Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 62

with the latitude of the point where the equation is evaluated. For applications in

coastal lagoons, the value of f varies little and therefore often is kept constant. In
this case

ϕ

is the average latitude of the basin.
As can be seen from the equation, the Coriolis force is always perpendicular to
the flow. For example, in a flow that has only a component in the x 1 direction ( u 2

ϕ

0),
the Coriolis force is only acting in the x 2 direction. For the northern hemisphere, f is
positive, and therefore the force is always to the right with respect to the direction
of flow.

=

3.4.2.4 The Reynolds Equations

As mentioned above, the Navier-Stokes equations describe all possible water motions
from scales on the order of the oceans down to scales where molecular friction is
active and dissipation is important. However, for modeling purposes this is not
acceptable. For example, the fact that the very small scales are not modeled directly
(because the computational grid is just too coarse) means that molecular friction
will never be important in the Navier-Stokes equations.
However, the fact that molecular friction is not important for the scales we are
considering is somehow misleading. As there will be input of energy in our basin
through solar heating, tides, and wind, there also must be some way to convert this
energy and eventually dissipate it, otherwise the total energy will continue to
increase. But the only way energy can be dissipated is through the molecular friction
term. All other terms only take part in redistributing the energy inside the basin.
Therefore, there must be some place in the basin where friction becomes important
and the molecular forces do their work.
This obvious paradox has been resolved by the British physicist O. Reynolds
who applied an averaging technique to the full Navier-Stokes equations. He
described the flow as one that can be divided into a slowly varying part and a ra nd om
fluctuat io n around this. So, for example, the velocity u is represented as
where

uuu =+ ′

is the fluctuations. If all variables are
represented like this and are introduced into the hydrodynamic equations and these
equations are averaged over a suitable time interval, then new equations result for
the slowly varying parts of all variables.
The structure of these averaged equations is very similar to that of the original
equations. However, due to the nonlinear nature of the advective terms that are
contained implicitly in the total derivative, some new terms appear in the conser-
vation equation for momentum, temperature, and salt. These new terms, called
Reynolds fluxes , are averages of products of fluctuating variables. Without going
into detail, these terms can be formally written in the same way as the diffusion
terms in the Navier-Stokes equations. The new terms now read

u is the slowly varying part and u ′

2 2

+

2

∂
∂

u
x

+ ∂
∂

u
x

∂
∂

u
x

t H V

Fv =

i i

v

i

i M 2 2 M 2

1 2 3

where two new parameters have been substituted for the molecular viscosity v .
A distinction has been made between the horizontal and the vertical component and the
subscript M indicates the momentum equation.

νν
M

H

,

V

M