2.1. Distribution function and jump conditions

The drift kinetic equation of the distribution function $f$ can be written in the form

(2.2) \begin{equation} \dot {\theta }\frac {\partial {f}}{\partial {\theta }}=C[f,f]+\unicode{x1D6F4} , \end{equation}

where $C[f,f]$ is the collision operator and $\unicode{x1D6F4}$ is a source term. The derivative with respect to poloidal angle $\theta$ is performed holding the magnetic moment $\mu \equiv v_\perp ^2/(2B)$ and the fixed- $\theta$ variables $v_{\parallel f}\equiv v_\parallel (\theta_{\!f})$ and $\psi_{\!f}\equiv \psi (\theta_{\!f})$ fixed, where $v_\perp$ is the perpendicular speed and $\theta_{\!f}$ is a reference angle (as in Trinczek et al. Reference Trinczek, Parra, Catto, Calvo and Landreman2023). To the order required, $\dot \theta =(v_\parallel +u)/qR$ with $f=f(\psi_{\!f},\theta ,v_{\parallel f},\mu )$ . The fixed- $\theta$ variables for electrons are derived and explained in detail in Appendix A. The source for electrons is assumed to be of order

(2.3) \begin{equation} \unicode{x1D6F4} _e\sim \sqrt {\epsilon }\delta ^2 \nu _{ee} f_e, \end{equation}

where $f_e$ is the electron distribution function.

In the banana regime, collisionality is small, and trapped particles complete their orbits many times before colliding. In this low collisionality limit, the drift kinetic equation to lowest order in $\nu _\ast$ is

(2.4) \begin{equation} \dot \theta \frac {\partial {f_e}}{\partial {\theta }}=0. \end{equation}

The distribution function in fixed- $\theta$ variables does not depend on poloidal angle.

The transit average of (2.2) eliminates the poloidal derivative and gives

(2.5) \begin{equation} \langle C_e\rangle _\tau =-\langle \unicode{x1D6F4} _e\rangle _\tau , \end{equation}

where $C_e$ is the collision operator capturing electron–electron and electron–ion collisions. The transit average of a function $\mathcal{F}$ is different for trapped and passing particles. For trapped particles, it is defined as

(2.6) \begin{align} \langle \mathcal{F}\rangle _\tau \equiv \frac {1}{\tau } \int _{-\theta _b}^{\theta _b}\frac {\mathrm{d}\theta }{\lvert \dot {\theta }\rvert }\mathcal{F}(\sigma =+1)+\frac {1}{\tau }\int _{-\theta _b}^{\theta _b}\frac {\mathrm{d}\theta }{\lvert \dot {\theta }\rvert }\mathcal{F}(\sigma =-1), \end{align}

where

(2.7) \begin{equation} \tau \equiv 2 \int _{-\theta _b}^{\theta _b}\frac {\mathrm{d}\theta }{\lvert \dot {\theta }\rvert }, \end{equation}

with $\sigma = v_\parallel /\lvert v_\parallel \rvert$ and where $\theta _b$ is the location of the bounce point, determined to lowest order in $\delta$ by $v_\parallel =0$ . It is clear from the definition (2.6) that the transit average of an odd function in $\sigma$ vanishes. The transit average for passing particles is

(2.8) \begin{align} \langle \mathcal{F}\rangle _\tau \equiv \frac {1}{\tau } \int _{-\pi }^{\pi }\frac {\mathrm{d}\theta }{\lvert \dot {\theta }\rvert }\mathcal{F}, \end{align}

where

(2.9) \begin{align} \tau \equiv \int _{-\pi }^{\pi }\frac {\mathrm{d}\theta }{\lvert \dot {\theta }\rvert }. \end{align}

The jump $\varDelta \mathcal{F}$ of a function $\mathcal{F}$ will be needed later and is defined as

(2.10) \begin{equation} \varDelta \mathcal{F}=\mathcal{F}^p(v_\parallel \rightarrow 0^+)-\mathcal{F}^p(v_\parallel \rightarrow 0^-)=\mathcal{F}^{bp}(v_\parallel\rightarrow \infty )-\mathcal{F}^{bp}(v_\parallel\rightarrow - \infty ), \end{equation}

where $\mathcal{F}^p$ and $\mathcal{F}^{bp}$ are defined in the passing and barely passing region, respectively.

The source term is small by $\delta ^2$ according to the ordering in (2.3). Thus, it follows from (2.5) that $\langle C_e\rangle _\tau =0$ and hence the distribution function is an isotropic Maxwellian to lowest order in $\delta$ . The electron–ion collision operator forces the electron Maxwellian to have the same flow as the ions. The ion flow is smaller than $v_{te}$ by $\delta$ , and hence the Maxwellian is isotropic to lowest order in $\delta$ . Equation (2.4) also imposes that $f_e$ be independent of $\theta$ when written in terms of $\psi_{\!f}$ , $v_{\parallel f}$ and $\mu$ . We choose

(2.11) \begin{align} f_e\simeq f_{Mef}=n_{e0}(\psi_{\!f})\left (\frac {m_e}{2\pi T_{e0}(\psi_{\!f})}\right )^{3/2}\exp\! \bigg \lbrace \!-\frac {m_e v_{\parallel f}^2}{2T_{e0}(\psi_{\!f})}-\frac {m_e\mu B(\theta_{\!f})}{T_{e0}(\psi_{\!f})} + \frac {e\phi _1(\psi_{\!f},\theta_{\!f})}{T_{e0}(\psi_{\!f})}\bigg \rbrace .\nonumber\\[-3pt] \end{align}

Here, $\theta_{\!f}$ is a reference angle further discussed in Appendix A. The electron density $n_{e0}$ and the temperature $T_{e0}$ in $f_{Mef}$ are only the lowest-order pieces of the full electron density and temperature, defined by

(2.12) \begin{align} n_e\equiv \int \mathrm{d}^3v\: f_e, \quad \frac {3}{2}n_eT_e\equiv \int \mathrm{d}^3v\: \frac {m_ev^2}{2}f_e. \end{align}

Density and temperature are hence flux functions to lowest order and can be written as

(2.13) \begin{equation} n_e(\psi ,\theta )=n_{e0}(\psi )+n_{e1}(\psi ,\theta ), \quad T_e(\psi ,\theta )=T_{e0}(\psi )+T_{e1}(\psi ,\theta ), \end{equation}

where $n_{e1}/n_{e0}\sim \epsilon$ because of the Maxwell–Boltzmann response explained below, and $T_{e1}/T_{e0}\sim \delta ^2$ because the first-order correction to the Maxwellian will be shown to be odd in $v_\parallel$ and hence does not contribute to temperature. Similarly, the electric potential $\varPhi =\phi (\psi )+\phi _1(\psi ,\theta )$ has a flux function piece $\phi$ with $e\phi /T\sim 1$ and a smaller piece $\phi _1$ that depends on poloidal angle. We showed in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023) that $\phi _1/\phi \sim \epsilon$ and, for circular flux surfaces, $\phi _1(\psi ,\theta )=\phi _c(\psi )\cos \theta$ . The poloidally varying part of the electric potential can cause electrostatic trapping and de-trapping, thus modifying the trapping condition and number of trapped particles in the system.

Using (A.2) and $\psi -\psi_{\!f}\sim \sqrt {\epsilon }\delta \psi \ll \psi \sim RB_p\rho _p$ , where $B_p$ is the poloidal magnetic field, the distribution function can be written as

(2.14) \begin{equation} f_e=f_{Mef}(v_{\parallel f}, \psi_{\!f}, \mu )+f_{e1f}(v_{\parallel f}, \psi_{\!f}, \mu )=f_{Me}(v_\parallel , \psi , \mu ,\theta )+f_{e1}(v_\parallel , \psi , \mu ,\theta ), \end{equation}

where

(2.15) \begin{equation} f_{Me}=n_{e0}(\psi )\left (\frac {m_e}{2\pi T_{e0}(\psi )}\right )^{3/2}\exp \bigg \lbrace -\frac {m_e v_\parallel ^2}{2T_{e0}(\psi )}-\frac {m_e\mu B(\theta )}{T_{e0}(\psi )} + \frac {e\phi _1(\psi ,\theta )}{T_{e0}(\psi )}\bigg \rbrace . \end{equation}

To lowest order, fixed- $\theta$ and particle variables are equivalent and thus the lowest-order distribution function is a Maxwellian in both the fixed- $\theta$ variables and the particle variables, except for the fact that we have made it explicit in the particle variables that the density $n_e=n_{e0}(\psi )\exp \lbrace e\phi _1(\psi ,\theta )/T_{e0}(\psi )\rbrace$ is not constant within flux surfaces. The relation between (2.11) and (2.15) is given in Appendix B in (B.3). The correction to the Maxwellian $f_{e1}$ will be shown to have two parts. One part is of order $\delta f_{Me}$ and one part is of order $\sqrt {\epsilon }\delta f_{Me}$ . An expression for $f_{e1}$ is derived in what follows.

