2.1 Phase space coordinates

In the kinetic description of low collisionality regimes, it is useful to employ spatial coordinates $\{r,\unicode[STIX]{x1D6FC},l\}$ , defined as follows. The radial coordinate $r\in [0,a]$ labels the magnetic surface and has dimensions of length, $a$ being the minor radius of the stellarator. For given $r$ , the angular coordinate $\unicode[STIX]{x1D6FC}\in [0,2\unicode[STIX]{x03C0})$ selects a magnetic field line on the surface. The coordinate $l\in [0,l_{\text{max}}(r,\unicode[STIX]{x1D6FC}))$ is the arc length along the field line. In these coordinates the magnetic field reads

(2.1) $$\begin{eqnarray}\boldsymbol{B}=\unicode[STIX]{x1D6F9}(r)\unicode[STIX]{x1D735}r\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$

As for the meaning of the flux function $\unicode[STIX]{x1D6F9}(r)$ , it is easy to check that $2\unicode[STIX]{x03C0}\int _{0}^{r_{0}}\unicode[STIX]{x1D6F9}(r)\,\text{d}r$ equals the magnetic flux across a surface defined by $l=\text{const.}$ and $0\leqslant r\leqslant r_{0}$ .

In these coordinates the flux-surface average of a function $f(r,\unicode[STIX]{x1D6FC},l)$ is defined as

(2.2) $$\begin{eqnarray}\langle f\rangle (r)=V^{\prime }(r)^{-1}\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D6FC}\int _{0}^{l_{\text{max}}}\text{d}l\,\unicode[STIX]{x1D6F9}B^{-1}f,\end{eqnarray}$$

where $\unicode[STIX]{x1D6F9}B^{-1}=[(\unicode[STIX]{x1D735}r\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC})\boldsymbol{\cdot }\unicode[STIX]{x1D735}l]^{-1}$ is the volume element (the coordinates $\{r,\unicode[STIX]{x1D6FC},l\}$ are chosen such that $\unicode[STIX]{x1D6F9}>0$ ), $B(r,\unicode[STIX]{x1D6FC},l)$ is the magnitude of $\boldsymbol{B}$ and

(2.3) $$\begin{eqnarray}V^{\prime }(r)=\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D6FC}\int _{0}^{l_{\text{max}}}\text{d}l\,\unicode[STIX]{x1D6F9}B^{-1},\end{eqnarray}$$

with $V(r)$ being the volume enclosed by the surface $r$ . We will use primes to denote differentiation with respect to $r$ .

In velocity space, we employ coordinates $\{v,\unicode[STIX]{x1D706},\unicode[STIX]{x1D70E}\}$ , where $v$ is the magnitude of the velocity $\boldsymbol{v}$ , $\unicode[STIX]{x1D706}=v_{\bot }^{2}/(Bv^{2})$ is the pitch-angle coordinate, $v_{\bot }$ is the component of $\boldsymbol{v}$ perpendicular to $\boldsymbol{B}$ and $\unicode[STIX]{x1D70E}=v_{\Vert }/|v_{\Vert }|$ is the sign of the parallel velocity, with

(2.4) $$\begin{eqnarray}v_{\Vert }(r,\unicode[STIX]{x1D6FC},l,v,\unicode[STIX]{x1D706},\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D70E}v\sqrt{1-\unicode[STIX]{x1D706}B(r,\unicode[STIX]{x1D6FC},l)}.\end{eqnarray}$$

Integrals over velocity space of gyrophase-independent functions (all functions in this paper are independent of gyrophase) read, in these coordinates,

(2.5) $$\begin{eqnarray}\int (\cdot )\,\text{d}^{3}v\equiv \mathop{\sum }_{\unicode[STIX]{x1D70E}}\int _{0}^{\infty }\text{d}v\int _{0}^{B^{-1}}\text{d}\unicode[STIX]{x1D706}\,\frac{\unicode[STIX]{x03C0}Bv^{3}}{|v_{\Vert }|}(\cdot ).\end{eqnarray}$$

2.2 Perturbed omnigenous magnetic fields

Trapped trajectories are those with $\unicode[STIX]{x1D706}\in [B_{\text{max}}^{-1},B_{\text{min}}^{-1}]$ , where $B_{\text{max}}(r)$ and $B_{\text{min}}(r)$ are, respectively, the maximum and minimum values of $B$ on the flux surface. For each trapped trajectory, the second adiabatic invariant reads

(2.6) $$\begin{eqnarray}J(r,\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D706})=2\int _{l_{b_{1}}}^{l_{b_{2}}}|v_{\Vert }|\text{d}l.\end{eqnarray}$$

Here, $l_{b_{1}}$ and $l_{b_{2}}$ are the bounce points of the trapped orbit, obtained by solving in $l$ the equation $1-\unicode[STIX]{x1D706}B(r,\unicode[STIX]{x1D6FC},l)=0$ (see figure 1 for a cartoon showing some of the quantities that describe trapped trajectories).

Figure 1. Cartoon showing some of the quantities (defined in the text) employed to describe a trapped trajectory.

The importance of $J$ comes from the relations

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{2}{\unicode[STIX]{x1D70F}_{b}}\int _{l_{b_{1}}}^{l_{b_{2}}}\frac{1}{|v_{\Vert }|}\boldsymbol{v}_{M}\boldsymbol{\cdot }\unicode[STIX]{x1D735}r\,\text{d}l=\frac{m}{Ze\unicode[STIX]{x1D6F9}\unicode[STIX]{x1D70F}_{b}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{2}{\unicode[STIX]{x1D70F}_{b}}\int _{l_{b_{1}}}^{l_{b_{2}}}\frac{1}{|v_{\Vert }|}\boldsymbol{v}_{M}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}\,\text{d}l=-\frac{m}{Ze\unicode[STIX]{x1D6F9}\unicode[STIX]{x1D70F}_{b}}\unicode[STIX]{x2202}_{r}J, & \displaystyle\end{eqnarray}$$

where

(2.9) $$\begin{eqnarray}\boldsymbol{v}_{M}=\frac{mv^{2}(1-\unicode[STIX]{x1D706}B)}{ZeB}\hat{\boldsymbol{b}}\times (\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\hat{\boldsymbol{b}})+\frac{mv^{2}\unicode[STIX]{x1D706}}{2ZeB}\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}B\end{eqnarray}$$

is the magnetic drift, $\hat{\boldsymbol{b}}=B^{-1}\boldsymbol{B}$ , $m$ is the particle mass, $Ze$ is the particle charge (we assume $Z\sim 1$ ), $e$ is the proton charge and

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{b}(r,\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D706})=2\int _{l_{b_{1}}}^{l_{b_{2}}}|v_{\Vert }|^{-1}\,\text{d}l\end{eqnarray}$$

is the bounce time; i.e. the time that the trapped particle needs to complete its orbit. Hence, $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J$ and $\unicode[STIX]{x2202}_{r}J$ contain the information on the motion of the trapped particle in the directions perpendicular to $\boldsymbol{B}$ , averaged over the motion in the direction parallel to $\boldsymbol{B}$ .

The magnetic field is called omnigenous (Cary & Shasharina Reference Cary and Shasharina1997; Parra et al. Reference Parra, Calvo, Helander and Landreman2015) if the average radial magnetic drift vanishes for all trapped particles. Due to (2.7), this is equivalent to saying that the second adiabatic invariant is independent of $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J=0$ , for all trapped trajectories.

From the definition of omnigeneity one can deduce that, for any function $q$ that depends on $\unicode[STIX]{x1D6FC}$ and $l$ only through an omnigenous magnetic field $B$ , one has (Cary & Shasharina Reference Cary and Shasharina1997; Helander & Nührenberg Reference Helander and Nührenberg2009; Calvo et al. Reference Calvo, Parra, Velasco and Alonso2017)

(2.11) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\int _{l_{b_{1}}}^{l_{b_{2}}}q(r,v,\unicode[STIX]{x1D706},B(r,\unicode[STIX]{x1D6FC},l))\,\text{d}l=0.\end{eqnarray}$$

This useful property is employed in the derivation of the neoclassical equations of § 2.3 and it will also be used in the analysis of their solutions.

In this paper, we will carry out the calculations in magnetic configurations that can be written as an exactly omnigenous magnetic field $\boldsymbol{B}_{0}$ plus a perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{1}$ ,

(2.12) $$\begin{eqnarray}\boldsymbol{B}=\boldsymbol{B}_{0}+\unicode[STIX]{x1D6FF}\boldsymbol{B}_{1},\end{eqnarray}$$

where $B_{0}:=|\boldsymbol{B}_{0}|\sim |\boldsymbol{B}_{1}|=:B_{1}$ and $0\leqslant \unicode[STIX]{x1D6FF}\ll 1$ . As advanced in § 1, and following Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017), we restrict ourselves to cases in which the characteristic scale of variation of $B_{1}$ on the flux surface is not much smaller than that of $B_{0}$ . Specifically, we require $|\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}B_{1}|/|\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}B_{0}|\sim |\unicode[STIX]{x2202}_{l}B_{1}|/|\unicode[STIX]{x2202}_{l}B_{0}|\ll \unicode[STIX]{x1D6FF}^{-1}$ . Thus, in terms of an expansion in $\unicode[STIX]{x1D6FF}\ll 1$ , we can assume that $B_{0}$ and $B_{1}$ have a similar characteristic variation length on the flux surface, that we can take to be $R_{0}$ , the major radius of the stellarator. This is what we mean when we say that the deviation from omnigeneity has small gradients. In particular, deviations with small gradients can deform the omnigenous magnetic wells, but not create new wells.

