2. Background

In a tokamak the transit averaged QL operator for electrons, when the applied wave field is at a frequency $\omega$ much smaller than the electron cyclotron frequency ${\varOmega _e} = eB/{m_e}c$ and the unperturbed electron distribution function is nearly the Maxwellian ${f_0}$, is

(2.1)\begin{equation}\bar{Q}\{ {f_0}\} = \mathop \sum \limits_{\boldsymbol{k}} \frac{1}{{{\tau _f}}}\frac{\partial }{{\partial E}}\left( {{\tau_f}{v^2}D\frac{{\partial {f_0}}}{{\partial E}}} \right),\end{equation}

with ${\tau _f} = \oint_f {\textrm{d}\tau }$ the time for a full ( f) poloidal passing $(\sigma = {v_{||}}/|{v_{||}}| = \pm 1)$ or trapped $(\sigma = 0)$ poloidal circuit, the sum over the applied rf wave vectors $\boldsymbol{k}$, and D the velocity space diffusivity (Catto & Tolman Reference Catto and Tolman2021a). Here e is the charge on a proton, c is the speed of light, ${m_e}$ is the electron mass and $E = v^{2}/2 - e\varPhi/m_{e}$ is the total energy, with $\varPhi$ the electrostatic potential, $v = |\boldsymbol{v}|$ the electron speed and $v_{||} = \boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{v}$ the parallel electron velocity along the tokamak magnetic field $\boldsymbol{B} = B\boldsymbol{n} = I\boldsymbol{\nabla }\zeta + \boldsymbol{\nabla }\zeta \times \boldsymbol{\nabla }\psi \;$. The unit vector along the magnetic field is $\boldsymbol{n}$, $\zeta \;$ is the toroidal angle variable, $\psi$ is the poloidal flux function and the flux function $I(\psi )$ is $I = R{B_t}$, where ${B_t}$ is the toroidal magnetic field, R is the major radius and ${B_p}$ is the poloidal magnetic field in $|\boldsymbol{\nabla }\psi |= R{B_p}$. The poloidal angle $\vartheta$ satisfies $\boldsymbol{\nabla }\zeta \boldsymbol{\cdot }\boldsymbol{\nabla }\psi = 0 = \boldsymbol{\nabla }\zeta \boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta$ and is chosen such that $\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta = |I|/q{R^2} = {q^{ - 1}}|\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\zeta |$, making the safety factor, $q = q(\psi )$, a flux function. Taking the toroidal current to be in the $\boldsymbol{\nabla }\zeta$ direction makes ${B_p} > 0$, $\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta > 0$ and $\vartheta$ increases in the ${\boldsymbol{B}_p} = \boldsymbol{\nabla }\zeta \times \boldsymbol{\nabla }\psi$ direction. Then the incremental time along a trajectory is $\textrm{d}{\tau} = \textrm{d}\vartheta/v_{||}\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla}\vartheta > 0$, with $\textrm{d}\vartheta$ and $v_{||}$ reversing signs together when a trapped electron reflects, such that $\textrm{d}\vartheta < 0$ for a passing electron with ${v_{||}} < 0$.

The QL diffusivity D in tokamak geometry for successive correlated interactions through the applied rf (Catto & Tolman 2021) is

(2.2)\begin{equation}D = \frac{{\mathrm{\pi }{e^2}}}{{2m_e^2{v^2}{\tau _f}}}\sum\limits_\ell {\delta ({\oint_f {\textrm{d}\tau \chi - 2\mathrm{\pi }\ell } } )} {\left|{\oint_f {\textrm{d}\tau {\boldsymbol{e}_k}\boldsymbol{\cdot }\left[ {\boldsymbol{n}{v_{||}}{J_0}(\eta ) - i\boldsymbol{n} \times \boldsymbol{k}\frac{{{v_ \bot }}}{{{k_ \bot }}}{J_1}(\eta )} \right]\,{\textrm{e}^{ - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\chi (\tau^{\prime})} }}} } \right|^2},\end{equation}

with the transit averaged resonance condition defined by

(2.3)\begin{equation}\oint_f {\textrm{d}\tau \chi } = \omega {\tau _f} - 2\mathrm{\pi }\sigma (qn - m),\end{equation}

and the exponential phase factor given by

(2.4)\begin{equation}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\chi (\tau^{\prime})}=\omega(\tau-\tau_{0})-\sigma(qn-m)\vartheta(\tau),\end{equation}

where ${\tau _0}$ is taken to be the trajectory time at the equatorial plane crossing where B is a minimum. In the preceding and what follows ${\boldsymbol{e}_k}$ is the Fourier amplitude of the applied electric field of wave vector $\boldsymbol{k} = {\boldsymbol{k}_ \bot } + {k_{||}}\boldsymbol{n}$ having ${\boldsymbol{k}_ \bot } = {k_ \bot }(\boldsymbol{\psi }\cos \varsigma + \boldsymbol{p}\sin \varsigma )$, ${k_{||}} = (qn - m)/qR$ and $\eta = {k_ \bot }{v_ \bot }/{\varOmega _e}$, with $\boldsymbol{\psi } = \boldsymbol{\nabla }\psi /|\boldsymbol{\nabla }\psi |= \boldsymbol{\nabla }\psi /R{B_p}$, and $\boldsymbol{p} = \boldsymbol{\nabla }\zeta \times \boldsymbol{\nabla }\psi /|\boldsymbol{\nabla }\zeta \times \boldsymbol{\nabla }\psi |= {\boldsymbol{B}_p}/{B_p}$. The orthonormal unit vectors satisfy $\boldsymbol{\psi } \times \boldsymbol{p} = \boldsymbol{n}$, n and m are the toroidal and poloidal mode numbers, respectively, and large aspect ratio is assumed to write $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta \approx 1/qR$. The integer $\ell$ denotes the resonant path in velocity space as electrons can experience other (usually less important) resonances besides $\ell = 0$ in toroidal geometry as their velocity changes along a field line as they cannot remain in resonance indefinitely in an inhomogeneous magnetic field. Indeed, they can have other resonant interactions by crossing the various resonant paths or curves in velocity space defined by $\oint_f {\textrm{d}\tau \chi } = 2\mathrm{\pi }\ell$.