The drift kinetic equation for the electrons is first expanded in $\delta$ and then in $\epsilon$ . We start by expanding the collision operator. Collisions of electrons with other electrons occur as frequently as electron–ion collisions. The collision operator for electrons has to account for both electron–electron and electronion collisions

(2.16) \begin{equation} C_e\equiv C_{ee}[f_e,f_e]+C_{ei}[f_e,f_i]\simeq C^{(l)}[f_{e1}]+\mathcal{L}\left [f_{e1}-\frac {m_e v_\parallel V_\parallel }{T_{e0}}f_{Me}\right ], \end{equation}

with $f_i$ the ion distribution function. The nonlinear terms of the collision operator are small to the order of interest and can be dropped. The self-collisions of electrons are captured by $ C^{(l)}[f_{e1}]$ , which for electrons is

(2.17) \begin{equation} C^{(l)}[f_{e1}]=\boldsymbol{\nabla }_v\boldsymbol{\cdot }\left [f_{Me} \boldsymbol{\mathsf{M}_{ee}} \boldsymbol{\cdot }\boldsymbol{\nabla }_v\left (\frac {f_{e1}}{f_{Me}}\right )-\lambda _e f_{Me}\int \mathrm{d}^3v'\: f'_{Me} \boldsymbol{\nabla }_\omega \boldsymbol{\nabla }_\omega \omega \boldsymbol{\cdot } \boldsymbol{\nabla }_{v'}\left (\frac {f_{e1}'}{f'_{M_e}}\right )\right ], \end{equation}

where

(2.18) \begin{equation} \boldsymbol{\mathsf{M}_{ee}}\equiv \lambda _e\int \mathrm{d}^3v'\: f'_{Me} \boldsymbol{\nabla }_\omega \boldsymbol{\nabla }_\omega \omega =\lambda _e\int \mathrm{d}^3v'\: f'_{Me} \frac {\omega ^2\mathsf{I}-\boldsymbol{\omega }\boldsymbol{\omega }}{\omega ^3}, \end{equation}

with $\boldsymbol{\omega }\equiv \boldsymbol{v}-\boldsymbol{v'}$ , $\lambda _e=2\pi e^4 \log \unicode{x1D6EC} /m_e^2$ and $\boldsymbol{v}$ is the particle velocity. Collisions of electrons and ions are approximately described by a Lorentz collision operator

(2.19) \begin{equation} \mathcal{L}\left [f_{e1}-\frac {m_ev_\parallel V_\parallel }{T_{e0}}f_{Me}\right ]= \boldsymbol{\nabla }_v\boldsymbol{\cdot }\left [f_{Me}\boldsymbol{\mathsf{M}_{ei}}\boldsymbol{\cdot }\boldsymbol{\nabla }_v \left (\frac {f_{e1}}{f_{Me}}-\frac {m_e v_\parallel V_\parallel }{T_{e0}}\right )\right ], \end{equation}

where

(2.20) \begin{equation} \boldsymbol{\mathsf{M}_{ei}}\equiv Z^2\lambda _e n_i \frac {v^2 \mathsf{I}-\boldsymbol{v}\boldsymbol{v}}{v^3}. \end{equation}

Here, $Z$ is the ion charge number and the ion mean parallel flow $V_\parallel$ is defined as

(2.21) \begin{equation} n_i V_\parallel \equiv \int \mathrm{d}^3v\: v_\parallel f_i, \end{equation}

where $n_i$ is the ion density. Just like density and temperature, the mean parallel flow has a lower-order flux surface piece and a higher-order piece that depends on the poloidal angle, $V_\parallel =V_{\parallel 0}(\psi )+V_{\parallel 1}(\psi ,\theta )$ . We separate the first-order electron distribution function into two pieces,

(2.22) \begin{equation} f_{e1}=g_e+\frac {m_e v_\parallel V_\parallel }{T_{e0}}f_{Me}. \end{equation}

The first piece will be shown to be of order $\sqrt {\epsilon }\delta f_{Me}$ and the second piece is of order $\delta f_{Me}$ . With this definition, $C^{(l)}[f_{e1}]=C^{(l)}[g_e]$ because $C^{(l)}[v_\parallel f_{Me}]=0$ . The combination of (2.17) and (2.19) gives (2.16) with the final collision operator treating both electron–electron and electron–ion collisions

(2.23) \begin{equation} C_e= \boldsymbol{\nabla }_v\boldsymbol{\cdot }\left [f_{M_e} \boldsymbol{\mathsf{M}_{e}}\boldsymbol{\cdot }\boldsymbol{\nabla }_v\left (\frac {g_e}{f_{M_e}}\right )-\lambda _e f_{M_e}\int \mathrm{d}^3v'\: f'_{M_e} \boldsymbol{\nabla }_\omega \boldsymbol{\nabla }_\omega \omega \boldsymbol{\cdot } \boldsymbol{\nabla }_{v'}\left (\frac {g_e'}{f'_{M_e}}\right )\right ]. \end{equation}

Here,

(2.24) \begin{equation} \boldsymbol{\mathsf{M}_{e}}\equiv \boldsymbol{\mathsf{M}_{ee}}+ \boldsymbol{\mathsf{M}_{ei}}. \end{equation}

At this point, we can perform the same large aspect ratio expansion as for the ion calculation. We need to solve (2.5). In the trapped–barely passing region, $v_\parallel \sim \sqrt {\epsilon }v_{te}$ and thus the derivative of $g^{t,bp}$ with respect to parallel velocity is larger than other velocity derivatives by $\sim 1/\sqrt {\epsilon }$ . Furthermore, $\nabla _v v_{\parallel f}\simeq v_\parallel /v_{\parallel f} \hat{\boldsymbol{b}}$ , so we find that to lowest order, in the trapped–barely passing region, (2.23) can be approximated by

(2.25) \begin{equation} C_e\simeq \frac {v_\parallel }{v_{\parallel f}}\frac {\partial }{\partial {v_{\parallel f}}}\left [\mathsf{M}_{\parallel e}\frac {v_\parallel }{v_{\parallel f}}\frac {\partial {g_{e0}^{t,bp}}}{\partial {v_{\parallel f}}}\right ], \end{equation}

where

(2.26) \begin{equation} \mathsf{M}_{\parallel e}\equiv \mathsf{M}_{\parallel ee}+\mathsf{M}_{\parallel ei}, \end{equation}

(2.27) \begin{equation} \mathsf{M}_{\parallel ee}\equiv \hat{\boldsymbol{b}}\boldsymbol{\cdot } \boldsymbol{\mathsf{M}_{ee}}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\simeq \frac {3}{2}\sqrt {\frac {\pi }{2}}\left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)\right ] \frac {T_e}{m_e}\frac {\nu _{ee}}{x_e} , \end{equation}

(2.28) \begin{equation} \mathsf{M}_{\parallel ei}\equiv \hat{\boldsymbol{b}}\boldsymbol{\cdot } \boldsymbol{\mathsf{M}_{ei}}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\simeq Z\frac {3}{2}\sqrt {\frac {\pi }{2}}\frac {T_e}{m_e}\frac {\nu _{ee}}{x_e}, \end{equation}

and $x_e^2=v^2/v^2_{te}\simeq 2\mu B/v_{te}^2$ . In the derivation of $\mathsf{M}_{\parallel ee}$ and $\mathsf{M}_{\parallel ei}$ , we have used that $u+V_{\parallel }\sim v_{ti}\ll v_{te}$ for electrons. The function $\unicode{x1D6E9} (x)=(2/\sqrt {\pi })\int _0^x\exp (-t^2)\mathrm{d}t$ is the error function and $\unicode{x1D6F9} (x)=(\unicode{x1D6E9} -x\unicode{x1D6E9} ')/(2x^2)$ is the Chandrasekhar function. We take a transit average of (2.25) and employ (2.3) and (2.5) to find

(2.29) \begin{equation} \Bigg \langle \frac {\partial }{\partial {v_{\parallel f}}}\left [ \tau {v_{\parallel f}}\left (\frac {v_\parallel }{v_{\parallel f}}\right )^2 \mathsf{M}_{\parallel e}\frac {\partial {g_{e0}^{t,bp}}}{\partial {v_{\parallel f}}}\right ]\Bigg \rangle _\tau =0. \end{equation}

The solution to (2.29) was calculated for ions by Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023). The derivation of the electron distribution function is similar and is presented in Appendix B. The results are

