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.

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 effect8–13 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 Upgrade3 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 TEXTOR26,27 and at Wendelstein 7-X.28 The anomalous gyrotron frequency sub-harmonics plasma emission with ECRH was also predicted in the theoretical model29 and then was experimentally discovered at ASDEX Upgrade.3,6

Since both a pronounced hollow density profile caused by the pump-out effect20 and anomalous effects such as fast ion generation,14,15 broadening of the electron cyclotron power deposition profile30 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.

The TCV facility is a medium-sized tokamak characterized by a strongly elongated vacuum chamber rectangular in toroidal cross section.14 The typical discharge major is R 0 = 0.88 m and its minor radius is a = 0.22 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 keV and T i 0 = 0.35 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 
D U H ( ω , q ) = q 2 + χ ( ω , q ) + ω 2 c 2 g ( ω ) 2 = 0 .
(1)
FIG. 1.

Density (dashed-dotted line) and magnetic field (solid line) profiles at on-axis X2-mode ECRH.

FIG. 1.

Density (dashed-dotted line) and magnetic field (solid line) profiles at on-axis X2-mode ECRH.

Close modal
FIG. 2.

The trajectory of the UH wave (solid thick line) in the poloidal section of the plasma column at the profiles given in Fig. 1. Thin solid lines are magnetic surfaces. The window in the upper left corner shows the projections of the ray trajectory at a major radius and at a vertical coordinate.

FIG. 2.

The trajectory of the UH wave (solid thick line) in the poloidal section of the plasma column at the profiles given in Fig. 1. Thin solid lines are magnetic surfaces. The window in the upper left corner shows the projections of the ray trajectory at a major radius and at a vertical coordinate.

Close modal
In Eq. (1), q 2 = q 2 + q / / 2, and the last term arises due to the presence of a small electromagnetic component of the UH wave in the vicinity of the UH resonance, g is the non-diagonal component of the cold-plasma dielectric tensor, and
χ ( ω , q ) = j = e , i 2 ω p j 2 υ t j 2 ( 1 + ω | q / / | υ t j p = Z ( ω p ω c j q / / υ t j ) exp ( q 2 υ t j 2 2 ω c j 2 ) I p ( q 2 υ t j 2 2 ω c j 2 ) )
(2)
is the linear plasma susceptibility. In Eq. (2), υ t i , e are the ion and electron thermal velocities, ω ci , e are the ion and electron cyclotron frequencies, ω p i , e are the ion and electron plasma frequencies, Z is the plasma dispersion function, I p is the modified Bessel function, and q and q / / are perpendicular and parallel components of the wavenumber, respectively. The trajectory of the UH wave in Fig. 2 appears to be two-dimensionally localized: along the flux variable in the vicinity of the local density maximum and along the poloidal direction on the magnetic surface as a result of a strong two-dimensional inhomogeneity of the magnetic field on the magnetic surface in a plasma core. The small window in the upper left corner shows the projections of the trajectory on the major radius and on the vertical coordinate. It can be seen that the trajectory is localized in the vicinity of the equatorial plane and the characteristic frequency of the ray oscillations in a two-dimensional waveguide along the flux variable is significantly larger than in the direction of the vertical coordinate. For the analytical description of 2D localized UH waves, we take into account that the region of localization of UH waves is situated in the vicinity of the local maximum of the hollow density profile and around the equatorial plane. We introduce a Cartesian coordinate system ( x , y , z ), in which the coordinate x is directed along the flux coordinate inside the plasma, y is the coordinate coinciding with the poloidal coordinate in the vicinity of the equatorial plane, z is a coordinate along the toroidal direction, and the origin of the coordinates coincides with the local maximum of the UH frequency profile. Although, in the dispersion Eq. (1) the variables are not separated, we can use the adiabatic procedure developed in Ref. 39. At its first step, we will neglect the dependence of Eq. (1) on y and q y, considering them fixed. Solving Eq. (1) numerically with respect to q x ±, where the upper index “+” corresponds to the “warm” branch of the dispersion curve, and the upper index “−” to the “cold” one, and using the quantization procedure, we arrive at the reduced dispersion equation for the UH wave,
D ̃ U H ( q y , y ) = x l x r q x + ( ω , ξ ; q y , y ) d ξ + x r x l q x ( ω , ξ ; q y , y ) d ξ π ( 2 m x + 1 ) = 0 ,
(3)
where x l and x r are the turning points of the ray trajectory. Integration in Eq. (3) can be performed only numerically, but as a result we obtain q y E = q y E ( y ). Carrying out again the quantization procedure,
2 y r y r q y E ( f m E , y , m x ) d ξ = π ( 2 m y + 1 ) ,
(4)
where y r and y r are turning points of the UH wave trajectory in the direction of the coordinate y, we obtain the eigenfrequency of the 2D localized wave, where the mode eigenvalue has two components, m = ( m x , m y ). Figure 3 shows the dispersion curve obtained by the ray-tracing procedure ( x = R R max, R max is the big radius at which the UH resonance profile has its local maximum). The numerical solution of Eq. (1) at q y = 0, y = 0 is completely overlapped (perfect matched) by the numerical solution obtained by the ray-tracing procedure. Figure 4 shows the dispersion curve q y E ( y ) obtained by the ray-tracing procedure and the numerical solution q y E ( y ) of Eq. (3). The eigenvalue and frequency found as a result of the quantization procedure (3) and (4) are equal to m = ( 30 , 1 ), f m E = 52.85 GHz. As it is shown below (see Fig. 10), this UH wave provides the strongest coupling to the pump wave and to additional extraordinary daughter wave at frequency satisfying the decay condition. We use expressions Eqs. (2)–(4) to present the eigenfunctions of the UH wave in the framework of the Wentzel—Kramers—Brillouin (WKB) approximation. The eigenfunction describing the localization of the UH wave along the direction x has the form23 
ϕ m x ( x ) = 1 L m x + ( x ) exp ( i x l x q x E + ( f m E , ξ ) d ξ i π 4 ) + 1 L m x ( x ) exp ( i x l x q x E ( f m E , ξ ) d ξ + i π 4 ) ,
(5)
where L m x ± ( x ) = | D q x ± ( x ) | x l x r d ξ ( | D q x + ( ξ ) | 1 + | D q x ( ξ ) | 1 ), D q x ± = D U H / q x | q x E ± ( f m E , x ) , q y = 0 , y = 0. The eigenfunction along the y direction has the following form:34 
ψ m y ( y ) = 2 L m y ( y ) cos ( y r y q y E ( f m E , ξ ) d ξ π 4 ) , L m y ( y ) = 2 | D q y ( y ) | y r y r d ξ | D q y ( ξ ) | ,
(6)
where D q y = D ̃ U H / q y | q y , y. The finite height of the hump on the hollow plasma density profile results in a finite height potential well for eigenmodes of UH waves. As an example, Fig. 5 shows one-dimensional dispersion curves of two eigenmodes of the UH wave at q y , z E = 0: m = ( 30 , 1 ) with eigenfrequency f m E = 52.85 GHz and n = ( 40 , 1 ) with eigenfrequency f n E = 52.31 GHz in the equatorial plane of the plasma column. The non-localized (running) wave with frequency f U H = 51.77 GHz is pointed by the arrow.
FIG. 3.

