2. Nonlinear electron kinetic equation

Consider an applied lower hybrid wave of frequency $\omega \ll \Omega ={eB}/{mc}$ in a uniform plasma with a constant magnetic field $\vec {{B}}={B}\vec {{z}}$ , where B is the magnitude of the magnetic field and $\vec {{z}}$ is the unit vector in the direction of the uniform field. For a lowest order Maxwellian distribution function,

(2.1) \begin{equation}{f}_{0}={n}({m}/2\pi {T})^{3/2}{e}^{-{m}{{v}^{2}}/2{T}},\end{equation}

the perturbed distribution function ${f}_{1}$ satisfies the nonlinear electron drift kinetic equation

(2.2) \begin{equation}\partial {f}_{1}/\partial {t}+{v}_{||}\partial {f}_{1}/\partial {z}-({e}/{m}){E}_{||}\partial {f}_{1}/\partial {v}_{||}-{C}\{{f}_{1}\}=-({e}/{T}){E}_{||}{v}_{||}{f}_{0}\end{equation}

in ${v}_{||}$ and ${v}_{\bot }$ velocity variables with ${v}^{2}={v}_{||}^{2}+{v}_{\bot }^{2}$ . Here n and T are the electron density and temperature, and ${f}_{0}+{f}_{1}={f}$ with ${f}_{0}\gg {f}_{1}$ . The perturbing parallel applied electric field of the lower hybrid wave is taken to be $\vec {{E}}={E}_{||}\vec {{z}}$ , with the unimportant unperturbed electric field neglected for simplicity.

The collision operator is needed to resolve singular behaviour at a resonance or in the vicinity of small scale velocity space structure. It is approximated by the usual like and unlike terms, namely the high speed expansion ( ${v}^{2}\gg {v}_{{e}}^{2}=2{T}/{m})$ , self adjoint operator

(2.3) \begin{align}{C}\{{f}\} & =\nabla _{{v}}\cdot \left\{\frac{\nu _{{e}}}{2{x}^{3}}\left[\Big({v}^{2}{\mathop{I}\limits^{\leftrightarrow}}-\vec {{v}}\vec {{v}}\Big)\cdot \nabla _{{v}}{f}_{1}+\frac{2{Tf}_{0}}{({Z}+1){m}}\nabla _{{v}}\left(\frac{{f}_{1}}{{f}_{0}}\right)\right]\right\}\nonumber\\& \approx \frac{\nu _{{e}}}{2{x}^{3}}\left[{v}_{\bot }^{2}+\frac{{v}_{{e}}^{2}}{({Z}+1)}\right]\frac{\partial ^{2}{f}_{1}}{\partial {v}_{||}^{2}},\end{align}

where ${x}={v}/{v}_{{e}}$ , $\nu _{{e}}=3\sqrt{\pi }({Z}+1)\nu _{{ee}}/4$ and ${\nu _{{ee}}}=\nu _{{ei}}/{Z}=4\sqrt{2\pi }{e}^{4}{n}\ell \mathrm{n}\,\Lambda _{{c}}/3{m}^{1/2}{T}^{3/2}$ for a quasineutral plasma, with Z the ion charge number and $\ell \mathrm{n}\,\Lambda _{c}$ the Coulomb logarithm. The final form of ${C}\{{f}\}$ is all that is required near the resonance. The high speed expansion of the collision operator was used for the original LHCD calculations (Karney & Fisch Reference Karney and Fisch1979, Reference Karney and Fisch1985) and was also used for a recent linearised treatment of LHCD in a tokamak (Catto & Zhou Reference Catto and Zhou2023) which found that $\omega ^{2}/{k}_{||}^{2}{v}_{{e}}^{2}\approx 5/2$ maximises current drive.

To make analytic progress the applied electric field is assumed to be monochromatic with a wave frequency $\omega \gt 0$ , and a parallel wave vector ${k}_{||}\gt 0$ :

(2.4) \begin{equation}{E}_{||}=\tilde{{E}}_{||}\sin ({k}_{||}{z}-\omega {t}).\end{equation}

If only the linearised equation for ${f}_{1}$ were to be solved as in standard QL theory, it would be adequate to let ${f}_{1}=\mathfrak{I}\mathrm{m}[\tilde{{f}}{\mathrm{e}}^{{i}({{k}_{||}}{z}-\omega {t})}]$ , with $\mathfrak{I}\mathrm{m}$ denoting imaginary part, and solve

