We investigate the impact of relativistic effects on upper hybrid (UH) waves in plasmas with thermal electrons, particularly focusing on modifications of the conditions under which UH wave trapping and related low-threshold parametric decay instabilities (PDIs) may occur. A moderately relativistic (MR) dispersion relation for UH waves, valid for electron temperatures up to 25 keV and wave frequencies up to twice the electron cyclotron frequency, is obtained from previous results and shown to reduce to the warm non-relativistic result commonly used for PDI studies at low electron temperatures. The conditions under which MR UH waves propagate are then determined and compared with warm and cold plasma theory, showing a general increase in the electron density and background magnetic field strength at which the UH resonance occurs for finite electron temperatures. We next investigate the impact of the MR corrections on the possibility of UH wave trapping for X-mode electron cyclotron resonance heated (ECRH) plasmas at the ASDEX Upgrade tokamak and scaled versions of the ASDEX Upgrade parameters with core electron temperatures resembling those expected in ITER X-mode ECRH plasmas. The MR UH wave trapping conditions are virtually unchanged for ASDEX Upgrade relative to warm theory, due to the low electron temperatures, while potentially important differences between warm and MR theory exist for ITER-like core electron temperatures; cold theory is found to be insufficient in both cases. Finally, the MR dispersion relation is shown to qualitatively reproduce the PDI thresholds from warm theory for previously studied ASDEX Upgrade cases.

## I. INTRODUCTION

Magnetized plasmas support a large number of wave modes, many of which possess cutoff and mode conversion layers where the incoming wave energy can be reflected.^{1} When reflecting layers enclose a region allowing propagation of the waves involved in the mode conversion process, the waves are trapped in that region.^{2,3} Such trapping regions form effective cavities that support eigenmodes, whose characteristics can be determined from Bohr–Sommerfeld conditions within a Wentzel–Kramers–Brillouin (WKB) framework.^{2,3} While the trapped modes cannot easily be excited directly under fusion-relevant conditions, they are of significant interest in connection with nonlinear phenomena, since their spatial localization greatly increases the time over which they are able to interact with strong coherent external electromagnetic waves, used for heating and current drive.^{4,5}

In this work, we particularly focus on the conditions under which trapping of upper hybrid (UH) waves may occur and the modification of these conditions due to relativistic effects. UH waves have frequencies in the electron cyclotron/plasma frequency range and consist of two wave branches, an electromagnetic slow X-mode-like branch and a quasi-electrostatic electron Bernstein-like branch.^{6} The two branches coincide at a mode conversion layer, commonly referred to as the upper hybrid resonance (UHR) due to its role in cold plasma theory,^{1} where energy is transferred between them. In underdense and weakly overdense plasmas, the mode conversion process is further accompanied by a reflection of the wave energy, allowing UH wave trapping when a region permitting propagation of both UH wave branches bounded by UHR layers exists.^{6–30} UH wave trapping has been a topic of significant interest in recent years, as it allows nonlinear phenomena, known as parametric decay instabilities (PDIs),^{31} to be excited at microwave power levels attainable by the gyrotron sources typically used for electron cyclotron resonance heating (ECRH) in magnetic confinement fusion experiments.^{6–31} By contrast, PDIs involving ECRH waves without wave trapping under similar circumstances are only expected in scenarios where a significant amount of the injected power reaches the UHR in X-mode^{32–42} or at extremely high power levels attainable by pulsed free-electron maser sources.^{43}

A PDI occurs when the amplitude of a coherent (pump) wave, e.g., used for ECRH, exceeds a certain threshold. Beyond the threshold, decay of the pump wave into two daughter waves becomes unstable, resulting in the daughter waves gaining energy at the expense of the pump wave until the instability saturates. While the occurrence of a PDI at a given point in the plasma is only determined by the nonlinear interaction strength between the three waves, and thus is independent of wave trapping, the amplification of a specific wave mode due to a PDI in an inhomogeneous plasma only occurs in a narrow region around the point where the frequency and wave number selection rules imposed by energy and momentum conservation are exactly satisfied.^{31,44–48} This means that PDIs in inhomogeneous plasmas without wave trapping are usually convective in nature and saturate at very low signal levels,^{31,44–48} except in cases with very large pump wave amplitudes.^{32–43} When the PDI-generated daughter waves are trapped, they repeatedly enter the region where amplification occurs, such that the instability becomes absolute once the gain in one round trip through the trapping region exceeds the losses due to convection and diffraction perpendicular to the direction of trapping,^{4} leading to significant signals at much lower pump power levels.^{4} Since energy conservation in the three-wave process requires the daughter waves excited by PDIs to be shifted in frequency relative to the pump wave, PDIs involving an ECRH pump wave can generate significant spectral power outside the filters used to protect microwave diagnostics from stray radiation around the ECRH frequency, making them a considerable risk to such diagnostics.^{17} Additionally, PDIs involving trapped UH daughter waves may lead to a significant fraction of the ECRH power being transferred to the UH waves,^{8–15} resulting in heating and current drive characteristics different from linear expectations.

Several different PDIs involving trapped UH waves have been investigated. The one which has been most thoroughly investigated involves the decay of an X-mode pump wave into two trapped UH waves near half the pump wave frequency.^{6–20} This type of PDI has been used to explain strong anomalous microwave signals observed at the TEXTOR^{49,50} and ASDEX Upgrade^{6,16,17} tokamaks, including microwave diagnostics damage in ASDEX Upgrade,^{17} as well as significant anomalous ECRH absorption in low-temperature plasma filament experiments.^{11} PDIs involving two trapped UH waves are also likely to occur in connection with second-harmonic X-mode ECRH scenarios planned for the early operation phase of ITER.^{17,51} The examples in this paper involve trapped UH wave PDIs of the above type and further assume both trapped UH waves to be at exactly half the ECRH frequency to simplify the analysis. We do, however, note that other PDIs involving only one trapped UH wave occur^{21–30} and that the results from the present paper are relevant to UH wave trapping for such PDIs as well. For instance, a PDI in which an O-mode pump wave decays to a trapped UH wave and a low-frequency lower hybrid wave^{21–28} may be relevant for fundamental O-mode ECRH in full-field scenarios at ITER.^{23,51} Furthermore, a PDI in which an X-mode pump wave decays to a fast X-mode wave above half the ECRH frequency and a trapped UH wave below half the ECRH frequency can explain strong microwave signals in the Wendelstein 7-X stellarator^{29,30} and low-density TEXTOR plasmas.^{49,50} Finally, the results presented here also have relevance for PDIs without wave trapping, where X-mode ECRH waves decay into UH waves and lower hybrid waves near the UHR due to enhancement of the electric field of the X-mode wave in this region.^{32–36}