Dispersion curve q x E ( x ), x = R R max obtained by the ray-tracing procedure. The numerical solution of Eq. (1) at q y , z = 0 , y = 0 is completely overlapped by the numerical solution obtained by the ray-tracing procedure; m = ( 30 , 1 ), f m E = 52.85 GHz.

FIG. 3.

Dispersion curve q x E ( x ), x = R R max obtained by the ray-tracing procedure. The numerical solution of Eq. (1) at q y , z = 0 , y = 0 is completely overlapped by the numerical solution obtained by the ray-tracing procedure; m = ( 30 , 1 ), f m E = 52.85 GHz.

Close modal
FIG. 4.

Dispersion curve q y E ( y ) obtained by the ray-tracing procedure and the dependence q y E ( y ) obtained by solving the Eq. (3) at q z = 0 is shown by the arrow. m = ( 30 , 1 ), f m E = 52.85 GHz.

FIG. 4.

Dispersion curve q y E ( y ) obtained by the ray-tracing procedure and the dependence q y E ( y ) obtained by solving the Eq. (3) at q z = 0 is shown by the arrow. m = ( 30 , 1 ), f m E = 52.85 GHz.

Close modal
FIG. 5.

The wavenumbers of two UH modes at q y , z E = 0 m = ( 30 , 1 ) with eigenfrequency f m E = 52.85 GHz, n = ( 40 , 1 ) with eigenfrequency f n E = 52.31 GHz—are labeled. The non-localized (running) wave with frequency f U H = 51.77 GHz is pointed by the arrow.

FIG. 5.

The wavenumbers of two UH modes at q y , z E = 0 m = ( 30 , 1 ) with eigenfrequency f m E = 52.85 GHz, n = ( 40 , 1 ) with eigenfrequency f n E = 52.31 GHz—are labeled. The non-localized (running) wave with frequency f U H = 51.77 GHz is pointed by the arrow.

Close modal
Figure 6 shows the IB wave trajectory ( f s I = 539.2 MHz) in a poloidal section of the plasma column for the profiles shown in Fig. 1. It is obtained by the ray-tracing procedure for the IB wave dispersion equation,41 
D I B ( ω , q ) = q 2 + χ ( ω , q ) = 0 ,
(7)
where the linear susceptibility χ ( ω , q ) is defined through Eq. (2). It can be seen that the trajectory is two-dimensionally localized: along the flux variable in the vicinity of the local maximum of the magnetic field; along the poloidal direction on the magnetic surface as a result of a strong two-dimensional inhomogeneity of the magnetic field on the magnetic surface. Thin solid lines are magnetic surfaces. The small window in the upper right corner shows the trajectory projections on the vertical coordinate (lower “fast” dependence) and on the major radius (upper dependence). It can be seen that the characteristic frequency of the ray oscillations in a two-dimensional waveguide along the flux variable is significantly smaller than in the direction of the vertical coordinate. Figure 7 shows the 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
D I B D I B ( q x , x ) | q y = 0 , y = 0 D q y q y ( q x , x ) q y 2 + 2 D y q y ( q x , x ) q y y D y y ( q x , x ) y 2 = 0 ,
(8)
where D q y q y ( x ) = | 2 D I B / 2 q y 2 | q y = 0 , y = 0, D y q y ( x ) = | 2 D I B / 2 q y y | q y = 0 , y = 0, and D y y = | 2 D I B / 2 y 2 | q y = 0 , y = 0. All the coefficients in Eq. (8) are taken at fixed ( q * , x * , ω I * ) obeying the equation D I B ( q * , x * , ω I * ) | q y = 0 , y = 0 = 0, D I B ( q * , x * , ω I * ) / q | q y = 0 , y = 0 = 0, and D I B ( q * , x * , ω I * ) / x | q y = 0 , y = 0 = 0. By solving Eq. (8), we obtain
q y I ± ( y ; q x , x ) = D y q y ( q x , x ) D q y q y ( q x , x ) y ± D I B ( q x , x ) | q y = 0 , y = 0 y 2 ( D y y ( q x , x ) D y q y ( q x , x ) 2 D q y q y ( q x , x ) ) .
(9)
FIG. 6.

