2. Formulation

The development of the QL description outlined here begins in a manner similar to Catto, Lee & Ram (Reference Catto, Lee and Ram2017) except for the retention of collisions, that are shown to be greatly enhanced in a narrow boundary layer for resonant particle interactions with the waves as in Catto (Reference Catto2020).

Consider the full kinetic equation

(2.1)\begin{equation}\partial f/\partial t + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }f + (Ze/M)[\boldsymbol{e} - \boldsymbol{\nabla }\varPhi + {c^{ - 1}}\boldsymbol{v} \times (\boldsymbol{B} + \boldsymbol{b})]\boldsymbol{\cdot }{\nabla _v}\,f = C\{ \,f\} ,\end{equation}

with $\boldsymbol{e}$ and $\boldsymbol{b}$ the applied rf wave fields, and $\boldsymbol{B} = B\boldsymbol{n}$ and $\boldsymbol{E} ={-} \boldsymbol{\nabla }\varPhi$ the unperturbed magnetic and electric fields, with $\boldsymbol{n}$ a parallel unit vector. Here, the unperturbed electric field drift is determined by toroidal angular momentum conservation (Parra & Catto Reference Parra and Catto2008). Writing

(2.2)\begin{equation}f = {f_0} + {f_1} + \cdots ,\end{equation}

with ${f_1} \ll {f_0}$ and ${f_0}$ independent of gyrophase $\varphi$, the fast time and space dependence of f are removed from ${f_0}$ by coarse grain (cg) averaging,

(2.3)\begin{equation}{f_0} = {\langle \,f\rangle _{cg}} = \frac{1}{{2T}}\int_{t - T}^{t + T} {\textrm{d}\tau } \frac{1}{{2\varDelta }}\int_{\psi - \varDelta }^{\psi + \varDelta } {\textrm{d}\psi ^{\prime}} \oint {\frac{{\textrm{d}\zeta }}{{2{\pi }}}\langle \,f} {\rangle _{\varphi ,\vartheta }}.\end{equation}

Here the integral over toroidal angle $\zeta$ ensures axisymmetry, while ${\langle \cdots \rangle _{\varphi ,\vartheta }}$ includes a gyroaverage ${\langle \cdots \rangle _\varphi } \equiv {(2{\pi })^{ - 1}}\oint_{\boldsymbol{r}} {\textrm{d}\varphi ( \ldots )}$ at fixed $\boldsymbol{r}$ and an average ${\langle \cdots \rangle _\vartheta }$ over poloidal angle $\vartheta$ that is used to remove the high poloidal mode number variation. The $\tau$ integral removes fast time variation by taking ${\omega ^{ - 1}} \ll T \ll \nu _{eff}^{ - 1}$, where ${\nu _{\textrm{eff}}}$ and $\omega$ are the effective collision frequency and typical rf wave frequency, respectively. The radial average over the poloidal flux function $\psi$ labelling the flux surface removes rapid variation by assuming the radial wavelengths $\lambda _{\textrm{rf}}^{\textrm{radial}}$ are much less than the radial cg width $\varDelta /R{B_p}$ which are much smaller than the minor radius a, $\lambda _{rf}^{radial} \ll \Delta /R{B_p} \ll a$, with R the major radius and ${B_p}$ the poloidal magnetic field. Further details on ${\langle \cdots \rangle _\vartheta }$ will be given soon, but for now it is sufficient to know that ${f_0}$ is allowed to be a slow function of $\vartheta$, as well as $\psi$ and t. The orderings adopted for ${f_0}$ are

(2.4)\begin{equation}\varOmega {f_0} \gg v \cdot \nabla {f_0} \gg \partial {f_0}/\partial t\sim C\{{{f_0}} \}\sim Q\{{{f_0}} \},\end{equation}

with $\varOmega = ZeB/ Mc$, C the full collision operator, and Q the QL operator to be derived by retaining trapped as well as passing particles.

Changing to the drift kinetic variables $\boldsymbol{r}$, $E = {v^2}/2 + Ze\varPhi /M$, $\mu = v_ \bot ^2/2B$, $\varphi$, and t, satisfying

(2.5)\begin{equation}\boldsymbol{v} = {\boldsymbol{v}_ \bot } + {v_{||}}\boldsymbol{n} = {v_ \bot }[{\boldsymbol{e}_1}(\boldsymbol{r})\cos \varphi + {\boldsymbol{e}_2}(\boldsymbol{r})\sin \varphi ] + {v_{||}}\boldsymbol{n}(\boldsymbol{r}),\end{equation}

and ${\boldsymbol{e}_1} \times {\boldsymbol{e}_2} = \boldsymbol{n} = \boldsymbol{B}/B$, the unperturbed QL equation in its drift kinetic form is

(2.6)\begin{equation}\partial {f_0}/\partial t + {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }{f_0} = C\{ {f_0}\} + Q\{ {f_0}\} ,\end{equation}

with the QL operator given by

(2.7)\begin{equation}Q\{ {f_0}\} ={-} {\langle {\nabla _v}\boldsymbol{\cdot }(\boldsymbol{a}{f_1})\rangle _{cg,\varphi }},\end{equation}

and the perturbed acceleration defined as

(2.8)\begin{equation}\boldsymbol{a} = (Ze/M)(\boldsymbol{e} + {c^{ - 1}}\boldsymbol{v} \times \boldsymbol{b}).\end{equation}

In the acceleration $\boldsymbol{a}$, the perturbed electric and magnetic fields are denoted by $\boldsymbol{e}$ and $\boldsymbol{b}$, respectively. The QL operator $Q\{ {f_0}\}$ is a slow function of $\vartheta$ in a tokamak so that ${\langle \cdots \rangle _\vartheta } \ne {(2{\pi })^{ - 1}}\oint {\textrm{d}\vartheta (\ldots )}$. The average ${\langle \cdots \rangle _\vartheta }$ is only allowed to operate on the poloidal variation of the wave phase to preserve entropy production (Catto et al. Reference Catto, Lee and Ram2017). Alpha particles can be treated by adding a source to the right side of (2.6).

The magnetic and electric drifts

(2.9)\begin{equation}{v_d} = {\varOmega ^{ - 1}}n \times ({\mu \nabla B + v_{||}^2n \cdot \nabla n} )+ c{B^{ - 1}}n \times \nabla \Phi \simeq {\varOmega ^{ - 1}}{v_{||}}\nabla \times ({{v_{||}}n} ),\end{equation}

are ignored by assuming ${v_{||}}n \cdot \nabla {f_0} \gg {v_d} \cdot \nabla \psi \partial {f_0}/\partial \psi $ or $1 \gg {\rho _p}f_0^{ - 1}\partial {f_0}/\partial r\sim {\rho _p}/a$, with ${\rho _p}$ the poloidal gyroradius ${\rho _p} = {v_{\textrm{th}}}B/{B_p}\varOmega = {v_{\textrm{th}}}/{\varOmega _p}$, ${v_{\textrm{th}}} = \sqrt {2T/M}$ the thermal speed, $I = R{B_t}$ a flux function, ${B_t}$ the toroidal magnetic field, and ${\boldsymbol{v}_d}\boldsymbol{\cdot }\boldsymbol{\nabla }\psi = {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }(I{\varOmega ^{ - 1}}{v_{||}})$ for an axisymmetric tokamak. It is convenient to let ${\boldsymbol{e}_1}$ be the unit vector in the $\boldsymbol{\nabla }\psi$ direction by taking $\boldsymbol{\nabla }\psi = R{B_p}{\boldsymbol{e}_1}$, with ${B_p}$ the poloidal magnetic field.

The drift kinetic treatment requires averaging the QL term over gyrophase and time, as well as performing a cg average, to make $Q\{ {f_0}\}$ a slow function of time, space, and velocity. To make the treatment analytically tractable in tokamak geometry, ${v_{||}}n \cdot \nabla {f_0} \gg C\{{{f_0}} \}\sim Q\{{{f_0}} \}\sim \partial {f_0}/\partial t$ is assumed. Then to lowest order ${v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\overline {{f_0}} = 0$ gives

(2.10)\begin{equation}\overline {{f_0}} = {f_0}(\psi ,E,\mu ,\sigma ,t),\end{equation}

where $\sigma = {v_{||}}/| {v_{||}} |$ is the sign of ${v_{||}}$. Transit or bounce averaging the equation for ${f_0}$ to annihilate the streaming term yields

(2.11)\begin{equation}\partial {f_0}/\partial t = \overline {C\{ {f_0}\} } + \overline {Q\{ {f_0}\} } ,\end{equation}

where the transit average over a full (f) poloidal circuit is defined as

(2.12)\begin{equation}\overline {( \ldots )} \equiv \oint_f {{{\textrm{d}\tau ( \ldots )} \Bigg/ {\left({\oint_f {\textrm{d}\tau } } \right)}}} ,\end{equation}

with $\textrm{d}\tau = \textrm{d}\vartheta /{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta$. The ordering here assumes ${v_{||}}n \cdot \nabla {f_0} \gg {v_d} \cdot \nabla {f_0}$ so no radial derivatives will appear in the QL operator.

An axisymmetric magnetic straight field line form

(2.13)\begin{equation}\boldsymbol{B} = I(\psi )\boldsymbol{\nabla }\zeta + \boldsymbol{\nabla }\zeta \times \boldsymbol{\nabla }\psi = \boldsymbol{\nabla }\alpha \times \boldsymbol{\nabla }\psi ,\end{equation}

is employed, with $\zeta$ the toroidal angle variable and $\vartheta$ satisfying $I\boldsymbol{\nabla }\zeta = \boldsymbol{\nabla }\psi \times \boldsymbol{\nabla }(q\vartheta )$ and

(2.14)\begin{equation}\alpha = \zeta - q(\psi )\vartheta ,\end{equation}

where the safety factor is given by

(2.15)\begin{equation}q(\psi ) = I(\psi )/{R^2}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta .\end{equation}

3. Perturbed kinetic equation

Many of the subtleties and complications are associated with the solution of the perturbed kinetic equation. In addition to retaining collisions, this equation must be solved by integrating over the unperturbed trajectories corresponding to the near periodic poloidal motion of the trapped and passing particles. To accomplish this task in toroidal geometry it is convenient to take full advantage of high frequency gyrokinetics (Lee et al. Reference Lee, Myra and Catto1983) by extending the treatment of Catto et al. (Reference Catto, Lee and Ram2017).

The perturbed distribution function ${f_1}$ is allowed to be rapidly varying in time and space by permitting

(3.1)\begin{equation}\varOmega \partial {f_1}/\partial \varphi \sim \partial {f_1}/\partial t\sim {\boldsymbol{v}_ \bot }\boldsymbol{\cdot }\boldsymbol{\nabla }{f_1}\sim {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }{f_1}.\end{equation}

It satisfies the linear equation

(3.2)\begin{align} & \partial {f_1}/\partial t + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }{f_1} + (Ze/M)( - \boldsymbol{\nabla }\varPhi + {c^{ - 1}}\boldsymbol{v} \times \boldsymbol{B})\boldsymbol{\cdot }{\nabla _v}{f_1} - C\{ {f_1}\} \notag\\ & \quad ={-} (Ze/M)(\boldsymbol{e} + {c^{ - 1}}\boldsymbol{v} \times \boldsymbol{b})\boldsymbol{\cdot }{\nabla _v}{f_0}, \end{align}

where collisions are retained as collisional boundary layers are known to develop about Landau (Johnston Reference Johnston1971; Auerbach Reference Auerbach1977) and other wave–particle (Catto Reference Catto2020) resonances. Even though the lowest order results obtained here appear insensitive to the collisional physics of the boundary layer, collisions are useful to interpret results and give a physical meaning to the interaction time associated with the delta functions.

By keeping the magnetic moment to higher order, Catto et al. (Reference Catto, Lee and Ram2017) were able to show that the resonance is modified in the expected way by the magnetic drift in tokamak geometry. As their calculation is rather involved, it is not repeated here. Instead, it is convenient to take advantage of their results by beginning with their form of the linearized kinetic equation with the drift as well as the non-resonant responses explicitly extracted. Keeping only the important drift terms, the equation to be solved is

(3.3)\begin{align}& \dfrac{{{\partial ^2}g }}{{\partial {t^2}}} + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }\dfrac{{\partial g }}{{\partial t}} + \dfrac{{Ze}}{M}( - \boldsymbol{\nabla }\varPhi + {c^{ - 1}}\boldsymbol{v} \times \boldsymbol{B})\boldsymbol{\cdot }{\nabla _v}\dfrac{{\partial g }}{{\partial t}} - C\left\{ {\dfrac{{\partial g }}{{\partial t}}} \right\}\notag\\ &\quad ={-} \dfrac{{Ze}}{M}\left\{ {\dfrac{{\partial \boldsymbol{e} }}{{\partial t}}\boldsymbol{\cdot }\boldsymbol{v}\dfrac{{\partial {f_0}}}{{ \partial E}} + \left[ {\dfrac{{\partial \boldsymbol{e}}}{{\partial t}} + ({v_{||}}\boldsymbol{n} + {\boldsymbol{v}_d})\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{e}} \right]\boldsymbol{\cdot }\dfrac{\boldsymbol{v}}{B}\dfrac{{\partial {f_0}}}{{\partial \mu }}} \right\}, \end{align}

where a non-resonant response proportional to $\partial {f_0}/\partial \mu$ is removed by defining g to satisfy

(3.4)\begin{equation}\frac{{\partial {f_1}}}{{\partial t}} = \frac{{Ze}}{{MB}}\boldsymbol{e}\boldsymbol{\cdot }({v_{||}}\boldsymbol{n} + {\boldsymbol{v}_d})\frac{{\partial {f_0}}}{{\partial \mu }} + \frac{{\partial g}}{{\partial t}}.\end{equation}

In the preceding, Faraday's law,

(3.5)\begin{equation}c\boldsymbol{\nabla } \times \boldsymbol{e} ={-} \partial \boldsymbol{b}/\partial t,\end{equation}

is used to eliminate the perturbed magnetic field. As ${f_0}$ does not vary significantly in the collisional boundary layer

(3.6)\begin{equation}C\left\{ {\frac{{\partial {f_1}}}{{\partial t}}} \right\} \simeq C\left\{ {\frac{{\partial g}}{{\partial t}}} \right\}.\end{equation}

The proof in Catto et al. (Reference Catto, Lee and Ram2017) did not keep the unperturbed electric drift. However, their procedure works when it is included as it is independent of velocity and, like all unperturbed quantities, it is assumed to contain only slow spatial variation. Therefore, when it is kept, their steps (4.13) – (4.16) continue to hold.