The main motivation for studying relativistic modifications of UH wave propagation and trapping in the present paper is the potential impact of such effects on the analysis of PDIs at ITER. While the ASDEX Upgrade discharges, where trapped UH wave PDIs leading to microwave diagnostics damage have been identified,^{17} had central electron temperatures $ \u223c 3 \u2009 keV$, the ITER second-harmonic X-mode ECRH scenarios are expected to have central electron temperatures $ \u223c 15 \u2009 keV$ (Ref. 51), making the effects of a finite electron temperature significantly stronger. Although electron temperatures $ \u223c 15 \u2009 keV$ are still far below the electron rest energy of 511 keV, relativistic effects do nevertheless play an important role in wave propagation perpendicular to the background magnetic field,^{1} which is the case of interest in connection with UH wave trapping. The relatively low electron temperatures do, however, allow us to base our analysis on the moderately relativistic (MR) framework,^{1} which has already been the subject of significant theoretical investigations relevant to UH waves.^{1,52–58} We specifically base our analysis on the MR dispersion relation studied by Ref. 58. This dispersion relation is accurate for electron temperatures up to 25 keV and UH wave frequencies below twice the electron cyclotron frequency,^{58} which is the range relevant to most ITER plasmas,^{51} while allowing the conditions for UH wave trapping to be analyzed in a relatively straightforward manner.

The remainder of this paper is organized as follows. Section II presents the MR UH wave dispersion relation of Ref. 58, shows that it reduces to the non-relativistic result at low electron temperatures, and analyzes the conditions under which a trapped UH wave pair exists. Section III investigates the impact of the MR modifications on UH wave trapping and PDIs in the ASDEX Upgrade discharges discussed in Refs. 16 and 17, as well as for scaled versions of the plasma parameters from Ref. 17 to investigate MR effects for ITER-like core electron temperatures. Finally, Sec. IV presents our conclusions and an outlook.

## II. THEORY

*ω*is the angular wave frequency, $ \omega c e = \u2212 e B / m e$ is the electron cyclotron frequency,

*T*is the electron temperature in energy units,

_{e}*B*is the magnitude of the background magnetic field

**B**,

*e*is the elementary charge, and

*m*is the electron (rest) mass. The dispersion relation is based on an expansion of the MR dielectric tensor of a plasma with thermal electrons for wave propagation perpendicular to

_{e}**B**to first order in $ r L e 2 k \u22a5 2$, as described in Sec. 4.7.4 of Ref. 1, with $ r L e = T e / m e / | \omega c e |$ being the thermal electron Larmor radius and $ k \u22a5$ being the magnitude of the wave vector (perpendicular to

**B**), $ k \u22a5$; the ion response is neglected due to the high frequencies of the waves. Rearranging Eq. (A5) of Ref. 58, the dispersion relation may be expressed as

*c*is the vacuum speed of light), $ A = S ( 1 ) [ 1 \u2212 T ( 1 ) ] + [ D ( 1 ) ] 2 , \u2009 B = S ( 0 ) [ 1 \u2212 S ( 1 ) \u2212 T ( 1 ) ] + 2 D ( 0 ) D ( 1 )$, and $ C = [ D ( 0 ) ] 2 \u2212 [ S ( 0 ) ] 2$, with

*n*is the electron density, and $ \epsilon 0$ is the permittivity of vacuum), $ \mu e = m e c 2 / T e , \varphi m = \mu e ( 1 \u2212 m Y )$, and

_{e}*q*(Refs. 52, 56, and 57); $ \Gamma ( 1 \u2212 q , \varphi m ) = \u222b \varphi m \u221e u \u2212 q \u2009 e \u2212 u \u2009 d u$ denotes the upper incomplete Gamma function with arguments $ 1 \u2212 q$ and $ \varphi m$ (Refs. 56 and 59). The numerical scheme used for computing

*F*is discussed in the Appendix.

_{q}*T*. To do this, we assume $ \mu e \u226b 1$, such that only the lowest order terms of $A$ and $B$ expanded in orders of $ 1 / \mu e$ need to be considered, yielding

_{e}^{1}The coefficients in Eq. (9) yield the non-relativistic warm dispersion relation

^{6–17}of UH waves propagating perpendicular to

**B**when inserted into Eq. (1), showing that the MR dispersion relation reduces to the non-relativistic one in the appropriate limit.

*j*(for which $ \omega = \omega j$, with

*ω*real, such that $ \varphi m$ is real) and $ \Delta j = B j 2 \u2212 4 A j C j$ is the discriminant for wave mode

_{j}*j*. In the region of interest to UH wave trapping, the + branch of Eq. (10) describes electron Bernstein-like waves, while the − branch of Eq. (10) describes slow X-mode-like waves. In a WKB framework, waves on the − branch are converted to waves on the + branch, and vice versa, when their dispersion relations coincide, i.e., at $ \Delta j = 0$ (Refs. 2 and 3). If a region, in which the ± branches are both propagating, exists between two mode conversion points (where $ \Delta j = 0$), UH wave trapping occurs.

