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Phys. Plasmas 27, 122303 (2020); https://doi.org/10.1063/5.0025861 27, 122303

© 2020 Author(s).

Influence of zonal flow and density on

resistive drift wave turbulent transport

Cite as: Phys. Plasmas 27, 122303 (2020); https://doi.org/10.1063/5.0025861

Submitted: 19 August 2020 . Accepted: 23 November 2020 . Published Online: 23 December 2020

Yanzeng Zhang, and Sergei I. Krasheninnikov

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Influence of zonal flow and density on resistive

drift wave turbulent transport

Cite as: Phys. Plasmas 27, 122303 (2020); doi: 10.1063/5.0025861

Submitted: 19 August 2020 .Accepted: 23 November 2020 .

Published Online: 23 December 2020

Yanzeng Zhang

a)

and Sergei I. Krasheninnikov

AFFILIATIONS

Mechanical and Aerospace Engineering Department, University of California San Diego, La Jolla, California 92093, USA

a)

Author to whom correspondence should be addressed: yaz148@eng.ucsd.edu

ABSTRACT

The generations of zonal ﬂow (ZF) and density (ZD) and their feedback on the resistive drift wave turbulent transport are investigated within

the modiﬁed Hasegawa-Wakatani model. With proper normalization, the system is only controlled by an effective adiabatic parameter, a,

where the ZF dominates the collisional drift wave (DW) turbulence in the adiabatic limit a>1. By conducting direct numerical simulations,

we found that the ZF can signiﬁcantly reduce the transport by trapping the DWs in the vicinities of its extrema for a>1, whereas the ZD itself

has little impact on the turbulence but can only assist ZF to further decrease the transport by ﬂattening the local plasma density gradient.

Published under license by AIP Publishing. https://doi.org/10.1063/5.0025861

I. INTRODUCTION

In the edge region of magnetically conﬁned plasmas, the drift

wave (DW) turbulence driven by the density and/or temperature

gradients plays an important role in determining the level of cross ﬁeld

particle and heat transport. Suppressing the edge DW turbulence is

crucial to reduce the losses of particles and energy from the bulk of

plasma and thus improve conditions for nuclear fusion. It has been

shown that the zonal ﬂow (ZF) can be a great candidate in regulating

the anomalous transport

1,2

since it drives no transport (from EB

drift) and thus can be a safe repository for the energy released by

gradient driven turbulence. The ZF here is referred to as the poloidally

and toroidally symmetric (kh¼kz¼0) and zero/low-frequency

vortex modes with ﬁnite radial scale (k

r

ﬁnite). Moreover, we focus on

the two-dimensional slab model for the edge plasmas and thus ignore

the toroidal effect, which can lead to the geodesic acoustic mode

(GAM or ﬁnite frequency ZFs) in the high-q tokamak edge region.

3,4

(Note that the suppression of turbulence by GAM can be different.

5,6

)

The generation of ZF can be seen as a “secondary” instability in

DW turbulence via Reynolds’ stresses, where the “primary” instability

occurs at a linear stability threshold with a zero background ﬂow and

causes growth of DWs. The secondary instability can be the modula-

tional instability,

7,8

an inverse energy cascade,

9

coherent hydrody-

namic instability,

10

or a resonant-type interaction between the ZF and

modulations of the small scale DW turbulence.

11

Once the ZFs are

fully developed, they can largely suppress the DW turbulence through

a widely assumed effect of shearing the eddies.

12–14

Taking into account the fact that the realistic models for the

DW/ZF system are complicated, it is useful to consider a simple non-

trivial model, namely, modiﬁed Hasegawa-Wakatani

15

(MHW) model

to study the feedback of ZF on the DW turbulence. Compared with

the original Hasegawa-Wakatani

16

(oHW) model, the MHW removes

the zonal-averaged components from the Boltzmann response, as the

zonal mode has no dependence on the direction parallel to the mag-

netic ﬁeld. As a result, the ZFs are not dissipated by the adiabatic term

and can be dominant. It is worth noting that even in the oHW model,

the generation of large ZF is also possible and can be important for an

extremely large adiabatic parameter,

17

which, however, makes the sim-

ulations time-consuming because of the small linear instability.

In the MHW model, a zonal density (ZD), in addition to the ZF,

will also be generated due to the non-zero non-linear advection of

density resulting from the non-adiabatic response of electrons.

