magn´ etohydrodynamique.
Application ` a l’effet dynamo
par
Thomas Lessinnes
Th`ese de Doctorat en Sciences
effectu´ee au sein du
Service de Physique Statistique et des Plasmas D´epartement de Physique
Facult´e des Sciences ULB Bruxelles
Directeur de th`ese:
Prof. Dr. Daniele Carati
Avril 2010
qui fait des merveilles,
A mon p` ere,
La magn´etohydrodynamique (MHD) est la science et le formalisme qui d´ecrivent les mouvements d’un fluide conducteur d’´electricit´e. Il est possi- ble que de tels mouvements donnent lieu `a l’effet dynamo qui consiste en la g´en´eration d’un champ magn´etique stable et de grande ´echelle. Ce ph´enom`ene est vraisemblablement `a l’origine des champs magn´etiques des plan`etes, des
´etoiles et des galaxies.
Il est surprenant qu’alors que les mouvements fluides `a l’int´erieur de ces ob- jets c´elestes sont turbulents, les champs magn´etiques g´en´er´es soient de grande
´echelle spatiale et stables sur de longues p´eriodes de temps. De plus, ils peu- vent pr´esenter une dynamique temporelle r´eguli`ere comme c’est le cas pour le champ magn´etique solaire dont la polarit´e s’inverse tous les onze ans.
D´ecrire et pr´edire les mouvements d’un fluide turbulent reste l’un des probl`emes les plus difficiles de la m´ecanique classique. Il est donc utile de construire des mod`eles aussi proches que possible du syst`eme de d´epart mais de moindre complexit´e de sorte que des ´etudes th´eoriques et num´eriques de- viennent envisageables.
Deux approches ont ´et´e consid´er´ees ici. D’une part, nous avons d´evelopp´e des mod`eles pr´esentant un tr`es petit nombre de degr´es de libert´e (de l’ordre de la dizaine). Une ´etude analytique est alors possible. Ces mod`eles ont une d´ependance en les param`etres physiques - nombres de Reynolds cin´etique et magn´etique et injection d’h´elicit´e - qualitativement similaire aux dynamos c´elestes et exp´erimentales.
D’autre part, les mod`eles en couches permettent de caract´eriser les trans- ferts d’´energie entre les structures de diff´erentes tailles pr´esentes au sein du champ de vitesse. Nous avons d´evelopp´e un nouveau formalisme qui permet d’´etudier aussi les ´echanges avec le champ magn´etique.
De plus, nous proposons une ´etude de la MHD dans le cadre de la d´ecom-
position h´elico¨ıdale des champs sol´eno¨ıdaux - une id´ee similaire `a la d´ecom-
position de la lumi`ere en composantes polaris´ees et que nous sommes les
premiers `a appliquer `a la MHD. Nous avons montr´e comment exploiter cette
approche pour d´eduire syst´ematiquement des mod`eles simplifi´es de la MHD.
En particulier, nos m´ethodes multiplient le nombre de situations descriptibles par les mod`eles en couche comme par exemple le probl`eme anisotrope de la turbulence en rotation. Elles permettent aussi de construire des mod`eles `a basse dimension en calquant les r´esultats de simulations num´eriques directes.
Ces mod`eles peuvent alors ˆetre ´etudi´es `a moindre coˆuts.
Magnetohydrodynamics (MHD) is both the science and the formalism that describe the motion of an electro-conducting fluid. Such motion may yield the dynamo effect consisting in the spontaneous generation of a large scale stationary magnetic field. This phenomenon is most likely the reason behind the existence of planetary, stellar and galactic magnetic fields.
It is quite surprising that also the fluid motion within these objects is tur- bulent, the generated magnetic fields present large spatial structures evolv- ing over long time scales. Moreover these fields can present a very regular non trivial dynamics like in the case of the Sun, the magnetic field of which switches polarity every eleven years.
To describe and predict the motion of a turbulent flow remains one of the most challenging problem of classical mechanics. It is therefore useful to build models as close to the initial system as possible but of a lesser complexity so that their theoretical and numerical analysis become tractable.
Two approaches have been considered here. Low dimensional models have been developed that present about ten degrees of freedom. An analytical study of the resulting dynamical system is then possible. Interestingly, the dependance of these models on the physical parameters - kinetic and magnetic Reynolds number as well as injection of kinetic helicity - qualitatively matches that of the cosmic and experimental dynamos.
On the other hand, shell models allow to characterise the energy transfers between structures of different sizes within the velocity field. A new formal- ism is presented which makes possible to also study the exchanges with the magnetic field.
Furthermore, a description of mhd in the helical decomposition is pro-
posed. I show how to use this decomposition to build new shell and low
dimensional models. The methods developed here allow to broaden the scope
of possible applications of the models. In particular, shell models are gen-
eralised in such a way that they can now describe anisotropic situations like
that of rotating turbulence.
Comme il se doit, je remercie tout d’abord le Professeur Daniele Carati, en sa qualit´e de directeur du groupe de Physique Th´eorique et Math´ematique et de directeur du Service de Pysique Statistique et des Plasmas. Je le remercie d’avoir accept´e que je r´ealise cette th`ese non seulement au sein de son service mais en plus sous sa direction. Evidemment, celui que je voudrais en fait avant tout remercier, c’est toi Daniele, en ta qualit´e d’Homme de qualit´e. Merci bien ´evidemment d’avoir accept´e de diriger cette th`ese mais surtout merci d’y avoir consacr´e autant de temps et d’attention. Je dois m’avouer vaincu en quelque sorte. Mes incessantes petites questions fastidieuses se sont heurt´ees `a une montagne de patience bienveillante. Cette forteresse est imprenable, tout simplement. Merci car pendant ces quatre ann´ees, il a toujours suffi de frapper
`a ta porte pour obtenir des r´eponses claires, directes et pr´ecises. Je sais que pendant ta th`ese tu as compar´e ton directeur `a un lion auquel tu pouvais donner en pˆature les probl`emes les plus ardus. Je tiens `a te dire que mon lion a ´et´e d’une efficacit´e monstrueuse. Evidemment, c’est dans le fond ton m´etier d’avoir toujours raison, de deviner `a chaque fois les bonnes directions et de trouver dans la demi-seconde la clef de probl`emes qui m’occupaient parfois depuis des semaines. Mais merci d’avoir aussi fait ton m´etier de cr´eer et de maintenir une ambiance de travail conviviale et stimulante. Combien de fois suis-je entr´e dans ton bureau en me demandant secr`etement si tout
¸ca a du sens ? R´eponse facile: exactement autant que de fois o`u j’en suis ressorti gonfl´e `a bloc, ayant hˆate de trouver les r´eponses, d’avancer, d’aller plus loin, d’en faire un peu plus. Merci aussi d’avoir toujours ´et´e l`a pour r´epondre `a mes questions inh´erentes `a la poursuite de ce m´etier. L`a aussi, tu as ´et´e d’une franchise et d’une efficacit´e dont je te suis tr`es reconnaissant. Ce paragraphe n’est pas en vers et `a la fin, je n’ai pas de “i” `a transformer en
“´e”. L`a non plus, je n’ai pas encore ´egal´e le Maˆıtre. J’esp`ere par contre que tu es convaincu que le coeur y est car toi, tu as transform´e un ing´enieur qui se cherchait en un passionn´e de recherche.
