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Bicoherence analysis of streamer dynamics induced bytrapped ion modes

Francesco Palermo, Xavier Garbet, Alain Ghizzo

To cite this version:Francesco Palermo, Xavier Garbet, Alain Ghizzo. Bicoherence analysis of streamer dynamics inducedby trapped ion modes. The European Physical Journal D : Atomic, molecular, optical and plasmaphysics, EDP Sciences, 2015, 69 (1), pp.8 - 8. �10.1140/epjd/e2014-50240-2�. �hal-01285763�

Eur. Phys. J. D (2015) 69: 8DOI: 10.1140/epjd/e2014-50240-2

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Bicoherence analysis of streamer dynamics inducedby trapped ion modes�

Francesco Palermo1,2,a, Xavier Garbet2, and Alain Ghizzo3

1 LMFA, Ecole Centrale de Lyon, Universite de Lyon, 69134 Ecully, France2 CEA, IRFM, 13108 Saint-Paul-Lez-Durance, France3 Institut Jean Lamour-UMR 7168, Universite de la Lorraine, 54506 Nancy, France

Received 25 March 2014 / Received in final form 25 August 2014Published online 8 January 2015 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2015

Abstract. High order spectral analyses have recently attracted a great deal of attention in the context ofexperimental studies for magnetic fusion. Among these techniques, bicoherence analysis plays an importantrole as it allows distinguishing between spontaneously excited waves and waves that arise from a couplingbetween different modes linked to specific physical mechanisms. Here, we describe and apply bicoherenceanalysis to kinetic simulations performed with reduced “bounce-averaged gyrokinetic” code, in order toinvestigate the nonlinear dynamics of trapped ion modes. This analysis shows a nonlinear wave couplingprocess linked to the formation of convective cells, thus furthers our understanding of the nonlinear energytransfer between turbulence structures in tokamaks.

1 Introduction

Cross-field transport of energy and particles in mag-netically confined plasmas represents a major issue forachieving magnetic fusion. This transport is commonly at-tributed to small-scale turbulence generated by microin-stabilities which are driven by temperature and densitygradients along the radial direction in a tokamak. In par-ticular, ion temperature-gradient driven modes, trappedelectron modes and electron temperature-gradient drivenmodes are considered as the most important micro-instabilities responsible for turbulent transport in mag-netised plasmas with low β = 2μ0P/B

2 values (whereP is the plasma pressure and B is the magnetic field).The use of simulation code that accounts for kinetic ef-fects is essential for studying turbulent transport. In fact,although a fluid treatment can give a first description,the nonlinear dynamics linked to these phenomena can bestrongly affected by specifically kinetic mechanisms suchas Landau Damping, finite orbit (FOW) effects and finiteLarmor radius (FLR) effects. In general such kinetic codehas to account for six-dimensions (6D), with 3 coordinatesfor positions, and 3 others for velocities. The disparatespatio-temporal length scales that are involved in toka-mak turbulence lead to costly simulations. Hence severalkinetic codes with reduced dimensionality have been de-veloped, usually by eliminating high-frequency phenom-

� Contribution to the Topical Issue “Theory and Applica-tions of the Vlasov Equation”, edited by Francesco Pegoraro,Francesco Califano, Giovanni Manfredi and Philip J. Morrison.

a e-mail: francesco.palermo@ec-lyon.fr

ena. Gyrokinetic codes are based on averaging the fastcyclotron time scale of particles and are now commonlyused [1]. This operation, licit when the frequency of fluc-tuations is smaller than cyclotron frequencies, is well jus-tified in most fusion devices. A further reduction step canbe taken when instabilities are driven by particles that aretrapped in the minima of the magnetic field. If the fre-quencies of modes are low enough, the dynamics can beaveraged over the particle bounce motion along the fieldlines. These numerical tools are called “bounce-averagedgyrokinetic” codes. Reduced kinetic codes have stronglycontributed to the understanding of physics of turbulenttransport in slab and toroidal geometries. Regarding thedynamics of turbulent transport driven by temperaturegradient instabilities, gyrokinetic simulations and exper-iments have shown the formation of structures whichplay an important role in turbulence self-organisation. Themost well-known structures are zonal flows that are gen-erated by fluctuations of turbulence and back-react viavortex shearing, thus reducing transport. Zonal flows canbe identified by means of a perturbed electric field withwave vector k ≈ (kr, 0, 0) pointing in the radial direc-tion. Structures called “streamers” are a counterpart ofzonal flows. Streamers are convective cells which are elon-gated in the radial direction and offer an efficient chan-nel for energy transport. As such they play an importantrole in turbulence self-organisation [2]. It is important tonote that streamers are commonly observed in simula-tions but no clear evidence of their existence has beenfound in fluctuation measurements. Although there is ageneral consensus on their existence, the precise nature of

