ATHESIS

presented by

Guilhem DIF-PRADALIER

in candidacy for the degree of

Doctor of Philosophy of the Universite´ de Provence (Aix-Marseille I), France Recommended for acceptance by the Institut de Recherche sur la Fusion par confinement Magnetique´

Speciality: Plasma physics

First-principle description of collisional gyrokinetic turbulence in tokamak plasmas

Board of examiners for public PhD viva voce on October, 30th 2008:

Peter BEYER Professor Thesis advisor Jean JACQUINOT High Commissioner scientific counsellor Chair Yanick SARAZIN Physicist (HDR) CEA advisor Eric SONNENDRUCKER¨ Professor Invited member Laurent VILLARD Professor Referee Howard WILSON Professor Referee ar ne rien savoir, ce n’est rien, ne rien vouloir savoir Cnon plus, mais ne rien pouvoir savoir, savoir ne rien pouvoir savoir, voila` par ou` passe la paix dans l’ameˆ du chercheur incurieux.

Samuel Beckett, Molloy, 1951

out jadis. Jamais rien d’autre. D’essaye.´ De rate.´ TN’importe. Essayer encore. Rater encore. Rater mieux.

Samuel Beckett, Cap au pire, 1982 Contents

Acknowledgements / Remerciements i

Forewords iii

Notice: directions for use vii

1 Some introductory words on nuclear fusion, plasma physics and around 1 1.1 The basic physics of nuclear fusion ...... 2 1.1.1 What is a fusion plasma ? ...... 2 1.1.2 Basic principle: mass–energy equivalence ...... 4 1.2 When applied to human enterprises ...... 5

2 The eternal problem of representation: which model for the plasma ? 11 2.1 Browsing the models ...... 11 2.2 The kinetic model ...... 15 2.3 The fluid model ...... 18 2.3.1 Collective behaviour ...... 20 2.3.2 A parametric closure ...... 25 2.3.3 A closure based on entropy production ...... 29

3 Calculating the plasma behaviour as an initial value problem 35 3.1 A reduction of the kinetic description: the gyrokinetic model ...... 36 3.1.1 A special set of gyrokinetic equations to study plasma turbulence . . . . 36 3.1.2 Numerics and the second life of first-principle modelling ...... 45 3.2 Initial state v.s. equilibrium: the central role of the electric field ...... 49 3.2.1 Non-canonical initial distribution functions generate poloidal sheared flows 52 3.2.2 Canonical equilibrium in GYSELA ...... 54 3.2.3 Early self-organising dynamics due to initial conditions ...... 56 3.2.4 Saturation processes for the poloidal electric field ...... 59 3.2.5 Nonlinear evolution: impact on transport in a steady state of turbulence . 61 3.2.6 Discussion and conclusion ...... 64

4 Adding binary interactions to collective behaviour 67 4.1 Towards a kinetic “neoclassical” theory of collisions ...... 68 4.1.1 General requirements for a kinetic theory of collisions ...... 68 4.1.2 Some essential models of collisions ...... 72 4.2 Incorporating neoclassical theory into gyrokinetic modeling ...... 77 4.2.1 Main neoclassical results ...... 77 4.2.2 A minimal collision operator containing conventional neoclassical theory 81 4.2.3 Discussion: which physics can we address ? ...... 82 4.2.4 Recovering the neoclassical results ...... 84 4.3 Gyrokinetic turbulence calculated with neoclassical effects ...... 91 4.3.1 Collisional enhancement of turbulent transport –a cartography in (R/LT , ν?) space ...... 92 4.3.2 Evidence of turbulent generation of poloidal momentum ...... 96

Conclusion 99

A Important features about Fourier–Laplace transforms 103

B Establishing the dispersion function 105

C The analytic continuation issue 107

D Establishing the Plemelj formulae 111

E Landau damping: the historical derivation 113

F Closures in real space 115

G Establishing the inverse Fourier Transform of sgn(k) 117

H The numerical impact of non locality 119

I Marginal stability of the drift kinetic instability 121

J Recovering the particle transverse fluid current 123

K Short-time dynamics governed by the curvature and grad-B drifts 125

L Analysis of the departure between the analytical prediction and simulations 127

M Onset of GAMs with qualitative agreement on the frequency 129

N Parallel flows, a robust saturation mechanism for the charge polarisation 131

O Numerical implementation of the collision operator in GYSELA 133

Bibliography 135

Index 140 Acknowledgements / Remerciements

UESOIENTICICHALHEUREUSEMENTREMERCIES´ ... Q . . . tout d’abord les membres du jury pour la disponibilite´ –elle rime ici avec amabilite,´ cel´ erit´ e´ et travail durant des conges–´ dont ils ont fait preuve : je pense a` Peter Beyer mon directeur de these,` jury le jour prec´ edant,´ jury encore quelques jours apres` ; Jean Jacquinot pour avoir accepte´ de presider´ la soutenance, malgre´ un emploi du temps charge´ et pour son appreciation´ declar´ ee´ du manuscrit ; Eric Sonnendrucker¨ pour etreˆ venu de loin ; mes rapporteurs pour leur relecture atten- tive, sous forte contrainte temporelle de ce memeˆ manuscrit : Laurent Villard pour son amabilite´ et un aller, puis retour immediat´ vers Lausanne ou` l’enseignement attendait ; Howard Wilson pour sa cravate et le rapprochement franco-anglais auquel il a consenti par sa venue appreci´ ee´ ; . . . l’Universite´ de Provence, en la personne d’Anne-Marie Bajetto pour un “carton-plein” de derogations´ accordees´ ; . . . le DRFC–IRFM, notamment en les personnes d’Alain Becoulet´ et Tuong Hoang, pour m’avoir accueilli ces trois annees´ de these,` quatre mois de stage, alors ignorant de plasma, pour leur sou- tien, leurs encouragements et leur cordialite,´ en entretien ou plus fortuite ; . . . honneur aux dames et a` Virginie (Grandgirard), si Precieuse´ pour Platine et Jade –pour ses nuits parfois, weeks-end souvent avec effraction chez des amis pour glanerˆ un WIFI, surtout vers La Fin : “jobs non chaines´ : Virginie re-soumet !”, pour ses gateaux,ˆ son sourire...... ET pour Glaciere` TM ; Chantal (Passeron) pour une grande aide informatique, malgre´ un oeil inquiet a` toutes mes, entrees´ ; Gloria (Falchetto) pour ses visites souriantes et son tiramisu –superbe ; Ma- rina (Becoulet),´ Emmanuelle (Tzitrone), Remi´ (Dumont), Maurizio (Ottaviani) pour avoir egay´ e´ le cafe´ ; Guido (Huysmans) pour sa patience ”ehhh, puis-je t’emprunter les noeuds 16 a` 32 de Norma ?” et ses indications vestimentaires pour le jury. Enfin, des remerciements particuliers a` Phlippe (Ghendrih) pour sa cravate le Jour J, son oreille attentive et ses tres` appreci´ ees´ discussions “mainstream”, et surtout autres ; George–Lazenby–Xavier (Garbet) pour la lueur d’inter´ etˆ gratifi- ante du 33e transparent ce memeˆ Jour J, parce qu’il a souvent des idees,´ qu’elles sont souvent tres` bonnes et qu’il les dispense avec classe ; Yanick enfin (Sarazin) qui comprend mon attachement au canard et aux ballons pas ronds, parce qu’il court vraiment tres` bien, memeˆ en montagne, memeˆ a` Varenna, mais surtout parce que j’ai beaucoup appris et qu’il a beaucoup donne.´ . . Pardon, pour le costard aussi, encore ce memeˆ Jour J ; . . . les thesard(e)s´ / post-doc cotoyˆ es´ ces trois annees.´ Encore honneur a` la dame : Chris- tine (Nguyen), l’“inspiratrice de mes jour(nee)s”´ , toute “patrie–science–et–gloire”, elle com- prendra ; Roland (Belessa), un autre “patrie–science–et–gloire”,, plus axe´ sur le “parfois brutal, toujours loyal” –j’espere` que tu es bien la` ou` tu es maintenant ; Stan’ (Pamela)–le Big Ben de 16h15, moi qui suis souvent / parfois / si peu en retard– pour son sourire, sa fraˆıcheur et pour ses cours de phonetique´ : Evaˆaˆaˆaˆ Pamelaˆaˆaˆaˆaˆ ; Emiel, Guillaume, Guigui, Manu, Paolo, Guido “der´ e-d´ er´ e”,´ Fred´ eric,´ Ale’, Alessandro, Sophie, Ozg¨ ur¨ ; des mentions speciales´ : a` mon ancien et regrette´ collegue` de bureau Nico’ (Dubuit) –non Christine, tu n’es pas visee–´ maˆıtre es-Linux,` docteur es-plasma` et l’air patibulaire. . . “mais presque” –quelqu’un qui fait aimer le Nord ; aux patouillades (Tamain) de l’adjudant, tres` appreci´ ees,´ trop breves,` memeˆ s’il ne fallait vraiment pas que l’on parle politique ; a` un grand footballeur (Nardon) ; a` un bon ami, longtemps un voisin, un–mec–bien–(Gerbaud) qui vit maintenant entoure´ d’eau, en milieu hostile (pour les papilles du moins), froid et blafardˆ –a` tres` bientotˆ au soleil de Californie ; et enfin a` mon coach-surfer pref´ er´ e,´ l’ami Steve (Jaeger), pour son dehanch´ e´ inimitable, ses rouflaquettes et ton intelligence –encore quelqu’un qui ferait aimer le Nord, et qui aime le Sud ; . . . Aix-en-Provence maintenant, charmante cite.´ Charmante par ses habitant(e)s et surtout ceux-la` : Nico, Hel´ ene,` Roy, Ben´ e,´ JC, Malika et Herve´ (Fontan) pour sa ”grande gu..le”, ses qualites´ de DJ et de tolerance´ , un autre Herve´ (Roche) pour sa sensibilite,´ son Mac, ses vins et surtout pour son amitie,´ Melissou,´ , Axel (Torre) “alors-la` mec je t’arreteˆ tout de suite” –prends soin de Manu, Seeeergiooooooo (Chiarenza) pour son tact dans la discussion en public, pour les ’xxx–Watts’, les ’xxx-Points’, pour ses anniversaires, parce qu’il a un coeur gros comme c¸a –c’est normal, c’est un sportif ! mais surtout pour Elsa ; Manu (Rouge),´ pour les “bocs-bocs”, pour tout ce que je ne peux pas ecrire´ ici, et bien bien bien plus encore –on se reverra l’ami, et enfin Elsa pour l’“avenir...a` venir” ? . . . mon lecteur –qui ne lira pas les 100 pages et quelques suivantes– et qui est dej´ a` fatigue,´ je le sens. Mais je ne peux oublier ma famille, son soutien, mes absences (prolongees),´ sa tolerance,´ l’Exemple vivant de ma mere,` de mon pere,` de ma soeur en devenir et de mon frere.` . . parce qu’il est mon Grand frere.` . . et que je compatis aux moments semblables de redaction´ que tu vis : courage.

ESREGRETS... D . . . pour la journee´ de RTT de Jean Jacquinot ce Jeudi 30 Octobre, pour n’avoir pas pu me servir de Glaciere` TM pour emmener outre Atlantique des produits civilises´ (foie gras, magret, aile, manchon, gesier,´ cou farc¸i –en un mot: canard, petite´ , aligot et vin) et enfin parce que trois annees,´ entoure´ de si bonnes gens, c’est bien court. Forewords

c An investment in knowledge always pays the best interest. d Benjamin Franklin (Richard Saunders), Poor Richard, 1732.

O PROFITABLY PRODUCE electric energy from thermonuclear fusion reactions, one must confine i.e. sustain far for thermodynamic equilibrium and over long periods of time a Trarefied reactive medium, house of an unrelenting turbulent motion. At thermonuclear tem- peratures, the ambient thermal energy is by orders of magnitude larger than the ionisation energy. The elemental charged constituents of this medium may be considered as separate; the medium itself is termed a plasma. Its confinement is allowed by strong magnetic fields, exerting a contain- ment pressure over it. These fields generate an intricate immaterial magnetic topology of which the plasma particles are aware. As charged constituents, they indeed flow along the magnetic field lines, which, while winding up in space, generate three-dimensional containment surfaces. In a first approach, should these magnetic surfaces remain closed and nested, particles would not es- cape the configuration otherwise than by colliding with one another. Also, collisions alone would allow for a heat conduction across these surfaces, leading to a net energetic loss for the plasma. A key concept to quantify the achievable quality of this confinement is the so-called energy confinement time τE, ratio of the energetic content of the plasma over the power losses. In a tokamak, these losses essentially occur by conduction across the confinement surfaces. The con- 2 finement time may therefore be understood as a diffusion time a /χT , where a is a typical size of the plasma and χT its thermal diffusivity. Experimental measurements in the past thirty years of actual heat fluxes in magnetically confined devices have led to the following situation: the measured diffusivities for the ions are at least one order of magnitude higher as compared to the collisional expectations, and two orders of magnitude higher for the electrons. Tradition found a name for the situation: these transport phenomena are anomalous. The blame for this detrimental situation is most certainly to be put on the small-scale turbu- lence which originates from the various small-scale plasma instabilities. These instabilities feed on the density, temperature, velocity or magnetic field gradients transverse to the confinement sur- faces. These gradients cannot but exist and stem from the very idea of magnetically confining over long periods of time a far for equilibrium reactive medium. A key question to be answered for present devices and predicted for future reactors is therefore the decay or sustenance of plasma density and energy content. Extrapolation techniques may be based on semi-empirical scaling laws; the idea being (i) to squeeze into a limited set of dimensionless parameters a hole range of the supposedly relevant physical phenomena. Then (ii) write critical quantities, such as the energy confinement time as a function of these parameters. This function is chosen such that it minimises the dispersion of experimental data gathered from existing machines. For instance, according to present interna- −2.83 −0.50 −0.10 0.97 −2.56 −0.55 −2.72 tional databases, τE scales as follows: τE ∝ ρ? β ν? M q  κ . The predicted confinement time for Iter should be around 6.0s. This law is represented in Fig.1. Here, ρ? is the normalised gyroradius and provides information on the required size of the device. In- formation on the achievable pressure is provided by the β parameter, ratio of the plasma kinetic iv Forewords

Figure 1: Thermal energy confinement time for ELMy data in the ITER H–mode database.

pressure over the magnetic pressure –thus smaller than unity. The collisionality of the medium is contained in the ν? parameter, a normalised frequency of collisions. Also, M provides informa- tion on the ion mass and thus on the dilution of the plasma, q on the ‘winding-up’ of the magnetic field lines,  on the compactness of the device and κ on the elongation of the plasma. A trend takes shape (iii) which allows one to foresee behaviours in ranges of parameters which actually do not exist –see the red Iter dot in Fig.1. Such laws are interesting as trends, are interesting for engineers and for design purposes. Genuine complexity (and beauty) of the underlying physics is not addressed though. Ab initio theories, while describing turbulence (ρ?), magnetohydrodynam- ics (β) or binary interactions (ν?) from first principles, may provide a comprehensive knowledge of the physics at play. They alone would validate the scaling law approach in the relevant foreseen ranges of parameters and e.g. quantitatively seek for enhanced confinement regimes. Above all, they do seek comprehension. The problem then arises of how to describe the plasma, this medium composed of a formidable number of interacting charged particles, across which large-scale electric and magnetic fields flow and characterised by unrelenting self-organising motions. Given a distribution of charges and currents, the dynamics of the electromagnetic field is well described by the Maxwell theory. On the other hand, as the fields fluctuate, they back react on the density of charges and currents within the plasma which then adjusts to the perturbation and further contributes to modify the fields. A self-consistent description of the dynamics of the plasma thus involves to conjointly calculate the Maxwell equations and the response of the plasma. Essentially, while only adopting distinct angles, two frameworks are widely considered so as to calculate this latter response. They rest on the idea that the macroscopic properties of the plasma may be inferred from the microscopic description of its elementary entities and proceed from one another by further projections. In that spirit, the considerable number of degrees of freedom of the system –associated with the very large number of particles– contracts into the statistical concept of distribution function, which in turn may be projected on a few thermodynamic macroscopic quantities such as temperature, density, pressure, heat flux or entropy. The temporal dynamics of the distribution function is described in the framework of the kinetic theory whereas the fluid theory focuses on the temporal evolution of these latter macroscopic quantities. These may all be deduced from projections of the distribution function in velocity space. Practically, a fluid model consists in an open set of equations for these thermodynamic quantities to which is adjoined an Forewords v auxiliary equation –the closure relation, which may be freely chosen such that it closes the latter set of equations and makes it self-consistent. The velocity information contained in a six-dimentional kinetic phase space is smoothed out in fluid models, though it may partly re-enter the model through an adequate ad hoc choice of the closure relation. As such, fluid models have special trouble self-consistently describing, e.g. weakly collisional systems, wave–particle collective resonant effects, local trapping phenomena, or finite orbit width effects. Nonetheless, it is worth trying to refine them by advanced closure choices or multi-fluid descriptions –though doing so one may easily reach a somehow inordinate level of complexity– since they are numerically less demanding and at present time they alone are able to simulate the plasma up to its confinement time. As a hot rarefied medium, a tokamak plasma is crucially affected by wave–particle resonances, trapping phenomena, orbit effects and is intrinsically low collisional. As such, any fully self- consisent description of its response is therefore kinetic, and includes collisions. Though they lead to usually smaller transport than the one generated by the turbulence, one may not conclude on collisions being negligible. The discovery of high-confinement regimes –which seem universal and reproducible features of fusion devices and as such are the foreseen operating scenari for Iter– has led to a renewed interest in collisional theories. These regimes are indeed characterised by a local strong reduction of the turbulent activity; in that case, the transport processes may get very close to being collisional. Binary interactions are also important features for a kinetic theory since they alone may provide a correct description of equilibrium flows. Such large-scale sheared flows are of special relevance for the saturation mechanisms of the turbulence, potentially leading to high-confinement regimes. At last, there seems to exist a non trivial interaction between collisions and turbulence in tokamak-relevant operating regimes. Through these processes, collisions might indirectly have a strong influence on the actual level of transport; elucidating this interplay is a subject of great current focus for first-principle modelling.

∴

This dissertation starts in chapter 1 with a comprehensive introduction to nuclear fusion, its basic physics, goals and means. It especially defines the concept of a fusion plasma and some of its essential physical properties. The following chapter 2 discusses some fundamental con- cepts of statistical physics. It introduces the kinetic and the fluid frameworks, compares them and highlights their respective strengths and limitations. The end of the chapter is dedicated to the fluid theory. It presents two new sets of closure relations for fluid equations which retain impor- tant pieces of physics, relevant in the weakly collisional tokamak regimes: collective resonances which lead to Landau damping and entropy production. Nonetheless, since the evolution of the turbulence is intrinsically nonlinear and deeply influenced by velocity space effects, a kinetic col- lisional description is most relevant. First focussing on the kinetic aspect, chapter 3 introduces the so-called gyrokinetic framework along with the numerical solver –the GYSELA code– which will be used troughout this dissertation. Very generically, code solving is an initial value problem. The impact on turbulent nonlinear evolution of out of equilibrium initial conditions is discussed while studying transient flows, self-organising dynamics and memory effects due to initial conditions. This dissertation introduces an operational definition, now of routine use in the GYSELA code, for the initial state and concludes on the special importance of the accurate calculation of the ra- dial electric field. The GYSELA framework is further extended in chapter 4 to describe Coulomb collisions. The implementation of a collision operator acting on the full distribution function is presented. Its successful confrontation to collisional theory (neoclassical theory) is also shown. GYSELA is now part of the few gyrokinetic codes which can self-consistently address the inter- play between turbulence and collisions. While investigating this interaction, new results on the generation of poloidal momentum are presented.

Directions for use:

Throughout the writing of this dissertation, my constant concern was to try to put into perspective practical aspects of the defence of a thesis with wider problems in physics that epitomise the elusive quality physicists would call beauty. Not pretending to epistemology, the openings of the different chapters start with general discussions; technicality is momentarily swept aside and bridges with more fundamental concepts are tried to be unveiled. As the reader though goes deeper into each chapter, he might notice the intricate links which exist between the ‘ethereal’ ideas and the more practical aspects of a day by day work. If this happens to be, my attempt was successful. The reader on the jump may at first reading skip the paragraphs in smaller fonts and concentrate on the ends of chapters. This manuscript was also written so as to be a working document. As such, I made a wide use of cross-references, appendices and index entries. At last, the reader is not required to be familiar with plasma physics, statistical mechanics or any of the epistemological disputes briefly mentioned throughout this manuscript to be able to read it; I have endeavoured to make this dissertation as self-contained as I could.

Chapter 1

Some introductory words on nuclear fusion, plasma physics and around

c Just a few months before, the Korean war had broken out, and for the first time I saw direct confrontation with the communists. It was too disturbing. The cold war looked as if it were about to get hot. I knew then I had to reverse my earlier position. If I didn’t work on the [H] bomb, somebody else would – and I had thought if I were around Los Alamos I might still be a force for disarmament. So I agreed to join in developing the H-bomb. It seemed quite logical. But sometimes I wish I were more consistent an idealist. d H. Bethe, reported by S. S. Schweber, In the Shadow of the Bomb: Bethe, Oppenheimer, and the Moral Responsibility of the Scientist, 2000.

c OW are we constituted as subjects of our own knowledge ? How are we constituted as subjects who exercise or submit to power relations ? How are we constituted as moral Hsubjects of our own actions ?” This is Foucault’s translation of Kant’s famous threefold question: “What can I know ? What should I will ? and, What may I reasonably hope for ?” The central question is the moral one, which makes Foucault add that Les Lumieres` or the Kantian Aufklarung¨ have to be considered not “as a theory, a doctrine, nor even as a permanent body of knowledge which is accumulating; it has to be conceived as an attitude, an ethos, a philosophical life in which the critique of what we are is at one and the same time the historical analysis of the limits that are imposed on us and an experiment with the possibility of going beyond them.” As a legacy from Les Lumieres` or from Aufklarung¨ , knowledge is now highly determinant of political relations and social life; as a legacy from recent history, modern physics –and especially physics related to massive energy production– is a cornerstone of such knowledge. With quantum theory, necessity intrinsically arose to criticise the knowledge being built, the mere presence of the physicist, or of its measuring apparatus, fundamentally influencing the result of the measurement. With nuclear physics came mighty power, and politics. This dissertation does not forget that it is part of a half-century long fostered research effort to understand, master and mass produce a reliable source of energy for peaceful purposes. As such, this research is prominently political and the researcher fundamentally in the world or, following Merlot-Ponty’s word, fundamentally “at the world”. Which is not, to a young mind, one of its less beauties. The present chapter is written for the non specialist and gives some comprehensive basic definitions and general ideas for a better comprehension of the whole dissertation. It especially provides the reader with the basic physics of nuclear fusion, as it is thought to happen in Nature 2 Some introductory words on nuclear fusion, plasma physics and around and by which means it is investigated on Earth.

1.1 The basic physics of nuclear fusion

S a basic physics concept, nuclear fusion essentially requires to beat the Coulomb repulsive Aforce between atomic particles so that the attractive, strong nuclear force can bind these into a new nucleus. Risking a definition:

Nuclear fusion (or simply fusion) denotes the elementary process, accompanied by the release or absorption of energy, by which multiple atomic particles join together to form a heavier nucleus. If the energy to initiate the reaction comes from accelerating one of these particles, the process is called beam–target fusion; if both nuclei are accelerated, it is beam– beam fusion. If the nuclei are part of a plasma sustained out of thermal equilibrium, one speaks of thermonuclear fusion, which represents today’s most promising approach for a controlled source of energy production on Earth.

Throughout this dissertation, fusion is only considered in this latter way and investigated through the lens of plasma physics. This field bubbles with a tremendous variety of phenomena (turbu- lence, intermittency, bifurcation, chaos) due to the intrinsic nature of the reactive medium –the plasma. Complexity further increases since these phenomena are also highly dependent on e.g. (i) the atomic composition of the plasma –deuterium, tritium, alpha particles, other– (ii) its geome- try, (iii) boundary conditions, (iv) external sources, etc. Inversely to self-sustained fission-like chain reactions, the essential physical problem with fusion consists in finding the means to sustain through time a far from equilibrium unstable turbulent plasma, i.e. tame its more violent insta- bilities, understand / control the transport processes which underlay its dynamics, and ultimately predict its statistical behaviour –essentially: predict its transport processes. Before proceeding, let us discuss the central concept of fusion plasmas.

1.1.1 What is a fusion plasma ? The fundamental forces in Nature tend to organise matter into structures, which are perennial as long as their binding energies are large as compared to ambient thermal energy. If this is not the case, these structures would ultimately decompose into their elementary charged constituents. By no means though, are these constituents free; they keep being fundamentally affected by each others’ electromagnetic fields, and at a lesser level by binary interactions between each others. Collective motions of great strength and complexity are intrinsic to such an ensemble, predomi- nantly characterised by the excitation of a very broad variety of collective dynamical modes. Such a medium develops an intrinsic magnetism, which is a mere consequence of the un- relenting motion of its interacting charges. In turn, magnetism creates anisotropy, meaning that the properties of this ensemble in the direction parallel to the magnetic field are different from those in the transverse direction. This anisotropy contributes to the multi-scale dynamics of such a medium, i.e. involves a broad range of spatially and temporally correlated different scales: (i) from the fast electron density plasma oscillations at ωp or the fast Larmor gyromotion ωc along a magnetic field line, to the intermediate bounce frequencies ωb of particles trapped in local inho- mogeneities of the magnetic field, collisional frequencies ν, and down to the inverse of the energy −1 confinement time τE which characterises the rate at which a system loses energy to its environ- ment. Concerning (ii) the spatial scales, they range from the nuclear level of the strong force λSF or the tunneling length λtunn. under the Coulomb potential barrier which allow fusion to occur up to the system’s own ones and to the large mean free path λmfp characteristic of weakly collisional media. An important intermediate scale λD for neutral plasmas is named after its first discoverer Debye and defines the typical length beyond which the medium is almost electrically neutral and 1.1 The basic physics of nuclear fusion 3

−1 ωp ωc,e ωc,i ωb,e ωturb ωb,i νii νei τE νee

5 1011 109 107 106 105 105 10 1 1 0.2 (s−1)

λSF λtunn. λD ρc,e δb,e ρc,i δb,i a R λmfp

10−15 10−12 10−4 10−4 5 10−4 10−3 5 10−3 1 3 103 (m)

the former collective effects dominant due to large-scale long-range electromagnetic interactions. This introduces the important property of quasi-neutrality, which is the macroscopic manifesta- tion of the existence of an electric sheath; at and above this Debye length, electric fields are low –explaining why λD is also often referred to as a ‘screening’ length– and the medium is electri- cally neutral, yet polarisable. Inversely, at shorter lengths1, back-pulling strong electric fields and binary collisional interactions are the rule. A further physical interpretation is deeply connected to collision phenomena and shall be discussed in much detail in the last chapter 4, p.72.

Such an ensemble is termed a magnetised thermonuclear plasma, or simply a plasma in the following of this dissertation. Though such a definition can be much refined, it has the practical benefit to provide the reader with a list of the essential characteristics I shall refer to in the remainder of this manuscript. A more general definition for a plasma is quite loose and difficult since it increasingly applies to media which increasingly deviate from the concept of gas (see Table 1.1), finally ending up formalising a distinct state of matter, of which a thermonuclear plasma can be seen as a fulfilment.

Table 1.1: Intrinsic complexity of the thermonuclear plasma state as compared to gas.

Property Gas Plasma Electrical Very low Usually very high (higher than copper at ther- Conductivity monuclear temperatures ∼ T 3/2) Independently One; all gas particles behave Two or three: electrons, ions, and neutrals; dis- acting species in a similar way, influenced by tinguished by their charge, they can often behave gravity and binary collisions independently, allowing for new instabilities Velocity Maxwellian (far from the Often non-Maxwellian; collisional interactions distribution boundaries) since collisions are weak, external forcing can drive the plasma are dominant; few hot tails of far from local equilibrium, with a significant pop- fast particles ulation of fast particles Interactions Binary; three-body collisions Binary: within a sphere of Debye; collective out- are extremely rare side with long-range interactions due to electric and magnetic forces.

This somewhat highly versatile state of matter seems to be widely prevalent throughout the universe which has the double merit to be extremely flattering to plasma physics, and quite im- possible to disprove (or verify). To our present knowledge, matter is always in the plasma state

1The phase space volume encompassed within this so-called Debye sphere still contains a sufficiently large number of particles to allow for a statistical approach. This result gives a further idea of the broad variety of scales involved. 4 Some introductory words on nuclear fusion, plasma physics and around whenever fusion is achieved. Following through the initial goal of this section to provide the reader with a comprehensive physical background, let us now discuss the basic physics of nuclear fusion.

1.1.2 Basic principle: mass–energy equivalence Quantum chromodynamics describe the way the strong interaction (also called strong force or colour force) holds quarks and gluons together to form nucleons (protons and neutrons). Inversely, the nuclear force2 (also called nucleon–nucleon interaction or residual strong force) refers to the force between two or more nucleons and is responsible for binding of protons and neutrons into atomic nuclei.

Nuclear binding energy is derived from the nuclear force and is the energy required to disassemble a nucleus into free unbound neutrons and protons, so that the relative distances of the particles from each other are infinite (or in practise far enough to postulate a scale separation) so that the nuclear force can no longer cause the particles to interact.

Given the famous mass–energy equivalence: E = mc2, E being the energy, m the rest mass and c the celerity of light, and since a bound system is at a lower energy level than its unbound con- stituents, its mass is less than the total mass of its unbound constituents. A common misconception of this relation would lead to the disturbing idea that mass could ‘disappear’ in an isolated system, which is false. Binding energy mass transforms into heat, and as such, the system needs to be thermodynamically ‘opened’ –cooled in other words– for the so-called ‘deficit’ mass to appear. This mass difference after binding may range from a small fraction of the total mass for systems with low binding energies –essentially the lightweight elements of the periodic table– to an easily measurable fraction for systems with high binding energies. Iron and nickel nuclei have the largest binding energies per nucleon of all nuclei. These results are summarised in Fig. 1.1. To illustrate this idea, let us recall that the atomic mass unit –1.000000 u– is defined to be 1/12 of the mass of a 12C atom. The atomic mass of an hydrogen atom is 1.007825 u; each nucleon in 12C has thus lost, on average, about 0.8% of its mass in the form of binding energy. From the latter discussion readily stems the following rule:

Nuclear transformations (fusion and fission) which tend to produce high binding energy nuclei –middle weight nuclei close to 56F e– are generally exoergic reactions.

Important for human energy production means, this principle also plays a central role in astro- physics, through the so-called process of nucleosynthesis, first intuited in the late 200s by Atkinson and Houtermans: Nucleosynthesis is the process of creating new atomic nuclei from preexisting nucleons. The very first process of primordial nucleosynthesis –nucleogenesis– is believed to have occurred during the quark-gluon plasma of the Big Bang. All subsequent nucleosynthesis of nuclei heavier than lithium or beryllium –including all carbon, oxygen, . . . – is believed to occur primarily in stars, either by nuclear fusion or fission. Without considering advanced capture processes of nucleons –e.g. S-process, R-process for neu- tron capture; Rp-process for rapid proton capture– or photodisintegration of existing nuclei, nucle- osynthesis could have hardly produced the vast observed quantities of nuclei heavier than nickel or iron, even in massive stars. The basis for these theories was coined by Hoyle in 1946. Be- fore him, Bethe (1936–1938) and von Weizsacker¨ (1938) successively pictured nuclear fusion

2It was thought before the advent of QCD that the nuclear force was a fundamental force directly acting on the nucleons. The verification of the quark model led to consider this phenomenon as a residual side-effect of the truly fundamental strong interaction acting directly on quarks and gluons inside the nucleons. 1.2 When applied to human enterprises 5

Figure 1.1: Average binding energy per nucleon (in MeV) against the number of nucleons in nucleus for abundant isotopes. A few important ones for the purposes of nuclear fusion and nuclear fission are marked (see Fig. 1.3), as well as iron 56F e, which sits at the top of this graph and cannot yield energy from either fusion or fission.

processes which could account for the heat, luminosity and gravitational balance of most main- sequence stars, including the sun. These latter processes, known as chains or cycles, are only able to create elements up to iron and nickel. Known as the ‘proton–proton’ chain (Bethe) or the CNO cycle (Bethe, von Weizsacker),¨ they are often refered to as hydrogen burning (see Fig. 1.2). They are complemented with a successive set of reactions involving heavier elements, namely helium burning, carbon burning, neon burning, oxygen burning and silicon burning.

1.2 When applied to human enterprises

c Quand maˆıtrisera-t-on la fusion nucleaire,´ en laboratoire ? En 1979, apres` dej´ a` 30 ans de recherche, l’ignition n’avait et´ e´ obtenue que dans la bombe a` hydrogene;` toutefois j’ecrivis´ dans Pour la Sci- ence que la fusion pourrait ˆetre obtenue en laboratoire en moins de dix ans et que la construction de centrales electriques´ a` fusion suiv- rait de peu. [. . . ] Vingt ans ont passe,´ les physiciens courent toujours apres` le Graal de la fusion, et l’ignition, pensent-ils, est encore pour dans dix ans. [. . . ] Je crois que la fusion thermonucleaire´ [l’ignition] sera declench´ ee´ en laboratoire dans moins de dix ans. d Gerold Yonas, Pour la Science, no252, 1998.

Artificial fusion on Earth cannot mimic celestial thermonuclear reactions since the critical mass for gravitational-induced fusion is approximately 1/10th of the sun’s mass, roughly explain- ing why lighter celestial bodies are not stars. A central problem in this human attempt rests on the question of confinement, since gravitation is irrelevant in the laboratory. 6 Some introductory words on nuclear fusion, plasma physics and around

Figure 1.2: The Bethe ‘proton–proton’ chain (a) and the Bethe–von Weizsacker¨ CNO cycle (b) (sometimes also referred to as the ‘carbon cycle’). The former is believed to be the main reaction chain in most mean-sequence stars lighter or comparable to the sun; the latter is thought to be dominant in stars more massive than the sun.

Confinement stands for the means by which a thermonuclear plasma is constrained into a chosen volume of phase space to allow, at times, the Coulomb repulsive force to be overcome and fusion processes to occur. It encompasses both a control of the loss channels of particles (geometric control) and a heat and energy control, the latter being absolutely crucial. NB: Most of the elementary fusion processes occur through quantum tunneling.

Building upon the nuclear transmutation experiments of Rutherford done a few years earlier, fu- sion of deuterium nuclei was first observed by Oliphant and Harteck in 1932. Research in fusion for military purposes began in the early 19400s, as part of the Manhattan Project, but was not suc- cessful until 1952. Research for civilian purposes remained scarce and classified until 1958 and the ‘Atoms for peace’ conference in Geneva. And with it a new lease on life for plasma physics. Through fifty years, many ideas on confinement strategies have arisen, former conventional ideas now being almost abandoned and inversely. The now ‘non conventional’ approaches in civilian world are essentially ‘cold’ or ‘locally cold’–based principles such as palladium-induced “cold” fusion, muon-catalised fusion, accelerator-based light-ion fusion, sonoluminescence-based “bubble” fusion, antimatter-initialised fusion and, to my knowledge, the latest-born: pyroelec- tric fusion. Most of the latter become irrelevant with respect to the criterion of a possible mass production of electric energy. More convenional methods involve thermonuclear plasmas and are essentially twofold: (i) a military-coloured (at least in the EU and the USA) inertial confinement strategy in which the driver is a laser, an ion or electron beam, a ‘Fusor’ or a ‘Z-pinch’ or (ii) a magnetically confined method which also has its own variations, e.g. the reversed-field pinch, field-reversed configuration, levitated dipole, Spheromak, Riggatron, Stellarator and Tokamak.

Tokamak is the Russian acronym for тороидальная камера с магнитными катуш- ками –Toroidalnaya kamera s magnitnymi katushkami, litterally “toric magnetic chamber”. The concept has been developed in the 500s in USSR, among other magnetic configurations. It is nowadays considered, along with the Stellarator, as the most promising approach for energy production. Its physics represents a cornerstone for a wide class of fundamental problems related to turbulence, chaos and magnetism. 1.2 When applied to human enterprises 7

Figure 1.3: Subplot (a)– Fusion reaction rate (average of cross-section times speed) as a function of temperature. The average is over Maxwellian ion distributions with the appropriate temperature (data from the NRL Plasma Formulary, 2006). Subplot (b)– The believed fusion process between deuterium and tritium, Eq.(1.1a); lithium is here, as in a reactor, so as to generate in situ the tritium.

Of course, with each of the latterly mentioned means of confinement, dedicated problems arise, but the basic nuclear reactions remain. Amongst the great variety of conceivable fusion reactions, preference will go3 to those with the highest reaction rates τ, i.e. reactions having the highest probability of fusion processes per unit of time and volume: τ = nAnBhσvi, hσvi being the probability of a fusion reaction for two reactant nuclei of respective densities nA and nB. Figure 1.3–(a) shows the fusion reaction rate as a function of temperature for three of the most common reactions. The maximal values of the respective reaction rates constrain for each reaction the optimal operating temperature. The deuterium (D) – tritium (T) reaction Eq.(1.1a):

[5He]? D + T −→ 4He(3.5 MeV ) + n(14.1 MeV ) (1.1a) n +6Li −→ 4He(2.1 MeV ) + T (2.7 MeV ) (1.1b) n +7Li −→ 4He + T + n − 2.5 MeV (1.1c) thus readily has both (i) a higher probability of fusion than other reactions commonly considered and (ii) at a lower temperature, which makes it the best elementary choice for practical use. Helium has an extremely low mass per nucleon and therefore is energetically favoured as a fusion product. As a noble gas, it is chemically almost inert and harmless in the case of a confinement accident. Tritium has a short radioactive half-life of 12.32 years, with a subsequent very low abundance, which requires its breeding from lithium4 to sustain the deuterium–tritium latter fuel cycle, Eqs.(1.1b) and (1.1c). In the former, the reactant neutron is supplied by Eq.(1.1a). which also produces most of the useful energy. The reaction with 6Li is exoergic, also providing a small energy gain. These processes are illustrated in Fig. 1.3–(b). The relative abundance of isotopes

3The straightforward practical requirements for fusion reactions consist in (i) being exoergic, (ii) involving low Z nuclei, (iii) being a two body reaction –three body collisions are too improbable (iv) conserving protons and neutrons since cross sections for the weak interaction are too small and (v) having at least two products to conserve momentum and energy without relying on the electromagnetic force. 4The supply of lithium is more limited than that of deuterium, but still large enough to supply the world’s energy demand at an extrapolated increased rate for thousands of years. 8 Some introductory words on nuclear fusion, plasma physics and around

6Li/7Li ≈ 0.08 also requires to consider the third reaction Eq.(1.1c), which is unfavourable by at least two means: it is endoergic and does not consume the neutron –with radio-protection problems associated, even though it also breeds tritium. The neutron and radioactivity problems open the way to advanced prospective fusion reactions. A criterion for a reaction being ‘advanced’ –as compared to the D–T reaction, is that it increasingly relaxes the two following constraints, regardless of the intrinsic difficulty of the reaction:

Neutronicity is the fraction of fusion energy released as neutrons and gives a precious indica- tion of the magnitude of problems associated with neutrons: radiation damage, biological shielding, remote handling and safety;

Electrical efficiency is an increasing function of the proportion of charged products since di- rect conversion to electricity by charge exchange is much more efficient (> 98%) –though difficult– than conventional electricity production via thermal cycles (< 50%).

In that spirit, the four most widely considered advanced fuel cycles read:

D + D −→ T (1.01 MeV ) + p+(3.02 MeV ) (50%) (1.2) −→ 3He(0.82 MeV ) + n(2.45 MeV ) (50%) D +3He −→ 4He(3.6 MeV ) + p+(14.7 MeV ) (1.3) p+ +6Li −→ 4He(1.7 MeV ) +3He(2.3 MeV ) (1.4) p+ +11B −→ 3 4He + 8.7 MeV (1.5)

Other cycles involving 3He would require its supply either from other nuclear reactions or from extraterrestrial sources, such as the surface of the moon or the atmosphere of the gas giant planets and are not considered here. Quantitative comparison between Eq.(1.1a) and Eqs.(1.2), (1.3) and (1.5) is displayed in Table 1.2.

Table 1.2: Increasingly ‘advanced’ fusion reactions with the criteria of decreasing neutronicity and increasing electrical efficiency.

Temperature Reaction rate Fusion Charged Reaction Neutronicity (keV) ≡ 1/Z energy products energy D + T 13.6 1 17.6 3.5 0.80 D + D 15 1 12.5 4.2 0.66 D + 3He 58 1/2 18.3 18.3 ∼ 0.05 p+ + 11B 123 1/5 8.7 8.7 ∼ 0.001

A man-induced fusion reaction should trivially (i) produce more energy than it is fuelled with and (ii) be self-sustained. To quantify these ideas, a useful concept is the fusion energy gain factor, usually expressed with the symbol Q and ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state.

Ignition refers to the conditions under which a plasma can be maintained by fusion re- actions without external energy input. Therefore, the goal of ignition corresponds to infinite Q and is not a necessary condition for a practical reactor.

The historical work which provides a criterion to reach ignition for a power producing thermonu- clear reactor was derived by Lawson in 1955, published in 1957 [77] and specifies a minimum 1.2 When applied to human enterprises 9 value on the so-called ‘triple product’ of density n, plasma temperature T and energy confinement time τE: 2 12kB T 21 −3 nT τE ≥ ≈ 10 keV sm (1.6) EFP hσvi where kB is the Boltzmann constant, EFP the energy of the charged fusion products and hσvi the probability of a fusion reaction for two reactant nuclei (cf. page 7). This so-called Lawson criterion can be physically interpretated as the ratio of the energetic contents of the plasma to its losses (or sources in a steady state). In practice, reaching ignition is not an absolute constraint for a viable power plant. Finite values for the fusion energy gain factor close to Q = 20 are often thought to be enough for a reactor. In Iter, whose objective is to demonstrate the practical feasability of a fusion-based production of energy on an industrial level, the goal is to reach Q = 10 and then demonstrate that a significant fraction of the heating power can come from fusion heating power.

Breakeven refers to the condition Q = 1 and means that 20% (for D–T fusion) of the heating power comes from fusion reactions. Above Q = 5 the fusion heating power is greater than the external heating power.

Chapter 2

The eternal problem of representation: which model for the plasma ?

c If you shut your door to all errors truth will be shut out. d Rabindranath Tagore, Stray Birds, 1916.

HYSICS essentially remains a back and forth confrontation of reality-induced man-forged models to man-probed ‘intrinsic’ reality. The discussion of whether this intrinsic, far-off P reality –the “truth” of Tagore– could be unveiled or does even exist is beyond the scope of the present dissertation. Such also applies to the question whether this reality is “written in a math- ematical language” and best described by Numbers. As a working hypothesis, let us postulate that indeed an ‘intrinsic’ reality does exist which can be accurately mathematised using differential equations, should our fate be that it shall always remain an unreachable ideal. A more immedi- ate problem for the physicist –some would say more relevant– is to build upon the transduction through perception or experience everyone does operate from the natural world and generate a representation. Even though this transduction process is very likely man-dependent, the represen- tation can become generic and a model emerge. The present chapter aims at providing the reader with the basics of some essentials models –the so-called kinetic and fluid models– which have proven most fecund over the years.

