Comparing spontaneous and pellet-triggered ELMs via non-linear extended MHD simulations
A. Cathey, M. Hoelzl, S. Futatani, P.T. Lang, K. Lackner, G.T.A. Huijsmans, S.J.P. Pamela, S. Günter, JOREK team, ASDEX Upgrade Team, EUROfusion MST1 Team
CComparing spontaneous and pellet-triggered ELMs via non-linearextended MHD simulations
A. Cathey ∗ , M. Hoelzl , S. Futatani , P.T. Lang , K. Lackner , G.T.A. Huijsmans ,S.J.P. Pamela , S. G¨unter , the JOREK team , the ASDEX Upgrade Team , and theEUROfusion MST1 Team Max Planck Institute for Plasma Physics, Boltzmannstr.2, 85748 Garching, Germany Physik Department, TUM, 85748 Garching, Germany Universitat Polit`ecnica de Catalunya, Barcelona, Spain CEA, IRFM, 13108 Saint-Paul-Lez-Durance, France Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands CCFE, Culham Science Centre, Abingdon, Oxon, OX14 3DB, United Kingdom see the author list of H. Meyer et al. 2019 Nucl. Fusion 59 112014 see the author list of B. Labit et al. 2019 Nucl. Fusion 59 0860020 Abstract
Injecting frozen deuterium pellets into an ELMy H-mode plasma is a well established scheme for trigger-ing edge localized modes (ELMs) before they nat-urally occur. Based on an ASDEX Upgrade H-mode plasma, this article presents a comparison ofextended MHD simulations of spontaneous type-IELMs and pellet-triggered ELMs allowing to studytheir non-linear dynamics in detail. In particular,pellet-triggered ELMs are simulated by injecting deu-terium pellets into different time points during thepedestal build-up described in [A. Cathey et al. Nu-clear Fusion 60, 124007 (2020)]. Realistic ExB anddiamagnetic background plasma flows as well as thetime dependent bootstrap current evolution are in-cluded during the build-up to capture the balance be-tween stabilising and destabilising terms for the edgeinstabilities accurately. Dependencies on the pelletsize and injection times are studied. The spatio-temporal structures of the modes and the resultingdivertor heat fluxes are compared in detail betweenspontaneous and triggered ELMs. We observe thatthe premature excitation of ELMs by means of pelletinjection is caused by a helical perturbation describedby a toroidal mode number of n = 1. In accordancewith experimental observations, the pellet-triggeredELMs show reduced thermal energy losses and nar-rower divertor wetted area with respect to sponta-neous ELMs. The peak divertor energy fluency isseen to decrease when ELMs are triggered by pelletsinjected earlier during the pedestal build-up. ∗ [email protected] Type-I edge localized modes (ELMs) are expected toinduce large losses and consequently excessive tran-sient divertor heat loads in large machines like ITER,which can affect the lifetime of the components [1].Low-frequency large type-I ELMs must therefore beavoided in ITER either by small-ELM scenarios [2],ELM-free scenarios like QH-mode [3], active con-trol via external resonant magnetic perturbations(RMPs) [4] or pellet ELM triggering [5, 6]. SinceRMPs may not be applicable in all phases of plasmaoperation (e.g., ramp-up and ramp-down) due to par-ticular constraints on the plasma parameters and theedge safety factor and the possibilities of access to no-ELM or small-ELM regimes remain uncertain, pelletELM triggering may offer a complementing approachand is foreseen as backup scheme in ITER. Addition-ally, the use of ELM pacing by means of pellet injec-tion is an option for ITER to avoid the accumulationof impurities in the plasma [1]. It has been shown ex-perimentally that pellets allow to increase the ELMfrequency and reduce the thermal energy losses as-sociated to an individual ELM crash [5]. For pel-let ELM triggering to be successful in mitigating theimpact of ELMs on the divertor lifetime, the proper-ties of spontaneous and pellet-triggered ELMs need tobe investigated in direct comparison because experi-mental observations suggest that a reduced extent ofthe wetted area can cancel out the beneficial effectsof the decreased energy expelled by pellet-triggeredELMs [7].In the present article, directly comparable simula-tions of spontaneous and pellet-triggered ELMs areperformed with the non-linear extended MHD code1 a r X i v : . [ phy s i c s . p l a s m - ph ] F e b OREK [8, 9, 10]. For this purpose, pellet injec-tions are considered during different time points ofthe pedestal build-up; the setup is thoroughly de-scribed in Ref. [11]. Realistic resistivity, heat dif-fusion anisotropy, and plasma background flows aretaken into account and two different pellet sizes arestudied. It has been observed experimentally that in-jecting pellets during the pedestal build-up does notalways lead to ELM triggering [12]. For the presentarticle, we solely concentrate on injections that man-age to trigger an ELM. Injections performed at ear-lier times during the pedestal build-up lie outside thescope of the present study and are investigated sepa-rately in Ref. [13], where the experimentally observedlag-time in the ELM cycle during which ELM trigger-ing is not possible is compared to simulation results.The first experiments on ELM pacing and mitiga-tion by pellet injection were performed at ASDEXUpgrade (AUG) [5] demonstrating an increase of theELM frequency by more than 50% in a reliable man-ner. With high pellet injection frequencies, undesiredfuelling effects were observed due to the compara-bly large pellet sizes available in this machine. Pel-let injection was shown to trigger an ELM when thepellet-induced seed perturbation was roughy half-waybetween the separatrix and the pedestal top ( ∼ ∼ . ∼
600 m / s) [14]. Key findings from AUGwith carbon wall (AUG-C) were confirmed at largermachines like DIII-D [6] and JET [15]. Investigationof pellet-triggered MHD events in AUG-C and JETshowed that triggered and spontaneous ELMs displayessentially the same features. In particular, the trig-gered ELM correlates to the spontaneous ELM typeoccurring at the same plasma conditions [16].Experiments with metal walls in JET with ITER-like wall (JET-ILW) and in ASDEX Upgrade withtungsten coated walls (AUG-W) showed that inject-ing pellets at different times during the ELM cycleare not always able to trigger ELMs [17, 12, 18]. Saidlag-time represents an important uncertainty in termsof pellet ELM pacing in future machines with metalwalls, like ITER. Another uncertainty that prevailsfor the feasibility of pellet injection as an ELM con-trol method is the wetted area of the heat flux de-position on the divertor targets. In particular, fromcurrent machines it is known that the wetted areafor pellet-triggered ELMs is smaller than that forspontaneously occurring ELMs [7]. Therefore, evenif pellet-triggered ELMs cause smaller energy lossesthan spontaneous ELMs, the fact that said energy isdeposited over a smaller area could mean that there isno