expression for the electron density as a
function of T that can be derived from the scaling laws contains the possibility for
124
Chapter 6 Discussion and conclusions
a singularity, which depends on the precise value of the exponents of ne and P . All
five scaling laws in the ITER physics basis happen to have their exponents very
close to the critical values, and the singularity lies well within the error margin of
the fit.
The singularity arises from the coupling of density and power in a burning
plasma, which is not present in the τE database on which the scaling laws are
based. The physical interpretation of the singularity is a decoupling of density
and temperature, meaning that the burn contours degenerate into two vertical
lines in the ne , T -plane. In other words: a reactor capable of ignition would either
burn at constant temperature, regardless of the plasma density.
Given the hypersensitive response of the operating points to changes in the
scaling law, it seems prudent to investigate this issue more thoroughly and put the
energy confinement scaling laws for burning plasmas on a firmer basis.
However, the database has to contain shots where the heating power, temperature and density correspond to an equilibrium in a burning plasma, even though
the absolute value of the confinement time is too low for real burn to occur should
the experiment have been carried out in a DT plasma. Because the energy confinement scaling fits the database rather well, it should also be able to describe
these shots.
With a new DT-campaign in JET in the works, it might be possible to perform
some simulated burn experiments, by coupling the heating power to the temperature and density, possibly with a gain to compensate for the fact that JET is too
small to achieve ignition. By performing such experiments at a range of densities, a
few data points could be acquired to investigate the effect of the coupling between
density and heating power in burning plasmas on the τE scaling in more detail.
125
Chapter 7
Outlook and
recommendations
From this work follow a few points of attention for reactor design. Looking at the
reactor and the burn conditions from an integral systems perspective, it became
clear in the course of writing this thesis that a successful fusion reactor needs
to meet a number of conflicting requirements, which on their own might seem
reasonable enough.
For instance, the need for a high tritium burn up fraction, which arises from the
need to keep the required tritium breeding ratio and the total tritium inventory
as low as possible, calls for a long particle confinement time. Maximum power
output, on the other hand, benefits from a low particle confinement time because
it reduces the helium accumulation in the plasma. A high particle confinement
time will also result in lower stability margins for the operating point, increasing
the need for control.
Similarly, we want a high power density, because the capital costs of a fusion
reactor scale with the plasma volume. This is in direct conflict with the need to
minimise the heat load on the divertor.
In a commercial power plant the recirculated power must be kept as low as
possible. Yet the power plant concepts foresee significant amounts of external
heating, typically 5-10% of the fusion power. This is primarily needed to drive
the plasma current, but does help to stabilize the burn and provide the operator
with some level of control. Still, unless radical improvements in the current drive
efficiency and/or the wall-plug efficiency of the current drive systems are made,
such a level of external power necessarily leads to a recirculated power of tens of
percents of the gross electric output power, to which all other power consumption
– such as the power needed to pump the coolant and run the cryo-plant – still
must be added.
126
Chapter 7 Outlook and recommendations
Simple models, such as the zero-dimensional one used in this thesis, are very
powerful in identifying such discrepancies, because they can easily be combined
and run for a large range of input parameters. The results can then be used to
target specific issues that are critical to the future success of fusion.
Furthermore, the detailed physics models that physicists develop to describe
more detailed phenomena that occur in fusion plasmas, are often of little use for
the engineers that are tasked with developing control systems to regulate the power
output or to stabilise the plasma. They are often looking for so-called ’OK’ models,
which are comparatively easy to understand, run fast and that capture just enough
of the physics to implement reliable model-based control systems.
From that perspective, an effort to model the effects of self-heating and helium
accumulation on reactor performance and energy confinement scaling, would be
well spent, in preparation for the ITER DT campaign. A start could be made
by developing a scaling law based on the points in the database that feature selfconsistent values of heating, density and plasma composition and see whether it
suffers from the same issues as the current ITER scalings. The next step could be
burning plasma simulation experiments in a JET DT-campaign, or the development of a scaling law from first principles, or maybe based on empirical relations
from gyro-kinetic simulations. Beyond ITER we need to address the conflicting requirements that arise from the different challenges that a successful fusion reactor
needs to overcome, using an integral systems perspective to find the compromise
that is optimal.
127
Appendix A
Partial derivatives for the
Jacobian
A.1
Reduced system
To write down an expression for the Jacobian, we need to know the partial derivatives of the confinement time with respect to nα and T . For the energy confinement
time we assume a power law of the form
τE = K ∗ Ak nle P −m ,
(A.1)
with K ∗ a constant depending on machine parameters. For the heating power P
we substitute the alpha power
Pα =
1
(ne − 2nα )2 hσviEα V,
4
(A.2)
where V is the plasma volume.
