Anomalous transport of multi-species edge plasma with the generalized Hasegawa–Wakatani model and the FLR effects Open Access

The physics of edge plasma in magnetic fusion devices is multifaceted and complex (e.g., see Ref. 1 and the references therein). Even though it has already been studied for a long time, there are still many open issues, including anomalous edge plasma transport, which plays a key role in edge plasma physics. The additional complexity of the study of anomalous edge plasma transport is related to the fact that the edge plasma is inherently multi-species, and apart from hydrogenic species and helium ash, it also contains impurities (e.g., neon, argon), injected into reactor edge plasma to radiate the heat flux, and the particles of eroded materials of plasma facing components.

Unfortunately, available experimental data on anomalous transport of different plasma species (see Refs. 1–6 and the references therein) are very sparse, do not have a general character, and, therefore, cannot be used for predictive modeling. Theoretical assessment of anomalous transport of plasma species (e.g., see Refs. 7–12) also has rather limited applicability and lacks predictive capability. Therefore, the simulations of edge plasma phenomena with two-dimensional (2D) edge transport codes like SOLPS-ITER, having very advanced modules for neutral transport, atomic physics, and classical transport of multi-species plasma along the magnetic field lines,^13–15 are still implementing rudimentary ad hoc models for cross-field transport, which often assume constant transport coefficients the same for all plasma species. Thus, it is clear that the issue of anomalous cross-field transport of multi-species edge plasma should be thoroughly addressed.

In a recent work, anomalous transport of multi-species plasma was considered with the generalized Hasegawa–Wakatani (HW) model (see Ref. 16 for details). It was found that within the HW framework, the turbulent motion of all ion species is described as a transport of a “passive scalar” in the fields of the electrostatic potential and electron density of the HW turbulence. The results of the numerical simulations have shown that within this model, the cross-field flux of all plasma species is not described by a linear function of the density gradient and thus cannot be treated as a conventional diffusion process. In addition, a strong accumulation (depletion) of impurity density was observed within very long-living self-sustained vortices.

In what follows we first discuss some features of the results obtained in Ref. 16 with the generalized multi-species HW model, then we incorporate the FLR effects into this model and consider their impact in some limiting cases.

In Sec. II, we present the main equations and outline the numerical method used in our studies. In Sec. III, we discuss some peculiarities of the results obtained in Ref. 16 with the generalized multi-species HW model in the “cold” ion approximation. In Sec. IV, we extend our generalized multi-species HW model¹⁶ to account for the FLR effects (in a long wavelength approximation) and present our results on the impact of these effects on ion transport related to the resistive drift waves (RDW) turbulence in some limiting cases. In Sec. V, we summarize our findings.

We consider a slab geometry designating “x” and “y” as the radial and poloidal coordinates assuming that the constant magnetic field is directed in z-direction, $B → = B e → z$⁠, where $e → z$ is the unit vector and B is the magnetic field strength.

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 both electron and ion diamagnetic currents for $B → = B e → z$ are divergence-free and 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 $∝ W → i$ and parallel electron flow contribute to the expression $∇ ⋅ j → = 0$⁠.

Finally, we will assume that ion, $N i ( r → , t )$⁠, and electron, $N e ( r → , t )$⁠, densities have a small departure 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 $| n ̂ e | / N ¯ e ≪ 1$⁠.

We notice that due to a small deviation of electron and ion densities from the equilibrium values in the expressions for both $Z eff$ and $ν e i$⁠, we will keep only unperturbed densities.

From Eq. (4), we find that for the case where $T i = 0$⁠, the polarization velocities of ion species $W → i$ only depend on $φ$ and are the same for all ions, $W → i ≡ U → ( φ )$⁠. Therefore, in the vorticity equation (10), we can introduce the effective averaged mass, $M eff = ∑ i M i N ¯ i / N ¯ e$⁠, and express the first term in Eq. (10) through $M eff$ and $∇ ⋅ U → ( φ )$⁠. As a result, no parameters characterizing the properties of individual ions enter separately into Eqs. (8) and (10). In addition, we can express $∇ ⋅ U →$ from Eq. (10) and substitute it into the ion continuity equation (9). 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.¹⁶ 

As it was noticed in Ref. 16, in the “cold” ion approximation, the ion transport [see Eq. (13)] is described as a transport of a passive scalar in the fields of the modified HW model¹⁹ for the electrostatic potential and electron density perturbation described by Eqs. (11) and (12).