reduction in the energy fluency for pellet-triggeredELMs.The article is structured as follows. Section 2provides essential information regarding the physics model and the pellet module in JOREK and it pro-vides a brief overview of previous pellet-triggeredELM simulations produced with JOREK. Section 3describes the simulation set-up and the different pel-let injection parameters used. A thorough descrip-tion of the temporal dynamics of spontaneous andtriggered ELMs, including detailed analysis of theobserved similarities and differences, is provided insection 4. Section 5 is focused on comparing severalspontaneous and pellet-triggered ELMs in terms ofELM sizes, toroidal mode spectra, divertor incidentpower, peak heat fluxes, and energy fluency. The ar-ticle ends with the main conclusions and outlook tofuture work. The JOREK code is a thoroughly tested MHD code(see Section 4 of Ref. [19] for an overview) that isused to non-linearly solve the reduced and full MHDequations. The reduced MHD model with variousextensions [20, 21] has been used for this work. Itevolves the coupled equations for poloidal magneticflux ψ , single fluid mass density ρ and temperature T (= T e + T i ), parallel plasma velocity v (cid:107) , and elec-trostatic potential Φ. Equations for the toroidal cur-rent density j φ and the vorticity ω are also solved fornumerical reasons. Further details of the model areoutlined below. The reduced MHD model solved with the JOREKcode relies on two simplifying assumptions on themagnetic field and on the plasma velocity. The latteris the zeroth order assumption for the plasma veloc-ity perpendicular to the magnetic field v ⊥ = v E × B ,and allows a potential formulation for v ⊥ throughthe electrostatic scalar potential, i.e. E = − ∇ Φ.The former is the consideration of a static toroidalmagnetic field that only varies with major radiusdescribed by B tor ( R ) = B axis R axis /R = F /R . Thisassumption simultaneously eliminates one dynamicvariable and the fast magnetosonic wave from the sys-tem.Introducing the assumptions above into the visco-resistive MHD equations, and considering diffusiveparticle and heat transport, results into the base re-duced MHD equations in JOREK [8]. The perpen-dicular diffusive particle and heat transport used inJOREK are ad-hoc diffusion profiles meant to rep-resent anomalous transport. The parallel heat dif-fusion is temperature dependent and it is set by therealistic Spitzer-H¨arm coefficients. It is possible to in-clude the two-fluid diamagnetic drift effect onto the2ase reduced MHD equations. In order to do so,the representation for the perpendicular plasma ve-locity is changed to v ⊥ = v ExB + v ∗ ion . Due to thisextension to reduced MHD, the radial electric fieldwell ubiquitous to the edge of H-mode plasmas (androughly proportional to the ion diamagnetic drift ve-locity v ∗ ion ∼ ∇ p i /n ) can be considered in JOREKsimulations [22, 23]. The presence of this radial elec-tric field has important consequences regarding thestability of the underlying instabilities that give riseto ELMs [24]. These extensions allowing to incor-porate realistic plasma flows are used in the presentwork.The reduced MHD model is further extended toinclude the self-generated neoclassical bootstrap cur-rent that is formed due to collisions between trappedand passing particles. The bootstrap current den-sity increases with increasing pressure gradient. Toinclude this neoclassical effect in JOREK, a sourceterm determined by the Sauter formula of the boot-strap current is considered in the induction equa-tion [25, 26]. The resulting equations may be foundin [21]. In order to study the influence of pellet injection ontothe plasma, JOREK features a so-called pellet mod-ule. The pellet module represents the pellet parti-cles that will be deposited to the bulk plasma as alocalised adiabatic 3D density source. The densitysource is poloidally localised to a narrow area, and itis stretched to span a user-defined toroidal arc. Theadiabatic density source moves with the pellet posi-tion, and its time-dependent amplitude results fromthe neutral gas shielding ablation model as describedin [27]. For a given time point this model describesthe number of particles that are ionised and becomepart of the bulk plasma.The pellet travels in a straight line following the di-rection of the predetermined injection velocity. Theadiabatic 3D density source locally increases theplasma density and, in turn, directly causes a re-duction in temperature. In the pellet ablation cloud(that stretches along the magnetic field lines), thepressure increases as the temperature is partiallyrestored by fast parallel electron heat conduction( τ χ (cid:107) = (2 πRq ) /χ (cid:107) ∼ . µ s) while the density redis-tribution within a flux surface happens on the longertime scale of parallel convection with the ion soundspeed ( τ s = 2 πRq/c s ∼ Edge localized mode physics was studied in variousways already using the JOREK code – first in Ref. [8]. Further relevant simulations with JOREK include:spontaneous ELMs with realistic plasma backgroundflows [20], RMP penetration [28, 29], investigation ofELM-RMP interactions [30], Quiescent H-Mode [31],triggering of ELMs by vertical magnetic kicks [32],and a direct comparison of the divertor heat fluencecaused by spontaneous ELMs to experimental scal-ing laws [21]. Pellet ELM triggering has also beenstudied with JOREK before, providing an explana-tion for the mechanism of pellet ELM triggering bya localised increase of the pressure in the re-heatedablation cloud and including experimental compar-isons to JET and DIII-D [33, 27, 34]. An overview ofELM related non-linear MHD simulations worldwideis given in Ref. [35]. However, the field has evolvedrapidly since the publication of that article. A recentoverview of ELM and ELM control simulations withJOREK is given in Section 5 of Ref. [19].For ASDEX Upgrade, in particular the localisedstructures forming during ELM crashes [36], non-linear mode coupling associated with an ELMcrash [37], the toroidal structure of an ELMcrash [38], and ELM control via resonant magneticperturbation fields [39] have been studied. Recently,type-I ELM cycles and the triggering mechanism re-sponsible for the violent onset of the ELM crash werestudied for the first time [11]. In the present arti-cle, pellet-triggered ELM simulations are comparedto spontaneous ELMs from the aforementioned arti-cle and new dedicated simulations.The present article goes beyond previous studiesof pellet-triggered ELMs by producing a direct com-parison between state-of-the-art simulations of type-I ELM cycles and pellet-triggered ELMs where bothare performed using the same extended MHD modelincluding ExB and diamagnetic background flows andthe same plasma conditions with realistic plasma pa-rameters.
