A mechanism of ion temperature peaking by impurity pellet injection in a heliotron plasma

The improvement of the ion energy confinement by seeding of different impurity species, such as nitrogen, neon, and argon, has been observed in diverse tokamak devices, e.g. ISX-B [1], ASDEX and ASDEX-U [2, 3], TEXTOR [4], JET [5, 6], DIII-D [7], and JT-60U [8]. The results of recent experiments performed on the heliotron Large Helical Device (LHD) [9, 10] have demonstrated that this phenomenon is not specific for the tokamak type of magnetic fusion devices: the injection of carbon pellets into the LHD led to a pronounced peaking of the ion temperature profile and a strong increase, up to a factor of 2, of the central T_i magnitude.

This allows reducing further the core anomalous transport in discharges where an internal transport barrier is generated by a high enough heating power [11]. The similarities of observations both on tokamaks and on helical devices reveal the fundamental nature of the confinement improvement induced with impurities in hot fusion plasmas. Therefore the understanding of mechanisms of impurity injection on the anomalous heat transport in the LHD are of significant importance for magnetic fusion investigations in general.

In [4, 12, 13] the results of experiments with impurity seeding in tokamaks have been interpreted by considering the impact of the effective plasma charge Z_eff on the toroidal ion temperature gradient (ITG) instability that is usually discussed as the main source of the anomalous ion transport. The curvature of magnetic field lines is the main cause of toroidal ITG modes and one has to expect the development of this instability and accompanying anomalous transport in helical devices as well. In the next section of the present paper a previous analysis of ITG instability is generalized for plasmas with two ion species, i.e. the main background one of a hydrogen isotope and one charge state of impurity. In the third section the method to numerically solve the ion heat transport equation with a heat conduction depending strongly non-linearly on the temperature gradient, as in the case of ITG, is outlined. In section 4 the results of the computations are compared with the experimental data from the LHD. Finally conclusions are drawn.

A significant peaking of the ion temperature T_i in the plasma core can be obtained in the LHD after the injection of carbon pellets, see figure 1 and reference [9]. The degree of the ion energy confinement improvement is most strongly related to the amount of the injected carbon, see figure 1, and approaches its maximum, where the central value of T_i is nearly doubled, at an impurity density n_Z of $5-10\cdot10^{17}m^{3}$, dependent on the injection scenario, see figure 2. The experiments in question have been done to achieve a maximum ion temperature [11]. In good agreement with the ISS04 energy confinement time scaling for stellarators [14], with a total heating power in the LHD of 20 MW it is possible to get a T_i≈ 3 − 4 keV at a relatively low plasma density of $1\div2\cdot10^{19}m^{-3}$ only, see figure 3. This is much lower than the level of $10^{20}m^{-3}$ required in a future reactor and a significant increase of the power as well as the usage of diverse means to reduce the anomalous transport, in particular, the impurity pellet injection discussed here, would allow to approach this level.

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Recent LHD experiments performed to study the isotope effect in the ion particle and energy confinement have demonstrated that the presence of impurity is of importance for the quality of the internal transport barrier (ITB) [15]: in deuterium plasma the ITB can be sustained longer than in a hydrogen one because the formation of the impurity hole requires more time due to lower impurity transport. Also the shape of the radial profile of the electron density n_e plays a role in the reduction of the ion heat conduction: a slightly hollow $n_{e}\left(r\right)$ results in a stronger improvement in confinement, see figure 3.

Under the conditions of the low plasma density in question in the experiments the increase of impurity radiation losses by the pellet injection does not contribute dramatically to the global power balance. As one can see in figure 4, even with the injection of pellets sufficient to double the central T_i the total power radiated did not exceed 10% of the input one. By extrapolating the impurity pellet scenario to a higher plasma density, one of the main concerns is, however, the plasma thermal collapse [16]. The application of resonant magnetic perturbations (RMPs), generating a broad magnetic island at the plasma edge, has been proven to be a promising approach to overcome this difficulty [17]. In the LHD experiments analyzed in the present paper the RMP has not been applied.

