Spectral broadening of lower hybrid waves produced by parametric instability in current drive experiments of tokamak plasmas

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION Nucl. Fusion 46 (2006) doi: / /46/4/007 Spectral broadening of lower hybrid waves produced by parametric instability in current drive experiments of tokamak plasmas R. Cesario 1, A. Cardinali 1, C. Castaldo 1, F. Paoletti 2, W. Fundamenski 3, S. Hacquin 4 and the JET-EFDA workprogramme contributors 5 1 Associazione EURATOM/ENEA sulla Fusione, Centro Ricerche Frascati c.p. 65, Frascati, Italy 2 East Windsor Regional School District, Hightstown, NJ 08520, USA 3 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK 4 Associação EURATOM/IST Centro de Fusão Nuclear, Lisbon, Portugal Received 16 December 2004, accepted for publication 30 December 2005 Published 16 March 2006 Online at stacks.iop.org/nf/46/462 Abstract In order to explain the results of the non-inductive current produced in the lower hybrid current drive (LHCD) experiments, a broadening of the radiofrequency (RF) power spectrum coupled to tokamak plasma needs to occur. The presented modelling, supported by diagnostic measurements, shows that the parametric instability (PI) driven by ion sound quasimodes, which occur in the scrape-off plasma layer located near the antenna mouth, produces a significant broadening of the launched LH spectrum. Considering the parameters of LHCD experiments of JET (Joint European Torus), and other machines as well, the PI growth rate is high enough for producing the compensation of the convective losses and, consequently, the broadening of a small fraction (of the order of 10%) of the launched power spectrum. Such a phenomenon is identified to be intrinsic to the RF power coupling in the LHCD experiments. As the principal implication of considering such spectral broadening in modelling the LH deposition profile, experiments of LHCD-sustained internal transport barriers in JET were successfully interpreted, which evidenced the effects of a well-defined LH deposition profile. The present work is important for addressing the long-lasting debate on the problem of the so-called spectral gap in LHCD. The design of LHCD scenarios relevant to the modern fusion research programme, an important requirement of which is the control of the plasma current profile in the outer half of plasma, can be properly achieved by considering PI-induced spectral broadening. PACS numbers: Hr (Some figures in this article are in colour only in the electronic version) 1. Introduction The lower hybrid current drive (LHCD) [1 3] can provide a key tool for controlling the current profile in tokamak experiments aimed at achieving important goals relevant to the modern plasma research on fusion plasmas, such as steadystate operation, improved confinement and quiescent MHD activity. In order to build such a tool, a deep understanding 5 See the appendix of J. Pamela et al 2004 Proc. 20th Int. Conf. on Fusion Energy 2004 (Vilamoura, Portugal, 2004) (Vienna: IAEA). of the physical mechanisms that determine the LH deposition profile in realistic operating conditions is necessary. Since the LHCD effect [1 5] is based on the quasilinear wave interaction with the tail of the electron distribution function of the plasma, the assessment of the LH power n spectra that effectively propagate in the experiments is a crucial issue [6]. Here, n is the refractive index component in a direction parallel to the confinement magnetic field. In this regard, a longlasting debate is still open on the so-called spectral gap in LHCD, i.e. on the causes that determine the broadening of the launched n spectrum, which is a necessary requirement /06/ $ IAEA, Vienna Printed in the UK 462

2 Lower hybrid waves to explain the available experimental data by means of the quasilinear theory [4 7]. Several mechanisms were proposed for filling the spectral gap in LHCD [4]. However, only the n upshift due to the effect of the toroidicity in the wave propagation [7] was widely considered in LHCD modelling. The LH deposition profile while retaining this effect was initially modelled in [8]. In a recent assessment, shown in [9], the model consists of two steps of calculations: the first for determining the wave domain by constraints for the slow wave propagation in tokamaks [10]; the second for performing assumptions on the quasilinear diffusion coefficient [11]. We consider such a approach, which we refer to as the standard approach, to be inadequate to model the LH deposition profile, for the following reasons. (i) The WKB approximation, which is the basic assumption for evaluating the LH spectral broadening produced by the effect of toroidicity, fails in the cut-off layers that are met by the rays near the plasma edge. At these layers, the LH waves are considered to be behaving like optic waves although they have much longer wavelengths [5]. Therefore, reasonable conclusions might not be drawn about the LH spectral broadening in the presence of many such reflections (especially when these reflections involve rays carrying a relatively high level of RF power density). (ii) The LH deposition profile is not evaluated taking into account consistently the effect of the quasilinear interaction in the wave propagation and of collisions as well. This means that the effect of tail of the LH propagating spectrum, which might be produced by toroidicity, or other effects, on the deposition of the main power spectrum might not be properly considered [6]. Moreover, several circumstances did not support the standard model. (a) By full wave analysis, a linear modelling of wave propagation and deposition can be provided, which is not affected by the lack of validity of the geometric optics at the LH cut-off layers. Considering the effect of the toroidicity consistently with a full-wave analysis, the LH deposition produced by the toroidicity resulted in being too small and not sufficient to interpret the LHCD experiments, without invoking the occurrence of some spectral broadening (possibly induced by non-linear effects at the plasma edge) [12]. (b) When utilizing the standard approach, it was difficult to interpret certain trends in the LH deposition with respect to some operating parameters observed in LHCD experiments [13, 14]. The spectral broadening is obtained by cumulative effects of the ray trajectories in the toroidal geometry, which in general involve the whole