Hamiltonian-Dirac Simulated Annealing

22 Pages·2010·5.63 MB·English

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PhysicaD240(2011)212–232
ContentslistsavailableatScienceDirect
Physica D
journalhomepage:www.elsevier.com/locate/physd
Hamiltonian–Dirac simulated annealing: Application to the calculation of
vortex states
G.R.Flierla,P.J.Morrisonb,∗
aDepartmentofEarth,Atmospheric,andPlanetarySciences,MassachusettsInstituteofTechnology,Cambridge,MA02139-4307,USA
bDepartmentofPhysicsandInstituteforFusionStudies,TheUniversityofTexasatAustin,Austin,TX,78712,USA
a r t i c l e i n f o a b s t r a c t
Articlehistory: Asimulatedannealingmethodforcalculatingstationarystatesformodelsthatdescribecontinuousmedia
Availableonline22September2010 isproposed.ThemethodisbasedonthenoncanonicalPoissonbracketformulationofmedia,whichis
usedtoconstructDiracbracketswithdesiredconstraints,andsymmetricbracketsthatcauserelaxation
Keywords:
withthedesiredconstraints.Themethodisappliedtotwo-dimensionalvortexdynamicsandavarietyof
Vortexdynamics
numericalexamplesisgiven,includingthecalculationofmonopoleanddipolevortexstates.
Hamiltonian
©2010ElsevierB.V.Allrightsreserved.
NoncanonicalPoissonbracket
Lie–Poissonbracket
Diracbracket
Simulatedannealing
1. Introduction generatedynamicsthatisanenergyextremizingrearrangement,
while preserving additional properties such as linear or angular
TheEuleriandescriptionofmediahasaHamiltonianformula- momentum.Here,thisisdoneforavarietyofnumericalexamples
tionintermsofnoncanonicalPoissonbrackets(see[1–6]andrefer- in the context of vortex dynamics. A preliminary version of our
method was reported in [17] where some results for contour
encestherein),bracketsthatarenotofthestandardformbecause
dynamicscalculationswerepresented.
Eulerianvariablesarenotcanonicalvariables.Incorporatingdissi-
The paper is organized as follows: in Section 2 we develop
pationintosomekindofgeneralformalismdatestoRayleigh,but
the needed theory, which includes a description of the various
theuseofsymmetricbracketsinconjunctionwithnoncanonical
bracketformulations.Thespecificcaseoftwo-dimensionalvortex
Poissonbracketsforthispurposewasproposedin[7–9].Unlikethe
dynamics isused to illustrate thegeneral ideas. This is followed
widelystudiedbutmoresimplisticCahn–Hilliardgradientflows
in Section 3 by numerical examples that illustrate various kinds
(seee.g.[10]),theformulationoftheseauthorswasconcernedwith
ofdynamics,includingvortexstateswithtwo-foldandthree-fold
brackets that generate a dynamics compatible with the laws of
symmetryanddipolestates.InSection4two-layerquasigeostropic
thermodynamics,aformulationthatwastermedmetriplecticdy-
dynamics is considered with monopolar vorticity in each layer.
namicsin[11].Thesesystemsarecompleteinthesensethaten-
Finally,inSection5weconclude.
ergyisconservedwhileentropyisproduced.Dissipativebrackets
havealsobeensuggestedforincompletesystemsthathavenono-
2. Theory
tionofentropybutdissipateenergywhilepreservingCasimirin-
variants[12–15].See[16]forageneraldiscussion.Thelatterwere AsmentionedinSection1,thetheorybehindourformulation
proposedasameansforcomputation,particularly[13–15]inthe has several basic parts: a noncanonical Poisson bracket defined
contextoffluidmechanics,wheretheformalismwasusedandpro- on functionals or observables (see e.g. [2,6,18]), a Dirac bracket
posedforcalculatingstationaryvortexstatesbyakindofsimulated basedonsuchanoncanonicalPoissonbracket(seee.g.[19,20]for
annealingdynamics. traditionalDiracbracketsandseee.g.[17,21]forthenoncanonical
We offer a new twist that overcomes difficulties obtained version), and a symmetric bracket that gives rise to relaxation
in [13–15]—we extremize energy at fixed Casimir invariants by akintocollisionoperatorsinkinetictheory(seee.g.[15,16]).We
using Dirac brackets to enforce constraints. In this way one can describetheseobjectshere.
2.1. Functionals
∗
Correspondingauthor.Tel.:[phone].