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The contribution of a toroidal ITG instability to the ion anomalous heat conduction κ_an is assessed by applying a mixing length approximation [18], which is less sophisticated but more transparent than normally done gyrokinetic simulations of turbulence in a high ion temperature plasma in the LHD [19]. By following [18], consider an ITG Fourier mode with the frequency ω, poloidal wave vector k, generating the perturbation $\widetilde{\varphi}$ of the electrostatic potential. Small perturbations of the density n_Z and temperature T_Z, $\widetilde{n}_{Z}$, and $\widetilde{T}_{Z}$, respectively, of ions with the charge Z are governed by the following linearized equations for particle and heat balances:

$$\begin{equation} \left(\bar{\omega}+\frac{\tau}{Z}\right)\frac{\widetilde{n}_{Z}}{n_{Z}}+ \left(\varsigma_{Z}\bar{\omega}-\epsilon_{Z}+1+\delta_{Z}\right) \frac{e\widetilde{\varphi}}{T_{e}}+\frac{\tau}{Z}\frac{\widetilde{T_{Z}}}{T_{Z}} = 0, \end{equation} \tag{ 1 }$$

$$\begin{equation} \left(\bar{\omega}+\frac{5}{3}\frac{\tau}{Z}\right)\frac{\widetilde{T_{Z}}}{T_{Z}}- \frac{2}{3}\bar{\omega}\frac{\widetilde{n}_{Z}}{n_{Z}}-\left(\epsilon_{T}-\frac{2}{3} \epsilon_{Z}\right)\frac{e\widetilde{\varphi}}{T_{e}} = 0, \end{equation} \tag{ 2 }$$

where

$$\begin{align*} \bar{\omega} & = \frac{\omega}{\omega_{De}},\:\omega_{De} = \frac{2kcT_{e}}{eBR}, \;\tau = \frac{T_{i}}{T_{e}},\;\epsilon_{Z} \nonumber\\ & = -\frac{d\ln n_{Z}}{dr}\frac{R}{2}, \;\epsilon_{T} = -\frac{d\ln T_{i}}{dr}\frac{R}{2}, \end{align*}$$

$$\begin{align*} \varsigma_{Z} = k^{2}\rho_{Z}^{2},\;\delta_{Z} = \varsigma_{Z}\left(\epsilon_{T}+ \epsilon_{Z}\right)\tau/Z, \end{align*}$$

with $\rho_{Z} = \sqrt{T_{e}m_{Z}}/\left(ZeB\right)$ being the Larmor radius of the Z-ions; henceforth it is assumed that the unperturbed temperature of the impurity ions is equal to that of the background hydrogen ions, T_i. The impact of ion parallel motion, which involves magnetic geometry characteristics, such as the safety factor q, the q shear, magnetic ripples etc is neglected in equations (Equation 1) and (Equation 2) so these are applicable both to tokamak and to stellarator configurations. For the conditions in question of hot core plasmas this does not lead to a large error in the growth rate of the ITG instability [20]. If one impurity ion species only, i.e. carbon nuclei in the case under consideration, is taken into account, the plasma quasi-neutrality condition is as follows:

$$\begin{equation} \widetilde{n}_{i}+Z\widetilde{n}_{Z} = \widetilde{n}_{e}. \end{equation} \tag{ 3 }$$

The relation between the perturbation of the electron density $\widetilde{n}_{e}$ and $\widetilde{\varphi}$ follows from the parallel force balance:

$$\begin{equation} T_{e}\widetilde{n}_{e} = en_{e}\widetilde{\varphi}. \end{equation} \tag{ 4 }$$

The perturbation frequency $\bar{\omega}$ is governed by a quartic polynomial algebraic equation, following from the requirement that the above system of linear equations for the amplitudes of small perturbations has a non-trivial solution:

$$\begin{equation} a_{4}\bar{\omega}^{4}+a_{3}\bar{\omega}^{3}+a_{2}\bar{\omega}^{2}+ a_{1}\bar{\omega}+a_{0} = 0, \end{equation} \tag{ 5 }$$

