Nondiffusive particle transport in the stellarator experiment TJ-K Open Access

In the edge, the scale separation breaks down. This, by the way, is even a problem for first-principle modeling. In the far SOL, where the mean gradients are found to be flattened out, turbulent structures cannot be generated.^7,8 Instead, the high fluctuation levels observed in this region arise from structures that emerge around the separatrix and are then transported into the far-SOL. This is a clear example of nonlocal transport. In the far-SOL, the turbulent structures have a similar size as the gradient decay lengths or are even larger in some cases.⁹ Scrape-off layer transport is nonlocal and nondiffusive.^10,11 Nevertheless, due to the compelling need for mean-field modeling, there are still methods to approximate diffusion coefficients,^12,13 partially accounting for different nondiffusive phenomena. The emergence and avalanching of meso-scale structures are also observed in the staircase in the core of the plasma.¹⁴ The breakdown of Fick's law is often attributed to avalanching,^15,16 and nonlocal theory is assumed to be important when l becomes particularly large to connect areas that are far apart. However, not only particularly shallow but also particularly steep gradients lead to the collapse of scale separation. Especially well-known is the breakdown of local transport theory for heat transport with steep temperature gradients. This can be addressed with nonlocal transport theory.^17–19 In magnetized plasmas, nonlocal modeling of transport is mainly used to capture the kinetic effects of the parallel electron heat flux along the magnetic field lines toward the targets.²⁰ The region around the separatrix is characterized by particularly steep gradients also across the magnetic flux surfaces. In this case, the ratio of the hybrid Larmor radius $ρ s = m i T e / ( e B )$ and the gradient decay length $L n = − ( ∇ r ln ⟨ n ⟩ t ) − 1$ is on the order of $ρ s / L n ∼ 0.1$⁠. At the same time, the driving scales l are at $k ⊥ ρ s ∼ 0.1$ (⁠ $k ⊥$ is the binormal wave number, i.e., normal with respect to parallel and radial direction), with $l ∼ 1 / k ⊥$⁠, such that $l / L n ∼ 1$⁠. Under these conditions, the conventional approach via Fick's law cannot be expected to describe the plasma realistically. Notably, modeling as such still provides satisfactory results.^21,22 Efforts have already been made to incorporate nonlocal transport effects into mean-field transport modeling of the edge, e.g., via $k − ϵ$ models.^23,24

In general, when the scales of the mean field become comparable to the scales of the perturbations, the mean field no longer varies slowly in space. Hence, the higher spatial derivatives need to be retained. In this case, one can either use a series expansion or replace the expressions by convolutions of the mean field with some integral kernels.

The paper is organized as follows: Sec. II describes the general experimental conditions as well as the particular setup. Subsequently, radial profiles of the turbulent particle transport are investigated through the convection–diffusion-based approach of Eq. (1) as well as the generic approach via Eq. (2) in Sec. III. Finally, Sec. IV discusses the emerging diffusive and nondiffusive contributions. A summary and conclusion are given in Sec. V.

Experiments have been carried out at the stellarator TJ-K.²⁶ The major plasma radius is $R 0 = 0.6$ m, and the minor plasma radius is $a = 0.1$ m. The plasmas are generated by microwaves at a frequency of 2.45 GHz and a power of 2 kW.²⁷ In this regime, typical electron temperatures are $T e ≈ 10$ eV, whereas ions are cold with temperatures of $T i ≤ 1$ eV.²⁸ The line-averaged density is on the order of $10 17 m − 3$ in all discharges. The average magnetic field strength was $B ≈ 68$ mT on the magnetic axis.

