Computer Physics Communications 141 (2001) 55–65www.elsevier.com/locate/cpc

Ideal MHD stability calculations for compact stellarators

R. Sancheza,∗, M.Yu. Isaevb, S.P. Hirshmanc, W.A. Cooperd, G.Y. Fue, J.A. Jimenezf,L.P. Kue, M.I. Mikhailov b, D.A. Monticelloe, A.H. Reimane, A.A. Subbotinb

a Departamento de Física, Universidad Carlos III de Madrid, Madrid, Spainb Nuclear Fusion Institute, RRC “Kurchatov Institute”, Moscow, Russia

c Oak Ridge National Laboratory, Oak Ridge, TN, USAd Centre de Recherches en Physique des Plasmas, Association Euratom-Suisse, Ecole Polytechnique Federale de Lausanne,

CRPP-PPB, Lausanne, Switzerlande Princeton Plasma Physics Laboratory, Princeton, NJ, USA

f Asociación Euratom-CIEMAT, Madrid, Spain

Received 9 March 2001; received in revised form 14 May 2001; accepted 24 May 2001

Abstract

Stability results for high-n ideal local pressure-driven instabilities (ballooning and interchange modes) are calculated fromthe COBRA and TERPSICHORE codes and compared for several low aspect ratio stellarators. Such a comparison is importantbecause of the predominant roles that these codes are playing in the design of compact stellarators at several laboratories aroundthe world. The code development required to reach the levels of convergence and accuracy needed for reliable operation at lowaspect ratios is also described. 2001 Elsevier Science B.V. All rights reserved.

PACS: 02.60.Lj; 52.35.Py; 52.55.Mc; 52.55.Kj

Keywords: Stellarators; Low aspect ratio; Magnetohydrodynamics; Ideal pressure-driven instabilities; Interchange and ballooning modes;Magnetic coordinates

1. Introduction

The stellarator concept was first proposed manyyears ago [1]. However, renewed interest in thisconcept as a possible means to a future fusion reactorhas increased during the last decade due to at least twofactors:(1) the possibility of reducing neoclassical losses

to levels comparable to equivalent tokamaks byusing magnetic fields exhibiting so-called quasi-symmetries [2] and

* Corresponding author.E-mail address: rsanchez@fis.uc3m.es (R. Sanchez).

(2) the increasing availability of faster and more so-phisticated numerical tools that can be used withina feasible numerical optimization scheme [3,4] toidentify those quasi-symmetric 3D configurationssimultaneously exhibiting other convenient phys-ical properties (including but not limited to equi-librium, stability, bootstrap current, and coil feasi-bility).

Compact stellarators have received a lot of atten-tion due to the smaller size (and cost) of the reac-tors that might originate from them. These stellara-tors have aspect ratiosA < 4, with A ≡ R0/a the ra-tio of the major to the plasma radius. In comparison,

0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0010-4655(01)00396-4

56 R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65

the state-of-the-art stellarators in the world, the Wen-delstein 7-X stellarator in Germany [5] and the LargeHelical Device (LHD) in Japan [6] have much largeraspect ratios,A∼ 12 andA∼ 8, respectively. Severallaboratories throughout the world are currently pur-suing compact stellarators: quasi-axisymmetric stel-larators (QAS) are being designed at both the Prince-ton Plasma Physics Laboratory (the National CompactStellarator Experiment or NCSX) [7,8] and at the Na-tional Institute for Fusion Science in Japan (the CHS-qa, a quasi-axisymmetric variation of the Compact He-lical System (CHS) already in operation) [9]); the OakRidge National Laboratory has instead developed a re-lated concept based on quasi-omnigeneity [10], andis designing a quasi-omnigeneous compact stellara-tor (QOS, recently renamed QPS since further opti-mization has made the design become quasi-poloidallysymmetric) [11,12].

The search for attractive compact configurationsmust be carried out numerically using fully three-dimensional (3D) codes, since the conditions that val-idate the use of analytical approximations [13] ortoroidally-averaged 2D codes [14], i.e.A � 1 andι/N � 1 (N is the periodicity andι the rotationaltransform), are strongly violated. In addition to themore extreme geometry compared with larger as-pect ratio stellarators, the nature of the optimizationmethod used in the design [4] also makes these calcu-lations even more demanding in terms of convergenceand accuracy, especially with respect to stability: aninaccurate assessment can point the optimizer towardsthe wrong (and more unstable) path in parameter spaceand render the optimization meaningless!

This situation has stimulated the development ofmany existing codes and the creation of some newones [4,7,8,12], not limited to the stability codes wewill focus on in this paper. As part of this effort,a new ideal ballooning code, COBRA [15,16], hasbeen created which is sufficiently fast and numericallyreliable to be included into the optimization process(see Section 3). It has been specifically developedto converge well at low aspect ratios and is able toavoid many of the convergence and accuracy problemsthat affect the other widely used stellarator stabilitycodes. These problems manifest themselves in severalways: (1) as a lack of convergence and/or accuracyof the equilibrium solution, usually obtained by theVMEC equilibrium solver [17] or/and (2) a lack

of convergence and/or accuracy in the conversionof the VMEC solution into the coordinate systemused by the stability code, usually that introducedby Boozer [18] (for instance, this is the case ofthe TERPSICHORE [19], CAS3D [20] or JMC [21]codes). COBRA is only affected by the first problem,since it uses the same magnetic coordinates as VMEC.But the recent improvements of the VMEC code,briefly reviewed in Section 2, have enhanced itsreliability even further for use within the optimizationprocess.

