fusion energy - burning questions2016/11/14 · nuclear fusion is the process in which two atomic...

Fusion energy : burning questions

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Fusion Energy - BurningQuestions

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. F.P.T. Baaijens, voor eencommissie aangewezen door het College voorPromoties 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 depromotiecommissie is als volgt:

voorzitter: prof.dr. K.A.H. van Leeuwen1e promotor: prof.dr. N.J. Lopes Cardozocopromotoren: dr. R.J.E. Jaspers

dr.ir. L.P.J. Kampleden: 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 inovereenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

The witches’ ride to the devil’s castle,where we meet only ourselves, ourselves, ourselves. . .

Dag HammarskjöldWaymarks

A catalogue record is available from the Eindhoven University of TechnologyLibrary.

Jakobs, MerlijnFusion Energy - Burning QuestionsEindhoven: Technische Universiteit Eindhoven, 2016.ISBN: 978-90-386-41-63-8NUR 926

Cover:Original image ‘The Wizard’ CC BY 2.0 by Sean McGrathSolar image by ESA/NASA/SOHOPhoto montage by SuperNova Studios

Typeset by the author using LATEX2ε.

c© 2016 Merlijn Jakobs

i

Summary

Nuclear fusion is the process in which two atomic nuclei are joined together toform a heavier one, thereby releasing a large amount of energy. It is the energysource of all stars in our universe. Its application as an energy source on earthwould have several appealing properties like a virtually inexhaustible fuel, inherentsafety and the absence of long-lived radioactive waste. It is therefore an attractivecandidate to contribute to the world energy supply.

Currently the first power producing fusion reactor ITER is under constructionin southern France, and, if successful, a first generation of electricity producingdemonstration reactors is foreseen to follow in the 2040-2050 time frame. Presentday fusion reactors require external heating power to achieve the high temperaturesneeded 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-tritiumfusion reaction.

A fusion reactor will therefore ’burn’, much like an ordinary wood-burningstove. You fill it with fuel, kindle it (i.e. inject heat until the ignition temperatureis 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 removethe ash (i.e. the helium that is produced in the fusion reaction). This thesis dealswith the properties of these burn equilibria, what determines their fusion powerand position in the operational space of the reactor, and how the system reacts toa perturbation of its equilibrium state.

There are several parameters that govern the burn equilibria in a burningplasma. One of the most important is the energy confinement time τE, a measurefor how fast energy is lost from the plasma. Because it is difficult to calculatethe energy transport in a fusion plasma from first principles, often scaling lawsare used which express the energy confinement time in engineering or physicsparameters. We have found an expression relating the electron density ne at theoperating points to the temperature, by eliminating τE from the equations usingsuch a scaling law.

We showed that the so-called burn contours, i.e. the contours in the operationalspace 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 whichcontains the design values of the reactor, such as its dimensions and magnetic fieldstrength. This finding implies that the results of the analyses of the burn equilibriaare generic, i.e. are of application to any reactor design that follows the same τEscaling.

One of the salient results of the analysis is that, for a given reactor, the poweroutput will generally not increase if the energy confinement is improved. Goodconfinement - one of the central goals of fusion research - is still a highly desir-able property as it allows smaller reactors to ignite and burn, but in the existingconceptual power plant designs an improvement of confinement does not bringany benefit. This also means that the fusion output power of such a reactor willrespond only weakly to (small) changes in τE, disqualifying it as a useful controlparameter. However, reducing τE too much, say by 30% or so, will quench thereactor.

This result is directly connected to a second parameter that has a big influ-ence on the operating points of a reactor, the ratio between energy and particleconfinement 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 theplasma, but they lose their energy in the process. This is called (edge) recyclingand 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 thefusion power output reacts strongly to variations in ρ. This makes it a candidate tocontrol the fusion power of a burning reactor, if a means can be found to effectivelychange the value of ρ, for instance by changing the rate at which particles arepumped from the reactor exhaust. It should be kept in mind, however, that theefficiency of the reactor is highest at low values of ρ (say < 5), while the burncan become unstable when ρ nears 10 (as we shall see) and no burn is possible forρ > 15.

This would suggest aiming for a high value of ρ, but it is not that simple unfor-tunately. A fusion reactor needs to breed the tritium it consumes from lithium, astritium does not occur naturally on earth. The tritium breeding ratio, the amountof tritium bred divided by the consumed amount, just exceeds one, requiring tri-tium losses to be minimised. One of the ways of doing this is reducing the numberof cycles tritium needs to make through the reactor before it fuses. The tritiumburn-up fraction, the amount of tritium that fuses before being exhausted fromthe plasma, therefore needs to be as high as possible, which requires a long particleconfinement time, or high value of ρ.

The first demonstration reactors will most likely still require some amount ofexternal power (to drive the plasma current, with heating only a side effect), andthis changes the shape and position of the burn contours in the reactor operatingspace. Most importantly, it increases the fusion power output, but in most casesnot 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 numberZ have a big impact on the fusion power, because they are very effective at dilutingthe 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 adouble effect: both the helium and the impurity concentration will increase, whichhas an even stronger impact on the fusion power. The upside to this effect is thatthe fusion power becomes more sensitive to the external heating power for highervalues of ρ and impurity content.

We have analysed the stability of the operating points and, although the systempossesses many interesting properties (including saddle points, several differentbifurcation points, limit cycles, and damped or growing oscillations), the upshotis that (virtually) all reactor relevant operating points are stable except for ρ >10. However, the addition of external heating also stabilises these equilibria, sostability considerations will most likely only have implications for the case of areactor design with little or no external heating.

Finally, we show that the current form of the τE scaling law can result inbizarre predictions when applied to burning plasmas. First of all, ignition shouldbe possible at arbitrarily low densities, arbitrarily low power and arbitrarily smallreactor size. Secondly, a small change in the density or power dependence of thescaling law, which has a negligible effect on the predicted value of τE, results inwildly different operating points and fusion power.

These unphysical results are the consequence of the coupling between the den-sity and the heating power in a burning plasma, which leads to a singularity inthe burn condition for a particular combination of the n- and P -dependence in theτE scaling. This might be a point of academic interest only, were it not for thestrange coincidence that the family of 5 scaling laws that are used in the ITERphysics 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 thatif for P the fusion power Pfus ∝ n2 is substituted, the well-known triple productnτET 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 havethis peculiar behaviour, the data base on which they are based does not containburning plasmas at all.

Summarising, this thesis shows that the particle confinement is an attractivecandidate for burn control, whereas the energy confinement is not. The operatingpoints for future reactors are stable and their stability is increased by the additionof 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. Theapplicability 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.

v

Contents

Summary i

1 Introduction 11.1 Ignition and burn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Burn stability and sensitivity . . . . . . . . . . . . . . . . . . . . . 41.3 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 72.1 The fusion reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Operational limits . . . . . . . . . . . . . . . . . . . . . . . 112.3 Transport and confinement . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Classical transport . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Neo-classical transport . . . . . . . . . . . . . . . . . . . . . 132.3.3 Anomalous or turbulent transport . . . . . . . . . . . . . . 142.3.4 L and H mode . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.5 Sawtooth crashes . . . . . . . . . . . . . . . . . . . . . . . . 172.3.6 Energy confinement time . . . . . . . . . . . . . . . . . . . 172.3.7 Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.8 Particle transport and confinement . . . . . . . . . . . . . . 19

2.4 Helium transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.1 Helium profile . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Tritium breeding and burn-up fraction . . . . . . . . . . . . . . . . 252.6 Power balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 Burn equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8 Reactor studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Stellarators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vi

3 Burn equilibria 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Burning plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Burn equilibria with impurities and Pext . . . . . . . . . . . . . . . 493.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.2 Temperature domain of a burning plasma . . . . . . . . . . 493.3.3 Helium fraction with external heating . . . . . . . . . . . . 513.3.4 Burn equilibria with external heating . . . . . . . . . . . . . 523.3.5 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.6 Power output with external heating and impurities . . . . . 603.3.7 The effect of Pext on net electric output . . . . . . . . . . . 613.3.8 Uncertainties in scaling laws . . . . . . . . . . . . . . . . . . 62

3.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 64

4 Burn stability 674.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 Burn equations . . . . . . . . . . . . . . . . . . . . . . . . . 694.2.2 Stability of a two-dimensional system . . . . . . . . . . . . 714.2.3 Bifurcation theory . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.2 Jacobian matrix of the reduced system . . . . . . . . . . . . 744.3.3 Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.4 Reduced system stability . . . . . . . . . . . . . . . . . . . 764.3.5 Physical interpretation . . . . . . . . . . . . . . . . . . . . . 784.3.6 Low temperature stability . . . . . . . . . . . . . . . . . . . 814.3.7 High temperature stability . . . . . . . . . . . . . . . . . . 824.3.8 Phase portrait . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.9 Stability for different scaling laws . . . . . . . . . . . . . . . 854.3.10 Stability with external heating . . . . . . . . . . . . . . . . 874.3.11 Reactor comparison . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.4.1 Jacobian matrix of the full system . . . . . . . . . . . . . . 924.4.2 Full system stability . . . . . . . . . . . . . . . . . . . . . . 954.4.3 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . 974.4.4 Low temperature stability . . . . . . . . . . . . . . . . . . . 984.4.5 High temperature stability . . . . . . . . . . . . . . . . . . 99

vii

4.4.6 Stability for different scaling laws . . . . . . . . . . . . . . . 1024.4.7 Reactor stability comparison with external heating . . . . . 102


5 Sensitivity of burn contours to form of scaling laws 1095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3.1 Operating contours . . . . . . . . . . . . . . . . . . . . . . . 1135.3.2 Density and power coupling . . . . . . . . . . . . . . . . . . 115


6 Discussion and conclusions 121

7 Outlook and recommendations 125

Appendix A Partial derivatives for the Jacobian 127A.1 Reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.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 joiningtwo elements into a heavier one has lit up our universe since the birth of the firststars and, in the case of our sun, enabled life to evolve on earth. Ever since weunderstood this seemingly unlimited source of energy, the harnessing of its powerhas stood as one of the great challenges of physics. And we are fortunate to live in atime when our collective efforts are about to culminate in the first demonstrationof controlled fusion as an energy source. Successful operation of the large testreactor ITER will hopefully lead to the construction of one or more demonstrationreactors, 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 reactorsdepends on the ability of the fusion process to maintain this temperature with littleor no external power, so the basic question is: can we create a fusion reactor thatworks like a stove? You put in fuel, heat it until it reaches the ignition temperatureand after that it will burn indefinitely, as long as you refuel on time and removethe ash.

When thinking about the design of such a reactor, several questions arise. Howhigh 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? Isthe system stable? What happens in case of a disturbance? Do we need to controlit? And if this is the case, can we?

1The electronvolt (eV) is a unit that is often used in plasma physics and corresponds toapproximately 11000 degrees Kelvin.

2 Chapter 1 Introduction

1.1 Ignition and burn

At the temperatures required for fusion, the fuel has become a plasma, the fourthstate of matter. In a plasma, the nuclei are stripped of their accompanying elec-trons and form a soup of charged particles. This has the advantage that it can becontained in a magnetic field that reduces the heat loss from the plasma by severalorders or magnitude. Also, it prevents the plasma from touching the walls of thereactor.

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 fieldlines were to touch the reactor wall, there would be excessive heat and particlelosses, so to avoid these the field lines in a fusion reactor are bent such that theyclose on themselves, resulting in a toroidally shaped magnetic field.

The sun creates energy by fusing hydrogen atoms into helium [1], but the fusionreaction 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 giventemperature. 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 ofconservation of momentum.

The neutron escapes the magnetic field unhindered and is absorbed in thewall, where its energy is converted into heat. This heat is then extracted and usedto power a generator. The alpha particle on the other hand, is confined by themagnetic field and will heat the plasma by transferring its kinetic energy throughcollisions with plasma particles. It is this process that will have to provide mostof the heating power in a fusion reactor.

The plasma loses energy through radiation and conduction, and at low tem-peratures 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 un-desirable from an economic point of view because it lowers the efficiency of theplant.

Fortunately, it turns out that for a large enough reactor, the fusion powerincreases faster with temperature than the radiation and conduction losses. Soat a certain temperature the external heating can be switched off and the plasmaheats itself.

This is a precarious balance, because the fusion power has a stronger responseto variations in temperature than the radiation and conduction losses. This meansthat a small temperature perturbation will grow, making this an unstable equi-librium. 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, notonly the fuel.

A positive temperature perturbation will be kept in check by the conductionlosses, that will ultimately outweigh the fusion power. There is therefore a sec-ond, stable, equilibrium at higher temperature, which corresponds to the naturalunderstanding of a ‘burn point’.

This dynamic behaviour is represented in figure 1.1, which shows the timederivative of the temperature (T ) as a function of temperature (T ) for a hypothet-ical reactor. For low temperatures T is negative, indicating the need for externalheating, until at the ignition temperature it crosses zero. Beyond that T is pos-itive, which means that the temperature in the reactor will increase on its ownaccord until the second zero crossing at the burn temperature.

0 5 10 15

−0.5

0

0.5ignition

burn

T (keV)

T(k

eV/s

)

Figure 1.1: The time derivative of the temperature (T ) plotted against the tem-perature for a hypothetical fusion reactor. For low temperatures, T is negativeand external heating is required. The curve then crosses the horizontal axis andT becomes positive, so the temperature of the plasma will increase by itself, untilthe 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 fusionpower 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 poweroutput, and therefore needs to be removed from the plasma after it has had timeto transfer its energy.

Because the helium and fuel are mixed, selective removal of one particle speciesis complicated. Consequently, rapid removal of helium results in a low burn-upfraction of the fuel, because it is exhausted from the plasma before it has had timeto fuse. The fuel can of course be separated from the helium and be recirculated,


but this process is inevitably accompanied by some losses. As tritium does notoccur naturally (it has to be made, or ’bred’, in the reactor) and the maximumtritium breeding ratio (defined as the average number of tritium atoms bred perfusion 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 inthe plasma longer, will result in a higher burn-up fraction. But this comes at thecost 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 asthe Troyon or β-limit [2]. The exact value of this limit depends on the shape ofthe plasma, but exceeding it inevitably leads to the development of a magneto-hydrodynamic (MHD) instability that changes the geometry of the magnetic fieldand 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 par-ticles, energy and current by the sawtooth instability, possible changes in transportdue 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 heliumash in the plasma core.

Such phenomena will nudge the plasma out of its burn point and the questionis: 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 operationallimits 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 acandle or will it make uncontrollable excursions in temperature and power? Thelatter is of course highly undesirable behaviour, since not only would the utilitycompanies not like it, it would also put harder requirements on the plasma facingcomponents and structural materials in the reactor.

Furthermore, the energy transfer from alpha particles to the plasma is notinstantaneous but happens gradually. The time scale of this transfer depends onplasma parameters such as density, temperature and composition and introduces atime delay between variations in density and temperature, and the heating powerdelivered to the plasma. This could introduce oscillatory behaviour or change thestability properties of the burn equilibria.

To model the performance of a fusion reactor, descriptions of the energy andparticle losses are needed. The common approach is to use scaling laws thatpredict the energy confinement time τE and particle confinement time τp (measuresof how fast the plasma loses its thermal energy and its particles, respectively),taking machine and plasma parameters as input. This allows the study of burnequilibria 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 reactorsin which alpha heating of the plasma was (almost) completely absent, and cautionneeds 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 tem-perature and density where stable burn is possible;

• the dependencies of these contours on parameters that are under operatorcontrol, such as the density, and those that are much less so, such as theparticle and energy confinement time and plasma purity;

• the stability of the burn under perturbation of these parameters, and thelevel 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 withrespect to these parameters?

To keep the cost of electricity down, we want to maximise the power outputof the reactor which requires operation close to the density limit, limiting itseffectiveness as an actuator for control of the power. In a burning plasma, theonly other parameters at the disposal of the operator are the energy and heliumremoval, leading to the question

How does the power output of a burning plasma respond to changesin 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 equi-librium might be stable, the evolution of the system in phase space in responseto a perturbation might still lead to a violation of an operational limit, be it afundamental physics limit for the plasma, or a material limit for the reactor. Itis therefore of importance to know how the system responds to perturbations ofthe equilibrium and whether this leads to a reduction in the accessible operatingspace:

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 offuture experiments, they are based on databases without burning plasma entries.


Applying them to burning plasmas might uncover sensitivities that are not presentin externally heated plasmas.

How sensitive are the burn equilibria to errors in the scaling lawsfor the energy confinement time?

This thesis is organised as follows. Chapter 2 provides the theoretical frame-work of burning plasmas based on existing literature, followed by an analysis ofburn equilibria - and the influence of density, particle and energy confinement onthese equilibria - and the effect of external heating and impurities in chapter 3.Subsequently, chapter 4 provides a linear stability analysis of burn equilibria, forboth a two dimensional and a four dimensional system. The sensitivity of burnequilibria with respect to scaling laws is investigated in chapter 5. The final chap-ters, 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 productsup to iron, the mass of the resulting nucleus is slightly smaller than the sum ofthe 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 inprinciple a lot of reactions could be used as an energy source, but there are somefactors that limit the choice to only one realistic candidate.

Firstly, there is a tradeoff between overcoming the Coulomb barrier and thetime the particles are close enough to interact. Atomic nuclei carry a positivecharge and repel each other. To overcome this repulsion, the particles need to haveenough kinetic energy1. Although a higher initial velocity will bring the particlescloser together, thereby increasing the chance that they will fuse, it also reducesthe 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 andthe particle energy at which this optimum occurs is reaction specific.

The repulsive force between two particles with charge Z scales with Z2, whilethe kinetic energy scales only with the mass of the nucleus, which ∝ Z. Particleswith higher charge need a higher velocity to overcome the Coulomb barrier, thusmaking it harder to fuse them. And indeed, low Z particles generally have highercross-sections. For a given element however, reactions with heavier isotopes arefavoured because for equal energies they have a lower velocity.

It is no surprise therefore that fusion reactions involving light elements likehydrogen and helium have the highest cross-sections, or reactivity. The reac-tivity is the integral of the product of velocity and cross-section of the reaction

1Another way of overcoming the Coulomb barrier is to create a very high pressure, which isthe case in the core of stars and for inertial confinement fusion. Because this thesis deals withmagnetic confinement fusion, we will not discuss this further.

8 Chapter 2 Theory

over a Maxwellian temperature distribution. This is relevant in case the reac-tions take place in a plasma where the energy of the individual particles followsa (Maxwellian) distribution function. Figure 2.1 displays the reactivity, denoted〈σv〉, for the deuterium-tritium (DT), the DD and the 3HeD reactions based onthe fitting formulas provided by Bosch and Hale [4].

100 101 102 10310−32

10−29

10−26

10−23

10−20

DTDD

3 HeD

T (keV)

〈σv〉(

m3/s

)

Figure 2.1: The reactivity of three fusion reactions involving hydrogen isotopes.The DT reaction has the highest reactivity for temperatures up to several hundredkeV. Please note that, although plotted here up to 1 keV, the parametrisation ofthe reactivities from [4] is only valid below 100 keV for the DT and DD reactions,and below 190 keV for the 3HeD reaction.

A second consideration when picking a fuel is availability. The 3HeD reactionhas the advantage that it is (mainly) aneutronic, which reduces the radioactiveactivation of the machines, increases the lifetime of components, enables moreneutron susceptible technologies and diminishes the need for neutron shielding.Unfortunately, 3He is exceedingly rare on earth and thus seems unlikely to beused for fusion on a commercial scale2. Moreover, the 3HeD reaction requirestemperatures that are an order of magnitude higher than the DT reaction, whichis problematic because of the β-limit (see section 2.2.1).

2There are significant resources of 3He on the moon though, which might become accessiblein the future [5].

2.2 The tokamak 9

The DD reaction has neither the advantage of being aneutronic, nor of havingthe highest reactivity. On top of that, the energy released per reaction is, at 3.70MeV, much lower than for DT (17.6 MeV) or 3HeD (18.3 MeV). This leaves theDT reaction as the only realistic candidate at this moment.

While deuterium is a naturally occurring isotope of water and there is plentyavailable on earth, this is not the case for tritium. Tritium has a half life of 12.3years 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 DTreactions, 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 fromthe walls of the reactor and while there are several ways in which this can beachieved, the most promising approach for reactor development relies on magneticfields to confine and position the plasma. Charged particles can move freely alongmagnetic field lines, but are restricted in their perpendicular motion due to theLorentz force. To avoid end losses, the magnetic field is usually bent into a toroidalshape.

The most successful reactor concept to date is the tokamak, invented in Russiain the 1950s by Sacharov and Tamm. It derives its name from the Russion acronymfor ’toroidal chamber with magnetic coils’: тороидальная камера с магнитнымикатушками (toroidal’naya kamera s magnitnymi katushkami). The results fromexperiments on the first tokamak, T1, were presented to the world at the secondGeneva Conference on the Peaceful Uses of Atomic Energy in 1958 [6], although thedevice was at that time still unnamed. A schematic representation of a tokamakcan be found in figure 2.2.

A tokamak consists of a toroidally shaped vacuum vessel, which is surroundedby coils that generate a toroidal magnetic field (Bφ, see figure 2.2). The curvednature of the field causes the particles to drift, necessitating a helical transform ofthe field lines. This is achieved by running a current through the plasma, whichinduces a poloidal magnetic field (Bθ). The resulting field lines have a helical shapeand form a set of nested flux surfaces which are isothermals and isobars (a detailedderivation of the magnetic equilibrium in a tokamak can be found in [7], but theintuitive picture is that particles are free to travel along the field lines, smoothingout variations in pressure and temperature). The safety factor q is defined as thenumber 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 windingof a transformer, the primary of which is the central solenoid placed in the centralopening of the vacuum vessel. A final set of coils generates a vertical field thatprevents 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 vesselomitted for clarity. The toroidal field coils are shown in light blue, the poloidalfield coils in silver, the central solenoid in green and the plasma in purple. Imagecourtesy 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 so-called divertor. Usually located at the bottom of the vacuum vessel, this is aregion where the field lines intersect the wall. The transition surface betweenclosed flux surfaces and open field lines is called the separatrix and the plasmaoutside this surface is referred to as the scrape-off layer.

Current tokamaks rely on external heating to create the necessary conditionfor fusion. The ratio between the fusion power Pfus and external heating powerPext

Q =PfusPext

, (2.1)

is often used to gauge reactor performance. In case of a burning plasma in whichthe 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 theLorentz 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 fieldcoils constitute a large fraction of the cost of a fusion reactor, ideally the ratiobetween plasma and magnetic pressure

β =p

B2/2µ0(2.3)

would be one, so there is no ’wasted’ magnetic pressure. Unfortunately this valueis unattainable due to the existence of MHD instabilities, or modes as they areoften referred to.

The maximum value of β that can be achieved in a tokamak, before large scaleMHD modes become unstable, can be expressed as

βmax = gIMaB

, (2.4)

and is generally referred to as the Troyon limit. Here IM is the plasma currentin megaAmperes, a the minor radius in meters and B the magnetic field on axisin Tesla. Extensive stability calculations for a wide range of pressure and currentprofiles by Troyon et. al [2] found the value of g to be 0.28 N/A2, in fusionliterature often used without units.

12 Chapter 2 Theory

Another limit that needs to be respected when operating a tokamak concernsthe electron density ne. Empirically it was established that above the Greenwalddensity (in units of 1020m−3) [8, 9]

nG =IMπa2

, (2.5)

a disruption, an event in which control over the plasma is lost and which can dam-age the reactor, becomes very hard to avoid. Advanced regimes allow operationup to ne ≈ 1.5nG [10] which results in approximately double fusion power outputcompared to operation at ne = nG.

2.3 Transport and confinement

The study of energy transport, and to a lesser extent, particle transport in mag-netically 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 dif-fusion across a straight B-field) into account. However, it was quickly realisedthat with the introduction of curved magnetic geometries, classical transport wasgreatly enhanced because of the drift motions and the trapping of particles (dueto the variation in field strength along a field line), and this realisation led to thestudy of neoclassical transport.

When the reactors became bigger en more advanced and temperatures in-creased, the predictions again turned out to be far off the mark and this timethere was no easy explanation, hence the name ’anomalous transport’. Increas-ing diagnostic capabilities and physical understanding led to the insight that thiswas in fact turbulent transport, which to this day is not completely understood,although advanced numerical models are reaching the point where experimentalresults can be reproduced and predicted. This section will briefly introduce thethree forms of transport and the way the resulting confinement is modelled atreactor level.

2.3.1 Classical transportPreviously it was stated that plasma particles stick to the field lines like ’beadson a string’. This picture is not entirely accurate. In a homogeneous, straightmagnetic field the particles gyrate around the field lines with the Lorentz forceacting as the centripetal force, with the cyclotron frequency ωc and gyroradius ρg(also referred to as Larmor radius) given by

ωc =qB

m(2.6)

ρg =mv⊥qB

, (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 andB the magnetic field strength.

In case of a collision, particles can ’hop’ onto a different field line which istypically one gyroradius away. This results in classical diffusion with the diffusioncoefficient of the order of χCL = νeiρg, where νei is the electron-ion collision rate.Assuming a Maxwellian velocity distribution it can be shown to be [7]

νei =

√2

12π3/2

e4

ε20m1/2e T 3/2

ln Λ, (2.8)

with ln Λ ≈ ln

(12πε

3/20 T 3/2

e

n1/2e e3

)≈ 15−20 the Coulomb logarithm. Classical transport

coefficients amount to χCLe = 4.8×10−3n20/B

2T1/2k ≈ 3.4×10−5 m2/s and χCL

i =

0.10n20/B2T

1/2k ≈ 7.2× 10−4 m2/s.