Recent simulations of type-I ELM cycles show thatthe seed perturbations out of which instabilities growprior to the ELM have an important effect on the dy-namics of the non-linear dynamics and properties ofthe crash. Namely, starting from arbitrary seed per-turbations, instead of self-consistent perturbations,causes larger ELMs when the pedestal build-up isconsidered. Simulating type-I ELM cycles circum-vents this problem since the seed perturbations forall ELMs (except the first) retain the “memory” ofthe previous existence of an ELM, i.e., have non-negligible amplitudes and peeling-ballooning modestructure [11]. The present study uses the first threetype-I ELMs from [11] to compare the dynamics of3pontaneous ELMs and pellet-triggered ELMs. Addi-tionally, new simulations of type-I ELM cycles withincreased toroidal resolution are performed to com-pare against the pellet-triggered ELMs.The spontaneous ELM simulations are set up inthe following way. An ideal MHD stable post-ELMequilibrium reconstruction for AUG shot B ) profiles with a well in the pedestalregion to model the H-mode edge transport barrieras well as stationary heat and particle sources suchthat the pedestal builds up with time and crosses thepeeling-ballooning stability boundary. This simpli-fied pedestal build-up does not take into account thedynamical response of anomalous transport, nor doesit include the physics of neutrals, which are key fora realistic consideration of the time-evolving parti-cle source profiles. Since we use a single fluid model,the disparate evolution timescales of the electron andion temperatures is neglected. Plasma resistivity andparallel heat transport are modelled with fully real-istic parameters, and with Spitzer and Spitzer-H¨armtemperature dependencies, respectively. Simulationsare performed with a high resolution finite elementgrid in the poloidal direction and the convergence ofthe results has been verified.With the density and temperature profiles steep-ening as the simulation evolves, the radial electricfield and the neoclassical bootstrap current evolve ac-cordingly. Four pressure profiles at the outer mid-plane are shown in fig. 1. These correspond tothe pre-ELM state of one spontaneous ELM (fullline labelled Sp318.1 – the naming conventions aredescribed later), and to the pellet injection times:12 , , and 15 ms (dashed lines in blue, black, andgray respectively). The AUG pellet injection system has an injection an-gle of α injection = 72 ◦ , and it can handle pellets of dif-ferent nominal sizes and velocities. These correspondto 1 . × , . × , and 3 . × D atoms perpellet and v p = 240 , , / s. How-ever, material is lost on the guide tube as the pel-let travels to the plasma and, ultimately, the pelletsizes arriving at the plasma show significant vari-ations from their nominal sizes. These variationsare most pronounced with increasing pellet injec-tion velocity and, on average, account for roughly50% of the nominal mass lost for v p = 560 m / s [40].For the pellets used in the present simulations, wechose v p = 560 m / s and two different pellet sizes of0 . × and 1 . × D atoms per pellet (“small”and “large”). These values reflect particle content af-ter losses in the guide tube and therefore correspond Figure 1: Pressure profiles at the outboard mid-plane in the pre-ELM stage of a spontaneous ELM(full lines) and at the different pellet injection times(dashed lines).approximately to the range of accessible pellet sizesin the experiment.The pellet module describes the pellet movingstraight with constant velocity. The simplifying as-sumption of a constant pellet velocity is motivated byAUG experimental observations [14]. Using the sameinitial position and injection velocity and angle, thetwo different pellet sizes are injected at 12 and 14 ms.The small pellet is additionally injected at 15 ms. Weadopt a naming convention for the pellet-triggeredELMs using the pellet size and injection time. Forexample, to refer to the ELM triggered by injectingthe small (0 . × D atoms) pellet at 14 ms, we usethe name
Tr-08-14ms . Accordingly, the large pelletinjected at 12 ms is named
Tr-15-12ms .Figure 2 shows poloidal planes (at the toroidal an-gle corresponding to the pellet injection) of the elec-tron number density n e at three different time pointsafter a large pellet has been injected at 14 ms, i.e. Tr-15-14ms . The pellet trajectory is also plottedwith a dashed black line. The first frame shows thedensity 60 µ s after the pellet was injected. At thistime, no observable non-axisymmetric perturbations(other than the pellet) are seen, and pellet is lo-cated inside of the separatrix in the radial position ψ N = 0 . .
09 ms and the pelletposition at the ELM onset is ψ N = 0 . n e during the pellet-triggered ELM crash can beobserved in the second and third frames of fig. 2 at14 .
22 ms and 14 .
38 ms, respectively. In these frames,strong ( δn e /n e ∼
1) peeling-ballooning modes at theedge of the confined region are observed.4igure 2: Three different time points of the poloidal profile of the density for
Tr-15-14ms – large (1 . × Datoms) pellet injection at 14 ms. The redistribution of the excess density introduced by the pellet takes placedue to parallel convection at the local sound speed. The pellet trajectory is shown as black dashed line. Thepellet location at the three different time points is ψ N = 0 . . .
749 respectively.
For the spontaneous ELMs simulation of [11], whichrequired resolving the fast timescales related to theELMs ( τ A ∼ . µ s) and the slow time-scales of thepedestal build-up ( τ ped ∼ /f ELM ∼
10 ms), the eventoroidal mode numbers until n = 12 were considered,i.e. the toroidal periodicity chosen for those simu-lations was n period = 2. This is equivalent to simu-lating only half of the torus, and it was chosen inorder to reduce the computational cost of the sim-ulations – while maintaining the cyclical dynamics.Recent code optimizations [41] allow to increase thetoroidal resolution for the present work. Thus, dedi-cated spontaneous ELM simulations were performedfor this article with higher maximum toroidal modenumbers, n max = 15, 18, and 20. For these dedi-cated simulations the ELM crashes become faster andmore violent. For this reason, we focus the compari-son on these new spontaneous ELM crash simulationswith increased n max . Where comparisons are basedon simulations from Ref. [11], we restrict them totime-integrated quantities as they are less dependenton toroidal resolution.The new spontaneous ELM simulations with in-creased n max include the toroidal mode numbersn = 0 − −
15, n = 0 − −
18 and n = 0 − − th spontaneous ELM simulated with the follow-ing toroidal mode numbers n = 0 − n period − n max isnamed Sp- n period n max .j . As an example, the first spontaneous ELM simulated with the toroidal modenumbers n = 0 − −
18 is named
Sp-318.1 . Finally,a simulation with a further increased resolution in-cluding the toroidal mode numbers n = 0 − − − −
15 and n = 0 − −
30 was found to be ∼ − −
12. As can be seenfrom the energy spectra shown in Ref. [13], the n = 12mode is strongly sub-dominant in all cases providinga justification for this choice. Higher toroidal reso-lutions will only be affordable after future code opti-mizations.
