Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 61

The vertical component of this equation may be written as

∂
∂

p
x

=− ρ g

3

This equation is called the hydrostatic equation because it is exactly valid in a static
fluid with no motion. If we integrate this equation from a depth z up to the surface
that is supposed to be at x 3 = 0, we have:

0 0

∂
∂

p
x

∫ ∫

dx =− =− p p z () g ρ dx

3 0 3

3

z z

or

0

∫

pz () =+ p g ρ dx (3.20)

0 3

z

where p 0 is the surface or atmospheric pressure. In this form the hydrostatic
equation states that the pressure at depth z is due to the weight of the water column
above it.
It turns out that this equation is also a very good approximation of the vertical
component of the momentum equation in case of a situation where the velocity
vector u i is not zero. Only in regions of very strong vertical convection due to strong
bathymetric gradients or cooling of surface water may the vertical acceleration have
appreciable effects on the total momentum equation.

3.4.2.3 The Coriolis Force

As specified above, the Coriolis force is an apparent force that is due to the rotation
of the Earth. The complete expression also takes into account the influence of rotation
due to and on vertical motion. However, it can be seen that only the horizontal part
of the Coriolis force is important. In fact, the vertical momentum equation has been
substituted by the hydrostatic assumption.
If the horizontal components are evaluated there is one term that is multiplied
with the vertical velocity u 3 . This term is nearly always much smaller than the other
terms due to the smallness of the vertical velocity. If this term is neglected the
remaining Coriolis vector can be written as

( ) =− +

C CCC

F = F , F , F ( fu , fu

1 0

, )

i 1 2 3 2

where f is called the Coriolis parameter and has already been defined as

f = 2 Ω sin ( ϕ )