plasma column. As a consequence, the broadening results in being strongly sensitive to details of the complex magnetic structure of the plasma, and to the minor changes, caused by experimental uncertainties, of the kinetic profiles utilized in the modelling. Therefore, assuming that toroidicity in multi-passes were the only cause of the spectral broadening, the LH deposition profile should not appear as a well-defined feature of an LHCD experiment and, thus, a robust tool for controlling the current profile should not be provided by LHCD. Conversely, there are clear indications that the LHCD produces a welldefined LH-deposition profile, and reproducible effects, with well-defined trends with respect to the operating conditions, as occurred in the LHCD-sustained internal transport barriers (ITBs) produced by JET [14 17]. In these experiments, the peak of the LH-driven current density profile results localized in the outer half of plasma, consistently with the observed ITB features [14]. (c) By considering only the toroidicity effect in multi-passes, as shown in [14], the precision of the modelled LHCD profile turned out to be insufficient to find consistency with both the measured current profile and the observed features of the LHCD-sustained ITBs of JET [14 17]. The present paper considers important for bridging the gap in LHCD the effect of parametric instability (PI), which is a non-linear wave interaction producing mode coupling of the LH wave with the background of the density fluctuations of the edge. A lot of literature is available on the PI produced by LH power coupled to tokamak plasmas [18 28]. By retaining such an effect of LH spectral broadening at the plasma edge, the modelling of the LH deposition profile is not affected by the aforementioned limits of the standard models. This new approach was utilized for the first time in [14] to model the LH deposition in the aforementioned LHCD-sustained ITB experiments of JET. Although the subject of wave propagation and damping is treated in a dedicated paper [29], it is useful to clarify the main differences with respect to the standard approach utilized for LHCD modelling. The ray tracing and Fokker Planck analyses were performed in parallel and consistently at each step of the calculation for each ray [30]. This way, the cooperating effects of PI-induced spectral broadening in the SOL and toroidicity in the quasilinear interaction can be properly evidenced. The evolution of the LH power distribution in the n spectrum consistently with the quasilinear effects is accurately calculated. This method provides results that are consistent with the general quasilinear analysis [6], for which a small fraction (of the order of 10%) of the LH wave spectrum can produce the full absorption of the coupled power in LHCD experiments. The long-lasting ITBs of JET were considered as plasma targets useful for properly testing the different LHCD model approaches (which retain, or not, the LH physics of the edge) [14]. This plasma has indeed a high electron temperature in the core and broad profiles (T e0 8 kev and about 3 kev at two thirds of the minor radius), and is supported by measurements of plasma current profiles. Results consistent with the measurements and with the ITB features were obtained only by retaining the effect of the physics of the edge. The same plasma parameters have been considered in the present paper for performing the analysis of the spectral broadening induced by PIs. It is useful to recall the evidence of occurrence of PIs observed for the first time during LH experiments on tokamaks, as these experiments evidenced also the link between the physics of the edge and the LH power propagation and damping in the plasma [25 27, 31, 32]. The behaviour of the PIs revealed a strong dependence on the operating plasma density, which is the key parameter for producing a certain experimental scenario. As a primary condition for LH power coupling, the density near the antenna should be above the cut-off of the slow electrostatic plasma wave (ω pe ω 0, 463

3 R. Cesario et al Figure 1. Typical frequency spectra of the RF probes obtained during the experiment aimed at heating the plasma bulk ions on the FT tokamak. (a) Broadband spectrum exhibiting the ion cyclotron sidebands (span filter bandwidth: 20 MHz) and (b) pump broadening (span filter bandwidth: 5 MHz). ω 0 is the operating frequency). Further conditions on the operating density are required for producing the ion heating or the current drive scenarios. In the former case, a high density is required in order to locate the cold LH resonant layer (ω LH ω 0 ) in the core. In the latter, the LH resonant layer is located outside the machine by means of relatively low operating densities (ω pe ω 0 ). The PI signatures represented the only effect of the LH power coupled in the early LH experiments (which operated at high densities, ω LH ω 0 ), and consisted of the frequency spectra of the signal detected by RF (radio-frequency) probes located outside the machines. These spectra showed an enormous broadening of the operating line frequency (orders of magnitude higher than the line width of the RF power sources) accompanied by many ion cyclotron sidebands, i.e. sidebands shifted by harmonics of the ion cyclotron frequency of the plasma edge, which exhibit a typical non-monotonic envelope [23, 25, 26, 31]. An example of such spectra observed during the LH experiments on FT (Frascati Tokamak) [31] is shown in figure 1. The broadband frequency spectrum, figure 1(a) shows the ion cyclotron sidebands (with maximum of the sidebands located at the tenth harmonic). The spectrum around the operating line frequency is shown in figure 1(b). Such spectra were interpreted in terms of sidebands of PIs in which the launched LH power acts as the pump, and low frequency modes in the ion sound and ion cyclotron frequency ranges drive the instabilities [26]. In summary, the RF probe spectra are the signature of a cascade of ion sound into ion cyclotron quasimode driven PIs, as the level and the shape envelop of the ion cyclotron sidebands could be produced only provided that, at the plasma edge, an LH pump with an n spectrum broader than that launched by the antenna is available. Such pump broadening, possibly produced by ion sound quasimode-driven PIs, is responsible for the behaviour of the cyclotron sidebands and of the deposition at the very periphery. Such a