Are ghost surfaces quadratic-flux-minimizing?

0.72 MB·English

by  S. R. Hudson

Are ghost surfaces quadratic-flux-minimizing?
S.R.Hudson
PrincetonPlasmaPhysicsLaboratory,POBox451,PrincetonNJ08543,USA
R.L.Dewar
PlasmaResearchLaboratories,ResearchSchoolofPhysics&Engineering,TheAustralianNationalUniversity,Canberra,ACT0200,Australia
0
Abstract
1
0Two candidates for “almost-invariant” toroidal surfaces passing through magnetic islands, namely quadratic-flux-minimizing
2
(QFMin)surfacesandghostsurfaces,usefamiliesofperiodicpseudo-orbits(i.e.pathsforwhichtheactionisnotexactlyextremal).

nQFMinpseudo-orbits,whicharecoordinate-dependent,arefieldlinesobtainedfromamodifiedmagneticfield,andghost-surface
(cid:72)
apseudo-orbitsareobtainedbydisplacingclosedfieldlinesinthedirectionofsteepestdescentofmagneticaction, A·dl. Agener-
J
alizedHamiltoniandefinitionofghostsurfacesisgivenandspecializedtotheusualLagrangiandefinition. AmodifiedHamilton’s
2
PrincipleisintroducedthatallowstheuseofLagrangianintegrationforcalculationoftheQFMinpseudo-orbits.Numericalcalcula-

tionsshowQFMinandLagrangianghostsurfacesgiveverysimilarresultsforachaoticmagneticfieldperturbedfromanintegrable
]
hcase,andthisisexplainedusingaperturbativeconstructionofanauxiliarypoloidalangleforwhichQFMinandLagrangianghost
p
surfacesarethesameuptosecondorder. Whilepresentedinthecontextof3-dimensionalmagneticfieldlinesystems,theconcepts
-
mareapplicabletodefiningalmost-invarianttoriinother11 degree-of-freedomnonintegrableLagrangian/Hamiltoniansystems.
2
s
Keywords: Toroidalmagneticfields,Hamiltoniandynamics,Lagrangiandynamics,almost-invarianttori
a
lPACS:05.45.-a,52.55.Dy,52.55.Hc
p
.
s
c
1. introduction
i
s
y The understanding of nonintegrable Hamiltonian systems is
h
greatlysimplifiedifonecanconstructacoordinateframework
p
based on a set of surfaces that are either invariant under the
[
dynamicsor,wherethisisimpossible,surfacesthatarealmost-

2
invariant. As invariant tori and cantori in nonintegrable sys-
v
temscanbeapproximatedbysequencesofperiodicorbits,the
6
9theoryofalmost-invariantsurfacesisbuiltaroundperiodicor-
0bits, whichconstitutetheremanentinvariantsetssurvivingaf-
2ter integrability is destroyed by symmetry-breaking perturba-
.
9tions. We consider two classes of almost-invariant surfaces,
0quadratic-flux-minimizing(QFMin)surfaces[1]andghostsur-
9faces[2,3].
0
Almost-invarianttoriareimportantinthetheoryofmagnetic
:
vconfinement of toroidal plasmas, in particular to the theory of
itransportinchaoticmagneticfields[4],andwesetthispaperin
X
thecontextofthenonintegrablemagneticfields,B,encountered
r
ain devices without a continuous symmetry. However, as mag-
neticfieldlinesareorbitsofa11 degree-of-freedomHamilto-
2
nian system, [5] the discussion is applicable, with appropriate
translations of terminology, to any such system—e.g. in this
paperweuse“magneticfieldline”and“orbit”interchangeably.
In Sec.2 we introduce our general, arbitrary background
Figure1: Asketchofthegeneralcurvilineartoroidalcoordinatesystemde-
toroidalcoordinatesystem s,θ,ζ,andanauxiliarypoloidalan-
scribedinthetext.
Emailaddresses:[email protected](S.R.Hudson),
[email protected](R.L.Dewar)
PreprintsubmittedtoarXiv January2,2010
gleΘ(s,θ,ζ)thatallowsustodefinethequadraticfluxinaform The directed infinitesimal area element on an arbitrary sur-
independentofthechoiceofθ. InSec.3weintroducethemag- face Γ is dS ≡ dθdζn/n·∇θ×∇ζ, where n is the unit normal
neticactionintegral,itsfirstandsecondvariationsandHamil- at any point on Γ. Thus the net magnetic flux crossing an ar-
ton’sprinciple, whileinSec.4weintroduceQFMinand(gen- bitrary torus Γ (which we assume to contain the θ-coordinate
eralized) ghost-surface pseudo-orbits as alternative strategies singularitycurve)is
forcontinuouslydeformingtheaction-minimaxorbitassociated (cid:90) 2π(cid:90) 2π n·B
withanislandchainintothecorrespondingaction-minimizing ϕ [Γ]≡ dθdζ . (1)
1 n·∇θ×∇ζ
orbit. 0 0
InSec.5wepresentnumericalresultsforfield-lineHamilto- This integral is independent of choice of coordinates. In fact
niansoftheformχ0(ψ)+(cid:15)χ1(ψ,θ,ζ),wherethefluxfunctionψ the absence of magnetic monopoles implies that ϕ1 vanishes
plays the role of a momentum canonically conjugate to θ, and identically,soitisindependentofthechoiceofΓalso,whether
(cid:15) parametrizes the strength of the perturbation away from the it be a magnetic surface (invariant torus of the field-line flow)
integrable case described by the action-angle Hamiltonian χ . orotherwise.
