Nonlinear saturation of kinetic ballooning modesby zonal fields in toroidal plasmas

Cite as: Phys. Plasmas 26, 010701 (2019); doi: 10.1063/1.5066583Submitted: 15 October 2018 . Accepted: 07 January 2019 . Published Online:23 January 2019

G. Dong,1 J. Bao,2 A. Bhattacharjee,1 and Z. Lin2

AFFILIATIONS1 Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08540, USA2 Department of Physics and Astronomy, University of California, Irvine, California 92697, USA

ABSTRACT

Kinetic ballooning modes (KBMs) are widely believed to play a critical role in disruptive dynamics as well as turbulent transport inmagnetic fusion and space plasmas. While the nonlinear evolution of the ballooning modes has been proposed as a mechanismfor “detonation” in various scenarios such as the edge localized modes in tokamaks, the role of the kinetic effects in suchnonlinear dynamics remains largely unexplored. In this work, global gyrokinetic simulation results of KBM nonlinear behavior arepresented. Instead of the finite-time singularity predicted by ideal magnetohydrodynamic theory, the kinetic instability is shownto develop into an intermediate nonlinear regime of exponential growth, followed by a nonlinear saturation regulated by sponta-neously generated zonal fields. In the intermediate nonlinear regime, rapid growth of localized current sheets, which can inducemagnetic reconnection, is observed.

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Ballooning instability (or its astrophysical counterpart, theParker instability) in a magnetized plasma is driven by local unfa-vorable magnetic curvature and a pressure gradient.1 The non-linear evolution of the instability has been a subject of greatinterest for a diverse range of eruptive phenomena such as sub-storms in the Earth’s magnetotail2,3 and edge-localized modes(ELMs)4 in toroidal fusion plasmas.5,6 Theoretical studies of non-linear ideal magnetohydrodynamic (MHD) ballooning modespredict explosive nonlinear growth.7 Finger-like structuresdevelop, forming a front with a steep pressure gradient whichcan nonlinearly destabilize the mode, and result in a finite-timesingularity (“detonation”).8 However, attempts at simulatingsuch an instability using the full MHD equations have not suc-ceeded in realizing a finite-time singularity. While finger-likestructures are indeed observed,9 the mode is seen to grow in thenonlinear regime exponentially with its linear growth rate. A newasymptotic regime, called the “intermediate” nonlinear regime ofexponential growth, has been formulated analytically to accountfor these simulations.10 During the intermediate regime, the modestructure becomes sufficiently narrow that the validity of theMHD model is questionable. In collisionless plasmas, kineticeffects intervene. This leads to considerations of the kinetic bal-looning mode (KBM) which is recognized to play an importantrole in the stability and transport of fusion plasmas near theplasma edge,11,12 as well as substorm dynamics in the Earth’s

magnetotail.13,14 However, the nonlinear dynamics of the KBM intoroidal plasmas is not well understood. Flux-tube gyrokineticsimulations of the KBM arrived at contradictory conclusions:zonal flows play a dominant role in KBM saturation inGENE simu-lations15 but not in GKV simulations where the zonal flows areseen to be much weaker than that in the ion-temperature-gradi-ent (ITG) turbulence.16,17 KBM saturation requires external flowshear in GYRO simulations18 beyond a critical beta value. InBOUTþþ gyrofluid simulations,19 KBM saturates via profilerelaxation.

Here, we demonstrate from a global gyrokinetic particle-in-cell simulation that after a linear regime, the KBM evolvesinto an intermediate regime, followed by a saturated nonlinearregime. In addition to features that are similar to its ideal MHDcounterpart,9 the kinetic intermediate regime also exhibits qual-itatively different features. The most important one is that thekinetic electromagnetic dynamics leads to the spontaneousgeneration of zonal flow (flux-surface-averaged electrostaticpotential hd/i) and zonal current (flux-surface-averaged vectorpotential hdAki). When the zonal flow shear exceeds the lineargrowth rate, zonal flow shearing suppresses the nonlinear insta-bility which in turn self-regulates the zonal fields (the zonal flowand the zonal current), leading to a saturated nonlinear regime.In the kinetic intermediate regime, thin current sheets developnear the mode rational surfaces, which can eventually exhibit

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tearing instability, but the resistive tearing mode growth rateappears to be too slow to have a strong effect on KBM nonlinearsaturation.

