2 Alpha background

The isotropic slowing down tail solution $f_{s}=f_{s}(\boldsymbol{r},v)$ satisfies

(2.1) $$\begin{eqnarray}{\displaystyle \frac{Ze}{Mc}}\boldsymbol{v}\times \boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{v}\,f_{s}=C\{f_{s}\}+{\displaystyle \frac{S\unicode[STIX]{x1D6FF}(v-v_{0})}{4\unicode[STIX]{x03C0}v^{2}}},\end{eqnarray}$$

where the coefficient of the $\unicode[STIX]{x1D6FF}(v-v_{0})$ function is the fusion birth rate for alphas:

(2.2) $$\begin{eqnarray}S=S(\unicode[STIX]{x1D713})=n_{T}(\unicode[STIX]{x1D713})n_{D}(\unicode[STIX]{x1D713})\langle \unicode[STIX]{x1D70E}v\rangle _{DT}.\end{eqnarray}$$

The alphas of mass $M$ and charge number $Z$ are born with birth speed $v_{0}$ . The alpha collision operator is $C$ , with $c$ the speed of light, $e$ the charge on a proton and $\boldsymbol{B}$ the magnetic field.

We expect alphas to be born on surfaces of constant pressure so that they have the usual slowing down distribution function on a flux surface to lowest order for small poloidal gyroradius and ripple ( $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D700}/qN$ ):

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

with $\unicode[STIX]{x1D713}$ the poloidal flux function, $H(v_{0}-v)$ the Heaviside step function, $\unicode[STIX]{x1D70F}_{s}$ the alpha slowing down time

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{s}=\unicode[STIX]{x1D70F}_{s}(\unicode[STIX]{x1D713})={\displaystyle \frac{3MT_{e}^{3/2}(\unicode[STIX]{x1D713})}{4(2\unicode[STIX]{x03C0}m)^{1/2}Z^{2}e^{4}n_{e}(\unicode[STIX]{x1D713})\ell n\unicode[STIX]{x1D6EC}}}\end{eqnarray}$$

and $v_{c}$ the critical speed defined by summing over background ions

(2.5) $$\begin{eqnarray}v_{c}^{3}=v_{c}^{3}(\unicode[STIX]{x1D713})={\displaystyle \frac{3\unicode[STIX]{x03C0}^{1/2}T_{e}^{3/2}(\unicode[STIX]{x1D713})}{(2m)^{1/2}n_{e}(\unicode[STIX]{x1D713})}}\mathop{\sum }_{i}{\displaystyle \frac{Z_{i}^{2}n_{i}(\unicode[STIX]{x1D713})}{M_{i}}}.\end{eqnarray}$$

Electron and bulk ion densities and temperatures are denoted by $n_{j}$ and $T_{j}$ , with $m$ the electron mass and $M_{i}$ the mass of a bulk ion of charge $Z_{i}$ . The density of slowing down alphas is

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

where we assume $v_{0}^{3}\gg v_{c}^{3}$ as is the case for the deuterium–tritium (D–T) fusion reaction. Self-collisions are unimportant for the alphas. They are effectively a trace population since $n_{s}/n_{e}\sim T_{i}/Mv_{0}^{2}\ll 1$ , based on the alpha heating estimate $n_{s}Mv_{0}^{2}\sim n_{e}T_{i}$ .

To obtain the slowing down distribution function the usual collision operator for alphas is employed (see Cordey Reference Cordey1976), namely,

(2.7) $$\begin{eqnarray}C\{f\}={\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}}}\unicode[STIX]{x1D735}_{v}\boldsymbol{\cdot }\left[\left({\displaystyle \frac{v^{3}+v_{c}^{3}}{v^{3}}}\right)\boldsymbol{v}f+{\displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}}{2v^{3}}}(v^{2}\boldsymbol{I}-\boldsymbol{v}\boldsymbol{v})\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{v}\,f\right],\end{eqnarray}$$

where

(2.8) $$\begin{eqnarray}v_{\unicode[STIX]{x1D706}}^{3}=v_{\unicode[STIX]{x1D706}}^{3}(\unicode[STIX]{x1D713})={\displaystyle \frac{3\unicode[STIX]{x03C0}^{1/2}T_{e}^{3/2}(\unicode[STIX]{x1D713})}{(2m)^{1/2}M_{\unicode[STIX]{x1D6FC}}n_{e}(\unicode[STIX]{x1D713})}}\mathop{\sum }_{i}Z_{i}^{2}n_{i}(\unicode[STIX]{x1D713}).\end{eqnarray}$$

To obtain this form and generalizations of it expansions of the collision operator are employed for $v_{e}\gg v\gg v_{i}$ , where $v_{e}=\sqrt{2T_{e}/m}$ and $v_{i}=\sqrt{2T_{i}/M_{i}}$ are the electron and typical bulk ion thermal speeds. Typically, $v_{0}\gtrsim v_{c}\sim v_{\unicode[STIX]{x1D706}}$ , with $v_{0}/v_{c}\sim 3$ for D–T fusion. Equation (2.3) is the solution of (2.1) with (2.7) inserted since a delta function sink resides in $\unicode[STIX]{x1D70F}_{s}^{-1}v_{c}^{3}\unicode[STIX]{x1D735}_{v}\boldsymbol{\cdot }(v^{-3}\boldsymbol{v}f)\rightarrow -4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{s}^{-1}v_{c}^{3}f(\unicode[STIX]{x1D713},v=0)\unicode[STIX]{x1D6FF}(\boldsymbol{v})$ as $v\rightarrow 0$ .

3 Drift kinetic formulation

For the Vlasov operator we use the drift kinetic equation of Hazeltine (Reference Hazeltine1973) in $\boldsymbol{r}$ , $v$ or

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle E=v^{2}/2+Ze\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D713})/M, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}=v_{\bot }^{2}/2B, & \displaystyle\end{eqnarray}$$

and gyrophase $\unicode[STIX]{x1D711}$ variables, with an electrostatic electric field

(3.3) $$\begin{eqnarray}\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D713})-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}(\boldsymbol{r}),\end{eqnarray}$$

where the total electrostatic potential is $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D713})+\unicode[STIX]{x1D719}(\boldsymbol{r})$ with $\unicode[STIX]{x1D719}(\boldsymbol{r})$ a periodic function of the poloidal, $\unicode[STIX]{x1D717}$ , and toroidal, $\unicode[STIX]{x1D701}$ , angles that also depends on the poloidal flux function $\unicode[STIX]{x1D713}$ . Then

(3.4) $$\begin{eqnarray}\displaystyle (v_{\Vert }\boldsymbol{b}+\boldsymbol{v}_{d})\boldsymbol{\cdot }\left(\unicode[STIX]{x1D735}f-{\displaystyle \frac{Ze}{Mv}}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}v}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\right) & = & \displaystyle {\displaystyle \frac{v_{\Vert }}{B}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left\{f\left[\boldsymbol{B}+{\displaystyle \frac{Mc}{Ze}}\unicode[STIX]{x1D735}\times (v_{\Vert }\boldsymbol{b})\right]\right\}\nonumber\\ \displaystyle & & \displaystyle -\,{\displaystyle \frac{Ze}{Mv}}v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}v}}=C\{f\}+{\displaystyle \frac{S\unicode[STIX]{x1D6FF}(v-v_{0})}{4\unicode[STIX]{x03C0}v^{2}}},\qquad\end{eqnarray}$$

where $f$ is the alpha distribution function, $C\{f\}$ is the linear collision operator (2.7) for alphas, and

(3.5) $$\begin{eqnarray}\boldsymbol{v}_{d}={\displaystyle \frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}}}\unicode[STIX]{x1D735}\times (v_{\Vert }\boldsymbol{b})={\displaystyle \frac{c}{B^{2}}}\boldsymbol{B}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}+{\displaystyle \frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D6FA}}}\boldsymbol{b}\times \unicode[STIX]{x1D735}B+{\displaystyle \frac{v_{||}^{2}}{\unicode[STIX]{x1D6FA}}}\boldsymbol{b}\times (\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{b})+{\displaystyle \frac{v_{\Vert }^{2}}{\unicode[STIX]{x1D6FA}}}\boldsymbol{b}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{b}.\end{eqnarray}$$

The preceding form of the drift kinetic equation is adequate for our purposes as it gives the correct electric and magnetic drifts. It does not give the proper small parallel velocity correction $(v_{\bot }^{2}/2\unicode[STIX]{x1D6FA})\boldsymbol{b}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{b}$ , to the parallel streaming term, as discussed in Boozer (Reference Boozer1980), Parra & Catto (Reference Parra and Catto2008), and Landreman & Catto (Reference Landreman and Catto2013). We neglect this small parallel streaming correction from here on as it is of no importance, and we assume $Ze\unicode[STIX]{x1D719}\ll Ze\unicode[STIX]{x1D6F7}\sim T_{i}\sim T_{e}$ . In addition, we may view $v=v(E,\unicode[STIX]{x1D713})$ by using

(3.6) $$\begin{eqnarray}v=\,\sqrt{2[E-Ze\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D713})/M]}\end{eqnarray}$$

whenever we need to keep non-radial components of the $(c/B)\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}$ drift. Moreover, we use

(3.7) $$\begin{eqnarray}v_{\Vert }^{2}=v^{2}-2\unicode[STIX]{x1D707}B=\,2[E-(Ze\unicode[STIX]{x1D6F7}/M)-\unicode[STIX]{x1D707}B].\end{eqnarray}$$

The turning points of a trapped particle are always at the same value of $B$ as it moves on a flux surface even in the presence of precession.

In the preceding, the magnetic field is

(3.8) $$\begin{eqnarray}\boldsymbol{B}=\boldsymbol{B}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701})=B\boldsymbol{b}=|\boldsymbol{B}|\boldsymbol{b},\end{eqnarray}$$

and the alpha gyrofrequency is

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}=ZeB/Mc.\end{eqnarray}$$

Using $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D717}$ , and $\unicode[STIX]{x1D701}$ as the variables, with

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D701}-q\unicode[STIX]{x1D717},\end{eqnarray}$$

and $q=q(\unicode[STIX]{x1D713})$ the safety factor, the Clebsch and Boozer (Reference Boozer1981) representations for the magnetic field are

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

with $K(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701})$ periodic in $\unicode[STIX]{x1D717}$ and $\unicode[STIX]{x1D701}$ . The preceding give

(3.12) $$\begin{eqnarray}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D717}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}\end{eqnarray}$$

and

(3.13) $$\begin{eqnarray}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}=q\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D717}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}=q\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717},\end{eqnarray}$$

as well as $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=0=\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}$ . In addition,

(3.14) $$\begin{eqnarray}B^{2}=(G+qI)\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717},\end{eqnarray}$$

with $G/qI\sim rB_{p}/qRB_{t}\sim \unicode[STIX]{x1D700}^{2}/q^{2}\ll 1$ , $\unicode[STIX]{x1D700}$ the inverse aspect ratio, and $B_{p}$ and $B_{t}$ the poloidal and toroidal magnetic fields. In addition, we are ignoring finite gyroradius corrections, but will retain finite poloidal gyroradius effects by assuming $q/\unicode[STIX]{x1D700}\gg 1$ .

The ripple $\unicode[STIX]{x1D6FF}$ due to $N$ toroidal field coils is defined as

(3.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=(B_{\max }-B_{\min })/(B_{\max }+B_{\min }),\end{eqnarray}$$

with $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D700}$ . We consider the ripple trapping wells as being due to the toroidal magnetic field that is produced by the $N$ toroidal field coils. We will often make use of the simple form

(3.16) $$\begin{eqnarray}B=B_{0}[1-\unicode[STIX]{x1D700}(\unicode[STIX]{x1D713})\cos \unicode[STIX]{x1D717}-\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D713})\cos (N\unicode[STIX]{x1D701})],\end{eqnarray}$$

to obtain explicit results. This simple form allows magnetic field minima away from $\unicode[STIX]{x1D717}=0$ . For ripple extending to the magnetic axis a form such as $\unicode[STIX]{x1D6FF}\simeq \unicode[STIX]{x1D6FF}_{0}+(\unicode[STIX]{x1D6FF}_{a}-\unicode[STIX]{x1D6FF}_{0})(r/a)^{p}$ can be used with $p$ an integer and poloidal variation of the ripple ignored. Forms with poloidally dependent ripple are not considered in order to obtain carefully evaluated analytic results. However, the evaluations in the $1/\unicode[STIX]{x1D708}$ are performed quite generally, with the simple form of (3.16) only being used to get explicit results in the final steps. Following Stringer (Reference Stringer1972) and Connor & Hastie (Reference Connor and Hastie1973), the extrema in the magnetic field are found from $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B=0$ at fixed $\unicode[STIX]{x1D6FC}$ to be given by $\unicode[STIX]{x1D700}\sin \unicode[STIX]{x1D717}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}+N\unicode[STIX]{x1D6FF}\sin (N\unicode[STIX]{x1D701})\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}=0$ or

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D700}\sin \unicode[STIX]{x1D717}+qN\unicode[STIX]{x1D6FF}\sin (N\unicode[STIX]{x1D701})=0.\end{eqnarray}$$

If $qN\unicode[STIX]{x1D6FF}\,\ll \unicode[STIX]{x1D700}$ then the ripple wells are only near $\unicode[STIX]{x1D717}=0$ and $\unicode[STIX]{x1D717}=\pm \unicode[STIX]{x03C0}$ . However, for $qN\unicode[STIX]{x1D6FF}\,>\unicode[STIX]{x1D700}$ ripple wells exist for all $\unicode[STIX]{x1D717}$ . To see this more clearly we note that because $N\gg 1$ the ripples have a short toroidal extent compared to the tokamak circumference so the variation $\unicode[STIX]{x1D717}$ of a trapped alpha during its bounce motion in a ripple is small while $N\unicode[STIX]{x1D701}$ varies by a full bounce period. To introduce magnetic field minima containing ripple trapped particles at all $\unicode[STIX]{x1D717}$ requires $qN\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D700}|\!\sin \unicode[STIX]{x1D717}|>\unicode[STIX]{x1D700}$ . Consequently, we will assume the ordering $qN\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D700}$ when possible, and use $qN\unicode[STIX]{x1D6FF}\,\gg \unicode[STIX]{x1D700}$ to further simplify expressions. Retaining a slow poloidally varying ripple introduces a small order $\unicode[STIX]{x1D6FF}$ correction to (3.17). We do not expect it to result in a qualitative change in our results.

In an axisymmetric tokamak of inverse aspect ratio $\unicode[STIX]{x1D700}$ there is a single well and the fraction of particles that are trapped is $\unicode[STIX]{x1D700}^{1/2}$ . In a tokamak with $N$ field ripples the trapped fraction in each ripple well is $\unicode[STIX]{x1D6FF}^{1/2}$ . Consequently, transit averages in simple ripple wells such as $B\simeq B_{0}[1-\unicode[STIX]{x1D6FF}\cos (N\unicode[STIX]{x1D701})]$ are very similar to those in standard axisymmetric tokamak wells $B\simeq B_{0}(1-\unicode[STIX]{x1D700}\cos \unicode[STIX]{x1D717})$ .

To retain axisymmetric and ripple transport systematically it is convenient to separate $f$ into two terms by introducing one function $f_{\ast }$ that only depends on what would be the constants of motion if we were in the axisymmetric limit and a remainder function $h$ . Therefore, we let

(3.18) $$\begin{eqnarray}f=f_{\ast }(\unicode[STIX]{x1D713}_{\ast },v)+h(\boldsymbol{r},v,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}),\end{eqnarray}$$

where $f_{\ast }\simeq f_{s}\gg h$ ,

(3.19) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=v_{\Vert }/|v_{\Vert }|,\end{eqnarray}$$

and a variable that is not the drift kinetic constant of the motion in the presence of ripple,

(3.20) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\ast }\equiv \unicode[STIX]{x1D713}-I(\unicode[STIX]{x1D713})v_{\Vert }/\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Because our tokamak is rippled $(v_{\Vert }\boldsymbol{b}+\boldsymbol{v}_{d})\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{\ast }\neq 0$ . Then we may write

(3.21) $$\begin{eqnarray}\displaystyle & & \displaystyle (v_{\Vert }\boldsymbol{b}+\boldsymbol{v}_{d})\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{\ast }+{\displaystyle \frac{v_{\Vert }}{B}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left\{h\left[\boldsymbol{B}+{\displaystyle \frac{Mc}{Ze}}\unicode[STIX]{x1D735}\times (v_{\Vert }\boldsymbol{b})\right]\right\}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,{\displaystyle \frac{Ze}{Mv}}v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}v}}=C\{f\}+{\displaystyle \frac{S\unicode[STIX]{x1D6FF}(v-v_{0})}{4\unicode[STIX]{x03C0}v^{2}}},\end{eqnarray}$$

where in going from (3.4) to (3.21) only a small term proportional $\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ to is neglected.

Recall that in an axisymmetric tokamak

(3.22) $$\begin{eqnarray}\boldsymbol{B}\rightarrow I(\unicode[STIX]{x1D713})\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}+\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713},\end{eqnarray}$$

and the alphas try to move on a surface of constant canonical angular momentum

(3.23) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\ast }\rightarrow \unicode[STIX]{x1D713}-(Mc/Ze)R^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}\boldsymbol{\cdot }\boldsymbol{v}=\unicode[STIX]{x1D713}-I(\unicode[STIX]{x1D713})v_{\Vert }/\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D6FA}^{-1}\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\boldsymbol{v}\simeq \unicode[STIX]{x1D713}-I(\unicode[STIX]{x1D713})v_{\Vert }/\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where we used $R^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}\rightarrow B^{-1}(I\,\boldsymbol{b}-\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713})$ and neglect gyromotion in the last form on the right.

