Kendl, A.

Wobig, H.

English

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27th EPS Conference on Contr. Fusion and Plasma Phys.  Budapest, 12-16 June 2000    ECA Vol. 24B (2000) 996-99999627th EPS Conference on Contr. Fusion and Plasma Physics, Budapest, Hungary, 12.-16- June, 2000Drift Waves and Test Particle Transport in a StellaratorA. Kendl, H. WobigMax-Planck-Institut fr Plasmaphysik, EURATOM Association,D-85748 Garching bei Mnchen, Germany1. IntroductionElectrostatic drift waves are a major candidate to cause anomalous transport in toroidaldevices. In assessing the effect of drift waves on particle orbits one must distinguish betweencirculating particles travelling around the torus and trapped particles localised in magneticmirrors. The present paper deals with circulating particles in stellarators and computes theorbits in guiding center approximation. Because of the small gyro radii drift surfaces andmagnetic surfaces differ only little and the drift rotational transform is close to the magneticrotational transform. Islands of magnetic surfaces will also lead to corresponding islands ofdrift surfaces. The parallel electric field of the drift surfaces modifies the energy of theparticles, which introduces the effect of Arnold diffusion [1] as the combined effect of driftislands and time-varying electromagnetic fields. Recently J.M. Kwon et al.[2] haveinvestigated the effect of drift waves on circulating particles in tokamaks using the mappingtechnique. Since the authors assume energy conservation particle orbits can be described bytwo coupled equations of an area preserving map. However, violation of energy conservationby time-varying electric field destroys this property and leads to four coupled equations [3].In the following we follow this line and develop a set of mapping equations describingcirculating particles in a Helias configuration [4]. Nonlinear drift waves in Heliasconfigurations have been computed applying the turbulence code of B. Scott [5].2. The modelGuiding center orbits in toroidal geometry can be described in terms of Hamiltonianmechanics in a 4-dimensional phase space. Reduction to 4 canonical variables has beendiscussed by Meiss and Hazeltine [6], who showed that a canonical coordinate system can befound where the covariant Br-component can be eleminated by a proper choice of the angularcoordinates. Neglecting the magnetic part of the drift waves leads to a time-independentcoordinate system, otherwise the canonical coordinate system is time-dependent (see ref. 3).The Hamiltionian H0+H1 can be written in the form  Hmu p p B q H h q020 12= ( ) + + = +q j q j m d d, , , ;F F    Eq. 1u is the parallel velocity, m is the magnetic moment. The poloidal canonical momentum pq isroughly proportional to the radial coordinate r if the poloidal magnetic field is smallcompared with the toroidal one. The time-dependent part of the electric potential dF is theeffect of the electrostatic drift waves, which in the following will be considered as a smallperturbation. dh is the time-independent perturbation, which causes islands of the driftsurfaces. The four canonical equations can be written in conjugate variables pq,pj,q and j.However, in case of circulating particles the toroidal velocity is not zero, which allows one tointroduce the energy E and the time t as conjugate variables and use the toroidal angle j asthe independent variable. To do so we invert the Hamiltonian and write the toroidalmomentum pj as function of energy E and the other variables p j = -K(E,t,pq,q,j), whichdefines a new Hamiltonian K. Since the time-dependent term in Eq. 1 is small we obtain thefollowing approximation of the new Hamiltonian27th EPS CCFPF 2000;   H. Wobig et al.: Drift waves and test particle transport in a stellarator 99727th EPS Conference on Contr. Fusion and Plasma Physics, Budapest, Hungary, 12.