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.
Various conservation laws are defined on a material volume
of a homogeneous substance such as water or air, moving
with the flow.
Conservation of mass
Within 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 gravitation
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 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 (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 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
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.
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.
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.
Dynamic shallow water equations
Finally, the dynamic equations are of the following form:
acceleration
advection
rotation
pressure
gradient
friction
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 density
Fx,Fy: friction terms
Decomposition of pressure gradient
The pressure gradient can be decomposed to three contributions:
pressure
gradient
=
surface
slope
+
density
gradient
atmospheric
+ pressure
gradient
Equation of state
Density 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 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
Channel flow
Inertial oscillations
Balance between rate of change and Coriolis rotation:
Inertial oscillation
(observations in the Western Baltic Sea)
Van der Lee and Umlauf (2011)
Geostrophic equilibrium
Balance between pressure gradient and Coriolis rotation:
Flow is 90° to the right of the pressure gradient.
Geostrophic equilibrium
Air flow around a low-pressure area is anti-clockwise
in the Northern hemisphere, and clockwise in the
Southern hemishere (=cyclonic).
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.
Ekman dynamics
Ekman spiral for constant eddy viscosity:
Kundu and Cohen (2002)
Ekman depth:
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.
Kelvin waves
Kelvin waves are long propagating waves which lean on a coast to
the right (Northern hemisphere):
Gill (1982)