Anomalous absorption in ECRH experiments due to the parametric low-threshold excitation of localized UH waves

We describe the theoretical model which interprets the anomalous phenomena, i.e. the generation of backscattering signal observed in the ECRH experiments at TEXTOR, TCV, TJ-II, ASDEX-UG, LHD and FTU, as a consequence of the excitation of the parametric decay instability (PDI) leading to anomalous damping of the pump wave. The PDI power-threshold is shown to be extremely low due to the localization of both or one daughter upper hybrid (UH) waves in presence of a nonmonotonic (hollow) density profile, which is often observed in the ECRH experiments due to the magnetic island or the density pump-out effect. In the case of the extraordinary wave pump the model predicts substantial (up to 25%) anomalous absorption in the electron channel and explains the anomalous ion acceleration by the generation of secondary low frequency waves which directly transfer the pump power to the ion component. The possibility of anomalous absorption of the O-mode pump in the ECRH experiment due to the parametric excitation of trapped UH wave is also discussed and the anomalous absorption rate at the 10% level is predicted.


Introduction
Electron cyclotron resonance heating (ECRH) and current drive is widely used in toroidal plasmas and is considered for application in ITER for heating and neoclassical tearing mode control. The parametric decay instabilities (PDIs), which can accompany the ECRH experiments, are believed to be deeply suppressed by the huge energy loss of daughter waves from the decay region, according to the predictions of theory developed in 80 th [1]. However, during the last decade numerous ECRH experiments performed at TEXTOR, TCV, TJ-II, ASDEX-UG, LHD, T-10 and FTU [2][3][4][5][6][7][8][9][10][11][12][13] have demonstrated excitation of the anomalous phenomena. The clearest evidence of the nonlinear phenomena excitation was obtained at TEXTOR [2,3] where the strong backscattering signals down-shifted in frequency and amplitude modulated at the frequency of magnetic island were observed. A convincing demonstration of the anomalous ion acceleration during the ECRH pulse under conditions when the energy exchange between the ion and electron components is negligible was obtained at TCV [4,5].
In the present paper we describe the theoretical model taking into account, as distinct from the conventional theory [1], the presence of a nonmonotonic density profile, which always exists on the discharge axis or may be present due to the magnetic island or the density pump-out effect. We interpret the generation of backscattering signal and the anomalous ion heating, as a result of secondary nonlinear processes that accompany a primary low -threshold two-upperhybrid (UH) -plasmon PDI of the pump X2-mode wave.
The threshold of primary PDI is shown [14,15] to be smaller than the one predicted in [1] due to the trapping of at least one UH wave in presence of the nonmonotonic density profile. The primary PDI appears to be absolute due to the finite-size of a pump beam. Its growth enhancing the UH wave fluctuations from the thermal noise level is saturated in our theory due to both the secondary decays of the daughter UH wave that leads to excitation of the secondary UH and ion Bernstein (IB) waves [16] and the pump wave depletion. The threshold of this spontaneous parametric frequency downconversion can be easily overcome for the radially trapped secondary UH waves. The couplings of different daughter UH waves are responsible in the theory for the generation of backscattering signals [17]. This mechanism appears capable of reproducing the fine details of the frequency spectrum of anomalously backscattered X wave and the absolute value of the observed backscattering signal in TEXTOR experiment. It also predicts substantial (up to 25%) anomalous absorption in the electron channel and explains the anomalous ion heating at TCV by the generation of secondary IB waves, which directly transfer the pump power to the ion component.
The results of a model experiment, demonstrating strong anomalous absorption of the X-mode pump in a plasma filament due to the two-plasmon decay [18] and thus confirming the theory, are discussed.
The possibility of anomalous absorption of the Omode pump in the ECRH experiment due to parametric excitation of the trapped UH wave [19,20] is discussed as well.

Model
We assume that the three-wave nonlinear interaction leading to the low-threshold PDI excitation occurs in a small volume in the vicinity of a local maximum of plasma density profile illuminated by the pump beam where the daughter UH waves/wave are localized. Therefore, it seems to be reasonable to utilize throughout the paper, describing different PDIs, the Cartesian coordinate system ( , , ) x y z with its origin located at the local maximum of the UH frequency profile, the x-axis directed along the flux coordinate and (y,z) imitating accordingly the coordinates perpendicular to and aligned with the magnetic field line on the magnetic surface.