phenomenology of PIs shows a strong analogy with the spectral gap problem in LHCD. All the LH experiments treated the problem of poor access of the coupled LH power in dense plasmas caused by strong PI activity at the edge. It was overcome by utilizing higher operating frequencies (so that ω pe ω 0 at the edge, and ω LH ω 0 in the whole plasma column). In this condition a pure interaction with the plasma electrons was met, useful for driving current. The highest frequency was utilized for the first time on FT, which increased the operating frequency from 2.45 to 8 GHz [33]. In the electron regime, the cyclotron sidebands of the RF probe spectra disappeared, though a significant pump broadening was still present. Furthermore, weaker LH effects on the core electrons occurred during experiments in which a more pronounced pump broadening was observed, attributable to some higher plasma density in the SOL [34]. A residual pump broadening was observed in the LHCD regime [25, 27, 34 36]. The spectral broadening turned out to be non-linearly dependent on the coupled LH power and the operating plasma density [27]. Incidentally, the early LH experiments operating at high densities identified a regime in which fast ions were observed at the plasma periphery. Also this phenomenon can be explained in terms of spectral broadening produced by PI, which makes possible the quasilinear LH power damping on the plasma particles of the periphery [37]. In summary, as the main hypothesis of the present paper, the PI broadens at the plasma edge the LH spectrum coupled by the antenna. We then establish the link between the LH physics at the edge and that of the bulk as being an important mechanism leading, in cooperation with the toroidicity effect, to the determination of the LH deposition profile in any LHCD experimental regime, by taking consistently into account the quasilinear effects in wave propagation and absorption [6, 14, 29]. We support such a hypothesis with modelling and experimental results. The PI model, widely utilized for the early LH experiments [19, 22, 26], has been now extended to the LHCD operating regimes, by taking into account realistic scrape-off plasma geometry. In addition, the plasma inhomogeneity effect has been retained, considering the convective nature of the instability. In the scrape-off plasma of an LH experiment (ω pe ω 0 ), the spectral broadening for n n peak 1.5 consists of the depletion of a small fraction of the pump power (roughly of the order of 10%, in the LHCD experiments operating at low density). The mode coupling is driven by ion sound quasimodes [19, 37], which are modes of plasma that exist only in the presence of a pump wave with a finite amplitude. The relevant quasimodes 464

4 Lower hybrid waves have frequencies (in the range of 0.1 MHz a few megahertz) consistent with the results of edge diagnostics performed in LHCD experiments. For the routinely coupled LH power densities ( 10 MW m 2 ), the growth rate is sufficiently high for balancing the convective loss due to the finite extent of the pump and plasma inhomogeneity. As a result, the PI amplification is consistent with the aforementioned PI phenomenology observed at the different regimes of operating plasma densities. Very high densities (ω LH ω 0 ) enhance the activity of PIs driven by both ion sound and ion cyclotron quasimodes; the relatively low densities of the LHCD experiments (ω pe ω 0, ω LH ω 0 ) reduce the PI activity, but that of the ion-sounddriven PIs producing spectral broadening remains significant. Consequently, the related spectral broadening should affect the LH absorption along the rays of propagation, as was shown in [14]. This could be true for ion absorption at high operating densities, and for electron absorption at low operating densities as well. The paper is organized as follows. In section 2, the phenomenology of the pump broadening observed on JET by RF probes and microwave reflectometry of the plasma edge is summarized. In section 3 the PI modelling is shown by considering the parameters of a discharge of JET with LHCD-sustained ITB [14, 15]. The PI analysis is performed by (i) considering the homogeneous plasma limit for identifying frequencies and growth rate of the coupled modes; (ii) performing an original calculation of the convective losses relevant to LH-induced PI driven by quasimodes and (iii) calculating the spatial amplification of the instability, the pump depletion and, finally, the spectral broadening. In section 4 the operating conditions necessary for producing the spectral broadening in LHCD experiments are identified and discussed. Section 5 is dedicated to comments and conclusions. 2. Pump broadening and scrape-off layer parameters in the lower hybrid current drive experiment of JET In this section a summary is given of the measurements of spectral broadening performed in LHCD experiments of JET by means of RF probes and by microwave reflectometry of the plasma edge. Similar measurements were performed on ASDEX, which found that the pump broadening exhibits a non-linear behaviour with the operating plasma density of the SOL and with the level of the coupled LH power [27]. Measurements of spectral broadening in the LHCD regime were also performed by RF probes on Alcator C [25], FT [31, 34] and JET [36]. The pump broadening turned out to be dependent mainly on the operating plasma density (for a given operating frequency). No significant dependence was found, however, on the launched n antenna spectrum. The spectral broadening of both in the LH frequency range (RF probes) and in that of the low frequency plasma density fluctuations of the edge (microwave reflectometry) exhibited a similar non-linear behaviour, with different thresholds in coupled LH power at different operating plasma densities [27]. This result shows that a mutual energy transfer occurs, as typical of the mode coupling, during LHCD operations Figure 2. Typical spectral broadening from a RF probe located inside the main vessel during an LHCD experiment of JET (span filter bandwidth: 0.2 MHz). Shot at t = s, coupled power P LH 5MW,n e m 3, B T 3T,I P 3 MA, n peak = The upper horizontal line is positioned roughly for excluding the line width (of the order of a few tens of kilohertz) intrinsic to the RF power source. from plasma waves at high and low frequencies. The broadening of the RF probe spectra is strongly enhanced when operating at lower plasma currents, for