2 Motivation

Before performing a detailed analysis, it is useful to consider a simple (but experimentally relevant) limit, where the importance of classical transport in a stellarator is apparent. For this purpose, we summarize results from earlier work (Braun & Helander Reference Braun and Helander2010; Helander et al. Reference Helander, Newton, Mollén and Smith2017; Buller et al. Reference Buller, Smith, Helander, Mollén, Newton and Pusztai2018).

At fusion-relevant temperatures, the bulk hydrogen species of the confined plasma will be in a low-collisionality regime. However, as the collisionality increases with charge, high- $Z$ impurities (with $Z$ being the charge number) can still have high collisionality. Such impurities can occur, for example, in experiments using tungsten plasma-facing components, which is the favoured material for the divertor of future fusion reactors (Bolt et al. Reference Bolt, Barabash, Federici, Linke, Loarte, Roth and Sato2002). These plasmas will thus be in a mixed-collisionality regime, with low-collisionality bulk and high-collisionality impurity ions.

In this regime, the ratio of classical to neoclassical impurity particle fluxes calculated from the mass-ratio expanded collision operator is given by a purely geometrical factor (Buller et al. Reference Buller, Smith, Helander, Mollén, Newton and Pusztai2018)

(2.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\langle \unicode[STIX]{x1D71E}_{z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle ^{\text{C}}}{\langle \unicode[STIX]{x1D71E}_{z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle ^{\text{NC}}}}={\displaystyle \frac{\langle \,j_{\bot }^{2}\rangle \langle B^{2}\rangle }{\langle \,j_{\Vert }^{2}\rangle \langle B^{2}\rangle -\langle \,j_{\Vert }B\rangle ^{2}}}. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D713}$ is a radial coordinate (a flux-surface label), $\langle \cdot \rangle$ is the flux-surface average, $\unicode[STIX]{x1D71E}_{z}$ is the flux of impurity ions, $\langle \unicode[STIX]{x1D71E}_{z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle ^{\text{(N)C}}$ is the radial (neo)classical impurity flux averaged over the flux surface, $\boldsymbol{B}$ is the magnetic field, $B=|\boldsymbol{B}|$ , and $\boldsymbol{j}$ is the current density, here defined by $\boldsymbol{j}\times \boldsymbol{B}=\unicode[STIX]{x1D735}p(\unicode[STIX]{x1D713})$ , $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{j}=0$ ; with $j_{\Vert }$ and $j_{\bot }$ being the current components parallel and perpendicular to $\boldsymbol{B}$ , and $p$ the total pressure.

Equation (2.1) also enters into the ratio of classical and neoclassical transport at yet higher collisionalities: in the Pfirsch–Schlüter regime, where both bulk and impurity ions are collisional. This can be shown using the expression for neoclassical transport derived by Braun & Helander (Reference Braun and Helander2010) together with the expression for classical transport in, for example, Buller et al. (Reference Buller, Smith, Helander, Mollén, Newton and Pusztai2018). For stellarators optimized for low $j_{\Vert }/j_{\bot }$ (such as W7-X), the (2.1) ratio will be large and classical transport will thus dominate at high collisionality. This will be verified by numerical simulations in § 4.

3 Linearized Fokker–Planck operator

In this section, we write down the classical particle and heat transport due to a linearized Fokker–Planck operator. The flux-surface averaged radial classical transport of particles and energy is given by

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{a}^{\text{C}}\equiv \langle \unicode[STIX]{x1D71E}_{a}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle ^{\text{C}}\equiv \left\langle {\displaystyle \frac{\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{Z_{a}eB}}\boldsymbol{\cdot }\boldsymbol{R}_{a}\right\rangle , & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{a}^{\text{C}}\equiv \langle \boldsymbol{Q}_{a}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle ^{\text{C}}\equiv \left\langle {\displaystyle \frac{\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{Z_{a}eB}}\boldsymbol{\cdot }\boldsymbol{G}_{a}\right\rangle , & \displaystyle\end{eqnarray}$$

where we have introduced the friction force and energy-weighted friction force

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{R}_{a}\equiv \int m_{a}\boldsymbol{v}C[f_{a}]\,\text{d}^{3}v, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{G}_{a}\equiv \int {\displaystyle \frac{m_{a}v^{2}}{2}}m_{a}\boldsymbol{v}C[f_{a}]\,\text{d}^{3}v. & \displaystyle\end{eqnarray}$$