2.3.2 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 fieldstrength that points towards the center of the torus. The curvature and gradientgive 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 anE × B drift. Together these drifts change the trajectories of the particles in theplasma; the radius of their gyration changes periodically over an orbit and they nolonger 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 canbecome ’trapped’. The magnetic moment µ = mv2

⊥/2B of a particle, with v⊥ itsvelocity perpendicular to the magnetic field, is conserved. Because the magneticfield is stronger on the inside of the torus than on the outside and the magneticfield lines are twisted, a particle starting at the outside and following a field linewill initially move along the gradient. To keep its magnetic moment constant,v⊥ has to increase and conservation of energy dictates that its parallel velocityv‖ decreases. If its initial parallel velocity was too small, at some point it willdecrease to zero and it will reverse direction. It now follows a field line in theopposite direction, until it again runs up against the magnetic ’hill’ and reversesdirection again. These particles are ’trapped’ and will bounce back and forth.

Looking in the poloidal plane, the centre of mass of particles traveling aroundthe torus in the direction of the magnetic field, describes a circle that is shiftedinwards and is slightly larger than the flux surface its associated with. For particlestraveling agains the magnetic field the opposite holds true: they are on a trajectorythat is slightly smaller and shifted outwards. This means that a trapped particle

14 Chapter 2 Theory

that reverses direction does not retrace its original path exactly, but follows a moreor less parallel trajectory. The resulting orbit looks like a banana in the poloidalprojection, and the step size for collisional transport of these particles is not theirgyroradius, but the width of their so-called banana orbit.

An approximate scaling for the resulting neoclassical diffusion coefficient isχNC ∝ q2ε−3/2χCL, with ε = a/R0 the inverse aspect ratio. Neoclassical trans-port is rock bottom: a tokamak cannot do better than this and although theproportionality factor to classical transport looks inconspicuous, it turns out to bea 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 neoclassicaltransport and this phenomenon is referred to as anomalous transport. It is causedby turbulence and a complete description is extremely complicated due to the non-linear nature of the turbulence. Turbulent transport is convective, which sets itapart 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 trans-port with an effective diffusion coefficient χT ∝ γmaxL2

c , where γmax is the growthrate 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 elec-trostatic drift wave instabilities are the major drivers of turbulent transport underfusion conditions. For ion thermal transport this is believed to be the ion tempera-ture gradient (ITG) instability [12, 13, 14], and the electron transport is dominatedby trapped electron modes (TEM) [12, 13, 15].

For drift wave turbulence, the value of Lc scales with the gyroradius ρg inthe limit of small ρ∗ = ρg/a, which results in so-called gyroBohm scaling witha diffusion coefficient χT ∝ ρ∗T/eB. This in contrast to the Bohm scaling thatapplies to modes with a size comparable to the plasma dimensions (or minorradius), which follows χT ∝ T/eB [16, 11].

Often there is a threshold gradient above which the turbulence growth rateincreases sharply. The result is a corresponding sharp increase in diffusion coeffi-cient, an effect which is referred to as profile stiffness, because above the thresholdthe gradient responds much less to changes in the heat flux. This phenomenonis illustrated in figure 2.3, where the heat flux is plotted as a function of thedimensionless temperature gradient length

R

LTi=R

Ti

dTidr

.

The diffusion coefficient in the stiff region of the plasma is often approximated


Threshold (κ)

χ 0

χ 1

Figure 2.3: Schematic representation of energy transport in a tokamak as a func-tion of the normalised temperature gradient. For low temperature gradients, theheat flux increases linearly with R/LT, until a critical value is reached. Beyondthis value, the gradient becomes stiff, i.e. it hardly responds to changes in heatflux anymore. Figure adapted from [18].

with a critical gradient model [13]

χT = χgB

[χs

(R

LTi− κc

)H

(R

LTi− κc

)+ χ0

], (2.9)

with χgB = q3/2Tρg/eBR the gyroBohm normalisation, χs the stiffness level, Hthe Heaviside step function, κc the threshold (with a value around 5 often foundfor reactor relevant tokamaks, although there are also parametrisations based ongyrokinetic simulations [17]) and χ0 the level of residual transport in the absenceof turbulence.

2.3.4 L and H modeIn the eighties a new regime of operation was discovered in the ASDEX toka-mak [19]. ASDEX was one of the first tokamaks equipped with a divertor, andwhen enough heating power was supplied, the plasma would ’jump’ to a state inwhich the confinement was roughly a factor two better than before. The new

16 Chapter 2 Theory

18

Figure 1: Schematic view showing regions with differenttransport characteristics in tokamak.

Figure 2: Link between central and edge ion temperaturefor a series of ASDEX-Upgrade discharges [30].

Figure 3: Ion temperature profile for a series of JET shotswith varying edge density and edge ion temperature [32].

Figure 4: Link between central and edge temperature ina series of JT60-U plasmas [33].

Sawteeth

10

1

Internal transport barrier(ITB)

Edge localized modes(ELMs)

Edge transportbarrier

in H-mode

Normalised radius r/a

JG98

.483

/35c

Pla

sma

Tem

pera

ture

L—mode

Core

Neutrals

JG03

.338

-1c

0

2

4

6

8Experiment (Scans)

Tiex

p (0.4

) (k

eV)

Tiexp (0.8) (keV)

10.0

8.0

6.0

4.0

2.0

3.0 3.1 3.2 3.3 3.4Major radius (m)

Pulse No: 47543, 47545, 47546

Ion

tem

pera

ture

(ke

V)

3.5 3.6 3.7 3.8

JG99

.238

/4bw

1.0

0.8

0.6

0.4 JG03

.338

-2c

0

5

10

15

0 1 2 3 4

Ti(0

) and

Te(

0) (

keV

)

Tiped and Teped (keV)

PNBI = 8-18MWType-I ELMs

IP = 1.8MA / Bt =3.0T

Electrons

Ions

Figure 2.4: Typical radial temperature profiles in a tokamak for different operatingregimes. When going from L to H mode, an edge transport barrier is created whichresults in very steep temperature and pressure gradients at the plasma edge, andelevates the core temperature. Figure courtesy of EFDA-JET.

regime was dubbed H-mode (high confinement) and the ’normal’ regime retroac-tively received the name L-mode (for low confinement).

The improved confinement originates from a transport barrier at the plasmaedge, where the pressure gradient creates a radial electric field that drives E ×Bshear flows that locally reduce the turbulent transport [20, 21]. This can be seenin a strong reduction in the balmer α radiation around the plasma [22], indicatinga reduction in outward particle flux. The results are steep temperature en densitygradients at the plasma edge, and because the core transport remains unaffected,it looks like their respective profiles are placed on a pedestal, which is illustratedin figure 2.4. Because the pedestal raises the temperature and density over thewhole cross section of the plasma, it has a large contribution to the total storedenergy W and therefore the confinement time.

The H-mode comes at a price though. The transport barrier at the plasmaedge is usually so strong, that the pressure gradient keeps increasing until it hits a(local) stability limit, which triggers an edge localized mode (ELM) that ejects upto 10% of the stored energy from the plasma [23]. This energy (and the particlesthat carry it) travel through the scrape-off layer to the divertor, where they hit thewall. The short timespan (≈1 ms) of these events results in transient heat loadsof up to 1 GW/m2 on the divertor surface [24], which may damage the divertor.For this reason, a reactor will require some form of ELM mitigation to protect it.


Analogous to the pedestal, the plasma can also develop transport barriers inthe core, often associated with a region of strong flow shear and the presence offlux surface with rational values of q [25]. These regions are referred to as internaltransport barriers (ITBs) and might be used in advanced reactor scenarios.

2.3.5 Sawtooth crashesAn important mechanism in (particle) and energy transport in the center of theplasma is the sawtooth mechanism. This takes its name from a sudden drop intemperature in the center of the plasma, followed by a gradual recovery, until theprocess repeats itself. When the central temperature is plotted as a function oftime, the resulting graph has a distinct sawtooth shape.

The sawtooth crash is caused by the central value of the safety factor q droppingbelow one. This triggers an MHD instability, in which the hottest, central partof the plasma is pushed outwards and replaced by cooler plasma. The result isan outward propagating heat flux and a flattening of the temperature, and to alesser extend, density profiles. Because the fusion power Pfus ∝ p2, the effect onthe fusion power can be significant.

Generally, sawteeth will not cause a disruption, but if they become too bigthey might destabilise other, more harmful, MHD modes, like neoclassical tearingmodes. They can also play a role in flushing accumulating impurities from theplasma core, but this is a double-edged sword as they can also help impuritiespenetrate into the plasma center [26].

2.3.6 Energy confinement timeFrom a reactor point of view, the overall transport properties of the plasma aremore relevant than the precise values of the transport coefficients at each radial(and poloidal) position. These global properties are reflected in the energy con-finement time τE which is a measure of how long the plasma retains its energy

τE =W

Pcond − dWdt

. (2.10)

Here, W = 3/2ntotT is the total internal energy of the plasma, with ntot = ne +ΣiniZi, and Pcond is the conducted power. In equilibrium dW/dt = 0, so thedefinition simply becomes

τE =W

Pcond. (2.11)

Because Pcond is hard to measure, often a different version (with a slightly differentnotation) of the confinement time is used

τE =W

Ploss, (2.12)

18 Chapter 2 Theory

in which case Ploss = Prad + Pcond, the total loss power. In experiments with onlyexternal heating, this makes determining τE simply a matter of measuring thetemperature and density, because in equilibrium Ploss = Pext.

2.3.7 Scaling lawsBecause of the complicated nature of the transport processes the expected con-finement time for a new experiment is often calculated using scaling laws, whichprovide τE as a function of engineering or physics parameters. The most commonapproach is to fit a function of the form τE ∝

∏i pαii , known as a power law, to

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 theBohm time

τB =a2B

T∝ ε2R2BT−1, (2.13)

the normalised ion gyroradius

ρ∗ =

(2eT

Mi

)1/2Mi

eBa∝√MiT

εRB, (2.14)

the ratio of plasma and magnetic pressure

β ∝ nTB−2, (2.15)

the normalised collision frequency (collision frequency divided by the bounce fre-quency of trapped particles)

ν∗ = νii

(Mi

eT

)1/2(R

a

)3/2

qR ∝ nRT−2qε−3/2 (2.16)

and the cylindrical safety number

qcyl =RB

ε2If(κ, δ) ∝ BRI−1ε2κ, (2.17)

with f(κ, δ) a function of the plasma triangularity δ, the elongation κ = b/a, (withb 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 theabove definitions, a linear transformation can be made between the engineeringand physics variables and their respective exponents.

The number of free parameters in the fit can be reduced by placing constraintson the exponents using the method developed by Kadomtsev [27] and Connor and


Taylor [28]. This relies on finding linear transformations of the physics variablesunder which the governing equations are invariant. Applying these transformationsto the general form of the scaling law then constrains the exponents.

For instance, the Kadomtsev, or high β, constraint demands that the exponentssatisfy 4αR − 8αn − αI − 3αP − 5αB = 5. The Bohm and gyro Bohm constrainedscalings 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 fromdifferent tokamaks was made in the eighties, which resulted in the ITER89P L-mode scaling [29]

τE = 0.048I0.85M R1.2a0.3κ0.5B0.2A0.5n0.1

20 P−0.5, (2.18)

where R is the reactor major radius, A the average ion mass in amu, n20 theelectron density in 1020 m−3 and P the heating power in MW. Until the first H-mode scalings were published in the nineties, the L-mode scaling was also used forH-mode discharges by multiplying the predicted confinement time with an H-factorfH = 2.

In reference [26] a comprehensive review of confinement data was made, result-ing in a set of closely related forms of the τE-scaling. The IPB98(y,2) is the mostcommonly used scaling law for H-mode plasmas and also the recommended scalinglaw for reactor extrapolations [31]. The value of τE that it predicts, is plotted infigure 2.5 against the measured value of τE, for experiments in the confinementscaling database [32, 33]. Its parameters are given in table 2.1, together with theother H-mode scalings presented in the ITER physics basis [26]. These scalingsdiffer in the definition of κ (κ = b/a vs κ = πa2/area and the database thatthey are based on (the differences between the databases lie mainly in the type ofexternal heating for the plasma discharges that they contain).

2.3.8 Particle transport and confinementFor a burning plasma it is of importance how fast the ash is removed, relative to theburn rate, as this determines the burn-up fraction. This ratio is governed by therelation between particle and energy transport. Particle transport is somewhatdifferent process from energy transport, but it works by the same mechanisms:diffusion and turbulence (convection).

Experimental observations put the particle diffusion coefficient at the sameorder 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 isconfirmed by gyro-kinetic simulations, that also reveal a convection term, the signof which depends on the ratio between electron and ion heat flux [39, 40].

20 Chapter 2 Theory

Plasma Phys. Control. Fusion 50 (2008) 043001 Review Article

COMPASSJT-60UPDXTFTRASDEXDIII–DMASTSTARTITERAUGJETNSTXTCVC-ModJFT-2MPBX-MTdeV

0.001

0.01

0.1

1.0

10.0

0.001 0.01 0.1 1 10τ th,IPB98(y,2) (s)

JG06

.455

-1c

Figure 34. Plot of the measured H-mode thermal energy confinement time versus that predicted bythe IPB98(y, 2) scaling expression. The symbols indicate data from different tokamaks as notedin the legend.

where ε is the inverse aspect ratio (ε ≡ a/R), B is the toroidal magnetic field in T, and n is theline-averaged density in 1019 m3. However, the anticipated operating regime for ITER is notL-mode, but ELMing H-mode. Therefore, an H-mode database was formed [132] containingboth ELM-free and ELMing H-mode data. A full description of the H-mode database and itsvariables 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 currentrecommended expressions to be used for the scaling of energy confinement time with selecteddimensional 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 scalingexpressions pertain to the thermal energy confinement time (τth) rather than the global energyconfinement time (τE), which includes the energy content in fast ions from auxiliary heating.The IPB98(y, 2) scaling expression, which is now a standard H-mode energy confinementbenchmark, has the form:

τth,IPB98(y,2) = 0.0562I 0.93B0.15P −0.69n0.41A0.19R1.97ε0.58κ0.78a . (4.3)

The prediction of the IPB98(y, 2) is plotted in figure 34 against the measured confinementtimes in the international H-mode database (DB3). Note that the present experimental dataspans three orders of magnitude in confinement time. The L-mode scaling ITER97-L hasthe form

τth,ITER97-L = 0.023I 0.96B0.03P −0.73n0.40A0.20R1.83ε−0.06κ0.64. (4.4)

In order to apply dimensional analysis to these scaling relations, a method for transformingthe dimensional variables into the chosen dimensionless parameters must be specified. The setρ∗, β, νC that was found in section 3.2 to describe the plasma behavior in identity experimentswill be used here. The database contains no information on the plasma rotation, so the Machnumber is omitted from the dimensionless parameter set; it can be considered a hidden variablein the database analysis. It is also assumed that radiation and charge exchange losses are

59

0.001 0.01 0.1 1 100.001

0.01

0.1

1

10

τ98(y,2)E

τex

pE

Figure 2.5: The predicted value of τE from the IPB98(y,2) scaling plotted againstthe measured values [30]. Figure courtesy of EFDA-JET.


Tab

le2.1:

The

expo

nentsof

thediffe

rent

parametersin

theIP

B98(y)an

dIP

B98(y,1–4)scalinglawsforτ E

inH-m

ode

plasmas

[26].The

IPB98(y)an

d(y,1)scalings

areba

sedon

theIT

ERH.D

B3da

tasetforELM

yH-m

odes,thefirst

usingκ

=b/aan

dthelatter

usingκ

=are

a/πa

2.Scalings

IPB98(y,2–4)also

useκ

=are

a/πa

2,bu

tareba

sedon

arestricted

dataset:

(y,2)on

ITERH.D

B3restricted

toNBIheated

discha

rges,(y,3)on

thesameas

(y,2)bu

twitho

uttheAlcator

C-M

OD

data

and(y,4)on

thesameas

(y,2)bu

tusingon

lyda

tafrom

thefiv

eIT

ER

simila

rdevices.

All

scalings

meettheKad

omtsev

constraint,e

xceptfor(y,3)which

isjust

afree

fitto

theda

ta.

RMSE

ITER

Scaling

C(10−

2)

IB

nP

Rκ

εA

N(%

)τ E

(s)

IPB98(y)

3.65

0.97

0.08

0.41

-0.63

1.93

0.67

0.23

0.20

1398

15.8

6.0

IPB98(y,1)

5.03

0.91

0.15

0.44

-0.65

2.05

0.72

0.57

0.13

1398

15.3

5.9

IPB98(y,2)

5.62

0.93

0.15

0.41

-0.69

1.97

0.78

0.58

0.19

1310

14.5

4.9

IPB98(y,3)

5.64

0.88

0.07

0.40

-0.69

2.15

0.78

0.64

0.20

1273

14.2

5.0

IPB98(y,4)

5.87

0.85

0.29

0.39

-0.70

2.08

0.76

0.69

0.17

714

14.1

5.1

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 reasonableto 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 particlecontent N divided by the net flux of particles Γp, leaving the plasma through theseparatrix:

τp,core =N

Γp. (2.20)

Although the definition looks innocuous, there is a caveat which complicates mat-ters somewhat. When a particle exits the plasma and hits the wall, it loses itsenergy and electric charge. The, now cold and neutral, particle either enters thepumping duct that leads to the pumps that maintain the vacuum, or it reentersthe 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 longerthan the primary particle confinement time τp,core. The value of Rcyc, definedas the ratio of the recycled particle flux and the total particle flux, depends onthe 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], butfor simplicity we will stick to the more general definition of the global particleconfinement time τp = τp,core/(1−Rcyc).

Experimental observations indicate that the ratio ρ between τE and τp is ap-proximately the same for different species, which makes it a suitable figure of meritfor helium transport [41, 24]. The energy and particle confinement times are thenrelated;

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-modeplasmas [42, 37, 43, 44]. For the remainder of this thesis, we will write τp for theparticle confinement time as defined in equation (2.21), unless explicitly statedotherwise.

Although we assume the same confinement time for all particle species, therecan be a difference due to the different transport and edge recycling coefficients,and different pumping efficiencies. All of these depend on the atomic mass andthe effective ion charge

Zeff =ΣjnjZ

2j

njZj, (2.22)

with nj the density of particle species j, and Zj its atomic number.

2.4 Helium transport 23

The figure of merit for the relative efficiency of the helium exhaust, is theenrichment ratio η, defined as the ratio of the helium concentration at the separa-trix to the helium concentration in the exhaust gases (that are removed from thesystem)

ηHe =ΓHe

2ΓDT

ncoreDTncoreHe

. (2.23)

Values of ηHe = 0.3 have been found in AUG with the original divertor [45], whileafter 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 ITERto be successful [48], but future reactors might require higher values to meet thetritium 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 theplasma 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 distributionthat matters. Helium in the plasma center will have a much larger effect on reactorperformance than helium in the plasma edge, because of the peaked fusion powerprofile that is expected.

The issue of ash removal from future fusion machines is a rather complicatedproblem that is difficult to investigate in present machines, due to the fact thatthere is no significant production of helium in the core. Furthermore, in contrast topresent day machines, the sawtooth period in ITER is expected to be significantlylonger than the energy confinement time, resulting in a different effect on thetransport of impurities from the core towards the edge of the plasma. Predictionsfor the helium transport and profile in future reactors therefore rely heavily onnumerical simulations.

2.4.1 Helium profileThe 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 theboundary condition at the edge, set by the recycling (which in itself is determinedlargely by the pumping efficiency).

The source profile can easily be determined if the fuel density and tempera-ture profiles are known. The alpha particles are created with an energy of 3.52MeV 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 profiletherefore is not exactly the same as the fusion power profile. Since this thesis onlydeals with volume averaged values for the temperature and particle densities, thisis of little interest at this point, but needs to be considered for a detailed reactordesign.

Already in the 1960s the issue of helium transport was investigated, but onlyin last decade of the century did people start to self-consistently study the effectof helium in burning plasmas [49, 50, 51, 52, 53]. The most popular approach wasto use a zero-dimensional (0D) model that only included the power balance andthe alpha particle balance, resulting in a cubic equation for the helium fraction inthe plasma, with the ratio between particle and energy confinement time ρ as afree parameter, which will be discussed in more detail in section 2.7.

Currently, a full understanding of helium (or impurity) transport in fusionplasmas 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 atransport equation of the form

ΓHe = −DHe∇nHe + VHenHe, (2.24)

with DHe and VHe the flux surface averaged diffusion coefficient and pinch velocityrespectively.

Both approaches have been used for ITER modeling. In [54], only the dif-fusive term is taken into account, using the 1.5D BALDUR code with both theMMM95 and the Mixed Bohm/gyro-Bohm transport models to obtain the diffu-sion coefficients for the core transport. Three different pedestal models providedthe boundary conditions at the top of the pedestal, while the helium density atthe edge was calculated using Zeff and a specified impurity (Beryllium) content.The resulting helium profiles are only slightly peaked, and the effect of sawtoothoscillations on the central helium density is rather limited. The sensitivity studyshows that the central helium density is strongly dependent on the helium fractionat the edge.

The approach of combining diffusion with a pinch has been taken in [55]. Us-ing PTRANSP (predictive TRANSP [56]) and calculating the helium profile fordifferent values of DHe and VHe, it was found that the dependance of the centralhelium density on the recycling increases with the inward pinch velocity, as wouldbe expected for a boundary value problem. The opposite holds for an outwardpinch velocity of course, in which case hollow helium profiles are expected. Theresulting predicted power for ITER ranges from 320 MW for the outward pinch(for both large and small diffusivity, independent of the recycling) and 170-240MW for the case of an inward pinch. This would give values of Q ranging from4.6 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 transportequation reads

∂nα∂t

+∇ (−DHe∇nHe + VHenHe) =nατ∗sd

, (2.25)

with τ∗sd the thermalization time for fast alpha particles. The helium concentrationin the plasma center was found to be of 5% for a recycling factor Rcyc = 0.97, andfor 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. fastcore particle transport compared to particle removal from the edge) is expectedfor global helium concentrations of 2% and higher. In that case the shape of thehelium 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 heliumcompared to the time it takes a helium particle to enter the pumping system fromthe plasma edge. Hence, the total helium concentration is mostly determined bythe efficiency of the pumping system [37, 45].

The precise value of ρ that still allows ignition is usually found to range from5 - 10 [53, 57], depending on the precise model used. Early simulation of ITERperformance with the BALDUR 1.5D code give similar results, but also put re-quirements on the recycling coefficient to obtain the desired values of ρ [50, 52].Although the precise knowledge of the transport processes involved has greatly in-creased over the past two decades, the fundamental requirements have not changedmuch. 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. Conse-quently, tritium does not occur naturally on earth, but it can be made by irradiat-ing lithium with the neutrons from the DT fusion reaction. The possible reactionswith both lithium isotopes, 6Li and 7Li, are given by:

6Li + n→ 4He + T + 4.8 MeV, (2.26)7Li + n + 2.5 MeV→ 4He + T + n′, (2.27)

where the neutron n′ released by the 7Li reaction has less energy than the originalneutron.

The 6Li reaction has a large cross section (940 barn) for thermal neutrons,resulting in nearly all of them being captured and breeding a tritium atom oncethey 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 afusion reactor. The 7Li 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 possibilityof breeding more than one tritium atom per neutron.

Because of the inevitable losses associated with tritium handling and the nat-ural 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 surplusis required to obtain a starting inventory for new reactors when they are underconstruction. The current world production of tritium (mostly from CANDUfission reactors) amounts to several kilograms per year [58], which is at best of thesame order as the start-up inventory off a single fusion power plant, depending onthe 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 multi-plication reactions [59].

The tritium breeding in a fusion reactor is foreseen to take place in a blanketsurrounding the plasma, between the plasma facing first wall and the vacuumvessel. Even so, not all neutrons released in the fusion reaction will end up reactingwith lithium, due to absorption by structural materials and openings in the blanketfor 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 + 9Be + 3 MeV→ 2He + n′ + n′′, (2.29)n + Pb + 10 MeV→ Pb + n′ + n′′. (2.30)

The cross sections for the lithium, beryllium and lead reactions are plotted infigure 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 reactorto test tritium breeding blanket modules), but is expected to lie between 1.08 and1.15 [60, 61, 62, 63].

Whether the achievable TBR values are sufficient depends strongly on thetritium burn-up fraction fb, which is the probability that a tritium atom injectedinto the plasma will actually fuse. This is defined as the fusion rate (since everyfusion reaction consumes one tritium atom) divided by the rate at which tritiumis lost from the plasma, which is the sum of the fusion and the loss rates:

fb =nDnT〈σv〉

nDnT〈σv〉+ nT/τ∗p, (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 throughthe system before it fuses. The time it takes for a given tritium atom to fuse isthe 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 3He produced in by the decay of tritium can be converted back intotritium by letting it absorb a neutron in the blanket, but this is still a net lossbecause that neutron can no longer breed tritium from lithium (in other words: ittakes two neutrons to make one tritium atom in that case).

Part of the tritium might adsorb onto the surfaces of tritium handling systemsin places where it would be difficult to recover. This needs to be taken into accountin a reactor design to prevent the buildup of a large tritium inventory, which wouldbe unacceptable both from a safety and a TBR point of view. Modelling of thetritium circulation puts the minimum at fb = 0.02 and above 0.05 if the reactoralso needs to breed the startup tritium inventory for a new reactor every 3 to 5years [64].

2.6 Power balance

The following section provides a brief overview of the energy balance of a fusionplasma, the derivation of the burn criterion and stable and unstable operatingpoints, 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, whichwill 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, theheat losses from the plasma need to be balanced by a heat source, a requirementwhich 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 heatingpower density, SΩ the Ohmic heating power density, Srad the power density ofradiation losses and Sκ the power density of the conduction losses.

Assuming no fast particle losses3, the alpha power density delivered to theplasma is given by

Sα = nDnT〈σv〉Eα, (2.33)

for a plasma with nD and nT the deuterium and tritium densities, 〈σv〉 the reac-tivity of the plasma and Eα = 3.52MeV the alpha particle energy. In case nD = nT

and no impurities, this reduces to

Sα =1

4n2

e〈σv〉Eα. (2.34)

The Ohmic heating power plays a role in case there runs a (large) current inthe plasma, for instance in a tokamak. The power density is given by

SΩ = ηJ2, (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 tempera-tures, we will neglect it from now on. Note however, that Ohmic heating plays animportant role during the first phase of the discharge when the plasma temperatureis still quite low.