a fixed density, which generally produce higher plasma densities and lower electron temperatures in the SOL [38, 39]. Such a result (and those further described below) indicates that pump broadening and sideband level depend on the plasma density and temperature occurring in the scrape-off layer. In JET, the RF probe is located inside the main vessel for collecting radiofrequency power around the operating frequency (3.7 GHz). The pump broadening is shown in figure 2. About 0.2 MHz (measured 10 db below the power peak, by excluding the line width intrinsic to the RF sources, roughly a few tens kilohertz) was observed during typical LHCD experiments (operating with n e m 3, B T 3T, I P 3 MA). Accordingly with the typical trend of pump broadening observed in different LHCD experiments [25, 27, 34 36], the pump broadening of JET increased up to about 1 MHz when the operating plasma density in the bulk (and, correspondently, in the SOL) was a factor of two higher than in the case of figure 2. The present paper considers as reference experiments the long-lasting ITBs of JET sustained by LHCD [15, 16], which utilized 2 MW of LHCD power in combination with the main heating power performed by neutral beam injection (NBI) and ion cyclotron resonant heating (ICRH). The main plasma parameters are summarized in table 1. Note from figure 3 that the electron temperature ramps to about 1 kev at 2 cm inside the LCMS. The profiles of the plasma density and the electron temperatures measured by Langmuir probes and edge spectroscopy during the experiments are shown in figure 3. Within the LCMS antenna distance ( 5 cm) plasma density and electron temperature range from n e m 3, T e 10 ev (at the radial position of the antenna plasma interface) to n e m 3, T e 30 ev (near the LCMS), respectively. During the experiment, the LHCD is switched on at t = 5.7 s. It is worth noting that no significant 465

5 R. Cesario et al Table 1. Plasma parameters of the JET shot at a time point t = 6.5 s (during the main heating phase) which are considered for performing the model analysis. B T = 3.4T,I P = 2.3 MA, n e m 3 ;LHpowerP LH = 2 MW, frequency f 0 = 3.7 GHz, the n spectrum has a peak at n 0 = 1.84 and a width of Antenna dimensions: 32 columns, 12 rows of waveguides with internal width: m, external width: m, internal height: m, external height: m, radiating area of m 2, antenna width: a z = m, a y = m RF, power density: p LH 10 MW m 2. Toroidal magnetic field in the SOL: B T 2.6T. Parameters of the SOL, in the radial layers from the LCMS (last closed magnetic surface) to the antenna plasma-interface (see also figure 3): LCMS antenna distance 5 cm; density and temperatures in the range (from Langmuir probes), n eantenna = m 3, n elcms = m 3, T eantenna = 5 10 ev, T elcms = ev, electron ion temperature ratio: T e /T i = 1 ± 0.5. Note from figure 3 that the electron temperature ramps to values 1 kev for radial location of the layers 2 cm inside the LCMS. Figure 3. Profiles of the plasma density and of the electron temperature of the scrape-off layer utilized for the analysis. The reference LHCD experiment of JET of table 1 is considered (shot during the LHCD phase, at t = 6.4 s). The star symbols refer to Langmuir probes (for density and electron temperature), the circles refer to the density and the squares to the electron temperature measured with the spectroscopy of the edge. The dashed and the dotted-dashed curves represent, respectively, the n e and T e profiles assumed for performing the PI analysis. change occurs in the scrape-off plasma parameters as an effect of the coupling of the NBI and ICRH powers, as these powers interact mainly with the plasma core. The measurements of microwave reflectometry of the edge of JET utilized the upgraded X-mode reflectometry [40]. One of the reflectometer systems works in the frequency range GHz, which is generally suitable for probing the plasma edge region. A proper detection allows getting the complex signal a(t) exp[iω(t)] where a(t) and ω(t) are the amplitude and phase of the reflected signal, respectively. The spectrum of the complex reflectometry signal is sensitive mainly to the turbulence (and then the density fluctuation spectrum) characteristics in the cut-off layer region [41]. For the considered plasma discharge performed with a magnetic field of 3.2 T, the cut-off layer for the probing wave at 76 GHz is located in the outer region of the plasma (around m, which corresponds to a flux radial coordinate ρ 0.98). A fast-acquisition system is used to acquire the reflectometry signal from t = 5.0 s up to 6.5 s with a sampling frequency of Figure 4. Comparison of the spectra of the reflectometry signal before ( s) and during ( s) the LHCD phase of an experiment of JET utilizing 2 MW of LH power in combination with the main heating phase (neutral beam injection and ion cyclotron resonant heating, shot 62515). A similar broadening of the density fluctuation spectrum is observed operating in different conditions of plasmas utilizing LHCD. 1 MHz. A comparison of the spectra of the reflectometry signal before ( s) and during ( s) the LHCD phase is depicted in figure 4. An enhancement of the plasma density fluctuation spectrum (of about 25% for f > khz) is produced during the phase of injection of LHCD power, in the plasma edge up to the maximum detectable limit value (0.5 MHz). As the noise of the reflectometer is identical before and after the LHCD switch-on, the change in the signal spectrum is not affected by the instrument noise but is really due to an increase in plasma turbulence. The broadening of the density fluctuation spectrum detected on ASDEX [27] was more pronounced. In this regard, it should be noted that the reflectometer was in this case located close to the LH antenna, in the same port; instead, in JET the distance of several toroidal ports separates this diagnostic from the LH antenna. Since the PI effect is expected to occur mainly in the plasma close to the antenna mouth (see the next section), it is reasonable to expect that the effect on the density fluctuations becomes weaker far away from the LH antenna. Although the data available for the case of JET are not sufficient for performing a complete analysis of the effect of the LH power on the pump and density fluctuation spectral 466

