Master thesis — April – July 2011

Department of Physics

Ecole Normale Superieure de Lyon

Universite Claude Bernard – Lyon 1 – France

Blob statistics in the scrape-off layer of tokamaks

Mickael Melzani

Contact: mickael.melzani@ens-lyon.fr

Project supervised by Prof. Odd Erik Garcia (odd.erik.garcia@uit.no)

Conducted at the Department of Science and Technology,University of Tromsø, Norway

UNIVERSITETETI TROMSØUiT

Abstract

We investigate the properties and statistics of blobs – coherent structures of high den-sity plasma – using sitting-probe measurements of unprecedented record length in the farscrape-off layer (SOL) of the TCV tokamak. This is complemented by an analysis of a runof the ESEL simulation program which shows that blobs are produced by the elongationand detachment of edge-based convective cells by the sheared poloidal flow. The blob ve-locity is the sum of its own E× B velocity and of the poloidal velocity of the backgroundplasma. The signature of the latter on probe potential measurements is a dipolar wave-form, that is exploited to estimate the mean poloidal velocity. The imprint of blobs onTCV particle density and flux signals is an almost invariant bursty shape. The propertiesof these peaks are explored by conditional window averaging. Both their amplitudes andwaiting-times follow a Poisson distribution and are independent of one another. This is afirst sign of the absence of correlation, and is not compatible with self-organized critical-ity. We find that blob amplitude, E × B velocity and spatial extent all increase together.We also investigate the probability distribution functions (PDFs) from TCV signals. Thelength of our data records allows to reject several suggested fitting models for the particledensity PDF. We explain the shape of the flux PDF by considering the blobs only, andderive an analytical expression characterized by the waiting-time distribution and singleburst PDF. We show that our TCV data present a self-similarity exponent of 0.55 at lowline-averaged density and 0.6 at high density, indicating very weak long-range temporalcorrelations. We compare this to ESEL outputs and find good agreement. Finally, thereliability of the rescaled range analysis applied to short time series is discussed.

Subject headings: plasma physics – turbulence – scrape-off layer – blobs and coherentstructures – long range correlations

Version: July 27, 2011

Contents

1 Introduction 1

1.1 Principles of fusion and tokamaks . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The problem of perpendicular transport . . . . . . . . . . . . . . . . . . . . 3

1.3 Contents of the present report . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Models and simulation 6

2.1 Interchange mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Self-organized criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Statistical analysis 11

3.1 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Conditional window averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Probability distribution functions . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Self-similarity 19

4.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Presentation of the concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Rescaled range and cumulated structure functions . . . . . . . . . . . . . . 20

4.3.1 Analysis of TCV data . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3.2 Analysis of ESEL data . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.3 Reliability of the rescaled range . . . . . . . . . . . . . . . . . . . . . 22

5 Conclusion 24

A Reminder of plasma physics 25

B A derivation of ESEL equations 28

C A quick overview of sheath physics 31

D Concepts of self-similarity 33

E Physical constants 37

Acknowledgments 38

Bibliography 42

i

1 Introduction

1.1 Principles of fusion and tokamaks

A source of energy

Since the first Russian tokamak fusion devices built in the 1950s to the building of ITER,the most expensive experiment ever designed, the idea behind the production of electricalenergy by fusion have been simple and audacious: it is to heat a gas of light nuclei up totemperatures comparable to those encountered in the centre of stars – 108 K – where thecross-section for fusion reactions becomes important. When fusing, two light nuclei forma heavier nucleus with a lower binding energy per nucleon, releasing significant kineticenergy∗. This thermal energy can then be converted in electricity with the help of classicalturbines. The advantages are numerous: huge instantaneous power (0.5 GW for ITER,equivalent to 1/2 fission reactor), safety of the device (no melt-down possible, for exampleITER will contain only 0.5 g of fuel gas), efficiency (energy amplification by a factor of10 for ITER), cost and availability of the combustible (at least if D-D reactions are used,since it can be extracted from water), and low environmental impact (the few radioactivematerials produced are short lived). Much more details on fusion and energy productioncan be found in the literature (e.g. in Freidberg [19, Part I]).

Confining the plasma: the magnetic field geometry

The fusion project is as appealing as challenging. The main difficulty is to confine thehot plasma far from the walls of the device. Most of the solutions envisaged so far consistin magnetic confinement and differ by the geometry of the field created. In a tokamak,the main magnetic field is toroidal (along −φ here, see figure 1.1 for notations). At thetemperatures required for fusion, the gas is ionized and the charged nuclei and electronsconstituting the plasma are then bound to follow the field-lines, thus never escaping (atleast in the absence of collisions and collective motions).

However, in a purely toroidal magnetic field, the centrifugal force and the gradient of thefield Bφ, both directed along −R, make the positive and negative particles (q = ±e) drift in

opposite directions along qR∧Bφ ∝ −qZ†. The resulting current polarizes the surdensity

perturbations with a field E ∝ +Z, and the resulting E × B drift advects the particlesradially outwards, making the plasma unstable‡. There is no equilibrium configuration.

To counter this effect, a poloidal magnetic field Bθ is added, so that the total field windsup in a helix around the torus. Following a field-line, the curvature and gradient vectorscontinuously change direction and the associated particle drifts average to zero. However,this field has to be weak enough so that the plasma column remains stable against twisting(the kink instability, see Bellan [5, chap.10]). The safety factor q ≡ aBφ/(RBθ) is typically3-4 in the edge and up to 10 in the core. This means that the field is mostly toroidal.

In tokamaks, the toroidal field Bφ (typically a few Tesla) is created by poloidal coils.The poloidal field Bθ is created inductively: the inductive current in the transformer coilgenerates a varying magnetic field in the transformer core, and thus a varying magnetic fluxthrough the torus. According to Faraday’s law (or −∂tB = ∇ ∧ E) an inductive toroidalelectric field is created in the plasma, that then creates a toroidal current. The situation

∗A good candidate for fusion devices, with a “low” temperature of sufficient cross-section of 15 keV =1.7 · 108 K, is a mixture of deuterium and tritium (21D and 3

1T ), that fuse to give a neutron and a Heliumnucleus, releasing in the process 17.6 MeV of kinetic energy.†The reader unfamiliar to plasma physics concepts is encouraged to read the short appendix A, where

the essential physics used in this report is briefly described.‡Note that this picture is equivalent to the usual MHD description of the interchange instability of a

z-pinch. See e.g. Freidberg [19, sec. 12.3].

1

1.1. PRINCIPLES OF FUSION AND TOKAMAKS 2

divertorcoil

poloidal plane

Edge

transf

orm

er

coil

transformer core

Figure 1.1: Left: Schematic description of a tokamak. Note that the poloidal cross sectionshown, as well as the coil, should be the same all around the main torus. Clarity (anddrawing skills) have prevented us to do so. Right (taken from Garcia et al. [27]): Part (a)is a more detailed view of the poloidal cross section for TCV. The curves are projectionsof the magnetic field-lines on that plane, i.e. Bpol. Part (b) is a zoom on the shaded partof (a). See the main text for explanations. Numerical values can be found in appendix E.

is that of a classical transformer, the plasma playing the role of the secondary coil. Thetoroidal current is finally at the origin of the poloidal field. This current is also used toheat the plasma via electron-ion Coulomb collisions§. However, its inductive nature makesit only temporary, and it is the main factor limiting the confinement duration in a tokamak(limited to a few seconds in present experiments).

The poloidal field only establishes equilibrium, and particles may still drift toward thewalls under the action of the E×B drift, collisions and other effects to be discussed in thefollowing, especially in the region where the magnetic field-lines are not closed. But all thewalls cannot be built to withstand the consequent heat fluxes involved, and besides, plasma-wall interactions have strong effects (neutral production, impurity) that have to be wellcontrolled in a small region far for the main plasma. For this reason, the field configurationis modified by external toroidal coils (Iext on figure 1.1). In the final configuration, theprojection of the field-lines on a poloidal plane is as sketched in figure 1.1 right. (i-plainred) In the central region (the core), the field-lines wind-up in a close way, as explainedpreviously. (ii-dotted blue) There is then the SOL (or scrape-off layer) where the field-lines are open and terminate on a surface called the divertor. The last closed flux surface(LCFS, black) separating these two regions is the separatrix. The edge is the region closeto but inside the separatrix. (iii-dotted red) There is finally the wall shadow, where thefield-lines intersect with the main chamber walls.

Up to the edge, where the field-lines are still closed, particles flow freely along them,the curvature and gradient drifts average to zero, and they are transported radially towardthe separatrix due to collisions and turbulent motions. They then enter the SOL wherethey propagate quickly (roughly at the sound speed) toward the divertor targets, whilealso being transported toward the main chamber walls: the plasma is “scraped-off”. Thereis a competition between parallel and perpendicular transport, and it is tremendouslyimportant that most of the particles reach the divertor before entering the wall shadowwhere they would inevitably end up on the main chamber walls. Particles hitting a solidsurface recombine and are back-scattered as neutrals, and also remove particles from thissurface. This results in erosion of the plasma facing components, and may cause severelifetimes limits for the next generation fusion experiments. Moreover, impurities releasedin the main plasma reduce the heat confinement. The divertor is specially built and isplaced far from the main torus precisely to control these phenomena.

Much more details on the subject of tokamaks can be found in Freidberg [19] or Stangeby[54, 55], and an overview of the TCV device is available at crppwww.epfl.ch/tcv/.

§This ohmic heating is however limited by the fact that collisionality decreases with increasing tem-perature (equation A.4), so that above 10 keV another mechanism has to be used. A possibility is toinject resonant electromagnetic waves, for example at the ion or electron cyclotron frequency, that will beabsorbed and heat the plasma.

3 CHAPTER 1. INTRODUCTION

1.2 The problem of perpendicular transport

Anomalous diffusion

The main concern of this report is perpendicular transport in the edge and SOL. It isindeed in these regions, where the transition from closed to open field-lines creates strongprofile gradients, that perpendicular transport is most important. A first approach wouldbe to assume that it has a diffusive nature: electron-ion collisions make the particles jumpfrom one field-line to another in a random walk¶. Fick’s law then relates the perpendicularparticle flux to the particle density gradient as Γr = −D∇n. The diffusion coefficient isexpressed as D = λ2ν up to a factor of order unity, where λ and ν−1 are the length andtime step of the diffusive process.

If we take for these parameters the gyration Larmor radius and the collision time, asis relevant for highly magnetized plasmas (i.e. where Ωi and e νei, see equations A.3and A.4), we obtain the classical diffusion coefficient Dcl = (ν/Ω)2(v2

t /ν), where vt is thethermal speed. A MHD or two-fluid approach also leads to Dcl (Bellan [5, sec. 2.8] or Chen[14, chap. 5]). But compared to experimental measurements, it predicts too low diffusionrates, meaning that the transport is not collisional (see e.g. Horacek [35, fig. 5.16]).

A more successful approach is to replace the steptime of the diffusive process by the cy-clotron period to end up with Bohm diffusion coefficient: DB ' (ν/Ω)(v2

t /ν) (∼ 16 m2 s−1

in the edge of our experiment), which is a factor Ω/ν higher than Dcl. This value is ofthe order of what is found experimentally (Dexp ∼ 0.1 − 10 m2 s−1, see Lipschultz et al.[41], Stangeby [54], Garcia et al. [29]), and this can be expected because Bohm coefficientreflects turbulent transport. To see that, consider a maximum in potential φ that inducesa E×B flow around the potential hill, with velocity magnitude E/B. If we expect the po-tential perturbation to be of the order of T/e‖ and if we take the length of the perturbationas the relevant stepsize for the diffusive process, we indeed find λv2 = T/(eB) = DB.

Turbulent convective transport

Actually, and as might be expected, the radial flux does not follow a phenomenological lawas simple as Fick’s law. A law emphasizing convective transport such as Γr = nveff givesa more coherent scaling [41, 29], but is not predictive either. The real situation is quitecomplex: the perpendicular flux is powered by sources of free energy (the profile gradients)that relax when a threshold for instability is reached, and is affected by the magnetic field,by shear flows, by impurities, etc. Wootton [57] indeed shows that many functional formsfor the flux can be found experimentally.

The consensus today is that perpendicular transport is mostly due to convective tur-bulent motions. Measurements in the edge show that particle density and electrostaticpotential fluctuate widely (see figure 1.2). The fact that fluctuations in the magnetic fieldare weaker indicates that the turbulence is mostly electrostatic (see [57] for a comparisonbetween electrostatic and magnetic turbulence), that is, E = −∇φ where fluctuations inthe vector potential are neglected. Fluctuations in E then naturally give rise to a turbulentE×B velocity field across the magnetic field-lines. This turbulence is powered by the heatand particle fluxes coming from the core, that maintain the gradients, and regulates thetransport across the edge and SOL toward the walls.

The objective of current research is to identify the instabilities at the origin of thisturbulence (there are several candidates), and to find the scaling involved for the flux infunction of relevant key parameters. Several strategies are possible. A first one is to startfrom model equations – often a two-fluid Braginskii-like model – that are then simplified tosingle out a relevant instability. One way to do so it is to derive a coherent set of equationsfor scalars (e.g. density, vorticity and temperature), and to take advantage of the 2D aspectof the problem∗∗. Analytical work can then be done to study various energy transfers, or

¶Because of momentum conservation, self-species collisions are far less efficient, see Chen [14, chap.5].‖Here and everywhere else, the temperature is expressed, via Boltzmann’s constant kB , in energy units.∗∗The strong magnetic field allows for the propagation of Alfven waves in the direction parallel to

the field-lines, that homogenize the physical quantities along B (similarly to the situation in rotatinghydrodynamics with inertial waves, see Davidson [15, chap. 10]).

1.3. CONTENTS OF THE PRESENT REPORT 4

a wave turbulence approach can explore the way the non-linearities cascade energy orenstrophy. Since the turbulence is fully developed, the essential physics is nonlinear andthe analytical work has often to be further explored by numerical simulations. This reportprecisely exploits such a two-dimensional scalar model.

Another strategy is to study the two-fluid models with less approximations, includingthe third dimension, complex geometries and fluctuations in the magnetic field. This isthe case of the model EMFDET (electromagnetic fluid drift turbulence, see [40, 47]), andalso of the 3D code BOUT. These models have some success, but the physical insight theyprovide is limited by their complexity.

Of course in both cases, results are to be compared to experimental findings. This isalso what will be done here.

