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Geodynamics456‐556

Problemset#3.IsostasyandRidgePush(Due in class on May 6)

A.Whydoonlyoceansgetsubducted?

Thestabilityof thecontinentsandthe tendency foroldoceanicplates tosubductback intothemantlecanbeexplainedbycalculatingtheaveragedensityofeachplate.Thedensity,ρ,ofrockisrelatedtotemperature,T,accordingto

ρ=ρ0(1−αT) (1)

where ρo is the rock density at surface temperature and pressure (S.T.P.) and α is the thermalexpansioncoefficientoftherock.Equation1saysthatrocksbecomelessdenseastheyareheatedbecausetheyexpand. In the followingquestions,refer to thetemplatesketched inFigure1andusethevalueslistedinTable1.

1.Determinetheaveragedensityofanoldoceanicplateusingthefollowingprocedure.

(a)First,calculatetheaveragetemperaturesofthecrustandtheaveragetemperatureofthesub‐crustaluppermantle(i.e.thelithosphericmantle).Assumealineargeothermalgradientfrom0◦

CatthesurfacetotemperatureTBatthebaseoftheplate.(b)Useequation1tocalculatetheaveragedensitiesofthecrustandthelithosphericmantlefrom

theirtemperatures.(c)Calculatetheaveragedensityoftheoceanicplatefromthetwodensitiesyoudeterminedin

part(b),bearinginmindtherelativethicknessesofthecrustandlithosphericmantle.

2.Nowcalculatetheaveragedensityofacontinentalplatebyrepeatingtheprocedureinpart(1).

3.Thirdly,calculatethedensityoftheconvectingmantleimmediatelybelowtheplate(i.e.theasthenosphere)usingequation1.

4.Comparetheaveragedensitiesyouhavejustcalculatedforanoceanicplate,acontinentalplateandtheasthenosphere.Youshouldfindthattheoldoceanicplateisdenserthantheconvectingmantlebeneath,whilstcontinentalplatesarelessdensethantheconvectingmantlebeneath.Thusoceanicplatesareunstableandwillsinkbackdownintothemantle,givenachance.Continentalplatesfloatstablyontopoftheconvectingmantle.

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B.Whyareoceansdeep?

Thedifferenceinelevationbetweencontinentsandoceanscanbeexplainedusingtheprincipleofisostasy.Isostasysaysthatifweconsidertwoblocksofdifferentdensityandthicknessfloatinginafluid,thepressureatsomereferencedepthbelowbothblocksmustbeequal.Thepressure,P,atthebaseofaunitcolumnofrockisgivenby

P=ρgh(2)

whereρisthedensityandhisthecolumnthickness.Theaimofthefollowingcalculationsistofindthedifferenceinheightbetweencontinentsandoceans,termedΔHinFigure2.

1. Usingequation2,writedownanexpressionforthepressureatthereferencedepthbeneathanoceanicplate.Thepressuresgeneratedbythewatercolumnandtherockcolumncanbeadded.UsetheaveragedensityfortheoceanicplateyoucalculatedinpartA(1).

2. Similarly,writedownanexpressionforthepressureatthereferencedepthbeneaththecontinentalplateillustratedinFigure2.

3. Setthetwoexpressionsforpressureatthereferencedepthequal.YoushouldfindthatΔHistheonlyunknownintheresultingequation.RearrangeyourequationandcalculateΔH.

4. Youshouldfindthatthedifferenceinheightbetweencontinentsandoceansisjustover6km.Infact,oceanicabyssalplainslieatadepthofbetween5and6kmandtheaverageheightofthecontinentsabovesea‐levelis0.8km.

C. Isostasy and ice sheets 1.ConsidertheLaurentideicesheetthatwas,atthelastglacialmaximum(LGM,about20Ka),3kmthickinitscenter.Calculatetheexpectedsubsidenceofthecrustifthisicesheetwasinplacelongenoughforisostaticequilibriumtohavebeenachieved.Assumedensitiesoficeandmantletobe910and3300kg/m3,respectively.Notethattheanswerisindependentofwhatyouassumeforthedensityofthecrust.

D. Average Continental crustal thickness 1.Ifthemeanaltitudeofcontinentsis0.8km,andthemeanoceanicdepthis3.5km,andthemeanoceaniccrustalthicknessis6km,howthickisthemeancontinentalcrust?Assumethatthedensitiesinvolvedare:seawater(1000kg/m3),continentalandoceaniccrust(2700and3000kg/m3,respectively)anduppermantle(3300kg/m3).

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E.Isostasy,potentialenergy,andtheforcesofridge‐push

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Forpartb/ofthefollowingproblem:Ifyoucannotintegrate,thenassumeaconstantdensityforthecompletelithosphere(similartothesketchesabove)toestimatetheGravitationalPotentialEnergyforeachcolumn(theareainthetriangle).

d/Assumingthatlithosphericrockshaveastrengthof70MPa,wouldthegravitationalforcesdeterminedinb/producelithosphericdeformation?