6 Lower hybrid waves broadening, this phenomenon indicates, as in ASDEX, that the LH pump produces at the edge a coupling between low and high frequency modes, typical for PIs as shown in the next section. In order to interpret the pump broadening, the linear wave scattering by density fluctuations, occurring in the full radial region of the plasma, was proposed as responsible for producing the LH spectral broadening [42, 43]. However, such a model should not be consistent with the experimental observations indicating that the spectral broadening is originated at the plasma edge and is more pronounced when operating with higher plasma density of the SOL [27]. In addition, the linear scattering model is not able to provide an interpretation of the linked phenomenology of RF probes and plasma edge microwave reflectometry, observed during the LHCD experiment on ASDEX, as well as the aforementioned non-linear behaviour of the pump broadening phenomenon [27]. The main hypothesis of the present paper is, instead, that the non-linear mode coupling phenomenon produces the LH spectral broadening at the edge, consistently with the microwave reflectometry of the edge performed on ASDEX [27] and in JET. In the next section it will be shown that the frequency range of both the enhanced plasma density fluctuations and the pump broadening corresponds to that of the quasimode frequencies driving the PIs. 3. Model of the parametric instability for lower hybrid current drive experiments The tools utilized for performing the analysis of the parametric instability (PI) relevant to the LHCD scheme for tokamak plasmas are summarized in this section, which is aimed at: (i) identifying the frequencies and growth rates of the coupled modes in the limit of the homogeneous plasma; (ii) calculating the amplification factor of the parametric instability, taking into account the convective losses due to effects of both finite extent of the pump and plasma inhomogeneity and the pump depletion which determines the broadening in the n spectrum launched by the antenna (section 3.2). The PI model relevant to the LHCD scheme described in the classical [19, 24] has been considered for performing the analysis of the instability, in the homogeneous and unbounded plasma limits. Such a work has been carried out utilizing the numerical tool described in [26], which achieved an interpretation consistent with the PI model for the RF-probe spectra observed during the early LH ion heating experiments. The summary of the utilized PI model and the growth rate results are shown in section 3.1. The convective losses have been taken into account by extending to PIs driven by quasimodes the analysis of [24], relevant to PIs driven by resonant modes. The pump depletion and the consequent spectral broadening have been calculated (in the section 3.3) considering the classic work of Chen and Berger [21]. It is important to note that for identifying the PI channel producing the LH spectral broadening, a search has been performed for the maximum growth rate of LH sidebands occurring in the same range of the operating frequency, imposing energy and momentum conservations of coupled modes. For a given set of plasma values, the Figure 5. Slab geometry and wavevectors of the coupled modes. concomitant behaviour of the growth rate has been considered for the following two important parameters: the component of the low frequency mode wavevector (k ) perpendicular to the magnetic field which maximizes the growth rate and the angle between the perpendicular wavevectors of sideband and pump (δ = [k, k 0 ]), which determines the group velocity direction of the pump and sideband. Consequently, this parameter is important for determining the maximum amplification factor due to convective loss (see the next subsection). Moreover, the search has been repeated in a similar way for a further set of parameters, considering the plasma gradients Identification of the mode-coupled frequencies and computation of the growth rate As the LH waves are electrostatic, the dispersion relation relevant for electrostatic coupled modes has been considered [19, 22]. The PI is determined by the non-linear coupling of LH waves with low frequency modes (ω ω 0 ) of the density fluctuation background. The parametric instability of a lower hybrid pump wave 0 [ i(ω 0 t k 0 r)] isdrivenby a low frequency mode [ i(ωt k r)] and growths by two sideband waves 1,2 [ i(ω 1,2 t k 1,2 r)], where k 2,1 = k±k 0, ω 2,1 = ω ± ω 0 are the selection rules provided by momentum and energy conservation of the coupled modes (subscripts 1, 2 refer to the lower and upper sidebands, respectively). We assume k 0 = k 0x x + k 0z z, k 1,2 = k 1,2x x + k 1,2z y + k 1,2z z, and utilize the relation n = kc/ω 0 between refractive indexes and wavevectors, figure 5. Moreover, the following notation is utilized: k k x x + k y y for k-values relevant to each mode. The plasma is modelled as a slab including the region of the edge close to the antenna mouth. The x direction coincides with the (radial) direction of the plasma gradients, and y,z correspond to the poloidal and the toroidal directions, respectively. The PI analysis is based on the solution of the Vlasov and Poisson equations for coupled electrostatic modes up to the second order. The following ordering is considered: the Maxwellian distribution function corresponds to the zero order, the variation produced by the pump wave to the first order and the perturbation of the low frequency modes to the second order ( 0, 1,2 ). The parametric dispersion relation of electrostatic coupled modes is [19, 22] ε(ω, k) µ 1(ω 1, k 1, k 0,E 0 ) ε(ω 1, k 1 ) µ 2(ω 2, k 2, k 0,E 0 ) ε(ω 2, k 2 ) = 0. (1) 467