Two-UH-plasmon parametric decay of an extraordinary wave
We start with consideration of the low-power threshold two-UH-plasmon decay of a beam of the extraordinary (X) waves propagating perpendicular to the magnetic surface along the inhomogeneity direction and its saturation owing to the pump depletion and the cascade of secondary decays.
where 1,2 ϕ are the potentials of primary UH waves, 1,2 ϕ′ are the potentials of secondary UH and ion Bernstein (IB) waves and 2 , p s κ are the coupling coefficients for both decays presented explicitly in [21]. In weakly inhomogeneous plasma the integral operators D are taking the following form:  χ standing for the familiar expressions of the  electron and ion susceptibilities   2  2 2  2 2  pj  cj  tj  tj  j  2  2  2  tj  tj  tj  cj  cj   2  1  exp  2 In (4) is the eigenfunction, n C const = , nx q ± is a root of the local UH wave dispersion relation, n L ± stands for the optical length of the wave path between two turning points Equation (6) describes the UH wave localized in the xdirection at the maximum of the UH frequency profile.
In the second step of the perturbation procedure we take into consideration the nonlinear interactions and the pump energy loss due to the UH plasmons excitation. We utilize the envelope amplitude method and keep in mind the boundary condition for the pump wave ( ) are the coupling coefficients [21] describing the primary ( 0 ν ) and secondary ( s ν ) decays, the pump wave depletion ( d ν ) and Further, we analyze (9) analytically and then solve numerically. We start with consideration of the growthrates of primary and secondary decays and get an estimation of the daughter wave saturation levels based on the energy-balance formulation. For the conditions under consideration the convective energy losses of the primary and secondary UH plasmons from the decay layer in the poloidal direction are much smaller than their energy losses in a form of diffraction in the toroidal direction, i.e. , , with ζ being the unstable eigenmode number. As soon as the energy of primary plasmons growing exponentially reaches a level which exceeds the threshold of secondary decay instability, the energy of secondary plasmons has to start growing as well. Eventually, the nonlinear system relaxes to the stationary regime in which the saturation level of primary plasmons number 2 | | s n a can be evaluated from a condition indicating saturation of the secondary decay described by the last equation in (9). Namely, the growth rate 2 1 | | s a ν of this decay should be compensated by the secondary UH plasmon diffractive loss rate from the pump beam spot, characterized by a typical time scale 2 / w τ π = Λ . The corresponding level is given by The number of the primary plasmons | | s m a in the saturation regime could be estimated using the second equation of (9) as The energy of secondary plasmons . This saturation level can be estimated as Then, we solve (9) numerically in a 2D box 2 2 B B y z × assuming the periodic boundary conditions and an initial thermal-noise-level of the UH waves. The imposed boundary conditions imitate the toroidal geometry and describe the multiple transitions of UH plasmons through the 2D box. However, it is worth noting that equations (9) were derived ignoring the magnetic field poloidal dependence. Thereby, the meaningful solution of (9) is an intermediate asymptotic dependence valid at the time much less than the transition time of the daughter waves through the 2D box. We assume the initial thermal noise level (12) of a pump wave is presented. In the early stage of PDI one can see an exponential growth of the energy of primary plasmons with the growth rate being adequately described by the analytical equation (13) (short dotted line). When the energy of primary plasmons is large enough so that the threshold of the secondary decay instability appears to be overcome, the energy of secondary plasmons starts to grow as well. Eventually, the energies of primary and secondary plasmons achieve their saturation levels. It is seen that the saturation levels of the daughter plasmons derived analytically (equation (14) -lower thin dashed horizontal line, equations (15) and (16)  . Therefore, we can conclude that the anomalously absorbed power should be proportional to the pump power and thus the anomalous absorption rate 0 / P P ∆ at large threshold excess should be constant. This conclusion is confirmed by the results of numerical solution which predicts at the pump power much larger than the PDI power threshold, 0 5 kW th P = , the anomalous absorption of X -mode wave at a level of 18%, independently on the pump power. Thus, for the parameters under consideration up to 110 kW are transferred to the daughter UH waves. Subsequently, this power fraction is deposited into the electron component far from the pump wave second harmonic electron cyclotron resonance layer. This can make the power deposition profile much wider than it is predicted by the linear theory and can create an illusion of the fast electron energy redistribution. In other words, this nonlinear absorption mechanism can be responsible, at least partly, for the so-called nonlocal electron transport in the on-axis X2-mode ECRH experiments at T-10 and LHD [12,13]. We can also estimate the pump power fraction absorbed by the daughter IB waves as  The nonlinear coupling of different UH waves can lead to the generation of microwave signals with the frequencies smaller than the pump frequency. In this way the proposed nonlinear mechanism explains in detail the anomalous backscattering effect observed at TEXTOR [2]. In particular, the frequency spectrum structure and the spectral power density of the backscattering signal predicted by this theoretical model appear to be in a reasonable agreement with those measured by the ECE and CTS techniques [2,3].