Functionalsmapasetofdynamicalvariablesintonumbers.In
E-mailaddresses:[email protected](G.R.Flierl),
[email protected](P.J.Morrison). general one may have M dynamical variables denoted by χi(z),
0167-2789/$–seefrontmatter©2010ElsevierB.V.Allrightsreserved.
doi:10.1016/j.physd.2010.08.011
Author's personal copy
G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232 213
wherei = 1,2,...,M,thatarefunctionsofz = (z1,z2,...,zN), Clearlythefunctionalderivativedependsingeneraluponspa-
coordinatesforanN-dimensionaldomain,D,andfunctionalsof tiallocation—attimeswewillneedtobeexplicitaboutthistodis-
thesevariableswillbedenotedbyF[χ]. tinguishxfromx′etc.;therefore,wewillusethenotation
Forexample,whenthedynamicalvariableisasinglevorticity- δF[q]
like variable, the case for the calculations of Section 3, the ,
δ ( )
functionalF[q]mapsthefunctionq(x)intoR.Herex = (x,y)is q x
apointinD,aplanarregion,takenforthenumericalcalculations explicitlyshowingthattheindependentvariableisx.Thus(7)takes
here to be a 2-torus. Examples of such observables include the theperspicuousform
angularmomentum, δ ( )
q x
L=−1∫ dx|x|2q(x), (1) δq(x0) =δ(x0−x),
2
D aquantitythatwillbeneededintheevaluationofPoissonbrackets
andtheHamiltonianfunctional, below.
∫ ∫
1
H =− dx dx′q(x)G(x,x′)q(x′), (2) 2.2. NoncanonicalPoissonbrackets
2
D D
where G is the relevant Green’s function; for two-dimensional AgeneralnoncanonicalPoissonbracketisabinaryoperatoron
vorticitydynamicsitsatisfies∇2G(x)=δ(x). functionalsoftheform
Note,thekineticenergyofthefluidonaninfinitedomaincan ∫ ∫ δF[χ] δG[χ]
haveaninfinitepartassociatedwiththe1/rfar-fieldwhenthenet {F,G}= dNz′ dNz′′ δχi(z′)Jij(z′,z′′)δχj(z′′), (10)
vorticityisnon-zero;sincethelatterisconserved,thiscomponent D D
χ
of the energy is fixed. Subtracting it from the total gives the where denotes a set of dynamical variables with M elements,
Hamiltonian,whichthusisnotquitethephysicalenergy.Asimilar i,j = 1,...,M, that depends on a coordinate z of a space of
situationarisesfortheangularmomentumwhereaboundaryterm dimension N. In (10), J is the cosymplectic operator that must
isremovedfromthequantityL.InthefollowingwewillrefertoH ensurethatthebracketsatisfies:antisymmetry,{F,G}=−{G,F},
astheenergyandLastheangularmomentum. andtheJacobiidentity,{{F,G},H}+{{G,H},F}+{{H,F},G}=0,
In addition to the functionals (1) and (2), there is q(x ), the forallfunctionalsF,G,H.Notethat,becauseoftheformof(10),
0
functional that is the evaluation of q at a particular point of its bilinearityandtheLeibnitzderivationpropertiesaremanifest.Itis
domain, notrequiredthatJbenondegenerate.Whenitisdegenerate,i.e.
∫ hasanontrivialnullspace,thereexistCasimirinvariants,C,that
q(x )= dxδ(x −x)q(x). (3) satisfy{C,G}=0forallfunctionalsG.(Seee.g.[6,17]forreview.)
0 0
D ThePoissonbracketforvortexdynamics[22,2,23]isthefollow-
This latter functional is needed when casting the equations of ing:
motionintoHamiltonianform,asisdoneinSection2.5below. ∫ [δF[q] δG[q]]
Thetimerateofchangeoffunctionalswillbegeneratedbymak- {F,G}= dx′q(x′) , , (11)
inguseofvariousbrackets,andtheseinvolveanotionoffunctional D δq(x′) δq(x′)
orvariationalderivative.Thiskindofderivativedescribeshowthe where[A,B]=zˆ·∇A×∇Bwithzˆaunitvectornormaltothepla-
valueofthefunctionalchangesaswemakesmallchangestoits narregion;thus,[A,B]isanordinarybracketthatcanbewritten
argument.Thusweconsiderthefirstvariation, inCartesiancomponents,oritcanbewrittenintermsofwhatever
δF[χ;η]= d F[χ +ϵη] =∫ dNz δF[χ]ηi, (4) tphleecitnicdeoppeenradteonrtJv=aria−b[lqe,s·m]aignhdttbhee.JFaocrobthiiisdeenxtaimtypclaenthbeecporsoyvmed-
dϵ ϵ=0 D δχi directlybythetechniquesdescribedin[2].Thewell-knownquan-
whererepeatedindicesaretobesummedhere(andwillbehence- titiesC[q]= dxC(q)withCarbitrary,afamilyofintegralsthat
D
forthunlessotherwisestated).Usingtheintegraloverzasaninner includesthenetvorticityandenstrophy,areCasimirinvariants.It
product,thequantityδF/δχ canbeinterpretedasagradientand iseasytoshow{C,G}=0forallG.
