Gyrokinetic simulations of nonlinearinteractions between magnetic islands andmicroturbulence

Kaisheng FANG (方凯生)1 , Jian BAO (包健)2,3 and Zhihong LIN (林志宏)4

1 Fusion Simulation Center, Peking University, Beijing 100871, People’s Republic of China2 Beijing National Laboratory for Condensed Matter Physics and CAS Key Laboratory of Soft MatterPhysics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China3University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China4Department of Physics and Astronomy, University of California, Irvine, CA 92697, United States ofAmerica

E-mail: zhihongl@uci.edu

Received 4 June 2019, revised 16 August 2019Accepted for publication 19 August 2019Published 13 September 2019

AbstractA conservative scheme of kinetic electrons for gyrokinetic simulations in the presence of magneticislands has been implemented and verified in the gyrokinetic toroidal code, where zonal and non-zonal components of all perturbed quantities are solved together. Using this new conservativescheme, linear simulation of kinetic ballooning mode has been successfully benchmarked with theelectromagnetic hybrid model. Simulations of nonlinear interactions between magnetic islands andthe ion temperature gradient (ITG) mode in a tokamak show that the islands rotate at the electrondiamagnetic drift velocity. The linear ITG structure shifts from the island O-point toward theX-point due to the pressure flattening effect inside the islands, and the nonlinear ITG structurepeaks along the magnetic island separatrix because of the increased pressure gradient there.

Keywords: gyrokinetic simulations, magnetic islands, microturbulence

(Some figures may appear in colour only in the online journal)

1. Introduction

Nonlinear interactions between macroscopic magnetic islandsand microturbulence are important for the stability andtransport in magnetic confinement plasmas. On one hand,magnetic islands generated by neoclassical tearing mode(NTM) [1, 2] or resonant magnetic perturbations [3, 4] canmodify plasma profiles, flows within and around the islands,therefore influence turbulent transport significantly. Recentexperiments [5, 6] on DIII-D tokamak and LHD stellaratorhave shown considerable reduction of cross-field thermaldiffusivity at the O-point of magnetic island. A global gyro-kinetic simulation [7] found that the reduction of micro-turbulence is related to the island flattening effect and thestrong shear flows formed around the islands. On the otherhand, microturbulence regulates plasma current and electronheat conductivity along and across the magnetic islands, thusaffecting the island dynamics. An experimental observation of

turbulence triggered fast recovery of NTM has recently beenreported in the DIII-D tokamak [8]. Previous fluid simulations[9, 10] found that broad-spectral microturbulence can transferenergy to low-n modes and thus induce seed island [9].Finally, turbulence-driven current can play an important rolein the determination of NTM onset [10].

Global kinetic simulation of the nonlinear interactionsbetween magnetic islands and microturbulence can be diffi-cult. Firstly, electron density and flow calculated from parti-cles cannot accurately satisfy electron continuity equation dueto particle noises. This inconsistency leads to an incompletecancellation of electrostatic and inductive electric fields, andgives rise to an unphysical component in parallel electric field[11, 12]. Secondly, it is difficult to analytically calculate theadiabatic response of electron in the presence of magneticislands, where the zonal and non-zonal components of per-turbed electromagnetic fields need to be defined with respectto the helical magnetic flux surface of the islands.

© 2019 Hefei Institutes of Physical Science, Chinese Academy of Sciences and IOP Publishing Printed in China and the UK Plasma Science and Technology

Plasma Sci. Technol. 21 (2019) 115102 (10pp) https://doi.org/10.1088/2058-6272/ab3c83

