USF Libraries
USF Digital Collections

Studies of the mechanics and structure of shallow magmatic plumbing systems

MISSING IMAGE

Material Information

Title:
Studies of the mechanics and structure of shallow magmatic plumbing systems
Physical Description:
Book
Language:
English
Creator:
Díez, Mikel
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Eruption triggering
Eruption dynamics
Magma ascent mechanisms
Volcano-tectonic interaction
Static stress changes
Dissertations, Academic -- Geology -- Doctoral -- USF   ( lcsh )
Genre:
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: Volcanic activity, and the resultant deposits and structures at the Earth's surface, are the outcome of the inner workings of underground magmatic plumbing systems. These systems, essentially, consist of magma reservoirs which supply magma to the surface through volcanic conduits feeding volcanic eruptions. The mechanics and structure of plumbing systems remain largely unknown due to the obvious challenges involved in inferring volcanic processes occurring underground from observations at the surface. Nevertheless, volcanologists are beginning to gain a deeper understanding of the workings and architecture of magmatic plumbing systems from geophysical observations on active volcanoes, as well as from geological studies of the erosional remnants of ancient volcanic systems. In this work, I explore the relationship between the structure and mechanics of shallow plumbing systems and the volcanic eruptions these systems produce.I attempt to contribute to the understanding of this complex relationship by linking geological and geophysical observations of an eroded basaltic subvolcanic system, and the eruptive and tectonic activity of an active volcano, with mathematical models of magma ascent and stress transfer. The remarkable exposures of the Carmel outcrop intrusions, near the San Rafael swell, southeast Utah, U. S. A., allow detailed geological and geophysical observations of the roots of volcanic conduits that emerge from a subhorizontal magma feeder reservoir. These observations reveal a new mechanism for magma ascent and eruption triggering through gravitational instabilities created from an underlying feeding sill, and shed light on the mechanics of sill emplacement.Geophysical and geological observations of the 1999 and xii 1992 eruptions of the Cerro Negro volcano, Nicaragua, are used to explore the coupling between changes in the stress field and the triggering of volcanic eruptions, and magma ascent through the shallow crust. Modeling results of stress transfer and conduit flow highlight the importance of the surrounding stress field and geometry of the volcanic conduits that comprise shallow plumbing systems.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
System Details:
Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Mikel Díez.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 211 pages.
General Note:
Includes vita.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001994157
oclc - 317406759
usfldc doi - E14-SFE0002436
usfldc handle - e14.2436
System ID:
SFS0026753:00001


This item is only available as the following downloads:


Full Text

PAGE 1

StudiesoftheMechanicsandStructureofShallowMagmaticPlumbingSystemsbyMikelDezAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofGeologyCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:CharlesB.Connor,Ph.D.OnnoBokhove,Ph.D.CostanzaBonadonna,Ph.D.SarahE.Kruse,Ph.D.PaulH.Wetmore,Ph.D.DateofApproval:April4,2008Keywords:Eruptiontriggering,eruptiondynamics,magmaascentmechanisms,volcano-tectonicinteraction,staticstresschanges,Rayleigh-Taylorinstabilityinmagmas,volcanoconduitgeometryandstructure,SanRafaelsubvolcaniceld,CerroNegroVolcano,wallrockrheologycCopyright2008,MikelDez

PAGE 2

DEDICATIONTomyparentsJulia&JoseLuis,myaititeAngel,mygirlfriendAna,andmysister,Maria,andbrother,Jon.

PAGE 3

ACKNOWLEDGMENTSIwouldliketothankmyadvisorProfessorCharlesB.Connorforlettingmeworkwithhimandallowingmetofreelyexplore,Dr.IvanP.Savovforeldassistance,forworking18hoursstraightwithmeinthegeochemistrylabtoanalyzeconduitrocks,forchallengingscienticandphilosophicaldiscussionsduringlongnightsaccompaniedbygoodrumandcigars,butaboveall,forbeingagreatfriend,LauraConnorforherinvaluablehelpwiththealwayschallengingcomputersciences,Dr.OnnoBokhoveforhishospitalityattheUniversityofTwente,TheNetherlands,itwasaveryrewardingexperienceworkingwithhimandstudentsoftheDepartmentofAppliedMathematics,Dr.SarahKruseandstudentsofthe2008geology/geophysicseldcourseforassistancewithgeophysicsintheSanRafaeldesert,Utah,Dr.JeRyanforuseofhisDCplasmaemissionspectrometerandotherequipment,Dr.PaulWetmoreforuseofhismicroscopesandeldassistance,Dr.ClintonConradattheUniversityofJohnsHopkinsinBaltimore,MarylandforinterestingdiscussionsandforlettingmeborrowhiscodetoadaptitforRayleigh-Taylorinstabilitiesinsills,Dr.PeterLaFemina,Dr.WilfriedStrauch,VirginiaTenorioandthestaofInstitutoNicaraguensedeEstudiosTerritorialesINETERfortheircollaborativeresearchinNicaragua,Dr.CostanzaBonadonna,Armando,MandieandNick,Alain,SophieandKojiforeldassistanceandcompanyinthevastandlonelydesertofSanRafael,Utah,Dr.BogdanOnacforinterestingdiscussionsonsedimentarypetrologyandCosminandSeanforhelpingoutwithmicrophotographsofthinsections.Especialmente,megustaraagradeceramichica,Ana,todalainconmensurablepacienciaycomprensionduranteestelargoperiplo,ysobretodoporanimarmecuandomaslohenecesitado.IamalsogratefultotheBasqueCountryGovernmentforafouryearBecaparaformaciondeinvestigadores"fellowshipwhichallowedmetomakealivinghereintheUS.

PAGE 4

TABLEOFCONTENTSLISTOFTABLESiiiLISTOFFIGURESivABSTRACTxiiCHAPTER1INTRODUCTION1CHAPTER2GEOLOGYANDGEOPHYSICSOFTHECARMELOUTCROP32.1Introduction32.2Geologicsetting42.2.1Overburdenthicknessestimation52.3GeologyoftheCarmeloutcrop92.3.1Dikeandbuds102.3.2Plug2anddomes152.3.3Plug3162.3.4Plug4202.4Geophysics262.5Mechanicsofsillemplacementandconstraintsondiapirascentvelocity292.6Triggeringofmagmaascentanderuptionbygravitationalinstabilities32CHAPTER3COOLINGANDCRYSTALLIZATIONOFATHINBASALTICSILL363.1Introduction363.2Thermalmodel373.2.1Kineticsofmagmacrystallization393.2.2Governingequationsandboundaryconditions413.3Results423.3.1Central-injectionscenario453.3.2Upper-injectionscenario463.4ImplicationsfortheCarmeloutcropsill483.5Conclusions51CHAPTER4GROWTHOFRAYLEIGH-TAYLORINSTABILITIESFROMASILL534.1Introduction534.2Rheologyoftheoverlyingwallrocks564.2.1Wallrockdeformationbydiusioncreep57i

PAGE 5

4.2.1.1Depth-dependentviscosity604.2.2Deformationofamicrofracturedwallrock634.3GeneralEquations644.4BoundaryConditions684.5NumericalSolutions724.6Modelverication754.7Results774.8ApplicationtotheCarmeloutcrop814.9Conclusions85CHAPTER5CERRONEGRO1999ERUPTIONTRIGGERINGBYSTATICSTRESSCHANGES885.1Introduction885.21999AftershocksequencesandEruptionofCerroNegro905.3FaultGeometryandSlip915.4Results945.5DiscussionandConclusions96CHAPTER6SOLUTIONANDPARAMETRICSENSITIVITYSTUDYOFACOUPLEDCONDUITANDERUPTIONCOLUMNMODEL986.1Introduction986.2MathematicalModels996.2.1TheConduitFlowModel996.2.2EruptionColumnModel1016.2.3TheDecompressionJetModel:CouplingtheConduitandEruptionColumnModels1016.3TheComputationalApproach1046.3.1TheFlowoftheConduitCode1046.3.2EruptionColumnCode1066.4ModelValidationUsingthe1992eruptionofCerroNegroVolcano1106.5Parametricanalysis1136.6Conclusions116REFERENCES118APPENDICES128AppendixABasicFlowintheRayleigh-Taylormodel129AppendixBNomenclatureandconduitowanderuptioncolumnmodelequations132AppendixCMatlabcodesfortheRayleigh-Taylorinstabilitymodel142AppendixDCcodes163ABOUTTHEAUTHOREndPageii

PAGE 6

LISTOFTABLESTable2.1Majorelementoxidecompositionofrocksamplestakenfromthedike,Bud2andPlug2,determinedbyDirectCoupledPlasmaEmissionSpectrometryDCP-ES.10Table2.2MajorelementoxidecompositionofrocksamplestakenfromPlugs3and4,determinedbyDirectCoupledEmissionSpectrometryDCP-ES.16Table3.1Valuesoftheparametersusedinthecoolingandcrystallizationmodel.44Table4.1Parametersusedinrheologicalmodelsdescribedinequations.5and.6.59Table4.2ParametersusedinmodelresultsshowninFigures4.5and4.679Table4.3ParametersforthemodelsappliedtotheCarmeloutcrop.83Table4.4Characteristicgrowthtimesandcorrespondingcharacteristicwave-lengthsforthemodelsappliedtotheCarmeloutcrop.83Table5.1Faultgeometryandslipforthethreeearthquakeson5August,1999Figure5.192Table6.1Inputandoutputparametersusedandobtainedinthecalculationsfor1992eruptionofCerroNegrovolcano.Note,forthisparticulareruptiontheconduitischoked.Thus,theconduitanderuptioncolumnmodelsarecoupledthroughadecompressionjetmodel.111iii

PAGE 7

LISTOFFIGURESFigure2.1SimpliedgeologicmapoftheSanRafaelsubvolcaniccomplex,UT,emphasizingintrusiverocksandstratigraphicgroupsandformations.6Figure2.2GeologicandstructuralmapoftheCarmeloutcrop.Areasinwhitearecoveredinalluviumand/orcolluvium.8Figure2.3GeologicalmapofBud2.SeeFigure2.2forlocation.aPeperiticlenseatthewestmarginofthebud;andbpeperiticcontactbetweentheinnerne-grainedsandstoneandtheouterdikebasalt.Seetextfordetails.11Figure2.4aMicrofoldwithinne-grainedsandstoneCDK-2sample,seeFigure2.3delineatedbyveryne-graineddarkmaterial.bRocksampleofaverticalsectiontakenfromapeperiticlenseFigure2.3a,withabandedstructurealternatingne-grainedsandstonebandswithbasan-itebands.Asaresultofmechanicalmixing,ahybridgraybandisformed,containingisolatedclinopyroxenecrystalswithinthegrayishpeperiticmatrix.Notethesimilarityofofanellipticalshapedspot"withinthene-grainedsandstonewiththeredspot"astormdevel-opedintheJovianatmosphereasaresultofsimpleshearingbetweenlayersofdierentcomposition.12Figure2.5aDetailedgeologicalmapofasectorcontainingPlug2andDomes1and2seeFigure2.2forlocation.bSchematiccrosssectionthroughPlug2A-A'ina.SeeTable2.1formajorelementanalysisofsamplesCAR-1andCSCB.14Figure2.6aDetailedgeologicalmapofPlug3seeFigure2.2forlocation.bSchematiccrosssectionthroughPlug3B-B'ina.Seetable2.2formajorelementanalysisofsamplesCAR-3andCBC-CORE.17iv

PAGE 8

Figure2.7aOverturnedstrataofbrecciatedne-grainedclasticrockatthesoutheastcontactofPlug3Figure2.6.bFoldedblockofthesamewallrocklithofaciesatalocationnearbya.Notethatthehingeofthefoldrunsnearlyorthogonalthroughthepictureplane.cHandspecimenofthewallrockshowninaandb.Notethemudstoneandsiltstonesubparallelclastsinadarkermatrix.dMicrophotographoftherockspecimeninc,showingevidenceofbrittledeformationanddisplacementlightermudstoneclastinthelowerleftcornerandplasticdeformationelongatedthinnerclasts.19Figure2.8DetailedgeologicalmapofasectorcontainingPlug4seeFigure2.2.SeetableformajorelementanalysisofsamplesCAR-9andCAR-6.21Figure2.9aExampleoflithofaciesCAR-9takenfromthecentralareaofthenorthernlobeofPlug4Figure2.8.Notethatsomesandstonexeno-lithsabovethepencilforscalearesubparallelandnearlyverticallyoriented,indicatingverticalmagmaowthroughtheconduit.bMicrophotographofathinsection,takenfromoutcropina,withplanepolarsshowingaglassydarkerdropletwithinamorecrys-tallinebasaniticaphiricmatrixlighter.cThinsectionofmatrixoflithofaciesCAR-9withplanepolarswithabundantarmoredlapilli,typicallyfoundinhydrovolcanicdeposits.22Figure2.10aHandspecimenoflithofaciesCAR-6withglobularjuvenileenclaveswithinasandymatrix.bHandspecimenoflithofaciesCAR-6withuidalandblockysandstoneenclaveswithinabasaniticmatrix.23Figure2.11ConcentrationofmajorelementsinlithofaciesCAR-6,CAR-3andabasaniteaveragebetweenCBSCandCDK-4.NotethatpeperiteCAR-3canbeformedbymixingthesandstonecomponentofCAR-6withtheaveragebasanite.Seetextfordetails.23Figure2.12aBrecciatedwallrockwithinternalrotationanddisplacementofblocksatthebasalpartofthewestcontactexposureofthenorthernlobeofPlug4Figure2.8.Notespalledblockswithinbasaniteabovethenotebookforscale.bVerticallyorientedsandstonexenolithsandelongatedbubblesatthewestrimofthesouthernlobeofPlug4Fig-ure2.8.24Figure2.13MagneticmapoftheCarmeloutcrop.Notethetwomagneticanomaliescorresponding,inthesoutharea,toashallowsillbeneaththeplugsanddomesdescribedpreviously,andinthenortharea,toadeepersillthatextendsafewtensofmetersfurthertotheeast.Thereddashedlinerepresentstheoutlineoftheinferredsills.27v

PAGE 9

Figure2.14GPRandresistivityprolesthroughthesectorwithPlug2andDomes1and2.aGPRproleG-G'acrossDome1.Notethepresenceofaburieddome4mwestofDome1.bResistivityproleW-W'acrossPlug2.NotetheresistivityanomalybeneaththedikeandPlug2.cResistivityproleN-N'throughtheeastsideofPlug2andDome1.Notethesubhorizontalresistivityanomalybeneaththeintrusionsindicatingthepresenceofaveryshallowsillofanaveragethicknessof4m.28Figure2.15Sketchofahypotheticalhybridsill-dikesystemsimilartotheoneintheCarmeloutcrop;aasanascendingdikeencountersabeddingplaneoranunconformity,abovewhichthewallrockshaveacomparativelyhighrigidity,itinitiateslateralinjectionofmagmathroughthisinterface,and;bsillpropagationismainlycontrolledbyviscousdissipation.Astheinjectionsintrudeincoolsedimentaryrock,thedecreaseintemper-ature,particularlyatthetipregion,increasesmagmaviscosityandthesillstartstoarrest.Theinuxofmagmaintothesilliscontinuous,thustheoverpressurecreatedbehindthetipregionisaccommodatedbytheupliftofoverlyingwallrocks.30Figure2.16ThreedimensionalillustrationshowingsaltdiapiricstructuresemergingfromsaltridgescreatedatthetopofamothersaltbedprintedfromRamberg,1981afterTrusheim,1960.33Figure2.17Sequenceofillustrationsofmagmaascentanderuptionbyagravita-tionalinstabilitytriggeredfromasill.aAveryshallowsillbecomesgravitationallyunstablebythedensitycontrastbetweenmagmaandtheoverlyingwallrocks.bTheseinstabilitiesgrowformingadiapiricstructure.cAsthetensilestrengthofthediapirroofisovercomeadikeformsandcontinuesascendingreachingunconsolidatedsedimentsandformingpeperites.dPeperitesbecomeunstableandahydrovol-caniceruptionistriggeredopeningacrateratthesurface.eAstheeruptionwanes,theconduitisinlledwithblocksfromtheventthatmixwithascendingdegassedmagma.Notethattheverticalscaleoftheguresisdiscontinuous.ReddashedlineonpanelerepresentsanapproximatetopographicprolesimilartotheNW-SEprolethroughPlug3seeFigure2.6.34Figure3.1EectivecrystallizationfunctionfTuversusundercoolingTu.Notethecrystallizationmaximumatlargeundercoolingsandthesmallarrowindicatingthenucleationdelay.SeeTable3.1forparametersused.40vi

PAGE 10

Figure3.2aSketchofthecoolingandcrystallizationmodelset-upforacentral-injectionscenario.Notethatthelargestarrowrepresentsmagmainuxandthesmalleronesindicaterelativeupwardanddownwardmotiontoaccommodatethenewinjection.bInitialtemperatureproleandtemperaturedistributionafterthefourthintrusionfortheupperpartofthewallrock/sillsystem.Notethatusehasbeenmadeofthesymmetryofthemodelset-up.43Figure3.3aSketchofthecoolingandcrystallizationmodelset-upforanupper-injectionscenario.Note,asinthecentral-injectionscenario,thatthelargestarrowrepresentsmagmainuxandthesmallerarrowsindicaterelativedownwardmotiontoaccommodatethenewlyinjectedmagma.bInitialtemperatureproleandtemperaturedistributionafterthefourthintrusionforthewholewallrock/sillsystem.Thetimebetweentherstandfourthinjectionis2400s.44Figure3.4Temperatureaandnon-crystallizedfractionbprolesforacentral-injectionscenarioafterfourinjectionsandwithatimebetweeninjec-tionsof3600s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.45Figure3.5Temperatureaandnon-crystallizedfractionbprolesforacentral-injectionscenarioafterfourinjectionsandwithatimebetweeninjec-tionsof600s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.46Figure3.6Temperatureprolesforacentral-injectionscenarioafterfourinjectionsandwithatimebetweeninjectionsof60s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.47Figure3.7Temperatureaandnon-crystallizedfractionbprolesforanupper-injectionscenarioafterfourinjectionsandwithatimebetweeninjec-tionsof3600s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.49Figure3.8Temperatureaandnon-crystallizedfractionbprolesforanupper-injectionscenarioafterfourinjectionsandwithatimebetweeninjec-tionsof3600s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.49vii

PAGE 11

Figure4.1SketchofamultilayermodelfortheRTinstability,whereiandiarethedensitiesandviscositiesofthewallrocklayersand,sandsarethesilldensityandviscosityrespectively.Thesill/wallrockinterfaceislocatedatz=0,thegroundsurfaceatz=hoandthebottomofthesillatz=-hs.Therearep+1layersincludingthesillandatotalmnumberofinnerinterfaces.Notethattheoverburdenlayersaremodeledasanon-NewtonianmaterialandtherheologyofthesillisassumedtobeNewtonian.Theconstitutivemodelisderivedsuchthatalineartemperatureprolethroughtheoverburdeninducesviscosityvariationsintheverticaldimensionseetextfordetails.Notethedensitycontrastwithlowersilldensitybetweenthesilllowerandtheoverburdenhigher,requiredtotriggerRTinstabilities.55Figure4.2Viscosityasafunctionoftemperatureforsedimentaryrocksofthreedierentgrainsizesfor:aCoblecreepandbNabarro-Herringcreep.SeeTable4.1forparametersusedinequations.5and.6.60Figure4.3Numericalsolutionsarecomparedtoanalyticalsolutionsplottingdi-mensionlesswavenumberversusdimensionlessgrowthrateforalayeroverahalf-spacewith:aequalviscosities,1=2=1019Pasand;blargeviscositycontrast,12,where1=1019and2=1016Pas.76Figure4.4Sketchoftheset-upforamultilayermodelofaRTinstabilitycon-strainedwithgeologicalandgeophysicalobservationsoftheCarmeloutcrop.Thenumberoflayersis3andinnerinterfaces2.Thedensityandviscosityofthewallrocklayersareiandirespectively,withi=1,2,andforthesill,sands,withs=3forthesill.Therheologicalexponentnis,n=1fortheNewtoniansilland,n=1fortheperfectlyplasticwallrocklayers.Thethicknessofthesillish3and,h2andh3oftheoverlyingwallrocksrespectively.78Figure4.5Dimensionlessgrowthrateq0versusdimensionlesswavenumberk0:avaryingdensityofthesill,3and,bvaryingthicknessofthesill,h3.79Figure4.6Dimensionlessgrowthrateq0versusdimensionlesswavenumberk0:avaryingtheviscosityoflayer2,2,bvaryingthethicknessofoverbur-den1,h1andcanddvaryingthethicknessoflayer2,h2.80Figure4.7Growthtimeversuswavelengthfor:avaryingthethicknessoflayer2,h2andbvaryingtheviscosityoflayer2,2.84viii

PAGE 12

Figure5.1LocationmapforCerroNegrovolcanoandnearbyQuaternaryvol-canoesLasPilas,Rota,CerroLaMulaandMomotomboBlacktri-angles.EpicenterlocationsblackstarsandfocalmechanismsareshownforMw>5earthquakeson5-7August,1999seeTable5.1.SolidblacklinesareknownQuaternaryfaults.NotefaultsstrikingNEleft-lateral,NWright-lateralandNSdip-slip.100mtopographiccontoursareshownaslightgraylines.Studyareashowninsetblackbox,withinCentralAmericavolcanicArchistoricallyactivevolca-noesshowninblacktriangles.89Figure5.2aMapofcalculatedchangein3resultingfromsliponthreefaultplanesTable5.1withepicentersshownaswhitestars.Dailyseismic-ityisindicatedfor5-8August.NotestressreductionatCerroNegro,consistentwithdikeinjectionfollowing5Augustseismicity.Othersym-bolsasinFigure5.1.CrosssectionFigure5.2bindicatedbyA-A'.bStresschangein3calculatedforaN-SplanethroughCerroNegrovolcanoFigure5.2a,usingfaultgeometryshowninTable5.1.Notestressreductionandstressgradientinthisplane,consistentwithdikeinjection.93Figure5.3CoulombfailurestresschangeCFSCcalculatedusingfaultgeometryandslipdataofthethreeearthquakesTable5.1.NotetheasymmetryoftheCFSCpattern,possiblyduetodip-slipcomponentsalongthefaultplanes.AnexceptionisseismicitynearCerroNegrovolcano,attributedtodikeinjectionanderuption.OthersymbolsasinFigure5.1.95Figure5.4ComparisonoftheCoulombfailurestresschangeCFSCdistributionovertheentireareaofstudyandtheCSFCdistributionwhereepi-centersintheaftershocksequencesoccurredAugust5,6,7and8.ThedistributionofepicentersisnotrandomwithrespecttotheCFSCmodelgreaterthan99%condenceKolmogorov-Smirnovtestforcu-mulativedistributions.Thatis,epicentersoftheaftershocksequenceshaveastatisticallysignicanttendencytooccurinareasofhighCFSC.96Figure6.1Schematicrepresentationofaavolcanicconduitandbaneruptioncolumn.Figuresarenotdrawntoscale.100ix

PAGE 13

Figure6.2Sketchshowingthecouplingbetweentheconduitanderuptioncolumnmodelsthroughthedecompressionregion,andtheverticalvariationofpressure,velocity,densityandradiusalongthevolcanicsystemforinputconditionsshowninTable6.1andaninitialvolatilefractionof4wt:%.Notethattheconduitmodeliscoupleddirectlytothedecom-pressedjet,anditsvelocityandradiusarecalculatedusingequationsderivedbyWoods&Bower1995,Eqs.B.12-B.18.Thevariationofvelocityandradiusisnotcomputedthroughthedecompressionre-gion,anditshypotheticalvariationisrepresentedbyadashedline.Seethedecompressionregionzoomedfortheverticalprolesofpressure,densityandradius.Note,despitejumpsinthesevariablesbetweentheventandthedecompressedjet,massisconservedEq.B.16.102Figure6.3Computationalsketchforatheconduitowmodel,andbtheerup-tioncolumnmodel.Ineachcase,inputparametersarespeciedatthebaseofthemodelandgradientsofdependentvariablesareusedtoes-timatevaluesofdependentvariableshigherintheconduitdp dzcandcolumndq dze,dm dze,de dze.Thresholdsoccurintheconduit<0:75andcolumna=ethatchangethecomputationEqs.B.10,B.19,B.20.Figuresarenotdrawntoscale.103Figure6.4Flowchartfortheconduitowmodel.Theequationsandnomencla-tureusedbytheconduitowmodelcodearepresentedinAppendix.Note,theODEsolveriteratesuptheconduituntilanyoftheboundaryconditionsaresatised.105Figure6.5Flowchartfortheeruptioncolumnmodel.Theequationsandnomen-clatureusedbytheeruptioncolumnmodelarepresentedinAppendix.Note,theODEsolveriteratesupwardthroughthecolumnuntiltheboundaryconditionissatised.107Figure6.6Flowcharthighlightingthecouplingbetweentheconduitanderuptioncolumnmodelsthroughadecompressionjetmodel.TheequationsandnomenclatureusedbythedecompressedjetmodelarepresentedinAppendix.Note,dependingontheboundaryconditionsatthevent,theinitialconditionsatthebaseoftheeruptioncolumnarepassedfromthedecompressionjetmodelordirectlyfromtheconduitmodel,conditionsaresubsonic.108Figure6.7ComparisonofmodelresultsofWoods,1988withresultsob-tainedwiththecodesforconduitowanderuptioncolumn,respec-tively.aComparisonofconduitowresultsfornc0=0.04,rc=20mandTm=1000K.bComparisonoferuptioncolumnresultsforne0=0.03,re0=100mandTe0=1000K.109x

PAGE 14

Figure6.8Variationofthevelocityalongtheconduitforthemodelcalculationsinwhichthemagma'sinitialvolatilemassfractionis0.03,0.04and0.05.Theconduitradiusis3mandthemagmatemperatureis1200K.Exitvelocitiesandmassowratesareconsistentwithobservationsandestimationsofthe1992CerroNegroeruption.112Figure6.9Modelcalculationsoferuptioncolumnradiusvs.heightwheremagma'sinitialgasmassfractionis0.03,0.04and0.05,initialtemperatureis1150Kinthethreesimulations,initialcolumnvelocityis239,288and328ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,andinitialcolumnradiusis5,7and9mvelocityandradiusofthedecompressedjet.Theseinitialconditionsproducearangeofheightsinagreementwithobservationsmadeduringthe1992CerroNegroeruption.112Figure6.10Numericalexperimentscarriedoutwiththeeruptioncolumnmodelx-ingsomeparametersandvaryingothersaslog-normalprobabilityden-sityfunctionstoinvestigatevariationsincolumnheight.aFixedpa-rameterne0=4:5wt:%;variableparametersue0range:100)]TJ/F15 10.9091 Tf 10.9091 0 Td[(350ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,re0range:20)]TJ/F15 10.9091 Tf 10.909 0 Td[(200mandTe0range:1000)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1250K.bFixedparame-tersue0=220ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,re0=60mandTe0=1000K;variableparameterne0range:3)]TJ/F15 10.9091 Tf 10.9091 0 Td[(7wt:%,cne0,ue0andTe0areheldconstantandre0range:20)]TJ/F15 10.9091 Tf 10.9091 0 Td[(200m,dalltheparametersarexedexceptforQwhichisallowedtovary.114Figure6.11NumericalexperimentsperformedusingtheconduitmodeltoexplorethesensitivityofmassowQwithchangingparametersduringsus-tainedexplosiveeruptions.AsinFigure6.10,certainparametersareheldconstantandothersarevariedaslog-normalprobabilitydensityfunctions.Thebaseoftheconduitisat3kmdepth.aVariableparameterne0range:3)]TJ/F15 10.9091 Tf 10.9091 0 Td[(7wt:%;xedparameterspc0=77MPa,c0=2250kgm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3,Tm=1000Kandrc=50m,ballparametersareheldconstant,nc0=4:5wt:%andrcrange:20)]TJ/F15 10.9091 Tf 10.9091 0 Td[(100m.115xi

PAGE 15

StudiesoftheMechanicsandStructureofShallowMagmaticPlumbingSystemsMikelDezABSTRACTVolcanicactivity,andtheresultantdepositsandstructuresattheEarth'ssurface,aretheoutcomeoftheinnerworkingsofundergroundmagmaticplumbingsystems.Thesesys-tems,essentially,consistofmagmareservoirswhichsupplymagmatothesurfacethroughvolcanicconduitsfeedingvolcaniceruptions.Themechanicsandstructureofplumbingsys-temsremainlargelyunknownduetotheobviouschallengesinvolvedininferringvolcanicprocessesoccurringundergroundfromobservationsatthesurface.Nevertheless,volcanol-ogistsarebeginningtogainadeeperunderstandingoftheworkingsandarchitectureofmagmaticplumbingsystemsfromgeophysicalobservationsonactivevolcanoes,aswellasfromgeologicalstudiesoftheerosionalremnantsofancientvolcanicsystems.Inthiswork,Iexploretherelationshipbetweenthestructureandmechanicsofshal-lowplumbingsystemsandthevolcaniceruptionsthesesystemsproduce.Iattempttocontributetotheunderstandingofthiscomplexrelationshipbylinkinggeologicalandgeophysicalobservationsofanerodedbasalticsubvolcanicsystem,andtheeruptiveandtectonicactivityofanactivevolcano,withmathematicalmodelsofmagmaascentandstresstransfer.TheremarkableexposuresoftheCarmeloutcropintrusions,neartheSanRafaelswell,southeastUtah,U.S.A.,allowdetailedgeologicalandgeophysicalobservationsoftherootsofvolcanicconduitsthatemergefromasubhorizontalmagmafeederreservoir.Theseobservationsrevealanewmechanismformagmaascentanderuptiontriggeringthroughgravitationalinstabilitiescreatedfromanunderlyingfeedingsill,andshedlightonthemechanicsofsillemplacement.Geophysicalandgeologicalobservationsofthe1999andxii

PAGE 16

1992eruptionsoftheCerroNegrovolcano,Nicaragua,areusedtoexplorethecouplingbetweenchangesinthestresseldandthetriggeringofvolcaniceruptions,andmagmaascentthroughtheshallowcrust.Modelingresultsofstresstransferandconduitowhigh-lighttheimportanceofthesurroundingstresseldandgeometryofthevolcanicconduitsthatcompriseshallowplumbingsystems.xiii

PAGE 17

CHAPTER1INTRODUCTIONThedynamicsofvolcaniceruptionsislargelygovernedbythephysicalpropertiesofmagmasandthestructureandmechanicsoftheplumbingsystemthatfeedstheerup-tions.Whereasagreatdealofattentionhasbeendirectedtowardstheunderstandingofphysicalmagmaproperties,theinnerarchitectureandmechanicsofmagmaticplumbingsystemsremainrelativelyunknown,asthesesystemsremaininaccessibleduringongoingvolcaniceruptionsanddirectobservationoftheirinnerworkingsisimpossible.Neverthe-less,geophysicaltechniques,seismicsinparticular,haveprovidedanapproximatepictureofmagmaticplumbingsystemsbeneathactivevolcanoese.g.,Ryanetal.,1981,andmorerecently,the3DstructureofaburiedfossilbasalticplumbingsystemintheNorthRockalltrough,northwestofScotlandThomsonandHutton,2004.ThroughaseriesofdierentworksIaddressafewtopicsthatrelatevolcaniceruptionswithshallowmagmaticplumbingsystems,suchasstructuralrelationsoffeederreservoirswithvolcanicconduits,eruptiontriggeringmechanismsanderuptiondynamics.Thesetop-icsareexploredbylinkinggeophysicalandgeologicaleldobservationswithmathematicalmodelsofmagmaascentandmodelsofheatandstresstransfer.Geologicalandgeophysicalobservationsofabasalticsub-volcaniceldexposedinre-markableconditionsintheSanRafaelSwellarea,southeastUtah,revealtheinnerstructureoftherootsofvolcanicconduitsandilluminatenewprocessesformagmaascent,sillem-placementanderuptiontriggeringintheshallowcrustseeChapter2.Inparticular,anewmechanismofmagmaascentanderuptionisdiscoveredbywhichtheuppercontactofasillbecomesgravitationallyunstableinitiatingthediapiricascentofmagmatowardsthesurface.Toquantitativelyexploretheviabilityofmagmadiapiricascentthroughtheshallowsubsurface,thermalandhydrodynamicinstabilitymodelsarederivedconstrainedbyeldgeologicalandgeophysicalobservationsseeChapters3and4.1

PAGE 18

Themechanicalrelationshipbetweenthemagmaticplumbingsystemandtheregionalstresseldisexploredonanactivevolcano,CerroNegroVolcano,NicaraguaseeChapter5.InAugust1999,CerroNegroeruptedrightaftertheoccurrenceofthreetectonicearthquakesthatoccurrednearthevolcano.Iproposethatthelow-volumessuraleruptionwastriggeredbystatic-stresschangesinducedbythethreeearthquakes.Inaddition,themodeledCoulombstresschange,afterthethreeearthquakesanderuption,isconsistentwiththeearthquakeaftershockdistributionrecordedduringthefollowingdays.Thedynamicsofmagmaascentthroughashallowvolcanicconduit,andintotheat-mospherethroughaneruptioncolumn,isstudiedbyderivingacoupledconduitowanderuptioncolumnmodelseeChapter6.Thismodelisconstrainedbygeologicaldatafromthe1992sustainedexplosivephaseoftheCerroNegroVolcanoeruption.Thecoupledmodelalsoindicatesthatchangesinthedimensionsoftheconduitcauselargevariationineruptionmassowrate,highlightingtherelevanceofunderstandingthestructureanddimensionsofrealvolcanicconduits.2

PAGE 19

CHAPTER2GEOLOGYANDGEOPHYSICSOFTHECARMELOUTCROP2.1IntroductionVolcaniceruptionsattheEarth'ssurfacearetheresultoftheinnerworkingsofanunderlyingmagmaticplumbingsystem.Inaverybroadsense,thisplumbingsystemmaybedescribedasmagmareservoirsfromwhichconduitsemergetotransportmagmatowardsthesurfaceandnallyfeederuptions.Forobviousreasons,activeplumbingsystemsremainhiddenduringongoingeruptionsandvolcanologistshavetorelyongeophysicalobservationstodelineatetheirinnerstructuree.g.Ryanetal.,1981.Directobservationoftheirinnerarchitectureisonlypossibleinareaswhereerosionhaspartiallyexposedtheplumbingsystem.Forsilicicvolcanicsystems,anumberofstudieshavebeenconductedAlmond,1971;Koronovsky,1971;EkrenandByers,1976;Reedmanetal.,1987;Woletal.,1989;Stasiuketal.,1996;Kanoetal.,1997;Sorianoetal.,2006andonlyafewofthem,duetothelimitationsoftheexposures,correlateconduitinllingsandstructurewithdepositsatthesurfaceAlmond,1971;Koronovsky,1971;EkrenandByers,1976;Sorianoetal.,2006.AlargernumberofworksJohnson,1906;Williams,1936;Hunt,1938;AppeldornandWright,1957;McGetchin,1968;DelaneyandPollard,1981;White,1991;DelaneyandGartner,1997;NemethandWhite,2003;RossandWhite,2006;McClintockandWhite,2006;ValentineandKrogh,2006;Keatingetal.,2007haveexploredthesubsurfacestructuresofbasalticsystems,suchasdikesandplugs,andinsomecasese.g.,White,1991;HootenandOrt,2002;NemethandWhite,2003,depositspreservedatthesurfacehavebeenrelatedtodepositswithinthefossilfeederconduits.Intheseoutcropsonlytheshallowerpartoftheplumbingsystemisexposedand,tomyknowledge,noaccounthasbeenreporteddescribingtherootsofconduits,thatis,theregionofthereservoirfromwhichconduitsemergetofeedvolcaniceruptions.ThePliocenedikes,sillsandplugsoftheSanRafaelsubvolcaniceld,Utah,US,areexposedin3

PAGE 20

starkreliefwithinmiddleJurassicclasticsedimentsoftheColoradoPlateauFigure2.1.Theirmodernexposurescorrespondto500-1500mofemplacementdepthDelaneyandGartner,1997andprovideremarkableoutcropstoexploredeeperpartsofacontinentalbasalticplumbingsystem.Thisworkfocusesononeoutcrop,theCarmelOutcrop,locatedintheeasternmarginoftheSanRafaeleld,southoftheSanRafaelswellFigure2.1.ThisoutcropconsistsofaN-Strendingdikewithplugsdistributedtobothsidesofthedike,scatteredoutcropsofanunderlyingsill,exposedonasmallcreek,andexposuresofsurroundingwallrocksFigure2.2.Idescribeandinterpretthegeologyandstructureoftheseplugsandwallrocks.Somelithofacieswithintheplugsindicatethattheyactedasvolcanicconduits.Dierentgeophysicalsurveysrevealashallowintrusioninterpretedasasillbeneaththeplugsandislikelycontinuouswiththebasaltexposures20mwestofthedike,exposedinadrycreekbed.Iinterpretthissillasfedbythedikeandasareservoirfromwhichtheplugsgrewasgravitationalinstabilitiesfeedingeruptionsatthesurface.Themechanicsofemplacementofthedikeandthesillarediscussedandaconceptualmodelfortheascentofmagmatriggeredbyagravitationalinstabilityisproposed.2.2GeologicsettingTheCarmeloutcropislocatedintheSanRafaeldesert,southeastUTFigure2.1onthewesternColoradoPlateau.Inthisdesert,aPliocenebasalticsubvolcaniccomplex,theSanRafaelSubvolcanicComplexSRSC,cropsoutinremarkableconditions.Thisintrusivecomplexisabout60kmlong,nowheremorethan30kmwide,andrepresentsanancientmagmaticplumbingsystemthatfedcindercones,maarsandturingsinthepast.ThismagmaplumbingsystemisnowexposedbecauseofthedeeperosionoftheColoradoPlateausedimentarysequenceDelaneyandGartner,1997.TheSRSCconsistsofapproximately2000dikesegments,severalvolcanicplugsandmorethantwelvesillsthatarenothickerthan30mDelaney,1982Figure2.1.Thedikerocksaredarkgray,locallyporphyriticalkalinediabasesGilluly,1927;Williams,1983and,chemically,correspondtotrachybasaltsandbasaniteswithsilicacontentsrangingfrom4