0
Plotsarepresentedcomparingtheuncorrected(i.e. withΘ=θ) Thus,tomeasuretheamountbywhichΓdepartsfrombeing
QFMin and Lagrangian ghost curves of Ref. 2,superposed on a magnetic surface, we are led to define the positive definite
field-line puncture plots in a Poincare´ surface of section. Two quadratic flux [1], defined with the aid of a new generalized
cases with different strengths of perturbation are shown, both poloidalangleΘ(s,θ,ζ),
quitestronglychaoticandbothshowingthatthedifferencesbe- 1(cid:90) 2π(cid:90) 2π n·B n·B
tweenevenuncorrectedQFMinandghostcurvesareverysmall ϕ [Γ]≡ dθdζ . (2)
2 2 n·∇θ×∇ζn·∇Θ×∇ζ
(except for some higher-order surfaces, in the more strongly 0 0
chaoticcase).Thissuggeststhatthetwo,seeminglyverydiffer- Thequadraticfluxϕ2 isindependentofthechoiceofbaseco-
ent,approachestodefiningalmost-invarianttorimaybeunified ordinatess,θ,ζ,butdependsonthechoiceofΘ.
by appropriate choice of Θ, and that this will differ from θ by In the numerical work presented in this paper, Θ has been
anamountsmallin(cid:15). chosen equal to the given angle θ. However in the formal de-
In Sec.6 we introduce a modified form of Hamilton’s Prin- velopmentwedistinguishitfromθsowecanexplorethecon-
ciplethatgivesQFMinpseudo-orbitsasextremizersofapseu- sequencesofmakingdifferentchoices,inparticularwhetherit
doaction. Section 7 gives the canonical, Hamiltonian form of canbechosensothatQFMintoricoincidewithghosttori.
thisactionprinciple, whileSec.8discussesthetransformation
totheLagrangianform. InSec.9wederiveaconsistencycon- 3. Magneticactionintegral
dition that Θ must satisfy for corrected QFMin surfaces to be
Lagrangianghostsurfaces,findinginSec.10anexpressionfor Thefield-lineactionS[6]isafunctionalofapathCinEu-
achoiceoftheauxiliaryangleΘthatsatisfiesthiscriterionup clidean 3-space, points on which we take to be labeled by the
tofirstorderin(cid:15). ThedifferencebetweenuncorrectedQFMin toroidal angle ζ, which thus takes on the role played by time
and ghost/corrected-QFMin pseudo-orbits is shown indeed to in a more conventional Hamiltonian system. In this paper we
beverysmall,O((cid:15)2). confine our attention to paths that are closed loops, with θ in-
In Sec.11 we sketch our finite-element variational method creasing by 2πp when ζ increases by 2πq (p and q > 0 being
for numerical construction of QFMin surfaces using the new mutually prime integers), so the average rate of increase of θ
Hamilton’sPrincipleintroducedinSec.6,andinSec.12wedis- along the path is the rational fraction p/q, where the angular
cussthenumericalconstructionofghostsurfacesviaGalerkin frequency -ιiscalledtherotationaltransform.
projectionontothefiniteelementbasis. Themagneticactionisdefinedby
Appendix A contains a derivation of the Euler–Lagrange (cid:90) (cid:90) 2πq
equation for QFMin pseudo-orbits in the canonical represen- S[C]≡ A·dl≡ A·r˙dζ, (3)
tation, and Appendix B shows the relation between the gener- C 0
wherethesingle-valuedfunctionA(r)isamagneticvectorpo-
alized definition of ghost pseudo-orbit given in Sec.4 and our
tential for the magnetic field, B = ∇×A, and dl ≡ r˙dζ is an
morestandardLagrangianform[2],usedinthenumericalwork
infinitesimallineelementtangentialtoC. Asuperscriptdotde-
andinSec.9.
notes the total derivative with respect to ζ, so that r˙·∇ζ ≡ 1.
Hamilton’sPrincipleisthestatementthatSisstationary,with
2. Coordinatesandfluxes
respecttovariationsδrofC,whenCisasegmentofaphysical
orbit(inourcaseamagneticfieldline). IfCisanopensegment
AsdepictedinFig.1,weassumeageneral,essentiallyarbi-
trary curvilinear toroidal coordinate system s(r),θ(r),ζ(r) has thevariationsaretobetakenholdingtheendpointsfixed,butif
been established, where r is a point in Euclidean 3-space and (asweassume)Cisaclosedloopthenthevariationsareuncon-
θ and ζ are respectively poloidal and toroidal angles label- strainedbecausetheendpointcontributionscancel. Then,after
ing points on the toroidal isosurfaces of s, nested around the integrationbyparts,wehavetheexpansionforthetotalchange
curvealongwhichθissingular(sincreasingoutward). Weas- inS
sumethenonorthogonalbasis{es,eθ,eζ}≡{∇s,∇θ,∇ζ}isright (cid:90) 2πq(cid:32) δS 1 δ2S (cid:33)
∆S= δr· + δr· ·δr+... dζ, (4)
handed,asisitsreciprocalbasis{es,eθ,eζ}≡{∂sr,∂θr,∂ζr}. 0 δr 2 δrδr
2
wherethefirstfunctionalderivativeisgivenby
δS δS δS δS
≡es +eθ +eζ =r˙×B. (5)
δr δs δθ δζ
Hamilton’s Principle is now readily verified: The Euler–
LagrangeequationδS/δr = 0issatisfiedifr˙ = B/Bζ,i.e. ona
magneticfieldline.
ThesymmetrizedHessianoperatoris
δ2S d d
2 = − B×I−I×B
δrδr dζ dζ
+r˙×(∇B)T−(∇B)×r˙ , (6)
whereI=e es+e eθ+e eζ =ese +eθe +eζe istheidentity
s θ ζ s θ ζ
dyadicandsuperscriptTdenotesthetranspose.