The first term on the right hand side of the energy balance is the radiationlosses. In the core of the plasma these consist of two parts: Bremsstrahlung andsynchrotron radiation. The Bremsstrahlung radiation power density is given by

SB =∑j

(21/2

3π5/2

)(e6

ε30c

3hm3/2e

)Z2njneT

1/2e , (2.36)

3Some fast particle losses are expected due to imperfections in the magnetic field and inter-actions with MHD instabilities (mostly Alfvénic modes), but the magnitude of these losses isexpected 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 temper-ature. For future references it is convenient to define

CB =

(21/2

3π5/2

)(e6

ε30c

3hm3/2e

).

Synchrotron radiation is emitted by charged particles because of the acceler-ation associated with their gyration around the magnetic field lines. Because ofre-absorption and reflection on metallic surfaces, quantification of net radiatedsynchrotron power requires an involved calculation, although simpler fitting for-mulas have been derived [67]. The emitted synchrotron power in ITER is negligi-ble compared to the Bremsstrahlung everywhere except in the core of the plasma,where they are approximately equal [68]. However, synchrotron radiation losses in-crease rapidly with temperature and at higher plasma temperatures can increaseto about 20% of the total energy losses from the plasma (or roughly twice theBremsstrahlung losses) [67]. Nevertheless, because synchrotron losses also dependstrongly on the temperature profile, first wall reflectivity and reactor geometry, wehave 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 mecha-nism. However, line radiation from the plasma core is only important during theso-called burn through phase at the beginning of the discharge. Once the plasmareaches fusion relevant temperatures line radiation from the core can be safelyneglected for our purposes.

Combining equations (2.34), (2.36) and (2.11), and setting Pext = 0, a criterionfor ignition can be derived for a 50:50 DT plasma:

neτE =3T

14 〈σv〉Eα − CBT 1/2

, (2.37)

which shows that the product of density and confinement time required for ignitioncan be expressed as a function of T . In fusion research it is common to multiplyboth sides by T , to obtain the triple product neτET on the left. The triple productis 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 energyconfinement time scales with machine size, both of which have a big share in thecost of the reactor. Increasing this triple product for a given reactor is thereforea good measure of progress. Figure 2.7 plots the minimum value of the tripleproduct as a function of T , where it can be seen that the Bremsstrahlung has theeffect of increasing the minimal ignition temperature, but doesn’t affect the burnequilibria at higher temperature.

The curves plotted in figure 2.7 are contours of T = 0 with the dot denotingdifferentiation with respect to time (t), but the graph provides no information onthe stability of these equilibria. In this case an equilibrium is stable if the second

30 Chapter 2 Theory

100 101 1020

10

20

30

40

50

60

With

Brem

sstrahlung

Without

Brem

sstrahlung

T (keV)

neτ

ET

(atm

s)

Figure 2.7: The triple product at ignition plotted against T . The dashed line isthe ignition curve without radiation losses, which only deviates significantly attemperatures below 10 keV.

2.7 Burn equilibria 31

derivate of T is negative, so T < 0, and unstable if T > 0, which can be seen in agraph that plots T as a function of T . To do this, the unknown confinement timeneeds to be eliminated from the left hand side of the burn equation, to which enda scaling law, such as the IPB98(y,2) scaling [26], introduced in section 2.3.7, canbe used.

Figure 2.7 plots T as a function of T for a hypothetical fusion reactor (theignition experiment presented by Freidberg in table 14.3 on page 520 in ref. [7]),at a density of 1.1 × 1020m−3. While it is necessary to choose specific values forthe engineering parameters, the graph would look similar for any reactor capableof reaching ignition.

For temperatures between 0 and roughly 5 keV T < 0, meaning that externalheating is required to keep the plasma stable at a temperature in that range.Between 5 and 17 keV T > 0, so once the plasma enters this range the temperaturewill increase by itself until it reaches 17 keV, since above 17 keV T < 0 again(without external heating). This also means that the equilibrium at 5 keV isunstable and the one at 17 keV is stable (albeit at a higher fusion power than at5 keV.

Given the characteristics of the equilibria it seems logical to refer to the unsta-ble equilibrium at lower temperature as the ignition point, since from that pointonward 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 burnpoint, since this is the operating point the plasma will converge to in the absenceof external control.

2.7 Burn equilibria

The assumption of a pure DT plasma is useful to understand the power balancein a fusion plasma and how this translates to ignition and stable burn, but isnot self-consistent because it neglects the alpha particles produced by the fusionreaction. Since these form the energy source that keeps the plasma at the requiredtemperature, a proper treatment of fusion plasmas needs to include the heliumash. In this section we will introduce the model used by Reiter et al. and Rebhanet al. [41, 49, 53, 65], which includes the helium concentration self-consistently andalso allows a (fixed) concentration of impurity ions.

Using the particle confinement time as defined above, the helium balance inthe plasma reads

nDnT〈σv〉 =nατp, (2.38)

with nα the helium density. Given the fact that the densities of the differention species are coupled through the electron density, it is convenient to define a

32 Chapter 2 Theory

dilution parameter

fi =ni

ne= 1− ZfZ − 2fα, (2.39)

ni = nD + nT being the fuel density, fα = nα/ne and fZ = nZ/ne where nZ is theimpurity density. The total particle density ntot then is

ntot = ne + ni + nα + nZ = neftot, (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

neτE =4fα

ρf2i 〈σv〉

(2.42)

and equation (2.37) changes to

neτE =6ftotT

f2i 〈σv〉Eα − 4Rrad(T )

, (2.43)

with

Rrad = fiRB,1 + fαRB,2 + fZRB,Z (2.44)

= CB

√T

[figff

(1

T

)+ 4fαgff

(4

T

)+ Z2fZ

(Z2

T

)]. (2.45)

These are the Bremsstrahlung losses written in a slightly different form, withgff(Z2/T ) the Gaunt factor. With respect to equation 2.4 in [41], we have usedthe 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 impuritieswith temperatures between 5 and 100 keV).

We have now two expressions for neτE, so combining equations (2.42) and (2.43)and substituting fi = 1− ZfZ − 2fα results in a cubic expression for fα:

a0 + a1fα + a2f2α + a3f

3α = 0, (2.46)

2.8 Reactor studies 33

with

a0 =− 3

2T[f3

Z(Z2 − Z3) + f2Z(4Z2 − 2Z) + fZ(1− 5Z) + 2

], (2.47a)

a1 =− 3

2T[f2

Z(4Z − 5Z2) + fZ(14Z − 4)− 9]

+Eαρ

[f2

ZZ2 − 2fZZ + 1

]+

4

ρ〈σv〉 [(fZZ − 1)RB,1(T )− fZRB,Z(T )] , (2.47b)

a2 =− 3

2T [fZ(4− 8Z) + 12] +

4Eαρ

[fZZ − 1]

+4

ρ〈σv〉 [2RB,1(T )− RB,2(T )] (2.47c)

and

a3 = 6T +4Eαρ. (2.47d)

Equation 2.46 can be solved to obtain the helium concentration as a functionof T. This result can then be inserted into equation 2.42 to plot the burn contoursin the neτE, 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 notyet been finished, most ITER partners are already developing preliminary reactordesigns for a commercial fusion power plant. For the European countries this isthe 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 powerplants. Their main design parameters are summarised in table 2.2

The PPCS A, AB and B designs rely on materials and technology that iscurrently being developed, and expect an improvement in plasma parameters,mainly density and pressure, of about 20% over the ITER values. For PPCSmodels C and D, advanced materials need to be developed and another gain ofabout 20% in plasma performance is required. In the remainder of this thesis thereactor designs from the PPCS will be used to investigate the burn equilibria andtheir stability.

34 Chapter 2 Theory

Table 2.2: Proposed parameters for the PPCS reactors. The plant efficiency isdefined as the ratio between net power and fusion power, with net power theelectric power delivered to the grid. Adapted from [70].

Parameter Model A Model AB Model B Model C Model DMajor radius (m) 9.55 9.56 8.6 7.5 6.1Minor radius (m) 3.18 3.19 2.87 2.5 2.03Aspect ratio 3 3 3 3 3Bφ (T) 7.0 6.7 6.9 6.0 5.6Ip (MA) 30.5 30.0 28.0 20.1 14.1Avg. ne (1020 m−3) 1.1 1.05 1.2 1.2 1.4ne/nG 1.2 1.2 1.2 1.5 1.5βN (thermal/total) 2.8/3.5 2.7/3.5 2.7/3.4 3.4/4.0 3.7/4.5H98 1.2 1.2 1.2 1.3 1.2Zeff 2.5 2.6 2.7 2.2 1.6Pfus (GW) 5.00 4.29 3.60 3.41 2.53Blanket gain 1.15 1.18 1.39 1.17 1.17Pnet (GW) 1.55 1.50 1.33 1.45 1.53Padd (MW) 246 257 270 112 71Plant efficiency 0.31 0.35 0.37 0.42 0.6

2.9 Stellarators 35

2.9 Stellarators

The stellarator is an alternative type of fusion reactor, which, like the tokamak, re-lies on magnetic confinement in a toroidal geometry. Contrary to tokamaks, whichare (quasi-) axisymmetric, stellarators generally have a complicated magnetic ge-ometry. As a measure of the helicity, or pitch, of the field lines, the rotationaltransform ι is used for stellarators, which is related to the safety factor q in atokamak in the following way:

ι =q

2π. (2.48)

The magnetic field in a stellarator is completely generated by external coilsand it therefore has no plasma current. This makes it an inherently steady statedesign, with the added benefit that it is immune to disruptions. The absenceof disruptions also means that there are no hard operational limits like the β anddensity 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 astable regime.

Due to the complicated structure of the magnetic field, stellarators always suf-fered from worse confinement than tokamaks. However, recently the constructionof the first optimised, large scale, stellarator has been completed in Greifswald,Germany [72]. If successful, this might lead to a stellarator reactor design, dubbedHELIAS [73].

The transport processes in stellarators are the same as in tokamaks and con-sequently, the same approach using a scaling law for the energy confinement timeis used [74]:

τE = 0.134a2.28R0.64B0.84ι0.41n0.54e P−0.61, (2.49)

where ι takes the place of the plasma current that is present in the scaling lawsfor tokamaks.

Apart from being inherently stable, stellarators have the benefit of requiring noexternal heating for current drive, thus having a lower recirculating power fractionthan a tokamak of comparable size. Although stellarators will not be treatedexplicitly in the remainder of this thesis, the analyses performed are also valid forstellarators since the scaling law for τE has the same form.

37

Chapter 3

Burn Equilibria

3.1 Introduction

When designing a power-producing fusion reactor, one needs to know how it willoperate exactly. Necessarily, future fusion reactors are thought to operate at, orvery close to, ignition, where the alpha particles provide essentially all heatingpower, otherwise they would never succeed in generating electricity at a com-petitive cost. However obvious this may be, it doesn’t say anything about thecomposition and temperature of the plasma in operating point. Yet this is cru-cial information for maximising performance within the operational and materiallimits of the reactor. Furthermore, knowledge of the stability and sensitivity tochanges in plasma and machine parameters of the operating point is important forburn control purposes.

A good starting point for the analysis of the operating space in a reactor isa global 0D model based on the energy balance of the plasma. The model canthen be expanded by including the particle balances of deuterium, tritium andhelium to investigate the effects of fuel burn up and ash accumulation. This hasbeen done ad hoc, by assuming the helium density is a certain fraction of the fueldensity, and self-consistently where the helium density is calculated using a particleconfinement time which is proportional to the energy confinement time, a methodwhich was first introduced by Reiter et al. [41]. Furthermore the effect of profileshaping was investigated in a follow-up paper by Reiter et al. [49], in which theshaping factors were defined by the volume averaged value of the correspondingvariables, and it was shown that a change in one of the profiles is equivalent to atranslation of the system in the ne, T plane.

Later, Rebhan et al. resolved the discrepancy introduced by the different defi-nitions of the confinement time (including or excluding radiation losses) [53], andinvestigated the burn stability of the old ITER design using the ITER89P L-modescaling 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 scal-ing 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 relation-ship between the density and temperature in a burning plasma and compare theobtained burn contours to those based on the ITER89P L-mode scaling. Further-more, we will use this expression to investigate and comment on the sensitivity ofthe 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 operat-ing contours extend over many orders of magnitude in electron density. Obviously,solutions outside the density range on which the scaling law is based should betreated with extreme caution, and solutions far outside this range have no physicalmeaning whatsoever.

In a real tokamak there are several mechanisms that will limit the density ofthe operating contours to more reasonable levels. The high density regime is notaccessible 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 scaleslinearly with density, as opposed to the Bremsstrahlung and fusion power whichhave a quadratic density dependence. We have neglected synchrotron radiationlosses because at power plant relevant densities they have only a minor effect onthe energy balance.

Then there is the LH-transition, which has a power threshold which scalesroughly linearly in ne [75], meaning that for very low fusion power the plasma willnot enter H-mode.

And finally, the alpha slowing down time depends on the density, and cannotbecome too high, otherwise the alpha particles will be lost before they can transfertheir energy. The relevant time scale here is τE and not τp, because alpha particlescannot be recycled at the edge while maintaining their energy.

For consistency with earlier work and because we are interested in the shapeof 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, wehave chosen an electron density ne = 1 × 1020m−3, or selected a reactor relevantdensity range.

Because all current reactor designs feature some amount of external heating, wewill investigate the effect this has on the operating contours. We will then extendthis analyses to include the effect of impurities and look at the sensitivity of thenet electric power to the external heating power. The chapter will conclude withthe effect of uncertainties in the confinement scaling on the operating contours andfusion power.

This chapter is a synthesis of two papers, which are complementary to eachother. Section 3.2 is included here as published in Nuclear Fusion, while an adaptedversion of section 3.3 will be submitted to Nuclear Fusion.

http://stacks.iop.org/0029-5515/54/i=12/a=122005?key=crossref.c1878ac97984fa0ae5493e8bb71dc865

3.2 Burning plasmas 39

3.2 Burning plasmas1;

3.2.1 IntroductionIn a fusion reactor of the type tokamak, a plasma of the hydrogen isotopes deu-terium and tritium is kept at a temperature of hundreds of millions Kelvin, confinedin a toroidal geometry by means of magnetic fields. To start the reactor, externalheating is applied to bring the plasma to the ‘ignition’ point: a combination of suf-ficiently high temperature and density at which the heating power delivered by thefusion reactions balances the heat loss. Once the plasma is ignited its temperature– and thereby the fusion power – increases autonomously until the stable ‘burntemperature’ (Tburn) is reached. Above this temperature the heat losses increasesfaster than the fusion power. For a given Tburn, the electron density ne is thereforedetermined by the reaction rate [4] and heat loss rate, which is the sum of the ra-diation and conduction losses. The latter are conveniently expressed by the energyconfinement time τE = W

Pcond, defined as the ratio of the kinetic energy content W

of the hot plasma and the conductive power losses Pcond. As the heat loss is the re-sult of complex turbulent processes, empirical scaling laws are used which expressτE 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 controlthat 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) andthe 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 obviousburn control parameter is the ratio of particle and energy transport

ρ =τpτE

(3.1)

In tokamak reactors, particle confinement is much better than energy confine-ment, 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 es-sential to reduce power losses, good particle confinement makes the reaction chokeon its own ash. The effect of particle confinement on the burn equilibrium is ev-ident from the contours in the neτE, T -plane (assuming T = Te = Ti) for whichthe fusion power heating balances the losses, an analysis already performed byReiter et al. [41]. Note that whereas the contours are open towards high energyconfinement 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 nosustained burn is possible. To complicate matters, a further constraint comes fromthe fuel cycle, which requires ρ to be sufficiently high as was shown by Sawan etal. [64].

1Published as: Jakobs, M.A., Lopes Cardozo, N.L.C. and Jaspers, R.J.E., Fusion burn equi-libria sensitive to the ratio between energy and helium transport, Nucl. Fusion 54 (2014) 122005

http://stacks.iop.org/0029-5515/54/i=12/a=122005?key=crossref.c1878ac97984fa0ae5493e8bb71dc865


10 1001014

1015

1016

1017

ρ=

0 ρ = 1

ρ = 5

ρ = 9ρ =

13

T (keV)

neτ

E(s

/cm

3)

Figure 3.1: The plasma operating contours (POPCON [77] in the neτE, T -planefor different values of ρ. For ρ = 0 the curve is open because there is no helium tochoke the reaction. For increasing values of ρ the operating range in T and neτEbecomes more and more limited, until it vanishes for ρ = 14.71.


For these reasons, we address the question how the Tburn and Pburn changeunder variation of H98 and ρ, as well as ne, while assuming that the fuel mix is50-50. 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, Reiteret 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 contoursand derive two new results for the influence of energy and particle confinement onthe 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 radiationand conduction, respectively. External and Ohmic heating are neglected as theyhave a minor influence on the burn equilibrium. The alpha power density is givenby Sα = nDnT〈σv〉Eα, where 〈σv〉 is the DT-reactivity [4], Eα = 3.52 MeV theenergy of the alpha particle that is produced in the DT-reation, while nj denotesthe number density of species j in units of m−3. The dominant radiation lossis due to the Bremsstrahlung, given by SB =

∑j CBZ

2njnegffT1/2e,keV with CB =

5.35×10−37Wm3, Te,keV the electron temperature in keV, Z the ion charge, gff theGaunt 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 fusionreactions we write nDnT〈σv〉 = nα/τp, thereby assuming that the confinement ofalpha particles is the same as that of other species. We further introduce the fueldilution parameter fi = (nD + nT)/ne and the alpha fraction fα = nα/ne. Usingthese notations, ref [41] finds

neτE = 4fαρf2

i 〈σv〉(3.3)

neτE = 6ftotTf2i 〈σv〉Eα−4Rrad(T )

, (3.4)

and by solving these equations arrives at burn contours, i.e. the contours inthe neτE, 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 fig-ure 3.1. Note that neτE must exceed a critical value for burn to occur. For givenneτE there are two solutions: the unstable ignition temperature (left hand branch)


and the stable burn temperature (right hand branch), in agreement with the intu-itive picture of ignition and burn.

In this calculation τE is an independent parameter, whereas in fact it dependson plasma parameters. Rebhan et. al [65] proposed a self-consistent analysisby using a scaling law which expresses τE = W/(Pcond + Prad) as a function ofplasma parameters. They used the ITER89 L-mode scaling [29] to find burncontours in the ne, T -plane for this specific scaling, for a specific choice of reactorparameters. We follow this approach, using the now more relevant scaling for H-mode confinement, the IPB98(y,2) -scaling [26], which is commonly used to predictthe performance of future fusion devices. Since the radiation losses are not includedin IPB98(y,2) , the method applied in ref [65] cannot be used. Instead, we insertedthe expression for the alpha heating power to eliminate the confinement time andderived an expression for ne as a function of T which is valid for all scaling lawsof the form τE = KAknleP

−m:

ne =

(4fαA

k

ρK

) 11−2m+l

[1

4f2

i 〈σv〉] m−1

1−2m+l

(EαV )m

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 (bythe alpha particle or external sources).

Since K and the plasma volume V are the only reactor specific parameters inthis equation, ne(V −mK)1/(1−2m+l) represents a normalised density that is thesame for all fusion reactors that follow the same scaling law.

Figure 3.2 shows the POCPONs for ITER ID [78] and 3 conceptual reactordesigns 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 theITER ID reactor. The solid curves for ITER ID and PPCS models A - C areisomorphic, which can be shown by applying the normalisation described above.

It is important to note that while the formalism using the scaling laws leadsto burn equilibria at values of ne and T that are far from the normal operatingconditions of a fusion reactor, these are probably artefacts due to the mathematicalform of the scaling laws. Reliable extrapolations can only be made in the parameterrange where the database on which the scaling laws are based is well populated,i.e. with ne in the range 1019 to 1021m−3.

To elucidate the role of particle confinement, while zooming in on the reactorrelevant 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 fromignition 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. Thefusion power at ignition and burn depends quadratically on ne, and therefore theGreenwald density limit nG = Ip/πa

2 is of fundamental importance. For all butthe lowest ρ-values this limit is more restrictive than the Troyon pressure limit


4 5 6 7 8 9 10 20 30 40 501019

1020

1021

1022

ITER ID 89L

ITER

ID

PP

CS

APP

CS

B

PP

CS

C

β limit

Density limit

T (keV)

ne

(m−3)

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 theIPB98(y,2) scaling, as is the solid blue ITER ID curve, whereas the dashed ITERID contour is created using the 89L-scaling following the procedure developedin [65]. The Greenwald density limit and the Troyon β limit for the ITER IDdesign are plotted. They also give a good indication of these limits for the PPCSdesigns, although the precise position is reactor specific.


4 5 6 7 8 9 10 20 30 40 501019

1020nG

β limit

ρ=

0

ρ=

5

ρ=

10

ρ=

14.71

100 MW

200 MW

500 MW

1 GW

2 GW

5 GW

T (keV)

ne

(m-3

)

Figure 3.3: The operating contours for ρ = 0, 5, 10 and 14.71 for the PPCS Adesign made with the IPB98(y,2) scaling. The colour indicates the fusion power ineach operating point and the dashed dotted lines indicate lines of constant fusionpower. The Greenwald density limit and the Troyon β limit are also indicated.


given by βmax = gTIpaB with Ip in MA and the Troyon factor gT = 0.03.

Calculating the β limit requires knowledge about the composition of the plasmaand that is only available on the equilibrium contours, so plotting it is not straight-forward. We have taken the following approach: for figs. 3.2 and 3.3 we havecalculated the β limit for a pure DT plasma, which results in a underestimationof neof at most 20%. For figs. 3.4 and 3.5 we have taken the value of fα and Tat the equilibrium and used that to calculate the value of neand subsequently thefusion power at the β limit. This results in a β limit that has two values a onevalue of ρ and H98 (because every point on an equilibrium contour is associatedwith a point on a β limit contour).

Figure 3.4 shows that the Tburn, and therefore Pburn, will change under variationof ρ. In other words, if by some process in the plasma or in the exhaust the ratiobetween particle and energy confinement changes, this will significantly affect theoutput power of the reactor. This may be a point of concern as it may lead tounwanted excursions of Pburn , but may also have potential as actuator for burncontrol. The dependence of fusion power (at burn equilibrium) on ρ is depictedin figure 3.4 for PPCS models A, B and C, for ne = 1 × 1020m−3. These curvesshow a lower (ignition) and upper (burn) branch that meet at ρmax, the highestvalue of ρ that can be tolerated at this particular density. For PPCS model Aρmax ≈ ρcrit, the fundamental limit on ρ set by the Bremsstrahlung and fusioncross section for the DT-reaction. For PPCS models B and C ρmax < ρcrit. Alongmost 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 powerbranch is not accessible because it exceeds the β limit, but the expected impactfor future reactor designs is minimal because constraints on the achievable tritiumbreeding 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, 10and ne = 1 × 1020m−3, for PPCS model A. We see that below H98 = 0.73 and0.83 for ρ = 5 and 10 , respectively, there is no ignition because the confinementis 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 theincreased 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 needsto be taken into account when choosing the operating point for a reactor.

On the unstable ignition branch (bottom half of the contour), an increase inH98 beyond its minimum value initially results in ignition at lower temperatureand 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 fuelhas become so diluted that the fusion power can no longer balance the conductionand radiation losses.


5 10 15

103

104

β limit

PPCS B

PPCS A

β limit

PPCS C

β limit

ρ

Fusi

onpo

wer

(MW

)

Figure 3.4: The fusion power at a constant electron density ne = 1× 1020m−3 asa function of ρ for PPCS models A, B and C. It can be clearly seen that thereabove 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 burncontours are shifted towards higher densities with respect to PPCS model A. Belowthe maximum value of ρ there are two equilibria: the unstable ignition branch atlower fusion power and the stable burn branch at higher power. The latter isespecially 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 notignite in the current model below ne ≈ 1× 1020m−3.


0.5 1 1.5 2 2.5 3 3.5 4

103

104

β limit

ρ = 5

β limit

ρ = 10

H98

Fusi

onpo

wer

(MW

)

Figure 3.5: The fusion power as a function of the H-factor H98 = τE/τE,98 forPPCS model A at a density of ne = 1× 1020m−3. While it is not surprising thata too large reduction in the H-factor will lead to a loss of burn, the result whenH98 increases deserves more attention. The fusion power initially increases butreaches 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 = 6.4and H98 = 2.8 respectively. The dash-dotted lines indicate the β limit for thetwo contours, and at low values of ρ this is the limiting factor of the fusion power(dotted part of the contour).


In short, for a given reactor there is no gain to be expected from improvementof energy confinement. Either the plasma exceeds the β limit, or the power outputdecreases. Rather, the reactor should be designed in such a way that its operatingpoint is at Hmax, provided it does not conflict with the β limit. Of course, betterconfinement does allow one to reach ignition in a reactor with smaller dimensionsand lower Pburn.

3.2.4 ConclusionsWe have derived an analytical expression (3.5) relating T and ne in a fusion reactorwith self-consistent treatment of fuel burn up and helium accumulation, using theIPB98(y,2) scaling law for confinement time. This expression is valid for all fusionreactors that obey the same energy confinement scaling if one takes into account ascale factor that depends on the reactor parameters. Using these results we haveplotted the burn contours of the PPCS A design in the ne, T plane, including thecurves of constant fusion power and the Greenwald and Troyon limits.

The fusion power at these equilibria was found to be very sensitive to changesin ne, ρ and H98. The fusion power scales quadratically with the density aroundthe Greenwald density, although this will be different for reactors that have aminimum density for ignition that is close to this limit. The dependence on ρ isespecially strong for intermediate to high values of ρ, and since the value of ρ isto a large extent determined by the helium exhaust at the plasma edge, this offerspossibilities for burn control using helium pumping [79, 80]. The effect of H98 onthe fusion power could have implications for advanced tokamak scenarios wherevalues of H98 well above 1 are expected.

