Influence of long-scale length radial electric field components on zonal flow-like structures in the TJ-II stellarator

Various works have reported the experimental detection of long-range correlations consistent with the theory of 'zonal flows', i.e. stable modes that are driven by turbulence and regulate turbulent transport [1] (and references therein). The amplification of zonal flows by externally imposed radial electric fields has been observed both in tokamaks and stellarator devices [2, 3]. The coupling between long range correlations of the floating potential fluctuations and radial turbulent transport has been studied in TJ-II edge plasmas in earlier works [4, 5] and the driving and damping mechanisms of the ${{E}_{\text{r}}}\times B$ velocity associated with the zonal flows were discussed [6]. At present, the clarification of the relation between the growth of zonal flows and the development of transport barriers is an active area of research, with the goal of identifying the triggering mechanisms of the transition to improved confinement regimes [7].

Radial electric fields play a key role in explaining transport in fusion plasmas. It is known that in quasi-axisymmetric or quasi-helically symmetric magnetic configurations, collisional plasma transport is intrinsically ambipolar. In non-quasi-axysimmetric magnetic field configurations, parallel viscosity and neoclassical (NC) non-ambipolar fluxes determine the NC radial electric field on scale length much larger than the gyroradius and plasma rotation [8]. On the other hand, recent simulation results for tokamak plasmas have shown that multiple radial scale lengths of the electric field and pressure may develop, together with the spontaneous growth of long-lived patterns of ${{E}_{\text{r}}}\times B$ flows [9]. Direct observation of fine scale structures in radial electric fields have also been reported in the JET tokamak consistent with stationary zonal flows [10].

Simulations have shown that the interplay between transport driven by turbulence and 3D drift-optimized configurations could be explained on the basis of the reduction of turbulent transport by zonal flow generation [11]. The amplification of low frequency ZF-structures in plasma scenarios with reduced NC viscosity has been observed in TJ-II [12]. In addition, gyrokinetic simulations have shown that radial electric fields may affect the residual level of zonal flows in stellarators [13, 14].

Experiments performed in the TJ-II stellarator have shown that long range correlations detected in potential fluctuations, consistent with zonal flows, are amplified either by externally imposed radial electric fields [2, 7] or when approaching the L–H confinement edge transition. In this article, we present experimental results regarding the spatial and temporal dynamics of radial electric fields in the edge region. For high values of NC E_r, long-range correlations (a proxy of ZFs) are enhanced. The electric field producing these effects was characterized by an oscillating structure with a short radial scale (of the order of a few times the ion Larmor radius), and with a magnitude that is linked to the long scale length radial electric field. The shearing rate associated with the short scale component of E_r reaches values of the order of the turbulence decorrelation rate, so that it is likely to be responsible for the observed reduction of the radial correlation length of plasma potential fluctuations.

The paper is organized as follows. In section 2 the experimental set up is described. Section 3 lists the approximations used to calculate the radial electric field and long-range correlations, and presents the corresponding experimental results. Section 4 contains experimental results regarding the radial correlation length of plasma fluctuations. The study of the radial electric field and its shear at different spatial scale lengths is presented in section 5. Finally, some conclusions are drawn in the last section.

Experiments were carried out in ion root (negative electric fields), pure neutral beam injection (NBI) heated plasmas (700 kW port-through power at 33 kV) in the TJ-II stellarator (toroidal magnetic field $B\approx 1\,\text{T}$, plasma minor radius $a\approx 0.20\,\text{m}$).

The results reported here were obtained by the use of a detection system consisting of two Langmuir probe arrays, denoted probe 1 and probe 2, located at two different toroidal/poloidal ports, as shown in figure 1. The use of multi-Langmuir probes at different locations allows measuring different plasma parameters simultaneously and performing global as well as local studies of these parameters. Rake probe 1 is installed on a fast reciprocating drive on top of the device. This probe consists of twelve Langmuir probe pins (measuring floating potential) with a radial separation of 3 mm, and three poloidally separated pins at the rake probe front edge, measuring the floating potential $\left({{V}_{\text{f}}}\right)$ and the ion saturation current (${{I}_{\text{sat}}}$). The second probe (probe 2) is also installed on a fast reciprocating drive, located in a bottom port entering the plasma in a region with higher flux compression than probe 1. This probe has eight probe pins separated by 2 mm, together with three poloidally separated pins at the top of the probe head. The sampling rate of the floating potential signals is 2 MHz. The two probe systems were used to study the temporal and spatial evolution of floating potential profiles, the radial correlation length of floating potential fluctuations and long-range correlations in the edge of hydrogen plasmas.

