Circular Basin

UNDER CONSTRUCTION

Analytical Solution

Dupont (2001) presented an analytical solution for a closed circular domain on an f-plane, with radius $R$, a linear bottom friction, and a spatially variable wind stress equal to ${\tau }_{Wx}=Wy/R$, ${\tau }_{Wy}=0$ where $W$ is the gradient of the wind forcing and $y$ is the vertical coordinate. The water surface elevation solution is given by

$\eta =\{\begin{array}{ll}\frac{W{r}^2\sin 2\theta }{4gHR},& \text{if }{f}_c=0\\ \frac{W{f}_c}{RgH\kappa }[\frac{R^2}{8}+\frac{r^2}{4}(\frac{\kappa }{f_c}\sin 2\theta -1)],& \text{if }{f}_c\ne 0\\ \end{array}$

(1)

The current velocities are independent of the Coriolis parameter and are given by

$u=\frac{Wy}{2R\kappa }$

(2)

$v=-\frac{Wx}{2R\kappa }$

(3)

Setup

Figure 1. Computational grid.

The model is run to steady state from zero current and water level initial conditions with Failed to parse (syntax error): {\displaystyle W = 10^-4 \text{m^2s^{-2}} } , Failed to parse (syntax error): {\displaystyle \kappa = 10^{-3} \text{s^{-1}} </math , and both <math> f_c = 0} and $f_c=10^{-4}{\text{s}}^{-1}$. Figure 1 shows the computational grid with 5 levels of refinement from 2000 m to 125 m.

Table 1. General Settings for Flow over a Bump

Parameter	Value
Time step	3600 s
Initial Water Depth	10 m
Linear Bottom Friction Coefficient	0.001

Results

Table 2. Goodness of fit statistics for the water elevation

Statistic	Value
RMSE	0.0074 m
RMAE	0.0068
R^2	0.991
Bias	0.0017 m

References

• Dupont, F., 2001. Comparison of numerical methods for modelling ocean circulation in basins with irregular coasts. Ph.D. thesis, McGill University, Montreal.

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Test Cases

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