^{17}Unlike the equivalent non-relativistic expression, Eq. (10) only satisfies the strict WKB mode conversion condition, $ \Delta j = 0$, at isolated points in $ ( X j , Y j 2 )$ space;

^{58}representations in $ ( X j , Y j 2 )$ space are referred to as CMA-like diagrams in this work. However, as noted by Ref. 58, the imaginary part of Δ

_{j}tends to be very small away from the (cold) electron cyclotron resonances (ECRs, $ \omega j = m | \omega c e |$) and the dispersion relations of the ± branches will thus almost coincide at $ Re ( \Delta j ) = 0$. We therefore expect mode conversion to occur at $ Re ( \Delta j ) = 0$ when allowing for effects beyond the WKB approximation or adopting a weak damping approximation (discussed below). A CMA-like diagram showing the $ Re ( \Delta j ) = 0$ contour for $ Y j 2 \u2208 [ 1 / 4 , 1 ]$ at $ T e = 15 \u2009 keV$ is plotted in Fig. 1. $ Re ( \Delta j ) < 0$ on the side of the $ Re ( \Delta j ) = 0$ contour containing the blue shaded regions in Fig. 1. In this region, a large imaginary part is added to $ ( N \u22a5 j \xb1 ) 2$ due to the general dominance of the real parts of $ A j$ and $ B j$ over the corresponding imaginary parts ( $ C j$ is purely real for $ Y j \u2264 1$). This means that the UH waves are evanescent for $ Re ( \Delta j ) < 0$ (Ref. 58), which is also shown to be consistent with the weak damping framework below. In order for mode conversion to occur at $ Re ( \Delta j ) = 0$, the ± branches must both be propagating on the side where $ Re ( \Delta j ) > 0$. Since the real parts of $ A j$ and $ B j$ are assumed to dominate, this condition can be described by the requirement $ Re [ ( N \u22a5 j \xb1 ) 2 ] > 0$, as a wave can generally be assumed to be evanescent when $ Re [ ( N \u22a5 j \xb1 ) 2 ] < 0$ and $ Re ( \Delta j ) > 0$. The contours of $ Re [ ( N \u22a5 j + ) 2 ] = 0$ and $ Re [ ( N \u22a5 j \u2212 ) 2 ] = 0$ are marked by orange and green lines, respectively, in Fig. 1; the regions where either $ Re [ ( N \u22a5 j + ) 2 ] < 0$ or $ Re [ ( N \u22a5 j \u2212 ) 2 ] < 0$ is satisfied are additionally shaded orange or green, respectively, while the regions where $ Re [ ( N \u22a5 j \xb1 ) 2 ] < 0$ for both branches are shaded red. When the ± mode conversion regions that cannot be reached by propagating waves from both branches are excluded, corresponding to the parts of the $ Re ( \Delta j ) = 0$ contour in the orange, green, and red shaded areas of Fig. 1, we see that ± mode conversion occurs near the second-harmonic ECR (just above $ Y j 2 = 1 / 4$) for $ X j < 0.6$ and along a diagonal line slightly above the cold UHR [ $ Y j 2 = 1 \u2212 X j$ (Ref. 1)] which we identify as the MR UHR. Just above $ Y j 2 = 1 / 4$, the imaginary parts of $ Re [ ( N \u22a5 j \xb1 ) 2 ]$ are generally comparable to the real parts, making the simple picture of ± mode conversion presented above inaccurate, as the waves can generally be assumed to experience strong damping in this region.

^{1,58}We can exclude the waves in this region and find the cutoffs of the main UH waves by considering the $ C j = 0$ contour, represented by the dotted purple lines in Fig. 1. The former point is seen from the fact that $ C j < 0$ for the waves just above $ Y j 2 = 1 / 4$ (the purple shaded region of Fig. 1), while $ C j > 0$ in the region of the main UH waves (the unshaded region of Fig. 1), thus considering only the region where $ C j > 0$ enables us to select the main propagating UH waves. The latter point is seen from the fact that $ N \u22a5 j \xb1 = 0$ for at least one branch in Eq. (10) when $ C j = 0$, which is the definition of a cutoff.

^{1}As noted by Ref. 58, the branches of the $ C j = 0$ contour at low and high

*X*represent the MR R- and L-cutoffs, respectively.

_{j}^{57}

^{,}Figure 1 additionally shows that the − branch of the UH waves becomes evanescent for

*X*above the L-cutoff, meaning that the L-cutoff represents a limit of the region where a trapped UH wave pair can exist, in agreement with the non-relativistic analysis.

_{j}^{17}For

*X*between the MR UHR and L-cutoff, the region permitting a trapped UH wave pair to exist is bounded from below by waves on the + branch becoming evanescent due to absorption just above the second-harmonic ECR ( $ Y j 2 = 1 / 4$, cf. the main orange region of Fig. 1) and from above by the fundamental ECR ( $ Y j 2 = 1$), above which both UH wave branches are damped significantly;

_{j}^{1,58}this is why the region with $ Y j 2 > 1$ is not plotted in Fig. 1.

By including only the parts of the curves bordering the unshaded area in Fig. 1, the region of CMA space in which a trapped UH wave pair may exist can be computed at various *T _{e}* values. Furthermore, comparisons with the corresponding regions from cold and warm non-relativistic theory

^{17}may be undertaken; Fig. 2 shows such comparisons for $ T e = 5 , \u2009 15 , \u2009 25 \u2009 keV$. Since we are interested in analyzing the conditions under which an X-mode ECRH pump wave with angular frequency,

*ω*

_{0}, decays into two UH waves with $ \omega j = \omega 0 / 2$, Fig. 2 is plotted in $ ( \omega p e 2 / \omega 0 2 , \omega c e 2 / \omega 0 2 ) = ( X j / 4 , Y j 2 / 4 )$ space. As is evident from Fig. 2, the warm and MR UHRs, marked by the thick lines, remain close to each other in most of CMA space up to the

*T*limit of the MR dispersion relation at 25 keV (Ref. 58). The MR UHR usually occurs at slightly higher $ ( \omega p e 2 / \omega 0 2 , \omega c e 2 / \omega 0 2 )$ than the warm UHR, although regions with the opposite trend also exist, e.g., in the $ T e = 5 \u2009 keV$ pane of Fig. 2. Near the fundamental and second-harmonic ECRs of the UH waves, which correspond to the second- ( $ \omega c e 2 / \omega 0 2 = 1 / 4$) and fourth-harmonic ( $ \omega c e 2 / \omega 0 2 = 1 / 16$) ECRs of the pump wave, respectively, MR and warm theory show greater deviations. At $ \omega c e 2 / \omega 0 2 = 1 / 4$, the MR UHR occurs at finite $ \omega p e 2 / \omega 0 2$, indicating that an UH wave pair does not propagate in a vacuum, while the warm UHR always approaches the cold UHR at $ \omega p e 2 / \omega 0 2 = 0$. The difference between warm and MR theory is more fundamental near $ \omega c e 2 / \omega 0 2 = 1 / 16$. While the warm UHR extends all the way to the L-cutoff, which is identical to the cold L-cutoff [ $ Y j = X j \u2212 1$ (Ref. 1)] and marked by the medium lines in Fig. 2, the MR UHR only extends partway to the MR L-cutoff,

