Anomalous transport of multi-species plasma related to the resistive ballooning and resistive drift wave turbulence is considered in a “cold” ion approximation. It is found that similar to the resistive drift wave turbulence [see A. R. Knyazev and S. I. Krasheninnikov, Phys. Plasmas 31, 012502 (2024); and S. I. Krasheninnikov and R. D. Smirnov, Phys. Plasmas (to be published)] the addition of the ballooning drive does not change the main features of anomalous transport of the multi-species plasma: (i) The transport of all ion species is described as a transport of the passive scalars in the turbulent field of the electrostatic potential and electron density perturbation; (ii) the density of ion species with a larger ratio of the mass to charge has the tendency to the accumulation/depletion in the vortices of plasma flow; and (iii) the cross-field transport of all plasma species (including electrons and ions) is described by the same anomalous transport coefficient.

The multi-species plasma is an inherent feature of plasma in magnetic fusion devices. This is in particular true for the edge region where, in addition to hydrogen species and helium (ash of D-T fusion reaction), plasma contains impurities due to both erosion of plasma facing components and impurities (such as neon and argon) deliberately injected into plasma volume to radiate the heat flux coming from the fusion core.1 

However, the basic theoretical understanding of anomalous transport of multispecies edge plasma is rather limited (e.g., see Refs. 2–10 and the references therein). As a result, for the simulations of edge plasma with two-dimensional (2D) transport codes, having sophisticated models for atomic physics and classical transport of multispecies plasma, very simple models are used for anomalous cross-field transport of plasma species (e.g., see Refs. 11–13).

It is widely accepted that the main ingredients of anomalous edge plasma transport are associated with the turbulence driven by the resistive ballooning (RB) and the resistive drift wave (RDW) turbulence (e.g., see Refs. 5, 14–17). Therefore, in this brief communication, we incorporate the physics related to the RB modes into our studies9,10 of anomalous transport of multi-species plasma transport driven by the RDW turbulence.

In this work, we consider a slab geometry designating “r” and “y” as the radial and poloidal coordinates, assuming that B = B ( r ) e ϑ, where e ϑ is the toroidal unit vector and B(r) is the magnetic field strength. We will assume that d n ( B ( r ) ) / d r = 1 / R = const ., where R is a tokamak major radius.

We adopt a “cold” ion approximation, which gives the following expressions for the cross-field ion velocities:
V i = V E × B + 1 Ω i W i ,
(1)
where
V E × B = c B 2 ( B × φ ) , W i = c B ( t φ + ( V E × B ) φ ) .
(2)

Here Z i and M i are the charge number and mass of the ion species “I,” respectively, φ φ ( r , t ) is the electrostatic potential, and Ω i = e Z i B / M i c, where e and c are the elementary charge and speed of light, respectively.

For the electrons, we consider cross-field E × B and diamagnetic flows and the flow along the magnetic field lines, which is governed by the electric field, electron pressure, and electron–ion friction force. As a result, we have the following expression for the electron velocity:
V e = V E × B c T e e B 2 { B × n ( N e ( r , t ) ) } + e z 1 m ν e i { e | | φ T e | | n ( N e ( r , t ) ) } ,
(3)
where m, T e = const ., and N e ( r , t ) are the electron mass, temperature, and density, respectively, ν e i 1.67 e 4 N e Z eff Λ / T e 3 / 2 m is the electron–ion collision frequency, Λ is the Coulomb logarithm,
Z eff = i N i ( r , t ) Z i 2 / N e ( r , t )
(4)
is the effective ion charge, and N i ( r , t ) is the density of the ions “i.”

We will assume that plasma is quasi-neutral, i Z i N i = N e, which implies that net electric current, j , is divergence-free, j = 0. Then, taking into account that the electron and ion currents associated with E × B drifts cancel each other due to quasi-neutrality conditions, we find that only ion polarization drift W i, parallel electron flow, and a part of the electron diamagnetic current, driven by the spatial dependence of the magnetic field strength, contribute to the expression j = 0. We also assume that ion, N i ( r , t ), and electron, N e ( r , t ), densities have a small departures from the equilibrium values, N ¯ i ( x ) and N ¯ e ( x ) so that N i = N ¯ i ( x ) + n ̂ i ( x , y , t ) and N e = N ¯ e ( x ) +  n ̂ e ( x , y , t ), where | n ̂ i | / N ¯ i 1 and | n ̂ e | / N ¯ e 1.