-16- June, 2000  K K p E K p E t KqH p= ( ) + ( ) =¢ ( )0 1 1 0q qjq j q jd, , , , , , , ;    FEq. 2K0 is defined by E = H0(pq,q,pj=-K0,j). The denominator in the perturbation K1 is the toroidalangular velocity of the particles, which shows that fast particles are less affected by the elec-trostatic waves than slow particles. The unperturbed Hamiltonian K0 describes unperturbeddrift surfaces characterised by the two ÒfrequenciesÓ  i pq , E( ) =¶K0¶pq   ;   T pq , E( ) =¶K0¶E  Eq. 3T(pq,E) is the toroidal transit time of circulating particles and it decreases with energy. i(pq,E)is the drift rotational transform. Particles circulating around the torus having initial valuesq0,t0,p0,E0 will change these values to q1,t1,p1,E1 after one toroidal transit j Ñ>j+2p. The mapdescribing this transformation has the generating function  S S p E t p E t K E p S p E t= ( ) = + + ( ) + ( )1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 0, , , , , , ,q q q Eq. 4The four mapping equations are (Eq. 5)  E EStt t T E pSEp pSp ESp0 1101 0 1 1110 1101 0 1 111= + = + ( ) + = + = + ( ) +¶¶¶¶¶¶qq q i¶¶  ;      ;      ;       , ,The perturbation S1 is uniquely determined by the perturbation of the Hamiltonian K1. K1 de-scribes all possible drift wave spectra in a plasma. Instead of specifying K1 one can as wellspecify the action S1 and study the generic behaviour of transit particles under the influence ofS1. Equations 5 are two couples area-preserving maps which in case of time-independent per-turbations are reduced to one area-preserving map. The general form of the perturbation  S S p E im in il tmnlmnl1 = ( ) + +[ ]åRe , exp j q w    Eq. 6w is the fundamental frequency of the drift wave. The relation between S1 and K1 is discussedin ref. 3. Due to the periodicity of the perturbation resonances occur and the resonancecondition is n i(p,E) + lwT(p,E) - m = 0. For given numbers m,n,l this is a curve in the p-E-plane and all curves together represent the Arnold web. Without time-dependent perturbations(wT=0) resonance only occurs on rational drift surfaces and enhanced radial losses can onlyarise if island overlap leads to a strong stochastisation of the particle orbits. Any time-dependent perturbation, however, can lead to a migration of the particles along the lines of theArnold web and to radial loss of the particles.3. Helias configurationsIn Helias configurations like Wendelstein 7-X the magnetic shear is very small and weapproximate the drift rotational transform by the magnetic transform. Furthermore, we adopt asimple model of the toroidal transit time, which is inversely proportional to the squre root ofthe energy.  The rotational transform varies between 0.83 in the center and 1.0 at theboundary. Islands of drift surfaces Ñ like in magnetic surfaces Ñ arise at i = 5/6 and i = 5/5.  n p l T EEEmi di w0300 0+( ) + ( ) - = Eq. 7di is a measure of the shear, which is small in Helias configurations.27th EPS CCFPF 2000;   H. Wobig et al.: Drift waves and test particle transport in a stellarator 99827th EPS Conference on Contr. Fusion and Plasma Physics, Budapest, Hungary, 12.-16- June, 20000.0 0.2 0.4 0.6 0.8 1.0E0.00.20.40.60.81.0P0.0 0.2 0.4 0.6 0.8 1.0E0.00.20.40.60.81.0PFig. 1: Left: Densely covered Arnold web with a number of resonances that may lead to rapidlosses for mode numbers around n=50, m=150 and lwT = 90. Steep lines in the Arnold webcan also originate from several low mode number perturbations (e.g. n=1, m=2, lwT = 1).Right: Arnold web for W7-X with mode numbers n=50, lwT =±1 and various numbers m =40...48.Numerical computations of the drift-wave turbulence in Helias configurations exhibit a richspectrum of mode numbers. The next figures show the Arnold web for high mode numbers,which are of relevance in drift wave turbulence. For numerical calculations we write eq.8 inthe form  S K p Emba a blmnlmn11= ( ) ( ) - +( )[ ]å , cos cos Eq. 8a = nq0+lwt0 and b = (-m+ni+lwT) and K1lmn as in eq.3. An additional mode number dependentamplitude factor f(m,n,l,p) represents