core.ac.uk · 39Y. Hamada, T. Watari, A. Nishizawa, K. Narihara, Y. Kawasumi, T. Ido, M. Kojima,...

General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from orbit.dtu.dk on: Dec 17, 2017

Intermittent convective transport carried by propagating electromagnetic filamentarystructures in nonuniformly magnetized plasma

Xu, G.S.; Naulin, Volker; Fundamenski, W.; Rasmussen, Jens Juul; Nielsen, Anders Henry; Wan, B.N.

Published in:Physics of Plasmas

Link to article, DOI:10.1063/1.3302535

Publication date:2010

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Xu, G. S., Naulin, V., Fundamenski, W., Juul Rasmussen, J., Nielsen, A. H., & Wan, B. N. (2010). Intermittentconvective transport carried by propagating electromagnetic filamentary structures in nonuniformly magnetizedplasma. Physics of Plasmas, 17(2), 022501. DOI: 10.1063/1.3302535

http://dx.doi.org/10.1063/1.3302535

http://orbit.dtu.dk/en/publications/intermittent-convective-transport-carried-by-propagating-electromagnetic-filamentary-structures-in-nonuniformly-magnetized-plasma(dcc636cd-5563-47e4-abda-bd99530c4241).html

Intermittent convective transport carried by propagating electromagneticfilamentary structures in nonuniformly magnetized plasma

G. S. Xu,1,2 V. Naulin,3 W. Fundamenski,1 J. Juul Rasmussen,3 A. H. Nielsen,3 andB. N. Wan2

1Euratom-UKAEA, Culham Science Centre, Abingdon OX14 3DB, United Kingdom2Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China3Association Euratom-Risø DTU, DK-4000 Roskilde, Denmark

�Received 11 November 2009; accepted 6 January 2010; published online 9 February 2010�

Drift-Alfvén vortex filaments associated with electromagnetic turbulence were recently identified inreversed field pinch devices. Similar propagating filamentary structures were observed in the Earthmagnetosheath, magnetospheric cusp and Saturn’s magnetosheath by spacecrafts. Thecharacteristics of these structures closely resemble those of the so-called mesoscale coherentstructures, prevailing in fusion plasmas, known as “blobs” and “edge localized mode filaments” inthe boundary region, and propagating avalanchelike events in the core region. In this paper thefundamental dynamics of drift-Alfvén vortex filaments in a nonuniformly and strongly magnetizedplasma are revisited. We systemize the Lagrangian-invariant-based method. Six Lagrangianinvariants are employed to describe structure motion and the resultant convective transport, namely,magnetic flux, background magnetic energy, specific entropy, total energy, magnetic momentum,and angular momentum. The perpendicular vortex motions and the kinetic shear Alfvén waves arecoupled through the parallel current and Ampere’s law, leading to field line bending. On thetimescale of interchange motion ��, a thermal expansion force in the direction of curvature radiusof the magnetic field overcomes the resultant force of magnetic tension and push plasma filament toaccelerate in the direction of curvature radius resulting from plasma inertial response, reacted tosatisfy quasineutrality. During this process the internal energy stored in the background pressuregradient is converted into the kinetic energy of convective motion and the magnetic energy of fieldline bending through reversible pressure-volume work as a result of the plasma compressibility inan inhomogeneous magnetic field. On the timescale of parallel acoustic response ��, part of thefilament’s energy is transferred into the kinetic energy of parallel flow. On the dissipation timescale�d��, the kinetic energy and magnetic energy are eventually dissipated, which is accompanied byentropy production, and in this process the structure loses its coherence, but it has already traveleda distance in the radial direction. In this way the propagating filamentary structures induceintermittent convective transports of particles, heat, and momentum across the magnetic field. It issuggested that the phenomena of profile consistency, or resilience, and the underlying anomalouspinch effects of particles, heat, and momentum in the fusion plasmas can be interpreted in terms ofthe ballistic motion of these solitary electromagnetic filamentary structures.�doi:10.1063/1.3302535�

I. INTRODUCTION

A. Filamentary structures in fusion plasmas

Plasma turbulence and the resultant anomalous transportoccupied a critical role in the physics of magnetically con-fined plasmas for thermonuclear fusion research from itsvery infancy.1 Its nonlinearity and complexity are even moreprominent in the plasma edge region where the fluctuationlevels are typically of the order of unity.2 In the last decadeaccumulating experimental evidences from tokamaks,3

stellarators,4 reversed field pinches �RFPs�,5 sphericaltokamak,6 simple magnetized torus,7 and linear devices8 re-vealed the presence of solitary coherent structures9 in theplasma turbulence, propagating in the radial direction as wellas in the azimuthal direction, commonly referred to as blobsor magnetic-field-aligned filaments,10 which lead to intermit-tent convective transport of particles, heat, momentum,charge, and current across the magnetic field. Recent obser-

vations of strong filamentary structures in the plasma bound-ary in association with edge localized mode �ELM� activityin high confinement mode �H-mode�11–18 have many simi-larities with the blob structures observed in low confinementmode �L-mode�, indicating that they might be governed bythe same physical mechanisms. By careful analysis of fastcamera and reciprocating Langmuir probe data, it was nowdemonstrated that turbulent transport in the inter-ELM peri-ods is also predominantly carried by filamentary structures,although with much smaller fluctuation amplitudes.6

The generation of these structures is generally thought tobe the result of strong turbulence at the plasma edge.19–23

Their two-dimensional �2D� �in the drift plane, i.e., the planeperpendicular to the magnetic field� electrostatic featureshave been extensively investigated numerically.20–25 Blob re-lated polarization, vorticity, and sheath closure were consid-ered in Refs. 19–23. The interchange dynamics for filament

PHYSICS OF PLASMAS 17, 022501 �2010�

1070-664X/2010/17�2�/022501/22/$30.00 17, 022501-1

Downloaded 24 Jun 2010 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp

http://dx.doi.org/10.1063/1.3302535

http://dx.doi.org/10.1063/1.3302535

http://dx.doi.org/10.1063/1.3302535

motion were highlighted in Refs. 24 and 25. Alfvén closuresfor 2D blob models and some electromagnetic features wereintroduced in Refs. 26–29. Transport momentum with blobswas studied in Ref. 30, and kinetic simulation of blob waspresented in Ref. 31. The current state of the art in this fieldwas reviewed in Refs. 32 and 33. Now, strong interest isarising around their three-dimensional �3D� characteristics,with strong emphasis on the parallel dynamics34 and electro-magnetic features.35–37 This interest is enhanced by someanalogies with the propagating avalanchelike events38 orstreamers39 previously observed in the plasma core region,where the plasma � �the ratio of thermal to magnetic pres-sure� is much higher. These events are generally thought tobe associated with some nonlinear electromagnetic structuresor fronts ballistically propagating in the radial direction.1

Partially motivated by this target, nonlinear kinetic simula-tions with electromagnetic effects are rapidly evolving in therecent years.40–42

The electromagnetic characters of ELM filaments werestudied in MAST spherical tokamak14,17 and ASDEX Up-grade tokamak15,16 experiments, ruling out purely electro-static dynamics. Hints of electromagnetic features of blobswere obtained in linear devices,43 and more recently, the firstdirect experimental measurement of the parallel current den-sity associated with blob structures44 and the first experimen-tal evidence showing the association of the propagating tur-bulent structures with the drift-Alfvén vortices was obtainedin the RFP device.45 Also, the first direct observation of cur-rent in ELM filaments was made on ASDEX Upgradetokamak.46 These new experimental activities aimed at iden-tifying the electromagnetic features of filamentary structuresin the fusion plasmas are partially motivated by recent ob-servation of drift-Alfvén vortices in space plasmas.

B. Filamentary structures in space plasmas

It is well know that propagating filamentary structuresare frequently found at boundary layers in astrophysical,geophysical, and solar atmospheric plasmas. In recent years,dipolar drift-Alfvén vortices have been detected both in themagnetospheric cusp47 and in the magnetosheath48–50 by thefour-spacecraft Cluster mission, and its relation to the cross-field transport was found.51 Similar Alfvén vortex filamentswere observed in Saturn’s magnetosheath by the Cassinispacecraft,52 indicating the universality of such structures inplanetary space. In the year 2007, Alfvén waves were de-tected in the solar corona,53 where the filamentary structuresassociated with Alfvén waves were suggested to explain howenergy is transferred to the solar corona, which is millions ofdegrees hotter than the solar surface, known as the photo-sphere. The connection between the erupting filamentarystructures in the fusion laboratory plasma and the solar flareswas summarized in a recent review paper.54 The auroralplasma is occasionally observed to evolve into highly coher-ent electromagnetic vortex structures,55 in which the per-turbed electric and magnetic fields exhibit regular rotationtogether with the particles trapped inside the structures. An-other example is the phenomenon that takes place duringionospheric irregularities where localized regions of plasma

depletions, often referred to as bubbles, move radially out-ward on the night side of the equatorial F layer ionosphere.56

C. Impacts of the filamentary transportin fusion plasmas

The similarity of the electromagnetic filaments in fusionplasmas and in space plasmas suggests that it could be auniversal phenomenon in plasma turbulence. As a conse-quence, the study of plasma blobs or filaments and the re-sultant intermittent convective transport is one of the mostactive research areas within plasma physics.2,3,10,33 In the fu-sion plasma boundary the propagating filaments are believedto dominate the transport across the scrapeoff layer �SOL� offusion devices and possibly lead to serious wall erosion, im-purity, and recycling problems for future fusion reactors.33

More seriously, the transient power loads associated with theELM filaments on plasma facing components �both divertorand limit tiles� pose stringer design criteria and operationallimits on the material for future fusion reactor, most notablythe planned ITER experiment.57

An important consequence of the filamentary transport isa strong ballooning nature of turbulence and a strong poloi-dal asymmetry of radial fluxes in toroidal geometry. Turbu-lence and flow measurements in the SOL indicate that trans-port is concentrated on a narrow sector near the outboardmidplane,58 and the radial fluxes on the outboard midplaneare nearly two orders of magnitude larger than on the inboardmidplane.59 There is also increasingly evidence for the exis-tence a B��B-independent subsonic parallel flow compo-nent which is driven by the strong ballooning in the radialturbulent transport, and thus direct from the outboard mid-plane region toward the divertor targets.60,61 These 3D fea-tures of turbulent transport have significant implications onITER, considering that currently ITER design is largelybased on predications from 1D transport modeling.57 Assuch, understanding the elementary electromagnetic filamen-tary structures which constitutes the plasma turbulence isrecognized as an issue of the highest priority with regard toITER.

D. Lagrangian-invariant-based method

In this contribution we intend to concentrate our discus-sions on elementary electromagnetic structures and funda-mental mechanism of turbulent transport in the plasma edge.Specifically, the dynamics of drift-Alfvén vortex filaments ina nonuniformly and strongly magnetized plasma are revis-ited. The physical parameters and derived quantities for atypical filamentary structure �using the Joint European Torus�JET� tokamak parameters� in the pedestal region �inH-mode� or periphery region �in L-mode�, in the vicinity ofseparatrix and in the SOL are summarized in Table I in theAppendix. The Lagrangian-invariant-based method was sys-temized and intensively used in this paper. Six Lagrangianinvariants are employed to describe the structure motion andthe resultant convective transport. They are magnetic flux,background magnetic energy, specific entropy, total energy,magnetic momentum, and angular momentum.

022501-2 Xu et al. Phys. Plasmas 17, 022501 �2010�


The Lagrangian-invariant-based method was proposedfor predicting the quasisteady profiles in tokamakplasmas.62–72 The basic assumption is that turbulence mixingcauses uniformly distribution of the Lagrangian invariantsover the accessible phase space, a state denoted turbulenceequipartition �TEP�.62 The mechanism behind the TEP pro-cess is the Rayleigh–Bénard convection, carried by struc-tures, known as the convective cells.73 The simplest case ofTEP is the adiabatic vertical temperature profile that resultsfrom large scale convection in a fluid heated from below.Note that this profile is not determined by the intensity of theturbulence, but by the uniform distribution of specific en-tropy, which is a Lagrangian invariant.

It is well known that TEP occurs in atmospherical con-vection, for instance, in the troposphere, and also appears inthe convection zone of the sun, a natural nuclear fusion re-actor with turbulent transport of energy to the surface, justlike in fusion devices. According to data for sunseismology,74 the Lagrangian invariant Tn−2/3 is constant towithin a factor of 10−2 in the convection zone. Here T is thetemperature and n is the particle density. This is generallyattributed to the conservation of the specific entropy, leadingto the isentropic atmosphere model.75 The same mechanismis responsible for the decrease in temperature with height inthe Earth’s atmosphere. The success of the TEP approach toturbulent transport in the sun makes it natural to apply thesame ideas to fusion devices.62–72

Recently, the TEP theory was extensively used to studythe anomalous pinch effects of toroidal momentum.76,77 Inthis paper the TEP theory is extended and systemized toinclude the electromagnetic effects. It will be demonstratedin this paper that the well-know phenomena of profile con-sistency, or resilience,78 and the underlying anomalous pinch�up-gradient transport� effects67 of particles, heat, and mo-mentum in fusion plasmas can be interpreted in terms of theballistic motion of electromagnetic filamentary structures.

E. Paper organization

This paper is organized as follows. In the following sec-tion, we show the relation between the intermittent convec-tive transport and the mesoscale structures using some ex-perimental data from JET tokamak. In Sec. III the orderingscheme for mesoscale structures in fusion plasmas and theconcept of drift-Alfvén vortex filaments are introduced. InSec. IV we highlight that the presence of compressibility inan inhomogeneous magnetic field is responsible for the en-ergy exchange between random thermal motion and collec-tive motion. In Sec. V the generation mechanism of kineticenergy of filament motion and magnetic energy of field linebending is analyzed. The energy transfer is through the re-versible pressure-volume work which can be interpreted interms of the fundamental thermodynamic relation and theentropy equation. The dynamics for filament acceleration andelectromagnetic vorticity generation are presented in Sec. VI.An equivalent circuit is used to illustrate the processes in adrift-Alfvén vortex filament. The acceleration of plasma fila-ment is induced by a force unbalance in the direction ofcurvature radius resulting from plasma inertial response on

the timescale of interchange motion, reacted to satisfyquasineutrality. A discussion on the fundamental kineticmechanism underlying the cross-field turbulent transport as-sociated with the filamentary process is given in Sec. VII.The filamentary structures present a channel for local energyexchange between particles and magnetic field perturbations,leading to breaking of the periodic orbits of particles and thetoroidal symmetry of magnetic field and resulting in the vio-lation of the adiabatic invariance associated with the poloidalmagnetic flux. In Sec. VIII six Lagrangian invariants aresummarized. The filament motion is largely controlled bythese Lagrangian invariants. The mode-independent part ofthe curvature-driven turbulent convective pinch of particles,heat, and toroidal momentum are briefly reviewed in Sec. IX.We employ the quasilinear method to present a qualitativeestimation of the intermittent convective transport inducedby the radial propagation of the filamentary structures. Fi-nally a discussion of the results is given in Sec. X, followedby a summary. This contribution can be generally regardedas a concept upgrade from electrostatic filamentarystructure19–25 to electromagnetic filamentarystructure26–29,35–37 in response to the recent experimentalprogress43–53 in the context of intermittent convective trans-port mediated by propagating coherent structures. This is acontinuation of our previous work presented in Refs. 24, 25,32, 34, 54, and 73.

