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 studies^{9,10} 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 \u2192 = B ( r ) e \u2192 \u03d1$, where $ e \u2192 \u03d1$ is the toroidal unit vector and B(r) is the magnetic field strength. We will assume that $ \u2212 d \u2113 n ( B ( r ) ) / d r = 1 / R = const .$, where R is a tokamak major radius.

Here $ Z i$ and $ M i$ are the charge number and mass of the ion species “I,” respectively, $ \phi \u2261 \phi ( r \u2192 , t )$ is the electrostatic potential, and $ \Omega i = e Z i B / M i c$, where e and c are the elementary charge and speed of light, respectively.

We will assume that plasma is quasi-neutral, $ \u2211 i Z i N i= N e$, which implies that net electric current, $ j \u2192$, is divergence-free, $ \u2207 \u22c5 j \u2192 = 0$. Then, taking into account that the electron and ion currents associated with $ E \u2192 \xd7 B \u2192$ drifts cancel each other due to quasi-neutrality conditions, we find that only ion polarization drift $ \u221d W \u2192 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 $ \u2207 \u22c5 j \u2192 = 0$. We also assume that ion, $ N i ( r \u2192 , t )$, and electron, $ N e ( r \u2192 , t )$, densities have a small departures from the equilibrium values, $ N \xaf i ( x )$ and $ N \xaf e ( x )$ so that $ N i = N \xaf i ( x ) + n \u0302 i ( x , y , t )$ and $ N e = N \xaf e ( x ) +\u2009 n \u0302 e ( x , y , t )$, where $ | \u200a n \u0302 i \u200a | / N \xaf i \u226a 1$ and $ | \u200a n \u0302 e \u200a | / N \xaf e \u226a 1$.

^{18}we find

Notice that we neglect here the small terms associated with the spatial dependence of the magnetic field strength in the expression for $ V \u2192 E \u2192 \xd7 B \u2192$.

From Eq. (1), we find that the expression for $ W \u2192 i$ only depends on $\phi $ and is the same for all ions, $ W \u2192 i \u2261 U \u2192 ( \phi )$. Therefore, in the vorticity equation (5), we can introduce the effective averaged mass, $ M eff = \u2211 i M i N \xaf i / N \xaf e$, and express the first term in Eq. (5) through $ M eff$ and $ \u2207 \u22c5 U \u2192 ( \phi )$. As a result, no parameters characterizing the properties of individual ions enter separately into Eq. (5). Therefore, we can express $ \u2207 \u22c5 U \u2192$ 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 $ \kappa e \Omega eff t \u2192 t$, coordinates $ r \u2192 \u22a5 / \rho eff \u2192 r \u2192 \u22a5$, where $ \rho eff 2 = T e / ( M eff \Omega eff 2 )$ and $ \Omega eff = e B / M eff c$, electrostatic potential $ \varphi = ( e \phi / T e ) / \kappa e$, and densities $ n e = ( n \u0302 e / N \xaf e ) / \kappa e$, $ n i = ( n \u0302 i / N \xaf i ) / \kappa i$, where $ \kappa e = \u2212 \rho eff d \u2113 n ( N \xaf e ( x ) ) / d x = const .$, $ \kappa i = \u2212 \rho eff d \u2113 n ( N \xaf i ( x ) ) / d x = const .$. In addition, we assume that $ \u2207 | | 2 = \u2212 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).

To describe the small-scale “dumping” effects, which are beyond our approximation, we add the dissipative terms $ D \u0302 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.

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 $ \u2207 \u22a5 2 \varphi $ [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 effects^{10} 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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**S. I. Krasheninnikov:** Conceptualization (equal).

## DATA AVAILABILITY

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

## REFERENCES

*On the Edge of Magnetic Fusion Devices*

*L-H*transition, and the formation of the edge pedestal