Modeling Concepts - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 294

6.5.1.2 Numerical Restrictions

Numerical methods are used in numerical models in order to approximate the
differential equations of motion and constituent transport into algebraic difference
forms. These are then solved for unknown values at incremental finite points in
space and time. However, the difference form approximations introduce numerical
constraints and, occasionally, numerical errors.
For example, truncation errors arise when high-order terms are neglected in the
approximations. These errors are directly proportional to the spatial grid size and
the integration time step. On the other hand, attempts to reduce the time step or the
grid size will introduce a number of calculations and thus round off errors. This
error arises because the model computations are performed with a fixed number of
decimal places, which is directly proportional to the number of calculations involved.
A good numerical model should sum all numerical errors and provide the user with
information on the space and/or time discretization limits that will minimize the
errors for a given domain.
Numerical algorithms used in a model have convergence and sensitivity char-
acteristics. Some widely used numerical schemes are known to converge toward
the real solution. This is not the case for some nonlinear problems or some
unconventional algorithms. Regarding sensitivity, it is suggested that the numerical
methods used should not enhance the physical sensitivity of the problem under
consideration, i.e., small variations in model parameters or boundary conditions
should not give rise to solutions outside the physical range or to unbounded
solutions.

6.5.1.3 Subgrid Processes Restrictions

Field measurements have shown that fluid motion in natural basins is mainly turbu-
lent. In open oceans, the turbulent energy cascades down from large-scale gyres to
small-scale turbulent eddies. In lagoons, however, only part of that turbulent spectrum
will be present. The spatial spectral window is limited, on the one hand, by the size
of the lagoon and, on the other hand, by molecular dissipation scales (~1 cm);
however, two important processes contributing kinetic energy at intermediate scales
can be found in lagoons. 43 These are the transformation of the potential energy into
kinetic energy following baroclinic instability at internal Rossby scales (~10 km)
and the transformation of the internal wave energy into turbulent kinetic energy due
to the Kelvin-Helmholtz instability. In the latter case, horizontal and vertical scales
are L H 20 cm, respectively.

In addition, numerical models can describe only that part of the spectrum energy
bounded in space at one extreme by the lagoon length scale and at the other extreme
by the Nyquist wave number imposed by the grid size. Therefore, processes with
wave numbers outside this range must be parameterized in the primitive equations
of motion by turbulent diffusion, dissipation, and inter-scale energy exchange terms
and are referred to as subgrid processes. In the ideal case, the parameterization of
subgrid processes should be based on a detailed analysis of the subgrid vertex
dynamics and its relationship with the large-scale processes directly resolved in the
primitive models.

= 1 m and L V =