Although the generation of ZF and its feedback on the DW turbulence

have been well studied, the effect of ZD on turbulent transport has

been overlooked, which can be important by modulating the equilib-

rium plasma density.

18

Moreover, there is no detailed study of the role

of ZF in the transport of radial particle ﬂux within the MHW model

yet. Therefore, in this paper, we will examine the effects of both the ZF

and ZD on the radial particle ﬂux. To identify their impacts separately,

we will not only consider the MHW model but also employ two of its

modiﬁcations by artiﬁcially removing the ZF and ZD, respectively.

By introducing proper normalization, the MHW model is only

controlled by the effective adiabatic parameter, ak2

zv2

th=eijxj,

Phys. Plasmas 27, 122303 (2020); doi: 10.1063/5.0025861 27, 122303-1

Published under license by AIP Publishing

Physics of Plasmas ARTICLE scitation.org/journal/php

where v

th

is the electron thermal velocity,

ei

is the electron-ion colli-

sional rate, and jxjis the characteristic DW frequency. It was shown

that the DW/ZF system has a bifurcation behavior

15

depending on a:

in the adiabatic limit a>1, the DW/ZF system is dominated by ZF,

whereas, in the hydrodynamic limit a<1, turbulence is dominant.

This reduction of the ZF effect is related to the enhancement of turbu-

lent viscosity and the reduction of residual ﬂux when the plasma

response passes from the adiabatic to hydrodynamic limits. The

underlying physics of the shear ﬂow collapse and DW turbulence

enhancement in the hydrodynamic limit was investigated in detail in

Ref. 19 from the point of view of Reynolds work, wave energy and

momentum ﬂuxes, potential vorticity mixing, and predator-prey

model. It is worth pointing out that in Ref. 19 the transition of the

adiabatic parameter from a>1toa<1 is linked to the approaching

to the density limit, which is accompanied by a weakening of the edge

shear lays and degradation of particle conﬁnement. From direct

numerical simulations (DNS), we ﬁnd that ZD itself has little impact

on the turbulence but can only assist ZF to reduce the transport.

Therefore, in this paper, we mainly focus on the adiabatic limit a>1,

where the ZF dominates the turbulence. Keep in mind that a>1

means that the standard adiabatic parameter ^

ak2

zv2

th=ei is greater

than the characteristic frequency of DW.

The rest of the paper is organized as follows. In Sec. II,we

describe the physical models that will be used. Their DNS results will

be presented in Sec. III.SectionIV will examine the underlying physics

of the suppression of particle ﬂux, and Sec. Vwill conclude and discuss

the main results.

II. MODELS

The models we will consider are based on the MHW equations,

which capture the essential characteristics of turbulent transport in

the DW/ZF system. If we consider a slab geometry with a constant

equilibrium magnetic ﬁeld B¼B0ez(where ezis a unit vector in the

z-direction), an exponentially decaying background plasma density

n0/expðx=LnÞin the x(radial)-direction, parallel electron resistiv-

ity, and cold ions, the MHW equations are in the quasi-two-dimensional

(2D) form

@

@tr2/þ/;r2/

¼að~

/~

nÞþr2ðr2/Þ;(1)

@

@tnþ/;nfg¼að~

/~

nÞ@/

@yþDr2n:(2)

Here, all physical quantities have been normalized as x=qs!x;

jxcit!t;e/=Tej!/;n1=n0j!n,where/and n

1

are, respec-

tively, the electrostatic potential and density ﬂuctuations, qs¼Cs=xci

is the ion sound gyroradius, Cs¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Te=M

pis the ion sound speed, T

e

is electron temperature, Mis ion mass, xci ¼eB0=Mc is the ion cyclo-

tron frequency, eis the elementary charge, cis the speed of light, and

j¼qs@lnðn0Þ=@xqs=Lnis a constant as the energy source for

DW instabilities. The effective adiabatic parameter is redeﬁned as

a^

a=xcij(xci jis the characteristic frequency of DW), which

controls the essential physical behavior. The only difference between

the MHW and oHW models is in the resistive coupling term: in the

MHW model, it is determined only by the non-zonal components

~

f¼fhfi,wherehfiÐLy

0fdy=Lydenotes the integration along the

poloidal line at a given radial location; whereas, in the oHW model, all

the components contribute to it. The non-linear advection terms are

expressed in the Poisson brackets fa;bg¼ð@a=@xÞð@b=@yÞ

ð@a=@yÞð@b=@ xÞ. The dissipation terms of the form r2Awith coef-

ﬁcients Dand are added to the equations for numerical stability.