First of all we take the partial derivatives of the different terms in (A.1) and
(A.2) with respect to nα , which we subsequently use for the derivative of the whole
system. Then the process is repeated for the ∂/∂T terms.
In the reduced system the electron density ne is kept constant, yielding the
trivial result
∂ne /∂nα = 0.
(A.3)
The average ion mass A changes with the composition of the plasma (equation (4.15)) and its derivative with respect to nα is
∂A
1.5ne
=
.
∂nα
(ne − nα )2
(A.4)
128
Appendix A Partial derivatives for the Jacobian
For the fuel density it reads
∂nD nT
1 ∂
=
(ne − 2nα )2 = −(ne − 2nα ).
∂nα
4 ∂nα
(A.5)
The reactivity only depends on the temperature and does not contribute to the
∂/∂nα terms.
Using the above expressions, the derivative of the confinement time with respect
to nα can now be written down explicitly
1.5ne
4m
K ∗ Ak nle
∂τE
k
m
+
.
(A.6)
= 1
2
∂nα
A (ne − nα )2
(ne − 2nα )
4 (ne − 2nα ) hσviEα V
The partial derivative of the confinement time with respect to T is less complicated
∂τE
=
∂T
1
4 (ne
K ∗ Ak nle
−m dhσvi
m
,
hσvi dT
− 2nα )2 hσviEα V
(A.7)
however, this includes the derivative of hσvi with respect to T . The often taken
approximation hσvi ∝ T 2 is not valid in our case because we want to cover the complete temperature axis. Instead for hσvi we use the Bosch and Hale parametrization [4], which is valid from 0.5 keV up to 550 keV.
r
ξ
hσvi = C1 θ
e−3ξ ,
(A.8)
mr c 2 T 3
2 1/3
BG
ξ=
,
(A.9)
4θ
T
.
(A.10)
θ=
T (C2 +T (C4 +T C6 ))
1 − 1+T (C3 +T (C5 +T C7 ))
Taking the derivative involves applying the chain rule a few times:
r
dhσvi
3
ξ
d ln θ 1 − 6ξ dξ
−3ξ
= C1 θ
+
−
,
e
dT
mr c 2 T 3
dT
2ξ dT
2T
1/3
dξ
dξ dθ
1 BG
dθ
ξ dθ
=
=−
=−
dT
dθ dT
3θ 4θ2
dT
3θ dT
(A.11)
(A.12)
and
1 + 2C3 T + C23 − C2 C3 + C4 + 2C5 T 2
+2(C3 C5 − C2 C5 + C6 + C7 )T 3
+ C25 − C4 C5 + C3 C6 − 3C2 C7 + 2C3 C7 T 4
dθ
+ 2(C5 − C4 )C7 T 5 + C7 (C7 − C6 )T 6
=
.
dT
(−1 + T (C2 − C3 + T (C4 − C5 + C6 T − C7 T )))2
(A.13)
A.2 Full system
129
For the radiation losses the partial derivatives with respect to T and nα are
given by
∂Srad
Srad
=
,
∂T
2T
∂Srad
5n2 Rrad (T, 1)
.
= e 3 2
∂nα
2 ntot
A.2
(A.14)
(A.15)
Full system
The derivatives in the Jacobian with respect to T are of course (almost) the same
as for the reduced system, but the derivatives with respect to nj are different. The
derivatives of the energy confinement time now read
∂τE
k ∂A
K ∗ Ak nle
l ∂ne
m ∂nD nT
=
+
−
(A.16)
m
∂nj
A ∂nj
ne ∂nj
nD nT ∂nj
(nD nT hσviEα V )
and
(j − 1)nD + (j − 2)nT + (j − 3)nα + (j + 1 − mZ )nZ
∂A
=
,
∂nj
(nD + nT + nα + nZ )2
2
∂ne
j − 3j + 4
,
=
∂nj
2


j = 1;
nT
∂nD nT
= nD
j = 2;

∂nj

0
j = 3.
(A.17)
(A.18)
(A.19)
The expression for the alpha heating power has changed to
Pα = nD nT hσviEα V.
Also the expression for the radiation losses needs to be modified:
X
Srad = ne
nj Rradj ,
(A.20)
(A.21)
j
√
Rradj = gff CB Zj2 T ,
(A.22)
and consequently its derivatives have taken a slightly different form:
∂Srad
Srad ∂ne
=
+ ne Rradj .