Here, $C[f_{a}]=\sum _{b}C_{ab}[f_{a},f_{b}]$ is the Fokker–Planck collision operator, accounting for the collisions of all species ‘ $b$ ’ with species ‘ $a$ ’; $f_{a}$ the distribution function of species ‘ $a$ ’, with mass $m_{a}$ and charge $Z_{a}e$ , with $e$ the elementary charge; the integral is over all velocities $\boldsymbol{v}$ . In a confined plasma, the distribution functions are close to a Maxwell–Boltzmann distribution $f_{a0}$ , such that $f_{a}=f_{a0}+f_{a1}$ , and $f_{a1}$ satisfies $f_{a1}/f_{a0}\ll 1$ . For later reference, we also define the classical conductive heat flux $q_{a}^{\text{C}}=Q_{a}^{\text{C}}-(5/2)T_{a}\unicode[STIX]{x1D6E4}_{a}^{\text{C}}$ , where $T_{a}$ is the temperature of species ‘ $a$ ’.

For a magnetized plasma, it is useful to separate out the dependence of the distribution function on the gyro-phase. Only the gyro-phase-dependent part of $f$ , which we denote by $\tilde{f}$ , contributes to $\boldsymbol{R}$ and $\boldsymbol{G}$ perpendicular to the magnetic field, and thus to the classical fluxes (3.1)–(3.2). For a magnetized plasma with an isotropic Maxwellian, it is well known that (Hazeltine Reference Hazeltine1973)

(3.5) $$\begin{eqnarray}\displaystyle \tilde{f}_{a1}=-\unicode[STIX]{x1D746}_{a}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{a0}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D746}_{a}=\boldsymbol{B}\times \boldsymbol{v}m_{a}/(Z_{a}eB^{2})$ is the gyro-radius vector.

With (3.5), we can readily evaluate the classical transport given by (3.1)–(3.4). Lately in stellarator research, the importance of flux-surface variation of the electrostatic potential has been recognized (García-Regaña et al. Reference García-Regaña, Beidler, Kleiber, Helander, Mollén, Alonso, Landreman, Maaßberg, Smith and Turkin2017); such effects can be incorporated into the classical transport by including the flux-surface varying part of the potential in the Maxwell–Boltzmann distribution $f_{0}$ (Hinton & Wong Reference Hinton and Wong1985)

(3.6) $$\begin{eqnarray}\displaystyle f_{0}=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D713})\left({\displaystyle \frac{m}{2\unicode[STIX]{x03C0}T}}\right)^{3/2}\exp \left(-{\displaystyle \frac{mv^{2}}{2T}}-{\displaystyle \frac{Ze\tilde{\unicode[STIX]{x1D6F7}}}{T}}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}$ is the electrostatic potential, $\tilde{\unicode[STIX]{x1D6F7}}=\unicode[STIX]{x1D6F7}-\langle \unicode[STIX]{x1D6F7}\rangle$ and we have introduced the pseudo-density

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}(\unicode[STIX]{x1D713})\equiv n\text{e}^{Ze\tilde{\unicode[STIX]{x1D6F7}}/T}, & & \displaystyle\end{eqnarray}$$

with $n$ the density. In terms of gradients of $\unicode[STIX]{x1D702}$ , $T$ and $\unicode[STIX]{x1D6F7}$ , the gradient in (3.5) thus becomes,

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}f_{0}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}{\displaystyle \frac{\unicode[STIX]{x2202}f_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}f_{0}\left[{\displaystyle \frac{\text{d}\ln \unicode[STIX]{x1D702}}{\text{d}\unicode[STIX]{x1D713}}}+{\displaystyle \frac{Z_{a}e}{T_{a}}}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}+{\displaystyle \frac{Z_{a}e\tilde{\unicode[STIX]{x1D6F7}}}{T_{a}}}{\displaystyle \frac{\text{d}\ln T_{a}}{\text{d}\unicode[STIX]{x1D713}}}+\left({\displaystyle \frac{m_{a}v^{2}}{2T_{a}}}-{\displaystyle \frac{3}{2}}\right){\displaystyle \frac{\text{d}\ln T_{a}}{\text{d}\unicode[STIX]{x1D713}}}\right]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