(2.30) \begin{align} \frac {\partial {g_{0e}^{t}}}{\partial {v_{\parallel f}}}=-\frac {v_{\parallel f}}{v_\parallel }\alpha _{0e}\quad \text{and}\quad \frac {\partial {g_{0e}^{bp}}}{\partial {v_{\parallel f}}}=\left (\frac {v_{\parallel f}}{\langle v_\parallel \rangle _\psi }-\frac {v_{\parallel f}}{v_\parallel }\right )\alpha _{0e}, \end{align}

where

(2.31) \begin{equation} \alpha _{0e} \equiv \frac {I}{\unicode{x1D6FA} _{e}}\left [\frac {\partial }{\partial {\psi }}\ln p_{e} +\left (\frac {m_e\mu B}{T_{e}}-\frac {5}{2}\right )\frac {\partial }{\partial {\psi }}\ln T_{e}+\frac {m_e(u+V_{\parallel })}{T_{e}}\frac {\unicode{x1D6FA} _{e} }{I}\right ]f_{Mef}. \end{equation}

The superscripts $t$ and $bp$ denote the distribution functions in the trapped and the barely passing regions, respectively. The electron pressure is $p_e=n_e T_e$ . We can set $T_e\simeq T_{e0}$ because the difference is small in $\delta ^2$ , and $n_e\simeq n_{e0}$ because the difference is small in $\epsilon$ . To simplify our notation, we dropped the distinction between the fixed- $\theta$ variables and ( $\psi$ , $v_\parallel$ , $\mu$ ), and the difference between quantities with and without the subscripts $f$ and $0$ where possible, as these differences are small. Note that the electron Larmor frequency $\unicode{x1D6FA} _{e}\equiv -eB/mc$ is by definition negative and the ion Larmor frequency $\unicode{x1D6FA} _i\equiv ZeB/mc$ is by definition positive. Integrating the expression for the electron distribution function over the trapped and barely passing region gives the height of the jump of the freely passing distribution function. The integration was carried out by Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023) and gives

(2.32) \begin{align} \varDelta g_e \equiv \bigg \langle \int _{V_{t,bp}}\mathrm{d}{v_{\parallel f}} \: \frac {\partial {g^{t,bp}_{e0}}}{\partial {v_{\parallel f}}}\bigg \rangle _\psi & = \int _{V_{t,bp}}\mathrm{d}{v_{\parallel f}} \: \frac {{v_{\parallel f}}\tau }{2\pi qR}\bigg \langle \frac {v_\parallel }{v_{\parallel f}}\frac {\partial {g^{t,bp}_{e0}}}{\partial {v_{\parallel f}}}\bigg \rangle _\tau \\ & = -2.758 \sqrt {\bigg \lvert {\left (\mu B \frac {r}{R}+\frac {e\phi _{c}}{m_e}\right )\bigg \rvert }}\alpha _{0e},\nonumber \end{align}

where $\langle \ldots \rangle _\psi \equiv 1/(2\pi )\int _{-\pi }^\pi \mathrm{d}\theta \:(\ldots )$ is the flux surface average. The symbol $V_{t,bp}$ denotes the trapped–barely passing region defined by $v_{\parallel f}\sim \sqrt {\epsilon }v_{te}$ . The modification of the trapping condition by the poloidal variation of the electric potential results in the appearance of $\phi _{c}$ in (2.32). The contributions from particles trapped on the low and high field sides were combined by choosing first $\theta_{\!f}=0$ and then $\theta_{\!f}=\pi$ to get to the result in (2.32).

2.2. Neoclassical electron particle flux

Now that the jump condition (2.32) is known, we can proceed to calculate the transport relations. The electron particle flux $\varGamma _e$ is defined by the particle conservation equation

(2.33) \begin{equation} \frac {\partial {\varGamma _e}}{\partial {\psi _{\!f}}}=\int \mathrm{d}^3 v_{\!f}\: \langle \unicode{x1D6F4} _e\rangle _\tau . \end{equation}

The integration over $\mathrm{d}^3v_{\!f}$ is an integration over velocity space in the fixed- $\theta$ variables, $\mathrm{d}^3v_{\!f}\equiv 2\pi B_{\!f}\mathrm{d}\mu \mathrm{d}v_{\parallel f}$ . Following the exact same steps as for the ion particle flux calculation, we integrate over the drift kinetic equation

(2.34) \begin{equation} -\int \mathrm{d}^3v_{\!f} \:\langle C_e\rangle _\tau =\int \mathrm{d}^3v_{\!f} \: \langle \unicode{x1D6F4} _e\rangle _\tau . \end{equation}

For the integration, it is useful to express the divergence in the collision operator in fixed- $\theta$ variables,

(2.35) \begin{align} \langle C_e\rangle _\tau & = \frac {1}{{v_{\parallel f}}\tau }\frac {\partial }{\partial {v_{\parallel f}}}\left[f_{Me}{v_{\parallel f}}\tau \left \langle \boldsymbol{\nabla }_v {v_{\parallel f}}{\boldsymbol \cdot }\left[ \boldsymbol{\mathsf{M}_{e}}{\boldsymbol \cdot } \boldsymbol{\nabla }_v\left (\frac {g_e}{f_{Me}}\right )\right.\right.\right.\nonumber\\& \qquad\qquad\qquad\left.\left.\left.-\lambda _e\int \mathrm{d}^3v'\: f'_{M_e} \boldsymbol{\nabla }_\omega \boldsymbol{\nabla }_\omega \omega {\boldsymbol \cdot } \boldsymbol{\nabla }_{v'}\left (\frac {g_e'}{f'_{M_e}}\right )\right]\right \rangle _\tau \right]\nonumber \\ & \quad +\frac {1}{{v_{\parallel f}}\tau }\frac {\partial }{\partial {\mu }}\left[f_{Me}{v_{\parallel f}}\tau \left \langle \boldsymbol{\nabla }_v \mu {\boldsymbol \cdot }\left[\boldsymbol{\mathsf{M}_{e}}{\boldsymbol \cdot } \boldsymbol{\nabla }_v\left (\frac {g_e}{f_{Me}}\right )\right.\right.\right.\nonumber\\& \qquad\qquad\qquad \left.\left.\left. -\lambda _e\int \mathrm{d}^3v'\: f'_{M_e} \boldsymbol{\nabla }_\omega \boldsymbol{\nabla }_\omega \omega {\boldsymbol \cdot } \boldsymbol{\nabla }_{v'}\left (\frac {g_e'}{f'_{M_e}}\right )\right ]\right\rangle _\tau \right]\nonumber \\ & \quad +\frac {1}{{v_{\parallel f}}\tau }\frac {\partial }{\partial {\psi _{\!f}}}\left[f_{Me}{v_{\parallel f}}\tau \left \langle \boldsymbol{\nabla }_v \psi_{\!f}{\boldsymbol \cdot }\left[\boldsymbol{\mathsf{M}_{e}} {\boldsymbol \cdot } \boldsymbol{\nabla }_v\left (\frac {g_e}{f_{Me}}\right )\right.\right.\right.\nonumber\\ & \qquad\qquad\qquad\left.\left.\left.-\lambda _e\int \mathrm{d}^3v'\: f'_{M_e} \boldsymbol{\nabla }_\omega \boldsymbol{\nabla }_\omega \omega {\boldsymbol \cdot } \boldsymbol{\nabla }_{v'}\left (\frac {g_e'}{f'_{M_e}}\right )\right]\right\rangle _\tau \right]. \end{align}

The integration over the collision operator can be divided into an integration over the freely passing region and the trapped–barely passing region. Multiplying by $v_{\parallel f}\tau /2\pi qR$ and integrating over the freely passing region yields, to lowest order in $\delta$ ,

(2.36) \begin{align} -\int _{V_{p}}\mathrm{d}^3v_{\!f}\: \frac {v_{\parallel f} \tau }{2\pi qR}\langle C_e\rangle _\tau \simeq \int \mathrm{d}\mu \: 2\pi B_{\!f} \varDelta \left [f_{M_e} \hat{\boldsymbol{b}}\boldsymbol{\cdot } \boldsymbol{\mathsf{M}_{e}} \boldsymbol{\cdot } \boldsymbol{\nabla }_v \left (\frac {g_e^p}{f_{M_e}}\right ) \right ]+\textit {O}\big(\delta ^2 \epsilon ^{3/2}n_e \nu _e\big), \end{align}

where we used (2.10). Note that in the freely passing region $v_{\parallel f}\tau \simeq 2\pi qR$ . The diffusion part of the collision operator contains the jump in the derivative of $g_e^p$ which needs to be kept. The term proportional to $\partial /\partial \mu$ in (2.35) vanishes when integrating over the freely passing region. The term proportional to $\partial /\partial \psi_{\!f}$ is of order $\delta ^2\epsilon ^{3/2}n_e\nu _e$ because $\nabla _v \psi_{\!f}\sim \epsilon \psi_{\!f}/v_{te}$ and has been dropped.

