//img.uscri.be/pth/15ce4ffd8816d2cadef5a6a71300a294f8ee0078
Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

Numerical resolution of a mono disperse model of bubble growth in magmas

De
31 pages
Numerical resolution of a mono-disperse model of bubble growth in magmas L. Forestier-Coste?, S. Mancini?,†, A. Burgisser†, F. James† Keywords: coupled ode system and diffusion equation ; mass preserving scheme ; bubble growth AMS Class.: 86A04, 8608, 65L20, 65M06 Abstract Growth of gas bubbles in magmas may be modeled by a system of differential equations that account for the evolution of bubble radius and internal pressure and that are coupled with an advection-diffusion equation defining the gas flux going from magma to bubble. This sys- tem of equations is characterized by two relaxation parameters linked to the viscosity of the magma and to the diffusivity of the dissolved gas, respectively. Here, we propose a numerical scheme preserving, by construction, the total mass of water of the system. We also study the asymptotic behavior of the system of equations by letting the relaxation parameters vary from 0 to∞, and show the numerical con- vergence of the solutions obtained by means of the general numerical scheme to the simplified asymptotic limits. Finally, we validate and compare our numerical results with those obtained in experiments. 1 Introduction All volcanic eruptions involve a decompression of the magma during its as- cent from the Earth's crust to the surface. This decompression causes the volatiles dissolved into the magma to come out of solution as gas bubbles.