In the oHW model, there are two different limits depending on

a: the adiabatic limit as a!1 for collisionless plasma, where the

density ﬂuctuations become enslaved to the electrostatic potential

ﬂuctuations through the Boltzmann relation and oHW equations are

reduced to Hasegawa-Mima equation;

20

and the hydrodynamic limit

for a!0, in which the equations are decoupled similar to the 2D

Navier-Stokes equations. In the MHW model, these two limits corre-

spond to, respectively, the ZF-dominated regime and the turbulence-

dominated regime,

15

where the ZF absorbs almost all the kinetic

energy from the DWs in the saturated state in the former case, while

the turbulence is more isotropic in the latter. In the following, we

mainly focus on the adiabatic limit, i.e., a>1, where the ZF domi-

nates in the system.

15

From Eqs. (1) and (2), we can see that, in the MHW model, the

ZF (ZD) can be generated via the non-linear advection of vorticity

(density)

@V

@tþ@h~

vx~

vyi

@x¼@2V

@x2;@N

@tþ@h~

vx~

ni

@x¼D@2N

@x2;(3)

where VðxÞ@h/i=@xis the ZF velocity, NðxÞhniis the ZD, and

~

v¼ezr

~

/.Equation(3) shows that the non-adiabatic response of

electrons is important in the generation of ZD. We are particularly

interested in the impacts of the ZF and ZD on the particle ﬂux in the

radial direction

Cnð~

vx~

ndx¼ð~

n@~

/

@ydx;(4)

where ÐfdxÐLx

0ÐLy

0fdxdy=LxLy. We will also monitor the total

energy and the energy stored in the zonal-components

E¼EnþEk¼1

2ððn2þjr/j2Þdx;

Ez¼Ez

nþEz

k¼1

2ððN2þj@h/i=@xj2Þdx:(5)

Moreover, in order to distinguish the effects of the ZF and ZD on C

n

,

we will not only use the oHW and MHW models but also consider the

modiﬁcations of MHW equations by artiﬁcially removing the ZF (we

will call it MHW w/o ZF) and ZD (referring as MHW w/o ZD) com-

ponents, respectively.

III. SIMULATION RESULTS

Equations (1) and (2) will be numerically solved by using a

pseudo-spectral Fourier code in a square box with doubly periodic

boundary condition in Dedalus.

21

The size of the box is Lx¼Ly

¼10pso that the lowest wavenumber is Dk¼0:2, while the number

of the modes is chosen as 256 256. The time integration algorithm is

the fourth-order Runge-Kutta method with a time step Dt¼103.

We mainly focus on the non-linearly saturated particle ﬂux with ﬁxed

dissipation coefﬁcients of ¼D¼104. For the oHW and MHW

models, we start the simulations from one small perturbation, while in

the cases of the MHW model w/o ZF and ZD, we employ a different

initial condition with zero zonal-averaged components.

Physics of Plasmas ARTICLE scitation.org/journal/php

Phys. Plasmas 27, 122303 (2020); doi: 10.1063/5.0025861 27, 122303-2

Published under license by AIP Publishing

In all the cases, we observe that the ﬂuctuations ﬁrst grow linearly

due to the resistive DW instability, where the modiﬁcations will not

affect the growth of the DW in the linear regime (e.g., see Fig. 1 for

a¼2). Note that, in the late linear stage, the simulations of oHW and

MHW are dominated by the mode ðkx;kyÞ¼ð0:2;1Þ, while those of

MHW w/o ZF/ZD are governed by ðkx;kyÞ¼ð0:2;0:6Þ.Thegrowth

rates from the simulations are, respectively, 0.025 and 0.057, which

agree with the analytic result of resistive DW in the adiabatic limit,

c¼k2

yk2=ð1þk2Þ3ain our normalized units with k2¼k2

xþk2

y.