2.3 Drift-kinetic and quasineutrality equations in stellarators close to omnigeneity

In drift kinetics (Hazeltine Reference Hazeltine1973; Parra & Calvo Reference Parra and Calvo2011; Calvo et al. Reference Calvo, Parra, Velasco and Alonso2013), the characteristic frequency of particle motion in the direction parallel to $\boldsymbol{B}$ is $O(v_{t}/R_{0})$ , whereas the motion perpendicular to the magnetic field has a much smaller typical frequency $O(\unicode[STIX]{x1D70C}_{\ast }v_{t}/R_{0})$ , where $\unicode[STIX]{x1D70C}_{\ast }=\unicode[STIX]{x1D70C}/R_{0}\ll 1$ is the normalized Larmor radius, with $\unicode[STIX]{x1D70C}=v_{t}/\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FA}=ZeB/m$ . Low collisionality regimesFootnote ³ are defined by $\unicode[STIX]{x1D708}_{\ast }\ll 1$ , where $\unicode[STIX]{x1D708}_{\ast }=\unicode[STIX]{x1D708}_{ii}R_{0}/v_{t}$ is the ion collisionality, $\unicode[STIX]{x1D708}_{ii}$ is the ion–ion collision frequency and $v_{t}=\sqrt{T/m}$ is the ion thermal speed. The condition $\unicode[STIX]{x1D708}_{\ast }\ll 1$ means that the characteristic parallel streaming frequency is much greater than the collision frequency $\unicode[STIX]{x1D708}_{ii}$ , and therefore the drift-kinetic ion equation can be averaged over the parallel motion. In this subsection, we give the neoclassical equations for low collisionality plasmas in stellarators close to omnigeneity where the deviation from omnigeneity has small gradients, but we do not include their derivation. For that, we refer the reader to Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017). No assumptions are made about the aspect ratio of the stellarator yet.

For $\unicode[STIX]{x1D6FF}\ll 1$ , one finds that the electrostatic potential, $\unicode[STIX]{x1D711}$ , is a flux function to lowest order,

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D711}(r,\unicode[STIX]{x1D6FC},l)=\unicode[STIX]{x1D711}_{0}(r)+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}^{(1)}(r,\unicode[STIX]{x1D6FC},l)+\cdots \,.\end{eqnarray}$$

The component of the electrostatic potential that is non-constant on the flux surface, denoted in § 1 by $\tilde{\unicode[STIX]{x1D711}}$ , is therefore given by

(2.14) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{0}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}^{(1)}+\cdots \,.\end{eqnarray}$$

In an expansion in $\unicode[STIX]{x1D6FF}$ , the ion distribution function, $F$ , can be written as

(2.15) $$\begin{eqnarray}F=F_{M}-\frac{Ze\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}^{(1)}}{T}F_{M}+\unicode[STIX]{x1D6FF}g^{(1)}+\cdots \,,\end{eqnarray}$$

where $F_{M}$ is a Maxwellian distribution constant on flux surfaces,

(2.16) $$\begin{eqnarray}F_{M}(r,v)=n(r)\left(\frac{m}{2\unicode[STIX]{x03C0}T(r)}\right)^{3/2}\exp \left(-\frac{mv^{2}}{2T(r)}\right),\end{eqnarray}$$

and $n$ and $T$ are the ion density and temperature. The non-adiabatic perturbation to the Maxwellian distribution, $g^{(1)}(r,\unicode[STIX]{x1D6FC},l,v,\unicode[STIX]{x1D706})$ , is independent of $l$ and $\unicode[STIX]{x1D70E}$ , and vanishes for passing trajectories of the omnigenous field, $0\leqslant \unicode[STIX]{x1D706}<B_{0,\text{max}}^{-1}$ , where $B_{0,\text{max}}(r)$ is the maximum value of $B_{0}$ on the flux surface. The function $g^{(1)}$ can be chosen such that

(2.17) $$\begin{eqnarray}\int _{0}^{2\unicode[STIX]{x03C0}}g^{(1)}\,\text{d}\unicode[STIX]{x1D6FC}=0.\end{eqnarray}$$

The equations that determine $g^{(1)}$ and $\unicode[STIX]{x1D711}^{(1)}$ are radially local and linear. First, one has the bounce-averaged drift-kinetic equation for trapped trajectories,

(2.18) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x2202}_{r}J^{(0)}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g^{(1)}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J^{(1)}\unicode[STIX]{x1D6F6}F_{M}=\mathop{\sum }_{\unicode[STIX]{x1D70E}}\frac{Ze\unicode[STIX]{x1D6F9}}{m}\,\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{1}{|v_{\Vert }^{(0)}|}C_{ii}^{\ell (0)}[g^{(1)}]\,\text{d}l, & & \displaystyle\end{eqnarray}$$

whereFootnote ⁴

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{r}J^{(0)}=-\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{\unicode[STIX]{x1D706}v\unicode[STIX]{x2202}_{r}B_{0}+2Ze/(mv)\unicode[STIX]{x1D711}_{0}^{\prime }}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}\,\text{d}l, & \displaystyle\end{eqnarray}$$

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle J^{(1)}=-\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{\unicode[STIX]{x1D706}vB_{1}+2Ze/(mv)\unicode[STIX]{x1D711}^{(1)}}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}\,\text{d}l, & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F6}=\frac{n^{\prime }}{n}+\frac{T^{\prime }}{T}\left(\frac{mv^{2}}{2T}-\frac{3}{2}\right)+\frac{Ze\unicode[STIX]{x1D711}_{0}^{\prime }}{T} & \displaystyle\end{eqnarray}$$

and $C_{ii}^{\ell }$ is the linearized ion–ion collision operator (Helander & Sigmar Reference Helander, Sigmar and Haines2002). The ion–electron collision term has been neglected because we assume $\sqrt{m_{e}/m}\ll 1$ , with $m_{e}$ the electron mass. The superindex $(0)$ in $C_{ii}^{\ell (0)}$ means that $B$ has been replaced by $B_{0}$ in the expression for the collision operator in coordinates $v$ and $\unicode[STIX]{x1D706}$ . Analogously,

(2.22) $$\begin{eqnarray}v_{\Vert }^{(0)}(r,\unicode[STIX]{x1D6FC},l,v,\unicode[STIX]{x1D706},\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D70E}v\sqrt{1-\unicode[STIX]{x1D706}B_{0}(r,\unicode[STIX]{x1D6FC},l)},\end{eqnarray}$$

and $l_{b_{10}}$ , $l_{b_{20}}$ are the solutions of the equation $1-\unicode[STIX]{x1D706}B_{0}(r,\unicode[STIX]{x1D6FC},l)=0$ for $\unicode[STIX]{x1D706}\geqslant B_{0,\text{max}}^{-1}$ . Finally, note that $\unicode[STIX]{x2202}_{r}J^{(0)}$ does not depend on $\unicode[STIX]{x1D6FC}$ because $B_{0}$ is omnigenous. Sometimes, it will be useful to write $J^{(1)}=J_{B}^{(1)}+J_{\unicode[STIX]{x1D711}}^{(1)}$ , where

(2.23) $$\begin{eqnarray}J_{B}^{(1)}=-\unicode[STIX]{x1D706}v\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{B_{1}}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}\,\text{d}l\end{eqnarray}$$

and

(2.24) $$\begin{eqnarray}J_{\unicode[STIX]{x1D711}}^{(1)}=-\frac{2Ze}{mv}\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{\unicode[STIX]{x1D711}^{(1)}}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}\,\text{d}l.\end{eqnarray}$$

Second, one has the quasineutrality equation. In the mass ratio expansion $\sqrt{m_{e}/m}$ $\ll 1$ , the non-adiabatic response of the electrons can be dropped, giving

(2.25) $$\begin{eqnarray}\displaystyle \left(\frac{Z}{T}+\frac{1}{T_{e}}\right)\unicode[STIX]{x1D711}^{(1)}=\frac{2\unicode[STIX]{x03C0}}{en}\int _{0}^{\infty }\text{d}v\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{v^{3}B_{0}}{|v_{\Vert }^{(0)}|}g^{(1)}, & & \displaystyle\end{eqnarray}$$

where $T_{e}(r)$ is the electron temperature. Note that the choice (2.17) implies that

(2.26) $$\begin{eqnarray}\langle \unicode[STIX]{x1D711}^{(1)}\rangle =0,\end{eqnarray}$$

where the flux-surface average is taken using $B_{0}$ . In addition, integration of (2.25) over trapped orbits, application of property (2.11) and of condition (2.17) implies

(2.27) $$\begin{eqnarray}\int _{0}^{2\unicode[STIX]{x03C0}}J_{\unicode[STIX]{x1D711}}^{(1)}\,\text{d}\unicode[STIX]{x1D6FC}=0.\end{eqnarray}$$

Equations (2.18) and (2.25) are correct for values of the collisionality that include the neoclassical transport regimes known as $1/\unicode[STIX]{x1D708}$ , $\sqrt{\unicode[STIX]{x1D708}}$ and superbanana-plateau regimes. In Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017), it was explained why the equations stop to be valid at low enough values of the collisionality, so that they should be modified in case one is interested in describing, for example, the $\unicode[STIX]{x1D708}$ or superbanana regimes. In § 2.4, we explain how some of the quantities entering the neoclassical equations (2.18) and (2.25) behave when a large aspect ratio expansion is taken. In § 2.5, the perturbed omnigenous magnetic field that will be later used in the numerical examples is described.

2.4 Conventions and some useful scalings for large aspect ratio stellarators

Stellarators in operation today have large aspect ratios, $\unicode[STIX]{x1D716}^{-1}=R_{0}/a$ . In Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017), the scaling with $\unicode[STIX]{x1D716}\ll 1$ was not determined, and we would like to address this refinement in the present paper. In this subsection, some conventions that are convenient for $\unicode[STIX]{x1D716}\ll 1$ are explained, and the scaling with $\unicode[STIX]{x1D716}$ of several quantities entering (2.18) and (2.25) is provided for later use.

In a given flux surface, we write magnetic fields in large aspect ratio stellarators close to omnigeneity as

(2.28) $$\begin{eqnarray}B=B_{0}+\unicode[STIX]{x1D6FF}B_{1},\end{eqnarray}$$

where

(2.29) $$\begin{eqnarray}B_{0}=B_{00}+\tilde{B}_{0},\end{eqnarray}$$

$B_{00}$ is constant and $\tilde{B}_{0}\sim \unicode[STIX]{x1D716}B_{00}$ . As for the non-omnigenous perturbation, we assume $B_{1}\sim \unicode[STIX]{x1D716}B_{00}$ . Note that if the aspect ratio is large, the meaningful measure of the size of the non-omnigenous perturbation is given by $\unicode[STIX]{x1D6FF}\sim |B-B_{0}|/|B_{0}-B_{00}|$ . This is why we have included a factor $\unicode[STIX]{x1D716}$ in our convention for the size of $B_{1}$ . The definition

(2.30) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\sim \frac{B-B_{0}}{\unicode[STIX]{x1D716}B_{0}}\end{eqnarray}$$

is valid both for large aspect ratio, $\unicode[STIX]{x1D716}\ll 1$ , and tight aspect ratio, $\unicode[STIX]{x1D716}\sim 1$ .