PAGE 21

44to48wt.%DelaneyandGartner,1997.ThedikesandsyeniticsillsareinterpretedascontemporaneousandcomagmaticGilluly,1927,andthesyeniteinthesillsasderivedfromthebasalticmeltbyfractionalcrystallizationWilliams,1983.DikerockshavebeendatedwithK-Armethodsandyieldagesbetween3.4and4.7MaDelaneyandGartner,1997.Interestingly,themagmaticactivityintheSRSCiscontemporaneouswith3.8to6.4Maoldtrachybasaltandbasalticandesitelavaowsmappedathigherelevations30kmtothesouthwestofCedarMountainDelaneyandGartner,1997Figure2.1.Mostofthedikes,sillsandplugsoftheSRSCintrudeintosubhorizontalMiddleJuras-sicsedimentaryclasticandne-grainedstrataoftheSanRafaelGroupFigure2.1.TheSanRafaelGrouphasanaveragethicknessof550mthroughouttheSRSCDelaneyandGartner,1997andconsistsoftheCarmelformation,EntradaSandstone,andCurtisandSumervilleformationsGilluly,1929;Smithetal.,1963.Theserockscorrespondmostlytone-grainedsandstones,siltstones,shalesandlimestonesdepositedinanearshoreshallow-seaenvironmentDelaneyandGartner,1997.Someofthedikesreachedupperstrati-graphiclevelsoftheunconformablyoverlyingLateJurassicMorrisonformation.OtherdikesareexposedwithintheunderlyingeolianTriassicandJurassicNavajoSandstoneoftheGlenGroup,whichisbeneaththeCarmelformationseparatedbyanunconformityFigure2.1.2.2.1OverburdenthicknessestimationThethicknessoftheoverburdenatthetimeofintrusion4MaagoisestimatedusingareconstructionoftheOligoceneca.30MapaleosurfaceoftheColoradoPlateauPedersonetal.,2002.Pedersonetal.calculateamapforthetotalexhumation,,bysubtractingthepresentlandsurfacefromthereconstructedOligocenepaleotopographyseetheirFigure7;Pedersonetal.,2002.Thethicknessoftheoverburden,e,then,canbecalculatedbye=)]TJ/F21 10.9091 Tf 10.9091 0 Td[(p;.15

PAGE 22

Figure2.1.SimpliedgeologicmapoftheSanRafaelsubvolcaniccomplex,UT,emphasizingintrusiverocksandstratigraphicgroupsandformations.6

PAGE 23

wherepistheerodedmaterialfromtheOligocenetothetimeoferuption.Beforecal-culatingp,andtheerosionrate,E,areneeded.ThevalueoffortheCarmeloutcropis1200mPedersonetal.,2002,andPedersonwrittenpersonalcomm.,however,theestimationoftheerosionrate,E,throughtheCenozoicdeservesabriefdiscussion.Signif-icanterosionoftheColoradoPlateaudidnotoccuruntil6MawhentheColoradoRiversystem,asweseeittoday,rstbegantoowotheedgeofthehigherColoradoPlateauandintothelowerBasinandRangenearthemouthofGrandCanyonPederson,personalcomm..AllindicationssuggestthatthisdrainageintegrationeventhasdriventhevastbulkoftheerosionthatnowisseeninthelandscapePederson,personalcomm..Thus,assumingmostoftheerosionhappenedoverthelast6Maandthatittookplacelinearlythroughtime,theerosionrateisgivenbyE= t6Ma)]TJ/F22 7.9701 Tf 6.5865 0 Td[(Present;.2wheret6Ma)]TJ/F22 7.9701 Tf 6.5865 0 Td[(PresentisthetimeintervalfromtheMioceneca.6Matothepresent.Assumingatotalexhumationof1200mfortheSanRafaeleld,E=200m/myr.TheerodedmaterialintheSanRafaeleld,sinceca.6MatothePlioceneca.4Maisthengivenbyp=Et6Ma)]TJ/F19 7.9701 Tf 6.5865 0 Td[(4Ma:.3Substitutingthisequationinto.1,andsolvingfore,IndthatthePliocenesurfacewasabout800mabovethepresentgroundsurface.ItisimportanttonotethatasthetotalexhumationvaluesareconservativeseePedersonetal.,2002,thisvaluerepresentsaminimumamountoferosion.Then,thevolcanicoutcropsnowseenatthesurfacewouldhavebeenformedbeneathaminimumof800mofsedimentarycover.7

PAGE 24

Figure2.2.GeologicandstructuralmapoftheCarmeloutcrop.Areasinwhitearecoveredinalluviumand/orcolluvium.8

PAGE 25

2.3GeologyoftheCarmeloutcropTheCarmeloutcropislocatedintheeasternedgeoftheSRSC,southoftheSanRafaelSwellFigure2.1.Theoutcropconsistsofseveralbasalticplugsanddomes1distributedatbothsidesofaN-StrendingdikeFigure2.2.SomeoftheinternallithofaciesinPlugs3and4,discussedinthenextsection,suggestthattheseplugsactedasvolcanicconduitsduringthetimeoftheiremplacement,4Maago.WallrocksexposedintheCarmeloutcropconsistofne-grainedrecrystallizedlimestones,siltstonesandshalesinterbeddedwithgypsumbedsofthebandedandgypsiferousmembersoftheCarmelformationDoelling,2004.TheexcellentexposureconditionsattheCarmeloutcroprevealaninterestingstructuralrelationshipbetweenthebasalticintrusionsandthesurroundingwallrocks.ItisclearfromtheCarmeloutcropgeologicalmapFigure2.2thatsomefeaturesalongthedike,suchasbuds,mayrepresentlocalizationoftheowofmagmaalongthedike.Ontheotherhand,somedomesandplugsareosetfromthedikeandreachedthelevelofcurrentexposureindependentofthedike.ThisfeatureoftheCarmeloutcropisnotconsistentwithclassicmodelsofssureeruptions,whichrelyonlocalizationofowtoformconduitsalongdikese.g.,Wylieetal.,1999.Rather,magmaascendedfromdepthtroughconduitscurrentplugsanddomesbyadierentmechanism.Inareassurroundingtheplugs,thewallrocklayersdipinwardwithrespecttotheintrusionsFigure2.2.Also,inPlug3,wallrocklayersatthecontactareoverturned.Theseobservationssuggestadiapiricoriginoftheseplugs.AlthoughtheexposureconditionofPlug1andabasaltoutcroproughlycircularinplanesectionsouthofDome2Figure2.2aresucientlygoodtobedelineated,theydonotprovideenoughgeologicalandstructuralinformationtobestudiedindetail.InthisstudyIfocusindierentsectors:atheN-Strendingdike;banareaincludingPlug2,Dome1andDome2,andtheirregularbasaltoutcrops20mwestofDome2;cPlug3; 1Thetermdome"referstoastructureinwhich,atthegroundsurface,acoreofbasaltissurroundedbyoutwarddippingwallrocklayersandwhere,inprinciple,itmaybeinferredthatthebasaltdidnotpiercethroughtheoverlyingwallrocks.9

PAGE 26

Table2.1.Majorelementoxidecompositionofrocksamplestakenfromthedike,Bud2andPlug2,determinedbyDirectCoupledPlasmaEmissionSpectrometryDCP-ES. Samples CSCBCAR-1CDK-4SSCDK-2Oxidewt.%BasaniteLimestoneDikebasaniteSiltstoneSandstone SiO248.90.5751.354.4476.74Al2O314.531.39315.679.568.25Fe2O310.170.629.473.221.74MgO12.430.9710.027.522.15MnO0.180.070.170.100.08CaO9.595.179.0222.728.44K2O0.410.740.470.620.23Na2O2.400.282.61.121.89TiO21.470.181.290.700.45 anddPlug4.Withthegoalofexploringdierentmechanismsofmagmaascent,thesesectorsaredescribedindetailinthefollowingsubsections.2.3.1DikeandbudsTheN-Strendingdikeconsistsofvesegmentsofvariablelengthandanaveragethicknessof1m,arrangedinanen-echelonpatternFigure2.2.Thedikerockisabasanite,withaSiO2contentof51wt.%seesampleCDK-4inTable2.1andFigure2.3anddiabasictexture.NowallrockxenolithsarepresentalongthedikeexceptwithinBud2asdescribedinthefollowing.BudsareanomalouslythickpartsthatemergefromdikesegmentsWentworthandJones,1940whereremovalofthedilationalcomponentoftheoverallformwouldnotbringadjacentdikewallsbackintocontactDelaneyandPollard,1981.NortheastofPlug2,atoneofthedikesegmentstip,emergesBud1,whichis2mwideFigure2.2.Notethatbothcontactsdonottbacktogetherbyremovingthebasaniteinbetween,suggestingthatBud1isanerosionalfeaturecreatedbytheremovalofwallrocksbytheupwardowingmagma.Bud2islocatednortheastofPlug3andemergeswithinadikesegmentFigure2.2.ThisbudhasamorecomplexstructurethanBud1,withasandstonebodysurrounded10

PAGE 27

Figure2.3.GeologicalmapofBud2.SeeFigure2.2forlocation.aPeperiticlenseatthewestmarginofthebud;andbpeperiticcontactbetweentheinnerne-grainedsandstoneandtheouterdikebasalt.Seetextfordetails.11

PAGE 28

Figure2.4.aMicrofoldwithinne-grainedsandstoneCDK-2sample,seeFigure2.3delineatedbyveryne-graineddarkmaterial.bRocksampleofaverticalsectiontakenfromapeperiticlenseFigure2.3a,withabandedstructurealternatingne-grainedsand-stonebandswithbasanitebands.Asaresultofmechanicalmixing,ahybridgraybandisformed,containingisolatedclinopyroxenecrystalswithinthegrayishpeperiticmatrix.Notethesimilarityofofanellipticalshapedspot"withinthene-grainedsandstonewiththeredspot"astormdevelopedintheJovianatmosphereasaresultofsimpleshearingbetweenlayersofdierentcomposition.12

PAGE 29

bybasaniteFigure2.3.Theinnersedimentarybodyconsistsofane-grainedwell-sortedsilicicsandstoneseesampleCDK-2inTable2.1withthinbandsofverynematerialunidentiedunderopticalmicroscope,butpossiblyvolcanicashthatdelineatemicrofoldswithinthesandstoneFigure2.4a.ThesurroundingbasaniteCDK-4containsfewbubbles<5%,someofthemelongatedintheverticaldirection,particularly,closetothewestcontact.Inthiscontact,thebasanitealsocontainssandstonexenoliths,mostofwhicharedisrupted,deningverticalowlineations.Inlocalizedareas,northandwestofthecontactwiththewallrock,lensesofpeperiticmaterialarepresent,withthelongestaxisverticallyorientedseeFigure2.3b.ApictureoftheinternalstructureofoneofthislensesisshowninFigure2.4a.Notethebandedstructure,indicatingviscouslaminarow.Graybandsformasaresultofthemechanicalmixingbetweenthesandstoneandthebasanite,asthepresenceofisolatedclinopyroxenecrystalsindicate.Theinterfacebetweentheinnersandstonebodyandthebasaniteexposedinthenorthwestpartofthecontact,ispeperitic,withjuvenileenclavesofglobularandblockybasaniteintrudingintothesandstoneseeFigure2.3b.ThepresenceofpeperitesindicatecontemporaneityofmagmatismandsedimentationSkillingetal.,2002.Inaddition,thestructureofthesandstoneinllingthebudshowsne-grainedmaterialbandsdelineatingmicrofolds,suggestingthatitwasunconsolidatedatthetimeofintrusion.Iinterprettheseobservationsasindicativeofmagma-sedimentinteractionatorclosetothePliocenesurface,weresedimentationwastakingplace.Asmagmaeruptedatthesurfacethroughthedike,atsomepointthepeperitesinlledtheconduit,andthelastmagmaticpulseforceditsascentthroughtheunconsolidatedbodycreatingsimpleshearingandmechanicalmixing,leadingtothedevelopmentofthepeperiticlensesobservedatthecontactofBud2.13

PAGE 30

Figure2.5.aDetailedgeologicalmapofasectorcontainingPlug2andDomes1and2seeFigure2.2forlocation.bSchematiccrosssectionthroughPlug2A-A'ina.SeeTable2.1formajorelementanalysisofsamplesCAR-1andCSCB.14

PAGE 31

2.3.2Plug2anddomesInthecentralareaoftheCarmeloutcropFigure2.2asectorcontainingPlug2,Dome1andDome2,andafewscatteredbasaniteoutcropswestoftheN-Strendingdike,croppingoutthroughadrycreekbed,ismappedindetailFigure2.5a.Plug2isroughlycircularinplanesectionwithanapproximatediameterof8mFigure2.5a.Theintrusiverockwithintheplugisabasanite,with48.8wt.%SiO2,anddiabasictexture,verysimilartothedikerockCDK-4Table2.1.Thisintrusiverockhasavesiclecontentof5to7%andcontainsnowallrockxenolithsexceptforspalledblocksnearthecontacts.Thesurroundingwallrocksareexposedincertainplacesalongthecontact.Thewallrockconsistsofbaked,banded,silica-andcarbonate-richsiltstonesseesampleSSinTable2.1withinterbeddedshalybeds,whichdelineatepre-existingpinchandswellstructures.Theirdipvariesalongthecontact,withnearlyatoutwardandinwarddippingbedsalongtheeastandnorthcontactsrespectively,andoverturnedbedsatthesouthcontact.Twodomes,Dome1andDome2,cropoutsouthofPlug2.ThedistancesbetweenPlug2andDome1,andDome1andDome2are,21and11m,respectivelyFigure2.5a.Dome1iscircularinplanesectionwithadiameterof3m,comparativelylargerthanDome1witha0.5mdiameter.ThesedomesarestratigraphicallyslightlylowerthanthegroundsurfaceexpressionofPlug2,andcropoutwithinanoutwarddipping40one-grainedrecrystallizedlimestoneseeCAR-1inTable2.1.Thislimestoneisexposednearby,dippingwestintheareasurroundingPlug2.Anumberofirregularbasaniteoutcropsaremappedthroughasmallcreek5mwestofDome1Figure2.5.Althoughthequalityofexposureisratherpoor,itisenoughtorevealthepresenceofanintrusionsillothedikeandbeneaththeplugsanddomesofthissectorintheintrusivecomplex.ThesegeologicalobservationssuggestthatPlug2andDomes1and2havegrownasgravitationalinstabilitiesfromanunderlyingsill,creatinganinwarddippingstratigraphyin15

PAGE 32

Table2.2.MajorelementoxidecompositionofrocksamplestakenfromPlugs3and4,determinedbyDirectCoupledEmissionSpectrometryDCP-ES. Samples CAR-3CBC-CORECAR-6CAR-9Oxidewt.%PeperiteBasaniteSandstoneVolcanoclastic SiO275.7951.1078.6561.3Al2O310.0515.439.3215.67Fe2O32.569.542.377.55MgO2.3510.653.006.58MnO0.060.170.050.12CaO5.898.213.145.53K2O0.340.550.350.35Na2O2.533.032.621.93TiO20.441.310.490.96 theareasurroundingPlug2.StructuraldatarevealadiapiricgeometryforPlug2Figure2.5b,andthedomesareinterpretedasfrozen"earlystagesofdiapiricdevelopment.2.3.3Plug3Plug3isexposedinanarea50msouthofPlug2and6mwestoftheN-StrendingdikeFigure2.2.Thisplugisroughlycircularinplanesection,withadiameterofabout16m,andshowsamorecomplexinnerstructurethanPlug2Figure2.6a.LithofaciesCAR-3seeTable2.2,noteitshighSiO2contentof75.9wt.%isinllingtheplug.ThislithofaciesresemblesthegraybandofpeperiticlensesFigure2.4aofBud2Figure2.3,whichformedasaresultofmechanicalmixingbetweenane-grainedsandstoneandthebasanite.InPlug3,theoriginofthisrockisrevealedbythepresenceofbothmixingpoles:aanoutcropofne-grainedsandstonewithtexturesthatindicateow,similartoCDK-2inBud1Figure2.3;andbjuvenilebasaniteenclaves.Thepresenceofabundantverticalelongatedbubblesindicatethattheunconsolidatedsandstonewaswetatthetimeofintrusion,andasaconsequencestartedtoboil.Italsoindicatesverticalmovementalongtheconduitasthesevesiclesaresignicantlyelongated,someofthemwithlengthsofseveralcentimeters.Inaddition,roundedsandstonexenolithsareconsistentwithdisplacementandsimpleshearingintheverticaldirection.Possibly,the16

PAGE 33

Figure2.6.aDetailedgeologicalmapofPlug3seeFigure2.2forlocation.bSchematiccrosssectionthroughPlug3B-B'ina.Seetable2.2formajorelementanalysisofsamplesCAR-3andCBC-CORE.17

PAGE 34

rotationalmovementinducedonthexenolithsbythesimpleshearing,desintegratethemprogressively,eventuallycreatingroundedshapes.Aslightlyo-centered,circularinplanesection,basaniteoutcropisexposedwithinlithofaciesCAR-3.Thisbasanite,CBC-COREseeTable2.2,hasaSiO2contentof51.10wt.%,isfreeofxenolithsandcomparativelydenserthanbasaniteCSBCwithinPlug2.Thisbasaniteoutcropisinterpretedasalateintrusionofdegassedmagma.ThecontactofPlug3iscomplex,withoverturnedstratigraphyFigure2.6aand2.7a.Basaniteintrusionsformthin1mlensesalongthecontact.Thewallrockisane-grainedclasticrock,withmudstoneandsiltstonefragmentsFigures2.7cand2.7d,andisbrecciatedshowingevidenceofplasticdeformationFigure2.7b.ExposuresofstratigraphicallylowerwallrockCAR-1cropout25mnorthwestofPlug3Figure2.6aanddipSE.AnellipticalbasaltexposureismappedafewmetersnorthofPlug3,however,thisplugispoorlyexposed.Plug3isinterpretedasaconduitthroughwhichmagmawastransportedduringaneruption.Isuggest,basedonthestructuraldataattheplug'scontactandattheexposedwallrockFigure2.6b,thatthisplugwasformedasagravitationalinstabilityfromasubhorizontalunderlyingintrusion.Plug3wouldrepresentamoreadvancedstageinthedevelopmentofthegravitationalstructuresdescribedintheprevioussection,asitfedvolcaniceruptionsanddevelopedamorecomplexstructure.Thepresenceofoverturnedlayersatthecontactsuggestsamushroom-likestructureforthisplug,however,innerfaciesindicatethatitactedasavolcanicconduittransportingmagmatothesurface.Theseobservationssuggestthatatransitionfromdiapirtodikelikelyhappened.Suchatransitioncanbeexplainedwhenthephreaticmechanismofbrecciationofthewallrocknolongeroperates,ifforexample,theascendingdiapirencounterswallrocklayerswithhigherpermeability.Astheuidinthesedimentarywallrockporescanescapeaftersuddenheating,porepressureisdramaticallyreducedandbrecciationoftherocks,andsubsequentviscositydecreasearenotachieved.Theoverlyingwallrocklayersthenbehaveelasticallyunderthestressexertedbytheascendingdiapir,eithercausingitsarrestor,ifthe18

PAGE 35

Figure2.7.aOverturnedstrataofbrecciatedne-grainedclasticrockatthesoutheastcontactofPlug3Figure2.6.bFoldedblockofthesamewallrocklithofaciesatalocationnearbya.Notethatthehingeofthefoldrunsnearlyorthogonalthroughthepictureplane.cHandspecimenofthewallrockshowninaandb.Notethemudstoneandsiltstonesubparallelclastsinadarkermatrix.dMicrophotographoftherockspecimeninc,showingevidenceofbrittledeformationanddisplacementlightermudstoneclastinthelowerleftcornerandplasticdeformationelongatedthinnerclasts.19

PAGE 36

strengthofthewallrocksisovercome,openingnewtensionalfractures.Thesenewfracturesmaybeinlledbymagmafromthediapirformingadike.Alternatively,magmawithinthediapircouldexploitpreexistingfractures.ThetransitionfromdiapirtodikehasbeentheoreticallyaddressedbyRubinwithinwallrocksdeformingunderaviscoelasticregime.2.3.4Plug4Plug4isexposedinthesouthernmostsectoroftheCarmeloutcropFigure2.2.Itisthelargestandmostcomplex,intermsofinnerlithofaciesandgeometry,ofthemappedplugs.Theoverallgeometryisbilobate,withanirregularsouthernlobeappendedtoamorecircularsectorthatintersectstheN-StrendingdikeFigure2.8.Thenorthlobeisinlledwithavolcanoclasticlithofacies,CAR-9seeTable2.2,whichconsistsofabasaniticaphiricmatrixandne-grainedsandstonexenolithsFigure2.9a,andlocallypeperiticdomains.Thisfaciesappeartoresemblethepeperite-likemagma-sedimentmixturesofLorenzetal..GlassydropletswithinthematrixrevealapyroclasticoriginofthislithofaciesFigure2.9b,andabundantarmoredlapilliinthematrixofCAR-9Figure2.9cindicatethatthisvolcanoclasticrockwaslikelyproducedduringhydrovolcaniceruptionse.g.,Macasetal.,1997.Thesouthernlobemainlyconsistsofbasanitefreeofxenoliths,similartoCBC-COREinPlug3,anditisdenserthanCSCBPlug2andCDK-4dikerock,likelyindicatingalateintrusionofdegassedmagma.ThislithofaciesisalsoexposedthrougharoughlycircularoutcropwithinCAR-9,inthenorthernlobe.AbodyofglobulartoblockypeperiteisexposedbetweenthesetwolobesFigure2.8.Texturalchangesareobservedthroughoutthebody,withglobularjuvenileenclaveswithinasandstonematrixseesampleCAR-6inTable2.2inthewestsectorFigure2.10agraduallychangingtoatexturewithuidalandblockysandstoneenclavessurroundedbyajuvenilebasaniticaphyricmatrixFigure2.10b.Interestingly,whenthemajorelementcompositionofCAR-6sandstonecomponentiscomparedwithanaveragecompositionbetweenlithofaciesCSBCPlug2,Figure2.5andCDK-4dikerock,Figure2.3,andwell-mixedpeperite20

PAGE 37

Figure2.8.DetailedgeologicalmapofasectorcontainingPlug4seeFigure2.2.SeetableformajorelementanalysisofsamplesCAR-9andCAR-6.CAR-3Plug3,Figure2.6,itcanbeobservedFigure2.11thatthewell-mixedpeperiteCAR-3maybeformedbythemixingofthetwootherlithofacies,CAR-6andtheaveragebasanite.Thissuggeststhatmagmabreachedthesurfacethroughthesedierentconduitsintruding,anderuptingthroughthesameunconsolidatedsedimentarylayers.InthewestrimofPlug4,ascreenofbandedsiltstone,stratigraphicallyhigherthanthewallrockinPlug2,isexposedona3to4mhighverticalcli,revealinganinwarddippingcontactbetweentheintrusiverockandthewallrockFigure2.8.Thebeddingisnearlysubhorizontalwhichsuggeststhattheupwarddivergingcontactinthispartoftheplugwascreatedbyerosioncausedbyowthroughtheconduit.Thebasalpartofthis21

PAGE 38

Figure2.9.aExampleoflithofaciesCAR-9takenfromthecentralareaofthenorthernlobeofPlug4Figure2.8.Notethatsomesandstonexenolithsabovethepencilforscalearesubparallelandnearlyverticallyoriented,indicatingverticalmagmaowthroughtheconduit.bMicrophotographofathinsection,takenfromoutcropina,withplanepolarsshowingaglassydarkerdropletwithinamorecrystallinebasaniticaphiricmatrixlighter.cThinsectionofmatrixoflithofaciesCAR-9withplanepolarswithabundantarmoredlapilli,typicallyfoundinhydrovolcanicdeposits.22

PAGE 39

Figure2.10.aHandspecimenoflithofaciesCAR-6withglobularjuvenileenclaveswithinasandymatrix.bHandspecimenoflithofaciesCAR-6withuidalandblockysandstoneenclaveswithinabasaniticmatrix. Figure2.11.ConcentrationofmajorelementsinlithofaciesCAR-6,CAR-3andabasaniteaveragebetweenCBSCandCDK-4.NotethatpeperiteCAR-3canbeformedbymixingthesandstonecomponentofCAR-6withtheaveragebasanite.Seetextfordetails.23

PAGE 40

Figure2.12.aBrecciatedwallrockwithinternalrotationanddisplacementofblocksatthebasalpartofthewestcontactexposureofthenorthernlobeofPlug4Figure2.8.Notespalledblockswithinbasaniteabovethenotebookforscale.bVerticallyorientedsandstonexenolithsandelongatedbubblesatthewestrimofthesouthernlobeofPlug4Figure2.8.24

PAGE 41

wallrockexposureisbrecciatedandfoldedbytheinternalrotationanddisplacementofindividualwallrockblocksFigure2.12a.ThecontactbecomesverticalthroughtheE-Wtrendingrimofthesouthernlobe.ThewallrockdoesnotcropoutalongtheN-SrimofthislobewhichisdelineatedbyathinbandofCAR-9.Locally,sandstonexenolithsandelongatedvesiclesareverticallyoriented,sub-paralleltotheerodedcontact,suggestingverticalmagmaticowneartheconduitwallFigure2.12.EastofPlug4,wallrockstrataconsistingofshaleswithinterbeddedthingypsum5to8cmlayers,cropoutdippingtowardthenorthwestFigure2.8.Thisstructuralob-servationindicatesthatthedeformationthatcreatedthewestwarddippingstratigraphyoccurredpriortodikeintrusion.ThisisconsistentwiththefactthattheN-StrendingdikecroppingoutatbothnorthandsouthintersectionswithPlug4andfurthernorthandsouthrespectively,isverticalandthereforewasnotaectedbythisdeformationevent.Thus,thepresentexposureofthedikemayrepresentalatedikeformedbyoverpressurefromtheunderlyingsill.Plug4isinterpretedastheremnantofavolcanicconduitwhich,aswithPlug2andPlug3,emergedasagravitationalinstabilityfromanunderlyingsubhorizontalintrusion.Asaresultofthedevelopmentofthisinstability,thesurroundingsedimentaryrocksac-commodatedthedeformationevincedbytheinwardtowardthecenterofplugsdippingstrataobservedintheoutcrop.Thisinstabilitygrewandascended,likelytransformingintoadikeatupperlevels,andmixingwithunconsolidatedwetsedimentsatthesurface,cre-atinghydrovolcaniceruptions.Asaconsequence,thisinitiallydiapiricconduitwaserodedbyowduringeruption,asPlug3,andeventually,astheeruptionwaneditwaslledinwithamixtureofwallrockfragmentsandjuvenilemagma.Alastpulseofmagmaowedthroughthevolcanoclasticfaciesandlledthesouthernlobe.ThecomplexgeometryofPlug4couldbeexplainedalsobytheinteractionoftwodiapiricinstabilitiesatearlierstagesofthemagmaticevent.25

PAGE 42

2.4GeophysicsAseriesofgeophysicalsurveys:magnetics,groundpenetratingradarGPRandresis-tivity,werecarriedoutattheCarmeloutcroptoexplorethesubsurfacestructurebeneaththeplugsanddomesdescribedintheprevioussection.ThemagneticsurveyFigure2.13revealsashallowintrusionbeneaththesectorcon-tainingPlug2anddomes,andthesectorcontainingPlug3,consistentwiththebasaniteexposureswestofDome2atthedrycreek.Thisintrusionisabout50mwideonaverageandisdelineatedfollowingareaswherethegradientintheintensityofthemagneticeldishighest.Interestingly,themagneticanomalyextendstobothsidesoftheN-Strendingdike.AnotherintrusionisrevealedthroughthenorthernareaoftheCarmeloutcropbyanirregularmagneticanomalythatextendsabout40meastoftheN-StrendingdikeFig-ure2.13.Thisintrusionisinterpretedtobeslightlydeeperasthemagneticeldgradientiscomparativelylow.NotethattheonlysurfaceexpressionoftheintrusioninthissectorisPlug1Figure2.2.TheGPRdata,alongproleG-G'Figure2.14a,delineatethegeometryofthesill/wall-rockcontactintheDome1area.Anotherdomeisrevealed4mwestofDome1,wherethesill/wallrockinterfaceliesbeneaththegroundsurface.Notethatpenetrationdepthis2mandtheresolutionoftheGPRdataisrelativelypoorintheareasouthofDome1.Resistivitydataalongtwoproles,W-W'Figure2.14bandN-N'Figure2.14c,revealresistivityanomaliesbeneaththesectorcontainingPlug2andDomes1and2.AlthoughtheW-W'proleshowsaroughlydiapiricstructurebeneathPlug2Figure2.14b,asthesurveyistwo-dimensional,thesedatacannotresolvethedetailedstronglythree-dimensionalgeometryofthisstructure.Theshapeofthisresistivityanomalycouldbegeneratedeitherbyacylindricaloradiapir-likethree-dimensionalbodyinversionresultsbyS.Kruse.TheN-N'proledelineatesasubhorizontalresistivityanomaly,whichsuggeststhepresenceofasillofabout4mofaveragethicknessbeneaththeintrusionsofthissectorFigure2.14c.26

PAGE 43

Figure2.13.MagneticmapoftheCarmeloutcrop.Notethetwomagneticanomaliescorresponding,inthesoutharea,toashallowsillbeneaththeplugsanddomesdescribedpreviously,andinthenortharea,toadeepersillthatextendsafewtensofmetersfurthertotheeast.Thereddashedlinerepresentstheoutlineoftheinferredsills.27

PAGE 44

Figure2.14.GPRandresistivityprolesthroughthesectorwithPlug2andDomes1and2.aGPRproleG-G'acrossDome1.Notethepresenceofaburieddome4mwestofDome1.bResistivityproleW-W'acrossPlug2.NotetheresistivityanomalybeneaththedikeandPlug2.cResistivityproleN-N'throughtheeastsideofPlug2andDome1.Notethesubhorizontalresistivityanomalybeneaththeintrusionsindicatingthepresenceofaveryshallowsillofanaveragethicknessof4m.28

PAGE 45

2.5MechanicsofsillemplacementandconstraintsondiapirascentvelocityThereisnocurrentmodelthatadequatelyexplainssillemplacement.EssentialconceptswererecentlyreviewedbyKavanaghetal..Twodierentmechanismshavebeeninvokedinthepasttoexplaintheformationofsills.Therstinvolvesabuoyancycontrolledascent,inwhichmagmaintrudeslaterallyintothewallrockformingasillwhentheneutralbuoyancylevelisreachedBradley,1965.Asecondbasicmechanisminvolvestectonicstresses.Anascendingdikeunderanextensionalstresseld,withthegreatestprincipalstress,1,verticalandtheleastprincipalstress,3horizontal,encountersashallowerlevelofcompressionalstresseldwith,3verticaland1and2horizontal,resultinginlateralinjectionintoasillRoberts,1970.TheneutralbuoyancylevelemplacementmechanismhasbeenproventobenotalwaysvalidasthereiseldevidenceofsillsemplacedinbothlowerFrancis,1982andhigherdensitywall-rocksGunn,1962.Inaddition,thesetwomechanismspredictsillmorphologiessubparalleltotheoverlyingcontemporaneoustopography,however,3-DseismicreectiondatafromtheNorthRockallTroughplumbingsystemThomsonandHutton,2004donotrevealsuchsillmorphologies.RecentlaboratoryexperimentsofsillformationKavanaghetal.,2006indicatethatunderinitiallyhydrostaticconditions,sillsareemplacedthroughinterfacesseparatingmorerigidmaterialslargerYoungModulusfromlessrigidlowerYoungModulusunderlyingmaterials.Theseexperimentsalsoshedlightonthedynamicsofsillpropagation;twoparametersareessentialinunderstandingthepropagationofdikesandsills:aThefracturepressure,Pf,whichisgivenbyListerandKerr,1991Pf=Kc l1=2;.4whereKcisthefracturetoughnesswhichcharacterizesthestrengthofthewall-rocksandlthesilllength.Whenthemagmaoverpressure,Pothepressureinexcessoftheambientpressure,islargerthanPf,acrackcanbeinitiated;andbtheviscouspressurelossListerandKerr,1991,Pv,whichcharacterizesthelossofkineticenergyduetoviscosityduring29

PAGE 46

Figure2.15.Sketchofahypotheticalhybridsill-dikesystemsimilartotheoneintheCarmeloutcrop;aasanascendingdikeencountersabeddingplaneoranunconformity,abovewhichthewallrockshaveacomparativelyhighrigidity,itinitiateslateralinjectionofmagmathroughthisinterface,and;bsillpropagationismainlycontrolledbyviscousdissipation.Astheinjectionsintrudeincoolsedimentaryrock,thedecreaseintemperature,particularlyatthetipregion,increasesmagmaviscosityandthesillstartstoarrest.Theinuxofmagmaintothesilliscontinuous,thustheoverpressurecreatedbehindthetipregionisaccommodatedbytheupliftofoverlyingwallrocks.laminarowwithinthefracturePv=l2 w2t;.5whereisthemagmaviscosity,wthesillthicknessandtistime.TheexperimentsbyKavanaghetal.showthatwhereasdikepropagationiscontrolledbyfractureme-chanics,thatisbyPfseeeq..4,silldynamicsismainlygovernedbyviscousdissipationorPvseeeq..5because,astheyintrudeparalleltoapreexistinginterface,onecanassumethatfracturingbetweentwoadjacentbedsrequireslessenergythanfracturingthroughcoherentmorecompetentwall-rocksKavanaghetal.,2006.Theresultsfromtheseexperimentscanbeappliedtothesill-dikesystemoftheCarmeloutcropFigure2.2.ThebasalpartoftheCarmelformationconsistsofshalesandsilt-30

PAGE 47

stoneswithinterbeddedlensesofmorecompetentlimestonesReches,1998.IntheCarmeloutcrop,theinterfacebetweenthesillandtheoverlyingrecrystallizedlimestonewall-rockscanbeobservedintheareaofscatteredoutcropsofbasalt15mwestoftheN-Strendingdike,andinDomes1and2Figure2.5.Thelowerinterfaceisnotexposed,however,thestratigraphyexposedintheCedarMountainareaReches,1998suggeststhatshalesandsiltstones,withgypsumintercalationsunderlietheintrusion.Consideringthesegeologicalobservations,Ihypothesizethatthesillemplacementwascontrolledbytherigiditycontrastbetweentheshalesandsiltstones,andtheoverlying,morerigidlimestone.ExperimentsbyKavanaghetal.reproduceahybridsill-dikesystem,roughlysimilartotheCarmeloutcrop,whenthedrivingpressureismarginallyinexcessofthefracturepressurerequiredtointrudethemorerigidupperlayer.TheCarmelsillinthesouthsectorhasarelativelyhighwidthh/lengthlaspectratioofr=h=l0.2Figures2.13and2.14c,comparedtocommonlymeasuredaspectratiosforsillsintheeldof10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3Shirley,1987,andexperimentalsillsKavanaghetal.,2006.Thisobservationcanbeexplainedtakingintoaccounttheheattransferfromthesillintothesurroundingwall-rocks;astheascendingdikeencountersthemorerigidlimestonelayeritinitiatesitslateralinjectionthroughtheinterfaceasathin1msill.Recallthatitspropagationismainlygovernedbyviscousdissipationseeeq..5andduetothelargetemperaturecontrastwiththecoldsedimentarywallrocks,theviscosityofmagmainthetipregionstartstoincrease.Asaconsequence,thelossofkineticenergybythegraduallyincreasingviscosityprovokesthearrestofthelaterallypropagatingsill.However,newmagmaisbeingcontinuouslyinjectedintothesill,increasingitsdrivingpressure.ThisexcesspressurebeginstobeaccommodatedbytheverticalexpansionofthesillFigure2.15eventuallyformingahigh-aspectratiosill,suchastheoneobservedattheCarmeloutcrop.However,othersillswithsmalleraspectratiosareexposedintheCedarMountainareawithintheEntradasandstoneFigure2.1.Thissuggeststhat,althoughviscousdissipationisagenerallygoverningfactorinsillpropagation,intheCarmeloutcrop,31

PAGE 48

thestrengthofthebeddingplanes,andthereforePfeq.2.4,likelyplayedanimportantrolealongwithviscousdissipation,Pveq..5,duetoviscosityincreasebyrapidcooling.Duringtheaccretionofthesill,anewdikeinitiatesitsascenttowardsthesurfacethroughtheoverlyingwall-rocks,asinthehybridsill-dikesystemofKavanaghetal.,whereasthesill/wall-rockupperinterfacestartstobecomegravitationallyunstabletrig-geringthediapiricascentofmagma.Thedrivingpressurewithinthedikeimposesanimportantconstraintonthetimescaleforthedevelopmentandascentofthediapiricin-stabilities;ifthedikereachesthesurfaceanderupts,itsdrivingpressurewilldecreaseconsiderably.Thus,itwillbemoreenergeticallyecientfornewmagmatocontinueitsascentthroughthedikeratherthanbeinginjectedintothesill,asmoreworkwouldbeneededtouplifttheoverlyingwall-rockstocreatethenecessaryspace.Fromthisobser-vationitnecessarilyfollowsthat,asthediapiricinstabilitiesneedtobecontinuouslyfedtogrow,theirascentvelocityofthefeederdikeabovethesillshouldhavebeenequalorfasterthanthedikeascentvelocity.ThereiseldevidencesuggestingthatboththedikeseeBud2,Figure2.3andthediapiricinstabilitiesseePlug3and4,Figures2.6and2.8reachedthesurfaceanderupted.Itremainsanopenquestionwhetherthediapiricinstabilitiesevolvedintodikesorcontinuedtheirascenttowardsthesurfacediapirically,becausetheoverlyingsedimentarysequenceiserodedaway.2.6TriggeringofmagmaascentanderuptionbygravitationalinstabilitiesGeologicalobservationsoftheplugsanddomesexposedattheCarmeloutcropsug-gestthattheseintrusionswerecreatedasgravitationalinstabilitiesfromanunderlyingsillwithintheCarmelsedimentaryformation,under800mofsedimentarycover.AdetailedgeologicalandstructuralstudyofthesectorcontainingPlug2andDomes1and2seeFigure2.5suggeststhatthesefeaturesrepresentthefrozen"earlierstagesofthegrav-itationalinstabilities.Plug2and3structuresareconsistentwithadiapiricorigin,andtheirinnerlithofaciesindicatethattheyactedasvolcanicconduitsduringthePliocene.GeophysicalsurveysdelineatesubhorizontalintrusionsbeneaththePlugsanddomesof32

PAGE 49

Figure2.16.ThreedimensionalillustrationshowingsaltdiapiricstructuresemergingfromsaltridgescreatedatthetopofamothersaltbedprintedfromRamberg,1981afterTrusheim,1960.theCarmeloutcrop.TheinwarddippingstratigraphyexposedatbothsidesoftheN-Strendingdike,atthenorthandsouthendsoftheCarmeloutcrop,suggeststhatthefeedersillsbecameunstablepriortotheupwarddikepropagation,whichisconsistentwithaconceptualanalysisofthemechanicsofsillemplacement.Thisanalysisalsohelpstoqual-itativelyconstraintheascentrateoftheinstabilities,indicatingthatitwasequalorfasterthantheascentrateoftheN-Strendingdike.Thethree-dimensionalstructureoftheplugsanddomesintheCarmeloutcropmaybeenvisagedasaseriesofparallellinearridges,generatedattheuppercontactofthesillandparalleltotheN-Strendingdike,fromwhichdiapiricstructuresemerge.AsimilardistributionofridgesanddiapiricstructureshasbeenproposedbyTrusheimwithinthecontextofsaltdiapirismFigure2.16,innorthernGermany,andhasalsobeenreproducedinlaboratoryexperimentsRamberg,1981;Talbotetal.,1991.Intheseexperiments,thelinearridgesarecreatedbytheeectofthecontainer'swallswhentheunstableuidlayerhasalargethickness/lenghtaspectratioTalbotetal.,1991.ThisissueisdiscussedinmoredetailinChapter4.Iproposeasequenceofevents,constrainedbygeologicalandgeophysicalobservationsoftheCarmeloutcrop,bywhich,asillemplacedinthesubsurfacebecomesgravitationallyunstable,initiatingtheascentofmagmatowardsthesurfaceandultimatelyeruptinginto33