COBRA’s performance is however a consequenceof its high degree of specialization: its domain of ap-plicability is limited to high-n local pressure-drivenmodes, namely ballooning and interchange modes.These are usually the limiting instabilities for nearlycurrentless configurations (this is the case of mostQOS designs). But NCSX configurations usually havetoroidal currents of hundreds ofkA flowing in theplasma, which can drive current-driven modes that setthe critical 〈β〉 in some cases. Some control of thekink instability during the optimization process is ac-complished by using an approximate analytic crite-rion [22], but quasiaxisymmetric cases usually requirethe use of the TERPSICHORE code within the op-timization for these global modes. In any case, withindependence of the underlying symmetry, a carefulpost-optimization analysis of the final optimized con-figuration is required to confirm the accuracy of its sta-bility properties. This is usually done using TERPSI-CHORE, which can carry out a very complete range ofstability calculations, including global and local analy-sis, low-n and high-n pressure-driven and current-driven modes and fixed- and free-boundary calcula-tions. But TERPSICHORE’s performance can also beaffected by the accuracy of the Boozer reconstruction(in addition to the accuracy of the VMEC equilibriumsolution), which seems to be critical at low aspect ra-tios as indicated by the appearance of unphysical (ornumerical) unstable modes in some of the NCSX con-figurations. A method for avoiding these modes hasbeen implemented and is described in Section 4.

The complementary roles that COBRA and TERP-SICHORE play within the compact stellarator designprocess thus makes the consistency of their resultsnot only reassuring but essential. Clarifying such acomparison is the main purpose of this paper, andit is carried out in Section 5 using two compact de-

R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65 57

signs obtained from this optimization effort: the quasi-axisymmetric C82 configuration of the NCSX projectat Princeton [7] and a high-ι case of the QOS project atOak Ridge [4]. The comparison is by no means trivialsince both codes address the high-n stability calcula-tions in different ways, using different normalizationsand different underlying equilibrium solutions due tothe intermediate mapping to Boozer coordinates re-quired by TERPSICHORE. In addition, the computedeigenvalues are not equivalent, which has caused someconfusion in the past when interpreting their results.But this paper will show that despite the vast differ-ences in approaches used by these stellarator stabilitycodes, the results from both of them can be reliablyused to compute high-n stability properties of low as-pect ratio stellarators. Finally, some conclusions willbe presented in Section 6.

2. Modifications of the VMEC code

The VMEC code [17], which uses a conjugate gra-dient method to solve the MHD inverse equilibriumequations, has been redifferenced to improve the con-vergence at lower aspect ratios and as well as for equi-libria with a wider range of rotational transform pro-files. At lower aspect ratios, the enhanced toroidal cou-pling of modes requires finer angular meshes (morepoloidal modes) than was feasible in earlier versionsof VMEC. In VMEC, the “inverse” equilibrium equa-tions are cast as second order equations (with radiusas the independent variable) for the Fourier compo-nents of the cylindrical coordinatesR andZ, andµ,the renormalization stream function [notice that thisstream function is referred to asλ in Ref. [17], but weprefer to useµ to avoid confusion with the eigenvaluesto be introduced later on]:

R(s, θ,φ)=∑mn

Rmn(s)cos(mθ − nφ),

Z(s, θ,φ)=∑mn

Zmn(s)sin(mθ − nφ), (1)

µ(s, θ,φ)=∑mn

µmn(s)sin(mθ − nφ),

where (s, θ,φ) is a set of magnetic coordinates inwhichφ coincides with the geometrical toroidal angle,s is a radial coordinate varying from 0 at the magneticaxis to 1 at the last closed magnetic surface (plasma

boundary), andθ is a poloidal angle determined toaccelerate the convergence inm-space of the Fourierseries in Eq. (1).

