2 Transit averaged formulation

The Boozer (Reference Boozer1981) and Clebsch representations for a stellarator magnetic field are

(2.1) $$\begin{eqnarray}\displaystyle \boldsymbol{B}=B\boldsymbol{b}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}=K(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701})\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}+G(\unicode[STIX]{x1D713})\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}+I(\unicode[STIX]{x1D713})\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}, & & \displaystyle\end{eqnarray}$$

with $K(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701})$ periodic in the poloidal, $\unicode[STIX]{x1D717}$ , and toroidal, $\unicode[STIX]{x1D701}$ , angle variables, and

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D701}-q\unicode[STIX]{x1D717}=\unicode[STIX]{x1D701}-\not \unicode[STIX]{x1D704}^{\,-1}\unicode[STIX]{x1D717}, & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D713}$ the poloidal flux function, $q=q(\unicode[STIX]{x1D713})$ the safety factor and $\not \unicode[STIX]{x1D704}=q^{-1}$ the rotational transform. The toroidal current within a flux surface is $2G/c$ , while the poloidal current outside the same flux surface is $2I/c$ . The preceding give

(2.3) $$\begin{eqnarray}\displaystyle \boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}=q\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D717}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}=q\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}=qB^{2}/(G+qI), & & \displaystyle\end{eqnarray}$$

as well as $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=0=\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}$ . Assuming a large aspect stellarator the approximations $I\simeq RB_{t}$ and $G\simeq rB_{p}$ may be employed ( $I$ and $G$ are interchanged here to more closely conform with tokamak notation). Then $\text{d}\boldsymbol{r}\times \boldsymbol{B}=0$ gives $Ir^{2}\,\text{d}\unicode[STIX]{x1D717}\simeq GR^{2}\,\text{d}\unicode[STIX]{x1D701}=qGR^{2}\,\text{d}\unicode[STIX]{x1D717}$ and $RG/rI\simeq B_{p}/B_{t}\sim r/qR\ll 1$ , where $B_{p}$ and $B_{t}$ are the poloidal and toroidal magnetic fields to lowest order.

The depth $\unicode[STIX]{x1D6FF}$ of the quasisymmetric helical well is

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}=(B_{\max }-B_{\min })/(B_{\max }+B_{\min }). & & \displaystyle\end{eqnarray}$$

The magnetic field departures from an optimized quasisymmetric stellarator of $N$ toroidal cells ( $N\neq 0$ ) and well depth $\unicode[STIX]{x1D6FF}$ are assumed to be weak and are written in the simple form

(2.5) $$\begin{eqnarray}\displaystyle B=B_{0}[1-\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D713})\cos (M\unicode[STIX]{x1D717}-N\unicode[STIX]{x1D701})-\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D6FF}_{m}^{n}(\unicode[STIX]{x1D713})\cos (m\unicode[STIX]{x1D717}-n\unicode[STIX]{x1D701})], & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{m}^{n}\ll \unicode[STIX]{x1D6FF}$ and the sum can be over $m$ and $n$ not equal to $M$ and $N$ , respectively. In this form a mod $B$ contour closes on itself after $M$ toroidal turns and $N$ poloidal turns. To avoid introducing ripple wells along the magnetic field $\boldsymbol{B}$ , requires $(qn-m)\unicode[STIX]{x1D6FF}_{m}^{n}\ll (qN-M)\unicode[STIX]{x1D6FF}$ since the minima of the field are given by

(2.6) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}\left.\left(\frac{B}{B_{0}}\right)\right|_{\unicode[STIX]{x1D6FC}}=(q^{-1}M-N)\unicode[STIX]{x1D6FF}\sin (M\unicode[STIX]{x1D717}-N\unicode[STIX]{x1D701})+\unicode[STIX]{x1D6F4}(q^{-1}m-n)\unicode[STIX]{x1D6FF}_{m}^{n}\sin (m\unicode[STIX]{x1D717}-n\unicode[STIX]{x1D701})=0. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

To streamline the analysis only a single departure term need be retained as the sum can be reinserted at the end of the evaluation. As a result, only

(2.7) $$\begin{eqnarray}\displaystyle B=B_{0}[1-\unicode[STIX]{x1D6FF}\cos (M\unicode[STIX]{x1D717}-N\unicode[STIX]{x1D701})-\unicode[STIX]{x1D6FF}_{m}^{n}\cos (m\unicode[STIX]{x1D717}-n\unicode[STIX]{x1D701})], & & \displaystyle\end{eqnarray}$$

is considered, with $mN=nM$ preserving quasisymmetry since then $N(m\unicode[STIX]{x1D717}-n\unicode[STIX]{x1D701})=n(M\unicode[STIX]{x1D717}-N\unicode[STIX]{x1D701})$ .

It is useful to introduce the quasisymmetric angle variable

(2.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}=M\unicode[STIX]{x1D717}-N\unicode[STIX]{x1D701}, & & \displaystyle\end{eqnarray}$$

and use $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D6FC}$ as the angle variables with $\unicode[STIX]{x1D717}=0=\unicode[STIX]{x1D701}$ a point at the equatorial or stellarator symmetry plane. Then for transit averages it is convenient to integrate over $\unicode[STIX]{x1D702}=M\unicode[STIX]{x1D717}-N\unicode[STIX]{x1D701}$ at fixed $\unicode[STIX]{x1D6FC}$ by using

(2.9) $$\begin{eqnarray}\displaystyle \overline{A}=\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D70F}A}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D70F}}=\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D717}A/v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D717}/v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}=\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}A/v_{||}B}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}/v_{||}B}\simeq \frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}A/v_{||}}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}/v_{||}}, & & \displaystyle\end{eqnarray}$$

where $A$ is arbitrary, $\text{d}\unicode[STIX]{x1D70F}=\text{d}\ell /v_{||}=\text{d}\unicode[STIX]{x1D717}/v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}$ , $q\,\text{d}\unicode[STIX]{x1D717}=\text{d}\unicode[STIX]{x1D701}$ for $\unicode[STIX]{x1D6FC}$ fixed (denoted by the subscript on the integral) for the trapped, and $q\not \unicode[STIX]{x1D704}=1$ with $q$ the safety factor and $\not \unicode[STIX]{x1D704}$ the rotational transform. The integrals are over a full bounce for trapped particles.

In terms of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D702}$ ,

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D717}=(N\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D702})/(M-qN), & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}=(M\unicode[STIX]{x1D6FC}+q\unicode[STIX]{x1D702})/(M-qN), & \displaystyle\end{eqnarray}$$

and

(2.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D702})\equiv m\unicode[STIX]{x1D717}-n\unicode[STIX]{x1D701}=[(mN-nM)\unicode[STIX]{x1D6FC}+(m-qn)\unicode[STIX]{x1D702}]/(M-qN). & & \displaystyle\end{eqnarray}$$

The flux surface average can be written using $\text{d}\unicode[STIX]{x1D717}\,\text{d}\unicode[STIX]{x1D701}=\text{d}\unicode[STIX]{x1D702}\,\text{d}\tilde{\unicode[STIX]{x1D6FC}}$ with $\tilde{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FC}/(M-qN)$ , then the new variables give

(2.13) $$\begin{eqnarray}\displaystyle \langle A\rangle =\frac{\displaystyle \oint \text{d}\unicode[STIX]{x1D701}\oint \text{d}\unicode[STIX]{x1D717}A/B^{2}}{\displaystyle \oint \text{d}\unicode[STIX]{x1D701}\oint \text{d}\unicode[STIX]{x1D717}/B^{2}}=\frac{\displaystyle \oint \text{d}\unicode[STIX]{x1D702}\oint \text{d}\tilde{\unicode[STIX]{x1D6FC}}A/B^{2}}{\displaystyle \oint \text{d}\unicode[STIX]{x1D702}\oint \text{d}\tilde{\unicode[STIX]{x1D6FC}}/B^{2}}=\frac{\displaystyle \oint \text{d}\tilde{\unicode[STIX]{x1D6FC}}\oint _{\unicode[STIX]{x1D6FC}}\text{d}\ell AB^{-1}}{\displaystyle \oint \text{d}\tilde{\unicode[STIX]{x1D6FC}}\oint _{\unicode[STIX]{x1D6FC}}\text{d}\ell B^{-1}}, & & \displaystyle\end{eqnarray}$$

with $\tilde{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FC}/(M-qN)\rightarrow 2\unicode[STIX]{x03C0}+\tilde{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FC}/(M-qN)$ for $M$ toroidal circuits and $N$ poloidal circuits on a constant $B$ curve.

The transit averaged equation

(2.14) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}+\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}=\overline{C\{\,f_{t}\}}, & & \displaystyle\end{eqnarray}$$

must be solved for the trapped correction $f_{t}=f_{t}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E})$ to the slowing down distribution

(2.15) $$\begin{eqnarray}\displaystyle f_{s}(\unicode[STIX]{x1D713},v)=\frac{S(\unicode[STIX]{x1D713})\unicode[STIX]{x1D70F}_{s}(\unicode[STIX]{x1D713})H(v_{0}-v)}{4\unicode[STIX]{x03C0}[v^{3}+v_{c}^{3}(\unicode[STIX]{x1D713})]}, & & \displaystyle\end{eqnarray}$$

where $f=f_{s}+f_{t}$ with $f_{t}\ll f_{s}$ and the density of slowing down alphas is

(2.16) $$\begin{eqnarray}\displaystyle n_{s}=\int \text{d}^{3}vf_{s}=S\unicode[STIX]{x1D70F}_{s}\int _{0}^{v_{0}}\frac{v^{2}\,\text{d}v}{(v^{3}+v_{c}^{3})}=\frac{S\unicode[STIX]{x1D70F}_{s}}{3}\ell n[1+(v_{0}^{3}/v_{c}^{3})]\simeq S\unicode[STIX]{x1D70F}_{s}\ell n(v_{0}/v_{c}), & & \displaystyle\end{eqnarray}$$

where $v_{0}^{3}/v_{c}^{3}\gg 1$ for the deuterium–tritium (D-T) fusion reaction of interest here. The passing correction to $f_{s}$ vanishes since $\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=0$ for the passing, with $\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}$ the radial magnetic drift due to the non-quasisymmetric portion of the magnetic field. Letting $\unicode[STIX]{x1D707}=v_{\bot }^{2}/2B$ be the magnetic moment, and $\unicode[STIX]{x1D70E}=v_{||}/|v_{||}|$ be the sign of the parallel velocity $v_{||}=\unicode[STIX]{x1D70E}\sqrt{\unicode[STIX]{x1D708}^{2}-2\unicode[STIX]{x1D707}B}$ , then the magnetic drift is

(2.17) $$\begin{eqnarray}\displaystyle \boldsymbol{v}_{m}=\unicode[STIX]{x1D6FA}^{-1}\boldsymbol{b}\times (\unicode[STIX]{x1D707}\unicode[STIX]{x1D735}B+v_{||}^{2}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{b})\simeq \unicode[STIX]{x1D6FA}^{-1}v_{||}\unicode[STIX]{x1D735}\times (v_{||}\boldsymbol{b}), & & \displaystyle\end{eqnarray}$$

where the final form is a useful approximation as the parallel velocity correction is negligible. For energetic alphas, the $\boldsymbol{E}\times \boldsymbol{B}$ drift is small and can be ignored. In the preceding equations, $C$ is the alpha collision operator, $S$ is the alpha birth rate, $\unicode[STIX]{x1D70F}_{s}$ is the slowing down time for the alphas, $v_{c}$ is the critical speed at which the drag of the background ions and electrons on the alphas is equal and $v_{0}$ is the alpha birth speed with $H$ a Heaviside step function vanishing for speeds $v>v_{0}$ . The magnetic drift term in a flux surface, $\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}$ , is dominated by the $\unicode[STIX]{x1D735}B$ drift as curvature drift is small for the trapped, and $\unicode[STIX]{x1D6FA}=ZeB/M_{\unicode[STIX]{x1D6FC}}c$ is the alpha gyrofrequency with $M_{\unicode[STIX]{x1D6FC}}$ and $Z$ alpha mass and charge, $c$ light speed and $e$ proton charge.