PAGE 50

Figure2.17.Sequenceofillustrationsofmagmaascentanderuptionbyagravitationalinstabilitytriggeredfromasill.aAveryshallowsillbecomesgravitationallyunstablebythedensitycontrastbetweenmagmaandtheoverlyingwallrocks.bTheseinstabilitiesgrowformingadiapiricstructure.cAsthetensilestrengthofthediapirroofisover-comeadikeformsandcontinuesascendingreachingunconsolidatedsedimentsandformingpeperites.dPeperitesbecomeunstableandahydrovolcaniceruptionistriggeredopeningacrateratthesurface.eAstheeruptionwanes,theconduitisinlledwithblocksfromtheventthatmixwithascendingdegassedmagma.Notethattheverticalscaleoftheg-uresisdiscontinuous.ReddashedlineonpanelerepresentsanapproximatetopographicprolesimilartotheNW-SEprolethroughPlug3seeFigure2.6.34

PAGE 51

theatmosphere.Thissequenceofeventsisillustratedthroughatwo-dimensionalsectionadjacenttotheN-Strendingdike,andalongoneoftheridgespreviouslymentioned.aAsthesillcontinuouslyintrudesintowetandporoussedimentarywallrocks,thesuddenporepressureincreaselowerstheeectivestressleading,ultimately,tothefailureandbrecciationoftheoverlyingwallrocksDelaney,1982.Thus,itsviscosityisdramaticallyreduced.Asthemagmaislessdensethantheoverburden,andtheviscosityissignicantlyreduced,agravitationalinstabilityistriggered.bThecontinuousinuxofmagmaintothesillsustainsthegrowthoftheinstabilitythatstartsitsascenttowardsthesurfaceasamagmaticdiapirbystopingandplasticowofverylocalizedareasofthesurroundingwall-rocks.cIfthetensilestrengthofthewallrocksisovercomeattheroofofthedevelopingdiapir,adikecanform.Thisdikecontinuesitsascent,reachingtheunconsolidatedsedi-mentarydepositsatthesurfaceandmixingthoroughlywiththeunconsolidatedsedimentsformingpeperites.dFluid-uidinstabilitieswithinthesepeperites,atsomepoint,triggerahydrovolcaniceruptionWhite,1991;HootenandOrt,2002,seeFigures2.9band2.9cformingacrateratthesurface.Duringtheeruption,fastowthroughtheconduitandshockwavesassociatedtothehydrovolcanicexplosionWhite,1991;Lorenzetal.,2002enlargetheupperpartsoftheconduitformingadiatreme.eAstheeruptionstartstowane,blocksfromtheenlargedvent,llintheconduitandmixwiththelatestascendingpulsesofdegassedbasalticmagma.Inconclusion,geologicalandgeophysicalobservationsoftheCarmeloutcropnotonlyrevealaneweruptiontriggeringmechanism,butalso,indicatethatmagmaticdiapiricascentthroughthebrittlesubsurfaceisaviablemechanismformagmaascent.35

PAGE 52

CHAPTER3COOLINGANDCRYSTALLIZATIONOFATHINBASALTICSILL3.1IntroductionGeologicalandgeophysicalstudiesoftheCarmeloutcropindicatethatasubhorizontalintrusionofapproximately4mthicknessandextendingabout20mwestofaN-Strendingdike,underliesanareafromPlug2toDome2seeFigure2.5inChapter2.Inthischapter,Istudythecoolingandcrystallizationofthisinferredsillandaddresstheheattransferintotheoverlyingsedimentarywallrocks.TheseheattransferprocessesareofgreatimportancetoconstrainthedevelopmentoftheCarmelplugsanddomesasRayleigh-TaylorRTinstabilitiesemergingfromanunderlyingsill.ThenumericalanalysisofthesillbecomingRTunstablewillbeexploredinthenextchapter.Essentially,developmentofachilledmarginatthetopofthecoolingsillwouldeventuallypreventRTinstabilityfromoccurring.Thus,thetimerequiredforsillinjection,comparedtothetimerequiredforRTinstabilitiestoform,mustbeconsidered.Overthelastfewdecades,twodierentapproacheshavebeenfollowedinmodelingthecoolingandcrystallizationofmagmabodiesofdierentsizesandgeometries,rangingfromdikesandsillstoplutons.Intherstapproach,thecoolingandsolidicationofmagmaismodeledusingtheone-dimensionalheatconductionequatione.g.,CarslawandJaegger,1959;Connoretal.,1997,oraslightlymorecomplicatedformulationincludingthekineticsofcrystallizatione.g.,BrandeisandJaupart,1987b;Spohnetal.,1988;Hort,1997.Thesecondapproachinvolves2Dand3Dnumericalsimulationofthemomentum,energy,massandspeciesconservationequationse.g.,Speraetal.,1995,seeBergantzforareview.Acommonfeaturetoallofthesemodelsistheassumptionthattheintrusiontakesplacebyonesinglemagmapulse.However,modalolivinedistributionsinalkalinebasicsillsledGibbsandHendersontoproposeamechanismforthegrowthofsmallto36

PAGE 53

medium-largesizedsillsbyaseriesofdiscreteinjectionsoracontinuouspulsedinuxduringaprotractedperiodoftime.Inaddition,eldevidenceinsheetedsillcomplexesMooreandLockwood,1973;ChapmanandRhodes,1992;Sissonetal.,1996;Tegneretal.,1993;BrownandMcClelland,2000alsosuggeststheaccretionofmagmaticreservoirsbysequentialinjections.Basedontheseobservations,MichautandJaupartderivedacoolingandcrystallizationmodelinwhichalarge,long-livedmagmabodygrowsbysmallincrementsofthinsillinjectionsoveranextendedperiodoftime.Also,studiesofmagmagenerationinthelowercrustBergantzandDawes,1992;PetfordandGallagher,2001recognizetheimportanceofmultiplemagmainjectionsforregionalmagmaproduction.Iderivea1Dheatconductionmodelthataccountsforthekineticsofcrystallizationofathinsill<10mandheattransferintotheoverlyingwallrocksfollowingMichautandJaupart.Themodelissolvedfortwoemplacementscenarioswheremagmaisinjectedinsequentialpulses;rstthroughtheuppercontactontopofthelatestintrusionand,second,centrallybetweenpreviousinjections.Asingleinjectionscenarioisruledoutfortworeasons:ageologicalobservationsofdierentsillsandmagmaticbodiesaroundtheworld,andrecentmodelingstudiespreviouslymentionedseethepreviousreferences,favorasequentialinjectionscenario;and,basexplainedinthepreviouschapter,theN-S1mdikeintheCarmeloutcropisinterpretedtofeedthesill.However,thesillisaboutfourtovetimesthickerthanthedike,makingthesingleinjectionscenariohighlyunlikely.Asinglemagmainjectionwouldresultintoathinsilland,hence,thethicknessoftheCarmelsillindicatesthatmultipleinjectioneventsmusthavetakenplace.Modelresultsarediscussedpayingspecialattentiontothedevelopmentofchilledmargins,astheseimposeamechanicalandtemporalconstrainttothedevelopmentoftheRTinstability.3.2ThermalmodelAmodelisderivedtostudythecoolingandcrystallizationofabasalticintrusionemplacedsequentiallybytheinjectionofthinnersillintrusionsintheshallowsubsurface.Whentherstinjectionsofhotbasalticmagmaintrudeintocoldrock,duetothelarge37

PAGE 54

undercooling,nucleationandgrowthofcrystalsareinhibitedandachilledmarginHuppertandSparks,1989formsatthecontact.Atsmallerundercooling,whenthetemperaturegradientisreducedduetorepeatedinjections,crystallizationstartstoreleaselatentheatofcrystallizationintothesystem.Assumingthatmagmaislosingheatpurelybyconduction,whichisjustiedforthinintrusions<10mGibbsandHenderson,1992,andnotingthatgeologicalandgeophysicalobservationsrevealthatthelateralextentofthesillisconsiderablylargerthanitsverticalextent,thecoolingandcrystallizationofthemagmaisdescribedbyKirkpatrick,1981mCp@T @t=m@Q @t+k@2T @z2;.1whereTisthetemperature,kisthethermalconductivity,mthedensityofmagmaandCptheheatcapacityofcrystalsplusmelt.Therateofgenerationoflatentheatofcrystallization,@Q=@t,canbeexpressedas@Q @t=L@ @t;.2where@=@tistherateofchangeofthevolumeofcrystalsortherateofbulkcrystalliza-tionperunitvolumeandListhelatentheatperunitvolumeofcrystalsformed.Notethatascrystallizationaddsheattothesystem,itreducestherateatwhichthetemperaturedecreasesbydiusion.Heattransferthroughthewallrockduetoheatlossfrombothsillcontactsismodeledbythe1DheatconductionequationCarslawandJaegger,1959rCp@T @t=k@2T @z2;.338

PAGE 55

whereriswallrockdensity,andCpandkarethesameasin.1.Meltingofthewallrockhasnotbeenconsideredsincenoevidenceofwallrockpartialmeltinghasbeenfoundintheeld.Forsimplicity,heatlossbyconvectiveporeuidisnotconsideredintheheattransfermodel.3.2.1KineticsofmagmacrystallizationMichautandJaupartandreferencesthereinoeragooddescriptionofthepresentempiricalknowledgeinregardtonucleationandgrowthofcrystalsinsilicatemeltsseetheirTable2.Formeltsofbasalticcomposition,nucleationandgrowthrateshavebeenreportedfromsamplesoflavalakesforolivine,pyroxeneandplagioclaseKirkpatrick,1977;CashmanandMarsh,1988;Mangan,1990.Forexample,growthratesforolivinerangefrom10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(11to10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(10cms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,andnucleationratesrangefrom10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(6to10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(5cm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3s)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1.Theseestimateshavebeenconstrainedeitherinsituorthroughcrystalsizedistributions,andforsmallundercoolingoflavasthathadalreadystartedtocrystallize.Fieldandnumericalstudiesinmacintrusionshavereportedcomparativelymuchfasternucleationandgrowthrateestimatesforlargerundercooling.Forexample,valuesbetween1cm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3s)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1and10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(7cms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1havebeenreportedforpyroxeneandplagioclasenucleationandgrowthrates,respectively.KineticdataforalargernumberofmineralphasesisstillessentialtounderstandthedetailsofcrystallizationMichautandJaupart,2006,particularlyinpetrologicalstudies.However,inthecoolingandcrystallizationmodeldevelopedhere,kineticsishighlysimpliedbyconsideringthecrystallizationkineticsofallthemineralsinasinglefunction.ThekineticsofcrystallizationisincorporatedinthemodelbylumpingtogetherthenucleationandgrowthofcrystalsintoaneectivefunctionMichautandJaupart,2006fTu=CTuexp)]TJ/F21 10.9091 Tf 30.4602 7.3801 Td[(K2 TuTu)]TJ/F15 10.9091 Tf 10.9091 0 Td[(12exp)]TJ/F21 10.9091 Tf 9.6804 7.3801 Td[(K3 Tu;.439

PAGE 56

Figure3.1.EectivecrystallizationfunctionfTuversusundercoolingTu.Notethecrys-tallizationmaximumatlargeundercoolingsandthesmallarrowindicatingthenucleationdelay.SeeTable3.1forparametersused.whereTu=T TListheundercooling,TListheliquidustemperature,K2=10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3,K3=30andCissuchthatmaxf=1seeMichautandJaupart,2006,fordetails.ThereisapeakofcrystallizationatlargevaluesofundercoolingandthenitdecreasesexponentiallytozeroforlowerundercoolingsseeFigure3.1.Therateofgenerationoflatentheat,in.2,maybeequivalentlyexpressedintermsofthemeltfraction,U,as@Q @t=)]TJ/F21 10.9091 Tf 9.6804 7.3801 Td[(@U @t:.5Thecrystallizationrate,@U=@t,maybedenedasafunctionofthekineticcrystalliza-tionfTuin.4as@U @t=)]TJ/F21 10.9091 Tf 8.4849 0 Td[(UfTu k;.640

PAGE 57

wherekisacharacteristictimeforcrystalnucleationandgrowth,andaccordingtoBran-deisandJaupartahasavalueof106sormoreforbasalticmagmas.NotethatdimensionlessfunctionfTuisnormalized,suchthatthemaximumcrystallizationrateis1=kMichautandJaupart,2006.3.2.2GoverningequationsandboundaryconditionsAftersubstituting.5in.1,theequationsgoverningthecoolingandcrystallizationoftheintrusionaregivenbymCp@T @t=k@2T @z2)]TJ/F21 10.9091 Tf 10.9091 0 Td[(mL@U @t;.7with@U @t=)]TJ/F21 10.9091 Tf 8.4849 0 Td[(UfTu kandfTu=CTuexph)]TJ/F22 7.9701 Tf 24.6487 4.4887 Td[(K2 TuTu)]TJ/F19 7.9701 Tf 6.5865 0 Td[(12iexph)]TJ/F22 7.9701 Tf 9.6804 4.4887 Td[(K3 Tui;.8suchthatmaxf=1.TheconductionofheatthroughthesurroundingwallrocksisgovernedbyrCp@T @t=k@2T @z2:.9Thesill/wallrockinterfaceisdenedatz=0.Equation.7issolvedwithinthesilldomain,whichisincrementedwitheachinjectionuntilacertainsillthicknessHisreachedH=4minthisworkbasedongeophysicalobservations,seeChapter2.Equation.9issolvedwithinthewallrockdomain,abovetheuppercontactz<0,andbelowthelowercontactz>ht.Theseequationsaresolvedfortwoemplacementscenarios.Inthesymmetricalsandwich"scenarioinwhichsuccessiveintrusionsareinjectedthroughthe41

PAGE 58

centerofthesillFigure3.2a,spaceismadethroughtheupwarddisplacementoftheuppersillandwallrocklayersanddownwardrelativemovementofthecorrespondinglowerlayers.Thesecondscenarioconsidersnewinjectionsofmagmaemplacedalongtheuppercontact.InthisscenariothesillgrowsbyaccommodatingthedeformationbydownwarddisplacementoftheunderlyinglayersFigure3.3a.Inbothscenarios,thedisplacementisassumedtooccurinstantaneously.Themechanicsofthedeformationassociatedwithemplacementisneglectedinordertoisolatethethermalprocess.MichautandJaupartdemonstratethatthemotionassociatedwithintrusionhasasmalleectonthetemperaturedistribution.Boundaryandinitialconditionsare:xedtemperatureT=Trattheundisturbedwallrockfarfromthecontacts,jzj=1,andthenewlyinjectedmagmaisatliquidustemperature,T=TL,atthebeginningofeverynewinjectionFigures3.2band3.3b.Heattransferbetweeninjectionsoccurspurelybyconduction.Continuityisassumedatthemagma/wallrockandmagma/magmainterfacesatz=0and,z=zicentral-injectionscenarioandz=ziupper-injectionscenario,respectively.Equations.7,.8and.9alongwiththespeciedboundaryandinitialconditionsaresolvedinniteelementsusingtheCOMSOLcommercialprogram.Atthetimeofeverynewinjection,temperatureisresettoTlwithinthisinjectionwhilepreservingthetemperatureelsewherewithinthesill.3.3ResultsThecoolingandcrystallizationmodelisappliedtothesilloftheCarmeloutcropforthetwoscenariosFigures3.2and3.3.Iassumeathicknessforeverydiscreteinjectionof1m,basedontheaveragethicknessmeasuredontheN-Sfeederdike,althoughtheemplacementofthesillcouldhavetakenplacethroughrelativelythinnerinjections.Atotalof4injectionsarerunasthestudiedsillisconstrainedtohaveathicknessof4m.Modelsarerunfortwodierenttimesbetweeninjectionsof600and3600srespectively.ThesetimesarecomparativelymuchshorterthaninjectiontimesconsideredinotherstudiesMichautandJaupart,2006;AnnenandSparks,2002withinthecontextoflarger-volumeandlonger-42

PAGE 59

Figure3.2.aSketchofthecoolingandcrystallizationmodelset-upforacentral-injectionscenario.Notethatthelargestarrowrepresentsmagmainuxandthesmalleronesindicaterelativeupwardanddownwardmotiontoaccommodatethenewinjection.bInitialtemperatureproleandtemperaturedistributionafterthefourthintrusionfortheupperpartofthewallrock/sillsystem.Notethatusehasbeenmadeofthesymmetryofthemodelset-up.livedmagmareservoirs.MichautandJaupartdeneadimensionlessparameter,=i=d,whichcomparesthetimebetweeninjections,i,andthecoolingtimed.For1,magmainjectioncanbetreatedascontinuous,thatis,chillingofthewholenewlyemplacedsingleinjectionisnotallowed;for1,magmahascrystallizedbythetimeanewinjectionoccurs,atypicalsituationinlargevolcanicsystemssuchascalderas.Clearly,assumingavalueforbasalticsillthicknessof1mandwiththeinjectiontimesof600and3600s,1forthesillintheCarmeloutcrop.Thus,thesillemplacementprocessmaybedenedascontinuous.ThevaluesfortherestofmodelparametersarepresentedinTable3.1.Theliquidusandsolidustemperaturesforthebasalticmagma,TlandTs,arecalculatedwiththeMELTScodeGhiorsoandSack,1995usingasinputthemajorelementanalysisofabasaltsampletakenfromPlug2seethepreviouschapteronthegeologyoftheCarmeloutcrop.43

PAGE 60

Figure3.3.aSketchofthecoolingandcrystallizationmodelset-upforanupper-injectionscenario.Note,asinthecentral-injectionscenario,thatthelargestarrowrepresentsmagmainuxandthesmallerarrowsindicaterelativedownwardmotiontoaccommodatethenewlyinjectedmagma.bInitialtemperatureproleandtemperaturedistributionafterthefourthintrusionforthewholewallrock/sillsystem.Thetimebetweentherstandfourthinjectionis2400s.Table3.1.Valuesoftheparametersusedinthecoolingandcrystallizationmodel. ParametersSymbolValues HeatcapacityaJKg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1K)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1Cp1.3102ThermalconductivityaWK)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1m)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1k2.5LatentheatofcrystallizationaJkg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1L4.18105Magmadensitykgm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3m2300Wallrockdensitykgm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1r2500LiquidustemperaturebCTL1204SolidustemperaturebCTs1000WallrockinitialtemperatureCTr50 aValuestakenfromMichautandJaupartbValuescalculatedwiththeMELTScode44

PAGE 61

Figure3.4.Temperatureaandnon-crystallizedfractionbprolesforacentral-injectionscenarioafterfourinjectionsandwithatimebetweeninjectionsof3600s.Notethethick-nessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.3.3.1Central-injectionscenarioTemperatureandnon-crystallizedfractionprolesthroughtheuppercontactofthesillareshowninFigure3.4foracentral-injectionscenariowithatimebetweeninjectionsof3600s.NotethedownwardprogressionofthetemperatureprolethroughtimeFigure3.4a.Bythetimethefourthinjectionhasoccurred,14400saftertherstintrusion,asolidmarginof15cmhasdevelopedatthesilluppercontact.Thismarginisglassy,withcrystalcontentsofnomorethan0.4wt.%Figure3.4band,thus,canbedescribedasachilledmargin.Notetheincreaseincrystallinitythroughthesolidmargintowardsthecenteroftheintrusionpredictedbythemodelandcommonlyobservedindikesandsillsintheelde.g.,BrandeisandJaupart,1987a.Notealsothattemperaturesinthewallrockareincreasedabove,forexample,400Cwithinanextentofwallrockof7cmabovethesill/wallrockinterfaceFigure3.4a.Whenthetotalinjectiontimeisshortenedto600sthepatternoftemperaturetemporalevolutionandcrystallinitydistributionisthesameasinthepreviousrun,however,arelativelythinchilledmarginof6cmisdevelopedFigure3.5.Astheamountofheat45

PAGE 62

Figure3.5.Temperatureaandnon-crystallizedfractionbprolesforacentral-injectionscenarioafterfourinjectionsandwithatimebetweeninjectionsof600s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.transferredtothewallrockisreduced,amorelimitedextentofoverlyingrock4cmissignicantlyheated.Chilledmarginsofsignicantthicknessaredevelopedinthepreviousmodels.However,ifthetimebetweeninjectionsisshortenedto60sFigure3.6,chilledmarginscompara-tivelythin,slightlylessthan2cm,aredeveloped.3.3.2Upper-injectionscenarioIncontrasttothecentral-injectionscenario,inanupperinjectionscenario,thetemper-atureprolespredictedbythemodelsforinjectiontimesof3600and600s,respectively,progressupwardswithtimeFigures3.7aand3.8a.Thisisexplainedbythecontinuousowofheatintothewallrockasnewmagmaisincontactwiththeoverlyingrockaftereverynewinjection.Consequently,agreatervolumeofwallrockisheatedinthisscenario.Forexample,temperatureswithinthewallrockreach400Catdepthsof12Figure3.7aand6cmFigure3.8a,forinjectiontimesof3600and600s,respectively.Itisalsoveryinterestingtonotethat,astheemplacementprocessprogressesandnewmagmainjec-tionsareintrudedattheuppersillcontact,theinitiallycreatedchilledmarginispartially46

PAGE 63

Figure3.6.Temperatureprolesforacentral-injectionscenarioafterfourinjectionsandwithatimebetweeninjectionsof60s.Notethethicknessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.47

PAGE 64

remelted.Forexample,whentheinjectiontimebetweenintrusionsis600s,thechilledmarginrecedesfromaninitialthicknessof3cmto1cmafterthefourthinjectionFigure3.8a.Notealsothatattheearlieststagesaftertherstintrusion,achilledmarginisformedatthelowersillcontact,however,thecontinuousheattransferredfromthebaseofthenewinjection,remeltsthemargincompletely.ThechilledmarginsformedattheuppersillcontactareinbothcasesveryglassyasshowninFigures3.7band3.8b.Modelresultsforsillsthatgrowbycentralinjectionsofthinnersill-likeintrusions,indicatethatthetemperatureatthecontactrisestoabout,T=Tr+Tl=2600C,andremainsatthistemperatureduringtheemplacementprocessFigures3.4aand3.5aandlongafteremplacement,inagreementwithotherunsteadyconductionmodelingeortsCarslawandJaegger,1959;Jaegger,1968;DelaneyandPollard,1982.However,whentheinjectionstakeplacethroughtheuppersillcontact,thisconditionisclearlynotsatisedFigures3.7aand3.8a.Itisveryinterestingtonotethatwhenthetimebetweeninjectionsis600s,afterthefourthinjection,achilledmarginof1cmisdeveloped.Itmaybeanticipatedthatforfastersillemplacements,thechilledmarginwillcompletelyremelt,orthintoapointwhereitcanbeneglected.3.4ImplicationsfortheCarmeloutcropsillGeologicalobservationsoftheCarmeloutcropimposeanimportantconstraintonthesillemplacementprocessandthermalhistory.Thepresenceofdomesandplugsatthesill/wallrockupperinterfaceimpliesthatmagmaattheuppercontacthadtobecapableofowingatthetimeoftheirdevelopment.If,onthecontrary,athickchilledmargin,considerablystierthantheuidmagma,haddeveloped,thegravitationalinstabilitywouldhavelikelybeeninhibited.Thethinnestchilledmarginsareobtainedforacentral-injectionscenariowithinjectiontimesof60sFigure3.6andanupper-injectionscenariowithtimesbetweeninjectionsof600sFigure3.8.ThesescenariosmaybeconsideredasapproximateupperboundsfortheCarmelsillbecause,forsignicantlylongeremplacement48

PAGE 65

Figure3.7.Temperatureaandnon-crystallizedfractionbprolesforanupper-injectionscenarioafterfourinjectionsandwithatimebetweeninjectionsof3600s.Notethethick-nessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole. Figure3.8.Temperatureaandnon-crystallizedfractionbprolesforanupper-injectionscenarioafterfourinjectionsandwithatimebetweeninjectionsof3600s.Notethethick-nessofthechilledmarginredbandafterthefourthintrusiondenedbytheintersectionofthesolidusblackdashedlinewiththeredtemperatureprole.49

PAGE 66

times,ofsay60min.,thedevelopmentofthickchilledmarginswouldinhibittheonsetofthegravitationalinstabilities.Thesefastemplacementtimes,intheorderoffewminutes,shedlightonthesillem-placementprocess;asmagmainuxthroughthedikeisconstrainedtobefastandcontinu-ous,oncethedikeinjectslaterallythroughabeddingplaneoranunconformityKavanaghetal.,2006,seesectiononMechanicsofsillemplacementinChapter2,itisenergeticallymoreecienttoexploitthealreadyopenconduitthanspendenergyinopeninganewone.Thus,Iconsiderasmorelikelythecentral-injectionscenario,inwhichmagmakeepsbeinginjectedthroughtherstintrusion,makingthesillgrowbyrelativelydisplacingthesurroundingwallrocksseeFigure2.15inChapter2.Thethicknessofsingleinjectionsandthetimeelapsedbetweenthemareperhapstheleastconstrainedassumptions.ThethicknessoftheN-Sfeederdikeprovidesanupperboundof1m,buttheemplacementprocesscouldhaveverywelltakenplacethroughcomparativelythinnerinjections.Obviously,ifthiswasthecase,intrusionshouldhaveoc-curredsucientlyfasttoavoidthecompletequenchingofthesingleinjectionsrightafteremplacement,toallowforthedevelopmentoftheinstabilitiesattheuppersillcontact.Thetimebetweeninjections,althoughxedinthismodel,mayhavechangedfromshortertolongerintervals,orvice-versa,withtimeasthesillemplacementprogressed.However,alimiting-casescenarioinwhichfastinjectionsoccuratearlieremplacementstagesandthen,towardstheend,intrusionshappenedattheuppersillcontactthroughlongerinter-valsoftimecanberuledout,astheseinjectionswouldhavecooledrapidly,blockingthedevelopmentofthegravitationalinstability.Theoppositelimiting-casescenarioandotherintermediatescenariosare,inprinciple,allpossible.Despiteallthesubtlecomplexitiesinvolvedduringtheemplacement,coolingandcrys-tallizationofthesillattheCarmeloutcrop,themodelresultsandthegeologicalobserva-tionsseeChapter2suggestthattheemplacementoccurredquickly,inamatteroffewminutes,throughcontinuoussill-likeinjectionsthroughthecentralpartofthesill.InthisworkIhaveassumedthatthesillbecomesgravitationallyunstableoncethe4mintrusion50

PAGE 67

isemplacedandcompletelyuid,withaverythinorabsentupperchilledmargin.Thesubsequentinjectionsthenfedtheinstabilitiesplugsanddomesintheoutcropintheirgrowthtowardsthesurface.Nevertheless,itisalsolikelythatthesillstartedtobecomeunstableduringtheemplacementprocessbeforethesillacquiredtheinferrednalthick-nessof4m.Forsimplicity,Idecoupledtheheattransferfromthehydrodynamicmodel,whichwillbeconsideredinthefollowingchapter.Inregardtotheheatingofthewallrocks,itisinterestingtonotethatresultsforbothscenariospredictverylimitedextentstowhichtheoverlyingwallrockisheated.Thethickestbandofwallrockheatedover400C,forexample,is12cmandisattainedforanupper-injectionscenariowithatimebetweeninjectionsof3600sFigure3.7.Thisresultsuggeststhatthedecreaseinviscosityoftheoverlyingwallrocknecessarytogeneratethegravitationalinstabilityonthetimescalesofminutestofewhoursisprobablynotthermallycontrolled.Analternativemechanismtoexplainthenecessarydecreaseinviscosityofthewallrockisproposedinthenextchapteronhydrodynamicinstability.3.5ConclusionsThecoolingandcrystallizationofathinbasalticsillandtheheatingoftheoverlyingwallrocksareexploredbythedevelopmentofathermalmodelinwhichintrusionoccursbyrepetitivesillinjections.Themodelisconstrainedwithgeologicalandgeophysicalob-servationsoftheCarmeloutcropsillandset-upfortwodierentemplacementscenarios;onewheretheemplacementtakesplacethroughthecentralpartofthesillwithineverypreviousintrusionand,anotheroneinwhichthenewmagmainjectionsareintrudedattheuppercontactofthesill,betweentheoverlyingwallrocksandtheunderlyingpreexistingintrusion.Aftersillemplacementinthecentral-injectionscenario,thickchilledmarginsaredevelopedandthetemperatureproleevolvestowardsthecenteroftheintrusion.Within-jectiontimesof60s,chilledmarginsthinnerthan2cmaredeveloped,whichareconsiderednegligibleforthedevelopmentofthegravitationalinstability.Intheupper-injectionsce-nario,thetemperatureprolesadvanceoutwardstowardstheoverlyingwallrocks,remelt-51

PAGE 68

ingtheinitiallycreatedchilledmargins.Inbothscenariostheextentofoverlyingwallrocksignicantlyheatedbytheintrusionisverylimited,about10cmcentimetersatmost,suggestingthatthedecreaseinviscosityneededtotriggerthegravitationalinstabilityisnotthermallyinduced.ModelingresultsandgeologicalobservationsoftheCarmelsillareconsistentwithasignicantlyfast,ontheorderoffewminutes,incrementalgrowthbysill-likeinjectionsthroughthecenterofthesill.Thedevelopmentofchilledmarginsintheuppersill/wallrockinterfaceconstrainsthetimescalesforgravitationalinstabilitygrowthontheorderoffewofminutesaswell.52

PAGE 69

CHAPTER4GROWTHOFRAYLEIGH-TAYLORINSTABILITIESFROMASILL4.1IntroductionGeologicalobservationsattheCarmeloutcrop,SanRafaeldesert,UtahrevealaseriesofPliocenebasalticplugsanddomes"emergingfromanunderlyingsillseeFigures2.2and2.5inChapter2.Overturnedlayersatthecontactofsomeplugsandinwarddippingstratigraphysurroundingthemsuggestadiapiricorigin.IproposeamodelinwhichtheinterfacebetweenthesillandtheoverlyingwallrockbecomessubjecttoRayleigh-TaylorRTinstability,triggeringtheascentofmagmatowardsthesurfacefromthehorizontalsill.Asomehowcuriouslyconverseshallowerscenario,incomparisonwiththeCarmelout-crop,hasbeendescribedinaremarkableProterozoicoutcropinnorthernAustraliaNeed-ham,1978.Inthisoutcropsanddiapirsemergefromanunconsolidatedsaturatedsanddepositasaconsequenceoftheloadingofabasalticlavaoverburden.Talbotetal.qualitativelyrelatesacharacteristicwavelengthwiththeobservedintra-diapirspacing.TodateIhavenotbeenabletondanyreferencesondiapiricascenttriggeredasRTinstabilitiescreatedontheuppercontactsofsills,whichindicatethattheCarmeloutcropisrelativelyunusualcomparedtoothersillsstudiedworldwide.Diapirismingeneral:amechanismbywhichintrusionsascendbydeformingtheirhostbyviscousorplasticowundertheactionofgravityisanimportantmechanismformagmaascente.g.,Marsh,1982;WeinbergandPodlachikov,1994;MillerandPaterson,1999,especiallyinthemantleandlowercrust.Magmaticdiapirsarelargelyinferredfromlargeigneousintrusionse.g.,theTenPeakPluton,WA;MillerandPaterson,1999.Becauseofthelargescaleoftheseplutons,itisimpossibletoseemorethanafractionoftheplutonicbody,andthus,theprocessesthathavegovernedtheascentarecompletelyinferred.Also,becauseofthescaleofthesebodies,itisimpossibletolinktheplutontothemagmasourceregion.53

PAGE 70

TheuniquenatureoftheCarmeloutcropallowsstudyofthegeometryofthesillandtheigneousplugsabovethissillinthesameoutcropseeFigures2.5,2.13and2.14inChapter2.Thisprovidesanexceptionalopportunitytomapthewholestructure,andtostudytheprocessesgoverningdiapiricascentinthebrittlecrustfromanumericalperspective.Regardingmagmaticbodiesotherthansills,adomedroofwithseveralsynmagmaticdomeshasbeenreportedintheRieserfernerpluton,EasternAlps,byWagneretal..TheyinterpretthisstructureasagravitationalinstabilityandusetheanalyticalsolutionderivedbySelig1965toconstraintheviscositycontrastbetweenmagmaandtheoverlyingwallrock.IconsiderRTinstabilityinamodelset-uptoexplorethepossibilityofasill,suchastheintrusionintheCarmeloutcrop,becominggravitationallyunstable.ThismodelisderivedfollowingavariationoftheChandrasekharformalismintroducedbyConradandMolnarfortheinnitesimalgrowthofaRTinstability.TheprocedureconsistsofintroducingsinusoidaldisturbancesofstressandstrainrateonabasicplaneStokesinertialtermsneglectedviscousow.Thissystemofequationsgovernstheevolutionofsmallperturbationsofthephysicalvariables.Thesystemisthenreducedtoafourthorderhomogeneousdierentialequationbyseekingsolutionswithsinusoidaldependenceonthespatialandtemporalvariables,xandt,oftheformexpikx+qt,wherekisthehorizontalwavenumberandqthegrowthrateoftheperturbations.Thestabilityofthesystemisthenstudiedbyndingthegrowthrateqasafunctionofwavenumberk.Thefastestgrowingmodeisassumedtodominatetheinstabilityandthereforegivesthecharacteristicwavelengthofinstabilities.Themodelisset-upforageologicalscenariointwodimensions.Althoughdiapirsarestronglythree-dimensionalstructures,Chandrasekhardemonstratedthatthelinearstabilityanalysisdescribedinthepreviousparagraph,isstillvalidinthreedimensions.ThisgeologicalscenarioconsistsofamagmaticsillwithacertainthicknessoverlainbysedimentaryoverburdenFigure4.1.ThesillhasNewtonianrheology,andthehost-rockoverburdencanbemultilayeredandistreatedasanon-Newtonianmaterial.The54

PAGE 71

Figure4.1.SketchofamultilayermodelfortheRTinstability,whereiandiarethedensitiesandviscositiesofthewallrocklayersand,sandsarethesilldensityandvis-cosityrespectively.Thesill/wallrockinterfaceislocatedatz=0,thegroundsurfaceatz=hoandthebottomofthesillatz=-hs.Therearep+1layersincludingthesillandatotalmnumberofinnerinterfaces.Notethattheoverburdenlayersaremodeledasanon-NewtonianmaterialandtherheologyofthesillisassumedtobeNewtonian.Theconstitutivemodelisderivedsuchthatalineartemperatureprolethroughtheoverburdeninducesviscosityvariationsintheverticaldimensionseetextfordetails.Notetheden-sitycontrastwithlowersilldensitybetweenthesilllowerandtheoverburdenhigher,requiredtotriggerRTinstabilities.55

PAGE 72

rheologicaldetailsarediscussedinthefollowingsectiondevotedtorheologyoftheoverlyingwallrocks.Viscositiesasafunctionoftemperatureanddepthcanbeimplementedwithintheoverburden,whichisrelevanttomodellong-livedmagmaticreservoirsorintrusionsemplaceddeeperinthemiddleorlowercrust.Here,however,themodelisimplementedtomodelthegrowthofRTinstabilityfromasmallsillemplacedintheupperlevelsofthecrust.4.2RheologyoftheoverlyingwallrocksThedynamicsoftheRTinstabilityislargelycontrolledbytherheologicalpropertiesoftheoverburdenWeijemarsetal.,1993.Anappropriaterheologicalmodelofthewallrockisneededtorealisticallycapturethecomplexmechanicalbehaviorofrocksunderrelevantgeologicalconditions.Forarockofgivenmineralogyandmicrostructure,thevariablesimportantindeterminingstrengtharepressure,temperature,strain,strainhistory,strainrate,poreuidpressure,grainsize,fugacitiesofwaterandothervolatiles,andchemicalactivitiesofthemineralcomponentsEvansandKohlstedt,1995.Mostofourknowledgeregardingmechanismsofrockdeformationcomesfromlaboratoryexperimentsfromwhichconstitutivelawsarederived.Duetothecomplexitymentionedaboveandtheimpos-sibilityofduplicatingtrulygeologicalconditionsinthelaboratory,manyoftherelevantparametersintheseconstitutiveequationsremainpoorlyconstrained.Inaddition,par-ticularlyforcalciterocks,undercertainlaboratoryconditions,diusionanddislocationcreepmechanismsmayoperatesimultaneouslyaddingcomplexitytothequanticationofowlawsforcalciterocksRenneretal.,2002.Acknowledgingallthesecomplications,Iattempttomodelthedeformationofcalciterocksusingsimpliedconstitutivemodels.ForthedevelopmentofbasaltdiapirsintheCarmelformation,whereabasalticsillat1100Cisheatingupanddeformingtheoverlyingsedimentaryrock,twodierentscenariosareexplored;1thewallrockisassumedtodeformbydiusioncreepashightemperatureandlowstressesareprevailingconditions;and2theoverlyingsedimentarysectionisrelativelywetandporous,butwithlowpermeability.Thus,Iassumethatthe56

PAGE 73

suddenriseoftemperaturecausesalargeincreaseinporepressure,aconsequentdecreaseineectivestressandthemicrofracturingorbrecciationofthewallrock.Therheologicalmodelisthatofaperfectplasticmaterialwithaconsiderablyreducedviscosity.ThisscenarioisjustiedbasedongeologicobservationsonrocksamplesatplugcontactsseeFigures2.7and2.12ainChapter2.Theconstitutivemodelsforthesetwoscenariosaredescribedindetailinthefollowingsection.4.2.1WallrockdeformationbydiusioncreepRocksaredeformedbydiusioncreepmechanismsatrelativelyhightemperaturewithrespecttothemeltingtemperatureoftheirconstituentminerals,withsmallgrain-sizesandunderlowstressesRanalli,1995;PasschierandTrouw,2005.Deformationbydiusioncreepoccursbyvacancymigration.Dependingonwherethisvacancydiusiontakesplacewithintherock,twodierenttypesofdiusioncreepmechanismsaredened.Ifgrainboundariesarethesourceandsinkofvacancies,deformationoccursbyCoblecreepRanalli,1995.Ontheotherhand,ifvacancydiusionoccursthroughthelatticeofminerals,deformationisaccommodatedbyNabarro-HerringNHcreep.TheCarmelsillatthetimeofemplacementcreatedasteeptemperaturegradientintheoverlyingwallrock.Temperaturesnearthecontactwereconsiderablyincreasedwithrespecttowallrocktemperaturepriortointrusion.Inaddition,thiswallrocknearthecontactwasstressedbytheonsetoftheRTinstability.Stressesinvolvedduringthedevelopmentoftheinstabilityareintheorderofp)]TJ/F21 10.9091 Tf 10.909 0 Td[(sghs,wherehsistheoverburdenthickness,pandsarethedensitiesofwallrockandmagmarespectively,andgisthegravitationalacceleration.Forexample,fordensitycontrastsintherangeofp)]TJ/F21 10.9091 Tf 10.9091 0 Td[(s100to500kg/m3andthicknessof1000m,stressesbetween1and5MPawillbecreated.Notethatthesestressesaresignicantlylowerthanstressesinvolvedinothertectonicprocessesdeeperinthecrustwheresolidstateowisusuallyinvoked.Undersuchlowstressandhightemperatureconditions,thewallrockislikelytodeformbyadiusioncreepmechanism,asdeformationmapsforcalciterocksreportedbyRutter,Figure6,p.212andEvans57

