Gantenbein, G.

Günter, S.

Maraschek, M.

Meskat, J.

Suttrop, W.

Yu, Q.

Zohm, H.

English

MPG.PuRe

27th EPS Conference on Contr. Fusion and Plasma Phys.  Budapest, 12-16 June 2000    ECA Vol. 24B (2000) 968-971968Analysis of the Dynamics of Tearing Modes in ASDEX Upgrade1J.P. Meskat, 1G. Gantenbein, 2S. Günter, 2M. Maraschek, 2W. Suttrop, 2Q. Yu, 2H. Zohm, 2ASDEXUpgrade Team1 Institut für Plasmaforschung, Pfaffenwaldring 31, D-70569 Stuttgart2 MPI f. Plasmaphysik, Boltzmannstr. 2, D-85748 Garching, EURATOM-Association1. IntroductionA tearing mode is a resistive magnetohydrodynamic (MHD) instability characterised by a helicalperturbation current on a resonant surface in a tokamak plasma. In a classical tearing mode the per-turbation current is due to an unfavourable equilibrium current profile. The associated perturbationof the magnetic configuration leads to reconnection of the flux surfaces and hence to the formation ofso-called magnetic islands. Due to the resulting dramatically reduced radial heat isolation the tem-perature and density profiles inside the islands are flattened. An important consequence is a loss ofconfined energy what can be observed by a drop in β occuring at mode onset. Another consequence isa reduction of helical bootstrap current which represents a destabilising perturbation current. In highβ discharges reaching the β limit this loss of bootstrap current associated with a small seed island gen-erated e.g. by a sawtooth crash may lead to further island growth although the seed island should beclassically tearing stable. These so-called neoclassical tearing modes (NTM) limit the energy contentof magnetically confined fusion plasmas at high β [1]. Thus, tearing modes deserve closer investi-gation of their magnetic structure [2] and dynamics. In this work the theoretically predicted islanddynamics given by the generalised Rutherford equation is compared with experimental observationsin ASDEX Upgrade. Thus, the free parameters of the Rutherford equation can be assessed.2. Theoretical description of magnetic island dynamicsThe growth of a neoclassical tearing mode can be described by the generalised Rutherford equationfor the magnetic island width W [3, 4]:τresrresdWdt  rres∆  W rresβpaBS  ε LqLp WW 2  W 20  aGGJ rresR20 L2qLp 1W  aPol 	 ρp LqLp  2 g  ε  1W 3  aECCD 12  π rresd Lq IECCDIP 1W (1)Here, τres   µ0r2res   1  22η  is the resistive time scale (η plasma conductivity), rres the minor ra-dius and ε   rres  R0 the inverse aspect ratio of the resonant surface. Lq   q  q  and Lp   p  p  de-note shear and pressure decay length. The values of βp   2µ0 p  B2p (Bp poloidal magnetic field)and the poloidal ion larmor radius ρp    2miT    eBp  (T in eV) are also to be taken at theresonant surface. The nonlinear stability parameter ∆ W  describes the stability of the equilib-rium current profile against tearing mode instability [5]. The following destabilising bootstrap termincludes the reduction of neoclassical drive by finite heat conductivity across the island throughW0   5  1  χ   χ  1  4  rresLqq   εm  1  2 that will lead to an incomplete flattening of Te  r  within theisland for small W [6]. The next two terms describe the stabilising effects of shaping and toroidicity(Glasser-Green-Johnson-effect) and of the polarisation currents induced by the motion of the island27th EPS CCFPF 2000;   J. P. Meskat et al.: Analysis of the Dynamics of Tearing Modes in ASDEX Upgrade 969through the plasma [7, 8]. In the cases considered here the plasma collisionality is low and hencegε    ε3  2 is used whereas the polarisation term becomes much more important in ’collisional’ plas-mas where gε    1. The constants aBS, aGGJ and aPol are the free numerical parameters of theRutherford equation whose values have to be determined in this work. The last term describes thestabilising effect of electron cyclotron current drive (ECCD). The deposition of microwave power inan area of width d (typically d   5cm) causes a noninductively driven current IECCD (IP is the plasmacurrent). The coefficient can be approximated