A study of $ρ s / L n$ has been performed by varying the ion mass m_i. Measurements were carried out for hydrogen (¹H), deuterium (²H), helium (⁴He), neon (²⁰Ne), and argon (⁴⁰Ar) at four nominal pressures 2.6, 3.9, 6, and 8 mPa each. No deuterium discharge at 8 mPa is available. The present setup does not allow for studies on nonlocal transport as it may appear in the core of high-temperature fusion plasmas.^14,31–33 This is because of the small system size a and low magnetic-field strength, the latter of which entails large Larmor radii. Such studies would require a small $ρ * = ρ i / a$⁠, the ratio of Larmor to minor radius, as usually examined for core turbulence. Plasma edge dynamics, on the other hand, can be controlled via the drift scale (ρ_s) as well as the plasma beta (β) and the collisionality (ν). These are normalized to geometrical constants in relation to the gradient decay length $L ⊥$ rather than to a. The gradient decay length refers to the steepest gradient. In TJ-K, this is the density gradient. At the separatrix of a high-temperature plasma, this is the electron temperature gradient. As can be inferred from Table I, discharges in TJ-K are dimensionally similar to those in the edge region of magnetically confined fusion plasmas with respect to $ρ * = ρ s / L ⊥ , β *$⁠, and $ν *$ as defined in Ref. 34. In particular, Table I demonstrates similarities to low- $β *$ L-mode discharges in ASDEX Upgrade. In this parameter range, turbulence is drift-wave dominated both in ASDEX Upgrade³⁵ and in TJ-K.^36,37

With respect to $ρ s / L n$⁠, for the hydrogen isotopes, the plasmas in TJ-K are similar to those around the separatrix or in the pedestal of a high-temperature tokamak fusion experiment. Helium plasmas might still be representative of extremely steep profiles. In tokamak plasmas, at low collisionalities and high temperatures, the electron temperature decay length approaches $q s ρ s$⁠.³⁰ For a typical ITER-like safety factor q_s = 3, this yields $ρ s / L ⊥ ≈ 0.33$⁠. This is similar to the values obtained for helium in TJ-K. Steeper gradients and higher values of $ρ s / L ⊥$ do not appear to be relevant to fusion plasmas. From now on, results are presented by the example of helium unless otherwise stated. This is justified, as the observations are representative of all lighter gases.

For the heavier gases, i.e., neon and argon, the discharges are characterized by $ρ s ∼ L n$⁠. Hence, these are outside the range of comparability to typical high-temperature magnetically confined plasmas. Furthermore, their discharges are increasingly dominated by coherent modes rather than broadband turbulence. This may be explained as follows: poloidal wave numbers correspond to poloidal mode numbers m in cylindrical geometry $k = m / r$ at radius r. Previous experiments report m = 4 for helium.³⁸ Due to the inverse energy cascade,³⁹ all lower mode numbers are then activated. Hydrogen and helium allow for complex and rich dynamics due to the mutual versatile mode coupling possibilities. On the other hand, neon and argon allow only m = 1 and m = 2 modes. If there are only a few modes in the system, this leads to phase synchronization, resulting in quasi-coherent modes.⁴⁰ A detailed analysis of the dynamics of quasi-coherent modes is still to be performed for TJ-K; we leave that for future work. Nevertheless, we still consider these heavy gases to be interesting limit cases, with $ρ s / L n ≈ 1$ and beyond.

Information from each discharge is captured as follows: a vertical three-pin Langmuir probe moves radially in the midplane of an up/down symmetric cross section. Ninety-six equidistant positions R are taken from $R − R 0 = − 5$ to 14 cm. The ion-saturation current is measured with the centered probe, while the floating potential $ϕ f$ is measured above and below at vertical distances of d = 5 mm each. Fluctuation data are sampled at 1 MHz over two²⁰ points at each radial position. Note that the subsequent analyses focus exclusively on the low-field side.