II. INTERMITTENT CONVECTIVE TRANSPORTAND MESOSCALE STRUCTURES

In this section, we use some reciprocating Langmuirprobe data from the JET tokamak to show the relation be-tween the intermittent convective transport and the mesos-cale structures. The existence and importance of the mesos-cale structures, known as blobs and holes, for cross-fieldtransport in tokamak edge have been demonstratedexperimentally,2,3,10 where blobs are observed as magnetic-field-aligned filaments of enhanced density and temperatureas compared with the background plasma, while holes arefilaments of reduced density and temperature. Recently, thefirst experimental evidence showing the formation of blobsand holes in the edge velocity shear layer and the transport ofazimuthal momentum by blobs was obtained on JETtokamak.79

The most direct indication of turbulence intermittency isreflected by the bursts emerging in the raw signal of the ionsaturation current Is�n�Te+Ti, as shown in Fig. 1. For de-tails about this discharge please see Ref. 79. Intermittentpositive bursts are prevailing in the SOL, see Fig. 1�a�, indi-cating the presence of blobs. Similar behavior of Is signal hasbeen observed in the SOL of almost all tokamaks, see refer-ences in Refs. 2, 3, and 10. At the plasma edge there is ashear layer of poloidal velocity �r=r−rLCFS=−15�−3 mm,79 where rLCFS is the minor radius of the last closedflux surface �LCFS�. Intermittent negative bursts are detectedslightly inside the shear layer, see Fig. 1�c�, suggesting theexistence of holes. The first report of holes was on DIII-Dtokamak.3 In the shear layer, positive and negative perturba-tions are nearly balanced, see Fig. 1�b�. It was suggested that

022501-3 Intermittent convective transport… Phys. Plasmas 17, 022501 �2010�


the shear layer is the generation region of blobs and holes.79

An important feature of turbulence intermittency is thenon-Gaussian probability distribution functions �PDFs� ofplasma density fluctuations.2,3,10 The PDFs of Is fluctuationsmeasured at four radial locations are plotted in Fig. 2. On thehorizontal axes, the fluctuation amplitudes have been nor-malized to the root-mean-square �rms� fluctuation levels ofIs. In the SOL the PDFs are positively skewed with a heavytail because of the positive bursts. The skewness �S� andkurtosis �K� of the PDFs, i.e., the deviation of the Is signalsfrom Gaussian statistics, increases from the near SOL to thefar SOL, see Figs. 2�a� and 2�b�, which was speculated to bedue to the reduction in background pressure toward farSOL.79 The skewness and kurtosis, defined as the third- andfourth-order moments of the PDF, give a measure of thedegree of “asymmetry” and “peakedness” of a distributionwith respect to its mean value, respectively. For a Gaussiansignal, S=0 and K=3, whereas for others the deviation from0 and 3 indicates a higher degree of non-Gaussianity. In theshear layer, the PDF is very close to a Gaussian distribution,as shown in Fig. 2�c�. Slightly inside the shear layer, a nega-tive tail appears on the PDF and the skewness changes sign,which can be seen in Fig. 2�d�, suggesting the presence ofnegative bursts.

When many propagating structures with different sizesand velocities pass by probe tips, low-frequency high-amplitude fluctuations, constituted by bursts, are detected.Figure 3 shows the power spectra ln S�k� , f� of floating po-tential fluctuations, where k� is the poloidal wavenumber.The black solid curves show the dispersion relations, which

is defined as k��f�=�k��k�S�k� , f� /�k�

S�k� , f��. Inside theshear layer the turbulence propagates in the electron diamag-

netic direction as shown in Fig. 3�a�, where �r=−2 cm.Outside the shear layer the turbulence propagates in the iondiamagnetic direction as shown in Fig. 3�b�, where �r=1 cm. The reversion of propagating direction is mainly due

100

140

180

40

80

120

20

60

100

0

140

1 2 3 4 5 6 7 8 90 10

I s(mA)

I s(mA)

I s(mA)

(a)

(b)

(c)

Time (ms)FIG. 1. Turbulence intermittency shown on the raw signal of the ion satu-ration current Is, �a� positive bursts are prevailing in the SOL �r�28 mm,�b� positive and negative perturbations are nearly balanced in the edge ve-locity shear layer �r�−10 mm, and �c� negative bursts are prevailing justinside the edge velocity shear layer �r�−18 mm.

10-210-1

10-3

10-210-1

10-3

100

100

10-210-1

10-3

100

10-210-1

10-3

100

0-2-4 2

(a) far SOL

(b) near SOL

(c) shear layer

(d) edge

4 6 8(Is-<Is>)/σ(Is)

I sPDF

Δr = 30~50mmS = 1.5, K = 6

Δr = 0~15mmS = 0.3, K = 2.5

Δr = -20~-17mmS = -0.47, K = 4

Δr = -14~-9mmS = 0, K = 3

FIG. 2. �Color online� Semilogarithmic plots of the PDFs of the ion satura-tion current signals Is measured in the �a� far SOL �r=30–50 mm, �b� nearSOL �r=0–15 mm, �c� edge velocity shear layer �r=−14 to −9 mm,and �d� just inside the edge velocity shear layer �r=−20 to −17 mm. Onthe horizontal axes, the fluctuation amplitudes have been normalized to therms fluctuation levels of Is. The solid red lines in plotS �c� and �d� are thebest Gaussian fit to the PDFs. The corresponding skewness �s� and kurtosis�k� of the PDFs are also shown in the figure.

0

-2

-4

-6

-8

0

-15

-10

-5

20

-2

246

8

0

-8

-4-2

02

-6

40 60 80 100 120 140 160 180 200

20 40 60 80 100 120 140 160 180 200

k θ(rad/cm

)k θ(rad/cm

)

f (kHz)

JET Pulse No: 45783

(b) ∆r = 1cm

(a) ∆r = -2cm

JG09.288-3c

FIG. 3. �Color� Power spectra ln S�k� , f� of floating potential fluctuations,�a� 2 cm inside LCFS and �b� 1 cm outside the LCFS. The spectral intensityis plotted in a logarithmic scale. The black solid curves show the dispersionrelations.



to the radial variation of the Er�B rotation velocity.79 FromFig. 3 one can see that most of the spectrum power distrib-utes in the low-frequency low-wavenumber regions �f �40 kHz and k��4 rad /cm�, which has been demon-strated to be in association with the mesoscale structures.79

The nature of turbulence intermittency can be furthercharacterized by the time behavior of the power spectra, forwhich the complex Gaussian wavelet and continuous wavelettransform are used. Figure 4 shows the time-resolved wave-let power spectra ln S�f , t� of �a� ion saturation current Is and�b� radial E�B convective velocity vr, in the vicinity ofLCFS. Consistent with Fig. 3, the spectrum power concen-trates in the low-frequency region. Some intermittent struc-tures can be identified around 10 kHz, possibly with someoverlapping of adjacent structures. Comparing Figs. 4�a� and4�b�, one can find the structures in the Is signal and the vr

signal are strongly correlated, indicating that these structuresare propagating in the radial direction driven by the E�Bdrift, suggesting the interchange drive as the underlyingmechanism governing structure motion, as depicted in Refs.24 and 25. The correlation was further confirmed by theconditional average analysis, as shown in Fig. 5 of Ref. 79,indicating that the fluctuations of the Is and the vr tend to bein phase, i.e., charge polarization is nearly at the center of thestructures, which maximizes the convective transport.

The cross-field convective transport is dominantly car-ried by these propagating structures. Figure 5 shows thewavenumber power spectra S�k�s� of density fluctuation n,

poloidal electric field fluctuation E�, and radial convective

particle flux = nE�� /B0, measured in the vicinity of LCFS,where s is the ion sound gyroradius and at this location s

�0.54 mm. From the figure one can see that the spectrumpower concentrates in the low-wavenumber region withk�s�1, implying that the structure size is much bigger thans, whereas, compared with the system size a, i.e., the minorradius of plasma, these structures are still small, thereforethey are usually called mesoscale coherent structure �versussmallscale turbulence�.2,3,10,33 The radial convective particleflux is dominated by these structures, as indicated in Fig. 5.We use these data as a brief introduction so that one can havea direct idea for what are mesoscale structures in fusionplasma edge and why we want to study these structures.

III. PHYSICS BEHIND THE MESOSCALESTRUCTURES

From the theoretical point of view it has been shown thatthe edge turbulence is electromagnetic even for low localvalue of plasma �.80–82 Fully electromagnetic nonlinear gy-rokinetic theory for edge turbulence has now come to bemature.83,84 Efforts dedicated to the development of kineticsimulation of edge turbulence are now underway.85 Concern-ing the basic mechanism, in the recent years substantialprogress has been made in understanding the radial propaga-tion of blobs in the plasma boundary.19–33,86–89 For toroidallymagnetized plasmas this was suggested to be due to the guid-ing center drifts caused by an inhomogeneous magnetic field,resulting in a vertical charge polarization90 and a resultantE�B radial convection reacted to satisfy quasineutrality,� · j=0.19–25

A. Quasineutrality

Employing the quasineutral condition implies c /vA

=�pi /�ci�s / D�1, where c is the speed of light, vA

�B02 /�0min�1/2 is the Alfvén speed, �pi �ne2 /mi�0�1/2 is

the ion plasma frequency, �ci eB0 /mi is the ion gyrofre-quency, cs ��Ti+Te� /mi�1/2��e

1/2vA is the ion sound speed,s cs /�ci is the ion sound gyroradius, D ��0Te /ne2�1/2

0-2

-4

-6

-8

-10

0-2

-4

-6

-8

-1020 4 6 8 10 12 14

20

100

100

101

101

102

102

4 6 8 10 12 14

f(kHz)

f(kHz)

Time (ms)

JET Pulse No: 45783(a) Ion saturation current

(b) Radial ExB convective velocity

JG09.288-4c

FIG. 4. �Color� Time-frequency wavelet power spectra ln S�f , t� of �a� ionsaturation current Is and �b� radial E�B convective velocity vr, in thevicinity of LCFS. The complex Gaussian wavelet and continuous wavelettransform are used to calculate the power spectra. The spectral intensity isplotted in a logarithmic scale.

FIG. 5. �Color online� Wavenumber power spectra S�k�s� of density fluc-

tuation n �solid black line�, poloidal electric field fluctuation E� �dashed red

line� and radial convective particle flux = nE�� /B0 �dotted blue line�, inthe vicinity of LCFS.



�cs /�pi is the Debye length, �e 2�0peB0−2��cs /vA�2

= �s /Lpi�2 is the ratio of electron pressure to magnetic pres-sure, and Lpi c /�pi is the ion inertial skin length. The per-mittivity in a quasineutral plasma is �= �B0

−2+�0��B0−2 and

the permeability in a low-� plasma is �=�0+�0�B /�0M−1�−1��0, where min is the mass density �note i is theion gyroradius� and M −B−1p�b is the magnetization. Withthe permittivity and the permeability, the Alfvén speed isformally defined as vA ��−1/2. Assuming singly chargedions, as herein, n refers to both the electron or ion density. Infusion plasmas, even in the SOL, the quasineutral conditionis well satisfied, please see Table I.

B. Ordering scheme for mesoscale structures

The typical cross-field size L� of the filamentary struc-tures was observed to be close to the ion poloidal gyroradiusand the ion inertial length L��i�Lpi�s,

2,3 which doesnot follow the standard gyrokinetic ordering of drift-wavemicroturbulence k�s�1,42,84 thereby they are usually calledmesoscale coherent structures, where k� is the perpendicularwavenumber, �i iB0 /B� is the ion poloidal gyroradius,i vthi /�ci is the ion gyroradius, and vthi �2Ti /mi�1/2 is theion thermal velocity.

The ion inertial length Lpi is typically of the same orderas the radial gradient lengths of background pressure p� andzonal potential ��, Lpi�Lp�L� in the plasma edge of to-kamaks, where usually steep pressure gradient and strongradial electric field present, please see Table I, here Lp

��r lnp��−1, L� ��r ln��−1, and ¯ � denotes averageover a flux surface, i.e., the surface with constant poloidalmagnetic flux �p. These structures are strongly elongatedalong the field lines,2 manifested as magnetic-field-alignedfilaments. In tokamak geometry the typical parallel scalelength of a filament L� is of the same order of the parallelgeometry length �qR, where q is the magnetic safety factorand R is the major radius.

Following Refs. 54 and 91, we employ the drift orderingfor small parallel gradient. The magnetization parameter � s /L��1 is used as a measure of the smallness of the E�B drift velocity uE /cs��, where uE B−2E�B. By thisordering the vortex turnover time �� L� /uE��−1�s

��−2�ci is much longer than the ion gyroperiod �ci �ci−1,

therefore only low-frequency dynamics are involved, where�s L� /cs is the perpendicular ion sound transit time.36 Inthis paper �� is referred to the timescale for the interchange�or convective� motion. The ratio between poloidal and totalmagnetic fields in a tokamak is � B� /B0= � /q=i /�i

��, where � r /R�1 is the local inverse aspect ratio.The ion guiding center drift velocity is composed of u

=u� +uE+uGi+up+uFLR, where u� u�b is the parallel flowvelocity, b B /B is the unit magnetic field vector, uGi

=e−1B−1�Ti�b�� ln B+Ti��b� is the curvature andgrad-B drift velocities, up B−1�cidtE� is the ion polariza-tion drift velocity, the total time derivative is given by dt

�t+u ·�, and uFLR= �1 /4�i2��

2 uE is due to the finite-Larmor-radius �FLR� effect. By this ordering we haveuGi /uE�ivthi /RuE�L� /R��, up /uE��ci /��2, anduFLR /uE�i

2 /L�2 ��2. Consequently, the E�B drift is the

only lowest order cross-field guiding center drift, which ismoreover the same for both positively and negativelycharged particles. To lowest order, the ion guiding centerdrift velocity is reduced to u�u� +uE.

C. In the finite-� plasmas

We only consider the case of strongly magnetizedplasma, where the plasma � is low, �e��cs /vA�2= �s /Lpi�2

��2. For the case of greatest current interest, �e�me /mi,i.e., in a finite-� plasma,42 the electromagnetic effects areimportant. In this case the Alfvén speed vA is smaller thanthe electron thermal velocity vthe �2Te /me�1/2, i.e., vA /vthe

��me /mi�e�1/2�1. From Table I one can see this is gener-ally the case inside the separatrix, i.e., in the closed field lineregion. In the vicinity of the separatrix, we have me /mi

��e��2 and hence e /i=vthi /vthe�Lpe /Lpi��me /mi�1/2

��, where Lpe c /�pe is the electron inertial skin length and�pe �ne2 /me�0�1/2 is the electron plasma frequency.

At fusion plasma edge, the vortex turnover time �� iscomparable with the shear Alfvén time �A L� /vA �see TableI�, so that the electrostatic vortex motions and the kineticshear Alfvén waves �KSAWs� are coupled, ��A. The cou-pling with the compressional Alfvén wave, i.e., the magneto-sonic wave, is negligible, since �� is much longer than thecompressional Alfvén time �A� L� /vA��s��2��ci,where cs /vA��e

1/2�� has been used. By this ordering thedegree of the spatial anisotropy is L� /L� �uE /vA��e

1/2

��2. Using the quasineutral condition � · j=0, we havej� / j� �B� /B��A� /A� �L� /L� ��2, where B� is the parallelmagnetic perturbation, and A� is the perpendicular compo-nent of the magnetic vector potential. B� and A� are ne-glected in this paper since they are associated with the com-pressional Alfvén wave. The timescale for parallel acousticresponse is �� L� /cs��−1��, indicating that the excitationof the parallel ion sound wave is a slow process comparedwith the perpendicular convective motion. This explains why2D approximation is frequently applied to the ion motion andelectrostatic vortex dynamics.9

When the filaments and the KSAW are coupled, the par-allel phase speed vph� � /k� =L� /�� is of the order of vA�1+k�

2 s2�1/2. The relative wave impedance � uF /uE=vph� /Z

depends not only on the plasma � but also on the scalek�s,

92,93 where uF is the perpendicular velocity of fieldlines, Z E� /B�=uEB0 /B�� /A� is the wave impedance,here � is the scalar potential, and A� is the parallel compo-nent of the magnetic vector potential. In a finite-� plasma�from Table I one can see this is generally the case inside theseparatrix�, for mesoscale structures �k�s�1�, vph� �vA, thewave impedance Z is of the same order of the Alfvén speedvA and structures move perpendicularly with a velocity uE

��cs, whereas the wave impedance Z of smallscale struc-tures �k�s�1� substantially exceed vA; they move withsmall velocities uE��cs and are strongly coupled with driftwaves. Note, for the smallscale structures, one should applythe standard gyrokinetic ordering instead of the orderingscheme for mesoscale structures.