where

$$\begin{align*} a_{4} & = 1+\xi_{i}\varsigma_{i}+Z\xi_{Z}\varsigma_{Z},\:a_{3} = \tau_{1}+ \frac{\tau_{1}}{Z}+\xi_{i}\left(\frac{\varsigma_{i}\tau_{1}}{Z}+b_{i}\right)\nonumber\\[3pt] & \quad+ Z\xi_{Z}\left(\varsigma_{Z}\tau_{1}+b_{Z}\right), \nonumber\\[3pt] a_{2} & = \frac{\tau_{1}^{2}}{Z}+\tau_{2}+\frac{\tau_{2}}{Z^{2}}+ \xi_{i}\left(\frac{b_{i}\tau_{1}}{Z}+\frac{\varsigma_{i}\tau_{2}}{Z^{2}}+ c_{i}\right) \nonumber\\[3pt] & \quad +Z\xi_{Z}\left(b_{Z}\tau_{1}+\varsigma_{Z}\tau_{2}+c_{Z}\right), \nonumber\\[3pt] a_{1} & = \frac{\tau_{1}\tau_{2}}{Z}\left(1+\frac{1}{Z}\right)+\frac{\xi_{i}}{Z} \left(c_{i}\tau_{1}+\frac{b_{i}\tau_{2}}{Z}\right)+Z\xi_{Z}\left(c_{Z}\tau_{1}+ b_{Z}\tau_{2}\right), \nonumber\\[3pt] a_{0} & = \tau_{2}\left(\frac{\tau_{2}+\xi_{i}c_{i}}{Z^{2}}+Z\xi_{Z}c_{Z}\right), \end{align*}$$

with $\xi_{i} = n_{i}/n_{e},\:\xi_{Z} = n_{Z}/n_{e}$ being the ion concentrations, $\tau_{1} = 10\tau/3,\:\tau_{2} = 5\tau^{2}/3$, $b_{Z} = 1+\delta_{Z}-\epsilon_{Z} +\frac{5}{3}\frac{\tau}{Z}\varsigma_{Z},\:c_{Z} = \frac{\tau}{Z}\left(\frac{5}{3} +\frac{5}{3}\delta_{Z}+\epsilon_{T}-\frac{7}{3}\epsilon_{Z}\right)$, $b_{i} = 1+\delta_{i}-\epsilon_{i}+\frac{5}{3}\tau\varsigma_i$, and $c_{i} = \tau\left(\frac{5}{3}+\frac{5}{3}\delta_{i}+\epsilon_{T}-\frac{7}{3}\epsilon_{i}\right)$; all values with the subscript i are related to the main ions of the hydrogen isotope with the charge Z = 1.

According to a mixing length approximation [18] the contribution to the heat conductivity from unstable drift modes can be estimated as $\chi_{an}\approx \left(\textrm{Im}\omega\right)_{\max}/k_{\max}^{2}$, where $\left(\textrm{Im}\omega\right)_{\max}$is the maximum value of the mode growth rate as a function of k and $k_{\max}$, the value of k at which $\left(\textrm{Im}\omega\right)_{\max}$ is approached. This approximation does not account for the reduction of the instability grows rate with the shear of the rotation induced by the radial electric field, $\Omega_{E\times B}$ [21]. The important role of $\Omega_{E\times B}$ for the temperature peaking in tokamaks induced by impurity injection has been demonstrated in [23] for JET L-mode discharges with neon seeding. It has been also shown that the main contribution to the time averaged radial electric field (do not mix with fluctuations triggered by ITG instability) is due to poloidal diamagnetic rotation induced by the radial temperature gradient, and $\Omega_{E\times B}\approx\frac{1}{eBr}\left|\frac{dT_{i}}{dr}\right|$. Finally, with the dimensionless wave number $\varsigma = k_{\max}^{2}\rho_{i}^{2}$ one gets the following assessment for the anomalous ion heat conduction due to the ITG instability:

$$\begin{equation} \kappa_{an} {=} \left(n_{i}+n_{Z}\right)\chi_{an} {=} \kappa_{*}\left(1- \frac{Z_{eff}-1}{Z}\right)\left[\frac{\textrm{Im}\bar{\omega} \left(\varsigma\right)}{\sqrt{\varsigma}}-\frac{\tau\psi\epsilon_{T}\rho_{i}}{\varsigma r}\right], \end{equation} \tag{ 6 }$$

Here $\kappa_{*} = 3n_{e}T_{e}^{1.5}\sqrt{m_{i}}/\left(e^{2}B^{2}R\right)$, $Z_{eff} = \left(n_{i}+Z^{2}n_{Z}\right)/n_{e}$ is the effective plasma charge, and the relations $n_{i} = n_{e}\left(Z-Z_{eff}\right)/\left(Z-1\right)$, $n_{Z} = n_{e}\left(1-\xi_{i}\right)/Z$ have been used; the factor $\psi = \xi_{i}+ \xi_{Z}\sqrt{A_{Z}/Z}$, with A_Z being the atomic weight of the impurity ions, takes into account the braking of the main ion diamagnetic rotation by collisions with the impurity ones.