0.1

1.0

10

20 m

-3

Density

-20 -10 0 1 (mm) wall shadow

10

100

eV

Electron temperature

separatrix

SOLedge

Rela

tive

flu

ctu

ati

on

lev

els

0.6 0.7 0.8 0.9 1.0 1.1 1.2

0.8

0.4

0.6

0.2

0.0

separatrix

Figure 1.2: (Left): Average profile of particle density and electron temperature in a typicalconfiguration. Taken from La Bombard et al. [40], where more profiles can be seen. Wenote the steep gradients in the SOL. (Right): Typical fluctuation levels in terms of rootmean square (rms). Taken from Wootton [57]. We see that δn/n and eδφ/T are between0.1 and 0.8 around the separatrix. The situation is much more quiet in the core, where thebackground gradients are weak (see on the left). See also Zweben et al. [61], Garcia et al.[29] for other profiles with again the same essential features.

Bursty transport by coherent structures

The turbulence in the edge and in the SOL regions is quite different. In the edge, itmainly consists in the saturation of instabilities (mainly the drift wave instability, see[14, 36, 46, 38]). The latters produce localized density perturbations that are released intothe SOL and that are believed to participate to a large extent to the SOL turbulence. Theyhave been observed experimentally as coherent structures of 0.5-5 cm size in the poloidaldirection (z), a little less in the radial direction, and strongly elongated along the field-lines (several meters). They are variously called filaments, elongated streamers, or blobs.They propagate radially outwards quasi-ballistically in the SOL at speeds of the order ofa fraction of the sound speed cs =

√Te/mi ∼ 3 · 104 m s−1 here, and transport most of the

radial flux. They have been observed either by conditional averaging on probe time signals(e.g. Garcia et al. [29] or the present report), or by direct imaging methods (Grulke et al.[33], Zweben et al. [61, 60, 59]††, and figure 1.3), and their signature on particle density orflux time signals can be seen as high peaks of fluctuations (as on figure 3.1).

The fact that most of the transport is realized by these blobs is a source of concern:they will cross the SOL and interact with the walls more easily than if the flux was steady.For a review on blob theory, simulation and experiment, see D’Ippolito et al. [16].

1.3 Contents of the present report

We have seen that the perpendicular flux is a key quantity for the design of a tokamak:the SOL has to be large enough so that most of the particles reach the divertor before

††Videos of blob propagation are available online at www.pppl.gov/~szweben/NSTX04/NSTX_04.html

5 CHAPTER 1. INTRODUCTION

Figure 1.3: Taken from Grulke et al. [33]. Imaging of blob-like plasma filaments (a puffof neutral deuterium gas is injected locally, the atoms are then ionized by the plasma ata rate proportional to n and T , and release photons in the process: the intensity recordedby the camera is a trace of density fluctuations). (a) Poloidal plane. A local densityperturbation (in red), or blob, is created outside of the separatrix (continuous line) andpropagates radially outwards, until it is dissipated in the wall shadow (after the dashedline). (b) View in the z-φ plane, showing that the blobs are elongated along the field-lines.

reaching the walls, and this minimal width can be estimated only if the radial particleflux is known. The problem is that there is presently no physical (nor empirical) modelthat can accurately predict this flux. This is a major concern for the design of ITERand other future devices, where the parameters will range in areas still unexplored. Sinceperpendicular transport in the SOL is dominated by blobs, it is natural to study theirformation, motion and statistical properties in order to grasp the main features of theparticle flux. This is the central motivation of this report.

In section 2 we introduce the interchange model and explain the blob propagationmechanism. We analyse a run of the ESEL simulation code to show that this instabilityproduces blobs by the detachment of the tips of the edge-based convective cells that areelongated by the sheared poloidal flow. We study the blob trajectories, and show that theirvelocity is the sum of their E × B velocity and of the poloidal mean flow present in theplasma. The poloidal component of the E×B velocity is determined only by the gradient ofthe sheared poloidal flow, and is always so as to enhance this gradient, in a way similar tothe tilting of convective cells. We also introduce the paradigm of self-organized criticality(SOC), that is often alluded to as a relevant model for turbulence in the edge and SOL.

In section 3, we describe the experimental data used in this report: two shots fromTCV of unprecedented record length that differ only by their mean particle density andallow for a reliable statistical analysis. We explore the blob statistics with the use ofconditional window averaging. We find that the signature of a blob passing the probe is acharacteristic peak for the particle density and flux signals, of shape roughly independentof amplitude (when normalized). The amplitude and waiting time statistics are found tobe Poisson distributions and independent of one another. This is in contradiction withthe SOC paradigm advocated in many previous works. We also show that blob amplitude,spatial extent and E×B velocity increase altogether. The potential associated with particledensity peaks presents a dipolar waveform, that we interpret as a trace of the dipolarpotential blob structure advected by the mean plasma poloidal velocity. The lag betweenthe two potential curves is used to estimate this poloidal velocity. We further analyse theprobability distribution functions (PDFs) of the signals. We reject several popular models(gamma, log-normal, ...) for the particle density PDF. The latter exhibits an exponentialtail and the flux PDF a stretched exponential tail. We explore the incidence of the blobstatistics on the PDF, and interpret the flux PDF as the product of an averaged peak PDFtimes the cumulative distribution of blob amplitudes, convoluted with a weak backgroundsignal. An analytical expression is given in a simple case.

In section 4, we explore the presence of long-range correlations in our signals by meansof a self-similar analysis. This approach as been widely used in the literature, wherelong-range correlations are interpreted as a signature of SOC. Here we find no long-rangecorrelations at low density, and weak long-range correlations at high density. We alsotest the ESEL data output by checking that its self-similar exponents agree with theexperimental ones. Finally, we discuss the reliability of the rescaled range analysis whenapplied to short time series, and show that the fitting range (and consequently the recordlength) controls the uncertainty and the bias of the method.

2 Models and simulation

2.1 Interchange mechanism

The drift wave instability [14, 36, 46, 38] is believed to greatly contribute to the edgeturbulence, but in this report we only focus on the interchange mechamism. The first reasonis that reduced fluid models based on this mechanism, such as the ESEL code, successfullydescribe many aspects of experimental observations (PDF shapes, profile gradients), andproduce and drive blobs. The second is that the central objects of our study, blobs, are setinto motion and driven by the interchange mechanism.

Interchange driven turbulence

Let us first discuss the physical picture behind this instability. Consider the geometry offigure 2.1a. Ions drift downwards whereas electrons drift upwards, due to the particle cur-vature and gradient drifts (T/qB)b∧ [∇ lnB+κ] (where κ = n/R is the curvature vector).We consider a background density profile n0 decreasing radially outwards, and a sinusoidalperturbation in density n. At point A, there are more electrons coming from the densityjust below than ions coming from the depletion just above. Said otherwise, the divergenceof the diamagnetic current at point A is strictly positive, and there is an accumulation ofnegative charges. Similarly, point B sees more ions coming from the hill above it than elec-trons from the crest below and becomes positively charged. The result is that the densitybetween A and B is polarized and the resulting E × B velocity, directed outwards, dragseven more particles in there (because ∇n0 ∝ −x): the perturbation amplitude grows.

A

B

electrondrift

ion drift

(a) (b)

inside of the torus

Figure 2.1: (a) Illustration of the interchange instability. (b) Illustration of the radialpropagation of an excess density. Note that the middle panel applies to both (a) and (b).

This phenomenology can be translated into a set of coupled equations for the particledensity n, the E× B vorticity Ω and optionally the electron temperature Te. This is donein appendix B. Here we just mention some important facts. The essential physics of thisinstability is grasped by considering electrons of zero mass and constant temperature, ionsof zero temperature, and no collisional effects nor parallel motions. All that remains forthe electrons is the E× B velocity vE and the diamagnetic velocity vDe; and for the ionsthe E× B velocity vE and the polarization velocity vPi = (mi/qB)b ∧ dvE/dt, where weintroduced d/dt = ∂t + vE · ∇. We recall that:

vE =E ∧ b

B=

b ∧∇φB

=1

B

∣∣∣∣∣∣∣−∂yφ∂xφ

0

. (2.1)

If we take the electron continuity equation ∂tn + ∇ · (nvE + nvDe) = 0, we see that∇ · nvDe and n∇ ·vE involve variations in the magnetic field (see equation 2.1 and vDe in

6

7 CHAPTER 2. MODELS AND SIMULATION

equation A.6), that are far smaller than variations in the particle density. To first order wethen have ∂tn + vE · ∇n = 0. This important result states that the density is essentiallyadvected by the E×B velocity alone. Looking back at equation 2.1, we note that φ acts asa stream-function: the velocity is tangent to iso-φ contours. Moreover, the E×B vorticitycan be simply expressed as Ω ≡ b·∇∧vE = ∇2

⊥φ/B up to small terms involving derivativesof B and parallel gradients.

Interchange driven blob motion

The interchange mechanism also provides an explanation for the propagation of the blobstructures, independently of the instability by which they are produced. The physicalpicture is described in figure 2.1b: here again, the gradient and curvature currents polarizethe blob, that is then advected outward by the E × B velocity. The vorticity drawn onfigure 2.1b follows from the relation Ω ∝ ∇2

⊥φ. The model equations are the same as for theturbulence. Analytical and numerical studies of such blob motions have been performed(Garcia et al. [23, 25], Krasheninnikov [39]), and the resulting dipolar potential structurehave been observed experimentally (Grulke et al. [33]) and in the following.

Parallel losses

Another point of view is to consider that the magnetic field gradient and curvature act asa current generator inside the blob. This current is closed by the ion polarization currentwhich is proportional to dΩ/dt and thus radially advects the blob. This picture showsthe importance of parallel currents: they can also close the current loop in place of thepolarization current. A parallel current is established in the SOL if the blob extends alongthe magnetic field up to the divertors. When this is the case, and as is shown in appendixC, the fact that the field-lines are connected to the solid surfaces of the divertor in the SOLinvolves a parallel flow and a parallel current given by equation C.5. The plasma adaptsits potential to cancel this current and preserve its quasi-neutrality, but fluctuations in φlead to an uncompensated parallel current that can be obtained by linearizing equationC.5 around φ = φse = φw + TeΛs/e, and shown to be to first order proportional to thepotential fluctuations. The divergence or this current is proportional to the gradient of φ,and once integrated along a field-line becomes proportional to φ.

However, this parallel current is established only if the blob feet are on the divertor,and so only if perturbation induced by the blob travel along the field-line to the divertorbefore the blob has crossed the SOL. It is argued in Fundamenski et al. [20] that this is notalways the case. Moreover, since the field-lines wind around the main torus (figure 1.1),and since a blob always propagates in the radial direction, the blob structure propagatingfrom the edge to the SOL will remain coherent along the parallel direction only on theouter side of the tokamak and will not reach the divertor. This leads to the considerationof parallel advection of the blob vorticity, simply by the parallel flow established by theblob surpression. The loss term in the vorticity equation is then Ω/τΩ.

Simulation code

In this report we use data coming from the ESEL code (Horacek [35], Fundamenski et al.[20], Garcia et al. [29], and for an older version Garcia et al. [22, 24, 28, 27]). It simulatesequations for n, Ω and Te describing the interchange mechanism and parallel advection,derived in appendix B. The simulation domain is divided in three regions that representthe edge, the SOL and the wall shadow, and that differ by the treatment of the paralleladvection of n, Ω and Te: it is set to zero in the edge, where the field-lines are closed, set toa non zero value in the SOL and to a higher value in the wall shadow where the connectionlength is shorter. The boundary conditions are the following: periodicity in the poloidaly direction; at x = 0, the turbulence is driven by a constant value of particle density andelectron temperature (n = T = 1), a no-slip condition is imposed (φ = 0, so that vx = 0),and Ω = 0 is enforced; at x = Lx, we impose vy = Ω = ∂xn = ∂xT = 0.

Sarazin [51] and Bisai et al. [7, 8] have solved a similar set of equations describing inter-change electrostatic turbulence, but with a parallel dynamic governed by sheath dissipation

2.1. INTERCHANGE MECHANISM 8

(as explained above). They obtain blobs that have a smaller spatial extent, especially forthe potential, which is to be expected since the sheath loss term ∝ φ scales as Ω/k2

⊥ wherek⊥ is a wavevector, and dissipates preferentially large structures. Outputs from thesesimulations have not proven to agree with experimental blob waveforms and PDFs.

Global dynamic: relaxation-oscillations

The global dynamic of the interchange turbulence have been largely studied (see e.g. [6, 26,30] and in the context of ESEL [28, 24]), partly because it is similar to thermal convectionin fluids (the interchange mechanism is similar to baroclinic generation of vorticity). Asthe Rayleigh number Ra is increased, the system evolves from a purely diffusive state to astate where convective cells appear. A Galerkin projection scheme similar to the one of theLorenz model, but including a shear flow mode [26], shows that as Ra increases further,the convective cells are unstable to tilting: they slightly tilt, and thus generate a zonal flow(because for example in figure 2.2 they transport positive (negative) poloidal velocity fromthe right to the left (left to the right), thus powering a poloidal flow increasing from rightto left), that will in turn tilt them even more.

It is interesting to consider the total kinetic energy contained in the fluctuations (con-vective cells, waves, turbulence) and in the mean poloidal flow

K =

∫V

dx1

2v2 and U =

∫V

dx1

2v2

0 . (2.2)

where we defined v0(x, t) = L−1y

∫dyvy(x, y, t) (note that due to periodic boundary con-

ditions, the mean radial velocity L−1y

∫dyvx(x, y, t) is zero), and the fluctuating velocity

as v(x, y, t) = v(x, y, t)− v0(y)y. It can then be shown (see equation B.11) that the fluc-tuations are powered by the instability and by the pressure gradient, and that there isa unidirectional transfer of energy, through the Reynolds stress tensor, from the fluctua-tions to the poloidal flow (the tilting discussed above). In the light of what is advancedin [6, 26, 30, 28, 24], and of the snapshots taken from the simulation, we are led to thefollowing picture.

(i) The boundary conditions at x = 0 always impose a n = T = 1, and these quantitiesare lost in the SOL: the profile gradient grows and triggers instabilities. The fluctuationsgrow, saturate in the form of convective cells that are unstable to tilting (if Ra is highenough): they eventually tilt, and transfer energy to the poloidal sheared flow∗. (ii) Thefluctuations are still driven and power the shear flow, where energy is efficiently dissipatedvia collisions. When the shear flow is strong enough, it dissipates more energy than isinjected in the fluctuations by the driving term: the fluctuations are suppressed (figure2.2-6). (iii) We enter a quiet phase, with weak fluctuations and a direct transfer of drivingenergy to the shear flow, that dissipates both this energy and its own energy. The shearflow slowly decays on a viscous time-scale (figure 2.2-1 to 2). (iv) When the shear flow isweak enough, it can no more dissipate the driving energy: the fluctuations then suddenlyrise, there is a burst (figure 2.2-3 to 5). We are then back to point (i)

During the quiet phase (iii), convective radial transport is strongly suppressed by theunidirectional transfer of energy from the fluctuations to a mode (the shear flow) thatcannot transport radially. This transfer is indeed unidirectional because the cells are tiltedby the shear flow itself. The fact that the shear flow decreases the wavelength of anyperturbation and thus tends to dissipate it more easily, or the fact that coherent structurescan be sheared and destroyed is not the main mechanism of radial confinement [6].