_{e}^{57}to which it is connected by quasi-horizontal thin lines (in Fig. 2) slightly above $ \omega c e 2 / \omega 0 2 = 1 / 16$, marking the points at which waves from the + branch become evanescent due to absorption at the second-harmonic ECR. From Fig. 2, we note that the $ \omega c e 2 / \omega 0 2$ value of the ECR absorption line increases with increasing

*T*due to the broadening of the electron distribution function, allowing absorption to occur further from the cold second-harmonic ECR,

_{e}^{1}marked by the thin dashed lines at $ \omega c e 2 / \omega 0 2 = 1 / 16$ in Fig. 2. MR theory further accounts for the shift of the L-cutoff at finite

*T*(Ref. 57). Finally, when $ T e \u2192 0$, the MR UHR approaches the cold UHR [ $ Y j 2 = 1 \u2212 X j$ (Ref. 1)] marked by the thick dashed line in Fig. 2, while the MR second-harmonic ECR connects the UHR and L-cutoff at $ \omega c e 2 / \omega 0 2 = 1 / 16$ in this case. By contrast, the warm UHR always extends from the cold UHR at $ ( \omega p e 2 / \omega 0 2 , \omega c e 2 / \omega 0 2 ) = ( 0 , 1 / 4 )$ to the cold L-cutoff at $ ( \omega p e 2 / \omega 0 2 , \omega c e 2 / \omega 0 2 ) = ( 3 / 8 , 1 / 16 )$ and simply approaches the line defined by the cold UHR and cold second-harmonic ECR without distinguishing between the two regions.

_{e}*T*resembling the values in ITER second-harmonic X-mode ECRH scenarios.

_{e}^{17,51}Apart from comparing the MR, warm, and cold UHR contours for the above equilibria, we shall also compare the PDI thresholds obtained with the warm dispersion relation in Figs. 9 and 10 of Ref. 17, as well as Fig. 15 of Ref. 16, to those obtained with the MR dispersion relation. In order to do this, we adapt the electrostatic PDI framework used with the warm dispersion relation in Refs. 6 and 16 to the MR dispersion relation. Such an adaptation is facilitated by noting that from Eq. (9), the coefficients of the warm dispersion relation can be converted to those of the MR dispersion relation through the substitutions $ \u2113 T e 2 \u2192 ( c 2 / \omega 2 ) A , \u2009 S \u2192 B$, and $ D 2 \u2212 S 2 \u2192 C$. Using this substitution in Eq. (2.58) of Ref. 6, we obtain the following effective electrostatic dispersion relation, $D$, for the MR expression:

*ν*being the electron–ion collision frequency

_{ei}^{1}and $ k \u2225$ ( $ k \u22a5$) being the component of the wave vector,

**k**, parallel (perpendicular) to

**B**. The term proportional to

*P*in Eq. (11) is responsible for diffraction of UH waves along

**B**at $ k \u2225 = 0$, while the two terms of

*I*in Eq. (12) represent collisional and electron Landau damping, respectively; electron Landau damping is generally negligible for UH waves.

^{6}In the linear limit, where $ D ( k , \omega ) = 0$, Eq. (11) is equivalent to Eq. (1) with the addition of electrostatic, non-relativistic corrections for diffraction along

**B**, collisional damping, and electron Landau damping. The additional effects are required for the computation of PDI thresholds using the weak damping/amplification framework of Refs. 6 and 16. Since the nonlinear coupling terms in Refs. 6 and 16 are based on cold plasma theory and MR effects are neglected for diffraction along

**B**, collisional damping, and electron Landau damping, the PDI model is only truly accurate at low

*T*. We do, however, note that the model of Refs. 6 and 16 employs an extremely simplified model of the ECRH beams (discussed in Sec. III C) and only considers daughter waves possessing $ \omega j = \omega 0 / 2$, with PDI thresholds being estimated exclusively for the cases in which convective losses perpendicular to

_{e}**B**or diffractive losses parallel to

**B**dominate. These simplifications mean that the PDI thresholds provided in the present paper should mainly be considered rough estimates, as also noted in Refs. 16 and 17. The main goal of recalculating the PDI thresholds from Refs. 16 and 17 is to confirm that the qualitative conclusions of these papers hold when the MR dispersion relation is used at relatively low

*T*, for which the simplified framework is considered sufficient. It will, however, be of future interest to carry out a detailed investigation of PDIs in the MR framework, e.g., using an approach resembling that of Ref. 60, and compare the results with particle-in-cell simulations.

_{e}^{18–20}

*ω*being the real wave vector and angular frequency of UH wave mode

_{j}*j*, respectively. We can then split $D$ into real and imaginary parts at $ ( k j , \omega j )$, written as $ D \u2032 ( k j , \omega j )$ and $ D \u2033 ( k j , \omega j )$, respectively; for $D$ from Eq. (11) and $ \omega j \u2265 | \omega c e |$,

*j*denotes a quantity evaluated at $ ( k j , \omega j )$. In the weak damping/amplification approximation, deviations of $ ( k , \omega )$ from $ ( k j , \omega j )$, due to $ D \u2033$ (which is proportional to the linear damping rate) and the nonlinear coupling responsible for PDIs,

^{6}are small. To lowest order, it therefore holds that $ D \u2032 ( k j , \omega j ) = 0$, which for Eq. (13) at $ k j \u2225 = 0$ gives

^{6,16}These terms add an imaginary part to

*ω*, which is generally negative for the damping, diffraction, and convection terms, and positive for the nonlinear coupling terms. The PDI threshold is defined as the smallest ECRH beam power that makes $ Im ( \omega ) > 0$, which is the requirement for an absolute instability. Details of the computation of the PDI threshold, as well as the expressions for the nonlinear coupling terms, are found in Refs. 6 and 16, where $ D \u2032$ and $ D \u2033$ should be replaced by Eqs. (13) and (14), respectively. We note that computation of the PDI threshold requires the determination of the derivative of $ D \u2032$ with respect to

*ω*, which involves derivatives of

*F*. Such derivatives can be evaluated using Eq. (19) of Ref. 56, $ d F q ( \varphi m ) / d \varphi m = F q ( \varphi m ) \u2212 F q \u2212 1 ( \varphi m )$, along with the methods for evaluating

_{q}*F*discussed in the Appendix.

