Environmental Engineering Reference
In-Depth Information
Applying now Eq. (
2.4
), the Lagrange identity and the well-known property of
the Dirac delta one can obtain
T
T
A
∗
g
drdt
N
−
∂
g
0
ˆ
t
+
=
Q
i
(
t
)
g
(
r
i
,
t
)
dt
+
g
(
r
,
0
)ˆ
(
r
)
dr
.
(2.17)
∂
i
=
1
0
D
0
D
In order to take advantage of Eq. (
2.17
), which explicitly relates the discharge
rates of nutrient
Q
i
(
t
)
with the concentration of nutrient
ˆ(
r
,
t
)
through the adjoint
function
g
, we consider the following adjoint dispersion model:
−
∂
g
∂
t
−
U
·∇
g
−∇·
μ
∇
g
+
˃
g
−∇·
g
s
=
p
(
r
,
t
),
(2.18)
g
s
=−
v
s
g
k
in
D
,
(2.19)
μ
∂
g
n
+
ʶ
·
=
0on
S
T
,
g
k
n
(2.20)
∂
μ
∂
g
0on
S
+
,
n
+
U
n
g
=
(2.21)
∂
μ
∂
g
0on
S
−
,
n
=
(2.22)
∂
μ
∂
g
n
=−
g
s
·
n
on
S
B
,
(2.23)
∂
g
(
r
,
T
)
=
0in
D
.
(2.24)
Note that the boundary conditions (
2.20
)-(
2.23
) and final condition (
2.24
)im-
posed on the solution are those that guarantee the fulfilment of the Lagrange identity.
Furthermore, one can see that the first, the second and the fifth terms of Eqs. (
2.4
)
and (
2.18
) have opposite signs. Thus, the comparison of the equations and boundary
conditions of the models (
2.4
)-(
2.12
) and (
2.18
)-(
2.24
) leads to the important result:
if the adjoint model (
2.18
)-(
2.24
) is solved backward in time (from
t
=
T
to
t
=
0)
then it also has a unique solution, which continuously depends on the forcing
p
(
r
,
t
)
.
This result can be immediately shown by the transformation of variable
t
=
T
−
t
,
cf. [
43
].
Moreover, the forcing
p
(
r
,
t
)
of Eq. (
2.18
) can be defined so that the mean con-
centration of nutrient
T
1
˄
|
ʩ
i
|
J
i
(ˆ)
=
ʩ
i
ˆ(
r
,
t
)
drdt
T
−
˄
in an oil-polluted zone
ʩ
i
ↂ
D
will be explicitly related with all the discharge rates
0
Q
j
(
t
)
,
j
=
1
,...,
N
, and initial concentration of nutrient
ˆ
(
r
)
through the adjoint
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