Note also that variations δr = r(ζ +δζ)−r(ζ) = r+r˙δζ +
1r¨(δζ)2+...thatsimplyrelabelthepathcanbeverifiedtoleave
2
S invariant for arbitrary δζ(ζ), as expected. Thus, to find a
uniqueminimizer,wemustsuppressthisdegreeoffreedomin
theallowedvariations. Tothisendweconstrainδrtothetan-
gent plane of the poloidal surface of section ζ = const at r,
denoting theconstrained variationby ∆r ≡ e δs+e δθ. (Pro- Figure2: AcomparisonoftheintersectionsofuncorrectedQFMinsurfaces
s θ (thickdashedlines)andLagrangianghostsurfaces(thinlines)withthesur-
vided B, es and eθ are linearly independent, the two compo- face ζ = 0, with the choices s = ψ, Θ = θ described in Sec. 7. The two
nents of the Euler–Lagrange equation, δS/δs ≡ e ·r˙×B = 0 almost-invariantsurfacedefinitionsarealmostindistinguishableinthismoder-
s
andδS/δθ ≡ e ·r˙×B = 0implyr˙×B = 0,sothethirdcompo- atelychaoticcase,describedinSec.5.APoincare´plot(reddots)isalsoshown.
θ
nentoftheEuler–Lagrangeequation, δS/δζ ≡ e ·r˙×B = 0, is
ζ
redundant.) nonnegative dyadic. The most natural choice of P might
ghost
seemtobeP ≡I−eζeζ/|eζ|2projectingontothepoloidaltan-
pol
gentplaneatr,asthisisindependentofthechoiceof sandθ.
4. GhostandQFMinsurfaces
However,asweshowinAppendixB,thisdoesnotcorrespond
Inanintegrablesystemacontinuousfamilyof(p,q)-periodic withthatrequiredtorecovertheusualLagrangiandefinition[2]
orbits, each extremizing the action, exists, defining an invari- ofghostsurfaces.
ant torus with rotational transform p/q. However such invari- At the periodic orbits, the action gradient is zero. Begin-
ant tori are not structurally stable—small perturbations to the ningfromtheminimaxorbit,weinitiallypushthecurveinthe
systemdestroyintegrability(seee.g. Fig.2),leavingonlyiso- decreasingdirection(providedbytheeigenfunctionoftheHes-
lated action-extremizing periodic orbits. In fact (assuming a sian with negative eigenvalue) and then evolve the curve ac-
twist condition holds), by the Poincare´–Birkhoff theorem [7], cordingtotheactiongradientflowEq.(7).
onlytwodistinctperiodicorbitssurviveinagiven(p,q)island UsingEq.(7)inEq.(4)wefind
chain, namely the action-minimizing orbit [8], and an action- DS (cid:90) 2πq δS δS
minimax orbit. The minimizing orbit is a hyperbolically un- =− ·P · dζ, (8)
Dτ δr ghost δr
stable “X-point” orbit in the chaotic separatrix region of the 0
islandchainwhiletheminimaxorbitthreadsthecentersofthe so the sequence of pseudo-orbits tends monotonically toward
islands. Theminimaxorbitmaybeellipticallystable,or,after the minimizing orbit, tracing out a surface, which we call the
aperiod-doublingbifurcation,becomeunstablebutcontinueas ghostsurface.
a(p,q)hyperbolicorbitaccompaniedbyadaughter(2p,2q)el- Ghost surfaces display several attractive properties (proved,
liptic orbit that does not affect the following discussion. An forLagrangianghostorbits,inthecaseofsymplecticmaps[3];
almost-invariant torus, the “ghostly remnant” of an invariant verifiednumericallyforcontinuoustime/magneticfieldsystems
torus, is formed from a family of pseudo-orbits, labeled by a [2]). Their intersections with a surface of section are graphs
parameter τ, that “fill in” the regions between the minimizing whenplottedincanonicalphase-spacecoordinates,sothateach
andminimaxorbitssoastoformatorusΓ. lineofconstantθ crossestheghostcurveonlyonce. Theyare
Thepseudo-orbitsofaghosttorusaredefined[2]bydeform- nonintersecting,andthusadiscreteselectionofghostsurfaces
ingtheminimaxorbitviaanaction-gradientflow maybeusedasaframeworkforageneralizedaction-angle-like
coordinate system for chaotic fields. Furthermore, there ap-
Dr δS
=− ·P , (7) pearstobeaclosecorrespondencebetweenghostsurfacesand
Dτ δr ghost
theisothermsresultingfromstronglyanisotropicheat-transport
where D/Dτ denotes the total τ-derivative at fixed ζ, i.e. inchaoticmagneticfields[4]. However,asitstands,thereisa
D/Dτ ≡ (Ds/Dτ)∂ + (Dθ/Dτ)∂ , and P is a symmetric significantdisadvantagetotheirconstruction: thegradientflow
s θ ghost
3
andaperturbation,
(cid:88)
χ = χ (ψ)cos(mθ−nζ). (10)
1 m,n
Weuseonlytwononzero,ψ-independent,perturbationharmon-
ics, χ =0.0010 and χ =−0.0005, to drive islands at the
2,1 3,2
-ι = 1/2 and -ι = 2/3 rational surfaces. The degree of chaos
induced by the perturbations is illustrated by a Poincare´ plot,
shownwithreddotsinFig.2. Quadratic-flux-minimizingsur-
faces (thick dashed lines) constructed as in Sec.11 and La-
grangianghostsurfaces(thinlines)constructedasinSec.12as-
sociatedwith31rationalsbetweenthesetwoislandsareshown.
