EFDA JET PR(3)58 S.D. Pinches, H.L. Berk, M.P. Gryaznevich, S.E. Sharapov and JET EFDA contributors Spectroscopic Determination of the Internal Amplitude of Frequency Sweeping TAE
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Spectroscopic Determination of the Internal Amplitude of Frequency Sweeping TAE + S.D. Pinches, H.L. Berk, M.P. Gryaznevich 3, S.E. Sharapov 3 and JET EFDA contributors* Max-Planck-Institut für Plasmaphysik, EURATOM-Assoziation, Boltzmannstraße, D-85748 Garching, Germany Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas, 787, USA 3 EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon, OX4 3DB, UK Based on an invited talk given at the 8 th International Atomic Energy Agency Technical Meeting on Energetic Particles in Magnetic Confinement Systems, San Diego, USA, 6 th -8 th October 3. * See annex of J. Pamela et al, Overview of Recent JET Results and Future Perspectives, Fusion Energy (Proc. 8 th Int. Conf. Sorrento, ), IAEA, Vienna (). Preprint of Paper to be submitted for publication in Plasma Physics and Controlled Fusion
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ABSTRACT. From an understanding of the processes that cause a marginally unstable eigenmode of the system to sweep in frequency, it is shown how the absolute peak amplitude of the mode can be determined from the spectroscopic measurements of the sweeping rate, e.g. with Mirnov coils outside the plasma. In the complex geometry of real tokamak plasmas, theoretical estimates are replaced with more realistic, and thus complex, numerical simulations performed using the HAGIS code [,]. Internal measurements of the perturbing field strength such as these, are important since they avoid the uncertainties associated with extrapolating from magnetic measurements at the plasma periphery and are an important way to independently and directly obtain values of the perturbing field strength.. INTRODUCTION Comparisons between numerical models and present day tokamak devices rely upon accurate diagnostic measurements of the key plasma parameters. For calculations of energetic particle-driven instabilities and their effect on the energetic particles this means the mode amplitude. In this paper we demonstrate how from the measured spectrum of frequency sweeping toroidal Alfvén eigenmodes (TAE) it is possible to calculate the absolute value of the maximum amplitude of these modes. The numerical simulations reported in this paper have been performed using the HAGIS code [, ]. This is a nonlinear drift-kinetic Hamiltonian δf code that models the interaction between a distribution of energetic particles and a set of Alfvén eigenmodes to obtain the linear growth rates, γ L, and the nonlinear behaviour of the mode amplitudes determined by kinetic wave-particle nonlinearities. The code uses realistic plasma geometry and allows for arbitrary particle distributions. The waves are described by their linear eigenfunctions that vary in amplitude and phase in response to the energetic particles. Through the simulation of marginally unstable TAEs, the HAGIS code has demonstrated self-consistent frequency sweeping, with the sweeping rate found to be.4γ L (γ d t) /, where γ d is the mode damping rate, in agreement with theoretical predictions [3, 4, 5]. In this paper the results obtained from these numerical simulations are presented together with the experimental observations for direct comparison. In particular, from a comparison of frequencysweeping modes observed on MAST [] absolutely calibrated mode amplitudes in the plasma core are computed.. FREQUENCY SWEEPING Numerical simulations of marginally unstable bump-on-tail kinetic instabilities [3] have shown how a hole and clump spontaneously appear in the particle distribution function and how this process supports a set of long-lived Bernstein, Greene, Kruskal (BGK) nonlinear waves that shift up and down in frequency. A similar nonlinear process of hole-clump production also occurs for modes driven by the radial gradient of the fast ion pressure, such as TAEs. Figure shows an experimental example of such frequency sweeping nonlinear waves in the JET tokamak [7]. In this case the frequency
