平均温度

InterpretingthetemperatureofwateratcoldspringsandtheimportanceofgravitationalpotentialenergyMichaelMangaandJamesW.KirchnerDepartmentofEarthandPlanetaryScience,UniversityofCalifornia,Berkeley,Berkeley,California,USAReceived25November2003;revised22March2004;accepted31March2004;published19May2004.[1]Circulatinggroundwatertransportsheat.Ifgroundwaterflowvelocitiesaresufficientlyhigh,mostofthesubsurfaceheattransportcanoccurbyadvection.Thisisthecase,forexample,intheCascadesvolcanicarcwheremuchofthebackgroundgeothermalheatistransportedadvectivelyandthendischargedwhenthegroundwateremergesatsprings.Thetemperatureofspringwatercanthusbeusedtoinferthegeothermalheatflux.Ifspringwatertemperatureismanydegreeswarmerthantheambienttemperature,asitisathotsprings,determiningtheheatdischargedatspringsisstraightforward.Atlarge-volumecoldsprings,however,thegeothermalwarmingofwaterissmallbecausetheaddedheatisdilutedinalargevolumeofwater.Weshowthatinordertointerpretthetemperatureofcoldspringswemustaccountforthreeprocesses:(1)conversionofgravitationalpotentialenergytoheatthroughviscousdissipation,(2)conductionofheattoorfromtheEarth’ssurface,and(3)geothermalwarming.UsingspringtemperaturedatafromthecentralOregonCascadesandMountShasta,California,weshowthatthewarmingduetosurfaceheatexchangeanddissipationofgravitationalpotentialenergycanbecomparabletothatduetogeothermalheating.Unlesstheseconfoundingsourcesofheatingaretakenintoaccount,estimatesofgeothermalheatfluxderivedfromtemperaturesofcoldspringscanbeincorrectbylargefactors.INDEXTERMS:1829Hydrology:Groundwaterhydrology;8130Tectonophysics:Heatgenerationandtransport;8424Volcanology:Hydrothermalsystems(8135);KEYWORDS:geothermalheat,gravitationalpotentialenergy,springCitation:Manga,M.,andJ.W.Kirchner(2004),Interpretingthetemperatureofwateratcoldspringsandtheimportanceofgravitationalpotentialenergy,WaterResour.Res.,40,W05110,doi:10.1029/2003WR002905.1.Introduction[2]Watermovingthroughagroundwatersystemtrans-portsheatandthuschangesthesubsurfacetemperaturedistribution.Temperaturecanthereforebeusedasatracerofhydrologicprocessesandfortestingconceptualhydro-geologicmodels[e.g.,Andrewsetal.,1982].Temperatureoffersseveraladvantagesasatracerbecauseitiseasy,quick,andinexpensivetomeasureaccuratelyinthefield.Thetemperatureofgroundwateralsoprovidesinsightintothesubsurfacegeologicalprocessesthatgenerateheat.[3]Severalpreviousstudieshavedevelopedmodelsforpredictingorinterpretingspringtemperatures.Theseincludeanalyticalmodelstointerpretseasonaltemperaturefluctua-tions[e.g.,Bundschuh,1993]andnumericalmodelstostudyregional-scaleflows[e.g.,ForsterandSmith,1989].Thegeneralconclusionofthesemodelsisthatthepredictedtemperatureofspringwaterdependsonthevolumefluxofwater.Foragivenaquifer,asthegroundwatervelocityincreases,theheataddedbygeothermalwarmingisdilutedintolargervolumesofwater,andconsequently,springtemperaturesbecomecolder.Infact,springtemperaturescanbecolderthanthemeanannualsurfacetemperatureatthedischargeelevationifthewaterisrechargedatmuchhigherelevations,asfirstnotedbyAlexandervonHumboldtin1844[Davis,1999].Incontrast,forverylowvelocities(typicallyaresultoflowpermeabilities),thesubsurfacetemperaturegradientisnearlyundisturbedbygroundwaterflowandthetemperatureofspringwaterwillbeclosetothemeanannualsurfacetemperatureatthedischargeelevation.Thewarmestspringtemperaturesoccurforanintermediaterangeofvelocitiessuchthatgroundwaterflowremovesmostofthegeothermalheatfluxadvectively,buttheaddedheatisnotdilutedbylargevolumesofwater[ForsterandSmith,1989].Inthislattercase,thetemperatureofspringwatercanbeusedtoinferthebackgroundgeothermalheatfluxprovidedtheareafromwhichheatiscollectedisknown[e.g.,Ingebritsenetal.,1989].[4]Inthispaperwedevelopananalyticalmodelthatincludesacontributiontothethermalbudgetofspringwaterthatisusuallyneglected(seeDomenico[1972,p.160]foracalculationinwhichitisincluded):theconversionofgravitationalpotentialenergy(hereinafterreferredtoasGPE)toheat.WethenapplythismodeltocoldspringsintheOregonandCaliforniaCascades.Forthesespringsweshowthatthechangeinelevationanddischargearesuffi-cientlylargethatthedissipationofgravitationalpotentialenergysometimesdominatestheinferredwarmingofdis-chargedspringwater.WearguethatignoringGPEcanleadtosignificanterrorsintheamountofgeothermalwarmingofwaterdischargedatcoldsprings.2.LocalEnergyBalance[5]ConsiderthesteadyflowofwaterthroughtheaquifershowninFigure1.ThethermalenergybalanceforaparcelCopyright2004bytheAmericanGeophysicalUnion.0043-1397/04/2003WR002905$09.00W05110WATERRESOURCESRESEARCH,VOL.40,W05110,doi:10.1029/2003WR002905,20041of8ofwaterwithheighthandmovingwithporevelocityuandDarcyvelocityuf,wherefistheporosity,isgivenbyrwfhCwdTdt|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}rateofchangeofthermalenergy¼rwghfusinq|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}gravitationalpotentialenergyGPEðÞdissipationþkrTsTðÞd|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}heattransferto=fromsurfaceþQb|{z}geothermalwarming:ð1ÞHeresubscriptswandrindicatepropertiesofwaterandsaturatedrock,respectively,kisthermalconductivity,risdensity,Cisspecificheat,gisgravitation