科里奥利效应.docx

科里奥利效应.docx

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科里奥利效应

Corioliseffect

Figure1:

Intheinertialframeofreference（upperpartofthepicture）,theblackobjectmovesinastraightline,withoutsignificantfrictionwiththedisc.However,theobserver（reddot）whoisstandingintherotating（non-inertial）frameofreference（lowerpartofthepicture）seestheobjectasfollowingacurvedpathduetotheCoriolisandcentrifugalforcespresentinthisframe.

Inphysics,theCorioliseffectisadeflectionofmovingobjectswhentheyareviewedinarotatingreferenceframe.Inareferenceframewithclockwiserotation,thedeflectionistotheleftofthemotionoftheobject;inonewithcounter-clockwiserotation,thedeflectionistotheright.ThemathematicalexpressionfortheCoriolisforceappearedinan1835paperbyFrenchscientistGaspard-GustaveCoriolis,inconnectionwiththetheoryofwaterwheels,andalsointhetidalequationsofPierre-SimonLaplacein1778.Andevenearlier,ItalianscientistsGiovanniBattistaRiccioliandhisassistantFrancescoMariaGrimaldidescribedtheeffectinconnectionwithartilleryinthe1651AlmagestumNovum,writingthatrotationoftheEarthshouldcauseacannonballfiredtothenorthtodeflecttotheeast.[1]Earlyinthe20thcentury,thetermCoriolisforcebegantobeusedinconnectionwithmeteorology.

Newton'slawsofmotiongovernthemotionofanobjectina（non-accelerating）inertialframeofreference.WhenNewton'slawsaretransformedtoauniformlyrotatingframeofreference,theCoriolisandcentrifugalforcesappear.Bothforcesareproportionaltothemassoftheobject.TheCoriolisforceisproportionaltotherotationrateandthecentrifugalforceisproportionaltoitssquare.TheCoriolisforceactsinadirectionperpendiculartotherotationaxisandtothevelocityofthebodyintherotatingframeandisproportionaltotheobject'sspeedintherotatingframe.Thecentrifugalforceactsoutwardsintheradialdirectionandisproportionaltothedistanceofthebodyfromtheaxisoftherotatingframe.Theseadditionalforcesaretermedeitherinertialforces,fictitiousforcesorpseudoforces.[2]TheyallowtheapplicationofNewton'slawstoarotatingsystem.Theyarecorrectionfactorsthatdonotexistinanon-acceleratingorinertialreferenceframe.

PerhapsthemostcommonlyencounteredrotatingreferenceframeistheEarth.TheCorioliseffectiscausedbytherotationoftheEarthandtheinertiaofthemassexperiencingtheeffect.BecausetheEarthcompletesonlyonerotationperday,theCoriolisforceisquitesmall,anditseffectsgenerallybecomenoticeableonlyformotionsoccurringoverlargedistancesandlongperiodsoftime,suchaslarge-scalemovementofairintheatmosphereorwaterintheocean.Suchmotionsareconstrainedbythe2-dimensionalsurfaceoftheearth,soonlythehorizontalcomponentoftheCoriolisforceisgenerallyimportant.ThisforcecausesmovingobjectsonthesurfaceoftheEarthtoveertotheright（withrespecttothedirectionoftravel）inthenorthernhemisphere,andtotheleftinthesouthern.Ratherthanflowingdirectlyfromareasofhighpressuretolowpressure,astheywouldonanon-rotatingplanet,windsandcurrentstendtoflowtotherightofthisdirectionnorthoftheequator,andtotheleftofthisdirectionsouthofit.Thiseffectisresponsiblefortherotationoflargecyclones（seeCorioliseffectsinmeteorology）.

Gaspard-GustaveCoriolispublishedapaperin1835ontheenergyyieldofmachineswithrotatingparts,suchaswaterwheels.[3]Thatpaperconsideredthesupplementaryforcesthataredetectedinarotatingframeofreference.Coriolisdividedthesesupplementaryforcesintotwocategories.Thesecondcategorycontainedaforcethatarisesfromthecrossproductoftheangularvelocityofacoordinatesystemandtheprojectionofaparticle'svelocityintoaplaneperpendiculartothesystem'saxisofrotation.Coriolisreferredtothisforceasthe"compoundcentrifugalforce"duetoitsanalogieswiththecentrifugalforcealreadyconsideredincategoryone.[4][5]Bytheearly20thcenturytheeffectwasknownasthe"accelerationofCoriolis".[6]By1919itwasreferredtoas"Coriolis'force"[7]andby1920as"Coriolisforce".[8]

In1856,WilliamFerrelproposedtheexistenceofacirculationcellinthemid-latitudeswithairbeingdeflectedbytheCoriolisforcetocreatetheprevailingwesterlywinds.[9]

UnderstandingthekinematicsofhowexactlytherotationoftheEarthaffectsairflowwaspartialatfirst.[10]Lateinthe19thcentury,thefullextentofthelargescaleinteractionofpressuregradientforceanddeflectingforcethatintheendcausesairmassestomove'along'isobarswasunderstood.[citationneeded]

[

Innon-vectorterms:

atagivenrateofrotationoftheobserver,themagnitudeoftheCoriolisaccelerationoftheobjectisproportionaltothevelocityoftheobjectandalsotothesineoftheanglebetweenthedirectionofmovementoftheobjectandtheaxisofrotation.