For LHCD the applied rf fields are tailored to make the first or ${v_{||}}{J_0}$ term dominate. For HCD the rf fields must be applied in a manner that makes the second or ${v_ \bot }{J_1}$ term dominate (Prater et al. Reference Prater, Moeller, Pinsker, Porkolab, Meneghini and Vdovin2014; Pinsker Reference Pinsker2015; Pinsker et al. Reference Pinsker, Prater, Moeller, Degrassie, Petty, Porkolab, Anderson, Garofalo, Lau, Nagy, Pace, Torreblanca, Watkins and Zeng2018; Yin et al. Reference Yin, Zheng, Gong, Yang, Yin, Song, Huang, Chen and Zhong2022), so ${J_1}$ will not be expanded by assuming its argument is small. Of course, both HCD and LHCD rely on there being a Landau resonance. Here the goal is to evaluate HCD by an adjoint technique (Antonsen & Chu Reference Antonsen and Chu1982) very similar to the one used recently to evaluate LHCD in full toroidal geometry to find the aspect ratio modifications to the driven current and efficiency (Catto Reference Catto2021). Unlike for LHCD, HCD does not seem to have an accepted analytic expression for the driven current or efficiency even without toroidal effects retained.

3. Adjoint technique summary

For an adjoint evaluation of HCD the preceding QL operator is all that is required along with the perturbed electron kinetic equation

(3.1)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }{f_1} = C\{ {f_1}\} + Q\{ {f_0}\} ,\end{equation}

and its adjoint equation for the adjoint function h associated with ${f_1}$

(3.2)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }h + C\{ h\} ={-} \left( {\frac{B}{I} - \frac{{{m_e}{\nu_u}}}{{{T_e}{x^3}}}{V_{||}}} \right){v_{||}}{f_0},\end{equation}

where the small term containing the parallel mean ion velocity ${V_{||}}$ is needed to account for the non-self-adjoint term in the electron-ion collision operator and ${f_0}$ is the Maxwellian

(3.3)\begin{equation}{f_0} = {f_0}(\psi ,E) = {n_e}(\psi ){\left( {\frac{{{m_e}}}{{2\mathrm{\pi }{T_e}}}} \right)^{3/2}}\,{\textrm{e}^{ - [{m_e}E + e\varPhi (\psi )]/{T_e}(\psi )}}.\end{equation}

The potential may be assumed to be a flux function to lowest order. In the electron kinetic equation ${f_1}$ is the perturbed electron distribution function with $f = {f_0} + {f_1}$, $Q\{ {f_0}\}$ is the QL operator prior to transit averaging, and $C\{ {f_1}\} = {C_{\textrm{ee}}}\{ {f_1}\} + {C_{\textrm{ei}}}\{ {f_1}\}$ is the sum of the self-adjoint linearized electron-electron collision operator plus the electron-ion collision operator

(3.4)\begin{equation}{C_{\textrm{ei}}}\{ {f_1}\} = \frac{{{\nu _u}}}{{{x^3}}}L\left\{ {{f_1} - \frac{{{m_e}}}{{{T_e}}}{V_{||}}{v_{||}}{f_0}} \right\},\end{equation}

where $L\{ h\}$ is the self-adjoint Lorentz or pitch angle scattering operator

(3.5)\begin{equation}L\{ h\} = \frac{{2{B_0}}}{B}\xi \frac{\partial }{{\partial \lambda }}\left( {\lambda \xi \frac{{\partial h}}{{\partial \lambda }}} \right),\end{equation}

with $x = v/{v_e}$ and ${v_e} = {(2{T_e}/{m_e})^{1/2}}$ the electron thermal speed. The pitch angle is defined as $\lambda = 2\mu {B_0}/B{v^2} = {B_0}v_ \bot ^2/B{v^2}$ and $\xi = {v_{||}}/v$, with ${B_0}$ a normalizing flux function to be defined shortly. The unlike collision frequency ${\nu _u}$ is defined as

(3.6)\begin{equation}{\nu _u} = 3\sqrt {\pi } {\nu _{\textrm{ei}}}/4,\end{equation}

where ${\nu _{\textrm{ei}}} = 4\sqrt {2{\pi }} {Z^2}{e^4}{n_i}\ell n{\varLambda _C}/3m_e^{1/2}T_e^{3/2} = Z{\nu _{\textrm{ee}}}$ for a quasineutral plasma with the ion, ${n_i}$, and electron, ${n_e}$, densities satisfying $Z{n_i} = {n_e}$, Z the ion charge number, ${\nu _{\textrm{ee}}}$ the electron-electron collision frequency and $\ell n{\varLambda _C}$ the Coulomb logarithm.

Defining the flux surface average of any quantity A by

(3.7)\begin{equation}\langle A\rangle = {{\left({\oint {\textrm{d}\vartheta A/\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } } \right)}\Bigg/ {\left({\oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } } \right)}},\end{equation}

using $\langle \boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }({B^{ - 1}}\int {{\textrm{d}^3}v{v_{||}}h{f_1}/{f_0}} \rangle = 0$, and the self-adjointness of both ${C_{\textrm{ee}}}$ (that requires ${f_0}$ to be Maxwellian) and L to combine the equations leads to the convenient adjoint relation

(3.8)\begin{equation}\left\langle {B\int {{\textrm{d}^3}v{v_{||}}{f_1}} } \right\rangle = {{I\left( {\int {{\textrm{d}^3}v\frac{{{v_{||}}\bar{h}}}{{B{f_0}}}{\tau_f}\bar{Q}\{ {f_0}\} } } \right)} \Bigg/ {\left({\oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } } \right)}},\end{equation}

where the ${V_{||}}\sim {v_i}{\rho _{\textrm{pi}}}/a$ term, with ${v_i}$ the ion thermal speed, ${\rho _{\textrm{pi}}}$ the poloidal ion gyroradius and a the minor radius, is negligible for the rf amplitudes of interest. Catto (Reference Catto2021) can be consulted for more details on this approximation and the treatment to follow. To write the final form of $\langle B\int {{\textrm{d}^3}v{v_{||}}{f_1}} \rangle$ the lowest-order result $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\bar{h} = 0$ is employed along with the transit average definition $\bar{A} = \oint_{} {\textrm{d}\tau A/{\tau _f}}$. It is convenient to let $B_0^2 = \langle {B^2}\rangle$.

