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HeatFlowacrosstheSanAndreasFaultErikAnderson&BrendanCych
FrictionAlongFaultsGeneratesHeat
• Energyfromslippartitionedinto:
1)Fracturecreation
2)Seismicradiation
3)Thermalenergy
• Whatcomponentplaysthelargestrole?
EarthquakesOccurAlongFaultsintheUpperCrust
ΔT=27oC/km Trouwetal.,2010
RadiatedSeismicEnergyMeasurementsUnderestimateTheory•
τsσn
µ
Rock1
Rock2
Settinguptheequationandboundaryconditions
• Needtoconsideralinesourceatz=atorepresentheating• Initialformulaforheatflowgivenby
• Boundaryconditions
• Wewillneedaheatsinktosatisfythese,placedatz=-a
𝛻"𝑇 =1𝑘𝑄 𝑥, 𝑧 =
1𝑘 𝛿(𝑥)𝛿(𝑧 + 𝑎)
𝑇 𝑥, 0 = 0 lim|5|→7
𝑇 𝑥, 𝑧 = 0
lim|8|→7
𝑇 𝑥, 𝑧 = 0
SolvingthedifferentialequationfordT/dz
• Wealreadysolvedourdifferentialequationforheatflowinclass.(NotesonFouriertransforms).• TaketheFouriertransformofbothsides,usingthederivativepropertyontheLHSanddefinitionofthedeltafunctiononRHS• TaketheinversetransformintheZdirection,usingtheCauchyresiduetheoremtomakethiseasier.• Taketheinversetransforminthexdirection,usingthederivativepropertywithrespecttoztogetdT/dz
𝛻"𝑇 =1𝑘 𝑄 𝑥, 𝑧 =
1𝑘 𝛿(𝑥)𝛿(𝑧 + 𝑎)
SolvingthedifferentialequationfordT/dz• Notethatoursolutionisdifferenttointhenotes,aswehaveaconductionterm
• Weneedtoaddthelinesinktotheequationsoweobtain
𝜕𝑇(𝑥, 𝑧)𝜕𝑍 = −
12𝜋𝑘
𝑧 + 𝑎(𝑥" + 𝑧 + 𝑎 ") −
𝑧 − 𝑎(𝑥" + 𝑧 − 𝑎 ")
𝜕𝑇(𝑥, 𝑧)𝜕𝑍 = −
12𝜋𝑘
𝑧 + 𝑎(𝑥" + 𝑧 + 𝑎 ")
FindingthesurfaceheatconductionusingFourier’sLaw• Aswehaveincludedconductioninourequation,wecansolveforthesurfaceheatflowusingFourier’slawofthermalconduction
𝑞 = −𝑘𝑑𝑇𝑑𝑧
𝑞(𝑥, 𝑧) =12𝜋
𝑧 + 𝑎(𝑥" + 𝑧 + 𝑎 ") −
𝑧 − 𝑎(𝑥" + 𝑧 − 𝑎 ")
ObtainingtheGreen’sFunction
• Foroursurfaceheatflow,wesolveforz=0
ThisgivesusaGreen’sfunctionwhichwecanthenconvolvewithanarbitrarysource.
𝐺 =1𝜋
𝑧𝑥" + 𝑧"
𝑞 𝑥, 0 =1𝜋
𝑎𝑥" + 𝑎"
ConvolvingwiththeGreen’sFunction
𝑞 𝑥 =𝑢𝜋B
𝑧𝜏(𝑧)𝑥" + 𝑧" 𝑑𝑧
E
F
• Ourheatsourceisnotalinesourceatdepth,it’saplane,wheretheheatflowatdepthzisgivenby
𝑞 𝑧 = 𝑢𝜏 𝑧
• WecanthenconvolvethiswithourGreen’sFunctiontogetoursurfaceheatflowforthissource.
ConvolvingwiththeGreen’sFunction
• Wecanuseourinitialformulaforthenormalstress,givenby𝜏 𝑧 = 𝜇𝜌I𝑔𝑧• Wecanthenplugthisintoourequationforthesurfaceheatflow.
𝑞 𝑥 =𝜇𝜌I𝑔𝑢𝜋 B
𝑧"
𝑥" + 𝑧"E
F𝑑𝑧
SolvingthisintegralWecanrearrangetheequationintheintegral
5K
8KL5K=5
KL8KM8K
8KL5K
= 1 −𝑥"
𝑥" + 𝑧"
𝑞 𝑥 =𝜇𝜌I𝑔𝑢𝜋 B 1 −
𝑥"
𝑥" + 𝑧"E
F𝑑𝑧
=𝜇𝜌I𝑔𝑢𝜋 𝐷 − 𝑥 tanMR
𝐷𝑥
SlipRateRed:20mm/yrBlue:40mm/yr
1HFU=41.8mW/m2
ComparingModelwithSurfaceObservationsofHeatFlow
• Lachenbruch andSass(1980)observenoperturbedsurfaceheatflowmeasurementspredictedbytheirmodel.
ChangingFrictionalCoefficientCannotProduceHeatFlowAnomaly
AccountingforHydrothermalCirculation
𝑞 𝑥 =𝜇(𝜌I−𝜌S)𝑔𝑢
𝜋 B 1 −𝑥"
𝑥" + 𝑧"E
T𝑑𝑧
=𝜇(𝜌I−𝜌S)𝑔𝑢
𝜋 𝐷 − 𝑑 + (𝑥 tanMR𝑑𝑥 −𝑥 tan
MR 𝐷𝑥)
HydrothermalCirculationBroadensHeatAnomalyProfile