Resonant wave–filament interactions as a loss mechanism for HHFW heating and current drive

Perkins et al (2012 Phys. Rev. Lett. 109 045001) reported unexpected power losses during high harmonic fast wave (HHFW) heating and current drive in the National Spherical Torus Experiment (NSTX). Recently, Tierens et al (2020 Phys. Plasmas 27 010702) proposed that these losses may be attributable to surface waves on field-aligned plasma filaments, which carry power along the filaments, to be lost at the endpoints where the filaments intersect the limiters. In this work, we show that there is indeed a resonant loss mechanism associated with the excitation of these surface waves, and derive an analytic expression for the power lost to surface wave modes at each filament.


Introduction
Reportedly [1,2], during high harmonic fast wave (HHFW) heating and current drive in the edge plasma of the National Spherical Torus Experiment (NSTX), a substantial fraction of the radiofrequency power fails to reach the core plasma. Instead, it flows along field lines and hits the divertors.
Recently, we proposed a mechanism that can explain why such losses are observed on NSTX but not elsewhere [9] (i.e. mainly in conditions with HHFW and small parallel wavenumber (k ∥ )). Knowing that small (non-ELM) filaments * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. are plentiful in the NSTX SOL [10,11], we used an analytic solution for wave scattering at plasma filaments [12], which shows that surface waves on such filaments can be resonantly excited and carry power along the filaments and thus along the field lines. This is more common in the HHFW heating scenarios used in NSTX than in generic ion cyclotron range of frequencies (ICRF) heating scenarios. We hypothesized that the power would be lost due to some endpoint loss mechanism (e.g. sheath rectification, imperfect reflection, surface resistivity) where the filament finally intersects a divertor.
In this work, we quantify the amount of power that is redirected to the surface waves in an idealized case (infinite magnetic field lines) where the end point mechanisms are not present. We follow an approach that should be generic to any resonant surface-wave-like oscillations. The main steps are as follows.
(a) Derive the resonance condition for surface wave excitation on a given filament. Specifically, find the parallel wavenumber k ∥ at which a resonance occurs for prescribed filament properties (density, radius). Several resonant k ∥ values may exist. (b) Introduce a causality parameter ν, formally similar to a collision frequency, as explained in [13]. (c) Calculate the total power dissipated in the filament from step (a), with ν > 0. (d) The resulting integrand has Lorentzian terms with peaks at the resonant k ∥ , growing higher and narrower, but remaining integrable, as ν decreases. (e) Take the limit ν → 0 + . If the filament is long enough, we show that the asymptotic dissipated power remains finite, independent of ν. We claim that the result is fairly independent of the physical dissipation mechanism described phenomenologically by ν (appendix A). (f) Average the dissipated power over the filament type. Filaments are inherently random structures in terms of density/radius, with each a specific probability to be present at a given time, and a specific resonant k ∥ value that matches a specific part of the launched k ∥ spectrum for the incident waves.
This work is organized as follows: in section 2 we analytically perform step (a) for one specific branch of resonant modes relevant in the HHFW regime. In section 3 we derive the power redirected due to this resonant surface wave excitation (steps (b)-(e)). Section 4 contains a discussion of the statistical nature of the filaments and the average power converted per filament (step (f)). An example is given in section 5, and the conclusion is given in section 6.