spectral and radial mode structure variations. Theresulting mapping equations are written in dimensionless form, with e.g. energies normalizedto thermal energy and frequencies w Ñ>wa0/cs. All free parameters are determined by linearor nonlinear drift wave calculations in Helias geometry. Typical linear mode numbers in [7]are m » n » 50 and frequencies in the order of the diamagnetic frequency of electrons. Forsimplicity we here assume a radially constant mode amplitude and no static radial electricfield. Particles are launched at specified radial position and initial energy and circulate Ntimes around the torus. For several sets of parameters (see captions) we show results of ourtest particle simulations with regard to confinement time and time-dependent radial excursion(N-p plot) in the following figures.   27th EPS CCFPF 2000;   H. Wobig et al.: Drift waves and test particle transport in a stellarator 99927th EPS Conference on Contr. Fusion and Plasma Physics, Budapest, Hungary, 12.-16- June, 2000Fig. 2: Left: Test particle excursions for different values of electric potential perturbationamplitude (EPS): An increase of EPS leads to an overlap of the trajectory with a resonance inthe Arnold web and rapid radial losses. All particles are launched at P0=0.5, w = 0.5 and E =1.0. Right: Comparison of test particle excursions pq(N) for a constant frequency electricpotential perturbation wa0/cs = 0.2 (black), and with a 1/w type spectrum (as found in drift-Alfvn turbulence simulations and also experiments in the plasma edge of stellarators) of thesame integrated amplitude (red). The presence of a whole spectrum of frequencies leads to ahigher probability of hitting an Arnold web resonance. The abcissa is the number of toroidaltransits N.   0 1 2 3 4E0 / Eth103104105106 / E1/2Fig. 3: Left: The influence of ergodization around static magnetic 5/6 islands on trajectoriescan be seen by launching the test particles from different radial positions with a drift waveperturbation amplitude that is sub-critical for most of the start positions P0. (wa0/cs =0.5).Right: Dependence of the test particle confinement time t(E0) in the W7-X map on initialenergy E0. Test particle orbits are launched at half radius and are subject to an electrostaticperturbation with intensity qdF/Eth = 0.1. The perturbation frequency wa0/cs = 0.65 has beenobtained by linear eigenmode calculation in Wendelstein 7-X geometry.4. SummaryNumerical calculations have been carried out for test particletransport in a low-shear stella-rator magnetic field with time-dependent electrostatic drift wave perturbations. Trajectoriesobtained by mapping techniques are not self-consistent, in the sense that particle motion doesnot act back on the perturbed electric potential. However, they allow for investigation of longtime scale test particle behaviour, when the perturbation is prescribed by predetermined linearor nonlinear drift modes. Time-dependent perturbations together with variable energy lead toenhanced diffusion described by migration on an Arnold web. Our simulations have assumedradially local mode structure, and global drift waves together with equilibrium radial electricfield will be treated in a future publication.                                                 [1] A.J. Lichtenberg, Phys. Fluids B4, (10) Oct. 1992, 3132[2] J.M. Kwon, W. Horton, P. Zhu, P.J. Morrison, H.B. Park, D.I. Choi, Phys. Plasmas, Vol. 7,No.4 (2000) 1169[3] H. Wobig , D. Pfirsch, IPP-report IPPIII/245[4] J. Nhrenberg and R. Zille, Phys. Lett. A 114, 129 (1986).[5] B.D. Scott, Plasma Phys. Control. Fusion 39, 471 (1997)[6] J.D. Meiss, R.D. Hazeltine, Phys. Fluids B2, (1990) 2563[7] A. Kendl and H. Wobig, Phys. Plasmas 6, 4714 (1999)

Drift Wave and Test Particle Transport in Stellarators

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Drift Wave and Test Particle Transport in Stellarators

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