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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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Coastal Ocean Dynamics First course: Hydrodynamics Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de
Whatmakesitmove? Someprinciplelawsofmechanicsandthermodynamics.
Variousconservationlawsaredefined on a material volume of a homogeneoussubstance such aswaterorair, moving withtheflow.
Conservationofmass Within a material body, massisconserved, i.e., thenumberofmoleculesandtheirmassremainthe same.
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.
Conservationof angular momentum Within a material body, thechangeof total angular momentum M isequaltosumofthetorqueoftheforces acting on thebody.
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.
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.
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 …
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
Material laws FluidslikewaterorairarecalledNewtonian because the viscous stresses that arise from its flow, are proportional to the local shear rate.
Incompressibilityconstraint In contrasttoair, waterisrelativelyincompressible. This hastheconsequencethathorizontallyconvergingwatertransportsleadto an increasingsealevel.
Hydrostaticassumption If all flowisatrest, thepressure p is in hydrostaticequilibrium, i.e. theverticalpressuregradientis proportional tothedensityofthewater (gravitationalacceleration g is theconstantofproportionality): In oceanmodelsweassumethatthepressureishydrostatic also whentheflowis not atrest.
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
Decompositionofpressuregradient The pressuregradientcanbedecomposedtothreecontributions: pressuresurfacedensityatmospheric = + + pressure gradientslopegradientgradient
Equationofstate Densityofseawateris a nonlinearfunctionof temperature, salinity S, pressure p: maximum density temperature freezing temperature
Letusnowstudyidealisedsituationswheretwoterms in thedynamicequationsbalanceandtheothersarezero.
Channel flow Balance betweenpressuregradientandfriction*. Solution forconstanteddyviscosity: Solution forparaboliceddyviscosity: *Weneedtomakehere a littleexcursionintothedefinitionofeddyviscosity
Inertialoscillations Balance between rate ofchangeand Coriolis rotation:
Inertialoscillation (observations in the Western Baltic Sea) Van der Lee and Umlauf (2011)
Geostrophicequilibrium Balance betweenpressuregradientand Coriolis rotation: Flow is 90° totherightofthepressuregradient.
Geostrophicequilibrium Air flowaround a low-pressureareais anti-clockwise in the Northern hemisphere, andclockwise in the Southern hemishere (=cyclonic).
Ekmandynamics Balance between Coriolis rotationandfriction: Verticallyintegratedtransport (U,V) is 90° totherightofthe wind stress (in Northern hemispere). This is also calledtheEkmantransport.
Ekmandynamics Ekman spiral forconstanteddyviscosity: Ekmandepth: Kunduand Cohen (2002)
Upwelling Ifthereis a coasttotheleft (Northern hemisphere) ofthecurrent, thentheEkmantransportiscompensatedbyupwellingwaterfromdepth: Downwellingresultsfrom a coast totherightofthe wind. upwelling Wind downwelling
Kelvin waves Kelvin wavesarelongpropagatingwaveswhichlean on a coasttotheright (Northern hemisphere): Gill (1982)