Derivation of the resonance condition
We consider a wave scattering problem much like that discussed in [9,12]: a cylindrical filament with radius r f and constant density n f , in a background plasma with constant density n b . The filament is along the confining magnetic field, which is along z. The parallel direction (subscript ∥) is always along z, parallel to B. The perpendicular direction (subscript ⊥) is perpendicular to B. We use a cylindrical coordinate system (r, θ, z) or (r, z) centered on the filament.
In [9], we identified multiple kinds of conditions under which surface waves exist on the filament. Here, we try to find an approximate analytical expression for the resonance condition for the ones we expect to be physically relevant in the HHFW regime, the ones which occur when n f n b is not much greater than 1.
We start by writing the dielectric tensor in the HHFW limit where , Ω i = qiB mi , ω = 2πf, with f the launched wave frequency, and q i and m i the ion charge and mass. For simplicity, we assume a single-ion plasma. In the range of plasma parameters of interest for this work, the slow wave is evanescent, and its perpendicular wavevector k ⊥ is purely imaginary, with imaginary part κ which is conveniently independent of the density, in the HHFW limit. Throughout this work, we will assume without loss of generality that k ∥ and κ are positive. Following [14], we introduce a spectral representation of the scattered SW field, in the filament and in the background with the amplitudes A f (k ∥ ), A b (k ∥ ) to be determined. We will see that the poles of A f (k ∥ ) and A b (k ∥ ) are the resonances for which we are looking. Next, we write E as the gradient of a potential [14] where I m , K m are Bessel functions. This approximation requires that ϵ ⊥ is negligible w.r.t. n 2 ∥ , which fails for k ∥ ≈ 0. In practice, ICRF antennas avoid exciting very low k ∥ modes, especially when operating in dipole, which is common. As will be clear soon, only one azimuthal mode can be resonant, m = ±1 depending on the the type of filament. Therefore, to simplify the notation in the following we omit the sum over m. For the fast wave, since k ⊥ r f ≪ 1, the perpendicular electric field of the incident FW is nearly constant across the radial scale of the filament. Mathematically, this is equivalent to approximating E ⊥ by ∇ ⊥ (E ⊥ · r) where E ⊥,i (k ∥ ) is the spectrum of E ⊥ of the incident FW launched by the antenna. Thus, the perpendicular component of the solution of the wave equation can be approximated by the perpendicular gradient of the following potential In the spirit of the Born approximation [15], we omit the contribution from the scattered fast wave since it is negligible to the lowest order. Although our approach should remain valid in the case where the scattered fast wave is evanescent, this approximation is especially well-justified in the case where it is propagative: the surface wave cannot radiate energy away from the surface, and thus cannot contain a significant contribution of a propagating fast wave. As a consequence of this approximation, we will see the surface waves on the filament gain energy from the incident fast wave, but we will not see the fast wave lose energy-an approximation that is only really valid in a perturbative regime where at most a small fraction of the incident power is lost to the surface waves. Within these approximations, of the six boundary conditions on the interface [12], only two are not trivially satisfied: the continuity of Φ (E ∥ from the fast wave is continuous by assumption since we neglect the scattered fast wave, and the continuity of E ∥ from the scattered slow wave follows from the continuity of Φ) and the continuity of D n = ϵ ⊥ E r − iϵ × E θ at the filament surface. These two boundary conditions suffice to constrain the unknown coefficients A f and A b . The first gives For the continuity of D n , we consider a single k ∥ mode, where we may assume without loss of generality that E ⊥,i points along the x axis, and |E ⊥, The matching condition for D n is and writing it all out: It is clear that only m = ±1 modes can be excited by the incident FW. Projecting (14) onto m = ±1 using cos(θ) → where ξ = κr f . The argument of the Bessel functions and their derivatives is from now on assumed to be ξ when not explicitly given. Collecting terms in the unknown amplitudes A f , A b , (15) becomes where The solution which obeys both continuity conditions, (10) and (16), is In the HHFW limit, ϵ ⊥ and ϵ ∥ are both proportional to the density. Normalizing everything to the background density (i.e. unless otherwise noted, ω i and ω e are calculated from the background density n b ), and using R ≡ n f n b and φ ≡ ω The resonance condition In figure 1 we see To determine the forward or backward nature of these waves, we can solve the resonance condition for φ (26) Figure 2 shows (25) for various values of φ. From both theory [16,17] and experiment [11], we know that filaments with n f n b not much greater than 1 are common, but filaments with much higher n f n b are rare. It becomes clear that this mechanism is relevant for HHFW (φ ≫ 1) but not for traditional minority and second harmonic ICRF heating: exciting the surface waves in the latter case requires much higher, hence much rarer, n f n b . While the approximations made in this section may seem extreme, we have previously determined the locations of these resonances within an exact electromagnetic full-wave solution [9]. In figure 3, we see that (25) is indeed a good approximation to the resonances found in that exact full-wave solution, which justifies our approximations. Numerical calculations in [18,19] and theoretical arguments in [20] also confirm that such resonances exist even for nonidealized filaments. Thus, for filaments whose density exceeds that of the background (so-called 'blob filaments', where R > 1), (25) predicts that resonances can only exist at m = 1, R > φ+1 φ−1 > 1, and ξ < ξ ∞ , where ξ ∞ is the positive root of φI 1 (ξ) − ξI ′ 1 (ξ). From (26), ∂ k ∥ ω > 0 for k ∥ > 0, so the surface waves resonantly excited on blob filaments are forward in the parallel direction.