The radial derivative of $B_{0}$ at the surface of interest will also be needed. In our models, we will assume

(2.31) $$\begin{eqnarray}\unicode[STIX]{x2202}_{r}B_{0}=\frac{\tilde{B}_{0}}{\unicode[STIX]{x1D716}R_{0}}.\end{eqnarray}$$

We proceed to explain how different quantities appearing in the neoclassical equations scale with $\unicode[STIX]{x1D716}$ . We denote by $\unicode[STIX]{x1D6E5}_{\text{trapped}}$ the size (measured in the coordinate  $\unicode[STIX]{x1D706}$ ) of the region of phase space corresponding to particles trapped in $B_{0}$ . Observing that the difference between the maximum and the minimum values of $B_{0}$ on the flux surface is $O(\unicode[STIX]{x1D716}B_{00})$ , we infer that

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\text{trapped}}\sim \unicode[STIX]{x1D716}B_{00}^{-1}.\end{eqnarray}$$

Analogous reasons lead to conclude that the size of the parallel velocity for trapped particles is

(2.33) $$\begin{eqnarray}v_{\Vert }^{(0)}\sim \unicode[STIX]{x1D716}^{1/2}v_{t}.\end{eqnarray}$$

The scaling of $\unicode[STIX]{x2202}_{r}J^{(0)}$ depends on the size of the radial electric field, obtained by imposing ambipolarity of the neoclassical radial particle fluxes. For typical values $\unicode[STIX]{x1D711}_{0}^{\prime }\sim \unicode[STIX]{x1D716}^{-1}T/(eR_{0})$ , the term containing the radial electric field dominates and

(2.34) $$\begin{eqnarray}\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-3/2}v_{t},\end{eqnarray}$$

where we have assumed that the typical length of the omnigenous wells is $R_{0}$ . If the radial electric field is small, $|\unicode[STIX]{x1D711}_{0}^{\prime }|\lesssim T/(eR_{0})$ , thenFootnote ⁵

(2.35) $$\begin{eqnarray}\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-1/2}v_{t}.\end{eqnarray}$$

As for $J^{(1)}$ , we have

(2.36) $$\begin{eqnarray}J_{B}^{(1)}\approx -B_{00}^{-1}v\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{B_{1}}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}\,\text{d}l\sim \unicode[STIX]{x1D716}^{1/2}v_{t}R_{0},\end{eqnarray}$$

whereas

(2.37) $$\begin{eqnarray}J_{\unicode[STIX]{x1D711}}^{(1)}\sim \unicode[STIX]{x1D716}^{-1/2}v_{t}R_{0}\frac{e\unicode[STIX]{x1D711}^{(1)}}{T}.\end{eqnarray}$$

Recall that $\unicode[STIX]{x1D711}^{(1)}$ is one of the unknowns of the neoclassical equations and its size is to be determined.

The function $\unicode[STIX]{x1D6F9}$ appearing in (2.1) scales as

(2.38) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}\sim \unicode[STIX]{x1D716}B_{00}R_{0}.\end{eqnarray}$$

Regarding the plasma profile gradients, it is expected that, generically,

(2.39) $$\begin{eqnarray}\unicode[STIX]{x1D6F6}\sim \frac{1}{\unicode[STIX]{x1D716}R_{0}}.\end{eqnarray}$$

In large aspect ratio stellarators, the collision term on the right-hand side of (2.18) is dominated by the pitch-angle scattering piece of the collision operator,

(2.40) $$\begin{eqnarray}\displaystyle C_{ii}^{\ell (0)}[g^{(1)}]\approx \unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}\frac{v_{\Vert }^{(0)}}{v^{2}B_{00}^{2}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\left(v_{\Vert }^{(0)}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}g^{(1)}\right), & & \displaystyle\end{eqnarray}$$

where

(2.41) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(v)=\frac{3\sqrt{\unicode[STIX]{x03C0}}}{2\unicode[STIX]{x1D70F}_{ii}}\,\frac{\text{erf}\left(v\sqrt{m/(2T)}\right)-\unicode[STIX]{x1D712}\left(v\sqrt{m/(2T)}\right)}{\left(v\sqrt{m/(2T)}\right)^{3}}\end{eqnarray}$$

is the ion–ion collision frequency,

(2.42) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ii}=6\sqrt{2}\unicode[STIX]{x03C0}^{3/2}\unicode[STIX]{x1D700}_{0}^{2}m^{1/2}T^{3/2}/(Z^{4}e^{4}n\ln \unicode[STIX]{x1D6EC}),\end{eqnarray}$$

$\ln \unicode[STIX]{x1D6EC}$ is the Coulomb logarithm, $\unicode[STIX]{x1D700}_{0}$ is the vacuum permittivity,

(2.43) $$\begin{eqnarray}\text{erf}(x)=\frac{2}{\sqrt{\unicode[STIX]{x03C0}}}\int _{0}^{x}e^{-y^{2}}\,\text{d}y\end{eqnarray}$$

and

(2.44) $$\begin{eqnarray}\unicode[STIX]{x1D712}(x)=\frac{\text{erf}(x)-(2x/\sqrt{\unicode[STIX]{x03C0}})\exp (-x^{2})}{2x^{2}}.\end{eqnarray}$$

Employing the estimates above for the scaling of $\unicode[STIX]{x1D6E5}_{\text{trapped}}$ and $v_{\Vert }^{(0)}$ with $\unicode[STIX]{x1D716}$ , we find

(2.45) $$\begin{eqnarray}\displaystyle C_{ii}^{\ell (0)}[g^{(1)}]\sim \frac{\unicode[STIX]{x1D708}_{\ast }}{\unicode[STIX]{x1D716}}\frac{v_{t}}{R_{0}}g^{(1)}, & & \displaystyle\end{eqnarray}$$

which is the typical size of the collision term across the trapped region. It is important to realize that the condition for low collisionality ions, that reads $\unicode[STIX]{x1D708}_{\ast }\ll 1$ for $\unicode[STIX]{x1D716}\sim 1$ , becomesFootnote ⁶   $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{3/2}$ for $\unicode[STIX]{x1D716}\ll 1$ . This result is derived by comparing the factor in front of $g^{(1)}$ on the right-hand side of (2.45) with the typical frequency of the parallel particle motion, which is $O(\unicode[STIX]{x1D716}^{1/2}v_{t}/R_{0})$ when $\unicode[STIX]{x1D716}\ll 1$ .

Finally, we point out that (see (2.40)), when $\unicode[STIX]{x1D716}\ll 1$ , we can define

(2.46) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{\ast }:=\frac{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}(v_{t})R_{0}}{v_{t}}\end{eqnarray}$$

as the expression for the collisionality and

(2.47) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\ast }:=\frac{\sqrt{mT}}{ZeB_{00}R_{0}}\end{eqnarray}$$

for the normalized Larmor radius.

2.5 Model magnetic field close to omnigeneity

In order to define a specific omnigenous field $B_{0}$ for the numerical examples provided in subsequent sections, we follow Cary & Shasharina (Reference Cary and Shasharina1997) and Landreman & Catto (Reference Landreman and Catto2012). The description is easier in Boozer coordinates (Boozer Reference Boozer1981). We denote by $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D701}$ the poloidal and toroidal Boozer angles, respectively. We use the construction explained in section V of Landreman & Catto (Reference Landreman and Catto2012) with $M=1$ and $N=4$ (which means that the contours of $B_{0}$ close on themselves after 4 poloidal turns and 1 toroidal turn) and with rotational transform .We takeFootnote ⁷

(2.48) $$\begin{eqnarray}\tilde{B}_{0}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D701})=\unicode[STIX]{x1D716}B_{00}\cos \unicode[STIX]{x1D702}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D701}),\end{eqnarray}$$

where the function $\unicode[STIX]{x1D702}$ is obtained as in equation (73) of Landreman & Catto (Reference Landreman and Catto2012). In figure 2 we show a plot of an omnigenous magnetic field of the form just introduced.

Figure 2. Plot of $B_{0}$ for the model omnigenous magnetic field described at the beginning of § 2.5; that is, $B_{0}=B_{00}+\tilde{B}_{0}$ , with $\tilde{B}_{0}$ of the form (2.48). Here, $\unicode[STIX]{x1D716}=0.1$ .

As for the non-omnigenous perturbation, we take

(2.49) $$\begin{eqnarray}B_{1}=\unicode[STIX]{x1D716}B_{00}\cos (2\unicode[STIX]{x1D703}).\end{eqnarray}$$

Note that $B=B_{0}+\unicode[STIX]{x1D6FF}B_{1}$ is symmetric under the transformation $(\unicode[STIX]{x1D703},\unicode[STIX]{x1D701})\mapsto (-\unicode[STIX]{x1D703},-\unicode[STIX]{x1D701})$ . Magnetic fields with this property are called stellarator symmetric (Dewar & Hudson Reference Dewar and Hudson1998). Stellarator designs typically impose this symmetry on the magnetic configuration, and from here on we always assume that our magnetic field satisfies this.

Given Boozer angles $\{\unicode[STIX]{x1D703},\unicode[STIX]{x1D701}\}$ , there are many different ways to define coordinates $\{\unicode[STIX]{x1D6FC},l\}$ . First, one has to choose a closed curve ${\mathcal{C}}$ that is not contractible to a point. A simple choice is to take the curve $\unicode[STIX]{x1D701}=0$ . Then,

(2.50)

As for the arc length $l$ ,

(2.51) $$\begin{eqnarray}l(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D701})=\int _{0}^{\unicode[STIX]{x1D701}}\frac{1}{\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}^{\prime }}\,\text{d}\unicode[STIX]{x1D701}^{\prime },\end{eqnarray}$$

where the integral on the right-hand side is taken at constant $\unicode[STIX]{x1D6FC}$ and $l_{\text{max}}$ is determined by the intersection of the field line with ${\mathcal{C}}$ when it goes all the way around the torus.

