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Microtearing turbulence: magnetic braiding anddisruption limitMarie-Christine Firpo

To cite this version:Marie-Christine Firpo. Microtearing turbulence: magnetic braiding and disruption limit. Physics ofPlasmas, American Institute of Physics, 2015, 22, pp.122511 �10.1063/1.4938273�. �hal-01248802�

Microtearing turbulence: magnetic braiding and disruption limitMarie-Christine Firpo1

Laboratoire de Physique des Plasmas, CNRS - Ecole Polytechnique, 91128 Palaiseau cedex,France

A realistic reduced model involving a large poloidal spectrum of microtearing modes is used to probe theexistence of some stochasticity of magnetic field lines. Stochasticity is shown to occur even for the low valuesof the magnetic perturbation δB/B devoted to magnetic turbulence that have been experimentally measured.Because the diffusion coefficient may strongly depend on the radial (or magnetic-flux) coordinate, being verylow near some resonant surfaces, and because its evaluation implicitly makes a normal diffusion hypothesis,one turns to another indicator appropriate to diagnose the confinement: the mean residence time of magneticfield lines. Their computation in the microturbulence frame points to the existence of a disruption limit,namely of a critical order of magnitude of δB/B above which stochasticity is no longer benign yet leads toa macroscopic loss of confinement in some tens to hundred of electron toroidal excursions. Since the level ofmagnetic turbulence δB/B has been measured to grow with the plasma electron density this would also be adensity limit.

I. INTRODUCTION AND OBJECTIVES

In the tokamak terminology, magnetic microturbulencerefers to the simultaneous excitation of a large spec-trum of magnetic modes with poloidal mode numbersm ≫ 1. It might however be thought that the small-ness of the relative magnitude of the magnetic pertur-bation δB/B devoted to magnetic turbulence as mea-sured in some tokamak experiments1,2 makes the ques-tion of the stochasticity of the magnetic field lines ratherirrelevant. Yet, several recent results coming from gy-rokinetic simulations have unveiled the ubiquity of themagnetic stochasticity under physical parameters rele-vant to fusion-oriented tokamak plasmas. In particular,electromagnetic numerical simulations in the conditionsof tokamak ion temperature gradient turbulence3 havedemonstrated that the magnetic field could be stochas-tic even at very low plasma pressure. Concurrently, an-other study8 aiming at unveiling the nature and mech-anisms behind turbulent transport in tokamak plasmasbased on ab initio gyrokinetic simulations demonstratedthat heat transport is dominated by the electron mag-netic component and pointed to microtearing modes as apossible candidate to explain turbulent transport in toka-mak plasmas. Moreover, other nonlinear gyrokinetic nu-merical simulations of microtearing mode turbulence us-ing the experimental parameters from a high-β dischargeof the spherical torus NSTX experiment9 indicated thatthe transport is almost entirely electromagnetic as a re-sult of the electrons diffusing in the stochastic magneticfield. Lastly, electromagnetic gyrokinetic simulations ofplasma microturbulence10 have also supported the near-ubiquitous character of magnetic stochasticity.Experimentally, there is increasing evidence of the mi-

crotearing nature of magnetic turbulence. Very recently,the first direct experimental verification of microtearingmode turbulence in the core region of tokamak plasmashas been reported in the JIPPT-IIU tokamak2, where lo-cal magnetic fluctuations were measured using a heavyion beam probe. In the ASDEX-Upgrade, edge mea-surements using electron cyclotron emission imaging just

detected also that the temperature fluctuations have fea-tures similar to microtearing modes11. Accordingly, thereis now a growing body of evidence both from numericalgyrokinetic simulations and novel experimental diagnos-tic approaches that magnetic fluctuations come under theform of microtearing modes that control electron heattransport.

Gyrokinetic numerical simulations are certainly use-ful since they should eventually provide a fine-scale self-consistent picture of tokamak plasma dynamics, yet theyare highly numerically-demanding and challenging. Itremains therefore desirable to continue to build and de-velop reduced and versatile models to improve the un-derstanding of the puzzling and critical tokamak trans-port issue and to test paths towards the improvementof the confinement. In order to reduce the complexityof the system formed by the strong equilibrium mag-netic field and the assemblies of charged particles self-consistently interacting with turbulent electromagneticperturbations, one may adopt a non self-consistent testparticle approach under suitable assumptions for the elec-tromagnetic field. A more fundamental approach is to fo-cus on the magnetic structure of the confinement. Indeedthe Maxwell equation divB = 0 forms a closed and exactequation enabling to isolate the effect of the magneticfield and study its properties. This does not mean at allthat one neglects the electric field or assumes the nullityof the electric perturbations. This does not mean eitherthat the trajectories of charged particles in a spatiallyintegrable magnetic field are integrable since the three-degrees of freedom Hamiltonian from which derive theparticle equations of motion may a priori allow chaoticmotions. Indeed in the case where the (integrable) mag-netic structure possesses an X-point, as in the divertorscheme, charged particles experience a chaotic motion inthe vicinity of the magnetic separatrices12. This is just areduction of the picture. It was notably shown by Caryand Littlejohn in a seminal paper13 that the zero diver-gence of the magnetic field, which may be interpreted asa condition of phase space conservation, makes the equa-tions of magnetic field lines derive from a Hamiltonian