δF[χ;η]astheFréchetderivativeactingonη(z). DynamicsisgeneratedfromthenoncanonicalPoissonbracket
Fromthisdefinition,thefunctionalderivativesoftheexamples accordingto
(1)–(3),arereadilyobtained, ∂
q
={q,H}, (12)
δL[q] |x|2 ∂t
=− , (5)
δq 2 theusualPoissonbracketHamiltonianform.Wewillrefertothis
δH[q] ∫ dynamicsasH-dynamics.With(7)and(11),wecanshowthat
δq =− Ddx′G(x,x′)q(x′), (6) {q(x),G} = ∫ dx′q(x′)[δq(x), δG[q]]
δ ( ) δq(x′) δq(x′)
q x D
δq0 =δ(x0−x), (7) = ∫ dx′q(x′)[δ(x−x′), δG[q]]
δq(x′)
wherein(6)thesymmetricnatureofGhasbeenused.Withthe D
streamfunctionψthatsatisfies,∇2ψ =q,wecanwrite [ δG[q]]
= − q(x), , (13)
∫ δq(x)
ψ(x)= Ddx′G(x,x′)q(x′), (8) using the identity Ddxf[g,h] = −Ddxg[f,h], obtained by
integrationbypartsandneglectofsurfaceterms.Wenowhave
andthereforethefunctionalderivativeoftheHamiltonianis
δH[q] ∂ [ δH[q]]
=−ψ(x). (9) q(x)={q(x),H}=− q(x), =[q,ψ], (14)
δq(x) ∂t δq(x)
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214 G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232
whichistheusualformforthetwo-dimensionalEulerequations. is,atmost,afunctionoftimebutnotx.Thereforefrom
Functionalsevolveaccordingto
{F,G} ={F,G}−Ai{F,C}, (21)
D i
∂
F
={F,H}; (15) weseethattheevolutionofF under(11)isgivenby
∂
t
∫ [δ δ ] ∫ [δ δ ]
F G F C
since{C,H} = 0fortheCasimirinvariants,theyareindeedcon- {F,G} = dxq , −Ai dxq , i
D δ δ δ δ
q q q q
stantintimeforanychoiceoftheHamiltonian. D D
∫ [δ δ δ ]
F G C
= dxq , −Ai i . (22)
δ δ δ
2.3. NoncanonicalDiracbrackets D q q q
Thus,theflowdevelopsaccordingto
From any bracket of the form of (10) and any choice of con-
straintfunctionalsCisuchthattheantisymmetricmatrix{Ci,Cj}is ∂ q=−[Ψ,q], (23)
∂
nonsingular,onecanconstructageneralizedDiracbracketasfol- t
lows: where
{F,G}D ={F,G}−{F,Ci}Cij{Cj,G}, (16) Ψ =ψ+Aici and
whereCij :={Ci,Cj}−1.Eq.(16)definesabracketthatisobviously Ai =−∫ dxq[c,c]−1∫ dxc[ψ,q]. (24)
antisymmetricandcanbeproventosatisfytheJacobiidentity(cf. i j j
D D
AppendixCof[21]).Underthisform,theconstraintsCi,areCasimir Thisformfollowsfrom{C,H}using(11)ordirectlyfrom
invariants, j
∂ ∫ ∂
{C,G} =0 forallG. (17) {C,H}= C = dxc q.
i D j ∂ j j∂
t t
D
Forthematrix{Ci,Cj}tobeinvertible,wemusthaveanevennum- Thus,Diracbracketdynamicsisthe‘‘advection’’ofqbyΨ,whichis
berofconstraints,andnoneoftheC’scanbeCasimirinvariants
i theusualstreamfunctionwiththeadditionoftheconstraintfields
oftheoriginalnoncanonicalbracket;todistinguishthosefromthe multipliedbythetimedependentcoefficientsAi.
onesusedintheDiracconstruction,wewillrefertothelatteras InSection3wewillusetheinvariantsC =−LandC equalto
1 2
Diracconstraints.
thestrainmoment,definedby
DiracconstructedsuchbracketsoutofcanonicalPoissonbrack-
∫
etswiththegoalofquantizingclassicalfieldtheories,butherewe S = dxxyq. (25)
usenoncanonicalPoissonbrackets,andtheaimisthepurelyclas-
D
sicalimpositionofconstraintsonthedynamics.Althoughonecan WiththesechoicesweseethatΨ obtainsarotationalpiecemul-
constructDiracbracketsoutofanyevennumberofconstraints,we
tipliedbyA ,aglobalquantity,andapiecethatisastrainingfield
willgenerallyuseonlytwoorfourglobalconstraints,i.e.onesthat 1
multipliedbyA ,alsoaglobalquantity,viz.