To treat correlated passes through resonance with finite orbit effects retained, the linearized equation is solved by a slightly more general procedure than in Catto et al. (Reference Catto, Lee and Ram2017). To do so, a different set of gyrokinetic variables is employed that takes advantage of the canonical angular momentum

(3.7)\begin{equation}{\psi _ \ast } = \psi - (Mc/Ze){R^2}\boldsymbol{\nabla }\zeta \boldsymbol{\cdot }\boldsymbol{v} = \psi + {\varOmega ^{ - 1}}\boldsymbol{v} \times \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\psi - I{v_{||}}/\varOmega ,\end{equation}

as in Kagan & Catto (Reference Kagan and Catto2008), while continuing to assume ${\rho _p} \ll a$ for unperturbed quantities, where ${\rho _p} = B\rho /{B_p}$, with ${B_p}$ the poloidal magnetic field and $\rho$ the gyroradius. In addition, the gyrokinetic angle variable

(3.8)\begin{equation}{\alpha _ \ast } = {\rm Z} - {q_ \ast }({\psi _ \ast })\varTheta ,\end{equation}

is employed along with

(3.9)\begin{gather}\varTheta = \vartheta + {\varOmega ^{ - 1}}\boldsymbol{v} \times \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta + \cdots ,\end{gather}

(3.10)\begin{gather}{\rm Z} = \zeta + {\varOmega ^{ - 1}}\boldsymbol{v} \times \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\zeta + \cdots ,\end{gather}

and to lowest order

(3.11)\begin{equation}\phi \simeq \varphi + \cdots .\end{equation}

The variables E, $\mu$, $\phi$, ${\psi _ \ast }$, ${\alpha _ \ast }$, and $\varTheta$ are chosen to follow the particle as it moves on a drift surface. The unperturbed Vlasov operator operating on any quantity Q is

(3.12)\begin{equation}\dot{Q} \equiv \frac{{\partial Q}}{{\partial t}} + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }Q + \frac{{Ze}}{M}( - \boldsymbol{\nabla }\varPhi + {c^{ - 1}}\boldsymbol{v} \times \boldsymbol{B})\boldsymbol{\cdot }{\nabla _v}Q.\end{equation}

For unperturbed trajectories, conservation of toroidal angular momentum

(3.13)\begin{equation}{\dot{\psi }_ \ast } = 0,\end{equation}

and total energy

(3.14)\begin{equation}\dot{E} = 0,\end{equation}

lead to

(3.15)\begin{equation}\dot{Q} \equiv \frac{{\partial Q}}{{\partial t}} + \dot{\phi }\frac{{\partial Q}}{{\partial \phi }} + \dot{\mu }\frac{{\partial Q}}{{\partial \mu }} + {\dot{\alpha }_ \ast }\frac{{\partial Q}}{{\partial {\alpha _ \ast }}} + \dot{\varTheta }\frac{{\partial Q}}{{\partial \varTheta }},\end{equation}

where

(3.16)\begin{gather}\dot{\varTheta } \simeq \left( {{v_{||}}\boldsymbol{n} + \frac{c}{B}\boldsymbol{n} \times \boldsymbol{\nabla }\varPhi } \right)\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }\left( {\frac{1}{\varOmega }\boldsymbol{n} \times \boldsymbol{\nabla }\vartheta } \right)\boldsymbol{\cdot }\boldsymbol{v} + \cdots ,\end{gather}

(3.17)\begin{gather}\dot{{\rm Z}} \simeq \left( {{v_{||}}\boldsymbol{n} + \frac{c}{B}\boldsymbol{n} \times \boldsymbol{\nabla }\varPhi } \right)\boldsymbol{\cdot }\boldsymbol{\nabla }\zeta + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }\left( {\frac{1}{\varOmega }\boldsymbol{n} \times \boldsymbol{\nabla }\zeta } \right)\boldsymbol{\cdot }\boldsymbol{v} + \cdots ,\end{gather}

(3.18)\begin{gather}\dot{\phi } \simeq{-} \varOmega (\boldsymbol{r}) + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }{\boldsymbol{e}_2}\boldsymbol{\cdot }{\boldsymbol{e}_1} - \frac{{{v_{||}}}}{{v_ \bot ^2}}\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{n} \times \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v} + \frac{{Ze}}{{Mv_ \bot ^2}}\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{\nabla }\varPhi + \cdots ,\end{gather}

and, as $\mu$ is an adiabatic invariant, $\dot{\mu }$ will have a vanishing gyroaverage,

(3.19)\begin{equation}{\langle \dot{\mu }\rangle _ \ast } = 0.\end{equation}

An asterisk subscript indicates the gyroaverage is to be performed holding ${\psi _ \ast }$, $\varTheta$, ${\rm Z}$, E and $\mu$ fixed as in Kagan & Catto (Reference Kagan and Catto2008). Slow temporal variation of unperturbed quantities is assumed negligible in the preceding.

Only the lowest order expressions for $\varTheta$, ${\rm Z}$, $\mu$, and $\phi$ are given here. However, $\dot{\varTheta }$, $\dot{{\rm Z}}$, $\dot{\mu }$ and $\dot{\phi }$ must be evaluated to one order higher to remove their gyrophase dependence and avoid gyro-harmonic coupling, as first demonstrated by Lee et al. (Reference Lee, Myra and Catto1983). Once this is done $\dot{\Theta } \simeq {\langle \dot{\Theta }\rangle _\ast }$, $\dot{{\rm Z}} \simeq {\langle \dot{{\rm Z}}\rangle _\ast }$, $\dot{\mu } \simeq {\langle \dot{\mu }\rangle _\ast } = 0$, and $\dot{\phi } \simeq {\langle \dot{\phi }\rangle _\ast }$ to the required order, where the gyrophase independent correction to $\mu$ must also be evaluated to ensure ${\langle \dot{\mu }\rangle _ \ast } = 0$. Details of the procedure are given in Lee et al. (Reference Lee, Myra and Catto1983), Kagan & Catto (Reference Kagan and Catto2008) and Parra & Catto (Reference Parra and Catto2008).

Drift speeds are small compared with the thermal speed, and parallel scale lengths are assumed long compared to perpendicular ones. These gyroaverages are straightforward to evaluate as the scale length for unperturbed magnetic field variation is assumed to be much larger than a gyroradius. Using the gyroaverage ${\langle \boldsymbol{vv}\rangle _\varphi } = (v_ \bot ^2/2)(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} - \boldsymbol{nn}) + v_{||}^2\boldsymbol{nn}$, with $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I}$ the unit dyad, it follows that to the requisite order

(3.20)\begin{equation}\dot{\phi } \simeq {\langle \dot{\phi }\rangle _\ast } \simeq {\langle \dot{\varphi }\rangle _\ast } \simeq{-} {\langle \varOmega \rangle _\ast } - {v_{||}}n \cdot \nabla {e_1} \cdot {e_2} - ({{v_{||}}/2} )n \cdot \nabla \times n \simeq{-} {\langle \varOmega \rangle _\ast },\end{equation}

where

(3.21)\begin{align} {\langle \varOmega \rangle _\ast } &= \langle \varOmega ({\psi ,\vartheta } )\rangle \simeq {\langle {\varOmega _\ast }({{\psi_\ast },\Theta } )+ ({\psi - {\psi_\ast }} )\partial {\varOmega _\ast }/\partial {\psi _\ast } + ({\vartheta - \Theta } )\partial {\varOmega _\ast }/\partial \Theta \rangle _\ast }\notag\\ &\simeq {\varOmega _\ast } + I{B^{ - 1}}{v_{||}}\partial B/\partial \psi .\end{align}

Next, using $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }({\varOmega ^{ - 1}}\boldsymbol{n} \times \boldsymbol{\nabla }\vartheta )\boldsymbol{\cdot }\boldsymbol{n} = {\varOmega ^{ - 1}}\boldsymbol{n} \times (\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{n})\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta$ yields

(3.22)\begin{equation}\begin{array}{l} \dot{\Theta } \simeq {\langle \dot{\Theta }\rangle _\ast } \simeq {\langle {v_{||}}n \cdot \nabla \vartheta \rangle _\ast } + \left( {\dfrac{{v_ \bot^2}}{{2\varOmega }}nn \cdot \nabla \times n + \dfrac{c}{B}n \times \nabla \Phi } \right) \cdot \nabla \vartheta +\\ \quad \dfrac{1}{\varOmega }n \times ({v_{||}^2n \cdot \nabla n + \mu \nabla B} )\cdot \nabla \vartheta \simeq {\langle {v_{||}}n \cdot \nabla \vartheta \rangle _\ast } + {v_d} \cdot \nabla \vartheta , \end{array}\end{equation}

where $\boldsymbol{n} \times (\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{n}) + \boldsymbol{\nabla } \times \boldsymbol{n} = \boldsymbol{nn}\boldsymbol{\cdot }\boldsymbol{\nabla } \times \boldsymbol{n}$ is employed, the parallel velocity correction term $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla } \times \boldsymbol{n}$ is negligible, and to lowest order $\mu = v_ \bot ^2/2B$. Similarly,

(3.23)\begin{equation}\begin{array}{l} \dot{{\rm Z}} \simeq {\langle \dot{{\rm Z}}\rangle _\ast } \simeq {\langle {v_{||}}n \cdot \nabla \zeta \rangle _\ast } + \left( {\dfrac{{v_ \bot^2}}{{2\varOmega }}nn \cdot \nabla \times n + \dfrac{c}{B}n \times \nabla \Phi } \right) \cdot \nabla \zeta +\\ \quad {\varOmega ^{ - 1}}n \times ({v_{||}^2n \cdot \nabla n + \mu \nabla B} )\cdot \nabla \zeta \simeq {\langle {v_{||}}n \cdot \nabla \zeta \rangle _\ast } + {v_d} \cdot \nabla \zeta , \end{array}\end{equation}

Combining these two expressions and using $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\alpha = 0$ yields

(3.24)\begin{align} \mathop {{\alpha _\ast }}\limits^ \cdot{\simeq} {\langle \mathop {{\alpha _\ast }}\limits^ \cdot \rangle _\ast } &= {\langle \dot{{\rm Z}}\rangle _\ast } - {q_\ast }{\langle \dot{\Theta }\rangle _\ast } \simeq {v_d} \cdot ({\nabla \zeta - q\nabla \vartheta } )- ({{q_\ast } - q} ){v_{||}}n \cdot \nabla \vartheta \notag\\ &\simeq \dfrac{{I{v_{||}}}}{{\varOmega {R^2}}}\dfrac{\partial }{{\partial \psi }}({R{v_{||}}} )+ \dfrac{{Iv_{||}^2}}{{\varOmega Rq}}\dfrac{{\partial q}}{{\partial \psi }} = \dfrac{{I{v_{||}}}}{{\varOmega {R^2}q}}\dfrac{\partial }{{\partial \psi }}({qR{v_{||}}} ). \end{align}

In the last two forms of (3.24), ${q_ \ast }$ is expanded about q by keeping drift, but ignoring gyromotion departures from flux surfaces in the streaming term $({q_ \ast } - q){v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta$.

In these high frequency gyrokinetic variables the kinetic equation for g becomes

(3.25)\begin{align}& {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \dfrac{\partial }{{\partial \varTheta }}\left( {\dfrac{{\partial g}}{{\partial t}}} \right) + \dfrac{{{\partial ^2}g }}{{\partial {t^2}}} - {\langle \varOmega \rangle _ \ast }\dfrac{\partial }{{\partial \varphi }}\left( {\dfrac{{\partial g}}{{\partial t}}} \right) + {{\dot{\alpha }}_ \ast }\dfrac{\partial }{{\partial {\alpha _ \ast }}}\left( {\dfrac{{\partial g}}{{\partial t}}} \right) - C\left\{ {\dfrac{{\partial g}}{{\partial t}}} \right\}\notag\\ & \quad ={-} \dfrac{{Ze}}{M}\left\{ {\dfrac{{\partial \boldsymbol{e}}}{{\partial t}}\boldsymbol{\cdot }\boldsymbol{v}\dfrac{{\partial {f_0}}}{{ \partial E}} + \left[ {\dfrac{{\partial \boldsymbol{e}}}{{\partial t}} + ({v_{||}}\boldsymbol{n} + {\boldsymbol{v}_d})\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{e}} \right]\boldsymbol{\cdot }\dfrac{\boldsymbol{v}}{B}\dfrac{{\partial {f_0}}}{{\partial \mu }}} \right\}.\end{align}

The response g still contains some non-resonant effects. In this form only the lowest-order expressions $\mu = v_ \bot ^2/2B$, $\phi \simeq \varphi $,

(3.26)\begin{equation}\dot{\Theta } \simeq {v_{||}}n \cdot \nabla \vartheta ,\end{equation}

and (being careful to take $\psi$ the derivative at fixed E)

(3.27)\begin{equation}{\omega _\alpha } \equiv \mathop {{\alpha _\ast }}\limits^ \cdot{\simeq} \frac{{I{v_{||}}}}{{\varOmega {R^2}q}}\frac{\partial }{{\partial \psi }}({qR{v_{||}}} )={-} \frac{{cI}}{{RB}}\frac{{\partial \Phi }}{{\partial \psi }} - \frac{{I\mu }}{{\varOmega R}}\frac{{\partial B}}{{\partial \psi }} + \frac{{Iv_{||}^2}}{{\varOmega {R^2}q}}\frac{\partial }{{\partial \psi }}({qR} ),\end{equation}

are needed. The form for the drift frequency ${\omega _\alpha }$ is obtained by using the last form of (2.9) along with

(3.28)\begin{equation}\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{A} = \boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \left[ {\frac{\partial }{{\partial \psi }}\left( {\frac{{\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{\nabla }\psi }}{{\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta }}} \right) + \frac{\partial }{{\partial \vartheta }}\left( {\frac{{\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta }}{{\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta }}} \right) + \frac{\partial }{{\partial \zeta }}\left( {\frac{{\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{\nabla }\zeta }}{{\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta }}} \right)} \right],\end{equation}

and ignoring parallel velocity corrections.

