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

$\eta ={\begin{cases}{\frac {W{{r}^{2}}\sin 2\theta }{4gHR}},&{\mbox{if }}f_{c}=0\\{\frac {W{{f}_{c}}}{RgH\kappa }}\left[{\frac {{R}^{2}}{8}}+{\frac {{r}^{2}}{4}}\left({\frac {\kappa }{{f}_{c}}}\sin 2\theta -1\right)\right],&{\mbox{if }}f_{c}\neq 0\\\end{cases}}$

(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 (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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.

---

Test Cases

Documentation Portal