There is a region of rapid $v_{\parallel f}$ variation for the trapped–barely passing particles. The integration gives to lowest order in $\delta$ and $\epsilon$

(2.37) \begin{align} -\int _{V_{t,bp}}\mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}}\tau }{2\pi qR}\langle C_e\rangle _\tau & \simeq -\int \mathrm{d}\mu \: 2\pi B_{\!f} \varDelta \left [f_{M_e} {\hat{\boldsymbol{b}}}\boldsymbol \cdot {\boldsymbol{\mathsf{M}_{e}}} \boldsymbol \cdot \boldsymbol \nabla _v \left (\frac {g_e^p}{f_{M_e}}\right ) \right ]\nonumber\\& -\frac {\partial }{\partial {\psi _{\!f}}}\int _{V_{t,bp}}\mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}}\tau }{2\pi qR} \frac {I}{\unicode{x1D6FA} _{e} }\mathsf{M}_{\parallel e}\bigg \langle \left (\frac {v_\parallel }{v_{\parallel f}}-1\right ) \frac {v_\parallel }{v_{\parallel f}}\frac {\partial {g^{t,bp}_{e0}}}{\partial {v_{\parallel f}}}\bigg \rangle _\tau . \end{align}

In the second term, we only kept terms to order $\delta ^2 \sqrt {\epsilon }n_e\nu _e$ . The first term is larger than the second term by order $\delta ^2$ . We keep the second term in the trapped–barely passing region because the jump terms cancel when we combine (2.36) and (2.37)

(2.38) \begin{align} -\int \mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}} \tau }{2\pi qR}\langle C_e\rangle _\tau & =-\int _{V_{t,bp}}\mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}} \tau }{2\pi qR}\langle C_e\rangle _\tau -\int _{V_{p}}\mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}} \tau }{2\pi qR}\ \langle C_e\rangle _\tau \nonumber \\& \simeq -\frac {\partial }{\partial {\psi _{\!f}}}\int _{V_{t,bp}}\mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}}\tau }{2\pi qR} \frac {I}{\unicode{x1D6FA} _{e} }\mathsf{M}_{\parallel e}\bigg \langle \left (\frac {v_\parallel }{v_{\parallel f}}-1\right ) \frac {v_\parallel }{v_{\parallel f}}\frac {\partial {g^{t,bp}_{e0}}}{\partial {v_{\parallel f}}}\bigg \rangle _\tau . \end{align}

The integral over $v_{\parallel f}$ gives the jump (2.32) such that

(2.39) \begin{align} -\int _{V_{p}}\mathrm{d}^3v_{\!f}\: \frac {{v_{\parallel f}} \tau }{2\pi qR} \langle C_e\rangle _\tau =-\frac {\partial }{\partial {\psi _{\!f}}}\left [2.758 \frac {I}{\unicode{x1D6FA} _{e} } \sqrt {\frac {r}{R}}\int \mathrm{d}\mu \: 2\pi B \mathsf{M}_{\parallel e}\sqrt {\bigg \lvert {\mu B+\frac {e\phi _{c}R}{m_e r}}}\bigg \rvert \alpha _{0e} \right ] ,\nonumber\\[-1pt]\end{align}

since (D.16) of Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023) shows $\langle (v_\parallel ^2/v_{\parallel f}^2)\partial g_{e0}^{t,bp}/\partial v_{\parallel f}\rangle _\tau =0$ . We can calculate the integral over $\mu$ using the expression for $\mathsf{M}_{\parallel e}$ (2.26) and $\alpha _{0e}$ (2.31)

(2.40) \begin{align} & \int \mathrm{d}\mu \: 2\pi B\mathsf{M}_{\parallel e}\sqrt {\bigg \lvert {\mu B+\frac {e\phi _{c}R}{m_e r}}}\bigg \rvert \alpha _{0e} =1.15 \frac {\nu _{ee}p_{e}}{m_e}\frac {I}{\unicode{x1D6FA} _{e} }\nonumber\\[5pt] & \qquad \times \left\lbrace \left [\frac {\partial }{\partial {\psi }}\ln p_{e} +\frac {m_e (u+V_{\parallel })}{T_{e}}\frac {\unicode{x1D6FA} _{e} }{I}\right ] G_{1e}(\phi _{c},Z)-1.39 G_{2e}(\phi _{c},Z)\frac {\partial }{\partial {\psi }}\ln T_{e}\right\rbrace . \end{align}

The function $G_{1e}$ is defined as

(2.41) \begin{align} G_{1e}(\phi _{c},Z) & =\frac {\int _0^\infty \mathrm{d}x_e\: \sqrt {\left\lvert x_e^2+\dfrac {e\phi _{c} R}{T_{e} r}\right\rvert }e^{-x_e^2} \left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)+Z\right ]}{\int _0^\infty \mathrm{d}x_e\: x_e e^{-x_e^2} \left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)+1\right ]}\nonumber\\[5pt]& \simeq 1.30 \int _0^\infty \mathrm{d}x_e\: \sqrt {\left\lvert x_e^2+\frac {e\phi _{c} R}{T_{e} r}\right\rvert }e^{-x_e^2} \left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)+Z\right ], \end{align}

and $G_{2e}$ is defined as

(2.42) \begin{align} G_{2e}(\phi _{c},Z)&=\frac {\int _0^\infty \mathrm{d}x_e\:\left (x_e^2-\dfrac {5}{2}\right ) \sqrt {\left\lvert x_e^2+\dfrac {e\phi _{c} R}{T_{e} r}\right\rvert }e^{-x_e^2} \left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)+Z\right ]}{\int _0^\infty \mathrm{d}x_e\:\left (x_e^2-\frac {5}{2}\right ) x_e e^{-x_e^2} \left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)+1\right ]}\nonumber\\[5pt] &\simeq -0.94 \int _0^\infty \mathrm{d}x_e\:\left (x_e^2-\dfrac {5}{2}\right ) \sqrt {\left \lvert x_e^2+\frac {e\phi _{c} R}{T_{e} r}\right \rvert }e^{-x_e^2} \left [\unicode{x1D6E9} (x_e)-\unicode{x1D6F9} (x_e)+Z\right ] .\end{align}

Combining (2.33), (2.34) and (2.40) gives the lowest-order neoclassical electron particle flux

(2.43) \begin{align} \varGamma _e&=-3.17\frac {\nu _{ee} I^2 p_{e}}{\unicode{x1D6FA} _{e}^2 m_e}\sqrt {\frac {r}{R}}\Bigg \lbrace \left [\frac {\partial }{\partial {\psi }}\ln p_{e} +\frac {m_e (u+V_{\parallel })}{T_{e}}\frac {\unicode{x1D6FA} _{e } }{I}\right ] G_{1e}(\phi _{c},Z)\\[5pt] &\!\!\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad -1.39G_{2e}(\phi _{c},Z)\frac {\partial }{\partial {\psi }}\ln T_{e}\Bigg \rbrace .\nonumber \end{align}

The neoclassical ion particle flux for strong gradient regions from Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023) is

(2.44) \begin{align} \varGamma _i &=-1.1 \sqrt {\frac {r}{R}} \frac {\nu I^2 p_i}{\lvert {S}\rvert ^{3/2} m_i\unicode{x1D6FA} _i^2}\Bigg \lbrace \bigg [\frac {\partial }{\partial {\psi }}\ln p -\frac {m_i (u+V_{\parallel })}{T_i}\left (\frac {\partial {V_{\parallel }}}{\partial {\psi }}-\frac {\unicode{x1D6FA} _i}{I}\right )\bigg ] G_{1}(u,V_{\parallel },\phi _c)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad -1.17 G_2(u,V_{\parallel },\phi _c)\frac {\partial }{\partial {\psi }}\ln T_i\Bigg \rbrace .\end{align}

For ions, the particle flux depends explicitly on the mean parallel flow gradient and the squeezing factor. The functions $G_1$ and $G_2$ are defined in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023), in (5.13) and (5.14), and depend on $u$ and $V_\parallel$ , which is not the case for the electrons. The neoclassical ion and electron particle fluxes do not have to be equal. For strong gradients where $L_{n,T,\varPhi }\sim \rho _p$ , the fluxes are not necessarily intrinsically ambipolar (Sugama & Horton Reference Sugama and Horton1998; Parra & Catto Reference Parra and Catto2009; Calvo & Parra Reference Calvo and Parra2012). The neoclassical electron particle flux is then smaller than the neoclassical ion particle flux by order $\delta$ . Thus, unless the turbulent particle flux compensates for the difference between $\varGamma _i$ and $\varGamma _e$ , we need to impose $\varGamma _i\simeq 0$ .