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Figure 2 shows the time evolution of line-averaged plasma density and edge fluctuations, as well as the edge floating potential profiles. The density is scanned in the range (1–7)  ×  10¹⁹ m⁻³. The level of edge fluctuations increases and edge plasma potential profiles become steeper as plasma density increases.

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The mean velocity of fluctuations perpendicular to the magnetic field (${{v}_{\bot}}$) has been calculated in earlier work using a two-point correlation technique, both in the plasma edge of tokamaks [15] and stellarators [16]. Previous analyses have shown that the perpendicular velocity of the fluctuations (${{v}_{\bot}}$) close to the outermost closed flux surface is dominated by the ${{E}_{r}}\times B$ velocity [17].

We have checked this assumption also in the present plasma conditions in the edge of the TJ-II stellarator. For this purpose, the mean velocity of fluctuations perpendicular to B and along the magnetic flux surface has been computed with the two-point correlation technique in the laboratory frame of reference, using floating potential signals from probe pins with a poloidal separation of 3 mm. In the plasma edge region, ${{v}_{\bot}}$ in the electron drift direction achieves values of up to a few km s⁻¹. The radial electric field deduced from the gradient of the floating potential matched the perpendicular velocity of the fluctuations within error, as shown in figure 3. This implies that the contribution of the pressure gradient to the perpendicular velocity is small under these conditions, consistent with the low beta of TJ-II plasmas [18]. Thus, gradients in floating potential have been used as a proxy for radial electric fields in the following.

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The NC ${{E}_{\text{r}}}$ have been measured from a linear fit of the radial profile of the floating potential calculated along a distance of the order ten times the ion Larmor radius. As shown in figure 2, floating potential profiles are strongly dependent on plasma density. The deduced ${{E}_{\text{r}}}$ magnitude reaches values up to about 3 kV m⁻¹, which is in quantitative agreement with NC calculations in the TJ-II stellarator [19].

Zonal flows are global (${{k}_{\text{poloidal}}}=~{{k}_{\text{toroidal}}}=0$) low- frequency fluctuations with a finite radial scale length (${{k}_{\text{r}}}\ne 0$) [20]. Thus, ZFs correspond to long range correlated (LRC) structures with a (poloidal/toroidal) phase shift close to zero and a finite radial scale length. Here, the long range correlation is defined as cross-correlation coefficient (1) between two Langmuir probes with a large toroidal and poloidal separation, considering only zero time lags computed for time intervals of 1 ms.

$$\begin{eqnarray}&&{{c}_{XY}}=\,\frac{\langle \left[x(t)-\bar{x}\right]\left[y(t+\tau )-\bar{y}\right]\rangle}{\sqrt{\langle {{\left[x(t)-\bar{x}\right]}^{2}}\rangle \langle {{\left[y(t)-\bar{y}\right]}^{2}}\rangle}}.\,\end{eqnarray} \tag{ 1 }$$

Here, x and y are the floating potential signals measured at the remote probe pins. Experimental results described in this report show the existence of an empirical correlation between the amplitude of LRCs (taken as a proxy of zonal flows) and the radial gradients of the floating potential (taken as a proxy of radial electric fields) in NBI plasmas (L-mode regime). As shown in figure 4, the amplitude of LRCs strongly increases with increasing values of the long scale length radial electric field, reaching saturation for ${{E}_{\text{r}}}$ values of the order of $2$ kV m⁻¹. In other words, there is an empirical coupling between long scale length (NC) radial electric fields and the amplitude of zonal flow-like structures.

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Long range correlations are dominated by frequencies below 20 kHz. Interestingly, the high toroidal cross-coherence appears also for higher frequencies as the radial electric field increases (in agreement with theoretical predictions [12]), i.e. zonal flows become more stationary (dominated by frequencies below 5 kHz) in plasma regimes with small E_r and more fluctuating (dominated by frequencies below 20 kHz) as |E_r| reaches values above $2~$kV m⁻¹, as shown in figure 5.

