Nonlinear evolution of beam driven waves on MAST

Transcription

1 Nonlinear evolution of eam driven waves on MAST M. K. illey, M. isak Department of Earth and Space Sciences, Chalmers University of Technology, 4196 Göteorg, Sweden, B. N. Breizman Institute for Fusion Studies, The University of Texas, Austin, Texas, 7871 USA EXW/P7-14 S. E. Sharapov, S.D. Pinches, M.P.Gryaznevich Euratom/CCFE Fusion Association, Culham Science Centre, Aingdon, Oxfordshire, OX14 3DB, UK Astract Experiments on Alfvénic instailities driven y super-alfvénic eams in the spherical tokamak MAST have exhiited a variety of modes excited in a road range of frequencies. The aim of the present work is to demonstrate that, in spite of the significant differences in the nature of the modes (e.g. shear Alfvén or CAE), and in the corresponding excitation mechanisms, the nonlinear evolution of these modes, when driven just aove their excitation threshold, is determined y a competition etween the electromagnetic field of the modes and the relaxation processes of the fast particles driving these modes. In particular, we validate the recent theoretical finding that drag encourages eam-driven waves near marginal staility to exhiit an explosive scenario. An efficient fully nonlinear numerical tool has now een created for this task, using a ump-on-tail model, the results of which show characteristic features typical of any near threshold energetic particle driven instaility. These universal features are oserved in recent and past MAST experiments, indicating the central role that the drag might play in the evolution of eam or alpha particle driven waves in tokamaks. Recent quantitative modelling of eam driven Alfvénic instailities on MAST, in the presence of drag using the HAGIS code, shows agreement with the main trends of the ump-on-tail analysis. Universal experimental signatures Energetic particle driven Alfvénic instailities are often oserved in present-day fusion experiments and they are of major importance for the next step urning plasmas. These instailities exhiit a variety of nonlinear scenarios for the mode amplitude and frequency evolution, from a steady-state saturated amplitude, to a ursting one. The nonlinear evolution, together with the linear staility criteria and the characteristic values of the amplitudes of the Alfvénic modes, are essential for understanding the gloal transport of the energetic particles and hence particle re-distriution and losses. Experiments on Alfvénic instailities driven y super-alfvénic eams in the spherical tokamak MAST [1] exhiit a variety of modes excited in a road range of frequencies from Alfvén Cascade (AC) eigenmodes (FIG. 1), Toroidal Alfvén Eigenmodes (TAE) (FIG. ), and chirping modes in the frequency range khz, to compressional Alfvén eigenmodes (CAE) in the frequency range MHz (FIG. 3), which is approaching the cyclotron frequency of plasma ions, (0.1 1) ci. In spite of the differences in the nature of the modes (shear Alfvén or CAE), and in the corresponding excitation mechanisms, the nonlinear evolution of these modes, as seen in their spectrograms, show similarities.

2 FIG. 1. Magnetic spectrogram showing AC and TAE FIG.. Modes sweeping up in frequency over the TAE frequency range khz FIG. 3. Magnetic spectrogram showing ursting CAE (B=0.5 T, I=750 ka, P NBI =3 MW) This is consistent with the theoretical concept that, regardless of the specifics of the energetic particle driven instaility, the nonlinear mode evolution just aove the excitation threshold is determined y the competition etween the wave field and the relaxation process restoring the unstale distriution function of the energetic particles. Moreover this competition reduces to a 1-D set of equations identical in form to that of the ump-on-tail prolem []. It should e noted however that for high frequency modes, although the trends do appear to e the same as for low frequency modes, the theory has not een fully developed. Modes driven y NBI vs those driven y ICRH In MAST, the energetic particle population is created y super-alfvénic neutral eam injection (NBI) with a maximum velocity V 0 satisfying V0 VA Vc rit. Here, V A is Alfvén velocity, and V crit is the critical speed at which the energy flow from energetic ions to thermal plasma electrons and ions ecomes equal. Since the region of Cherenkov resonance, V, is well aove V crit, the Coulom electron drag dominates over velocity space V A diffusion in the resonance region. Experimentally, on MAST, as on some other machines with NBI, a hard nonlinear regime is oserved more often for NBI-driven Alfvénic modes, especially at low frequencies (e.g FIG. 4), resulting in ursting amplitude evolution and in rapid frequency sweeping [3-7]. Spontaneous formation of phase space holes and clumps [8] is typical of these NBI-driven scenarios [3]. The holes and clumps in the energetic particle distriution function correspond to resonant particles that are trapped in the field of the wave. These nonlinear structures can e viewed as long living Bernstein-Greene-Kruskal (BGK) modes [9], whose associated frequency can deviate far away from the original linear eigenfrequency. The theory was recently extended to include the case when the range of frequency sweeping is comparale to the frequency itself [10]. In contrast, Alfvén eigenmodes (AEs) excited y ICRH (ion cyclotron resonance heating)- produced ions show usually the soft excitation regimes [11-16] (e.g FIG. 5). In these nonlinear scenarios, the mode frequency remains close to the linear AE eigenfrequency and the structure of the mode remains similar to the linear one.