involveintegrationanddonotdependexplicitlyonz.Forpoint- 2
wiseconstraints,Ci(∫z),thege∫neralbrackethasthefollowingform: A1 = Dddxx(xx2y[−ψ,y2q)]q and
{F,G}D = ×{F,{FG,}C−i(z′D)}CdNij(zz′′,Dz′′d)N{Czj′(′z′′),G}. A2 =−D21Ddxdx(x(2x2+−y2y)2[)ψq,q]. (26)
D
We will refer to dynamics generated by Dirac brackets as HD- The time dependence in the Ai applies the two fields in just the
dynamics. right combination to keep the C constant. Although the angular
i
ForthenumericalexamplestreatedinSections3and4wewill momentumisalreadyconservedbyvortexdynamics,thisisnot
useglobalDiracconstraintsoflinearform, necessarily the case when quantities evolve according to Dirac
brackets(cf.Section3.1andAppendixDof[21]).Choosing−Las
∫
C = dxc(x)q(x), (18) oneofourDiracconstraintsguaranteesthatitremainsconserved.
i i
D Although (21) resembles the procedure suggested in [15] of
addingtermstoHwithLagrangemultipliers,itisdifferentinthat
wherethec andtheinitialconditionwillbechosenbyintuition
i thecoefficientschangeintimetoensurethattheDiracconstraints
andbyconsideringthedesiredsymmetriesofthefinalstate.With
remainconstant.Furthermore,atleastoneoftheconstraintsmust
this form for the Dirac constraints, the Dirac bracket is readily
evaluatedusingδCi/δq(x)=ci(x)whichgivese.g. nthoattbteheavcoenctsotran{Cto,fHm}oistinoonnu-nzedreor.theoriginaldynamicstoensure
j
∫ ∫ [δ ]
F
{C,C}= dxq[c,c] and {F,C}= dxq ,c .
i j i j i δq i 2.4. Symmetricbrackets
D D
Inthecaseoftwoconstraints,therequiredmatrixissimple: Ageneralsymmetricbracketanalogousto(10)isgivenby
C=1/{C01,C2} −1/{C01,C2}. (19) ((F,G))G =∫DdNz′∫DdNz′′ δδχFi[(χz′])Gij(z′,z′′)δδχGj[(χz′]′), (27)
Togeneratedynamics,wenotethatforaglobalfunctionalG, where the metric operator G is chosen to ensure ((F,G))G =
(( , ))
theterm G F G andtobesemidefinite.Wemayalsowanttobuildde-
generaciesintoGsothatthereexistdistinguishedfunctionalsD
Ai =Cij{Cj,G} (20) thatsatisfy((D,G))G =0forallG.
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G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232 215
Herewewillbeinterestedinaspecificformof(27)thatisagen- using ∇2Φ = [q,ψ]. Therefore, because ((H,H)) ≥ 0, when
α > , α <
eralizationofthesymmetricbracketsgiveninpreviouswork[11, 0 Htendstomaximize,whileif 0,Htendstominimize.
13,15,16],viz. Finally,theDSAformisgivenby
∫ ∫ ∫
((F,G))= dNz′ dNz′′{F,χi(z′)}Kij(z′,z′′){χj(z′′),G}, (28) ((q(x),H))D = dx′{q(x),q(x′)}DΦD(x′) (33)
D D D
withKadefinitesymmetrickernelthatcanbechosenatwill,e.g., with
toeffectsmoothing.Withthisform,theCasimirinvariantsof{F,G} ∫
willautomaticallybedistinguishedfunctionalsD.Wewillreferto ΦD(x) = dx′K(x,x′){q(x′),H}D
dynamicsinvolvingthiskindofbracketasSA-dynamics,whereSA D
∫
standsforsimulatedannealing. = dx′K(x,x′)[q(x′),ψ(x′)+Aic(x′)]. (34)
Thesymmetricbracketof(28)canbeevenfurthergeneralized i
D
asfollows:
UsingthedefinitionoftheDiracbracket(16)and(13)gives
∫ ∫
((F,G)) = dNz′ dNz′′{F,χi(z′)} K (z′,z′′){χj(z′′),G} , {q(x),q(x′)} = {q(x),q(x′)}−{q(x),C}Cij{C,q(x′)}
D D ij D D i j
D D = [q(x′),δ(x−x′)]−[c(x),q(x)]Cij[q(x′),c(x′)].
