2. Adjoint method and collision operators

Using Gaussian cgs units, and the drift kinetic variables of spatial location $\boldsymbol{r}$, total energy $E = {v^2}/2 - e\Phi /{m_e}$, magnetic moment $\mu = v_ \bot ^2/2B$ and gyrophase $\varphi$, such that the velocity is

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

with $v_{||}^2 = {v^2}{\kern 1pt} - 2\mu B$, the QL equation for the unperturbed electron distribution function f is

(2.2)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }f = C\{ f\} + Q\{ f\} .\end{equation}

In the preceding, Q denotes the QL operator, C is the collision operator, the three orthonormal spatial unit vectors are related by ${\boldsymbol{e}_1} \times {\boldsymbol{e}_2}{\kern 1pt} = \boldsymbol{n} = \boldsymbol{B}/B$, and the unperturbed magnetic field is

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

where ${m_e}$ and e are the mass and magnitude of the charge on an electron, and $\Phi $ is the electrostatic potential. The spatial variables employed are the poloidal flux function $\psi$, and the poloidal, $\vartheta$, and toroidal, $\zeta$, angles, where q is the safety factor and $I = R{B_t}$, with ${B_t}$ the toroidal magnetic field and R the major radius. The detailed forms of Q and C will be given as needed.

Linearizing about an electron Maxwellian of density ${n_e}$ and temperature ${T_e}$,

(2.4)\begin{equation}{f_0} = {f_0}(\psi ,E) = {n_e}(\psi ){\left[ {\frac{{{m_e}}}{{2{\rm \pi}{T_e}(\psi )}}} \right]^{3/2}}{\textrm{e}^{ - {m_e}{v^2}/2{T_e}(\psi )}} = {n_e}(\psi ){\left[ {\frac{{{m_e}}}{{2{\rm \pi}{T_e}(\psi )}}} \right]^{3/2}}{\textrm{e}^{ - [{m_e}E + e\Phi (\psi )]/{T_e}(\psi )}},\end{equation}

by writing

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

and using

(2.6)\begin{equation}C\{ f\} = {C_{ee}}\{ f\} + {C_{ei}}\{ f\} ,\end{equation}

with $C\{ {f_0}\} = 0$, and ${C_{ee}}$ and ${C_{ei}}$ the electron--electron and electron--ion collision operators, leads to the unperturbed linearized equation to be solved by the adjoint method, namely

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

The electron--ion collision operator (Hinton & Hazeltine Reference Hinton and Hazeltine1976; Helander & Sigmar Reference Helander and Sigmar2002) is

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

with $x = v/{v_e}$ and ${v_e} = {(2{T_e}/{m_e})^{1/2}}$ the electron thermal speed. The Lorentz operator L is self-adjoint and defined here as

(2.9)\begin{equation}L\{ h\} = \frac{1}{2}{\nabla _v} \cdot [({v^2}\boldsymbol{I} - \boldsymbol{vv})\boldsymbol{\cdot }{\boldsymbol{\nabla }_v}h] = \frac{{2{B_0}}}{B}\xi \frac{\partial }{{\partial \lambda }}\left( {\lambda \xi \frac{{\partial h}}{{\partial \lambda }}} \right),\end{equation}

with $L\{ {v_{\|}}{f_0}\} = - {v_{\|}}{f_0}$, $\lambda = 2\mu {B_0}{\kern 1pt} /{\kern 1pt} {v^2} = {B_0}v_ \bot ^2/B{v^2}$, $\xi = {v_{\|}}/v$ and ${B_0}$ a normalization constant to be made explicit shortly. The parallel ion mean velocity is ${V_{\|}}$ and the unlike collision frequency is

(2.10)\begin{equation}{\nu _u} \equiv \sqrt 2 {\rm \pi}{Z^2}{e^4}{n_i}\ell n\varLambda /m_e^{1/2}T_e^{3/2} \to 3\sqrt {\rm \pi}{\nu _{ei}}/4,\end{equation}

where ${\nu _{ei}} = 4\sqrt {2{\rm \pi}} {Z^2}{e^4}{n_i}\ell n\varLambda /3{m^{1/2}}{T^{3/2}} = Z{\nu _{ee}}$ for a quasineutral plasma with $Z{n_i} = {n_e}$, ${\nu _{ee}}$ the electron--electron collision frequency, and Z the ion charge number.

The electron--electron collision operator is self-adjoint. As only a lowest-order solution is desired, the standard high speed expansion of the electron--electron collision operator in its self-adjoint form is employed, namely

(2.11)\begin{align}{C_{ee}}\{ h\} &= {\nu _\ell }\left\{ {\frac{1}{{{x^3}}}L\{ h\} + {\nabla_v} \cdot \left[ {\frac{{{T_e}{f_0}}}{{{m_e}{x^3}}}{\nabla_v}\left( {\frac{h}{{{f_0}}}} \right)} \right]} \right\}\nonumber\\ &= \frac{{2{B_0}{\nu _\ell }\xi }}{{B{x^3}}}\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{align}

where

(2.12)\begin{equation}{\nu _\ell } \equiv \sqrt 2 {\rm \pi}{Z^4}{e^4}{n_e}\ell n\varLambda /m_e^{1/2}T_e^{3/2} \to 3\sqrt {\rm \pi}{\nu _{ee}}/4,\end{equation}

where ${\nu _{ee}} = 4\sqrt {2{\rm \pi}} {e^4}{n_e}\ell n\varLambda /3{m^{1/2}}{T^{3/2}}$ and h is a perturbed distribution function. The preceding like particle collision operator is just the usual non-momentum conserving high speed expansion of the Rosenbluth potentials for collisions with a Maxwellian (Karney & Fisch Reference Karney and Fisch1979).

The adjoint method provides an explicit means of evaluating the parallel current provided the adjoint equation is easier to solve than the original kinetic equation. It has been used for LHCD by Antonsen & Chu (Reference Antonsen and Chu1982), Karney & Fisch (Reference Karney and Fisch1985) and Cohen (Reference Cohen1987). It has also been used to calculate the bootstrap current in stellarators (Helander, Geiger & Maaβberg Reference Helander, Geiger and Maaßberg2011), and in the plateau regime to evaluate the bootstrap current in a tokamak (Pusztai & Catto Reference Pusztai and Catto2010).