1009-0630/19/115102+10$33.00 1

In this paper, we extend the conservative scheme of kineticelectrons in gyrokinetic simulation [11, 12] to incorporatemagnetic islands and verify this new method in the gyrokinetictoroidal code (GTC) [13–15]. Our conservative scheme useselectron continuity equation and the Ampere’s law to evolveelectron density and parallel flow, thus enforcing the con-sistency in the calculation of parallel electric field. The zonaland non-zonal components of all perturbed quantities aresolved together, as required in the presence of magnetic island.By default, our model uses gyrokinetic ions and drift-kineticelectrons to incorporate kinetic effects, and the model can alsobe reduced to a fluid electron model if desirable. As a first steptowards simulating the nonlinear interactions between NTMand microturbulence, we verify our model for drift-Alfvénicwaves with kinetic electrons, and then use it to study theinteractions between magnetic islands and microturbulence. Inparticular, simulation results of the kinetic Alfvén wave(KAW) in a slab-geometry agree well with the theoreticaldispersion relation. We also demonstrate superior numericalproperties of our conservative scheme when simulation gridsize is larger than electron skin depth. Secondly, we benchmarkthe conservative model with the electromagnetic hybrid model[14] for linear kinetic ballooning mode (KBM) [16]. In thefluid electrons simulations, we find that the two models getvery good agreement for the KBM growth rates and real fre-quencies. For the kinetic electrons simulations, the two modelsalso get reasonable agreement. Finally, we use the conservativemodel with fluid electrons to study the interactions between amagnetic island and the ion temperature gradient mode (ITG)[17, 18]. In our simulations, we find that the magnetic islandrotates at the electron diamagnetic drift velocity, which isconsistent with the theory of the drift-tearing mode [19, 20].We also observe that the magnetic island can significantlymodify the ITG mode structures. In the linear simulation, theITG structure shifts from the island O-point toward the islandX-point due to the pressure flattening inside the island. In thenonlinear simulation, the ITG structure peaks along themagnetic island separatrix because of the increased pressuregradient there. Our results are in qualitative agreement withprevious theoretical and numerical studies of the magneticisland effects on the ITG mode [21–25].

The remainder of this paper is organized as follows:section 2 describes the new conservative model, the ver-ification of conservative model is given in the appendix. Inthe section 3, we use the model to study the interactionsbetween the magnetic island and ITG turbulence. Section 4gives a brief conclusion.

2. Gyrokinetic simulation model

2.1. Nonlinear gyrokinetic equations

GTC uses the gyrokinetic equation to study low frequencyplasma waves, the nonlinear gyrokinetic equation [26, 27] is:

⎛⎝⎜

⎞⎠⎟( ) ˙ · ˙ ( )

m =

¶¶

+ +¶¶

- =R Rtf v t

tv

vC f

d

d, , , 0. 1

The distribution function ( ) mRf v t, , , for ion or electron isdescribed by five dimensional independent variables: thegyrocenter position R, the parallel velocity vP, and themagnetic momentum μ. The gyrocenter velocity is denotedby R, the parallel acceleration is v , and the collision operatoris C. We do not take any collision effects into account in thiswork and simply set C=0. The equations for R and v are:

˙

( )

d df

m

=+

+´

+W

´ +W

´

RB B b

b b

vB

c

B

v

mB , 2

v

v v

0 0

0 0

0 0

2

0

E

c g

˙ · ( ) ( )d

m dfd

= -+

+ -¶

¶B B

vm B

B ZZ

mc

A

t

1, 30

00

*

where d df dB B A, , ,0 are background and perturbed magn-etic field, perturbed electrostatic potential and parallelmagnetic potential, respectively; º = +b B B,B

B0 0 00

0*

´W

bB v

00 and ( )d d= ´B bA ;0 W =m Z c, , , ZB

mc0

denote particle mass, electric charge, light speed and cyclo-tron frequency, respectively; v v v, ,E c g are E×B drift velo-city, magnetic curvature drift velocity, and magnetic gradientdrift velocity, respectively.

In the δf method [28, 29], both the distribution function fand the propagator

t

d

dare separated into an equilibrium part

and a perturbed part: dº +L Lt

d

d 0 and f=f0+δf. Corre-spondingly, the equation (1) can be expressed as

( )( )d d= = + + =f Lf L L f f 0t

d

d 0 0 . Where:

⎛⎝⎜⎜

⎞⎠⎟⎟

· · ( ) ( )

m

m

=¶¶

+ +W

´ +W

´

- ¶¶

b b b

B

Lt

vv

mB

m BB

v

1, 4

0 0 0

0

0

2

0

00

*

⎛⎝⎜

⎞⎠⎟ ·

·

· ( )

d

d

d

ddf

df

md

= +´

-+

¶¶

- ¶¶

-¶

¶¶¶

B b

B B

B

L vB

c

B

m BZ

v

m BB

v

Z

mc

A

t v

1

1. 5

0

0

0 0

0

00

*

The equilibrium is defined by =L f 00 0 , and can beapproximated by a local Maxwellian for calculating perturbeddistribution function:

