Quantum chaos analysis of the ideal interchange spectrum in a stellarator

Transcription

1 4th Asia-Pacific Dynamics Days Conference, DDAP04 July 12 14, 2006 Quantum chaos analysis of the ideal interchange spectrum in a stellarator R. L. Dewar The Australian National University, Canberra C. Nührenberg Max Planck Institute for Plasma Physics, Greifswald T. Tatsuno University of Maryland, College Park

2 Plan of this talk n Introduction to COSNet n Evidence for quantum chaos in normal mode spectrum of an ideal-mhd interchange-unstable 3D plasma confinement system (stellarator) n Comparison with data from cylindrical model (i.e. integrable) with same q profile n Toy model explaining non-generic interchange spectrum found in integrable case [Dewar et al., Phys. Rev. E 70, (2004)] n Conclusion 2

3 Searchable: If you would like to make yourself known to the Australian complex systems community you are invited to go to and join!

4 COSNet Themes (help develop in the Wiki!) Irreversibility and emergence in nonequilibrium systems Turbulence and coherent structures, control and computation Dynamics and statistics of multi-scale systems Network theory Cellular automata, agent-based modelling and simulation 4

5 COSNet Application Areas 1. Complex Physical Systems 2. Complex Biological Systems 3. Complex Computational Systems 4. Complex Socio-Economic Systems 5

6 COSNet: Some Big Questions How did life emerge from primordial chaos? Can we tame a turbulent, burning fusion plasma to power our civilisation in coming centuries? Can we design an economic system that works without cyclic booms and busts? Will machines ever develop intelligence? 6

7 Tokamaks and Stellarators Tokamaks ~ axisymmetric toroidal plasmas e.g. KSTAR, Daejeon Stellarators ~ deliberately break axisymmetry e.g. H-1NF, Canberra 7

8 Plasmas complex fluids n Strongly nonequilibrium n Multiple time/space scales n Turbulent transport (quasi-2d in strong magnetic field) n However, will ignore all this and study ideal magnetohydrodynamic (MHD) model: no dissipation no displacement current adiabatic equation of state linearize about static equilibrium spectrum of unstable normal modes 8

9 History of quantum chaos n Wigner ( 30s) nuclear spectrum Wigner conjecture n Dyson & Mehta ( 60s Canberra Summer School 1967) improved on Wigner using eigenvalues of ensemble of random matrices n Berry & Tabor ( 70s), relation to semiclassical chaos; genericity in integrable and chaotic systems 9

10 Statistical characterization of spectra: probability distribution for eigenvalue separation P(s) sensitive test for chaos NB We suppose the spectrum has first been unfolded by transforming eigenvalues so that d<n(e)>/de ~ 1 for large E. Let P(s)ds be the probability of finding two consecutive eigenvalues E n a distance s apart: Generic integrable systems give Poisson distribution, as if random! (Eigenvalues uncorrelated) No avoidance of degeneracies Level repulsion Generic chaotic systems give dist n. like random matrices from a Gaussian Orthogonal Ensemble 10

11 Integrable WKB ray dynamics/separable wave equation Typical (?) generic case waves in a box: w 2 = (2p) 2 c s 2 ( m2 a 2 + n2 b 2 ) b w = const a Many accidental neardegeneracies fi most probable separation s = 0 fi Poisson distribution P(s) = exp(-s) 11

12 Caveat: Waves in box not (quite) generic Casati, Chirikov & Guarneri, Phys Rev Lett 54, 1350 (1985) 12

13 Quantum chaos in a stadium McDonald & Kaufman Phys Rev Lett 42, 1189 (1979) 13

14 Is MHD interchange instability spectrum generic?: Similarities & differences Like quantum, microwave & acoustics spectral problems, ideal MHD on static equilibrium is Hermitian fi real eigenvalues l (= w 2 unstable modes have w 2 < 0, w = ig). Unlike typical spectral problems the MHD spectrum is to be found from a generalized eigenvalue problem, Lj = lmj, in which L and M contain spatial derivatives of the same order. 14

15 3-D Geometry: numerical study of MHD spectrum of W7-X stellarator, Greifswald W7-X helias being built in Greifswald Mode families: 5-fold symmetry couples toroidal Fourier indices n±5. Restriction to real eigenfunctions couples ±n. Net effect is only 3 uncoupled subspaces. C. Nührenberg using CAS3D Galerkin code 15

16 To get many modes, choose Mercier (interchange) unstable case 16

17 Galerkin studies on Mercierunstable, W7X-like equilibrium Analysis of data set of CAS3D eigenvalues using m,n tableaux each with a full spectrum in n corresponding to the mode families N fam = 0, 1, 2 and running up to n =n max. For each n, sufficient m s are used to capture all m + nq = 0 possibilities for q = 1/i range within plasma, plus some more. i axis = i min = i edge = Resonance condition is n + m i = 0. This tableau is for n max = 21, 22, 25, resp. 17