2.1 Browsing the models

c It is fair to say that plasma turbulence lacks the elements of simplicity, clarity and universality which have attracted many re- searchers to the study of high Reynolds number fluid turbulence. In contrast to the famous example, plasma turbulence is a problem in the dynamics of a multi-scale and complex system. d P. H. Diamond, S. I. Itoh, K. Itoh, Series: Self-Organization and Structure Formation in Turbulent Plasmas, to be published.

Arguably, a model for the plasma should at least satisfy the following four primal conditions, regardless of any of its later technical details: (i) have steady states which are special solutions of the time-independant evolution equation. This allows one to describe e.g. Langmuir waves, to define a thermodynamical equilib- rium characterised by temperatures, energies, . . . 12 The eternal problem of representation: which model for the plasma ?

(ii) be linear in time and space, i.e. satisfy a superposition principle to allow for undu- latory phenomena (interference, diffraction, resonances. . . ) (iii) be first order in time so as to satisfy causality. In the kinetic framework –which is of primal interest to us throughout the remainder of this dissertation– a quantitative explanation of this idea can be found in section 2.2, page 17. It is deeply linked with the Born interpretation of quantum mechanics.

So as not to renounce trying to understand Nature (even as an unreacheable ideal) rather than being satisfied with the increasing power physics can provide over it is highly dependent on the applicability of a causality principle in physics. Following Grete Hermann, it could be enounced as: Nothing is produced in Nature which is not due to anterior processes – i.e. which does not follow these processes with necessity. This statement applies to all physical characteristics which could be acknowledged.

Such a principle does not overlap with the concept of physical determinism since it would hastily assimilate the possibility to know that causal chains do underlay all perceived phe- nomena and the possibility to formulate predictions on them from primal hypotheses. Deter- minism is not contained in the concept of cause and some authors wrongly foretold the death of causality because of the emergence of chance. Going even further, Cournot did show that strict determinism does not even identify with absolute prediction capabilities as those of a Laplace demiurge for instance. Being able to predict a succession of events is not contained in the idea of causality and is only a means by which causal chains postulated by a theory can be justified or discarded in the light of experiments. The causality principle does only emphasise the necessary suc- cession of events and not the human possibility to evaluate them beforehand.

(iv) satisfy a ‘Correspondence’-like principle, i.e. encompass the Boltzmann kinetic theory of gases in the low temperature and high collisionality limits or the Navier–Stokes fluid mechanics in the limit of a vanishing electromagnetic field and highly collisional medium.

The fundamental idea of projection underlying the whole following discussion is very simple, has been known for long and is very likely better enunciated elsewhere. The consequences and re- sulting concepts though are non trivial. May the expert reader forgive their unsatisfactory wording; the goal of the following discussion is to provide the reader with some very basic ideas to better apprehend the profound unity behind the apparent great amount of available distinct models. The following ten pages could easily be skipped at first reading.

Three traditional approaches While only adopting distinct angles, three great possible frameworks are often introduced. They rest on the idea that the macroscopic properties of the plasma can be inferred from the microscopic description of its elementary entities. Building upon this idea, the successive models which will be briefly discussed are increasingly simplified (this simplification can go well beyond the fluid de- scription as enunciated in the following pages). They are successively obtained from one another by further projections, i.e. (i) a further decrease of the number of free parameters necessary to describe central concepts, (ii) a further cristallisation around a few central concepts which in turn (iii) often get increasingly abstract. Following this idea of projection, a disputable goal of physics is perhaps to explore the difficult (and somehow contradictory) focal point of (i) being techni- cally increasingly simple and (ii) conceptually increasingly abstract so as to put in one, or a few dense concepts an increasing proximity with the experiment. It is in that spirit that by successive 2.1 Browsing the models 13 projections the immensely large number of degrees of freedom of the system –associated with its very large number of particles– contracts into the statistical concept of distribution function which in turn can be projected on a few thermodynamic macroscopic quantities (temperature, density, pressure, volume, entropy. . . ). New fundamental concepts appear, deeply connected to the definition taken for the system: is it isolated, closed or open? is its evolution irreversible? is coarse graining a relevant concept (only a few macroscopic variables from the detailed microscopic state of the system are relevant for its physical description)?

An open system is a system with input, which changes its behavior in response to conditions outside its boundaries. Inversely, an isolated system is a system without input, whose behaviour is entirely explainable from within its boundaries. Stricto sensu, only the universe is rigorously isolated; isolation is often postulated –in plasma physics, core plasmas which this dissertation is about are always assumed to have no interactions with e.g. the coil system which generates the magnetic field. This assumption is somehow crude since work must be exchanged between the plasma and the coils, should it only be to sustain the former out of equilibrium. Inbetween, a closed system can exchange heat and work, but not matter with its environment. Very generally, systems may be variously closed to matter or energy and/or to work, to information and/or to organisation. It is sometimes stated that systems closed to energy are autark, systems closed to information are independent and systems closed to organisation are autonomous. At last, as emphasised by Krippendorff, whether or not a system has outputs does not enter the distinction between open and closed systems. Systems without output are non-knowable by an external observer (e.g. black holes) and systems without inputs are not controllable.

These questions are fundamental since thermodynamics and statistical mechanics occupy centre stage in the edifice of modern physics. Since their inception in the 19th century, they have seen spectacular developments, extending their range of application far beyond what their founding fathers had envisaged. However, at the same time it has also been well-known ever since that both theories face serious conceptual difficulties. Many of these became most palpable when the relation between the two theories was studied. Thermodynamics was (and is) believed to reduce to statistical mechanics, but Loschmidt’s reversibility objection and Zermelo’s recurrence objection soon showed that the two theories do not enter into any simple reductive relationship. This idea will not be discussed any further though it very interestingly emphasises how crucial the problem of representation is. The first model which shall now be described is at the same time the more intuitive and the more untractable. Each particle is individually described and its evolution intrinsically coupled to all other particles. The complete dynamical description of a such system therefore requires to solve the N coupled equations for the N particles the system is composed of and thus embrace at any given time the 6N degrees of freedom of the associated phase space.

As shown e.g. by Poincare´ [94], the description of geometrical space requires a three- dimensional basis q = (e1, e2, e3), defining what is usually referred to as real space. Con- jointly, velocity space v = (v1, v2, v3) –or momentum space p in the general case– can be defined, which allows one to introduce the six-dimensional (q, p) phase space. First introduced by Gibbs in 1901, it is a dimensionaly minimal space where all the possible trajectories of a dynamical system are represented.

As a many body problem and though making use of the most up-to-date computational techniques, it remains untractable for a system involving over a billion particles –which is far from the require- ments of the physical world since, typically, N ∼ 1023 in Iter. 14 The eternal problem of representation: which model for the plasma ?

The other approaches are statistical. Let us first introduce the notion of a system, i.e. a given ensemble of plasma particles. Within such an ensemble, given the great complexity of the inter- actions exerted on e.g one of its sub-systems by all the other parts of it, the statistical approach postulates that during a sufficiently long time, this sub-system will meet all its possible states a great number of times. More precisely, let dqdp be the infinitesimal volume of phase space occu- pied during the interval of time δt by this sub-system. The statistical hypothesis says that during a sufficiently long interval of time T the complex and entangled nature of the phase trajectory of this sub-system will make it loopback this elementary volume of phase space dqdp a great number of times. When T goes to infinitiy, δt/T is finite and defines the elementary probability dP to find at an arbitrary time the sub-system in the elementary volume dqdp: δt lim = dP = f(q, p) dqdp (2.1) T →∞ T where the fundamental quantity f is a function of all phase space coordinates and represents the density of probability in phase space. It is the physical analogue of the mathematical concept of a measure. As such it is positive definite and satisfies the normalisation condition: R f dqdp = 1. More commonly, f is referred to as the distribution function of the system and does not depend on the initial state of any of its sub-systems when T is sufficiently large. Indeed, during this interval of time, any of the states our sub-system has been through can be called the initial state since statistically equivalent to all the others. This very important property allows one to find the statistical distribution function for sub-systems without solving the initial mechanical problem with initial values given for the whole system. The fundamental problem of statistical physics therefore reduces for any physical system to the determination of its distribution function. At this stage, the reader could be concerned the temporal dynamics has vanished within this statistical description. It has not in a most interesting way. To plainly explain this idea, let us shortly go back to classical mechanics. In this theory, the knowledge of its initial conditions entirely determine the later dynamics of a system. For systems with a great number of degrees of freedom, these initial conditions are incompletely known –and even though they were, their mere number would hardly allow for an analytical solution of the mechanical problem. Statistical mechanics allows one to go beyond these latter imperfections since its mere concern for a given system is to determine its behaviour on average and does not depend on an exhaustive calculation of all its possible phase space trajectories. By definition of the distribution function, any average quantity O¯ of O(q, p) is simply obtained as the ensemble average over all possible states: Z O¯ = O(q, p) f(q, p) dqdp (2.2)

This formulation allows one to avoid following in time the temporal evolution of the physical quantity O(q, p). Statistical physics proceeds as follows:

(i) time is frozen; at t = t0, a snapshot of the system is taken (ii) an ensemble average is performed over all possible states which allows one to define at t0 the macroscopic averaged quantity O¯ of the physical quantity O(q, p) (iii) time is advanced infinitesimally further to t0 + δt and then frozen again, etc. In this approach, the ensemble average is the fundamental concept to define a macroscopic dynam- ical function and is not to be related to any other concept considered more fundamental. To answer a possible source of puzzlement for the reader, let us emphasise that this is the most fundamental approach of statistical physics since the ergodic assumption is sidestepped and time re-enters the problem rather as a parameter than as a true dynamical variable. For more details on this special interpretation of statistical mechanics, see e.g. Refs.[2], vol.2, p.386 or [76], p.14. From a practical point of view though, it is often very difficult –if not impossible, see e.g. discussion p.50 and subsequent definition– to define average quantities or equilibrium states via 2.2 The kinetic model 15 ensemble averaging procedures (e.g. numerically, this would require to perform way too many costly almost identical simulations, see following chapters). So practically, one would favour time averages. To rigourously be able to do this, since one does want to follow in time the variations of the physical quantity O(q, p), it is possible to construct a time-dependant quantity O(t) such that: 1 Z T O¯ = lim O(t)dt (2.3) T →∞ T 0 since from Eq.(2.1), the statistical average is equivalent to the temporal average. Output quantities of e.g. numerical simulations are such constructed O(t) functions, which explains why such a temporal approach of statistical mechanics is still of some special relevance. To finish with gen- eral principles, let us also emphasise that inversely to univocal classical mechanics, the statistical approach has an inbuilt randomness which is not due to the very nature of the physical objects which are considered but only to the fact that the macroscopic quantities are obtained from a lim- ited set of data as compared to the complete mechanical problem. Nonetheless, in practice, given the√ typical sizes of the considered systems, this randomness is totally negligible (it decreases as 1/ N , N being the number of particles) and the predictions of statistical physics univocal. According to the latter discussion, a first-principle description of the plasma dynamics is achieved when writing down: (i) an evolution equation for the distribution function coupled to (ii) self-consistent evolution equations for the electromagnetic fields E and B The laws of electrodynamics coined by Maxwell –step (ii)– were the essential discovery of the XIXth century and since then remain a well-defined problem. On the other hand, a clear descrip- tion of step (i) i.e. an unambiguous characterisation of the response of the plasma is of more recent history and to some extent is still undergoing. Throughout the two following sections, I shall briefly describe with decreasing complexity the essential theories coined to characterise the response of the plasma.

2.2 The kinetic model

An interesting starting point is the Liouville equation which is valid in the absence of dissipation. In such a framework, it is easy to show using hamiltonian dynamics that (i) the number of systems dN in an infinitesimal region of phase space is constant through time and (ii) using the integral invariants of Poincare´ (e.g. see [49], p.394) the latter infinitesimal volume of phase space dV itself dN is constant. Therefore, the density of systems D = dV is also constant and its total derivative vanishes: dD   = 0 ⇔ ∂ − [H, ] D = 0 (2.4) dt t which constitutes the Liouville theorem. The formal solution of this equation is:

−H(t−t0) D(t) = e D(t0) (2.5) where t0 is an arbitrary time (initial time). The hamiltonian H is the propagator and satisfies the Hamilton equations:

∂tq ≡ q˙ = ∂pH (2.6)

∂tp ≡ p˙ = −∂qH (2.7)

The exponential form of the solution readily stems from the causality principle, as will be empha- sised page 17. A hand-waving argument why evolution equations must be first order in time is the following: starting from t0 = 0 < t1 < t2, one must satisfy: H(t2 − t1) + H(t1) = H(t2), 16 The eternal problem of representation: which model for the plasma ? stating that the final state is statically independant of the chosen evolution path. This idea is deeply connected to the Maupertuis’ principle of least action. From this property, there exists a function ℵ(H) such that: D(t2) = ℵ [H(t2)] D(0) and D(t2) = ℵ [H(t2 − t1)] ℵ [H(t1)] D(0). Therefore: ℵ is the exponential function as in Eq.(2.5) and the Liouville equation is first order in time. An equivalent and perhaps more geometric way to understand this theorem rests upon volume transformations of a region D(t = t0) → D(t) of phase space. Let us introduce x = (q, p) t t 2 and g the one parameter group of transformation such that: g (x) =t→0 x + v(x) t + O(t ). The function v(x) represents a velocity in phase space and the volume D(t) transforms such t R that: D(t) = g D(0). The elementary volume V(t) of phase space reads: V(t) = D(t) Idτ = R t R t
D(0) Iog Jgt dx = D(0) 1. det ∂g /∂x dx, where I represents the unit tensor and Jgt the Jaco- t
2 bian of the transformation, which takes the simple form: det ∂g /∂x =t→0 1 + t ∇ · v + O(t ). The rate of change of an elementary volume of phase space then reads: Z dV = ∇ · v dx (2.8) dt t=t0 D(t0) which is the so-called Reynolds lemma. Since v represents the velocity of the elementary domain

(q, p), the divergence ∇ · v = ∂pi p˙i + ∂qi q˙i vanishes and so does the right-hand side of the latter equation. In the light of Eqs.(2.4) and (2.8) comes the following interpretations of the Louville theorem. In the absence of dissipation:

the volume of phase space occupied by an ensemble of trajectories is con- served through time. the distribution function of an ensemble of particles is constant along its or equivalently1: phase trajectories.

Liouville’s Theorem shows that, for closed systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble –i.e. the total or convective time derivative vanishes. Thus, if the microstates are uniformly dis- tributed in phase space initially, they will remain so at all times. Liouville’s theorem ensures that the notion of time average makes sense, but ergodicity does not follow from Liouville’s theorem. This theorem does not hold in the presence of dissipation and, strictly speaking, is only valid fo closed systems or on short periods of time as compared to the typical time of energy exchange with the outside. Therefore, this approach must be extended to more general systems. Different approaches have been considered to go beyond the Liouville theorem –the so-called Klimontovitch or Lenard–Balescu approaches, or the BBGKY hierarchy which starts from the Liouville theorem. I shall not describe these ideas here, a complete derivation can be found e.g. in [91]. Essentially, these approaches all converge to what is known as a kinetic equation:

N df X = C(f) ⇔ ∂ f + (q˙ ∂ + p˙ ∂ ) f = C(f) dt t i qi i pi i=1

⇔ ∂tf − [H, f] = C(f) (2.9) where H is the hamiltonian Eqs.2.4–(2.7) and C is the collision operator. Though at this stage H and C are still unspecified, Eq.(2.9) is the generic form for a kinetic equation2 Such an equa- tion is sometimes referred to as a master equation –or the Botzmann equation in physics– i.e.

1This conservation property of f along the phase space trajectories is an important property which I shall come back to in section 3.1.2, page 46, since it has a practical numerical use. 2It is possible to go beyond the already very general hypotheses presiding over the derivation of this equation and to write even more general non-Markovian evolution equations; this aspect is far beyond the scope of the present dissertation and will not be discussed here. 2.2 The kinetic model 17 a phenomenological set of first-order differential equations describing the time evolution of the probability of a system to occupy each one of a discrete set of states. In probability theory, this identifies the evolution as a continuous-time Markov process. Over the past century, this equation has given rise to a considerable amount of works, herewith giving rise to an almost as large amount of limit models, often named after their discoverers and according to which version of the collision operator it contains (see section 4.1.2, p.72 for further details on the collision operator). Some of which are of great significance and have provided at a given time new physical insight. (i) A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles which are assumed to be uncorrelated prior to the col- lision. This assumption was referred to by Boltzmann as the Stosszahl Ansatz, and is also known as the molecular chaos assumption; (ii) The Fokker–Planck equation was first coined to statistically describe the Brownian motion of a particle in a fluid, focusing on the time evolution of the probability density function of the position of this particle. It has since been generalised to other observables, especially the distribution function, and in its latter form is the fundamental equation for a collisional plasma. According to the form the collision operator it contains, Eq.(2.9) is either referred to a the Bolt- mann equation or the Fokker–Planck equation. Since his 1931 work on Markov processes, this same equation is also known as the Kolmogorov forward equation. (iii) In hot plasmas, it is of wide use not to consider collisions since the mean free path of a particle is usually very large and its collision rate very low. This simplified model was first studied by Vlasov in the 500s. For obvious reasons, the so-called Vlasov equation df/dt = 0 is also sometimes referred to as the collisionless Boltzmann equation. (iv) This Vlasov equation, though simplified, is significantly difficult to handle and still beyond today’s most up-to-date computational possibilities. In strongly magnetised plasmas, as in the case of fusion, it is possible to further simplify the Vlasov equation into the gyrokinetic equation. Since the past ten years essentially, this latter approach has triggered a considerable amount of interest in the plasma community. It also is the central focus of this dissertation. A more precise description of this model is especially given in section 3.1. (v) As detailed in the last two chapters of this dissertation, Coulomb binary collisions can have a significant impact on the response of the plasma and therefore must be modelled3. The collisional gyrokinetic equation, when coupled to the Maxwell equations, provides the state-of- the-art first-principle description of the evolution of a magnetised plasma. All the mentioned equations do satisfy the latter four primal conditions: they all (i) have steady states f0: [H, f0] + C(f0) = 0 or simply [H, f0] = 0. Also, since (ii) the Poisson bracket and the collision operator are bilinear forms, the superposition principle is straightforwardly met. All equations are (iii) first order in time, thus satisfying a causality principle. This idea needs some more explanation; I show below the weak form: non(∂t) ⇒ non(% > 0) ⇒ non physical.

An equation of the form: ∂tf − [H, f] = C(f) has a positive definite probability density, which is not the case for equations with higher order time derivatives.

Main line of the proof: In action-angle coordinates (α, J), equation: ∂tf −[H, f] = C, averaged over all angles and integrated by part can be recast into a continuity equation: ∂t% + ∇ · j = D with D a dissipative term, % = hfiα and ∇ · j = ∂JhH∂αfiα. Since f is positive definite, % is positive definite. Inversely, higher order time derivatives lead to probability densities written as time derivatives of hfiα, which can be negative. Nota bene: this has some deep connections with the Born interpretation of quantum mechanics. Let Ψ? be the ? ? complex conjugate of Ψ, the quantum state vector. Then ∂tΨ = G ⇒ Ψ (∂tΨ) = Ψ G and ∂tΨ = G ⇒ ? ? ? ? ? ? ? ∂tΨ = G . Thus: (∂tΨ ) Ψ = G Ψ. Summing the two contributions: ∂t (Ψ Ψ) = Ψ G + G Ψ. With very reasonable assumptions on G, the latter equation can be recast into a continuity equation: ∂t% + ∇ · j = 0 with ? ? a positive definite probability density % = Ψ Ψ and a probability current j = ~/m = (Ψ ∇Ψ). Inversely, a 2nd ? ? second-order equation in time ∂ttΨ = K(t) leads to a probability density % ∼ (Ψ ∂tΨ + Ψ∂tΨ ) which can take negative values. A more complete discussion can be found in the works of Born.

? 3In particular, an open question is especially whether lim (collisionality → 0) ≡ (collisionality = 0)? In particular, there always exists an ‘effective collisionality’ in numerical models, see discussions pp.26, 39 and 74. 18 The eternal problem of representation: which model for the plasma ?

At last, the problem (iv) of the ‘correspondence’ principle is non trivial and most interesting since it provides a greater insight on the true richness embedded within the kinetic description of a magnetised plasma as compared to a fluid. I shall now detail this idea.

2.3 The fluid model

This latter kinetic approach is still both analytically and numerically sorely tractable. The third possible description proceeds from a further simplification of the above model. The so-called fluid approach projects the kinetic equation Eq.(2.9) on the infinite polynomial velocity basis 2 k 4 {1, v, v , . . . , v ,... } and focuses on the moments of the kinetic equation . The moment Mk of order k is an integral over velocity space of v⊗kf, where ⊗k denotes a tensorial product of order i ≤ k. The first moments are the density (k = 0), the flow velocity (k = 1), the pressure (k = 2) and the heat flux (k = 3): Z +∞ n = d3v f (2.10) −∞ 1 Z +∞ u = d3v vf (2.11) n −∞ Z +∞ ¯ ¯ 3 P¯ ≡ p I + Π¯ = m d v (v − u) ⊗ (v − u) f (2.12) −∞ m Z +∞ q = d3v |v − u|2 (v − u) f (2.13) 2 −∞ where p = T r(P¯ )/3 is the isotropic scalar pressure and Π¯ the viscous stress tensor which embeds the off-diagonal contribution of the pressure tensor and its anisotropy. The result of this projection is an infinite set of equations, formally written: Z 3 ⊗k h i ∀k ∈ N, d v v Eq.(2.9) = g(Mk−1, Mk, Mk+1) (2.14) in which every kth–order fluid equation couples the order k and k − 1 moments to the following one k + 1, leading to the infinite fluid hierarchy. Should this infinite set of equations be solved, the approach would be equivalent to the kinetic model. As a greater simplicity is seeked, the fundamental fluid problem lies on how to truncate this infinite hierarchy, i.e. close the set of Eqns.(2.14) at a finite order kc < ∞, yet retaining the fairest possible amount of physics. This is referred to as the ‘closure problem’ and is detailed in the following sections.

It is not by chance that the fluid terminology has here been chosen. This model is indeed very closely related to what people usually refer to as fluids, i.e. systems which satisfy some form of Navier–Stokes equation. The first odd moment equation of Eq.(2.9), describing the momentum flux: ¯ m∂t (nu) + m∇ · (nu ⊗ u) + ∇ · P¯ − en (E + u × B) = F (2.15) displays the analogue of the usual convective derivative of a fluid particle and the latter equation can be recast in the usual Navier–Stokes way (with no particle sources):

mn (∂tu + u · ∇) u = −∇p + en (E + u × B) + F + R (2.16)

4Other projective basis are possible, especially those based on Hermite or Laguerre polynomials. The convergence is faster in the sense that for a given accuracy as compared to the kinetic model, less moments are needed. However, the physical interpretation of these moments is less clear than with the velocity basis. This idea will not be disussed in further detail here. The interested reader could e.g. refer to Ref.[48], p.104 for Hermite polynomials and to Ref.[106] for the Laguerre projection basis. 2.3 The fluid model 19 where F is the collisional friction force: Z F = m d3v v C (2.17) and R contains the viscous stresses. Though it is not the usual means by which the Navier– Stokes equation is usually derived, Eq.(2.16) shows that the ‘correspondance’ principle is indeed satisfied. The kinetic statistical approach does embed the usual fluid conservation principles of mass, momentum and energy. The derivation of the equations of fluid mechanics is usually done via the Reynolds transport theorem which states that the changes of some intensive property L defined over a control volume D must be equal to what is exchanged through the boundaries of the volume ∂D plus what is created (or consumed) by sources and sinks S inside the control volume:

d Z Z Z dτ L = − dσ L u · n + dτ S (2.18) dt D ∂D D

From this relation, readily stems the following equation on L:

∂tL + ∇ · (Lu) = S (2.19) which is the generic form of a continuity equation. Taking L = n, the density, gives the first even moment equation of the kinetic equation Eq.(2.9), describing the evolution of density:

∂tn + ∇ · (nu) = S (2.20)

In fluid mechanics, fluxes are often written in an isotropic way: Df ∆n. Here arises an essen- tial difference between a hydrodynamic fluid and a magnetised plasma since the magnetic field p creates an intrinsic anisotropy, which favors a transverse transport, often written: S = D ∆⊥n (transverse and parallel refer to the direction of the magnetic field lines). The most elemental form of the Navier–Stokes equation is obtained when Eq.(2.19) is applied to momentum L = ρu, with ρ = nm. Behind this idea are the hypotheses that: (i) it is possible to define a so-called fluid particle –composed of all particles inside the control volume– which, at any given time, is (ii) at thermodynamical equilibrium. This elementary volume is highly collisional –the mean free path is very small (∼ 10−9 m) as compared to the size of this pseudo-‘particle’– and therefore local thermodynamical equilibrium is always postulated. Inversely, in a plasma, the Debye length is often comparable to the electron Larmor radius (see page 3). It is therefore delicate to define a plasma particle in the same way than in a fluid, though the closely related notion of quasi-particle is at the root of the Particle In Cell (PIC) approach –see p.46. It is also rather inaccurate to postulate that a plasma is always close to local thermodynamical equilibrium (for a dedicated discussion on how to define a kinetic equilibrium, see next chapter). To briefly summarise this discussion, as compared to fluid mechanics:

• the ‘plasma mechanics’ lack two essential elements of simplicity, namely:

– the high collisionality of fluids which allows one to define local thermodynamical equilibrium inside a ‘fluid particle’; – the isotropy of fluids –with the notable exception of highly stratified geophysical fluids– which may be linked to their low or vanishing electro-magnetic field;

• . . . while containing the Navier–Stokes nonlinearities and therefore hydrodynamic turbu- lence and Kolmogorov cascade theory. This is the classic paradigm to approach the ‘prob- lem of turbulence’ as enounced by Kolmogorov in 1941, namely the question of the spectral 20 The eternal problem of representation: which model for the plasma ?

energy transfer and distribution between scales5.

The theory of hydrodynamic turbulence puts most emphasis on the question of spectral transfer and excitation; plasma kinetic turbulence theory complements this approach while also featuring the central questions of (i) the back-reaction of small-scale turbulence on all scales of the dis- tribution function and (ii) the response of the plasma to a seed perturbation. The fluid models of plasma physics derived from this kinetic description describe plasmas in terms of smoothed quantities in real space. All the velocity information contained in a phase space kinetic descrip- tion is smoothed out, though it can partly re-enter the model through an adequate ad hoc choice of the closure relation. Especially, fluid models have special trouble self-consistently describing systems which are (i) weakly collisional (ii) far from thermodynamic equilibrium; problems arise when also trying to (iii) resolve wave–particle resonance effects (e.g. collective behaviour such as Landau damping), (iv) local trapping phenomena (crucial for a weakly collisional description (i), see chapter 4), (v) finite orbit width effects or (vi) capturing velocity space structures like beams or double layers. Nonetheless, fluid models remain of wide use since: (i) they are numerically less demanding –until now they alone would allow to reach the plasma confinement time, given the available numerical resources, (ii) they allow for an interesting ‘physical intuition’ and as such give considerable insight on plasma physics, (iii) they interestingly model magnetohydrody- namic instabilities (the plasma is treated as a single fluid governed by a combination of Maxwell and Navier–Stokes equations) which are crucial for plasma stability and at last (iv) they can be much refined by advanced closure choices or multi-fluid descriptions, though possibly reaching a somehow inordinate level of complexity.

Before moving to the true heart of this dissertation based on a kinetic description of the plasma, I shall now detail some advanced closure ideas which allow to incorporate the Landau damping kinetic effect within a fluid model. On this specific question, the interested reader could also refer to e.g. Refs.[28, 55]. The following section 2.3.1 aims to provide the reader with the physical characteristics of the Landau damping mechanism. A new fluid closure embedding this latter mechanism is proposed in specified limits in section 2.3.2 and associated appendices while section 2.3.3 tackes a similar problem from the standing point of the entropy: it is intented to derive a fluid closure which intrinsically satisfies an H–theorem. For further details on this theorem, see discussion p.73 and Eq.(4.7).

2.3.1 Collective behaviour In this section, let us consider the simple Landau model very commonly studied so as to emphasise some basic and fundamental properties of a plasma kinetic description. It deals with the collision- less dynamics of non relativistic electrons, embedded in a vanishing magnetic field. The problem is further assumed to be electrostatic. Such a supposition is usual in tokamak plasmas, where most of the turbulent transport is governed by the electric potential. In the end, the ions are supposed at rest, with a constant density n0. The electron distribution function f and the electric potential φ satisfy the following normalised set of equations:

∂tf + v ∂xf + ∂xφ ∂vf = 0 (2.21a) Z +∞ 2 ∂xφ = fdv − 1 (2.21b) −∞

5Navier–Stokes turbulence is one of the last great open problems of classical physics; it is also of great interest in a purely mathematical sense. Somewhat surprisingly, given the wide range of practical uses for the Navier–Stokes equation, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist they do not contain any infinities, singularities or discontinuities (smoothness). These are called the Navier– Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics, and offered a $1, 000, 000 prize for a solution or a counter-example. 2.3 The fluid model 21

q 0kB T where distances are normalised to the Debye screening length λ = 2 and time is nor- D 4πnee −1 malised to ωp = λ/vT . The problem is two-dimensional in phase space (x, v). It is nonlinear, due to the last term in the Vlasov equation. In the remainder of this section, we shall restrict our- selves to the linear characteristics of Eqs.(2.21a) and (2.21b). For historical reasons, this system has deserved much attention since Landau first showed that it captures some essential features of the plasma kinetic properties, namely the resonant interactions between waves and particles [75]. He has solved Eqs.(2.21a) and (2.21b) for small-amplitude perturbations near an homogeneous 6 Maxwellian equilibrium , characterised by a vanishing electric field φeq = 0. Doing so, the Vlasov equation constrains the equilibrium distribution function feq to depend on velocity v only:  f(x, v, t) = feq(v) + f˜(x, v, t) φ(x, t) = φ˜(x, t) where hf˜i = 0 by definition; the brackets referring to time and space averages. Moreover, in the limit of small perturbations, namely for f˜  feq, the nonlinear term can be restricted to ∂xφ ∂vfeq, the nonlinearity ∂xφ ∂vf˜ being of the second order. In this case Eqs.(2.21a) and (2.21b) reduce to their linearised expressions:

∂tf˜+ v ∂xf˜+ ∂xφ ∂vfeq = 0 (2.22a) Z +∞ 2 ˜ ∂xφ = fdv (2.22b) −∞

Consistently with the φeq = 0 hypothesis, the electron and ion equilibrium densities are equal.

The Fourier–Laplace issue: an initial value problem Physically, the idea is to look at the time and space response of a small-perturbed plasma, given an initial homogeneous equilibrium. Mathematically, since Eqs.(2.22a) and (2.22b) are linear and feq only depends on velocity, plane waves of the form exp{i(kx − ωt)} are eigenvectors of the linear problem. Therefore, in this linear framework, the following Fourier expansion seems natural:

 P P ˆ ı(kx−ωt) f(x, v, t) = feq(v) + ∗ ∗ f (v) e k∈R ω∈R k,ω P P ı(kx−ωt) (2.23) φ(x, t) = ∗ ∗ φˆ e k∈R ω∈R k,ω and in the linear framework, the intricate form for ∂xφ ∂vf:

X 0 ık0x X ık00x 0 00 ˆ 0 00 ∂xφ ∂vf k = ı k φk e ∂vfk e δ(k − k − k ) k0 k00 can be written in a simple way: ˜ ˜
˜ ˆ ∂xφ ∂vf |k = ∂xφ ∂vfeq + ∂vf |k ≈ ∂xφ ∂vfeq |k = ı k φk ∂vfeq

In the linear regime, quadratic fluctuating terms can be neglected and each mode can be treated separately.

What Landau first emphasised is that causality is only satisfied when performing a Fourier transform over space and a Laplace transform over time. His argument is that plane waves are not physically acceptable solutions, since they carry infinite energy. Introducing causality in this problem especially states that one should account for the necessary vanishing character of the wave

6Van Kampen, a few years later, proposed a different way for solving this system, the so-called ‘normal modes of singular integral equations’. The two methods lead to the same result, each having special advantages and drawbacks. 22 The eternal problem of representation: which model for the plasma ? at t → −∞. By considering a Laplace transform in time, Landau emphasised this problem should be understood and treated as an initial value problem. For more explanations, the curious reader can refer to appendix A. Equations (A.1) and (A.2) rigorously define the way to perform Fourier– Laplace transforms (F LT s), replacing the expansion in Eq.(2.23). Since formally: ∂xf˜ → ıkfˆ and ∂tf˜ → −ıωfˆ− fˆ(t = 0), the linearised system Eq.(2.22a) and (2.22b) yields for each mode:

( ˆ ˆ ˆ −ı(ω − kv)fk,ω + ıkφk,ω ∂vfeq = fk(t = 0) 2 ˆ R +∞ ˆ (2.24) −k φk,ω = −∞ fk,ωdv

Given the definition of the Laplace transform, it should be noticed that ω has to be in the upper half of the complex plane, i.e. with a postive imaginary part (see discussion page 104). It follows from (2.24) the following dispersion relation:

ˆ R +∞ fk(t=0) dv ˆ −∞ ω−k v φk,ω = (2.25) n R +∞ ∂vfeq o ı k k + −∞ ω−k v dv

This work can rigorously be found in appendix B. At this point, it is important to remind that the latter equation is a priori strictly defined for positive values of Im(ω) ≡ ωi. Two important features derive from (2.25):

• The two integrals cannot be defined in the whole complex plane unless a so-called analytic continuation is performed. Indeed, when ω moves from the upper to the lower plane of the complex plane, ωi becomes zero. In this case, the integrand exhibits a singularity for the resonant condition ω = kv, and the integral is subsequently not defined.

• Provided the analytic continuation is performed, the only remaining resonance is for van- ˆ ishing values of the denominator. In this case, the dynamics of φk,ω is governed by the following relation that consequently can be referred to as the dispersion relation:

Z +∞ ∂ f D(k, ω) ≡ k + v eq dv = 0 (2.26) −∞ ω − kv

Ultimately, the dispersion relation first obtained by Vlasov is recovered, with an additional ingre- dient: integrals in (2.25) have to be analytically continued for the dispersion relation Eq.(2.26) to hold; Landau damping is a straightforward consequence of this continuation and leads to a drastic change in the physical behavior of the system. The following section summarises the main results of appendix C, where the analytic continuation technique is carefully addressed, and the classical Landau results rigorously derived.

The analytic continuation issue

This section aims at defining both integrals that appear in (2.25) [or (B.5)] whatever ω ∈ C –while initially only defined for ωi > 0. The theory of analytic functions ensures that such integrals are analytically continued if the poles do not cross the integration contour. Differently speaking, the continuation consists in deforming the integration contour when the pole ω = kv, belonging to the complex plane, moves from the upper half-plane to the lower one. Since the integral is performed on the velocity v, the contour choice will depend on the sign of wavevector k. The correct mathematics for analytic continuation stands on two basic theorems for analytic functions, namely Cauchy’s theorem7 and the residues theorem. The former tells us how to deform a given contour without changing the integral value, whereas the latter essentially accounts for the

7 R or more precisely its integral form: C f(z)dz = 0, with C a closed path and f analytic in and on C. 2.3 The fluid model 23 singularity’s presence. Appendix C describes how to rigorously recast both integrals which appear in Eq.(2.25) in such a way that they remain valid for ω in the whole complex plane. While doing so, the dispersion relation Eq.(2.26) now contains an imaginary part:

Z +∞ ∂vfeq ı π D(k, ω) = k + P dv − ∂vfeq|v=ω/k = 0 (2.27) −∞ ω − kv |k| which is often referred to as the “ı ε” Landau prescription. It can be further understood as equiva- lent to the physical argument saying that the denominator of the integrand should read ω−kv+ı ε, + ω ∈ R, ε → 0 so as to remain definite, with respect to the causality principle. Such an expres- sion Eq.(2.27) explicitly reminds of the necessity for analytic continuation to be performed on the integrals in Eq.(2.25). A comprehensive reformulation of the 1946 historical derivation by Landau [75] can be found in appendix E.

A consequence of analytic continuation: Landau damping

The imaginary part in Eq.(2.27) is responsible for the well-known linear Landau damping γL. Let Dr (resp. Di) stand for the real (resp. imaginary) part of the dispersion relation, namely for the two first terms (resp. the last term) in Eq.(2.27). Similar notations are used for the complex frequency ω = ωr + ı ωi. We will further assume that the imaginary part is much smaller than the frequency itself: ωi  ωr. This latter assumption can be checked a posteriori as a necessary yet sufficient requirement. The dispersion relation Eq.(2.27) can then be expanded as a function of this small parameter. At first order:

D(k, ω) ≈ Dr(k, ωr) − ωi ∂ωDi(k, ω)|ωr + ı {ωi ∂ωDr(k, ω)|ωr + Di(k, ωr)} = 0 (2.28)

It turns out that the imaginary part of the dispersion relation Di is smaller than the real part Dr. In the limit Di  Dr, the imaginary (resp. real) part of the frequency ωi (resp ωr) then derives from the imaginary (resp. real) part of (2.28), namely:

Di(k, ωr) ωi = − (2.29) ∂ωDr(k, ω)|ωr

∂ωDi(k, ω)|ωr Dr(k, ωr) = −Di(k, ωr) ≈ 0 (2.30) ∂ωDr(k, ω)|ωr

2 Close to local thermodynamical equilibrium feq = exp(−v /2), Eqs.(2.29) and (2.30) become:

2 − 1 ( ωr ) π ωr e 2 k γL = − (2.31) |k| k R +∞ ve−v2/2 2 dv P −∞ (ωr−kv) 2 Z +∞ ve−v /2 k − P dv = 0 (2.32) −∞ ωr − kv This second equation Eq.(2.32) leads to the real frequency historically obtained by Vlasov and describes an undamped propagating wave. Its analytic approximation can be obtained by consid- ering the expansion of the denominator as a function of the small parameter ωr/k v. Inversely, since γL in Eq.(2.31) is proportional to Di, it is obvious that it directly derives from the analytic continuation. Moreover, it always remains negative8 since the principal value integral in Eq.(2.31) has the same sign as ωr/k. Therefore, it can be referred to as a damping rate. Any wave launched

8Landau damping was only experimentally measured for the first time in 1964, 18 years after Landau’s original paper. Physicians also have experimentally acknowledged the opposite effect: positive perturbation growth rate –see e.g. the so-called ‘bump-on-tail’ effect. 24 The eternal problem of representation: which model for the plasma ?

Figure 2.1: A simple phase mixing illustration with four waves possessing a small exponential phase discrepancy. The resulting pseudo-wave is exponentially damped while the elementary Langmuir waves are not.

in the system will be damped away by transferring energy to the particles. The detailed mecha- nism of such a transfer is beyond the scope of the present work and is still debated. A modern interpretation of this phenomenon is based on phase mixing processes. To briefly illustrate this idea let us consider, consistently with the primal linear hypothe- ses, small perturbations in a uniform plasma, namely high-frequency electron oscillations. The ions, heavier, remain at rest and are modelled as a continuous background of positive charges so that the whole medium is globally electrically neutral. These displacements cause local electric fields and their corresponding forces −e δE to exist on each electron, leading to density perturba- tions. Such perturbations are known as Langmuir oscillations at the electron plasma frequency: 2 2 ωpe = nee /me0. The plasma oscillations are somewhat artificial since the thermal electrons self-organise; Langmuir oscillations then become traveling electrostatic waves. These waves feed upon small density perturbations and back-react on the particles, contributing to create the very oscillations on which they have fed. This dynamics self-sustains through collective effects. A large spectrum of Langmuir waves do exist at any time in a plasma, with a random phasing between one another. It is well known that small phase differences of even initially coherent waves lead to a global ‘damping’ of the resulting mixed wave, sum of all the elementary waves in the medium. This process is known as phase mixing and illustrated in Fig.2.1. For a finite number of waves, in the absence of other dissipative phenomena, a time always exists where all the elementary waves are once more in phase. The resulting mixed wave then increases, back to its initial value (the well-known numerical ‘bounce’). This bounce is even a numerical test of how much the consid- ered numerical scheme is dissipative. In a real plasma though, this bounce is never seen, due to (i) the infinite number of Langmuir waves at any time and (ii) to irreversible dissipative phenomena.

This dissipation problem is essential –and highlights an intrinsic discrepancy between the fluid and the kinetic approaches– since the Landau damping process is well-known to generate small scales structures, which in turn must be somehow dissipated. This dissipation is often ad hoc in fluid models –see discussion in the forthcoming section 2.3.2. In the following chapter 4, collisions will introduce a physical dissipative mech- anism accounting for the entropy increase of the Second Law of Thermodynamics, this irreversibility being due to collisional dissipation of the small-scale structures in velocity space –see p.73 for an introduction to Boltzmann’s H–theorem. An attempt to incorporate within a fluid description this evolution principle is carried out in the following section 2.3.3. We shall now describe another advanced closure idea which allows to recover in a fluid approach the correct kinetic Landau damping mechanism. 2.3 The fluid model 25

Landau damping is thought to as the exponential relaxation of Langmuir waves towards the thermal level. It is important to notice that this interpretation in terms of phase mixing does not imply that the elementary Langmuir waves are eventually damped. Indeed, so as to test this Landau prediction, one shall launch a probe wavepacket W(ω0, k0) inside the plasma in which exists a very large amount of Langmuir waves at every wavevector and frequency. The mere presence of W selects among all existing waves the ones close to (ω0 + δω0, k0 + δk0) which will resonantly interact with the probe wave as described by Eq.(2.26). All waves have different phases and their integrated combination –the resulting mixed pseudo-wave– is then damped by phase mixing processes while the energy of the probe wave spreads over all the other elementary resonant Langmuir waves which still exist whatsoever, undamped.

The Landau damping mechanism from wave-particle resonant interaction is an important ef- fect to be modelled so as to get the correct plasma response. Indeed, without appropriate closure choices as described below (i) fluid models would not even linearly reproduce the correct kinetic damping Eq.(2.31). Also, (ii) from a spectral point of view, Landau damping contributes to make the “inertial” range relatively narrow in wave-vector k space. As such, plasma turbulence differs from fluid turbulence [42] in which the inertial range exists over several decades in k space and for which the Reynolds number Re –which characterises the ratio between nonlinear coupling and classical dissipation– is much greater than unity: Re  1. Since this damping is fundamentally a resonance in velocity space, it cannot intrinsically be self-consistently described by a fluid model. Nonetheless, it is possible to incorporate it into a fluid description in an ad hoc way. This is the matter of the following section.