(2.5) \begin{equation}\mathrm{i}\big(\omega -{k}_{||}{v}_{||}\big)\tilde{{f}}+\nu {v}_{\bot {z}}^{2}\partial ^{2}\tilde{{f}}/\partial {v}_{||}^{2}=({e}/{T})\tilde{{E}}_{||}{v}_{||}{f}_{0}\approx \big({e}\omega \tilde{{E}}_{||}/{Tk}_{||}\big){f}_{0},\end{equation}

where only pitch angle scattering need be retained to resolve the singularity at $\omega ={k}_{||}{v}_{||}$ , ${v}_{\bot {z}}^{2}={v}_{\bot }^{2}+{v}_{{e}}^{2}/({Z}+1)$ and $\nu =\nu _{{e}}/2{x}^{3}$ . The solution is (Su & Oberman Reference Su and Oberman1968; Johnston Reference Johnston1971; Auerbach Reference Auerbach1977; Catto Reference Catto2020; Catto & Tolman Reference Catto and Tolman2021b )

(2.6) \begin{equation}\tilde{{f}}=-\frac{{e}\omega \tilde{{E}}_{||}{f}_{0}}{{T}\Big({k}_{||}^{5}{v}_{\bot {z}}^{2}\nu \Big)^{1/3}}\int _{0}^{{\infty }}\mathrm{d}\tau \mathrm{e}^{-{\mathrm{is}}\tau -{\tau ^{3}}/3},\end{equation}

with ${s}=({k}_{||}/{v}_{\bot {z}}^{2}\nu )^{1/3}({v}_{||}-\omega /{k}_{||})$ and ${s}=1$ giving a velocity space collisional boundary layer width of ${v}_{||}-\omega /{k}_{||}=(\Delta {v}_{||})_{\nu }=\big|\nu {v}_{\bot {z}}^{2}/{{k}_{||}}\big|^{1/3}$ .

When the applied lower hybrid wave amplitude becomes large enough, the nonlinear term cannot be neglected and the full nonlinear drift kinetic equation must be considered by letting ${f}={{f}_{0}}+{f}_{1}$ . Fortunately, ${f}_{0}$ is slowly varying as is the ${v}_{\bot }$ dependence of ${f}_{1}$ because the resonance involves only ${v}_{||}$ . Therefore, the ordering

(2.7) \begin{equation}\partial {f}_{1}/\partial {v}_{||}\sim \partial {f}_{0}/\partial {v}_{||}\end{equation}

is employed. Assuming a homogeneous plasma such that

(2.8) \begin{equation}{f}_{1}={f}_{1}(\phi ,{v}_{||},{v}_{\bot }),\end{equation}

with ${v}_{\bot }$ entering as a parameter, and defining

(2.9) \begin{equation}\phi ={k}_{||}{z}-\omega {t}\end{equation}

leads to

(2.10) \begin{equation}({k}_{||}{v}_{||}-\omega )\partial {f}_{1}/\partial \phi -({e}\tilde{{E}}_{||}/{m})\sin \phi \partial ({f}_{0}+{f}_{1})/\partial {v}_{||}={C}\{{f}_{1}\}.\end{equation}

Now only the last form of (2.3) matters. Letting ${v}_{||}=\omega /{k}_{||}+{u}$ , with u small, leads to

(2.11) \begin{equation}{k}_{||}{u}\frac{\partial {f}_{1}}{\partial \phi }-\frac{{e}\tilde{{E}}_{||}}{{m}}\sin \phi \left(\frac{\partial {f}_{0}}{\partial {v}_{||}}+\frac{\partial {f}_{1}}{\partial {u}}\right)=\nu {v}_{\bot {z}}^{2}\frac{\partial ^{2}{f}_{1}}{\partial {u}^{2}}.\end{equation}

To cast this equation into the form considered by Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023), let

(2.12) \begin{equation}{f}_{1}={g}({u},\phi )-({u}-\sigma \alpha )\partial {f}_{0}/\partial {v}_{||},\end{equation}

where $\sigma ={u}/|{u}|=\pm 1$ or 0, and $\alpha$ is a constant to be determined, to find

(2.13) \begin{equation}{k}_{||}{u}\frac{\partial {g}}{\partial \phi }-\frac{{e}\tilde{{E}}_{||}}{{m}}\sin \phi \frac{\partial {g}}{\partial {u}}=\nu {v}_{\bot {z}}^{2}\frac{\partial ^{2}{g}}{\partial {u}^{2}}.\end{equation}