To evaluate the drifts the divergence of an arbitrary vector $\boldsymbol{A}$ is written as

(2.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{A}=\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\left(\frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\left(\frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\left(\frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)\right], & & \displaystyle\end{eqnarray}$$

giving

(2.19) $$\begin{eqnarray}\displaystyle \boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC} & \simeq & \displaystyle \frac{v_{||}}{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times (v_{||}\boldsymbol{b})\nonumber\\ \displaystyle & = & \displaystyle \frac{v_{||}}{\unicode[STIX]{x1D6FA}}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\left(\frac{Bv_{||}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\left(\frac{v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)\right],\end{eqnarray}$$

and

(2.20) $$\begin{eqnarray}\displaystyle \boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713} & \simeq & \displaystyle \frac{v_{||}}{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times (v_{||}\boldsymbol{b})\nonumber\\ \displaystyle & = & \displaystyle -\frac{v_{||}}{\unicode[STIX]{x1D6FA}}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\left(\frac{Bv_{||}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\left(\frac{v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\right)\right].\end{eqnarray}$$

As a result, neglecting curvature drift and using $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}=(M-qN)\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}$ ,

(2.21) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\simeq \frac{M_{\unicode[STIX]{x1D6FC}}cv^{2}(qN-M)}{Ze\oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\left(\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}}{qN-M}\right), & & \displaystyle\end{eqnarray}$$

and

(2.22) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\simeq -\frac{M_{\unicode[STIX]{x1D6FC}}cv^{2}}{Ze\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}=2\unicode[STIX]{x1D707}B_{0}/v^{2}$ and $\unicode[STIX]{x1D709}^{2}=v_{||}^{2}/v^{2}=1-2\unicode[STIX]{x1D707}B/v^{2}=1-\unicode[STIX]{x1D706}B/B_{0}$ .

The assumption that the departure from quasisymmetry is very small means, of course, that to lowest order

(2.23) $$\begin{eqnarray}\displaystyle B\simeq B_{0}(1-\unicode[STIX]{x1D6FF}\cos \unicode[STIX]{x1D702}), & & \displaystyle\end{eqnarray}$$

can be employed in many places. Indeed, for the purpose of the calculation to be performed here the departure from quasisymmetry only matters for the radial magnetic drift terms. Therefore, except for a very small radial drift due to asymmetry, the alphas try to move on surfaces of constant drift kinetic canonical angular momentum (Boozer Reference Boozer1983; Landreman & Catto Reference Landreman and Catto2011)

(2.24) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D713}}_{\ast }=\unicode[STIX]{x1D713}_{h}-I_{h}(\unicode[STIX]{x1D713})v_{||}/\unicode[STIX]{x1D6FA}, & & \displaystyle\end{eqnarray}$$

where the helical current $I_{h}$ is

(2.25) $$\begin{eqnarray}\displaystyle I_{h}=MI+NG, & & \displaystyle\end{eqnarray}$$

and helical combination of the poloidal flux $2\unicode[STIX]{x03C0}\unicode[STIX]{x1D713}$ and toroidal flux $2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6F9}$ is defined as

(2.26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{h}=M\unicode[STIX]{x1D713}-N\unicode[STIX]{x1D6F9}, & & \displaystyle\end{eqnarray}$$

with $\not \unicode[STIX]{x1D704}\,\text{d}\unicode[STIX]{x1D6F9}=d\unicode[STIX]{x1D713}$ , and $\unicode[STIX]{x1D6FA}=ZeB/M_{\unicode[STIX]{x1D6FC}}c$ the alpha gyrofrequency. The assumption of near quasisymmetry is most nearly satisfied for HSX and QIPC in figures 5 and 8, respectively, of Beidler et al. (Reference Beidler, Allmaier, Isaev, Kasilov, Kernbichler, Leitold, Maaßberg, Mikkelsen, Murakami and Schmidt2011). For these configurations the $B$ field maxima are all nearly the same as assumed here so there are very few transitional particles on collisionless orbits that trap and detrap.

The departure from quasisymmetry is unimportant in the tangential drift term, and the radial variation of $qN-M$ compared to that of $\unicode[STIX]{x1D709}$ is small by roughly $\unicode[STIX]{x1D709}^{2}/(qN-M)\sim \unicode[STIX]{x1D6FF}/(qN-M)\ll 1$ . Therefore, using $2\unicode[STIX]{x1D709}\unicode[STIX]{x2202}\unicode[STIX]{x1D709}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}=-\unicode[STIX]{x1D706}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\cos \unicode[STIX]{x1D702}$ leaves

(2.27) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\simeq \frac{M_{\unicode[STIX]{x1D6FC}}cv^{2}\unicode[STIX]{x1D706}}{2Ze}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}\cos \unicode[STIX]{x1D702}}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}}. & & \displaystyle\end{eqnarray}$$

In addition, the departure from quasisymmetry does not matter in the collision operator. Neglecting curvature drift it may be written as

(2.28) $$\begin{eqnarray}\displaystyle \overline{C\{\,f_{t}\}}\simeq \frac{2v_{\unicode[STIX]{x1D706}}^{3}}{\unicode[STIX]{x1D70F}_{s}v^{3}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}/\unicode[STIX]{x1D709}\right)}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}\left[\unicode[STIX]{x1D706}\left(\oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\right)\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}\right], & & \displaystyle\end{eqnarray}$$

where

(2.29) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\unicode[STIX]{x1D706}}^{3}=\frac{3\unicode[STIX]{x03C0}^{1/2}T_{e}^{3/2}}{(2m)^{1/2}M_{\unicode[STIX]{x1D6FC}}n_{e}}\mathop{\sum }_{i}Z_{i}^{2}n_{i}, & \displaystyle\end{eqnarray}$$

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle v_{c}^{3}=\frac{3\unicode[STIX]{x03C0}^{1/2}T_{e}^{3/2}}{(2m)^{1/2}n_{e}}\mathop{\sum }_{i}\frac{Z_{i}^{2}n_{i}}{M_{i}}, & \displaystyle\end{eqnarray}$$

and

(2.31) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{s}=\frac{3M_{\unicode[STIX]{x1D6FC}}T_{e}^{3/2}}{4(2\unicode[STIX]{x03C0}m)^{1/2}Z^{2}e^{4}n_{e}\ell n\unicode[STIX]{x1D6EC}}, & & \displaystyle\end{eqnarray}$$

with the electron and bulk ion densities and temperatures denoted by $n_{j}$ and $T_{j}$ , and $m$ the electron mass and $M_{i}$ the mass of an ion of charge $Z_{i}$ . For the D-T reaction $v_{\unicode[STIX]{x1D706}}^{3}/v_{c}^{3}=3/5$ .

However, for the radial drift term the departure from quasisymmetry matters. Using $\unicode[STIX]{x1D709}^{2}=1-\unicode[STIX]{x1D706}B/B_{0}$ with $2\unicode[STIX]{x1D709}\unicode[STIX]{x2202}\unicode[STIX]{x1D709}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}|_{\unicode[STIX]{x1D702}}=(mN-nM)(qN-M)^{-1}\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D706}\sin \unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D702})$ yields

(2.32) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\simeq \frac{M_{\unicode[STIX]{x1D6FC}}c(mN-nM)\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D706}v^{2}}{2Ze(qN-M)}\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}\sin \unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D702})}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}}. & & \displaystyle\end{eqnarray}$$

The quasisymmetric radial $\unicode[STIX]{x1D735}B$ drift term is proportional to $\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}\propto \sin \unicode[STIX]{x1D702}$ so results in a vanishing bounce averaged contribution to $\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}$ .

3 Evaluation of the terms in the kinetic equation

Ignoring the departure from quasisymmetry, the full bounce, large aspect ratio results for the transit averages appearing in the tangential drift and collision operator terms are

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}/\unicode[STIX]{x1D709}=8(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706})^{-1/2}\int _{0}^{\unicode[STIX]{x03C0}/2}\text{d}x/\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}x}=8(2\unicode[STIX]{x1D6FF})^{-1/2}\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}K(\unicode[STIX]{x1D705}),\qquad & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}=\frac{8\sqrt{2\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D705}^{2}}{\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}}\int _{0}^{\unicode[STIX]{x03C0}/2}\frac{\text{d}x\cos ^{2}x}{\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}x}}=\frac{8\sqrt{2}\unicode[STIX]{x1D6FF}[E(\unicode[STIX]{x1D705})-(1-\unicode[STIX]{x1D705}^{2})K(\unicode[STIX]{x1D705})]}{\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}},\qquad & \displaystyle\end{eqnarray}$$

and

(3.3) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\cos \unicode[STIX]{x1D702}/\unicode[STIX]{x1D709} & = & \displaystyle 8(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706})^{-1/2}\int _{0}^{\unicode[STIX]{x03C0}/2}\text{d}x(1-2\unicode[STIX]{x1D705}^{2}\sin ^{2}x)/\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}x}\nonumber\\ \displaystyle & = & \displaystyle 8(2\unicode[STIX]{x1D6FF})^{-1/2}\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}[2E(\unicode[STIX]{x1D705})-K(\unicode[STIX]{x1D705})],\end{eqnarray}$$

where $\unicode[STIX]{x1D705}\sin x=\sin (\unicode[STIX]{x1D702}/2)$ and the integrals are performed at fixed $\unicode[STIX]{x1D6FC}$ . Here $K(\unicode[STIX]{x1D705})$ and $E(\unicode[STIX]{x1D705})$ are complete elliptic integrals of the first and second kind, respectively, with

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}^{2}=[1-(1-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D706}]/2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}, & & \displaystyle\end{eqnarray}$$

where the $\unicode[STIX]{x1D705}=0$ alphas are deeply trapped, while the $\unicode[STIX]{x1D705}=1$ ones are barely trapped. The treatment here will assume the $\unicode[STIX]{x1D713}$ dependence of $\unicode[STIX]{x1D705}$ (which also depends on the adiabatic invariant $\unicode[STIX]{x1D706}$ ) is unimportant. As $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D713})$ this means $\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\gg (\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\unicode[STIX]{x2202}f_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D705}$ is assumed – a serious restriction at small collision frequency arises, as will be seen later.