Recall the Ohmic current is in the positive toroidal direction. Therefore, the helicon waves must drive the current in the same direction, requiring $\langle B\int {{\textrm{d}^3}v{v_{||}}{f_1}} \rangle < 0$.

4. Solution of the adjoint equation

The advantage of the adjoint method is that only the simpler adjoint equation

(4.1)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }h + {C_{\textrm{ee}}}\{ h\} + {\nu _u}{x^{ - 3}}L\{ h\} ={-} {I^{ - 1}}B{v_{||}}{f_0},\end{equation}

need be solved instead of the more complicated electron kinetic equation. To solve the preceding equation, it is adequate to approximate the electron-electron collision operator by its standard high speed $({v^2} \gg v_e^2)$ expansion. Then its self-adjoint form becomes

(4.2)\begin{equation}{C_{\textrm{ee}}}\{ h\} = \frac{{2{\nu _\ell }{B_0}\xi }}{B}\frac{\partial }{{\partial \lambda }}\left( {\lambda \xi \frac{{\partial h}}{{\partial \lambda }}} \right) + \frac{{{\nu _\ell }{T_e}}}{{{m_e}{v^2}}}\frac{\partial }{{\partial v}}\left[ {\frac{{{v^2}{f_0}}}{{{x^3}}}\frac{\partial }{{\partial v}}\left( {\frac{h}{{{f_0}}}} \right)} \right],\end{equation}

where the like collision frequency is defined as

(4.3)\begin{equation}{\nu _\ell } = 3\sqrt {\pi } {\nu _{\textrm{ee}}}/4,\end{equation}

with ${\nu _{\textrm{ee}}} = 4\sqrt {2\mathrm{\pi }} {e^4}{n_e}\ell n{\varLambda _C}/3m_e^{1/2}T_e^{3/2}$. This like particle operator is the usual non-momentum conserving, high speed expansion of the Rosenbluth potentials for collisions with a Maxwellian used by Karney & Fisch (Reference Karney and Fisch1979, Reference Karney and Fisch1985). Recent estimates (Catto Reference Catto2021; Catto & Tolman Reference Catto and Tolman2021a,Reference Catto and Tolmanb) indicate ${f_0}$ must be nearly Maxwellian for QL theory to remain valid.

The adjoint equation is solved using the Cordey eigenfunctions (Cordey Reference Cordey1976; Hsu et al. Reference Hsu, Catto and Sigmar1990; Xiao et al. Reference Xiao, Catto and Molvig2007; Parker & Catto Reference Parker and Catto2012) by writing $h = \bar{h} + \tilde{h}$ with $\partial \bar{h}/\partial \vartheta = 0$ to obtain, upon annihilating ${v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\tilde{h}$ the term, the transit average equation

(4.4)\begin{equation}{\bar{C}_{\textrm{ee}}}\{ \bar{h}\} + {\nu _u}{x^{ - 3}}L\{ \bar{h}\} ={-} {I^{ - 1}}\overline {B{v_{||}}} {f_0}.\end{equation}

Integration over a full trapped (t) bounce gives $\overline {B{v_{||}}} = 0$ implying that ${\bar{h}_t} = 0$. For the passing (p) electrons flux surface averages are used to rewrite the adjoint equation as

(4.5)\begin{equation}2(Z + 1)\frac{\partial }{{\partial \lambda }}\left[ {\lambda \langle \xi \rangle \frac{\partial }{{\partial \lambda }}\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right] + \; \frac{{{T_e}{x^3}\langle B/{v_{||}}\rangle }}{{{m_e}{B_0}{v^2}{f_0}}}\frac{\partial }{{\partial v}}\left[ {\frac{{{v^2}{f_0}}}{{{x^3}}}\frac{\partial }{{\partial v}}\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right] ={-} \frac{{\langle {B^2}\rangle v{x^3}}}{{I{B_0}{\nu _\ell }}},\end{equation}

where $\overline {B{v_{||}}} \oint_p {\textrm{d}\tau } = \langle {B^2}\rangle \oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta }$.

For the purposes here, the recent solution of (4.5) by Catto (Reference Catto2021) is adequate and convenient. Ignoring order $\varepsilon = r/R \ll 1$ terms it is given by

(4.6)\begin{align} \begin{aligned} \dfrac{{{{\bar{h}}_p}}}{{{f_0}}} & \approx \dfrac{{v{x^3}}}{{R{\nu _\ell }}}\left\{ {\dfrac{{(1 + 0.62\sqrt \varepsilon ){\varLambda_1}(\lambda ) - 1.02[(Z + 5)/(7Z + 11)]\sqrt \varepsilon {\varLambda_2}(\lambda )}}{{[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]}}} \right\}\\ & \equiv \dfrac{{v{x^3}}}{{R{\nu _\ell }}}\dfrac{{{\varLambda _{1 + 2}}(\sqrt \varepsilon ,Z,\lambda )}}{{\left[ {(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4} \right]}}, \end{aligned}\end{align}

where the Cordey (Reference Cordey1976) eigenfunctions ${\varLambda _j}$ with eigenvalues ${\kappa _j}$ satisfy the Sturm--Liouville differential equation obtained by transit averaging,

(4.7)\begin{equation}\frac{\partial }{{\partial \lambda }}\left( {\lambda \langle \xi \rangle \frac{{\partial {\varLambda_j}}}{{\partial \lambda }}} \right) = {\kappa _j}\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }}{\varLambda _{j\; }}.\end{equation}

In the preceding, $\partial \langle \xi \rangle /\partial \lambda ={-} \langle B/2{B_0}\xi \rangle$, ${\varLambda _j}(\lambda = 0) = 1$,

(4.8)\begin{gather}{\varLambda _{1 + 2}}(\sqrt \varepsilon ,Z,\lambda = 0) = 1 + [0.62 - 1.02(Z + 5)/(7Z + 11)]\sqrt \varepsilon ,\end{gather}

(4.9)\begin{gather}\langle \xi \rangle = 2\sqrt {2\varepsilon } E(k)/\mathrm{\pi }\sqrt {(1 - \varepsilon ){k^2} + 2\varepsilon } ,\end{gather}

where E(k) is the elliptic integral of the second kind, ${k^2} = 2\varepsilon \lambda /[1 - (1 - \varepsilon )\lambda ]$ and ${\varLambda _j} = 0$ at k = 1. The response ${\bar{h}_p}$ increases with speed because of the v dependence of $C\{ h\}$. For the passing

