2.1. Wave scattering by filaments in the spectral domain

In an infinite spatial domain with vanishing scattered fields at infinity transverse to B ₀, one can most easily solve the scattering problem in the spectral domain, by splitting the incident and scattered waves into independent cylindrical modes oscillating as ${\rm exp} (\textrm{i}m\theta + \textrm{i}{k_{/{/}}}z)$. This subsection briefly summarizes this method used Ram & Hizanidis (Reference Ram and Hizanidis2016) and Tierens et al. (Reference Tierens, Zhang and Myra2020a, Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022a). Tierens et al. (Reference Tierens, Zhang and Manz2020b) investigated numerically non-cylindrical FAMC modes in more realistic geometries and with more realistic density distributions. The RF fields inside and outside the filament are the superposition of cylindrical wave modes with slow and fast wave polarizations, of the form

(2.8)\begin{equation}\hat{\boldsymbol{E}}(r,m,{k_\parallel }) = \textrm{exp}(\textrm{i}m\theta + \textrm{i}{k_\parallel }z)\left\{ {\begin{array}{*{20}{@{}c}} {{{\hat{\boldsymbol{E}}}_{\boldsymbol{f}}}(m,{k_\parallel }){I_m}({k_{ \bot f}}r),\quad r < {r_f}}\\ {{{\hat{\boldsymbol{E}}}_{\boldsymbol{b}}}(m,{k_\parallel }){K_m}({k_{ \bot b}}r),\quad r > {r_f}} \end{array}} \right..\end{equation}

Here, I_m and K_m are modified Bessel functions, where we have assumed evanescent modes both inside and outside the filament. The perpendicular wave vectors k _⊥b and k _⊥f are eigenvalues of the dispersion relation outside and inside the filaments, for prescribed k _//

(2.9)\begin{equation}|\boldsymbol{k} \times \boldsymbol{k} - \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{kI} - k_0^2\boldsymbol{\varepsilon }|= 0;\quad \boldsymbol{k} = \left[ {\begin{array}{*{20}{@{}c@{}}} {\textrm{i}{k_ \bot }}\\ 0\\ {{k_\parallel }} \end{array}} \right].\end{equation}

The field polarizations ${\hat{\boldsymbol{E}}_{\boldsymbol{b}}}(m,{k_\parallel })$ and ${\hat{\boldsymbol{E}}_{\boldsymbol{f}}}(m,{k_\parallel })$ are the eigenvectors associated with k _⊥b and k _⊥f. Once the incident spectral RF electric field ${\hat{\boldsymbol{E}}_0}({k_\parallel })$ is prescribed, the general scattering problem consists of ensuring continuous total tangential RF electric and magnetic fields across the filament boundary, with an adequate linear combination of scattered fast and slow waves inside and outside the filament. One describes this matching by a 4 × 4 linear system of equations, whose unknows are the four complex amplitudes of the scattered modes and the right-hand side is a drive from the incident wave.

Tierens et al. (Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022a) investigated analytically a simplified scattering problem under the following assumptions:

1. (i) Transverse to B ₀, the incident spectral RF electric field ${\hat{\boldsymbol{E}}_0}({k_\parallel })$ is homogeneous over the filament boundary. Parallel to B ₀, it still oscillates as ${\rm exp} ( + \textrm{i}{k_{/{/}}}z)$.

2. (ii) The scattered fast wave is negligible, both inside and outside the filament.

3. (iii) Simplified dispersion properties are considered in the HHFW range of frequencies.

4. (iv) The scattered slow mode is considered as electrostatic.

Assumption (i) implies that only azimuthal modes m  =  ±1 can be excited. Assumptions (iii) and (iv) yield the simplified dispersion relation for the slow mode in the absence of dissipation

(2.10)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} { - \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{D} = {\varepsilon_\parallel }{\partial_{zz}}\phi + {\varepsilon_ \bot }{\Delta _ \bot }\phi = 0}\\ {{k_{ \bot f}}\sim {k_{ \bot b}} = {k_ \bot } = |{k_\parallel }|\sqrt {\dfrac{{{M_i}}}{{{m_e}}}}} \end{array}} \right\}.\end{equation}

The associated slow wave RF field is of the form

(2.11)\begin{gather}{\hat{\boldsymbol{E}}_{\boldsymbol{s}}}(r,m,{k_\parallel }) ={-} \boldsymbol{\nabla }{\hat{\phi }_s}(r,m,{k_\parallel }),\end{gather}

(2.12)\begin{gather}{\hat{\phi }_s}(r,m,{k_\parallel }) = {\hat{\phi }_{s0}}\textrm{(}m,{k_\parallel }\textrm{)exp}(\textrm{i}m\theta + \textrm{i}{k_\parallel }z)F({k_ \bot }r).\end{gather}

(2.13)\begin{gather}F(x) \equiv \left\{ {\begin{array}{*{20}{@{}c}} {\dfrac{{{I_m}(x)}}{{{I_m}({k_ \bot }{r_f})}},\quad x < {k_ \bot }{r_f}}\\ {\dfrac{{{K_m}(x)}}{{{K_m}({k_ \bot }{r_f})}},\quad x > {k_ \bot }{r_f}} \end{array}} \right.;\quad {k_ \bot }\sim |{k_\parallel }|\sqrt {\frac{{{M_i}}}{{{m_e}}}} .\end{gather}

In the HHFW domain the complex spectral amplitude ${\hat{\phi }_{s0}}(m,{k_\parallel })$ writes (Tierens et al. Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022a)

(2.14)\begin{equation}{\hat{\phi }_{s0}}(m,{k_\parallel }) ={-} \frac{\textrm{i}}{2}{r_f}{\hat{E}_{ \bot 0}}({k_\parallel })\frac{{(R - 1)(1 - \textrm{i}{\nu ^\mathrm{\ast }} - m\varphi )}}{{\xi (1 - \textrm{i}{\nu ^\mathrm{\ast }})\left( {R\frac{{I^{\prime}}}{I} - \frac{{K^{\prime}}}{K}} \right) - m\varphi (R - 1)}}.\end{equation}

In this expression, $\xi = {k_ \bot }{r_f},\;I = {I_m}(\xi ),\;I^{\prime} = \textrm{d}{I_m}(x)/\textrm{d}x$ evaluated at $x = {k_ \bot }{r_f},\;I^{\prime}/I = \textrm{d}\log [{I_m}(x)]/\textrm{d}x,\;K = {K_m}(\xi ),\;K^{\prime} = \textrm{d}K_m(x)/\textrm{d}x$ evaluated at $x = {k_ \bot }{r_f},\;K^{\prime}/K = \textrm{d}\log [{K_m}(x)]/\textrm{d}x$.

2.2. Nearly resonant wave fields in the spatial domain

Nearly resonant HHFW scattering occurs when the real part of the denominator cancels in expression (2.14)

(2.15)\begin{equation}{\xi _{\textrm{res}}}\left( {R\frac{{I^{\prime}}}{I} - \frac{{K^{\prime}}}{K}} \right) = m\varphi (R - 1).\end{equation}

This dispersion relation defines a resonant parallel wave vector k _//res, once all other parameters (R, φ, r_f, m) are fixed in the model. As the left-hand side of relation (2.15) is positive, resonant m  =  +1 FAMC modes appear for over-density inside the filament (n_f > n_b), which is the most frequent case for turbulent fluctuations in the SOL of tokamaks. The m  =  −1 FAMC modes correspond to a local density depletion (n_f < n_b). Close to k _//  =  k _//res, and for ν* values small enough, one can approximate the complex spectral amplitude of the scattered wave as

(2.16)\begin{equation}{\hat{\phi }_{s0}}(m,{k_\parallel }) \approx \frac{{\textrm{i}{\phi _0}}}{{2\mathrm{\pi }}}\frac{1}{{{k_{{\parallel} \,\textrm{res}}} - {k_\parallel } + \textrm{i}{\nu ^\ast }{k_i}}},\end{equation}

where

(2.17)\begin{gather}{\phi _0} \equiv \mathrm{\pi }\frac{{{\xi _{res}}{{\hat{E}}_{ \bot 0}}({k_{{\parallel} \,\textrm{res}}})}}{{m\varphi }},\end{gather}

(2.18)\begin{gather}{k_i} \equiv \textrm{Re}{\left( {\frac{{{\phi_0}}}{{2\mathrm{\pi }{\nu^\ast }{{\hat{\phi }}_{s0}}(m,{k_\parallel })}}} \right)_{\xi = {\xi _{\textrm{res}}}}} = \frac{{{k_{{\parallel} \,\textrm{res}}}}}{{1 + {\xi _{\textrm{res}}}{\partial _\xi }\textrm{log}\left( {R\frac{{I^{\prime}}}{I} - \frac{{K^{\prime}}}{K}} \right)}}.\end{gather}

While spectral calculations are convenient for deriving dispersion relations, they are not well suited for implementing parallel boundary conditions. One can, however, approximate from relation (2.16) the nearly resonant RF scattered fields in the spatial domain. Let us first focus on positive values of k _//res and apply an inverse Fourier transform to the spectral waves (2.16)