Letting

(2.14) \begin{equation}{j}=\big|{m}/{e}\tilde{{E}}_{||}{k}_{||}\big|^{1/2}{k}_{||}{u}=\big|{m}/{e}\tilde{{E}}_{||}{k}_{||}\big|^{1/2}\big({k}_{||}{v}_{||}-\omega \big)\end{equation}

and

(2.15) \begin{equation}\unicode{x1D6E5} =\nu {k}_{||}^{2}{v}_{\bot {z}}^{2}\left|\frac{{m}}{{e}\tilde{{E}}_{||}{k}_{||}}\right|^{3/2}\gt 0\end{equation}

gives the equation to be the same as that solved numerically by Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023):

(2.16) \begin{equation}{j}\frac{\partial {g}}{\partial \phi }-\sin \phi \frac{\partial {g}}{\partial {j}}=\Delta \frac{\partial ^{2}{g}}{\partial {j}^{2}}.\end{equation}

For j ∼ 1, (2.14) gives the velocity space island width to be

(2.17) \begin{equation}\big(\Delta {v}_{||}\big)_{\mathrm{is}}=\left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}}\right|^{1/2}\ll \frac{\omega }{{k}_{||}}.\end{equation}

QL theory assumes that the collisional boundary layer width

(2.18) \begin{equation}\big(\Delta {v}_{||}\big)_{\nu }=\big|\nu {v}_{\bot {z}}^{2}/{{k}_{||}}\big|^{1/3}\end{equation}

is wider than this velocity space island structure, requiring $\unicode{x1D6E5} =[(\Delta {v}_{||})_{\nu }/$ $(\Delta {v}_{||})_{\mathrm{is}}]^{3}\gg 1$ . However, for a strong applied RF field at low enough density the QL assumption (2.17) fails. In the next section an analytic solution to (2.16) is found in the intense-field limit $\unicode{x1D6E5} \ll 1$ .

3. Solution of the kinetic equation in the intense field limit

The limit where the velocity space island and collisional boundary layer widths are comparable ( $\unicode{x1D6E5} \sim 1$ ) is not analytically tractable. However, when the velocity space island width is small, | $(\Delta {v}_{||})_{\mathrm{is}}/{v}_{{e}}|^{3}\ll 1$ , but larger than the collisional boundary layer width,

(3.1) \begin{equation}1\gg \left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|^{3/2}\gg \frac{\nu }{{{k}_{||}}{v}_{{e}}},\end{equation}

it becomes possible to analytically investigate the failure of QL treatments of LHCD at low density, high temperature and large applied wave amplitudes.

The preceding nonlinear kinetic equation (2.16) with a temporal evolution term has been solved numerically by Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023) for an astrophysical application. They also provided a partial analytic steady solution for $\unicode{x1D6E5} \ll 1$ , which was completed by Catto (Reference Catto2024) for a stellarator transport evaluation. The solution is skew symmetric, satisfying

(3.2) \begin{equation}{g}({j},\phi )=-{g}({-}{j},-\phi ).\end{equation}

The analytic treatment starts by introducing the reduced constant of the motion or Hamiltonian h defined by

(3.3) \begin{equation}{h}({j},\phi )={j}^{2}/2-\cos \phi\end{equation}

with h = 1 the separatrix between the bound or librating $({-}1 \lt h \lt 1)$ and the unbound or circulating $(h > 1)$ electrons (to avoid confusion, the terminology trapped and passing is avoided and reserved for magnetic wells). The reduced Hamiltonian allows the kinetic equation to be rewritten in terms of $h,\phi$ variables as

(3.4) \begin{equation}\left.\frac{\partial {g}}{\partial \phi }\right| _{{h}}=\Delta \left.\frac{\partial }{\partial {h}}\right| _{\phi }\left({j}\left.\frac{\partial {g}}{\partial {h}}\right| _{\phi }\right).\end{equation}

Then $\unicode{x1D6E5} \ll 1$ suggests a solution of the form ${g}={g}_{1}({h})+{g}_{2}({h},\phi )+\cdots$ to satisfy $\partial {g}_{1}/\partial \phi |_{{h}}=0$ to lowest order. The next order equation,

(3.5) \begin{equation}\left.\frac{\partial {g}_{2}}{\partial \phi }\right| _{{h}}=\Delta \left.\frac{\partial }{\partial {h}}\right| _{\phi }\left({j}\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }\right),\end{equation}

then leads to the solubility constraint from the collision operator, namely

(3.6) \begin{equation}\left.\frac{\partial }{\partial {h}}\right| _{\phi }\left[\left(\oint _{{h}}\mathrm{d}\phi {j}\right)\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }\right]=0.\end{equation}

Therefore, ${g}_{1}$ is independent of collision frequency, but its form is constrained by collisions.