Page 2 of 6 Eur. Phys. J. D (2015) 69: 8

streamers is still being discussed. As indicated by Hollandand Diamond [2] two distinct groups can be identified:“isolated intermittent streamers”, which are caused bythe nonlinear dynamics, and “streamer arrays”, which aregenerated from linear and nonlinear processes. Streamerscan thus be identified as perturbations rapidly changingin the poloidal direction but almost constant in the radialdirection k ≈ (0, kθ, 0). Streamers have been studied in-tensively because of their scientific impact, and becauseof their implications in Ion Temperature Gradient (ITG)turbulence and also as a channel of transport for Elec-tron Temperature Gradient turbulence. Although theseconvective structures have been addressed in an exten-sive overview [3], some important aspects of their forma-tion and interaction still need to be clarified. Consequentlynew methods of analysis become necessary to investigatethem. High order spectral analyses have recently attracteda great deal of attention in the context of experimen-tal studies for magnetic fusion [4–7]. The most popularspectral analysis tool is probably the bicoherence method,which was proposed by Kim and Powers [8]. Bicoherenceallows one to investigate the degree of phase correlationamong three waves and is related to the quadratic nonlin-earities. It sheds light on the exchange of energy betweenmodes that is due to specific physical mechanisms. Thispaper presents results from a bicoherence analysis toolthat has been developed on the basis of the computationalprocedure developed by Kim and Powers. The objectiveis to elucidate the energy transfer mechanisms in the dy-namics of turbulence, and to identify the structures thatplay a key role in kinetic simulations. Here we applied thisanalysis to simulations performed with a bounce-averagedgyrokinetic code that is specifically designed for studyinglow frequency instabilities driven by trapped ion (TIM).In toroidal geometry, the specific dynamics of TIM be-comes important when the frequency of ITG modes fallsbelow the ion bounce frequency, allowing one to aver-age on both the cyclotron and bounce motion fast timescales. This reduction of the number of degrees of freedomleads to a strong reduction of the requested computer re-sources (memory and computation time). Therefore, longsimulation runs compared to the ones that could be ob-tained with 5D gyrokinetic codes can be performed, whilekeeping the essential physical ingredients. In our model wehave focused the analysis on the large scale dynamics ofinterchange-like instabilities to assess the nonlinear evolu-tion of convective cells. We show and discuss results ob-tained by using a bicoherence analysis. The present articleis organised as follows. In Section 2 we briefly recall theset of equations integrated in the bounce-averaged gyroki-netic code. In Section 3 we summarise the characteristicsof the bicoherence analysis tool. In Section 4 we presentand discuss results on three-wave coupling modes observedin the simulations. Finally, conclusions follow in Section 5.

2 Numerical model

The present model describes the dynamics of TIM modes,which are mainly driven by the gradient of temperature.

The derivation of the equations and their implementationin TIM code have been discussed in previous works [9–11].Hence only the main expressions are summarised anddiscussed. We consider an axi-symmetric toroidal geom-etry, with a magnetic axis located at a major radius R0.The modulus of the magnetic field is B(R0 + r, θ) ≈B0[1 + ε sin2(θ/2)] in which ε = r/R0 is the inverse ofthe aspect ratio and B0 is the minimum magnetic field atthe angle θ = 0. The motion of the particle can be sepa-rated into a fast cyclotron motion, with a time scale ω−1