(2.19)\begin{equation}\begin{aligned} {\phi _{s0}}(m,z) & = \int_{ - \infty }^{ + \infty } {{{\hat{\phi }}_{s0}}} \textrm{(}m,{k_\parallel }\textrm{)exp}(\textrm{i}{k_\parallel }z)\,\textrm{d}{k_\parallel } \approx -\dfrac{{\textrm{i}{\phi _0}}}{{2\pi }}\textrm{exp}(\textrm{i}{k_{{\parallel} \,\textrm{res}}}z)\int_{ - \infty }^{ + \infty } {\dfrac{{\textrm{exp}(\textrm{i}{k_{{\parallel} }}z)\textrm{d}{k_\parallel }}}{{ {k_\parallel } - \textrm{i}{\nu ^\ast }{k_i}}}} \\ & = {\phi _0}H(z)\textrm{exp}(\textrm{i}{k_{{\parallel} \,\textrm{res}}}z - {\nu ^\ast }{k_i}z);\quad {k_{{\parallel} \,\textrm{res}}} > 0. \end{aligned}\end{equation}

Here, H(z) is the Heaviside function. The above formula applies to positive values of k _//res, for which k_i > 0 and the amplitude of the RF fields decays exponentially for large positive values of z. Figure 1 plots $|{\phi _{s0}}(m,z)|/|{\phi _0}|$ from (2.19) versus k_iz over a scan of the dissipation parameter ν*.

Figure 1. Value of $|{\phi _{s0}}(m,z)|/|{\phi _0}|$ from (2.19) and (2.25) versus k_iz over a scan of ν*.

As ν* decreases to 0⁺, the RF fields do not diverge in the spatial domain, unlike their Fourier transform. However, the RF fields exhibit a discontinuity in z  =  0, even for finite ν*. In addition, the characteristic decay length $1/{\nu ^\ast }{k_i}$ gets very large and the RF fields exhibit nearly harmonic oscillations for z > 0. One can extrapolate formula (2.19) to ν* = 0. However, the resulting RF fields are not square-integrable anymore, and therefore one cannot apply the Fourier treatments (e.g. Parseval's theorem) in a standard way. Throughout the document, we will assume that the FAMC mode is weakly damped, i.e. the ordering ${\nu ^\ast }{k_i} \ll {k_{/{/}\,\textrm{res}}}$ applies, so that the FAMC modes possess a well-defined resonant parallel wave vector. Section 5.1 discusses this ordering.

Associated with the above RF electric fields one can define a Poynting flux P _RF(z) across each plane z  =  const. For quasi-static cylindrical FAMC modes, Appendix A shows that the Poynting flux takes the form (A13)

(2.20)\begin{equation}\begin{array}{ccccc} {P_{\textrm{RF}}}(z) & \equiv \dfrac{1}{2}\int_0^{2\mathrm{\pi }} {\textrm{d}\theta } \int_0^{ + \infty } {\textrm{Re}[({\boldsymbol{E}^\ast }(r,\theta ,z) \times \boldsymbol{H}(r,\theta ,z))\boldsymbol{\cdot }{\boldsymbol{e}_{\boldsymbol{z}}}]r\,\textrm{d}r} \\ & ={-} \mathrm{\pi }{\varepsilon _0}{\omega _0}\int_0^{ + \infty } {\textrm{Im}[{\varepsilon _\parallel }{\phi ^\ast }(r,z){\partial _z}\phi (r,z)]r\,\textrm{d}r} . \end{array}\end{equation}

For one isolated weakly damped FAMC mode this simplifies to leading order in the small parameter ${\nu ^\ast }{k_i}/{k_{/{/}\,\textrm{res}}}$

(2.21)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {{P_{\textrm{RF}}}(z) = {Y_{\textrm{FAMC}}}|{\phi_0}{|^2}\,H(z)\textrm{exp}( - 2{\nu^\ast }{k_i}z);\quad {k_{{\parallel} \,\textrm{res}}} > 0}\\ {{Y_{\textrm{FAMC}}} \equiv \dfrac{\mathrm{\pi }}{{{Z_0}|{n_{{\parallel} \,\textrm{res}}}|}}\dfrac{{\omega_{\textrm{pib}}^2}}{{\omega_0^2}}\left[ {\int_0^{{\xi_f}} {R|F(\xi ){|^2}\xi \,\textrm{d}\xi } + \int_{{\xi_f}}^{ + \infty } {|F(\xi ){|^2}\xi \,\textrm{d}\xi } } \right]} \end{array}} \right\}.\end{equation}

In this expression, Y _FAMC has the dimension of an admittance, and depends only on the Hermitian part of the dielectric tensor; ${Z_0} \equiv {({\mu _0}/{\varepsilon _0})^{1/2}}$ is the characteristic impedance of vacuum and n _//res is the resonant parallel refractive index k _//res/k ₀. As already noticed in Tierens et al. (Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022a), the power flux of the FAMC mode is proportional to the local plasma density and is a fraction of the HHFW spectral power launched by the antenna at k _// = k _//res. For z > 0, P _RF(z) decreases with z, with a decay length $1/2{\nu ^\ast }{k_i}$. Using Poynting's theorem, Appendix A shows that the decay is due to power dissipation in the plasma volume. The total loss over the magnetic field line is

(2.22)\begin{equation}{P_{\textrm{RF}}}({0^ + }) - {P_{\textrm{RF}}}( + \infty ) = {Y_{\textrm{FAMC}}}|{\phi _0}{|^2}.\end{equation}

This coincides with the power loss estimated in Tierens et al. (Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022a) using a spectral approach. As already noticed in Tierens et al. (Reference Tierens, Bilato, Bertelli, Shiraiwa, Myra and Colas2022a), the total loss becomes independent of ν* in the weak dissipation limit. This counter-intuitive result stresses the need to regularize the initial scattering problem and to take the friction-less limit only at the end of the calculations, in order to obtain valid results. The local power loss in a thin parallel layer at position z, per unit of parallel length, is $- {\partial _z}{P_{\textrm{RF}}}(z)$. From (2.21)

(2.23)\begin{equation}{P^{\prime}_V}(z) \equiv{-} {\partial _z}{P_{\textrm{RF}}}(z) = 2{\nu ^\ast }{k_i}{P_{\textrm{RF}}}(z) - {P_{\textrm{RF}}}({0^ + })\delta (z).\end{equation}

Let us first focus on z ≠ 0 and discuss the first term on the right-hand side. As ${\nu ^\ast } \to 0$ at fixed z, ${P_{\textrm{RF}}}(z) \to {P_{\textrm{RF}}}({0^ + })$ and the local loss scales as ${\nu ^\ast }{P_{\textrm{RF}}}({0^ + })$. In the global power balance, this decrease with lower ν* is compensated for by the fact that the power is dissipated over a larger parallel decay length. Formula (2.23) shows that the ratio ${\mathop{\rm Im}\nolimits} ({k_{/{/}}}) \equiv {P^{\prime}_V}(z)/2{P_{\textrm{RF}}}(z)$, together with the expressions of ${P^{\prime}_V}(z)$ and P _RF(z) in Appendix A, provide an alternative definition of the parallel decay length of the FAMC mode, for any form of the anti-symmetric part of the dielectric tensor, in the weak damping regime. Then, Im(k _//)/k _//res writes

(2.24)\begin{equation}\frac{{\textrm{Im(}{k_\parallel }\textrm{)}}}{{{k_{{\parallel} \,\textrm{res}}}}} = \frac{{\frac{{\omega _0^2}}{{\omega _{\textrm{peb}}^2}}\int_0^{ + \infty } {\textrm{Im(}{\varepsilon _\parallel }\textrm{)|}F(\xi ){\textrm{|}^2}\xi \,\textrm{d}\xi } + \frac{{\omega _0^2}}{{\omega _{\textrm{pib}}^2}}\int_0^{ + \infty } {\textrm{Im(}{\varepsilon _ \bot }\textrm{)}\left[ {|{\partial_\xi }F{|^2} + {{\left|{\frac{{mF(\xi )}}{\xi }} \right|}^2}} \right]\xi \,\textrm{d}\xi } + \frac{m}{\mathrm{\pi }}[[\textrm{Im(}{\varepsilon _ \times }\textrm{)}]]\frac{{\omega _0^2}}{{\omega _{\textrm{pib}}^2}}}}{{2\left[ {\int_0^{{\xi_f}} {R|F(\xi ){|^2}\xi \,\textrm{d}\xi } + \int_{{\xi_f}}^{ + \infty } {|F(\xi ){|^2}\xi \,\textrm{d}\xi } } \right]}}.\end{equation}

This could be a way to incorporate other dissipation processes than the present phenomenological friction, as Raether (Reference Raether1988) and Girka & Thumm (Reference Girka and Thumm2022) did for other types of surface waves. We can generalize most of the formulas below by substituting ${\nu ^\ast }{k_i} \to {\mathop{\rm Im}\nolimits} ({k_{/{/}}})$.

