A sandpile model with tokamak-like enhanced confin

arXiv:physics/0010032v2[physics.plasm-ph]2Feb2001HEP/123-qedAsandpilemodelwithtokamak-likeenhancedconfinementphenomenologyS.C.Chapman1∗,1PhysicsDept.Univ.ofWarwick,CoventryCV47AL,UKR.O.Dendy22EURATOM/UKAEAFusionAssociation,CulhamScienceCentre,Abingdon,OxfordshireOX143DB,UnitedKingdomB.Hnat1(February9,2008)AbstractConfinementphenomenologycharacteristicofmagneticallyconfinedplas-masemergesnaturallyfromasimplesandpilealgorithmwhentheparametercontrollingredistributionscalelengthisvaried.Closeanaloguesarefoundforenhancedconfinement,edgepedestals,andedgelocalisedmodes(ELMs),andforthequalitativecorrelationsbetweenthem.Theseresultssuggestthattoka-makobservationsofavalanchingtransportaredeeplylinkedtotheexistence∗[e-mail]1ofenhancedconfinementandELMs.52.55.Dy,52.55.Fa,45.70.Ht,52.35.RaTypesetusingREVTEX2Theintroductionofthesandpileparadigm[1]-[3]intomagnetizedplasmaphysics(fusion[4]-[12],magnetospheric[13]-[15],andaccretiondisk[16]-[18]:forrecentreviewsseeRef.[19,20])hasopenednewconceptualavenues.Itprovidesaframeworkwithinwhichobserva-tionsofrapidnondiffusivenonlocaltransportphenomenacanbestudied;recentexamplesincludeanalysesofauroralenergydepositionderivedfromglobalimaging[15],andofelec-trontemperaturefluctuationsintheDIII-Dtokamak[12],bothofwhichinvolveavalanching.Insofarassuchphenomenaresemblethoseinexperimentalsandpilesormathematicallyide-alizedmodelsthereof,theysuggestthattheconfinementphysicsofmacroscopicsystems(plasmaandother)mayreflectunifyingunderlyingprinciples.Inthispaperwepresentresultssuggestingthatthisunitymayextendtosomeofthemostdistinctivefeaturesoftoroidalmagneticplasmaconfinement:enhancedconfinementregimes(“H-modes”),edgelocalisedmodes(“ELMs”),steepedgegradients(“edgepedestal”),andtheirobservedphenomenologicalandstatisticalcorrelations–forrecentquantitativestudies,seeforexampleRefs.[21–23]andreferencestherein.AnimportantquestioniswhethertheLtoHtransitionnecessarilyreflectsacatastrophicbifurcationofconfinementproperties,orcanbeassociatedwithamonotonicchangeinthecharacteroftheturbulence[23].Transi-tionalbehaviourresemblingthatobservedinfusionplasmashasbeenfoundinothersandpilealgorithms:forexampleinRef[4]alocalshearedflowregionisimposedwhichreducesthecharacteristicavalanchelengththerebyaffectingconfinement;andinRef[6],changesintheredistributionruleleadtochangesinprofilestiffness.Weshowthatkeyelementsoftheob-servedphenomenologyemergenaturallyfromasimpleone-dimensionalsandpilemodel,thatofChapman[24],whichincorporatesotherestablishedmodels[1,25]aslimitingcases.Thiscentrallyfueled(atcelln=1)model’sdistinctivealgorithmicfeaturerelatestothelocalredistributionofsandatacell(sayatn=k)wherethecriticalgradientzcisexceeded:thesandpileisinstantaneouslyflattenedbehindtheunstablecelloveralengthLf,embracingthecellsn=k−(Lf−1),k−(Lf−2),...,k;andthissandisconservativelyrelocatedtothecellatn=k+1.Lfdefinesthelengthscaleoverwhichthemostrapidredistributionoccurs;inaplasmacontextthiscouldbeconsideredaproxyforturbulentcorrelationlengthoreddy3size.InRef.[24]thesandpileisexploredforallregimes1LfNforbothconstantandfluctuatingcriticalgradientzc.Hereweconsiderthedynamicsofthemorerealisticcasewithrandomfluctuationsinzc;thesystemisrobustinthatoncefluctuationsareintroducedinthecriticalgradient,thebehaviorisessentiallyinsensitivetoboththeirlevelandspectralproperties[24],seealsoRef.[19].ThelimitLf=1isthefixedpointcorrespondingtothecentrallyfueledalgorithmofRef.[1]inonedimension.InthelimitLf=N(whereNisthenumberofcellsinthesandpile)thesandpileisflattenedeverywherebehindanunstablecellasinRefs.[25,19].Arealspacerenormalizationgroupapproach[26]showsthattherobustscalefreedynamicsforthelimitingcaseLf=Ncorrespondstoanontrivial(repulsive)fixedpoint(seee.g.Ref.[3]).TheessentialresultofRef.[24]isthatdifferentregimesofavalanchestatisticsemerge,resemblingatransitionfromregulartointermittentdynamicsreminiscentofdeterministicchaos.ThecontrolparameteristhenormalizedredistributionscalelengthLf/NwhichspecifieswhetherthesystemisclosetothenontrivialLf=Nfixedpoint.Heightprofilesforthesandpilewith512cells,timeaveragedovermanythousandsofavalanches,areshowninFig.1forthreedifferentvaluesofthefluidizationlengthLfintherange50Lf250.Thesandpileprofileshape,storedgravitationalpotentialenergy,andedgestructure(smoothdeclineorpedestal)correlatewitheachotherandwithLf.AsLfisreduced,theedgepedestalsteepensandthetimeaveragedstoredenergyrises;multiple“barriers”(regionsofsteepgradient)arevisibleintrace(a)andtosomeextenttrace(b)ofFig.1.TimeevolutionofthesandpileforLf=50,150,and250respectivelyisquantifiedinFigs.2-4.Thetoptracesshowtotalstoredenergy;themiddletracesshowthepositionoftheedgeofthesandpile(thelastoccupiedcell);andthebottomtracesshowthemagnitudeandoccurrencetimesofmasslossevents(hereafterMLEs)inwhichsandislostfromthesystembybeingtransferredbeyondthe512thcell.Timeisnormalizedtothemeaninter-avalanchetimeΔτ(proportionaltothefuelingrate).Thesandpileisfueledonlyatthefirstcell,sothatthegreatmajorityofavalanchesterminatebeforereachingthe512thcell(theseareclassifiedasinternal).Whileinte