4.1 The $1/\unicode[STIX]{x1D708}$ regime

For tight aspect ratio stellarators, the $1/\unicode[STIX]{x1D708}$ regime corresponds to $\unicode[STIX]{x1D70C}_{\ast }\ll \unicode[STIX]{x1D708}_{\ast }\ll 1$ . In this situation, the first term on the left-hand side of (2.18) is negligible compared to the term on the right-hand side,

(4.1) $$\begin{eqnarray}\left|\unicode[STIX]{x2202}_{r}J^{(0)}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g^{(1)}\right|\ll \left|\mathop{\sum }_{\unicode[STIX]{x1D70E}}\frac{Ze\unicode[STIX]{x1D6F9}}{m}\,\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{\text{d}l}{|v_{\Vert }^{(0)}|}C_{ii}^{\ell (0)}\left[g^{(1)}\right]\right|.\end{eqnarray}$$

The correction to the Maxwellian distribution in the $1/\unicode[STIX]{x1D708}$ regime is therefore found by integrating the expression

(4.2) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D70E}}\frac{Ze\unicode[STIX]{x1D6F9}}{m}\,\int _{l_{b_{10}}}^{l_{b_{20}}}|v_{\Vert }^{(0)}|^{-1}C_{ii}^{\ell (0)}\left[g_{1/\unicode[STIX]{x1D708}}^{(1)}\right]\,\text{d}l=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J^{(1)}\unicode[STIX]{x1D6F6}F_{M}. & & \displaystyle\end{eqnarray}$$

Below, we will find useful to distinguish between $g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]$ and $g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]$ , where

(4.3) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D70E}}\frac{Ze\unicode[STIX]{x1D6F9}}{m}\,\int _{l_{b_{10}}}^{l_{b_{20}}}|v_{\Vert }^{(0)}|^{-1}C_{ii}^{\ell (0)}\left[g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]\right]\,\text{d}l=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J_{B}^{(1)}\unicode[STIX]{x1D6F6}F_{M} & & \displaystyle\end{eqnarray}$$

and

(4.4) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D70E}}\frac{Ze\unicode[STIX]{x1D6F9}}{m}\,\int _{l_{b_{10}}}^{l_{b_{20}}}|v_{\Vert }^{(0)}|^{-1}C_{ii}^{\ell (0)}\left[g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]\right]\,\text{d}l=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J_{\unicode[STIX]{x1D711}}^{(1)}\unicode[STIX]{x1D6F6}F_{M}, & & \displaystyle\end{eqnarray}$$

and therefore $g_{1/\unicode[STIX]{x1D708}}^{(1)}=g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]+g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]$ . We note that

(4.5) $$\begin{eqnarray}g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{n}{v_{t}^{3}}\end{eqnarray}$$

and

(4.6) $$\begin{eqnarray}g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{e\unicode[STIX]{x1D711}^{(1)}}{T}\frac{n}{v_{t}^{3}}.\end{eqnarray}$$

The equation for $\unicode[STIX]{x1D711}^{(1)}$ reads (recall (2.25))

(4.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-28.45274pt}\unicode[STIX]{x1D711}^{(1)}=\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\frac{2\unicode[STIX]{x03C0}B_{0}}{en}\int _{0}^{\infty }\text{d}vv^{3}\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\,\text{d}\unicode[STIX]{x1D706}|v_{\Vert }^{(0)}|^{-1}\left(g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]+g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]\right).\end{eqnarray}$$

Using (4.6) and $\unicode[STIX]{x1D70C}_{\ast }\ll \unicode[STIX]{x1D708}_{\ast }$ , and comparing the size of the left-hand side of (4.7) with the size of the term on the right-hand side containing $g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]$ , we conclude that the latter can be dropped. Hence, $\unicode[STIX]{x1D711}^{(1)}$ is calculated as

(4.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{(1)}=\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\frac{2\unicode[STIX]{x03C0}B_{0}}{en}\int _{0}^{\infty }\text{d}vv^{3}\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}|v_{\Vert }^{(0)}|^{-1}g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]. & & \displaystyle\end{eqnarray}$$

Then, in the $1/\unicode[STIX]{x1D708}$ regime (see (4.5) and (4.6)),

(4.9) $$\begin{eqnarray}g^{(1)}\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{n}{v_{t}^{3}}\end{eqnarray}$$

and

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{T}{e}.\end{eqnarray}$$

If $B_{0}$ and $B_{1}$ are stellarator symmetric, $J_{B}^{(1)}$ is symmetric and, consequently, $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J_{B}^{(1)}$ is antisymmetric. Since the collision operator does not change the symmetry, the solution $g_{1/\unicode[STIX]{x1D708}}^{(1)}$ of (4.2) is antisymmetric. Finally, $\unicode[STIX]{x1D711}^{(1)}$ , obtained from (4.8), is stellarator antisymmetric.

4.1.1 The $1/\unicode[STIX]{x1D708}$ regime in a large aspect ratio stellarator

As indicated by the end of § 2.4, when $\unicode[STIX]{x1D716}$ is small the low collisionality condition reads $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{3/2}$ . How small $\unicode[STIX]{x1D708}_{\ast }$ can be before the tangential drifts start to count depends on their size, as will be discussed mainly in §§ 4.2.1 and 4.2.2.

Let us refine the estimates (4.9) and (4.10) for the distribution function and the electrostatic potential when $\unicode[STIX]{x1D716}\ll 1$ . The drift-kinetic equation for $\unicode[STIX]{x1D716}\ll 1$ is

(4.11) $$\begin{eqnarray}\displaystyle \frac{2Ze\unicode[STIX]{x1D6F9}}{mB_{00}^{2}}\,\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\left[\left(\int _{l_{b_{10}}}^{l_{b_{20}}}\sqrt{1-\unicode[STIX]{x1D706}B_{0}}\,\text{d}l\right)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}g_{1/\unicode[STIX]{x1D708}}^{(1)}\right]=\frac{v}{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J^{(1)}\unicode[STIX]{x1D6F6}F_{M}. & & \displaystyle\end{eqnarray}$$

Employing the scalings of § 2.4, it is easy to infer that

(4.12) $$\begin{eqnarray}g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{n}{v_{t}^{3}}\end{eqnarray}$$

and

(4.13) $$\begin{eqnarray}g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]\sim \unicode[STIX]{x1D716}^{-1/2}\frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{J_{\unicode[STIX]{x1D711}}^{(1)}}{v_{t}R_{0}}\frac{n}{v_{t}^{3}}.\end{eqnarray}$$

Next, we apply the $\unicode[STIX]{x1D716}\ll 1$ expansion to (4.7). We must treat two cases separately. If

(4.14) $$\begin{eqnarray}\unicode[STIX]{x1D716}^{-1/2}\unicode[STIX]{x1D70C}_{\ast }\ll \unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{3/2},\end{eqnarray}$$

then the first term on the right-hand side of (4.7) is negligible with respect to the left-hand side. Using that $\unicode[STIX]{x1D6E5}_{\text{trapped}}\sim \unicode[STIX]{x1D716}B_{00}^{-1}$ , we find

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \unicode[STIX]{x1D716}^{1/2}\frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{T}{e}.\end{eqnarray}$$

And noting that $g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]\ll g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]$ ,

(4.16) $$\begin{eqnarray}g^{(1)}\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{n}{v_{t}^{3}}.\end{eqnarray}$$

Then, both $\unicode[STIX]{x1D711}^{(1)}$ and $g^{(1)}$ scale as $\unicode[STIX]{x1D708}_{\ast }^{-1}$ . The fact that $\unicode[STIX]{x1D711}^{(1)}$ is antisymmetric is also clear. In figure 4, a contour plot of $\unicode[STIX]{x1D711}^{(1)}$ on the magnetic surface is provided for a set of values such that (4.14) holds.

Figure 4. Contour plot of $\unicode[STIX]{x1D711}^{(1)}$ for $\unicode[STIX]{x1D716}=1.1\times 10^{-4}$ , $\unicode[STIX]{x1D708}_{\ast }=10^{-7}$ and $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ .

However, if $\unicode[STIX]{x1D708}_{\ast }\sim \unicode[STIX]{x1D716}^{-1/2}\unicode[STIX]{x1D70C}_{\ast }$ or smaller, the first term on the right-hand side of (4.7) can become comparable to the left-hand side. Let us study the asymptotic regime

(4.17) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{-1/2}\unicode[STIX]{x1D70C}_{\ast }.\end{eqnarray}$$

From (4.7) we learn that $\unicode[STIX]{x1D711}^{(1)}$ depends on $l$ only through $B_{0}$ . This will be important in a moment. When (4.17) holds, the left-hand side of (4.7) is small compared to the term on the right-hand side that involves $\unicode[STIX]{x1D711}^{(1)}$ . Then, we can write

(4.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{0}^{\infty }\text{d}vv^{2}\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{1}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}\,g_{1/\unicode[STIX]{x1D708}}^{(1)}=0,\end{eqnarray}$$

where $g_{1/\unicode[STIX]{x1D708}}^{(1)}$ is the solution of (4.11). For the argument that follows, we find it convenient to write $J^{(1)}$ as

(4.19) $$\begin{eqnarray}J^{(1)}=-\int _{B_{0,\text{min}}}^{B_{0}}\frac{G}{\sqrt{1-\unicode[STIX]{x1D706}\hat{B}}}\,\text{d}\hat{B}.\end{eqnarray}$$

Here, $B_{0,\text{min}}$ is the minimum of $B_{0}$ ,

(4.20) $$\begin{eqnarray}G(r,\unicode[STIX]{x1D6FC},\hat{B},\unicode[STIX]{x1D6FD},v):=\mathop{\sum }_{\unicode[STIX]{x1D6FD}=-1}^{1}\left(\frac{B_{1}v}{B_{00}}+\frac{2Ze\unicode[STIX]{x1D711}^{(1)}}{mv}\right)\frac{1}{\left|\unicode[STIX]{x2202}_{l}B_{0}\right|}\end{eqnarray}$$

and we have used the magnitude of $B_{0}$ as the integration variable in (4.19). Since there are two valuesFootnote ⁸ of $l$ corresponding to each value $B_{0}=\hat{B}$ , we introduce the discrete coordinate $\unicode[STIX]{x1D6FD}=\pm 1$ . The value $\unicode[STIX]{x1D6FD}=-1$ labels the branch of the magnetic well where $\unicode[STIX]{x2202}_{l}B_{0}<0$ and the value $\unicode[STIX]{x1D6FD}=1$ labels the branch where $\unicode[STIX]{x2202}_{l}B_{0}\geqslant 0$ . On the right-hand side of (4.20), every function is understood to be expressed using $\hat{B}$ and $\unicode[STIX]{x1D6FD}$ instead of $l$ . In particular, we note that, in general, $|\unicode[STIX]{x2202}_{l}B_{0}|(\hat{B},\unicode[STIX]{x1D6FD})\neq |\unicode[STIX]{x2202}_{l}B_{0}|(\hat{B},-\unicode[STIX]{x1D6FD})$ .

