http://fvcom.smast.umassd.edu/wiki/index.php?title=Special:Contributions/Gcowles&feed=atom&limit=50&target=Gcowles&year=&month=FVCOM Wiki - User contributions [en]2020-10-29T04:13:58ZFrom FVCOM WikiMediaWiki 1.16.5http://fvcom.smast.umassd.edu/wiki/index.php/AboutAbout2012-08-08T14:53:44Z<p>Gcowles: </p>
<hr />
<div>FVCOM is a prognostic, unstructured grid, finite-volume, free-surface, three-dimensional (3-D) primitive equations ocean model developed by Chen et al. (2003a). The original version of FVCOM consists of momentum, continuity, temperature, salinity and density equations and is closed physically and mathematically using the Mellor and Yamada level 2.5 turbulent closure scheme for vertical mixing and the Smagorinsky turbulent closure scheme for horizontal mixing. The irregular bottom topography is represented using a σ-coordinate transformation, and the horizontal grids are comprised of unstructured triangular cells. FVCOM solves the governing equations in integral form by computing fluxes between non-overlapping horizontal triangular control volumes. This finite-volume approach combines the best of finite-element methods (FEM) for geometric flexibility and finite-difference methods (FDM) for simple discrete structures and computational efficiency. This numerical approach also provides a much better representation of mass, momentum, salt, and heat conservation in coastal and estuarine regions with complex geometry. The conservative nature of FVCOM in addition to its flexible grid topology and code simplicity make FVCOM ideally suited for interdisciplinary application in the coastal ocean. <br />
<br />
The initial development of FVCOM was started by a team effort led by C. Chen in 1999 at the University of Georgia with support from the Georgia Sea Grant College Program. This first version was designed to simulate the 3-D current and transports within the estuary/tidal creek/inter-tidal salt marsh complex and was written in Fortran 77 in 2001. In 2001, C. Chen moved to the School of Marine Science and Technology at the University of Massachusetts-Dartmouth (SMAST/UMASS-D) and established the Marine Ecosystem Dynamics Modeling (MEDM) Laboratory where work on FVCOM has continued with funding from several sources under Chen’s leadership. The scientific team led by C. Chen and R. C. Beardsley built the original structure of FVCOM and conducted a series of model validation experiments. G. Cowles joined the MEDM group in 2003 and directly contributed to converting FVCOM to Fortran 90/95, modularized the coding structure and added the capability for parallel computation. The present version of FVCOM includes a nudging data assimilation module added by H. Liu, an improved 3D wet/dry point treatment module modified and tested by J. Qi, several choices for freshwater discharge and groundwater input and turbulence modules by C. Chen, H. Liu and G. Cowles, a tracer-tracking module by Q. Xu, and a 3-D Lagrangian particle tracking code (originally written by C. Chen and L. Zheng, modified by H. Liu to fit the FVCOM, and corrected by G. Cowles). A spherical coordinate version of FVCOM is also available for a basin or global scale application. The conversion from the local Cartesian coordinates to the spherical coordinates was first made based on the original Fortran 77 code of FVCOM by J. Zhu, a visiting scholar from the East China Normal University in Shanghai, P.R. China in 2002, and was subsequently migrated to Fortran 90/95 by G. Cowles in 2003. During 2004 and 2005 G. Cowles parallelized FVCOM using MPI with a Single Program Multiple Domain approach (Cowles, 2008). <br />
<br />
In the early stage of the FVCOM development, D. Chapman at the Wood Hole Oceanographic Institution (WHOI) gave many valuable suggestions and comments on the code structure and model validation. F. Dupont (while a postdoctoral investigator at the Bedford Institute of Oceanography (BIO) added an ice formation model into FVCOM. J. Pringle at the University of New Hampshire (UNH) was one of the first users and contributed to including the wind-induced water transport input at the upwind open boundary condition in the model application to the Gulf of Maine/Georges Bank region.<br />
Many people in the MEDM group have contributed to FVCOM validations and applications, including the Mount Hope Bay (Massachusetts) modeling by L. Zhao, the Okatee Estuary (South Carolina) by H. Huang, the Satilla River (Georgia) by J. Qi, the Ogeechee River (Georgia) by H. Lin and J. Qi, the South China Sea by Q. Xu and H. Lin, and dye experiments on Georges Bank by Q. Xu. <br />
Several types of companion finite-volume biological models have been developed and coupled into FVCOM by the MEDM ecosystem model team led by C. Chen. They include: a) a nutrient-phytoplankton-zooplankton (NPZ) model developed by Franks and Chen (1996; 2001), b) an 8-component phosphorus-limited, lower trophic level food web model (nutrients, two sizes of phytoplankton, two sizes of zooplankton, detritus and bacteria: NPZDB) developed by Chen et al. (2002), c) a state-of the art water quality model with inclusion of the benthic flux developed by Zheng and Chen (Zheng et al. 2004), and d) a 9-component coastal ocean NPZD model developed by Ji and Chen (Ji, 2003). A 3-D sediment model developed by Zheng and Chen (Zheng et al., 2003b) was also added into FVCOM.<br />
