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We investigate ion temperature gradient (ITG) mode in hydrogen-isotope plasmas under a radial electric field in Large Helical Device (LHD) using the global gyrokinetic code, XGC-S. The radial electric field is taken into account by additional $E \times B$ drift motion in the poloidal direction. The present multi-mode linear simulations indicate the following properties of ITG mode and resulting heat flux. (i) In the absence of the radial electric field, mass-number dependencies of maximum growth rate and heat flux are in agreement with the quasi-linear theory. (ii) The radial electric field stabilizes the ITG mode and also affects the dominant wavelengths. The heat flux in heavy-hydrogen plasmas is lower than that predicted from the theoretical mass number dependency because of this modification of the mode structures. (iii) The radial electric field selectively stabilizes the ITG modes relevant to the light-hydrogen component in multi-component plasma. This selective contribution might limit the influence of the radial electric field on heavy-hydrogen heat flux.

ITG mode has been considered a primary cause of anomalous transports in magnetic confinement devices. Neoclassical transport accompanied by a radial electric field also plays a critical role in LHD. The radial electric field generates plasma rotation in the poloidal direction, and potentially affects the ITG mode and resulting turbulent transport. Recently, isotope effects in the anomalous transport are considered in deuterium experiments in the LHD. In this work, we investigate ITG mode properties in hydrogen-isotope plasmas under a radial electric field in a magnetic equilibrium relevant to LHD. The present work is the first step of global simulation study on the interaction between turbulent and neoclassical transport phenomena, which is not fully understood for hydrogen-isotope plasmas in stellarators.

We employ the global gyrokinetic code, XGC-S, developed for non-axisymmetric fusion devices$^{a}$. Benchmark studies on basic plasma phenomena in stellarators have been carried out for this code$^{b,c}$. Linear ITG simulations for a light-hydrogen plasma in the present equilibrium are also benchmarked with the previous EUTERPE simulations$^{d}$. An additional $E \times B$ drift term in a given radial electric field, $E_{r}$, is included in the ion equations of motion. We assume a uniform number density, $n=10^{19} m^{-3}$, and radial electric field related to the ion-root branch. In Figure 1(A), the temperature and radial electric field profiles are shown. We use unstructured meshes to represent perturbed electrostatic fields. Typical poloidal and toroidal resolutions are $\sim 1200$ / toroidal cross section / flux surface and $144$ / toroidal periodicity, respectively. Approximately $50$ delta-f marker particles are initially loaded per one mesh vertex. Only adiabatic response is assumed for the electron dynamics. The hydrogen mass number, $A$, is varied from $A=1$ to $A=3$. We also consider a multi-component plasma with $A=1$ ($50\%$) and $A=3$ ($50\%$) components.

In Figure 1(B), the maximum growth rates are summarized as functions of average mass number with ($\times$) and without ($+$) $E_{r}$. The growth rates are normalized to $V_{ti}/a$, where $V_{ti}$ and $a$ are ion thermal velocity for average mass number ($i=H,D,T$ for $A=1,2,3$) and typical minor radius, respectively. The green and red points indicate the growth rates in single- and multi-component plasmas, respectively. In the absence of $E_{r}$, the normalized growth rates are almost constant, exhibiting a dependency on the square root of mass number. The same property is observed in the presence of $E_{r}$ with lower growth rates. Therefore, the poloidal rotation is considered to stabilize the ITG mode with a similar stabilization rate independent of the hydrogen mass.

In Figure 2, mode amplitudes as a function of wavenumber in the early linear phase are shown for the $A=1$ (red), $3$ (blue), and multi-component (green) cases without (A) and with (B) $E_{r}$. The mode amplitudes are normalized to their maximum values. In the absence of $E_{r}$, the dominant wavenumber $k_{max}$ depends on the square root of average mass number in both single- and multi-component plasmas. The wavenumbers normalized to ion Larmor radius $\rho_{i}$ are roughly constant ($k_{max}\rho_{i}\sim0.3$). The theoretical linear growth rate, $\gamma_{k}$, is proportional to the wavenumber, independently of the hydrogen mass$^{e}$. Therefore, the maximum growth rate $\gamma_{k_{max}}$ is expected to be proportional to $V_{ti}/a$, namely, $(1/A)^{1/2}$. The maximum growth rates shown in Figure 1(B) agree with this theoretical prediction.

