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Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde

Coastal Ocean Dynamics First course: Hydrodynamics. Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de. What makes it move ? Some principle laws of mechanics and thermodynamics .

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Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde

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  1. Coastal Ocean Dynamics First course: Hydrodynamics Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de

  2. Whatmakesitmove? Someprinciplelawsofmechanicsandthermodynamics.

  3. Variousconservationlawsaredefined on a material volume of a homogeneoussubstance such aswaterorair, moving withtheflow.

  4. Conservationofmass Within a material body, massisconserved, i.e., thenumberofmoleculesandtheirmassremainthe same.

  5. Conservationofmomentum Momentum: density X velocity Newton‘sSecond Law: Within a material body, thechangeofmomentum isequaltosumoftheforcesacting on thebody F maybe due to a bodyforce (typicallygravitationalforce) or due to a force on thesurfaceofthebody.

  6. Conservationof angular momentum Within a material body, thechangeof total angular momentum M isequaltosumofthetorqueoftheforces acting on thebody.

  7. Actio = Reactio Newton‘s Third Law: If a body A excerts a force on a secondbody B, then B excertsthe same force on A but withthe different sign.

  8. Law ofgravitation The body B1 has mass m1, and a second body, B2 has mass m2, and they have the distance r along the unit vector, n, connecting the two. Then, the gravity force, G, between the two bodies given by where g is the universal constant of gravity.

  9. First lawofthermodynamics Balance ofenergy The changeof total energyof a material bodyisequaltotherate ofworkdonebythemechanicalforcesacting on thebody (PV) anditssurface (PA), theinternalheatsupply(R) andthe total heatflux Q throughtheboundary: 4 waystoincreasetheenergy of an apple …

  10. Second lawofthermodynamics Entropy* cannotdecreaseexceptforexternalforcing. This meansforexample … … Heat always flows from high to low temperature. … Mechanical energy can be converted into heat via friction, but not the other way around. *Measure for disorder

  11. Material laws FluidslikewaterorairarecalledNewtonian because the viscous stresses that arise from its flow, are proportional to the local shear rate.

  12. Incompressibilityconstraint In contrasttoair, waterisrelativelyincompressible. This hastheconsequencethathorizontallyconvergingwatertransportsleadto an increasingsealevel.

  13. Hydrostaticassumption If all flowisatrest, thepressure p is in hydrostaticequilibrium, i.e. theverticalpressuregradientis proportional tothedensityofthewater (gravitationalacceleration g is theconstantofproportionality): In oceanmodelsweassumethatthepressureishydrostatic also whentheflowis not atrest.

  14. Dynamic shallowwaterequations Finally, thedynamicequationsareofthefollowing form: x,y,z: westward, northwardandupwardcoordinate(m/s) u,v,w: westward, northwardandupwardvelocitycomponent (m/s) t: time (s) p: pressure (N/m2=kg/(s2m) f: Coriolis parameter (2w sin(f), f latitude, w Earth rotation rate) g: gravitationalacceleration (=9.81 m/s2) r0: referencedensity Fx,Fy: frictionterms pressure gradient rotation acceleration advection friction

  15. Decompositionofpressuregradient The pressuregradientcanbedecomposedtothreecontributions: pressuresurfacedensityatmospheric = + + pressure gradientslopegradientgradient

  16. Equationofstate Densityofseawateris a nonlinearfunctionof temperature, salinity S, pressure p: maximum density temperature freezing temperature

  17. Letusnowstudyidealisedsituationswheretwoterms in thedynamicequationsbalanceandtheothersarezero.

  18. Channel flow Balance betweenpressuregradientandfriction*. Solution forconstanteddyviscosity: Solution forparaboliceddyviscosity: *Weneedtomakehere a littleexcursionintothedefinitionofeddyviscosity

  19. Channel flow

  20. Inertialoscillations Balance between rate ofchangeand Coriolis rotation:

  21. Inertialoscillation (observations in the Western Baltic Sea) Van der Lee and Umlauf (2011)

  22. Geostrophicequilibrium Balance betweenpressuregradientand Coriolis rotation: Flow is 90° totherightofthepressuregradient.

  23. Geostrophicequilibrium Air flowaround a low-pressureareais anti-clockwise in the Northern hemisphere, andclockwise in the Southern hemishere (=cyclonic).

  24. Ekmandynamics Balance between Coriolis rotationandfriction: Verticallyintegratedtransport (U,V) is 90° totherightofthe wind stress (in Northern hemispere). This is also calledtheEkmantransport.

  25. Ekmandynamics Ekman spiral forconstanteddyviscosity: Ekmandepth: Kunduand Cohen (2002)

  26. Upwelling Ifthereis a coasttotheleft (Northern hemisphere) ofthecurrent, thentheEkmantransportiscompensatedbyupwellingwaterfromdepth: Downwellingresultsfrom a coast totherightofthe wind. upwelling Wind downwelling

  27. Kelvin waves Kelvin wavesarelongpropagatingwaveswhichlean on a coasttotheright (Northern hemisphere): Gill (1982)

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