4. Perturbed formulation with collisions

The solution procedure for the response begins as in Catto et al. (Reference Catto, Lee and Ram2017) by Fourier decomposing the electric field in time and toroidal and poloidal angles, as

(4.1)\begin{align} \boldsymbol{\varepsilon } & \equiv \dfrac{{Ze}}{M}\boldsymbol{e}(\boldsymbol{r},t) = \sum\limits_{\omega ,n,m,S} {{\boldsymbol{\varepsilon }_m}(\omega ,n,s)} {\textrm{exp}} \textrm{(} - \textrm{i}\omega t + \textrm{i}n\zeta - \textrm{i}m\vartheta + \textrm{i}S(\psi )\textrm{)}\notag\\ & = \sum\limits_{\omega ,n,m,S} {{\boldsymbol{\varepsilon }_m}} {\textrm{exp}} ( - \textrm{i}\omega t + \textrm{i}n({\alpha _ \ast } + {q_ \ast }\varTheta ) - \textrm{i}m\varTheta + \textrm{i}S({\psi _ \ast }) - \textrm{i}L + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi ), \end{align}

where all flux surface dependence of the perturbed electric field has been relegated to the exponential ${\textrm{exp}} (\textrm{i}S)$, with

(4.2)\begin{align}{S_\ast } &\equiv S({{\psi_\ast }} )\simeq S(\psi )+ ({{\psi_\ast } - \psi } )\partial S/\partial \psi + \cdots \notag\\ &= S(\psi )+ {\varOmega ^{ - 1}}v \times n \cdot \nabla S - I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi + \cdots . \end{align}

The poloidal and toroidal angle dependence of the electric field is periodic, and the radial mode structure is assumed to be Fourier transformable as denoted by the sum over S. The notation is meant to indicate that for a full wave treatment as formulated in § 8 it is convenient to replace the exponential radial eikonal form S(r) by a radial Fourier transform. The final form of the perturbed electric field is written in gyrokinetic variables by making use of the definitions

(4.3)\begin{gather}\boldsymbol{k} \equiv \boldsymbol{\nabla }S - m\boldsymbol{\nabla }\vartheta + n\boldsymbol{\nabla }\zeta ,\end{gather}

(4.4)\begin{gather}{\boldsymbol{k}_ \bot } \equiv {k_ \bot }({\boldsymbol{e}_1}\cos \varsigma + {\boldsymbol{e}_2}\sin \varsigma ),\end{gather}

and

(4.5)\begin{equation}L \equiv {\varOmega ^{ - 1}}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v} \times \boldsymbol{n} = ({k_ \bot }{v_ \bot }/\varOmega )\sin (\varphi - \varsigma ) \equiv \eta \sin (\varphi - \varsigma ).\end{equation}

The resonant response contained in g cannot be Fourier decomposed in poloidal angle because of the trapped particles (and the passing particles are not Fourier decomposed to avoid a hierarchy of coupled equations). However, g is easily Fourier decomposed in gyrophase. Recall that all the gyrophase dependence of the left side of the kinetic equation has been removed by using the gyrokinetic change of variables to second order where needed. As g must depend on time, the gyrokinetic ${\alpha _ \ast }$ and $\varTheta$ angle variables, and the canonical angular momentum ${\psi _ \ast }$, it has the convenient representation

(4.6)\begin{align}g({\psi _ \ast },{\alpha _ \ast },\varTheta ,E,\mu ,\sigma ,t) =\! \sum\limits_{\omega ,n,p,S} {{g_p}} (\varTheta ,E,\mu ,\sigma ,\omega ,n,S)\, \textrm{exp}\,( - \textrm{i}\omega t + \textrm{i}n{\alpha _ \ast } - \textrm{i}p\varphi + \textrm{i}S({\psi _ \ast })).\end{align}

The variables ${\psi _ \ast }$ and ${\alpha _ \ast }$ label drift orbits whereas $\psi$ and $\alpha$ label magnetic field lines. The response g is not Fourier transformable in poloidal angle because the periodic trapped particles are unable to access the equatorial plane on the inboard side. Moreover, as ${\alpha _ \ast }$ follows particles, it introduces a secular poloidal angle dependence for the passing particles as they trace out surfaces of constant ${\psi _ \ast }$ for fixed E and $\mu$.

Inserting the preceding expressions into the kinetic equation (3.25), multiplying by ${\textrm {exp}} (\textrm{i}p^{\prime}\varphi )$, and integrating over $\varphi$ from 0 to $2{\pi }$, leads to

(4.7)\begin{align}& {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \dfrac{{\partial {g_p}}}{{\partial \varTheta }} - \textrm{i}(\omega - p{\langle \varOmega \rangle _ \ast } - n{\omega _\alpha }){g_p} - C\{ {g_p}\} \notag\\ &\quad ={-} \sum\limits_m {\left\{ {\dfrac{{\partial {f_0}}}{{ \partial E}} + \left[ {1 - \dfrac{{({k_{||}}{v_{||}} + n{\omega_\alpha })}}{\omega }} \right] \dfrac{1}{B}\dfrac{{\partial {f_0}}}{{\partial \mu }}} \right\}}\notag\\ & \quad\quad \boldsymbol{\varepsilon} \boldsymbol{\cdot }{\langle \boldsymbol{v}{\textrm {exp}} (\textrm{i}p\varphi - \textrm{i}L)\rangle _ \ast }{\textrm {exp}} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi ). \end{align}

Making use of the Bessel generating function

(4.8)\begin{equation}{\textrm{e}^{ - \textrm{i}L}} = {\textrm {exp}} ( - \textrm{i}\eta \sin (\varphi - \varsigma )) = \sum\limits_p {\textrm {exp} ( - \textrm{i}p(\varphi - \varsigma ))} {J_p}(\eta ),\end{equation}

results in the lowest-order result

(4.9)\begin{equation}{\langle \boldsymbol{v}\,{\textrm{e}^{\textrm{i}p^{\prime}\varphi - \textrm{i}L}}\rangle _ \ast }\,{\textrm{e}^{ - \textrm{i}p\varsigma }} = \boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot } \equiv \boldsymbol{d},\end{equation}

with

(4.10)\begin{equation}{\boldsymbol{\varUpsilon }_p} \equiv \frac{p}{{{k_ \bot }\eta }}{J_p}(\eta ){\boldsymbol{k}_ \bot } + \frac{i}{{{k_ \bot }}}{J^{\prime}_p}(\eta )\boldsymbol{n} \times \boldsymbol{k} = \frac{1}{2}[{\boldsymbol{e}_ + }\,{\textrm{e}^{ - \textrm{i}\varsigma }}{J_{p - 1}}(\eta ) + {\boldsymbol{e}_ - }\,{\textrm{e}^{\textrm{i}\varsigma }}{J_{p + 1}}(\eta )],\end{equation}

${J^{\prime}_p}(\eta ) = \partial {J_p}(\eta )/ \partial \eta$, ${\boldsymbol{e}_ \pm } \equiv {\boldsymbol{e}_1} \pm \textrm{i}{\boldsymbol{e}_2}$, and ${\boldsymbol{k}_ \bot } = {k_ \bot }({\boldsymbol{e}_ + }\,{\textrm{e}^{ - \textrm{i}\varsigma }} + {\boldsymbol{e}_ - }\,{\textrm{e}^{\textrm{i}\varsigma }})/2$. The more compact form

(4.11)\begin{align}& {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \dfrac{{\partial {g_p}}}{{\partial \varTheta }} - \textrm{i}(\omega - p{\langle \varOmega \rangle _ \ast } - n{\omega _\alpha }){g_p} - C\{ {g_p}\}\notag\\ & \quad ={-} \sum\limits_m {\textrm {exp} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi ){W_{p,m}}} , \end{align}

is found by defining

(4.12)\begin{equation}{W_{p,m}} \equiv {\textrm{e}^{\textrm{i}p\varsigma }}{\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}{v_ \bot }]\left[ {\frac{{\partial {f_0}}}{{\partial E}} + \frac{{(\omega - {k_{||}}{v_{||}} - n{\omega_\alpha })}}{{\omega B}}\frac{{\partial {f_0}}}{{\partial \mu }}} \right].\end{equation}

Apart from small details to improve the notation and the retention of collisions, this result corresponds to (4.51) in Catto et al. (Reference Catto, Lee and Ram2017) with the important difference that by using ${\psi _ \ast }$and ${\alpha _ \ast }$ as the variables, additional finite poloidal gyroradius corrections and shear modifications are retained by the $\partial S/ \partial \psi$ and ${q_ \ast }$ terms, respectively. The only finite orbit effect retained previously was from expanding $\varOmega$ about ${\psi _ \ast }$ to obtain

(4.13)\begin{equation}{\langle \varOmega \rangle _\ast } \simeq {\varOmega _\ast } + I{B^{ - 1}}{v_{||}}\partial B/\partial \psi .\end{equation}

The difference between ${\langle \varOmega \rangle _ \ast }$ and ${\varOmega _ \ast }$ is odd in ${v_{||}}$ and thereby introduces an asymmetry in ${v_{||}}$ in the response ${g_p}$ that has been shown to allow symmetric spectrum current drive (Catto Reference Catto2018). Spatial variation of $\eta$ is ignored as weak. One further detail of the treatment here is that

(4.14)\begin{equation}\frac{{\partial e}}{{\partial t}} + ({{v_{||}}n + {v_d}} )\cdot \nabla e \simeq \frac{{\partial e}}{{\partial t}} + \dot{\Theta }\frac{{\partial e}}{{\partial \Theta }} + {\dot{\alpha }_\ast }\frac{{\partial e}}{{\partial {\alpha _\ast }}},\end{equation}

is used in (3.25) to obtain the $\partial {f_0}/\partial \mu$ coefficients in (4.7) and (4.12).

In the final form of the kinetic equation for ${g_p}$ only diffusive collisions enter and they can normally be assumed to be due to pitch angle scatter (see Appendix A and (3.2) of Catto (Reference Catto2020) for more details) in the form

(4.15)\begin{equation}C\{ {g_p}\} = \frac{{4\nu (v){v_{||}}}}{{{v^2}}}\frac{\partial }{{\partial \lambda }}\left( {\lambda {v_{||}}\frac{{\partial {g_p}}}{{\partial \lambda }}} \right) = \nu (v)\frac{B}{{{B_0}}}\frac{\partial }{{\partial {v_{||}}}}\left( {v_ \bot^2\frac{{\partial {g_{p,m}}}}{{\partial {v_{||}}}}} \right),\end{equation}

with $\lambda = 2\mu {B_0}/{v^2}$ and ${B_0} = {B_0}(\psi )$. The second form of the collision operator is obtained by changing the velocity variable to ${v_{||}} = \sigma v\sqrt {1 - \lambda B/{B_0}}$ and $2{v_{||}}\partial {v_{||}}/\partial \lambda ={-} B{v^2}/{B_0}$. For scattering in and out of the boundary layers momentum conservation in like particle collisions is unimportant. With this collision operator inserted (4.11) becomes

(4.16)\begin{align}& {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \dfrac{{\partial {g_p}}}{{\partial \varTheta }} - \textrm{i}(\omega - p{\langle \varOmega \rangle _ \ast } - n{\omega _\alpha }){g_p} - \nu (v)\dfrac{B}{{{B_0}}}\dfrac{\partial }{{\partial {v_{||}}}}\left( {v_ \bot^2\dfrac{{\partial {g_{p,m}}}}{{\partial {v_{||}}}}} \right)\notag\\ & \quad ={-} \sum\limits_M {\textrm {exp} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi ){W_{p,m}}} , \end{align}

which contains no information about the rf fields in the unperturbed trajectories used on the left side of the equation.

5. Helical magnetic field in a cylinder with pitch angle scattering

The procedure of Catto (Reference Catto2020) for a uniform field cylinder is next extended to illustrate how the collisional boundary layer alters QL theory for a helical field cylinder.

For a helical magnetic field in a cylinder of length $2{\pi }R$,

(5.1)\begin{equation}\boldsymbol{B} = {B_0}\boldsymbol{\nabla }z + \boldsymbol{\nabla }z \times \boldsymbol{\nabla }\psi = B\boldsymbol{n} = \boldsymbol{\nabla }(z - {B_0}r\vartheta /{B_p}) \times \boldsymbol{\nabla }\psi ,\end{equation}

with ${B_0}$ constant, $\psi = \psi (r)$, and $\boldsymbol{\nabla }\psi = {B_p}(r)\boldsymbol{\nabla }r$, there are no finite drift orbit effects. Making use of the cylindrical $\alpha$ variable, ${\alpha _c} \equiv z - {B_0}r\vartheta /{B_p}$, and using $\partial / \partial z = {R^{ - 1}} \partial /\partial \zeta$, gives

(5.2)\begin{align} n{\omega _\alpha } & = n{\boldsymbol{v}_d}\boldsymbol{\cdot }\boldsymbol{\nabla }{\alpha _c} = n{v_{||}}{\varOmega ^{ - 1}}\boldsymbol{\nabla }\boldsymbol{\cdot }[{v_{||}}{B^{ - 1}}({B_0}\boldsymbol{\nabla }z + r{B_p}\boldsymbol{\nabla }\vartheta ) \times \boldsymbol{\nabla }{\alpha _c})]\notag\\ & ={-} \dfrac{{nc}}{{R{B_p}}}\dfrac{{\partial \varPhi }}{{\partial r}} + \dfrac{n}{{\varOmega Rr}}\left( {v_{||}^2\dfrac{{{B_p}}}{B} - \mu r\dfrac{{\partial {B_p}}}{{\partial r}}} \right) \equiv n{\omega _E} + n{\omega _{||}} - n{\omega _ \bot }, \end{align}

where the electric field, curvature, and magnetic gradient drift frequencies are defined. To obtain the large aspect ratio tokamak form let $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta = {B_p}/r{B_z} \equiv 1/ qR$ with qR the connection length and R constant, then

(5.3)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \partial {g_p}/ \partial \varTheta = \textrm{i}{k_{||}}{v_{||}}{g_p},\end{equation}

with

(5.4)\begin{equation}{k_{||}} \equiv ({nq - m} )n \cdot \nabla \vartheta \simeq ({nq - m} )/qR.\end{equation}

To carefully estimate the dimensionless velocity space boundary layer width w and the effective collision time ${\nu _{\textrm{eff}}}$ (or the interaction time ${\tau _{{\mathop{\rm int}} }} \equiv \nu _{\textrm{eff}}^{ - 1}$), the width is defined as

(5.5)\begin{equation}{v_ \bot }w \equiv {v_{||}} - (\omega - p\varOmega )/{k_{||}},\end{equation}

where drifts are neglected. Then a careful estimate of the diffusion follows from

(5.6)\begin{equation}\nu \frac{\partial }{{\partial {v_{||}}}}\left( {v_ \bot^2\frac{{\partial {g_p}}}{{\partial {v_{||}}}}} \right)\sim \nu v_ \bot ^2\frac{{{\partial ^2}{g_p}}}{{\partial v_{||}^2}}\sim \frac{\nu }{{{w^2}}}{g_p}.\end{equation}

Balancing these terms in (4.16) gives a narrow collisional boundary layer width about the resonance with

(5.7)\begin{equation}w\sim {\left( {\frac{\nu }{{{k_{||}}{v_ \bot }}}} \right)^{1/3}}\sim {\left( {\frac{\nu }{{{k_{||}}{v_{th}}}}} \right)^{1/3}} \ll 1,\end{equation}

where ${v_{\textrm{th}}}$ is the species thermal speed, and ${v_ \bot } \sim {v_{\textrm{th}}}$. The width w is normally narrow as tokamak plasmas are in the long mean free path limit ${k_{||}}{v_{th}}/\nu \gg 1$ (except near the separatrix and in the SOL).

The diffusive form of the collision operator gives the effective collision frequency

(5.8)\begin{equation}{\nu _{\textrm{eff}}} \equiv \frac{1}{{{\tau _{{\mathop{\rm int}} }}}}\sim \frac{\nu }{{{w^2}}}\sim \nu {\left( {\frac{{{k_{||}}{v_{\textrm{th}}}}}{\nu }} \right)^{2/3}},\end{equation}

where consistent with definition (5.5) the effective interaction time is the inverse of ${\nu _{\textrm{eff}}}$. Then (5.8) implies that a narrow collisional boundary exists whenever

(5.9)\begin{equation}{k_{||}}{v_{th}}/\nu \gg 1.\end{equation}

To estimate how many poloidal circuits are made by a charge during an interaction time, $d\vartheta /d\tau \simeq {v_{||}}/qR$ is integrated to find the estimate

(5.10)\begin{equation}\Delta \vartheta \sim \frac{{{v_{||}}{\tau _{{\mathop{\rm int}} }}}}{{qR}}\sim \frac{{{{({v_{\textrm{th}}}/\nu qR)}^{1/3}}}}{{{{({k_{||}}qR)}^{2/3}}}}.\end{equation}

Consequently, charges make many bounces or poloidal circuits in an interaction time as long as the mean free path ${v_{\textrm{th}}}/\nu$ remains much larger than a connection length qR, such that $\Delta \vartheta \gg 1$. As a result, successive resonant interactions with the rf in the core are well correlated even for ${k_{||}}qR \gg 1$. For ions at 10 keV with $ n\sim {10^{14}}\;\textrm{c}{\textrm{m}^{ - 3}}$ and R ~ 10 m, $ {v_i}\sim {10^8}\;\textrm{cm}\;{\textrm{s}^{ - 1}}$ and $ {\nu _i}\sim 100\;{\textrm{s}^{ - 1}}$ giving $ {v_i}/{\nu _i}qR\sim {10^3}$ for q ~ 1. Therefore, charges are continuously interacting with the applied rf over a few to many closed poloidal transits.