ThevectorformulaforthemagnitudeanddirectionoftheCoriolisaccelerationis

where（hereandbelow）

istheaccelerationoftheparticleintherotatingsystem,

isthevelocityoftheparticleintherotatingsystem,andΩistheangularvelocityvectorwhichhasmagnitudeequaltotherotationrateωandisdirectedalongtheaxisofrotationoftherotatingreferenceframe,andthe×symbolrepresentsthecrossproductoperator.

TheequationmaybemultipliedbythemassoftherelevantobjecttoproducetheCoriolisforce:

.

Seefictitiousforceforaderivation.

TheCorioliseffectisthebehavioraddedbytheCoriolisacceleration.TheformulaimpliesthattheCoriolisaccelerationisperpendicularbothtothedirectionofthevelocityofthemovingmassandtotheframe'srotationaxis.Soinparticular:

▪ifthevelocityisparalleltotherotationaxis,theCoriolisaccelerationiszero.

▪ifthevelocityisstraightinwardtotheaxis,theaccelerationisinthedirectionoflocalrotation.

▪ifthevelocityisstraightoutwardfromtheaxis,theaccelerationisagainstthedirectionoflocalrotation.

▪ifthevelocityisinthedirectionoflocalrotation,theaccelerationisoutwardfromtheaxis.

▪ifthevelocityisagainstthedirectionoflocalrotation,theaccelerationisinwardtotheaxis.

Thevectorcrossproductcanbeevaluatedasthedeterminantofamatrix:

wherethevectorsi,j,kareunitvectorsinthex,yandzdirections.

[Causes

TheCorioliseffectexistsonlywhenoneusesarotatingreferenceframe.Intherotatingframeitbehavesexactlylikearealforce（thatistosay,itcausesaccelerationandhasrealeffects）.However,Coriolisforceisaconsequenceofinertia,andisnotattributabletoanidentifiableoriginatingbody,asisthecaseforelectromagneticornuclearforces,forexample.Fromananalyticalviewpoint,touseNewton'ssecondlawinarotatingsystem,Coriolisforceismathematicallynecessary,butitdisappearsinanon-accelerating,inertialframeofreference.Forexample,considertwochildrenonoppositesidesofaspinningroundabout（carousel）,whoarethrowingaballtoeachother（seeFigure1）.Fromthechildren'spointofview,thisball'spathiscurvedsidewaysbytheCorioliseffect.Supposetheroundaboutspinscounter-clockwisewhenviewedfromabove.Fromthethrower'sperspective,thedeflectionistotheright.[11]Fromthenon-thrower'sperspective,deflectionistoleft.ForamathematicalformulationseeMathematicalderivationoffictitiousforces.

Anobserverinarotatingframe,suchasanastronautinarotatingspacestation,veryprobablywillfindtheinterpretationofeverydaylifeintermsoftheCoriolisforceaccordsmoresimplywithintuitionandexperiencethanacerebralreinterpretationofeventsfromaninertialstandpoint.Forexample,nauseaduetoanexperiencedpushmaybemoreinstinctivelyexplainedbyCoriolisforcethanbythelawofinertia.[12][13]SeealsoCorioliseffect（perception）.Inmeteorology,arotatingframe（theEarth）withitsCoriolisforceprovesamorenaturalframeworkforexplanationofairmovementsthananon-rotating,inertialframewithoutCoriolisforces.[14]Inlong-rangegunnery,sightcorrectionsfortheEarth'srotationarebaseduponCoriolisforce.[15]Theseexamplesaredescribedinmoredetailbelow.

TheaccelerationenteringtheCoriolisforcearisesfromtwosourcesofchangeinvelocitythatresultfromrotation:

thefirstisthechangeofthevelocityofanobjectintime.Thesamevelocity（inaninertialframeofreferencewherethenormallawsofphysicsapply）willbeseenasdifferentvelocitiesatdifferenttimesinarotatingframeofreference.Theapparentaccelerationisproportionaltotheangularvelocityofthereferenceframe（therateatwhichthecoordinateaxeschangedirection）,andtothecomponentofvelocityoftheobjectinaplaneperpendiculartotheaxisofrotation.Thisgivesaterm

.Theminussignarisesfromthetraditionaldefinitionofthecrossproduct（righthandrule）,andfromthesignconventionforangularvelocityvectors.

Thesecondisthechangeofvelocityinspace.Differentpositionsinarotatingframeofreferencehavedifferentvelocities（asseenfromaninertialframeofreference）.Inorderforanobjecttomoveinastraightlineitmustthereforebeacceleratedsothatitsvelocitychangesfrompointtopointbythesameamountasthevelocitiesoftheframeofreference.Theeffectisproportionaltotheangularvelocity（whichdeterminestherelativespeedoftwodifferentpointsintherotatingframeofreference）,andtothecomponentofthevelocityoftheobjectinaplaneperpendiculartotheaxisofrotation（whichdetermineshowqui