Fluctuations of ion-saturation current and floating potential are reasonably interpreted as density and plasma potential fluctuations (⁠ $I ̃ sat ∼ n ̃ e$ and $ϕ ̃ f ∼ ϕ ̃ p$⁠), respectively. For the low-temperature plasmas at TJ-K, an adverse effect on this interpretation could originate from finite nonzero correlations of the floating potential and the ion-saturation current, each with fluctuations in the electron temperature (⁠ $T ̃ e$⁠).⁴¹ Neither direct measurements of fluctuations in the plasma potential (utilizing emissive probes) in comparison to those in the floating potential⁴² nor conditional sampling of probe characteristics with respect to coherent events in the ion-saturation current revealed a strongly distorting coherent coupling with $T ̃ e$⁠. Hence, the poloidal electric-field fluctuations at the center position $E ̃ θ$ can be estimated from the outer pins of the triple probe according to $E ̃ θ ≈ ( ϕ ̃ f , 2 − ϕ ̃ f , 1 ) / ( 2 d )$⁠. The ion-saturation current (⁠ $I i , sat ∼ n e T e$ with n_e the electron density) is acquired at the same time.

To this end, the radial profiles of Γ and $∇ r n e$ are Fourier transformed on the low-field side in the range between the magnetic axis (⁠ $R − R 0 ≈ 4$ cm) and the separatrix (⁠ $R − R 0 ≈ 13$ cm). According to Eq. (3), the kernel spectra are calculated as their negative ratio. The Nyquist limit is determined by the radial step size of $Δ R = 2$ mm.

In the first step, radial profiles of the equilibrium density gradient and transport are examined. Subsequently, the different contributions to turbulent particle transport are investigated through the convection–diffusion equation and the generic approach via kernel functions. Moreover, three scale-separated regimes will be identified, whose dependence on the gas type and pressure will be analyzed.

The radial density profiles are centrally peaked around the magnetic axis in all discharges. The inverse density decay lengths are shown in Fig. 1. They take maximum absolute values where the gradients are steepest. Even small irregularities in the profile lead to large variations in the derivatives. In order to overcome these variations, the density profiles were found to be best represented by fifth-order polynomials. Their derivatives smoothly reflect the gradients, as shown in the foreground of Fig. 1.

Drift waves are predominantly driven in regions of strong density gradients. Figure 1 indicates that these are located roughly halfway between the magnetic axis and the separatrix, around $R − R 0 = 9$ cm. As such, this region is of particular interest for the analysis of turbulence in TJ-K. Here, the density fluctuation levels—as ratio of standard deviation σ_n to mean value n₀—are found to be around 30%. Moreover, $σ n / n 0 ∼ e σ ϕ p / T e$⁠, which is in line with drift-wave turbulence.⁴³ Additionally, former studies verified further drift-wave features of the turbulence, like broadband low-density potential cross-phases,^34,36 finite parallel elongation with $k | | ≪ k ⊥$⁠,⁴⁴ and a small magnetic component due to the coupling to Alfvén waves.⁴⁵ In addition, coherent drift-wave structures were observed to propagate into the electron-diamagnetic drift direction, i.e., contrary to the weaker poloidal E × B rotation at TJ-K.⁴⁶ Therefore, lengths and velocities are presented here in units of the drift scale ρ_s and cold-ion sound speed $c s ≈ T e / m i$⁠, respectively, in accordance with common drift-wave models.

Normalized turbulent transport levels are displayed for the outboard midplane in Fig. 2. They decay to zero near the core and the scrape-off layer (not shown here). Emanating from the maximum gradient region, transport increases toward the separatrix, where it reaches maximum values. The transport values in Fig. 2 compare well with estimates of the particle fluxes from particle balance considerations.⁴⁷ Hence, it is reasonable to assume that the total particle flux is primarily governed by turbulent fluxes.

While transport coefficients can vary radially, this does not explain the large scattering of the data: the diffusion coefficient of drift-interchange turbulence mainly follows the collisionality $D ∝ ν$⁠,^48,49 besides dependencies in ρ_s and c_s. In TJ-K, the electron temperature profiles are hollow and rather flat, so there is a dependence primarily on density $D ∝ n e$⁠. In the quasi-linear case, this would, at worst, account for 60% deviations of the profile mean. Moreover, the collisionality in TJ-K plasmas—unlike in high-temperature magnetized plasmas—is higher at the center. Thus, the quasi-linear diffusion coefficients should be large in the center rather than in the edge.