∂nj
ne ∂nj
(A.23)
130
131
Appendix B
Derivation of ne as function
of T
The idea of eliminating the confinement time from the power balance by means of
a scaling law was presented and carried out for the ITER89 scaling by Rebhan et.
al. [65], which yielded the following result:
m
τ̃E = Knl+1
e T .
(B.1)
It is possible to take the approach presented in [65] one step further and derive
an analytical expression for ne as a function of T .
Starting with the definition of τ̃E and equating that to the scaling law:
τ̃E =
W
=
P
3
2 ftot ne T
P
2π 2 κa2 R = Knle P −m ,
(B.2)
= Knle .
(B.3)
which can be written as
W
P 1−m
Raising both sides to the power 1/1 − m and using the definition of τ̃E again
m
1
l
τ̃E W 1−m = K 1−m ne1−m ,
(B.4)
and solving for τ̃E , the following expression is found
τ̃E =
K
Em
1
1−m
l−m
ne1−m ,
(B.5)
where we have defined E = 23 ftot T 2π 2 κa2 R = 32 ftot T V , with V the plasma volume.
Appendix B Derivation of ne as function of T
132
Using expression (B.5), we can eliminate τ̃E from the burn criterion in the case
the confinement time includes radiation losses, and be left with an expression for
ne (T ). Using the burn criterion derived from the helium balance, equation (2.42),
and substituting the expression for τ̃E yields
#
"
1
l−m
4fα
K 1−m 1−m
.
(B.6)
ne
=
ne
Em
ρfi2 hσvi
From this it follows that
ne =
Em
K
1
1−2m+l
4fα
ρfi2 hσvi
1−m
1−2m+l
(B.7)
.
The above approach only works in case the definition of the confinement time
includes the radiation losses. If not, these need to be taken into account explicitly,
which makes it impossible to eliminate the heating power using the definition of
the confinement time analytically. Instead, for the IPB98(y,2) scaling, it can be
done by inserting the expression for the alpha particle heating (plus any external
heating if appropriate) into the scaling law. The expression for the alpha power
reads
Pα =
1 2 2
1
2
n f hσviEα = n2e (1 − 2fα − ZfZ ) hσviEα V,
4 e i
4
(B.8)
with fi = (nD + nT )/ne = 1 − 2fα − ZfZ the fraction of fuel ions in the plasma
and V the plasma volume. Inserting this in the equation for the confinement time
yields
τE =
Knl−2m
e
1
2
(1 − 2fα − ZfZ ) hσviEα V
4
−m
.
(B.9)
4fα
.
ρfi2 hσvi
(B.10)
Again inserting this expression into equation (2.42), we find
ne τE = Kn1+l
e
1 2
2
n (1 − 2fα − ZfZ ) hσviEα V
4 e
−m
=
Some rearranging and eliminating fi from the last term, the gives the desired
expression for ne (T )
ne =
fα
ρK
1
1−2m+l
1
2
(1 − 2fα − ZfZ ) hσvi
4
m−1
1−2m+l
m
(Eα V ) 1−2m+l .
(B.11)
133
Appendix C
Neoclassical confinement
time
Since the scaling law for the energy confinement time does not include an explicit
description of neoclassical transport, we expected that setting a lower limit to
the confinement time equal to the value predicted by neoclassical transport would
increase the minimum density on a given burn contour.
A simple estimate for the neoclassical energy confinement time is [7]
τENC ≈
a2
,
χNC
i
(C.1)
with a the plasma minor radius and χNC
the neoclassical temperature diffusion
i
coefficient for ions, which is given by [93]
χNC
= 0.68q 2
i
R
r
3/2
χCL
i ,
(C.2)
with R and r the major and minor radius, χCL
= 0.10n20 /B 2 T 1/2 [7] and q the
i
safety factor. Since we use a 0D-model, we evaluate χNC
at the plasma edge, so
i
we adopt the definition of q95 from the ITER physics basis [94]:
q95 =
5a2 BT
f,
RIM
with f a form factor to account for the shaping of the plasma
1 + κ2 1 + 2δ 2 + 1.2δ 3 1.17 − 0.65ε−1
f=
.
2
2
(1 − ε−2 )
(C.3)
(C.4)
134
Appendix C Neoclassical confinement time
Here κ and δ are the plasma elongation and triangularity at q95 and ε = R/a is
the aspect ratio.