(29) i j
whichisoftheformof(28)butwithaDiracbracketreplacing{,}. ThereforetheDSAbracketbecomes
Whenthisisthecase,thesetofdistinguishedfunctionalswillbe ((q(x),H)) = [Φ (x),q(x)]+[Bic(x),q(x)]
D D i
tDhieraucncioonnstorfatinhtesCuasseidmiinrtinhveacroiannsttsruocftitohneobfr{aFck,eGt}{F.W,Ge}wwiiltlhretfheer = [ΦD+Bici,q] (35)
D
todynamicsgeneratedbythiskindofbracketasDSA-dynamicsfor with
Diracsimulatedannealingdynamics. ∫
IfaHamiltonian-likefunctionalF isusedtogeneratedynamics, Bi =Cij dx′c(x′)[q(x′),Φ (x′)]. (36)
j D
thenallthreeofthebrackets(27)–(29)havethepropertydF/dt ≥ D
0.Forexample,underSAdynamics, Again,theqfieldevolvesbyadvection,withthe‘flow’including
correctionstomaintaintheDiracconstraints.
dF
=α((F,F))≥0, (30)
dt 2.5. Variousdynamics
whenK isnegativeandα,aparameterthatmeasuresthe‘dissi-
ij
pative’timescale,ispositive.Thisinequalityfollowsfromthesym- Variouskindsofdynamicalsystemsthatcanbegeneratedby
metrybuiltintothebracket.Infact,thesignofαcanbechosento thebracketsdescribedabovearelistedbelow:
makethedynamicsminimizeormaximizeF.Thuswehaveafor- ∂
mal‘H-theorem’,akintothatoftheBoltzmannequation,andthe Hamiltonian:∂ F ={F,F}, (37)
t
rudimentsforpossiblybuildingrigoroustheoremsaboutasymp-
∂
toticstability. HamiltonianDirac: F ={F,F} , (38)
ForourvortexexamplewiththeHof(2), ∂t D
∫ ∫ ∂
((q(x),H)) = dx′ dx′′{q(x),q(x′)}K(x′,x′′){q(x′′),H} SimulatedAnnealing:∂ F =σ{F,F}+α((F,F)), (39)
t
D D
∫ ∂
= − dx′[δ(x−x′),q(x′)] DiracSimulatedAnnealing:∂ F =σ{F,F}D+α((F,F))D. (40)
t
D
∫ IntheseequationsF representsanarbitraryobservableandF isa
× dx′′K(x′,x′′)[q(x′′),ψ(x′′)] singlefunctionalthatisusedtogeneratethetimeadvancement.
D Eqs. (37) and (38) are ideal and conserve energy in principle,
= [Φ(x),q(x)],
although this will not be precisely the case when numerically
σ α
where we have used (13) and (14) and defined an effective solved. In (39) and (40) the parameters and can be used to
streamfunctionby weight the contributions of the ideal and dissipative dynamics,
respectively. For our runs we will choose σ ∈ {0,1} and α ∈
∫
Φ(x)= dx′K(x,x′)[q(x′),ψ(x′)]. (31) {−1,1}.ForthefunctionalF,weshallusetheHamiltonian,H,but
remarkthatitcanmoregenerallybeassumedtohavetheform
D
Aspointedoutin[13,14]forthespecialcasewhereK(x,x′)= F =H+−C +λiP, (41)
δ(x − x′), the resulting dynamical equation is still an advective i i
i
equation,butwithanalterednon-divergentflowfield,sothatqis
where the Cs are Casimir invariants and the Ps are dynamical
conservedpointwiseandallthemomentsofqareconserved.The
invariants that commute with the particular Hamiltonians H of
specificformofinequality(30),whichreliesonthesymmetryof
interest—we also require that the Ps commute with any Dirac
K,canbeshowndirectly.WedosoforthecasewhenK isthe
Green’sfunctionGthatsatisfies∇2G(x)=δ(x).Weobtain constraintsemployed.ThepresenceofthePsinF generatesdy-
namicsinmovingframes.Forexample,ifoneaddsthelinearor
∫
((H,H)) = − dx′Φ(x′){q(x′),H} angularmomentumtoH,then(39)generatesdynamicsinauni-
formly translating or rotating frame, respectively. Consequently,
D
∫ extremalpointsof(41)arestationarystatesinmovingframes.Un-
= − dx′Φ(x′)[q(x′),ψ(x′)] likethePsof(41),theHandtheDiracconstraintsinDSA-dynamics
D areusedtoholdthevorticesinplace.
∫ ∫
WiththeaboveassumptionsonF,thedynamicsgeneratedby
= − dx′Φ(x′)∇2Φ(x′)= dx′|∇Φ(x′)|2, (32)
(37) clearly conserves all Casimir invariants for any H, and the
D D
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216 G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232
and for C ̸= 0 it is easy to show that k > 0. The eigenvalues
0
for this problem are λ = 1/k = n2, where n ∈ N, and the
n n
correspondings√etoforthonormaleigenfunctionsiscomposedof
u = sin(nx)/ π. Choosing u = au , for some m ∈ N, gives
n m
C = a2/k , which determines a, and H = C /λ . Thus, the
0 m 0 m
maximum value of H occurs for the smallest eigenvalue, λ =
1
1. Had we chosen u to be a linear combination of two or more
eigenfunctions,thenitisnotdifficulttoshowtheresultingvalueof
Hwouldbesmaller.Thus,thisexampleconstitutesawell-defined
extremizationproblem.
