Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 63

It is important to note the different physical processes that lead to these equations.
In the case of the Navier-Stokes equations, the friction term was due to the molecular
(Brownian) motion of the fluid particles. This motion was statistical in nature and
the diffusivity was only a function of the fluid static properties.
In the case of the Reynolds equations, the additional term is now due to the
fluctuating part of the variable that is under consideration (here, the velocity). This
fluctuating part is also called the turbulent part. It is not actually statistical in nature
because the Navier-Stokes equations could at least in principle be used to compute all
the small-scale motions. However, as in thermodynamics, it would in principle be
possible to describe the motion of 10 23 molecules that are in one mole of air; practically,
it is not possible and only a statistical representation of the motion is given.
Therefore, the diffusion of the fluid particles is now due to the turbulence that
is acting on scales smaller than the ones that have been retained after the averaging
procedure. This turbulent motion is similar to the molecular motion of the fluid
particles. However, there are two main differences. Because the turbulent motions
(fluctuation) depend on the averaging scale, the turbulent viscosities will
also depend on the scale of averaging. Even worse, because the turbulence depends
on the dynamic state of the fluid, the turbulent viscosities also depend on the
dynamics of the fluid. Various factors, such as the local buoyancy or the velocity
shear, will influence these parameters.
It is clear now that there is a fundamental difference between the molecular and
the turbulent diffusion term. The first one depends only on the static properties of
the fluid (type of fluid, temperature, salinity), whereas the second one also depends
on the dynamic flow field itself. The molecular viscosity can be given a very accurate
value that can be measured or computed in statistical mechanics. However, the
turbulent parameter varies over several orders of magnitude depending crucially on
the flow field itself. Because of the dynamic nature of the turbulent motion, it is
expected that the values for the viscosities will be different in the horizontal and the
vertical directions. This fact has been accounted for by having the viscosity take
different values for the horizontal and vertical dimensions.
Comparing the range of values for the two types of parameters, it can be seen
that the turbulent diffusion term is some orders of magnitude bigger than its molec-
ular counterpart. Therefore, the molecular friction is almost always neglected and
only the turbulent terms are retained. The value for the turbulent viscosity parameter
is often set to a constant average, one that best represents the physical processes to
be described. In a more general case a turbulence closure model must be used that
will actually compute the parameter

νν
M

H

,

V

M

νν
M

H

,

V

M

ν M in every point of the water body. The
description of these turbulence closure models (e.g., k - ε , Mellor-Yamada) is beyond

the scope of this topic.

3.4.2.5 The Primitive Equations

In this section the simplified three-dimensional equations are given one more time
as a reference for the next section. These equations are also called primitive equa-
tions, not because they are easy to solve, but because only basic simplifications have
been applied to them, and they are presented in their “primitive” structure.