PAGE 74

andKohlstedt,Figure3,p.158suggest.ItisalsointerestingtonotethatthesedeformationmapsindicatethatCoblecreepoperatesatcomparativelylowertemperaturesthanNHcreep.ThetransitionfromCobletoNHcreepdependsontherateofactivationenergiesforeachcreepmechanism,Ec=Enc,whereEcandEncaretheactivationenergiesforCobleandNHcreeprespectively,andongrainsize.Thus,decreasingthisratioorthegrainsizeraisesthetransitiontemperatureconsiderablyRutter,1976.Theconstitutiveequationfordiusioncreepcanbedescribedbyanequationofgeneralforme.g.,EvansandKohlstedt,1995:=AnD dm;.1wheretheexponentialfactorn=1orn1,disthegrainsize,m=3forCoblecreepandm=2forNHcreep.DiusioncoecientDisgivenbyD=D0exp)]TJ/F21 10.9091 Tf 8.4849 0 Td[(E RT;.2whereD0isafrequencyfactor,EisanactivationenergyandRisthegasconstant.TheconstantAfortheCoblecreepisdescribedasA=24Va RT;.3whereVaistheactivationvolumeandisthegrainboundarywidth,andasA=24Va RT.4fortheNHcreep.IdenethefollowingexpressionderivedbyTurcotteandSchuberttoexplorethetemperaturedependenceoftheviscosityofacalcitewallrockthatisbeingdeformed58

PAGE 75

Table4.1.Parametersusedinrheologicalmodelsdescribedinequations.5and.6. ParametersSymbolsValues ActivationenergyforCobblecreepkJmol)]TJ/F19 7.9701 Tf 6.5866 0 Td[(1Ec166ActivationenergyforNabarro-HerringcreepkJmol)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1Enc250GasconstantJmol)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1K)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1R8.31GrainsizemdvariedActivationvolumecm3Va37Grainboundarywidthnm1Frequencyfactorm2s)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1D05x10)]TJ/F19 7.9701 Tf 6.5865 0 Td[(6 byCoblecreep=RTd3 24VaD0expEc RT;.5andforNHcreep,=RTd2 24VaD0expEnc RT;.6wheretheeectofpressureisneglected.ResultsareshowninFigure4.2forCoblecreepFigure4.2aandNHcreepFigure4.2b,andthreedierentgrainsizes:anesandstone,d=0.1mm,bsiltstone,d=0.005mm;and,cclay,d=510)]TJ/F19 7.9701 Tf 6.5865 0 Td[(5mm.OtherparametersarepresentedinTable4.1valuestakenfromRutter,1976.Aspreviouslydiscussed,thetransitiontemperatureisdependentontheratioofactivationenergiesandgrainsize.AsRutter1976doesnotprovidearangeofvariationforthistemperature,IassumethatforgrainsizesinFigure4.2thistransitiontemperatureliesbetween500and700C.Notethatfortemperaturechangesof600Cviscositycanbedecreased8to15ordersofmagnitude.Likewise,foragiventemperature,variationsingrainsizefromnegrainsandstonetoclayinducechangesinviscosityof8ordersofmagnitude.Notealsothat,fortemperaturesof200to300C,theviscosityofthewallrockisintheorderof1017to1015PasFigure4.2a.59

PAGE 76

Figure4.2.Viscosityasafunctionoftemperatureforsedimentaryrocksofthreedierentgrainsizesfor:aCoblecreepandbNabarro-Herringcreep.SeeTable4.1forparametersusedinequations.5and.6.TheseresultshighlighttherelevanceofheattransferthroughthewallrockabovethesillcontactinthedevelopmentoftheRTinstabilityandindicatethatasignicantwallrockviscositydecreaseisneededtotriggertheinstability.ThisissuewillbeaddressedinthefollowinginthesectionApplicationtotheCarmeloutcrop.4.2.1.1Depth-dependentviscosityDuetothethermalgradientcreatedinthewallrockbythesill,averticalviscositygradientwillbecreatedintheverticaldimension.Adepth,z-dependentviscositycanbeformulatedfollowingtheConradandMolnarapproach.TheconstitutiveequationforCoblecreep,4.1with.2,canbereformulatedas=Anexp)]TJ/F21 10.9091 Tf 8.4849 0 Td[(E RT;.7whereAisdenednowasA=24VaD0 RTd3.Relation.7isderivedfromlaboratoryex-perimentswithcylindricalloadingsymmetriesunderverticalaxialstresses,1,anduni-formradialorcircumferentialstresses,3.Intheseexperiments,nin.7,isgivenby1)]TJ/F21 10.9091 Tf 11.5032 0 Td[(3n,however,amoregeneralformulationoftheconstitutiveequationsisneeded60

PAGE 77

whereallquantitiesareindependentoftheorientationofthecoordinateaxeswithrespecttothematerial.Thus,amoreconvenientisotropicnon-linearformoftheconstitutiverelationsisPollardandFletcher,2005ij=AJn)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1=22sij;J2=1 2sklskl:sij=ij)]TJ/F15 10.9091 Tf 12.1046 7.3801 Td[(1 3kkij:.8HerethescalarJ2isthesecondisotropicinvariantofthedeviatoricstress,andforplaneowszz=)]TJ/F21 10.9091 Tf 8.4849 0 Td[(sxx=xx)]TJ/F21 10.9091 Tf 10.9091 0 Td[(zz=2andsxz=xz.J2isthengivenbyJ2=2xz+1 4xx)]TJ/F21 10.9091 Tf 10.9091 0 Td[(zz2;.9andAisdenedasA=Aexp)]TJ/F21 10.9091 Tf 8.4849 0 Td[(E RT:.10InanalogywithaNewtonianviscousuid,Ideneij=1 2sij;.11andthenusingtherstequationin.8Ind=1 2AJn)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1=22)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1:.12Combiningtheincompressibilitycondition,xx)]TJ/F21 10.9091 Tf 10.3531 0 Td[(zz=0with.11,andassumingapuresheareldgivesxx)]TJ/F21 10.9091 Tf 10.9091 0 Td[(zz=4xxandxz=0:.1361

PAGE 78

Insertingtheserelationsinto4.12yields=1 2A)]TJ/F19 7.9701 Tf 10.8207 0 Td[(1sxx 21)]TJ/F22 7.9701 Tf 6.5865 0 Td[(n=1 2A)]TJ/F19 7.9701 Tf 10.8207 0 Td[(1=nxx)]TJ/F22 7.9701 Tf 6.5865 0 Td[(n=n;.14wheresxx=xx)]TJ/F21 10.9091 Tf 10.9091 0 Td[(zz=2.Heattransferbyconductionthroughsolidsisgovernedbyanon-linearanalyticalso-lutione.g.,CarslawandJaegger,1959,however,tomaketheanalysismathematicallytractablealineartemperaturegradientisassumed.Throughthewallrock:Tz=Tm)]TJ/F21 10.9091 Tf 10.909 0 Td[(z;.15whereTmisthemagmatemperatureandisthethermalgradient.Substitutionofthisexpressioninto.14using.10gives=1 2A)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1=nexpE nRTm)]TJ/F21 10.9091 Tf 10.9091 0 Td[(zxx)]TJ/F22 7.9701 Tf 6.5865 0 Td[(n=n:.16ThisequationcanbeapproximatedbyanexpressionoftheformFletcherandHallet,1983=0expz.17where0=1 2A)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1=nexpE nRTmxx)]TJ/F22 7.9701 Tf 6.5865 0 Td[(n=nand=E nRT2m=1 L:.18Thus,alinearincreaseintemperaturethroughawallrocklayerwillproduceanexponentialdecayofviscositywithcharacteristicdecaylengthL.62

PAGE 79

4.2.2DeformationofamicrofracturedwallrockAdierentmodel,basedongeologicalobservations,maybemoreapplicabletotherheologicalbehaviorofthewallrock.Incertainplaces,thesedimentarywallrockclosetothesillcontactisdenselymicrofracturedorevenmicrobrecciatedseeFigure2.7inChapter2.Aplausiblemechanismtoexplaintheseobservationsinvolvesawet,porousandlowpermeabilitywallrock.Whensubjecttohightemperaturesduetothemagmaticintrusion,theporepressureincreases,reducingtheeectivestressandleadingtofailureofthewallrock.ThismechanismwasproposedbyMcBirneyandlaterpartiallyquantiedbyDelaney.Delaneycalculatedchangesinporepressureinducedbyasuddenmagmaintrusionbutdidnotmodelthebrecciationorfragmentationprocess.EvidenceofbrecciatedwallrockhasalsobeenfoundinbasalticdikesandplugscroppingoutintheShipRockarea,NewMexicoDelaneyandPollard,1981andthesamplesintheirFigures14band14cofbrecciatedMancosshaleshowsaremarkableresemblancewithsamplesfoundintheCarmeloutcropseeFigure2.7cinChapter2.Rockswithporositieshigherthan5%,permeabilitieslowerthan1mdarcyandheated500Kaboveambient,undergopressureincreasesof10MPaforconditionstypicalof1kmwatertabledepthDelaney,1982.Asaresult,thewallrockismicrofractured,andcouldalsobecomeuidizedKokelaar,1982;Thomson,2007.TheseconditionsdescribedabovewereusedbyDelaneyinhiscalculations,however,theymayverywellbeappliedtotheCarmeloutcropwallrockspriortotheinitiationofthemagmaticevent.Atpresent,somerocksarerecrystallizedandshowsecondaryporosityduetohydrothermalactivity.Othersshowevidenceofcontactmetamorphismandrecrystallization.Thus,isextremelydiculttoestimatetheporositiesandpermeabilitiesoftheserocksbeforetheywereintruded.Inthecaseofadenselymicrofracturedwallrock,deformationisaccommodatedbyfrac-tureswithmanydierentorientations,allowingtherocktoowplastically.Ifbrecciationoccurs,therockisassumedtoowinaquasi-staticplasticregime,which,representsalimitingcaseinthetraditionalwaytotreatgranularows,andiscommonlyusedinsoilmechanicsSavage,1998.Forbothsituationstherheologicalmodelisthatofaperfect63

PAGE 80

plasticmaterial,whichdoesnotexhibitworkhardening,andinwhichdynamicfrictionalongmicrofracturesorclastsboundariesdoesnotdependonstrainrate.TheconstitutiveequationofaperfectplasticmaterialobeysthevonMisesequationsPragerandHodge,1951sij=ij I2;.19wheresijisthestresstensor,ijisthestrainratetensor,istheyieldlimitandI2=1=2klklisthesecondinvariantofthestrainrate.Comparingthisconstitutiveequationwiththeoneforapower-lawuidPollardandFletcher,2005sij=BI)]TJ/F22 7.9701 Tf 6.5865 0 Td[(n=n2ij;.20wherenow,I2=klkl1=2,thelimitingcase,n!1,ofapower-lawrheologyreducestotheperfectplasticmodelgivenby.19,with=BConradandMolnar,1997;Ismail-Zadehetal.,2002.Notealsothatwithn=1,4.20reducestotheconstitutiveequationforNewtonianuids.4.3GeneralEquationsIdenetheviscousowproblemin2DwhereasillofthicknesshsisoverlainbyanoverburdenoftotalthicknesshothatmightbecomposedofdierentsubhorizontallayersFigure4.1.IbuilddirectlyontheanalysisdevelopedbyConradandMolnarbasedonlinearstabilitytheory.ThisapproachisacceptableasIaminterestedinthecharacteristicwavelengths,whicharelockedattheearlieststagesoftheinstabilities,thus,onlyinnitesimalamplitudesareconsidered.Therelationshipbetweentheoreticalcharacteristicwavelengthsandintra-plug/domespacingsobservedintheeldisdiscussedintheApplicationtoCarmeloutcropsection4.8.Theproblemisformulatedsuchthatinstabilities,intheirinitialstages,canbetreatedasthegrowthofsmallperturbationsto64

PAGE 81

abasicbackgrounddeformationseeAppendixA.Thisbackgrounddeformation,intheformoftectonicforcesactingalongthex-axis,inducesahorizontalstrain,Exx,andabasicstateofpureshearinthelayeredmedium.Assumingplanestrainandincompressibility,Exx)]TJ/F21 10.9091 Tf 10.909 0 Td[(Ezz=0;Exz=0:.21ThemodelisconsideredgeneralinthesensethattheRTproblemisaspecialcaseofpuresheardeformationinwhichbothExxandEzzaresmall.DependingonthemagnitudeofExxtheperturbationscangrowintofoldingcompressionorboudinageextensionwhenExxislarge,oranRTinstabilitydiapirswhenExxissmall.ToisolatetheRTinstabilityandexplorethedetailsofthephysicsgoverningit,IassumethatExxwassmallatthetimeofintrusionduringthePlioceneattheCarmeloutcrop.ThisisconsistentwiththefactthatthestratigraphyintheColoradoPlateauissubhorizontalandnoevidenceforsignicanttectonicstrainisobserved.First,IdeneabasicoworstaterepresentedbyaplaneStokesowAppendixA.Second,thereisatotalowDrazin,2002governedbythesameequationsandboundaryconditionswithdependentvariablesu,w,p,,ijandij.Then,perturbationquantitiesaredenedas~u=u)]TJ/F21 10.9091 Tf 10.9091 0 Td[(U;~w=w)]TJ/F21 10.9091 Tf 10.909 0 Td[(W;~p=p)]TJ/F21 10.9091 Tf 10.9091 0 Td[(P;~=)]TJ/F21 10.9091 Tf 10.909 0 Td[(R;.22~ij=ij)]TJ/F15 10.9091 Tf 10.9091 0 Td[(ij;~ij=ij)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Eij;wherevariablesdenotedbyatilderepresentperturbationstothebasicowandupper-casevariablesdenotethevariablesthatdescribethebasicowAppendixA.65

PAGE 82

Expandingthestresscomponentsofthetotalow,ij,andretainingtherst-orderterms,andusingequation.16forthedetailsofthederivationsseeConradandMolnar,1997,AppendixA,yieldsthefollowingperturbedconstitutiverelationships~xx=2 n~xx)]TJ/F15 10.9091 Tf 11.8352 0 Td[(~p;~zz=2 n~zz)]TJ/F15 10.9091 Tf 11.8352 0 Td[(~p;~xz=2~xz;.23whichareusedinthefollowingtoformulatethegrowthofperturbationstothebasicow.ThegoverningequationsforaplaneStokesowintheperturbedvariablesareseee.g.,PollardandFletcher,2005:@~xx @x+@~xz @z=0.24@~xz @x+@~zz @z)]TJ/F15 10.9091 Tf 11.9109 0 Td[(~g=0:Notethattheseequations,althoughpresentedforonelayer,equallyapplywithineverylayerofthemodel.Combining.23with.24weobtaintheequationsthatgoverntheperturbationsinequation.22:2 n@xx~u)]TJ/F21 10.9091 Tf 10.9091 0 Td[(@x~p+@zz~u+@xz~w+@z~u+@x~w=0.25@xz~u+@xx~w+2 n@zz~w+2 n@z~w)]TJ/F21 10.9091 Tf 10.909 0 Td[(@z~p)]TJ/F15 10.9091 Tf 11.911 0 Td[(~g=0;66

PAGE 83

wheresubscriptsindicatetheindependentvariableofthepartialderivativesandistheinverseofthecharacteristicviscositydecaylength,describedinequation.18.Assumingnormalmodesolutionsoftheseequationswithadependenceonxsuchthatexpikx;.26wherekisawavenumber,equations.25canbereducedtoseeChandrasekhar,1961;ConradandMolnar,1997,fordetailsofthederivationsD4~w+2D3~w+2)]TJ/F15 10.9091 Tf 10.9091 0 Td[(2k22 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1D2~w)]TJ/F15 10.9091 Tf 10.909 0 Td[(22 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1k2D~w+k2)]TJ/F21 10.9091 Tf 5 -8.8364 Td[(k2+2~w=)]TJ/F21 10.9091 Tf 8.4849 0 Td[(k2g~=;.27whereD=d=dz.Thecontinuityequationrequires@~ @t+u@R @x+w@R @z=0:.28Ifperturbationsareallowedtopropagateintimewithgrowthrateq,onlythroughtheverticaldimensionz,suchthat~w=expqt,equation.28canbesolvedas~=)]TJ/F15 10.9091 Tf 10.7185 0 Td[(~wdR dz:.29Applying.29to.27givesaafourthorderhomogeneousordinarydierentialequationontheverticalvelocitycomponent~w:D4~wi+2D3~wi+2)]TJ/F15 10.9091 Tf 10.9091 0 Td[(2k22 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1D2~wi)]TJ/F15 10.9091 Tf 10.9091 0 Td[(22 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1k2D~wi+k2k2+2)]TJ/F21 10.9091 Tf 10.9091 0 Td[(gD~ q~wi=0:.3067

PAGE 84

Notethatthisequationapplieswithinpwallrocklayers,thusi=1;2;:::;pFigure4.1.Thisequationisvalidfornon-Newtonianrheologieswithaviscositydependentondepthandtemperature.AssumingNewtonianow,n=1,andconstantviscositywithinalayer,=0,equation.30reducesto)]TJ/F21 10.9091 Tf 5 -8.8364 Td[(D2)]TJ/F21 10.9091 Tf 10.9091 0 Td[(k22~ws=0;.31whichmodelsowwithinthesill,asdenotedbysubscriptsFigure4.1.Equations.30and.31havebeenfoundthroughananalysisintwodimensions.StructuresgeneratedbyaRTinstability,however,arestronglythree-dimensional,withnopreferredorientation,andathree-dimensionalanalysiswouldberequiredConradandMolnar,1997.Chandrasekhar,p.431showedthatbycombiningtwohorizontaldimensions,k2=k2x+k2z,equa-tions.30and4.31canbeappliedinthreedimensionswhenn=1in.30.Forthenon-Newtoniancase,ConradandMolnarshowedthatthewavelengthofmaximumgrowthrateoftheinstabilitiesisnotaectedbytheinclusionofn6=1,provingthisanalysisisstillvalidinthreedimensions.4.4BoundaryConditionsInthefollowing,theboundaryconditionsoftheperturbedsystemareconsidered.AgeologicalmodelfortheCarmeloutcropconsistsofaplanarsillwiththicknesshsoverlainbyamultilayeredoverburdenoftotalthicknesshoFigure4.1.Thebottomofthesillislocatedatz=)]TJ/F21 10.9091 Tf 8.4849 0 Td[(hs,theinterfacesill-wallrockisatz=0andthesurfaceoftheoverburdenatz=ho.Therearepwallrocklayers,plusthesilllayer,makingatotalofp+1layers.Atotalofminterfacesbetweenwallrocklayerswithdierentrheologicalpropertiesaredenedatarbitrarydepthsj>0.AsinthebasicowmodelAppendixA,therearetwovelocitycontinuityboundaryconditionsontherigiduppersurface;onefortheverticalvelocity,~w,andtheotherforthehorizontalvelocity,~u.Atintermediateperturbedordeformableinterfacestherearefour68

PAGE 85

boundaryconditions:twocontinuityvelocityconditions,andanothertwocontinuitystressconditions,forthenormalandshearstress.Asatthetopoftheoverburden,attherigidbottomofthesilltherearetwovelocityboundaryconditions.Fortherigiduppersurfacetheboundaryconditionsforthevertical,~w,andhorizontal,~u,velocitiesare~w1=0and~u1=0atz=ho;.32wheresubscript1referstotheuppermostlayer.Thecontinuityequationandtheincom-pressibilityassumptiongiveChandrasekhar,1961;ConradandMolnar,1997r~u=ik~u+D~w=0.33fromwhich~u1=D~w1=0.Atanarbitraryinterfacebetweentwonon-Newtonianwallrocklayers,z=zj,conti-nuityoftheverticalandhorizontalvelocitiesgives~wi=~wi+1andD~wi=D~wi+1atz=zj;.34wheresubscriptsiandi+1refertotheupperandlowerlayers,respectively.Continuityofshearstressacrossaperturbedinterface,torstorder,requiresseeConradandMolnar,1997,AppendixB~i;xz)]TJ/F15 10.9091 Tf 11.4943 0 Td[(~i+1;xz=4@ @xi)]TJ/F15 10.9091 Tf 11.7714 0 Td[(i+1atz=zj;.35where=coskxrepresentsasinusoidalperturbationinthez-coordinateoftheboundarybetweenthetwolayers.Thenotationi;xzdenotesshearstressintheupperlayer.~xzcan69

PAGE 86

bedeterminedfrom.23and.32byseeChandrasekhar,1961,p.432~xz=)]TJ/F15 10.9091 Tf 10.5428 7.3801 Td[( k)]TJ/F21 10.9091 Tf 5 -8.8364 Td[(D2+k2~w:.36Continuityofnormalstress,zz,isperturbedbythedisplacementoftheinterfacebetweentwolayersofdierentdensityseeConradandMolnar,1997,AppendixB~i;zz)]TJ/F15 10.9091 Tf 11.4943 0 Td[(~i+1;zz=i)]TJ/F21 10.9091 Tf 10.909 0 Td[(i+1gatz=zj:.37Ifi>i+1,thisboundaryconditiongeneratesagravitationalinstability.~zzcanbedeterminedfromConradandMolnar,1997~zz=)]TJ/F15 10.9091 Tf 12.3306 7.38 Td[(1 k2D3)]TJ/F21 10.9091 Tf 14.3551 7.3801 Td[( k2D2+4 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1D)]TJ/F21 10.9091 Tf 10.909 0 Td[(~w:.38in.35canbeeliminatedbytakingitstimederivative,whichistheverticalvelocity@=@t=~wi=~wi+1,and4.35becomes)]TJ/F21 10.9091 Tf 8.4849 0 Td[(D3)]TJ/F21 10.9091 Tf 10.9091 0 Td[(D2)]TJ/F26 10.9091 Tf 10.9091 15.3819 Td[(4 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1k2D)]TJ/F21 10.9091 Tf 10.9091 0 Td[(k2i~wi)]TJ/F15 10.9091 Tf 11.7714 0 Td[(i+1~wi+1+i)]TJ/F21 10.9091 Tf 10.9091 0 Td[(i+1gk2~wi=q=0:.39AttheinterfacebetweentheNewtoniansillandthewallrock,z=0,continuityoftheverticalandhorizontalvelocitygives~wp=~wsandD~wp=D~wsatz=0;.40wheresubscriptspandsrefernowtotheupperlayerandunderlyingsillrespectively.Continuityofshearstressacrosstheperturbedsill-wallrockinterface,torstorderrequires~p;xz)]TJ/F15 10.9091 Tf 11.4943 0 Td[(~s;xz=4@ @xp)]TJ/F15 10.9091 Tf 11.7715 0 Td[(satz=0;.4170

PAGE 87

asin.35.Continuityofnormalstress,asin.37,requires~p;zz)]TJ/F15 10.9091 Tf 11.4943 0 Td[(~s;zz=p)]TJ/F21 10.9091 Tf 10.9091 0 Td[(sgatz=0:.42Here,asin.38,~p;zzisgivenby~p;zz=p)]TJ/F15 10.9091 Tf 12.3306 7.3801 Td[(1 k2D3)]TJ/F21 10.9091 Tf 14.3551 7.3801 Td[( k2D2+4 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1D)]TJ/F21 10.9091 Tf 10.909 0 Td[(~wp:.43Ontheotherhand,asthesilllayerisdenedasNewtonian,n=1andviscosityisassumedconstantwithinit,=0,equation.43yields~s;zz=s1 k2D3)]TJ/F15 10.9091 Tf 10.9091 0 Td[(3D~ws:.44Aftereliminating,asbefore,equation.42becomesp)]TJ/F15 10.9091 Tf 12.3306 7.3801 Td[(1 k2D3)]TJ/F21 10.9091 Tf 14.3551 7.3801 Td[( k2D2+4 n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1D)]TJ/F21 10.9091 Tf 10.9091 0 Td[(~wp+s1 k2D3)]TJ/F15 10.9091 Tf 10.9091 0 Td[(3D~ws+p)]TJ/F21 10.9091 Tf 10.9091 0 Td[(sgk2~wp=q=0:.45Atthebottomofthesill,z=hs,wedenetwovelocityboundaryconditions.Continuityofthevertical,~w,andhorizontal,~u,velocitycomponentsrequire~ws=0andD~ws=0atz=hs:.46Equations.30and.31withtheboundaryconditions.32,.34,.35,.39,.40,.41,.45and.46makeupaboundaryvalueproblemfortheeigenvalueqandeigenfunction~w.Asequation.30andequation.31are4thorderdierentialequations,acompletedescriptionofowineachinterfacerequiresfourboundaryconditionswithfourundeterminedcoecients.Thetwoboundaries,theupperrigidsurfaceandthebottomofthesillrequireanotherfourboundaryconditions.Thus,formintermediateinterfacesweseek4m+4unknowns,determinedby4m+4boundaryconditions.71

PAGE 88

4.5NumericalSolutionsFormodelswithtwolayersandoneintermediateinterface,thesolutiontotheboundaryvalueproblemcanbefoundanalyticallyseee.g.,ConradandMolnar;Ismail-Zadehetal..However,formodelswithseverallayersandinterfaceswehavetorelyonnumericalsolutionssincethetaskofndinganalyticalsolutionsbecomestoocumbersome.WendnumericalsolutionstotheboundaryvalueproblemfollowingthemethodbyBassiandBonnin.Westartbyndinggeneralsolutionsto.30and.31oftheform~w=expksz+qt;.47wheres=+i.Theconstantsandaregiveninthefollowing.Forowwithinanon-Newtonianlayer,BassiandBonninsuggestasolutionto.30:Wn=Ancoskzexp0kz+Bnsinkz kexp0kz+Cncoskzexp00kz+Dnsinkz kexp00kz;.48whereWgivesthezdependenceof~wwithinanon-Newtonianlayer,An,Bn,CnandDnareundeterminedintegrationcoecients,and=r a;0=a)]TJ/F22 7.9701 Tf 12.3878 4.2952 Td[(c 2;00=)]TJ/F21 10.9091 Tf 8.4848 0 Td[(a)]TJ/F21 10.9091 Tf 12.4714 7.3801 Td[(c 2r=c2 4+n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1 n21=2;.49a=c2 8+1 n)]TJ/F15 10.9091 Tf 12.1046 7.38 Td[(1 2+1 2c4 16+c2 22 n+1+11=21=2;c= k=1 kL;72

PAGE 89

where=1=L,andparameterLisaparameterrelatedtorheologydenedinequation.18.ForowwithinaNewtonianlayerweuseageneralsolutionto.31oftheformWs=Asexpkz+Bszexpkz+Cs expkz+zDs expkz;.50whereWsgivesthezdependenceof~wwithinthesillandAs,Bs,CsandDsareundeter-minedintegrationcoecientsasin.48.FromsolutionsWnandWswecalculate~u,~xzand~zzfornon-NewtonianwallrocklayersandtheNewtoniansillusingequations.33,.36,.38or.43and.44,andapplythemtotheboundaryconditions.34,.39,.40,.41,.45and.46.TheseboundaryconditionscanbearrangedinalinearequationasMC=R;.51whereMisa4m+4xm+4ordermatrixconsistingofeigenfunctionsevaluatedattheboundaries,Cistheintegrationcoecientsvector,andRisavectorconsistingoftheright-handsidesoftheboundaryconditions.Notethatalltheright-handsidesoftheboundaryconditionsarezeroexceptfor.35,.39,.41and4.45.Theright-handsidesoftheseequationsallowtheinclusionofperturbationsduetobothgravityandhorizontalstrainrate,xx.ToisolatetheRTinstabilitywedenexxtobeverysmallsotheperturbationisuniquelycreatedbygravity.Keepingthisformulationisinterestingbecauseitallowsformodelingoffoldingorboudinageinfuturestudies.Idenetheamplitude,i,ofaperturbationtoeachinterfacei,usingavectorH:Ht=nXi=1i0Vjexpqjt;.52wherei0istheinitialperturbationofeachinterface,Vjisanormalizedvectorconsistingoftherelativeamplitudesofperturbationstoeachlayer,andqjisthecorrespondinggrowthrate;i0,Vjandqjarefunctionsofthewavenumber,k.Thetimederivativeofthe73

PAGE 90

amplitudeistheverticalvelocity,givenbyW=dH dt;.53whereWisthevectorofgenerictermwi=~wt;zi.However,WcanbeexpressedasalinearfunctionofCbyW=QC;.54orW=QM)]TJ/F49 7.9701 Tf 6.5865 0 Td[(1R;.55whereQisanm+1xm+4matrix.Theright-handsidesofboundaryconditions.35,.39,.41and.45arelinearini,ifiisasinusoidalfunctionofx.Thus,RisalinearfunctionofH:R=PH;.56wherePisam+4xm+1matrix.Combining.51,.53,4.55and.56yieldsdH dt=QM)]TJ/F49 7.9701 Tf 6.5865 0 Td[(1PH:.57ThestandardwayofsolvingthissystemconsistsofcomputingtheeigenvaluesofthematrixQM)]TJ/F49 7.9701 Tf 6.5865 0 Td[(1P,whichcorrespondtogrowthratesq,andassociatedeigenvectorsVj,whichdescribetherelativeamplitudesofdeformationateachinterface.Then,thesolutionto4.57isgivenby4.52.Theeigenvalues,orgrowthratesq,arecalculatednumericallyforarangeofwavenumbersk,thus,afunctionalrelationshipbetweencharacteristicgrowthrateandwavenumber,qk,isfound.Thisfunctionusuallyhasamaximumataunique74

PAGE 91

wavenumberk,thismaximumcorrespondingtothelargestgrowthrateq.Therefore,perturbationsassociatedtothiswavenumberwillgrowthemostrapidly.Attheearlieststagesoftheinstability,whentheamplitudesareinnitesimalrecallIamusinglineartheory,andassumingthatperturbationswithallwavenumbersarepresent,theonewiththelargestgrowthratewilldominatetheonsetoftheRTinstability.Thewavelengthassociatedwiththismaximumwavenumberandfastestgrowthrateisthecharacteristicwavelengthoftheinstabilitye.g.,JacksonandTalbot,1994,andisdenedasc==kmax.4.6ModelvericationThereisnoanalyticalsolutionavailableforthemultilayerRTinstabilityproblemwithdierentrheologies.Thederivationoftheanalyticalsolutionoftheboundaryvalueproblemwouldinvolveverylargeboundaryconditionmatrices,and,althoughpossibleinprinciple,theextremelylonganalyticalsolutionwouldbe,practically,ofnouse.Ontheotherhand,forproblemsinvolvingalayeroverahalfspace,ConradandMolnarderivedanalyticalsolutions.Thenumericalsolutionsobtainedinthisstudyarecomparedtotheseanalyticalsolutions.Iconsidermodelsconsistingofasillofnitethicknessoverlainbyamultilayeroverbur-denofacertaintotalthickness.However,tocomparethismodelset-upwiththatofConradandMolnarIapproximatethesillasahalfspace,bysettingitsthicknessverylargeanddeningtheoverburdenasonesinglelayer.Thegrowthrate,q,andwavenumber,k,areadimensionalizedasq0=lq l)]TJ/F21 10.9091 Tf 10.909 0 Td[(ughandk0=kh;.58wherelandlaretheviscosityandthedensityofthelowerhalfspace,respectively,hthethicknessoftheupperlayer,anduthedensityoftheupperlayer.75

PAGE 92

Figure4.3.Numericalsolutionsarecomparedtoanalyticalsolutionsplottingdimensionlesswavenumberversusdimensionlessgrowthrateforalayeroverahalf-spacewith:aequalviscosities,1=2=1019Pasand;blargeviscositycontrast,12,where1=1019and2=1016Pas.Whentheviscositiesinthetwolayersareequal,1=2,theanalyticalsolutionisgivenbyq0=2n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1sinhbexpb)]TJ 10.9091 8.5697 Td[(p n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1sinc)]TJ/F15 10.9091 Tf 10.9091 0 Td[(2sin2c 4bn)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1[sinhb+coshb];.59wherebandcaredenedasb=k0 p nandc=k0p n)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1 p n:.60Theresultsofthenumericalsolutionswiththeanalyticalsolution4.59areplottedinFigure4.3awheretheexponent,n,issetton1.0001totestthenon-Newtonianbehaviorofthemodel.Notethatthenumericalsolutionpredictsveryaccuratelytheanalyticalsolutionsforlayerswithequalviscosities.Iftheviscositycontrastbetweenthewallrockandthesillislarge,1>>2,andn=1,theanalyticalsolutionisgivenbyq0=1 2k0coshk0sinhk0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(k0 k02+cosh2k0:.6176

PAGE 93

Resultsofthenumericalsolutionwiththeanalyticalsolution4.61areplottedinFigure4.3bwhere1>>2andn1.Notethatthetbetweenthetwosolutionsisalsoverygoodwithlargeviscositycontrastsbetweenthetwolayers.4.7ResultsTheresultsofthecoolingandcrystallizationmodelsinthepreviouschapterindicatethattemperaturesriseover300Cwithinaverylimiteddistanceintotheoverlyingwall-rock.AsnotedinsubsectionWallrockdeformationbydiusioncreep,theviscosityofacalciterockdeformedbydiusioncreepfortemperaturesof300Corlower,isintheorderof1015Pasorhigher.IconsidertheseviscositiestoohightoallowthedevelopmentoftheRTinstabilityonthetimescalessuggestedbythesillcoolingmodel.Astemperatureshigherthan300Careonlyattainedwithinafewcentimetersfromthecontact,IruledoutdiusioncreepasthemechanismcontrollingthedeformationofthewallrockduringthedevelopmentoftheRTinstabilityfromashallowsill.Alternatively,Iassumethatthewallrockviscosityisdecreasedbymicrofracturingormicrobrecciation,asdescribedinsectionDeformationofamicrofracturedwallrock,thusitsrheologicalbehavioristhatofaperfectplasticmaterialwithn=1.Iconsideramodelofasillwiththicknessh3overlainbytwolayersofwallrockofthicknessesh1andh2,respectivelyFigure4.4.Thetopofthelayer1isthegroundsurfaceandtheinterfacebetweenthesillandthelayer2istheinterfacethatbecomesunstable.ThesillismodeledasaNewtonianuidwithn=1andbothwallrocklayersaretreatedasperfectplasticmaterialswithn=1.AsheatingoftheoverlyingwallrockisconstrainedtobeverylimitedseeChapter2,aconstantambienttemperatureforthewallrocks,Twr=50K,isassumed.Consequently,viscositywithinwallrocklayersisdenedasconstant,thatis,=0inequation.18.Likewise,temperatureandviscositythroughthesillareassumedconstantforsimplicity.ToexplorewhatparametersaregoverningthedevelopmentoftheRTinstabilityatthesill-wallrockcontact,aseriesofrunsareperformedvaryingsomeparametersandxing77

PAGE 94

Figure4.4.Sketchoftheset-upforamultilayermodelofaRTinstabilityconstrainedwithgeologicalandgeophysicalobservationsoftheCarmeloutcrop.Thenumberoflayersis3andinnerinterfaces2.Thedensityandviscosityofthewallrocklayersareiandirespectively,withi=1,2,andforthesill,sands,withs=3forthesill.Therheologicalexponentnis,n=1fortheNewtoniansilland,n=1fortheperfectlyplasticwallrocklayers.Thethicknessofthesillish3and,h2andh3oftheoverlyingwallrocksrespectively.othersseeTable4.2.Resultsarepresentedfordimensionlessgrowthrate,q0,versusdimensionlesswavenumber,k0.Resultsforvaryingsilldensity,3,andthickness,h3,areshowninFigure4.5.Interestingly,thegrowthrateandthespacingbetweeninstabilitiestheinverseofthewavenumberareinsensitivetodensitycontrastsofupto400kg/m3andchangesinsillthicknessrangingfrom2to6m.Notealsotheoscillationsinthesolution,thisisduetothehyperboliccharacteroftheequationgoverningtheRTinstabilityforperfectplasticmaterialsIsmail-Zadehetal.,2002.Aspreviouslymentioned,itisassumedthatthemodewiththefastestgrowthratewilldominatetheonsetoftheinstability.AschangesinsillthicknessanddensitydonotsignicantlyimpactthedynamicsoftheRTinstabilityforgeologicconditionsrelevanttotheCarmeloutcrop,h3and3areheldconstantinthefollowingsimulations.Byincreasingtheviscosityoflayer2,2,twoordersofmagnitudethegrowthrateoftheinstabilityisalsoincreasedbythesameamount,whereasthewavenumberremainsessentiallyunchangedFigure4.6a.Theimpactofvaryinglayer1thicknessisexploredin78

PAGE 95

Table4.2.ParametersusedinmodelresultsshowninFigures4.5and4.6 ParametersFig.4.5aFig.4.5bFig.4.6aFig.4.6bFig.4.6cFig.4.6d Layer11kg/m3250025002500250025002500h1m800800800varied8008001Pas101910191019101910191019Layer22kg/m3250025002500250025002500h2m4varied44variedvaried2Pas10121012varied101210121012Sill3kg/m3avaried22002200220022002200h3m4varied44443Pas100100100100100100 aThedensityofthebasaltwascalculatedinthelaboratoryusingsamplesfromlithofaciesCSCBseeTable2.1inChapter2withinPlug2. Figure4.5.Dimensionlessgrowthrateq0versusdimensionlesswavenumberk0:avaryingdensityofthesill,3and,bvaryingthicknessofthesill,h3.79

PAGE 96

Figure4.6.Dimensionlessgrowthrateq0versusdimensionlesswavenumberk0:avaryingtheviscosityoflayer2,2,bvaryingthethicknessofoverburden1,h1andcanddvaryingthethicknessoflayer2,h2.80