2

system.This line of research connected years ago to the intro-

duction of the paradigmatic standard map14 that had aconsiderable importance in the understanding of the on-set and phase-space manifestations of Hamiltonian chaos.This captured the universal behaviour of area-preservingmaps with divided phase space when integrable islandsof stability are surrounded by a chaotic component. Yet,this model is not meant to address the quantitative trans-port issues posed by magnetic confinement fusion. Us-ing the present computing capacities and the improvedknowledge of the magnetic turbulence coming from ex-perimental measurements and recent gyrokinetic results,it has become possible to consider more realistic modelsof magnetic field lines under microturbulence conditionsand compute their transport properties. The microtur-bulence magnetic framework used in the present studywill be exposed in Section II. Some numerical results ob-tained within such microtearing turbulence models hav-ing a large spectrum of poloidal modes will be presented.It will be shown that a value of the magnitude of themagnetic perturbation as low as 10−5 does not implythat magnetic field lines are close to integrability.The problem of the disruptions in toroidal devices for

magnetic confinement fusion must be tamed to ensurethe viability of industrial fusion reactors. Contrarilyto macroscopic dramatic phenomena, such as the saw-tooth crash, that may be related to the onset of thechaos of magnetic field lines due to some subset of long-wavelength modes with different helicities (See e.g.4–7),the consequences of the loss of the integrity of mag-netic surfaces in the presence of plasma microturbu-lence have been observed in some gyrokinetic numericalsimulations3 to be normally benign or moderate. Yet,there is certainly some limit in the braiding of the mag-netic field lines above which magnetic confinement breaksand a disruption occurs. Section III will be devoted tothe exploration of disruption limits associated to mag-netic microturbulence. This study will involve the evalu-ation of some probability distribution functions (pdf) ofmagnetic confinement times. These are obtained throughrather demanding numerical simulations within the re-duced magnetic turbulence model. Yet, it is useful topoint out the benefit of using a reduced approach bynoting that the numerical cost of analogous simulationswithin a full gyrokinetic frame would have been muchhigher and possibly prohibitive. A short discussion con-cludes the work.

II. THE HAMILTONIAN MICROTEARINGTURBULENCE FRAMEWORK

A. Presentation

The canonical representation of an arbitrary magneticfield in a toroidal magnetic confinement device is writtenin Eq. (1). Let us consider a set of variables (ρ, θ, ϕ)

where ρ is a radius-like variable that vanishes along themagnetic axis. Then, there exist15 two single-valuedfunctions ψ(ρ, θ, ϕ) and Φ(ρ, θ, ϕ), such that the magneticfield can be written in the form

B = ∇ψ ×∇θ +∇ϕ×∇Φ. (1)

Eliminating the variable ρ by using Φ(ψ, θ, ϕ), the equa-tions for the magnetic field lines read

dψ

dϕ=

B.∇ψB.∇ϕ

= −∂Φ∂θ

, (2)

dθ

dϕ=

B.∇θB.∇ϕ

=∂Φ

∂ψ. (3)

In this representation, Φ plays the role of a genericallyone-and-a-half degrees of freedom Hamiltonian16–18, thetoroidal angle ϕ is a time-like variable and ψ is the mo-mentum conjugated to the poloidal angle θ. In a non-steady state, the field-line Hamiltonian depends also onthe real time t, that plays the role of a parameter, butnot of a canonical variable, and will be noted Φt.

In tokamaks, the equilibrium configuration is axisym-metric. The equilibrium magnetic field B0(r) has itsfield lines spiralling on perfect nested magnetic surfacesaround the magnetic axis and these are defined by a con-stant ψ. This amounts to say that the magnetic field-line Hamiltonian associated to B0 is integrable, beingΦ0(ψ). It identifies with the poloidal magnetic flux andψ with the toroidal magnetic flux. From the equilibriumpoloidal flux Φ0(ψ), one defines the very important quan-tity called the safety factor profile q(ψ), or equivalentlyits inverse, the winding profile w (ψ), through

dΦ0(ψ)

dψ=

1

q (ψ)= w(ψ). (4)

Yet, axisymmetry is bound to be broken by external ef-fects, such as the toroidal ripple in the magnetic field,which appears in tokamaks due to the finite number oftoroidal magnetic field coils, or as the edge magnetic per-turbations due to additional coils serving to control par-ticle and heat loads on tokamak walls19 or by some in-trinsic instability phenomena such as MHD activity ormicroturbulence. The generic form of the magnetic field-line Hamiltonian reads then

Φt(ψ, θ, ϕ) = Φ0(ψ) + δΦt(ψ, θ, ϕ). (5)