_{q}## III. ANALYSIS OF ASDEX UPGRADE AND SCALED EQUILIBRIA

### A. ASDEX Upgrade equilibria

Figure 3 shows the $ \omega 0 = 2 \pi \xd7 140 \u2009 GHz$ third-harmonic X-mode ECRH scenarios of ASDEX Upgrade, where microwave diagnostics were damaged due to trapped UH wave PDIs, labeled cases 1–3 for reference.^{17} The details were discussed in connection with Fig. 3 of Ref. 17, which showed the same equilibria, along with the warm and cold second-harmonic UHRs. As seen from Fig. 3, the MR second-harmonic UHR is virtually identical to the warm second-harmonic UHR for the ASDEX Upgrade equilibria, while the cold second-harmonic UHR occurs at noticeably lower $ ( \omega p e 2 / \omega 0 2 , \omega c e 2 / \omega 0 2 )$ values in the plasma core. This is expected since $ T e \u2272 3 \u2009 keV$ in all cases; *n _{e}* and

*T*profiles vs the normalized poloidal flux coordinate, $ \rho pol$, are seen in Fig. 4 for all cases ( $ \rho pol = 0$ at the plasma center, 1 at the last closed flux surface, and approximately proportional to the distance from the plasma center). We also note that while Fig. 2 showed a greater deviation between the MR and the cold/warm second-harmonic UHRs near the second-harmonic ECR compared with other regions at fixed

_{e}*T*, all three second-harmonic UHRs usually coincide close to the second-harmonic ECR in practice, due to the generally low

_{e}*T*values in this region.

_{e}From Fig. 3, we conclude that the analysis of the trapping regions in ASDEX Upgrade based on the warm second-harmonic UHR from Ref. 17 is still accurate when MR effects are taken into account. In Sec. III C, we shall, nevertheless, show that the precise PDI thresholds obtained from MR theory are not quantitatively identical to those obtained from warm theory. However, we first investigate the UH wave trapping regions for scaled versions of the ASDEX Upgrade equilibria from Fig. 3 to study ITER-like core plasma parameters.

### B. Scaled equilibria

Figure 5 shows equilibria scaled to have core *n _{e}*,

*T*, and

_{e}*B*values resembling those of the one-third field ITER scenarios. These scenarios have an on-axis $ B = 1.8 \u2009 T$ (Ref. 51), which is identical to the ASDEX Upgrade equilibria in Fig. 3, and

*B*has therefore been left unchanged. On the other hand, H-mode helium plasmas in the one-third field ITER scenarios are expected to have $ n e ( 0 ) \u2248 2 \xd7 10 19 \u2009 m \u2212 3$ and $ T e ( 0 ) \u2248 15 \u2009 keV$ based on Fig. F-6 of Ref. 51. We therefore scale the

*n*and

_{e}*T*profiles from Fig. 4 to approximately obtain the above $ n e ( 0 )$ and $ T e ( 0 )$ values; specifically,

_{e}*n*is scaled by 5/8, 1/2, and 1/2, while

_{e}*T*is scaled by 5, 4, and 6, for cases 1–3, respectively. The second-harmonic UHRs and ECRs plotted in Fig. 5 assume $ \omega 0 = 2 \pi \xd7 110 \u2009 GHz$, which is one of the proposed gyrotron frequencies for second-harmonic X-mode ECRH in one-third field ITER scenarios

_{e}^{51}that has been identified as being prone to trapped UH wave PDIs.

^{17}Clear deviations between the MR and warm second-harmonic UHRs are visible in the plasma core in Fig. 5, showing that an MR approach is necessary to confidently identify UH wave trapping regions in ITER. As expected from Fig. 2, the MR and warm second-harmonic UHRs follow each other at low

*T*, while the MR second-harmonic UHR is displaced toward higher $ ( \omega p e 2 / \omega 0 2 , \omega c e 2 / \omega 0 2 )$ values than the warm second-harmonic UHR at high

_{e}*T*; the cold second-harmonic UHR only approaches the MR and warm ones near the plasma edge due to the high

_{e}*T*in the plasma core. While the deviation of the MR second-harmonic UHR from the warm one observed in Fig. 5 is generally small compared with the deviation of both the MR and warm ones from the cold one, we note that this deviation may still have important consequences for the occurrence of PDIs requiring UH wave trapping. In case 1, for example, the MR second-harmonic UHR intersects the (1, 1) mode identified in the core of the ASDEX Upgrade plasma (marked by the shaded area),

_{e}^{17}while the warm second-harmonic UHR occurs on the low-field side of the mode. Thus, the

*n*perturbation associated with the (1, 1) mode would be likely to allow UH wave trapping within MR theory, but unlikely to do so within warm theory. While the cold second-harmonic UHR in case 1 occurs near the plasma edge and thus also fails to predict the possibility of UH wave trapping in connection with the (1, 1) mode, it is at least topologically similar to the MR and warm contours; in cases 2 and 3, even the topology of the cold second-harmonic UHR is different from that of the MR and warm ones. For MR and warm theory, cases 2 and 3 have closed regions without a propagating UH wave pair inside the main region, where both UH wave branches propagate, while cold theory yields a simply connected region with propagating UH waves. UH wave trapping is possible in ECRH beams launched from the low-field side and intersecting the closed region without a propagating UH wave pair, meaning that MR and warm theory allow such trapping, while cold theory does not. To understand why cold theory does not permit trapping, while MR and warm theory do, it is instructive to consider the origins of the trapping regions in cases 2 and 3. In case 2, the

_{e}*n*profile is monotonic and

_{e}*T*only acquires a significant value for $ \rho pol < 0.8$, as seen in Fig. 4. The trapping region on the low-field side of the plasma center is therefore a consequence of the

