Electron thermal internal transport barriers triggered by the effect of ion shielding

Anomalous transport [1–3] and self-organized criticality (SOC) [4–6] are generally acknowledged and studied in both space and laboratory plasmas. In magnetic confined fusion devices, such as tokamaks and stellarators, anomalous transport is mainly driven by the drift-wave like micro-instabilities [3] including, but not limited to, ion-temperature-gradient driven mode, trapped electron mode (TEM), and electron-temperature-gradient (ETG) driven mode, which can tap the free energy contained in the thermal plasmas and cause the degradation of energy confinement with increasing heating power. However, the plasma confinement can be improved with the formation of internal transport barriers (ITBs), which are kinds of important transport phenomena with SOC dynamics [7]. The electron ITBs (eITBs) have been observed in many tokamak experiments under different operation scenarios, especially with strong external electron heating [8–10]. Understanding the underlying physical mechanisms for the formation and dynamics of the eITB in modern tokamaks is of significant importance to improve fusion performance for future fusion devices, such as ITER or DEMO, which are self-sustained by dominant electron heating provided by fusion α-particles [11].

The formation of eITB is commonly accompanied by the suppression/mitigation of ETG and TEM instabilities, which are believed to be two main candidates for the electron energy loss in tokamak plasmas [11]. Relevant experiments have verified that turbulence induced electron thermal transport can be effectively reduced by the negative/weak magnetic shear [12–14], which depends on a hollow current density profile induced by an off-axis externally driven current or self-driven bootstrap current, or by increasing the magnetohydrodynamic (MHD) α-parameter (or Shafranov shift) in high-β_p experiments [15, 16]. The E × B flow shear driven by zonal flows or the momentum injection from neutral beam are less effective for short wavelength turbulence like ETG [17] but nevertheless can suppress TEM turbulence independently [18] or combined with precession shear [19]. Meanwhile, it has long been recognized that there exist (linear or nonlinear) critical gradients for ETG and TEM, above which the turbulence flux can increase dramatically with the normalized temperature (density) gradient $R/{L}_{T\left(n\right)}$. Here, ${L}_{T\left(n\right)}=T\left(n\right)/\nabla T\left(n\right)$ is the scale length of the temperature (density) profile. In this case, the gradient of plasma profiles tend to stay close to the critical value so that the micro-instabilities are marginally unstable [20]. The turbulence threshold depends on many factors, such as safety factor, magnetic shear and electron–ion temperature ratio (τ = T_e/T_i), etc. In this letter, we report the first experimental evidence from EAST tokamak that the electron–ion temperature ratio can elevate turbulence threshold for the electron micro-instabilities and hence trigger eITB formation in hot electron mode with low torque injection and positive magnetic shear. This type of eITB has therefore typical SOC dynamics in magnetized confined plasma with dominant electron heating.

EAST is a superconducting tokamak with a major radius of R = 1.75 m and a minor radius a = 0.45 m, respectively. The toroidal magnetic field at plasma center is B = 2.5 T. In the eITB experiments, the low hybrid wave (LHW) is applied for current drive and heating. The heating power of LHW deposit on electrons dominantly in the core region while ions are heated by thermal exchange with electrons through collisions. The discharge waveform of a typical eITB shot #104102 are shown in figure 1. The average density is about  1.5 × 10¹⁹ m⁻³. The electron density and effective charge number are measured by polarimeter-interferometer (POINT) [21] and visible bremsstrahlung [22], respectively. The plasma current reaches I_p = 0.35 MA. The radial profile of safety factor is monotonic with the minimum safety factor q_min ≈ 1.3 and the edge safety factor q₉₅ ≈ 7.2, which is shown in figure 1(b). The q-profile is obtained by MHD equilibrium reconstruction constrained by POINT data [23] with EFIT code [24]. The equilibrium configuration can be found in figure 2(a). The electron temperature is measured by Thomson scattering (TS) [25]. As shown in figure 2(b), the T_e profile can be constructed by numerically fitting experimental measurements with a global analytical function including six free parameters (A₀, A₁, A₂, r₀, Δr, ν) as

