Lecture Notes in Physics 946 Gary Webb Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws Lecture Notes in Physics Volume 946 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard M.Bartelmann,Heidelberg,Germany P.Hänggi,Augsburg,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLöhneysen,Karlsruhe,Germany A.Rubio,Hamburg,Germany M.Salmhofer,Heidelberg,Germany W.Schleich,Ulm,Germany S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany J.D.Wells,AnnArbor,USA G.P.Zank,Huntsville,USA The Lecture Notes in Physics TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelop- ments in physics research and teaching – quickly and informally, but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) to be a source of advanced teaching material for specialized seminars, courses andschools Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. Editedvolumesshould,however,consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. VolumespublishedinLNParedisseminatedbothinprintandinelectronicfor- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposalsshouldbe sent to a memberof the EditorialBoard, ordirectly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] Moreinformationaboutthisseriesathttp://www.springer.com/series/5304 Gary Webb Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws 123 GaryWebb CSPAR TheUniversityofAlabamainHuntsville Huntsville,Alabama USA ISSN0075-8450 ISSN1616-6361 (electronic) LectureNotesinPhysics ISBN978-3-319-72510-9 ISBN978-3-319-72511-6 (eBook) https://doi.org/10.1007/978-3-319-72511-6 LibraryofCongressControlNumber:2017963310 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The motivation for the present book originated in the quest to understand wave- waveinteractionsinmagnetohydrodynamics(MHD)inanon-uniformbackground flow(thisprocessissometimesreferredtoaswavemixinginthesolarwindandin cosmicraymodifiedshocks).ThevariationalapproachtoWKBwavepropagation in a non-uniform background plasma flow was developed by Dewar (1970). My initialaimwastounderstandlinear,non-WKBwavepropagationinthesolarwind. The problem of wave mixing has also been identified as an important process in the evolution of turbulenceand Alfvénic fluctuationsin the solar wind (e.g. Zhou andMatthaeus1990a,b;Zanketal.2012).Wavesinnon-uniformflowsalsoplayan importantroleinLagrangianaveragedEuler-Poincaréequations(LAEPequations) of wave-mean flow interactions and the so-called alpha model of turbulence (e.g. Holm2002). Another motivation for the book was to understand the elegant non-canonical Hamiltonian formalism for MHD and fluids developed by Morrison and Greene (1980, 1982), Holm and Kupershmidt (1983a,b) and Marsden et al. (1984). The connection between a Clebsch variable action principle for MHD and the non- canonical Poisson bracket of Morrison and Greene (1980, 1982) and the Clebsch variational approach is developed by Zakharov and Kuznetsov (1997) (see also Zakharov and Kuznetsov (1971) for the canonical form of Hamilton’s equations forMHDusingClebschvariables).InparticulartheworkofPadhyeandMorrison (1996a,b) shows the connection between Noether’s second theorem and the con- servationofpotentialvorticityin idealfluidmechanicsandMHD,duetothefluid relabellingsymmetryoftheequations(seealsoSalmon(1982,1988)foranaccount ofthe fluidrelabellingsymmetryin idealfluids).Thefluid relabellingsymmetries are due to the invariance of the action, in which the Lagrangian fluid labels can change (i.e. there are transformations or maps of the fluid labels onto new fluid labelsthatare diffeomorphisms)butthe usualphysicalvariablesremaininvariant. TherearerelationshipsbetweenthefluidrelabellingsymmetriesandtheCasimirsof thenon-canonicalMHDPoissonbracket,whichareexploredinthepresentlecture notes. v vi Preface Yet another motivation for the book is applications of topological methods in fluid dynamics and MHD. In the book we give examples of magnetic helicity conservation(e.g.Woltjer1958;KruskalandKulsrud1958;BergerandField1984; Finn and Antonsen 1985; Moffatt 1969, 1978; Moffatt and Ricca 1992) in solar physicsandinsolarwindphysics.InChap.2,Sect.2.5,wedescribetheinvestigation ofToroketal.