As the FVCOM development team leader, Dr. Changsheng Chen reserves all rights of this product. The University of Massachusetts-Dartmouth and the Georgia Sea Grant Program share the copyright of the software of this model. All copyrights are reserved. Unauthorized reproduction and distribution of this program are expressly prohibited. This program is only permitted for use in non-commercial academic research and education. The commercial use is subject to a fee charge. Modification is not encouraged for users who do not have a deep understanding of the code structures and finite-volume numerical methods used in FVCOM. Contributions made to correcting and modifying the program will be credited, but not affect copyrights. For public use, all users should name this model as "FVCOM". In any publications with the use of FVCOM, acknowledgement must be included.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:43:59Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right] \;\;\; Eq \; 7.1<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
Defining ''h'' as the height of dike or groyne, we divide a TCE into two element, calculate the flux individually, and then combine them. Applying (7.1) to each element, we have</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:43:49Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right] \;\;\; \textmf{Eq} \; 7.1<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
Defining ''h'' as the height of dike or groyne, we divide a TCE into two element, calculate the flux individually, and then combine them. Applying (7.1) to each element, we have</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:43:19Z<p>Gcowles: /* Introduction */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right] \;\;\; Eq \; 7.1<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
Defining ''h'' as the height of dike or groyne, we divide a TCE into two element, calculate the flux individually, and then combine them. Applying (7.1) to each element, we have</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:42:59Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right] \;\;\; Eq \; 7.1<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
Defining ''h'' as the height of dike or groyne, we divide a TCE into two element, calculate the flux individually, and then combine them. Applying (7.1) to each element, we have</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:42:27Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right] \;\;\; Eq 7.1<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
Defining ''h'' as the height of dike or groyne, we divide a TCE into two element, calculate the flux individually, and then combine them. Applying (7.1) to each element, we have</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:41:57Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
Defining ''h'' as the height of dike or groyne, we divide a TCE into two element, calculate the flux individually, and then combine them. Applying (7.1) to each element, we have</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:40:52Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). [[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]] In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:40:27Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
[[Image:groyne_stencil.png|thumb|100px|right|alt=Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs)]]</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/File:Groyne_stencil.pngFile:Groyne stencil.png2012-01-09T17:39:40Z<p>Gcowles: Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs).</p>
<hr />
<div>Sketch of the separation of the control element at dikes or groynes. The shaded regions indicate the tracer control elements (TCEs).</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:38:38Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.<br />
<br />
[[Image:GoM_8Way_Partition.jpg|thumb|100px|right|alt=8-Way Partition of the Gulf of Maine using METIS| 8-Way Partition of the Gulf of Maine using METIS]]</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:38:04Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). <math>\Omega</math> is the area of the TCE.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:35:37Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and ''v'' are the ''x'' and ''y'' components of the horizontal velocity, ''D'' is the total water depth defined as <math>H+ζ</math>, and ''H'' is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which ''u'' and ''v'' are placed at centroids and all scalars (i.e, <math>\zeta</math>, H, ) are placed at nodes. ''u'' and ''v'' at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). is the area of the TCE.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:34:17Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math><br />
<br />
where <math>\zeta</math> is the free-surface elevation, ''u'' and v are the x and y components of the horizontal velocity, D is the total water depth defined as H+ζ, and H is the mean water depth. In FVCOM, an unstructured triangle is comprised of three nodes, a centroid, and three sides, on which u and v are placed at centroids and all scalars (i.e, ζ, H, ) are placed at nodes. u and v at centroids are calculated based on the net flux through the three sides of that triangle (shaded regions in Fig.7.1, hereafter referred to as the Momentum Control Element: MCE), while scalar variables at each node are determined by the net flux through the sections linked to centroids and the middle point of the sideline in the surrounding triangles (shaded regions in Fig.7.1), hereafter referred to as the Tracer Control Element: TCE). is the area of the TCE.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:33:36Z<p>Gcowles: /* The Unstructured-Grid Dike-Groyne Algorithm */</p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of<br />
<br />
<math><br />
\frac{\partial \zeta}{\partial t} = - \frac{1}{\Omega}\left[ \oint \left(\bar{u}D\right) \mathrm{d}y - \oint \left( \bar{v}D\right) \mathrm{d}x \right]<br />
</math></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:31:05Z<p>Gcowles: </p>
<hr />
<div>== Introduction ==<br />
<br />
It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.<br />
<br />
<br />
== The Unstructured-Grid Dike-Groyne Algorithm ==<br />
<br />