Interestingly, this oscillation-relaxation dynamic can be caught by a Lokta-Vokterrapredator-prey model, where the fluctuations are the linearly driven predators (that growat a rate ∝ ∆p, the food reserve, and are eaten at a rate ∝ KU) and the shear flow the preys(that grow by eating the fluctuations at a rate ∝ KU and are linearly damped at a rate∝ −U representing overpopulation). An equation for profile relaxation by the fluctuationscan be added for ∆p. This is another source of oscillation-relaxation not mentioned above.See Bian and Garcia [6], Garcia et al. [26].

∗In the simulation, this shear flow is not necessarily null in average because momentum can escapefrom the boundaries and by parallel dissipation. In a tokamak, poloidal momentum can also be lost to thedivertor plates.

9 CHAPTER 2. MODELS AND SIMULATION

Blobs dynamic

With the help of the snapshots from the simulation, we can add new material to this picture.First, we see that small blobs are still ejected during the quiet phase (iii) (see figure 2.2-1and 2), and that the bigger ones are ejected during the bursty phase (ii). Sometimes,the whole pair of convective cells detaches to form a blob, as on figure 2.2-5. For moremoderate events, the process seems to be a tilting and elongation of the convective cells bythe shear flow, followed by the detachment of their tips to form an independent blob thatpropagates into the SOL. Note that in the absence of shear flow, simulations by Sarazin[51] show radially elongated structures that do not detach from the edge. It indicates thatthe shear flow, as well as sheath dissipation, are key elements to create blobs.

A consequence of this formation process is that when the blobs detach, their tiltingmakes them propagate at an angle to the radial axis. Note that this direction is determinedonly by the gradient of the shear flow, not by its sign. Note also that in doing so, theysustain the shear flow in the same way as tilted cells (e.g. in the situation of the simulation,they remove negative poloidal momentum at small x to transport it at large x). The blobvelocity is the sum of the background plasma E × B velocity and of the E × B velocitycreated by its dipolar structure. When they enter into the SOL, the excess density is partlydissipated by parallel advection, so that the strength of their dipolar structure dwindles andtheir poloidal velocity becomes dominated by the background velocity. This loss of poloidalmomentum to the divertor plates explains the curvy path observed in the simulation.

2.2 Self-organized criticality

In this subsection we present a concept believed by some authors to apply to turbulencein the edge and SOL regions. It is not a precise model but a paradigm, that could help tograsp some key features. We will discuss its relevance throughout this report, and showthat it does not apply to the data from TCV analysed here.

The concept of Self-Organized Criticality (SOC) was introduced by Bak et al. [3, 4, 2] toexplain the frequent occurrence in Nature of 1/f noise and of self-similarity. The canonicaltoy model of a system exhibiting SOC is that of a 2D pile of sand where grains are addedat random positions. At a given point, when the slope of the pile reaches a given threshold,four grains of sand are taken from that point and distributed to its four neighbours. Thisredistribution can in turn make the slope of one or more of the neighbours reach thethreshold, and so this cascading process can reach a large number of points. Such anavalanche can be stopped only by more-than-minimally stable points, i.e. points that arewell below the threshold. Note that such points are actually produced by an avalanche,because four grains are removed from a cascading point.

The system will become stable when the network of more-than-minimally stable pointsis extended enough to stop an avalanche to propagate to the boundaries. It then reaches astationary state where avalanches of all scales occur. Their sizes actually follow a power-law distribution, which means that the clusters surrounded by more-than-minimally stablepoints form all together a self-similar fractal object. Another consequence is that the du-rations of the avalanches are also power-law distributed. It follows that the signal recordedat one point will be a succession of peaks of power-law distributed durations, that can beshown to possess a power-law power spectrum and so to be a 1/f noise (in a large sense:with arbitrary PDF and self-similar exponent for the cumulated signal, see later).

The term “critical” here has the same meaning as in a critical point for a phase tran-sition: the correlation length diverges, fluctuations extend to all scales and there is nolength scale remaining. The essential difference with phase transition is that reaching thiscritical point does not require any fine tuning. Instead, the system possesses an attractorwhere the dynamics is critical and where it eventually ends up, in a self-organized way.The situation is somehow similar to the one in turbulence, where injection of energy atlarge scales is redistributed equally to all scales.

This last analogy have pushed several authors (e.g. Carreras et al. [11, 13, 10, 9],Sanchez et al. [50], Newman et al. [45]) to explain the situation in the edge and SOL withSOC. The random driving would be particles and heat from the core, the slope would

2.2. SELF-ORGANIZED CRITICALITY 10

(1)

1 1.05 1.1 1.2 1.25 1.30

50

(2)

(3)

(4)

(5)

(6)

Color: density levels.

White lines:iso-fluctuatingpotential(negative ifdashed).

Arrows:total velocity field. Note thatthe scale is different beforeand after theseparatrix (magnificationby a factor 10 after theseparatrix).

Color:fluctuatingpotential levels,i.e.:

It is thestream-functionof the fluctuatingvelocity, i.e. totalvelocity minus mean poloidalvelocity, or :

Separatrix

Begining ofwall shadow

Probe

12

3 6

:

between potentialprobes

(or poloidal)

(or radial)

Figure 2.2: Snapshots from ESEL. Quantities are in Bohm units (see [24]).

be the profile gradients, the slope threshold would be a threshold for an instability, andthe avalanches would be the relaxation of the gradients triggered by the instability andturbulent motions. They then look for long-range temporal correlations.

If the avalanches in the edge are at the origin of the blobs released in the SOL, thenSOC has strong consequences for their statistics: their amplitude and duration (linked tothe avalanches) should be power-law distributed. We will use these facts to discuss thepertinence of SOC. As for the waiting time statistics (times between bursts), it is influencedby the driving of the edge, i.e. the flux from the core, and not by the edge dynamic. Forexample a randomly driven pile of sand produces Poisson distributed waiting times [50].

3 Statistical analysis

In this section, we study the blob statistics and the PDF properties of the TCV probemeasurements. We show that large amplitude peaks are present in the particle density andflux signals, with both amplitudes and waiting times Poisson distributed and uncorrelated;we show that blob amplitude, velocity and spatial extent increase together; we interpret theconditional dipolar waveform of the electrostatic potential signal as a trace of the dipolarblob structure advected by the poloidal flow, and use it to estimate this mean flow; we putaside several functional forms for the density PDF and find that the best description forits tail is an exponential; we fit the flux PDF by a stretched exponential and show that itis dominated by the blobs statistics and shapes.

3.1 Experimental data

The data studied in this report come from the TCV device (Tokamak a ConfigurationVariable) situated at the EPFL in Lausanne. Some of its characteristics are presented inappendix E, and figure E.1 is actually a precise view of its poloidal cross section, where themagnetic field-lines have been reconstructed to their actual values. Our attention is focusedon two discharges that differ only by their line averaged density. In both experiments, aLangmuir probe head was kept at a fixed position in the far SOL (at 22 mm from the LCFSand 3 mm from the wall). These data sets are outstanding by there record lengths of ∼ 1 sof exploitable signal (or ∼ 6 · 106 points sampled at 6 MHz). The line averaged density isne = 4.5 · 1019 m−3 for discharge #27601 and ne = 8.5 · 1019 m−3 for #27602.

A Langmuir probe can give access either to the particle density, the electrostatic poten-tial, or the electron temperature. The probe used here actually has three Langmuir probes,as reproduced on the simulation snapshot (figure 2.2, probe 10), aligned in the poloidaldirection. The central one is biased very negatively, so that it attracts all the ions and noelectron. The current collected (called the ion saturation current) is then proportional tothe plasma particle density. The upper and lower probe tips are left floating and adjusttheir potential in order to equalize the electron and ion loss. Such a potential, called thefloating potential, is equal to the plasma potential plus a term depending on the electrontemperature that is usually assumed to be constant. More details on Langmuir probe the-ory can be found in appendix C, but we will retain that the central probe measures theparticle density whereas the two others measure the electrostatic potential. Furthermore,the probe is not believed to perturb significantly the plasma collective dynamic because ithas a small parallel-to-the-field extent.

The plasma column was actually slowly drifting toward the probe during the discharges.We removed the induced trend in particle density and electrostatic potential by subtractinga linear regression performed on the whole signal. That the trend was indeed linear waschecked by comparison with the signal detrended using a sliding window averaging method.The fact that the density was increasing also involves an increasing level of turbulence, andconsequently an increasing root mean square value with time (of 20% for 27601 and 40%for 27602). We corrected for this trend by dividing each signal by the rms computed on asliding window of length 33 ms (a time greater than the autocorrelation time).

In the electrostatic turbulence considered here, the cross-field motion of the plasma ismostly the E×B drift (equation 2.1). This fact allows to estimate the radial velocity of theplasma simply by approximating the gradient in φ by the difference in potential betweenthe upper and lower pins of the probe divided by their separation (1 cm). The accuracyof this method will be tested in the following. We can then compute the radial particleflux as Γ = nvr where here and everywhere else a tilde on a quantity indicates a centredand normalized signal: X = (X −X)/Xrms. X is the temporal mean of X, and Xrms itsstandard deviation. See figure 3.1.

11

3.2. STRUCTURE FUNCTIONS 12

0

2

4

6

−2

−2

0

2

4

−5

0

5

10

807.4 807.6 807.8 808 808.2 808.4−500

0

500

1000

rms u

nits

rms u

nits

rms

un

its

autocorrelationtime:

thresholdwaiting time

burstduration

Figure 3.1: Short part of the normalizedsignals for the particle density, the po-tentials at upper and lower probes, andthe particle flux, for shot #27601. Theradial velocity is kept in real units. Theautocorrelation times shown are com-puted from the autocorrelation func-tions – that are exponentially decreas-ing at short time lags – as the time atwhich they cross exp(−1). The rms val-ues of the raw ion saturation and float-ing potential signals are: 4.9 A, 5.0 Vand 4.7 V for #27601; 13 A, 5.9 V and5.2 V for #27602.

3.2 Structure functions

The structure function of order q for a stochastic signal X(t) is defined as Sq(t0, τ) =〈|X(t0 + τ) − X(t0)|q〉, where 〈...〉 is an ensemble average. If the signal is stationary, orif its increments are stationary, then Sq(t0, τ) does not depend on t0 and can be easilyestimated by

Sq(kδt) =

N−k∑i=1

|Xi+k −Xi|q

N − k, (3.1)

where N is the total number of points and δt the sampling time. The saturation after theautocorrelation time is a sign that the signal becomes stationary at these temporal scales[58]. We also note that high values of q > 1 enhance the strong fluctuations of X, sothat structure functions of increasing order q reveal the statistics of increasing fluctuationlevels. A consequence is that when q is high, the number of high fluctuations contributingto Sq is low and the estimate is prone to errors. When 0 < q < 1, fluctuations smaller thanunity are increased, whereas fluctuations higher than unity are decreased. When q < 0,increasing |q| explores smaller and smaller fluctuations.

10−1

100

101

102

103

104

105

100

102

104

106

108

q=0.51

2

3

4

5

6

7

autocorrelation time

for

100

101

102

103

104

105

101

103

105

107

109

q=0.51

2

3

4

5

6

7

autocorrelation time

forFigure 3.2: Structure func-tions for the normalized den-sity and potential of #27601.For the flux, they have exactlythe same shape, without ap-parent oscillations.

3.3 Conditional window averaging

Method and results

The method of conditional window averaging has been widely used (e.g. Garcia et al.[29], Horacek [35]) and allows to identify the signature of blobs on probe measurements.This signature on n or Γ are sharp peaks, as identified on figure 3.1.

13 CHAPTER 3. STATISTICAL ANALYSIS

The method is the following. We choose a threshold value in rms units of the signal andexplore the signal (see figure 3.1). Every time it passes above this threshold, we find themaximum value reached by the signal before it passes back below the threshold. We thenstore the signal recorded over a window of fixed size centred on the maximum. In the end,we average all such events to obtain the average waveform. We also eliminate overlappingwindows. We can also calculate the waiting time between two events and the durationspent above the threshold for each event, as defined on figure 3.1, or various quantitiessuch as the amplitude of the event. The trigger can be done either on the density or onthe flux, and the potential and radial velocity corresponding to each event can also berecorded.

Our long record lengths allow for a solid statistical analysis, with for #27601/#27602:2674/2836 events for a threshold on density above 2.5rms, and 1927/1958 events for thresh-old on flux above 4 rms. Important results are the facts that for both shots, the waitingtimes between density peaks as well as flux peaks, and the amplitudes of the density peaksas well as flux peaks, all follow Poisson statistics (figure 3.3d-e). Moreover, we computedthe distribution of waiting times by taking only those following a peak restricted betweentwo thresholds. This distribution does not depend on the thresholds, which shows thatamplitudes and waiting times are uncorrelated.

We have also performed triggering for peaks restricted between two rms values. Varyingthese two values produces a scan of various properties as function of event amplitude. Thisanalysis has been performed with trigger either on the density or on the flux. We findthat in both cases, the averaged waveforms have the same functional shape (figure 3.3a-b);that the mean waiting time increases exponentially with the threshold; that the number ofevents decreases exponentially with increasing threshold; that the mean peak duration firstincreases with increasing threshold and then stabilizes at a roughly constant value (figure3.3c); and that the amplitude of the corresponding potential increases with increasingthreshold (figure 3.3f). A Poisson statistic for the waiting times is also found when onlybursts between two thresholds are kept (4-6, 6-8, 8-10 for the flux).

The results for #27601 are shown in figure 3.3. They are the same for #27602, withminor differences: (a) and (b): similar shapes of the peaks; (c): same shape of the durationcurve for density peaks but with duration ranging from 15µs at 2 rms to 35µs at 8 rms, andduration almost identical for the flux peaks; (d): Poisson distribution and mean waitingtime of 0.34 ms for n and 0.49 ms for Γ; (e): Poisson distribution and mean amplitude of3.8 for n and 8.2 for Γ; (f): same linear-like increase but up to higher values (∆φ = 4.1 at8.5 rms for n and 3.7 at 6.5 rms for Γ).

