xbm h1Cxt Fig12Fr1Interfaceevolutionxtinmfromitsrestpositionh1forseveraltimesduringtheslowdownoftheboatintheframeoftheboatwhichisrepresentedbytheblackrectangularboxSameparametersasinF ID: 416686
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200M.J.Mercieretal.:Resurrectingdead-waterphenomenon xb(m) h1C.x;t/ Fig.12.Fr1.Interfaceevolution.x;t/(inm)fromitsrestpositionh1forseveraltimesduringtheslowdownoftheboat,intheframeoftheboatwhichisrepresentedbytheblackrectangularbox.SameparametersasinFig. 11 . theleftsideofthetankanditsshaperemainsalmostun-changed(t�75s).Althoughnotpresentedhere,wehaveveriedthatthissolitarywavepropagatingupstreamiswelldescribedbyaKorteweg-deVriesmodelsinceitsamplitudeisalwayssmallerthan0:4h1( Grueetal. , 1999 ).Whentheboatstartsagain,theprocessisrepeatedandanewdepres-sionisgenerated.3.2.2IntheframeoftheboatMoreinformationcanbeobtainedwhenfollowingthedy-namicsintheframeassociatedwiththeboat.Wearemorespecicallyinterestedinwhatsetstheamplitudeandfre-quencyoftheoscillationsoftheboatwhenFr1.InFig. 12 ,wehavesuperposedtheinterfacialdisplace-mentsintheframeoftheboat(representedasablackrect-angle)fordifferenttimesextractedfromFig. 11 .Wecanobservethatthewavesgetclosertothesternoftheboatastheiramplitudegrows.ThisismainlyduetothedecreaseofspeedoftheboatsincewecanseeinFig. 11 thatthewavecrestsevolvewithaconstantspeed.Thenonlinearnatureoftheinterfacialwavesisalsoclearlyvisibleasthewavefrontssteepen,thelimitofgrowthbeingsetherebythetimewhenthewaveshitthehull(27t32s).Thesedownstreamfea-turesarefullynonlinear(amplitudeslargerthan0:6h1)andhavesimilaraspectstoobservationsmadeinthetranscriti-calregimeby MelvilleandHelfrich ( 1987 )andreproducednumericallyby Grueetal. ( 1997 ),althoughcorrespondingheretowavesinanon-galileanframe.Itismoreoversur-prisingsinceasobservedonthegreencurveinFig. 7 ,theFroudenumberdoesnotexceed0.7.Theseobservationsaddtothecommentsmadepreviouslyconcerningtheinadequacyofmodelsbasedonsteadyobjectinstratieduid.Hereagainweseeupstreamthesolitarydepressionbelowtheboatescapingatthebowasasingleoscillation.Fortimest32s,onecanobservethatthedecapitatedrstcrestdisappearsbelowtheboat,thenextwavedepres-siontakespositionbelowtheboatandtheinitiallysecondel-evationofthepacketbecomestheclosesttothestern.Hence, (a) v.t/ t(s) x(m)xb(m)(b)(c) t(s) .x;t/(m)Fig.13.Oscillatingregime(Fr1).Panel(a)presentsthespeedoftheboatversustime,whilepanels(b)and(c)showsthespatio-temporaldiagramsoftheinterfacialdisplacementsinthelabora-toryframeandintheoneoftheboatrespectively.Theblacklinesrepresentthebowandstern.Experimentalparameters:h1D5:0cm,h2D14:0cm,1D0:9980gcm�3,2D1:0157gcm�3,SbD24cm2,FtD16:3mN. duringanoscillationoftheboat,thewavetrainslidesofonewavelengthcomparedtotheboat,repeatingitself.Nochangeinthedominantwavelengthofthewavetrainisvisible.Wecannowestimatethefrequencyoftheoscillationsoftheboattobe fbDcg (2) withcgthegroupvelocityofthewavetrainattherearoftheboatanditsdominantwavelength,correspondingapproxi-matelytothedistancebetweenthersttwowavecrests.TheexperimentpresentedinFig. 11 allowsanobservationofthesteepeningprocessbutdoesnotpresentmanyoscil-lations.Inordertoconrmthevalidityofformula( 2 ),weconsideranotherexperimentwithFtD16:3mNinFig. 13 .FromFig. 13 a,onecanestimatethefrequencyoftheoscilla-tionsoftheboattobefb0:07�0:08Hz.ThegroupspeedandwavelengthofthewavetrainobtainedfromthediagraminFig. 