2.3. The bootstrap current

The strong gradient effects on the electrons modify the neoclassical bootstrap current $j_\parallel ^B$ , which is defined as

(2.45) \begin{align} j_\parallel ^B&\equiv Ze\int \mathrm{d}^3v\: v_\parallel f_i - e\int \mathrm{d}^3v \: v_\parallel f_e\nonumber\\ & \quad =Zen_iV_\parallel -en_eV_{\parallel } -e\int \mathrm{d}^3v\: v_\parallel g_e = -e\int \mathrm{d}^3v\: v_\parallel g_{e}, \end{align}

where we have used quasineutrality. The trapped–barely passing region is small in velocity space. The main contribution to the integration for the bootstrap current comes from the freely passing region where $v_\parallel =v_{\parallel f}+\textit {O}(\epsilon v_{te}^2/v_{\parallel f})$

(2.46) \begin{equation} \big\langle j^B_\parallel \big\rangle _\psi \simeq -e\bigg \langle \int \mathrm{d}^3v \: v_\parallel g_{e} \bigg \rangle _\psi \simeq -e \bigg \langle \int _{V_p}\mathrm{d}^3v_{\!f}\: v_{\parallel } g_{e} \bigg \rangle _\psi . \end{equation}

Here, $V_p$ denotes the freely passing region. We can calculate this integral using the Spitzer–Härm function $f_{e,SH}$ which satisfies

(2.47) \begin{equation} v_\parallel f_{Me}=C_e[f_{e,SH}]. \end{equation}

The Spitzer–Härm function is a known function

(2.48) \begin{equation} f_{e,SH}=\frac {v_\parallel }{\sqrt {2}\nu _{ee}}f_{Me}A_{SH}\!\left (x_e^2\right ). \end{equation}

Here,

(2.49) \begin{equation} A_{SH}\big(x_e^2\big)=\sum _{i} a_i L_i^{(3/2)}\left (x_e^2\right ), \end{equation}

where $L_i^{(3/2)}$ are generalised Laguerre polynomials and the coefficients $a_i$ depend on $Z$ and are tabulated. For example, the first three coefficients for $Z=1$ are $a_0=-1.975$ , $a_1=0.558$ and $a_3=0.015$ . One can use the property of self-adjointness of the collision operator in velocity space to calculate the bootstrap current. Starting with (2.47) inserted in (2.46), self-adjointness gives

(2.50) \begin{equation} \big\langle j_\parallel ^B\big\rangle _\psi =-e\bigg \langle \int \mathrm{d}^3v\: \frac {g_{e}}{f_{Me}}C_e[f_{e,SH}]\bigg \rangle _\psi =-e\bigg \langle \int \mathrm{d}^3v\: \frac {f_{e,SH}}{f_{Me}}C_e[g_{e}]\bigg \rangle _\psi . \end{equation}

We can write the expression for the bootstrap current as

(2.51) \begin{equation} \big\langle j_\parallel ^B\big\rangle _\psi \simeq -\bigg \langle \int \mathrm{d}^3v\: \frac {e}{\sqrt {2}\nu _{ee}} v_\parallel A_{SH} C_e[g_e]\bigg \rangle _\psi , \end{equation}

where we used the explicit form of the Spitzer–Härm function in (2.48). The largest contribution comes from the lowest-order term in the trapped–barely passing region

(2.52) \begin{equation} \big\langle j_\parallel ^B\big\rangle _\psi \simeq -\frac {e}{\sqrt {2}\nu _{ee}}\Bigg \langle \int _{V_{t,bp}}\mathrm{d}^3v\: v_\parallel A_{SH} \frac {v_\parallel }{v_{\parallel f}}\frac {\partial }{\partial {v_{\parallel f}}}\left [ \mathsf{M}_{\parallel e} \frac {v_\parallel }{v_{\parallel f}}\frac {\partial {g_{0e}^{t,bp}}}{\partial {v_{\parallel f}}}\right ]\Bigg \rangle _\psi \sim \sqrt {\epsilon }\delta n_ie v_{te}. \end{equation}

Using $\mathrm{d}^3v=(v_{\parallel f}/v_\parallel )\mathrm{d}^3v_{\!f}$ , we can make a change to fixed- $\theta$ variables

(2.53) \begin{align} \big\langle j_\parallel ^B\big\rangle _\psi &\simeq -\frac {e}{\sqrt {2}\nu _{ee}}\Bigg \langle \int _{V_{t,bp}}\mathrm{d}^3v_{\!f}\: v_\parallel A_{SH} \frac {\partial }{\partial v_{\parallel f}}\left [ \mathsf{M}_{\parallel e} \frac {v_\parallel }{v_{\parallel f}} \frac {\partial g_{0e}^{t,bp}}{\partial v_{\parallel f}}\right ]\Bigg \rangle _\psi \nonumber\\[5pt] &=\frac {e}{\sqrt {2}\nu _{ee}}\bigg \langle \int _{V_{t,bp}}\mathrm{d}^3v_{\!f}\: A_{SH}\mathsf{M}_{\parallel e} \frac {\partial g_{0e}^{t,bp}}{\partial v_{\parallel f}}\bigg \rangle _\psi , \end{align}

where we integrated by parts in the second step and we employed $\partial v_\parallel /\partial v_{\parallel f}=v_{\parallel f}/v_\parallel$ . For trapped and barely passing particles, $m_ev^2/(2T_e)\simeq m_e\mu B/T_e$ , so $A_{SH}$ is independent of $v_{\parallel f}$ . The integration over $v_{\parallel f}$ gives the jump $\varDelta g_e$ , which is given in (2.32), and we arrive at,

(2.54) \begin{equation} \big\langle j^B_\parallel \big\rangle _\psi =-2.758\sqrt {\frac {r}{R}}\frac {e}{\sqrt {2}\nu _{ee}}2\pi B \int \mathrm{d}\mu \: \sqrt {\left\lvert \mu B +\frac {e\phi _{c} R}{m_e r}\right\rvert }\alpha _{0e} \mathsf{M}_{\parallel e}A_{SH}\left (\frac {m_e\mu B}{T_e}\right ). \end{equation}

The expression for $\alpha _{0e}$ is given in (2.31). Appendix C gives a derivation that treats the discontinuities more carefully but demonstrates that our procedure presented here is completely consistent with the jump conditions that we calculated in § 2.1. The neoclassical bootstrap current including strong gradient effects is

(2.55) \begin{align} \big\langle j_\parallel ^B\big\rangle _\psi & =-2.43 \frac {\textit{cIp}_{e}}{B}\sqrt {\frac {r}{R}}\bigg [\left (\frac {\partial }{\partial \psi } \ln p_{e}+\frac {m_e(u+V_{\parallel })}{T_{e}}\frac {\unicode{x1D6FA} _{e} }{I}\right ) J_{1e}(\phi _{c},Z) \nonumber\\[5pt] & \qquad\qquad\qquad\qquad -0.71J_{2e}(\phi _{c},Z)\frac {\partial }{\partial \psi }\ln T_{e} \bigg ], \end{align}

where

(2.56) \begin{align} J_{1e}(\phi _c,Z) & =\frac {\int _0^\infty \mathrm{d}x\: \sum _{i} a_iL_i^{3/2}(x^2)\left [ \unicode{x1D6E9} (x)-\unicode{x1D6F9} (x) +Z\right ]e^{-x^2}\sqrt {\left\lvert x^2+\dfrac {e\phi _{c} R}{T_{e} r}\right\rvert }}{\int _0^\infty \mathrm{d}x\: \sum _{i} a_{i,Z=1}L_i^{3/2}(x^2)\left [ \unicode{x1D6E9} (x)-\unicode{x1D6F9} (x) +1\right ]e^{-x^2}x}\nonumber\\[5pt]& \simeq -1.2 \int _0^\infty \mathrm{d}x\: \sum _{i} a_iL_i^{3/2}(x^2)\left [ \unicode{x1D6E9} (x)-\unicode{x1D6F9} (x) +Z\right ]e^{-x^2}\sqrt {\left\lvert x^2+\frac {e\phi _{c} R}{T_{e} r}\right\rvert }, \end{align}

and

(2.57) \begin{align} J_{2e}(\phi _c,Z)&=\frac {\int _0^\infty \mathrm{d}x\:\left (x^2-\dfrac {5}{2}\right ) \sum _{i} a_iL_i^{3/2}(x^2)\left [ \unicode{x1D6E9} (x)-\unicode{x1D6F9} (x) +Z\right ]e^{-x^2}\sqrt {\Big {\lvert } x^2+\dfrac {e\phi _{c} R}{T_{e} r}\Big {\rvert }}}{\int _0^\infty \mathrm{d}x\: \left (x^2-\dfrac {5}{2}\right )\sum _{i} a_{i,Z=1}L_i^{3/2}(x^2)\left [ \unicode{x1D6E9} (x)-\unicode{x1D6F9} (x) +1\right ]e^{-x^2}x}\nonumber\\[5pt] &\simeq 1.7 \int _0^\infty \mathrm{d}x\: \sum _{i} a_iL_i^{3/2}(x^2)\left [ \unicode{x1D6E9} (x)-\unicode{x1D6F9} (x) +Z\right ]e^{-x^2}\sqrt {\left \lvert x^2+\frac {e\phi _{c} R}{T_{e} r}\right\rvert }. \end{align}

The weak gradient expression for the bootstrap current is modified by the poloidal variation of the electric potential, which is captured by the modification of the coefficients via the functions $J_{1e}$ and $J_{2e}$ . The bootstrap current also depends on the ion flow which can be different in the strong gradient region as it is no longer determined through flow damping, see Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023). The origin of the poloidal variation and its effects on the electron particle flux, the bootstrap current and the mean parallel flow are further discussed in the next sections.

