In this paper, we analyze theoretically the low-threshold parametric decay instability (PDI) that can be excited at the tokamak à configuration variable (TCV tokamak, Lausanne, Switzerland) during X2 electron cyclotron resonance heating experiments producing a two-dimensionally localized upper hybrid (UH) wave and a daughter extraordinary wave running from the decay layer outward to the plasma edge. The primary instability is then saturated due to the cascade of secondary decays into two-dimensionally localized UH and ion Bernstein waves. The level of plasma microwave emission in the frequency range substantially below half the pump wave frequency and of anomalous power losses of the pump are predicted.

## I. INTRODUCTION

The physics of burning plasma is getting closer to being demonstrated in the International Thermonuclear Experimental Reactor (ITER). This shifts the focus of fusion research in the current toroidal devices toward the details of methods and techniques required for successful reactor discharge control. Electron cyclotron resonance heating (ECRH) by powerful microwave beams has become an important tool for the localized modification of plasma parameters, e.g., current profile tailoring, neoclassical tearing mode (NTM) stabilization, or sawtooth control. The energy and space selectivity thought to be characteristic of ECRH, low antenna interaction with plasma, and availability of high-power sources have made it the main technique for plasma heating in ITER. However, by now a large data bank has been accumulated in ECRH experiments that indicates nonlinear phenomena in microwave propagation. These are anomalous microwave backscattering,^{1–7} an evident broadening of the ECRH power deposition profile as well as the missing power effect^{8–13} and the plasma emission at sub-harmonics of the gyrotron frequency.^{3,6,7} A remarkable phenomenon, also observed in ECRH mode, is the acceleration of ions under conditions where the energy transfer from electrons to ions due to collisions is negligible.^{14–17} Most of these observations correlate with a change in the density profile from smooth monotonic to non-monotonic (hollow) due to magnetic islands,^{18,19} edge localized modes (ELMs), or the pump-out effect.^{20}

As has been predicted analytically,^{21–24} shown numerically,^{25} and confirmed in recent X2-mode ECRH experiments on ASDEX Upgrade^{3} and Wendelstein 7-X,^{5} a pump wave propagating through any local maximum of a non-monotonic density profile, including the core of a plasma column, is unstable to parametric decays, which lead to the excitation of daughter localized upper hybrid (UH) waves. Within the developed theoretical model,^{21–24} this instability is saturated due to the pump wave depletion and by the cascade of consecutive decays of the primary trapped UH daughter wave, yielding secondary trapped UH waves and ion Bernstein (IB) waves. This scenario allowed explaining and reproducing the fine details of the frequency spectra of anomalous plasma radiation with red- and blue-shifted bands with respect to the gyrotron frequency and its level in X2-mode ECRH experiments at TEXTOR^{26,27} and at Wendelstein 7-X.^{28} The anomalous gyrotron frequency sub-harmonics plasma emission with ECRH was also predicted in the theoretical model^{29} and then was experimentally discovered at ASDEX Upgrade.^{3,6}

Since both a pronounced hollow density profile caused by the pump-out effect^{20} and anomalous effects such as fast ion generation,^{14,15} broadening of the electron cyclotron power deposition profile^{30} and the electron cyclotron current drive (ECCD) profile,^{31} and the pump frequency sub-harmonic emission,^{7} were observed in on-axis X2 ECRH experiments at TCV, it makes sense to revisit those results, analyze the possibility of excitation of low-threshold parametric decay instability (PDI) under the specific TCV conditions, and consider its consequences. For the conditions under consideration,^{14,15} the two-dimensional (2D) localization of IB waves in the equatorial plane of the plasma column is possible, as was previously found.^{32,33} Although two-dimensional UH wave localization in a magnetic island has been found previously,^{34} in this paper we demonstrate for the first time that UH waves can be two-dimensionally localized at a hollow density profile: along the major radius due to the local maximum of the hollow density profile and in the vertical direction due to the strong 2D inhomogeneity of the magnetic field on the magnetic surface in the plasma core. Focusing on the typical conditions of on-axis X2-mode ECRH experiments at TCV, where the pump beam was launched from the low magnetic field side in the equatorial plane of the plasma column, we will analyze the decay of the pump X2 mode into a daughter X1 mode running from the decay layer outward to the plasma edge and a two-dimensionally localized upper hybrid (UH) wave. The primary instability is then saturated due to the pump wave depletion and a cascade of secondary decays into two-dimensionally localized UH and ion Bernstein (IB) waves. We demonstrate that the two-dimensional localization of UH and IB waves leads to a decrease in the instability threshold and instability saturation level compared to the case of one-dimensional (1D) localization of daughter UH and IB waves.^{35,36}

Since the power deposition of all daughter UH waves due to electron cyclotron damping is far from the region predicted for the pump wave, the anomalous absorption should broaden the total ECRH power deposition profile. Based on the results of the investigation, it seems possible to explain, at least qualitatively, the observations of groups of suprathermal ions at ECR heating as a consequence of the strong damping of daughter IB waves. We can also predict the plasma emission in the frequency range below half the pump wave frequency as a result of the escape of the primary extraordinary wave from the plasma volume toward the low-field side in the equatorial plane.