13 b,correspondstocg0:035�0:040ms�1and0:5m.Indeed,weobtaincg=0:07�0:08Hzshow-ingverygoodagreementwithfb.AfterrevisitingexperimentssimilartotheonesperformedbyEkman,wehaveconsiderednewsituations.Indeed,dead-waterphenomenoncanbestudiedinmorecomplex(andreal-istic)stratications,whentheuidhasmorethantwolayersofdifferentdensities.Wenowturntonewexperimentswithathree-layeruidorwithacontinuouslystratieduidwithapycnocline,wherethecomplexdynamicshasalsobeenob-served. Nonlin.ProcessesGeophys.,18,193– 208 ,2011www.nonlin-processes-geophys.net/18/193/2011/ M.J.Mercieretal.:Resurrectingdead-waterphenomenon201 Table2.Experimentalparametersusedfortheexperimentswithathree-layeruid. Parameterssymbolsvaluesunits Fluid1density10:9967gcm�3depthh15:0cmFluid2density21:0079gcm�3depthh23:0cmFluid3density31:0201gcm�3depthh35:5cmInterface12meandensityN12D2C1 21:0023gcm�3densityjump112D2�1 N120:0112maximumphasespeedcm ;12Dq 112gh1h2 h1Ch20:0147ms�1FroudenumberFr12DU cm ;124.1–7.5Interface23meandensityN23D3C2 21:014gcm�3densityjump123D3�2 N230:012maximumphasespeedcm ;23Dq 123gh2h3 h2Ch30:0150ms�1FroudenumberFr23DU cm ;234.0–7.3Modess/amaximumphasespeedcms=a0:073=0:044ms�1FroudenumberFrs=aDU cms=a1.5/2.4–0.6/0.9 zx - 6123 12.x;t/ 23.x;t/Fig.14.Pictureillustratingthedead-waterphenomenonforathree-layeruid. 4Experimentswithathree-layeruidWeconsiderathree-layeruidofdepthhianddensityi,iD1;2;3.Twointerfacesmustbeconsiderednow,ij.x;t/correspondingtotheinterfacialdisplacementsbetweenlay-ersiandj.ApicturesummarizesthesetupinFig. 14 andthemainparametersaregiveninTable 2 .Onlyonegivenstraticationwillbediscussedhere.Weusetechnique1againtoextractthepositionsofthetwointerfaces12and23,alongwiththepositionoftheboat.4.1AnalyticsAlthoughthethree-layercaseisasimpleextensionofthewelldocumentedtwo-layeronediscussedinprevioussec-tion,weherereproduceexplicitexpressionsforthephasespeedofthetwoeigenmodesolutionsofthisproblem,whichcanalsobendestablishedinthemostgeneralformby Rus asandGrue ( 2002 )or Xiao-GangandJin-Bao ( 2006 )forexample.Perturbationsofathree-layeruidcanbedescribedbytwotypesofharmonicinterfacialwaves,verifyingthedispersionrelationexpressedasthefollowingdeterminant 1 2D0; (3) with,foriD1;2 iDg.i�iC1/C!2 ki tanh.khi/CiC1 tanh.khiC1/; (4) D�2!2 ksinh.kh2/: (5) Thefourthorderpolynomialin!associatedwithEq.( 3 )leadstotwotypesofinterfacialwaves.Usingthenotations www.nonlin-processes-geophys.net/18/193/2011/Nonlin.ProcessesGeophys.,18,193– 208 ,2011 M.J.Mercieretal.:Resurrectingdead-waterphenomenon203 Fig.16.Oscillatingregime(Frs1).Spatio-temporaldiagramofinterfacialdisplacementss.x;t/anda.x;t/(inm).Thesolidblacklinesarethebowandsternoftheboat.Themagenta(resp.white)dottedlineisanindicationofthespeedofpropagationof0:0325ms�1(resp.0:06ms�1).ExperimentalparametersaregivenininTable 2 andSbD24cm2,FtD20:6mN. Asobservedpreviously,bothmodesaregeneratedwhentheboatstarts.Mode-sremainstwotimeslargerthanmode-abuttheyareofnoticeableamplitude.Thesymmetricmodeevolveswiththeboatandreproducestheamplicationandsteepeningprocessesobservedinthetwo-layercase.Themode-sbreaksontheboatwhenitsamplitudeisthelargest,andadepression(symmetricforbothinterfaces)isexpelledatthebow.Thecharacteristicsofthemode-s,wavelengths'0:4mandgroupvelocitycg;s'0:06ms�1leadtoanoscillatingfrequencyof0:15Hz,tooslowforthisvisualiza-tion.Concerningmode-a,itsspatialstructureisclosetoasoli-tarywavetrain,withagroupvelocitycg;a'0:0325ms�1,closetocma(being0:044ms�1).Eachtimeitisgenerated,astrongaccelerationoftheboatoccurs.4.4DiscussionTheunsteadybehaviorassociatedwithdead-waterisstillob-servedinthethree-layeruid,withstronganalogytothetwo-layercase.Thestraticationconsideredbeingmorecomplex,wemustconsidertwobaroclinicmodesassociatedwithsymmetricandanti-symmetricoscillationsoftheinter-facesandthatarealsoreferredtoasmode-1andmode-2intheliterature. Rus