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

Denis Poisson

Description

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

Sujets

• Poisson
• Magma (algebra)
• Convection?diffusion equation
• Types of volcanic eruptions

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Numericalresolutionofamono-dispersemodel ofbubblegrowthinmagmas L.Forestier-Coste, ∗ S.Mancini ∗ , † ,A.Burgisser, † F.James † Keywords:coupledodesystemanddiffusionequation;masspreserving scheme;bubblegrowth AMSClass.:86A04,8608,65L20,65M06 Abstract Growthofgasbubblesinmagmasmaybemodeledbyasystemof differentialequationsthataccountfortheevolutionofbubbleradius andinternalpressureandthatarecoupledwithanadvection-diffusion equationdefiningthegasfluxgoingfrommagmatobubble.Thissys- temofequationsischaracterizedbytworelaxationparameterslinked totheviscosityofthemagmaandtothediffusivityofthedissolved gas,respectively.Here,weproposeanumericalschemepreserving,by construction,thetotalmassofwaterofthesystem.Wealsostudy theasymptoticbehaviorofthesystemofequationsbylettingthe relaxationparametersvaryfrom0to ∞ ,andshowthenumericalcon- vergenceofthesolutionsobtainedbymeansofthegeneralnumerical schemetothesimplifiedasymptoticlimits.Finally,wevalidateand compareournumericalresultswiththoseobtainedinexperiments. 1Introduction Allvolcaniceruptionsinvolveadecompressionofthemagmaduringitsas- centfromtheEarth’scrusttothesurface.Thisdecompressioncausesthe volatilesdissolvedintothemagmatocomeoutofsolutionasgasbubbles. Thewaythesebubblesaregrowing,whethertheycoalescewithoneanother ortravelfasterthanorwiththemagma,areallconditioningthewaythe ∗ Fe´de´rationDenisPoisson(FR2964),MAPMO(UMR6628),BP.6759,Universityof Orle´ansandCNRS,F-45067Orle´ans,France † InstitutdesSciencesdelaTerred’Orle´ans,CNRS/INSU,Universite´d’Orle´ans,Uni- versite´Franc¸oisRabelais-Tours,1AruedelaFe´rolerie,Orle´ans,F-45071cedex2,France 1

volcaniceruptionwillunfold.Bubblesthatremaintrappedwiththemagma theyoriginallygrewfromwillaccumulategaspressureuntilfailureofthe magmareleasesitsuddenlytoproduceanexplosiveeruption.Suchscenario ismostlikelywhenthemagmaishighlyviscousandpreventsbubblemotion. Thissituationispropitioustomodelingbecausebubblescanbeconsidered asimmobilewithrespecttothemagmaandtheresultingsphericalgeome- tryallowsonetoreducebubblegrowthtoasystemofdifferentialequations describingtheevolutionofpressureandgasmassinabubblecoupledwith anadvection-diffusionequationdescribingthedrainageofthedissolvedgas towardsthebubble.Afurtherassumptionisthatbubblesareexclusively madeofwatervapor,whichcanbejustifiedbythefactthatwateris,byfar, themostabundantvolatilespeciesinsuchviscousmagmas. Sincetheseminalworkdonein[1]severalnumericalschemesthatsolve suchsystemofdifferentialequationshavebeenproposedinthecontextof visco-elasticfluids(see[2,3,4,5,6]).Applicationtogasbubbleinmagmas isslightlymorerecent(see[7,8,10,11,12,13]).Alltheseschemeshave incommonadiscretizationoftheadvection-diffusionequationthatisnot conservativebyconstructionwithrespecttothediffusedspecies.Infact, theyinvolveuser-defineddiscretizationparametersthathavetobeempiri- callyadjustedtoensuresufficientconvergenceand/oraccuracyofthescheme. Developingalternate,robustschemeswouldallowincludingthedynamicsof bubblegrowthintomoresophisticatedmodelthattakeintoaccount,forin- stance,thatbubblehavedifferentsizes,orthat,ifmagmaviscosityislow enough,bubblemayrisewithrespecttothemagma.Inthispaperwepresent anewnumericalscheme,inwhichthefluxintheadvection-diffusionequation iscomputedinordertoconservethetotalwatermassinthebubble-magma systematadiscretelevel,andthisdespitethemeshdiscretizationweapply. Moreover,undertheassumptionofconstantintimediffusioncoefficient,we givesomeexplicitsolutionsoftheproposedmodelwhentheviscosityorthe diffusionareverylarge(infinity)orverysmall(zero),weshallcallthese asymptoticslimitregimes;wealsonumericallyverifytheconvergenceofthe proposedschemetowardstheselimitregimes. Thepresentworkisdevelopedasfollows.Insection2,werecallthe differentialequationsdescribingtherespectiveevolutionofbubbleradiusand mass,togetherwiththeadvection-diffusionequationdescribingthebehavior ofthewaterconcentrationinthemagma.Following[7,8]and[12]wewrite theproblemindimensionlessform,introducingtworelaxationparametersΘ V andΘ D .Section3,isdevotedtothenumericalapproximationofthemodel. Themainnoveltyisthediscretizationoftheadvection-diffusionequation, 2