(4.10)\begin{equation}{\tau _p} = \oint_p {\textrm{d}\tau } \approx 4qR\sqrt {(1 - \varepsilon ){k^2} + 2\varepsilon } K(k)/v\sqrt {2\varepsilon } ,\end{equation}

with K(k) the elliptic integral of the first kind. The ${\varLambda _2}$ term has a somewhat small numerical coefficient. It was ignored in Catto (Reference Catto2021), but is retained here as it modestly alters the $\sqrt \varepsilon$ correction of interest (order $\varepsilon$ corrections are ignored). More eigenfunction details are available in Catto (Reference Catto2021), Hsu et al. (Reference Hsu, Catto and Sigmar1990), Xiao et al. (Reference Xiao, Catto and Molvig2007) and Parker & Catto (Reference Parker and Catto2012).

5. Helicon driven current in a tokamak compared with LHCD

Only the passing electrons contribute to helicon drive current, giving

(5.1)\begin{align}\left\langle {B\int {{\textrm{d}^3}v{v_{||}}{f_1}} } \right\rangle = \frac{{\langle {B^2}\rangle }}{{2\mathrm{\pi }q}}\int {{\textrm{d}^3}v\frac{{{v_{||}}{{\bar{h}}_p}}}{{B{f_0}}}{\tau _p}\bar{Q}\{ {f_0}\} } = \frac{{\langle {B^2}\rangle }}{{2\mathrm{\pi }q}}\sum\limits_{\boldsymbol{k}} {\int {{\textrm{d}^3}v\frac{{{v_{||}}{{\bar{h}}_p}}}{{Bv{f_0}}}{{\left. {\frac{\partial }{{\partial v}}} \right|}_\mu }\left( {{\tau_p}vD{{\left. {\frac{{\partial {f_0}}}{{\partial v}}} \right|}_\mu }} \right)} } ,\end{align}

at large aspect ratio, with the QL diffusivity for helicon waves simplifying to

(5.2)\begin{equation}D = \frac{{\mathrm{\pi }{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}{\tau _p}v_ \bot ^2J_1^2(\eta )}}{{8m_e^2k_ \bot ^2{v^2}}}\sum\limits_\ell {\delta [\omega {\tau _p} - 2\mathrm{\pi }\sigma (qn - m) - 2\mathrm{\pi }\ell ]\varTheta (v,\lambda ,n,m)} .\end{equation}

Here,

(5.3)\begin{align}\varTheta = \frac{1}{{\tau _p^2}}{|{\oint_p {\textrm{d}\tau \,{\textrm{e}^{ - \textrm{i}[\omega (\tau - {\tau_0}) - \sigma (qn - m)\vartheta (\tau )]}}} } |^2} = \frac{{{q^2}{R^2}}}{{\tau _p^2{v^2}}}{\left|{\int_{ - \mathrm{\pi }}^\mathrm{\pi } {\frac{{\textrm{d}\vartheta }}{\xi }\,{\textrm{e}^{ - \textrm{i}\left[ {\omega qR\int_0^\vartheta {\textrm{d}\vartheta /{v_{||}} - \sigma (qn - m)\vartheta } } \right]}}} } \right|^2} \le 1,\end{align}

and ${\textrm{d}^3}v \to 2\mathrm{\pi }B\,\textrm{d}E\,\textrm{d}\mu /{v_{||}} \to 2\mathrm{\pi }B{v^3}\,\textrm{d}v\,\textrm{d}\lambda /{B_0}{v_{||}}$. Integrating by parts in $E,\mu$ variables, yields

(5.4)\begin{equation}\left\langle {B\int {{\textrm{d}^3}v{v_{||}}{f_1}} } \right\rangle = \frac{{{m_e}\langle {B^2}\rangle }}{{2\mathrm{\pi }q{T_e}}}\sum\limits_{\boldsymbol{k}} {\int {{\textrm{d}^3}v\frac{{{v_{||}}}}{B}{\tau _p}vD{f_0}{{\left. {\frac{\partial }{{\partial v}}} \right|}_\mu }\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right)} } .\end{equation}

Catto (Reference Catto2021) has an extra ${v^2}$ multiplying ${\bar{h}_p}$ in his (4.7), implying the factors of $v$ in his (4.27) for the lower hybrid driven parallel current need to be corrected, as will be found shortly.

To sustain or enhance the poloidal magnetic field the parallel current driven by the helicon waves must be positive, requiring $\langle B\smallint {\textrm{d}^3}v{v_{||}}{f_1}\rangle < 0$. Therefore, passing electrons with ${v_{||}} < 0$ must drive the current implying $\sigma ={-} 1$ and $\ell + m - qn > 0$ in the argument of the delta function (requiring ${k_{||}} < 0$ for $\ell = 0$). As a result, writing it in a form allowing the speed integral to be performed gives

(5.5)\begin{equation}\delta [\omega {\tau _p} - 2\mathrm{\pi }(\ell + m - qn)] = v_{\omega /{k_{||}}}^2\delta (v - {v_{\omega /{k_{||}}}})/\omega v{\tau _p},\end{equation}

with

(5.6)\begin{equation}{v_{\omega /{k_{||}}}} = \omega v{\tau _p}/2\mathrm{\pi }(\ell + m - qn) > 0.\end{equation}

Then the exponential factor in the Maxwellian becomes

(5.7)\begin{equation}x_{\omega /{k_{||}}}^2 = {m_e}v_{\omega /{k_{||}}}^2/2{T_e},\end{equation}

where recall a high speed expansion of the like collision operator is used so $x_{\omega /{k_{||}}}^2 \gg 1$ in ${f_0}$. In addition, the barely passing do not contribute significantly to the current since $v{\tau _p}/4qR\mathop \to \limits_{k \to 1} {(2\varepsilon )^{ - 1/2}}\ell n(4/\sqrt {1 - {k^2}} ) \gg 1$. Therefore, because of the exponential from ${f_0}$, the freely passing $({k^2}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }\varepsilon \ll 1)$ electrons make the dominant contribution. As a result, only $\ell = 0$ need be retained in