Je tiens `a remercier Prof. L´eon Br´enig, et Prof. Michel Mareschal d’avoir accept´e de constituer mon comit´e d’accompagnement de th`ese. Merci aussi
`a L´eon pour nos discussions concernant les syst`emes hamiltoniens, encore un faiseur de pluie qui fait des merveilles! Merci `a eux deux ainsi qu’aux Dr.
Thomas Gilbert, Dr. Franck Plunian et Dr. Jean-Fran¸cois Pinton d’avoir accept´e de constituer mon Jury de th`ese et donc de lire ce travail, de le d´ecortiquer et mˆeme, de me faire part de leurs avis. Merci en particulier au Dr. Franck Plunian. Ce sont ses articles qui m’ont fait d´ecouvrir les mod`eles en couches. Merci surtout pour l’invitation `a passer une semaine au LGIT et pour nos discussions concernant les spectres d’h´elicit´e. Sa capacit´e
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`a synth´etiser, `a isoler les points clefs et `a ´eviter la dispersion inh´erente `a ma fougue juv´enile sont pour beaucoup dans l’existence mˆeme du chapitre 6. Enfin, merci `a lui et au Dr. Jean-Fran¸cois Pinton d’avoir accept´e de se d´eplacer pour assister `a mes d´efenses priv´ees et publiques.
Je remercie bien ´evidemment l’ensemble des membres du groupe de Physi- que Th´eorique et Math´ematique de l’ULB. En particulier, merci `a Maxime de m’avoir si sympathiquement accueilli dans son antre. Merci aussi de m’avoir appris le Poker sans m’avoir rien pris. En plus, j’ai pu cˆotoyer en chaire et en os quelqu’un qui pense que la course `a pied est un sport vraiment passionnant.
Et mˆeme, tu as finis par me le faire comprendre. Chapeau bas l’ami! Merci
`a Sara pour nos beaucoup trop rares discussions philosophico-´economiques.
Merci `a Stijn pour avoir partag´e sa science de la vie politique de notre pays.
Merci aussi pour les quelques probl`emes que nous avons attaqu´es ensemble.
Je tiens absolument `a le remercier ici en mˆeme temps qu’Axelle pour les incessants encouragements lors de la r´edaction de ce travail. Vous sentir `a ce point compatissants et tristes que je ne puisse pas profiter du seul week- end ensoleill´e de l’ann´ee a ´et´e un r´eel moteur et un parfait baume au coeur.
Peut-ˆetre plus sinc`erement, merci `a tous deux pour la sympathie pendant ces quatre ann´ees. Ca a toujours ´et´e un r´eel plaisir de vous croiser. Oui, oui, mˆeme quand il s’agissait de vous extorquer du temps de calcul sur le cluster...
Merci aussi `a Chiara, Paolo, Xavier, Michael, Pierre, Oleg, Carlos et Marc.
Merci `a Jean Wallenborn pour le plaisir que j’ai eu `a discuter avec lui d’une multitude de sujets. Je voudrais aussi remercier Thomas Erneux qui m’a aid´e
`a comprendre le syst`eme dynamique li´e `a l’interaction en triades h´elico¨ıdales.
Avoir le point de vue d’un math´ematicien a men´e, `a chaque fois, `a des angles d’attaque diff´erents et fructueux. Merci `a Mustapha Tlidi pour les discussions autour d’un caf´e ainsi que pour les nombreux encouragements. Merci aussi
`a Gr´egory Kozyreff qui par l’exemple m’a rappel´e `a une certaine forme de rigueur scientifique. Merci `a Paul Mandel de m’avoir toujours prˆet´e une oreille attentive, de m’avoir fait red´ecouvrir la musique classique, et d’avoir guid´e certaines lectures. Merci enfin `a Bernard pour ses encouragements et sa constante bonne humeur. Merci aussi pour le cours de golf particulier. Et non, je n’oublie pas que je te dois la pareille `a l’escrime.
Many thanks to every single member of the group of Prof. Mahendra K.
Verma at IIT Kanpur, India. Thanks also to Prof. Pankaj Wahi. It has been
wonderfull to work and stay at IITK for a month. Prof. Verma taught me
almost everything I know about low dimensional models and Prof. Wahi cared
for the rest of it! Prof. Verma’s wonderful calmness, tremendous knowledge
and profound gentleness made it an always reiterated pleasure to work with
him. I hope we keep collaborating and eventually build a low dimensional
model reaching up to our now common expectations. It has been a true
wonder for me to discover what welcoming a foreigner means in India. I hope he knew, during his previous long stay in Brussels, that it was rather a cultural difference than rudeness on my part. I also want to seize the opportunity to particularly thank Dr. Supriyo Paul, Rakesch Yadav and Mani Chandra with whom I directly collaborated while I was at IITK. Thanks also to Owen and Antje for the kind invitation. I hope I get the chance to return the favour.
Finally, special thanks to Mani and Meena. I will not forget Panna nor the neighbouring waterfalls. Sharing time with you two was simply ‘awesome’.
Thanks again for the chess games too!
Ce travail a ´et´e support´e financi`erement par le Fond National de la Recher- che Scientifique F.R.S.-FNRS grˆace auquel il a ´et´e simplissime de r´epondre
`a la premi`ere question pos´ee. This work has been partly supported by the contract of association EURATOM - Belgian state. Its content is the sole responsibility of the author and it does not necessarily represent the views of the Commission or its services.
Je voudrais aussi profiter de l’occasion pour remercier chacune des Flying Marmottes et cela inclut bien ´evidemment Marmotta Marmotta mˆeme si elle nous snobe. Grimper au sommet de ce tiers l`a fˆut un r´eel plaisir. Que dis-je un plaisir ? Ce fˆut un honneur !
Merci `a Niels de m’avoir prouv´e que la Libert´e existe, qu’il suffit de la saisir. Merci `a Sam et `a Olivier pour tant de bon moments pass´es et `a venir.
Merci aux trois rockeuses d’avoir fait `a elles seules que Paris tint ses promesses.
Merci `a Amandine et Nicolas, car c’est chaque jour un peu plus charmant d’apprendre `a les connaˆıtre. Merci pour vos passages r´ep´et´es par mon bureau quand vous saviez que ¸ca faisait un peu trop longtemps que j’´etais en tˆete-`a- tˆete avec mon ´ecran.
Merci `a Sarah pour ces ann´ees partag´ees, pour ces points de vues diff´erents, surprenants et toujours enrichissants. Merci aussi pour la confiance et la con- nivence qui persistent...
Et puis, il y a les ‘depuis toujours’, et ceux qui le deviennent, ceux qui comme par hasard d´ebarquent pile les soirs o`u c’´etait vraiment n´ecessaire, ceux qui font que la vie a un sens profond et flagrant, qu’elle flamboie et qu’elle est belle. Les phares d’amiti´e qui guident et ram`enent au port lorsque la tempˆete gronde et d´echaˆıne l’oc´ean d’incertitudes. Alex, C´edric, Christine, Jessy et Laurent, je vous dois plus que je ne pourrai jamais l’exprimer.
Je remercie bien ´evidemment absolument chaque membre de ma famille.