Calculations of coefficients over the entire profile would result in inaccurate estimates due to the vast spreads. Instead, the point arrangement suggests bifurcations in most discharges of Fig. 3. These emerge most clearly in the light gases, i.e., hydrogen and deuterium. On the other hand, bifurcations are less pronounced in heavier gases, i.e., neon and argon.

Interestingly, the values in these different branches are also spatially separated. This is indicated by the transparency of markers in Fig. 3, which increases from the magnetic axis to the separatrix. The upper branches lead from low to high transport values that are located near the separatrix. In Ref. 10, such high fluxes in the absence of density gradients are associated with strongly intermittent, nondiffusive transport. On the other hand, the lower branches contain comparably small transport values and resemble straight lines, in line with Eq. (6). Hence, these clusters could be attributed to diffusive contributions.

The scatter plot created from observed particle transport in TJ-K shows a similar structure as results from numerical simulations in Ref. 10 and experimental investigations in the scrape-off layer of the TCV tokamak from Ref. 11. In summary, the convection–diffusion model can describe the diffusive contributions, which are present in the low-flux branches near the plasma core of TJ-K. However, the high-flux branches toward the separatrix contain contributions that should be classified as strongly intermittent and nondiffusive.¹⁰ These cannot be modeled through effective diffusion and convection coefficients. A different model is required to incorporate the full turbulent particle transport in this regime of TJ-K.

The convolution-based method is now applied to analyze the experimental turbulent transport in a generic manner. The kernel spectra $| F ( K r ) |$ can be calculated via Eq. (3), as described in Sec. II. They are shown in terms of the dimensionless wave number $k r ρ s$ in Fig. 4(a).

First and foremost, the nonconstant shape of observed kernel spectra confirms the existence of nonlocal, nondiffusive contributions. Although the kernel spectra in Fig. 4(a) are subject to considerable scattering, three scale-separated regimes can be identified within the spectrum of all discharges for all gases:

• At large scales (low k_r), the kernel strength plateaus. This plateau expands toward higher wave numbers at increased pressures. A constant kernel over the entire spectrum relates to conventional diffusion, i.e., a description via Eq. (1) with a single diffusion coefficient. In terms of Eq. (2), this corresponds to $K r ( r , r ′ ) = δ ( r − r ′ )$⁠.

• At intermediate k_r, the amplitude exhibits a descending trend. Note that in Fig. 4, the kernel spectra are plotted on a logarithmic scale. The Fourier transform of a kernel spectrum with exponential decay toward larger wave numbers is given by Eq. (4). In particular, the slope of the logarithmic spectrum or, correspondingly, its respective width in real space can be interpreted as a radial influence length,¹⁴ reflecting the nonlocality of the transport.

• An ascending trend is identified going from intermediate wave numbers toward large k_r, i.e., small scales. To date, the meaning of this trend is not entirely clear. Enhanced contributions at small scales might be related to an enhanced retention time at these scales and could physically be explained by trapping effects, as discussed in more detail in Sec. IV.

Note that the mirror symmetry around k_r = 0 is a characteristic of the Fourier transformation and reflects no physical property. Furthermore, while the interpretation of individual kernel regimes is possible, there is no unique composition for such a spectrum.

In Fig. 4(b), piece-wise linear functions are fitted to all observed spectra with three independent sections. The piece-wise linear function is chosen on purpose: it has minimal degrees of freedom and mostly preserves a good match with the original data. Thus, a quantitative analysis becomes possible, and the aforementioned regimes are emphasized. Note that the linear least squares method was used with an automated procedure. The fit converges for all but the 3.9 mPa and 6 mPa argon discharges and mostly aligns with the visual intuition.

Figure 5 displays the dependency on the ion mass m_i for the constant, descending, and ascending regimes in a compact manner for all gases. In particular, regime transitions from Fig. 4 are marked by a single point. For each gas, the same transitions at different pressures lie at similar wave numbers. Thus, straight lines are chosen to guide the eye to the regime transitions on the log-log scale in Fig. 5. The scattering in the kernel spectra translates into a few larger outliers at the transition from the constant to the descending regime.