Using these results in the barely trapped limit, $E(\unicode[STIX]{x1D705})\rightarrow 1+(1-\unicode[STIX]{x1D705}^{2})\{\ell n[16/(1-\unicode[STIX]{x1D705}^{2})-1]\}/4+\cdots \,$ and $K(\unicode[STIX]{x1D705})\rightarrow \ell n(4/\sqrt{1-\unicode[STIX]{x1D705}^{2}})+\cdots \,,$ giving

(3.5) $$\begin{eqnarray}\displaystyle \overline{C\{\,f_{t}\}}\simeq \frac{v_{\unicode[STIX]{x1D706}}^{3}}{4\unicode[STIX]{x1D70F}_{s}v^{3}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}K(\unicode[STIX]{x1D705})}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}\left\{\!\frac{[E(\unicode[STIX]{x1D705})-(1-\unicode[STIX]{x1D705}^{2})K(\unicode[STIX]{x1D705})]}{\unicode[STIX]{x1D705}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}\!\right\}\underset{\unicode[STIX]{x1D705}\rightarrow 1}{\rightarrow }\frac{(v_{\unicode[STIX]{x1D706}}^{3}/2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}_{s}v^{3})}{\ell n[8/(1-\unicode[STIX]{x1D705})]}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}},\qquad & & \displaystyle\end{eqnarray}$$

and

(3.6) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\simeq \frac{M_{\unicode[STIX]{x1D6FC}}cv^{2}[2E(\unicode[STIX]{x1D705})-K(\unicode[STIX]{x1D705})]}{2ZeK(\unicode[STIX]{x1D705})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\underset{\unicode[STIX]{x1D705}\rightarrow 1}{\rightarrow }-\frac{M_{\unicode[STIX]{x1D6FC}}cv^{2}}{2Ze}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}. & & \displaystyle\end{eqnarray}$$

To obtain this last result, $(K-2E)/K\rightarrow 1$ is employed.

The numerator of $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}$ vanishes at $\unicode[STIX]{x1D705}_{0}^{2}\simeq 0.83$ when

(3.7) $$\begin{eqnarray}\displaystyle K(\unicode[STIX]{x1D705}_{0})=2E(\unicode[STIX]{x1D705}_{0}). & & \displaystyle\end{eqnarray}$$

Expanding the numerator of the tangential drift about this zero gives

(3.8) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}} & \simeq & \displaystyle \frac{M_{\unicode[STIX]{x1D6FC}}cv^{2}[(1-2\unicode[STIX]{x1D705}_{0}^{2})E(\unicode[STIX]{x1D705}_{0})-(1-\unicode[STIX]{x1D705}_{0}^{2})K(\unicode[STIX]{x1D705}_{0})]}{2Ze\unicode[STIX]{x1D705}_{0}(1-\unicode[STIX]{x1D705}_{0}^{2})K(\unicode[STIX]{x1D705}_{0})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D705}-\unicode[STIX]{x1D705}_{0})\nonumber\\ \displaystyle & = & \displaystyle -\frac{B_{0}v^{2}(\unicode[STIX]{x1D705}-\unicode[STIX]{x1D705}_{0})}{4\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D705}_{0}(1-\unicode[STIX]{x1D705}_{0}^{2})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}},\end{eqnarray}$$

as well as

(3.9) $$\begin{eqnarray}\displaystyle \overline{C\{\,f_{t}\}}\simeq \frac{v_{\unicode[STIX]{x1D706}}^{3}[E(\unicode[STIX]{x1D705}_{0})-(1-\unicode[STIX]{x1D705}_{0}^{2})K(\unicode[STIX]{x1D705}_{0})]}{4\unicode[STIX]{x1D70F}_{s}v^{3}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}_{0}^{2}K(\unicode[STIX]{x1D705}_{0})}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}}=\frac{v_{\unicode[STIX]{x1D706}}^{3}(2\unicode[STIX]{x1D705}_{0}^{2}-1)}{8\unicode[STIX]{x1D70F}_{s}v^{3}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}_{0}^{2}}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}}. & & \displaystyle\end{eqnarray}$$

For the radial drift term, using

(3.10) $$\begin{eqnarray}\displaystyle \sin \unicode[STIX]{x1D717}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D702}) & = & \displaystyle -\text{sin}\left[\frac{(mN-nM)\unicode[STIX]{x1D6FC}}{(qN-M)}\right]\cos \left[\frac{(qn-m)\unicode[STIX]{x1D702}}{(qN-M)}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\cos \left[\frac{(mN-nM)\unicode[STIX]{x1D6FC}}{(qN-M)}\right]\sin \left[\frac{(qn-m)\unicode[STIX]{x1D702}}{(qN-M)}\right],\end{eqnarray}$$

and noticing that the term odd in $\unicode[STIX]{x1D702}$ vanishes, leaves

(3.11) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}} & \simeq & \displaystyle -\frac{M_{\unicode[STIX]{x1D6FC}}c(mN-nM)\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D706}v^{2}}{2Ze(qN-M)}\sin \left[\frac{(mN-nM)\unicode[STIX]{x1D6FC}}{(qN-M)}\right]\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}\cos [(qn-m)\unicode[STIX]{x1D702}/(qN-M)]}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\text{d}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}^{-1}}.\end{eqnarray}$$

The asymptotic expansion of the right side for $\unicode[STIX]{x1D705}\rightarrow 1$ is found as in Catto (Reference Catto2019) by carefully expanding the $\unicode[STIX]{x1D702}$ dependence of $\unicode[STIX]{x1D709}$ about the turning point $\unicode[STIX]{x1D702}_{t}$ , where

(3.12) $$\begin{eqnarray}\displaystyle \sin (\unicode[STIX]{x1D702}_{t}/2)=\sqrt{[1-(1-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D706}]/2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D705}. & & \displaystyle\end{eqnarray}$$

Using $\sin (\unicode[STIX]{x1D702}/2)=\unicode[STIX]{x1D705}\sin x$ leads to the form

(3.13) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\frac{\text{d}\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D709}}\cos \left[\frac{(qn-m)\unicode[STIX]{x1D702}}{qN-M}\right] & = & \displaystyle \frac{8}{(2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706})^{1/2}}\text{Re}\int _{0}^{\unicode[STIX]{x03C0}/2}\frac{\text{d}x\text{e}^{\text{i}(qn-m)\unicode[STIX]{x1D702}/(qN-M)}}{\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}x}}\nonumber\\ \displaystyle & = & \displaystyle \frac{8\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}}{(2\unicode[STIX]{x1D6FF})^{1/2}}\text{Re}\{\text{e}^{\text{i}(qn-m)\unicode[STIX]{x1D702}_{t}/(qN-M)}\tilde{K}(\unicode[STIX]{x1D705})\},\end{eqnarray}$$

where

(3.14) $$\begin{eqnarray}\displaystyle \widetilde{K}(\unicode[STIX]{x1D705})\equiv \int _{0}^{\unicode[STIX]{x03C0}/2}\frac{\text{d}x\text{e}^{\text{i}(qn-m)(\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}_{t})/(qN-M)}}{\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}x}}. & & \displaystyle\end{eqnarray}$$

Then changing to $\unicode[STIX]{x1D702}$ by using $\cos (\unicode[STIX]{x1D702}/2)\,\text{d}\unicode[STIX]{x1D702}=2\unicode[STIX]{x1D705}\cos x\,\text{d}x$ , gives

(3.15) $$\begin{eqnarray}\displaystyle \widetilde{K}(\unicode[STIX]{x1D705})\equiv \frac{1}{2}\int _{0}^{\unicode[STIX]{x1D702}_{t}}\frac{\text{d}\unicode[STIX]{x1D702}\text{e}^{\text{i}(qn-m)(\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}_{t})/(qN-M)}}{\sqrt{\unicode[STIX]{x1D705}^{2}-\sin ^{2}(\unicode[STIX]{x1D702}/2)}}=K(\unicode[STIX]{x1D705})+O\left(\frac{qn-m}{qN-M}\right), & & \displaystyle\end{eqnarray}$$

where for $\unicode[STIX]{x1D705}\rightarrow 1$ , $\unicode[STIX]{x1D702}_{t}\rightarrow \unicode[STIX]{x03C0}$ (see the appendix of Catto Reference Catto2019 for detailed evaluation procedure). Assuming $K\rightarrow \ell n(4/\sqrt{1-\unicode[STIX]{x1D705}^{2}})\gg (qn-m)/(qN-M)\sim 1$ ,

(3.16) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\simeq -\frac{M_{\unicode[STIX]{x1D6FC}}c(mN-nM)\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D706}v^{2}}{2Ze(qN-M)}\cos \left[\frac{(qn-m)\unicode[STIX]{x03C0}}{qN-M}\right]\sin \left[\frac{(mN-nM)\unicode[STIX]{x1D6FC}}{(qN-M)}\right], & & \displaystyle\end{eqnarray}$$

may be used (recall $qN\neq M$ ). This approximation is equivalent to assuming that $\unicode[STIX]{x1D702}$ does not vary significantly as the barely trapped alphas reflect.