(5.8)\begin{equation}D \approx \frac{{\mathrm{\pi }{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}\lambda J_1^2(\eta )}}{{8m_e^2k_{ \bot \; }^2|{k_{||}}|}}\delta (v - {v_{\omega /{k_{||}}}}) \equiv {D_{\vec{k}}}\; \lambda J_1^2(\eta )\delta (v - {v_{\omega /{k_{||}}}}).\end{equation}

Defining the exponential factor of ${f_0}$ for $\ell = 0$ as ${X^2}(\lambda ) \equiv x_{\omega /{k_{||}}}^2(\ell = 0)$, gives

(5.9)\begin{equation}{X^2}(\lambda ) \approx ({\omega ^2}/k_{||}^2v_e^2)(1 + {k^2}/2\varepsilon ) \approx ({\omega ^2}/k_{||}^2v_e^2)[\lambda /(1 - \lambda )],\end{equation}

as $v{\tau _p}\mathop \to \limits_{{k^2} \ll 1} 2\mathrm{\pi }qR\sqrt {1 + {k^2}/2\varepsilon }$, ${v_{\omega /{k_{||}}}}\mathop \to \limits_{\ell = 0} \omega \sqrt {1 + {k^2}/2\varepsilon } /{k_{||}}$ and

(5.10)\begin{equation}\varTheta \mathop \to \limits_{\varepsilon \ll 1} {\left|{\int_{ - \mathrm{\pi }}^\mathrm{\pi } {\frac{{\textrm{d}\vartheta }}{{2\mathrm{\pi }}}\,{\textrm{e}^{ - \textrm{i}\ell \vartheta }}} } \right|^2} = {\delta _{0\ell }} = \left\{ {\begin{array}{@{}ll} 1& {\ell = 0}\\ 0& {\ell \ne 0} \end{array}} \right..\end{equation}

Then, integrating over ${v_{||}} < 0$ only, as D vanishes for ${v_{||}} > 0$, leads to

(5.11)\begin{align}\left\langle {\frac{B}{{{B_0}}}\int {{\textrm{d}^3}v{v_{||}}{f_1}} } \right\rangle ={-} 4\mathrm{\pi }R\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{\omega ^2}}}{{k_{||}^2v_e^2}}{D_{\boldsymbol{k}}}} \int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda \lambda J_1^2(\eta )} \int_0^\infty {\textrm{d}v\delta (v - {v_{\omega /{k_{||}}}})} v{\left. {{f_0}\frac{\partial }{{\partial v}}} \right|_\mu }\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right).\end{align}

In $v,\lambda$ variables

(5.12)\begin{align}v{\left. {{f_0}\frac{\partial }{{\partial v}}} \right|_\mu }\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right) = \; v{f_0}\left[ {{{\left. {\frac{\partial }{{\partial v}}} \right|}_\lambda }\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right) + {{\left. {\frac{{\partial \lambda }}{{\partial v}}} \right|}_\mu }{{\left. {\frac{\partial }{{\partial \lambda }}} \right|}_v}\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right] = \left( {4 - \frac{{2\lambda }}{{{\varLambda_{1 + 2}}}}\frac{{\partial {\varLambda_{1 + 2}}}}{{\partial \lambda }}} \right){\bar{h}_p},\end{align}

since the lowest-order solution is ${\bar{h}_p}/{f_0} \propto {v^4}{\varLambda _{1 + 2}}(\lambda )$. The exponential in the Maxwellian makes the evaluation of the $\lambda$ integral insensitive to its upper limit $(\lambda < 1 - \varepsilon )$ as will be shown shortly. The pitch angle derivative of ${\varLambda _{1 + 2}}$ is finite at $\lambda = 0$ making $\varLambda _{1 + 2}^{ - 1}\partial {\varLambda _{1 + 2}}/\partial \lambda \sim 1$.

Only the $\lambda = 0$ limit of this term will matter when the pitch angle integral is performed in (5.18) by integrating by parts, leaving just

(5.13)\begin{equation}\left\langle {\frac{B}{{{B_0}}}\int {{d^3}v{v_{||}}{f_1}} } \right\rangle ={-} 16\mathrm{\pi }R\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{\omega ^2}}}{{k_{||}^2v_e^2}}{D_{\boldsymbol{k}}}} \int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda \lambda J_1^2(\eta ){{\bar{h}}_p}} ,\end{equation}

to be evaluated, where at resonance $\eta = \sqrt \lambda {k_ \bot }\omega /{k_{||}}{\varOmega _e}$ and

(5.14)\begin{equation}{\bar{h}_p} = \; \frac{{4{n_e}{\omega ^4}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}{\varLambda _{1 + 2}}(\sqrt \varepsilon ,Z,\lambda )\,{\textrm{e}^{ - {X^2}(\lambda )}}}}{{3{\mathrm{\pi }^2}R{\nu _{\textrm{ee}}}k_{||}^4v_e^6[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]}}.\end{equation}

Inserting ${\bar{h}_p}$ leads to the expression for the parallel current driven by helicon waves to be

(5.15)\begin{align}\begin{aligned} J_{||}^H & ={-} e\left\langle {\dfrac{B}{{{B_0}}}\int {{\textrm{d}^3}v{v_{||}}{f_1}} } \right\rangle = \dfrac{{8e{n_e}}}{{(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4}}\\ & \quad \times\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\dfrac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}}}{{3m_e^2k_{ \bot \; }^2}}} \dfrac{{{\omega ^6}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{{\nu _{\textrm{ee}}}|k_{||}^7|v_e^8}}\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda \lambda J_1^2(\eta ){\varLambda _{1 + 2}}(\lambda )\,{\textrm{e}^{ - {X^2}(\lambda )}}} , \end{aligned}\end{align}

where $\eta \sim \sqrt \lambda {k_ \bot }{v_e}/{\varOmega _e} = \sqrt \lambda {k_ \bot }{\rho _e}$. As a result, ${X^2}(\lambda )$ provides a cut-off before $\lambda \to 1 - \varepsilon$. However, the procedure is more complex in the HCD case than in the LHCD case because of the Bessel function behaviour due the helicon wave interacting with gyromotion of the electron, while the lower hybrid wave only interacts with its parallel streaming.

