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A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey,...
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![Page 1: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/1.jpg)
A Finite Volume Coastal Ocean Model
Peter C Chu and Chenwu Fan
Naval Postgraduate School
Monterey, California, USA
![Page 2: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/2.jpg)
Four Types of Numerical Models
• Spectral Model (not suitable for oceans due to irregular lateral boundaries)
• Finite Difference (z-coordinate, sigma-coordinate, …)
• Finite Element
• Finite Volume
![Page 3: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/3.jpg)
Finite Volume
![Page 4: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/4.jpg)
Finite Volume Model
• Transform of PDE to Integral Equations
• Solving the Integral Equation for the Finite Volume
• Flux Conservation
![Page 5: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/5.jpg)
Dynamic and Thermodynamic Equations
• Continuity
• Momentum
• Thermodynamic
![Page 6: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/6.jpg)
Integral Equations for Finite Volume
• Continuity
• Momentum
• Thermodynamic
![Page 7: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/7.jpg)
Time Integration of Phi-Equation
![Page 8: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/8.jpg)
Comparison Between Finite Difference (z- and sigma-coordinates) and Finite
Volume Schemes
![Page 9: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/9.jpg)
Seamount Test Case
![Page 10: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/10.jpg)
Initial Conditions
• V = 0
• S = 35 ppt
• HT = 1000 m
![Page 11: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/11.jpg)
Known Solution
• V = 0
• Horizontal Pressure Gradient = 0
![Page 12: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/12.jpg)
Evaluation
• Princeton Ocean Model
• Seamount Test Case
• Horizontal Pressure Gradient (Finite Difference and Finite Volume)
![Page 13: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/13.jpg)
Numerics and Parameterization
• Barotropic Time Step: 6 s
• Baroclinic Time Step: 180 s
• Delta x = Delta y = 8 km
• Vertical Eddy Viscosity: Mellor-Yamada Scheme
• Horizontal Diffusion: Samagrinsky Scheme with the coefficient of 0.2
![Page 14: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/14.jpg)
Error Volume Transport Streamfunction
![Page 15: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/15.jpg)
Temporally Varying Horizontal Gradient Error
• The error reduction by a factor of 14 using the finite volume scheme.
![Page 16: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/16.jpg)
Temporally Varying Error Velocity
• The error velocity reduction by a factor of 4 using the finite volume scheme.
![Page 17: A Finite Volume Coastal Ocean Model Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California, USA.](https://reader036.fdocuments.net/reader036/viewer/2022062320/56649d965503460f94a7f30c/html5/thumbnails/17.jpg)
Conclusions
• Use of the finite volume model has the following benefit:
(1) Computation is as simple as the finite difference scheme.
(2) Conservation on any finite volume.
(3) Easy to incorporate high-order schemes
(4) Upwind scheme