3.3 Burn equilibria with impurities and Pext 49

3.3 Burn equilibria with impurities and Pext

3.3.1 Introduction

The analysis in section 3.2 deals exclusively with equilibria in burning plasmaswithout external heating, which is unlikely to be achieved in a tokamak becauseof the need for non-inductive current drive. Furthermore, we have only lookedat ’pure’ plasmas so far, containing only deuterium, tritium and helium. A realfusion plasma always contains a non-zero amount of impurities, be it beryllium ortungsten from the reactor wall, or for example neon or nitrogen to increase theradiated power in the divertor.

This section therefore investigates the burn equilibria in PPCS model A withexternal heating. First we derive a way to determine the minimum and maximumtemperature on a POPCON, after which we present a procedure to determine thehelium fraction in a plasma with external heating. Then we analyse the operatingcontours with external heating, and describe the effect of impurities on the oper-ating contours and the power output at a given operating point. Finally, we studythe effect that external heating has on the net electrical power delivered to thegrid.

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 tosay something about the solution. In fact, we can already determine the allowedtemperature domain of a burning plasma by looking only at the determinant

∆ = 18a3a2a1a0 − 4a32a0 + a2

2a21 − 4a3a

31 − 27a2

3a20, (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 = 0.5corresponds to a pure helium plasma and fα = 0 is only possible in case τp =ρτE = 0, which for finite ρ would mean that neτE = 0 which has no physicalrelevance.

It turns out that there is a real root fα > 0.5 for 1 keV < T < 1000 keV that wecan discard. Because we are looking for physically meaningful solutions (fα has tobe real), this restricts us to the temperature domain where ∆ ≥ 0. In figure 3.6 thediscriminant 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 haveno physical meaning. This leaves us with ∆ > 0 in a temperature window rangingfrom 5 to several hundred keV, depending on the value of ρ, where the solutions


100 101 102 103−2

−1

0

1

2

ρ=

0.1

ρ=

1

ρ=

5

ρ=

9

ρ=

13

T (keV)

∆/∆

pea

k

Figure 3.6: The discriminant ∆ of equation (2.46) plotted as a function of T fordifferent values of ρ. The values are normalised to the peaks around 10 keV.


satisfy 0 < fα < 0.5. Thus, the second and third zero crossing in figure 3.6 are thelower limit Tmin and upper limit Tmax of the accessible temperature window, andby definition the two roots coincide at these points.

3.3.3 Helium fraction with external heatingThe derivation of a cubic equation for the helium fraction by means of equa-tions (2.42) and (2.43) no longer works for a plasma with external heating becausethe product of ne and Pext shows up in the resulting equation. Instead we candivide both equation (2.42) and (2.43) by ne and equate the two expressions forτE that we obtain that way.

Solving for fα again yields a cubic equation

0 = a0 + a1fα + a2f2α + a3f

3α, (3.7)

with coefficients

a0 =

(−3

2T[f3

Z(Z2 − Z3) + f2Z(4Z2 − 2Z) + fZ(1− 5Z) + 2

])n2

e , (3.8a)

a1 = (−3

2T[f2

Z(4Z − 5Z2) + fZ(14Z − 4)− 9]

+Eαρ

[f2

ZZ2 − 2fZZ + 1

]+

4

ρ〈σv〉

[(fZZ − 1)RB,1(T )− fZRB,Z(T ) +

Sext

n2e

])n2

e , (3.8b)

a2 =1

n2e

(−3

2T (fZ(4− 8Z) + 12) +

4Eαρ

(fZZ − 1)− 8RB,1(T )

ρ〈σv〉

), (3.8c)

a3 =

(6T + 4

Eαρ

)n2

e , (3.8d)

with Sext the external power density. These differ from the coefficients in equa-tion (2.47) by a factor of n2

e and have an extra term 4Sext/ρ〈σv〉 in a1. In thiscase only one of the roots satisfies 0 ≤ fα ≤ 0.5, and the solution is

fα = −a2 + C + ∆0

C

3a3, (3.9)

with the discriminant ∆ defined by equation (3.6) and

∆0 = a22 − 3a1a3, (3.10)

∆1 = 2a32 − 9a1a2a3 + 27a0a

23, (3.11)

C =

(∆1 +

√−27a2

3∆

2

)(1/3)

. (3.12)


All terms on the right hand side of equation (3.9) are either a free parameter ora function of T , so we have the desired expression for fα. The catch is that fromequation (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 andinsert this in the scaling law to compute τE. This value of τE can then be equatedto the value of τE obtained from equation (2.42). Solving for T then yields thedesired equilibrium temperature. This cannot be done analytically, but is easilyachieved numerically and this procedure is much faster than finding the equilibriumby 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 with-out external heating. This provides insight in the operational space of an ignitedplasma, but future reactors are designed to operate with some external heating forcurrent drive, and possibly control purposes. This leaves us with density, temper-ature, ρ and Pext as parameters, of which three can be chosen ’freely’ (respectingmachine and plasma limits of course).

First, we want to know how the addition of external heating changes the oper-ating contours that were obtained previously. Figure 3.7 shows this for the PPCSA reactor design with different levels of external heating. The solid lines representthe operating contours for ρ = 5 (blue) and ρ = 10 (red), and the dashed linesrepresent operating contours with different levels of external heating.

As expected, the equilibria with external heating almost coincide with thecontours obtained previously at high densities, because in this region the externalheating is insignificant compared to the fusion power. These equilibria are wellabove the Greenwald density or the β limit and therefore of little meaning forreactor design, except for a reactor so large that the operational contours areshifted towards lower densities.

The equilibria at low density and intermediate temperature (between 5 and 20keV) have disappeared for reasonable amounts of external heating for PPCS modelA, because there the external heating outweighs the power losses from the plasma.This is of little consequence, since these equilibria are mainly a mathematicalartefact and are not relevant for realistic reactor designs because of the (extremely)low densities.

When following a burn contour toward lower temperatures, the curves withexternal heating start to deviate and they converge on a minimum density that isonly a function of external heating and independent of ρ. This density is deter-mined by the balance between radiation losses and external heating because thereis little alpha heating at these low temperatures.

After reaching this (local) minimum in the density, the curve will turn upwardsagain when the temperature decreases even further. Since our model does not


4 5 6 7 8 9 10 20 30 40 501018

1019

1020

1021

1022 ρ=

5

ρ=

5

ρ=

10

ρ=

10

nG

β limit10 MW

100MW

200MW

200 MW100 MW

10 MW

50 MW

50 MW

T (keV)

ne(m

−3)

Figure 3.7: POPCON plot for PPCS model A with fixed external heating powerfor ρ = 5 ( ) and ρ = 10 ( ). The solid lines represent contours withoutexternal heating. For high densities, the curves almost coincide with the burncurves because the external heating is only an insignificant fraction of the fusionpower. The equilibria at low densities have disappeared because in that case boththe conduction and radiation losses are very small and outweighed by the externalheating. On the low temperature side, the equilibrium is mainly determined bythe radiation losses because there is hardly any fusion power due to the low crosssection at these temperatures. On the high temperature side, the equilibrium isalso set by the balance between radiation losses and external heating, but the curvehas a downward slope because an increase in temperature has to be compensatedby a reduction in density to maintain equilibrium.


include (impurity) line radiation and Ohmic heating, the results for temperaturesbelow a few keV should be treated with extreme caution.

On the high temperature (stable) branch, the external heating curves alsodeviate from the burn contours for decreasing temperature. In this case the curvestrend towards an asymptote determined by the degradation of confinement withincreasing power and the increase in radiation losses with

√T . In this temperature

range, around 70 keV, the reactivity is almost temperature independent and thefusion power is only sensitive to fuel density.

From a reactor perspective, the interesting area are the stable operating pointsaround the Greenwald density and just below the β-limit. For a given electrondensity, adding external heating shifts the operating points to a higher temperature(and consequently higher fusion power). The shift becomes larger for increasingvalues of ρ, and causes the curves of different values of ρ to converge on the sameasymptote.

This means that depending on the precise design of the reactor and the cor-responding positioning of the operating contours in the ne, T -plane, either ρ, theexternal heating power Pext, or a combination of both could serve as an actuatorfor burn control.

Finally, adding external heating changes the sign of dne/dT : when the curvestarts to deviate from the operating contour without external heating, it entersa regime where the temperature increases with decreasing density on the stablebranch (and vice versa on the unstable branch). This needs to be taken intoaccount when designing a controller for the reactor.

In contrast to the PPCS A, B and C designs, the ITER reactor is not antic-ipated to ignited. Hence there is always a non-zero amount of external heatingrequired, and the operating contours look very different from those for the PPCSA reactor, which is illustrated in figure 3.8.

3.3.5 ImpuritiesAlthough all expressions presented so far allow for the presence of impurities in theplasma, the previous sections have neglected their effect on the operating contours.This is of course not realistic and therefore this section presents operating contoursfor several impurities at different concentrations in the plasma. It turns out thatthe main effect of impurities is a contraction of the operating contours for thereactor, and therefore a reduction in fusion power on the stable burn branch,and an increase in fusion power on the unstable ignition branch. The maximumallowable value of ρ is lowered.

Every impurity has a critical concentration above which the burn can no longersustain 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 Bremsstrahlunglosses scale with Zeff which is a quadratic function of Z, whereas the fuel dilution


5 10 15 20 25 30

0.5

1

1.5

2·1020

nG

5

10

20

20

30

30

50

50

70

70

100

100

100

β-limit

T (keV)

ne(

m−3)

Figure 3.8: POPCON plot for ITER at ρ = 5 with constant levels of externalheating (labels are in MW). There are no equilibria for Pext = 0, i.e. ITER doesnot ignite in our model.


scales linearly in Z. However, this is only a minor effect compared to the overalcontraction of the burn contours.

4 5 6 7 8 9 10 20 30 40 501010

1015

1020

1025

1030

1035

1040

1045

nG

β limit

ρ = 5

ρ = 10

1% Be

1% Be

2% Be

2% Be

0.01%W

0.01%W

0.5% Ne

0.5% Ne

1%N

1%N

T (keV)

ne(m

−3)

Figure 3.9: PPCS model A POPCON plots for plasmas containing different im-purities (beryllium, nitrogen, neon and tungsten) at different concentrations. Thepresence of impurities dilutes the fuel and enhances the radiation losses by increas-ing Zeff. The result is a large reduction in the accessible density and temperaturerange 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 differentway (because the dilution is linear in Z whereas Zeff scales with Z2), this is onlya minor effect compared to the reduction in operating range.


Of more importance to a particular reactor design is the effect of impurities inthe plasma on the fusion power and energy gain factor Q. Figure 3.10 shows thefusion power and helium fraction (left plot), and Q and Zeff as a function of theberyllium fraction fBe = nBe/ne in the plasma, for ρ = 5 and 10 in PPCS modelA at a density of ne = 1× 1020m−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 relativeeffect is fairly similar, with a decrease 22% and 23% at Zeff = 1.5 respectively. Itis clear that the PPCS estimate of Zeff ≈ 2.5 yields unacceptable results for thefusion power if helium and beryllium are the only impurities.

0 0.05 0.1 0.150

2.5

5

7.5

10

Pfus,ρ=

5

Pfus,ρ=

10

fBe

Pfu

s(G

W)

0.05

0.1

0.15

0.2

fα,ρ=5

fα,ρ=

10

f α

0 0.05 0.1 0.150

10

20

30

40

Qρ=

5

Qρ=10

fBe

Q

1.5

2

2.5

3

Z eff,ρ=5Z eff

,ρ=10

Zeff

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 ofberyllium concentration fBe for ρ = 5 ( ) and 10 ( ) for the high temperatureequilibrium at ne = 1 × 1020m−3. The relative decrease of fusion power is moreor less independent of the value of ρ. Even a moderate value of Zeff = 1.5 alreadyleads 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 pre-cisely of course, but JET reported Zeff = 1.2 in the core plasma during divertoroperation with the ITER like wall [81]. This was in a divertor plasma withoutsignificant fusion power and therefore no helium content. Neglecting the tungstencontribution (so assuming a ’pure’ D-Be plasma), this amounts to a beryllium con-centration fBe = 0.017, which corresponds to a 17% decrease in fusion power forPPCS model A relative to a plasma without beryllium pollution. The erosion ratesreported in [81] are probably on the low side for a fusion power plant, because ofthe presence of fast He particles impacting on the first wall.


The effect of tungsten in the plasma on the fusion power is much weaker thanthat of beryllium for similar values of Zeff, as can be seen in figure 3.11, whichdisplays the fusion power and helium fraction fα on the left, and Q and Zeffon the right, as a function of the tungsten concentration in the plasma. BothJET and ASDEX report tungsten concentrations up to 10−4, but we have takenfW = 3 × 10−4 as the upper limit to obtain Zeff values that are foreseen in thePPCS.

0 1 2 3

·10−4

0

2.5

5

7.5

10Pfus,ρ=5

Pfus,ρ=10

fW

Pfu

s(G

W)

0.1

0.15

0.2

fα,ρ=5

fα,ρ=10

f α

0 1 2 3

·10−4

0

10

20

30

40

Qρ=5

Qρ=10

fW

Q

0 1 2 3

·10−4

1.5

2

2.5

3

Z eff,ρ=5

Z eff,ρ=10 Z

eff

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 oftungsten concentration fW for ρ = 5 ( ) and 10 ( ). Also for tungsten, therelative decrease of fusion power is more or less independent of the value of ρ, butthe 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 bythe quadratic dependence of Zeff on Z, and the quadratic dependence of the fusionpower on the fuel density, an effect which was already reported in [82]. The atomicnumber of beryllium is 4, and it takes quite a lot of it to obtain the same valueof Zeff compared to tungsten with an atomic number of 74, for which a smallconcentration already leads to a rather large change in Zeff.

Please note that the effect of high Z impurities is not limited to fuel dilutionand an increase in Zeff which results in higher Bremsstrahlung losses. They willmost likely not ionise completely and therefore emit line radiation, which can havea non-negligible effect on the (local) energy balance in the plasma. The resultingtemperature decrease might very well result in a significant loss in fusion power,but a complete treatment is outside the scope of this thesis.


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 willbe accompanied by a change in impurity content (assuming the impurity sourcestays the same of course). Figure 3.12 shows contours of constant Pfus in thefBe, ρ-plane, as well as contours of constant beryllium influx SBe (dark blue lines).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

·10−2

2

4

6

8

10

0.001

0.002

0.005

0.01

0.02

0.05

4,000

5,000

5,000

6,000

6,000

7,000

7,000

8,000

8,000

9,000

9,000

10,00011,000

12,000

fBe

ρ

Figure 3.12: Contour plot of the fusion power (coloured lines) in MW for PPCSmodel A with Pext = 246 MW as a function of the beryllium concentration fBeand ρ, at a density of ne = 1× 1020. The impurity concentration is of course alsoa function of ρ, and to indicate the effect of this coupling the contours of constantberyllium influx are also plotted (dark blue lines). The labels denote particles persecond per m3 in units of 1020.

If the impurity source stays constant during a change in ρ, the plasma willmove parallel to one the SBe = constant contours. This would have a significanteffect on the fusion power: for instance, if ρ increases from 8 to 9, and the originalberyllium 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 thefusion power changes by such a large fraction, but the example nevertheless shows


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 ofthe burn equilibria in operating space. Obviously, when the equilibrium changes, sodoes the fusion power, and figure 3.13 displays Pfus (left y-axis) and the derivativedPfus/dPext (right y-axis) as a function of Pext for ρ = 5 and 10 (blue and red curvesrespectively) for fBe = 0 and fBe = 0.025 for the PPCS A design at ne = 1020m−3.

0 50 100 150 2000

0.25

0.5

0.75

1·104

Pfus,fBe=0

Pfus,fBe=0

Pfus,fBe=0.025

Pfus,fBe=0.025

Pext (MW)

Pfu

s(M

W)

0

5

10

15

20

dPfusdPext

, fBe=0

dPfusdPext , fBe=0

dPfusdPext

, fBe=0.025

dP

fus

dP

ext , f

Be=

0.025

dP

fus

dP

ext

Figure 3.13: The fusion power Pfus (left axis) and its derivative with respect tothe external heating power dPfus/dPext (right axis) for PPCS model A at ne =1×1020m−3 for ρ = 5 ( ) and ρ = 10 ( ), with fBe = 0 and 0.025. The fusionpower increases with Pext, but this effect decreases for increasing Pext. Equilibriaat 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 inrelative terms.

With the addition of Pext the fusion power increases, and this effect is strongerfor higher values of ρ and for higher impurity concentrations. For ρ = 10 andfBe = 0.025 the fusion power increases by 1.8 GW when Pext goes from 0 to 246MW, 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


is stronger, both in relative and absolute terms, for higher values of ρ and fZ: thevalue 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 correspondingratio is 70%.

So for higher values of ρ and fZ external heating becomes a more effective toolto increase the fusion power output of the reactor. But the ultimate goal is notthe fusion power, but the power delivered to the grid.

3.3.7 The effect of Pext on net electric outputThe 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 externalheating power. (Obviously the plant also uses a significant amount of power forthe cryostat, coolant pumps, and a host of other auxiliary systems, but their powerconsumption is relatively insensitive to the fusion power and we will neglect it fornow.)

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 plasmawith 100% efficiency.

Combining these two expressions, we get

Pnet = ηPfus −(1− ξ)ξ

Pext. (3.15)

On the face of it, any amount of external heating will reduce the power deliveredto 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, soan increase in Pext also results in an increase in Pfus.

To determine what happens to Pnet if Pext changes, we take the derivative ofequation (3.15) with respect to Pext

dPnet

dPext= η

dPfus

dPext− 1− ξ

ξ. (3.16)

If dPnet/dPext is greater than zero, an increase in Pext will result in an increase inPnet. Some algebra allows us to transform this into a condition on dPfus/dPext:

dPfus

dPext>

1− ξηξ

. (3.17)


Using η = 0.31 as anticipated for PPCS model A [71], and ξ = 0.35 (refer-ence [71] gives a value of ξ = 0.6, but this is based on negative ion source neutralbeam injection, which seems highly unlikely for a commercial fusion reactor.) thecondition is dPfus/dPext > 6.0. In figure 3.13 it can be seen that for ρ = 5 thismeans that the addition of any amount of external heating leads to a reduction inPnet, while for ρ = 10 the limit lies somewhat above 100 MW, depending on theimpurity concentration in the plasma.

In the analysis above we have neglected the power gain from tritium breedingin the blanket, which changes the numbers but not the argument. If we assumeall tritium breeding is done by the 6Li + n → 4He + T + 4.8 MeV reaction, thetotal thermal power Pth is given by

Pth =

(1 + TBR

4.8

17.6

)Pfus, (3.18)

with TBR the tritium breeding ratio, which has a maximum value of approximately1.15 [60, 61, 62, 63]. We can absorb this power multiplication factor in η and seethat it relaxes the requirement on dPfus/dPext by roughly 30%.

So depending on the values of η and ξ that can be achieved, adding externalheating 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 forthe first generation of fusion power plants is optimal from a cost of electricity pointof view.

3.3.8 Uncertainties in scaling lawsThe scaling laws for the energy confinement time are based on a database con-taining several thousand experiments, and a scaling law is just a fit through thedatapoints, not an exact representation. The coefficients of the different param-eters in the scaling law come with an uncertainty, which is also apparent in thedifferences between the respective scaling laws in the ITER physics basis. Theseare all based on different subsets of the same data, sometimes with different physicsrestrictions. The variation in predicted energy confinement time is of the order ofone second, or roughly 25%, and the 95% log-nonlinear confidence interval for theIPB98(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 exponentsshould only have a small effect on the burn contours. This is indeed the case forchanges to the exponents of all parameters, except ne and P . The burn contoursare very sensitive to variations in the scaling of τE with ne and P , as is shownin figure 3.14. Reducing the P exponent compresses the burn contours along thedensity axis, and increasing the exponent of P results in a stretching. The reverseis true for changes in the exponent of ne. Already changes as small as one percentresult in a change of several decades in the predicted density range for the burn.The temperature range remains unaffected as explained before.


4 5 6 7 8 9 10 20 30 40 50

1015

1020

1025

1030

1035

1040

1045

1050

ρ =5

ρ =5

nG

β limit

P−0.68

31

P−0.6831

P−0.6969

P−0.6969

n0.4

059

e

n0.4059

e

n 0.4141e

n0.4141e

T (keV)

ne

(m−3)

Figure 3.14: Operating contours for PPCS model A for ρ = 5 and ρ = 10 applyingthe normal IPB98(y,2) scaling, and the IPB98(y,2) scaling with a ± 1% changein the exponents of ne and P respectively. These relatively small changes to thescaling law result in comparatively large changes in the operating contours. Theaccessible temperature range stays the same, since this is only a function of thefusion and radiation cross sections combined with the transport modelling using aconfinement time, but the density range is heavily affected.


This strong sensitivity of the operating contours to minor variations in theenergy confinement time scaling are clearly undesirable, because the exact positionof the operating point and the corresponding fusion power output have a majorimpact on reactor design. Further investigation of this phenomenon is outside ofthe 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 thenecessary heating. If a fixed ratio between the energy and particle confinementtime is assumed, it can be shown that the resulting equilibria form closed contoursin the neτE, T -plane [41]. We have derived an analytical expression for the tem-perature range that is spanned by such contours, based on the discriminant of thecubic equation for the helium fraction in the plasma. This result is valid for allreactors (not only tokamaks), for which the energy and particle confinement timehave a fixed ratio ρ.

Following the approach presented in [65], we also derived an expression for theelectron 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 contoursobtained with the IPB98(y,2) scaling are shifted towards higher densities in thene, T -plane. This is because the electron density of the equilibria is extremelysensitive to changes in the exponents of ne and P in the scaling law, something thatwe 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-modefactor fH, which in this case is taken to be fH = 2.

The contours obtained using the IPB98(y,2) scaling also have a shape thatdiffers from the ones found using the ITER89P L-mode scaling. This is becausethe radiation losses are not included in the IPB98(y,2) scaling, as opposed to theITER89P L-mode scaling that does include them. An explicit treatment of theradiation losses is thus required, and this does away with the artificial ’radiationlimit’ [53], eliminating the ’bump’ that is present on contours obtained using ascaling law for τE.

The use of scaling laws to eliminate τE from the burn equations results in burncontours that span many orders of magnitude in density. Care has to be takenwhen interpreting equilibria outside of the density range on which the scalinglaws are based, and often the contours extend to densities that have no physicalmeaning. This is purely a mathematical artefact, which originates in the form ofthe equations.

Burn contours in the neτE, T -plane are universal since they only depend on thereactivity and the radiation cross sections, but this property appears to be lostwhen they are transformed to the ne, T -plane. This can be resolved by dividing theexpression 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 factoris arbitrary, and the resulting isomorphic burn contours still extend over severalorders of magnitude in normalised electron density, depending on the scaling lawfor τE.

The main benefit of this discovery lies in the fact that a sensitivity analysis ofthe burn equilibria is also valid for all reactors: the results can simply be scaledby the ratio of the reactor specific part of τE. We performed a sensitivity analysisof the fusion power with respect to the three parameters that are under operatorcontrol in a burning plasma ne, ρ and H98.

The fusion power scales quadratically with electron density, apart from theregion around the minimum and maximum operating density on a contour. Thisstrong dependence makes the density a powerful actuator for control of the fusionpower. However, due to the desired operation close to the density limit the actualachievable variation in density might be too small for practical use.

The fusion power is also very sensitive to H98 around the minimum and max-imum values of H98. In between, the power output reaches a maximum (andminimum) and the different PPCS designs project H98 right at the value wherethese 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 theburn branch, the sensitivity on the ignition branch is rather weak. Although actingon the slowest time scale of the three parameters that were investigated, ρ mightbe the most suitable actuator for power control in a fusion reactor operating on theburn branch since it can be varied over a wide range without the risk of crossing astability limit. One needs to take the tritium breeding requirements into accountwhen considering this approach however, as values of ρ < 5 might result in a toolow tritium burn up fraction [64].

When including external heating in the model, several things change. Forreactors capable of ignition and high (≈ 25 keV) or low (<5 keV) temperatures, theoperating contours converge to a single contour that depends only on the amountof external heating and not on the value of ρ. For intermediate temperatures,the low density solutions have disappeared, because they feature such low fusionpowers that for realistic values of Pext the system is completely dominated by theexternal heating. Only the high density solutions remain, dominated by the alphaheating and consequently the curves with external heating almost coincide withthe original operating contours. In between the intermediate and the low and hightemperature regimes there is a smooth transition. The fact that contours withdifferent values of ρ but the same level of Pext converge to the same curve meansthat 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 valueof ρ decreases for increasing impurity concentrations. There is a small difference


between low and high Z impurities, due to the different scaling of fuel dilution andconduction and radiation losses with Z. This changes the shape of the operatingcontours, but it is insignificant compared to the contraction of the contours.

When looking at the effect of impurities on fusion power, the difference be-tween high and low Z impurities becomes more pronounced. Because the requiredconcentration of low Z impurities is much higher for a given Zeff, they dilute thefuel much more than high Z impurities. Since Pfus scales with the fuel densitysquared, 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 thanit is for low Z impurities.

Most likely a change in ρ will be accompanied by a change in the impurity con-tent of the plasma, and the fusion power is very sensitive to this. This sensitivityincreases for higher impurity concentrations.

The addition of external heating has a positive effect on the fusion poweroutput of the reactor, and Pfus is more sensitive to Pext for higher values of ρ andhigher impurity fractions. Whether the addition of Pext is beneficial for the netelectricity production of the plant depends on the thermal and heating efficiencyof the reactor, but a reduction in external heating power would most likely resultin a larger Pnet for the PPCS A design.

Finally, we looked at the sensitivity of the operating contours to small changesin the scaling laws for the energy confinement time. The predicted value of τE atthe ITER operating points varies relatively little between the different scalings inthe ITER physics basis and the predictions are robust against small changes inthe exponents of the individual scaling laws [26], but this is not the case for theoperating 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 compressedalong the density axis by many orders of magnitude. Although the correspondingshifts in operating points around the Greenwald density and just below the β limitare far less severe, we consider this an unphysical and also unwanted effect. Amore 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 essentialto know the nature of the operating point. What will the plasma do when leftalone. Will it wander off to some unknown destination in phase space? Or will itremain happily where it is. And if shaken or rattled by some external event, howwill it respond? In other words: is the operating point stable and if so, what isthe 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 tempera-ture [7, 65]. For a simple DT plasma without conduction losses this follows fromthe fact that the reactivity has a maximum, whereas the radiation losses increasemonotonically with temperature. There are therefore two temperatures at whichthe radiation losses are equal to the alpha heating power from the fusion reaction.