It is interesting to note that according to Eq. (21) for a large $A i$⁠, the fluctuations of the density $n i$ are associated mostly with the vorticity $∇ ⊥ 2 ϕ$⁠, which, however, does not contribute to the flux $⟨ Γ i ⟩ x , y$⁠, where only a small part of $n i$⁠, associated with the electron density fluctuations $n e$⁠, matters.

Thus, depending on the problem setup, the evolution of density and flux of the “fictitious” ion species can be different. For the case where we start the solution of Eqs. (11)–(13) from equilibrium conditions, which implies virtually zero values of initial $ϕ$⁠, $n e$⁠, and $n i$⁠, we have $A E n i ( t = 0 ) = 0$⁠. Therefore, in the course of the development of RDW turbulence, we have $A E n i ( t > 0 ) = 0$ and the relations (21) and (22) hold all the time. However, if we “add” our ion species instantaneously into developed RDW turbulence, it will take some time for the $A E n i$ integral to be cascaded into small scales and be dumped there. Before it happens, neither relation (21) nor equality (22) is fulfilled.

Recalling that $A i = Z i − 1 ( M i / M eff ) ( κ e / κ i )$⁠, from Eqs. (21) and (22), we arrive to the main findings of Ref. 16, which state that for the “cold” ion approximation the generalized HW model demonstrates (i) the heavier impurity (which implies larger $A i$⁠) exhibits a stronger accumulation/depletion in long-living vortices, $∇ ⊥ 2 ϕ$ [see Eq. (21)] and (ii) the normalized anomalous flux of all ions equals to that of electrons [see Eq. (22)].

To simulate the RDW turbulent dynamics of multi-species plasma, we solve Eqs. (11)–(13) and others, using spectral code Dedalus.²³ Dedalus and other spectral codes were extensively used for the studies of the HW-like models of plasma turbulence (e.g., see Refs. 16, 20, and 24–26) owing to their rapid convergence properties.

In Fig. 1, one can see the time variation of the standard deviation of vorticity obtained numerically from the solution of Eqs. (11) and (12) for $α = 0.01$ and $α = 0.1$⁠. As we see, following some linear perturbation growth and transitional turbulence stages, the vorticity amplitude saturates starting at the dimensionless time $t ≈ 10 3$ for both $α$⁠. Therefore, in further statistical studies related to the FLR effects, we will use the time interval $t = 10 3 ÷ 10 4$⁠.

Figure 2 demonstrates the time evolution of the fluxes $⟨ Γ e ⟩ x , y$ and $⟨ Γ i ⟩ x , y$ for $A i = 91.3$ [which for $Z i = 1$ and $κ e = κ i$ correspond to the tungsten to deuterium mass ratio in Eq. (13)] for the cases where all parameters $ϕ$⁠, $n e$⁠, and $n i$ are set close to zero at t = 0 (a) and where the “fictitious” ion species instantaneously introduced into developed RDW turbulence (b). As one can see, in agreement with our expectations, for the case (a), the fluxes $⟨ Γ e ⟩ x , y$ and $⟨ Γ i ⟩ x , y$ are virtually indistinguishable, whereas for the case (b), they are very different just after the “injection” of ion species into developed RDW turbulence and then relax toward each other with time.

Our numerical simulations also confirm the persistence of long-living vortices and accumulation/depletion of heavy impurities in these vortices in agreement with Eq. (21). One can see it in Fig. 3, where the results of the simulations for the time about tenfold longer, than in Ref. 16, are shown.

Now we consider the impact of the FLR effects. For $T i ≠ 0$⁠, the polarization velocities $W → i$ start depending on the density perturbations $n ̂ i$⁠, which enter the vorticity equation (10) as the additive terms. As a result, both plasma turbulence, described by Eqs. (8) and (10), and the dynamics of individual ion species, described by Eq. (9), become interconnected.

However, the resulting system of equations turns out to be rather complex and the results of their numerical simulation are difficult to interpret. Therefore, in what follows, we will assume that all ion species, but one, are “cold.” In addition, we assume that the “cold” ions are the same single charged ions, whereas the density of the “warm” ion species is small and we can neglect its contribution to the vorticity equation. We will call that ion species “trace impurity” and use the subscript “tr” to distinguish its parameters from other ion species. Then, the vorticity equation (10) and the electron and ion continuity equations (8) and (9) become identical to that used in Ref. 16.

