SEDLOB and PATLOB
125
without producing ripples and spikes of sediment transport capacity in the adjacent grid points. Additional smoothing of other fields (bottom topography,
sediment concentration near bottom, etc.) is therefore not required.
2.8
Numerical design
SEDLOB uses a staggered Arakawa-B-grid with a half grid distance's shift between T -5 points and u-v points (Cox 1984; Mesinger and Arakawa 1976), and a
Cartesian coordinate system with the vertical axis directed downward. The time
integration is carried out using an "upstream differencing" scheme (Mesinger
and Arakawa 1976; Struve 1978). This scheme is known to be rather effective
(Dube et al. 1986), and may become essential for long-term integrations. It
should be noted that this scheme, which is also called the donor cell, upward, or
upwind scheme, is positive definite: positive values, like concentration, always
remain positive during integration ["possitivity"; ( Cf ?: 0 for all i) ~ ( C i
N ?: 0 for
all i and N)] (Eppel1977178; Smolarkiewicz 1983). This is an important feature
for mass transports, e.g., the transport of marine and eolian sediments, vapor,
or gas in the atmosphere all are positive definite. Using second-order or higherorder integration accuracy schemes can introduce some difficulties because of a
negative solution of the equation results (Smolarkiewicz 1983). Furthermore, a
given disturbance is transported in the direction of physical advection and not,
as in other discretization schemes, in the opposite direction (Mesinger and Arakawa 1976; Struve 1978). Furthermore, the "upstream differencing" scheme is
mass conservative. All these requirements are only satisfied if the Courant-Friedrichs-Levy criterion is not violated (Eppel 1977178; Mesinger and Arakawa
1976; Smolarkiewicz 1983; Struve 1978):
(22)
where cx' cy,c z are Courant numbers (cx,y,z::; i).Thus, the maximum time step in
the model is:
{
dx·· dy..
dz J
M < . __ I,J _ _ _ I,J _ ___ k_
_m! IUi,j,kl'h,j,kl'lw g .. I'
I,J,k
(23)
Here dXi,j' dYi,j' dZ k denote the grid steps. The three velocity components are
by Ui,j,k' Vi,j,k and w gj . k'
Because a normar'~upwind" -scheme has a strong implicit diffusion, we modify it in the 2-D submodel. There are different schemes to overcome the deficiencies of the simple upwind formulation. For example, the self-adjusting hybrid
scheme (SASH), or the flux correction technique (FCT) offer better functionality
while retaining the advantages of the upward scheme. However, the usefulness
125
without producing ripples and spikes of sediment transport capacity in the adjacent grid points. Additional smoothing of other fields (bottom topography,
sediment concentration near bottom, etc.) is therefore not required.
2.8
Numerical design
SEDLOB uses a staggered Arakawa-B-grid with a half grid distance's shift between T -5 points and u-v points (Cox 1984; Mesinger and Arakawa 1976), and a
Cartesian coordinate system with the vertical axis directed downward. The time
integration is carried out using an "upstream differencing" scheme (Mesinger
and Arakawa 1976; Struve 1978). This scheme is known to be rather effective
(Dube et al. 1986), and may become essential for long-term integrations. It
should be noted that this scheme, which is also called the donor cell, upward, or
upwind scheme, is positive definite: positive values, like concentration, always
remain positive during integration ["possitivity"; ( Cf ?: 0 for all i) ~ ( C i
N ?: 0 for
all i and N)] (Eppel1977178; Smolarkiewicz 1983). This is an important feature
for mass transports, e.g., the transport of marine and eolian sediments, vapor,
or gas in the atmosphere all are positive definite. Using second-order or higherorder integration accuracy schemes can introduce some difficulties because of a
negative solution of the equation results (Smolarkiewicz 1983). Furthermore, a
given disturbance is transported in the direction of physical advection and not,
as in other discretization schemes, in the opposite direction (Mesinger and Arakawa 1976; Struve 1978). Furthermore, the "upstream differencing" scheme is
mass conservative. All these requirements are only satisfied if the Courant-Friedrichs-Levy criterion is not violated (Eppel 1977178; Mesinger and Arakawa
1976; Smolarkiewicz 1983; Struve 1978):
(22)
where cx' cy,c z are Courant numbers (cx,y,z::; i).Thus, the maximum time step in
the model is:
{
dx·· dy..
dz J
M < . __ I,J _ _ _ I,J _ ___ k_
_m! IUi,j,kl'h,j,kl'lw g .. I'
I,J,k
(23)
Here dXi,j' dYi,j' dZ k denote the grid steps. The three velocity components are
by Ui,j,k' Vi,j,k and w gj . k'
Because a normar'~upwind" -scheme has a strong implicit diffusion, we modify it in the 2-D submodel. There are different schemes to overcome the deficiencies of the simple upwind formulation. For example, the self-adjusting hybrid
scheme (SASH), or the flux correction technique (FCT) offer better functionality
while retaining the advantages of the upward scheme. However, the usefulness