## II. TWO-DIMENSIONAL LOCALIZATION OF UPPER HYBRID WAVES IN THE POLOIDAL SECTION OF THE TCV TOKAMAK

^{14}The typical discharge major is $ R 0 = 0.88 \u2009 m$ and its minor radius is $ a = 0.22 \u2009 m$. In the X2-mode ECRH experiments discussed below, a plasma column was formed and held by nested magnetic surfaces that had an ellipse shape with an elongation factor of 1.5 and a triangularity factor of 0.6.

^{14}There was no Shafranov shift at this magnetic equilibrium. Figure 1 shows the density and magnetic field profiles in the equatorial cross section of the plasma column, similar to those observed in the on-axis X2-mode ECRH experiments.

^{14,15}The central electron and ion temperatures were $ T e 0 = 1.9 \u2009 keV$ and $ T i 0 = 0.35 \u2009 keV$. The density profile was hollow due to the strong on-axis electron heating leading to the pump-out effect.

^{20}The magnetic field was flat due to the high poloidal beta.

^{37}One would expect two-dimensional localization of upper hybrid waves in the vicinity of the local maximum of the hollow density profile in the equatorial plane of the plasma column due to strong poloidal inhomogeneity of the magnetic field on a magnetic surface in the plasma core, as it has been discovered theoretically with electron Bernstein waves in typical on-axis ECRH mode at TCV.

^{38,39}Figure 2 shows the ray trajectory of the UH wave (solid thick line) in the poloidal cross section of the plasma column for the profiles given in Fig. 1. The thin solid lines are magnetic surfaces. Figure 2 is obtained with the ray-tracing technique for the UH-wave dispersion equation valid in the vicinity of the UH resonance,

^{40}

^{23}

^{34}

## III. TWO-DIMENSIONAL LOCALIZATION OF ION BERNSTEIN WAVES IN THE POLOIDAL SECTION OF THE TCV TOKAMAK

^{41}

*y-*dependence of the wavenumber

*y-*component (solid line). It should be noted that the area bounded by the slowly precessing trajectory is preserved, but its angle with respect to the vertical coordinate has a slight change. Figure 8 shows the dependence of the wavenumber

*x-*component along the flux variable

*x*on the major radius (solid line). A substantial difference in the characteristic frequencies of the ray oscillations in both directions makes it possible to quasi-analytically describe the eigen IB modes following the adiabatic procedure developed in Ref. 39. Next, we use the Cartesian coordinate system $ ( x , y , z )$ and take advantage of the large difference in the characteristic frequencies of the ray oscillations along the vertical and horizontal directions. This allows decomposing Eq. (7) into a series

## IV. EQUATIONS DESCRIBING A CASCADE OF LOW-THRESHOLD SECONDARY DECAYS

^{23}we can arrive at a reduced equation for the daughter extraordinary wave,

*z*-direction.

*n*[see Eq. (20)] generates a running UH wave, whose dispersion curve is pointed out in Fig. 5 by an arrow, and contributes to the same eigenmode number

*p*of the IB wave [see Eq. (21)] that participates in the secondary instability. Figure 12 shows the dispersion curves illustrating the tertiary instability of the secondary UH wave. The difference in the numbers of running and secondary UH waves and the number of IB wave involved in the secondary instability ( $ f p I = 539.2 \u2009 MHz$ corresponding to the 49th ion Bernstein harmonic) is shown. By means of the envelope algorithm to solve Poisson's equation with the nonlinear plasma susceptibility $ \chi e n l$ describing the coupling of three electrostatic waves,

^{42}we can derive the reduced equation for the amplitude $ C U H$ of the running UH wave with frequency $ f U H = f n E \u2212 f p I$,

^{23}We use Eq. (22) in equations describing the tertiary instability. We analyze the secondary and tertiary instabilities following the procedure employed earlier in Ref. 35. Omitting routine mathematical calculations and referring the reader to Refs. 23 and 35 for details, we write out the following coupled differential equations describing the pump wave cascade decays resulting in the excitation of the 2D-localized UH and IB waves:

*x*-coordinate, $ l m y = 2 w \u222b \u2212 y r y r d y L m y ( y ) \u2212 1 \u2009 exp ( \u2212 y 2 / w 2 )$ is the length of the decay layer along the

*y*-coordinate. The averaging procedure over the mode localization, which provides the diffraction coefficients $ \Lambda j z$

*, j = m,n,s*and the group velocities' projections onto the toroidal direction $ u j$,

*j = n,p*, reads $ \u27e8 A \u27e9 | m , n = \u222b \u2212 \u221e \u221e d x | \phi m , n x ( x ) | 2 \u222b \u2212 \u221e \u221e d y | \psi m , n y ( y ) | 2 A ( x , y )$ and $ \u27e8 A \u27e9 | p = \u222b \u2212 \u221e \u221e d x | \Phi p x ( x ) | 2 \u222b \u2212 \u221e \u221e d y | \Psi p y ( y ) | 2 A ( x , y )$. Furthermore, $ \xi \u2009 sec \u2009 \u221d \chi e n l$ is the nonlinear coupling coefficient describing the secondary decay of the primary UH wave with the amplitude $ a m$ into the secondary modes of the UH wave with the amplitude $ a n$ and the eigenmode of the IB wave with the amplitude $ b p$; $ \xi ter \u221d | \chi e n l | 2$ is the nonlinear coupling coefficient describing the tertiary decay of the secondary eigenmode of the UH wave with the amplitude $ a n$, leading to the eigenmode (

*p*) of the IB wave with the amplitude $ b p$ and the running UH wave with the amplitude Eq. (22); $ \nu d = \pi H ( b p \u2212 b p t h ) \omega p I ( \upsilon p / \upsilon t i ) 3 \u2009 exp ( \u2212 \upsilon p 2 / \upsilon t i 2 )$ is the coefficient, which describes amplitude-dependent “stochastic” damping of IB waves,

^{43}

^{,}$ H ( \u2026 )$ is the Heaviside function, $ \upsilon p = \omega p I / q p x I$ is the phase velocity, $ b p t h = B 0 L x I L y I w \pi 1 / 2 T e ( \omega c i \omega p I ) 1 / 3 \upsilon p c$ is a threshold value indicating transition into the stochastic mode.

^{43}The squares of the moduli of the amplitudes multiplied by the coefficient $ T e / ( \pi w )$ are equal to their one-dimensional energy density. The coupling coefficients for all the decays are complex values. Their real components describe the corresponding instability. The imaginary components lead to a nonlinear shift of the corresponding eigenfrequency. For a detailed derivation of Eq. (23), we can refer readers to Refs. 23, 24, 35, and 36.

Furthermore, we solve the system of Eq. (23) numerically in an one-dimensional box $ 2 z B = 2 \pi R 0$, assuming the thermal-noise level of all UH waves and periodic boundary conditions. The system of Eq. (23) with the periodic boundary conditions can describe multiple transitions of the UH plasmons through the plasma waveguide. Following the conditions of the on-axis X2-mode ECRH in the TCV tokamak, we assume a beam with a power of 500 kW and a width of 2 cm. The results of solving Eq. (23) are shown in Figs. 13–15. Figure 13 demonstrates the energy densities averaged over the pump beam for both UH waves $ \epsilon m , n E = ( \pi w ) \u2212 1 \u222b \u2212 \u221e \u221e | a m , n | 2 \u2009 exp ( \u2212 z 2 / w 2 ) d z$ and the IB wave $ \epsilon p I = ( \pi w ) \u2212 1 \u222b \u2212 \u221e \u221e | b p | 2 \u2009 exp ( \u2212 z 2 / w 2 ) d z$ on a semi-logarithmic scale. The dashed horizontal line shows the IB-wave level $ | b p t h | 2$ above which the ion behavior becomes stochastic. As can be seen, $ | b p t h | 2$ determines the saturation level of the IB wave. The threshold of excitation of the instability, found as a result of numerical solution, is equal to $ P 0 t h = 124 \u2009 kW$. In the case of only one-dimensional localization of all daughter UH and IB waves,^{35,36} the threshold would be higher $ P 0 t h = 153 \u2009 kW$. When the amplitude of the primary UH wave exceeds the threshold of the secondary instability, the decay of this wave occurs, which leads to an increase in the amplitude of the secondary UH mode *n* and IB mode *p*.

