2. Electron kinetic equation

Consider an applied low frequency rf wave of frequency $\omega \ll \varOmega = eB/mc$ in an axisymmetric tokamak magnetic field

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

with B the magnitude of the magnetic field, $\boldsymbol{n}$ the unit vector in the magnetic field direction, $\zeta$ the toroidal angle variable, $\psi$ the poloidal flux function, $I = I(\psi ) = R{B_t}$ with R the major radius and ${B_t}$ the toroidal magnetic field, m and e the mass and magnitude of the charge on an electron and c the speed of light. Then for a lowest order Maxwellian distribution function

(2.2)\begin{equation}{f_0} = n{(m/2T)^{3/2}}\,{\textrm{e}^{ - m{v^2}/2T}},\end{equation}

quasilinear rf treatments solve the linearized electron kinetic equation

(2.3)\begin{equation}- \textrm{i}\omega {f_1} + \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }{f_1} - \varOmega \boldsymbol{v} \times \boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{\nabla }_v}{f_1} - C\{ {\,f_1}\} =- (e/T)\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{v}{f_0},\end{equation}

using unperturbed particle trajectories. Here, C is the electron collision operator, $\boldsymbol{e} \propto {\textrm{e}^{ - \textrm{i}\omega t}}$ is the applied rf field, ${f_1} \propto {\textrm{e}^{ - \textrm{i}\omega t}}$ is the perturbed electron distribution function, $\varOmega = eB/mc$ is the electron cyclotron frequency and T is the electron temperature. To simplify the presentation the unperturbed electric field is neglected, and only a single applied frequency $\omega$ and toroidal mode number n are considered.

The gyrokinetic variables (Catto Reference Catto1978, Reference Catto2019) $E = {v^2}/2$ (kinetic energy), $\mu = v_ \bot ^2/2B$ (magnetic moment), $\varphi =$ gyrophase and $\boldsymbol{R} = \boldsymbol{r} - {\varOmega ^{ - 1}}\boldsymbol{v} \times \boldsymbol{n}$ (guiding centre location) are used to derive the electron kinetic equation, with $\boldsymbol{v} = {\boldsymbol{v}_ \bot } + {v_{||}}\boldsymbol{n}$, $v_{||}^2 = {v^2} - 2\mu B$, ${\boldsymbol{v}_ \bot } = {v_ \bot }({\boldsymbol{e}_\psi }\cos \varphi + {\boldsymbol{e}_ \times }\sin \varphi )$, and the orthonormal unit vectors satisfying $\boldsymbol{n} = {\boldsymbol{e}_\psi } \times {\boldsymbol{e}_ \times }$, where ${\boldsymbol{e}_\psi } = \boldsymbol{\nabla }\psi /|\boldsymbol{\nabla }\psi |= \boldsymbol{\nabla }\psi /R{B_p}$. The poloidal magnetic field ${B_p}$ is assumed positive, making ${B_t} > 0$ for co-current operation and ${B_t} < 0$ when operation is counter-current.

To lowest order, $\partial {f_1}/\partial \varphi = 0$ in the lower hybrid and helicon wave limit of interest. Gyroaveraging the next order equation holding $\boldsymbol{R}$ fixed, and Fourier decomposing by writing

(2.4)\begin{equation}\boldsymbol{e} = {\Sigma _m}{\boldsymbol{e}_m}\,{\textrm{e}^{\textrm{i}n\zeta - \textrm{i}m\vartheta + \textrm{i}S(\psi ) - \textrm{i}\omega t}} = {\textrm{e}^{\textrm{i}n\zeta - \textrm{i}\omega t}}{\Sigma _m}{\boldsymbol{e}_m}\,{\textrm{e}^{ - \textrm{i}m\vartheta + \textrm{i}S(\psi )}},\end{equation}

leads to (Catto Reference Catto2020; Catto & Tolman Reference Catto and Tolman2021a)

(2.5)\begin{equation}- \textrm{i}\omega {f_1} + (\textrm{i}qn{f_1} + \partial {f_1}/\partial \vartheta ){v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta - C\{ {\,f_1}\} = {\Sigma _m}{W_m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }},\end{equation}

with ${f_1} \propto {\textrm{e}^{\textrm{i}n\zeta - \textrm{i}\omega t}}$ to streamline the notation and

(2.6)\begin{equation}{W_m} =- (e/T)[{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}\; {J_0}(\eta ){v_{||}} + \textrm{i}k_ \bot ^{ - 1}{v_ \bot }{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{k} \times \boldsymbol{n}\; {J_1}(\eta )]{f_0}\,{\textrm{e}^{ - \textrm{i}L}},\end{equation}

where $\boldsymbol{k} = \boldsymbol{\nabla }S(\psi ) - m\boldsymbol{\nabla }\vartheta + n\boldsymbol{\nabla }\zeta$, ${k_{||}} = (qn - m)\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \approx (qn - m)/qR$, $\eta = {k_ \bot }{v_ \bot }/\varOmega$ and the poloidal angle $\vartheta$ defined such that $q = q(\psi ) = |I(\psi )|/{R^2}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta$. To find the preceding, write ${\boldsymbol{k}_ \bot } = {k_ \bot }({\boldsymbol{e}_\psi }\cos \beta + {\boldsymbol{e}_ \times }\sin \beta )$, $L = {\varOmega ^{ - 1}}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v} \times \boldsymbol{n} = \eta \sin (\varphi - \beta )$, and ${\textrm{e}^{ - \textrm{i}L}} = {\Sigma _p}\,{\textrm{e}^{ - \textrm{i}p(\varphi - \beta )}}{J_p}(\eta )$ to obtain ${(2{\rm \pi})^{ - 1}}\oint {\textrm{d}\varphi } \,{\textrm{e}^{\textrm{i}L}} = {J_0}(\eta )$ and ${(2{\rm \pi})^{ - 1}}\oint {\textrm{d}\varphi } \,\; {\textrm{e}^{\textrm{i}L}}{\boldsymbol{v}_ \bot } =- \textrm{i}k_ \bot ^{ - 1}{v_ \bot }\boldsymbol{k} \times \boldsymbol{n}{J_1}(\eta )$. The result is similar to that of a homogenous plasma, where it corresponds to using ${\textrm{e}^{\textrm{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}}} = {\textrm{e}^{\textrm{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{R} + \textrm{i}L}}$ prior to gyroaveraging at fixed $\boldsymbol{R}$. Additional Bessel functions will appear when the density and current are formed as they are evaluated in $\boldsymbol{r}$ and $\boldsymbol{v}$ variables, requiring $\boldsymbol{R} = \boldsymbol{r} - {\varOmega ^{ - 1}}\boldsymbol{v} \times \boldsymbol{n}$ to be employed once again. To streamline the notation the ${\textrm{e}^{ - \textrm{i}L}}$ factor that results from this change back is displayed in ${W_m}$ since the distinction between $\boldsymbol{r}$ and $\boldsymbol{R}$ is unimportant elsewhere in the kinetic equation. As time evolution and the departure from axisymmetry do not result in any temporal or toroidal mode coupling, a monochromatic wave is considered.

The narrow boundary layers that must be considered are assumed to be due to pitch angle scattering collisions of parallel streaming for a couple of reasons. First, pitch angle scattering dominates for collisional boundary layers at a trapped–passing boundary at large aspect ratio. Second, for lower hybrid heating (Bonoli Reference Bonoli2014) and lower hybrid and helicon current drive (Catto & Zhou Reference Catto and Zhou2023), a high-speed expansion of the electron–electron collision operator is normally adequate and tends to make energy scatter less important. Consequently, the replacement

(2.7)\begin{equation}C\{ {\,f_1}\} \to \frac{{2{\nu _e}{B_0}\xi }}{{B{x^3}}}\frac{\partial }{{\partial \lambda }}\left( {\lambda \xi \frac{{\partial {f_1}}}{{\partial \lambda }}} \right),\end{equation}

is normally adequate, where ${\nu _e} = 3\sqrt {\rm \pi}(Z + 1){\nu _{ee}}/4$, ${\nu _{ee}} = 4\sqrt {2{\rm \pi}} {e^4}n\ell n{\varLambda _C}/3{m^{1/2}}{T^{3/2}}$, $x = v/{v_e}$, ${v_e} = {(2T/m)^{1/2}}$ is the electron thermal speed, and $\ell n{\varLambda _C}$ is the Coulomb logarithm. In addition, $v_{||}^2 = {v^2}{\xi ^2} = {v^2} - 2\mu B = {v^2}(1 - \lambda B/{B_0})$, with ${B_0}$ a flux function at most. When lower hybrid heating and current drive due to lower hybrid and helicon waves is evaluated, the second or energy scatter term appearing in (4.2) of Catto & Zhou (Reference Catto and Zhou2023) is required (unfortunately, the ${x^3}$ is missing in the denominator of their first term) since when the collision operator acts on ${f_0}$, instead of ${f_1}$, there are no boundary layers in the ${f_0}$ equation.