0 1 2 310

0

101

102

103

Cumulated distributionof waiting times

(number of events)

, threshold , threshold

mean:

0 5 10 20 2510

0

101

102

103

, threshold

, threshold

Cumulated distributionof peaks amplitude (number of events)

mean:

2 4 6 8 10 12 14 16 18 205

10

15

20

25

30

2364

Mean peak duration

number of events

centre th of threshold interval

in , for thresholdbetween two values

th-1/2,th+1/2

trigger on

trigger on

1 2 3 4 5 6 7 8 9

1

2

3

Potential versus density amplitudes

trigger on

trigger on

maximum of the density averaged peak

−50 −30 −10 0 10 30 50

0

0.2

0.4

0.6

0.8

1 triggeron between(in rms)

2-34-56-78-9

Averaged density peak(normalized to 1)

−20 −10 0 10 20

0

0.2

0.4

0.6

0.8

1Averaged flux peak(normalized to 1)

triggeron between(in rms)

2-36-710-1114-1518-19

(a) (b)

(c)

(d) (e) (f)

Figure 3.3: Conditional window analysis for #27601. It is very similar for #27602, seemain text.

3.3. CONDITIONAL WINDOW AVERAGING 14

Interpretation

A Poisson distribution for waiting times between large amplitude events can be derived ifthe probability of emission of a blob per unit time is constant, exactly as in the case ofradioactive decay. We also find the same Poisson statistic for waiting times selected as fol-lowing a waiting time comprised between two thresholds, independently of the thresholds.We are thus led to the conclusion that the mechanism producing blobs in the edge releasesa blob independently of the preceding or following ones. The fact that we also find the samePoisson statistic for waiting times selected as following a peak of amplitude comprised be-tween two thresholds, independently of the threshold, also suggests that waiting times andamplitudes are uncorrelated. This was not obvious from our discussion on blob generation,where we could have expected periodicities and correlations in the waiting times, and thathuge blobs formed by the detachment of the edge-based convective cells may be followedby quiet periods. Antar et al. [1] also found uncorrelated waiting times and amplitudes inseveral devices, and conclude that it is not compatible with SOC.

For all cases, the amplitude are also Poisson distributed. This fact is in contradictionwith a SOC dynamic producing the blobs in the edge, because as was discussed previously,the avalanche amplitudes in a SOC-governed system are power-law distributed. This is akey element, because at equal total radial particle flux, power-law distributed amplitudeswould involve far more large amplitude blobs than an exponential distribution, and wouldconsequently require a larger SOL to stop the propagation of these huge blobs.

We now turn to figure 3.3f. Remembering equation 2.1 and the fact that our twopotential probes are separated by a fixed distance, we see that the radial velocity of a blobis directly proportional to the amplitude of the potential (defined here as maximum minusminimum of the averaged potential corresponding to density or flux peaks). Our findinghere is that it increases with increasing amplitude. Consequently, in order to achieve aroughly constant duration (fig. 3.3c) when passing the probe, the blobs must also have aspatial size increasing with increasing amplitude. The conclusion is that amplitude, spatialextent and radial velocity all increase together. Two scalings for the blob E×B velocity infunction of its spatial extent l have been derived in the literature (e.g. Garcia et al. [25])and observed experimentally (Furno et al. [21], Theiler et al. [56]): one where v ∝ 1/l2 inthe limit where the sheath current dominates over the polarization current, and one wherev ∝

√l in the limit where there are no parallel losses and where the interchange current

is balanced by polarization currents. Our analysis indicates that we are nearest to thelimiting case of no parallel losses. It is in favour of the approach followed by ESEL.

Interpretation of the potential dipolar structure

The potential waveform obtained by averaging the electrostatic potential signal when theparticle density (or the flux) is above a given threshold value is shown on figure 3.4 forTCV and probe 10 of ESEL. It has an evident dipolar structure, that deserves someexplanation. If the blobs were only driven by their E × B velocity, then a single blobwould give either a completely positive or a completely negative signal when passing bythe probes. Consequently, on average over all the blobs, the net contribution of thisE× B velocity will vanish. All that remains is a possible poloidal motion of the blob, notaligned with its dipolar structure.

In ESEL simulations, the blobs do have a poloidal motion that is, on average, given bythe poloidal flow +y. As shown in figure 3.4-b, the probe then sees first the negative partof the dipolar blob structure, and then the positive part, exactly as found on the averagedpotential (fig. 3.4b-down). Moreover, the upper probe tip passes closer to the minimum ofthe locally negative potential, and the lower probe closer to the maximum of the positivepotential, which is again coherent with the averaged potential recorded. Also note thatthe lower probe crosses the zero potential line before the upper probe.

In the SOL of TCV (figure 3.4a), we know that the temperature decreases radiallyoutwards. The fact that we are in a sheath connected area (for the background plasma)implies that φ is also decreasing outwards (see appendix C). The resulting poloidal flowis then directed along B ∧ ∇φ ∝ −z ∧ (−x) = +y. Since the magnetic field is directedalong −z, the outward propagating blobs are polarized as indicated on figure 3.4a, and the

15 CHAPTER 3. STATISTICAL ANALYSIS

probe path

TCV

probe path

ESEL

−50 −30 −10 0 10 30 50−1

−0.5

0

0.5

1

−50 −30 −10 0 10 30 50

−0.5

0

0.5

0 50 100 150

0

0.1

0.2

0.3

ESEL, estimatedESEL, real

TCV, estimated

(a) (b) (c)

Radial distance (50 is separatrix and100 the wall shadow)

Figure 3.4: (a), up TCV case. Situation of a blob passing in front of the probe. Forconvenience, we have drawn the path of the probe as if it was moving toward the blob.Red (blue) is positive (negative) potential, and the shaded circles are density contours.(a), dn Averaged potential from TCV#27601 with trigger on Γ > 3.5rms. (b) is the sameas (a) but for ESEL probe 10. (c) represents poloidal velocities.

resulting averaged potential is exactly as predicted.

It is interesting to notice that the lag between the two probe signals can be exploitedto estimate the mean poloidal velocity of the flow. The time difference between the zerocrossings of the upper and lower probes is equal to the mean poloidal velocity multipliedby the probe separation, or vy = ∆lprobe/∆tup dn. We have performed this calculationfor all 14 probes of ESEL, with a potential waveform averaged from events correspondingto a trigger at 3.5 rms on the flux signal. The results are shown on figure 3.4c, wherethe true mean poloidal velocity is also shown. We see that this method overestimatesvy by 0 to 0.07 cs. This difference arises because the situation when there is a blob isnecessarily different from the averaged situation. It is however an interesting and simpleway to estimate the poloidal velocity from three-points probe measurements. We note thatit is also relevant in the edge where there are no blob structures, because there the lag stillreflects the time spent by the zero potential to travel from the lower to upper probe.

On TCV and for #27601, we obtain a velocity of 0.05 cs, but we do not have any directmeasurement to compare with.

3.4 Probability distribution functions

We now turn our attention toward the probability distribution functions (PDFs) of theelectrostatic potential, particle density and flux time signals. We computed all the PDFwith 1000 bins.

Potential

For the low density case (#27601), the PDF of potential fluctuations is nearly Gaussian(figure 3.5a), and the discrepancies to normality are probably due to the fact that Langmuirprobe measurements give access to the floating potential, that is dependent on Te: φprobe =φplasma − TeΛs/e (see appendix C). The effect of blobs on the particle density and on theelectron temperature signals are very similar, so that the PDF for Te is expected to have thesame shape as the particle density PDF. This effect is even stronger for the high density case(#27602, figure 3.5b) because the fluctuation level is higher. Here the potential PDF clearlybears the shape of the PDF of −Te, and it questions the approximation φprobe = φplasma.

3.4. PROBABILITY DISTRIBUTION FUNCTIONS 16

Density

The PDF for particle density fluctuations is presented in figure 3.5c to h. It presents thesame universal shape in various devices, and in the literature several attempts have beenmade to fit this PDF, some justified by theoretical arguments. The time series are howeveroften too short to allow an acceptable test, and here we take advantage of our long signalsto provide new insights.

Sattin et al. [52] have tried a Gaussian distribution at small values, extended by anexponential tail. Here we find that the PDF foot at small values is not Gaussian. Moreover,such a function has a discontinuous derivative, which is not supported by our data.

A gamma function has been considered by several authors (Horacek [35], Graves et al.[32]). In the low density case, we find here (figure 3.5c) that the tail is well fitted by thisfunction, but any attempt to fit the whole PDF fails. To fit the left part of the PDFrequires c > 1 (a fact often hidden by the exclusive use of log-lin plot, see here figure 3.5f),which is then incompatible with the behaviour of the tail: the gamma distribution cannotfit the whole density PDF. For the high density case, the tail can also be well fitted bythis function (in a smaller range than for #27601), but with b = 1 and c = −0.026. Sucha small value of c questions the relevance of the gamma function, and as we will see, anexponential tail provides a similar fit, but with only one fitting parameter (c = 0).

A log-normal function can be justified by assuming a normally distributed potential,linked to the density by Boltzmann relation ne ∝ exp(φ/eTe) (which is obviously not oftenvalid, see figure 1.2 and [57, 35, 32]). We find that the tail is well fitted by this function.A fit extended to the whole PDF shows a good visual agreement for #27601 and #27602(figure 3.5d-e-f), but the compensated PDF indicates that this is not the true functionalform of the PDF. In particular for #27602, the behaviour of the tail at large values isclearly wrong: the log-normal distribution has a too heavy tail. Besides, the peak of thePDF is too sharp to be described by this function. In conclusion, the log-normal functionis not appropriate for the density PDF.

Sattin et al. [52] have proposed a generalization of the log-normal distribution, justifiedby replacing the Boltzmann relation by a balance between the polarization, sheath andinterchange driven currents. We found here that this expression does slightly better thanthe log-normal for #27601. However, it adds a new fitting parameter, and the improvementis not good enough to justify it. For #27602, the peak is badly described (too smooth forSattin distribution). Here again, the density PDF escapes from an accurate description.

Finally, we find that the best fit for the tail in the low and high density cases is a simpleexponential (figure 1.2g).

Flux

The PDF of the flux is presented in figure 3.5i to l. No attempt of theoretical descriptionhas been found in the literature. We find here that in both cases the tail is very wellfitted by a stretched exponential (figure 3.5i), with almost identical parameters for the twodischarges. Also, the function f(x) = a exp(−x/b)/x is a qualitatively good description ofthe PDF from the peak to the tail (once the peak have been centred on zero), see figure 3.5j-k-l. Interestingly, this last expression can be derived from a signal consisting of randomlydistributed peaks of individual shape ai exp(−t/τ) (for t = 0..m × τ), of amplitudes airandomly distributed according to Poisson statistic, that is, a PDF g(a) = A−1 exp(−a/A),and separated by periods where the signal is null.

To see this, we first note that since the peaks are not overlapping, the total number of

points of the signal that are between x and x+∆x is: #tot(x, x+∆x) =∑Npeak

i=1 #peak i(x, x+∆x). Here Npeak is the total number of peaks. For a single peak i, that we assume dis-cretized with a steptime ∆t, the number of points between x and x+ ∆x is: #peak i(x, x+

∆x) = |f−1(x) − f−1(x + ∆x)|/∆t = |f−1′(x)|∆x/∆t = (τ/x)(∆x/∆t). We see thatthe exponential shape has the simplifying property of having a PDF independent of itsamplitude. Of course, #peak i(x, x + ∆x) is zero when x > ai, so that we arrive at thefactorized expression #tot(x, x + ∆x) = #ai / ai > x × (τ/x)(∆x/∆t). The first factoris the number of ai above x, i.e. the cumulated distribution of amplitudes. To obtain the

17 CHAPTER 3. STATISTICAL ANALYSIS

PDF of the signal, we divide #tot(x, x+ ∆x) by the total number of points, T/∆t, whereT is the total length. We arrive at:

PDF (x)∆x =#ai / ai > x

T/∆t× τ

x

∆x

∆t= e−x/A

τ/(T/Npeak)

x∆x. (3.2)

The PDF is zero for x below the value at which we cut the shape of the peaks (exp(−m)).

We see that at small x, the behaviour of the PDF is dominated byτ

x

∆x

∆t, i.e. by the

PDF of a single peak. This means that for the flux PDF, the peak that we see at smallpositive x would be the PDF of an individual averaged peak. The tail however, at largex, is dominated by the cumulative distribution of peak amplitudes, which is a decreasingexponential in the case of the synthetic signal and also of the flux. We learn however thatthe exact dependence of the tail is not given only by the distribution of peak amplitudes,but is corrected by a factor related to the single peak PDF, ∝ 1/x here.

A fit of the flux PDF centred on zero by f(x) = (a/x) exp(−x/b) is presented on figure3.5k-l. For the low/high density case we found b = 3.8/4.6, which is to be compared withA, the slope of the cumulated distribution of peak amplitudes (in rms value here), inde-pendently found (fig. 3.3e) to be 4.8/5.2, a close value. The value of the fitting parametera = 4.5/2.8 · 10−3 should be compared to τ/(T/Npeak), which is the mean duration of a

peak divided by the mean duration between peaks. From conditional averaging on Γ withthreshold 4 rms we have τ/(T/Npeak) = 10µs/500µs = 2 · 10−2 in both cases.

There is essentially two approximations made that can explain the above discrepancies.First, the peaks in the flux signal have a varying duration τ , especially the small ones.These small peaks are the more numerous and will greatly influence the shape of the PDFat small positive x values. The fact that τ varies prevents the factorization made whencomputing the synthetic PDF and changes its global shape. Secondly, the peaks in theflux signal are not as simple as decaying exponentials, so that their PDFs depend on theiramplitudes, which again prevents the factorization leading to equation 3.2; and gives adifferent PDF shape and thus a behaviour different from 1/x at small positive values.

Considering these facts, the above comparisons show a reasonable agreement, at leastin order of magnitude, between what would be expected if the flux PDF was explainedby a series of peaks, and what is found independently with conditional averaging. Itsuggests that the PDF is indeed dominated by the blob shapes and statistics. Of course,a third major difference between the two signals is the background signal on Γ, i.e. thefluctuations that are not the large amplitude peaks. If we model Γ by the synthetic signalplus a background signal, then its PDF would be equation 3.2 convoluted with the PDF ofthe background signal, say, PDFback. The effect on expression 3.2 is a moving local averageweighted by PDFback. We immediately see that this background signal is not significantfor the flux PDF, because its peak at x = 0 is sharp, which means that PDFback behavealmost as a Dirac function, and that its standard deviation is small.