Gyrokinetic simulation of KBM.—In the simulations usingthe gyrokinetic toroidal code (GTC),20 ions are treated by thegyrokinetic Vlasov equation, while electrons are described usingthe nonlinear fluid equations: the electron perturbed densitydne is calculated by time-advancing the continuity equation21

including the diamagnetic (pressure gradient) term which pro-vides the interchange drive, and the electron parallel flow duke iscalculated by inverting the parallel Ampere’s Law.22 The gyroki-netic Poisson’s equation is solved to obtain the perturbed elec-trostatic potential d/. For the completeness of the model, theparallel magnetic perturbation dBk

23 and the equilibrium currentdensity,21 which provide an additional linear drive, are kept inthe simulation. The parallel vector potential dAk ¼ dAadik þ dAnakis solved for the adiabatic and non-adiabatic parts. Integratingthe electron drift kinetic equation to the momentum order, wecan derive the linear Ohm’s law for adiabatic dAadik and the non-linear Ohm’s law24 for non-adiabatic dAnak as follows:

@dAadik@t

¼ cB0

B0 � rd/ind (1)

and

1c

@dAnak@t

¼ dB?B0� rd/ind �

men0e2

r � dukecPe0B0 �rdBk

B30

!

þ Pe0en0

dB?B20� rdBk; (2)

where B0 is the equilibrium magnetic field and dB? is the per-

turbed perpendicular magnetic field. Here, d/ind ¼ Teednen0

�� dw

adi

n0@n0@w0Þ � d/ is the inductive potential, dwadi is the adiabatic

component of the perturbed poloidal flux, defined as rdwadi

�ra ¼ rdAadik � B0=B0; a ¼ qðw0Þh� f is the field-line labelwith the Boozer poloidal angle h and toroidal angle f, and thesafety factor q(w0) is a function of the equilibrium poloidal fluxw0. Also, Te is the electron equilibrium temperature, n0 is theplasma equilibrium density, and Pe0 ¼ n0Te is the electron equi-librium pressure. The first term on the right-hand-side of Eq. (2)represents the so-called nonlinear ponderomotive force25 in thefluid electron momentum equation. The nonlinear drive fromfinite dBk is obtained in the second and third terms, which aresmall compared with the nonlinear ponderomotive drive due tothe smallness of b. A complete form of the generalized Ohm’slaw is presented in Ref. 24. In future work, if we consider colli-sionless micro-tearing mode dynamics or cases with large flowat the plasma edge,26 the terms associated with electron inertiain the generalized Ohm’s law need to be kept. The flux-surface-averaged component of the Poisson’s equation and Eq. (2) aresolved for the zonal flow and the zonal current, respectively.

In the simulations, Cyclone Base Case parameters are usedfor the background plasmas: the major radius is R0 ¼ 83.5 cm,the inverse aspect ratio is a/R0 ¼ 0.357. At r¼0.5a, the plasmaparameters are B0 ¼ 2.01T, Te ¼ 2223eV, R0/LT ¼ 6.9, R0/Ln ¼

2.2, and q¼ 1.4. The first order s–a model21 is used for the equilib-rium magnetic field. With these parameters and be ¼ 2%, theKBM is linearly unstable.23,27 In the linear simulations for a singlen¼ 10 toroidal mode, the mode exhibits ballooning mode char-acteristics, with real frequency xlinr ¼ 0:77cs=a and growth rateclin ¼ 0.63cs/a. In the nonlinear simulations, we simulate n¼ 10toroidal mode (keeping all the poloidal harmonics m), and itsnonlinear interaction with the zonal mode (m¼0, n¼0). TheGTC global field-aligned mesh has 32, 400, and 200 grids in theparallel, poloidal, and radial direction, respectively. Convergencestudies show that the physical results in the linear and nonlinearsimulations are not sensitive to the grid size, time step size, ornumber of particles per cell.

Intermediate regime and saturation by zonal fields.—A timehistory for the nonlinear KBM simulation is shown in Fig. 1. Theperturbed electrostatic potential, parallel vector potential, andparallel magnetic field are normalized as ed/=Te; cdAk=vAB0R0,and dBk=B0, respectively, where vA ¼ B0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pn0mip

. The per-turbed electrostatic potential d/10,14, the parallel vector poten-tial dAk10;14, and the parallel magnetic field dBk10;14 of thedominant (10, 14) mode are measured at the mode rationalsurface with q¼ 1.4 at the center of the simulation domain. Thezonal flow hd/i and the zonal current hdAki amplitude are aver-aged over the simulation domain. Before t � 11a/cs, d/10,14 isseen to grow more than two orders of magnitudes at the lineargrowth rate clin after a brief transient stage. dAk10;14 remainsmuch lower than d/10,14, as shown by the diamond solid red linein Fig. 1, since the linear adiabatic component dAadik10;14 is zero atthe rational surface, as constrained by Eq. (1). A linear phaseshift between dAk10;14 and d/10,14 (measured at q¼ 1.36) is about

FIG. 1. Time history of the normalized perturbed electrostatic potential d/, parallelvector potential dAk, parallel magnetic field dBk for the mode (m¼ 14, n¼ 10)measured at the (14, 10) rational surface, and radial averaged zonal flow andzonal current amplitude. Comparison of a linear growth at c ¼ clin (grey dashedline) and the d/ evolution (black solid line) in the intermediate regime is shown inthe zoom-in plot.

