Heat Flow across the San Andreas Fault

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Transcript of Heat Flow across the San Andreas Fault

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