⎛⎝⎜

⎞⎠⎟( )

( )p

= -fn

v

E

T2exp . 60

0

th2 3 2

0

Here m= = +v T m E mv B,th 01

22

0 are thermal velocityand kinetic energy, n0, T0 are equilibrium density and temp-erature, respectively. Defining a weight variable as w=δf/f,the dynamic equation for w can be expressed as:

2

Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

⎜ ⎟

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎛⎝

⎞⎠

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

( )

( )

·

·

·

·

( )

k

k

d

df

df

=-

=- -

´

+ +

+ ´ + ´

+

d

d

df

d

m

´

´

+ ¶

¶

W Wb b

tw

fLf

w

B

v

d

d

1

1

, 7

b

B B

B

c

B

E B

v Z

T B c

A

t

Z

T

v

m

B

0 0

0

1

parallel

0

magnetic drift

flutter

0

0

0

0 0

0

2

0

where ⎜ ⎟⎛⎝

⎞⎠

k = + + -m .n

n

B

T

mv

T

T

T2

3

20

0

0

0

2

0

0

0

The E×B term gives the drive for drift-wave instabil-ities, the parallel term determines the landau damping phy-sics, the last term consists of the magnetic flutter motion. Thefluid moments are calculated by integrating the distributionfunction over the velocity space. Defining the integraloperator as ò ò mº pv vd d dB

m

2 0 , we have: òd d= vn fd ,

òd d= vn u v fd0 , òd d= vP mv fd 2 , òd m d=^ vP B fd 0 . Toclose the system, electrostatic potential and parallel magneticpotential are solved by the gyrokinetic Poisson equation [26]and the parallel Ampere’s law.

( ˜ ) ( )df df d d- = +Z n

TZ n Z n 8i

20i

0ii i e e

( ) ( ) dp

d d = - +^ Ac

Z n u Z n u4

. 92e e e i i i

Equations (1)–(9) form a closed system for gyrokineticsimulation. When the wavelength is longer than electrongyroradius, the above gyrokinetic model reduces to driftkinetic electrons model for electrons species.

2.2. Conservative scheme

In the particle-in-cell method, the electrons’ perturbed densitydne and the perturbed flow du e are calculated with largenumerical noises, so dne and du e fail to satisfy the electroncontinuity equation accurately. As a result, the inductive andelectrostatic parallel electric fields do not cancel with eachother well, leading to an unphysical component in the parallel

electric field ·d df= - -

d¶

¶bE

c

A

t 01 . To solve this, we

use the conservative scheme [11, 12], where the particlegathering process is only taken for the electrons pressureterms dP e and dP e. dne and du e are solved from electroncontinuity equation and the Ampere’s law for consistency.

Using the equation (7), δfe is time advanced and used tocalculate the electron pressure terms dP e and dP e. For theelectron density, we integrate the equation (7) to get the

electron continuity equation:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

· ·

( ) ·

·

·( )

( ) ·( )

d d

d d d

d

d d d d

d df

¶¶

=- - +

+ +

-

+

- ´

+

- -

´ + + ´

*

^

B v

v v B

B

b

B

n

t

n u

BB

n n

B

nB

B

n u n u

B

c

B

P

Z

P P B

Z B

c n

B

B n nc

B.

10

0 E

e E

0

0

0

e e e

00

0e e

0

0e0

0

0e 0e 0e e

0 02

e

e

e e 0

e 0

e

02

0 0e e02

Here d *v e is the perturbed electron diamagnetic drift velocity

defined as ( )d d d=W

´ +* ^v bn m

P P1

e 00e e e

e e . The elec-

tron parallel flow dn ue e is calculated by inverting theAmpere’s law equation (9):

( ) dp

d d= - -^Z n uc

A Z n u4

. 11e 0e e2

i 0i i

The electrostatic potential δfis solved by the gyrokineticequation (8):

( ˜ ) ( )df df d d- = +Z n

TZ n Z n . 12i

20i

0ii i e e

Taking the time derivative for the Ampere’s law (9), one can

get the Ohm’s law for d¶¶t

A :

