Hans Burchard

Leibniz Institute for Baltic Sea Research Warnemünde

hans.burchard@io-warnemuende.de

Coastal Ocean Dynamics

First course: Hydrodynamics

What makes it move?

Some principle laws of mechanics and thermodynamics.

Various conservation laws are defined on a material volumeof a homogeneous substance such as water or air, moving

with the flow.

Conservation of massWithin a material body, mass is conserved, i.e.,

the number of molecules and their mass remain the same.

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.

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.

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.

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.

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 …

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

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.

Incompressibility constraintIn contrast to air, water is relatively

incompressible.

This has the consequence that horizontally converging water transports lead to an

increasing sea level.

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.

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

Decomposition of pressure gradientThe pressure gradient can be decomposed to three contributions:

pressure surface density atmospheric = + + pressuregradient slope gradient gradient

Equation of stateDensity r of seawater is a nonlinear function of

temperature , salinity S, pressure p:

maximum density temperature

freezing temperature

Let us now study idealised situations where two terms in the

dynamic equations balance and the others are zero.

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

Channel flow

Inertial oscillationsBalance between rate of change and Coriolis rotation:

Inertial oscillation (observations in the Western Baltic Sea)

Van der Lee and Umlauf (2011)

Geostrophic equilibriumBalance between pressure gradient and Coriolis

rotation:

Flow is 90° to the right of the pressure gradient.

Geostrophic equilibriumAir flow around a low-pressure area is anti-clockwise

in the Northern hemisphere, and clockwise in the Southern hemishere (=cyclonic).

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.

Ekman dynamicsEkman spiral for constant eddy viscosity:

Ekman depth:Kundu and Cohen

(2002)

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

Kelvin wavesKelvin waves are long propagating waves which lean on a

coast to the right (Northern hemisphere):

Gill (1982)