However,whenweseektominimizeH[u]subjecttoC[u] =
C , we see that C plays no role. In this case we have the same
0
Euler–Lagrangeequation,eigenvalues,andeigenfunctions,butwe
wishtominimizeH = C /λ .Becauseλ = m2,withm ∈ N,
0 m m
canbemadearbitrarilylarge,infH[u] = 0,whichisofcoursethe
minimum of H[u] without the constraint C. A rule of thumb for
minimumproblemsisthattheconstraintwillonlycontributeifit
containsfewerderivativesthanthefunctionalthatisdesiredtobe
Fig. 1. H–L diagram showing the relationship between energy and angular
minimizedandviceversaformaximumproblems.
momentum for various vortex states, including the Kirchhoff elliptical vortex
parameterizedbyaspectratio,a‘‘coarse-grained’’circularvortexparameterizedby Thesimplemaximumexampleabovewaschosentorepresent
size,andafewvorticeswithacentralcoresurroundedbyanannularbandalso thesituationforvortexdynamicswhentheenergyismaximizedat
parameterizedbysizeThepoint◦markstheaspectratiothreestabilitythreshold fixedCasimirinvariants.ThusitisnotasurprisethatSA-dynamics
fortheKirchhoffellipses.
shouldproducenicecircularvortexstates,andthisisevidenced
by the successful numerical examples of [13,14]. However, if
H chosenisitselfaninvariant.Similarly,thedynamicsgenerated
by (38) conserves the Casimir invariants of {,} and the Dirac additional constraints are imposed one may obtain other states.
Forexample,asstatedinSection2.5,itisknownthatequilibriain
constraints for any choice of H, and the chosen H (cf. [21]). It is rotatingframesareextremaofH+λLandthus,assuggestedin[15],
well-known that finite degree-of-freedom Hamiltonian systems
onemightbeabletouseSA-dynamicstoobtainnonaxisymmetric
cannot have asymptotically stable fixed points, for this would
vortexstates.Inourteststhisapproachtendedtobesusceptible
violateLiouville’stheorem.Thus,wedonotexpect(37)and(38)to λ
toeitherformingverythinellipses,maximizing Lattheexpense
possessasymptoticallystableequilibriainanyframeofreference.
ofH,ortocreatingasponge-likestructure,minimizingH atthe
However,thesesystemsareinfinite-dimensionalsystemsinwhich λ
expenseof L.WewillseeinSection3thatweobtainbetterresults
filamentationisknowntooccur(seee.g.Fig.4).Filamentationgives
byusingDSA-dynamicswithadditionalconstraints.
risetosteepgradientsandthusasmallamountofdissipationcan
Asnotedabove,oursimpleminimizationproblemrepresents
causerelaxation.Similarly,averagingoverfilamentationgivesrise
the situation of vortex energy minimization at constant Casimir
torelaxationbythemechanismofphasemixing,theessenceofthe
invariants.TheCasimirinvariantsareaconsequenceofthefactthat
Riemann–LebesguelemmaandLandaudamping.Thus,itwillnot
vortexdynamicsisarearrangement,i.e.thattheformalsolutionis
beasurprisetoseerelaxationinournumericalexperiments.
givenby
AsnotedinSection2.4,byconstructionthedynamicsof(39)
and(40)conservetheCasimirinvariants,andthelatterconserves q(x,y,t)=q (x (x,y,t),y (x,y,t)),
0 0 0
tHhoewiemvepro,stehdecDhoirsaecncHonosrtrHain+ts∑, bλoitPhifsonroalnoyngHeracmoinltsoenrviaend,bHut. where x0(x,y,t) and y0(x,y,t) represent the inverted solutions
satisfiesH˙ ≥ 0orH˙ ≤ 0dependingionthesignofα.Therefore, tothecharacteristicequations.Thusifonehasanisolatedsingle-
signed patch of vorticity surrounded by zero vorticity, and one
H canserveasaLiapunovfunctionalforasymptoticstability(see
wantstominimizetheenergyofthispatchsubjecttocontinuous
e.g.[24]).ThusasindicatedinSection1,thesebracketsgiveriseto
rearrangement, then this can be accomplished by the spreading
dynamicsthatisanenergyextremizingrearrangementsubjectto
out of the patch concomitantly with the stretching of the patch
theDiracconstraints,anditisthisdynamicsthatwefindusefulfor
boundary,insuchawayastointerpenetrateregionsofzeroand
numericallyobtainingequilibriumstates.