In the ideal magnetohydrodynamic �MHD� limit, �=1,the field lines are exactly frozen in fluid elements and mov-



ing with them. In the electrostatic limit, ��1. For mesoscalestructures, � is of order unity,92,93 implying that the electro-magnetic effects are more important for the mesoscale dy-namics than the smallscale dynamics. This gives the orderingof the perpendicular magnetic field fluctuations B� /B0

=uF /vph� ��e1/2��2, which is typically very small due to

the low �. But, the magnetic fluctuation levels increase withplasma �, so that field line bending in high-� plasmas isgenerally stronger than in low-� plasmas. Recent experi-ments on blobs from a RFP device44,45 and ELM filamentsfrom ASDEX Upgrade tokamak46 suggested some MHD be-haviors, ��1.

The parallel wavelength of the shear Alfvén wave L�

=vA�A is comparable with the typical parallel scale length�qR in a tokamak, implying that the induced field line bend-ing is global in the parallel direction,33 but is localized in thedrift plane with the cross-field size L�, manifested in a toka-mak by a finite toroidal mode number. The coupling withKSAW allows a perpendicular displacement of the field linesby an amount of r� L�B� /B0=uF�A��L�, which definesthe radius of an Alfvén vortex. Inside an Alfvén vortex theperturbed electric and magnetic fields are coupled with eachother through Maxwell’s equations and exhibit regular rota-tion together with the particles trapped inside the structures.9

In such a way the plasma could evolve into highly coherentelectromagnetic vortex structures.

The electrostatic fluctuation levels are of order unity atthe plasma edge, e� /Te� n / n�� pe / pe��1, where n n

− n�, pe pe− pe�, and � �− ��. Here, �� is the zonallyaveraged potential. The electrostatic fluctuation levels areconsistent with the ordering, e� /Te�uE /cs��1.

D. Alfvén vortex filaments

The strong mobility of the electrons along the field linesallows a parallel current j� to arise as a response to the chargepolarization induced by curvature and grad-B drifts in thedrift plane.24,25 This current provides a channel to couple theelectrostatic vortex dynamics with the KSAW.44–52 The cou-pling is through Ampere’s law �0j� =��B�=−b��

2 A�,where the total magnetic field is B=B0+B�, B0 is the back-ground �static equilibrium� magnetic field, z B0 /B0 is theunit vector along B0, B� ��A� =−z��A� is the perpen-dicular �to B0� magnetic perturbation, with B2=B0

2+B�2 and

� ·B=� ·B0=� ·B�=0.These more general vortices are denoted Alfvén vortex

filaments94–96 and are characterized by both an electrostaticvorticity � ��uE=bB−1��

2 � and a magnetic vorticity�A �0j�, where uE B−2E�B�B0

−1z�� and the elec-tric field is E −��−�tA�. The perpendicular electric field isnearly electrostatic E�=−��, while the parallel electricfield involves magnetic perturbation through E� =−��−�tA�

=��j� −��, where � ��j� −E��dl� is the parallel electro-motive force �emf�; here we defined l� as the distance alongthe perturbed field line, i.e., �� /�l� =b ·�=�z+B0

−1�A�

� z ·�, �t � /�t, and �z z ·�. The parallel derivatives carrynonlinearity entering through A�. Then, the parallel potentialgradient and the parallel electric field can be written as ��

=�z�+uE ·�A� and E� =−�z�− dtA�, where dt �t+uE ·� isthe transverse advective derivative.

When an Alfvén vortex arises in a plasma with strongbackground pressure gradient, which is a typical case in fu-sion plasmas, it will propagate in the azimuthal direction andcouple to the drift wave, so that it is usually called drift-Alfvén vortex.97 The concept of drift-Alfvén vortex was re-cently applied to the space plasma to interpret the Clusterobservations in the Earth’s magnetosphere.92,93

E. Ordering in generalized Ohm’s law

In the vicinity of the separatrix, the inertial parameter

�=me /mi�qR /Lp�2�1, the inductive parameter �=�e�qR /Lp�2�1, and the resistive parameter C=0.51�Lp /cs�ei�1 are of the same order �see Table I�,36

which implies strong nonadiabatic electron activity due tothe inertial, inductive, and resistive parallel electron re-sponses. With qR /vthe, Lp /cs, and �ei all comparable �seeTable I�, the situation is referred to as transcollisional, where�ei�Te

3/2n−1 is the electron-ion collisional time.The parallel component of generalized Ohm’s law is

given by42

�tA� + ��j� + e−2medt�j�n−1�

= e−1n−1��pe� − �pe� − pe�� ln B� − �� , �1�

where �� is the parallel resistivity and it is about two timessmaller than the perpendicular resistivity ��=1.96��

=me / �ne2�ei�, pe� nTe� is the electron parallel pressure andpe� nTe� is the electron perpendicular pressure. Equation�1� is also known as the electron parallel force balance equa-tion. With the parallel emf, it can be written as �tA� +��j�

=��−��. The first term on the left-hand side �LHS� andthe two terms on the right-hand side �RHS� of Eq. �1� are ofthe same order. Using �� as a reference, we have the order-ing of the first terms on the LHS ��A /��1 and the firstterm on the RHS �peTe / pee��1. The second term �theJoule dissipation term ��ALpe

2 /�eiL�2 �me�� /mi�ei� and the

third term �the electron inertial term ��ALpe2 /��L�

2 �me /mi�on the LHS are small compared to the other terms by a factorof me /mi.

From Table I one can see that in the whole boundaryregion ��ei, indicating that the Joule dissipation termdominates over the electron inertial term. Only in the nearlycollisionless plasma core region where ��ei, the electroninertial term is more important. For simplifying the discus-sion, the electron inertia and the electron viscosity are omit-ted in this paper, since we are mainly interested in the plasmaedge region, otherwise the electron kinetic energy must betaken into account in the energy conservation.

From the above ordering analysis we find the first termon the LHS, i.e., the magnetic induction terms, is of the sameorder of the terms on the RHS, indicating that the magneticinduction effects are important for mesoscale dynamics. Thephysics reflected by Eq. �1� is commonly interpreted as theresponse of the parallel current j� to the net parallel gradientforce en��−�� on electrons.36 Any force imbalance repre-sented by a nonvanishing RHS will excite a j� and the result-



ant magnetic fluttering A�. When this force is zero, the twogradients balance and the electrons are said to be “adiabatic.”Here, the adiabatic means these electrons do not exchangeenergy with the magnetic field. The “adiabatic electrons” areexpected to follow the Boltzmann distribution n / n�=e� / Te�.

98 When all the electrons are adiabatic electrons,the drift waves are in the electrostatic limit. The parallelphase velocity of a drift wave is expected to be in the rangevthi�vph��vthe.

In the SOL the electron temperature is so low that usu-ally vA�vthe and ��

e��, where ��e L�

2 /��e is the electron

parallel thermal conductive time and ��e 3.2�eiTe /me is the

electron parallel thermal conductivity �see Table I�, implyingthat the mobility of electrons along field lines is relativelyweak. The drift-wave phase velocity condition vph��vthe isunsatisfied if vph� �vA, thereby for mesoscale structures thecoupling to drift wave could be weak and there could be asignificant number of “nonadiabatic electrons.” This explainswhy the interchange turbulence is prevailing in the SOL�Ref. 34� and the SOL turbulence is electromagnetic even forlow local value of plasma �.37

Inside the separatrix, although vph��vthe is generally sat-isfied and ��

e��, the electron temperature isotropizationtime �Te becomes comparable with or even longer than ��

�see Table I�, so that the trapped electron effects becomeimportant. The trapped electrons do not obey the Boltzmannrelation and they generally contribute to the so-called nona-diabatic response. Therefore, it could be everywhere �fromthe SOL to the plasma core� that a significant percentage ofnonadiabatic electrons exists and the turbulence is electro-magnetic in nature.

When the RHS of Eq. �1� does not vanish, reflecting animbalance in the parallel force on electrons, the parallel cur-rent j� and magnetic perturbations A� will arise as a result ofthe magnetic induction, driven by the so-called nonadiabatic

part of the density fluctuations h n−e�n� / Te�.36 It is

called nonadiabatic, because during this process the energyof particle system is not conserved. This nonadiabatic part ofdensity fluctuations provides a channel to exchange internalenergy of particles with the magnetic energy of field linebending. The second term on the LHS of Eq. �1�, i.e., theJoule dissipation term, is responsible for the irreversiblemagnetic energy dissipation, governing the magnetic diffu-sion process. It is irreversible since it is accompanied byentropy production.

F. Alfvén’s frozen-in law and magnetic diffusion

In the ideal MHD limit ��=1�, Alfvén’s frozen-in law isan accurate law: the field lines are exactly frozen in fluidelements and moving with them, and the parallel emf �vanishes. In the resistive MHD case ��1�, only collisionalresistivity can break Alfvén’s frozen-in law. When kineticeffects, such as Landau damping and/or trapped particles,42

are taken into account ��1�, Alfvén’s frozen-in law canalso be broken by the kinetic effects and generating the par-allel emf � �note in this paper the definition of the parallelemf excludes the contribution from the collisional resistiv-ity�. The kinetic effects are alternatives to resistive diffusion

for decoupling the magnetic field and plasma. In collisionlessplasmas the former dominates over the latter. The presenceof the parallel emf and/or collisional resistivity allowsplasma to drift across the field lines.

With the help of generalized Ohm’s law �1�, the electricfield can be expressed as

E = − �� − �� + �j . �2�

Note ��j� /��j� �L� /L� ��2. Taking the curl of the electricfield expression, we obtain the induction equation, i.e., thedifferential form of Alfvén’s frozen-in law,

�tB� = �� uE� B� + �� + Dm�2B� − �� j ,

�3�

where Dm �0−1��Te

3/2 is the magnetic diffusivity. The firstterm on the RHS of Eq. �3� can be written as −��. TheLHS and the first term �frozen term� and the second term�drift term� on the RHS are of the same order. The third term�magnetic diffusion term� and the last term �resistance gradi-ent term� ��Lpe

2 /�eiL�2 ��me /�eimi. Combining the first

two terms on the RHS, we have ��−��, where��−�� is the net parallel gradient force on electrons,thereby it is the net parallel gradient force that drives thefield line bending.

The electric field in Eq. �2� can be rewritten as42

E = B� uF − �� + �j , �4�

where uF B−1b��−��=vph�B� /B0=�uE is the velocityof field lines when the resistive magnetic diffusion is absent.Its radial component is the velocity of the magnetic surfaces��p�.

From Faraday’s law, we rewrite Alfvén’s frozen-in lawas

�tB� = �� uF� B� + Dm�2B� − �� j . �5�

The difference between the E�B drift velocity and the fieldline velocity �u uE−uF=B−1b�� is a function of theparallel emf �. The parallel emf is generally related to somekinetic effects.42

It follows that magnetic flux � ��B ·d�=B� is con-served in the zero electron mass limit,99 where � is the cross-section area of a field-aligned fluid element. The integralform of Alfvén’s frozen-in law is

��t + uF · �� = 0, �6�

showing that the magnetic flux � is a Lagrangian invariant.The E�B convection of a plasma filament is thus performedin the form of interchange of flux tubes on the timescalesmaller than the magnetic diffusion time �m Dm

−1L�2

��eiL�2 /Lpe

2 ��ei�−2.

The last two terms on the RHS of Eq. �5� are smallcompared to the other terms except in the far SOL wherecollisionality is very high. From Table I one can see that theplasma parameters vary significantly across the plasmaboundary in a tokamak, which typically involves one order



of magnitude variations in density and two orders in tem-perature. As a consequence the collisionality �� L� / e

changes by more than two orders of magnitude, where e

vthe�ei is the electron mean free path. The SOL are usuallyhigh-collisionality ��10� region, and the magnetic diffu-sion time �m is comparable to or even shorter �in the farSOL� than ��, so that the SOL generally belongs to the un-frozen or dissipation region. In the SOL, filaments quicklydisplace away from the frozen-in flux tubes and drift acrossfield lines due to the magnetic diffusion.

Inside the separatrix ��10�, the magnetic diffusiontime �m is much longer than ��, the magnetic diffusionlength Lm is much shorter than L�, the magnetic diffusivityDm�1 m2 /s, and the magnetic Reynolds number Rm

uEL� /Dm�1 �see Table I�, indicating that the magneticdiffusion, i.e., field lines diffuse across the width L� of thefilamentary structure, is typically a slow process compared tothe transverse convective motion, so that on the timescale of�� the magnetic diffusion effect is negligible. We can thusrefer to the edge as the frozen region. The field lines aredragged away from the unperturbed magnetic field by theplasma filaments at a speed of uF, where � uF /uE�1.

G. Parallel ion sound wave

When a plasma filament drifts across the field lines, itmay pass through a succession of many magnetic fluxtubes.54 Such drifting motion creates transient, local distur-bance of pressure within the encountered flux tubes, initiat-ing transient parallel transport, i.e., by launching ion soundwaves and/or Alfvén waves, away from the position of dis-turbance. On the timescale of �� , the pressure per-turbations associated with the filament motion drive parallelacoustic response through the parallel motion equation,

dtu� = − ��p� + �p� − p�� ln B − ��u� , �7�

where �� is the kinematic viscosity of the parallel flow, p�

= pi� + pe� and p�= pi�+ pe�. From Eq. �7�, we have the time-scale of the parallel acoustic response �pu�L� / pcs

2��

��−1��, where u� �cs. As a consequence, on the timescaleof �� the parallel acoustic response is negligible.

IV. COMPRESSIBILITY IN AN INHOMOGENEOUSMAGNETIC FIELD

Plasma is different from the incompressible neutral fluidin that it is compressible in the directions both parallel andperpendicular to the ambient magnetic field. The parallelcompressibility is due to the coupling with ion sound wave;34

the perpendicular compressibility is due to the inhomogene-ity of the magnetic field,71 whereas the magnetic field isincompressible in the case of low-�, i.e., neglecting the cou-pling with compressional Alfvén wave. The equation of ioncontinuity is dt ln n=−� ·u�−� ·u� �2� ·u�−B ·��u�B−1�,where we defined the field line curvature � �b ·��b=−b� ��b�.

A. Poynting’s theorem

In a quasineutral plasma, using �0j=��B, we have

� · �B2uE� = �0j · �� = �0 � · ��j� . �8�

It can be rewritten as �� B�=�0�j−B2uE. �0�j andB2uE are of the same order. We will show that relation �8� isconsistent with Poynting’s theorem,

�tW = − � · S − j · E , �9�

where W �1 /2��0−1B2+ �1 /2��0E2��1 /2��0

−1B2 is the elec-tromagnetic energy density which is dominated by the mag-netic energy density since the electrostatic energy density isnegligible in a quasineutral plasma, S �0

−1E�B=�0−1B2uE

is the Poynting vector and its divergence is � ·S=�0

−1� · �B2uE�. The Joule dissipation term, i.e., the rate atwhich the electromagnetic fields do work, is j ·E=−j ·��

− j ·�tA�. Substituting �tW, � ·S, and j ·E into the Poynting’stheorem �9�, we obtain the energy conservation equation forthe electrostatic field �1 /2��0�tE

2+�0−1� · �B2uE�= j ·��.

With the quasineutral condition, we finally get the same re-lation as Eq. �8�.

With the help of the electric field expression �4�, wehave the equation of fluctuating magnetic energy,

�tW� = − �0−1 � · �E� B�� − j�E�

= − �0−1 � · ��B� uF + �j�� B�� − ��j�

2, �10�

where W� �1 /2��0−1B�

2 . The first term on the RHS is thedivergence of the Poynting vector of the KSAW. The lastterm is the Joule dissipation term. From �0j0=��B0, wehave

� · �B02uE� = �0j0 · �� = � · ��0�j0� . �11�

The gradient length on the LHS of Eq. �11� is very long�qR / � ��−1R, but the LHS does not vanish.