The preceding estimate is very different than that of Stix (Reference Stix1975). He introduces a diffusive collisional ‘irreproducibility’ time $\Delta {\tau _p} = {\tau _p}\Delta {v_{||}}/{v_{||}}\sim {\tau _p}{(\nu qR/ {v_{\textrm{th}}})^{1/2}}$, with $\Delta {v_{||}}/{v_{||}}\sim {(\nu {\tau _p})^{1/2}}$ and ${\tau _p}\sim qR/{v_{\textrm{th}}}$. He then finds a decorrelation condition for successive passes by proposing ‘that ions forget the cyclotron phase between transits’ of the resonance if $\varOmega \Delta {\tau _p} \gg 1$. Then, the loss of relative phase between the wave and the gyrating ion occurs when ${(\nu qR/{v_{th}})^{1/2}} \gg {v_{th}}/q\varOmega R$ or ${\nu ^{1/2}}\varOmega \tau _p^{3/2} \gg 1$. However, this estimate ignores that it is actually the velocity space dependence of the resonance ${v_{||}} - (\omega - p\varOmega )/{k_{||}}$ that matters for collisions and in it $\varOmega$ is independent of gyrophase to lowest order. Therefore, $\varOmega \Delta {\tau _p} \gg 1$ should presumably be replaced by the more restrictive condition ${k_{||}}{v_{th}}\Delta {\tau _p} \gg 1$ or ${\nu ^{1/2}}{k_{||}}{v_{th}}\tau _p^{3/2} \gg 1$, giving $\Delta \vartheta \ll 1$ in (5.10) and requiring shorter mean free paths as one might expect. Catto & Myra (Reference Catto and Myra1992) and Belikov & Kolesnichenko (Reference Belikov and Kolesnichenko1994) slightly refined the Stix estimate for minority heating in a related way, but like Stix they did not consider the possibility of collisional boundary layers about each resonant curve in velocity space. Alternately, when gyrophase dependence enters the resonance because of the difference $\varOmega (R )- \varOmega (r )\simeq {\varOmega ^{ - 1}}v \times n \cdot \nabla \varOmega \sim {v_{th}}/R$, it leads to the replacement of $\varOmega \Delta {\tau _p} \gg 1$ with an even more restrictive condition ${R^{ - 1}}{v_{th}}\Delta {\tau _p} \gg 1$or ${\nu ^{1/2}}{R^{ - 1}}{v_{th}}\tau _p^{3/2} \gg 1$ from gyrophase scattering. The preceding boundary layer estimates of (5.5)–(5.10) do not support any of these prior collisional estimates.

The constant magnetic field solution (Catto Reference Catto2020) is easily extended to a helical cylindrical field if ${B_p}$ varies on the scale of the radius of the cylinder. As Fourier decomposition in $\varTheta$ is allowed

(5.11)\begin{equation}{g_p} = \sum\limits_m {{g_{p,m}}\ {\textrm {exp}}\ (\textrm{i}(nq - m)\varTheta )} ,\end{equation}

is employed to obtain the helical magnetic field form

(5.12)\begin{equation}\textrm{i}(\omega - p\varOmega - {k_{||}}{v_{||}} - n{\omega _E} - n{\omega _B}){g_{p,m}} + \nu (v)\frac{\partial }{{\partial {v_{||}}}}\left( {v_ \bot^2\frac{{\partial {g_{p,m}}}}{{\partial {v_{||}}}}} \right) = {W_{p,m}},\end{equation}

where for the moment curvature drift is neglected. Solving by letting

(5.13)\begin{equation}{s_h} = {({k_{||}}/v_ \bot ^2\nu )^{1/3}}[{v_{||}} - (\omega - p\varOmega - n{\omega _E} - n{\omega _ \bot })/{k_{||}}],\end{equation}

leads to

(5.14)\begin{equation}{\partial ^2}{g_{p,m}}/\partial s_h^2 - i{s_h}{g_{p,m}} = {(k_{||}^2v_ \bot ^2\nu )^{ - 1/3}}{W_{p,m}}.\end{equation}

The Su & Oberman (Reference Su and Oberman1968) solution vanishing at large $|{s_h}|$ is

(5.15)\begin{align} {g_{p,m}} & ={-} \dfrac{{{W_{p,m}}}}{{{{(k_{||}^2v_ \bot ^2\nu )}^{1/3}}}}\int_0^\infty {\textrm{d}z\ {\textrm {exp}}\,( - \textrm{i}{s_h} z - {z^3}/3)} \notag\\ & \quad \mathop \to \limits_{{s_h} \to \pm \infty } \dfrac{{{W_{p,m}}}}{{\textrm{i}(\omega - p\varOmega + n{\omega _E} - n{\omega _B} - {k_{||}}{v_{||}})}}.\end{align}

Using (5.5), ${s_h} = {({k_{||}}{v_ \bot }/\nu )^{1/3}}w = 1$, is consistent with the estimate (5.7) that a narrow collisional boundary layer requires ${(\nu /{k_{||}}{v_ \bot })^{1/3}} \ll 1$.

As ${k_{||}} \to 0$ the collisional boundary layer widens to cover all of velocity space as w ~ 1 when $ {k_{||}}\sim \nu /{v_ \bot }$, and the effective collision frequency approaches $\nu$ as these small $ {k_{||}}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel\lt \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }\nu /{v_ \bot }$ are in the short mean free path regime. To phenomenologically capture this feature, it is easiest to modify the definition of s to include bulk collisions by taking

(5.16)\begin{equation}{s_h} = {({k_{||}}/v_ \bot ^2\nu )^{1/3}}[{v_{||}} - (\omega - p\varOmega - n{\omega _E} - n{\omega _ \bot } + \textrm{i}\nu |{v_{||}}|v_ \bot ^{ - 1})/{k_{||}}].\end{equation}

This change corresponds to letting $\omega \to \omega + \textrm{i}\nu |{v_{||}}|/ {v_ \bot }$ in (5.12) and (5.15) and, of course, is not a rigorous procedure. However, with these changes, when $\omega - p\varOmega - n{\omega _E} - n{\omega _ \bot } \to 0$ and $ {k_{||}} \to \nu / {v_ \bot }$, then ${s_h} \to {v_{||}} /{v_ \bot } \sim 1$ and $w \to {v_{||}}/{v_ \bot }\sim 1$. Moreover, $|{s_h}|\to \infty$ when $\omega - p\varOmega - n{\omega _E} - n{\omega _ \bot } \to 0$ and ${k_{||}} \to 0$, giving ${g_{p,m}} \to - {v_ \bot }{W_{p,m}}/\nu |{v_{||}}|$; a well behaved result. Therefore, no singular behaviour arises when this simple change is made.

When $n{\omega _ \bot } \gg {k_{||}}{v_{||}} \gt \nu $, any collisional boundary layer is due to $n{\omega _ \bot }$. Then, it is the $\mu$ or, more conveniently, $\lambda = 2\mu {B_0}/{v^2}$ velocity space variation that becomes rapidly varying. In this limit (5.12) is replaced by an alternate Su & Oberman (Reference Su and Oberman1968) form

(5.17)\begin{equation}i(\omega - p\varOmega - n{\omega _E} - \lambda n{\omega _\nabla }){g_{p,m}} + 4\nu \frac{{{v_{||}}}}{{{v^2}}}\frac{\partial }{{\partial \lambda }}\left( {\lambda {v_{||}}\frac{{\partial {g_{p,m}}}}{{\partial \lambda }}} \right) = {W_{p,m}},\end{equation}

with

(5.18)\begin{equation}{\omega _\nabla } = ({v^2}/2{B_0}\varOmega R)\partial {B_p}/\partial r.\end{equation}

The drift ${\omega _{||}}$ can also be retained, but for the discussion here there is no need to do so.

Defining the pitch angle boundary layer width as

(5.19)\begin{equation}{w_\lambda } \equiv \lambda - (\omega - p\varOmega - n{\omega _E}) / n{\omega _\nabla },\end{equation}

using $\partial / \partial \lambda \sim 1/{w_\lambda }$, and balancing gives

(5.20)\begin{equation}{w_\lambda }n{\omega _\nabla }\sim \nu v_{||}^2/{v^2}w_\lambda ^2,\end{equation}

To recover a tokamak estimate of the magnetic drift, $B_0^{ - 1}\partial {B_p}/\partial r\sim {B_0}/R{B_p}$ is employed in ${\omega _\nabla }$ to obtain the boundary layer width estimate

(5.21)\begin{equation}{w_\lambda }\sim {(\nu v_{||}^2/{v^2}n{\omega _\nabla })^{1/3}}\sim {( R/n{\rho _p})^{1/3}}{(\nu qR/ {v_{\textrm{th}}})^{1/3}}\end{equation}

Making ${w_\lambda } \ll 1$, requires $n{\rho _p}/R \gg \nu qR/{v_{th}}$, which is consistent with $n{\omega _ \bot } \gg {k_{||}}{v_{||}} \gt \nu $ for the same drift estimate. However, retention of another boundary layer for the ions leads to more complex expressions. Consequently, only the collisional shift inserted in (5.16) is used.

Thus, in the cylindrical limit, where the resonant interaction is continuous as $B = B(\psi )$, the boundary layer analysis suggests that a reasonably precise solution is

(5.22)\begin{equation}g_p^{\textrm{cyl}} ={-} \sum\limits_m {\frac{{\textrm {exp}\,(\textrm{i}(qn - m)\varTheta ){W_{p,m}}}}{{{{(k_{||}^2v_ \bot ^2\nu )}^{1/3}}}}\int_0^\infty {\textrm{d}z\,{\textrm {exp}}\,( - \textrm{i}{s_h}z - {z^3}/3)} } ,\end{equation}

based on (5.15), but with the full drift from (5.2) being resonant, and (5.16) becoming

(5.23)\begin{equation}{s_h} = {({k_{||}}/v_ \bot ^2\nu )^{1/3}}[{v_{||}} - (\omega - p\varOmega - n{\omega _\alpha } + \textrm{i}\nu |{v_{||}}|v_ \bot ^{ - 1})/{k_{||}}].\end{equation}

The $\nu |{v_{||}}|v_ \bot ^{ - 1}$ shift is inserted to remove singular behaviour associated with ${k_{||}} \to 0$.

In standard QL formulations, collisions do not enter at all. Instead, a delta function appears whenever the replacement ${(\omega - p\varOmega - {k_{||}}{v_{||}})^{ - 1}} \to - \mathrm{i}\pi \delta (\omega - p\varOmega - {k_{||}}{v_{||}})$ is made. The preceding discussion suggests the improved replacements as resonant functions:

(5.24)\begin{equation}\frac{1}{{\textrm{i}(\omega - p\varOmega - {k_{||}}{v_{||}} - n{\omega _\alpha })}} \to \frac{{ - \displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - \textrm{i}{s_h}z - {z^3}/3}}} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||}^{2/3}}},\end{equation}

and

(5.25)\begin{equation}\delta (\omega - p\varOmega - {k_{||}}{v_{||}} - n{\omega _\alpha } ) \to \frac{{{\pi }\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - {z^3}/3}}\cos ({s_h}z)} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||}^{2/3}}}.\end{equation}

These replacements will be employed to conjecture as to the tokamak geometry form in the next section when it is no longer possible to rigorously treat a collisional boundary layer.

6. Solution procedure in a tokamak for correlated orbits and weak collisions

Collisions do not appear in standard QL treatments but they (or some other mechanism) implicitly enter to decorrelate successive passes through resonance and to resolve singularities to lowest order. Therefore, the replacement $C\{ {g_p}\} \to - {\nu _{\textrm{eff}}}{g_p}$ is normally adequate, and leads to the perturbed kinetic equation

(6.1)\begin{align}& {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \dfrac{{\partial {g_p}}}{{\partial \varTheta }} - \textrm{i}(\omega - p{\langle \varOmega \rangle _ \ast } - n{\omega _\alpha } + \textrm{i}{\nu _{\textrm{eff}}}){g_p}\notag\\ & \quad ={-} \sum\limits_m {\textrm {exp} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi ){W_{p,m}}} , \end{align}

that need only be solved in a lowest order plateau limit. Then the final results are independent of the details of collisional boundary layer physics required to resolve singular behaviour. The solution procedure for this correlated poloidal passes limit is presented next, but it should be kept in mind the insertion of ${\nu _{\textrm{eff}}}$ is only allowed to lowest order and when the details of collisional boundary layer physics do not matter.