To rewrite the kinetic equation we use

(3.24) $$\begin{eqnarray}(v_{\Vert }\boldsymbol{b}+\boldsymbol{v}_{d})\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{\ast }=\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(\unicode[STIX]{x1D713}-Iv_{\Vert }/\unicode[STIX]{x1D6FA})-v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(Iv_{||}/\unicode[STIX]{x1D6FA}).\end{eqnarray}$$

In an axisymmetric tokamak $\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rightarrow v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(Iv_{\Vert }/\unicode[STIX]{x1D6FA})$ and $\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(Iv_{||}/\unicode[STIX]{x1D6FA})\rightarrow 0$ (see appendix B of Parra & Catto (Reference Parra and Catto2010)). Due to ripple we must keep the difference $\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}-v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(Iv_{\Vert }/\unicode[STIX]{x1D6FA})$ . However, we can neglect $\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(Iv_{\Vert }/\unicode[STIX]{x1D6FA})$ as small since it can only depend on the departure from axisymmetry and it is also small in the drift. As a result, we find

(3.25) $$\begin{eqnarray}\displaystyle & & \displaystyle v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}h+\left[\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}-v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left({\displaystyle \frac{Iv_{\Vert }}{\unicode[STIX]{x1D6FA}}}\right)\right]{\displaystyle \frac{\unicode[STIX]{x2202}f_{\ast }}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{\ast }}}+{\displaystyle \frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }[h\unicode[STIX]{x1D735}\times (v_{\Vert }\boldsymbol{b})]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,{\displaystyle \frac{Ze}{Mv}}v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}{\displaystyle \frac{\unicode[STIX]{x2202}f_{\ast }}{\unicode[STIX]{x2202}v}}=C\{f\}+{\displaystyle \frac{S\unicode[STIX]{x1D6FF}(v-v_{0})}{4\unicode[STIX]{x03C0}v^{2}}},\end{eqnarray}$$

where we have also neglected $v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}v$ as small since we assume both $h$ and $\unicode[STIX]{x1D719}$ are small.

It is convenient to be able to relate $f_{s}$ and $f_{\ast }$ by Taylor expansion by defining

(3.26) $$\begin{eqnarray}f_{\ast }=f_{s}(\unicode[STIX]{x1D713}\rightarrow \unicode[STIX]{x1D713}_{\ast },v)={\displaystyle \frac{S(\unicode[STIX]{x1D713}_{\ast })\unicode[STIX]{x1D70F}_{s}(\unicode[STIX]{x1D713}_{\ast })H(v_{0}-v)}{4\unicode[STIX]{x03C0}[v^{3}+v_{c}^{3}(\unicode[STIX]{x1D713}_{\ast })]}}=f_{s}(\unicode[STIX]{x1D713},v)+(\unicode[STIX]{x1D713}_{\ast }-\unicode[STIX]{x1D713}){\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}+\cdots \,,\end{eqnarray}$$

from which it follows that our kinetic equation becomes

(3.27) $$\begin{eqnarray}\displaystyle & & \displaystyle v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}h+\left[\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}-v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left({\displaystyle \frac{Iv_{\Vert }}{\unicode[STIX]{x1D6FA}}}\right)\right]{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}+{\displaystyle \frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }[h\unicode[STIX]{x1D735}\times (v_{\Vert }\boldsymbol{b})]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,{\displaystyle \frac{Ze}{Mv}}v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}v}}=C\{f_{\ast }-f_{s}+h\},\end{eqnarray}$$

where we use the approximations

(3.28) $$\begin{eqnarray}\unicode[STIX]{x2202}f_{\ast }/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{\ast }\simeq \unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\end{eqnarray}$$

and

(3.29) $$\begin{eqnarray}\unicode[STIX]{x2202}f_{\ast }/\unicode[STIX]{x2202}v\simeq \unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}v,\end{eqnarray}$$

that are consistent with assuming $(Iv_{\Vert }/\unicode[STIX]{x1D6FA})f_{s}^{-1}\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\ll 1$ . In an axisymmetric tokamak the left side of (3.27) reduces to

(3.30) $$\begin{eqnarray}v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(h-{\displaystyle \frac{Ze\unicode[STIX]{x1D719}}{Mv}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}v}}\right)\simeq v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}h,\end{eqnarray}$$

as the streaming term is larger than the drift term and the $\unicode[STIX]{x1D719}$ term is unimportant. We always assume that to lowest order the slowing distribution function $f_{s}$ holds. In the next section and in the summary we will discuss that the consistency of this assumption requires that the radial diffusion is weak compared to slowing down and results in an important constraint.

We will often make use of $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D717}$ , and $\unicode[STIX]{x1D701}$ independent variables. In these variables, the divergence of an arbitrary vector $\boldsymbol{A}$ can be written as

(3.31) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{A}=\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}\left[{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left({\displaystyle \frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}}\right)+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}}\left({\displaystyle \frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}}\right)+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\left({\displaystyle \frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}}\right)\right].\end{eqnarray}$$

Then, for example,

(3.32) $$\begin{eqnarray}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times (v_{\Vert }\boldsymbol{b})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(v_{\Vert }\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713})=\boldsymbol{B}\;\,\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}\,\left[\,{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}}\left({\displaystyle \frac{I\,v_{\Vert }}{B}}\right)-\,{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\left({\displaystyle \frac{Gv_{\Vert }}{B\,}}\right)\right].\end{eqnarray}$$

Using these variables and assuming streaming dominates for the alphas gives the lowest order equation to be

(3.33) $$\begin{eqnarray}v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\bar{h}=v_{||}\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}\left({\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}}+q{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\right)=0.\end{eqnarray}$$

For the passing alphas, $h\rightarrow h_{p}$ and it must be a periodic flux function to lowest order as the passing trace out a flux surface. Therefore, we write

(3.34) $$\begin{eqnarray}h_{p}=\bar{h}_{p}(\unicode[STIX]{x1D713},v,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E})+\tilde{h}_{p}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701},v,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}).\end{eqnarray}$$

The lowest order trapped alpha distribution function, $h\rightarrow \bar{h}_{t}$ , must be even in $v_{\Vert }$ and need not be periodic in $\unicode[STIX]{x1D717}$ and $\unicode[STIX]{x1D701}$ . For it we must allow

(3.35) $$\begin{eqnarray}h_{t}=\bar{h}_{t}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707})+\tilde{h}_{t}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D717},v,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}).\end{eqnarray}$$

To annihilate the streaming term to next order we introduce the transit average along the exact $\boldsymbol{B}$ :

(3.36) $$\begin{eqnarray}\bar{A}={\displaystyle \frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\ell \,A/v_{\Vert }}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\ell /v_{\Vert }}}={\displaystyle \frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}\,A}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}\,}}={\displaystyle \frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D717}\,A/v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D717}\,/v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}}={\displaystyle \frac{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,A/v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,/v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}},\end{eqnarray}$$

with $A$ arbitrary, $\text{d}\unicode[STIX]{x1D70F}=\text{d}\ell /v_{||}=\text{d}\unicode[STIX]{x1D717}/v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}=\text{d}\unicode[STIX]{x1D701}/v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}$ , and $q\,\text{d}\unicode[STIX]{x1D717}=\text{d}\unicode[STIX]{x1D701}$ for $\unicode[STIX]{x1D6FC}$ fixed (as denoted by the subscript on the loop integral) for the trapped. Here the integrals are over a full bounce for trapped particles and over all $\unicode[STIX]{x1D717}$ or $\unicode[STIX]{x1D701}$ for passing particles as they trace out a flux surface. We transit average by allowing $B=B(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701})$ and $v_{||}=v_{\Vert }(\unicode[STIX]{x1D713},\unicode[STIX]{x1D717},\unicode[STIX]{x1D701})$ with $\unicode[STIX]{x1D717}$ and $\unicode[STIX]{x1D701}$ at fixed $\unicode[STIX]{x1D6FC}$ related by $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D701}-q\unicode[STIX]{x1D717}$ .

Then, using

(3.37) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}h}=0, & \displaystyle\end{eqnarray}$$

(3.38) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left({\displaystyle \frac{Iv_{\Vert }}{\unicode[STIX]{x1D6FA}}}\right)}=0, & \displaystyle\end{eqnarray}$$

and

(3.39) $$\begin{eqnarray}\overline{v_{\Vert }\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}}=0,\end{eqnarray}$$

the transit averaged equation to the requisite order becomes

(3.40) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}+\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D701}}=\overline{C\{f_{\ast }-f_{s}+\bar{h}_{p}+\bar{h}_{t}\}},\end{eqnarray}$$

for the trapped and the passing. Our transit averaged equation retains all the ripple neoclassical transport effects we are interested in as well as the standard axisymmetric neoclassical banana regime transport. In the axisymmetric limit $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\rightarrow 0$ and $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\rightarrow 0$ giving $\bar{h}_{t}\rightarrow 0$ . The linearity of the collision operator allows (3.40) to be split into the axisymmetric/passing and non-axisymmetric/trapped equations:

(3.41) $$\begin{eqnarray}\overline{C\{f_{\ast }-f_{s}+\bar{h}_{p}\}}=0,\end{eqnarray}$$

and

(3.42) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}+\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D701}}=\overline{C\{\bar{h}_{t}\}}.\end{eqnarray}$$

The axisymmetric equation for the passing contains only a drive $f_{\ast }-f_{s}$ and response $\bar{h}_{p}$ odd in $v_{||}$ , while the non-axisymmetric equation has a drive $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}$ and response $\bar{h}_{t}$ even in $v_{\Vert }$ . When non-axisymmetric effects are present $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=0$ for the passing along with $\unicode[STIX]{x2202}\bar{h}_{p}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}=0$ . Consequently, there is no drive term for the passing and we are left with the homogeneous equation $\overline{C\{\bar{h}_{p}\}}=0$ with ripple response $\bar{h}_{p}=0$ . For the trapped, however, $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\neq 0$ , so there is a drive term to make $\bar{h}_{t}\neq 0$ . For the trapped it is convenient to use $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D6FC}$ , and $\unicode[STIX]{x1D701}$ variables for $\bar{h}_{t}$ . As a result, we employ

(3.43) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{A}=\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}\left[{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left({\displaystyle \frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}}\right)+\left.{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D701}}\left({\displaystyle \frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}}\right)+\left.{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\right|_{\unicode[STIX]{x1D6FC}}\left({\displaystyle \frac{\boldsymbol{A}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}{\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}}\right)\right].\end{eqnarray}$$

We have also assumed $\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\gg \unicode[STIX]{x2202}\bar{h}_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ .

For the passing it is useful to introduce the flux surface average

(3.44) $$\begin{eqnarray}\langle A\rangle ={\displaystyle \frac{\displaystyle \oint \text{d}\unicode[STIX]{x1D717}\,\text{d}\unicode[STIX]{x1D701}\,A/\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}{\displaystyle \oint \text{d}\unicode[STIX]{x1D717}\,\text{d}\unicode[STIX]{x1D701}/\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}}={\displaystyle \frac{\displaystyle \oint \text{d}\unicode[STIX]{x1D717}\,\text{d}\unicode[STIX]{x1D701}\,A/B^{2}}{\displaystyle \oint \text{d}\unicode[STIX]{x1D717}\,\text{d}\unicode[STIX]{x1D701}/\,B^{2}}}\end{eqnarray}$$

to rewrite the passing kinetic equation as

(3.45) $$\begin{eqnarray}\left\langle {\displaystyle \frac{B}{v_{\Vert }}}C\{f_{\ast }-f_{s}+\bar{h}_{p}\}\right\rangle =0.\end{eqnarray}$$

The passing collisional constraint determines the velocity space dependence required of the transit averaged $\bar{h}_{p}$ in response to the poloidal as well as velocity space variation of the radial drift drive term

(3.46) $$\begin{eqnarray}f_{\ast }-f_{s}=(\unicode[STIX]{x1D713}_{\ast }-\unicode[STIX]{x1D713})\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}+\cdots \simeq -(I\,v_{\Vert }/\unicode[STIX]{x1D6FA})\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}.\end{eqnarray}$$

In this regime the radial magnetic drift of the passing across flux surfaces is disrupted by collisions, but the collisional fluxes in velocity space across the trapped–passing boundary must balance to maintain a vanishing perturbed axisymmetric trapped distribution function $\bar{h}_{t}\,=0$ . The trapped response vanishes because $\overline{v_{\Vert }/B\,}=0$ , thereby removing the drive term

(3.47) $$\begin{eqnarray}\overline{C\{v_{\Vert }f_{s}/B\}}=C\{\overline{(v_{\Vert }/B)}f_{s}\}=0.\end{eqnarray}$$

In (3.42) collisions cause radial steps that prevent the ripple trapped from returning to their starting point making $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\neq 0$ (unless the departure from axisymmetry is omnigeneous – see Landreman & Catto (Reference Landreman and Catto2012), Calvo et al. (Reference Calvo, Parra, Alonso and Velasco2014) and Helander (Reference Helander2014), for example). The radial steps randomized by collisions generate the trapped response $\bar{h}_{t}$ . When drifts within a flux surface are negligible ( $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\rightarrow 0$ ) the $1/\unicode[STIX]{x1D708}$ radial diffusivity regime is recovered. When the drifts within a flux surface become significant the radial drift decreases and the radial diffusivity is reduced to roughly a $\sqrt{\unicode[STIX]{x1D708}}$ dependence. Normally only the $\boldsymbol{E}\times \boldsymbol{B}$ is retained in the $\sqrt{\unicode[STIX]{x1D708}}$ regime.

An alternate form of (3.42) can be found by introducing the second adiabatic invariant $J$ ,

(3.48) $$\begin{eqnarray}J=J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707})={\displaystyle \frac{Mc}{Ze}}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\ell \,v_{||}={\displaystyle \frac{Mc}{Ze}}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}\,v_{||}^{2}={\displaystyle \frac{Mc}{Ze}}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D701}\,v_{\Vert }}{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}}={\displaystyle \frac{Mc}{Zeq}}(G+qI)\displaystyle \oint _{\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D701}\,v_{\Vert }}{B}},\end{eqnarray}$$

with the last form using Boozer coordinates. Then for $\bar{f}=\bar{f}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707})$

(3.49) $$\begin{eqnarray}\displaystyle & & \displaystyle \overline{{\displaystyle \frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\,[\bar{f}\,\unicode[STIX]{x1D735}\,\times \,(v_{||}\boldsymbol{b})]}={\displaystyle \frac{Mc}{Ze\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}}}\!\left[\left.{\displaystyle \frac{\unicode[STIX]{x2202}\bar{f}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\right|_{\unicode[STIX]{x1D6FC}}\!\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D701}v_{\Vert }}{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}}\right)-{\displaystyle \frac{\unicode[STIX]{x2202}\bar{f}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D713}}\!\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D701}v_{\Vert }}{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}}}\!\right)\!\right]\nonumber\\ \displaystyle & & \displaystyle \qquad ={\displaystyle \frac{1}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}}}\left(\left.{\displaystyle \frac{\unicode[STIX]{x2202}\bar{f}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}{\displaystyle \frac{\unicode[STIX]{x2202}J}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\right|_{\unicode[STIX]{x1D6FC}}-{\displaystyle \frac{\unicode[STIX]{x2202}\bar{f}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}J}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D713}}\right).\end{eqnarray}$$

The $J$ integral is to be performed at fixed $\unicode[STIX]{x1D6FC}$ such that $J=J(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707})$ . Neglecting the $\unicode[STIX]{x2202}\bar{h}_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ term as small compared to the $\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ term leads to our trapped kinetic equation with

(3.50) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=-{\displaystyle \frac{1}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}J}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D713}}\end{eqnarray}$$

and

(3.51) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}={\displaystyle \frac{1}{\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D70F}}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}J}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\right|_{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

The $\unicode[STIX]{x1D6FC}$ dependence of $J$ acts as the drive for the ripple transport, or said another way, it is responsible for the departure from omnigenity.

In the next section we will use the preceding expressions to make estimates of the neoclassical and ripple transport levels.

4 Phenomenological transport estimates

In the following subsections we modify the standard diffusivity estimates to make them appropriate for alphas in the axisymmetric and ripple transport regimes. In making these estimates we treat $\ell n(v_{0}/v_{c})\sim 1$ for simplicity.

4.1 Axisymmetric regime

For an axisymmetric tokamak, collisional alpha transport is evaluated by solving the passing equation $\overline{C\{f_{\ast }-f_{s}+\bar{h}_{p}\}}=0$ with $\bar{h}_{t}\,=0$ for the trapped. For the passing $f_{\ast }-f_{s}\sim (\unicode[STIX]{x1D70C}_{p0}/a_{\unicode[STIX]{x1D6FC}})f_{s}$ , with $\unicode[STIX]{x1D70C}_{p0}\simeq q\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D700}$ the poloidal gyroradius at birth, $\unicode[STIX]{x1D70C}_{0}=v_{0}/\unicode[STIX]{x1D6FA}$ the gyroradius at birth, $\unicode[STIX]{x1D700}\sim a/R$ in the ripple region of minor radius $a$ , and $\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}r\sim f_{s}/a_{\unicode[STIX]{x1D6FC}}$ , where the radial scale of the alpha density $a_{\unicode[STIX]{x1D6FC}}$ is allowed to be less than the minor radius a for near axis burn or within the pedestal. Assuming pitch angle scattering dominates gives the trapped fraction as $\unicode[STIX]{x1D700}^{1/2}$ , the step size as a banana width $q\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D700}^{1/2}$ , so that $\bar{h}_{p}\sim f_{s}q\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D700}^{1/2}a_{\unicode[STIX]{x1D6FC}}$ , and an effective collision time of $\unicode[STIX]{x1D700}\unicode[STIX]{x1D70F}_{s}$ . The diffusivity ( $D\sim \mathscr{F}\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D70F}$ with $\mathscr{F}$ the trapped fraction, $\unicode[STIX]{x1D6E5}$ the step, $\unicode[STIX]{x1D70F}$ the correlation time) is then

(4.1) $$\begin{eqnarray}D_{\text{axi}}^{\text{pas}}\sim (q\unicode[STIX]{x1D70C}_{0})^{2}/\unicode[STIX]{x1D700}^{3/2}\unicode[STIX]{x1D70F}_{s}=\unicode[STIX]{x1D700}^{1/2}\unicode[STIX]{x1D70C}_{p0}^{2}/\unicode[STIX]{x1D70F}_{s},\end{eqnarray}$$

as in Catto (Reference Catto1988). The pitch angle scattering pre-factor $(v_{\unicode[STIX]{x1D706}}/v)^{3}$ plays no role because in this limit it only enters as a multiplier in the solubility constraint.