This section is divided into four parts. In subsec-tion 4.1, we present a detailed description of the tem-poral dynamics of a representative spontaneous ELM5rash (
Sp-318.1 ), and in subsection 4.2 of a repre-sentative pellet-triggered ELM (
Tr-08-14ms ). Eachsubsection details the evolution of the energies of thenon-axisymmetric perturbations present in the simu-lations, the power that is incident on the simplifieddivertor targets, the time evolving reconnecting mag-netic field, and other dynamical quantities of interest.Thereafter, in subsection 4.3 a side-by-side compar-ison in terms of the heat fluxes between the rep-resentative pellet-triggered and spontaneous ELMsis presented. Finally, subsection 4.4 describes thedifferences and similarities of the non-axisymmetricmode activity present during the ELM crash between
Sp-318.1 and
Tr-08-14ms . A systematic comparisonof all cases is described later in section 5.
Here, we provide a detailed look at the dynamics ofa representative spontaneous ELM crash. In this sec-tion, we focus on the first type-I ELM from the sim-ulation with n = 0 − −
18. This ELM crash takesplace at 16 ms, and the pre-ELM pressure profile wasshown in fig. 1. The magnetic energy of the non-axisymmetric perturbations, the incident power onthe inner and outer divertors, and the toroidally av-eraged outboard midplane maximum edge pressuregradient are shown in fig. 3(a-c). During the ELMcrash the thermal energy stored inside the separa-trix is reduced from 421 kJ to 388 kJ, i.e. the ELMsize is ∆ W ELM = 33 kJ and the relative ELM size is∆ E ELM = ∆ W ELM /W preELM ≈ . . . . . P div , in / out return to the inter-ELMvalues within roughly 0 . q ( t, s, φ ), where t is time, s is the distance alongthe target, and φ is the toroidal angle. The totalpower that reaches a given divertor target is definedby P div = (cid:82) π (cid:82) s max s q ( t, s, φ ) R ds dφ , where R is themajor radius. During the peak of the ELM crash, thepower that reaches the outer divertor target P div , out is roughly twice the power reaching the inner target P div , inn . This power asymmetry is observed for all thesimulated spontaneous and pellet-triggered ELMs. Inexperiments under similar conditions, i.e., with theion B × ∇ B direction pointing to the active X-point,the inner divertor receives more energy than the outerdivertor [42]. Without a proper inclusion of ExBand diamagnetic background flows, this discrepancywould be more pronounced [43]. The main reason forthe remaining differences lies in the simplified modelfor the SOL used here. Separate efforts are underwayto address this issue, but are not part of the present Figure 3: Time evolution of the magnetic energies(a), incident power on the inner and outer divertors(b), and change in the toroidally averaged outboardmidplane maximum edge pressure gradient (c) duringa spontaneous ELM.work.The maximum (toroidally averaged at the outermidplane) pressure gradient crashes rapidly as a re-sult of the ELM as shown in fig. 3(c). This reductioncauses the drive for the underlying instabilities to beremoved and for the ELM crash to conclude shortlythereafter. This allows the maximum ∇ p to recoverquickly until it transiently stagnates at ∼ . .
20 ms) and after (17 . , . , and 17 .
30 ms) theELM crash are shown in fig. 4, where it is evidentthat the spontaneous ELM is significantly reducingthe edge pedestal pressure. Together with the de-pletion of the pressure pedestal, the radial electricfield and the bootstrap current density are similarlyreduced.Poincar´e plots for this spontaneous ELM are shownin fig. 5 at different times for the radial region from ψ N = [0 . − . φ = 0 are plotted. Differ-ent colors are used for each field line such that the“mixing of colors” visualizes the radial diffusion offield lines during stochastisation. The y-axis corre-sponds to the geometrical poloidal angle θ with theoutboard midplane located at θ = 0. The time ofmaximum heat flux P outer div to the outer divertor is6 O u t boa r d m i dp l ane p r e ss u r e [ k P a ] ψ N Figure 4: Outer midplane pressure profiles before andafter a spontaneous ELM crash (
Sp-318.1 ). Thepressure at the radial location ψ N = 0 .
95 goes from ∼ . ∼ . t = 16 .
57 ms, and the different times correspond to t − t = [ − .
22 : 0 .
04 : 0 .
02] ms.At the earliest time plotted, t − t = − .
22 ms, theenergy of the non-axisymmetric perturbations is al-ready large, as can be seen in fig. 3(a), and thedensity perturbations are comparable to the back-ground plasma ( δn/n ∼ ψ N (cid:38) . ψ N ≈ [0 . − .
88] three sets of magnetic is-lands located at the q = 11 / , / , and 13 / t − t = − .
22 ms.The amplitude of the peeling-ballooning modes in-creases further thereby causing the stochastic mag-netic topology to erode further inwards. The in-creased stochastisation around the time of maximumdivertor heat flux (the two bottom plots of fig. 5)is clearly visible. Field lines in the outer plasma re-gions ψ N (cid:38) .
87 reach the divertor targets with a veryshort connection length in this phase, as reflected bythe strongly reduced number of points visible close tothe plasma edge.Figure 5 features characteristic structures in theoutermost edge. These lobe structures representsplitting of the strike lines that hit the divertor tar-gets (and are shown clearly later on in fig. 12). Suchsplitting is a common experimental observation, andevaluating quasi-toroidal mode numbers has been ac- Figure 5: Seven Poincar´e plots at different times withrespect to t = 16 .
57 ms. The chosen times are sep-arated by 0 .
04 ms. Precursor modes cause the non-axisymmetric topology observed in the earliest plots(top panels). The growing perturbations cause theregion with short connection length (white region)to penetrate further inward until ψ N ≈ .