With this $\unicode[STIX]{x1D735}f_{0}$ , the resulting classical fluxes can be calculated using Braginskii matrices (as in, for example, Newton & Helander (Reference Newton and Helander2006)), resulting in

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{a}^{\text{C}} & = & \displaystyle {\displaystyle \frac{m_{a}}{Z_{a}e^{2}}}\mathop{\sum }_{b}{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{ab}n_{b}}}\left\{\left\langle n_{a}n_{b}{\displaystyle \frac{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}}{B^{2}}}\right\rangle M_{ab}^{00}\left({\displaystyle \frac{T_{a}}{Z_{a}}}{\displaystyle \frac{\text{d}\ln \unicode[STIX]{x1D702}_{a}}{\text{d}\unicode[STIX]{x1D713}}}-{\displaystyle \frac{T_{b}}{Z_{b}}}{\displaystyle \frac{\text{d}\ln \unicode[STIX]{x1D702}_{b}}{\text{d}\unicode[STIX]{x1D713}}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left\langle n_{a}n_{b}{\displaystyle \frac{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}}{B^{2}}}e\tilde{\unicode[STIX]{x1D6F7}}\right\rangle M_{ab}^{00}\left({\displaystyle \frac{\text{d}\ln T_{a}}{\text{d}\unicode[STIX]{x1D713}}}-{\displaystyle \frac{\text{d}\ln T_{b}}{\text{d}\unicode[STIX]{x1D713}}}\right)\nonumber\\ \displaystyle & & \displaystyle \left.+\left\langle n_{a}n_{b}{\displaystyle \frac{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}}{B^{2}}}\right\rangle \left[(M_{ab}^{00}-M_{ab}^{01}){\displaystyle \frac{T_{a}}{Z_{a}}}{\displaystyle \frac{\text{d}\ln T_{a}}{\text{d}\unicode[STIX]{x1D713}}}-\left(M_{ab}^{00}-{\displaystyle \frac{m_{a}T_{b}}{m_{b}T_{a}}}M_{ab}^{01}\right){\displaystyle \frac{T_{b}}{Z_{b}}}{\displaystyle \frac{\text{d}\ln T_{b}}{\text{d}\unicode[STIX]{x1D713}}}\right]\right\},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle q_{a}^{\text{C}} & = & \displaystyle -{\displaystyle \frac{T_{a}m_{a}}{Z_{a}e^{2}}}\mathop{\sum }_{b}{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{ab}n_{b}}}\left\{\left\langle n_{a}n_{b}{\displaystyle \frac{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}}{B^{2}}}\right\rangle M_{ab}^{01}\left({\displaystyle \frac{T_{a}}{Z_{a}}}{\displaystyle \frac{\text{d}\ln \unicode[STIX]{x1D702}_{a}}{\text{d}\unicode[STIX]{x1D713}}}-{\displaystyle \frac{T_{b}}{Z_{b}}}{\displaystyle \frac{\text{d}\ln \unicode[STIX]{x1D702}_{b}}{\text{d}\unicode[STIX]{x1D713}}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left\langle n_{a}n_{b}{\displaystyle \frac{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}}{B^{2}}}e\tilde{\unicode[STIX]{x1D6F7}}\right\rangle M_{ab}^{01}\left({\displaystyle \frac{\text{d}\ln T_{a}}{\text{d}\unicode[STIX]{x1D713}}}-{\displaystyle \frac{\text{d}\ln T_{b}}{\text{d}\unicode[STIX]{x1D713}}}\right)\nonumber\\ \displaystyle & & \displaystyle \left.+\left\langle n_{a}n_{b}{\displaystyle \frac{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}}{B^{2}}}\right\rangle \left[(M_{ab}^{01}-M_{ab}^{11}){\displaystyle \frac{T_{a}}{Z_{a}}}{\displaystyle \frac{\text{d}\ln T_{a}}{\text{d}\unicode[STIX]{x1D713}}}-(M_{ab}^{01}+N_{ab}^{11}){\displaystyle \frac{T_{b}}{Z_{b}}}{\displaystyle \frac{\text{d}\ln T_{b}}{\text{d}\unicode[STIX]{x1D713}}}\right]\right\},\end{eqnarray}$$

where $M_{ab}^{jk}$ are the Braginskii matrix elements (Braginskii Reference Braginskii1958), defined in appendix A, using the same notation as Helander & Sigmar (Reference Helander and Sigmar2005); the collision time $\unicode[STIX]{x1D70F}_{ab}$ is defined as