  • free magma

  • model when

  • equation defining

  • general numerical

  • advection-diffusion equation

  • perfect gas constant

  • magma

  • constant c0

  • volcanic eruption


Voir plus Voir moins
Numericalresolutionofamono-dispersemodelofbubblegrowthinmagmasL.Forestier-Coste,S.Mancini,,A.Burgisser,F.JamesKeywords:coupledodesystemanddiffusionequation;masspreservingscheme;bubblegrowthAMSClass.:86A04,8608,65L20,65M06AbstractGrowthofgasbubblesinmagmasmaybemodeledbyasystemofdifferentialequationsthataccountfortheevolutionofbubbleradiusandinternalpressureandthatarecoupledwithanadvection-diffusionequationdefiningthegasfluxgoingfrommagmatobubble.Thissys-temofequationsischaracterizedbytworelaxationparameterslinkedtotheviscosityofthemagmaandtothediffusivityofthedissolvedgas,respectively.Here,weproposeanumericalschemepreserving,byconstruction,thetotalmassofwaterofthesystem.Wealsostudytheasymptoticbehaviorofthesystemofequationsbylettingtherelaxationparametersvaryfrom0to,andshowthenumericalcon-vergenceofthesolutionsobtainedbymeansofthegeneralnumericalschemetothesimplifiedasymptoticlimits.Finally,wevalidateandcompareournumericalresultswiththoseobtainedinexperiments.1IntroductionAllvolcaniceruptionsinvolveadecompressionofthemagmaduringitsas-centfromtheEarth’scrusttothesurface.Thisdecompressioncausesthevolatilesdissolvedintothemagmatocomeoutofsolutionasgasbubbles.Thewaythesebubblesaregrowing,whethertheycoalescewithoneanotherortravelfasterthanorwiththemagma,areallconditioningthewaytheFe´de´rationDenisPoisson(FR2964),MAPMO(UMR6628),BP.6759,UniversityofOrle´ansandCNRS,F-45067Orle´ans,FranceInstitutdesSciencesdelaTerred’Orle´ans,CNRS/INSU,Universite´d’Orle´ans,Uni-versite´Franc¸oisRabelais-Tours,1AruedelaFe´rolerie,Orle´ans,F-45071cedex2,France1
volcaniceruptionwillunfold.Bubblesthatremaintrappedwiththemagmatheyoriginallygrewfromwillaccumulategaspressureuntilfailureofthemagmareleasesitsuddenlytoproduceanexplosiveeruption.Suchscenarioismostlikelywhenthemagmaishighlyviscousandpreventsbubblemotion.Thissituationispropitioustomodelingbecausebubblescanbeconsideredasimmobilewithrespecttothemagmaandtheresultingsphericalgeome-tryallowsonetoreducebubblegrowthtoasystemofdifferentialequationsdescribingtheevolutionofpressureandgasmassinabubblecoupledwithanadvection-diffusionequationdescribingthedrainageofthedissolvedgastowardsthebubble.Afurtherassumptionisthatbubblesareexclusivelymadeofwatervapor,whichcanbejustifiedbythefactthatwateris,byfar,themostabundantvolatilespeciesinsuchviscousmagmas.Sincetheseminalworkdonein[1]severalnumericalschemesthatsolvesuchsystemofdifferentialequationshavebeenproposedinthecontextofvisco-elasticfluids(see[2,3,4,5,6]).Applicationtogasbubbleinmagmasisslightlymorerecent(see[7,8,10,11,12,13]).Alltheseschemeshaveincommonadiscretizationoftheadvection-diffusionequationthatisnotconservativebyconstructionwithrespecttothediffusedspecies.Infact,theyinvolveuser-defineddiscretizationparametersthathavetobeempiri-callyadjustedtoensuresufficientconvergenceand/oraccuracyofthescheme.Developingalternate,robustschemeswouldallowincludingthedynamicsofbubblegrowthintomoresophisticatedmodelthattakeintoaccount,forin-stance,thatbubblehavedifferentsizes,orthat,ifmagmaviscosityislowenough,bubblemayrisewithrespecttothemagma.Inthispaperwepresentanewnumericalscheme,inwhichthefluxintheadvection-diffusionequationiscomputedinordertoconservethetotalwatermassinthebubble-magmasystematadiscretelevel,andthisdespitethemeshdiscretizationweapply.Moreover,undertheassumptionofconstantintimediffusioncoefficient,wegivesomeexplicitsolutionsoftheproposedmodelwhentheviscosityorthediffusionareverylarge(infinity)orverysmall(zero),weshallcalltheseasymptoticslimitregimes;wealsonumericallyverifytheconvergenceoftheproposedschemetowardstheselimitregimes.Thepresentworkisdevelopedasfollows.Insection2,werecallthedifferentialequationsdescribingtherespectiveevolutionofbubbleradiusandmass,togetherwiththeadvection-diffusionequationdescribingthebehaviorofthewaterconcentrationinthemagma.Following[7,8]and[12]wewritetheproblemindimensionlessform,introducingtworelaxationparametersΘVandΘD.Section3,isdevotedtothenumericalapproximationofthemodel.Themainnoveltyisthediscretizationoftheadvection-diffusionequation,2
seesection3.2,inwhichweexplainhowtocomputethemeshandfluxateachiterationinsuchawaythatthetotalmassisconserved.Insection4wedealwiththeasymptoticsofthedimensionlessproblem,whentheratiobetweentherelaxationparametersvariesfrom0to.Threemainregimesareunderlined:viscous,diffusive,andequilibrium.Foreachlimit,wealsoproposeawaytodiscretizeit.Numericalresults,convergenceofthesolutiontowardsthesimplifiedasymptoticlimits,comparisonswithexperimentsandwiththecodeofreference[7],arediscussedinsection5.Finally,insection6,wesummarizeourstudyandsuggestpossibleextensionsofthemodelingofbubblesgrowthinmagma.2ThemodelWeareinterestedinthemodelingofbubblegrowthinahighlyviscous,crystal-freemagma.Thishastwomainconsequencesonthemodel.Thefirstoneisthatweassumethatbubblesdonotinteractwitheachother,inparticulartherearenocoalescenceeffects.Thisisstronglylimitativeforthesimulationofamagmaticconduit,butthepresenceorabsenceofcoalescencecanbecontrolledinlaboratoryexperiments,seesection5.2.Theotherpointisthat,duetothehighviscosityofthemagma,bubblestravelalongwiththesamevelocityasthemelt.Inotherwords,theycanbeconsideredasimmobilewithrespecttothemelt.Atthisstage,wecanconsiderthatabubblecanbedescribedwithtwoparameters,itsvolumeVˆanditsgaseousmassMˆ.Inthissection,wedenotewithahatthedimensionalvariables.Takingintoaccountthatthebubbleismadeonlyofwateringaseousform,wecanwritetheperfectgaseslawinsidethebubbleinordertorelatethegaspressurePˆtothegasdensityρˆ:GTρˆ=PˆMw,(1)withMwthemolarmassforwater,GtheperfectgasconstantandTthegastemperature.Next,following[7,8]and[12]weassumethatthebubbleisspherical,withradiusRˆ,sothatVˆ=4πRˆ3/3,andwesetforfutureconvenienceMˆ=ρˆRˆ3,sothatthebubblemassis4πMˆ/3.ThuswecanchoosetheradiusRˆ=Rˆ(t)andthevariableMˆ=Mˆ(t),proportionaltothemass,todescribetheevolutionofthebubble,andseekforasystemofdifferentialequationsforthesevariables.Noticethatin[12]anequationonthepressurePˆ(t)isgiven,wechooseheretotrackthebubblemassbecauseitleadstoabetterhandlingofmassconservationatthenumericallevel.Suchamodelgivesadescriptionofthegrowthofasinglebubble,orforapopulationofidentical,non-interactingbubbles:thisistheso-calledmono-dispersecase.3