2.3.2 The closure problem –(i) a parametric closure

As have been discussed, the projection of the kinetic equation on the fluid polynomial infinite velocity basis {1, v, v2, . . . , vk,... } leads to an infinite set of coupled differential equations, the so-called fluid hierarchy. To close this set of equations at an arbitrary order kc requires to coin at least a new equation –the closure equation– relating the moment Mkc+1 to the lower order moments (M ) . This equation is central in fluid theory since the accuracy and robustness of i i≤kc the subsequent fluid description will strongly depend on this latter closure choice. In this section will be explained a rather simple closure idea which allows analytic comparison with a linear fully kinetic result, near an homogeneous, electrostatic, Maxwellian equilibrium. The goal is to embed the kinetic Landau damping into the fluid description. This wave-particle interaction indeed plays a central role both in the linear stabilisation of small structures and in the nonlinear saturation mechanism of turbulence. This approach, as the following pages will explain, consists in adjusting a set of free parameters so as to minimise the discrepancy between the fluid and the kinetic linear response functions. It will be referred to as the parametric closure. Starting from the system of Eqs.(2.21a) and (2.21b), the first three moment equations of the Vlasov equation read:

∂tn + ∇x (nu) = 0 (2.33)  p  eE ∂ (nu) + ∇ + n = 0 (2.34) t x m m  p   q  eE ∂ + ∇ + 2 nu = 0 (2.35) t m x m m where n, u, p and q are the scalar versions of Eqs.(2.13)–(2.13). 26 The eternal problem of representation: which model for the plasma ?

Let us halt for a short while on these three equations. To be physical, this problem would require some kind of small-scale viscous dissipation –see discussions in both the latter section and in the following one 3.1.1, p.39; from a numerical point of view it does requires ‘sub-grid’ dissipation so as to avoid filamentation and facilitate code convergence. For drift-wave and ITG turbulence –see section 3.1.1 for further details– the importance of nonlinear Landau damping is that it provides a nonlinear saturation mechanism. This problem is reminiscent of the old ‘Euler v.s. Navier–Stokes’ debate in fluid mechanics, relative to the importance of the viscosity for both (i) physical sense and (ii) numerical purposes. Regularisation phenomena do always occur through the intrinsic dissipation present in any numerical scheme; yet important phenomena such as collisions –see the following chapter 4, p.74– or wave-particle damping mechanisms –this section– are crucial to accurately describe e.g. cascade phenomena. For the sake of simplicity, let us assume a simple dissipative Kolmogorov-like term in the right-hand side of Eqs.(2.33)–(2.35) which reads: 2 2 D∂xφ. The fluid dispersion relation would then read: Dk + D0, where D0 is the dispersion relation which comes out from Eqs.(2.33)–(2.35) (with no dissipation, i.e. a vanishing right-hand side). This form would never allow for an imaginary part depending on the absolute value |k| as in the kinetic case, which alone makes the transport non local, i.e. non diffusive in real space.

According to the former discussion, the closure relation will express the higher order heat flux q as a linear combination of the whole set of (known) fluid moments of lower order n, u and p. Doing so, the Vlasov Eq.(2.21a) remains self-consistent. Here, the closure is performed on the heat flux but the entire idea can be extended to closures on higher order moments. Linearly, Eqs.(2.33)–(2.35) can be recast into: n k n˜ = 0 u˜ (2.36) ω 1
u˜ = k p˜ + en0k φ˜ (2.37) n0ωm 2ken φ˜ k p˜ = 0 u˜ + q˜ (2.38) ω ω where the tilde ˜· refers to the fluctuations, e.g. n = n0 +n ˜ with n˜  n0. To remain consistent, the closure itself must be linear and therefore only involve the fluctuating terms of the moments. The most general form of a such closure is thus9: q˜ = a n˜ + b u˜ + c p˜ (2.39) The {a, b, c} set of parameters refers to initially unconstrained complex linear operators. The whole idea is to constrain them while trying to match the fluid response function R to its kinetic analogue. This function is defined as follows: eφ˜ n˜ = −n0 R(ζ) (2.40) T0 √ where ζ = ω/ 2 vT |k| is the normalised frequency (k 6= 0). Injecting Eq.(2.39) into Eqs.(2.36)– 10 (2.38), one can easily derive the fluid response function Rf : √ m n v2 ( 2 v ζ − c sgn(k)) R = √ 0 T T √ (2.41) f 3 3 2 2
−2 2 mn0vT ζ + 2cmn0vT sgn(k)ζ + 2 b + Γp0 vT ζ + an0sgn(k) where sgn(k) = k/|k| is the sign of the wavevector k.

We shall now derive the kinetic response function. The Vlasov equation can be linearised into: ∂tf˜ + v ∂xf˜ + ∂xφ˜ ∂vfeq = 0. Integrating the latter equation over velocity and expanding it in Fourier space leads to: Z ∞ ˆ ∂vfeq nˆk,ω = φk,ω ω dv (2.42) −∞ k − v

9 One could prefer to express the closure in terms of temperature. This is of course possible since p˜ = n0 T˜ + T0 n˜ 10Eq.(2.41) is a based upon slightly generalised definition for the moments and follows [55] when introducing the Γ parameter, ratio of specific heats in order to compare with different Landau damping models. We shall find that Γ = 3, as expected for a collisionless one dimensional gas. 2.3 The fluid model 27

√ mv2 neq(x) m − 2 T (x) Assuming feq = √ e eq is a Maxwellian equilibrium distribution function, the ki- 2πTeq(x) netic response function Rk is derived from Eq.(2.42):

Rk = 1 + ζ Z(ζ) (2.43) where Z is the Fried and Conte function: 1 Z +∞ e−t2 Z(ζ) = √ dt (2.44) π −∞ t − ζ and the latterly defined ζ normalised frequency naturally arises in the kinetic approach.

The two response functions Eqs.(2.41) and (2.43) shall now be adjusted in the most demanding kinetic limit ζ → 0. Given Eq.(2.42), for large ζ, i.e. for a phase velocity vϕ = ω/k  vT , the R +∞ ω
considered velocities are in the tail of the distribution function, where −∞ ∂vfeq/ k − v dv, thanks to the value of the exponential factor ∂vfeq, vanishes. Therefore, one looses the effects of resonances which govern the physics of Landau damping. Therefore, the ζ → ∞ limit corresponds to the fluid limit. Although the kinetic approach should be taken everywhere into account, the ζ → ∞ limit can be accurately described with a fluid approach only. Therefore, the so-called ‘kinetic limit’ is rather a limit in which the kinetic effects are essential. It corresponds to the opposite limit where ω/k  vT . In other words where the exponential factor ∂vfeq remains large, i.e. when ζ → 0. An open question is whether at mid values of the ζ parameter this approach will remain valid; it will be highlighted in what follows. t→0 The expansion of the Fried and Conte function near the origin is well-known [40]: Z(t) = √ −t2 2t2 4t4
6 ı π e − 2t 1 − 3 + 15 + ◦(t ). Therefore, one can Taylor-expand Rk and Rf in the ζ → 0 limit up to the second order in ζ. The identification of the same order expansion terms in both the kinetic and fluid approach leads to a (3 × 3) Kramer system which solutions are11: √ ı m 2π v3 sgn(k) a = − T (2.45) π − 4 4 p b = −Γ p − 0 (2.46) 0 π − 4 √ ı 2π v sgn(k) c = T (2.47) π − 4 If one were to carry the same series expansions up to the third order –i.e. in ◦(ζ3)– one would find that the upper system does not have any solutions; in other words, this approach is valid up to the second order in ◦(ζ2). In this limit: √ √ ı m 2π v3 sgn(k) 4 p ı 2π v sgn(k) q˜ = − T n˜ − (Γ p + 0 )u ˜ + T p˜ (2.48) π − 4 0 π − 4 π − 4 Inversely, at first order in ζ, one of the three parameters a, b and c would be vanishing. Focusing on Eq.(2.41): a and c play an important role while no term would vanish if b was to be canceled out (its effect can be included in the Γ p0 term). Therefore, taking b = 0 leads to a (2 × 2) Kramer system which solutions are: √ a = ı 2 mv3 χ sgn(k) √ T L c = −ı 2 vT χLsgn(k)

11This set of parameters, as will be shown below, depends on the choice of the distribution function and are thus rigorously only valid close to a Maxwellian distribution function. Nonetheless, the scope of this approach is not drastically shortened since the study which is displayed here has only been performed in this limit for analytical reasons and can easily be extended numerically to a non Maxwelian case. 28 The eternal problem of representation: which model for the plasma ?

Figure 2.2: Real and imaginary parts of the response functions R plotted against ζ. In red the kinetic response; in blue our parametric closure Eq.(2.48) and in green the historic closure of Hammett [55] Eq.(2.49).

˜ Since p˜ = n0 T + T0 n˜, √ q˜ = −n0 ı 2 vT sgn(k) χL T˜ (2.49) √ in which χL = 2/ π . This latter relation Eq.(2.49) is the historical result [55]. The parametric approach here described embeds this classical result and extends the associated closure. It is also generalisable, while [55] postulated a priori the form of the closure. However, the authors did nei- ther detail how they were deriving this form, nor explain why they were assuming q˜ proportional to T˜ only. The parametric closure provides some physical insight on these questions (no velocity fluctuations u˜ = 0). Also, it leads to an accuracy up to the second order in ζ. In this case, a more general dependency of the heat flux q˜ on both the temperature and the fluid velocity u˜ is found. The closure relations Eqs.(2.48) and (2.49) are determined in wave-number space. They can be expressed in real space –the detailed calculation can be found in appendix F:

Z ∞ 2T0vT n˜(z + s) − n˜(z − s) n 4p0 o q˜(z) = ds − Γp0 + u˜(z) π − 4 0 s π − 4 2v Z ∞ p˜(z + s) − p˜(z − s) − T ds (2.50) π − 4 0 s This description in real space leads to non-local operators, i.e. a local fluid temperature (or particle density, pressure. . . ) in real space requires so as to be satisfactorily described a global knowledge of the temperature (resp. the density or the pressure) throughout the entire system. The impact of this non locality from a numerical point of view is briefly described in appendix H. As displayed in Fig.2.2, the kinetic response is indeed accurately met when ζ → 0 and ζ → ∞. On average, the second order parametric closure remains closer to the kinetic result than the first order Hammett and Perkins closure. Yet, for the imaginary part , the Hammett–Perkins closure enables a fastest convergence towards the kinetic response function at intermediate values of ζ. Therefore, which closure is of greatest interest between those two cannot be settled when only looking at the linear theory. Comparing the kinetic and fluid approaches in the nonlinear regime should help in discriminating the best closure. Also, it could show the necessity for a higher order 2.3 The fluid model 29 closure, i.e. involving more than only the first three fluid moments. Such an investigation has not been performed yet though it could lead to a reasonable chance to improve the fluid description. Now taking a different perspective, incorporating the kinetic Landau damping effects into a fluid description is not the only way to improve it. Another interesting idea consists in coining a closure relation such that the fluid system intrinsically satisfies an H–theorem. This idea is developed in the following section. For further details on this theorem, see discussion p.73 and Eq.(4.7).

2.3.3 The closure problem –(ii) a closure based on entropy production An easy thermodynamical calculation readily shows that the (collisionless) Vlasov equation is isentropic. Yet, in a quasi-linear approach, the entropy is no longer conserved and a non-vanishing positive definite quasi-linear entropy production rate can be coined. This quasi-linear approach is interesting since (i) it has been shown to be surprisingly close, at least in the main trends, to both experimental results [117] and nonlinear gyrokinetic simulations [66], (ii) while remaining easily tractable analytically. A new prescription for the fluid heat flux is derived while comparing the fluid and kinetic quasi-linear entropy production rates. As will be shown, this approach is particularly interest- ing since (i) the entropy production rate is readily linked to the turbulent fluxes and constitutes (ii) an interesting quantification of the information displayed in nonlinear resonance phenomena. The goal of this section is to provide the reader with a precise idea of the method. The detailed calculation can be found in Ref.[39] and will give rise to a forthcoming publication [?].

The model equations In this section, we shall start from the so-called drift-kinetic equation in slab geometry x = (r, θ, z), z being the axis of the cylinder, (r, θ) being the polar coordinates in the circular cross- section and f the ion distribution function: 1 e ∂ f + [φ, f] + v∂ f − ∂ φ∂ f = 0 (2.51) t B r,θ z m z v B is constant in norm and direction. We derive a linear perturbative fluid closure. For so, f and φ are expanded over their equilibrium and perturbative parts:

2 −mv /2Teq(r) f = feq(r, v) + f˜(r, θ, z, t) with feq = f0(r) exp (2.52) φ = φ˜(r, θ, z, t) (2.53) the latter reading in Fourier space:

kθ ˜ ˜ X ˆ ˆ i( θ+kzz−ωt) f, φ = fk,ω(r), φk,ω(r)e r + c.c (2.54) k,ω

2 p where Teq ≡ mvT is the temperature, vT the thermal speed, f0 = neq/ 2πTeq/m and c.c refers ? ˜? ˜? to the conjugate complex g of complex g for which the following relations hold: fk,ω, φk,ω = ˜ ˜ ˜ ? f−k,−ω, φ−k,−ω. We shall also introduce the following normalisations: Φ ≡ eφk,ω/T¯ and ω = ? ? 2 2
? ωn + ωT v /2vT − 1/2 , with ωx ≡ kθρivT ∂rx/¯ x¯, x being either n, T or p and ωk ≡ kzvT . Since f0 does not depend on θ, the Fourier modes are the eigenmodes of the linear system and the dispersion easily reads:  ?  ˆ ω − ω ˆ fk,ω = − 1 − feq φk,ω (2.55) ω − kzv The assumed boundary conditions for the perturbations are respectively vanishing boundary con- ditions in the radial direction and periodic conditions in θ and z. 30 The eternal problem of representation: which model for the plasma ?

The kinetic entropy production rate

The kinetic entropy Sk is defined as: ZZ Sk = − dτdv f log f (2.56) where dτ ≡ rdrdθdz is the integral over the volume. And thus the kinetic entropy production rate S˙k reads in its quasi-linear approximation: Z ˙QL Sk = − dτdv (1 + log feq) ∂tfeq (2.57)

We adopt the usual drift-kinetic ordering in plasma turbulence k⊥ρi  1, which is satisfied in a strong magnetically confined plasma. We also classically assume that ∂vf˜ = 0, although it is sometimes reported both in experiments [95] and in flux driven numerical simulations [14, 46, 7] that the distribution function can significantly depart from a Maxwellian. The approach presented in this section, simplified on purpose, can easily be renewed without this assumption being made.

 1 h i e  ∂tfeq = − φ,˜ f˜ + v∂zf˜− ∂zφ∂˜ vf˜ B m θ,z,t  1 h i = − φ,˜ f˜ B θ,z,t  Z 2π −1 dθ n ˜ ˜ ˜ ˜o h ˜ ˜i = ∂rθφ f + ∂θφ∂rf − ∂rφf rB 2π 0 z,t   1 D ˜E 1 ˜ = − ∂r r v˜Er f where v˜Er = − ∂θφ r θ,z,t rB 1 = − ∂ (rΓ ) (2.58) r r k

R T dt R 2π dθ R L dz where h·iθ,z,t ≡ 0 T 0 2π 0 L and Γk is the kinetic turbulent flux which can be written in Fourier space:   1 X Γ = −= k φˆ? fˆ (2.59) k B θ k,ω k,ω k,ω where =(g) refers to the imaginary part of complex g; similarly, <(g) is its real part. Given Eq.(2.58), the kinetic entropy production rate can be written in terms of Γk: Z Z   ˙QL 1 ˜ ˜ Sk = − dτ dv (1 + log feq) ∂r ∂θφ f B θ,z,t * ( )+ ZZ Z 1 + log f +R Z 1 = − dθdz dv eq f∂˜ φ˜ − dr f˜ ∂ φ˜ ∂ log f B θ B θ r eq −R θ,z,t ZZ = − dτdv Γk ∂r log feq (2.60)

The latter equation (2.60) can now be written in Fourier space: ZZ ∂r log feq X S˙QL = = dτdv k φˆ? fˆ k B θ k,ω k,ω k,ω Z ¯ Z  ?  T X ? ω − ω = −= dτ ∂r log feq kθΦ Φ dv 1 − feq (2.61) eB ω − kzv k,ω 2.3 The fluid model 31 where we made use of the dispersion relation Eq.(2.55). The imaginary part of the velocity integral in Eq.(2.61) can be classically estimated a` la Landau, assuming the solution of the dispersion relation is damped: ω → ω + ıε with ε > 0. In that case, the Plemelj formula Eq.(D.6) reads: Z  ω − ω?  Z  ω − ω?  dv 1 − feq = P dv 1 − feq ω − kzv ω − kzv 2 " 2 ! # − ω ıπ ω 1 2ω2 ? ? k − ωn + ωT 2 − − ω f0 exp (2.62) |kz| 2ωk 2 P being Cauchy’s principal part of the integral. The kinetic entropy production rate has now an explicit expression: Z ˙QL X 2 QL Sk = dτ |kθΦ| Wk (2.63) k,ω where 2 r ? − ω π Ω 2ω2 QL ? k Wk =n ¯ (Ω − ω) exp (2.64) 2 |ωk|

?  2  ? ? ωT ω and Ω = ωn + 2 2 − 1 . ωk

The fluid entropy production rate In this section we shall now calculate the fluid counterpart of the kinetic entropy production rate Eqs.(2.63) and (2.64). The time evolution of the first three fluid moments Eqs.(2.10)–(2.13) reads: 1 ∂ n + [φ, n] + ∂ (un) = 0 (2.65) t B z 1 ∂ p e ∂ u + [φ, u] + u∂ u + z + ∂ φ = 0 (2.66) t B z nm m z 1 ∂ p + [φ, p] + ∂ (up) + ∂ q + 2p∂ u = 0 (2.67) t B z z z Since we derived a kinetic linear perturbation calculation, we shall also linearise the latter equa- tions (2.65)-(2.67). Consistently with the previous sections, we write: n =n ¯ +n ˜, u =u ˜ and p =p ¯ +p ˜, which is consistent with the kinetic calculation where the equilibrium (with a bar) refers to the Maxwellian distribution function Eq.(2.52) with no mean velocity. We shall also in- troduce the following normalisations: N ≡ n˜k,ω/n¯, U ≡ u˜k,ω/vT and P ≡ p˜k,ω/p¯. In Fourier space, Eqs.(2.65)-(2.67) assume the simple following form: ? ωk ω N = U − n Φ (2.68) ω ω ωk U = (P + Φ) (2.69) ω  ?  ωk q˜ ω P = k,ω + 3U − p Φ (2.70) ω pv¯ T ωk In a linearised system, the most general form q˜ can assume is –see also Eq.(2.39):

q˜k,ω = a n˜k,ω + b u˜k,ω + c p˜k,ω (2.71) where the most general a, b and c are complex operators, e.g.: a ≡ ar + ıai. Given the closure equation (2.71), the density and the pressure fluctuations can be expressed similarly as in Eq.(2.40) in terms of the electric potential and the equilibrium profiles only:

N = −n¯ Φ Rn (2.72)

P = −p¯Φ Rp (2.73) 32 The eternal problem of representation: which model for the plasma ?

The calculation of the response functions Ri is straightforward but somehow fastidious; it shall not be reported here since, as latterly stated, the aim here is to provide the reader with the method; the interested reader could soon refer to Ref.[?] for the complete details. Coming back to the entropy, the fluid analogue of the quasi-linear kinetic entropy Eq.(2.56) is defined as follows: Z QL QL ¯
Sf = dvSk f = fM (2.74) Z n¯ n o = dτ log p¯n¯−γ
+ log 2π + 1 (2.75) 2 where in this case, γ = 3. The entropy production rate involves the time derivatives of both the equilibrium density and pressure, which take the following general form in the limit r k⊥  1: 1 ∂ n¯ = − ∂ (rΓ ) (2.76) t r r f 1 ∂ p¯ = − ∂ (rQ ) (2.77) t r r f

D 1 ˜ E D 1 ˜ E where Γf ≡ − rB ∂θφ n˜ and Qf ≡ − rB ∂θφ p˜ . The brackets are defined as in section 2.3.3, Eq.(2.58). Both Γf and Qf read in Fourier space:   1 X ˆ? Γf = −=  kθφ nˆk,ω  (2.78) B k,ω k,ω   1 X ˆ? Qf = −=  kθφ pˆk,ω  (2.79) B k,ω k,ω

Given Eqs.(2.76) and (2.77), the fluid entropy production rate simplifies into:

1 Z  S˙QL = dτ −∂ Γ log p¯n¯−γ
+ log 2π + 1 f 2 r f n¯  +γ ∂ Γ − ∂ Q (2.80) r f p¯ r f 1 Z  p¯ 1  = dτ Γ ∂ log + Q ∂ (2.81) 2 f r n¯γ f r T¯ where p¯ =n ¯T¯. At this point, for the sake of tractability, the remainder of this section will focus on a plasma with constant radial density. The radial dependence of the pressure is kept. Thus Eq.(2.81) simplifies: 1 Z  Q  S˙QL = dτ∂ logp ¯ Γ − f (2.82) f 2 r f T¯ which can be written in Fourier space, given Eqs.(2.78) and (2.79): Z 1 ∂rp¯ X S˙QL = dτ k φˆ? = (P − N) (2.83) f 2 BT¯ θ k,ω k,ω

At this point, we recast the latter equation following the kinetic pattern Eq.(2.63): Z ˙QL X 2 QL Sf = dτ |kθΦ| Wf (2.84) k,ω 2.3 The fluid model 33

Figure 2.3: Comparison between the kinetic and the fluid weights, Eqs.(2.64) and (2.85) when both the instability threshold and the Second Law of thermodynamics are satisfied. where ¯ QL 1 T n¯  ? ?  Wf = = ωT [Rn(ω) − Rp(ω)] − 2ωnRn(ω) (2.85) 2 kθB The fluid weight function Eq.(2.85) is now expressed in terms of the equilibrium quantities n¯, p¯ and the closure a, b and c quantities only, Eq.(2.71). Matching the fluid and kinetic weights, Eqs.(2.64) and (2.85), constrains the choice of the closure equation.

Matching the fluid and kinetic weights Let us here consider the usual case where the density profile is much flatter than the temperature ? ? ? profile, i.e. ωn  ωT . For the sake of simplicity, we shall here take ωn ≈ 0. Since the kinetic weight is an even function of ω, the fluid weight should also be such. Enforcing this property, one finds that:

a = −(3 + b)c with <(c) = 0 and b ∈ R (2.86) At this stage, two degrees of freedom remain; provided we further constrain b = −3 + ℵ, with ℵ = 1 + 2 C > 1 k ρ  1 ? 2 and since ⊥ s , it is straightforward to show that this 1+4C(1−C)(ωk/ωT ) ? ? closure gives the correct general instability threshold ωT = (2 + b)ωn. In this case, the closure coefficients read: r 2 a = ı2 sgn(ω ) (2.87) π k b = −3 + ℵ (2.88) 2 r 2 c = −ı sgn(ω ) (2.89) ℵ π k which yields the closure equation:

r 2 2 r 2 q˜ = ı2 sgn(ω )n ˜ + (ℵ − 3)u ˜ − ı sgn(ω )p ˜ (2.90) π k ℵ π k Similarly to the previous section in which we derived the parametric closure Eq.(2.48), this special choice encompasses the Hammett and Perkins result [55], which is recovered in the limit of a vanishing temperature gradient. Furthermore, it is also immediate to show that in the case where b ? does not depend on ωT , the system intrinsically satisfies an H–theorem in the two following limit cases: ℵ = 1 (b = −2) and ℵ = 3 (b = 0). The comparison in these two cases between the kinetic and the fluid weight functions is respectively displayed in Fig.2.3–(a) and (b). 34 The eternal problem of representation: which model for the plasma ?

Without nonlinear simulations being performed implementing the two methods here displayed –either parametric Eq.(2.48) or entropy-based Eq.(2.90)– to derive very general closure rela- tions, a further discussion assessing the accuracy of these closure proposals should prove difficult. Nonetheless, general conclusions can be drawn [86] which are the following: Landau-fluid mod- els such as the one which shall be derived below (i) can generally, provided an adequate choice for the closure relation, meet some success while attempting to describe linear kinetic properties but (ii) have difficulty describing the nonlinear dynamics associated with resonances such as in Eq.(2.42). In particular, Landau-fluid models may only accurately describe nonlinear Landau res- onances when the interacting waves are not deeply resonant. Since this is especially not the case near instability threshold, this is an underlying reason why the Landau-fluid models fail to capture the nonlinear kinetic threshold for ITG turbulence [34]. Chapter 3

Calculating the plasma behaviour as an initial value problem – or the self-consistent description of the electric field in a gyrokinetic full–f code

c [. . . ] These considerations justify the view that a considerable mathematical effort towards a detailed understanding of the mech- anism of turbulence is called for. The entire experience with the subject indicates that the purely analytical approach is beset with difficulties, which at this moment are still prohibitive...Under these conditions there might be some hope to ‘break the deadlock’ by ex- tensive, but well-planned, computational efforts. It must be admit- ted that the problems in questions are too vast to be solved by a direct computational attack, that is, by an outright calculation of a representative family of special cases. There are, however, strong indications that one could name certain strategic points in this com- plex, where relevant information must be obtained by direct calcu- lations. . . This should, in the end, make an attack with analytical methods, that is truly more mathematical, possible. d John von Neumann, on turbulence, 1949.

NTHEREMAINDER of this dissertation, the plasma shall be described from the kinetic point of view. Plasmas lie on the boundary between ordered and disordered systems and cannot Ibe described either by simple, smooth mathematical functions, or by pure randomness. The spontaneous formation of spatio-temporal structures on a wide range of scales, as the remainder of this dissertation will emphasise, is one manifestation of plasma complexity. In order to address this full complexity, the description of the entire phase space is mandatory, disqualifying fluid models: the extraordinarily varied plasma behaviour is only fully captured in a kinetic context. The price to be paid is on the underlying complexity. Kinetic models are drastically more demanding on resources than fluid models are. They make an intensive use of massively paral- lel supercomputing and as such have greatly benefited from the dramatic enhancement of both the hardware capabilities –at present time, petaflop calculators are being built which are able to process 1015 floating point operations per second– and from new computational techniques based on fast solvers, massive parallelisation protocols, etc. Nonlinear behaviour which was far beyond investigation a decade ago now starts to be unveiled. This old area of physical sciences –plasma physics– which is deeply connected with the primal problems of (out of equilibrium) thermody- namics, turbulence, self-organisation within chaotic dynamics, etc. knows a new lease on life 36 Calculating the plasma behaviour as an initial value problem with the promise of Iter and the start of a systematic investigation of its nonlinear regimes by first-principles models. The turbulent fluctuations which must accurately be described are at much lower frequency than the high frequency cyclotron motion. It is therefore possible to incorporate part of this small scale temporal behaviour into the larger scales temporal dynamics of both the distribution function and the fields. The useful part of the distribution function now evolves in a five dimensional space generated by four slow variables and an adiabatic invariant. This model is known as the gyroki- netic model. In a first approximation, the gyrokinetic distribution function can be thought of as deduced from its six-dimensional kinetic analogue by an average procedure over the fast-varying gyrophase angle. This model consensually provides today’s deepest insight on plasma behaviour: an excellent example in which the nonlinear gyrokinetic formulation has applied well and has made a deep and long-lasting impact is the theoretical study of microturbulence in tokamaks and stellarators. The remainder of this dissertation shall make a vast use of this description and explore some important pending problems especially related to collisions and their interaction with the gy- rokinetic turbulence. At first though, the present chapter is restricted to collisionless plasmas. The following section 3.1 shall briefly introduce the gyrokinetic model and the numerical tool to solve it: GYSELA. Through the intervention of numerics, the problem intrinsically becomes an initial value problem for which specific initialisation techniques must be coined such that the late dy- namics of the plasma does not depend on the special choice of the initial state. The discussion of these techniques and more generally of the equilibrium can be found in section 3.2. It especially emphasises the central role of the electric field to capture the plasma dynamics.

3.1 A reduction of the kinetic description: the gyrokinetic model

Charged particles confined by a strong magnetic field with weak inhomogeneity execute three types of quasiperiodic motion [73, 92]: (i) a rapid Larmor gyromotion about a single magnetic field line, (ii) an intermediate bounce motion along a magnetic-field line driven by the parallel gradients and (iii) a slow drift motion across the magnetic field lines driven by the perpendicu- lar gradient and the magnetic curvature. The development of low-frequency gyrokinetic theory was motivated by the need to describe complex plasma dynamics over time scales that are long compared to the gyromotion time scale. This theory developed as a generalisation of guiding- centre theory [92, 83, 112] and goes beyond the guiding-centre phase-space coordinates histori- cally introduced by constructing a new set of gyrocentre phase space coordinates which describe gyroangle-averaged perturbed guiding-centre dynamics. Two successive steps are required to derive a self-consistent, energy-conserving set of equa- tions describing the nonlinear turbulent evolution of low-frequency, short perpendicular wave- length electromagnetic fluctuations in nonuniform magnetised plasmas: the initial phase-space particle coordinates become guiding-centre coordinates and then gyrocentre coordinates. This procedure involves the asymptotic decoupling of the fast gyromotion from (i) the kinetic Vlasov equation p.17 and (ii) the Maxwell equations and introduces a polarisation density and a magneti- sation current which are in turn properly accounted for by an exact gyrokinetic energy conservation law for the self-consistent newly built set of gyrokinetic Vlasov–Maxwell equations. The present dissertation shall not further describe the construction of modern nonlinear gyrokinetic theory; the interested reader may have a look at the review paper, Ref.[11]. The following section introduces the gyrokinetic model which shall be used in the remainder of this dissertation.

3.1.1 A special set of gyrokinetic equations to study plasma turbulence The magnetic geometry –i.e. the geometry of the magnetic field lines– is central for a correct description of the plasma instabilities and its nonlinear behaviour. Before explicitly writing the 3.1 A reduction of the kinetic description: the gyrokinetic model 37

Figure 3.1: Internal view of the JET tokamak (top) where the cold radiating plasma is visible (top right) since the hot core plasma does not radiate in the visible spectrum. At the bottom, the corresponding idealised toroidal magnetic geometry and its adopted notations. gyrokinetic set of equations, let us first briefly describe the geometry and the notations we shall use.

A foreword on the tokamak configuration The tokamak magnetic field configuration (see page 6) is axially symmetric at lowest order with closed toroidal magnetic surfaces in which a toroidal field is produced by currents in external coils and a poloidal field by a current in the plasma. Figure 3.1 (top) displays an internal view of the Joint European Torus (JET) tokamak superimposed with an image of a plasma taken with a visible spectrum video camera. The bottom diagram represents its schematic idealisation. Given this configuration, the general form of the magnetic field in a tokamak is:

B = I∇ϕ + ∇ϕ × ∇χ (3.1) where χ is the opposite of the poloidal flux (χ = −ψ in conventional notations), ϕ is the toroidal angle, and I is a flux function. Since it is not convenient to manipulate a flux as a variable, we 38 Calculating the plasma behaviour as an initial value problem moreover introduce a radius r such that χ is a function of r only. We define here the poloidal angle such that the safety factor is a flux function, i.e.:

B · ∇ϕ = q(r) (3.2) B · ∇θ so that an alternative form of the field is B = χ0∇r × ∇(qθ − ϕ), where the primes denote a derivative with respect to r: χ0 = dχ/dr. All quantities can then be expressed in e.g. the (eˆr, eˆθ, eˆϕ) orthonormal basis or in the widely used non orthogonal decomposition of space based on the magnetic field (e⊥, ek) = (eˆr, eˆθ, b = B/B) is also of wide use: parallel refers to the direction of the total magnetic field –sum of the strong toroidal field and the weak poloidal field and thus ek ≈ eˆϕ, whereas transverse refers to the poloidal cross section. The strong toroidal field provides the confinement in the toroidal direction and the weaker poloidal field provides the rotational transform –the winding up of the magnetic field lines, quan- tified by the safety factor q1. A ‘vertical’ magnetic field is necessary to control the tendency of the plasma column to expand in the direction of the major radius R = R0 + r cos θ. The poloidal field is stronger on the outside of the torus at (r, θ) = (a, 0), a being the minor radius and weakens on the inside where (r, θ) = (a, π). In modern tokamaks, the plasma position is controlled with an active feedback system using external coils to generate these transverse magnetic fields. It is also useful to compute the components of the plasma current J. The current in a tokamak must satisfy the constraints ∇ × B = µ0J (and therefore ∇ · J = 0), and J × B = ∇P . Hence it is of the form: 0 0 0 2 χ µ0J = −I B − µ0P R ∇ϕ (3.3)

The gyro-‘average’ operation

The nonlinear gyrokinetic Vlasov–Maxwell equations are traditionally derived through a multi- ple space-time-scale expansion which relies on the existence of one or more small dimensionless ordering parameters [41]: B = ρc/L  1 and δ = |δf/f| ∼ |δB|/B  1. These order- ing parameters are defined in terms of the characteristic physical parameters associated with the background magnetised plasma –represented by the distribution function f and the magnetic field B– and the fluctuation fields –represented by the perturbed distribution function δf and the per- turbed electric and magnetic fields δE and δB. Since we are not interested in resolving the fast Larmor motion, both the distribution function and the electromagnetic field shall be averaged over the rapidly varying gyroangle α. Let us explicitly consider how the generic quantity ℵ(r) defined for a particle changes for a guiding-centre. Since the position of the particle during its ¯ Larmor gyration is unknown, the corresponding value ℵ(xG) for the guiding-centre is such that: ¯ R 2π dα ℵ(xG) = 0 2π δ (|r − xG| − ρs) ℵ(r), which written in Fourier space reads:

Z 2π dα Z ¯ 3 ˆ ık·(r−xG+xG) ℵ(xG) = d k ℵk δ (|r − xG| − ρs) e (3.4) 0 2π Z Z 2π dα  3 ık·ρs ˆ ık·xG = d k e ℵke (3.5) 0 2π | {z } J0(k⊥ρs) = J (r) ~ ℵ(r) (3.6)

1The tokamak plasma current is usually driven by a toroidal electric field, induced by transformer action. Most of the available flux change is used in setting up the magnetic configuration and driving the current in the early stages when the plasma is cold and resistive. Thereafter, relatively little flux is consumed in maintaining a steady current in a highly conducting hot plasma –allowing tokamaks to be driven inductively for some time and the current to be maintained using non-inductive drive schemes. 3.1 A reduction of the kinetic description: the gyrokinetic model 39

Figure 3.2: Representation of the exact gyroaverage operator J0 and of one of its approximations in Fourier space.

where the bar over ℵ denotes the gyroaverage operation, xG is the position of the guiding centres: r = xG + ρs cos α and k · ρs = k⊥ρs cos α. Also, J0(k⊥ρs) is the Bessel function of first order, J is the gyroaverage operator –defined as the inverse Fourier transform of the Bessel function of ˆ first order J0(k⊥ρs), and ℵk is the Fourier transform of ℵ. It can be shown that the general form for the gyrokinetic equation then reads –compare to Eq.(2.9):   ∂tf¯− H(E¯, B¯ ), f¯ = C(f¯) (3.7)

In real space, the gyroaverage operation is a non-local convolution product: x(z) ~ y(z) = R +∞ −∞ x(u)y(z − u)du which thus reduces in Fourier space (k, ω) to a multiplication by the Bessel function of first order J0(k⊥ρs). This function contains the effect of Larmor averaging over the slow spatial variation of the perturbed electromagnetic fields.

This point importantly reminds of the previous discussion for fluid models in section 2.3.2, p.26 concerning the importance of dissipative mechanisms. A way to limit the filamentation problem in numerics is to cut off the small scales. This can be done (and is rather unphysical) by adding a numerical dissipation, e.g. a white noise. The drawback of this kind of dissipation is that its impact during the nonlinear regime is difficult to control –cf. for instance the debate [90] concerning particle-in-cell (PIC) codes, see p.46. A physical filter which consists in taking into account the finite Larmor radius effects; this is the role endorsed by the gyroaverage operator on the small scales in geometric space. As far as the velocity space is concerned, small scales are present in the bulk of the distribution function, and their regularisation requires the adjunction of a collision operator. This point shall be highlighted in the following chapter 4, p.74. Figure 3.2 displays the exact Bessel function of the first order. It can especially be seen that it has a long-lasting oscillating behaviour which requires, so as to be accurately described, the non-local knowledge of a significant part of the Fourier transverse space, which means that the mesoscopic calculation of the finite Larmor radius effects would require a very precise description of the sub-Larmor scales. In particular, the electromagnetic fields can, in this approach, significantly vary across a particle’s Larmor orbit. Slightly anticipating on the numerical point of view of the following sections, this property makes the exact calculation of the gyroaverage operator in finite differences overwhelmingly time-consuming in its exact Bessel form. A somewhat crude approximation to J0 consists in replacing it by its Pade´ approximation: 1 J0(k⊥ρs) ≈ 2 (3.8) 1 + (k⊥ρs/2) 40 Calculating the plasma behaviour as an initial value problem where sub-Larmor scales are strongly overdamped, as discussed in [106]. However, it has the advantage of being valid when k⊥ρs  1 and keeps J0 finite in the opposite limit k⊥ρs → ∞. The Frieman–Chen traditional approach [41] is valid up to second order in  ∼ B ∼ δ and involves a true averaging procedure as being described –with the associated loss of the informa- tion associated to the fast gyromotion. In contrast to these conventional methods used for deriving nonlinear gyrokinetic equations based on a regular perturbation expansion parameters and a direct gyrophase average, the modern nonlinear gyrokinetic derivation [11] pursues a reduction of dy- namical dimensionality via phase-space coordinate transformations. This latter derivation of the nonlinear gyrokinetic Vlasov equation is based on the construction of a time-dependent phase- space transformation from old particle coordinates to the new gyrocenter phase-space coordinates such that the new gyrocenter equations of motion are independent of the fast gyromotion time scale at arbitrary orders in B and δ. This transformation effectively decouples the fast gyromo- tion time scale from the slow reduced time scales and allows for a clearer ordering. Yet, it still contains the short-scale fast information associated with the Larmor motion. At lowest orders in B and δ, these two approaches are equivalent.

Charged particle motion

c Le chaos est impossible aux yeux de la raison, car il est impossible que, l’intelligence etant´ eternelle,´ il y ait jamais eu quelque chose d’oppose´ aux lois de la l’intelligence ; car le chaos est precis´ ement´ l’oppose´ de toutes les lois de la nature. d Voltaire, Le philosophe ignorant, 1766

All descriptions of plasma motion are based, ultimately, on the motions of the constituent particles. For the case of an unmagnetised plasma, the motions are fairly trivial, since the con- stituent particles move essentially in straight lines between collisions. The motions are also trivial in magnetised plasmas where the collision frequency ν greatly exceeds the Larmor gyrofrequency ωc = eB/m: in this case the particles are scattered after executing only a small fraction of gyro-orbit and therefore essentially move in straight lines between collisions. The situation of primal interest is that of a low collisional (ν  ωc) magnetised plasma primarily occurring in magnetic fusion and space plasma physics contexts. In these cases, the Larmor gyroradiusradius ρc = mv⊥/eB of the particle is also much smaller than the typical variation length-scale L of the −1 magnetic field and the gyroperiod ωc is much smaller than the typical time-scale τ on which these fields change. In such a plasma, the motion of the constituent particles consists, at leading order, of a rapid gyromotion perpendicular to the magnetic field lines, combined with a free-streaming parallel to the field lines. In addition to this, it is of special interest to calculate how this motion is perturbed by the spatial and temporal gradients in the electric E and magnetic B fields. The mo- tion of a charged particle in an arbitrarily complicated electromagnetic external field is analytically intractable in the general case. In some special cases though it is not so, especially in the so-called fusion-relevant adiabatic limit where a strong quasi-static magnetic field is externally imposed −1 and the assumed smallness of the parameters ρc/L and (ωcτ) can be exploited. This limit shall be assumed in what follows. It states that the external electromagnetic fields are slowly varying in both space and time at the scale of the Larmor motion of the particle. More quantitatively, ‘slowly varying’ means that:

• the typical evolution scale of the magnetic field is large as compared to the Larmor radius of the particle: ∇B ρ ρ ∼ c ∼ ρ ∼ 10−3  1 c B R ? 3.1 A reduction of the kinetic description: the gyrokinetic model 41

• the electromagnetic field felt by the particle is slowly varying as compared to the cyclotron motion: ∂ B ∂ E ω−1 t ∼ (ω τ )−1 1 ; ω−1 t 1 c B c D ≪ c E ≪ where τD is the characteristic current diffusion time. Accordingly, the general motion of a particle is composed of a fast Larmor rotating motion plus slow drifts of its guiding-centre. In the frame- work of the gyrokinetic theory [11], the evolution equations of the gyrocentre coordinates can be written in the very general form: dx 1 B∗ = v B∗ + b × ∇Ξ (3.9) k dt k e dvk B∗m = −B∗ · ∇Ξ (3.10) k dt (3.11) where: Ξ = eφ¯ + µB (3.12) mvk B∗ = B + ∇ × b (3.13) e mvk B∗ = B + b·∇ × b (3.14) k e

An alternative expression can be found by using the relation: ∇ × b = b × ∇B/B + µ0J/B: dx = v b∗ + v + v + v (3.15) dt k E×B ∇B c dvk mvk m = −µb∗ · ∇B − eb∗ · ∇φ¯ + v · ∇B (3.16) dt B E×B where

∗ B mvk µ0J b = ∗ + ∗ (3.17) Bk eBk B mvk B∗ = B + µ b · J (3.18) k eB 0 In Eqs.(3.15) and (3.16) appear the three drift velocities evaluated at the gyrocentre, namely: • the ‘E × B’ drift: 1 ¯ vE×B = ∗ b × ∇φ (3.19) Bk • the ‘grad–B’ drift: b  ∇B  v∇B = ∗ × µB (3.20) eBk B • the ‘curvature’ drift:   b 2 N vc = ∗ × mvk (3.21) eBk R

2 where N is locally transverse to the magnetic field and N/R = ∇⊥B/B + ∇⊥p/(B /µ0). 2 Introducing the dimensionless number β = p/(B /2µ0), ratio of the plasma kinetic pressure over the magnetic pressure which is usually very low in tokamaks: β ≈ 0.01, the second term is negligible and N/R ≈ ∇B/B. In this limit, the grad–B and curvature drifts can combine into the compact form: 2 mvk + µB ∇B vD = ∗ b × (3.22) eBk B 42 Calculating the plasma behaviour as an initial value problem

Interestingly, the great force of this latter approach is to provide the drift expressions at all orders in the gyrokinetic expansion, or differently speaking, these latter equations Eqs.(3.19)–(3.21) hold at all orders in the expansion of the gyroaverage operator, see previous section, p.38. It is a generalisation of the classical calculation to derive the drift solutions by iteratively solving the Lorentz equation of motion: dv m = e (E + v × B) (3.23) dt which, written for guiding-centres, reads: dv m G = e (E + v × B) − µ∇B (3.24) dt G

2 Here all quantities are evaluated at the guiding-centre position and µ = mv⊥/2B is the magnetic moment, often referred to as the adiabatic invariant. This last term embeds the effect on the motion of the particle at the Larmor scale of the large-scale fluctuations B˜ of the magnetic field: µ∇B = −ehv⊥×B˜ i but at a limited order in the gyrokinetic expansion (here, v⊥ is evaluated at the position of the particle, the fluctuations B˜ represent the discrepancy between the total magnetic field and that evaluated at the guiding-centre position and h·i is the average over a cyclotron motion). The classic expressions usually found in textbooks are based on the latterly mentioned iterative calculation while starting from Eq.(3.24). Essentially, the result given in textbooks is the existence of four drifts: • the ‘E × B’ drift: E × B v = (3.25) E×B B2 • the ‘grad–B’ drift: B „ ∇B « v = × µB (3.26) ∇B eB2 B • the ‘curvature’ drift: B „ N « v = × mv2 (3.27) c eB2 k R • and the higher order ‘polarisation’ drift:

m » d – v = B × (v + v + v ) (3.28) p eB2 dt E×B ∇B c

These expressions above, including the polarisation drift, are contained in the three general drift expressions Eqs.(3.19)–(3.21).