The order of magnitude of the relative perturbation|δB|/B0 quantifies the deviation from axisymmetry andintegrability. Using the poloidal and toroidal periodici-ties, this can be Fourier decomposed as

δΦ(ψ, θ, ϕ) =∑m,n

εm,nδΦmn(ψ) cos(mθ−nϕ+χmn), (6)

where from now on, the time dependence index has beendropped to simplify the notations. A rough model forthe behavior of microtearing waveforms was proposed by

3

Stix in28. In the present work, a more refined model ofmicrotearing turbulence described in27 will be used. Thisis an empirical experimentally-based model taking

δΦmn(ψ) ≡[cosh

(1

∆ln

ψ

ψmn

)]−m∆2

. (7)

The (m,n) waveform is maximal on the rational surfaceψmn, defined as the value of the toroidal magnetic flux forwhich q(ψmn) = m/n. The physically realistic value ofthe parameter ∆ is about 0.01. This ensures the correctminimally-smooth peaking of the waveforms. In numer-ical simulations, the poloidal modes with 1 < m < 40have been retained and the associated toroidal modesare such that the resonances (m,n) are present. Figure1 shows these modes. A standard winding profile is usedwith

w(ψ) = 0.2(2− ψ)(ψ2 − 2ψ + 2), (8)

corresponding to a monotonously growing q-profile. It

5 10 15 20 25 30 35 40n

5

10

15

20

25

30

35

40m

FIG. 1. The (m,n) modes retained in this study are withinthe limits imposed by the minimal and maximal q-valuesdefining respectively the lower and upper lines.

should be noted that the present model considers thewhole spectrum of magnetic modes, not the only thepart with the largest m modes that may be prop-erly defined as the turbulent part. This is consistentwith the experimental reality where relatively large scalemagnetohydrodynamic-like modes coexist with smallscale perturbations.

B. The Hamiltonian for a single mode: estimation of theresonance widths

Let us first characterize the Hamiltonian model inthe case of a single arbitrary mode (m0, n0). Us-ing a canonical transform with the generating functionF2(θ,Ψ, ϕ) = (m0θ − n0ϕ)Ψ, that amounts to moving tothe (m0, n0) wave frame, the single mode Hamiltonian

reads Φ(Ψ,Θ, ϕ) = Φ(ψ, θ, ϕ) + ∂F2/∂ϕ with ψ = m0Ψ

and Θ = m0θ − n0ϕ. This yields

Φ(Ψ,Θ) = Φ0(m0Ψ)− n0Ψ+ εm0,n0δΦm0n0(m0Ψ) cosΘ.(9)

Figure 2 and Figure 3 represent respectively the per-turbed Hamiltonian waveform and the topology of theHamiltonian phase space for the same one-wave inte-grable case. The fixed points of the Hamiltonian (9)

0.5 0.6 0.7 0.8 0.9 1Ψ

0.2

0.4

0.6

0.8

1

∆F33,8HΨL

FIG. 2. Perturbed Hamiltonian waveform δΦm0n0 for themode m0 = 33, n0 = 8 and for the parameter ∆ = 0.01. Theq-profile considered here is the same as in Figure 3.

-3 -2 -1 0 1 2 30.023

0.0235

0.024

0.0245

0.025

0.0255

0.026

0.0265

Q

Y

FIG. 3. Phase space portrait of magnetic field lines in thecase where a single microtearing mode is retained, displayedin the wave frame in the (Θ,Ψ) space. Here this is the modem0 = 33, n0 = 8. The q-profile used in the figure is q(ψ) =1.5 + 3.2ψ which gives ψm0=33,n0=8 ≃ 0.82 that corresponds,in the new wave frame variables, to Ψ ≃ 0.248. The amplitudeof the mode is εm0,n0 = 5.10−4.

are given by Θ = 0 or π. For Θ = π, the action ψ of thefixed points satisfies

0 =1

q(ψ)− n0m0

+εm0,n0

m0

2ψδΦm0n0(ψ) tanh

(1

∆ln

ψ

ψm0n0

).

(10)This equations admits an obvious solution ψ = ψm0n0

corresponding to an elliptic fixed point and it admitstwo other roots ψ+ > ψm0n0 and ψ− < ψm0n0 in the

4

case where the q-profile is monotonously increasing thatcorrespond to hyperbolic fixed points. We can then es-timate the resonance width associated to the (m0, n0)single wave Hamiltonian. The largest toroidal flux cor-responding to the upper half-width of the resonance, de-noted by ψ+, may be estimated from the energy conser-

vation on the upper separatrix so that Φ(ψ+/m0, π) =

Φ(ψ+/m0, 0). Under the assumption that ψ+ ≃ ψ+, thisyields

ψ+ − ψ+ ∼ εm0,n0

δΦm0n0(ψ+) + δΦm0n0(ψ

+)n0

m0− 1

q(ψ+)

. (11)

Finally, for the consistent hypotheses ensuring the small-ness of ψ+ − ψ+, namely εm0,n0

sufficiently small and anon-vanishing magnetic shear at the rational surface, itis possible to further simplify Eq. (11) by making ψ+ ≃ψm0n0 , δΦm0n0(ψ

+) ≃ δΦm0n0(ψ+) ≃ 1, and by using

q(ψ+

)−m0/n0 ≃ q′(m0/n0) (ψ

+ − ψm0n0). The expres-

sion of the resonance half-width, δψm0n0 = ψ+ − ψm0n0 ,amounts then to

δψm0n0 =

(2εm0,n0

|w′(ψm0n0)|

)1/2

=m0

n0

(2εm0,n0

q′(ψm0n0)

)1/2

.