Nous avons une histoire si int´eressante! Merci `a ma grand-m`ere pour son
infinie gentillesse, sa bont´e de coeur et ses attentions permanentes. Merci
`a mon grand-p`ere pour tout ce que nous avons bˆati ensemble. Le peu que je sais faire de mes dix doigts, c’est `a toi que je le dois. Merci `a Hubert pour nos discussions mais surtout merci parce qu’elles sont vari´ees. Merci `a Anne de m’avoir rappel´e au bon moment: “choisir c’est renoncer”. Merci `a Madame Ma Tante pour l’aristocratique plaisir de la rencontrer. Merci aussi pour les conseils linguistiques. Merci `a ma M`ere pour cet incroyable travail de Cheyenne, mais surtout pour cette sollicitude permanente et infaillible.
Merci `a Antoine d’avoir toujours ´et´e le rayon de soleil de la famille. Merci de
m’avoir fait voler. Merci d’ˆetre venus `a Bruxelles pendant les derniers week-
ends. Ce travail te doit plus que tu crois. Merci `a Mathieu pour sa relecture
de l’introduction. Merci aussi parce que chaque fois que je pense `a toi, je me
dis que que l’intransigeance paie et que satisfaisant n’existe pas. Merci pour
Brel, juste au bon moment. Merci enfin `a Bernard et `a mon P`ere de m’avoir
appris la droiture, l’honnˆetet´e et l’amiti´e. Qui d’autre peut, `a 26 ans encore,
se vanter d’avoir un p`ere g´eant ?
R´ esum´ e v
Abstract vii
Acknowledgements ix
Table of Contents xiii
Introduction 1
I Theoretical aspects 9
1 Navier-Stokes and MHD turbulence 11
1.1 Navier-Stokes equations . . . 12
1.1.1 Exact and statistical symmetries . . . 14
1.1.2 Boundaries . . . 17
1.1.3 Invariants . . . 18
1.1.4 Cascade processes: tea cup, clouds and fleas, a glance at Richardson, and Kolomogorov . . . 19
1.2 Magnetohydrodynamics (MHD) . . . 20
1.2.1 Maxwell equations and the MHD approximation . . . . 20
1.2.2 MHD invariants and cascade . . . 24
1.3 Introduction to the dynamo effect . . . 25
2 NS and MHD equations in Fourier space 33 2.1 Definitions . . . 34
2.2 NS and MHD in Fourier Space . . . 35
xiii
2.3 Conservation laws . . . 38
2.4 Spectra and scale by scale budgets . . . 43
2.5 Energy fluxes and mode-to-mode energy transfers . . . 49
2.6 Helical decomposition . . . 56
2.6.1 Definition of the helical basis . . . 57
2.6.2 NS and MHD equations . . . 59
2.6.3 Helical triadic interaction for NS and MHD . . . 61
2.6.4 Scale independence of g . . . 62
2.6.5 2D-3C fows . . . 64
II Modelling fluid dynamics 69 3 Low Dimensional Models 71 3.1 Derivation of low dimensional models . . . 72
3.2 Analysis of the model . . . 75
3.2.1 Nonhelical model . . . 75
3.2.2 Helical model . . . 78
3.3 Discussion . . . 80
4 Shell models for NS and MHD turbulence 83 4.1 Pre-existing shell models of Navier-Stokes turbulence . . . 84
4.1.1 GOY and SABRA models . . . 85
4.1.2 Helical shell models . . . 91
4.1.3 Zimin model . . . 94
4.2 Shell models for MHD . . . 94
4.2.1 A general formalism . . . 103
4.2.2 Evolution equations for the shell energies . . . 106
4.2.3 Energy Fluxes . . . 108
4.2.4 Shell-to-shell energy exchanges . . . 110
4.2.5 GOY shell model for MHD turbulence . . . 111
5 The helical decomposition: a machinery for model building 115 5.1 Helical shell models for MHD turbulence . . . 116
5.1.1 Shell models based on triadic dynamical systems . . . . 117
5.1.2 An example: retrieving the GOY model . . . 119
5.1.3 Scale invariance . . . 121
5.1.4 Properties of the coupling constants . . . 124
5.1.5 A user-friendly formulation . . . 126
5.1.6 Definition of the coupling constants of the local model . 129
5.2 Ring models . . . 132
5.2.1 Motivation: rotating turbulence . . . 132
5.2.2 Regional models . . . 133
5.2.3 A particular case: ring models . . . 134
5.2.4 Reality matters: Sabra model revisited . . . 136
5.3 Helical low dimensional models . . . 138
5.3.1 One triad NS system . . . 139
5.3.2 A simple dynamo transition . . . 141
5.3.3 Getting rid of the stable fixed points . . . 142
III Numerical Results 145 6 Helicity cascade in Navier-Stokes turbulence 147 6.1 Ditlevsen and Giuliani’s argument . . . 148
6.2 Reinterpretation . . . 150
6.3 Helical shell model analysis . . . 154
6.4 Conclusion . . . 162
7 Energy Fluxes in MHD tubrulence 163 7.1 Energy spectra . . . 164
7.2 Energy fluxes . . . 164
7.3 Π B< B> and Reynolds number effects . . . 168
8 Helical shell models for MHD 171 8.1 Navier-Stokes turbulence . . . 173
8.2 Influence of helicity injection on the onset of dynamo . . . 173
9 Discussion 179
A Helical MHD equations 183
B Conservation of cross helicity in the helical decomposition 187
C Symmetries of the g factor 191
D Automatic conservation of kinetic helicity in the GOY model
for MHD 197
E Another definition of the circulating term 199
Bibliography 201
Note: A quick and easy way to get a first insight into this work is to read the introductions of the various chapters that are announced by this particular font.
Furthermore, the figures of the first two parts are mostly sketches representing the concepts detailed in the text. Hence, a second level of reading can be done by browsing through these sketches and reading the corresponding captions.
This work is dealing with the modelling of turbulent flows of electro-conducting fluids. These systems are known to generate the planetary and stellar magnetic fields like those of the Earth and the Sun. Two types of models have been considered.
Shell models describe the exchanges of physically relevant quantities - like the energy or the helicities - between structures of various sizes. A new approach to shell models is proposed. It leads to a formulism which encompasses most of the pre-existing models. More importantly, it allows to easily develop a wide range of new models which can for instance tackle anisotropic problems like that of rotating turbulence. A clear and unambiguous description of the energy exchanges is also proposed. It then becomes apparent that the shell models can predict the occurrence of interesting phenomena in ranges of parameters unreachable by direct numerical simulations but relevant to both astrophysical objects and experimental devices.
Another modelling approach consists in projecting the velocity and magnetic fields on few large scale modes and then describing the interplay of these large structures. Interestingly, this very simple, analytically tractable approach leads to a phenomenology in qualitative agreement with the astrophysical and experimental observations.
This document is divided into three parts. The first one presents the theoretical developments that have been required in order to build the models. A second part focuses on the modelling techniques themselves. Finally, numerical results illustrating the previous developments are reported.
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and wondered: ‘Will I have something to eat tomorrow?’. I am afraid this thesis does not help in this most important matter.
It however tackles other essential interrogations like that of understand- ing the existence and the dynamics of the magnetic field of the Earth, of the Sun and, in general, of other celestial bodies which remains one of the most challenging problems of classical physics. Astronomical and geophys- ical observations have provided many insights into these phenomena [1–3].
Laboratory experiments [4–6] have recently confirmed the so-called dynamo effect: magnetic fields can be generated by the motion of conducting fluids, leading to a variety of complex behaviours. Yet, analytical approaches of this problem are extremely complicated while numerical efforts are limited to a range of parameters that is often quite distant from realistic systems.
