J. Plasma Physics (2006), vol. 72, part 6, pp. 1239–1242.

c 2006 Cambridge University Press

1239

doi:10.1017/S0022377806006015 Printed in the United Kingdom

Quantum chaos analysis of the idealinterchange spectrum in a stellarator
R. L. D E W A R,1 C. N Ü H R E N B E R G2 and T. T A T S U N O3
1

Department of Theoretical Physics and Plasma Research Laboratory,
Research School of Physical Sciences and Engineering, The Australian
National University, Canberra ACT 0200, Australia(robert.dewar@anu.edu.au)
2Max-Planck-Institut für Plasmaphysik, Teilinstitut Greifswald IPP-EURATOM
Association, D-17489 Greifswald, Germany
3
Center For Scientiﬁc Computation and Mathematical Modeling, University of Maryland,College Park, MD 20742-3289, USA(Received 17 August 2005 and accepted 2 February 2006)

Abstract. The eigenmode spectrum is a fundamental starting point for the analysisof plasma stability and the onset of turbulence. Quantum chaos theory provides
tools for characterizing the spectrum statistically, from the regular spectrum of theseparable case (integrable semiclassical dynamics) to that where the semiclassicalray dynamics is so chaotic that no simple classiﬁcation of the individual eigenvalues is possible (quantum chaos). Using the CAS3D code, a data set of several
hundred growth-rate eigenvalues has been calculated for a Mercier-unstable threedimensional stellarator equilibrium with a rather ﬂat, non-monotonic rotationaltransform proﬁle. Statistical analysis of eigenvalue spacings for individual modefamilies shows evidence of quantum chaos, strongest for the N = 0 family, but totest this we compare it with the distribution of eigenvalue spacings in a similar
separable case—ideal interchange modes in a Suydam-unstable plasma cylinder—using a similar rotational transform proﬁle to the stellarator case. The statistics inthe cylindrical model appear Poissonian, as expected for generic integrable systems
and in clear contrast to the three-dimensional stellarator results.

1. Introduction
Numerical matrix eigenvalue programs, such as the three-dimensional TERPSICHORE [1] and CAS3D [2] codes, are routinely used to assess the ideal magnetohydrodynamic (MHD) stability of candidate conﬁgurations for proposed fusionrelevant experiments with complicated geometries. An example is in the designof the ﬁve-fold-symmetric Wendelstein 7-X (W7-X) stellarator, currently underconstruction, where CAS3D was used [3] to study a number of different cases.
The three-dimensional nature of stellarator equilibria breaks all continuous symmetries, coupling both poloidal (m) and toroidal (n) Fourier harmonics. This precludes separation of variables and simple classiﬁcation of the eigenvalues. As aconsequence, the nature of the spectrum is still imperfectly understood, making
both the physical interpretation of stability calculations and the rigorous numericalanalysis of convergence difﬁcult.

1240

R. L. Dewar, C. Nührenberg and T. Tatsuno
Mercier criterion

rotational transform

5/4
10/9

5/5

00.5
normalized toroidal flux s(a)1–1–5–10–50–100

00.5

1

normalized toroidal flux s(b)Figure 1. (a) The rotational transform ι- = ι-(s) proﬁle versus normalized toroidal ﬂux s (s ∝ r2
near the magnetic axis). (b) A measure of the Mercier stability versus normalized toroidal
ﬂux for our unstable W7-X variant equilibrium. (A negative value indicates instability.)