In spite of its simplicity the present model for ITG instability reproduces well important features of calculations with the gyrokinetic Vlasov code GS2 [22]. In particular, if the stabilizing effect of $\Omega_{E\times B}$ is not taken into account both approaches provide similar values of 0.5-0.6 for $k_{\max}\rho_i$. In future studies the present model will be extended to calculate the impurity transport, see [20], to do a more thorough comparison with the results of the GS2 modeling.

In figure 5 the ratio $\kappa_{an}/\kappa_{*}$ is shown versus Z_eff for hydrogen plasmas with carbon nuclei with Z = 6 as the dominant impurity species, $n_{e} = 1.5\cdot10^{19}m^{-3}$, τ = 1, _T = 5, and several combinationsof _i and _Z. One can see that the ion heat conduction reduces significantly with Z_eff. This is due to the decrease of the diamagnetic drift, induced by the density and temperature perturbations, with the ion charge [18, 24], which is mimicked in equations (Equation 1) and (Equation 2) by the terms proportional to τ/Z. Since $\widetilde{n}_{Z}, \widetilde{T}_{Z}$ change mostly in the poloidal direction, this type of drift motion is radially directed and dominates the anomalous heat transfer generated by ITG modes.

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Thus the ITG growth rate reduces with increasing impurity density because the impurity ion charge Z is significantly larger than that of the main ions, and thus, the terms in equations (Equation 1) and (Equation 2) responsible for the instability are smaller. However, with the approach of Z_eff to Z the anomalous ion heat conduction starts to grow with the impurity density. This can be explained by (i) the change in the ITG instability threshold with Z, (ii) the reduction of $\Omega_{E\times B}$, and (iii) the increasing $k_{\max}$ which is increasingly dominated by the Larmor radius of the impurity ions and $k_{\max}\sim 1/\rho_{Z}\sim Z/\sqrt{A_{Z}}\sim \sqrt{Z}$. Therefore the factor ζ increases and this affects the former term in relation (Equation 6), which is weaker than the latter one, by reducing additionally the impact of the E × B shear.

In the core of the LHD plasma the poloidal cross sections of the magnetic flux surfaces are close to elliptic ones. The plasma parameters are nearly homogeneous on the surfaces and can be described as functions of the effective radius $r = \sqrt{yz}$, with y and z being the minor and major axes of the cross section. The ion temperature is governed by the following heat conduction equation:

$$\begin{equation} \frac{1}{r}\frac{d}{dr}\left(-r\kappa_{i}\frac{dT_{i}}{dr}\right) = Q_{heat}^{i}-Q_{ie}, \end{equation} \tag{ 7 }$$

where $\kappa_{i} = \kappa_{0}+\kappa_{an}$ is the ion perpendicular heat conduction with the value κ₀ defined from the requirement that the magnitude of $T_{i}\left(r = 0\right)$ calculated for the state with the maximum peaking of the temperature profile in the presence of carbon impurity reproduces the experimental one; $Q_{heat}^{i}$ is the ion heating power and $Q_{ie} = \alpha\left(T_{i}-T_{e}\right)$ that of the heat transfer from ions to electrons through Coulomb collisions. The anomalous contribution to the ion heat conduction due to ITG unstable modes, κ_an, depends itself on the ion temperature gradient. Figure 6 displays κ_an versus _T for _i,Z = 1 and different magnitudes of Z_eff, with other parameters assumed the same by calculating the results presented in figure 4. One can see that at some critical values of _T the anomalous heat conduction changes abruptly. As it has been demonstrated in [25], by solving equation (Equation 7) with a standard approach, e.g. a finite difference one, this can lead to numerical instabilities. This situation is exaggerated by the fact that the radial position of such a discontinuity in κ_an is unknown in advance.