3 FIG. 4. Magnetic spectrogram of a MAST experiment with constant NBI power, showing rapid frequency sweeping FIG. 5. Magnetic spectrogram of a JET experiment with slowly increasing ICRH power, showing a variety of soft non-linear regimes The disparity etween experiments on AE excitation y NBI and y ICHR has recently een attriuted to the role of dynamical friction (drag) as a relaxation process for resonant particles [17]. This oservation was made in the ump-on-tail model for a near threshold instaility with d d where is the energetic particle contriution to the wave growth rate and d is the wave damping rate due to dissipation in the ulk plasma. In the past such a model was successfully used to explain nonlinear ifurcations and frequency sweeping events oserved in various AE experiments [3,11,1]. Dynamical friction For energetic ions produced via neutral eam injection (NBI), the unstale distriution function is formed y Coulom collisions, with dynamical friction (drag) and velocity space diffusion dominating in different regions of phase space. In contrast the relaxation processes for ICRH are dominated y velocity space diffusion due to the ICRH wave field. In the ump-on-tail prolem with an unstale wave with wave numer k and linear eigenfrequency, the collision operator surrounding the wave particle resonance can e represented as pe df 3 F F0 F F0 (1) dt coll u u where F is the distriution function of resonant fast particles, F0 is the equilirium distriution function, ukv pe, and and are constants characterising the dynamical friction and velocity space diffusion respectively. It was found [17] that the mode exhiits no steady state and in fact grows explosively when dynamical friction (drag) dominates over velocity space diffusion in the vicinity of the wave-particle resonance, indicating the onset of chirping. Translating the ump-on-tail results to eam driven TAEs reveals the ratio of diffusion to drag to e [17] 3/ 4 9/ EA 7 m T e 0 64 e A TAE c ms () TAE eb r m E Where m is the poloidal mode numer (of order unity), m is the mass of the eam species, r is the minor radius, B 0 and T e are the magnetic field and the electron temperature, E A is

4 the resonant Alfvénic energy, is the eam angle to the magnetic field, S is the magnetic 3/ 1/ 4 shear, EA m / Z e neln. For MAST conditions we find TAE (3) TAE showing that drag can e dominant in experiments, suggesting that drag could e responsile for the ursting ehaviour of the TAEs on MAST. The same analysis for ITER parameters with alpha particles gives TAE 1.4 (4) TAE showing that, although drag should not dominate, its effects should certainly e taken into account. Fully nonlinear effect of dynamical friction The previous analysis was limited to a weakly nonlinear regime with a perturative treatment of the energetic particle response to the wave field. An extension to the analysis of [17], to descrie the full nonlinear ehaviour in drag dominated scenarios, has now een performed [18] and is largely ased on numerical modelling using the BOT tool, which generalises an earlier code [19] to include the effect of drag. We oserve that the drag continues to play a destailising role in the fully nonlinear prolem [18]. It gives rise to the generation of holes and clumps (shown schematically in FIG. 6) in the fast ion distriution function, whose associated frequencies chirp asymmetrically from the original linear eigenfrequency as can e seen in FIG. 7. The drag enhances a phase space hole and acts to weaken, or even suppress, a phase space clump. The universal feature of all simulations including drag is the aforementioned asymmetry, which is also seen in experiments. FIG. 6. Cartoon illustrating the initial motion of holes and clumps and the wake (lue line) that acts to steepen the distriution function, creating a favourale environment for instaility. FIG. 7. Pure drag shows indefinite upward chirping holes and suppressed downward chirping clumps. Spectrogram shows the electric field increases indefinitely in time for the holes The drag collision operator consists of two pieces, the first is a so called dynamic part ( F/ u) that enters the kinetic equation formally in the same way as a DC electric field and the second is a so called sink of particles ( F / u 0 ) that allows the formation of a steady-