We conclude that the PDF of the flux is indeed dominated by the blobs. The densityPDF, on another hand, is compatible with a tail in exp(−x/b)/x (figure 3.5h, with a similaragreement for #27602), but does not present a sharp peak at x = 0. It means that it hasbeen smoothed by the background signal, which is important for n. The peak is howeversharper for #27602 than for #27601 (compare 3.5g and h), which is to be expected ifthe ratio of blob amplitude over background signal amplitude (which is independent of thenormalization used for the signal) is higher in the high density case. The background signalis the turbulence created by the passing blobs, their trailing wakes, and also by other sourcesof fluctuations (the “usual” turbulence). This turbulence results in a Gaussian potential,that is usually the starting point of the derivation of analytical PDF expressions (e.g.log-normal, Sattin). Here, we underline that our point of view is the opposite: we modelthe PDF shapes by starting from the blob properties and distributions, that can then becorrected for effects of this background signal. The blobs are absent from a descriptionstarting from a Gaussian potential, because they do not appear in the potential PDF (orvery weakly, since their conditional waveform is not a peak), so that such descriptionscould not explain the flux PDF shape.

Finally, the negative part of the flux PDF can be similarly explained by the inward fluxevents that result, on the flux time signal, in negative peaks.

3.4. PROBABILITY DISTRIBUTION FUNCTIONS 18

−6 −4 −2 0 2 4 6

10−6

10−4

10−2

normal law

#27602

(b)

−4 −2 0 2 4 6

10−6

10−4

10−2

normal law

#27601

(a)

−2 0 2 4 6 8 10

10−6

10−4

10−2

2

1

0.5

5

#27601(c)

Gamma

−2 0 2 4 6 8 10

10−6

10−4

10−2

2

1

0.5

5

#27601(e)

log-normal

−2 0 2 4 6 8 100

2

4

6

x 10−3

#27601(f)

log-normal

−2 0 2 4 6 8 10

10−6

10−4

10−2

1

2

#27601(g)

exponential

(for #27602, )

−2 0 2 4 6 8 10

10−6

10−4

10−2

1

2

#27601(h)

exp over x

−10 0 10 20 30 40

10−6

10−4

10−2

100

1

2

0.5

#27601(k)

exp over x

−10 0 10 20 30 40

10−6

10−4

10−2

100

12#27602

(l)

exp over x

−2 0 2 4 6 8 10

10−6

10−4

10−2

210.5

#27602(d)

log-normal

−10 0 10 20 30 40

10−6

10−4

10−2

100

1

2

0.5

#27601(j)

exp over x

−10 0 10 20 30 40

10−6

10−4

10−2

100

1

2

0.5

#27601(i)

stretchexp.

(for #27602,

)

Figure 3.5: PDF of normalized signals with some fits. The fits are performed over the areabetween the vertical black lines. Also shown is the compensated PDF, i.e. (data)/(fittingfunction).

4 Self-similarity

In this section, we explore the self-similar properties of our signals. The concept of self-similarity is briefly introduced in appendix D, and here we first focus on the motivationsthat led to this concept in plasma physics. We then show the absence of long-rangecorrelations in the low-density case, and the presence of weak long-range correlations forthe high-density case; compare the self-similar properties of TCV and ESEL data and findgood agreement; and end up with a discussion on the accuracy of the R/S analysis.

4.1 Motivations

The concept of self-similarity has been a success story for turbulence in neutral fluids,where the velocity increments are self-similar with respect to space (see e.g. Falkovich andSreenivasan [17]). It has led to the concept of mono and multi-fractality, and has helpedfor a statistical understanding of the flow. There have been some attempts to apply thesetools to plasma physics (e.g. Carreras et al. [12], where a multifractal behaviour have beenfound for the local turbulent range and a monofractal one for the meso-scales), but thesituation here is even more complex than in hydrodynamics. The 2D situation allows for adirect and inverse cascade, and the forcing is not localized in k-space. Dissipation occursby collisions at small scales, but also by parallel damping at all scales. These two factsreduce a possible inertial range with power law scalings. And even if an inertial rangeis present, there are several non-linearities (drift waves, interchange) that can cascadeseveral conserved quantities such as energy and enstrophy. Another difficulty is the failureof Taylor hypothesis: turbulence structures can pass in front of the probe, but also begenerated here. Finally, dimensional analysis is no more univocal because the magneticfield introduces both another time-scale through the Alfven velocity and an anisotropy;and a wide range of dimensional models can be thought of [53].

This is not the path that we will follow. Our time signals are stationary, and cannotbe self-similar. We will simply use some tools to look for long range temporal correlations,as described in the following. This approach has been followed by several authors thathave found a self-similar exponent H ranging from 0.62 to 0.75 in the edge (Carreras et al.[10, 9, 11]) and from 0.52 to 0.92 in the SOL (Carreras et al. [13, 10, 9]) of various devices,and have interpreted this presence of long-range correlations as a signature of a dynamicgoverned by self-organized criticality.

4.2 Presentation of the concepts

A discussion of the tools used to search for self-similarity and some demonstrations, as wellas a definition of self-similarity, are presented in appendix D. Here we only sumarize theessential ideas. We want to study a stationary temporal signal X(i δt) (for example particle

density or flux), and to do so we form the cumulated signal Y (kδt) =∑ki=1X(iδt). Y is

then a walk starting from 0 (Y (0) = 0). The study of Y then gives us informations on itsincrements X. For example, if Y is a random walk or Brownian motion, then we know thatX is uncorrelated. More generally, and as is shown in appendix D, if Y is H-self-similarwith 0 < H < 1 then we know that its standard deviation increases as tH and that: (i) IfH = 1/2, Y is a random walk. (ii) If H > 1/2, Y (t) will tend to deviate more than therandom walk, which means that its increments have a tendency to stay positive when theywere positive, and vice versa: X is positively long range correlated, and its autocorrelationfunction decreases as t2H−2. (iii) If H < 1/2, Y will tend to stay in an area of smallerextent than if it was a random walk, which means that its increments have a tendencyto become positive when they were negative, and vice versa: X is negatively long range

19

4.3. RESCALED RANGE AND CUMULATED STRUCTURE FUNCTIONS 20

correlated. Its autocorrelation function also decreases as t2H−2.

The fact that X has long range correlations is called Joseph effect by Mandelbrot [43], inreference of the seven years of rain followed by seven years of drought in the biblical event.It has no incidence on the PDF of the signal. For other biblical allusions, high amplitudeincrements during short periods of time are called Noah effect, and result in heavy tails.These two effects are independent. Our particle density and flux signals clearly exhibit theNoah effect, but we do not know for the Joseph effect.

We will mostly use two diagnostics to search for self-similarity in the cumulated signalY . The first one is the rescaled range, or R/S analysis, widely utilized in plasma physics.If Y is H-self-similar, then R/S(τ) ∝ τH . This method is very robust against the presenceof coherent periodic modes in the signal, and insensitive to large amplitude peaks [44]. Thesecond is the structure functions of Y , that follow Sq(τ) ∝ τ ζ(q) with ζ(q) = Hq. A plot ofζ(q) then gives H. When the signal is multi-fractal (ζ(q) curved), the exponent H to beused in the autocorrelation expression for X is H = ζ(2)/2 (appendix D for more details).

4.3 Rescaled range and cumulated structure functions

4.3.1 Analysis of TCV data

We performed the R/S analysis on the TCV signals (figure 4.1 and 4.3). The result is aself-similar range extending from 1 ms to 100 ms for all signals and both particles densitycases. For the low-density case, the H-exponent is around 0.55 and indicates the absenceof long-range correlations. For the high density case, it is around 0.6 and points towardweak long-range correlations. See table 4.1. The linear part at times shorter than 0.3 mshas a slope close to one. The breaking point between these two regimes is around 450µs.

The different time-scales indicated by an R/S curve are however to be taken withcaution. Figure 5 of Gilmore et al. [31] shows that the breaking point above the part whereH ∼ 1 is overestimated by a factor ∼ 10, which brings us closer to the autocorrelation timeof the signals. It indicates that the region at small lags represents the local turbulencetime-scales, from times up to the autocorrelation time, where the signal is not stationary.This non-stationnarity induces a H-exponent close to one. A breaking point at the end ofthe self-similar range where H ∼ 0.5 would also be overestimated by a factor ∼ 2.

101

102

103

104

105

101

102

103

104

105

mean waiting time

Figure 4.1: #27601. R/S curves for poten-tial, particle density and flux. The indicatedmean waiting time between bursts is for trig-ger on the density at 2.5 rms (0.45 ms). Theself-similar range, where the linear regressionis performed, extends from 1 ms to 100 ms.Numbers indicate the corresponding slopes.Except for their slopes, reported in table 4.1,the R/S curves for #27602 are very similar.

The structure functions do not possess this memory effect and give accurately thedifferent breaking points [31]. The structure functions for the cumulated signals are shownon figures 4.2 and 4.3. They are quite smooth for the density and the potential, where theyare composed of three parts: (i) the first, from lags 0 to ∼ 40µs ∼ 2τ , gives H ∼ 1 andis the local turbulence region. Note that the breaking point is indeed 10 times less thanit was on the R/S curves. (ii) The second, from 40µs ∼ 2τ to ∼ 40 ms gives H around0.55 at low density and around 0.6 at high density. It is the meso-scale self-similar rangewhere there is no or only weak long-range correlations. (iii) The third, above ∼ 40 ms, isthe global range where the signal is no more self-similar.

In the case of #27601, the structure functions for the cumulated flux and radial velocityare not smooth and do not allow to make a satisfying linear regression (figure 4.3). The

21 CHAPTER 4. SELF-SIMILARITY

10−1

100

101

102

103

104

105

100

108

1016

1024

for

q=0.512

3

4

5

6

7

100

101

102

103

104

105

100

108

1016

1024

q=0.51

2

3

4

5

6

for

100

101

102

103

104

105

100

108

1016

1024

q=0.512

3

4

5

6for

0 2 4 60

1

2

3

4Slopes of the structure functions

10−1

Figure 4.2: #27601.Structure functions ofthe cumulated signals,and (bottom-right)slopes ζ(q) of the linearregressions shown inorange on the threeother plots. Here thethree numbers are theslopes of the linearregressions performedon the curves ζ(q) (inthe range q = 0.5-5 forn, 0.5-7 for φ, and 0.5-2only for Γ). The verti-cal black lines are theautocorrelation timesand mean waiting times.Similar for #27602.

27601, R/S SF∑

27602, R/S SF∑

n H = 0.55 0.55 H = 0.58 0.62

φ H = 0.55 0.57 H = 0.56 0.56vr H = 0.46 H = 0.58 0.48

Γ H = 0.52 0.53 H = 0.59 0.53

Table 4.1: Self-similarity exponentsobtained with R/S curve and cumu-lated structure functions for the low(#27601) and high (#27602) densitycases.

case of vr is however interesting because it presents the clear signature of a coherent modeof period 0.63 ms. This fact is confirmed by a zoom in the R/S curve of vr. Yu et al.[58] have superposed a sinusoidal signal to a fractional Gaussian noise and indeed find thispattern for the cumulative structure functions and the R/S curve. See also Mandelbrotand Wallis [44] for R/S. The irregular behaviour of the cumulated flux structure functioncan then be explained by the fact that Γ ∝ vrn: the coherent mode is mixed by the particledensity signal. Note however that the R/S curves are not really perturbed and are stillconclusive. The origin of this coherent mode is not really known.

10−1

100

101

102

103

104

105

100

104

108

1012

1016

1020

1024

10−1

100

101

102

103

104

105

100

101

102

103

104

105

106

107

108

q=0.51

2

3

4

5

6

7

autocorrelation time

for

for

q=0.51

2

3

4

5

6

101

102

103

104

101

102

103

104

103

103

for

Figure 4.3: #27601. Case of the radial velocity. For #27602, the problematic oscillationsare not present.

In conclusion, the analysis of the rescaled range and of the cumulated structure functionsare coherent and indicates no long range correlations at low density (H ∼ 0.55), and weaklong range correlations at high density (H ∼ 0.6). We notice that the mean waiting timefalls at the beginning of the self-similar range. It seems to indicate that the physics of thisrange is governed by the blobs, and more specifically by their waiting times. If it is indeedthe case, and since the blobs are uncorrelated with each others, then it is not surprising tofind almost no long range temporal correlations. If it is not the case, then the weak longrange correlations that we see come from the background signal and are not really relevantfor transport problems.

4.3. RESCALED RANGE AND CUMULATED STRUCTURE FUNCTIONS 22

4.3.2 Analysis of ESEL data

We also performed the R/S and cumulated structure function analysis for the potential,particle density and flux and radial velocity of each of the fourteen probes of the ESELsimulation run 116. The results are presented in figure 4.4.

We first note a discrepancy for the potential between results from R/S and structurefunctions. The structure function and R/S curves both present a satisfactory range wherethe linear regressions can be safely performed, and ζq is indeed linear, but it is unclearto decide which method is right. There is however a real problem with ESEL potentialstructures: they appear to be larger and smoother compared to TCV measurements. Itcan be seen either on the potential time signals, or on the conditionally averaged potentialstructures of figure 3.4. It may result from the fact that sheath dissipation is not includedin ESEL: it would then imply a damping of large scales and lead to smaller potentialstructures, as can be seen in Sarazin [51], Garcia et al. [25], Bisai et al. [8].

Secondly, we note an excellent agreement between ESEL and TCV #27601 for the H-exponent of the particle density, flux and radial velocity. This is in favour of the dynamicdescribed by ESEL.

Thirdly, the flux and radial velocity in ESEL can be obtained either directly from theprogram output, or estimated in the same way as in the experiment by subtracting thepotential from two probes. These two probes are represented around probe 10 on figure2.2, and are distant from 1 cm in real units (same distance as is the experiment). Theanalysis have then been performed on these estimated flux and radial velocity signals, andare reported in orange on figure 4.4. We see a very good agreement. We note however thatbecause the potential has a finer structure in the experiment, this estimation may be lessprecise for TCV.

Finally and for information, the black squares represent the range of values found byCarreras et al. [11, 13, 10, 9] with R/S analysis of data in the edge of several devices.

structure function

R over S

ESEL

structure function

R over S

TCV

same, but flux of radialvelocity estimated bypotential difference

Range of values foundby Carreras et al.

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

separatrix wall shadow

SOL

H-exponents

probe numberprobe number

Figure 4.4: Radial variation of H-exponents. See legend inside the figure.

4.3.3 Reliability of the rescaled range

The length of our time series allows for a test of the accuracy of the rescaled-range analysiswhen it is used on short time series. This is often done in the literature (e.g. [11, 13, 10, 9]use time series of at most 20 ms) for technical reasons. We have cut the density signal of#27601 in smaller parts of fixed size ∆t and performed an R/S analysis on each sample.The results are shown in figure 4.5.