WetermtheseQFMinsurfacesuncorrectedbecausenoattempt
attransformingthecoordinatesystemtoimproveagreementbe-
tweenQFMinandghostsurfaceshasbeenmade(i.e. wehave
takenΘ=θ).Onthescaleofthefigure,thequadratic-fluxmin-
imizingsurfacesandtheghostsurfacesareindistinguishable.
As the degree of chaos increases, the quadratic-flux min-
imizing surfaces and the ghost surfaces appear to deviate,
particularly those associated with the higher order rationals.
Increasing the perturbation amplitude to χ =0.0020 and
2,1
Figure3: AcomparisonofuncorrectedQFMincurves(thicklines)andghost χ =−0.0010,thesurfacesareshowninFig.3.
curves(thinlines)foramorestronglychaoticcasedescribedinthetext.Some 3,2
caseswhereQFMincurvesviolatethegraphpropertyareseen. The close agreement between QFMin curves and ghost
curvesinthecaseofmoderatechaosforthecoordinatechoice
used(action-anglecoordinatesintheunperturbedsystem)sug-
vanishesasoneapproachesintegrability. Inthislimit,thecon- gestsacoordinatesysteminwhichtheybecomeidenticalmay
structionoftheghostsurfacesbecomesarbitrarilyslow! notbestronglyperturbedawayfromaction-anglecoordinates.
A QFMin surface is one that minimizes ϕ2 under deforma- Belowwepointthewaytoachievingthisunification.
tionsofΓ. InRef.1(seealsoAppendixA)itisshownthatthe
Euler–Lagrange equation for this variational principle implies
that Γ is composed of pseudo-orbits tangential to the pseudo 6. ModifiedHamilton’sPrinciple
field
It has long been known [10, 2] that the quadratic flux ϕ
2
B ≡B−ν∇Θ×∇ζ, (9) canbeexpressedintermsofactiongradient,thusgeneralizing
ν
theQFMinconcepttogeneralLagrangian/Hamiltoniansystems
where ν ≡ n·B/n·∇Θ×∇ζ is constant on each such pseudo- andprovidingalinkwiththeghostsurfaceapproach. However,
orbit. Intuitively, the pseudo field is constructed from the true neitherapproachsofarhasprovidedavariationalprinciplefor
fieldbyaddingaradialfieldthatcancelstheradialfieldcaused individualpseudo-orbits. Inthissectionwepresentanewand
by a perturbation away from a neighboring integrable field, very useful variational principle for the QFMin pseudo-orbits,
while the poloidal field is unchanged [9]. Although QFMin modifyingHamilton’sPrinciplebyadjoiningtheconstraint
curves in a Poincare´ section are not guaranteed to have the
graph property, QFMin pseudo-orbits are easier to construct (cid:90) (cid:90) 2πq
A≡ Θ∇ζ·dl≡ Θdζ =const. (11)
than ghost orbits and thus it would be advantageous to find a
C 0
QFMinformulationwhosepseudo-orbitsarealsoghostorbits.
This associates with a path r = r(ζ) an “area” A under the
To construct a rational-rotational-transform QFMin surface
corresponding curve in the (ζ,Θ) plane. By writing Θ =
we find periodic pseudo-orbits with rotational transform p/q.
(p/q)ζ +Θ +Θ(cid:101)(ζ), where Θ(cid:101) is a periodic function averaging
Wevaryνcontinuouslyovertherangeforwhichsolutionscan 0
to 0 and Θ is a constant approximately equal to the value of
be found, the corresponding pseudo-orbits sweeping out rib- 0
bons that may be joined to form the entire surface Γ. As the Θ at which the pseudo-orbit cuts the section ζ = 0, we get
range includes ν = 0, Γ includes the closed field lines, both A = 2πqΘ0 + 2π2pq. Thus, constraining, A to be constant
allowsonetoselectapseudo-orbitlabelledbyΘ ,and
action-minimizingandminimax, associatedwiththemagnetic 0
islandchainwiththegivenrotationaltransform. r=r (ζ|Θ ), Θ∈[0,2π), (12)
p,q 0
definesafamilyofpathscoveringatorusembeddedinR3.
5. Comparison
ToimplementtheconstraintEq.(11)usingaLagrangemul-
tiplierνwedefinethepseudoaction
For illustration we use a model magnetic Hamiltonian (see
Sec. 7), χ + χ , consisting of an integrable part, χ =ψ2/2, S ≡S−νA, (13)
0 1 0 ν
4
whichisthesameasthephysicalactionEq.(3)withAreplaced 8. Lagrangianformulation
by the vector pseudopotential A ≡ A − νΘ∇ζ. Thus, the
ν
Anarbitrarytrialcurve,C,requiresboththe“positioncurve,”
verification of Hamilton’s Principle for QFMin pseudo-orbits
θ(ζ),andthe“momentumcurve,”ψ(ζ),tobespecified.
