48
D. Parra-Guevara and Y.N. Skiba
Fig. 2.3 Adjoint functions
g i j = g i (r ∗
j , t)
corresponding to zone Ω 1
(i = 1) when they are
restricted to the optimal
discharge points r ∗
j
( j = 1, 2, 3)
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.0025
0.005
0.0075
0.01
0.0125
0.015
t
g 12
g 11
g 13
Fig. 2.4 Adjoint functions
g i j = g i (r ∗
j , t)
corresponding to zone Ω 2
(i = 2) when they are
restricted to the optimal
discharge points r ∗
j
( j = 1, 2, 3)
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
t
g 22
g
21
g 23
A similar result follows from Fig. 2.5, since the adjoint functions g 31 and g 32 are
almost zero in the time interval [2.25, 4.0], and therefore the discharge of nutrients at
points r ∗
1 and r ∗
2 practically has no influence on its concentration in zone Ω 3 . However,
it follows from Fig. 2.4 that function g 21 is positive in the time interval [2.25, 4.0], and
hence, the discharge of nutrient at point r ∗
1 influences its concentration in zone Ω 2 ,
as it was expected. Finally, the temporal behaviour of adjoint function g 23 allows us
to conclude that the discharge at point r ∗
3 does not affect the concentration of nutrient
in Ω 2 .
Thus, the polluted zones are not independent with respect to the dispersion process,
since the release of nutrient in a particular zone can affect the concentration in
D. Parra-Guevara and Y.N. Skiba
Fig. 2.3 Adjoint functions
g i j = g i (r ∗
j , t)
corresponding to zone Ω 1
(i = 1) when they are
restricted to the optimal
discharge points r ∗
j
( j = 1, 2, 3)
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.0025
0.005
0.0075
0.01
0.0125
0.015
t
g 12
g 11
g 13
Fig. 2.4 Adjoint functions
g i j = g i (r ∗
j , t)
corresponding to zone Ω 2
(i = 2) when they are
restricted to the optimal
discharge points r ∗
j
( j = 1, 2, 3)
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
t
g 22
g
21
g 23
A similar result follows from Fig. 2.5, since the adjoint functions g 31 and g 32 are
almost zero in the time interval [2.25, 4.0], and therefore the discharge of nutrients at
points r ∗
1 and r ∗
2 practically has no influence on its concentration in zone Ω 3 . However,
it follows from Fig. 2.4 that function g 21 is positive in the time interval [2.25, 4.0], and
hence, the discharge of nutrient at point r ∗
1 influences its concentration in zone Ω 2 ,
as it was expected. Finally, the temporal behaviour of adjoint function g 23 allows us
to conclude that the discharge at point r ∗
3 does not affect the concentration of nutrient
in Ω 2 .
Thus, the polluted zones are not independent with respect to the dispersion process,
since the release of nutrient in a particular zone can affect the concentration in