When the linearized kinetic equation is solved numerically it may be convenient to use $v$ and ${v_{||}}$ variables. Then, the $\mu \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }B$ mirror force term explicitly appears in the equation

(2.8)\begin{align}- \textrm{i}\omega {f_1} + \left( {\textrm{i}qn{f_1} + \frac{{\partial {f_1}}}{{\partial \vartheta }}} \right){v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta - \mu \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }B\frac{{\partial {f_1}}}{{\partial {v_{||}}}} - \frac{{{\nu _e}}}{{2{x^3}}}\frac{\partial }{{\partial {v_{||}}}}\left[ {({v^2} - v_{||}^2)\frac{{\partial {f_1}}}{{\partial {v_{||}}}}} \right] = {\Sigma _m}{W_m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}.\end{align}

In $v$ and $\mu$ variables the mirror force is implicit in the $\vartheta$ variation of $v_{||}^2 = {v^2}(1 - \lambda B/{B_0})$.

3. Collisional boundary layer for resonant electrons without geometrical effects

In a cylindrical plasma immersed in a homogenous magnetic field ${B_0}$, $v_{||}^2 = {v^2}(1 - \lambda ) = {v^2}{\xi ^2}$. It is then convenient to use $v$, ${v_{||}}$ as velocity variables in place of $v$, $\lambda$ as there is no poloidal mode coupling or mirror force. The kinetic equation is then simply

(3.1)\begin{equation}- \textrm{i}(\omega - {k_{||}}{v_{||}}){f_m} - \bar{\nu }(v)\frac{\partial }{{\partial {v_{||}}}}\left( {v_ \bot^2\frac{{\partial {f_m}}}{{\partial {v_{||}}}}} \right) = {W_m},\end{equation}

for ${f_1} = {\textrm{e}^{\textrm{i}n\zeta - \textrm{i}\omega t}}{\Sigma _m}{f_m}\,{\textrm{e}^{ - \textrm{i}m\vartheta + \textrm{i}S}}$ and where $\bar{\nu }(v) = 2{\nu _e}/{x^3}$. To see that a narrow collisional boundary layer of width $\Delta {v_{||}}$ arises when ${k_{||}} \ne 0$, the estimates ${k_{||}}{v_{||}} - \omega = {k_{||}}\Delta {v_{||}}$ and $\partial /\partial {v_{||}}\sim 1/\Delta {v_{||}}$ are used to balance the resonance term with collisions, ${k_{||}}\Delta {v_{||}}\sim \bar{\nu }v_ \bot ^2/{(\Delta {v_{||}})^2} \equiv {\bar{\nu }_{\textrm{eff}}}$, to obtain $\Delta {v_{||}}/{v_e}\sim {(\bar{\nu }v_ \bot ^2/{k_{||}}v_e^3)^{1/3}}\sim {(\bar{\nu }/{k_{||}}{v_e})^{1/3}} \ll 1$ and the effective collision frequency ${\bar{\nu }_{\textrm{eff}}}\; \sim \; \bar{\nu }{(k_{||}^2v_ \bot ^2/{\bar{\nu }^2})^{1/3}} \gg \bar{\nu }$. Consequently, the resonant electrons are in a collisionally broadened boundary layer, and have a wave–particle interaction time of ${\bar{\tau }_{{\mathop{\rm int}} }} = 1/{\bar{\nu }_{\textrm{eff}}}$. Since the mean free path of an electron is $\mathrm{\sim }{v_e}/{\nu _e}\sim {10^4}$ metres, for $1/{k_{||}}\sim 10$ metres, this gives $\Delta {v_{||}}/{v_e}\sim 1/10$, ${\bar{\nu }_{\textrm{eff}}}/\bar{\nu }\sim 100$, and a resonant electron mean free path of ${v_e}{\bar{\tau }_{{\mathop{\rm int}} }}\sim 100$ metres.

To see this more rigorously, Catto (Reference Catto2020) uses the narrowness of the boundary layer in ${v_{||}}$ space to ignore the ${v_{||}}$ dependence of $v_ \bot ^2 = {v^2} - v_{||}^2$. Then, introducing the new variable

(3.2)\begin{equation}s = ({k_{||}}{v_{||}} - \omega )/{(k_{||}^2v_ \bot ^2\bar{\nu })^{1/3}},\end{equation}

finds the non-singular solution

(3.3)\begin{equation}{f_m} = \frac{{{W_m}}}{{{{(k_{||}^2v_ \bot ^2\bar{\nu })}^{1/3}}}}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{ - \textrm{i}st - {t^3}/3}}} \mathop \to \limits_{s \to \pm \infty } \frac{{ - {W_m}}}{{\textrm{i}(\omega - {k_{||}}{v_{||}})}} \to {W_m}{\rm \pi}\delta (\omega - {k_{||}}{v_{||}}).\end{equation}

Setting $s = {k_{||}}\Delta {v_{||}}/{(k_{||}^2v_ \bot ^2\bar{\nu })^{1/3}}\sim 1$ recovers the earlier estimates and implies

(3.4)\begin{equation}\frac{{ - 1}}{{\textrm{i}(\omega - {k_{||}}{v_{||}})}}\; \sim \; {\bar{\tau }_{{\mathop{\rm int}} }} = 1/{\bar{\nu }_{\textrm{eff}}}\sim {\rm \pi}\delta (\omega - {k_{||}}{v_{||}}),\end{equation}

thereby providing a physical meaning for the delta function that cannot be obtained by considering a steady state causal Landau resonance. Notice the function

(3.5)\begin{equation}\bar{P} = \frac{1}{{{\rm \pi}{{(k_{||}^2v_ \bot ^2\bar{\nu })}^{1/3}}}}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{ - {t^3}/3}}} \cos (st),\end{equation}

now plays the role of a delta function for $s \gg 1$.

The next section starts to incorporate tokamak geometry.

4. Resonant electrons in tokamak geometry without collisions

The preceding section indicates that collisions are very weak, but become significant for the resonant electrons in or near collisional boundary layers. As a result, it is useful to briefly examine the collisionless resonant condition in tokamak geometry, where $\omega - {k_{||}}{v_{||}}$ is replaced by its transit-averaged form (Catto & Tolman Reference Catto and Tolman2021a)

(4.1)\begin{equation}\oint_f {\textrm{d}\tau (\omega - {k_{||}}{v_{||}})} = \omega \oint_f {\textrm{d}\tau } - 2{\rm \pi}\sigma (qn - m),\end{equation}

with $\textrm{d}\tau = \textrm{d}\vartheta /{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \approx qR\,\textrm{d}\vartheta /{v_{||}} > 0$ and the subscript f denoting an integration over a full $2{\rm \pi}$ for the passing $(\sigma = {v_{||}}/|{v_{||}}|={\pm} 1)$ and a full bounce for the trapped $(\sigma = 0)$.

For a small inverse aspect ratio $(\epsilon \ll 1)$ tokamak with

(4.2)\begin{equation}B = {B_0}(1 - \epsilon \cos \vartheta ),\end{equation}

and $v_{||}^2 = {v^2}(1 - \lambda B/{B_0})$, letting ${k^2} = 2\mathrm{\epsilon }\lambda /[1 - (1 - \mathrm{\epsilon })\lambda ] = {\kappa ^{ - 2}}$ leads to

(4.3)\begin{equation}\oint_f {\textrm{d}\tau } = \frac{{4qR}}{{v\sqrt {2\epsilon } }}\left\{ {\begin{array}{*{20}{@{}ll}} {\sqrt {(1 - \epsilon ){k^2} + 2\epsilon } K(k)}& {\textrm{passing}\;(0 \le k < 1)}\\ {2K(\kappa )}& {\textrm{trapped}\;(0 \le \kappa < 1)} \end{array}} \right.,\end{equation}

with K the complete elliptic integral of the first kind.