To understand the procedure and generalize the LHCD results, it is helpful to first consider the LHCD case in more detail than was given in Catto (Reference Catto2021). To obtain the parallel current driven by the LH waves, $J_{||}^{\textrm{LH}}$, the preceding steps are repeated starting with the replacement $k_ \bot ^{ - 2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}v_ \bot ^2J_1^2(\eta ) \to |{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}v_{||}^2J_0^2(\eta )$ and then using $v_ \bot ^2 \approx \lambda {v^2} \approx \lambda {\omega ^2}/k_{||}^2 \to v_{||}^2 \approx {\omega ^2}/k_{||}^2$. As a result, the driven parallel LH current is

(5.16)\begin{align}J_{||}^{\textrm{LH}} = \frac{{8e{n_e}}}{{(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4}}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}}}{{3m_e^2}}} \frac{{{\omega ^6}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{{\nu _{\textrm{ee}}}|k_{||}^7|v_e^8}}\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda J_0^2(\eta ){\varLambda _{1 + 2}}(\lambda )\,{\textrm{e}^{ - {X^2}(\lambda )}},} \end{align}

where again $\eta = \sqrt \lambda {k_ \bot }\omega /{k_{||}}{\varOmega _e}$. Then, an integration by parts is performed by using ${\textrm{e}^{ - {X^2}(\lambda )}} ={-} [{(1 - \lambda )^2}/z]\partial \,{\textrm{e}^{ - {X^2}(\lambda )}}/\partial \lambda$ to obtain an asymptotic expansion in inverse powers of $z = {\omega ^2}/k_{||}^2v_e^2 \gg 1$ to find

(5.17)\begin{equation}J_{||}^{\textrm{LH}} \approx \frac{{8e{n_e}{\varLambda _{1 + 2}}(\sqrt \varepsilon ,Z,0)}}{{(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4}}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}}}{{3m_e^2}}} \frac{{{\omega ^4}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{{\nu _{\textrm{ee}}}|k_{||}^5|v_e^6}},\end{equation}

as $\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda J_0^2{\varLambda _{1 + 2}}\,{\textrm{e}^{ - {X^2}}}} \approx {\varLambda _{1 + 2}}(\lambda = 0)/z$ since ${\varLambda _j}(0) = 1 = J_0^2(0)$. This result is 2/3 of the value given by Catto (Reference Catto2021) when ${\varLambda _2}$ is neglected. The difference is due to the extra power of ${v^2}$ multiplying ${\bar{h}_p}$ inside the v derivative beginning of his (4.7). Notice that only terms that remain finite at $\lambda = 0$ will contribute, as assumed in (5.13).

In the helicon case three integrations by parts in $\lambda$ are required. Again, only terms finite at $\lambda = 0$ contribute. Ignoring exponentially small terms

(5.18)\begin{align}\begin{aligned} & \int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda \lambda J_1^2{\varLambda _{1 + 2}}\,{\textrm{e}^{ - {X^2}}}} \approx \dfrac{1}{{{z^2}}}\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda \,{\textrm{e}^{ - {X^2}}}} \dfrac{\partial }{{\partial \lambda }}\left\{ {{{(1 - \lambda )}^2}\dfrac{\partial }{{\partial \lambda }}[{{(1 - \lambda )}^2}\lambda J_1^2{\varLambda _{1 + 2}}]} \right\}\\ & \quad \approx \dfrac{1}{{{z^3}}}\dfrac{\partial }{{\partial \lambda }}{\left. {\left\{ {{{(1 - \lambda )}^2}\dfrac{\partial }{{\partial \lambda }}[{{(1 - \lambda )}^2}\lambda J_1^2{\varLambda _{1 + 2}}]} \right\}} \right|_{\lambda = 0}} = \dfrac{1}{{{z^3}}}{\left. {\dfrac{{\partial J_1^2}}{{\partial \lambda }}} \right|_{\lambda = 0}} = \dfrac{{k_ \bot ^2\rho _e^2}}{{2{z^2}}}. \end{aligned}\end{align}

Therefore, for HCD

(5.19)\begin{equation}J_{||}^H = \frac{{4e{n_e}{\varLambda _{1 + 2}}(\sqrt \varepsilon ,Z,0)}}{{(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4}}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}}}{{3m_e^2k_{ \bot \; }^2}}} \frac{{{\omega ^2}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{{\nu _{\textrm{ee}}}|k_{||}^3|v_e^4}}k_ \bot ^2\rho _e^2.\end{equation}

The stronger ${T_e}$ dependence of $J_{||}^H$ compared with $J_{||}^{\textrm{LH}}$ was previously noted by Li et al. (Reference Li, Ding, Dong and Liu2020a, Reference Li, Li and Liu2021). Helicon waves are also sensitive to the density. If ${n_e}$ increases substantially, (A14) indicates $k_{||}^2$ will increase, causing a less effective rf to interaction with the bulk electrons.

Letting $z = {\omega ^2}/k_{||}^2v_e^2 = {c^2}/v_e^2n_{||}^2$, then HCD and LHCD depend on ${z^{3/2}}{e^{ - z}}$ and ${z^{5/2}}{e^{ - z}}$, respectively, and are maximized at z = 3/2 and z = 5/2, consistent with assuming ${v^2} > v_e^2$ in ${C_{\textrm{ee}}}$. Consequently, a slightly higher parallel refractive index is desirable for HCD compared with LHCD, as noted by Prater et al. (Reference Prater, Moeller, Pinsker, Porkolab, Meneghini and Vdovin2014). In addition to penetrating into the central core plasma, the current driven by helicon waves can be comparable to or larger than that driven by lower hybrid waves for the same ${\omega ^2}/k_{||}^2v_e^2$ based on the estimate of (A15) and simulations (Vdovin Reference Vdovin2013; Prater et al. Reference Prater, Moeller, Pinsker, Porkolab, Meneghini and Vdovin2014; Lau et al. Reference Lau, Jaeger, Bertelli, Berry, Green, Murakami, Park, Pinsker and Prater2018, Reference Lau, Berry, Jaeger and Bertelli2019; Li et al. Reference Li, Ding, Dong and Liu2020a,Reference Li, Liu, Xiang and Lib, Reference Li, Li and Liu2021).