The replacement of the collision operator by ${\nu _{\textrm{eff}}}$ allows geometrical complications to be retained in a streamlined manner because it ignores the poloidal variation of the collisional boundary layer. By integrating from $\tau ={-} \infty$ to $\tau = 0$, the perturbed kinetic equation is solved using the unperturbed trajectories of

(6.2)\begin{equation}d\Theta (\tau )/d\tau = {v_{||}}(\tau )n \cdot \nabla \vartheta \simeq {v_{||}}(\tau )/qR,\end{equation}

with $\varTheta (\tau = 0) = \varTheta$, and requiring that the result be independent of the initial distribution by taking ${g_p}(\tau \to - \infty ) = 0$. Note that for passing and trapped particles $\textrm{d}\varTheta (\tau )$ and ${v_{||}}(\tau )$ have the same sign, and for trapped particles $\textrm{d}\varTheta (\tau )$ and ${v_{||}}(\tau )$ change sign together at turning points. These trajectories are also used to evaluate (2.12). Integration then gives

(6.3)\begin{align} {g_p} & ={-} \sum\limits_m {\textrm {exp} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi )} \notag\\ & \quad \int_{ - \infty }^0 {\textrm{d}\tau } {W_{p,m}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} + {\nu_{\textrm{eff}}}\tau } \right), \end{align}

where

(6.4)\begin{equation}\varLambda (\tau ) \equiv \left\{ {\omega - p{{\langle \varOmega \rangle }_ \ast } - n{\omega_\alpha }(\tau ) - {v_{||}}(\tau )\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\left[ {({q_ \ast }n - m)\vartheta (\tau ) + \frac{{I{v_{||}}(\tau )}}{{\varOmega (\tau )}}\frac{{\partial S}}{{\partial \psi }}} \right]} \right\}.\end{equation}

For the correlated case considered here it is necessary to depart from the procedure of Catto et al. (Reference Catto, Lee and Ram2017) by removing another non-resonant contribution. To do so, integration by parts is required to obtain

(6.5)\begin{align}&\int_{ - \infty }^9 d\tau {B^{ - 1}}{\textrm {exp}} \left( { - i\int_0^\tau d\tau^{\prime}\Lambda^{\prime} + {\nu_{eff}}\tau } \right)({\omega - {k_{||}}{v_{||}} - n{\omega_\alpha }} ){e^{ip\varsigma }}{\boldsymbol{\varepsilon} _m} \cdot [{n{v_{||}}{J_p}(\eta )+ {\boldsymbol{\varUpsilon}_p}{v_ \bot }} ]\notag\\ &\quad \simeq i{B^{ - 1}}{e^{ip\varsigma }}{\boldsymbol{\varepsilon}_m} \cdot [{n{v_{||}}{J_p}(\eta )+ {\boldsymbol{\varUpsilon}_p}{v_ \bot }} ]\notag\\ &\quad \quad + \int_{ - \infty }^0 d\tau {B^{ - 1}}{\textrm {exp}} \left( { - i\int_0^\tau d\tau^{\prime}\Lambda^{\prime} + {\nu_{eff}}\tau + ip\varsigma } \right)\notag\\ &\quad \quad {\boldsymbol{\varepsilon}_m}\cdot [{n{v_{||}}{J_p}(\eta )+ {\boldsymbol{\varUpsilon}_p}{v_ \bot }} ]\left[{p{{\langle \varOmega (\tau )\rangle }_\ast } + \dfrac{{n({{q_\ast } - q} ){v_{||}}(\tau )}}{{qR}} + I\dfrac{{\partial S}}{{\partial \psi }}\dfrac{d}{{d\tau }}\left( {\dfrac{{{v_{||}}(\tau )}}{{\varOmega (\tau )}}} \right)} \right] + \cdots , \end{align}

where the slow variation of ${e^{\textrm{i}p\varsigma }}{\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}{v_ \bot }]$ is used to ignore small terms. The last two terms of (6.5) are small compared with $\varOmega$ by $(n{\rho _p}/qR)(\rho \partial q/ \partial r)$ and $(\rho /qR)(\varepsilon {k_\psi }{\rho _p})$, whereas the first term on the right is the non-resonant response that is normally extracted (and retained) by using the local resonance condition $(\omega - n{\omega _\alpha } - {k_{||}}{v_{||}} = p\varOmega )$ that only holds in a cylinder. Using the lowest order results from (6.5) leads to

(6.6)\begin{align}&{g_p} \simeq g_p^{nr} - \mathop \sum \limits_m {\textrm {exp}} (i({n{q_\ast } - m} )\Theta + iI{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi )\notag\\ &\quad \int_{ - \infty }^0 d\tau W_{p,m}^{res}(\tau ){\textrm {exp}} \left( { - i\int_0^\tau d\tau^{\prime}\Lambda ({\tau^{\prime}} )+ {\nu_{eff}}\tau } \right), \end{align}

where the resonant form is

(6.7)\begin{equation}W_{p,m}^{\textrm{res}} \equiv {\textrm{e}^{\textrm{i}p\varsigma }}{\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}{v_ \bot }]\left( {\frac{{\partial {f_0}}}{{ \partial E}} + \frac{{p{{\langle \varOmega \rangle }_ \ast }}}{{\omega B}}\frac{{\partial {f_0}}}{{\partial \mu }}} \right).\end{equation}

and the non-resonant contribution is

(6.8)\begin{align}g_p^{nr} \simeq \frac{{ - i}}{{\omega B}}\frac{{\partial {f_0}}}{{\partial \mu }}\mathop \sum \limits_m \textrm {exp} (i({n{q_\ast } - m} )\Theta + iI{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi + ip\varsigma ){\boldsymbol{\varepsilon}_m} \cdot [{\boldsymbol{n}{v_{||}}{J_p}(\eta )+ {\boldsymbol{\varUpsilon}_p}{v_ \bot }} ].\end{align}

Letting ${\tau _f}$ be the time to complete a full bounce for the trapped or a full poloidal circuit for the passing,

(6.9)\begin{align}{\tau _f} = \mathop {\oint }\limits_f d\tau = \mathop {\oint }\limits_f d\vartheta /{v_{||}}n \cdot \nabla \vartheta \simeq \left\{ {\begin{array}{*{20}{@{}cl}} {{\tau_p} = 4qR\sqrt {({1 - \varepsilon } ){k^2} + 2\varepsilon } K(k )/v\sqrt {2\varepsilon } }&{passing}\\ {{\tau_t} = 8qRK(\kappa )/v\sqrt {2\varepsilon } }&{trapped} \end{array},} \right.\end{align}

and using the near periodicity of the poloidal motion, yields

(6.10)\begin{align} &\int_{ - \infty }^0 \textrm{d} \tau W_{p,m}^{\textrm{res}} \textrm {exp} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}[\varLambda (\tau^{\prime}) + \textrm{i}{\nu_{\textrm{eff}}}]} } \right)\notag\\ &\quad = \left[ {\int_{ - {\tau_f}}^0 + \int_{ - 2{\tau_f}}^{ - {\tau_f}} + \cdots \int_{ - (j + 1){\tau_f}}^{ - j{\tau_f}} + \cdots } \right]\textrm{d}\tau W_{p,m}^{\textrm{res}} \textrm {exp} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime}) + {\nu_{\textrm{eff}}}\tau } } \right)\notag\\ &\quad = [{1 + {\textrm {exp}} \left({ - {\nu_{\textrm{eff}}}{\tau_f} + \textrm{i}\oint_f {\textrm{d}\tau \varLambda } } \right)+ \cdots + {\textrm {exp}} \left({ - j{\nu_{\textrm{eff}}}{\tau_f} + \textrm{i}j\oint_f {\textrm{d}\tau \varLambda } } \right)+ \cdots } ]\notag\\ &\quad \quad\times\int_{ - {\tau _f}}^0 {\textrm{d}\tau W_{p,m}^{\textrm{res}}} {\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime}) + {\nu_{\textrm{eff}}}\tau } } \right), \end{align}

where K is the elliptic integral of the first kind and ${\kappa ^2} = [1 - (1 - \varepsilon )\lambda ]/ 2\varepsilon \lambda = {k^{ - 2}}$. As a result,

(6.11)\begin{align} {g_p} - g_p^{nr} \equiv g_p^{\textrm{res}} &={-} \sum\limits_m {\dfrac{{{\textrm {exp}} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi )}}{{1 - {\textrm {exp}} \left({ - {\nu_{\textrm{eff}}}{\tau_f} + \textrm{i}\displaystyle\oint_f {\textrm{d}\tau \varLambda } } \right)}}} \notag\\ &\quad\times \int_{ - {\tau _f}}^0 {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime}) + {\nu_{\textrm{eff}}}\tau } } \right). \end{align}

Expanding about the zeros using $\oint_f {\textrm{d}\tau \varLambda } = 2{\pi }\ell$, with $\ell = 0, \pm 1, \pm 2, \ldots$ gives

(6.12)\begin{align}g_p^{\textrm{res}} &= \sum\limits_{m,\ell } {\frac{{ - \textrm{i}\ {\textrm {exp}} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi )}}{{\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu _{\textrm{eff}}}{\tau _f}} }}} \notag\\ &\quad \times \int_{ - {\tau _f}}^0 {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ) {\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right),\end{align}

with collisions used to resolve the singularity, ${\nu _{eff}} \ll \omega $ assumed in the integral over a single full transit, and $\ell$ the bounce or poloidal transit harmonic or more descriptively the index of the resonant path in velocity space. The full result for ${f_1}$ is the combined resonant and non-resonant response needed for a full wave code to evaluate the perturbed density and current when ${\nu _{eff}}{\tau _f} \ll 1$ holds.

The usual local resonance condition is now replaced by a transit averaged resonance condition because the successive resonant interactions (denoted by the bounce resonant path index $\ell \ne 0$) are highly correlated and as a particle’s velocity changes it may experience other resonances due to toroidal geometry. In toroidal geometry particles do not remain in resonance indefinitely (as they do in a cylinder or slab), but instead particles have resonant interactions by crossing the various resonant paths or curves in velocity space that are defined by the vanishing of the denominator in (6.12) for ${\nu _{\textrm{eff}}} \to 0$. All spatial localization is contained in the exponential phase factor of the final phase integral in (6.12), which depends on the strongly spatially varying fields via the integral of phase function $\varLambda$, and the more slowly varying coefficient $W_{p,m}^{\textrm{res}}$.

Recalling (5.24) and (5.25), the lowest order collisional resolution of the singularity results in the resonant delta function contribution needed for the QL operator and follows from

(6.13)\begin{equation}\frac{1}{{\textrm{i}\left({\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu_{\textrm{eff}}}{\tau_f}} } \right)}} \to - {\pi }\delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right).\end{equation}

This replacement leads to the desired correlated form needed for the QL operator,

(6.14)\begin{align} g_p^{\textrm{res}} & ={-} {\pi }\sum\limits_{m,\ell } {{\textrm {exp}} (\textrm{i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi )} \delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)\notag\\ & \quad\times \oint_{{\tau _f}} {\textrm{d}\tau W_{p,m}^{\textrm{res}}(\tau )} {\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

with a phase factor in the exponential and the integral of this phase factor over a full poloidal circuit as denoted by the loop integral in the argument of the delta function. The final result for ${g_p}$ contains the anticipated delta function, suggesting that the details of the collisional boundary layer are unimportant to lowest order. As already noted, the argument of the delta function is now a transit averaged resonance rather than the local resonance that is obtained by assuming uncorrelated kicks.

To simplify the analysis it is convenient to discard the less important finite orbit effects. Equation (4.13) suggests the distinction between $\varOmega = \varOmega (\psi ,\vartheta )$ and ${\varOmega _ \ast } = \varOmega ({\psi _ \ast },\varTheta )$ is unimportant as normally p ~ 1 and

(6.15)\begin{equation}I{B^{ - 1}}{\varOmega ^{ - 1}}{v_{||}}\partial B/\partial \psi \sim {\rho _p}/R \ll 1.\end{equation}

Consequently,

(6.16)\begin{equation}{\langle \varOmega \rangle _\ast } \simeq {\varOmega _\ast } \simeq \varOmega ,\end{equation}

is assumed. The $\partial S/ \partial \psi$ term does not enter $\oint_f {\textrm{d}\tau \varLambda }$. It only enters $\int_0^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})}$, but there it can be comparable to the finite orbit effect implicit in the difference

(6.17)\begin{equation}{q_ \ast } - q ={-} I{\varOmega ^{ - 1}}{v_{||}}\partial q/\partial \psi ,\end{equation}

as

(6.18)\begin{equation}\frac{{nI{\varOmega ^{ - 1}}{v_{||}}\partial q/ \partial \psi }}{{I( \partial S/\partial \psi ){\varOmega ^{ - 1}}{v_{||}}}}\sim \frac{m}{{r {k_\psi }}}\frac{{r\partial q}}{{q\partial r}}\sim 1,\end{equation}

for ${k_\psi }\sim m/r$.

Using the preceding assumptions

(6.19)\begin{equation}\oint_f {\textrm{d}\tau \varLambda } \simeq \oint_f {\textrm{d}\tau (\omega - p\varOmega - n{\omega _\alpha }) - 2{\pi }\sigma ({q_ \ast }n - m)} ,\end{equation}

where $\oint_f $ is over a complete banana orbit for the trapped and over a single poloidal circuit for the passing, with $\sigma = {v_{||}}/|{v_{||}}|$ for the passing and $\sigma = 0$ for the trapped in (6.19). In addition,

(6.20)\begin{align} \int_0^\tau d\tau^{\prime}\Lambda ({\tau^{\prime}})&\simeq \int_0^{\tau} d\tau^{\prime}({\omega - p\varOmega^{\prime} - n\omega_\alpha^{\prime}} )- ({n{q_\ast } - m} )[{\vartheta (\tau )- \vartheta } ]\notag\\ & \quad - I\dfrac{{\partial S}}{{\partial \psi }}\left\{ {\dfrac{{{v_{||}}[{\vartheta (\tau )} ]}}{{\varOmega [{\vartheta (\tau )} ]}} - \dfrac{{{v_{||}}(\vartheta )}}{{\varOmega (\vartheta )}}} \right\}, \end{align}

where no $\sigma$ is inserted as $\varTheta (\tau ) - \varTheta$ has the same sign as ${v_{||}}$. In these last two forms the gyroradius and drift distinctions between $\varTheta$ and $\vartheta$ no longer matter.

The drift term in the preceding expressions is

(6.21)\begin{equation}{\omega _\alpha } \simeq{-} \frac{{cI}}{{RB}}\frac{{\partial \Phi }}{{\partial \psi }} - \frac{{I\mu }}{{\varOmega R}}\frac{{\partial B}}{{\partial \psi }} + \frac{{Iv_{||}^2}}{{\varOmega {R^2}q}}\frac{\partial }{{\partial \psi }}({qR} )\equiv {\omega _E} + {\omega _B},\end{equation}

with

(6.22)\begin{equation}{\omega _E} \equiv{-} ({cI/RB} )\partial \Phi /\partial \psi \simeq{-} c\partial \Phi /\partial \psi \end{equation}

The curvature drift term in ${\omega _B}$ now contains the non-local nature of shear that particles experience as they drift off flux surfaces. The lowest order transit averages of the magnetic drifts at large aspect ratio with concentric circular flux surfaces yield

(6.23)\begin{align} {{\bar{\omega }}_B} &\equiv \tau _f^{ - 1}\oint_f d \tau \,{\omega _B} \notag\\ &\simeq \;\dfrac{{{v^2}}}{{2{\varOmega _p}{R^2}}}\left\{ {\begin{array}{*{20}{@{}c}} \begin{array}{l} [2E(k)\, - (2\, - {k^2})K(k)] / K(k)\\ +\,4{q^{ - 1}}r(\partial q\,/ \partial r)E(k) / [(1\, - \varepsilon ){k^2} + \,2\varepsilon ]K(k) \end{array}&{\textrm{passing}}\\ \begin{array}{l} [2E(\kappa ) - K(\kappa )]/K(\kappa ) + 4{q^{ - 1}}r(\partial q\,/ \partial r)[E(\kappa )\\ - (1 - {\kappa^2})K(\kappa )]/K(\kappa ) \end{array}&{\textrm{trapped}} \end{array}} \right.\end{align}

For the evaluations the curvature drift is small for the trapped and there is some cancelation between the $\boldsymbol{\nabla }B$ and curvature drifts for the passing. Shear is retained by the $\partial q/ \partial r$ terms and E is the complete elliptic integral of the second kind.