No resonance occurs for the trapped electrons. However, for the passing the collisionless resonance condition becomes

(4.4)\begin{equation}\frac{{|{k_{||}}|v}}{\omega } = \frac{{\sqrt {{k^2} + 2\epsilon } K(k)}}{{({\rm \pi}/2)\sqrt {2\epsilon } }} \to \left\{ {\begin{array}{*{20}{@{}ll}} {\sqrt {{k^2} + 2\epsilon } /\sqrt {2\epsilon } }& {{k^2}\lesssim 2\epsilon }\\ {\ell n[16/(1 - {k^2})]/{\rm \pi}\sqrt {2\epsilon } }& {{k^2} \to 1} \end{array}} \right.,\end{equation}

where $1 \le |{k_{||}}|v/\omega = |qn - m|v/\omega qR < \infty$, and no passing resonance occurs for ${k_{||}} = 0$. For lower hybrid and helicon current drive, the freely passing $({k^2}\lesssim 2\epsilon )$ are of most interest (Catto & Zhou Reference Catto and Zhou2023).

5. Collisional boundary layer in tokamak geometry without poloidal mode coupling

The results of the preceding section are next extended to consider a collisional boundary layer in tokamak geometry when no poloidal mode coupling occurs. In this slightly artificial, but useful to understand, limit the perturbed passing distribution function is periodic in poloidal angle. Using

(5.1)\begin{equation}{f_1} = {f_m}\,{\textrm{e}^{\textrm{i}n\zeta - \textrm{i}\omega t - \textrm{i}m\vartheta + \textrm{i}S(\psi )}},\end{equation}

and transit averaging, the passing electron kinetic equation (2.5) then yields

(5.2)\begin{equation}\textrm{i}{f_m}\left[{\omega \oint_f {\textrm{d}\tau } - 2{\rm \pi}\sigma (qn - m)} \right]+ \oint_f {\textrm{d}\tau } C\{ {\,f_m}\} =- \oint_f {\textrm{d}\tau } {W_m},\end{equation}

where

(5.3)\begin{equation}\oint_f {\textrm{d}\tau } C\{ {\,f_m}\} = \frac{{2{\nu _e}{B_0}}}{{{x^3}}}\frac{\partial }{{\partial \lambda }}\left[ {\left( {\lambda \oint_f {\textrm{d}\tau } \frac{{{\xi^2}}}{B}} \right)\frac{{\partial {f_m}}}{{\partial \lambda }}} \right],\end{equation}

with

(5.4)\begin{equation}{B_0}\oint_f {\textrm{d}\tau } \frac{{{\xi ^2}}}{B} = \frac{{4qR\sqrt {2\epsilon } E(k)}}{{v\sqrt {(1 - \epsilon ){k^2} + 2\epsilon } }} \to \frac{{4qR}}{v}\left\{ {\begin{array}{*{20}{@{}ll}} {{\rm \pi}\sqrt {2\epsilon } /2\sqrt {{k^2} + 2\epsilon } }& {{k^2}\lesssim 2\epsilon }\\ {\sqrt {2\epsilon } /\sqrt {1 + \epsilon } }& {{k^2} \to 1} \end{array}} \right.,\end{equation}

and E the complete elliptic integral of the second kind. For a resonant interaction when $qn > m\, (qn < m)$ a passing electron must have ${v_{||}} > 0\, ({v_{||}} < 0)$ since $\omega \oint_f {\textrm{d}\tau } > 0$.

Taylor expanding about the resonant pitch angle ${\lambda _{\textrm{res}}}$

(5.5)\begin{equation}{\tau _f} = \oint_f {\textrm{d}\tau } = \tau _f^0 + (\lambda - {\lambda _{\textrm{res}}})\partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}} + \cdots ,\end{equation}

where $\tau _f^0 = 2{\rm \pi} |qn - m|/\omega > 0$, gives the Airy form

(5.6)\begin{equation}\frac{{{\partial ^2}{f_m}}}{{\partial {u^2}}} + \textrm{i}u{f_m} \approx- \frac{{{{ {\oint_f {\textrm{d}\tau } {W_m}} |}_{\lambda = {\lambda _{\textrm{res}}}}}}}{{{{(\nu \tau _f^0)}^{1/3}}{{(\omega \partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}})}^{2/3}}}},\end{equation}

with $\oint_f {\textrm{d}\tau } {W_m} \approx \oint_f {\textrm{d}\tau } {W_m}{|_{\lambda = {\lambda _{\textrm{res}}}}}$ due to the narrowness of the boundary layer, and

(5.7)\begin{equation}u = (\lambda - {\lambda _{\textrm{res}}}){(\omega \partial {\tau _f}/\partial \lambda )^{1/3}}/{(\nu \tau _f^0)^{1/3}} = (\lambda - {\lambda _{\textrm{res}}})/w.\end{equation}

In the preceding, the resonance is used to define $k_{\textrm{res}}^2 = 2\epsilon {\lambda _{\textrm{res}}}/[1 - (1 - \epsilon ){\lambda _{\textrm{res}}}]$ in (4.4) and its narrowness is used to approximate

(5.8)\begin{equation}\oint_f {\textrm{d}\tau } C\{ {\,f_m}\} \approx \frac{{2{\nu _e}{B_0}}}{{{x^3}}}\left( {{\lambda_{\textrm{res}}}{{\left. {\oint_f {\textrm{d}\tau } \frac{{{\xi^2}}}{B}} \right|}_{\textrm{res}}}} \right)\frac{{{\partial ^2}{f_m}}}{{\partial {\lambda ^2}}} \equiv \nu \tau _f^0\frac{{{\partial ^2}{f_m}}}{{\partial {\lambda ^2}}},\end{equation}

with $\nu \tau _f^0 \approx 2{\nu _e}{x^{ - 3}}{\lambda _{\textrm{res}}}\oint_f {\textrm{d}\tau } {\xi ^2}$.

Assuming ${k^2}\lesssim 2\epsilon$, then $\lambda \approx {k^2}/({k^2} + 2\epsilon )$ and ${k_0} = {k_{\textrm{res}}}$ give

(5.9)\begin{equation}\partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}} = {\rm \pi}qR{(k_0^2 + 2\epsilon )^{3/2}}/v{(2\epsilon )^{3/2}}\sim qR/v\sim \tau _f^0.\end{equation}

Then, making use of $\partial /\partial \lambda \sim 1/(\lambda - {\lambda _{\textrm{res}}})$ leads to the boundary layer width estimate

(5.10)\begin{equation}\lambda - {\lambda _{\textrm{res}}}\sim \; {[\nu \tau _f^0/(\omega \partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}})]^{1/3}}\sim {(\nu /\omega )^{1/3}},\end{equation}

and effective collision frequency estimate

(5.11)\begin{equation}{\nu _{\textrm{eff}}} \approx \nu /{(\lambda - {\lambda _{\textrm{res}}})^2} \approx \nu {[\omega (\partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}})/\nu \tau _f^0]^{2/3}}\sim \nu {(\omega /\nu )^{2/3}}.\end{equation}

Use of $({\partial ^2}/\partial {u^2} + \textrm{i}u)\int_0^\infty {\textrm{d}t\,{\textrm{e}^{\textrm{i}ut - {t^3}/3}}} = \int_0^\infty {\textrm{d}t} (\partial /\partial t)\,{\textrm{e}^{\textrm{i}ut - {t^3}/3}} =- 1$ leads to the solution

(5.12)\begin{equation}f_m^{\textrm{pass}} = \frac{{{{ {\oint_f {\textrm{d}\tau } {W_m}} |}_{\lambda = {\lambda _{\textrm{res}}}}}}}{{(\omega \partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}})w}}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{\textrm{i}ut - {t^3}/3}}} \mathop \to \limits_{u \to \pm \infty } \frac{{{{ {\textrm{i}\oint_f {\textrm{d}\tau } {W_m}} |}_{\lambda = {\lambda _{\textrm{res}}}}}}}{{\omega \oint_f {\textrm{d}\tau } - 2{\rm \pi} |qn - m|}},\end{equation}

for the passing, with $w = {(\nu \tau _f^0)^{1/3}}/{(\omega \partial {\tau _f}/\partial \lambda {|_{\lambda = {\lambda _{\textrm{res}}}}})^{1/3}}\sim {(\nu /\omega )^{1/3}} \ll 1$ in $u = (\lambda - {\lambda _{\textrm{res}}})/w$, as expected from $u \approx 1$. In addition, $f_m^{\textrm{pass}} \propto \; {\nu ^{ - 1/3}}$ for $|u|\ll 1$.