Indeed, the flows leading to astrophysical dynamos are turbulent and a correct analysis of turbulence itself is yet another open (one million dollar) problem of classical mechanics and mathematics.
Furthermore, when the fluid is electro-conducting, as required for the dy- namo effect, the Navier-Stokes ( ns ) equations must be supplemented by an equation of evolution for the magnetic field, leading to the magnetohydro- dynamics (mhd) formalism. The state of the system is then determined by two vectorial fields. These velocity and magnetic fields interact non-linearly which leads to exchanges between kinetic and magnetic energies. There are furthermore two dissipative processes. The viscous effects damp the velocity field while the magnetic energy is dissipated by the Joule effect.
In certain astrophysical bodies as well as in laboratory experiments, the kinematic viscosity ν of the fluid is six orders of magnitude smaller than its magnetic diffusivity η. The two dissipation processes therefore take place at very different time scales which makes direct numerical simulations of dynamo a particularly thorny problem.
The two most important non-dimensional parameters for the dynamo studies are the Reynolds number R e = U L/ν and the magnetic Prandtl num- ber P m = ν/η, where U and L are typical velocity and length-scale of the system respectively. Another non-dimensional parameter used in this field is the magnetic Reynolds number R m = U L/η. Clearly R m = R e P m , hence only two among the above three parameters are independent. While the two Reynolds numbers depend on both the type of fluid and the particular flow, the Prandtl number is a property of the fluid only. Note that galaxies, clus- ters, and the interstellar medium have large P m & 10 8 , while stars, planets,
2
small m . 10 [1, 2].
In a typical simulation, the conducting fluid is somehow forced into motion and the dynamo transition is considered to be observed when a nonzero self- sustained magnetic field is maintained in the steady-state laminar solution or in the statistically stationary turbulent solution, depending on the regime.
Typically, dynamos occur for forcing amplitudes beyond a critical value which defines the critical Reynolds number Re c and the critical magnetic Reynolds number R c m . One of the objectives of both the numerical simulations and the experiments [5, 6] is the determination of this critical magnetic Reynolds number R c m . It has been found that R c m depends on both the type of forcing and the Prandtl number (or Reynolds number). The range of R c m observed in numerical simulations varies from 10 to 500. Note that in the Von-Karman- Sodium (VKS) experiment [5, 6], R c m is around 30.
There are many attempts to understand these observations. For large Prandtl numbers, Schekochihin et al. [7, 8] suggested that the growth rate of the magnetic field is higher in the small scales because stretching is faster at these scales. This kind of magnetic field excitation is referred to as small-scale turbulent dynamo. For low P m , Stepanov and Plunian [9] also report growth at small scales.
In this picture, energy is injected in the system by some large scale me- chanical stirring. A turbulent flow develops and the energy is transferred to structures of smaller and smaller sizes until friction becomes strong enough to dissipate it. As this cascade of energy proceeds, energy reaches scales of motion which can generate a magnetic field. A part of the injected kinetic energy is thus transformed into magnetic energy at these small scales. How- ever, the magnetic fields of cosmic objects are known to present very large scales structures (the Earth is within the magnetic field generated by the Sun!). There must therefore be a process that transfer the magnetic energy back to the large scales. The direct numerical simulations realised to date do not observe such a backward energy transfer.
A spatial Fourier analysis is a very convenient way to describe such a system where energy spreads among structures of various sizes. Indeed, in this decomposition, every variable is associated to a particular wavelength.
The energy content of the different Fourier modes therefore directly relates to the amount of energy in structures of corresponding sizes in the flow.
The equation of evolution of the Fourier modes can always be decomposed into triadic interactions. This is a key concept throughout the present work.
3
sition of interactions. Each such interaction implies three Fourier modes that exchange physically relevant quantities which generally correspond to quanti- ties that are conserved during the motion in absence of dissipative processes - the so called ‘ideal case’.
In ns turbulence, the kinetic energy and the kinetic helicity are conserved by the ideal motion. The latter measures the alignment of the velocity field with its curl and relates to how knotted the flow is. In mhd , owing to the Lorentz force, kinetic helicity is no longer conserved. However, the cross and magnetic helicity are themselves quadratic ideal invariants. The former measures the alignment between the velocity and the magnetic fields while the latter corresponds to the alignment of the magnetic field and the vector potential. In mhd , each triadic interaction generates exchanges of these three quantities.
In the case of planetary and stellar dynamos as well as in experiments, the magnetic Prandtl number is very low: P m . 10 −4 . The principal limitation of numerical studies of these flows comes from the turbulent regime itself. The dynamo condition R c m ∼ 10 implies kinetic Reynolds numbers R e & 10 5 which corresponds to extremely well developed turbulence. It is well known that the number of grid points required to capture all the turbulent scales increases like R 9/4 e which makes the direct simulation of high R e flows prohibitive [10].
Shell models have been introduced to cope with this difficulty. First devel- oped for hydrodynamic turbulence [11–14], they represent the entire velocity field by a limited number of interacting variables. This is achieved by pro- viding the Fourier space with a set of spheres of growing radii and centred at the origin. All the Fourier modes with a wave vector lying between two such sphere - i.e. within a shell - are replaced by one complex variable. As this shell partition of the Fourier space corresponds to the definition of intervals of wave numbers, each variable in the model is associated to a typical wave number and thus to a typical size of structures. The equations of evolution of these variables are then written in such a way that the Navier-Stokes ideal invariants are conserved by the nonlinear couplings in the model.
This approach has been extended to mhd [9, 15–19]. It correctly repro- duces well known properties of mhd flows: energy spectra - i.e. the distri- bution of energy among length scales - as well as the spontaneous generation of magnetic field in 3D simulations and the impossibility of such an effect in 2D turbulence. As aforementioned, the dynamo process involves growth of magnetic energy that is supplied from the kinetic energy by the non-linear interactions. Furthermore, energy is injected at large scales and dissipated
4
state, this energy must somehow flow through the different sizes of structures and between the two fields. Hence, we must describe not only how energy is distributed among scales, but also, and perhaps more importantly, what are the energy fluxes across scales and the energy exchanges between the veloc- ity and the magnetic fields. A clear and unambiguous understanding of this dynamical picture is very important in the study of the dynamo effect.
Hence, a part of my work consisted in a systematic and consistent defi- nition of these energy exchanges in magnetohydrodynamic turbulence. This scheme is then applied to study energy transfers in a shell model of mhd.
My approach follows quite closely the previous efforts of [20, 21] where en- ergy exchanges between two degrees of freedom have been defined up to an indeterminate circulating energy transfer within each triad. The strategy adopted here is somewhat different. The various energy transfers are derived from the energy equations by identifying advection terms which are known to redistribute the energy of the advected field without altering it as a whole.
In particular, the energy transfers from the magnetic field in a shell to the magnetic field in another shell is driven by the advective term of the induction equation. Such an energy exchange between structures of different sizes within the magnetic field must naturally conserve the total magnetic energy. The identification of the advective term in the shell model is one of the main practical improvements of my approach. It is actually required to define physically meaningful shell to shell energy transfers.
A quite interesting result is obtained when applying this formalism to the well known GOY shell model. Indeed, the model agrees with direct numerical simulations: in the range of Reynolds numbers reachable by the latter, there is no cascade of energy from the small structures towards the large scales of the magnetic field. However, when the physical parameters are set to values corresponding to actual cosmic and experimental dynamos, the model does exhibit a reverse cascade from the small scales of the magnetic field towards its large scales. These results were obtained in collaboration with Prof. Mahendra K. Verma [19].