_{e}*T*gradient around $ \rho pol \u2248 0.7$ and no trapping is possible according to cold theory, since the cold UHR does not depend on

_{e}*T*. As the gradients in Fig. 5 are approximately four times larger than those expected in ITER, since the linear dimensions of ASDEX Upgrade are roughly 1/4 of those of ITER,

_{e}^{66}it is unclear whether trapping regions solely due to

*T*gradients will occur in practice. We do, however, note that the

_{e}*B*gradient, which is the main factor responsible for the generally monotonic decrease in the UH frequency at larger major radii in ASDEX Upgrade that prevents UH wave trapping in most cases, is also roughly four times larger in the current examples than in ITER, reducing the

*n*and

_{e}*T*gradients required for UH wave trapping at ITER. In case 3, an

_{e}*n*spike is present near the last closed flux surface, which was attributed to the occurrence of a locked mode in Ref. 17, and

_{e}*T*has a pedestal around the last closed flux surface as well (cf. Fig. 4). Here, UH wave trapping occurs around the

_{e}*n*maximum in MR and warm theory, while no trapping occurs in cold theory. This may be attributed to the

_{e}*n*spike being too small to allow UH wave trapping on its own within cold theory, due to the reduced

_{e}*n*, which is scaled by 1/2 compared with Fig. 3. When finite

_{e}*T*effects are included, the spike is, however, sufficiently large to allow UH wave pair propagation around its center in a region that would otherwise be excluded in MR and warm theory. Since the closed region without UH wave pair propagation is somewhat larger in MR theory than in warm theory, UH wave trapping is more likely to occur in ECRH beams from off-midplane launchers in case 2 and near the plasma center in case 3 for the former theory. If

_{e}*ω*

_{0}is changed to $ 2 \pi \xd7 104 \u2009 GHz$, which is another gyrotron frequency proposed for second-harmonic X-mode ECRH in one-third scenarios at ITER,

^{51}the second-harmonic UHRs for the plasma parameters in Fig. 5 all occur close to the plasma edge, such that UH wave trapping is only likely in connection with edge phenomena, as described in Ref. 16. Since trapped UH wave PDIs near the plasma edge have not resulted in microwave diagnostic damage at ASDEX Upgrade, the present analysis favors the use of 104 GHz gyrotrons for second-harmonic X-mode ECRH of helium H-mode plasma in one-third field scenarios at ITER.

We next consider scaled equilibria with core *n _{e}*,

*T*, and

_{e}*B*values resembling those of the ITER half-field scenarios in Fig. 6. These scenarios have an on-axis $ B = 2.65 \u2009 T$, so

*B*of the ASDEX Upgrade equilibria from Fig. 3 is scaled by a factor of 3/2 in all cases. The average core

*n*for L–H transition experiments at 2.65 T will be in the range from $ 2.4$ to $ 4.8 \xd7 10 19 \u2009 m \u2212 3$ at ITER

_{e}^{51}(40%–80% of the Greenwald density,

^{67}cf. Tables 2–13 and 2–14 from Ref. 51), which is similar to the values in Fig. 4, so

*n*is not scaled. For

_{e}*T*, we use the same scaling factors as in the one-third field scenarios, i.e., 5, 4, and 6 for cases 1–3, respectively, giving

_{e}*T*values similar to those expected in ITER H-mode deuterium–tritium plasmas at 2.65 T, cf. Figs. 2.6–5 and 2.6–6 in Ref. 51. The second-harmonic UHRs and ECRs in Fig. 6 are plotted for $ \omega 0 = 2 \pi \xd7 170 \u2009 GHz$, which is the gyrotron frequency of the main ECRH system at ITER that will be used for second-harmonic X-mode ECRH in the half-field scenarios.

_{e}^{51}The qualitative behavior of the different second-harmonic UHRs in Fig. 6 is similar to Fig. 5. In case 1, the MR second-harmonic UHR occurs slightly on the high-field side of the plasma center and has significant overlap with the (1, 1) mode identified in the ASDEX Upgrade discharge, while the warm second-harmonic UHR occurs very close to the plasma center, indicating that different PDI characteristics would be expected for ECRH beams launched from the outboard midplane; the cold second-harmonic UHR occurs on the low-field side outside the region where PDIs relying on wave trapping are expected to occur. In case 2, the MR second-harmonic UHR occurs at the plasma center, while the warm second-harmonic UHR occurs slightly on the low-field side of the center, but they should both allow wave trapping near the plasma center; the cold second-harmonic UHR is once again far from the region potentially allowing wave trapping. Finally, in case 3, the MR and warm second-harmonic UHRs also occur close to the plasma center and could thus permit UH wave trapping under similar conditions. However, the cold second-harmonic UHR is seen to have a different topology than the MR and warm ones. Unlike the discussion of case 3 in connection with Fig. 5, it is the cold second-harmonic UHR that allows wave trapping near the plasma edge around the outboard midplane for case 3 of Fig. 6, while the MR and warm second-harmonic UHRs indicate that the regions allowing UH wave propagation and trapping near the plasma edge are only present in the upper and lower parts of the plasma. This can be explained by the

*n*bump being too small to permit wave propagation near the plasma edge around the outboard midplane for the MR and warm theories due to the high

_{e}*T*, which further illustrates the importance of including finite-

_{e}*T*effects when analyzing UH wave trapping for ITER-relevant

_{e}*T*values.