18 CAS3D data for n max = 19 Analysis of data set of 96 CAS3D eigenvalues found for case n max = 19 using mode tableau with 88 Fourier harmonics. N = 1 is most unstable eigenvalue (most ve l), N = 96 is the least unstable. fit to d 0 expd 1 l +exp(c 0 + c 1 l + c 2 l 2 + c 3 l 3 ) Log-linear plot of raw data After unfolding using double-exponential fit 18

19 Level spacing PDFs for the three mode families Strong level repulsion in N = 0 and N = 1 families. Is N = 2 spectrum mixed? n max = 25 N modes = 137 n max = 22 N modes = 178 n max = 21 N modes = 214 Combining spectra from independent subspaces greatly reduces level 19 repulsion

20 Refined Grid: 451 radial grid points N = 1 case little changed still close to Wigner conjecture N = 2 case now shows level repulsion now close to Wigner conjecture 20

21 Results robust wrt unfolding method 21

22 So 3D MHD interchange spectrum is QC-Generic? Why this is surprising n Our recent work on integrable case indicated non-genericity (see later in this talk) n Our earlier work on a different stellarator indicated a regular spectrum (next slide) 22

23 WKB quantization for ideal ballooning modes in LHD TERPSICHORE (dots)/wkb (lines) comparison for LHD extended modes (Cooper, Singleton & Dewar PP 96) Ray eqns. ~integrable for tokamaklike modes Agrees with EBK quantization (with k a = n) 23

24 MHD separable case: Interchange modes in (effectively) cylindrical geometry Averaging over field periods and introducing stream function j with dependence j(r,t) = j(r)exp(gt + imq - inz /R 0 ) eigenmode equation: is Suydam instability criterion: Denote radial quantum no. by l 24

25 Accumulation points very different from generic waves in a box spectrum Eigenvalue g depends on 3 quantum nos., l, n, m. But asymptotically at large m, n, it depends only on l and m n/m fi accumulation points at each rational m as n, m Æ m = i(1) - m = i(0) - m n/m Over finite range, only l = 0 contributes to spectrum m max Accumulation point at g = 0 as l Æ m, n lattice l, m dependence 25

26 Finite m, n spectral plot for l = 0 accumulation point Lower accumulation sequence from n and m in exact 4/7 ratio S 0 - S 0 + upper accumulation sequence from approximants of 4/7 26

27 Spacing distribution for l = 0 is non-poissonian (bimodal) 27

28 Toy interchange spectrum model for a given radial mode number l y = expi(mq + nf) E n,m = n m E n,m = n m = const In limit m max Æ, the spectrum is infinitely dense and infinitely degenerate fi essential spectrum fi need to regularize by using cutoff m max or equivalent. 28

29 Farey sequences of rationals: drop duplicates (use p, q instead of n, m) F 1 F 8 F 8 = 0 1, 1 8, 1 7, 1 6, 1 5, 1 4, 2 7,1 3, 3 8, 2 5, 3 7, 1 2, 4 7, 3 5, 5 8, 2 3, 5 7, 3 4, 4 5, 5 6, 6 7, 7 8,1 1 Gaps occur around low-order rationals due to slow (1/Q) convergence of nearest approximants Each term in a given Farey sequence is the mediant,, of its neighbors: p 1 Number of terms ~ 3Q 2 /p 2 p 2 p 1 + p 2 q 1 q 2 q 1 + q 2 29

30 Devil s staircase and levelspacing distribution Renormalize (unfold) energy levels to make average slope unity: 2 0 n < m m max E n,m = m max 2 2 E n,m Level-spacing distribution: degeneracy fi delta function at origin. We can understand the tail by considering PDF for Farey sequences 30

31 Farey spectrum contd. 31

32 Back to stellarators: Cylindrical model with same rotational transform profile as 3D case studied (Vol. averaged beta 5%) Eigenfunction for m = 16, n = 17 mode 32

33 Unfolding cylindrical interchange spectrum Log-lin. plot of data with doubleexponential fit similar to W7X spectrum near origin (accumulation point as radial mode no. l Æ ) After unfolding using double-exponential fit 33

34 Spacing PDFs with all radial mode numbers included now look Poissonian 34

35 Conclusion n When all unstable eigenvalues are included, ideal-mhd interchange spectrum appears to generate a generic eigenvalue-spacing probability distribution function Poisson in cylindrical (integrable) case Wigner/GOE in strongly nonaxisymmetric stellarator case fi MHD modes can exhibit typical quantum chaos behaviour 35

36 To dos n Investigate 3D mode structure in W7X to understand better why it is strongly quaotic, despite extended nature of interchange modes along field lines Plots Semiclassical (WKB ballooning) analysis n Understand why double exponential fit works for density of states n Analyze 3D drift wave spectrum how does quantum chaos interact with turbulence? n Develop chaotic toy model, find Riemannium Hamiltonian! 36