In order to examine the mechanism of anomalous absorption described in this section the model experiment on the linear device "Granite" in Minsk was proposed and then performed [18]. The huge anomalous absorption of microwave power was observed in the plasma column when the density was close to the UHR value for half the pump frequency which validates the main theoretical predictions concerning the possibility of the low-power-threshold two-UH-plasmon PDI of extraordinary wave and its role in the energy balance.
Summarizing, we would like to emphasize that there are two key points of the model developed. The first point is the trapping of the UH waves at the local maximum of a nonmonotonic density profile. The second one of them is the possibility of an additional localization of the UH waves propagating in the opposite directions on the magnetic surface within the spot of a finite-width pump beam. In the forthcoming section we consider the possibility of low-power-threshold ordinary wave decay into a low hybrid wave (LH) and a radially trapped UH wave. We demonstrate that even one radially trapped UH daughter wave can be nonlinearly localized on the magnetic surface due to the finite-width pump, that leads to excitation of the most dangerous absolute PDI.

Low-threshold parametric decay of an ordinary wave
In this section we analyse the low-threshold PDI of an obliquely propagating ordinary wave. The parametric decay leads to the excitation of the obliquely propagating LH wave and the UH wave. The latter appears to be 3D localized due to the nonmonotonic density profile and the finite-width beam. We analyze also the transition of primary PDI to the saturation regime due to the secondary decay instability of primary UH wave. We focus on the case of the one-step down-conversion leading to the low-threshold excitation of the IB and radially trapped UH waves. We derive a set of equations describing the cascade of decays and then solve it numerically under the conditions similar to the on-axis O1-mode ECRH experiments at FTU [6] in which the nonmonotonic density profile arises due to the density pump-out effect from the ECR layer.
The pump ordinary wave can be represented by means of the WKB approximation as follows   [20] and [24], correspondingly. Assuming the nonlinear coupling is weak we treat the set of equations (18) using the perturbation procedure [23]. In its first step we neglect the nonlinear pump. The solutions of homogeneous equations describing the trapped UH waves can be presented in the following form   (6) and (7). The choice 1y, 0 z q = , 1y 0 q′ = allows us to minimize the energy losses of UH plasmons from the decay region. Then, we treat equations in (18) (20) where the amplitudes I I In the second step of the perturbation procedure we take into account the nonlinear wave coupling. The nonlinear three-wave interactions lead to the variation of amplitudes ( ) where I , Substituting (25) into the right-hand-sides of (22), (24) and introducing the dimensionless functions ( )   .
Zoom gives the comparison of the function derived numerically with the analytical dependence (30).
The PDI power-threshold 0 th P corresponding to the imaginary part of the eigen frequency being equal to zero is determined by the balance between the diffractive loss of UH waves ( holds and the power threshold of secondary PDI appears to be overcome. Then, the UH wave n b starts not only to grow due to the primary instability but at the same time to transfer a part of pump power to the secondary wave. When the secondary PDI power threshold is exceeded significantly, the temporal gain coefficient can be estimated as follows where the averaging procedure over the spot in which the decay occurs is defined as in equation (11). If the PDI saturation results in a stationary regime, the saturation levels of daughter waves can be estimated assuming the nonlinear pumping in the primary decay is balanced by the energy losses due to the secondary decay Then, we can determine the fraction of anomalous power absorbed by the daughter decay waves as follows In (38) we have assumed that at the dominant losses of primary and secondary waves from the pump spot are along the magnetic field. It is worth noting that the pump power absorbed by secondary plasmons (the second term in (38)) should be proportional to the pump power, whereas the power fraction absorbed by primary plasmons is not dependent on it. Therefore well above the PDI threshold 0 In this regime the primary daughter wave retains its spatial distribution invariable (see figure 3). As it is shown in a zoom in figure 3 it is well described by the analytical dependence (30). When the secondary PDI power threshold is exceeded the energy of secondary wave (dashed curve) starts to grow. The growth is reasonably described by the analytical dependence (35). Then, we can see the temporal relaxation of the strong nonlinear phenomenon and the transition of the instability into the saturation regime. The saturation levels of the daughter waves are quite well described by the analytical estimations (36) and (37) . This anomalous absorption can lead to the generation of a population of the suprathermal ions. In figure 4 the anomalous absorption rate derived numerically (scattered symbols) and predicted by the asymptotic equation (38) valid at 0 0 th P P >> , th 0 125kW P = are depicted. As it is seen, the model developed in this paper predicts at the high pump power anomalous absorption rate of 8.4 %. This value is lower than in the case of the X-mode decay producing UH plasmons analysed in [21] due to the specific direction of the pump electric field (almost parallel to the magnetic field). The low value of the anomalous absorption rate provides an argument in favour of omitting the pump depletion in our analysis.