In previous versions of VMEC,µ was differencedradially (in s) on a mesh centered betweenR, Znodes, which greatly improved the radial resolution.This could be done to second order accuracy (inhs ≡ 1/(Ns − 1), with Ns the number of magneticsurfaces in the computational grid), since no radialderivatives ofµ appear in its determining equation,J · ∇s = 0. Near the magnetic axis, however, a typeof numerical interchange instability occurred as theangular resolution was refined (i.e. as the maximumpoloidal mode number increased). This behavior hasprevented the temporal convergence of 3D solutionswith large numbers of poloidal (m) and toroidal (n)modes (typically,m ∼ 6–8 was the practical limita-tion). It has also produced convergence problems forequilibria with very low rotational transforms, wherefield lines must encircle the magnetic axis many timesto adequately resolve a magnetic surface. The new dif-ferencing scheme computes the stream function on thesame mesh asR andZ (although the output values ofµ continue to be on the centered-grid for backwardscompatibility), which leads to numerical stabilizationof the origin interchange. To avoid first order errors(in hs ) near the plasma boundary resulting from thenew representation ofµ, the radial currentJs contin-ues to be internally represented (in terms ofµ) on thecentered-grid. This maintains the good radial spatialresolution associated with the original half-grid rep-resentation forµ. As a result, computation of conver-gent solutions with substantially higher mode numbersis now possible in VMEC (m < 20), correspondingto much finer spatial resolution and significantly im-proved force balance in the final equilibrium state. Italso results in convergence for equilibria with low val-ues of rotational transform, which was difficult to ob-tain with the previous differencing scheme.

An additional improvement in the output fromVMEC includes a recalculation (once the VMEC equi-librium has been obtained) of the magnetic force bal-anceF = (J × B − ∇p)= 0. The radial (∇s) compo-nent ofF is solved in terms of the non-vanishing con-travariant components ofB (Bθ andBφ) and the met-ric elements determined by VMEC, as a magnetic dif-ferential equation forBs . An angular collocation pro-cedure (with grid points matched to the Nyquist spa-

58 R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65

tial frequency of the modes) is used to avoid aliasingarising from nonlinear mode coupling of Fourier har-monics ofR andZ in the inverse representation ofthe equilibrium equation. The accurate determinationof Bs , together with the improved angular resolutionafforded by the larger limits on the allowable(m,n)spectra, permits an accurate assessment for the parallelcurrent (which contains angular derivatives ofBs , as afunction of poloidal mode number, to be performed.Studies of the Hamada condition, and its impact onMercier stability, near low order rational surfaces andcomparison with the PIES code are presently under-way [8].

3. The COBRA ballooning code

The COBRA code can in principle solve the stan-dard ideal ballooning equation [30] in any set of mag-netic coordinates(s, θ,φ):[

d

dφ

[P(φ)

d

dφ

]+Q(φ)+ λR(φ)

]F(φ)= 0 (2)

with P = Bφ |k⊥|2/B2, R = P/(Bφ)2 and Q =ε2β0p

′κs/Bφ . The field line curvature is given by�κ ,the pressure gradient isp′ andε = a/R0 is the inverseaspect ratio (a is the minor radius andR0 the majorradius). Magnetic fields are normalized toB0, paralleland perpendicular lengths respectively toR0 and aand times to the Alfvèn time.β0 ≡ 2µ0p0/B

20, with

p0 the axis pressure. In contrast to TERPSICHORE(see Section 4), COBRA uses the toroidal angleφto parametrize the field line (instead of the poloidalangle). The first term on the left-hand side of Eq. (2)corresponds to the stabilizing influence of the bendingof magnetic field lines, the last one is the stabilizingcontribution of inertia, while theQ-term, proportionalto the pressure gradient, drives the instability in thoseregions with bad curvature when the pressure gradientexceeds some threshold. This threshold is responsiblefor theβ stability limit due to ballooning modes.P(φ)is written in a way that it satisfies positiveness in allcases, independently of the accuracy of the VMECsolution, to avoid the appearance of unphysical modes(see Section 4).

Integrability of the eigensolutionF along the fieldline determines the eigenvalue,λ, that is related to theballooning growth rate,γ , normalized to the inverse

Alfvèn time, byλ = −γ 2. COBRA solves Eq. (2) onany set of magnetic surfaces of the configuration, onany set of initial locations on those surfaces and forany set of values of the arbitraryφk parameter (similarto theθk parameter in TERPSICHORE), whose secu-lar dependence is imbedded in both�k⊥ andκs .

COBRA computesλ very quickly by taking advan-tage of the Stürm–Lioville character of Eq. (2). Thisproperty allows an estimate for the eigenvalue to 4th-order accuracy (in the mesh step size along the mag-netic field line) by variationally refining a previous2nd-order accurate estimation of the eigensolution,F ,obtained by standard matrix methods:

λ= −〈F,LF 〉〈F,RF 〉 , (3)

where the following definitions have been used for thedifferential operatorL and the inner product〈·, ·〉 inL2(−∞,+∞):L(φ)≡ d

dφ

[P(φ)

d

dφ

]+Q(φ),

〈F,G〉 ≡+∞∫

−∞G∗(φ)F (φ)dφ

(4)

and withP , R andQ being the functions appearingin Eq. (2). The size of the numerical integration boxis set by choosing how many “helical wells” alongthe magnetic field line are included in the calculation.Each helical well approximately corresponds to atoroidal displacement of the order of 2π/M, with Mthe periodicity of the configuration. The number of“helical wells”, Kw , to be used will depend on thedegree of localization of the eigenmode along the fieldline (see Section 5 for some examples).