Integration over velocity space requires an integration over pitch angle $({\textrm{d}^3}v \approx 2{\rm \pi} B{v^3}\,\textrm{d}v\,\textrm{d}\lambda /{B_0}{v_{||}})$. In this case

(5.13)\begin{equation}P = {({\rm \pi} w)^{ - 1}}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{ - {t^3}/3}}} \cos (ut),\end{equation}

is a delta function representation in pitch angle as shown in figure 1. To see this, let $\varsigma \gg 1$ and recall any integral over velocity space will involve pitch angle, resulting in

(5.14)\begin{equation}\int_{{\lambda _{\textrm{res}}} - \varsigma w}^{{\lambda _{\textrm{res}}} + \varsigma w} {\textrm{d}\lambda P} = \frac{1}{\rm \pi}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{ - {t^3}/3}}} \int_{ - \varsigma }^\varsigma {\textrm{d}u} \cos (ut) = \frac{2}{\rm \pi}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{ - {t^3}/3}}} \frac{{\sin (\varsigma t)}}{t} = 1.\end{equation}

The function that contains P as its real part will be denoted by

(5.15)\begin{equation}U(u) = {({\rm \pi} w)^{ - 1}}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{\textrm{i}ut - {t^3}/3}}} \mathop \to \limits_{u \to \pm \infty } i/{\rm \pi}(\lambda - {\lambda _{\textrm{res}}}).\end{equation}

Notice that the imaginary part (ImU) is odd in u and dominates asymptotically, while the $P(u) = Re U(u)$ exhibits the delta function behaviour. Therefore

(5.16)\begin{equation}{\mathop{\rm Im}\nolimits} \int_{{\lambda _{\textrm{res}}} - \varsigma w}^{{\lambda _{\textrm{res}}} + \varsigma w} {\textrm{d}\lambda U} = \frac{1}{\rm \pi}\int_0^\infty {\textrm{d}t\,{\textrm{e}^{ - {t^3}/3}}} \int_{ - \varsigma }^\varsigma {\textrm{d}u\sin (ut)} = 0,\end{equation}

indicating ImU no longer plays a role once the pitch angle integral is performed

Figure 1. A plot of the function P versus pitch angle $\lambda$ to demonstrate its delta function behaviour in the vicinity of the resonant pitch angle ${\lambda _{\textrm{res}}}$ (reprinted with permission from Catto & Tolman Reference Catto and Tolman2021b).

For the trapped electrons $(\sigma = 0)$ in (5.2), implying no resonant boundary layers occur and collisions are negligible. As a result, the expected trapped response is

(5.17)\begin{equation}f_m^{\textrm{trap}} \approx \textrm{i}{\omega ^{ - 1}}{\Sigma _m}{W_m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}.\end{equation}

However, there will be an additional collisional boundary layer about the trapped–passing boundary since both $f_m^{\textrm{trap}}$ and $f_m^{\textrm{pass}}$ must vanish there, as they are different functions of $\lambda$. More details will be given in the following sections.

Better solutions for the trapped and passing will be given in the subsequent sections when poloidal mode coupling is retained for the trapped as well as the passing alphas. And the trapped–passing boundary layer will also be addressed further in § 8.

6. Poloidal mode coupling with an effective Krook model resolution of singularities

An insightful solution of (2.5) retaining poloidal mode coupling is obtained by making the Krook replacement $C\{ {\,f_1}\} \to - {\nu _{\textrm{eff}}}{f_1}$ with ${\nu _{\textrm{eff}}}\sim \nu {(\omega /\nu )^{2/3}}$. Introducing the trajectory time variable $\tau$ by letting $\textrm{d}\tau = \textrm{d}\vartheta /{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta > 0$ with $\textrm{d}\vartheta > 0$ for ${v_{||}} > 0$, $\textrm{d}\vartheta < 0$ for ${v_{||}} < 0$, and $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta > 0$, then integrating from $\tau \to - \infty$ where ${f_1} = 0$ to $\tau = 0$ with $\vartheta (\tau = 0) = \vartheta$ gives

(6.1)\begin{equation}{f_1} = {\textrm{e}^{ - \textrm{i}m\vartheta }}\int_{ - \infty }^0 {\textrm{d}\tau {W_m}(\tau )} \,{\textrm{e}^{ - (\textrm{i}\omega - {\nu _{\textrm{eff}}})\tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}.\end{equation}

To retain trapped and passing electrons ${f_1}$ is written in $\psi ,\vartheta ,\zeta ,v,\lambda$ and $\sigma$ variables. Consequently, successive passes through resonance are strongly correlated, and poloidal variation entering ${v_{||}}$ by its B dependence is fully retained. Collisional boundary layer modifications are neglected in (6.1). However, they are considered further in subsequent sections once poloidal variation is properly evaluated in (6.2) and (6.3).

The trajectory integral form (6.1) for ${f_1}$ is similar to many expressions in the rf literature (Bernstein & Baxter Reference Bernstein and Baxter1981; Belikov & Kolesnichenko Reference Belikov and Kolesnichenko1982, Reference Belikov and Kolesnichenko1994; Brambilla Reference Brambilla1989, Reference Brambilla1994, Reference Brambilla1999; Kasilov et al. Reference Kasilov, Pyatak and Stepanov1990; Catto & Myra Reference Catto and Myra1992; Lamalle Reference Lamalle1993, Reference Lamalle1997; Jaeger et al. Reference Jaeger, Berry, D'azevedo, Batchelor and Carter2001, Reference Jaeger, Harvey, Berry, Myra, Dumont, Phillips, Smithe, Barret, Batchelor, Bonoli, Carter, D'azevedo, D'ippolito, Moore and Wright2006, Reference Jaeger, Berry, D'azevedo, Barret, Ahern, Swain, Batchelor, Harvey, Myra, D'ippolito, Phillips, Valeo, Smithe, Bonoli, Wright and Choi2008; Brambilla & Bilato Reference Brambilla and Bilato2020). The KINETIC-J treatment of Green & Berry (Reference Green and Berry2014) includes a function $\alpha (t^{\prime})$ under its trajectory integral in their eq. (11) that plays a role similar to a very large ${\nu _{\textrm{eff}}}$. Taking full advantage of the periodicity of unperturbed orbits and ${W_m}$, and defining the time for a full poloidal circuit as ${\tau _f} = {{\oint_f {\textrm{d}\vartheta } } / {{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta }} > 0$ leads to (Bernstein & Baxter Reference Bernstein and Baxter1981; Belikov & Kolesnichenko Reference Belikov and Kolesnichenko1982; Tolman & Catto Reference Tolman and Catto2021)

(6.2)\begin{equation}{f_1} = {\Sigma _m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}\frac{{\int_{ - {\tau _f}}^0 {\textrm{d}\tau } {W_m}(\tau )\,{\textrm{e}^{ - (\textrm{i}\omega - {\nu _{\textrm{eff}}})\tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}}}{{1 - {\textrm{e}^{(\textrm{i}\omega - {\nu _{\textrm{eff}}}){\tau _f} - \textrm{i}2{\rm \pi}\sigma (qn - m)}}}}.\end{equation}

Only for a homogenous magnetic field does the phase factor $\textrm{i}(\omega - {k_{||}}{v_{||}})\tau$ in the exponential vanish at resonance to allow ${f_1} \to \textrm{i}{\Sigma _m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}{W_m}/(\omega - {k_{||}}{v_{||}})$ in the collisionless limit.

The transit average in the denominator (implied by the appearance of ${\tau _f}$) indicates successive poloidal circuits are correlated. Each poloidal mode m may couple to other poloidal modes. This behaviour enters through the exponential phase factor in the preceding integral appearing in the numerator and through the denominator. Taylor expanding the denominator by introducing the mode coupling index $\ell = 0, \pm 1, \pm 2, \ldots$ via $1 = {\textrm{e}^{ - \textrm{i}\sigma 2{\rm \pi}\ell }}$, allows each m to couple to neighbouring poloidal modes for each $n \gg 1$

(6.3)\begin{equation}{f_1} = \textrm{i}{\Sigma _m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}\frac{{\int_{ - {\tau _f}}^0 {\textrm{d}\tau } {W_m}(\tau )\,{\textrm{e}^{ - (\textrm{i}\omega - {\nu _{eff}})\tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}}}{{(\omega + \textrm{i}{\nu _{\textrm{eff}}}){\tau _f} - \textrm{i}2{\rm \pi}\sigma (qn - m - \ell )}}.\end{equation}

This type of coupling only matters for the lower $|\ell |$ since for $|\ell |\gg 1$ the rapidly oscillating phase factor in the integral of the numerator reduces the contributions of higher $|\ell |$. Solution (6.3) is valid for trapped and passing electrons. The transit average behaviour of the denominator and drive term alters the spatial and velocity space behaviour of the full wave solution. Inserting the attenuation factor of Lamalle (Reference Lamalle1993, Reference Lamalle1997) decorrelates successive poloidal transits through resonance and removes this poloidal coupling.

