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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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Coastal Ocean Dynamics

First course:

Hydrodynamics

Hans Burchard

Leibniz Institute for

Baltic Sea Research Warnemünde

[email protected]


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

Densityofseawateris 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)


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