Because the energy balance contains only cross sections that follow from atomicphysics, the temperatures at these equilibria are independent of engineering pa-rameters and can be determined from

〈σv〉 = CB

√T . (4.1)

The addition of conduction losses to the energy balance introduces a reactordependence in the system, but in general two equilibria remain. Again the oneat low temperature is unstable and the other is stable. Irrespective of the reactordesign, some general observations can be made. For instance, the temperatureof the stable equilibrium decreases drastically because at high temperatures con-duction is the dominant energy loss mechanism (because of turbulent and, to alesser 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 thattemperature range the radiation losses dominate.

If the conduction losses are higher than the alpha heating power for all temper-atures (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 atonly 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 stabilityof 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 becausethe fusion reaction produces helium which has a finite residence time in the plasma.The helium particles change the equilibrium by diluting the fuel, increasing radi-ation losses and even affecting the conduction losses. So any realistic descriptionof a burning plasma needs to take this into account.

Linear stability of the burn point is not the only consideration for the designof a future fusion reactor. Of even greater importance is the convergence radiusof the equilibrium, or the size of the basin of attraction. The burn equations arehighly non-linear and although the Poincare-Lyapunov theorem states that a non-linear 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 theplasma respond to a disturbance of the burn equilibrium? Does it return to theoriginal equilibrium, or does it find a new one (possibly at T = 0)? And in doingso, does it cross any operational limits? And what happens to the fusion powerduring these excursions? Does the system have bifurcations and associated limitcycles? In short: can we rely on the plasma to regulate the burn by itself, or doesit need to be controlled by external means?

First we will present the system of differential equations that describe a burningplasma, and subsequently introduce a reduced system (with only two variables)which is able to reproduce the most important dynamics. We will study thestability of this system as a function of the free parameters by deriving an analyticalexpression for the Jacobian and applying planar bifurcation theory [83].

Using these results we will investigate the effect of changes in the τE scaling onthe stability and compare different PPCS designs, both with and without externalheating and reflect on the implications for operating point selection and controlrequirements.

Then we will return to the full system, derive the Jacobian, present the stabilityof the system, and compare the respective reactor designs.

4.2 Theory 69

4.2 Theory

4.2.1 Burn equationsThe burn dynamics can be described by four coupled, non-linear ordinary differ-ential equations (ODEs). Rebhan and Vieth [65] performed a first analysis of thissystem and determined the linear stability by deriving the Jacobian and evaluat-ing its eigenvalues for the equilibria that they found using the ITER98P L-modescaling.

In their analysis the effect of helium ash accumulation and fuel dilution on theion mass was omitted, resulting in a symmetry in the fD and fT terms in theJacobian matrix. This symmetry enables a reduction of the number of dimensionsby developing the Jacobian with respect to the first row or column, which yieldsnD = −nT to be an eigenfunction of the system. However, since a completeanalysis 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= sD − nDnT〈σv〉 −

nD

τp, (4.2a)

dnT

dt= sT − nDnT〈σv〉 −

nT

τp, (4.2b)

dnαdt

= nDnT〈σv〉 −nατp, (4.2c)

dT

dt=nDnT32ntot

〈σv〉(

Eα +3

2T

)− T

τE+T

τp− Srad

32ntot

− 2(sD + sT)T

ntot. (4.2d)

Here 〈σv〉 is the reactivity of the plasma as a function of T , for which we havetaken the Bosch and Hale parametrisation [4], Eα = 3.52 MeV the energy of analpha particle, Srad denotes the radiation losses due to Bremsstrahlung and theparticle and energy confinement time are related through τp = ρτE, with ρ a freeparameter with a typical value in the range 5-10, although values up to 30 havebeen 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(3/2ntotT ) and solving for dT/dt. Consequently, equa-tion (4.2d) contains some terms with an interpretation that may not be immedi-ately obvious. Their meaning is most easily understood when considering that achange in internal energy can be brought about by a change of temperature (atconstant density) or a change of density (at constant temperature). A change intemperature is then equal to the total change in internal energy minus the changein 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).


The nDnT〈σv〉T/ntot term denotes the temperature change due to the changein density when two particles fuse. Since a fusion reaction doesn’t exchange energywith the environment, the process has to be adiabatic (apart from the energygained from the change in binding energy obviously), so the kinetic energy of thereacting particles has to be retained in the plasma.

The T/τp term compensates for the difference between particle and energytransport. 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 temperatureis equal to the difference between particle and energy transport, with τp > τE ingeneral. Of course the use of energy and particle confinement times neglects theclose relationship between particle and energy transport, but this is unavoidablein such a simple model.

Finally the 2(sD +sT)T/ntot term represents the cooling effect of refuelling. Inthis case the fuel is assumed to be at zero kelvin, which is a good approximationin case of pellet fuelling and gas puffing. Only in the case of NBI heating does thenew 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)

the total particle density

ntot = 2nD + 2nT + 3nα + (Z + 1)nZ, (4.4)

and the average ion mass

A =2nD + 3nT + 4nα +mZnZ

nD + nT + nα + nZ. (4.5)

This burning plasma model implicitly assumes that dnZ/dt = 0, which is of coursemost 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 inplasma wall interaction or impurity seeding, which are expected to be the mainsources of impurities in future reactors. From here onwards we will assume nZ = 0for simplicity unless explicitly stated otherwise. However, relaxing this assumptiondoes neither change the analysis nor the conclusions.

The above system is four-dimensional, and its properties are determined bythe three free parameters sD, sT and ρ. Although the stability of the system caneasily be determined by calculating the eigenvalues of the linearised system, it iscomplicated to investigate the behaviour of a four-dimensional system around theequilibria and the dynamics of the system in general.

In the next section we will therefore present a reduced system that has onlytwo degrees of freedom and is determined by two parameters. This will allow theapplication of planar stability theory, and we can study the transition betweendifferent stability regions using bifurcation theory.

4.2 Theory 71

4.2.2 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 asymptoticallystable 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 deter-mining the Jacobian matrix J , and subsequently calculating its eigenvalues at theequilibria. Positive eigenvalues correspond to an unstable equilibrium and negativeeigenvalues to a stable point.

The eigenvalues of any (linear) system of equations can be determined by solv-ing

det(J − λI) = 0. (4.6)

In the two-dimensional case this can be written as

(j11 − λ) (j22 − λ)− j12j21 = 0. (4.7)

Defining p = j11 + j22 and q = j11j22 − j12j21 transforms this to

λ2 − pλ+ q = 0, (4.8)

so that the solutions are

λ1,2 =1

2

(p±√

∆), (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 ofthe equilibrium. The different possibilities, unstable (US), stable (S) and asymp-totically 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 intable 4.1). However, by looking at the different elements of J we can still obtaininformation about the stability. If J = ( 0 0

0 0 ) is the system stable, and if J 6= ( 0 00 0 )

it is unstable.


Table 4.1: Stability properties of a two-dimensional system as a function of theelements of the Jacobian matrix J , with p = j11 + j22 and q = j11j22 − j12j21.

q > 0 q = 0 q < 0

p > 0 US US USp = 0 S UK USp < 0 AS S US

4.2.3 Bifurcation theory

In order to fully appreciate the changes in behaviour between different regions inthe stability diagram of a burning plasma, we will briefly introduce the accompa-nying bifurcations. We will encounter them again in section 4.3.4.

The definition of a bifurcation is the division of something into two branchesor parts, or, in the study of dynamical systems, a sudden change in the qualitativeor topological structure of the system. This abrupt change is brought about by asmall, continuous change in the bifurcation parameter(s) of the system.

From bifurcation theory we know that there can be local bifurcations in thesystem at points where one or both eigenvalues has a real part that is equal tozero [83]. This property can be exploited when looking for bifurcations: we willfind a single zero eigenvalue at points where q = 0 (λ1 = p and λ2 = 0), and adouble zero eigenvalue when p = q = 0 (λ1,2 = 0). In case p = 0 and q > 0 thereare two completely imaginary eigenvalues, and if p = 0 and q < 0 both eigenvaluesare real.

The type of bifurcation depends on the nature of the eigenvalues at the bifurca-tion point. For a single zero eigenvalue, the result is a saddle-node bifurcation, alsoknown as a fold, or limit point bifurcation [83, 85]. On one side of the bifurcationthere are two equilibria: a saddle point (two real eigenvalues: one positive andone negative), and either a source (an unstable equilibrium) or a sink (a stableequilibrium). At the bifurcation, these two equilibria meet and annihilate eachother, and on the other side of the bifurcation they have disappeared.

At the point where both eigenvalues are completely imaginary, the system goesthrough a (Poincaré–Andronov–) Hopf bifurcation [83, 86], which signifies thebirth of a limit cycle. The bifurcation can be either supercritical, in which caseit 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 sourceor a sink), which annihilate in a saddle-node bifurcation. An Andronov-Hopfbifurcation 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 thispoint.

The saddle-homoclinic bifurcation occurs when a limit cycle collides with asaddle point and connects it with itself, which does not depend on the local valueof the eigenvalues, but instead is determined by the global properties of the sys-tem [83].

There are several other bifurcations that can occur in systems with two or moredimensions, but they are of little relevance in our case.

4.3 Reduced system

4.3.1 DerivationThere are several ways of reducing the dimensionality of the system. The mostlogical reduction follows from the observation that the fusion power output hasa maximum at, or very close to, nD = nT, so it is reasonable to take this as aproperty of the system. Because the transport of deuterium and tritium in ourmodel is exactly the same, this also means that sD = sT, hence this assumptionreduces 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 instantaneousfeedback system on the particle sources sD and sT, it enables us to isolate theeffect of helium accumulation on the burn dynamics at a constant density and ithas the added effect of making the system a bit less complicated.

The other possible relation between ni and nα would be through Zeff, but thiswould also imply a coupling with T because in equilibrium fα = f(T ).

Combining the condition nD = nT with equation (4.3), allows us to expressthe 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 thatcontains only one unknown, nα:

dni

dt= si −

1

2n2

i 〈σv〉 −ni

τp

= si −1

2(ne − 2nα)

2 〈σv〉 − ne − 2nατp

= −2dnαdt

, (4.11)

where si = 2sD = 2sT. We can derive an expression for the refuelling rate

si = −2dnαdt

+1

2(ne − 2nα)

2 〈σv〉+ne − 2nα

τp, (4.12)


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 con-stant 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 ofelectrons unchanged, the only way an electron can leave the plasma is by transportto the wall. The refuelling only needs to compensate for these losses, hence theintuitive form of expression (4.13).

Also the expressions for the total particle density

ntot = 2ne − nα, (4.14)

and the average ion mass

A =2.5ne − nαne − nα

(4.15)

take on a simpler form, as do the radiation losses

Srad = ne [(ne − 2nα) Rrad(T, 1) + nαRrad(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

dnαdt

=1

4(ne − 2nα)

2 〈σv〉 − nατp, (4.17)

dT

dt=

(ne − 2nα)2

6ntot〈σv〉

(Eα +

3

2T

)− nαT

ntotτp− T

τE− Srad

32ntot

, (4.18)

where in the latter the refuelling term has been combined with the particle trans-port term. The properties of the system are determined by the choice of the freeparameters ne and ρ.

4.3.2 Jacobian matrix of the reduced systemWriting xi = nα, T, the burn equations (4.17) and (4.18) can be expressed as

dxidt

= fi(x1, x2) i = 1, 2 (4.19)

We can linearise the system around an equilibrium point xj0 as

dxidt

=

2∑j=1

∂fi∂xj

∣∣∣∣xj0

(xj − xj0) , i = 1, 2. (4.20)


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 areevaluated at the equilibrium points xj0. Determining J comes with some extensivealgebra, and the full derivation can be found in section A.1. Here we will restrictourselves to the final result:

j11 = (2nα − ne)〈σv〉 − 1

ρτE+

nαρτ2

E

∂τE∂nα

, (4.21)

j12 =(ne − 2nα)2

4

d〈σv〉dT

+nαρτ2

E

∂τE∂T

, (4.22)

j21 =(ne − 2nα)(2nα − 7ne)

6n2tot

〈σv〉(

Eα +3

2T

)− T

τ2E

[2neτEρn2

tot

−(

1 +nαρntot

)∂τE∂nα

]− 5n2

eRrad(T, 1)32n

2tot

, (4.23)

j22 =(ne − 2nα)2

6ntot

[(Eα +

3

2T

)d〈σv〉dT

+3

2〈σv〉

]−(

nαρntot

+ 1

)(1

τE− T

τ2E

∂τE∂T

)− Srad

3Tntot. (4.24)

Although consisting of long expressions, numerical evaluation of the Jacobianmatrix is straightforward and very quick, because of the analytical expressions forall derivatives which makes computationally expensive numerical derivates unnec-essary.

4.3.3 Normalisation

The density spans a range of several orders of magnitude, and consequently anumerical treatment of the problem is prone to precision and rounding errors. Thiscan be solved by normalising the density, which we have done by expressing thehelium density as a fraction of the electron density, fα = nα/ne for the numericalcalculations. To a lesser extent the same holds true for the temperature, whichcan be normalised to the equilibrium temperature Teq to obtain T ∗.

So for the different terms of the Jacobian, we need to make the following


substitutions:

nα = nefα, (4.25)T = TeqT

∗, (4.26)dnαdt

= nedfαdt, (4.27)

∂

∂nα=

1

ne

∂

∂fα, (4.28)

dT

dt= Teq

dT ∗

dt, (4.29)

∂

∂T=

1

Teq

∂

∂T ∗. (4.30)

With these transformations, the relations between the elements of the Jacobian Jand the normalised Jacobian J∗ take the form

j11 = j∗11, (4.31)

j12 =ne

Teqj∗12, (4.32)

j21 =Teq

nαj∗21, (4.33)

j22 = j∗22. (4.34)

4.3.4 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 (orp = q = 0), dfα/dt = 0 and dT ∗/dt = 0 simultaneously. In reference [41] anexpression relating fα and T is presented, which can be used to express ne as afunction of T for different forms of the τE scaling law [65, 88]. Although an analyticsolution 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 insightinto 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 shapeof the equilibrium contours presented in references [65] and [88], we see that theequilibria form nested, closed contours in the ne, T -plane, with increasing valuesof ρ resulting in smaller (contracted) curves.

The question now is what happens to the stability when we change either ne orρ. Changing ρ at constant ne means moving parallel to the T -axis, whereas varyingne at constant ρ means moving along a burn contour (black curves in figure 4.1).


In doing so, the system will pass through regions of different stability, which wewill discuss in the following paragraphs.

Figure 4.1 shows the nature of the equilibria in the reduced system in thene, T -plane for the PPCS A reactor design [70]. On the low temperature side, theequilibrium is a saddle point having both a stable (green in figure 4.1) and an un-stable (red) eigenvector. Moving towards higher temperatures, both eigenvectorsfirst become unstable and subsequently acquire an imaginary part (blue), whichintroduces 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 isagain encountered.

5 10 20 50 100 2001010

1020

1030

1040

1050

1060

1070

T (keV)

ne

(m−3)

λ1

5 10 20 50 100 200T (keV)

λ2

Figure 4.1: The stability of the two eigenvectors of the reduced burn system forPPCS model A [70]. The black lines indicate contours of constant ρ, starting withρ = 1 on the outside and ending with ρ = 14 for the innermost curve. Unstablebehaviour for a given eigenvector is denoted , means oscillatory unstablebehaviour, oscillatory stable behaviour and stable behaviour.

Starting at the lowest temperature on a contour of constant ρ, we find that theequilibria are saddle points, with a stable and an unstable eigenvector. Followingthe contour in the clockwise direction, the stable eigenvector also becomes unstable,which happens exactly at the maximum density of the contour. This transition isaccompanied by a saddle-node bifurcation, which occurs when a system has oneeigenvalue equal to zero. In this case the source point meets the saddle point andthey annihilate each other.

For constant ρ, the bifurcation parameter is ne and we can understand the


physical reason for the bifurcation as follows. When increasing the density, thetemperature difference between the stable an unstable equilibria is reduced, untilat the maximum density both equilibria coincide. A further increase in ne is notpossible without also changing the value of ρ.

Further along the contour, the eigenvalues acquire an imaginary part, resultingin unstable oscillating behaviour of the equilibrium (blue points in figure 4.1). Thereal parts of the eigenvalues now become increasingly smaller, until at some pointthey become negative, corresponding to a stable oscillatory equilibrium (greenpoints). At the change from unstable to stable oscillations, both eigenvaluesare purely imaginary, which means that the system goes through a (Poincaré–Andronov–) Hopf bifurcation. In this case the bifurcation is subcritical, resultingin the birth of an unstable limit cycle.

Moving along, the limit cycle increases in size until it degenerates into a homo-clinic orbit to the saddle equilibrium at the same density, but lower temperature,where it disappears in a (saddle-)homoclinic bifurcation. This is a global bifurca-tion, meaning that it does not depend on the local parameters of the system, butinstead arises from the properties of the system at different points, contrary to thefold and Hopf bifurcations, which are local bifurcations.

Traversing the contour further in the clockwise direction, for ρ ≤ 10.5, theimaginary part of the eigenvalues can decrease to zero yielding a stable equilibrium.Approaching the low density side of the contour, the equilibrium once more entersthe stable oscillatory part, before moving into the saddle region at the minimumdensity 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 hightemperature side, but for ρ ≥ 14 there are no stable equilibria anymore, effectivelylowering the upper limit on the accessible value of ρ for a fusion reactor if stableburn 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 aBogdanov-Takens bifurcation [87]. A Bogdanov-Takens bifurcation occurs whenthe system has a zero eigenvalue of multiplicity two. There are two nearby equi-libria: a saddle point and a node (sink or source), which annihilate via a saddlenode bifurcation. The non-saddle equilibrium undergoes a Hopf bifurcation thatgenerates a limit cycle that connects to the saddle point in a saddle-homoclinicbifurcation (which we encountered a little while back).

4.3.5 Physical interpretationBesides looking at the stability properties by themselves, we can also try to under-stand the physical mechanisms behind the stability or instability of the differenteigenvectors. This section will interpret the local stability properties in terms ofthe physical mechanisms that drive them. Because the parameter space is too


large to be covered in detail, we will discuss the eigenvectors and eigenvalues atdifferent relevant points, all at a density of ne = 1 × 1020m−3 for the PPCS Adesign with the IPB98(y,2) scaling for τE.

The Jacobian matrix consists of the partial derivatives of the burn equationswith respect to the different variables. Evaluating it at an equilibrium point tellsus something about the sensitivity of the equilibrium to changes in both variables.Looking at equation (4.7), it is clear that

j∗11 < −j∗22 (4.35)

is a necessary, but not sufficient condition for stability. The other requirement is

j∗11j∗22 ≥ j∗12j

∗21. (4.36)

So the stability of the system depends on all four elements in J∗.Looking at the physical meaning, the top left term j∗11 represents the derivative

of dfα/dt with respect to fα. A negative value here means that an increase in fαwill result in a negative value for dfα/dt, driving the system back to equilibrium.

The bottom right term j∗22 corresponds to the sensitivity of the rate of changeof temperature to changes in temperature, and again, a negative value has a sta-bilising effect.

When taking the top right and bottom left terms into account, the picturebecomes more complicated. They describe the change in dfα/dt and dT ∗/dt, causedby variations in T ∗ and fα respectively. Their effect is stabilising if they haveopposite signs, and destabilising in case they have the same sign, but the overallstability 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 thoughj∗11 could be negative and drive the helium concentration back to the equilibriumlevel, the temperature increase might promote an increase in helium content. If thislatter effect is stronger than the former, the system will be unstable. Notice thatthis effect can take place even when j∗22 < 0, it really depends on the magnitudeof the different terms.

We will take a closer look at the different terms that make up the different el-ements of the Jacobian matrix, and try to determine on physical grounds whetherthey will be positive or negative. This will provide insight in the driving mecha-nisms behind the instabilities that are present in the system.

The first element, j11, consists of three parts. The first

(2nα − ne) 〈σv〉is always negative, because ne ≥ 2nα, and corresponds to the reduction in reactionrate when the fuel dilution increases.

The second term

− 1

ρτE


is always negative by definition. This leaves us with the last part

nαρτ2

E

∂τE∂nα

,

which is always positive. This can be seen by looking at equation (A.6) andrealising that nα < 0.5ne.

The second element, j12, has only two terms. The first can be positive ornegative, depending on the sign of d〈σv〉/dT . Because the maximum of 〈σv〉lies at 67 keV, this term will be positive under reactor relevant conditions. Thisrepresents the increase in helium production because of a higher reactivity if thetemperature rises.

The sign of second term of j12 is also determined by d〈σv〉/dT (equation (A.7)),but has the opposite sign to the first term, because a higher temperature resultsin a higher fusion power, which decreases the (particle) confinement time.

The third element j21, turns out to be the term with the largest magnitudefor all temperatures at ne = 1× 1020m−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 whichis

(ne − 2nα)(2nα − 7ne)

6n2tot

〈σv〉(

Eα +3

2T

).

This is the derivative of heating power per particle with respect to nα. An in-crease in helium content leads to lower fuel concentration and hence lower fusionpower. The ntot term in the denominator becomes slightly smaller for increasingnα, but since nα < 0.5ne it can easily be seen that this term is always negative bysubstituting ne − 2nα > ne − ne = 0 and realising that 2nα − 7ne < 0.

The second part reads

− T

ρτ2E

∂τE∂nα︸︷︷︸1

+2neT

ntotρτ2E

∂τE∂nα︸︷︷︸

2

− 2neT

τpn2tot︸︷︷︸

3

− T

τ2E

∂τE∂nα︸︷︷︸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 inthe 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 respectto the helium content. Part 2 represents the changes in refuelling that accompany achange 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 heliumcontent changes. In the latter case the refuelling itself is not directly affected, butthe cooling effect is divided among more or less particles. This term has a minussign because a higher helium content means a lower number of particles.


Part 4 is due to a change in energy confinement caused by a change in he-lium content, which is negative because a larger helium content leads to a longerconfinement time and hence lower energy losses.

Finally, the third term in equation (4.23),

−5n2eRrad(T, 1)

32n

2tot

,

represents temperature decrease due to the change in radiation losses, which in-crease when nα increases due to the Z2

eff 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 thatthe 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

6ntot

[(Eα +

3

2T

)d〈σv〉dT

+3

2〈σv〉

],

is positive and corresponds to the temperature increase due to the increased alphaheating for a rising temperature.

The second term

−(

nαρntot

+ 1

)(1

τE− T

τ2E

∂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. Determiningthe sign of this term is not difficult, because equation (A.7) tells us that ∂τE/∂Tis 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 signof j22 thus depends on whether the increased fusion power outweighs the increasedlosses and this changes with temperature.

4.3.6 Low temperature stabilityWe can apply the results from the previous section to different equilibria in thephase plane. Starting with the low temperature equilibrium at ρ = 5, which has atemperature of 6.24 keV, we can evaluate the normalised Jacobian matrix:

J∗ =

(−0.011693 0.000094−2.542608 0.158415

).

The (normalised) eigenvectors for the low temperature equilibrium are

v1 = (0.066202, 0.997806)


and

v2 = (0.000556, 1.000000),

with v1 stable and v2 unstable. Although in both eigenvectors the alpha en tem-perature components have the same sign, the v2 has a much larger temperaturecomponent than v1

1. The temperature component in both v1 and v2 is destabil-ising, but it is the larger helium component that stabilises v1.

4.3.7 High temperature stabilityFor the high temperature equilibrium at ρ = 5 and T = 29 keV, the Jacobianmatrix is

J∗ =

(−0.086266 0.000082−17.932381 −0.255725,

).

Compared to the low temperature equilibrium, we can observe that the only qual-itative difference can be found in the bottom right term. This term has becomenegative, reflecting the fact that at the high temperature equilibrium the lossesincrease faster with temperature than the fusion power. The derivative of thechange in helium density with respect to the temperature is still positive, whichis caused by the fact that d〈σv〉/dT , which is always positive for reactor relevanttemperatures, is larger than ∂τE/∂T , which is negative because the increase ofpower with increasing temperature results in a lower value of τE. The major quan-titative difference is in the bottom left element, which shows that the sensitivity ofthe temperature with respect to changes in the helium concentration at constantρ increases with temperature, which finds its origin in the strong temperaturedependence of 〈σv〉.

In this case the eigenvectors are

v1 = (0.066202,−0.997806)

and

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 ofthe Jacobian change when going to higher or lower densities, or when changingthe value of ρ, but the overal picture stays the same.

1Even 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. Sincea 2x2 matrix is always in Hessenberg form, balancing does not help in this case (in fact balancinga matrix in Hessenberg form changes the eigenvalues and vectors).


4.3.8 Phase portrait

We can divide the ne, T -plane into regions with different behaviour. Figure 4.2shows the phase portraits for PPCS model A for different values of ρ at a densityof ne = 1020m−3. The red markers indicate the equilibria, whose position inthe ne, T -plane is indicated in the top middle graph. The red lines indicate theseparatrix between regions that converge towards a stable equilibrium and regionsthat 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 hasρ = 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 thetemperature. The stable equilibrium at higher temperature is a stable impropernode, meaning that all orbits approach the node from opposite directions alongthe same line, except for two orbits which come in from opposite directions witha certain angle to the above mentioned line.

Both axis can be found by finding the matrix T ∈ R2×2 that satisfies TAT−1 =J , with J the Jordan canonical form of A. Then define y(t) to satisfy the system

y′ = Jy (4.37)

and all orbits come in along the direction of the y1 vector, except for the two orbitsthat come in along the y2 direction.