To account for the non-self-adjointness of the electron--ion collision operator it is necessary to use the modified adjoint equation

(2.13)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }h + C\{ h\} ={-} \left( {\frac{B}{I} - \frac{{m{\nu_u}}}{{T{x^3}}}{V_{||}}} \right){v_{||}}{f_0}.\end{equation}

Then, defining the flux surface average of any quantity A as

(2.14)\begin{equation}\langle A\rangle \equiv {{[{\oint {\textrm{d}\vartheta A/\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } } ]} / {[{\oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } } ]}},\end{equation}

using

(2.15)\begin{equation}\left\langle {\int {{\textrm{d}^3}v} f_0^{ - 1}{v_{||}}(h\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }{f_1} + {f_1}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }h)} \right\rangle = \left\langle {\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\left( {{B^{ - 1}}\int {{\textrm{d}^3}} v{v_{||}}h{f_1}/{f_0}} \right)} \right\rangle = 0,\end{equation}

and the self-adjointness of the electron--electron collision operator,

(2.16)\begin{equation}\left\langle {\int {{\textrm{d}^3}v} f_0^{ - 1}({f_1}{C_{ee}}\{ h\} - h{C_{ee}}\{ {f_1}\} )} \right\rangle = 0,\end{equation}

and the Lorentz operator

(2.17)\begin{equation}\left\langle {\int {{\textrm{d}^3}v} f_0^{ - 1}x_e^{ - 3}({f_1}L\{ h\} - hL\{ {f_1}\} )} \right\rangle = 0,\end{equation}

yields the desired adjoint relation

(2.18)\begin{equation}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle = I\left\langle {\int {{\textrm{d}^3}} vh\left( {f_0^{ - 1}Q\{ {f_0}\} - \frac{{m{\kern 1pt} {\nu_u}}}{{T{\kern 1pt} x_e^3}}{V_{||}}{v_{||}}} \right)} \right\rangle .\end{equation}

Therefore, only the adjoint equation need be solved to evaluate the parallel electron current driven by the lower hybrid waves. It is convenient to define $B_0^2 \equiv \langle {B^2}\rangle$.

Based on the direction of the poloidal magnetic field, the Ohmic current is in the positive toroidal direction. Consequently, the lower hybrid (LH) parallel electron flow is to be driven in the opposite or negative direction to make $\left\langle {B\int {{\textrm{d}^3}} v{v_{\|}}{f_1}} \right\rangle < 0$.

The parallel ion flow term in (2.18) is normally ignored as

(2.19)\begin{equation}\frac{{m{\nu _u}\left\langle {{V_{||}}\int {{\textrm{d}^3}} v{x^{ - 3}}h{v_{||}}} \right\rangle }}{{T\left\langle {\int {{\textrm{d}^3}} vhf_0^{ - 1}Q\{ {f_0}\} } \right\rangle }}\sim \frac{{{V_{||}}{\nu _{ee}}{f_0}}}{{{v_e}Q}}\sim {\left( {\frac{{{m_e}}}{M}} \right)^{1/2}}\frac{{{\rho _{pi}}}}{a}\frac{{{\nu _{ee}}{f_0}}}{{Q\{ {f_0}\} }} \ll 1,\end{equation}

where M is the ion mass, and ${V_{\|}}\sim{v_i}{\rho _{pi}}/a$, with ${v_i}$ the ion thermal speed, ${\rho _{pi}}$the poloidal ion gyroradius and a the minor radius. Consequently, ${V_{\|}}$ can be ignored and only

(2.20)\begin{align}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle = I\int {{\textrm{d}^3}} v\frac{{{v_{||}}h}}{{B{f_0}}}\left\langle {\frac{B}{{{v_{||}}}}Q\{ {f_0}\} } \right\rangle = I{{\left( {\int {{\textrm{d}^3}} v\frac{{{v_{||}}h}}{{B{f_0}}}{\kern 1pt} {\tau_f}\overline {Q\{ {f_0}\} } } \right)} / {({\oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } } )}},\end{align}

need be evaluated, where the solution for h is shown in the next section to satisfy $\partial h/{\kern 1pt} \partial \vartheta = 0$ to lowest order. The transit average over a full (f) poloidal circuit is defined using $\textrm{d}\tau = \textrm{d}\vartheta /{v_{\|}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta$ to be

(2.21)\begin{equation}\bar{A} \equiv {{\oint_f {\textrm{d}\tau A} } / {({\oint_f {\textrm{d}\tau } } )}}.\end{equation}

3. Eigenfunction solution procedure in tokamak geometry

Rewriting the adjoint equation yields a Spitzer & Härm (Reference Spitzer and Härm1953) equation in toroidal geometry,

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

Writing $h = \bar{h} + \tilde{h} + \cdots$ with $\tilde{h} \ll \bar{h}$ then to lowest order

(3.2)\begin{equation}{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\bar{h} = 0,\end{equation}

giving

(3.3)\begin{equation}\bar{h} = \bar{h}(\psi ,\alpha ,E,\mu ,\sigma ),\end{equation}

where $\sigma = {v_{\|}}/| {v_{\|}} |$ for the passing and $\sigma = 0$ for the trapped. Then transit averaging the next order equation,

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

leads to

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

Integration over a full bounce for the trapped (t) electrons gives $\overline {B{v_{\|}}} = 0$, implying that the trapped response ${\bar{h}_t}$ vanishes,

(3.6)\begin{equation}{\bar{h}_t} = 0.\end{equation}

For the passing electrons

(3.7)\begin{equation}\overline {B{v_{||}}} \oint_f {\textrm{d}\tau } = \oint {\textrm{d}\vartheta {B^2}/\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } = \langle {B^2}\rangle \oint {\textrm{d}\vartheta /\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } .\end{equation}

Using the flux surface average to rewrite the passing (p) adjoint equation leads to

(3.8)\begin{equation}I\left\langle {\frac{B}{{{v_{||}}}}{C_{ee}}\{ {{\bar{h}}_p}\} } \right\rangle + \frac{{I{\nu _u}}}{{x_e^3}}\left\langle {\frac{B}{{{v_{||}}}}L\{ {{\bar{h}}_p}\} } \right\rangle ={-} \langle {B^2}\rangle {f_0},\end{equation}

where

(3.9)\begin{equation}\left\langle {\frac{B}{{{v_{||}}}}L\{ {{\bar{h}}_p}\} } \right\rangle = \frac{{2{B_0}}}{v}\frac{\partial }{{\partial \lambda }}\left( {\lambda \langle \xi \rangle \frac{{\partial {{\bar{h}}_p}}}{{\partial \lambda }}} \right),\end{equation}

and

(3.10)\begin{equation}\left\langle {\frac{B}{{{v_{||}}}}{C_{ee}}\{ {{\bar{h}}_p}\} } \right\rangle = \frac{{2{B_0}{\nu _\ell }}}{{{x^3}v}}\frac{\partial }{{\partial \lambda }}\left( {\lambda \langle \xi \rangle \frac{{\partial {{\bar{h}}_p}}}{{\partial \lambda }}} \right) + \frac{{{\nu _\ell }{T_e}}}{{{m_e}{v^3}}}\left\langle {\frac{B}{\xi }} \right\rangle \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].\end{equation}

The last term of (3.10) proportional to ${v^{ - 3}}\partial {\bar{h}_p}/\partial v$ is drag and the remainder is energy scattering.