B. Compressibility and Lagrangian invariant

With the low-� approximation,76,77 i.e., B� /B0��2 and� · �B2uE��B2� ·uE, regarding B2uE and �j as incompress-

ible, � · �WuE�= �1 /2�j ·��0, we can thus write the com-pression of electric drift as

� · uE = B−1�b� � ln B + �� b� · ��

= 2R + �0B−2j · ��

� 2R , �12�

where R −uE ·� ln B0=g ·��, g B0−1z�� ln B0,

�� ln B0�R−1, and ��b=Bg+�0B−1j. The compressibilityis induced by magnetic field curvature and gradient.

In a low-� plasma the most unstable perturbations arethose inducing minimal magnetic energy variation, i.e., mostweakly bending of field lines. On the timescale of ��, al-

though the local magnetic energy density W varies with timedue to coupling with KSAW, the energy exchange betweenparticles and the equilibrium magnetic field B0 is negligiblein a frame moving with the fluid element. This is essentially



equivalent to say that there is nearly no coupling to compres-sional Alfvén waves in a low-� plasma, hence W0 is a La-grangian invariant,

dtW0 = V � · �W0u� = 0, �13�

where we defined the magnetic energy of the equilibrium

magnetic field as W0 VW0, V n−1 is the volume, and W0

�1 /2��0−1B0

2 is the magnetic energy density of the equilib-

rium magnetic field, note �tW0=0.With Eq. �13� we regard nB0

−2 as a Lagrangian invariantand B0

2u as an incompressible flow,

� · �B02u� = B0

2�� · u − 2R + 2J� = 0, �14�

where J u� ·� ln B0 denotes the parallel compression,�z ln B0�L�

−1. For simplicity here we only keep the leadingorder perpendicular guiding-center velocity u��uE. UsingEq. �14�, the continuity equation is written as

dt ln n = − � · u = 2�J − R� . �15�

The induction equation �frozen-in equation� can be writ-ten in another form, dt�n−1B�= �n−1B ·��u. Since the RHSusually does not vanish, nB−1 is generally not a Lagrangianinvariant. Only in 2D models, where magnetic field curva-ture is neglected, nB−1 can be regarded as a Lagrangian in-variant and BuE can be regarded as an incompressiblequantity.71,72

C. Compressibility in toroidal geometry

Now, let us specify this question in the circumstance oftoroidal geometry. For simplicity we consider a axisymmet-ric tokamak with the magnetic field, given by B0=BcR0��+��p= �e�+�e��BcR0R−1, where Bc and R0 are con-stants, Bc is the center magnetic field value, and � and � arethe toroidal and poloidal angles of a torus, respectively, ��=R−1e�, �p is the poloidal flux function, �=B� /B� is theratio between the poloidal and toroidal magnetic fields, andthe major radius is given by R=R0+r cos �. With such ge-ometry, the time rate of the perpendicular compression is�� R�uER−1, and the parallel gradient is given by �z

=kc��, where kc �qR�−1 and �� /��. The time rate of theparallel compression is �� J��u�R−1 sin �. Note uE /u�

��.On the outboard �low field side ��0� or inboard �high

field side �� midplanes, sin ��0, the parallel compres-sion �� and the parallel advection u� ·�n nearly vanish; theperpendicular compression thus dominates over the parallelcompression �� and the continuity equation is reducedto

dt ln n = − 2R . �16�

It means that, when a filament is displaced toward regions ofweaker background magnetic field, its volume expands at arate of ��, and as a result of expansion its density is reduced.

At the top �� /2 or bottom ��−� /2 of the torus, theparallel and perpendicular compressions are of the same or-der �� . The perpendicular compression could be par-tially canceled by the parallel compression, so that the in-

duced density variations in a frame moving with the fluidelement could be much smaller than those on the outboardmidplane. The spatial dependence of �� in a torus is oneof the origins of ballooning in toroidal geometry, and as aconsequence interchange instabilities mainly arise in the un-favorable curvature region and filaments in the toroidal ge-ometry generally manifest themselves as ballooning struc-tures.

By taking the scalar product with u� of the parallel mo-mentum Eq. �7�, we obtain the equation of parallel kineticenergy evolution,

ndtK� = − u� · �p� + �p� − p��J − 2n��K� , �17�

where K� �1 /2�miu�2 is the kinetic energy of parallel flows.

The second term on the RHS depicts the parallel compres-sion induced by mirror force. The first two terms on the RHSdrives ion sound waves along field lines.

On the outboard or inboard midplanes, the parallel com-pression nearly vanishes, dtK�→0, indicating that the energytransferred into the parallel flows is negligible. Therefore, thedynamics on the outboard or inboard midplanes are nearly2D. At the top or bottom of the torus, the parallel compres-sion is important, but the mirror force usually counteracts thepressure gradient.

D. Coupling with kinetic shear Alfvén waves

The shear Alfvén wave is a transverse wave with electricfield and magnetic field perturbations perpendicular to B0.When coupling with KSAW, the field lines are dragged awayfrom the unperturbed magnetic field by the plasma filamentsat a speed of uF, where � uF /uE�1. In a tokamak, theKSAW are launched from the unfavorable curvature region,i.e., the wave source is located on the low field side, whereinterchange modes are unstable, � ·�p��0. The wavespropagate along field lines with a phase speed vph�. On thehigh field side interchange modes are stable, � ·�p��0. Theinterchange motion perturbs the field lines in the perpendicu-lar direction, inducing field line bending. A restoring forceassociated with the resultant force of magnetic tension doeswork, generating the magnetic energy of field line bending.

In the ideal MHD limit �=1, the parallel emf � van-ishes, the electric field and magnetic field perturbations in ashear Alfvén wave divide equally the total energy of theelectromagnetic wave.100 When ��1, we have the magneticenergy of field line bending,

W� = �2K�, �18�

where K� �1 /2�miu�2 ��1 /2�miuE

2 is the perpendicular ki-

netic energy and W� VW�. For mesoscale structures, � isof order unity ��1�,92,93 and W� and K� are of the sameorder.

Using Eq. �18�, we obtain a relation between the parallelcurrent density and the electrostatic vorticity,

j� � ��0−1vA

−1B0 , �19�

showing that the parallel current carried by filaments is pro-portional to the electrostatic vorticity. The amplitudes ofmagnetic field perturbation and parallel current density mea-



sured in recent experiments associated with blobs,44,45 ELMfilaments,46 and Alfvén vortex filaments49–52 are generallyconsistent with Eqs. �18� and �19�. The importance of themagnetic components was stressed and the relation betweenthe magnetic and electrostatic fluctuation levels was verifiedin a recent electromagnetic simulation of edge resistive bal-looning turbulence.101 With the help of Eqs. �13� and �18�,we have the energy exchange between particles and the totalmagnetic field B in a frame moving with the fluid element,

dtW = dtW� = �2dtK�, �20�

where W VW=W0+W� is the total magnetic energy.

V. GENERATION OF KINETIC ENERGYAND MAGNETIC ENERGYTHROUGH THERMAL EXPANSION

In this section, a general discussion is given of the gen-eration mechanism of drift-Alfvén vortex structures in fusionplasmas.

A. Fundamental thermodynamic relation

Free energy is stored in the background pressure gradi-ent. The generation of structures requires an effective mecha-nism to release the free energy and do work so that the ther-mal energy of particles can be converted into kinetic energyof structure motion and magnetic energy of field line bend-ing. The most important energy transfer channel for a low-�plasma is due to the presence of compressibility in an inho-mogeneous magnetic field. This process can be interpreted interms of the fundamental thermodynamic relation

dtU = Tdts − pdtV , �21�

and the entropy equation102

pdts = �� · �� · u − � · q + j · E , �22�

where U CvT= �1 /2�T� +T�= �1 /2�mw2 is the plasma inter-nal energy, T Ti+Te is the plasma temperature, s is thespecific entropy, p nT is the plasma pressure, � is the off-diagonal tensor of the stress tensor P −p�I− �p� − p��bb+�, and q is the conductive heat flux. Cv= ��−1�−1 is thespecific heat at constant volume, Cp=��−1�−1 is the spe-cific heat at constant pressure, and � Cp /Cv�1 is the ratioof specific heats. The particle motion velocity is composed ofcollective motion velocity and thermal velocity, v=u+w.The thermal velocity in a magnetic field is composed of w2

=w�2+w�

2 . We defined the parallel temperature T� mw�2, the

perpendicular temperature T� �1 /2�mw�2 , the parallel pres-

sure p� nT�, and the perpendicular pressure p� nT�,where w2 means average over velocity space.

The first term on the RHS of Eq. �22� describes theviscous dissipation which is dominated by ion viscosity, thesecond term describes the dissipation due to heat conduction,and the last term describes the Joule dissipation. Note herethat the dissipation effect associated with the magnetic dif-fusion has already been included in Eq. �22�.

B. Conservation of specific entropy

There are several dissipation processes, such as viscos-ity, heat conduction, and Joule dissipation, which can lead toentropy production, and there are also several different dis-sipation mechanisms taking place in different situations. Inthe nearly collisionless plasma core region the entropy pro-duction is dominated by nonlinear wave-particle interactions�linear and nonlinear Landau damping� and wave-waveinteractions.9 In the plasma boundary it could be dominatedby collisional dissipation �see Table I�, or in the far SOLdominated by sheath dissipation.19–23

Whatever the detailed dissipation processes or mecha-nisms are, we can use a dissipation timescale �d to describethe entropy production,

Tdts = �d−1�K� + W�� , �23�

indicating that the effect of dissipation is to consume thekinetic energy of convective motion and the magnetic energyof field line bending and convert them back into the internalenergy. This process is irreversible since it is accompaniedby entropy production. In many situations of magnetizedplasma, the dissipation timescale is much longer than thetimescale for perpendicular dynamics, �d��. For instancein a tokamak, except in the far SOL the magnetic diffusiontime �m is much longer than �� see Table I�.

Therefore, on the timescale of ��, the dissipation effectsare negligible, and the specific entropy is a Lagrangian in-variant,

dts = 0. �24�

Substituting Eq. �24� into Eq. �21�, we have

dtU = − pdtV . �25�

It means that the internal energy stored in the backgroundpressure gradient can be tapped off through reversiblepressure-volume work. The presence of compressibility in aninhomogeneous magnetic field renders the expansion orcompression available. Using Eq. �15�, we can rewrite Eq.�25� as

dtU = 2�J − R�T . �26�

On the outboard midplane the parallel compression J van-ishes, Eq. �26� is thus reduced to

dtU = − 2RT , �27�

indicating that when a filament is displaced toward regionsof weaker background magnetic field and/or in the directionof curvature radius, its internal energy decreases. The re-duced internal energy will be converted into kinetic energyand magnetic energy as a result of energy conservation.

C. Conservation of total energy

The total energy H �Hamiltonian� is composed of mag-netic energy W, thermal energy U, kinetic energy K �1 /2�miu

2=K� +K�, and electrostatic energy WE

�1 /2��0E2V. In a low-� plasma generally speaking W�U�K�WE. In a quasineutral plasma, since B0

−2��0, thekinetic energy of E�B motion is much larger than the elec-



trostatic energy K��WE, hence we will hereafter neglect theelectrostatic energy. On the timescale of ��, if �d��, alldissipation effects can be neglected.

Now, let us focus on the outboard midplane, where theparallel compression nearly vanishes dtK�→0, thus the con-servation of total energy requires

dtH = 0, �28�

where H W+U+K�. Combining Eqs. �27� and �28�, wehave

dt�K� + W�� = 2RT . �29�

This formula depicts the generation of perpendicular kineticenergy and magnetic energy through thermal expansion.From Eq. �29�, we get the filament acceleration in the direc-tion of curvature radius gint�cs

2 /R, the interchange accelera-tion time �int�cs

−1�RL��1/2, and the interchange velocityuint�cs�L� /R�1/2�10%cs. The interchange velocity is theupper limit of filament transverse motion velocity24,25 �seeTable I�.

We can use a simple model to include the dissipationeffects,

dt�K� + W�� = 2RT − �d−1�K� + W�� . �30�

On the timescale of �� the parallel kinetic energy K� willparticipate in the energy partition. In that case part of thefilament’s energy is transferred into the kinetic energy ofparallel flow through the parallel compression, depicted byEq. �17�. With the help of Eq. �16�, Eq. �27� can be rewrittenin the following forms:

dt ln p = − 2�R , �31�

dt ln T = 2�1 − ��R . �32�

With the help of Eq. �13�, Eq. �25� can be rewritten in theconservation forms

dt�pn−�� = dt�Tn1−�� = dt�pB0−2�� = dt�TB0

2�1−�� = 0. �33�

We recognize that dt�pn−��=0 is just the adiabatic equation.

VI. FILAMENT ACCELERATION AND VORTICITYGENERATION

In the last section, we explained how the free energystored in background pressure gradient is released and con-verted into the kinetic energy of convective motion and themagnetic energy of field line bending associated with thedrift-Alfvén vortices. In this section, we will review this pro-cess in a dynamic perspective. The acceleration of plasmafilament is induced by a force unbalance in the direction ofcurvature radius resulting from the plasma inertial responseon the timescale of ��, reacted to satisfy quasineutrality. Arestoring force associated with the resultant force of mag-netic tension does work, generating the magnetic energy offield line bending.

A. Vorticity equation

The plasma current j= j f + jp+ jb is composed of the freecurrent j f, the polarization current jp, and the magnetization�bound� current jb, where the free current j f = j� + jG can befurther decomposed into the parallel current j� and themagnetic-drift current jG. The polarization current jp enup

is dominated by the contribution from the ion polarizationdrift. The magnetization current is defined as jb ��M= jd− jB,103 where jd B−1b��p� is the diamagnetic currentand jB B−1p��b�� ln B+��b� is a component of themagnetic-drift current. The magnetic-drift current is inducedby the oppositely directed guiding center drifts of ions andelectrons in an inhomogeneous magnetic field jG B−1�p�b�� ln B+ p��b�= jB+ ja, where ja B−1�p� − p��b isinduced by anisotropy. The total perpendicular current den-sity can be approximately written in the form j�� jb+ jG

+ jp= jd+ ja+ jp.104

Noting � · jb=0, we have the quasineutral condition

� · j� + � · j� = � · jp + � · jG + B��j�B−1� = 0, �34�

where the compression of magnetic-drift current is given by

� · jG = g · ��p� + p��

+ �0B−2j · ��p� + �p� − p�� ln B�

� g · ��p� + p�� . �35�

With the Boussinesq approximation105 the compression ofpolarization current can be written as

� · jp � dt� B−1� . �36�

The compression of diamagnetic current is given by

� · jd = � · jB = 2g · �p� + �0B−2j · �p� � 2g · �p�.

�37�

Inserting Eqs. �35� and �36� into the quasineutral condition�34�, we arrive at the vorticity equation24,25

dt� B−1� = B��j�B−1� + g · ��p� + p�� . �38�

Another approach to get the vorticity equation is by tak-ing curl of the motion equation,

dtu = j� B − ��p� − ��p� − �p� − p�� − �� ln B�

+ � · � , �39�

and forming the scalar product with b.106 The field line cur-vature is �=�� ln B+�0B−2j�B and b��= ��b��

=Bg+�0B−1j�. With the Boussinesq approximation,105 weobtain the same vorticity equation as Eq. �38�.24,25

The last term in Eq. �38� is induced by the perpendicularcompressibility. This term is the same term as that on theRHS of Eq. �29�. This term releases the free energy stored inbackground pressure gradient and converts them into the ki-netic energy of convective motion resulting in vorticity gen-eration, and into the magnetic energy of field line bendingthrough the parallel current. The parallel current is containedin the first term on the RHS of Eq. �38�. One can see that thephysics described by Eq. �29� is essentially the same as thosein the vorticity Eq. �38�.