7. Poloidal mode coupling and collisional boundary layers in tokamak geometry

The Krook solution (6.3) resolves singularities, but does not capture collisional boundary layer physics of the passing electron solution (5.12) for $\ell = 0$, which gives

(7.1)\begin{align}f_1^{\textrm{pass}}\mathop \to \limits_{\ell = 0} {\Sigma _m}\frac{{{\textrm{e}^{ - \textrm{i}m\vartheta }}{{\left. {\int_{ - {\tau_f}}^0 {\textrm{d}\tau } {W_m}} \right|}_{\lambda = {\lambda _{\textrm{res}}}}}}}{{(\omega \partial {\tau _f}/\partial \lambda )w}}\int_0^\infty {\textrm{d}t} \,{\textrm{e}^{\textrm{i}ut - {t^3}/3}} = {\rm \pi}{\Sigma _m}\frac{{{\textrm{e}^{ - \textrm{i}m\vartheta }}{{\left. {\int_{ - {\tau_f}}^0 {\textrm{d}\tau } {W_m}} \right|}_{\lambda = {\lambda _{\textrm{res}}}}}}}{{(\omega \partial {\tau _f}/\partial \lambda )}}U(u).\end{align}

This form of the solution suggests a satisfactory modification to the Krook model to account for the resonance shift due to the mode coupling index $\ell$, is found by letting $qn - m \to qn - m - \ell$ in (5.2). Then the mode couple shifted resonance occurs at

(7.2)\begin{equation}\tau _f^\ell = 2{\rm \pi}\sigma (qn - m - \ell )/\omega \ge 2{\rm \pi} qR/v,\end{equation}

and the expansion about the resonant pitch angle $\lambda _{\textrm{res}}^\ell$ becomes

(7.3)\begin{equation}{\tau _f} = \oint_f {\textrm{d}\tau } = \tau _f^\ell + (\lambda - \lambda _{\textrm{res}}^\ell )\partial {\tau _f}/\partial \lambda {|_\ell } + \cdots .\end{equation}

Consequently, neglecting the ${\nu _{\textrm{eff}}}$ in the numerator, since collisions are negligible for a single poloidal transit, the linearized passing solution with mode coupling retained might be expected to have close to the following behaviour:

(7.4)\begin{equation}f_1^{\textrm{pass}} \approx {\rm \pi}{\Sigma _{m,\ell }}\frac{{{\textrm{e}^{ - \textrm{i}m\vartheta }}{{\left. {\int_{ - {\tau_f}}^0 {\textrm{d}\tau } {W_m}(\tau )\,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}} \right|}_{\lambda = \lambda _{\textrm{res}}^\ell }}}}{{\omega \partial {\tau _f}/\partial \lambda {|_\ell }}}U({u_\ell }),\end{equation}

with $U({u_\ell }) = {({\rm \pi}{w_\ell })^{ - 1}}\int_0^\infty {\textrm{d}t} \,{\textrm{e}^{ - {t^3}/3 - \textrm{i}{u_\ell }t}}$, ${w_\ell } = {(\nu \tau _f^\ell )^{1/3}}/{(\omega \partial {\tau _f}/\partial \lambda {|_{\lambda = \lambda _{\textrm{res}}^\ell }})^{1/3}}$, $P({u_\ell }) = Re U({u_\ell })$ and ${u_\ell } = (\lambda - \lambda _{\textrm{res}}^\ell )/{w_\ell }$.

The pitch angle variation due to k allows additional $\ell \ne 0$ resonances in (7.2). Defining $k_{||}^\ell = (qn - m - \ell )/qR$, the collisionless mode coupled resonances are then at

(7.5)\begin{equation}\frac{{|k_{||}^\ell |v}}{\omega } = \frac{{\sqrt {k_\ell ^2 + 2\epsilon } K({k_\ell })}}{{({\rm \pi}/2)\sqrt {2\epsilon } }} \to \left\{ {\begin{array}{*{20}{@{}ll}} {{{[(k_\ell^2 + 2\epsilon )/2\epsilon ]}^{1/2}}}& {k_\ell^2\lesssim 2\epsilon }\\ {\ell n[16/(1 - k_\ell^2)]/{\rm \pi}\sqrt {2\epsilon } }& {k_\ell^2 \to 1} \end{array}} \right.,\end{equation}

where $1 \le |k_{||}^\ell |v/\omega < \infty$, making $\ell = \ell ({k_\ell })$ with

(7.6)\begin{equation}k_\ell ^2 = 2\epsilon \lambda _{\textrm{res}}^\ell /[1 - (1 - \epsilon )\lambda _{\textrm{res}}^\ell ].\end{equation}

For example, if $\ell ({k_0} = 0) = 0$ at $|k_{||}^0|v/\omega = 1$, then there can be other resonances at $\ell ({k_\ell } \ne 0) \ne 0$. Notice no passing resonances occur for $|k_{||}^\ell |v/\omega \le 1$.

To clarify the notation, consider an applied rf wave driving a positive current. The rf has to act on the ${v_{||}} < 0$ passing electrons, requiring $\tau _f^\ell = 2{\rm \pi}(m - qn + \ell )/\omega > 0$. As a result, for $k_\ell ^2\lesssim 2\epsilon$, $\tau _f^\ell = 4qR{(k_\ell ^2 + 2\epsilon )^{1/2}}/v\sqrt {2\epsilon } \ge \tau _f^0$. Also, ${v_{||}} < 0$ means $\textrm{d}\vartheta < 0$ to keep $\textrm{d}\tau \propto \textrm{d}\vartheta /{v_{||}} > 0$, making $\vartheta (\tau ) - \vartheta < 0$. Letting $\varTheta = \vartheta (\tau )$ to simplify the notation, then $\tau = qR{{\int_\vartheta ^\varTheta {\textrm{d}\vartheta ^{\prime}} } / {{v_{||}}(\vartheta ^{\prime};\lambda _{\textrm{res}}^\ell )}}\;$ giving

(7.7)\begin{align} & \int_{ - {\tau _f}}^0 {\textrm{d}\tau } {W_m}(\tau )\,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}{|_{\lambda = \lambda _{\textrm{res}}^\ell }}\notag\\ & \quad =- qR\int_\vartheta ^{\vartheta + 2{\rm \pi}} {\textrm{d}\varTheta } \dfrac{{{W_m}(\varTheta )}}{{{v_{||}}(\varTheta ;\lambda _{\textrm{res}}^\ell )}}\,{\textrm{e}^{\textrm{i}\omega qR\int_\varTheta ^\vartheta {\textrm{d}\vartheta ^{\prime}} /{v_{||}}(\vartheta ^{\prime};\lambda _{\textrm{res}}^\ell ) - \textrm{i}(m - qn)(\vartheta - \varTheta )}}{|_{\lambda = \lambda _{\textrm{res}}^\ell }}. \end{align}

This form retains poloidal mode coupling, which occurs since ${v_{||}}$ cannot be constant in a tokamak so the phase factor under the integral cannot vanish as it does for a uniform magnetic field. It uses the resonance condition to write $v_{||}^2(\varTheta ;\lambda _{\textrm{res}}^\ell ) = {v^2}[1 - \lambda _{\textrm{res}}^\ell B(\varTheta )/{B_0}]$. A ${v_{||}} < 0$ electron starts from $\vartheta + 2{\rm \pi}$ and moves to $\vartheta$ in the time ${\tau _f}$ for a passing poloidal circuit.

The solution (6.3) is also valid for the trapped electrons when ${\tau _f}$ is the time for a full bounce and $\sigma = 0$. When the integration in denominator is also over a full bounce

(7.8)\begin{equation}f_1^{\textrm{trap}} = {\Sigma _m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}\frac{{\int_{ - {\tau _f}}^0 {\textrm{d}\tau } {W_m}(\tau )\,{\textrm{e}^{ - (\textrm{i}\omega - {\nu _{\textrm{eff}}})\tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}}}{{1 - {\textrm{e}^{(\textrm{i}\omega - {\nu _{\textrm{eff}}}){\tau _f}}}}}.\end{equation}

Recalling the trapped form of ${\tau _f}$ and using $1 = {\textrm{e}^{\textrm{i}2{\rm \pi}\ell }}$, a collisionless resonance requires

(7.9)\begin{equation}\omega {\tau _f} = \frac{{8\omega qR}}{{v\sqrt {2\epsilon } }}K(\kappa ) = 2{\rm \pi}\ell \ge \frac{{4{\rm \pi}\omega qR}}{{v\sqrt {2\epsilon } }}.\end{equation}

The high frequency limit is made tractable by assuming $\omega {\tau _f}/2{\rm \pi} = \ell \gg 1$. Then integration by parts and the periodicity of the trapped motion over a full bounce leads to

(7.10)\begin{equation}f_1^{\textrm{trap}} \approx \textrm{i}{\Sigma _m}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}{W_m}/\omega + \cdots .\end{equation}

The same result is obtained by solving the high-frequency limit of (2.5).