asandGrue ( 2002 )presentsolutionsofthenonlinearequationsthathavestrongsimilaritieswithourobservations.Morespecically,thespatio-temporaldiagramofmode-aex-hibitsolitarywavesofmode-2withoscillatoryshortmode-1wavessuperimposed.Thisisinagreementwithclosevaluesoftheexperimentswiththenumericalcalculations(Boussi-nesqlimit,h1=h3'1andh1=h2'1:7).Perturbationsgeneratedbytheboatgivebirthtobothmodes,especiallywhentheaccelerationoftheboatisim-portant.Thisresultisconsistentwiththestudyof Nico-laouetal. ( 1995 ),veriedexperimentallyby Robey ( 1997 ),statingthatanacceleratedobjectinacontinuouslystratieduid,withaBrunt–V¨ais¨al¨afrequencyN.z/,excitesacontin-uumofmodeswhoseverticalprolew.z/isdescribedby d2w dz2.z/Ck2x N2.z/ !2�1!w.z/D0; (11) alongwiththeboundaryconditionsfortheverticalvelocitytobezeroatthetopandbottom.Finally,wehaveobservedthatthemode-1isstronglycou-pledtothedynamicsoftheboat,whichcorrespondstothefastestwavepropagatinginthisstratication.Aweakmode-2isassociatedtonoticeableaccelerationoftheboat,butevolvesfreelyfromtheothermode.5ContinuouslystratieduidwithapycnoclineWehaveobservedthatseveralwavesaregeneratedwhentheboatevolvesinacomplexstratication.Inthecaseofalin-earlystratieduid,aninnitenumberofmodescanpropa-gate.Wehaveactuallyconsideredthecaseofalinearlystrat-ieduidwithapycnocline(seeFig. 4 ).Severalreasonscanbeinvoked.Itkeepsthestraticationundisruptedwhentheboatevolvesinthetoplayer,itallowslargerverticaldis-placementsatthedensityjumpleadingtolargeramplitudes,itcanbemodeledasacouplingbetweeninterfacialandin-ternalwaves,anditcorrespondstoamorerealisticsetupincomparisonwithobservationsmadeinnaturalenvironment. www.nonlin-processes-geophys.net/18/193/2011/Nonlin.ProcessesGeophys.,18,193– 208 ,2011 206M.J.Mercieretal.:Resurrectingdead-waterphenomenon Fig.20.Frh1.Instantaneoushorizontaldensitygradients@x.x;z;t0/attimes(a)t0D32sand(b)t0D61s,ingcm�4. Fig.21.Frh1.AmplitudesOan.x0;t/withtimeoftheprojectedmodalstructureatseveralx0-locationsformodesnD1(a),2(b)and3(c)ofthehorizontaldensitygradientinarbitraryunits.Differentcolorscorrespondtodifferentfrequenciesassociatedwiththemodalbasis.Thetilteddashedlinescorrespondstothecharacteristicspeedofpropagationcm D7:2cms�1.Theblackrectanglerepresentstheboatpassingthroughaverticalcross-sectionateachx0-locationstudied. Fig. 21 bisofcomparableamplitudeatthelowestfrequencies(!D0:25and0:50rads�1)butisabsentforlargervaluesof!.Highermodes(n3)areevenweakerinamplitudesandonlypresentatlowfrequencies(Fig. 21 c).WenoticethattheinitialperturbationsfortheselowmodesaregeneratedbelowtheboatsincetheblackrectanglesinFig. 21 aandbareassociatedwitharampingamplitudeofmodes1and2.Thelongertherectangleis,thesmallerthespeedoftheboatis,leadingtolargerinitialperturbationswhichareverysimilartowhathasbeenobservedinthetwo-layercase(seeFig. 12 ).Furthermore,theapparentlycno¨dal(sharpcrestsandattroughs)oscillationsassociatedwiththemode-1internalwavesobservedinFig. 21 aasthemodespropagate,areasignatureofthenonlineardynamicsoftheinternalwaves.Finally,bygivinginallimagesthemaximumphasespeedof Nonlin.ProcessesGeophys.,18,193– 208 ,2011www.nonlin-processes-geophys.net/18/193/2011/