In our further simulations, we mostly will take $α = 0.01$⁠, will assume that the main ions are deuterium, and will consider two different trace impurities: (i) light (single charged neon, corresponding to $M t r / M i = 10$⁠) and (ii) heavy (single charged tungsten, corresponding to $M t r / M i = 91.3$⁠). Since the magnitude of R depends not only on the masses and charge of involved particles but also on their temperatures, we will assume that R is an independent parameter and in our simulations we take R = 3, 10, and 30.

By observing Eq. (25), we conclude that the FLR effects should cause both the delocalization and the amplitude reduction of the trace impurity density perturbations. Indeed, from Fig. 4, one can see smaller amplitude and more diffuse impurity density distributions for R ≠ 0 in comparison with that in Fig. 3.

The reduction of the fluctuation amplitudes for both light and heavy impurities with the increase in R can also be seen from their probability distribution functions (PDFs) shown in Fig. 5. The standard deviation for light (heavy) impurity PDFs are 24, 5.0, 4.5, and 2.8 (120, 77, 41, 16) for, correspondingly, R = 0, 3, 10, and 30.

However, the reduction of the trace impurity density fluctuations does not necessarily result in the reduction of impurity flux and its fluctuations. Indeed, from Eq. (21), we learned that even though for large A the main contribution to the density comes from the vorticity, it does not contribute to impurity flux at all. Something similar we found for the impact of the FLR effects. Whereas the fluctuations of impurity density go down with increasing R, the fluctuations of impurity flux can experience a non-monotonic dependence (see Fig. 6). Indeed, in Fig. 6(b), one can find a strong increase in the heavy trace impurity flux fluctuations for R = 3, whereas Fig. 5(b) shows the monotonic reduction of heavy trace impurity density fluctuations for increasing R.

In addition to that, the mean flux of heavy trace impurity also exhibits a non-monotonic dependence on R. One can see this in Fig. 7, where the ratio of electron to trace impurity fluxes are shown for different $α$ and both light and heavy impurity. Also, Fig. 7 shows that the FLR effects have a stronger impact on the impurity flux for larger $α$ due to the smaller spatial scale of turbulent perturbations, see Eq. (27).

In this work, the generalized HW model of RDW turbulence in multispecies plasma¹⁶ is extended to incorporate the FLR effects. It shows that the dynamics of electrons and different ion species become coupled, and all ion species affect plasma turbulence. This is unlike “cold” ion approximation, where ion dynamics can be treated as the transport of passive scalars on a background of electrostatic potential and electron density turbulence.

We have shown that with no FLR effects, one can introduce the “associated” enstrophy [Eq. (19)], which is monotonically dissipated due to cascade to small spatial scale [see Eq. (20)]. As a result, it drives the density perturbations of all ion species to the state defined by Eq. (21). Notice that the expression (21) automatically explains the equality of normalized fluxes of electrons and all ion species as well as the strong accumulation/depletion of heavy ion species in long-living vortices, discussed in Ref. 16.

The simplified version of the extended generalized HW equations was used to study the impact of the FLR effects for a small fraction of “warm” impurity. Such “trace” impurity model shows that the manifestation of the FLR effects can be different for “light” (neon) and “heavy” (tungsten) impurities. Whereas both density fluctuations and anomalous flux of “light” impurity are decreasing with increasing impurity Larmor radius, for “heavy” impurity an impact of the FLR effects on the anomalous flux is non-monotonic. However, in both cases, the accumulation/depletion of impurity density in long-living vortices is reduced with the FLR effects included.

The obtained results demonstrate that turbulent transport including FLR effects is substantially different for various ion species and electrons. In addition, the anomalous impurity flux is not a linear function of the impurity density gradient and thus cannot be described as a conventional diffusion process neither without nor with the FLR effects included. This questions the validity of global plasma transport simulations with fluid codes, such as SOLPS and UEDGE, which use diffusion–convection cross-field transport model and often assume the same anomalous cross-field transport coefficients for main plasma and various impurity ion species.

S.K. acknowledges useful discussions with Dr. P. J. Catto and Professor A. I. Smolyakov. 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 (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). R. D. Smirnov: Data curation (lead); Formal analysis (supporting); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal).

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