Although it displays such an interesting phenomenology in agreement with both DNS and the picture proposed by Schekochihin et al. [7, 8] and Stepanov and Plunian [9], this model, making use of only one complex variable per shell, suffers from a very crude description of the kinetic and magnetic helicities:
in each shell, the helicity and the energy of either fields are not independent quantities as they should be.
In order to solve this difficulty, Benzi et al. [22] have introduced shell models for Navier-Stokes turbulence that use two complex variables per shell.
5
terms of eigenvectors of the curl operator [23]. It was shown that each three- dimensional velocity Fourier modes can be represented by a superposition of two eigenvectors corresponding respectively to maximal and minimal helicity.
The models developed by Benzi et al. [22] identify the two complex variables per shell to the amplitude of these eigenvectors. Although these models pro- pose a very elegant framework for representing helicity in shell models, they have the drawback to contain several free parameters. Indeed, the number of constraints imposed by the energy and the helicity conservation is too small to prescribe the amplitude of all the nonlinear couplings between the two variables per shell, even when only local interactions are considered.
In regards of these developments, I extended the helical mode analysis to the mhd equations. Rather than simply generalising the models of [22] to mhd , I propose here (see also [24, 25]) a new method to derive shell models.
It is based on a close match between the triadic interactions of the helical components in Fourier space and the triadic interactions of the model’s vari- ables. This procedure allows to build at once a variety of different models which all automatically conserve the relevant ideal invariants. It furthermore allows to fix the free parameters of the helical shell models previously men- tioned. Most importantly this method allows to envisage a wide range of new applications like for example the case of rotating turbulence which is by essence anisotropic and was difficult to model before, because of the isotropic structure of the shells. A ring model, that is well suited to this rotating case is presented in chapter 5.
In parallel to these shell models, another, complementary modelling tech- nique has also been considered. While the shell models discard the fine dy- namics of the particular Fourier modes to concentrate on the energy cascade across the scales, this other approach focuses on the exact dynamics of very few large scales modes. To this end, the fields are projected onto these par- ticular modes in such a way that the state of the system can be described by only few ( ∼ 10) degrees of freedom. The resulting dynamical systems can be studied analytically. Quite interestingly they display behaviours in qualita- tive agreement with the experimental results: dependency of R m c on P m and on the presence of kinetic helicity in the flow.
The structure of this document is as follows. It contains three parts. The first one corresponds to theoretical developments and consists of two chapters.
The first of them contains an introduction to Navier-Stokes turbulence, a derivation of the magnetohydrodynamic approximation and an introduction to the most important analytical results concerning the dynamo effect. The second chapter is focused on a detailed description of the dynamics in the
6
in mhd and the helical decomposition of the mhd equations.
The second part concerns the modelling of ns and mhd turbulence. The third chapter describes our first low dimensional models as published in [26].
The fourth chapter is divided into two sections. The first one corresponds to an introduction to pre-existing shell models, how they were build and how they differ from each other. The second section is dedicated to the definition of the energy transfers within a shell model while making minimal assumptions on its structure. These results can therefore be applied to any of the models defined previously. The fifth chapter is the keystone of this thesis.
It first describes the new method to derive shell models. The announced generalisation to ‘regional’ models is then discussed. Finally, low dimensional models are build by projection on the helical modes. These last models display a simply understandable and even controllable dynamo transition. In this framework, chaotic regimes can also be obtained, which was not observed in the models of chapter 3.
Finally, a last part presents numerical integrations of shell models. The sixth chapter focuses on the cascade of kinetic helicity within ns flows; a work we are about to publish in collaboration with Franck Plunian and Rodion Stepanov [27]. Chapter 7 contains an analysis of the energy fluxes within the GOY shell model for mhd . It is intended as an illustration of the techniques developed in sections 2.5, 4.2.1-4.2.4. In particular, it discusses the presence of the backward cascade of magnetic energy at high R m only. Chapter 8 presents a test of the helical model of section 5.1 in the ns case. Furthermore as proof of the usefulness of the afore construction, it proposes a study of the influence of the injection of kinetic helicity on the onset of dynamo. Interestingly, when kinetic helicity is injected, a magnetic helicity develops as well. This issue needs further investigation; studying the cascade of the helicities in a mhd flow would be a very interesting extension of my work. The shell models of chapter 5 are particularly well suited to this task.
A last chapter contains a summarised discussion of the main results.
7
Theoretical aspects
9
Navier-Stokes and MHD turbulence
This chapter presents a quick derivation of the Navier-Stokes equations of motion of Newtonian fluids. I introduce the notions of statistical symmetry, homogeneity and isotropy as well as that of a statistically stationary flow. A glance at the turbulent cascade process is also proposed at the occasion of which I state the universality assumptions behind Kolmogorov’s theory of turbulence.
The case of conducting fluids is then described. This thesis is concerned with motions that are slow relatively to the electrostatic relaxation time. By definition, this is the scope of application of Magnetohydrodynamics (MHD). The fluid is sup- posed to be neutral and a closed description can be made in terms of the velocity and magnetic fields only. The MHD invariants are then introduced although proofs of the invariance are delayed to the end of chapter 2.
Finally, I introduce the dynamo effect focusing on the properties that are most relevant to my work. A turbulent electro-conducting flow can indeed amplify and maintain a large scale magnetic field. It is now widely accepted that this phenomenon is responsible for the present magnetic field of the Earth, the Sun, an more gener- ally, planets and stars. It has been proven that axisymmetric magnetic fields are complicate to maintain. Furthermore, two dimensional magnetic fields can not be sustained via MHD effects. In the case of moderate inertial forces (in comparison to Joule dissipation effects), the importance of having a helical flow to trigger the dynamo effect is also highlighted.
11
1.1 Navier-Stokes equations
The Navier-Stokes equations consist in an expression of the conservation of mass and momentum applied to a continuous medium that has very little resistance to deformations. Such a medium is generally called a fluid. On earth, this is most often a liquid or a gas. But larger systems can also be described in this way like the movements of mass within a galaxy for instance.
The conservation of mass states that considering a mass M initially en- closed in a volume V . As the material points move with the flow, the volume V might be strongly unformed but the mass within it remains constant:
D
Dt M = D Dt
Z
V
ρdV,
= Z
V
∂ t ρ + ∇ · (ρu) dV = 0, (1.1) where u is the velocity of the fluid, ρ its mass density and Dt D stands for Lagrangian or material derivative. As this relation must hold for any volume V , it has the following local form:
∂ t ρ + ∇ · (ρu) = 0. (1.2)
My work exclusively concerns incompressible flows which change the shape of the volume V but not its size:
∇ · u = 0. (1.3)
The mass conservation (1.2) together with the divergence free condi- tion (1.3) implies that the mass density is itself constant while followed along field lines:
D
Dt ρ = 0. (1.4)
This relation (1.4) can be developed as
∂ t ρ + (u · ∇ )ρ = 0. (1.5)
It has a simple solution ρ = Constant for which both ∂ t ρ and ∂ i ρ are zero.
Although it is not the only possible solution, we will restrain here to this simple case 1 .
1
I do not consider convective systems.