In the low frequency limit $\omega {\tau _f}/2{\rm \pi} \ll 1$ there is no resonance and the trapped response remains small since

(7.11)\begin{equation}f_1^{\textrm{trap}} \approx \frac{{\textrm{i}q{{Re }^{ - \textrm{i}qn\vartheta }}}}{{\omega {\tau _f}}}{\Sigma _m}\oint {\textrm{d}\varTheta } {W_m}(\varTheta )\,{\textrm{e}^{\textrm{i}(qn - m)\varTheta }}/{v_{||}}(\varTheta )\mathop \to \limits_{|qn - m|\gg 1} 0.\end{equation}

Consequently, only for low $(\ell \sim 1)$ mode coupling indices can the trapped have a significant response. To simplify the presentation in subsequent sections, $f_1^{\textrm{trap}} \approx 0$ is assumed from here on. The analysis in the next section adds further support to this assumption.

8. The effect of the collisional boundary layer at the trapped–passing boundary

It is not possible to obtain a rigorous analytic solution retaining pitch angle scattering collisions as well as poloidal mode coupling associated with the mode coupling index $\ell$ of (6.3). In fact, even the approximate passing solution already presented in § 7 is incomplete. It does not account for the trapped–passing boundary where $f_1^{\textrm{pass}}$ must vanish.

In this section, the passing solution is reconsidered for a resonance near the trapped–passing boundary where ${\tau _f} \propto K(k) \propto- \ell n(1 - {k^2})$ causes the Taylor expansion to fail since $\partial {\tau _f}/\partial \lambda \to 1/[2\epsilon (1 - {k^2})]$. Recall from § 4 that the collisionless resonance condition for the barely passing electrons $({k^2} \to 1)$ in the absence of poloidal mode coupling is

(8.1)\begin{equation}\frac{{|{k_{||}}|v}}{\omega } = \frac{{|qn - m|v}}{{\omega qR}} \approx- \frac{{\ell n(1 - {k^2})}}{{{\rm \pi}\sqrt {2\epsilon } }},\end{equation}

since $\oint_f {\textrm{d}\tau } \approx- (2qR/v\sqrt {2\epsilon } )\ell n(1 - {k^2})$. The trapped–passing boundary layer requires very large $|{k_{||}}|$ at resonance (a large $|{k_{||}}|$ upshift) due to resonant m being very different in trapped–passing boundary layer than in the more freely passing boundary layers.

Using $2\epsilon \,\textrm{d}{k^2} \approx \textrm{d}\lambda$, the barely passing limit of the collision operator becomes

(8.2)\begin{equation}\oint_f {\textrm{d}\tau C} \approx \frac{{8qR{\nu _e}}}{{{{(2\epsilon )}^{3/2}}v{x^3}}}\frac{{{\partial ^2}{f_m}}}{{\partial {{({k^2})}^2}}}.\end{equation}

Therefore, the boundary layer limit of (5.1)–(5.4) yields

(8.3)\begin{align}- \textrm{i}[(2\omega qR/v\sqrt {2\epsilon } )\ell n(1 - {k^2}) + 2{\rm \pi}\sigma (qn - m)]{f_m} + \frac{{8qR{\nu _e}}}{{{{(2\epsilon )}^{3/2}}v{x^3}}}\frac{{{\partial ^2}{f_m}}}{{\partial {{({k^2})}^2}}} \approx- \oint_f {\textrm{d}\tau {W_m}} ,\end{align}

with $\sigma (qn - m) > 0$ required for a passing resonance. Letting

(8.4)\begin{equation}\gamma = {e^\Pi },\end{equation}

with

(8.5)\begin{equation}\Pi = \frac{{{\rm \pi}\sqrt {2\epsilon } \sigma (qn - m)v}}{{\omega qR}}\gtrsim 1,\end{equation}

the preceding boundary layer equation becomes

(8.6)\begin{equation}\frac{{2{\nu _e}}}{{\epsilon \omega {x^3}}}\frac{{{\partial ^2}{f_m}}}{{\partial {{({k^2})}^2}}} - \textrm{i}\ell n[\gamma (1 - {k^2})]{f_m} =- \frac{{v\sqrt {2\epsilon } }}{{2\omega qR}}\oint_f {\textrm{d}\tau } {W_m}.\end{equation}

This form of the kinetic equation allows two types of collisional responses. The first response is the one already considered. It is sometimes referred to as a resonant plateau response, indicating it will give a result independent of collision frequency upon integration over pitch angle (as in the plateau regime of neoclassical theory). The second response is collisionally dependent and proportional to $\sqrt {{\nu _e}}$. To verify this behaviour analytically, $\gamma \gg 1$ must be assumed. Then $y = \gamma (1 - {k^2})$ can be expanded about y = 1 while keeping ${k^2} \approx 1$. Keeping only $\ell n(y) \approx y - 1$, and letting $z = (y - 1)/\delta$, with

(8.7)\begin{equation}\delta = {\gamma ^{2/3}}{x^{ - 1}}{(2{\nu _e}/\epsilon \omega )^{1/3}} \ll 1,\end{equation}

then

(8.8)\begin{equation}\frac{{{\partial ^2}{f_m}}}{{\partial {z^2}}} - \textrm{i}z{f_m} =- \frac{{v\sqrt {2\epsilon } }}{{2\omega qR\delta }}\oint_f {\textrm{d}\tau } {W_m}.\end{equation}

To solve, define

(8.9)\begin{equation}{f_m} = \left( {\frac{{v\sqrt {2\epsilon } }}{{2\omega qR\delta }}\oint_f {\textrm{d}\tau {W_m}} } \right)\varUpsilon (z),\end{equation}

where $\varUpsilon (z)$ is the desired solution to the more familiar inhomogeneous Airy equation

(8.10)\begin{equation}{\partial ^2}\varUpsilon /\partial {z^2} - \textrm{i}z\varUpsilon =- 1.\end{equation}

In the absence of a trapped–passing boundary condition needing to be satisfied, the solution is as before, namely the nearly freely passing resonant plateau (rp) solution

(8.11)\begin{equation}{\varUpsilon _{rp}} = \int_0^\infty {\textrm{d}\tau } \,{\textrm{e}^{ - \textrm{i}z\tau - {\tau ^3}/3}}\mathop \to \limits_{|z|\gg 1} \frac{{ - \textrm{i}}}{z} = \frac{{ - \textrm{i}\delta }}{{y - 1}}.\end{equation}

To make $\varUpsilon$ vanish at the trapped–passing boundary $z =- 1/\delta$, as well as at $|z|\gg 1$, a homogeneous solution is required. Proceeding as in Catto, Tolman & Parra (Reference Catto, Tolman and Parra2023), but for the passing rather than the trapped, gives the desired solution

(8.12)\begin{equation}\varUpsilon = {\varUpsilon _{rp}}(z) - {\varUpsilon _{rp}}(z =- 1/\delta )Ai(z\,{\textrm{e}^{\mathrm{i}{\rm \pi}/6}})/Ai( - {\textrm{e}^{\mathrm{i}{\rm \pi}/6}}/\delta ) \equiv {\varUpsilon _{rp}}(z) + {\varUpsilon _{\surd \nu }}(z),\end{equation}

where the Ai are Airy functions (Abramowitz & Stegun Reference Abramowitz and Stegun1964). The ${\varUpsilon _{\surd \nu }}(z)$ contribution arises from the need to make the passing response vanish at the trapped–passing boundary.