However, when the amplitudes of our normalized DWs become

the order of unity, they will undergo secondary instabilities and thus

generate the ZF and/or ZD (as an example, the growth rate of ZF can

be 0:1 in our normalized unit

22,23

for a¼2, i.e., czf 0:1jxci). The

ZF and/or ZD will, in turn, affect the DW, and the system will be satu-

rated eventually when the particle ﬂux C

n

approximately balances the

dissipation, Da¼a1Ð½^

að~

n~

/Þ2dx, due to the parallel resistivity.

From Fig. 1 we see that, the saturated particles ﬂuxes are quite differ-

ent, where the time-averaged C

n

at the saturated state from t¼400 to

t¼1000 are 0.36 (oHW), 7:4103(MHW), 2:2102(MHW

w/o ZD), and 0.4 (MHW w/o ZF), respectively. This is consistent with

the standard recognition that the ZF can largely suppress the anoma-

lous transport, whereas it illustrates that the ZD plays an important

role in the MHW model (as seen from the plots of MHW and MHW

w/o ZD) but not in the oHW model (e.g., see the comparison between

the oHW and MHW w/o ZF). Therefore, we conclude that the ZD

can reduce the transport only via its synergy with the ZF. This conclu-

sion has also been conﬁrmed by the simulations for other adiabatic

parameters a>1. For example, for a¼4, the time-averaged C

n

in the

non-linear saturated stage are, respectively, 0.21 (oHW), 2:6103

(MHW), 3:8103(MHW w/o ZD), and 0.24 (MHW w/o ZF).

Note that the larger adiabatic parameter means that the electron

response to the electrostatic potential is more adiabatic and, thus, the

overall particle ﬂux is smaller.

19

However, on one hand, in the hydrodynamic limit a1, even

the MHW model is strongly turbulent and thus both the ZF and ZD are

small. On the other hand, the generation of ZD largely depends on the

non-adiabatic response of electrons, ~

hk;xð

~

n~

/Þk;x¼iðxjkyÞ

~

/k;x=ðaixÞ, which is small for larger a1. Therefore, it is

expected that the ZD effect on the transport begins to drop when ais

above some threshold values (e.g., the ZD reduces the time-averaged

particle ﬂux by a factor of 3 for a¼2 but only by a factor of 1.5 for

a¼4). Therefore, the role of ZD in the suppression of transport can be

important only for a large but ﬁnite range of ain the MHW model.

However, searching for such a range of adiabatic parameter needs time-

consuming simulations to scan the parameter space and is beyond the

scope of this paper.

To see the different spatial behavior of the ZD in the oHW and

MHW models, we plot the saturated density in Fig. 2. (The different

spatial structures of the electrostatic potential are similar and can be

found, e.g., in Ref. 15.) It shows that the ZD is more prominent in the

MHW, where the potential energy stored in the ZD, Ez

n,isapproxi-

mately half of the total potential energy, E

n

, whereas in the oHW

model, Ez

n=En0:1andsoisEz

k=Ek. This demonstrates that although

the ZF/ZD can still be generated in the oHW model, their effects are

negligibly small. We notice that in the MHW w/o ZF, the saturated

density is similar to Fig. 2(a), but the ZD stores less energy

Ez

n=En0:03. Moreover, Fig. 2 illustrates that the saturated level of ~

n

(and thus ~

/~

nin the adiabatic limit) is largely reduced by the ZF.

However, this suppression of turbulence is not simply due to the con-

centration of energy to the ZF since it was also observed in the resistive

DW system with a constant externally applied ZF.

24

In Sec. IV,wewill

show that the reduction of transport is because of the trapping of

turbulence near the extrema of ZF.

IV. ZONAL FLOW AND DENSITY IN THE MHW MODEL

In the last section, we showed that the ZD can assist the ZF in

suppressing the saturated C

n

in the MHW model, and we will examine

the underlying physics in detail in this section. We will focus on a line-

arized problem of DW with quasi-static ZF velocity and ZD in simple

sinusoidal proﬁles

VðxÞ¼V0cosðqvxÞ;NðxÞ¼N0cosðqnxþDuÞ;(6)

where q

v

and q

n

are the products of integer numbers and Dkso that

V(x)andN(x) are periodic, and Duis a constant phase shift needed to

be determined from the simulation results. Then the governing equa-

tion for ~

/¼^

/ðxÞexpðixtþikyyÞis

d2^

/

dx21þk2

yky

jeff þV00

~

xþi

~

xkyjeff

ai~

x

^

/¼0;(7)

where jeff ¼1N0,and ~

x¼xkyV.Here,xxrþicis com-

plex with cbeing the growth rate.

