Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 43

or management. For these purposes the system has to be divided into homogeneous
parts (control volumes) and fluxes between them have to be calculated. This
approach requires a numerical model. The number of space dimensions required
to describe the control volumes equals the number of dimensions of the model.
The resolution of the transport equations in practical situations has been made
possible using numerical methods and computers (see Chapter 6 for details).
Before the advent of computers transport processes had to be studied using empir-
ical formulations (derived from experiments) or analytical solutions in simple
geometries or boundary conditions.
This chapter presents a general transport equation (also called an evolution equa-
tion) based on the concepts of (1) control volume, (2) advective flux, and (3) diffusive
flux. Based on this generic equation, equations for hydrodynamics, temperature, salin-
ity, and suspended sediments are also introduced. Special flows and simplification of
dimensionality and boundary processes and conditions, particularly for coastal
lagoons, are described in detail for use in lagoon modeling studies.

3.2 FLUXES AND TRANSPORT EQUATION

3.2.1 V D L T

ELOCITY AND IFFUSIVITY IN AMINAR AND URBULENT

F N M

LOWS AND IN A UMERICAL ODEL

For transport purposes, fluids are considered a continuum system. Velocity is defined
in a macroscopic way based on the concept of continuum system. Because fluids
are not a real continuum, system velocity cannot describe transport processes at the
molecular scale. The nonrepresented processes are represented by diffusion.
Although the concept of velocity is well known, it is reconsidered for modeling
purposes.
Diffusion in laminar flows occurs from movements at a molecular scale not
represented by the velocity. In turbulent flows, velocity, as defined for laminar flows,
becomes time dependent, changing at a frequency that is too high to be represented
analytically. As a consequence time average values must be considered, following
the Reynolds approach (see Section 3.4.2.4 for details). Transport processes, not
described by this average velocity, are represented by turbulent diffusion (using an
eddy diffusivity, which is several orders of magnitude higher than molecular diffu-
sivity). More information on this topic is given in Section 3.4.
Most numerical models use grids with spatial and time steps larger than those
associated with turbulent eddies. Again, processes not resolved by velocity computed
by models have to be accounted for by diffusion (subgrid diffusion).
The box represented in Figure 3.1 is commonly used to illustrate the concept of
diffusion in laminar flows. The same box could be used to illustrate the concept of
eddy diffusion in turbulent flows or subgrid diffusion in numerical models. In
molecular diffusion white and black dots represent molecules, while in other cases
they represent eddies. In the initial conditions (stage (a) in Figure 3.1) two different
fluids are kept apart by a diaphragm. Molecules inside each half-box move randomly,
with velocities not described by our model (Brownian or eddy). When the diaphragm
is removed, particles from each side keep moving, resulting in the possibility of