PAGE 97

Figure4.6b.Forthickerh1overburdens,theinstabilitygrowsmoreslowlyandtheintra-instabilityspacing,orwavelength,becomessmaller.Conversely,notethatbyincreasingoverburdenh2thickness,rightabovethecontact,theinstabilitygrowsfasterandtheintra-instabilityspacingbecomeslargerFigures4.6cand4.6d.Thiseectmaybeexplainedbythecloserpositionoftheinterfacebetweenlayer1and2forthinnerh2;aslayer1'sviscosity,1,iscomparativelylargerthan2,deformationinlayer1mustbeaccommodatedearlierthaninthecasewithathickerlayer2,thusthedevelopmentoftheRTinstabilitytakesalongertimetodevelop.InFigure4.6d,resultsforthinnerh2,rangingfrom0.1to1mareshown.Notethatforthicknessessmallerthan0.5mtheintra-instabilityspacingbecomesextremelysmallfork010000,0.5mandgrowscomparativelymoreslowly.4.8ApplicationtotheCarmeloutcropGeologicalobservationsoftheCarmeloutcrop,suchasoverturnedbedsatthecontactofplugs,andinwarddippingstratigraphysurroundingtheintrusions,suggestthattheseplugsanddomeswereformedasaresultofanRTinstabilitythatgrewfromanunder-lyingfeedingsill.Iapplythe2DRTinstabilitymodelFigure4.4toascenariosimilartothatoftheCarmeloutcroptolinkobservationsoftheintra-plug/domespacingFigure2.5,Chapter2withtheoreticallypredictedcharacteristicwavelengths.Thisobservationsuggeststhatthewalleect"createdbythefeederdikeonthesill,excitedaninitiallyin-nitesimalwavewithacrestorientedN-S.Instabilitiesgrewalongthisridgewithacertaincharacteristicwavelengthevolvingwithtimeintodiapiricstructures.Thelinearpatternjustdescribed,consistingofaninitialridgefromwhichinstabilitiesgrowasdiapiricstruc-tures,hasbeenreportedin3DRTlaboratoryexperimentsRamberg,1981;Talbotetal.,1991andinterpretedascreatedbytheeectoftherigidverticalwallsofthecontainer.Talbotetal.notedthatwhentheexperimentswereperformedincontainerswithawidth-to-thicknessaspectratio>6,lateralboundariesinducedwallsorrollsofngersonlytwoorthreewavelengthsawayfromsuchboundaries.Inotherexperiments,withlargeraspectratios>10theboundaryeectswerenegligibleTalbotetal.,1991.It81

PAGE 98

isinterestingtonotethatintheCarmeloutcropFigure2.5,Chapter2,thesillhasanapproximateaspectratioof5,andtheplugsanddomesoccuratafewmetersothedike,intheorderofoneortwopredictedcharacteristicwavelengthsseediscussionbelow,remarkablysimilartotheresultsdescribedintheexperiments.Geologicalandgeophysicalobservationsprovideinformationaboutcertainmodelpa-rameters,suchasthethicknessoftheoverburden,h1,thesillthickness,h3,andintra-plug/domedistances.Theoverburdenthicknessisapproximatelyh1=800mbasedonstratigraphyandpaleogeographicalcalculationsseeChapter2.Thesillthicknessiscon-strainedfromaN-Sresistivityprolewhichrevealsahighresistivityareaof4mthatIinterpretasasubhorizontalintrusion.Thissillcropsoutinariverbed15mwestofDome1seeFigure2.5inChapter2.Theintra-instabilitydistanceismeasuredatthegroundsurfacebetweenplugsanddomes;thedistancebetweenPlug2andDome1is21mandbe-tweenDome1andDome2is11m.AN-SgroundpenetratingradarGPRprolerevealspartofthesubsurfaceinterface,andanotherdomeemergesfromthecontactatadistanceof4msouthofDome1seeFigure2.14ainChapter2.Itisimportanttonotethatasthesill/wallrockinterfaceisonlypartiallyexposed,thesmallestobservedintra-domespacing,4m,representsanupperboundforthemodeledcharacteristicwavelength.Ihaveshownintheprevioussectionthatthemostimportantparametersgoverningthegrowthandintra-instabilityspacingoftheRTinstabilityareoverburdenthicknesses,h1andh2,andviscosity2.Ash1isrelativelywellconstrainedfromgeologicalobservations,andh2and2areunconstrained,Iruntwodierentmodelsvaryingh2and2.Therestofparametersarexed.Alltheparameters,andthefastestresultantcharacteristicgrowthtimes,calongwiththeircharacteristicwavelengths,c,arelistedinTables4.3and4.4,respectively,foreveryrun.Afterdimensionalizingq0bymultiplyingbytimes1)]TJ/F21 10.9091 Tf 9.0326 0 Td[(3gh=1,thegrowthtimeiscalculatedas=1=q.Similarly,afterdimensionalizingk0bydividingitbyhc,thewavelengthiscalculatedasc=2=k.Figure4.7showstheresultsofversusforthetwomodels.82

PAGE 99

Table4.3.ParametersforthemodelsappliedtotheCarmeloutcrop. ParametersModelvaryingh2Modelvarying2 Layer11kg/m325002500h1m8008001Pas10191019Layer22kg/m325002500h2mvaried0.52Pas7x106variedSill3kg/m322002200h3m443Pas102102 Table4.4.CharacteristicgrowthtimesandcorrespondingcharacteristicwavelengthsforthemodelsappliedtotheCarmeloutcrop. Modelschrscm Modelvaryingh2m0.53.11.411.52.9300.04.2min83.8Modelvarying2Pas1060.311.41073.11.4108311.4 83

PAGE 100

Figure4.7.Growthtimeversuswavelengthfor:avaryingthethicknessoflayer2,h2andbvaryingtheviscosityoflayer2,2.Inthemodelwhereh2isvariedFigure4.7,thethicknessesarechosentobesmallbecauseonlyathinlayerofwallrockisheatedduringintrusion.Therefore,thephreaticmechanismthatmicrofracturethewallrock,discussedpreviously,wouldoperateonalim-itedvolumeofwallrockabovethecontact.However,Ihavenotattemptedtomodelthedetailsofthisprocessbecauseamodeltoincorporatesuchacomplexprocessisbeyondthescopeofthiswork.Alternatively,Ihavearbitrarilyassumedalowerboundfrom0.5to1mandanupperboundof30m,theinferredthicknessoftheerodedoverlyingCarmelformation.Withathicknessofh2of30m,thewavelengthofthefastestgrowingmodeisc=83.8mTable4.4,whichislargerthantheN-Sextensionofthesilland,therefore,notgeologicallyreasonable.TheresultisnotshowninFigure4.7atoeasethevisualizationtheresultsforsmallerh2thicknesses.Byvaryingh2from0.5to1mthecassociatedwiththefastestcrangesfrom1.4to2.9mTable4.4andFigure.7a.Intheory,inanyofthesetwomodels,characteristicwavelengthslongerthan3marenotallowed,consistentwithobservedintra-domespacingupper-boundof4m.InFigure4.7bviscosity,2,isvariedfrom106to108PasandtherestofparametersarexedseeTable4.3.AsnotedinrelationwithFigure4.6a,aoneorderofmagnitudeincreaseoflayer2viscosityproducesaoneorderofmagnitudeincreaseofthecharacteristic84

PAGE 101

growthtimeoftheinstability.Notealsothatchangesinviscositywithinlayer2donotsignicantlyimpactthecharacteristicwavelengthsoftheinstability.ResultsfromthecoolingmodelindicatethatfastinjectiontimesintheorderofminutesarenecessarytopreventthedevelopmentofachilledmarginwhichwouldinhibittheonsetoftheRTinstabilityseeChapteronsillcoolingandcrystallizationattheupperinterface.Thus,thisinjectiontimerepresentsatimereferenceforthegrowthtimeoftheRTinstabilities.ResultsinTable4.4andFigure4.7suggestthat,fortheseparametervalues,characteristicgrowthtimesintherangeofminutesassuggestedbythermalmodelsinChapter3arepossiblewithviscositiesofthesedimentaryoverburdenh2ontheorderof106Pasorlower.Asareference,thisviscosityvalueisintermediatebetweenabasalt,10-102Pasandarhyolite,1010Pas.Notethatlithiedsedimentshaveviscositiesintheorderof1019Pas,however,aspreviouslydiscussed,viscositycanbesignicantlyreducedbybrecciation.Basedontheseresults,IsuggestamodelfortheCarmeloutcropinwhichthesill-wallrockinterfacebecomesgravitationallyunstable,forminganinitialinnitesimalN-Strendingridge,afewmetersawayfromthedike,fromwhichinstabilitieswithacharac-teristicwavelengthsrangingfrom1.5to3mstarttogrow.Then,tosustainthegrowthoftheinstabilitiesacontinuousinuxofmagmaintothesillisneeded.Thismagmainuxislikelynotgoingtobehomogeneousalongthesillentrance,thus,someareaswillbefavoredfordiapiricgrowthcomparedtoothers.Inaddition,spatialchangesinporosityandpermeabilityinthewallrockwouldimpactthethegrowthtimealongtheinterface.Eventually,asthemagmainuxwanes,thediapiricintrusionsstarttofreezetoeventuallystoptheirascenttowardsthesurface.4.9ConclusionsGeologicalstudiesandgeophysicalsurveysoftheCarmeloutcroprevealstructures,suchasplugsanddomes,emergingfromasubhorizontalmagmaticintrusionorsill.IinterpretthesestructuresastheresultofaRayleigh-TaylorRTinstabilityattheupperinterface85

PAGE 102

betweenthesillandtheoverlyingwallrock.A2DmathematicalmodelfortheinnitesimalgrowthofanRTinstabilityisderivedtogaininsightintothephysicsoftheinstabilityundergeologicalconditionsrelevanttotheCarmeloutcrop.Inthismodel,thesillistreatedasaNewtonianlayerandtheoverburdenisdenedasamultilayerperfectplasticmaterial.Theoverburdencanalsobespeciedasamaterialfollowinganon-Newtoniandiusioncreeprheology,whichwillbeusedinfutureworkswithdeeper,largerandlonger-livedsubhorizontalmagmaticreservoirs.ToadaptthemodeltoconditionsrelevanttotheCarmeloutcrop,Idenetheoverbur-denbytwolayersofdierentthicknessesandviscosities.Thewallrocklayerrightabovethesill-wallrockinterfaceiscomparativelythinandhasamuchlowerviscositythantheoverlyinglayer.Thislastassumptionisbuiltontheideathataphreaticmechanism,astheoneproposedbyMcBirneyandDelaney,operatedinthewallrockabovethesillrightaftertheintrusionofmagmaintoacold,wetandporousrock.Thesud-denincreaseintemperatureinducesanexpansionoftheporeuidsreducingtheeectivestressandleadingtofailureofthewallrock.Asaconsequencetheviscosityisdramaticallyreduced.Thishypothesisissupportedbymicrofracturedwallrockobservedintheoutcrop.Asensitivitytestperformedwiththemodelindicatesthatthemostimportantparam-etersgoverningthegrowthoftheRTinstabilityandtheintra-instabilitywavelengtharethethicknessandviscosityoftheoverburdenlayers.Characteristicintra-instabilitywave-lengthspredictedbythemodelareconsistentwithintra-domespacingsmeasuredintheeld.Themodelalsosuggestsviscositiesfortheoverlyingmicrofracturedwallrocklowerthan106Pas.Insummary,thedevelopmentofRTinstabilitiesatthesill-wallrockinterfaceandthesubsequentevolutionoftheseintomagmaticdiapirsrequiresaseriesofconditions:aashallowintrusionbuiltupbyfastsequentialinjectionsofmagma;bhighporosity,lowpermeabilitysedimentarywallrocksand;cacontinuousinuxofmagmaintothesilltosustainthesteadygrowthandascentoftheinstability.Thislatterconditionexplainswhy86

PAGE 103

thesediapiricstructuresarelocatedincloseproximitytothefeederdikeintheCarmeloutcrop.87

PAGE 104

CHAPTER5CERRONEGRO1999ERUPTIONTRIGGERINGBYSTATICSTRESSCHANGES25.1IntroductionRecentobservationsatnumerousvolcanoeshavesuggestedcouplingbetweenvolcanicactivityandincreasedtectonicearthquakese.g.,Nostroetal.,1998;Todaetal.,2002.ThiscouplingmayexistacrosslargespatialandtemporalscalesHilletal.,2002.Trigger-ingoferuptiveactivitybyearthquakes,orviceversa,reliesontheideathatsmallchangesinstaticanddynamicstresses1baror0.1MPainthecrustcanpromptsuchactivitybecausethesesystemsexistatapointnearfailure.Thetriggeringmechanismsarenotcompletelyunderstood,butpossiblemechanismsforeruptiontriggeringinclude;arelax-ationoftheminimumprincipalstress3suchthatmagmaconduitsopentomagmaowNostroetal.,1998;bincreasein3compressingmagmachambersorsimilarreservoirsBautistaetal.,1996;Nostroetal.,1998,andcrectieddiusionofvolatilesinthemagmaandothermagmachamberprocessescausedbypassageofseismicwaves,resultinginhighermagmaticpressuresLindeetal.,1994;Sturtevantetal.,1996.Inaddition,clearevidenceexistsforinitiationofearthquakeswarmsinresponsetomagmaintrusionHill,1977;Todaetal.,2002.The1999eruptionofCerroNegrovolcano,Nicaraguaandregionalseismicity,providearemarkableexampleofhowthreesmallMw5earthquakesincloseproximitytoavolcanocantriggerasmallvolumeeruption0.001km3,DREandregionalaftershocksequencesLaFeminaetal.,2004Figure5.1.LaFeminaetal.demonstratethatconduitowmodelsofthe1999eruptionareconsistentwithlowmagmaticoverpressures,indicatingthattheeruptioncouldhavebeentriggeredbydilationalongtheCerroNegro-LaMulavolcanicalignment.Here,wepresentresultsofstaticstresschangecalculationsthatindicatethethreeearthquakesdecreasedtheminimumhorizontalprincipalstress 2ThecontentofthischapterhavebeenpublishedinDezetal.88

PAGE 105

Figure5.1.LocationmapforCerroNegrovolcanoandnearbyQuaternaryvolcanoesLasPilas,Rota,CerroLaMulaandMomotomboBlacktriangles.EpicenterlocationsblackstarsandfocalmechanismsareshownforMw>5earthquakeson5-7August,1999seeTable5.1.SolidblacklinesareknownQuaternaryfaults.NotefaultsstrikingNEleft-lateral,NWright-lateralandNSdip-slip.100mtopographiccontoursareshownaslightgraylines.Studyareashowninsetblackbox,withinCentralAmericavolcanicArchistoricallyactivevolcanoesshowninblacktriangles.89

PAGE 106

andincreasedtheCoulombfailurestressintheregionpromotingeruptionandtriggeredseismicity,respectively.AsstaticstresschangescalculatedareontheorderofthosefoundtotriggerearthquakeselsewhereZivandRubin,2000,ourobservedcorrelationssuggestthatevensmallmagnitudeevents,incloseproximitytooneanother,maybelinkedbystaticstresschanges.5.21999AftershocksequencesandEruptionofCerroNegroCerroNegrovolcanoisanactivecindercone,mostrecentlyeruptingin1992,1995,and1999Roggensacketal.,1997;McKnightandWilliams,1997;Hilletal.,1998.ThevolcanoislocatedontheQuaternaryvolcanicarcofwesternNicaraguaFigure5.1.ItisthemostrecentvolcanoformedinaeldofbasalticcinderconesandmaarsonthenorthandwestanksoftheElHoyovolcanocomplex.CerroNegroformsthesouthernportionofthenorthtrending,3.5-km-longCerroLaMula-CerroNegroventalignment.VentsformedontheediceofCerroNegroandjustoitssouthankin1968,1995,and1999extendthisoveralltrend.Continuingdevelopmentofthisalignmentisconsistentwiththeinferredorientationofthestresseldinthisarea.InthispartofNicaragua,trenchparalleldextralshearofapproximately8-14mm/yrDeMets,2001;Turneretal.,2003resultsinamaximumhorizontalprincipalstressorientedbetweenN350EandN10E.Thisstressisaccommodatedlargelybyslipalongnortheast-trendingleft-lateralfaultsandbyeast-westextensionalongvolcanicalignments,ratherthandirectlybydextralslipontrenchparallelfaultsLaFeminaetal.,2002.Thistectonicsettingsetsthestageforinteractionbetweencloselyspacedactivevolcanoesandfaults.OnAugust5,1999threeearthquakes,twoMw5.2andoneMw5.1,occurredduringaperiodofthreehours,1-2kmeast-northeastofCerroNegroINETER,1999;Dziewonskietal.,2000.Followingtheseevents,seismicitywasincreasedintheregionforvedaysearthquakesandincludedanMw5.2earthquakeonAugust6andMl4.6onAugust7.Aftershocksweretriggerednorthwestandsoutheastoftheinitialepicentersandclus-teredonalignmentsthatareconsistentwithmappednortheast-trendingleft-lateralfaults.90

PAGE 107

TheaftershocksequencesexhibitapowerlawdecreasewithtimethatmatchesmodelsofseismicswarmbehaviorTodaetal.,2002.CerroNegroerupted11hoursaftertheinitialearthquake,alonga200mlongnorthtrendingfracturesouthofthevolcano'sedice.Firefountainingassociatedwiththisactivityoccurredtoheightsof300mfromtwoscoriaconesthatcoalescedalongtheinitialfracture.Real-timeseismicamplitudemeasurementsRSAMwereatbackgroundlevelspriortothethreeearthquakes.TherewerenoothergeophysicalsignsofprecursoryvolcanicactivityLaFeminaetal.,2004.CerroNegrodoesnottypicallyhavelongperiodsofgeophysicalunrestpriortoeruptiveactivity,however,forlargervolumeandlongerdurationeruptions,suchas1995,eruptiveactivitymayprogressfromphreatictomagmaticactivityoverdaystomonths.Cumulativelythisisthesmallestandshortestdurationeruptiondaysversusameanof18daysforeruptionswithknowndurationstohaveoccurredatCerroNegrosinceitsformationin1850Hilletal.,1998.5.3FaultGeometryandSlipWeuseHarvardCMTfocalmechanismsolutionsandearthquakerelocationsfromtheINETERseismicnetworkINETER,1999;Dziewonskietal.,2000tocalculatestaticstresschangesinducedbythethreeearthquakes.TheformerdataareusedtoderivefaultgeometryandslipparametersWellsandCoppersmith,1994.AllfaultandstresseldparametersrequiredtoestimatethestresschangeassociatedwiththeseearthquakesareprovidedinTable5.1.StresschangesarecalculatedusingmethodsproposedbyChinnery,1963andimplementedinthecomputercodePOLY3DThomas,1993.Inthesemodelstheuppercrustisrepresentedasanelastichalf-spaceandfaultsarerectangulardislocationsurfacesinthishalf-space.Weconsiderchangesin3Figures5.2aand5.2bandCoulombfailurestresschangesassociatedwithslipononlythenortheast-trendingfaultplanes.Weconsiderdilation3perpendiculartotheN-Strendingvolcanoalignment.CoulombfailurestresschangesCFSCarecalculatedforplaneswithanoptimalorientationforfailurexedbytheregionalstresseldKingetal.,1994,andmapthetendencyforN30Efaultstoslip91

PAGE 108

Table5.1.Faultgeometryandslipforthethreeearthquakeson5August,1999Figure5.1 080599C080599E080599F Date8/5/19998/5/19998/5/1999TimeGMT4:35:555:31:527:11:20Latitude12.522N12.512N12.506NLongitude86.695W86.693W86.697WDepthkmtocenteroffaultplane101010Mw5.25.25.1Strike17o44o20oDipwest87o78o75oRake-51o-33o-43oSRLakm4.54.54RAbkm2181814.8ADccm2.22.21.8Dxdcm1.41.81.3Dyecm1.71.21.2 aSurfacerupturelengthbRuptureareacAveragedisplacementdHorizontalcomponentofdisplacementdVerticalcomponentofdisplacement92

PAGE 109

Figure5.2.aMapofcalculatedchangein3resultingfromsliponthreefaultplanesTable5.1withepicentersshownaswhitestars.Dailyseismicityisindicatedfor5-8August.NotestressreductionatCerroNegro,consistentwithdikeinjectionfollowing5Augustseismicity.OthersymbolsasinFigure5.1.CrosssectionFigure5.2bindicatedbyA-A'.bStresschangein3calculatedforaN-SplanethroughCerroNegrovolcanoFigure5.2a,usingfaultgeometryshowninTable5.1.Notestressreductionandstressgradientinthisplane,consistentwithdikeinjection.93

PAGE 110

Figure5.3.TheCFSCwascalculatedforeachofthethreeearthquakes,indicatingthatthersteventcouldhavetriggeredthesecondeventandthattherstandsecondinturncouldhavetriggeredthethird.Theauxiliarynodalfaultplanesi.e.,northwest-trendingdextralfaultsweremodeled,however,theresultswerenotconsistentwithobservations.5.4ResultsModelresultsindicatethatthestaticstresschangeresultingfromthethreeMw5earthquakesreduced3,andincreasedCoulombfailurestressonnortheastorientedfaultplanesintheregion.NearCerroNegrovolcano,3wasreducedbyupto0.08MPaatadepthof7kmFigure5.2a.ThesevaluesarewithintherangeofstresschangesobservedtoprecedeeruptionsinothervolcanicsystemsNostroetal.,1998.Thegreatestreductionin3isestimatednorthofCerroNegro;however,volcanoesinthisregione.g.,CerroLaMulaareinactiveandnotlikelyunderlainbyamagmareservoir.Themodelpredictsthathighstressgradientswilloccurwithinanorth-southorientedplane,locatedbeneaththeCerroLaMula-CerroNegroalignmentFigure5.2b.Partsofthisplaneexperiencestressreductioninresponsetotheseismicity,aidingthetendencyforfracturestodilate.ThedistributionofepicentersassociatedwiththeaftershocksequencesinrelationtoCFSCisconsideredinFigure5.3.WeusedtheKolmogorov-SmirnovtestDavis,2002tocalculatetheprobabilityoftheepicentersoccurringrandomlyversusbeinglocatedinzonesofpositiveCFSCFigure5.4.HypocenterclusterstendtooccurinregionsofpositiveCFSContheorderof0.01-0.04MPa.Specically,aftershocksincludinganMw5.2earthquakeoccurredAugust6nearRotavolcano,approximately5kmNNWofCerroNegrovolcano.ThisaftershocksequenceislocatedwithinazoneofpositiveCFSC.EpicentersnorthofCerroNegroareinazoneofnegativeCFSC.ModelsofCFSCfordikeopeningindicatethattheseearthquakesoccurinregionsofnegativeCFSC.Twodayslater,anaftershocksequenceoccurredatLaPazCentrofollowingaMl4.6earthquake,wheremodeledCFSC<0.001MPaFigure5.4.94

PAGE 111

Figure5.3.CoulombfailurestresschangeCFSCcalculatedusingfaultgeometryandslipdataofthethreeearthquakesTable5.1.NotetheasymmetryoftheCFSCpattern,possiblyduetodip-slipcomponentsalongthefaultplanes.AnexceptionisseismicitynearCerroNegrovolcano,attributedtodikeinjectionanderuption.OthersymbolsasinFigure5.1.95

PAGE 112

Figure5.4.ComparisonoftheCoulombfailurestresschangeCFSCdistributionovertheentireareaofstudyandtheCSFCdistributionwhereepicentersintheaftershocksequencesoccurredAugust5,6,7and8.ThedistributionofepicentersisnotrandomwithrespecttotheCFSCmodelgreaterthan99%condenceKolmogorov-Smirnovtestforcumulativedistributions.Thatis,epicentersoftheaftershocksequenceshaveastatisticallysignicanttendencytooccurinareasofhighCFSC.5.5DiscussionandConclusionsThisstudyreliesontheclosetemporalandspatialassociationofearthquakesandthevolcaniceruptiontoinferstresstriggeringoftheseevents.EruptiveactivityatCerroNegrobefore1999occurredin1995.The1995eruption,likeothersfromCerroNegro,waslargervolumethanthe1999eruptionandnotprecededbytectonicearthquakes.Norecordexistsofaregionaltriggeredmagma-tectonicactivityliketheonethatoccurredAugust5-8,1999,everoccurringinthisareabefore.Inlightofthiscomparativepaucityofpriorcoupledactivity,wedonotregardthecoincidenceofthethreeearthquakes,theregionalaftershocksequencesanderuption,aspossiblyoccurringbyrandomchance.Ourmodelofstaticstresschangeindicatesthatreductionin3ontheorderof0.01-0.1MPaissucienttotriggertheeruption,andsuggeststhatthesestresschangescanaccompanyevensmallmagnitudeearthquakes,iftheyarelocatedsucientlyclosetothemagmareservoir.Stressreduction,upto0.08MPaFigure5.2aand5.2b,isparticularlyeectiveatdepthsof5-10km.Otherpartsofthisplanedepth>10kmexperience96

PAGE 113

compression.CompressionatdepthanddilationclosertothesurfacemaydriveupwardmagmaowNostroetal.,1998.Inconclusion,theremarkablegeophysicaleventsintheareaaboutCerroNegrovolcanoonAugust5-8,1999,clearlyindicatethatanelybalancedstateofstressexistsalongthevolcanicarc.Inthisoneexample,smallchangesinthisstateresultedindramaticeects.97

PAGE 114

CHAPTER6SOLUTIONANDPARAMETRICSENSITIVITYSTUDYOFACOUPLEDCONDUITANDERUPTIONCOLUMNMODEL36.1IntroductionTheowofmagmainavolcanicconduitandtheowofthemixtureofgas,solid,andliquidthroughaplinianeruptioncolumnarehighlycomplexphysicalprocessesgov-ernedbyalargenumberofmechanismsoperatingatdierentspatialandtemporalscales.Nevertheless,undercertainconditions,steady-statehomogeneousuidmechanicalmod-elsprovidevolcanologistswithleadingorderapproximationsofsuchcomplexphenomena.Thesetypesofmodels,forconduitowWilsonetal.,1980;BurestiandCasarosa,1989;Woods,1995;Mastin,1995andforplinianeruptioncolumnsWilsonetal.,1978;Sparks,1986;WilsonandWalker,1987;Woods,1988,1995arewelldevelopedinthevolcanologicalliteratureandhelpvolcanologiststogaininsightintotheimportantparametersgoverningrealvolcaniceruptions.Agoodknowledgeoftheseparametersisessentialnotonlyincharacterizingancienteruptionsbutalsoinimprovingforecastsoffutureeruptions.Thisparameterexplorationisusuallyaccomplishedthroughaparametricsensitivityanalysis,inwhichthesensitivityofthemodelstovariouschangesinparametervaluesisstudied.Aparametricsensitivityanalysiscanalsobeusedtodeterminehowwelloutputparameters,suchaseruptioncolumnheight,canbedeterminedfrominputparameterrangesthatarepoorlyconstrained,andtoinvestigatetheoriginofshapesofparameterdistributionsob-servedinnaturee.g.log-normaldistributionsoferuptioncolumnheights.Thiscanbeaccomplishedbyintroducinginputparametersinconduitanderuptioncolumnmodelsasprobabilitydensityfunctions.Inthischapterthemodellingprocess,fromthenumericalsolutionofthedierentialequationsdescribingthemodelstotheparametricanalysis,isdescribedindetail.Awaytocodetheone-dimensionalconduitanderuptioncolumnmathematicalmodelsisprovided. 3ThecontentsofthischapterhavebeenpublishedinDez00698

PAGE 115

Amethodlinkingconduitowanderuptioncolumnmodelsispresented.Thisisusefulbecauseitbecomespossibletolinkcommonobservationsofmagmaproperties,suchasinitialvolatilecontent,tocommonlyreportederuptionparameters,suchaseruptioncol-umnheight.Todemonstratethevalidityofthesenumericalmodelsasquantitativetoolsforvolcanologists,simulationshavebeenmadeofthemostenergeticsustainedphaseoftheApril1992eruptionofCerroNegro.Aparametricanalysisisperformedtoexploretherelationshipbetweeninputparametersandtheheightoftheeruptingcolumnduringsustainedexplosiveeruptions.Theappendixattheendofthechapterdescribestheequationsgoverningtheconduitowanderuptioncolumnmodelsanddenesthevariablesusedbythemodelsandinthecode.6.2MathematicalModels6.2.1TheConduitFlowModelAmathematicalmodelforconduitowduringsustainedexplosiveeruptionsisderivedfollowingWoodsandMastin1995.Itisassumedthatthemixtureascendsadiabat-icallysincethetimescaleforheatconductionawayfromthewallrocksismuchlongerthanthetimescaleoftheowWoods,1995.Duringexplosiveeruptions,asmagmaascendsalongavolcanicconduit,atsomeheight,volatilesstarttonucleateandformbubblesduetodecompression.Abovethislevelbubblegrowthcausestheowtoexpandresultinginanupwardlyacceleratingmixture.Atacertainleveltheliquidisfragmentedintodiscretepar-ticlesreachingtheventasagas-particleowFigure6.1a.Steady-stateowconditions,constantstressandsteadyshear,areassumedforsimplicity.Homogeneousowisalsoassumedthroughouttheconduit,thatis,multiphasemagmaistreatedasasingle-phaseliquidwiththeaveragepropertiesofthemixture.Bubblesandparticlesdonotappearexplicitlyinthemodel.Alongthebubbly-owregion,relativemovementbetweenbubblesandmeltislimitedbecausethemagmaascentrateduringsustainedexplosiveeruptionsishighenoughtopreventbubblecoalescenceandtwo-phaseowdevelopment.Flowalong99

PAGE 116

Figure6.1.Schematicrepresentationofaavolcanicconduitandbaneruptioncolumn.Figuresarenotdrawntoscale.thegas-particleregioncanbeapproximatedbyhomogeneousowbecausetheparticlesareassumedtobesucientlysmallthatgasandsolidsmovewiththesamespeed.SeeMader,thisvolume,foradetaileddescriptionofthephysicalprocessesoccurringduringmagmaascentthroughtheconduit.Viscosityisallowedtovarywithbubblecontent,alsoknownasthevoidfraction,andthecondition,voidfraction75%,isadoptedasafragmentationcriterionDobran,1992;Mastin,1995;Woods,1995.Thenumericalapproachusedsolvesforspecicpressureandowconditionsincludingmeanowvelocity,volatilemassfraction,Machnumber,anddensityofthemixture,ateachstepalongthevolcanicconduit,usingmassandmomentumconservationequationscombinedwithconstitutiverelationshipsfordensity,viscosity,soundvelocityinthemixture,andafrictionfactorEquationsB.1-B.3andB.4-B.11.Theboundaryconditionsattheventaresubsonicorchoked.Ifthemagma-bubblemixtureisatsubsonicconditions,thatis,themixturevelocityislowerthanthesonicspeedinthemixture,thenconduitpressureisequaltoatmosphericpressure.Otherwise,thevelocityofthemixtureisequaltothesonicspeedinthemixturechokedconditionandtheconduitpressurewillbehigherthanatmosphericpressure.100

PAGE 117

6.2.2EruptionColumnModelTheeruptioncolumnmodelisbasedonamodelforPlinianeruptioncolumnsderivedfromrstprinciplesbyWoods.Plinianeruptionsconsistofasustained,quasi-steadydischargeoffragmentedmagmaandgas.Theydeveloperuptioncolumnsthatinjectashandparticleshighintotheatmosphere.Theeruptioncolumnisdividedintworegions:thegas-thrustandtheconvectiveregionFigure6.1b.Thegas-thrustregionisamomentum-dominatedregioninthelowerpartofthecolumn.Velocitiesarehighandaninnerdensercoresurroundedbyalessdenseouterregiondevelops.Betweentheseregionsshearstressesdevelopandairentrainmentoccurs.Asaresultthebulkdensityoftheeruptioncolumndecreases.Whenthecolumndensityislessthanthedensityofthesurroundingair,thecolumnbecomesbuoyantandaconvectiveregiondevelops.Thecolumncontinuestoriseuntilalevelofneutralbuoyancyisreached,thenitexpandsradiallyveryrapidlyintotheumbrellaregion.DuetothesustainedcharacteroftheseeruptionsthePliniancolumnbehaviourismodelledasasteady-stateprocess.Theow,althoughamultiphasemixtureofgasandparticles,canbetreatedashomogeneousowbecause,astheaverageparticlegrainsizeissmall,thermalequilibriumbetweengasandparticlescanbeassumedSparksandWilson,1976;Woods,1988.Usingmass,momentumandenergyconservationequations,B.19-B.21,aconstitutiverelationshipforbulkdensityEquationB.26,andexpressionsforgasmassfraction,bulkgasconstantatanyheightandbulkheatcapacityofthecolumnmaterialEquationsB.27,B.28andB.29,thevelocity,temperature,radiusanddensitycanbecalculatedalongbothgas-thrustandconvectiveregions.6.2.3TheDecompressionJetModel:CouplingtheConduitandEruptionColumnModelsDuringchokedconditions,thegas-magmamixtureiseruptedintotheatmosphereatpressureshigherthanatmosphericpressure.Asaresult,rapiddecompressionofthismix-turetakesplace.Theradiusoftheascendingmixture,nowcalledaneruptioncolumn,rapidlyincreasestoaradiuslargerthantheventWoodsandBower,1995.Experimental101

PAGE 118

Figure6.2.Sketchshowingthecouplingbetweentheconduitanderuptioncolumnmodelsthroughthedecompressionregion,andtheverticalvariationofpressure,velocity,densityandradiusalongthevolcanicsystemforinputconditionsshowninTable6.1andaninitialvolatilefractionof4wt:%.Notethattheconduitmodeliscoupleddirectlytothedecompressedjet,anditsvelocityandradiusarecalculatedusingequationsderivedbyWoods&Bower,Eqs.B.12-B.18.Thevariationofvelocityandradiusisnotcomputedthroughthedecompressionregion,anditshypotheticalvariationisrepresentedbyadashedline.Seethedecompressionregionzoomedfortheverticalprolesofpressure,densityandradius.Note,despitejumpsinthesevariablesbetweentheventandthedecompressedjet,massisconservedEq.B.16.102

PAGE 119

Figure6.3.Computationalsketchforatheconduitowmodel,andbtheeruptioncolumnmodel.Ineachcase,inputparametersarespeciedatthebaseofthemodelandgradientsofdependentvariablesareusedtoestimatevaluesofdependentvariableshigherintheconduitdp dzcandcolumndq dze,dm dze,de dze.Thresholdsoccurintheconduit<0:75andcolumna=ethatchangethecomputationEqs.B.10,B.19,B.20.Figuresarenotdrawntoscale.investigationsofhigh-speedgasjetsandnumericalmodelssuggestthatthejetwilldecom-presswithinashortdistancefromtheventValentineandWohletz,1989;Dobranetal.,1993;WoodsandBower,1995.Forsimplicity,thedecompressiondistanceisneglected.Theconduitmodeliscoupledtothetopofthedecompressingjet,wherethemixturehasdecompressedtoatmosphericpressureFigure6.2.Massisconserveddespitethejumpsinpressure,velocity,densityandradius,betweentheconduitventandthebaseoftheeruptioncolumnEquationB.16.ThevelocityandradiusofthedecompressedjetarecalculatedfollowingWoodsandBowerEquationsB.12-B.18andusedastheini-tialvelocityandradiusoftheeruptioncolumnmodel.Forsimplicity,thepresenceofthecraterisdisregarded.Ontheotherhand,duringatmosphericconditions,theexitvelocitycalculatedbytheconduitmodel,andtheconduitradiusareusedastheinitialvelocityandradiusfortheeruptioncolumnmodel.TheinputandoutputparametersfortheconduitandcolumnmodelsarespeciedinFigures6.3,6.4,6.5.103

PAGE 120

6.3TheComputationalApproachThecoreofthecomputationalapproachforboththeconduitanderuptioncolumnmodelsisperformedbytheordinarydierentialequationODEsolver.ForadetaileddescriptionofeachofthealgorithmsinvolvedintheODEsolverseePressetal..ThefunctionoftheODEsolveristocarryouttheintegrationofthedierentialequationsthatdescribethemodels.TheODEsolverestimatesthegradientofthedependentvariableatthebeginningoftheintegrationpathandextrapolatesanewvalueataslightlyhigherlevelwithanoptimalstepsize.ThecodeiteratesusingthisnewvaluetocalculatepropertiesatsuccessivelyhigherpositionsuntilboundaryconditionsaresatisedFigure6.3.TheODEsolvercanbesummarizedbythefollowingfourfunctions:aitcalculatesvaluesfordierentowpropertiesandgradientsofthedependentvariables;busingthesevaluesitcalculatesthedependentvariablesviaa5thorderRunge-Kuttaalgorithm;citeratingthesetwofunctions,thedependentvariablesatthenewposition,givenbyanoptimalstepsize,arecalculated;dapossiblestepsizeforthenextiterationisestimated.Thesefourfunctionsworktogethertoupdatephysicalconditionsalongthevolcanicconduitanderuptioncolumn,givenspeciedboundaryconditions.6.3.1TheFlowoftheConduitCodeTheconduitcodeexecutesasfollowsseeFigure6.4.Duringinitialization:ainitialconditionsandparametersarespeciedatthebaseoftheconduitmodel;bdensityiscalculatedandmassowisestimatedEquationB.1;cowpropertiessuchas,viscosity,vesicularity,Reynoldsnumber,frictionfactor,soundvelocity,Machnumberandtheini-tialpressuregradient,arecalculatedusingconstitutiverelationshipsandthemomentumequationsB.2-B.11.Atsuccessivelyhigherconduitpositions,conduitpressure,owproperties,anewpressuregradient,etc.arerecalculated.Thefollowingthreestepsiterateuntilanoptimalstepsizeisfound:aestimatethephysicalpropertiesoftheowandthepressuregradient;busingthesevaluesandtheinitialstepsize,calculatepressure;citeratethesetwostepsuntilthestepsizeisoptimalwithinaspeciedtolerance.In104

PAGE 121

Figure6.4.Flowchartfortheconduitowmodel.TheequationsandnomenclatureusedbytheconduitowmodelcodearepresentedinAppendix.Note,theODEsolveriteratesuptheconduituntilanyoftheboundaryconditionsaresatised.105

PAGE 122

otherwords,inregionswherepressuregradientsarehigh,smallstepsizespreventlossofinformation,whileinregionswherepressuregradientsarelow,longerstepsizesspeedthecomputation.Thestepsizetobeattemptedinthenextiterationisalsocalculated.Atthetopoftheconduit,i.e.,atthevent,boundaryconditionsarechecked.Executionstopswhenboundaryconditionsatthevent,eitherchokedorsubsonic,aresatisedandresultsformassowrateandexitvelocityareoutput.Inchokedconditions,theveloc-ityandradiusofthedecompressedjetabovetheventarecalculatedEquationsB.12-B.18andpassedtothecolumnmodelasinitialcolumnvelocityandradius.Insubsonicconditions,exitvelocityandradiusaredirectlyusedbytheeruptioncolumnmodel.Casesarisewhenneitherboundaryconditioncanbesatised.Thesecasesinclude:athemixtureachievessupersonicconditionsM>1,orbpressuresomewhereintheconduitdropsbelowatmosphericpressurepcatmosphericpressure,initialcolumnradiusandvelocityvaluesareobtainedfromthede-compressionjetmodel.Initialvaluesformass,momentumandenergyuxesarecalculatedfromtheseinitialparameters.Variousowpropertiesliketemperature,density,velocityandradiusarecalculatedusingtheconstitutiverelationshipsEquationsB.26-B.27.PhysicalpropertiesofthesurroundingatmospherearecalculatedEquationsB.22-B.25toapplyadensitycriterionbetweentheeruptingmixtureandtheatmosphere.Usingthe106