Consider a submerged dike or groyne case. For this case, the water column connected to the structure is characterized by two layers: an upper layer in which the water can flow freely across the structure, and a lower layer in which flow is blocked (with no flux into the wall). In general, the width of a dike or groyne is on the order of 2-5 m. For a numerical simulation with a horizontal resolution of > 20-100 m, these dikes or groynes can be treated as lines without width. Under this assumption, we can construct the triangular grid along dikes and groynes, with a single control volume above the structure and two separate control volumes beneath it (Fig. 7.1). The mathematic equation A detailed description of mathematics is given in Ge et al. (2011), and a brief summary <br />
<br />
In the Cartesian coordinate system, the vertically integrated continuity equation can be written in the form of</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Dike_and_Groyne_ModelDike and Groyne Model2012-01-09T17:29:47Z<p>Gcowles: Created page with "It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the construct..."</p>
<hr />
<div>It is a challenge for a terrain-following coordinate ocean model to simulate the flow field in an estuarine or coastal system with dikes and groynes. In most cases, the constructions are usually submerged during high tide but out of the water during low tide. If a vertical wall is placed within the computational domain, then terrain-following coordinate transformation can fail. Adding a slope on the surface of a dike or groyne could make the topographic coordinate transformation work, but it changes the fluid dynamics. Instead of solid blocking (no flux towards the wall) in the lower column with the dike or groyne and free exchange in the upper column above the construction, the model makes the water flow along the submerged construction under the dynamical of the sloping bottom boundary layer. As a result, this slope treatment could overestimate vertical and lateral mixing and thus produce unrealistic circulation around the construction.<br />
<br />
Coastal inundation, which is defined as coastal flooding of normally dry land caused by heavy rains, high river discharge, tides, storm surge, tsunami processes, or some combination thereof, has been received intense attention in model applications to coastal and estuarine problems. In many coastal regions, dams are built around the area where the height of land is lower or close to the mean sea level to protect the land from flooding. An inundation forecast system is aimed at making warming of coastal flooding on an event timescale in order to facilitate evacuation and other emergency measures to protect human life and property in the coastal zone, and to accurate estimating the statistics of coastal inundation in order to enable rational planning regarding sustainable land-use practices in the coastal zone. A model used for this application must produce accurate, real-time forecasts of water level at high spatial resolution in the coastal zone and must have the capability to resolve the overtopping process of dams. These dams are like a solid wall boundary when the water level is lower than it, but become submerged constructions like dikes when flooding occurs. The wet/dry treatment technology is capable of resolving coastal flooding (Chen et al., 2006a,b), but cannot since handle a vertical seawall in the computational domain. <br />
<br />
We have developed an unstructured grid dike and groyne treatment algorithm in a terrain-following coordinate system to calculate the velocity and tracer concentration in the coastal or estuarine system with emerged or submerged dikes and groynes. This algorithm has been coded into FVCOM with MPI parallelization, and validated for idealized cases with dike-groyne construction where analytical solutions or laboratory experiment results are available. The FVCOM with inclusion of dike and groyne treatment module has been applied to simulate the flow field off the Changjiang Estuary where a dike-groyne structure was constructed in the Deep Waterway channel in the inner shelf of the East China Sea and also to simulate the coastal inundation in Scituate Harbor, Massachusetts. The validation and application results were written up for a paper and submitted to Ocean Modeling (Ge et al. 2011). A brief description of this module is given below.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/FVCOM_DocumentationFVCOM Documentation2012-01-09T17:28:34Z<p>Gcowles: </p>
<hr />
<div>==Background and Legal==<br />
# [[About]]<br />
# [[Introduction]]<br />
# [[User Agreement]]<br />
# [[Acknowledgements]]<br />
# [[List of FVCOM related publications]]<br />
<br />
==Technical Documentation==<br />
# [[Governing Equations]]<br />
# [[Discretization]]<br />
# [[Parallelization]]<br />
# [[Dike and Groyne Model]]<br />
<br />
==User Guides==<br />
# [[Installing FVCOM]]<br />
# [[Model Setup]]<br />
# [[Running FVCOM]]<br />
# [[Postprocessing and Visualization]]<br />
<br />
<br />
==Sample Model Setups==<br />
<br />
# [[Idealized Estuary]]<br />
<br />
== Getting started ==<br />
Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]<br />
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/Postprocessing_and_VisualizationPostprocessing and Visualization2011-11-17T17:23:18Z<p>Gcowles: /* Visit */</p>
<hr />
<div>==Visit==<br />
<br />
FVCOM output can be visualized using the free [https://wci.llnl.gov/codes/visit/ VisIt Visualization software] from LLNL using a plugin developed by D. Stuebe. Binaries for OS X and Windows and some distributions of Linux [https://wci.llnl.gov/codes/visit/executables.html are available]. You may also build visit yourself from source. VisIt is a very powerful visualization engine designed to handle very large datasets.<br />
<br />
==Datatank==<br />
<br />