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We have also studied the influence of NC ${{E}_{\text{r}}}$ on the radial scale length of the zonal flow-like structures. As shown in figure 5(b), the radial scale length of LRC structures decreases slightly with increasing ${{E}_{\text{r}}}$. Here, the radial scale length of the LRC structures is determined as follows. One Langmuir pin is selected on probe 1 and the cross-correlation coefficient is calculated using all pins of probe 2. This yields a radial LRC profile. The radial scale length is taken to be the distance for the LRC to drop by a factor of 1/e.

The radial correlation length of plasma potential fluctuations (${{L}_{\text{r}}}$) has been calculated using the twelve floating potential signals of probe 1, with a radial separation of 3 mm. This yields a spatial resolution of 3 mm and allows a maximum radial scale of 36 mm for the measurement of ${{L}_{\text{r}}}$. The radial correlation length of fluctuations is defined as the distance at which the cross-correlation coefficient (1) decays below ${{e}^{-1}}$. The radial cross-correlation was calculated for different frequency bands. It was found to be clearly dominated by low frequencies (⩽$20\,\text{kHz}$). Experimental results show that the magnitude of the radial correlation length decreases by about 30–40% when the absolute (NC) radial electric field increases, as seen in figure 6(a). The clear decrease of the radial correlation length is motivated by the decrease of the cross-correlation function as a function of radial distance, as shown in figure 6(b).

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However, when selecting only data pertaining to the highest values of ${{E}_{\text{r}}}$ (above 2 kV m⁻¹) one observes that the radial scale of LRCs is comparable to the radial scale of potential fluctuations, whereas for lower values of NC ${{E}_{\text{r}}}$ (below 1 kV m⁻¹) the radial scale of potential fluctuations is larger (figure 7). This means that the radial correlation length of potential fluctuations for low frequencies ($1\leqslant f\leqslant 20\,\text{kHz}$) is determined both by edge turbulence and the radial size of zonal flows.

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The shearing rate of radial electric fields has been computed along two different radial scale lengths for frequencies below 20 kHz: (a) a global or long-range shearing rate scale length (of the order of several tens times the ion Larmor radius), which represents the shearing rate of the NC ${{E}_{\text{r}}}$ and (b) a short-range shearing rate scale length (of the order of a few times the ion Larmor radius).

5.1. Calculation of the shearing rate

The (${{E}_{\text{r}}}\times B$)/${{B}^{2}}$ shearing rate has been calculated as the second radial derivative of the floating potential measured by the twelve radially disposed electric pins, mutually separated by 3 mm, divided by the magnitude of the magnetic field ($B\approx 1\,\text{T}$). In principle, for the local calculation of the short scale length shearing rate of ${{E}_{\text{r}}}$, three radial probe pins are sufficient. However, in this case, the objective was to study the influence of the spatial scale length on the shearing rate magnitude. So, the radial distance over which the second derivative is calculated is adjusted by using a discrete Gaussian kernel (see expression 2). In the calculation of the second derivative, the space scale is modulated by the use of a Gaussian kernel (2), where ${{V}_{\text{f}}}(x)$ is the floating potential signal at a given position x, K is the first order modified Bessel function, n is the relative spatial index of the probe pins, t is the scale parameter of the Kernel and $L(x,t)$ is the second spatial derivative of ${{V}_{\text{f}}}(x)$ at scale t.

In this case, the second derivative was calculated approximately in the center of probe 1, so x was taken as the sixth radial pin of probe 1 and its value was obtained by applying the kernel to the range of pins $-5\leqslant n\leqslant 5~$ [21].

$$\begin{eqnarray}&&L(x,t)=\underset{n=-5}{\overset{n=5\,}{\sum}}\,{{V}_{\text{f}}}(x-n)K(n,t)=K(x,t)\ast {{V}_{\text{f}}}.\end{eqnarray} \tag{ 2 }$$

The shearing rate was calculated at $\rho \approx 0.9$. During these experiments, rake probe 1 was fixed radially so that position of the sixth pin of the rake probe corresponds to $\rho \approx 0.9$. By modifying the value of the scale parameter of the Gaussian kernel, t, the radial distance along which the second derivative of the floating potential is calculated is adjusted, as shown in figure 8(a).