For the bound or librating electrons ${g}_{1}=0=\alpha =\sigma$ , giving ${f}_{1}={u}\partial {f}_{0}/\partial {v}_{||}$ as the only acceptable and well behaved solution. For the freely circulating or unbound electrons, far from the resonance, a solution with ${g}_{1}\rightarrow ({u}-\sigma \alpha )\partial {f}_{0}/\partial {v}_{||}$ is required, where $\sigma =\pm {u}/|{u}|=\pm {j}/|{j}|$ and $\alpha$ is a constant still to be determined. Introducing the complete elliptic function of the second kind, E, leads to (Hamilton et al. Reference Hamilton, Tolman, Arzamasskiy and Duarte2023; Catto Reference Catto2024)

(3.7) \begin{align}{g}_{1} & =\sigma \pi \left|\frac{{eE}_{||}}{{mk}_{||}}\right|^{1/2}\frac{\partial {f}_{0}}{\partial {v}_{||}}\int _{{k}}^{1}\frac{\mathrm{d}t}{{t}^{2}{E}\left({t}\right)}\underset{{k}\rightarrow 0}{\rightarrow }\sigma \left|\frac{{eE}_{||}}{{mk}_{||}}\right|^{1/2}\frac{\partial {f}_{0}}{\partial {v}_{||}}\left[\frac{2}{{k}}-1.379-\frac{{k}}{2}+{O}({k}^{3})\right]\nonumber\\& \approx \left|\frac{{eE}_{||}}{{mk}_{||}}\right|^{1/2}\frac{\partial {f}_{0}}{\partial {v}_{||}}\left[{j}-1.379\sigma -\frac{\cos \phi }{{j}}\right]=\left({u}-\sigma \alpha -\frac{{e}\tilde{{E}}_{||}\cos \phi }{{mk}_{||}{u}}\right)\frac{\partial {f}_{0}}{\partial {v}_{||}},\end{align}

with ${j}=\sqrt{2({h}+\cos \phi )}=(2/{k})\sqrt{1-{k}^{2}\mathrm{sin}^{2}(\phi /2)}$ , ${k}=\sqrt{2/({h}+1)}$ , $\int _{-\pi }^{\pi }\mathrm{d}\varphi {j}=\sigma 8{k}^{-1}{E}({k})$ and $\alpha =1.379$ now determined. The unbound solution satisfies ${g}_{1}\rightarrow 0$ at the separatrix (h = 1), but $\partial {g}_{1}/\partial {h}|_{\phi }$ and $f_1$ step across it. The narrow collisional boundary layer about the separatrix provides the smooth matching as in the banana regime of neoclassical theory. Moreover, far from the resonance layer ${f}_{1}({k}\rightarrow 0)\rightarrow 0$ .

Summarising, the full solution to lowest order (Hamilton et al. Reference Hamilton, Tolman, Arzamasskiy and Duarte2023; Catto Reference Catto2024) is

(3.8) \begin{equation}{f}_{1}=\left|\frac{{eE}_{||}}{{mk}_{||}}\right|^{1/2}\frac{\partial {f}_{0}}{\partial {v}_{||}}\left[\sigma \pi \int _{{k}}^{1}\frac{\mathrm{d}\tau }{\tau ^{2}{E}\left(\tau \right)}-\left({j}-1.379\sigma \right)\right],\end{equation}

but now with $\sigma =\pm {u}/|{u}|=\pm 1$ for the unbound or circulating electrons and $\sigma =0$ for the bound or librating electrons. Even though the solution is independent of the collision frequency, its form is determined by the solubility constraint from the collision operator. The $1.379\sigma$ step in the solution ${f}_{1}$ at the separatrix is not cancelled by the piecewise continuous behaviour of the derivative $\partial {g}_{1}/\partial {h}|_{\phi }$ there. Instead, the step is smoothed by a narrow collisional boundary layer at the separatrix that need not be resolved by the procedure here. The behaviour in the boundary layer does not play a role in the results that follow next.