Trajectory of the IB wave ( f s I = 539.2 MHz) in a poloidal cross section of the plasma column for the profiles shown in Fig. 1. Thin solid lines are magnetic surfaces. The window in the upper right corner shows the trajectory projections on the vertical coordinate (lower “fast” dependence) and on the major radius (upper dependence).

FIG. 6.

Trajectory of the IB wave ( f s I = 539.2 MHz) in a poloidal cross section of the plasma column for the profiles shown in Fig. 1. Thin solid lines are magnetic surfaces. The window in the upper right corner shows the trajectory projections on the vertical coordinate (lower “fast” dependence) and on the major radius (upper dependence).

Close modal
FIG. 7.

Dependence of the wavenumber y-component on the vertical coordinate (solid line). The dotted line is the solution of Eq. (9). The eigen number is p y = 2.

FIG. 7.

Dependence of the wavenumber y-component on the vertical coordinate (solid line). The dotted line is the solution of Eq. (9). The eigen number is p y = 2.

Close modal
FIG. 8.

Dependence of the wavenumber x-component on the major radius (solid line). The dotted line is the solution of Eq. (10). The eigen number is p x = 14.

FIG. 8.

Dependence of the wavenumber x-component on the major radius (solid line). The dotted line is the solution of Eq. (10). The eigen number is p x = 14.

Close modal
In Fig. 7, the dotted line is the solution of Eq. (9). It can be seen that the quasi-analytical dependence adequately describes the results of the ray-tracing procedure. Carrying out the quantization procedure between two turning points ± y r at fixed values of q x and x, we obtain the modified IB wave dispersion equation,
D ̃ I B = D I B ( q x , x ) | q y = 0 , y = 0 D q y q y ( q x , x ) D y y ( q x , x ) D y q y ( q x , x ) 2 ( 2 s y + 1 ) = 0 .
(10)
Solving Eq. (10) numerically, we find q x I = q x I ± ( x , ω s I ). Carrying out numerically the quantization procedure for q x I, i.e., x l s x r s ( q x I + ( x , ω I s ) q x I ( x , ω I s ) ) = π ( 2 s x + 1 ) where x l s and x r s are the wave turning points, we obtain the eigenfrequency ω s of the eigenmode s = ( s x , s y ). The dashed line in Fig. 8 shows the solution of Eq. (10), which corresponds to the eigenmode s = ( 2 , 14 ) with frequency f s I = 539.2 MHz. Thus, the eigenfunction along the y direction has the following form:
Ψ p y ( y ) = 1 L p y + ( y ) exp ( i y r p y q y I + ( ξ ) d ξ i π 4 ) + 1 L p y ( y ) exp ( i y r p y q y I ( ξ ) d ξ + i π 4 ) ,
(11)
where L p y ± ( y ) = | q y ± I ( y ) | y r p y r p 1 | q y I ( ξ ) | + 1 | q y + I ( ξ ) | d ξ. The eigenfunction that describes the IB-wave localization along the direction x has the form
Φ p x ( x ) = 1 L p x + ( x ) exp ( i x l p x q x I + ( ξ ) d ξ i π 4 ) + 1 L p x ( x ) exp ( i x p s x q x I ( ξ ) d ξ + i π 4 ) ,
(12)
where L p x ± ( x ) = | D I q x ± ( x ) | x l p x r p d ξ ( | D I q x + ( ξ ) | 1 + | D I q x ( ξ ) | 1 ), D I q x ± = | D ̃ I B / q x | q x I ±. Because of the finite height of the hump on the hollow plasma density profile, there exists a potential well of finite height for the IB-wave eigenmodes. Figure 9 shows one-dimensional dispersion curves of three eigenmodes of the IB wave in the equatorial plane of the plasma column at q y , z I = 0: p = ( 14 , 2 ), r = ( 9 , 1 ), and t = ( 0 , 0 ).
FIG. 9.

Wavenumbers of three IB modes at q y , z I = 0: p = ( 14 , 2 ), r = ( 9 , 1 ), and t = ( 0 , 0 ).

FIG. 9.

Wavenumbers of three IB modes at q y , z I = 0: p = ( 14 , 2 ), r = ( 9 , 1 ), and t = ( 0 , 0 ).