In the large aspect ratio limit

(3.11)\begin{equation}\langle \xi \rangle = \frac{{2\sqrt {2\varepsilon } E(k)}}{{{\rm \pi} \sqrt {(1 - \varepsilon ){k^2} + 2\varepsilon } }},\end{equation}

and

(3.12)\begin{equation}\frac{1}{{{B_0}}}\left\langle {\frac{B}{\xi }} \right\rangle ={-} 2\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }},\end{equation}

with E an elliptic integral of the second kind, ${k^2} = 2\varepsilon \lambda /[1 - (1 - \varepsilon )\lambda ]$, and $\varepsilon = r/R$ with r the minor radius. The equation to be solved is first rewritten as

(3.13)\begin{align}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}}}{{{m_e}{B_0}{v^2}{f_0}}}\left\langle {\frac{B}{\xi }} \right\rangle \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{align}

Only the lowest -order $v$ dependence is required. Inserting

(3.14)\begin{equation}\frac{{{{\bar{h}}_p}}}{{{f_0}}} = \frac{{\langle {B^2}\rangle v{x^3}}}{{I{B_0}{\nu _\ell }}}\varLambda (\lambda ),\end{equation}

leads to

(3.15)\begin{equation}2(Z + 1)\frac{\partial }{{\partial \lambda }}\left( {\lambda \langle \xi \rangle \frac{{\partial \varLambda }}{{\partial \lambda }}} \right) + \frac{{{T_e}\varLambda }}{{{m_e}{B_0}{v^3}{f_0}}}\left\langle {\frac{B}{\xi }} \right\rangle \frac{\partial }{{\partial v}}\left[ {\frac{{{v^2}{f_0}}}{{{x^3}}}\frac{\partial }{{\partial v}}(v{x^3})} \right] ={-} 1.\end{equation}

Using

(3.16)\begin{align}\frac{{{T_e}}}{{{m_e}{v^3}{f_0}}}\frac{\partial }{{\partial v}}\left[ {\frac{{{v^2}{f_0}}}{{{x^3}}}\frac{\partial }{{\partial v}}(v{x^3})} \right] = \frac{1}{{2{x^3}{e^{ - {x^2}}}}}\frac{\partial }{{\partial x}}\left( {\frac{{{e^{ - {x^2}}}}}{x}\frac{{\partial {x^4}}}{{\partial x}}} \right) = \frac{2}{{{x^3}{e^{ - {x^2}}}}}\frac{\partial }{{\partial x}}({x^2}{e^{ - {x^2}}}) ={-} 4 + \frac{4}{{{x^2}}},\end{align}

then for $x \gg 1$ drag dominates over energy scattering and the equation to be solved reduces to

(3.17)\begin{equation}\frac{\partial }{{\partial \lambda }}\left( {\lambda \langle \xi \rangle \frac{{\partial \varLambda }}{{\partial \lambda }}} \right) + \frac{4}{{Z + 1}}\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }}\varLambda ={-} \frac{1}{{2(Z + 1)}}.\end{equation}

The Cordey (Reference Cordey1976) eigenfunctions ${\varLambda _j}$ and associated eigenvalues ${\kappa _j}$ of the Sturm--Liouville differential equation

(3.18)\begin{equation}\frac{\partial }{{\partial \lambda }}\left( {\lambda \langle \xi \rangle \frac{{\partial {\Lambda_j}}}{{\partial \lambda }}} \right) = {\kappa _j}\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }}{\varLambda _j} ={-} \frac{{{\kappa _j}}}{2}\left\langle {\frac{B}{{{B_0}\xi }}} \right\rangle {\varLambda _j},\end{equation}

are used to obtain a solution. Expanding in the eigenfunctions ${\varLambda _j}$, which satisfy ${\varLambda _j}(\lambda = 0) = 1$, ${\varLambda _j}(\lambda = {B_0}/{B_{\textrm{max}}}) = 0$ and, for $j \ne k$, the orthogonality condition

(3.19)\begin{equation}\int_0^{{B_0}/{B_{max}}} {\textrm{d}\lambda } {\varLambda _k}{\varLambda _j}\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }} = 0,\end{equation}

by inserting

(3.20)\begin{equation}\varLambda = \sum\limits_{j = 1}^\infty {{A_j}{\varLambda _j}} ,\end{equation}

into the differential equation leads to

(3.21)\begin{equation}\sum\limits_{j = 1}^\infty {{A_j}\left( {{\kappa_j} + \frac{4}{{Z + 1}}\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }}} \right){\varLambda _j}} ={-} \frac{1}{{2(Z + 1)}}.\end{equation}

Multiplying by ${\varLambda _k}$ and integrating over $\lambda$, yields the coefficients ${A_j}$ to be given by

(3.22)\begin{equation}2{\alpha _j}[(Z + 1){\kappa _j} + 4]{A_j} = {\beta _j},\end{equation}

where

(3.23)\begin{equation}{\alpha _j} ={-} \int_0^{{B_0}/{B_{max}}} {\textrm{d}\lambda } \varLambda _j^2\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }}\mathop \to \limits_{\varepsilon \to 0} \frac{1}{{4j - 1}},\end{equation}

and

(3.24)\begin{equation}{\beta _j} = \int_0^{{B_0}/{B_{max}}} {\textrm{d}\lambda } {\varLambda _j}\mathop \to \limits_{\varepsilon \to 0} \frac{2}{3}.\end{equation}