If the plasma is incompressible, the RHS of Eq. �38� willvanish, dt� B−1�=0. With the help of the magnetic flux in-variant, dt�=0, we have I �� ·d�= �=const. It is justthe integral form of Kelvin’s frozen-in law for incompress-ible fluid.68 Its differential form is �t�=�� u��, imply-ing that vortices are frozen in fluid elements and movingwith them. In a compressible fluid this law is violated.

The interchange mechanism leading to radial motion of afilamentary structure is illustrated in Fig. 6. In tokamak ge-ometry g is in the vertical direction, so that the charge po-larization induced by the magnetic guiding center drifts is inthe vertical direction, leading to a radial electric drift at thecenter of the filamentary structure,19–25 and a net force due tothe vertical polarization current drives the filament to accel-erate in the direction of curvature radius, which is followedby the formation of dipolar vorticity24,25 and field linebending.33

B. Equivalent circuit

From the quasineutral condition �34�, one can see thatthe curvature and grad-B drifts function as a local currentsource. The currents generated by this current source flowalong and across the field lines compete through their effec-tive impedance and establish current loops,27,107 as illustratedin the following equivalent circuit diagram.

Plasma inertia plays the role of a capacitor. The currentflowing through the capacitor is the polarization current. Thiscurrent is responsible for filament acceleration and accumu-lation of perpendicular kinetic energy. All dissipation pro-cesses, such as the Joule dissipation, viscous damping, non-linear wave-particle Landau interaction, or sheathdissipation, can be represented by an equivalent resistor, asshown in Fig. 7. The dissipation processes are irreversiblesince they are accompanied by entropy production. Parallelcurrent provides a channel to couple the electrostatic vortexdynamics with the KSAW. The coupling process is similar tothat occurring in an inductor, as illustrated in Fig. 7. Mag-netic energy is accumulated in the inductor during this pro-cess. The inductance of a filament is of the order of �0L�.

In the electrostatic limit ��1�, there is nearly no field

line bending. This special case corresponds to a very smallequivalent inductance, in a tokamak it implies L��qR andL��i, showing that the electrostatic limit is more appli-cable to smallscale structures. For mesoscale structures, ��1, L� ��qR, and L��s, therefore the mesoscale struc-tures are essentially electromagnetic, and compared withsmall scale structures they are more close to the MHD limit.

In the ideal MHD limit ��=1�, the parallel emf � van-ishes and the induction part of parallel electric field cancelsthe electrostatic part, leading to vanishing parallel electricfield E� =−��−�tA�→0. The parallel streaming of adiabaticelectrons is strongly impeded by the magnetic induction sothat the neutralization along the field lines is incomplete. Fordetailed discussion about the magnetic induction effects,please see Sec. III E.

C. Plasma equilibrium

On the timescale of plasma equilibrium �eq��, theplasma inertial term �� · jp0��eq

−1� is negligible, then thequasineutral condition is reduced to � · jG0+� · j�0=0, i.e.,g ·��p�0+ p�0�=−B0��j�0B0

−1�. In tokamak plasmas it meansthat an electric charge separation in the vertical directioninduced by magnetic guiding center drifts �see Fig. 6� isneutralized by a parallel return current, i.e., the so calledPfirsch–Schlüter current. The rotation transformation of fieldlines in tokamak geometry guarantees the current loop isclosed and the neutralization is nearly accomplished �excepta small neoclassical parallel electric field E�0=��j�0�. The

equilibrium equation is given by ��W0�1+��=�0

−1�B0 ·��B0, indicating that the gradient of magnetic pres-sure plus thermal pressure is balanced by the resultant force�in the curvature direction� of the magnetic tension �alongfield line� due to the field line curvature �0=�� ln B0

+�0B0−2��p0��1+�� ln B0. This force acts as a restoring

force in the filamentary dynamics.

D. Filament acceleration

Plasma has a natural tendency to expand in the directionof curvature radius in an inhomogeneous magnetic field. Thethermal expansion force is usually referred to as an effectivegravity,33

B

κ B

E jp uE Fp

g = B-1 b × κκ

FIG. 6. �Color� Illustration of the interchange mechanism leading to radialmotion of a filamentary structure in tokamak geometry. The vertical polar-ization and vorticity due to magnetic guiding center drifts results in a radialelectric drift at the center of the filamentary structure and a net force due tothe vertical polarization current drives the filament to accelerate in the di-rection of curvature radius, which is followed by the formation of dipolarvorticity and field line bending.

j|| = -µ0-1 ∇⊥2A||

jG jp

Dissipa

tion

φ

FIG. 7. The equivalent circuit diagram for an Alfvén vortex filament.



FG − �p� + p�� ln B − p��0B−2j� B

� − �p� + p�� . �40�

On the equilibrium timescale this thermal expansion force isbalanced by a magnetic force, i.e., the resultant force of themagnetic tension FB0= jG0�B0=−FG0.

On the timescale of interchange motion �� the plasmainertia term � · jp becomes important. In tokamak geometrythe magnetic guiding center drifts induce a local charge po-larization in the vertical direction �see Fig. 6�. There areseveral mechanisms impeding the parallel motion of freeelectrons, such as magnetic induction, magnetic trapping,electron-ion collisions, electron inertia, and wave-electronLandau interaction, so that the parallel motion of electronscannot completely neutralize the charge polarization on theshort timescale ��. Thus, a polarization current jp in the ver-tical direction �see Fig. 6� will react to satisfy the localquasineutrality. The vertical charge polarization and the re-sultant vertical electric field leads to a radial E�B drift atthe center of the filamentary structure �see Fig. 6�, whichdrives plasma filament to move in the direction of curvatureradius. This interchange mechanism for filament motionplays a central role in the 2D theoretical models of blob19–23

and was highlighted in our recent work.24,25,86–89

From the dynamic point of view, the filament accelera-tion is induced by a force unbalance in the direction of cur-vature radius. The magnetic guiding center drift current jG islocally partially canceled by the vertical polarization currentjp, so that the local magnetic force is reduced FB= �jG+ jp��B, which can no longer balance the thermal expansionforce FG, and then the net force

Fp = FG + FB = jp� B �41�

will push plasma to accelerate in the direction of curvatureradius. During this process the thermal expansion force doeswork,

uE · FG = jG · �� = ndt�K� + W�� = 2Rp , �42�

where Eq. �29� has been used. As a consequence, the thermalenergy stored in the background pressure gradient is tappedoff. Note that only part of the work is converted into thekinetic energy of convective motion, leading to the filamentacceleration and electrostatic vorticity generation. This partof work is done by the net force.

uE · Fp = ndtK�. �43�

The other part of the work is converted into the magneticenergy of field line bending, and resulting in the generationof magnetic vorticity �A ��B�. This part of work is doneby the restoring force, i.e., the resultant force of the magnetictension

uE · FB = − ndtW�. �44�

VII. CROSS-FIELD TRANSPORTDUE TO THE VIOLATION OF THE THIRDADIABATIC INVARIANCE

In this section we discuss the fundamental kineticmechanism responsible for the cross-field transport associ-ated with the filamentary dynamics. If the particle system ofa magnetized plasma is completely adiabatic, that is to say,there is no energy exchange between particles and electro-magnetic field and no collision or other dissipation process,particle orbits are exactly determined by several adiabaticinvariants. In tokamak geometry there are three such adia-batic invariants: � −�1 /2�bB−1mv�

2 is the magnetic mo-ment �the action of the gyromotion�, J �mv�dl� is the lon-gitudinal invariant �the action of parallel bounce�, and �p isthe poloidal magnetic flux �the action of procession� whichdefines magnetic flux surfaces. If the particle system is com-pletely adiabatic, the three actions are conserved and no par-ticle can escape from the geometry space confined by themagnetic field. Each adiabatic invariant corresponds to atype of periodic motion. The period of the gyromotion is �c,the period of the bounce motion is �b, the period of toroidalprocession is �p, and usually �p��b��c.

In a quasineutral plasma the electrostatic energy WE is sosmall that the energy exchange between particles and elec-trostatic field is negligible. If the timescale for energy ex-change between particles and magnetic field is shorter thanor comparable with the period of an adiabatic motion, thecorresponding adiabatic invariant will be violated.

From Eqs. �20�, �29�, and �44�, we have known that thetimescale for the energy exchange between particles andmagnetic field, i.e., the timescale for field line bending, is thetimescale for interchange motion ��. This timescale is typi-cally short compared to the period of toroidal procession �p,implying that the magnetic flux invariant �p is not conservedduring this process, and, as a consequence, particles can es-cape across the nested magnetic flux surfaces.

Note that filaments are localized structures with finitetoroidal mode number. When a filament is generated, it willinduce local field line bending, which is accompanied by Btoroidal symmetry breaking on the timescale of ��. This isone of the fundamental kinetic mechanisms for cross-fieldturbulent transport in a toroidal magnetic confinement sys-tem.

VIII. LAGRANGIAN INVARIANTS

In this section we summarize the involved Lagrangianinvariants. The filament motion is controlled by six Lagrang-ian invariants. The first Lagrangian invariant is the magneticflux, introduced in Eq. �6�. The second is the magnetic energyof the equilibrium magnetic field B0, introduced in Eq. �13�.The third is the specific entropy, introduced in Eq. �24�. Thefourth is the total energy �Hamiltonian�, introduced in Eq.�28�. The other two are magnetic momentum and toroidalangular momentum.

A. Conservation of magnetic momentum

The timescale for interchange motion is much longerthan the ion gyroperiod ��ci, so that the adiabatic condi-



tion for gyromotion is satisfied and the magnetic moment �is an adiabatic invariant. Also, the ion-ion collision time �ii,the ion temperature isotropization time �Ti, and the electron-ion energy exchange time �ex are much longer than �� seeTable I�, thus the magnetic momentum of ion is conserved.But for electrons, in most of the boundary region, theelectron-ion collision time �ei and the electron temperatureisotropization time �Te are shorter than ��. Therefore, in theplasma boundary region, the average magnetic momentum�=T�B−1 is generally a Lagrangian invariant for ions butnot for electrons, where the average is over velocity space.

dt� = 0. �45�

With the help of Eq. �33�, we find when the temperatureis isotropic, T� =T�=T, the ratio of specific heats is �=5 /3.The corresponding degrees of freedom is N=2��−1�−1=3.Using Eq. �25�, �33�, and �45�, we obtain the Chew, Gold-berger, and Low �CGL� double adiabatic equations,108

dt�p�B−1n−1� = dt�p�B2n−3� = 0. �46�

On the outboard midplane, with the help of Eqs. �16�, �27�,�31�, and �32�, Eq. �46� can be rewritten in the followingforms:

dtU = − �T� + T��R , �47�

dt ln T� = − R , �48�

12 dt ln T� = − R , �49�

dt ln p� = − 3R , �50�

12 dt ln p� = − 2R . �51�

B. Conservation of toroidal angular momentum

In toroidal configuration the toroidal angular momentumdensity L� is a conserved quantity.109 Similar to the equationof continuity �15�, we have the conservation equation ofL�,

76,77

dt ln L� = − � · u = 2�J − R� . �52�

Combining the continuity Eqs. �15� and �52�, we find that the

toroidal angular momentum �� VL�= Ru� is a Lagrangianinvariant,

dt�� = dt�L�B0−2� = dt�Ru�� = 0, �53�

where R R2� / R� is an effective major radius and it is anincreasing function of r �minor radius�, and u� is the toroidalrotation velocity. For a simple torus with concentric circular

flux surfaces, R2�R02�1+ �3 /2��2�.76 The moment of inertia

density is related to the effective major radius through I

R2, which suggests that the core is less inert, i.e., lighter,than the edge, so if the toroidal angular momentum �� ishomogenized by turbulence, the rotation near the core willbe faster than at the edge. Because of this r dependence ofthe moment of inertia density in toroidal geometry, the linear

toroidal momentum density P� u�=L� / R is not con-served. With the definition of the angular velocity �� R−1u�, the toroidal angular momentum density can be ex-pressed as L�= I��. From Eq. �53�, using the relation be-tween the toroidal angular momentum density and the linear

toroidal momentum density, we have dt�P�RB0−2�=0.

Assuming filaments carry toroidal momentum, Eq. �53�implies that, when a filament is displaced outwards, R in-creases, and thus the toroidal rotation velocity u� will bereduced. On the contrary, when a filament is displaced in-wards, u� will increase. This is an anomalous pinch effect,which has recently been proposed to explain the phenomenaof the so-called intrinsic/spontaneous rotation in the plasmacore.76,77 It was observed in recent experiments110 that therotation profiles were peaked in the plasma core without ex-ternal momentum injection. In general, an anomalous“pinch” effect is required to explain this observation.

On the outboard midplane, similar to Eq. �16�, Eq. �52�is reduced to

dt ln L� = − 2R . �54�

One can see that the behaviors of toroidal angular momen-tum density are very similar to the particle density.76,77

IX. CURVATURE-DRIVEN CONVECTIVE TRANSPORTAND ANOMALOUS PINCH EFFECTS

A strongly intermittent nature of cross-field transport ofparticles and heat in the boundary region of fusion plasmashas been recognized for more than one decade.2,3 There arestrong indications that this is caused by field-aligned fila-mentary structures in the form of plasma blobs propagatingradially far into the SOL.33 In this paper we suggested thatthese structures can be associated with the drift-Alfvén vor-tices. The intermittent convective transports of particles,heat, and momentum across magnetic field can be interpretedin terms of the ballistic motion of these solitary filamentarystructures. To quantitatively calculate the resultant transportfluxes, we need nonlinear electromagnetic turbulencesimulations,35–37,101 but for a qualitative estimation we canuse the quasilinear method.76 The mode-independent part ofthe curvature-driven turbulent convective pinch of particles,heat, and momentum111 are briefly reviewed in this section.The phenomena of profile consistency, or resilience, are gen-erally thought to be associated with these anomalous pincheffects.112 We divide the discussion below into three subsec-tions according to different transport categories.

A. Particle transport

To simplify the further analytic process we use the toka-mak configuration described in Sec. IV C. The radial gradi-ent of the background magnetic field B0=Bc�1+ � cos ��−1

is �rB0!=−!B0

!R−1 cos �. The symbol for perpendicular com-pressibility R��1 /2�� ·uE can be expressed as R=uRR−1

=urR−1 cos �, where uR is a component of uE in the direction

of curvature radius and ur is the radial component. Assumingthe perpendicular kinetic energy associated with the filamen-tary structures bears ballooning distribution like K�= f��



�K��, where f�� "−1�−1/2 exp�−�2 /"2� is the poloidaldistribution function and " is a factor characterizing balloon-ing degree. We note that the poloidal distribution function isa weighting function, then we have K�R−1 cos �� / K��Rout

−1 , where Rout R0+r denotes the major radius on theoutboard side �low-field side�.

Following the quasilinear method,76 separating n= n�+ n, substituting it into Eq. �13�, we have �tn+B0

2ur�r�n�B0−2�=0, indicating that nB0

−2 is conserved in theconvective process. Note the timescale for density variationis ��. Then, the density perturbation can be expressed as

n = − ��ur�rn� − 2��n�urR−1 cos � . �55�

Substituting the density perturbation �55� into the radial con-vective particle flux,

n nur� = − D�rn� + Vnrn� , �56�

we get the turbulent diffusivity D mi−1��K�� and the

curvature-driven pinch velocity Vnr −2Rout−1 D, where we

rendered K�=miur2. This anomalous pinch effect is induced

by curvature-driven convection, as a consequence the pinchvelocity is proportional to Rout

−1 . Both the turbulent diffusivityand the anomalous pinch velocity are proportional to thefluctuation intensity K��.

Note that there is no offdiagonal term in Eq. �56�, suchas the temperature gradient term. According to the principleof Onsager symmetries there should be no off diagonal termin the temperature transport equations. It was suggested that,when the FLR effects are taken into account, off diagonalterms will appear in the transport matrix.72 Substituting theradial particle flux �56� into the particle transport equation,

�tn� + r−1�r�rn� = 0, �57�

we finally arrive at the density evolution equation. With thisequation one can calculate density profile and its evolution.