Because of the local RF field discontinuity at z  =  0 in (2.19), the Poynting flux in (2.21) also exhibits a discontinuity there, that is responsible for the second term on the right-hand side of (2.23). The power step ${P_{\textrm{in}}} \equiv {P_{\textrm{RF}}}({0^ + }) - {P_{\textrm{RF}}}({0^ - })$ coincides with the total power dissipated over the domain z ≠ 0. Since it is the only power source in our model, it is tempting to interpret P _in as the HHFW power fraction ‘redirected’ into the FAMC mode by the scattering of the incident HHFW wave. Since z  =  0 is the only location where some RF power is ‘injected’ into the FAMC mode, it is also tempting to identify it with a typical parallel position of the HHFW antenna. Finite element modelling of the HHFW scattering process supports this interpretation (Zhang et al. Reference Zhang, Tierens, Bobkov, Cathey, Cziegler, Griener, Hoelzl and Cardaun2021); large modifications of the parallel Poynting fluxes are observed near the toroidal position of the radiating straps.

2.3. Parallel symmetry properties of the RF electric fields

While we have so far considered positive values of k _//res, resonant parallel wave vectors may also be negative. For symmetry reasons, if k _//res fulfils the FAMC dispersion relation (2.15), so does −k _//res. In this process, k _⊥ remains invariant while k_i transforms into –k_i. In order to fulfil the boundary conditions at the extremities of open magnetic field lines, we will need to combine FAMC modes with opposite parallel wave vectors. Therefore, we need to extend the previous formulas. While the spectral results up to (2.18) are valid for both signs of k _//res, the spatial representation (2.19) of the RF fields needs to be adapted to ensure the regularity of the solution when k_i < 0. For k _//res < 0, the inverse Fourier transform of (2.16) yields

(2.25)\begin{equation}{\phi _{s0}}(m,z) ={-} {\phi _0}H( - z)\textrm{exp(i}{k_{{\parallel} \,\textrm{res}}}z - {\nu ^\ast }{k_i}z\textrm{)};\quad {k_{{\parallel} \,\textrm{res}}} < 0,\end{equation}

i.e. when ${k_{/{/}\,\textrm{res}}} \to - {k_{/{/}\,\textrm{res}}},\;{\phi _{s0}}(m,z) \to - {\phi _{s0}}(m, - z)$ and the scattered electric field components transform as ${E_{/{/}}}(r,\theta ,z) \to + {E_{/{/}}}(r,\theta , - z)\;\textrm{and}\;{E_ \bot }(r,\theta ,z) \to - {E_ \bot }(r,\theta , - z)$. Now the RF fields for the FAMC modes are non-zero in the half-space z < 0. Figure 1 also applies to k _//res < 0, with k_iz > 0 corresponding to z < 0. The damping factor ${\nu ^\ast }{k_i}z$ is positive for k _//res < 0 and z < 0. This ensures the regularity of solution (2.25) in the spatial domain. Formula (2.20) is still valid, but instead of (2.21) the Poynting flux now reads

(2.26)\begin{equation}{P_{\textrm{RF}}}(z) ={-} {Y_{\textrm{FAMC}}}|{\phi _0}{|^2}H( - z)\textrm{exp(} {-} 2{\nu ^\ast }{k_i}z\textrm{)};\quad {k_{{\parallel} \,\textrm{res}}} < 0.\end{equation}

In this case P _RF(z) < 0, i.e. the power now flows towards negative z and is null for z > 0. The power step ${P_{\textrm{in}}} = {P_{\textrm{RF}}}({0^ + }) - {P_{\textrm{RF}}}({0^ - })$ (‘redirected HHFW power’) is still positive, and is still equal to the total dissipated power ${P_{\textrm{RF}}}( - \infty ) - {P_{\textrm{RF}}}({0^ - })$. Formula (2.23) remains valid, and since k_i < 0, the local power loss ${P^{\prime}_V}(z)$ is positive, as it should. But the volume dissipation now occurs for z < 0.

Finally, when two FAMC modes with opposite k _//res are excited simultaneously, P _RF(z) is the sum of the Poynting fluxes by each FAMC mode taken individually. Therefore, the redirected powers and total dissipated powers add up in the weak damping regime. One can see this as the manifestation of the Parseval theorem, valid for infinite parallel domains in the weak dissipation limit. This property will need revision in bounded geometry.

3.1. Outline of the model

Figure 2 sketches the model studied in § 3. We consider a semi-infinite magnetic field line extending from z  =  –∞ to z  =  z ⁺ > 0. We perform here for the weakly damped FAMC modes a similar analysis as Myra & Kohno (Reference Myra and Kohno2019) did for propagating electrostatic plane waves with the slow mode polarization. Consistent with (2.19) we excite, via the HHFW scattering process, an incident FAMC mode of the form

(3.1)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {{\phi^ + }(r,\theta ,z) = {\phi^ + }(z)F({k_{ \bot \textrm{res}}}r)\textrm{exp}(\textrm{i}m\theta )}\\ {{\phi^ + }(z) = \phi_0^ + H(z)\textrm{exp(} + \textrm{i}{k_{{\parallel} \textrm{res}}}z - {\nu^\mathrm{\ast }}{k_i}z\textrm{)};\quad {k_{{\parallel} \textrm{res}}} > 0} \end{array}} \right\}.\end{equation}

Figure 2. Sketch of the model studied in § 3. The two nearly resonant FAMC modes in the model have opposite parallel wave vectors and propagate along the same filament. The colour shades are representative of the amplitudes for the isolated modes. In practice, the modes interfere and the total RF field amplitudes oscillate spatially, but this is not represented. Inset: equivalent model of TEM mode reflection by loaded transmission line.

By using (2.19), we assume that the FAMC excitation is not disturbed by the presence of a field line extremity. In the semi-infinite geometry, exciting a FAMC mode with k _//res < 0 yields a similar result as on infinite field lines in § 2. Throughout the rest of this document, we will therefore assume k _//res > 0, k_i > 0. When needed we will add a minus sign explicitly. This incident mode is excited at z  =  0 and extends to z  =  z ⁺ > 0. At z  =  z ⁺, the mode reaches a field line extremity and interacts with the sheaths. As the sheath widths in SOL plasmas are far smaller than any other characteristic parallel scale length in the system (magnetic field line extension, parallel decay length of the FAMC mode, parallel wavelength) it is legitimate to model the sheaths as boundary conditions (BCs) that the RF fields need to fulfil at z  =  z ⁺ (Myra Reference Myra2017). In order to keep the calculations tractable, we simplify the modelled geometry; we assume that the field lines intercept the walls at normal incidence. Section 5.2 will discuss this assumption. The RF sheath BCs then write (Myra Reference Myra2017)

(3.2)\begin{equation}{\boldsymbol{E}_ \bot } = {\nabla _ \bot }{V_{\textrm{shRF}}} = {\nabla _ \bot }\textrm{(}{z_{\textrm{sh}}}{j_{\textrm{sh}}}\textrm{)};\quad {j_{\textrm{sh}}} \equiv{-} \textrm{i}{\omega _0}{\varepsilon _0}{\varepsilon _\parallel }{E_\parallel }.\end{equation}

Here, the ⊥ subscript refers to the directions both normal to B ₀ and tangential to the boundary. The // subscript refers to the direction both parallel to B ₀ and normal to the boundary. Also, V _shRF and j _sh are the sheath oscillating voltage and the RF current density at the sheath entrance and z _sh is the sheath RF impedance. Myra (Reference Myra2017) and Myra et al. (Reference Myra, Elias, Curreli and Jenkins2021) describe its parametric dependence. In order to keep the calculations tractable, we assume below that the sheath impedance is prescribed and that the product z _shε _// is independent of r. Section 5.3 will discuss this assumption.