Simplified circular geometry

As the picture of the magnetic chamber Fig.3.1 (top) would suggest, there are advantages for confinement and achievable pressure with plasmas which are vertically elongated and slightly D shaped. In this dissertation, the effects of elongation and so-called triangularity are not addressed and the plasma magnetic surfaces are considered nested, circular and concentric2. It is therefore possible to greatly simplify the magnetic configuration and thus the equations of motion. Writing ζ = r/qR0, the magnetic field now reads:

R B = B 0 (e + ζe ) (3.29) 0 R ϕ θ

R0 p 2 The field modulus is therefore B = B0 R b0, where b0 = 1 + ζ . Quantities with a subscript 0 are computed at the magnetic axis (r = 0). It is straightforward to notice that since the modulus of the magnetic field decreases with the major radius, a tokamak has a ‘low field side’ on the outside, at (r, θ) = (+a, 0) and a ‘high field side’ on the inside, at (r, θ) = (−a, 0). Let us introduce B such that: R  2 ε q0 cos θ  ∇ × B = − r − (3.30) R0 qR q q R

2In a real tokamak, vertical elongation renders the plasma unstable against vertical displacement and additional coils are required to shape the plasma and control its position. 3.1 A reduction of the kinetic description: the gyrokinetic model 43

where εr = r/R. Then the generalised parallel gradient operator reads:   ∗ B R mvk ∂ϕ b · ∇ ≡ ∇k = ∗ ∂ϕ + ∂θ + ∗ (∇ × B) (3.31) b0Bk R qR0 eBk R and: ∗ mvk Bk = B + ∇ × B (3.32) eb0 Finally: mv mv   k k ∂rB 1 ¯ 1 ∂θB ¯ ∂rB 1 ¯ vE×B · ∇B = ∗ − ∂θφ + ∂rφ + ζ ∂ϕφ (3.33) B Bk b0 B r r B B R

The collisionless gyrokinetic equation3 then reads:

∂f¯ dvk + (v + v + v ) · ∇f¯+ v ∇ f¯+ ∂ f¯ = 0 (3.35) ∂t E×B ∇B c k k dt vk The first five terms of the gyrokinetic equation Eq.(3.35) contain the Lagrangian time derivative convecting along the guiding-centre trajectories and represent the essential nonlinearities associ- ated to micro-instabilities which are the possible underlying mechanisms for plasma turbulence. Let us end this introduction to gyrokinetics by briefly discussing the plasma micro-instabilities which we shall address with this latter equation Eq.(3.35).

Micro-instabilities: possible underlying mechanisms for plasma turbulence As already noticed in section 2.3, p.19, plasma turbulence differs in many ways from fluid tur- bulence, starting from the structure of sources and sinks. Plasma turbulence is driven by the free energy sources of the many plasma micro-instabilities, essentially the gradients of the density and temperature. The tokamak spectrum of instabilities is quite rich, though most micro-instabilities belong either to the pressure-driven modes or to the current-driven modes. The former is essen- tially related to the transverse current through the MHD equilibrium relation: ∇p = j⊥ × B whereas the latter is driven by the parallel current. These current-driven modes shall not be ad- dressed any further; they essentially set important limits on the maximum toroidal current that can flow in the plasma (so-called kink mode) or on the achievable pressure and current (so-called ballooning-kink mode). Their study is traditionally performed in the framework of the fluid MHD theory in which the magnetic field is a genuine primal quantity which allows to evaluate the elec- tric field and the sources. This dissertation shall not address the coupling between MHD and kinetic theories, namely how to embed within the kinetic approach the self-consistent fluctuations of the magnetic field and a microscopic Ohm’s law (with the induction and Hall effects and the central question of the plasma conductivity / resistivity). Though still in its early stages, this task is an ongoing active current research program. When speaking of possible underlying mechanisms for plasma turbulence, most micro-insta- bilities belong, in essence, to the pressure-driven family of interchange modes, with the notable exception though of the so-called slab ITG modes which shall not be further discussed. An inter- change mode is unstable when the gradient of the magnetic field is aligned with the equilibrium pressure gradient. In this case the exchange of two flux tubes around a field line releases free energy. Such a situation occurs in a tokamak on the low field side whereas the plasma is locally

3For consistency, it appears readily that in this simple geometry the laplacian, useful for the Poisson equation, reads: 1 1 1 ∇2φ = ∂ (rR∂ φ) + ∂ (R∂ φ) + ∂ φ (3.34) rR r r r2R θ θ R2 ϕϕ 44 Calculating the plasma behaviour as an initial value problem

Figure 3.3: Formal analogy between the hydrodynamic Rayleigh–Benard´ instability and the pressure-driven plasma interchange instability. stable with respect to interchange on the high field side. There is a formal analogy between the in- terchange instability and the hydrodynamic Rayleigh–Benard´ instability. The magnetic topology is crucial since the curvature plays the formal role of the gravity and the electric potential is the formal analogue of the hydrodynamic stream function. As pictured in Fig.3.3, in the Rayleigh- Benard´ instability, the temperature gradient and the gravity field point in the same direction; the resulting Archimedes force can overcome viscosity forces and an upwards instability appears: the convective cells (iso-contours of the stream function ψ) stretch upwards. Inversely, with the inter- change mode, when the pressure gradient and the gradient of the magnetic field are aligned and point in the same direction, the E × B, grad–B and curvature drifts Eqs.(3.19)–(3.21) combine to destabilise the convective cells (iso-contours of the electric potential φ) and an outwards instability appears on the low field side4. Since locally stable and unstable regions exist at any time in a tokamak, and since magnetic field lines do connect these regions, this process leads to modes which tend to be aligned along the equilibrium magnetic field. Differently speaking, the typical transverse size of a mode (k⊥ρi ∼ 0.1) is much smaller than its wavelength along the magnetic field (kk ∼ 1/qR). This feature persists in the nonlinear regime. Hence, even though 3D effects cannot be ignored due to magnetic shear, the turbulence in tokamak plasmas is quasi-2D. As compared to the hydrodynamic case, this remark is in the same vein than in section 2.3, p.19: an intrinsic anisotropy exists in plasma physics between the parallel and the transverse directions which bears some analogies with the geostrophic (stratified) fluid dynamics. Amongst this family of interchange instabilities, in the core of the plasma do exist micro- instabilities driven by the Ion Temperature Gradient (ITG). These are the electrostatic ITG drift modes and trapped ion modes. These modes are often referred to as ηi modes since they could be characterised by a value ηi = d(log Ti)/d(log n). The ITG family of instabilities plays a central role in plasma turbulence and represents one of von Neumann’s “representative family of special cases”, p.35. Other core instabilities are driven by the electron temperature gradient: electrostatic trapped electron and shorter wavelength electromagnetic ηe drift modes, and micro-tearing modes. To complete an already rather intricate picture, there are also fluid-like instabilities driven by pressure gradients: the current diffusive ballooning and neoclassical tearing modes. At last, at

4Also, since the trapped particles are localised on the low field side as it corresponds to the zone of minimum field along the field lines, these are expected to play a prominent role in the interchange process. 3.1 A reduction of the kinetic description: the gyrokinetic model 45 the collisional plasma edge, there is a range of fluid instabilities driven by gradients in pressure, resistivity and current. A brief overview of the main plasma instabilities is displayed in Table.3.1. The present dissertation shall restrict to the low wavenumber electrostatic ITG instability as a

Table 3.1: Main instabilities contributing to the turbulent transport in tokamaks. [after Ref.[93]]

major possible underlying mechanism for core plasma turbulence. Electrostatic means here that perturbations of the magnetic field are ignored, so that only the perturbed electric field matters; this approximation is valid in the core plasma, at low values of the β parameter (see p.41). Through this instability, the plasma drives couple directly to a broad range of turbulence scale lengths; in general, one cannot separate between the driving and inertial ranges, as is done in hy- drodynamic turbulence. Research on tokamak plasma turbulence has proceeded at several levels: (i) renormalisation of simple sets of equations which provide models for the turbulence; (ii) scale- invariance and dimensional analysis techniques and (iii) numerical solution of turbulence mod- els. While the renormalisation approach starts from perturbation theory and the scale-invariance approach is an alternative approach to renormalisation for the determination of the transport coef- ficients, the numerical solution of the primitive equations allows one to cover a broader parameter range than by the analytical approaches. It is a promising method since over the past decade, computational capabilities have increased such that tokamak-relevant resolutions are starting to be reached. A new generation of gyrokinetic codes are arriving to maturity; amongst them the GYrokinetic SEmi-LAgrangian code: GYSELA has some unique features. These shall be discussed in the remainder of this dissertation.

3.1.2 Numerics and the second life of first-principle modelling Kinetic models describe the particle velocity distribution function at each point in the plasma phase space. From a numerical point of view, there are two common approaches to tackle this problem, named after Euler and Lagrange and their two famous complementary yet radically different repre- sentations of fluid dynamics. The so-called Eulerian approach represents the distribution function once and for all discretised on a fixed grid in velocity and position. The other, known as the La- 46 Calculating the plasma behaviour as an initial value problem grangian, or particle-in-cell (PIC) technique, follows the trajectories in time of a large number of individual particles. Both techniques have their pros and cons, their defenders and (strong) detrac- tors. This aspect of codes has triggered much debate amongst the community over the last couple of years. A third way exists which is later born and tries to take advantage of both approaches. The semi-Lagrangian method [20] (which could easily have been called semi-Eulerian) discretises the five dimensional phase space on a mesh grid fixed in time, allowing for a low numerical noise as typical of Eulerian schemes, while at the same time the Vlasov equation is integrated along the trajectories using the invariance of the distribution function along its phase space characteristics, as latterly evoked in section 2.2, p.16. New mathematical techniques have been developed to perform local cubic spline interpolations [23] to evaluate the new value of the distribution function on the grid points. The integration along the trajectories is performed with a time-splitting algorithm which allows to split the 4D advection equation into a sequence of 2D and 1D advections. The global numerical scheme is second order accurate in time by using a symmetrical time-splitting scheme and a leap-frog algorithm. Further details can be found in Ref.[50]. In the history of plasma kinetic codes, PIC schemes, which could benefit from Monte-Carlo sampling techniques have come first. They face strong constraints related to numerical noise [90] at times smaller than the energy confinement time, though this difficulty is cured by techniques of ‘optimal loading’ and filtering [57, 10] or weight spreading [12, 19]. Eulerian codes do not face such difficulty yet require time-consuming high order numerical schemes to overcome intrinsic numerical dissipation. Semi-Lagrangian schemes were coined to avoid these drawbacks while not having any theoretical intrinsic Courant [or Courant–Friedrichs–Lewy (CFL)] condition –though in practice the time step cannot be chosen arbitrarily, this point remaining under investigation. The harsh constraint in semi-Larangian schemes lay in the interpolation process, the verification of energy conservation –ensured if small scales are filtered– and the large amount of computational memory it requires.

The GYrokinetic SEmi-LAgrangian code: GYSELA Building upon the results of the previous section, the present following paragraphs shall detail the adopted gyrokinetic model solved using the GYSELA code which shall now be adopted throughout this dissertation. Essentially, the following simplifying hypotheses are made: • (i) Simplified toroidal magnetic geometry: magnetic surfaces are circular and concen- tric; the analytical form for the magnetic field B is given by Eq.(3.29). • (ii) Electrostatic approximation: it rests on the assumption that the electric field is ir- rotational which implies from Faraday’s law that time-varying magnetic fluctuations are ignored. The magnetic field is externally imposed; induction, resistivity and more generally MHD phenomena are ignored. The dynamics of the plasma is governed by the perturbed electric potential φ. This approximation is valid in the core plasma, at low values of the β parameter (see p.41). The gyroaveraged E × B drift vE×B reads: B × ∇φ¯ v = (3.36) E B2 • (iii) Low β approximation: the efficiency of confinement of plasma pressure by the mag- netic field is moderate, but the plasma current is also moderate. The latterly evoked current- driven modes (see former section, p.43) are negligible and the core plasma turbulence is believed to be driven by interchange-like modes. In this approximation, the groaveraged v∇B and vc drifts combine into the following drift: 2 mvk + µB B × ∇B v = (3.37) D eB B2 3.1 A reduction of the kinetic description: the gyrokinetic model 47

which, for simplicity, will be generically called the curvature drift in the remainder of this dissertation.

• (iv) Adiabatic response of the electrons: while the ions are treated kinetically, the elec- trons are assumed to respond adiabatically to the low frequency fluctuations. Quasi-neutrality is assumed (see p.3): the electron and ion density perturbations densities are almost always exactly equal although electrons and ions are governed by entirely different physics. Within this framework, electron instabilities are not considered –e.g. the Trapped Electron Modes, see Table 3.1; the only investigated instabilities are the ITG modes.

• (v) Gyroaverage operator: the exact Bessel fuction J0 is approximated by its Pade´ ra- tional version which strongly overdamp sub-Larmor scales [106], Eq.(3.8).

• (vi) General validity: the following gyrokinetic model equations are fully consistent with the general gyrokinetic theory [11] at first order in  –see p.40– or equivalently, first order in ρ? = ρi/a, ρi being the ion gyroradius.

Within this framework, the gyrokinetic equation Eq.(3.35) reads:

∂f¯ dvk + (v + v ) · ∇f¯+ v ∇ f¯+ ∂ f¯ = 0 (3.38) ∂t E D k k dt vk where vE and vD are respectively given [hypotheses (ii) and (iii)] by Eqs.(3.36) and (3.37), ∇k is defined in Eq.(3.31) and the evolution equation of the parallel velocity dtvk is given by Eq.(3.16), provided Eqs.(3.31)–(3.33)5. Well above the Debye length –see discussions pp.3 and 72, self-consistency is ensured by the plasma neutrality constraint niZqi + neqe = 0, subscript ‘i’ (resp. ‘e’) referring to the ions (resp. electrons). For simplicity, let us consider the case of single charged ions Z = 1. Since the plasma is polarisable, though electrically neutral –see p.3– local fluctuations of the ion and electron density lead to local violations of this strict neutrality condition, which is recast into the weak quasi-neutrality form: δne/neq = δni/neq, which traduces the fact that δne − δni  neq ≈ ne ≈ ni. The electronic response is adiabatic [hypothesis (iv)]: δne/neq = e (φ − φ00) /Te(r), where RR 2 φ00 ≡ dθdϕ/4π φ accounts for the magnetic flux-surface average. Due to this hypothesis, the electron density fluctuations vanish for axisymmetric zonal modes. Inversely, the ion response  init pol reads: δni/neq = nGi − nGi /neq + δn /neq, where the ion guiding-centre density is nGi = R R ¯ init 2πB(r, θ)/m dµ dvk J· f, m is the ion mass and nGi refers to the initial ion guiding-centre density. This last term corresponds to the polarisation term and accounts for the difference between

5Though this simplification is not done in GYSELA, the gyrokinetic equation can be further simplified for large aspect ratio tokamaks, or equivalently at low values of εr = r/R. In this limit, the parallel gradient ∇k simply reads:

1 „ 1 « ∇ = ∂ + ∂ (3.39) k R ϕ q(r) θ

∗ ∗ ` ´ ? The modified magnetic field B also has a simpler form: B = B + mvk/e ∇ × b whose norm reads: B = ` ´ B + mvk/e b · ∇ × b. Then, straightforwardly, Eq.(3.16), at first order in ρ? [hypothesis (vi)], can be recast into: dv B∗ m k = − `µ∇B + e∇φ¯´ · (3.40) dt B∗ or equivalently, in terms of the magnetic field B, the parallel gradient Eq.(3.39) and the electric drift Eq.(3.36): dv mv m k = −µ∇ B − e∇ φ¯ + k v · ∇B (3.41) dt k k B E 48 Calculating the plasma behaviour as an initial value problem the guiding-centre and the particle density. This is reminiscent of the discrepancy which exists between the particle distribution function and the guiding-centre one. In the long wavelength limit (small k⊥ρi) and assuming that the ion distribution function is Maxwellian in µ, the Bessel function J can be expanded, leading to:

pol   δn 1 neq ≈ ∇⊥ · ∇⊥φ (3.42) neq neq B0 ωci

1 with ∇⊥ ≡ (∂r, r ∂θ). The Laplacian in the right-hand side comes from the Taylor expansion of the double gyroaveraged operator [35] and validly describes a quasi-neutral plasma in the k⊥ρs . 1 limit. Such an approximation is consistent with the Pade´ approximation [hypothesis (v)] of Eq.(3.8) in the latter limit k⊥ρs . 1 limit. Obviously, small scales should be treated more accurately when non-adiabatic electrons are included, either trapped or passing. The gyroaverage operator will be optimised in this perspective. The quasi-neutrality equation therefore reads:   δne δni e   1  init 1 neq = ⇔ φ − φ00 = nGi − nGi + ∇⊥ · ∇⊥φ (3.43) neq neq Te neq neq B0 ωci

Written in terms of the ion distribution function, this quasi-neutrality equation couples the gyroav- eraged f¯ to the particle electric potential φ:

  Z ∞ Z +∞ e   1 neq(r) 2πB  ¯ ¯  φ − φ00 − ∇⊥ · ∇⊥φ = dµ dvk J· f − finit Te(r) neq(r) B0 ωci mneq(r) 0 −∞ (3.44) The ion cyclotron pulsation is ωci, finit refers to the initial gyroaveraged distribution function for the ions and initially, the electric potential φ is a perturbation [hypothesis (ii)]. At last, as there is not more ambiguity, in the remainder of this dissertation, the bars over f¯ in Eqs.(3.38) and (3.44) and φ¯ in Eq.(3.38) shall be dropped.

The GYSELA code, at present time, solves this latter set of equations, with the exception of Coulomb collisions which shall be addressed in the following chapters. More than for its unique semi-Lagrangian numerical scheme, the GYSELA code [50, 51] has novel features in the small world of big gyrokinetic codes [roughly ∼ 19 such codes throughout the world, essentially in the US (∼ 9), Europe (∼ 6) and Japan (∼ 4)]: namely this code is both global and full–f which makes it highly attractive. These two concepts shall periodically appear throughout the dissertation; it is therefore desirable to start specifying them now:

• (i) Full–f capabilities is one of the most attractive features of GYSELA. Only two or three other codes have such capabilities. In full–f codes, the full distribution function f is calculated, without any scale-separation assumption between the reference background distribution function fref –often taken as a local Maxwellian– and its fluctuations, the per- turbed distribution function δf = f − fref . Being full–f is not a matter of advecting f and not δf in the gyrokinetic equation; it is more of assuming that the latter assumption δf  f which may initially be made as a mathematical choice for convenience remains valid at any further time. Most codes make this assumption6 and therefore artificially suppress the back- reaction of small-scales on the large-scale reference distribution function. Full–f codes do not do so and allow for non-Maxwellian background equilibrium; they also intrinsically model large-scale time-fluctuating axisymmetric so-called zonal flows (ZF) and their time- independant counter-part, the so-called mean flows (MF). Both flows are well-known to play a crucial role in the regulation of the turbulence [81, 67]. Such full–f codes are known to

6In contrast to the standard PIC method, the semi-Lagrangian method has no constraint on the computation time related to the condition δf ≤ f. 3.2 Initial state v.s. equilibrium: the central role of the electric field 49

be especially sensitive to a correct calculation of the radial electric field. Special difficulties also arise with neoclassical (collisional) equilibrium. The latter shall be addressed in the following chapter 4 whereas the former shall be addressed right now.

• (ii) Global codes solve the gyrokinetic equation over the whole volume of the plasma, i.e. the entire poloidal cross section (see Fig.3.1) for part or the totality of the toroidal di- rection. They differ from so-called flux-tube codes which concentrate the resolution of the gyrokinetic equation at the vicinity of a magnetic flux tube. While this flux-tube winds up

toroidally, it describes a complete annular section, yet poloidally localised. An example of the flux-tube GS2 code is given above. Straightforwardly, at similar mesh discretisations, global codes are numerically much more demanding; yet, beyond this numerical challenge, they alone can describe radially elongated structures, large-scale fast transport events and reactor-relevant boundary conditions. This latter condition is central in global codes since the mean temperature, pressure, . . . relax, as in real life, and an external heating must be supplied to make durable the tokamak configuration. A present and crucial topic is there- fore to numerically mimic tokamak-relevant so-called heat flux-driven conditions (see sec- tion 3.2.5, p.62 and Refs.[24, 46, 103]) which alone can account for a true ‘steady state’. Approximately half of the 19 codes are global, some having both flux-tube and global capa- bilities.

To conclude, since 2005, GYSELA has been thoroughly benchmarked either in linear and non- linear regimes against analytical predictions (growth rates, damping mechanisms, residual flows, level of turbulence and transport with and without regulating flows, . . . ) and against other compa- rable gyrokinetic codes, though differing by their resolution method and some of their character- istics (essentially full–f v.s. δf). All these are background results for the present dissertation and the interested reader could e.g. refer to Refs.[38, 47, 51, 52, 105].

3.2 Initial state v.s. equilibrium: the central role of the electric field

Because of individual and collective interactions between the particles which comprise a plasma, individual motions are intricate; the existence of resonances7 (e.g. Landau damping, see sec- tion 2.3.1, or collisional resonant interaction between trapped and passing particles, see following chapter 4) further complicate these motions and make the underlying physics time-irreversible. As already evoked in section 2.1, the classical notion of trajectories is thus often replaced by its statistical counterpart via the introduction of the distribution function f(x, t), probability to re- alise state x at time t. One can still evoke the concept of trajectories but the dynamical way to equilibrium requires a probabilistic representation [96]. This equilibrium problem founds a deep echo with the modern attempt to solve the complete Vlasov equation ab initio. Numerically speaking, both the questions of the equilibrium and its

7Poincare´ was deeply disturbed by his own works on resonance phenomena showing that most dynamical systems (and especially the Solar system) are non integrable. He would consider the problem of resonances as the problem of classical mechanics. 50 Calculating the plasma behaviour as an initial value problem counterpart the initial state are of extreme importance for gyrokinetic codes such as GYSELA which resolves the full distribution function. Defining the concept of an equilibrium is of special difficulty –this question highlights the general discussion of section 2.1, p.14– in a never resting medium which develops an intrinsic electromagnetic field such as a plasma. Indeed, if equilibrium stands for a static property of invariance in time and/or in space, strictly speaking, a plasma can never be at equilibrium; it can converge towards an hypothetic equilibrium, or fluctuate around it. Hence, this concept is either only pure mathematics or has to be broadened into a statistical con- cept. This debate shall not be further evoked here, the goal was only to superficially point out the deep conceptual difficulties one would face trying to define it the traditional (and intuitive) way8 in terms of characteristic averages over lengths and time. Here, the adopted point of view so as to define equilibrium tries to be less reliant on the existence of such characteristic averages: it rests on the existence of dynamical motion invariants which connect to the very general Hamiltonian or Lagrangian formulations of plasma physics through action principles. These actions –Lagrangian variable actions, Hamiltonian-Jacobi actions or kinetic Clebsch actions– are of special importance to highlight the relationship between symmetries and constants of motion; used with Noether’s theorem, they provide a powerful and reliable way to physically identify energy and other invari- ants. Though such formalism shall not be displayed here, a central result for the present discussion is the existence of three motion invariants in a integrable and quasi-perodic system, which, in the current framework are:

mv2 the adiabatic invariant: µ = ⊥ (3.45) 2B 1 the total energy: H = mv2 + µB + eφ(r, θ) (3.46) 2 k the toroidal kinetic momentum: Pϕ = mRvϕ + eψ (3.47) here, ψ is the poloidal flux. In more intuitive ways: the Hamiltonian H remains a motion invariant while the forces exerted on the system are conservative; also, while the magnetic topology is axisymmetric –which is relevant at lowest order in a tokamak– the toroidal kinetic momentum Pϕ is also an invariant. In the limit of adiabatic theory (assuming both that the characteristic evolution times of the system are much faster than the evolution time of the magnetic field and the spatial excursion of the particles is small compared to the space-evolution of this field), which is here assumed, µ is also a motion invariant9. Coming from the Liouville theorem Eq.(2.4), the distribution function of a quasi-isolated sys- tem remains constant in time along the phase space trajectories, which states that the only com- binations of coordinates q and momenta p the equilibrium distribution function can bear must remain invariant during the system’s motion. This way, without e.g. collisions, an equilibrium it- self is a motion invariant and the concept of equilibrium contains the concept of motion invariance.

8With good approximation, the tokamak is an integrable and quasi-periodic system in which an action-angle (α, J) coordinate system can be built. One can then traditionally define equilibrium in these coordinates by an averaging process of the distribution function f over time and angles so as to ‘suppress’ fluctuations. In the quasi-linear approxi- mation, the time evolution on the axisymmetric (o, o) Fourier component of f can be put into a diffusion-like equation ¯ 2 ¯ ∂tfoo − ∂JD¯ ∂Jfoo = 0. Doing so, one introduces a typical evolution time τD ∼ a /T r(D¯), larger than the intrinsic characteristic time related to the Larmor motion: τL ∼ 1/Ωc ∼ a/vT , where a, the minor radius, denotes a character- istic length of the system. Only when a time scale separation is possible between τD and τL and when the quasi-linear approximation is legitimate is the time-averaging procedure to define equilibrium valid. The statistical description also constrains upstream a minimal descriptible interval of time τstat to exist, enforcing the system has explored all micro- scopic allowed states a great number of times. Thus, all accessible phenomena must have much larger characteristic times, and especially τstat  τL. 9This latter adiabatic hypothesis deeply constrains the Hamiltonian to be slowly varying in time. To make it rigorous, one usually sets forth a slowness parameter  [73, 78]. With no special conditions imposed on the Hamiltonain, the constructed adiabatic invariants explicitely depend (slowly) on time. Inversely, provided the Hamiltonian is a smooth function of time, the well-known exact adiabatic invariance of µ to arbitrarily large orders in  is recovered [89]. 3.2 Initial state v.s. equilibrium: the central role of the electric field 51

This is an exact result for the microcanonical distribution and thus only valid for short times re- garding the characteristic relaxation time of the global system. For larger times, the quasi-isolated assumption is no longer valid. Different parts of the system can reach local equilibrium with dif- ferent characteristic times. In a Markovian-like framework, these characteristic times do remain shorter than the global relaxation time of the whole system; in a more general context though where the problem could be non-local in time, these different time scales can become compara- ble –this happens in a tokamak e.g. while zonal flows, feeding on the turbulence can nonlinearly back-react on the background ‘equlibrium’ state. Thus, equilibrium is no longer as easily defined as in the microcanonical case and becomes intrinsically a dynamical process. One can then try the following characterisation:

An equilibrium is a global ordered-like state, dynamically building up and perennating from many other possibly evolving sub-states. Each of the latter could be on its own either stationary, fluctuating or even chaotic –here, order would refer to a quantity which could be described in terms of smoothed mathematical functions, e.g. functions in the L2 Lebesgue space. We choose to define (operationally –a general highlight to this problem can be found in section 2.1, p.14) an equilibrium solution of the collisionless gyrokinetic system as any L2 arbitrary function of the latter three invariants Eqs.(3.45)–(3.47). A such function will be called canonical. Importantly, both (i) in the Scrape Off Layer (SOL) and (ii) with colli- sions C, see following chapter 4, this definition breaks since (i) it is difficult to coin motion invariants in the SOL, leading to great difficulties to initialise a full–f edge gyrokinetic code [74] –see discussion in the following sections and (ii) during a collision the local energy is not conserved for a single particle and the general problem of how to define equilibrium / average quantities arises again. In particular, the only collisional equilibrium solution is the Galilean-invariant shifted Maxwellian –see items (ii) and (iii) p.71. To be more precise, though collisions conserve the total energy –see Eq.(4.21): R |v|2C = 0, collisions do not satisfy: C(|v|2) = 0. On even longer times, some kind of dissipation must be taken into account and collisions ac- counted for. In an infinitely-collisional limit, the exact solution at long times of the Fokker–Planck equation is the classical shifted Maxwellian fSM :

fSM (v) = fLM (v − U) (3.48) 2 n(r) − mv fLM (v) = e 2T (r) (3.49) 2πT (r)/m
3/2 which cannot be expressed as a function of the motion invariants since dr/dt 6= 0. Neither can the R 3 local Maxwellian distribution function fLM . Here, U = 1/n d v vf is the fluid mean velocity. In the tokamak-relevant weakly collisional limit, the equilibrium is no longer either a shifted (or local) Maxwellian or an arbitrary function of the motion invariants rather than some in-between (and in the general case non analytical) function, see following chapter, section 4.2.4 p.89 for further details.

A reliable prediction of the turbulent transport is of uttermost importance in fusion devices. It is known to strongly depend on the interplay between microturbulence and large scale flows. The major importance of a correct treatment of the the zonal flows has been widely recognised [26]. Direct numerical simulation using gyrokinetic modelling is now able to address such issues. In collisionless models, an important question has emerged on the choice of the initial state. Follow- ing [65], we will distinguish between the local and the canonical Maxwellian initial distribution functions. The latter, since it is a function of the motion invariants only, is an equilibrium solu- tion of the gyrokinetic equation. Inversely, the former is not, and depends on the label of the flux surface. 52 Calculating the plasma behaviour as an initial value problem

When starting with a local Maxwellian, several authors [65, 1, 47, 29], including us, reported the spurious generation of large-scale sheared electric flows which hinder the turbulent modes and are likely to quench the onset of the turbulence. This chapter will show that this result, correct at short simulation times (twice the linear growth, at most), breaks down in a fully developed steady turbulent state (at least five to ten times the linear growth). In such a state, the statistical properties of the turbulence are independent of the choice of the initial state: no memory of its previous history is held within the system. More precisely, the transport level is the same since similar temperature profiles and turbulent heat fluxes are obtained. The turbulence itself also exhibits the same statistical properties, as e.g. Fig. 3.12 will exemplify. On the other hand, when starting with a canonical Maxwellian, no such flows are generated and the turbulence develops first hand. It is acknowledged that a proper definition for an initial equilibrium state is a crucial piece of physics for gyrokinetic linear studies. Without flux-driven gyrokinetic simulations, it also avoids a detrimental delay since (i) long transients are avoided and (ii) profile relaxation in density and temperature is reduced. These two points are especially important for a new generation of global and full–f codes like the GYSELA code [50, 51] since (i) large scale flows are intrinsically modelled and (ii) the mean profiles can evolve freely. The main issues the remainder of this chapter will address are twofold: (i) how does the sys- tem evolve from its non equilibrium initial state to an equilibrium and (ii) how does this evolution towards an equilibrium impact the onset of underlying instabilities, and ultimately the turbulent transport. The first point is crucial to discriminate between instability-driven fluctuations and time evolution towards the equilibrium. The second axis clarifies whether a non-equilibrium initial state can durably impact the dynamics of the turbulence itself. After showing that a non canonical initial state shall generate large-scale sheared flows which could hinder [9, 116, 54] the growth of unstable modes (section 3.2.1) we shall define an operational canonical initial state which avoids this drawback (section 3.2.2). In this section, we shall also show the consistency of this latter canonical choice with the fluid model. Section 3.2.3 provides an analytical prediction of the tem- poral dynamics, the poloidal structure and the saturation time of the flows generated due to a non canonical initial state. These results are successfully compared to dedicated numerical simulations with GYSELA. The saturation processes for these flows are discussed in section 3.2.4 and their nonlinear impact on the transport and the turbulence is reserved for section 3.2.5. A summary and discussion of the important results can be found in section 3.2.6.

3.2.1 Non-canonical initial distribution functions generate poloidal sheared flows A full–f turbulent simulation is initialised with an initial choice of a distribution function plus a bath of small amplitude modes. These small perturbations eventually grow and nonlinearly saturate, provided the initial equilibrium is above the linear stability threshold. Inversely, if the initial distribution function is not an equilibrium function, the system will first evolve towards an equilibrium. The consequences are twofold: the time evolution of the initial perturbations will be hard to discriminate from that of the search of an equilibrium, and the equilibrium found by the system, if any, may turn out to be stable with regard to perturbations. To start quantifying this idea, let us investigate what happens when the initial state is a local Maxwellian Eq.(3.49), i.e. the usual choice for most collisionless gyrokinetic codes. This choice leads to the onset of a large-scale electric field. The fundamental reason is that such a non-canonical distribution function does not carry any parallel flow which would ensure the compensation of the vertical charge polarisation governed by the curvature and grad-B drifts in a toroidal geometry. The fluid analogous of such a physics is the fact that the divergence of the diamagnetic velocity has to be balanced by the divergence of the parallel flow [58]. Let us look for equilibrium solutions at vanishing electric field of the following general form: −E/T (r) 2 feq = gµ(r, θ, vk)e , where E = mvk/2 + µB is the energy and gµ is a given function of 3.2 Initial state v.s. equilibrium: the central role of the electric field 53

r, θ and vk parametrised by µ. The gyrokinetic equation Eq.(3.38) then reads:  ∂ T ∂ T cos θ  v g µB r sin θ + g v2m r sin θ + ∂ g sin θ + ∂ g g µ T 2 µ k 2T 2 r µ θ µ r µBr sin θ vk +∂ g − ∂ g = 0 (3.50) vk µ mqR2 θ µ qR where vg is the modulus of vD Eq.(3.37). We concentrate on an axisymmetric equilibrium, i.e. ∂ϕ ≡ 0 –consistently with Pϕ Eq.(3.47) being an invariant. We intend to show that gµ cannot be an even function of vk and must be such that any equilibrium at vanishing electric field requires a finite parallel flow. To proceed, let us assume by contradiction an equilibrium distribution function 2 P∞ 2n can exist, with a fixed even parity in vk, such that gµ(r, θ, vk) = n=0 an(r, θ) vk . Projecting  2p 2p+1 Eq.(3.50) on the vk , vk (p ∈ N) basis allows one to separate the odd and even parities in vk: ∂ T D E ∂ T D E µB r sin θ g v v2p + m sin θ r g v v2p+2 T 2 µ g k 2T 2 µ g k D E cos θ D E + sin θ ∂ g v v2p + ∂ g v v2p = 0 (3.51a) r µ g k r θ µ g k µBr sin θ D E D E ∂ g v2p+1 − ∂ g v2p+2 = 0 (3.51b) mR vk µ k θ µ k R +∞ 2 where the brackets stand for hGi ≡ −∞ dvk G exp(−mvk/2T0). Using the expression of gµ, Eq.(3.51b) reads as follows: ∞  n     X 2T0 1 2 µBr sin θ Γ n + p + na − ∂ a = 0 (3.52) m 2 n m R θ n−1 n=0 R ∞ n−1 −t with Γ(n) ≡ 0 dt t e . Since Eq.(3.52) must remain valid for all integer p, the following relationship holds, for all integers n ≥ 1: mR ∂ a a = θ n−1 (3.53) n 2µBrsinθ n

Injecting such a relationship in Eq.(3.51a) yields for the a0 term: µB∂ B ∂ T [B, a ] = a θ r (3.54) 0 0 r T 2 1 1 2 where [g, h] ≡ r (∂rg∂θh−∂rh∂θg) is a Poisson bracket. Equation (3.54) implies that sin θ [B, a0] = 2 2µB ∂rT 2 R T 2 a0. Integrated over r and θ, the latter equation can be recast as follows: Z 2 Z r a0 n 1 ∂rT o h 2 B i max dr dθ 2 2 − 2µ 2 = dθ a0 2 (3.55) R rsin θ T R rmin The left-hand side integrand is strictly positive. The right-hand side integrand depends on θ only. Therefore, when differentiating Eq.(3.55) with respect to r, and for any a0 and T regular enough, the only physical solution constrains a0 to be identically vanishing. Equation (3.53) then implies that all the other coefficients an are also identically zero. Thus, with a vanishing electric potential, the only even solution in vk of the gyrokinetic equation is null everywhere. In other words, any possible equilibrium solution which is even in vk –which is especially the case if gµ is independent of vk– is characterised by a non-vanishing electric field. Alternatively, equilibrium solutions at vanishing electric field are asymmetric in vk, such that there exists a finite parallel flow analogous to the Pfirsch–Schluter¨ current. Notice that this flow may be vanishing when averaged on a flux surface, similarly to the Pfirsch–Schluter¨ current. As will be shown in the following section, such a property is intrinsically fulfilled when the initial state is canonical. 54 Calculating the plasma behaviour as an initial value problem

3.2.2 Canonical equilibrium in GYSELA Any arbitrary function of the motion invariants Eqs.(3.45)–(3.47) represents a canonical equilib- rium of the collisionless gyrokinetic system. Let fCM be the canonical Maxwellian: n(¯r) f = e−H/T (¯r)e+eφ(¯r)/T (¯r) (3.56) CM (2πT (¯r)/m)3/2 where r¯ stands for an effective radial coordinate derived from the third invariant: r¯ = r0 − q0Pϕ/r0eB0 + h(µ, H), such that fCM is a function of the motion invariants only (the quantities with a label 0 are defined at half-radius of the simulation box and h is an arbitrary function of the motion invariants µ and H). The density n and temperature T are no longer radial profiles, but also functions of the motion invariant Pϕ. Since the parallel dynamics is crucial to equilibrium, r¯ is written in terms of Pϕ so as to bear this dependence. Following [1], a good choice appears to be: q0 h i mq0 h i r¯ = r0 − ψ(r) − ψ(r0) − Rvk − R0v¯k (3.57) r0 eB0r0 R r r0dr0 where ψ(r) = 0 q . Also, the last term in Eq.(3.57) is by definition a motion invariant, which ? will be assumed to scale like ρ , the ratio of the ion gyroradius ρi over the minor radius: r 2 p v¯ = Sign(v ) E − µB Y(E − µB ) (3.58) k k m max max with Y the Heaviside function and Bmax the maximum of the magnetic field on the whole sim- ulation box. It is worth noticing that, due to the explicit dependence in vk of the invariant r¯, the canonical function fCM exhibits an asymmetry in vk. Such a property allows for the compensa- tion of the vertical polarisation by a finite parallel flow, as discussed in the previous section. This justifies a posteriori the use of the invariant r¯ as an effective radial coordinate. So as to minimise ? the parallel velocity, v¯k Eq.(3.58) is chosen such that (Rvk − R0v¯k) scales like ρ . It is important to notice that even though the difference between r and r¯ is small, it carries a new piece of physics: the fact that the guiding-centres deviate from a magnetic field line by a few Larmor radii (a banana width at most). This radial excursion leads to a parallel flow.

The impact of the local Maxwellian fLM Eq.(3.49) as compared to this latter canonical Max- wellian Eq.(3.56) on the early and late evolution of the plasma is successively investigated in the following sections. Before proceeding, it is important to check that, as expected, the system of equations and the method we use here are fully consistent with the more intuitive fluid model. In particular, the gyrokinetic approach does embed the fluid magnetising current and we do recover the well-known force balance equation.

Consistency with the fluid description The consistency between fluid and gyrokinetic descriptions is a matter of current work, as ex- emplified in Ref.[97] for the coupling between the gyrokinetic and the MHD frameworks. The present section focuses on some aspects of this consistency, restricted to the simplified equations solved by the GYSELA code. Namely, we only consider the calculation restricted to the first order in ρ?, consistently with the latter reference. It is worth noticing that Eqs.(3.57) and (3.58) belong to the following larger class of equilibrium distribution functions:     n(ψ¯) H eφ(ψ¯) mω¯kκ¯ f = exp − + 1 + (3.59) CM (2πT (ψ¯)/m)3/2 T (ψ¯) T (ψ¯) T (ψ¯)

¯ ¯ Pϕ mRvϕ ¯ Here, ψ is derived from the third invariant: ψ ≡ e = ψ + e . ψ is analogous to r¯ in that it represents an effective radial coordinate in the case of a more general class of axisymmetric 3.2 Initial state v.s. equilibrium: the central role of the electric field 55

Figure 3.4: In flux coordinates, the field lines eˆk are straight lines. magnetic equilibria of the form: B = I∇ϕ + ∇ψ × ∇ϕ. Here, I is a flux function I(ψ) and the ¯ ¯ ‘density’ n(ψ) and ‘temperature’ T (ψ) profiles are motion invariants. At last, ω¯k and κ¯ are arbi- ¯ ? trary functions of the motion invariants such that mω¯kκ/T¯ (ψ) ∝ ρ . This section aims at showing the simple relationship between κ¯ and fluid quantities such that GYSELA shall intrinsically satisfy the force balance equation. This latter equation states that:

e n (E + V × B) = ∇p (3.60)

Summing over all species, this equation implies B·∇p = 0, further implying E·B = −B·∇φ = 0. This means that the pressure p and the electrostatic potential φ are flux functions: they only depend on ψ. The general solution of Eq.(3.60) then reads:

B B  p0  B V = V + V? + V = −∇ψ × φ0 + + V (3.61) E k B B2 ne k B where p0 and φ0 stand for the derivatives of p and φ with respect to ψ. Uppercase letters V refer to fluid velocities. The cross-product can be derived from the general expression of the magnetic 0 2 2 κf −I  0 p  Vk field: ∇ψ × B = IB − B R ∇ϕ. Let us define ≡ 2 φ + + . With these new Bmax(ψ) B ne B notations, the fluid velocity V Eq.(3.61) can be recast into:

κ  p0  V = f B + φ0 + R2∇ϕ (3.62) Bmax(ψ) ne where Bmax is the maximum magnetic field on each magnetic surface: Bmax(ψ) ≡ B(ψ, θ = π). Figure 3.4 illustrates the different ways of expressing the fluid velocity V: field aligned and transverse coordinates are used in Eq.(3.61), whereas Eq.(3.62) expresses V as a difference between two adjacent directions, namely ∇ϕ and B. In this latter system of coordinates, the matching between the fluid and the gyrokinetic approaches is easily tackled. In the limit of large aspect ratios, incompressibility can be reasonably assumed: ∇ · V = 0. In this case, B · ∇ κf Bmax is vanishing, such that κf is a flux function κf (ψ). The parallel and transverse components of V then read: 0 B  0 p  I(ψ) Vk = κf (ψ) + φ + (3.63) Bmax(ψ¯) ne B B × ∇p V = V + V? = V + (3.64) ⊥ E E enB2 56 Calculating the plasma behaviour as an initial value problem

Let us now express the fluid-like velocity as derived from the general kinetic equilibrium Eq.(3.59). The standard definitions for the parallel and transverse guiding-centre fluid velocities 1 R 1 R of such an equilibrium are: Vk = n vkfCM and V⊥ = n (vg + vE)fCM , where integration is R R ∞ R +∞ carried out over normalised phase space ≡ 0 (2πB/m)dµ −∞ dvk. It is easy to show that both kinetic and fluid transverse flows exhibit the same divergence. Indeed, let us first recall that R B ∇B  R 2  e vg ·∇fCM = B2 × B ·∇ (mvk +µB)fCM . The MHD equilibrium imposes j×B = ∇p, B×∇B B×∇p such that the later term can be recast into 2 B3 ·∇p = ∇ · ( B2 ). The divergence of V⊥  B×∇p  then reads: ∇ · V⊥ = ∇ · VE + enB2 . One recovers both the electric and the diamagnetic drifts obtained in the fluid description. Up to a rotational term V⊥ and V⊥ are then equal. The calculation of the magnetisation current from the transverse particle fluid velocity is given in appendix J. As far as the parallel flow is concerned, first notice that the difference between ψ¯ and ψ, ? namely mRvϕ/e, remains small and scales like ρ . One can then Taylor expand fCM , Eq.(3.59):

¯ ¯ dfCM fCM (ψ) ' fCM (ψ) + (ψ − ψ) ¯ dψ ψ  0 0 0  mωkκ mIvk n T hE 3i eφ ' f (ψ) + f (ψ) + f (ψ) + − + (3.65) LM LM T (ψ) LM e n T T 2 T ¯ ¯ ¯ ωk and κ correspond to ω¯k(ψ) and κ¯(ψ) where ψ is replaced by ψ, respectively and fLM is the local Maxwellian Eq.(3.49); therefore, fLM (ψ) does not carry a parallel flow. Only do the two last terms in Eq.(3.65) contribute to Vk:

mκ I(ψ) p0  V = hhv ω ii + φ0 + (3.66) k T k k B ne

1 R ∞ R +∞ where hhGii ≡ n 0 2πB/mdµ −∞ dvkGfLM (ψ). The parallel fluid velocities Eqs.(3.66) and (3.63) are equal provided ω¯k reads:

r 2 q ω¯ = Sign(v ) E − µB (ψ¯) Y(E − µB (ψ¯)) (3.67) k k m max max

Y being the Heaviside function. In this case: mκ hhv ω ii = κ B and V and V are recon- T k k f Bmax k k ¯ ciled provided κ = κf , or equivalently κ¯ = κf (ψ). This shows the deep coherency between the two widely used gyrokinetic and fluid approaches.