However, if electron drag dominates then this estimate must be modified in accordance with the Nocentini, Tessarotto & Engelmann (Reference Nocentini, Tessarotto and Engelmann1975) evaluation. In this case the trapped fraction and banana width step remain unchanged, but the effective collision time is just the slowing down time, giving the $\unicode[STIX]{x1D700}$ smaller result

(4.2) $$\begin{eqnarray}D_{\text{axi}}^{\text{drag}}\sim (q\unicode[STIX]{x1D70C}_{0})^{2}/\unicode[STIX]{x1D700}^{1/2}\unicode[STIX]{x1D70F}_{s}=\unicode[STIX]{x1D700}^{3/2}\unicode[STIX]{x1D70C}_{p0}^{2}/\unicode[STIX]{x1D70F}_{s}.\end{eqnarray}$$

The $v_{\Vert }>0$ passing alphas are moving to larger $\unicode[STIX]{x1D713}$ surfaces to keep $\unicode[STIX]{x1D713}_{\ast }$ constant as $v$ is reduced by drag, while the $v_{\Vert }<0$ passing alphas move to smaller $\unicode[STIX]{x1D713}$ surfaces. More alphas move out then in, resulting in transport due to electron drag.

Although the ratio

(4.3) $$\begin{eqnarray}D_{\text{axi}}^{\text{drag}}/D_{\text{axi}}^{\text{pas}}\sim \unicode[STIX]{x1D700}\end{eqnarray}$$

is small, in practice the neoclassical diffusivity is between $D_{\text{axi}}^{\text{pas}}$ and $D_{\text{axi}}^{\text{drag}}$ as shown by Hsu et al. (Reference Hsu, Catto and Sigmar1990). Their plots compare their exact results with the Nocentini et al. (Reference Nocentini, Tessarotto and Engelmann1975) and Catto (Reference Catto1988) results, given by their equations (33a) and (33b), and in rough agreement with the preceding estimates.

To maintain a slowing down distribution function we need the slowing down time to be short compared to the time for diffusive losses to takes to place, that is, we need $\unicode[STIX]{x1D70F}_{s}D_{\text{axi}}^{\text{pas}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ or

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D700}^{1/2}\unicode[STIX]{x1D70C}_{p0}^{2}\ll \,a_{\unicode[STIX]{x1D6FC}}^{2}.\end{eqnarray}$$

Consequently, we need to keep the alpha poloidal gyroradius at birth comparable or less than the radial scale length of the alpha density.

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

To evaluate transport in the $1/\unicode[STIX]{x1D708}$ regime we solve the trapped equation with no toroidal rotation

(4.5) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}=\overline{C\{\bar{h}_{t}\}},\end{eqnarray}$$

when the passing response is zero, $\bar{h}_{p}=0$ . Pitch angle scattering dominates when $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\ll 1$ , where the boundary layer width due to ripple is of order $\unicode[STIX]{x1D6FF}^{1/2}$ . Using $\overline{C\{\bar{h}_{t}\}}\sim \bar{h}_{t}v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{3}\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D6FF}$ this equation gives the estimate

(4.6) $$\begin{eqnarray}{\displaystyle \frac{\bar{h}_{t}}{f_{s}}}\sim {\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}v_{0}^{4}\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D6FF}}{a_{\unicode[STIX]{x1D6FC}}Rv_{\unicode[STIX]{x1D706}}^{3}}}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{\ast }\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D708}_{\ast p}}}\ll 1,\end{eqnarray}$$

with $\unicode[STIX]{x1D70C}_{\ast }=\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}\ll 1$ , $\boldsymbol{v}_{d}\sim v_{0}\unicode[STIX]{x1D70C}_{0}/R$ , and a normalized pitch angle scattering frequency $\unicode[STIX]{x1D708}_{\ast p}$ of

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D708}_{\ast p}=Rv_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D70F}_{s}v_{0}^{4}\ll 1.\end{eqnarray}$$

The $N$ toroidal field coils result in ripple wells of depth $\unicode[STIX]{x1D6FF}$ resulting in a trapped fraction of $\mathscr{F}\sim \unicode[STIX]{x1D6FF}^{1/2}$ . The pitch angle scattering time is $(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\unicode[STIX]{x1D70F}_{s}$ is larger than the slowing down time, but the effective ripple trapped collision time $\unicode[STIX]{x1D70F}\sim (v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D6FF}$ is smaller. The radial magnetic drift estimate for the alphas, $v_{0}\unicode[STIX]{x1D70C}_{0}/R$ , is insensitive to the number of coils since it is due to the nearly axisymmetric toroidal field. It gives the radial $\unicode[STIX]{x1D735}B$ drift as $V\sim \;v_{0}\unicode[STIX]{x1D70C}_{0}/R$ (the poloidal gyroradius does not enter in the absence of axisymmetric banana motion). Then a smaller step $\unicode[STIX]{x1D6E5}\sim V\unicode[STIX]{x1D70F}\sim \,\unicode[STIX]{x1D6FF}\,(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\,\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}/R$ occurs because $\unicode[STIX]{x1D6FF}\,(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\ll 1$ . As a result, we expect an alpha diffusivity $D\sim \mathscr{F}\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D70F}$ of

(4.8) $$\begin{eqnarray}D_{1/\unicode[STIX]{x1D708}}^{\text{pas}}\sim \mathscr{F}\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D6FF}^{3/2}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\,\unicode[STIX]{x1D70F}_{s}v_{0}^{2}\,\unicode[STIX]{x1D70C}_{0}^{2}/R^{2}\end{eqnarray}$$

due to pitch angle scattering.

Our estimate assumes large ripple $\unicode[STIX]{x1D6FF}\gg \unicode[STIX]{x1D700}/qN$ and is the same as Galeev et al. (Reference Galeev, Sagdeev, Furth and Rosenbluth1969), and also the same as the Ho & Kulsrud (Reference Ho and Kulsrud1987) estimate for electrons when their power of $\unicode[STIX]{x1D700}_{h}=\unicode[STIX]{x1D6FF}$ in (3) is corrected to read $\unicode[STIX]{x1D700}_{h}^{1/2}$ as would be expected by their argument. This estimate is consistent with the large ripple tokamak limit of Stringer (Reference Stringer1972) and the more general treatment of Connor & Hastie (Reference Connor and Hastie1973). Interestingly, it is smaller than the very near quasisymmetry stellarator estimate from equation (7) of Calvo et al. (Reference Calvo, Parra, Alonso and Velasco2014) that implies $D_{1/\unicode[STIX]{x1D708}}^{\text{CPAV}}\sim \unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D70C}_{0}^{2}v_{0}^{2}/R^{2}$ , but larger than the Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2013) less quasisymmetric stellarator estimate that suggests $D_{1/\unicode[STIX]{x1D708}}^{CPVA}\sim \,\,\unicode[STIX]{x1D6FF}^{2}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D70C}_{0}^{2}v_{0}^{2}/R^{2}$ .

To keep $1/\unicode[STIX]{x1D708}$ ripple losses due to pitch angle scattering from depleting the lowest order slowing down distribution during a slowing down time we need the ripple to be small enough to keep $\unicode[STIX]{x1D70F}_{s}D_{1/\unicode[STIX]{x1D708}}^{\text{pas}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ . Needing $\bar{h}_{t}\ll f_{s}$ with $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\ll 1$ this gives a condition that is difficult to satisfy,

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{3/2}(v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{3})\,\unicode[STIX]{x1D70C}_{\ast }^{2}\ll \,\unicode[STIX]{x1D708}_{\ast p}^{2}\ll 1.\end{eqnarray}$$

The ratio of pitch angle scattering $1/\unicode[STIX]{x1D708}$ ripple transport to neoclassical,

(4.10) $$\begin{eqnarray}D_{1/\unicode[STIX]{x1D708}}^{\text{pas}}/D_{\text{axi}}^{\text{pas}}\sim \,\,\unicode[STIX]{x1D6FF}^{3/2}(v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{3})(\unicode[STIX]{x1D700}^{3/2}/\unicode[STIX]{x1D708}_{\ast p}^{2}q^{2}),\end{eqnarray}$$

implies that $1/\unicode[STIX]{x1D708}$ regime ripple transport dominates at small collisionalities when $D_{1/\unicode[STIX]{x1D708}}^{\text{pas}}\;>\;D_{\text{axi}}^{\text{pas}}$ or

(4.11) $$\begin{eqnarray}q\unicode[STIX]{x1D708}_{\ast p}<\unicode[STIX]{x1D700}^{3/4}\unicode[STIX]{x1D6FF}^{3/4}(v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{3})^{1/2}.\end{eqnarray}$$

Our estimate does not account for electron drag, which does not trap and de-trap alphas, but does change their radial step size as they slow. Electron drag dominates when $\unicode[STIX]{x1D6FF}\,(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\gg 1$ . Using $\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D70F}_{s}$ , $V\sim v_{0}\unicode[STIX]{x1D70C}_{0}/R$ , $\unicode[STIX]{x1D6E5}\sim V\unicode[STIX]{x1D70F}\sim \,\,\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}/R$ , and $\mathscr{F}\sim \unicode[STIX]{x1D6FF}^{1/2}$ gives the estimate

(4.12) $$\begin{eqnarray}D_{1/\unicode[STIX]{x1D708}}^{\text{drag}}\sim \mathscr{F}\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D6FF}^{1/2}\unicode[STIX]{x1D70F}_{s}v_{0}^{2}\,\unicode[STIX]{x1D70C}_{0}^{2}/R^{2}.\end{eqnarray}$$

Electron drag ripple loss and pitch angle scattering of ripple trapped are comparable when

(4.13) $$\begin{eqnarray}D_{1/\unicode[STIX]{x1D708}}^{\text{pas}}/D_{1/\unicode[STIX]{x1D708}}^{\text{drag}}\sim \unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\sim 1.\end{eqnarray}$$

For D–T fusion $v_{0}/v_{c}\sim 3$ with $v_{\unicode[STIX]{x1D706}}\sim v_{c}$ , indicating that for a ripple $\unicode[STIX]{x1D6FF}>1/30$ , electron drag could dominate.

Keeping $1/\unicode[STIX]{x1D708}$ ripple losses due to electron drag from depleting the lowest order slowing down distribution during a slowing down time requires $\unicode[STIX]{x1D70F}_{s}D_{1/\unicode[STIX]{x1D708}}^{\text{drag}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ or

(4.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\,\unicode[STIX]{x1D70C}_{\ast }^{2}\ll \,R^{2}/v_{0}^{2}\unicode[STIX]{x1D70F}_{s}^{2}=\unicode[STIX]{x1D708}_{\ast d}^{2}\ll 1.\end{eqnarray}$$

This inequality is difficult to satisfy in this electron drag dominated limit since it requires

(4.15) $$\begin{eqnarray}{\displaystyle \frac{\bar{h}_{t}}{f_{s}}}\sim {\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}v_{0}\unicode[STIX]{x1D70F}_{s}}{a_{\unicode[STIX]{x1D6FC}}R}}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{\ast }}{\unicode[STIX]{x1D708}_{\ast d}}}\ll 1\end{eqnarray}$$

to find a self-consistent solution.

Electron drag ripple loss compares and axisymmetric neoclassical compare as

(4.16) $$\begin{eqnarray}D_{1/\unicode[STIX]{x1D708}}^{\text{drag}}/D_{\text{axi}}^{\text{drag}}\,\sim \,\,\unicode[STIX]{x1D6FF}^{1/2}\,\unicode[STIX]{x1D700}^{1/2}\,(v_{0}\unicode[STIX]{x1D70F}_{s}/qR)^{2}=\unicode[STIX]{x1D6FF}^{1/2}\,\unicode[STIX]{x1D700}^{1/2}/q^{2}\unicode[STIX]{x1D708}_{\ast d}^{2}.\end{eqnarray}$$

Consequently, $D_{1/\unicode[STIX]{x1D708}}^{\text{drag}}\;>\;D_{\text{axi}}^{\text{drag}}$ when

(4.17) $$\begin{eqnarray}q\unicode[STIX]{x1D708}_{\ast d}<\unicode[STIX]{x1D6FF}^{1/4}\unicode[STIX]{x1D700}^{1/4},\end{eqnarray}$$

which differs by $\unicode[STIX]{x1D6FF}^{1/2}\unicode[STIX]{x1D700}^{1/2}v_{0}^{3/2}/v_{\unicode[STIX]{x1D706}}^{3/2}$ from the pitch angle scattering inequality.

Once the ripple is large enough to satisfy $q\unicode[STIX]{x1D708}_{\ast d}<\unicode[STIX]{x1D6FF}^{1/4}\unicode[STIX]{x1D700}^{1/4}$ electron drag transport enters the $1/\unicode[STIX]{x1D708}$ ripple loss regime only if (4.15) is satisfied. Similarly, when $q\unicode[STIX]{x1D708}_{\ast }<\unicode[STIX]{x1D700}^{3/4}\unicode[STIX]{x1D6FF}^{3/4}(v_{\unicode[STIX]{x1D706}}^{3}/v_{0}^{3})^{1/2}$ the ripple is large enough for pitch angle scattering to be in the $1/\unicode[STIX]{x1D708}$ ripple loss regime provided (4.6) is satisfied.

4.3 The $\surd \unicode[STIX]{x1D708}$ regime

As the toroidal rotation increases we enter the $\sqrt{\unicode[STIX]{x1D708}}$ regime. Once again only the trapped matter since $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=0$ provides no drive for the passing. For the trapped we must solve

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

where we expect

(4.19) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}\bar{h}_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\sim \unicode[STIX]{x1D714}\bar{h}_{t}/q,\end{eqnarray}$$

with $\unicode[STIX]{x1D714}=\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}$ the toroidal $\boldsymbol{E}\times \boldsymbol{B}$ drift frequency in a flux surface. In the $\sqrt{\unicode[STIX]{x1D708}}$ regime $\overline{C\{\bar{h}\}}\ll \overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}\unicode[STIX]{x2202}\bar{h}_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}$ except in a narrow boundary layer requiring the toroidal drift frequency to be fast compared to the effective pitch angle scattering frequency,

(4.20) $$\begin{eqnarray}\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\gg q.\end{eqnarray}$$

Moreover, away from the boundary layer

(4.21) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\sim \overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\end{eqnarray}$$

gives

(4.22) $$\begin{eqnarray}\bar{h}_{t}/f_{s}\sim qv_{0}\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D714}R\,a_{\unicode[STIX]{x1D6FC}}\ll 1.\end{eqnarray}$$

A boundary layer narrower than $\unicode[STIX]{x1D6FF}^{1/2}$ is only possible if pitch angle scattering is the dominant collisional process. Therefore, in the boundary layer we must balance

(4.23) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\sim \overline{C\{\bar{h}_{t}\}}.\end{eqnarray}$$

This balance between the strong $\boldsymbol{E}\times \boldsymbol{B}$ drift within the flux surface and collisions, further reduces the width $w$ in pitch angle $\unicode[STIX]{x1D706}$ of the boundary layer by enhancing the pitch angle scattering time $\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}$ :

(4.24) $$\begin{eqnarray}\overline{C\{\bar{h}\}}\sim {\displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}}{\,\unicode[STIX]{x1D70F}_{s}v_{0}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}\bar{h}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}^{2}}}\sim {\displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}\bar{h}}{\,\unicode[STIX]{x1D70F}_{s}v_{0}^{3}w^{2}}}.\end{eqnarray}$$

The balance between collisions and $\boldsymbol{E}\times \boldsymbol{B}$ drift then gives the normalized width of the boundary layer $w$ to be

(4.25) $$\begin{eqnarray}w\sim (q/\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s})^{1/2}(v_{\unicode[STIX]{x1D706}}/v_{0})^{3/2}\ll \unicode[STIX]{x1D6FF}^{1/2},\end{eqnarray}$$

indicating that the alphas must $\boldsymbol{E}\times \boldsymbol{B}$ drift on a flux surface faster than they pitch angle scatter off the ions. The effective trapped fraction is estimated from this boundary layer width to be

(4.26) $$\begin{eqnarray}\mathscr{F}\sim w,\end{eqnarray}$$

with $w\ll \unicode[STIX]{x1D6FF}^{1/2}$ giving (4.20). Then the effective correlation time to move the alphas out of the ripple traps is then the pitch angle scattering time multiplied by the fraction squared,

(4.27) $$\begin{eqnarray}\unicode[STIX]{x1D70F}\sim \mathscr{F}^{2}(v_{0}/v_{\unicode[STIX]{x1D706}})^{3}\unicode[STIX]{x1D70F}_{s}\sim q/\unicode[STIX]{x1D714}.\end{eqnarray}$$

The effective step size in the presence of $\boldsymbol{E}\times \boldsymbol{B}$ drift is

(4.28) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}\sim v_{d0}\unicode[STIX]{x1D70F}\sim q\unicode[STIX]{x1D70C}_{0}v_{0}/\unicode[STIX]{x1D714}R,\end{eqnarray}$$

as the toroidal $\boldsymbol{E}\times \boldsymbol{B}$ drift now limits the radial step. The narrower boundary layer results in $\mathscr{F}$ , $\unicode[STIX]{x1D70F}$ , and $\unicode[STIX]{x1D6E5}$ being independent of $\unicode[STIX]{x1D6FF}$ . Therefore, the pitch angle scattering diffusivity in the $\sqrt{\unicode[STIX]{x1D708}}$ regime is

(4.29) $$\begin{eqnarray}D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{pas}}\sim \mathscr{F}\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D70F}=(qv_{\unicode[STIX]{x1D706}}/v_{0})^{3/2}(\unicode[STIX]{x1D70C}_{0}v_{0}/\unicode[STIX]{x1D714}R)^{2}(\,\unicode[STIX]{x1D714}/\unicode[STIX]{x1D70F}_{s})^{1/2}.\end{eqnarray}$$

These estimates are consistent with the Galeev et al. (Reference Galeev, Sagdeev, Furth and Rosenbluth1969) and Ho & Kulsrud (Reference Ho and Kulsrud1987) estimates for ions except they did not keep $q$ . Like them, we assume large ripple, $\unicode[STIX]{x1D6FF}\gg \unicode[STIX]{x1D700}/qN$ . A more detailed boundary layer analysis presented later modifies this estimate slightly because of boundary layer subtleties found by Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) that introduce a $[\ell n(\unicode[STIX]{x1D6FF}^{1/2}/w)]^{1/2}$ factor in the denominator of $D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{pas}}$ .