We ﬁrst consider a long-lasting problem of linear resistive DW

with ZF only, i.e., N0¼0. For the inhomogeneous proﬁle of ZF, the

ﬂuctuations can be trapped near the extrema

24–26

of V(x), e.g., see

Fig. 3(a). These trapped structures move along the positive (negative)

y-direction near the maximum (minimum) of V(x) and are reﬂected

back and forth from the “slopes” of ZF in the x-direction. (It is worth

noting that the propagation direction of trapped structures depends

on the sign of equilibrium magnetic ﬁeld so that if we ﬂip the sign of

magnetic ﬁeld, the propagation direction of trapped structures will

also change.) In fact, the trapped structures near the extrema of V(x)

can be interpreted as eigenmode solutions of Eq. (7) with zero bound-

ary conditions, which is a product of traveling wave in y-direction and

standing wave in x-direction. Near the maximum and minimum of

V(x), the eigenmode solutions are growing and decaying, respectively.

FIG. 1. Time evolution plots of radial particle ﬂux from oHW (black), MHW (green),

MHW w/o ZF (green), and MHW w/o ZD (red) for a¼2.

Physics of Plasmas ARTICLE scitation.org/journal/php

Phys. Plasmas 27, 122303 (2020); doi: 10.1063/5.0025861 27, 122303-3

Published under license by AIP Publishing

Therefore, if the ZF is externally applied, DWs will be trapped only

near the maximum of V(x),whilethoseneartheminimumwilldisap-

pear. However, if the ZF is self-generated as in our simulations,

trapped DWs also appear near the negative peak of V(x)asshownin

Fig. 3, because fast self-generated ZF needs strong turbulence as an

energy supplier.

27

The eigenmode solution can be interpreted within the eikonal

approximation, which assumes that the DW wavelength is much

smaller than that of ZF. Although this assumption limits the range of

both DW and ZF parameters, this approach can provide some insights

of the characteristics of non-linear interaction of DW and ZF. In this

approach, the wave packet near the maximum of V(x) could be con-

sidered as an effective “particle,” dynamic of which is described by the

“Hamiltonian”

xðkx;xÞ¼kyVðxÞþjeff þV00ðxÞ

1þk2

yþk2

x

"#

;(8)

and the canonical variables xand k

x

(e.g., see Refs. 1,11,28,and29).

Here, we ignored the growth rate cin xand the last term in the

bracket Rið~

xjeff Þ=ðai~

xÞin Eq. (7) for the adiabatic limit.

To have an eigenmode (trapped) solution of the wave packet while

keeping xðkx;xÞas constant, the motion of the particle should be

bounded by two turning points corresponding to k

x

¼0. This is only

possible in the vicinity of the extrema of V(x) in the form of Eq. (6).

Equation (7) is solved numerically both near the maximum and

minimum of Vfor a¼2, where the results of growth rates and

contours of the equipotential are plotted in Fig. 4. The size of the

computation box ~

Lxis chosen that there is only one extrema of V(x)

for given q

v

, while we vary k

y

in the simulations. In fact, the trapped

ﬂuctuations in Fig. 3(a) both near the maximum and minimum of V

are superpositions of different modes (k

y

), where the mode ampli-

tude linearly decreases to zero when ky!1 in the former case, but it

follows a slower decreasing curve till ky3 in the latter. The aver-

aged wavenumber,

kyÐkyEkdk=ÐEkdkwhere Ek¼j

~

/kj2,of

the ﬂuctuations near the maximum and minimum of V(x) are,

respectively,

ky0:4 and

ky1, which match the visualization in

Fig. 3(a). These averaged wavenumbers are used in the numerical

simulation of Eq. (7).

From Fig. 4(a), we see that the growth rates of the modes near

the maximum of V(x) in the top panel only slightly change (either

increase or decrease) concerning V

0

.However,increasingV

0

can cause

the trapping of the DWs. Particularly, for V0¼0, the plasma is homo-

geneous and the DW is not localized at all, whereas, for large V0Ɀ1,

FIG. 2. Contour plots of nin the saturated state of (a) oHW model and (b) MHW model for a¼2.