The helical cylinder results retain the details of the boundary layer. However, when ${k_{||}} = (qn - m)/qR \to 0$ in a tokamak, finite orbit effects make

(6.24)\begin{equation}{k_{||\ast }} \equiv \frac{{({{q_\ast }n - m} )}}{{qR}} \to - \frac{{nI{v_{||}}}}{{qR\varOmega }}\frac{{\partial q}}{{\partial \psi }} \simeq{-} \frac{{n{v_{||}}}}{{qR{\varOmega _p}}}\frac{{\partial q}}{{\partial r}}.\end{equation}

Comparing the preceding with ${k_{||}}\sim \nu / {v_{\textrm{th}}}$ gives this finite orbit magnetic shear term as being unimportant only if $n{\rho _p}\partial q/\partial r \lt \nu qR/ {v_{\textrm{th}}}$.

The helical cylinder replacements (5.24) and (5.25) suggest the following tokamak resonant function replacements may be useful:

(6.25)\begin{equation}\frac{{{\tau _f}}}{{\textrm{i}\left({\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)}} \to \frac{{ - \displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - \textrm{i}sz - {z^3}/3}}} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||\ast }^{2/3}}},\end{equation}

and

(6.26)\begin{equation}{\tau _f}\delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)\to \frac{{{\pi }\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - {z^3}/3}}\cos (sz)} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||\ast }^{2/3}}},\end{equation}

with $\oint_f {\textrm{d}\tau \varLambda } \to \oint_f {\textrm{d}\tau \varLambda + i\nu {\tau _f}}$ in s,

(6.27)\begin{equation}s = \frac{{\left(2{\pi }\ell - \displaystyle\oint_f \textrm{d}\tau \varLambda - \textrm{i}\nu {\tau _f}\right) }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||\ast }^{2/3}{\tau _f}}},\end{equation}

where no attempt is made to distinguish between ${v_ \bot }$ and ${v_{||}}$ in the $i\nu {\tau _f}$ term. With these substitutions (6.8) becomes

(6.28)\begin{align} g_p^{\textrm{res}} & ={-} \sum\limits_{m,\ell } {{\textrm {exp}}{(i}(n{q_ \ast } - m)\varTheta + \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi \textrm{)}} \left( {\dfrac{{\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - \textrm{i}sz - {z^3}/3}}} }}{{{\nu^{1/3}}v_ \bot^{2/3}k_{||\ast }^{2/3}{\tau_f}}}} \right)\notag\\ & \quad\times \int_{ - {\tau _f}}^0 {\textrm{d}\tau W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right)} . \end{align}

These last three expressions are, of necessity, speculative as there appears to be no analytically tractable way of solving the collisional boundary layer problem in tokamak geometry for the correlated limit for which the boundary layer width presumably depends on poloidal angle as well. The rapid variation of the phase in the exponential of the $\tau$ integral implies that only a few $\ell$ near $\ell = 0$ contribute for the passing, while there is additional rapid variation due to m – qn for the trapped (Tolman & Catto Reference Tolman and Catto2020). To better understand this behaviour note that the argument of the delta function is just the resonance condition for the passing, implying that $\varLambda \propto \ell$ for the larger $|\ell |$. Consequently, the phase integral in the exponential of the $\tau$ integral becomes more rapidly varying as $|\ell |$ increases thereby reducing their contributions to (6.28). The ${k_{||}} \to 0$ behaviour for the correlated limit considered in (6.28) is different than the standard uncorrelated case for which the resonance is viewed as spatially local so the singularity need only be resolved during each pass through the local resonance (Brambilla Reference Brambilla1994, Reference Brambilla1999; Berry et al. Reference Berry, Jaeger, Phillips, Lau, Bertelli and Green2016).

7. QL operator

To complete the QL treatment the QL diffusion coefficient and operator need to be evaluated for correlated successive poloidal transits. Catto et al. (Reference Catto, Lee and Ram2017) find that as usual the out of phase non-resonant terms do not contribute so the QL operator reduces to their (5.14), which in E, $\mu$ variables with the incorrect parentheses removed and small terms neglected, is

(7.1)\begin{align}Q\{ {f_0}\} ={-} {\langle {\nabla _v}\boldsymbol{\cdot }(\boldsymbol{a}g)\rangle _{cg,\varphi }} ={-} {v_{||}}\left[ {\frac{\partial }{{\partial E}}\left( {\frac{{{{\langle g\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{v}\rangle }_{cg,\varphi }}}}{{{v_{||}}}}} \right) + \frac{1}{B}\frac{\partial }{{\partial \mu }}\left( {\frac{{{{\langle g\boldsymbol{a}\boldsymbol{\cdot }({\boldsymbol{v}_ \bot } - {\boldsymbol{v}_d})\rangle }_{cg,\varphi }}}}{{{v_{||}}}}} \right)} \right],\end{align}

with

(7.2)\begin{equation}g = {g^{nr}} + {g^{\textrm{res}}} = \sum\limits_{\scriptstyle\omega ,n \atop \scriptstyle p,S } {(g_p^{nr} + g_p^{\textrm{res}}){\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}n{\alpha _ \ast } - \textrm{i}p\varphi + \textrm{i}S({\psi _ \ast }))} .\end{equation}

To get this form recall that for harmonic fields $\boldsymbol{\varepsilon }$ and $\boldsymbol{\chi }$

(7.3)\begin{equation}{\langle \boldsymbol{\varepsilon \chi }\rangle _{cg}} = \frac{1}{4}\sum\limits_{\omega ,m,n.\kappa } {({\boldsymbol{\varepsilon }_{\omega mn\kappa }}\boldsymbol{\chi }_{\omega mn\kappa }^ \ast{+} \boldsymbol{\varepsilon }_{\omega mn\kappa }^ \ast {\boldsymbol{\chi }_{\omega mn\kappa }})} .\end{equation}

Then, as ${f_1} - g$ from (3.4) has no gyrophase dependence and is out of phase with $\boldsymbol{\varepsilon }$, using (7.1) and (7.3) gives ${\langle ({\,f_1} - g)\boldsymbol{\varepsilon } \boldsymbol{\cdot }\boldsymbol{n}{v_{||}}\rangle _{cg}} = 0$, ${\langle ({\,f_1} - g)\boldsymbol{a}\boldsymbol{\cdot }{\boldsymbol{v}_d}\rangle _{cg}} = 0$, and ${\langle ({\,f_1} - g)\boldsymbol{a}\boldsymbol{\cdot }{\boldsymbol{v}_ \bot }\rangle _{cg}} = 0$. Similarly, recalling (4.9), there is no contribution from the out of phase non-resonant term $g_p^{nr} \propto {\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }\boldsymbol{d}/\textrm{i}\omega$, as ${\langle {g^{nr}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{v}\rangle _{cg,\varphi }} \propto \boldsymbol{d}\boldsymbol{\cdot }{\langle \textrm{i}\boldsymbol{\varepsilon \varepsilon }\rangle _{cg}}\boldsymbol{\cdot }\boldsymbol{d} = 0$, ${\langle {g^{nr}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{n}\rangle _{cg,\varphi }} = 0$ and ${\langle {g^{nr}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }{\boldsymbol{v}_d}\rangle _{cg,\varphi }} = 0$. Hence, (7.1) becomes

(7.4)\begin{align} Q\{ {f_0}\} & ={-} {\langle {\nabla _v} \cdot (\boldsymbol{a}{g^{\textrm{res}}})\rangle _{cg,\varphi }} ={-} {v_{||}}\left[ {\dfrac{\partial }{{\partial E}}\left( {\dfrac{{{{\langle {g^{\textrm{res}}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{v}\rangle }_{cg,\varphi }}}}{{{v_{||}}}}} \right)} \right.\notag\\ &\quad + \left. \dfrac{1}{B}\dfrac{\partial }{{\partial \mu }}\left( {\dfrac{{{{\langle {g^{\textrm{res}}}\boldsymbol{a}\boldsymbol{\cdot }({\boldsymbol{v}_ \bot } - {\boldsymbol{v}_d})\rangle }_{cg,\varphi }}}}{{{v_{||}}}}} \right) \right]. \end{align}

Then using the transit average over the drifting trajectory as defined by (2.12) yields the transit averaged Q operator

(7.5)\begin{align}\overline {Q\{ {f_0}\} } ={-} \tau _f^{ - 1}\left[ {\frac{\partial }{{\partial E}}\left( {\oint_{{\tau_f}} {\textrm{d}\tau } {{\langle {g^{\textrm{res}}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{v}\rangle }_{cg,\varphi }}} \right) + \frac{\partial }{{\partial \mu }}\left( {\oint_{{\tau_f}} {\textrm{d}\tau } {B^{ - 1}}{{\langle {g^{\textrm{res}}}\boldsymbol{a}\boldsymbol{\cdot }({\boldsymbol{v}_ \bot } - {\boldsymbol{v}_d})\rangle }_{cg,\varphi }}} \right)} \right].\end{align}

Defining ${\tau _0}$ as the trajectory time at the equatorial plane crossing at $\vartheta = 0$, then

(7.6)\begin{equation}\int_0^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} = \int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} - \int_{{\tau _0}}^0 {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} ,\end{equation}

(4.9), (6.14), (7.2) and the complex conjugate of (4.9) are used to first write

(7.7)\begin{align} {\langle {g^{\textrm{res}}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{v}\rangle _{cg,\varphi }} & ={-} {\pi }\sum\limits_{\scriptstyle\omega ,n,m \atop \scriptstyle S,p,\ell } {\delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)} {\left\langle {{\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }{\boldsymbol{d}^ \ast }\,{\textrm{e}^{ - \textrm{i}p\varsigma }}\oint_{{\tau_f}} {\textrm{d}\tau W_{p,m}^{\textrm{res}}(\tau )} } \right\rangle _{cg}}\notag\\ &\quad\times {\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

with $W_{p,m}^{\textrm{res}}$ given by (6.7), and then obtain

(7.8)\begin{align} \oint_{{\tau _f}} {\textrm{d}\tau } {\langle {g^{\textrm{res}}}\boldsymbol{\varepsilon }\boldsymbol{\cdot }\boldsymbol{v}\rangle _{cg,\varphi }} &={-} \dfrac{{\pi }}{2}\sum\limits_{\scriptstyle\omega ,n,m \atop \scriptstyle S,p,\ell } {\delta \left({\oint_f {\textrm{d}\tau \varLambda } - 2{\pi }\ell } \right)} \notag\\ & \quad {\left|{\oint_{{\tau_f}} {\textrm{d}\tau } {\textrm {exp}} \left( { - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right){\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }\boldsymbol{d}} \right|^2}\left( {\dfrac{{\partial {f_0}}}{{\partial E}} + \dfrac{{p\varOmega }}{{\omega B}}\dfrac{{\partial {f_0}}}{{\partial \mu }}} \right). \end{align}

The equatorial plane is chosen to be where $\partial ({v_{||}}/B)/ \partial \vartheta = 0$. Following the procedure in Catto et al. (Reference Catto, Lee and Ram2017) from their (5.18)–(5.20) without assuming the local resonance condition is satisfied leads to

(7.9)\begin{align} &\langle g^{\textrm{res}}\boldsymbol{a}\boldsymbol{\cdot} ({\boldsymbol{v}_ \bot } - {\boldsymbol{v}_d})\rangle_{cg,\varphi}\notag\\ &\quad ={-} \pi\sum_{\scriptstyle\omega ,n,m \atop \scriptstyle S,p,\ell } {\delta \left(\displaystyle{\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)} \left\langle \left\{ {\dfrac{{p\varOmega }}{\omega }{\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + \boldsymbol{\varUpsilon }_p^ \ast {v_ \bot }]}\right.\right.\notag\\ &\qquad +\left. \dfrac{{\omega - p\varOmega - {k_{||}}{v_{||}} - \boldsymbol{k}\boldsymbol{\cdot }{\boldsymbol{v}_d}}}{\omega }({\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }\boldsymbol{\varUpsilon }_p^ \ast {v_ \bot })\right\}\notag\\ &\qquad \times \left.{\textrm {exp}} \left( { - \textrm{i}p\varsigma + \textrm{i}\int_{{\tau_0}}^0 {\textrm{d}\tau \varLambda (\tau )} } \right)\oint_{{\tau_f}} {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ) \right\rangle _{cg} {\textrm {exp}}\left( { - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right). \end{align}

The last term in the preceding equation does not appear in (5.20) of Catto et al. (Reference Catto, Lee and Ram2017) as it vanishes when the local resonance condition is satisfied. Transit averaging will eliminate it upon integrating by parts in $\vartheta$ to introduce an out-of-phase term that makes no contribution to the real part, and a small term proportional to $\partial (\boldsymbol{\varUpsilon }_p^ \ast {v_ \bot })/\partial \vartheta$ that can be neglected because of scale separation. As a result,

(7.10)\begin{align}& \oint_{{\tau _f}} {\textrm{d}\tau {B^{ - 1}}} {\langle {g^{\textrm{res}}}\boldsymbol{a}\boldsymbol{\cdot }({\boldsymbol{v}_ \bot } - {\boldsymbol{v}_d})\rangle _{cg,\varphi }}\notag\\ &\quad ={-} \dfrac{{\pi }}{2}\sum\limits_{\scriptstyle\omega ,n,m \atop \scriptstyle S,p,\ell } {\delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)} \dfrac{{p\varOmega }}{{\omega B}}{\left|{\oint_{{\tau_f}} {\textrm{d}\tau } \,{\textrm{e}^{ - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} }}{\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }\boldsymbol{d}} \right|^2}\left( {\dfrac{{\partial {f_0}}}{{ \partial E}} + \dfrac{{p\varOmega }}{{\omega B}}\dfrac{{\partial {f_0}}}{{\partial \mu }}} \right). \end{align}

Therefore, the correlated transit averaged QL diffusivity is found to be

(7.11)\begin{align}D \equiv \frac{{{\pi }{Z^2}{e^2}}}{{2{M^2}{v^2}{\tau _f}}}\sum\limits_\ell {\delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)} {\left|{\oint_f {\textrm{d}\tau } {\boldsymbol{e}_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }]\,{\textrm{e}^{ - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} }}} \right|^2},\end{align}

the transit average QL operator for correlated passes is

(7.12)\begin{equation}\overline {Q\{ {f_0}\} } = \sum\limits_{\scriptstyle\omega ,p \atop \scriptstyle m,n,S } {\frac{1}{{{\tau _f}}}} \left( {\frac{\partial }{{\partial E}} + \frac{{p\varOmega }}{{\omega B}}\frac{\partial }{{\partial \mu }}} \right)\left[ {{\tau_f}{v^2}D\left( {\frac{{\partial {f_0}}}{{\partial E}} + \frac{{p\varOmega }}{{\omega B}}\frac{{\partial {f_0}}}{{\partial \mu }}} \right)} \right],\end{equation}

and for fixed ${\psi _ \ast }$

(7.13)\begin{equation}\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} = \int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}} (\omega - p\varOmega ^{\prime} - n{\omega ^{\prime}_\alpha }) - \sigma ({q_ \ast }n - m)\vartheta - \frac{{I{v_{||}}}}{\varOmega }\frac{{\partial S}}{{\partial \psi }},\end{equation}

where $\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\nabla }({v_{||}}/B){|_{{\tau _0}}} = 0$ and $\sigma = {v_{||}}/|{v_{||}}|$ in (7.13). Equation (6.19) gives $\oint_f {\textrm{d}\tau \varLambda }$ in D. Importantly, entropy production is negative definite for the correlated QL diffusivity as

(7.14)\begin{equation}\left\langle {\int {{\textrm{d}^3}v} \ell n{f_0}\overline {Q\{ {f_0}\} } } \right\rangle ={-} \left\langle {\sum\limits_{\scriptstyle\omega ,p \atop \scriptstyle m,n,s } {\int {{\textrm{d}^3}v{v^2}} } \frac{D}{{{f_0}}}{{\left|{\frac{{\partial {f_0}}}{{\partial E}} + \frac{{p\varOmega }}{{\omega B}}\frac{{\partial {f_0}}}{{\partial \mu }}} \right|}^2}} \right\rangle \le 0,\end{equation}

where $\langle \cdots \rangle \equiv [\oint {\textrm{d}\vartheta ( \ldots )/\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } ]/[\oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } ]$ denotes the flux surface average.

