The complex processes in edge tokamak plasma are affected (among others) both by resonant magnetic perturbation (RMP) and by plasma turbulence. RMP is nowadays considered to be a candidate for the mitigation of the edge-localized modes (ELMs). The effect of plasma turbulence inthe edge region has not been studied yet. Since both RMP and plasma turbulence should influence plasma dynamics, studies of their simultaneous effect have potential practical implications. Using a simplified model of the turbulence and single-particle simulations, we discovered that its effect at realistic amplitudes changes the ion dynamics significantly. We expect that the turbulence has a similar effect on electrons, thus potentially influencing the ELM mitigation mechanism

Cahyna, Pavel

Krlín, Ladislav

Kurian, Matúš

Pánek, Radomír

Acta Polytechnica

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica 53(4):359–364, 2013 © Czech Technical University in Prague, 2013available online at http://ctn.cvut.cz/ap/PARTICLE DYNAMICS IN THE RMP ERGODIC LAYER UNDERTHE INFLUENCE OF EDGE PLASMA TURBULENCEMatúš Kuriana,c,∗, Ladislav Krlínb, Pavel Cahynab, Radomír Pánekba Faculty of Mathematics and Physics, Charles Universityb Institute of Plasma Physics AS CR, v.v.i.c Gymnázium Jura Hronca, Bratislava∗ corresponding author: kurian@gjh.skAbstract. The complex processes in edge tokamak plasma are affected (among others) both byresonant magnetic perturbation (RMP) and by plasma turbulence. RMP is nowadays considered to bea candidate for the mitigation of the edge-localized modes (ELMs). The effect of plasma turbulence inthe edge region has not been studied yet. Since both RMP and plasma turbulence should influenceplasma dynamics, studies of their simultaneous effect have potential practical implications. Usinga simplified model of the turbulence and single-particle simulations, we discovered that its effect atrealistic amplitudes changes the ion dynamics significantly. We expect that the turbulence has a similareffect on electrons, thus potentially influencing the ELM mitigation mechanism.Keywords: resonant-magnetic perturbation, plasma turbulence, non-linear dynamics.1. IntroductionIt has been observed that RMP (Resonant MagneticPerturbation), created by auxiliary coils, can mitigatethe ELM instability on several tokamaks, includingthe DIII-D [1]. Earlier explanations state that thestochasticity of the magnetic field lines enables fastescape of electrons, which results in a change of thedensity gradient. Experiments cast doubt upon thissimple explanation, and as yet no convincing explana-tion of the mitigation of ELM has been found. Thiscalls for a discussion of the roles of other mechanismsthat can appear in this region, and also of the effectof all sorts of particles involved in the process. Thegeneration of a radial electric field in the ergodic layerseems to play an important role.The usual approach to the generation of a radial elec-tric field considers quiet plasma without any chaoticperturbation of the density of the plasma in this re-gion. However, RMP will surely work in a plasma withsome level of electrostatic turbulence, which seems tobe a generic phenomenon of an edge tokamak plasma.This turbulence causes the generation of blobs madeof potential hills and valleys in the potential land-scape [2]. It has been observed that the dynamics ofelectrons and ions is different in such potential. Whileelectrons tend to be trapped in its structures, ionsmay also be trapped, or – depending on their massand energy – they may be subject to anomalous diffu-sion [3]. It was proposed in [4] that such differencescan result in the generation of an electrostatic field.Furthermore, considering the effect of the ergodiclayer, plasma particles will be influenced not only bythe electrostatic turbulence, but also by the toroidalmagnetic field and the magnetic field of overlappingmagnetic islands. Thus there are two potential mech-anisms for generating of a radial electric field in thesystem.However, a self-consistent simulation of the concur-rent effect of the ergodic layer and the electrostaticturbulence on plasma particles presents an extremelydifficult problem that has not been yet solved. Perhapsthe first important step towards a solution appearsin [5]. There, the influence of RMP on electrostaticturbulence is discussed in the framework of fluid dy-namics and the extended Hasegawa-Wakatani model.In our opinion, the effect on all types of particles alsoneeds to be investigated.The first step can be formulated as follows: Underwhich conditions will the effect of electrostatic turbu-lence overshadow (or, at least be comparable with) theeffect of the ergodic layer of RMP? To answer thisbasic question, it is necessary to study the particledynamics in the complicated system of electrostaticturbulence potential and in the ergodic layer. Thecomplications center around the turbulent electro-static potential, the form of which has to be estimated.A full, self-consistent solution can therefore be ex-pected only in a theoretical