We learn that when the sample size is short, the mean of the H values computed forseveral samples gives an accurate estimate of the real H, but that the standard deviationcan be large. In particular, a single estimate for samples of short length is very prone toerrors. A histogram of the H-exponents obtained from an analysis of several samples of the

23 CHAPTER 4. SELF-SIMILARITY

101

102

103

0.4

0.5

0.6

0.7 86

5628 36

3331 24 221

50%60%70%80% 90%

4.3 5.1 9.7 14 18 25 43 75 210

andquantiles

sample size

overlapping

# of samples

end of fiting area (ms)

Figure 4.5: H-exponent as function of samplelength. The blue curve is the mean performedover all the R/S analysis, the red ones themean plus or minus the standard deviation,and the black ones the 10% and 90% quan-tiles. We also show the number of samples,the overlapping between samples (0% whennot indicated), and the end of the fitting areaon the R/S curves (it always starts at 1 ms,as on fig. 4.1). The variance has always beenchecked to converge by a plot of variance ver-sus number of samples. It may however bereduced by overlapping between samples.

same length shows that it is normally distributed. Another issue is the range over whichthe fit is performed on the R/S curve. The upper value is typically the total sample lengthdivided by ∼ 5, but the choice of the lower value is not evident for short sample length,and it is often tempting to push the fit down in the local turbulence range, which resultsin an overestimate of H.

We have also tested the effect of a reduction of the sampling frequency of the signal.There is no significant changes when it ranges from 6 MHz to 300 kHz. This is not surpris-ing, because the meso-scale is concerned with long time scales, and it shows that the keyparameter is the sample length. More specifically, the parameter controlling the uncer-tainty on H is the length of the fitting area on the R/S curve, a length that is constrainedby the sample duration. For example, we computed the H-exponent from the rescaledrange analysis performed on 100 white noise realizations generated by Matlab. The lengthof the signals (∼ 3 · 106 points) was equal to that of the experimental data. A fit for eachrealization on the whole R/S curve gives H = 0.55 with a standard deviation of 7.9 · 10−4;but a fit on the upper half curve (as is actually done for TCV data, see figure 4.1) givesH = 0.50 with a standard deviation of 1.3 · 10−2. The differences between the mean valuesof H may be due to imperfections in the white noise algorithm and are not really relevant,but the standard deviations support our argument.

Finally, the rescaled-range function R/S(∆t) can be computed with or without overlap-ping between the sub-samples of length ∆t. We have found that a 50% overlapping givessmoother R/S curves, but without changing the final result, and no further ameliorationfor higher overlaps. We have computed all our R/S functions with this value.

5 Conclusion

We now conclude with a summary of the essential findings of this report, and discuss someperspectives.

The study of the ESEL snapshots has allowed for an understanding of blob propagation:detachment from the edge convective cells, and drive by the E × B velocity and plasmapoloidal flow. We have seen that the poloidal component of the blob E×B velocity is givenby the sign of the gradient of the poloidal background plasma flow and that it is always soas to support the shear. A study of fast camera imaging from the edge and SOL regionswould be useful to see whether this dynamics persists in experiments.

We have interpreted the dipolar waveform obtained by conditional window averaging ofthe electrostatic potential on the upper and lower probes: it is a trace of the dipolar poten-tial blob structure advected by the plasma poloidal flow. We have deduced a new methodto estimate this flow. It results in a slight overestimation for ESEL data, and it would bevery interesting to compare it to other existing tools for poloidal flow measurements.

Conditional window averaging has shown that the bursts in particle density and fluxsignals have a shape and duration roughly independent of the amplitude threshold. A linkbetween blob velocity, amplitude and spatial extent has also been found: they increasealtogether, which indicates that sheath currents are not dominant.

Waiting-times are Poisson distributed and not correlated with burst amplitudes orbetween them. It indicates that the blobs are produced independently. This absence ofcorrelations or of clustering has implications for the design of SOLs and plasma facingcomponents.

We have shown that gamma, log-normal or generalized log-normal distributions are notsuited to fit the particle density PDF. We have interpreted the flux PDF as shaped bythe blob waveforms and by their amplitude distribution. More specifically, it is to firstapproximation the product of an averaged blob PDF and of the cumulated distribution ofblob amplitudes. Since the flux is the ultimate quantity relevant for transport, this newapproach shows that derivations of PDFs from relations between physical quantities and aGaussian potential, where the blobs are absent, are of limited interest.

At low and high line averaged densities, the facts that blob amplitudes are exponentiallydistributed and that neither the particle density PDF and nor the flux PDF in the SOLhave power-law tails preclude dynamics governed by the SOC paradigm in the edge, atleast under the assumption that the blobs are a product of this dynamics. If this lastassumption does not hold, then there could be a SOC-like dynamics in the edge; but if theSOL blobs, responsible for most of the transport, are not linked to this part of the edgephysics, then the SOC behaviour of the edge would be of little importance. Said otherwise,if the Noah signature of a possible SOC edge physics is lost in the SOL (by an unknownprocess), then the SOC dynamics is not relevant for transport problems and plasma-wallinteractions.

We underline once more that the PDF tails and blob statistics are of vital concern forthe design of future devices, because a given fixed amount of heat and particles released by aconstant flux (no blobs), or by exponentially distributed blobs, or by power-law distributedblobs (SOC), will lead to completely different instantaneous fluxes in the SOL.

The Joseph signature of SOC could be long-range temporal correlations for the timesignals. We have found weak long-range correlations for both low and high line-averageddensity, a fact not conclusive and that actually has little influence on the flux.

Finally, we have shown that the range and length of the fitting area on R/S curves giverespectively an eventual bias and the uncertainty for H. We performed an estimation ofthis uncertainty that may be used in future works.

24

A Reminder of plasmaphysics

We here briefly summarize the essential physics needed for the understanding of this report.The interested reader may refer to the books of Chen [14, chap. 1-3, 5], of Bellan [5, chap.1-3], or to the excellent online course of Fitzpatrick [18, chap. 1-3].

Screening and quasi-neutrality

A plasma is a gas of electrons, ions, and neutrals. We will only consider fully ionized andelectrically neutral plasmas with one ion species, that is, plasmas with no neutral particlesand where the electron and ion particle densities are equal on average: ne ' ni. Themeaning of this ”almost equal” is that the difference between ne and ni indeed creates theelectric field responsible for part of the dynamics of the plasma through Poisson’s equation,but that this difference, if taken into account, leads to terms of the order of (λDk)2 thatare negligible (k is a wave vector and λD is Debye length, defined below)∗. We will denoteboth densities by n.

One of the essential features of a plasma is its ability to stay neutral. If a charge isinserted in the plasma, the particles of the plasma will immediately react to form a cloudsurrounding the charge, such as to cancel the charge difference. The plasma will then benon neutral only inside that shielding cloud, which can be shown to be of radius†

λD =

√ε0T

ne2= 10µm×

(T/20eV

n/1019m−3

)1/2

. (A.1)

The screening is not perfect (λD > 0) because of the thermal motion of the particles:potentials of the order of T/e will not been screened because they are too weak to maintainthe cloud in place.

The plasma will be quasi-neutral only if this shielding is effective. If L is a characteristicvariation length-scale, then this requires L ≥ λD.

Another required condition for the validity of the theory used here is that the plasmamust behave collectively. This is true if there is a large number of particles in the De-bye sphere participating to the shielding process. This number is given by the plasmaparameter:

Λ ≡ n 4

3πλ3

D = 5 · 104 × (T/20eV)3/2

(n/1019m−3)1/2 1. (A.2)

Relevant frequencies

There are two relevant frequencies for our work. The cyclotron frequencies are defined by

Ωi =ZeB

mi= 5.7 · 107Hz×B/1.2T and Ωe =

eB

me= 2.1 · 1011Hz×B/1.2T. (A.3)

The period of revolution of a particle s around a field-line is then 2π/Ωs.

In plasmas where Λ 1, the dominant collision process is grazing Coulombic collisions.The mean time for a 90° deviation of an electron by an accumulation of grazing collisions

∗To see this, one has to write Poisson equation with ∇2 ∼ k2 and Φ ∼ T/e (which is indeed the orderof magnitude of the potential that leaks out of the Debye cloud, see text).†The numerical values used in this appendix approximately match the conditions of the experiments

studied in this report. In particular, mi = 2mp = 3.35 · 10−27kg and Z = 1. More values can be found inappendix E. Note also that here and everywhere else the temperature is expressed in energy units.

25

26

with ions is given by (with Braginskii closure scheme):

νei =

√2Z2e4 ln Λ

12π3/2ε20

n

m1/2e T

3/2e

= 3.5 · 106Hz× n/1019m−3

(Te/20eV)3/2. (A.4)

We also have νee ∼ νei, νii ∼ νei/43 and νie ∼ νee/1836. We see that our plasma is verymagnetized, because a particle gyrates many times between two collisions (Ωe, Ωi νei).

Description of a plasma with the two-fluid model

A plasma can be described either with a fluid description or by considering the individualparticle motions. The two pictures are roughly equivalent, see e.g. Chen [14], or in thecontext of tokamaks Garcia [30]. Here we will focus on the fluid description, sometimesmentioning the equivalent individual motions.

There are essentially three levels of fluid description of a plasma. The first one is thekinetic description, where an equation on phase space is considered (Vlasov equation com-pleted with an appropriate collision operator), along with Maxwell’s equations where thefields are self-consistently created by the particles. This description is precise, complicated,and needed only when details on the particle velocity distribution are relevant.

The second level is the two-fluid equations, obtained by taking the three first momentsof the kinetic equation closed by a given closure scheme. Braginskii closure is often used (seee.g. Fitzpatrick [18, chap. 3]), and is valid when the velocity distribution is Maxwellian.This is the level that catches the essential physics of our problems while being reasonablysimple. We briefly describe it here.

The two-fluid continuity and momentum equations are (with (s, s′) = (i, e) or (e, i)):

0 = ∂tn+∇ · nvs, (A.5a)

msn(∂t + vs · ∇)vs = qsnE + qsnvs ∧B−∇ps −∇ · πs −msnsνss′(vs − vs′). (A.5b)

Here vs is the fluid velocity of species s, ps = nTs is the pressure and πs is the gyroviscousstress-tensor that represents the deviation of the pressure tensor from isotropy. It accountsfor self-collisions as in a neutral gas, and also for the fact that particles are sort of delocalizedby their gyration motions around the field-lines, an effect that tends to homogenize thefluid on a scale of a Larmor radius and acts as a collisionless viscosity (called gyroviscosity).We will mostly ignore it in the following.

The last term in A.5b accounts for collisions between ions and electrons. It acts as afriction term. It is normally completed by other terms due to temperature gradients, suchas the thermal force, but we neglect them in this simple description (see [18, chap. 3]).

The parallel and perpendicular motions are solved for separately. We see that theparallel dynamic is unaffected by B simply by taking the scalar product of A.5b with b (aunit vector in the direction of B). To obtain the perpendicular dynamic we take the crossproduct of A.5b with b and use b ∧ (vs ∧ b) = vs⊥:

v⊥s =E ∧ b

B︸ ︷︷ ︸vE

+b ∧∇psqsnB︸ ︷︷ ︸vDs

+msνss′

qsBb ∧ (vs⊥ − vs′⊥) +

ms

qsBb ∧ (∂t + vs · ∇)vs︸ ︷︷ ︸

vPs

. (A.6)

The fluid is not restricted to move only along b, but has also perpendicular drifts.

The first term on the right-hand side is the E× B drift. It is the same for electronsand ions, and is responsible for most of the perpendicular displacement of the plasma.It also arises in the particle picture and follows from the acceleration or decelerationby E of a particle during its gyration around b, that results in a varying gyroradiusand a net drift.

The second term is the diamagnetic drift. It can be shown to be the sum of amagnetization drift, uM = (∇∧M)/qsn where M is the density of magnetic moments,and of the gradient and curvature drifts.

27 APPENDIX A. REMINDER OF PLASMA PHYSICS

The magnetization drift is here because if, at a given point, there are more gyratingpositively charged particles to the right than to the left, then we see a net flux ofparticles going up. But it is a purely fluid effect: no gyrating centres are moving.The consequence is that it does not transport matter (∇·(nuM ) = 0) nor momentum((uM · ∇)vs cancels with non-collisional terms of the gyroviscous stress tensor) norheat (see appendix B) nor anything in any moment of the kinetic equation. This factis called diamagnetic cancellation.

The gradient and curvature drifts arise in the particle picture as:

u∇B =T⊥qsB

b ∧∇ lnB and uCB =T‖

qsB∇∧ b =

T‖

qsBb ∧ κ, (A.7)

where κ = n/R is the inverse of the curvature vector of the field-line, T⊥ and T‖are the perpendicular and parallel temperatures. The gradient drift results fromthe variation of the gyroradius (with B) along the motion, and the curvature driftis similar to the E × B drift where the electric force is replaced by the centrifugalacceleration.

The third term is a drift due to collisions: when they collide, particles jump from afield-line to another and drift perpendicularly. Contrary to the case of a neutral gasor to parallel diffusion, perpendicular diffusion is enhanced by collisions.

The fourth term is the polarization drift, and accounts for inertia effects.

Equation A.6 is implicit in vs, and to obtain the velocity we use an iteration scheme.If ω is the characteristic pulsation of the dynamic, then the collision and polarizationdrifts are respectively a factor ω/Ωs and νei/Ωs smaller than the E × B and diamagneticdrifts. We thus express them using vE + vDs wherever vs⊥ appears. This is called thedrift approximation, valid only in magnetized plasmas where particles revolve many timesaround a field-line during a significant evolution of vs and between two collisions. Note thatvs‖ also appears in the polarization drift, so that the parallel and perpendicular dynamicsare actually coupled.

Finally, an energy equation must be considered to describe the temperature. See ap-pendix B.

Magnetohydrodynamics

Let us finally have a word on the third level of description: magnetohydrodynamics (orMHD). Here the two-fluid momentum equations are added to obtain an equation for thecentre-of-mass fluid, and subtracted to obtain an equation on the current. The two-fluidcontinuity equations are also subtracted. Under specific assumptions (ω Ωi and vD vE), we arrive at:

ηJ = E + U ∧B, (A.8a)

mn(∂t + U · ∇)U = −∇P + J ∧B, (A.8b)

0 = ∇ · J, (A.8c)

with P = pe+pi, m = me+mi, mU = (meve+mivi), J = ne(vi−ve), and η = meνei/ne2

(the resistivity). We see by crossing the first equation with B that the perpendicularvelocity is given, except for a small collisional diffusive term, by the E×B drift. The MHDordering thus consists in setting vs⊥ ' vE . By crossing the second equation with B welearn that the MHD current is the sum of the diamagnetic current b ∧ ∇P/B and of theE× B polarization current (mn/B)b ∧ dU/dt.