(3.11) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{ab}n_{b}}}\equiv {\displaystyle \frac{\sqrt{2}Z_{a}^{2}Z_{b}^{2}e^{4}\ln \unicode[STIX]{x1D6EC}}{12\unicode[STIX]{x03C0}^{3/2}\unicode[STIX]{x1D716}_{0}^{2}m_{a}^{1/2}T_{a}^{3/2}}}, & & \displaystyle\end{eqnarray}$$

where $\text{ln}\,\unicode[STIX]{x1D6EC}$ is the Coulomb logarithm, and $\unicode[STIX]{x1D716}_{0}$ the vacuum permittivity. These expressions are valid for all collisionalities. In (3.9) and (3.10), the effect of $\tilde{\unicode[STIX]{x1D6F7}}$ is to induce a weighting over the flux surface due to the flux-surface variation of $n_{a}$ and its radial gradient. Note that the radial electric field (from $\langle \unicode[STIX]{x1D6F7}\rangle$ and $\tilde{\unicode[STIX]{x1D6F7}}$ ) does not contribute in the above expression, even when $\text{d}\ln \unicode[STIX]{x1D702}/\text{d}\unicode[STIX]{x1D713}$ is expressed in terms of (3.7).

In (3.9)–(3.10), the $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}$ factors correspond to the $j_{\bot }$ factor in (2.1), while the $j_{\Vert }$ factor in (2.1) arises due to the neoclassical transport (Braun & Helander Reference Braun and Helander2010; Helander et al. Reference Helander, Newton, Mollén and Smith2017). In the following section, we will evaluate the above expression for example magnetic configurations.

4 Comparison to neoclassical calculations

In this section, we will compare the classical transport from (3.9) to the neoclassical transport calculated with the drift-kinetic solver Sfincs. Unlike analytical calculations of the neoclassical transport (Buller et al. Reference Buller, Smith, Helander, Mollén, Newton and Pusztai2018; Calvo et al. Reference Calvo, Parra, Velasco, Alonso and Na2018), this procedure is not limited to a specific collisionality regime, which will let us assess the importance of classical transport for any collisionality.

For this study, we will look at two stellarator configurations, where the neoclassical transport coefficients have been calculated across a wide range of collisionalities. Specifically, we will look at a simulated W7-X standard configuration case at the radial location $r_{N}=0.88$ , with $T=1~\text{keV}$ and impurity parameters $Z=6$ , $Z_{\text{eff}}=2.0$ , studied by Mollén et al. (Reference Mollén, Landreman, Smith, Braun and Helander2015). The normalized radius is defined as $r_{N}=\sqrt{\unicode[STIX]{x1D713}_{\text{t}}/\unicode[STIX]{x1D713}_{\text{t},\text{LCFS}}}$ , with $\unicode[STIX]{x1D713}_{t}$ the toroidal flux and $\unicode[STIX]{x1D713}_{\text{t},\text{LCFS}}$ its value at the last-closed flux surface. Since W7-X has been optimized for a low parallel current, and the standard configuration has low neoclassical transport compared to other configurations, we here expect the classical transport to dominate at high collisionality, as indicated at the end of § 2. In addition, we will look at a scenario based on an impurity hole plasma ( $\#113208$ , $t=4.64~\text{s}$ , $r_{N}=0.6$ , $T=3.2~\text{keV}$ ) of the Large Helical Device (LHD), where we replaced the mixture of helium and carbon impurities with purely carbon ( $Z_{\text{eff}}=3.44$ ) for the sake of comparison. This magnetic configuration has been investigated in several studies, using both neoclassical (Velasco et al. Reference Velasco, Calvo, Satake, Alonso, Nunami, Yokoyama, Sato, Estrada, Fontdecaba and Liniers2017; Mollén et al. Reference Mollén, Landreman, Smith, García-Regaña and Nunami2018) and turbulence codes (Nunami et al. Reference Nunami, Sugama, Velasco, Yokoyama, Sato, Nakata and Satake2016).