Applying Newton’s law ma = f to a fluid element δV leads to an expres- sion for the evolution of its velocity:
ρδV Du
Dt = −∇ p δV + ∇ · τ δV + f ρδV, (1.6) where p is the pressure which is by definition isotropic so that all the pressure forces around the volume element are given by − H
∂δV p dS = − R
δV ∇ p dV =
−∇ p δV . The last equality is to be understood in the limit of vanishing δV . The second term in (1.6) describes the stresses applied on the small vol- ume due to the entrainment by the neighbouring fluid - i.e. viscous effects.
The global force at stake here corresponds to the difference between stresses (τ ) applied to opposing faces of a vanishingly small cube of fluid, see for in- stance [28] or any fluid mechanic textbook. Finally f accounts for all external body forces that could be applied.
To close the system, a constitutive law must be assumed. This thesis is solely focused on Newtonian fluids for which the viscous stresses applied to a fluid element are proportional to its relative velocity with its neighbours. In the incompressible case (1.3), this consitutive law takes the following form [28, 29]:
τ ij = ρν
∂ i u j + ∂ j u i
. (1.7)
The coefficient of proportionality ν is called the kinetic viscosity and is an intrinsic parameter of the fluid. Replacement of (1.7) in (1.6) and straight- forward algebra yield the Navier-Stokes equations:
Du
Dt = −∇ p ρ
+ ν ∆u + f . (1.8)
and everything would be simple (boring?) if it was not for this Lagrangian derivative Dt D in the left hand side. Indeed, applying Newton’s law to a fluid element provides an equation for the velocity of that element. It is however a formidable task to follow the trajectories of every single bits of a turbulent fluid. We could, on the other hand, look at one particular place and have an equation for the velocity of the fluid at that particular position. In other words, work in the Eulerian picture 2 . The equality between operators: Dt D =
∂ t + u i ∂ i yields the translation.
2
Let me clearly state here that the Eulerian picture contains exactly as much information as the Lagrangian one. Going from one to the other can be done at any time. The ‘best’
approach simply depends on what is to be achieved. To my purpose, the Eulerian picture
has the advantage to make the non-linearity of the Navier-Stokes equations manifest which
is desirable since my work is an attempt at modelling this non-linearity.
As we assumed constant mass density, our final step will therefore be to normalise p in term of ρ. In this case, the pressure can be seen as a quantity that ensures the velocity field to remain solenoidal regardless of the applied body forces. In the Eulerian picture, the incompressible Navier-Stokes equations are recast as:
∂ t u = − (u · ∇ ) u − ∇ p + ν ∆u + f , (1.9a)
∇ · u = 0, (1.9b)
where the non-linearity is manifest.
Note that the mass totally disappeared from this description. There are therefore only two physical dimensions: space L and time T .
Another presentation of (1.9) will ease upcoming calculations. It is based on the vorticity which is defined as the curl of the velocity: ω = ∇ × u.
Simple algebraic manipulations (cf. beginning of appendix A) yield:
∂ t u = u × ω − ∇ p 0 + ν ∆u + f (1.10) where p 0 = p + | u | 2 . For completeness, the equation for the evolution of vorticity itself is:
∂ t ω = ∇ × (u × ω) + ν∆ω + ∇ × f . (1.11) When dealing with turbulent states characterised by the importance of the non-linear term in (1.9), it is useful to characterise how strong the turbulence is. To do so, we roughly evaluate the ratio of the non-linear term and the viscous one. If the flow has typical length L and velocity U , the non-linear term is of order U 2 l −1 while the dissipative term is of order νU L −2 . The (kinetic) Reynolds number is defined as the ratio of these two quantities:
R e = U L
ν ; (1.12)
the stronger the non-linear term, the larger R e .
1.1.1 Exact and statistical symmetries
In this section, I discuss the symmetries of the NS equations (1.9) disregard-
ing any boundaries or forcing (f = 0). More precisely, we assume that all
Figure 1.1: Flow past a cylinder with respective Reynolds number R e =
1.54, 26, 140, and 1770. The pictures are taken from Van Dyke’s ‘Album
of fluid motion’ [30]. The three first pictures are from Taneda and the last
one is from Falco.
integrals on the flow boundary vanish. This very natural assumption for no slip boundaries, will be further backed up in the next section. It is important to recognise the difference between symmetries of a system of equations and symmetries of some particular solutions. We shall also define the notion of symmetry in (statistical) law.
An exact symmetry G of the Navier-Stokes equations is a transforma- tion G which applied to any solution u(t, r) of (1.9) yields another solution G[u(t, r)]. Here is the list of known such symmetries [10]: time-translation, Galilean transformation (u → u+U , r → r+U t), parity (r → − r, u → − u), rotations, space-translations and scaling (t → λ 1−h t, r → λr, u → λ h u with λ ∈ R + , h ∈ R). This later symmetry must be handled with care. Indeed, only the case h = − 1 is an actual symmetry whenever viscosity is finite. Note that indeed, the Reynolds number is invariant for h = − 1 only. In the case of zero viscosity, an infinite set of scaling symmetries arise. This situation is however known to be very different from actual flows.
On the other hand, there are the symmetries of particular solutions. In other words, some solutions of the Navier-Stokes equations, can be matched on themselves applying particular symmetries. A very interesting phenomenon is that of symmetry breaking.
In Figure 1.1, the Reynolds number increases from panel to panel. In the two first ones, the flow is symmetric under time-translation, and mirror reflection through an horizontal axis. In the two last panels, these symmetries have been broken. In the last panel, a turbulent wake appears which seems to be uniformaly blur (at least when remaining close enough to a central horizontal line). In a sense, we have the feeling that the flow retrieved many of the previously lost symmetries. In particular, it does not seem to matter when we start looking at it (time-translation) nor in which sense we look at it (orthogonal symmetry through the horizontal axis). However, if for instance, the orthogonal symmetry is applied at a given time, it is clear that it will not lead back to the very same solution. To mathematically describe this type of symmetry, we need to move on to a statistical description.
In a statistical description of turbulent flows, the velocity is no longer
described by three components real vectors depending on both time and po-
sition. The measure of the velocity at a given position and time is now the
result of a random process which has a particular distribution function also
called a statistical law. The velocities at two different positions in space-time
are said to be equal in law whenever the distribution function of their mea-
surements are the same at both positions. In the last panel of figure 1.1,
the solution is symmetric in the statistical sense that the distribution func-
tions are symmetric under time translation and reflectional symmetry. In this
statistical description, the various physical quantities are quantitatively de- scribed by averaging their measures over a large number of realisations. Such an averaging of a quantity f will be noted as h f i av .
Obviously, whenever a particular flow is considered, both the classical and statistical description can be used. Given an initial condition, the velocity field still evolves as dictated by the Navier-Stokes equations (1.9). However, this fine description is not always the one bringing most pertinent information.
In this statistical sense, a solution which is symmetric under time trans- lations is said to be a stationary or a steady state. In such a flow, averaged quantities are constant in time: ∂ t h f i av = 0. This does not mean that the time derivative vanish when looking at one particular solution. In stationary state, I systematically assumed ergodicity and ensemble averaging has always been replaced by averaging one particular solution over a long periods of time in the simulation presented in the third part of this manuscript.
A flow is isotropic whenever all quantities are symmetric in law under the group of rotations SO(3).
A flow is homogeneous whenever all quantities are symmetric in law under the group of translations.