( ) d p d

¶

¶= -

¶

¶^A

t cZ

n u

t

4. 132

e0e e

Note thatd¶

¶

u

t

iis much small than

d¶

¶

u

t

edue to the small

mass ratio m m 1e i and is thus dropped for simplicity. Forthe RHS of the equation (13), we can integrate the driftkinetic equation (7) with vP to get the electron momentumequation:

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎫

⎬

⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪

· ( )

· ( )

[ ( ) ] ·

·( )

· ( )

· ( )

( )

d

d

d d d

d df

df

¶

¶= -

+ -

+ + -

+ + +

+ +

-

+ ´

+ ´

+

-

d

d

d

+^

^

¶

¶

B B

v

b

B

n u

t

P P P

P P P

n n n

n u

n u

B n T u

n u Z

n T u

n T u

3

4

2

3

,

14

B B

B

m B

B

B

m B

B

B

Z

m B

Z n

m c

A

t B

B

c

B Z

c

B ZB

B

0

E

0

0

0e e

e e e

0e 0e 0e

0e 0e e

0e e

0e e

0 0e 0e e

0e e e

0e 0e e

0e 0e e

0

e 0

0

0

e 0

0

0

e

e 0

e 0e

e0

0

02

e

02

e0

0

where ( ) d= +u u ue e 0e . The energy flux terms are muchsmaller than other terms in the equation (14), so we ignorekinetic effects in these terms to reduce particle noises by

3

Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

using the following closures [30]: ò d =vm v fd e3

e

òd m d d=vn T u v B f n T u3 , d0e 0e e 0 e 0e 0e e. Here we define:

( ) pw

d dD = -

¶

¶-

¶

¶cZ n u

t

A

t

4, 15e

pe2

0e e

where w = pZ n

mpe4 e

20e

eis the electron plasma frequency, the

Ohm’s law (13) can be written as:

⎛⎝⎜⎜

⎞⎠⎟⎟ ( )w d w

-¶

¶= D^

c

A

t c. 162 pe

2

2

pe2

2

Combining the equations (7), (10)–(16) the system is closedand can be used for nonlinear electromagnetic simulations.

The kinetic electron effects in the conservative schemecome from the pressure terms: d d ^P P,e e, which can be reducedto the fluid electron model if desirable. For example, one cantake isothermal approximation for the electrons pressure

⎛⎝⎜

⎞⎠⎟ ( )d d d

ydy

ada= = +

¶¶

+¶¶

^P P n T nT T

, 17e e e 0e 0e0e

0

0e

0

where δα, δψare used for the Clebsch representation ofthe magnetic field: y a= ´ B0 0 0 and d+ =B B0

( ) ( )y dy a da + ´ +0 0 .This conservative model is different from the previous

conservative model [11]. The previous model splits the elec-tron response into an adiabatic part plus a non-adiabatic cor-rection and solves the zonal and non-zonal components ofperturbations separately, the adiabatic part is treated with fluidequations and the non-adiabatic is solved dynamically. On theother hand, our model solves the zonal and non-zonal com-ponents together, which is required for simulations withmagnetic islands. Finally, the isothermal approximation for theelectrons pressure in the previous conservative model [11] is

d d df dy da= = + +y a^

¶¶

¶¶

P P en P Pe e 0e

eff A A0e

0e

0e

0, where dfeff

is an effective potential defined with ·d df= - bE 0eff and

dy da,A A are the adiabatic components of dy da, ,respectively.

3. The interactions between an n=1, m=2magnetic island and ITG

Previous studies based on slab-geometry have suggested thatmagnetic islands can modify the ITG mode structures[21–25]. In particular, the linear eigenfunction of the ITGmode is shifted towards the X-point of the magnetic island[21, 25], some nonlinear works found that a magnetic islandinduced ITG mode (MITG) [22–24] can be excited in thenonlinear stage, where the ITG structure is spread along themagnetic island separatrix. In this section, we use our modelto verify these observations in a tokamak geometry. Theconservative model using the kinetic electrons still hasnumerical issues when the plasma beta is small (i.e. ITGcase), and this is being investigated and will be reported in afuture publication. For simplicity, we use fluid electrons inthis part, where the isothermal approximation (17) is used.