Close modal
Consider a monochromatic extraordinary (X) pump wave beam propagating across the external magnetic field along the equatorial plane. By means of the WKB approximation, it can be represented as
E 0 = e 0 X C 0 2 n 0 x ( x ) exp ( y 2 2 w 2 z 2 2 w 2 ) exp ( i x k 0 x ( x ) d x i ω 0 t ) + c . c . ,
(13)
where P 0 and w are the beam power and radius, respectively, e 0 X = e X ( ω 0 ) = e y i e x g 0 / ε 0 is the polarization vector, e x , y are components of the unit vector in the corresponding directions, n 0 x = c k 0 x / ω 0 = c k x ( ω 0 ) / ω 0 = ε 0 g 0 2 / ε 0 is the refraction index, g 0 = g ( ω 0 ) and ε 0 = ε ( ω 0 ) are perpendicular components of the dielectric tensor of the “cold” plasma. When a microwave beam passes through the local maximum of the density profile in the equatorial plane of the plasma column, it can decay into an extraordinary microwave and an eigenmode of 2D localized UH wave. Figure 10 shows the radial coordinate dependence of the wavenumber of this primary UH eigenmode m = ( 30 , 1 ) shifted downward by the pump wavenumber ( f 0 = 82.3 GHz, f 0 = 2 f c e ( 0 ), solid line). The dashed-dotted line in the figure corresponds to the wavenumber of the extraordinary daughter wave. The wave's frequencies obey the decay resonance condition ω 0 = ω s + ω m. In the vicinity of the point where the two thin curves merge, the decay of the pump wave with the largest nonlinear coupling coefficient becomes possible (because of the tangency of the dashed and solid dispersion curves the length of the coupling region is maximal). Thus, the primary decay leads to the excitation of a particular UH mode trapped in the vicinity of the UH frequency local maximum, for which the nonlinear coupling with the pump wave and scattered extraordinary wave is the strongest and thus the excitation threshold is the lowest. As seen in Fig. 10, the daughter extraordinary wave escapes the decay layer and can leave the plasma volume propagating along the equatorial plane to the low-field side. It worth stressing that for other UH eigenmodes, the parametric excitation is as well possible; however, the pump power threshold is higher and the growth rate is lower.
FIG. 10.

The solid line is the number of the primary UH eigenmode m = ( 30 , 1 ) shifted downward by the pump ( f 0 = 82.3 GHz, f 0 = 2 f c e ( 0 )) wavenumber. The dashed-dotted line is the number of the extraordinary daughter wave.

FIG. 10.

The solid line is the number of the primary UH eigenmode m = ( 30 , 1 ) shifted downward by the pump ( f 0 = 82.3 GHz, f 0 = 2 f c e ( 0 )) wavenumber. The dashed-dotted line is the number of the extraordinary daughter wave.

Close modal
In what follows, we will use the WKB approximation to describe the waves involved in the cascade of decays. This allows representing the extraordinary wave in the following form:
E s = e s X C s 2 n s x ( x ) exp ( i x d p x k s x ( x ) d x + i ω s t ) + c . c . .
(14)
In its turn, the UH eigenmode potential is represented as follows:
φ u h prim = C m 2 ϕ m x ( x ) ψ m y ( y ) exp ( i ω m E t ) + c . c . .
(15)
In Eq. (14) x d p is the point at which the decay condition for the wavenumbers of the pump and daughter waves is nearly zero Δ K p = k 0 x + k s x q m x E 0 and has its local minimum d Δ K p / d x | x d p = 0. The amplitude of the pump wave incident onto the decay region is determined by the beam power,
C 0 = 2 P 0 / ( c w 2 ) ,
(16)
whereas the amplitude of the extraordinary daughter wave incident onto the decay region is determined by the thermal-noise level,
C s T e / ( π w 2 ) , C m T e / ( ω m E | D U H / ω | ω m E π w 2 ) .
(17)
Furthermore, we hold the amplitude of the pump wave constant, assuming the depletion is weak.
Using the envelope algorithm to solve Maxwell's equations with nonlinear current density, which describes the nonlinear coupling between the daughter waves and the pump wave,23 we can arrive at a reduced equation for the daughter extraordinary wave,
x C s = i ω s 4 c κ B exp ( y 2 2 w 2 z 2 2 w 2 ) 1 n 0 x ( x ) n s x ( x ) C 0 * φ U H exp ( i x Δ K p d ξ ) ,
(18)
with the coupling coefficient, κ, given by
κ = q m x ω p e 2 | ω c e | ( Θ 0 + Θ s ) ( ω 0 2 ω c e 2 ) ( ω s 2 ω c e 2 ) ( ( ω 0 ω s ) 2 ω c e 2 ) , Θ s = n s x ( ( 2 ω s ω 0 ) | ω c e | ( g s ε s ω s | ω c e | ) + ω 0 ( ω c e 2 ω s 2 ) ) ,
Θ 0 = n 0 x ( ω 0 ( g s ε s ω s | ω c e | ) | ω c e | + ω 0 2 ( g s ε s | ω c e | ω s ) + | ω c e | ( g s ε s ( ω s ( ω 0 ω s ) | ω c e | 2 ) + ( 2 ω s ω 0 ) | ω c e | ) ) .
The boundary conditions for Eq. (18) are given through Eq. (17) at x . Integrating Eq. (18), we obtain
C s = i ω s 4 c κ B exp ( y 2 2 w 2 z 2 2 w 2 ) ψ m y ( y ) C m ( z , t ) x d x C 0 * ( x ) ϕ m x ( x ) n 0 x ( x ) n s x ( x ) exp ( i x ( k 0 x + k s x ) d ξ ) .
(19)
Furthermore, we will use Eq. (19) in Poisson's equation describing the daughter two-dimensionally trapped UH mode.
As the instability develops and the amplitude of the localized UH wave grows exponentially, they may experience subsequent decays into 2D-localized UH and IB waves. The secondary UH and IB waves potentials by means of the WKB approximation can be represented as follows:
φ u h sec = C n 2 ϕ n x ( x ) ψ n y ( y ) exp ( i q z z + i ω n E t ) + c . c . ,
(20)
φ I B sec = B p 2 Φ p x ( x ) Ψ p y ( y ) exp ( i q z z i ω p I t ) + c . c . .
(21)
For the density profile under consideration, there is only one decay of the primary eigenmode of the UH wave, leading to the emergence of the secondary eigenmodes of the UH and IB waves. The dispersion curves of the daughter UH waves representing the eigenmodes of a finite height potential well are shown in Fig. 5. Figure 11 shows the dispersion curves illustrating the secondary instability of the primary UH wave. In the vicinity of the points where a curve representing the sum of the numbers of the primary and secondary UH waves and the dispersion curve of the IB wave intersect, the decay resonance conditions are fulfilled. The eigenfrequency of the primary eigenmode of the UH wave is higher and its number is smaller than for the secondary eigenmode n = ( 40 , 1 ) of the UH wave at frequency f n E, i.e., m x < n x , f m E > f n E. The decay of the UH wave into two eigenmodes, whose frequencies are not independent due to the decay condition for coupled wave frequencies, requires a finite q z for both daughter waves. This causes the phase factor in Eqs. (20) and (21) to describe the spatial structure of the wave along the z-direction.
FIG. 11.