Hole filaments (R < 1)
For filaments whose density is below that of the background (so-called 'hole filaments', where R < 1, see [21]), (25) , so the surface waves resonantly excited on hole filaments are forward in the parallel direction.

Derivation of the filament losses
We now introduce the causality parameter ν, formally similar to a collision frequency, in the dielectric tensor The case where ν plays the role of a physical collision frequency, and the electrons and ions have different collision frequencies, will be treated in appendix A. The introduction of ν The power loss density P, in SI units, is [22] The total power absorbed within r < ϱ is We want to calculate the total power in the ν → 0 + limit Taking the limit ϱ → ∞ is a mathematical convenience, in practice the integrands vanish within a few SW decay lengths away from the filament surface. It will be useful to reparametrize (4) and (5) in terms of the dimensionless quantity ξ = k ∥ r f m i me . Then Let us now integrate over z (i.e. along B, along the filament). The length of the filament is much larger than its radius. We approximate the finite filament by taking the integral from −∞ to ∞. In reality, the filaments have finite length, and the ones Integrating also over the azimuthal angle, and inserting (28) (recall ω i , ω e are background quantities, hence the factor R), Let the resonance be at ξ R , i.e. ξ = ξ R obeys (25). We linearize the denominator of (20) around the resonance, where the prime is the derivative w.r.t ξ at constant R. The dimensionless quantities α and β defined in (40) and (41) are shown in figures 4 and 5. We will soon see that the power P 0 scales with 1 αβ . From these figures, we see that for blob filaments at constant k ∥ , α and β decrease with increasing ω Ωi , so the power increases with increasing ω Ωi . Thus, resonant interactions not only become rarer at lower ω Ωi , they also lose less power.
At this point, we note that ν|A f | 2 behaves, in a neighbourhood of the resonance/pole, like a Lorentzian. As we approach the ν → 0 + limit, The factor ν a 2 (ξ−ξR) 2 +b 2 ν 2 becomes higher but more narrowly peaked around ξ = ξ R as ν approaches 0, such that its integral remains finite even in the ν → 0 + limit. We find this integral by complex contour integration. The poles are at ξ = ξ R ± i b a ν,  and the corresponding residues, ∓ i 2ab , do not depend on ν.
We have the following 'Dirac delta-like' behaviour Thus, in the filament Similarly, in the background P 0 , the total power converted to the surface wave mode, is then simply From (6) and (7) we have Thus, Inserting S from (24) and α and β from (40) and (41) into (52): Recall that |E ⊥,i | 2 is the power spectrum of the perpendicular component of the incident FW electric field, in units of V 2 . Thus, P 0 has units of power. There is also a resonance at −ξ R , and the total power converted by the filament is the sum of P 0 over both resonances. If |E ⊥,i | 2 is a symmetric power spectrum, this amounts to doubling P 0 .