When $\varUpsilon$ is integrated over pitch angle, or more conveniently z, and exponentially small terms are ignored because $\delta \ll 1$, then

(8.13)\begin{equation}Re \int_{ - 1/\delta }^{(\gamma - 1)/\delta } {\textrm{d}z} {\varUpsilon _{\sqrt \nu }}(z) \approx \frac{{{\delta ^{3/2}}}}{{{2^{1/2}}}},\end{equation}

with ${\delta ^{3/2}} = \gamma {(2{\nu _e}/\epsilon \omega {x^3})^{1/2}}$ and ${x^3} > 1$ (due to the high-speed expansion of C), and

(8.14)\begin{equation}Re \int_{ - 1/\delta }^{(\gamma - 1)/\delta } {\textrm{d}z} \varUpsilon (z) = {\rm \pi}\left( {1 + \frac{{{\delta^{3/2}}}}{{{2^{1/2}}{\rm \pi}}}} \right).\end{equation}

As a result, the $\sqrt {{\nu _e}}$ term from the trapped–passing boundary condition is expected to give a small contribution in the core. The trapped are expected to give similar $\sqrt {{v _e}}$. However, at the edge of a tokamak, larger $\delta \sim 1$ may occur so the analysis here may need to be extended. In the earlier sections, the $\sqrt {{\nu _e}}$ contribution has been treated as negligible by assuming $\delta \ll 1$.

The preceding sections suggest the KINETIC-J code (Green & Berry Reference Green and Berry2014) can be improved by using (6.3) as the solution, along with the passing particle replacement

(8.15)\begin{equation}\frac{\textrm{i}}{{(\omega + \textrm{i}{\nu _{\textrm{eff}}}){\tau _f} - \textrm{i}2{\rm \pi}\sigma (qn - m - \ell )}} \to \frac{1}{{{\rm \pi} w_{\textrm{res}}^\ell }}\int_0^\infty {\textrm{d}t} \,{\textrm{e}^{\textrm{i}{u_\ell }t - {t^3}/3}},\end{equation}

to remove any undesirable singular behaviour. Here, ${u_\ell } = [\omega {\tau _f} - \textrm{i}2{\rm \pi}\sigma (qn - m - \ell )]/w_{\textrm{res}}^\ell$ with $w_{\textrm{res}}^\ell \equiv {w_\ell }\omega \partial {\tau _f}/\partial \lambda {|_{\lambda = \lambda _{\textrm{res}}^\ell }}$ except very near the trapped–passing boundary.

9. Comments on electron (and ion) cyclotron heated plasmas

The material in the preceding sections focuses on the lower hybrid and helicon wave regimes for simplicity. However, cyclotron resonances are only slightly more involved for the electrons. For the ions similar treatments are also valid, although finite drift departures from flux surfaces may add complications. In this section, only a simple electron cyclotron resonance is considered as an illustration.

To simplify the treatment further, effects associated with the trapped–passing boundary condition are ignored and large aspect ratio is assumed. Moreover, on axis heating is assumed by letting $\omega = {\varOmega _0} = e{B_0}/mc$ to write

(9.1)\begin{equation}\varOmega = {\varOmega _0}(1 - \epsilon \cos \vartheta ) = \omega (1 - \epsilon \cos \vartheta ).\end{equation}

Then the collisionless resonance conditions for the passing and trapped electrons from

(9.2)\begin{equation}\oint_f {\textrm{d}\tau } (\omega - \varOmega - {k_{||}}{v_{||}}) = \oint_f {\textrm{d}\tau } (\epsilon \omega \cos \vartheta - {k_{||}}{v_{||}}) = 0,\end{equation}

are

(9.3)\begin{align}2{\rm \pi}\sigma (qn - m) = \epsilon \omega \oint_f {\textrm{d}\tau } \cos \vartheta = \frac{{4\epsilon \omega qR}}{{v\sqrt {2\epsilon } }}\left\{ {\begin{array}{*{20}{@{}ll@{}}} {{k^{ - 2}}\sqrt {({k^2} + 2\epsilon )} [{2E(k) - (2 - {k^2})K(k)} ]}& {\textrm{passing}}\\ {2[2E(\kappa ) - K(\kappa )]}& {\textrm{trapped}} \end{array}} \right..\end{align}

The passing condition for a resonance is slightly more involved than before, but once again Taylor expansions about the resonant $\lambda$ or k can be employed to find behaviour similar to what has already been discussed.

At first glance it might appear there is not a resonance for the trapped, however, $2E(\kappa ) = K(\kappa )$ at $\kappa = 0.91$ to give resonant plateau behaviour (Tolman & Catto Reference Tolman and Catto2021). Consequently, the trapped are heated, but they are unable to drive current as ${v_{||}}$ does not enter. Poloidal mode coupling will occur for the trapped as well as the passing as before.

Similar behaviour will occur for ion cyclotron heated ions, with magnetic drifts included in the resonance to retain finite drift departures off flux surfaces. The transit averaging of the wave–particle resonances may alter some of the velocity space structure observed in AORSA simulations (Jaeger et al. Reference Jaeger, Harvey, Berry, Myra, Dumont, Phillips, Smithe, Barret, Batchelor, Bonoli, Carter, D'azevedo, D'ippolito, Moore and Wright2006, Reference Jaeger, Berry, D'azevedo, Barret, Ahern, Swain, Batchelor, Harvey, Myra, D'ippolito, Phillips, Valeo, Smithe, Bonoli, Wright and Choi2008).

10. Improved quasilinear operators

The general expression for the rf quasilinear operator retaining the full transit averaged correlated resonance condition and including poloidal mode coupling was derived by Catto & Tolman (Reference Catto and Tolman2021a) in the resonant plateau limit and is given by their (7.11)–(7.13). It leads to the correct entropy production as shown by their (7.14). In the lower hybrid and helicon limit considered here, for a single applied wave frequency and toroidal mode number, their quasilinear operator acting on a Maxwellian ${f_0}$ has the form

(10.1)\begin{equation}Q\{ {\,f_0}\} = {\Sigma _{m,S}}\frac{1}{{v{\tau _f}}}\frac{\partial }{{\partial v}}\left( {v{\tau_f}D\frac{{\partial {f_0}}}{{\partial v}}} \right),\end{equation}

with the quasilinear diffusivity D given by

(10.2)\begin{align} D & = \dfrac{{{\rm \pi}{e^2}}}{{2{m^2}{v^2}{\tau _f}}}{\Sigma _\ell }\; \delta \left({\oint_f {\textrm{d}\tau \varLambda } - 2{\rm \pi}\sigma \ell } \right)\left|{{J_0}(\eta ){\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}\int_{ - {\tau_f}}^0 {\textrm{d}\tau } {v_{||}}\,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}} \right.\notag\\ & \quad + \left. \textrm{i}{J_1}(\eta )\dfrac{{{\lambda^{1/2}}}}{{{k_ \bot }}}{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{k} \times \boldsymbol{n}\int_{ - {\tau_f}}^0 {\textrm{d}\tau } v\,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}\right|^2, \end{align}

for ${v_ \bot } \approx {\lambda ^{1/2}}v$ and $\eta \approx {k_ \bot }{\lambda ^{1/2}}v/{\varOmega _0}$. The subscripts on the sums indicate poloidal (m) and radial (S) modes are to be summed over, and for each m, the $\ell$ modes it couples to must also be summed over. In addition, $\oint_f {\textrm{d}\tau \varLambda } = \omega {\tau _f} - 2{\rm \pi}\sigma (qn - m)$, and the trapped electron response has been assumed negligible based on the estimates of § 7. If the lowest electron cyclotron resonance is retained then $\oint_f {\textrm{d}\tau \varLambda } = \oint_f {\textrm{d}\tau } (\omega - \varOmega ) - 2{\rm \pi}\sigma (qn - m)$, and the term inside $|\cdots {|^2}$ needs to be generalized as found by Catto & Tolman (Reference Catto and Tolman2021a).