7 R. Cesario et al Equation (1) is expressed in terms of the complex frequency of the low frequency driving mode, ω +iγ, where γ is the growth rate of the PI and the frequency of the sidebands are given by ω 2,1 = ω ± ω 0. ε is the dielectric function, µ 1 and µ 2 are the coupling coefficients referring to the lower and the upper sidebands respectively, and magnetized ions are considered for the calculation. The expression of the coupling coefficients is [22]: µ 1,2 = χ e(ω) ε(ω) ωpi 2 ω 2 ( ) pi ω 2 χ e (ω) ω Z 4k 2 cs 2 k 2 v the sin 2 u 2 δ 1,2. (2) c 2 s The expression of the coupling coefficient is derived in the limit ω k v the, which is satisfied by all the solutions of equation (1) obtained considering typical parameters of the plasma edge of LHCD experiments performed in tokamak plasmas. In equation (2), χ e is the electron susceptibility, c s is the sound speed, Z is the plasma function which retains, in the performed calculations, a sum over many ( 100) ion cyclotron frequency harmonic terms for achieving an accurate evaluation of the growth rate [20]. δ 1,2 are the angles between the perpendicular wavevector of the lower and upper sidebands and the perpendicular wavevector of the pump, δ 1,2 = (k 1,2, k 0 ), and u = ek 0 0 /m e ω ce. The angle δ 1,2 is one of the aforementioned important parameters for determining the strongest PI channels, since it affects the convective loss (see section 3.2). For the geometry relevant to the wavevector components of the coupled mode, see figure 5, in which only the lower sideband is considered for the sake of simplicity. For the computation of frequency and growth rate we search for solutions of equation (1) that have positive values of γ, which is the primary condition for having a PI. These solutions represent functions of the complex frequency, ω +iγ, in the independent variable k. The parametric dispersion relation (equation (1)) is numerically solved by setting five of the six parameters, k, δ 1,2 = (k 1,2, k 0 ), B 0, n e, T e, T i, as the input. In this way, different solution functions can be obtained for a parameter of interest. For a given experiment, the quantities relevant to the pump wave: k 0 and 0 are considered to be fixed parameters. The linear LH wave dispersion relation, ε Re (ω 0, k 0 ) = 0, equivalent to k 0,1x ω LH mi k 0,1/z (3) ω 0 m e gives the value of k 0. The subscript Re has the usual meaning of real part. As a necessary condition for strong PI, the highest growth rate is identified by means of the mode exhibiting the highest growth rate in the range of the maximum γ obtained in the different runs of solutions of equation (1). Wavenumbers and corresponding frequencies of the high frequency coupled modes are obtained by the selection rules (four equations for each sideband). In this way, a map of the PI channels potentially important for an LH experiment is obtained. These results will be considered in the next subsection for carrying out the computation of the convective effects. Several LHCD experiments have been considered [29,37]. As a general result, the obtained solutions Figure 6. The real part of the dielectric function of the lower sideband ε 1Re ε Re (ω 1, k 1 )in(a), and the frequencies and growth rates, in (b) are plotted against the perpendicular wavenumber of the quasimode, as solutions of equation (1). The parameters of table 1 and a set of parameters roughly occurring around the middle of the scrape-off plasma layer (n e = m 3, T e = 25 ev, see figure 3) have been considered. The n of the pump is n 0 = 1.84, that of the lower sideband is n 1 = 3. exhibit common features in the wide range of frequencies of the LH experiments (from about 0.5 GHz 8 GHz) and plasma densities ( n e = m 3 ). Such circumstance is due to the necessity of obtaining a good antenna coupling in an LHCD experiment, which requires similar values of ω pe /ω 0 near the antenna plasma interface. Moreover, the PI appears to be a phenomenon intrinsic to the physics of the LH wave in tokamak plasmas. The results of the PI analysis in the homogeneous plasma limit, obtained for a set of plasma parameters in the inner SOL (shown in table 1 and figure 3), are shown in figure 6. This figure represents, for the solutions of equation (1), the trend of ε Re (ω 1, k 1 ) (in (a)), and frequencies and growth rates (γ > 0) of the low frequency modes (ω ω 0 ), with respect to the perpendicular wavenumber of quasimode, k c/ω 0, which is considered a free variable (in (b)). The trend of ε Re (ω 1, k 1 ) indicates the condition of propagation of the lower sideband as LH wave when ε Re (ω 1, k 1 ) 0. Both the conditions γ > 0 and ε Re (ω 0, k 1 ) 0 are necessary for the growth of the instability. The range k c/ω corresponds to very low frequency (ion sound) PI driving modes (ω/ω corresponding to the range MHz, with ω k z v thi ). Such modes are strongly damped on the plasma particles, i.e. they are quasimodes, as the conditions ω k z v thi or ω k z v the generally occur for tokamak plasma, due to T e T i. In the same range, also the upper sideband propagates (i.e. ε Re (ω 2, k 2 ) 0, not shown in the figure). Consequently, the PI can occur by the growth of both the LH sideband waves within a small frequency shift. At higher values of k c/ω 0 (10.65 k c/ω ), the PIs driven by ion cyclotron quasimode are indicated (with ω/ω ) For this channel, only the lower sideband results in an LH propagating wave, while the upper sideband is evanescent (i.e. ε Re2 1). This channel will not further be considered 468

8 Lower hybrid waves Figure 7. Behaviour of the growth rates (maximized with respect to k ) plotted against the local plasma density. Different results are compared by keeping the electron temperature constant (25 ev, ), and changing accordingly with the reference scrape-off layer profile of figure 3 ( ). n 0 = 1.84, n 1 = 2.1, δ 1 = 10. Other parameters as in table 1. in the paper, as, for the relatively low plasma densities of the LHCD experiments, it exhibits generally lower growth rates with respect to the case of ion sound-quasimode-driven PIs. In addition, higher convective losses occur due to the bigger shifts in frequency and k of the sidebands (see the next subsection). It is to be noted that for the growth rates values around the maximum (γ /ω ) the variation of the wavenumber is very small (about 0.5%). However, further ranges of k are found, which satisfy the aforementioned conditions necessary for the PI occurrence, when considering the changes of the input parameters due to the plasma profiles (changes up to more than a factor two have been considered). Such behaviour of frequency and growth rate of the coupled modes is thus typical of the plasma edge of LHCD experiments in tokamaks. Therefore, the parametric dispersion relation (equation (1)) should be considered the relevant equation for LH waves launched in tokamaks rather than the linear dispersion relation, equation (3). This last equation should be considered the relevant equation for LH waves only provided that significant effects of mode coupling do not occur over the plasma column. As an important conclusion of the present paper, such a circumstance favourable for the PI onset occurs only in the SOL. For determining the conditions for the PI occurrence by considering the convective nature of