3.1. Origin of poloidal variation

The amplitude $\phi _c$ of the part of the electric potential that depends on poloidal angle was derived by Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023). The final result reads

(3.1) \begin{align} & \Bigg \lbrace \frac {en_e}{T_e} - \frac {Z^2n_ie I}{T_i\unicode{x1D6FA} _i} \left [ \sqrt {\frac {2T_i}{m_i}} J \left ( \frac {\partial }{\partial \psi }\ln p_i - \frac {3}{2}\frac {\partial }{\partial \psi }\ln T_i\right ) \right.\nonumber\\[5pt]& \quad \left. + \left [1 - 2\sqrt {\frac {m_i}{2T_i}}(V_\parallel +u)J\right ] \Bigg(\frac {\partial V_\parallel }{\partial \psi } - \frac {\unicode{x1D6FA} _i}{I} -\frac {(V_\parallel +u)}{2}\frac {\partial }{\partial \psi }\ln T_i\Bigg )\right]\Bigg \rbrace \phi _c\nonumber\\[5pt]& \quad =-Zn_i\frac {Ir}{\unicode{x1D6FA} _i R}\Bigg \lbrace \sqrt {\frac {2T_i}{m_i}}J \Bigg [\left (\frac {m_iV_\parallel ^2}{T_i}+1\right )\left (\frac {\partial }{\partial \psi }\ln p_i-\frac {3}{2}\frac {\partial }{\partial \psi }\ln T_i\right )\nonumber\\[5pt] & \quad +\frac {\partial }{\partial \psi }\ln T_i\Bigg ]+ \left [1 - 2\sqrt {\frac {m_i}{2T_i}}(V_\parallel +u)J\right ] \Bigg [ (V_\parallel - u) \left (\frac {\partial }{\partial \psi } \ln p_i - \frac {3}{2}\frac {\partial }{\partial \psi } \ln T_i\right ) \nonumber\\[5pt] & \quad +\left (\frac {\partial V_\parallel }{\partial \psi } - \frac {\unicode{x1D6FA} _i}{I}\right ) \left (\frac {m_iu^2}{T_i} + 1 - \frac {m_i(V_\parallel +u)^2 }{T_i}\right ) -\frac {V_\parallel +u}{2}\left (\frac {m_iV_\parallel ^2}{T_i}+1\right )\frac {\partial }{\partial \psi }\ln T_i\Bigg ]\nonumber\\[5pt] & \quad +\left [1+2\frac {m_i}{2T_i}(V_\parallel +u)^2-4\left (\frac {m_i}{2T_i}\right )^{3/2}(V_\parallel +u)^3J\right ]\nonumber\\[5pt] & \qquad\qquad\qquad \times \left (\frac {\partial V_\parallel }{\partial \psi }-\frac {\unicode{x1D6FA} _i}{I}+\frac {V_\parallel -u}{2}\frac {\partial }{\partial \psi }\ln T_i\right )\Bigg \rbrace -2Zn_i\frac {r}{R}, \end{align}

where

(3.2) \begin{equation} J\equiv \frac {\sqrt {\pi }}{2}\exp \left [-\frac {m(u+V_\parallel )^2}{2T}\right ]\textrm{erfi}\left [{\sqrt {\frac {m}{2T}}(u+V_\parallel )}\right ], \end{equation}

and $\textrm{erfi}(x)\equiv (2/\sqrt {\pi })\int _0^x\exp (t^2)\: \mathrm{d}t$ . The strong gradients cause poloidal variation in four different ways: passing particle number asymmetry, centrifugal forces, orbit width asymmetry and mean parallel flow gradient. The first effect, passing particle number asymmetry, was presented in detail in § 4.4 and figure 5 in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023). In this paper, we want to explain the other three strong gradient effects that cause poloidal variation and were not explicitly mentioned in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023).

We start by reminding the reader that there is a passing particle number asymmetry, and that this asymmetry causes poloidal variation in the density, flow, temperature and potential. For $V_\parallel \neq -u$ , the passing particle region is no longer symmetric around the trapped-particle region. This causes an asymmetry in the number of passing particles circulating in the positive and negative poloidal direction. For example, for $V_\parallel \gt -u$ more particles circulate poloidally in the positive direction than in the negative one. For any flux surface of interest, there are two groups of particles at the outboard and at the inboard side: one group with positive and one with negative poloidal velocity. The average radial position of the particles with positive poloidal velocity on the outboard side lies inside the flux surface of interest, that is, in the high density region, whereas the average radial position of the other group of particles, the group with negative poloidal velocity on the outboard side, is in the region of slightly lower density. Due to the asymmetry in passing particle numbers, this creates a point of slightly higher density on the outboard sign. For the inboard side, this picture reverses and a point of slightly lower density is created.

Setting $V_\parallel +u=0$ eliminates the asymmetry in passing particle number, yet the poloidal variation of the electric potential does not vanish,

(3.3) \begin{align} \left [\frac {en_e}{T_e}-\frac {Z^2n_ie I}{T_i\unicode{x1D6FA} _i}\left (\frac {\partial V_\parallel }{\partial \psi }-\frac {\unicode{x1D6FA} _i}{I}\right )\right ]\phi _c & = -Zn_i\frac {Ir}{\unicode{x1D6FA} _i R}\bigg [-\frac {\unicode{x1D6FA} _i}{I}\frac {m_i V_\parallel ^2}{T_i}+2\frac {\partial V_\parallel }{\partial \psi }\left (1+\frac {m_iV_\parallel ^2}{2T_i}\right )\nonumber\\& \qquad\qquad\qquad +2V_\parallel \frac {\partial }{\partial \psi }\ln n_i\bigg ].\end{align}

The first term, which is proportional to $m_iV_\parallel ^2/T_i$ , is the centrifugal force. The centrifugal force pushes ions to the outboard side. The electrons are lighter and less affected by the centrifugal force, so an electrostatic potential is created that is positive on the outboard side and negative on the inboard side to ensure quasineutrality. This effect vanishes in the low flow limit of weak gradient theory because $V_\parallel$ is small.

The last term is proportional to the density gradient and $V_\parallel$ . This term is related to the asymmetry in orbit widths. Looking back at the ion orbit equations for passing particles derived by Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023), particles with negative poloidal velocity have a slightly larger orbit width than particles with positive poloidal velocity, that is, the orbit widths are not symmetric in $v_\parallel +u$ . The asymmetry is caused by the curvature drift which is symmetric in $v_\parallel$ but not with respect to $v_\parallel +u=0$ . We explain this effect in figure 1, where we assume $u\gt 0$ and hence $V_\parallel \lt 0$ . Particles with parallel velocity $v_{\parallel -}\lt V_\parallel =-u$ in figure 1 experience a stronger curvature drift than particles with parallel velocity $v_{\parallel +}\gt V_\parallel =-u$ . In other words, the red particles have a larger orbit width and move away from their flux surface further than the blue particles, see figure 1. On the outboard side, the average radial position of the red particles is deeper into the low density region than the average radial position of the blue particles is in the high density region. The outboard side turns into a point of slightly lower density. The picture again reverses for the low field side, where the inward going particles travel further in radius such that the high field side has slightly higher density than the low field side and poloidal variation occurs. This effect depends on the sign of $V_\parallel$ . In this argument and in figure 1, we assumed that $u\gt 0$ and thus $V_\parallel =-u\lt 0$ . If $V_\parallel \gt 0$ , $\lvert v_{\parallel -}\rvert \lt \lvert v_{\parallel +}\rvert$ , so the orbit width of the blue particles would be bigger. In the limit of weak gradients, this effect vanishes because $V_\parallel$ and the density gradient are small.

Figure 1.

The orbit width of co- and counter-circulating particles is asymmetric because of curvature drift. In this figure, we assume $V_\parallel =-u\lt 0$ . On the low field side, red particles have a larger orbit width so their average radial position locates them deeper in the low density region than blue particles are located in the high density region (a). The opposite happens on the high field side (b). This creates a higher density on the high field side than on the low field side. This effect depends on the sign of $V_\parallel$ and reverses for positive $V_\parallel$ .