Fortunately, aspect ratio expansions of the preceding eigenvalues and coefficients are available (Hsu, Catto & Sigmar Reference Hsu, Catto and Sigmar1990; Xiao, Catto & Molvig Reference Xiao, Catto and Molvig2007; Parker & Catto Reference Parker and Catto2012). In particular, the lowest eigenvalue is ${\kappa _1} \simeq 1 + 1.46\sqrt \varepsilon + 1.48\varepsilon + 0.13{\varepsilon ^{3/2}}$ so always of order unity, with the others increasing in size as for all Sturm--Liouville problems. Moreover, as $\varepsilon$ increases so do all of the ${\kappa _j}$ and the ${\beta _j}/{\alpha _j}$. These prior investigations find that only the leading few eigenfunctions are required, as might be expected from how quickly the eigenvalues increase. Being more explicit by using results from Hsu et al. (Reference Hsu, Catto and Sigmar1990),

(3.25)\begin{align}\kappa _j - (2{j^2} - j) = 1.46\sqrt \varepsilon \frac{{4j - 1}}{3}\left[\frac{{(2j - 1)!!}}{{(2j - 2)!!}}\right]^2 = 1.46\sqrt \varepsilon \left\{\begin{array}{@{}ll} 1&{j = 1}\\ {21/4}&{j = 2}\\ {825/64}&{j = 3} \end{array} \right.,\end{align}

where $(2j - 1)!! = 1 \cdot 3 \cdot 5 \ldots (2j - 1)$ and $(2j - 2)!! = 2 \cdot 4 \cdot 6 \ldots (2j - 2)$ (and both equal 1 for j = 1). Consequently, only the first couple of ${\beta _j}$ and ${\alpha _j}$ are required. For ${\beta _j}$

(3.26)\begin{align}\begin{aligned} {\beta _j} & = \int_0^{{B_0}/{B_{\max }}} {\textrm{d}\lambda } {\varLambda _j}{\kern 1pt} = \dfrac{2}{3}(1 - \varepsilon )\left[ {{a_{j1}} - {{( - 1)}^{j + 1}}\dfrac{{(2j - 1)!!}}{{(2j - 2)!!}}\varepsilon } \right]\\ & \quad + O({\varepsilon ^{3/2}}) \to \dfrac{2}{3}(1 - \varepsilon )\left\{ {\begin{array}{@{}ll} {{a_{11}} - \varepsilon }&{j = 1}\\ {{a_{21}} + 3\varepsilon /2}&{j = 2} \end{array}} \right., \end{aligned}\end{align}

where based on appendix B and the fits in table 1 of Hsu et al. (Reference Hsu, Catto and Sigmar1990)

(3.27)\begin{equation}{a_{jj}} = 1 - \sum\limits_{j \ne m} {{a_{jm}}} \mathop \to \limits_{j = 1} {a_{11}} = 1 - 0.62\sqrt \varepsilon + 1.33\varepsilon ,\end{equation}

and

(3.28)\begin{align}\begin{aligned} {a_{jm}} & \simeq{-} \dfrac{{{{( - 1)}^{j + m}}(4m - 1)1.46\sqrt \varepsilon }}{{3[2{m^2} - m - (2{j^2} - j)]}}\left[ {\dfrac{{(2j - 1)!!(2m - 1)!!}}{{(2j - 2)!!(2m - 2)!!}}} \right]\\ & \quad + O(\varepsilon )\mathop \to \limits_{\scriptstyle j = 2 \atop \scriptstyle m = 1 } {a_{21}} ={-} 0.44\sqrt \varepsilon - 0.13\varepsilon . \end{aligned}\end{align}

In addition,

(3.29)\begin{align}{\alpha _j} &={-} \int_0^{{B_0}/{B_{\max }}} {\textrm{d}\lambda } \varLambda _j^2\frac{{\partial \langle \xi \rangle }}{{\partial \lambda }} = \sum\limits_{k = 1}^\infty {\frac{{a_{jk}^2}}{{4k - 1}} - \frac{4}{3}\varepsilon {{\left[ {\frac{{(2j - 1)!!}}{{(2j - 2)!!}}} \right]}^2}}\nonumber\\ &\quad + O({\varepsilon ^{3/2}}) = \left\{\begin{array}{@{}ll} {a_{11}^2/{\kern 1pt} 3 + O(\varepsilon )}&{j = 1}\\ {1/7 + O(\varepsilon )}&{j = 2} \end{array} \right.,\end{align}

giving

(3.30)\begin{equation}{\beta _1}/{\alpha _1} \simeq 2/{a_{11}} + O(\varepsilon ) = 2/(1 - 0.62\sqrt \varepsilon ) + O(\varepsilon ),\end{equation}

and

(3.31)\begin{equation}{\beta _2}/{\alpha _2} \simeq 14{a_{21}}{\kern 1pt} /3 + O(\varepsilon ) ={-} 2.04\sqrt \varepsilon + O(\varepsilon ).\end{equation}

Using the preceding results gives the lowest -order adjoint solution to be

(3.32)\begin{align} \frac{{{{\bar{h}}_p}}}{{{f_0}}} &= \frac{{\langle {B^2}\rangle v{x^3}}}{{2I{B_0}{\nu _\ell }}}\sum\limits_{j = 1}^\infty {\frac{{{\beta _j}{\varLambda _j}(\lambda )}}{{[(Z + 1){\kappa _j} + 4]{\alpha _j}}}}\nonumber\\ &\simeq \frac{{v{x^3}}}{{R{\nu _\ell }}}\left\{ {\frac{{(1 + 0.62\sqrt \varepsilon ){\Lambda_1}(\lambda )}}{{[(Z + 1)(1 + 1.46\sqrt {2\varepsilon } ) + 4]}} - \frac{{1.02\sqrt \varepsilon {\Lambda_2}(\lambda )}}{{[(Z + 1)(7 + 7.66\sqrt {2\varepsilon } ) + 4]}}} \right\},\end{align}

which will be used to evaluate the lower hybrid driven current. Notice that even the j = 2 term is small and can be ignored. The preceding solution also ignores ${x^{ - 3}} \ll 1$ corrections as small.

As ${\bar{h}_t}{\kern 1pt} = 0$ only the passing electrons contribute to 2.20, it simplifies to

(3.33)\begin{equation}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle \simeq \frac{{{B_0}I}}{{2{\rm \pi} qR}}\int {{\textrm{d}^3}} v\frac{{{v_{||}}{{\bar{h}}_p}}}{{B{f_0}}}{\tau _p}\overline {Q\{ {f_0}\} } ,\end{equation}

where large aspect ratio is assumed and the passing transit time for a full poloidal circuit is

(3.34)\begin{equation}{\tau _p} = \oint_p {\textrm{d}\tau } = \oint_p {\textrm{d}\vartheta /{v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } \simeq 4qR\sqrt {(1 - \varepsilon ){k^2} + 2\varepsilon } K(k)/v\sqrt {2\varepsilon } ,\end{equation}

with $\varepsilon = r/R$, ${k^2} = 2\varepsilon \lambda /[1 - (1 - \varepsilon )\lambda ]$, and K an elliptic integral of the first kind. The parallel current will next be formed by evaluating (3.33).