87 (bottompanels).complished through such observations [42, 44]. Atthe time of maximum incident power on the outer di-vertor, t = t , for instance, the heat flux impingingon different toroidal angles of the outer divertor isshown in fig. 6, and the strike line splitting can be The range of toroidal angles is limited to φ = [0 − ◦ because this simulation only considers toroidal mode numberswhich are multiples of 3, i.e. one third of the tokamak. P outer div , t = 16 .
57 ms. The footprint corresponding to sev-eral strike lines can be observed. The wetted area atthis time point is A wet ≈ .
95 m .inferred from the slightly slanted stripes in the heatflux profile.An important feature in terms of the heat flux ontothe divertor target, which is related to the strike linesplitting, is the area over which the energy is de-posited, i.e. the wetted area A wet = (cid:82) π (cid:82) q ( s, t = t ) dsRdφ max( q ( s, t = t )) = P div ( t = t )max( q ( s, t = t )) . For the type-I ELM crash discussed here (
Sp-318.1 ),the wetted area of the outer divertor at the time cor-responding to fig. 6 is A wet ≈ .
95 m . Now, we shift the focus to the temporal dynamics rel-evant to pellet-triggered ELMs. To do so, we presenta detailed description of a representative ELM trig-gering case where a pellet of 0 . × D atoms isinjected at 14 ms (
Tr-08-14ms ), i.e. 2 ms beforethe spontaneous ELM crash would appear. Like allthe pellet-triggered ELMs present in this work, thetoroidal mode numbers included in the simulationare n = 0 − −
12. At 14 ms, the thermal energystored inside the plasma at the time of injection is( ∼ Tr-08-14ms is roughly 18 kJ,which corresponds to a relative ELM size of 4 . . × D atoms at 14 ms.This energy ultimately reaches the divertor targets,as observed in fig. 7(b). The peak incident power ontothe inner and outer divertors caused by this pellet-triggered ELM are 10 . . . , . ψ N = 1 . ψ N = 0 . . v p = 560 m / s, the timerequired to reach the maximum pressure gradient lo-cation is 80 µ s. The pre- and post-ELM pressureprofiles for Tr-08-14ms , and post-ELM for
Sp-318.1 and
Sp-318.2 , at the outer midplane are shown infig. 8. The position of the pellet in terms of ψ N for6 different time points is also plotted. It is observedthat the post-ELM profiles for the spontaneous ELMsconsidered here have lower pedestals than the pellet-triggered ELM.As the pellet enters the confined region, the growthrates of the high-n perturbations begin to increaserapidly. This increase takes place due to the excita-tion of ballooning modes in the high-field side (HFS)and low-field side (LFS) regions. In the absence ofpellet injection, ballooning modes are localised at theLFS because the magnetic field curvature (which sta-bilises ballooning modes and always points to the cen-tre of the device) is parallel to the pressure gradi-ent (which destabilises ballooning modes and alwayspoints to the magnetic axis) in the LFS, and it is anti-parallel to ∇ p in the HFS. However, in the presenceof pellet injection, as the pellet enters the confined8igure 8: Outer midplane pressure profiles before(in dark red) and after Tr-08-14ms (in yellow andgreen), and after
Sp-318.1 and
Sp-318.2 (in greyand dark grey, respectively). The post-ELM profilesfor the spontaneous ELMs have lower pedestals than
Tr-08-14ms . The position of the pellet in terms ofthe normalised poloidal flux is also shown for 6 dif-ferent points in time.region there is a significant local increase of the den-sity taking place due to the pellet ablation. Withan adiabatic source, the density increase is relatedto a temperature decrease. The local temperature israpidly increased again due to the fast electron par-allel heat transport (simulations performed with therealistic Spitzer-H¨arm parallel heat diffusion). Theexcess density is also transported along the magneticfield lines, but on the much slower time scale of theion sound speed, as described in section 2.2. As a re-sult of the fast temperature recovery and slow excessdensity propagation, a strong pressure perturbation isobtained along the magnetic field lines affected by thepellet source. The pressure gradient is both paralleland anti-parallel to the pellet trajectory (dependingon which side of the pellet position is being consid-ered). The localised pressure gradient to the right ofthe pellet position is parallel to the field line curva-ture and, therefore, may excite ballooning modes inthe HFS.The time of maximum outer divertor incidentpower is t = 14 .
22 ms. Seven different times withequal spacing 0 .
04 ms are chosen with respect tothe time of maximum outer divertor incident power t − t = − .
22 : 0 .
04 : 0 .
02 ms in order to analysehow the edge magnetic topology changes with pel-let injection. For this pellet-triggered ELM, Poincar´eplots at the times mentioned before are shown infig. 9. At t − t = − .
22 ms, i.e. at the time of pel-let injection, the edge magnetic field is axisymmetric. Forty microseconds later, as the pellet crosses the sep-aratrix (see fig. 8), the magnetic topology outwardsof ψ N ≈ .
90 has become non-axisymmetric, but noreconnection has yet taken place.Figure 9: Seven Poincar´e plots at different timeswith respect to the time of maximum P outer div , t = 14 .
22 ms. From top to bottom, the chosen timesare separated by 0 .
04 ms. The initially axisymmet-ric topology (top panel) becomes perturbed when thepellet crosses the separatrix (second panel), and laterstarts to reconnect (third panel). The ELM is thentriggered (fourth panel), and ultimately the regionwith short connection length (white region) pene-trates until ψ N ≈ .
87 (bottom panels).In the next time slice ( t − t = − .
14 ms, i.e. t = t inj . + 0 .
08 ms) the pellet has reached the radial9igure 10: Outer divertor heat flux profile at thetime of maximum P outer div , t = 14 .
22 ms. Thefootprint corresponding to two main strike lines canbe observed. The wetted area at this time point is A wet ≈ .
65 m , which is ∼
31 % lower than the spon-taneous ELM
Sp-318.1 .position of ψ N ≈ .
96. At this point, some magneticreconnection has taken place and a stochastic edgeis formed from the last closed flux surface until thepellet position. This time slice is the last to clearlyrepresent the pellet-induced magnetic perturbation.The following time slices ( t ≥ t inj. + 0 .
12 ms) featurea stochastic magnetic topology mostly caused by theMHD response to the pellet perturbation. This canbe distinguished because the stochastic layer has pen-etrated significantly further inwards than the pellethas. Namely, at t = t inj . + 0 .
12 ms the pellet locationis ψ N ≈ .
95 while the stochastic region has pene-trated until ψ N ≈ .