4. Driven current and power absorbed in the intense field limit

The skew symmetric perturbed solution ${f}_{1}({u},\phi )=-{f}_{1}({-}{u},-\phi )$ cannot lead to a spatiotemporal averaged density as seen by letting ${u}\rightarrow -{u}$ and $\phi \rightarrow -\phi$ , then applying $\langle \cdots \rangle _{\phi }=\oint \mathrm{d}\phi (\cdots )/2\pi$ at fixed ${v}_{||}$ and employing ${d}^{3}{v}=2\pi {v}_{\bot }{\mathrm{d}v}_{\bot }{\mathrm{d}v}_{||}\propto \mathrm{d}u\propto \mathrm{d}j$ , to form

(4.1) \begin{equation}\left\langle \int _{-{\infty }}^{{\infty }}{\mathrm{d}uf}_{1}({u},\phi )\right\rangle _{\phi } =-\left\langle \int _{-{\infty }}^{{\infty }}{\mathrm{d}uf}_{1}({-}{u},-\phi )\right\rangle _{\phi }=-\left\langle \int _{-{\infty }}^{{\infty }}{\mathrm{d}uf}_{1}({u},\phi )\right\rangle _{\phi }=0,\end{equation}

where the limits of the u integration are for the outer limit of the inner-boundary-layer region and, of course, not actually infinite. However, the parallel current requires the u moment

(4.2) \begin{equation}\left\langle \int _{-{\infty }}^{{\infty }}{\mathrm{d}uuf}_{1}({u},\phi )\right\rangle _{\phi }=-\left\langle \int _{-{\infty }}^{{\infty }}{\mathrm{d}uuf}_{1}({-}{u},-\phi )\right\rangle _{\phi }=2\left\langle \int _{0}^{{\infty }}{\mathrm{d}uuf}_{1}({u},\phi )\right\rangle _{\phi }\neq 0\end{equation}

and the extra u means that skew symmetry allows a parallel current! As a result, the driven parallel current is evaluated using

(4.3) \begin{equation}\left\langle {J}_{||}\right\rangle _{\phi }=-{e}\left\langle \int {d}^{3}{vv}_{||}{f}_{1}\right\rangle _{\phi }=-{e}\left\langle \int {d}^{3}{vuf}_{1}\right\rangle _{\phi }.\end{equation}

Only the circulating electrons carry current. Moreover, integrating by parts (recalling ${{j}^{2}}{f}_{1}\propto 1/{j}\rightarrow 0$ far from the separatrix as shown in (3.7)) results in

(4.4) \begin{align}\left\langle {J}_{||}\right\rangle _{\phi } & =-\frac{{e}}{2}\left\langle \int {d}^{3}{vf}_{1}\frac{\partial {u}^{2}}{\partial {u}}\right\rangle _{\phi }=\frac{{e}}{2}\left\langle \int {d}^{3}{vuj}\left.\frac{\partial {f}_{1}}{\partial {j}}\right| _{\phi }\right\rangle _{\phi }\nonumber\\& =\frac{2\pi {e}^{2}}{{m}}\left|\frac{\tilde{{E}}_{||}}{{k}_{||}}\right|\left\langle \int _{0}^{{\infty }}{\mathrm{d}v}_{\bot }{v}_{\bot }\int _{0}^{{\infty }}{\mathrm{d}jj}^{2}\left.\frac{\partial {f}_{1}}{\partial {j}}\right| _{\phi }\right\rangle _{\phi },\end{align}

where both signs of sigma from j are summed over. Inserting

(4.5) \begin{equation}\left.\frac{\partial {f}_{1}}{\partial {j}}\right| _{\phi }=-\left|\frac{{eE}_{||}}{{mk}_{||}}\right|^{1/2}\frac{\partial {f}_{0}}{\partial {v}_{||}}\left[\frac{\sigma \pi }{{k}^{2}{E}\left({k}\right)}\left.\frac{\partial {k}}{\partial {j}}\right| _{\phi }+1\right],\end{equation}

leads to

(4.6) \begin{equation}\left\langle {J}_{||}\right\rangle _{\phi }=\frac{2{en}\omega \mathrm{e}^{-{\omega ^{2}}/{k}_{||}^{2}{v}_{{e}}^{2}}}{\pi ^{1/2}|{k}_{||}|}\left|\frac{{eE}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|^{3/2}\left\langle \int _{0}^{{\infty }}{\mathrm{d}jj}^{2}\left[\left.\frac{\partial {k}}{\partial {j}}\right| _{\phi }\frac{\pi }{{k}^{2}{E}\left({k}\right)}+1\right]\right\rangle _{\phi }.\end{equation}

