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Influence of zonal flow and density on resistive drift wave turbulent transport

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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, α, where the ZF dominates the collisional drift wave (DW) turbulence in the adiabatic limit α>1 α > 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 α>1 α > 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.
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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/expðx=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, Cs¼ffiffiffiffiffiffiffiffiffiffiffi
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
j¼qs@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
~
f¼fhfi,wherehfiÐLy
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, NðxÞhniis the ZD, and
~
v¼ezr
~
/.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~
ndx¼ð~
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, Da¼a1н^
að~
n~
/Þ2dx, 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;xð
~
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^
/ðxÞexpðixtþikyyÞis
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 x0ðxÞxðkx¼0;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 N00. 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
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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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