90. Similarly to the spontaneousELM crash (fig. 5), field lines at Ψ N (cid:38) .
87 exhibit avery short connection length to the divertor targetsclose to the time of maximum divertor heat flux.The energy expelled by the pellet-triggered ELMis then non-axisymmetrically deposited on the diver-tor targets. Similar to the spontaneous ELM shownbefore, the pellet-triggered ELM deposits more en-ergy (roughly twice) onto the outer divertor thanthe inner divertor. This is in fact a feature of allthe pellet-triggered ELMs described in the presentwork. Using the time of maximum outer divertor in-cident power, t = 14 .
22 ms, the heat flux onto theouter target at different toroidal angles is shown infig. 10. This profile shows two distinct regions withlarge heat deposition: one located in the originalstrike-line position ( ∼ φ ≈ ◦ ).This non-axisymmetric secondary deposition regionrotates slowly along the toroidal direction and isclosely linked to lobe structures.Comparing the instantaneous heat flux profilesalong the toroidal angle for the spontaneous (fig. 6)and pellet-triggered ELMs (fig. 10), it is clear thatthe wetted area for the latter is significantly reduced. As stated before, the wetted area at the time ofmaximum outer divertor incident power is A wet ≈ .
95 m for the spontaneous ELM ( Sp-318.1 ). Onthe other hand, for the pellet-triggered ELM anal-ysed in this section (
Tr-08-14ms ), the wetted area at t is reduced by approximately 31% with respect to Sp-318.1 , i.e. A wet ≈ .
65 m . Sp-318 and
Tr-08-14ms
The ELM outer target energy fluency, ε target ( s, φ ) = (cid:90) t ELM q ( s, φ, t ) dt, (1)for Sp-318.1 (dark yellow lines),
Sp-318.2 (dark yel-low lines with symbols), and
Tr-08-14ms (black lines)is shown in fig. 11. The full lines for each ELM cor-respond to the target fluency profile at φ = 0 and thesmall dots correspond to other toroidal angles. Thefigure clearly shows the reduction in wetted area be-tween the pellet-triggered ELM and the spontaneousELM that was described in the previous paragraph.Figure 11: Outer divertor ELM fluency for two spon-taneous ELMs ( Sp-318.1 and
Sp-318.2 ) and anELM triggered by a pellet containing 0 . × Datoms (
Tr-08-14ms ). The wider wetted area asso-ciated to the spontaneous ELMs with respect to thetriggered ELM can be observed.The second spontaneous ELM,
Sp-318.2 , is in-cluded so that the fluency of a spontaneous ELMborne out of self-consistent seed perturbations canalso be observed. There is a reduction in the peaktarget fluency for the triggered ELM with respect toboth spontaneous ELMs:
Sp-318.1 and
Sp-318.2 .The second spontaneous ELM (full lines with sym-bols in fig. 11) shows a lower peak fluency with re-spect to the first spontaneous ELM. This happensbecause
Sp-318.1 is borne out of noise-level pertur-bations, while
Sp-318.2 is borne out of self-consistentseed perturbations [11]. The reduction of peak tar-get fluency between
Tr-08-14ms and
Sp-318.1 maybe attributed to the lower ELM size of the triggeredELM ( ∼ . ∼ . E . . Scaling the peaktarget fluency of the spontaneous ELM ( ∼ . kJm )with the pellet-triggered ELM size results in a peakvalue of 15 . kJm (cid:0) . . (cid:1) . ≈ . kJm , which is compa-rable to the peak fluency of ∼ . kJm measured forthe pellet-triggered ELM. The remaining difference isattributed to the fact that the pedestal pressure rightbefore the spontaneous ELM is larger than before thepellet-triggered ELM, as may be observed from fig. 1.A side-by-side comparison between Sp-318.1 and
Tr-08-14ms in terms of the heat fluxes to the in-ner and outer divertors, and of the non-axisymmetrictopologies (portrayed by Poincar´e maps in real spacewith a colour scale that reflects the temperature atthe starting position of the magnetic field line associ-ated with each point of the Poincar´e map) is shownin fig. 12 at the times of maximum incident powerto the outer divertor. The wider wetted area andhigher peak values of the spontaneous ELM (left),with respect to the pellet-triggered ELM (right), canbe clearly observed. It appears that the reason be-hind the wider wetted area in the spontaneous ELMis a larger number of secondary strike lines, whichcarry thermal energy from the bulk plasma, acrossthe magnetic separatrix, towards the divertor targets.A slight difference is also observed in terms of the ini-tial temperature of the magnetic field lines traced forthe Poincar´e map – the stochastic region seems toreach further inwards for the spontaneous ELM thanfor the pellet-triggered ELM.
In this subsection we present different quantitiesrelated to the structure of the modes involved inthe ELM crashes described before,
Sp-318.1 and
Tr-08-14ms . In particular, we pick the time point re-lated to the peak incident power onto the outer diver-tor for each event (the same times as figs. 6 and 10 forthe spontaneous and pellet-triggered ELMs, respec-tively). The density (in 10 m − ), poloidal mag-netic flux (in Wb), and the total plasma tempera-ture (in eV) for the spontaneous ELM and the pellet-triggered ELM are shown in the top and bottom rowsof fig. 13. All plots also indicate, with gray lines, sur-faces with different values of normalised poloidal flux( ψ N = [0 . , . , . , . T n=0 and a non-axisymmetric component T n > to form the total tem-perature T = T n=0 + T n > , where n is the toroidalmode number.The spontaneous ELM Sp-318.1 (top row), clearlyfeatures peeling-ballooning structures inside the lastclosed flux surface. Ballooning structures predom- inantly on the LFS are observed in all quantities,but particularly in the perturbations to the poloidalmagnetic flux (centre plot). The spontaneous ELMclearly features large temperature fluctuations on thescrape-off layer (SOL), which maintain the structureobserved in the Poincar´e plot of fig. 12 (left). Inter-estingly, the pellet-triggered ELM shows faint hints ofpeeling-ballooning structures (the most obvious indi-cations are shown in the HFS density perturbationsenclosed between ψ N ≈ . . ψ N ≈ . . Tr-08-14ms than for
Sp-318.1 . The large (in sizeand amplitude) density perturbation induced by thepellet ablation is observed to influence the size of theperturbations related to the ELM crash, i.e. the den-sity perturbations observed for the pellet-triggeredELM appear to be larger than those for the sponta-neous ELM. This is perhaps most clearly evidencedby comparing the plots of the magnetic flux pertur-bations (top and bottom centre plots). The temper-ature perturbations obtained for the pellet-triggeredELM appear to be generally weaker than those for thespontaneous ELM. These perturbations also demon-strate the structures observed in fig. 12 (right). Fi-nally, it is interesting to note that while the sponta-neous ELM features weak (density, flux, and temper-ature) perturbations in the confined region near themagnetic X-point, the pellet-triggered ELM showslarge perturbations (particularly of density and flux)in this region.