Filament statistics
Since all the RF calculations in this work are performed within linear electrodynamics, the powers dissipated by two filament modes with different resonant k ∥ values are additive. Thus, we can calculate the individual power P 0 for isolated filaments and then perform a statistical average ⟨P 0 ⟩ of the individual powers when ∼10 2 filaments [11] are present simultaneously in front of the antenna. The most important averaging is over R = n f n b , the statistics of which have been studied theoretically by [16,17], and empirically by [10,11]. Given some probability density function P(R), a corresponding probability distribution over the resonant ξ R , P(ξ R ), can be readily derived using (25) Then Equation (57) is a bilinear operator from the power spectrum and the probability density to the expected power lost per fila- Garcia et al [16,17] predict that n−⟨n⟩ σn , where n is the fluctuating density, ⟨n⟩ is the time-averaged background density, and σ n is the standard deviation of the fluctuating density, obeys a Gamma distribution The parameter γ is the so-called 'intermittency parameter' γ = . From all this, we can derive the cumulative distribution function Identifying n f n b with n ⟨n⟩ , the probability distribution function is .
The probability that a given filament is a blob filament (R > 1) is The probability that a given filament is a hole filament (R < 1) is The probability that a given filament interacts resonantly with the incident HHFW (that is, , the band where (25) has no solutions) is For completeness, we note that very recently, Biswas et al [23] argued that P(R, r f ) should be given by a bivariate skewnormal distribution, with a positive correlation between R and r f .

Example
To show that the power involved in this mechanism is not in general negligible, we consider some NSTX-like values. First, we must assume a power spectrum. Let the spectrum of the vertical component of the incident FW be a simple dipole of amplitude A i , peaking at k ∥ = ±k ∥,0 The power spectrum we need is that of the perpendicular component of the incident fast wave. Within the HHFW limit In figure 6, we show P 0 P(ξ R ), the curves whose integral gives ⟨P 0 ⟩, assuming deuterium plasma with n = 10 17 m −3 , k ∥,0 = 3 m −1 , r f = 1-3 cm, and A i = 10 4 V, numbers roughly representative of the NSTX SOL [3,11]. We see that ⟨P 0 ⟩ is typically in the order of multiple kW. Considering the presence of tens or hundreds of filaments in the SOL [10,11] at all times, a considerable fraction of the incident ∼1 MW of power can be lost to surface wave excitation on filaments in NSTX-like conditions. Furthermore, the power lost due to this mechanism is proportional to ω 2 i i.e. to the density, and indeed, the HHFW heating in NSTX is more efficient at lower edge densities [24]. Thus, we expect this mechanism to be among the factors responsible for the observed edge losses on that machine [1,2].

Conclusion
We have discussed a resonant loss mechanism for high harmonic fast waves in tokamak edge plasmas. This loss mechanism is associated with the resonant excitation of surface waves on naturally occurring plasma filaments.
We have derived this loss mechanism for the specific case of resonant surface waves in HHFW-heated plasmas, where the resonant surface wave excitation is approximately described by the result of section 2. The derivation of this loss mechanism does not depend on details of the resonance condition. Analogous loss mechanisms likely exist for any resonantly excited surface wave. If resonant wave-filament interactions occur for heating systems other than HHFW, such as lower hybrid [25], it is likely that similar loss mechanisms are relevant there as well.
Throughout this paper, we interpret P 0 as the power lost from the incident fast wave, and transferred to the surface wave along the filament. In this interpretation, it is a transfer of power from one wave mode to another, which is seen as a loss from the point of view of using the incident wave for heating or current drive, but physically the power is not dissipated, it ends up in the other mode. The ultimate fate of this mode-converted power cannot be described by our model (see appendix B). It likely involves loss mechanisms at the filament endpoints.
Finally, from the estimate we provide for NSTX-like parameters, we find that in a HHFW heating or current drive scenario, this mechanism can easily redirect several kW of power per filament. Further work will focus on the detailed application of this theory to the case of the HHFW edge losses observed in NSTX.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). We acknowledge fruitful discussions with Nicola Bertelli and Syun'ichi Shiraiwa of PPPL, and thank Vladimir Bobkov for careful reading and comments.

