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TopographyT.Perron–12.001–April2,2010

We’llspendalargefractionofthesecondhalfofthecoursediscussingEarth’ssurface.Todaywe’lldotwothings:First,we’lldiscussthewaystopographyiscommonlyrepresentedinmaps,andhowitinteractswithgeologicstructuresatdepth.Thiswillhelpprepareyoufortoday’slab,aswellasthegeologicmappinglabyouwillbeginnextweek.Second,we’lldiscussfirst‐ordercontrolsontheshapeofEarth’ssurface,startingatthelargestscales.I.TopographicmapsThisismostlycommonsenseandgeometry,butitcanbetrickytovisualize.1.Mapprojections:waysofrepresentingasphericalsurfacein2D

• Geographic/“PlateCarree”/SimpleCylindrical:convenient(x=lon,y=lat),butgeometricallyunfortunate[PPT]

• MosttopomapsofsmallregionswillbeinUniversalTransverseMercator(UTM),orsomevariationthereon.

o Mercator:projectionontocylinderparalleltorotationaxiso UTM:projectionontocylinderperpendiculartorotationaxis,rotated

aroundEarthin60incrementstomake60slices(“zones”)[PPT]2.ContourMaps

• Contourlinesonmapso Needtoplot3Ddataon2Dsurface.Contoursarelinesindicating

constantvalueof“z”coordinate–canbeelevation,oranythingelseo Howdocontoursshow:

Slope:contourspacingreflectsgradientofz Valleys:contoursmakeVspointingupslope(upvalley) Ridges:contoursmakeVspointingdownslope(downridge) Peaks:concentricclosedcontours,withcontourvalues

increasinginward Sinks(localminima):concentricclosedcontours,withcontour

valuesdecreasinginward.Usuallymarkedwithhatches.(Whythespecialnotation,i.e.,whyaresinksuncommon?)

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• Drainageo Ingeneral,waterflowsdownslope,soflowpathsusuallyarenormal

tocontourso Riversgenerallyfollowvalleys.Riverpathscanbeidentifiedby

followingthechainofVsthatpointupslopeo Watersheds

Regionsthatalldraintoacommonpoint. Everypointonthemapfallswithinawatershed Drainagedividesaretheboundariesbetweenwatersheds.

Theyfollowridgelines,wheretheflowdiverges.SodrainagedividescanbeidentifiedbyfollowingthechainofVsthatpointdownslope,startingattheoutletofawatershedandmovingupslopeuntilthedividereachesapeak(localmaximum).

• Cross‐sectionso Canbeconstructedbydrawingatransectacrossacontourmapand

measuringhorizontalpositionwheretransectintersectseachcontour.o Thenwecandrawaplotofz(elevation)vs.x(horizontaldistance)o Verticalexaggeration

Theratioofthelengthofagivenunitonthezscaletothelengthofthesameunitonthexscale.

Topographyissubtle,soverticalexaggeration>1isoftenused,especiallyoverlongdistances.

Usuallyreportedas,e.g.,2:13.Geologiccross‐sections

• Howcanweinfertheorientationsofplanesfromthegeometryatwhichtheyintersectthesurface?

• RuleofVso Horizontalbedsfollowcontourlines[PPT:GrandCanyon]o Ifdippingbedintersectsvalley,itssurfaceoutcropwillmakeaVin

mapviewthatpointsinthedirectionofdip.[PPT:Blockmodels]o ThesharpertheV,theshallowerthedip(verticalbedsmakenoV)o Onlyexception:ifabedisveryshallowlydippingatananglelessthan

thevalleyslope,itwillmakeaVupstream,likeahorizontalbedwould• Apparentthicknessofbeds[PPT:Blockmodels]

o Themorenearlyparallelthebedistothesurface,thethickerthebedappears

o Themoreperpendicular,thethinnerthebedappears4.Howdowemeasuretopography?

• Levelingsurveys• Stereophotos• RADAR

o Altimetryo Interferometric

• Laseraltimetry

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II.Large­scaletopography0.Tozerothorder,Earthisspherical.Why?

• Earthisheldtogetherbyitsowngravity• Overlongtimescales,Earthbehavesasafluid.Fluidsseektheirownlevel;

thatis,theytendtoevolvetowardhydrostaticequilibriumunlessactedonbyotherforces.