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5.2. Result: the magnitude of the shearing rate depends on the spatial scale length

Figure 8(b) shows how the magnitude of the shearing rate of ${{E}_{\text{r}}}$ depends on the spatial scale along which it is calculated, according to the scale parameter of the kernel. The results show that for absolute values of ${{E}_{\text{r}}}$ (NC) smaller than about $2\,\text{kV}\,{{\text{m}}^{-1}}$ the scale parameter does not have an influence in the value of the shearing rate of the radial electric field. On the other hand, when |E_r| reaches higher absolute values (2.5 kV m⁻¹), the magnitude of the second derivative of the floating potential depends strongly on the spatial scale length along which it is calculated. In other words, the shearing rate of the short scale length radial electric field increases when the NC E_r increases. The second derivative of the floating potential was computed over time intervals of the order of the lifetime of the short scale radial structures of the electric field, about 1 ms.

Figure 9 shows the evolution of the radial correlation length as well as local and global ${{E}_{\text{r}}}\times B$ shearing rates versus the magnitude of the NC (long scale length) E_r. In addition, the turbulence de-correlation rate is directly estimated as $\frac{1}{{{\tau}_{\text{c}}}}$, where ${{\tau}_{\text{c}}}$ denotes the auto-correlation time of the potential fluctuations, of the order of ${{10}^{5}}\,{{\text{s}}^{-1}}~$ [22, 23]. Interestingly, the global shear rate is always well below ${{10}^{5}}\,{{\text{s}}^{-1}}~$, whereas the local shearing values exceed ${{10}^{5}}\,{{\text{s}}^{-1}}~$ concomitant with the reduction in the radial correlation length of fluctuations. These findings point to the importance of different radial scale lengths in the radial electric field, so that both NC (long radial scales) and turbulent (short radial scales) mechanism control the radial correlation length of fluctuations.

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In this work, we have studied the influence of long scale length radial electric fields on zonal flow-like structures in the TJ-II stellarator.

It was found that the amplitude of LRCs increases strongly with increasing radial electric fields but saturates for absolute values of ${{E}_{\text{r}}}$ around 2 kV m⁻¹. The frequencies associated with the ZF-like structures increase with increasing NC ${{E}_{\text{r}}}$. For high values of ${{E}_{\text{r}}}$, LRCs are dominated by low frequencies.

A strong relation between the magnitude of electric field structures with long and short radial scales was found. The calculated $\boldsymbol{E}_{{\mathbf{r}}}\times \boldsymbol{B}$ shearing rate corresponding to the short scale length structures of the radial electric field may be sufficient to regulate turbulence.

The radial correlation length of fluctuations is determined by both edge turbulence and zonal flows. This observation is particularly relevant in order to unravel the dependence of the characteristic scale length of turbulent structures with plasma gyro-radius and the isotope physics [24]. In particular, once ${{L}_{\text{r}}}$ is fully controlled by zonal flows, an increase in ${{L}_{\text{r}}}$ does not imply a deleterious effect on transport. Therefore, both NC and turbulent radial scale length should be taken into account when determining the impact of ${{E}_{\text{r}}}\times B$ sheared flows on the radial correlation length of fluctuations and transport.

These experimental findings are consistent with previous experiments that show that the amplitude of LRCs are amplified by external radial electric fields [4] and in the proximity of the L–H transition [25] as well as with GK simulations showing the influence of radial electric fields in the residual level of zonal flows [12, 13]. The observed interplay between NC ${{E}_{\text{r}}}\times B$ shear flows and the development of low frequency zonal flow-like structures could be explained by considering that electric fields may act as a turbulence symmetry-breaking mechanism [26–28] or/and that radial electric fields may affect particle orbits [19, 29, 30]. This is an issue that is still under study at the moment.

The NC radial electric field [31, 32] arises in 3D magnetic configurations as a consequence of the ambipolarity condition. At high density, this electric field (with a negative mean value) satisfies the ion-root solution for ambipolar particle fluxes, in accordance with measurements in various stellarator/heliotron devices [33]. The amplification of ZF structures in plasma scenarios with reduced NC viscosity [12], their coupling with the NC radial electric field reported in this paper, and the scaling with the ion Larmor radius [34] are of special relevance for stellarators with reduced NC transport, since the shearing produced by such structures can determine the saturation level of turbulent transport.

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research under grant agreement No. 633053. This work has been partially funded by the Spanish Ministry of Economy and Competitiveness under contract numbers ENE2012-38620-C02-01. The views and opinions expressed herein do not necessarily reflect those of the European Commission.