Dispersion curves illustrating the secondary instability of the primary UH wave. The sum of the radial wavenumbers of primary and secondary UH waves and the wavenumber of the IB wave mode ( f p I = 539.2 MHz) is shown.

FIG. 11.

Dispersion curves illustrating the secondary instability of the primary UH wave. The sum of the radial wavenumbers of primary and secondary UH waves and the wavenumber of the IB wave mode ( f p I = 539.2 MHz) is shown.

Close modal
The following decay of the secondary UH eigenmode with number 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 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 χ 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 f p I,
x C U H = i χ e n l C n B p | e | ϕ n x ( x ) * ψ n y ( y ) * Φ p x ( x ) Ψ p y ( y ) 2 | D q x + ( x ) | 1 / 2 T e exp ( i x q x E + d ξ i 2 q z z ) .
FIG. 12.

Dispersion curves illustrating the tertiary decay of the secondary UH wave. The difference in the wavenumbers of the tertiary (running) and secondary UH waves and the wavenumber of IB wave involved in the secondary instability ( f p I = 539.2 MHz) is shown.

FIG. 12.

Dispersion curves illustrating the tertiary decay of the secondary UH wave. The difference in the wavenumbers of the tertiary (running) and secondary UH waves and the wavenumber of IB wave involved in the secondary instability ( f p I = 539.2 MHz) is shown.

Close modal
Solving it, we obtain
C U H = i χ e n l C n B p | e | ψ n y ( y ) * Ψ p y ( y ) 2 T e x d ζ ϕ n x ( ζ ) * Φ p x ( ζ ) | D q x + ( ζ ) | 1 / 2 exp ( i ζ q x E + d ξ i 2 q z z ) .
(22)
To describe the cascade decay scenario shown in Figs. 10–12, we will follow the procedures developed in Refs. 23, 24, and 35. We use the amplitude of the primary extraordinary wave Eq. (19) in the equation describing the primary UH mode and treat it using the perturbation theory procedure.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:
{ a m t + i Λ m z 2 a m z 2 = γ p exp ( z 2 w 2 ) a m + i ω m ξ sec * a n * b p , a n t + u n a n z i Λ n z 2 a n z 2 = i ω n ξ sec * a m * b p ω n ξ ter | b p | 2 a n , b p t + u p b p z + i Λ p z 2 b p z 2 + ν d ( | b p | 2 ) b p = i ω p ξ sec a m a n + ω p ξ ter | a n | 2 b p .
(23)
The nonlinear coupling coefficient for the primary instability has the form γ p = D U H ω | ω m , q m x E ± 1 ω s n 0 x n s x c l d p 2 L m x κ 2 P 0 2 c w 2 B 0 2 | x d p l m y w ξ d ξ d ξ exp ( i ξ 3 i ξ 3 i Δ K p ( x d p ) l d p ( ξ ξ ) ), where l d p = | Δ K p / 6 | x d p 1 / 3 is the length of the decay layer along the x-coordinate, l m y = 2 w y r y r d y L m y ( y ) 1 exp ( 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 Λ j z, j = m,n,s and the group velocities' projections onto the toroidal direction u j, j = n,p, reads A | m , n = d x | φ m , n x ( x ) | 2 d y | ψ m , n y ( y ) | 2 A ( x , y ) and A | p = d x | Φ p x ( x ) | 2 d y | Ψ p y ( y ) | 2 A ( x , y ). Furthermore, ξ sec χ 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; ξ ter | χ 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); ν d = π H ( b p b p t h ) ω p I ( υ p / υ t i ) 3 exp ( υ p 2 / υ t i 2 ) is the coefficient, which describes amplitude-dependent “stochastic” damping of IB waves,43, H ( ) is the Heaviside function, υ p = ω p I / q p x I is the phase velocity, b p t h = B 0 L x I L y I w π 1 / 2 T e ( ω c i ω p I ) 1 / 3 υ 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 / ( π 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 π 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 ε m , n E = ( π w ) 1 | a m , n | 2 exp ( z 2 / w 2 ) d z and the IB wave ε p I = ( π w ) 1 | b p | 2 exp ( 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 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 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.