Because the j derivative of k is at fixed $\phi$ , the $\phi$ integral can be performed at fixed k or h by using $\langle {j}^{2}\rangle _{\phi }=2{h}$ and noting $\partial {k}/\partial {j}|_{\phi }\lt 0$ to find

(4.7) \begin{equation}\left\langle \int _{0}^{{\infty }}{\mathrm{d}jj}^{2}\left.\frac{\partial {k}}{\partial {j}}\right| _{\phi }\frac{\pi }{{k}^{2}{E}\left({k}\right)}\right\rangle _{\phi }=-\pi \int _{0}^{1}{\mathrm{d}k}\frac{\langle {j}^{2}\rangle _{\phi }}{{k}^{2}{E}\left({k}\right)}=-2\pi \int _{0}^{1}{\mathrm{d}k}\frac{(2-{k}^{2})}{{k}^{4}{E}\left({k}\right)}.\end{equation}

This procedure removes the awkwardness of the upper limit of the j integral in the outer limit of the inner region. A similar procedure and use of $\langle {j}\rangle _{\phi }=4\pi ^{-1}{k}^{-1}{E}({k})$ gives

(4.8) \begin{equation}\left\langle \int _{0}^{{\infty }}{\mathrm{d}jj}^{2}\right\rangle _{\phi }=\int _{1}^{{\infty }}{\mathrm{d}h}\left\langle {j}\right\rangle _{\phi }=4\int _{0}^{1}{\mathrm{d}kk}^{-3}\left\langle {j}\right\rangle _{\phi }=\frac{16}{\pi }\int _{0}^{1}{\mathrm{d}kk}^{-4}{E}({k}).\end{equation}

As a result, the parallel current is negative (since $\omega /{k}_{||}\gt 0$ ) and given by

(4.9) \begin{equation}\left\langle {J}_{||}\right\rangle _{\phi }=-\frac{16{en}\omega }{\sqrt{\pi }|{k}_{||}|}\left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|^{3/2}{\mathrm{e}}^{-{\omega ^{2}}\big/{k}_{||}^{2}{v}_{{e}}^{2}}\int _{0}^{1}\frac{{\mathrm{d}k}}{{k}^{4}}\left[\frac{\pi (2-{k}^{2})}{4{E}\left({k}\right)}-\frac{2{E}\left({k}\right)}{\pi }\right],\end{equation}

where

(4.10) \begin{equation}{C}=\int _{0}^{1}\frac{{\mathrm{d}k}}{{k}^{4}}\left[\frac{\pi (2-{k}^{2})}{4{E}\left({k}\right)}-\frac{2{E}\left({k}\right)}{\pi }\right]=0.047675.\end{equation}

Therefore, the lower hybrid driven current for an intense applied field is

(4.11) \begin{equation}\left\langle {J}_{||}\right\rangle _{\phi }=-0.430{env}_{{e}}\left|\frac{\omega }{{k}_{||}{v}_{{e}}}\right|\left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|^{3/2}{\mathrm{e}}^{-{\omega ^{2}}/{k}_{||}^{2}{v}_{{e}}^{2}}.\end{equation}

Unlike the QL limit for which $\langle {J}_{||}\rangle _{\phi }\propto 1/\nu _{{e}}$ , the expression for $\langle {J}_{||}\rangle _{\phi }$ is independent of the collision frequency and scales as $\tilde{{E}}_{||}^{3/2}$ .

The RF power absorbed by the electrons is proportional to $\nu _{{e}}$ , as can be seen by first using $2\sin \phi =-{\mathrm{d}j}^{2}/{d}\phi |_{{h}}$ at fixed h to write

(4.12) \begin{align}{P} & =\left\langle {E}_{||}{J}_{||}\right\rangle _{\phi }=-{e}\tilde{{E}}_{||}\left\langle \sin \phi \int {d}^{3}{vv}_{||}{f}\right\rangle _{\phi }\approx \frac{\omega {e}\tilde{{E}}_{||}}{2{k}_{||}}\left\langle \int {d}^{3}{vf}\left.\frac{{\mathrm{d}j}^{2}}{\mathrm{d}\phi }\right| _{{h}}\right\rangle _{\phi }\nonumber\\& =-\frac{\omega {e}\tilde{{E}}_{||}}{2{k}_{||}}\left\langle \int {d}^{3}{vj}^{2}\left.\frac{\partial {g}_{2}}{\partial \phi }\right| _{{h}}\right\rangle _{\phi }=-\frac{\omega {e}\tilde{{E}}_{||}}{2{k}_{||}}\left\langle \int {d}^{3}{v}\Delta {j}^{2}\left.\frac{\partial }{\partial {h}}\right| _{\phi }\left({j}\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }\right)\right\rangle _{\phi }.\end{align}