c ,and a slower “gyrocenter motion”. The analysis of thegyrocenter trajectories allows one to identify two differentclasses of particles called “passing particles” and “trappedparticles”, respectively. Circulating particles have a largeenough parallel velocity to move on a magnetic field line onits entire length. Trapped particles follow a bouncing mo-tion on the low field side, thus describing a “banana” orbitshape with a width δb in the radial direction. The trappedparticle motion is characterised by a bounce frequency ωband a slower magnetic drift around the torus with a fre-quency ωd [12]. Deeply trapped particles spend most oftheir time in the “bad” curvature zone of the tokamak,that is the zone where interchange-like modes are locallyunstable. This aspect emphasises their possible role in tur-bulent transport. In collisionless plasmas, these instabili-ties are energised via Landau resonances between particlesand waves in a range of frequencies that is well below theparallel transit frequency. This feature allows us to decou-ple the dynamics of trapped ions from passing ions. Thedensity response of passing ions is then simply adiabatic.The existence of three motion invariants in the unper-turbed case ensures that the system is integrable. Theseinvariants are the magnetic momentum μ (also known asthe adiabatic invariant), proportional to the magnetic fluxacross a cyclotron orbit, the particle energy E, and thetoroidal canonical momentum Mk, which for a trappedparticle coincides with the flux of the poloidal magneticfield ψ. Moreover the quasi-periodicity allows the con-struction of a set of action angle variables which are canon-ically conjugated (Jk, αk) (with k = 1, 2, 3). This set ofcoordinates is well adapted to describe the motion of par-ticles and implement the gyro- and bounce-averaging pro-cedures. The system at equilibrium can be described by anunperturbed Hamiltonian H = H0(Jk) that is a functionof the actions only, with J1, J2, J3 related to the three in-variants (μ,E, ψ). By means of the motion equations, theangle variables α1, α2, α3 are related respectively to thecyclotron ωc, bounce ωb and precession ωd frequencies.The low frequency response for TIM is obtained by makinga phase-angle average over the cyclotron and bounce mo-tions (i.e. over the angles α1 and α2). Only the precessionmotion is explicitly taken into account in the model. Fi-nally the two important variables are the precession phaseα3 (hereafter indicated by α) that is a function of toroidaland poloidal coordinates, and the poloidal flux ψ relatedto J3 which plays the role of a radial coordinate.

These directions define a plane (α, ψ) that is orthog-onal to the magnetic field. The code evolves a bounce-averaged distribution function f(α, ψ) by solving a Vlasov

Eur. Phys. J. D (2015) 69: 8 Page 3 of 6

equation∂f

∂t+ [φ, f ] + Eωd

∂

∂αf = 0 (1)

where [., .] is the usual Poisson Bracket [g, f ] = ∂ψg∂αf −∂αg∂ψf and φ is the bounce-averaged electric potential.In this expression the last term represents the interchangeterm that drives the main instability. Trapped ion tur-bulence develops on length scales of the order of the ba-nana orbit width δb which is much larger than the Debyelength. In this case, self-consistency is ensured by thequasi-neutrality condition, instead of the Poisson equa-tion. Thus assuming an adiabatic response for electrons,the electro-neutrality condition is written as:

Ce (φ− 〈φ〉α) = Ci∇2φ+ ni − n0 (2)

where the left hand side indicates that the electron densityvanishes when the electric potential equals its flux average〈φ〉α. The latter represents the potential averaged on amagnetic surface (constant ψ surface) by means of theoperator 〈.〉α. In equation (2), n0 is the total ion densityand ni represents the ion guiding centre density obtainedby integrating the distribution function on the velocityspace expressed in action-angle variables. The polarisation∇2φ term is an approximation of the difference of particle

and guiding centre densities. Ce = (1 + τ)/fp and Ci =Cefp/τ are constants accounting for the ratio τ of ionto electron temperatures, and for the fraction of trappedparticles fp = 2

√2ε/π which is mainly determined by the

inverse aspect ratio ε. Finally, we note that the electronpolarisation can be neglected because of their small mass.In summary, our kinetic trapped-ion model is given by areduced Vlasov equation, self-consistently coupled to theelectric potential φ. In the code, the time t is normalised tothe inverse drift frequency ω−1

d,0 of reference, the poloidalflux ψ is counted in Δψ = πr20B0/q units, the energy Eis normalised to a temperature T0, and the potential φ isgiven in ωd,0Δψ units.