With consistency checks completed, we now address the interesting question of the dynamics of the plasma when initial state finit is not an equilibrium solution of the collisionless gyrokinetic equation (3.38).

3.2.3 Early self-organising dynamics due to initial conditions

The system is initialised with the local Maxwellian, Eq.(3.49), namely finit ≡ fLM . We write:

f(r, θ, vk, µ, t) = fLM (r, E) + δfLM (r, θ, vk, µ, t) (3.68)

2 where E stands for the total energy mvk/2 + µB and the axisymmetry of the problem imposes f to be independent of the toroidal angle. Initially, δfLM (t = 0) = 0. We call such a case out of equilibrium. Characterising the dynamics of such a system essentially consists in determining the radial component of the electric field which builds up. At vanishing electric field, an equilibrium flow along the parallel direction must develop to balance the vertical polarisation associated to the 3.2 Initial state v.s. equilibrium: the central role of the electric field 57 ion curvature drift. It is analogous to the Pfirsch–Schluter¨ current carried by the electrons. With a local Maxwellian initial state which is even in vk, no such flow can exist initially or develop on short times (see section 3.2.1). The remaining possibility for the system consists in generating a large-scale polarisation flow in the poloidal plane which in turn can be seen as resulting from a large-scale driven electric potential. In what follows, we calculate this electric potential and give a prediction of the level, time growth and saturation time of such flows. These results are detailed in Ref.[33].

In this framework, at short times, δfLM is a perturbation. We restrict its Fourier expansion to the first modes only, which appear to be dominant:

s c δfLM ≈ f00(r, E, t) + f10(r, E, t) sin θ + f10(r, E, t) cos θ (3.69)

s c where f00 corresponds to the axisymmetric mode: (m, n) = (0, 0), f10 and f10 correspond to the sidebands: (m, n) = (±1, 0), m and n being respectively the poloidal and toroidal mode numbers. As far as analytical calculations are concerned, in this section, only n = 0 modes are considered, which allows one to focus on the dynamics of equilibrium axisymmetric flows. Inversely, no such constraint is made on the numerical results where all possible (n, m) modes are allowed to develop and interact. The remarkable agreement which shall be displayed in the remainder of this section between the two approaches emphasises the crucial role of the equilibrium n = 0 modes. With fLM as an initial state, the quasi-neutrality equation is satisfied for a vanishing electric potential φ(t = 0) = 0. Therefore, initially at least, φ is a perturbation; we also expand it on its first Fourier modes: s c φ(r, θ, t) ≈ φ00(r, t) + φ10(r, t) sin θ + φ10(r, t) cos θ (3.70) s c s c No ordering is here presupposed between respectively f00, f10, f10 and φ00, φ10, φ10. The small parameter  of this study is:  f f s f c   = max 00 , 10 , 10 (3.71) fLM fLM fLM which will be shown to scale like ρ?. The sidebands f10 build up because of the polarisation due to the curvature drift. It follows that at short times, Eq.(3.38) writes at leading order in :

∂tf + vD · ∇f = 0 (3.72)

As far as analytical calculations are concerned, in the remainder of this section, gyroaveraged quantities will be assumed to be equal to the quantities themselves, namely: J ≡ Identity. Such a simplification is valid in the limit of large gradient lengths with regard to ρs, as in the case of the present analysis. Eq.(3.72), coupled to the quasi-neutrality equation (3.44), enable to calculate the time evolution of both the perturbed distribution function and the electric potential. It is possible to selectively derive the dynamics of each perturbed part, given its parity in the poloidal angle. The main equation on the distribution function which governs the dynamics of the system at short times is: s vdfLM f10 = t (3.73) Lp

 2  2 with vd = mvk + µB /eR0B0, Teq/eB0 = ρscs and cs = Teq/m. For what follows, we note −1 −1 −1 −1 LX ≡ ∂rX/X and κX ≡ ∂rrX/X. At last: Lp ≡ Ln + LT (E/Teq − 3/2). When coupled s to the quasi-neutrality equation, this latter equation, yields the time evolution of φ10 at first order in ρ? –see the detailed calculation in appendix K:

e s s φ10 ≈ Ω10 t (3.74) Teq s 2ρscs Ω10 = (3.75) R0Lp 58 Calculating the plasma behaviour as an initial value problem

Figure 3.5: Time evolution of the transient electric potential when the plasma is initially out of equilibrium. Solid line: analytical estimation of the growth rate; dotted line, the complete simulation with GYSELA (ρ? = 1/400). See section 3.2.4 for discussion about τE and τk.

The detailed calculation shows that Eq.(3.74) is the key equation which allows one to calculate the time evolution for all the other perturbed quantities. The vertical charge polarisation due to the curvature drift is the driving effect which explains the growth of an electrostatic potential at s c short times. Coupling between φ10 and f00 (resp. f10) generates the axisymmetric (resp. cosine) component of the electric potential:

e c c 2 2 φ10 ≈ Ω10 t (3.76) Teq 2 c 2 (ρscs) Ω10 = (3.77) rR0LpLn e 2 2 ∂rrφ00 ≈ Ω00 t (3.78) Teq      2 2 7 1 4 12 1 1 16 cs Ω00 = − + + + + κp − κn (3.79) 4 Ln 7LpLT 7Lp Lp r 7 R0 Equations (3.74), (3.76) and (3.78) are predicted to characterise the dynamics of the system in response to the vertical polarisation drift.

These analytical predictions can be checked with GYSELA. The code is initialised with the local Maxwellian and solves the full self-consistent problem –Eqs.(3.38) and (3.44). The tempera- ture gradient is taken well below the ITG threshold to prevent the onset of any instability. The case above the threshold is detailed in section 3.2.5. Figure 3.5 compares the full numerical solution to the analytical predictions, Eqs.(3.74), (3.76) and (3.78). s c Those predictions are twofold: (i) on the temporal growth of φ00, φ10 and φ10 and (ii) on their respective amplitudes at the first saving time step ∆t. On the one hand, the slopes which capture −1 the time exponent agree remarkably well until approximately 700 ωc . At this time, the electric drift and the parallel dynamics are brought into play. The forthcoming section 3.2.4 will quanti- −1 tatively discuss this point. On the other hand, the prediction on the amplitudes at ∆t = 15ωc cannot be straightforwardly compared: a mismatch does exist for numerical reasons between the analytical prediction of the initial amplitudes and their numerical counterparts. Appendix L de- tails the following result: this mismatch remains well below the accessible numerical error bars. 3.2 Initial state v.s. equilibrium: the central role of the electric field 59

s c Figure 3.6: Dependence with ρ? of the amplitudes of φ00, φ10 and φ10, when initial state is fLM . −1 Values are taken at ∆t = 15 ωc .

s 2 c 2 Although a direct test of the prediction on the amplitudes |Ω10|, |Ω00| and |Ω10 | in Eqs.(3.74), (3.76) and (3.78) is difficult with present discretisation, it is possible to check their parametric dependencies. Those equations give a strong dependence with ρ?. Eqs.(3.74) and (3.78) predict that: s 2 2 |Ω10| , |Ω00| ∝ ρ? (3.80) and similarly, Eq.(3.76) predicts that: c 2 4 |Ω10 | ∝ ρ? (3.81) Five simulations were run, which were initialised with the local Maxwellian and well below the ITG instability threshold, see Figure 3.6. The only varying parameter was ρ? in the set: ρ? = {1/800, 1/400, 1/200, 1/100, 1/50}. The expected parametric dependencies in Eqs.(3.80) and (3.81) are indeed very well recovered numerically. The physical picture underlying those results can now be drawn. As expected, the first dom- s inant term at short times is the up–down φ10 dipolar electric field. It is linearly driven in time by the up–down asymmetry of the vertical curvature drift. The cosine component is orders of s magnitude smaller. Quasi-linear coupling of φ10 to f00 generates via the quasi-neutrality equation an axisymmetric flow associated with the ∂rrφ00 term. Differently speaking, a poloidal velocity s vpol ∝ dφ10/dt cos θ builds up in time to enforce quasi-neutrality and is responsible for large scale sheared φ00 flows in the poloidal plane.

3.2.4 Saturation processes for the poloidal electric field

The strong dependencies with ρ? contained in Eqs.(3.74), (3.76) and (3.78) do not mean that these flows remain weak. Indeed, an upper boundary of the time growth is given by the time τsat at which either the parallel dynamics or the electric drift velocity are sufficient to counterbalance the charge polarisation. The five simulations in Figure 3.6 show that the saturation level of φ00 (close to 0.17 Teq/e) remains unchanged for each value of ρ? in the five runs. Therefore, given Eqs.(3.78) −1 and (3.80), τsat should also scale with ρ?: τsat ∝ ρ? such that, in the end, the saturation level of these polarisation flows remains significant, whatever the value of ρ?. Two physical mechanisms can be invoked to keep the radial electric field from growing, re- spectively associated with the electric drift and the parallel dynamics [33]. It is well known that 60 Calculating the plasma behaviour as an initial value problem

s c Figure 3.7: Fourier time spectrum analysis of φ00, φ10 and φ10. Frequencies are normalised to the GAM frequency. (ρ? = 1/400).

without collisions, an axisymmetric φ00 flow is linearly damped towards a non vanishing resid- ual value [99]. It saturates since it couples to Landau damped m = ±1 sidebands and oscillates p at the Geodesic Acoustic Mode (GAM) frequency [110]: ωGAM = 7/4 + τ vT /R. Here, τ = Te/Ti(= 1) for adiabatic electrons and vT is the electron thermal velocity. We plot in Figure s c 3.7 the Fourier spectral analysis of φ10, φ00 and φ10. As expected, they are peaked at the ωGAM frequency. The width of the peaks comes from the finite time resolution available in this simula- tion. This piece of physics appears as soon as the electric drift is included in the model, with a characteristic time τE: −1 τE ∼ ωGAM (3.82) −1 And as expected, we find τE ∝ ρ? . Details of the calculation of τE are given in appendix M. At −1 ρ? = 1/400, τE ≈ 670 ωc , which is in good agreement with Figure 3.5. At that time, the electric potential starts indeed to depart from the power law predicted by Eqs.(3.74), (3.76) and (3.78). The second saturation mechanism for the electric potential deals with the parallel dynamics. It consists in building up a parallel flow which will compensate the ever existing polarisation due to the curvature drift. No further electric field is then required to satisfy quasi-neutrality. Appendix N contains the detailed calculation. When introducing the parallel dynamics in Eq.(3.72), Eq.(3.73) is modified into: s vdfLM µB r f10 = t − vk 2 fLM t (3.83) Lp Teq qR The first term is due to the inhomogeneity of the magnetic field and triggers the growth of the electric potential already discussed in the previous section. The second term is new, due to the parallel dynamics. Importantly, at short times, it does not modify the dynamics of the electric s potential discussed in section 3.2.3. Indeed, one may recall that in order to calculate φ10 –and thus c s also φ00 and φ10– f10 is integrated over phase space. Since the new term in Eq.(3.83) is odd in s vk, it does not contribute to φ10. Nonetheless, because of this symmetry-breaking term in vk, the distribution function keeps departing from a local Maxwellian. Now, when keeping the second order terms in , a new term –even in vk– is added to Eq.(3.83). With this new term detailed in appendix N, an important new piece of physics is added to the model, allowing for a parallel flow to build up. The back reaction of this flow on the level of generated electric potential can be quantified. It is found that the introduction of the parallel dynamics results in a t3 correction term 3.2 Initial state v.s. equilibrium: the central role of the electric field 61

Figure 3.8: Initial and final temperature profiles when starting from the canonical and the local Maxwellian (initial temperature is the same). The ad-hoc radial diffusion coefficient is non-zero in buffer zones only, at r < 0.28 a and r > 0.72 a.

s 4 c for φ10 (and a t correction term for φ00 and φ10): s   eφ10 2ρscs 2ρscs Teq 1 1 3 ≈ t − 2 2 + t (3.84) Teq R0Lp R0 mq R Lp LT

c The quadratic corrections in time for φ00 and φ10 are straightforwardly calculated from the above expression. The first term describes the growth of the electric potential at short times as discussed in section 3.2.3. As expected, the negative sign in front of the second term shows that the de- velopment of a parallel flow leads to a decrease of this growth. The perturbative analysis of the latter section breaks when the cubic term balances the linear term. It gives an estimate for the characteristic building-up time of the parallel flow: r qR Lp τk ∼ 1 + (3.85) vT LT

−1 where vT is the thermal ion velocity. At ρ? = 1/400, τk ≈ 2160 ωc , which is a very good estimate of the saturation time of the flows, as exemplified in Figure 3.5. Its necessary parametric −1 dependence with ρ? is also right since indeed τk ∝ ρ? .

3.2.5 Nonlinear evolution: impact on transport in a steady state of turbulence

c We are not matter which remains rather than structures which endure. d after N. Wiener, Cybernetics, or Control and Communications in the Animal and the Machine, 1948

Two simulations at ρ? = 1/200 and with the same initial temperature gradient R/LT ≈ 12 –well above the threshold– have been compared up to the nonlinear regime [33]. They only differ by their initial distribution function, either canonical or local Maxwellian. These global simulations used fixed boundary conditions with φ = 0 enforced at the radial boundaries, periodic boundary 62 Calculating the plasma behaviour as an initial value problem

Figure 3.9: Time evolution of the total amount of parallel flow across the midplane. The inset is a zoom at early times (ρ? = 1/200).

conditions in (θ, ϕ) and no-perturbation conditions in the non-periodic r and vk directions for the distribution function. Two thermal baths lay at r ≤ 0.2 a and r ≥ 0.8 a, complemented with two radially-localised diffusive regions inside the simulation box (see Fig. 3.8) in which a source term S(r) = ∂r [D(r)∂rf] is added. In these buffer zones, the gyrokinetic equation reads: df/dt = S(r). The two diffusive buffer regions at the edge boundary layers are essential to reach a quasi-steady state. Indeed, they lead to an effective coupling of the system with the radial boundary conditions where the temperature is prescribed. The effect of the diffusive buffer zones is twofold: (i) they damp the fluctuations towards zero, consistently with the boundary conditions, and (ii) they allow for the diffusive exchange of energy between the thermal baths and the system. It is worth noticing that in such a configuration, the injected energy flux in the system is not constant in time: it is a priori unknown and is an output of the simulation. Alternatively, one could think of simulations at constant (or at least controlled) incoming energy flux. In this case, the temperature freely evolves at one of the radial boundary conditions at least. Such situations would be close to so-called “flux-driven” fluid simulations [46]. −1 In both nonlinear simulations, we calculated 3 200 time steps ∆t ≈ 5−10 ωc on half a torus. The grid mesh is regular and consists in ∼ 2 109 grid points in the 5D phase space, more precisely (r, θ, ϕ, vk, µ) = (256, 256, 128, 32, 8). A broad spectrum of low amplitude, random phasing, toroidal and poloidal perturbation modes is added to these initial states. Since the diagnostic R R was not available, we choose to focus on the following quantity: dr dvkvkf , which we call total amount of parallel flow across the midplane (at poloidal angle θ = 0). Figure 3.9 plots this quantity as a function of time (the absolute value enables to take into account Pfirsch–Schluter-like¨ flows whose average over a flux surface would otherwise vanish by symmetry). As expected, in the non canonical case, a parallel flow builds up on the typical time τk, Eq.(3.85) (inset of Fig.3.9). It saturates at the same level as the initial magnitude in the canonical case: both systems end up R R with a comparable level of parallel flows. The quantity dr dvkvkf provides an estimate of the parallel flows, including up-down asymmetric ones.

At shorter times than τk, the strong sheared poloidal flows discussed in section 3.2.3 play a dominant role. Their influence on the turbulence is shown on Figure 3.10. The time evolution 3.2 Initial state v.s. equilibrium: the central role of the electric field 63

Figure 3.10: Time evolution of the heat flux and the temperature gradient length in both canonical and non canonical cases. The inset is a zoom at early times, its y-axis is in log scale (ρ? = 1/200).

Figure 3.11: Comparison of the levels of φRMS and of the mean flows |φ00| in both the canonical fCM and non canonical fLM cases (ρ? = 1/200).

of the heat flux averaged over the central 70% of the simulation box –corresponding to the width of the global mode– is plotted for the two cases. The turbulent transport is delayed when starting from an initial local Maxwellian. As illustrated on the inset of Figure 3.10, the shear flow turns out to reduce the effective growth rate, by a factor of about 35%. Such an effect qualitatively agrees 64 Calculating the plasma behaviour as an initial value problem

Figure 3.12: Comparison of the radial and temporal correlation lengths of the electric potential over the whole nonlinear phase in both the canonical and non canonical cases (ρ? = 1/200).

with the theoretical predictions [9, 116, 54], although the reduction appears to be much smaller than expected. The turbulence overshoot at the end of the linear phase is also less pronounced. Note that the system is almost in a statistical steady state, in the sense that the mean temperature gradient relaxes adiabatically as compared to the fluctuations. Indeed, it has relaxed (i) similarly for the two different initial states and (ii) less than 33% for the canonical case and 34% for the non −1 canonical case between 10 000 and 25 000 ωc , while the heat flux has remained fairly constant. Nonetheless, the temperature profile keeps on relaxing on longer times, as illustrated on Fig. 3.10. The incoming energy flux should be maintained in order to avoid such a long-term relaxation in global and full–f simulations. After the turbulence overshoot (for times larger than 2 to 3 τk), the heat flux levels for the canonical and non canonical cases are comparable. This global result does not allow for a straightforward prediction on the saturation levels of the electric potential in both cases. Figure 3.11 shows that the levels of the axisymmetric flow |φ00| and the root mean q 2 square φRMS ≡ hδφ iθ,ϕ in the two cases are alike over the whole nonlinear phase. As well, the statistical properties of the turbulence are almost identical in both cases. Figure 3.12 compares the time and radial correlation functions calculated in the fully developed turbulent state, since the end −1 of the overshoot (from 5 000 to 25 000 ωc ). The typical size of a turbulent eddy is independent of the initialisation and of the order of 6 ρs. Similarly, the correlation time is also independent of −1 the initial state and of the order of 800 ωc . In the collisionless regime, provided the inevitable profile relaxation is very slow (no flux- driven conditions), after 2 to 3 times τk the turbulence develops and washes out very efficiently any possible memory the plasma could retain of its initial state.

3.2.6 Discussion and conclusion Let us start from a linearly unstable initial state in which a small initial perturbation exponentially grows in time. Two generic and very different cases can be considered. First of all, (i) the initial state is a (quasi) stationary state which evolves on a time scale τs which is large as compared to −1 the inverse of the linear growth rate γlin of the instability: τsγlin  1. This case is exemplified by the choice of the canonical Maxwellian, Eq.(3.56) as an initial distribution function. Secondly, (ii) the initial state is non stationary and its characteristic self-organising time τs is comparable −1 to γlin : τsγlin ∼ 1. The local Maxwellian, Eq.(3.49), is a representative initial state for (ii). In this latter case, to the instability-driven evolution of the plasma superimposes the self-organising dynamics due to the non equilibrium initial state. A priori, this can lead to very different dynamics as compared to case (i) in which the evolution of the plasma is only due to the instability. 3.2 Initial state v.s. equilibrium: the central role of the electric field 65

Transiently indeed, sheared poloidal flows (including zonal flows) are generated in the colli- sionless regime by the non stationary local Maxwellian initial state [29]. They develop over the whole simulation box on time scales comparable to the ITG growth rate. Such flows are several orders of magnitude smaller when the initial state is a stationary canonical distribution function. This chapter has clarified the underlying physical mechanism: these flows emerge as a result of the vertical plasma polarisation, which is due to the inhomogeneity of the magnetic field [33]. The poloidally sinusoidal (resp. axisymmetric) component of the electric potential grows linearly (resp. quadratically) in time. Such electric flows saturate due to both the onset of GAMs and the development of parallel flows, analogous to the Pfirsch–Schluter¨ current. These latter flows effi- ciently counterbalance the vertical charge polarisation. We analytically predict their symmetries, time evolution, initial amplitudes (or at least their parametric dependencies) and saturation time up to the beginning of the nonlinear phase (t < τk). These results are in very good agreement with simulations. In this transient regime, the E × B shearing rate effectively competes with the linear growth rate of the instability and the onset of the turbulence is delayed, as qualitatively already noticed in [1, 47, 29]. At long times though, irrespective to the choice of the initial state, the turbulence robustly develops, with identical statistical properties. This is in marked contrast [33] with the claim of different final states, depending on the choice of the initialisation. A new ingredient is necessary to understand this difference. As expected from the ITG context, the mean temperature gradient appears to be that crucial element. In global simulations, the problem consists in its unavoidable relaxation (no flux-driven conditions). In order to minimise it, small diffusive buffer zones at the radial edge of the simulation box have been implemented in GYSELA. They efficiently allow the coupling of the plasma to a free energy reservoir, in our case modelled by two thermal baths. With these conditions, the mean temperature gradient relaxes (i) very slowly and (ii) identically for both the two different considered initial states (see Figure 3.10). These results allow to run simulations during much larger times well above the threshold, which leads to crucial differences as compared to the latter works. Indeed, the more the mean temperature gradient relaxes, the weaker the drive on the turbulent modes. When the system is not coupled to an external source of free energy, as in [1, 47, 29], this relaxation leads to a marginally stable state on time scales comparable to the linear growth rate. In this case, the turbulent modes are hindered and the onset of the turbulence can be quenched. Inversely, when the system can feed on an external source of free energy, the turbulence keeps existing on larger times. In this case, the turbulence is only delayed and its nature is ultimately unaltered: the same level of parallel and axisymmetric flows, the same level of transport and the same correlation times and lengths are obtained (see respectively Figures 3.9, 3.11, 3.10 and 3.12). The shearing rate due to the axisymmetric flows (including the zonal flows) which develop in the case of a local Maxwellian initial state is not sufficient to quench the onset of the turbulence. Indeed, these flows develop over the whole simulation box on large radial scales, generating a large-scale shear as compared to the typical size of the turbulent modes and the turbulence is ultimately almost unaffected. As gyrokinetic simulations are numerically heavily demanding, the choice of an initial canon- ical state in global collisionless simulations is nonetheless important. Indeed, it allows one to address the relevant physics of turbulence growth and nonlinear saturation right from the start, avoiding long transients. In this case, the gain in terms of CPU time can be appreciable. Besides, the local Maxwellian case could prevent the onset of the instability in some specific cases, if the transient polarisation electric field turns out to be strong enough to hinder the growth of the unsta- ble modes. This could be all the more critical for those simulations run close to the linear stability threshold. Such a physics is expected to bear some similarities with the so-called Dimits upshift, when the onset of self-generated zonal flows prevents the development of the instability [34]. In collisional plasmas, the distribution function is forced towards a local Maxwellian and this prob- lem is less marked (see following chapters). Still, in order to genuinely sustain a steady turbulent 66 Calculating the plasma behaviour as an initial value problem state on a confinement time, flux-driven conditions with a prescribed and controlled heat source should be implemented [103]. Chapter 4

Adding binary interactions to collective behaviour – or the theory of Coulomb collisions in a turbulent plasma with curved magnetic geometry

c If you can look into the seeds of time, And say which grain will grow and which will not, Speak then to me [. . . ] d Shakespeare, MacBeth (Act I, scene 3)

MAJOR difficulty in the way of confinement has come from plasma instability, triggered either through violent disruptions due to large-scale MHD modes or small-scale turbu- A lence. The investigation of mechanisms responsible for the degradation of confinement –especially mechanisms responsible for the detrimental increase of heat transport– has thus given rise to a considerable amount of interest over the past forty years. Amongst them, a special role is endorsed by the Coulomb collisional processes: thermal fluctuations do exist in even a per- fectly stable and quiescent plasma, leading to diffusion and heat conduction across the confining magnetic surfaces. Nonetheless, in axisymmetric systems as is a tokamak, a pure collisional transport regime has long been recognised to be too low to offer a serious impediment to controlled fusion. Sub- dominant as compared to turbulent transport, collisional transport is the minimal level of transport. Obtaining such low levels in actual devices is a highly important milestone to understand and mas- ter the physics of the foreseen high-confinement operating regimes of a reactor. These regimes are characterised by the local suppression –or at least the impediment– of the turbulence either by triggering a so-called Transport Barrier to access a bifurcated state of higher confinement (the so-called H–mode [115], VH–mode [98, 85] or ITB). One result has been a renewed interest in the theory of transport in a magnetically-confined toroidal plasma in the perspective of a coherent description of both turbulence and collisions at a kinetic level. The theory of collisional transport goes back to the late 600s with the pioneering work of Galeev and Sagdeev [44] and has since been extensively studied. A successful attempt to incorpo- rate binary Coulomb collisions in a first-principle kinetic framework was first achieved in 1995, without turbulence, in a modern gyrokinetic δf PIC formulation [82]. The end of 1999 saw the first dedicated study of a turbulent collisional plasma [80]. This result, conjointly with the one which shall be presented here would both favour the existence of a synergetic-like interaction –or at least an interplay– between the turbulence and the collisions either through (i) the collisional enhancement of the turbulent heat flux [80, 30], through (ii) the turbulent generation of poloidal momentum [30] or through (iii) a stabilisation of Trapped Ion Modes. This latter mechanism 68 Adding binary interactions to collective behaviour could prove a promising candidate to account for the surprisingly large poloidal rotation which has been increasingly reported in the Joint European Torus (JET) [22] and in Alcator C–Mod [88] since its first observation in the late Tokamak Fusion Test Reactor (TFTR) [3]. The present chapter is organised as follows: the following section 4.1 discusses some essential characteristics which collision operators must bear. It describes milestone model operators with decreasing complexity, introducing in section 4.2 to the special choice we made in GYSELA to include binary Coulomb collisions in a full–f gyrokinetic framework [45, 31, 32]. Remarkably, given the simple form of the considered collision operator, a close agreement is shown with the most accurate transport predictions at fusion-relevant low collisionalities. Turbulence and colli- sions are investigated simultaneously throughout the last section 4.3, especially focusing on both the collisional enhancement of turbulent transport and the turbulent generation of poloidal mo- mentum.

4.1 Towards a kinetic “neoclassical” theory of collisions

The term neoclassical is a reminder of the historical development of the theory of transport in a magnetised plasma. In the earlier, classical phase, a modern form for the Fokker–Planck colli- sion operator was established; a characteristic result of classical transport theory being that the magnetic topology has no influence on the transport properties. A departure from these classical results came from recognising the intrinsic inhomogeneity of the magnetic field in a toroidally confined device [44]. It was soon to be noticed that in the long mean-free-path regime appropriate at thermonu- clear temperatures, the spatial variation of the magnetic field which a particle would see during its motion had crucial effects, namely (i) it leads to local trapping phenomena and (ii) to a grad- ual distortion of the Larmor orbits leading to the curvature and grad–B drifts –see Eqs.(3.20) and (3.21), page 41. Differently speaking, toroidicity leads to perpendicular drifts and thus to a weak- ening of the confinement while trapped particles are responsible for a significant enhancement of transport as compared to the former classical calculations.

So were lain the foundations of the neoclassical theory [59], i.e. the theory which describes the effects of binary Coulomb collisions in an inhomogeneous magnetic field in the presence of trapping. Neoclassical theory is essentially an equilibrium theory –see general discussion in section 2.1, p.14– which modifies the equilibrium properties of the collisionless kinetic equation –see previous chapter, p.51– especially leading to transport at equilibrium and to irreversibility through physical dissipation –see following sections. While classical and neoclassical transport processes are additive, neoclassical effects are typ- ically much larger. Still, experimentally-deduced transport levels are in most of the cases orders of magnitude larger than the predicted neoclassical ones. For this reason, the transport in actual devices is often qualified as anomalous and is widely believed to be due to the turbulence. This question is investigated in the last section 4.3 of the present chapter; we now focus on the basic ideas for a theory of pure collisional transport.

4.1.1 General requirements for a kinetic theory of collisions

c A serious debate on fusion research in the House of Lords, the upper chamber of the Brtish Parliament, would be interrupted by the question “How do they measure [the russians] a temperature of 300 million degrees?” which provoked the response “I expect that they use a very long thermometer”. d 4.1 Towards a kinetic “neoclassical” theory of collisions 69

Figure 4.1: The trajectory of a ‘test’ charged particle (solid line) in a background medium com- posed of both other charged particles (red and cyan) and neutrals (dark blue) exhibits continuous small-angle deflections or scatterings of its direction of motion while a neutral (dash-dotted line) experiences abrupt atomic collisions.

Collisions at thermonuclear temperatures

The characteristics and effects of Coulomb collisions between charged particles are rather subtle in a plasma as compared to the more commonly understood collisions of neutral particles which move along straightline trajectories between distinct collision events. Collisions then occur when these neutrals come within about an atomic radius –typically one Angstrøm: 1A–˚ of another par- ticle (a neutral or a charged particle) and the repulsive electric force associated with the atomic potential –typically one electron-Volt: 1eV– is operative. The resultant interaction is inelastic, causing the neutral to be statistically scattered at random (see Fig.4.1). Inversely, in a plasma, as a charged ‘test’ particle moves through a background ‘sea’ of other charged particles and neutrals, it simultaneously experiences the weak Coulomb electric forces surrounding all the nearby charged particles, and its direction of motion is deflected as it passes by each of them, with the closest encounters producing the largest deflections. This test particle thus experiences a sequence of random collisions which alter both its momentum and energy. Let us now consider a many-particle point of view: a mono-energetic beam of particles im- pinges upon a background plasma. Several effects are expected to occur due to collisions between the beam and the background plasma. First, there will be a slowing down of the beam as it looses momentum due to collisions. Second, there should be a spreading out of the velocity distribu- tion of the particles in the beam since collisions will tend to diffuse their velocity within the beam. Meantime, the background plasma should be heated and also gain momentum due to the collisions. Eventually, the beam particles become indistinguishable from the hotter background plasma. So as to quantify these intuitive ideas, let us classically define the impact parameter b as the minimal distance along the trajectory between the test particle and any other field (background) particle. Solution of the Rutherford scattering problem leads to two great classes of scattering events, with respect to the deflection angle θ represented in Figs.4.1 and 4.2, namely: (i) large- angle collisions where θ ≈ π/2 and (ii) small-angle grazing collisions where θ  1. Since the Coulomb electric force decays exponentially in space for distances larger than the Debye length λD, the maximum impact parameter b0 is very close to this length: b0 = λD. In this case, the deflection angle θ vanishes. Inversely, an estimate for the minimum impact parameter bπ/2 in Fig.4.2 is provided when the Coulomb potential energy becomes comparable to the electron kinetic 70 Adding binary interactions to collective behaviour

Figure 4.2: Differential scattering cross sections for large and small deflections.

energy, i.e. the net velocity change along the parallel direction (sometimes noted |∆vk|) becomes comparable to the perpendicular deflection (noted |∆v⊥|). In this case, the typical large-angle 2 cross section reads: σπ/2 = πbπ/2. Eventually, grazing collisions occur while bπ/2 < b < λD, i.e. when the test particle impinges outside the shaded circle in Fig.4.2. Although each small- angle collision does not scatter the test particle by much, there are far more grazing collisions than large-angle collisions. An important question for a quantitative theory of collisions is thus to compare the relative cumulative effect of grazing collisions with the cumulative effect of large- angle collisions. To clearly stress out the importance of this question, let us remind the reader that the quasi-universally used Fokker–Planck collision model is a restriction of the general Boltzmann model to the special case of grazing collisions (see the following section 4.1.2) and is characterised by a moderate transfer of momentum during an elementary collisional process. Validity of this approach requires to find the grazing collisions as the dominant collisional mechanism in a plasma. Since the effective cross-section of grazing collisions depends on impact parameter –the larger b is, the smaller the scattering– random walk statistics must therefore be used to describe the cumulative effect of small-angle scatterings. A quantitative way to answer the question: awhat time τg do we have to wait for the cumulative effect of the grazing collisions on a test particle to give an effective scattering equivalent to a single large angle scattering?b consists in taking this weighting of impact parameters into account as follows: if the test particle impinges on the differential cross-section having radii between b and b + db, then the test particle will be scattered by an angle lying between θ(b) and θ(b + db). The area of the differential cross-section is 2πb db which is therefore the effective cross-section for scattering between θ(b) and θ(b + db). Doing so, the area outside the shaded circle is subdivided into a set of concentric annuli, the so-called differential cross-sections. The cumulative cross-section σΣ is computed as:

Z λD  λ  σ = 2πb db [θ(b)]2 = 8 log D σ (4.1) Σ b π/2 bπ/2 π/2

−1 2 3 Since bπ/2 ∝ nλD, the condition σΣ  σπ/2 is equivalent to the condition: nλD  1. A complete description of this interesting but somewhat fastidious calculation can be found in Ref.[4], pp.11– 14. For a typical magnetic fusion plasma, nλ3 ≈ 106 and thus the Coulomb logarithm log Λ ≡
D log λD/bπ/2 ≈ 17 is much greater than unity (see also p.76 for further discussion). The answer to the latter question is consequently that τg νπ/2  1, where νπ/2 is the frequency of large- 4.1 Towards a kinetic “neoclassical” theory of collisions 71 angle scattering events. Eventually, it means that Debye shielding is important and that grazing collisions dominate large-angle collisions as soon as the Debye sphere contains a sufficiently large number of particles. This criterion is also at the root of both the definition of a plasma in terms of a quasi-neutral medium (see section 1.1.1, p.3) and the validity of the statistical approach. A different way of seeing this result is to recall that the collisionality depends on the velocity of the particle, the slowest being more likely to endure large-angle scattering. Thus, the impact parameter is connected to the structure of the distribution function in velocity space. At thermonu- clear temperatures, an outcome of the latter calculation is the following: the contribution from all velocities (including the sub-thermal more collisional particles) corresponds on average to a typ- ically large value of the impact parameter: b  bπ/2. As a result, the Coulomb electric forces produced by individual field particles are small and can be assumed to be experienced randomly by the test particle as it passes close to individual field particles, leading to a random walk –or Brownian motion– process. The effects of many cumulative grazing elastic Coulomb collisions are (i) diffusion of the test particle’s direction of motion (at constant energy in the center-of- momentum frame) and consequently (ii) transfer of momentum from the test particle to the field particles –e.g. in a tokamak, transfer of trapped banana particles’ momentum to passing parti- cles (bootstrap current, see section 4.2.1). Exploration of the collision operators which describe Coulomb collisional processes (momentum loss, velocity space diffusion and energy exchange) between particles in a plasma is the main subject of this section.

General properties a Collision operator should satisfy A collision operator C as in Eq.(2.9) provides a statistical account for the Coulomb collisions. Very generally, it can be expressed in the form: X Cs = Css0 (fs, fs0 ) (4.2) s0 where Css0 describes the effects on the scattered species s of collisions with the field particles of species s0. Whatever the basic scattering processes, physical requirements constrain the form any such operator Cs should satisfy. These requirements are the following: (i) Bilinearity ensures that any collisional process is dominated by binary collisions; 1 tertiary collisions being highly improbable. This property means that mathematically Css0 is a bilinear form and Css is quadratic. (ii) The Vlasov equation (see discussion in section 2.2) is isentropic; collisions ensure dissipation in the form of an H–theorem:

d d S ≥ 0 and also: S = 0 ⇔ Cs = 0 (4.3) dt c dt c where S measures the entropy of the plasma and d/dt|c represents the collisional rate of entropy production, excluding local changes due to entropy flow. An important necessary and sufficient condition for the collision operator to vanish is that local thermodynamic equilibrium is reached:  Cs = 0 ⇔ fs = fM,s and fs0 = fM,s0 and Ts = Ts0 (4.4) where fM,s is the classical Maxwell-Boltzmann distribution function –see Eq.(3.48). For further details, see discussion p.73 and Eq.(4.7). (iii) Galilean invariance states that the collisional process should not be altered when a common uniform velocity U is imparted alike to both species s and s0:

Css0 (fs(v − U), fs0 (v − U)) = Css0 (fs(v), fs0 (v)) (4.5)

1 In general, there is no immediate relation between Css0 and Cs0s. 72 Adding binary interactions to collective behaviour

Figure 4.3: Spatial symmetry of the collisional processes: in (a) collisions have spherical sym- metry and the corresponding operator is isotropic; in (b) the collision processes have cylindrical symmetry and the corresponding operator is anisotropic.

Essentially, combined with prescription (ii), this property states that any general equilibrium of the collision operator has the form of a shifted Maxwellian Eq.(3.48). (iv) The collision operator should reflect the spatial symmetry of the collisional pro- cesses, namely: • translation symmetry: Cs does only depend on the position through fs and fs0 • spherical or cylindrical symmetry, whether the collisional processes are isotropic or not. This idea is discussed in Fig.4.3. While the thermal gyroradius ρ remains larger than the Debye screening length: ρ ' λD (see section 1.1.1, p.3), colliding orbits are almost unaffected by the direction of the magnetic field lines and collisions have spherical symmetry as displayed in Fig.4.3–(a). The corresponding collision operator is isotropic, i.e. describes collisions which are equally effective in all directions. Inversely, when ρ  λD, the scattering dynamics is sensitive to the direction of the magnetic field since orbits are dominated by gyromotion; the underlying symmetry is thus rather cylindrical and the corresponding operator is anisotropic2. (v) Many processes in fusion devices can be non-locally conserving (e.g. wave-particles effects can convert particle momentum into electromagnetic work). By convention, these pro- cesses are excluded from the collision operator; any operator should thus locally conserve the mechanical quantities density, momentum and energy. These five properties are central for a collision model; the following section now describes some of the more widely used model operators, with decreasing complexity.

4.1.2 Some essential models of collisions c Boltzmann did not fully succeed in proving the tendency of the world to go to final equilibrium state. . . The very important irre- versibility of all observable processes can be fitted into the picture: the period of time in which we live happens to be a period in which the H-function of the part of the world accessible to observation de- creases. This coincidence is not really an accident, it is a precondition for the existence of life. d P. Ehrenfest and T. Ehrenfest, 1912

There is no exact form for the collision operator, but only a sequence of more accurate (and less tractable) forms. Discussion over collision operators is the tip of an iceberg which fundamentally

2...and immensely complicated; it shall not be addressed in the remainder of this dissertation. A reason is that we shall mainly consider ion–ion collisions which are genuinely isotropic in a fully magnetised plasma [58], p.174. 4.1 Towards a kinetic “neoclassical” theory of collisions 73 is the investigation of the elementary processes responsible for breaking time-reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour.

The Boltzmann approach The historical operator is due to Boltzmann, in his founding contributions to the kinetic theory. The key insight was to determine the form of the collision operator resulting from two-body collisions and write it as a momentum-space integral over the product of one-particle distribution functions. This possibility rests on a fundamental assumption that particles are uncorrelated prior to the collision and that this process is independent from their spatial location; it was referred to by Boltzmann himself as the Stosszahl Ansatz and is now known as the molecular chaos assumption. This assumption is a key element, although initially unrecognised, in Boltzmann’s H–theorem of 1872 Eq.(4.7), which attempted to derive the thermodynamical evolution principle from the kinetic theory; it gave rise to a long and public dispute with Loschmidt over the so-called Loschmidt paradox that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism: for every phase space trajectory there exists a conjugate time reversed antitrajectory [84] which is also a solution of the equations of motion. If the initial phase space distribution is symmetric under time reversal symmetry (which is the case for all the usual statistical mechanical ensembles) then the Boltzmann H–function could not decrease monotonically as predicted by the H–theorem. A heuristic solution to this paradox was proposed in 1895 by recognising that the velocities of two particles after a collision cannot be genuinely uncorrelated, hypothesis which Boltzmann had asserted. Doing so, he had introduced an element of time asymmetry through the formalism of his calculation. Thus, the molecular chaos hypothesis intrinsically breaks time-reversal symmetry and Boltzmann’s apparent success with the H–theorem only emanated from probabilistic arguments which made Maxwell write in 1878 in a book review for Nature: athe truth of the second law is. . . a statistical, not a mathematical truth, for it depends on the fact that the bodies we deal with consist of millions of molecules. . . Hence the second law of thermodynamics is continually being violated, and that to a considerable extent, in any sufficiently small group of molecules belonging to a real body.b However, Loschmidt’s observation does not deny the possibility of deriving the Second Law of Ther- modynamics from reversible microscopic equations of motion. The rationalisation of the Second Law, i.e. a satisfactory answer to the question of ‘how irreversible macroscopic behaviour can be derived from re- versible microscopic equations of motion’ has long been a central open question since the foundation of thermodynamics. The Fluctuation theorem [36] of 1993 provides an elegant answer to this question; it does more than merely prove that in large systems described by microscopic reversible equations of motion and observed for long periods of time, the Second Law of Thermodynamics is likely to be satisfied with exponen- tial likelihood. It essentially quantifies the probability of observing Second Law violations in small systems observed for a short time, while remaining fully consistent with Loschmidt’s observation that for time re- versible dynamics, every dynamical phase space trajectory and its conjugate time reversed ‘anti-trajectory’ are both solutions of the underlying equations of motion. Furthermore, it proves that an ensemble of non- dissipative purely Hamiltonian systems will, with exponential likelihood, relax from any arbitrary initial (non-equilibrium) distribution towards the appropriate equilibrium distribution. This point is the analogue of Boltzmann’s H–theorem and can be thought of as a proof of Le Chatelier’s Principle [18]. At last, from a more physical point of view, we now know that the Second Law of Thermodynamics can be derived assuming ergodicity at equilibrium, and causality –see section 2.1, p.12– and that it is causality which ultimately is responsible for breaking time-reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. This point shall not be further discussed here; the interested reader could especially refer to the review paper [37]. Another very fundamental dispute was lead by Zermelo and based on the recurrence theorem of 1890 by Poincare´ which states that if the system has a fixed total energy that restricts its dynamics to bounded subsets of its phase space, the system will eventually return an infinite number of time in an arbitrary small vicinity of any given initial set of molecular positions and velocities. If the entropy is determined by these variables, then it must also return to its original value. This Zermelo–Poincare´ paradox –or recurrence paradox– pinpoints an apparent contradiction between the behavior of a deterministic mechanical system of particles and the Second Law of Thermodynamics. On these grounds, Poincare´ especially wrote that a[he] could not recommend anyone reading the works of Boltzmann since [he] cannot recommend studying scien- tific arguments in which the conclusions contradict the initial assumptions.b. This objection was discarded by Boltzmann himself on the basis –which still has its defenders– that, though recurrences are completely consistent with the statistical viewpoint, they are fluctuations which may occur on characteristic times com- N parable to τrec ∼ 10 , N being the number of particles the system is composed of. Making N the Avogadro number, τrec is by far superior to the age of the universe; an objection which we fail to be able to observe cannot constitute an objection to the theory. 74 Adding binary interactions to collective behaviour

The Boltzmann work is central in modern formulations of kinetic theory, though the deep connection between statistical mechanics and thermodynamics still gives rise to active research programmes. Interpreted in terms of the gyrokinetic theory, the adjunction of a collision operator and its subsequent collisional dissipation has been argued to be crucial (i) to recover physical energy cascades [107] and (ii) is indispensable to realising a steady-state of turbulence [119].