The optimized conﬁgurations used in designing W7-X are MHD stable, but inthe present paper we are concerned with a variant, unstable equilibrium fromthis study, a high-mirror-ratio, high-rotational transform case. Its less-pronounced
shaping makes it quite unstable, thus allowing us to perform a numerical experiment designed to explore some fundamental properties of the MHD spectrum instellarators without the added complication of the stable Alfvén spectrum.
The eigenvalues, ω 2 ≡ −γ 2 , are real due to the self-adjointness of the force andkinetic energy operators in ideal MHD, linearized about a static equilibrium. This isanalogous to the Hermitian nature of quantum mechanics. Thus, we study the W7X Mercier (interchange)-unstable case mentioned above using statistical techniquesfrom the theory of quantum chaos (see, e.g., [4]).
As a non-chaotic reference case we also study, following [5], the interchange
unstable spectrum in an effectively cylindrical model of a stellarator. In the cylindrical case the eigenvalue problem is separable into three one-dimensional eigenvalue problems, with radial, poloidal, and toroidal (axial) quantum numbers
l, m, and n, respectively. If the spectrum falls within the generic quantum chaostheory universality class for integrable, non-chaotic systems, then the probability
distribution function for the separation of neighboring eigenvalues is a Poisson
distribution [4].
In [5] we found that the spectrum for the most unstable ideal MHD interchange modes in a cylinder, those with radial mode number l = 0, gave highly nonPoissonian statistics (when we used a sharp truncation in m and n). However, it islikely that the statistics of the full spectrum, including the many weakly unstablemodes with l > 0 near the interchange accumulation point at the origin, is closer toPoisson statistics because of the occurrence of many accidental degeneracies.
In the present paper we compare the results of the quantum chaos analysis of
the W7-X N = 1 mode family [2] with eigenvalues from a cylindrical model withsimilar rotational transform proﬁle.

2. Numerical results
The magnetic ﬁeld lines of the W7-X variant equilibrium studied are taken tolie on nested toroidal ﬂux surfaces labelled by s ∈ [0, 1], the enclosed toroidal
magnetic ﬂux divided by the toroidal ﬂux enclosed by the plasma boundary. The
magnetic ﬁeld is characterized on each ﬂux surface by its winding number ι-(s). As
seen in Fig. 1, the rotational transform proﬁle is non-monotonic and has low shear

Quantum chaos analysis in a stellarator

1241

Figure 2. (a) Unfolded eigenvalue spacing distributions for W7-X mode family N = 1,calculated using two different unfolding methods. (b) The same statistic from the cylindricalcomparison case.

(ι-axis = 1.1066, ι-min = 1.0491, ι-edge = 1.0754) so it is close to, but greater than, unity
over the whole plasma. As also seen from Fig. 1, the equilibrium is interchange
unstable because the Mercier stability criterion is violated over the whole plasma.
In the CAS3D code, the radial dependence of the perturbation functions is treatedby a hybrid ﬁnite-element approach, using a linear interpolation for the normal
displacement and piecewise constant interpolations for the scalar components that
describe the MHD displacement within the magnetic surfaces. Fixed boundary conditions were used with incompressible perturbations and a non-physical kinetic energy normalization. (As the eigenvalues are renormalized—‘unfolded’ in quantum
chaos language—to give an average separation of unity, the precise normalization
used in the code is not important for the statistics.) In the calculations discussed
here, 301 radial grid points were used, but a run with 451 grid points gave thesame qualitative results. For the plot in Fig. 2, 99 Fourier harmonics were used,
distributed with half-bandwidth 5 about n/m = 1, where n and m are the toroidal
and poloidal Fourier indices, respectively, with nmax = 21. There were 214 unstableeigenvalues found in this case, and all were used in the statistical analysis.
Owing to the ﬁve-fold symmetry of the equilibrium, any toroidal Fourier harmonic n in an eigenfunction is coupled to toroidal harmonics n ± 5. With thepoloidal harmonics chosen to be positive, m ≥ 0, there are just three uncoupledmode families N = 0, 1, 2 (cf. [2]).
After unfolding the eigenvalues so that their average separation is unity, wecharacterize the statistics of the ensembles of eigenvalues within the N = 1 modefamily using a standard statistic from quantum chaos theory [4, 6], the probability
distribution function, P (x), for the eigenvalue separation x, shown in Fig. 2.
As seen from Fig. 2(a), the eigenvalue spacing distribution function for the W7-X
case is closer to the form found in generic chaotic systems [4] than to the Poisson
distribution for separable systems, giving support for the quantum chaos predictionof [7]. The same is true for the other two mode families. (Although the N = 2 caseappeared to deviate signiﬁcantly from the Wigner-conjecture distribution using
301 grid points, when the number was increased to 451 grid points, giving 402
eigenvalues, the distribution became much closer to quantum chaotic.)
To test the sensitivity to the precise method of unfolding chosen, we calculated
the statistics using two different methods. They are the Gaussian unfolding method[6] and a ﬁt with exponentials. The results, shown in Fig. 2(b) are seen to be1242