A fast and accurate evaluation of the eigenvalueis then achieved by coupling this evaluation processto a Richardson’s extrapolation scheme, that willextrapolate, using a quartic function, to zero mesh-stepsize from a few previous eigenvalue evaluations onvery coarse (and therefore rapidly evaluated) meshes.For comparison, TERPSICHORE integrates a similarequation (see Section 4) using a fixed-step shootingalgorithm. Accuracy is then a function of the numberof mesh points used, which makes it a relatively moretime-consuming calculation.

The original version of COBRA was based onstraight magnetic (Boozer) coordinates. It was re-

R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65 59

cently modified to carry out calculations directly inVMEC coordinates to further enhance computationalefficiency and to prevent convergence and accuracyproblems common to ALL codes based on Boozer co-ordinates. In this version, the magnetic field line mustbe followed numerically at the same time that Eq. (2)is solved in a way that does not interfere with the ef-ficiency of the Richardson’s scheme. Use is made ofthe fact that any magnetic line on any given magneticsurface, labeled bys, satisfies an equation of the type:

θ − ιφ +µ(s, θ,φ)= α (5)

for some constant valueα, that is used as line label,and with the stream functionµ provided by the VMECcode (see Section 2). Therefore,θ can be obtainedalong the magnetic line by solving the followingequation forθ(φ):

G(θ)≡ θ − ιφ +µ(s, θ,φ)− α = 0 (6)

using a Newton–Raphson scheme that iterates:

θk+1 = θk − G(θk)

dG(θk)/dθ. (7)

The overall speed has in this way been increased morethan a hundred times relative to most standard codes.[Many more details about the solution scheme heresketched and its mathematical and numerical basis canbe found in Refs. [15,16] and references there.]

4. Modifications of the TERPSICHORE code

The ideal MHD stability code TERPSICHORE [19]was developed in the late 80s and early 90s. It hasbeen routinely used for the numerical assessment ofglobal and local stability properties of many stellara-tors (for instance, the ATF stellarator at Oak RidgeNational Laboratory [23], the W-7X advanced heliascurrently under construction in Germany [24] or theHSX quasi-helical stellarator recently built at the Uni-versity of Wisconsin [25]). TERPSICHORE’s perfor-mance regarding ideal global stability has also beenbenchmarked against the well-known CAS3D stabil-ity code [20] for several large aspect ratio configura-tions [26].

TERPSICHORE (as almost all MHD stability codes)carries out all its computations in Boozer coordi-nates [18],(s, θb,φb), since magnetic field lines are

then simply defined asθb − ιφb = α, for some realnumberα, and the magnetic field vector has very sim-ple representations:

Bi = (Bs, J,−I),√gBi = (0,Ψ ′,Φ ′).

(8)

This considerably simplifies analytic manipulations(Ψ and Φ are the toroidal and poloidal magneticfluxes,J andI the toroidal and poloidal current fluxesand

√g is the Jacobian of the coordinate transforma-

tion). Instead of just mapping the VMEC equilibriumto these coordinates, TERPSICHORE carries out a re-calculation from the VMEC solution to guarantee thatthe plasma current satisfies∇ · J = 0 also at rationalsurfaces. This causes the parallel current to diverge,which is missed by the VMEC solution (even whenthis has been somewhat relieved by the changes de-scribed in Section 2), causing the typical spikes ap-pearing at rational surfaces in the Mercier criterion forinterchange instability [24].

To estimate high-n ballooning stability, TERPSI-CHORE solves for the eigenvalue of a modified Euler–Lagrange equation derived from the MHD energyprinciple. When expressed in Boozer coordinates, ittakes the form [28]:

d

dθb

(Cb

dF

dθb

)+ (1− λ)CaF = 0, (9)

whereF determines the eigenvalueλ when forcedto be an integrable function along the field line. TheCb term, related to the stabilizing energy associatedwith bending of magnetic field lines, is proportionalto k2⊥/(

√gB2) with k⊥ = ∇(φb − qθb) + q ′θk∇s,

and is strictly positive. Here,q is the inverse of therotational transform andθk is an arbitrary parameterrelated to the radial mode number. Finally,Ca = dp +ds(θb − θk) is the destabilizing term, proportional tothe pressure gradient and the local magnetic curvature(the exact expression fordp andds can be found inRef. [24]).λ is not the growth rate of the mode (themode inertia, proportional to the squared growth rate,is not included when the energy principle is minimizedin this fashion) but when it is positive, the mode isunstable.

Regarding the numerical method used to integrateEq. (9), TERPSICHORE uses a 2nd-order accurateshooting method. It integrates the solution from the

60 R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65

two end points of the integration box towards thecenter, being the eigenvalueλ determined by requiringthe first derivative of the solution to be continuous atthe middle point. The size of the integration box is setby the number of poloidal circuits along the field line:the poloidal angle is varied from−2πS0 to +2πS0,with S0 the number of transits. Convergence of theeigenvalue therefore requires a sufficiently large valueof S0 (depending on the degree of localization of themode) and a sufficiently dense mesh along the line.