$$\begin{equation}{T}_{\mathrm{e}}\left(r\right)={A}_{1}\,{\mathrm{e}}^{-\frac{{\left(r-{r}_{0}\right)}^{2}}{{\Delta}r}}+{A}_{2}{(1-{r}^{2})}^{\nu }+{A}_{0},\end{equation} \tag{ 1 }$$

where the exponential part describes the ITB of the electron temperature and the polynomial part describes the electron temperature outside of the ITB. The ion temperature is measured by x-ray crystal spectrometers (XCS) [26]. The XCS can provide only the ion temperature data in the core region. Therefore, we apply 1D transport model to extend the T_i profile to the outer region. The one-dimensional thermal diffusion equations of electron and ion temperature can be written as

$$\begin{equation}\frac{3}{2}n\frac{\partial {T}_{\mathrm{e}}}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left[r\left({q}_{\mathrm{e}}+\frac{5}{2}{T}_{\mathrm{e}}{{\Gamma}}_{\mathrm{e}}\right)\right]={Q}_{\mathrm{e}},\end{equation} \tag{ 2 }$$

$$\begin{equation}\frac{3}{2}n\frac{\partial {T}_{\mathrm{i}}}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left[r\left({q}_{\mathrm{i}}+\frac{5}{2}{T}_{\mathrm{i}}{{\Gamma}}_{\mathrm{i}}\right)\right]={Q}_{\mathrm{i}},\end{equation} \tag{ 3 }$$

where ${{\Gamma}}_{\mathrm{e}\left(\mathrm{i}\right)}$, ${q}_{\mathrm{e}\left(\mathrm{i}\right)}$ and ${Q}_{\mathrm{e}\left(\mathrm{i}\right)}$ are particle flux, heat flux and heat source of electron (ion), respectively. The LHW heating power deposition, which is calculated by GENRAY/CQL3D code [27, 28], is shown in figure 2(b). A constant ion thermal diffusivity is assumed and the value can be determined by fitting the measurement of T_i in the core region. The electron thermal diffusivity χ_e can be obtained by solving these equations after electron and ion temperature profiles and heat source are given [29]. The LHW was launched in the I_p flat-top phase in ascending three steps for the analysis of the process of ITB formation. The LHW power is initialized with 1.0 MW at 4.0 s and then rises to the maximum value of 1.68 MW at 5.1 s. As the LHW injected, the electron temperature in the core region increases quickly while the ion temperature decreases slightly as can be seen in figures 3(a) and 4. This is due to the lack of direct ion heating and reduced collision between electrons and ions as T_e increases. The opposite trends of temperature evolution for electrons and ions after LHW injection indicate the decoupling of thermal transport channels between electrons and ions in the core region. Similar results for ion temperature have also been reported in recent experiments on AUG and W7-X with dominant electron heating by electron cyclotron wave [30], where the clamping of T_i profiles are observed owing to the strong turbulence transport. In this EAST discharge, the maximum electron temperature can reach to ${T}_{\mathrm{e}}^{\text{Max}}\sim 9\enspace \mathrm{k}\mathrm{e}\mathrm{V}$ with LHW heating only while ${T}_{\mathrm{i}}^{\text{Max}}< 0.8\enspace \mathrm{k}\mathrm{e}\mathrm{V}$ for ions. Thus a large value of temperature ratio, τ = T_e/T_i ≫ 1, can be achieved in such experiments, which can be referred to as the hot electron mode. Through 1D transport analysis it can be found in figure 3(b) that the effective thermal diffusivity of electron χ_e drops heavily down toward magnetic axis, which indicates the formation of eITB in the core plasma. This eITB is established within  0.2 s after LHW injection accompanied by the rapid rise of on-axis electron temperature from ${T}_{\mathrm{e}}\left(0\right)\sim 1.8\enspace \mathrm{k}\mathrm{e}\mathrm{V}$ to ${T}_{\mathrm{e}}\left(0\right)\sim 4.3\enspace \mathrm{k}\mathrm{e}\mathrm{V}$. The location of eITB is close to the magnetic axis at r/a ∼ 0.155 at t = 4.2 s. The ITB location can be defined by [31] r_ITB = (r_shoulder + r_foot)/2 with r_{shoulder(foot)} the ITB shoulder (foot) point. In this analysis, r_shoulder is set to zero since there is no clear shoulder observed. After that, T_e as well as the gradient ∇T_e grow relatively slowly while the eITB moves radially outward gradually to r_ITB ∼ 0.185. During this process, χ_e decreases significantly around r/a ∼ 0.3 accordingly, as shown in figure 3(b). Note that although the local shape of the χ_e profile depends on the specific numerical scheme for constructing the T_e profiles, the relatively small χ_e in the inner region can always be observed, which confirms the improved electron energy confinement and the formation of eITB.