by aECCD   34d  W  exp   d  W  if d  W  0  5 andaECCD   10  311 if d  W  0  5 [9].As a necessary but not sufficient condition for dW  dt  0 and hence instability of a neoclassicaltearing mode βp must exceed a critical value βp  crit . In this case the generation of a seed island e.g.due to a sawtooth crash is sufficient for further growth of the mode if the seed island width exceeds acritical width Wseed  crit . The island grows until its width is equal the saturated island width Wsat whichis in good approximation proportional to βp. During the increase of magnetic island size βp dropsfrom its initial value due to the flattening of the pressure profile.3. Fit of the Rutherford equationThe magnetic structure of an island can be visualised by its electron temperature profile measuredby electron cyclotron emission (ECE) spectroscopy [2]. Further analysis can be performed with theFourier transform of the temperature profile [2, 6] in order to assess rres (taken as constant) and Wfor some timepoints. Together with the electron density (measured by interferometry, DCN deconvo-lution) Tet  from ECE gives the plasma pressure and thus βp  rres  t  . The pressure profile gives Lp(taken as constant). From Tet  also τR  t  can be derived using the neoclassical resistivity for small ε[10] and Ze f ft  . For ρp an averaged value of Te  rres  is used. Lq   0  3 m is taken from a q-profile us-ing a modelled current density. A typical value for W0 is 8mm (with χ   χ    1010). For the function∆ W  a constant value ∆0    m  rres is used (m is the poloidal mode number) [3].The island width evolution W  t  is reconstructed from its relative change measured by Mirnov coilsignals (which measure any change of the poloidal magnetic field that also contains the perturbedfield dB1  dt of the rotating island) and the absolute values measured by ECE. The Mirnov signalaccording to the observed (3,2)-modes is separated by singular value decomposition. The signal isintegrated and the envelope is taken in order to get the amplitudeB̂1t  W  t  2. ELM activity andsawteeth which show up in the Mirnov signal have to be removed. The seed island width can beassessed by the signal at mode onset.The Rutherford equation has to be fitted to the experimentally measured island dynamics in orderto obtain the values of the free parameters aBS, aGGJ and aPol . Since dW  dt cannot be taken from themeasurements of Wt  with sufficient accuracy due to signal noise the Rutherford equation has beennumerically solved for varying sets of parameters aBS, aGGJ and aPol using the measured βpt  . Thebest fits have been determined by application of a least squares fit neglecting the island growth phase(where Wt  cannot be measured with sufficient accuracy) and giving the same weight to the islanddecay phase as to the rest. Thus, aPol can be determined with more confidence since the polarisationterm is most effective with small W . Fitting has been done for four different β limit discharges inASDEX Upgrade (plasma current 800 kA, line averaged density 6  1019m  3, heating power 10 MW,toroidal magnetic field 2.05-2.17 T, q95  4  5). An example is shown in fig. 1. Fig. 2 shows theprojection of the fifty best parameter sets (aBS, aGGJ , aPol) for those discharges to the aBS-aGGJ andthe aBS-aPol plane. All these parameter sets yield very similar solutions of the Rutherford equation.The linear dependence of aGGJ on aBS is due to the 1  W dependence of both bootstrap and Glasser-Green-Johnson effect when W is large and W0 can be neglected. Thus, regarding the unevitableuncertainties of the different measurements entering into the plasma parameters of the Rutherford27th EPS CCFPF 2000;   J. P. Meskat et al.: Analysis of the Dynamics of Tearing Modes in ASDEX Upgrade 970equation it is not reasonable to look for the very best fit and hence only a range of the parameters canbe obtained.4.0 t [s]3.0 5.00.00.1W[m]βpBest fit:aBS=3.7  aGGJ=37  aPol=0.65#12238Best general fit: aBS=1.2  aGGJ=1  aPol=0.83.5 4.5Figure 1: Best fit of the Rutherford equation for discharge 12238 of ASDEX