The appearance of unphysical ballooning modes inthe NCSX cases is related to the line bending term,which can be written explicitly as follows:

Cb = Cp +Cs(θb − θk)+Cq(θb − θk)2 (10)

with:

Cp = gss√g

− B2s

B2√g ,

Cs = 2q ′Ψ ′

Φ ′

(JBs

B2√g − gsθb√g

),

Cq = q ′2Ψ ′2|∇s|2B2√g .

(11)

Analytically, Cb is strictly positive. However, itcan become (numerically) negative at some locationsalong the magnetic field line if the VMEC equilibriumsolution and/or the Boozer reconstruction are not suffi-ciently accurate. Numerical simulations seems to sug-gest that this happens more frequently at lower aspectratios, most likely due to the larger number of modesrequired both in the equilibrium solution as well as inthe Boozer transformation. When this is the case, thepotential in the Schrödinger equation obtained fromEq. (9) by a variable transformation [29] exhibits infi-nite barriers at the locations whereCb = 0. These arecapable of trapping an unphysical unstable mode. Toprevent these unphysical solutions, the line bending-term in TERPSICHORE has been rewritten using arepresentation that ensures positiveness independentlyof the equilibrium accuracy [21]:

Cb = C1(1+ [

C2 +C3(θb − θk)]2), (12)

with

C1 = 1√g|∇s|2 ,

C2 = −IgsθbB

√g

+ Jgsφb

B√g, (13)

C3 = q ′Ψ ′|∇s|2B

,

where several relationships between the Boozer metriccoefficients and magnetic fluxes have been used. Itis straightforward to relate the newCi with the oldcoefficients:

Cq = C1C23,

Cs = 2C1C2C3,

Cp = C1(1+C2

2

).

(14)

5. COBRA-TERPSICHORE benchmarking

Ideal MHD pressure-driven instabilities compriseboth interchange and ballooning modes [31]. Theyboth satisfyk⊥ � k‖, being therefore amenable of alocal analysis when represented by means of the bal-looning formalism [32]. Using this representation, sta-bility β-limits against both interchange and ballooningmodes can be studied by solving an ordinary differen-tial equation (Eqs. (2) or (4)) along the magnetic fieldline. The properties of the resulting eigensolution ishowever very different for interchanges and balloon-ing modes. Interchanges extend along the magneticfield line for many helical magnetic wells (strictlyspeaking, they are only defined in the limitk‖ → 0,but here, any solution of the ballooning equation satis-fying k‖R0 � 1 will be referred to as an interchange).As a result of this extended structure, they are usuallyfound linked to the lower-order rational surfaces of theconfiguration. In contrast, ballooning modes are typi-cally highly localized (k‖R0 � 1) [15,33], extendingfor only a few helical wells.

Taking advantage of their large extension alongthe field lines, an asymptotic analysis can be usedto derive a stability criterion for interchanges (see,for instance, Ref. [28]), completely analogous to thatoriginally derived by Mercier [27]. From this analysis,it is clear that interchange modes are destabilizedon a magnetic surface if the surface-average of theline-curvature and parallel currents is large enoughas to oppose the stabilizing effect of the surface-averaged magnetic well or shear. Since the evaluation

R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65 61

of this criterion only involves the computation ofthese few surface-averaged quantities, it is the usualmethod to determine the stabilityβ-limits of theconfigurations against interchange modes, and it isthus routinely included in the compact stellaratoroptimization loop (in fact, it has been included in thelatest release of VMEC). This asymptotic method ismuch faster than trying to evaluate their exact growthrates by solving Eqs. (2) or (4), since it is thennecessary to include in the numerical box along thefield line a very large number of helical wellsKwin COBRA (see Section 3) or poloidal transitsS0 inTERPSICHORE (see Section 4). This fact, togetherwith the requirement of keeping a sufficiently densemesh of points along the line to get a well-convergedeigenvalue, turns the computation very slow.

On the other hand, localized ballooning modesare rather insensitive to any of the aforementionedsurface-averaged quantities, being however driven ordamped by the local magnetic shear, local current den-sities and local curvatures. This implies that: (1) a de-tailed knowledge of these local quantities is neces-sary to estimate their growth rates, which implies thatthe equilibrium solution must be locally accurate andwith good local force balance to yield reliable resultsand (2) no ballooning version of the Mercier crite-rion exists. The fast ballooning growth rate estimationthat COBRA produces is the closest to such a crite-rion we have been able to achieve for localized bal-looning modes. Since the growth rate can be obtainedat several locations and several surfaces very quickly,COBRA can be used to generate a positive target func-tion (for instance, by using the sum of all positivegrowth rates) whose minimization would eventuallyincrease the criticalβ above which the configurationbecomes unstable to ballooning modes [34].