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An approach to handle transport models, predicting abrupt changes of the heat conduction with the temperature gradient, has been elaborated in [25]. This is based on the introduction of the new variable:

$$\begin{equation} \Theta\left(r\right) = \frac{1}{r^{2}}\intop_{0}^{r}\Omega T_{i}\rho d\rho, \end{equation} \tag{ 8 }$$

where the function $\Omega\left(r\right)$ is chosen by analyzing the T_i dependence on the right-hand side (rhs) of equation (Equation 7). One can straightforwardly get:

$$\begin{equation} T_{i} = \frac{1}{\Omega}\left(r\frac{d\Theta}{dr}+2\Theta\right), \end{equation} \tag{ 9 }$$

$$\begin{equation} \frac{dT_{i}}{dr} = \frac{r}{\Omega}\left[\frac{d^{2}\Theta}{dr^{2}}+ \left(\frac{3}{r}-\frac{d\ln\Omega}{dr}\right)\frac{d\Theta}{dr}- \frac{2}{r}\frac{d\ln\Omega}{dr}\Theta\right]. \end{equation} \tag{ 10 }$$

By integrating equation (Equation 7) from r = 0, one gets an equation for $\Theta\left(r\right)$. In the present case, where T_i is involved in the rhs of (Equation 7) only through Q_ie, Ω = α and:

$$\begin{equation} \frac{d^{2}\Theta}{dr^{2}}+\left(\frac{3}{r}-\frac{d\ln\alpha}{dr}\right) \frac{d\Theta}{dr}-\left(\frac{2}{r}\frac{d\ln\alpha}{dr}+ \frac{\alpha}{\kappa_{i}}\right)\Theta = -F, \end{equation} \tag{ 11 }$$

with free term $F = \frac{\alpha}{\kappa_{i}r^{2}}\intop_{0}^{r}\left(Q_{heat}^{i}+\alpha T_{e}\right)\rho d\rho$.

One can see that on the contrary to the equation for T_i the one above for Θ does not contain dκ_i/dr, becoming infinite in the points of κ_an, and thus, of κ_i discontinuity. Therefore, as it was shown in [25] it can be integrated without numerical difficulty by standard numerical methods for the second order ODE. From relations (Equation 9) and (Equation 10) it follows that the boundary condition at the plasma axis for T_i, $dT_{i}/dr\left(0\right) = 0$, transforms into $d\Theta/dr\left(0\right) = 0$. The boundary condition at the outer border of the computational region, r = r_*, corresponds that the T_i- profile does not change with impurity injection for r > r_*, and thus, $T_{i}\left(r_{*}\right)$ is fixed.

5.1. LHD experiment

Computations have been performed for the conditions of the LHD shot #106455 where the amount of impurity has been varied by injecting two carbon pellets with a certain time interval; in the calculations this variation has been mimicked by changing Z_eff. The input parameters for the transport model have been assessed by analyzing the experimental profiles: $\kappa_{0} = \left(5.7-2.5r^{2}/r_{*}^{2}\right)10^{19}m^{-1}s^{-1}$ with r_* = 0.4 m; the plasma density is slightly hollow, $n_{e}\left(r\right) = \left(1.2+0.3r^{2}/r_{*}^{2}\right)10^{19}m^{-3}$; according to the transport analysis [9] the deposition power changes very slightly with the pellet injection and henceforth $Q_{heat}^{i} = 0.8(1-0.66r/r_{*}) MW\,m^{-3}$ is adopted for all conditions. Figure 7 shows the assumed electron temperature profile, which practically did not change with the impurity injection, and the ion temperature profiles calculated with different magnitudes of Z_eff. One can see that the computed T_iprofiles mimic quantitatively and qualitatively well the experimental ones, see figure 1. The central temperature value increases with the impurity content up to $Z_{eff}\gtrsim3$ but drops noticeably if Z_eff approaches a level of 4. This is in line with the results shown in figure 2 predicting the deteriorating effect of a too high impurity concentration on the ion energy confinement. The optimum carbon density of $8 \times 10^{17} m^{-3}$, corresponding to the maximum central T_i, is in rough agreement with the experimental data presented in figure 2. The agreement between the calculations and measurements becomes better if one takes into account that in the discharges a significant contribution to Z_eff was from He seeded into H plasma for other experimental aims.