The simulation parameters are as follows: inverse aspectratio a/R0=0.42, where a is minor radius and R0 is majorradius, ion gyro-radius ρi= ´ - R2.86 10 3

0. At the mag-netic axis, Te=Ti=2.22 keV, ne=ni=1.13×1013 cm−3.The electron temperature profile is uniform but the ITG iskept for the ITG drive. At the magnetic surface with

ˆ= = =r a q s0.5 , 2, 0.54. Defining the characteristiclengths of the density and temperature gradients as:

=Lnn

n rd d0 , =LT

T

T rd d0

0, at = =r a R L0.5 , 1.9n0 i ,

=R L 6.0T0 i and = =R L R L1.9, 0.0n T0 e 0 e . We keep insimulation only 4 toroidal modes with n=1, n=12, 13, 14,where the n=1 component relates to the magnetic islandharmonic and the n=12, 13, 14 parts are the most unstableITG harmonics.

We introduce a prescribed n= 1, m= 2 magnetic islandin our simulation by adding an external parallel vectorpotential ( ) ( ) d q z= - - ´ -A A R B h r rcos 2IS

0 0 0 0 to theequilibrium. The perturbed magnetic field is:

( )d d= ´B bA . 18IS0

IS

Here, AP0 is a parameter that controls the island’s size,( )-h r r0 is a radial envelope function used to suppress dA IS

at the simulation boundary, and =r a0.50 . ( )-h r r0 has thefollowing form:

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟( )

( )

- = +-

+-

h r rr r

r

r r

r1 tanh 2 1 tanh 2 .

19

00

0

0

0

The width of the magnetic island is estimated by:

ˆ( )=W

qA R

B s

8. 20

0 0

0

For our case, the calculated magnetic island width isr= =W a0.13 22 i. The island topology can be viewed from

Figure 1. Poloidal contour plot of the helical flux ψhe of the (2,1)magnetic island at the ζ=0 poloidal plane.

4

Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

a magnetic helical flux function [25] defined as:

( )y yy d

= - -A

Bg

2, 21he

tIS

0

where ψ, ψt are the magnetic poloidal flux and the toroidalflux functions, and g is a coefficient from the covariantrepresentation of the equilibrium magnetic field: =B0

q z + I g . The contour plot of ψhe on the poloidal plane atζ=0 is shown in figure 1.

The (2,1) magnetic island flattens the plasma profile byfollowing equation, where the ITG drive is removed:

⎛⎝⎜

⎞⎠⎟( ) · ( )

d k= - - ´B

tw w v

B

d

d1 . 22IS

0

We use the equation (22) to evolve the system for R C20 0 s

until it achieves a flattened state, =C T

ms0e

iis the ion sound

speed. In figure 2 (panel (a)), when =t R C20 0 s, we plot theradial profile of the ion density at the strong field side (θ=π,ζ=0). The magnetic island boundary is marked by the greendashed lines, and it is clear that the total ion density profile(red line) is flattened inside the magnetic island. In figure 2(panel (b)), we also plot the radial gradient of the ion densityat the magnetic island O-point (θ=π, ζ=0, r=0.5a),which is gradually decreased to zero in the flattening process.

After we get the flattened profiles at =t R C20 0 s, we setit as the initial state of the system and use the conservativescheme to evolve the system self-consistently. In this stage,turbulence drive is turned on in the weight equation (7) andthe magnetic island island is evolved using the Ohm’s law(16). Note that in figure 2 the perturbations of particle den-sities are large for the flattened state (d ~ -n n 100

1), toensure the quasineutral condition, we enforce d d=n ne i

at =t R C20 0 s.The magnetic islands can rotate with electrons due to the

frozen-in-line effect. The characteristic rotation frequency isthe electron diamagnetic frequency [19, 20]. This phenom-enon is successfully verified both in our linear and nonlinearsimulations. In figure 3, we plot the poloidal contour plot ofδAP at =t R C20 0 s and =t R C60 0 s. It is clear that δAP is

dominated by the magnetic island induced component whichhas a m=2 structure. The magnetic island is found to rotateat the electron diamagnetic direction, from figures 3(a) to (b),the island structure rotates about π/3 in R C40 0 s, with a

measured poloidal drift velocity ofp

=vr

C R120

s 0. The

theoretical rotation frequency can be estimated from the Ohmlaw (16) by keeping the dominant linear terms:

⎛⎝⎜

⎞⎠⎟·

· ( )

d ddf

d

¶

¶=- +

+ ´

b

b

A

tc

P

Z n

cP

Z n BA . 23

0

0

e

e 0e

0e

e 0e 0

Note that = -* ´

v cbP

Z n Be00e

e 0e 0is the electron diamagnetic drift

velocity. If we take the limit of long-wavelength to the par-allel direction, we have:

· ( )

dd

¶

¶» - *v

A

tA . 24e

Taking = +z qz qe ek ki i and w= -¶¶

it

for the perturbedquantities. Where ze and qe are the unit vectors of the toroidaldirection and the poloidal direction. We have

∣ ∣ ( )w » q *vk . 25e

The estimated theoretical rotation frequency is w =C R0.057 s 0, which is close to the measured rotation fre-

quency is w¢ = =qk v C R0.052 s 0.We further find that the ITG mode structures are sig-

nificantly modified by the magnetic island. Due to the initiallylarge perturbations, we cannot observe a linear to nonlineartransition in a single simulation, so we separately carry outlinear and nonlinear simulations. The nonlinear terms in theweight equation (7) and the Ohm’s law (16) are closed in thelinear simulation, while the island impact in the equations ofmotion (2), (3) is kept in both two simulations. In the linearsimulation, at =t R C60 0 s we plot the poloidal structure ofdf df df= -> =n n1 1, which represents the high n compo-nents of the ITG electrostatic potentials. In figure 4, in thelinear case (panel (a)), the ITG poloidal mode structure isdecreased at the island O-point and peaked at the X-point.

Figure 2. (a) Equilibrium ion density profile (black solid line) and total ion density profile (red solid line) at the strong field side (θ=π,ζ=0). Radial position of the magnetic island separatrix is marked with the green dashed lines. (b) Time history of the ion density gradient atthe O-point on the high field side (θ=π, ζ=0, r=0.5a) during the flattening process.

5

Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

Here the ITG structure is still located around the q=2resonant surface (r=0.5a). Our result agrees with a lineartheoretical study, which argues that the magnetic island flat-tening effects can decrease the ITG drive at the island O-point[21]. Consequently, the ITG mode structure is shifted to theX-point.

In the nonlinear case (panel (b)), we also find that theITG fluctuation is decreased at the island O-point, but moreinterestingly, the nonlinear ITG structure is spread along themagnetic island separatrix. Previous fluid simulations [22–24]based on a slab-geometry configuration also found that MITG

can be nonlinearly excited and mainly locates at the islandseparatrix. Note that in figure 2(a), the flattened ion densityprofile has a strong gradient at the island separatrix, whichstrengthens the ITG drive at the separatrix and can be animportant factor leading to the ITG structure spreading alongthe separatrix. Previous slab simulations [22–24] also sug-gested that the magnetic island induces a broadened dis-tribution of the rational surface (i.e. kP=0 at the separatrix),which causes the ITG mode excitation at the island boundary.In this paper, we used our model to verify the magnetic islandeffects on ITG turbulence both linearly and nonlinearly

Figure 4. Poloidal contour plots for the n>1 components of the electrostatic potential δf n>1. (a) The linear simulation, (b) the nonlinearsimulation.

Figure 3. Poloidal contour plot of perturbed magnetic vector potential dA at (a) =t R C20 0 s and (b) =t R C60 0 s.

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Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

mainly through the change of local instability drive (i.e.change of ion pressure profile) and the existence of islandmagnetic surfaces. However, it should be emphasized thatmagnetic islands and microturbulence can interact with eachother in other ways. For example, recent experiments andglobal gyrokinetic simulations both demonstrate that largescale mode and vortex shear flow can be excited aroundmagnetic islands [4, 7, 31], which play an important role inthe regulation of turbulence and radial transport near theisland. On the other hand, turbulence spreading [32] fromoutside of magnetic island into the O-point of the island canmodify electron temperature profile and bootstrap currentinside the island, thus influence NTM dynamics [33]. In thefuture, we will use kinetic electrons in our model to furtherstudy those important physics to comprehensively understandnonlinear interactions between magnetic islands andmicroturbulence.