It is clear that

(4.21) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }v^{2}g_{1/\unicode[STIX]{x1D708}}^{(1)}\,\text{d}v=0 & & \displaystyle\end{eqnarray}$$

gives the solution of (4.18) and that, by imposing (4.21), we obtain a relation between the orbit integrals of $B_{1}$ and $\unicode[STIX]{x1D711}^{(1)}$ . The orbit integral of a function can be interpreted as a transformation that replaces the coordinate $l$ by the coordinate $\unicode[STIX]{x1D706}$ . This can be viewed as an Abel transform, and an explicit inversion formula is given in appendix A. In principle, the inversion cannot distinguish between two points of the magnetic well with the same value of $B_{0}$ (this is quite intuitive when we use the magnitude of $B_{0}$ as the integration variable, as we have done in (4.19)). Carrying out the integral in $v$ on the left-hand side of (4.21), we arrive at

(4.22) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FD}=-1}^{1}\frac{Ze\unicode[STIX]{x1D711}^{(1)}}{T}\frac{1}{|\unicode[STIX]{x2202}_{l}B_{0}|}=-2.83\,\frac{n^{\prime }/n+Ze\unicode[STIX]{x1D711}_{0}^{\prime }/T+2.33T^{\prime }/T}{n^{\prime }/n+Ze\unicode[STIX]{x1D711}_{0}^{\prime }/T+1.33T^{\prime }/T}\mathop{\sum }_{\unicode[STIX]{x1D6FD}=-1}^{1}\frac{B_{1}}{B_{00}}\frac{1}{|\unicode[STIX]{x2202}_{l}B_{0}|}. & & \displaystyle\end{eqnarray}$$

Recalling that $\unicode[STIX]{x1D711}^{(1)}$ depends on $l$ only through $B_{0}$ (see (4.7)), we finally obtain the explicit solution

(4.23) $$\begin{eqnarray}\displaystyle \frac{Ze\unicode[STIX]{x1D711}^{(1)}}{T}=-2.83\,\frac{n^{\prime }/n+Ze\unicode[STIX]{x1D711}_{0}^{\prime }/T+2.33T^{\prime }/T}{n^{\prime }/n+Ze\unicode[STIX]{x1D711}_{0}^{\prime }/T+1.33T^{\prime }/T}\,\frac{\mathop{\sum }_{\unicode[STIX]{x1D6FD}=-1}^{1}(B_{1}/B_{00})|\unicode[STIX]{x2202}_{l}B_{0}|^{-1}}{\mathop{\sum }_{\unicode[STIX]{x1D6FD}=-1}^{1}|\unicode[STIX]{x2202}_{l}B_{0}|^{-1}}. & & \displaystyle\end{eqnarray}$$

From (4.23) we infer that in this regime $\unicode[STIX]{x1D711}^{(1)}$ is stellarator symmetric and that its size, in general, is

(4.24) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \unicode[STIX]{x1D716}\frac{T}{e}.\end{eqnarray}$$

We note that for particular combinations of the profile gradients, $\unicode[STIX]{x1D711}^{(1)}$ can get larger because the denominator of (4.23) can become small. This will be studied elsewhere.

In figure 5, a contour plot of $\unicode[STIX]{x1D711}^{(1)}$ on the magnetic surface is provided for a set of values that satisfy (4.17).

Finally, we have (see (4.12) and (4.13))

(4.25) $$\begin{eqnarray}g_{1/\unicode[STIX]{x1D708}}^{(1)}\sim g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{\unicode[STIX]{x1D711}}^{(1)}]\sim g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}\frac{n}{v_{t}^{3}}.\end{eqnarray}$$

In figure 6, and for $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ , we illustrate the scaling of $\unicode[STIX]{x1D711}^{(1)}$ and of

(4.26) $$\begin{eqnarray}n_{B}:=\int g_{1/\unicode[STIX]{x1D708}}^{(1)}[J_{B}^{(1)}]\,\text{d}^{3}v\sim \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast }}n\end{eqnarray}$$

in a range of collisionality values such that both regimes, (4.14) and (4.17), are observed. Whereas $\unicode[STIX]{x1D711}^{(1)}$ and $n_{B}$ have the same scaling with $\unicode[STIX]{x1D708}_{\ast }$ in the regime (4.14), they scale differently in the regime (4.17).

Figure 5. Contour plot of $\unicode[STIX]{x1D711}^{(1)}$ for $\unicode[STIX]{x1D716}=1.1\times 10^{-4}$ , $\unicode[STIX]{x1D708}_{\ast }=5.8\times 10^{-11}$ and $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ .

Figure 6. Scaling of $\unicode[STIX]{x1D711}^{(1)}$ and $n_{B}$ (evaluated at the point of the flux surface where they reach their maximum values) with $\unicode[STIX]{x1D708}_{\ast }$ and $\unicode[STIX]{x1D716}$ . Here, $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ . The regimes (4.14) and (4.17) are confirmed numerically. The central value $\unicode[STIX]{x1D716}=1.1\times 10^{-4}$ has been used to draw the thin dotted lines indicating where the collisionality values $\unicode[STIX]{x1D70C}_{\ast }/\unicode[STIX]{x1D716}^{1/2}$ and $\unicode[STIX]{x1D716}^{3/2}$ are located.

4.2 The $\sqrt{\unicode[STIX]{x1D708}}$ regime

We turn to discuss collisionality regimes with $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D70C}_{\ast }\ll 1$ . In this situation, as explained at length in Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017), the behaviour of both the radial neoclassical fluxes and $\unicode[STIX]{x1D711}^{(1)}$ depends on the zeroes of $\unicode[STIX]{x2202}_{r}J^{(0)}$ .

When $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D70C}_{\ast }$ , the collision term in (2.18) is negligible compared to the first term on the left-hand side,

(4.27) $$\begin{eqnarray}\left|\mathop{\sum }_{\unicode[STIX]{x1D70E}}\frac{Ze\unicode[STIX]{x1D6F9}}{m}\,\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{\text{d}l}{|v_{\Vert }^{(0)}|}C_{ii}^{\ell (0)}[g^{(1)}]\right|\ll \left|\unicode[STIX]{x2202}_{r}J^{(0)}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g^{(1)}\right|.\end{eqnarray}$$

Hence, to lowest order in an expansion in $\unicode[STIX]{x1D708}_{\ast }/\unicode[STIX]{x1D70C}_{\ast }$ ,

(4.28) $$\begin{eqnarray}g^{(1)}=g_{0}+\cdots \,,\end{eqnarray}$$

where $g_{0}$ is the solution of

(4.29) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x2202}_{r}J^{(0)}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g_{0}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J^{(1)}\unicode[STIX]{x1D6F6}F_{M}=0. & & \displaystyle\end{eqnarray}$$

This equation can be readily solved, obtaining an explicit expression for $g_{0}$ ,

(4.30) $$\begin{eqnarray}\displaystyle g_{0}=\frac{1}{\unicode[STIX]{x2202}_{r}J^{(0)}}\left(J^{(1)}-\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}J^{(1)}\text{d}\unicode[STIX]{x1D6FC}\right)\unicode[STIX]{x1D6F6}F_{M}, & & \displaystyle\end{eqnarray}$$

where we have chosen $\int _{0}^{2\unicode[STIX]{x03C0}}g_{0}\text{d}\unicode[STIX]{x1D6FC}=0$ to fix the integration constant. From (4.30), it is clear that the solution $g_{0}$ is not valid at phase space points where $\unicode[STIX]{x2202}_{r}J^{(0)}$ vanishes, and we devote § 4.3 to those points. However, if $\unicode[STIX]{x2202}_{r}J^{(0)}$ does not vanish at any point of phase space (or if it vanishes only for values of the velocity $v\gg v_{t}$ , that are irrelevant because their effect is supressed by the smallness of the Maxwellian distribution), (4.30) is enough to correctly determine $\unicode[STIX]{x1D711}^{(1)}$ via the quasineutrality equation

(4.31) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{(1)}=\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\frac{2\unicode[STIX]{x03C0}B_{0}}{en}\int _{0}^{\infty }\text{d}vv^{3}\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}|v_{\Vert }^{(0)}|^{-1}g_{0}. & & \displaystyle\end{eqnarray}$$

In this case, $\unicode[STIX]{x1D711}^{(1)}$ is stellarator symmetric (due to the fact that the expression (4.30) for $g_{0}$ does not involve derivatives of $B$ along the flux surface) and does not scale with collisionality,

(4.32) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \frac{T}{e}.\end{eqnarray}$$

As proven in Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017), $g_{0}$ does not contribute to the radial neoclassical fluxes. They are determined by corrections to $g_{0}$ in (4.28) that are localized in a region of phase space that has a typical size $\unicode[STIX]{x1D6E5}_{\sqrt{\unicode[STIX]{x1D708}}}\sim \sqrt{\unicode[STIX]{x1D708}_{\ast }/\unicode[STIX]{x1D70C}_{\ast }}$ in the coordinate $\unicode[STIX]{x1D706}$ . This small layer produces the scaling with collisionality that justifies the name of this regime, the $\sqrt{\unicode[STIX]{x1D708}}$ regime.

If the aspect ratio is large, it is interesting to distinguish two cases (mostly in connection to the solutions of the quasineutrality equation), to which we devote §§ 4.2.1 and 4.2.2.

4.2.1 Large aspect ratio and large tangential drift, $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-3/2}v_{t}R_{0}$

If $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-3/2}v_{t}R_{0}$ , the inequality (4.27) holds provided that $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{-1}\unicode[STIX]{x1D70C}_{\ast }$ . Hence, the $1/\unicode[STIX]{x1D708}$ regime takes place in the range

(4.33) $$\begin{eqnarray}\unicode[STIX]{x1D716}^{-1}\unicode[STIX]{x1D70C}_{\ast }\ll \unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{3/2}.\end{eqnarray}$$

In particular, the regime defined by (4.17) is not observed because the tangential drift starts to count for collisionalities much larger than $\unicode[STIX]{x1D716}^{-1/2}\unicode[STIX]{x1D70C}_{\ast }$ . It can be observed, however, when the tangential drift is smaller (see § 4.2.2).

