Geoscience Reference
In-Depth Information
3.4.3
S
PECIAL
F
LOWS
AND
S
IMPLIFICATIONS
IN
D
IMENSIONALITY
The equations derived above explain the whole spectrum of dynamic behavior that
may be expected in the coastal seas and lagoons. Quite often, however, the flow
shows some peculiar characteristics that make it possible to simplify the equations
even more. In this section some often-made simplifications are presented that allow
the reduction of the dimensions of the problem.
An example may be a flow that is constant in one direction. This means that all
derivatives in this direction vanish. Therefore, the solution in this equation is trivial
and the component of this direction may be eliminated, resulting in a lower dimen-
sional problem that is easier to track.
3.4.3.1
Barotrophic 2D Equations
A very important simplification can be achieved when the key variables can be
considered constant along the water column. This is normally the case when no
stratification occurs and the water is well mixed.
In this case, the equations can be integrated over the water column and the two-
dimensional barotropic (vertically integrated) equations result. These equations are
also called
shallow water equations
, even if they are not applicable only to shallow
basins and not all shallow lagoons allow for this simplification.
A better-suited continuity equation may be deduced for this vertically integrated
flow. This can be done by applying the kinematic boundary conditions (explained
below) to the continuity equation, or it can also be easily derived directly. Consider
a control volume over the whole water column with rectangular area
x
2
. The
total inflow in
x
1
direction is . A similar equa-
tion holds in the
x
2
direction and the total water depth is called
H
. The total inflow
must result in more water in the control volume and because of incompressibility
the only way for water to accumulate is through the rise of the water level
∆
x
1
∆
Huxx
(, )
∆
x
−
Hux
(
+
∆
xx
, )
∆
x
112
2
11
12
2
η
. The
rate of increase in the volume can therefore be written as
(/
∂∂
η
t
)
∆
12
xx
. If we
divide this conservation equation by the area of the control volume
∆
x
1
∆
x
2
we have
Hu
(
x
+
∆
x
,
x
)
−
Hu
(,
x
x
)
Hu
(,
x
x
+−
∆
∆
x
)
Hu
(,
x
x
)
∂
∂
t
+
11
12
112
+
212
2
212
=
0
∆
x
x
1
2
and in the limit
+
∂
∂
Hu
x
+
∂
∂
Hu
x
∂
∂
t
1
2
=
0
1
2
Because the variables are considered constant throughout the water column the
density
that is changing mainly in the vertical direction may be considered constant
all over and is denoted as
ρ
ρ
0
. With this simplification the hydrostatic equation can
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