Environmental Engineering Reference
In-Depth Information
in the zone and the optimal release rate Q i (
)
, also named as basic or preliminary
discharge rate. We prove that this problem has unique solution.
In the second stage, we consider the process of dispersion of nutrient in all zones
together. Due to advection by currents, the nutrient released in one zone can reach
other polluted zones. Therefore we must specify (modulate) the strength of all release
rates Q i (
t
in order to fulfil the critical mean concentrations c i in all the polluted zones
during the time interval
t
)
[
T
˄,
T
]
. To this end, we introduce positive coefficients
ʳ i to replace all release rates Q i (
. These coefficients are chosen as
the solution of a quadratic programming problem where the objective function for
minimizing is the mass of nutrient introduced by the new discharge rates
t
)
by
ʳ i Q i (
t
)
ʳ i Q i (
t
)
.
Also, we prove the existence and uniqueness of this optimization problem.
Both stages of this remediation strategy use the adjoint solutions to assess the
mean concentration of nutrient in the oil-polluted zones. Such approach is quite
useful. Indeed, in the first stage, the optimal release point for a specific oil-polluted
zone is found by maximizing a continuous non-linear function of three real variables.
The function is built with the adjoint problem solution corresponding to the selected
zone. In addition, the respective basic discharge rate is determined as a multiple of
the adjoint solution which is evaluated at the optimal discharge point. Of course, the
basic discharge rate also depends on the critical concentration for the respective oil-
polluted zone. And in the second stage, the adjoint solutions, evaluated at the optimal
discharge points, are also used to pose the constraints for the quadratic programming
problem.
Thus, this new remediation method is strongly based on the adjoint estimates.
Nevertheless, it also uses the direct concentration estimates of nutrient in the pol-
luted zones when various discharges of the nutrient are needed. Therefore, the two
equivalent (direct and adjoint) estimates complement each other well in the assess-
ment of nutrients and control of pollutants. The direct estimates, utilizing the solution
of the advection-diffusion problem, enable making the comprehensive analysis of
ecological situation in the whole area. On the other hand, the adjoint estimates use
solutions of the adjoint problems and explicitly depend on the positions of sources,
their discharge rates, and also on the initial distribution of nutrient in the region. Be-
sides, the solutions of adjoint problem serve as influence (weight) functions, which
show the impact of the location of discharge source and its intensity on the con-
centration of nutrient in each oil-polluted zone. Therefore, the adjoint estimates are
effective and economical in the sensitivity study of the concentrations of nutrient to
variations in the model parameters.
Owing to special boundary conditions, both the main and adjoint problems are
well-posed according to Hadamard, that is the solution of each problem exists, is
unique and stable to initial perturbations. These conditions are reduced to the well-
known and natural boundary conditions in the non-diffusion limit (pure advection
problem) and also in the case of a closed sea basin (when the boundary is the coast
line).
Finite difference schemes for the solution of the main and adjoint transport prob-
lems are also given. The schemes are balanced, unconditionally stable, of second-
order approximation, and are based on using the splittingmethod andCrank-Nicolson
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