(12)This is the classical expression of the microtearing reso-nance half-width.Unless otherwise specified, we define ε such that

εm,n ≡ εn/m, so that the mode amplitudes εm,n areindeed all of the same order ε. The expression ofthe resonance half-widths reduces then to δψm0n0 =

[2εq(ψm0n0)/q′(ψm0n0)]

1/2. The magnetic field pertur-

bation δBmn relates to εm,n through

δBmnB0ϕ

=1

2√2

mεm,n

A√ψmn

, (13)

where A denotes the tokamak aspect ratio (A = R0/a).Although the knowledge of the magnetic spectrum wouldbe necessary to make quantitative estimations, which re-mains a really challenging task due to the experimentaldifficulties to measure internal magnetic perturbations,the objective of the present study is more qualitativethan quantitative so that we shall mainly retain for ourpurposes that δB/B ∝ ε.

C. Characterization of the local magnetic stochasticity

The previous expression of the microtearing resonancehalf-widths may serve to estimate the local Chirikovparameter that quantifies the local chaoticity of theHamiltonian magnetic field lines. The expression ofthe Chirikov parameter relative to any couple of modes(m,n) and (m′, n′) reads

s(m,n),(m′,n′) =δψmn + δψm′n′

|ψmn − ψm′n′ |.

Figure 4 represents the local maximal Chirikov parameterassociated to the modes (m,n). More precisely, for eachresonant surface ψmn is plotted the maximal Chirikovparameter s(ψmn) = max(m′,n′) s(m,n),(m′,n′) divided by√ε that the mode (m,n) realizes with some other mode

(m′, n′). The extent of the local maximal stochasticityzone is represented by an horizontal segment betweeneach couple (ψmn, ψm′n′). From Figure 4, it is clear that

0.2 0.4 0.6 0.8 1Ψ

500

1000

1500

2000

2500

3000

3500

¶-1�2s

FIG. 4. For each resonant surface ψmn on the X-axis is plottedby a point the maximal Chirikov parameter s divided by

√ε

realized with some mode (m′, n′) among the available modes.The segments between the ψmn and their associated ψm′n′

are marked but are only wide enough to be visible about theresonances associated to the integer values of q. The q-profileused in the calculations corresponds to Eq. (8).

there are local minima of stochasticity in the close vicin-ity of the resonances associated to integer values of theq-profile while there are local maxima of stochasticity justin the periphery of these zones. This result is valid forq-profiles having a non-vanishing magnetic shear, that isa condition for the validity of Eq. (12), and is consis-tent with experimental results indicating that transportbarriers preferentially form about surfaces where q is aninteger.

Another consequence of the results shown on Figure 4is that, apart from those neighborhoods of the q = integerand low rational surfaces, some mid scale stochasticityis realized even at extremely low values of the relativemagnetic perturbation amplitude ε, in our case even atε ∼ 10−7 − 10−6. Experimentally, ε has been reported1

to be of the order of 10−5 to 10−4 (in the L mode) inTore Supra and, very recently, to be of order 10−4 inthe JIPPT-IIU tokamak plasmas2. Therefore, within themicrotearing turbulence frame, the stochasticity of themagnetic field lines under experimentally relevant valuesof ε is generic. This is in agreement with the seminalStix’s analysis28 and the aforementioned recent gyroki-netic results3,8–10.

5

D. Numerical integration of the equations for themagnetic field lines

The Hamiltonian dynamics associated to the equations(2)-(3) has been integrated using a symplectic integratorwith toroidal angle ’time’-step 2π/3200.Figures 5 and 6 represent the radial-type component,