3 Bicoherence analysis

As mentioned in the introduction, bicoherence representsa useful tool to investigate the nonlinear dynamics, as itallows one to distinguish between spontaneously excitedwaves and coupled waves in a fluctuation spectrum. Thisinformation is not contained in the energy spectra of a sig-nal based on the linear analysis of the Fourier decomposi-tion such as the power spectrum P (k) = |Xk|2 (with Xk

k-fourier coefficient). In a signal, nonlinear wave-wave in-teraction can be observed when the conditions k3 = k1+k2

(f3 = f1+f2) and ϕ3 = ϕ1+ϕ2 are satisfied between threewaves with wavenumbers k1, k2, k3 (or frequencies f1, f2,f3) and phases ϕ1, ϕ2, ϕ3. In the case where ϕ3 behaves ina random way, the associated wave evolution is not linkedto other two waves. In summary, the bicoherence mea-sures the degree of coherence of the three wave phases. It

is quantified by the following expression:

b(k, l) =| 1M

∑Mj=1X

(j)k X

(j)l X

∗(j)k+l |

[ 1M

∑Mj=1 |X(j)

k X(j)l |2]1/2[ 1

M

∑Mj=1 |X(i)

k+l|2]1/2,

(3)where M represents the number of independent datarecords of a signal. It is not straightforward to calculatethe bicoherence because of the variability of the denomi-nator that can produces spurious spikes in the associatedspectrum [8,13]. However, neglecting the variability of thedenominator, Kim and Powers [8] have shown that thevariance, which measures the degree of accuracy of thesquared bicoherence, can be approximated by the expres-sion var(b2) ≈ (1 − b2)1/M . We note that var(b2) � 1/Mand consequently the variance approaches 0 as M be-comes infinite. The numerator of equation (3) is knownas the bispectrum. Bicoherence is thus a normalised bis-pectrum that varies between 1 (for a three wave coher-ence relation) and 0 (corresponding to no-coherence phaserelation). Bicoherence can be represented graphically inthe (k, l)-plane by the inner triangular region defined by0 ≤ l ≤ a/2 and l ≤ k ≤ (a − l) where a = kN/Δk withkN Nyquist limit number and Δk = 1/T bandwidth ofthe signal. Indeed, for a single regular signal it is suffi-cient to compute the bicoherence over this region becausethe information is redundant in all the other regions of theplane. In the case where bicoherence is calculated betweenmultiple quantities, the non-redundant information regionhas the shape of a peculiar polygonal domain. Taking asreference the work of Kim and Powers [8], we have devel-oped a bicoherence tool on the basis of equation (3). Forthe sake of discussion, we show an application of this toolon the following signal:

g(x) = R(x) + cos(k1x+ ϕ1) + cos(k2x+ ϕ2)+ cos(k1x+ ϕ1) cos(k2x+ ϕ2) (4)

with k1 = 0.22kN , k2 = 0.37kN and where R(x) is arandom signal. The last term in equation (4) generatestwo signals of type 0.5 cos(k±x + ϕ±) with k±=k1 ± k2

corresponding to the sum and the difference of originalwavenumbers k1, k2 and ϕ±=ϕ2 ± ϕ1. Squared bicoher-ence in Figure 1 shows two peaks at the points [k2, k1]and [k1, k−] and the background noise is evident in the tri-angular spectrum domain. The relationship of the phasebetween the components of the signal in equation (4) aresummarised in Table 1. We note that the bicoherence anal-ysis can only confirm the existence of mode coupling be-tween three waves, but cannot determine the causal rela-tionship between the waves or which wave is generated bythe other two waves.

4 Results and discussion

In this section, we applied a bicoherence analysis to asimulation performed with a bounce-averaged gyrokineticcode in order to investigate the nonlinear dynamics of TIMinstabilities. We recall that the most important feature of

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Fig. 1. Squared bicoherence spectrum corresponding to signalof equation (4).

Table 1. Frequency relation for the signal in equation (4).Considering phase relations, two peaks are expected in the bi-coherence spectrum.

k relations Phase relations Peak locations

1 k1 + k2 = k+ ϕ1 + ϕ2 = ϕ+ [k2, k1]2 k1 + k− = k2 ϕ1 + ϕ− = ϕ2 [k1, k−]

the TIM model is that high-frequency phenomena such asthe cyclotron motion and the bounce motion of trappedions have been eliminated, while essential kinetic effectsfor low frequency instabilities are kept. Modes such asTIM are a prototype of kinetic instability since they aredriven through the resonant interaction between wavesand trapped ions via the toroidal precession of bananaorbits. The starting point for the investigation of TIMinstability is the gyrokinetic Vlasov model coupled to thequasi-neutrality condition initiated in an equilibrium statewith a perturbed term. The initial distribution functionand electrostatic potential perturbations are as follows:

f (α, ψ) = F0 (ψ) +∑

kα

δfkα (ψ) ei(kαα) (5)