Without collisions, the gyrokinetic (Vlasov) equation is isentropic, i.e. dS/dt = 0. As discussed in sec- tion 3.1.1, p.39, it could be argued that numerical noise (dissipation) may lead to a positive definite entropy production rate, while cutting off the small scales. This ‘recipe’ is both unphysical and difficult to control in a nonlinear steady turbulent state. On the other hand, collisions alone physically ensure an H–theorem, or equivalently a positive definite entropy production rate. This problem in intimately linked to the regu- larisation of the small scales. Indeed, in the following gyrokinetic framework: turbulent fluctuations are anisotropic with respect to the mean field and that their frequency is low compared to the ion cyclotron frequency, it has been argued [107] that if the turbulence is assumed to be forced at some system-specific outer scale, the energy injected at this scale has to be dissipated into heat, which ultimately cannot be ac- complished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. What we shall consider in this chapter is the regularisation of these velocity scales alone, though e.g. the gyroaverage operator introduces spatial variations within the collision operator. Also, based on dedicated numerical investigation, it has been shown [119] that statistical steady states of the ITG-driven turbulence in fusion-relevant weak collisionality regimes are such that in the saturated state of the entropy variable, the ion heat transport balances with the collisional dissipation. This latter thus being indispensable to realising a steady-state of turbulence. In a modern (isotropic) formulation using a local system of spherical coordinates with axis z = n vs − vs0 and spherical angle σ, the Boltzmann collision operator reads in general R space: Z Z h i Css0 (fs, fs0 ) = dvs0 dσ B(v − vs0 , σ) fs(us)fs0 (us0 ) − fs(vs)fs0 (vs0 ) (4.6) n n−1 R S n−1 As the spherical angle σ varies in the unit sphere S , us and us0 describe all possible postcol- lisional velocities of two particles colliding with respective velocities vs and vs0 . This operator does model both grazing and large-angle collisions. Though Boltzmann did not succeed in deriv- ing the Second Law of Thermodynamics from his kinetic collision theory –see discussion above– the collision operator Eq.(4.6) has the interesting and widely used property to satisfy what Boltz- mann called the H–theorem Eq.(4.7). This relation has proved an interesting milestone to deduce macroscopic irreversibility from microscopic time-reversible basic equations. Taking the entropy R functional to be −f log f and defining h = n dvs fs log fs, this theorem is expressed as: R Z Z d ∂fs h = dvs (1 + log fs) = dvs Cs(fs) log fs ≤ 0 (4.7) dt n ∂t n R R

The only distribution function fs for which the H–function h is statistically constant is the shifted Maxwellian fSM,s Eq.(3.48). This solution alone is the general equilibrium solution of the colli- sional problem Cs(fs) = 0. With the assumption of elastic collisions, the problem further simplifies:

us + us0 = vs + vs0 (4.8) 2 2 2 2 us + us0 = vs + vs0 (4.9) such that:

v + v 0 |v − v 0 | u = s s + σ s s (4.10) s 2 2 vs + vs0 |vs − vs0 | u 0 = − σ (4.11) s 2 2 2 and σ = I (4.12) 4.1 Towards a kinetic “neoclassical” theory of collisions 75

Eq.(4.6) leaves the weight function B unspecified. On the physical grounds of particle indiscern- ability and Curie symmetry principle –the effects have at least the symmetry of their causes– the form of B is assumed: (i) positive definite and (ii) depending only on the absolute value of |z| and on the deflection angle θ ∈ [0, π] –the same deflection angle as in Figs.4.1 and 4.2– and defined as the angle between z and σ. The problem further simplifies in the limit N = 3 (usual Euclidian space) and considering an interaction between two particles in the form of an inverse power law 1/rγ, γ ≥ 2. In this limit, B reads:

γ−5 B(z, σ) = |z| γ−1 ζ(θ) (4.13)

The Coulomb case γ = 2 is of course of special interest and will be especially discussed.

In the special case of grazing collisions: the Fokker–Planck model At last, in the case of grazing collisions, ζ is a smooth function of θ and can be specified:

− γ+1 ζ(θ) = C θ γ−1 (4.14) where C > 0. Given Eqs.(4.13) and (4.14), the Boltzmann operator Eq.(4.6) is nonintegrable. It is possible, though, to give a distributional sense to Css0 under weak physical assumptions provided that the total cross-section for momentum transfer be finite, condition which reads [114]: Z π θ ℵ = dθ ζ(θ) sin2 < ∞ (4.15) 0 2 In that case, considering soft collisions, it is a classical result that the Boltzmann operator can be recast into the following form:   ∂ ¯  ∂fs   Css0 (fs, fs0 ) = · D fs0 (vs0 ) · − V fs0 (vs0 ) fs(vs) (4.16) ∂vs ∂vs which is the classical generic form of a Fokker–Planck operator. Here D¯ represents a diffusion tensor in velocity space while V is a dynamical friction term which models collisional drag. In this representation, the increase of entropy is satisfied through the combination of both drag and diffusion. With grazing collisions, the typical collision time is classically defined as the ‘90o scattering time’ τg, see p.70. The differential structure of Eq.(4.16) makes this model very efficient for analytic and numerical purposes. This point will be further discusses in the following section; let us now further continue the simplification process and specify the Boltzmann operator to the central case of Coulombian long-range interactions.

Further restriction to long-range Coulomb interaction: the asymptotic Landau model In this limit, ζ(θ) ∝ θ−3 and the latter integral Eq.(4.15) diverges logarithmically. In that case, the Boltzmann operator Eq.(4.6) which now reads (recall that z = vs − vs0 ): Z Z C h i Css0 (fs, fs0 ) = dvs0 dσ 3 fs(us)fs0 (us0 ) − fs(vs)fs0 (vs0 ) (4.17) n n−1 3 R S θ |z| must be asymptotically continuated so as to circumvent the slow divergence of Eq.(4.15) in the case of long-range Coulombian interactions [79]. This idea was first proposed by Landau which found an especially compact form of the latter Fokker–Planck operator: Z    ∂ ℵ zizj hfs0 (vs0 ) ∂fs fs(vs) ∂fs0 i Css0 (fs, fs0 ) = dvs0 δij − 2 − (4.18) ∂v n |z| m ∂v m 0 ∂v 0 s,i R |z| s s,j s s ,j 76 Adding binary interactions to collective behaviour where the Einstein sum convention is used. In this spirit, the cross section for momentum transfer which is connected to the ℵ quantity can be expressed in terms of the Coulomb logarithm log Λ3: 2 2 4 ℵ ∝ Zs Zs0 e log Λ/ms and yields the estimate for the typical collision time τss0 between species s and s0: 3 2 3 msvT msvT 0 τss ∝ ∝ 2 2 4 (4.20) nℵ nZs Zs0 e log Λ

It is also straightforward to show that all the latter collision operators Css0 do satisfy the five requirements discussed in section 4.1.1, p.71 and especially the conservation laws for density, momentum and energy –see also definition p.51:

Z Z   3 X 3 msvs d v Css0 = 0 ; d v 2 Css0 = 0 (4.21) ms|vs| s s0 Z Z h 3 3 2 i weak form: d v msvsCss = 0 ; d v ms|vs| Css = 0 (4.22)

Further restriction to the scattering of light particles in a background of heavy immovable field particles: the Lorentz model

Proceeding one step further, we now come to the model we shall use in the remainder of this chapter. This so-called Lorentz model has some very attractive features: (i) it is reasonably simple from both an analytical and a numerical point of view to be highly attractive while remaining (ii) accurate –especially regarding neoclassical transport predictions [82, 45, 31]– as compared to more complete models, e.g. the Landau model and its various versions derived by Rosenbluth [101], Hirshmann [63] and others. This model considers the scattering of light test particles as they move through a background of heavy immovable field charged particles. It is well acknowledged that to the lowest order in the mass ratio expansion, ion–ion collisions dominate either electron–electron or electron–ion 1/2 collisions by a factor ∼ 60 which is roughly the square root of the mass ratio (mi/me) . We shall thus focus on these ion–ion collisions in the remainder of this dissertation and neglect other higher-order collisional processes.

The following section details the successful recovery of conventional neoclassical predictions within the gyrokinetic GYSELA framework using a minimal Landau-like collision operator. In par- ticular, this operator is shown to accurately describe some essential features of neoclassical theory, namely the neoclassical transport, the poloidal rotation and the linear damping of axisymmetric flows while interestingly preserving a high numerical efficiency. Its form makes it especially adapted to Eulerian or Semi–Lagrangian schemes.

3It is called so since it represents the cumulative effects of all Coulomb collisions within a Debye sphere for impact parameters ranging from bπ/2 to λD, see section 4.1.1, p.70. The logarithmic factor reflects Debye shielding, limiting the range of the Coulomb interaction at about the Debye length. Above λD, the Coulomb electric force decays expo- nentially; at thermonuclear temperatures though, quantum effects slightly complicates the expression of Λ as compared to the approximate value given p.70:

 23.4 − 1.15 log n + 3.45 log T for T < 50 eV log Λ = e e (4.19) 25.3 − 1.15 log n + 2.3 log Te for Te > 50 eV

The detailed treatment of hard collisions with impact parameters b ≤ bπ/2 yields order unity corrections to the value of the Coulomb Logarithm. Thus, the Coulomb collision momentum loss frequency which follows, Eq.(4.20), and the other Coulomb collision processes calculated in this chapter should be assumed to be accurate to within factors of order 1/ log Λ ∼ 5 − 10%; evaluation of Coulomb collision processes and their effects to greater accuracy is unwarranted. 4.2 Incorporating neoclassical theory into gyrokinetic modeling 77

4.2 Incorporating neoclassical theory into gyrokinetic modeling

It has been emphasised in the previous chapter that collisionless gyrokinetic theory consensu- ally provides today’s deepest insight on turbulence-related problems in fusion plasma physics, especially allowing for a very accurate first-principle description of turbulent transport. On the other hand, the neoclassical theory was earlier coined to describe the effects of binary Coulomb collisions in an inhomogeneous magnetic field, when trapped particles are present [59]. As com- pared to the highly-collisional edge region of a tokamak, core conditions exhibit (i) typical col- lision frequencies orders of magnitude smaller than the ones associated with the turblence and (ii) experimentally-measured transport levels, close to the turbulent transport levels and orders of magnitude larger than the predicted neoclassical ones. Thus, core plasmas are widely believed to be ‘collisionless’. Recent evidences from both the experimental side [13, 3, 22] and from gyrokinetic modelling [80] are contributing to modify this classical idea. The adjunction of neoclassical theory to gy- rokinetic models is therefore of great current interest: the onset and control of the improved con- finement regimes necessary to operate e.g. Iter require the access to advanced regimes where the turbulence can be locally suppressed. In such regimes, (i) neoclassical transport becomes dominant. Also, an increasing number of experimental observations [88] tends to emphasise the fundamental role of plasma rotation for the onset and the control of these improved regimes, a special focus being held on (ii) poloidal rotation. This latter quantity, even with turbulence, is widely believed to be set by neoclassical theory. The generation of axisymmetric (zonal) flows by turbulence, as recently emphasised, could however drastically modify this picture [81, 26]. These flows have been shown to survive the linear Landau damping process in the collisionless regime [99]. As a result, the level of transport can be underestimated when collisions are not taken into account [80]. The calculation of the collisional processes in a hot plasma is notoriously complex. The con- cept alone of collision in such a medium is rather subtle and since collisions between particles depend on their relative velocities, the overall collisional result is an integrated effect of non-local interactions between particles of all velocities. Also, as evoked previously p.74, in the framework of the gyrokinetic theory, collisions would operate on the gyroaveraged distribution function, and as such introduce spatial variations in the collision operator. As a consequence, model operators which allow to recover the main neoclassical results while preserving some interesting simplicity are especially attractive. In this dissertation, we report the implementation of a such simplified op- erator in the global and full–f gyrokinetic semi–Lagrangian GYSELA code and its successful con- frontation against theoretical neoclassical predictions. The main results of this theory are briefly evoked in the following section 4.2.1. The model equations are detailed in section 4.2.2, along with the adopted collision operator. While the accessible physical features with such a simplified operator are discussed in great detail in section 4.2.3, the last section 4.2.4 displays a quantitative confrontation to conventional neoclassical predictions. We especially accurately recover the pre- dicted neoclassical transport, the collisional zonal flow damping. We also report the reverse of the poloidal rotation with the collisionality regime, as predicted analytically over a decade ago [70].

4.2.1 Main neoclassical results Neoclassical theory was coined as a theory of transport processes taking both collisions into ac- count and the spatial inhomogeneity of the magnetic field. This theory may be considered to be born in 1968 with the work of Galeev and Sagdeev [44] in which the particle and heat fluxes were first calculated from the kinetic point of view and the expressions for diffusion and thermal conductivity coefficients for axially symmetric tokamak-type systems were found. Since then, transport processes in a strong magnetic field have shown to conceal much unexpected complex- ity as compared to earlier ideas on the question. Nonetheless, the physical processes underlying 78 Adding binary interactions to collective behaviour

Figure 4.4: Poloidal cross section showing banana orbits corresponding to different kinds of trapped particles, given the relative importance of their parallel velocity as compared to their transverse velocity. These orbits are located on the low field side Bmin of the torus.

neoclassical theory are now relatively well understood and the mathematical apparatus used for describing them has, in general, been worked out in great detail; in that sense, neoclassical theory is a standard against which predictive capabilities must confront. Importantly, the transport coefficients appear to depend not only on the magnetic field magni- tude but also on its spatial structure; in a tokamak-like magnetic configuration, the inhomogeneity of the toroidal magnetic field causes local magnetic wells to exist. One major outcome of this theory has been to predict the existence of particles trapped in these local wells and to foresee the central role they might play in determining the transport coefficients. These trapped particles are represented in Fig.4.4. These particles are called banana particles for the obvious reason that the projection of their orbit on a poloidal cross section displays a characteristic ‘banana shape’ as the right-hand side of figure 4.4 shows. Inversely, passing particles in an axisymmetric system are such that the projection of their orbit on a poloidal cross section is a circle. Trapped particles differ from passing particles since their parallel velocity is small as compared to their transverse velocity, which yields the trapping condition:

vk √ ≤ 2 (4.23) v⊥ θ=0 where  = r/R0. In velocity space (vk, v⊥), the region of space which makes the transition from trapped to passing is called the loss cone and is represented in Fig.4.5. Friction forces which arise due to collisional interaction between the trapped and the passing particles are especially located at the boundary layer of the loss cone (dotted lines on Fig.4.5). They represent the main physical dissipation mechanism due to collisions. It is responsible for the regularisation of the small scales in velocity space –see discussions pp.26, 39 and 74. As such, a correct modelling of this boundary layer is a crucial feature for a collision operator –we shall further emphasise this idea in the following section 4.2.4, p.89. The neoclassical theory has already predicted many interesting new phenomena associated with the presence of trapped particles which we shall briefly mention in the following paragraphs. Amongst them the non-linear dependence of diffusion and thermal conductivity coefficients on collisions frequency is a central result. It was not until 1962 that Pfirsch and Schluter,¨ working 4.2 Incorporating neoclassical theory into gyrokinetic modeling 79

o Figure 4.5: Velocity space showing a typical particle with vk ∼ v⊥ undergoing a 90 collision reversing the sign of vk. In shaded grey, the loss cone.

in the context of the hydrodynamic approximation with a number of simplifying assumptions, obtained expressions for the transverse diffusion coefficient in axisymmetric toroidal systems and three years later, in 1965, that Shafranov determined the coefficient of thermal conductivity. They turned to be q2-fold higher than the corresponding classical coefficients. The elaboration of a kinetic theory of transport processes in inhomogeneous magnetic fields thus became an essential task for fusion. It led to the discovery by Galeev and Sagdeev of the existence of three paradigmatic transport regimes, schematically displayed in Fig.4.6: namely the banana, plateau and Pfirsch–

Figure 4.6: Qualitative dependence of neoclassical thermal conductivity χ on the colisionality ν?. This schematic view is to be compared to the forthcoming Fig.4.12. The dash-dotted line corresponds to earlier classical predictions in which the magnetic topology has no influence on the transport properties.

Schluter¨ regimes. Let us construct a dimensionless parameter ν? from the collision frequency –a precise expression for this number Eq.(4.29) can be found in the following section. While ν? is smaller than unity, trapped particles almost fully determine the transport coefficients [59]; this regime is thus named after them the banana regime. The thermal conductivity monotonically increases with the collisionality, the proportionality factor being q2/3/2. Such is also the case 80 Adding binary interactions to collective behaviour

2 in the asymptotic collision-dominated Pfirsch–Schluter¨ regime where χ ∝ q ν?. In-between, the plateau regime4 is characterised by a weak dependence of the transport on the collisionality of the plasma. Core tokamak conditions are typically in the banana regime, while cold edge plasma conditions range from the banana to the plateau or even to the Pfirsch–Schluter¨ collisionalities. To complete this brief tour of neoclassical theory, it is worth mentioning the following works: • Hinton and Oberman [61] showed that at low collisionalites the longitudinal conductivity in tokamaks turns out to be less than the classical Spitzer conductivity [109]. This result readily comes from the fact that, in the presence of trapping, the number of ‘free’ charge carriers, equal to the number of transit particles, is only a fraction of the tokamak particles. • Galeev and independently Ware [118] discovered that the existence of a longitudinal elec- tric field in tokamaks results for the electrons in an extra pinch effect whose velocity may substantially exceed that of a classical pinches. • One more new effect was discovered by Galeev [43] and independently by Bickerton, Con- nor and Taylor [8] and has turned out to be one of the most important predictions of neoclas- sical theory. The so-called bootstrap current is a natural current generated by the Coulomb friction between trapped and passing particles which always flows in the same direction as the Ohmic current and hence can be, in principle, used for maintaining the plasma in steady state operation with the addition of only a small amount of extra external current drive power. For this reason, it is currently viewed as a critical element on the path to an economically viable tokamak reactor. • In accordance with the results obtained earlier by Kovrizhnykh [71], Rutherford indepen- dently concluded [102] that, to the lowest order in the Larmor radius, the diffusion fluxes of electrons and ions are automatically equal to each other and do not depend on the ambipolar electric field. • The central role of the radial electric field and its connection with the parallel velocity and its influence on the transport processes was examined in papers by Kovrizhnykh [72] and by Rosenbluth, Rutherford, Taylor, Frieman and Kovrizhnykh [100]. These papers showed the necessity for incorporation of viscosity in deriving the plasma velocity; the first expression for the neoclassical viscosity tensor was proposed. Later, Kim, Diamond and Groebner [70] derived a prediction for the poloidal velocity of the plasma for both the main ions and the impurities. We shall come back on this result in great detail in what follows. • Connor [21] examined the influence of impurities on transport processes in a tokamak. • Hinton and Rosenbluth [62] predicted the collisional damping of the axisymmetric mean and zonal flows, which may play a central role for transport –see the last section.

The most striking difference between neoclassical and cylindrical transport theories is the ef- fect of trapped particle banana orbits. Classical and neoclassical predictions differ by a factor which corresponds for each species to nearly two orders of magnitude. Though experimental data show that the micro-turbulence-driven ion thermal conductivity represents the fastest loss of en- ergy and is anomalously large by at least one order of magnitude with respect to the neoclassical prediction, it has also been emphasised experimentally that in enhanced confinement regimes with Internal Transport Barriers (ITBs), the ion transport approaches its neoclassical value. This is a fundamental point to be addressed for Iter and is the subject of the last section 4.3 of this chap- ter. First, let us now describe how the GYSELA code has been upgraded to perform gyrokinetic simulations of neoclassical transport.

4As will be seen in Fig.4.12, this ‘plateau’ regime is more of an ideal and a smooth transition from the banana to the Pfirsch–Schluter¨ regime is more relevant. 4.2 Incorporating neoclassical theory into gyrokinetic modeling 81

Figure 4.7: Parallel velocity dependance of the diffusive and convective operators D and V in Eqs.(4.26) and (4.27).

4.2.2 A minimal collision operator containing conventional neoclassical theory A full Coulomb collisional transport has notoriously considerable implementation difficulties in Eulerian-like numerical schemes and a simplified version is therefore highly attractive, provided it keeps the important properties of the complete operator. The precise derivation of this simplified model can be found in [45] where it is especially shown that it embeds the exact neoclassical trans- port as would be calculated by a full Fokker–Planck operator while preserving some simplicity. A simplified Landau-type ion–ion collision operator C conserving the number of particles is implemented in the right-hand side of the gyrokinetic equation Eq.(3.38) –see appendix O for the numerical details– which, as evoked previously p.76, accounts for the dominant collisional contribution in neoclassical transport: d f = C(f) (4.24) dt ∂  ∂f  where C(f) = D − Vf (4.25) ∂vk ∂vk Self-consistency is achieved for the electric field by solving the quasi-neutrality condition Eq.(3.44). Operators D and V (see Fig. 4.7) respectively model a diffusion and a drag:

√ 3 3/2 π vT  Φ(v) − G(v) D = 3 ν? (4.26) 2 qR0 2v vk V = − 2 D (4.27) vT

2 The first piece accounts for diffusion in parallel velocity D = h∆vki/2 generated by the Coulomb binary interactions. It is essentially responsible for the neoclassical diffusive transport and models the resonant enhancement of collisional effects in the presence of local trapping. The second piece is often referred to as ‘dynamic friction’ and is also reminiscent of the Fokker–Planck structure of the collision operator: V = h∆vki. It is an average time rate of change of parallel velocity due to the scattering effects of collisions and accounts for the poloidal flow damping obtained from neoclassical theory. Both operators readily depend on the parallel velocity vk, the radial coordinate 82 Adding binary interactions to collective behaviour

2 r through the temperature profile T and the energy E = mvk/2 + µB through the total velocity 2 p v = E/T . The thermal velocity is vT = T/m , m being the ion mass. The safety factor is q, R0 is the major radius evaluated on the midplane and  = r/R. The ion–ion collision frequency: √ 4 π n e4 log Λ νii = 2 2 3 (4.28) 3 (4π0) m vT is here expressed in terms of the dimensionless ion–ion collisionality parameter ν?: qR ν ν = 0 ii ? 3/2 (4.29) vT  where 0 is the permittivity of free space, log Λ ≈ 17 the Coulomb logarithm (see also pp.70 and 76 for further details) and n the ion density. Note that even though the collision operator Eq.(4.25) does depend on the energy, the advection is only performed in the parallel direction. We shall illustrate this idea in the following section 4.2.4, but let us just state at this point that such an operator is numerically extremely efficient since its parallelisation over velocity space (vk, µ) is intrinsic: each processor, or set of processors, has its own value for µ and performs its own collisions in the parallel direction. The explicit expressions for Eqs.(4.26) and (4.27) involve the error function Φ: v 2 Z 2 Φ(v) = √ e−x dx (4.30) π 0 2 2 Φ0(v) = √ e−v (4.31) π and the Chandrasekhar function G: Φ(v) − vΦ0(v) G(v) = (4.32) 2v2 Using Eq.(4.27), one can trivially see that the solution for the full–f collisional problem is the local 3/2 Maxwellian fLM = n/(2πT/m) exp(−E/T ) for both the bulk equilibrium and the fluctua- tions, which allows one to write the particle-conserving collision operator Eq.(4.25) in the compact form: ∂  ∂  f  C(f) = DfM (4.33) ∂vk ∂vk fM D still being given by Eq.(4.26). Such an operator pertains to the wider class of Landau opera- tors, see previous section, which have been extensively used in neoclassical theory. This operator trivially satisfies requirements (i) and (iv) as discussed in section 4.1.1, p.71, namely bilinearity and spatial symmetry. It has been coined so as to satisfy requirement (ii), i.e. a positive definite collisional entropy production rate. The following section aims at discussing the physics this min- imal model can address before assessing its accuracy for computing neoclassical transport. The two remaining points (iii) and (v), namely Galilean invariance and conservation properties, are there further discussed.

4.2.3 Discussion: which physics can we address ? This point is enlightening to be discussed in the framework of the fluid theory since it proposes a simple, yet considerable insight into plasma behaviour. In the absence of source of both particles and momentum, the general momentum flux conservation equation reads: ¯ mn dtVi = en(E + Vi × B) − ∇pi − ∇ · Π¯ NC − mn νie(Vi − Ve) (4.34) 4.2 Incorporating neoclassical theory into gyrokinetic modeling 83

Here, fluid quantities are written in capital letters, inversely to lowercase kinetic quantities. The ¯ electron-related quantities are labelled by subscript ‘e’, ion quantities by subscript ‘i’ and Π¯ NC = R d3v m(vv − v2/3¯I)f denotes the viscous stress tensor, crucial in toroidal geometry and which reduces at leading order to the parallel viscous stress. Focussing on the poloidally-symmetric part of the latter equation, its equilibrium parallel projection simply reads:

 ∇T hB i enE − mn νNC V − K ϕ − mn ν (V − V ) = 0 (4.35) k θi 1 ehB2i ie ki ke where h·i denotes the average over flux surfaces, Bϕ is the toroidal component of the magnetic field   ¯ NC 2
and hb · ∇ · ΠNC iθ ≈ mn ν Vθi − K1∇T hBϕi/ehB i . Let us first concentrate on this term. Here, νNC is the neoclassical ion viscous damping frequency, which can be approximated NC √ in the low-collisionality banana and plateau regimes [69] by: ν ≈ 0.78  νii/(1 + 0.44ν?). We adopt the notation used in [70] for the K1 dimensionless coefficient, which sign is predicted to depend on the collisionality regime. This result has a practical relevance when confronting e.g. to L–H transition theories since poloidal rotation is increasingly thought to play at crucial role in the onset of high-confinement regimes. Moreover, experimental measurements of the ion poloidal velocity have been performed over the years at and after the transition [13]. Practically, one correctly gets the poloidal flow damping as soon as the viscous stress tensor is correctly modelled. Physically, it means that (i) magnetic pumping due to the poloidal inhomogeneity of the magnetic field and (ii) Coulomb collisional interaction between trapped and passing particles are correctly being described. Both effects combine to enslave the ion poloidal velocity on the temperature gradient, or equivalently, on the ion diamagnetic velocity. This physics is an essential part of neoclassical theory and is intrinsically embedded in this model since this model operator Eq.(4.33) allows to exactly recover the neoclassical viscous stress tensor in the banana and plateau regimes as demonstrated in [45].

Moving to the last term in Eq.(4.35), attention is drawn to the momentum conservation prop- erties of the collision operator. This friction term accounts for the inter species electron–ion transfer of momentum. Since momentum conservation between species is central in neoclassi- cal theory, the implementation of the conventional neoclassical equilibrium is delicate while the electron response remains adiabatic. In this case, this inter species friction needs to be replaced by a like-species friction: mn νiiVki, which tends to drive the parallel flow to zero. Provided Galilean invariance is allowed, i.e. the solution of the collisional problem is a shifted Maxwellian: fSM (vk) = fLM (vk − Vki), the correct amount of parallel friction forces is accounted for. In this spirit, a slightly more complicated collision operator as compared to Eq.(4.33) is constructed in [45] which satisfies both momentum and energy conservation and, from a practical point of view, consists in performing the collisions locally, in the rest frame of the moving ions. It also means that when running a gyrokinetic code, Vki and T must be calculated at each time step, and fuelled back in the collision operator. Differently speaking, operators without momentum conservation create an artificial friction force and overestimate the momentum loss of circulation ions and elec- trons which is especially important in the banana regime. Nonetheless, as clearly shown from [82], satisfying Galilean invariance is important not to (i) overestimate the particle fluxes and (ii) un- derestimate the bootstrap current. Inversely, (iii) it has almost no influence on the heat transport. Bootstrap current cannot be modelled with adiabatic electrons and since neoclassical theory is am- bipolar, no particle flux is allowed with a single species collisions, which we accurately verify in GYSELA. So in the perspective of studying at the same time and with the same code both neoclas- sical theory and turbulence, this latter neoclassical friction is not relevant since overwhelmingly dominated by the turbulent friction. Therefore, we adopt a different approach, appropriate to an adiabatic electron response. The transverse equilibrium of Eq.(4.34) –the classical force balance 84 Adding binary interactions to collective behaviour equation:

∇p E − v B + v B = (4.36) r ϕ θ θ ϕ nZe connects the radial electric field and the plasma rotation with the thermodynamic forces. This equation separately applies to each constituent of the plasma and remains valid in the turbulent regime since the ion inertia term mv · ∇v is much smaller than the pressure gradient term and can be neglected [13]. Since in neoclassical theory, collisions tie poloidal rotation to the temper- ature gradient, the neoclassical equilibrium prediction is therefore on the sum Er − vϕBθ which is degenerate in axisymmetric systems with no source of toroidal momentum. At this stage, we do not wish to investigate the effects of toroidal rotation on neoclassical equilibrium and seek to relax this degeneracy. The accurate and self-consistent calculation of the electric field is a known central problem for a full–f electrostatic gyrokinetic code [33] and therefore the restriction of conventional neoclassical equilibrium to its sub-class associated with a vanishing mean toroidal velocity allows to unambiguously relate the radial electric field to the poloidal rotation and the thermodynamic forces while the exact transport is modelled as more general Galilean-invariant Fokker–Planck operators would. Differently speaking, this operator addresses a sub-class of neo- classical equilibria associated with a mean vanishing toroidal rotation, yet modelling the exact poloidal rotation and heat transport in the banana and plateau regimes. On the other hand, this col- lision operator is not suitable for addressing problems such as the toroidal momentum generation since it impedes the spin-up by overdamping rotation in the toroidal direction. Another crucial point which makes collisions especially important rests with the idea that collisions alone can linearly damp the mean and zonal parts of the axisymmetric φ00 flows. By this means, ion–ion collisions have been reported to crucially impact the level of ITG turbulent transport in zonal flow dominated regimes [80], which also corresponds to the expected operating regime of Iter. Therefore, a collision operator must accurately model the collisional damping rate of such flows, as predicted by Hinton and Rosenbluth [62]. As Fig.4.14 will show, this effect is indeed very accurately modelled. At last, the collisional system does intrinsically satisfy an H–theorem since the collision op- erator Eq.(4.33) is associated to an entropy extremalisation principle [45]. Due to this entropy principle, the full distribution function is guaranteed to relax towards a Maxwellian while most model operators only act on the perturbed distribution function. As such they are inappropri- ate when calculating the complete collisional equilibrium, including both the perturbed and the unperturbed problem. The net positive entropy production rate is reminiscent of the dissipation processes occurring in velocity space at the interface between the trapped and the passing regions of phase space. As a diffusive boundary layer, this interface gets increasingly localised as ν? diminishes. Within it, the dynamics of the collisions is accurately modelled by a friction force between the trapped and the untrapped particles with a favoured direction along the magnetic field lines. These aspects are quantitatively discussed in Fig.4.13. Briefly summarising this discussion, this operator Eq.(4.33) should very accurately describe (i) the collisional damping of the mean and the zonal flows, (ii) the neoclassical heat transport processes and (iii) the plasma poloidal rotation while remaining numerically very efficient. It is not suited to study toroidal momentum generation. Inversely, the expressions of the poloidal velocity and thermal fluxes in the banana and plateau regimes are correctly reproduced, even in extreme cases where the electron response is adiabatic.

4.2.4 Recovering the neoclassical results Recovering neoclassical results can reveal very subtle. So as not to generate artificial equilibrium flows which could prove detrimental for neoclassical equilibrium, it is important with full–f codes 4.2 Incorporating neoclassical theory into gyrokinetic modeling 85

Figure 4.8: Test of the accuracy of the force balance equation Eq.(4.36) in the banana (a) and the plateau (b) regimes. The poloidal velocity is evaluated by two means: either consistently evolved within GYSELA: vθBϕ or as the sum: ∇p/ne − Er + vϕBθ. Excellent agreement is found, regardless of the precise moment in the simulation. to accurately compute the radial electric field [33], which we do not externally impose but self- consistently evolve. Therefore, simple tests can reveal powerful in assessing the accuracy of the model and its numerical implementation. In this framework, let us consider the following simple yet non trivial equilibrium state: N f = e−H/T0 0 3/2 (4.37) (2πT0/m) eφ0 n0  = − log + c1K0 + c2I0 (4.38) T0 N 2 Here H = mvk/2 + µB + eφ0 is the hamiltonian, N is a constant, so is T0 the temperature, φ0 is the initial (and equilibrium) electric potential depending on the radial coordinate only, n0 is the density profile, K0 and I0 are the modified Bessel functions of the first type and c1 and c2 are complex coefficients such that φ0 vanishes at the radial boundaries. This special choice Eq.(4.37) is obviously (i) a motion invariant and (ii) satisfies C(f0) = 0. Neoclassical theory predicts 86 Adding binary interactions to collective behaviour

Figure 4.9: Neoclassical poloidal velocity normalised to the temperature gradient (coefficient K1 in [70]) as a function of the collisionality ν? ( = 0.17 and θ = π). The definition of the collision- ality parameter in [70] is slightly different to the one we use here and has thus been renormalised to the value in Eq.(4.29). This result is robustly obtained for various values of ρ?.

vθ ∝ ∇T/eB, as we shall discuss in the following paragraphs, thus in the case of a constant tem- perature T0, the usual force balance equation Eq.(4.36) trivially reduces to: e∇φ/T0 = −∇n/n, the electric field balancing the density gradient. When initialising GYSELA with Eqs.(4.37) and (4.38), the equilibrium solution we find for the electric field is indeed given by Eq.(4.38), thus emphasising on the stability of its calculation. In the remainder of this section, the results which are displayed have been checked to be stationary and independent of the initial state: should GYSELA be either started from a local Maxwellian or a canonical Maxellian –see section 3.2– the same results are robustly obtained. With these elementary checks performed, we now investigate the general case of a non vanishing temperature gradient. In the remainder of this chapter, GYSELA is initialised with the canonical equilibrium. The problem is axisymmetric. Temperature and density gradients are chosen such that the system remains below the linear instability threshold: R0/LT = 3 and R0/Ln = 2. As a result, all the simulations reported in this section do not exhibit any turbulence nor turbulent transport. Other typical parameters at mid radius for all the following simulations read:  = 9 r/R = 0.15 and q = 1.4. The typical grid size at ρ? = ρi/a = 1/256 involves over 10 grid points on a half-torus (r, θ, φ, vk, µ) = (256, 256, 8, 128, 16) mesh. The electric potential then adjusts so as to satisfy Eq.(4.36), as illustrated in Fig.4.8. The poloidal velocity vθ is displayed, calculated by two means: (i) self-consistently within GYSELA G vθ , as an output of the gyrokinetic–Poisson problem and (ii) by means of the force balance equa- FB tion: vθ Bϕ = ∂rφ + ∇p/ne + vϕBθ, each of the right hand side terms coming from the simula- tion. All these quantities are plotted. As expected from the previous section, toroidal rotation is all the lower as the ion–ion parallel collisional friction increases from the banana regime Fig.4.8–(a) to the plateau regime Fig.4.8–(b). The response of the electric potential is therefore different in the two collisionality regimes since the poloidal velocity itself strongly depends on ν? [70]. In the banana regime, the radial electric field mainly compensates the density gradient, whereas in the plateau regime, the poloidal velocity is smaller and the radial electric field increases so as to mainly compensate the pressure gradient. Hence the different behaviours observed for Er in Fig.4.8–(a) and (b), while the global force balance remains satisfied within a few percent precision. 4.2 Incorporating neoclassical theory into gyrokinetic modeling 87

Figure 4.10: Neoclassical poloidal velocity normalised to the temperature gradient (coefficient K1) as a function of  (ν? = 0.08).

The plasma poloidal rotation plays a crucial role in the transition from the Low to High con- finement regimes both in modelling [108, 56, 87] and experimentally [53, 68, 22], should it be as a precursor of this transition [13] or even troughout the H mode [88]. The neoclassical predic- tion that vθ = K1(ν?, )∇T/eB varies with the ion temperature gradient is essentially twofold: (i) the analytical prediction on K1 is accurate in the asymptotic banana and Pfirsch–Schluter¨ regimes, while only approximate in the plateau regime and (ii) regardless of the precise transition between regimes, vθ should reverse sign with the collisionality: it is predicted to rotate in the electron diamagnetic direction in the Pfirsch–Schluter¨ regime and in the ion diamagnetic direction in the banana regime [70]. A very consistent agreement with the neoclassical prediction is found with GYSELA in the asymptotic banana regime in which this prediction is valid. The collisional dependance for the ion poloidal rotation is confirmed [5], rotating from the electron to the ion dia- magnetic direction as collisionality decreases. Nonetheless, the theory in [70] is rather inaccurate b−p in predicting the actual values of (i) the transition ν? from the banana to the plateau regime and c (ii) the critical value ν? for which the ion poloidal rotation vanishes on average. Given the results b−p c from Fig.4.9, we would predict ν? ≈ 0.2 and a critical ν? ≈ 40. In the case of a large shear of radial electric field, the poloidal flow velocity is well-known to possibly increase well above these neoclassical values and become a significant fraction of the ion diamagnetic velocity [60]. This orbit squeezing correction is plotted in Fig.4.9 (green curve) but since the radial electric shear in all simulations is low, its effect here is subdominant. A widely used expression for the poloidal rotation in the banana regime is vθ = 1.17 ∇T/eB. This result is indeed the neoclassical prediction in the limit  → 0 i.e. for very large aspect ratio tokamaks. At intermediate values of , it should be noted that a more accurate prediction exists in the banana regime [70], as shown in Fig.4.10. To investigate this, we performed three different neoclassical simulations at ν? = 0.08 corresponding to three different values of . In each simulation, five different radii are considered. The results show a good agreement with the neoclassical prediction, confirming the significant importance of the aspect ratio on the evaluation of the neoclassical poloidal rotation. This parameter also influences transport, as displayed in Fig.(4.11). The same three simula- tions at ν? = 0.08 show the normalised ion heat diffusivity L¯i [111] plotted against : χi L¯i = (4.39) 2 χb 88 Adding binary interactions to collective behaviour

Figure 4.11: Normalised neoclassical ion heat diffusivity (coefficient L¯i) as a function of  (ν? = 0.08).

Figure 4.12: Neoclassical heat diffusivity normalised to its Bohm value as a function of collision- ality ν? ( = 0.17).

2 where χb denotes a banana-like ion thermal diffusivity: χb = (mvT /ehBi) qvT ν?/R0 where 1/2 vT = (T/m) and the brackets stand for an average over the flux surfaces. While the reduced Hirshman–Sigmar operator [63] underestimates the ion energy flux in all aspect ratio regimes, we confirm that in the intermediate aspect ratio regime, we confirm (i) that in the intermediate aspect ratio regime, the widely-used Chang–Hinton analytical model [15] overestimates the ion thermal diffusivity χi and we find (ii) a good agreement with the more accurate Taguchi [111] banana regime model. At low values of , the ion thermal diffusivity as derived by Chang and Hinton [16] provides an interpolation between the banana, plateau and Pfirsch–Schluter¨ regimes in the limit of a vanishing 4.2 Incorporating neoclassical theory into gyrokinetic modeling 89

Figure 4.13: Collisional regularisation of the full distribution function at the boundary layer be- tween the trapped and untrapped regions of phase space. This regularisation is essentially occuring along the magnetic field lines. The island in (θ, vk) space represents iso-contours of the full dis- tribution function. impurity contribution and no Shafranov shift: √ CH √   χi qvT 0.66 + 1.88  − 1.54  0.59  ν? 2 = 2 2 ν? √ Ib−p + √ (Ib−p − IPS) (4.40) ρi ωc ωcR 1 + 1.03 ν? + 0.31ν? 1 + 0.74  ν?

√ 1/2 The 2 2 coefficient comes from our definition of thermal velocity: vT = (T/m) . This relation is displayed in Fig.4.12. A very accurate agreement is found with GYSELA throughout the entire banana and plateau regimes. The Ib−p term represents the neoclassical contributions from the ba- 2 nana and plateau regimes and describes the transition between them: Ib−p = 1 + 3 /2. Similarly, 3/2 2
1/2 the IPS term represents the Pfirsch–Schluter¨ contribution when ν? > 1: IPS = 1 −  . In simulations with  = 0.17, the discrepancy observed at low collisionalities between the GY- SELA results and the Chang–Hinton prediction is consistent with the above discussion in Fig.4.11. We recover that at intermediate aspect ratios, the Taguchi model is more accurate in the banana regime. As compared to early ‘classical’ theories [17] in which the spatial variation of the magnetic field has no influence on the transport, neoclassical theory is deeply connected to the resonance phenomena due to particle trapping arising in an inhomogeneous magnetic field and has some crucial features on the level of collisional transport. Collisions indeed contribute to regularise the trapping singularities, broadening the highly localised region of phase space which delineates the trapped region from the untrapped. The broadening of this region with increasing collisionality is displayed in Fig.4.13. Due to the spatial nonuniformity of the magnetic field, iso-contours of the distribution function display the typical ‘cat-eye’ shape in phase space which traduces particle trapping. Starting from the canonical Maxwellian [33], the initial distribution function is a motion invariant, i.e. an exact solution of the gyrokinetic equation but not a solution of the collisional problem. It is singular at the vicinity of the loss cone, as clearly illustrated by the transverse plot of the distribution function at θ = 0 (red curve). The dissymmetry in vk comes from the choice of the canonical Maxwellian. Let us concentrate on this right-hand side of Fig.4.13. The two simulations at ν? = 10 and ν? = 0.01 are compared at the same time, i.e. after 10 collision times 90 Adding binary interactions to collective behaviour

Figure 4.14: Collisional damping of the mean and the zonal flows confronted to the analytical prediction by Hinton and Rosenbluth [62] without the GAM contribution (ν? = 0.04,  = 0.18). The frequency obtained with GYSELA agrees within the percent with the precise GAM frequency calculation [110].

and 0.1 collision time respectively. As expected, as the collisionality increases the full distribution function (i) is increasingly regularised and (ii) relaxes towards a local Maxwellian. At low colli- sionalities (here in the banana regime at ν? = 0.01), the form of the distribution function after a few collisions is inbetween the dotted and the diamond curves, achieving a compromise between a function of the motion invariants and a local Maxwellian. On the iso-contour plot, as best seen close to (θ, vk) = (0, 3), the initial highly-localised region delineating the trapped domain from the untrapped broadens as collisionality increases (diamond contours at ν? = 10). This regular- isation occurs predominantly transversely to the island, providing a hand-waving argument why physically the correct transport could be recovered while writing the collision operator Eq.(4.33) with vk only. In the banana regime, it is well-known that this boundary layer gives the dominant contribution to the heat transport coefficient [59]. In the presence of a perturbed hamiltonian –i.e. of particle trapping, the enhancement of collisional transport, which is at the root of neoclassical transport, is due to the resonant interaction between the particles and this perturbation [45]; it dominantly occurs in this region. A correct modelling of this boundary layer is therefore crucial to recover the correct neoclassical transport as displayed in Fig.4.12.