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The shear of the drift rotation induced by the radial electric field E_r, $\Omega_{E\times B}$, is traditionally considered as very importance for the suppression of drift micro-instabilities triggering anomalous transport in magnetic fusion devices [21]. The analysis in [23] has shown that under the conditions in question the main contribution to E_r is from the diamagnetic rotation component induced by the ion temperature gradient. In the present model this is taken into account by the last term in the square brackets in the relation for κ_an. With increasing density of impurity the flow of the main hydrogen ions is broken by the friction through Coulomb collisions with the impurity ions whose rotation velocity is smaller by a factor of 1/Z. This is of importance for the deterioration of the ion heat transport at high n_Z. In the relation (Equation 6) for κ_an this is described by the reduction of the factor ψ with decreasing ξ_i. To demonstrate the significance of the $\Omega_{E\times B}$ effect computations have been done with ψ = 1 corresponding to Z_eff= 1. The results presented in figure 8 demonstrate that in such a case, in disagreement with observations [9], there is no deterioration in the ion heat transport, at least up to Z_eff = 4.

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One of the main peculiarities of the plasma parameter profiles in the heliotron LHD is an essentially flat or even slightly hollow electron density profile. Under the conditions in question with carbon pellet injection plasmas with a hollower n_e(r) reveals a stronger reduction in the ion heat transport channel, see figure 3. This is in line with the reduction of the ITG growth rate with a hollow density profile observed in recent gyrokinetic Vlasov simulations [26]. To also prove that the model elaborated in the present paper reproduces these experimental findings calculations were performed with an ideally flat density, $n_{e}\left(r\right) = 1.5\cdot10^{19}m^{-3}$, and the T_i profiles obtained for different Z_eff are shown in figure 9. One can see that the improvement in the confinement in this case is noticeably smaller than for hollow density profiles and deteriorates very fast with Z_eff > 3. The main effect of the hollow n_e profile is a smaller plasma density in the core. This results in a lower κ_*, and see equation (Equation 6), a smaller κ_an

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Finally, we consider the role of the ion component cooling through Coulomb collisions with electrons given by the term Q_ie. Due to the mass relation collisions between the main and impurity ions being much more effective for heat transfer than those with electrons, the collisions of impurity ions with electrons is of importance for both ion components. If as above these collisions are taken into account, $Q_{ie}\sim\left(1+\xi_{i}\right)/2$. On the contrary, by neglecting the contribution of the Z − e collisions $Q_{ie}\sim \xi_{i}$ and figure 10 shows the ion temperature profiles obtained in this case. They very marginally differ from those shown in figure 7 found by taking the Z − e collisions in Q_ie into account. This fact does not mean, however, that the factor in question is not important generally. Its role may be more significant under reactor conditions with a much higher electron density.

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5.2. Extrapolation to reactor conditions

Many experimental facts both presented and cited, in particular, in this paper show a positive influence of impurities on the anomalous ion heat transport, by leading to the suppression of ITG induced turbulence. The fact that these findings were done on tokamak and helical fusion devices suggest a fundamental nature of these phenomena being of value by itself. Nonetheless, its practical usefulness for future reactors is still a matter of debate. The main arguments against impurity injection are the suspicions that the increase of Z_eff will lead to (i) the plasma dilution, and thus, the reduction of the fusion output and (ii) the growth of the edge radiation provoking disruption in tokamaks and thermal collapse in helical devices.