3.2. The RF field spatial structure, wave reflection coefficient

Under the above simplifying assumptions, the wave reflection problem at the sheaths is linear. In these conditions, it is possible to fulfil the sheath BCs (3.2) using a linear combination of the incident FAMC mode (3.1) and a reflected FAMC mode $\phi _0^ -$ with the same azimuthal mode number m and the opposite parallel wave vector, propagating along the same filament

(3.3)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {{\phi^ - }(r,\theta ,z) = {\phi^ - }(z)F({k_{ \bot \textrm{res}}}r)\textrm{exp} ({+} \textrm{i}m\theta \textrm{)}}\\ {{\phi^ - }(z) = \phi_0^ - \textrm{exp} ({-} \textrm{i}{k_{{\parallel} \,\textrm{res}}}z + {\nu^\mathrm{\ast }}{k_i}z\textrm{)};\quad {k_{{\parallel} \,\textrm{res}}},\;{k_i} > 0} \end{array}} \right\}.\end{equation}

Unlike the incident mode, the reflected fields are present from z  =  −∞ to z  =  z ⁺. In addition, they are continuous at z  =  0, since they are not directly excited by the HHFW scattering process. In the presence of the two modes, (3.2) writes

(3.4)\begin{equation}\begin{aligned} {\phi ^ + }({z^ + }) + {\phi ^ - }({z^ + }) & ={-} \textrm{i}{\omega _0}{\varepsilon _0}{z_{\textrm{sh}}}{\varepsilon _\parallel }(\textrm{i}{k_{{\parallel} \,\textrm{res}}} - {\nu ^\mathrm{\ast }}{k_i})[{\phi ^ + }({z^ + }) - {\phi ^ - }({z^ + })]\\ & ={-} {z_{\textrm{sh}}}{y_w}[{\phi ^ + }({z^ + }) - {\phi ^ - }({z^ + })] \end{aligned}\end{equation}

where we have introduced a complex wave admittance ${y_w} \equiv{-} {\omega _0}{\varepsilon _0}{\varepsilon _{/{/}}}{k_{/{/}\textrm{res}}}(1 + \textrm{i}{\nu ^\ast }{k_i}/{k_{/{/}\textrm{res}}}) ={-} {\varepsilon _{/{/}}}{n_{/{/}\textrm{res}}}(1 + \textrm{i}{\nu ^\ast }{k_i}/{k_{/{/}\textrm{res}}})/{Z_0}$. In the weak damping regime, the wave admittance is mainly real. From (3.4) we express the complex amplitude reflection coefficient in the region z > 0

(3.5)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {R_\phi^ + (z) \equiv \dfrac{{{\phi^ - }(z)}}{{{\phi^ + }(z)}} = R_\phi^ + \textrm{exp}\left[ {2\textrm{i}{k_{{\parallel} \,\textrm{res}}}\left( {1 + \textrm{i}{\nu^\mathrm{\ast }}\dfrac{{{k_i}}}{{{k_{{\parallel} \,\textrm{res}}}}}} \right)({z^ + } - z)} \right]}\\ {R_\phi^ + \equiv \dfrac{{{\phi^ - }({z^ + })}}{{{\phi^ + }({z^ + })}} = \dfrac{{{z_{\textrm{sh}}}{y_w} - 1}}{{{z_{\textrm{sh}}}{y_w} + 1}}} \end{array}} \right\}.\end{equation}

The FAMC mode reflection at the extremity of the filament is formally analogous to that of a transverse electro-magnetic (TEM) mode in a (lossy) transmission line of characteristic impedance 1/y_w, loaded by the sheath impedance z _sh (see inset of figure 2). In the absence of dissipation in the plasma volume and in the sheaths, y_wz _sh is pure imaginary. In these conditions the reflection coefficient at z  =  z ⁺ is of amplitude 1: $|{\phi ^ - }({z^ + })|= |{\phi ^ + }({z^ + })|$ and the mode reflection only introduces a phase shift. In the general case

(3.6)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {|R_\phi^ + {|^2} = 1 - 4\dfrac{{\textrm{Re(}{z_{\textrm{sh}}}{y_w}\textrm{)}}}{{|{z_{\textrm{sh}}}{y_w} + 1{|^2}}}}\\ {\textrm{Arg}(R_\phi^ + ) = \textrm{Arctan}\left[ {\dfrac{{2\textrm{Im(}{z_{\textrm{sh}}}{y_w}\textrm{)}}}{{|{z_{\textrm{sh}}}{y_w}{|^2} - 1}}} \right]} \end{array}} \right\}.\end{equation}

The case z _sh  =  0 (metallic BCs) yields ${\phi ^ - } ={-} {\phi ^ + }$, i.e. E _⊥  =  0 at z  =  z ⁺, as it should. The opposite limit $|{z_{\textrm{sh}}}{y_w}|\gg 1$ (insulating BCs) yields ${\phi ^ - } ={+} {\phi ^ + }$ i.e. j _sh  =  0. As in a transmission line, the reflection is null if the sheath impedance matches the wave impedance.

3.3. Sheath oscillating properties and power dissipation

In (3.4) the quantity ${\phi ^ - } + {\phi ^ + }$ represents a sheath oscillating voltage at r  =  r_f that we can express as a function of the incident mode amplitude

(3.7)\begin{gather}{V_{\textrm{shRF}}}(r,\theta ) = {V_{\textrm{shRF}}}({r_f})F({k_{ \bot \,\textrm{res}}}r)\textrm{exp(i}m\theta \textrm{)},\end{gather}

(3.8)\begin{gather}\begin{aligned} {V_{\textrm{shRF}}}({r_f}) & = {\phi ^ + }({z^ + }) + {\phi ^ - }({z^ + }) = (1 + R_\phi ^ + ){\phi ^ + }({z^ + })\\ & = \dfrac{2}{{1 + 1/{z_{\textrm{sh}}}{y_w}}}{\phi ^ + }({z^ + }). \end{aligned}\end{gather}

The RF voltage is null for metallic BCs. It is maximal in amplitude at r  =  r_f. The amplitude is independent of θ. In terms of the original excitation, the sheath RF voltage scales as $|\phi _0^ + |{\rm exp} ( - {\nu ^\ast }{k_i}{z^ + })$. Its amplitude is maximal for ${\nu ^\ast }{k_i}{z^ + } \ll 1$ and vanishes for ${\nu ^\ast }{k_i}{z^ + } \gg 1$. This is a first indication that the model recalled in § 2 is valid when the parallel extent of the FAMC mode on infinite field limes is far smaller than the parallel distance to the field line extremities. Since this model assumes fixed k_i, this implies either that ${z^ + } \to + \infty$ or that ν* cannot go below some critical value. Below we will call ‘short field line limit’ the regime ${\nu ^\ast }{k_i}{z^ + } \ll 1$, and ‘long field line limit’ the regime ${\nu ^\ast }{k_i}{z^ + } \gg 1$. The RF current through the sheath is

(3.9)\begin{gather}{j_{\textrm{sh}}}(r,\theta ) = {j_{\textrm{sh}}}\textrm{(}{r_f}\textrm{)exp(i}m\theta \textrm{)}\left\{ {\begin{array}{*{20}{@{}l}} {RF({k_{ \bot \,\textrm{res}}}r),\quad r < {r_f}}\\ {F({k_{ \bot \,\textrm{res}}}r),\quad r > {r_f}\; } \end{array}} \right.,\end{gather}

(3.10)\begin{gather}\begin{aligned} {j_{\textrm{sh}}}({r_f}) & = {y_w}[{\phi ^ + }({z^ + }) - {\phi ^ - }({z^ + })]\\ & = \dfrac{{{V_{\textrm{shRF}}}({r_f})}}{{{z_{\textrm{sh}}}}} = \dfrac{{2{y_w}}}{{{z_{\textrm{sh}}}{y_w} + 1}}{\phi ^ + }({z^ + }), \end{aligned}\end{gather}

where the quantities j _sh(r_f), y_w, z _sh are evaluated in the background plasma. The sheath RF current is null in the insulating limit. It is maximal in amplitude at r  =  r_f. The amplitude is independent of θ. We quantify the local RF power dissipation in the sheaths as

(3.11)\begin{gather}\textrm{d}{P_{\textrm{shRF}}}(r,\theta ) = \textrm{d}{P_{\textrm{shRF}}}({r_f})\left\{ {\begin{array}{*{20}{@{}l}} {R|F({k_{ \bot \,\textrm{res}}}r){|^2}}& {r < {r_f}}\\ {|F({k_{ \bot \,\textrm{res}}}r){|^2}}& {r > {r_f}} \end{array}} \right.,\end{gather}

(3.12)\begin{gather}\begin{array}{ccccc} & \textrm{d}{P_{\textrm{shRF}}}({r_f}) = \dfrac{1}{2}\textrm{Re}[{j_{\textrm{sh}}}({r_f})V_{\textrm{shRF}}^\ast ({r_f})]\\ & \quad = \dfrac{1}{2}{y_w}|{\phi ^ + }({z^ + }){|^2}(1 - |{R_\phi }{|^2}) = 4\textrm{Re(}{z_{\textrm{sh}}}\textrm{)}{\left|{\dfrac{{{y_w}{\phi^ + }({z^ + })}}{{{z_{\textrm{sh}}}{y_w} + 1}}} \right|^2}. \end{array}\end{gather}

Here again, dP _shRF(r_f), y_w, z _sh are evaluated in the background plasma. The total RF power dissipation in the sheaths is then

(3.13)\begin{equation}\begin{aligned} {P_{\textrm{shRF}}} & = \mathrm{\pi }\dfrac{{{y_w}|{\phi ^ + }({z^ + }){|^2}(1 - |{R_\phi }{|^2})}}{{k_{ \bot \,\textrm{res}}^2}}\left[ {\int_0^{{\xi_f}} {Rx{F^2}(x)\,\textrm{d}x} + \int_{{\xi_f}}^\infty {x{F^2}(x)\,\textrm{d}x} } \right]\\ & = {Y_{\textrm{FAMC}}}|{\phi ^ + }({z^ + }){|^2}[1 - |R_\phi ^ + {|^2}]. \end{aligned}\end{equation}

3.4. Power balance

In the half-plane z < 0, only the reflected FAMC mode $\phi _0^ -$ is present. In this region the Poynting flux is similar to (2.26)