As a result, using the Boussinesq approximation (e.g., see Ref. 18), the equation j = 0 (so-called vorticity equation) can be expressed as
c B i M i N ¯ i N ¯ e W i c T e B 2 R y ( n ̂ e N ¯ e ) e T e m ν e i | | 2 ( e φ T e n ̂ e N ¯ e ) = 0 .
(5)
Correspondingly, using Eqs. (1)–(3) from the continuity equations for the electron and ion species in the Boussinesq approximation,18 we find
n ̂ e t + V E × B ( N ¯ e + n ̂ e ) + 2 c T e B N ¯ e R y ( n ̂ e N ¯ e ) + T e N ¯ e m ν e i | | 2 ( e φ T e n ̂ e N ¯ e ) = 0 ,
(6)
n ̂ i t + V E × B ( N ¯ i + n ̂ i ) + N ¯ i M i c e Z i B W i = 0 .
(7)

Notice that we neglect here the small terms associated with the spatial dependence of the magnetic field strength in the expression for V E × B .

From Eq. (1), we find that the expression for W i only depends on φ and is the same for all ions, W i U ( φ ). Therefore, in the vorticity equation (5), we can introduce the effective averaged mass, M eff = i M i N ¯ i / N ¯ e, and express the first term in Eq. (5) through M eff and U ( φ ). As a result, no parameters characterizing the properties of individual ions enter separately into Eq. (5). Therefore, we can express U from Eq. (5) and substitute it into the ion continuity equation (7). Then, we find that the ion dynamics is described by the transport of a passive scalar in the turbulence fields of electrostatic potential and the electron density perturbation similar to that of Ref. 9.

Following Ref. 9, we will use the following normalized units: time κ e Ω eff t t, coordinates r / ρ eff r , where ρ eff 2 = T e / ( M eff Ω eff 2 ) and Ω eff = e B / M eff c, electrostatic potential ϕ = ( e φ / T e ) / κ e, and densities n e = ( n ̂ e / N ¯ e ) / κ e, n i = ( n ̂ i / N ¯ i ) / κ i, where κ e = ρ eff d n ( N ¯ e ( x ) ) / d x = const ., κ i = ρ eff d n ( N ¯ i ( x ) ) / d x = const .. In addition, we assume that | | 2 = k | | 2, where k | | is the component of the wave number of plasma parameter perturbations parallel to the magnetic field lines (e.g., see Ref. 19).

As a result, from Eqs. (5)–(7), we find
t 2 ϕ + { ϕ , 2 ϕ } = G ̂ ( ϕ ̃ , n ̃ e ) + D ̂ ϕ ( 2 ϕ ) ,
(8)
n e t + { ϕ , n e } = G ̂ ( ϕ ̃ , n ̃ e ) ϕ ̃ y + D ̂ n e ( n e ) ,
(9)
n i t + { ϕ , n i } = A i G ̂ ( ϕ ̃ , n ̃ e ) ϕ y + D ̂ n i ( n i ) ,
(10)
where
G ̂ ( ϕ ̃ , n ̃ e ) = α R B n ̃ e y + α RDW ( ϕ ̃ n ̃ e ) , A i = M i Z i M eff κ e κ i .
(11)
{ a , b } x a y b y a x b is the Poisson bracket, and a ̃ = a a y, a y means the averaging over coordinate y, and the parameters α R B and α RDW, governing, respectively, the resistive ballooning and resistive drift wave instabilities (e.g., see Refs. 14–16), are defined as
α R B = 2 ρ eff / ( R κ e ) , α RDW = ( T e / m ) k | | 2 / ( ν e i Ω eff κ e ) .
(12)

To describe the small-scale “dumping” effects, which are beyond our approximation, we add the dissipative terms D ̂ a ( a ) into Eqs. (8)–(10).

As we see from Eqs. (8)–(10), similar to Ref. 9, the anomalous transport of “cold” ions, driven by the combination of the resistive ballooning and resistive drift waves turbulence, is described as a transport of the passive scalars in the fluctuating fields of the electrostatic potential and electron density.