In the preceding, ${v}_{||}=\omega /{k}_{||}+{u}$ and skew symmetry give $\langle \int _{-{\infty }}^{{\infty }}\text{d}\textit{uu}\sin\phi {f}_{1}({u},\phi )\rangle _{\phi }=0$ and

(4.13) \begin{align}\left\langle \int _{-{\infty }}^{{\infty }}\text{d}u\sin\phi {f}_{1}({u},\phi )\right\rangle _{\phi } & =-\left\langle \int _{-{\infty }}^{{\infty }}\text{d}u\sin\phi {f}_{1}({-}{u},-\phi )\right\rangle _{\phi }\nonumber\\& =2\left\langle \int _{0}^{{\infty }}\text{d}u\sin\phi {f}_{1}({u},\phi )\right\rangle _{\phi }\neq 0.\end{align}

Then ${u}\propto {j}$ , ${j\mathrm{d}j}=\mathrm{d}h$ at fixed $\phi$ and ${h}\underset{{h}\rightarrow {\infty }}{\longrightarrow}2/{k}^{2}$ yield

(4.14) \begin{align}\left\langle \int _{-{\infty }}^{{\infty }}\mathrm{d}jj^{2}\left.\frac{\partial }{\partial {h}}\right| _{\phi }{j}\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }\right\rangle _{\phi } & =2\left\langle \int _{1}^{{\infty }}\mathrm{d}hj\left.\frac{\partial }{\partial {h}}\right| _{\phi }{j}\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }\right\rangle _{\phi }\nonumber\\& =2\int _{1}^{{\infty }}\mathrm{d}h\left[\left.\frac{\partial }{\partial {h}}\right| _{\phi }\langle {j}^{2}\rangle _{\phi }\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }-\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{\phi }\right]\nonumber\\& =4{h}\frac{\partial {g}_{1}}{\partial {h}}\left| _{\phi ,{h}=1}^{{h}\rightarrow {\infty }}-2{g}_{1}\right| _{{h}=1}^{{h}\rightarrow {\infty }}\nonumber\\& =2\left|\frac{{e}\tilde{{E}}_{||}}{m{k}_{||}}\right|^{1/2}\frac{\partial {f}_{0}}{\partial {v}_{||}}\left[\frac{\pi }{2}-\frac{2}{{k}} +\left(\frac{2}{{k}}-1.379\right)\right]\nonumber\\& =0.384\left|\frac{{e}\tilde{{E}}_{||}}{m{k}_{||}}\right|^{1/2}\left.\frac{\partial {f}_{0}}{\partial {v}_{||}}\right| _{{u}=0}\end{align}

Consequently, using ${\mathrm{d}v}_{||}={\mathrm{d}u}=|{e}\tilde{{E}}_{||}/{m}{{k}_{||}}|^{1/2}{\mathrm{d}j}$ yields

(4.15) \begin{align}{P} & =-0.384\pi {m}\omega \left|\frac{{e}\tilde{{E}}_{\left| \right| }}{mk_{\left| \right| }}\right|^{2}\int _{0}^{{\infty }}{\mathrm{d}v}_{\bot }{v}_{\bot }\Delta \left.\frac{\partial {f}_{0}}{\partial v_{\left| \right| }}\right| _{u=0}\nonumber\\& =0.384\frac{{mn}{\omega ^{2}}\nu _{{e}}{\mathrm{e}}^{-{\omega ^{2}}/{k}_{||}^{2}{v}_{{e}}^{2}}}{\pi ^{1/2}k_{||}^{2}{v}_{{e}}^{2}}\left|\frac{{e}\tilde{{E}}_{||}}{{m}}\right|^{1/2}\int _{0}^{{\infty }}{\mathrm{d}v}_{\bot }\frac{{v}_{\bot }{v}_{\bot {z}}^{2}\mathrm{e}^{-{v}_{\bot }^{2}/{v}_{{e}}^{2}}}{\left({v}_{\bot }^{2}+\omega ^{2}/k_{||}^{2}\right)^{3/2}}.\end{align}

Recalling that the collision operator is valid for ${{x}^{2}}={v}^{2}/{v}_{{e}}^{2}\gg 1$ , while $\omega ^{2}/{k}_{||}^{2}{v}_{{e}}^{2}\approx 5/2\gg 1$ and ${v}_{\bot }^{2}\sim {v}_{{e}}^{2}$ , the resulting integral is approximately