4. Current in a tokamak for a correlated QL diffusivity

The recently derived form for the correlated diffusivity (Catto & Tolman Reference Catto and Tolman2021) for the electrons in a tokamak is

(4.1)\begin{equation}\bar{D} = \frac{{2{{\rm \pi}^3}{q^2}{R^2}{e^2}|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}}}{{m_e^2{v^2}{\tau _p}}}\sum\limits_\ell {\delta \left({\oint_p {\textrm{d}\tau \varLambda - 2{\rm \pi}\ell } } \right)\Theta (v,k)} ,\end{equation}

where the argument of the delta function is transit averaged with

(4.2)\begin{equation}\oint_p {\textrm{d}\tau \varLambda } \simeq \omega {\tau _p} - 2{\rm \pi}\sigma (qn - m),\end{equation}

allowing aspect ratio modifications to be properly evaluated. The phase integral $\Theta $ is defined as

(4.3)\begin{equation}\Theta \equiv {\left|{\oint_p {\frac{{\textrm{d}\vartheta }}{{2{\rm \pi}}}} \,{\textrm{e}^{ - i\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}\varLambda (\tau^{\prime})} }}} \right|^2} \le 1,\end{equation}

with

(4.4)\begin{equation}\int_{{\tau _0}}^\tau {\textrm{d}\tau ^{\prime}\varLambda } (\tau ^{\prime}) = \omega (\tau - {\tau _0}) - \sigma (qn - m)\vartheta (\tau ) = \omega qR\int_0^\vartheta {\textrm{d}\vartheta /{\kern 1pt} {v_{||}}} - \sigma (qn - m)\vartheta ,\end{equation}

where $\vartheta ({\tau _0}) = 0$ and

(4.5)\begin{equation}\textrm{d}\vartheta /\textrm{d}\tau = {v_{||}}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta .\end{equation}

For notational simplicity, only the poloidal mode number (m) is shown as a subscript on the Fourier coefficients of the applied rf parallel electric field ${\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}$. The frequency $(\omega )$, toroidal mode number (n) and radial mode index (s) subscripts are suppressed.

For a Maxwellian the transit averaged QL operator is

(4.6)\begin{equation}\overline {Q\{ {f_0}\} } = \sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{1}{{{\tau _p}}}\frac{\partial }{{\partial E}}\left( {{\tau_p}{v^2}\bar{D}\frac{{\partial {f_0}}}{{\partial E}}} \right)} = \sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,S } {\frac{1}{{{\tau _p}v}}{{\left. {\frac{\partial }{{\partial v}}} \right|}_\mu }} \left( {{\tau_p}v\bar{D}\frac{{\partial {f_0}}}{{\partial v}}} \right).\end{equation}

with $\ell$ the bounce harmonic index as sucessive passes are correlated. As only the passing contribute, the large aspect ratio form of the flow becomes

(4.7)\begin{align}\begin{aligned} \left\langle {B\int {{d^3}} v{v_{||}}{f_1}} \right\rangle & \simeq \dfrac{{{B_0}I}}{{2{\rm \pi} qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {{\textrm{d}^3}v} \dfrac{{{v_{||}}{{\bar{h}}_p}}}{{B{f_0}v}}{\left. {\dfrac{\partial }{{\partial v}}} \right|_\mu }\left( {v{\tau_p}\bar{D}{{\left. {\dfrac{{\partial {f_0}}}{{\partial v}}} \right|}_\mu }} \right)\\ & = \dfrac{{{m_e}{B_0}I}}{{2{\rm \pi}{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\int {{\textrm{d}^3}v} } \dfrac{{{v_{||}}{\tau _p}}}{{Bv}}\bar{D}{f_0}{\left. {\dfrac{\partial }{{\partial v}}} \right|_\mu }\left( {\dfrac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right).\end{aligned}\end{align}

To proceed it is convenient to perform the v integral first by writing

(4.8)\begin{equation}\bar{D} = {v^{ - 2}}\tau _p^{ - 1}\sum\limits_\ell {{d_\ell }\Theta (v,k)\delta ({\oint_p {\textrm{d}\tau \varLambda - 2{\rm \pi}\ell } } )} ,\end{equation}

with

(4.9)\begin{equation}{d_\ell } \equiv 2{{\rm \pi}^3}{q^2}{R^2}{e^2}m_e^{ - 2}|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}.\end{equation}

Then

(4.10)\begin{equation}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle = \frac{{{m_e}{B_0}I}}{{2{\rm \pi}{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {{d_\ell }} \int {{\textrm{d}^3}} v\frac{{{v_{||}}{f_0}}}{{B{v^3}}}\Theta (v,k)\delta ({\oint_p {\textrm{d}\tau \varLambda - 2{\rm \pi}\ell } } ){\left. {\frac{\partial }{{\partial v}}} \right|_\mu }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right).\end{equation}

Taylor expanding the argument of the delta function leads to

(4.11)\begin{equation}\delta ({\oint_p {\textrm{d}\tau \varLambda - 2{\rm \pi}\ell } } )= \frac{{\delta (v - {v_k})}}{{\omega |\partial {\tau _p}/\partial v{\kern 1pt} {|_{{v_k}}}}} = \frac{{v_k^2\delta (v - {v_k})}}{{\omega v{\tau _p}}},\end{equation}

with the speed at resonance defined as

(4.12)\begin{equation}{v_k} \equiv \omega v{\tau _p}/2{\rm \pi}(\ell + |qn - m|).\end{equation}

As a result,

(4.13)\begin{equation}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle = \frac{{{m_e}{B_0}I}}{{2{\rm \pi} {T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {{d_\ell }} \int {{\textrm{d}^3}v} \frac{{{v_{||}}{f_0}}}{{Bv}}\frac{{\Theta (v,k)}}{{\omega v{\tau _p}}}\delta (v - {v_k}){\left. {\frac{\partial }{{\partial v}}} \right|_\mu }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right).\end{equation}