1.1.2 Boundaries
In practice, a flow is always bounded to a finite volume which strongly influ- ences the dynamics of the large scales. In a highly turbulent flow, assuming that the small scales are largely independent of these boundaries yields cor- rect phenomenology [10, 31, 32]. Physicists have therefore concentrated lots of their efforts to those cases where the boundaries generate no difficulties.
If the boundaries are simply removed and the fluid is supposed to fill the full R 3 -space, some mathematical difficulties arise linked to the appearance of infinities in the theory. A very successful way to circumvent those issues is to look at what is happening in one bounded box of that full R 3 -space while supposing that the rest of the flow is a repetition of this particular box.
Mathematically speaking, after dividing the physical space into an infinite mesh of boxes, it is assumed that all quantities have a value at a position r which only depends on the distance to the walls of the box encompassing r. This periodic boundary condition state that any function of space f(r) is assumed to verify:
f (x + L x , y + L y , z + L z ) = f (x, y, z) ∀ x, y, z ∈ R (1.13)
where L x , L y , and L z are the period along the three axis. This restriction has the tremendous advantage to allow to easily work in Fourier space (see next chapter) where the different spatial scales of the flow are easily separated.
To my modelling purposes, it has been enough to restrict to the case of L 3 - periodic functions: L x = L y = L z = L. Note that the full R 3 space can be retrieved at any time by taking the limit L → ∞ .
1.1.3 Invariants
The average of a quantity f over the L 3 box will be written:
h f i = 1 L 3
Z
L
3f dV. (1.14)
Here is a set of linear and quadratic conservation laws for the L 3 -periodic Navier-Stokes equations:
• Momentum: ∂ t h u i = 0,
• Energy: ∂ t 1 2 h u · u i = − 1 2 ν D P
ij (∂ i v j + ∂ j v i ) 2 E
= − ν h ω · ω i ,
• Helicity: ∂ t 1 2 h u · ω i = − ν h ω · ∇ × ω i .
Note that all boundary terms vanished due to the periodic boundary con- dition. Proofs can be found in [10] in this L 3 -periodic case. The conservation of momentum plays little role here. In fact, it is linked to space-translation symmetry by Noether’s theorem. This symmetry is of little importance when periodic boundary conditions are assumed as all boxes are a priori identical 3 The two other conservation laws, on the other hand are central to my work and will be discussed throughout. I will show at the end of the next chapter that they are indeed ideally 4 conserved quantities.
3
Although this latter symmetry is discrete while the translation is symmetry of the boundary-less ns equations is continuous.
4
The ns equations with zero viscosity and no forcing term are usually called the Euler
or ideal equations. I will use both designations.
1.1.4 Cascade processes: tea cup, clouds and fleas, a glance at Richardson, and Kolomogorov
This hand waving section is committed to give a hint of how energy is spread among the various scales of the flow. A more mathematical approach will be given in the next chapter after the Fourier space has been introduced. Indeed, it will then be possible to give a mathematical sense to what we call structures of a given spatial scale. For now, I simply describe a turbulent cascade partly as discussed in [10, 28] and partly as I came to understand it.
The experience goes as follows: stir a tea cup with a spoon. After a while remove the spoon and observe. At first, large eddies are rotating that will eventually break apart into smaller eddies. These smaller eddies keep decaying to yet smaller structures, until the liquid finally stops.
Richardson came to realising this in his very early work [33]. His inspira- tion was not a tea cup but clouds and Jonathan Swift’s verses:
So, nat’ralists observe, a flea
Hath smaller fleas that on him prey;
And these have smaller yet to bite ’em, And so proceed ad infinitum.
Thus every poet, in his kind, Is bit by him that comes behind.
When the tea cup is stirred, energy is injected via the motion of the spoon which corresponds to a large scale movement. This injection of energy must be eventually compensated by an equivalent amount of dissipation due to the ν ∇ 2 u term. As ν is generally a small quantity, this term can only significantly contribute if the second derivatives of the velocity field are large, thereby requiering sharp spatial variations of the flow. Hence, the cascading picture: energy is injected through the generation of large scale motions. This energy must somehow be transferred to the small scale structures where it is eventually allowed to dissipate. In this picture, a larger injection of energy implies that yet smaller scale structures must form.
The assumption was later made by Kolmogorov [31] that this cascade pro- cess, from large structures to smaller ones, is actually universal - i.e. inde- pendent from the source of the movement or the geometry of the boundaries.
As will shall see in the next chapter, he could predict how energy organises between the different scales of the flow.
Two universality assumptions were made which I quote here from [10].
At very high but not infinite Reynolds numbers, all the small-scale statis-
tical properties [of phenomena associated to structure of size `] are uniquely
and universally determined by the scale `, the mean energy dissipation rate ε and the viscosity ν.
In the limit of infinite Reynolds number, all the small-scale statistical properties [of phenomena associated to structure of size `] are uniquely and universally determined by the scale ` and the mean energy injection rate ε.
Under this later assumption, he derived his famous ‘four-fifths’ law which relates the correlator appearing in the flux of energy across scales to the scale itself and the rate of energy dissipation in steady state:
D
δu || (r, `) 3 E
av = − 4
5 ε`, (1.15)
where
δu || (r, `) = (u(r + `) − u(r)) · `
` , (1.16)
is the velocity increment in the parallel direction. A detailed proof can be found in [10].
1.2 Magnetohydrodynamics (MHD)
Magnetohydrodynamics ( mhd ) discribes fluids that conduct electricity but the movements of which are much slower than electrostatic relaxation pro- cesses. Hence the fluid is assumed to be neutral everywhere although intense electric currents may be present. As proofed below, it is then possible to fully predict the motion of the flow with the only knowledge of the velocity and magnetic fields at a given time.
1.2.1 Maxwell equations and the MHD approximation
The eldest study of such a flow dates back to Faraday’s work (1832 5 ) where he attempted to measure the electric current induced when both ends of a wire are in contact with the water on either sides of the Tames. The river’s flow, interacting with the earth’s magnetic field must generate a current as we shall soon see (1.19).
This introduction is inspired by the books of Davidson [28, 34], Priest [35]
and Biskamp [36]. Here Maxwell’s equations (1.17) are postulated a pri- ori, refer to [34, 37] for a discussion; the MHD approximation follows. I
5
Not that old, Belgium already existed!
hence gather Faraday’s, Amp`ere-Maxwell’s and Gauss’ laws together with the solenoidal condition on the magnetic field B:
∇ × E = − ∂ t B, (1.17a)
∇ · E = ρ e /ε 0 , (1.17b)
∇ × B = µ(J + ε 0 ∂ t E), (1.17c)
∇ · B = 0, (1.17d)
where E and B are electric and magnetic fields, J is the current density through the medium, µ and ε 0 are free space’s permeability and permittivity and ρ e is the electrical density of charge.
These can be completed by
• an expression for the force - electrostatic + Lorentz - experienced by a single particle of charge q and velocity v:
f e = q(E + v × B), (1.18)
• the empirical Ohm’s law relating the current density to the electromag- netic fields, and the medium’s velocity u and conductivity σ:
J = σ(E + u × B), (1.19)
Note that (1.17) implies a charge conservation law: the current through a closed surface equate minus the charge variation within the enclosed volume:
∇ · J = − ∂ t ρ e . (1.20)
In mhd, we are concerned with those cases for which there is no clear separation of charge within the medium. Hence, the charge density ρ e is assumed to be vanishingly small. Please, note that this does not cause the current density J to vanish 6 .