The separatrix between the stable and unstable region starts on the T -axis alittle above T = 5 keV in this case and increases more or less linearly until T = 10keV and fα = 0.2, at which point the curve starts to flatten and disappears towardsT =∞. Lowering the value of ρ will lift the asymptote of the separatrix towardsfα = 0.5, but it always stays below the physical limit fα = 0.5, which correspondsto a complete helium plasma. In the scope of our model this would be a anunrecoverable scenario without external heating, because a pure helium plasmameans no fusion power and consequently infinite energy and particle confinementtimes.

If the value of ρ is increased to eleven, we see that the separatrix starts toflatten a bit more strongly, but still approaches an asymptote that extends towardsT = ∞, albeit at a lower value of fα. The unstable equilibrium has moved upa bit and the stable equilibrium is closer to the separatrix. At this point theeigenvalues of the system at the stable equilibrium have acquired an imaginarypart and consequently this has become a stable spiral point. The pitch of thespiral 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 resultsin the separatrix revolving around the stable equilibrium and connecting with itselfin the unstable equilibrium. This also means that the lower part of the separatrixhas disappeared from the system and the stable region of the phase space has beenreduced to the area within the homoclinc orbit.


0

0.1

0.2

0.3

f α

ρ = 5 ρ = 11

0

0.1

0.2

0.3

f α

ρ = 13.8375

10 20 30 40

ρ = 14.4

10 20 30 400

0.1

0.2

0.3

T (keV)

f α

ρ = 14.7

5 10 20 50T (keV)

ne

Figure 4.2: Phase portraits for PPCS model A in the fα, T -plane of the reducedburn system for ne = 1020m−3 for different values of ρ. The equilibria are indicatedby corresponding red markers in the phase portrait and in the ne, T -plane (bottomright), and the red line indicates the separatrix between stable and unstable regionsin the phase plane. The top left image at ρ = 5 shows a saddle point at lowtemperature and a stable equilibrium (sink) at high temperature. When the valueof 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 ahomoclinic orbit that connects the saddle point with itself, orbiting the (hightemperature) stable equilibrium. For even higher values of ρ the stability of thehigh temperature equilibrium changes: it becomes unstable with an imaginarypart, until for very high values of ρ the high temperature equilibrium also becomesa source. Another effect of increasing ρ is that the two equilibria approach eachother, until they coincide at the maximum allowed value of ρ.


The homoclinic orbit becomes an unstable limit cycle which shrinks in size witha further increase of ρ, until the system hits the Hopf bifurcation where the stableequilibrium changes to an unstable one. Note that right at the Hopf bifurcationthere is an infinite number of limit cycles around the equilibrium, that at thatpoint is stable, but not asymptotically stable.

Towards the upper limit of ρ the imaginary part of the eigenvalues at the hightemperature equilibrium disappears again, changing this point into an impropernode, and the system is left with two unstable equilibria.

An interesting observation that can be made when comparing the 5 phaseportraits in figure is that the overall picture looks remarkably similar, with mostorbits converging towards a curve that resembles a skewed parabole. The equilibriaare located somewhere along this curve, and depending on the value of ρ, theseparatrix partially runs along this curve as well.

This observation could have implications for reactor start up or the design ofburn control systems, because some orbits might be highly undesirable as they willcross the β limit, or put too much heat load on the first wall.

4.3.9 Stability for different scaling lawsThe ITER physics basis contains five different scaling laws, (IPB98(y) andIPB98(y,i), with i = 1, 2, 3, 4 [26]), and although their predictions for τE in ITER donot differ much, it has already been shown that the operating contours they predictfor a burning plasma show large differences in density range [88]. In the followingsection we will investigate the changes in stability for the equilibria between thedifferent scalings in the ITER physics basis.

Figure 4.3 shows the different stability regions in the ne, T -plane for the fivedifferent energy confinement time scalings in the ITER physics basis. The globalpicture looks similar for the first four scalings: y, and y(1,2,3), with an unstablesaddle 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 stabilityregions, 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 verysimilar to the other four. The reason for this is that the value of 1− 2m+ l, withm 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. Thefact that this value is very close to zero for all scalings explains the large spreadin 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 maindifferences in stability can be found when looking at the intersections of the curvesof constant ρ with the stability boundaries. For the IPB98(y) scaling, the ρ = 14contour extends to about two/thirds of the width of the unstable (dark blue) regionon the high temperature side. The same contour for subsequent scalings reaches


1015

1020

1025

1030

ne

(m−3)

98(y)

1016

1020

1024

1028

1032

ne

(m−3)

98(y,1)

1010

1025

1040

1055

1070

ne

(m−3)

98(y,2)

5 10 20 50 100 200100

1025

1050

1075

10100

T (keV)

ne

(m−3)

98(y,3)

5 10 20 50 100 20010−130

10−80

10−30

1020

1070

T (keV)

ne

(m−3)

98(y,4)

Figure 4.3: Stability regions for the PPCS A design for the different τE scalingsin the ITER physics basis [26]. The area is asymptotically stable, asymp-totically stable with an oscillation, is unstable with a oscillation, is unstableand is a saddle point (unstable).


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 significantpart of this curve that traverses the asymptotically stable (green) area of the plot.

The scaling laws for τE contain three parameters that change when movingthrough the phase space of a burning plasma; the average ion mass A, the electrondensity ne and the heating power P . Looking at the exponents k, l and m ofthese parameters in the scaling law, we see that l decreases and m increases forthe subsequent scalings, whereas there is no clear pattern for k.

The expansion of the unstable and oscillatory region therefore correlate withan increase in density and a decrease in power dependence, but we have not beenable to identify a physical mechanism for this.

For all scalings the point where both eigenvalues are zero lies roughly on theρ = 8 contour, but the density for this point varies greatly. Depending on the signof 1−2m+ l, it can be found below or above the Greenwald density, but it is neverin the reactor relevant density range.

Figure 4.4 zooms in on the reactor relevant density and temperature range forthe PPCS A design, plotting the stability of the equilibria on curves of constantρ as well as the Greenwald density and β limit. In all subplots we see an unstablesaddle point on the low temperature side, and a, mostly stable, equilibrium at thehigh temperature side. Only for ρ = 14 does this equilibrium become unstable,but it already acquires an imaginary eigenvalue part above ρ ≈ 10, depending onthe scaling.

In the two plots on the top row, the ρ = 2 curve shows an oscillatory region (inthe bottom right of the plot). This is the intersection of this particular contourwith the long ’tail’ of the stable oscillating region, which could already be seenin figure 4.1. The fact that it is not visible in the other plots is an artefact fromplotting only integer values of ρ.

Concluding we can say that for the scaling laws in the ITER physics basis thereare only minor differences in stability, both on a global level and in the reactorrelevant domain. Those differences manifest themselves at the high temperatureequilibrium, and correlate with the values of the power en density exponents inthe scaling law: a weaker power dependence and a stronger density dependenceresult in an expansion of the unstable area to lower values of ρ. However, basedon the results of present day tokamaks [34], it seems unlikely that this will be anissue for future reactors, although much higher values of ρ have also been reportedin limiter plasmas [76].

4.3.10 Stability with external heatingSo far we have only considered ignited plasmas without external heating. Thisapproach allows for an analytical treatment of the system, but it is an unrealistic


1

2

ρ=

5

nG

β limit

ρ=

9

ρ=

13ρ

=1

ρ=

13

ne(102

0m

−3)

IPB98(y)

1

2

ne(102

0m

−3)

IPB98(y,1) IPB98(y,2)

5 10 15 20 25 30 35 40

1

2

T (keV)

ne(102

0m

−3)

IPB98(y,3)

5 10 15 20 25 30 35 40T (keV)

IPB98(y,4)

Figure 4.4: Stability for the reduced system burn contours of PPCS model Aat different values of ρ for the different τE scalings in the ITER physics basis.The asymptotically stable part of the contours is denoted , whereas isasymptotically stable with an oscillation, is unstable with an oscillation,is unstable and is a saddle point (unstable).


scenario for a tokamak reactor design, if only because some form of non-inductivecurrent drive will be needed. The external heating power required for this will alsoaffect the stability of the equilibria.

In the case of external heating, equation (4.18) changes to

dT

dt=

(ne − 2nα)2

6ntot〈σv〉

(Eα +

3

2T

)− nαT

ntotτp− T

τE+Sext − Srad

32ntot

, (4.38)

where Sext is the external heating power density in keV/m3. Since Sext is inde-pendent of nα or T , it does not show up in the Jacobian, which consequently hasthe same eigenvalues as before. Adding external heating does therefore not affectthe stability of the system directly.

22 24 26 28 30 320.1

0.15

0.2

0.25

0M

W50

MW

100

MW

150

MW

200

MW

250

MW

0M

W

50M

W10

0M

W15

0M

W20

0M

W25

0M

W

T (keV)

f α(%

)

Figure 4.5: The position of the high temperature equilibrium in the fα, T -planeas a function of Pext, at a density of ne = 1 × 1020m−3 for the PPCS A designfor ρ = 5 ( ) and ρ = 10 ( ). For increased levels of external heating, theequilibrium temperature increases and the helium fraction decreases. Whereas therelative decrease in helium fraction is the same at roughly 5% at both ρ values, therelative 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 todifferent values of fα and T , and the eigenvalues of the Jacobian evaluated at thesepoints are different. The shift in fα and T is illustrated in figure 4.5, which showsthe equilibrium position in the fα, T -plane as a function of Pext for the PPCS Adesign at ρ = 5 and ρ = 10.

Increasing Pext results in a shift of the equilibrium to higher temperaturesand lower helium fraction, which can be understood as follows. Adding externalheating increases the temperature and consequently the reactivity of the plasma.This increase, together with the resulting increase in fusion power, lowers theenergy confinement time, which results in a faster exhaust of helium ash.


The relative shift in helium fraction is roughly 5%, and this value is more orless independent of the value of ρ. The relative temperature shift, on the otherhand, increases for higher values of ρ. The reason this effect is stronger for highervalues of ρ lies in the ratio between alpha heating power and external heating. Athigher values of ρ the equilibrium temperature is lower and the helium fractionhigher, which implies a lower fusion power output. Adding a certain amount ofexternal heating will therefore have a bigger effect at high ρ values.

An interesting observation is that the effect on the fusion power output of thereactor, is almost linear in T , with the gradient dPfus/dT a function of ρ. For highvalues of ρ it might therefore be interesting to consider external heating to increasethe power output of the reactor. As long as the gain in fusion power output islarger than the additional external heating divided by the plant efficiency timesthe heating efficiency, the net effect is positive.

To determine the stability of the system with Pext 6= 0, we first solve the burnequations (4.17) and (4.38) to obtain the equilibrium values of nα and T . Theseare used to obtain the Jacobian of the system at the equilibrium and subsequentlywe determine the eigenvalues of the system.

Figure 4.6 plots the stability of the system in the ne, T plane for the PPCS Areactor with different levels of external heating, at different values of ρ. The ploton the top left is identical to the bottom left in figure 4.4.

It is immediately apparent that the green area at high temperature increasesfor increasing levels of external heating and that the blue segment on the ρ = 14curve disappears. At the same time the addition of Pext results in a change fromclosed contours to open contours, which manifests itself in a change in the sign ofthe slope of the high temperature equilibrium curves. Instead of having a positivevalue of dne/dT , this now becomes negative.

On the low temperature side the equilibrium curves also change. Additionalheating first of all changes their slope from negative to positive, and secondlycauses them to curve upwards after hitting a local minimum in density, whichintroduces a second stable equilibrium at low temperatures. Around the mini-mum density point, the system goes through a transition from unstable to stable,passing through an oscillating area, with the same bifurcations as described insection 4.3.4. The stability of the equilibrium to the right of the local density min-imum 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 densityon the low temperature side upwards, and at some point they disappear from theplot (bottom middle plot). The relevance of these equilibria for reactor purposesis minimal because of the very low fusion power output, but it basically rules outa low power startup scenario at high density.

Concluding we can say that the addition of external heating has a beneficialeffect on the stability of the high temperature equilibrium, for which the stable


1

2

nG

β limit

ρ=

1

ρ=

5

ρ=

9

ρ=

13ρ

=1

ρ=

13

ne(102

0m

−3)

Pext = 0MW Pext = 50MW

1

2

ne(102

0m

−3)

Pext = 100MW Pext = 150MW

5 10 15 20 25 30 35 40

1

2

T (keV)

ne(102

0m

−3)

Pext = 200MW

5 10 15 20 25 30 35 40T (keV)

Pext = 250MW

Figure 4.6: Stability for the reduced system burn contours of PPCS model A atdifferent values of ρ for different levels of external heating. The asymptoticallystable part of the contours is denoted , whereas is asymptotically stablewith an oscillation, is unstable with an oscillation, is unstable andis a saddle point (unstable).


area increases for increasing Pext. The stability of the low temperature equilibriumis unaffected for higher densities, and a stable, third equilibrium is introducedat an even lower temperature. The transition between the stable and unstableequilibrium at low temperature occurs at the local density minimum.

4.3.11 Reactor comparison

There are currently four different prototype reactor designs under consideration inEurope [71, 70], with PPCS models A and B being more conservative and PPCSmodels C and D applying riskier technology extrapolations. This section analysesand compares the stability for the four different designs at their designed level ofexternal heating.

Figure 4.7 plots the stability of the burn contours for the PPCS model A, B, Cand D designs for different values of ρ with the amount of external heating specifiedin 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 equilibriumvalues of nα and T for the given values of ρ and Pext and subsequently determin-ing the eigenvalues of the Jacobian. The resulting stability curves are plotted infigure 4.7, in which the top left plot closely resembles the bottom right plot infigure 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 theirdesignated level of external heating, but the type of stability differs between thedifferent reactors at different operating points. The operating space for the PPCSA and B designs are almost completely asymptotically stable, the PPCS C and Ddesigns show considerable areas with oscillatory stable behaviour. For the PPCSC 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 systemof burn equations, as presented in section 4.2.1. First we will linearise the systemaround the equilibria and derive the Jacobian matrix, and use this result to in-vestigate the stability of the system in the ne, T -plane for different reactor designsand energy confinement scaling laws. Finally, we will discuss the stability of thesystem 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

dxidt

= fi(x1, x2, x3, x4) i = 1, 2, 3, 4 (4.39)

4.4 Full system 93

1

2

ρ=

1

ρ=

5

ρ=

9

ρ=

13

nG

β limit

ne(102

0m

−3)

PPCS A, Pext = 246MW PPCS B, Pext = 270MW

5 10 15 20 25 30 35 40

1

2

T (keV)

ne(102

0m

−3)

PPCS C, Pext = 112MW

5 10 15 20 25 30 35 40T (keV)

PPCS D, Pext = 71MW

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 specifiedin [71]. The asymptotically stable part of the contours is denoted , whereas

is asymptotically stable with an oscillation.


We can linearise the system around an equilibrium point xj0 as

dxidt

=

4∑j=1

∂fi∂xj

∣∣∣∣xj0

(xj − xj0) , i = 1, 2, 3, 4. (4.40)

To determine the linear stability of the system requires determining the eigen-values (and eigenvectors) of the Jacobian matrix jij = ∂fi/∂xj evaluated at theequilibrium points xj0.

As with the reduced system, determining the derivatives used in the Jacobian isnot difficult, but involves some extensive algebra which can be found in section A.2.Here we simply present the Jacobian of the full system:

j11 = −nT〈σv〉 −1

ρτE+nD

ρτ2E

∂τE∂nD

, (4.41)

j12 = −nD〈σv〉+nD

ρτ2E

∂τE∂nT

, (4.42)

j13 =nD

ρτ2E

∂τE∂nα

, (4.43)

j14 = −nDnTd〈σv〉dT

+nD

ρτ2E

∂τE∂T

, (4.44)

j21 = −nT〈σv〉+nT

ρτ2E

∂τE∂nD

, (4.45)

j22 = −nD〈σv〉 −1

ρτE+nT

ρτ2E

∂τE∂nT

, (4.46)

j23 =nT

ρτ2E

∂τE∂nα

, (4.47)

j24 = −nDnTd〈σv〉dT

+nT

ρτ2E

∂τE∂T

, (4.48)

j31 = nT〈σv〉+nαρτ2

E

∂τE∂nD

, (4.49)

j32 = nD〈σv〉+nαρτ2

E

∂τE∂nT

, (4.50)

j33 = − 1

ρτE+

nαρτ2

E

∂τE∂nα

, (4.51)

j34 = nDnTd〈σv〉dT

+nαρτ2

E

∂τE∂T

, (4.52)

4.4 Full system 95

j41 =ntotnT − 2nDnT

32n

2tot

〈σv〉(

Eα +3

2T

)+

(ρ− 1)T

ρτ2E

∂τE∂nD

− 132ntot

∂Srad

∂nD+

2Srad32n

2tot

+4(sD + sT)T

n2tot

, (4.53)

j42 =ntotnD − 2nDnT

32n

2tot

〈σv〉(

Eα +3

2T

)+

(ρ− 1)T

ρτ2E

∂τE∂nT

− 132ntot

∂Srad

∂nT+

2Srad32n

2tot

+4(sD + sT)T

n2tot

, (4.54)

j43 = −2nDnT32n

2tot

〈σv〉(

Eα +3

2T

)+

(ρ− 1)T

ρτ2E

∂τE∂nα

− 132ntot

∂Srad

∂nα+

3Srad32n

2tot

+6(sD + sT)T

n2tot

, (4.55)

j44 =nDnT32ntot

[3

2〈σv〉+

(Eα +

3

2T

)∂〈σv〉∂T

]+

1− ρρ

(1

τE− T

τ2E

∂τE∂T

)− 1

32ntot

Srad

2T− 2(sD + sT)

ntot. (4.56)

This result differs from the Jacobian presented by Rebhan and Vieth [65] for tworeasons. Firstly, our result is valid for scaling laws for the energy confinement timethat do not include radiation losses, whereas ref [65] uses the ITER89P scaling [29],which includes the radiation losses in the energy confinement time. This introducessome extra terms in our Jacobian to account for this.

The second difference arises because ref [65] assumes a constant ion mass of2.5 amu, which is only correct in case of a pure DT plasma. A burning plasma bydefinition contains helium and this needs to be taken into account in the averageion mass, which consequently will be ≥ 2.5 amu.

4.4.2 Full system stabilityUsing the expression for the Jacobian presented in section 4.4.1, we can determinethe stability of the different eigenvectors of the system for each equilibrium bydetermining their respective eigenvalues. In figure 4.8 the stability of the differenteigenvectors of the system is plotted. Again, red is unstable, blue is oscillatoryunstable, purple is oscillatory stable and green is stable.

We can see that on the high temperature side the system is stable, whereas onthe low temperature side there is one eigenvector that turns the system unstable.In the central region of the operating space their are several transitions betweenstable and unstable behaviour for the different eigenvectors.

A transition between different stability regimes means that either the real or theimaginary part of the eigenvalues has a zero crossing. There is only one point (near


1010

1020

1030

1040

1050

1060

1070

ne

(m−3)

5 10 20 50 100 2001010

1020

1030

1040

1050

1060

1070

T (keV)

ne

(m−3)

5 10 20 50 100 200T (keV)

Figure 4.8: The stability of the different eigenvectors for a burning plasma inthe PPCS A reactor using the IPB98(y,2) scaling for τE. Unstable behaviour isindicated by , signifies oscillatory unstable behaviour, oscillatory stablebehaviour and stable behaviour. The irregularities in the boundaries betweenthe different colours, for instance on the green-purple boundary in the two plotson the right is caused by a an interchange of two or more eigenvectors.Please notethat 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 botheigenvalues goes to zero, which corresponds to the Bogdanov-Takens bifurcationdescribed in section 4.2.3. Hence the fact that blue always borders red and purpleand never borders green. Similarly, purple is never adjacent to red. Note thatalthough it appears otherwise, there is actually a small slither of red between thegreen and blue areas in the bottom right plot.

The irregularities on the border between purple and green that can be seenin the two plots on the right of figure 4.8 are caused by a numerical difficultysorting the eigenvectors. When determining the eigenvectors, most algorithms willdetermine the largest eigenvalue and corresponding eigenvector first, then reducethe dimension of the matrix and repeat. If two (or more) eigenvalues happen toapproach each other closely enough, both in argument and absolute value, it canbe difficult to know which is which at the next point in parameter space. This canof course be solved by looking closely at the trajectory of the different eigenvaluesand vectors, but developing a fail safe algorithm is rather involved and doesn’tlead to new physical insight.

Although this plot allows us to investigate the stability of each eigenvectorin 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, withred the unstable and green the stable part. The system is unstable on the low, andstable on the high temperature side. The transition between the two regions lies atthe 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 highervalues of ρ and at the high density part of the contour, the transition occurs at ahigher temperature and a higher respectively lower density.

Comparing this picture to figure 4.1 the overal stability of the four- and two-dimensional systems looks very similar, lending greater credibility to the claimthat the reduced system captures the essential physics.

4.4.3 Eigenvectors and eigenvaluesApart from looking at the stability of the individual eigenvectors and of the systemas a whole, we can also try and interpret the eigenvectors of the system in a physicalsense. Although new for the ITER physics basis scalings, a similar approach wastaken by Rebhan and Vieth for the ITER89P L-mode scaling [65].

They investigated three cases: one where they assumed that τE and τp kepttheir equilibrium values during a perturbation, one where τE and τp followed theITER89P scaling law but the heating power was equal to the loss power, and onewhere they equated the heating power to Sα = nDnT〈σv〉Eα2π2κa2R, again withτE and τp following the scaling law.

We have only investigated the latter case, of course using the IPB98(y,2) scal-ing, since this is of most relevance for future reactors. Besides the fact that theITER89P scaling includes radiation losses and the IPB scalings do not, we have


also included the presence of helium in the plasma on the average ion mass, some-thing 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 forthe reduced system, at a density of ne = 1020m−3 and ρ = 5 for the PPCS Areactor 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 temperatureequilibrium for a burning plasma in the PPCS A reactor using the IPB98(y,2)scaling at a density of ne = 1020m−3 and ρ = 5. The first eigenvector also has avery weak fα dependence which doesn’t show up at this level of accuracy.

fD fT fα T ∗ λv1 0.01 0.01 0 −1.00 0.195v2 0.21 0.21 −0.03 −40.96 −0.018v3 −0.14 −0.14 0.05 0.98 −0.014v4 −0.48 0.29 0.05 0.83 −0.012

For the low temperature equilibrium all four eigenvectors have four non-zeroelements, which agrees with our finding that the dimensionality of the systemcannot be reduced without discarding physics information, as was done in ref [65].

The first eigenvector v1 is unstable and is predominantly a temperature per-turbation. This corresponds to the temperature instability that is present in thesimplest model for a burning plasma that contains only Bremsstrahlung lossesand alpha particle heating. At the low temperature equilibrium the reactivityof the plasma has a stronger temperature dependence than the radiation losses.Therefore a temperature perturbation will quickly grow until the plasma eitherextinguishes (in case of a negative perturbation) or the temperature reaches thestable equilibrium. Because a temperature perturbation also has an effect on theplasma composition, v1 also has non-zero deuterium, tritium and helium compo-nents.

The second and third eigenvectors differ from the first in that they show muchlarger deuterium, tritium and helium components. They both correspond to anin-phase deuterium and tritium density fluctuation coupled to a temperature andhelium density fluctuation.

The fourth eigenvector v4 differs from the previous three in that the deuteriumand 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 theeigenvector (nD,−nT, 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 bythe fact that we have included the ion mass dependence in the τE scaling law, whichcauses a coupling between the temperature and particle densities that cannot beremoved.

Concluding we can say that it is the thermal part that causes the linear in-stability of the low temperature equilibrium in a burning plasma. The particledominated perturbations are all stable.

4.4.5 High temperature stability

The eigenvectors at the high temperature equilibrium (T = 21.5 keV) look a littledifferent and are listed in table 4.3. The most obvious difference is the fact thatthe first two eigenvectors have acquired an imaginary part. Secondly the particlecomponents are a lot smaller (for almost the same normalisation: the particledensities for both the high and low equilibrium are normalised to ne = 1020m−3

and the temperature is normalised to the equilibrium temperature of 6.22 and 21.5keV respectively).

Table 4.3: The normalised eigenvectors and eigenvalues (λ) at the high temper-ature equilibrium for a burning plasma in PPCS model A using the IPB98(y,2)scaling at a density of ne = 1020m−3 and ρ = 5. The first eigenvector also has avery weak fα dependence which doesn’t show up at this level of accuracy.

fD(10−4) fT(10−4) fα(10−4) T ∗ λv1 2 + 68i 2 + 68i 2 − 9i 1.0000 −0.210 − 0.120iv2 2 − 68i 2 − 68i 2 +9i 1.0000 −0.210 + 0.120iv3 −1747 1642 -102 0.9708 −0.074v4 116 116 328 0.9993 −0.060

The unstable temperature dominated fluctuation at the low temperature equi-librium, is stable at the high temperature equilibrium, but has acquired an imagi-nary part which causes an oscillating behaviour. The magnitude of the imaginarypart relative to the real part of the eigenvalue increases with ρ, giving rise to higherratio of oscillation period to damping time.

The third eigenvector corresponds to the (nD,−nT, 0, 0) eigenvector found byRebhan and Vieth with eigenvalue λ3 = −1/τp, but in our case this is againcoupled to a temperature and (weak) helium fluctuation.

The fourth and final eigenvector is a combination of temperature en densityfluctuations, but it stabilises slower than the other three eigenmodes.


Because of the lack of symmetry in the Jacobian, both at the high and lowtemperature equilibria all eigenvectors are coupled density and temperature fluc-tuations. This makes a simple interpretation in terms of a pure temperature ordensity perturbations difficult, but nevertheless we can identify the underlyingmechanisms 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 pureDT plasma. The eigenvalue of this mode changes with T : for low values of T itis 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 lossesaccount for a large fraction of the total energy losses and these are not includedin τE, which is consequently much larger than the timescale of the fluctuation. Athigher temperature the importance of the radiation losses decreases, bringing theenergy confinement time and the eigenvalue of the mode closer together.