At last, Rosenbluth and Hinton predicted [99] the existence of linearly undamped axisym- metric flows φ00(r) which survive the collisionless linear Landau damping process. It is widely believed that the mean [68] and the zonal [81] components of these flows play a significant role in the regulation of the turbulence, ultimately determining the nonlinear level of transport. With col- lisions though, this residual flow is slowly damped [62], as illustrated in Fig.4.14. This is mainly due to the collisional friction force the trapped particles exert on the passing particles at the vicin- ity of the loss cone. Initially, we introduce a φ00 perturbation and follow its damping in time. The plasma parameters are those in [62]: ν? = 0.04,  = 0.18 and q = 1.4 at mid radius. The observed decay accurately matches the latter Hinton et al. calculation; in the analytical model though, time scales are larger than the ion bounce period so that the transient oscillations displayed by the gy- rokinetic result are not modelled. A fourier transform of these ocsillations shows as expected a very close agreement (< 1%) with the Geodesic Acoustic Mode frequency [110]. 4.3 Gyrokinetic turbulence calculated with neoclassical effects 91

Summary of the main results obtained while confronting to conventional neoclassical theory A minimal ion–ion collision operator has been designed and successfully implemented in global and full–f gyrokinetic simulations using the semi-Lagrangian GYSELA code. This operator is minimal in the sense that it reproduces the essential results of neoclassical theory while avoiding much of the parallelisation cost inherent to more general operators. This approach (i) features a very efficient parallelisation with respect to the adiabatic variable and (ii) takes advantage of a fixed grid discretisation and as such is especially attractive in Eulerian or semi-Lagrangian-based numerical schemes. As a special mandatory feature of full–f codes, without an external forcing, the full distribu- tion function has to relax towards a Maxwellian. Such a constraint is not mandatory for reduced collision operators acting on the perturbed distribution function only. Inversely, this condition is intrinsically fullfilled in the collision operator displayed here. It is especially shown that the correct neoclassical transport, poloidal rotation and mean and zonal flow damping are accurately repro- duced. Building upon the advantages of the present approach, the self-consistent –and significant– interplay between turbulence and collisions is investigated in the following section.

4.3 All ingredients in one: towards a synergy turbulence–collisions? – or gyrokinetic turbulence calculated with neoclassical effects

There has been many breadcrumb trails throughout this dissertation, but none perhaps has proven as central as the delicate problem of how to accurately calculate the radial electric field. This problem rises once more in this section; we shall here link it with plasma rotation, widely believed to be set by neoclassical theory. Elucidating the interplay between turbulence and collisions is a subject of great current focus for first-principle modelling since some recent evidence from both the experimental side [113, 13, 3, 22, 88] and from modelling [80] has started to emphasise its special relevance for the onset and the control of enhanced confinement regimes in the next-generation fusion devices like Iter. As this dissertation has emphasised, the GYSELA code now allows to address within the same tool state-of-the-art first-principle gyrokinetic ion turbulence and neoclassical physics. We report the evidence from nonlinear gyrokinetic simulations in global realistic magnetic geometry and representative turbulent core plasma parameters of a significant deviation of the poloidal rotation from its expected neoclassical value [64]. In particular, this can lead to a significant increase in the magnitude of the radial electric field. As also noticed in the previous section 4.2, p.77, it is widely believed that core plasmas are ‘collisionless’ and that the transport level in a tokamak is set by the turbulence alone. This classical assumption is in marked contrast with recent gyrokinetic simulations of Ion Temperature Gradient turbulence including the effects of like-species ion–ion collisions [80]. These authors indeed report in a tokamak-relevant operating regime a dramatic enhancement of the turbulent transport with the ion–ion collisionality, through the collisional damping [62] of axisymmetric non stationary turbulence-induced zonal flows (ZF), which back-react on the turbulence and regulate it –this damping mechanism has been confirmed with GYSELA [31], see Fig.4.14. A mirror problem to this collisional enhancement of turbulent transport is the turbulent gen- eration of poloidal momentum. Differently speaking, the impact of turbulence on the poloidal rotation of the plasma –see Fig.4.10 and the associated discussion p.87– which is widely believed to be set by neoclassical processes. An increasing interest for this quantity is arising, which is due to the crucial role it could play to design advanced operational scenari for Iter. Indeed, triggering and controlling enhanced confinement regimes –regimes with turbulence suppression leading to the formation of Internal Transport Barriers (ITBs) or leading to a bifurcated state of high confine- ment, the H–mode in which the plasma self-organises– requires a deep understanding (i) of the 92 Adding binary interactions to collective behaviour formation and the dynamics of ITBs and (ii) of the physical mechanism underlying the transition from Low to High confinement regimes. Such enhanced confinement regimes seem a universal and reproducible feature of fusion devices and as such trigger a considerable amount of interest in the community. In such regimes, an increasing amount of experimental observations are starting to report sig- nificant deviations of the poloidal velocity from its expected neoclassical prediction, should it be in ITB operating regimes [3, 22], before and at the L–H transition [13] or even throughout the duration of the H–mode [88]. The fundamental importance of the radial electric field in such tran- sitions has been widely emphasised [113, 13]. Changes in Er are associated through the force balance equation Eq.(4.36) with changes in the ion poloidal rotation and pressure gradient –see further discussion in section 4.2.3, p.84. Charge exchange recombination measurements using carbon impurities have determined that Er can change prior to any significant change in the ion pressure gradient. Accordingly, even though the question of causality between the ITB formation [22] or the L–H transition [13] and the sudden increase of poloidal rotation cannot be unambigu- ously settled due to intrinsic limitations on the time resolution of the experimental apparatus, a promising idea is that the change in the poloidal rotation is a precursor for the observed evolution of the radial electric field. Other interpretations exist though which emphasise a possible role for the edge rational security factor surfaces. Furthermore, the contribution of vθ to the observed level of radial electric field can even remain dominant throughout the duration of the H–mode [88]. In the following section 4.3.1, we discuss evidence of a collisional enhancement of the tur- bulent diffusivity in the banana regime parametrised by the distance towards instability threshold. In the last section 4.3.2, we report evidence from state-of-the-art first-principle modelling of a significant enhancement of the ion poloidal rotation in the presence of turbulence as compared to its neoclassical prediction [30, 64]. The results in these two sections are based on a series of non- linear gyrokinetic simulations including neoclassical effects in global realistic magnetic geometry and representative core plasma parameters. These simulations have been performed in both neo- classical and turbulent regimes using the same tool. A convergence study has been performed, at different values of ρ?.

4.3.1 Collisional enhancement of turbulent transport –a cartography in (R/LT , ν?) space

We performed a scan in both the strength of the turbulence drive and in collisionality to assess the possible synergetic effects of collisions on turbulence, transport and poloidal rotation. Four different values of the parameter R/LT have been investigated, either inside the so-called ‘Dimits upshift’ i.e. below the ITG nonlinear threshold [34] (R/LT = 5.6), at this threshold (R/LT = 6.3) or above (R/LT = 7.6 and 10.9). Each value corresponds to a different simulation. Inside each of these simulations, three different values of ν? in the banana regime are scanned: ν? = 0.5, 0.1, 0.05. Four additional simulations at the latter values of R/LT have been run at ν? = 0. At last, one more simulation at ν? = 0.01 and R/LT = 7.6 has been performed. Each of these were run on a half-torus (r, θ, ϕ, vk, µ) = (256, 256, 64, 64, 8) grid, at ρ? = 1/128. Furthermore, two additional simulations only differing by the initial drive of the turbulence have also been run on a full-torus high-resolution (r, θ, ϕ, vk, µ) = (512, 512, 128, 128, 16) grid at the tokamak-relevant small value of ρ? = 1/256. In each of them, the two values of the collisionality ν? = 0.5 and 0.05 are scanned. Within the Dimits upshift, turbulent transport is expected to be most sensitive to zonal flow regulation, via shear decorrelation of turbulent eddies. Since zonal flows are linearly damped by collisions alone [62] –see Fig.4.14, the triad interaction between zonal flows, turbulence and collisions may be foreseen as possibly particularly interesting for weak drives of the turbulence. Such regimes also happen to be all the more interesting since representative of tokamak operating 4.3 Gyrokinetic turbulence calculated with neoclassical effects 93

Figure 4.15: Temporal evolution of the radial average of the drive of the turbulence (left) and of the turbulent heat diffusivity (right) at ρ? = 1/128.

Figure 4.16: Influence of the collisionality on the turbulent heat flux χturb at different values of the turbulence drive R/LT –a cartography in (R/LT , ν?) space (the right-hand side figure is a close-up of the left-hand side figure).

regimes: realistic normalised temperature gradients are indeed in the range R/LT ∈ [5; 7.5].A typical simulation is shown in Fig.4.15: the system is initialised with a canonical Maxwellian and a target temperature profile which, in this case, is initially such that in the centre of the simulation box R/LT = 9.2. All quantities displayed here are radially averaged between 0.4 and 0.7 in −1 normalised radius, which corresponds to the source-free region. After 20000 ωc , a statistical steady-state is reached, corresponding to a mean constant value of the R/LT parameter. The −1 simulation is initially run at ν? = 0.1 until 37000 ωc ; this state is then saved. Starting again from it, the simulation is now split into two simulations: (i) the ν? = 0.1 one is prolonged −1 until 76000 ωc while (ii) another simulation is run while changing the collisionality to ν? = 0.01. Doing so, we eliminate the uncertainty during the turbulence growth phase and connected to the impact of collisions on the linear instability. The radially averaged heat diffusivity for this simulation is displayed in Fig.4.15, right-hand side. The result from all simulations is summarised in Fig.4.16 where small markers represent the simulations at ρ? = 1/128 and large markers the high-resolution ρ? = 1/256 ones. Each of these symbols corresponds to a radial and temporal average of both the turbulent heat diffusivity and 94 Adding binary interactions to collective behaviour

Figure 4.17: Root mean square fluctuations and shearing rate of the electric potential as a function of both the collisionality (left-hand side) and the distance towards instability threshold (right-hand side).

the drive of the turbulence. The radial averages are as previously mentioned performed in the source-free region between 0.4 and 0.7 and the temporal averages are performed over one to three collision times ∆t of steady turbulence, which represents state-of-the-art capability in gyrokinetic simulations and requires flux-driven boundary conditions now of routine use in GYSELA.

Focussing on the mean values of the turbulent heat diffusivity in Fig.4.16 –as exemplified in Fig.4.15, right-hand side, one may argue of a significant enhancement of turbulent transport with the collisionality, both within the Dimits upshift –in quantitative agreement with the only previous work on the subject by Lin and co-workers [80]– and even beyond this regime for larger drives of the turbulence. The dramatic enhancement they reported of the turbulent transport with the ion– ion collisionality in the range ν? = 0 − 0.4 was interpreted as a result of the collisional damping the non stationary turbulence-induced zonal flows. We investigate this idea, especially looking in Fig.4.17 at the root mean square δφRMS and shearing rate γE [116] of the electric potential. Even in weak turbulence regimes where the drive of the turbulence is low (R/LT = 5.6 − 6.3) and thus where turbulence may be expected to be extremely sensitive to the zonal flow intensity or shear flows, ion–ion collisionality does not seem to importantly impact both δφRMS and γE. While these quantities monotonically increase with the distance towards instability threshold, no clear dependance with the collisionality can be foreseen. This result is consistent with the calculation of the radial and temporal autocorrelation functions for the electric potential fluctuations presented in Fig.4.18. Regardless of the precise value of the collisionality, the radial and temporal correlation 4.3 Gyrokinetic turbulence calculated with neoclassical effects 95

Figure 4.18: Autocorrelation lengths and time of the electric potential as a function of the distance towards instability threshold for various values of the collisionality.

Figure 4.19: Influence of the collisionality on the linear growth rate of the most unstable mode, parametrised by the distance to instability threshold. The dash-dotted line is the linear growth rate of the most unstable mode evaluated at ν? = 0. The dotted line is the fit given in Eq.(4.41).

functions are very similar in shape, suggesting a weak influence of ν? on the radial and temporal structure of the E × B flows. Furthermore, since collisions may affect the linear growth of the instability and possibly endure throughout the nonlinear phase, affecting transport, we investigate this idea through a set of thirty-three simulations pictured in Fig.4.19. The linear growth rate γlin of the most unstable mode (m, n) = (−25, 18) (corresponding to the lowest-order resonant rational) is unaffected in the range of collisionalities ν? = 0 − 0.5. Here, m and n are respectively the poloidal and toroidal mode numbers and the safety factor q = 1.4 in the centre of the simulation box. A remarkably good fit to these results is given by:

γlin,0 γlin = lin (4.41) (R/LT ) 1 + 0.03 ν? R/LT lin where (R/LT ) = 4 is the linear instability threshold and for each value of R/LT , γlin,0 repre- 96 Adding binary interactions to collective behaviour sents the linear growth rate of the most unstable mode (−25, 18) at vanishing collisionality. The turb enhancement of χ with the collisionality does not seem to be imputable to an effect of ν? on the linear growth of the instability, though a deeper investigation of this idea for the whole spectrum of the instability (and not the only most unstable mode) is needed to totally discard this idea. This effect is nonetheless unlikely since collisions in this perspective are rather expected to contribute to reduce the transport. To summarise the results obtained so far, though the transport phenomenology looks similar to the one reported in Ref.[80] –see Fig.4.16, right-hand side, the underlying phenomena and their possible interpretation seem quite different. The result by Lin features the central role of the zonal flows with a strong dependance on the turbulence regime (distance towards instability threshold): the result in [80] indeed holds within the Dimits upshift region (R/LT = 5.3) while no notable effect of the ion–ion collisionality on the turbulent diffusivity was found at R/LT = 6.9.

In the light of these results, we deepen our analysis and go beyond the mere radial and temporal averaging procedures displayed in Fig.4.16 and investigate the different sources of ‘uncertainty’ in these actual averaged values. They are twofold: (i) temporal fluctuations δχt of the turbulent heat diffusivity during ∆t and (ii) radial fluctuations δhR/LT i in the drive of the turbulence must be considered. Indeed, at each radial position, as fronts propagate radially, the local drive seen by the turbulence fluctuates by δhR/LT i around its mean value. These fluctuations traduce the intrin- sically global character of the simulation and the existence of avalanche-like large-scale transport events associated to large heat outflows radially propagating through time on large distances (sev- eral correlation lengths) and at fractions of the diamagnetic velocity v? = ρ?vT [104]. These ‘avalanche bars’ are plotted in Fig.4.16: the horizontal bars representing the variance δhR/LT i and the vertical bars capturing the temporal fluctuations δχt. Interestingly, with the same set of parameters and initial conditions as compared to the ρ? = 1/128 runs, the high resolution ρ? = 1/256 simulations prominently highlight self-organised near-critical transport, featuring the crucial role of temperature profile stiffness near marginal sta- bility. The system self-organises sitting close to criticality, so that large discrepancies in transport can occur with a very modest change in the turbulence drive, behaviour which bears some close similarities with the large stiffness of the temperature profile observed in experiments. This fact has important implications for current open issues such as the predictive use of transport codes based on a fixed gradient approach or the validation of the underlying physical transport models. Since flux-driven conditions in a global simulation exhibit some of the characteristic properties of self-organised criticality, a central yet still unaddressed question is the influence of collisions on the statistical properties of the system and on the drive of the instability. These points should give rise to forthcoming work.

4.3.2 Evidence of turbulent generation of poloidal momentum

As a mirror problem to the collisional enhancement of turbulent transport, we now address the question of the turbulent generation of poloidal rotation. Poloidal rotation is indeed widely be- lieved to be set by neoclassical processes, though some recent theories [27] have pointed out the possible generation of poloidal momentum from turbulent Reynolds stress in a bifurcation scenario in which poloidal flows could act as a precursor in the transition from Low to High con- finement. Such calculations require the self-consistent calculation of Reynolds stresses and their contribution to large-scale poloidal flows; as such they may only be consistently addressed by full–f computations. We apply the same latter set of simulations to assess the validity of the usual neoclassical picture: that collisional interaction between trapped and passing particles in banana regime and magnetic pumping in all collisionality regimes poloidal heavily damp poloidal flow towards its 4.3 Gyrokinetic turbulence calculated with neoclassical effects 97

Figure 4.20: Time and flux-surface averaged computation of the poloidal velocity, its neoclassical and turbulent components at low collisionality in the source-free region.

neoclassical value. From the ion momentum equation, the time evolution of the flux-surface aver- NC
aged poloidal velocity reads: ∂thvθi = ∂thvEθi − µii hvθi − hvθ i , where vEθ is the poloidal component of the E × B velocity, such that the long-time equilibrium may be recast into:

NC 1 1 ∂ 2
hvθi = hvθ i − 2 r hv˜θv˜ri (4.42) µii r ∂r

Out of the scan in (R/LT , ν?) space, we highlight in Fig.4.20 the results in the source-free re- gion from one of the high resolution ρ? = 1/256 simulations, in the banana regime ν? = 0.05 and close to marginality R/LT = 6.3. The twenty other simulations run for this scan robustly predict a similar behaviour. Each term in Eq.(4.42) is displayed: the left-hand side term is la- GY S belled vθ and corresponds to the self-consistent computation of the poloidal velocity within GYSELA in the different turbulent regimes, including collisions. The second term in Eq.(4.42) NC corresponds to the neoclassical prediction and is labelled vθ . We evaluate this quantity using the following procedure: (i) the K1 coefficient (with its radial dependance) is evaluated through dedicated simulations in neoclassical regime, below the instability threshold (R/LT = 3) –see GY S Figs.4.9 and 4.10. Then, (ii) from the turbulent simulations from which we have evaluated vθ , we compute the actual ∇T value –these simulations have the exact same parameters than the neo- classical ones apart from the actual value of this temperature gradient. We then (iii) compute NC vθ = K1∇T/eB as to be the neoclassical prediction in this turbulent regime. The last term in Eq.(4.42) is labelled −∇ · ΠRS/µii in Fig.4.20 and corresponds to the Reynolds stress-driven √ GY S NC poloidal velocity, µii ≈ 0.78  νii being the ion viscosity. At last, the difference vθ − vθ is turb labelled vθ . For each of these fifteen simulations, the radial force balance equation is accurately satisfied within a few percent, as displayed in Fig.4.21. turb The very strong correlation between vθ and the Reynolds stress-driven poloidal velocity GY S sorely suggests that the observed discrepancy between the complete computation vθ and its neoclassical estimate does come from the turbulence. This discrepancy, depending on the radial position, can range from no discrepancy at all up to locally a factor of two in the actual magnitude of the rotation. The calculation reported here may apply to current tokamaks, though a reliable prediction for Iter would require to go two orders of magnitude lower in collisionality (ν? ≈ 5 10−4), meaning roughly two orders of magnitude longer gyrokinetic simulations, which would Figure 4.21: Accurate verification of the force balance equation Eq.(4.36) in the banana regime (ν? = 0.01) and in the presence of turbulence. The poloidal velocity is evaluated by two means: either consistently evolved within GYSELA: vθBϕ or as the sum: ∇p/ne − Er + vϕBθ.

push the envelope beyond current capabilities. Nonetheless, these simulations may all the more be relevant since as the collisionality decreases, the ion viscosity comparably decreases and the turbulent Reynolds stress-induced poloidal velocity may accordingly increase. As it opens the way to investigate these relevant low-collisionality regimes, the current work should be extended in forthcoming work in such direction. Conclusion

c Tout jadis. Jamais rien d’autre. D’essaye.´ De rate.´ N’importe. Essayer encore. Rater encore. Rater mieux. d Samuel Beckett, Cap au pire, 1982

OLLOWINGPERSONALINTERESTS in statistical physics and fundamental properties of non- linear, self-organised systems, microturbulence, as one of the last vast open problems of Fclassical physics, was extremely appealing. This thesis was undertaken in interesting times, while international interest or disputes on fusion over the perspective of Iter, the maturity of the- ory and the not so old possibility to access massively parallel supercomputing would promise an exciting and renewed branch of physics. This thesis aimed at presenting three years of dedicated theoretical research in the field of fusion plasma physics. As such, it is primarily concerned with the investigation of transport phe- nomena in magnetically confined systems. Two main goals have been pursued: (i) develop and implement techniques for making quantitative predictions for state-of-the art first-principle trans- port description and (ii) push the envelope of today’s knowledge on turbulence-related phenomena in tokamak-relevant regimes. The understanding of the confinement properties of such systems has made much progress due to a combination of dedicated experiments, analytical results and numerical simulations. Theory and simulations have now reached some maturity and a rather clear picture, though incomplete, has emerged with time, essentially based on two great model frameworks. In the first place, turbulence simulations have long been performed by solving fluid equations. As part of this work, two linear fluid closures in the collisionless regime are proposed. One is a generalisation of an existing re- sult; the other is original. They allow one to incorporate within the fluid approach (i) the Landau damping phenomenon, an intrinsically kinetic resonant effect between waves and particles which the fluid cannot normally describe and (ii) an operational form of the Second Law of Thermody- namics based on entropy production rates. However, since reactor-grade plasmas are weakly collisional (characterised by long mean free paths) and since fluctuations do cover a wide range of spatial scales which can be smaller than the ion gyroradius, a kinetic description is way more precise and rigourous. Indeed, this description alone allows one to self-consistently describe fusion-relevant weak collisional regimes with strong resonant collective interactions between waves and particles, finite orbit width effects, small-scale velocity structures and local trapping phenomena. Understanding the nonlinear dynamics of magnetically confined plasmas in a kinetic frame- work is a formidable task. This thesis is committed to a better understanding of microturbulence and its impact on transport in tokamak fusion plasmas. In the framework of the gyrokinetic theory, focussing on the paradigmatic model of the ITG instability, it has developed over two main axes, connected with the exploitation and the development of the GYSELA code. At first, the impact of the choice of an initial state on the late development of the turbulence and the level of transport has been investigated. This question is particularly crucial in a full–f code where the equilibrium and the fluctuations are solved simultaneously. Indeed, strong sheared 100 Conclusion poloidal flows, corresponding to a polarisation velocity, are generated when the initial state is not an equilibrium solution of the collisionless gyrokinetic equation. The transient dynamics of these flows has been analytically calculated along with their poloidal structure and saturation time. All predictions agree well with dedicated numerical simulations. Inversely, these flows –which delay or might even hinder the growth of the turbulence when initially close enough to the instability threshold– play no role when the initial state is a function of the motion invariants. The reason being that in the latter case, the Pfirsch–Schluter-like¨ parallel dynamics is correctly accounted for, right from the start. The coherence with the fluid description of such a choice for an initial state is proved. Besides, once turbulence develops, provided the drive of the turbulence is sustained, the system does not seem to hold memory of its initial state –canonical or not– in the long time limit of simulation runs. The second axis deals with Coulomb collisions. Taking these binary interactions into account leads to crucial novel features for confinement, inaccessible to collisionless theories: essentially, the correct description of equilibrium transport and of equilibrium flows. The former can become important in high-confinement regimes, characterised by a weak turbulent activity. The latter con- tribute to the saturation of the turbulence. This work has comprised the numerical implementation of a simplified ion–ion collision operator in the full–f GYSELA code. The implemented version lacks Galilean invariance and overdamps the parallel velocity; the impact of such choices as well as ways to overcome them are discussed. From a numerical point of view, this approach features a very efficient parallelisation with respect to the adiabatic variable and takes advantage of a fixed grid discretisation. As such, it is especially attractive in Eulerian or semi-Lagrangian-based nu- merical schemes. Physically, this operator reproduces in the reactor-relevant low and intermediate collisionality regimes the essential results of neoclassical theory –namely neoclassical transport, poloidal rotation and collisional damping of axisymmetric mean and zonal flows. The GYSELA code now allows to address within the same tool state-of-the-art first-principle gyrokinetic ion turbulence and conventional neoclassical theory. Building upon these advantages, the self-consistent interplay between turbulence and collisions has been investigated. A result had been previously reported in recent years, emphasising the dramatic enhancement of the turbulent transport with the ion–ion collisionality, through the collisional damping of axisymmetric non sta- tionary turbulence-induced zonal flows. This result is confirmed; we further emphasise the crucial role of the distance to instability threshold: at strong drives of the turbulence, this latter result is subdominant. It mainly holds close to marginality –which is often the operating regimes of actual and foreseen devices, close to the so-called Dimits upshift, in a zonal flow-dominated regime. As a miror problem, the poloidal rotation of the plasma is widely believed to be set by neoclas- sical processes. An increasing interest for this quantity is arising, which is due to the crucial role it could play to design advanced operational scenari for Iter. Indeed, triggering and controlling enhanced confinement regimes –regimes with turbulence reduction leading to the formation of In- ternal Transport Barriers (ITBs) or leading to a bifurcated state of high confinement, the H–mode in which the plasma self-organises– requires a deep understanding of the physical mechanisms underlying these transitions. An increasing amount of experimental observations are starting to report significant deviations of the poloidal velocity from its expected neoclassical prediction, should it be in ITB operating regimes, before and at the Low to High transition or even throughout the duration of the H–mode. This thesis defends the idea that non trivial interactions between turbulence and collisions may prove important in the way to understand, master and predict such foreseen high-confinement operational regimes. We report evidence from nonlinear gyrokinetic simulations in global realistic magnetic geometry and representative turbulent core plasma parameters of a significant deviation (around 40%) of the poloidal rotation from its expected neoclassical value. In particular, this can lead to a significant increase in the magnitude of the radial electric field, which is the widely acknowledged crucial parameter which triggers the transitions to enhanced confinement regimes. Conclusion 101

Future research, open questions Open questions for future research remain, which are essentially threefold. First of all, in any fusion device, fluxes are externally prescribed and profiles do fluctuate around their time-averaged value. Implementing genuine ‘flux-driven’ conditions in first-principle solvers may lead to self- organised criticality. Avalanche-like intermittent effects would occur, featuring bursts of turbu- lence, much like a domino effect. In these conditions, how would the system self-organise; how far would it be from criticality? –the answer to this latter question being especially important. Secondly, collisions are well-known to importantly impact the linear behaviour; one may in- vestigate the departure from linear zoology and ask for the impact of resonances on the nonlinear behaviour. For instance, how would collisions nonlinearly contribute to modify the distribution function? Similarly, while allowing the collision operator to be Galilean invariant, one may study the important problem of the generation of toroidal momentum. At last, a strong controversy does exist for Trapped Electron Modes on the role of collisions on the nonlinear instability threshold, especially connected to the role of zonal flows or zonal density on the turbulent transport. One may look forward to perform a similar study of the interaction between turbulence and collisions on TEMs since the study with ITG modes has highlighted as crucial the effect of zonal flows and more generally of the distance towards marginality.

Also, a` la Prevert´ , one may ask in the following list for: the (spatial) structure of flows and their connection to collisional dissipation in flux-driven systems, the respective role of collisions, zonal flows and mean flows in drift-wave turbulence, the appreciation of the impact of particle transport and turbulence spreading on flow evolution and poloidal spin-up, the problem of spectral transfer and small-scale dissipation in velocity space, . . .

Appendix A

Important features about Fourier–Laplace transforms

SMENTIONED in section 2.3.1, in the near-equilibrium case, considering a linear problem, A Fourier-Laplace transforms (referred to as F LT s) manage to make Eqs.(2.21a) and (2.21b) algebraic. Eqs.(2.21a) and (2.21b) can then be easily solved in (k, ω)-space. To outline the true physical meaning it embedded in Fourier–Laplace space, one can perform an inverse Fourier- Laplace transform (referred to as IF LT ), so as to obtain the real space and time-dependency of the fields. Doing so arise many mathematical problems, embedded in the generic ‘analytic continuation’ terminology. The forthcoming appendix C will give an overview of them and at least partially solve them. This appendix C discussion is not only of interest for the physician’s ‘mathematical sake’ (if one possesses so) for three good reasons. First one. Because the problem we are going to deal with is very general, representing an archetype of what is often encountered when writing a dispersion function. Second. Because it can nicely be solved once and for all and last but not at all least, not doing so properly makes one miss important physical phenomenon : Landau [75], some years after Vlasov, while performing the analytic continuation of the plasma dispersion function (2.26) discovered the very important so-called ‘Landau damping’ Vlasov had missed.

Considering a function g depending on real space through the x dimension, through time t and that can also depend on velocity v, one can define as follows the direct Fourier-Laplace transform (from now on referred to as F LT ):

Z +∞ Z ∞ g(v, k, ω) = dx e−ıkx dt eıωt g(v, x, t) (A.1) −∞ 0 It is important to note that, performing a Laplace transform, we integrate over time from 0 to +∞. This makes the problem an initial value problem, breaking the time symmetry and introduc- ing in the very mathematical formalism the causality question, which Landau, while integrating the dispersion function (2.26) invoked explicitly. This causality question has become known as the ‘Landau prescription’. When only performing Fourier transforms, one has to reintroduce at the end this physical demand that is automatically satisfied from the beginning when performing the ‘rigorous’ F LT . One can also define the inverse Fourier-Laplace transform (referred to as IF LT ) as follows :

Z dk Z dω g(v, x, t) = eıkx e−ıωt g(v, k, ω) (A.2) F 2π L 2π where L stands for ‘Laplace contour’ and obviously F for ‘Fourier contour’. A–104 Important features about Fourier–Laplace transforms

For the sake of future analytic continuation understanding it is very important to rigorously define the validity domains of integration in which the L and F contours make sense, i.e. in which domains of complex k-space and ω-space equation (A.2) can be validly written (this constrains, by feedback, the k and ω values in equation (A.1)). We will see that this will introduce the causality principle and justify that the F LT is the good transformation to be considered rather than a Fourier transform. These domains will be referred to as Domains of Absolute Convergence (DAC). In the most general case, both k and ω are complex. They can therefore be written into the following form : k = kr + ıki and ω = ωr + ıωi. The choice of the DAC is based upon the causality principle : for t < 0 (and that includes t → −∞) the real-space and time fields must vanish. Consequently :

In k space, the DAC in the complex k-space must include the real k-axis since at any finite time the physical response must be limited in space. Since the response in real space x can be directional (i.e. for x > 0 or x < 0), and since at any finite time (but finite or not x) the physical response must remain limited in space, the complex DAC must be a strip in k-space 1 (cf figure A.1).

In ω space, the problem no longer is to conserve a finite physical response at finite times but to match the causal principle, assuming that the response function after an IF LT must vanish for any x and for any t < 0 (especially t → −∞). Consequently, the DAC is a fraction of the semi upper-plane in ω-space. Moreover, in order to allow both stable and unstable fields to exist and grow (as physics leads to consider both these situations), ωi must be strictly positive 2.(cf figure A.1).

Figure A.1: Domains of Absolute Convergence in the k-space and ω-space.

1Containing the real k-axis allows one to represent modulus integrable fields ; the strip structure allows to describe exponentially decreasing fields for x → ±∞. [6] has shown that the causality principle connects the k-space DAC and the ω-space DAC. 2Doing so, this settles a lower boundary for unstable field growth velocity (the fields can only grow more quickly than the minimal value of ωi allows them to). Therefore, if ωi were to be null, there could not exist any unstable field growth, which is non physic. If ωi were to be negative (despite causality) unstable growing fields would not be allowed to exist which is non realistic either. Appendix B

Establishing the dispersion function

THESYSTEM one wants to solve is Eqs.(2.21a) and (2.21b): ∂tf + v ∂xf + ∂xφ ∂vf = 0 (B.1a) Z +∞ 2 ∂xφ = fdv − 1 (B.1b) −∞ As discussed in section 2.3.1, one can want to solve it near a Maxwellian equilibrium and will preform an F LT . See appendix A for more details. Let’s define, in agreement with equations (A.1) and (A.2) the following functions : Z +∞ −ıkx gk(k) = dx e g(x) (B.2) −∞ Z ∞ ıωt gω(ω) = dt e g(t) (B.3) 0 I Integrating the normalized Vlasov Eq.(2.21a) over x, from −∞ to +∞ and multiplying by e−ıkx (thus performing a Fourier transform), leads to: Z +∞ Z +∞ Z +∞ −ıkx −ıkx −ıkx dx e ∂tf˜+ v dx e ∂xf˜ + ∂vfeq dx e ∂vφ˜ = 0 −∞ −∞ −∞ | {z } | {z } ˜ −ıkx +∞ R +∞ −ıkx ˜ ˜ −ıkx +∞ R +∞ −ıkx ˜ v [f e ]−∞+ıkv −∞ dx e f ∂vfeq [φ e ]−∞+ık∂vfeq −∞ dx e φ ˜ ˜ ˜ ∂tfk + v ı k fk + ı k ∂vfeq φk = 0

ıωt I Now integrating the latter equation over t, from 0 to ∞ and multiplying by e (thus performing a Laplace transform), leads to: Z ∞ Z ∞ Z ∞ ıωt ˜ ıωt ˜ ıωt ˜ dt e ∂tfk + dt e v ı k fk + dt e ı k ∂vfeq φk = 0 0 0 0 | ∞ {z } f˜ eıωt −ıω R ∞ dt eıωt f˜ k 0 0 k ˜ ˜ ˜ ˜ −fk(t = 0) − ı ω fk,ω + v ı k fk,ω + ı k ∂vfeq φk,ω = 0 (B.4)

It is time to also Fourier–Laplace transform the Poisson Eq.(2.21b). To do so, let’s multiply both R +∞ −ıkx R ∞ ıωt ˜ R +∞ ˜ sides of this equation with −∞ dx e 0 dt e . One shall obtain −k φk,ω = −∞ dv fk,ω. Combining this transformed Poisson equation with equation (B.4) allows to find the following relation (B.5) : ˜ R +∞ fk(t=0) dv ˜ −∞ ω−k v φk,ω = (B.5) n R +∞ ∂vfeq o ı k k + −∞ ω−k v dv B–106 Establishing the dispersion function

At this stage, some important mathematical existence problems occur. They will be carefully addressed in what follows. The analytic continuation issues are discussed in appendix C in greater details. Before handling these problems, let’s reformulate the method. If I had defined (A.1) and (A.2) the following way:

Z +∞ Z ∞ g(v, k, p) = dx e−ıkx dt e− p t g(v, x, t) (B.6) −∞ 0 Z dk Z dp g(v, x, t) = eıkx e p t g(v, k, p) (B.7) F 2 π L 2 ı π where L is a vertical contour in the right half-plane, this would have seemed less physical (even if mathematically equivalent) but that could have avoided a great trap in which the physician reader can easily fall. The reason why I prefer this formulation is that, this way, p is thought to be from the beginning a very general complex number, free from the reader’s physical background that would make him ask: “why consider ω to be complex ? If it is a pulsation, it is of no use, and this can be as confusing as artificial. . . ”. What is presently being done is trying to solve a partial derivatives system. It definitely is a mathematical step and has to be done properly, as already latterly mentioned. Good physics will emerge from good mathematical work. So, if one is ready to free himself from his supposedly ‘physical’ constrain upon ω, he is ready to consider those two approaches as exactly equivalent, both ω and p being complex numbers. With this ‘p’-formulation, Laplace DAC then turns to be the right half-plane Re p > 0 [upper half-plane Im ω > 0 in figure A.1 was the ‘ω’-one], and (B.5) becomes :

˜ R +∞ fk,p(t=0) dv ˜ −∞ p+ı k v φk,p = (B.8) n R +∞ ∂vfeq o k − k + ı −∞ p+ı k v dv

R ∞ −p t where gp(p) = 0 dt e g(t) [see (B.3) for instance]. ˜ ˜ At this stage, φk,p is a function of p [φk,ω is a function of ω], and thus (B.8) [(B.5)] does only make sense when Re p > 0 [Im ω > 0]. Differently saying, domain definition of (B.8) [(B.5)] is the complex right half-plane [upper half-plane]. As appendix C will establish, the two integrals that appear in (B.8) [(B.5)] can be analytically continued using the Plemelj formulae (D.5) or ˜ ˜ (D.6). Therefore, the only singularities that φk,p [φk,ω] has are the zeros of the denominator.

Whether these resonances do/can exist is of primal interest for the way waves can behave in a plasma. Indeed, the existence of these resonances governs the physics of Landau damping. Following the Landau historical work [75], its mathematics can be found in appendix E. The latter work will then be brought to an end. Appendix C

The analytic continuation issue

R HEGOAL of this section is to clearly settle how to choose the integration path Γ of Γ f when T the integrand f is a very general function of a complex variable and to explicitly calculate the above integral. This may seem too technical a problem but as mentioned in section 2.3.1 and appendix A, this will be of very general interest, especially dealing with dispersion functions, and will settle on a breeding ground the important discussion on Landau damping of section 2.3.1. Of course not all physical integrands f are analytic functions everywhere, but at least to be of physical interest f is analytic is some region of the complex plane. Therefore, the theory of analytic functions and analytic continuation will be of great interest for what will be discussed from now on.

An analytic function is a differentiable function of a complex variable. The famous Cauchy theorem settles the integration contour choice for any analytic function that neither do pos- sess singularities within the integration contour nor on the integration path. The also famous residues theorem allows one to calculate a complex integral of a function analytic ‘nearly everywhere’ within a closed contour but can’t tell anything if the integration path actually goes through a singularity of the integrand. The aim of this section is to clearly settle the last possibil- ity : f is such that Γ passes through a f-singularity 1. Consequently, the latter problem can be expressed as follows : evaluate I = R f(z) dz, where C is a closed or not contour of the complex C z−z0 plane passing through z0.

Rigorously, I can’t be straightforwardly evaluated. The natural idea is to define I+ = R f(z) − R f(z) + − + dz and I = − dz, where C and C are represented on figure C.1. C z−z0 C z−z0

These integrals are defined everywhere, assuming from the latter discussion that f is a regular, ‘physical’ function, that do possess an analycity domain containing the infinitely close C+ and C− ∓ıθ contours to C. For on the semi-circle z = z0 + ε e , θ ∈ [0, π], the residues theorem leads to :

Z z0−ε Z ∞+ı Im z0 + n f(z) f(z) o I = lim dz + dz − ı π f(z0) (C.1) ε→0 −∞+ı Im z0 z − z0 z0+ε z − z0 Z z0−ε Z ∞+ı Im z0 − n f(z) f(z) o I = lim dz + dz + ı π f(z0) (C.2) ε→0 −∞+ı Im z0 z − z0 z0+ε z − z0 From this analysis, four major results can be driven :

1. An integral whose integration contour goes through a singularity of its integrand can- not –rigorously– be defined. Indeed, it is impossible to continuously go from the (definite)

1[25] is one of the best presented books that helped to make what follows clear enough. . . hopefully. And useful. C–108 The analytic continuation issue

Figure C.1: Given the C contour (leading to an undefined integration), C+ and C− are two natural choices in both avoiding the singularity (and thus leading to a defined integration) and remaining infinitely close to the nominal contour (ε can be arbitrary small).

value of an integral on an infinitely close contour to the nominal C one (without singulari- ties) to a (definite) other value of the same integrand on any other infinitely close contour without singularities 2. Therefore, I can’t be mathematically 3 rigorously defined. And that leads us to the second point.

2. The latter discussion allows one to feel the need for the monodromic theorem (or analytic continuation theorem, or analytic continuation principle):

Theorem 1 (Analytic continuation) Given f and g two analytic functions in a domain (a simply-connected open) V in the complex plane. If an open U exists 4 such as U ⊂ V and ∀z ∈ U f(z) = g(z), then ∀z ∈ V f(z) = g(z).

This very nice theorem, more fundamental than Cauchy’s theorem, allows the continuous contour deformations that have been latterly mentioned to be rigorously defined. The trick will consist in deforming a contour that would have initially gone through a singularity in order to have a definite integral (that can be evaluated thanks to the residues theorem) and doing so one will make sure the continuously deformed contour now in use remains (and has remained during all the deformation process) part of the DAC (this settles the mathematically cornelian choice of using C− rather than C+, for instance.).

Figure C.2 illustrates this idea : assuming the whole drawn space to be a domain of analyt- + + icity for the function we are interested in (except in z0), the Γ , C and C + ı ε contours are

2In other words : because of the change in sign when choosing C+ rather than C− or vice versa, the residues theorem states that the very (definite) value of an integral is not a continuous function of a continuous deformation of the integration contour as soon as the continuous contour deformation has to go through an integrand singularity ; I do not have one but at least two different possible values, with respect to the contour choice (to go round the singularity by above or by below, in that case). 3It is very important to state here that, physically, the contour choice is constrained. Physically, the contour defor- mation will have to round the singularity by above [or by the right] or (an exclusive or) by below [or by the left]. This is indeed good news for the physical theory to remain unique in its consequences. . . This can be so thanks to the causality principle taken into account while performing F LT s, leading to the notion of DAC, as commented in section 2.3.1 and appendix A. 4Less demanding hypothesis do exist. The analytic continuation issue C–109

+ + Figure C.2: Assuming z0 is the only singularity, all Γ , C and C + ı ε contours are equivalent.

equivalent : for a given integrand, using one or the other will lead to the same value of the integral. One can start to understand that with some work it can be possible to extend the (undefined) I integral (with contour C) to some other definite integral that can be evaluated, without ambiguity. One would then say that I can be analytically continued using one of the three (or more) contours on figure C.2. Doing so requires the Plemelj formulae and before that a definition of the principal part of an integral. This will be introduced in what follows.

3. In order to get rid of the problematic choice of which contour to choose to avoid the sin- gularity, is is convenient to split the integral that is to be evaluated into two parts : a part that will embed the presence of the singularity on the integration contour (this will need an adequate continuation choice) and a ‘free-singularity’ part, i.e. the value of the integral ev- erywhere else. This is the so-called principal part of an indefinite integral. This principal part must be defined, as the above explanation makes it clear, on a contour free from the singularity’s existence. Given equations (C.1) and (C.2), this can be done by defining as follows the principal value of the integral :

Z 1n Z Z o princ F (z) dz ≡ lim F (z) dz + F (z) dz (C.3) C ε→0 2 C+ C−

where the C+ and C− contours are those of figure C.1. It is interesting to note that, combined with the analytic continuation theorem that allows continuous deformations of an integral contour inside the DAC, the principal value can as well be defined, if needed, as follows :

Z 1n Z Z o P F (z) dz ≡ lim F (z) dz + F (z) dz (C.4) C ε→0 2 C+ı ε C−ı ε

where the C+ and C− contours are those of figure C.3 5.