(3.14)\begin{equation}{P_{\textrm{RF}}}(z) ={-} {Y_{\textrm{FAMC}}}|\phi _0^ - {|^2}\,\textrm{exp(}2{\nu ^\mathrm{\ast }}{k_i}z\textrm{)};\quad z<0;{k_i}>0.\end{equation}

In this region the Poynting flux is negative: the power flows from the FAMC excitation point towards negative z. The power dissipation in this half-plane is due to the collisional losses by the reflected mode in the plasma volume and amounts to

(3.15)\begin{equation}{P_{\textrm{RF}}}( - \infty ) - {P_{\textrm{RF}}}({0^ - }) = {Y_{\textrm{FAMC}}}|\phi _0^ +{|^2}|R_\phi ^ + {|^2}\,\textrm{exp}( - 4{\nu ^\mathrm{\ast }}{k_i}{z^ + }).\end{equation}

Since the half-plane is of infinite parallel extent, the volume power dissipation remains finite in the collision-less limit. In the region 0 < z < z ⁺ the two FAMC modes interfere. This affects the Poynting fluxes. From (2.20)

(3.16)\begin{equation}{P_{\textrm{RF}}}(z) ={-} \mathrm{\pi }{\varepsilon _0}{\omega _0}\textrm{Im}[{\varepsilon _\parallel }{\phi ^\ast }(z){\partial _z}\phi (z)]\frac{1}{{k_{ \bot \,\textrm{res}}^2}}\int_0^\infty {x{F^2}(x)\,\textrm{d}x} ,\end{equation}

with

(3.17)\begin{gather}\phi (z) = {\phi ^ + }(z) + {\phi ^ - }(z) = {\phi ^ + }(z)[1 + R_\phi ^ + (z)],\end{gather}

(3.18)\begin{gather}\begin{aligned} {\partial _z}\phi (z) & = \textrm{i}{k_{{\parallel} \,\textrm{res}}}\left( {1 + \textrm{i}{\nu^\mathrm{\ast }}\dfrac{{{k_i}}}{{{k_{{\parallel} \,\textrm{res}}}}}} \right)[{\phi ^ + }(z) - {\phi ^ - }(z)]\\ & = \textrm{i}{k_{{\parallel} \,\textrm{res}}}\left( {1 + \textrm{i}{\nu^\mathrm{\ast }}\dfrac{{{k_i}}}{{{k_{{\parallel} \,\textrm{res}}}}}} \right){\phi ^ + }(z)[1 - R_\phi ^ + (z)], \end{aligned}\end{gather}

(3.19)\begin{gather}{\varepsilon _\parallel } = \textrm{Re(}{\varepsilon _\parallel }\textrm{)(}1 - \textrm{i}{\nu ^\mathrm{\ast }}\textrm{)}.\end{gather}

This yields, to first order in ν*,

(3.20)\begin{equation}\begin{aligned} {P_{\textrm{RF}}}(z) & = {Y_{\textrm{FAMC}}}|{\phi ^ + }(z){|^2}\left[ {1 - |R_\phi^ + (z){|^2} + 2{\nu^\mathrm{\ast }}\left( {\dfrac{{{k_i}}}{{{k_{{\parallel} \,\textrm{res}}}}} - 1} \right)\textrm{Im}(R_\phi^ + (z))} \right] \\ & = {Y_{\textrm{FAMC}}}|\phi _0^ + {|^2}[{\textrm{exp}( - 2{\nu^\mathrm{\ast }}{k_i}z) - |R_\phi^ + {|^2}\textrm{exp}(2{\nu^\mathrm{\ast }}{k_i}(z - 2{z^ + }))} \\ & \quad + \left. {2{\nu^\mathrm{\ast }}\left( {\dfrac{{{k_i}}}{{{k_{{\parallel} \,\textrm{res}}}}} - 1} \right)\textrm{|}R_\phi^ + ({0^ + })\textrm{|sin}[2{k_{{\parallel} \,\textrm{res}}}({z^ + } - z) + \textrm{arg}(R_\phi^ + )]} \right];\quad z>0;{k_i}>0. \end{aligned}\end{equation}

To leading order in ν*, the Poynting flux in the presence of the two FAMC modes is the sum of a positive flux that would be obtained with the incident FAMC mode alone, and a negative contribution corresponding to the reflected mode alone. In the presence of volume dissipation, an additional oscillatory term of order ν* appears, due to the interference of the two modes. As $|R_\phi ^ + |< 1$ and 0 < z < z ⁺, the leading contribution to (3.20) is positive: in this region the power flows from the FAMC excitation point towards the sheath. In addition,

(3.21)\begin{equation}\begin{aligned} {\partial _z}{P_{\textrm{RF}}}(z) & ={-} 2{\nu ^\mathrm{\ast }}{k_i}{Y_{\textrm{FAMC}}}|{\phi ^ + }(z){|^2}\left[{1 + |R_\phi^ + (z){|^2}} \right.\\ & \quad + 2\left. {\left( {1 - \dfrac{{{k_{{\parallel} \,\textrm{res}}}}}{{{k_i}}}} \right)\textrm{|}R_\phi^ + (z)\textrm{|cos[}2{k_{{\parallel} \,\textrm{res}}}({z^ + } - z) + \arg (R_\phi^ + )\textrm{]}} \right],\quad z > 0. \end{aligned}\end{equation}

The first two contributions to (3.21) are the sum of the volume losses by each FAMC mode taken individually. The third oscillatory term results from the interference of the two modes, and can be of the same order as the other ones. As ${\nu ^\ast } \to {0^ + }$ at fixed z, the local loss scales as ν*. However, now the parallel domain is of finite parallel extent, so that the cumulated volume losses ${P_{\textrm{RF}}}(0) - {P_{\textrm{RF}}}({z^ + })$ vanish in the collision-less limit. To leading order in ν*

(3.22)\begin{equation}{P_{\textrm{in}}} = {P_{\textrm{RF}}}({0^ + }) - {P_{\textrm{RF}}}({0^ - }) = {Y_{\textrm{FAMC}}}|\phi _0^ + {|^2}.\end{equation}

The redirected power is the same with the semi-infinite as with the infinite field line model. Remarkably, P _in is independent of the collisionality, of the parallel distance z ⁺, of the sheath properties and more generally independent of which physical process dissipates the FAMC modes. To leading order in ${\nu ^\ast }{k_i}/{k_{/{/}\textrm{res}}}$, the Poynting flux at z  =  z ⁺ amounts to

(3.23)\begin{equation}{P_{\textrm{RF}}}({z^ + }) = {P_{\textrm{in}}}\textrm{exp}( - 2{\nu ^\mathrm{\ast }}{k_i}{z^ + })[1 - |R_\phi ^ + {|^2}].\end{equation}

${P_{\textrm{RF}}}({z^ + })$ from (3.23) is equal to the sheath power dissipation P _shRF from (3.13). Formula (3.23) provides the fraction of redirected power lost in the sheaths. Figure 3 maps ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{in}}}$ versus $|R_\phi ^ + {|^2}$ and ${\rm exp} ( - 2{\nu ^\ast }{k_i}{z^ + })$. ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{in}}}$ increases with decreasing $|R_\phi ^ + {|^2}$ and decreasing ${\nu ^\ast }{k_i}{z^ + }$. Contour lines are hyperbolas with asymptotes $|R_\phi ^ + {|^2} = 1$ and ${\rm exp} ( - 2{\nu ^\ast }{k_i}{z^ + }) = 0$. The RF sheaths do not dissipate all the redirected power, even when the anti-Hermitian part of the dielectric tensor vanishes. The sheath power fraction can be low for two reasons:

1. (1) ${\nu ^\ast }{k_i}{z^ + } \gg 1$ (long field line limit): most of the power from the incident FAMC mode is damped in the plasma volume before reaching the sheath. In this regime, the reflected FAMC mode hardly exists, the infinite field line model is fully valid.

2. (2) $|R_\varphi ^ + |\sim 1$: most of the power carried by the incident FAMC mode at z  =  z ⁺ is reflected into the mode ϕ ⁻, and subsequently dissipated in the (infinite) plasma volume −∞ < z < z ⁺.

Figure 3. Colour map of ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{in}}}$ from (3.23) versus $|R_\phi ^ + {|^2}$ and ${\rm exp} ( - 2{\nu ^\ast }{k_i}z)$, together with contour lines (one every 5 %). To be compared with figure 7 in a bounded geometry.