The other two eigenvectors are hybrid modes that consequently have timescalessomewhere in between −1/τE and −1/τp. Their eigenvalues also depend on theprecise location of the equilibrium.

We can plot the stability of the system in the ne, T -plane like in figures 4.8and 4.1, or look at the composition of the eigenvectors in individual equilibria, butthis doesn’t tell us how the stability is affected by changing the parameters of thesystem that are under control of the operator.

In the model that we used, the properties of the system are determined bythree parameters which, within limits, can be chosen freely: sD, sT and ρ. In ourcalculations so far we have made the assumption that sD = sT, effectively givingus two inputs that we can adjust.

The standard approach in this case would be to plot the stability boundary (orthe eigenvalues) as a function of sD and ρ-plane. However, because for each valueof ρ there are two equilibria that have the same refuelling rate and the boundaryof stability is near the maximum or minimum density on a ρ = const contour, sucha plot would not be very instructive.

Instead, we have plotted the lines of constant ρ and constant sD (or sT for thatmatter) in the stability overview for PPCS model A with the IPB98(y,2) scalingin figure 4.9. At low density the stability boundary intersects the contours ofconstant ρ at the minimum density for ρ < 9. For higher values of ρ the stabilityboundary deviates towards higher temperatures and consequently higher densityat the intersection with the ρ iso-contours. This trend continues at the high densityside of the contours, where the intersection lies below the maximum density.

The dashed lines that indicate constant fuelling rate are slightly curved down-ward and are just a bit lower at the high temperature side. The reason for this itthat in the center the value of ρ increases, which means a better particle confine-ment and consequently a less need for refuelling. At the low and high temperature

4.4 Full system 101

5 10 20 50 100 200

1070

1060

1050

1040

1030

1020

1010

T (keV)

ne

(m-3

)

Figure 4.9: Stability plot for the PPCS A reactor with IPB98(y,2) scaling, includ-ing 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.


sides the particle confinement is lower, so for a given density there is a higherparticle transport. Finally the reason for the slope in the lines lies in the factthat the fusion power increases with temperature, so the confinement time de-creases towards higher temperatures. This effect weakens when the temperatureapproaches the maximum in the reactivity around 70 keV, which can be seen inthe lines of constant refuelling rate at high density and temperature, which runalmost horizontally at high temperature.

Looking at the intersections between the lines of constant refuelling rate andthe stability boundary, it is obvious that it is only possible to cross the stabilityboundary by changing the refuelling rate while keeping ρ constant for high densitiesor high values of ρ. For ρ < 10 at low densities the stability boundary intersectsthe constant ρ contours at the point where they are tangent to the lines of constantrefuelling rate.

4.4.6 Stability for different scaling lawsThe ITER physics basis contains five different scaling laws, each based on a fit toa different subset of the database, or using slightly different fit restrictions. Theeffect of using a different scaling on the stability boundary in a burning plasma isillustrated in figure 4.10 for the PPCS A reactor design.

Again green indicates stable behaviour and red unstable. The stability bound-ary hardly changes between the different scalings, except for the region around theminimum density. There the position of the critical point, which corresponds tothe 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, wesee that only at very high values of ρ there is a difference in stability between thedifferent scalings at the high temperature equilibrium. It is interesting to notehowever, that although basically all high temperature equilibria are stable, theyall have at least two imaginary eigenvalues. A perturbation of these equilibria willresult in damped oscillations of the plasma parameters. From these plots there isno way of telling how strong the damping is compared to the oscillatory part, butin general higher values of ρ exhibit weaker damping (which makes sense becausethese equilibria are closer to the unstable equilibria at low temperature).

4.4.7 Reactor stability comparison with external heatingMost reactor designs, even though capable of ignition, still employ some level ofexternal heating, mostly for current drive and control purposes [70]. Figure 4.12shows the stability of the PPCS A, B, C and D reactor designs with their respectivelevels 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 thestability properties are somewhat different. The addition of two extra degrees offreedom has resulted in the addition of an imaginary component to the eigenvalues

4.4 Full system 103

1015

1020

1025

1030

ne

(m−3)

98(y)

1016

1020

1024

1028

1032

ne

(m−3)

98(y,1)

1010

1025

1040

1055

1070

ne

(m−3)

98(y,2)

5 10 20 50 100 200100

1025

1050

1075

10100

T (keV)

ne

(m−3)

98(y,3)

5 10 20 50 100 20010−130

10−80

10−30

1020

1070

T (keV)

ne

(m−3)

98(y,4)

Figure 4.10: The stability regions for a burning plasma in the PPCS A reactor forthe five scalings in the ITER physics basis using the full system. Please note thechange in density range between the different plots, and be aware that the densityand β limits are not taken into account.


1

2

ρ=

5

nG

β limit

ρ=

9

ρ=

13ρ

=1

ρ=

13

ne(102

0m

−3)

IPB98(y)

1

2

ne(102

0m

−3)

IPB98(y,1) IPB98(y,2)

5 10 15 20 25 30 35 40

1

2

T (keV)

ne(102

0m

−3)

IPB98(y,3)

5 10 15 20 25 30 35 40T (keV)

IPB98(y,4)

Figure 4.11: Stability for the full system burn contours of PPCS model A at dif-ferent values of ρ for the different τE scalings in the ITER physics basis. Theasymptotically stable part of the contours is denoted , whereas is asymp-totically stable with an oscillation, is unstable with an oscillation and isunstable.

4.4 Full system 105

1

2

ρ=

1

ρ=

5

ρ=

9

ρ=

13

nG

β limit

ne(102

0m

−3)

PPCS A, Pext = 246MW PPCS B, Pext = 270MW

5 10 15 20 25 30 35 40

1

2

T (keV)

ne(102

0m

−3)

PPCS C, Pext = 112MW

5 10 15 20 25 30 35 40T (keV)

PPCS D, Pext = 71MW

Figure 4.12: Stability of the full system burn contours of the PPCS A, B, C and Ddesigns [70], for ρ ranging from 1 to 14. At low density and high temperatures, mostequilibria are stable ( ). At higher densities, they are stable, but the solutions inthis region will oscillate ( ), and the PPCS D design has a region with unstableoscillating behaviour ( ). Finally, at high density and low temperatures, thereare unstable equilibria ( ).


for most of the high temperature equilibria, compared to the stability of the re-duced system. Only at lower densities do equilibria without oscillatory behaviourstill exist.

Again, the PPCS D design is the odd one out, and in this case the systemis unstable for higher densities at low temperatures, which were stable for thereduced system with external heating. Unfortunately, the anticipated operatingpoint for PPCS model D (T = 12 keV and ne = 1.4 × 1020m−3) lies within theunstable density range, although the equilibrium temperature that our simulationspredict is somewhat lower than the value from the PPCS.


We have derived a simple two-dimensional system to study the operating pointstability of a burning plasma, by assuming a constant electron density and equaldeuterium and tritium concentrations. This allowed us to use planar bifurcationtheory to describe the transitions between regions with different stability in thephase plane. Furthermore, we have analysed the physical mechanism behind thestabilising or destabilising effect of the different elements of the Jacobian matrix.

In general, a burning plasma has two equilibria at a given density, one ata lower, and one at a higher temperature. The low temperature equilibrium isalways unstable, and the stability of the high temperature equilibrium depends onthe density and ρ. For low values of ρ, the high temperature equilibrium is stablefor all but the highest densities (which are inaccessible anyway because they are farabove the Greenwald and Troyon limits). For high values of ρ the high temperatureequilibrium is stable for intermediate densities, and the stable range shrinks forincreasing values of ρ and it disappears completely when approaching ρcrit.

At the boundaries of the different stability regions, the system features differentbifurcations. In the two dimensional system these can be easily distinguished bylooking at the eigenvalues. Generally speaking, there is a local bifurcation in thesystem when either one, or both, of the eigenvalues is zero or has a real part equalto zero. These zeros occur at the boundary between different stability regions.

It turns out that the reduced system contains five bifurcations: two saddle nodebifurcations at the maximum and minimum density on a contour respectively, asub-critical Hopf bifurcation at the stable-unstable transition which results in thebirth of a limit cycle and a saddle homoclinic bifurcation where the homoclinicorbit collides with the saddle point at the unstable equilibrium and disappears. Thefinal bifurcation is a Bogdanov-Takens bifurcation which occurs at the point whereboth eigenvalues are equal to zero. This is a point where two fold bifurcations, aHopf bifurcation and a saddle-homoclinic bifurcation meet. The Hopf bifurcationcreates a limit cycle, which grows and collides with the low temperature saddlepoint in the saddle-homoclinic bifurcation.

For low values of ρ both the reduced and the full system show linearly stable


behaviour at high temperature and linearly unstable behaviour at low tempera-tures, while the density has negligible influence on the stability properties. Forintermediate temperatures, which are only accessible at higher values of ρ forintermediate densities, the system goes through a transitional phase, where theeigenvalues acquire an imaginary part which gives rise to oscillatory behaviour.

The linear stability properties of a system only provide (very) limited infor-mation about its non-linear stability. For the reduced system we made streamlineplots that show the temporal evolution of the system in the ne, T -plane which canbe used to identify safe operating regimes, where a perturbation of the system willnot grow to violate the β-limit. The Greenwald limit cannot be exceeded in thereduced system because we assume the density to be fixed.

For the four dimensional system it is not possible to make such streamlineplots. Also there is no guarantee that the orbits in phase space map out contiguousvolumes; there is a real possibility that some of the trajectories of the system passthrough a saddle point or even form interlocking loops.

While we haven’t found any trajectories starting at a stable, high temperatureequilibrium that cross the β or density limit, we cannot exclude that such orbitsexist. A perturbation of an unstable equilibrium at low temperature on the otherhand will almost certainly cross the β-limit in case the trajectory converges tothe high temperature equilibrium. In case the trajectory leads to an extinguishingof the plasma there is the risk of crossing the density limit, since the particleconfinement increases and the refuelling rate stays the same.

In the intermediate temperature range with imaginary eigenvalues there is alsothe possibility that a perturbation of a stable equilibrium leads to a trajectory thatends up in an unstable part of phase space, which can lead to either an extinguish-ing of the plasma and crossing the density limit, or an increase in temperaturewhich can lead to a violation of the β-limit.

The addition of external heating power lifts the low temperature equilibria toinaccessible densities, and has a stabilising effect on the high temperature equilibriaat high values of ρ. Only for high levels of Pext, ρ close to ρcrit and ne ≈ 2nG doesexternal 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 Dis not capable of ignition and requires external heating to maintain the requiredtemperature, the position of the operating points deviates significantly from thoseof the other three reactors. However, also for PPCS model D the operating pointsare stable.

Although the dynamics of the four-dimensional system of burn equations aremuch more complex than those of the two-dimensional system, the overal stabilitylooks very similar. The main difference being that all operating points for thePPCS A, B and C designs lie in the stable region of phase space where at leastone of the eigenvalues has an imaginary component. In most cases the imaginary


part is small compared to the real part, so possible oscillations will experience astrong damping, but nevertheless this might need to be taken into account in adetailed reactor design.

The inclusion of the ion mass dependence in our derivation breaks the sym-metry between deuterium and tritium that was present in the results presented inreference [65]. Consequently, the eigenvectors that describe a pure density and apure temperature perturbation that they identified do not exist anymore. Never-theless, we were able to identify the corresponding thermal and particle dominatedeigenvectors in the full system. The other eigenvectors are hybrid modes for whichwe have not been able to find a simple physical interpretation.

Concluding we can say that the stability properties of a two-dimensional burnsystem with constant ne and D/T-ratio are in good agreement with those of thefull four-dimensional system. Operating points in the reactor relevant density andtemperature range are mostly stable, with the exception of those of the PPCSD design. The use of different scaling laws yields only slightly different stabilityproperties at the operating points, and the addition of external heating has astabilising effect in all reactor relevant scenarios.

One point of concern that remains is the temperature evolution during thestart-up of the reactor. After heating the plasma to the ignition point, the tem-perature and helium content will evolve until they settle at their stable values.However, when starting with a pure DT plasma, the temperature can overshootthe equilibrium temperature by more than 10 keV, which will most likely violatethe β-limit.

To prevent this from happening the plasma could be started with a non-zerohelium concentration, but this would require a higher heating power. A bettersolution might be to start with a pure DT plasma, but replace (part of) thefuelling by helium injection once ignition has been reached. That way the reactorcould be started with a minimum amount of external heating while still preventingthe dangerous temperature overshoot.

109

Chapter 5

Sensitivity of burn contoursto form of scaling laws

5.1 Introduction

Unlike present day experiments, an economically viable fusion reactor cannot relyon external heating to keep the plasma at the required temperature. While somelevel of external heating might still be required, for example for non-inductivecurrent drive, the main source of power has to come from the alpha particlesproduced by the fusion reaction. This constitutes a radical change to the dynamicsof the plasma, because it creates a strong link between the density n, temperatureT and fusion power Pfus.

The common method for evaluating reactor designs is to make use of a scalinglaw for the energy confinement time τE, because a complete description of thetransport properties of the plasma is too complicated for this purpose. For lackof a better alternative, this is also the common approach for reactors capable ofignition (or in any case have predominant alpha heating). The implicit assumptionis that the plasma doesn’t know what form of heating is applied. From a transportpoint of view, this seems reasonable, because the main drive for turbulent transportis the temperature gradient which should be independent of the precise heatingmethod.

From the Lawson criterion [89] for ignition follows that to good approximationthe triple-product nτET must exceed a critical value, (which is why the tripleproduct is commonly used as a metric of progress in fusion research). In a burningplasma Pfus ∝ n2, and the commonly used scaling law IPB98(y,2) [26] predicts anenergy confinement time τE ∝ n0.41P−0.69. Combining these two proportionalities,we find that τE ∝ n−0.97, according to which the triple product for an ignitedplasma 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 mechanismssuch as the alpha slowing down time, synchrotron radiation losses or the powerthreshold for the LH-transition (all linearly dependent on density), which will put alower limit on the density. Nevertheless, these only become a factor below electrondensities ne < 1019m−3. So either ignition at such low densities is possible, or thereis 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 researchin an attempt to compare the results of different reactors and develop a basis onwhich to design new (and better) experiments. In this regard they have beenhighly successful, but care has to be taken when applying them outside the rangeof plasma parameters present in the dataset on which they are based. This isespecially important in the case of burning plasmas, as they will explore parameterregimes that are currently inaccessible.

A second problem arises from the assumption underlying all current τE scalinglaws: that the electron density ne and heating power P are independent of eachother. However, in a burning plasma density and power are strongly coupled andconsequently a scaling law should not treat them as such.

In this chapter we investigate the application of current scaling laws to burningplasmas, and show that there is a singularity in the system. We will explore theconsequences of this singularity for the predicted operating contours and concludewith some suggestions that might help resolve the issue.

5.2 Theory

In a burning plasma, the only parameters that are under direct control of theoperator are the electron density and fuel ratio. Given these two, the plasmamight find an equilibrium. The existence, stability and position in phase spacehas been the topic of several studies. Early work on burning plasmas was doneby Kolesnichenko et al. [90], and Houlberg et al. [77] were the first to introducethe plasma operation contour (POPCON) plot, that displays the operating con-tours 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 ofhelium on the burn equilibria when it is taken into account self-consistently. Reb-han et al. studied the stability of the operating points for ITER ID ([78]) withself-consistent helium treatment and the ITER89P L-mode scaling ([29] for theenergy confinement time τE (including an H-mode factor fH = 2 to mimic H-modebehaviour).

All these studies apply a scaling law for τE in their calculation of the operatingpoints (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 eliminatingτE 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 wasexpanded by Rebhan et al. in [53, 65], and largely adopt the notation introducedtherein.

The power balance in a burning plasma is

Pα = nDnTEα〈σv〉 =W

τE+ n2

eRrad = Pcond + Prad, (5.1)

where nD and nT are the deuterium and tritium density, Eα is the alpha particleenergy, 〈σv〉 the reactivity, ne the electron density, W = 3/2neftotT the internalenergy of the plasma and Rrad can be interpreted as the Bremsstrahlung reactivitymultiplied with the energy per interaction, defined by

Rrad = CBT12

[figff

(1

T

)+ 4fαgff

(4

T

)+ ZfZgff

(Z2

T

)]. (5.2)

Here fi = ni/ne = (nD + nT)/ne, fα = nα/ne, fZ = nZ/ne, T is the temperatureand 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 theorder of 10%, but does not affect the stability or the dynamics of the system.

From equation (5.1) the burn criterion can be derived:

neτE =32ftotT

14f

2i Eα〈σv〉 − Rrad

, (5.3)

with

ftot =1

ne(ne + ni + nα + nZ) = 1 + fi + fα + fZ. (5.4)

Note that ftot = 2 in case of a pure hydrogen plasma and limZ→∞ ftot = 1. Asimilar condition can be derived for the alpha particle fraction fα. Since fi =1− ZfZ − 2fα, the alpha particle balance is

1

4n2

ef2i 〈σv〉 =

nefαρτE

, (5.5)

which translates to

neτE =4fα

ρf2i 〈σv〉

, (5.6)

where we used the definition τp = ρτE. Equating (5.3) and (5.6) results in a cubicequation for fα, which can be solved to obtain fα as a function of T . The resultis 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 ofthe burn criterion and the value of neτE can now be plotted as a function of T


10 1000

0.1

0.2

0.3

0.4

ρ = 1ρ=5ρ

=9ρ=13

T (keV)

f α

10 1001014

1015

1016

1017ρ=

0

ρ = 1

ρ = 5ρ =

9ρ=13

T (keV)

neτ

E(s

/cm

3)

Figure 5.1: Helium fraction (left) and neτE plotted as a function of T for differentvalues of ρ.

to obtain so-called plasma operating point contour (POPCON) plots, as shown inthe right half of figure 5.1.

For reactor design purposes, it is desirable to plot the operating points in thene, T -plane instead of the neτE, T -plane. This requires elimination of τE from theburn criterion and for want of a good description of the transport in a tokamak thiscan only be done by means of a scaling law. Most scaling laws for the confinementtime in a tokamak take the general form

τE = KnleP−m, (5.7)

with P the external heating power delivered to the plasma and K a factor whichis obtained from several machine parameters. For the well-known ITER89P andIPB98(y,2) scaling laws K is given by

K89 = 0.048I0.85M R1.2a0.3κ0.5B0.2A0.5 (5.8)

K98 = 0.145I0.93M R1.39a0.58κ0.78B0.15A0.19. (5.9)

Here R is the major radius, a the minor radius, IM the plasma current in MA, κthe plasma elongation, B the applied toroidal field on axis and A the plasma ionmass in amu. The exponents of density and power are l = 0.1 and m = 0.5 for theITER89 scaling and l = 0.41 and m = 0.69 for the IPB98(y,2) scaling. Note thatthe average ion mass A depends on the plasma composition, and is therefore notconstant.

5.3 Results 113

There are two significant differences between the ITER89 and IPB98(y,2) scal-ing laws. Firstly, the former is based on L-mode plasmas, whereas latter is de-veloped for H-mode. However, the most important difference for the problem athand is that the IPB98(y,2) scaling law does not include radiation losses in thedefinition of the confinement time, where the ITER89 scaling does include these.This has implications for the form the power balance takes, and consequently forthe calculation of the alpha particle content and the determination of the operat-ing points. Henceforth we will use τE to denote the confinement time includingradiation losses.

In [88] we presented an analytical expression for ne as a function of T usingthe IPB98(y,2) scaling:

ne =

(fαρK

) 11−2m+l

[1

4(1− 2fα − ZfZ)

2 〈σv〉] m−1

1−2m+l

(EαV )m

1−2m+l , (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 infigure 5.2 for the PPCS A design at ρ = 5 and 10. Simply following the math hasresulted in nicely closed operating contours, but they extend to either very highor very low densities. The high density points can be discarded on the grounds ofbeing above the Greenwald density, the β-limit or both, but this is not the casefor the low density points.

Looking at equation (B.9), it is apparent that the confinement time scaleswith nl−2m

e which is n−0.97e for the IPB98(y,2) scaling law. Of course there is

some effect from the variation in helium concentration over a burn contour, butthe main trend is determined by the density. Lower densities therefore result inlonger confinement times which, in combination with the alpha concentration atintermediate temperatures, results in burn equilibria that extend to extremely lowdensities.

The obvious thing to try to put a lower limit on the accessible density is to lookat the neo-classical confinement time, because this is the absolute upper limit onconfinement in a tokamak. So if with decreasing density the value of τE predictedby the scaling laws at some point exceeds the value of τNCE , this puts a lower limiton the density of the operating contours. However, as shown in section C, τNCEalso scales linearly with density and exceeds τE for all densities.

A second thing to look at is the LH-transition power threshold: when theheating power becomes too low, the plasma will lose the H-mode confinement.


4 5 6 7 8 910 20 30 40 50 70

1015

1020

1025

1030

1035

1040

1045

ρ=5

ρ =10

nG

β-limit

T (keV)

ne

(m−3)

Figure 5.2: The operating contours for the PPCS A design [70, 71], for ρ = 5and 10. The contours extend to extremely high and low densities. The solutionmight 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 thenear-degeneration of the solution leads to virtually vertical contours, and thereobvious way to tell where the solutions are no longer valid. In other words, fromthe point of view of the scaling law, there is no good reason why burn could notbe achieved at densities of 1019m−3 or even lower.

5.3 Results 115

This threshold has a roughly linear dependence on density [75], as opposed to thealpha power which scales with density squared. Hence this puts a lower limit onthe accessible density range.

Another factor to take into account is the alpha slowing down time, whichalso depends on density. At first glance, the relevant time scale appears to beτp, but because τp is mainly determined by the edge recycling and alpha particlescannot be recycled without losing their energy, it is actually τE that matters. Inother words: the alpha particles need to transfer their energy to the plasma beforehitting the wall.

5.3.2 Density and power coupling

The more fundamental problem stems from the coupling between power en den-sity in a burning plasma. In present day fusion reactors, the heating power anddensity can be chosen independently. Consequently, the confinement database ispopulated 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 powerthe temperature cannot be influenced directly, only the density is under control ofthe operator.

When using the expression for the alpha power to eliminate the energy confine-ment time from the power balance (with the purpose of expressing ne as a functionof 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 inthe scaling law for τE.

Looking at equation (B.9), it is immediately obvious that there is a singularityat z = 1− 2m+ l = 0. At this point equation (5.10) is no longer valid, and has tobe replaced by an expression from which the density dependence has disappeared:

K

(1

4f2

i 〈σv〉EαV)−m

=4fα

ρf2i 〈σv〉

. (5.11)

In this expression, fi, fα and 〈σv〉 are all functions of T , so this is one equationfor one variable, T . It can be rearranged to

K

Ak

(4

V Eα

)−m=

4fαρAk

(f2

i 〈σv〉)m−1

, (5.12)

where the left hand side is constant (the factor A−k gets rid of the ion massdependence which was included in K) and determined by the reactor parameters.The right hand side is reactor independent and only a function of T . Dependingon the reactor properties, this equation has two, one or no solutions, becausechanging the reactor parameters changes the value of V and K, which determinesthe intersections with the closed contours described by the expression on the right


hand side. If the solutions exist, these are independent of density, meaning thatthe 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 theITER89P scaling law (see Appendix B), yielding:

ne =

(EmK

) 11−2m+l

(4fα

ρf2i 〈σv〉

) 1−m1−2m+l

, (5.13)

with E = 32ftotT2π2κa2R = 3

2ftotTV .In this case the expression at the singularity reads(

K

Em) 1

1−m

=4fα

ρf2i 〈σv〉

, (5.14)

which can also be rewritten to have only reactor dependent, constant term on theleft and reactor independent terms on the right hand side:(

K

V m

) 11−m

=

(3

2ftotT

)m(4fα

ρf2i 〈σv〉

)1−m. (5.15)

It turns out that the different scaling laws in the ITER physics basis are allclose to the singularity. As a matter of fact, the IPB98(y,4) scaling is located onthe other side of the singularity than the other four scaling laws, which resultsin 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 τEIPB98(y) 6.0IPB98(y,1) 5.9IPB98(y,2) 4.9IPB98(y,3) 5.0IPB98(y,4) 5.1

The predicted confinement times from the different τE scalings in the ITERphysics basis are very similar (see table 5.1). Yet the fact that the scaling laws areso close to this singularity means that small variations in the exponents l orm havea major impact on the operating contours. Approaching the singularity results ina stretching of the burn contours along the density axis, until at the singularitythe 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 axisis determined by the solutions to equation (5.12) or (5.15).

Because the right hand side of these equations is reactor independent, thepositions of the solutions is determined by the left hand side, i.e. the reactorparameters. For large enough reactors there will be two solutions and for reactorsthat are not capable of ignition there are none. In between there are reactors thathave a ’critical size’, where both solutions coincide and there is only one operatingtemperature.

Ergo, for a large enough reactor, a hypothetical scaling law for τE with thevalues 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, andignition could be achieved at any power and density between the LH-transitionthreshold, and the Troyon and Greenwald limits.

While the scaling laws yield similar values for the operating temperature aroundthe Greenwald density, which is to be expected because the data they are based oncontain mostly points in this region. However, the value of dne/dT varies greatlybetween the different scalings and changes sign when crossing the singularity. Thisis an issue because the required response of a control system depends on the valueof dne/dT : in the case of IPB98(y,4) an increase in density will result in a decreasein temperature on the stable burn branch, which is the opposite of what is currentlyexpected.

A further illustration of the problem can be seen in figure 5.3, which displaysthe burn contours of the ITER ID design for ρ = 3 using the IPB98(y,2) scaling law,but for two different exponents l of the electron density: the original value n0.41

e anda slight different value n0.35

e . We deliberately chose the ITER ID design becausethe IPB98(y,2) scaling law predicts that it will not ignite. However, choosing adifferent 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 minimumdensity on the contour lies above the Greenwald and Troyon limits. However, forl = 0.35 the situation looks completely different: all of a sudden the ITER IDdesign does ignite, and the maximum density at which it ignites lies well below theGreenwald and Troyon limits. And the change in predicted confinement (for thesame density and power of course) is just a few percent, depending on the exactvalue of the density.