5 state solution of the unpertured distriution function. The dynamic part provides energy to the wave in the form of fresh particles from high velocity. This is similar to the effect of chirping of holes and clumps. Both processes allow the wave to e sustained in the presence of ackground dissipation. The sink part acts to continually deepen a phase space hole, which prevents a steady state from forming. Drag and diffusion In reality, a system will exhiit oth drag and diffusion. The effect of diffusion, unlike drag, is always to act to destroy a hole and a clump y filling the hole and depleting the clump. We oserve [18] that a steady hole can e estalished when oth drag and diffusion are present, as can e seen in FIG. 8. With somewhat more diffusion the steady state is lost and there is a tendency for the frequency and the electric field to undulate. With a further increase in diffusion the frequency spectrum exhiits intermittent hooks (FIG. 9). It is noteworthy that these hooks resemle experimentally oserved chirping patterns [0]. FIG. 8. Estalishment of a steady state hole in the presence of oth drag and diffusion FIG. 9. Hooked frequency spectrum with drag and an increase in diffusion from that in Fig 9 These ehaviours can e understood y recognising the competition that now exists etween the sink in the drag collision operator that acts to deepen the hole, and the diffusion that acts to fill the hole. This competition can e captured using a simplified model that treats the wave as an ideal BGK mode [18]. More specifically the amplitude of a BGK wave defines a so called separatrix in phase space (of characteristic width B / k where B eke / m is the ounce frequency for the particles trapped in the wave field) inside of which particles are trapped and mixed y the potential of the wave. It is these particles that form either a hole or a clump. For a wave whose frequency remains close to the plasma frequency one can write the wave equation in the form [18] 16 B 3 g (5) where, is the frequency deviation and g is a measure of the relative depth of a hole to the F F / v g / k 0 (similar to that in FIG. 7). For early times, or in the asence collisions g, ut in general we should write [18] amient distriution, defined y / g B t g t (6)

6 so that the relative depth evolves in time due to the comined effect of drag, diffusion and chirping. Any flow of particles, either from drag or from the motion of the separatrix due to a changing wave frequency, cannot enter the separatrix and so make a jump to a lower velocity, which releases energy for a hole structure. By alancing the power released from drag and from chirping with the dissipation due to d 3 3 B g (7) 48 t (noting that it is assumed that the wave is driven close to the threshold so that d ) a complete set of equations is formed allowing the frequency and electric field evolution to e calculated. Eqs (5) (7) can e comined into a single parameter set of coupled nonlinear equations where c c 4 3 B 1.84 x /, 1/3 y x y( 1) xy y y a (1 ) x d1.8 4 y /, t d /, 1/ 3 3 / 48 and d 3 / / / d (8) a with. The ehaviours in FIGS. 8 and 9 are reproduced y this reduced model, with the staility oundary of Eqs. (8) ( a 1) marking a dramatic change in ehaviour from a steady state to one with hooked chirping. The full scale simulations agree with the trends of this model, namely parameters from Fig 8 give a.0 (i.e. stale) whereas the parameters from FIG. 9 give a 0.9 (i.e. unstale). A numerical solution of Eqs (8) (shown in FIG. 10) for the unstale case reproduces the evolution of a single hook. The frequency and time scales are in good quantitative agreement with the well isolated hooks seen in FIG. 9 as shown in FIG. 11. FIG. 10. Hooked chirp solution of Eqs 1.8, the solid line and dashed lines show the ounce frequency and the deviation of the frequency from the plasma frequency respectively. FIG. 11. Quantitative agreement etween hooks seen in Fig 11 and Fig 1 (white line) is shown.

