Modeling Concepts - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 268

mixed with V 0 and that an equivalent mixed volume leaves without returning into
the system.
In reality, mixing of incoming waters at either boundary of the lagoon will more
likely occur with waters located near that boundary and the local flushing time will
progressively increase as the distance away from this boundary increases. Hence,
the integral and local flushing times are in fact functions of space. For nonstationary
Q , it is a function of time as well. As such, we can also characterize a coastal system
by its local flushing time distribution where x , y , z are the spatial coor-
dinates and t is time. More generally, the integral flushing time is simply the spatial
integration of the local flushing time over the area of the system, under stationary
conditions.
While the integral flushing time is useful for comparative coastal ecosystems
studies, the local flushing time is useful for selecting the most appropriate sites for
localized human interventions in a lagoon, such as an aquaculture site, an outfall
diffuser site, etc.
Spatial distribution estimates of the local flushing time normally require the use
of hydrodynamic-numerical models. In this case, two approaches can be used: the
Lagrangian particle tracking approach or the Eulerian dissolved tracer advection-
dispersion approach. In both methods, a hydrodynamic model forced by most rep-
resentative forcing functions at its open boundaries is first developed for the lagoon.
Once the model has been calibrated and validated with measurements, coupled
simulations are then carried out. For the Lagrangian method, a Lagrangian particle
model is coupled to the hydrodynamic model and particles are introduced at each grid
point in the lagoon at the beginning of the simulation. The simulation is then performed
for a long period of time relative to the expected integral time and the time at which
each particle leaves the lagoon is recorded. For the Eulerian method, an advection-
dispersion model is coupled to the hydrodynamic model and a dissolved tracer con-
centration of 100% is set as an initial condition in the lagoon and 0% elsewhere. The
coupled simulation is then performed for the same period as above and the time at
which the initial concentration in each cell drops below is recorded.
In both methods, the result is a two-dimensional distribution of the local flushing
time of the lagoon. A more formal discussion of the flushing time and related time
scales can be found in Bolin and Rodhe, 10 Takeoka, 11 and Zimmerman. 9 An appli-
cation of the tracer method to estimate local renewal times is found in Koutitonsky
et al. 12

τ (, ,,) xyzt

C
e 0

C 0

6.3.3.2 Surface and Bottom Friction Layers

Both the maximum and the average wind-induced upper layer currents may be
estimated from Ekman's (1905) theory. 13 A more recent development is presented
by Madsen. 14 These theories describe currents in the upper layer of the open ocean,
in equilibrium under the effect of shear stress, Coriolis force, and the local pressure
gradient. Momentum is transmitted from the surface to deeper layers by vertical
turbulent mixing. The vertical mixing is constant for Ekman's theory and variable
for Madsen's theory. In both theories, nonlinear convective terms and horizontal
turbulent mixing as well as temporal acceleration term are neglected. The resulting