Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 56

also of the additional Coriolis force, and the new equations (not deduced) can be
written as

du
dt

−=− ∂
∂

1 p

x

1

fu (3.14a)

2

ρ

1

du
dt

+=− ∂
∂

1 p

x

2

fu (3.14b)

1

ρ

2

du
dt

1 ∂

∂

p
x

3

=− − g (3.14c)

ρ

3

where f is the Coriolis parameter with f = 2 Ω sin( ϕ ), Ω is the angular frequency of

rotation of the earth (2 denotes the geographical latitude. For ease of

notation, we have expanded the index notation into the three equations, one for each
coordinate direction. In this form the equations are called Euler equations in a
rotating frame of reference .
Besides the gravitational force, the Coriolis force is the only other important
volume force that acts in a fluid body. Because of the vector product, the Coriolis
force always acts perpendicular to the current velocity, in the northern hemisphere to
the right and in the southern hemisphere to the left of the fluid flow. It is the Coriolis
force that is responsible for all the meso-scale structure we can see on the weather
charts that contain cyclones and anticyclones. However, the Coriolis force is
important only for large-scale circulations and, therefore, may not be important
for many coastal lagoons of the world.

π /24 h ), and ϕ

3.4.1.2.3 The Navier-Stokes Equations
The above-derived Euler equations describe the flow of a fluid without friction. Often
this may be a good approximation, especially if no material boundaries (lateral and
vertical) are close to the area of investigation. Internal friction in a fluid is normally
very small and the Euler equations provide a satisfactory simplification. However,
once the fluid is close to a boundary, friction becomes more important and another
area force, the stress tensor, has to be introduced. A moving fluid layer exerts a force
on the neighboring fluid layers. The strength of this force is directly proportional to
the area of the fluid layer and the velocity difference between these layers, and
inversely proportional to the distance of the layers:

FAux = µ ∆∆

1

/

3

where A is the area, ∆ u 1 is the velocity difference from one layer to the other, and

∆ is the constant of propor-

tionality and is called the dynamic viscosity coefficient . It is a parameter that depends
only on the type of fluid and its temperature and salinity contents, but not on the
fluid dynamics.

x 3 is the distance of the fluid layers. The parameter µ