Having shown in detail the differing dynamics ofa representative spontaneous and pellet-triggeredELM, the attention is now turned towards comparingseveral key quantities between more simulations ofspontaneous and pellet-triggered ELMs. At first, insubsection 5.1, a comparison in terms of the thermalenergy losses caused by the different types of ELMsis described. Later, an analysis of the toroidal modespectrum is discussed in subsection 5.2. Finally, insubsection 5.3, we study and detail how the respective11igure 12: Heat flux to the inner and outer divertor targets (MW / m ) and real space Poincar´e plots with acolour scheme that reflects the temperature (keV) at the starting location of the associated magnetic fieldlines for (left) Sp-318.1 and (right)
Tr-08-14ms at the respective times of maximum P outer div . The lownumber of secondary strike lines of the pellet-triggered ELM can explain the narrower wetted area whencompared to the spontaneous ELM that splits the strike line into several secondary strike lines.heat fluxes and resulting energy fluencies compare. The energy released by the different ELMs (∆ W ELM )and the relative ELM size (∆ E ELM ) are plottedagainst the pre-ELM pedestal stored energy (pre-ELM W th , ped is obtained with a volume integral from ψ N = 0 . . × Datoms, while the triangles correspond to pellets with1 . × D atoms. Squares and circles representspontaneous ELMs, and different colours are used forsimulations with different sets of toroidal mode num-bers included, e.g. dark yellow for n = 0 − − th , ped ) result insmaller ELM losses with respect to similar-sized pel-lets injected at a later time (larger pre-ELM W th , ped ).For example, the blue pentagons have a smaller ab-solute, and relative, ELM size than their black andgray counterparts. For a given injection timing, thelarger pellets induce a slightly larger triggered ELMsize (triangles lie above pentagons for constant pre-ELM W th , ped ).Comparing the spontaneous and pellet-triggeredELMs reveals only a small difference in ELM size,and such difference is most pronounced when compar-ing the earliest pellet injection. With the large pellet,ELMs may be triggered at even earlier injection times( t inj ≥ t inj ≥
12 ms (further details found inRef. [13]). Finally, it is worth pointing out that thesmall pellet injected at the latest time (only 1 ms be-fore the spontaneous ELM crash would take place)triggers an ELM with a comparable size to the spon-taneous ELMs simulated in these studies. The smalldifferences between spontaneous and pellet-triggeredELMs appear to be in agreement with experimentalobservations at AUG [12].We observe that spontaneous ELMs typically ex-pel more thermal energy than pellet-triggered ELMs,except when the pellet is injected very briefly beforethe spontaneous ELM would appear, i.e. at roughlythe same pedestal conditions. For the pellet-triggeredELMs simulated, we also observe that large pellets in-duce slightly larger thermal energy losses than ELMs12igure 13: Density (in 10 m − ), flux (in Wb), and temperature (in eV) perturbations for Sp-318.1 (toprow), and for
Tr-08-14ms (bottom row), at the respective times of maximum P outer div . Gray lines in allplots indicate surfaces with normalised poloidal flux of ψ N = [0 . , . , . , . Tr-08-14ms than for
Sp-318.1 .triggered by small pellets injected at the same time.
Time-averaged spectra of the perturbations associ-ated to the spontaneous and pellet-triggered ELMsare shown in fig. 15 for the different toroidalmode numbers. The averaging is performed over0 . . (cid:80) n max n > W mag , n is maximised. Excluding n = 1 fromthe summation is done in order to separate, tosome extent, the pellet-induced perturbation fromthe pellet-triggered ELM crash.For all the pellet-triggered ELMs, the most en-ergetic perturbation is the n = 1, followed by thenext low n modes. Such observation appears tobe in agreement with experimental observations atJET [46] and is consistent with previous pellet-13igure 14: Energy losses (a) and relative ELM size(b) of spontaneous (circles denote the first simulatedELM in a series, while squares represent the subse-quent ELMs) and pellet-triggered ELMs (pentagonsfor pellets containing 0 . × D atoms and trian-gles for pellets containing 1 . × D atoms). Spon-taneous ELMs borne out of self-consistent seed per-turbations have similar absolute and relative ELMsizes. Early enough injection can reduce the ELMsize with respect to spontaneous ELMs.Figure 15: Time averaged magnetic (top) and ki-netic (bottom) energy spectrum for spontaneous(full lines) and pellet-triggered ELMs (symbols).The average is performed over 0 . t ( W = W mag , max ) − . Tr-08-14ms (a) and
Sp-318.1 (b). The timeranges chosen correspond to those of figs. 3 and 7,and the colour map is in logarithmic scale.triggered ELM simulations with JOREK [27, 34].The energy spectrum of the pellet-triggered ELMsis broad since the pellet-induced perturbation is de-scribed by low-n modes, and it excites high-n bal-looning modes (as described in section 4.2). On theother hand, for the spontaneous ELMs simulated forthis study, the most unstable toroidal mode num-bers during the ELM crash are 2 , , Sp-318.1 and
Tr-08-14ms ).In fig. 16 we show the time-evolving magnetic en-ergy spectrum, which highlights different dynamicsfor these two different types of events. In partic-ular, the precursor phase of the spontaneous ELM(fig. 16(b) is clearly visible for roughly 1 ms prior tothe ELM onset. On the other hand, the energy of thenon-axisymmetric perturbations related to the pellet-induced perturbation and pellet-triggered ELM be-come abruptly excited roughly 0 . − −
12 are too slowdue to too low toroidal resolution. Simulations withn max ≥
15 are properly converged.