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To investigate the underlying mechanisms of the formation and movement of the eITB in such experiments, the critical gradient (CG) of T_e for ETG and TEM is calculated for the core plasma. The analysis is focused on the time window of  4.0 s < t < 4.5 s, when the ITB is triggered by LHW heating. An explicit formula for the threshold of the linear ETG instability has been given based on the gyrokinetic numerical simulation as [32]

$$\begin{align}\hfill {\left(R/{L}_{{T}_{\mathrm{e}}}\right)}_{\mathrm{c}}^{\text{ETG}}& =\mathrm{max}\left\{\left(1+\tau {Z}_{\text{eff}}\right)\left(1.33+1.91\hat{s}/q\right)\left(1-1.5{\epsilon}\right)\right.\hfill \\ \hfill & \quad \left.\times \left[1+0.3{\epsilon}\left(\mathrm{d}\kappa /\mathrm{d}{\epsilon}\right)\right],0.8R/{L}_{n}\right\}.\hfill \end{align} \tag{ 4 }$$

Here, Z_eff is the effective ion charge number. q and $\hat{s}$ are safety factor and magnetic shear. = r/R and $\kappa \left(r\right)$ is the plasma elongation. It can be seen that for τZ_eff ≫ 1 the CG is almost proportional to τ for given Z_eff. On the other hand, the CG of TEM has been derived in reference [33]

$$\begin{align}\hfill {\left(R/{L}_{{T}_{\mathrm{e}}}\right)}_{\mathrm{c}}^{\text{TEM}}& =\left(1+\alpha \tau \right)\frac{0.357\sqrt{{\epsilon}}+0.271}{\sqrt{{\epsilon}}}\left[4.90-1.31\frac{R}{{L}_{n}}\right.\hfill \\ \hfill & \quad \left.+2.68\hat{s}+\mathrm{ln}\left(1+20{\nu }_{\text{eff}}\right)\right],\hfill \end{align} \tag{ 5 }$$