The interaction between the medium’s velocity and magnetic field can be understood as follows:
• the presence of both a velocity and a magnetic fields ( u and B) intro- duces a current J via Ohm’s law (1.19),
6
As an example, consider an electric cable long after any battery switching on or off
events occurred. There is no charge separation within the medium, yet an electric current
might flow.
• this current leads to the induction of a secondary magnetic field B i via Amp`ere’s law (1.17c),
• finally, the current J and the magnetic field B acts on the flow via the Lorentz force (1.26).
In the following, the MHD approximation ρ e ' 0 is justified and applied to derive the induction equation describing the evolution of the magnetic field. The extra forcing term appearing in the Navier-Stokes equations for the conducting fluid is also deduced. A close formulation results which only implies the velocity and magnetic fields.
Taking the divergence of Ohm’s law (1.19), and applying Gauss’ law (1.17b) and charge conservation (1.20), leads to:
∂ t ρ e + σ
ε 0 ρ e + σ ∇ · (u × B) = 0. (1.21) The ratio appearing in front of the second term has the unit of an inverse time and the relaxation time of the system is τ relax = ε 0 /σ. Indeed, in absence of flow u = 0, (1.21) reduces to a linear equation, the solution of which is a decaying exponential:
ρ e = ρ e (t = 0)exp( − t/τ relax ). (1.22) The MHD approximation applies by definition to processes occurring on a much slower time scale than that of relaxation. Hence, the first term in (1.21) can be neglected in front of the second one and we are left with a static equation:
ρ e = − ε 0 ∇ · (u × B), (1.23)
showing that a small charge can persist in a conductor which is in movement through a magnetic field. This charge however turns out to be extremely small and negligible to any practical purposes.
When the Lorentz force acts on the charged particles within a fluid, it is useful to recast it in a volumetric form, the sum over all charges P
q is then replaced by the electric density while the sum over the charged particles appearing in the second term P
qv yields the current density J . Hence, the fluid experiences a body force F given by (1.18) summed over the charges q and recast in a volumetric form:
F = (ρ e E + J × B). (1.24)
The order of magnitude of the ρ e E term can be evaluated as:
ρ e E ∼ ε 0 uB/l × J/σ ∼ τ relax τ mech
JB, (1.25)
where u, B, l, and J are the characteristic velocity, magnetic field, length scale and current density of the system. The characteristic time scale for mechanical movement of the fluid is defined as τ mech = l/u. In conducting fluids moving at moderate speed, τ mech ∼ 1s 7 while τ relax ∼ 10 −18 s, and typical values for the times ratio are τ relax /τ mech ∼ 10 − 18 so the (volumetric) Lorentz force applying on the fluid is extremely well approximated by
F = J × B. (1.26)
Furthermore, as we neglected the variation of fluid charges in our ap- proach, the charge conservation (1.20) simplifies to
∇ · J = 0. (1.27)
Finally, the displacement current is negligible in (1.17c):
ε 0 ∂ t E ∼ ε 0
σ ∂ t J ∼ τ relax
characteristic time of evolution of J J J (1.28) and the pre-Maxwell version of Amp`ere’s law is enough:
∇ × B = µJ . (1.29)
The induction equation, describing the evolution of the magnetic field is derived as follows.
∂ t B = −∇ × E
= −∇ × J
σ − u × B
= ∇ × (u × B) + η∆B, (1.30)
where the first line is Faraday’s equation (1.17a), the second one is insured by Ohm’s law (1.19). Ampere’s law is behind the third one and the solenoidal condition on B finally comes in play. The new coefficient η = 1/µσ is the magnetic diffusivity.
7
In astrophysical situations l yielding τ
mechand the ratio τ
relax/τ
mech.
On the other hand, the electromotive force applied on the fluid is described by the body force (1.26). Using Amp`ere’s law (1.29), yields:
J × B = − B × ( ∇ × B)/µ
= (B · ∇ )B/µ − ∇ (B 2 /(2µ)), (1.31) and the equation for the evolution of the fluid velocity field (1.9) becomes
∂ t u = − (u · ∇ ) u + (B · ∇ ) B ρµ − ∇
p + B 2 2ρµ
+ ν ∆u + f , (1.32) where f stands for other body forces.
Eventually, normalising b = B/ √ ρµ and lumping the b 2 term into the pressure, the MHD equations for an incompressible flow are:
∂ t u = − (u · ∇ ) u + (b · ∇ )b + −∇ p + ν ∆u + f , (1.33a)
∂ t b = ∇ × (u × b) + η∆b, (1.33b)
∇ · u = 0, (1.33c) ∇ · b = 0. (1.33d)
In ns turbulence, the Reynolds number R e was introduced as a ratio between inertial and viscous effects. In mhd turbulence, an other such ratio comes also into play. The induction equation presents two terms. There is a non-linear part which may lead complex evolution of the magnetic field and a linear part that tend always to oppose the field. The former has a time scale τ mech = L/U and the later a dissipative time scale τ η = L 2 /η. The effect with the fastest time scale is expected to dominate the evolution of B . The magnetic Reynolds number is therefore defined as the ratio of these time scales:
R m = U L
η . (1.34)
The ratio of the two non-dimensional Reynolds numbers (1.12,1.34) defines the magnetic Prandtl number
P m = R m
R e = ν
η . (1.35)
It is a property of the fluid while the Reynolds number depends on both the fluid and the flow.
1.2.2 MHD invariants and cascade
In the 3 dimensional ns case, two ideal quadratic invariants are present: ki-
netic energy and kinetic helicity. In mhd, the former is generalised to the
total energy encompassing a magnetic contribution:
E tot = 1 2L 3
Z
L
3( | u | 2 + | b | 2 )dV. (1.36) The latter is no longer conserved owing to the presence of the Lorentz force.
On the other hand, two new quadratic ideal invariants appear: the mag- netic helicity H b and the cross helicity H c . The first one corresponds to the alignment between the magnetic field and its potential vector a defined 8 by
∇ × a = b:
H b = 1 L 3
Z
L
3a · b dV. (1.37)
The second one measures the alignment between the velocity and the magnetic fields:
H c = 1 L 3
Z
L
3u · b dV. (1.38)
Proofs that these quantities are conserved in the ideal limit are postponed to the end of chapter 2.
The question of how these quantities behave within the turbulent cascade is very interesting and probably crucial if we are to correctly describe turbu- lent dynamos (see below). Note that the magnetic helicity only depends on the magnetic field. As long as mechanical forcing are concerned, it is there- fore not possible to inject it in the system. It is however a non positively defined quantity. It could therefore be that a kind of symmetry breaking oc- curs within the mhd flow that transfer for example positive magnetic helicity to large scales structures while the negative helicity would be cascaded to small scales. As dissipation is most efficient at small scales, such a symme- try breaking in the flow together with Joule dissipative effect could very well generate magnetic helicity.
1.3 Introduction to the dynamo effect
This section is an introduction to the dynamo effect with emphasis on the phenomenology which is most relevant to my work. It is by no mean an attempt at a review of this very vast and fascinating subject. Such a review can however be found in [38] and references therein. The dynamo effect is the process by which a magnetic field can be spontaneously created and maintained by the motion of a conducting fluid. The main official motivation for its study is that it seems to happen about everywhere in the observable
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