B. Thermal transport

Following the same procedure as in Sec. IX A, we sepa-

rate p= p�+ p and T= T�+ T. Substituting them into Eq.

�33�, we have �tp+B02�ur�r�p�B0

−2��=0 and �tT+B0

2��−1�ur�r�T�B02�1−��=0, indicating that pB0

−2� andTB0

2�1−�� are conserved in the convection process. The pres-sure perturbation and the temperature perturbation can bewritten as

p = − ��ur�rp� − 2��p�urR−1 cos � , �58�

T = − ��ur�rT� − 2�� − 1��T�urR−1 cos � . �59�

Substituting the temperature perturbation �59� into theradial conductive thermal flux,

T Tur� = − D�rT� + VTrT� , �60�

where D is still the turbulent diffusivity and VTr −2��−1�Rout

−1 D is the conductive thermal pinch velocity. Substitut-ing the pressure perturbation �58� into the radial thermal flux�convective plus conductive thermal flux�,

p pur� = − D�rp� + Vprp� = T�n + n�T, �61�

where Vpr −2�Rout−1 D is the anomalous thermal pinch veloc-

ity.Equations �31� and �32� can be rewritten as

�tp + � · �puE� = 2�1 − ��pR , �62�

�tT + � · �TuE� = 2�2 − ��TR . �63�

Note Eqs. �31� and �32� are written on the outboard mid-plane. Inserting Eqs. �58� and �59� into Eqs. �62� and �63�and taking average over a flux surface, we obtain the thermaltransport equations,

�tp� + r−1�r�rp� = �1 − ��Vnr��rp� − VprD−1p�� , �64�

�tT� + r−1�r�rT� = �2 − ��Vnr��rT� − VTrD−1T�� . �65�

Using these thermal transport equations, one can calculatethe self-consistent profiles of pressure and temperature.

C. Toroidal momentum transport

It was pointed out in a recent research76,77 that the be-haviors of toroidal angular momentum density L� are verysimilar to the particle density since they are both conservedquantities. Similar to the procedure in Sec. IX A, we obtainthe L� flux,

L = − D�rL�� + VnrL�� , �66�

and the L� transport equation,

�tL�� + r−1�r�rL� = 0. �67�

As noted,76,77 it is the toroidal flow velocity u�=L�−1R−1

rather than the toroidal angular momentum density L� that ismeasured in experiments. The toroidal velocity profiles canbe directly measured by charge exchange recombinationspectroscopy diagnostics.110 In order to write the evolution oftoroidal flow, we need to disentangle the flow and the den-sity.

Separating u�= u��+ u�, substituting it into Eq. �53�, we

have �tu�+ R−1ur�r�Ru��=0. Then, the toroidal velocityperturbation is

u� = − ��ur�ru�� − ��uru��R−1�rR . �68�

Substituting the toroidal velocity perturbation �68� into theradial convective flux

� u�ur� = − D�ru�� + V�ru�� , �69�

where V�r −mi−1��K�R−1�rR� is the anomalous pinch ve-

locity of the toroidal velocity. Rewriting Eq. �53� as

�tu� + � · �u�u� = − u�urR−1�rR + 2u�R . �70�

Averaging Eq. �70� over a flux surface, and with the help ofEq. �68�, we get the transport equation for the toroidal ve-locity,



�tu�� + r−1�r�r�� = �Vnr − V�r��ru��

+ mi−1��K�R−2��rR�2

− 2K�R−1 cos �R−1�rR�u�� , �71�

where the last term on the RHS is a high order term. Usingthis transport equation, one can calculate the radial profile oftoroidal velocity.

X. DISCUSSIONS AND SUMMARY

In the last decade, the basic physics picture of transportin the plasma boundary of tokamaks and other magneticallyconfined fusion devices has shifted.113 This basic picture wasbased on the cross-field diffusive transport driven by localturbulent fluctuations,114 and in the SOL this cross-field dif-fusion competes with parallel classical transport toward ma-terial surfaces to establish a dynamic balance and determinethe SOL radial width.115 However, in the recent years accu-mulating experimental evidences have shown that the cross-field transport of plasma particles, heat, and momentum isdominated by intermittent convection mediated by radiallypropagating filamentary structures rather than diffusion.2,3,10

The presence of ballistic motion of solitary coherent objectsand bursty transport events break the linear flux-gradient re-lationship and make the cross-field transport exhibit nonlocalcharacter,32 namely, transport is not determined by local pa-rameters but is induced by propagating structures generatedsomewhere else.

Significant progress has been made in understanding the2D electrostatic dynamics of blobs in the plasmaboundary.19–25 Now, strong interest is arising on their 3D andelectromagnetic features.26–29,35–37 This interest is enhancedby some analogies with the ELM filaments in H-mode11–18

and the avalanchelike events in the plasma core region.38

Recently, the first experimental evidence showing the asso-ciation of the propagating plasma turbulent structures withthe drift-Alfvén vortices was obtained in the RFP device.44,45

Moreover, dipolar drift-Alfvén vortices were identified bothin the magnetospheric cusp47 and in the magnetosheath48–50

by the four-spacecraft Cluster mission. In this contributionthe generation mechanism and fundamental dynamics ofdrift-Alfvén vortex structures in a nonuniformly and stronglymagnetized plasma are revisited. This contribution can begenerally regarded as a concept upgrade from electrostaticfilamentary structure19–25 to electromagnetic filamentarystructure26–29,35–37 in response to the recent experimentalprogress43–53 in the context of intermittent convective trans-port mediated by propagating coherent structures. This is acontinuation of our previous work presented in Refs. 24, 25,32, 34, 54, 73, and 86–89.

The main points in this paper are summarized as fol-lows:

�A� The mesoscale structures �k�s�1 see Table I� do notobey the standard gyrokinetic ordering of microturbu-lence �k�s�1�, so that a special set of orderingscheme is employed. This ordering scheme is consis-tent with order-unity electrostatic fluctuation levels atthe plasma edge.

�B� The turbulence at fusion plasma boundary is essentiallyelectromagnetic even for low local value of plasma �.The perpendicular dynamic timescale �� is comparablewith the shear Alfvén time �A �see Table I�, as a resultthe electrostatic vortex motions and the KSAWs arecoupled, through the parallel current and Ampere’s law,leading to field line bending.

�C� The induction part �tA� of parallel electric field E� is ofthe same order as the electrostatic part �� and theparallel emf gradient ��, indicating the importance ofthe electromagnetic effects for mesoscale dynamics.Any imbalance in the parallel gradient force en��−�� on electrons will allow the parallel current j� andmagnetic perturbations A� to arise, driven by the so-called nonadiabatic part of the density fluctuations,which provides a channel to exchange internal energyof particles with the magnetic energy of field line bend-ing.

�D� The relative wave impedance � uF /uE=vph� /Z de-pends not only on the plasma � but also on the scalek�s. In a finite-� plasma, for mesoscale structuresk�s�1, � is of order unity ��1�,92,93 implying thatthe electromagnetic effects are more important for themesoscale dynamics than the smallscale dynamics. Inthe ideal MHD limit ��=1�, the field lines are exactlyfrozen in fluid elements and moving with them, and theparallel emf � vanishes. In the resistive MHD case��1�, only collisional resistivity can break Alfvén’sfrozen-in law. In progressing from the MHD limit ��=1� to the kinetics limit ��1�, the contribution ofparallel emf � increases and kinetic effects becomealternatives to resistive diffusion for breaking Alfvén’sfrozen-in law and decoupling the magnetic field andplasma. The mesoscale dynamics is somewhere in be-tween. Recent experiments on blobs from a RFPdevice44,45 and ELM filaments from ASDEX Upgradetokamak46 suggested some MHD behaviors ��1.

�E� In the SOL ��10 see Table I�, plasma filamentsquickly displace away from the frozen-in flux tube anddrift across field lines due to the magnetic diffusion.Inside the separatrix ��10 see Table I�, the magneticdiffusion effects are much weaker. The field lines aredragged away from the unperturbed magnetic field bythe plasma filaments at a speed of uF, where �

uF /uE�1.�F� In a low-� plasma the background magnetic energy is a

Lagrangian invariant and B02u is an incompressible

flow. In toroidal geometry the parallel compression ��

competes with the perpendicular compression ��. Onthe outboard midplane the parallel compression ��

nearly vanishes, and the perpendicular compressionthus dominates the density variation. At the top or bot-tom of the torus, the parallel and perpendicular com-pressions are of the same order. The spatial dependenceof �� /�� in a torus is one of the origins of ballooningstructures associated with the filamentary phenomenain toroidal geometry.



�G� In a tokamak, when the perpendicular interchange mo-tions and the KSAW are coupled, the KSAWs arelaunched from the unfavorable curvature region. Thegenerated magnetic energy of field line bending andperpendicular kinetic energy of convective motion, orthe parallel current density and the electrostatic vortic-ity, are related through Eqs. �18�–�20�.

�H� In fusion plasmas the free energy is stored in the back-ground pressure gradient. The generation of structuresrequires an effective mechanism to release the free en-ergy and do work so that the thermal energy of par-ticles can be converted into kinetic energy of structuremotion and magnetic energy of field line bending. Fora low-� plasma the energy transfer is through the re-versible pressure-volume work, which can be inter-preted in terms of the fundamental thermodynamic re-lation and the entropy equation. When the dissipationtimescale is much longer than the timescale for perpen-dicular convective motion �d�� which is a typicalcase in fusion plasmas except in the far SOL, see TableI�, the specific entropy is a Lagrangian invariant on thetimescale of ��. The description of the energy transferprocess is closed by the conservation of total energy.

�I� The acceleration of plasma filament is induced by aforce unbalance in the direction of curvature radius re-sulting from the plasma inertial response on the time-scale of ��, reacted to satisfy quasineutrality. A restor-ing force associated with the resultant force ofmagnetic tension does work, generating the magneticenergy of field line bending. The interchange mecha-nism leading to radial motion of filaments is illustratedin Fig. 6.

�J� An equivalent circuit �Fig. 7� is used to illustrate theprocesses in a drift-Alfvén vortex filament. The curva-ture and grad-B drifts function as a local currentsource. The currents generated by this current sourceflow along and across the field lines compete throughtheir effective impedance and establish current loops.Plasma inertia plays the role of a capacitor. All dissi-pation processes can be represented by an equivalentresistor. The process of coupling to the KSAW is simi-lar to that occurring in an inductor.

�K� Since plasma filaments are localized structures with fi-nite toroidal mode number. The filamentary structurespresent a channel for local energy exchange betweenparticles and magnetic field perturbations, leading tobreaking of the periodic orbits of particles and the tor-oidal symmetry of magnetic field and resulting in theviolation of the adiabatic invariance associated with thepoloidal magnetic flux. This is one of the fundamentalkinetic mechanisms for cross-field turbulent transportin a toroidal magnetic confinement system.

�L� The structure motions are controlled by six Lagrangianinvariants, namely, magnetic flux, background mag-netic energy, specific entropy, total energy, magneticmomentum, and angular momentum. The conservation

of magnetic momentum is consistent with the CGLdouble adiabatic equations. The behaviors of toroidalangular momentum density are very similar to the par-ticle density. Because of the radial dependence of themoment of inertia in toroidal geometry, an anomalouspinch effect emerges. This effect has recently been pro-posed to explain the phenomena of the so-calledintrinsic/spontaneous rotation in the plasma core.

�M� The intermittent convective transports of particles,heat, and momentum across magnetic field can be in-terpreted in terms of the ballistic motion of these soli-tary filamentary structures. The mode-independent partof the curvature-driven turbulent convective pinch ofparticles, heat, and momentum are briefly reviewed inSec. IX. The phenomena of profile consistency, or re-silience, are generally thought to be associated withthese anomalous pinch effects.

The quasilinear calculation presented in Sec. IX is onlyqualitative. For quantitative transport prediction one needsnonlinear electromagnetic turbulence simulations. Currentlythere have been several attempts in this direction, see thereferences in Refs. 40 and 41. Moreover, fully electromag-netic nonlinear gyrokinetic theory for edge turbulence hasnow come to be mature.83,84 A shift to a kinetic formulationmay be required to capture the kinetic effects, such as theneoclassical flow equilibrium. Efforts dedicated to the devel-opment of such gyrokinetic models of the plasma edge arenow underway.40,41,85

In summary, in this paper the ordering scheme and somefundamental aspects of filamentary structures at fusionplasma edge are reviewed. The Lagrangian-invariant-basedmethod was systemized and extended to include the electro-magnetic effects. The similarity of the electromagnetic fila-ments in fusion plasmas and in space plasmas suggests that itcould be a universal phenomenon in plasma turbulence. Theimportance of such phenomenon has been widely recog-nized. It provides a fundamental mechanism for cross-fieldtransport at the fusion plasma edge.116 The understanding ofthe plasma filamentary phenomena is rapidly evolvingthrough the combined numerical and experimental efforts,and we expect that progress in this field will be rapid in thenext several years.

ACKNOWLEDGMENTS

This work, supported by the European Communities un-der the contract of association between EURATOM/UKAEAand the National Natural Science Foundation of China underGrant Nos. 10605028, 10721505, and 10725523, was carriedout within the framework of the European Fusion Develop-ment Agreement. The views and opinions expressed hereindo not necessarily represent those of the European Commis-sion.

APPENDIX: PHYSICAL PARAMETERS AND DERIVEDQUANTITIES

See Table I.



TABLE I. Physical parameters and derived quantities for a typical filamentary structure �using the JET tokamakparameters� in the pedestal region �in H-mode� or periphery region �in L-mode�, in the vicinity of separatrix andin the SOL, assuming a pure deuterium plasma. We define the region between the pedestal �or periphery� andthe separatrix as the plasma edge, and the region outside the separatrix as the SOL. See text for explanation anddiscussion.