(2010,2014)ontheevolutionofthetwistandwrithecomponentsof magnetic helicity in the evolution of the kink instability for solar magnetic flux ropes, and its role in coronal mass ejections (CMEs). Other applications to the magnetic helicity of the Parker interplanetary,Archimedeanspiral magnetic field, to nonlinear Alfvén waves in the solar wind, and the MHD topological soliton solutionsaredescribedinChap.6. ConservationlawsobtainedbyLie draggingadvectedinvariantsinmagnetohy- drodynamics(MHD)andgasdynamicsorhydrodynamics(HD)wereinvestigated by Moiseev et al. (1982), Sagdeev et al. (1990), Tur and Yanovsky (1993), Volkov et al. (1995), Kats (2001, 2003, 2004) and Webb et al. (2014a). The ten Galilean, Lie point symmetries of the action give rise to the energy conservation, momentumconservation,angularmomentumandcentreofmassconservationlaws, via Noether’s first theorem. The advected invariants are due to fluid relabelling symmetries, or diffeomorphisms associated with the Lagrangian map. There are differentclassesofgeometricalquantitiesthatareadvectedorLiedraggedwiththe flow. Examplesare the entropyS (a 0-form)andthe conservationof the magnetic flux (B(cid:2)dS which is an invariant advected two-form), moving with the flow (i.e. Faraday’sequation).AdvectedinvariantsareobtainedbyusingtheEuler-Poincaré approach to Noether’s second theorem. Some of the invariants are important in topological fluid dynamics and MHD. We discuss different variants of helicity including kinetic helicity, cross helicity, magnetic helicity, Ertel’s theorem and potential vorticity, the Hollman invariant and the Godbillon Vey invariant. Lie dragged invariants or Cauchy invariants play an important role in describing the dynamicsofvortexandmagneticfieldlinesinidealhydrodynamicsandMHD(e.g. KuznetsovandRuban1998,2000;Kuznetsov2006;BesseandFrisch2017). The multi-symplectic and multi-momentum approach to Hamiltonian systems was originally developed by de Donder (1930) and Weyl (1935). They studied generalizedHamiltonianmechanicsinwhichtheLagrangianLDL.x;'i;@'i=@x(cid:2)/ where x(cid:2).1 (cid:3) (cid:2) (cid:3) n/ are the independent variables and 'k .1 (cid:3) k (cid:3) m) are the dependentvariables.Forthe case wheren (cid:4) 2 onecan define multi-momenta (cid:3)(cid:2) D @'i=@x(cid:2) corresponding to each x(cid:2) (in the usual Hamiltonian formulation i x0 Dtistheevolutionvariable).Themulti-symplecticapproachhasbeendeveloped infieldtheoryinthesearchforamorecovariantformofHamiltonianmechanics(in the usual Hamiltonian formulation, there is only one evolution variable). Bridges et al. (2005,2010),Marsden and Shkoller (1999),Hydon (2005)and Cotter et al. (2007) describe multi-symplectic systems. Our aim is to present both Eulerian and Lagrangian variational principles for ideal fluids and MHD obtained by, e.g. Newcomb(1962),HolmandKupershmidt(1983a,b),Dewar(1970)andWebbetal. (2005a,b, 2014a,b). Both Eulerian and Lagrangian multi-symplectic forms of the equations can be obtained. In this book we concentrate on the Eulerian multi- Preface vii symplectic form of the equations (the Lagrangian, multi-symplectic ideal fluid equationsare described by Webb (2015)and Webb and Anco (2016)).The multi- symplectic Noether’s theorem and symplecticity and pullback conservation laws areobtained.Nonlocalconservationlaws,foranon-barotropicequationofstatefor the fluid, in which the time integral of the temperature back along the fluid path plays an importantmemory role, are obtained (see also Mobbs(1981) for similar conservationlawsforhelicityinnon-barotropicfluids).Yahalom(2016a,2017a,b) exploresthephysicalandtopologicalmeaningofthenon-barotropiccrosshelicity andcrosshelicityperunitmagneticfieldflux,usingaClebschpotentialformulation (seealsoWebbandAnco2017).Theconnectionofthemulti-symplecticapproach withCartan’stheoryofdifferentialequationsusingdifferentialformsisdeveloped. A potential vorticity type conservation law is derived for MHD using Noether’s secondtheorem. The motivation is to provide both local and nonlocal conservation laws of the fluid and MHD equations that give insight into the physics. Conservation laws are useful for the testing numerical codes and reveal new aspects of the physics (e.g. nonlocal conservation laws associated with potential symmetries and fluid relabelling symmetries, reveal the time history of the fluid elements can play an importantrole in understandingfluid vorticity).For example,the barocliniceffect leadstothecreationofvorticityinfluids(e.g.intornadoes),butthecorresponding nonlocal conservation law for fluid helicity is not usually discussed. Casimirs (i.e. quantities with zero Poisson bracket with other functionals of the physical variables) are important in describing the stability of steady flows and equilibria. The knowledge of new conservation laws is important in fusion plasmas, space plasmas, fluid dynamics and atmospheric physics. New conservation laws are also important in mathematics in elucidating the symmetries responsible for the conservationlaws(e.g.Liepseudogroupsaremostlikelyrelatedtofluidrelabelling symmetries). WhatIs NotIncludedinthe Book The abstract geometrical mechanics aspects of fluid mechanics and MHD are not developed in the present approach. Detailed descriptions of the geometrical mechanicsapproachtothetheoryaredescribedinMarsdenetal.