ψ, of different magnetic field lines as a function of thetoroidal angle ϕ for the microturbulence model for twovalues of the relative magnetic perturbation, namely ε =2 · 10−5 and ε = 10−4. The non-integrable character ofthe magnetic field lines is quite manifest from the figures.However, whereas the behavior of the magnetic field

lines looks stochastic for ε = 10−4, this is much weakerfor ε = 2 · 10−5 where the magnetic field trajectoriescorrelate with the (m,n) resonances. As the magneticbraiding gets smaller, the stochasticity in the ψ-domainmay become locally quite low so that the resonances arenot fully disaggregated behaving as cantori. As apparenton the left plot of Figure 5, magnetic field lines can be re-flected by remnants or almost intact resonances, possiblyof a high order, which induces some memory effect.In Figure 4, the relative minimal values of the two-wave

overlap Chirikov parameter divided by ε1/2 are close to124 about the (4, 1) resonance and to 133 about the (3, 1)resonance. Being obtained for ε = 2·10−5, Figure 5 corre-sponds then to a situation where the Chirikov parameterabout the resonances associated to the integer values ofq, where the chaos is minimal, is about 0.6, so noticeablybelow 1. One could therefore be surprised to observesome (at least partial) breaking of the KAM tori aboutthe integer surfaces.However, it must be emphasized that the classical cri-

terion for chaos – that the Chirikov overlap parameters1,2 between two resonances waves be larger than one – isa crude one and many refinements have been proposed29.In particular, even if we had a Hamiltonian system withjust two waves, taking into account the secondary res-onances arising from the nonlinear beating between thetwo primary resonances would reduce the effective dis-tance between resonances30. A rule of thumb is thatconnected chaos occurs when s1,2 ≃ 2/3 instead of 1 dueto this mechanism. The most sophisticated treatment forthe paradigmatic case of two primary resonances consistsin taking into account the creation of infinitely manysecondary resonances and using renormalization theoryto compute the chaos threshold31,32. Since this effectof secondary resonances couples here to the existence ofa spectrum of primary modes, it is not surprising to ob-serve the onset of chaos for values of the Chirikov overlapparameter below 2/3. Moreover there is some source foradditional chaos since the resonances are not purely pen-dulum sinusoidal waves. This is apparent in the phasespace portrait of a microtearing mode that has two closex-points instead of one in the pendulum sinusoidal case(see Figure 3).The situation of Figure 5 with ε = 2 · 10−5 is certainly

above but very close to the threshold for connected chaos

since one can observe that two magnetic field lines re-main contained between the (3, 1) and (4, 1) resonanceradial locations whereas the other two magnetic fieldlines happen to cross these. For low enough values ofε, the Chirikov parameter between the magnetic reso-nances eventually becomes small enough in macroscopicdomains of the ψ-space impeding (normal) diffusion.

III. MAGNETIC CONFINEMENT TIMES

A. The magnetic microturbulence picture

In Sect. IID, from the integration of the magnetic fieldline trajectories at experimentally relevant values of therelative magnetic perturbation parameter ε emerges thepicture of the microturbulent magnetic transport. Somelocally stochastic domains of the phase space are sepa-rated by partial transport barriers (cantori) limiting thelarge scale magnetic diffusion. This suggests that char-acterizing the transport through a magnetic diffusion co-efficient, Dm, may not be a universally appropriate ap-proach. Indeed, the presence of cantori should induce astrong locality of the diffusion coefficient: namely Dm de-pends on the action ψ, with locally almost vanishing min-ima about the ψmn associated to the cantori. Moreover,the diffusive nature of the transport is unclear. Com-puting a diffusion coefficient implicitly means that thediffusion is normal, neither sub- nor super-diffusive. Analternative physical indicator, that is a natural indicatorin the fusion context where Lawson’s criterion is central,would rather be given by the characteristic time spentby the magnetic field lines to visit the phase space, thatmay be viewed as a confinement time. The benefit of thisindicator is that it remains valid even in the absence of(normal) diffusion.

B. First exit times

A physically meaningful indicator of the magnetic con-finement for fusion applications is the residence time ofthe magnetic field lines inside some inner volume of thedevice. Indeed, in magnetic confinement devices, elec-trons are strongly magnetized, so that a breakdown ofthe magnetic confinement may result in the terminationof the electron confinement, or in other words, in a dis-ruption.

To probe the magnetic transport properties within themicroturbulence framework, extensive numerical simula-tions have been performed. Starting from magnetic fieldlines in the core of the tokamak, more precisely suchthat ψ(ϕ = 0) be in the range [0.05; 0.1], the numberof toroidal turns ϕexit/(2π) after which ψ(ϕexit) becomeslarger than 0.9 was computed for a large number of fieldlines and at different values of the stochasticity parame-ter ε. Figure 7 represents the distribution of the first exittimes ϕexit in units of toroidal turns for two values of ε.

6

0.555

0.56

0.565

0.57

0.575

0.58

13900 13920 13940 13960 13980 14000

ψ

φ/(2 π)

(35,12)(38,13)(41,14)(44,15)(47,16)(50,17)(79,27)(59,20)(53,18)(62,21)

FIG. 5. (Left) Four magnetic field line traces ψ(ϕ) for the magnetic microturbulence model with ε = 2 · 10−5. (Right) Focuson one of the magnetic field line traces ψ(ϕ) being transiently exploring a narrow ψ-domain below ψ31 ≃ 0.5816. The locationsin the ψ-space of some secondary resonances are indicated.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 200 400 600 800 1000

ψ

φ / (2 π)

FIG. 6. Three magnetic field line traces ψ(ϕ) for the magneticmicroturbulence model with ε = 10−4.