φ = φ0 (ψ) +∑

kα

δφkα (ψ) ei(kαα) (6)

where δφkα(ψ) = sin(nπψ) with n = 1. The F0(ψ) andφ0(ψ) equilibrium functions are equal to:

F0 (ψ) = e−E[

1 +Δτsωd

(

E − 32

)

ψ

]

(7)

φ0 (ψ) = 0 (8)

where Δτs is normalised temperature gradient. TIM in-stabilities are driven by the temperature gradient with thepossibility of a resonance arising between trapped wavesand ions through their precession motion. Density N (ψ)and temperature profile T (ψ) can be obtained as fluidmomenta of the distribution function F0. In this way, weobtain an almost flat profile of density N = 1 and thefollowing profile of temperature:

T (ψ) = 1.5 (1 +Δτsψ ωd) (9)

Fig. 2. Isocontours of potential at t = 15.5 showing formationof about 22 streamer structures.

Fig. 3. kα spectrum plot at t = 15.5 (black line), 19.5 (greenline), 35 (red line).

with ωd = 0.9. We have considered nψ × nα = 128 × 256phase space sampling. We have chosen an ion-electrontemperature ratio τ = 1 and we have used the follow-ing parameter values: Ce = 0.61, Ci = 0.5; the timestep is Δt = 0.005 and the initial perturbed potentialamplitude is δφ ∼ 10−5. The ion banana orbit widthδb = 0.09 is three times the Larmor radius. With the se-lected Ce and δb values, the stability threshold tempera-ture is Δτ0(Ce, δb) = 0.6 (for detail see Ghizzo et al. [11]).The value Δτs = 0.8 has been chosen in equation (9) inorder to get a sufficient drive. It is interesting to considerthe behaviour of the system in the first phase of the dy-namics, when TIM modes are dominant. In Figure 2 weshow the isocontour of the potential perturbations in the(α, ψ)-box and we observe nonlinear streamers formationin the first phase at t = 15.5. These structures appearin the simulation as an effect of the TIM instability thatprovides evidence for the dominance of interchange modesalong the ψ direction. The formation of streamers is wellidentified also in the spectrum performed along the α di-rection. In Figure 3 we show the principal modes thatgrow in the simulation at three different times t = 15.5(black line), 19.5 (green line), and 35.0 (red line). Theenergy is initially injected in the system in the range10 � kα � 15. At time t = 15.5 we observe that themodes around kα,S ≈ 22 become dominant. In order toinvestigate the onset of modes we perform a bicoherenceanalysis in the plane (ψ, α). For this purpose we consider

Eur. Phys. J. D (2015) 69: 8 Page 5 of 6

Fig. 4. Zoom of squared bicoherence spectrum on the first 30modes at t = 15.5.

Fig. 5. “Summed bicoherence” spectrum 0 ≤ kα ≤ 50 att = 15.5.

profiles of the potential between ψ1 = 0.2 and ψ2 = 0.8along the α direction. These profiles correspond to datarecords of the signal. Considering M = 80 profiles we findvar(b2) � 0.01. In Figure 4 the squared bicoherence anal-ysis performed at t = 15.5 shows a significant bicoherencevalue (b2 ≈ 0.31) for modes (k, l) ≈ (12, 10). In order tointerpret this signal, it is useful to calculate a “summedbicoherence” defined as Sb2(kα) = 1/s

∑b2(k, l) where

the sum is taken over all k and l such as k + l = kαand where s is the number of terms in the sum for agiven value of kα. Figure 5 shows this summed bicoher-ence. A peak is observed around kα,S ≈ 22. Then, astrong coupling appears, of the type k+ l = kα for modes(12 + 10) ≈ (11 + 11) ≈ 22. We emphasise that a bico-herence analysis can only confirm the existence of modecoupling between three waves, but cannot determine thecausal relationship between the modes or which mode isgenerated by the other two modes. For this purpose, sev-eral methods have been proposed to determine the direc-tion of energy transfer [6,14]. Here we use the profile ofthe spectrum at different times to estimate this transferdirection. When doing so, the bicoherence shows whichmodes are involved in the nonlinear coupling and the tem-poral evolution of the spectrum provides an informationon the chronology and the direction of the energy transferbetween these modes. Streamers can be identified as thefirst harmonic of modes around kα,F ≈ 11 that develop inthe system. In Figure 6 we show the squared bicoherencespectrum calculated at t = 19. We observe again the cou-

Fig. 6. Zoom of squared bicoherence spectrum on the first30 modes at t = 19.