Using ${\nabla _v}v = \boldsymbol{v}/v$, ${\nabla _v}\lambda = (2{B_0}{\kern 1pt} /B{v^2}){\boldsymbol{v}_ \bot } - (2\lambda /{v^2})\boldsymbol{v}$, and ${\nabla _v}\varphi = v_ \bot ^{ - 2}\boldsymbol{n} \times \boldsymbol{v}$, gives ${\textrm{d}^3}v = \textrm{d}v\,\textrm{d}\lambda \textrm{d}\varphi /{\nabla _v}v \times {\nabla _v}\lambda \boldsymbol{\cdot }{\boldsymbol{\nabla }_v}\varphi = \textrm{d}v\,\textrm{d}\lambda \,\textrm{d}\varphi B{v^3}/{\kern 1pt} 2{B_0}{v_{\|}}$. The LH parallel electron flow must be driven in the negative direction to sustain the poloidal magnetic field, making $\left\langle {B\int {{\textrm{d}^3}} v{v_{\|}}{f_1}} \right\rangle < 0$. Hence, a minus sign must be inserted as the velocity space integration is only over ${v_{\|}} < 0$ electrons with $\bar{D}$ vanishing for ${v_{\|}} > 0$. Consequently, in the resonance condition, $\sigma = - 1$ and m > qn. Therefore, the magnitude of the parallel wave number $|{k_{\|}}| \equiv |qn - m|/qR$, is used as ${k_{\|}} < 0$.

Multiplying the velocity integral by 1/2 and inserting the diffusivity

(4.14)\begin{align}\begin{aligned} \left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle & = \dfrac{{ - {m_e}I}}{{4{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {{d_\ell }} \int_0^{{B_0}/{B_{\textrm{max}}}} {\textrm{d}\lambda } \int_0^\infty {\textrm{d}v\Theta (v,k)\delta (v - {v_k})} \dfrac{{{v^2}{f_0}}}{{\omega v{\tau _p}}}{\left. {\dfrac{\partial }{{\partial v}}} \right|_\mu }\left( {\dfrac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right)\\ & ={-} \dfrac{{{m_e}I}}{{4{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {{d_\ell }} \int_0^{{B_0}/{B_{\textrm{max}}}} {\textrm{d}\lambda } \Theta ({v_k},k)\dfrac{{{v^2}{d_\ell }{f_0}}}{{\omega v{\tau _p}}}{\left. {\dfrac{\partial }{{\partial v}}} \right|_\mu }{\left. {\left( {\dfrac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right|_{v = {v_k}}}. \end{aligned}\end{align}

The preceding equation is a convenient form of the parallel flow and can be used to evaluate the driven parallel electron current for a more general like particle collision operator than was considered in the preceding section.

Defining

(4.15)\begin{equation}x_k^2 = {m_e}v_k^2/2{T_e} = {m_e}{(\omega v{\tau _p})^2}/8{{\rm \pi}^2}{(\ell + |qn - m|)^2}{T_e},\end{equation}

with $x_k^2 \gg 1$ in ${f_0} \propto {\textrm{e}^{ - x_k^2}}$, the $v{\tau _p}{\kern 1pt} \propto K(k)\mathop \to \limits_{k \to 1} \infty$ indicates that the freely passing will dominate the driven current. Consequently, $x_k^2 \gg 1$ is used to integrate by parts by first noting

(4.16)\begin{equation}{\left. {\frac{{\partial {f_0}}}{{\partial \lambda }}} \right|_v} ={-} {f_0}{\left. {\frac{{\partial x_k^2}}{{\partial \lambda }}} \right|_v} ={-} {f_0}\frac{{{m_e}{\omega ^2}v{\tau _p}}}{{4{{\rm \pi}^2}{T_e}{{(\ell + |{\kern 1pt} qn - m|)}^2}}}{\left. {\frac{{\partial (v{\tau_p})}}{{\partial \lambda }}} \right|_v}.\end{equation}

Then ignoring $x_k^{ - 2} \ll 1$ corrections

(4.17) \begin{align}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle &={-} \frac{{{{\rm \pi}^2}I}}{{qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {\frac{{{d_\ell }{{(\ell + |{\kern 1pt} qn - m|)}^2}{f_0}}}{{\omega {{(\omega {\tau _p})}^2}\partial (v{\tau _p})/\partial \lambda }}} \Theta ({x_k},k = 0)\nonumber\\&\quad \times{\left. {\left[{{{\left. {\frac{\partial }{{\partial v}}} \right|}_\lambda }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right) - {{\left. {2\lambda v\frac{\partial }{{\partial \lambda }}} \right|}_v}\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right]} \right|_{\scriptstyle v = {v_k} \atop \scriptstyle\lambda = 0 = k }} + \cdots .\end{align}

Changing to v and $\lambda$ velocity variables using

(4.18)\begin{equation}{\left. {\frac{\partial }{{\partial v}}} \right|_\mu } = {\left. {\frac{\partial }{{\partial v}}} \right|_\lambda } + {\left. {\frac{{\partial \lambda }}{{\partial v}}} \right|_\mu }{\left. {\frac{\partial }{{\partial \lambda }}} \right|_{{v_{||}}}} = {\left. {\frac{\partial }{{\partial v}}} \right|_\lambda } - \frac{{2\lambda }}{v}{\left. {\frac{\partial }{{\partial \lambda }}} \right|_v},\end{equation}

gives

(4.19)\begin{equation}{\left. {\frac{\partial }{{\partial v}}} \right|_\mu }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right) = {\left. {\frac{\partial }{{\partial v}}} \right|_\lambda }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right) - {\left. {2\lambda v\frac{\partial }{{\partial \lambda }}} \right|_v}\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right).\end{equation}

Therefore,

(4.20) \begin{align} \left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle &={-} \frac{{{{\rm \pi}^2}I}}{{qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {\frac{{{d_\ell }{{(\ell + |{\kern 1pt} qn - m|)}^2}{f_0}}}{{\omega {{(\omega {\tau _p})}^2}\partial (v{\tau _p})/\partial \lambda }}} \Theta ({x_k},k = 0)\nonumber\\&\quad \times{\left. {\left[ {{{\left. {\frac{\partial }{{\partial v}}} \right|}_\lambda }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right) - {{\left. {2\lambda v\frac{\partial }{{\partial \lambda }}} \right|}_v}\left( {\frac{{{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right]} \right|_{\scriptstyle v = {v_k} \atop \scriptstyle\lambda = 0 = k }} + \cdots .\end{align}

