Geoscience Reference
In-Depth Information
Together with the continuity equation
+
∂
∂
Hu
x
+
∂
∂
Hu
x
∂
∂
t
1
2
=
0
(3.26)
1
2
they form a closed set of equations for the variables
η
,
u
1
,
u
2
, provided that the
boundary conditions for the stress
b
and the values of are specified. The
values for the velocities
u
1
,
u
2
represent average values over the whole water column.
Equations for the transport and diffusion of the temperature and salinity may
also be derived as above. After vertical integration of the conservation equations,
we have
τ
s
,
τ
v
H
+
2
2
Q
H
∂
∂
T
t
∂
∂
T
x
∂
∂
T
x
∂
∂
T
x
+
∂
∂
T
x
1
ρ
H
+
u
+
u
=
v
s
(3.27)
1
2
T
2
2
c
1
2
1
2
p
and
2
2
∂
∂
S
t
∂
∂
S
x
∂
∂
S
x
∂
∂
S
+
∂
∂
S
H
+
u
+
u
=
v
(3.28)
1
2
S
2
2
x
x
1
2
1
2
where the fluxes at the surface and bottom have been set to zero. If these fluxes are
important they can easily be included in the equations. The values
T
and
S
represent
average values of the temperature and salinity over the water column.
It should be noted that these equations could be used to compute the transport
and diffusion of temperature or salinity or any other conservative dissolved substance
C
by the velocity field
u
i
; however, they are not necessary for the solution of the
hydrodynamic equations. This is due to the fact that the hydrodynamic equations
do not depend on the unknown density
ρ
but on
ρ
0
, which is constant and does not
depend on
T
and
S
.
3.4.3.2
1D Equations (Channel Flow)
A further simplification can be adopted for the hydrodynamic equations if the flow
is mainly in one direction. This is true for a flow in a channel where flow perpen-
dicular to the channel axis can be neglected or is of no interest. In this case, the 2D
equations may be integrated over the width of the channel.
If this integration is done the continuity equation reads
+
∂
BHu
x
∂
∂
η
B
1
=
0
(3.29)
t
∂
1
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