To analyze the Z_eff effect on the fusion power we start from the plasma power balance:

$$\begin{equation*} \frac{\left(n_{e}+n_{i}+n_{Z}\right)T}{\tau_{E}}\sim P_{fus}, \end{equation*}$$

where T is the plasma temperature assumed the same for all charged components; for the energy confinement time τ_E we assume the ISS04 scaling [14] and modify this to take into account the effect of impurity on the anomalous transport, which according to the present study, decreases roughly as $1/\sqrt{Z_{eff}}$, thus

$$\begin{equation*} \tau_{E}\sim\frac{n_{e}^{0.54}}{P_{fus}^{0.61}}\sqrt{Z_{eff}}, \end{equation*}$$

and in the important temperature range of $5\ {\rm keV}\lesssim T\lesssim20keV$ the fusion power is

$$\begin{equation*} P_{fus}\sim n_{i}^{2}T^{2}. \end{equation*}$$

By combining the relations above and using the plasma quasi-neutrality condition, providing $n_{i} = n_{e}\frac{Z-Z_{eff}}{Z-1},\;n_{Z} = \frac{n_{e}}{Z}\frac{Z_{eff}-1}{Z-1}$, one gets:

$$\begin{equation} T\sim f_{ZT}n_{e}^{1.45},~ P_{fus}\sim f_{ZP}n_{e}^{4.9}, \end{equation} \tag{ 12 }$$

with $f_{ZT} = \varphi_{Z}\psi_{Z}^{3.54},~ f_{ZP} = \psi_{Z}^{9.1}, ~\psi_{Z} = \varphi_{Z}\frac{Z-Z_{eff}}{Z-1}$ and $\varphi_{Z} = \frac{2Z\sqrt{Z_{eff}}}{2Z+1-Z_{eff}}$.

Figure 11 shows the factor f_ZP as a function of the effective plasma ion charge for three impurity species, namely, helium, carbon, and neon. One can see that, on the one hand, the accumulation of He ash will, as expected, lead to a drop in the fusion power. On the other hand, the presence of seeded C and Ne impurities results in a significant increase, in spite of the plasma dilution, in the power input, at least if the ISS04 energy confinement time scaling can be extrapolated to reactor conditions. The increase of f_ZP with Z_eff for the effective ion charge not much larger than 1 is through the factor ϕ_Z and reveals the dependence of τ_E on the heating power. The decrease of f_ZP for larger Z_eff is the dilution effect. Although it is a very qualitative indication it motivates continuing the exploration of the impurity influence on the energy confinement in the plasma of fusion devices.

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The role of impurity radiation energy losses has been neglected in the analysis above. As has been mentioned in the LHD experiments of a low plasma density discussed in the present paper these did not exceed 10−15% of the heating power. However, it is important to consider what will be happened by going to a higher density required in a reactor, of $10^{20}m^{-3}$, by simultaneously maintaining Z_eff at a fixed level. The radiation power from the plasma core, $P_{rad}^{core}$, is mostly due to Bremsstrahlung and recombination of impurity nuclei and increasing as $n_{Z}n_{e}\approx n_{e}^{2}/Z^{2}$. By using the ADAS-data [27] one gets that for C and Ne impurities $P_{rad}^{core}/n_{e}^{2}V_{pl}\approx 3\cdot10^{-36}Wm^{3}$ at a plasma temperature of 15 keV, i.e. by a factor of 30 smaller than $P_{fus}/n_{e}^{2}V_{pl}\approx 10^{-34}Wm^{3}$. Thus, also under the reactor conditions the core radiation of injected impurity should not be a large problem. Nonetheless, it is wellknown that in modern devices the line radiation of incompletely stripped impurity ions dominate radiation losses and for the impurity species in question these are Li-like ion states, C^3 + and Ne^7 +, respectively. These species reside at the plasma edge and are not in a corona equilibrium, being essentially affected, in particular, by transport phenomena, see, e.g. [28]. A firm assessment of their radiation, $P_{rad}^{edge}$, is not so trivial for impurity nuclei in the plasma core and requires special consideration which is beyond the scope of the present study. Here we note only that a simple proportion of $P_{rad}^{edge}\sim n_{e}^{2}$ obviously cannot be applied in this case. For example, the measurements on LHD [29] have demonstrated that by increasing the line averaged density n_e from $2\cdot10^{19}$–$7\cdot10^{19}m^{-3}$ the total radiation losses, dominated by $P_{rad}^{edge}$, have grown by a factor of 2 only. At the same time, being enhanced to a certain fraction of the input power, the edge impurity radiation can provoke thermal instabilities resulting in a plasma collapse. To avoid such a problem in the LHD the application of a resonant magnetic perturbation at the plasma edge has been verified as an effective tool [17]. It has allowed obtaining a stable detached plasma at a high reactor relevant density of $10^{20}m^{-3}$ with the radiation losses approaching close to the input power. This method deserves further deeper study and elaboration, especially, by taking into account that RMP is now widely exploited on magnetic fusion devices of different concepts. In future experiments one has to try to apply RMP also by attempting to explore the impurity injection scenario for the improvement of the confinement at a higher plasma density.