FIG. 3. A snapshot of contour of ~

/and ZF, for a¼2, in the (a) MHW w/o ZD and (b) MHW at t¼400 and (c) MHW at t¼940 corresponding to the lowest particle ﬂux in

Fig. 1. In the MHW model (b) and (c), we also show j

eff

(blue) and x0ðxÞxðkx¼0;xÞ(red) in Eq. (8) for ky¼0:6(k

y

only affects the shape of x

0

slightly).

Physics of Plasmas ARTICLE scitation.org/journal/php

Phys. Plasmas 27, 122303 (2020); doi: 10.1063/5.0025861 27, 122303-4

Published under license by AIP Publishing

the DWs are localized as seen in Fig. 4(b). This is consistent with the

simulation observations that the trapping of ﬂuctuations begins once

the large ZFs, V0Ɀ1, are generated. Moreover, the linear solutions

illustrate that the real frequency of these eigenmodes is positive

approaching kyV0for V0Ɀ1 as indicated in Eq. (8).ForV(x)havinga

negative peak, the least stable eigenmode is found and plotted in the

bottom panel. We note that the real frequency for these modes also

approaches kyV0but with negative sign, agreeing with the numerical

observations for Fig. 3(a). It is also worth noting that the ZF does not

tilt the eddies of the contour of equipotential as shown in Fig. 4(b),

which is consistent with the case, where the DWs are trapped by the

inhomogeneity of the equilibrium plasma density.

30

Therefore, we

conjecture that the suppression of the particle ﬂux by the ZF is mainly

duetothetrappingoftheDWsratherthanbyshearingtheeddies.

We then consider the impact of ZD on the linear DW and thus

the non-linear turbulent transport by taking N06¼ 0. As a result, the

structure of the Hamiltonian xðkx;xÞin Eq. (8) and thus the trapped

structures will be changed by N0through j

eff

, where the ﬂuctuations

accumulate near the extrema of xðkx¼0;xÞ. These trapped

structures have been observed in the simulations, e.g., see Fig. 3(b).

From Fig. 3(b), we see that the maximum (minimum) of Vcoexists

with the minimum (maximum) of j

eff

, which results in a relatively

smaller growth rate of DW compared with the case of N0¼0and

thus indicates a reduced transport. From Eq. (3),weseethatZD

evolves faster than ZF due to the viscous damping since ZD has larger

effective wavenumber

26

as seen from Fig. 3(b). This has been observed

in the simulations, e.g., see Fig. 5(a). In such a process, ZD will

exchange energy with DWs, which alters the structure of x0ðxÞin

Eq. (8) and thus the trapping of DWs. As a result, trapping DWs can

penetrate radially and accumulate at the locations, where the maxima

of ZF and ZD coexist (and thus smallest j

eff

), e.g., see Fig. 3(c) com-

pared with Fig. 3(b). To see how the ZD affects the energy division in

the turbulent and zonal components, we plot Edr

k¼EkEz

k;Edr

n

¼EnEz

n;Ez

k=Ek,andEz

n=Enin Fig. 5. It shows that the presence of

ZD can slightly increase Ez

k=Ekthat the ZF is more dominant and

thus reduces the ﬂuctuations of DWs (Edr

k;n). The shifts of Ez

k=Ekcan

be also understood by considering a decreased

15

j

eff

near the maxi-

mum of V.

FIG. 4. (a) Growth rates of the most unstable (decaying) eigenmodes near the maximum (minimum) of Vin the top (bottom) panel for a¼2, where we also plot the growth

rates of the zeroth modes (dotted curves) near the maximum of V, and (b) schematic view of contours of the eddies of the equipotential near the maximum of V(center with

ky¼0:4) and near the minimum of V(left with ky¼1:2 and right with k

y

¼1) for V0¼3.

FIG. 5. Evolution of Edr

k;Ez

k=Ek;Edr

n, and Ez

n=Enfor (a) MHW and (b) MHW w/o ZD for a¼2.

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Phys. Plasmas 27, 122303 (2020); doi: 10.1063/5.0025861 27, 122303-5

Published under license by AIP Publishing

V. DISCUSSIONS AND CONCLUSIONS

In this paper, the generations of ZF and ZD and their roles in

suppression of the radial particle ﬂux in the resistive DWs turbulence

are examined based on the MHW model and its modiﬁcations. (Here,

we focus on the adiabatic limit for the resistive DWs, where the ZF

can largely regulate the turbulence.) Numerical simulations of these

models are conducted, from which we observe that when the ampli-

tudes of DWs become the order of unity as a result of growth from the

linear state, the secondary instabilities will lead to the generation of the

ZF and/or ZD, which will, in turn, suppress the DW turbulence and

thus reduce the radial particle ﬂux. We ﬁnd that the ZF itself can sig-

niﬁcantly reduce the particle ﬂux as shown in Fig. 1,whereastheZD

can reduce the transport only through its synergy with ZF.

