Geoscience Reference
In-Depth Information
∂
q
x
∂
gh
∂
z
s
=−
x
+
D
x
(7.29)
∂
t
∂
q
y
∂
gh
∂
z
s
∂
=−
+
D
y
(7.30)
t
y
where
q
x
and
q
y
are the depth-integrated specific discharges in the
x
- and
y
-directions,
and
D
x
and
D
y
include all the remaining terms in the momentum equations.
The perturbation (not necessarily infinitesimal) velocities are defined as
u
x
ˆ
=
u
x
h
. Subtracting the vertically-integrated momentum
equations (7.29) and (7.30) from the 3-D momentum equations (7.26) and (7.27)
yields
−
q
x
/
h
and
u
y
ˆ
=
u
y
−
q
y
/
h
1
h
∂(
h
u
x
ˆ
)
D
x
h
+
∂
∂
t
∂
∂
u
x
ˆ
+
q
x
=
B
x
−
ν
(7.31)
∂
t
z
z
h
h
1
h
∂(
h
u
y
ˆ
)
D
y
h
+
∂
∂
t
∂
∂
u
y
ˆ
+
q
y
=
B
y
−
ν
(7.32)
∂
t
z
z
h
Eqs. (7.31) and (7.32) are the governing equations of the internal mode. They do
not contain the surface-slope terms, and thus, a large time step (much larger than the
limit imposed by the gravity wave propagation) may be used in the computation of
the internal mode.
The vertical
-coordinate transformation (4.91) and a horizontal stretching coor-
dinate transformation are adopted to handle the variation of free surface and the
complexity of horizontal geometry. The water level and velocity are stored on a stag-
gered grid. In the external mode, all terms in the continuity equation (7.28) and the
time-derivative and surface-slope terms in the momentum equations (7.29) and (7.30)
are treated implicitly. The resulting discretized equation system is factorized in the
x
-
and
y
-directions and solved consecutively by inversion of tridiagonal matrices. In the
internal mode, Eqs. (7.31) and (7.32) are discretized by a two-level scheme with the
vertical diffusion terms treated implicitly. The vertical implicit scheme is essential, as it
increases the time step significantly. The bottom friction terms are also treated implic-
itly for unconditional numerical stability in shallow water. After the depth-averaged
and perturbation velocities are solved in the external and internal modes, the local
velocities
u
x
and
u
y
are obtained using
u
x
σ
h
. The vertical
local velocity
u
z
is then determined using the discretized 3-D continuity equation.
In Blumberg and Mellor's model, the external mode also solves Eqs. (7.28)-(7.30)
to obtain the depth-averaged quantities of tidal motions, but the internal mode directly
solves Eqs. (7.26) and (7.27) to compute the local velocity rather than the perturbation
velocity. Explicit schemes are used in both internal and external modes except that
the vertical diffusion terms in Eq. (7.26) and (7.27) are treated implicitly. Because
numerical stability is controlled by surface gravity waves in the external mode but
by advection and diffusion in the internal mode, the time step in the external mode is
much shorter than that in the internal mode. To enhance the computational efficiency,
a special time-marching strategy is adopted. The external mode solutions are first
obtained with the terms
D
x
and
D
y
in Eqs. (7.29) and (7.30) held fixed in time, and
=ˆ
u
x
+
q
x
/
h
and
u
y
=ˆ
u
y
+
q
y
/