As pointed out in § 2.4, $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-3/2}v_{t}R_{0}$ corresponds to $\unicode[STIX]{x1D711}_{0}^{\prime }\sim \unicode[STIX]{x1D716}^{-1}T/(eR_{0})$ , which is the standard size of the radial electric field in a large aspect ratio stellarator. The derivation of the radial fluxes in this regime can be found, for example, in Galeev & Sagdeev (Reference Galeev, Sagdeev and Leontovich1979), Ho & Kulsrud (Reference Ho and Kulsrud1987).

Let us start the discussion about $\unicode[STIX]{x1D711}^{(1)}$ by writing (4.31) more explicitly. Inserting in (4.31) the expression for $g_{0}$ (see (4.30)) and rearranging, we get

(4.34) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D711}^{(1)}-\frac{2\unicode[STIX]{x03C0}}{en}\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\int _{0}^{\infty }\text{d}v\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{v^{3}B_{0}}{|v_{\Vert }^{(0)}|}\frac{\unicode[STIX]{x1D6F6}F_{M}}{\unicode[STIX]{x2202}_{r}J^{(0)}}\widetilde{J}_{\unicode[STIX]{x1D711}}^{(1)}\nonumber\\ \displaystyle & & \displaystyle \qquad =\frac{2\unicode[STIX]{x03C0}}{en}\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\int _{0}^{\infty }\text{d}v\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{v^{3}B_{0}}{|v_{\Vert }^{(0)}|}\frac{\unicode[STIX]{x1D6F6}F_{M}}{\unicode[STIX]{x2202}_{r}J^{(0)}}\widetilde{J}_{B}^{(1)},\end{eqnarray}$$

where

(4.35) $$\begin{eqnarray}\widetilde{J}_{B}^{(1)}=J_{B}^{(1)}-\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}J_{B}^{(1)}\text{d}\unicode[STIX]{x1D6FC}\end{eqnarray}$$

and

(4.36) $$\begin{eqnarray}\widetilde{J}_{\unicode[STIX]{x1D711}}^{(1)}=J_{\unicode[STIX]{x1D711}}^{(1)}-\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}J_{\unicode[STIX]{x1D711}}^{(1)}\text{d}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$

In (4.36) we have emphasized that only the component of $J_{\unicode[STIX]{x1D711}}^{(1)}$ that fluctuates in $\unicode[STIX]{x1D6FC}$ enters (4.34) although, strictly speaking, it is superfluous due to (2.27).

Recalling the estimates in § 2.4, it is straightforward to see that the second term on the left-hand side of (4.34) is smaller than the first one by a factor $\unicode[STIX]{x1D716}^{1/2}$ . Hence, the size of $\unicode[STIX]{x1D711}^{(1)}$ is determined by the size of the right-hand side of (4.34). That is,

(4.37) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \unicode[STIX]{x1D716}^{3/2}\,\frac{T}{e}.\end{eqnarray}$$

This scaling is consistent with (4.15) when $\unicode[STIX]{x1D708}_{\ast }\sim \unicode[STIX]{x1D716}^{-1}\unicode[STIX]{x1D70C}_{\ast }$ . In figure 7, a contour plot of $\unicode[STIX]{x1D711}^{(1)}$ corresponding to $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-3/2}v_{t}R_{0}$ and $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{-1}\unicode[STIX]{x1D70C}_{\ast }$ is shown. In figure 8, the collisionality-independent scaling (4.37) for $\unicode[STIX]{x1D711}^{(1)}$ is numerically checked.

Figure 7. Contour plot of $\unicode[STIX]{x1D711}^{(1)}$ for $\unicode[STIX]{x1D716}=1.1\times 10^{-4}$ , $\unicode[STIX]{x1D708}_{\ast }=1.1\times 10^{-10}$ and $\unicode[STIX]{x1D711}_{0}^{\prime }=2\unicode[STIX]{x1D716}^{-1}T/(eR_{0})$ .

Figure 8. Scaling of $\unicode[STIX]{x1D711}^{(1)}$ with $\unicode[STIX]{x1D708}_{\ast }$ and $\unicode[STIX]{x1D716}$ (evaluated at the point of the flux surface where it reaches its maximum value) in a range of values of the collisionality where (4.37) is observed. Here, $\unicode[STIX]{x1D711}_{0}^{\prime }=2\unicode[STIX]{x1D716}^{-1}T/(eR_{0})$ .

4.2.2 Large aspect ratio and small tangential drift, $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-1/2}v_{t}R_{0}$

If $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-1/2}v_{t}R_{0}$ , (4.27) is satisfied when $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D70C}_{\ast }$ . Therefore, for such typical size of the tangential drifts, the $1/\unicode[STIX]{x1D708}$ regime happens for collisionality values

(4.38) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\ast }\ll \unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{3/2}.\end{eqnarray}$$

To be precise, in the range $\unicode[STIX]{x1D70C}_{\ast }\ll \unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D716}^{-1/2}\unicode[STIX]{x1D70C}_{\ast }$ , the subregime (4.17) of the $1/\unicode[STIX]{x1D708}$ regime will be observed.

When $\unicode[STIX]{x2202}_{r}J^{(0)}\sim \unicode[STIX]{x1D716}^{-1/2}v_{t}R_{0}$ , the left-hand side of (4.31) can be neglected compared to the term on the right-hand side containing $\unicode[STIX]{x1D711}^{(1)}$ , so that we have

(4.39) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }\text{d}vv^{2}\int _{B_{0,\text{max}}^{-1}}^{B_{0}^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{g_{0}}{\sqrt{1-\unicode[STIX]{x1D706}B_{0}}}=0, & & \displaystyle\end{eqnarray}$$

which can be solved by finding the solution of

(4.40) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }\text{d}vv^{2}g_{0}=0. & & \displaystyle\end{eqnarray}$$

One could formally proceed as done after equation (4.17), but it seems difficult to find a completely explicit solution for $\unicode[STIX]{x1D711}^{(1)}$ due to the relatively complicated dependence of $\unicode[STIX]{x2202}_{r}J^{(0)}$ on $v$ and $\unicode[STIX]{x1D706}$ . In any case, it is clear that imposing (4.40) implies

(4.41) $$\begin{eqnarray}J_{\unicode[STIX]{x1D711}}^{(1)}\sim \unicode[STIX]{x1D716}^{1/2}v_{t}R_{0}\end{eqnarray}$$

and, therefore,

(4.42) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \unicode[STIX]{x1D716}\frac{T}{e}.\end{eqnarray}$$

In addition, from (4.40) we can infer that $\unicode[STIX]{x1D711}^{(1)}$ is stellarator symmetric.

Note that for $\unicode[STIX]{x1D708}_{\ast }\sim \unicode[STIX]{x1D70C}_{\ast }$ , expression (4.42) matches (4.24).

4.3 Superbanana-plateau regime

In this subsection, we treat cases with $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D70C}_{\ast }\ll 1$ and such that $\unicode[STIX]{x2202}_{r}J^{(0)}$ vanishes for $v\lesssim v_{t}$ . Sections 4.1 and 4.2, although brief, were self-contained to a large extent. In this subsection we will refer the reader to Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) more frequently for the derivation of particularly technical results.

The zeroes of $\unicode[STIX]{x2202}_{r}J^{(0)}$ correspond to ‘resonant’ points of phase space where the orbit-averaged tangential component of the magnetic and $E\times B$ drifts cancel each other.

Recall equation (2.19). The condition $\unicode[STIX]{x2202}_{r}J^{(0)}=0$ can be compactly expressed as

(4.43) $$\begin{eqnarray}\unicode[STIX]{x1D706}\overline{\unicode[STIX]{x2202}_{r}B_{0}}(r,\unicode[STIX]{x1D706})=-\frac{2Ze\unicode[STIX]{x1D711}_{0}^{\prime }(r)}{mv^{2}},\end{eqnarray}$$

where

(4.44) $$\begin{eqnarray}\overline{(\cdot )}=\frac{1}{\unicode[STIX]{x1D70F}_{b}^{(0)}}\mathop{\sum }_{\unicode[STIX]{x1D70E}}\int _{l_{b_{10}}}^{l_{b_{20}}}(\cdot )|v_{\Vert }^{(0)}|^{-1}\,\text{d}l\end{eqnarray}$$

denotes orbit average and

(4.45) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{b}^{(0)}(r,v,\unicode[STIX]{x1D706})=2\int _{l_{b_{10}}}^{l_{b_{20}}}|v_{\Vert }^{(0)}|^{-1}\,\text{d}l\end{eqnarray}$$

is the time that it takes for the particle to complete the orbit in the magnetic field $B_{0}$ . A necessary (but not sufficient) condition for this equation to have solutions for $v\lesssim v_{t}$ is

(4.46) $$\begin{eqnarray}\frac{Ze|\unicode[STIX]{x1D711}_{0}^{\prime }|}{T}\lesssim \frac{1}{R_{0}}.\end{eqnarray}$$

That is, the effect of the resonances is expected to be important for relatively small values of the radial electric field. When $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D70C}_{\ast }\ll 1$ and $Ze|\unicode[STIX]{x1D711}_{0}^{\prime }|/T\gg R_{0}^{-1}$ , the treatment of subsection (4.2) applies.

From here on, let us assume $\unicode[STIX]{x1D708}_{\ast }\ll \unicode[STIX]{x1D70C}_{\ast }\ll 1$ and condition (4.46) in the rest of this subsection, and let us also assume that, for $v\lesssim v_{t}$ , there exists one solution in $\unicode[STIX]{x1D706}$ of the resonance condition (4.43), that we denote by $\unicode[STIX]{x1D706}_{r}(r,v,\unicode[STIX]{x1D711}_{0}^{\prime })$ . In general, $\unicode[STIX]{x1D706}_{r}$ will be defined only for some values of $r$ , $v$ and $\unicode[STIX]{x1D711}_{0}^{\prime }$ . For fixed $r$ and $\unicode[STIX]{x1D711}_{0}^{\prime }$ , we denote by $I$ the interval in $v$ for which $\unicode[STIX]{x1D706}_{r}$ is defined.