Quantity, symbol �unit� Pedestal or periphery Separatrix SOL

Particle density, n �m−3� 1�1020 3�1019 5�1018

Temperature, Te, Ti �eV� Te=Ti=1000 Te=Ti=100 Te=Ti /2=10

Magnetic field, B �T� 3 3 3

Major radius, R �m� 3 3 3

Safety factor, q 3 4 5

Parallel length, L� ��qR �m� 30 40 30

Background pressure gradient length, Lp �mm� �50 �50 �100

Typical vortex width, L� �mm� �10 �10 �10

Transverse motion velocity, u� �km/s� �1 �1 �1

Experimental cross-field diffusivity, Deff �m2 /s� �0.1 �1 �2

Vortex lifetime or eruption time, �life ��s� �100 �100 �100

Ion gyrofrequency, fci=eB /2�mi �MHz� 23 23 23

Ion plasma frequency, fpi=�pi /2� �MHz� 1.5�103 810 330

Debye length, D= ��0Te /ne2�1/2 �mm� 2.4�10−2 1.4�10−2 1�10−2

Ion gyroradius, i=vthi /�ci �mm� 2.2 0.7 0.3

Ion poloidal gyroradius, �i=iB /B� �mm� 43 13.6 6.1

Ion sound gyroradius, s=cs /�ci �mm� 2.2 0.7 0.26

Quasineutrality, i / D�c /vA=�pi /�ci 92 50 29

Magnetic curvature, i /R 7�10−4 2.3�10−4 1�10−4

Magnetization parameter, �=s /L� 0.22 0.07 0.026

Magnetic diffusion length, Lm= ��Dm�1/2 �mm� 0.6 3.3 17

Electron inertial length, Lpe=c /�pe �mm� 0.5 1 2.4

Ion inertial length, Lpi=c /�pi �mm� 32 59 144

Electron mean free path, e=vthe�ei �m� 140 5 0.35

Ion mean free path, i=vthi�ii �m� 170 7 2

Collisionality, ��=L� / e 0.2 7.7 86

Spatial anisotropy, L� /L� ��e1/2 3.3�10−4 2.5�10−4 3.3�10−4

Specific pressure, �e=2�0pe /B2��cs /vA�2= �s /Lpi�2 4.5�10−3 1.3�10−4 2.2�10−6

Mass ratio for deuterium, me /mi 2.7�10−4 2.7�10−4 2.7�10−4

Alfvén speed, vA= �B2 /�0nmi�1/2 �km/s� 4.6�103 8.5�103 2�104

Electron thermal speed, vthe= �2Te /me�1/2 �km/s� 2�104 6�103 2�103

Ion thermal speed, vthi= �2Ti /mi�1/2 �km/s� 310 98 44

Ion sound speed, cs��1/2vA �km/s� 310 98 38

Transverse Mach number, M =u� /cs 0.003 0.01 0.03

Interchange velocity, uint=cs�L� /R�1/2 �km/s� 18 6 2

Froude number, Fr=u� /uint 0.06 0.2 0.5

Magnetic diffusivity, Dm=� /�0=Lpe2 /�ei �m2 /s� 0.04 1 30

Classical cross-field diffusivity, Dcl=�eDm �m2 /s� 1.7�10−4 1.4�10−4 6.8�10−5

Bohm diffusivity, DB=Te /16B �m2 /s� 20 2 0.2

Kinematic viscosity, �=i2 /�ii �m2 /s� 8.4�10−3 6.6�10−3 2.2�10−3

Electron thermal conductivity, ��e=3.2�eiTe /me �m2 /s� 4�109 5�107 1�106

Ion thermal conductivity, ��i =3.9�iiTi /mi �m2 /s� 1�108 1.3�106 1.6�105

Vortex turnover time, ��=L� /u� ��s� �10 �10 �10

Shear Alfvén time, �A=L� /vA ��s� 6.5 4.7 1.4

Compressional Alfvén time, �A�=L� /vA ��s� 2�10−3 1�10−3 5�10−4

Parallel convective time, �� =L� /cs ��s� 100 410 790

Perpendicular sound transit time, �s=L� /cs ��s� 0.03 0.1 0.26

Ion transit time, �i� =L� /vthi ��s� 100 410 690

Electron transit time, �e� =L� /vthe ��s� 1.6 6.7 16

Magnetic diffusion time, �m=L�2 /Dm ��s� 2600 93 3

Magnetic reconnection time, �K= ��m�A�1/2 ��s� 130 20 2

Viscous dissipation time, ��=L�2 /� ��s� 1.2�104 1.5�104 4.6�104



1B. A. Carreras, IEEE Trans. Plasma Sci. 25, 1281 �1997�.2S. J. Zweben, J. A. Boedo, O. Grulke, C. Hidalgo, B. LaBombard, R. J.Maqueda, P. Scarin, and J. L. Terry, Plasma Phys. Controlled Fusion 49,S1 �2007�.

3J. A. Boedo, J. Nucl. Mater. 390–391, 29 �2009�.4O. Grulke, T. Klinger, M. Endler, A. Piel, and the W7-AS Team, Phys.Plasmas 8, 5171 �2001�.

5M. Spolaore, V. Antoni, E. Spada, H. Bergsåker, R. Cavazzana, J. R.Drake, E. Martines, G. Regnoli, G. Serianni, and N. Vianello, Phys. Rev.Lett. 93, 215003 �2004�.

6N. Ben Ayed, A. Kirk, B. Dudson, S. Tallents, R. G. L. Vann, H. R.Wilson, and the MAST team, Plasma Phys. Controlled Fusion 51, 035016�2009�.

7I. Furno, B. Labit, M. Podestà, A. Fasoli, S. H. Müller, F. M. Poli, P. Ricci,C. Theiler, S. Brunner, A. Diallo, and J. Graves, Phys. Rev. Lett. 100,055004 �2008�.

8T. A. Carter, Phys. Plasmas 13, 010701 �2006�.9W. Horton and Y.-H. Ichikawa, Chaos and Structures in Nonlinear Plas-mas �World Scientific, Singapore, 1996�.

10G. Serianni, M. Agostini, V. Antoni, R. Cavazzana, E. Martines, F. Sattin,P. Scarin, E. Spada, M. Spolaore, N. Vianello, and M. Zuin, Plasma Phys.Controlled Fusion 49, B267 �2007�.

11W. Fundamenski, W. Sailer, and JET EFDA contributors, Plasma Phys.Controlled Fusion 46, 233 �2004�.

12M. Endler, I. García-Cortés, C. Hidalgo, G. F. Matthews, ASDEX Team,and JET Team, Plasma Phys. Controlled Fusion 47, 219 �2005�.

13J. A. Boedo, D. L. Rudakov, E. Hollmann, D. S. Gray, K. H. Burrell, R. A.Moyer, G. R. McKee, R. Fonck,P. C. Stangeby, T. E. Evans, P. B. Snyder,A. W. Leonard, M. A. Mahdavi, M. J. Schaffer, W. P. West, M. E. Fen-stermacher, M. Groth, S. L. Allen, C. Lasnier, G. D. Porter, N. S. Wolf, R.J. Colchin, L. Zeng, G. Wang, J. G. Watkins, T. Takahashi, and DIII-DTeam, Phys. Plasmas 12, 072516 �2005�.

14A. Kirk, N. Ben Ayed, G. Counsell, B. Dudson, T. Eich, A. Herrmann, B.Koch, R. Martin, A. Meakins, S. Saarelma, R. Scannell, S. Tallents, M.Walsh, H. R. Wilson, and the MAST Team, Plasma Phys. Controlled Fu-sion 48, B433 �2006�.

15A. Herrmann, A. Kirk, A. Schmid, B. Koch, M. Laux, M. Maraschek, H.W. Mueller, J. Neuhauser, V. Rohde, M. Tsalas, E. Wolfrum, and theASDEX Upgrade Team, J. Nucl. Mater. 363–365, 528 �2007�.

16A. Schmid, A. Herrmann, H. W. Mueller, and the ASDEX Upgrade Team,Plasma Phys. Controlled Fusion 50, 045007 �2008�.

17B. D. Dudson, N. Ben Ayed, A. Kirk, H. R. Wilson, G. Counsell, X. Xu,M. Umansky, P. B. Snyder, B. Lloyd, and the MAST Team, Plasma Phys.Controlled Fusion 50, 124012 �2008�.

18C. Silva, W. Fundamenski, A. Alonso, B. Gonçalves, C. Hidalgo, M. A.Pedrosa, R. A. Pitts, M. Stemp, and JET EFDA Contributors, Plasma Phys.Controlled Fusion 51, 105001 �2009�.

19S. I. Krasheninnikov, Phys. Lett. A 283, 368 �2001�.20D. A. D’Ippolito, J. R. Myra, and S. I. Krasheninnikov, Phys. Plasmas 9,

222 �2002�.21N. Bian, S. Benkadda, J.-V. Paulsen, and O. E. Garcia, Phys. Plasmas 10,

671 �2003�.22O. E. Garcia, V. Naulin, A. H. Nielsen, and J. Juul Rasmussen, Phys. Rev.

Lett. 92, 165003 �2004�.23A. Y. Aydemir, Phys. Plasmas 12, 062503 �2005�.24O. E. Garcia, N. H. Bian, V. Naulin, A. H. Nielsen, and J. J. Rasmussen,

Phys. Plasmas 12, 090701 �2005�.25O. E. Garcia, N. H. Bian, and W. Fundamenski, Phys. Plasmas 13, 082309

�2006�.26S. I. Krasheninnikov, D. D. Ryutov, and G. Yu, J. Plasma Fusion Res. 6,

139 �2004�.27J. R. Myra and D. A. D’Ippolito, Phys. Plasmas 12, 092511 �2005�.28D. D. Ryutov, Phys. Plasmas 13, 122307 �2006�.29J. R. Myra, Phys. Plasmas 14, 102314 �2007�.30J. R. Myra, D. A. Russell, and D. A. D’Ippolito, Phys. Plasmas 15,

032304 �2008�.31S. Ishiguro and H. Hasegawa, J. Plasma Phys. 72, 1233 �2006�.32V. Naulin, J. Nucl. Mater. 363–365, 24 �2007�.33S. I. Krasheninnikov, D. A. D’Ippolito, and J. R. Myra, J. Plasma Phys.

74, 679 �2008�.34W. Fundamenski, O. E. Garcia, V. Naulin, R. A. Pitts, A. H. Nielsen, J.

Juul Rasmussen, J. Horacek, J. P. Graves, and JET EFDA Contributors,Nucl. Fusion 47, 417 �2007�.

35V. Naulin, Phys. Plasmas 10, 4016 �2003�.36B. D. Scott, Plasma Phys. Controlled Fusion 49, S25 �2007�.37T. T. Ribeiro and B. Scott, Plasma Phys. Controlled Fusion 50, 055007

�2008�.38P. A. Politzer, Phys. Rev. Lett. 84, 1192 �2000�.

TABLE I. �Continued.�

Quantity, symbol �unit� Pedestal or periphery Separatrix SOL

Electron thermal conductive time, ��e=L�

2 /��e ��s� 0.2 32 860

Ion thermal conductive time, ��i =L�

2 /��i ��s� 9 1.2�103 5.7�103

Electron-ion collisional time, �ei�Te3/2 /n ��s� 7 0.9 0.2

Ion-ion collisional time, �ii�Ti3/2 /n ��s� 550 70 42

Electron-ion energy exchange time, �ex=�eimi /3me ��s� 9000 1100 230

Electron temperature isotropization, �Te ��s� 26 3 0.7

Ion temperature isotropization, �Ti ��s� 1400 180 100

Interchange acceleration time, �int= �RL��1/2 /cs ��s� 0.6 2 4.6

Magnetic Reynolds number, Rm=u�L� /Dm 260 9.3 0.3

Lundquist number, Sm=RmvA /u� 1.2�106 8�104 7�103

Magnetic Prandtl number, Pm=Rm /Re=� /Dm 0.2 6�10−3 7�10−5

Reynolds number, Re=u�L� /� 1.2�103 1.5�103 4.6�103

Schmidt number, Sc=� /Dcl 49 46 32

Péclet number, Pe=Re Sc=u�L� /Dcl 6�104 7�104 1.5�105

Interchange acceleration, g=cs2 /R �m /s2� 3�1010 3�109 5�108

Rayleigh number, Ra=gL�3 /�2 4.5�108 7.3�107 1�108

Normalized blob size, a�=s�L�2 /Rs�1/5 �mm� 23 10 4.3

Normalized blob velocity, v�=cs�a� /R�1/2 �km/s� 27 5.7 1.4

Inertial parameter, �= �me /mi��qR /Lp�2 8.8 15.7 24.5

Inductive parameter, �=�e�qR /Lp�2 145 7.7 0.2

Resistive parameter, C=0.51�Lp /cs�ei 0.1 4.7 89



http://dx.doi.org/10.1109/27.650902

http://dx.doi.org/10.1088/0741-3335/49/7/S01

http://dx.doi.org/10.1016/j.jnucmat.2009.01.040

http://dx.doi.org/10.1063/1.1418021

http://dx.doi.org/10.1063/1.1418021

http://dx.doi.org/10.1103/PhysRevLett.93.215003


http://dx.doi.org/10.1088/0741-3335/51/3/035016


http://dx.doi.org/10.1063/1.2158929

http://dx.doi.org/10.1088/0741-3335/49/12B/S25

http://dx.doi.org/10.1088/0741-3335/49/12B/S25

http://dx.doi.org/10.1088/0741-3335/46/1/015

http://dx.doi.org/10.1088/0741-3335/46/1/015

http://dx.doi.org/10.1088/0741-3335/47/2/002

http://dx.doi.org/10.1063/1.1949224

http://dx.doi.org/10.1088/0741-3335/48/12B/S41

http://dx.doi.org/10.1088/0741-3335/48/12B/S41


http://dx.doi.org/10.1088/0741-3335/50/4/045007

http://dx.doi.org/10.1088/0741-3335/50/12/124012

http://dx.doi.org/10.1088/0741-3335/50/12/124012

http://dx.doi.org/10.1088/0741-3335/51/10/105001

http://dx.doi.org/10.1088/0741-3335/51/10/105001

http://dx.doi.org/10.1016/S0375-9601(01)00252-3

http://dx.doi.org/10.1063/1.1426394

http://dx.doi.org/10.1063/1.1541021



http://dx.doi.org/10.1063/1.1927539

http://dx.doi.org/10.1063/1.2044487

http://dx.doi.org/10.1063/1.2336422

http://dx.doi.org/10.1063/1.2048847

http://dx.doi.org/10.1063/1.2403092

http://dx.doi.org/10.1063/1.2776900

http://dx.doi.org/10.1063/1.2889419

http://dx.doi.org/10.1017/S0022377806006003


http://dx.doi.org/10.1017/S0022377807006940

http://dx.doi.org/10.1088/0029-5515/47/5/006

http://dx.doi.org/10.1063/1.1605951

http://dx.doi.org/10.1088/0741-3335/49/7/S02

http://dx.doi.org/10.1088/0741-3335/50/5/055007


39Y. Hamada, T. Watari, A. Nishizawa, K. Narihara, Y. Kawasumi, T. Ido,M. Kojima, and K. Toi, Phys. Rev. Lett. 96, 115003 �2006�.

40D. A. Batchelor, M. Beck, A. Becoulet, R. V. Budny, C. S. Chang, P. H.Diamond, J. Q. Dong, G. Y. Fu, A. Fukuyama, T. S. Hahm, D. E. Keyes,Y. Kishimoto, S. Klasky, L. L. Lao, K. Li, Z. Lin, B. Ludaescher, J.Manickam, N. Nakajima, T. Ozeki, N. Podhorszki, W. M. Tang, M. A.Vouk, R. E. Waltz, S. J. Wang, H. R. Wilson, X. Q. Xu, M. Yagi, and F.Zonca, Plasma Sci. Technol. 9, 312 �2007�.

41G. L. Falchetto, B. D. Scott, P. Angelino, A. Bottino, T. Dannert, V.Grandgirard, S. Janhunen, F. Jenko, S. Jolliet, A. Kendl, B. F. McMillan,V. Naulin, A. H. Nielsen, M. Ottaviani, A. G. Peeters, M. J. Pueschel, D.Reiser, T. T. Ribeiro, and M. Romanelli, Plasma Phys. Controlled Fusion50, 124015 �2008�.

42F. L. Hinton, M. N. Rosenbluth, and R. E. Waltz, Phys. Plasmas 10, 168�2003�.

43O. Grulke, S. Ullrich, T. Windisch, and T. Klinger, Plasma Phys. Con-trolled Fusion 49, B247 �2007�.

44M. Spolaore, N. Vianello, M. Agostini, R. Cavazzana, E. Martines, P.Scarin, G. Serianni, E. Spada, M. Zuin, and V. Antoni, Phys. Rev. Lett.102, 165001 �2009�.

45N. Vianello, M. Spolaore, E. Martines, R. Cavazzana, G. Serianni, M.Zuin, E. Spada, and V. Antoni, “Drift-Alfvén vortex structures in the edgeregion of a fusion relevant plasma,” Phys. Rev. Lett. �submitted�.

46E. Martines, N. Vianello, D. Sundkvist, M. Spolaore, M. Zuin, M. Agos-tini, V. Antoni, R. Cavazzana, C. Ionita, M. Maraschek, F. Mehlmann, H.W. Müller, V. Naulin, J. J. Rasmussen, V. Rohde, P. Scarin, R. Schrit-twieser, G. Serianni, E. Spada, the RFX-mod Team, and the ASDEX Up-grade Team, Plasma Phys. Controlled Fusion 51, 124053 �2009�; N. Vi-anello, V. Naulin, R. Schrittwieser, H. W. Muller, M. Zuin, C. Ionita, J. J.Rasmussen, F. Mehlmann, V. Rohde, R. Cavazzana, M. Maraschek, andthe ASDEX Upgrade Team, “Direct observation of current in type I ELMfilaments on Asdex Upgrade,” Phys. Rev. Lett. �submitted�.