patchvorticity(cf.Fig.15).Thusalthoughtheconstraintdoesnot
preclude the march to zero energy, it does enforce a manner in
2.6. Kelvin’ssponge whichitcantakeplace.ThisisthemechanismofKelvin’ssponge.
InSection3wewillseehowDSA-dynamicsaccomplishesthiswith
Itiswell-knowninthecalculusofvariationsthatconstraints theinclusionofadditionalconstraints.
aresometimesineffective[25].Forvortexdynamicsthissituation
mayarisewhenoneattemptstominimizeenergyatfixedCasimir 3. Vortexexamples
invariants. As anticipated by Kelvin [26], vorticity invariants do
not preclude the minimum zero energy state, but may produce Inthissectionweconsideravarietyofexamples.InSection3.1
asponge-likestructurewithinterpenetratingregionsofvorticity we explore the various dynamics of Section 2.5 for two-fold
(see [13] for another discussion and examples). Here we give a symmetricinitialconditionswiththevorticitybeingofsinglesign
simple example that contains a central idea of when this effect and,whereapplicable,thetwo-foldsymmetricDiracconstraints
occurs,anexamplewefindusefulforinterpretingournumerical of(1)and(25).InSection3.2,three-foldsymmetricevolutionwith
results. single-signedvorticityisconsideredunderDSA-dynamics.Finally,
Considerareal-valuedfunctionu(x)definedon[0,2π].First, inSection3.3weconsiderevolutionunderdipoleinitialconditions.
suppose we wish to maximize the ‘energy’ functional H[u] = Forallthesingle-signed,i.e.monopolar,runstheunitoftimeis
2πdxu2subjecttotheconstraintC[u]=2πdxu2 =C ,withthe determinedbythevalueoftheintegratedvorticity.Forexample,
b0oundaryconditionsu(0) = u(2π) = 0.T0hevariaxtiona0lproblem thistendstobearoundπforthemonopolarruns,givingaparticle
δ(H − kC) = 0 givesthe Euler–Lagrangeequationku = −u, circulation time of about 4π. A Kirchhoff ellipse with an aspect
xx
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G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232 217
condition being on the order of one unit. A pseudospectral code
wasusedwithintegralsevaluatedassumsandtimeadvancement
accomplishedbysecondorderAdams–Bashforth.
Togivesomeinsightintothewaysenergyandangularmomen-
tumchangeasavortexshapeschange,wehaveplottedH–Ldia-
gramsforseveralvortexstructures:theKirchhoffellipticalvortex,
a‘‘coarse-grained’’circularvortexwithalargerradiusbutsmaller
valueofq(q=a−2forr <a),andcasesinwhichsomeofthevor-
ticityhasbeenspreadintoanannularbandaroundasmallercore
(q=1forr <band(1−b2)/(a2−b2)forb<r <a).Fig.1shows
theenergy-angularmomentumrelationshipsasparametriccurves
withaspectratiooravaryingalongthecurves.ThemaximumH
andLoccurforallcaseswhenthevortexiscircularwithunitra-
diusandq=1,correspondingtotheaspectratioorabeingunity.
Forreference,wehavemarkedwithacirclethewell-knownstabil-
itythresholdontheellipsecurve,wheretheaspectratioisequalto
three.Ellipsesbelowthismarkarelinearlyunstable.Wewillfindit
convenienttointerpretsomeofourresultsintermsofthisfigure.
Fig.2. Theinitialconditionof(42).Thisvorticity(shading)andcorresponding
streamfunction(contours)isusedfortherunsofFigs.3–18.
3.1. Single-signedvorticitywithtwo-foldsymmetry
ratio2to1rotatesatΩ = 2/(2 + 1)2 = 2/9,givingaperiod
ofabout28.3.Therunslastinguntilt =200,then,representabout Formostoftherunsofthissubsectionweuseaninitialcondi-
16revolutionsoftheparticlesand7oftheellipses. tionofthefollowingform:
All runs were done at a resolution of 256 × 256 points with
a total domain size of 8 or 16 units with the scale of the initial q=e−(r/r0)10 withr =1+ϵcos(2θ), (42)
0
(a)t=40. (b)t=80.
(c)t=120. (d)t=200.
Fig.3. Vorticity(shading)andstreamfunction(contours)undertheH-dynamicsof(37)withthetwo-foldsymmetricinitialconditionof(42)atthetimesshown.
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218 G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232
√
(a)q. (b) q.
Fig.4. Depictionoffilamen√tsformingundertheH-dynamicsof(37)withthesameparametersandinitialconditionasthoseofFig.3.Thesnapshotisatt=20.Panel(a)is
thevorticityq,while(b)is qinordertoaccentuatethefilaments.