4. Now is time to finish the work and extract an operational formula from all the interesting latter stuff. Most often, it is of use to consider complex integration even though the physical integration variable is real. It seemed to me it was an important and non trivial feature to clear up the sometimes highly risky business often to be read when shuttling back and forth between real and complex variables, between epsilons that sometimes vanish, sometimes

5The contour orientations are of course very important in the two definitions (C.3) and (C.4) C–110 The analytic continuation issue

Figure C.3: Another possible choice for C+ and C−. Same legend as for figure C.1.

not, and invoking at the end ‘physical sense’ as safeguards. It is my personal view that such an attitude sometimes destructively interferes with the rigorous reader’s sake. As, in our case, the business can be rigorously settled considering the so-called Plemelj Formulae,I think it is in good taste that such would be done. None demonstration was to be found in the books I checked ; never mind, I have tried to perform one myself. It can be found in appendix D. It rests on the contour equivalence shown on figure C.2 and the principal value definition. Appendix D

Establishing the Plemelj formulae

S THE LATTER APPENDIX started to allude, the Plemelj formulae are important so as to rig- A orously perform analytic continuation and find Landau damping. The following lines are a personal attempt to demonstrate them.

R R Thanks to appendix C, it has been established that C+ F (z) dz = Γ+ F (z) dz and similarly R R ± ± C− F (z) dz = Γ− F (z) dz, where the C and Γ contours are those shown in figures C.1, C.2 R R 1 and C.3. Moreover, the residues theorem leads to C− F (z) dz − C+ F (z) dz = 2ıπ Res(F, z0) . Combining this with the principal value definition of equation (C.3), leads to : Z Z Z F (z) dz = F (z) dz = P F (z) dz ± ıπ Res(F, z0) (D.1) Γ∓ C∓ C Moreover, for the same reasons,

Z Z Z +∞+ı Im z0+ı ε F (z) dz = lim F (z) dz = lim F (z) dz (D.2) + ε→0 ε→0 Γ C+ı ε −∞+ı Im z0+ı ε

Performing the variable change z = x + ı Im z0 + ı ε, the latter equation becomes :

Z Z +∞ f(x + ı Im z + ı ε) F (z) dz = lim 0 dx (D.3) Γ+ ε→0 −∞ x − Re z0 + ı ε where x is Real 2.

What causes problem is that physically f at the numerator is the distribution function and can 2 n(x) − v typically be written, for a Maxwellian equilibrium as f(v) = √ e 2 T (x) . Therefore, in the T (x) case of a real singularity (z0 ∈ R), f(v + ı ε) can be evaluated and remains real. But this can’t be done when the distribution function isn’t close to a Maxwellian. It would be of great interest for the sake of generality to be able to remove the ε limit from the distribution function’s argument. Hopefully, this can be done, assuming f is analytic. It can therefore be expanded in Taylor series (referred to as TSE):

(n) X f (x + ı Im z0) f(x + ı Im z + ı ε) = f(x + ı Im z ) + (ı ε)n (D.4) 0 0 n! n≥1

1 In particular, one shall always consider z0 ∈ C as the only singularity of the integrand F . Thus, F can be written F (z) = f(z) , f being analytic. z−z0 2Here, x is a generic variable. For direct application, x stands for velocity v [see (2.27) for instance]. D–112 Establishing the Plemelj formulae

(n) f (x+ı Im z0) n→∞ Because TSE lead to an absolute convergent series, n! −→ 0. Consequently, (n) (n) f (x+ı Im z0) f (x+ı Im z0) R +∞ as f is analytic, ∀n ∈ N, is bounded and so is also n! dx , let’s n! −∞ x−Re z0+ı ε say, by A. A is independent of ε, which is non-trivial. It is possible thanks to the analytic con- tinuation of the latter integral 3 : the principal value is well-defined. . . and bounded independently of ε. Therefore, as the integral is linear :

Z +∞ f(x + ı Im z + ı ε) Z +∞ f(x + ı Im z ) lim 0 dx = lim 0 dx ε→0 −∞ x − Re z0 + ı ε ε→0 −∞ x − Re z0 + ı ε f (n)(x+ı Im z ) X Z +∞ 0 + lim (ı ε)n n! dx ε→0 x − Re z0 + ı ε n≥1 −∞ | {z } 1 = 1−ı ε −1 | {z } 1 | · | ≤ A ( 1−ı ε −1)

It has been possible to invert the sum and integral for the Taylor series is absolutely convergent. The infinite sum can be evaluated as it is geometric. In the limit ε → 0, the latter term vanishes. Only remains :

Z +∞ f(x ∓ ı Im z ) Z +∞ f(x ∓ ı Im z ∓ ı ε) lim 0 dx = lim 0 dx ε→0 −∞ x − Re z0 ∓ ı ε ε→0 −∞ x − Re z0 ∓ ı ε Z f(z) = dz Γ∓ z − z0 Z f(z) = dz C∓ z − z0 Z f(z) = P dz ± ıπ f(z0) (D.5) C z − z0 which constitutes the complex Plemelj formula.

+ − Now assuming that C is the real axis and z0 = Re z0 = x0 ∈ R, the choice of the C or C contours (and thus the DAC and causality principle) will decide on which sign stands in front of the ı ε at the denominator. Therefore : Z f(x) Z +∞ f(x) dx = lim dx x − x ε→0 x − x ∓ ı ε R 0 −∞ 0 Z f(x) = dx ± ıπ f(x ) (D.6) P x − x 0 R 0 which constitutes the real Plemelj formula.

3 R +∞ dx R +∞+ı ε du R du R +∞ dx for = ≡ + ≡ − ıπ · 1 −∞ x−Re z0+ı ε −∞+ı ε u−Re z0 C u−Re z0 P −∞ x−Re z0 Appendix E

Landau damping: the historical derivation

HISAPPENDIX will be based upon Landau’s historical paper [75] and will settle the damping T existence. Although [75] is quite clear and pedagogue, some important questions were to appear while I studied it, and remain presently under investigation. They will also be presented.

This appendix happens to be the continuation of appendix B. I will assume the reader is famil- iar with the notations, as they are the same. Before assuming the dispersion function D (see (2.27), for instance) can have solutions, one has to make sure the zeros of the denominators of (2.25) [or (B.5) or (B.8)] can be properly defined, i.e. there can exist resonances. These zeros cannot be defined straightforwardly. This is a consequence of the Laplace transform that imposes constraints on the sign of Re p [or Im ω]. Indeed, these zeros have to lie in the complex right half-plane for the p-description and the upper half-plane for the ω-description. An important question is whether they do lie in the latter half-planes or not. The answer seems to be that they do not lie in these half-planes. If this was to be true, this is why a decent mathemat- ical work is necessary. The reader should understand that, at this stage, these resonances cannot be mathematically defined. This is the true physical argument why an analytic continuation is necessary. As it is be important, I shall first point out why the zeros of the denominators of (2.25) –namely the poles of D– are respectively thought to be in the left half-plane and lower half-plane. I will settle this result using the p-description [ω-description]. Reminding page 106 [equation (B.3)], ˜ R ∞ −p t ˜ ˜ R ∞ ıωt ˜ ˜ ˜ ˜ φp(p) = 0 dt e φ(t) [φω(ω) = 0 dt e φ(t)], if φp [φω] has resonances it implies that φp ˜ [φω] is infinite for at least one pk [ωk] value. Thus the latter integral is not defined. Since this integral is always defined when Re p > 0 [Im ω > 0](φ˜ is thought to be analytic), a resonance cannot occur when Re p > 0 [Im ω > 0]. Assuming, as previously discussed that φ˜p [φ˜ω] is an entire function (i.e. an (analytic) function that has no singularities at finite values of p), one can extend the existence domain of φ˜p [φ˜ω] to the left half-plane [lower half-plane] by deforming the initial real-axis contour such that pk remains in the DAC. In both p-case and ω-case, it would consist in circling the singularity by below when k > 0, by above when k < 01. As appendix C has shown, such contour deformations are possible, lead to the same value of the integrals and enables to define φ˜p [φ˜ω] as an analytic function in the whole plane.

1 pi ı pr ωr ı ωi Indeed poles are obtained when v = − k + k and v = + k + k . Therefore the DACs are respectively the above plane when k > 0, the lower plane when k < 0. Which leads us to displace the real axis nominal integration contour far into the lower plane for the singularity to keep lying above it (k > 0). And conversely. E–114 Landau damping: the historical derivation

Now that resonances can exist it is possible, thanks to (B.7) and (A.2), to investigate the true ˜ physical sense whey would embed, namely the φk dependance on time. In the inversion formula Z  Z  ˜ dp +p t ˜ ˜ dω −ıωt ˜ φk(t) = e φω,k φk(t) = e φp,k (E.1) L 2 ıπ L 2π ˜ ˜ the Laplace contour is a vertical [horizontal] line in the right [upper] half-plane. As φp,k [φω,k] is analytic everywhere, it is possible to displace this integration path into the left [lower] half-plane, going around by the right [by above] all the poles it meets. Why do so ? Because pr [ωi] becomes negative and thus (E.1) can take a very simple form for large values of time. From which the asymptotic behavior of the field in time can be derived. Thus, for large values of time, as Re p < 0 ˜ pk t ˜ −ıωk t [Im ω < 0], (E.1) states that φk ∼ e [φk ∼ e ], where pk [ωk] are zeros of D. For complex p = pr + ıpi [ω = ωr + ıωi], one can easily see that the field has a periodical part and an exponentially decreasing part with time, namely a damping with time. This is the so-called Landau damping which is more physically discussed in section 2.3.1. The dispersion relation determines pk [ωk] and thus the frequency and the damping decrement of the wave vibrations. In this context, both integrals that appear in Eq.(2.25) can be rigorously recast in such a way R +∞ ∂vfeq that they remain valid for ω in the whole complex plane and −∞ ω−kv dv, for instance, can be read as follows2:  h i Z +∞ 1 R +∞ ∂vfeq ω ∂vfeq  − k P −∞ v− ω dv + ı π ∂vfeq( k ) = 0 if k > 0 dv = h k i ω − kv 1 R +∞ ∂vfeq ω −∞  − P ω dv − ı π ∂vfeq( ) = 0 if k < 0 k −∞ v− k k Z +∞ 1 h ∂vfeq ω i = − P ω dv + sgn(k) ı π ∂vfeq( ) k −∞ v − k k Z +∞ ∂vfeq ı π = P dv − ∂vfeq|v=ω/k −∞ ω − kv |k| Note the absolute value of wavevector k in the last term. This imaginary part of the dispersion relation is often referred to as the “ı ε” Landau prescription. It is indeed equivalent to the phys- ical argument saying that the denominator of the integrand should read ω − kv + ı ε, ω ∈ R, ε → 0+ so as to remain definite, with respect to the causality principle3. As shown in ap- R +∞ g(v) pendix D, in the limit of vanishing positive ε, the following relation holds: −∞ ω−kv dv = R +∞ g(v) R +∞ g(v) R +∞ lim + dv = dv − ı π g(v)δ(ω − kv) dv, assuming g is an- ε→ 0 −∞ ω−kv+ı ε P −∞ ω−kv −∞ |{z} Landau prescription 1 ω
alytic. The scaling property of the Dirac distribution then states δ(ω − kv) = |k| δ k − kv , in agreement with the absolute value already reported. Finally, the dispersion relation Eq.(2.26) is to be understood as follows: Z +∞ ∂vfeq ı π D(k, ω) = k + P dv − ∂vfeq|v=ω/k = 0 (E.2) −∞ ω − kv |k|

2Thanks to the appendix A discussion and the Plemelj formulae Eqs.(D.5) or (D.6), the ± signs comes from “ Im ω ” sgn k = sgn(k). 3See appendix A on Laplace transforms for deeper understanding. Appendix F

Closures in real space

The closure relations Eqs.(2.48) and (2.49) are determined in wave-number space. Expressing them in real space requires to perform inverse Fourier transforms (referred to as IFT ). The generic closure form Eq.(2.39) can be recast into: qˆk,ω = −ı A sgn(k)n ˆk,ω − B uˆk,ω + ı C sgn(k)p ˆk,ω where A, B and C read: √ 2π T v A = 0 T (F.1) π − 4 4 p B = Γ p + 0 (F.2) 0 π − 4 √ 2π v C = T (F.3) π − 4 The real space form leads to:     q˜(z) = −BIFT (˜u(k)) + lim −ıAIFT (sgn(k)e−|k|εn˜(k)) + ıCIFT (sgn(k)e−|k|εp˜(k)) | {z } →0+ | {z } | {z } I2  I1 I3  (F.4)   +∞ where IFT f(k) ≡ √1 R dk f(k) eıkz. 2π −∞ In order to perform this calculation, the convolution theorem Eq.(F.5) is important. So is the following relation Eq.(F.6). Proof is given in appendix G:

IFT (f.g) = IFT (f) ~ IFT (g) (F.5) r 2 z IFT (sgn(k)e−|k|ε) = ı (F.6) π z + ε R +∞ where ~ refers to the convolution product x(z) ~ y(z) = −∞ x(u)y(z − u)du

I2 calculation: The Fourier-transform is straightforward, leads to −B u˜(z);

I1 calculation: Thanks to the latter relations: r 2 z r 2 Z ∞ s ı ~ n˜(z) = ı n˜(z − s)ds (F.7) π z + ε π −∞ s + ε       r 2 Z ∞ s Z 0 s  = ı n˜(z − s)ds + n˜(z − s)ds π  0 s + ε −∞ s + ε   | {z }  R ∞ s  − 0 s+ε n˜(z+s)ds F–116 Closures in real space

+ ε→0 2T0vT R ∞ n˜(z+s)−n˜(z−s) Consequently, I1 −→ π−4 0 s ds;

+ ε→0 −2vT R ∞ p˜(z+s)−p˜(z−s) I3 calculation: A very similar result leads to I3 −→ π−4 0 s ds. And thus: Z ∞ 2T0vT n˜(z + s) − n˜(z − s) n 4p0 o q˜(z) = ds − Γp0 + u˜(z) π − 4 0 s π − 4 2v Z ∞ p˜(z + s) − p˜(z − s) − T ds (F.8) π − 4 0 s Appendix G

Establishing the inverse Fourier Transform of sgn(k)

OASTO be able to evaluate the inverse Fourier transform of sgn(k)n ˜(k) or sgn(k)p ˜(k) one S can naturally think about using the convolution theorem (F.5). In order to do so, it is important to calculate the inverse Fourier transform of sgn(k). But this is not straightforward, as the sign function is not absolutely sommable and does not vanish at infinity. But it can be made so if multiplied with the exponential factor e−|k|ε, ε > 0. We define f(k) = sgn(k) e−|k|ε. It is important to see that limε→0 f(k) = sgn(k). As the inverse Fourier transform of f can easily be obtained as detailed in the following lines, the only thwarting in our attempt is whether limε→0 R +∞ and −∞ can be commuted. Assuming this, it is possible to finish the work : 1 Z +∞ IFT (sgn(k)) = √ lim f(k)
eıkz dk 2π −∞ ε→0 1 Z +∞ = √ lim sgn(k) e−|k|ε
eıkz dk 2π −∞ ε→0 1 n Z ∞ Z 0 o = √ lim sgn(k) e−kε+ıkz
dk + lim sgn(k) ekε+ıkz
dk 2π 0 ε→0 −∞ ε→0 1 Z ∞   = lim √ e−kε+ıkz − e−kε−ıkz dk ε→0 2π 0 1 2ız = √ 2π z2 + ε2 It is now necessary to justify the above limit–integral interversion. This will be done using the Lebesgue domination theorem. Let’s set ε > 0 and z ∈ R. One can define :

( −(k+ 1 )(ε−ız) e n k > 0 gn(k) = +(k− 1 )(ε+ız) e n k < 0

For n ∈ NF : 1 • gn ∈ L (R) ;

• the gn are integrable over R ;
n→∞ −|k|ε ıkz • for k ∈ R, gn(k) −→ g(k) = sgn(k) e e ;

−|k+1|ε •∀ n ∈ NF, ∀k ∈ R, |gn(k)| 6 e . | {z } ∈L1 G–118 Establishing the inverse Fourier Transform of sgn(k) Appendix H

The numerical impact of non locality

The closure relations Eqs.(2.48) and (2.49) are determined in wave-number space. They can be expressed in real space –the detailed calculation can be found in appendix F: Z ∞ 2T0vT n˜(z + s) − n˜(z − s) n 4p0 o q˜(z) = ds − Γp0 + u˜(z) π − 4 0 s π − 4 2v Z ∞ p˜(z + s) − p˜(z − s) − T ds (H.1) π − 4 0 s This description in real space leads to non-local operators; in other words, a local fluid temperature (or particle density, pressure. . . ) in real space requires so as to be satisfactorily described a global knowledge of the temperature (resp. the density or the pressure) throughout the entire system. This represents a serious constraint in non-spectral fluid codes: the integro-differential equation which p p is to be solved cannot be developed using a truncated Taylor expansion –e.g. (s − z) ∂z T (z)– since the high-order terms are not vanishing when p → ∞. They can even become the dominant p terms, with respect to ‘good’ regularity properties of the derivatives ∂z T (z). Moreover, this expansion requires the explicit calculation of all derivatives, for all orders, which can become quite time-consuming. Another idea could consist in merely evaluating the non-local integrals. R ∞ F (z+s)−F (z−s) The following graphs highlight this idea; this direct evaluation of G(z) = 0 s ds is compared to the one starting from Fourier space Gˆk ∼ ı sgn(k)Fˆk and using fast Fourier transforms (FFT s). The left-hand graph shows the initial simple power law choice for the F

profile in k-space. The F profile in real space is reported in dotted line in the right-hand graph. The ‘exact function’, in red, stands for the Fourier transform using FFT . It can be seen that the blue curve, the direct evaluation, does not perfectly match the red one. Moreover, the computation times are not comparable: the FFT is instantaneous whereas the direct, non local evaluation is closer to the tens of seconds. This latter classical result is very important in a realistic code where many of such evaluations are to be done at every time step. As one would have expected, this H–120 The numerical impact of non locality simple study only confirms that the choice method (more precise and faster), when one has to evaluate such integro-differential non local operators is to start from Fourier space and use fast Fourier transform algorithms. Appendix I

Marginal stability of the drift kinetic instability

The quasi-neurality equation for adiabatic electrons reads: 1 Z 1 − ρ2∇2
Φ = dv f − 1 (I.1) i ⊥ n¯

2 1 1 1  ˆ  where ∇⊥ ≡ r ∂r (r∂r) + r2 ∂θθ. Without any lack of generality, we shall write r ∂r r∂rφk,ω = ˆ α(r)φk,ω. Combining the latter equation (I.1) with eq.(2.55) yields the dispersion relation D(k, ω) ≡ Dr + ıDi = 0, which can be split, using eq.(2.62) into its real and imaginary parts: 1 Z ω − ω? Dr = L − P dv feq (I.2) n¯ ω − kzv 2 " ? ? # − ω πf ω ω 2ω2 0 T ? 2 T k Di = − ωn + ω − ω 2 exp (I.3) n¯|kz| 2 2ωk

2 2
where L = 2 + ρi kθ − α . At the instability threshhold, Di = 0. The pulsations ω± of a linear ? ?  2 2  eigenmode are real and satisfy: ω± − ωn − ωT ω± /2ωk − 1/2 = 0, or equivalently:

ω2 ω k k q ?2 ? ? 2 ω± = ? ± ? ωT − 2ωnωT + ωk (I.4) ωT ωT

?2 ? ? 2 with ωT − 2ωnωT + ωk ≥ 0. At the threshold, Dr vanishes and the integral in eq.(I.2) can be evaluated: ?  Z Z  ? ωT feq vfeq ωT L = 2 ω± dv + kz dv = 2 ω± (I.5) 2ωk n¯ n¯ 2ωk Combined with eq.(I.4), the latter equation leads to the kinetic marginal stability diagram in the ? ? (ωn, ωT ) plane: ? 2ω2 ωT ? k − ωn + ? L (1 − L) = 0 (I.6) 2 ωT I–122 Marginal stability of the drift kinetic instability Appendix J

Recovering the particle transverse fluid current

We shall here emphasise on the classical result which enables to derive the correct particle trans- verse fluid velocity. It is well known that the transverse current J⊥ reads: J⊥ = J⊥gc + Jmagn, where J⊥gc is the guiding-centre component, Jmagn = ∇ × M is the magnetisation current, M = − R µbf the plasma magnetisation and integration is carried out over normalised phase R R ∞ R +∞ space ≡ 0 2πB/mdµ −∞ dvk. The following calculation is performed taking into account 2 mvk the ‘low β-drift’, namely vD = vg + eB2 (∇ × B)⊥. Z J⊥ = e (vE + vD) fCM + (∇ × M)⊥ B × ∇B (∇ × B) Z  Z  ⊥ 2 ∇ = enVE + p 3 + 2 mvkfCM − ∇ × µbfCM B B ⊥ B × ∇p B × ∇B (∇ × B) = enV + ⊥ + (p − 2p ) + p − p
⊥ (J.1) E B2 ⊥ B3 k ⊥ B2 which is the transverse fluid current up to the first order in ρ?, the pressure being defined as follows: R 2 R pk ≡ mvkfCM , p⊥ ≡ µBfCM and p = pk + p⊥. In the case of an isotropic pressure tensor, pk = p⊥, the last two terms vanish and the expression for ∇ · V⊥ is recovered. J–124 Recovering the particle transverse fluid current Appendix K

Short-time dynamics governed by the curvature and grad-B drifts

We start with the following equation:

∂tf + vD · ∇f = 0 (K.1)

We concentrate on the leading order contributions in . The latter equation reads:

 cos θ  ∂ δf = v sin θ∂ f + ∂ f (K.2) t LM d r r θ

 2  where vd = mvk + µB /eR0B0. Similarly, the quasi-neutrality equation Eq.(3.44) reads:

  Z ∞ Z +∞ s c 2 ∂θθφ Teq 2πB φ10 sin θ + φ10 cos θ − ρs ∂rrφ + 2 = dµ dvk δfLM (K.3) r eneq m 0 −∞

For the sake of simplicity, we assume Ti = Te = Teq. At leading order in , the sin θ component reads: s fLM f10 ≈ vd t (K.4) Lp

−1 −1 −1 −1 with Lp ≡ Ln + LT (E/Teq − 3/2). We recall that LX ≡ ∂rX/X and κX ≡ ∂rrX/X. In c particular, Lp is the pressure gradient length. Similarly, the f10 component is such that:

s c φ10 fLM ∂tf10 ≈ (K.5) rB0 Lp

At last, the evolution equation for the axisymmetric perturbation f00 reads:

2    s  vd fLM 1 ∂rLp 1 µfLM s φ10 ∂ttf00 ≈ − + + ∂t ∂rφ10 − (K.6) 2 Lp Lp Lp r 2R0Teq r

s At this stage, φ10 remains unknown. The quasi-neutrality equation then allows one to solve R +∞ R ∞ explicitly Eqs.(K.5) and (K.6). We define: hGi ≡ 2πB/mneq −∞ dvk 0 dµ G fLM . And thus:

D 2E h1i = 1 and m vk = B hµi = Teq (K.7) 2 2Teq 2 7Teq hvdi = and vd = 2 2 2 (K.8) eB0R0 e B0 R0 K–126 Short-time dynamics governed by the curvature and grad-B drifts

As a consequence, hE/Teq − 3/2i = 0. Taking the time derivative of Eq.(K.3) yields the time evolution of φs : 10   e s vd 2Teq ∂rpeq ∂tφ10 ≈ = (K.9) Teq Lp eB0R0 peq 2 s s 2
s Since the ρs ∂rrφ10 − φ10/r term is a ρ? correction to φ10, it has been neglected. At first order in ρ?: e s 2ρscs φ10 ≈ t (K.10) Teq R0Lp 2 s with Teq/eB0 = ρscs and cs = Teq/m. Injecting the expression of φ10, Eq.(K.10), into eqs.(K.5) and (K.6) leads to:

c fLM 2 f10 ≈ (ρscs t) (K.11) rR0LpLp (  2   2  ) µB0 fLM ρscs 2 1 1 vd fLM 1 ∂rLp 1 2 f00 ≈ − + Lp κp − + − + (K.12)t Teq 2Lp R0 LT Lp r 4 Lp Lp Lp r

c Similarly, one can obtain the evolution equations for φ10 and φ00 from Eq.(K.3):

 2 2 2  2 e c ρscs t (ρscs) 2 φ10 ≈ = t (K.13) Teq rR0LpLp rR0LpLn   2 e f00 −ρs ∂rrφ00 ≈ (K.14) Teq fLM The latter integral can be estimated:

     2 e 7 1 4 12 1 1 16 cs t ∂rrφ00 ≈ − + + + + κp − κn (K.15) Teq 4 Ln 7LpLT 7Lp Lp r 7 R0 Equations (K.10), (K.13) and (K.15) respectively correspond to eqs.(3.74), (3.76) and (3.78). They are predicted to characterise the dynamics of the system in response to the vertical polarisation drift. Appendix L

Analysis of the departure between the analytical prediction and simulations

Eqs.(3.74), (3.76) and (3.78) are obtained after integrations over phase space. In GYSELA, these integrals are calculated using cubic splines and have a typical precision int ≈ 10−4 in our sim- ulations, which can be defined as the relative error between the exact solution of the full prob- s lem and the numerical result. To explain it more plainly, let us concentrate on Ω10, Eq.(3.75). s,N s,E s,A At least three different values of it exist, written Ω10 , Ω10 and Ω10 . The first one is given by the numerical simulation; the second one would be the exact solution of the full problem without any source of numerical imprecision and the last one is the analytical prediction we want to check (summary in Table L.1). What this appendix is going to show for φs is that: 10 s,N s,E s,A s,E c log Ω10 − log Ω10  log Ω10 − log Ω10 . The same results are also found for φ00 and φ10. Though we cannot conclude the amplitudes predicted in eqs.(3.74), (3.76) and (3.78) are recov- ered numerically, they remain well inside the accessible numerical error bars. In equation (3.74), the relation:   2 vd 2Teq ∂rpeq = 2 (L.1) Lp e B0R0 peq is exact analytically but only approached on a numerical point of view. When integrating over vk and µ, an error s is generated:   e s 2ρscs s φ10 ≈ +  t (L.2) Teq R0Lp c This error enters the f10 and f00 calculation:  s  c fLM ρscs ρscs  2 f10 ≈ + t (L.3) rLp R0Lp 2

 2       s  vd fLM 1 ∂rLp 1 µfLM 2ρscs s 2ρscs  2 f00 ≈ − + + ∂r +  − + t 4 Lp Lp Lp r 4eR0 R0Lp rR0Lp r (L.4) The integration of the former equation (L.3) leads to:  2 2 s    e c ρscs  ρscs 1 2 φ10 ≈ + t (L.5) Teq rR0Lp 2r Lp c Let us introduce  the error for h1/Lpi. The first term of Eq.(L.5) can be written as follows: 2 2 c
2 s c 2 ρscs/rR0LpLn +  t and the second: ( ρscs/2rLn +  ) t . Thus:  s  e c c c  ρscs 2 φ10 ≈ φ10 + 2 + t (L.6) Teq s=0 2rLn L–128 Analysis of the departure between the analytical prediction and simulations

Table L.1: Summary of the different sources of imprecision for the estimate of the electric poten- s tial, e.g. φ10 (∆t is the first saving time step).

s s,A analytical: log φ10 = log Ω10 + log ∆t s s,N numerical: log φ10 = log Ω10 + log ∆t s s,E exact: log φ10 = log Ω10 + log ∆t

Proceeding the same way for φ00:

2 Teq 2πB − ρs∂rrφ00 ≈ x dµdvk f00 (L.7) eneq m

s=0 s 0 0 Let us define f00 given by Eq.(L.4) when  = 0. Let us introduce 1 and 2 which represent s=0 the errors for respectively f00 /fLM and hµi. Thus, Eq.(L.7) reads:

 s  e e 1 0 0  ρscs 2 ∂rrφ00 ≈ ∂rrφ00 −  +  − t (L.8) s 2 1 2 Teq Teq  =0 ρs 4rR0

The smaller the ρ? parameter, the more important those errors. At ρ? = 1/400, the mismatch between analytics and numerics is most penalising:

s ≈ −2.78 10−6 (L.9) sρ c 2c + s s ≈ −2.24 10−10 (L.10) 2rLn 2 s 0 0
h r i  ρscs (r log r − r) −8 2 1 + 2 − 2 ≈ 8.52 10 ρ? (L.11) a 4a R0 And thus:

s ≈ −2.78 10−6 (L.12) c ≈ −1.09 10−10 (L.13) 0 0 −13 1 + 2 ≈ 1.16 10 (L.14)

Yet, all three errors eqs.(L.12), (L.13) and (L.14) are well below int. Appendix M

Onset of GAMs with qualitative agreement on the frequency

We calculate the modification of the prediction on the components of the electric potential eqs.(3.74), (3.76) and (3.78) when the electric drift is taken into account. We find an oscillating modulation, close to the GAM frequency, as expected from the Rosenbluth and Hinton theory [?]. Let us consider:

∂tf + (vD + vE) · ∇f = 0 (M.1)

We will concentrate on the first order terms in  since section 3.2.3 already discussed the leading order at short times. Using eqs(3.68)–(3.70), Eq.(M.1) reads:

    cos θ µfLM cos θ ∂θφ ∂tδfLM ≈ vd sin θ∂rδfLM + ∂θδfLM + sin θ∂rφ − ∂θφ + fLM r RTeq r rBLp (M.2) In the following, we will neglect the terms with a cos θ dependence as compared to those in sin θ. Proceeding the same way than in section 3.2.3, the first Fourier modes of the distribution function read:

vd s 0 µfLM s 0 ∂tf00 ≈ f10 + φ10 (M.3) 2 2RTeq

s 0 µfLM 0 ∂tf10 ≈ vdf00 + φ00 (M.4) RTeq

c fLM s ∂tf10 ≈ φ10 (M.5) rBLp

s 0 s where φ10 ≡ ∂rφ10. Eqs.(M.3) and (M.4) can be recast into:

2  s 0  vd 00 µfLM 00 ∂tφ10 ∂ttf00 ≈ f00 + φ00 + (M.6) 2 RTeq 2 2  s 00  s vd s 00 µfLM φ10 0 ∂ttf10 ≈ f10 + + ∂tφ00 (M.7) 2 RTeq 2

When coupled to the quasi-neutrality equation, the first two terms in eqs.(M.6) and (M.7) due to 4 2 s s the curvature drift lead to high-order terms ∝ ∂r φ00 (resp. ∂r φ10) for φ00 (resp. φ10) which are M–130 Onset of GAMs with qualitative agreement on the frequency negligible. The electric potential reads:

 00   s 0  2 eφ00 1 00 ∂tφ10 ∂tt −ρs ≈ φ00 + (M.8) Teq R0B0 2  s   s 00  eφ10 1 φ10 0 ∂tt ≈ + ∂tφ00 (M.9) Teq R0B0 2  c  s   s eφ10 φ10 1 φ10 ∂t ≈ = (M.10) Teq rB Lp rBLn

s where we made use of Eq.(K.7). Since we are interested in times when φ00 or φ10 start to satu- s 0 00 rate, the ∂tφ10 /2 term in Eq.(M.8) is neglected as compared to φ00. This equation has a time- oscillating solution close to the GAM frequency: c  φ00 ∝ cos s t + ℵ (M.11) 00 R with ℵ an arbitrary phase. Given Eq.(M.11), a function ∝ sin (cs t/R + ℵ) is a solution of c 00 Eq.(M.9). And then, Eq.(M.10) leads to: φ10 ∝ φ00. All three components of the electric po- tential have a time-oscillating behaviour close to the GAM frequency, as found indeed in Figure 3.7. It also clearly scales with ρ? since cs/R ∝ ρ?. Appendix N

Parallel flows, a robust saturation mechanism for the charge polarisation

With the parallel dynamics, Eq.(3.72) reads:

dvk ∂ f + v · ∇f + v ∇ f + ∂ f = 0 (N.1) t D k k dt vk The sine component equation has the simple form:

s vdfLM µ vk c ∂tf10 = + ∇kB∂vk fLM + f10 (N.2) Lp m qR

At leading order in , it can be integrated over time:

s vdfLM µB r f10 = t − vk 2 fLM t (N.3) Lp Teq qR

The cosine component equation can also be estimated. To do so and include the contribution of the parallel dynamics, we assume it is initially a perturbation. Equation (K.5) is modified into:

c fLM s vk s ∂tf10 = φ10 − f10 (N.4) rB0Lp qR

s s c with φ10 and f10 given respectively by Eqs.(3.74) and (N.3). The odd part in vk of f10 –written c O(f10)– reads: c vk vdfLM O(f10) = t (N.5) qR Lp s It allows one to determine φ10:

 s   ∂ f c  eφ10 vd µ vk LM vk f10 ∂t = + ∇kB + Teq Lp m fLM qR fLM   * 2 +  2 v 1 vkvd t = d − (N.6) Lp 2 Lp qR

After a little algebra:

s   eφ10 2ρscs 2ρscs Teq 1 1 3 = t − 2 2 + t (N.7) Teq R0Lp R0 mq R Lp LT N–132 Parallel flows, a robust saturation mechanism for the charge polarisation

The introduction of the parallel dynamics results in a t3 correction term, which is negligible at short times. It becomes significant when the linear term in this equation is balanced by the cubic term. An upper boundary for this time is: r qR Lp τk ∼ 1 + (N.8) vT LT c where vT is the thermal velocity of the ions. Since the φ00 and φ10 components of the elec- s 4 2 tric potential are evaluated from φ10, the parallel dynamics introduce a t –correction to their t – −1 dependence. Also, τk clearly scales with ρ? since R/vT ∝ ρ? . Appendix O

Numerical implementation of the collision operator in GYSELA

We solve the following equation: n o ∂tf = C(f) = ∂vk D∂vk f − Vf (O.1) in which the collision operator C Eq.(4.25)–(4.27) is implemented in GYSELA using the semi- n implicit, second-order Crank-Nicolson scheme. Classically, let us write: fj ≡ f(vj, tn), where subscript j ∈ {1 ...N} refers to the discretised index in parallel velocity space and superscript n refers to the time. For each value of µ, i.e. on each processor associated with an MPI world, each term in Eq.(O.1) reads:

f n+1 − f n ∂ f → j j (O.2) t ∆t h n+1 n i h n+1 n i Vj+1 fj+1 + fj+1 − Vj−1 fj−1 + fj−1 ∂vk Vf → (O.3) 4∆vk h n+1 n+1 n ni h n+1 n+1 n n i D 1 f − f + fj+1 − fj − D 1 f − f + fj − fj−1 j+ 2 j+1 j j− 2 j j−1 ∂vk D∂vk f → 2 (O.4) 2∆vk which can be rewritten in the compact tridiagonal form:   B1 C1  f n+1   Rn   ..  1 1  A1 B2 .  . .    .  =  .  (O.5)  .. ..       . . CN  n+1 n fN RN AN BN where:

D 1 j− 2 Vj−1 Aj = − 2 − (O.6) 2∆vk 4∆vk

D 1 + D 1 1 j+ 2 j− 2 Bj = + 2 (O.7) ∆t 2∆vk

D 1 j+ 2 Vj+1 Cj = − 2 + (O.8) 2∆vk 4∆vk O–134 Numerical implementation of the collision operator in GYSELA and  2  −B1 + −C1  Rn  ∆t  f n  1  2 ..  1 .  −A1 −B2 + .  .  .  =  ∆t   .  (O.9)    .. ..    Rn  . . −CN  f n N 2 N −AN −BN + ∆t The above values are univocal provided adequate boundary conditions, which are chosen such that (vanishing Laplacian at the edges):

for A0 and B0 ⇒ D 1 = 2D0 − D 1 and V−1 = 2V0 − V1 (O.10) − 2 2 for BN and CN ⇒ D 1 = 2DN − D 1 and VN+1 = 2VN − VN−1 (O.11) N+ 2 N− 2 The tridiagonal system is parallelised using MPI and OpenMP and solved by an LU algorithm.

Conservation properties The basic conservation properties of this scheme are tested, while varying: • the numerical time step: ∆t

• the number of discretisation grid points in the vk direction: Nvk

• the integration interval in vk: [−4, 4] or [−6, 6] To do so, Eq.(O.1) is solved which solution is well-known to be a Maxwellian. The system is thus initialised with this equilibrium solution and we look at the evolution of the density: δn/n. Ideally, this quantity is vanishing; its departure from the null solution provides an elementary upper bound estimate for the conserving properties of the scheme (i.e. its precision). The values −1 reported below are stationary values, after a total simulation time of 1500 ωc . Table O.1 shows that a correct treatment of the tail of the distribution function is mandatory. We shall thus always discretise vk in [−6, 6]. Conversely, this simplified test also tends to show –see Table O.2– that

o Table O.1: Case n 1 : influence of the integration interval (∆t = 1ωc)

δn/n Nvk = 32 Nvk = 64 Nvk = 128 Nvk = 256 Nvk = 512 vk ∈ [−4, 4] 0.2% 0.76% 1.8% 2.5% 2.9% −7 −5 −5 vk ∈ [−6, 6] 3.3 10 % − 1.1 10 % − 4.7 10 %

o Table O.2: Case n 2 : influence of the time step & of the number of grid points (vk ∈ [−6, 6])

δn/n Nvk = 32 Nvk = 128 Nvk = 512 −5 −5 −7 ∆t = 0.25 ωc 4.7 10 % 1.1 10 % 3.3 10 % −5 −5 −7 ∆t = 1 ωc 4.7 10 % 1.1 10 % 3.3 10 % −5 −5 −7 ∆t = 10 ωc 4.5 10 % 1.1 10 % 4 10 % the accuracy of the scheme seems weakly dependent on both the time step and the number of grid points. Classical parameters in GYSELA typically correspond to time steps of the order of a few units and 128 discretisation grid points in vk. Furthermore, the additional cost introduced by the collision operator is around 7% of the total calculation time, in both linear and turbulent regimes. Bibliography

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A grad–B ...... 41, 42 Adiabatic polarisation...... 42 invariant ...... see Invariant limit...... 40, 50 E Electrostatic approximation ...... 46 B Elongation...... 42 Banana orbit ...... 78 Equation Bessel function ...... 39, 85 Fokker–Planck ...... 17 Bootstrap current ...... 80, 83 Boltzmann ...... 16 Breakeven ...... 9 Euler ...... 26 Brownian motion ...... 17, 71 gyrokinetic ...... 47 kinetic ...... 16 C Kolmogorov ...... 17 Canonical ...... see Maxwellian master ...... 16 Cascade theory...... 19 Navier–Stokes ...... 12, 18, 26 Cat eye structures ...... see Trapping quasi-neutrality ...... 48 Causality principle ...... 12, 21, 73 Vlasov ...... 17 Chain (proton–proton) ...... 5 Equilibrium Chaos collisional ...... 51 molecular chaos assumption ...... 17, 73 collisionless ...... 51 Collisions ...... 67 weakly collisional ...... 51, 89 Confinement ...... 67 Ergodic assumption ...... 14 Confinement ...... 6 Eulerian codes ...... 45 gravitational ...... 5 Coulomb force ...... 2 F Coulomb logarithm ...... 70, 76, 82 Flows Courant–Friedrichs–Lewy (CFL) ...... 46 mean ...... 48 Curie principle ...... 75 zonal ...... 48 Cycle Fluid carbon ...... 5 consistency with ...... 18, 54 CNO...... 5 limitations ...... 20, 24, 34 model...... 18 D Flux-driven boundary conditions ...... 49, 62 Debye screening...... 2, 21, 69 Flux-tube codes ...... 49 Dimensionless number Force balance equation...... 55, 84, 86 β ...... 41, 46 Full–f code ...... 48 ν? ...... 82 Fusion ...... 2 ρ? ...... 47 devices ...... 6 Dissipation15, 24, 26, 39, 46, 51, 68, 74, 78, 84 gain factor Q ...... 8 Distribution function ...... 14 Drift velocities G E × B ...... 41, 42, 46 Galilean invariance ...... 51, 71, 83 curvature ...... 41, 42, 46 Global code ...... 49 INDEX O–141

Guiding-centre cordinates ...... 36 mass–energy equivalence ...... 4 Gyrocentre cordinates ...... 36 Maxwellian Gyrokinetic theory ...... 36 canonical ...... 51, 54 GYSELA approximations...... 46 local ...... 51 shifted ...... 51 H Microcanonical ...... see System H–theorem ...... 20, 24, 29, 71, 73, 84 Molecular chaos assumption...... see Chaos Heat flux (coll. enhancement of) . . . . 67, 91, 92 High-confinement regimes N H–mode ...... 67, 91 Neoclassical theory...... 68 ITB ...... 67, 91 Neutronicity ...... 8 VH–mode ...... 67 Noether’s theorem...... 50 Hydrodynamic turbulence ...... 19 Nuclear binding energy ...... 4 I force ...... 4 Ignition ...... 8 fusion, see Fusion ...... 2 Instability ...... 43 Nucleosynthesis ...... 4 slab ITG ...... 43 balloning-kink ...... 43 O Electron Temperature Gradient (ETG). .44 Ordered state ...... 51 interchange ...... 43 Ordering ...... 30 Ion Temperature Gradient (ITG) ...... 44 Ordering (gyrokinetic) ...... 38 kink ...... 43 Rayleigh–Benard´ ...... 44 P threshold ...... 33, 59, 86 Pade´ approximation ...... 47 Trapped Electron Mode (TEM) ...... 44 Paradox Trapped Ion Modes (TIM) ...... 67 recurrence ...... see Zermelo–Poincare´ Invariants Zermelo–Poincare...... 73´ adiabatic ...... 36, 42 Paradox of motion ...... 50 Loschmidt...... 73 Invariants Phase mixing ...... 24 of Poincare...... 15´ Phase space...... 13 PIC codes...... see Lagrangian K Plasma K41 theory ...... 19, 26 binary interaction ...... 3 collective effects ...... 2 L definition ...... 2 Lagrangian codes...... 19, 39, 46 magnetism ...... 2 Landau damping ...... 23 quasi-neutrality ...... 3, 47 Larmor Poloidal momentum (turb. generation of) . . 67, gyrofrequency ...... 40 91, 96 gyroradius...... 40 Lawson criterion ...... 8 Q Lebesgue space L2 ...... 51 Quantum Liouville theorem ...... 15 chromodynamics ...... 4 Loss cone ...... 78, 89 tunneling...... 2, 6 Low to High (L–H) transition ...... 83, 87, 92 Quasi-particle...... 19

M R Magnetisation current ...... 56 Random walk ...... 71 Mass Reaction rate...... 7 critical ...... 5 Recurrence theorem ...... 73 O–142 INDEX

Regime Pfirsch–Schluter¨ ...... 80 banana ...... 79 plateau ...... 80 Reynolds lemma ...... 16 Reynolds number ...... 25 Reynolds transport theorem ...... 19

S Scale separation ...... see Full–f Scrape Off Layer ...... 51 Self-organised criticality (SOC) ...... 96, 101 Semi-Lagrangian codes ...... 46 Spitzer conductivity ...... 80 Stosszahl Ansatz...... see Chaos Stress tensor fluid ...... 18 neoclassical ...... 80, 83 System closed ...... 13, 51 isolated ...... 13 open ...... 13

T Tokamak ...... 6 Transport anomalous ...... 68 classical...... 68 neoclassical ...... 68 Trapping phenomena ...... 89 Triangularity...... 42

V Viscosity ...... see Dissipation

W Ware pinch ...... 80