Inserting the preceding $\unicode[STIX]{x1D705}\rightarrow 1$ results in the transit averaged kinetic equation yields the trapped–passing boundary layer equation that must be solved

(3.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}}{2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}_{s}v^{3}\ell n[8/(1-\unicode[STIX]{x1D705})]}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}}+\frac{B_{0}v^{2}}{2\unicode[STIX]{x1D6FA}_{0}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{(mN-nM)B_{0}v^{2}\unicode[STIX]{x1D6FF}_{m}^{n}}{2\unicode[STIX]{x1D6FA}_{0}(qN-M)}\cos \left[\frac{(qn-m)\unicode[STIX]{x03C0}}{(qN-M)}\right]\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\text{Im}[\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}],\end{eqnarray}$$

where $\text{Im}$ denotes imaginary part and $\unicode[STIX]{x1D6FA}_{0}=ZeB_{0}/M_{\unicode[STIX]{x1D6FC}}c$ . The kinetic equation to be solved about the zero of the tangential drive is

(3.18) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}(2\unicode[STIX]{x1D705}_{0}^{2}-1)}{8\unicode[STIX]{x1D70F}_{s}v^{3}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}_{0}^{2}}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}}+\frac{B_{0}v^{2}(\unicode[STIX]{x1D705}-\unicode[STIX]{x1D705}_{0})}{4\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D705}_{0}(1-\unicode[STIX]{x1D705}_{0}^{2})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{B_{0}(mN-nM)\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D706}v^{2}}{2\unicode[STIX]{x1D6FA}_{0}(qN-M)}\cos \left[\frac{(qn-m)\unicode[STIX]{x03C0}}{qN-M}\right]\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\text{Im}[\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}].\qquad\end{eqnarray}$$

In the preceding expressions $v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{3}\unicode[STIX]{x1D70F}_{s}\equiv \unicode[STIX]{x1D708}$ is the pitch angle scattering frequency. For alphas the regime of interest is $\unicode[STIX]{x1D70C}_{0}/R\gg qRv_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{4}\unicode[STIX]{x1D70F}_{s}=qR\unicode[STIX]{x1D708}/v_{0}$ so $1/\unicode[STIX]{x1D708}$ transport is never of interest.

In the next section the condition for a narrow collisional boundary layer is estimated for the $\sqrt{\unicode[STIX]{x1D708}}$ regime, as is the $\sqrt{\unicode[STIX]{x1D708}}$ diffusivity. Then in § 5 the estimates are made for the superbanana plateau case.

4 Phenomenological estimates for $\surd \unicode[STIX]{x1D708}$ regime

To get a boundary layer narrower than the more familiar $\unicode[STIX]{x1D6FF}^{1/2}$ width, pitch angle scattering must be the dominant collisional process. A balance between the strong $\unicode[STIX]{x1D735}B$ drift of the alphas tangential to the flux surface and collisions is only possible if the width $w$ in pitch angle $\unicode[STIX]{x1D706}$ of the boundary layer narrows to decrease the effective pitch angle scattering time $\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}$ . Using

(4.1) $$\begin{eqnarray}\displaystyle \overline{C\{\,f_{t}\}}\sim \frac{v_{\unicode[STIX]{x1D706}}^{3}}{\unicode[STIX]{x1D70F}_{s}v_{0}^{3}}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}^{2}}\sim \frac{v_{\unicode[STIX]{x1D706}}^{3}f_{t}}{\unicode[STIX]{x1D70F}_{s}v_{0}^{3}w^{2}}\sim \frac{(mN-nM)B_{0}v_{0}^{2}}{(qN-M)\unicode[STIX]{x1D6FA}_{0}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}f_{t}\sim \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x2202}f_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\sim f_{t}(mN-nM)/(qN-M)$ , gives the normalized width of the boundary layer $w$ to be

(4.2) $$\begin{eqnarray}\displaystyle w\sim [(qN-M)/(mN-nM)B_{0}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}]^{1/2}(v_{\unicode[STIX]{x1D706}}/v_{0})^{3/2}\ll \unicode[STIX]{x1D6FF}^{1/2}\sim (r/R)^{1/2}, & & \displaystyle\end{eqnarray}$$

indicating that the alphas must $\unicode[STIX]{x1D735}B$ drift on a flux surface much faster than they are pitch angle scattered by the ions. This condition for the boundary layer analysis to be valid is normally satisfied. For D-T with $R=10~\text{m}$ , $B=5T$ , $T_{i}\simeq 10~\text{keV}$ and $n_{e}\simeq 10^{14}~\text{cm}^{-3}$ , $R/\unicode[STIX]{x1D70C}_{0}\sim 10^{2}$ and $v_{0}\unicode[STIX]{x1D70F}_{s}/R\sim 10^{6}$ , while $v_{0}/v_{c}\simeq 3.25$ , and $R^{2}B_{p}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\sim 1$ , so a narrow boundary layer will occur.

The effective barely trapped fraction is estimated from this boundary layer width to be

(4.3) $$\begin{eqnarray}\displaystyle F\sim w\sim [(qN-M)rR/q(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}]^{1/2}(v_{\unicode[STIX]{x1D706}}/v_{0})^{3/2}. & & \displaystyle\end{eqnarray}$$

Then the effective drift decorrelation time for the alphas is the pitch angle scattering time multiplied by the fraction squared,

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}\sim F^{2}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}\sim (qN-M)Rr/q(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}. & & \displaystyle\end{eqnarray}$$

The radial speed $V$ of the alphas is

(4.5) $$\begin{eqnarray}\displaystyle V\simeq \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}/RB_{p}\sim q(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D6FF}_{m}^{n}/(qN-M)r, & & \displaystyle\end{eqnarray}$$

giving the effective radial step size to be

(4.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}\sim V\unicode[STIX]{x1D70F}\sim R\unicode[STIX]{x1D6FF}_{m}^{n}. & & \displaystyle\end{eqnarray}$$

Consequently, the weak ripple diffusivity is

(4.7) $$\begin{eqnarray}\displaystyle D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{weak}}\sim F\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D70F} & {\sim} & \displaystyle \left[\frac{q(mN-nM)}{qN-M}\right]^{1/2}\frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}\unicode[STIX]{x1D70C}_{0}^{1/2}v_{\unicode[STIX]{x1D706}}^{3/2}R^{3/2}}{r^{1/2}v_{0}\unicode[STIX]{x1D70F}_{s}^{1/2}}\nonumber\\ \displaystyle & {\sim} & \displaystyle (\unicode[STIX]{x1D6FF}_{m}^{n})^{2}R^{2}\left[\frac{q(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{\unicode[STIX]{x1D706}}^{3}}{(qN-M)Rrv_{0}^{2}\unicode[STIX]{x1D70F}_{s}}\right]^{1/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}$ is the pitch angle scattering time.

This result will be justified in more detail by solving a boundary layer problem that allows the trapped distribution function $f_{t}$ to vanish at the trapped–passing boundary. It is very much of the same form as for a tokamak (Catto Reference Catto2019), but harder to satisfy in a stellarator as precise quasisymmetry only occurs at an isolated rational surface (Garren & Boozer Reference Garren and Boozer1991a ,Reference Garren and Boozer b ; Landreman & Sengupta Reference Landreman and Sengupta2018; Landreman et al. Reference Landreman, Sengupta and Plunk2019). The proportionality of $D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{weak}}$ to $\unicode[STIX]{x1D70F}_{s}^{-1/2}$ is the key characteristic of what is referred to as $\sqrt{\unicode[STIX]{x1D708}}$ regime transport (Galeev et al. Reference Galeev, Sagdeev, Furth and Rosenbluth1969; Ho & Kulsrud Reference Ho and Kulsrud1987; Calvo et al. Reference Calvo, Parra, Velasco and Alonso2017) and the existence of a narrow boundary layer.

Departures from quasisymmetry must be kept small to avoid depleting the slowing down distribution function during $\sqrt{\unicode[STIX]{x1D708}}$ regime transport since $\unicode[STIX]{x1D70F}_{s}D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{weak}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ gives

(4.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{m}^{n}<\left(\frac{a_{\unicode[STIX]{x1D6FC}}}{R}\right)\left(\frac{rRv_{\unicode[STIX]{x1D706}}^{3}}{\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s}}\right)^{1/4}\sim 10^{-2}, & & \displaystyle\end{eqnarray}$$

with $a_{\unicode[STIX]{x1D6FC}}$ the radial scale length of the alphas. Since $\unicode[STIX]{x1D6FF}_{m}^{n}\propto r/R$ , this threshold condition weakly favours high magnetic field and high plasma density operation in large $R$ devices.

A detailed boundary layer evaluation performed in § 6 confirms the preceding estimates.

5 Superbanana plateau estimates

In the superbanana plateau regime the tangential drift vanishes at some pitch angle $\unicode[STIX]{x1D706}_{0}$ , making $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{0}\sim w$ . The boundary layer width estimate is therefore modified to become

(5.1) $$\begin{eqnarray}\displaystyle \overline{C\{\,f_{t}\}}\sim \frac{v_{\unicode[STIX]{x1D706}}^{3}}{\unicode[STIX]{x1D70F}_{s}v_{0}^{3}}\frac{\unicode[STIX]{x2202}^{2}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}^{2}}\sim \frac{v_{\unicode[STIX]{x1D706}}^{3}f_{t}}{\unicode[STIX]{x1D70F}_{s}v_{0}^{3}w^{2}}\sim w\frac{(mN-nM)B_{0}v_{0}^{2}}{(qN-M)\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D6FF}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}f_{t}\sim \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

where again $\unicode[STIX]{x2202}f_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\sim f_{t}(mN-nM)/(qN-M)$ and $\unicode[STIX]{x1D706}=1/(1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2})$ gives $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{0}\sim (\unicode[STIX]{x1D705}-\unicode[STIX]{x1D705}_{0})\unicode[STIX]{x1D6FF}$ . The normalized boundary layer width is then

(5.2) $$\begin{eqnarray}\displaystyle w\sim (v_{\unicode[STIX]{x1D706}}/v_{0})[(qN-M)\unicode[STIX]{x1D6FF}/(mN-nM)B_{0}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}]^{1/3}\ll \unicode[STIX]{x1D6FF}^{1/2}\sim (r/R)^{1/2}, & & \displaystyle\end{eqnarray}$$

which gives the fraction that are trapped in the drift boundary layer to be

(5.3) $$\begin{eqnarray}\displaystyle F\sim w/\unicode[STIX]{x1D6FF}^{1/2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}^{1/2}$ is the total trapped fraction. The alphas are still $\unicode[STIX]{x1D735}B$ drifting on a flux surface much faster than they are pitch angle scattered by the ions. This condition is easily satisfied by alphas, primarily because $\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}\gg R/v_{0}\unicode[STIX]{x1D70F}_{s}\sim 10^{-6}$ .

Using the effective drift decorrelation time for the alphas of $\unicode[STIX]{x1D70F}\sim w^{2}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}$ , then for the radial speed $V$ of the alphas remains

(5.4) $$\begin{eqnarray}\displaystyle V\simeq \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}/RB_{p}\sim q(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D6FF}_{m}^{n}/(qN-M)r, & & \displaystyle\end{eqnarray}$$

but the effective radial step size,

(5.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}\sim V\unicode[STIX]{x1D70F}\sim Vw^{2}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}, & & \displaystyle\end{eqnarray}$$

changes because the boundary layer width $w$ has changed. As a result, the superbanana plateau diffusivity for $R^{2}B_{p}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\sim 1$ is estimated from this altered boundary layer width to be

(5.6) $$\begin{eqnarray}\displaystyle D_{\text{sbp}}^{\text{weak}}\sim FV^{2}\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D6FF}^{-1/2}w^{3}V^{2}v_{0}^{3}\unicode[STIX]{x1D70F}_{s}/v_{\unicode[STIX]{x1D706}}^{3}\sim \left[\frac{q(mN-nM)}{qN-M}\right](\unicode[STIX]{x1D6FF}_{m}^{n})^{2}\unicode[STIX]{x1D6FF}^{-1/2}\unicode[STIX]{x1D70C}_{0}v_{0}. & & \displaystyle\end{eqnarray}$$

As the result is independent of collisionality it is referred to as the superbanana plateau regime.