_{e}### C. ASDEX Upgrade PDI thresholds

As a final point, we investigate whether the MR dispersion relation from Eq. (11) yields PDI power thresholds similar to those of the warm dispersion relation when applied to the ASDEX Upgrade equilibria from Figs. 9 and 10 of Ref. 17 and the JOREK simulation^{68} from Fig. 15 of Ref. 16, within the framework discussed in Sec. II. Figure 7 shows the plasma and MR UH wave parameters along the ECRH beam suspected of driving PDIs in case 3 based on the ASDEX Upgrade profiles from Figs. 3 and 4; the same plot using the warm dispersion relation is seen in Fig. 9 of Ref. 17. In Figs. 7–9, the pump wave is taken to be a 140 GHz X-mode wave propagating along the *x* direction with a wave number of $ k 0 > 0$. The *x* direction is assumed to be perpendicular to **B**, which is directed along the *z* direction, and to be the only direction along which the plasma parameters vary, giving a slab-like geometry. The daughter waves, referred to by subscripts $ j \u2208 { 1 , 2}$, have $ \omega j = \omega 0 / 2$ and wave vectors $ k j \u22a5 = ( k j x \xb1 , k j y , 0 )$, with $ k 1 x \xb1 = \u2212 k 2 x \xb1 > 0$ and $ k 1 y = \u2212 k 2 y = k y > 0$. We determine $ k 1 x \xb1 = ( k 1 \u22a5 \xb1 ) 2 \u2212 k y 2$ and *k _{y}* by Eq. (15), along with the requirement that the trapped daughter waves, whose existence is indicated by the closed loops in Fig. 7 (and Fig. 9), satisfy the Bohr–Sommerfeld condition with the smallest possible

*k*to minimize convective losses.

_{y}^{6,16}Using the above definitions, the PDI wave vector selection rule is satisfied at points where $ k 1 x + = k 0 \u2212 k 2 x \u2212$, which are marked by the intersections of the solid and dashed lines of Fig. 7 (and Fig. 9). PDI power thresholds are calculated based on the model from Refs. 6 and 16 with the MR modifications described in Sec. II. In the model, the ECRH beams are assumed to have a square cross section with a constant electric field amplitude inside the square and zero amplitude outside the square. The electric field amplitude is set equal to the one at the center of the ECRH beams, based on a Gaussian beam model,

^{6}and the side length of the square is determined by setting the power of the square beam equal to that of the ECRH beam. In Figs. 7 and 8, a side length of 3.309 cm is used, while Fig. 9 utilizes a side length of 3.556 cm, as the PDI region occurs further from the ECRH beam foci in the latter case. PDI thresholds are calculated based on two sets of assumptions, the first one (called the convective estimate) assumes that convection losses perpendicular to

**B**dominate, while the second one (called the diffractive estimate) assumes that diffraction losses parallel to

**B**dominate.

^{6,16}The threshold of the PDI involving daughter waves with $ \omega j = \omega 0 / 2$ may then be estimated by the larger of the convective or diffractive PDI threshold estimates. When using the above model to estimate the overall PDI threshold for larger trapping regions, we must consider that a combination of UH wave modes satisfying the Bohr–Sommerfeld condition with $ k y \u2248 0$, resulting in negligible convection losses, is likely to exist, even if this is not the case for the daughter waves with $ \omega j = \omega 0 / 2$. In such cases, the diffractive PDI threshold estimate can be considered representative of the overall PDI threshold.

^{17}

As expected from Fig. 3, the MR (Fig. 7) and warm (Fig. 9 of Ref. 17) theories both predict the existence of trapped UH waves with $ \omega j = \omega 0 / 2$ near the plasma edge in case 3. $ | k j x \xb1 |$ of the trapped UH waves are further seen to be similar, with the MR ones being slightly larger than the corresponding warm ones. While the linear wave parameters are very similar, the PDI threshold obtained from MR theory is significantly smaller than that obtained from warm theory: the MR convective and diffractive PDI threshold estimates are 7.4 MW and 44 kW, compared with estimates from warm theory of 72.8 MW and 333 kW, respectively.^{17} This may be explained by the sensitivity of the PDI threshold of the modes with $ \omega j = \omega 0 / 2$ on the precise parameters of the cavity allowing UH wave trapping. We do, however, note that the MR convective and diffractive PDI threshold estimates are still, respectively, well above and well below the nominal gyrotron power of 889 kW, meaning that the qualitative conclusions are similar for MR and warm theory. In order to further investigate the qualitative similarity between MR and warm theory, we plot the convective and diffractive PDI threshold estimates for the time points of case 3 from Fig. 10 of Ref. 17 in Fig. 8. As found in Fig. 10 of Ref. 17, the convective PDI threshold estimates in Fig. 8 are generally significantly larger than the diffractive estimates. The convective estimates are further generally larger than the nominal gyrotron power, while the diffractive estimates generally are smaller than the nominal gyrotron power. Unlike Fig. 10 of Ref. 17, Fig. 8 does, however, have time points at which the diffractive estimate exceeds the convective estimate. This occurs when the cavity has a length enabling the Bohr–Sommerfeld condition to be fulfilled for waves with $ \omega j = \omega 0 / 2$ with a small *k _{y}* and supports the claim from Ref. 17 that the diffractive estimate is representative of the overall PDI power threshold in case 3, due to the relatively large trapping region. The qualitative conclusions obtained from Fig. 8 are thus similar to those obtained from Fig. 10 of Ref. 17, meaning that the MR and warm dispersion relations yield qualitatively similar results for case 3 from ASDEX Upgrade.

We finally plot the MR dispersion curves for the plasma profiles found along two 140 GHz X-mode ECRH beams in a JOREK simulation^{68} of an edge-localized mode at ASDEX Upgrade in Fig. 9; the corresponding warm dispersion curves are plotted in Fig. 15 of Ref. 16. MR effects are found to have a small impact on the shape of the dispersion curves because of the low *T _{e}* values. The PDI thresholds, given by the convective estimate, are modified somewhat, from warm values of 175 and 171 kW, to MR values of 128 and 310 kW in Panes (a) and (b), respectively. This can be explained by differences in the

*k*values satisfying the Bohr–Sommerfeld condition in the warm and MR cases. We note that the estimated PDI thresholds remain on the same order of magnitude and close to the experimentally estimated PDI threshold of 250 kW from Ref. 16, showing that MR theory yields qualitatively similar results to warm theory and has a similar level of agreement with the experimental results, as expected for the low

_{y}*T*near the ASDEX Upgrade plasma edge.