goes through exactly as that in Sec. 3 for physical magnetic
It is advantageous to reduce the number of dependent vari-
field lines (except that we must use the restricted variations,
δr=∆r,sothattheendpointcontribution2πpδr·∇ζ vanishes): ables by transforming from the Hamiltonian phase-space de-
δS /δs = δS /δθ = 0 implies r˙×B = 0, where B is the scription to the Lagrangian configuration-space description in
ν ν ν ν
thestandardway,eliminatingthemomentumψinfavorofthe
pseudo magnetic field Eq.(9), as required for QFMin pseudo-
orbits,forwhichr˙ =B /Bζ. velocityθ˙givenbyEq.(16). [Weassumethe“twistcondition,”
ν χ(cid:48)(cid:48)(ψ)>0,toallowuniqueinversiontogiveψ=Ψ (θ˙,θ,ζ),so
0 ν
thistransformationmaynotbepossibleforsystemswithnon-
7. Hamiltonianformulation monotonicrotationaltransform.]
Thevelocitycurvethendefinesthemomentumcurve, mak-
By exploiting gauge freedom we may write ingS afunctionalofthepositioncurvealone,whilepartially
A=ψ∇θ−χ(ψ,θ,ζ)∇ζ, for which S takes the familiar ν
extremizing
form
(cid:90) 2πq
(cid:90) S = dζL (θ,θ˙,ζ), (18)
S= (ψdθ−χdζ). (14) ν ν
0
C
where the pseudo-Lagrangian L ≡ Ψ θ˙ − χ (Ψ ,θ,ζ). (The
ν ν ν ν
Thisisthecanonicalformforthevectorpotential,astheequa-
physical field-line Lagrangian L and action S are obtained as
tionsdescribingthefieldlinesareθ˙ =∂χ/∂ψandψ˙ =−∂χ/∂θ. thespecialcaseν=0.)
TheseareHamilton’sequations,withχthemagnetic-field-line
Nowthetotalperturbationofthepseudoactionisoftheform
Hamiltonian. Henceforthweremovethearbitrarinessinthera-
dialcoordinatesbychoosingittobethefluxfunctionψ,sothat (cid:90) 2πq δS
δS = δθ ν dζ , (19)
sandθarenowcanonicallyconjugatephase-spacevariablesfor ν δθ
0
thegivenmagneticfield,andthereturnmapgeneratedbyfield
wheretheLagrangianpseudoactiongradientis
linesintersectingthePoincare´ sectionζ =0isarea-preserving.
(cid:32) (cid:33)
FromHamilton’sequations,weseethatifχdependsonlyon δS ∂L d ∂L
ν = ν − ν . (20)
ψ, e.g. χ=χ0(ψ), then ψ and θ are action-angle coordinates δθ ∂θ dζ ∂θ˙
[11]: ψ is constant, and θ increases linearly with ζ according
to θ˙ = -ι(ψ). If the transform is rational, θ˙ = p/q for inte-
9. ReconcilingQFMinandLagrangianGhostsurfaces
ger p and q, there exists a continuous family of periodic field
lines. Each periodic field line may be identified, for example, We now seek to choose Θ so that QFMin pseudo-orbits are
bythepoloidalanglewherethefieldlineintersectsthePoincare´ alsoLagrangianghostpseudo-orbitsasdefinedinAppendixB.
section ζ = 0. Together, the periodic field lines form a ratio- First note that, to reconcile Eq.(16) and Eq.(47), we require
nal surface. However, in a nonintegrable system no transfor- ∂ Θ=0. Thatis,
ψ
mation to action-angle coordinates exists and the Hamiltonian
has the more general form χ=χ (ψ)+χ (ψ,θ,ζ). No matter Θ≡Θ(θ,ζ). (21)
0 1
how small χ1 is, the continuous family of periodic field lines ThenΨν = Ψ,Lν = L−νΘ,∂Lν/∂θ˙ = ∂L/∂θ˙,andmembersof
isdestroyed,beingreplacedbyisolatedaction-minimizingand ourfamilyofQFMinpseudo-orbits
minimaxorbitsinanislandchain.
IncanonicalcoordinatestheQFMinpseudoaction, Eq.(13), θ=θ(ζ|Θ0), (22)
(cid:72)
is,fromEq.(14),Sν = (ψθ˙−χ−νΘ)dζ,uptoO(∆r), where Θ is an as yet arbitrary label [cf. Eq.(12)], satisfy the
0
(cid:90) 2πq (cid:104)(cid:16) (cid:17) Euler–Lagrangeequation
δS = dζ θ˙−∂ χ−ν∂ Θ δψ (15)
ν ψ ψ δS δS
0 ν = −ν(Θ )Θ (θ,ζ)=0, (23)
δθ δθ 0 θ
(cid:16) (cid:17) (cid:105)
− ψ˙ +∂θχ+ν∂θΘ δθ , withΘ (θ,ζ)≡∂Θ(θ,ζ)/∂θ.
θ
ToreconcileQFMinandghostorbitswerequirethatthefam-
Settingthefirstvariationtozero,wefindHamilton’sequations ily of pseudo-orbits defined by Eqs (22) and (23) is the same
forQFMinpseudo-orbits family as is generated by Eq.(48). Thus the labels Θ and τ
0
must be functionally dependent: τ = τ(Θ ), dτ = τ(cid:48)(Θ )dΘ .
θ˙ = ∂ χ (16) 0 0 0
ψ ν Eliminating δS/δθ between Eq.(23) and Eq.(48) and observ-
ψ˙ = −∂θχν , (17) ing that DΘ/DΘ ≡ (Dθ/DΘ )Θ we find the reconciliation
0 0 θ
whereχ ≡χ+νΘ. Thatis,νΘactsasascalarpseudopotential condition
ν
thatdisplacespseudo-orbitspoloidally,awayfromtheperiodic DΘ µ(ζ) (cid:32) Dθ (cid:33)2
orbitsfoundwhenν=0. DΘ =−τ(cid:48)(Θ )ν(Θ ) DΘ . (24)
0 0 0 0
5
WenowdefineΘ sothat,forallΘ , where δ¯ ≡ 1 − δ deletes the resonant components
0 0 mp,nq mp,nq
(which have been absorbed by ν —unlike the KAM problem,
τ(cid:48)(Θ )ν(Θ ) ≡ −1, 1
0 0 perturbation theory for almost-invariant tori is not inherently
θ(ζ|Θ +2π) ≡ θ(ζ|Θ )+2π, (25) afflictedbysmalldenominators, atleastwhen -ιisalow-order
0 0
rational).