To avoid depleting the slowing down distribution function during $\sqrt{\unicode[STIX]{x1D708}}$ regime ripple transport requires $\unicode[STIX]{x1D70F}_{s}D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{pas}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ or very strong toroidal rotation satisfying

(4.30) $$\begin{eqnarray}\unicode[STIX]{x1D714}R/qv_{\unicode[STIX]{x1D706}}\gg (\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}})^{4/3}(v_{0}\unicode[STIX]{x1D70F}_{s}/R)^{1/3}.\end{eqnarray}$$

As $v_{\unicode[STIX]{x1D706}}\sim v_{c}\gg v_{i}$ , sonic rotation seems required even though $\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}\ll 1$ since $v_{0}\unicode[STIX]{x1D70F}_{s}/R\gg 1$ .

Comparing $\sqrt{\unicode[STIX]{x1D708}}$ to axisymmetric banana regime neoclassical transport yields

(4.31) $$\begin{eqnarray}D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{pas}}/D_{\text{axi}}^{\text{pas}}\sim (v_{0}\unicode[STIX]{x1D70F}_{s}/qR)^{1/2}(v_{0}/\unicode[STIX]{x1D714}R)^{3/2}(\unicode[STIX]{x1D700}v_{\unicode[STIX]{x1D706}}/v_{0})^{3/2},\end{eqnarray}$$

which will be larger than unity as we expect

(4.32) $$\begin{eqnarray}(v_{0}\unicode[STIX]{x1D70F}_{s}/qR)^{1/3}\,>\,\unicode[STIX]{x1D714}R/\unicode[STIX]{x1D700}v_{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

Normally the toroidal rotation frequency will be strong enough that the weaker $\sqrt{\unicode[STIX]{x1D708}}$ regime transport replaces the stronger $1/\unicode[STIX]{x1D708}$ regime transport. Even when $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\ll 1$ , the transition occurs at about $D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{pas}}\sim D_{1/\unicode[STIX]{x1D708}}^{\text{pas}}$ or

(4.33) $$\begin{eqnarray}\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\sim q,\end{eqnarray}$$

as might be expected from (4.20). As $\unicode[STIX]{x1D70F}_{s}\sim 1~\text{s}$ , we expect there will normally be enough toroidal rotation to make $\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\gg q$ unless $\unicode[STIX]{x1D6FF}$ is tiny.

If $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\gg 1$ , then drag dominates in the $1/\unicode[STIX]{x1D708}$ regime and the transition condition becomes $D_{\sqrt{\unicode[STIX]{x1D708}}}^{\text{pas}}\sim D_{1/\unicode[STIX]{x1D708}}^{\text{drag}}$ , giving a lower toroidal rotation for the transition to the pitch angle scattering dominated $\sqrt{\unicode[STIX]{x1D708}}$ regime of

(4.34) $$\begin{eqnarray}\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}(\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})^{1/3}\sim q.\end{eqnarray}$$

In the $\sqrt{\unicode[STIX]{x1D708}}$ regime we assume the $\boldsymbol{E}\times \boldsymbol{B}$ dominates over the $\unicode[STIX]{x1D735}B$ by taking $Ze\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}>M\unicode[STIX]{x1D707}\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ or

(4.35) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D714}\,R}{v_{0}}}>{\displaystyle \frac{q\unicode[STIX]{x1D70C}_{0}}{a}},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}\;\sim c\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\sim c|\boldsymbol{E}|/RB_{p}$ . Consequently, to enter the $\sqrt{\unicode[STIX]{x1D708}}$ regime with $\boldsymbol{E}\times \boldsymbol{B}$ the dominant drift requires a large electric field and/or the birth alpha gyroradius to be small compared to the minor radius. In $H$ mode pedestals (Kagan & Catto Reference Kagan and Catto2008)

(4.36) $$\begin{eqnarray}\unicode[STIX]{x1D714}R\sim c|\boldsymbol{E}|/B_{p}\sim v_{i},\end{eqnarray}$$

so that using $v_{0}/v_{\unicode[STIX]{x1D706}}\sim 3$ and $v_{i}/v_{\unicode[STIX]{x1D706}}\sim (m/M)^{1/6}\sim 1/4$ , with $v_{i}=(2T_{i}/M_{i})^{1/2}$ , we expect $q\unicode[STIX]{x1D70C}_{0}/a<1/10$ or less to be required. If the $\boldsymbol{E}\times \boldsymbol{B}$ and $\unicode[STIX]{x1D735}B$ drifts compete such that $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}$ vanishes at isolated flux surfaces for $\unicode[STIX]{x1D707}\simeq \unicode[STIX]{x1D706}v_{0}^{2}/2B_{0}$ with $\unicode[STIX]{x1D706}\simeq 1$ to be in the narrow boundary layer, then superbanana-plateau transport can occur in small regions of phase space as discussed in detail in Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017). We ignore such effects. Also, we do not consider direct orbit losses or collisionless stochastic diffusion in the vicinity of the separatrix since the difficult issue of the collisional filling of the unconfined loss regions would need to be evaluated. Instead, we simply assume the birth flux surface departure is small compared to the alpha radial scale. This assumption also avoids the issue of stochastic diffusion in the vicinity of the separatrix.

5 Neoclassical alpha transport simplifications and notation

For $H$ mode operation the plasma density and ion temperature profiles are rather flat until they approach the pedestal, where a rapid drop in the density occurs with an often weaker drop in the ion temperature. Moreover, the radial variation of $v_{c}^{3}$ is unimportant since $v_{c}^{3}\ll v_{0}^{3}$ for D–T ( $v_{c}^{3}\sim v_{0}^{3}$ for D– $\text{He}^{\text{3}}$ ). The ion and electron densities enter as ratios in $v_{c}^{3}$ , with the electron and ion temperatures equilibrated. As a result, for D–T we can assume the strongest radial variation of the alphas is due to $S\unicode[STIX]{x1D70F}_{s}$ and we may approximate $f_{s\ast }$ by

(5.1) $$\begin{eqnarray}f_{\ast }\simeq {\displaystyle \frac{n_{s}(\unicode[STIX]{x1D713}_{\ast })}{n_{s}(\unicode[STIX]{x1D713})}}f_{s}(\unicode[STIX]{x1D713},v).\end{eqnarray}$$

If we write the $\unicode[STIX]{x1D713}$ dependence of the slowing down density as an exponential

(5.2) $$\begin{eqnarray}n_{s}(\unicode[STIX]{x1D713})\propto \text{e}^{-\unicode[STIX]{x1D705}(\unicode[STIX]{x1D713})},\end{eqnarray}$$

then for $\unicode[STIX]{x1D705}$ a slowly varying function we may use

(5.3) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\ast }=\unicode[STIX]{x1D713}-Iv_{\Vert }/\unicode[STIX]{x1D6FA}\end{eqnarray}$$

to write

(5.4) $$\begin{eqnarray}n_{s}(\unicode[STIX]{x1D713}_{\ast })/n_{s}(\unicode[STIX]{x1D713})=\text{e}^{-\unicode[STIX]{x1D705}(\unicode[STIX]{x1D713}_{\ast })+\unicode[STIX]{x1D705}(\unicode[STIX]{x1D713})}\simeq \text{e}^{-(\unicode[STIX]{x1D713}_{\ast }-\unicode[STIX]{x1D713})\,\text{d}\unicode[STIX]{x1D705}/\text{d}\unicode[STIX]{x1D713}}=\text{e}^{Q}\end{eqnarray}$$

to retain finite orbit effects, by defining

(5.5) $$\begin{eqnarray}Q=\unicode[STIX]{x1D705}^{\prime }Iv_{\Vert }/\unicode[STIX]{x1D6FA}\,\;\sim \unicode[STIX]{x1D70C}_{p}/a_{\unicode[STIX]{x1D6FC}}\lesssim 1,\end{eqnarray}$$

with $RB_{p}\unicode[STIX]{x2202}n_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\sim n_{s}/a_{\unicode[STIX]{x1D6FC}}$ and $a_{\unicode[STIX]{x1D6FC}}$ the radial scale length of the alpha density profile. Based on the preceding, for D–T we can use

(5.6) $$\begin{eqnarray}f_{\ast }-f_{s}\,=(\text{e}^{Q}-1)f_{s}.\end{eqnarray}$$

We assume $Q\ll 1$ so we may use

(5.7) $$\begin{eqnarray}n_{s}(\unicode[STIX]{x1D713}_{\ast })/n_{s}(\unicode[STIX]{x1D713})=1+Q+Q^{2}/2+\cdots \,.\end{eqnarray}$$

We are, of course, assuming that gyroradius corrections from the full expression for the canonical angular momentum will be small, that is, we assume that at birth

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}\,\ll 1.\end{eqnarray}$$

The drag and pitch angle scattering collision operators are

(5.9) $$\begin{eqnarray}C_{\text{drag}}\{g\}={\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}}}\unicode[STIX]{x1D735}_{v}\boldsymbol{\cdot }\left[\left({\displaystyle \frac{v^{3}+v_{c}^{3}}{v^{3}}}\right)\boldsymbol{v}g\right]={\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}v^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[(v^{3}+v_{c}^{3})g]\end{eqnarray}$$

and

(5.10)

where $\unicode[STIX]{x1D706}=B_{0}v_{\bot }^{2}/v^{2}B$ and $\unicode[STIX]{x1D709}=v_{||}/v$ , with $B_{0}$ any convenient normalization constant or flux function such as the on axis value or $\sqrt{\langle B^{2}\rangle }$ . Transit averaging gives

(5.11) $$\begin{eqnarray}\overline{C_{\text{drag}}\{g\}}={\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}v^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[(v^{3}+v_{c}^{3})\bar{g}]\end{eqnarray}$$

and

(5.12) $$\begin{eqnarray}\overline{C_{\text{pas}}\{g\}}={\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}B_{0}}{\unicode[STIX]{x1D70F}_{s}v^{3}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}B}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}}{B^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right)\right],\end{eqnarray}$$

with the subscribe a reminder that $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D713}$ are to be held fixed when transit averaging.

6 Axisymmetric neoclassical alpha transport: a brief summary

The $v_{0}^{3}\gg v_{c}^{3}\sim v_{\unicode[STIX]{x1D706}}^{3}$ limit of axisymmetric neoclassical alpha transport is briefly summarized to indicate how it relates to the neoclassical ripple transport of alphas.

Keeping pitch angle scatter as well as drag, and expanding for $Q\ll 1$ , we must solve

(6.1) $$\begin{eqnarray}\overline{C\{\bar{h}_{p}+Qf_{s}\}}=0.\end{eqnarray}$$

However, $\bar{h}_{t}=0$ for the axisymmetric trapped response so we may employ flux surface averages to obtain the passing equation

(6.2) $$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}v^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[(v^{3}+v_{c}^{3})(\bar{h}_{p}+\bar{Q}f_{s})]+{\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}B_{0}}{\unicode[STIX]{x1D70F}_{s}v^{5}\langle B/v_{\Vert }\rangle }}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left\langle v_{\Vert }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}(\bar{h}_{p}+Qf_{s})\right\rangle \right]=0.\end{eqnarray}$$

For this axisymmetric portion of the solution the ripples may be assumed small since $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D700}\ll 1$ and the flux surface average simplifies to

(6.3) $$\begin{eqnarray}\langle A\rangle \rightarrow {\displaystyle \frac{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D717}\,A/\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D717}/\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D717}}}={\displaystyle \frac{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D717}\,A/B^{2}}{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D717}/B^{2}}}.\end{eqnarray}$$

To relate this equation to that of Hsu et al. (Reference Hsu, Catto and Sigmar1990) we continue to use the odd function

(6.4) $$\begin{eqnarray}Q=\unicode[STIX]{x1D70E}v\unicode[STIX]{x1D709}Q_{0}B_{0}/Bv_{0}\end{eqnarray}$$

and define the poloidal angle independent dimensionless function $P$ via

(6.5) $$\begin{eqnarray}\bar{h}_{p}=(\unicode[STIX]{x1D70E}Q_{0}vf_{s}/2v_{0})P(\unicode[STIX]{x1D713},v,\unicode[STIX]{x1D706}),\end{eqnarray}$$

with

(6.6) $$\begin{eqnarray}Q_{0}=v_{0}I\,\unicode[STIX]{x1D705}^{\prime }/\unicode[STIX]{x1D6FA}_{0}=v_{0}I\unicode[STIX]{x1D6FA}_{0}^{-1}\,\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713},\end{eqnarray}$$

then

(6.7) $$\begin{eqnarray}\bar{h}_{p}+Qf_{s}=(\unicode[STIX]{x1D70E}Q_{0}vf_{s}/2v_{0})[P+2\unicode[STIX]{x1D709}(B_{0}/B)],\end{eqnarray}$$

and our axisymmetric passing equation becomes

(6.8) $$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{(v^{3}+v_{c}^{3})}{2v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[v(P+2B_{0}\overline{\unicode[STIX]{x1D709}/B})H(v_{0}-v)]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,{\displaystyle \frac{B_{0}H(v_{0}-v)}{\langle B/\unicode[STIX]{x1D709}\rangle }}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left\langle \unicode[STIX]{x1D709}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}(P+2\unicode[STIX]{x1D709}B_{0}/B)\right\rangle \right]=0,\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D709}/B}\,\langle B/\unicode[STIX]{x1D709}\rangle =1$ . Using $\unicode[STIX]{x1D709}^{2}=1-\unicode[STIX]{x1D706}B/B_{0}$ gives $2\unicode[STIX]{x1D709}\unicode[STIX]{x2202}\unicode[STIX]{x1D709}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}=-B/B_{0}$ so this becomes

(6.9) $$\begin{eqnarray}{\displaystyle \frac{(v^{3}+v_{c}^{3})}{v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}\left[v\left(1-P{\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D709}\rangle }{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right)H(v_{0}-v)\right]+H(v_{0}-v){\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\langle \unicode[STIX]{x1D709}\rangle {\displaystyle \frac{\unicode[STIX]{x2202}P}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}-1\right)\right]=0,\end{eqnarray}$$

since

(6.10) $$\begin{eqnarray}\langle B/\unicode[STIX]{x1D709}\rangle =1/(\overline{\unicode[STIX]{x1D709}/B})=-2B_{0}\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D709}\rangle /\unicode[STIX]{x2202}\unicode[STIX]{x1D706}.\end{eqnarray}$$

The preceding equation for $P$ is in agreement with equation (7) of Hsu et al. (Reference Hsu, Catto and Sigmar1990) in the large drag limit. For the passing in the large aspect limit

(6.11) $$\begin{eqnarray}\langle \unicode[STIX]{x1D709}\rangle ={\displaystyle \frac{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D703}\,\unicode[STIX]{x1D709}/B^{2}}{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D703}/B^{2}}}\simeq {\displaystyle \frac{\displaystyle \oint \,\text{d}\unicode[STIX]{x1D703}\,\unicode[STIX]{x1D709}}{2\unicode[STIX]{x03C0}}}={\displaystyle \frac{4\sqrt{2\unicode[STIX]{x1D700}}E(k)}{2\unicode[STIX]{x03C0}\sqrt{(1-\unicode[STIX]{x1D700})k^{2}+2\unicode[STIX]{x1D700}}}},\end{eqnarray}$$

with $\unicode[STIX]{x1D700}=r/R$ and $k^{2}=2\unicode[STIX]{x1D700}\unicode[STIX]{x1D706}/[1-(1-\unicode[STIX]{x1D700})\unicode[STIX]{x1D706}]$ .

The drag only result of Nocentini et al. (Reference Nocentini, Tessarotto and Engelmann1975) is

(6.12) $$\begin{eqnarray}P(\unicode[STIX]{x1D713},v,\unicode[STIX]{x1D706})|_{\text{drag}}\rightarrow 1/(\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D709}\rangle /\unicode[STIX]{x2202}\unicode[STIX]{x1D706}),\end{eqnarray}$$

while the pitch angle scattering only solution of Catto (Reference Catto1988) is

(6.13) $$\begin{eqnarray}\unicode[STIX]{x2202}P(\unicode[STIX]{x1D713},v,\unicode[STIX]{x1D706})/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}|_{\text{pas}}\rightarrow 1/\langle \unicode[STIX]{x1D709}\rangle ,\end{eqnarray}$$

with $P=0$ for the trapped. Hsu et al. (Reference Hsu, Catto and Sigmar1990) point out that the drag solution fails at the trapped–passing separatrix, and the pitch angle scattering solution does not satisfy the jump condition obtained by integrating (6.8) across the delta function alpha birth source.