Depletion of the alpha tail for superbanana plateau transport occurs when $\unicode[STIX]{x1D70F}_{s}D_{\text{sbp}}^{\text{weak}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ . To avoid depletion for $qN-M\sim mN-nM$ requires

(5.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{m}^{n}<\frac{a_{\unicode[STIX]{x1D6FC}}}{R}\left(\frac{rR}{\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}}\right)^{1/2}\sim 10^{-3}; & & \displaystyle\end{eqnarray}$$

a restriction that is more of a design challenge than for the $\sqrt{\unicode[STIX]{x1D708}}$ regime transport since

(5.8) $$\begin{eqnarray}\displaystyle (R^{2}/\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s})^{1/4}<(Rv_{\unicode[STIX]{x1D706}}^{3}/rv_{0}^{3})^{1/4}<1. & & \displaystyle\end{eqnarray}$$

Superbanana plateau transport is normally more important than $\sqrt{\unicode[STIX]{x1D708}}$ transport for $qN-M\sim mN-nM$ because

(5.9) $$\begin{eqnarray}\displaystyle D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{weak}}/D_{\text{sbp}}^{\text{weak}}\sim (R^{2}v_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s})^{1/2}\ll 1, & & \displaystyle\end{eqnarray}$$

since the $\sqrt{\unicode[STIX]{x1D708}}$ boundary layer width is smaller than the usual trapped boundary layer width of $\unicode[STIX]{x1D6FF}^{1/2}$ . Keeping the superbanana plateau boundary layer width small compared to the trapped width is only slightly less restrictive, requiring

(5.10) $$\begin{eqnarray}\displaystyle (R^{2}v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{4}\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D70C}_{0})^{1/3}\ll (R/r)^{1/6}. & & \displaystyle\end{eqnarray}$$

As a result, when the two transport regimes compete, $(R^{2}v_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s})^{1/2}\sim 1\ll (R/r)^{1/4}$ , the $\sqrt{\unicode[STIX]{x1D708}}$ regime is at its upper limit of validity. Only at small aspect ratio, high density, and high magnetic field, will $\sqrt{\unicode[STIX]{x1D708}}$ transport approach superbanana plateau transport.

As superbanana plateau dominates for alphas, it is useful to re-examine the assumption $\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\gg (\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\unicode[STIX]{x2202}f_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D705}$ . Using (5.2) gives $f_{t}/f_{s}\ll Rw/a_{\unicode[STIX]{x1D6FC}}\sim (R/a_{\unicode[STIX]{x1D6FC}})(r^{2}v_{\unicode[STIX]{x1D706}}^{3}/q\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s})^{1/3}\ll 1$ as being assumed. The violation of this inequality is not responsible for the diffusivities linear in $\unicode[STIX]{x1D708}$ seen on the mono-energetic diffusivity plots of Beidler et al. (Reference Beidler, Allmaier, Isaev, Kasilov, Kernbichler, Leitold, Maaßberg, Mikkelsen, Murakami and Schmidt2011) and Henneberg et al. (Reference Henneberg, Drevlak, Nührenberg, Beidler, Turkin, Loizu and Helander2019). These plots are with $\boldsymbol{E}\times \boldsymbol{B}$ drift only, and the largest value shown is more than an order of magnitude smaller than the $\unicode[STIX]{x1D735}B$ drift ( $cE/B_{0}\sim qRv_{i}\unicode[STIX]{x1D70C}_{i}/a^{2}$ versus $q\unicode[STIX]{x1D70C}_{0}v_{0}/2a$ ).

6 Boundary layer analysis and transport fluxes for $\surd \unicode[STIX]{x1D708}$ regime

The limit of a small departure from quasisymmetry with the $\unicode[STIX]{x1D735}B$ drift retained and the zero in the tangential drift ignored is completely analytically tractable, as will now be demonstrated.

Defining

(6.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}\equiv (1-\unicode[STIX]{x1D705})/8, & \displaystyle\end{eqnarray}$$

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle f_{t}\equiv \text{Im}[H(\unicode[STIX]{x1D712})\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}], & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle k\equiv \frac{32(mN-nM)B_{0}\unicode[STIX]{x1D70F}_{s}v^{5}\unicode[STIX]{x1D6FF}}{(qN-M)v_{\unicode[STIX]{x1D706}}^{3}\unicode[STIX]{x1D6FA}_{0}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\gg 1, & \displaystyle\end{eqnarray}$$

and

(6.4) $$\begin{eqnarray}\displaystyle L\equiv \frac{32\unicode[STIX]{x1D6FF}_{m}^{n}(mN-nM)B_{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}_{s}v^{5}}{(qN-M)\unicode[STIX]{x1D6FA}_{0}v_{\unicode[STIX]{x1D706}}^{3}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\cos \left[\frac{(qn-m)\unicode[STIX]{x03C0}}{(qN-M)}\right], & & \displaystyle\end{eqnarray}$$

the boundary layer equation becomes of the form first considered by Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017):

(6.5) $$\begin{eqnarray}\displaystyle \frac{1}{\ell n(\unicode[STIX]{x1D712})}\frac{\unicode[STIX]{x2202}^{2}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}^{2}}-2\text{i}kH=2L. & & \displaystyle\end{eqnarray}$$

This boundary layer equation is of the exact same form as in Catto (Reference Catto2018). There it is shown that the matched asymptotic solution vanishing at the trapped–passing boundary is

(6.6) $$\begin{eqnarray}\displaystyle H=-\frac{\text{i}L}{k}[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D712}\sqrt{2k\ell n(\sqrt{2k})}}-1]. & & \displaystyle\end{eqnarray}$$

Consequently,

(6.7) $$\begin{eqnarray}\displaystyle f_{t}=-\frac{\unicode[STIX]{x1D6FF}_{m}^{n}\cos [(qn-m)\unicode[STIX]{x03C0}/(qN-M)]}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\text{Re}\{[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D712}\sqrt{2k\ell n(\sqrt{2k})}}-1]\,\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}\}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $f_{t}/f_{s}\sim \unicode[STIX]{x1D6FF}_{m}^{n}/\unicode[STIX]{x1D6FF}\ll 1$ is required.

To determine the alpha flux we need to evaluate

(6.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}=\left\langle \int \text{d}^{3}v(M_{\unicode[STIX]{x1D6FC}}v^{2}/2)^{d}f_{t}\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\right\rangle , & & \displaystyle\end{eqnarray}$$

where $\langle \cdots \rangle$ is the flux surface average, and $d=0$ and $d=1$ indicate particle and energy fluxes, respectively, and the integral is over the trapped alphas. Ignoring curvature drift, using $d^{3}v\rightarrow 2\unicode[STIX]{x03C0}(Bv^{2}/B_{0}\unicode[STIX]{x1D709})\,\text{d}v\,\text{d}\unicode[STIX]{x1D706}$ (summed over both signs) with $\unicode[STIX]{x1D706}^{-2}\,\text{d}\unicode[STIX]{x1D706}=4\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}\,\text{d}\unicode[STIX]{x1D705}$ , performing the transit average first by holding $\unicode[STIX]{x1D6FC}$ fixed and integrating over a full bounce by noting

(6.9) $$\begin{eqnarray}\displaystyle \int _{-\unicode[STIX]{x1D702}_{t}}^{\unicode[STIX]{x1D702}_{t}}\frac{\text{d}\unicode[STIX]{x1D702}}{\boldsymbol{ B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\frac{B}{\unicode[STIX]{x1D709}}\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713} & = & \displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\int _{-\unicode[STIX]{x1D702}_{t}}^{\unicode[STIX]{x1D702}_{t}}\frac{\text{d}\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D709}\boldsymbol{ b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}}\nonumber\\ \displaystyle & \simeq & \displaystyle \frac{2B_{0}\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}}{\unicode[STIX]{x03C0}\sqrt{2\unicode[STIX]{x1D6FF}}}\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}K(\unicode[STIX]{x1D705})\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\frac{\text{d}\unicode[STIX]{x1D702}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D702}},\end{eqnarray}$$

with $\unicode[STIX]{x1D709}(\pm \unicode[STIX]{x1D702}_{t})=0$ , gives the alternate and more useful form of the fluxes to be

(6.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d} & = & \displaystyle \left\langle \int \text{d}^{3}v(M_{\unicode[STIX]{x1D6FC}}v^{2}/2)^{d}f_{t}\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\right\rangle \nonumber\\ \displaystyle & \simeq & \displaystyle 4\sqrt{2\unicode[STIX]{x1D6FF}}\int _{0}^{v_{0}}\text{d}vv^{2}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v^{2}}{2}\right)^{d}\left\langle \int _{0}^{1}\text{d}\unicode[STIX]{x1D705}f_{t}\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\ell n\left(\frac{8}{1-\unicode[STIX]{x1D705}}\right)\!\right\rangle ,\end{eqnarray}$$

where $f_{t}\propto \exp [\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)]$ is held fixed during the transit average and the last form is for $\unicode[STIX]{x1D705}\rightarrow 1$ . Inserting $f_{t}$ and $\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}$ , and using

(6.11) $$\begin{eqnarray}\displaystyle \frac{\ell n(v_{0}/v_{c})}{n_{s}}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\gg \frac{1}{v_{c}}\frac{\unicode[STIX]{x2202}v_{c}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\sim \frac{1}{n_{i}}\frac{\unicode[STIX]{x2202}n_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}, & & \displaystyle\end{eqnarray}$$

leaves

(6.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d} & \simeq & \displaystyle \frac{4B_{0}(2\unicode[STIX]{x1D6FF})^{1/2}(mN-nM)(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}\unicode[STIX]{x2202}n_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x03C0}(qN-M)\unicode[STIX]{x1D6FA}_{0}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\ell n(v_{0}/v_{c})}\cos ^{2}\left[\frac{(qn-m)\unicode[STIX]{x03C0}}{(qN-M)}\right]\int _{0}^{v_{0}}\frac{\text{d}vv^{4}}{v^{3}+v_{c}^{3}}\nonumber\\ \displaystyle & & \displaystyle \times \left(\frac{M_{\unicode[STIX]{x1D6FC}}v^{2}}{2}\right)^{d}\text{Re}\left[\text{i}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D712}\text{e}^{-(1-\text{i})\unicode[STIX]{x1D712}\sqrt{k\ell n(2k)}}\ell n\unicode[STIX]{x1D712}\right],\end{eqnarray}$$

where $\langle \sin [(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)]\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}\rangle \simeq \text{i}/2$ is used along with the variable $\tilde{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FC}/(M-qN)$ . Letting $x=\unicode[STIX]{x1D712}\sqrt{k\ell n(2k)}\propto v^{5/2}$ and defining