approach. Nevertheless,an approach employing a simple model of the tur-bulence can provide us at least a glimpse of whathappens in this complicated region.2. ModelWe start with a discussion of the effect of low frequencyelectrostatic turbulent potential perturbation, whichcan compete with the effect of a system of stochasticfield lines. This turbulence is the result of the instabil-ity of resistive drift waves (e.g. in [2]). We modelledthe potential by a time-stationary and spatially peri-odical function, which we had used in [3, 4]. The time359M. Kurian, L. Krlín, P. Cahyna, R. Pánek Acta Polytechnica(a) C, U = 20 V, not trapped (b) C, U = 20 V, trapped(c) e, U = 20 V, not trapped (d) e, U = 20 V, trappedFigure 1. Comparison of the trajectories of carbon ions and electrons – projection: a trapped trajectory, as wellas a non-trapped trajectory, is presented in each case. The trajectories in the non-trapped case are also different:the electron tends to follow a path parallel to the separatrix (see Fig. 1c), whereas the ion demonstrates a fairlycomplicated random walk (see Fig. 1a).stationarity is justified by well-based experimentalevidence (recently in [5]) of the long correlation ofpotential perturbation along magnetic field lines. Thespatial periodicity models the appearance of “hills”and “blobs” in the electron density distribution. In theliterature, this potential is called “egg-crate potential”— see an analogy in solid state physics [6]. Obviously,both assumptions — the time independence and thespatial periodicity of blobs — represent strong approx-imations. However, they enable us to obtain someresults, which can later be verified on more realisticmodels.We describe the particle motion by means of a single-particle Hamiltonian, which includes the toroidal andpoloidal magnetic fields, the resonant magnetic pertur-bation, and a corresponding electrostatic perturbation,respecting the helical character of the field lines. Thepresence of two rows of magnetic islands of RMP isincluded as a perturbation of the main tokamak field.The electrostatic potential follows the helicity, givenby the safety factor qf . To avoid the effect of drifts incurvilinear field lines in toroidal representation, andthe generation of an additional sort of chaos result-ing from such geometry, we limit our procedure to acylindrical system with periodic boundary conditions.The final form of the Hamiltonian is (using [7, 8])H = 12M[p2r +1r2(pθ − qAθ)2 + (pz − qAz)2]+ U0q cos(kr) cos[krd(θ − zR0qf)], (1)where Aθ = ψ =12B0r2and Az = − 1R0∫qf−1dψ − LψR0[cos(2θ − z/R0)+ cos(3θ − z/R0)].3. MethodWe follow the dynamics of particles separately forelectrons, protons, and heavier ions. The canonicalequations corresponding to the Hamiltonian (1) were360vol. 53 no. 4/2013 Particle Dynamics in the RMP Ergodic Layer(a) U = 0 V (b) U = 0.1 V (c) U = 0.5 V(d) U = 5 V (e) U = 20 VFigure 2. A close-up of the surface of section of lithium ions 36Li+: confined trajectories are marked in red,unconfined trajectories in black. It appears that the frequency of the confined trajectories increases with U0: there areno confined trajectories for U0 = 0 V (see Fig. 2a), and their number increases slowly with non-zero U0 (Figs. 2b–2d).For amplitudes of 5 V and above (Figs. 2d–2e), a majority of the trajectories get confined. Overall 1000 trajectorieswere run, and there are approximately 15000–20000 points in each surface of section. Parameter L, governing theergodic layer, was set to 0.0007.solved by means of the Adams (predictor-corrector)method for particles of different mass at 20 eV. Apartfrom the energy, which is kept constant, three otherparameters are to be considered: the charge-massratio, L, which modulates the ergodic layer, and U0,which modulates the electrostatic perturbation. Giventhat we are interested in behaviour of the most rele-vant particles of the system – electrons and protons(as well as heavier ions) – the choice of the massesand charges is straightforward, and we are left with a2-dimensional parametric space (L,U0).The first glimpse into the system is provided bya study of the individual trajectories. Although theinformation gained is qualitative rather than quan-titative, it often provides a powerful demonstration.Calculating the dynamics of a batch of electrons overtime intervals of 50-100µs is a very challenging numer-ical task. Hence, we follow the individual trajectoriesand draw our conclusions from the prevailing trends.When dealing with a potentially chaotic system,Poincaré sections (surfaces of section) and diffusionare often employed. In our case, the surface of sectionis the plane z = piR0 and its periodic extensions.We mark the intersections of trajectories with thesurface of section over a period of approximately 20revolutions. We consider the case L,U0 = 0 to bethe “ground state”, and visually compare its Poincarésection with the Poincaré sections corresponding tonon-zero values of U0. In this way we will be able toestablish the effect of the electrostatic perturbation.Since this procedure can be performed for