Coastal Ocean Dynamics
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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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Hans burchard leibniz institute for baltic sea research warnem nde

Coastal Ocean Dynamics

First course:

Hydrodynamics

Hans Burchard

Leibniz Institute for

Baltic Sea Research Warnemünde

hans.burchard@io-warnemuende.de


Hans burchard leibniz institute for baltic sea research warnem nde

Whatmakesitmove?

Someprinciplelawsofmechanicsandthermodynamics.


Hans burchard leibniz institute for baltic sea research warnem nde

Variousconservationlawsaredefined on a material volume

of a homogeneoussubstance such aswaterorair, moving

withtheflow.


Hans burchard leibniz institute for baltic sea research warnem nde

Conservationofmass

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


Hans burchard leibniz institute for baltic sea research warnem nde

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.


Hans burchard leibniz institute for baltic sea research warnem nde

Conservationof angular momentum

Within a material body, thechangeof

total angular momentum M

isequaltosumofthetorqueoftheforces

acting on thebody.


Hans burchard leibniz institute for baltic sea research warnem nde

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.


Hans burchard leibniz institute for baltic sea research warnem nde

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.


Hans burchard leibniz institute for baltic sea research warnem nde

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 …


Hans burchard leibniz institute for baltic sea research warnem nde

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


Hans burchard leibniz institute for baltic sea research warnem nde

Material laws

FluidslikewaterorairarecalledNewtonian

because

the viscous stresses that arise from its flow,

are proportional to the local shear rate.


Hans burchard leibniz institute for baltic sea research warnem nde

Incompressibilityconstraint

In contrasttoair, waterisrelativelyincompressible.

This hastheconsequencethathorizontallyconvergingwatertransportsleadto an increasingsealevel.


Hans burchard leibniz institute for baltic sea research warnem nde

Hydrostaticassumption

If all flowisatrest, thepressure p is in

hydrostaticequilibrium, i.e. theverticalpressuregradientis proportional tothedensityofthewater (gravitationalacceleration g is

theconstantofproportionality):

In oceanmodelsweassumethatthepressureishydrostatic also whentheflowis not atrest.


Hans burchard leibniz institute for baltic sea research warnem nde

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


Hans burchard leibniz institute for baltic sea research warnem nde

Decompositionofpressuregradient

The pressuregradientcanbedecomposedtothreecontributions:

pressuresurfacedensityatmospheric

= + + pressure

gradientslopegradientgradient


Hans burchard leibniz institute for baltic sea research warnem nde

Equationofstate

Densityofseawateris a nonlinearfunctionof

temperature, salinity S, pressure p:

maximum

density

temperature

freezing

temperature


Hans burchard leibniz institute for baltic sea research warnem nde

Letusnowstudyidealisedsituationswheretwoterms in thedynamicequationsbalanceandtheothersarezero.


Hans burchard leibniz institute for baltic sea research warnem nde

Channel flow

Balance betweenpressuregradientandfriction*.

Solution forconstanteddyviscosity:

Solution forparaboliceddyviscosity:

*Weneedtomakehere a littleexcursionintothedefinitionofeddyviscosity



Hans burchard leibniz institute for baltic sea research warnem nde

Inertialoscillations

Balance between rate ofchangeand Coriolis rotation:


Hans burchard leibniz institute for baltic sea research warnem nde

Inertialoscillation

(observations in the Western Baltic Sea)

Van der Lee and Umlauf (2011)


Hans burchard leibniz institute for baltic sea research warnem nde

Geostrophicequilibrium

Balance betweenpressuregradientand Coriolis rotation:

Flow is 90° totherightofthepressuregradient.


Hans burchard leibniz institute for baltic sea research warnem nde

Geostrophicequilibrium

Air flowaround a low-pressureareais anti-clockwise

in the Northern hemisphere, andclockwise in the

Southern hemishere (=cyclonic).


Hans burchard leibniz institute for baltic sea research warnem nde

Ekmandynamics

Balance between Coriolis rotationandfriction:

Verticallyintegratedtransport (U,V) is 90° totherightofthe wind stress (in Northern hemispere). This is also calledtheEkmantransport.


Hans burchard leibniz institute for baltic sea research warnem nde

Ekmandynamics

Ekman spiral forconstanteddyviscosity:

Ekmandepth:

Kunduand Cohen (2002)


Hans burchard leibniz institute for baltic sea research warnem nde

Upwelling

Ifthereis a coasttotheleft (Northern hemisphere) ofthecurrent, thentheEkmantransportiscompensatedbyupwellingwaterfromdepth:

Downwellingresultsfrom a coast

totherightofthe wind.

upwelling

Wind

downwelling


Hans burchard leibniz institute for baltic sea research warnem nde

Kelvin waves

Kelvin wavesarelongpropagatingwaveswhichlean on a coasttotheright (Northern hemisphere):

Gill (1982)