in the limit of α = 0, where α represents the effect of temperature ratio on TEM threshold. Here, ν_eff is the effective collisionality. An approximate expression for α can be obtained from reference [34]. It has been found that, rather than that of ETG, the CG of TEM increases with τ for τ < 3 but saturates with τ for τ > 3. In figure 5, the theoretically predicted CGs and experimental gradients of T_e are plotted and compared for four snapshots during the formation of eITB. Before the ITB emergence (figure 5(a), t = 4.02 s) the CG of ETG is much lower than that of TEM and regulates the electron temperature profile. In this case, the T_e profile should be very stiff in the core region ρ < 0.2, since toroidal ETG can induce strong electron thermal flux [35]. This also illustrates the relatively small value of $R/{L}_{{T}_{\mathrm{e}}}$ and large value of χ_e in the plasma core. As τ increases quickly after LHW injection, the CG of ETG can be greatly increased, especially near the magnetic axis. This can be seen in figure 5(b) for t = 4.22 s. As a result, the electron temperature as well as its gradient can be effectively elevated by electron heating provided by LHW in the inner region, while $R/{L}_{{T}_{\mathrm{e}}}$ is still kept below the ETG threshold. An eITB at r_ITB = 0.15 is thus generated with maximum $R/{L}_{{T}_{\mathrm{e}}}\sim 18$. The theoretical evaluation of the ETG threshold inside the ITB is in accordance with the experimental observed small value of χ_e there, as shown in figure 3(b), which is owing to being free of anomalous thermal transport induced by ETG turbulence. It is also interesting to note that ETG are marginally unstable near the ITB foot around r/a = 0.3 where dT_e/dr drops dramatically. This marginal instability is mainly due to the reduced CG of ETG with τ and actually determines the width of eITB. Therefore, one identifies that the physical mechanism of eITB formation in this hot electron mode is the suppression of ETG turbulence due to the ion shielding effect. The ion shielding effect can be easily understood by noting the fact that the ion response to the ETG turbulence is essentially adiabatic response [32], which is determined by the ion temperature.

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After t = 4.2 s the value of τ increases further (see figures 3(a) and (c)) and the CG of ETG also keeps growing with τ, which can be seen in figures 5(c) and (d). The stable region for ETG is radially extended so that the location of ITB moves outward from r/a = 0.155 to r/a = 0.185 consequently. ETG remains to be stable inside the eITB and marginally unstable around the ITB foot. However, since the CG of TEM approach to saturation with τ, the TEM become more and more unstable in the ITB region so that the thermal flux induced by TEM turbulence should dominate electron transport in the plasma core. In this condition, the gradient $R/{L}_{{T}_{\mathrm{e}}}$ can only be increased gradually by external heating. This also indicates that the TEM turbulence has a relatively weak degradation with heating power in comparison with ETG so that the eITB is not terminated. This speculation can be verified by experimental measurement of TEM turbulence fluctuations. Shown in figure 6 is the integrated turbulence power given by the CO₂ laser collective scattering diagnostics, which measures the turbulent density fluctuation within 0 < r/a < 0.4 at fixed wavenumber k_θ = 10 cm⁻¹ [36, 37]. It can be seen that the turbulence fluctuations decrease before t = 4.28 s and rise up later. This is consistent with the threshold analysis given above; the high-k turbulence is decreased at first due to the up-shift of CG of ETG and increased later due to the increase of $R/{L}_{{T}_{\mathrm{e}}}$ as well as the saturation of CG of TEM with τ. Therefore, the height of eITB is determined by balancing the RF heating power against the TEM induced turbulence flux inside the eITB region, though the width of ITB is determined by the ETG threshold.

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In summary, the formation of eITB with high electron temperature has been observed in EAST with dominant electron heating by RF waves. The χ_e sharply drops in the plasma core within about 1/3 minor radius. The radial movement of eITB is also observed clearly. By the CG analysis of electron turbulence in experiments, it has been identified for the first time that electron thermal transport barriers can be triggered by electron turbulence suppression through the ion shielding when the electron–ion temperature ratio increases to a threshold value. It should be pointed out that no sawtooth oscillations have been observed in this type of experiment due to q_min > 1, which excludes the modulation of electron turbulence by the internal kink modes [38]. This discovery of the underlying mechanism for eITB in the hot electron mode has potential benefits for tokamak reactors which will be operated with dominant electron heating by fusion α-particles. The value of the electron–ion temperature ratio will be decreased with higher particle density due to higher collisional energy transfer. However, the effects of τ on the eITB dynamics should be taken into consideration forpredictions of transport properties in burning plasmas.

This work is supported by National MCF Energy R&D Program of China (Nos. 2018YFE0302100, 2019YFE03060000 and 2019YFE03040004) and the National Natural Science Foundation of China under Grant Nos. 11875254 and 12075240.