particles ofvarious mass, it is possible to compare the effect ofelectrostatic perturbation on heavy and light particles.Later the amplitude of the stochastic magnetic field Lcan be changed, while U0 is kept constant to determinethe influence of the ergodic layer.It should be noted that the (average) computationtime for particles of different mass (and hence the phys-ical time) differs greatly, as it is roughly the distancetravelled in the z-direction that is kept constant. Dueto the different particles that are used, the physicaltime of the evolution may exceed the stationary life-span of the electrostatic perturbation. However, themethod provides a long-term insight into the system,which would otherwise be impossible.An important feature of the method is that it iden-361M. Kurian, L. Krlín, P. Cahyna, R. Pánek Acta Polytechnica(a) ions (b) protonsFigure 3. Figure showing the logarithm of the mean radial diffusion coefficient α for carbon ions (Fig. 3a) and forprotons (Fig. 3b) as a function of parameters Υr∗ and L/r∗. A total of 250 trajectories were run; α is defined as theslope of the dependence [r(t)− r(0)]2. On the whole, the system shows that both an increase in L and an increase inΥ generally lead to increased diffusion. Further, for small values of Υ (mainly in region I) L is as important for anincrease in diffusion as Υ. This does not seem to be the case in region II, where the diffusion is determined mainly byΥ. The mesoregion is very interesting: first, L is of much importance in volatile region Q, where relatively smallchanges in L result in transitions from a diffusive state to a non-diffusive state, and vice versa. The region expandswith the decrease in particle mass.tifies the trajectories of the trapped particles. Theseare the trajectories whose projection onto the surfaceof section forms a set with a very small diameter. Thisis consistent with the notion of confinement of theparticle in the (r, θ) plane.In addition to the Poincaré sections, we study thediffusion in the system. Given that radial diffusion isby far the most relevant in the system, we focus onthe radial diffusion coefficient. This was calculatedfor a bundle of 200 trajectories run over 50µs, whichis the estimated lifetime of the perturbation. Thecoefficient was calculated as the slope of the meansquare displacement: D = [r(t)− r(0)]2. We alsoattempted to identify different diffusion regimes basedon the time evolution of the square displacement,namely: normal, hyper, strange, and none.In order to study the scaling of the diffusion coeffi-cient with respect to particle mass we replace L,U0by L/r∗,Υr∗, where Υ ≡ U0/E, and r∗ ≡ rgyr,0/a.This choice is essentially phenomenological, albeitmotivated by ideas about relevant dimensionless pa-rameters of the system; it also appears to be partlyjustified by the results.4. Results4.1. TrajectoriesParticle trajectories were first computed in the inertialsystem. However, drawing an entire 3-D trajectory ona plane is impractical, so we intend to make use of aprojection into the (r, θ) plane. Still, we wish to studythe trajectories with respect to the potential structure,which is also z-dependent. Fortunately, based on thepotential form of equation (1), it is possible to find atransformation that eliminates the axial dependence,and then to use the projection.In general, two types of trajectories can be observed.The first are trapped trajectories, which appear ei-ther periodic, or quasi periodic, see the examples inFigures 1b and 1d. The others are non-trapped trajec-tories, which either follow a potential separatrix (seeFigure 1c), or resemble random-walk trajectories (seeFigure 1a).The effect of the parameters introduced in section 3can be summed up on the basis of the trajectory plots.First, the trajectories of nucleon-mass particles aretrapped much more often than electron trajectories;other factors being constant. Similarly, the increasein the potential amplitude results in both the lightand heavy particles being trapped more frequently.Last, no effect based on the change of RMP amplitudecould be observed.4.2. Surfaces of SectionSurfaces of section (see Figure 2) were obtained usingthe procedures described above. As expected, surfacesof section change with increasing amplitude of theelectrostatic perturbation U0. The increase also leadsto (2-D) confinement of individual trajectories. Thisis demonstrated in the surfaces of section for lithiumsee Figure 2. The lower threshold of the effect ofelectrostatic perturbation on lithium is at 0.1-0.5V,and this matches well with the results obtained forcarbon ions and protons. The fraction of confinedlithium ions is 0.81 at 20 V. We plan to perform a362vol. 53 no. 4/2013 Particle Dynamics in the RMP Ergodic Layer(a) ions (b) protonsFigure 4. Figure showing different types of diffusion as a function of Υr∗ and L/r∗ for carbon ions (Fig. 4a), andprotons (Fig. 4b). The different colours indicate the type of function which best approximates the time evolution ofthe average square radial deviation, and thus essentially the type