choosingµ(ζ)sothatEq.(24)satisfiesthesolvabilitycondition As the two Fourier coefficients in Eq.(31) are the same,
thattheintegralofbothsideswithrespecttoΘ0overtheinterval θ (ζ|Θ ) = Θ (-ι ζ +Θ ,ζ). However, Θ is not used in the
1 0 1 p,q 0 1
[0,2π]mustbe2π,giving
calculationofθ : tofirstorder, ghostanduncorrectedQFMin
1
µ(ζ)=(cid:90)02π d2Θπ0 (cid:32)∂θ∂(ζΘ|Θ00)(cid:33)2−1 . (26) pitnhsgeesueodftow-µoo1r,abliimtssocasorten-isniidvsetaenrnitaitcnawtls,iutwhrfhatihcceehs,nicenoaFmrigbin.ind2ei.sdtiwngituhisthhaebvilaitnyisho-f
AtO((cid:15)2)wefind
10. PerturbativeconstructionofQFMin-ghostsurfaces θ(2) = δ¯ (cid:88)(cid:48)im(cid:48)(m+m(cid:48))2Vm+m(cid:48),n+n(cid:48)Vm∗(cid:48),n(cid:48)
m,n mp,nq (m-ι −n)2(m(cid:48)-ι −n(cid:48))2
For example, to approach this task perturbatively, consider m(cid:48),n(cid:48) p,q p,q
the Lagrangian of the class corresponding to the case studied − δ¯mp,nq (cid:88)(cid:48)ν(1) im(cid:48)Θ(1)∗ , (32)
numericallyinSec.5 (m-ι −n)2 m+m(cid:48),n+n(cid:48) m(cid:48),n(cid:48)
p,q m(cid:48),n(cid:48)
θ˙2 (cid:88)∞ where the prime on the sum over m(cid:48) and n(cid:48) indicates that the
L= −(cid:15) V exp(imθ−inζ), (27)
2 m,n resonant terms, m(cid:48)p = n(cid:48)q, are to be deleted, and ν(1) ≡
m,n=−∞ −imV δ . The term containing Θ(1) gives the O((cid:15)2m),ndif-
m,n mp,nq
with the reality condition V∗ = V (superscript ·∗ de- ference between ghost pseudo-orbits and uncorrected QFMin
m,n −m,−n
noting complex conjugation), and (cid:15) the expansion parameter. pseudo-orbits.
(The cases studied in Sec.5 thus correspond to the choices WealsoseefromEq.(26)thatµ =−2(cid:15)2(cid:104)(∂θ /∂Θ )2(cid:105),where
2 1 0
V = χ /2, V = χ /2, and V = 0 for {m,n} (cid:60) (cid:104)·(cid:105)denotesaveragingoverΘ ,sothecorrectionfactorµinthe
2,1 2,1 3,2 3,2 m,n 0
{{2,1},{−2,−1},{3,2},{−3,−2}}.) definitionofghostsurfacesrequiredforreconciliationdoesde-
Astheunperturbedsystemisintegrable,theexpansionsofν, viatefromunitybysmallamount,O((cid:15)2).Inthenumericalwork
µ,andΘ(θ,ζ)areoftheform presentedinSec.12wetakeµ=1.
ν = (cid:15)ν +(cid:15)2ν +... ,
1 2
11. NumericalconstructionofQFMinsurfaces
µ = (cid:15)µ +(cid:15)2µ +... , (28)
1 2
(cid:88)(cid:16) (cid:17)
Θ = θ+ (cid:15)Θ(1) +(cid:15)2Θ(2) +... expi(mθ−nζ), Previously [9], periodic pseudo field lines were found by
m,n m,n
integrating the Hamiltonian equations of motion Eqs (16–17).
m,n
Theconstrainedactionprincipleinsteadallowsperiodicpseudo
andofthe(p,q)QFMinpseudo-orbitsareoftheform
orbits to be found variationally. This is of great benefit for
(cid:88)(cid:16) (cid:17) strongly chaotic fields, as a defining characteristic of chaos is
θ(ζ|Θ ) = -ι ζ+Θ + (cid:15)θ(1) +(cid:15)2θ(2) +...
0 p,q 0 m,n m,n the exponential separation of initially nearby trajectories at a
m,n
rategivenbytheLyapunovexponent. Consequently,theintrin-
(cid:104)(cid:16) (cid:17) (cid:105)
×expi m-ιp,q−n ζ+mΘ0 ,sic numerical error associated with field-line-following meth-
odsisguaranteedtoincreaseasthetrajectorybecomeslonger.
