2 A Strategy for Bioremediation of Marine Shorelines …
45
Due to Eq. (2.58) and condition (2.60), Eq. (2.61) can be written as
m(Q 0 ) − m(Q
∗
) = λ
T
0
g 1 (r 1 , t)δ Q dt +
1
2
T
0
δ
2 Q dt =
1
2
T
0
δ
2 Q dt > 0
where λ is given by (2.59). Thus, m(Q 0 ) > m(Q ∗ ), and hence, Q ∗ defined by (2.58)
and (2.59) is the global minimum of variational problem (2.50)–(2.52). Note that
Theorem 2.2 from the previous section ensures the uniqueness of this minimum.
On the other hand, the mass of nutrient introduced into the aquatic system by
means of the discharge rate Q ∗ is assessed as
m(Q
∗
) =
c 2
1
2
T
0 g 2
1 (r 1 , t) dt
(2.62)
so that, in order to minimize the amount of mass, the integral
I (r 1 ) =
T
0
g
2
1 (r 1 , t) dt
(2.63)
must take its maximum value. Thus, the optimal discharge point r ∗
1 is chosen so as to
maximize the area under the function g 2
1 (r 1 , t), t ∈ (0, T ). Note that I (r 1 ) defined
by (2.63) is a continuous non-linear function of three real variables r 1 = (x, y, z),
which has a global maximum in the closed set Ω 1 . Indeed, according to the definition
of the adjoint model forcing p(r, t), the greatest values of the adjoint function are
always achieved at the points of domain Ω 1 .
Because all these results can successively be applied to each oil-polluted zone,
we conclude that during the first stage of the remediation strategy, the method allows
us to determine the discharge points r ∗
i , one in each oil-polluted zone Ω i , as well
as to define with Eqs. (2.58) and (2.59) the corresponding basic discharge rates of
nutrient:
Q
∗
i (t) = λ i g i (r
∗
i , t) =
c i
T
0 g 2
i (r ∗
i , t)dt
g i (r
∗
i , t),
i = 1, . . . , N .
(2.64)
Note that all the discharge parameters are calculated by using the adjoint model
solutions.
2.6.2 Second Stage: Modulation of Basic Discharge Rates
In the second stage of the remediation strategy, we determine positive parameters
γ 1 , γ 2 . . . , γ N such that the new discharge rates of nutrient
Q i (t) = γ i Q
∗
i (t),
1 ≤ i ≤ N
(2.65)
45
Due to Eq. (2.58) and condition (2.60), Eq. (2.61) can be written as
m(Q 0 ) − m(Q
∗
) = λ
T
0
g 1 (r 1 , t)δ Q dt +
1
2
T
0
δ
2 Q dt =
1
2
T
0
δ
2 Q dt > 0
where λ is given by (2.59). Thus, m(Q 0 ) > m(Q ∗ ), and hence, Q ∗ defined by (2.58)
and (2.59) is the global minimum of variational problem (2.50)–(2.52). Note that
Theorem 2.2 from the previous section ensures the uniqueness of this minimum.
On the other hand, the mass of nutrient introduced into the aquatic system by
means of the discharge rate Q ∗ is assessed as
m(Q
∗
) =
c 2
1
2
T
0 g 2
1 (r 1 , t) dt
(2.62)
so that, in order to minimize the amount of mass, the integral
I (r 1 ) =
T
0
g
2
1 (r 1 , t) dt
(2.63)
must take its maximum value. Thus, the optimal discharge point r ∗
1 is chosen so as to
maximize the area under the function g 2
1 (r 1 , t), t ∈ (0, T ). Note that I (r 1 ) defined
by (2.63) is a continuous non-linear function of three real variables r 1 = (x, y, z),
which has a global maximum in the closed set Ω 1 . Indeed, according to the definition
of the adjoint model forcing p(r, t), the greatest values of the adjoint function are
always achieved at the points of domain Ω 1 .
Because all these results can successively be applied to each oil-polluted zone,
we conclude that during the first stage of the remediation strategy, the method allows
us to determine the discharge points r ∗
i , one in each oil-polluted zone Ω i , as well
as to define with Eqs. (2.58) and (2.59) the corresponding basic discharge rates of
nutrient:
Q
∗
i (t) = λ i g i (r
∗
i , t) =
c i
T
0 g 2
i (r ∗
i , t)dt
g i (r
∗
i , t),
i = 1, . . . , N .
(2.64)
Note that all the discharge parameters are calculated by using the adjoint model
solutions.
2.6.2 Second Stage: Modulation of Basic Discharge Rates
In the second stage of the remediation strategy, we determine positive parameters
γ 1 , γ 2 . . . , γ N such that the new discharge rates of nutrient
Q i (t) = γ i Q
∗
i (t),
1 ≤ i ≤ N
(2.65)