Conclusions
We have described the theoretical model, which interprets the anomalous phenomena experimentally observed in the X2-mode and O1-mode ECRH experiments, such as the generation of backscattering signals and the fast ion generation, as a consequence of the low-threshold PDI of a pump wave leading to generation of localized UH daughter waves in presence of a nonmonotonic (hollow) density profile. The natural consequence of this model is also the absolute instability excitation in presence of a finite-width pump beam and its saturation at a substantially high level due to the secondary decay phenomena and the pump depletion. The developed model also results in the anomalous pump wave absorption at a level of 10% or higher.
We have investigated the X2-mode wave decay into two 3D localized UH waves and analysed the transition of primary instability into the saturation regime due to both the pump wave depletion and the secondary decays of daughter UH wave, leading to the excitation of secondary IB/LH and UH waves. The threshold of this spontaneous parametric frequency down-conversion is shown to be easily overcome for the secondary UH waves trapped at the local maximum of a density profile. The model predicts substantial RF power transfer to the UH waves (up to 25%). Subsequently, the UH waves power is deposited into the electron component far from the pump wave electron cyclotron resonance layer. This makes the power deposition profile much wider than it is predicted for the pump wave itself. The anomalous absorption of the pump power by the IB or LH waves leads to the generation of a population of the suprathermal ions, which can explain the experimental observations of the ion acceleration in the X2-mode ECRH experiments at TCV and TJ-II [7][8][9][10].
Nonlinear coupling of different UH waves can lead to the generation of microwave signals at frequencies smaller than the pump frequency. Thus, the nonlinear mechanism allows explaining the anomalous backscattering effect observed at TEXTOR [2] in detail. In particular, the frequency spectrum structure and the spectral power density of the backscattering signal predicted by this theoretical model appear to be in a reasonable agreement with those measured by the ECE and CTS techniques [2,3].
We have also investigated the pump O-mode wave anomalous absorption due to the parametric excitation of trapped UH waves. The instability power threshold ( th 0 125 kW P = ) can be exceeded by the MW microwave beams utilized usually for auxiliary heating in the O1mode ECRH experiments. However, its growth-rate is much smaller than the growth-rate of X-mode PDI. This is due to the weak coupling of the ordinary wave to the electrostatic waves. Nevertheless, the derived growthrate of O-mode PDI is large enough allowing this nonlinear phenomenon to play an important role in the ECRH power balance. In order to estimate the fraction of pump power absorbed anomalously we have analyzed the saturation regime of primary instabilty. We assumed the UH wave secondary decay into the localized UH waves and IB waves to be a presumably most important nonlinear mechanism responsible for the transition to saturation regime. We have estimated the power fraction gained by the daughter UH waves in the O1-mode ECRH experiments at FTU, in which the 600 kW pump beams are utilized. Numerical solution gives the power fraction E 52kW P ∆ ≈ absorbed via the UH daughter waves. This power fraction is deposited into the electron component far from the pump wave absorption region predicted by linear theory.
It is worth noting that the powerful ECRH utilizing ordinary waves is planned for application in ITER. It should allow the neoclassical tearing mode control leading to reduction of its influence on the confinement in this thermonuclear reactor. According to the analyses performed in this paper, if the nonmonotonic density profile will arise in ITER at the edge of the discharge, the low-threshold parametric decay of the pump wave to the 3D localized UH wave could be possible playing a role in the ECRH power balance. This stimulates further analysis of the O1-mode parametric decay under the conditions which will be typical for ITER.
The financial support of the Ioffe Institute is acknowledged.