We can expand the drift-kinetic equation (2.18) around the position of the resonance. Defining $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{r}(r,v,\unicode[STIX]{x1D711}_{0}^{\prime })$ and writing the drift-kinetic equation in coordinates $v$ and $\unicode[STIX]{x1D709}$ , we have

(4.47) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x2202}_{r}J_{r}^{(0)}\unicode[STIX]{x1D709}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g_{\text{rl}}+k\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}g_{\text{rl}}=(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J_{B,r}^{(1)}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}})\unicode[STIX]{x1D6F6}F_{M}, & & \displaystyle\end{eqnarray}$$

where $k(r,v)$ is given by

(4.48) $$\begin{eqnarray}k=\frac{2Ze\unicode[STIX]{x1D6F9}}{mv}\left[\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D706}_{r}\int _{l_{b_{10}}}^{l_{b_{20}}}B_{0}^{-1}\sqrt{1-\unicode[STIX]{x1D706}_{r}B_{0}}\,\text{d}l+\frac{\unicode[STIX]{x1D708}_{v}v^{2}}{2}(\unicode[STIX]{x2202}_{v}\unicode[STIX]{x1D706}_{r})^{2}\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{1}{\sqrt{1-\unicode[STIX]{x1D706}_{r}B_{0}}}\,\text{d}l\right],\end{eqnarray}$$

subindices $r$ indicate that the corresponding quantity is evaluated at $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{r}(r,v,\unicode[STIX]{x1D711}_{0}^{\prime })$ and $\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}}$ is an approximation of $J_{\unicode[STIX]{x1D711}}^{(1)}$ around $\unicode[STIX]{x1D706}_{r}$ . In particular, $\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}}$ is a function of $v$ and $\unicode[STIX]{x1D709}$ . We will see shortly why, in general, we cannot simply evaluate $J_{\unicode[STIX]{x1D711}}^{(1)}$ at $\unicode[STIX]{x1D706}_{r}$ . The second term in (4.48) comes from the energy diffusion piece of the collision operator, which must be included because the coordinate across the layer (the coordinate in which $g_{\text{rl}}$ varies fast), $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{r}$ , mixes $v$ and $\unicode[STIX]{x1D706}$ . The collision frequency $\unicode[STIX]{x1D708}_{v}$ is given by

(4.49) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{v}(v)=\frac{3\sqrt{\unicode[STIX]{x03C0}}\,\unicode[STIX]{x1D712}\left(v\sqrt{m/(2T)}\right)}{2\unicode[STIX]{x1D70F}_{ii}\left(v\sqrt{m/(2T)}\right)^{3}}.\end{eqnarray}$$

In Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017), the energy diffusion piece of the collision operator was incorrectly neglected. Note that none of the scalings derived in Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) are affected by the inclusion of energy diffusion.

Using that $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x2202}_{r}J_{r}^{(0)}=O(B_{0}v_{t})$ and that $k(r,v)=O(\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}B_{0}^{-2}R_{0}\unicode[STIX]{x1D70C}_{\ast }^{-1})$ , and balancing the two terms on the left-hand side of (4.47), we deduce that $g_{\text{rl}}$ is localized in a layer whose size in the coordinate $\unicode[STIX]{x1D709}$ is

(4.50) $$\begin{eqnarray}B_{0}\unicode[STIX]{x1D6E5}_{\text{sb-p}}\sim (\unicode[STIX]{x1D708}_{\ast }/\unicode[STIX]{x1D70C}_{\ast })^{1/3}.\end{eqnarray}$$

Then, balancing the left- and right-hand sides of (4.47), we find that

(4.51) $$\begin{eqnarray}g_{\text{rl}}\sim (B_{0}\unicode[STIX]{x1D6E5}_{\text{sb-p}})^{-1}F_{M}.\end{eqnarray}$$

This resonant layer, via $g_{\text{rl}}$ , is responsible for superbanana-plateau transport.

The solution outside the resonant layer is given by (4.30). Since $g_{0}$ diverges at $\unicode[STIX]{x1D706}_{r}$ , we remove the divergence so that we obtain a function $g_{0}^{\text{out}}$ which is finite everywhere and asymptotically coincides with $g_{0}$ far from the resonant layer. Namely,

(4.52) $$\begin{eqnarray}\displaystyle \hspace{-18.49411pt}g_{0}^{\text{ o ut}}=g_{0}-\frac{1}{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{r})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x2202}_{r}J_{r}^{(0)}}\left(\widetilde{J}_{B,r}^{(1)}+\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}}-\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}}\text{d}\unicode[STIX]{x1D6FC}\right)\unicode[STIX]{x1D6F6}F_{M}, & & \displaystyle\end{eqnarray}$$

where $\widetilde{J}_{B,r}^{(1)}$ is the quantity defined in (4.35) evaluated at $\unicode[STIX]{x1D706}_{r}$ .

In principle, both pieces, $g_{0}^{\text{out}}$ and $g_{\text{rl}}$ , contribute equally to the quasineutrality equation (although only $g_{\text{rl}}$ contributes to radial transport). Hence, one needs to solve

(4.53) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{(1)}=\frac{2\unicode[STIX]{x03C0}B_{0}}{en}\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\left[\int _{0}^{\infty }\text{d}v\int _{B_{0,\text{max}}^{-1}}^{B^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{v^{3}g_{0}^{\text{out}}}{|v_{\Vert }^{(0)}|}+\int _{I}\text{d}v\int _{K\unicode[STIX]{x1D6E5}_{\text{sb-p}}}\text{d}\unicode[STIX]{x1D706}\frac{v^{3}g_{\text{r l}}}{|v_{\Vert }^{(0)}|}\right], & & \displaystyle\end{eqnarray}$$

with $K\gg 1$ . An explicit, asymptotically correct expression for the quasineutrality equation is given by (see Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) for the details)

(4.54) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{(1)} & = & \displaystyle \frac{2\unicode[STIX]{x03C0}B_{0}}{en}\left(\frac{Z}{T}+\frac{1}{T_{e}}\right)^{-1}\left\{\int _{0}^{\infty }\text{d}v\int _{B_{0,\text{max}}^{-1}}^{B^{-1}}\text{d}\unicode[STIX]{x1D706}\frac{v^{3}}{|v_{\Vert }^{(0)}|}g_{0}^{\text{out}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\int _{I}\text{d}v\,v^{2}\int _{-\infty }^{\unicode[STIX]{x1D706}_{L}(l)}\text{d}\unicode[STIX]{x1D706}\frac{g_{\text{rl}}}{\sqrt{\unicode[STIX]{x1D706}_{r}|\unicode[STIX]{x2202}_{l}B_{0}(l_{L})|(l-l_{L})-(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{r})B_{0}(l_{L})}}\nonumber\\ \displaystyle & & \displaystyle +\,\int _{I}\text{d}v\,v^{2}\int _{-\infty }^{\unicode[STIX]{x1D706}_{R}(l)}\text{d}\unicode[STIX]{x1D706}\frac{g_{\text{rl}}}{\sqrt{\unicode[STIX]{x1D706}_{r}|\unicode[STIX]{x2202}_{l}B_{0}(l_{R})|(l_{R}-l)-(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{r})B_{0}(l_{R})}}\nonumber\\ \displaystyle & & \displaystyle +\,\int _{I}\text{d}v\,v^{2}\left[\frac{1}{\sqrt{1-\unicode[STIX]{x1D706}_{r}B_{0}(l)}}-\frac{1}{\sqrt{\unicode[STIX]{x1D706}_{r}|\unicode[STIX]{x2202}_{l}B_{0}(l_{L})|(l-l_{L})}}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.\left.\frac{1}{\sqrt{\unicode[STIX]{x1D706}_{r}|\unicode[STIX]{x2202}_{l}B_{0}(l_{R})|(l_{R}-l)}}\right]\int _{-\infty }^{\infty }\text{d}\unicode[STIX]{x1D706}\,g_{\text{rl}}\right\},\end{eqnarray}$$

where $l_{L}$ and $l_{R}$ denote the bounce points of the trajectory $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{r}$ ,

(4.55) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{L}(l)-\unicode[STIX]{x1D706}_{r}=\frac{\unicode[STIX]{x1D706}_{r}|\unicode[STIX]{x2202}_{l}B_{0}(l_{L})|(l-l_{L})}{B_{0}(l_{L})}\end{eqnarray}$$

and

(4.56) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{R}(l)-\unicode[STIX]{x1D706}_{r}=\frac{\unicode[STIX]{x1D706}_{r}|\unicode[STIX]{x2202}_{l}B_{0}(l_{R})|(l_{R}-l)}{B_{0}(l_{R})}.\end{eqnarray}$$

Expression (4.54) is useful in order to explain why we did not evaluate $J_{\unicode[STIX]{x1D711}}^{(1)}$ at $\unicode[STIX]{x1D706}_{r}$ in (4.47). First, note that the first term in brackets in the fourth line of (4.54) diverges at $l=l_{L}$ and $l=l_{R}$ . We distinguish two cases:

1. (i) If

   (4.57) $$\begin{eqnarray}\left|\frac{2Ze\unicode[STIX]{x1D711}_{0}^{\prime }}{mv^{2}\unicode[STIX]{x1D706}_{r}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\overline{\unicode[STIX]{x2202}_{r}B_{0}}(\unicode[STIX]{x1D706}_{r})}\right|\gg \unicode[STIX]{x1D6E5}_{\text{sb-p}},\end{eqnarray}$$
   the dependence of $\unicode[STIX]{x1D706}_{r}$ on $v$ is strong enough for the integral over $v$ to smooth out the divergence of $1/\sqrt{1-\unicode[STIX]{x1D706}_{r}B_{0}(l)}$ at $l=l_{L}$ and $l=l_{R}$ . Then,
   (4.58) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \frac{T}{e}.\end{eqnarray}$$
   In this case, $\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}}$ can be replaced by $J_{\unicode[STIX]{x1D711},r}^{(1)}$ in (4.47), where
   (4.59) $$\begin{eqnarray}J_{\unicode[STIX]{x1D711},r}^{(1)}=-\frac{2Ze}{mv}\int _{l_{b_{10}}}^{l_{b_{20}}}\frac{\unicode[STIX]{x1D711}^{(1)}}{\sqrt{1-\unicode[STIX]{x1D706}_{r}B_{0}}}\,\text{d}l,\end{eqnarray}$$
   

   Figure 9. If (4.60) is satisfied, the resonant value $\unicode[STIX]{x1D706}_{r}$ is approximately independent of $v$ . The accumulated effect of the resonance for all possible values of $v$ produces a larger $\unicode[STIX]{x1D711}^{(1)}$ in a neighbourhood of the bounce points of the resonant trajectory, $l_{L}$ and $l_{R}$ .

   and the solution of (4.47) can be found analytically, with $g_{\text{rl}}$ approaching a Dirac delta distribution when the collision frequency tends to zero. This gives the standard superbanana-plateau regime in which the radial neoclassical fluxes are strictly independent of the collisionality. This regime was treated in Shaing (Reference Shaing2015) for tokamaks of finite aspect ratio with broken symmetry.