(6.13) $$\begin{eqnarray}\displaystyle k_{0}\equiv \frac{32(mN-nM)B_{0}\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D6FF}}{(qN-M)v_{\unicode[STIX]{x1D706}}^{3}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\gg 1, & & \displaystyle\end{eqnarray}$$

then the fraction of barely trapped that contribute is proportional to

(6.14) $$\begin{eqnarray}\displaystyle \text{Re}\left[\text{i}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D712}\text{e}^{-(1-\text{i})\unicode[STIX]{x1D712}\sqrt{2k\ell n(\sqrt{2k})}}\ell n\unicode[STIX]{x1D712}\right]\simeq \frac{\ell n(\sqrt{k\ell n(2k)})}{2\sqrt{k\ell n(2k)}}, & & \displaystyle\end{eqnarray}$$

with the speed weighting

(6.15) $$\begin{eqnarray}\displaystyle \int _{0}^{v_{0}}\text{d}vv^{4}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v^{2}}{2}\right)^{d}\frac{\ell n[\sqrt{k\ell n(2k)}]}{(v^{3}+v_{c}^{3})\sqrt{k\ell n(2k)}}\simeq \frac{v_{0}^{2}\ell n[\sqrt{k_{0}\ell n(2k_{0})}]}{\sqrt{k_{0}\ell n(2k_{0})}}\left\{\begin{array}{@{}ll@{}}\ell n(v_{0}/v_{c}) & d=0\\ M_{\unicode[STIX]{x1D6FC}}v_{0}^{2}/3 & d=1,\end{array}\right. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

thereby yielding the particle and energy fluxes

(6.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d} & \simeq & \displaystyle -\frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}(mN-nM)^{1/2}v_{\unicode[STIX]{x1D706}}^{3/2}\unicode[STIX]{x1D70C}_{0}^{1/2}B_{0}^{1/2}\ell n[k_{0}\ell n(2k_{0})]\cos ^{2}[(qn-m)\unicode[STIX]{x03C0}/(qN-M)]}{4\unicode[STIX]{x03C0}(qN-M)^{1/2}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})^{3/2}v_{0}\sqrt{\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\ell n(v_{0}/v_{c})}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\left\{\begin{array}{@{}ll@{}}\ell n(v_{0}/v_{c}) & d=0\\ M_{\unicode[STIX]{x1D6FC}}v_{0}^{2}/3 & d=1.\end{array}\right.\end{eqnarray}$$

The sum over all $m$ and $n$ can be reinserted in this form if there is more than one error field to consider. This result is consistent with the Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) estimate that first predicted the appearance of $\sqrt{\ell n(2k_{0})}\sim \ell n[k_{0}\ell n(2k_{0})]/\sqrt{\ell n(2k_{0})}$ in the $\sqrt{\unicode[STIX]{x1D708}}$ result. Large $k_{0}$ is $w\ll \unicode[STIX]{x1D6FF}^{1/2}$ .

The fluxes imply that the particle diffusivity of the alphas is

(6.17) $$\begin{eqnarray}\displaystyle D_{0}^{\sqrt{\unicode[STIX]{x1D708}}}=\frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}(mN-nM)^{1/2}v_{\unicode[STIX]{x1D706}}^{3/2}\unicode[STIX]{x1D70C}_{0}^{1/2}B_{0}^{1/2}\ell n[k_{0}\ell n(2k_{0})]\cos ^{2}[(qn-m)\unicode[STIX]{x03C0}/(qN-M)]}{4\unicode[STIX]{x03C0}R^{2}B_{p}^{2}(qN-M)^{1/2}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})^{3/2}v_{0}\sqrt{\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

while the alpha energy diffusivity is

(6.18) $$\begin{eqnarray}\displaystyle D_{1}^{\sqrt{\unicode[STIX]{x1D708}}}=\frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}(mN-nM)^{1/2}v_{\unicode[STIX]{x1D706}}^{3/2}\unicode[STIX]{x1D70C}_{0}^{1/2}B_{0}^{1/2}\ell n[k_{0}\ell n(2k_{0})]\cos ^{2}[(qn-m)\unicode[STIX]{x03C0}/(qN-M)]}{6\unicode[STIX]{x03C0}R^{2}B_{p}^{2}(qN-M)^{1/2}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})^{3/2}v_{0}\sqrt{\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\ell n(v_{0}/v_{c})}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\text{d}\unicode[STIX]{x1D713}\sim RB_{p}\,\text{d}r$ and $B_{p}/B_{0}\sim \unicode[STIX]{x1D6FF}/q$ .

The first expression is the same as the earlier estimate within numerical and logarithmic factors. Consequently, a weak departure from quasisymmetry is expected to be an essential consideration for the alphas, but one that can be dealt with by maintaining small quasisymmetry departures of $\unicode[STIX]{x1D6FF}_{m}^{n}<10^{-2}$ . Not surprisingly, the details of the boundary layer analysis indicate that it is the departure from quasisymmetry experienced by the barely trapped that matters most as they dominate $\sqrt{\unicode[STIX]{x1D708}}$ regime transport. The fit in figures 7 and 8 of Beidler & D’haeseleer (Reference Beidler and D’haeseleer1995) retains $\sqrt{\unicode[STIX]{x1D708}}$ transport.

For a tokamak ( $M=1$ , $N=0=m$ , $\unicode[STIX]{x1D6FF}\rightarrow \unicode[STIX]{x1D700}$ ) with $n$ toroidal field coils, the flux reduces to

(6.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}^{\text{tok}}\simeq -\frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}(qn)^{1/2}v_{\unicode[STIX]{x1D706}}^{3/2}\unicode[STIX]{x1D70C}_{0}^{1/2}\ell n[k_{0}\ell n(2k_{0})]\cos ^{2}(qn\unicode[STIX]{x03C0})}{4\unicode[STIX]{x03C0}v_{0}\sqrt{\unicode[STIX]{x1D700}\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\ell n(v_{0}/v_{c})}R^{3}B_{p}^{2}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\left\{\begin{array}{@{}ll@{}}\ell n(v_{0}/v_{c}) & d=0\\ M_{\unicode[STIX]{x1D6FC}}v_{0}^{2}/3 & d=1,\end{array}\right. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which is missing $1/(4qn)^{4}$ in the argument of the logarithm in the numerator as $qN-M\sim mN-nM$ is assumed here, while Catto (Reference Catto2019) assumes $qn\gg 1$ . This result is also a factor of 2 smaller than Catto (Reference Catto2019) due to an error going from his (4.8) to (4.10).

7 Superbanana plateau boundary layer analysis and transport fluxes

The limit of a small departure from quasisymmetry with the $\unicode[STIX]{x1D735}B$ drift retained and a zero in the tangential drift is also analytically tractable.

Defining $\unicode[STIX]{x1D712}\equiv (1-\unicode[STIX]{x1D705})/8$ and $f_{t}$ as before, and using the new definitions

(7.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{0}\equiv (1-\unicode[STIX]{x1D705}_{0})/8, & \displaystyle\end{eqnarray}$$

(7.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F6}=\frac{4L\unicode[STIX]{x1D706}_{0}\unicode[STIX]{x1D705}_{0}^{2}}{(2\unicode[STIX]{x1D705}_{0}^{2}-1)}, & \displaystyle\end{eqnarray}$$

and

(7.3) $$\begin{eqnarray}\displaystyle s^{3}=\frac{32\unicode[STIX]{x1D705}_{0}k}{(1-\unicode[STIX]{x1D705}_{0}^{2})(2\unicode[STIX]{x1D705}_{0}^{2}-1)}\simeq 8k\gg 1, & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D706}_{0}=1/(1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}_{0}^{2})\simeq 1$ , the boundary layer equation becomes of the form (Calvo et al. Reference Calvo, Parra, Velasco and Alonso2017),

(7.4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}^{2}}-\text{i}s^{3}(\unicode[STIX]{x1D712}-\unicode[STIX]{x1D712}_{0})H=-2\unicode[STIX]{x1D6F6}. & & \displaystyle\end{eqnarray}$$

Letting $z=s(\unicode[STIX]{x1D712}-\unicode[STIX]{x1D712}_{0})$ , the resulting equation

(7.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}H}{\unicode[STIX]{x2202}z^{2}}-\text{i}zH=-\frac{2\unicode[STIX]{x1D6F6}}{s^{2}}, & & \displaystyle\end{eqnarray}$$

has the solution (Su & Oberman Reference Su and Oberman1968)

(7.6) $$\begin{eqnarray}\displaystyle H=\frac{2\unicode[STIX]{x1D6F6}}{s^{2}}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\text{e}^{-\text{i}z\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}^{3}/3}\underset{z\rightarrow \infty }{\rightarrow }-\frac{\text{i}2\unicode[STIX]{x1D6F6}}{s^{2}z}=-\frac{\text{i}2\unicode[STIX]{x1D6F6}}{s^{3}(\unicode[STIX]{x1D712}-\unicode[STIX]{x1D712}_{0})}\underset{\unicode[STIX]{x1D712}\ll \unicode[STIX]{x1D712}_{0}}{\rightarrow }\frac{\text{i}2L(1+\unicode[STIX]{x1D705}_{0})}{k}. & & \displaystyle\end{eqnarray}$$

The asymptotic form goes to a constant as the trapped–passing boundary layer is approached.

To make this $\unicode[STIX]{x1D712}\ll \unicode[STIX]{x1D712}_{0}$ solution match to that of the trapped–passing boundary layer, a WKB region satisfying

(7.7) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}=-\frac{\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}}{\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}, & & \displaystyle\end{eqnarray}$$

can be inserted. The solution in this WKB region is just

(7.8) $$\begin{eqnarray}\displaystyle f_{t}=\frac{\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D706}K(\unicode[STIX]{x1D705})\cos [(qn-m)\unicode[STIX]{x03C0}/(qN-M)]}{[K(\unicode[STIX]{x1D705})-2E(\unicode[STIX]{x1D705})]\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\cos \left[\frac{(mN-nM)\unicode[STIX]{x1D6FC}}{(qN-M)}\right], & & \displaystyle\end{eqnarray}$$

which matches to the trapped–passing boundary layer for $\unicode[STIX]{x1D705}\rightarrow 0$ , and to the vanishing tangential drift boundary layer as $\unicode[STIX]{x1D705}\rightarrow \unicode[STIX]{x1D705}_{0}$ . Therefore, the $\sqrt{\unicode[STIX]{x1D708}}$ and superbanana plateau regime boundary layers are well separated, with no transport occurring in the connecting WKB region as $f_{t}\propto \cos [(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)]$ , while $\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\propto \sin [(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)]$ .