[http://www.visualdatatools.com/ Datatank] is a commercial product for OS X. The author has developed a plugin to visualization directly from the FVCOM NetCDF output files. Using scripts, Datatank has practically all the capabilities as VisIt with higher quality graphics and movies. These include transects, contours, point probes, line probes, etc. A tarfile containing a sample script and simple NetCDF file can be downloaded here. [[Image:Datatank_screenshot.png|thumb|100px|right|alt=Screenshot of Datatank in Use| Screenshot of Datatank in Use]]</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:58:12Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<br />
Let us consider the continuity equation first. Integrating Eq. (2.30) over a given triangle area yields:<br />
<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} \mathrm{d}x\mathrm{d}y = - \iint \limits_{\Omega}\left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right] \mathrm{d}x\mathrm{d}y = - \oint \limits_{s'} \bar{v}_n D \mathrm{d}s'<br />
</math><br />
<br />
where <math>\bar{v}_n</math> is the velocity component normal to the sides of the triangle and ''s''' is the closed trajectory comprised of the three sides. This equation is integrated numerically using a modified fourth-order Runge-Kutta time-stepping scheme. This is a multi-stage time-stepping approach with second-order temporal accuracy. The detailed procedure for this method is described as follows:</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:57:05Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<br />
Let us consider the continuity equation first. Integrating Eq. (2.30) over a given triangle area yields:<br />
<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} \mathrm{d}x\mathrm{d}y = - \iint \limits_{\Omega}\left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right] \mathrm{d}x\mathrm{d}y = - \oint \limits_{s'} \bar{v}_n D \mathrm{d}s'<br />
</math><br />
<br />
where <math>\bar{v}_n</math> is the velocity component normal to the sides of the triangle and ''s''' is the closed trajectory comprised of the three sides. Eq. (3.3) is integrated numerically using the modified fourth-order Runge-Kutta time-stepping scheme. This is a multi-stage time-stepping approach with second-order temporal accuracy. The detailed procedure for this method is described as follows:</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:56:00Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} \mathrm{d}x\mathrm{d}y = - \iint \limits_{\Omega}\left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right] \mathrm{d}x\mathrm{d}y = - \oint \limits_{s'} \bar{v}_n D \mathrm{d}s'<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:54:46Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} \mathrm{d}x\mathrm{d}y = - \iint \limits_{\Omega}\left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right] \mathrm{d}x\mathrm{d}y = <br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:54:32Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} \mathrm{d}x\mathrm{d}y = - \iint \limits_{\Omega}\left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right]<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:54:10Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} = - \iint \limits_{\Omega}\left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right]<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:53:54Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \limits_{\Omega} \frac{\partial \zeta}{\partial t} = - \iint \left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right]<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:53:15Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint_{\Omega} \frac{\partial \zeta}{\partial t} = - \iint \left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right]<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:52:39Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \frac{\partial \zeta}{\partial t} = - \iint \left[\frac{\partial(\bar{u}D)}{\partial x} + \frac{\partial(\bar{v}D)}{\partial y} \right]<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:52:17Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \frac{\partial \zeta}{\partial t} = - \iint \left[\frac{\partial(\bar{u}D)}{\partial x} \right]<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:52:06Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \frac{\partial \zeta}{\partial t} = - \iint \left[\frac{\partial(\bar{u}D)}{\partial x}<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:51:27Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint \frac{\partial \zeta}{\partial t}<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:50:43Z<p>Gcowles: /* 2D External Mode */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<math><br />
\iint_{a}^b x dx<br />
</math><br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:42:03Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<br />
<br />
===2D External Mode===<br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy </math>--></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:40:43Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
<amsmath><br />
\label{e:barwq}\begin{split}<br />
H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}<br />
\sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\<br />
&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\\<br />
&\quad\cdot<br />
\Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].<br />
\end{split}<br />
</amsmath><br />
<br />
===2D External Mode===<br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy --><br />
<br />
</math></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:39:43Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
===2D External Mode===<br />
<br />
<!-- <math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy --><br />
<br />
</math></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:34:57Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