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0.8p. dBk is much smaller than d/ due to small plasma b. Att � 11a=cs; dAk10;14 starts to grow faster than exponential, indi-cating that the mode evolves into a nonlinear regime whereponderomotive effects become important. From t � 11a/cs to t� 15a/cs, d/10,14 and dBk10;14 grow slightly faster than exponential,with an effective growth rate cint ¼ 1.1clin. A comparison of d/10,14evolution with a pure linear growth is shown in the zoom-in plotin Fig. 1. During this regime, the field quantities retain their linearpoloidal mode structure. These features are qualitatively similarto those in the intermediate regime found in compressible MHDsimulations.9 The growth of dominant field quantities at a ratefaster than the linear growth rate indicates that the perfect can-cellation between nonlinear destabilization due to enhancedpressure gradients and stabilization due to field-line bendingthat occurs in the ideal MHD dynamics10 does not occur in thiskinetic intermediate regime. We characterize the intermediateregime of the KBM by the rapid growth of the tearing compo-nent of dAk at the rational surface (starting around t¼ 11a/cs inthis case), and the close-to-exponential growth of d/ and dBk.Mode saturation (at around t¼ 15a/cs in this case) indicates theend of the intermediate regime. In the linear regime and theintermediate regime, hd/i and hdAki both grow exponentially ata growth rate czonal �2clin. This suggests that the zonal fields inKBM are passively generated by three-wave coupling, in con-trast to the zonal flow excitation by modulational instabilities inelectrostatic ITG, where hd/i grows as a double exponentialfunction.28

At t � 15a/cs, the dominant mode and the zonal fields satu-rate nonlinearly. As shown by the diamond dotted blue line inFig. 1, the steady state zonal flow amplitude is around 5 timeslarger than the dominant d/10,14 component. The ion energytransport reaches steady state at the gyro-Bohm level with vi� vGB, as shown by the black solid line in Fig. 2, wherevGB ¼ q2i vi=a; vi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiTi=mi

p, and qi ¼ vimic/eB0. The ion heat

conductivity vi ¼ 1n0rTiÐdvð12miv2 � 32TiÞvrdf is defined as the

volume averaged ion energy flux normalized by the local tem-perature gradient, where vr is the radial drift velocity includingthe E�B drift and the magnetic flutter drift.24 In the simulation

where the zonal flow and the zonal current are both artificiallysuppressed, the nonlinear ion heat conductivity becomes oneorder of magnitude larger, as shown by the diamond red line inFig. 2. d/10,14 also saturates at a magnitude around 3 times higherthan that in the case with the zonal fields. In two other simula-tions where only the zonal current or the zonal flow is artificiallysuppressed, d/ and vi saturation levels also see a significantincrease, indicating that both zonal flow and zonal current regu-late ion energy transport and KBM saturation. A comparison ofthe d/ nonlinear poloidal structure between simulations withand without the zonal fields is shown in Fig. 3. In the simulationwith self-consistently generated zonal flow and zonal current,the zonal fields break up the radially elongated eigenmodestructure into microscale and mesoscale structures as in Fig.3(a), reducing radial transport. The radial variation scale lengthof the zonal fields is on the order of the distance between therational surfaces. In the simulation with the zonal fields artifi-cially suppressed, although the non-zonal nonlinear E�B termalso shears the mode structure, some macroscale radial fila-ments of streamers survive. These results show that the KBMsaturation is governed by the zonal fields, including both thezonal flow and the zonal current. In two additional simulationswhere be ¼ 1.74% and be ¼ 1.55% (near the KBM instabilitythreshold), we observe similar nonlinear saturation features. Insimulations with be ¼ 2%, but without dBk and equilibrium cur-rent,we also observe similar nonlinear KBM dynamics.

Onset of nonlinear rapid growth of the localized currentsheet.—As shown by the diamond solid red line in Fig. 1, dAk10;14 atthe mode rational surface first grows faster than exponentialand then grows more than one order of magnitude exponen-tially with a nonlinear growth rate cnl �3clin during the interme-diate regime. This growth rate can be explained by the couplingbetween the zonal current and non-zonal inductive potentialthrough the first term in Eq. (2). The poloidal dAk structureevolves from the linear eigenmode structure at t¼ 11cs/a, asshown in Fig. 4(a), tomesoscale structures at t¼ 17cs/a, as shownin Fig. 4(b). The mode structure becomes thin in the radial direc-tion. This corresponds to the rapid growth of current sheetslocalized at the rational surfaces, excited by the nonlinear pon-deromotive force terms in Eq. (2). In the simulation where thenonlinear ponderomotive force terms are not included(dAnak ¼ 0), although zonal flows still break the linear mode intomesoscale structures nearly isotropic in radial and poloidaldirections, as shown in Fig. 4(c), the radial correlation length ofthe turbulence eddies is much longer than that in the case withthe self-consistent ponderomotive force.