Hsu et al. (Reference Hsu, Catto and Sigmar1990) solve the general problem for arbitrary aspect ratio concentric circular flux surfaces by separation of variables and a Sturm–Liouville eigenvalue procedure similar to that of Cordey (Reference Cordey1976) by writing

(6.14) $$\begin{eqnarray}P(\unicode[STIX]{x1D713},v,\unicode[STIX]{x1D706})=\mathop{\sum }_{k=1}^{\infty }\unicode[STIX]{x1D6EC}_{k}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D706})V_{k}(\unicode[STIX]{x1D713},v).\end{eqnarray}$$

The eigenfunctions $\unicode[STIX]{x1D6EC}_{k}$ satisfy the eigenvalue equation (Cordey Reference Cordey1976; Hsu et al. Reference Hsu, Catto and Sigmar1990)

(6.15) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left(\unicode[STIX]{x1D706}\langle \unicode[STIX]{x1D709}\rangle {\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6EC}_{k}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right)=\unicode[STIX]{x1D705}_{n}{\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D709}\rangle }{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\unicode[STIX]{x1D6EC}_{k},\end{eqnarray}$$

with $\unicode[STIX]{x1D705}_{n}$ the $n\text{th}$ eigenvalue associated with the normalization

(6.16) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{k}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D706}=0)=1\end{eqnarray}$$

and the trapped–passing separatrix $(\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c})$ boundary condition

(6.17) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{k}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c})=0.\end{eqnarray}$$

The associated orthogonality condition is

(6.18) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x1D706}_{c}}\text{d}\unicode[STIX]{x1D706}\unicode[STIX]{x1D6EC}_{k}\unicode[STIX]{x1D6EC}_{j}\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D709}\rangle /\unicode[STIX]{x2202}\unicode[STIX]{x1D706}=0\end{eqnarray}$$

for $j\neq k$ . Continuing to follow Hsu et al. (Reference Hsu, Catto and Sigmar1990) gives for our simplified version of their (12a),

(6.19) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[v(\unicode[STIX]{x1D70E}_{k}-V_{k})H(v_{0}-v)]={\displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}(\unicode[STIX]{x1D70E}_{k}-\unicode[STIX]{x1D705}_{k}V_{k})}{(v^{3}+v_{c}^{3})}}H(v_{0}-v),\end{eqnarray}$$

where integrating across $v_{0}$ using

(6.20) $$\begin{eqnarray}V_{k}(\unicode[STIX]{x1D713},v>v_{0})=0\end{eqnarray}$$

yields the jump condition

(6.21) $$\begin{eqnarray}V_{k}(\unicode[STIX]{x1D713},v_{0})=\unicode[STIX]{x1D70E}_{k}\equiv {\displaystyle \frac{\displaystyle \int _{0}^{\unicode[STIX]{x1D706}_{c}}\,\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D6EC}_{k}}{\displaystyle \int _{0}^{\unicode[STIX]{x1D706}_{c}}\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D6EC}_{k}^{2}\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D709}\rangle /\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}.\end{eqnarray}$$

For the D–T case considered here the limiting solution for $v_{0}^{3}\gg v_{c}^{3}\sim v_{\unicode[STIX]{x1D706}}^{3}$ is simply

(6.22) $$\begin{eqnarray}V_{k}\simeq \unicode[STIX]{x1D70E}_{k}\left[1-(\unicode[STIX]{x1D705}_{k}-1){\displaystyle \frac{v_{\unicode[STIX]{x1D706}}^{3}(v_{0}^{2}-v^{2})}{2v_{0}^{3}v^{2}}}\right],\end{eqnarray}$$

with the more general solution given in Hsu et al. (Reference Hsu, Catto and Sigmar1990). The first eigenvalue $\unicode[STIX]{x1D705}_{1}$ is typically order unity and the other $\unicode[STIX]{x1D705}_{n}$ are typically large.

The axisymmetric neoclassical alpha radial particle ( $d=0$ ) and radial heat ( $d=1$ ) fluxes are evaluated as in Hsu et al. (Reference Hsu, Catto and Sigmar1990) from

(6.23) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{neo}}=-(I/\unicode[STIX]{x1D6FA}_{0})\left\langle \int \,\text{d}^{3}v(Mv^{2}/2)^{d}v_{\Vert }C\{Qf_{s}+\bar{h}_{p}\}\right\rangle .\end{eqnarray}$$

The results are typically above the Nocentini et al. (Reference Nocentini, Tessarotto and Engelmann1975) and below the Catto (Reference Catto1988) results.

Notice that for the axisymmetric neoclassical particle fluxes ambipolarity is automatically satisfied because if we sum over all species momentum conservation gives

(6.24) $$\begin{eqnarray}\left\langle \mathop{\sum }_{\text{all}}M\int \,\text{d}^{3}v\,v_{\Vert }C\{f\}\right\rangle =0.\end{eqnarray}$$

As a result,

(6.25) $$\begin{eqnarray}\mathop{\sum }_{\text{all}}Z\,\unicode[STIX]{x1D6E4}_{0}^{\text{neo}}=0,\end{eqnarray}$$

where here $\unicode[STIX]{x1D6E4}_{0}^{\text{neo}}$ , $Z$ , and $M$ are the species particle flux, charge number and mass.

7 $1/\unicode[STIX]{x1D708}$ ripple transport of alphas

In the $1/\unicode[STIX]{x1D708}$ regime $\bar{h}_{p}=0$ for the passing, while for the trapped we must solve

(7.1) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}=\overline{C\{\bar{h}_{t}\}}\end{eqnarray}$$

for $\bar{h}_{t}=\bar{h}_{t}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707})$ . We consider the two limits of pitch angle scattering and electron drag dominating. In all situations considered for $1/\unicode[STIX]{x1D708}$ transport both sets of turning points ( $v_{\Vert }=0$ ) for a charge of fixed $\unicode[STIX]{x1D707}$ are at the same value of $B$ as the motion is on a fixed field line of constant $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6FC}$ . For the simple form of (3.16) this means the poloidal angle of the two turning points must be at the same value of $\unicode[STIX]{x1D700}\cos \unicode[STIX]{x1D717}+\unicode[STIX]{x1D6FF}\cos [N(\unicode[STIX]{x1D6FC}-q\unicode[STIX]{x1D717})]$ . In the $1/\unicode[STIX]{x1D708}$ regime trapped alphas in local wells cannot be detrapped or trapped without collisions.

7.1 Pitch angle scattering dominates: $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\ll 1$

Our pitch angle scattering evaluation ignores drag in the alpha collision operator so that

(7.2) $$\begin{eqnarray}\overline{C\{\bar{h}_{t}\}}={\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}B_{0}}{\unicode[STIX]{x1D70F}_{s}v^{3}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right]\end{eqnarray}$$

and uses

(7.3) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=-{\displaystyle \frac{B_{0}v^{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B\right)}{\unicode[STIX]{x1D6FA}_{0}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}.\end{eqnarray}$$

This limit is valid when $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\ll 1$ .

As a result of the preceding, we must solve

(7.4) $$\begin{eqnarray}{\displaystyle \frac{3v_{\unicode[STIX]{x1D706}}^{3}}{\unicode[STIX]{x1D70F}_{s}v^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right]\simeq {\displaystyle \frac{B_{0}v^{2}}{\unicode[STIX]{x1D6FA}_{0}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)\right],\end{eqnarray}$$

where we use

(7.5) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}={\displaystyle \frac{3\unicode[STIX]{x1D709}}{2}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}=-{\displaystyle \frac{3B\unicode[STIX]{x1D709}}{2B_{0}}}.\end{eqnarray}$$

To keep $\unicode[STIX]{x2202}\bar{h}_{t}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}$ well behaved in the deeply trapped limit, the constant of integration must vanish. This also avoids a jump at the trapped–passing boundary where we make $\bar{h}_{t}=0$ to match $\bar{h}_{p}=0$ for the passing that must satisfy $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=0$ . Integrating once gives

(7.6) $$\begin{eqnarray}\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\simeq {\displaystyle \frac{B_{0}\unicode[STIX]{x1D70F}_{s}v^{5}}{3v_{\unicode[STIX]{x1D706}}^{3}\unicode[STIX]{x1D6FA}_{0}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right).\end{eqnarray}$$

Integrating again from the trapped–passing boundary associated with the maximum ripple magnetic field  (where ) to $\unicode[STIX]{x1D706}$ we obtain the pitch angle scattering solution

(7.7) $$\begin{eqnarray}\bar{h}_{t}={\displaystyle \frac{B_{0}\unicode[STIX]{x1D70F}_{s}v^{5}}{3\unicode[STIX]{x1D6FA}_{0}v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\int _{B_{0}/\underset{B}{\frown }}^{\unicode[STIX]{x1D706}}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}.\end{eqnarray}$$

We will continue to work with this general expression for now, rather than ignoring curvature drift and using (3.16) to find the approximate results

(7.8) $$\begin{eqnarray}\left.{\displaystyle \frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\right|_{\unicode[STIX]{x1D701}}=-{\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}}{q}}\sin [(\unicode[STIX]{x1D701}-\unicode[STIX]{x1D6FC})/q]\simeq {\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}}{q}}\sin (\unicode[STIX]{x1D6FC}/q)\end{eqnarray}$$

and

(7.9) $$\begin{eqnarray}\unicode[STIX]{x2202}\unicode[STIX]{x1D709}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}|_{\unicode[STIX]{x1D701}}\,\simeq -(\unicode[STIX]{x1D700}\unicode[STIX]{x1D706}/2q\unicode[STIX]{x1D709})\sin (\unicode[STIX]{x1D6FC}/q).\end{eqnarray}$$

These approximate results if used here give a slight difference with a final result smaller by a factor 3/4. The approximate results of (7.8) and (7.9) follow the procedure of Stringer (Reference Stringer1972) and Connor & Hastie (Reference Connor and Hastie1973) that take advantage of the slow variation of $\unicode[STIX]{x1D717}$ while $\unicode[STIX]{x1D701}$ varies between its two values at the lower of the magnetic field maximums  for fixed $\unicode[STIX]{x1D6FC}$ . Their procedure notes that $-\unicode[STIX]{x03C0}<N\unicode[STIX]{x1D701}<\unicode[STIX]{x03C0}$ implies $-\unicode[STIX]{x03C0}/Nq<\unicode[STIX]{x1D717}+(\unicode[STIX]{x1D6FC}/q)<\unicode[STIX]{x03C0}/Nq$ so that $\unicode[STIX]{x1D717}$ varies very little along a ripple trapped orbit at fixed $\unicode[STIX]{x1D6FC}$ for $N\gg 1$ . It will be used shortly to simplify (7.20).

The alpha particle ( $d=0$ ) and heat ( $d=1$ ) ripple fluxes are calculated from

(7.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{d}^{\text{rip}} & = & \displaystyle \left\langle \int \,\text{d}^{3}v(Mv^{2}/2)^{d}\bar{h}_{t}\boldsymbol{v}_{m}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\right\rangle \nonumber\\ \displaystyle & \simeq & \displaystyle (1/2\unicode[STIX]{x1D6FA}_{0})\left\langle \int \,\text{d}^{3}v(Mv^{2}/2)^{d}v^{2}\unicode[STIX]{x1D706}\bar{h}_{t}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\ell nB\right\rangle ,\end{eqnarray}$$

where $\boldsymbol{v}_{m}$ is the magnetic drift and we use $\langle \int \!\text{d}^{3}v\,f_{s}\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\rangle =\langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }[\unicode[STIX]{x1D6FA}^{-1}\boldsymbol{b}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\int \text{d}^{3}v\,f_{s}v_{\Vert }^{2}]\rangle =0$ . The last form of the ripple flux is obtained by using $v_{\Vert }$ small to ignore the curvature drift compared to the $\unicode[STIX]{x1D735}B$ drift. The $\,\boldsymbol{E}\times \boldsymbol{B}$ drift does not enter even though it has been implicitly retained.

Using the Boozer representation for the field

(7.11) $$\begin{eqnarray}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\times \boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\ell nB=(G+qI)^{-1}(G\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}-I\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D717})\end{eqnarray}$$

and

(7.12) $$\begin{eqnarray}\text{d}^{3}v\rightarrow 2\unicode[STIX]{x03C0}(Bv^{2}/B_{0}\unicode[STIX]{x1D709})\,\text{d}v\,\text{d}\unicode[STIX]{x1D706},\end{eqnarray}$$

where we sum over both signs of $\unicode[STIX]{x1D70E}$ , yields

(7.13) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{rip}}={\displaystyle \frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FA}_{0}(G+qI)}}\left\langle \left(G{\displaystyle \frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}-I{\displaystyle \frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}}\right){\displaystyle \frac{B}{B_{0}}}\int _{0}^{v_{0}}\text{d}v\,v^{4}\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D706}{\displaystyle \frac{\bar{h}_{t}}{\unicode[STIX]{x1D709}}}\right\rangle .\end{eqnarray}$$

To evaluate the fluxes we use $\unicode[STIX]{x2202}\unicode[STIX]{x1D709}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}=-B/2\unicode[STIX]{x1D709}B_{0}$ to integrate by parts to find

(7.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}}\left[\int _{B_{0}/\underset{B}{\frown }}^{\unicode[STIX]{x1D706}}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad ={\displaystyle \frac{2B_{0}}{B}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D709}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\int _{B_{0}/\underset{B}{\frown }}^{\unicode[STIX]{x1D706}}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad ={\displaystyle \frac{2B_{0}}{B}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D709}\left[\int _{B_{0}/\underset{B}{\frown }}^{\unicode[STIX]{x1D706}}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left[{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}\right].\end{eqnarray}$$

Integrating the first term on the right side by parts again using $\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}=-3B\unicode[STIX]{x1D709}/2B_{0}$ yields

(7.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}}\left[\int _{B_{0}/\underset{B}{\frown }}^{\unicode[STIX]{x1D706}}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad ={\displaystyle \frac{2B_{0}}{B}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D709}\,\left(1+{\displaystyle \frac{2B_{0}\unicode[STIX]{x1D709}^{2}}{3\unicode[STIX]{x1D706}B}}\right){\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}.\end{eqnarray}$$

Neglecting curvature drift and other $\unicode[STIX]{x1D709}^{2}$ terms compared to order unity terms and assuming $(\oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})/(\oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709})$ is a slow function of $\unicode[STIX]{x1D706}$ , we integrate by parts one final time to obtain

(7.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}}\left[\int _{B_{0}/\underset{B}{\frown }}^{\unicode[STIX]{x1D706}}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right)}}\right]\simeq {\displaystyle \frac{4B_{0}^{2}}{3B^{2}}}\underset{\unicode[STIX]{x1D709}}{\frown }^{3}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }^{3}\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\underset{\unicode[STIX]{x1D709}}{\frown }\right)}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\cdots \simeq -2\underset{\unicode[STIX]{x1D709}}{\frown }^{3}{\displaystyle \frac{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}|_{\unicode[STIX]{x1D701}}\right)}{\underset{B}{\frown }\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }\right)}},\end{eqnarray}$$

with . Inserting the preceding, noticing $G/I\simeq rB_{p}/RB_{t}\simeq \unicode[STIX]{x1D700}^{2}/q$ , and using $\unicode[STIX]{x2202}(B/B_{0})/\unicode[STIX]{x2202}\unicode[STIX]{x1D703}\simeq \unicode[STIX]{x1D700}\sin \unicode[STIX]{x1D703}$ and $\unicode[STIX]{x2202}(B/B_{0})/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}\simeq N\unicode[STIX]{x1D6FF}\sin (N\unicode[STIX]{x1D701})$ , gives the ripple driven radial particle ( $d=0$ ) and heat ( $d=1$ ) fluxes associated with pitch angle scattering to be

(7.17) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{pas}}\simeq -{\displaystyle \frac{2\unicode[STIX]{x03C0}B_{0}^{2}\unicode[STIX]{x1D700}\unicode[STIX]{x1D70F}_{s}}{3\unicode[STIX]{x1D6FA}_{0}^{2}qv_{\unicode[STIX]{x1D706}}^{3}}}\left(\int _{0}^{v_{0}}\text{d}v\,v^{9}\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\right)\left\langle \sin \unicode[STIX]{x1D717}\underset{\unicode[STIX]{x1D709}}{\frown }^{3}{\displaystyle \frac{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}|_{\unicode[STIX]{x1D701}}\right)}{\underset{B}{\frown }\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }\right)}}\right\rangle .\end{eqnarray}$$

Using $v_{0}\gg v_{c}$ to integrate we find

(7.18) $$\begin{eqnarray}\displaystyle \int _{0}^{v_{0}}\text{d}v\,v^{9}\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}} & \simeq & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left({\displaystyle \frac{S\unicode[STIX]{x1D70F}_{s}}{4\unicode[STIX]{x03C0}}}\right)\int _{0}^{v_{0}}\text{d}v\,v^{6}\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{(Mv_{0}^{2}/2)^{d}v_{o}^{7}}{4\unicode[STIX]{x03C0}(7+2d)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left[{\displaystyle \frac{n_{s}}{\ell n(v_{0}/v_{c})}}\right].\end{eqnarray}$$

Assuming

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

we obtain our most general form for the $1/\unicode[STIX]{x1D708}$ fluxes due to pitch angle scattering to be

(7.20) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{pas}}\simeq -{\displaystyle \frac{(Mv_{0}^{2}/2)^{d}\unicode[STIX]{x1D700}^{2}B_{0}^{2}\unicode[STIX]{x1D70F}_{s}v_{0}^{7}(\unicode[STIX]{x2202}n_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})}{6(7+2d)q^{2}\unicode[STIX]{x1D6FA}_{0}^{2}v_{\unicode[STIX]{x1D706}}^{3}\ell n(v_{0}/v_{c})}}\left\langle \sin \unicode[STIX]{x1D717}\,\underset{\unicode[STIX]{x1D709}}{\frown }^{3}{\displaystyle \frac{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}|_{\unicode[STIX]{x1D701}}\right)}{(\unicode[STIX]{x1D700}B_{0}/q)\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\underset{\unicode[STIX]{x1D709}}{\frown }\right)}}\right\rangle .\end{eqnarray}$$

