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Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de 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 volume of a homogeneous substance such as water or air, moving with the flow.

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Conservation of mass Within 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 gravitation The body B1 has mass m 1, and a second body, B2 has mass m 2, 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 is the universal constant of gravity.

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First law of thermodynamics Balance 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 (P V ) and its surface (P A ), 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 thermodynamics Entropy* cannot decrease except for external forcing. This means for example … … 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

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Material laws Fluids 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 constraint In 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 assumption If 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 equations Finally, 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/(s 2 m) f: Coriolis parameter (2 sin( ), latitude, Earth rotation rate g: gravitational acceleration (=9.81 m/s 2 ) 0 : reference density F x,F y : friction terms accelerationadvection rotation pressure gradient friction

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Decomposition of pressure gradient The pressure gradient can be decomposed to three contributions: pressure surface density atmospheric = + + pressure gradient slope gradient gradient

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Equation of state Density 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 flow Balance 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 oscillations Balance 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 equilibrium Balance between pressure gradient and Coriolis rotation: Flow is 90° to the right of the pressure gradient.

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Geostrophic equilibrium Air 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 dynamics Balance 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 dynamics Ekman spiral for constant eddy viscosity: Ekman depth: Kundu and Cohen (2002)

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Upwelling If 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 downwelling upwelling

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Kelvin waves Kelvin waves are long propagating waves which lean on a coast to the right (Northern hemisphere): Gill (1982)

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