The development of the localized current sheet in theintermediate and nonlinear regime in KBM is analogous to thenonlinear process in the ideal MHD theory. However, in this sce-nario where the kinetic effects become important during theintermediate regime, the mode saturates at the spatial scalecomparable to the ion gyroradius with a transport level con-trolled by the zonal fields. In contrast, the mode structure in theMHD theory tends to become singular until the pressure profileflattens by transport. The radial profiles of (n,m) harmonic of dAkat t¼ 11cs/a and t¼ 17cs/a are shown in Figs. 4(d) and 4(e). Thelinear mode structure has exact odd parity at the rational

FIG. 2. Time history is shown for the ion heat conductivity vi and the perturbedelectrostatic potential d/ for the mode (m¼ 14, n¼ 10) in a simulation with self-consistently generated zonal flow and zonal current and in a simulation with artifi-cially suppressed zonal fields.

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surfaces, and the nonlinear mode structure contains even paritycomponent at the rational surfaces driven by the nonlinear elec-tromagnetic ponderomotive force. For comparison, Fig. 4(f)shows the (n,m) harmonic of dAk after saturation in the

simulationwith dAnak ¼ 0. In this case, each (n,m) harmonic is stillzero at the q¼m/n surface. Because of the formation of a thincurrent layer near rational surfaces, we conducted simulationswith finite resistivity in the generalized Ohm’s law to test the

FIG. 3. Poloidal contour of the perturbed electrostatic potential d/ at the nonlinear regime. Panel (a) shows broken radial filaments in the simulation with self-consistently gen-erated zonal flow and zonal current. Panel (b) shows macroscale radial filaments in the simulation with the zonal fields artificially suppressed. To clearly illustrate the differencein radial filaments, the hd/i component is not plotted in (a).

FIG. 4. Poloidal contour of the parallel vector potential dAk linear structure before the intermediate regime in panel (a), and dAk nonlinear structure after the intermediateregime in panel (b) in the simulation with self-consistent ponderomotive force. Panel (c) shows poloidal contour of nonlinear dAk in the simulation without the ponderomotiveforce terms but with zonal flows. Panels (d), (e), and (f) show the radial profile of (n, m) harmonic of dAk in (a), (b), and (c).

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role of resistive tearing physics29 in the saturation of KBM.Withresistivity 100 times the Spitzer resistivity, no significant tearinginstability is observed within the time scale of KBM nonlinearsaturation. In this case, the KBM linear growth rate and real fre-quency are increased significantly by the resistive drive, and thezonal fields still saturate the mode with a radially smoother non-linear mode structure.

Conclusions and future work.—In summary, we have pre-sented global gyrokinetic simulation results of KBM nonlinearbehavior. The instability develops into an intermediate regime,followed by nonlinear saturation regulated by spontaneouslygenerated zonal fields. In the intermediate regime, rapid growthof the localized current sheet is observed. These qualitative fea-tures appear to be robust consequences of our work and havepotentially important consequences for space and fusion plas-mas. In the Earth’s magnetotail, where there has been significantcontroversy regarding the relative importance of ballooningmodes and magnetic reconnection in causing substorm onset,our studies suggest that nonlinear KBMs, which are self-regulated by zonal flows, can produce thin current sheets thatcan be unstable to secondary tearing instabilities, thus enablingboth mechanisms to play important roles at various stages oftime-evolution in causing substorm onset. This perspective issimilar to that presented in a recent MHD study,30 except thatthe mechanism driving ballooning modes in our simulationsis inherently kinetic. The simulations do not seem to exhibitplasmoid instabilities31,32 which might be suppressed or sta-bilized due to diamagnetic effects. On a longer time scale, thecurrent sheet near the rational surfaces might induce colli-sionless tearing instabilities, which can provide seed islandsfor the neoclassical tearing mode or plasmoid instabilities.33

For future work, we plan to explore the consequences of cou-pling nonlinear KBM instabilities with magnetic reconnectionin Earth’s dipole magnetic field for the magnetotail and intokamak edge configuration.

This research was supported by U.S. DOE Grant Nos. DE-AC02-09CH11466 and DE-FG02-07ER54916 and (DOE) SciDACISEP Center and used resources of the Oak Ridge LeadershipComputing Facility at the Oak Ridge National Laboratory(DOE Contract No. DE-AC05-00OR22725) and the NationalEnergy Research Scientific Computing Center (DOE ContractNo. DE-AC02-05CH11231).

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