2. (ii) If

   (4.60) $$\begin{eqnarray}\left|\frac{2Ze\unicode[STIX]{x1D711}_{0}^{\prime }}{mv^{2}\unicode[STIX]{x1D706}_{r}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\overline{\unicode[STIX]{x2202}_{r}B_{0}}(\unicode[STIX]{x1D706}_{r})}\right|\ll \unicode[STIX]{x1D6E5}_{\text{sb-p}},\end{eqnarray}$$
   then $\unicode[STIX]{x1D706}_{r}$ is approximately independent of $v$ and no such smoothing happens. The effect of the accumulation of the resonances at the same $\unicode[STIX]{x1D706}_{r}$ for every $v$ means that, close to the bounce points of the resonant trajectory, the electrostatic potential becomes very large. Specifically, one finds that
   (4.61) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \left\{\begin{array}{@{}ll@{}}(\unicode[STIX]{x1D708}_{\ast }/\unicode[STIX]{x1D70C}_{\ast })^{-1/6}T/e\quad & |l-l_{j}|\sim B_{0}\unicode[STIX]{x1D6E5}_{\text{sb-p}}R_{0}\\ T/e\quad & |l-l_{j}|\gg B_{0}\unicode[STIX]{x1D6E5}_{\text{sb-p}}R_{0},\end{array}\right.\end{eqnarray}$$
   where $j=L,R$ (see figure 9). This subregime of the superbanana-plateau regime was discovered in Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) and we will focus on it in the next subsection, devoted to illustrating it numerically. When (4.60) holds, the radial neoclassical fluxes depend logarithmically on collisionality. The estimation (4.50) for the size of the layer also incorporates logarithmic corrections.

Figure 10. Plot of $\unicode[STIX]{x2202}_{r}J^{(0)}$ as a function of $\unicode[STIX]{x1D706}$ in our model omnigenous magnetic field, with $\unicode[STIX]{x1D716}=1.1\times 10^{-4}$ and $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ . The value of $\unicode[STIX]{x1D706}$ at which $\unicode[STIX]{x2202}_{r}J^{(0)}$ vanishes determines the resonant value $\unicode[STIX]{x1D706}_{r}$ . Here, $(B_{0,\text{max}}\unicode[STIX]{x1D706}_{r}-1)/(B_{0,\text{max}}/B_{0,\text{min}}-1)=0.04$ .

Figure 11. Contour plot of $\unicode[STIX]{x1D711}^{(1)}$ for $\unicode[STIX]{x1D716}=1.1\times 10^{-4}$ , $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ and three values of the collisionality. From left to right, $\unicode[STIX]{x1D708}_{\ast }=1.2\times 10^{-12}$ , $\unicode[STIX]{x1D708}_{\ast }=1.3\times 10^{-13}$ , $\unicode[STIX]{x1D708}_{\ast }=1.3\times 10^{-14}$ . The black lines are the two curves determined by the left and right bounce points of the resonant trajectory, $l_{L}$ and $l_{R}$ , respectively. When $\unicode[STIX]{x1D708}_{\ast }$ decreases, the increase of $\unicode[STIX]{x1D711}^{(1)}$ around $l_{L}$ and $l_{R}$ , as predicted by (4.67), is confirmed. The colour scale has been kept constant in the three panels, so that the increase in $\unicode[STIX]{x1D711}^{(1)}$ becomes clearer; that is, in (b,c), $e|\unicode[STIX]{x1D711}^{(1)}|/(T\unicode[STIX]{x1D716})$ reaches larger values than the largest one indicated in the colour scale bar.

Figure 12. Size of $\unicode[STIX]{x1D711}^{(1)}$ as a function of $\unicode[STIX]{x1D708}_{\ast }$ and $\unicode[STIX]{x1D716}$ . Here, $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ . The two scalings given in (4.67) are checked. Full squares give the size of $\unicode[STIX]{x1D711}^{(1)}$ at generic points, $|l-l_{j}|\gg B_{00}\unicode[STIX]{x1D6E5}_{\text{sb-p}}R_{0}$ , whereas empty squares give the size of $\unicode[STIX]{x1D711}^{(1)}$ at a point that is very close to a bounce point of the resonant trajectory, $|l-l_{j}|\sim B_{00}\unicode[STIX]{x1D6E5}_{\text{sb-p}}R_{0}$ (that is, empty squares give approximately the maximum of $\unicode[STIX]{x1D711}^{(1)}$ on the flux surface).

Finally, we point out that, as clearly inferred from inspection of (4.47), $g_{\text{rl}}$ has no definite parity under stellarator symmetry transformations, and neither does $\unicode[STIX]{x1D711}^{(1)}$ .

4.3.1 Superbanana-plateau regime in a large aspect ratio stellarator

We start by giving some estimates that are needed below. First, note that $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x2202}_{r}J_{r}^{(0)}\sim \unicode[STIX]{x1D716}^{-3/2}v_{t}B_{00}$ . Second, observe that for large aspect ratio $\unicode[STIX]{x2202}_{v}\unicode[STIX]{x1D706}_{r}\sim \unicode[STIX]{x1D716}/(B_{00}v_{t})$ so that the term proportional to $\unicode[STIX]{x1D708}_{v}$ in (4.48) can be neglected compared to the term proportional to $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ , giving $k\sim \unicode[STIX]{x1D716}^{3/2}\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}B_{00}^{-2}R_{0}\unicode[STIX]{x1D70C}_{\ast }^{-1}$ . Then, balancing the two terms on the left-hand side of the drift-kinetic equation (4.47), we find that, for $\unicode[STIX]{x1D716}\ll 1$ ,

(4.62) $$\begin{eqnarray}B_{00}\unicode[STIX]{x1D6E5}_{\text{sb-p}}\sim \unicode[STIX]{x1D716}\,\left(\frac{\unicode[STIX]{x1D708}_{\ast }}{\unicode[STIX]{x1D70C}_{\ast }}\right)^{1/3}.\end{eqnarray}$$

For convenience, define $g_{\text{rl}}[J_{B}^{(1)}]$ and $g_{\text{rl}}[J_{\unicode[STIX]{x1D711}}^{(1)}]$ as the solutions of

(4.63) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x2202}_{r}J_{r}^{(0)}\unicode[STIX]{x1D709}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g_{\text{rl}}[J_{B}^{(1)}]+k\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}g_{\text{rl}}[J_{B}^{(1)}]=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}J_{B,r}^{(1)}\unicode[STIX]{x1D6F6}F_{M} & & \displaystyle\end{eqnarray}$$

and

(4.64) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x2202}_{r}J_{r}^{(0)}\unicode[STIX]{x1D709}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}g_{\text{rl}}[J_{\unicode[STIX]{x1D711}}^{(1)}]+k\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}g_{\text{rl}}[J_{\unicode[STIX]{x1D711}}^{(1)}]=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\widehat{J_{\unicode[STIX]{x1D711}}^{(1)}}\unicode[STIX]{x1D6F6}F_{M}, & & \displaystyle\end{eqnarray}$$

respectively. Of course, $g_{\text{rl}}=g_{\text{rl}}[J_{B}^{(1)}]+g_{\text{rl}}[J_{\unicode[STIX]{x1D711}}^{(1)}]$ . Then,

(4.65) $$\begin{eqnarray}g_{\text{rl}}[J_{B}^{(1)}]\sim \left(\frac{\unicode[STIX]{x1D708}_{\ast }}{\unicode[STIX]{x1D70C}_{\ast }}\right)^{-1/3}\frac{n}{v_{t}^{3}}\end{eqnarray}$$

and

(4.66) $$\begin{eqnarray}g_{\text{rl}}[J_{\unicode[STIX]{x1D711}}^{(1)}]\sim \unicode[STIX]{x1D716}^{-1/2}\left(\frac{\unicode[STIX]{x1D708}_{\ast }}{\unicode[STIX]{x1D70C}_{\ast }}\right)^{-1/3}\frac{J_{\unicode[STIX]{x1D711}}^{(1)}}{v_{t}R_{0}}\frac{n}{v_{t}^{3}}.\end{eqnarray}$$

One can deal with the quasineutrality equation (4.54) by employing again the procedure followed after (4.17) and also in § 4.2.2. The result is that

(4.67) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{(1)}\sim \left\{\begin{array}{@{}ll@{}}(\unicode[STIX]{x1D708}_{\ast }/\unicode[STIX]{x1D70C}_{\ast })^{-1/6}\unicode[STIX]{x1D716}\,T/e\quad & |l-l_{j}|\sim B_{00}\unicode[STIX]{x1D6E5}_{\text{sb-p}}R_{0}\\ \unicode[STIX]{x1D716}\,T/e\quad & |l-l_{j}|\gg B_{00}\unicode[STIX]{x1D6E5}_{\text{sb-p}}R_{0}.\end{array}\right.\end{eqnarray}$$

Let us check numerically the scalings of this subsection. First, in figure 10, we show the dependence of $\unicode[STIX]{x2202}_{r}J^{(0)}$ on $\unicode[STIX]{x1D706}$ for $\unicode[STIX]{x1D711}_{0}^{\prime }=0$ in our model magnetic configuration and we give, in particular, the location in phase space of the resonant value $\unicode[STIX]{x1D706}_{r}$ . In figure 11, contour plots of $\unicode[STIX]{x1D711}^{(1)}$ are shown for three different values of the collisionality, so that one can see how $\unicode[STIX]{x1D711}^{(1)}$ becomes larger and larger around $l_{L}$ and $l_{R}$ as the collisionality decreases. Figure 12 provides a precise numerical check of the two scalings given in (4.67).