(a)HandC1,2. (b)H−H(0)andC1,2−C1,2(0).
Fig.5. Evolution(a)andrelativeevolution(b)oftheHamiltonianandthequantitiesC1 =−Lof(1)andC2 =Sof(25),underthepureH-dynamicsof(37)fortheinitial
conditionofFig.2.Note,forclarityinpanel(b)H−H(0)hasbeenamplifiedby105andC1−C1(0)by103.
(a)t=20. (b)t=100.
Fig.6. Vorticity(shading)andstreamfunction(contours)underthepure(σ =0)SA-dynamicsof(39)withthetwo-foldsymmetricinitialconditionof(42)forthetimes
shown.Theparameterα=1.ThecodemaximizesenergywhilepreservingtheCasimirinvariants(rearrangementofvorticity)arrivingatthecircularstate.
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G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232 219
definedby(25).Withthesechoicesfortheconstraintsandinitial
condition,weexpectthesystemtoevolvetowardastatewithtwo-
foldsymmetry.
Fig.3showsthevorticityandcontoursofthestreamfunctionfor
theH-dynamicsof(37).Observethatthesystemshedsfilaments,
evolves toward an elliptical vortex state close to the Kirchhoff
vortexinatimet &20,andthenrotatesuniformlyforalongperiod
oftime.Fig.4showsanexampleoffilamentsheddingatanearly
time.However,Fig.5showsthattheinvariantsHandC =−Lare
1
well-conserved,asseeninFig.5(b),butHdecreasesmonotonically
atasmallrateandC hasanoscillatorybehaviorthatislikelydue
1
tothefactthatthe2-torusisnonaxisymmetric.Thestrainmoment
C =SisnotaninvariantofH-dynamics,andthisisevidentfrom
2
the figure. The vortex may continue to axisymmetrize slowly as
vorticityislostintofilamentsandspreadaroundthecore;interms
oftheH–LdiagramofFig.1,thiswouldcorrespondtonumerical
dissipationdecreasingHwhilepreservingL,goingfromtheellipse
Fig.7. PlotsoftheHamiltonian,H,theangularmomentum,L=−C1,andthestrain towardsanannularstate.
moment,C2 =S,(relativetotheirinitialvalues)vs.time,correspondingtoFig.6, Fig. 6 shows evolution with the same initial condition but
forthepureSA-dynamicsof(39).TheenergyHapproachesamaximum,whilethe now under the pure SA-dynamics of (39), i.e. with σ = 0 and
angularmomentumincreases. α = 1.Inafairlyshorttime,t ≈ 20,thesystemformsalong-
livedaxisymmetricvortexstate.ForSA-dynamicstheHamiltonian
wherer = x2+y2andθ = tan−1(y/x).Wechooseϵ = 0.4for should be maximized at fixed Casimir invariants, as described
allsuchruns.ThisinitialconditionisdepictedinFig.2.Also,when in Section 2.5, but there is no reason the angular momentum
Diracconstraintsareused,wewilltakeC1 =−L,wheretheangu- should be conserved. Indeed, in Fig. 7 it is seen that the system
larmomentumLisdefinedin(1)andC = S,thestrainmoment reaches the axisymmetric vortex state as the energy grows to a
2
(a)t=40. (b)t=80.
(c)t=120. (d)t=200.
Fig.8. Vorticity(shading)andstreamfunction(contours)undertheHD-dynamicsof(38)withthetwo-foldsymmetricinitialconditionof(42)forthetimesshown.The
DiracconstraintsC1andC2,minustheangularmomentumandthestrainmoment,of(1)and(25),respectively,wereused.
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220 G.R.Flierl,P.J.Morrison/PhysicaD240(2011)212–232
(a)HandC1,2. (b)H−H(0)andC1,2−C1,2(0).
Fig.9. Evolution(a)andrelativeevolution(b)oftheHamiltonianandtheDiracconstraintsC1,2vs.time,correspondingtothepureHD-dynamicsofFig.8.
(a)t=80. (b)t=120.
(c)t=160. (d)t=200.
Fig.10. Vorticity(shading)andstreamfunction(contours)underthepure(σ =0)DSA-dynamicsof(40)withthetwo-foldsymmetricinitialconditionof(42)forthetimes
shown.Theparametersσ =0andα=1;thusthebracketseekstomaximizetheenergywhilestrivingtokeeptheC1andC2of(1)and(25)fixed.Observeforintermediate
timesqapproachesafunctionofψ,butforlatetimeshorizontaltinesdevelop,andatlatertimesyetthesystembeginstoaxisymmetrize.Thelightercolorofthebackground
att=200isduetotheproductionofnegativevorticityvianumericalerror.