As the pitch angle derivative of ${\bar{h}_p}$ is well behaved at $\lambda = 0$ the last term vanishes leading to a lower hybrid driven current ${J_{\|}}$ of

(4.21) \begin{equation}{J_{||}} \equiv{-} eB_0^{ - 1}\left\langle {B\int {{\textrm{d}^3}} v{v_{||}}{f_1}} \right\rangle \simeq{-} \frac{{{{\rm \pi}^2}e}}{q}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {\frac{{{d_\ell }{{(\ell + |{\kern 1pt} qn - m|)}^2}{f_0}}}{{\omega {{(\omega {\tau _p})}^2}\partial (v{\tau _p})/\partial \lambda }}} \Theta ({x_k},k = 0){\left. {{{\left. {\frac{\partial }{{\partial v}}} \right|}_\lambda }\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right|_{\scriptstyle v = {v_k} \atop \scriptstyle\lambda = 0 = k }}.\end{equation}

To simplify further

(4.22)\begin{equation}{(v{\tau _p})_{k = 0}}{\kern 1pt} = 2{\rm \pi} qR,\end{equation}

is used along with

(4.23) \begin{align}\frac{1}{{{\tau _p}}}{\left. {\frac{{\partial {\tau_p}}}{{{\kern 1pt} \partial \lambda }}} \right|_{\lambda = 0 = k}} &= \frac{{{{[(1 - \varepsilon ){k^2} + 2\varepsilon ]}^2}}}{{4\varepsilon {k^2}}}{\left. {\left[ {\frac{E}{{(1 - {k^2})K}} - K + \frac{{(1 - \varepsilon ){k^2}}}{{(1 - \varepsilon ){k^2} + 2\varepsilon }}} \right]} \right|_{k = 0}}\nonumber\\ &= \frac{\varepsilon }{{{k^2}}}\left[ {\frac{{{k^2}}}{2} + \frac{{(1 - \varepsilon ){k^2}}}{{2\varepsilon }}} \right] \simeq \frac{1}{2}.\end{align}

Moreover, only $\ell = 0$ contributes as

(4.24) \begin{align}\Theta (k = 0,{x_k} = {x_{k = 0}}) &\equiv {\left|{\oint_p {\frac{{\textrm{d}\vartheta }}{{2{\rm \pi}}}\,} {\textrm{e}^{ - \textrm{i}\int_{{\tau_0}}^\tau {\textrm{d}\tau^{\prime}} \varLambda (\tau^{\prime})}}} \right|^2}\nonumber\\ &\simeq {\left|{\oint_p {\frac{{\textrm{d}\vartheta }}{{2{\rm \pi}}}\,} {\textrm{e}^{ - \textrm{i}(\omega qR/v - |qn - m|)\vartheta }}} \right|^2} \simeq {\left|{\oint_p {\frac{{d\vartheta }}{{2{\rm \pi}}}\,} {\textrm{e}^{ - \textrm{i}\ell \vartheta }}} \right|^2} = {\delta _{0\ell }},\end{align}

giving

(4.25)\begin{equation}{v_{k = 0}} = \omega {(v{\tau _p})_{k = 0}}/{\kern 1pt} 2{\rm \pi} |qn - m|= \omega qR/|qn - m|.\end{equation}

Consequently, the LH current becomes

(4.26)\begin{equation}{J_{||}} \simeq \frac{{{{\rm \pi}^{1/2}}{n_e}R}}{{2v_e^3}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{{e^3}|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}}}{{m_e^2\omega }}} \,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}{\left. {\frac{\partial }{{\partial v}}} \right|_\lambda }{\left. {\left( {\frac{{{v^2}{{\bar{h}}_p}}}{{{f_0}}}} \right)} \right|_{\scriptstyle v = {v_k} \atop \scriptstyle\lambda = 0 = k }}.\end{equation}

Keeping only the j = 1 term in ${\bar{h}_p}$ yields the final result for the driven lower hybrid current in a tokamak to be

(4.27)\begin{equation}{J_{||}} \simeq \frac{{4e{n_e}(1 + 0.62\sqrt \varepsilon )}}{{[(Z + 1)(1 + 2.06\sqrt \varepsilon ) + 4]{\nu _{ee}}}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{{e^2}|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^4}}}{{m_e^2|{\kern 1pt} {k_{||}}{\kern 1pt} {|^5}v_e^6}}} \,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}.\end{equation}

Importantly, the key result in the preceding expression is that it retains the proper leading order analytic dependence on the electron trapping parameter $\sqrt \varepsilon $ for the first time. Notice in particular that the aspect ratio dependence of the Z + 1 pitch angle scattering factor differs from that of the electron-electron energy scattering factor 4. Moreover, the overall current drive efficiency is reduced somewhat by the inverse aspect ratio dependences as the increase in the numerator is unable to overcome the decrease from the denominator. Presumably similar behaviour persists for more general cross sections and will make off-axis current profile control slightly more difficult in spherical tokamaks. Perhaps not surprisingly, energy scattering is less affected by toroidal geometry that pitch angle scattering, which is the reason only the leading Cordey (Reference Cordey1976) eigenfunction need be retained. As the freely passing dominate the electron response, (4.27) reduces to the standard $\sqrt \varepsilon {\kern 1pt} \to 0$ result. The driven parallel current (4.27) will be used in the next section to evaluate the current drive efficiency.

5. RF power and current drive efficiency

The rf power required to drive the lower hybrid current, ${P_{cd}}$, is found from

(5.1)\begin{equation}{P_{cd}} = \frac{{{m_e}}}{2}\left\langle {\int {{\textrm{d}^3}v{v^2}} \overline {Q\{ {f_0}\} } } \right\rangle = \frac{{{m_e}}}{2}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,S } {\left\langle {\int {{\textrm{d}^3}v{v^2}} \frac{1}{{{\tau_p}v}}{{\left. {\frac{\partial }{{\partial v}}} \right|}_\mu }\left( {{\tau_p}v\bar{D}\frac{{\partial {f_0}}}{{\partial v}}} \right)} \right\rangle } .\end{equation}

Only the passing electrons contribute as there is no resonance for the trapped since

(5.2)\begin{equation}\oint_t {\textrm{d}\tau } \varLambda \simeq \omega {\tau _t},\end{equation}

and

(5.3)\begin{equation}\Theta \propto {|{\oint_t {\textrm{d}\tau {v_{||}}} \,{\textrm{e}^{ - \textrm{i}\omega (\tau - {\tau_0})}}} |^2} \to 0.\end{equation}