the instability (considered in the next subsection), the values of maximum growth rate obtained at a certain k for plasma parameters of different radial layers should be determined. In the following, we indicate such growth rate maxima simply as growth rate. Its trend with respect to the plasma density is shown in figure 7, considering the electron temperature, respectively, constant (=25 ev) or following the realistic gradient of figure 3. Radial Figure 8. Growth rates of the quasimode-ion-sound-driven parametric instability (maximized with respect to k ) plotted versus the major radius. The scrape-off layer between the launcher mouth and a few centimetres insider the LCMS layer is indicated. n 0 = 1.84, n 1 = 2.1, δ 1 = 10. Other parameters as in table 1 and kinetic profiles as in figure 3. layers up to 2 cm inside the LCMS have been considered. The effect of reduction of the growth rates of the ion sound quasimode-driven PIs when increasing the local density and, more so, the electron temperature is evident. Indeed, such a PI channel disappears at high temperatures (T e 0.2 kev, with n e m 3 ). Therefore, the PI relevant to the LH spectral broadening is expected to occur mainly only in the plasma of the outer half of the SOL. Considering the reference SOL profiles (of figure 3), the trend of the growth rate and ion sound quasimode frequency (corresponding to the sideband shift from the pump) with respect to the radial distance from the antenna plasma interface is shown in figure 8. (Incidentally, this figure improves figure 1 of [14] containing similar data. The pump power density and the scale of the abscissa were not correct. ρ = 1 corresponds to R = 3.85 m, in place of R 4 m). The PI has the highest growth rate for plasma parameters close to the antenna plasma interface (γ /ω ) and frequencies of a few hundred kilohertz (ω/ω ). The error bars in the growth rates take into account the experimental uncertainties due to the scrape-off layer profiles (a factor of two on the plasma density and temperature in a given layer) and reveal a relatively small dependence of the growth rate on the input parameters. Such behaviour is due to the fact that quasimodes driving strong PI are however available for realistic LHCD operating conditions when considering reasonable changes of the input parameters. In figure 8 is also shown the effect of reduction of the growth rates obtained by hypothesizing a steeper electron temperature profile with respect to the reference case of figure 3, with 20 ev in place of 7 ev at x 0 cm from the antenna and 150 ev in place of 35 ev at x 4 cm. It is worth noting that the effect of the temperature is stronger in deeper layers with higher plasma densities, due to the aforementioned effect of depression of the temperature on the growth rate. 469

9 R. Cesario et al Figure 9. Trend of the growth rate with respect to the parallel refractive index of the lower sideband, keeping fixed the other parameters. Two different values of plasma densities, corresponding to radial positions of the outer and inner SOL (see figure 3), have been considered as parameters, respectively, n e = m 3 and n e = m 3. The electron (and ion) temperature is kept constant (10 ev). A similar behaviour is found for PI involving the upper sideband. Other parameters as for figure 8. Now, we focus on the spectral features of the LH sidebands that grow in the PI, as they might produce the searched broadening of the LH spectrum that penetrates into the bulk. In this regard, the behaviour of the growth rate with respect to the change in only the n of the lower sideband is plotted in figure 9. Two values of plasma densities typical of the SOL at radial positions of about one-third and two-thirds of the SOL, respectively, have been considered as parameters ( m 3 and m 3, respectively). The electron and ion temperatures are kept constant (10 ev). A similar trend (not shown here) is found considering the upper sideband. This figure indicates that LH sidebands with significant growth rates (γ /ω ) and small frequency shift with respect to the operating frequency (ω/ω ) occur over a wide range of sideband n (up to the maximum considered value, corresponding to about twice the pump n ). In the next subsection, it will be shown that the obtained growth rates are sufficient for compensating the convective losses and for producing a PI-induced spectral broadening of the launched LH spectrum Convective loss and spatial amplification of the parametric instability In the present subsection we identify the conditions that determine the occurrence of the parametric instability considering its nature of convective instability, in the realistic case of an LHCD experiment. We focus here on the determination of the amplification factor of the parametric instability, which is necessary for the computation of the fraction of pump power going to the sidebands. The results of the consequent spectral broadening will be presented in the next subsection. Figure 10. Scheme of the LH resonant cone regions in the scrape-off layer of the pump and LH waves. The approximations of unbounded and uniform plasma, utilized in the growth rate homogeneous analysis of section 3.1 should be removed for taking into account the convective losses. As a first step, the LH wave energy flux contained in well-defined resonant cones in a magnetized plasma [24, 44] is considered for obtaining the PI amplification factor in the limit of finite pump extent (A FPE ) [24]. The characteristic scalelength of the PI is thus determined, and the effect of the plasma inhomogeneity can be included. As main result, we will show that the plasma inhomogeneity inhibits the growth of the sidebands, especially for those with a higher n shift from the pump. The present analysis has been performed following [24] and is organized as follows: (i) the finite extent of the pump region is considered in slab geometry for evaluating the threshold of the pump power density and the radial extension of the region of PI, (ii) the effect of plasma inhomogeneity producing a phase mismatching between coupled modes is considered. In the analysis, only the lower sideband is considered for the sake of simplicity; however, the extension of the analysis to the upper sideband is straightforward. We consider the slab geometry, shown in figure 10, which indicates the group velocities of pump and sideband (for simplicity only the lower sideband), and the