The blue line denotes the resolution limit of the current experimental setup at which $k r = k r max = 0.5$ mm⁻¹. Both intermediate regime transitions result from linear regressions on the markers of the same color. Interestingly, their scaling exponents b match the exponent of the resolution limit $b ≈ 0.5 ± 0.1$ within one standard deviation. Hence, there is no clear dependence on the ion mass.

Finally, the normalized inverse of the smallest gradient decay length $ρ s / L n min$⁠, with $L n min = min ( L n )$⁠, of each discharge is indicated (green cross). A straight line is again fitted to guide the eye. The values are close to the plateau border and tend to get closer for heavier gases, whose structures in the present setup are limited by the system size.

In the following, the features previously discovered in Sec. III are discussed. First, possible explanations for the observed features of the scale-separated regimes in the spectra of Fig. 4 are proposed:

• At large scales, the observed kernels plateau. The boundary of this regime is found near the gradient decay length L_n for all discharges. This is clearly shown in Fig. 5. For the analyzed discharges in TJ-K, it is reasonable to consider $L n − 1$ to be a lower boundary of the descending regime because L_n is on the order of the minor plasma radius. Moreover, particularly for heavier gases, the turbulent structures present at low magnetic field strength in 2.45 GHz discharges become comparable to the system size.⁵⁰ The occurrence of diffusive transport on large scales is reminiscent of the telegraph equation.¹³ It describes a wave equation on short time scales below the correlation time, which leads to ballistic transport. On long time scales far above the correlation time, the appearance of stationary conditions is associated with diffusive transport. In turbulence, essentially, large spatial structures are associated with large temporal scales and vice versa. By comparison, in Fig. 4(a), nonlocal transport (likely ballistic transport) is found below a diffusive region at large scales. As such, this corresponds to the picture of the telegraph equation when wave numbers and frequencies behave in a proportional way.

• Over a small range at intermediate scales (⁠ $ρ s / L n ≲ k r ρ s ≲ 1$ for helium), observed spectra show a descending trend. An exponential scaling of the kernel function $F ( K r ) ∝ exp ( − Δ | k r | )$ may point to ballistic transport events, in analogy to simulations of core heat transport.¹⁴ The typical structures that carry ballistic transport are called blobs or holes in the edge. Blobs and holes are driven by the interchange mechanism and tend to appear near the separatrix of TJ-K.^51,52 This strongly intermittent anomalous transport is commonly found at intermediate sizes⁵³ and is in line with superdiffusive transport. Note that while the transport by plasma blobs leads to nonlocal transport,⁸ this can be considered weakly nonlocal compared to the staircases studied in Ref. 14.

• On the smallest measured scales, below the drift scale (⁠ $k r ρ s ≳ 1$ for helium), an ascending trend, from low to high wave numbers k_r, is identified. It could be interpreted as follows: when external perturbations are absent, charged particles orbit around their guiding center perpendicular to the magnetic field. The motion remains inside the ion or electron Larmor radius; such particles can be considered trapped. Using the ion-to-electron temperature ratio $τ i = T i / T e$⁠, the ion Larmor radius $ρ i = 2 τ i ρ s$⁠. For argon, the typical ion temperature for similar discharges is up to $T i ≈ 1 eV$⁠.²⁸ We expect $k ρ i = 1$ to be around $k ρ s ≈ 3$⁠. This is a rough estimate; the region of the ascending trend is approximately in that range. Trapping by Larmor radii can be a possibility for heavy gases. For helium, the typical ion temperature for similar discharges is much lower $T i ≈ 0.1 eV$⁠,²⁸ thus $τ i ≈ 0.01$ and $ρ i ≪ ρ s$⁠. Trapping by ion Larmor radii is likely not responsible for trapping effects in the light gases. Turbulence can also have a similar effect. Smaller $E → × B →$ vortices that emerge in the drift-wave turbulence may act as traps. Particles follow the $E → × B →$ velocity within the vortex. When the vorticity (inverse eddy turn-over time) is much higher than the growth rate of the instability, the particles are bound to the vortex much longer than transport across magnetic flux surfaces can occur. As the vorticity scales with $k ⊥ 2$⁠, smaller eddies exhibit a large inherent self-rotation and can trap particles much better than large eddies. As such, traps can constrain the mean squared displacement to small characteristic sizes. In the literature,^54,55 trapping effects are associated with subdiffusive behavior. It is likely that both mechanisms are important, but their contribution could only be distinguished after additional measurements of vorticity and ion temperature.