_{e}## IV. CONCLUSION AND OUTLOOK

In this paper, we have investigated the impact of relativistic effects on UH wave propagation and trapping, as well as their impact on low-threshold PDIs requiring UH wave trapping during ECRH of magnetic confinement fusion plasmas. An MR dispersion relation for UH waves propagating perpendicular to **B**, valid for $ \omega j < 2 | \omega c e |$ in plasmas with thermal electrons at $ T e \u2272 25 \u2009 keV$, was obtained from Ref. 58 (or following the procedure in Sec. 4.7.4 of Ref. 1). Next, the conditions under which a propagating UH wave pair existed based on the MR dispersion relation were identified. The MR dispersion relation was also shown to reduce to the warm non-relativistic dispersion relation, commonly used for trapped UH wave PDI studies,^{6–17} at low *T _{e}*. MR effects were found to generally increase the

*n*and

_{e}*B*at which the UHR occurred, compared with warm and cold non-relativistic theory.

^{17}A modification of the electrostatic framework used for studying trapped UH wave PDIs was further proposed to enable the effects of the MR dispersion relation to be assessed, without adopting an MR description of the three-wave process, collisional damping, electron Landau damping, and diffraction parallel to

**B**. We then applied the MR UH wave theory to the PDI plasma scenarios at the ASDEX Upgrade tokamak studied using a non-relativistic warm framework in Ref. 17, as well as to scaled versions of the ASDEX Upgrade plasma parameters, to study core

*T*values resembling those of the X-mode ECRH scenarios planned for the early operation phase of ITER.

_{e}^{51}For the ASDEX Upgrade plasma parameters, MR theory yielded almost identical results to warm theory because of the low

*T*. With the scaled ITER-like core

_{e}*T*, several differences with potentially important implications for the nature of the expected PDIs were observed between MR and warm theory. For instance, the plasma regions allowing UH wave trapping in MR theory were generally larger than those of warm theory, making a larger set of ECRH beam settings susceptible to low-threshold trapped UH wave PDIs. However, qualitative features, such as the topology of the UHR harmonic layers and their approximate location in the plasmas, were found to be similar between MR and warm theory. Cold theory was found to be insufficient for the analysis of UH wave trapping with both ASDEX Upgrade and ITER-like plasma parameters, as it did not reproduce the qualitative features of either MR or warm theory in a number of cases. We finally showed that the modified electrostatic PDI framework using the MR dispersion relation could reproduce the qualitative features of the electrostatic warm PDI framework for the ASDEX Upgrade plasma parameters from Refs. 16 and 17, as expected due to the low

_{e}*T*.

_{e}Several points of interest remain for future studies of relativistic effects on UH wave propagation and trapping, as well as related low-threshold PDIs. First, the MR dispersion relation studied in this work only includes first-order finite Larmor radius effects; i.e., terms beyond first order in $ k j \u22a5 2 r L e 2$ are neglected. This is sufficient for studying the basic features of UH wave trapping and propagation in the underdense and weakly overdense plasmas found in conventional tokamaks, including PDIs in which X-mode waves decay to UH waves and lower hybrid waves near the UHR due to the enhancement of the X-mode wave electric field.^{32–36} However, terms of higher order in $ k j \u22a5 2 r L e 2$ are important for determining the precise trapped UH wave modes with the highest PDI growth rate,^{19} as well as for the study of UH waves in strongly overdense plasmas where $ \omega j > 2 | \omega c e |$, which are relevant for PDIs involving O-X-B heating^{37,42} in spherical tokamaks.^{38–42} Within the MR framework used in this paper, such terms can be obtained using the expansion of the dielectric tensor elements described in Sec. 4.7.4 of Ref. 1. Second, an assessment of fully relativistic effects^{1,57,62,63,69} on UH wave propagation, e.g., the disappearance of the evanescent region between the R-cutoff and the UHR at high *T _{e}*(Refs. 57 and 69), would be desirable. To this end, it would be interesting to analyze the characteristics of the fully relativistic dispersion relation for $ k j \u2225 = 0$ from Sec. 4.7.5 of Ref. 1. Third, it is of interest to investigate the trapped UH wave PDIs within a self-consistent relativistic, electromagnetic framework, rather than by the

*ad hoc*electrostatic framework adopted in this paper. This would enable the impact of different plasma and ECRH parameters, e.g., wider beams which lead to simultaneous lower ECRH power density and convective/diffractive losses, on trapped UH wave PDIs to be investigated with greater confidence and could be facilitated by the approach of Ref. 60. Finally, the experimental realization of plasma scenarios requiring the inclusion of MR effects in order to account for the observed PDI characteristics will be interesting. Significant MR effects are unlikely to be observed in connection with trapped UH wave PDIs at ASDEX Upgrade, as these PDIs generally occur near the plasma edge

^{16}or in the plasma core of scenarios with reduced confinement,

^{17}where

*T*is low. It may, however, be possible to observe MR modifications of the UH wave dispersion relation in connection with PDIs involving 105 GHz X-mode radiation reaching the UHR at ASDEX Upgrade

_{e}^{33–36}for plasmas with simultaneous low

*n*and high

_{e}*T*values. At ITER, MR effects should be significant for most trapped UH wave PDIs in the core plasma, including the ones expected for X-mode

_{e}^{17}and O-mode

^{23}ECRH scenarios, making ITER well suited to study MR effects on UH wave PDIs.

## ACKNOWLEDGMENTS

S. K. Hansen acknowledges support by an Internationalisation Fellowship (No. CF19–0738) from the Carlsberg Foundation. This work was supported by a research grant (No. 15483) from VILLUM FONDEN. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200—EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Søren Kjer Hansen:** Conceptualization (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **Stefan K. Nielsen:** Funding acquisition (lead); Supervision (equal); Writing – review & editing (equal). **Jörg Stober:** Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Raw data were generated at the ASDEX Upgrade large-scale facility. Derived data supporting the findings of this study are available from the authors upon reasonable request.

### APPENDIX: COMPUTATION OF DNESTROVSKII FUNCTION VALUES

*q*(Ref. 59). In order to avoid problems related to the evaluation of $ e \varphi m$ and $ \Gamma ( 1 \u2212 q , \varphi m )$ at large $ | \varphi m |$, we approximate $ F q ( \varphi m )$ by the first four terms of the asymptotic expansion from Eq. (49) of Ref. 56 when $ | \varphi m | \u2265 500$,

*q*, we may obtain its value at different

*q*through the recurrence relation from Eq. (14) of Ref. 56,

## REFERENCES

*Handbook of Mathematical Functions*