propagating regions of their energy fluxes. The ξ direction is that perpendicular to the pump group velocity, and x,z indicate, as used, the directions of the plasma gradients and confinement magnetic field, respectively. The wave potential can be considered uniform in the region illuminated by the antenna, 0 = P 0 for L/2 < ξ < L/2 and 0 = 0 otherwise. At the boundary, 0 = P 0 for 0 < z < 2a and 0 = 0 otherwise, where 2a is the dimension of the antenna in the toroidal direction. The sideband potential, which corresponds to thermal fluctuations, is 1 = P 1 for all z. The onset of the convective instability is conditioned by the occurrence of a positive balance between the rate of energy gained by the sideband from the pump and the one lost in convection. The exponential amplification factor characterizing the spatial growth of the parametric instability considering the effect of the finite extension of the pump region is [24] A FPF = γ(k 1,E 0,ω 0,ω 1 )L v g1ξ, (4) 470

10 Lower hybrid waves Figure 11. Behaviour of the amplification factor, integrated in the SOL, with respect the angle δ 1 = / (k 1, k 0 ). n 1 = 2.1. The amplification factor has been integrated considering the profile of the growth rates and the changes of the group velocities of pump and sideband in the SOL. where γ is the growth rate, L is the width of the pump region in the x, z plane, v g1ξ is the group velocity component of the sideband in the direction perpendicular to that of the pump [21, 25]. The PI cannot grow in time, but only in space propagating in the plasma, as it is a convective instability [19]. The occurrence of significant PI requires that the amplification factor is A FPE π [20, 24]. Such a condition determines the threshold in the electric field of the pump and indicates that the power density fluxes escaping away are sufficiently balanced by that injected by the pump for producing spatial amplification of PI. As a general feature of the PI, the homogeneous growth rate is maximized by values of the angle δ 1 (k 1, k 0 ) π/2; instead, small angles that make the pump almost aligned and sideband group velocities reduce the convective loss. The trend of the PI amplification factor, calculated considering the homogeneous growth rate obtained in the previous subsection, is plotted against angle δ 1 in figure 11, and allows the determination of the optimum angle for higher amplification factor. The maximum amplification factor occurs for small angles of the pump and sideband k- vectors (δ 1 10 ). In this condition we have obtained, referring to the geometry in figure 9: v g1ξ /c (with, typically, v g0,1x /c 0.15, v g0,1z /c 0.4). Some indicative values obtained in the middle of the scrape-off layer and utilized for calculating the PI amplification factor, by solving equation (3), are γ/ω , L m and v g1ξ /c , which give A FPE 9. For determining the amplification factor A FPE we have solved equation (3) assuming δ 1 = 8 and repeated the calculation for plasma parameters relevant to different radial positions in the scrapeoff layer, in order to determine its averaged value, which we however indicate as A FPE. The radial extension of the PI interaction region can be defined, in the limit of homogeneous plasma, by the radial size x in which the condition for exceeding the threshold for the finite pump extension effect is satisfied (A FPE = π). Such a region should be identified by considering the changes in the growth rate occurring in this layer, due to the profiles of the plasma parameters. We have also considered the effect of the ray-tracing in the radial size of the interaction region by taking into account that the n 1 (and the frequency) of the sideband is slightly different from that of the pump. However due to the small extension of the PI interaction layer with respect to the distances of ray propagation, this effect is sufficiently small ( 5% for n 1 5) to be negligible. For typical LHCD experiment parameters, the resulting interaction region is x 5 cm and coincides roughly with the layer between the antenna mouth and the LCMS. It means that only in the scrapeoff layer can the PI growth occur, as it is strongly reduced by the temperature step gradient located near the LCMS. In the following, the contribution of the convective losses due to plasma inhomogeneity will be discussed, which makes the extent of the interaction region dependent on the shift in n (and in frequency) of pump and sidebands. To retain the effect of the plasma inhomogeneity allows, for a given spectral component of the LH sideband, determining the size of the interaction region and, consequently, the realistic condition of the PI growth. The contribution of a weak plasma inhomogeneity to the convective loss has been determined by extending the analysis of Liu and Tripathi, who considered PIs involved only resonant modes [24]. The same formalism of [24] has been utilized. Considering the effect of the plasma inhomogeneity (occurring for simplicity only in the x direction) the k-matching condition of the coupled modes, M k k 0 k 1 = 0, holds only at the interaction point x = 0 (see figure 9). Expanding away from x = 0, M y M z 0, M = M x k x k 0x k 1x can be expressed as M dm dx x. (5) x=0 Away from the point x = 0, the condition of frequency matching ω ω 0 ω 1 0 is conserved, while a phase mismatching M 0 due to the plasma inhomogeneity increasing linearly with distance from the interaction point is produced. As the pump and sideband waves propagate radially inwards in regions with higher densities, the expression of the x component of the wavevector is given by the relationship in equation (3). For the low frequency quasimode, instead, no limitation (like the one imposed by a dispersion relation) occurs, which has only the limitation of existing in the presence of a pump with a finite amplitude. In particular, a proper quasimode is available in the plasma which minimizes the phase mismatching, M 0 for sidebands with spectral characteristics close to that of the pump, i.e. ω 1 ω 0 0 and k 1z k 0z, since for such a sideband the detuning produced by the propagation away from the interaction point in the weakly inhomogeneous plasma is negligible. Such a condition is produced by quasimodes with k x 2k 0x, which give M x k 0x k 1x. The expression of the k-matching condition will be utilized for carrying out the WKB analysis, relevant for quasimode-driven PIs, which is necessary for determining the 471