From this discussion, it seems obvious that, in thecontext of ideal pressure-driven instabilities, there aretwo essential results that need to be benchmarkedbetween COBRA and TERPSICHORE: (1) agree-ment on the determination of the regions of the con-figuration where unstable interchange (also referredto as extended-ballooning modes throughout the pa-per) appear for any prescribed value of〈β〉 (notic-ing that these regions should coincide with those sur-faces where the equilibrium violates the Mercier cri-terion for stability) and (2) agreement on the deter-mination of the regions of the configuration where

localized-ballooning modes turn unstable for any〈β〉(notice that these regions may now as well be Mercierstable as Mercier unstable; if Mercier unstable, thiswould mean that more than one unstable eigenvalueexists in the discrete spectrum of Eqs. (2) or (4); themost unstable one would correspond to the localizedmode and the others, to more extended interchange-like modes).

But before beginning the benchmarking process, itwill prove useful to point out now the differences be-tween the eigenvaluesλ calculated by COBRA andTERPSICHORE: Eq. (9) is recovered from Eq. (2)(apart from normalization issues) if the inertial termR is set to zero and if the destabilizingQ-term is mul-tiplied by (1 − λ). Therefore, both equations are onlyidentical in the limitλ→ 0, i.e. for marginal stabil-ity. This implies that COBRA and TERPSICHOREwill predict the same instability regions and the samethresholds forβ , but the user should not expect thesame eigenvalues for any unstable surface. TERPSI-CHORE will yield a positiveλ, while COBRA willyield a positive growth rate and thus a negativeλ.More unstable surfaces will yield larger positiveλ’sin TERPSICHORE and smaller (more negative)λ’sin COBRA, corresponding to larger positive growthrates. On top of this, we should also keep in mind thatCOBRA uses the VMEC equilibrium solution whileTERPSICHORE uses its own Boozer reconstruction,which implies that the two equilibria are not identi-cal (for instance, they will specially differ at ratio-nal surfaces, where∇ · J is small but nonzero forthe VMEC solution. This was already made apparentwhen comparing the results obtained from evaluatingthe Mercier criterion directly in VMEC coordinatesand in Boozer coordinates for the W7-X stellarator inRef. [24]). At the same time, extra inaccuracies arepresent in the equilibrium reconstruction carried outby TERPSICHORE, whose relative importance willstrongly depend of the number of Boozer modes in-cluded. Finally, it is good to keep in mind that to usenumerical boxes of the same length in both codes, thenumber of poloidal transits in TERPSICHORE mustbe set toS0 ∼ Kwι/M, with Kw the number helicalwells used in COBRA.

We will then begin by benchmarking the firstpoint by choosing a compact stellarator case whereinterchange-like modes exist above some critical valueof β , but no localized-ballooning mode is found. This

62 R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65

state of things can be ensured by choosing a con-figuration where the Mercier criterion predicts insta-bility but no unstable solution is found when solv-ing the ballooning equation until the numerical boxexceeds a minimum length. As an example of sucha case we use C82, a previous reference configura-tion of the NCSX project [8]. This is aM = 3 quasi-axisymmetric equilibrium with aspect ratioA ∼ 3.4,magnetic fieldB ∼ 2T and almost 50% of the ro-tational transform,ι, provided by the bootstrap cur-rent. Its rotational transform increases monotonicallyfrom 0.26 at the axis to 0.47 at the edge atβ = 3%.The numerical equilibria for this case have been com-puted for values of〈β〉 ranging from 3 to 5% using97 radial points, and with 8 poloidal and 5 toroidalVMEC modes (83 modes in total), keeping the totaltoroidal current profile fixed, and lettingι be modi-fied by the increasingβ . The reason for this choice,is that it will help to identify the interchange charac-ter of the unstable modes, since they tend to be lo-cated near rational surfaces and will therefore try tofollow them as theι profile is modified. The most un-stable modes are usually found at the locations withthe most unfavorable curvature that, in this configu-ration, is found around(θ, ζ )= (0,0) and its periodsand semiperiods. Therefore, the ballooning equationhas been solved for eigensolutions centered at this lo-cation.

We have used COBRA to build a map of the unsta-ble regions inβ–s space (remember thats is the mag-netic surface label, that in VMEC corresponds to thetoroidal magnetic flux). This map is shown in Fig. 1,with the shaded regions corresponding to those re-gions where unstable solutions are found. The inter-change character of the unstable modes is readily con-firmed by the close alignment of the unstable regionswith the lower-order rationals present in the config-uration: ι = 3/7,3/8,3/9,3/10 and 3/11. TERPSI-CHORE can be seen to reproduce the same instabilityregions for all values of〈β〉. As an example, dark solidlines have been superimposed to the map, correspond-ing to the radial regions where TERPSICHORE findsunstable solutions for the equilibria with〈β〉 = 3.25,3.85 and 4.25%. A more detailed comparison is carriedout for the equilibrium with〈β〉 = 3.85% in Fig. 2.The eigenvalueλ obtained by TERPSICHORE and thegrowth rateγ = −λ2 obtained by COBRA are thereshown as a function of the toroidal flux. The gray-

Fig. 1. Contour map in〈β〉–s space of the growth-rates obtained byCOBRA for the QAS configuration C82. The unstable regions areshaded in gray. Thick solid lines show the regions where unstablemodes are found by TERPSICHORE for a subset of selectedequilibria. The radial location of the lower-order rational surfacesis also shown using dot-dashed lines.