• Considerperturbationsintheshapeofaspherical,non‐rotatingbody.Theseperturbationswillinducepressuregradients,whichwilldriveflowthatrestoresthesphericalshape.

• Whethersphericalshapeisachieveddependsonpressuregradient,viscosity,

andtime.[PPT:Earthvs.Hyperion]o ViscositydependsonP,Tlargerbodiesdeformfasterevenfor

samepressuregradiento LargestirregularlyshapedmoonisHyperion(Saturn),R~150kmo Smallerones(Deimos,Phobos,asteroids,someKBOs)allirregular

1.Earthisalmostspherical.

• Oblatespheroidwithflattening(a‐c)/a=1/298.26• Ifa~6370km,thena‐c~21km• LargesttopofeaturesonEarth:

o Mt.Everest=8.8kmo MarianaTrench=‐11km

• SoEarth’sflatteningisitslargesttopographicfeature• WhyisEarthoblate?

o Theansweronceagainishydrostaticequilibrium,andthecontributionofrotationtointeriorpressures

o Atanypointintheinterior,Pnet=Pgravitation–Pcentrifugalo BecausePcentrifugalissmallerathigherlatitudes,arotatingsphere

developsapressuregradientthatdrivesflowtowardtheequator,andplanetsdeveloparotationalbulge.

o Thefirst‐ordermodelofEarth’sshapeisthereforeanellipsoid.[SKETCH:XSofellipsoid,showingequatorialradiusa>polarradiusc]

2.Earthisalumpyellipsoid

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• Anotherwaytothinkaboutthisellipsoidalsurfaceisthatitisasurfaceofapproximatelyequalgravitationalpotential.Gravitationalpotentialisafieldwhosegradientisthegravitationalaccelerationvector,g(g=‐grad(potential)=GM/r2r_hat).

• Thisellipsoidofapproximatelyequalgravitationalpotentialisthe“referencegeoid”.

• IfwemeasureactualgravitationalpotentialovertheEarth’ssurface,therearedeviationsfromthereferencegeoid(“anomalies”)ofasmuchas100m.[PPT:Geoidanomalymap].Someoftheseanomalieshaveaknownexplanation:

o SubductingslabscreatepositiveanomaliesoverAndes,Indonesiao Remnantdepressionsfromicesheetscreatesnegativeanomalyover

HudsonBayo Someanomalies(negativeoverSouthernIndia)unexplained

• Anomaliesingravitationalacceleration(thegradientofthegeoid)highlightshort‐wavelengthfeaturesevenbetter.[PPT:GravityanomaliesfromGRACE]

3.Globaldistributionofelevations

• Earth’ssurfaceelevationsarebimodal[PPT:Shadedrelief,thenPDFandCDFofelevations].Thesedistributionsareknownasthe“hypsometry”.

• Meanelevations:o Continents:+0.8km(oftencalled“freeboard”)o Oceanbasins:‐3.8kmo Whatcontrolstheseelevations?Isostaticbalance:

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o So,tofirst,order,bimodalelevationsreflectbimodalityof

compositionandcrustalthickness4.Isostasyandtopography

• Whateffectwillerosionofcontinentalcrusthaveonelevations?

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• Lesson:non‐uniformerosioncancauseuplift• ThismaybeoneofthereasonswhymarginsofTibetanPlateauarehigher

thaninterior[PPT:Satelliteimageofplateau]5.Topographiceffectsofmantleprocesses

• Isostaticrebound:time‐dependentisostaticreponse.Mantleisviscoelastic.o Upliftinresponsetoremovaloficeloads[PPT:HudsonBay

shorelines]o Ongoingsubsidenceinresponsetosedimentdepositioninlargeriver

deltas,despiterecentchangesinload(e.g.Mississippi)• Flexure:Lithosphererespondstoloadsasanelasticshell

o Troughandforebulgearoundoceanislandso Differential“warping”ofglaciallakeshorelinesfollowingremovalof

icesheetloads• Dynamictopography:deflectionofEarth’ssurfaceduetoflowinupper

mantle(=topography–isostasy).Amplitudecanbeasmuchas1km–aslargeasfreeboard!

o Mitrovica:HudsonBayupliftislessthanexpectedforpostglacialrebound,presumablyb/cofdownwellingbeneathCanada

o UpwellingbeneatheastcoastofNorthAmericamayexplainwhyAppalachiansarestillhigh,despitelackoforogeniceventsforhundredsofMyr.