Fig. 7. Zoom of squared bicoherence spectrum on the first30 modes at t = 21.

pling of harmonics kα,S ≈ 2kα,F and at the same time weobserve the development of a strong interaction betweendifferent (k, l) modes whose sum is equal to kα,S ≈ 22 suchas (k + l) ≈ (12 + 10) ≈ (15 + 7) ≈ (17 + 5). Comparingthe spectral profile at t = 15, 5 and t = 19 in Figure 3, wededuce that the energy initially transferred on the specificmode kα,S ≈ 22 is distributed on different lower modesby means of nonlinear coupling interaction. These newmodes interact between themselves, and in particular themode kα,I = 16 is generated by nonlinear interaction, asshown by the spectral profile and the squared bicoher-ence at t = 19 in which we observe an important peak(b2 ≈ 0.34) around (k, l) ≈ (11, 5). After a growing phase,the kα,I = 16 mode saturates and releases its energy onlower kα modes. A second nonlinear energy transfer, sim-ilar to that of kα,S = 22, is found for kα,I = 16 in thebispectrum at t = 21 shown in Figure 7, in which we ob-serve the coupling of different modes whose sum is equalto kα,I = 16 such as (k+l) ≈ (12+4) ≈ (13+3) ≈ (14+2).At the same time, we observe an interaction of modes lo-calised between 5 � k � 10 for l ≈ 5. The spectrum shownin Figure 3 indicates that lower modes are amplified att = 35, in particular the modes kα ≈ 5, 8. The nonlinearmechanism that determines this sequential inverse energycascade can be qualitatively found in Figure 8, in whichisocontours of potential are shown at t = 39. In this figurewe observe the interaction of streamers characterised bynonlinear merging, which are able to generate more andmore low wavenumber modes along the poloidal direction.At the same time, we can identify kψ modes that develop

Page 6 of 6 Eur. Phys. J. D (2015) 69: 8

Fig. 8. Isocontours of potential at t = 39 in the region 2.4 ≤α ≤ 3.5.

along the radial direction. This observation has importantimplications for zonal flows formation and their interac-tion with streamers. The detailed mechanism of streamerinteractions and zonal flow formation will be the subjectof a future work (in preparation). Here we would like toemphasise the dynamics of streamers. We have observedthat energy is injected in the system via selected modesthat determine the size of streamers. These structures sat-urate after a growing phase. By means of a bicoherenceanalysis we demonstrated that a large fraction of their en-ergy is released on scales larger than the initial one. Sonew localised modes, at large scale, increase in time andsubsequently saturate by transferring in turn their energyon larger scales. This process defines an interesting inter-mittent inverse energy cascade.

5 Conclusions

In this paper we have described and applied a bicoherenceanalysis to simulations performed with a bounce-averagedgyrokinetic code that can be considered as a toy modelthat is able to describe the basic structures of turbulenttransport in tokamaks such as streamers and zonal flows.The advantage of this code is the possibility to investi-gate the dynamics of these structures on longer times,compared to those that could be reached with full 5Dgyrokinetic codes. We emphasise that understanding thedynamics of streamers and zonal flows is a quite impor-tant problem, since the confinement in a tokamak dependson the respective level and life time of these structures.

With our code, we have observed the development of TIMinstabilities through the formation of streamer structures.This instability is akin to a Rayleigh-Benard instabilityand is able to transport large quantities of energy towardsthe exterior zone of tokamak, by means of a convectivetransport mode regulated by streamers. Here, we haveshown that streamers can saturate by transferring energyto large scale structures. In particular, by using a bico-herence analysis tool and analysing the time evolution ofthe spectral profile, we have shown that the transfer of en-ergy is due in large part to a nonlinear coupling of modesalong the α direction. Thus, most of the energy initiallyinjected on modes that determine the size of streamers istransferred on lower modes via an intermittent mechanismof inverse energy cascade. Consequently, the transport ofenergy along the ψ direction is affected. Moreover, we haveobserved that at the same time kψ modes develop alongthe radial ψ direction. This has potential implications onthe interaction between zonal flows and streamers that areinvolved in the generation of zonal flows. This aspect couldbe very relevant for the control of turbulence suppressionand consequently for the development of magnetic fusion.

This work has been supported by French National ResearchAgency under contract ANR GYPSI, ANR-10 Blanc-941,SIMI9 2010.

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