hans burchard leibniz institute for baltic sea research warnemünde
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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 . - PowerPoint PPT PresentationTRANSCRIPT
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Hans Burchard
Leibniz Institute for Baltic Sea Research Warnemünde
Coastal Ocean Dynamics
First course: Hydrodynamics
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What makes it move?
Some principle laws of mechanics and thermodynamics.
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Various conservation laws are defined on a material volumeof a homogeneous substance such as water or air, moving
with the flow.
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Conservation of massWithin a material body, mass is conserved, i.e.,
the number of molecules and their mass remain the same.
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Conservation of momentum
Momentum: density X velocity
Newton‘s Second Law:
Within a material body, the change of momentum
is equal to sum of the forces acting on the body
F may be due to a body force (typically gravitational force) or due to a force on the
surface of the body.
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Conservation of angular momentum
Within a material body, the change of total angular momentum M
is equal to sum of the torque of the forces acting on the body.
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Actio = Reactio
Newton‘s Third Law:
If a body A excerts a force on a second body B,then B excerts the same force on A
but with the different sign.
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Law of gravitationThe 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.
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First law of thermodynamicsBalance of energy
The change of total energy of a material body is equal to the rate of work done by the
mechanical forces acting on the body (PV) and its surface (PA), the internal heat supply (R)
and the total heat flux Q through the boundary:
4 ways to increase the energy of an apple …
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Second law of thermodynamicsEntropy* cannot decrease except for external
forcing.
This means for example …
… Heat always flows from high to low temperature.
… Mechanical energy can be convertedinto heat via friction,
but not the other way around.
*Measure for disorder
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Material lawsFluids like water or air are called Newtonian
because
the viscous stresses that arise from its flow, are proportional to the local shear rate.
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Incompressibility constraintIn contrast to air, water is relatively
incompressible.
This has the consequence that horizontally converging water transports lead to an
increasing sea level.
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Hydrostatic assumptionIf all flow is at rest, the pressure p is in hydrostatic equilibrium, i.e. the vertical pressure gradient is proportional to the
density of the water (gravitational acceleration g is
the constant of proportionality):
In ocean models we assume that the pressure is hydrostatic also when the flow is not at rest.
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Dynamic shallow water equationsFinally, the dynamic equations are of the following form:
x,y,z: westward, northward and upward coordinate (m/s) u,v,w: westward, northward and upward velocity component (m/s) t: time (s)p: pressure (N/m2=kg/(s2m)f: Coriolis parameter (2w sin(f), f latitude, w Earth rotation rate)g: gravitational acceleration (=9.81 m/s2)r0: reference densityFx,Fy: friction terms
acceleration advection rotation pressuregradient
friction
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Decomposition of pressure gradientThe pressure gradient can be decomposed to three contributions:
pressure surface density atmospheric = + + pressuregradient slope gradient gradient
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Equation of stateDensity r of seawater is a nonlinear function of
temperature , salinity S, pressure p:
maximum density temperature
freezing temperature
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Let us now study idealised situations where two terms in the
dynamic equations balance and the others are zero.
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Channel flowBalance between pressure gradient and friction*.
Solution for constant eddy viscosity:
Solution for parabolic eddy viscosity:
*We need to make here a little excursion into the definition of eddy viscosity
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Channel flow
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Inertial oscillationsBalance between rate of change and Coriolis rotation:
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Inertial oscillation (observations in the Western Baltic Sea)
Van der Lee and Umlauf (2011)
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Geostrophic equilibriumBalance between pressure gradient and Coriolis
rotation:
Flow is 90° to the right of the pressure gradient.
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Geostrophic equilibriumAir flow around a low-pressure area is anti-clockwise
in the Northern hemisphere, and clockwise in the Southern hemishere (=cyclonic).
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Ekman dynamicsBalance between Coriolis rotation and friction:
Vertically integrated transport (U,V) is 90° to the right of the wind stress (in Northern hemispere). This is also called the
Ekman transport.
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Ekman dynamicsEkman spiral for constant eddy viscosity:
Ekman depth:Kundu and Cohen
(2002)
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UpwellingIf there is a coast to the left (Northern hemisphere) of the
current, then the Ekman transport is compensated by upwelling water from depth:
Downwelling results from a coast to the right of the wind.
Wind
downwelli
ng
upwelli
ng
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Kelvin wavesKelvin waves are long propagating waves which lean on a
coast to the right (Northern hemisphere):
Gill (1982)