To simplify we now use approximation (7.8) and assume the poloidal variation of $\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}$ is slow so we can use $\sin (\unicode[STIX]{x1D717}-q^{-1}\unicode[STIX]{x1D701})\simeq \sin \unicode[STIX]{x1D717}$ . The details are presented in the appendix A and yield

(7.21) $$\begin{eqnarray}\displaystyle \left\langle \sin ^{2}\unicode[STIX]{x1D717}\sqrt{[1-(B/\underset{B}{\frown })]^{3}}\right\rangle & = & \displaystyle {\displaystyle \frac{1}{4\unicode[STIX]{x03C0}^{2}}}\int _{\text{all}}\,\text{d}\unicode[STIX]{x1D717}\sin ^{2}\unicode[STIX]{x1D703}\int _{\text{all}}\text{d}(N\unicode[STIX]{x1D701})\sqrt{[1-(B/\underset{B}{\frown })]^{3}}\qquad \nonumber\\ \displaystyle & \equiv & \displaystyle {\displaystyle \frac{4\sqrt{2}}{3\unicode[STIX]{x03C0}}}\unicode[STIX]{x1D6FF}^{3/2}W(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}),\end{eqnarray}$$

with the ripple scattering coefficient defined as a function of a single parameter by

(7.22) $$\begin{eqnarray}\displaystyle & & \displaystyle W(\unicode[STIX]{x1D6FE})={\displaystyle \frac{3}{16\unicode[STIX]{x03C0}\sqrt{2}\,\unicode[STIX]{x1D6FE}^{2}}}\int _{0}^{U}{\displaystyle \frac{\text{d}X\cos X\sin ^{2}X}{\sqrt{\unicode[STIX]{x1D6FE}^{2}-\sin ^{2}X}}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\int _{X}^{Y_{1}(X)}\text{d}Y[\cos X-\cos Y+(X-Y)\sin X]^{3/2}\rightarrow \;\;\left\{\begin{array}{@{}ll@{}}1-3\unicode[STIX]{x1D6FE} & \unicode[STIX]{x1D6FE}\ll 1\\ 0.02/\unicode[STIX]{x1D6FE}^{3} & \unicode[STIX]{x1D6FE}\gg 1.\end{array}\right.\end{eqnarray}$$

The upper limit $U$ of the $X$ integral is $\unicode[STIX]{x03C0}/2$ when $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}>1$ , while for $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}<1$ it is $\sin ^{-1}(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})$ . The upper limit of the $Y$ integral is the zero of $\cos X-\cos Y_{1}+(X-Y_{1})\sin X=0$ adjacent to $Y=X$ . This monotonically decreasing function is plotted as $G$ in figure 1 in Connor & Hastie (Reference Connor and Hastie1973), where it is compared to the similar, but somewhat less accurate result of Stringer (Reference Stringer1972). Only accessible values of $\unicode[STIX]{x1D717}$ and $\unicode[STIX]{x1D701}$ are to be integrated over in the flux surface average. In particular,  are not accessible bythe ripple trapped so  is the value of $\unicode[STIX]{x1D701}$ for which . For weak ripple, $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}>1$ , some values of $\unicode[STIX]{x1D717}$ will not be accessible by the ripple trapped. The relevant details of the Stringer (Reference Stringer1972) and Connor & Hastie (Reference Connor and Hastie1973) treatments are given in appendix A.

Using the preceding we obtain

(7.23) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{pas}}\simeq -{\displaystyle \frac{2\sqrt{2}\;\unicode[STIX]{x1D6FF}^{3/2}(Mv_{0}^{2}/2)^{d}\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D70F}_{s}B_{0}^{2}v_{0}^{7}W(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})}{9\unicode[STIX]{x03C0}(7+2d)q^{2}v_{\unicode[STIX]{x1D706}}^{3}\unicode[STIX]{x1D6FA}_{0}^{2}\ell n(v_{0}/v_{c})}}{\displaystyle \frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}},\end{eqnarray}$$

yielding an alpha particle diffusivity due to pitch angle scattering of

(7.24) $$\begin{eqnarray}D_{0}^{\text{pas}}\simeq {\displaystyle \frac{2\sqrt{2}\;\unicode[STIX]{x1D6FF}^{3/2}}{63\unicode[STIX]{x03C0}}}\left({\displaystyle \frac{\unicode[STIX]{x1D70F}_{s}v_{0}^{3}}{v_{\unicode[STIX]{x1D706}}^{3}}}\right){\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}^{2}v_{0}^{2}W(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})}{R^{2}\ell n(v_{0}/v_{c})}},\end{eqnarray}$$

with $W(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})\leqslant 1$ . The preceding is in agreement with our phenomenological estimate when $W/\ell n(v_{0}/v_{c})\sim 1$ , but with a rather small coefficient that seems to be due to the slowing down distribution based on (7.18).

The weak ripple limit reduces $D_{0}^{\text{pas}}$ by order $(qN\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D700})^{3}\ll 1$ . In this limit, when $(\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D700})^{5/8}\sim 1/qN$ , $D_{0}^{\text{pas}}$ becomes comparable to the result of Linsker & Boozer (Reference Linsker and Boozer1982) who find a diffusivity of roughly $(\unicode[STIX]{x1D6FF}^{2}/\unicode[STIX]{x1D700}^{1/2}qN)(\unicode[STIX]{x1D70F}_{s}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})(\unicode[STIX]{x1D70C}_{0}v_{0}/R)^{2}$ due to the effect of very weak ripple on the turning points (at $\unicode[STIX]{x1D717}\simeq \pm \unicode[STIX]{x03C0}$ ) of the non-ripple trapped banana regime alphas in the nearly axisymmetric limit (also see Davidson (Reference Davidson1976) and Tsang (Reference Tsang1977)). By assuming $\unicode[STIX]{x1D700}N\unicode[STIX]{x1D6FF}/q\ll 1$ , that is, $I\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D717}\gg G\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}$ , we are ignoring the radial drift effect evaluated by Davidson (Reference Davidson1976), Tsang (Reference Tsang1977), and Linsker & Boozer (Reference Linsker and Boozer1982). This radial drift is of order $\unicode[STIX]{x1D6FF}Na\unicode[STIX]{x1D70C}_{0}v_{0}/qR^{2}$ and appears to be negligible unless $(\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D700})^{5/8}<1/qN$ . Stochastic effects due to ripple may play a role at low collisionalities and were considered by Goldston, White & Boozer (Reference Goldston, White and Boozer1981).

The ripple departure from axisymmetry means that ambipolarity,

(7.25) $$\begin{eqnarray}\mathop{\sum }_{\text{all}}Z\,\unicode[STIX]{x1D6E4}_{0}^{\text{rip}}=0,\end{eqnarray}$$

is no longer intrinsic. However, the alpha density $n_{s}$ is so small they may be treated as a trace population. Consequently, the radial electric field adjusts the ion and electron particle transport to maintain ambipolarity as discussed for a plasma without alphas by Galeev et al. (Reference Galeev, Sagdeev, Furth and Rosenbluth1969).

7.2 Electron drag dominates: $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\gg 1$

To make the electron drag estimate we ignore pitch angle scatter and ion drag by keeping only

(7.26) $$\begin{eqnarray}\overline{C\{\bar{h}_{t}\}}\rightarrow \,\;{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}v^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}(v^{3}\bar{h}_{t}).\end{eqnarray}$$

In this limit $\unicode[STIX]{x1D6FF}(v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})\gg 1$ and we must solve

(7.27) $$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}v^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}(v^{3}\bar{h}_{t})=\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}=-{\displaystyle \frac{B_{0}v^{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B\right)}{\unicode[STIX]{x1D6FA}_{0}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}.\end{eqnarray}$$

By performing the bounce averages in $\unicode[STIX]{x1D701}$ at fixed $\unicode[STIX]{x1D6FC}$ we will again be able to take advantage of the slow variation $\unicode[STIX]{x1D703}$ of while $\unicode[STIX]{x1D701}$ varies between its two values at the lower of the magnetic field maxima. Again, the $\,\boldsymbol{E}\times \boldsymbol{B}$ does not enter even though it is implicitly retained.

When solving for $\bar{h}_{t}$ we must avoid a step at $v_{0}$ . Consequently, we integrate from $v$ to $v_{0}$ , where $\bar{h}_{t}(v>v_{0})=0$ , to find the piecewise continuous electron drag solution

(7.28) $$\begin{eqnarray}\bar{h}_{t}={\displaystyle \frac{B_{0}\unicode[STIX]{x1D70F}_{s}(v_{0}^{2}-v^{2})}{2\unicode[STIX]{x1D6FA}_{0}\;\;}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}\{H[\unicode[STIX]{x1D706}-(B_{0}/\underset{B}{\frown })]-H[\unicode[STIX]{x1D706}-(B_{0}/B)]\},\end{eqnarray}$$

where we insert Heaviside step functions as a reminder that the trapped must satisfy . In this case the radial alpha flux occurs because electron drag acts to reduce flux surface departure of the larger (smaller) number of alphas inside (outside) the flux surface. The result is outward particle and heat fluxes. The boundary layer about  is not expected to appreciably change this result. A non-trivial boundary layer analysis is required to make  and will be discussed in the next subsection. The procedure is more involved than one used in a later section since a separable solution of the homogeneous equation that vanishes above the birth speed does not exist.

Neglecting curvature drift corrections by taking $\,\unicode[STIX]{x1D709}^{2}\ll 1$ , and assuming slow poloidal variation of $\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}$ so we can use (7.9), gives

(7.29) $$\begin{eqnarray}\bar{h}_{t}={\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}\unicode[STIX]{x1D70F}_{s}(v_{0}^{2}-v^{2})\unicode[STIX]{x1D706}}{4q\unicode[STIX]{x1D6FA}_{0}\;\;}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\sin (\unicode[STIX]{x1D717}-q^{-1}\unicode[STIX]{x1D701})\{H[\unicode[STIX]{x1D706}-(B_{0}/\underset{B}{\frown })]-H[\unicode[STIX]{x1D706}-(B_{0}/B)]\},\end{eqnarray}$$

Using $\sin \unicode[STIX]{x1D717}\sin (\unicode[STIX]{x1D717}-q^{-1}\unicode[STIX]{x1D701})\simeq \sin ^{2}\unicode[STIX]{x1D717}$ in the flux expression (7.13) gives the electron drag result

(7.30) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{drag}}\simeq -{\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D70F}_{s}B_{0}^{2}}{4q^{2}\unicode[STIX]{x1D6FA}_{0}^{2}}}\left[\int _{0}^{v_{0}}\text{d}v\,v^{4}(v_{0}^{2}-v^{2})\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\right]\left\langle \sin ^{2}\unicode[STIX]{x1D717}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{B\unicode[STIX]{x1D706}^{2}}{B_{0}\unicode[STIX]{x1D709}}}\right\rangle .\end{eqnarray}$$

Then

(7.31) $$\begin{eqnarray}\int _{0}^{v_{0}}\text{d}v\,v^{4}(v_{0}^{2}-v^{2})\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\simeq {\displaystyle \frac{v_{0}^{4}(Mv_{0}^{2}/2)^{d}}{8\unicode[STIX]{x03C0}(1+d)(2+d)\ell n(v_{0}/v_{c})}}{\displaystyle \frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}},\end{eqnarray}$$

and

(7.32) $$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{B}{B_{0}}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}^{2}}{\unicode[STIX]{x1D709}}}=-2\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D706}^{2}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\nonumber\\ \displaystyle & & \displaystyle \quad =2{\displaystyle \frac{B_{0}^{2}}{\underset{B}{\frown }^{2}}}\left\{\sqrt{1-B/\underset{B}{\frown }}+{\displaystyle \frac{2+3B/\underset{B}{\frown }}{15}}\sqrt{(1-B/\underset{B}{\frown }^{3})}\right\}\simeq 2\sqrt{1-B/\underset{B}{\frown }}\end{eqnarray}$$

give

(7.33) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{drag}}\simeq -{\displaystyle \frac{(Mv_{0}^{2}/2)^{d}\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D70F}_{s}B_{0}^{2}v_{0}^{4}}{16(1+d)(2+d)q^{2}\unicode[STIX]{x1D6FA}_{0}^{2}\ell n(v_{0}/v_{c})}}{\displaystyle \frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left\langle \sin ^{2}\unicode[STIX]{x1D717}\sqrt{[1-(B/\underset{B}{\frown })]}\right\rangle ,\end{eqnarray}$$

where

(7.34) $$\begin{eqnarray}\displaystyle \left\langle \sin ^{2}\unicode[STIX]{x1D717}\sqrt{[1-(B/\underset{B}{\frown })]}\right\rangle & = & \displaystyle {\displaystyle \frac{1}{4\unicode[STIX]{x03C0}^{2}}}\int _{\text{all}}\,\text{d}\unicode[STIX]{x1D717}\sin ^{2}\unicode[STIX]{x1D703}\int _{\text{all}}\text{d}(N\unicode[STIX]{x1D701})\sqrt{[1-(B/\underset{B}{\frown })]}\nonumber\\ \displaystyle & \equiv & \displaystyle {\displaystyle \frac{4\sqrt{2}}{3\unicode[STIX]{x03C0}}}\unicode[STIX]{x1D6FF}^{1/2}D(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})\end{eqnarray}$$

is the drag coefficient derived in appendix A to be

(7.35) $$\begin{eqnarray}\displaystyle D\,(\unicode[STIX]{x1D6FE}) & = & \displaystyle {\displaystyle \frac{\sqrt{2}}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}^{2}}}\int _{0}^{U}{\displaystyle \frac{\text{d}X\cos X\sin ^{2}X}{\sqrt{\unicode[STIX]{x1D6FE}^{2}-\sin ^{2}X}}}\int _{X}^{Y_{0}}\,\text{d}Y[\cos X-\cos Y+(X-Y)\sin X]^{1/2}\nonumber\\ \displaystyle & \rightarrow & \displaystyle \left\{\begin{array}{@{}ll@{}}1-{\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}}{3}}\left[\ell n\left({\displaystyle \frac{2}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}}}\right)+{\displaystyle \frac{17}{6}}\right] & \unicode[STIX]{x1D6FE}\ll 1\\ O(1/\unicode[STIX]{x1D6FE}^{3}) & \unicode[STIX]{x1D6FE}\gg 1.\end{array}\right.\end{eqnarray}$$

Using this result we obtain the flux due to drag to be

(7.36) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{drag}}\simeq -{\displaystyle \frac{\sqrt{2}\,\unicode[STIX]{x1D6FF}^{1/2}(Mv_{0}^{2}/2)^{d}\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D70F}_{s}B_{0}^{2}v_{0}^{4}D\,(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})}{12\unicode[STIX]{x03C0}(1+d)(2+d)q^{2}\unicode[STIX]{x1D6FA}_{0}^{2}\ell n(v_{0}/v_{c})}}{\displaystyle \frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}},\end{eqnarray}$$

giving the alpha particle diffusivity due to drag of

(7.37) $$\begin{eqnarray}D_{0}^{\text{drag}}\simeq {\displaystyle \frac{\sqrt{2}\;\unicode[STIX]{x1D6FF}^{1/2}\unicode[STIX]{x1D70F}_{s}}{24\unicode[STIX]{x03C0}}}{\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}^{2}v_{0}^{2}D\,(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})}{R^{2}\ell n(v_{0}/v_{c})}},\end{eqnarray}$$

with $D\,(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})\leqslant 1$ . The diffusivity due to drag is consistent with our earlier phenomenological estimate when $D/\ell n(v_{0}/v_{c})\sim 1$ and again has a small coefficient due to (7.31).

As expected, the ratio of the pitch angle scattering to electron drag results is

(7.38) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D6E4}_{d}^{\text{pas}}}{\unicode[STIX]{x1D6E4}_{d}^{\text{drag}}}}\simeq {\displaystyle \frac{4(1+d)(2+d)\unicode[STIX]{x1D6FF}v_{0}^{3}W(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})}{(7+2d)v_{\unicode[STIX]{x1D706}}^{3}D(\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF})}}\sim {\displaystyle \frac{\unicode[STIX]{x1D6FF}v_{0}^{3}}{v_{\unicode[STIX]{x1D706}}^{3}}},\end{eqnarray}$$

which is larger (smaller) than unity when electron drag (pitch angle scatter) ripple loss dominates.

The pitch angle scattering and electron drag solutions in the $1/\unicode[STIX]{x1D708}$ regime are not exact since in reality both effects will enter in combination while we have calculated them separately. However, these results are a useful indication of when ripple starts to become a problem.

7.3 Pitch angle scattering and drag

With both pitch angle scattering and drag the full equation must be solved:

(7.39) $$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}B_{0}}{\unicode[STIX]{x1D70F}_{s}v^{3}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right]+{\displaystyle \frac{1}{\unicode[STIX]{x1D70F}_{s}v^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[(v^{3}+v_{c}^{3})\bar{h}_{t}]\nonumber\\ \displaystyle & & \displaystyle \quad ={\displaystyle \frac{2B_{0}^{2}v^{2}}{3\unicode[STIX]{x1D6FA}_{0}}}{\displaystyle \frac{(\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}B\,\right)}}\simeq -{\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}v^{2}}{q\unicode[STIX]{x1D6FA}_{0}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\sin \left({\displaystyle \frac{\unicode[STIX]{x1D6FC}}{q}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706})\left(\unicode[STIX]{x1D706}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,\right)}}.\end{eqnarray}$$

However, this general $1/\unicode[STIX]{x1D708}$ does not seem to be an analytically tractable problem. To see the reason we seek a separable solution to the homogeneous equation

(7.40) $$\begin{eqnarray}{\displaystyle \frac{2B_{0}}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right]+{\displaystyle \frac{v}{v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[(v^{3}+v_{c}^{3})H]=0,\end{eqnarray}$$

by taking

(7.41) $$\begin{eqnarray}H=\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D706})V(v),\end{eqnarray}$$

then for the separation constant $\unicode[STIX]{x1D70E}$ we would find

(7.42) $$\begin{eqnarray}{\displaystyle \frac{v}{v_{\unicode[STIX]{x1D706}}^{3}V}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}}[(v^{3}+v_{c}^{3})V]=\unicode[STIX]{x1D70E}=-{\displaystyle \frac{2B_{0}}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)\unicode[STIX]{x1D6EC}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6EC}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right].\end{eqnarray}$$

But demanding $V(v>v_{0})=0$ means there is only the trivial solution $V=0$ .

