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__NOTOC__
__NOTOC__
<font color=red>'''UNDER  CONSTRUCTION'''</font>
<font color=red>'''UNDER  CONSTRUCTION'''</font>
=Analytical Solution=
=Problem=
Dupont (2001) presented an analytical solution for a closed circular domain on an f-plane, with radius , a linear bottom friction, and a spatially variable wind stress equal to ,  where  is the gradient of the wind forcing and  is the vertical coordinate. The water surface elevation solution is given by
Dupont (2001) presented an analytical solution for a closed circular domain on an f-plane with a linear bottom friction. The governing equations are
{{Equation|<math>
  \frac{\partial H U}{\partial x} + \frac{\partial H V}{\partial y} = 0
</math>|1}}


{{Equation| <math> \eta =  
{{Equation|<math>
\begin{cases}  
-f_c V + g \frac{\partial \eta}{\partial x} =  \kappa U + \frac{W y}{R H}
\frac{W{{r}^{2}}\sin 2\theta }{4gHR}, & \mbox{if } f_c=0  \\  
</math>|2}}
\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 \ne0 \\
 
\end{cases}  </math> |2=1}}
{{Equation|<math>
f_c U + g \frac{\partial \eta}{\partial y} =  \kappa V
</math>|3}}
 
where <math>U</math> and <math>V</math> are the depth-averaged current velocities in the <math>x</math> and <math>y</math> directions respectively, <math>g</math> is the gravitational constant, <math>\eta</math> is the water surface elevation with respect to mean sea level, <math>\kappa</math> is a linear bottom friction coefficient, <math>R</math> is the radius of the domain, <math>H</math> is the water depth, and <math>W</math> is a constant equal to the gradient of the wind forcing.
 
=Solution=
The analytical solution for water surface elevation solution is given by
{{Equation|<math>
\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 \ne0 \\
  \end{cases}
</math>|4}}
 
The current velocities are independent of the Coriolis parameter and are given by
{{Equation|<math>
  U = \frac{W y }{2 R H \kappa}
</math>|5}}
 
{{Equation|<math>
V = -\frac{W x }{2 R H \kappa}
</math>|6}}


= Setup =
= Setup =
The model is run to steady state from zero current and water level initial conditions with ,   , and both , and . Figure 1 shows the computational grid with 5 levels of refinement from 2000 m to 125 m.
[[Image:Grid_CB3.png|thumb|right|600px| Figure 1. Computational grid.]]
The model is run to steady state from zero current and water level initial conditions with <math> W = 10^-4 \text{m}^2 \text{s}^{-2} </math>, <math> \kappa = 10^{-3} \text{s}^{-1} </math> , and <math> f_c = 0</math>. 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.


[[Image:Grid_CB3.png|thumb|right|600px| Figure 1. Computational grid.]]
<br style="clear:both" />
'''Table 1. General Settings for Wind-driven flow in a circular basin'''
{|border="1"
|'''Parameter''' ||'''Value'''
|-
|Time step || 3600 s
|-
|Simulation Duration || 72 hrs
|-
|Ramp Period || 24 hrs
|-
|Initial Water Depth || 100 m
|-
|Mixing Terms || Off
|-
|Wall Friction || Off
|-
|Linear Bottom Friction Coefficient || 0.001
|}


   
   
<br style="clear:both" />
<br style="clear:both" />
= Results =
== No Coriolis ==
[[Image:CB3_Analytical_No_Coriolis_V4.png|thumb|left|600px| Figure 1. Analytical current velocities and water levels for the case without Coriolis.]]
[[Image:CB3_Calculated_No_Coriolis_V5.png|thumb|right|600px| Figure 2. Calculated current velocities and water levels for the case without Coriolis.]]
<br style="clear:both" />
'''Table 2. Goodness of fit statistics for the current velocity and water level.'''
{|border="1"
|'''Variable''' ||'''NRMSE, %'''||'''NMAE, %'''||'''R^2'''||'''Bias'''
|-
|U-Velocity || 2.52 || 0.37 || 0.999 || -8.5e-8 m/s
|-
|V-Velocity || 2.53 || 0.38 || 0.999 || 7.26e-8 m/s
|-
|Water Level|| 0.03 || 0.02 || 0.999 || -3.5e-7 m
|}
* For a definition of the goodness of fit statistics see [[Statistics |  Goodness of fit statistics]].


= Results =
<br style="clear:both" />


== With Coriolis ==
[[Image:CB3_Analytical_Coriolis_V4.png|thumb|left|600px| Figure 3.  Analytical current velocities and water  levels for the case with Coriolis.]]
[[Image:CB3_Calculated_Coriolis_V5.png|thumb|right|600px| Figure 4. Calculated current velocities and water levels for the case with Coriolis.]]


Table 2. Goodness of fit statistics for the water elevation
<br style="clear:both" />
'''Table 3. Goodness of fit statistics for the current velocity and water level.'''
{|border="1"
{|border="1"
|'''Statistic''' ||'''Value'''
|'''Variable''' ||'''RRMSE, %'''||'''RMAE, %'''||'''R^2'''||'''Bias'''
|-
|RMSE || 0.0074 m
|-
|-
|RMAE || 0.0068
|U-Velocity || 2.53 || 0.37 || 0.999 || -8.5e-8 m/s
|-
|-
|R^2 || 0.991
|V-Velocity || 2.56 || 0.37 || 0.999 || 6.5e-8 m/s
|-
|-
|Bias || 0.0017 m
|Water Level|| 0.03 || 0.02 || 0.999 || -3.0e-7 m
|}
|}
<br style="clear:both" />


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


----
----
[[Test_Cases  | Test Cases]]
[[Test_Cases  | Test Cases]]


[[CMS#Documentation_Portal  |  Documentation Portal]]
[[CMS#Documentation_Portal  |  Documentation Portal]]

Latest revision as of 20:07, 18 April 2013

UNDER CONSTRUCTION

Problem

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

  (1)
  (2)
  (3)

where and are the depth-averaged current velocities in the and directions respectively, is the gravitational constant, is the water surface elevation with respect to mean sea level, is a linear bottom friction coefficient, is the radius of the domain, is the water depth, and is a constant equal to the gradient of the wind forcing.

Solution

The analytical solution for water surface elevation solution is given by

  (4)

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

  (5)
  (6)

Setup

Figure 1. Computational grid.

The model is run to steady state from zero current and water level initial conditions with , , and . 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 100 m
Mixing Terms Off
Wall Friction Off
Linear Bottom Friction Coefficient 0.001



Results

No Coriolis

Figure 1. Analytical current velocities and water levels for the case without Coriolis.
Figure 2. Calculated current velocities and water levels for the case without Coriolis.


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

Variable NRMSE, % NMAE, % R^2 Bias
U-Velocity 2.52 0.37 0.999 -8.5e-8 m/s
V-Velocity 2.53 0.38 0.999 7.26e-8 m/s
Water Level 0.03 0.02 0.999 -3.5e-7 m


With Coriolis

Figure 3. Analytical current velocities and water levels for the case with Coriolis.
Figure 4. Calculated current velocities and water levels for the case with Coriolis.


Table 3. Goodness of fit statistics for the current velocity and water level.

Variable RRMSE, % RMAE, % R^2 Bias
U-Velocity 2.53 0.37 0.999 -8.5e-8 m/s
V-Velocity 2.56 0.37 0.999 6.5e-8 m/s
Water Level 0.03 0.02 0.999 -3.0e-7 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