seesection3.2,inwhichweexplainhowtocomputethemeshandfluxat eachiterationinsuchawaythatthetotalmassisconserved.Insection4 wedealwiththeasymptoticsofthedimensionlessproblem,whentheratio betweentherelaxationparametersvariesfrom0to ∞ .Threemainregimes areunderlined:viscous,diffusive,andequilibrium.Foreachlimit,wealso proposeawaytodiscretizeit.Numericalresults,convergenceofthesolution towardsthesimplifiedasymptoticlimits,comparisonswithexperimentsand withthecodeofreference[7],arediscussedinsection5.Finally,insection 6,wesummarizeourstudyandsuggestpossibleextensionsofthemodeling ofbubblesgrowthinmagma. 2Themodel Weareinterestedinthemodelingofbubblegrowthinahighlyviscous, crystal-freemagma.Thishastwomainconsequencesonthemodel.The firstoneisthatweassumethatbubblesdonotinteractwitheachother,in particulartherearenocoalescenceeffects.Thisisstronglylimitativeforthe simulationofamagmaticconduit,butthepresenceorabsenceofcoalescence canbecontrolledinlaboratoryexperiments,seesection5.2.Theotherpoint isthat,duetothehighviscosityofthemagma,bubblestravelalongwith thesamevelocityasthemelt.Inotherwords,theycanbeconsideredas immobilewithrespecttothemelt. Atthisstage,wecanconsiderthatabubblecanbedescribedwithtwo parameters,itsvolumeV ˆ anditsgaseousmassM ˆ .Inthissection,wedenote withahatthedimensionalvariables.Takingintoaccountthatthebubble ismadeonlyofwateringaseousform,wecanwritetheperfectgaseslaw insidethebubbleinordertorelatethegaspressureP ˆ tothegasdensityρˆ: GTρˆ=P ˆ M w ,(1) withM w themolarmassforwater,GtheperfectgasconstantandTthe gastemperature.Next,following[7,8]and[12]weassumethatthebubble isspherical,withradiusR ˆ ,sothatV ˆ =4πR ˆ 3 /3,andwesetforfuture convenienceM ˆ =ρˆR ˆ 3 ,sothatthebubblemassis4πM ˆ /3.Thuswecan choosetheradiusR ˆ =R ˆ (t)andthevariableM ˆ =M ˆ (t),proportionalto themass,todescribetheevolutionofthebubble,andseekforasystemof differentialequationsforthesevariables.Noticethatin[12]anequationon thepressureP ˆ (t)isgiven,wechooseheretotrackthebubblemassbecauseit leadstoabetterhandlingofmassconservationatthenumericallevel.Sucha modelgivesadescriptionofthegrowthofasinglebubble,orforapopulation ofidentical,non-interactingbubbles:thisistheso-calledmono-dispersecase. 3