A Large Spectrum of Free Oscillations of the World Ocean Including the Full Ocean Loading and Self-attraction Effects

ThisbookdependsonadissertationpreparedattheDepartementofGeosciencesat theUniversityHamburg. ItwasacceptedbytheDepartementwiththegradesumma cumlaudein2008. IwouldliketothankmyacademicadvisorProf. Dr. WilfriedZahelforhisconstant supportandforthelongandconstructivediscussions. Further,IwouldliketothankProf. Dr. Jur ¨ genSundermann ¨ forintroducingmeto theInternationalMaxPlanckResearchSchool. TheInternationalMaxPlanckResearchSchoolforMaritimeAffairsandinparticu larProf. Dr. Dr. h. c. Jur ¨ genBasedowandhisco directorsarethankedforgivingme theopportunitytoperformthisstudyinHamburg. IacknowledgethecomputationalsupportoftheDKRZandNEC,especiallythe helpfulcommentsofKlausKetelsenandJens OlafBeismann. LastbutnotleastmanythankstomywifeJanaSillmannandmysonDariusfor givingmethetimeIneededforthisstudyandprovidingajoyfulandlovinghome. ThisworkhasbeenfundedbytheInternationalMaxPlanckResearchSchoolfor MaritimeAffairsattheUniversityHamburg. Hamburg,August2008 MalteMuller ¨ Contents Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 TheoryandModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. 1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. 1. 1 SecondaryForces:TheLoadingandSelf AttractionEffect . 8 2. 1. 2 TheEquationsofMotionandtheEquationofContinuity. . . 11 2. 1. 3 EnergyBalance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2. 1. 4 ParameterizationoftheLSA AnAnalyticalApproach . . . . 16 2. 2 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2. 2. 1 TheImplicitlyRestartedArnoldiMethod. . . . . . . . . . . . . . . . 19 2. 2. 2 TheParallelizationwithMPI . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2. 2. 3 ThePerformanceoftheModel. . . . . . . . . . . . . . . . . . . . . . . . . 21 3 TheFreeOscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3. 1 GravitationalModes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3. 1. 1 The? value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3. 1. 2 TheIn?uenceoftheLSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3. 1. 3 TheAntarcticKelvinWave. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3. 1. 4 NewModes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3. 1. 5 TheSlowestModes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3. 2 VorticityModes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. 2. 1 TopographicalVorticityModes. . . . . . . . . . . . . . . . . . . . . . . . . 32 3. 2. 2 PlanetaryVorticityModes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 SynthesisofForcedOscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4. 1 TidalDynamicsandtheIn?uenceofLSA. . . . . . . . . . . . . . . . . . . . . . 40 4. 1. 1 TheProcedureofTidalSynthesis. . . . . . . . . . . . . . . . . . . . . . . 40 4. 1. 2 LSA effectonForcedOscillations. . . . . . . . . . . . . . . . . . . . . . 45 4. 1. 3 TheSynthesisoftheSemidiurnalandDiurnalTides. . . . . . . 50 xi xii Contents 4. 2 IntegrationoftheSolutionsofaTidalModelwithAssimilationof Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4. 2. 1 NewExpansionCoef?cients. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4. 2. 2 NewFrequenciesandAdjointEigenfunctions . . . . . . . . . . . . 59 4. 2. 3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4. 2. 4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4. 2. 5 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Abstract AnewsetofbarotropicfreeoscillationsoftheWorldOceaniscomputedwithex plicitconsiderationofdissipativetermsandthefulloceanloadingandself attraction effect(LSA). Thissetcontainsfreeoscillationsthatdidnotappearinthespectraof previousstudies. Furthermore,theexpansiontowardslongerperiods(165hours) yieldsnewglobalplanetarymodes. Altogether169freeoscillationsarecomputed withperiodslongerthan7. 7hours. Ofthese,71aregravitationalmodes,92areto pographicallycontrolledvorticitymodes,and6areplanetaryvorticitymodes. The in?uenceoftheLSAisinvestigatedforallthreekindsofmodeswithrespectto changesintheperiodsandinthespatialstructureoftheseasurfaceelevationand thehorizontalmasstransports. Inparticular,formodesinthesemi diurnalanddi urnalperiodrange,theparameterizationoftheLSAisanalyzed. Forthefreeoscillationsintheperiodrangefrom9to40hoursthecorresponding adjointsolutionsarecomputedandusedtosynthesizesemidiurnalanddiurnaltides ofseconddegree. Sincethesefreeoscillationsaredeterminedwithandwithoutcon siderationofthefullLSA effect,thisstudyallowsforadetailedanalysisoftheLSA onthedynamicsofoceantides,e. g. anphysicalexplanationisgivenfortheinduced phasedelaycomputedbyoceantidemodels. Further,thesynthesisgivesaspectral compositionofcertainwellknowntidalfeaturesandpairsoffreeoscillationsare identi?ed,diminishingtheircontributioneitheronaglobalorlocalscale. Further,semidiurnalanddiurnaltidalsolutionsofatidalmodelwithassimilation ofdataareintegratedintheprocedureofsynthesizingtides.