In Figure 8 is plotted for some values of ε above 10−4

the mean exit times computed by launching as many assome tens of thousand initial conditions for the magneticfield lines in the tokamak core. Figure 7 validates themagnetic microturbulence picture depicted in Sect. IIIA.For values of ε roughly below 10−4, the values of the exittimes become very large, in the sense that it becomesnumerically too demanding to compute them. Moreoveras ε decreases, there is an increasingly growing fractionof the trajectories that never attain the tokamak border,getting glued about low order resonances. As ε increases,the local Chirikov parameter eventually raises above onein the whole ψ domain and the conditions for the usualdiffusion in ψ space are fully satisfied. Indeed, for therange of ε values considered in Fig. 8, the mean exit timesscale with ε as ε−2. This is the scaling expected from aquasilinear diffusion approach33, since DQL ∝ (δB/B)2

and δB/B ∝ ε.

The Brownian nature of the behaviour of magneticfield lines may be assessed by comparing the numericallyobtained distribution of first exit times with its predic-tion in the case of Brownian motion. The first passagetime distribution of Brownian motion with positive drifttakes the form

f(τ ;µ, λ) =

(λ

2πτ3

)1/2

exp

[− λ

2µ2τ(τ − µ)2

]. (14)

It can be seen on Figure 9 that the distribution of thenumerical exit times in the case ϵ = 10−3 is fairly wellapproximated by a function of the form (14). The nextSection will be devoted to some consequences for toka-mak physics of the present results.

IV. INDICATIONS FOR A DISRUPTION LIMIT DUETO MAGNETIC MICROTURBULENCE

A. Preliminary justification for a static frame

The magnetic model presented in Section II may bemade time-dependent to account for the time variationsof field perturbations in real plasmas. Typical fluctuationfrequencies have been reported to be in the range 104 to105 Hz. In this Section, our interest will lie howeveron disruptive-type phenomena taking place on the timescale of tens to hundreds of electron toroidal excursions.For such a short time scale, it is possible to neglect thediffusion effects induced by phase and restrict to a staticapproach.

Moreover, it is a well-known experimental fact thatphase-locking between the rotating modes is one ’favor-able’ ingredient for disruptions34–36. Considering thatmodes rotate in phase is equivalent in the frame of themodes to consider constant phases. One just proceedsto a Galilean transform to the wave frame. Disruptions

7

number of toroidal turns

Pro

babi

lity

dens

ity fu

nctio

n

0 1000 2000 3000 4000 5000

0e+

002e

−04

4e−

046e

−04

8e−

04

0 500 1000 1500 2000

0.00

000.

0005

0.00

100.

0015

0.00

20

FIG. 7. Probability density function (PDF) of the first exit times for two different values of the stochasticity parameter ε inthe magnetic microturbulence model: (left) for ε = 3.10−4, (right) for ε = 5.10−4.

0 0.002 0.004 0.006 0.008¶

0

1000

2000

3000

MeanexittimesHintoroidalexcursionsL

0.000150.0002 0.0003 0.0005 0.0007 0.001

100

200

500

1000

2000

slope=-2

FIG. 8. Mean exit times in number of toroidal turns as afunction of ε. The inset shows a log-log representation of thesame points together with a line of slope −2.

have also been observed to take place with slowly ro-tating modes. Then, with regard to the brevity of thedisruptions, treating the disruption scenario in a staticframe can be a good approximation.

In any case, the static situation should be the moredeleterious, limit scenario for the loss of confinementsince one can easily imagine that some differentialpoloidal rotation in the modes contrarily makes the fieldlines spend more time moving in the poloidal directionwhich increases their mean exit times.

ε=10−3

number of toroidal turns

Pro

babi

lity

Den

sity

Fun

ctio

n

0 100 200 300 400 500 600

0.00

00.

002

0.00

40.

006

0.00

8

FIG. 9. Probability density function of the first exit times forε = 10−3 in the magnetic microturbulence model (histogramand plain line continuous representation). In dotted line issuperimposed the first passage time distribution of Brownianmotion with positive drift given in Eq. (14) having the samemean and variance as the data.

8

B. Magnetic turbulence-driven disruption limit

In tokamak physics, there exist some dramaticallyswift phenomena taking place on the lapse of time of sometens to a couple of hundreds of microseconds. These arethe collapse phases of sawteeth, edge localized modes anddisruptions. All these phenomena may be interpreted astransient, benign or severe losses of the magnetic confine-ment and are mediated by the lighter charged particles ofthe plasma, namely by the electrons. In a tokamak suchas JET, the electron thermal velocity at a temperatureof 10 keV is about 4.107m.s−1 so that the characteristictime spent by electrons to make a toroidal excursion is0.5µs. Consequently, a lapse of time of 100µs amounts to200 electron toroidal excursions. It is quite instructive tosee on Figure 8 that 200 toroidal excursions is the orderof magnitude of the qualitative threshold mean exit timeseparating a quick loss of the magnetic confinement forε > εth from a much better magnetic confinement pro-vided the perturbation parameter ε becomes some frac-tion of εth. From the Figure, one may retain the value5.10−4 for εth. The magnetic perturbation δB/B is oforder ε with the spectrum of modes used here. Its valueassociated to ε = 5.10−4 could be derived using Eq. (13).

