Geoscience Reference
In-Depth Information
yy
in Eqs. (7.22) and (7.23) are determined using
the Boussinesq assumption with the eddy viscosity determined by a turbulence model.
The shear stresses at layer interfaces are determined by
Reynolds stresses
τ
xx
,
τ
xy
,
τ
yx
, and
τ
t
∂
u
x
∂
t
∂
u
y
∂
τ
=
ρν
z
,
τ
=
ρν
(7.24)
xz
yz
z
The wind driving force determined by Eq. (6.5) is applied at the water surface, and
the bed shear stress at the channel bottom is determined by
c
fb
u
bx
u
bx
+
c
fb
u
by
u
bx
+
u
by
,
u
by
τ
=
ρ
τ
=
ρ
(7.25)
bx
by
where
u
bx
and
u
by
are the horizontal components of velocity at the bottom layer, and
c
fb
is the bottom friction coefficient.
The water level can be determined using the kinematic condition (2.71), the depth-
integrated continuity equation (6.1), or the Poisson equation (6.47).
Eqs. (7.21)-(7.23) can be discretized using the finite difference method, finite
element method, or finite volume method. Normally, the convection terms should be
discretized using upwind schemes, and the other spatial derivatives can be discretized
using the central difference schemes or other similar schemes. The time derivatives can
be discretized explicitly or implicitly. Examples of this layer-integrated model can be
found in Lin and Falconer (1996) and Shanhar
et al
. (2001).
7.2.2 Splitting of internal and external modes
Sheng (1983) and Blumberg andMellor (1987) developed 3-Dmodels for estuarine and
coastal systems based on the hydrostatic pressure assumption. The significant feature
of these models is the splitting of internal and external modes. The internal mode han-
dles the slower vertical baroclinic flow structures, while the external mode computes
the depth-integrated quantities that are governed by the fast barotropic dynamics.
In Sheng's model, the momentum equations (7.19) and (7.20) are rewritten as
ν
∂
u
x
∂
g
∂
z
s
x
+
∂
t
∂
u
x
∂
=−
+
B
x
(7.26)
∂
∂
t
z
z
ν
∂
u
y
∂
g
∂
z
s
+
∂
∂
t
∂
u
y
∂
=−
+
B
y
(7.27)
t
∂
y
z
z
where
B
x
and
B
y
include all the remaining terms.
As described in Section 2.4.1, vertically integrating Eqs. (7.18)-(7.20) leads to
Eqs. (2.79), (2.82), and (2.83), which are the governing equations of the external
mode. They are rewritten as
∂
+
∂
+
∂
q
y
∂
h
q
x
∂
=
0
(7.28)
∂
t
x
y