The remaining term is proportional to the mean parallel flow gradient. If the mean parallel flow varies radially, particles with different average radial positions belong to different ion Maxwellian distributions. The difference in mean parallel flow translates into a difference in number of particles. The shift of the Maxwellians between the different flux surfaces are shown in figure 2 for $V_\parallel =-u$ . If, for example, $\partial V_\parallel /\partial \psi$ is positive, the mean parallel flow is smaller on the inside of a flux surface than on the outside. There are fewer particles with positive poloidal velocity $v_{\parallel +}\gt V_\parallel =-u$ on the low field side (blue particles in figure 2(a,c)) because their average radial position is inside the flux surface of interest, where the average flow is smaller than the one in the flux surface of interest. The average radial position of particles with negative poloidal velocity $v_{\parallel -}\lt V_\parallel =-u$ on the low field side (red particles in figure 2 a,c) locates them in a region of larger mean parallel flow and hence with fewer particles with $v_{\parallel -}$ . The low field side develops a region of slightly lower density on the flux surface. On the high field side, the picture reverses. The positively circulating particles with velocities $v_{\parallel +}$ (blue particles in figure 2 b,d) are, on average, in the region where the mean parallel flow is larger. There are more particles with parallel velocities close to $v_{\parallel +}$ . The particles with negative poloidal velocity $v_{\parallel -}$ (red particles in figure 2 b,d) belong to a distribution with less particles with $v_{\parallel -}$ . The inboard side turns into a region of slightly higher density. Overall, the Boltzmann response of the electrons creates a poloidal potential variation where the outboard side has slightly smaller potential than the inboard side. This effect vanishes in the limit of weak gradients because the mean parallel flow gradient is small.

Figure 2.

In this figure, we assume $\partial V_\parallel /\partial \psi \gt 0$ . A blue (red) passing particle with parallel velocity $v_{\parallel +}\gt V_\parallel =-u$ ( $v_{\parallel -}\lt V_\parallel =-u$ ) on the low field side (a,c) or the high field side (b,d) is circulating in the positive (negative) sense in the poloidal direction. The solid lines represent the Maxwellian on the flux surface of interest. The dashed lines indicate the shifted Maxwellians radially inwards or outwards from the flux surface of interest. On the low field side, blue (red) particles complete their orbits through a region with smaller (larger) mean parallel flow (c), so their average radial position locates them in a region with fewer particles that have a parallel velocity close to $v_{\parallel +}$ ( $v_{\parallel -}$ ) (a). A point of slightly lower density develops on the outboard side. On the high field side, blue (red) particles complete their orbits through a region with larger (smaller) mean parallel flow (d), so their average radial position locates them in a region with more particles that have a parallel velocity close to $v_{\parallel +}$ ( $v_{\parallel -}$ ) (b). A point of slightly higher density develops on the high field side.

We discussed the centrifugal force, orbit width asymmetry and flow gradient effects for $V_\parallel =-u$ , but they also exist for $V_\parallel \neq -u$ . Allowing for $V_\parallel \neq -u$ introduces cross-terms that could be attributed to either of the four physical effects: passing particle number asymmetry, centrifugal force, orbit width asymmetry and mean flow gradient. Appendix E has a full list of expressions for the four effects for the purpose of the discussion in § 3.2. For example, we choose to attribute all terms that vanish as $V_\parallel =-u$ to the passing particle number asymmetry.

We now turn our attention to the left-hand side of (3.1) and (3.3). The right-hand side describes the potential in relation to the magnetic drifts whereas the left-hand side is related to the $E\times B$ -drift due the poloidal electric field. The origin of the different terms in (3.1) can be traced back to equation (E1) in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023), where the poloidal variation is calculated. An integration is carried out over an expression including the orbit width $\psi_{\!f}-\psi$ of particles, which is given in (A6) in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023), and contains terms proportional to $(v_\parallel ^2+\mu B)\cos \theta /(v_\parallel +u)$ (the magnetic dirft terms) and terms proportional to $ZeR\phi _1/mr(v_\parallel +u)$ (the $E\times B$ -drift terms). Since the $E\times B$ -drift also contributes to the orbit widths of passing particles, it is equally subject to the passing particle number asymmetry and the mean parallel flow gradient effect. Hence we find contributions to the ion density that are proportional to $\phi _c$ in addition to the usual adiabatic response term, i.e. $en_e/T_e+Z^2n_ie/T_i$ , that are also proportional to the gradient of the mean parallel flow or $V_\parallel +u$ . The orbit width asymmetry effect is intrinsically a curvature drift effect and unaffected by the poloidal electric field. The centrifugal force is unrelated to drifts and does not appear on the left-hand side either. For $V_\parallel +u=0$ , the term multiplying $\phi _c$ in (3.3) can go negative if $\partial V_\parallel /\partial \psi$ is large enough. When this happens, the $E\times B$ -drift induced poloidal variation of the electric potential is strong enough to overcome the effects of the magnetic drifts and the sign of $\phi _c$ reverses.

The poloidal variation of the electric potential enters the transport equations via the four modification functions $J_{1e}$ , $J_{2e}$ , $G_{1e}$ and $G_{2e}$ which are shown in figure 3.The four functions are all larger than 1 for $\phi _c\gt 0$ in which case the electric potential is slightly higher on the low field side than on the high field side. Consequently, electrons are pushed to the low field side and trapping by the magnetic field is increased. Trapped particles are the main drive of transport, so an increased number of trapped particles gives roughly speaking an enhancement of particle transport and bootstrap current. For a small poloidal variation amplitude, the electrostatic force weakens the magnetic force and less particles are trapped on the low field side. When the poloidal variation becomes negative enough, electrostatic trapping of electrons on the high field side dominates, the number of trapped particles increases again and the electron transport and bootstrap current are enhanced.

Figure 3.

Modifications $G_{1e}$ , $G_{2e}$ , $J_{1e}$ and $J_{2e}$ as a function of $\bar {\phi }_c=Ze\phi _cR/T_0r$ as defined in (2.41), (2.42), (2.56) and (2.57).

3.2. A case study

Not only do we know how poloidal variation modifies neoclassical transport properties but we also know where this variation is coming from and how to calculate it self-consistently inside strong gradient regions. To demonstrate this procedure, we first introduce a set of normalised equations and then compare two examples of a pedestal.

The poloidal variation, the neoclassical electron flux and the bootstrap current can be calculated for a given set of profiles for density, temperature and mean parallel flow. For this purpose, we introduce the normalised variables

(3.4) \begin{align} \bar {u}&=\sqrt {\frac {m_i}{2T_{i0}}}u, \quad \bar {V}=\sqrt {\frac {m_i}{2T_{i0}}}V_{\parallel },\quad \bar {T}=\frac {T_{i}}{T_{i0}}, \quad \bar {n}=\frac {n_i}{n_{i0}},\nonumber\\[5pt] \bar {\phi }_c&=\frac {Ze\phi _{c} R}{T_{i0} r},\quad \frac {\partial }{\partial \bar {\psi }}=\frac {I}{\unicode{x1D6FA} _{i}}\sqrt {\frac {2T_{i0}}{m_i}}\frac {\partial }{\partial \psi }, \quad \bar {\varGamma }_i=\frac {\varGamma _i}{n_{i0} I\sqrt {\frac {2T_{i0} r}{m_iR}}\frac {\nu _{0}}{\lvert \unicode{x1D6FA} _{i}\rvert }},\nonumber \\[5pt] \bar {n}_e&=\frac {n_{e}}{Zn_{i0}},\quad \bar {T}_e=\frac {T_{e}}{T_{i0}}, \quad \bar {j}^B=\frac {\big\langle j_\parallel ^B\big\rangle _\psi }{Ze n_{i0} \sqrt {\frac {2T_{i0}r}{m_iR}}},\quad \bar {\varGamma }_e=\frac {\varGamma _e}{Zn_{i0} I\sqrt {\frac {2T_{i0} r}{m_iR}}\frac {\nu _{ee,0}}{\lvert \unicode{x1D6FA} _{e}\rvert }}, \end{align}

where $n_{i0}$ , $T_{i0}$ , $\nu _0$ and $\nu _{ee,0}$ are ion density, temperature and ion and electron collision frequencies at a reference flux surface $\bar {\psi }=0$ . The electron particle flux in these normalised variables is

(3.5) \begin{equation} \bar {\varGamma }_e=-1.59 \frac {Z\bar {n}_e^2}{\bar {T}_e^{3/2}}\bigg \lbrace \left [\bar {T}_e \frac {\partial }{\partial \bar {\psi }}\ln \bar {p}_e-\frac {2}{Z}(\bar {u}+\bar {V})\right ]G_{1e}(\bar {\phi }_c,Z)-1.39G_{2e}(\bar {\phi }_c,Z)\frac {\partial \bar {T}_e}{\partial \bar {\psi }}\bigg \rbrace, \end{equation}

and the bootstrap current is

(3.6) \begin{equation} \bar {j}^B=-1.21\bar {n}_e\bigg \lbrace \left [\bar {T}_e\frac {\partial }{\partial \bar {\psi }}\ln \bar {p}_e-2\left (\bar {u}+\bar {V}\right )\right ]J_{1e}(\bar {\phi }_c,Z)-0.71J_{2e}(\bar {\phi }_c,Z)\frac {\partial \bar {T}_e}{\partial \bar {\psi }}\bigg \rbrace . \end{equation}

The point $\bar {\psi }=0$ is not the magnetic axis but a point where gradients are sufficiently small that usual neoclassical theory can be used. The radial electric field in the pedestal is often assumed to be mostly determined by the pressure gradient (Kagan & Catto Reference Kagan and Catto2008; McDermott et al. Reference McDermott2009; Viezzer et al. Reference Viezzer2013). For this example, we assume

(3.7) \begin{equation} Zen_i\frac {\partial \varPhi }{\partial \psi }+\frac {\partial p_i}{\partial \psi }=0, \end{equation}

from which $u$ can be calculated as shown in figure 4.