The correlated form of the diffusivity given by (7.11) is similar in form to (12) in Belikov & Kolesnichenko (Reference Belikov and Kolesnichenko1994) with some differences in addition to the treatment of finite orbit effects. They do not consider the ${v_{||}}$ term in $\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }$ so can pull ${\boldsymbol{\varUpsilon }_p}{v_ \bot }$ through the $\tau$ integral as it no longer depends sensitively on the parallel velocity. The earlier paper (Belikov & Kolesnichenko Reference Belikov and Kolesnichenko1982) does not make the same simplification, but no explicit expression for the diffusivity is ever given, and it uses poloidal rather than toroidal component of angular momentum as they note in their later paper (Belikov & Kolesnichenko Reference Belikov and Kolesnichenko1994). In fact, both papers keep radial derivatives in the QL operator by allowing the unperturbed distribution function to depend on the canonical angular momentum as well as minor radius. However, both papers ultimately fail to keep the necessary sharp distinction between radial variation depending only on the poloidal flux function variation and that due to variation of the canonical angular momentum. Neither paper considers entropy production, but certainly the latter one could easily have done so.

The preceding expressions (7.11)–(7.14) for the correlated diffusivity D, the QL operator, and entropy production are key results. Even though they look very similar to the standard results of Kennel & Engelmann (Reference Kennel and Engelmann1966) that were generalized to a tokamak by Catto et al. (Reference Catto, Lee and Ram2017), namely

(7.15)\begin{equation}{D_{KE}} = \frac{{{\pi }{Z^2}{e^2}}}{{2{M^2}{v^2}}}\delta (\omega - p\varOmega - {k_{||}}{v_{||}} - n{\omega _\alpha })|{\boldsymbol{e}_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }]{|^2},\end{equation}

and

(7.16)\begin{equation}Q\{ {f_0}\} = \sum\limits_{\scriptstyle\omega ,p \atop \scriptstyle m,n,S } {{v_{||}}\left( {\frac{1}{v}\frac{\partial }{{\partial v}} + \frac{{p\varOmega }}{{\omega B}}\frac{\partial }{{\partial \mu }}} \right)} \left[ {{D_{KE}}\frac{{{v^2}}}{{{v_{||}}}}\left( {\frac{1}{v}\frac{{\partial {f_0}}}{{\partial v}} + \frac{{p\varOmega }}{{\omega B}}\frac{{\partial {f_0}}}{{\partial \mu }}} \right)} \right],\end{equation}

they have a very different interpretation.

The most obvious difference between (7.11) and (7.15) is the argument of the delta function. This change will allow the leading square root of the inverse aspect ratio modifications to be rigorously evaluated for the first time. However, when the new QL operator is combined with an improved full wave treatment as outlined in § 8, the spatial details of the perturbed density and current density are bound to be modified in ways that cannot be predicted by simply examining the new QL operator. The modified spatial behaviour of the applied rf fields may be quite different and will alter the detailed spatial behaviour of the diffusivity.

Moreover, in the correlated form (7.11) some spatial localization is provided by the phase factor in ${\textrm {exp}} [ - \textrm{i}\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} ]$ and the ${v_{||}}$ in $\boldsymbol{d} = \boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }$. The argument of the delta function is localized in velocity space as it contains a trajectory average over a complete poloidal circuit of an unperturbed trajectory rather than a poloidal angle integral over the local spatial resonance condition $({\omega \simeq p\varOmega + {k_{||}}{v_{||}}} )$. In addition, the resonant path index $\ell$ in (7.11) is summed over to account for particles passing though multiple correlated resonances as their velocity changes. It enters (7.11) in the resonance condition argument of the delta function, but not in the exponential phase factor in the trajectory integral product $|\oint_f {\textrm{d}\tau ( \ldots ){|^2}}$. Therefore, for the larger $|\ell |$ it must be that $|\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} |\sim 2{\pi }|\ell |$ for the passing. Then the Reimann–Lebesgue lemma means that the larger $|\ell |$ can only make a small contribution in (7.11) as the delta function is multiplied by $|\oint_f {\textrm{d}\tau ( \ldots )} {|^2} \propto {\ell ^{ - 2}}$. Consequently, the most important contributions to the passing integrals $\oint_f {\textrm{d}\tau } ( \ldots )$ are expected to be from $\ell = 0$ and $|\ell |\sim 1$, that is, from the lower $\ell$ resonant paths for which $\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})}$ is more slowly varying. For the trapped $|\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda (\tau ^{\prime})} |\propto |\ell |+ |m - qn|$ might be expected, suggesting low $|\ell |$ and $|m - qn |$ contributions will be larger. As a result, the correlated diffusivity leads to a very different interpretation of how QL theory works in a tokamak. In the correlated picture the particles can experience multiple resonances throughout their poloidal motion as their velocity changes, but only the lowest resonant path indices $\ell$ contribute significantly due to the lack of $\ell$ dependence in the exponential phase factors. In the stationary phase picture the passes through resonance are spatially localized and assumed, without justification, to be decorrelated from all other passes through resonance.

The stationary phase (or uncorrelated) form of the diffusivity becomes valid when $\nu \gt {k_{||}}{v_{\textrm{th}}}$ and it is obtained as indicated in the appendix and as in Bernstein & Baxter (Reference Bernstein and Baxter1981), Catto & Myra (Reference Catto and Myra1992) and Belikov & Kolesnichenko (Reference Belikov and Kolesnichenko1994). In the absence of finite drift departures off of flux surfaces it is of the form

(7.17)\begin{equation}{\overline D _{\textrm{sp}}} = \frac{{2{\pi }{Z^2}{e^2}}}{{{M^2}{v^2}| p\dot{\varOmega } + {k_{||}}{{\dot{v}}_{||}}{|_{\textrm{res}}}{\tau _f}}}|{\boldsymbol{e}_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }]|_{\textrm{res}}^2,\end{equation}

where the subscript ‘res’ denotes evaluation at $\omega = p\varOmega + \sigma {k_{||}}{v_{||}}$, with $\mathop {( \ldots )}\limits^\cdot{\equiv} {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }( \ldots )$ and $\sigma = 0$ for the trapped, and the $\vartheta$ dependence of $| p\dot{\varOmega } + {k_{||}}{\dot{v}_{||}}{|_{\textrm{res}}}$ must be negligible. It is only valid when collisions are strong enough to satisfy (A 4) by making ${({k_{||}}qR)^{2/3}}{(\nu qR/{v_{th}})^{1/3}} \gg 1$. This condition is unlikely to be satisfied in the core, and if satisfied in the pedestal would likely require the retention of finite orbit effects and the separatrix.

The correlated QL diffusivity of (7.11) contains ${q_ \ast }$ as part of its diffusive step as indicated in (7.13). The difference ${q_ \ast } - q$ does not vanish for a sheared field as drift departures from a flux surface allow ions to experience the magnetic field shear. For trapped particles ${q_ \ast } = q$ at the banana tip where ${\psi _ \ast } = \psi$. The passing and other trapped particle contributions to this tip flux surface must satisfy $\psi \simeq {\psi _\ast } + I{v_{||}}/\varOmega $ giving $q \simeq {q_\ast } + I{\varOmega ^{ - 1}}{v_{||}}\partial q/\partial \psi $, where both ${\psi _ \ast }$ and ${q_ \ast }$ are drift kinetic constants of the motion (constants of the motion in the absence of gyration). The ${q_ \ast }$ parameter automatically keeps track of the contributions from nearby flux surfaces to the QL diffusion. Clearly, this non-local behaviour can only be avoided by assuming weak shear.

No singular behaviour occurs in the expression for the correlated diffusivity as the argument of the delta vanishes only along well defined curve in velocity space even when ${k_{||}} = 0$ as long as there is velocity space dependence in $\varLambda$. However, the delta function in the correlated diffusivity of (7.11) might on occasion be awkward to deal with. In such situations collisional boundary layer effects can be phenomenologically retained by using (6.26) and (6.27) to obtain the alternate correlated boundary layer form

(7.18)\begin{align}D &= \dfrac{{{Z^2}{e^2}}}{{2{M^2}{v^2}\tau _f^2}}\sum\limits_\ell {\dfrac{{\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - {z^3}/3}}\cos (sz)} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||\ast }^{2/3}}}} \notag\\ &\quad\times {\left|{\oint_f {\textrm{d}\tau \,} {\boldsymbol{e}_m} \boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }]{\textrm {exp}} \left( { - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right)} \right|^2}, \end{align}

which also leads to a negative definite entropy production due to the integration over $\boldsymbol{v}$. Equation (7.18) is phenomenological as the actual collisional boundary layer has $\vartheta$ dependence that is ignored as unimportant. In the helical cylinder limit there is no azimuthal variation. Then the correlated diffusivities (7.18) and (7.11) reduce to

(7.19)\begin{align}{D^{\textrm{cyl}}} = \frac{{{Z^2}{e^2}}}{{2{M^2}{v^2}}}|{\boldsymbol{e}_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }]{|^2}\left\{ {\begin{array}{*{20}{@{}l}} {{\nu^{ - 1/3}}v_ \bot^{ - 2/3}k_{||}^{ - 2/3}\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - {z^3}/3}}\cos (sz)} }\\ {{\pi }\delta (\omega - p\varOmega - {k_{||}}{v_{||}} - n{\omega_\alpha })} \end{array}} \right..\end{align}

It is informative to plot the resonance paths associated with (7.11) for a symmetric spectrum. To simplify the plots the drift term is ignored, ${q_ \ast }$ is set to q, and $\varOmega = {\varOmega _0}(1 + \varepsilon \cos \vartheta )$ is used with $\omega = {\varOmega _0}$. Making use of

(7.20)\begin{align}\overline {\cos \vartheta } \equiv \frac{{\mathop {\displaystyle\oint }\nolimits_{{\tau _f}} d\tau \cos \vartheta }}{{\mathop {\displaystyle\oint }\nolimits_{{\tau _f}} d\tau }} \simeq \left\{ {\begin{array}{*{20}{@{}c}} {1 - \dfrac{2}{{{k^2}}} + \dfrac{{2E(k )}}{{{k^2}K(k )}}}&{passing}\\ {\dfrac{{2E(\kappa )}}{{K(\kappa )}} - 1}&{trapped,} \end{array}} \right.\end{align}

the resonant paths of (6.19) lead to

(7.21)\begin{equation}\frac{v}{{{v_{\textrm{th}}}}} = \frac{{\sqrt {2\varepsilon } qR{\varOmega _0}}}{{{\pi }{v_{\textrm{th}}}}}\left\{ {\begin{array}{*{20}{@{}l}} {\dfrac{{(2 - {k^2})K(k) - 2E(k)}}{{|m - qn \pm \ell |k}}}&{\textrm{passing}}\\ {\dfrac{2}{{|\ell |}}|2E(\kappa ) - K(\kappa )|}&{\textrm{trapped}} \end{array}} \right..\end{equation}

Only the lowest $\ell$ are of interest and normally $|{m - qn} |\gg |\ell |\sim 1$. As a result, plotting the lowest three indices $\ell$ for the passing leads to figure 1 for a symmetric spectrum, where the barely passing must have very high speed to resonate. For the same parameters the trapped with $|\ell |= 1$ are unable to resonate except at uninterestingly high speeds unless they have pitch angles very close to where 2E = K or $\kappa \simeq 0.91$ as shown in figure 2. There is no $\ell = 0$ trapped resonance unless 2E = K. In the usual $\omega = {\varOmega _0}$ case the trapped receive an rf kick as long as they make it to $\vartheta ={\pm} {\pi }/2$, but for correlated kicks the trapped only get an rf kick if $\overline {\cos \vartheta } = 0$ when $\omega = {\varOmega _0}$.

Figure 1.

Passing resonance paths in velocity space from (7.21) for $\omega = {\varOmega _0}$ and $\ell = 0, \pm 1$. The parameters of $|m - qn|= 11/2$, $qR{\varOmega _0}/{v_{\textrm{th}}} = 30$, and $\varepsilon = 1/10$ are chosen to schematically display the separation between resonance paths.

Figure 2.

Trapped resonance paths in velocity space from (7.21) for $\omega = {\varOmega _0}$ and $\ell ={\pm} 1$. The $\ell = 0$ resonance is only possible for 2E = K or $\kappa \simeq 0.91$. The $\ell ={-} 1(\ell = 1)$ resonance occurs for $\kappa \lt 0.91(\kappa \gt 0.91)$. The parameters are again chosen to be $|m - qn|= 11/2$, $qR{\varOmega _0}/{v_{\textrm{th}}} = 30$, and $\varepsilon = 1/10$.

A full wave QL treatment also requires solving for the perturbed density and current density. For completeness these expressions are presented in the next section.

8. Perturbed densities and currents

Full wave descriptions require the perturbed particle density and current density employed in the Maxwell equations be consistent with the QL operator. Consequently, these moments must be written in $\psi ,\vartheta ,\zeta ,v,\mu ,\varphi$ variables using the correlated forms of (6.8) and (6.12).