From equation (C.2) and the definition of χi CL we find that neoclassical transport scales linearly with ne , and therefore τENC ∝ n−1
e . Coincidentally, this is very
close to the density dependence of τE ∝ n−0.97
in a burning plasma according to
e
the IPB98(y,2) scaling law (from equation(B.9) with l = 0.41 and m = 0.69). Because of this, τENC exceeds τE for every point on a burn contour and implementing
neoclassical transport to be the lower limit for transport doesn’t resolve the issue.
135
Appendix D
Alternative scaling for the
confinement time
The coupling between density and power in a burning plasma makes it desirable
to have a scaling law for the confinement time that includes only one of these two,
preferably the density. Such a scaling law would have to be relatively simple, but
would ideally be based on a deeper understanding of the transport. Given the fact
that future reactors are foreseen to operate in H-mode, we will focus our attention
on these types of plasmas.
In H-mode plasmas, there is a distinct difference between the edge transport
barrier that is responsible for the pedestal and the transport in the core (be it with
or without ITB). It has therefore been attempted to separate the pedestal from
the core, by developing a two term scaling model that includes expressions for the
energy content of the pedestal and of the core plasma. Cordey et. al. [92] have
presented two different models for the thermal energy content of both the pedestal
and the core (making 2 x 2 different combinations between them). The thermal
conduction model, where it is assumed that the dominant loss term in the pedestal
is heat conduction down the gradient, provides the following parametrisation for
the pedestal energy:
Wped = 0.000643I 1.58 R1.08 P 0.42 n−0.08 B 0.06 κ1.81 ε−2.13 A0.2 Fq2.09 ,
(D.1)
where Fq ≡ q95 /qcyl and qcyl = 5κa2 B/RI. Even though in this model the density
and power still appear together, the dependence on the density is rather weak.
The other pedestal model presented in [92] considers the MHD stability limits
to be the limiting factor for the pressure gradient in the edge, which has the
same coefficient for the density, but exchanges power dependence for temperature
dependence:
∗−0.08 0.2 2.29 −2.56 2.48
βped = 0.000833ρ∗0.27
A Fq ε
κ ,
ped ν
(D.2)
136
Appendix D Alternative scaling for the confinement time
1/2
2
with βped = Wped /RI 2 , ρ∗ped = Tpav /I, ν ∗ = nped R/Tpav
, Cv = 0.92 the fraction
of the total volume occupied by the pedestal (taking into account the pedestal
width), Tpav = 2 × 102 Wped /Cv Vnped , and V the device volume.
However, dropping ν ∗ from the scaling doesn’t significantly affect the results
for reactor relevant machines and yields an root mean square error value of 25%,
which is comparable to the 24% of the conduction model [92], and for our case has
the benefit of removing the density dependence from the scaling altogether. We
can therefore use the following scaling law for the pedestal pressure
0.2 2.18 −2.67 2.27
βped = 0.000643ρ∗0.3
Fq ε
κ .
ped A
(D.3)
Defining γ = 6.43 × 10−4 A0.2 Fq2.18 Rε−2.67 κ2.27 and using the definitions for ρ∗ped ,
Tpav and βped this can be written as
2
0.15 1.7
Wped = γρ∗0.3
=γ
ped I = γTpav I
2 × 102 Wped
Cv Vnped
0.15
I 1.7 .
(D.4)
Solving for Wped yields
1
Wped = γ 0.85
2 × 102
Cv Vnped
0.15
0.85
I2
(D.5)
and using Wped = 3nped Tped Cv V as defined by Thomsen et. al. [95], this gives
the following scaling for the pedestal temperature
Tped =
0.15
1 1
− 0.6
−1.85
.
γ 0.85 2 × 102 0.85 (Cv V) 0.85 I 2 nped
3
(D.6)
Using this result, we can write down an expression for the energy confinement time
in the pedestal that depends only on the density, and not on the power.
While this is possible for the pedestal scaling, the core scalings presented in
[92] do not allow this and therefore suffer from the same problem as the IPB98(y,2)
scaling law, even though the exact value of the coefficients might be a bit further
removed from the singularity. In order to completely remove this issue from the
scaling law, a different approach to confinement in the core needs to be taken.
137
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147
Acknowledgements
Writing a PhD thesis is a long and arduous road, which I most likely wouldn’t
have completed without the support of colleagues, friends and family. Although it
is impossible to list everyone who has made a contribution in one form or another,
there are some people who I think deserve a special mention here.
First of all I would like to express my gratitude and great appreciation to
my promotor, Niek Lopes Cardozo. Niek, throughout my professional, and also
personal, struggles of the last few years, you always remained positive and supportive. I especially enjoyed the ’thinking out loud’ sessions we had in front of a
white board, and the many discussions on issues that were not necessarily related
to fusion.