6. The rf power and current drive efficiency

To form the HCD efficiency requires evaluating the rf power absorbed by the passing electrons. Integrating over ${v_{||}} < 0$ and ignoring the negligible power into the barely passing leads to

(6.1)\begin{align}\begin{aligned} P_{\textrm{cd}}^H & = \dfrac{{{m_e}}}{2}\left\langle {\int {{\textrm{d}^3}v{v^2}Q} } \right\rangle \approx \dfrac{{{m_e}{B_0}}}{{4\mathrm{\pi }qR}}\int {{\textrm{d}^3}v\dfrac{{{v_{||}}}}{B}{v^2}{\tau _p}\bar{Q}} \\ & = \dfrac{{{m_e}{B_0}}}{{4\mathrm{\pi }qR}}\int {{\textrm{d}^3}v\dfrac{{{v_{||}}}}{B}v} {\left. {\dfrac{\partial }{{\partial v}}} \right|_\mu }\left( {{\tau_p}vD{{\left. {\dfrac{{\partial {f_0}}}{{\partial v}}} \right|}_\mu }} \right) = \dfrac{{m_e^2}}{{qR{T_e}}}\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda } \int_0^\infty {\textrm{d}v{\tau _p}{v^5}D{f_0}} \\ & = \dfrac{{{\mathrm{\pi }^{1/2}}}}{2}{m_e}{n_e}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\dfrac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}{\omega ^4}}}{{m_e^2k_{ \bot \; }^2|k_{||}^5|v_e^5}}\,} {\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda \lambda J_1^2(\eta )\,{\textrm{e}^{ - {X^2}(\lambda )}}} . \end{aligned}\end{align}

Similarly, fixing the ${v^2}$ and the numerical factor in Catto (Reference Catto2021) for LHCD, gives

(6.2)\begin{equation}P_{\textrm{cd}}^{\textrm{LH}} = \frac{{{\mathrm{\pi }^{1/2}}}}{2}{m_e}{n_e}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^4}}}{{m_e^2|k_{||}^5|v_e^5}}\,} {\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}\int_0^{1 - \varepsilon} {\textrm{d}\lambda J_0^2(\eta )\,{\textrm{e}^{ - {X^2}(\lambda )}}} .\end{equation}

Again using ${\textrm{e}^{ - {X^2}(\lambda )}}$ to integrate by parts gives

(6.3)\begin{equation}P_{\textrm{cd}}^H = \frac{{{\mathrm{\pi }^{1/2}}}}{4}{m_e}{n_e}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}}}{{m_e^2k_{ \bot \; }^2|{k_{||}}|{v_e}}}} k_ \bot ^2\rho _e^2\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}},\end{equation}

and

(6.4)\begin{equation}P_{\textrm{cd}}^{\textrm{LH}} = \frac{{{\mathrm{\pi }^{1/2}}}}{2}{m_e}{n_e}\mathop \sum \limits_{\vec{k}}^{{k_{||}} < 0} \frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^2}}}{{m_e^2|k_{||}^3|v_e^3}}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}.\end{equation}

The numerical coefficient of $P_{\textrm{cd}}^{\textrm{LH}}$ is half that of (5.7) in Catto (Reference Catto2021).

The current drive efficiency is defined by the ratio ${J_{||}}/{P_{\textrm{cd}}}$. Consequently

(6.5)\begin{equation}\frac{{J_{||}^H}}{{P_{\textrm{cd}}^H}} = \frac{{16e\{ 1 + [0.62 - 1.02(Z + 5)/(7Z + 11)]\sqrt \varepsilon \} \sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}{\omega ^2}}}{{{\nu _{\textrm{ee}}}m_e^2k_{ \bot \; }^2|k_{||}^3|v_e^4}}} \; k_ \bot ^2\rho _e^2\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{3{\mathrm{\pi }^{1/2}}{m_e}[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}}}{{m_e^2k_{ \bot \; }^2|{k_{||}}|{v_e}}}} k_ \bot ^2\rho _e^2\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}},\end{equation}

while

(6.6) \begin{equation}\frac{{J_{||}^{\textrm{LH}}}}{{P_{\textrm{cd}}^{\textrm{LH}}}} = \frac{{16e\{ 1 + [0.62 - 1.02(Z + 5)/(7Z + 11)]\sqrt \varepsilon \} \sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0}{{\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^4}}}{{{\nu _{\textrm{ee}}}m_e^2|k_{||}^5|v_e^6}}\,} } {\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{3{\mathrm{\pi }^{1/2}}{m_e}[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\frac{{{e^2}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^2}}}{{m_e^2|k_{||}^3|v_e^3}}} \,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}.\end{equation}

In the terms in the sums in these two general forms and in (5.17), (5.19), (6.1) and (6.2), the Fourier mode amplitudes $|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}$ and $|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}$ and the wavenumbers ${k_ \bot }$ and ${k_{||}}$ are to be obtained from a full wave code for precise evaluations.

Remarkably, for any (not necessarily the same) single $\omega$ and $\boldsymbol{k}$ the preceding forms are identical

(6.7)\begin{align}\frac{{J_{||}^H/e{n_e}{v_e}}}{{P_{\textrm{cd}}^H/{n_e}{m_e}v_e^2{\nu _{\textrm{ee}}}}} = \frac{{16\{ 1 + [0.62 - 1.02(Z + 5)/(7Z + 11)]\sqrt \varepsilon \} {\omega ^2}}}{{3{\mathrm{\pi }^{1/2}}[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]k_{||}^2v_e^2}} = \frac{{J_{||}^{\textrm{LH}}/e{n_e}{v_e}}}{{P_{\textrm{cd}}^{\textrm{LH}}/{n_e}{m_e}v_e^2{\nu _{\textrm{ee}}}}}.\end{align}

Approximation (6.7) suggests that comparable current drive efficiencies are possible with helicon and lower hybrid waves, with HCD providing core access as well as profile control. In addition, based on the cold plasma estimate (A15) of the Appendix, it seems possible to drive comparable parallel currents with helicon and lower hybrid waves, as suggested by the simulations of Prater et al. (Reference Prater, Moeller, Pinsker, Porkolab, Meneghini and Vdovin2014) and Lau et al. (Reference Lau, Jaeger, Bertelli, Berry, Green, Murakami, Park, Pinsker and Prater2018). The $\varepsilon = 0$ form of (6.7) is essentially the same as the non-relativistic, $z = v_p^2 \gg 1$ limit (31) in Karney & Fisch (Reference Karney and Fisch1985).