To begin, inserting (3.4) gives

(8.1)\begin{gather}\frac{\partial }{{\partial t}}\left( {\int {{\textrm{d}^3}v{f_1}} } \right) = \frac{{Ze}}{{MB}}\boldsymbol{e}\boldsymbol{\cdot }\int {{\textrm{d}^3}v} \left[ {({v_{||}}\boldsymbol{n} + {\boldsymbol{v}_d})\frac{{\partial {f_0}}}{{\partial \mu }}} \right] + \frac{\partial }{{\partial t}}\left( {\int {{\textrm{d}^3}vg} } \right),\end{gather}

(8.2)\begin{gather}\frac{\partial }{{\partial t}}\left( {\int {{\textrm{d}^3}v{v_{||}}{f_1}} } \right) = \frac{{Ze}}{{MB}}\boldsymbol{e}\boldsymbol{\cdot }\int {{\textrm{d}^3}v} \left[ {({v_{||}}\boldsymbol{n} + {\boldsymbol{v}_d}){v_{||}}\frac{{\partial {f_0}}}{{\partial \mu }}} \right] + \frac{\partial }{{\partial t}}\left( {\int {{\textrm{d}^3}v{v_{||}}g} } \right),\end{gather}

and

(8.3)\begin{equation}\frac{\partial }{{\partial t}}\left( {\int {{\textrm{d}^3}v{\boldsymbol{v}_ \bot }{f_1}} } \right) = \frac{\partial }{{\partial t}}\left( {\int {{\textrm{d}^3}v{\boldsymbol{v}_ \bot }g} } \right).\end{equation}

To write g in $\psi ,\vartheta ,\zeta ,v,\mu ,\varphi$ variables, Taylor expand in (4.6) or (7.2) to obtain

(8.4)\begin{align} n{\alpha _\ast } + S({{\psi_\ast }} )&= ({m - {q_\ast }n} )\Theta + n{\rm Z} - m\Theta + S({{\psi_\ast }} )\notag\\ &\simeq ({m - {q_\ast }n} )\Theta + [{n\zeta - m\vartheta + S(\psi )} ]+ L - I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi , \end{align}

and then write (7.2) as

(8.5)\begin{align} g & = {g^{nr}} + {g^{\textrm{res}}} = \sum\limits_{\scriptstyle\omega ,n \atop \scriptstyle p,S } {{g_p}(\varTheta ,v,\mu ,\sigma )} {\textrm {exp}} ( - \textrm{i}\omega t - \textrm{i}p\varphi + \textrm{i}[n\zeta - m\vartheta + S(\psi )]\notag\\ &\quad + \textrm{i}(m - {q_ \ast }n)\varTheta + \textrm{i}L - \textrm{i}I{\varOmega ^{ - 1}}{v_{||}}\partial S/\partial \psi ). \end{align}

Using the preceding, the correlated $({{\nu_{eff}}{\tau_f} \ll 1} )$ collisional boundary layer responses (6.8) and (6.12) in $\psi ,\vartheta ,\zeta ,v,\mu ,\varphi$ variables are

(8.6)\begin{align} {g^{\textrm{res}}} & = \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell ,p } {\dfrac{{ - \textrm{i}{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varphi + \textrm{i}L)}}{{\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu _{\textrm{eff}}}{\tau _f}} }}} \notag\\ &\quad\times \oint_f {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

and

(8.7)\begin{align} {g^{nr}} & = \dfrac{{ - \textrm{i}}}{{\omega B}}\dfrac{{\partial {f_0}}}{{\partial \mu }}\sum\limits_{\scriptstyle\omega ,n,m \atop \scriptstyle S,p } {{\textrm {exp}}{(} - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) + \textrm{i}L - \textrm{i}p\varphi + \textrm{i}p\varsigma \textrm{)}}\notag\\ &\quad {\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}{v_ \bot }]. \end{align}

The non-resonant density and current moments are

(8.8)\begin{align} & \int {{\textrm{d}^3}v} {g^{nr}}\left\{ {\begin{array}{*{20}{@{}l}} 1\\ {{v_{||}}}\\ {{\boldsymbol{v}_ \bot }} \end{array}} \right. = \frac{{ - \textrm{i}}}{{\omega B}}\sum\limits_{\scriptstyle\omega ,n,m \atop \scriptstyle S,p } {{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S))} \notag\\ &\quad {\boldsymbol{\varepsilon }_m}\boldsymbol{\cdot }\int {{\textrm{d}^3}v\frac{{\partial {f_0}}}{{\partial \mu }}} [\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {\boldsymbol{\varUpsilon }_p}(\eta ){v_ \bot }]\left\{ {\begin{array}{*{20}{@{}l}} {{J_p}(\eta )}\\ {{v_{||}}{J_p}(\eta )}\\ {{\boldsymbol{v}_ \bot }\boldsymbol{\varUpsilon }_p^ \ast (\eta )} \end{array}} \right.. \end{align}

The correlated resonant function result with the phenomenological collisional boundary layer retained is

(8.9)\begin{align} {g^{\textrm{res}}} & ={-} \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell ,p } {{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varphi + \textrm{i}L)} \notag\\ & \quad\times \dfrac{{\displaystyle\int_0^\infty {\textrm{d}z\ {\textrm {exp}} ( - \textrm{i}sz - {z^3}/3)} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||\ast }^{2/3}{\tau _f}}}\oint_f {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau )\,{\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

with s given by (6.27), and it may be used in place of (8.6).

Gyroaveraging both resonant forms leads to

(8.10)\begin{align} \langle {g^{\textrm{res}}}\rangle & = \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell } {\dfrac{{{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varsigma ){J_p}(\eta )}}{{\textrm{i}\left({\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu_{\textrm{eff}}}{\tau_f}} } \right)}}} \notag\\ &\quad\times \oint_f {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

(8.11)\begin{align} \langle {g^{\textrm{res}}}\rangle & ={-} \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell } {{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varsigma )} \left[ {\dfrac{{{J_p}(\eta )\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - \textrm{i}sz - {z^3}/3}}} }}{{{\nu^{1/3}}v_ \bot^{2/3}k_{||\ast }^{2/3}{\tau_f}}}} \right]\notag\\ &\quad\times \oint_f {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

(8.12)\begin{align} \langle \boldsymbol{v}{g^{\textrm{res}}}\rangle & = \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell } {\dfrac{{{{\textrm {exp}}(} - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varsigma \textrm{)}[\boldsymbol{n}{v_{||}}{J_p}(\eta ) + {v_ \bot }\boldsymbol{\varUpsilon }_p^ \ast (\eta )]}}{{\textrm{i}\left({\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu_{\textrm{eff}}}{\tau_f}} } \right)}}} \notag\\ &\quad\times \oint_f {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right), \end{align}

and

(8.13)\begin{align} \langle \boldsymbol{v}{g^{\textrm{res}}}\rangle & ={-} \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell } {{{\textrm {exp}}(} - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varsigma \textrm{)}} [\boldsymbol{n}{v_{||}}{J_p} + {v_ \bot }\boldsymbol{\varUpsilon }_p^ \ast ]\notag\\ &\quad\times \left[ {\dfrac{{\displaystyle\int_0^\infty {\textrm{d}z\ {\textrm {exp}} ({ - \textrm{i}sz - {z^3}/3} )} }}{{{\nu^{1/3}}v_ \bot^{2/3}k_{||\ast }^{2/3}{\tau_f}}}} \right]\oint_f {\textrm{d}\tau } W_{p,m}^{\textrm{res}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right). \end{align}

The resonant particle and current densities are found by using (8.10) and (8.12) to obtain

(8.14)\begin{align}& \int {{\textrm{d}^3}} v{g^{\textrm{res}}}\left\{ {\begin{array}{*{20}{@{}c}} 1\\ {{v_{||}}}\\ {{\boldsymbol{v}_ \bot }} \end{array}} \right. = \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell } {{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varsigma )} \notag\\ &\quad\times \int {{\textrm{d}^3}} v\left\{ {\begin{array}{*{20}{@{}c@{}}} {{J_p}(\eta )}\\ {{v_{||}}{J_p}(\eta )}\\ {{v_ \bot }\boldsymbol{\varUpsilon }_p^ \ast } \end{array}} \right\}\frac{{\displaystyle\oint_f {\textrm{d}\tau } {W_{p,m}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\displaystyle\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right)}}{{\textrm{i}\left({\displaystyle\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu_{\textrm{eff}}}{\tau_f}} } \right)}}, \end{align}

where ${\nu _{\textrm{eff}}}{\tau _f} \to 0$ is a reminder to move any pole off the real axis in velocity space. In the preceding and following the sum over both signs of ${v_{||}}$ for the trapped is implicit in the bounce average. For the passing the transit average is for a single sign of ${v_{||}}$.

The velocity space singularity that occurs at the resonance $\oint_f {\textrm{d}\tau \varLambda = 2{\pi }\ell }$ can be avoided entirely by using the collisional boundary layer forms (8.11) and (8.13) to evaluate the density and current density. These yield the phenomenological results

(8.15)\begin{align} & \int {{\textrm{d}^3}} v{g^{\textrm{res}}}\left\{ {\begin{array}{*{20}{@{}c}} 1\\ {{v_{||}}}\\ {{\boldsymbol{v}_ \bot }} \end{array}} \right. ={-} \sum\limits_{\scriptstyle\omega ,n,S \atop \scriptstyle m,\ell } {{\textrm {exp}} ( - \textrm{i}\omega t + \textrm{i}(n\zeta - m\vartheta + S) - \textrm{i}p\varsigma )} \notag\\ &\quad\times \int {{\textrm{d}^3}} v\frac{{\displaystyle\int_0^\infty {\textrm{d}z\,{\textrm{e}^{ - \textrm{i}sz - {z^3}/3}}} }}{{{\nu ^{1/3}}v_ \bot ^{2/3}k_{||\ast }^{2/3}{\tau _f}}}\oint_f {\textrm{d}\tau } {W_{p,m}}(\tau ){\textrm {exp}} \left( { - \textrm{i}\int_0^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} } \right)\left\{ {\begin{array}{*{20}{@{}c}} {{J_p}(\eta )}\\ {{v_{||}}{J_p}(\eta )}\\ {{v_ \bot } \boldsymbol{\varUpsilon }_p^ \ast (\eta )} \end{array}} \right., \end{align}

with s given by (6.27).

The preceding expressions allow the density and current density to be formed in the Maxwell equations for a full wave treatment. Equation (6.20) is to be used to evaluate the phase integral in the exponential. The collisional boundary layer forms may be used to avoid the lowest order resonant denominator forms containing ${\nu _{\textrm{eff}}}{\tau _f}$ which merely resolves the singularity and should not appear in the final results.

9. Limitations of QL theory

Recent estimates (Catto Reference Catto2020) of when QL theory is expected to become less reliable can be refined a bit based on the results herein.

Defining the quiver speed ${v_{\textrm{quiv}}}$ by

(9.1)\begin{equation}{v_{\textrm{quiv}}} \equiv Ze|\boldsymbol{e}|/M\omega ,\end{equation}

then the linearized kinetic equation (4.11) and (5.7) imply

(9.2)\begin{equation}\frac{g}{{{f_0}}}\sim \frac{{{v_{\textrm{quiv}}}\omega }}{{{k_{||}}v_{\textrm{th}}^2w}}\sim \frac{{{v_{\textrm{quiv}}}\omega }}{{{v_{\textrm{th}}}{\nu ^{1/3}}{{({k_{||}}{v_{\textrm{th}}})}^{2/3}}}}.\end{equation}

Combining $g/{f_0} \ll 1$ with the need to be in the long mean free path limit of (5.9) $({\nu \ll {k_{||}}{v_{th}}} )$ with $\Delta \vartheta \gg 1$ in (5.10) leads to the following bounds on the collision frequency for the validity of a resonant plateau treatment of QL theory:

(9.3)\begin{equation}\frac{{{v_{quiv}}}}{{{v_{th}}}} \ll \frac{{{\nu ^{1/3}}k_{||}^{2/3}v_{th}^{2/3}}}{\omega } \ll \frac{{{v_{th}}}}{{qR\omega }}.\end{equation}

In addition, (5.5) suggests that

(9.4)\begin{equation}{\left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell + \textrm{i}{\nu_{\textrm{eff}}}{\tau_f}} } \right)^{ - 1}}\sim \delta \left({\oint_f {\textrm{d}\tau \varLambda - 2{\pi }\ell } } \right)\sim {({k_{||}}{v_{\textrm{th}}}{\tau _f}w)^{ - 1}},\end{equation}

which along with (7.11) leads to

(9.5)\begin{equation}D\sim \frac{{v_{\textrm{quiv}}^2{\omega ^2}}}{{{k_{||}}{v_{\textrm{th}}}w}}\sim \frac{{v_{\textrm{quiv}}^2{\omega ^2}}}{{{\nu ^{1/3}}{{({k_{||}}{v_{\textrm{th}}})}^{2/3}}}}.\end{equation}

It is useful to realize that even though (7.11) appears to be independent of collisions, the delta function arises because of the narrow collisional boundary layer so it has an implicit width that depends on collisions. It is only when it is integrated over that the $\nu$ dependence is removed. For example, the energy per mass moment of the QL operator for a Maxwellian gives upon integration by parts a result that is truly independent of collisions, namely

(9.6)\begin{equation}\frac{1}{2}\int {{\textrm{d}^3}v{v^2}} \overline {Q\{ {f_0}\} } = \frac{M}{{2T}}\sum\limits_{\scriptstyle\omega ,p \atop \scriptstyle m,n,S } {\int {{\textrm{d}^3}v} {v^2}D{f_0}} .\end{equation}

Neglect of the nonlinear term $\boldsymbol{a}\boldsymbol{\cdot }{\nabla _v}g$ in the perturbed (4.11) compared with $\boldsymbol{a}\boldsymbol{\cdot }{\nabla _v}{f_0}$ suggests $g/{f_0} \ll w$. Using (9.2) then leads to assuming

(9.7)\begin{equation}\frac{{{v_{quiv}}}}{{{v_{th}}}} \ll {w^2}\left( {\frac{{{k_{||}}{v_{th}}}}{\omega }} \right)\sim \frac{{{\nu ^{2/3}}{{({k_{||}}{v_{th}})}^{1/3}}}}{\omega }.\end{equation}

Therefore, the neglect of the nonlinear term implies a slightly stronger restriction than the left most inequality in (9.3).

The QL and collision operators give $Q\{ {f_0}\} \sim D{f_0}/v_{\textrm{th}}^2$ and $C\{ {f_0}\} \sim \nu {f_0}$. For ${f_0}$ to depart appreciably from Maxwellian requires $Q\{ {f_0}\} \sim \; C\{ {f_0}\}$ or $D / v_{\textrm{th}}^2\sim \nu$, leading to the condition

(9.8)\begin{equation}\frac{{{v_{\textrm{quiv}}}}}{{{v_{\textrm{th}}}}}\sim \frac{{{\nu ^{2/3}}{{({k_{||}}{v_{\textrm{th}}})}^{1/3}}}}{\omega }.\end{equation}

Comparing this last estimate to inequality (9.7) that must be satisfied to use unperturbed trajectories leads to a contradiction. As a result, linear response (6.14) or (6.28) of QL theory begins to fail as the departure from Maxwellian becomes significant, as indicated by earlier estimates (Catto Reference Catto2020). Therefore, the associated QL operator of (7.11) or (7.18) is altered, suggesting that large departures from Maxwellian may not be allowed by a narrow collisional boundary layer treatment of QL rf heating and current drive.

The applied rf field estimate (9.7) is an indication of when unperturbed trajectories are no longer adequate and a full nonlinear treatment of the perturbed gyrokinetic equation becomes necessary. Alternatively, it may provide a rough estimate of when the stochastic boundary layer width about each wave–particle resonance curve in velocity space becomes as wide as the collisional boundary layer. However, such a statement is difficult to prove. Both the treatment herein and treatments of field line stochasticity (Becoulet, Gambier & Samain Reference Becoulet, Gambier and Samain1991; Helander & Lisak Reference Helander and Lisak1992) lead to what appears to be the same lowest order QL operator, but the Hamiltonian formulations are in three canonical action angle variables that have not been recast into the separate configuration space and velocity space variables needed to form the density and current density for a full wave treatment or a boundary layer width estimate from a perturbed kinetic equation. To make use of unperturbed trajectories, the treatment here assumes the rf field amplitude is well below the threshold for full stochasticity.