Upgrade. Also the bestgeneral parameter set for discharges 11657, 11671 and 12238 yields a good fit. For comparison βpt is shown (dotted).1.0 2.0 3.0 4.0aBS010203040aGGJ#11657 (50 best fits)#11659 (50 best fits)#11671 (50 best fits)#12238 (50 best fits)20 best general fits for#11657, 11671, 122381.0 2.0 3.0 4.0aBS02.0aPol1.01.50.5Figure 2: Best fits of the free parameters of the Rutherford equation for discharges 11657, 11659,11671 and 12238 of ASDEX Upgrade in projection to the aBS-aGGJ and aBS-aPol plane. Also shownare the best general parameter sets for discharges 11657, 11671 and 12238.For the determination of the free parameters of the Rutherford equation a general parameter set forall discharges has to be found. This has been done by minimisation of the averaged fitting errors ofthe fits for three of the discharges. As result we get some best general parameter sets also yieldingvery similar results. For an example and comparison of the parameters see figs. 1 and 2. The bestgeneral parameter set also gives fair fits for the discharges 11657 and 11671 which however do notmodel the island decay.Fig. 3 shows the stabilising effect of ECCD in the solution of eq.(1) for the same example asabove. Using the experimentally measured βpt  a value of IECCD  IP   0  8% is needed for modellingcomplete stabilisation. However, βp increases with decreasing W in real stabilisation experimentsrequiring higher values of IECCD  IP for complete stabilisation. Thus, a model of βp  W  should beimplemented for consistent simulation of ECCD stabilisation. Complete ECCD stabilisation of NTMshas been experimentally performed in ASDEX Upgrade with IECCD  IP   1  2% as calculated by27th EPS CCFPF 2000;   J. P. Meskat et al.: Analysis of the Dynamics of Tearing Modes in ASDEX Upgrade 971TORAY [11].4.0 t [s]3.0 5.00.00.1W[m]aBS=3.7aGGJ=37aPol=0.65#122383.5 4.5IECCD/IP=0.7%Modelled ECCD stabilisationIECCD/IP=0.8%IECCD/IP=0.0%, experimentFigure 3: Simulation of ECCD stabilisation by the numerical solution of the Rutherford equationwith the best fitting parameters and the measured βpt  for the discharge. Using IECCD  IP   0  7% theisland size is reduced, using IECCD  IP   0  8% complete stabilisation is attained.4. ConclusionsThe free parameters of the Rutherford equation have been assessed by fitting its solution to theexperimentally observed island width evolution for four experiments in ASDEX Upgrade. Becauseof the inaccuracies of experimental measurements only ranges for the parameters can be determined.Also a range of general parameters can be found that gives fair fits for three of the experiments. Thedetermination of the free parameters of the Rutherford equation is important for the computation ofthe required microwave power for ECCD stabilisation of neoclassical tearing modes in existing andplanned fusion experiments. With the determined parameters the Rutherford equation can be used inorder to simulate ECCD stabilisation.References[1] Zohm, H. et al., Plasma Phys. Control. Fusion 37 (1995) A313[2] Meskat, J. P. et al., 26th European Physical Society Conference on Controlled Fusion andPlasma Physics (Maastricht, 1999) P4.002[3] Sauter, O. et al., Phys. Plasmas 4 (1997) 1654[4] Zohm, H. et al., Plasma Phys. Controlled Fusion 39 (1997) B237[5] Rutherford, P. H., Phys. Fluids 16 (1973) 1903[6] Fitzpatrick, R., Phys. Plasmas 2 (1995) 825[7] Glasser, A. H. et al., Phys. Fluids 12 (1975) 875[8] Wilson H. et al., Phys. Plasmas 3 (1996) 248[9] Morris A. W. et al., Controlled Fusion and Plasma Physics, Proceedings of the 19th EuropeanConference, Innsbruck I (1992) 423[10] Connor J.W. et al., Nucl. Fusion 13 (1973) 211[11] Gantenbein G. et al. ’Complete suppression of NTMs with current drive at the electron cy-clotron resonance frequency in ASDEX Upgrade tokamak’, accepted for publication in Phys. Rev.Lett. (2000)

Analysis of the Dynamics of Tearing Modes in ASDEX Upgrade

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Analysis of the Dynamics of Tearing Modes in ASDEX Upgrade

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