R. L. Dewar, C. Nührenberg and T. Tatsuno

consistent to within statistical error, indicating little sensitivity to the unfolding
method.
The calculations for the cylindrical reference case were done with a shooting
code, so the radial resolution was essentially inﬁnite. To keep the set of eigenvalues
ﬁnite, and to model the ﬁnite radial resolution of CAS3D, only at most 101 radial
eigenmodes were kept for each unstable (m, n), giving a dataset of 907 eigenvalues,
200 of which are non-resonant. Figure 2(b) shows the statistics to be close toPoissonian.

3. Conclusion
Although space has precluded us from showing the results for the N = 0 and 1 modefamilies, in all three families the spacing distributions in the W7-X case are close tothat expected from quantum chaos theory, while those for the cylindrical referencecase are close to the Poisson distribution. We therefore conclude that interchange
instabilities in the W7-X helias are strongly three-dimensional.
This contrasts with earlier work on the 10-ﬁeld-period Large Helical Device
heliotron/torsatron [8, 9] where the interchange spectrum was found to be rather
regular, which was interpreted as being due to the averaging effect of the interchange eigenfunctions being extended over a large number of ﬁeld periods.
Acknowledgement
RLD acknowledges support by the Australian Research Council.

References[1] Anderson, D. V., Cooper, W. A., Gruber, R., Merazzi, S. and Schwenn, U. 1990 Int.
J. Supercomput. Appl. 4, 34.
[2] Schwab, C. 1993 Ideal magnetohydrodynamics: global mode analysis of threedimensional plasma conﬁgurations. Phys. Fluids B 5, 3195–3206.
[3] Nührenberg, C. 1996 Global ideal magnetohydrodynamic stability analysis for theconﬁgurational space of Wendelstein 7-X. Phys. Plasmas 3, 2401–2410.
[4] Haake, F. 2001 In: Quantum Signatures of Chaos, 2nd edn. Berlin: Springer.
[5] Dewar, R. L., Tatsuno, T., Yoshida, Z., Nührenberg, C. and McMillan, B. F. 2004
Statistical characterization of the interchange-instability spectrum of a separable idealmagnetohydrodynamic model system. Phys. Rev. E 70, 066409.
[6] Bruus, H. and d’Auriac, J.-C. A. 1997 Energy level statistics of the two-dimensional
Hubbard model at low ﬁlling, Phys. Rev. B 55, 9142–9159.
[7] Dewar, R. L., Cuthbert, P. and Ball, R. 2001 Strong ‘quantum’ chaos in the global
ballooning mode spectrum of three-dimensional plasmas. Phys. Rev. Lett. 86, 2321.
[8] Cooper, W. A., Singleton, D. B. and Dewar, R. L. 1996 Spectrum of ballooning instabilities
in a stellarator. Phys. Plasmas 3, 275–280 (Erratum: 1996 Phys. Plasmas 3, 3520).
[9] Cuthbert, P., Lewandowski, J. L. V., Gardner, H. J., Persson, M., Singleton, D. B., Dewar,R. L., Nakajima, N. and Cooper, W. A. 1998 Toroidally localized and nonlocalizedballooning instabilities in a stellarator. Phys. Plasmas 5, 2921–2931.