An important metric for the comparison between thesimulated spontaneous and pellet-triggered ELMs isthe dynamical heat deposition profiles. As it wasmentioned in section 4.1, the splitting of the heatload between inner and outer divertor does not matchexperimental observations due to the simplified SOLmodelling. Nevertheless, it is still of interest to anal-yse the differences and similarities between diver-tor heat fluxes for spontaneous and pellet-triggeredELMs. The outer divertor incident power, P outer div ,is plotted for several spontaneous and triggered ELMsin fig. 17. The time axis has been shifted suchthat the maximum of P outer div is at 0 . Sp-212.2 shows a peak diver-tor incident power roughly half that of
Sp-315.2 or Sp-318.2 . If we compare the pellet-triggered ELMswith Sp-212.2 we observe very similar peak divertorincident powers. However, if we compare the pellet-triggered ELMs with the more realistic spontaneousELMs (those with sufficient toroidal resolution), weobserve a clear reduction in the peak divertor inci-dent power by means of pellet injection. Indeed thelatter conclusion is the one that should be drawn.At this stage it is worth pointing out thatwe have performed one simulation comparable to
Tr-08-14ms , but with even higher toroidal modenumbers (n = 0 − − We consider the simulations with n = 0 − −
18 to be con-verged because they show only ∼
3% lower peak outer divertorincident power with respect to simulations with n = 0 − − ( ∼ . × D atoms at 12 ms leads to a muchreduced peak divertor incident power with respectto the other four pellet-triggered ELMs. These re-maining four pellet-triggered ELMs have similar peak P outer div , which is weaker than all the spontaneousELMs (excluding Sp-212.2 which is not yet properlyconverged). This significant reduction in the peak di-vertor incident power does not translate directly tothe target fluency (fig. 11) because of the narrower de-position area intrinsic to the pellet-triggered ELMs.We choose the time of maximum outer divertorincident power to clearly show the heat flux pro-file narrowing for the pellet-triggered ELMs. Fig-ure 18 shows the heat flux profile at φ = 0 (pel-let injection angle) as solid or dashed lines, and atall other toroidal angles as small coloured points.This is done for six different ELMs: four pellet-triggered ELMs (triggered by a small pellet (a) andby a large pellet (b)) and two spontaneous ELMs (c).The pellet-triggered ELMs show much narrower heatflux deposition profiles with respect to the sponta-neous ELMs. It is interesting to note that the non-axisymmetry observed for the pellet-triggered ELMsis most prominent in the vicinity of the maximumheat flux ( s ≈ ε , defined in eqn. 1. For each of the simulationspresented here, but excluding those with an insuffi-cient toroidal resolution ( Sp-212 ), we show the peaktarget fluency as a function of the pre-ELM pedestalstored thermal energy in fig. 19. Following the con-vention from fig. 14, spontaneous ELMs are symbol-ised with circles (for the first ELM in a series) andwith squares (for the subsequent ELMs in a series),and pellet-triggered ELMs are symbolised with pen-tagons and triangles for pellets with 0 . × and1 . × D atoms, respectively.Four out of five pellet-triggered ELMs show a de-crease in the peak target fluency when comparedagainst the converged spontaneous ELMs. In partic-ular, when the ELM crash is triggered significantly15igure 18: Outer divertors heat flux profile at t = t ( max ( P out , div )) for pellet-triggered ELMs (smalland large pellets injected at 12 and 14 ms) and spon-taneous ELMs. Pellet-triggered ELMs (a) and (b)show a narrower deposition profile with respect tospontaneous ELMs (c).Figure 19: Peak fluency for spontaneous (in cir-cles and squares) and pellet-triggered ELMs (in pen-tagons for small pellets and triangles for large pel-lets). Pellet-triggered ELMs show a moderate de-crease in the peak fluency with respect to all but onespontaneous ELMs. early in the ELM cycle the peak fluency can beclearly reduced ( Tr-08-12ms and
Tr-08-14ms ). Fur-thermore, the pellet size is not observed to play animportant role in the peak target fluency. The onlypellet-triggered ELM that does not see a reduction ofthe peak fluency corresponds to pellet injection onlyone millisecond before the spontaneous ELM wouldtake place,
Tr-08-15ms . For this ELM, the pre-ELMpedestal stored thermal energy is comparable to therest of the spontaneous ELMs.
We present detailed comparisons between sponta-neous and pellet-triggered ELMs simulated withJOREK. The spontaneous ELMs simulated for thisstudy extend recent work of modelling full type-IELM cycles [11]. Exploiting recent code optimiza-tions, we perform computations at higher toroidalresolution such that simulations are now convergednot only in terms of 0D quantities like the energylosses, but also regarding time scales of the crashesand peak divertor heat fluencies. The pellet-triggeredELMs are simulated by introducing pellets at differ-ent stages of the pedestal build up. Pellets of twodifferent sizes are considered covering the experimen-tally accessible range approximately. This novel ap-proach at simulating pellet-triggered ELMs has qual-itatively reproduced the lag-time that exists in metalwalled devices like AUG-W and JET-ILW as thor-oughly described in Ref. [13]. All the simulationspresented have been performed with realistic plasmaflows (ion velocity comprised of ExB plus diamag-netic flow, but neglecting higher order terms like thepolarisation drift). For the spontaneous ELMs, tak-ing into account these realistic plasma flows is a nec-essary requirement to recover the cyclical dynam-ics [43, 11]. For the pellet-triggered ELMs, thesesimulations consider, for the first time, such realis-tic plasma flows. Additionally, the pellet-triggeredELMs presented here push the previous state-of-the-art further by considering realistic values for thepedestal resistivity.At first, a detailed analysis of the non-linear dy-namics for one spontaneous ELM and one pellet-triggered ELM was presented to describe the simi-larities and differences between the two events. Inparticular, the spontaneous ELMs shown in this workfeature a precursor phase that lasts roughly one mil-lisecond, while the pellet-triggered ELMs are ex-ited abruptly ( ∼ . , , ∼ − ∼ Acknowledgements
The authors would like to thank Thomas Eich andDavide Silvagni for fruitful discussions. This workhas been carried out within the framework of theEUROfusion Consortium and has received fundingfrom the Euratom research and training program2014-2018 and 2019-2020 under grant agreement No633053. The views and opinions expressed herein donot necessarily reflect those of the European Com-mission. In particular, contributions by EUROfusionwork packages Enabling Research (EnR) and MediumSize Tokamaks (MST) is acknowledged. We acknowl-edge PRACE for awarding us access to MareNostrumat Barcelona Supercomputing Center (BSC), Spain.Some of the simulations were performed using theMarconi-Fusion supercomputer.
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