-
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.
Published under license by AIP Publishing.
https://doi.org/10.1063/1.5066583
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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