sweep, δω/ω ~ 5% in δt ~ ms, where ω v A /qr is the TAE s natural frequency and v A is the Alfvén speed, q is the safety factor and R is the major radius at the magnetic axis. The rate at which the frequency sweeps is uniquely determined by the wave-particle nonlinearity and this depends upon the wave amplitude through the following relationships. The first relates the nonlinear bounce frequency of a particle trapped in the potential well of the wave, ω b, to the frequency sweep [8], δω = C ω b 3/ δt / () with C being a constant of proportionality. For a simple mode structure with a single resonance C can be estimated to be C = π/( ). The second relation is that which determines the dependence of the nonlinear bounce frequency upon the mode amplitude and is of the form [9], ω b = C A / () where the mode amplitude is defined in relation to the toroidal magnetic field on axis, B, as, A = δb r /B and C is a second constant of proportionality to be determined by numerical simulation. For the case of a single-resonance TAE, C may be estimated from the relationship [9], v ωb ω n R S v ωr m A / δb r B (3) where m, n are the poloidal and toroidal mode numbers respectively, S is the magnetic shear at the position of the mode, r, v and v A are the parallel velocity of the resonant particles and the Alfvén speed and B and db r are the equilibrium magnetic field and the radial component of the perturbed magnetic field. Combining equations () and () provides an expression for how the absolutely calibrated mode amplitude in the core of the plasma can be derived from the observed sweeping rate of the TAE frequency, δ B B r = C δω Cδt /3 (4) Whilst the analystic estimates for C and C may be used to give rough estimates of the mode amplitudes, numerical simulations are necessary to obtain accurate values.. NUMERICAL MODELLING Frequency sweeping occurs close to marginal stability, i.e., when the competing effects of fast ion drive and damping almost balance such that the mode is marginally unstable. This situation has been numerically modelled in the HAGIS code [,] by introducing an additional external damping
mechanism, represented by the damping rate, g d such that the equations that determine the wave evolution become: X& k = Ek Y& k = E k np j= np δf Γ j= j j ( p) j δf Γ ( p) j ( k mv j ωk) m ( k mv j ωk) C jkm + Ykγ d m S jkm + X γ k d (5) where X k and Y k represent the real and complex components of the k th wave s amplitude, j is the summation index over numerical markers and m is the summation over poloidal harmonics. Simulations are initially performed with γ d = to obtain the mode s natural linear growth rate, γ L, before they are repeated with γ d γ L to obtain marginal instability and thus frequency sweeping. As a first example, we consider the case of a circular cross-section plasma with an inverse aspect ratio, a/ R =.3 and a monotonically increasing safety factor from an on-axis value of q =. to an edge value, q a = 3.5, as shown in Figure. A linear eigenmode analysis of this equilibrium using the MISHKA code [] and the HAGIS code to simulate the case of a distribution of a-particles reveals that it is unstable to an n = 3 core-localised TAE mode shown in Figure 3. The kinetic drive for this mode was provided by a centrally peaked slowing-down distribution of a-particles described by: f ( s, E) = C exp ψ ψ ψ + E 3 + E 3 c E E Erfc E with ψ =., ψ =.7, E = 3.5 MeV, E c = 494 kev and E = 4 kev and shown in Figure 4. The constant C was chosen by specifying the volume averaged fast particle beta, β f = 3-4. The background plasma density was chosen to be 9 m -3 with a 5:5 D:T isotope mix meaning that an a-particle with energy 3.5MeV has a velocity.34 v A. In the modelling, all the particles had l = v /v =, i.e. were deeply co-passing. In the absence of any additional external damping mechanisms, γ d =, a simulation using 5,5 simulation markers determines the linear growthrate for this mode is found to be γ L /ω =.7 ±. and finds that the mode saturates at an amplitude of δb/b -3 as shown in Figure 5. The mode is then made marginally unstable by adding an artificial external damping mechanism such that γ d /ω =. and the simulation repeated with, markers and run for 7 inverse growth times which equates to 5 wave periods. In this case, the mode amplitude saturates at a much lower level, δb/b -4. Producing a Fourier spectrogram of the evolving mode reveals a predominantly down-shifted frequency sweeping branch as shown in Figure 6. The reduced amplitude of the up-shifting frequency branch is attributed to the fact that the primary resonance, v = v A, is very close to the birth energy of the a-particles and there are significantly less resonant particles at higher energies to drive the mode if it shifts up in frequency. Reducing the kinetic a-particle drive by reducing their pressure so that β f = 7.5-5 results in a linear growth rate of γ L /ω = 4.5-3 ±.5-3. Introducing an external damping rate, 3