8 $\surd \unicode[STIX]{x1D708}$ ripple transport of alphas

In the $\surd \unicode[STIX]{x1D708}$ regime $\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}=0$ and $\unicode[STIX]{x2202}\bar{h}_{p}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}=0$ for the passing, giving $\overline{C\{\bar{h}_{p}\}}=0$ and therefore $\bar{h}_{p}=0.$ Here we must be careful to keep the $v\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}$ term to retain the $\boldsymbol{E}\times \boldsymbol{B}$ drift within a flux surface. This toroidal drift results in a very narrow collisional boundary layer in which alphas detrap and retrap.

For the ripple trapped we must solve

(8.1) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}+\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}=\overline{C\{\bar{h}_{t}\}}\end{eqnarray}$$

for the transit average trapped response $\bar{h}_{t}=\bar{h}_{t}(\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FC},v,\unicode[STIX]{x1D707})$ . For simplicity we consider only the case when the $\boldsymbol{E}\times \boldsymbol{B}$ drift dominates and define its associated toroidal rotation frequency $\unicode[STIX]{x1D714}$ via

(8.2) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}}={\displaystyle \frac{Mc}{Ze}}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D713})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,v_{\Vert }\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/v_{\Vert }\right)}}\simeq -c{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\equiv -\unicode[STIX]{x1D714}.\end{eqnarray}$$

In a rippled tokamak the $\boldsymbol{E}\times \boldsymbol{B}$ drift in a flux surface is mostly toroidal. If $\unicode[STIX]{x1D735}B$ drifts dominate then $\unicode[STIX]{x1D714}\rightarrow -(Mc\unicode[STIX]{x1D707}/Ze)\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}\unicode[STIX]{x1D713}\simeq (q\unicode[STIX]{x1D706}v^{2}/2\unicode[STIX]{x1D6FA}aR)\cos (\unicode[STIX]{x1D6FC}/q)$ and a similar procedure can be employed. If $\boldsymbol{E}\times \boldsymbol{B}$ and $\unicode[STIX]{x1D735}B$ drifts cancel superbanana-plateau transport must retained (Calvo et al. Reference Calvo, Parra, Velasco and Alonso2017).

We also use the more complete expressions for

(8.3) $$\begin{eqnarray}\overline{\boldsymbol{v}_{d}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}\,\,={\displaystyle \frac{2B_{0}^{2}v^{2}}{3\unicode[STIX]{x1D6FA}_{0}}}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706})(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}^{3}/B^{2}\right)}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}},\end{eqnarray}$$

and

(8.4) $$\begin{eqnarray}\overline{C\{\bar{h}_{t}\}}={\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}B_{0}}{\unicode[STIX]{x1D70F}_{s}v^{3}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\,B\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right].\end{eqnarray}$$

As a result, we are led to consider

(8.5) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D714}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}} & = & \displaystyle {\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}B_{0}}{\unicode[STIX]{x1D70F}_{s}v^{3}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}B\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{\unicode[STIX]{x1D706}\left[\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}/B^{2}\right){\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}-{\displaystyle \frac{B_{0}\unicode[STIX]{x1D70F}_{s}v^{5}}{3\unicode[STIX]{x1D6FA}_{0}v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,{\displaystyle \frac{\unicode[STIX]{x1D709}^{3}}{B^{2}}}\right)\right]\right\}.\end{eqnarray}$$

Notice that this form reduces to the solution in the $1/\unicode[STIX]{x1D708}$ regime when $\boldsymbol{E}\times \boldsymbol{B}$ drift is unimportant. We begin with this form to insure that the $1/\unicode[STIX]{x1D708}$ regime is being properly retained.

Neglecting curvature drift and assuming weak $\unicode[STIX]{x1D717}$ variation during a ripple trapped bounce leads to the simplified form

(8.6) $$\begin{eqnarray}-\unicode[STIX]{x1D714}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}={\displaystyle \frac{2v_{\unicode[STIX]{x1D706}}^{3}}{\unicode[STIX]{x1D70F}_{s}v^{3}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left\{\left(\unicode[STIX]{x1D706}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709}\right)\left[{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}+{\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}\unicode[STIX]{x1D70F}_{s}v^{5}}{2q\unicode[STIX]{x1D6FA}_{0}v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\sin \left({\displaystyle \frac{\unicode[STIX]{x1D6FC}}{q}}\right)\right]\right\}.\end{eqnarray}$$

Ho & Kulsrud (Reference Ho and Kulsrud1987) assume any field line between two mirror points is at fixed poloidal angle (see their comments below equation (13)). Consequently, they are assuming large ripple with $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}\ll 1$ based on the Stringer (Reference Stringer1972) and Connor & Hastie (Reference Connor and Hastie1973) treatments. Using this same assumption we can replace (3.16) by

(8.7) $$\begin{eqnarray}B=\tilde{B}_{0}[1-\unicode[STIX]{x1D6FF}\cos (N\unicode[STIX]{x1D701})],\end{eqnarray}$$

with

(8.8) $$\begin{eqnarray}\tilde{B}_{0}\simeq B_{0}(1-\unicode[STIX]{x1D700}\cos \unicode[STIX]{x1D717})\simeq B_{0},\end{eqnarray}$$

and then let $\unicode[STIX]{x1D700}\rightarrow \;\,\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D717}\rightarrow N\unicode[STIX]{x1D701}$ in the usual tokamak bounce averages (see, for example, the Appendix of Catto, Parra & Pusztai (Reference Catto, Parra and Pusztai2017)). Introducing the new independent variable $\unicode[STIX]{x1D705}$ by

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

with $\unicode[STIX]{x1D705}=1$ the ripple trapped–passing boundary and $\unicode[STIX]{x1D705}=0$ the deeply ripple trapped, we use

(8.10) $$\begin{eqnarray}\text{d}\unicode[STIX]{x1D706}/\unicode[STIX]{x1D706}^{2}\,=-\,4\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}\,\text{d}\unicode[STIX]{x1D705}\end{eqnarray}$$

to obtain

(8.11) $$\begin{eqnarray}{\displaystyle \frac{(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D706})(\unicode[STIX]{x1D706}\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\,\unicode[STIX]{x1D709})}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\right)}}\simeq -{\displaystyle \frac{1}{2\unicode[STIX]{x1D705}K(\unicode[STIX]{x1D705})}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}}[E(\unicode[STIX]{x1D705})-(1-\unicode[STIX]{x1D705}^{2})K(\unicode[STIX]{x1D705})]\,\substack{ \longrightarrow \\ \unicode[STIX]{x1D705}\rightarrow 1}\,-{\displaystyle \frac{1}{2}},\end{eqnarray}$$

where we use the full bounce results

(8.12) $$\begin{eqnarray}N\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}={\displaystyle \frac{8\sqrt{2\unicode[STIX]{x1D6FF}}}{\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}}}[E(\unicode[STIX]{x1D705})-(1-\unicode[STIX]{x1D705}^{2})K(\unicode[STIX]{x1D705})]\end{eqnarray}$$

and

(8.13) $$\begin{eqnarray}N\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;/\unicode[STIX]{x1D709}={\displaystyle \frac{8}{\sqrt{2\unicode[STIX]{x1D6FF}}}}\sqrt{1-\unicode[STIX]{x1D6FF}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}^{2}}\,K(\unicode[STIX]{x1D705}),\end{eqnarray}$$

with $E(\unicode[STIX]{x1D705})\rightarrow 1+(1-\unicode[STIX]{x1D705}^{2})[\ell n(4/\sqrt{1-\unicode[STIX]{x1D705}^{2}})-(1/2)]/2+\cdots \,$ and $K(\unicode[STIX]{x1D705})\rightarrow \ell n(4/\sqrt{1-\unicode[STIX]{x1D705}^{2}})+\cdots$ . As a result, using

(8.14) $$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{1}{\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}/\unicode[STIX]{x1D709}\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\left[\unicode[STIX]{x1D706}\left(\displaystyle \oint _{\unicode[STIX]{x1D6FC}}\,\text{d}\unicode[STIX]{x1D701}\;\unicode[STIX]{x1D709}\right){\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}}}\right]\simeq {\displaystyle \frac{1}{8\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D705}K(\unicode[STIX]{x1D705})}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \left\{{\displaystyle \frac{[E(\unicode[STIX]{x1D705})-(1-\unicode[STIX]{x1D705}^{2})K(\unicode[STIX]{x1D705})]}{\unicode[STIX]{x1D705}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}}\right\}\,\substack{ \longrightarrow \\ \unicode[STIX]{x1D705}\rightarrow 1}\,{\displaystyle \frac{1}{4\unicode[STIX]{x1D6FF}\ell n\left({\displaystyle \frac{16}{1-\unicode[STIX]{x1D705}^{2}}}\right)}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}}}\end{eqnarray}$$

the equation for $\bar{h}_{t}$ becomes

(8.15) $$\begin{eqnarray}{\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}v^{2}}{2q\unicode[STIX]{x1D6FA}_{0}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\sin \left({\displaystyle \frac{\unicode[STIX]{x1D6FC}}{q}}\right)-\unicode[STIX]{x1D714}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\,\simeq {\displaystyle \frac{(v_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D70F}_{s}v^{3})}{2\unicode[STIX]{x1D6FF}\ell n[16/(1-\unicode[STIX]{x1D705}^{2})]}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}}}.\end{eqnarray}$$

Continuing to take $\unicode[STIX]{x1D705}\rightarrow 1$ and defining

(8.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}=(1-\unicode[STIX]{x1D705})/8, & \displaystyle\end{eqnarray}$$

(8.17) $$\begin{eqnarray}\displaystyle & \displaystyle k\equiv {\displaystyle \frac{64\unicode[STIX]{x1D6FF}\,\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}v^{3}\,}{qv_{\unicode[STIX]{x1D706}}^{3}}}\gg 1, & \displaystyle\end{eqnarray}$$

and

(8.18) $$\begin{eqnarray}L\equiv {\displaystyle \frac{\,32\unicode[STIX]{x1D6FF}\,B_{0}\unicode[STIX]{x1D700}\,\unicode[STIX]{x1D70F}_{s}v^{5}\,}{q\unicode[STIX]{x1D6FA}_{0}v_{\unicode[STIX]{x1D706}}^{3}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}},\end{eqnarray}$$

gives the boundary layer form of the equation to be

(8.19) $$\begin{eqnarray}{\displaystyle \frac{1}{\ell n(\unicode[STIX]{x1D702})}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}}-2kq{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{t}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\simeq -2L\sin \left({\displaystyle \frac{\unicode[STIX]{x1D6FC}}{q}}\right)=-2L\text{Im}(\text{e}^{\text{i}\unicode[STIX]{x1D6FC}/q}),\end{eqnarray}$$

which is valid when $k\gg 1$ . Here Im denotes the imaginary part.

To solve the preceding equation in the boundary layer we take

(8.20) $$\begin{eqnarray}\bar{h}_{t}=\text{Im}[H(\unicode[STIX]{x1D702})\text{e}^{\text{i}\unicode[STIX]{x1D6FC}/q}]\end{eqnarray}$$

to obtain

(8.21) $$\begin{eqnarray}{\displaystyle \frac{1}{\ell n(\unicode[STIX]{x1D702})}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}}-2\text{i}kH=-2L,\end{eqnarray}$$

which has the particular solution

(8.22) $$\begin{eqnarray}H|_{p}=-\text{i}L/k.\end{eqnarray}$$

As this solution does not vanish at $\unicode[STIX]{x1D702}=0$ we must use the homogeneous solutions to satisfy this boundary condition.

For $1\ll \ell n(1/\unicode[STIX]{x1D702})\ll 1/k\unicode[STIX]{x1D702}^{2}$ one homogeneous solution starts off as

(8.23) $$\begin{eqnarray}H|_{\text{he}}\simeq \text{i}\,(L/k)\{1+\text{i}k\,\unicode[STIX]{x1D702}^{2}[\ell n(\unicode[STIX]{x1D702})-{\textstyle \frac{3}{2}}]+\cdots \,\},\end{eqnarray}$$

where the coefficient is chosen to cancel the particular solution at $\unicode[STIX]{x1D702}=0$ . The other begins as

(8.24) $$\begin{eqnarray}H|_{\text{ho}}\simeq C\unicode[STIX]{x1D702}\{1+\text{i}{\textstyle \frac{32}{3}}k\unicode[STIX]{x1D702}^{2}[\ell n(\unicode[STIX]{x1D702})-{\textstyle \frac{5}{6}}]+\cdots \,\},\end{eqnarray}$$

with $C$ an unknown constant (see appendix D of Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017)). These represent possible inner solutions. Defining a new variable

(8.25) $$\begin{eqnarray}z\equiv \unicode[STIX]{x1D702}\sqrt{2k}\sim 1,\end{eqnarray}$$

the homogeneous equation becomes

(8.26) $$\begin{eqnarray}{\displaystyle \frac{1}{\ell n(z/\sqrt{2k})}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}H|_{h}}{\unicode[STIX]{x2202}z^{2}}}\simeq \,\text{i}H|_{h},\end{eqnarray}$$

which slightly away from $\unicode[STIX]{x1D702}=0$ can be approximated by

(8.27) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}H|_{h}}{\unicode[STIX]{x2202}z^{2}}}+\text{i}\ell n(\sqrt{2k})H|_{h}\simeq 0\end{eqnarray}$$

since $k\gg 1$ . The decaying solution to this equation is

(8.28) $$\begin{eqnarray}H|_{h}\simeq \text{i}(L/k)\text{e}^{-\sqrt{-\text{i}\ell n(\sqrt{2k})}\,z}\simeq \text{i}(L/k)\text{e}^{-(1-\text{i})\sqrt{2k\ell n(\sqrt{2k})}\,\unicode[STIX]{x1D702}},\end{eqnarray}$$

where the coefficient is chosen to cancel the particular solution as $\unicode[STIX]{x1D702}\rightarrow 0$ to satisfy the boundary condition. Taking the inner limit of this outer solution gives

(8.29) $$\begin{eqnarray}H|_{h}\simeq \text{i}(L/k)[1-(1-\text{i})\sqrt{2k\ell n(\sqrt{2k})}\,\unicode[STIX]{x1D702}+\cdots \,],\end{eqnarray}$$

which allows us to match the terms linear in $\unicode[STIX]{x1D702}$ to determine the constant $C$ to be

(8.30) $$\begin{eqnarray}C=-(1+\text{i})L\sqrt{{\displaystyle \frac{2\ell n(\sqrt{2k})}{k}}}.\end{eqnarray}$$

Consequently, a adequate matched asymptotic solution for the trapped alphas up to the trapped–passing boundary and away from it is

(8.31) $$\begin{eqnarray}H={\displaystyle \frac{\text{i}L}{k}}[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}\,}-1],\end{eqnarray}$$

where both $k$ and $L$ are proportional to $\unicode[STIX]{x1D70F}_{s}$ . Forming $\bar{h}_{t}$ gives

(8.32) $$\begin{eqnarray}\bar{h}_{t}={\displaystyle \frac{B_{0}\unicode[STIX]{x1D700}v^{2}}{2\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D714}}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\text{Re}\{[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}\,}-1]\,\text{e}^{\text{i}\unicode[STIX]{x1D6FC}/q}\},\end{eqnarray}$$

with Re denoting the real part is to be taken.