7 Drag in toroidal systems HAGIS FIG. 1. Sliding Fourier transform showing frequency evolution of marginally unstale TAE mode in response to kinetic α-particle drive (γ /ω 0 = 0.061) and external damping ( γ d /ω 0 = 0.06). FIG. 13. This ehaviour is very similar to that seen in the 1-D model introduced aove and is more importantly, very similar to the mode evolution oserved experimentally in e.g FIG. 1. This provides confidence in the use of simple models to investigate the nonlinear ehaviour of modes near marginal staility and provides guidance on the physics aspects that need to e encapsulated in more sophisticated codes. High frequency modes on MAST The effect of drag in realistic tokamak (i.e. toroidal) geometry has een investigated using a modified version of the HAGIS code [1] which includes the effect of drag and Krook relaxation upon the fast ions. FIG. 1. shows the results in the asence of drag for a core-localised n = 3 TAE interacting with a centrally peaked slowing-down distriution of α-particles which provide a linear growth rate of γ /ω 0 = 0.061, in the presence of an extrinsic damping rate of γ d /ω 0 = Introducing an electron-ion drag of ν ei /γ d = and a Krook relaxation rate of ν eff /γ d = results in the spectrogram shown in FIG. 13. Spectrogram showing influence of electron-ion drag and Krook relaxation upon mode evolution in tokamak geometry. High frequencies modes ( ~ ci ) are also seen in some MAST discharges. These modes have een identified as compressional Alfvén eigenmodes (CAEs) [1] driven y anisotropy and energy space gradients in the fast ion distriution function. Ion cyclotron resonances k V p c kv (9) have to e considered for the wave-particle interaction, where p 1 corresponds to normal Doppler resonance, p 1 corresponds to the anomalous Doppler resonance, and the term k V descries the drift of the energetic ions due to the curvature of magnetic field. The experimental investigation of high-frequency modes is of special interest when studying nonlinear mode evolution e.g the competition etween drag and diffusion, since the resonant region of the fast particle distriution function is different from that of TAEs due to the Doppler shift in Eq 1.9. Experimentally, CAEs oserved in [1], where the modes are counter-propagating with respect to the eam on MAST (negative toroidal mode numers), are driven via the normal Doppler resonance. The asence of co-propagating modes was attriuted to the inaccessiility of the anomalous resonance in those MAST discharges []. In order to access the anomalous resonance a critical eam speed V / V is required []. MAST performed an experiment with a decreasing magnetic field, down to A B T. FIG. 14 shows the spectrum of modes otained in this experiment. In addition to TAEs in

8 the low-frequency range, and CAE similar to those descried in [1] seen in the frequency range 400 khz- MHz, a set of modes is oserved in the range MHz 4 MHz, with the frequency separation aout 50 khz are also oserved. In contrast to the CAEs, these newly oserved modes have positive mode numers n 9, 10, 11,... suggesting the mode is FIG. 14. Beam-driven high-frequency modes oserved in MAST discharge # 641 with reduced magnetic field, B 0 =0.35 T. driven via the anomalous Doppler resonance. The nonlinear evolution of these modes does not show any sweeping-frequency phenomena, in contrast to the modes descried in [1] indicating that a different phase space region is affected y the wave in this case, which further supports the relation of n>0 modes to the anomalous resonance in MAST. This work was funded jointly y the y RCUK Energy Programme, the Swedish Research Council, EURATOM and y the U.S Department of Energy Contract No. DE-FG03-96ER The views and opinions expressed herein do not necessarily reflect those of the European Commission References [1] GRYAZNEVICH. M.P., SHARAPOV. S. E., et al., Nucl.Fusion (008) [] BREIZMAN. B. N., BERK. H.., et al., Phys. Plasmas 4, 1559 (1997) [3] PINCHES. S. D., BERK. H.., et al., Plasma Phys. Controlled Fusion 46, S47 (004) [4] FREDRICKSON. E. D., GOREENKOV. N. N., et al., Nucl. Fusion 46, S96 (006). [5] WONG. K.., FONCK. R. J., et al.,phys. Rev. ett. 66, 1874 (1991) [6] HEIDBRINK. W., STRAIT. E., et al., Nucl. Fusion 31, 1635 (1991) [7] SHINOHARA. K., KUSAMA. Y., et al., Nucl. Fusion 41, 603 (001) [8] BERK. H.., BREIZMAN. B. N., Phys. ett. A 34, 13 (1997) [9] BERNSTEIN. I. B., GREENE. J. M., Phys. Rev. 108, 546 (1957). [10] BREIZMAN. B. N., Nuclear Fusion 50, (010). [11] FASOI, A., BREIZMAN. B. N., et al., Phys. Rev. ett. 81, 5564 (1998) [1] HEETER. R. F., FASOI. A. F., Phys. Rev. ett. 85, 3177 (000) [13] WONG. K.., WISON. J. R., et al., Plasma Phys. Controlled Fusion 36, 879 (1994) [14] SAIGUSA. M., KIMURA. H., et al., Plasma Phys. Controlled Fusion 37, 95 (1995) [15] BERNABEI. S., BUDNY. R. V., et al., Nucl. Fusion 41, 513 (001) [16] SNIPES. J. A., BASSE. N., et al., Phys. Plasmas 1, (005) [17] IEY. M. K., BREIZMAN. B. N., Phys. Rev. ett. 10, (009) [18] IEY. M. K., BREIZMAN. B. N., Accepted in Phys. Plasmas (Sept 010) [19] PETVIACHVII. N., Ph.D. thesis, University of Texas at Austin, [0] BERK. H.., BOSWE. C., et al., Nucl. Fusion 46, S888 (006) [1] PINCHES. S. D. et al., Comp. Phys. Comm. 111 (1998) 133 [] IEY. M. K., Ph.D. thesis, Imperial College ondon, 009.