In the presence of $E_{r}$, as shown in Figure 2(B), $k_{max}$ are small compared to those in the absence of $E_{r}$. This reduction of wavenumber tends to be evident in the light-hydrogen plasma. In the multi-component plasma, the poloidal rotation mainly stabilizes higher wavenumber modes relevant to the light-hydrogen component. Consequently, the mode profile becomes similar to that in the heavy-hydrogen plasma. This result indicates selective stabilization of higher wavenumber modes due to poloidal rotation in multi-component plasmas.

The graphs in Figure 3 represent heat flux, $Q$, divided by the squared amplitude of (A) electrostatic potential $\Phi$, and (B) perturbed electric field $E$, in single-component plasmas as a function of the mass number. These values are normalized to that obtained for $A=1$ without $E_{r}$. Points $+$ and $\times$ represent the results obtained without $E_{r}$ and with $E_{r}$, respectively. $Q/\Phi^{2}$ decreases in heavy-hydrogen plasmas, inversely proportional to the square root of mass number. This result agrees with the estimate of heat flux in the quasi-linear theory$^{e}$, i.e., $Q/\Phi^{2}\propto k_{max}\sim (\rho_{i}/a)^{-1}$. The observed mass number dependencies of ${k_{max}}$, $\gamma_{k_{max}}$, and $Q/\Phi^{2}$ are also consistent with Gyro-Bohm scaling, where spatial and temporal scales are characterized by $\rho_{i}$ and $a/V_{ti}$, respectively.

In contrast, $Q/E^{2}$ increases as the mass number increases in the absence of $E_{r}$, as shown in Figure 3(B), because $E/\Phi\sim k_{max}$ decreases in the heavy-hydrogen plasmas. In the presence of $E_{r}$, however, $Q/E^{2}$ is roughly constant because the decrease in $k_{max}$ due to poloidal rotation is more evident in light-hydrogen plasmas. In the multi-component plasma, $Q/E^{2}$ of the heavy-hydrogen component (not shown in the graphs) is observed to be $\sim 0.42$ (without $E_{r}$) and $\sim 0.39$ (with $E_{r}$) in the same normalization as that in Figure 3(B). $E_{r}$ has little effect on these values, likely because of the selective contribution of poloidal rotation to higher wavenumber modes relevant to the light-hydrogen component.

Although these simulations do not include nonlinear evolutions, the mixing length estimation gives typical amplitudes of electrostatic potential, namely $e\Phi/T \propto\lambda_{max}/a$, where $\lambda_{max}$ is the wavelength of the dominant mode. According to this estimation, $Q/E^{2}$ has the same mass number dependency as $Q$, because $Q/E^{2}\sim Q/(\Phi/\lambda_{max})^{2}\propto QA^{0}$. The present simulation results indicate that the heat flux due to the ITG mode under the effects of the poloidal rotation in heavy-hydrogen plasmas might be small compared to that predicted from the mass ratio dependency in Gyro-Bohm scaling, $Q\propto A^{1/2}$.

[a] T.Moritaka ${\it et~al}$., Plasma ${\bf~2}$ (2019) 179-200.

[b] M.Cole ${\it et~al}$., Phys.Plasmas ${\bf~26}$ (2019) 032506.

[c] M.Cole ${\it et~al}$., Phys.Plasmas ${\bf~26}$ (2019) 082501.

[d] J.Riemann ${\it et~al}$., Plasma Phys.Control.Fusion ${\bf~58}$ (2016) 074001.

[e] F.Romanelli, Phys.Fluid ${\bf~B1}$ (1989) 1018.

Affiliation | National Institute for Fusion Science |
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Country or International Organization | Japan |