Finally, the missing ion-mass scaling in Fig. 5 shall be discussed: under the assumption that drift waves are driven near $k ρ s ≲ 1$⁠. Vortices at $k ρ s ≳ 1$ lead to trapping. Meso-scale structures (blobs) form at $k ρ s < 1$⁠, which leads to ballistic transport. On large scales (⁠ $k < ρ s / L n$⁠), the transport would be diffusive. Consequently, one would expect these lines to scale with the ion mass (⁠ $ρ s ∼ m i$⁠). However, they do not. Using the typical scale ρ_s and velocity $ρ s c s / L n$ (electron-diamagnetic velocity) of the drift wave, the typical diffusion coefficient scales gyro-Bohm-like $ρ s 2 c s / L n = D B ρ s / L n$⁠, whereas the Bohm-like scaling $D B = ρ s c s$ has no remaining ion-mass dependence. In fact, a transition from the natural gyro-Bohm scaling of the drift-wave turbulence to the Bohm scaling can be expected in the limit $ρ s / L n → 1$⁠. In summary, a gyro-Bohm-like behavior could be hidden within the light gases, where $ρ s / L n < 0.3$ but deteriorated by the heavy gases. Indeed, this expected transition compares well with estimates on the ion-mass scaling of the transport: here, a gyro-Bohm-like scaling $⟨ Γ ⟩ t max / ( | ∇ r n | D B ) ∼ ρ s ( ∼ m i )$ is found for the lighter gases (H, D, and He) with steep gradients, which is in agreement with Ref. 50. This ρ_s dependence then becomes weak if the heavier gases (Ne, Ar) are included. However, the variation in the remaining control parameters is too large for a conclusive evaluation at this point.

The plasmas in the TJ-K stellarator are characterized by a very weak separation between fluctuation and background scales $ρ s / L n$⁠, similar to the edge of magnetically confined fusion plasmas. Due to the lack of scale separation, local transport theory, as described by Fick's laws, is assumed to break down. The issue of scale separation in mean-field transport modeling can be approached with nonlocal transport theory.

The convection–diffusion model, which obeys a strictly local paradigm, fails to fully describe the radial turbulent particle fluxes that are present in the discharges. In contrast, the generic kernel-based approach enabled the identification of three regimes at large, intermediate, and small scales that are in line with conventional, superdiffusive and/or subdiffusive contributions, respectively.

While in the present work, the spatial features of transport have been studied, an unambiguous identification of anomalous diffusion regimes (sub or superdiffusive) can only be accomplished through the additional consideration of the temporal decay characteristics.⁵⁵ This is left for future investigations.

We would like to thank Professor Dr. Brüggen from the University of Hamburg for making it possible for N. Müller to carry out his Bachelor's thesis at the University of Stuttgart, on which this work is based.

The authors have no conflicts to disclose.

Nils Müller: Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Software (lead); Validation (lead); Visualization (lead); Writing—original draft (lead); Writing—review and editing (equal). Peter Manz: Conceptualization (lead); Methodology (lead); Project administration (supporting); Supervision (lead); Writing—review and editing (equal). Mirko Ramisch: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Investigation (lead); Methodology (lead); Project administration (equal); Software (lead); Supervision (lead); Writing—review and editing (equal).

The data that support the findings of this study was produced in the stellarator TJ-K and is available from M. Ramisch upon reasonable request.