shaded regions correspond now to those radial loca-tions where the Mercier criterion predicts interchangeinstability for that equilibrium. Both codes can be seento detect these unstable extended modes in very goodagreement with the Mercier criterion. The good con-vergence of the growth-rates is also shown in Fig. 3,using a contour map inKw–s space of the growthrates obtained by COBRA for the same equilibrium.Notice that a growth rate obtained at any arbitrarymagnetic surface would be well converged only fora number of wells above which, the contour passingthrough that surface becomes a straight vertical line(this would imply that the same growth rate wouldbe obtained for any larger box!). Notice that, for thisequilibrium, this is not the case untilKw ∼ 65, whichcorresponds to usingS0 ∼ 10–12 poloidal transits inTERPSICHORE. The growth rates and eigenvalues in-

R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65 63

Fig. 2. Comparison of the eigenvalueλ obtained by TERPSICHOREand the growth rateγ computed by COBRA for the C82 equilibriumwith 〈β〉 = 3.85%. Mercier unstable regions are shaded in gray.

cluded in Fig. 2 have therefore been computed usingrespectivelyKw = 70 andS0 = 12. The good conver-gence of the TERPSICHORE results have also beenensured independently by rerunning on all surfaceswith a higher value ofS0. As a final comment, the con-tour map in Fig. 3 confirms again the interchange-likecharacter of the modes: it can be appreciated that nounstable mode is located ifKw < 20 is used!

In order to compare now COBRA and TERPSI-CHORE when detecting localized-ballooning modes,we have turned to a high-ι compact quasi-omnigeneous(QOS) case that has been altered on purpose to reduceits ballooning stabilityβ-limits. This QOS case hasalso three periods (M = 3), aspect ratioA∼ 3.5,B ∼1T and a small total bootstrap current (Ip < 50kAfor 〈β〉 = 3%). Its rotational transformι steadily in-creases from an axis value of 0.55 to close to 0.65 at

Fig. 3. Contour map inKw–s space of the growth rates obtained byCOBRA for the c82 equilibrium with〈β〉 = 3.85%.

the plasma edge [4]. We have computed a series ofequilibria with values of〈β〉 ranging from 0 to 3%.A fixed ι profile has now be chosen, and 97 radial sur-faces, 8 VMEC poloidal modes and 5 VMEC toroidalmodes have been used. Analogously to what we didin the NCSX case, we build a contour map in〈β〉–sspace of the most unstable growth rate obtained byCOBRA, which is shown in Fig. 4. Again, the shadedareas correspond to unstable solutions. The compar-ison with TERPSICHORE is done in the same wayas before: the dark solid lines correspond to the re-gions where unstable solutions are found by TERP-SICHORE for those equilibria with〈β〉 = 1, 2, 2.5and 3%. The agreement with COBRA is again re-markable. A more detailed comparison for the equilib-rium with 〈β〉 = 3% is also shown in Fig. 5, with thegray-shaded regions again representing those locationswhere the Mercier criterion for stability is violated.It can now be appreciated that all modes encounteredoutside these gray-shaded regions are truly localized

64 R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65

Fig. 4. Contour map in〈β〉–s space of the growth-rates obtained byCOBRA for the QOS configuration. The unstable regions are shadedin gray. Thick solid lines show the regions where unstable modes arefound by TERPSICHORE for a subset of selected equilibria. Theradial location of the lower-order rational surfaces is also shownusing dot-dashed lines.

solutions by looking at the growth-rate contour mapin Kw–s space for this equilibrium: the growth rateis well converged in the Mercier-stable region evenfor Kw < 5 all throughout that region (as a matter offact they can be obtained even if settingKw = 2!). Inthe Mercier-unstable shaded region, highly-localizedmodes still exist untils ∼ 0.75, indicating the coexis-tence of a less unstable extended mode, that is readilyrevealed when looking for the second most unstablemode with COBRA. Fors > 0.75, the most unstablemode becomes now an extended-ballooning mode, re-quiring at leastKw ∼ 30 helical wells for its growthrate to converge ats = 0.95. To ensure convergenceof growth rates and eigenvalues included in Fig. 5 forALL surfaces, they have been computed respectivelyusingKw = 40 andS0 = 12 in COBRA and TERPSI-CHORE.

Fig. 5. Comparison of the eigenvalueλ obtained by TERPSICHOREand the growth rateγ computed by COBRA for the QOS equilib-rium with 〈β〉 = 3%. Mercier unstable regions are shaded in gray.