4.1. Outline of the model

Figure 4 sketches the model studied in § 4. The bounded magnetic field line now consists of two segments [−z ⁻,0] and [0,z ⁺], denoted respectively left and right. In each segment there are two FAMC modes with positive wave vector +k _//res > 0 (respective complex amplitudes $\phi _{0l}^ +$ and $\phi _{0r}^ +$ at z  =  0) and opposite wave vector −k _//res < 0 (complex amplitudes $\phi _{0l}^ -$ and $\phi _{0r}^ -$ at z  =  0). The mode excitation occurs at z  =  0. Consistent with expressions (2.19) and (2.25), we model it as a kick on the complex amplitudes of the two modes

(4.1)\begin{gather}\phi _{0r}^ + = \phi _{0l}^ + + \Delta \phi _0^ + ,\end{gather}

(4.2)\begin{gather}\phi _{0r}^ - = \phi _{0l}^ - + \Delta \phi _0^ - .\end{gather}

Figure 4. Sketch of the bounded field line model in § 4. The two nearly resonant FAMC modes in the model have opposite parallel wave vectors and propagate along the same filament. The colour shades are representative of the amplitudes for the isolated modes. In practice, the modes interfere and the total RF field amplitudes oscillate spatially, but this is not represented. Inset: decomposition of the solution into partial waves.

The prescribed excitation terms $\Delta \phi _0^ +$ and $\Delta \phi _0^ -$ are respectively due to nearly resonant scattering of incoming HHFW spectral components at +k _//res and −k _//res. They are expressed as a function of the incident wave using formula (2.17).

We describe the propagation of the modes in segments [−z ⁻,0] and [0,z ⁺] and their reflection at z  =  z ⁺ or z  =  −z ⁻, using formulas similar to (3.15) in semi-bounded geometry

(4.3)\begin{gather}\phi _{0r}^ - = \phi _{0r}^ + R_\phi ^ + \textrm{exp(}2\textrm{i}{k_{{\parallel} \,\textrm{res}}}{z^ + } - 2{\nu ^\mathrm{\ast }}{k_i}{z^ + }\textrm{)} = \phi _{0r}^ + {R^ + },\end{gather}

(4.4)\begin{gather}\phi _{0l}^ + = \phi _{0l}^ - R_\phi ^ - \textrm{exp(}2\textrm{i}{k_{{\parallel} \,\textrm{res}}}{z^ - } - 2{\nu ^\mathrm{\ast }}{k_i}{z^ - }\textrm{)} = \phi _{0l}^ - {R^ - },\end{gather}

and we use formulas analogous to (3.5) to express $R_\phi ^ \pm$.

4.2. RF field structure, wave multi-reflections

Equations (4.1)–(4.4) fully define the system in terms of the excitations $\Delta \phi _0^ +$ and $\Delta \phi _0^ -$. One can directly solve the system, or alternatively proceed in an iterative way as in Renk (Reference Renk2017),

(4.5)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {\phi_{0r}^{ + (0)} = \Delta \phi_0^ + \quad (\textrm{excitation}\;\textrm{for}\;n = \textrm{0}\;\textrm{on}\;\textrm{the}\;\textrm{right}\;\textrm{side,}\;\textrm{for}\;{k_{/{/}}} ={+} {k_{/{/}\,\textrm{res}}})}\\ {\phi_{0r}^{ - (n)} = {R^ + }\phi_{0r}^{ + (n)}\quad (\textrm{reflection}\;\textrm{at}\;\textrm{right}\;\textrm{boudary})}\\ {\phi_{0l}^{ - (n)} = \phi_{0r}^{ - (n)} - \Delta \phi_0^ - {\delta_{n,0}}\quad (\textrm{Kronecker}\;\textrm{symbol}\;{\delta_{n,0}}\;\textrm{means}\;\textrm{kick}\;\textrm{for}\;n = 0)}\\ {\phi_{0r}^{ + (n + 1)} = \phi_{0l}^{ + (n)} = {R^ - }\phi_{0l}^{ - (n)}\quad (\textrm{reflection}\;\textrm{at}\;\textrm{left}\;\textrm{boundary,}\;\textrm{then}\;\textrm{continuity})} \end{array}} \right\}.\end{equation}

In the iteration rules (4.5) one can identify the generic terms, e.g. $\phi _{0r}^{ + (n)}$, with the complex amplitudes of ‘partial waves’ at z  =  0, after n reflections on the left and right boundaries. Within this interpretation the kicks at n  =  0 represent the amplitudes of the FAMC modes excited by the HHFW scattering process, before the first double reflection. These are responsible for the RF field discontinuity at z  =  0. One subsequently obtains the solution as a ‘global wave’, i.e. the superposition of partial wave complex amplitudes, allowing for interference

(4.6)\begin{equation}\left. {\begin{aligned} \phi_{0r}^ + & = \sum\limits_{n = 0}^\infty {\phi_{0r}^{ + (n)}} = \dfrac{{\Delta \phi_0^ +{-} {R^ - }\Delta \phi_0^ - }}{{1 - {R^ + }{R^ - }}}\\ \phi_{0r}^ - & = {R^ + }\phi_{0r}^ + = \dfrac{{{R^ + }}}{{1 - {R^ + }{R^ - }}}(\Delta \phi_0^ +{-} {R^ - }\Delta \phi_0^ - )\\ \phi_{0l}^ - & = \sum\limits_{n = 0}^\infty {\phi_{0l}^{ - (n)}} ={-} \dfrac{{\Delta \phi_0^ -{-} {R^ + }\Delta \phi_0^ + }}{{1 - {R^ + }{R^ - }}}\\ \phi_{0l}^ +& = {R^ - }\phi_{0l}^ - = - \dfrac{{{R^ - }}}{{1 - {R^ + }{R^ - }}}(\Delta \phi_0^ -{-} {R^ + }\Delta \phi_0^ + )\end{aligned}} \right\}.\end{equation}

For n > 1, each double reflection multiplies the partial wave amplitudes by R ⁺R ⁻. One can decompose the coefficient R ⁺ into an amplitude attenuation factor and a phase shift on the right part of the bounded magnetic field line

(4.7)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {|{R^ + }|= |R_\phi^ + |\textrm{exp} ({-} 2{\nu^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}}\\ {\textrm{arg(}{R^ + }\textrm{)} = 2{k_{{\parallel} \,\textrm{res}}}{z^ + } + \textrm{arg(}R_\phi^ + \textrm{)}} \end{array}} \right\}.\end{equation}

A similar formula applies on the left side. In relation to the prototype Fabry--Perot resonator (Renk Reference Renk2017), $|R_\phi ^ + R_\phi ^ - |$ quantifies the reflectivity of the cavity extremities, while ${\rm exp} ( - 2{\nu ^\ast }{k_i}({z^ + } + {z^ - }))$ quantifies the ‘gain’ (actually a loss <1) of the medium. The phase shift between successive partial waves is arg(R ⁺R ⁻), while the ‘single-pass attenuation’ due to dissipation is |R ⁺R ⁻|. In the presence of dissipation, |R ⁺R ⁻| < 1 and the series converges. The behaviour of 1/(1−R ⁺R ⁻) is therefore a direct consequence of the interference between multiple partial waves.

(4.8)\begin{equation}|1 - {R^ + }{R^ - }{|^2} = {[1 - |{R^ + }{R^ - }|]^2} + 4|{R^ + }{R^ - }|\textrm{si}{\textrm{n}^2}[\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2].\end{equation}

Figure 5 plots |1−R ⁺R ⁻|² versus |R ⁺R ⁻| and $\textrm{si}{\textrm{n}^2}[\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2]$. The number of partial waves to take into account (number of roundtrip transits for FAMC mode photons) is typically 1/1−|R ⁺R ⁻| (Renk Reference Renk2017). In the long field line limit ${\nu ^\ast }{k_i}{z^ - } \to + \infty$, one partial wave is enough: R ⁻→0 and one recovers on the right side of the field line the semi-bounded model in § 3. This semi-bounded model also applies to the left side if $\Delta \phi _0^ -{=} 0$ (no direct excitation of the FAMC mode at –k _//res). The opposite limit is more interesting to study. The interference between partial waves in (4.6) manifests in oscillatory terms on the mode amplitudes, persisting in the short field line limit ${\nu ^\ast }{k_i}{z^ - } \to 0$ and ${\nu ^\ast }{k_i}{z^ + } \to 0$. As arg(R ⁺R ⁻) spans $[0,\mathrm{\pi }],\;|1 - {R^ + }{R^ - }{|^2}$ ranges between ${(1 - |{R^ + }{R^ - }|)^2} < 1$ and ${(1 - |{R^ + }{R^ - }|)^2} > 1$. The factor is smaller than 1 if

(4.9)\begin{equation}\textrm{si}{\textrm{n}^2}[\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2] < (\textrm{|}R_\phi ^ + R_\phi ^ - \textrm{|exp[} - 2{\nu ^\mathrm{\ast }}{k_i}({z^ + } + {z^ - })\textrm{]} - 2)/4,\end{equation}

so 1/|1−R ⁺R ⁻| can grow very large if the ‘single-pass attenuation’ is weak (|R ⁺R ⁻| close to 1) and simultaneously the partial waves interfere constructively $({\sin ^2}(\arg ({R^ + }{R^ - })/2) \ll 1)$. This situation is characteristic of a resonant cavity with a large quality factor (Renk Reference Renk2017). In conditions of weak dissipation one can approximate (4.8) as