The mean parallel flow no longer follows from the neoclassical ion particle flux equation in strong gradient regions because for strong radial electric fields the mechanism of flow damping is no longer dominant (Trinczek et al. Reference Trinczek, Parra, Catto, Calvo and Landreman2023). The parallel flow has to be determined via a balance between flow damping and momentum transport. We discuss the intricacy of the calculation of the mean parallel flow in the next section. In this section, we compare two different, sensible cases for the mean parallel flow. In the first case

(3.8) \begin{equation} V_\parallel =-\frac {IT_i}{m_i\unicode{x1D6FA} _i}\left (\frac {\partial }{\partial \psi } \ln p_i +\frac {Ze}{T_i}\frac {\partial \varPhi }{\partial \psi }-1.17 \frac {\partial }{\partial \psi }\ln T_i\right ). \end{equation}

We call this case ‘low flow’ because this expression is the typical result for the mean parallel flow in the low flow regime of weak gradient neoclassical theory. We call the second case ‘high flow’ because of the choice

(3.9) \begin{equation} V_\parallel =-u, \end{equation}

which is the result for the mean parallel flow in the high flow regime of weak gradient neoclassical theory. Thus, the system is not studied in the usual low flow or high flow limits when comparing the two example cases – rather, we choose these two $V_\parallel$ profiles as reasonable assumptions for the ion mean parallel flow.

Figure 4.

Input profiles for density, temperature, mean parallel flow and radial electric field.

The example profiles for temperatures and density for $Z=1$ displayed in figure 4 are taken from Viezzer et al. (Reference Viezzer2016). The analytical formulas of the profiles are given in Appendix D. The poloidal variation of the electric potential as well as ion transport profiles were calculated by Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023) for the set of input profiles in figure 4. We replicate the results for the neoclassical ion particle flux and $\bar {\phi }_c$ in figure 5. For weak gradients, $\bar {\phi }_c$ is very small, as expected. The different contributions to the potential amplitude as discussed in § 3.1 are plotted individually for the ‘high flow’ and ‘low flow’ case in figure 6. The effect of passing particle number asymmetry vanishes exactly in the ‘high flow’ case because $V_\parallel +u=0=J$ and gives only a small contribution in the ‘low flow’ case. Interestingly, centrifugal force and orbit width asymmetry effects balance each other such that the main contribution to the total poloidally varying part of the potential is derived from the mean parallel flow gradient effect in both cases. The cancellation between centrifugal force and orbit width asymmetry is partially due to our choice of force balance in (3.7).

Figure 5.

Amplitude of the poloidal variation of the electric potential and neoclassical ion particle flux for the example profiles in figure 4.

Figure 6.

The poloidal potential variation amplitude can be split up into four different contributions, associated with the effect of passing particle number asymmetry, centrifugal force, mean parallel flow gradient and orbit width asymmetry. The mathematical expressions we used for this figure are summarised in Appendix E. The blue line shows the normalised contribution from the passing particle number asymmetry, the red line shows the piece due to the centrifugal force, the yellow line shows the normalised contribution from the asymmetry in the orbit width and the purple line shows the normalised contribution from the mean parallel flow gradient. Panel (a) shows the individual contributions in the ‘high flow’ example and panel (b) shows the individual contributions in the ‘low flow’ example.

Using the set of input profiles in figure 4, we find the profile of the neoclassical electron flux and bootstrap current as shown in figure 7.

Figure 7.

Neoclassical electron flux and bootstrap current for the example profiles in figure 4.

The overall particle fluxes of ions and electrons have to balance each other to satisfy ambipolarity. The total fluxes consist of a turbulent and a neoclassical contribution. Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023) showed that, due to the strong gradients considered, the neoclassical ion and electron particle fluxes need not balance each other. That is why the ‘high flow’ ion flux can be large as shown in figure 5(b). Note that $\varGamma _i$ and $\varGamma _e$ are normalised differently in (3.4) when comparing the neoclassical particle fluxes for ions and electrons. This is different from weak gradient neoclassical theory, where the neoclassical ion and electron fluxes have to be equal and the lowest-order ion particle flux has to vanish. As pointed out in Trinczek et al. (Reference Trinczek, Parra, Catto, Calvo and Landreman2023), a non-zero lowest-order neoclassical ion transport in a strong gradient system requires a source of parallel momentum that could be provided via interactions with turbulence.

The neoclassical electron particle flux grows significantly in the strong gradient region that is bracketed between the dashed lines in both the ‘low flow’ case and the ‘high flow’ case. Interestingly, the neoclassical particle flux of electrons in the ‘high flow’ case is smaller than in the ‘low flow’ case as opposed to the neoclassical ion flow, where the picture was reversed and the larger particle flux was found in the ‘high flow’ case. Similarly, the bootstrap current grows significantly in the strong gradient region and is smaller in the ‘high flow’ case than in the ‘low flow’ case. The difference between the ‘high flow’ and the ‘low flow’ cases can be traced back to the difference in the coefficients $G_{1e}$ , $G_{2e}$ , $J_{1e}$ and $J_{2e}$ . For positive $\bar {\phi }_c$ , $G_{2e}\gt G_{1e}$ and $J_{2e}\gt J_{1e}$ . From (3.5) and (3.6), one can see that the term proportional to $G_{2e}$ ( $J_{2e}$ ) decreases the neoclassical electron flux (bootstrap current), whereas the terms proportional to $G_{1e}$ ( $J_{1e}$ ) increase it. The poloidal variation of the potential is stronger in the ‘high flow’ case and thus the difference between $G_{1e}$ ( $J_{1e}$ ) and $G_{2e}$ ( $J_{2e}$ ) is larger. The modifications due to large gradients reduce the electron particle flux (bootstrap current) in the ‘high flow’ case more strongly than in the ‘low flow’ case. Additionally, the term multiplying $G_{1e}$ ( $J_{1e}$ ) is smaller in the ‘high flow’ case because $\bar {V}+\bar {u}$ vanishes exactly.

A comparison of the ‘low flow’ and ‘high flow’ cases with the respective results in the weak gradient limit (F.5) and (F.6) shows the significance of the poloidal variation modification. All four comparisons are shown in figure 8. The equations for the weak gradient limit that we use are those in Appendix F. In the weak gradient high flow limit, the poloidal variation reduces to the contribution from centrifugal forces (F.2). In the ‘low flow’ case, the differences between strong gradient and weak gradient neoclassical theory are small. In the ‘high flow’ case, the difference between weak gradient and strong gradient theory are significant. The respective maxima of the particle flux and bootstrap current are reduced by a factor of $\bar {\varGamma }_e/\bar {\varGamma }_e^{wg,hf}\simeq 0.36$ and $\bar {j}^B/\bar {j}^{B,wg,hf}\simeq 0.68$ .

Figure 8.

Comparison of strong gradient and weak gradient neoclassical electron particle fluxes and bootstrap current for ‘high flow’ and ‘low flow’.

These results are not universal and they are highly dependent on $V_\parallel$ (see § 4). In fact, for $T_e=T_i$ , using the $T_e$ and $n_i$ profiles in figure 4, the ‘low flow’ case gives an increase in bootstrap current of the order of 5 %. It is thus possible to construct cases that predict a higher or lower bootstrap current in comparison with weak gradient neoclassical theory. Less current drive might be required if the bootstrap current in the pedestal is in fact larger than assumed. However, a larger bootstrap current might also lead to more instabilities and is not necessarily favourable.

It is clear from figure 7 that the strong gradient modifications are very different in the ‘high flow’ and ‘low flow’ examples. The strong dependence on the mean parallel flow profile is reflected in the amplitude of the poloidal variation of the electric potential in figure 5. The mean parallel flow is not only relevant for the enhancement or reduction of fluxes and bootstrap current but it also leads to qualitative differences as can be seen from the sign change of $\phi _c$ in the ‘high flow’ case (see figure 5). More work is needed to accurately determine the mean parallel flow and its impact on neoclassical transport in strong gradient regions.