To evaluate the flux for

(7.9) $$\begin{eqnarray}\displaystyle f_{t}=\frac{2\unicode[STIX]{x1D6F6}}{s^{2}}\text{Im}\left[\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\text{e}^{-\text{i}z\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}^{3}/3}\right], & & \displaystyle\end{eqnarray}$$

with $z=s(\unicode[STIX]{x1D705}_{0}-\unicode[STIX]{x1D705})/8$ , use $\text{d}\unicode[STIX]{x1D706}\simeq 4\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}_{0}\,\text{d}\unicode[STIX]{x1D705}$ in $d^{3}v\rightarrow 2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}(Bv^{2}/B_{0}\unicode[STIX]{x1D709})\,\text{d}v\,\text{d}\unicode[STIX]{x1D706}$ , and note that

(7.10) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\text{e}^{-\unicode[STIX]{x1D70F}^{3}/3}\int _{0}^{1}\text{d}\unicode[STIX]{x1D705}\text{e}^{-\text{i}z\unicode[STIX]{x1D70F}} & \simeq & \displaystyle \frac{8}{s}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\text{e}^{-\unicode[STIX]{x1D70F}^{3}/3}\left[\int _{-s\unicode[STIX]{x1D705}_{0}/8}^{s\unicode[STIX]{x1D705}_{0}/8}\text{d}z\text{e}^{-\text{i}z\unicode[STIX]{x1D70F}}+\int _{-\infty }^{-s\unicode[STIX]{x1D705}_{0}/8}\text{d}z\text{e}^{-\text{i}z\unicode[STIX]{x1D70F}}\right]\nonumber\\ \displaystyle & \simeq & \displaystyle 16\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\text{e}^{-\unicode[STIX]{x1D70F}^{3}/3}\frac{\sin (s\unicode[STIX]{x1D705}_{0}\unicode[STIX]{x1D70F}/8)}{s\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

Only the leading order approximation is retained in the $\unicode[STIX]{x1D705}$ integral. The Taylor expansion about $\unicode[STIX]{x1D705}_{0}$ fails well before $\unicode[STIX]{x1D705}=0$ and $\unicode[STIX]{x1D705}=1$ , but the result will be insensitive to the limits of the $\unicode[STIX]{x1D705}$ integral as only the limit $s\gg 1$ is of interest. Then $\sin (t)/t\rightarrow \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(t)$ with $t=s\unicode[STIX]{x1D705}_{0}\unicode[STIX]{x1D70F}/8$ gives

(7.11) $$\begin{eqnarray}\displaystyle \int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\text{e}^{-\unicode[STIX]{x1D70F}^{3}/3}\int _{0}^{1}\text{d}\unicode[STIX]{x1D705}\text{e}^{-\text{i}z\unicode[STIX]{x1D70F}}=\frac{16}{s}\int _{0}^{\infty }\text{d}t\text{e}^{-(512)t^{3}/3(\unicode[STIX]{x1D705}_{0}s)^{3}}\frac{\sin (t)}{t}\simeq \frac{16\unicode[STIX]{x03C0}}{s}, & & \displaystyle\end{eqnarray}$$

where $s\propto \unicode[STIX]{x1D70F}_{s}^{1/3}v^{5/3}$ . Consequently, upon using $\langle \!\sin [(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)]$ $\text{e}^{\text{i}(mN-nM)\unicode[STIX]{x1D6FC}/(qN-M)}\!\rangle \simeq \text{i}/2$ , the flux

(7.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}\simeq 2\sqrt{2\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D705}_{0}K(\unicode[STIX]{x1D705}_{0})\int _{0}^{v_{0}}\text{d}vv^{2}(M_{\unicode[STIX]{x1D6FC}}v^{2}/2)^{d}\left\langle \int _{0}^{1}\text{d}\unicode[STIX]{x1D705}f_{t}\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\right\rangle , & & \displaystyle\end{eqnarray}$$

becomes

(7.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}\simeq -\frac{64\unicode[STIX]{x03C0}\sqrt{2\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FF}_{m}^{n}(mN-nM)B_{0}\unicode[STIX]{x1D705}_{0}^{3}}{(qN-M)\unicode[STIX]{x1D6FA}_{0}(2\unicode[STIX]{x1D705}_{0}^{2}-1)}\cos \left[\frac{(qn-m)\unicode[STIX]{x03C0}}{qN-M}\right]\int _{0}^{v_{0}}\text{d}vv^{4}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v^{2}}{2}\right)^{d}\frac{L}{s^{3}}.\qquad & & \displaystyle\end{eqnarray}$$

Using

(7.14) $$\begin{eqnarray}\displaystyle \frac{L}{s^{3}}=\frac{\unicode[STIX]{x1D6FF}_{m}^{n}(1-\unicode[STIX]{x1D705}_{0}^{2})(2\unicode[STIX]{x1D705}_{0}^{2}-1)}{32\unicode[STIX]{x1D705}_{0}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\cos \left[\frac{(qn-m)\unicode[STIX]{x03C0}}{(qN-M)}\right], & & \displaystyle\end{eqnarray}$$

and

(7.15) $$\begin{eqnarray}\displaystyle \int _{0}^{v_{0}}\text{d}vv^{4}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v^{2}}{2}\right)^{d}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\simeq \frac{\unicode[STIX]{x2202}n_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{4\unicode[STIX]{x03C0}\ell n(v_{0}/v_{c})}\int _{0}^{v_{0}}\frac{\text{d}vv^{4}}{v^{3}+v_{c}^{3}}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v^{2}}{2}\right)^{d}\simeq \frac{v_{0}^{2}(M_{\unicode[STIX]{x1D6FC}}v^{2}/2)^{d}}{8\unicode[STIX]{x03C0}\ell n(v_{0}/v_{c})}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

the superbanana plateau particle ( $d=0$ ) and energy ( $d=1$ ) fluxes are given by

(7.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}\simeq -\frac{\sqrt{2\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D705}_{0}^{2}(1-\unicode[STIX]{x1D705}_{0}^{2})(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}(mN-nM)B_{0}\unicode[STIX]{x1D70C}_{0}v_{0}}{4(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})(qN-M)\ell n(v_{0}/v_{c})}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v_{0}^{2}}{2}\right)^{d}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\cos ^{2}\left[\frac{(qn-m)\unicode[STIX]{x03C0}}{qN-M}\right]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Once again the sum over $m$ and $n$ can be reinserted if needed.

The associated particle and energy diffusivities of the alphas are the same:

(7.17) $$\begin{eqnarray}\displaystyle D_{d}^{\text{sbp}}=\frac{B_{0}\sqrt{2\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D705}_{0}^{2}(1-\unicode[STIX]{x1D705}_{0}^{2})(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}\cos ^{2}[(qn-m)\unicode[STIX]{x03C0}/(qN-M)]}{4RB_{p}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x2202}r)(qN-M)\ell n(v_{0}/v_{c})}. & & \displaystyle\end{eqnarray}$$

This result is in agreement with the earlier estimate within logarithmic factors and factors that depend on the location of the pitch angle at which the tangential drift vanishes ( $\unicode[STIX]{x1D705}_{0}\simeq 0.9$ , $\unicode[STIX]{x1D712}_{0}\simeq 10^{-2}$ ). The coefficient $\unicode[STIX]{x1D705}_{0}^{2}(1-\unicode[STIX]{x1D705}_{0}^{2})\simeq 0.14$ is somewhat small because $\unicode[STIX]{x1D705}_{0}$ is close to one. Near superbanana plateau behaviour is apparent in figure 8 of Beidler & D’haeseleer (Reference Beidler and D’haeseleer1995) and retained in their fit.

For a tokamak ( $M=1$ , $N=0=m$ , $\unicode[STIX]{x1D6FF}\rightarrow \unicode[STIX]{x1D700}$ ) with $n$ toroidal field coils, the flux reduces to

(7.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}^{\text{tok}}\simeq \frac{\unicode[STIX]{x1D705}_{0}^{2}(1-\unicode[STIX]{x1D705}_{0}^{2})(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}qn\unicode[STIX]{x1D70C}_{0}v_{0}\cos ^{2}(qn\unicode[STIX]{x03C0})}{2\sqrt{2\unicode[STIX]{x1D700}}[\ell n(v_{0}/v_{c})]}\left(\frac{M_{\unicode[STIX]{x1D6FC}}v_{0}^{2}}{2}\right)^{d}R^{2}B_{p}^{2}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}, & & \displaystyle\end{eqnarray}$$

and gives a diffusivity of

(7.19) $$\begin{eqnarray}\displaystyle D_{\text{tok}}^{\text{sbp}}=\frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}\unicode[STIX]{x1D705}_{0}^{2}(1-\unicode[STIX]{x1D705}_{0}^{2})nq\unicode[STIX]{x1D70C}_{0}v_{0}\cos ^{2}(qn\unicode[STIX]{x03C0})}{2\sqrt{2\unicode[STIX]{x1D700}}\ell n(v_{0}/v_{c})}. & & \displaystyle\end{eqnarray}$$

The tokamak limit of the superbanana plateau regime adds to the $\sqrt{\unicode[STIX]{x1D708}}$ result of Catto (Reference Catto2019). It is most sensitive to the ripple near $\unicode[STIX]{x1D717}_{0}\simeq \pm 3\unicode[STIX]{x03C0}/4$ corresponding to $\unicode[STIX]{x1D705}_{0}=\sin (\unicode[STIX]{x1D717}_{0}/2)$ .

8 Summary and discussion

A fully self-consistent evaluation of alpha particle transport fluxes for weak departures from quasisymmetry has been performed in the $\sqrt{\unicode[STIX]{x1D708}}$ and superbanana plateau regimes. The tokamak limits are easily recovered from the expressions. Both regimes exist for essentially the same parameter regime. Indeed, their transport contributions are additive with the superbanana plateau limit always expected to dominate for alphas. The detailed boundary layer evaluations herein confirms that the $\sqrt{\unicode[STIX]{x1D708}}$ regime diffusivity is enhanced by the logarithmic factor $\sqrt{\ell nk_{0}}\gg 1$ from the phenomenological estimate such that the diffusivity ratio is larger than expected:

(8.1) $$\begin{eqnarray}\displaystyle \frac{D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{weak}}}{D_{\text{sbp}}^{\text{weak}}}\sim \left[\frac{(qN-M)R^{2}v_{\unicode[STIX]{x1D706}}^{3}}{(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s}}\ell nk_{0}\right]^{1/2}\sim \left(\frac{\ell nk_{0}}{k_{0}}\right)^{1/2}, & & \displaystyle\end{eqnarray}$$

where typically

(8.2) $$\begin{eqnarray}\displaystyle k_{0}=\frac{32(mN-nM)B_{0}\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D6FF}}{(qN-M)v_{\unicode[STIX]{x1D706}}^{3}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\sim \frac{32(mN-nM)q\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s}}{(qN-M)R^{2}v_{\unicode[STIX]{x1D706}}^{3}}\sim 10^{7}. & & \displaystyle\end{eqnarray}$$

The $\sqrt{\unicode[STIX]{x1D708}}$ result verifies in detail the logarithmic dependence of the diffusivity as first carefully estimated by Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017). The asymptotic dominance of the superbanana plateau limit is also consistent with their estimates based on general analytic considerations for very large $k_{0}$ .