===2D External Mode===<br />
<br />
<math>\int\int\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy<br />
<br />
</math></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:29:00Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
===2D External Mode===<br />
<br />
<math>\iint\frac{\partial \zeta}{\partial t} = - \iint\left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy<br />
<br />
</math></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-14T14:28:25Z<p>Gcowles: </p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Computational Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.<br />
<br />
==Discretization in Cartesian Coordinates==<br />
<br />
===2D External Mode===<br />
<br />
<math>\int\int\frac{\partial \zeta}{\partial t} = - \int\int \left[\frac{\partial(\bar{u}D}\partial x} +\frac{\partial(\bar{v}D}\partial y} \right] dx dy<br />
<br />
</math></div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:32:00Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM horizontal stencil| FVCOM horizontal stencil]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:30:34Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
[[Image:FVCOM_horizontal_stencil.png|thumb|100px|right|alt=FVCOM vertical discretization|]]<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/File:FVCOM_horizontal_stencil.pngFile:FVCOM horizontal stencil.png2011-11-13T20:29:59Z<p>Gcowles: Horizontal Stencil for the FVCOM discretization showing overlapping control volumes for vertex and element based quantities.</p>
<hr />
<div>Horizontal Stencil for the FVCOM discretization showing overlapping control volumes for vertex and element based quantities.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:28:26Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
[[Image:FVCOM_vertical_discretization.png|thumb|100px|right|alt=FVCOM vertical discretization|FVCOM vertical discretization]]<br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:27:34Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
FVCOM_vertical_discretization.png<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/File:FVCOM_vertical_discretization.pngFile:FVCOM vertical discretization.png2011-11-13T20:26:46Z<p>Gcowles: Location of variables in the FVCOM vertical coordinate</p>
<hr />
<div>Location of variables in the FVCOM vertical coordinate</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:24:38Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer. There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:24:23Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''. <br />
<br />
To provide a more accurate estimation of the sea-surface elevation, currents and salt and temperature fluxes, ''u'' and ''v'' are placed at centroids and all scalar variables, such as ζ, H, , ω, S, T, ρ, are placed at nodes. Scalar variables at each node are determined by a net flux through the sections linked to centroids and the mid-point of the adjacent sides in the surrounding triangles (called the “tracer control element” or TCE), while u and v at the centroids are calculated based on a net flux through the three sides of that triangle (called the “momentum control element” or MCE). <br />
Similar to other finite-difference models such as POM and ROMS, all the model variables except ω (vertical velocity on the sigma-layer surface) and turbulence variables (such as and ) are placed at the mid-level of each σ layer (Fig. 3.2). There are no restrictions on the thickness of the σ-layer, which allows users to use either uniform or non-uniform σ-layers.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:23:26Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i</sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:23:12Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i<\sub>(m)'' where ''m'' is counted clockwise from 1 to ''NT(j)''.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:22:50Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers defined as <math>NBE_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. At open or coastal solid boundaries, <math>NBE_i(\hat{j})</math> is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as ''NT(j)'', and they are counted using integral numbers ''NB<sub>i<\sub>(m) where ''m'' is counted clockwise from 1 to ''NT(j)''.</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:18:57Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math> where <math>\hat{j}</math> is counted clockwise from 1 to 3. The surounding triangles that have a common side are counted using integral numbers defined as where is counted clockwise from 1 to 3. At open or coastal solid boundaries, is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as , and they are counted using integral numbers where m is counted clockwise from 1 to .</div>Gcowleshttp://fvcom.smast.umassd.edu/wiki/index.php/DiscretizationDiscretization2011-11-13T20:18:28Z<p>Gcowles: /* Discretization Stencil */</p>
<hr />
<div>The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly. <br />
<br />
==Discretization Stencil==<br />
<br />
Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:<br />
<br />
<math>[X(i),Y(i)], \; \; i=1:N</math><br />
<br />
and the location of the nodes can be specified as:<br />
<br />
<math>[Xn(j),Yn(j)], \;\; j=1:M</math><br />
<br />
Since none of the triangles in the grid overlap, ''N'' should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as <math>N_i(\hat{j})</math>where \hat{j} is counted clockwise from 1 to 3. The surounding triangles that have a common side are counted using integral numbers defined as where is counted clockwise from 1 to 3. At open or coastal solid boundaries, is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as , and they are counted using integral numbers where m is counted clockwise from 1 to .</div>Gcowles