Appendix A. Independent electron and ion collision frequencies
If we interpret ν as a physical collision frequency, we may want to use a different collision frequency for ions and electrons. Let the ion frequency be ν and the electron frequency ν e , usually with ν e ≫ ν, from momentum conservation considerations νe ν ≈ m i me . Then, we replace (28) by We take the collisionless limit keeping the ratio νe ν constant. This has two main consequences. First, it changes (54) to Note that Ψ f , Ψ b are now first degree polynomials in νe ν . Second, it modifies ξ which affects the linearisation (39) so a remains as it was in (41), but b gets an extra term, it becomes a first degree polynomial in νe ν . Thus, both the numerator and denominator in P 0 (52) are now first degree polynomials in νe ν . Remarkably, the polynomial in the numerator is proportional to the polynomial in the denominator: the dependence on νe ν cancels out. P 0 (52) does not depend on νe ν . We prove the case of blob filaments (m = 1). To show that the ratio a νe ν +b c νe ν +d is independent of νe ν , we need to show that b/a = d/c. We apply this observation to the numerator polynomial P N , obtained by working out the integrals (A.2) and (A.3) and the denominator polynomial P D which follows from (A.8) Making use of Bessel function identities, and eliminating φ using (26), we can simplify the expression for α to and the constant term in (A.9) to The condition for independence of νe ν , the ratio of the constant coefficient over the linear coefficient for the numerator polynomial, must equal that of the denominator polynomial, becomes (A.14) Insert (A.11) Working out the derivatives on the lhs and −I 2 0 + I 2 2 = (I 2 + I 0 )(I 2 − I 0 ) = −(I 2 + I 0 ) 2I1 ξR and −K 2 0 + K 2 2 = (K 2 + K 0 )(K 2 − K 0 ) = (K 2 + K 0 ) 2K1 ξR on the rhs, I 2 1 (K 0 + K 2 )K 1 + RK 2 1 (I 0 + I 2 )I 1 = I 2 1 (K 2 + K 0 )K 1 + RK 2 1 (I 2 + I 0 )I 1 (A. 16) which is clearly true.

Appendix B. Finite filaments
Our model is only valid in a strict sense for infinite magnetic field lines, where the parallel integration (37)  Our model is an asymptotic theory valid for ν 'small enough', such that ∆k ∥ is far smaller than any other spectral scalelength in the problem (this is the condition for the 'Dirac deltalike' relations (44) and (45) to hold). Spatially, the nearlyresonant wave-filament modes should have a large yet finite parallel extent ∆z ≈ 1 ∆k ∥ . This is the length scale over which the SW fields have non-negligible amplitude, as well as the length scale over which the mode-converted RF power is fully dissipated in the plasma (filament) volume. It is reasonable to think that our approach remains valid for bounded domains if the parallel extent of the nearly-resonant modes is smaller than the typical connection length L ∥ of open magnetic field lines. This leads to the criterion which defines a lower bound for the parameter ν. It is possible that in a more complete treatment of this problem, a small contribution of the scattered propagative fast wave (which we neglected in this work) will provide a natural lower bound to the losses. If (B.2) does not hold, if ∆k ∥ L ∥ < 1, we expect that not all the estimated mode-converted power can be dissipated in the finite filament in a single pass. Non-negligible RF fields can then reach the field line endpoints. There, they may be partially reflected back to the plasma, partially absorbed by lossy metallic walls, and/or excite sheath oscillations and enhance heat loads to the wall via sheath rectification. However, our model does not capture what happens at these endpoints and is less reliable in assessing the lost power in this regime.