PAGE 123

Figure6.5.Flowchartfortheeruptioncolumnmodel.TheequationsandnomenclatureusedbytheeruptioncolumnmodelarepresentedinAppendix.Note,theODEsolveriteratesupwardthroughthecolumnuntiltheboundaryconditionissatised.107

PAGE 124

Figure6.6.Flowcharthighlightingthecouplingbetweentheconduitanderuptioncolumnmodelsthroughadecompressionjetmodel.TheequationsandnomenclatureusedbythedecompressedjetmodelarepresentedinAppendix.Note,dependingontheboundarycon-ditionsatthevent,theinitialconditionsatthebaseoftheeruptioncolumnarepassedfromthedecompressionjetmodelordirectlyfromtheconduitmodel,conditionsaresubsonic.108

PAGE 125

Figure6.7.ComparisonofmodelresultsofWoods,1988withresultsobtainedwiththecodesforconduitowanderuptioncolumn,respectively.aComparisonofconduitowresultsfornc0=0.04,rc=20mandTm=1000K.bComparisonoferuptioncolumnresultsforne0=0.03,re0=100mandTe0=1000K.mass,momentumandenergyconservationequationsB.19-B.21,andthedensitycrite-rionatmosphericdensitybulkdensityinthecolumntodierentiatebetweengas-thrustandconvectiveregions,initialmass,momentumandenergyuxgradientsarecalculated.Atsuccessivelyhighercolumnpositions,thesameoperationsasdescribedforowintheconduitmodelareperformed.Theonlydierenceisthatfortheeruptioncolumnmodel,theODEsolverdealswithmass,momentumandenergyuxesinsteadofpressure.Thesethreeuxes,thegradientsoftheseuxes,thenextstepsizetobeattempted,andowandatmosphericpropertiesarecalculatedateachstepupthecolumn.Thecodeiteratesuntiltheboundarycondition,uecolumnbulkverticalvelocity0:5ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,issatised.Thisvalueforthebulkverticalvelocityinthecolumnischosenarbitrarily.Othervaluescanbechosenasaboundaryconditionaslongastheyremainclosetozero.Resultsforcolumnheight,andradiusasafunctionofcolumnheight,areoutput.Toverifythattheconduitowandtheeruptioncolumncodesareworkingproperly,coderesultswerecomparedwithresultsobtainedbyWoodsfortheconduitandWoodsfortheeruptioncolumn,respectivelyFigure6.7.AlthoughtheagreementbetweenthemodelsisreasonablygoodinbothcasesFigure6.7aand6.7b,itisinterestingtonotethedierenceinthefragmentationdepthbetweentheconduitmodelsFigure6.7a.ThisdierenceisduetothefactthatWoods1995usesdierentrela-tionshipstocalculatemixtureviscosityalongthebubblyowregionseeWoods109

PAGE 126

andWoodsandBowerfromthoseusedintheconduitmodeldescribedaboveEquationsB.9andB.10.6.4ModelValidationUsingthe1992eruptionofCerroNegroVolcanoNumericalmodelsmustbevalidatedusingreal-worldobservations.Inthiscaseitisnotpossibletodirectlyobserveeruptionconditionswithinanactiveconduit.Nevertheless,geologicobservationscanhelpreconstructlikelyeruptionconditionsandthisinformationcanbeusedtoconstrainmodelparameters.Awell-documentedeventisthe1992eruptionofCerroNegrovolcanoinNicaragua.CerroNegrovolcanoisasmallvolumecinderconelocatedintheQuaternaryvolcanicarcofwesternNicaraguaseeConnorandConnor,thisvolume,foralocationmapofCerroNegrovolcano.CerroNegrohashadfrequenteruptionssinceitsformationin1850McKnightandWilliams,1997;Hilletal.,1998.In1992,after21yearsoferuptivequiescencethevolcanoeruptedfor3.6days.ThiseruptionwasparticularlyenergeticforaCerroNegroeruption,generatingsustainedashcolumnsupto7kmhighduringtherstphaseofactivityApril9-12.Followingashortperiodofquiescence,thevolcanoenteredasecond,energetically-lowerphaseApril13-14,withstrombolianeruptionsandashcolumnsabout3.5kmhigh.TheintensitydecreasedthroughoutApril14untilactivityceasedbytheendofthedayGVN,1992.CouplingtheconduitowanderuptioncolumnmodelscansimulatethesustainedactivityoftheApril1992CerroNegroeruptionFigure6.2.Parameterssuchastheinitialvolatilecontent,density,temperatureandconduitlengtharerelativelywellconstrainedRoggensacketal.,1997,basedonthecompositionofbasalt,analysisofvolatilecontentspreservedinmineralstructures,andlimitedseismicdata.Inputandoutputparametersusedforthemodelsfor3runsarelistedinTable6.1.ResultsofowconditionsduringmixtureascentareshowninFigure6.8.Notethattheascentvelocitiesforthemixturefromdeptharerelativelyhigh)]TJ/F15 10.9091 Tf 10.9091 0 Td[(6ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1inagreementwithpetrologicconstraintsfromRoggensacketal..Exitvelocitiesrangefrom110

PAGE 127

Table6.1.Inputandoutputparametersusedandobtainedinthecalculationsfor1992eruptionofCerroNegrovolcano.Note,forthisparticulareruptiontheconduitischoked.Thus,theconduitanderuptioncolumnmodelsarecoupledthroughadecompressionjetmodel.ConduitModelParametersUnits Value inputs Run1Run2Run3ConduitLengthm 600060006000ConduitRadiusm 333InitialPressureMPa 173173173TemperatureofMagmaK 120012001200DensityofMagmakgm)]TJ/F20 5.9776 Tf 5.7561 0 Td[(3 280028002800InitialVolatileFractionwt:% 345 outputs ExitVelocityms)]TJ/F20 5.9776 Tf 5.7562 0 Td[(1 130150165MassFlowRatekgs)]TJ/F20 5.9776 Tf 5.7562 0 Td[(1 1:41052:21053:3105ExitPressureMPa 0.61.22.0VolatileFractionatExitwt:% 2.93.94.9 DecompressionJetModelParameters inputs PressureatVentMPa 0.61.22.0VolatileFractionatVentwt:% 2.93.94.9MassFlowRatekgs)]TJ/F20 5.9776 Tf 5.7562 0 Td[(1 1:41052:01053:3105VelocityatVentms)]TJ/F20 5.9776 Tf 5.7562 0 Td[(1 130150165 outputs Velocityms)]TJ/F20 5.9776 Tf 5.7562 0 Td[(1 239288330Radiusm 579 ColumnModelParameters inputs InitialRadiusm 579VolatileFractionwt:% 345InitialTemperatureK 115011501150InitialVelocityms)]TJ/F20 5.9776 Tf 5.7561 0 Td[(1 239288330VolatileFractionwt:% 2.93.94.9 outputs ColumnHeightkm 6.37.17.9Radiusm 181022502820BulkTemperatureK 249243237 111

PAGE 128

Figure6.8.Variationofthevelocityalongtheconduitforthemodelcalculationsinwhichthemagma'sinitialvolatilemassfractionis0.03,0.04and0.05.Theconduitradiusis3mandthemagmatemperatureis1200K.Exitvelocitiesandmassowratesareconsistentwithobservationsandestimationsofthe1992CerroNegroeruption. Figure6.9.Modelcalculationsoferuptioncolumnradiusvs.heightwheremagma'sinitialgasmassfractionis0.03,0.04and0.05,initialtemperatureis1150Kinthethreesimula-tions,initialcolumnvelocityis239,288and328ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,andinitialcolumnradiusis5,7and9mvelocityandradiusofthedecompressedjet.Theseinitialconditionsproducearangeofheightsinagreementwithobservationsmadeduringthe1992CerroNegroeruption.112

PAGE 129

130)]TJ/F15 10.9091 Tf 10.909 0 Td[(165ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1inagreementwiththoseestimatedbyConnoretal.fromballistics100ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1.Themassowratescalculatedbythemodelrangefrom1)]TJ/F15 10.9091 Tf 10.9091 0 Td[(3:3105kgs)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,oneorderofmagnitudehigherthanthemassowestimatedforentireeruptiveperiodfromeldmeasurements:3104kgs)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1.Unfortunatelynodirectmeasurementsofmassowduringtherstphaseweremade.Thisisreasonabletakingintoaccountthattheactivityduringthe3.6dayeruptiveperiodwasneitherasenergeticnorassustainedasthepeakphasemodelledhere.TheeruptioncolumnmodelresultsareshowninFigure6.9.Notethatthecalculatedcolumnheights.3-7.9kmareinagreementwiththeobservedvalues-7.5kmduringthemostenergeticphaseoftheeruptionGVN,1992.Despitethecomplexityoftheexplosivevolcaniceruptionprocess,themodelsareabletodescribethemostenergeticsustainedphaseofthe1992CerroNegroEruptionreasonablywell.6.5ParametricanalysisModelvalidationresultssuggestthatthemodelsworkwellundercertainconditions.Toexploretherelationshipbetweeninputparametersandtheheightoftheeruptioncolumnduringsustainedexplosiveeruptions,aseriesofnumericalexperimentswereconductedusingtheeruptioncolumncode.ThistopichasreceivedconsiderableattentionWilson,1976;Settle,1978;Wilsonetal.,1978;Sparks,1986;WilsonandWalker,1987;Woods,1988,1995.Whentheinitialgasmassfractionne0isxedandinitialcolumnvelocityue0,columnradiusre0andcolumntemperatureTe0vary,awiderangeoferuptioncolumnheightswascalculatedFigure6.10a.AnalogousresultsareobservedwhenvaryinginitialcolumnradiusandxingtheotherparametersFigure6.10c.Anarrowrangeincalculatedcolumnheightsisobservedkm-19kmwhenvaryinginitialgasmassfractionandholdingallotherparametersconstantFigure6.10b.ThisisexplainedbythefactthatthegasmassfractionintheeruptioncolumnmodeldoesnothaveastronginuenceonbulkcolumndensityEquationsB.26-B.27.Similarresultsareobtainedbysystematicallyvaryinginitialcolumnvelocityandcolumntemperature.Theseresults113

PAGE 130

Figure6.10.Numericalexperimentscarriedoutwiththeeruptioncolumnmodelxingsomeparametersandvaryingothersaslog-normalprobabilitydensityfunctionstoinvestigatevariationsincolumnheight.aFixedparameterne0=4:5wt:%;variableparametersue0range:100)]TJ/F15 10.9091 Tf 10.9091 0 Td[(350ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,re0range:20)]TJ/F15 10.9091 Tf 10.9091 0 Td[(200mandTe0range:1000)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1250K.bFixedparametersue0=220ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1,re0=60mandTe0=1000K;variableparameterne0range:3)]TJ/F15 10.9091 Tf 10.909 0 Td[(7wt:%,cne0,ue0andTe0areheldconstantandre0range:20)]TJ/F15 10.9091 Tf 10.9091 0 Td[(200m,dalltheparametersarexedexceptforQwhichisallowedtovary.114

PAGE 131

Figure6.11.NumericalexperimentsperformedusingtheconduitmodeltoexplorethesensitivityofmassowQwithchangingparametersduringsustainedexplosiveeruptions.AsinFigure6.10,certainparametersareheldconstantandothersarevariedaslog-normalprobabilitydensityfunctions.Thebaseoftheconduitisat3kmdepth.aVariableparameterne0range:3)]TJ/F15 10.9091 Tf 10.909 0 Td[(7wt:%;xedparameterspc0=77MPa,c0=2250kgm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3,Tm=1000Kandrc=50m,ballparametersareheldconstant,nc0=4:5wt:%andrcrange:20)]TJ/F15 10.9091 Tf 10.9091 0 Td[(100m.indicatethattheheightoftheeruptioncolumniscomparativelyinsensitivetochangesininitialgasmassfraction,columnvelocity,andcolumntemperatureatthebaseoftheeruptingcolumn.Themostimportantparametergoverningeruptioncolumnheightisinitialcolumnradius.Furtherinvestigationscomparetheinuenceofmassowqoncolumnheight.ByvaryingmassowandxingallotherparametersFigure6.10d,therangeincolumnheightsvariesbetween7kmand21km,indicatingastrongdependenceofcolumnheightonmassow.ThehighsensitivityoferuptioncolumnheighttochangesinmassowsuggestthatvariationsincolumnheightcouldbeusedtomonitorchangesindischargerateduringthecourseofaneruptionSparksetal.,1997.Similarly,theeruptioncolumnheightsofpasteruptionscouldbeestimatedfrommassowratescalculatedfromwellpreserveddeposits.Itisinterestingtonotethatlog-normaldistributionsincolumnheightareobtained,asinnature,fromlog-normaldistributionsofinputparametersSimkinandSiebert,1994.115

PAGE 132

TheconduitowcodewasrunundervaryingconditionstoexploretheparametersgoverningmassowrateQ.Themostinterestingresultsareobservedvaryinginitialvolatilemassfractionnc0andconduitradiusrcFigure6.11Massowratevariationofgreaterthanoneorderofmagnitudeareobtained.Itisimportanttonotethat,asmentionedabove,volatilemassfractioninthecolumnabovetheventdoesnothaveastronginuenceoncolumnheight,butdoeshaveacontrollinginuenceonmassowatthevent.Theseresultssuggestthatchangesinmassowand,ultimately,changesinthedynamicsofsustainederuptionsaremainlygovernedbychangesininitialvolatilecontentandconduitradiusduringsustainedexplosiveeruptionsWilsonetal.,1980;Macedonioetal.,1994.6.6ConclusionsNumericalsimulationsofvolcanicphenomenacannotbecompletewithouteachofthefollowingsteps:adevelopmentofthephysical-mathematicalmodele.g.Woods,1988,1995;Dobran,1992;Mastin,1995,babstractionofthephysicstofacilitatenumericalsolutions,cvalidationofthemodelasthoroughlyaspossibleusinggeologicobservations,anddparametricstudiesofthemagmapropertiesandrelatedphysicalconditionsthatgiverisetothephenomenaweobserve.Inthischapter,codesimplementingmathematicalmodelsofconduitanderuptioncolumnowhavebeendescribedindetail.Amethodhasbeenpresentedcouplingthetwomodels.Withchokedconditionsatthevent,themodelsarelinkedthroughadecompressionjetmodel,passingthevelocityandradiusofthisjettotheeruptioncolumnmodelasinitialconditions.Withsubsonicconditionsatthevent,themodelsarelinkedpassingconduitradiusandtheexitvelocityasinitialconditionstotheeruptioncolumnmodel.Theconduit-columncoupledmodelhasbeenvalidatedusingparameterestimatesfromthersteruptivephaseoftheApril1992eruptionofCerroNegrovolcano.Resultsofthemodellingintermsofascentvelocities,exitvelocities,massowratesandcolumnheightsshowgoodagreementwithpetrologicconstraintsanddirectobservationsoftheeruption.116

PAGE 133

Thisisofgreatinterestforvolcanichazardstudiessinceitbecomespossibletolinkcommonobservationsofmagmaproperties,suchasinitialvolatilecontent,tocommonlyreportederuptionparameters,suchaseruptioncolumnheight.Aparametricanalysiswasperformedthroughnumericalexperimentscarriedoutwiththeconduitanderuptioncolumnmodels.Resultsfromthenumericalexperimentsper-formedwiththeeruptioncolumnmodelshowthatcolumnheightchangedbyapproxi-matelyafactoroftwowithchangeintheinitialgasfractionattheventne0,initialvelocityue0,andtemperatureTe0ofthecolumn,andbyafactorofthreeormorewithchangeinmassowq0atthevent.ResultsfromnumericalexperimentsperformedusingtheconduitmodelshowthatvariationsinthemassowrateQattheventaregovernedbychangesinmagma'sinitialvolatilefractionnc0andconduitradiusrc.Essentially,theparametricanalysishasshownthateruptioncolumnheightduringsustainedexplosiveeruptionsismainlydeterminedbythemagma'sinitialvolatilefractionandconduitdi-mensions,whichprovidetheultimatecontrolsonmassow.Itisalsoshownthatvaryingcertaininputparametersaslog-normalprobabilitydensityfunctionsproduceslog-normaloutputdistributionsforcolumnheights,asobservedinnature.117

PAGE 134

REFERENCESAlmond,D.C..IgnimbriteintheSabalokacauldron,sudan.GeologicalMagazine,108:159{176.Annen,C.andSparks,R.J.S.2002.Eectsofrepetitiveemplacementofbasalticintrusionsonthermalevolutionandmeltgenerationinthecrust.EarthandPlanetaryScienceLetters,203:937{955.Appeldorn,C.R.andWright,H.E.,J..VolcanicstructuresintheChuskaMoun-tains,NavajoReservation,Arizona-NewMexico.GeologicalSocietyofAmericaBulletin,68:445{467.Bassi,G.andBonnin,J..Rheologicalmodellinganddeformationinstabilityoflithosphereunderextension.GeophysicalJournal,93:485{504.Bautista,B.C.,Bautista,M.L.P.,Stein,R.S.,Barcelona,E.S.,Punongbayan,R.S.,Laguerta,E.P.,Rasdas,A.R.,Ambubuyog,G.,andAmin,E.Q..RelationshipofregionalandlocalstructurestoMountPinatuboactivity.InNewhall,C.G.andPunongbayan,R.S.,editors,The1991-1992EruptionsofMountPinatubo,Philippines,pages351{370.Univ.ofWash.Press.Bergantz,G.W..Changingparadigmsandtechniquesfortheevaluationofmag-maticprocesses.JournalofGeophysicalResearch,100:17603{17613.Bergantz,G.W.andDawes,R..Aspectsofmagmagenerationandascentincontinentallithosphere.InRyan,M.P.,editor,MagmaticSystems,pages219{317.AcademicPress.Bradley,J..Intrusionofmajordoleritesills.TransactionsoftheRoyalSocietyofNewZealand,3:27{55.Brandeis,B.andJaupart,C.a.Crystalsizesinintrusionsofdierentdimensions:constraintsonthecoolingregimeandthecrystallizationkinetics.InMysen,B.O.,editor,MagmaticProcesses:PhysicochemicalPrinciples,volume1,pages307{318.GeochemicalSociety,SpecialPublications.Brandeis,G.andJaupart,C.b.Thekineticsofnucleationandcrystalgrowthandscalinglawsformagmaticcrystallization.ContributionstoMineralogyandPetrology,96:24{34.Brown,E.H.andMcClelland,W.C..PlutonemplacementbysheetingandverticalballooninginpartofthesoutheastCoastPlutonicComplex,BritishColumbia.GSABulletin,112:708{719.118

PAGE 135

Buresti,G.andCasarosa,C..One-dimensionaladiabaticowequilibriumgasparti-clemixturesinlongverticalductswithfriction.JournalofFluidMechanics,203:251{272.Carslaw,H.S.andJaegger,J.C..ConductionofHeat.Oxford:ClarendonPress.Cashman,K.V.andMarsh,B.D..Crystalsizedistributioninrocksandthekineticsanddynamicsofcrystallization.II.MakaopuhiLavaLake.ContributionstoMineralogyandPetrology,99:401{405.Chandrasekhar,S..HydrodynamicandHydromagneticStability.OxfordUniversityPress,Oxford.Chapman,M.andRhodes,J.M..CompositelayeringintheIsleauHautIgneousComplex,Maine:evidenceforperiodicinvasionofamacmagmaintoanevolvingmagmareservoir.JournalofVolcanologyandGeothermalResearch,51:41{60.Chinnery,M.A..Thedeformationofthegroundaroundsurfacefaults.BulletinoftheSeismologicalSocietyofAmerica,50:355{372.Chinnery,M.A..Thestresschangesthataccompanystrike-slipfaulting.BulletinoftheSeismologicalSocietyofAmerica,53:921{931.Connor,C.B.,Lichtner,P.C.,Conway,F.M.,Hill,B.E.,Ovsyannikov,A.A.,Fed-erchenko,I.,Doubik,Y.,Shapar,V.N.,andTaran,Y.A..Coolingofanigneousdike20yrafterinntrusion.Geology,25:711{714.Connor,C.B.,Powell,L.,Strauch,W.,Navarro,M.,Urbina,O.,andRose,W..The1992eruptionofCerroNegro,Nicaragua:andexampleofPlinian-styleactivityatasmallbasalticcindercone.EOSTransactions,AmericanGeophysicalUnion,74:640.Conrad,C.P.andMolnar,P..ThegrowthofRayleigh-Taylor-typeinstabilitiesinthelithosphereforvariousrheologicalanddensitystructures.GeophysicalJournalInternational,129:95{112.Davis,J.C.2002.StatisticsandDataAnalysisinGeology.JohnWiley,Hoboken,NJ,3rdedition.Delaney,P.T..Rapidintrusionofmagmaintowetrock:groundwaterowduetoporepressureincreases.JournalofGeophysicalResearch,87:7739{7756.Delaney,P.T.andGartner,A.E..Physicalprocessesofshallowmacdikeem-placementneartheSanRafaelSwell,Utah.GSABulletin,109:1177{1192.Delaney,P.T.andPollard,D.D..Deformationofhostrocksandowofmagmaduringgrowthofminettedikesandbreccia-bearingintrusionsnearShipRock,NewMexico.USGeologicalSurveyProfessionalPaper,1202:1{61.Delaney,P.T.andPollard,D.D..Solidicationofbasalticmagmaduringowinadike.AmericanJournalofScience,282:856{887.119

PAGE 136

DeMets,C..AnewestimateofpresentdayCocos-Caribbeanplatemotion:Impli-cationsforslipalongtheCentralAmericanvolcanicarc.GeophysicalResearchLetters,21:4043{4046.Dez,M..Solutionandparametricsensitivitystudyofacoupledconduitanderup-tioncolumnmodel.InMader,H.M.,Coles,S.G.,Connor,C.,andConnor,L.J.,editors,StatisticsinVolcanology,number1,pages185{200.GeologicalSocietyofLondon.Dez,M.,LaFemina,P.C.,Connor,C.B.,Strauch,W.,andTenorio,V..Evidenceforstaticstresschangestriggeringthe1999eruptionofCerroNegroVol-cano,Nicaraguaandregionalaftershocksequences.GeophysicalResearchLetters,32:doi:10.1029/2004GL021788.Dobran,F..Non-equilibriumowinvolcanicconduitsandapplicationstotheeruptionsofmt.st.helensonmay1980,andvesuviusinad79.JournalofVolcanologyandGeothermalResearch,49:285{311.Dobran,F.,Neri,A.,andMacedonio,G..Numericalsimulationsofcollapsingvolcaniccolumns.JournalofGeophysicalResearch,98:4231{4259.Doelling,H.H..InterimgeologicmapoftheeasternhalfoftheSalina30'x60'quadrangle,Emery,Sevier,andWayneCounties,Utah.UtahGeologicalSurveyOpen-FileReport,438:1{12.Drazin,P.G..IntroductiontoHydrodynamicStability.Cambridgetextsinappliedmathematics.CambridgeUniversityPress,Cambridge,2ndedition.Dziewonski,A.M.,Ekstrom,G.,andMaaternovskaya,N.N.2000.Centroid-momenttensorsolutionsforJuly-September,1999.PhysicsoftheEarthandPlanetaryInteriors,119:311{319.Ekren,E.B.andByers,F.M..Ash-owssureventinthewest-centralNevada.Geology,4:247{251.Evans,B.andKohlstedt,D.L..Rockrheology.InAhrens,T.J.,editor,Handbookofphysicalpropertiesofrocks,volume3,pages148{165.Washington,DC:AmericanGeophysicalUnion.Fletcher,R.C.andHallet,B..Unstableextensionofthelithosphere:AmechanicalmodelforBasin-and-Rangestructure.JournalofGeophysicalResearch,88:7457{7466.Francis,E.H..Magmaandsediment:1.EmplacementmechanismoflateCarbonif-eroustholeiitesillsinnorthernBritain.GeologicalSocietyofLondon,139:1{20.Ghiorso,M.S.andSack,R.O..Chemicalmasstransferinmagmaticprocesses.iv.arevisedandinternallyconsistentthermodynamicmodelfortheinterpolationandextrapolationofliquid-solidequilibriainmagmaticsystemsatelevatedtemperaturesandpressures.ContributionstoMineralogyandPetrology,119:197{212.120

PAGE 137

Gibbs,F.G.F.andHenderson,C.M.B..Convectionandcrystalsettinginsills.ContributionstoMineralogyandPetrology,109:538{545.Gilluly,J..AnalcitediabaseandrelatedalkalinesienitefromUtah.AmericanJournalofScience,14:199{211.Gilluly,J..GeologyandoilandgasprospectsofpartoftheSanRafaelSwellandsomeadjacentareasineasternUtah.USGeologicalSurveyBulletin,806-C:69{130.Gunn,B.N..DierentiationinFerrardolerites,Antarctica.NewZealandJournalofGeologyandGeophysics,5:820{863.GVN.ReportontheAprileruptionofCerroNegrovolcano,Nicaragua.GlobalVolcanismNetworkBulletin,9:4.Hill,B.E.,Connor,C.B.,Jarzemba,M.S.,LaFemina,P.C.,Navarro,M.,andStrauch,W..1995eruptionsofCerroNegrovolcano,Nicaragua,andriskassessmentforfutureeruptions.GeologicalSocietyofAmericaBulletin,110:1231{1241.Hill,D.P..Amodelforearthquakeswarms.JournalofGeophysicalResearch,82:1347{1352.Hill,D.P.,Pollitz,F.,andNewhall,C.G.2002.Earthquake-volcanotectonicinteraction.PhysicsToday,55:41{47.Hooten,J.A.andOrt,M.H..Peperiteasarecordofearly-stagephreatomagmaticfragmentationprocesses:anexamplefromtheHopiButtesvolcaniceld,NavajoNation,Arizona,USA.JournalofVolcanologyandGeothermalResearch,114:95{106.Hort,M..Coolingandcrystallizationinsheet-likemagmabodiesrevisited.JournalofVolcanologyandGeothermalResearch,76:297{317.Hunt,C.B..IgneousgeologyofandstructureoftheMountTaylorvolcaniceld,NewMexico.USGeologicalSurveyProfessionalPaper,181:51{80.Huppert,H.E.andSparks,R.S.J..Chilledmarginsinigneousrocks.EarthandPlanetaryScienceLetters,92:397{405.INETER.SismosyvolcanesdeNicaragua.BoletnSismologicoMensual,Aug.Ismail-Zadeh,A.T.,Huppert,H.E.,andLister,J.R..Gravitationalandbucklinginstabilitiesofarheologicallylayeredstructure:implicationsofsaltdiapirism.Geophys-icalJournalInternational,148:288{302.Jackson,M.P.A.andTalbot,C.J..Advancesinsalttectonics.InHancock,P.L.,editor,ContinentalDeformation,pages159{179.PergamonPress.Jaegger,J.C..Coolingandsolidicationofigneousrocks.InHess,H.H.andPolder-vaart,A.,editors,Basalts:ThePoldervaartTreatiseonRocksofBasalticComposition,pages503{536.Interscience.121

PAGE 138

Johnson,D.W..VolcanicnecksoftheMountTaylorregion,NewMexico.GeologicalSocietyofAmericaBulletin,18:303{324.Kano,K.,Matsuura,H.,andYamauchi,S..Miocenerhyoliticweldedtuinllingafunnel-shapederuptionconduitShiotani,southeastofMatsue,SwJapan.BulletinofVolcanology,59:125{135.Kavanagh,J.L.,Menand,T.,andSparks,R.S.J..Anexperimentalinvestigationofsillformationandpropagationinlayeredelasticmedia.EarthandPlanetaryScienceLetters,245:799{813.Keating,G.N.,Valentine,G.A.,Krier,D.J.,andPerry,F.V..Shallowplumbingsystemsforsmall-volumebasalticvolcanoes.BulletinofVolcanology,pagesDOI10.1007/s00445{007{0154{1.King,G.C.P.,Stein,R.S.,andLin,J..Staticstresschangesandthetriggeringofearthquakes.BulletinoftheSeismologicalSocietyofAmerica,84:935{953.Kirkpatrick,R.J..Nucleationandgrowthofplagioclase,MakaopuhiLavaLakes,KilaueaVolcano,Hawaii.GeologicalSocietyofAmericaBulletin,89:799{800.Kirkpatrick,R.J..Kineticsofcrystallizationofigneousrocks.ReviewsinMineralogyandGeochemistry,250:38{52.Kokelaar,B.P..Fluidizationofwetsedimentsduringtheemplacementandcoolingofvariousigneousbodies.JournaloftheGeologicalSocietyofLondon,139:21{33.Koronovsky,N.V..ThestructureofthefeedingchannelsoftheignimbriteandtulavacomplexesoftheNorhternCaucasus.BulletinofVolcanology,34:639{647.LaFemina,P.C.,Connor,C.B.,Hill,B.H.,Strauch,W.,andSaballos,J.A..Magma-tectonicinteractionsinNicaragua:The1999seismicswarmanderuptionofCerroNegrovolcano.JournalofVolcanologyandGeothermalResearch,137:187{199.LaFemina,P.C.,Dixon,T.H.,andStrauch,W..Book-shelffaultinginNicaragua.Geology,30:751{754.Linde,A.T.,Sacks,I.S.,andJohnston,M.J.S..Increasedpressurefromrisingbubblesasamechanismforremotelytriggeredseismicity.Nature,371:408{410.Lister,J.R.andKerr,R.C..Fluid-Mechanicalmodelsofcrackpropagationandtheirapplicationtomagmatransportindykes.JournalofGeophysicalResearch,96:10049{10077.Lorenz,V.,Zimanowski,B.,andBuettner,R..Ontheoriginofdeep-seatedsubter-raneanpeperite-likemagma-sedimentmixtures.JournalofVolcanologyandGeothermalResearch,114:107{118.Macedonio,G.,Dobran,F.,andNeri,A..Erosionprocessesinvolcanicconduitsandapplicationtothead79eruptionofVesuvius.EarthandPlanetaryScienceLetters,121:137{152.122

PAGE 139

Macas,J.L.,Sheridan,M.F.,andEspndola,J.M..Reappraisalofthe1982eruptionsofElChichonVolcano,Chiapas,Mexico:newdatafromproximaldeposits.BulletinofVolcanology,58:459{471.Mangan,M.T..Crystalsizedistributionsystematicsandthedeterminationofmagmastoragetimes:the1959eruptionofKilaueaVolcano,Hawaii.JournalofVol-canologyandgeothermalResearch,44:295{302.Marsh,B.D..Onthemechanicsofigneousdiapirism,stoping,andzoneremelting.AmericanJournalofScience,282:808{855.Mastin,L.G..Anumericalprogramforsteady-stateowofhawaiianmagma-gasmixturesthroughverticaleruptiveconduits.USGeologicalSurveyOpen-FileReport,pages95{756.McBirney,A.R..Factorsgoverningemplacementofvolcanicnecks.AmericanJournalofScience,257:431{448.McClintock,M.andWhite,J.D.L..LargephreatomagmaticventcomplexatCoombsHills,Antarctica:Wet,explosiveinitiationofoodbasaltvolcanismintheFerrar-KarroLIP.BulletinofVolcanology,68:215{239.McGetchin,T.R..TheMosesRockdike:Geology,petrology,andmodeofemplace-mentofakimberlite-bearingbrecciadike,SanJuanCounty,Utah.PhDthesis,CaliforniaInstituteofTechnology,Pasadena.McKnight,S.B.andWilliams,S.N..Oldcinderconeoryoungcompositevolcano?ThenatureofCerroNegro,Nicaragua.Geology,25:339{342.Michaut,C.andJaupart,C..Ultra-rapidformationoflargevolumesofevolvedmagma.EarthandPlanetaryScienceLetters,250:38{52.Miller,R.B.andPaterson,S.R..Indefenseofmagmaticdiapirs.JournalofStructuralGeology,21:1161{1173.Moore,J.G.andLockwood,J.P..Originofcomb-layeringandorbicularstructure,SierraNevadaBatholith,California.GeologicalSocietyofAmericaBulletin,84:1{20.Needham,R.S..Giant-scalehydroplasticdeformationstructuresformedbytheloadingofbasaltontowater-saturatedsand,MiddleProterozoic,NorthernTerritory,Australia.Sedimentology,25:285{295.Nemeth,K.andWhite,J.D.L..ReconstructingeruptionprocessesofaMiocenemonogeneticvolcaniceldfromventremnants:WaipataVolcanicField,SouthIsland,NewZealand.JournalofVolcanologyandGeothermalResearch,124:1{21.Nostro,C.,Stein,R.S.,Cocco,M.,Belardinelli,M.E.,andMarzocchi,W..Two-waycouplingbetweenVesuviuseruptionsandsouthernAppenineearthquakes,Italy,byelasticstresstransfer.JournalofGeophysicalResearch,103:24487{24504.123

PAGE 140

Passchier,C.W.andTrouw,R.A.J..Microtectonics.Springer-Verlag,BerlinHeidelberg,Germany,2ndedition.Pederson,J.L.,Mackley,R.D.,andEddleman,J.L.2002.ColoradoPlateauUpliftanderosionevaluatedusingGIS.GSAToday,12:4{10.Petford,N.andGallagher.Partialmeltingofmacamphiboliticlowercrustbyperiodicinuxofbasalticmagma.EarthandPlanetaryScienceLetters,193:483{499.Pollard,D.D.andFletcher,R.C..FundamentalsofStructuralGeology.CambridgeUniversityPress,Cambridge,UK.Prager,W.andHodge,P..Theoryofperfectlyplasticsolids.Wiley&Sons.Press,W.H.,Flannery,B.P.,Teulolsky,S.A.,andVetterling,W.T..NumericalRecipes,theArtofScienticComputing.CambridgeUniversityPress.Ramberg,H..Gravity,DeformationandtheEarth'sCrust.AcademicPress,Lon-don,2ndedition.Ranalli,G..RheologyoftheEarth.ChapmanandHall.Reches,Z..Tensilefracturingofstirocklayersundertriaxialcompressivestressstates.JournalofRockMechanicsandMiningSciences,35:456{457.Reedman,A.J.,Park,K.H.,Merriman,R.J.,andKim,S.E.1987.WeldedtuinllingavolcanicventatWelseong,RepublicofKorea.BulletinofVolcanology,49:541{546.Renner,J.,Evans,B.,andSiddiqui,G..Dislocationcreepofcalcite.JournalofGeophysicalResearch,107:2364{2379.Roberts,J.L..Theintrusionofmagmaintobrittlerocks.InNewall,G.andRast,N.,editors,Mechanismofigneousintrusion,volume2ofJournalofGeology,SpecialIssue,page380.LiverpoolLetterpressLimited.Roggensack,K.,Hervig,R.L.,McKnight,S.B.,andWilliams,S.N..ExplosivebasalticvolcanismfromCerroNegrovolcano:Inuenceofvolatilesoneruptivestyle.Science,277:1639{1642.Ross,P.S.andWhite,J.D.L..Debrisjetsincontinentalphreatomagmaticvol-canoes:AeldstudyoftheirsubterraneandepositsintheCoombsHillsventcomplex,Antarctica.JournalofVolcanologyandGeothermalResearch,149:62{84.Rubin,A.M..Dikesvsdiapirsinviscoelasticrock.EarthandPlanetaryScienceLetters,117:653{670.Rutter,E.H..Thekineticsofrockdeformationbypressuresolution.PhilosophicalTransactionsoftheRoyalSocietyofLondon,283:203{219.Ryan,M.P.,Koyanagi,R.Y.,andFiske,R.S..Modelingthethree-dimensionalstructureofmacroscopicmagmatransportsystems:applicationtoKilaueavolcano,Hawaii.JournalofGeophysicalResearch,86:7111{7129.124

PAGE 141

Savage,S.B..Analysesofslowhigh-concentrationowsofgranularmaterials.JournalofFluidMechanics,377:1{26.Selig,F.1965.Atheoreticalpredictionsofsaltdomepatterns.Geophysics,30:633{643.Settle,M..Volcaniceruptioncloudsandthethermalpoweroutputofexplosiveeruptions.JournalofVolcanologyandGeothermalResearch,3:309{324.Shirley,D.N..DierentiationandcompactioninthePalisadessill,NewJersey.JournalofPetrology,28:835{865.Simkin,T.andSiebert,L.1994.VolcanoesoftheWorld.GeosciencePresswiththeSmithsonianInstitutionGlobalVolcanismProgram.Sisson,T.W.,Grove,T.L.,andColeman,D.S..HornblendegabbrosillcomplexatOnionvalley,California,andmixingoriginfortheSierraNevadaBatholith.Contri-butionstoMineralogyandPetrology,126:81{108.Skilling,I.P.,White,J.D.L.,andMcPhie,J..Peperite:areviewofmagma-sedimentmingling.JournalofVolcanologyandGeothermalResearch,114:1{17.Smith,J.F.,Hu,L.C.,Hinrichs,E.N.,andLuedke,R.G..GeologyoftheCapitolReefarea,WayneandGarteldCounties,Utah.U.S.GeologicalSurveyProfessionalPaper,363:102.Soriano,C.,Galindo,I.,Mart,J.,andWol,J.2006.Conduit-ventstructuresandrelatedproximaldepositsintheLasCa~nadasCaldera,Tenerife,CanaryIslands.BulletinofVolcanology,69:217{231.Sparks,R.S.J..ThedimensionsanddynamicsofPlinianeruptioncolumns.BulletinofVolcanology,48:3{15.Sparks,R.S.J.,Bursik,M.I.,Carey,S.N.,Gilbert,J.S.,Glaze,L.,Sigurdsson,H.,andWoods,A..VolcanicPlumes.Wiley.Sparks,R.S.J.andWilson,L..Amodelofignimbritesbygravitationalcolumncollapse.JournaloftheGeologicalSociety,London,132:441{451.Spera,F.J.,Oldenburg,C.M.,Christiensen,C.,andTodesco,M..SimulationsofconvectionwithcrystallizationinthesystemKAlSi2O6-CaMgSi2O6:Implicationsforcompositionallyzonedmagmabodies.AmericanMineralogist,80:1188{1207.Spohn,T.,Hort,M.,andFischer,H..Numericalsimulationofthecrystallizationofmulticomponentmeltsinthindikesorsills1.Theliquidusphase.JournalofGeophysicalResearch,93:4880{4894.Stasiuk,M.V.,Barclay,J.,Carroll,M.R.,Jaupart,C.,Ratte,J.C.,Sparks,R.S.J.,andTait,S.R..DegassingduringmagmaascentintheMuleCreekventUSA.BulletinofVolcanology,58:117{130.125