(cid:88) (cid:16) (cid:17)(cid:16) (cid:17)
ψ(ζ|Θ ) = -ι + i m-ι −n (cid:15)θ(1) +(cid:15)2θ(2) +... The method of Lagrangian integration, also called variational
0 p,q p,q m,n m,n
m,n integration[12],avoidstheseproblems. AGalerkinexpansion
×expi(cid:104)(cid:16)m-ι −n(cid:17)ζ+mΘ (cid:105) , (29)ofthetrialcurve,
p,q 0
(cid:88)N−1
where -ι ≡ p/qandψ≡∂L/∂θ˙ =θ˙ =θ (ζ|Θ ). θ= -ι ζ+ au(ζ), (33)
p,q ζ 0 p,q i i
AtO((cid:15))wefindµ1 =0and i=0
in, say, N continuous, 2πq-periodic basis functions u, is in-
(cid:88)∞ i
ν1(Θ0)=− imδmp,nqVm,nexp(imΘ0), (30) sertedintotheactionprincipleandtheunknownamplitudesai
m,n=−∞ arevaried,
where the Kronecker delta δ selects Fourier coefficients (cid:88)N−1
suchthatmp=nq,resonantwmitph,nq(p,q)pseudo-orbits. δθ= δaiui(ζ), (34)
i=0
TheO((cid:15))contributionstothepseudo-orbitandpseudopoten-
tialaregivenby giving,fromEq.(19),
θm(1,)n =Θ(m1,)n = i(mmδ¯-ιmp,pq,n−qVnm),2n , (31) δSν = (cid:88)Ni=−01(cid:42)ui,δδSθν(cid:43)ai
6
1(cid:88)N−1(cid:88)N−1 (cid:42) δ2S (cid:43) 13. Conclusion
+ a u, νu a +... , (35)
2 i i δθδθ j j
i=0 j=0 In this paper we have given generalized definitions of
wheretheinnerproduct(cid:104)f,g(cid:105), f(ζ)andg(ζ)arbitrary,isdefined quadratic-flux-minimizing (QFMin) surfaces and ghost sur-
by faces, formulatingthemintermsofthemagneticfield-lineac-
(cid:90) 2πq tion. We have gone further by introducing a new constrained
(cid:104)f,g(cid:105)= dζ fg. (36) Hamilton’sPrincipleforQFMinpseudo-orbitsthata)facilitates
0 a reconciliation between QFMin surfaces and ghost surfaces,
Extremizingcurvesarethenobtained,forexample,byfind-
and b) provides a better algorithm than o.d.e. integration for
ing a zero of the N-dimensional gradient (cid:104)δS /δθ,u(cid:105) using
ν i calculationofQFMinorbitsinstronglychaoticregions.
Newton’s method. This approach allows very high order pe-
QFMin and ghost orbits have been computed for a model
riodicorbitstobefound, evenforquitestronglychaoticfields
magnetic field perturbed away from an integrable system de-
[13].
scribed in action-angle coordinates, and found almost, but not
Fornumericalwork,wemustdescribeθ(ζ)withafinitesetof
quite, to coincide. This is explained by finding a slight gen-
parameters. Itissimplesttouseapiecewiselineardescription,
eralization of our previous ghost surface definition that allows
i.e. afinite-elementexpansionofθ(ζ)intentfunctions:
QFMin and ghost surfaces to be fully reconciled, in principle,
1 (cid:2) the change in the ghost surfaces being O((cid:15)2). Using pertur-
u(ζ) ≡ H(ζ−ζ )H(ζ −ζ)(ζ−ζ )
i ∆ζ i−1 i i−1 bation theory we find a choice of poloidal angle that unifies
+H(ζ−ζi)H(ζi+1−ζ)(ζi+1−ζ)(cid:3). (37) QFMin and ghost surfaces up to second order in (cid:15), with the
differencebetweenghostsurfacesanduncorrectedQFMinsur-
where H(·) is the Heaviside step function, ζ is to be evalu-
facesbeingO((cid:15)2).
ated mod 2πq to enforce periodicity [thus splitting the sup-
Itremainsforfutureworktoimplementthisreconciliationat
port of u ≡ u between (0,ζ ) and (ζ ,ζ )], ζ ≡ i∆ζ,
0 N 1 N−1 N i alllevelsofnonlinearityandtoexplorewhetheritisbeneficial
and ∆ζ ≡ 2πq/N. At first, the piecewise-linear approxima-
to use further generalizations of the ghost surface definition.
tionseemscrude,butthegreatbenefitisthattheactionintegral
Alsorequiringfurtherinvestigationisthequestionofwhyghost
can be calculated analytically in each interval [13]. The dis-
surfaces coincide so closely with temperature isosurfaces [4]
cretized pseudoaction S becomes a rapidly computable func-
ν forheattransportinchaoticmagneticfields.
tion of the N independent parameters that describe the curve,
The innovations in this paper should also be applicable in
{θ ,θ ,...,θ }, where θ ≡ (p/q)ζ + a. Extremal curves
0 1 N−1 i i i general11-degree-of-freedomHamiltoniansystemsandinthe
are found as zeros of the constrained action gradient, which 2
theoryofarea-preservingmaps.
are efficiently found using an N + 1 dimensional Newton’s
method. Note that the Hessian matrix (cid:104)u,(δ2S /δθδθ)u (cid:105) as-
i ν j
sociated with the variations in the curve geometry is a cyclic A. QFMinEuler–Lagrangeequation
tridiagonalmatrix.
