Geoscience Reference
In-Depth Information
after a large number of time steps, of the order of 100, an internal mode calculation
is performed. The external mode provides
y for insertion into the
internal mode equations, which are then solved with a much longer time step. Once
the vertical structures have been determined, the terms D x and D y in Eqs. (7.29) and
(7.30) are updated and another external mode solution begins.
In both models described above, there may be a slow tendency that the vertical
integral of the internal mode velocity differs from the external mode velocity. This
arises because of different truncation errors in each mode. To prevent accumulated
mismatch, the vertical mean of the internal velocity is replaced at every time step by
the external mode velocity.
z s
/∂
x and
z s
/∂
7.2.3 Projection method
Casulli and Cheng (1992) proposed an algorithm that uses an implicit scheme in the
vertical direction and a semi-implicit scheme in the horizontal directions. The grid is
staggered. The momentum equations (7.19) and (7.20) are rewritten as
u x
g
z s
+
ADVU
+
x =
0
(7.33)
t
u y
g
z s
+
ADVV
+
=
0
(7.34)
t
y
where ADVU and ADVV include all the remaining terms.
Time differencing of Eqs. (7.33) and (7.34) leads to
z n + 1
s
t
u n + 1
x
=
F
(
u x
)
g
(7.35)
x
z n + 1
s
t
u n + 1
y
=
F
(
u y
)
g
(7.36)
y
In the above equations, F
(
u x
)
and F
(
u y
)
represent the solutions to Eqs. (7.33) and
(7.34) at time level n
+
1 where the contribution from the water surface gradient terms
has been deferred.
The depth-integrated continuity equation is written as
z s
z s
z s
t +
+
u n + x dz
u n + 1
dz
=
0
(7.37)
y
x
y
z b
z b
Substituting the momentum equations (7.35) and (7.36) into Eq. (7.37) and ignoring
the term
t 2
z n + 1
s
z n + 1
s
g
(∂
h
/∂
x
·
/∂
x
+
h
/∂
y
·
/∂
y
)
leads to the Poisson equation
for the water level at time level n
+
1:
t
dz (7.38)
z s
z s
+
t 2
2
z n + 1
s
z s
(
1
gh
)
=
F
(
u x
)
dz
F
(
u y
)
x
y
z b
z b
Eq. (7.38) is symmetric and positive definite and can be solved by many methods.
 
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