PAGE 142

Sturtevant,B.,Kanamori,H.,andBrodsky,E.E..Seismictriggeringbyrectieddiusioningeothermalsystems.JournalofGeophysicalResearch,101:25269{25282.Talbot,C.J.,Ronnlund,P.,Schmeling,H.,Koyi,H.,andJackson,M.P.A..Diapiricspokepatterns.Tectonophysics,188:187{201.Tegner,C.,Wilson,J.R.,andBrooks,C.K..IntraplutonicquenchzonesintheKapEdvardHolmlayeredgabbrocomplex,EastGreenland.JournalofPetrology,681:681{710.Thomas,A.L..POLY3D:Athree-dimensional,polygonalelement,displacementdiscontinuityboundaryelementcomputerprogramwithapplicationstofractures,faults,andcavitiesintheearth'scrust.Master'sthesis,StanfordUniversity,Stanford,Califor-nia.Thomson,K.2007.Determiningmagmaowinsills,dykesandlaccolithsandtheirimplicationsforsillemplacementmechanisms.BulletinofVolcanology,70:183{201.Thomson,K.andHutton,D..Geometryandgrowthofsillcomplexes:insightsusing3DseismicfromtheNorthRockallTrough.BulletinofVolcanology,66:364{375.Toda,S.,Stein,R.S.,andSagiya,T..Evidencefromthea.d.2000IzuIslandsearthquakeswarmthatstressingratesgovernsseismicity.Nature,419:58{61.Trusheim,F..MechanismofsaltmigrationinnorthernGermany.AssociationofAmericanPetroleumGeologistsBulletin,44:1519{1540.Turcotte,D.andSchubert,G.2002.Geodynamics.CambridgeUniversityPress,2ndedition.Turner,H.L.,LaFemina,P.E.,Saballos,J.A.,Mattiolli,G.E.,Jansma,P.E.,andDixon,T.H..GPSvelocityeldinNicaraguaforearc:Resultsfrom2000-2003.EosTrans.AGU,46:FallMeet.Suppl.,AbstractT52B{0264.Valentine,G.A.andKrogh,K.E.C..Emplacementofshallowdikesandsillsbeneathasmallbasalticvolcaniccenter-Theroleofpre-existingstructure.EarthandPlanetaryScienceLetters,246:217{230.Valentine,G.A.andWohletz,K.H..NumericalmodelsofPlinianeruptioncolumns.JournalofGeophysicalResearch,94:1867{1887.Wagner,R.,Rosenberg,C.L.,Handy,M.R.,Mobus,C.,andAlbertz,M..Fracture-drivenintrusionandupwellingofamid-crustalplutonfedfromatranpressiveshearzone-theRieserfernerPlutonEasternAlps.GSABulletin,118:219{237.Weijemars,R.,Jackson,M.P.A.,andVendeville,B.1993.Rheologicalandtectonicmodelingofsaltprovinces.Tectonophysics,217:143{174.Weinberg,R.F.andPodlachikov,Y..Diapiricascentofmagmasthroughpowerlawcrustandmantle.JournalofGeophysicalResearch,99:9543{9559.126

PAGE 143

Wells,D.L.andCoppersmith,K.J.1994.Newempiricalrelationshipsamongmagnitude,rupturelength,rupturewidth,rupturearea,andsurfacedisplacement.BulletinoftheSeismologicalSocietyofAmerica,84:974{1002.Wentworth,C.K.andJones,A.E..IntrusiverocksoftheleewardslopeoftheKooluaRange,Oahu,Hawaii.JournalofGeology,48:975{1006.White,J.D.L..Maar-diatremephreatomagmatismatHopiButtes,NavajoNationArizona,USA.BulletinofVolcanology,53:239{258.Williams,H..PliocenevolcanoesoftheNavajo-Hopicountry.GeologicalSocietyofAmericaBulletin,47:111{171.Williams,J.D..ThepetrographyanddierentiationofacompositesillfromtheSanRafaelSwellregion,Utah.Master'sthesis,ArizonaStateUniversity,Tempe.Wilson,L..Explosivevolcaniceruptions-III.Plinianeruptioncolumns.GeophysicalJournaloftheRoyalAstronomicalSociety,45:543{556.Wilson,L.,Sparks,R.S.J.,Huang,T.C.,andWatkins,N.D..Thecontrolofvolcaniccolumnheightdynamicsbyeruptionenergeticsanddynamics.JournalofGeophysicalResearch,83:1829{1836.Wilson,L.,Sparks,R.S.J.,andWalker,G.P.L..Explosivevolcaniceruptions-IV.thecontrolofmagmapropertiesandconduitgeometryoneruptioncolumnbehaviour.GeophysicalJournaloftheRoyalAstronomicalSociety,63:117{148.Wilson,L.andWalker,G.P.L..Explosivevolcaniceruptions-VI.EjectadispersalinPlinianeruptions:thecontroloferuptionconditionsandatmosphericproperties.GeophysicalJournaloftheRoyalAstronomicalSociety,89:651{679.Wol,J.A.,Elwood,B.B.,andSachs,S.D..Anisotropyofmagneticsusceptibilityinweldedtus:applicationtoawelded-tudykeintheTertiaryTrans-PecosTexasvolcanicprovince,USA.BulletinofVolcanology,51:299{310.Woods,A.W..Theuiddynamicsandthermodynamicsoferuptioncolumns.BulletinofVolcanology,50:169{193.Woods,A.W.1995.Thedynamicsofexplosivevolcaniceruptions.ReviewsofGeophysics,33:495{530.Woods,A.W.andBower,S..Thedecompressionofvolcanicjetsincraters.EarthandPlanetarySciencesLetters,131:189{205.Wylie,J.J.,Helfrich,K.R.,Dade,B.,Lister,J.R.,andSalzig,J.F.1999.Flowlocalizationinssureeruptions.Bulletinofvolcanology,60:432{440.Ziv,A.andRubin,A.M..Staticstresstransferandearthquaketriggering:Nolowerthresholdinsight?JournalofGeophysicalResearch,105:13631{13642.127

PAGE 144

APPENDICES128

PAGE 145

AppendixABasicFlowintheRayleigh-TaylormodelAppendixBasicowThebasicstateforthestabilityproblemtreatedinChapter4isgivenbythemotionequationsofaviscousuidinertialtermsneglectedintwodimensionsas0=@xx @x+@zx @z;=@P @x)]TJ/F21 10.9091 Tf 10.9091 0 Td[(r2U;0=@zz @z+@xz @x)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Rg;A.1=@P @z)]TJ/F21 10.9091 Tf 10.9091 0 Td[(r2W)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Rg;wherepisthepressure,UandWthehorizontalandverticalvelocitycomponents,respec-tively,Rtheuiddensityandgthegravityacceleration.Thestresscomponentsaregivenbythefollowingconstitutiverelationship,ij=2Eij)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Pij;A.2whereistheuidviscosityandEijthestrainrate.Forplaneowwith,zandxastheverticalandhorizontaldimensions,respectively,equationA.2canbeexpandedintostresscomponentsasxx=2Exx)]TJ/F21 10.9091 Tf 10.9091 0 Td[(P;A.3129

PAGE 146

AppendixAContinuedzz=2Ezz)]TJ/F21 10.9091 Tf 10.9091 0 Td[(P;A.4xz=2Exz;A.5wherexxandzzaretheviscousnormalstresses,andxzistheviscousshearstress.ThestrainratesaredenedasExx=@U @x,Ezz=@W @zandExz=1 2@U @z+@W @x.Thesystemisclosedwiththecontinuityequationas@U @x+@W @z=0:A.6Notethatthebasicowequationsarepresentedasappliedtoonelayerforclaritypurposes,however,theygovernowwithinbothwallrockandsilllayers.BoundaryconditionsTheboundaryconditionsforthebasicstateattheupperrigidsurface,z=)]TJ/F21 10.9091 Tf 8.4848 0 Td[(ho,seeFigure4.1are:no-slipconditions;U=0andW=0:A.7Attheinterfacesbetweentwoarbitrarywallrocklayers,atz=j,andthesillandoverlyingwallrocklayer,thefollowingvelocityandstressconditionsarerequiredcontinuityofhorizontalandverticalvelocities;Ui=Ui+1andWi=Wi+1;continuityofshearstress;i;xz=i+1;xz;A.8continuityofnormalstress;i;zz=i+1;zz;130

PAGE 147

AppendixAContinuedand,thebottomofthesill,z=hs,isalsodenedasarigidsurfacewithboundaryconditions:no-slipconditions;U=0andW=0:A.9131

PAGE 148

AppendixBNomenclatureandconduitowanderuptioncolumnmodelequa-tionsNomenclatureSubscriptscevaluatedalongtheconduitdevaluatedataheightabovetheventwherethepressurehasadjustedtoatmo-sphericpressureinchokedconditionseevaluatedalongtheeruptioncolumnaevaluatedinairmevaluatedinmagmagevaluatedingaspevaluatedatconstantpressure0initialcondition,evaluatedatbaseofconduitorbaseoferuptioncolumnConduitModeldenedconstantsggravitationalacceleration,9:8ms)]TJ/F19 7.9701 Tf 6.5865 0 Td[(2Rcgasconstant,462Jkg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1K)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1ssaturationconstant,6:810)]TJ/F19 7.9701 Tf 6.5865 0 Td[(10Pa0:7forbasalticmagma,4:310)]TJ/F19 7.9701 Tf 6.5865 0 Td[(6Pa0:5forrhyoliticmagmacoecientinHenry'slawdependentoncomposition,0.7forbasalticmagmas,0.5forrhyoliticmagmasggasviscosity,5:310)]TJ/F19 7.9701 Tf 6.5865 0 Td[(5PasPa0atmosphericpressure,0:1MPaattheventvariablesAconduitcross-sectionalarea132

PAGE 149

AppendixBContinuedDconduitdiameterffrictioncoecient,0.0025dimensionlessMMachnumberncvolatilemassfractionexsolvedfrommagma,i.e.inbubbles,e.g.0.04dimen-sionlessnmvolatilemassfractiondissolvedinmagmapcconduitpressureQmassowrateReReynoldsnumberrcconduitradiusTmmagmatictemperature,e.g.1200KucmeanowvelocityintheconduitusspeedofsoundinthemixturezcdepthalongaxiscoordinateincrementalheightupconduitLcconduitlength,e.g.6000mmmagmaviscositymixtureviscositycdensityoftheconduitmixturemdensityofmagma,e.g.2700kgm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3voidfractionDecompressionJetModelddensityofthedecompressedjetrddecompressedjetradiusuddecompressedjetvelocityAdcross-sectionalareaofthedecompressedjet133

PAGE 150

AppendixBContinuedEruptionColumnModeldenedconstantsCapspecicheatatconstantairpressure,998JK)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1kg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1Htheightofthetropopause,11kmHsheightofthestratosphere,20kmkentrainmentconstant,0.09dimensionlessRagasconstantfortheair,285Jkg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1K)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1Rebulkgasconstant,Re0=462JK)]TJ/F19 7.9701 Tf 6.5866 0 Td[(1kg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1temperaturegradientinthetroposphere,6:5Kkm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1!temperaturegradientinthestratosphere,2:0Kkm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1densityofthesolidpyroclasts,1000kgm)]TJ/F19 7.9701 Tf 6.5865 0 Td[(3variablesCepbulkspecicheatofcolumnmaterialatconstantpressure,Ce0=1617JK)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1kg)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1recolumnradiusnegasmassfractioninthecolumnPaatmosphericpressureabovethevent,Pa0=0:1MPaattheventTaambientatmospherictemperature,Ta0=293KattheventuebulkverticalcolumnvelocityzetheheightalongaxiscoordinateincrementalheightupthecolumnatheambientatmosphericdensityebulkcolumndensityTebulkcolumntemperaturehmaxmaximumcolumnheightqmassowmmomentumux134

PAGE 151

AppendixBContinuedeenergyuxGoverningEquationsforConduitFlowModelMassandMomentumConservationFluidmotioninvolcanicconduitsisgovernedbyequationsfortheconservationofmassuxcucA=QB.1andmomentumcucduc dzc=)]TJ/F21 10.9091 Tf 9.6804 7.38 Td[(dpc dzc)]TJ/F21 10.9091 Tf 10.9091 0 Td[(cg)]TJ/F21 10.9091 Tf 12.1046 7.38 Td[(cu2cf rc:B.2Thepressuregradientalongtheconduitcanthenbecomputedbycombiningbothmassandmomentumconservationequationsdpc dzc)]TJ/F21 10.9091 Tf 12.1046 7.3801 Td[(u2c u2s=)]TJ/F21 10.9091 Tf 8.4849 0 Td[(cg)]TJ/F21 10.9091 Tf 12.1046 7.3801 Td[(cu2cf rc:B.3ConstitutiveRelationshipsThedensityofthemixtureisdescribedby1 c=ncRcTm pc+1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(nc mB.4wherethemassofvolatilesexsolvedfromthemagma,nc,isgivenbync=nc0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(nm 1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(nmB.5135

PAGE 152

AppendixBContinuedwherenc0istheinitialvolatilecontentandnmisthevolatilemassfractiondissolvedinmagma,givenbyHenry'sLawnm=spcB.6wheresandareparametersrelatedtomagmacomposition.Thefrictionfactor,f,iscalculatedusingf=16 Re+f0=16 cucD+f0B.7wheref0isanempirically-derivedfactorrelatedtotheroughnessoftheconduitwalls.Vesiclecontentisexpressedasvoidfraction=1 1+)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ncpc=ncRcTmm:B.8Magmaviscosity,m,isdependentontemperaturefollowingtheempiricalrelationshiplogm=)]TJ/F15 10.9091 Tf 8.4849 0 Td[(10:737+1:818310000 Tm:B.9Viscosityofthemixture,,variesasafunctionofvesiclecontent1=8>>>><>>>>:m 1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(;for0:75gh1)]TJ/F26 10.9091 Tf 10.9091 12.1092 Td[(1)]TJ/F22 7.9701 Tf 6.5865 0 Td[( 0:62i)]TJ/F19 7.9701 Tf 6.5865 0 Td[(1:56;for>0:75B.10wheregisthegasviscosity. 1RecentworksonbubblyliquidrheologysuggestthattheelongationofbubbleswouldcauseareductioninviscositywithincreasinggascontentRustandManga2002;Llewellinetal.2002,thereforeamorecomplexexpressioncouldbeincludedtoaccountforthisprocessseeLlewellinandManga2005.136

PAGE 153

AppendixBContinuedThespeedofsoundinthemixtureisgivenbyus=RcTm nc1 2nc+)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ncpc mRcTmB.11andisusedtocomputetheMachnumberofthemixture.TheDecompressedJetModelFollowingWoodsandBower,theconditionsinthejetatthatpointabovetheventwherematerialhasdecompressedtoatmosphericpressurecanbecalculated.ThedensityofthedecompressedmaterialisapproximatedbydPa nc0RcTmB.12andevaluatedataheightabovetheventwherethematerialhasdecompressedtoatmo-sphericpressure.Thevelocitybeyondthedecompressionregioniscalculatedusingthefollowingequationud=nc0RcTm1 21+nc nc021)]TJ/F21 10.9091 Tf 12.1046 7.3801 Td[(Pa pcB.13where=1+)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ncpc ncRcTmmB.14=nc0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(spc n1 2c0nc0)]TJ/F22 7.9701 Tf 12.1046 4.9315 Td[(spc 21+pc RcTmm1 2:B.15137

PAGE 154

AppendixBContinuedFrommassconservationbetweentheventandthedecompressedregion,cucA=Q=dudAdB.16theareaofthedecompressedjetisgivenbyAd=Aucc udd=Q uddB.17andtheradiusisgivenbyrd=Q udd1 2:B.18SeeWoodsandBowerfordetails.GoverningEquationsforEruptionColumnModelMassandMomentumConservationMassisconservedbyd dze)]TJ/F21 10.9091 Tf 5 -8.8364 Td[(euer2e=8>>>><>>>>:uere 8p ae;whene>a2kuerea;wheneainthegas-thrustregionande
PAGE 155

AppendixBContinuedMomentumisconservedbyd dze)]TJ/F21 10.9091 Tf 5 -8.8364 Td[(eu2er2e=8>>>><>>>>:)]TJ/F22 7.9701 Tf 6.5865 0 Td[(eu2ere 8q a e+ga)]TJ/F21 10.9091 Tf 10.9091 0 Td[(er2e;whene>aga)]TJ/F21 10.9091 Tf 10.909 0 Td[(er2e;wheneainthegas-thrustregionande
PAGE 156

AppendixBContinuedTheAtmosphereTheeruptioncolumnmodelassumesathermallylayeredatmosphere.ThetemperatureproleisgivenbythreeequationsTa=8>>>>>>>>>><>>>>>>>>>>:Ta0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ze;forzeHtTa0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Ht;forHtzeHtTa0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Ht+!ze)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Hs;forzeHsB.22wherezeHtreferstothetemperatureproleinthetroposphere,HtzeHt,thetemperatureproleinthetropopause,andzeHs,thetemperatureproleinthestratosphere.Ht;Hs,,and!aretheheightsandtemperaturegradientsofthetropopauseandstratosphere,respectively.FollowingGreen,thechangeofatmosphericpressurewithheightisdescribedby1 PadPa dze=)]TJ/F21 10.9091 Tf 19.2082 7.3801 Td[(g RaTaB.23assumingthatairbehaveslikeaperfectgaswherePa=RaaTa.EquationB.23canbeintegratedtoobtainthepressureproleintheatmospherePa=Pa0exp)]TJ/F21 10.9091 Tf 14.4584 7.38 Td[(gze RaTa:B.24Theatmosphericdensityprolecanbecalculatedbya=Pa RaTaB.25140

PAGE 157

AppendixBContinuedusingPafromB.24andTafromB.22.ConstitutiveRelationshipsListedbelowareequationsfortheinverseofdensityB.26,thegasmassfractionB.27,thegasconstantB.28andthebulkspecicheatatconstantpressureB.291 e=)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ne1 +neReTe PaB.26ne=1+ne0)]TJ/F15 10.9091 Tf 10.9091 0 Td[(1r2e0ue0e0 r2eueeB.27Re=Ra+Re0)]TJ/F21 10.9091 Tf 10.9091 0 Td[(Ra1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ne nene0 1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ne0B.28Cep=Cap+Cep0)]TJ/F21 10.9091 Tf 10.909 0 Td[(Cap1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ne 1)]TJ/F21 10.9091 Tf 10.9091 0 Td[(ne0:B.29Thesubscript0denotesavalueevaluatedattheventandisthedensityofsolidpyro-clasts.141

PAGE 158

AppendixCMatlabcodesfortheRayleigh-Taylorinstabilitymodel%instability.m,adaptedfromClintonConradbyMikelDiez,25/11/07%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Thisprogram,instability.m,isamatlabroutinetocalculatethe%characteristicspacingbetweenRayleigh-Taylorinstabilitiescreatedat%theinterfacebetweenasillandtheoverburden.Thewallrockoverburden%canbedefinedwithseverallayersofdifferentnon-newtonianrheologies,%suchasdislocationorduffusioncreep.Themodelisbasedona%paperbyBassi&Bonnin,inGeophysicalJournal,93,485-504.The%programusespropagatormatricestocalculatethefastestgrowing%wavelengthofRTinstabilities.%Thisprogramassumesarigidupperboundaryandarigidbottomofthe%sill,thusw=v=0atbothsurfaces%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Variables:%%mnumberofintermediateinterfaces%%exxstrainrate-positiveforextension,negativeforcompression.%ForRTinstabilitiyIassumeitverysmall~10^-30%%kwavenumber=2*pi/l%%lwavelengthm%%Mm+4x4m+4matrixcontainingboundaryconditionsMC=R%%Cm+4x1vectorofintegrationcoefficients%%Rm+4x1vectorofrighthandsideofboundaryconditions%%Qm+1x4m+4matrixcontainingthetransformationW=QC%%Wmx1vectorwithverticalvelocities,w,ofeachinterface.142

PAGE 159

AppendixCContinued%Includingtheupperonebutnotthebottomofthesill%%Hmx1vectorwithamplitudesofsinusoidalundulations,h%%Pm+4xm+1matrixcontainingthetransformationR=PH%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Subroutinesassociatedmatlabprograms:%%param.msetsthevariousparametersforthelayeredmodel%%findM.mcalculatesthematrixM%%findQ.mcalculatesthematrixQ%%findP.mcalculatesthematrixP%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%setparametersclear;carmel_exph2;%getparametervalueskmax=500;%kmaxisthemaximumvalueofk'calculated.jnum=75;%jnumisthenumberofpointstocalculatebetweenk'=0andkmax.forjj=1:jnum%setsizesofimportantmatricesM=zeros*m+4,*m+4;%C=zeros*m-2,1;%R=zeros*m-2,1;Q=zerosm+1,*m+4;%W=zerosm,1;%H=zerosm,1;143

PAGE 160

AppendixCContinuedP=zeros*m+4,m+1;%setwavenumber,k=2*pi/l,l=wavelengthLKjj=jj/jnum*kmax;k=LKjj/hc;%calculateimportantmatrices:findM_mikel;findQ_mikel;findP_mikel;%%Computequadrupleprecisionfloatingpoints,otherwisebigerrorinthe%%solutionofthelinearsystem.Itgiveswarningaboutpossiblesingular%%matrixandinaccurateresults.Thisisfixedusingthesymbolicalgebra%%toolboxfunctionvpa,thatallowscomputationwithanyprecision.You%%actuallyspecifyit:e.g:vpa'pi',25,givesyoupiwith25digits.%%Bearinmindthatthelongerthefloatingpointtheslowerthe%%computationsF=vpaQ,17/vpaM,17*vpaP,17;[V,E]=eigF;%%Transformbacktodoubleprecisiononcetheinversionhasbeencomputed%%tospeedupcomputations144

PAGE 161

AppendixCContinuedE1=doubleE;q:,jj=E1*onesm+1,1;%q1jj=q,jj;%q2jj=q,jj;endqmax=maxq;qpmax=qmax*at/.8*r-r*hc;%qpmaxisthedimensionlessgrowthratecsvwrite'car_h2_30.txt.',qpmaxcsvwrite'xcar_axis.txt',LKplotLK,qpmax;setgca,'XTick',0:50:500ylabel'DimensionlessGrowthRate,q''';xlabel'DimensionlessWavenumber,k''';%%Dimensionalizeandtransformtocharacteristicwavelengthxp=LK/800;lmbdac=2*pi./xp;%%dimensionalizegrowthrateandtransformtocharacteristictime145

PAGE 162

AppendixCContinuedat10p=qpmax*.8*300*800/1e19;tau10c=1./at10p*3600;%%Findthefastestgrowthtimeanditsassociatedcharacteristic%%wavelength'growthtime'growth_time_blue=mintau10c[c,i]=mintau10c;'characteristicwavelength'characteristic_wavelength_blue=lmbdaci%findM.m,byMikelDiez,adaptedfromK.Conradversion,11/12/07%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CalculatesthematrixM,whichisthetransformationMC=Randcontains%%theboundaryconditionsforeachinterface,andthetopandbottom.%%Startatthetop,with2conditions,thenfourforeachlayerinterface,%%andthentwoatthebottomofthesill.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Variables:%%mnumberofintermediateinterfaces%%kwavenumber=2*pi/l%%lwavelengthm%%Mm+4x4m+4matrixcontainingboundaryconditionsMC=R%%mci,aci,rci,bci,a1i,a2i,M1i,M2i,M3i,M4i%%areallvariablesdefinedbelowtomakecalculationseasier%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%146

PAGE 163

AppendixCContinued%Parameterswhichmakethecalculationssimplier:fori=1:m%Onlyneedpropertiesform+1-1=mlayersnon-newtonianmci=gi/k;aci=sqrtmci^2/8+1/ni-0.5+sqrtmci^4/64++mci^2*/ni+1/8+0.25;rci=sqrtmci^2/4+ni-1/ni^2;bci=rci/aci;a1i=aci-mci/2;a2i=-aci-mci/2;M1i=/ni-1*a1i-mci*+a1i^2-bci^2--a1i*a1i^2-3*bci^2;M2i=bci*/ni-1-2*a1i*mci++bci^2-3*a1i^2;M3i=/ni-1*a2i-mci*+a2i^2-bci^2--a2i*a2i^2-3*bci^2;M4i=bci*/ni-1-2*a2i*mci++bci^2-3*a2i^2;end%Rigidsurface:twoconditions,theverticalandhorizontalvelocities%arezero.%non-newtonianviscosity147

PAGE 164

AppendixCContinuedM,1=expa1*k*z*cosbc*k*z;M,2=expa1*k*z*sinbc*k*z/bc*k;M,3=expa2*k*z*cosbc*k*z;M,4=expa2*k*z*sinbc*k*z/bc*k;M,1=expa1*k*z*-a1*cosbc*k*z++bc*sinbc*k*z;M,2=expa1*k*z*-cosbc*k*z/k--a1*sinbc*k*z/bc*k;M,3=expa2*k*z*-a2*cosbc*k*z++bc*sinbc*k*z;M,4=expa2*k*z**-cosbc*k*z/k-a2*sinbc*k*z/bc*k;%normalizebymultiplyingtimeswavenumber.Recallthatw%hasdimensionsoflengthm,bymultiplplyingtimesk%wemakeitnon-dimensionalseee.g.eq.%C1pag.110Conrad&Molnar.Thatis,%weremovekfromthe%denominator.forj=1:2Mj,2=Mj,2*k;Mj,4=Mj,4*k;148

PAGE 165

AppendixCContinuedend%Intermediateinterfaces.Therearem%deformableintermediateinterfaces.Thereare%fourboundaryconditionsforeach.%Notethatatm+2isthebottomofthesill%andm+1isthewallrock-sillinterfacefori=1:m%upperlayer:%non-newtonianviscosityM*i-1,4*i-1+1=expa1i*k*zi+1*cosbci*k*zi+1;M*i-1,4*i-1+2=expa1i*k*zi+1*sinbci*k*zi+1/bci*k;M*i-1,4*i-1+3=expa2i*k*zi+1*cosbci*k*zi+1;M*i-1,4*i-1+4=expa2i*k*zi+1*sinbci*k*zi+1/bci*k;M*i,4*i-1+1=expa1i*k*zi+1**-a1i*cosbci*k*zi+1+bci*sinbci*k*zi+1;M*i,4*i-1+2=expa1i*k*zi+1**-cosbci*k*zi+1/k-a1i*sinbci*k*zi+1/bci*k;M*i,4*i-1+3=expa2i*k*zi+1**-a2i*cosbci*k*zi+1+bci*sinbci*k*zi+1;149

PAGE 166

AppendixCContinuedM*i,4*i-1+4=expa2i*k*zi+1**-cosbci*k*zi+1/k-a2i*sinbci*k*zi+1/bci*k;M*i+1,4*i-1+1=ati*k*expa1i*k*zi+1**+a1i^2-bci^2*cosbci*k*zi+1--2*a1i*bci*sinbci*k*zi+1;M*i+1,4*i-1+2=ati*k*expa1i*k*zi+1***a1i/k*cosbci*k*zi+1+++a1i^2-bci^2/bci*k*sinbci*k*zi+1;M*i+1,4*i-1+3=ati*k*expa2i*k*zi+1**+a2i^2-bci^2*cosbci*k*zi+1--2*a2i*bci*sinbci*k*zi+1;M*i+1,4*i-1+4=ati*k*expa2i*k*zi+1***a2i/k*cosbci*k*zi+1+++a2i^2-bci^2/bci*k*sinbci*k*zi+1;M*i+2,4*i-1+1=ati*k*expa1i*k*zi+1**M1i*cosbci*k*zi+1-M2i*sinbci*k*zi+1;M*i+2,4*i-1+2=ati*k*expa1i*k*zi+1**M2i/bci*k*cosbci*k*zi+1++M1i/bci*k*sinbci*k*zi+1;M*i+2,4*i-1+3=ati*k*expa2i*k*zi+1**M3i*cosbci*k*zi+1-M4i*sinbci*k*zi+1;M*i+2,4*i-1+4=ati*k*expa2i*k*zi+1**M4i/bci*k*cosbci*k*zi+1++M3i/bci*k*sinbci*k*zi+1;150

PAGE 167

AppendixCContinued%normalizebydividingbyviscosity*wavenumber:forj=1:4M*i+1,4*i-1+j=M*i+1,4*i-1+j/at*k;M*i+2,4*i-1+j=M*i+2,4*i-1+j/at*k;end%Multiplytimesktorendertheequationsadimensional.Notethat%M*i+1,2,4*i-1+2,4isoperatedtwice,firstdivide1/eta*kandthen%multiplytimesk.forj=-1:2M*i+j,4*i-1+2=M*i+j,4*i-1+2*k;M*i+j,4*i-1+4=M*i+j,4*i-1+4*k;end%lowerlayer:ifi~=m%lowerlayerisnotthesillbutanotherwallrocklayer.%non-newtonianviscosity151

PAGE 168

AppendixCContinuedM*i-1,4*i+1=-expa1i+1*k*zi+1*cosbci+1*k*zi+1;M*i-1,4*i+2=-expa1i+1*k*zi+1*sinbci+1*k*zi+1/bci+1*k;M*i-1,4*i+3=-expa2i+1*k*zi+1*cosbci+1*k*zi+1;M*i-1,4*i+4=-expa2i+1*k*zi+1*sinbci+1*k*zi+1/bci+1*k;M*i,4*i+1=-expa1i+1*k*zi+1**-a1i+1*cosbci+1*k*zi+1+bci+1*sinbci+1*k*zi+1;M*i,4*i+2=-expa1i+1*k*zi+1**-cosbci+1*k*zi+1/k-a1i+1*sinbci+1*k*zi+1/bci+1*k;M*i,4*i+3=-expa2i+1*k*zi+1**-a2i+1*cosbci+1*k*zi+1+bci+1*sinbci+1*k*zi+1;M*i,4*i+4=-expa2i+1*k*zi+1*-cosbci+1*k*zi+1/k--a2i+1*sinbci+1*k*zi+1/bci+1*k;M*i+1,4*i+1=-ati+1*k*expa1i+1*k*zi+1**+a1i+1^2-bci+1^2*cosbci+1*k*zi+1--2*a1i+1*bci+1*sinbci+1*k*zi+1;M*i+1,4*i+2=-ati+1*k*expa1i+1*k*zi+1***a1i+1/k*cosbci+1*k*zi+1+++a1i+1^2-bci+1^2/bci+1*k*sinbci+1*k*zi+1;M*i+1,4*i+3=-ati+1*k*expa2i+1*k*zi+1**+a2i+1^2-bci+1^2*cosbci+1*k*zi+1--2*a2i+1*bci+1*sinbci+1*k*zi+1;M*i+1,4*i+4=-ati+1*k*expa2i+1*k*zi+1***a2i+1/k*cosbci+1*k*zi+1+152

PAGE 169

AppendixCContinued++a2i+1^2-bci+1^2/bci+1*k*sinbci+1*k*zi+1;M*i+2,4*i+1=-ati+1*k*expa1i+1*k*zi+1**M1i+1*cosbci+1*k*zi+1-M2i+1*sinbci+1*k*zi+1;M*i+2,4*i+2=-ati+1*k*expa1i+1*k*zi+1**M2i+1/bci+1*k*cosbci+1*k*zi+1++M1i+1/bci+1*k*sinbci+1*k*zi+1;M*i+2,4*i+3=-ati+1*k*expa2i+1*k*zi+1**M3i+1*cosbci+1*k*zi+1-M4i+1*sinbci+1*k*zi+1;M*i+2,4*i+4=-ati+1*k*expa2i+1*k*zi+1**M4i+1/bci+1*k*cosbci+1*k*zi+1++M3i+1/bci+1*k*sinbci+1*k*zi+1;%normalizebydividingbyviscosity*wavenumber:forj=1:4M*i+1,4*i+j=M*i+1,4*i+j/at*k;M*i+2,4*i+j=M*i+2,4*i+j/at*k;end%Multiplytimesktorendertheequationsadimensional.Notethat%M*i+1,2,4*i+2,4isoperatedtwice,firstdivide1/eta*kandthen%multiplytimesk.forj=-1:2153

PAGE 170

AppendixCContinuedM*i+j,4*i+2=M*i+j,4*i+2*k;M*i+j,4*i+4=M*i+j,4*i+4*k;endelse%thesillisreachedandthenewtonianequationsmustapply.%newtonianviscosity,sill%NoteIamusingziinsteadofzi+1isbecauseithastomatchthe%interfacewiththelayerrightabove.Form=2,weareatinterfacez%andusingpropertiesoflayer3sill,e.g.:atviscosity,r....M*i-1,4*i+1=-expk*zi+1;M*i-1,4*i+2=-zi+1*expk*zi+1;M*i-1,4*i+3=-exp-k*zi+1;M*i-1,4*i+4=-zi+1*exp-k*zi+1;M*i,4*i+1=expk*zi+1;M*i,4*i+2=/k+zi+1*expk*zi+1;M*i,4*i+3=-exp-k*zi+1;M*i,4*i+4=--1/k+zi+1*exp-k*zi+1;M*i+1,4*i+1=-ati+1*k*2*expk*zi+1;M*i+1,4*i+2=-ati+1*k*/k+2*zi+1*expk*zi+1;154

PAGE 171

AppendixCContinuedM*i+1,4*i+3=-ati+1*k*2*exp-k*zi+1;M*i+1,4*i+4=-ati+1*k*-2/k+2*zi+1*exp-k*zi+1;M*i+2,4*i+1=-ati+1*k*2*expk*zi+1;M*i+2,4*i+2=-ati+1*k*2*zi+1*expk*zi+1;M*i+2,4*i+3=-ati+1*k*-2*exp-k*zi+1;M*i+2,4*i+4=-ati+1*k*-2*zi+1*exp-k*zi+1;%Maketheespressionsnon-dimensional:%normalizebydividingbyviscosity*wavenumber:forj=1:4M*i+1,4*i+j=M*i+1,4*i+j/at*k;M*i+2,4*i+j=M*i+2,4*i+j/at*k;endforj=-1:2M*i+j,4*i+2=M*i+j,4*i+2*k;M*i+j,4*i+4=M*i+j,4*i+4*k;endendend%Endoftheloop155

PAGE 172

AppendixCContinued%%Bottomofthesill:twoconditions,thevertical%andhorizontalvelocities%%arezero.Thisclosesthe4m+4x4m+4Mmatrix.%iis2%NotethatIusezi+2=zm+2=zforourm=2%particularcase,asweare%atthebottomofthesillM*i+3,4*i+1=expk*zi+2;M*i+3,4*i+2=zi+2*expk*zi+2;M*i+3,4*i+3=exp-k*zi+2;M*i+3,4*i+4=zi+2*exp-k*zi+2;M*i+4,4*i+1=-expk*zi+2;M*i+4,4*i+2=-/k+zi+2*expk*zi+2;M*i+4,4*i+3=exp-k*zi+2;M*i+4,4*i+4=-1/k+zi+2*exp-k*zi+2;forj=3:4M*i+j,4*i+2=M*i+j,4*i+2*k;M*i+j,4*i+4=M*i+j,4*i+4*k;end156

PAGE 173

AppendixCContinued%findP_mikel.m,modifiedbyMikelDiezfromClintonConrad,25/11/07%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CalculatesthematrixP,whichisthetransformationR=PH.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Variables:%%mnumberofintermediateinterfaces%%kwavenumber=2*pi/l%%Pm+4xm+1matrixcontainingthetransformationR=PH%%Hmx1vectorwithamplitudesofsinusoidalundulations,h%%Rm+4xm+1vectorofrighthandsideofboundaryconditions%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%toplayer-thisisarigidsurface-thisisrows1and2%thesurfaceofthebottomlayersillhaszerovelocity-lasttworows%Notethattheonlynon-zerobc'sarethoseattheinterfacethataccount%forthecontinuityoftheshearandnormalstressesrespectivelyP,1=0;P,1=0;fori=1:m%misintermediateinterfaces.Asmanynon-zerointerfacesas%intermediateinterfacesm.157

PAGE 174

AppendixCContinuedP*i+1,1+i=4*exx*k*ati-ati+1/at*k;P*i+2,1+i=ri+1-ri*9.8/at*k;end%ThelasttwoconditionsareaddedwhenmemoryallocatedforP,P=%zerosm+4,m+1.%findQ_mikel.m,adaptedbyMikelDiez,25/11/07%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CalculatesthematrixQ,whichisthetransformationW=QC%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Variables:%%mnumberofintermediateinterfaces%%kwavenumber=2*pi/l%%Qmx4m+4matrixcontainingthetransformationW=QC%%Cm+4x1vectorofintegrationcoefficients%%Wmx1vectorwithverticalvelocities,w,ofeachinterface%%mci,aci,rci,bci,a1i,a2i%%areallvariablesdefinedbelowtomakecalculationseasier%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Parameterswhichmakethecalculationssimplier:fori=1:mmci=gi/k;158

PAGE 175

AppendixCContinuedaci=sqrtmci^2/8+1/ni-0.5+sqrtmci^4/64++mci^2*/ni+1/8+0.25;rci=sqrtmci^2/4+ni-1/ni^2;bci=rci/aci;a1i=aci-mci/2;a2i=-aci-mci/2;end%thereareminterfaces,eachwithavelocity,w.%Notethatnoweveryinterfacecounts,%itsvelocityisspecifiedinthebc's.Weexpress%thisvelocityinWintermsofthecoefficients%inC,usingthethematrixQ.Notewe%findthisatx=0.fori=1:m%upperlayers%ifgi~=0|ni~=1%non-newtonianviscosity%Notethatthisistheuppersurfacez.Inthebc'sit'sstatedthat%w=0,asit'sarigidsurface%WeusetheexpressionWzfornon-newtonianlayerQi,4*i-1+1=expa1i*k*zi*cosbci*k*zi;159

PAGE 176

AppendixCContinuedQi,4*i-1+2=expa1i*k*zi*sinbci*k*zi/bci*k;Qi,4*i-1+3=expa2i*k*zi*cosbci*k*zi;Qi,4*i-1+4=expa2i*k*zi*sinbci*k*zi/bci*k;endQi,4*i-1+2=Qi,4*i-1+2*k;Qi,4*i-1+4=Qi,4*i-1+4*k;i=m+1;%NewtonianFluid%Bottomlayer%WeusetheexpressionWz=Aiexpkz+Bizexpkz+Ciexp-kz+Di%zexp-kzQi,4*i-1+1=expk*zi;Qi,4*i-1+2=zi*expk*zi;Qi,4*i-1+3=exp-k*zi;Qi,4*i-1+4=zi*exp-k*zi;Qi,4*i-1+2=Qi,4*i-1+2*k;Qi,4*i-1+4=Qi,4*i-1+4*k;160