## V. ANOMALOUS POWER ABSORPTION

As can be seen in Fig. 13, the transition time to the saturation regime is about $ 1 \u2009 \mu s$. The temporal evolution of the total energy of the eigenmodes *m* and *n* of UH waves, defined through the equation $ W m p r = ( \pi w ) \u2212 1 \u222b \u2212 \u221e \u221e | a m | 2 d z$, $ W n \u2009 sec \u2009 = ( \pi w ) \u2212 1 \u222b \u2212 \u221e \u221e | a n | 2 d z$, after that is shown in Fig. 14. A linear approximation of these dependencies gives estimates of the pump power fraction transferred to each of them: about 14 kW by the primary UH mode *m* (upper dependence, the absolute error of the linear approximation determining the wave power is 0.7 kW) and 16 kW by the secondary UH mode *n* (lower dependence, the absolute error of the linear approximation is 1.57 kW). Propagating around the torus along the two-dimensional waveguide the UH waves suffers from the collisional damping and nonlinear coupling. When the damping due to these effects is weak, a dynamical regime can arise leading to toroidal UH eigenmodes. The parametric excitation of the 3D plasma cavity for daughter UH waves and its influence on the saturation regime will be analyzed elsewhere.

Using Eqs. (19) and (22), we can also calculate the pump energy fractions transmitted to the running primary extraordinary wave and the running tertiary UH wave. These dependences are shown in Fig. 15. As it is seen, about 20 kW of the pump power is gained by the running primary X wave (upper dependence, the absolute error of the linear approximation determining the wave power is 2.3 kW), and about 12 kW is carried out by the running tertiary UH wave (lower dependence, the absolute error of the linear approximation is 0.93 kW). The EC resonance conditions are not satisfied in the case under consideration for the primary extraordinary wave possessing extremely low frequency to be absorbed by thermal electrons. However, electrons accelerated by the pump in the second harmonic ECR region can take part in the primary X wave absorption. The tertiary running UH wave, arising from the coupling of the secondary UH mode and the ion Bernstein mode, taking from the pump power a piece of about 12 kW, propagates inward into the plasma volume. It encounters the surface of the fundamental EC resonance on the high magnetic field side at $ R \u2212 R 0 \u2248 \u2212 8 \u2009 cm$. Since the power deposition regions of all daughter waves due to the electron cyclotron damping or weak collisions are far from the region predicted for the pump wave, this can broaden the total ECRH power deposition profile. The power gained by the IB wave is smaller by a factor of about $ \omega s I / \omega n E$ than the power gained by UH waves. Altogether, the anomalous absorption level at the instability saturation can be estimated using the sum of contributions of all daughter waves as about 12.5% (or 62 kW of the pump power).

For background plasma parameters at which the cascade decay of the primary UH wave has an even number of steps, a stronger anomalous absorption should occur.^{24} In this case, the power deposition broadening should be much stronger. The fraction of power transferred to IB waves should also be much larger, which can explain, at least qualitatively, the emergence of groups of accelerated ions in the TCV ECRH experiment. Among the important consequences of the pump wave decay is the microwave emission from the plasma of a daughter extraordinary wave propagating along the major radius in the equatorial plane. In the forthcoming section, we consider this radiation.

## VI. ANOMALOUS MICROWAVE EMISSION AT FREQUENCY MUCH SMALLER THAN HALF THE GYROTRONS FREQUENCY

## VII. CONCLUSIONS

The possibility of the two-dimensional localization of UH waves in the vicinity of the maximum of the hollow density profile in the equatorial plane of the plasma column for typical conditions of on-axis X2-mode ECRH at TCV was shown. The considered effect leads to a decrease in the instability threshold and instability saturation level compared to the case of 1D localization of the daughter UH waves.

A scenario based on the excitation of low-threshold PDI of pump microwaves in on-axis X2-mode ECRH experiments on TCV was analyzed. Transition of the instability into the saturation mode was assumed to be a result of a cascade of decays leading to the excitation of the two-dimensionally localized UH and IB waves. Since the power deposition of all the daughter waves is far from the region predicted for the pump wave, this should broaden the total ECRH power deposition profile. We also predict the plasma emission in the frequency range substantially below half the pump wave frequency as a result of the escape of the primary X1-mode from the plasma volume at the low-field side in the equatorial plane of the plasma column. It seems possible to explain, at least qualitatively, the observations of groups of suprathermal ions at ECR heating on the TCV tokamak as a consequence of the strong damping of daughter IB waves.

## ACKNOWLEDGMENTS

The analytical treatment is supported under Grant No. RSF 22-12-00010, the numerical modelling is supported under the Ioffe Institute state Contract No. 0040-2024-0028, whereas the code for the PDI modelling was developed under the Ioffe Institute state Contract No. 0034-2021-0003.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**A. Yu. Popov:** Conceptualization (lead); Formal analysis (lead); Writing – original draft (lead). **E. Z. Gusakov:** Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Writing – original draft (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.