Effects of $\tilde{\unicode[STIX]{x1D6F7}}$ and the radial electric field are not included in this demonstration (they are zero in the simulations), as this would make the drift-kinetic equation nonlinear, and add the complexity of finding the ambipolar electric field at each step. These effects are not expected to strongly affect the classical transport, which is independent of the radial electric field, and typically not as sensitive to $\tilde{\unicode[STIX]{x1D6F7}}$ as the neoclassical transport (Buller et al. Reference Buller, Smith, Helander, Mollén, Newton and Pusztai2018). The neoclassical transport can be both enhanced or reduced by these effects, which thus would affect the relative importance of classical transport. These effects will be touched upon in § 5. As $\tilde{\unicode[STIX]{x1D6F7}}=0$ in this section, the density is a flux function, and $\unicode[STIX]{x1D702}_{a}=n_{a}$ .

We scan the collisionality by artificially scaling the collision frequency. For each point in the collisionality scan, we calculate the neoclassical and classical transport coefficients of the hydrogen bulk ion and the carbon impurity. The transport coefficients for the (neo)classical fluxes are defined such that

(4.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{a}^{\text{(N)C}}=-n_{a}\left(D_{a,ni}^{\text{(N)C}}{\displaystyle \frac{\text{d}\ln n_{i}}{\text{d}\unicode[STIX]{x1D713}}}+D_{a,nz}^{\text{(N)C}}{\displaystyle \frac{\text{d}\ln n_{z}}{\text{d}\unicode[STIX]{x1D713}}}+D_{a,T}^{\text{(N)C}}{\displaystyle \frac{\text{d}\ln T}{\text{d}\unicode[STIX]{x1D713}}}\right), & & \displaystyle\end{eqnarray}$$

where $a=i,z$ for ions and impurities. We have neglected the effects of electron collisions on the ion fluxes due to the small electron-to-ion mass ratio, and assumed that the bulk and impurity ions have the same temperature $T\equiv T_{i}=T_{z}$ .

The results of the collisionality scan are shown in figure 1, with the collisionality defined as

(4.2) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D708}}_{ab}={\displaystyle \frac{G+\unicode[STIX]{x1D704}I}{B_{00}\sqrt{2T_{a}/m_{a}}}}{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{ab}}}, & & \displaystyle\end{eqnarray}$$

where $B_{00}$ , $G$ and $I$ are related to the Boozer representation of the magnetic field (see, for example, Mollén et al. (Reference Mollén, Landreman, Smith, García-Regaña and Nunami2018)) and $\unicode[STIX]{x1D704}$ is the rotational transform. As seen in the left panels of figure 1, the impurity transport coefficients in the W7-X geometry are dominantly classical already for $\hat{\unicode[STIX]{x1D708}}_{CC}\gtrsim 1$ . The cross-species contributions are dominantly classical at even lower collisionality, $\hat{\unicode[STIX]{x1D708}}_{CC}\gtrsim 0.1$ , for both the bulk and the impurity ions. On the other hand, in LHD – which has not been optimized for low $|j_{\Vert }|/|\boldsymbol{j}_{\bot }|$ – the classical transport for both species at most collisionalities is at least an order of magnitude smaller than the neoclassical transport. An exception to this is the $D_{a,T}$ coefficient, where the classical transport becomes comparable to or greater than the neoclassical transport at very high collisionalities ( $\hat{\unicode[STIX]{x1D708}}_{CC}\gtrsim 100$ ). Another exception occurs in the collisionality range $\unicode[STIX]{x1D708}_{CC}\sim [0.1,1]$ , where the cross-species neoclassical $D_{z,ni}$ and $D_{i,nz}$ coefficients transition between different signs.

Figure 1. The neoclassical (——) and classical (- - -) transport coefficients as defined in (4.1), plotted against the impurity–impurity collisionality. (a) W7-X standard case. (b) LHD impurity–hole case. The classical coefficients were calculated using (3.9), while the neoclassical coefficients were calculated using Sfincs. Note the symmetric logarithmic scale of the $y$ -axis; the shaded region has a linear $y$ -axis scale.