(4.10)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {|1 - {R^ + }{R^ - }{|^2}\approx4{{\left[ {\dfrac{{\textrm{Re}({z_{\textrm{shr}}}){y_w}}}{{|{z_{\textrm{shr}}}{y_w} + 1{|^2}}} + \dfrac{{\textrm{Re(}{z_{\textrm{shl}}}\textrm{)}{y_w}}}{{|{z_{\textrm{shl}}}{y_w} + 1{|^2}}} + {\nu^\mathrm{\ast }}{k_i}({z^ + } + {z^ - })} \right]}^2}+ 4\textrm{si}{\textrm{n}^2}[\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2]}\\ {\dfrac{{\textrm{Re(}{z_{\textrm{shr}}}\textrm{)}{y_w}}}{{|{z_{\textrm{shr}}}{y_w} + 1{|^2}}} \ll 1;\quad \dfrac{{\textrm{Re(}{z_{\textrm{shl}}}\textrm{)}{y_w}}}{{|{z_{\textrm{shl}}}{y_w} + 1{|^2}}} \ll 1;\quad {\nu^\mathrm{\ast }}{k_i}({z^ + } + {z^ - }) \ll 1} \end{array}} \right\}.\end{equation}

Figure 5. Colour map (log scale) of $|1 - {R^ + }{R^ - }{|^2}$ versus $|{R^ + }{R^ - }|$ and ${\sin ^2}(\arg ({R^ + }{R^ - })/2)$, together with contour lines. The straight line corresponds to $|1 - {R^ + }{R^ - }|= 1$ ((4.9)), the dashed curves are contour lines for $|1 - {R^ + }{R^ - }{|^2} < 1$.

4.3. Redirected Power

On the parallel segment [0,z ⁺], a formula similar to (3.20) applies for the Poynting flux, where one replaces $\phi _0^ +$ with $(\Delta \phi _0^ +{-} {R^ - }\Delta \phi _0^ - )/(1 - {R^ + }{R^ - })$

(4.11)\begin{equation}\begin{aligned} {P_{\textrm{RF}}}(z) & = {Y_{\textrm{FAMC}}}{\left|{\dfrac{{\Delta \phi_0^ +{-} {R^ - }\Delta \phi_0^ - }}{{1 - {R^ + }{R^ - }}}} \right|^2} [\textrm{exp(} - 2{\nu ^\mathrm{\ast }}{k_i}z\textrm{)} - |R_\phi ^ + {|^2}\textrm{exp(}2{\nu ^\mathrm{\ast }}{k_i}\textrm{(}z - 2{z^ + }\textrm{))}];\quad z > 0. \end{aligned}\end{equation}

As already noticed in (3.20) and (3.21), P _RF(z) > 0 and ∂_zP _RF(z) < 0 on the right side of the magnetic field line. Similarly on the left side [−z ⁻,0].

(4.12)\begin{equation}\begin{aligned} {P_{\textrm{RF}}}(z) & ={-} {Y_{\textrm{FAMC}}}{\left|{\dfrac{{\Delta \phi_0^ -{-} {R^ + }\Delta \phi_0^ + }}{{1 - {R^ + }{R^ - }}}} \right|^2}\textrm{exp(}2{\nu ^\mathrm{\ast }}{k_i}z\textrm{)} - |R_\phi ^ - {|^2}\textrm{exp(} - 2{\nu ^\mathrm{\ast }}{k_i}\textrm{(}z + 2{z^ - }\textrm{))}];\quad z < 0. \end{aligned}\end{equation}

On this segment P _RF(z) < 0 and ∂_zP _RF(z) < 0. We deduce the redirected power

(4.13)\begin{equation}\begin{aligned} {P_{\textrm{in}}} & = {P_{\textrm{RF}}}({0^ + }) - {P_{\textrm{RF}}}({0^ - }) \\ & = {Y_{\textrm{FAMC}}}\left[ {{{\left|{\dfrac{{\Delta \phi_0^ +{-} {R^ - }\Delta \phi_0^ - }}{{1 - {R^ + }{R^ - }}}} \right|}^2}(1 - |{R^ + }{|^2}) + {{\left|{\dfrac{{\Delta \phi_0^ -{-} {R^ + }\Delta \phi_0^ + }}{{1 - {R^ + }{R^ - }}}} \right|}^2}(1 - |{R^ - }{|^2})} \right]\textrm{.} \end{aligned}\end{equation}

In order to compare with previous models, let us first take one single FAMC mode excitation and assume $\Delta \phi _0^ - = 0$. Then

(4.14)\begin{equation}{P_{\textrm{in}}} = {Y_{\textrm{FAMC}}}|\Delta \phi _0^ + {|^2}\frac{{1 - |{R^ + }{R^ - }{|^2}}}{{|1 - {R^ + }{R^ - }{|^2}}};\quad \Delta \phi _0^ - = 0.\end{equation}

As already noticed in § 3, the redirected power depends only on R ⁺R ⁻, regardless of which physical process dissipates the power. Compared with the ‘infinite field line model’, the redirected power is multiplied by the factor

(4.15)\begin{equation}\begin{aligned} \dfrac{{1 - |{R^ + }{R^ - }{|^2}}}{{|1 - {R^ + }{R^ - }{|^2}}} & = \dfrac{{1 - |{R^ + }{R^ - }{|^2}}}{{{{(1 - |{R^ + }{R^ - }|)}^2} + 4|{R^ + }{R^ - }|\textrm{si}{\textrm{n}^2}(\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2)}}\\ & = \dfrac{{1 + |{R^ + }{R^ - }|}}{{1 - |{R^ + }{R^ - }|}}\dfrac{1}{{1 + \dfrac{{4|{R^ + }{R^ - }|}}{{{{(1 - |{R^ + }{R^ - }|)}^2}}}\textrm{si}{\textrm{n}^2}(\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2)}}. \end{aligned}\end{equation}

In this expression, $1 - |{R^ + }{R^ - }{|^2}$ quantifies the power dissipated in one pass by a partial wave of amplitude 1, while 1/|1−R ⁺R ⁻| quantifies the enhancement of amplitude for the global wave due to the multi-reflections. Figure 6 plots the multiplication factor (4.15) versus |R ⁺R ⁻| and ${\sin ^2}(\arg ({R^ + }{R^ - })/2)$. At fixed |R ⁺R ⁻|, figure 6 displays Airy curves characteristic of a Fabry--Perot cavity (Renk Reference Renk2017): as arg(R ⁺R ⁻) increases from 0 (constructive interference) to π (destructive interference), the factor decreases from $(1 + |{R^ + }{R^ - }|)/(1 - |{R^ + }{R^ - }|) > 1$ (amplification) to $(1 - |{R^ + }{R^ - }|)/(1 + |{R^ + }{R^ - }|) < 1$ (reduction). The factor is larger than 1 if

(4.16)\begin{equation}|{R^ + }{R^ - }|< 1 - 2\textrm{si}{\textrm{n}^2}(\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2);\end{equation}

(straight line in contour plot).

Figure 6. Colour map (log scale) together with contour lines of the multiplication factor for the total dissipated power (formula (4.15)) versus $|{R^ + }{R^ - }|$ and ${\sin ^2}(\arg ({R^ + }{R^ - }))$. The straight line corresponds to factor 1 ((4.16)).

The range of the variation broadens as |R ⁺R ⁻| increases. For $|{R^ + }{R^ - }|\ll 1$ one recovers the infinite field line model (factor 1), regardless of the dissipation process. This also applies to the semi-bounded field line model, for which R ⁻  =  0. When |R ⁺R ⁻| is close to 1 and arg(R ⁺R ⁻) close to 0, the multiplication factor becomes very sensitive to small changes in the parameters: formula (4.15) behaves locally as the ratio $(1 - |{R^ + }{R^ - }|)/2\textrm{si}{\textrm{n}^2}(\textrm{arg(}{R^ + }{R^ - }\textrm{)}/2)$. In figure 6 the contour curves appear locally as straight lines converging at the critical point. This behaviour is characteristic of a high-Q resonant cavity, with a quality factor proportional to 1/(1−|R ⁺R ⁻|) (Renk Reference Renk2017). The half-width of the resonant peak scales as $|{R^ + }{R^ - }{|^{1/2}}/(1 - |{R^ + }{R^ - }|)$.

When the HHFW scattering process simultaneously excites two counter-propagative modes $\Delta \phi _0^ +$ and $\Delta \phi _0^ -$ in the system, the dissipated power is NOT the sum of the dissipated powers by each mode excited separately with the same amplitude. In other words the Parseval theorem, as formulated on infinite field lines, does not apply to the bounded magnetic field line. This is also a consequence of the mode reflections. Instead, (4.11) shows that ${P_{\textrm{rf}}}({0^ + })$ is proportional to $|\Delta \phi _0^ +{-} {R^ - }\Delta \phi _0^ - {|^2}$, while (4.12) shows that ${P_{\textrm{rf}}}({0^ - })$ is proportional to $|\Delta \phi _0^ -{-} {R^ + }\Delta \phi _0^ + {|^2}$.

