5.7 Exercise 11: Eulerian Advection
107
5.7.2 Task Description
We use the steady-state f ow fiel computed in Exercise 10 to predict the movement
pattern of Eulerian tracer being introduced at a concentration of unity in a certain
region of the model domain. In this exercise, tracer is released in the northwestern
part of the lake in a quadratic box with side lengths of 0.5 km. With an inspection
of the steady-state circulation established in the lake (see Fig. 5.8), we expect that
this tracer is initially advected southward and separates into westward and eastward
fl wing branches near the southern boundary. With “frozen” dynamics, a time step
much greater compared with that in Exercise 10 can be used. I used Δt = 200 s,
which satisfie the CFL condition (5.4). Advection schemes being tested are the
upstream scheme, the Lax-Wendroff scheme, the Superbee scheme and the Super-C
scheme.
5.7.3 Results
Figure 5.9 shows results employing the upstream scheme. This scheme is extremely
numerically diffusive and triggers substantial artificia decrease of the maximum
concentration by 85% after 9 h of simulation corresponding to 162 simulation steps.
Owing to this numerical diffusion, tracer concentration is vigorously mixed horizontally.
The Lax-Wendroff scheme appears to be less diffusive (Fig. 5.10), but has other
disadvantages. This scheme produces numerical oscillations, leading to slightly
negative concentrations in some regions and concentrations exceeding the initial
concentration in other regions.
The Superbee scheme produces more convincing results (Fig. 5.11) void of
numerical oscillations and far less diffusive compared with the upstream scheme.
Fig. 5.9 Exercise 11. Snapshots of contours of tracer concentration (coloured lines) using the
upstream scheme. The contour interval is maximum tracer concentration divided by 10. The header
displays maximum concentration relative to initial concentration in per cent. Thin black lines are
bathymetric contours. The square indicates the release area of the tracer
107
5.7.2 Task Description
We use the steady-state f ow fiel computed in Exercise 10 to predict the movement
pattern of Eulerian tracer being introduced at a concentration of unity in a certain
region of the model domain. In this exercise, tracer is released in the northwestern
part of the lake in a quadratic box with side lengths of 0.5 km. With an inspection
of the steady-state circulation established in the lake (see Fig. 5.8), we expect that
this tracer is initially advected southward and separates into westward and eastward
fl wing branches near the southern boundary. With “frozen” dynamics, a time step
much greater compared with that in Exercise 10 can be used. I used Δt = 200 s,
which satisfie the CFL condition (5.4). Advection schemes being tested are the
upstream scheme, the Lax-Wendroff scheme, the Superbee scheme and the Super-C
scheme.
5.7.3 Results
Figure 5.9 shows results employing the upstream scheme. This scheme is extremely
numerically diffusive and triggers substantial artificia decrease of the maximum
concentration by 85% after 9 h of simulation corresponding to 162 simulation steps.
Owing to this numerical diffusion, tracer concentration is vigorously mixed horizontally.
The Lax-Wendroff scheme appears to be less diffusive (Fig. 5.10), but has other
disadvantages. This scheme produces numerical oscillations, leading to slightly
negative concentrations in some regions and concentrations exceeding the initial
concentration in other regions.
The Superbee scheme produces more convincing results (Fig. 5.11) void of
numerical oscillations and far less diffusive compared with the upstream scheme.
Fig. 5.9 Exercise 11. Snapshots of contours of tracer concentration (coloured lines) using the
upstream scheme. The contour interval is maximum tracer concentration divided by 10. The header
displays maximum concentration relative to initial concentration in per cent. Thin black lines are
bathymetric contours. The square indicates the release area of the tracer