(4.16) \begin{equation}\int _{0}^{{\infty }}{\mathrm{d}v}_{\bot }\frac{{v}_{\bot }{v}_{\bot {z}}^{2}e^{-{v}_{\bot }^{2}/{v}_{{e}}^{2}}}{\left({v}_{\bot }^{2}+\omega ^{2}/k_{||}^{2}\right)^{3/2}}\approx \frac{|k_{||}|^{3}{v}_{{e}}^{4}}{2\omega ^{3}}\left(\frac{Z+2}{Z+1}\right).\end{equation}

As a result, the RF power absorbed to drive the current for an intense applied lower hybrid wave is

(4.17) \begin{equation}{P}=0.108\frac{Z+2}{Z+1}{mnv}_{{e}}^{2}\nu _{{e}}\left|\frac{{k}_{||}{v}_{{e}}}{\omega }\right|\left|\frac{{e}\tilde{{E}}_{||}}{{k}_{||}{mv}_{{e}}^{2}}\right|^{1/2}{\mathrm{e}}^{-{\omega ^{2}}/{k}_{||}^{2}{v}_{{e}}^{2}}.\end{equation}

The driven current of (4.11) only depends on ${g}_{1}$ as seen by (4.4). It only depends on the reduced constant of the motion ${h}$ from (3.4). The form for ${g}_{1}$ is constrained by the collision operator to satisfy (3.6), but the collision frequency does not enter. However, to drive current there must be dissipation. It enters through the power absorbed by the electrons from the applied lower hybrid wave. Collisions enter ${g}_{2}$ , as seen by (3.5), and it is required in (4.12).

Using $\nu _{{e}}=3\pi ^{1/2}({Z}+1)\nu _{{ee}}/4$ , the current drive efficiency in the intense field limit is

(4.18) \begin{equation}\frac{\left|\left\langle {J}_{||}\right\rangle _{\phi }\tilde{{E}}_{||}\right|}{{P}}=\frac{2.99\omega }{(Z+2)\nu _{{ee}}}\left|\frac{\omega }{{k}_{||}{v}_{{e}}}\right|\left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|^{2}\end{equation}

leading to the normalised form

(4.19) \begin{equation}\frac{\big|\big\langle {J}_{||}\big\rangle _{\phi }\big|/{env}_{{e}}}{{P}/{mnv}_{{e}}^{2}\nu _{{ee}}}=\frac{2.99\omega ^{2}}{({Z}+2){k}_{||}^{2}{v}_{{e}}^{2}}\left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|.\end{equation}

Ignoring aspect ratio modifications (Catto & Muni 2023) , the usual QL result (Fisch Reference Fisch1978) is

(4.20) \begin{equation}\frac{{J}_{||}^{{LH}}/{env}_{{e}}}{{P}_{{cd}}^{{LH}}/{mnv}_{{e}}^{2}\nu _{{ee}}}=\frac{3.01\omega ^{2}}{({Z}+5){k}_{||}^{2}{v}_{{e}}^{2}}.\end{equation}

Both pitch angle and energy scatter matter in the intense field limit. Importantly, the intense field limit depends on $\tilde{{E}}_{||}$ , and is smaller by roughly

(4.21) \begin{equation}1\gg \frac{\big(\Delta {v}_{||}\big)_{\mathrm{is}}^{2}}{{v}_{{e}}^{2}}\sim \left|\frac{{e}\tilde{{E}}_{||}}{{mk}_{||}{v}_{{e}}^{2}}\right|,\end{equation}

when $(\Delta {v}_{||})_{\mathrm{is}}^{2}\gg (\Delta {v}_{||})_{\nu }^{2}$ . Consequently, intense field LHCD is expected to be less efficient than QL predictions indicate. The result here, namely (4.19), as well as plasma edge turbulence likely explain some of the decrease in efficiency observed in experiments (Bonoli Reference Bonoli2014). The nonlinear reduction by $(\Delta {v}_{||})_{\mathrm{is}}^{2}/{v}_{{e}}^{2}\ll 1$ occurs because the bound electrons are unable to carry current. The normalised efficiency is reduced, even though less power is absorbed in the narrow collisional boundary layers, because the bound region containing the non-current carrying electrons becomes much wider as the applied lower hybrid wave amplitude becomes larger. Additional island structures are expected to result in further reductions.