of diffusion in the system.similar analysis for electrons in future and – basedon a comparison of the trajectories – we expect theamplitude threshold to be higher.The investigation of the effect of L on the systemis inconclusive. There is no measurable difference athigh amplitudes of U0, which confirms our assumptionthat the ergodic layer plays very little role at such am-plitudes. However, in the onset range 0.1-0.5V, thereappear to be marginally more confined trajectoriesin the absence of the ergodic layer. This conjecturecould be further investigated in future.4.3. Phase space plotsBased on our simulations, we observe that both theincrease in the amplitude of the electrostatic potential,represented by Υ, and the increase in the amplitudeof the ergodic layer L in general tend to increase thediffusion, see Figure 3. This is the case for both thecarbon ions (Fig. 3a) and the protons (Fig. 3b). Wealso studied the diffusion type based on the sameparameters (see Fig. 4). It is remarkable to observethat the regions of dominant diffusion types matchthe regions of a certain diffusion rate, cf. Figures 4aand 4b.Our simulations also suggest that the diffusion ofheavy particles follows a scaling pattern, i.e. thepositions of the distinct regions remain approximatelyconstant for particles of different mass, see Figure 3.The mappings of the diffusion types also show distinctsimilarities: it is possible to see transitions from onediffusion regime into another in the same span ofL/r∗,Υr∗ for particles of different mass, see Figure 4.Unfortunately, we were unable to complete the pro-cedure for electrons due to the enormous time require-ments. We have seen, however, that the diffusionof test particles of muon mass is considerably differ-ent from the diffusion of ions. This remains to beinvestigated in more detail in future.5. SummaryIn our paper, we have attempted to ascertain whetherit is electrostatic perturbation or the RMP layer thatdetermines the particle dynamics. Based on the resultspresented above, we can conclude that for particlesof nucleon mass at total energy 20 eV, electrostaticperturbation is the most influential factor in the sys-tem. Even at relatively low amplitudes of 0.1–0.5Velectrostatic perturbation affects the dynamics, andobviously dominates it at 5V. The increase of theamplitude of the potential results in two main ef-fects. First, the trajectories of the particles becometrapped in the potential structures. Second, the ra-dial diffusion increases globally, which suggests thatnon-trapped trajectories are diffusing much more inthe radial direction.We also scanned the parametric space of the am-plitudes of the electric and perturbed magnetic fieldin the system and calculated the radial diffusion co-efficient for heavy particles. We discovered that fora particular choice of parameters L/r∗, U0qr∗/E thecoefficient scales well with particle mass. We alsoobserved transitions between regimes dominated bythe first or second factor. In addition, we were ableto study the diffusion type and its dependence on theparameters of the system, where we saw that changesin the rate of diffusion and its type are closely related.Last, we were able to compare the effect of electro-static perturbation and the RMP layer on electronsand heavier particles. Based on a study of their tra-jectories, we observed that electrostatic perturbationaffects electrons as well, but its effect appears smallerthan it is for heavy particles. The range of parame-ters where electrostatic perturbation dominates theparticle dynamics is different for electrons. This sug-gests that the electron dynamics in the system is alsodifferent. We are planning to study electron dynam-ics in more detail in future. Eventually, our resultsshould be checked by employing a more realistic form363M. Kurian, L. Krlín, P. Cahyna, R. Pánek Acta Polytechnicaof the electrostatic potential, such as the Hasegawa-Wakatani model.AcknowledgementsThis work was supported by the Grant Agency ofthe Academy of Sciences of the Czech Republic No.IAA100430502 and by the Grant Agency of the CzechRepublic No. GA202/07/004, and partly supported byMSMT Project No.LM2011021.We would like to express our thanks to J. Seidl andR. Papřok for their opinions and suggestions, which theyshared with us freely.References[1] Evans T. E., et al. Nature Physics 2 (2006), 419-423[2] Naulin V., Nielsen A. H., Rasmussen J. J. Phys.Plasmas, 6 (1999) 4575.[3] Krlín L., Stöckel J., Svoboda V. Plasma Physics andControlled Fusions, 41 (1999) 339.[4] Krlín L., Papřok R., Svoboda V. European PhysicalJournal D, Vol. 48 (2008), 95-109[5] Bleuel J. et al. New Journal of Physics, Vol. 4 38(2002)[6] Geisel T. in Lectures Notes in Physics Vol. 450,(Editors M.F. Shlesinger, G.M. Zaslavsky, U. Frisch)Springer (1995), 151-173[7] Cahyna P., Krlín L. Czech. J. Phys. Vol. 56 (2006) 367[8] Krlín L., Cahyna P. Czech. J. Phys. Vol. 56 (2006)Suppl. B B111.364

Particle Dynamics in the RMP Ergodic Layer under the Influence of Edge Plasma Turbulence

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