6. Conclusions

It is difficult to carry out a benchmarking betweencodes that, even when addressing in principle the sameproblem (local pressure-driven modes), use differentsolution techniques, as it is the case for COBRA andTERPSICHORE. They use different normalizations,slightly different forms of the MHD equilibria, differ-ent coordinates and even compute different eigenval-ues from different equations. In spite of these differ-ences, it is reassuring to have shown that both CO-BRA and TERPSICHORE predict the same regionsof interchange and ballooning instability, and similarcritical β , for the low aspect ratio stellarators we haveinvestigated in this paper. This successful benchmark-ing ensures that ideal stability calculations are reliable

R. Sanchez et al. / Computer Physics Communications 141 (2001) 55–65 65

Fig. 6. Contour map inKw–s space of the growth rates obtained byCOBRA for the QOS equilibrium with〈β〉 = 3%.

in the complex optimization process that need to beundertaken to find attractive low-A stellarator config-urations.

Acknowledgements

This work was sponsored by Spanish DGES projectFTN2000-0924-C03-01, the U.S. Department of En-ergy under contracts DE-AC02-76CH03073 and DE-AC05-00OR22725, EURATOM and the Fonds Na-tional Suisse pour la Recherche Scientifique. Russianauthors are grateful to Prof. J. Nührenberg for pro-viding us with access to a NEC-SX5 supercomputer(RZG, IPP, FRG). This work was made in the frameof CRPP-IPM collaboration and also supported by IN-MARCON (Moscow, Russia).

References

[1] L. Spitzer Jr., U.S. Atomic Energy Commission ReportNo. NYO-993 (PM-S-1), 1951; U.S. Atomic Energy Commis-sion Report No. NYO-995 (PM-S-3), 1951.

[2] J. Nührenberg, R. Zille, Phys. Lett. A 129 (1988) 113.[3] J. Nührenberg, W. Lotz, P. Merkel et al., Trans. Fusion

Technology 27 (1995) 71.[4] S.P. Hirshman et al., Phys. Plasmas 6 (1999) 1858.[5] C. Beidler et al., Fus. Technol. 17 (1961) 149.[6] A. Iiyoshi, M. Fujiwara, O. Motojima, N. Ohyabu, K. Ya-

mazaki, Fus. Technol. 17 (1990) 169.[7] G.H. Neilson et al., Phys. Plasmas 7 (2000) 1911.[8] A.H. Reiman et al., Phys. Plasmas 8 (2001) 2083.[9] S. Okamura et al., in: Proc. 18th IAEA Conference on Fusion

Energy, Sorrento, Italy, 2000.[10] J.R. Cary, S.G. Shasharina, Phys. Rev. Lett. 78 (1997) 674.[11] S.P. Hirshman et al., Phys. Rev. Lett. 80 (1998) 528.[12] D.A. Spong et al., Nucl. Fusion 41 (2001) 711.[13] J.M. Greene, J.L. Johnson, Phys. Fluids 4 (1961) 871.[14] L. García, B.A. Carreras, N. Domínguez, J.N. Leboeuf, V.E.

Lynch, Phys. Fluids B 2 (1990) 2162.[15] R. Sánchez, S.P. Hirshman, J.C. Whitson, A.S. Ware, J. Com-

put. Phys. 161 (2000) 576.[16] R. Sánchez, S.P. Hirshman, H.V. Wong, Comput. Phys.

Comm. 135 (2001) 82.[17] S.P. Hirshman, W.I. van Rij, P. Merkel, Comput. Phys.

Comm. 43 (1986) 143.[18] A.H. Boozer, Phys. Fluids 23 (1980) 904.[19] D.V. Anderson, W.A. Cooper, R. Gruber, S. Merazzi,

U. Schwenn, J. SuperComput. Appl. 4 (1990) 34.[20] C. Schwab, Phys. Fluids 5 (1993) 3195.[21] J. Nührenberg, R. Zille, in: Theory of Fusion Plasmas, Proc. of

the 1987 Varenna Workshop, Editrice Compositori, Bologna,1986, p. 3.

[22] G.Y. Fu et al., Phys. Plasmas 7 (2000) 1809.[23] W.A. Cooper, S.P. Hirshman, D.K. Lee, Nucl. Fusion 29

(1989) 617.[24] R. Moeckli, W.A. Cooper, Nucl. Fusion 33 (1993) 1899.[25] J. Talmadge, W.A. Cooper, Phys. of Plasmas 3 (1996) 3713.[26] J. Nührenberg et al., Plasm. Phys. Contr. Fus. 35 (1993) B115.[27] C. Mercier, Nucl. Fusion 1 (1960) 47.[28] W.A. Cooper, Plasm. Phys. Contr. Fus. 34 (1992) 1011.[29] P.M. Morse, H. Feshbach, Methods for Theoretical Physics,

McGraw-Hill, New York, 1953.[30] D. Correa-Restrepo, Zeit. Naturf. 37 (1982) 848.[31] J.P. Freidberg, Rev. Mod. Phys. 54 (1982) 801.[32] J.W. Connor, R.J. Hastie, J.B. Taylor, Proc. Roy. Soc. London

A 365 (1979) 1.[33] N. Nakajima, Phys. Plasmas 3 (1996) 4556.[34] R. Sánchez, S.P. Hirshman, A.S. Ware, L.A. Berry, D.A.

Spong, Plasm. Phys. Contr. Fusion 42 (2000) 641.