Next we evaluate the particle and heat fluxes

(8.33) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\text{rip}}\simeq {\displaystyle \frac{-\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FA}_{0}q}}\left\langle {\displaystyle \frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}\unicode[STIX]{x1D717}}}\int _{0}^{v_{0}}\text{d}v\,v^{4}\left({\displaystyle \frac{Mv^{2}}{2}}\right)^{d}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\text{d}\unicode[STIX]{x1D706}\,\unicode[STIX]{x1D706}{\displaystyle \frac{\bar{h}_{t}}{\unicode[STIX]{x1D709}}}\right\rangle .\end{eqnarray}$$

To perform the pitch angle integral we let

(8.34) $$\begin{eqnarray}\sin (N\unicode[STIX]{x1D701}/2)=\unicode[STIX]{x1D705}\sin \unicode[STIX]{x1D711}\end{eqnarray}$$

so that

(8.35) $$\begin{eqnarray}\unicode[STIX]{x1D709}=\sqrt{1-\unicode[STIX]{x1D706}[1-\unicode[STIX]{x1D6FF}\cos (N\unicode[STIX]{x1D701})]}=\unicode[STIX]{x1D705}\sqrt{2\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF}}\cos \unicode[STIX]{x1D711}\end{eqnarray}$$

and

(8.36) $$\begin{eqnarray}\unicode[STIX]{x1D705}\cos \unicode[STIX]{x1D711}\,\text{d}\unicode[STIX]{x1D711}=\cos (N\unicode[STIX]{x1D701}/2)\text{d}(N\unicode[STIX]{x1D701}/2)=\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D711}}\,\text{d}(N\unicode[STIX]{x1D701}/2).\end{eqnarray}$$

Recalling that $\unicode[STIX]{x1D6FC}/q\simeq -\unicode[STIX]{x1D717}$ and noting that only $\sin (\unicode[STIX]{x1D6FC}/q)\simeq -\sin \unicode[STIX]{x1D717}$ terms contribute when evaluating flux surface averages, we find

(8.37) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Re}\left\langle \sin \unicode[STIX]{x1D717}\,\text{e}^{-\text{i}\unicode[STIX]{x1D717}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}}[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}\,}-1]\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad \simeq -{\displaystyle \frac{1}{2}}\text{Re}\left\langle \text{i}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}}[\text{e}^{-8(1-\text{i})(1-\unicode[STIX]{x1D705})\sqrt{2k\ell n(\sqrt{2k})}}-1]\right\rangle .\end{eqnarray}$$

Then using

(8.38) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D709}^{-1}\rangle & = & \displaystyle {\displaystyle \frac{\displaystyle \int _{0}^{\sin ^{-1}\unicode[STIX]{x1D705}}\unicode[STIX]{x1D709}^{-1}\text{d}(N\unicode[STIX]{x1D701}/2)}{\displaystyle \int _{0}^{\unicode[STIX]{x03C0}/2}\text{d}(N\unicode[STIX]{x1D701}/2)}}={\displaystyle \frac{\sqrt{2}}{\unicode[STIX]{x03C0}\sqrt{\unicode[STIX]{x1D6FF}}}}\int _{0}^{\unicode[STIX]{x03C0}/2}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D711}}{\sqrt{1-\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D711}}}}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{\sqrt{2}\,K(\unicode[STIX]{x1D705})}{\unicode[STIX]{x03C0}\sqrt{\unicode[STIX]{x1D6FF}}}}\simeq {\displaystyle \frac{\ell n[16/(1-\,\unicode[STIX]{x1D705}^{2})]}{\unicode[STIX]{x03C0}\sqrt{2\unicode[STIX]{x1D6FF}}}},\end{eqnarray}$$

we obtain

(8.39) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Re}\left\langle \sin \unicode[STIX]{x1D717}\,\text{e}^{-\text{i}\unicode[STIX]{x1D717}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}{\displaystyle \frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}}[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}\,}-1]\right\rangle \nonumber\\ \displaystyle & & \displaystyle \quad \simeq -{\displaystyle \frac{\sqrt{2\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x03C0}}}\text{Re}\left\{\text{i}\int _{0}^{1}\,\text{d}\unicode[STIX]{x1D705}\,\unicode[STIX]{x1D705}[\text{e}^{-8(1-\text{i})(1-\unicode[STIX]{x1D705})\sqrt{2k\ell n(\sqrt{2k})}\,}-1]\ln \left({\displaystyle \frac{8}{1-\unicode[STIX]{x1D705}}}\right)\right\}\nonumber\\ \displaystyle & & \displaystyle \quad \simeq -{\displaystyle \frac{\sqrt{\unicode[STIX]{x1D6FF}}\ell n[64\sqrt{2k\ell n(\sqrt{2k})}]}{8\unicode[STIX]{x03C0}\sqrt{k\ell n(\sqrt{2k})}}}\text{Re}\left[\text{i}\int _{0}^{8\sqrt{2k\ell n(\sqrt{2k})}}\,\text{d}\unicode[STIX]{x1D712}\text{e}^{-(l-\text{i})\unicode[STIX]{x1D712}}\right]\nonumber\\ \displaystyle & & \displaystyle =\sqrt{\unicode[STIX]{x1D6FF}}{\displaystyle \frac{\ell n[64\sqrt{2k\ell n(\sqrt{2k})}]}{16\unicode[STIX]{x03C0}\sqrt{k\ell n(\sqrt{2k})}}},\end{eqnarray}$$

where the imaginary integral does not contribute and $\unicode[STIX]{x1D712}=8(1-\unicode[STIX]{x1D705})\sqrt{2k\ell n(\sqrt{2k})}$ .

We also make use of

(8.40) $$\begin{eqnarray}\int _{0}^{v_{0}}\!\!\text{d}v\,v^{9/2}(Mv^{2}/2)^{d}\frac{\ell n[64\sqrt{2k\ell n(\sqrt{2k})}]}{\sqrt{\ell n(\sqrt{2k})}}{\displaystyle \frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\simeq {\displaystyle \frac{2v_{0}^{5/2}(Mv_{0}^{2}/2)^{d}\ell n[64\sqrt{2k_{0}\ell n(\sqrt{2k_{0}})}]}{(5+4d)\sqrt{\!\ell n(\sqrt{2k_{0}})}\,\ell n(v_{0}/v_{c})}}{\displaystyle \frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}}\!,\end{eqnarray}$$

where

(8.41) $$\begin{eqnarray}k_{0}\equiv {\displaystyle \frac{64\unicode[STIX]{x1D6FF}\,\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}v_{0}^{3}\,}{qv_{\unicode[STIX]{x1D706}}^{3}}}\gg 1.\end{eqnarray}$$

From the preceding we obtain

(8.42) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{d}^{\sqrt{\unicode[STIX]{x1D708}}}\simeq -{\displaystyle \frac{\unicode[STIX]{x1D700}^{2}B_{0}^{2}\unicode[STIX]{x1D70F}_{s}v_{\unicode[STIX]{x1D706}}^{3/2}v_{0}^{5/2}(Mv_{0}^{2}/2)^{d}[\ell n(64\sqrt{2k_{0}\ell n(\sqrt{2k_{0}})})]}{128(5+4d)\,q^{1/2}\unicode[STIX]{x1D6FA}_{0}^{2}\unicode[STIX]{x1D714}\sqrt{\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\,\ell n(v_{0}/v_{c})}}{\displaystyle \frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}},\end{eqnarray}$$

which is valid in the limit of $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}\ll 1$ . Notice that the diffusivity is of order

(8.43) $$\begin{eqnarray}D_{0}^{\sqrt{\unicode[STIX]{x1D708}}}\simeq {\displaystyle \frac{(qv_{\unicode[STIX]{x1D706}}/v_{0})^{3/2}[\ell n(64\sqrt{k_{0}\ell n(2k_{0})})](\unicode[STIX]{x1D70C}_{0}v_{0}/R)^{2}}{640\,\unicode[STIX]{x1D714}\sqrt{\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\,\ell n(v_{0}/v_{c})}}.\end{eqnarray}$$

Within logarithmic factors as in Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017) this result agrees with our phenomenological estimate, but notice that the coefficient is small due to the collisional boundary layer analysis. Interestingly, $D_{0}^{\sqrt{\unicode[STIX]{x1D708}}}$ only depends logarithmically on the ripple $\unicode[STIX]{x1D6FF}$ because (8.17) or (8.41) require a very narrow boundary layer of width $(qv_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}v_{0}^{3})^{1/2}\ll \unicode[STIX]{x1D6FF}^{1/2}$ . These boundary layers become comparable, $(qv_{\unicode[STIX]{x1D706}}^{3}/\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}v_{0}^{3})^{1/2}\sim \unicode[STIX]{x1D6FF}^{1/2}$ , at the transition between the $\sqrt{\unicode[STIX]{x1D708}}$ and $1/\unicode[STIX]{x1D708}$ regimes.

9 Summary

We formulate and solve for the neoclassical transport of alphas in a rippled tokamak by evaluating the $1/\unicode[STIX]{x1D708}$ and $\sqrt{\unicode[STIX]{x1D708}}$ regime modifications of Galeev et al. (Reference Galeev, Sagdeev, Furth and Rosenbluth1969), Stringer (Reference Stringer1972), Connor & Hastie (Reference Connor and Hastie1973), and Ho & Kulsrud (Reference Ho and Kulsrud1987) associated with alpha birth and slowing. The formulation also retains the standard axisymmetric neoclassical effects of Nocentini et al. (Reference Nocentini, Tessarotto and Engelmann1975), Catto (Reference Catto1988), and Hsu et al. (Reference Hsu, Catto and Sigmar1990). The ripple transport of alphas differs from that of the bulk ions and electrons because of its slowing down tail distribution function and the need to consider electron drag as well as pitch angle scatter. Moreover, the approximate variational treatment of Galeev et al. (Reference Galeev, Sagdeev, Furth and Rosenbluth1969) and boundary layer analysis of Ho & Kulsrud (Reference Ho and Kulsrud1987) in the $\sqrt{\unicode[STIX]{x1D708}}$ regime is avoided by a more complete and rigorous boundary layer analysis similar to that of Calvo et al. (Reference Calvo, Parra, Velasco and Alonso2017). Our $\sqrt{\unicode[STIX]{x1D708}}$ regime evaluation assumes large ripple ( $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}\gg 1$ ) and the results are given by (8.42) and (8.43). In this $\sqrt{\unicode[STIX]{x1D708}}$ regime drag is unimportant. The $1/\unicode[STIX]{x1D708}$ regime evaluations presented here allow general ripple ( $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}\sim 1$ ) and the results with only pitch angle scattering ( $\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\ll 1$ ) are given by equations (7.23) and (7.24), while those keeping only electron drag ( $\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\gg 1$ ) are given by equations (7.36) and (7.37). When pitch angle scatter and electron drag compete in the $1/\unicode[STIX]{x1D708}$ regime ( $\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\sim 1$ ) and the ripple is strong ( $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}\ll 1$ ), the transition from the axisymmetric banana regime to the $1/\unicode[STIX]{x1D708}$ occurs when the time for an birth alpha to travel a connection length is substantially smaller than the slowing down time, roughly $qR/\unicode[STIX]{x1D70F}_{s}v_{0}\sim \unicode[STIX]{x1D6FF}^{1/4}\unicode[STIX]{x1D700}^{1/2}$ , or more precisely

(9.1) $$\begin{eqnarray}qR/\unicode[STIX]{x1D70F}_{s}v_{0}\simeq 0.3\unicode[STIX]{x1D6FF}^{1/4}\unicode[STIX]{x1D700}^{1/2}(\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3})^{1/2}\sim 0.3\unicode[STIX]{x1D6FF}^{1/4}\unicode[STIX]{x1D700}^{1/2}\ll 1.\end{eqnarray}$$

For the axisymmetric banana regime we simplify by using $\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\sim 1$ and taking

(9.2) $$\begin{eqnarray}D_{\text{axi}}^{\text{ban}}\simeq 0.25q^{2}\unicode[STIX]{x1D70C}_{0}^{2}/\unicode[STIX]{x1D700}\unicode[STIX]{x1D70F}_{s}\ell n(v_{0}/v_{c})\end{eqnarray}$$

to obtain an estimate between the Nocentini et al. (Reference Nocentini, Tessarotto and Engelmann1975) and Catto (Reference Catto1988) limits to be in better agreement with the precise results of Hsu et al. (Reference Hsu, Catto and Sigmar1990). Our $1/\unicode[STIX]{x1D708}$ calculation assumes small alpha birth gyroradius,

(9.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{1/4}\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}\ll R/\unicode[STIX]{x1D70F}_{s}v_{0}\ll 1,\end{eqnarray}$$

to keep the alpha distribution function near the slowing down distribution that is assumed to have a radial scale length of $a_{\unicode[STIX]{x1D6FC}}$ . To keep $\unicode[STIX]{x1D70F}_{s}D_{\text{axi}}^{\text{ban}}/a_{\unicode[STIX]{x1D6FC}}^{2}\ll 1$ we need $\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}\ll \unicode[STIX]{x1D700}^{1/2}/q\ll 1$ , which is typically less restrictive than (9.3).

It is unlikely that (9.3) can be satisfied for any feasible $\unicode[STIX]{x1D6FF}$ . However, the inability to satisfy (9.3) is good news! It means that for all practical purposes there is no significant $1/\unicode[STIX]{x1D708}$ regime so that axisymmetric neoclassical banana regime transport transitions almost directly into less dangerous $\sqrt{\unicode[STIX]{x1D708}}$ regime for alpha ripple transport!

Moreover, for $\unicode[STIX]{x1D6FF}v_{0}^{3}/v_{\unicode[STIX]{x1D706}}^{3}\sim 1$ and $\unicode[STIX]{x1D700}/qN\unicode[STIX]{x1D6FF}\ll 1$ , the transition from the $1/\unicode[STIX]{x1D708}$ to $\sqrt{\unicode[STIX]{x1D708}}$ regime happens when a significant fraction of a full toroidal rotation happens in a slowing down time, roughly $\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\sim q$ , or more carefully

(9.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}}{q}\sim \left[\frac{\ell n(64\sqrt{k_{0}\ell n(2k_{0})})}{40\sqrt{\ell n(2k_{0})}}\right]^{2/3},\end{eqnarray}$$

where $2k_{0}\simeq 128\,\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}\unicode[STIX]{x1D6FF}v_{0}^{3}/qv_{\unicode[STIX]{x1D706}}^{3}\sim 128\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}/q\gg 1$ is required for our boundary layer analysis to be valid. Normally, $\unicode[STIX]{x1D714}\,\unicode[STIX]{x1D70F}_{s}\gg \,q$ .

To remain near a slowing down distribution the $\sqrt{\unicode[STIX]{x1D708}}$ regime requires the alpha birth gyroradius to satisfy the much stricter condition of

(9.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}R}{qv_{\unicode[STIX]{x1D706}}}\gg \left[\frac{\ell n(64\sqrt{k_{0}\ell n(2k_{0})})}{640\ell n(v_{0}/v_{c})\sqrt{\ell n(2k_{0})}}\right]^{2/3}\left(\frac{\unicode[STIX]{x1D70C}_{0}}{a_{\unicode[STIX]{x1D6FC}}}\right)^{4/3}\left(\frac{v_{0}\unicode[STIX]{x1D70F}_{s}}{R}\right)^{1/3}.\end{eqnarray}$$

Even with the small coefficient, rotation larger than the $\unicode[STIX]{x1D735}B$ drift level $(\unicode[STIX]{x1D714}R>\unicode[STIX]{x1D70C}_{0}v_{0}/R)$ and small $\unicode[STIX]{x1D70C}_{0}/a_{\unicode[STIX]{x1D6FC}}$ may be required to avoid depletion of the alpha slowing down tail distribution function.

To estimate the size of quantities for D–T fusion we use $v_{0}\simeq 1.3\times 10^{9}~\text{cm}~\text{s}^{-1}$ for a 3.5 MeV alpha, and $\unicode[STIX]{x1D70F}_{s}=\unicode[STIX]{x1D70F}_{ee}M/Z^{2}m\simeq 0.63~\text{s}$ for $T_{e}\simeq 10~\text{keV}$ and $n_{e}\simeq 10^{14}~\text{cm}^{-3}$ , with $\unicode[STIX]{x1D70F}_{ee}$ the electron collision time. We note that even if the alpha pressure was comparable to the plasma pressure that the slowing down alpha density $n_{s}$ would be small compared to the plasma density $n_{s}/n_{e}\sim 1/300$ . Consequently, the alphas are not expected to play a role in ambipolarity so any radial electric field due to ripple is due to the balance of $1/\unicode[STIX]{x1D708}$ regime electron transport and $\sqrt{v}$ regime ion transport or the radial electric field associated with an axisymmetric tokamak and determined by convservation of toroidal angular momentum (see Parra & Catto Reference Parra and Catto2008, Reference Parra and Catto2009, Reference Parra and Catto2010, must be smaller in this situation), giving an inadequate rotation level of $\unicode[STIX]{x1D714}R\sim q\unicode[STIX]{x1D70C}_{i}v_{i}/\unicode[STIX]{x1D700}a$ . Then the $\unicode[STIX]{x1D735}B$ drift will dominant and is expected to play a role similar to the $\boldsymbol{E}\times \boldsymbol{B}$ . The assumptions used to derive the alpha collision operator are satisfied since $v_{0}\ll v_{e}=(2T_{e}/m)^{1/2}\simeq 6\times 10^{9}~\text{cm}~\text{s}^{-1}$ and $v_{0}\gg v_{i}=(2T_{i}/M_{i})^{1/2}\simeq 10^{8}~\text{cm}~\text{s}^{-1}$ . Moreover, for equal amounts of D and $T$ , $v_{0}/v_{c}\simeq 3.25$ and $v_{\unicode[STIX]{x1D706}}^{3}/v_{c}^{3}\simeq 3/5$ , so our assumptions that $v_{0}^{3}\gg v_{c}^{3}\sim v_{\unicode[STIX]{x1D706}}^{3}$ are satisfied. In addition, we note that for $B_{0}=5~\text{T}$ , $\unicode[STIX]{x1D6FA}_{0}\simeq 2.4\times 10^{8}~\text{rad}~\text{s}^{-1}$ , giving $\unicode[STIX]{x1D70C}_{0}\simeq 5.4~\text{cm}$ , which must be small compared to the radial scale length $a_{\unicode[STIX]{x1D6FC}}$ of the alphas.

From the preceding numbers and $R\simeq 10~\text{m}$ , we obtain $R/\unicode[STIX]{x1D70F}_{s}v_{0}\simeq 1.2\times 10^{-6}$ . Consequently, based on (9.3), we expect that tokamaks will operate with the alphas well into the $\sqrt{\unicode[STIX]{x1D708}}$ ripple transport regime with $\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\gg q$ . Indeed, if we assume $\unicode[STIX]{x1D714}R\simeq v_{i}$ , then we find $\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\simeq 6.3\times 10^{4}$ , giving $\sqrt{\unicode[STIX]{x1D708}}$ regime ripple transport. Then the key issue becomes avoiding the depletion of the alpha slowing down distribution by satisfying (9.5).

In summary, ripple transport of alphas will be in the $\sqrt{\unicode[STIX]{x1D708}}$ regime, rather than the $1/\unicode[STIX]{x1D708}$ regime, thereby suppressing ripple transport well below the $1/\unicode[STIX]{x1D708}$ level. However, alpha ripple transport in the $\sqrt{\unicode[STIX]{x1D708}}$ regime will be a serious issue for tokamak reactors as it will be well above the axisymmetric neoclassical level and can deplete the alpha slowing down distribution function unless toroidal rotation is strong.