γ d /ω = 4-3 makes this mode marginally unstable and again leads to a frequency sweeping regime but with smaller dw as shown in Figure 7. 3. RESULTS In this section the HAGIS model is used to determine the mode amplitude of a frequency sweeping global mode in the analytically demanding geometry of the tight aspect-ratio MAST tokamak. The experimentally observed frequency sweeping measured with magnetic coils near the edge of the plasma boundary is shown in Figure 8. A mode number analysis using a toroidally distributed array of magnetic pick-up coils around of the inside of the vessel identifies this mode as an n = mode. This discharge was heated by neutral beam injection (NBI) with the injected deuterons having an energy of 4keV. The magnetic field strength at the magnetic axis was.5t, the major radius was.77m and the minor radius.55m. A careful linear stability analysis using the MISHKA code together with the EFIT [] reconstructed equilibrium at this time and the HAGIS code identifies this mode as a global n = TAE as shown in Figure 9. To identify the constant of proportionality between the nonlinear bounce frequency and the mode amplitude, simulations were performed with the HAGIS code. For a single particle moving in an axisymmetric toroidal equilibrium and trapped in the TAE wave there is a symmetry between the toroidal and temporal variation of the wave field associated with the ability to describe it using the ansatz, ξ(r,t) = ξ(r,q) exp(inφ- iωt). In this case, energy, E, and toroidal canonical momentum, P φ, are no longer conserved quantities, however it is possible to construct a new system invariant, H = E - (ω/n)p φ By launching a set of particles at various plasma radii but with all the same value of H it is possible to map out the island structure of fast ions trapped in the TAE. Such an example is shown for the MAST case under consideration in Figure where the value of H = kev was chosen such that the parallel velocity of fast ions at the centre (and the peak of the eigenfunction) was equal to the Alfvén velocity, v = v A. The amplitude of the mode was arbitrarily chosen to be A = δb r /B = -3. By evaluating the nonlinear bounce frequency for ions that are deeply trapped in the TAE wave field and varying the TAE amplitude it is possible to obtain the scaling of ω b with A as shown in Figure. As expected theoretically, the nonlinear bounce frequency scales with the square root of the mode amplitude. It should be noted however, that when repeating this scaling for earlier time points in the discharge when the safety factor profile was interpreted by EFIT to be reversed in the plasma core, this scaling broke-down for amplitudes larger than, A > -4. We now have sufficient information to estimate the amplitude of the TAE mode that is seen to sweep in frequency in the MAST discharge discussed above. Substituting the observed frequency sweep of 8kHz in a time of.8ms, together with the theoretical estimate of C. and the numerical calculation of C =. 6 into equation (4), we obtain, A = db r /B 4-4. This corresponds to an absolute amplitude of δb r -4 T. The measured amplitude of this mode, δb r, at the position of the mid-plane Mirnov coil (R =.85m) was -5 T. Using the MISHKA code with an ideally conducting wall at R/a = to calculate the linear mode structure in the plasma and vacuum regions allows the peak amplitude to be 4
determined to be δb r 5-4 T. This value is in good agreement with that obtained above using the spectroscopic observation of the mode s frequency sweeping rate. CONCLUSIONS A spectroscopic technique has been formulated for determining the absolute amplitudes of frequency sweeping modes that start from the TAE frequency and remain in the TAE gap during the sweep. The example presented is that of a frequency sweeping TAE mode in the core of the MAST tokamak. In this case (MAST Pulse No: 5568 at t = 65ms) the n = global TAE amplitude was determined to be δb r /B 4-4. A more complete simulation of the discharge conditions for this case, including a realistic description of the beam distribution of energetic ions and the effects of collisionality, is the subject of future work. Such a simulation should also be capable of capturing the repetitive nature of the frequency sweeping TAE observed. ACKNOWLEDGEMENTS The UKAEA authors were jointly funded by the United Kingdom Engineering and Physical Sciences Research Council and by EURATOM. The authors are grateful to R. Martin, T. Pinfold V. Shevchenko and the MAST teams for help in operating the tokamak and diagnostics during these