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# Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde - PowerPoint PPT Presentation

Coastal Ocean Dynamics First course: Hydrodynamics. Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde [email protected] What makes it move ? Some principle laws of mechanics and thermodynamics .

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Presentation Transcript

First course:

Hydrodynamics

Hans Burchard

Leibniz Institute for

Baltic Sea Research Warnemünde

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.

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

rotation

acceleration

friction

pressuresurfacedensityatmospheric

= + + pressure

Equationofstate

Densityofseawateris a nonlinearfunctionof

temperature, salinity S, pressure p:

maximum

density

temperature

freezing

temperature

Letusnowstudyidealisedsituationswheretwoterms in thedynamicequationsbalanceandtheothersarezero.

Channel flow

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

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)

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)