Looking at Figure 8 comes the impression of a thresh-old effect in the behaviour of the mean exit time as afunction of ε. We shall here examine its origin.

In the regime where the resonance overlap is sufficientto ensure the complete stochasticity of the magnetic fieldlines and the validity of the quasilinear approximation,the mean exit ’times’ ⟨ϕexit⟩ scales as ε−2: there existssome constant C > 0 such that ⟨ϕexit⟩ = C2ε−2. Con-verting the toroidal angle to a real time τ through the useof the electron parallel velocity as we have just done andusing the fact that δB/B is proportional to ε, this meansthat the characteristic exit time scales with respect to therelative magnetic perturbation as ⟨τexit⟩ ∝ (δB/B)−2.

Consequently, all other things being equal, a smallchange in the magnetic perturbation ∆(δB/B) is asso-ciated to a much larger change in the mean exit time∆⟨τexit⟩ since

∆⟨τexit⟩⟨τexit⟩

= −2

(δB

B

)−1

∆

(δB

B

), (15)

where δB/B is a (very) small parameter. From this itfollows that a relatively small change in the magneticperturbation is associated to a much larger relative vari-ation in the mean confinement time. This sensitivity ef-fect is beneficial when the magnitude of the magneticperturbation is reduced since the confinement time maydrastically increase but this effect is deleterious when themagnetic perturbation is increased, by intrinsic or extrin-sic routes, as the confinement time may drop to the pointwhere a macroscopic (major) disruption occurs and con-finement is lost.

V. DISCUSSION

In the present study, the magnetic perturbation δB/B,or equivalently the parameter ε, is a control parameterwhich may take arbitrary values. In the framework ofmagnetic microturbulence, magnetic field lines becomestochastic in a macroscopic fraction of the tokamak vol-ume when the Chirikov parameter computed for anytwo resonant waves (m,n) and (m′, n′) is roughly above2/329. This has been shown to take place for the val-ues of δB/B that have been experimentally measuredby cross polarization scattering in Tore Supra1 or by us-ing the poloidal sweep of a heavy ion probing beam re-cently in JIPPT-IIU2. These tokamaks therefore operatein regimes where the stochastic hypothesis necessary toderive a Fokker-Planck diffusive approach, like the quasi-linear theory, is satisfied. Consistently, both the ToreSupra and JIPPT-IIU teams have verified that the exper-imentally measured levels of magnetic perturbation werecompatible with their experimentally measured electronheat diffusivities χe when using the quasilinear expres-sion for χe

33.

Predicted here is the existence of some qualitativethreshold on δB/B in this stochastic regime above whichthe mean confinement exit times become too low to besustainable by the tokamak plasma. The sensitivity inδB/B is reflected by Eq. (15). One may wonder whetherthis disruptive-like limit may actually be reached. In-deed, once there is an overall stochasticity of magneticfield lines, there is some self-healing process coming intoplay in the sense that the stochasticity of magnetic fieldlines quenches the thermal gradients that are a source ofinstability for the microtearing modes and may regulatethe magnetic microturbulence. However, collisionalitymay remain as a source of destabilization of microtearingmodes.

These results should be put in perspective with the re-cent achievements on runaway electron suppression abovesome magnetic turbulence threshold reported by Zeng etal.37. These experimental measurements done on TEX-TOR have shown that magnetic turbulence is mainly con-tributed from the background plasma and that its leveldoes strongly dependent on the toroidal magnetic fieldand plasma density. Combining the two scaling laws ob-tained by the authors comes the experimental scaling forthe level of magnetic turbulence

δB

B∝ B−3√ne, (16)

where ne stands for the line averaged central density.It follows from Eq. (16) that the previous threshold-like limit on magnetic turbulence for disruption shouldalso be a density limit and that a high magnetic field isbeneficial.

9

ACKNOWLEDGMENTS

Preliminary discussions with S.S. Abdullaev and D.Constantinescu and assistance on parallel computing byA.F. Lifschitz are gratefully acknowledged. This workwas carried out within the framework the French Re-search Federation for Fusion Studies.

1X. L. Zou, L. Colas, M. Paume, J. M. Chareau, L. Laurent, P.Devynck, and D. Gresillon Phys. Rev. Lett. 75, 1090 (1995).

2Y. Hamada, T. Watari, A. Nishizawa, O. Yamagishi, K. Narihara,K. Ida, Y. Kawasumi, T. Ido, M. Kojima, K. Toi and the JIPPT-IIU Group, Nucl. Fusion 55, 043008 (2015).

3W. M. Nevins, E. Wang, and J. Candy, Phys. Rev. Lett. 106,065003 (2011).