Recent experiments on the heliotron LHD with the injection of carbon pellets have demonstrated that the reduction of anomalous heat transport in the presence of impurities, found previously in diverse tokamak devices, is not specific for this type of fusion device only. At an optimal level of a carbon density of $5-10\cdot10^{17}m^{-3}$ a pronounced peaking of the ion temperature profile and strong increase, up to a factor of 2, of the central T_i magnitude has been observed in the LHD. The similarity of the impurity influence both in tokamaks and in helical devices testify to the fundamental nature of this phenomenon. Therefore the approach used before to interpret the improvement in the confinement in TEXTOR and JET has been elaborated further and applied here to explain the observations on the LHD. It is based on the impact of the ion charge on the growth rate of a toroidal ion temperature instability considered usually as the main trigger of the anomalous transport in magnetic fusion devices.

The results of the calculations are generally in line with the experimental findings. They reproduce both qualitatively and quantitatively the drop in the ion anomalous heat conduction and the peaking of the ion temperature with Z_eff, increasing up to a critical level, and the drop in the ion confinement if this level is exceeded. Our computations also demonstrate the significant importance of the shear of the drift rotation induced by the radial electric field. Under the conditions in question E_r is mostly driven by the diamagnetic flow induced through the ion temperature gradient. If the braking of this flow by friction with impurity ions is neglected the ion temperature peaking with increasing impurity concentration is significantly higher than in the experiments. Moreover, there is no critical Z_eff, by overcoming which the energy confinement starts to deteriorate. The model elaborated reproduces also the stronger impurity effect on the T_i peaking for hollow density profiles compared to flat ones as is observed in the LHD. Calculations show that Coulomb collisions between impurities with electrons do not play a noticeable role in the ion heat balance. This cooling channel may become more important by going to a higher plasma density relevant to a fusion reactor. The present consideration demonstrates that both experiments on helical devices, such as the LHD, allow for good control and reproduction of experimental conditions, and their interpretation can make an important contribution to fusion studies in general, in particular, by investigating the impact of impurities on the plasma performance.

Although there are qualitative indications that the impurity injection approach can also work under reactor relevant conditions, for a firm extrapolation of the LHD results further development of the present model is necessary. In particular time dependent calculations for the entire plasma volume, by including a description of the impurity source through pellet ablation and transport of the material ablated, the evolution of the electron density seems to be of importance. This has been demonstrated in the LHD experiments with carbon pellets of different radii, see [9].

To validate the present model further comparison with different measurement data have to be done in future investigations. This includes, e.g. turbulence spectrum, the mechanism of the generation of the radial electric field leading to $\Omega_{E\times B}$ stabilization of the ITG modes. Here it is assumed as being generated mostly by the ion temperature gradient and this assumption has to be checked by comparing the measurements of the charge-exchange diagnostics providing the contributions from the toroidal, poloidal, and diamagnetic rotation components to E_r separately. In a reactor the α-particles will be intrinsically presented and an elaboration of our approach for the case of three or more ion species is of interest. It is also of importance to consider of heavier impurities such as neon, argon, and tungsten, studied in the present experiments.

The present study shows that the impurity injection can be a promising scenario to improve the energy confinement in heliotron fusion devices. Nonetheless skepticism among many researchers regarding a deliberate introduction of impurities into a fusion reactor is understandable. This motivates us to further investigate this approach in diverse respects such as:

• (a)  
  optimization of the impurity pellet injection method, e.g. by doing this at the high field side to use the polarization drift;
• (b)  
  combination with other successful schemes to improve ion confinement, e.g. the hydrogen pellet injection in W7-X [30] which provides a peaked profile of the plasma density (the density peaking has already been proven to be an effective method to suppress ITG instabilities in the TEXTOR RI-mode [4]);
• (c)  
  application of RMP at the plasma edge to control the radiation losses and prevent a plasma thermal collapse [17, 29]