4. Conclusion

In conclusion, we extend the conservative scheme of kineticelectrons for gyrokinetic simulations [11, 12] in the presenceof magnetic islands in the GTC, where zonal and non-zonalcomponents of all perturbed quantities are solved together.Our conservative model uses electron continuity equation andthe Ampere’s law to evolve electron density and parallel flow,thus enforcing the consistency in the calculation of parallelelectric field. By default, our model uses gyrokinetic ions anddrift-kinetic electrons, and the model can be reduced to a fluidelectron model if desirable. We verify this new conservativemodel for drift-Alvénic waves with kinetic electrons and useit to study the interactions between magnetic islands andmicroturbulence. In particular, simulation results of the KAWin a slab-geometry agree well with the theoretical dispersionrelation. We also demonstrate superior numerical propertiesof our conservative scheme when simulation grid size is lar-ger than electron skin depth. Secondly, we benchmark theconservative model with the electromagnetic hybrid model[14] for linear KBM. In the fluid electrons simulations, wefind that the two models get very good agreement for theKBM growth rates and real frequencies. For the kineticelectrons simulations, the two models also get reasonableagreement. Finally, we use the conservative model with fluidelectrons to study the interactions between a magnetic islandand the ITG. In our simulations, we find that the magneticisland rotates at the electron diamagnetic drift velocity, whichis consistent with the theory of the drift-tearing mode [19, 20].We also observe that the magnetic island can significantlymodify the ITG mode structures. In the linear simulation, theITG structure shifts from the island O-point toward the islandX-point due to the pressure flattening inside the island. In thenonlinear simulation, the ITG structure peaks along themagnetic island separatrix because of the increased pressuregradient there. Our results are in qualitative agreement withprevious theoretical and numerical studies of the magneticisland effects on the ITG mode [21–25].

Acknowledgments

The authors would like to thank useful discussions with P FLiu, X J Tang, J Y Fu, Z Y Li and GTC team. This work wassupported by the China National Magnetic ConfinementFusion Science Program (Grant No. 2018YFE0304100), theUS Department of Energy, Office of Science, Office ofAdvanced Scientific Computing Research and Office ofFusion Energy Sciences, Scientific Discovery throughAdvanced Computing (SciDAC) program under AwardNumber DE-SC0018270 (SciDAC ISEP Center), and theChina Scholarship Council (Grant No. 201306010032). Thiswork used resources of the Oak Ridge Leadership ComputingFacility at the Oak Ridge National Laboratory (DOE ContractNo. DE-AC05-00OR22725) and the National EnergyResearch Scientific Computing Center (DOE Contract No.DE-AC02-05CH11231).

Appendix A. Verification of KAW

We first use the conservative scheme to simulate the KAW ina shear-less cylindrical geometry in uniform plasmas. For thesimulations in this sub-section ions only provide polarizationdensity for simplicity. In the linear case, the dynamicequation (7) for electron can be simplified as:

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟· ·

( )

d df

d¶¶

+ = - +¶

¶¶¶

b bt

v fe

m c

A

t

f

v

1.

26

0 0ee

0e

Equations (8)–(10), (16) become:

( )p

df d =^c

eVn

427

2

A2

2e

( ) dp

d= ^n uc

eA

4280e e

2

· ( ) ( )d

d¶¶

= - bn

tn u 290

e0e e

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟· ( ) w d w

dfd

-¶

¶= -^ b

c

A

t c

P

n e. 300

2 pe2

2

pe2

e

0e

Here, =p

V B

m nA 40

i 0idenotes the Alfvén speed. Applying

· w= - = =¶¶ ^ ^ bk ki , i , i

t 0 to the above equations, wecan get the linear dispersion relation of KAW as:

⎛⎝⎜⎜

⎞⎠⎟⎟[ ( )] ( )

wx x r- + = ^

k VZ k1 1 . 31

2

2A2 e e

2s2

Here,

x = wk ve 2 the

, r = WT

ms ci0e

iand ( ) òx =

p x-¥

+¥

-

-Z xd

xe1 e x2

e

is the plasma dispersion function.The parameters used to simulate the KAW with kinetic

electrons are set as follows: the electron temperature=T 5.0 keV0e , the magnetic field B0=1.5 T, and the ratio

between the perpendicular wave vector and the parallel wavevector is =^k k 0.01. Figure A1 shows that the simulation

7

Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

results from the conservative scheme (red cross) agree wellwith the theoretical predictions (solid lines) both for the realfrequency ω (panel (a)) and the damping rate γ (panel (b))when we change the equilibrium b = pn T

Be8 0e 0e

02 .