I also want to thank my co-promotor Roger Jaspers, who has supervised me
since I did my internship in China nine years ago and taught me many of the
things I know about fusion. Roger, you have been a constant factor throughout
this long journey, and were always prepared to make time to help me, even though
I had long since abandoned my original topic.
Leon Kamp, my other co-promotor, always found time for my questions and
greatly helped me during my brief foray into liquid metal flows. Leon, I greatly
appreciate your unwavering focus on the physics questions and your rigorous approach for every problem.
A great thank you to all my (former) colleagues who made my time at Eindhoven University of Technology a pleasant one: Clazien and Hélène for their helpful
support in all practical matters, Herman for his assistance with the liquid metal
experiment and his limitless supply of interesting facts, Hans for his encouraging
enthusiasm, Maarten for sharing his extensive knowledge on plasma physics in
general and plasma rotation in particular, and Mark for his probing questions and
unfailing ICT support.
I want to express my gratitude towards the PhD students, both at DIFFER
and at the TU/e, that I had the privilege of calling my peers: Ephrem, Geert
Willem, Menno, Willem, Wolf, Bram, Rianne, Pieter Willem, Matthijs, Vitor and
others, thank you for your help and the constructive discussions we had. The
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Acknowledgements
same goes for the staff at DIFFER, or FOM Rijnhuizen as it was called when
I started: Tony, Marco, Hugo, Peter, Egbert and Dick, who helped sharpen my
ideas and understanding of fusion. At the TU/e my appreciation also extends to
Maarten Steinbuch for coaching me when needed. Furthermore, I want to thank
my students Stefan, Laszlo, Kevin, Benjamin, Selwyn, Wouter and Peter for all
their hard work, and all other students who shared my office for their company
and entertainment.
Then there are the people I am lucky to call my friends. Athina, Jonathan and
Thijs, thank you for all the professional and personal talks we shared. Gillis, thank
you for the many lunches, bike rides and holidays, and for taking the initiative to
meet when I didn’t. Gar, you made some of the lonely working hours less lonely.
Hjalmar, I enjoyed the many talks on topics that no one else seems to care about.
Margit, I took great pleasure in discussing the big question of life with you. Eveline,
your encouragement was of great help in the final push to complete this thesis.
Saskia, Leon and Jisse, you have been great friends despite the long periods of,
sometimes one-sided, radio silence. Jeroen, Erik, Eric, Toine: the shared holidays,
bike rides and discussions are unforgettable. Maaike, you have taught me many
things with your energy and lust for life, and I cherish fond memories of our
shared experiences. Mirja, of all my friends you probably understand me best.
Your continuous support, both close and from further away, means a lot to me
and I admire your caring and selfless attitude.
Finally, I want to thank my parents Anthonie and Marleen, and my sisters
Arwen en Niniane for their unwavering support. Arthur, thank you for being not
only my brother, but also my best friend. The many hours we spent on our bikes,
the many holidays together, but especially the many deep conversations we had
have helped me reach this point.
149
Curriculum Vitae
I was born on the 28th of April 1982 in the city of Eindhoven, the Netherlands,
and moved to Bakel shortly after my fourth birthday. I attended primary school
at the Vrije School Peelland in Helmond, and completed my Waldorf education
at the Vrije School Brabant in Eindhoven and, after moving to Terhorst, at the
Bernard Lievegoed School in Maastricht. After obtaining my VWO diploma at the
Montessori College in Maastricht, I started studying Applied Physics and Applied
Mathematics at Eindhoven University of Technology, but quit the latter after
completing the propedeuse (first year). In 2008 I obtained my BSc in Applied
Physics, for which I made sequential images on the breakdown of electric discharges
in different gases at different conditions in a non-planar geometry.
For my Master in Applied Physics I chose the plasma and radiation technology
track and did an internship at the South Western Institute of Physics in Chengdu,
China, where I investigated the hydrogen/deuterium ratio in the HL-2A tokamak.
Other work included the design of optics for the charge exchange recombination
spectroscopy system and assisting in installing the neutral beam injection system. My graduation project, on the interaction between magnetosonic Whistler
waves and runaway electrons during disruptions in tokamaks, I did at Chalmers
University in Gothenburg, Sweden.
Upon obtaining my master’s degree I started my PhD project at Eindhoven
University. After initial forays into the interaction between plasma shear flow and
turbulence, and free surface liquid metal MHD flows, this culminated in the thesis
you are currently reading, which focusses on equilibria in burning fusion plasmas.
150