experiments. REFERENCES []. Pinches S. D., 996 The University of Nottingham Ph.D. Thesis []. Pinches S. D., Appel L. C., Candy J., Sharapov S. E., Berk H. L., Borba D., Breizman B. N., Hender T. C., Hopcraft K. I., Huysmans G. T. A. and Kerner W. 998 Computer Physics Communications, 3 [3]. Berk H. L., Breizman B. N. and Petviashvili N. V., 997 Physics Letters A, 34, 3 [4]. Berk H. L., Breizman B. N. and Petviashvili N. V., 998 Physics Letters A, 38, 48 [5]. Vernon Wong H. and Berk H. L., 998 Physics of Plasmas 5 78 [6]. Sykes A., Akers R. J., Appel L. C., Arends E. R., Carolan P. G., Conway N. J., Counsell G. F., Cunningham G., Dnestrovskij A., Dnestrovskij Yu. N., Field A. R., Fielding S. J., Gryaznevich M. P., Korsholm S., Laird E., Martin R., Nightingale M. P. S., Roach C. M., Tournianski M. R., Walsh M. J., Warrick C. D., Wilson H. R., You S., MAST Team and NBI Team, Nuclear Fusion 4 43 [7]. Wesson J. A., 997 Tokamaks, nd edition p.58 (Oxford Science Publication, Oxford, UK) [8]. Berk H. L., Breizman B. N., Candy J., Pekker M. and Petviashvili N. V., 999 Physics of Plasmas 6 3 [9]. Berk H. L., Breizman B. N. and Ye H., 993 Physics of Fluids B 5 56 []. Mikhailovskii A. B., Huysmans G. T. A., Kerner W., Sharapov S. E. 997 Plasma Physics Reports 3 844 []. Lao L. L., St. John H., Stambaugh R. D., Kellman A.G., and Pfeiffer W. 985 Nuclear Fusion 5 6 5
Pulse No: 494 Magnetic Spectrogram Log( δb ) Log( δb ) Safety Factor Frequency (khz) 95 95 9 9 85 85 8 8 q 3..5. 75 75 7 5.94 5.945 5.95 5.955 (s) Figure : Example of frequency sweeping mode during a shear optimised D-T pulse in the JET tokamak. In this case, δω/ω ~ 5% in δt ~ ms. JG3.733-c.5..4.6.8. Figure : Safety factor profile for circular cross-section plasma with a/r =.3. JG3.733-c φ(x 8 ) -. -.4 -.6 -.8 -. -. -.4 Scalar Potential (m, n) = (, 3) (m, n) = (, 3) (m, n) = (, 3) (m, n) = (3, 3) (m, n) = (4, 3) (m, n) = (5, 3) (m, n) = (6, 3) (m, n) = (7, 3) (m, n) = (8, 3)..4.6.8. Figure 3: Core localised n = 3 TAE in circular crosssection plasma. Real components are given by the solid lines, imaginary components by the dashed lines. JG3.733-3c (a.u.) (a.u.)(x 6 ).8.6.4...4.6.8.. Energy Variation of f.5..5 Radial Variation of f 3 4 Energy (ev)(x 6 ) Figure 4: Distribution of a-particles used in simulations: Centrally peaked in radius and slowing down in energy from a D-T fusion birth energy of 3.5MeV. JG3.733-4c 6
- Amplitude Evolution JG3.733-5c.4-3.3 δb/b -4-5.. -6 4 6 -. 8 Time (ωt/π) Figure 5: In blue is shown the evolution of the mode amplitude and in red is shown the instantaneous mode growth rate calculated from the changing mode amplitude. A linear growth phase is clearly seen up to around 4 wave periods whereupon it is replaced by a saturated final state. γ L /ω Frequency (ω)....9.8 4 6 Time (γ L t) Figure 6: Sliding Fourier spectrum showing frequency evolution of marginally unstable TAE mode in response to kinetic a-particle drive (γ L /ω =.7) and external damping (γ d /ω =.). The over-plotted white line shows the theoretically predicted frequency sweeping rate of.4g L (g d t) /. JG3.733-6c 4 4 Frequency (ω)....99.98 3 4 Time (γ L t) 5 JG3.733-7c 6 7 Frequency (khz) 8 8 64 64 66 66 68 68 7 7 7 7 Time (ms) JG3.733-8c Tim Figure 7: Sliding Fourier spectrum showing frequency evolution of marginally unstable TAE mode in response to kinetic a-particle drive (γ d /ω = 4.5-3 ) and external damping (γ d /ω = 4-3 ). The over-plotted white line shows the theoretically predicted frequency sweeping rate of.4γ L (γ d t) /. Figure 8: Magnetic spectrogram showing a frequency sweeping n = core-localised mode in MAST Pulse No: 5568. The first event is seen to sweep by 8 khz in a time of.8ms. 7
Scalar Potential Z (m)..5 -.5 -..4.8. R (m) φ q..4.6.8. Safety Factor..4.6.8. Figure 9: Equilibrium mesh and safety factor profile constructed for linear stability calculations of MAST Pulse No:5568 at t = 65ms. Also shown is the radial mode structure of the global n = TAE (real components are shown with solid lines, imaginary components with dashed lines.) 5-5 - 6 4 8 6 4 JG3.733-9c.7 Radial Position Versus Phase.6.5.4.3. 4 6 Phase [ξ-ω / n*t] Figure : Poincaré plot for energetic deuterons (H = kev) in MAST Pulse No: 5568 at t = 65ms interacting with a global n = TAE. The amplitude of the mode is A = db r /B = -3. JG3.733-c 4. 3.5 HAGIS code Linear fit 3..5 ω b (x 4 )..5..5.5..5..5 3. 3.5 (δb r /B)(x - ) JG3.733-c Figure : Scaling of the nonlinear bounce frequency of energetic deuterons (H = kev) in MAST #5568 at t = 65ms with the square root of the n = global TAE amplitude. 8