7. Combined helicon and lower hybrid

The preceding idealized analytic evaluations of the currents driven and the form of the QL operator suggest it is possible to combine HCD with LHCD for the same applied resonant wave frequency (Yin et al. Reference Yin, Zheng, Gong, Yang, Yin, Song, Huang, Chen and Zhong2022). Helicon and lower hybrid waves then drive current by simultaneously acting on the perpendicular and parallel electron distribution function, respectively. Indeed, an antenna used for LHCD can drive some level of helicon waves and vice versa. Retaining only $\ell = 0$, using the approximation ${\textrm{e}^{ - \textrm{i}\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\chi (\tau ^{\prime})} }} \approx 1$ and inserting the delta function, the combined QL diffusivity for the passing electrons becomes

(7.1)\begin{equation}D = \frac{{\mathrm{\pi }{e^2}}}{{2m_e^2{v^2}{\tau _p}}}{\left|{\oint_p {\textrm{d}\tau } {\boldsymbol{e}_k}\boldsymbol{\cdot }\left[ {\boldsymbol{n}{v_{||}}{J_0}(\eta ) - \textrm{i}\boldsymbol{n} \times \boldsymbol{k}\frac{{{v_ \bot }}}{{{k_ \bot }}}{J_1}(\eta )} \right]} \right|^2}\frac{{v_{\omega /{k_{||}}}^2\delta (v - {v_{\omega /{k_{||}}}})}}{{\omega v{\tau _p}}},\end{equation}

with

(7.2)\begin{equation}{v_{\omega /{k_{||}}}} = \; \omega v{\tau _p}/2\mathrm{\pi (}m - qn\textrm{)} > 0.\end{equation}

The combined parallel current that can be driven is therefore

(7.3)\begin{align}\begin{aligned} J_{||}^{H + \textrm{LH}} & = \dfrac{{8{e^3}{n_e}\{ 1 + [0.62 - 1.02(Z + 5)/(7Z + 11)]\sqrt \varepsilon \} }}{{3m_e^2{\nu _{\textrm{ee}}}[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]}}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\dfrac{{{\omega ^6}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|k_{||}^7|v_e^8}}} \left\{\vphantom{\dfrac{{k_{||}^2v_e^2\rho_e^2}}{{2{\omega^2}}}}{|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}} \right.\\ & \quad \left. { + \,i\dfrac{{|{k_{||}}|{v_e}{\rho_e}}}{{2\omega }}[({\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n})(\boldsymbol{e}_k^\ast \boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}) - (\boldsymbol{e}_k^\ast \boldsymbol{\cdot }\boldsymbol{n})({\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k})] + \dfrac{{k_{||}^2v_e^2\rho_e^2}}{{2{\omega^2}}}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}} \right\}, \end{aligned}\end{align}

where the integrals are performed by integration by parts as before, but with one new integral appearing in the cross terms, namely

(7.4)\begin{align}\int_0^{\lambda < 1 - \varepsilon } {\textrm{d}\lambda {\lambda ^{1/2}}{J_1}(\eta ){J_0}(\eta ){\varLambda _{1 + 2}}(\lambda )\,{\textrm{e}^{ - {X^2}(\lambda )}}} \approx \frac{1}{z}{\left. {\frac{{\partial ({\lambda^{1/2}}{J_1})}}{{\partial \lambda }}} \right|_{\lambda = 0}} = \frac{{{k_ \bot }{\rho _e}}}{{2{z^{3/2}}}}{\varLambda _{1 + 2}}(\sqrt \varepsilon ,Z,\lambda = 0).\end{align}

The rf power absorbed by passing electrons is

(7.5)\begin{align}\begin{aligned} P_{\textrm{cd}}^{H + \textrm{LH}} & = \dfrac{{{\mathrm{\pi }^{1/2}}{e^2}{n_e}}}{{2{m_e}}}\sum\limits_{\boldsymbol{k}}^{{k_{||}} < 0} {\dfrac{{{\omega ^2}}}{{|k_{||}^3|v_e^3}}\,} {\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}\left\{\vphantom{\dfrac{{k_{||}^2v_e^2\rho_e^2}}{{2{\omega^2}}}}{|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n}{|^2}} \right.\\ & \quad \left. { + \,i\dfrac{{|{k_{||}}|{v_e}{\rho_e}}}{{2\omega }}[{({\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n})(\boldsymbol{e}_k^\ast \boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}) - (\boldsymbol{e}_k^\ast \boldsymbol{\cdot }\boldsymbol{n})({\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k})} ]+ \dfrac{{k_{||}^2v_e^2\rho_e^2}}{{2{\omega^2}}}|{\boldsymbol{e}_k}\boldsymbol{\cdot }\boldsymbol{n} \times \boldsymbol{k}{|^2}} \right\}. \end{aligned}\end{align}

As a result, the efficiency is the ratio $J_{||}^{H + \textrm{LH}}/P_{\textrm{cd}}^{H + \textrm{LH}}$, which for a single frequency and wavenumber recovers the same expression as given at the end of § 6. Moreover, the cross term may allow more current to be driven and more power to be absorbed, perhaps as observed in the EAST numerical simulations of Yin et al. (Reference Yin, Zheng, Gong, Yang, Yin, Song, Huang, Chen and Zhong2022). Synergy is possible, but not assured, since helicon and lower hybrid waves independently act on perpendicular and parallel motion, respectively, to make the electrons less collisional.

8. Summary

The new results are the analytic expressions for the parallel current that can be driven in a tokamak by a helicon wave, (5.19), and the associated efficiency of HCD, (6.5) and (6.7). In addition, the numerical coefficients of the corresponding tokamak expressions for LHCD (Catto Reference Catto2021), (5.17), (6.4), (6.6) and (6.7), are generalized as well as corrected, by using a more systematic derivation. Interestingly, for any single applied frequency and wave vector the efficiency of HCD and LHCD are shown to be the same, (6.7). In addition, HCD and LHCD can be combined for the same applied frequency as shown in (7.3) and (7.5) of § 7, and, of course, recover the same single wave vector efficiency, (6.7). If a combination of HCD and LHCD is possible from the same antenna and rf source, more current than the sum might be driven and steady state tokamak operation might be slightly more feasible.