(1984),Marsden andRatiu(1994),Holmetal.(1998)andHolm(2008a,b).HolmandKupershmidt (1983a,b),Marsdenetal.(1984)andHolmetal.(1998)describetheroleofsemi- direct product Lie algebras and Lie groups inherent in the non-canonicalPoisson bracket of Morrison and Greene (1980, 1982). Morrison (1982) gives a direct algebraic method to derive the Jacobi identity. Olver (1993) uses the variational complextodevelopmethodstocheckifagivenco-symplecticdifferentialoperator used to define the Poisson bracket is a Hamiltonian operator (i.e. the bracket is skewsymmetricandsatisfies theJacobiidentity).Chandreetal.(2012,2013)and Chandre (2013) derived Dirac brackets for MHD to obtain well-behaved brackets viii Preface thatsatisfytheJacobiidentity.Bridgesetal.(2010)usethevariationalbi-complex todescribemulti-symplecticsystems.TheanalysisofLiesymmetriesofdifferential equationsystemsusingLie’salgorithm(e.g.BlumanandKumei1989;Olver1993; Ovsjannikov 1962, 1982; Ibragimov 1985; Bluman et al. 2010) can be used to derive analytical solutions of the equations. We do not study conservation laws and symmetries for special and general relativistic MHD (see, e.g. Lichnerowicz 1967; Beckenstein and Oron 1978; Bekenstein 1987; Anile 1989; Achterberg 1983; D’Avignon et al. 2015). Pshenitsin (2016) has derived infinite classes of conservation laws for incompressible viscous MHD by using the so-called direct methoddevelopedbyAncoandBluman(see,e.g.Blumanetal.2010).Thismethod ofdeterminingconservationlawsisillustratedforthecaseoftheKdVequationin Chap.4.However,wehavenotusedthismethodtoderiveMHDconservationlaws inthepresentbook. We discuss topological invariants in fluids and plasmas, using Lie dragged invariantsin idealfluidsandMHD(see,e.g.ArnoldandKhesin1998;Bergerand Field 1984; Berger 1999a,b; Moffatt and Ricca 1992; Besse and Frisch 2017 for detailedanalysis).ThepapersbyKuznetsovandRuban(1998,2000)andKuznetsov et al. (2004) give an account of vortex lines and magnetic field lines, using a mixed Eulerian and Lagrangian approach,which shows how one may resolve the degeneracyofthenon-canonicalPoissonbrackets,byusingWebertransformations andLagrangianrepresentationsoftheequations.TheyalsoshowhowtheHasimoto transformation arises from their analysis. Euler potential representations of the magnetic field and its use in fusion and space plasmas are another large area of research not covered in our treatment (see, e.g. Stern (1966) for applications in spaceplasmas,andBoozer(2004)infusionplasmas). RecentworkbyWebb(2015)andWebbandAnco(2016)onLagrangian,multi- symplectic fluid mechanics and work on MHD gauge field theory by Webb and Anco(2017)areomittedfromthepresentexposition.ItisworthnotingthatCalkin (1963)developeda versionof gaugefield theoryfora polarizedversionofMHD. Both Calkin (1963) and Webb and Anco (2017) identified the gauge symmetry responsibleforthemagnetichelicityconservationlawinMHD.Thesedevelopments liebeyondthescopeofthepresentbook. Acknowledgements The material in this book originated in a series of papers on MHD with my colleaguesattheIGPP,UniversityofCaliforniaRiverside(2002–2008),andatthe CSPARattheUniversityofAlabamainHuntsville(G.P.Zank,Q.Hu,B.Dasgupta, N.P.PogorelovandJ.F.McKenzie).IamindebtedtoProfessorDarrylHolm(Math- ematicsDept.,ImperialCollegeLondon)andProfessorPhil.Morrison(Department ofPhysicsandInstituteofFusionStudiesoftheUniversityofTexasatAustin)for discussionsonthemathematicsandphysicsofconservationlawsandsymmetries, Poisson brackets, Casimirs, Euler-Poincaré equations, multi-symplectic systems, Preface ix LiesymmetriesandNoether’stheoremsinMHDandfluiddynamics.Iamindebted toG.P.ZankforsuggestingtowriteabookonMHDandfluidsandfordiscussions of the compressible,incompressibleand reducedMHD equationsand symmetries of the equations. I am indebted to Stephen Anco for discussions on some of the more obscure conservation laws of the MHD and fluid equations and the use of thedirectmethodto obtainconservationlawsforequationsystems, whicharenot necessarilyassociatedwithvariationalprinciples(e.g.Cheviakov2014;Cheviakov andOberlack2014;Pshenitsin2016).IamgratefultoDr.Q.Hu(whodrewmanyof thefigures)fordiscussionsonmagnetichelicityandmagneticclouds,andtoDr.B. Dasguptaforhisdetailedknowledgeofmagneticfieldsinplasmaphysics(e.g.the KamchatnovMHDtopologicalsolitonandchaoticversusintegrablemagneticfield lines).IamindebtedtoE.D.FackerellandC.B.G.McIntoshfortheirlecturesonLie symmetries and differential equations, whilst at Monash University in the period 1974–1977. I also acknowledge discussions on fluid relabelling symmetries with Nikhil Padhye and discussions with R.B. Sheldon on the importance of nonlocal conservationlaws.Theerrorsaremine. Huntsville,AL,USA GaryWebb January16,2018