The underlying physics has been examined, and it is shown that

the generation of large ZF can trap the DWs in the vicinities of its

extrema despite the sheared ﬂow. We conjecture that such trapping of

ﬂuctuations is the reason for the reduction of the radial particle ﬂux

rather than the conventional picture of shearing of eddies by ZF. First

of all, Fig. 4 (aswellasRef.30) shows that the velocity shear itself can-

not largely change the growth rate of the eigenmode solution of resis-

tive DW and does not tilt the eddies. Therefore, the conventional

picture of shearing eddies by ZF does not apply here. Second, the trap-

ping of DWs in this paper is due to the inhomogeneity of ZF, which

growsfromDWsinthenon-linearstage.WhenZFisstrongenough,

the DW ﬂuctuations will be transited into the trapping regime. During

such a transition, the conventional picture of shearing eddies can

work, where the strong ZFs can break big DW eddies into small ones.

However, this is only because our ZF is self-generated when DWs are

large. If the ZF is externally applied, the ﬂuctuations will also be

trapped as shown in Fig. 4 (see also, e.g., Ref. 25). In such a case, eigen-

mode solutions will grow from small ﬂuctuations without breaking of

large eddies, where the eigenmode with the largest growth rate as

shown in Fig. 4 will ﬁnally dominate. Therefore, the suppression of the

turbulence is not due to the shearing of DW eddies but through the

trapping of ﬂuctuations. Last but not least, we notice that the turbulent

level is largely suppressed by the ZF, either externally applied or self-

generated.

24

Therefore, the suppression of turbulent transport is also

not simply due to the concentration of ﬂuctuation energy to ZFs.

On the other hand, the ZD can ﬂatten the local effective plasma

density gradient (instability driver) in the regions, where the DWs are

trapped [notice that the ﬂattening of ZD on the background density is

small N=n02j1asshowninFig. 2(b)], which results in a

smaller linear growth rate and thus reduction of turbulence. However,

in the absence of ZF, the system is strongly turbulent that the impact

of ZD on the DW instability and eventual non-linear transport is neg-

ligible. Our conclusion of the ZD impact on the transport holds for

large but ﬁnite ain the MHW model. When a!1,theZDisnegli-

gible due to the adiabatic response of electrons, while for small aⱿ0:1

in the hydrodynamic limit, both the ZF and ZD become unimportant

since the turbulence is dominant. Therefore, there should exist a range

of athat the impact of the ZD on the particle ﬂux is important in the

MHW model. Based on this characteristic, it is interesting to discuss

the ZD effect in the oHW model. Recalling that ZD can affect the par-

ticle ﬂux only via its synergy with ZF, it can be important only when

ZF is dominant for an extremely large adiabatic parameter, aⱿ1000,

in the oHW model.

17

However, for such large a, the ZD is negligibly

small and thus hardly makes any impact. Therefore, it is conceivable

that ZD has little impact on the turbulent transport for the whole a

space in the oHW model.

Note that both the oHW and MHW models employ the

Boussinesq approximation and assume a constant logarithm equilib-

rium plasma density gradient. However, considering that we are study-

ing the ZD impact on the particle transport downhill to the plasma

density gradient, a relaxation of these assumptions may be important.

As an example, the resistive DW instability with non-Boussinesq

approximation was considered in Ref. 31 when dealing with the edge

plasma with a very steep background density gradient. A more reﬁned

model and a systematic study are needed and will lead to interesting

directions for future works.

ACKNOWLEDGMENTS

The authors thank H. Zhu for bringing to the authors’ attention

the Dedalus and an input script. This work was supported by the U.S.

Department of Energy, Ofﬁce of Science, Ofﬁce of Fusion Energy

Sciences under Award No. DE-FG02-04ER54739 at UCSD.

DATA AVAILABILITY

The data that support the ﬁndings of this study are available

from the corresponding author upon reasonable request.

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