A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

140 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
Continuing with comparisons of test simulations with those of Leendertse and Liu (1977), a second simulation of the viscous
seiching problem was computed using a slightly shorter basin length of 44 km. The simulation covered 20 hours using the same
model parameters as those used to simulate the longer basin. The calculations used the semi-implicit, leapfrog scheme with a time
step of 45 seconds. The vertical profiles of the u velocity component at mid-basin were saved to a file at a constant time interval
of 3.125 hours (250 time steps). A plot of these profiles is shown in figure 5.31. The same solution profiles, digitized from the
report by Leendertse and Liu (1977, fig. 12), are also shown. Although the two sets of profiles are in reasonably close agreement,
the semi-implicit solution predicts larger absolute magnitudes for the velocities in each profile than was computed by Leendertse
and Liu. It was suspected that the cause of the difference was that equation 5.4 underestimated the magnitude of the eddy viscosities
computed by the turbulence model of Leendertse and Liu. Leendertse and Liu (1977, fig. 6D) graphically present a distribution
of the magnitudes of the turbulent kinetic energy per unit mass calculated from their turbulence model after 60 time steps of
simulation; using equation 5.3 to translate these values into a distribution of the eddy viscosity, a depth-averaged eddy viscosity
at mid-basin was calculated as approximately 300 cm 2 /s. 53 The depth-averaged eddy viscosity calculated during the semi-implicit
simulation after 60 time steps (using eq. 5.4) was only 200 cm 2 /s. To decrease the differences between the two sets of velocity
profiles in figure 5.31, another simulation with the semi-implicit, leapfrog scheme was made using a constant value of 300 cm 2 /s
for the eddy viscosity. The two sets of profiles are in closer agreement (fig. 5.32); some differences still exist but are most likely
attributable to the difference in turbulence parameterizations or to inexact digitizing of profiles from the report by Leendertse and
Liu. Note that the computed profiles using the constant eddy viscosity exhibit less vertical variation in velocity (shear) than in the
previous solution in which the eddy viscosity varied over the depth.
One final simulation of the seiching problem was made to test the semi-implicit scheme for conditions using variable density.
A linear vertical density gradient of Δρ / Δz = 0.035 kg/m 3 /cm was introduced with freshwater at the free surface. Equation 5.4
was used to estimate the eddy viscosity under homogeneous density conditions, and the effects of stratification were incorporated
using the stability functions (eqs. 2.27 and 2.36) proposed by Munk and Anderson (1948). The longer basin length of 46 km was
used. The oscillation in this test showed greater vertical shear in the velocity profile than the test using the homogeneous density
because the vertical mixing was reduced by the stratification (fig. 5.33). The results were very similar to those of Leendertse and
Liu (1977, fig. 9), but they were not identical because of differences in the turbulence parameterizations and stability functions
used in the two models.
53 A depth-averaged eddy viscosity of 300 cm 2 /s is quite high for a basin 12 meters deep. A standard estimate for the depth-averaged eddy viscosity in
constant density flow in which the turbulence is generated by bottom shear stress is 0.067Hu* (Fischer and others, 1979). This estimate corresponds to a
maximum of about AV0 = 100 cm 2 /s. A simulation using A = 100 cm
V0
2 /s did not agree well with the solution of Leendertse and Liu (1977).