MHD is well suited for studying equilibrium in complex geometries, such as the globalequilibrium of the plasma in a tokamak. However, it does not describe correctly the paralleldynamic‡, which can be very important in the edge of tokamaks, nor various instabilitiesinvolving velocities smaller than the sound speed. This is why we do not use it here.

‡A first glimpse into this fact is to assume η = 0 and to notice that A.8a involves E‖ = 0, thus forbiddingparallel currents (which may be needed even by A.8c).

B A derivation of ESELequations

We present a derivation of the equations used in ESEL (see also Garcia et al. [24]), andprove equations on energy integrals relevant for the dynamic discussed in section 2.1.

Curvature operator

We introduce the curvature operator, which represents the curvature and gradient drifts:

C(f) = B−1(∇∧ b−∇ lnB ∧ b) · ∇f. (B.1)

This expression can be simplified if the diamagnetic response of the plasma is neglected,using Faraday’s law ∇∧ (Bb) = µ0J: the major contribution to the current is the diamag-netic current (this can also be seen from the equilibrium relation ∇p = J ∧ B), of orderp/(BLp); whereas the left-hand part of Faraday’s law is of order B/LB . Thus, if the plasmais strongly magnetized (β ≡ p/(B2/2µ0) 1), then∇∧(Bb) ' 0, i.e. ∇ lnB∧b = −∇∧b.The gradient and curvature effects add up to give

C(f) = 2B−1(∇∧ b) · ∇f = 2B−1(b ∧ κ) · ∇f = − 2

BR

∂f

∂y. (B.2)

We have used ∇∧ b = b ∧ κ, κ = −x/R, and y as the poloidal direction (see figure 2.2).

Continuity equation

Simple calculations then give (with s = i or e):

∇ · vE = C(φ) and ∇ · nvDs = C(ps)/qs. (B.3)

In the second equation we see the first effect of diamagnetic cancellation: only termsinvolving curvature of B appear. We take the continuity equation A.5a for electrons, withto first order ve = vE + vDe, and using B.3 we easily arrive at our first equation:(

∂

∂t+ vE · ∇

)n+ nC(φ) + C(nTe)/e = 0. (B.4)

The first term tells that particle density is advected by the E× B flow alone. The secondand third terms account for compressive effects.

Energy equation

We consider the electron fluid. The general energy equation is derived by taking the secondmoment of the kinetic equation, and the heat flux is obtained by Braginskii closure schemefor a collisional magnetized plasma (Bellan [5], Fitzpatrick [18]):

3

2n

(∂

∂t+ ve · ∇

)Te + nTe∇ · ve + πe · ∇ve +∇ · qe⊥ = W, (B.5a)

qe⊥ = −κ⊥∇⊥Te −5nTe2eB

b ∧∇⊥Te. (B.5b)

We momentarily ignore the first term of B.5b (self-collisional diffusion). The third termof B.5a involves the gyroviscous stress tensor and represents heating due to Coulombicself-collisions among electrons (collisionless terms are orthogonal to ve and do not produceheat). We neglect it. The term W is heating by ion-electron collisions, and is also ignored.

28

29 APPENDIX B. A DERIVATION OF ESEL EQUATIONS

We then use the following exact equations:

nvDe · ∇Te = −(Te/eB)b · ∇n ∧∇Te,nTe∇ · vDe = −(Te/e)C(nTe)− (Te/eB)b · ∇n ∧∇Te,∇ · qe⊥ = −(5nTe/2e)C(Te) + (5Te/2eB)b · ∇n ∧∇Te,

(B.6)

and inject them into B.5 to obtain equation B.7. We see that diamagnetic cancellationis here again: every contribution of vDe not due to magnetic curvature cancel with thenon-collisional parts of the heat-flux qe⊥.

3

2(∂t + vE · ∇)Te + TeC(φ)− Te

enC(nTe)−

5Te2e

C(Te) = 0. (B.7)

Vorticity equation

Here we need some approximations to have a simple picture. We assume that the electronshave zero mass, so that their polarization drift is zero, and that the ions are cold, sothat their diamagnetic drift is zero. We keep their polarization drift, expressed with theE× B velocity. Equating the ion and electron continuity equations then leads to:

∇ · (nvE + nvDe) = ∇ · (nvE + nvPi). (B.8)

The E×B velocities cancel. We use equation B.3 for the left-hand side. For the right-handside, we first neglect the variations in density, because they are not linked to a relevantprocess. Next, in ∇·vPi appears ∇⊥ · [(D∧∇⊥φ) ·∇⊥](D∧∇⊥φ) with D ≡ b/B 6= cst.It has a non trivial expression that would require another annexe but, if D = cst, itis not very difficult to show the exact equality ∇⊥ · [(D ∧ ∇⊥φ) · ∇⊥](D ∧ ∇⊥φ) =[(D ∧ ∇⊥φ) · ∇⊥](D ∧ ∇2

⊥φ). It means that if D is not constant, terms involving one ormore derivatives of D and three or less derivatives of φ will appear. Since the magnetic fieldis slowly varying (k⊥R 1), these terms will be negligible compared to the first one (thatinvolves four derivatives in φ). Consequently, we arrive at ∇ · vPi = −(mi/eB

2)d∇2φ/dt.Now with Ω ≡ ∇2φ/B we obtain:(

∂

∂t+ vE · ∇

)Ω− eB

nmiC(nTe/e) = 0. (B.9)

We have grasped the essential physics, where the divergence of the diamagnetic current(the second term), via magnetic inhomogeneities, accumulates charges. Quasi-neutrality isthen insured by a compensating divergence of the ion polarization current (the first term),which creates vorticity and thus advects the polarized density perturbations.

Perpendicular diffusion and parallel advection

Finally, a right-hand side accounting for collisional perpendicular diffusion and paralleladvection is added to equations B.4, B.7 and B.9, of the form ΛX = κX∇2X −X/τX . Asdiscussed in section 2.1, only parallel advection – and not sheath dissipation – is consideredfor Ω.

Remark

One may wonder why only the E× B velocity advects the scalar fields, and not a velocitylinked to the inhomogeneous magnetic field that indeed appears in the particle description(equation A.7). The reason is that an inhomogeneous magnetic field contributes to thefluid velocity only if the pressure tensor is anisotropic. This can be seen by replacing ∇psin equation A.5 by a simple anisotropic pressure tensor: (p‖ − p⊥)bb + p⊥I. The samereasoning than that following equation A.5 then leads to equation A.6 with a supplementaryterm (qnB)−1(p‖−p⊥) b∧κ, which is a drift due to magnetic field curvature, present onlyif p‖ 6= p⊥. Also, as noted by Chen [14], the ∇B drift never involves a fluid drift. Theparticle description would actually lead to the same result: there are cancellations whenadding uM + u∇B + uCB that indeed give the diamagnetic drift plus the new term inp‖ − p⊥ introduced here. See Garcia [30] for more details.

30

Energy integrals and oscillation-relaxation dynamic

In order to arrive at equations describing the evolution of the energy integrals K and Uintroduced in equation 2.2 and to have a better understanding of the discussion there, wehave to use some approximations. We first assume Te = cst, so that the heat equationis not needed anymore. We reduce equation B.4 to (∂t + vE · ∇)n = 0, and linearizen = n0(x) + n(x, y, t) with n0(x) ≡ n00 exp(−x/Ln). We also explicit C in equation B.9.We then obtain: (

∂

∂t+ vE · ∇

)n+

n00

LnB

∂φ

∂y= 0,(

∂

∂t+ vE · ∇

)Ω +

2Ten00miR

∂n

∂y= 0.

(B.10)

Manipulations of B.10 then lead to the evolution equations for the energy integrals:

dK

dt= ζ

∫dxnTevx +

∫dxv0

∂

∂x(vxvy)0 −

∫dxφB−1ΛΩ, (B.11a)

dU

dt= −

∫dxv0

∂

∂x(vxvy)0 −

∫dxφ0B

−1ΛΩ. (B.11b)

The last term of equations B.11 are just dissipation by collisional diffusion. The first termin the right hand side of equation B.11a means that the fluctuations are driven by radialtransport of heat, at a rate proportional to the instability parameter ζ ∝ 1/R. The secondterm, always negative and present with an opposite sign in B.11b, is a transfer of kineticenergy from the fluctuations to the poloidal flow via the Reynold stress.

C A quick overview ofsheath physics

Sheath physics is important to understand both measurements by Langmuir probes andthe influence of the divertor targets on turbulent motions in the SOL. These two elementsare indeed the same situation, where a solid surface is in contact with the main plasma.When colliding with it, charged particles will tend to stick to it for a moment, recombineon its surface, and be re-emitted in the plasma as neutrals. These neutrals will then beionized again, but far from the sheath, and we will ignore them in the following. In otherwords, the surface acts as a sink for charged species. Two situations are possible:

The surface is isolated, and then its potential can be adjusted by the plasma bygiving charges to it (it is said to be floating). Since electrons have a higher thermalvelocity than the ions (for Te ≥ Ti/1836), the flux of leaving electrons will tend tobe higher than the flux of leaving ions. The surface will thus charge negatively andits floating potential be adapted in order to equalize the two leaving fluxes, settingthe leaving net electric current to zero, and keeping the main plasma neutral. Therewill be a small non neutral area in front of the surface, called the sheath.

The surface is maintained at a given negative potential relative to the plasma. Asusually in a plasma, this charge imbalance will be screened. The screening area infront of the solid surface where quasi-neutrality is not respected is the sheath.

We will come back to the distinction between this two cases later, and in the meantime we denote φw the potential of the surface (w for wall). We consider a one dimensionalsituation, as depicted in fig.C.1. The potential of the plasma is denoted by φ(x).

A thin area where ni > ne is formed in front of the surface, such that the positive chargespresent here exactly shield the negative potential of the surface. However, the shieldingis not perfect: potentials of the order of −T/e can escape and create a pre-sheath electricfield throughout the whole plasma. This field accelerates ions and slow down electrons.

Moreover, the depletion in the sheath causes both species to be accelerated. Thus,electrons are accelerated by this gradient and slowed down by the electric field: theycan be in force balance. With the simplest momentum fluid equation, meneve∂xve =−∂xpe+nee∂xφ, neglecting the left-hand side (ve vt,e) and assuming Te = cst, we obtainBoltzmann equilibrium (the subscript se denotes values taken at the sheath entrance):

ne(x) = nse exp [e(φ(x)− φse)/Te] . (C.1)

Ions on the other hand are accelerated by both forces, and cannot be in force bal-ance. The continuity equation involves ni(x)vi(x) = cst = nsevi,se. Along with energyconservation for cold ions (mivi(x)2/2 + eφ(x) = miv

2i,se/2 + eφse), we obtain:

ni(x) = nse

(1− 2e(φ(x)− φse)

mv2i,se

)−1/2

. (C.2)

We thus reach a dynamic equilibrium where ions are accelerated and electrons repelled.We see that both densities decrease when going toward the solid surface: ions because ofa rarefaction effect, electrons because they are repelled.

Situation in the sheath. We must have ni(x) > ne(x) throughout the sheath toshield the negative bias of the surface. Poisson’s equation can be written as

d2φ

dx2= e(ni(x)− ne(x)) ' e2nse((miv

2i,se)

−1 − T−1e )(φ(x)− φse) (C.3)

31

32

This equation has non-oscillatory solutions (that have never been observed) only if v0i > cs.This condition of supersonic flow at the sheath entrance is called the Bohm criterium. Itcan also be obtained by imposing ni(x) > ne(x) throughout the sheath. A dimensionalanalysis of Poisson equation would show that the thickness of the non neutral area is oforder λD.

Situation in the plasma. For x > someλD the plasma is quasi neutral, and isaccelerated only by the pressure depression created at the sheath. Since the situation hereis similar to a freely expanding fluid under a pressure force – a situation that is likely toarise at the sound speed – and since the fluid must reach at least the sound speed at theentrance of the sheath, we will suppose that this is indeed the case:

vi,se = cs. (C.4)

This can be shown more rigorously by solving the equations of continuity and momentumfor the whole plasma (e.g. in Stangeby [54, 55]).

We can now obtain the contribution of ions and electrons to the current going to thesolid surface. For ions, since the velocity at the entrance of the sheath is cs, we simplyhave a current Ii = Aen0cs (A is the area of the surface). For electrons, we assume thatthere is enough collisions so that they have a Maxwellian velocity distribution, modifiedby the electric potential: fe(x,v, t) = a exp−mv2/Te + e(φ(x) − φse)/Te). The currentis then obtained by integrating −evfe over v at x = 0. One then obtain the total currentgoing to the surface:

I = Ii + Ie = Aen0cs −Aen0cs exp

Λs −

e(φse − φw)

Te

,

Λs = 0.5 lnmi/(2πme) = 3.2 (for deuterium).

(C.5)

More advanced models show that the potential at the sheath entrance is almost equal tothe plasma potential far from the sheath: φp ' φse.

If the surface is isolated, then it will be charged up to the floating potential needed tocancel I: φw = φp − TeΛs/e, with φp ∼ φse is the plasma potential far from the surface.A Langmuir probe is a simple wire immersed in the plasma. If it is left floating, then ameasure of its potential gives access to the potential of the plasma.

In the case of a tokamak plasma, the divertor targets and the main chamber are groundconnected at a fixed potential (say, 0). It is then the plasma that is floating at a valueφp = TeΛ/e. One important consequence is that in a sheath connected area, φp ∝ Te.

If a Langmuir probe is maintained at a given potential, it will drive a current I givenby equ.C.5. If this potential is strongly negative, all the electrons are repelled so that thecurrent saturates to I = n0Aecs. This saturation current gives the particle density.

If the voltage of the probe is swept from very negative value to positive ones, the curveI(φw) gives access to the electron temperature.

We finally remark that the surface collects all the ions but only the hot electrons ableto overcome the potential barrier. The consequence is that whereas the particle fluxes areequal, the heat flux given to the surface is more important for electrons, and this is whySOL plasmas tend to have Ti ≥ Te.

or ( )

in plasma

sheathpre-sheath

Figure C.1: Situation in the plasma and in the sheath along a field-line. There are solidsurfaces at both ends. Adapted from Stangeby [55].