To demonstrate the effect of variations in l or m, figure 5.4 shows the minimaldensity at which the ITER ID design ignites for the IPB98(y,2) scaling, as a

1The 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 thisdepends on the trajectory in phase space taken by the plasma.


5 10 20 50 10010−25

10−10

105

1020

1035

1050

1065

τE ∝ n 0.35e P −0.69

nG

β-limit

τE∝ n

0.41

eP−0.6

9

T (keV)

ne

(m−3)

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). Thissmall change in density dependence τE has a dramatic effect on the predicted burnequilibria.


function of the density exponent l (left plot) and power exponent m (right plot).

0.3 0.35 0.4 0.45 0.51015

1020

1025

l=

0.38

l

ne

0.6 0.65 0.7 0.75 0.8

m=

0.705

m

Figure 5.4: The maximum (red curves) and minimum (blue curves) density atignition as function of l (left plot) andm (right plot). Small changes in the values ofl andm lead to large changes in the predicted density at ignition. The red and bluecurves represent the maximum and minimum of the dashed and solid contours infigure 5.3 respectively. The left and right plot are not quite mirror images becauseof 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, thesolutions 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 wellwithin the error margins, can have such enormous effects on the operating pointsin a burning plasma.


When applying the current τE scaling laws to burning plasmas, this leads to pre-dictions for the operating points at extremely high and low densities. The highdensity operating points can be discarded because they lie above the Greenwaldand 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 slow-ing down time will put a lower limit on the density, but this only happens belowne ≈ 1019m−3. We suspected that including neoclassical transport explicitly mightsolve 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 conse-quence for real world applications, the predictions for the operating points are alsoextremely sensitive to small variations in the exponents l and m of ne and P in


the scaling law. We have used the expression for ne as a function of T in a burningplasma to show that small changes in l and m lead to big changes in the predictedminimum (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, whichmeans that the operating points have become independent of density. In otherwords, ignition is possible at arbitrarily low densities and fusion power.

Unfortunately, we have not been able to identify a physical mechanism thatcould be the reason for this problem. In a burning plasma, the density and tem-perature (and therefore the fusion power) are coupled, and one cannot be changedwithout affecting the other. In fact, al else staying constant, the operator canonly change the temperature in a burning plasma by changing the density. Thiscoupling is of course absent in the confinement database.

In recent years considerable effort has been spent on the development of twoterm scaling laws [92], combining a core and pedestal scaling. However, these stilltreat density and power as independent parameters and will therefore fundamen-tally suffer from the same problems, although the precise value of the density andpower exponents might be further removed from the singularity than is the casefor the ITER scaling laws. For a more detailed treatment of these two term scalinglaws see section D.

We have therefore, unfortunately not been able to find a solution to this prob-lem. It would be interesting to have shots in the confinement database that havedensities, 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 op-erating points if a hypothetical fusion reaction was used that delivers more energyto 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 thatwere the case might shed some light on the expected confinement scaling in burningplasmas.

A second suggestion is to mimic a burning plasma in, for instance, JET, bycoupling 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. Thischapter will provide an answer to these questions using the results presented in theprevious chapters, and subsequently discuss these answers in the broader context ofdeveloping electricity producing fusion reactors. The overarching research questionof 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 burnequilibria.

What parameters determine the position of the burn equilibria inoperating space and how sensitive is the system with respect to theseparameters? The burn criterion for a pure DT plasma is a well known result inliterature, but little work has been done on burning plasmas with a self consistenthelium treatment. Taking helium accumulation into account leads to closed burncontours in the neτE, T -plane which are completely determined by the tempera-ture [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 -planeare exactly the same for all reactors that obey the same τE scaling law, apart froma scaling factor that is a function of the engineering parameters of the reactor.Consequently, the burn equilibria in different reactors will coincide when plottedon a normalised density scale. Only the position relative to the Greenwald en βlimits will differ because these depend on the engineering parameters.

Figure 6.1 plots two such universal operating contours in the normalised densityand T -plane. Note that we cannot indicate the Greenwald density and β-limit inthis plot, since these are reactor specific. Also, the contours extend over manyorders of magnitude in the density, which is an artefact of the mathematical formof the scaling law.

122 Chapter 6 Discussion and conclusions

1 10 1001

1010

1020

1030

1040

ρ=5

ρ=10

T (keV)

norm

alis

edde

nsity

Figure 6.1: The generic operating contours for any fusion reactor that follows theIPB98(y,2) scaling for τE for ρ = 5 and 10.

How does the power output of a burning plasma respond to changesin energy confinement or particle transport?

There are several parameters that affect the position of the equilibria and thefusion power at these points. The most interesting parameters from a reactordesign perspective are the H-factor and ρ = τp/τE. The density also plays a roleof course, but since the fusion power scales quadratically with density over mostof 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 increas-ing 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) scal-ing is only 10 to 30 percent below the value at which maximum fusion power isachieved. Increasing it beyond that value will result in lower power output for agiven reactor design, although further improvements in energy confinement willallow the construction of smaller reactors.

The power output of a reactor also depends strongly on the helium accumula-tion in the plasma, which depends on both the temperature and ρ. For a givendensity, the output power on the high temperature side of a burn contour scalesroughly inversely with ρ, creating an incentive to keep ρ as low as possible. Sincethe 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 burncontrol through the particle confinement time τp.

Minimising τp might seem desirable from a power balance point of view, but itcomes 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 thisbecomes too low, the tritium has to be recycled too often, resulting in unacceptablyhigh tritium losses which cannot be replaced because there is an upper limit onthe tritium breeding ratio that can be achieved.

In our model we have assumed that the confinement time for all particle speciesis the same, which is not necessarily true. As already mentioned, a major factoris the recycling at the plasma edge and this might well be different for differentparticle species. Identifying or creating a mechanism that allows a controlled in-crease in tritium recycling (or decrease in helium recycling) would offer interestingpossibilities to increase the power production of a fusion reactor without impedingits 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 energyconfinement time of the form τE = KAknleP

m, which is the form of all scalings inthe 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 andthe temperature. This requires the assumption of a constant electron density andnD = nT. The properties of the reduced system are governed by ne and ρ, whereasfor the full system it is sD, sT and ρ.

While the system exhibits a complex stability diagram and features interest-ing transitions and bifurcations, the reactor relevant operating points are stable,except for very high values of ρ. The addition of a significant amount of externalheating, as foreseen for the reactors in the PPCS, stabilises also these operatingpoints. Only for PPCS model D does the stability of the operating point remaina concern. Using a different scaling law has a negligible effect on the stabilityproperties 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 likelya stable and robust operating point can be found for all values of ρ, where even asizeable disturbance of the equilibrium does not lead to a violation of an operationallimit 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 requiredamount of external heating, a pure DT start is desirable, but this will result in anovershoot of the temperature that could cause the plasma to exceed the Troyonlimit. This might be circumvented by injecting helium or changing the DT fuellingratio once the ignition temperature has been reached.

How sensitive are the burn equilibria to errors in the energy con-finement time scaling laws?

As is shown in chapter 5, great care has to be exercised in using the currentτE scaling laws for burning plasmas. The expression for the electron density as afunction 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 . Allfive scaling laws in the ITER physics basis happen to have their exponents veryclose to the critical values, and the singularity lies well within the error margin ofthe fit.

The singularity arises from the coupling of density and power in a burningplasma, which is not present in the τE database on which the scaling laws arebased. The physical interpretation of the singularity is a decoupling of densityand temperature, meaning that the burn contours degenerate into two verticallines in the ne, T -plane. In other words: a reactor capable of ignition would eitherburn at constant temperature, regardless of the plasma density.

Given the hypersensitive response of the operating points to changes in thescaling law, it seems prudent to investigate this issue more thoroughly and put theenergy confinement scaling laws for burning plasmas on a firmer basis.

However, the database has to contain shots where the heating power, temper-ature and density correspond to an equilibrium in a burning plasma, even thoughthe absolute value of the confinement time is too low for real burn to occur shouldthe experiment have been carried out in a DT plasma. Because the energy con-finement scaling fits the database rather well, it should also be able to describethese shots.

With a new DT-campaign in JET in the works, it might be possible to performsome simulated burn experiments, by coupling the heating power to the tempera-ture and density, possibly with a gain to compensate for the fact that JET is toosmall to achieve ignition. By performing such experiments at a range of densities, afew data points could be acquired to investigate the effect of the coupling betweendensity and heating power in burning plasmas on the τE scaling in more detail.

125

Chapter 7

Outlook andrecommendations

From this work follow a few points of attention for reactor design. Looking at thereactor and the burn conditions from an integral systems perspective, it becameclear in the course of writing this thesis that a successful fusion reactor needsto meet a number of conflicting requirements, which on their own might seemreasonable enough.

For instance, the need for a high tritium burn up fraction, which arises from theneed to keep the required tritium breeding ratio and the total tritium inventoryas low as possible, calls for a long particle confinement time. Maximum poweroutput, on the other hand, benefits from a low particle confinement time becauseit reduces the helium accumulation in the plasma. A high particle confinementtime will also result in lower stability margins for the operating point, increasingthe need for control.

Similarly, we want a high power density, because the capital costs of a fusionreactor scale with the plasma volume. This is in direct conflict with the need tominimise the heat load on the divertor.

In a commercial power plant the recirculated power must be kept as low aspossible. Yet the power plant concepts foresee significant amounts of externalheating, typically 5-10% of the fusion power. This is primarily needed to drivethe plasma current, but does help to stabilize the burn and provide the operatorwith some level of control. Still, unless radical improvements in the current driveefficiency 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 ofpercents 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 – stillmust be added.

126 Chapter 7 Outlook and recommendations

Simple models, such as the zero-dimensional one used in this thesis, are verypowerful in identifying such discrepancies, because they can easily be combinedand run for a large range of input parameters. The results can then be used totarget specific issues that are critical to the future success of fusion.

Furthermore, the detailed physics models that physicists develop to describemore detailed phenomena that occur in fusion plasmas, are often of little use forthe engineers that are tasked with developing control systems to regulate the poweroutput 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 enoughof the physics to implement reliable model-based control systems.

From that perspective, an effort to model the effects of self-heating and heliumaccumulation on reactor performance and energy confinement scaling, would bewell spent, in preparation for the ITER DT campaign. A start could be madeby developing a scaling law based on the points in the database that feature self-consistent values of heating, density and plasma composition and see whether itsuffers from the same issues as the current ITER scalings. The next step could beburning plasma simulation experiments in a JET DT-campaign, or the develop-ment of a scaling law from first principles, or maybe based on empirical relationsfrom gyro-kinetic simulations. Beyond ITER we need to address the conflicting re-quirements that arise from the different challenges that a successful fusion reactorneeds to overcome, using an integral systems perspective to find the compromisethat is optimal.

127

Appendix A

Partial derivatives for theJacobian

A.1 Reduced system

To write down an expression for the Jacobian, we need to know the partial deriva-tives of the confinement time with respect to nα and T . For the energy confinementtime we assume a power law of the form

τE = K∗AknleP−m, (A.1)

with K∗ a constant depending on machine parameters. For the heating power Pwe substitute the alpha power

Pα =1

4(ne − 2nα)2〈σv〉EαV, (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 wholesystem. Then the process is repeated for the ∂/∂T terms.

In the reduced system the electron density ne is kept constant, yielding thetrivial result

∂ne/∂nα = 0. (A.3)

The average ion mass A changes with the composition of the plasma (equa-tion (4.15)) and its derivative with respect to nα is

∂A

∂nα=

1.5ne

(ne − nα)2. (A.4)

128 Appendix A Partial derivatives for the Jacobian

For the fuel density it reads

∂nDnT

∂nα=

1

4

∂

∂nα(ne − 2nα)2 = −(ne − 2nα). (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 respectto nα can now be written down explicitly

∂τE∂nα

=K∗Aknle[

14 (ne − 2nα)2〈σv〉EαV

]m ( kA 1.5ne

(ne − nα)2+

4m

(ne − 2nα)

). (A.6)

The partial derivative of the confinement time with respect to T is less com-plicated

∂τE∂T

=K∗Aknle(

14 (ne − 2nα)2〈σv〉EαV

)m −m〈σv〉 d〈σv〉dT, (A.7)

however, this includes the derivative of 〈σv〉 with respect to T . The often takenapproximation 〈σv〉 ∝ T 2 is not valid in our case because we want to cover the com-plete temperature axis. Instead for 〈σv〉 we use the Bosch and Hale parametriza-tion [4], which is valid from 0.5 keV up to 550 keV.

〈σv〉 = C1θ

√ξ

mrc2T 3e−3ξ, (A.8)

ξ =

(B2

G

4θ

)1/3

, (A.9)

θ =T

1− T (C2+T (C4+TC6))1+T (C3+T (C5+TC7))

. (A.10)

Taking the derivative involves applying the chain rule a few times:

d〈σv〉dT

= C1θ

√ξ

mrc2T 3e−3ξ

(d ln θ

dT+

1− 6ξ

2ξ

dξ

dT− 3

2T

), (A.11)

dξ

dT=dξ

dθ

dθ

dT= − 1

3θ

(BG

4θ2

)1/3dθ

dT= − ξ

3θ

dθ

dT(A.12)

and

dθ

dT=

1 + 2C3T +(C2

3 − C2C3 + C4 + 2C5

)T 2

+2(C3C5 − C2C5 + C6 + C7)T 3

+(C2

5 − C4C5 + C3C6 − 3C2C7 + 2C3C7

)T 4

+ 2(C5 − C4)C7T5 + C7(C7 − C6)T 6

(−1 + T (C2 − C3 + T (C4 − C5 + C6T − C7T )))2. (A.13)

A.2 Full system 129

For the radiation losses the partial derivatives with respect to T and nα aregiven by

∂Srad

∂T=Srad

2T, (A.14)

∂Srad

∂nα=

5n2eRrad(T, 1)

32n

2tot

. (A.15)

A.2 Full system

The derivatives in the Jacobian with respect to T are of course (almost) the sameas for the reduced system, but the derivatives with respect to nj are different. Thederivatives of the energy confinement time now read

∂τE∂nj

=K∗Aknle

(nDnT〈σv〉EαV )m

(k

A

∂A

∂nj+

l

ne

∂ne

∂nj− m

nDnT

∂nDnT

∂nj

)(A.16)

and

∂A

∂nj=

(j − 1)nD + (j − 2)nT + (j − 3)nα + (j + 1−mZ)nZ

(nD + nT + nα + nZ)2, (A.17)

∂ne

∂nj=

(j2 − 3j + 4

2

), (A.18)

∂nDnT

∂nj=

nT j = 1;

nD j = 2;

0 j = 3.

(A.19)

The expression for the alpha heating power has changed to

Pα = nDnT〈σv〉EαV. (A.20)

Also the expression for the radiation losses needs to be modified:

Srad = ne

∑j

njRradj , (A.21)

Rradj = gffCBZ2j

√T , (A.22)

and consequently its derivatives have taken a slightly different form:

∂Srad

∂nj=Srad

ne

∂ne

∂nj+ neRradj . (A.23)

131

Appendix B

Derivation of ne as functionof T

The idea of eliminating the confinement time from the power balance by means ofa scaling law was presented and carried out for the ITER89 scaling by Rebhan et.al. [65], which yielded the following result:

τE = Knl+1e Tm. (B.1)

It is possible to take the approach presented in [65] one step further and derivean analytical expression for ne as a function of T .

Starting with the definition of τE and equating that to the scaling law:

τE =W

P=

32ftotneT

P2π2κa2R = KnleP

−m, (B.2)

which can be written as

W

P 1−m = Knle. (B.3)

Raising both sides to the power 1/1−m and using the definition of τE again

τEWm

1−m = K1

1−mnl

1−me , (B.4)

and solving for τE, the following expression is found

τE =

(K

Em) 1

1−m

nl−m1−me , (B.5)

where we have defined E = 32ftotT2π2κa2R = 3

2ftotTV , with V the plasma volume.

132 Appendix B Derivation of ne as function of T

Using expression (B.5), we can eliminate τE from the burn criterion in the casethe confinement time includes radiation losses, and be left with an expression forne(T ). Using the burn criterion derived from the helium balance, equation (2.42),and substituting the expression for τE yields

ne

[(K

Em) 1

1−m

nl−m1−me

]=

4fαρf2

i 〈σv〉. (B.6)

From this it follows that

ne =

(EmK

) 11−2m+l

(4fα

ρf2i 〈σv〉

) 1−m1−2m+l

. (B.7)

The above approach only works in case the definition of the confinement timeincludes 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 ofthe confinement time analytically. Instead, for the IPB98(y,2) scaling, it can bedone by inserting the expression for the alpha particle heating (plus any externalheating if appropriate) into the scaling law. The expression for the alpha powerreads

Pα =1

4n2

ef2i 〈σv〉Eα =

1

4n2

e (1− 2fα − ZfZ)2 〈σv〉EαV, (B.8)

with fi = (nD + nT)/ne = 1 − 2fα − ZfZ the fraction of fuel ions in the plasmaand V the plasma volume. Inserting this in the equation for the confinement timeyields

τE = Knl−2me

(1

4(1− 2fα − ZfZ)

2 〈σv〉EαV)−m

. (B.9)

Again inserting this expression into equation (2.42), we find

neτE = Kn1+le

(1

4n2

e (1− 2fα − ZfZ)2 〈σv〉EαV

)−m=

4fαρf2

i 〈σv〉. (B.10)

Some rearranging and eliminating fi from the last term, the gives the desiredexpression for ne(T )

ne =

(fαρK

) 11−2m+l

[1

4(1− 2fα − ZfZ)

2 〈σv〉] m−1

1−2m+l

(EαV )m

1−2m+l . (B.11)

133

Appendix C

Neoclassical confinementtime

Since the scaling law for the energy confinement time does not include an explicitdescription of neoclassical transport, we expected that setting a lower limit tothe confinement time equal to the value predicted by neoclassical transport wouldincrease the minimum density on a given burn contour.

A simple estimate for the neoclassical energy confinement time is [7]

τNCE ≈ a2

χNCi, (C.1)

with a the plasma minor radius and χNCi the neoclassical temperature diffusioncoefficient for ions, which is given by [93]

χNCi = 0.68q2

(R

r

)3/2

χCLi , (C.2)

with R and r the major and minor radius, χCLi = 0.10n20/B2T 1/2 [7] and q the

safety factor. Since we use a 0D-model, we evaluate χNCi at the plasma edge, sowe adopt the definition of q95 from the ITER physics basis [94]:

q95 =5a2BT

RIMf, (C.3)

with f a form factor to account for the shaping of the plasma

f =1 + κ2

(1 + 2δ2 + 1.2δ3

)2

(1.17− 0.65ε−1

)(1− ε−2)

2 . (C.4)

134 Appendix C Neoclassical confinement time

Here κ and δ are the plasma elongation and triangularity at q95 and ε = R/a isthe aspect ratio.

From equation (C.2) and the definition of χiCL we find that neoclassical trans-port scales linearly with ne, and therefore τNCE ∝ n−1

e . Coincidentally, this is veryclose to the density dependence of τE ∝ n−0.97

e in a burning plasma according tothe IPB98(y,2) scaling law (from equation(B.9) with l = 0.41 and m = 0.69). Be-cause of this, τNCE exceeds τE for every point on a burn contour and implementingneoclassical transport to be the lower limit for transport doesn’t resolve the issue.

135

Appendix D

Alternative scaling for theconfinement time

The coupling between density and power in a burning plasma makes it desirableto 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, butwould ideally be based on a deeper understanding of the transport. Given the factthat future reactors are foreseen to operate in H-mode, we will focus our attentionon these types of plasmas.

In H-mode plasmas, there is a distinct difference between the edge transportbarrier that is responsible for the pedestal and the transport in the core (be it withor without ITB). It has therefore been attempted to separate the pedestal fromthe core, by developing a two term scaling model that includes expressions for theenergy content of the pedestal and of the core plasma. Cordey et. al. [92] havepresented two different models for the thermal energy content of both the pedestaland the core (making 2 x 2 different combinations between them). The thermalconduction model, where it is assumed that the dominant loss term in the pedestalis heat conduction down the gradient, provides the following parametrisation forthe pedestal energy:

Wped = 0.000643I1.58R1.08P 0.42n−0.08B0.06κ1.81ε−2.13A0.2F 2.09q , (D.1)

where Fq ≡ q95/qcyl and qcyl = 5κa2B/RI. Even though in this model the densityand power still appear together, the dependence on the density is rather weak.

The other pedestal model presented in [92] considers the MHD stability limitsto be the limiting factor for the pressure gradient in the edge, which has thesame coefficient for the density, but exchanges power dependence for temperaturedependence:

βped = 0.000833ρ∗0.27ped ν∗−0.08A0.2F 2.29

q ε−2.56κ2.48, (D.2)

136 Appendix D Alternative scaling for the confinement time

with βped = Wped/RI2, ρ∗ped = T

1/2pav /I, ν∗ = npedR/T

2pav, Cv = 0.92 the fraction

of the total volume occupied by the pedestal (taking into account the pedestalwidth), Tpav = 2× 102Wped/CvVnped, and V the device volume.

However, dropping ν∗ from the scaling doesn’t significantly affect the resultsfor 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 hasthe benefit of removing the density dependence from the scaling altogether. Wecan therefore use the following scaling law for the pedestal pressure

βped = 0.000643ρ∗0.3ped A0.2F 2.18

q ε−2.67κ2.27. (D.3)

Defining γ = 6.43× 10−4A0.2F 2.18q Rε−2.67κ2.27 and using the definitions for ρ∗ped,

Tpav and βped this can be written as

Wped = γρ∗0.3ped I2 = γT 0.15

pav I1.7 = γ

(2× 102Wped

CvVnped

)0.15

I1.7. (D.4)

Solving for Wped yields

Wped = γ1

0.85

(2× 102

CvVnped

) 0.150.85

I2 (D.5)

and using Wped = 3npedTpedCvV as defined by Thomsen et. al. [95], this givesthe following scaling for the pedestal temperature

Tped =1

3γ

10.85

(2× 102

) 0.150.85 (CvV)

− 0.60.85 I2n−1.85

ped . (D.6)

Using this result, we can write down an expression for the energy confinement timein 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 furtherremoved from the singularity. In order to completely remove this issue from thescaling 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’thave completed without the support of colleagues, friends and family. Although itis 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 tomy promotor, Niek Lopes Cardozo. Niek, throughout my professional, and alsopersonal, struggles of the last few years, you always remained positive and sup-portive. I especially enjoyed the ’thinking out loud’ sessions we had in front of awhite board, and the many discussions on issues that were not necessarily relatedto fusion.

I also want to thank my co-promotor Roger Jaspers, who has supervised mesince I did my internship in China nine years ago and taught me many of thethings I know about fusion. Roger, you have been a constant factor throughoutthis long journey, and were always prepared to make time to help me, even thoughI had long since abandoned my original topic.

Leon Kamp, my other co-promotor, always found time for my questions andgreatly helped me during my brief foray into liquid metal flows. Leon, I greatlyappreciate your unwavering focus on the physics questions and your rigorous ap-proach for every problem.

A great thank you to all my (former) colleagues who made my time at Eind-hoven University of Technology a pleasant one: Clazien and Hélène for their helpfulsupport in all practical matters, Herman for his assistance with the liquid metalexperiment and his limitless supply of interesting facts, Hans for his encouragingenthusiasm, Maarten for sharing his extensive knowledge on plasma physics ingeneral and plasma rotation in particular, and Mark for his probing questions andunfailing ICT support.

I want to express my gratitude towards the PhD students, both at DIFFERand at the TU/e, that I had the privilege of calling my peers: Ephrem, GeertWillem, Menno, Willem, Wolf, Bram, Rianne, Pieter Willem, Matthijs, Vitor andothers, thank you for your help and the constructive discussions we had. The

148 Acknowledgements

same goes for the staff at DIFFER, or FOM Rijnhuizen as it was called whenI started: Tony, Marco, Hugo, Peter, Egbert and Dick, who helped sharpen myideas and understanding of fusion. At the TU/e my appreciation also extends toMaarten Steinbuch for coaching me when needed. Furthermore, I want to thankmy students Stefan, Laszlo, Kevin, Benjamin, Selwyn, Wouter and Peter for alltheir hard work, and all other students who shared my office for their companyand entertainment.

Then there are the people I am lucky to call my friends. Athina, Jonathan andThijs, thank you for all the professional and personal talks we shared. Gillis, thankyou for the many lunches, bike rides and holidays, and for taking the initiative tomeet 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 manythings with your energy and lust for life, and I cherish fond memories of ourshared experiences. Mirja, of all my friends you probably understand me best.Your continuous support, both close and from further away, means a lot to meand I admire your caring and selfless attitude.

Finally, I want to thank my parents Anthonie and Marleen, and my sistersArwen en Niniane for their unwavering support. Arthur, thank you for being notonly 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 hadhave 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 schoolat the Vrije School Peelland in Helmond, and completed my Waldorf educationat the Vrije School Brabant in Eindhoven and, after moving to Terhorst, at theBernard Lievegoed School in Maastricht. After obtaining my VWO diploma at theMontessori College in Maastricht, I started studying Applied Physics and AppliedMathematics at Eindhoven University of Technology, but quit the latter aftercompleting the propedeuse (first year). In 2008 I obtained my BSc in AppliedPhysics, for which I made sequential images on the breakdown of electric dischargesin different gases at different conditions in a non-planar geometry.

For my Master in Applied Physics I chose the plasma and radiation technologytrack 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 recombinationspectroscopy system and assisting in installing the neutral beam injection sys-tem. My graduation project, on the interaction between magnetosonic Whistlerwaves and runaway electrons during disruptions in tokamaks, I did at ChalmersUniversity in Gothenburg, Sweden.

Upon obtaining my master’s degree I started my PhD project at EindhovenUniversity. After initial forays into the interaction between plasma shear flow andturbulence, and free surface liquid metal MHD flows, this culminated in the thesisyou are currently reading, which focusses on equilibria in burning fusion plasmas.

fusion energy - burning questions2016/11/14 · nuclear fusion is the process in which two atomic...

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