In Ref. 10, it was shown that to shed light on the ion transport, it is convenient to introduce the “associated” enstrophy of individual ion species, A E n i, defined as
A E n i = 1 2 ( ( 1 A i ) 2 ϕ n e + n i ) 2 ,
(13)
where means the space averaging. Then, from Eqs. (8)–(10), we find that
dAE n i d t = D ̂ A E n i ,
(14)
where D ̂ A E n i is the dissipative term determined by D ̂ ϕ ( 2 ϕ ), D ̂ n e ( n e ), and D ̂ n i ( n i ).
As a result of the A E n i dissipation following from Eq. (13), we arrive to the relation identical to that obtained in Ref. 10,
n i = n e + ( A i 1 ) 2 ϕ ,
(15)
and, therefore, from Eq. (15), we find the following relation between anomalous ion and electron radial fluxes:
Γ i n i y ϕ = Γ e n e y ϕ .
(16)
In dimensional units, such averaged anomalous flux of ion species, Γ ̂ i, can be expressed as
Γ i ̂ = Γ i κ e α 1 3 T e M eff 1 2 d N ¯ i x d x D d N ¯ i x d x .
(17)
Since from Eq. (16) we have Γ i = Γ e, from Eq. (17), we find that the averaged anomalous flux of all ion species can be described by the same effective anomalous transport coefficient D.

Thus, we find that similar to the RDW turbulence (see Refs. 9 and 10), the addition of the RB drive does not change the main features of anomalous transport of the multi-species plasma: (i) The transport of all ion species is described as a transport of the passive scalars in the turbulent field of the electrostatic potential and electron density perturbation [see Eq. (10)]; (ii) the density of ion species with larger ratio of M i / Z i, which imply larger value of A i (11), has the tendency to the accumulation/depletion in the vortices 2 ϕ [see Eq. (15)]; and (iii) the transport of all plasma species (including electrons and ions) is described by the same anomalous transport coefficient [see Eqs. (16) and (17)]. The latter one to some extent justifies the usage of the same anomalous transport coefficients of all ion species in the edge plasma simulations with 2D plasma transport codes in Refs. 11–13. However, we should keep in mind that in this work, we use a very simple model of edge plasma turbulence, and further studies incorporating the FLR effects10 show that depending on plasma parameters such a model may be incomplete. Further studies, going beyond the Boussinesq approximation and including the FLR effects, are needed for more accurate analysis of anomalous transport of ion species in edge plasma. An impact of the FLR effects (in a long wavelength approximation) is considered in Ref. 10.

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.

The authors have no conflicts to disclose.