D Concepts ofself-similarity

We provide here a somehow informal glimpse to self-similarity, to some of its consequencesand to the different tools available to study experimental signals. The discussion will notbe precise in a mathematical sense. We consider a stochastic signal Y (t) and its incrementsX(t) where t can either be a continuous variable or a discrete variable with unit stepsize.The increments are defined as X(t) = Y (t+1)−Y (t), without concern about the exactitudeand approximations involved by this definition. A rigorous approach to self-similarity, fBmand fGn can be found in Mandelbrot and van Ness [42].

Definition and interpretation of self-similarity

A stochastic process Y (t) is said to be H-self-similar with stationary increments (that wewill abbreviate to H-self-similar) if the following properties hold for all λ, t0 and τ [42, 49]:

Y (τ + t0)− Y (t0)d= Y (τ)− Y (0),

Y (τ)− Y (0)d= λ−H(Y (λτ)− Y (0)).

(D.1)

If we take λ = 1/τ , and Y (0) = 0, we successively obtain:

Y (τ)− Y (0)d= τH(Y (1)− Y (0)) and Y (τ)

d= τHY (1). (D.2)

Hered= means equal in distribution: all the statistical quantities formed from these signals

are the same. In the following, we will consider that Y (0) = 0 because it is the case for ageneral walk starting at the origin, and also of the signals that we will study. Note alsothat the property of stationary increments involves (with t = 1) the usual stationarity for

X(t0) ≡ Y (t0 + 1)− Y (t0), i.e. X(t0)d= X(1).

For a function f , the curve y = ζ(λ)f(λx) is obtained from the curve y = f(x) by acompression of the x-axis of a factor λ and of the y-axis of a factor ζ(λ) around the origin.That the two curves are equal means that zooming around the origin with the right factorsfor x and y leaves the curve invariant, which explain the term self-similar. We can see bydoing two successive zoom-in that ζ(λ) = λ−H is actually the only functional form possiblefor the scaling factor. The only solutions of f(λx) = λHf(x) are power-laws (f ∝ xη).

Of course, that does not mean that a self-similar stochastic process is a power law,but rather that its envelop is. Consider first its mean: Y (t) =

∑X = tX(1) and Y (t) =

tHY (1), so that a linear trend in Y is not compatible with self-similarity, and we musthave Y (t) = X = 0. Consider then its variance. We have Y 2(t) = t2HY 2(1), so that theenvelop of several realizations starting from zero will be a power-law of exponent H∗. Ifwe zoom-in with a factor λ for x and λH for y, then all these realizations will globally lookthe same. Note that self-similar processes are not stationary.

For H < 0, the process would diverge as t→ 0. Following [42, sec. 3.1], an upper valuefor H can be found by considering Minowski’s inequality 〈[Y (t + τ1 + τ2) − Y (t)]2〉1/2 ≤〈[Y (t+ τ1 + τ2)− Y (t+ τ1)]2〉1/2 + 〈[Y (t+ τ1)− Y (t)]2〉1/2 and by translating it under thehypothesis of H-self-similarity as: (τ1 + τ2)H ≤ τH1 + τH2 . This is true for all τ1, τ2 only ifH ≤ 1. Consequently, we have 0 ≤ H ≤ 1.

∗It appears that 0 is a privileged point. It is not for an individual realization, but it is if all therealizations start from here.

33

34

Walks

A well known self-similar process is the random walk, or Brownian motion. Its incrementsare independent, so that its variance is found to be Y (t)2 = (

∑X)2 = tX(1)2. We conclude

that its self-similar exponent is H = 1/2. As we will see, H = 1/2 is the signature ofuncorrelated increments. More precisely, the Brownian motion is the sum of uncorrelatedstationary Gaussian increments: Y (t) =

∑t1Xi. It is somehow the integral of a white

noise.

This notion have been extended by Mandelbrot to provide a class of Gaussian self-similar processes with arbitrary self-similarity exponents between 0 and 1. They are calledfractional Brownian motions (fBm), and are intuitively defined as a fractional integral ofa white noise: a fBm of self-similarity exponent H is a white noise integrated 2H times.They represent all the Gaussian self-similar processes [42, sec. 3.1], and we underline thatself-similar process with arbitrary PDF can be found [44].

Noises

The increments of fBm form a class of processes, also introduced by Mandelbrot, calledfractional Gaussian noises (fGn). They are a generalization of the white noise, and allpresent Gaussian statistics. We emphasize that it is the walks that are self-similar, not thenoises (actually, H = 0 for fGn, because since they are stationary, their shape is preservedunder a zoom that does not magnify the y direction). They can be characterized by theirspectral index βX , related to H by −βX = 2H − 1. For H = 1/2 to 1, βX ranges from 0(white noise) to 1 (pink noise or 1/f noise).

Again, fGn are only a special class of noises, that are Gaussian. The stationary incre-ments of any self-similar process can also be studied.

Long range correlations

A key feature of self-similar processes is that their increments exhibit long-range correla-tions. To see this, we follow [49] and we first develop the covariance function of the processY :

〈Y (t)Y (s)〉 = 0.5〈Y (t)2 + Y (s)2 − (Y (t)− Y (s))2〉 = 0.5〈Y (t)2 + Y (s)2 − Y (t− s)2〉= 0.5〈Y (1)2〉(t2H + s2H − |t− s|2H).

(D.3)

We can then compute the autocorrelation function of the increments X:

CX(n) = 〈XnX1〉 = 〈(Yn − Yn−1)Y1〉 = 〈YnY1〉 − 〈Yn−1Y1〉= 0.5〈Y 2

1 〉([n2H + 1− (n− 1)2H ]− [(n− 1)2H + 1− (n− 2)2H ])

= 0.5〈Y 21 〉(n2H − 2(n− 1)2H + (n− 2)2H)

=n→∞

1

2〈Y 2

1 〉d2

dn2n2H

=n→∞

〈Y 21 〉H(2H − 1)n2H−2.

(D.4)

First, we see that if H = 1/2, then CX(n) = 0, which was expected since then Y is arandom walk with uncorrelated increments. If H 6= 1/2, then we have the important resultthat the autocorrelation decreases slowly as τ−αX at large time lags, with αX = 2− 2H.

If H > 1/2, CX(τ) is positive definite for all τ and its integral diverges: there are longrange correlations in the sense of Taylor.

If H < 1/2, then CX(τ) is negative at high lags, so that the process is anticorrelated.CX(τ) is however positive at short time-lags and crosses zero once. Its integral is exactlynull [42, sec.6], so that it is not long-range correlated in the sense of Taylor, but it stillexhibits long-range correlations in the sense of an algebraically slow decay of CX .

35 APPENDIX D. CONCEPTS OF SELF-SIMILARITY

Spectral index

The Wiener–Khinchin theorem can be applied to the stationary increments X. It statesthat the power spectrum is the Fourier transform of the autocorrelation function, so thatwe have the asymptotic relation:

PX(f) =f→0

cst× f−βX , βX = 2H − 1. (D.5)

It may appear strange that for 0 < H < 1/2 we have βX < 0, but it is not so becausethis relation is for small frequencies. For 1/2 ≤ H ≤ 1, we have 0 ≤ βX ≤ 1, and ifthe increments X are Gaussian, we indeed find the fGn class ranging from white noise atβX = 0 to pink noise at βX = 1.

Since Y is the cumulated signal obtained from X, its spectral index at small frequenciesis βY = βX + 2 = 2H + 1. It ranges from 1 to 3.

We remark that fBm and fGn are sometimes defined through the spectral index, andthen shown to be self-similar with H = (βY − 1)/2, for example by Hergarten [34].

Structure functions

It is immediate to show that if Y is H-self-similar then:

Sq(τ) ≡ 〈|Y (τ)− Y (0)|q〉 = 〈|Y (1)− Y (0)|q〉 τ ζq , ζq = qH. (D.6)

Note that Y is not stationary, but Y (τ+t0)−Y (t0) is, so that we can still evaluate equationD.6 with Sq(τ) defined in equation 3.1.

Another remark is that the power spectrum, the autocorrelation function and the secondorder structure function all carry the same information†, but generally only the structurefunction is smooth enough to be conclusive. Furthermore, when the scalings ζq are notlinear in q, it shows that the relevant exponent characterizing long-range correlations isH = ζ2/2.

Rescaled range

We consider the signal X between k0 and k0 + ∆k, and define:

δ(i) = Y (k0 + i)− Y (k0)− (i/∆k)[Y (k0 + ∆k)− Y (k0)] (see fig.D.1),

R(k0,∆k) = max0≤i≤∆k

δ(i)− min0≤i≤∆k

δ(i) (the range),

R/S(k0,∆k) = R(k0,∆k)/S(k0,∆k) (the rescaled range),

(D.7)

where S(k0,∆k) is the standard deviation of the subsignal X(i = k0 + 1..k0 + ∆k). Thephysical idea behind R/S is illustrated on figure D.1. For a H-self-similar process, it hasbeen widely observed that R/S(∆k) ∝ ∆kH [44]. This method was first suggested byHurst, and for this reason the H exponent is often called the Hurst exponent.

Other tools

Other tools are available to test for self-similarity. We can mention the variogram, whichis actually the second order structure function, of slope 2H in a log-log plot. It shouldbe used when the signal is multifractal. Another method is the aggregated variance, thatexplores the way the variance of sub-samples of the signal increases with the sample size.See e.g. Rypdal and Ratynskaia [48].

†By Wiener–Khinchin theorem and by the relation S2(t) = 〈|Y (τ) − Y (0)|q〉 =∫+∞0 df PY (f)[1 −

cos(2πft)] [34, 37].

36

(a) (b) (c)

1 year

water levelin a tank

precipitationover 1 year

Figure D.1: (a) We consider the part of the signal X from k0 to k0 + ∆k. The broken

line is the cumulated signal Y (k0 + i) − Y (k0) =∑k0+∆kk=k0+1Xk, whereas the straight line

is (i/∆k)[Y (k0 + ∆k) − Y (k0)]. The range R(k0,∆k) is the maximum minus minimumdifference between these two curves. For example, if the increments Xk are positivelycorrelated, then we expect the cumulated signal to largely deviate from the straight line,and the range to be great. (b) and (c) We represent water level in a tank. In case(b), the increments (the yearly precipitations) are uncorrelated and a situation where oneyear of rain is followed by one year of drought is shown. The total level (dotted curve)evolves close to the straight line, so that whatever the scale considered, the range (bluearrow) will be roughly the same. In case (c), there is four years of flood followed by fouryears of drought (upper diagram), so that the increments present long range correlations(Joseph effect). We see that the range (blue arrows) increases when we look at the signalfrom small to higher scales (smaller diagrams are subparts of the big one): ∼ 1/2 over twoyears, ∼ 2/3 over three years, etc. We also remark that even if the correlations last forfour years, the range still increases up to period of eight years. If there were one year ofhigh precipitations (Noah effect, a huge increment), then it would be erased when dividingthe range by the standard deviation of the increments.

E Physical constants

Fundamental constantsVelocity of light c = 2.998 · 108 m s−1

Boltzmann constant kb = 1.381 · 10−23 J K−1

Electron charge magnitude e = 1.602 · 10−19 cElectron mass me = 9.11 · 10−31 kg mp/me = 1836Proton mass mp = 1.67 · 10−27 kgPermittivity of free space ε0 = 8.854 · 10−12 A2 s4 kg−1 m−3 ε0 ≡ 1/µ0c

2

Permeability of free space µ0 = 4π · 10−7 N A−2

1 eV = 1.602 · 10−19 J 1 eV/kB = 11605 K 1 G = 10−4 T

Experimental parameters for TCV shots #27601/#27602Tokamak major radius R 87.5 cmMinor radius (poloidal plane) a 24 cmProbe position 22 mm from the separatrix and 3 mm from the wallSampling 6 MHz during 0.94 s/0.96 s

Plasma parameters Estimate in TCV SOL: #27601 #27602Ion mass (deuterium) mi = 2mp 3.35 · 10−27 kg sameIonization Z 1 sameToroidal plasma current IT 341 kA sameLine averaged density ne 4.5 · 1019 m−3 8.5 · 1019 m−3

Mean electronic temperature Te of the order of 20 eV or 2.3 · 105 K for bothToroidal magnetic field B 1.2 T sameCyclotron frequency (electron) Ωe = eB/me 2.1 · 1011 Hz sameCyclotron frequency (ion) Ωi = eB/mi 5.7 · 107 Hz same

Electron-ion collision frequency νei ∝ (ln Λ)n/(m1/2e T

3/2e ) 1.6 · 107Hz 2.9 · 107Hz

Plasma frequency (electron) Πe =√ne2/ε0me 3.8 · 1011 Hz 5.2 · 1011 Hz

Sound speed (Ti = 0) cs =√Te/mi 3.1 · 104ms−1 same

Cold ion Larmor radius ρs = cs/Ωi 0.54 mm samePlasma β β = nT/(B2/2µ0) 6.3 · 10−5 1.2 · 10−4

Debye length (Ti = 0) λD =√ε0Te/ne2 5.0µm 3.6µm

Plasma parameter Λ = (4/3)πnλ3D 2.3 · 104 1.7 · 104

Alfven velocity vA = B/(µ0min)1/2 2.8 · 106ms−1 2.0 · 106ms−1

E× B velocity vE = E ∧ b/BDiamagnetic velocity vD = b ∧∇(nT )/(neB)

Figure E.1:Magnetic field configuration during the two shots #27601 and#27602. The black thick line is the actual position of the sittingprobe. Axis units are in meters.

37

Acknowledgments

I first have to thank Odd Erik for many things: for the very interesting subject of thisthesis, for a lot of clear explanations and availability, for the various trips around Tromsø,for the photographic advices, and for the rest. To anyone wanting to work on the subjectof plasma physics, he is a really great supervisor. Just be careful when he suggests amountain hike.

It has been very nice to meet some of the students here, especially Ole and Henry (Ole,I know that you might read this some day, so I would like to say that it’s not a problemif in the end we couldn’t have played this ping-pong match with Henry. I guess that youtwo were afraid of my amazing level :)), Felix, Ralph, and others that have a much toocomplicated Norwegian name for me to risk to write it here.

I thank Njal for the morning spent at experimenting with the plasma device of thelaboratory. It is really worthwhile to actually see the object one is working on.

Je remercie aussi ceux restes (trop) au chaud en France. La bande des physicienslyonnais, dans le bon ordre : Etienne, Antoine, Xavier, David, Loren et Jeremie, pour lesdiscussions par mails, pour les trucs et astuces (Antoine pour l’histoire des fits), et pourle moins serieux. Et il y a z aussi Charlotte, pour les coups de fil, pour les livres, pour lescartes. Enfin, je termine en ajoutant Thibaut, et en oubliant certainement du monde.

38

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