4.3. Sheath oscillating properties and power partitioning.

On the bounded field line model, the sheath RF voltages are analogous to expression (3.8) for the isolated sheath in § 3, upon the substitution $\phi _0^ + \to \phi _{0r}^ +{=} (\Delta \phi _0^ +{-} {R^ - }\Delta \phi _0^ - )/(1 - {R^ + }{R^ - })$ or $\phi _{0l}^ -{=} - (\Delta \phi _0^ -{-} {R^ + }\Delta \phi _0^ + )/(1 - {R^ + }{R^ - })$

(4.17)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {\left|{\dfrac{{{V_{\textrm{shRFr}}}({r_f})}}{{\Delta \phi_0^ +{-} {R^ - }\Delta \phi_0^ - }}} \right|= \left|{\dfrac{{1 + R_\phi^ + }}{{1 - {R^ + }{R^ - }}}} \right|\textrm{exp} ({-} {\nu^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}}\\ {\left|{\dfrac{{{V_{\textrm{shRFl}}}({r_f})}}{{\Delta \phi_0^ -{-} {R^ + }\Delta \phi_0^ + }}} \right|= \left|{\dfrac{{1 + R_\phi^ - }}{{1 - {R^ + }{R^ - }}}} \right|\textrm{exp} ({-} {\nu^\mathrm{\ast }}{k_i}{z^ - }\textrm{)}} \end{array}} \right\}.\end{equation}

Depending on the excitation and on the interference patterns, the amplitudes of sheath oscillations can differ at the two extremities of the bounded field line. As ${\nu ^\ast }{k_i}{z^ - } \to + \infty$, the sheath oscillations on the left side tend to 0, and one recovers exactly expression (3.8) on the right sheath. The sheath voltages are proportional to the local wave amplitudes; when $\Delta \phi _0^ -{=} 0$, the voltage on the right sheath is multiplied by a factor $1/|1 - {R^ + }{R^ - }|$ compared with (3.8). The power losses at the sheaths also exhibit similarity with previous results

(4.18)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {\dfrac{{{P_{\textrm{RF}}}({z^ + })}}{{{P_{\textrm{RF}}}({0^ + })}} = \textrm{exp} ({-} 2{\nu^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}\dfrac{{1 - |R_\phi^ + {|^2}}}{{1 - |R_\phi^ + {|^2}\textrm{exp} ({-} 4{\nu^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}}}}\\ {\dfrac{{{P_{\textrm{RF}}}({z^ - })}}{{{P_{\textrm{RF}}}({0^ - })}} = \textrm{exp} ({-} 2{\nu^\mathrm{\ast }}{k_i}{z^ - }\textrm{)}\dfrac{{1 - |R_\phi^ - {|^2}}}{{1 - |R_\phi^ - {|^2}\textrm{exp} ({-} 4{\nu^\mathrm{\ast }}{k_i}{z^ - }\textrm{)}}}} \end{array}} \right\}.\end{equation}

These power fractions are independent of arg(R ⁺) and arg(R ⁻). In addition, the power fractions on the two sides of the magnetic field line are independent of each other. Figure 7 maps ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{RF}}}({0^ + })$ versus $|R_\phi ^ + {|^2}$ and ${\rm exp} ( - 2{\nu ^\ast }{k_i}{z^ + })$. ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{RF}}}({0^ + })$ increases with decreasing $|R_\phi ^ + {|^2}$ and decreasing ${\nu ^\ast }{k_i}{z^ + }$. As already observed on the right segment in § 3, in the presence of dissipative sheaths

(4.19)\begin{equation}\left. {\begin{aligned} {\mathop {\lim }\limits_{{\nu^\mathrm{\ast }}{k_i}{z^ + }\rightarrow0} {P_{\textrm{RF}}}({z^ + }) = {P_{\textrm{RF}}}({0^ + });\quad |R_\phi^ + |< 1}\\ {\mathop {\lim }\limits_{{\nu^\mathrm{\ast }}{k_i}{z^ - }\rightarrow0} {P_{\textrm{RF}}}( - {z^ - }) = {P_{\textrm{RF}}}({0^ - });\quad |R_\phi^ - |< 1} \end{aligned}} \right\}.\end{equation}

Figure 7. Colour map of ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{RF}}}({0^ + }))$ from (4.18) versus $|R_\phi ^ + {|^2}$ and ${\rm exp} ( - 2{\nu ^\ast }{k_i}{z^ + })$, together with contour lines (one every 5 %). To be compared with figure 3 in semi-infinite geometry.

Consequently, in the short field line limit ${\nu ^\ast }{k_i}{z^ - } \to 0$ and ${\nu ^\ast }{k_i}{z^ + } \to 0$, the sheaths dissipate all the redirected power. One could anticipate this result; the magnetic field line has a finite parallel extent, so that the volume dissipation vanishes in the collision-less limit, despite a possible RF field enhancement by multi-reflection. The only exception to property (4.19) occurs when one replaces both sheaths by perfect metallic walls. In that case $|R_\phi ^ + |= 1$ and $|R_\phi ^ - |= 1$, all the redirected power is dissipated in the plasma volume even in the short field line limit. When $|R_\phi ^ + |$ is close to 1 and ${\nu ^\ast }{k_i}{z^ + }$ is small (upper-right corner of figure 7), ${P_{\textrm{RF}}}({z^ + })/{P_{\textrm{RF}}}({0^ + })$ is very sensitive to small changes in the parameters. The power ratio locally behaves as

(4.20)\begin{equation}\frac{{{P_{\textrm{RF}}}({z^ + })}}{{{P_{\textrm{RF}}}({0^ + })}} \approx {1 / {\left( {1 + \frac{{4{\nu^\mathrm{\ast }}{k_i}{z^ + }}}{{1 - |R_\phi^ + {|^2}}}} \right)}};\quad 4{\nu ^\mathrm{\ast }}{k_i}{z^ + } \ll 1,\quad |R_\phi ^ + |\approx 1.\end{equation}

One can see the fraction of redirected power lost in the sheaths as a weighted average of formulas (4.18) on the two sides of the magnetic field line

(4.21)\begin{equation}\frac{{{P_{\textrm{RF}}}({z^ + }) - {P_{\textrm{RF}}}( - {z^ - })}}{{{P_{\textrm{in}}}}} = \; \frac{{{P_{\textrm{RF}}}({0^ + })}}{{{P_{\textrm{in}}}}}\frac{{{P_{\textrm{RF}}}({z^ + })}}{{{P_{\textrm{RF}}}({0^ + })}} + \frac{{|{P_{\textrm{RF}}}({0^ - })|}}{{{P_{\textrm{in}}}}}\frac{{{P_{\textrm{RF}}}( - {z^ - })}}{{{P_{\textrm{RF}}}({0^ - })}}.\end{equation}

When the field line geometry is left--right symmetric, i.e. $|R_\phi ^ + |= |R_\phi ^ - |$ and z ⁺  =  z ⁻, one finds a result similar to (4.18)

(4.22)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {\dfrac{{{P_{\textrm{RF}}}({z^ + }) - {P_{\textrm{RF}}}( - {z^ - })}}{{{P_{\textrm{in}}}}} = \textrm{exp} ({-} 2{\nu^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}\dfrac{{1 - |R_\phi^ + {|^2}}}{{1 - |R_\phi^ + {|^2}\textrm{exp(} - 4{\nu^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}}}}\\ {|R_\phi^ + |= |R_\phi^ - |,\quad {z^ + } = {z^ - }} \end{array}} \right\}.\end{equation}

This fraction is larger than the previous result (3.23) for semi-bounded magnetic field lines, by a factor $1/[1 - |R_\phi ^ + {|^2}\textrm{exp}({-} 4{\nu ^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}]$. When the magnetic field line is asymmetric but $\Delta \phi _0^ -{=} 0$, the power fraction lost in the sheaths amounts to

(4.23)\begin{equation}\begin{array}{ccccc} & \dfrac{{{P_{\textrm{RF}}}({z^ + }) - {P_{\textrm{RF}}}( - {z^ - })}}{{{P_{\textrm{in}}}}} = \dfrac{{\textrm{exp(} - 2{\nu ^\mathrm{\ast }}{k_i}{z^ + }\textrm{)}}}{{1 - |R_\phi ^ + {|^2}|R_\phi ^ - {|^2}\textrm{exp(} - 4{\nu ^\mathrm{\ast }}{k_i}\textrm{(}{z^ + } + {z^ - }\textrm{))}}}\\ & \times [1 - |R_\phi ^ + {|^2}[1 - (1 - |R_\phi ^ - {|^2})\textrm{exp(} - 2{\nu ^\mathrm{\ast }}{k_i}\textrm{(}{z^ + } + {z^ - }\textrm{))}]],\quad \Delta \phi _0^ -{=} 0. \end{array}\end{equation}

One recovers formula (3.23) in the long field line limit on the left side $({\nu ^\ast }{k_i}{z^ - } \gg 1)$, as well as (4.22) for a left--right symmetric system. It simplifies to ${\rm exp} ( - 2{\nu ^\ast }{k_i}{z^ + })$ when $|R_\phi ^ + |= 0$.