The singular behaviour of the delta function can be removed by the replacement

(10.3) \begin{equation}\delta \left({\oint_f {\textrm{d}\tau \varLambda } - 2{\rm \pi}\ell } \right)\to \; \frac{{\int_0^\infty {\textrm{d}t} \,{\textrm{e}^{ - {t^3}/3}}\cos ({u_\ell }t)}}{{{{ {{{{\rm \pi}{w_\ell }\partial ({\oint_f {\textrm{d}\tau \varLambda } } )} / {\partial \lambda }}} |}_{\lambda = \lambda _{\textrm{res}}^\ell }}}},\end{equation}

with

(10.4)\begin{equation}P({u_\ell }) = {({\rm \pi}{w_\ell })^{ - 1}}\int_0^\infty {\textrm{d}t} \,{\textrm{e}^{ - {t^3}/3}}\cos ({u_\ell }t),\end{equation}

acting as a delta function when integrated over pitch angle. Here, ${u_\ell } = (\lambda - \lambda _{\textrm{res}}^\ell )/{w_\ell }$ and ${w_\ell } = {(\nu \tau _f^\ell )^{1/3}}/{[\partial (\oint_f {\textrm{d}\tau \varLambda } )/\partial {\lambda _{\lambda = \lambda _{\textrm{res}}^\ell }}]^{1/3}}$. For each $\ell$ the preceding expressions give

(10.5) \begin{equation}\int {\textrm{d}\lambda } \delta \left({\oint_f {\textrm{d}\tau \varLambda } - 2{\rm \pi}\ell } \right)\to \; 1 / \partial \left(\oint_f \textrm{d}\tau \varLambda \right) / \partial \lambda |_{\lambda = \lambda _{\textrm{res}}^\ell },\end{equation}

as $\int {\textrm{d}\lambda } P({u_\ell }) \to 1$ when integrated over $\lambda$ region about $\lambda _{\textrm{res}}^\ell$ a few ${w_\ell }$ wide. The replacement (10.3) is valid away from the trapped–passing boundary. This substitution improves on the one suggested in Tolman & Catto (Reference Tolman and Catto2021) by more precisely accounting for tokamak geometry. In the vicinity of the trapped–passing boundary the resonant plateau response ${\varUpsilon _{rp}}$ has to be replaced by the full response ${\varUpsilon _{rp}} + {\varUpsilon _{\surd \nu }}$, as discussed in § 6.

11. Improved linear response to include poloidal mode coupling

Evaluating the perturbed density and current in tokamak geometry is complicated by the presence of the $\tau$ or $\varTheta$ integral associated with the trajectory integration over the near periodic motion in (7.4). However, some simplification is possible since the trapped are expected to be unimportant and only ${\mathop{\rm Im}\nolimits} U({u_\ell }) = P({u_\ell })$ survives the $\lambda$ integration over ${\textrm{d}^3}v \to B{v^2}\,\textrm{d}v\,\textrm{d}\lambda \,\textrm{d}\varphi /{B_0}\sqrt {1 - \lambda B/{B_0}}$. Moreover, since $P({u_\ell })$ behaves as a delta function to lowest order the replacement $P({u_\ell }) \to \delta (\lambda - \lambda _{\textrm{res}}^\ell )$ can be employed. For example, using (7.4), the perturbed charge density and parallel current are given by

(11.1)\begin{align}e\int_{\textrm{pass}} {{\textrm{d}^3}vv_{||}^jf_1^{\textrm{pass}}} \approx {\rm \pi}e{\Sigma _{m,\ell }}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}\int_{\textrm{pass}} {{\textrm{d}^3}vv_{||}^j} \frac{{\delta (\lambda - \lambda _{\textrm{res}}^\ell )}}{{\omega \partial {\tau _f}/\partial \lambda {|_\ell }}}\int_{ - {\tau _f}}^0 {\textrm{d}\tau {W_m}(\tau )} \,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}},\end{align}

with j = 0 or 1 and where, by approximating ${v_ \bot } \approx {\lambda ^{1/2}}v$ and $\eta \approx {k_ \bot }{\lambda ^{1/2}}v/{\varOmega _0}$, their poloidal angle dependence can be ignored. In addition, the gyrophase integral can be performed to recover another Bessel function from the ${\textrm{e}^{ - \textrm{i}L}}$ factor in ${W_m}$ to find

(11.2)\begin{align} & e\int_{\textrm{pass}} {{\textrm{d}^\textrm{3}}vv_{||}^j} f_1^{\textrm{pass}} \approx- \dfrac{{{\rm \pi}{e^2}}}{T}{\Sigma _{m,\ell }}\,{\textrm{e}^{ - \textrm{i}m\vartheta }}\int_{\textrm{pass}} {{\textrm{d}^3}vv_{||}^j} {f_0}\dfrac{{\delta (\lambda - \lambda _{\textrm{res}}^\ell )}}{{\omega \partial {\tau _f}/\partial \lambda {|_\ell }}}{J_0}(\eta )\{{{J_0}(\eta ){\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}} \notag\\ & \quad \left. {\int_{ - {\tau_f}}^0 {\textrm{d}\tau {v_{||}}} \,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}} + \textrm{i}{J_1}(\eta )\dfrac{{{\lambda^{1/2}}}}{{{k_ \bot }}}{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{k} \times \boldsymbol{n}\int_{ - {\tau_f}}^0 {\textrm{d}\tau v} \,{\textrm{e}^{ - \textrm{i}\omega \tau + \textrm{i}(qn - m)[\vartheta (\tau ) - \vartheta ]}}} \right\}. \end{align}

Due to the presence of the delta function, only a speed integral over $v$ and the trajectory integral over a period need be performed, as $\int {\textrm{d}\varphi } = 2{\rm \pi}$. The speed integral could be performed in the usual manner to obtain a modified Bessel function form if it were not for the exponential coupling factor. Its existence highlights the need to sum over all the poloidally coupled modes as indicated by the $\ell$ sum for each m summed. A similar expression holds for the perpendicular flow, but with the overall ${J_0}(\eta )$ multiplier in front of $\{\cdots\}$ in the j = 0 form of (11.2) replaced by $\textrm{i}{v_ \bot }k_ \bot ^{ - 1}{J_1}(\eta )\boldsymbol{k} \times \boldsymbol{n}$.

Once the replacement $P({u_\ell }) \to \delta (\lambda - \lambda _{\textrm{res}}^\ell )$ is made, the collision frequency no longer appears. Its absence does not mean rf quasilinear theory is collisionless. Instead, the absence of the collision frequency is a signature of the resonant plateau behaviour of the boundary layers associated with the wave–particle resonance. Fortunately, the details of the boundary layers about the trapped–passing boundary seem to be relatively unimportant as long as (8.7) is satisfied.

12. Limitations of a quasilinear treatment

It is possible to make some simple estimates of when quasilinear treatments of rf might fail (Catto Reference Catto2020; Catto & Tolman Reference Catto and Tolman2021a,Reference Catto and Tolmanb). To begin, notice that the quasilinear operator will compete with the collision operator to make ${f_0}$ depart from Maxwellian when

(12.1)\begin{equation}Q\{ {\,f_0}\} /C\{ {\,f_0}\} \sim D/v_e^2\nu \sim 1.\end{equation}

Recalling from (5.11) that ${\nu _{\textrm{eff}}}\sim \nu {(\omega /\nu )^{2/3}}$, $\delta (\oint_f {\textrm{d}\tau \varLambda } - 2{\rm \pi}\ell )\sim \omega /{\nu _{\textrm{eff}}}\sim {(\omega /\nu )^{1/3}}$ giving

(12.2)\begin{equation}D\sim {(e|{\boldsymbol{e}_m}|/m)^2}\; \delta {{\left({\oint_f {\textrm{d}\tau \varLambda } - 2{\rm \pi}\ell } \right)} / \omega }\sim {(e|{\boldsymbol{e}_m}|/m)^2}/{\nu _{\textrm{eff}}}.\end{equation}

Therefore, substantial distortion is expected when

(12.3)\begin{equation}e|{\boldsymbol{e}_m}|/m{v_e}\sim \nu {(\omega /\nu )^{1/3}}.\end{equation}

Quasilinear theory assumes

(12.4)\begin{equation}1 \gg \frac{{|{\nabla _v}{f_1}|}}{{|{\nabla _v}{f_0}|}}\sim \frac{{{f_1}/w{v_e}}}{{{f_0}/{v_e}}},\end{equation}

where $w\sim {(\nu /\omega )^{1/3}} \ll 1$ from § 5. Therefore, it requires

(12.5)\begin{equation}{f_1}/{f_0} \ll {(\nu /\omega )^{1/3}}.\end{equation}

However, from § 7

(12.6)\begin{equation}{f_1}\sim \frac{{{W_m}}}{\omega }\delta (\lambda - {\lambda _{\textrm{res}}})\sim \; \frac{{e|{\boldsymbol{e}_m}|{\,f_0}}}{{m{v_e}\omega w}},\end{equation}

requiring

(12.7)\begin{equation}e|{\boldsymbol{e}_m}|/m{v_e} \ll \nu {(\omega /\nu )^{1/3}},\end{equation}

and, thereby, a negligible distortion of the Maxwellian according to (12.1).

These estimates suggest that quasilinear theory is failing once the unperturbed distribution function becomes significantly distorted from Maxwellian (as it does in most non-adjoint treatments of rf). When this happens the nonlinear term needs to be retained in the perturbed kinetic equation to account for island formation.