PAGE 177

AppendixCContinued%Parameters_mikel.mcreatedbyMikelDiez,25/11/07%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Inputfilewithparameters%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%numberofintermediateinterfacesm=1;%densityoftheoverlyingmaterialr0=0;%Depths,zz=10000;z=0.0;z=-40000;%z=-20;hc=z-z;%hc=1*1000;%Stressexponent,n,forthewallrocknon-newtonianlayersn=1.0001;%n=1.5;161

PAGE 178

AppendixCContinued%Inverseviscositydecaylength,gammag=0.0;%g=0.0;%Densityofthelayer,rhor=3300;r=3250;%r=2200;%Layerviscosity,atthetopofthelayer,aetaat=10^19;at=10^16;%at=10^10;%Layerviscosity,atthebottomofthelayer,aeta%ab=at;%Strainrate,epsilonxxexx=10^-30;162

PAGE 179

AppendixDCcodesconduit_column_coupled.ccreatedbyMikelDiez2004-2005/*****************************************************************************************************************MAINPROGRAMThisroutinesolvesforflowalongthevolcanicsystemthroughtheconduitanderuptioncolumncalling:conduit_routines.candcolumn_routines.c*******************************************************************************************************************/#include#include#include#defineNRANSI#include"cdflib.h"#include"conduit_routines.h"#include"column_routines.h"#definepat100000.0//AtmosphericpressurePa#definet293.0//Atmospherictemperature#definepi3.14159#defineg9.8#defineR462.0/*enJkg-1K-1,asiquelaPenPascals*/#defines6.8e-10/*sctefromHenry'slawforbasalt*/intim=0,ip=0,i0=0;//Indexestostoreinitialvelocityin163

PAGE 180

AppendixDContinuedeachiterationofthefirstloopofCONDUITMODULEintmain//AsweareusingaC++compiler"int"insteadof"void"isneeded{/***************Conduitvariables****************************************************************/doublez,p,dpdz,htry,h,eps,yscal,hdid,hnext,mfg0,T,rhom,diam,f0,radio,mdot,area,xsarea,rhomix,vfgas,v,vel,mach,rhomixture,yerr,exit_vel,mfgc;doublepout[500],zout[500],vout[500],mout[500],hout[500],nout[500],rhout[500];doubleitermax;intentadj,flgend,j,i;/***********Freejetvariables********************************************************************/doubletrmalfa,trmbita,trm1,trm2,trm3,trm4,trm5,trm6,radius_decomp,velocity_decomp,densa,peta;/************Columnvariables*****************************************************************/164

PAGE 181

AppendixDContinueddoublen0,l0,u0,theta0,cp0,rg0,sigma,kappa,invdensity0,density0,u,l,theta,density_e,alpha,alpha0;doubley[3],dydx[3],yscale[3],yout0[500],yout1[500],yout2[500],zoutc[500],uout[500],thetaout[500],lout[500],densout[500],alphaout[500];doubleze,epse,htrye,hdide,hnexte,li;intnd,k,cr,w;/**************************************************************************************************************************************************************CONDUITMODULE*****************************************************************************************************************************************************************/FILE*fp;/*Initialconditions*/p=173.0;/*ojo!!,penmegapascals*/v=1.0;165

PAGE 182

AppendixDContinuedmfg0=0.03;//PassingaparameterfromaPDFforinitialvolatilecontentT=1300.0;rhom=2800.0;z=-6000.0;ifz>0.0z=-z;diam=5;//PassingaparameterfromaPDFforinitialventdiameterf0=0.0025;/*typicalvalueforwallroughedconduitsWilson*/p=p*1e+6;/*presionapascales*/eps=1.0e-8;/*tolerancia*/radio=diam/2.0;area=pi*powradio,2.0;xsarea=area;//printf"hey!!radio%f%fn",diam,mfg0;/*Checkforthepressureatthebaseoftheconduit*//*lithp=rhom*g*-z;ifp
PAGE 183

AppendixDContinuedvout[0]=v;zout[0]=z;/*Calculateparametersatthebaseoftheconduit*///T=T+273.15;/*TemperaturaentraencentigradosperotransformamosaKelvin*/itermax=0.0;do{//printf"v0entrando1erloopes%fn",vout[0];i=0.0;flgend=0;v=vout[0];z=zout[i];p=pout[i];htry=-z/100.0;/*stepsizetobeattempted*//*Calculatemassflux*/densityp,mfg0,T,rhom,&rhomix,&vfgas,&mfgc;mdot=rhomix*xsarea*v;/*massflux*/167

PAGE 184

AppendixDContinued//printf"mdot%f%f%f%.8fn",mdot,rhomix,xsarea,v;yscal=p;vout[0]=v;zout[i]=z;pout[i]=p;/*Calculatesthepressuregradientatthebottomoftheconduit*/derivscz,p,mfg0,T,rhom,mdot,xsarea,diam,f0,&vel,&mach,&rhomixture,&dpdz,&mfgc;//printf"gradenderivs%fn",dpdz;mout[i]=mach;rhout[i]=rhomixture;/*beginsthelooptosolveforthepressure*///printf"%f%f%f%f%dn",zout[i],pout[i],vout[i],mout[i],i;do{z=zout[i];p=pout[i];/*Calculatethepressureatahigherposition*/168

PAGE 185

AppendixDContinuedrkqsc&p,dpdz,&z,htry,eps,yscal,&hdid,&hnext,mfg0,T,rhom,mdot,xsarea,diam,f0,derivsc;hout[i+1]=hdid;zout[i+1]=z;pout[i+1]=p;htry=hnext;/*Calculateparametersandthepressuregradientatthisnewpressure*/derivscz,p,mfg0,T,rhom,mdot,xsarea,diam,f0,&v,&mach,&rhomixture,&dpdz,&mfgc;vout[i+1]=v;mout[i+1]=mach;nout[i+1]=mfgc;rhout[i+1]=rhomixture;//printf"v0trasentrar1erloopes%f%dn",vout[0],i;/*IfZ>0,adjusthandcallRK4tocalculatepatz=0*/ifzout[i+1]>0.0{h=-zout[i];rkckcpout[i],dpdz,zout[i],h,&p,&yerr,mfg0,T,rhom,169

PAGE 186

AppendixDContinuedmdot,xsarea,diam,f0,derivsc;pout[i+1]=p;zout[i+1]=0.0;}/*ifp<1atmorm>1,adjusttheinputvelocity*//*printf"lapaquies%fn",pout[i+1];*/ifpout[i+1]<1.013e+05||mout[i+1]>1.0&&zout[i+1]<-0.05{velbcvout,pout,zout,mout,eps,i,&entadj,&flgend;ifflgend==1break;}/*Ifm>0.99999andz>-0.05,callitgood*/ifzout[i+1]>-0.05&&mout[i+1]>0.99999{flgend=1;printf"ngoodsolution!!nn";break;}/*ifz=0andp>1atm,adjusttheinputvelocityandstartagainwithanewinitialvelocity*/170

PAGE 187

AppendixDContinuedifpout[i+1]>1.014e+05&&zout[i+1]==0.0{velbcvout,pout,zout,mout,eps,i,&entadj,&flgend;ifflgend==1break;}/*Ifpressureisbetween0.1012and0.1014callitgood*/ifpout[i+1]<=1.014e+05&&pout[i+1]>=1.012e+05{flgend=1;break;}i=i+1;}whileentadj!=1;entadj=0;ifflgend==1break;/*salecompletamentedelprograma*/itermax=itermax+1;//printf"nIterationnumber%fn",itermax;171

PAGE 188

AppendixDContinued//printf"nTryingnewinitialvelocity%lfn",vout[0];//printf"nv0enmainis%fn",vout[0];ifitermax>=500.0{//printf"nmax.numberofiterationsreachedn";break;}}whileitermax<1000.0;/*itermaxeselnumeromaximodeiteraciones*/fp=fopen"conduit.txt","w";//Createoutputfile//printf"DepthttPressuretVelocitytMachnumbernn";forj=0;j<=i+1;j++{ifmout[j]>=1.0break;pout[j]=pout[j]/1e+06;printf"%ft%ft%ft%fn",zout[j],pout[j],vout[j],mout[j];fprintffp,"%f%f%f%f%fn",zout[j],172

PAGE 189

AppendixDContinuedpout[j],vout[j],mout[j],rhout[j];}fclosefp;//printf"Massflow:%fn",mdot;exit_vel=vout[j-1];printf"exitvelocity%fn",exit_vel;//printf"mach%fn",mout[j-1];//return;printf"massflowcondis%fn",mdot;ip=0;i0=0;im=0;peta=pout[j-1]*1000000;//transformtopascals/********************************************************functiontoobtainthemaximumoftwonumbers********************************************************/doublemaxdoublea,doubleb{ifa>breturna;173

PAGE 190

AppendixDContinuedifabreturnb;returna;}/*********************************************************************************************************************VELBC:routinetoadjustinitialvelocityinordertosatisfy174

PAGE 191

AppendixDContinuedtheimpossedboundaryconditionsfromMastin,1995**********************************************************************************************************************/voidvelbcdoublevout[],doublepout[],doublezout[],doublemout[],doubleeps,inti,int*entadj,int*flgend{doubleinter,slope,v1m[1000],v10[1000],v1p[1000],rmaxo,rminv,z1m[1000],z1p[1000];intjv;//printf"i0is%dn",i0;*entadj=1;/*entadj=1letmeknowifIhaveenteredadjustRoutine.Ifthisisthecasetheinnerdo-whileloopstopsandtheouterdo-whileloopbeginswithanotherinitialvelocity*///printf"v0entrandoenadjustes%fn",vout[0];/*SOLUTIONSFORZ=0*/ifzout[i+1]==0.0{//printf"nExitpressure>1atmandM<1n";175

PAGE 192

AppendixDContinuedifpout[i+1]>1.014e+05{v10[i0]=vout[0];/*Findminimumofv1pandv1m*/rminv=5.0e+05;/*printf"imis%ldn",im;*/forjv=0;jv
PAGE 193

AppendixDContinued}/*Iftheminimumofv1porv1mandthemaximumofv10don'tdifferbymorethanabout10.0*eps,thencallthesolutiongood*///printf"rmaxois%lf,rminvis%lfn",rmaxo,rminv;ifrmaxo>0.0&&rminv-rmaxo/rmaxo
PAGE 194

AppendixDContinuedrmaxo=maxrmaxo,v10[jv];}vout[0]=rmaxo+0.5*vout[0]-rmaxo;ifvout[0]<0.001{//printf"nPressureinsufficienttoproduceeruptionn";exit;return;}}i0=i0+1;}/*SOLUTIONSFORZ<0*//*printf"machnumberis%lfn",mach[i];*/ifmout[i+1]>1.0{//printf"nmachnumber>1,adjustinginitialvelocityn";z1m[im]=zout[i+1];v1m[im]=vout[0];//printf"v1m%f%f%dn",v1m[im],v1m[im-1],im;ifim>=2{178

PAGE 195

AppendixDContinuedslope=z1m[im]-z1m[im-1]/v1m[im]-v1m[im-1];//printf"den.slopeenadjust%fn",slope;inter=z1m[im]-slope*v1m[im];vout[0]=-inter/slope+0.1*minv1m[im],v1m[im-1]+inter/slope;//printf"v0is%fn",vout[0];}elsevout[0]=-vout[0]*zout[i+1]-zout[0]/zout[0];im=im+1;}ifpout[i+1]<1.013e+05{//printf"npressurelessthan1atm,adjustinginitialvelocityn";z1p[ip]=zout[i+1];v1p[ip]=vout[0];ifip>=2{slope=z1p[ip]-z1p[ip-1]/v1p[ip]-v1p[ip-1];inter=z1p[ip]-slope*v1p[ip];if-inter/slope<=0.0||-inter/slope>vout[0]{vout[0]=vout[0]/2.0;179

PAGE 196

AppendixDContinued}elsevout[0]=-inter/slope;}elsevout[0]=-vout[0]*zout[i+1]-zout[0]/zout[0];ip=ip+1;}ifvout[0]<=0.001vout[0]=0.001;//printf"v0esjoder%fn",vout[0];/*limpiezadearraysyaquecomienzaunnuevocalculoconunanuevavelocidadinicial*/forjv=1;jv
PAGE 197

AppendixDContinuedforjv=0;jv
PAGE 198

AppendixDContinuedInchokedconditions,M=1andP>Patm,theeruptingjetdecompressesintotheatmosphereasafreejetuntilatmosphericpressureisachieved.WoodsandBower,derivedequationstocalculatethevelocityandradiusofthedecompressedjetatthelevelwherethepressureequalstheatmospheric.Theconduitmodelandthecolumnmodelcanbecoupledusingthesetwoparametersasinitialconditionsintheeruptioncolumnmodel.Thenmassisconservedintheinterphaseconduit-column.****************************************************************/ifmout[j-1]<0.99l0=radio;//susbsonicconditionselse{//chokedconditions,p>atmosphericpressureattheventtrmalfa=1+-mfg0*peta/mfg0*R*T*rhom;//printf"ntrmalfa%fn",trmalfa;trm1=s*powpeta,0.7/2.0/+pout[j-1]/R*T*rhom;//printf"ntrm1%lfn",trm1;trm2=mfg0-trm1;//printf"ntrm2%fn",trm2;trm3=powtrm2,0.5*powmfg0,0.5;//printf"ntrm3%fn",trm3;182

PAGE 199

AppendixDContinuedtrmbita=mfg0-s*powpeta,0.7/trm3;//printf"ntrmbita%fn",trmbita;trm4=nout[j-1]/mfg0*trmalfa*powtrmbita,2**-.0/peta;//printf"ntrm4%fn",trm4;trm5=mfg0*R*T;//printf"ntrm5%fn",trm5;velocity_decomp=powtrm5,0.5*trmalfa*trmbita*+trm4;printf"nvelocityofthedecompressedjet%lfn",velocity_decomp;printf"ngascontentatthevent%lfn",nout[j-1];densa=100000.0/mfg0*R*T;trm6=mdot/pi*velocity_decomp*densa;radius_decomp=powtrm6,0.5;printf"ndiameterofthedecompressedjet%lfn",radius_decomp*2;//velocityandradiusofthedecompressedjetenterasinitialconditions//inthecolumnmodel183

PAGE 200

AppendixDContinuedl0=radius_decomp;u0=velocity_decomp;}n0=mfg0;//Gascontentatthebaseoftheconduittheta0=1250;//Initialtemperature=mixturetemperaturethroughttheconduitisothermalascent/***********************************************************/cp0=1617;//Initialbulkspecificheatofthecolumnrg0=462;//Bulkgasconstantforthecolumnsigma=1000;//Densityofthesolidpyroclastskappa=0.09;//Entrainmentcoeff.epse=1e-8;htrye=5.0;//Initialstepsizeinmeters//printf"%f%f%f%f%fn",n0,sigma,rg0,theta0,p;invdensity0=-n0/sigma+n0*rg0*theta0/pat;//printf"invdensity0is%fn",invdensity0;184

PAGE 201

AppendixDContinueddensity0=1.0/invdensity0;//printf"density0is%fn",density0;ze=0.0;y[0]=density0*u0*powl0,2;y[1]=density0*powu0,2*powl0,2;y[2]=cp0*theta0*density0*u0*powl0,2;zoutc[0]=ze;yout0[0]=y[0];yout1[0]=y[1];yout2[0]=y[2];alphaout[0]=alpha0;uout[0]=u0;thetaout[0]=theta0;lout[0]=l0;densout[0]=density0;cr=0;//Changeregimecr=0forthrustregionfork=0;k
PAGE 202

AppendixDContinued&theta,&density_e,&u,&l,&alpha,dydx;//Beginloopk=0;for;;{ifalpha>=density_ecr=1;//Changeregimecr=1forconvectiveregion.//Criterion:whenthecolumnfirstbecomebuoyantze=zoutc[k];y[0]=yout0[k];//y[0]isMassy[1]=yout1[k];//y[1]isMomentumy[2]=yout2[k];//y[2]isEnergyrkqsey,dydx,nd,&ze,htrye,epse,yscale,&hdide,&hnexte,n0,l0,u0,theta0,cp0,rg0,sigma,kappa,cr,derivse;//printf"step%fn",x;zoutc[k+1]=ze;htrye=hnexte;yout0[k+1]=y[0];yout1[k+1]=y[1];186

PAGE 203

AppendixDContinuedyout2[k+1]=y[2];derivseze,y,n0,l0,u0,theta0,cp0,rg0,sigma,kappa,cr,&theta,&density_e,&u,&l,&alpha,dydx;//printf"airdensity%f%fn",alpha,u0;ifu<=5.0break;//Momentumoftheplumeisalmost0thetaout[k+1]=theta;densout[k+1]=density_e;uout[k+1]=u;lout[k+1]=l;alphaout[k+1]=alpha;k+=1;}fp=fopen"ecolumn.txt","w";//Createoutputfile//printf"nvelocitytradiustttemperaturetdensitytheighttnn";forw=0;w<=k;w++{//printf"%ft%ft%ft%ft%fn",uout[w],lout[w],alphaout[w],densout[w],zoutc[w];//fprintffp,"%f%f%f%f%fn",uout[j],lout[j],thetaout[j],densout[j],zoutc[j];//Writefile187

PAGE 204

AppendixDContinuedprintf"%ft%ft%ft%ft%f%fn",uout[w],lout[w],alphaout[w],densout[w],zoutc[w],thetaout[w];fprintffp,"%f%f%f%fn",lout[w],uout[w],densout[w],zoutc[w];//Writefile}fclosefp;return0;//Asthemainfunctionisexpectingtoreturnaninteger,0isreturnedtoavoidawarningwhilecompiling}Conduit_routines.ccreatedbyMikelDiez2004-2005/********************************************************Theseroutinesarecalledbyconduit_column_coupled.ctosolvetheflowthroughtheconduit********************************************************/#include#include#include#include"nrutil.h"188

PAGE 205

AppendixDContinued#defineSAFETY0.9#definePGROW-0.2#definePSHRNK-0.25#defineERRCON1.89e-4/*ThevalueERRCONequals/SAFETYraisedtothepower/PGROW,seeusebelow.*/#defineR462.0/*enJkg-1K-1,asiquelaPenPascals*/#defines6.8e-10/*sctefromHenry'slawforbasalt*/#definebeta0.7/*betactefromHenry'slawforbasalt*/#defineg9.8#definepi3.14159/*********************************************************SOUNDV:routinetocalculatesonicvelocity**********************************************************/voidsoundvdoublep,doubleT,doublemfg0,doublerhom,double*c{doublemfg,c1,c2,c3,c4,c5;mfg=mfg0-s*powp,beta/-s*powp,beta;c1=R*T/mfg;c2=powc1,0.5;189

PAGE 206

AppendixDContinuedc3=p/1000000.0/rhom*R*T;c4=-mfg*c3;c5=mfg+c4;*c=c2*c5;}/***********************************************************VISCOSITY:routinetocalculatetheviscosity***********************************************************/voidviscositydoubleT,doublevfgas,double*eta{doubleexp,vb;exp=-10.737+1.8183*.0/T;*eta=pow.0,exp;/*RyanandBlevinsmagmaviscosityasafunctionofT*/ifvfgas<=0.75{190

PAGE 207

AppendixDContinued*eta=*eta/-vfgas;/*equationsfromDobran*/}else{vb=1.0-.0-vfgas/0.62;*eta=5.3e-5*powvb,-1.56;}}/**********************************************************Density:routinetocalculatedensity**********************************************************/voiddensitydoublep,doublemfg0,doubleT,doublerhom,double*rhomix,double*vfgas,double*mfg{//printf"pindensityis%fn",p;//doublemfg;/*gasmassfraction-Henry'slaw*/*mfg=mfg0-s*powp,beta/-s*powp,beta;//printf"mfgis%f",mfg;191

PAGE 208

AppendixDContinuedif*mfg>0.0{*rhomix=1.0/*mfg*R*T/p+.0-*mfg/rhom;*vfgas=*mfg*R*T/p**rhomix;}/*else{*rhomix=rhom;*vfgas=0.0;printf"ole!!!n";}*/}/*****************************************************************DERIVSC:routinetocalculatepressuregradient*****************************************************************/voidderivscdoublex,doubley,doublemfg0,doubleT,doublerhom,doublemdot,doublexsarea,doublediam,doublef0,double*vel,192

PAGE 209

AppendixDContinueddouble*mach,double*rhomixture,double*dydx,double*mfgc{doublerhomix,vfgas,eta,v,reynolds,f,m,c,mfg;/*Calculatemixturedensity*/densityy,mfg0,T,rhom,&rhomix,&vfgas,&mfg;*rhomixture=rhomix;*mfgc=mfg;/*Calculateviscosity*/viscosityT,vfgas,η/*Calculatevelocity,reynoldsnumber,frictionfactor*/v=mdot/rhomix*xsarea;reynolds=rhomix*v*diam/eta;f=16.0/reynolds+f0;*vel=v;/*Calculatesonicvelocityandmachnumber*/soundvy,T,mfg0,rhom,&c;m=v/c;193

PAGE 210

AppendixDContinued*mach=m;/*Calculategradientinpressure*/*dydx=rhomix*-g-rhomix*f*powv,2.0/diam/2.0/.0-powm,2;}/**************************************************************************************************************RKCK:Givenvalueforyanditsderivativedydxknownatx,usethefifth-orderCash-KarpRunge-Kuttamethodtoadvancethesolutionoveranintervalhandreturntheincrementedvariableasyout.Alsoreturnanestimateofthelocaltruncationerrorinyoutusingembeddedfourthordermethod.Theusersuppliestheroutinederivsx,y,dydx,whichreturnsderivativesdydxatx.************************************************************/***************************************************************************************194

PAGE 211

AppendixDContinuedModifiedbyM.Diez2004-2005fromPressetal.,1997,"NumericalRecipesinC",p.719-720*************************************************************************************/voidrkckcdoubley,doubledydx,doublex,doubleh,double*yout,double*yerr,doublemfg0,doubleT,doublerhom,doublemdot,doublexsarea,doublediam,doublef0,void*derivscdoublex,doubley,doublemfg0,doubleT,doublerhom,doublemdot,doublexsarea,doublediam,doublef0,double*vel,double*mach,double*rhomixture,double*dydx,double*mfgc{staticdoublea2=0.2,a3=0.3,a4=0.6,a5=1.0,a6=0.875,b21=0.2,b31=3.0/40.0,b32=9.0/40.0,b41=0.3,b42=-0.9,b43=1.2,b51=-11.0/54.0,b52=2.5,b53=-70.0/27.0,b54=35.0/27.0,b61=1631.0/55296.0,b62=175.0/512.0,b63=575.0/13824.0,b64=44275.0/110592.0,b65=253.0/4096.0,c1=37.0/378.0,c3=250.0/621.0,c4=125.0/594.0,c6=512.0/1771.0,dc5=-277.00/14336.0;doubledc1=c1-2825.0/27648.0,dc3=c3-18575.0/48384.0,dc4=c4-13525.0/55296.0,dc6=c6-0.25;doubleak2,ak3,ak4,ak5,ak6,ytemp,vel,mach,rhomixture,mfgc;195

PAGE 212

AppendixDContinued//printf"%f%f%fn",y,dydx,h;/*Firststep*/ytemp=y+b21*h*dydx;*derivscx+a2*h,ytemp,mfg0,T,rhom,mdot,xsarea,diam,f0,&vel,&mach,&rhomixture,&ak2,&mfgc;/*Secondstep*/ytemp=y+h*b31*dydx+b32*ak2;*derivscx+a3*h,ytemp,mfg0,T,rhom,mdot,xsarea,diam,f0,&vel,&mach,&rhomixture,&ak3,&mfgc;/*Thirdstep*/ytemp=y+h*b41*dydx+b42*ak2+b43*ak3;*derivscx+a4*h,ytemp,mfg0,T,rhom,mdot,xsarea,diam,f0,&vel,&mach,&rhomixture,&ak4,&mfgc;/*Fourthstep*/ytemp=y+h*b51*dydx+b52*ak2+b53*ak3+b54*ak4;*derivscx+a5*h,ytemp,mfg0,T,rhom,mdot,xsarea,diam,f0,&vel,&mach,&rhomixture,&ak5,&mfgc;/*Fifthstep*/ytemp=y+h*b61*dydx+b62*ak2+b63*ak3+b64*ak4+b65*ak5;*derivscx+a6*h,ytemp,mfg0,T,rhom,mdot,xsarea,diam,f0,&vel,&mach,&rhomixture,&ak6,&mfgc;/*Sixthstep*/196

PAGE 213

AppendixDContinued/*Accumulateincrementswithproperweights*/*yout=y+h*c1*dydx+c3*ak3+c4*ak4+c6*ak6;*yerr=h*dc1*dydx+dc3*ak3+dc4*ak4+dc5*ak5+dc6*ak6;/*Estimateerrorasdifferencebetweenfourthandfifthordermethods*/}/****************************************************************************************************RKQS:Fifth-orderRunge-Kuttastepwithmonitoringoflocaltruncationerrortoensureaccuracyandadjuststepsize.Inputisthedependentvariableyanditsderivativedydxatstartingvalueoftheindependentvariablex.Alsoinputarethestepsizetobeattemptedhtry,therequiredaccuracyeps,andthevalueyscalagainstwhichtheerrorscaled.Onoutput,yandxarereplacedbytheirnewvalues,hdidisthestepsizethatwasactuallyaccomplished,andhnextistheestimatednextstepsize.derivsistheuser-suppliedroutinethatcomputestheright-handsidederivatives.***************************************************************197

PAGE 214

AppendixDContinued*************************************//***************************************************************************************************ModifiedbyM.Diez,2004-2005fromPressetal.,1997,"NumericalRecipesinC",p.719****************************************************************************************************/voidrkqscdouble*y,doubledydx,double*x,doublehtry,doubleeps,doubleyscal,double*hdid,double*hnext,doublemfg0,doubleT,doublerhom,doublemdot,doublexsarea,doublediam,doublef0,void*derivscdoublex,doubley,doublemfg0,doubleT,doublerhom,doublemdot,doublexsarea,doublediam,doublef0,double*vel,double*mach,double*rhomixture,double*dydx,double*mfgc{doubleerrmax,h,htemp,xnew,yerr,ytemp,ysav,xsav;//printf"yandx%f%f%f%fn",*y,*x,htry,dydx;xsav=*x;198

PAGE 215

AppendixDContinuedysav=*y;h=htry;/*Setstepsizetotheinitialtrialvalue*/for;;{/*printf"stepsizeandgradientinrkqs%f%fn",h,dydx;*/rkckcysav,dydx,xsav,h,&ytemp,&yerr,mfg0,T,rhom,mdot,xsarea,diam,f0,derivsc;/*Takeastep*/errmax=0.0;/*Evaluateaccuracy*/errmax=FMAXerrmax,fabsyerr/yscal;/*Scalerelativetorequiredtolerance*/errmax/=eps;iferrmax<=1.0break;/*Stepsucceded.Computesizeofnextstep*/htemp=SAFETY*h*powerrmax,PSHRNK;/*Truncationerrortoolarge,reducestepsize*//*Nomorethanafactorof10*/h=h>=0.0?FMAXhtemp,0.1*h:FMINhtemp,0.1*h;xnew=*x+h;ifxnew==*xnrerror"stepsizeunderflowinrkqs";}iferrmax>ERRCON*hnext=SAFETY*h*powerrmax,PGROW;else*hnext=5.0*h;/*Nomorethanafactorof5increse*/*x+=*hdid=h;*y=ytemp;//printf"hnext%fn",*hnext;199

PAGE 216

AppendixDContinued}column_routines.ccreatedbyM.Diez2004-2005/********************************************************Theseroutinesarecalledbyconduit_column_coupled.ctosolvetheflowthroughtheeruptioncolumn********************************************************/#include#include#defineNRANSI#include"nrutil.h"#defineSAFETY0.9#definePGROW-0.2#definePSHRNK-0.25#defineERRCON1.89e-4#defineca998//Specificheatatconstantpressurefortheair#definera287//Thegasconstantfortheair#defineg9.8//gravityacceleration#definet293//AtmospherictempreratureK#definepat100000.0//AtmosphericpressurePa#definemu6.5//temperaturegradientinthetroposphereK/km#defineomega2.0//temperaturegradientinthestratosphereK/km200

PAGE 217

AppendixDContinued/*********************************************************************************ATMOSPT:Calculatesthedensityprofilefortheatmosphere*********************************************************************************/voidatmosptdoublez,double*ta,double*pa{doubleH1,H2,g1;H1=11.0;//theheightofthetropopauseH2=20.0;//theheightofthestratospherez=z/1000;//ztokmg1=9.8*0.001;//gravitationalaccelerationtokm/s2//Calculateatmospherictemperatureifz<=H1*ta=t-mu*z;ifH1<=z&&z<=H2*ta=t-mu*H1;ifz>=H2*ta=t-mu*H1+omega*z-H2;201

PAGE 218

AppendixDContinued//Calculateatmosphericpressure//tr1=g1/ra*mu;Woods,1988eqs.forpressuredon'tworkproperlyforastandardatmosphere//tr2=t-mu*z;//tr3=t-mu*H1;//tr4=-g1*z-H1/ra*t-mu*H1;//tr5=-g1*H2-H1/ra*t-mu*H1;//tr6=t-mu*H1+omega*z-H2;*pa=pat*exp-g*z*1000/ra**ta;//ifz<=H1*pa=p*powtr2,tr1;//ifH1<=z&&z<=H2*pa=p*powtr3,tr1*exptr4;//printf"airteperature%fn",*pa;//ifz>=H2*pa=p*powtr3,tr1*exptr5*powtr6,-tr1;}/****************************************************DERIVS****************************************************/202

PAGE 219

AppendixDContinuedvoidderivsedoublez,doubley[],doublen0,doublel0,doubleu0,doubletheta0,doublecp0,doublerg0,doublesigma,doublekappa,intcr,double*theta,double*density,double*u,double*l,double*alpha,doubledydx[]{doubleinvdensity0,density0,n,cp,rg,ta,pa;invdensity0=-n0/sigma+n0*rg0*theta0/pat;density0=1.0/invdensity0;atmosptz,&ta,&pa;//CallatmospttocalculateatmosphericP,Tatdifferentheights*alpha=pa/ra*ta;//Calculatesatmosphericdensityatdifferentlevels//printf"%fn",*alpha;//Gasmassfractionn=1+n0-1*powl0,2*u0*density0/y[0];cp=ca+cp0-ca*-n/-n0;rg=ra+rg0-ra*-n/n*n0/-n0;*theta=y[2]/y[0]/cp;//Temperature//Density*density=1.0/-n/sigma+n*rg**theta/pa;203

PAGE 220

AppendixDContinued*u=y[1]/y[0];//Velocity*l=sqrty[0]/*density**u;//Radiusifcr==0{//Massfluxgradientforthrustregiondydx[0]=*u**l*sqrt*alpha**density/8;dydx[1]=-*density*pow*u,2**l/8.0**sqrt*alpha/*density;//Momentumfluxgradient}else{//Massfluxgradientforconvectiveregiondydx[0]=2*kappa**u**l**alpha;//Momentumfluxgradientforconvectiveregiondydx[1]=g**alpha-*density*pow*l,2;}dydx[2]=ca*ta+pow*u,2/2*dydx[0]--*alpha**u*pow*l,2*g;//Energyfluxgradient}/********************************************************204

PAGE 221

AppendixDContinued***********************RKCK*******************************************************************************/voidrkckedoubley[],doubledydx[],intnd,doublex,doubleh,doubleyout[],doubleyerr[],doublen0,doublel0,doubleu0,doubletheta0,doublecp0,doublerg0,doublesigma,doublekappa,intcr,void*derivsedoublez,doubley[],doublen0,doublel0,doubleu0,doubletheta0,doublecp0,doublerg0,doublesigma,doublekappa,intcr,double*theta,double*density,double*u,double*l,double*alpha,doubledydx[]{inti;staticdoublea2=0.2,a3=0.3,a4=0.6,a5=1.0,a6=0.875,b21=0.2,b31=3.0/40.0,b32=9.0/40.0,b41=0.3,b42=-0.9,b43=1.2,b51=-11.0/54.0,b52=2.5,b53=-70.0/27.0,b54=35.0/27.0,b61=1631.0/55296.0,b62=175.0/512.0,b63=575.0/13824.0,b64=44275.0/110592.0,b65=253.0/4096.0,c1=37.0/378.0,c3=250.0/621.0,c4=125.0/594.0,c6=512.0/1771.0,dc5=-277.00/14336.0;doubledc1=c1-2825.0/27648.0,dc3=c3-18575.0/48384.0,dc4=c4-13525.0/55296.0,dc6=c6-0.25;205

PAGE 222

AppendixDContinueddouble*ak2,*ak3,*ak4,*ak5,*ak6,*ytemp,u,theta,density,l,alpha;ak2=vector,nd;ak3=vector,nd;ak4=vector,nd;ak5=vector,nd;ak6=vector,nd;ytemp=vector,nd;fori=0;i
PAGE 223

AppendixDContinuedfori=0;i
PAGE 224

AppendixDContinued**************************************************************************************/voidrkqsedoubley[],doubledydx[],intnd,double*x,doublehtry,doubleeps,doubleyscal[],double*hdid,double*hnext,doublen0,doublel0,doubleu0,doubletheta0,doublecp0,doublerg0,doublesigma,doublekappa,intcr,void*derivsedoublez,doubley[],doublen0,doublel0,doubleu0,doubletheta0,doublecp0,doublerg0,doublesigma,doublekappa,intcr,double*theta,double*density,double*u,double*l,double*alpha,doubledydx[]{/*voidrkckdoubley[],doubledydx[],intnd,doublex,doubleh,doubleyout[],doubleyerr[],void*derivsdouble,double[],double[];*/inti;doubleerrmax,h,htemp,xnew,*yerr,*ytemp;yerr=vector,nd;ytemp=vector,nd;h=htry;for;;{rkckey,dydx,nd,*x,h,ytemp,yerr,n0,l0,u0,theta0,cp0,rg0,sigma,kappa,cr,derivse;208

PAGE 225

AppendixDContinuederrmax=0.0;fori=0;i=0.0?FMAXhtemp,0.1*h:FMINhtemp,0.1*h;xnew=*x+h;ifxnew==*xnrerror"stepsizeunderflowinrkqs";}iferrmax>ERRCON*hnext=SAFETY*h*powerrmax,PGROW;else*hnext=5.0*h;*x+=*hdid=h;fori=0;i
PAGE 226

AppendixDContinued*****************************************************CC=/usr/bin/gccconduit_column_coupled:conduit_column_coupled.oconduit_routines.ocolumn_routines.onrutil.oipmpar.odcdflib.o$CC-g-Wall-oconduit_column_coupled-lmconduit_column_coupled.onrutil.oconduit_routines.ocolumn_routines.oipmpar.odcdflib.oconduit_column_coupled.o:conduit_column_coupled.c$CC-Wall-cconduit_column_coupled.cconduit_routines.o:conduit_routines.c$CC-Wall-cconduit_routines.ccolumn_routines.o:column_routines.c$CC-Wall-ccolumn_routines.c210

PAGE 227

AppendixDContinuednrutil.o:nrutil.c$CC-Wall-cnrutil.cipmpar.o:ipmpar.c$CC-Wall-cipmpar.cdcdflib.o:dcdflib.c$CC-Wall-cdcdflib.cclean:rm*.o*~conduit_column_coupled211

PAGE 228

ABOUTTHEAUTHORMikelDezwasborninBilbao,BasqueCountry,Spain.HeobtainedhisB.S.ingeologyin2001attheUniversityofBasqueCountry.InAugust2003hecametoUSAandjoinedtheGeologyDepartmentattheUniversityofSouthFloridaUSF.ShortlyafterhewasawardedafouryearfellowshipgrantedbytheBasqueCountryGovernmenttoworkonhisPhDwithProfessorCharlesB.Connor.BesidesworkingatUSF,heparticipatedineldtripsandconferencesinUS,Nicaragua,Mexico,Chile,EnglandandEcuador.HealsovisitedDr.OnnoBokhoveattheUniversityofTwente,TheNetherlandsinthespringof2006.MikelhaspublishedseveralpapersininternationaljournalsandhasbeenawardedaCASinternalawardandtheRichardDavisfellowshipintheSpringof2007.


xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001994157
003 fts
005 20090330141823.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 090330s2008 flu s 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002436
035
(OCoLC)317406759
040
FHM
c FHM
049
FHMM
090
QE26.2 (Online)
1 100
Dez, Mikel.
0 245
Studies of the mechanics and structure of shallow magmatic plumbing systems
h [electronic resource] /
by Mikel Dez.
260
[Tampa, Fla] :
b University of South Florida,
2008.
500
Title from PDF of title page.
Document formatted into pages; contains 211 pages.
Includes vita.
502
Dissertation (Ph.D.)--University of South Florida, 2008.
504
Includes bibliographical references.
516
Text (Electronic dissertation) in PDF format.
520
ABSTRACT: Volcanic activity, and the resultant deposits and structures at the Earth's surface, are the outcome of the inner workings of underground magmatic plumbing systems. These systems, essentially, consist of magma reservoirs which supply magma to the surface through volcanic conduits feeding volcanic eruptions. The mechanics and structure of plumbing systems remain largely unknown due to the obvious challenges involved in inferring volcanic processes occurring underground from observations at the surface. Nevertheless, volcanologists are beginning to gain a deeper understanding of the workings and architecture of magmatic plumbing systems from geophysical observations on active volcanoes, as well as from geological studies of the erosional remnants of ancient volcanic systems. In this work, I explore the relationship between the structure and mechanics of shallow plumbing systems and the volcanic eruptions these systems produce.I attempt to contribute to the understanding of this complex relationship by linking geological and geophysical observations of an eroded basaltic subvolcanic system, and the eruptive and tectonic activity of an active volcano, with mathematical models of magma ascent and stress transfer. The remarkable exposures of the Carmel outcrop intrusions, near the San Rafael swell, southeast Utah, U. S. A., allow detailed geological and geophysical observations of the roots of volcanic conduits that emerge from a subhorizontal magma feeder reservoir. These observations reveal a new mechanism for magma ascent and eruption triggering through gravitational instabilities created from an underlying feeding sill, and shed light on the mechanics of sill emplacement.Geophysical and geological observations of the 1999 and xii 1992 eruptions of the Cerro Negro volcano, Nicaragua, are used to explore the coupling between changes in the stress field and the triggering of volcanic eruptions, and magma ascent through the shallow crust. Modeling results of stress transfer and conduit flow highlight the importance of the surrounding stress field and geometry of the volcanic conduits that comprise shallow plumbing systems.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
590
Advisor: Charles B. Connor, Ph.D.
653
Eruption triggering
Eruption dynamics
Magma ascent mechanisms
Volcano-tectonic interaction
Static stress changes
690
Dissertations, Academic
z USF
x Geology
Doctoral.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2436