FIG. 13.

Temporal evolution of the energy densities of excited two-dimensional UH modes m = ( 30 , 1 ), n = ( 40 , 1 ) and two-dimensional IB mode p = ( 14 , 2 ) on a semi-logarithmic scale. The dashed horizontal line shows the IB-wave level above which the ion behavior becomes stochastic.

FIG. 13.

Temporal evolution of the energy densities of excited two-dimensional UH modes m = ( 30 , 1 ), n = ( 40 , 1 ) and two-dimensional IB mode p = ( 14 , 2 ) on a semi-logarithmic scale. The dashed horizontal line shows the IB-wave level above which the ion behavior becomes stochastic.

Close modal
FIG. 14.

Temporal evolution of the total energy of the eigenmodes m (upper dependence, the absolute error of the linear approximation determining the wave power is 0.7 kW) and n (lower dependence, the absolute error of the linear approximation is 1.57 kW) of UH waves.

FIG. 14.

Temporal evolution of the total energy of the eigenmodes m (upper dependence, the absolute error of the linear approximation determining the wave power is 0.7 kW) and n (lower dependence, the absolute error of the linear approximation is 1.57 kW) of UH waves.

Close modal
FIG. 15.

Temporal evolution of the total energy of the running primary X wave (upper dependence, the absolute error of the linear approximation determining the wave power is 2.3 kW) and the running tertiary UH wave (lower dependence, the absolute error of the linear approximation determining the wave power is 0.93 kW) evaluated using Eqs. (19) and (22), correspondingly.

FIG. 15.

Temporal evolution of the total energy of the running primary X wave (upper dependence, the absolute error of the linear approximation determining the wave power is 2.3 kW) and the running tertiary UH wave (lower dependence, the absolute error of the linear approximation determining the wave power is 0.93 kW) evaluated using Eqs. (19) and (22), correspondingly.

Close modal

As can be seen in Fig. 13, the transition time to the saturation regime is about 1 μ 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 = ( π w ) 1 | a m | 2 d z, W n sec = ( π w ) 1 | 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 R 0 8 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 ω s I / ω 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.

Now, we use the representation Eq. (14), the second equation in Eqs. (18) and (19) to determine the power of the daughter extraordinary wave escaping the decay layer and leaving the plasma volume in the equatorial plane on the low-field side of the torus. In the saturation mode, the power fraction gained by this daughter wave is equal to
P s = R s 0 P 0 ,
(24)
where R s 0 = 2 π ω s 2 L x 2 c 2 | κ | 2 ω m E | D U H / ω m E | | a m ( 0 , t ) | 2 T e L x L y L z B 2 is the nonlinear reflection coefficient, which describes the efficiency of power transfer from the pump wave to the daughter extraordinary wave,
L x = d x ϕ m x * ( x ) n 0 x ( x ) n s x ( x ) x d x ϕ m x ( x ) n 0 x ( x ) n s x ( x ) exp ( i x x ( k 0 x + k s x ) d ξ ) , 1 L y = exp ( y 2 w 2 ) | ψ m y ( y ) | 2 d y π w , 1 L z = exp ( z 2 w 2 ) | a m ( z , t ) | 2 | a m ( 0 , t ) | 2 d z π w .
For the conditions under consideration, the daughter wave frequency is equal to f s = 29.65 GHz. According to Eq. (24), its power is about 19.7 kW. For the experimental parameters considered, reducing the density value in the local maximum to 1.3 × 10 13 cm 3 leads to a decay in which the daughter extraordinary wave encounters EC and upper hybrid resonances at the plasma edge, which prevents it from leaving the plasma volume. If, on contrary, the density value in the local maximum rises to a value of 3 × 10 13 cm 3, a cutoff for the extraordinary wave occurs in the pump wave decay region, making the primary instability impossible. Thus, the anomalous emission of this type can be observed in the considered TCV experiment in the local maximum density range from 1.3 × 10 13 cm 3 till 3 × 10 13 cm 3.

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.

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.

The authors have no conflicts to disclose.

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).