The advantage of the conservative scheme [11, 12] is thatit can avoid the requirement of grid size being larger thanelectron skin depth by using the continuity equation toguarantee the consistency between the electron density andflow. To illustrate this, we compare the original method usingkinetic markers to calculate electron density and the con-servative scheme in this paper. For the two models, we use anantenna with the theoretical frequency to excite the KAW andvary the grid size from =D 0.38x

deto 7.6, here w=d ce pe is

electron skin depth size and Δx is perpendicular grid size. Infigure A2, panel (a) shows that when the grid size is smaller

than the electron skin depth ( )=D 0.38x

de, the two models

both recover KAW oscillations. However, when we increasethe grid size =D 3.8x

de(panel (b)) and =D 7.6x

de(panel (c)),

the conservative scheme can recover KAW correctly, but theoriginal method without the continuity equation fails due tothe inconsistency between the electron density and flow.

Appendix B. Verification of KBM

We next verify the conservative model for linear KBM intokamak geometry, the simulation parameters used in this partare set as follows: inverse aspect ratio a/R0=0.42, where ais the minor radius and R0 is major radius, and ion gyro-radiusρi= ´ - R2.86 10 3

0. The electron temperature profile isuniform but the ITG is kept for the KBM drive. At themagnetic axis, Te=Ti=2.22 nkeV, e=ni=9.0×1013 cm−3. On the magnetic flux surface with r=0.5a,q=2, and the magnetic shear ˆ =s 0.54. Defining the char-acteristic lengths of the density and temperature gradients as:

= =L L,nn

n r TT

T rd d d d0 0

0, at the surface r=0.5a, for the ion

= =R L R L1.9, 6.0n T0 i 0 i and for the electron =R Ln0 e

=R L1.9, 0.0T0 e .The electromagnetic hybrid model [14] in GTC is used

for benchmark. In linear case, our model keeps linear terms inthe equation (14) and uses the equation (16) to calculate theinductive electric field, while the hybrid model [14] uses

· ( ) df df= -d¶

¶bc

A

t 0eff , where dfeff is an effective

potential defined with ·d df= - bE 0eff . First we show that

the two models would get very close results when the fluid

Figure A1.Comparisons between theoretical KAW dispersion relation and simulation results: (a) real frequency ω versus βe, (b) damping rateγ versus βe.

Figure A2. Time history of the electrostatic potential from simulations of antenna excitation of the KAW using the model with (blue solidlines) and without (orange dashed lines) the electron continuity equation.

8

Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

electrons are used. In the fluid limit, the isothermal approx-imation (17) is used for the conservative model. We scan thenormalized perpendicular wave number rk i from 0.1 to 0.8to calculate the KBM real frequency and linear growth rate. Infigure B1, we observe that the fluid electron conservativemodel (red dot) gets nearly the same KBM real frequency(panel (a)) and linear growth rate (panel (b)) with the fluidhybrid model (blue dot).

When the kinetic electrons are included (red and bluestars), the KBM real frequencies (figure B1 panel (a)) are veryclose to the fluid results, this is because that the KBM real

frequency is mainly decided by ions dynamics. In figure B1(panel (b)), the conservative model (red star) and the hybridmodel (blue star) get similar results, the linear KBM growthrates are all decreased compared to the fluid case, which isconsistent with the previous study that the trapped electronscan reduce the growth rate of the KBM [34]. The δfand δAP

poloidal mode structures of r =k 0.42i mode are shown infigure B2 for the kinetic conservative model. The agreementsof the two models on the linear KBM physics verify that ourconservative scheme can be reliably used to study the drift-wave instability physics.

Figure B1. Simulation results from the conservative scheme (red) and the hybrid model (blue) for KBM real frequency ω (panel (a)) andlinear growth rate γ (panel (b)) when k⊥ρi is changed.

Figure B2. Poloidal contour plots of the KBM poloidal mode structures in the conservative scheme simulation with k⊥ρi=0.42,(a) perturbed electrostatic potential δfand (b) perturbed electromagnetic potential δAP.

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Plasma Sci. Technol. 21 (2019) 115102 K Fang et al

ORCID iDs

Kaisheng FANG (方凯生) https://orcid.org/0000-0002-5274-762X

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