4A.J. Lichtenberg, Nucl. Fus. 24, 1277 (1984).5V. Igochine, O. Dumbrajs, H. Zohm, and ASDEX Upgrade Team,Nucl. Fusion 48, 062001 (2008).

6M.-C. Firpo, W. Ettoumi, R. Farengo, H. Ferrari, P.L. Garcıa-Martınez, and A.F. Lifschitz, Phys. Plasmas 20, 072305 (2013).

7W. Ettoumi, Ph-D thesis, Ecole Polytechnique (2013).8H. Doerk, F. Jenko, M. J. Pueschel, and D. R. Hatch, Phys. Rev.Lett. 106, 155003 (2011).

9W. Guttenfelder, J. Candy, S. M. Kaye, W. M. Nevins, E. Wang,R. E. Bell, G. W. Hammett, B. P. LeBlanc, D. R. Mikkelsen, andH. Yuh, Phys. Rev. Lett. 106, 155004 (2011).

10D. R. Hatch, M. J. Pueschel, F. Jenko, W. M. Nevins, P. W.Terry, and H. Doerk, Phys. Rev. Lett. 108, 235002 (2012).

11P Manz, J E Boom, E Wolfrum, G Birkenmeier, I G J Classen,N C Luhmann Jr, U Stroth and the ASDEX Upgrade Team,Plasma Phys. Control. Fusion 56, 035010 (2014).

12B. Cambon, X. Leoncini, M. Vittot, R. Dumont and X. Garbet,Chaos 24, 033101 (2014).

13J.R. Cary and R.G. Littlejohn, Annals of Physics 151, 1-34(1983).

14B.V.Chirikov, Phys. Rep. 52, 263 (1979).15W.D. D’haeseleer, W.N.G. Hitchon, J.D. Callen and J.L. Shohet,Flux coordinates and magnetic field structure (Springer, Berlin,1991).

16A. H. Boozer, Phys. Fluids 26, 1288-1291 (1983).17P. J. Morrison, Phys. Plasmas 7, 2279 (2000).18M.-C. Firpo and D. Constantinescu, Phys. Plasmas 18, 032506(2011).

19T. E. Evans, R. A. Moyer, K. H. Burrell, M. E. Fenstermacher,I. Joseph, A. W. Leonard, T. H. Osborne, G. D. Porter, M. J.Schaffer, P. B. Snyder, P. R. Thomas, J. G. Watkins, and W. P.West, Nat. Phys. 2, 419 (2006).

20S. S. Abdullaev, Construction of Mappings for Hamiltonian Sys-tems and Their Applications (Springer, 2006).

21K.-H. Spatschek, High Temperature Plasmas: Theory and Math-ematical Tools for Laser and Fusion Plasmas (Wiley, Weinheim,2012).

22H. Wobig, Z. Naturforsch. 42a, 1054 (1987).23A. Punjabi, A. Verma, and A. Boozer, Phys. Rev. Lett. 69, 3322(1992).

24S. S. Abdullaev and G. M. Zaslavsky, Phys. Plasmas 2, 4533(1995).

25S. S. Abdullaev, K. H. Finken, A. Kaleck, and K. H. Spatschek,Phys. Plasmas 5, 196-210 (1998).

26R. Balescu, M. Vlad, and F. Spineanu, Phys. Rev. E 58, 951 -964 (1998).

27S.S. Abdullaev, Nucl. Fusion 50, 034001 (2010).28T. H. Stix, Phys. Rev. Lett. 30, 833 (1973).29J. D. Meiss, Differential Dynamical Systems, SIAM Philadelphia(2007).

30A. J. Lichtenberg and M. A. Lieberman, Regular and ChaoticMotion, New York, Springer-Verlag (1992).

31D. F. Escande, Phys. Rep. 121, 165-261 (1985).32R. S. MacKay, Renormalization in area-preserving maps, Singa-pore, World Scientific (1993).

33A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38(1978).

34J.A. Snipes, D.J. Campbell, P.S. Haynes, T.C. Hender, M.Hugon, P.J. Lomas, N.J. Lopes Cardozo, M.F.F. Nave and F.C.Schuller, Nucl. Fusion 28, 1085 (1988).

35P.C. de Vries, M.F. Johnson, B. Alper, P. Buratti, T.C. Hen-der, H.R. Koslowski, V. Riccardo and JET-EFDA Contributors,Nuclear Fusion 51, 053018 (2011).

36F. A. Volpe, A. Hyatt, R. J. La Haye, M. J. Lanctot, J. Lohr, R.Prater, E. J. Strait, and A. Welander, to appear in Phys. Rev.Lett. (2015).

37L. Zeng, H. R. Koslowski, Y. Liang, A. Lvovskiy, M. Lehnen,D. Nicolai, J. Pearson, M. Rack, H. Jaegers, K. H. Finken, K.Wongrach, Y. Xu, and the TEXTOR team, Phys. Rev. Lett.110, 235003 (2013).