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

1.
E.
Westerhof
,
S. K.
Nielsen
,
J. W.
Oosterbeek
,
M.
Salewski
,
M. R.
De Baar
,
W. A.
Bongers
,
A.
Bürger
,
B. A.
Hennen
,
S. B.
Korsholm
,
F.
Leipold
,
D.
Moseev
,
M.
Stejner
, and
D. J.
Thoen
,
Phys. Rev. Lett.
103
,
125001
(
2009
).
2.
S. K.
Nielsen
,
M.
Salewski
,
E.
Westerhof
,
W.
Bongers
,
S. B.
Korsholm
,
F.
Leipold
,
J. W.
Oosterbeek
,
D.
Moseev
,
M.
Stejner
, and
the TEXTOR Team
,
Plasma Phys. Controlled Fusion
55
,
115003
(
2013
).
3.
S. K.
Hansen
,
S. K.
Nielsen
,
J.
Stober
,
J.
Rasmussen
,
M.
Stejner
,
M.
Hoelzl
,
T.
Jensen
, and
the ASDEX Upgrade Team
,
Nucl. Fusion
60
,
106008
(
2020
).
4.
S. K.
Hansen
,
A. S.
Jacobsen
,
M.
Willensdorfer
,
S. K.
Nielsen
,
J.
Stober
,
K.
Höfler
,
M.
Maraschek
,
R.
Fischer
,
M.
Dunne
, and
the ASDEX Upgrade Team
,
Plasma Phys. Controlled Fusion
63
,
095002
(
2021
).
5.
A.
Tancetti
,
S. K.
Nielsen
,
J.
Rasmussen
,
E. Z.
Gusakov
,
A.
Yu. Popov
,
D.
Moseev
,
T.
Stange
,
M. G.
Senstius
,
C.
Killer
,
M.
Vecséi
,
T.
Jensen
,
M.
Zanini
,
I.
Abramovic
,
M.
Stejner
,
G.
Anda
,
D.
Dunai
,
S.
Zoletnik
, and
H. P.
Laqua
, and
the W7-X Team
,
Nucl. Fusion
62
,
074003
(
2022
).
6.
A.
Tancetti
,
S. K.
Nielsen
,
J.
Rasmussen
,
D.
Moseev
, and
T.
Stange
, and
the W7-X Team
,
Plasma Phys. Controlled Fusion
65
,
015001
(
2023
).
7.
A.
Clod
,
M. G.
Senstius
,
A. H.
Nielsen
,
R.
Ragona
,
A. S.
Thrysøe
,
U.
Kumar
,
S.
Coda
, and
S. K.
Nielsen
,
Phys. Rev. Lett.
132
,
135101
(
2024
).
8.
Y.
Dnestrovskij
,
A. V.
Danilov
,
A.
Dnestrovskij
,
S. E.
Lysenko
,
A. V.
Melnikov
,
A. R.
Nemets
,
M. R.
Nurgaliev
,
G. F.
Subbotin
,
N. A.
Solovev
,
D.
Sychugov
, and
S. V.
Cherkasov
,
Plasma Phys. Controlled Fusion
63
,
055012
(
2021
).
9.
Y.
Dnestrovskij
,
A. V.
Melnikov
,
D.
Lopez-Bruna
,
A.
Dnestrovskij
,
S. V.
Cherkasov
,
A. V.
Danilov
,
L. G.
Eliseev
,
P. O.
Khabanov
,
S. E.
Lysenko
, and
D.
Sychugov
,
Plasma Phys. Controlled Fusion
65
,
015011
(
2023
).
10.
A. I.
Meshcheryakov
,
I. Y.
Vafin
, and
I. A.
Grishina
,
Plasma Phys. Rep.
46
,
1144
(
2020
).
11.
E. Z.
Gusakov
,
A.
Yu. Popov
,
A. I.
Meshcheryakov
,
I. A.
Grishina
, and
M. A.
Tereshchenko
,
Phys. Plasmas
30
,
122112
(
2023
).
12.
M. W.
Brookman
,
M. E.
Austin
,
C. C.
Petty
,
R. J.
La Haye
,
K.
Barada
,
T. L.
Rhodes
,
Z.
Yan
,
A.
Keohn
,
M. B.
Thomas
,
J.
Leddy
, and
R. G. L.
Vann
,
Phys. Plasmas
28
,
042507
(
2021
).
13.
J. H.
Slief
,
R. J. R.
van Kampen
,
M. W.
Brookman
,
J.
van Dijk
,
E.
Westerhof
, and
M.
van Berkel
,
Nucl. Fusion
63
,
026029
(
2023
).
14.
C.
Schlatter
, “
Turbulent ion heating in TCV tokamak plasmas
,” Ph.D. thesis (
Ecole Polytechnique Federale de Lausanne (EFPL)
,
Lausanne, Switzerland
,
2009
).
15.
S.
Coda
and
TCV Team
.
Nucl. Fusion
55
,
104004
(
2015
).
16.
B.
Zurro
,
A.
Baciero
,
V.
Tribaldos
,
M.
Liniers
,
A.
Cappa
,
A.
Lopez-Fraguas
,
D.
Jimenez-Rey
,
J. M.
Fontdecaba
, and
O.
Nekhaieva
, and
the TJ-II Team
,
Nucl. Fusion
53
,
083017
(
2013
).
17.
M.
Martínez
,
B.
Zurro
,
A.
Baciero
,
D.
Jiménez-Rey
, and
V.
Tribaldos
,
Plasma Phys. Controlled Fusion
60
,
025024
(
2018
).
18.