The preceding estimate indicates that $\sqrt{\unicode[STIX]{x1D708}}$ effects will only be apparent for smaller values of $k_{0}\sim 10$ or $\unicode[STIX]{x1D708}_{\ast }\equiv qRv_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{4}\unicode[STIX]{x1D70F}_{s}\sim \unicode[STIX]{x1D70C}_{0}/R\equiv \unicode[STIX]{x1D70C}_{\ast }$ , where the $\unicode[STIX]{x1D70C}_{\ast }\gg \unicode[STIX]{x1D708}_{\ast }$ assumption is failing. As $\unicode[STIX]{x1D70C}_{\ast }/\unicode[STIX]{x1D708}_{\ast }\sim k_{0}\sim 10^{7}$ for alphas, superbanana plateau transport will dominate and $\sqrt{\unicode[STIX]{x1D708}}$ transport will be negligible.

The explicit evaluations are obtained by performing boundary layer analyses retaining collisions and the careful treatment of the tangential magnetic $\unicode[STIX]{x1D735}B$ drift, including its reversal in direction. A WKB treatment demonstrates that these two boundary layers do not overlap and the transport from them is additive. The $\sqrt{\unicode[STIX]{x1D708}}$ boundary layer in the trapped region at the trapped–passing boundary is most sensitive to the departures from quasisymmetry experienced by the barely trapped, that is, any perturbation of the maximum constant $B$ contour. The wider boundary layer about the trapped pitch angle for which the tangential drift vanishes is insensitive to these details as it is further into the trapped region and separated from the narrower boundary layer at the trapped–passing boundary.

To avoid depletion of the alpha tail by superbanana plateau transport $\unicode[STIX]{x1D70F}_{s}D_{\text{sbp}}^{\text{weak}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ is required, with $a_{\unicode[STIX]{x1D6FC}}$ the radial scale length of the alpha density. For $qN-M\sim mN-nM$ this inequality requires that the normalized non-quasisymmetric magnetic field departure $\unicode[STIX]{x1D6FF}_{m}^{n}$ satisfy

(8.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{m}^{n}<\frac{a_{\unicode[STIX]{x1D6FC}}}{R}\left(\frac{r^{1/2}R^{3/2}}{\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}}\right)^{1/2}\sim 10^{-3}. & & \displaystyle\end{eqnarray}$$

Therefore, high magnetic field and high density along with lower temperature operation in a large $R$ stellarator is more forgiving for weak departures from quasisymmetry.

As $\unicode[STIX]{x1D708}_{\ast }\rightarrow 0$ , however, a regime with diffusivities linear in $\unicode[STIX]{x1D708}_{\ast }$ is expected to become important. In this regime the radial drift of alphas removes them from the tangential drift resonance at $\unicode[STIX]{x1D714}=0$ to reduce the transport further. The radial drift reduction occurs unless

(8.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{m}^{n}/\unicode[STIX]{x1D6FF}\sim f_{t}/f_{s}\ll \unicode[STIX]{x1D6FF}^{2/3}(R/a_{\unicode[STIX]{x1D6FC}})(\unicode[STIX]{x1D708}_{\ast }/\unicode[STIX]{x1D70C}_{\ast })^{1/3}\sim 10^{-2}. & & \displaystyle\end{eqnarray}$$

A rigorous analytic treatment of this superbanana limit does not seem to be available even though it seems likely that to be the alpha regime of most interest. A crude estimate of the diffusivity begins by balancing the perturbed radial and nearly vanishing tangential drift terms,

(8.5) $$\begin{eqnarray}\displaystyle w\frac{(mN-nM)B_{0}v_{0}^{2}}{(qN-M)\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D6FF}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}f_{t} & {\sim} & \displaystyle \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\sim \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}\frac{\unicode[STIX]{x2202}f_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}\nonumber\\ \displaystyle & {\sim} & \displaystyle \frac{RB_{p}q(mN-nM)\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D6FF}_{m}^{n}}{(qN-M)r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{f_{t}}{w},\end{eqnarray}$$

with $w$ the width in $\unicode[STIX]{x1D706}$ space, $(\unicode[STIX]{x2202}\unicode[STIX]{x1D706}/\unicode[STIX]{x2202}\unicode[STIX]{x1D705})(\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF})\sim 1$ and $\unicode[STIX]{x2202}f_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}\sim f_{t}/w$ . The preceding gives

(8.6) $$\begin{eqnarray}\displaystyle w\sim (\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D6FF})^{1/2}\ll \unicode[STIX]{x1D6FF}^{1/2}, & & \displaystyle\end{eqnarray}$$

and an effective radial step of

(8.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}\sim \overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}/RB_{p}\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\sim Rw\sim R(\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D6FF})^{1/2}. & & \displaystyle\end{eqnarray}$$

Therefore, the effective resonant trapped fraction is

(8.8) $$\begin{eqnarray}\displaystyle F\sim w/\unicode[STIX]{x1D6FF}^{1/2}\sim (\unicode[STIX]{x1D6FF}_{m}^{n})^{1/2}. & & \displaystyle\end{eqnarray}$$

As in the usual axisymmetric banana regime (Galeev & Sagdeev Reference Galeev, Sagdeev and Leontovich1979), this implies an effective pitch angle scattering time of

(8.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}\sim F^{2}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}\sim \unicode[STIX]{x1D6FF}_{m}^{n}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}. & & \displaystyle\end{eqnarray}$$

These estimates lead to a superbanana diffusivity of roughly

(8.10) $$\begin{eqnarray}\displaystyle D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}=0}\sim (\unicode[STIX]{x1D6FF}_{m}^{n})^{1/2}[R(\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D6FF})^{1/2}]^{2}/[\unicode[STIX]{x1D6FF}_{m}^{n}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}]\sim (\unicode[STIX]{x1D6FF}_{m}^{n})^{1/2}\unicode[STIX]{x1D6FF}R^{2}v_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D70F}_{s}v_{0}^{3}, & & \displaystyle\end{eqnarray}$$

as in Galeev & Sagdeev (Reference Galeev, Sagdeev and Leontovich1979). The preceding is a crude estimate, but it does indicate that why diffusivities linear in $\unicode[STIX]{x1D708}_{\ast }$ arise in GSRAKE simulations (Beidler & D’haeseleer Reference Beidler and D’haeseleer1995).

The linear $\unicode[STIX]{x1D708}_{\ast }$ behavior associated with the $\sqrt{\unicode[STIX]{x1D708}}$ regime arises for a somewhat different reason. The radial drift of the barely trapped alphas from a $\sqrt{\unicode[STIX]{x1D708}}$ boundary layer causes them to detrap and retrap to become transitional trapped alphas that never stay trapped in a single well. These transitional alphas move among various $\sqrt{\unicode[STIX]{x1D708}}$ boundary layers associated with other local maxima caused by the departures from quasisymmetry. This behaviour increases the number of trapped alphas to include the transitional trapped alphas. The quasisymmetric trapped fraction $\unicode[STIX]{x1D6FF}^{1/2}$ expands to include the transitional trapped alphas that experience trapping in multiple wells, increasing the trapped fraction to roughly $(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D6FF}_{m}^{n})^{1/2}$ . The increase $(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D6FF}_{m}^{n})^{1/2}-\unicode[STIX]{x1D6FF}^{1/2}\sim \unicode[STIX]{x1D6FF}_{m}^{n}/\unicode[STIX]{x1D6FF}^{1/2}$ is the new effective trapped fraction. The effective collision time associated with each error well is $\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}$ as in a usual banana regime approximation. These estimates give a superbanana diffusivity linear in $\unicode[STIX]{x1D708}$ of roughly

(8.11) $$\begin{eqnarray}\displaystyle D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}\neq 0}\sim (\unicode[STIX]{x1D6FF}_{m}^{n}/\unicode[STIX]{x1D6FF}^{1/2})(R\unicode[STIX]{x1D6FF}_{m}^{n})^{2}/(\unicode[STIX]{x1D6FF}_{m}^{n}\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\sim (\unicode[STIX]{x1D6FF}_{m}^{n}R)^{2}v_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D6FF}^{1/2}\unicode[STIX]{x1D70F}_{s}v_{0}^{3}, & & \displaystyle\end{eqnarray}$$

where the radial step is $V/\overline{\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\sim R\unicode[STIX]{x1D6FF}_{m}^{n}$ as in the $\sqrt{\unicode[STIX]{x1D708}}$ regime. This behaviour is related to the linear diffusivity regime seen in Beidler & D’haeseleer (Reference Beidler and D’haeseleer1995), Beidler et al. (Reference Beidler, Allmaier, Isaev, Kasilov, Kernbichler, Leitold, Maaßberg, Mikkelsen, Murakami and Schmidt2011), Henneberg et al. (Reference Henneberg, Drevlak, Nührenberg, Beidler, Turkin, Loizu and Helander2019), and is in agreement with Mynick (Reference Mynick1983). It is smaller than the prior superbanana drift diffusivity by $(\unicode[STIX]{x1D6FF}_{m}^{n}/\unicode[STIX]{x1D6FF})^{3/2}\ll 1$ .

Therefore, avoiding alpha depletion in the linear regimes in a slowing down time only requires $\unicode[STIX]{x1D70F}_{s}D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}\neq 0}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll \unicode[STIX]{x1D70F}_{s}D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}=0}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ , or

(8.12) $$\begin{eqnarray}\displaystyle \frac{a_{\unicode[STIX]{x1D6FC}}^{2}v_{0}^{3}}{rRv_{\unicode[STIX]{x1D706}}^{3}}\gg (\unicode[STIX]{x1D6FF}_{m}^{n})^{1/2}\gg \frac{(\unicode[STIX]{x1D6FF}_{m}^{n})^{2}}{\unicode[STIX]{x1D6FF}^{3/2}}, & & \displaystyle\end{eqnarray}$$

even when $a_{\unicode[STIX]{x1D6FC}}^{2}/rR\sim \unicode[STIX]{x1D6FF}$ as $v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\sim 50$ . Accurate numerical simulations of these linear in $\unicode[STIX]{x1D708}_{\ast }$ regimes are essential as they are much more reliable than the rough estimates given here.