Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 60

T 0 and the reference salinity S 0 . The parameter α T is called the thermal expansion

coefficient and α S , the expansion coefficient, changes due to salinity. Both coeffi-

cients are positive. As can be seen, denser (heavier) water may result by either
lowering the temperature or raising the salinity.

3.4.2 S IMPLIFICATION AND S CALE A NALYSIS

The equations given above, especially the equation for momentum, are still too
complicated to be used for modeling. This is true mainly because of the variability
of the spatial and temporal scales to which these equations are applicable.
Although we did not make any assumptions, these equations describe the flow for
the global ocean circulation as they describe the flow down to scales where
molecular effects become important. Therefore, these equations must describe
scales from 10 6 m on the ocean scale down to about 10 -6 m ( µ m) where dissipation

becomes important.
The same is true concerning the time scales to which the equations apply.
These time scales range from years for the general circulation down to micro-
seconds if we deal with dissipation effects due to friction. Because we are
interested in describing motion that takes place in coastal lagoons or estuaries,
not all processes included in the equations above are equally important and
simplifications have to be made.

3.4.2.1 Incompressibility

Compared to air, water is a relatively incompressible medium. That there is a certain
compressibility of the water is clearly seen by the fact that acoustic waves can travel
in water. These waves depend completely on the compressibility of the medium.
However, it can be shown that, excluding acoustic waves, the effect of compress-
ibility on the dynamics of the oceans is negligible.
Therefore, if we assume that water is an incompressible medium, this can be
written mathematically as d

0. If we substitute this into the continuity equation,
we have the simplified continuity equation:

ρ / dt =

∂
∂

u
x

i

= 0 (3.19)

i

In this version the continuity equation states that the divergence of the mass flow is
zero. This is the form of the mass conservation normally used in oceanography.

3.4.2.2 The Hydrostatic Approximation

If we consider a basin of water at rest ( u i = 0), the Navier-Stokes equations reduce to

1 ∂

∂

p
x

0 =− + g

i

ρ

i