Here we rederive in canonical coordinates the Euler–
12. NumericalconstructionofLagrangianghostsurfaces Lagrange equation for quadratic-flux-minimizing surfaces
found by Dewar et al. [1]. Consider a toroidal magnetic
We discretize Eq.(48) using the Galerkin method. That is, field in canonical form, B = ∇×(ψ∇θ − χ∇ζ), with Jaco-
wesubstitutetheansatzEq.(33)intoEq.(48)andprojectonto bian 1/√g = ∇ψ·∇θ×∇ζ. A toroidal surface is described by
thefinitebasis{ui}, ψ= P(θ,ζ). Wedefinethetangentialdynamicsusingtheequa-
(cid:88)N−1 Da (cid:42) δS(cid:43) tionθ˙ = Bθ/Bζ asaconstraint,andrequirethe“radial”dynam-
(cid:104)ui,uj(cid:105) Dτj = − ui, δθ . (38) ics to be confined to the surface, so that ψ˙ = Pθθ˙+Pζ, where
i=0 P ≡ ∂ P, α ∈ {θ,ζ}. The angle parametrization is arbitrary,
α α
Forabasiswithglobalsupportthematrix(cid:104)ui,uj(cid:105)isfulland and so let Θ = Θ(θ,ζ) be a new poloidal angle. We define
itmustbeinvertednumericallytoget Daj/Dτ. Forourfinite- ν as the projection, ν ≡ B·N¯, of the magnetic field onto the
elementbasisthematrixistridiagonal, vector N¯ ≡(eΘ+Pθeψ)×(eζ +Pζeψ) normal to Γ, where here
(cid:104)ui,uj(cid:105)=∆ζ(cid:34)δi,j+ (∆6ζ)2δi+1,j−(2∆δζi,)j2+δi−1,j(cid:35), (39) (sapnedctotnolyΘ,hζerea)t ceoΘn,setaζnatrζe,tΘherdeseprievcattiivveelsy:ofeΘpo≡sit∂ioxn/∂wΘi|tζharned-
eζ ≡ ∂x/∂ζ|Θ. Combiningtheseexpressions,thetangentialdy-
where the Kronecker δi,j is the identity matrix. The term namicsmaythenbewrittenθ˙ =∂χ/∂ψandψ˙ =−∂χ/∂θ−νΘ ,
(δi+1,j−2δi,j+δi−1,j)/(∆ζ)2isafinite-differenceapproximation whichmayberecognizedasthepseudofield,Eq.(9). Thegenθ-
to d2/dζ2, so, assuming the action gradient is a smooth func-
eralizedquadratic-fluxfunctionalEq.(2)canbewritten
tion, the second term on the RHS is two orders in ∆ζ smaller
(cid:90)(cid:90)
than the first and so can be neglected. Thus Eq.(38) can be 1 (cid:16) (cid:17)
ϕ = dθdζ (B·N) B·N¯ , (41)
solvedanalyticallytogive 2 2
(cid:42) (cid:43)
DDaτi ≡ DDθτi =−∆1ζ ui,δδSθ ≡−∆1ζ∂∂θSi . (40) nwohwereeζN≡≡∂(xe/θ∂+ζ|θP.θeTψh)e×fi(erζst+vParζieaψti)o,nwihnetrheiseθfu≡nct∂ixo/n∂alθ|dζ,ueantod
7
variationsinthesurfaceis C. Erratum
(cid:90)(cid:90)
√ (cid:16) (cid:17)
δϕ = dθdζδP g Bθ∂ +Bζ∂ ν, (42) The corrected version above includes the er-
2 θ ζ
rata for Phys. Letts. A 373, 4409–4415 (2009),
√ √ http://dx.doi.org/10.1016/j.physleta.2009.10.005 :
whe√re we have used ∂( gBβ)/∂α|ψ=P = ∂ψ( gBβ)Pα + On p. 4413, in the fifth line below Eq. (23), replace “dτ =
∂α( gBβ),forα ∈ {θ,ζ}andβ ∈ {θ,ζ},toreflecttheconstraint τ(cid:48)(Θ )dθ”with“dτ=τ(cid:48)(Θ )dΘ ”.
0 0 0
thatwhenθorζvary,ψmustalsovarytoremainonthesurface.
Onp. 4414,onthesecondlineofEq.(32),replace
imΘ(1)
B. Generalizedghostsurfaces +ν (Θ ) m,n ,
1 0 (m-ι −n)2
p,q
IncanonicalcoordinatesEq.(4)becomes,toO(δr), with
δ¯ (cid:88)
δS = (cid:90) 2πdqζ(cid:16)δψeψ+δθeθ+δζeζ(cid:17)·δδSr (43) − (m-ιpm,qp,−nqn)2 m(cid:48),n(cid:48)(cid:48)νm(1+)m(cid:48),n+n(cid:48)im(cid:48)Θ(m1(cid:48)),∗n(cid:48) ,
0
(cid:90) 2πq (cid:104)(cid:16) (cid:17) (cid:16) (cid:17) (cid:105) andreplacethesentenceafterthisequation,
= dζ θ˙−∂ψχ δψ− ψ˙ +∂θχ δθ+χ˙δζ , “wheretheprimeonthesumoverm(cid:48)andn(cid:48)indicatesthatthe
0 resonantterms,m(cid:48)p=n(cid:48)q,aretobedeleted.”
wherethesecondformfollowsdirectlyfromEq.(14). Identify- with
ingcoefficientsofδψandδθbetweenthetwoforms, “where the prime on the sum over m(cid:48) and n(cid:48) indicates that
the resonant terms, m(cid:48)p = n(cid:48)q, are to be deleted, and ν(1) ≡
m,n
δS =(cid:16)θ˙−∂ χ(cid:17)eψ−(cid:16)ψ˙ +∂ χ(cid:17)eθ+χ˙eζ . (44) −imVm,nδmp,nq.”
δr ψ θ
A slightly generalized definition of the Lagrangian definition Acknowledgements
[2] of ghost surfaces can be found by choosing the projection
operatorinEq.(7)tobe SomeofthisworkwassupportedbytheAustralianResearch
Council (ARC) and U.S. Department of Energy Contract No.
P = 1e e + 1e e , (45) DE-AC02-76CH03073 and Grant No. DE-FG02-99ER54546.
ghost (cid:15) ψ ψ µ θ θ WeacknowledgeausefuldiscussionwithProf. J.D.Meisson
bifurcationofaction-minimaxorbits.
where (cid:15) → 0 is a switching factor required to obtain the La-
grangiandescription,asdescribedbelow,andµ(ζ) = O(1)isa
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8