Recalling ${\textrm{d}^3}v = \textrm{d}v\,\textrm{d}\lambda \,\textrm{d}\varphi B{v^3}/2{B_0}{v_{\|}}$, and using $\oint {\textrm{d}\vartheta /\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\vartheta } \simeq 2{\rm \pi} qR$

(5.4)\begin{equation}{P_{cd}} = \frac{{{B_0}{m_e}}}{{4{\rm \pi} qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,S } {\int {{\textrm{d}^3}v} } \frac{{{v_{||}}v}}{B}{\left. {\frac{\partial }{{\partial v}}} \right|_\mu }\left( {{\tau_p}v\bar{D}\frac{{\partial {f_0}}}{{\partial v}}} \right) = \frac{{{B_0}m_e^2}}{{{\rm \pi}{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,S } {\int {{\textrm{d}^3}v} } \frac{{{v_{||}}}}{B}{\tau _p}{v^2}\bar{D}{f_0}.\end{equation}

Multiplying by 1/2 since only ${v_{\|}} < 0$ contribute, the preceding becomes

(5.5)\begin{equation}{P_{cd}} = \frac{{m_e^2}}{{2{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,S } {\int_0^{{B_0}/{B_{\textrm{min}}}} {\textrm{d}\lambda } } \int_0^\infty {\textrm{d}v} {\tau _p}{v^5}\bar{D}{f_0}.\end{equation}

Inserting $\bar{D}$

(5.6) \begin{align}\begin{aligned} {P_{cd}} & = \dfrac{{m_e^2}}{{2{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {\dfrac{{{d_\ell }}}{\omega }} \int_0^{{B_0}/{B_{\textrm{max}}}} {\dfrac{{\textrm{d}\lambda }}{{v{\tau _p}}}} \int_0^\infty {\textrm{d}v} {v^5}{f_0}\Theta (v,k)\delta (v - {v_k})\\ &= \dfrac{{m_e^2}}{{2{T_e}qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s,\ell } {\dfrac{{{d_\ell }}}{\omega }} \int_0^{{B_0}/{B_{\textrm{max}}}} {\dfrac{{\textrm{d}\lambda }}{{v{\tau _p}}}} {v^5}{ {{f_0}\Theta (v,k)} |_{v = {v_k}}}\\ & ={-} \dfrac{{2{{\rm \pi}^2}{m_e}}}{{qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\dfrac{{{d_{\ell = 0}}|{\kern 1pt} qn - m{|^2}}}{{{\omega ^3}}}} \int_0^{{B_0}/{B_{\textrm{max}}}} {\dfrac{{\textrm{d}\lambda }}{{{{(v{\tau _p})}^2}}}} \dfrac{{{v^5}\Theta (v,k)}}{{\partial (v{\tau _p}){\kern 1pt} /{\kern 1pt} \partial \lambda }}{\left. {\dfrac{{\partial {f_0}}}{{\partial \lambda }}} \right|_{v = {v_k}}}. \end{aligned}\end{align}

Integrating by parts and noting that only $\ell = 0{\kern 1pt}$ contributes, the rf power density required to drive the lower hybrid current ${J_{\|}}$ is just

(5.7)\begin{equation}{P_{cd}} \simeq \frac{{2{{\rm \pi}^2}{m_e}}}{{qR}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{{d_{\ell = 0}}|qn - m{|^2}{v^5}}}{{{\omega ^3}{{(v{\tau _p})}^2}\partial (v{\tau _p})/\partial \lambda }}} { {{f_0}} |_{v = {v_k}}} = {{\rm \pi}^{1/2}}{n_e}{m_e}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{{e^2}|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^2}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{m_e^2|{k_{||}}{|^3}v_e^3}}} .\end{equation}

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

(5.8) \begin{equation}\frac{{{J_{||}}}}{{{P_{cd}}}} = \frac{\displaystyle{4e(1 + 0.62\sqrt {2\varepsilon } )\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^4}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|{k_{||}}{|^5}v_e^6}}} }}{\displaystyle{\sqrt {\rm \pi} {m_e}[(Z + 1)(1 + 2.06\sqrt \varepsilon ) + 4]{\nu _{ee}}\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^2}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|{k_{||}}{|^3}v_e^3}}} }},\end{equation}

or in its normalized form

(5.9) \begin{equation}\frac{{{J_{||}}/e{n_e}{v_e}}}{{{P_{cd}}/{n_e}{m_e}v_e^2{\nu _{ee}}}} \equiv \eta = \frac{\displaystyle{4(1 + 0.62\sqrt {2\varepsilon } )\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^4}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|{k_{||}}{|^5}v_e^5}}} }}{\displaystyle{\sqrt {\rm \pi} [(Z + 1)(1 + 2.06\sqrt \varepsilon ) + 4]\sum\limits_{\scriptstyle\omega ,m \atop \scriptstyle n,s } {\frac{{|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^2}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|{k_{||}}{|^3}v_e^3}}} }}.\end{equation}

Various normalizations of ${J_{\|}}$ and ${P_{cd}}$ (often because of a $\sqrt 2$ difference in the definition of the electron thermal speed), and differing definitions of collision frequency appear in the LHCD literature, as well as incompletely defined notation.

For a single frequency and single toroidal mode number (5.9) leads to the dimensionless current drive efficiency with electron trapping retained of

(5.10) \begin{align} \eta &= \frac{\displaystyle{4(1 + 0.62\sqrt {2\varepsilon } )\sum\limits_{m,s} {\frac{{|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^5}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|{k_{||}}{|^5}v_e^5}}} }}{\displaystyle{\sqrt {\rm \pi} [(Z + 1)(1 + 2.06\sqrt \varepsilon ) + 4]\sum\limits_{m,s} {\frac{{|{\boldsymbol{e}_m}\boldsymbol{\cdot }\boldsymbol{n}{|^2}{\omega ^3}\,{\textrm{e}^{ - {\omega ^2}/k_{||}^2v_e^2}}}}{{|{k_{||}}{|^3}v_e^3}}} }}\nonumber\\ &\to \frac{{4(1 + 0.62\sqrt {2\varepsilon } ){\omega ^2}}}{{\sqrt {\rm \pi}[(Z + 1)(1 + 2.06\sqrt \varepsilon ) + 4]|{\kern 1pt} {k_{||}}{|^2}v_e^2}},\end{align}

where the sums over poloidal mode number and radial mode structure (either as a Fourier or eikonal representation) are retained except in the final form.