A. J. H.
Donne
,
J. C.
van Gorkom
,
V. S.
Udintsev
,
C. W.
Domier
,
A.
Krämer-Flecken
, Jr.
,
N. C.
Luhmann
, and
F. C.
Schüller
,
Phys. Rev. Lett.
94
,
085001
(
2005
).
19.
M.
Yu. Kantor
,
A. J. H.
Donne
,
R.
Jaspers
, and
H.
van der Meiden
, and
TEXTOR Team
,
Plasma Phys. Controlled Fusion
51
,
055002
(
2009
).
20.
H.
Weisen
,
I.
Furno
, and
T.
Goodman
, and
TCV Team
, in
Proceedings of 18th IAEA Fusion Energy Conference IAEA-CN-77/PDP/6
(Sorrento, Italy, 4–10 October 2000).
21.
A.
Yu. Popov
and
E. Z.
Gusakov
,
Plasma Phys. Controlled Fusion
57
,
025022
(
2015
).
22.
A.
Yu. Popov
and
E. Z.
Gusakov
,
Europhys. Lett.
116
,
45002
(
2016
).
23.
E. Z.
Gusakov
and
A.
Yu. Popov
,
Phys. Usp.
63
,
365
(
2020
).
24.
E. Z.
Gusakov
and
A. Y.
Popov
,
Nucl. Fusion
60
,
076018
(
2020
).
25.
M. G.
Senstius
,
E. Z.
Gusakov
,
A. Y.
Popov
, and
S. K.
Nielsen
,
Plasma Phys. Controlled Fusion
64
,
115001
(
2022
).
26.
E. Z.
Gusakov
and
A.
Yu. Popov
,
Phys. Plasmas
23
,
082503
(
2016
).
27.
E. Z.
Gusakov
and
A. Y.
Popov
,
Plasma Phys. Rep.
49
,
949
(
2023
).
28.
E. Z.
Gusakov
and
A. Y.
Popov
,
Plasma Phys. Rep.
49
,
194
208
(
2023
).
29.
E. Z.
Gusakov
,
A. Y.
Popov
, and
P. V.
Tretinnikov
,
Nucl. Fusion
59
,
106040
(
2019
).
30.
J.
Cazabonne
,
S.
Coda
,
J.
Decker
,
O.
Krutkin
,
U.
Kumar
,
Y.
Peysson
, and
the TCV Team
,
Nucl. Fusion
64
,
026019
(
2024
).
31.
H.
Weisen
,
S.
Alberti
,
C.
Angioni
,
K.
Appert
,
J.
Bakos
,
R.
Behn
,
P.
Blanchard
,
P.
Bosshard
,
R.
Chavan
,
S.
Coda
,
I.
Condrea
,
A.
Degeling
,
B. P.
Duval
,
D.
Fasel
,
J.-Y.
Favez
,
A.
Favre
,
I.
Furno
,
P.
Gomez
,
T. P.
Goodman
,
M. A.
Henderson
,
F.
Hofmann
,
R. R.
Kayruthdinov
,
P.
Lavanchy
,
J. B.
Lister
,
X.
Llobet
,
A.
Loarte
,
V. E.
Lukash
,
P.
Gorgerat
,
J.-P.
Hogge
,
P.-F.
Isoz
,
B.
Joye
,
J.-C.
Magnin
,
A.
Manini
,
B.
Marlétaz
,
P.
Marmillod
,
Y.
Martin
,
A.
Martynov
,
J.-M.
Mayor
,
E.
Minardi
,
J.
Mlynar
,
J.-M.
Moret
,
P.
Nikkola
,
P. J.
Paris
,
A.
Perez
,
Y.
Peysson
,
Z. A.
Pietrzyk
,
V.
Piffl
,
R. A.
Pitts
,
A.
Pochelon
,
H.
Reimerdes
,
J. H.
Rommers
,
O.
Sauter
,
E.
Scavino
,
A.
Sushkov
,
G.
Tonetti
,
M. Q.
Tran
, and
A.
Zabolotsky
,
Nucl. Fusion
41
,
1459
(
2001
).
32.
E. Z.
Gusakov
and
A.
Yu. Popov
,
Phys. Rev. Lett.
105
,
115003
(
2010
).
33.
E. Z.
Gusakov
and
A.
Popov
,
Nucl. Fusion
51
,
073028
(
2011
).
34.
A.
Yu. Popov
,
E. Z.
Gusakov
, and
N. V.
Teplova
,
Plasma Phys. Rep.
50
,
35
(
2024
).
35.
E. Z.
Gusakov
and
A.
Yu. Popov
,
Plasma Phys. Controlled Fusion
63
,
125017
(
2021
).
36.
E. Z.
Gusakov
and
A. Yu.
Popov
,
Plasma Phys. Rep.
49
,
936
(
2023
).
37.
S.
Coda
, privet communication (
2012
).
38.
E.
Gusakov
and
A.
Popov
,
Europhys. Lett.
99
,
15001
(
2012
).
39.
E. Z.
Gusakov
,
A.
Yu. Popov
, and
A. N.
Saveliev
,
Plasma Phys. Controlled Fusion
56
,
015010
(
2014
).
40.
V. E.
Golant
and
A. D.
Piliya
,
Sov. Phys. Usp.
14
,
413
(
1972
).
41.
T. H.
Stix
,
Theory of Plasma Waves
(
McGraw-Hill
,
New York
,
1962
).
42.
E. Z.
Gusakov
,
A.
Yu. Popov
, and
P. V.
Tretinnikov
,
Plasma Phys. Controlled Fusion
61
,
085008
(
2019
).
43.
C. F. F.
Karney
,
Phys. Fluids
21
,
1584
(
1978
).