47M. L. Goldstein, Nature �London� 436, 782 �2005�.48D. Sundkvist, V. Krasnoselskikh, P. K. Shukla, A. Vaivads, M. Andre, S.

Buchert, and H. Reme, Nature �London� 436, 825 �2005�.49O. Alexandrova, A. Mangeney, M. Maksimovic, N. Cornilleau-Wehrlin, J.

M. Bosqued, and M. André, J. Geophys. Res. 111, A12208, doi:10.1029/2006JA011934 �2006�.

50O. Alexandrova, Nonlinear Processes in Geophysics 15, 95 �2008�.51D. Sundkvist and S. D. Bale, Phys. Rev. Lett. 101, 065001 �2008�.52O. Alexandrova and J. Saur, Geophys. Res. Lett. 35, L15102,

doi:10.1029/2008GL034411 �2008�.53S. Tomczyk, S. W. McIntosh, S. L. Keil, P. G. Judge, T. Schad, D. H.

Seeley, and J. Edmondson, Science 317, 1192 �2007�.54W. Fundamenski, V. Naulin, T. Neukirch, O. E. Garcia, and J. Juul Ras-

mussen, Plasma Phys. Controlled Fusion 49, R43 �2007�.55J. E. Wahlund, P. Louarn, T. Chust, H. de Feraudy, A. Roux, B. Holback,

P. O. Dovner, and G. Holmgren, Geophys. Res. Lett. 21, 1831,doi:10.1029/94GL01289 �1994�.

56A. A. Pimenta, Y. Sahai, J. A. Bittencourt, and F. J. Rich, Geophys. Res.Lett. 34, L02820, doi:10.1029/2006GL028529 �2007�.

57A. Loarte, B. Lipschultz, A. S. Kukushkin, G. F. Matthews, P. C.Stangeby, N. Asakura, G. F. Counsell, G. Federici, A. Kallenbach, K.Krieger, A. Mahdavi, V. Philipps, D. Reiter, J. Roth, J. Strachan, D.Whyte, R. Doerner, T. Eich, W. Fundamenski, A. Herrmann, M. Fenster-macher, P. Gendrih, M. Groth, A. Kirschner, S. Konoshima, B. LaBom-bard, P. Lang, A. W. Leonard, P. Monier-Garbet, R. Neu, H. Pacher, B.Pegourie, R. A. Pitts, S. Takamura, J. Terry, E. Tsitrone, and the ITPAScrape-off Layer and Divertor Physics Topical Group, Nucl. Fusion 47,S203 �2007�.

58J. P. Gunn, C. Boucher, M. Dionne, I. Ďuran, V. Fuchs, T. Loarer, I.Nanobashvili, R. Pánek, J.-Y. Pascal, F. Saint-Laurent, J. Stöckel, T. VanRompuy, R. Zagórski, J. Adámek, J. Bucalossi, R. Dejarnac, P. Devynck,P. Hertout, M. Hron, G. Lebrun, P. Moreau, F. Rimini, A. Sarkissian, andG. Van Oost, J. Nucl. Mater. 363–365, 484 �2007�.

59N. Smick, B. LaBombard, and C. S. Pitcher, J. Nucl. Mater. 337–339, 281�2005�.

60S. K. Erents, R. A. Pitts, W. Fundamenski, J. P. Gunn, and G. F. Matthews,Plasma Phys. Controlled Fusion 46, 1757 �2004�.

61N. Asakura and ITPA SOL and Divertor Topical Group, J. Nucl. Mater.363–365, 41 �2007�.

62V. V. Yankov, JETP Lett. 60, 171 �1994�.63J. Nycander and V. V. Yankov, Phys. Plasmas 2, 2874 �1995�.

64M. B. Isichenko, A. V. Gruzinov, and P. H. Diamond, Phys. Rev. Lett. 74,4436 �1995�.

65M. B. Isichenko and N. V. Petviashvili, Phys. Plasmas 2, 3650 �1995�.66M. B. Isichenko, A. V. Gruzinov, P. H. Diamond, and P. N. Yushmanov,

Phys. Plasmas 3, 1916 �1996�.67J. Nycander and V. V. Yankov, Phys. Scr. T63, 174 �1996�.68V. V. Yankov and J. Nycander, Phys. Plasmas 4, 2907 �1997�.69M. B. Isichenko and V. V. Yankov, Phys. Rep. 283, 161 �1997�.70J. Nycander and J. J. Rasmussen, Plasma Phys. Controlled Fusion 39,

1861 �1997�.71V. Naulin, J. Nycander, and J. J. Rasmussen, Phys. Rev. Lett. 81, 4148

�1998�.72X. Garbet, N. Dubuit, E. Asp, Y. Sarazin, C. Bourdelle, P. Ghendrih, and

G. T. Hoang, Phys. Plasmas 12, 082511 �2005�.73O. E. Garcia, N. H. Bian, V. Naulin, A. H. Nielsen, and J. Juul Rasmussen,

Phys. Scr. T122, 104 �2006�.74J. Harvey, Phys. Today 48, 32 �1995�.75E. Fermi, Thermodynamics �Prentice-Hall, New York, 1937�.76Ö. D. Gürcan, P. H. Diamond, and T. S. Hahm, Phys. Rev. Lett. 100,

135001 �2008�.77T. S. Hahm, P. H. Diamond, O. D. Gurcan, and G. Rewoldt, Phys. Plasmas

15, 055902 �2008�.78F. Wagner and U. Stroth, Plasma Phys. Controlled Fusion 35, 1321

�1993�.79G. S. Xu, V. Naulin, W. Fundamenski, C. Hidalgo, J. A. Alonso, C. Silva,

B. Gonçalves, A. H. Nielsen, J. J. Rasmussen, S. I. Krasheninnikov, B. N.Wan, and M. Stamp, Nucl. Fusion 49, 092002 �2009�.

80B. D. Scott, Plasma Phys. Controlled. Fusion 39, 1635 �1997�.81B. D. Scott, Plasma Phys. Controlled. Fusion 45, A385 �2003�.82B. D. Scott, Plasma Phys. Controlled. Fusion 48, B277 �2006�.83H. Qin, R. H. Cohen, W. M. Nevins, and X. Q. Xu, Phys. Plasmas 14,

056110 �2007�.84T. S. Hahm, L. Wang, and J. Madsen, Phys. Plasmas 16, 022305 �2009�.85R. H. Cohen and X. Q. Xu, Contrib. Plasma Phys. 48, 212 �2008�.86O. E. Garcia, V. Naulin, A. H. Nielsen, and J. J. Rasmussen, Phys. Plasmas

12, 062309 �2005�.87O. E. Garcia, V. Naulin, A. H. Nielsen, and J. J. Rasmussen, Phys. Scr.

T122, 89 �2006�.88O. E. Garcia, J. Horacek, R. A. Pitts, A. H. Nielsen, W. Fundamenski, J. P.

Graves, V. Naulin, and J. J. Rasmussen, Plasma Phys. Controlled Fusion48, L1 �2006�.

89O. E. Garcia, J. Plasma Fusion Res. 4, 019 �2009�.90Here we use the term “charge polarization” instead of “charge separation,”

since in a quasineutral plasma the net charge is negligible, #=#b+# f

=�0� ·E, where the net charge # is composed of the bound charge #b=−� ·Pb and the free charge # f =� ·D. Here, Pb is the polarization per unitvolume and D=Pb+�0E�Pb is the electric displacement field. The con-servation of charge requires �t#+� · j=�t#b+� · jp=�t# f +� · j f =0, wherethe plasma current j= j f + jp+ jb is composed of the free current j f, thepolarization current jp, and the magnetization �bound� current jb. Noting� · jb=0. The details about these current components are addressed in Sec.VI.

91J. J. Ramos, Phys. Plasmas 14, 052506 �2007�.92O. G. Onishchenko, V. V. Krasnoselskikh, and O. A. Pokhotelov, Phys.

Plasmas 15, 022903 �2008�.93O. G. Onishchenko, O. A. Pokhotelov, V. V. Krasnoselskikh, and S. I.

Shatalov, Ann. Geophys. 27, 639 �2009�.94A. B. Mikhailovskii, V. P. Lakhin, G. D. Aburdzhaniya, L. A.

Mikhailovskaya, O. G. Onishchenko, and A. I. Smolyakov, Plasma Phys.Controlled. Fusion 29, 001 �1987�.

95V. I. Petviashvili and O. A. Pokhotelov, Solitary Waves in Plasmas and inthe Atmosphere �Gordon & Breach, London, 1992�.

96V. I. Petviashvili, Plasma Phys. Rep. 19, 256 �1993�.97P. K. Shukla, M. Y. Yu, and R. K. Varma, Phys. Fluids 28, 1719 �1985�.98W. Horton, Rev. Mod. Phys. 71, 735 �1999�.99P. B. Synder, Ph.D. thesis, Princeton University, 1999.100D. Biskamp, Nonlinear Magnetohydrodynamics �Cambridge University

Press, Cambridge, 1993�.101G. Fuhr, P. Beyer, S. Benkadda, and X. Garbet, Phys. Rev. Lett. 101,

195001 �2008�.102S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontov-

ich �Consultants Bureau, New York, 1965�, Vol. I, p. 205.103O. E. Garcia, Eur. J. Phys. 24, 331 �2003�.




http://dx.doi.org/10.1088/1009-0630/9/3/13

http://dx.doi.org/10.1088/0741-3335/50/12/124015

http://dx.doi.org/10.1063/1.1524630

http://dx.doi.org/10.1088/0741-3335/49/12B/S23

http://dx.doi.org/10.1088/0741-3335/49/12B/S23


http://dx.doi.org/10.1088/0741-3335/51/12/124053

http://dx.doi.org/10.1038/436782a

http://dx.doi.org/10.1038/nature03931

http://dx.doi.org/10.1029/2006JA011934


http://dx.doi.org/10.1029/2008GL034411

http://dx.doi.org/10.1126/science.1143304

http://dx.doi.org/10.1088/0741-3335/49/5/R01

http://dx.doi.org/10.1029/94GL01289

http://dx.doi.org/10.1029/2006GL028529

http://dx.doi.org/10.1029/2006GL028529

http://dx.doi.org/10.1088/0029-5515/47/6/S04



http://dx.doi.org/10.1088/0741-3335/46/11/006


http://dx.doi.org/10.1063/1.871186


http://dx.doi.org/10.1063/1.871064

http://dx.doi.org/10.1063/1.871987

http://dx.doi.org/10.1088/0031-8949/1996/T63/027

http://dx.doi.org/10.1063/1.872422

http://dx.doi.org/10.1016/S0370-1573(96)00058-0

http://dx.doi.org/10.1088/0741-3335/39/11/007


http://dx.doi.org/10.1063/1.1951667

http://dx.doi.org/10.1088/0031-8949/2006/T122/014

http://dx.doi.org/10.1063/1.881453


http://dx.doi.org/10.1063/1.2839293

http://dx.doi.org/10.1088/0741-3335/35/10/002

http://dx.doi.org/10.1088/0029-5515/49/9/092002

http://dx.doi.org/10.1088/0741-3335/39/10/010

http://dx.doi.org/10.1088/0741-3335/45/12A/025

http://dx.doi.org/10.1088/0741-3335/48/12B/S27

http://dx.doi.org/10.1063/1.2472596

http://dx.doi.org/10.1063/1.3073671

http://dx.doi.org/10.1002/ctpp.200810038

http://dx.doi.org/10.1063/1.1925617

http://dx.doi.org/10.1088/0031-8949/2006/T122/013

http://dx.doi.org/10.1088/0741-3335/48/1/L01

http://dx.doi.org/10.1585/pfr.4.019

http://dx.doi.org/10.1063/1.2717595

http://dx.doi.org/10.1063/1.2844744

http://dx.doi.org/10.1063/1.2844744

http://dx.doi.org/10.1088/0741-3335/29/1/001

http://dx.doi.org/10.1088/0741-3335/29/1/001

http://dx.doi.org/10.1063/1.864964

http://dx.doi.org/10.1103/RevModPhys.71.735


http://dx.doi.org/10.1088/0143-0807/24/4/351

104H. Qin, W. M. Tang, G. Rewoldt, and W. W. Lee, Phys. Plasmas 7, 991�2000�.

105G. Q. Yu, S. I. Krasheninnikov, and P. N. Guzdar, Phys. Plasmas 13,042508 �2006�.

106J. F. Drake and T. M. Antonsen, Phys. Fluids 27, 898 �1984�.107J. R. Myra, D. A. Russell, and D. A. D’Ippolito, Phys. Plasmas 13,

112502 �2006�.108G. L. Chew, M. L. Goldberger, and F. E. Low, Proc. R. Soc. London, Ser.

A 236, 112 �1956�.109F. L. Hinton and S. K. Wong, Phys. Fluids 28, 3082 �1985�.110J. E. Rice, A. Ince-Cushman, J. S. deGrassie, L.-G. Eriksson, Y. Saka-

moto, A. Scarabosio, A. Bortolon, K. H. Burrell, B. P. Duval, C. Fenzi-Bonizec, M. J. Greenwald, R. J. Groebner, G. T. Hoang, Y. Koide, E. S.Marmar, A. Pochelon, and Y. Podpaly, Nucl. Fusion 47, 1618 �2007�.

111T. S. Hahm, P. H. Diamond, O. D. Gurcan, and G. Rewoldt, Phys. Plas-mas 14, 072302 �2007�.

112N. H. Bian and O. E. Garcia, Phys. Plasmas 12, 042307 �2005�.113B. A. Carreras, J. Nucl. Mater. 337–339, 315 �2005�.114P. C. Liewer, Nucl. Fusion 25, 1281 �1985�.115P. C. Stangeby, The Plasma Boundary of Magnetic Fusion Devices �In-

stitute of Physics, Bristol, 2000�.116E. J. Doyle, W. A. Houlberg, Y. Kamada, V. Mukhovatov, T. H. Osborne,

A. Polevoi, G. Bateman, J. W. Connor, J. G. Cordey, T. Fujita, X. Garbet,T. S. Hahm, L. D. Horton, A. E. Hubbard, F. Imbeaux, F. Jenko, J. E.Kinsey, Y. Kishimoto, J. Li, T. C. Luce, Y. Martin, M. Ossipenko, V.Parai, A. Peeters, T. L. Rhodes, J. E. Rice, C. M. Roach, V. Rozhansky, F.Ryter, G. Saibene, R. Sartori, A. C. C. Sips, J. A. Snipes, M. Sugihara, E.J. Synakowski, H. Takenaga, T. Takizuka, K. Thomsen, M. R. Wade, H.R. Wilson, ITPA Transport Physics Topical Group, ITPA ConfinementDatabase and Modelling Topical Group, and ITPA Pedestal and EdgeTopical Group, Nucl. Fusion 47, S18 �2007�.



http://dx.doi.org/10.1063/1.873898

http://dx.doi.org/10.1063/1.2193087

http://dx.doi.org/10.1063/1.864680

http://dx.doi.org/10.1063/1.2364858

http://dx.doi.org/10.1098/rspa.1956.0116

http://dx.doi.org/10.1098/rspa.1956.0116

http://dx.doi.org/10.1063/1.865350

http://dx.doi.org/10.1088/0029-5515/47/11/025

http://dx.doi.org/10.1063/1.2743642

http://dx.doi.org/10.1063/1.2743642

http://dx.doi.org/10.1063/1.1867994


http://dx.doi.org/10.1088/0029-5515/47/6/S02

core.ac.uk · 39Y. Hamada, T. Watari, A. Nishizawa, K. Narihara, Y. Kawasumi, T. Ido, M. Kojima,...

Documents

Transcript of core.ac.uk · 39Y. Hamada, T. Watari, A. Nishizawa, K. Narihara, Y. Kawasumi, T. Ido, M. Kojima,...