S. I. Krasheninnikov: Conceptualization (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
S.
Krasheninnikov
,
A.
Smolyakov
, and
A.
Kukushkin
,
On the Edge of Magnetic Fusion Devices
, Plasma Science and Technology (
Springer
,
2020
).
2.
M.
Priego
,
O. E.
Garcia
,
V.
Naulin
, and
J.
Juul Rasmussen
, “
Anomalous diffusion, clustering, and pinch of impurities in plasma edge turbulence
,”
Phys. Plasmas
12
,
062312
(
2005
).
3.
S.
Futatani
,
S.
Benkadda
,
Y.
Nakamura
, and
K.
Kondo
, “
Multiscaling dynamics of impurity transport in drift-wave turbulence
,”
Phys. Rev. Lett.
100
,
025005
(
2008
).
4.
Z.
Lin
,
T.
Maurel-Oujia
,
B.
Kadoch
,
P.
Krah
,
N.
Saura
,
S.
Benkadda
, and
K.
Schneider
, “
Synthesizing impurity clustering in the edge plasma of tokamaks using neural networks
,”
Phys. Plasmas
31
,
032505
(
2024
).
5.
S. I.
Krasheninnikov
,
D. A.
D'Ippolito
, and
J. R.
Myra
, “
Recent theoretical progress in understanding coherent structures in edge and SOL turbulence
,”
J. Plasma Phys.
74
,
679
717
(
2008
).
6.
H.
Hasegawa
and
S.
Ishiguro
, “
Impurity transport caused by blob and hole propagations
,”
Nucl. Fusion
57
,
116008
(
2017
).
7.
A.
Coroado
and
P.
Ricci
, “
A self-consistent multi-component model of plasma turbulence and kinetic neutral dynamics for the simulation of the tokamak boundary
,”
Nucl. Fusion
62
,
036015
(
2022
).
8.
S.
Raj
,
N.
Bisai
,
V.
Shankar
, and
A.
Sen
, “
Effects of nitrogen seeding in a tokamak plasma
,”
Phys. Plasmas
27
,
122302
(
2020
);
S.
Raj
,
N.
Bisai
,
V.
Shankar
,
A.
Sen
,
J.
Ghosh
,
R. L.
Tanna
,
M. B.
Chowdhuri
,
K. A.
Jadeja
,
K.
Assudani
,
T. M.
Macwan
,
S.
Aich
, and
K.
Singh
, “
Studies on impurity seeding and transport in edge and SOL of tokamak plasma
,”
Nucl. Fusion
62
,
036001
(
2022
);
S.
Raj
,
N.
Bisai
,
V.
Shankar
, and
A.
Sen
, “
Argon, neon, and nitrogen impurity transport in the edge and SOL regions of a tokamak
,”
Phys. Plasmas
30
,
062302
(
2023
).
9.
A. R.
Knyazev
and
S. I.
Krasheninnikov
, “
Resistive drift wave turbulence and anomalous transport of multi-species plasma
,”
Phys. Plasmas
31
,
012502
(
2024
).
10.
S. I.
Krasheninnikov
and
R. D.
Smirnov
, “
Anomalous transport of multi-species edge plasma with the generalized Hasegawa-Wakatani model and the FLR effects
,” Phys. Plasmas (to be published).
11.
S.
Wiesen
,
D.
Reiter
,
V.
Kotov
,
M.
Baelmans
,
W.
Dekeyser
,
A. S.
Kukushkin
,
S. W.
Lisgo
,
R. A.
Pitts
,
V.
Rozhansky
,
G.
Saibene
,
I.
Veselova
, and
S.
Voskoboynikov
, “
The new SOLPS-ITER package
,”
J. Nucl. Mater.
463
,
480
(
2015
).
12.
S. O.
Makarov
,
D. P.
Coster
,
E. G.
Kaveeva
,
V. A.
Rozhansky
,
I. Y.
Senichenkov
,
I. Y.
Veselova
,
S. P.
Voskoboynikov
,
A. A.
Stepanenko
,
X.
Bonnin
, and
R. A.
Pitts
, “
Implementation of SOLPS-ITER code with new Grad–Zhdanov module for D–T mixture
,”
Nucl. Fusion
63
,
026014
(
2023
).
13.
F.
Wang
,
Y.
Liang
,
Y.
Xu
,
X.
Zha
,
F.
Zhong
,
S.
Mao
,
Y.
Duan
, and
L.
Hu
, “
SOLPS-ITER drift modeling of neon impurity seeded plasmas in EAST with favorable and unfavorable toroidal magnetic field direction
,”
Plasma Sci. Technol.
25
,
115102
(
2023
).
14.
B. N.
Rogers
,
J. F.
Drake
, and
A.
Zeiler
, “
Phase space of tokamak edge turbulence, the L-H transition, and the formation of the edge pedestal
,”
Phys. Rev. Lett.
81
,
4386
4399
(
1998
).
15.
B. D.
Scott
, “
Drift wave versus interchange turbulence in tokamak geometry: Linear versus nonlinear mode structure
,”
Phys. Plasmas
12
,
062314
(
2005
).
16.
B. D.
Scott
, “
Tokamak edge turbulence: Background theory and computation
,”
Plasma Phys. Controlled Fusion
49
,
S25
S41
(
2007
).
17.
T.
Eich
and
P.
Manz
, and
the ASDEX Upgrade Team
. “
The separatrix operational space of ASDEX Upgrade due to interchange-drift-Alfven turbulence
,”
Nucl. Fusion
61
,
086017
(
2021
).
18.
G. K.
Batchelor
,
H. K.
Moffart
, and
M. G.
Worster
, “
Perspectives in fluid dynamics: A collective introduction to current research
,”
J. Fluids Eng.
125
,
93
(
2003
).
19.
A.
Hasegawa
and
M.
Wakatani
, “
Plasma edge turbulence
,”
Phys. Rev. Lett.
50
,
682
(
1983
).