Circular Basin

UNDER CONSTRUCTION

Problem

Dupont (2001) presented an analytical solution for a closed circular domain on an f-plane, with radius $R$, a linear bottom friction. The governing equations are

$gh{\frac {\partial \eta }{\partial x}}+{\frac {\partial (U_{j}hC_{tk})}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}$

(1)

$\tau _{bx}=\kappa hU,$, 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.

Solution

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 $W=10^{-}4{\text{m}}^{2}{\text{s}}^{-2}$, $\kappa =10^{-3}{\text{s}}^{-1}$ , and $f_{c}=0$. Table 1 shows the general settings used for CMS-Flow. Figure 1 shows the computational grid with 5 levels of refinement from 2000 m to 125 m.

Table 1. General Settings for Wind-driven flow in a circular basin

Parameter	Value
Time step	3600 s
Simulation Duration	72 hrs
Ramp Period	24 hrs
Initial Water Depth	10 m
Mixing Terms	Off
Wall Friction	Off
Linear Bottom Friction Coefficient	0.001

Results

Figure 1. Analytical current velocities and water levels.

Figure 1. Calculated current velocities and water levels.

Table 2. Goodness of fit statistics for the current velocity and water level

Variable	RRMSE, %	RMAE, %	R^2	Bias
U-Velocity	3.88	0.64	0.997	-4.06e-5
V-Velocity	3.87	0.64	0.997	4.06e-5
Water Level	0.16	0.13	1.000	-3.56e-6

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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