# UGRID Conventions (v1.0)

## Introduction

This page describes a proposal for storing unstructured (or flexible mesh) model data in the Unidata Network Common Data Form (NetCDF) file. Our focus is on data for environmental applications and hence we start from the Climate & Forecast (CF) Metadata Conventions. The CF Conventions have been the standard in climate research for many years, and are being adopted by others as the metadata standard (e.g. NASA,Open Geospatial Consortium). The CF conventions allow you to provide the geospatial and temporal coordinates for scientific data, but currently assumes that the horizontal topology may be inferred from the i,j indices of structured grids. This proposal adds conventions for specifying the topology for unstructured (e.g. triangular) grids.

In its most basic form unstructured data may be stored as data defined at a series of points, the CF-conventions are then sufficient. However, it is often useful or even necessary to also know the topology of the underlying unstructured mesh: is it a one dimensional (1D) network, a two dimensional (2D) triangular mesh or more flexible mixed triangle/quadrilateral mesh, a 2D mesh with vertical layers, or a fully unstructured three dimensional (3D) mesh. This document describes the attribute conventions for storing the mesh topology and for associating variables with (specific locations on) the mesh topology. The conventions have been designed to store the output data of a combined 1D-2D-3D flow model with staggered data, but the metadata for a simple 1D network or 2D triangular mesh doesn’t suffer from the genericity needed for the most complex models.

Due to the complexity in unstructured mesh models, some concepts have not yet been worked out in this version.

Known issues left for future versions include: adaptive mesh topology (this could be supported by defining a time_concatenation attribute for a time-series of mesh topologies) higher order element data; for an idea how such data could be stored see this other proposal. subgrid data; the NetCDF pages by the Bundesanstalt für Wasserbau (BAW) contain some proposals on this topic (see their pages (in German)). 3D fully unstructured meshes (some concepts are included here, but still somewhat limited in scope). multiply-connected domains ghost elements

## Topology

### Naming conventions for geometrical elements

Inspired by Wikipedia’s definition of network topology, we define the mesh topology here as the interconnection of various geometrical elements of the mesh. The pure interconnectivity is independent of georeferencing the individual geometrical elements, but for the practical applications for which we are defining this CF extension, we’ll always add coordinate data. Within a mesh, one can distinguish 0-, 1-, 2- and 3-dimensional elements. We need some names to identify these four types of elements; after discussion we propose the following names:

0 node A point, a coordinate pair or triplet: the most basic element of the topology. The word “node” seems to be more commonly used than the alternative “vertex”.
1 edge A line or curve bounded by two nodes.
2 face A plane or surface enclosed by a set of edges. In a 2D horizontal application one may consider the word “polygon”, but in the hierarchy of elements the word “face” is most common.
3 volume A volume enclosed by a set of faces. The alternative word “cell” was considered, but “cell” is often used in the community to describe 2 dimensional structures.

In favor of simpler code for interpreting compliant files, we have dropped to use of the locations attribute which allowed the user to specify his/her own names for nodes, edges, faces and volumes.

### 1D network topology

The topology information is stored as attributes to a dummy variable (in the example below called “Mesh1”) with cf_role mesh_topology.

Required topology attributes Value
cf_role mesh_topology
topology_dimension 1
node_coordinates
edge_node_connectivity
Optional attributes
edge_coordinates

The attribute topology_dimension indicates the highest dimensionality of the geometric elements; for a 1D network this should be 1. The attribute node_coordinates points to the auxiliary coordinate variables representing the node locations (latitude, longitude, and optional elevation or other coordinates). These auxiliary coordinate variables will have length nNodes. The attribute edge_node_connectivity points to an index variable identifying for every edge to the indices of its begin and end nodes. The connectivity array will thus be a matrix of size nEdges x 2. For the indexing one may use either 0- or 1-based indexing; the convention used should be specified using a start_index attribute to the index variable (i.e. Mesh1_edge_nodes in the example below). Consistent with the CF-conventions compression option, the connectivity indices are 0-based by default.

The option to support both 0- and 1-based indexing was introduced to be able to support existing files with 1-based index tables using ncML. See this section on zero or one-based indexing for more details.

The mesh_topology may optionally include an edge_coordinates attribute which points to the auxiliary coordinate variables associated with the characteristic location of the edge (commonly the midpoint). These auxiliary coordinate variables will have length nEdges, and may have in turn a bounds attribute that specifies the bounding coordinates of the edge (thereby duplicating the data in the node_coordinates variables).

This use of the bounds attribute is consistent with the CF-convention on the use of bounds for multi-dimensional coordinate variables with p-sided cells, but it may not strictly be supported by the CF-convention right now.

Example:

dimensions:
nMesh1_node = 5 ; // nNodes
nMesh1_edge = 4 ; // nEdges

Two = 2;

variables:
// Mesh topology
integer Mesh1 ;
Mesh1:cf_role = "mesh_topology" ;
Mesh1:long_name = "Topology data of 1D network" ;
Mesh1:topology_dimension = 1 ;
Mesh1:node_coordinates = "Mesh1_node_x Mesh1_node_y" ;
Mesh1:edge_node_connectivity = "Mesh1_edge_nodes" ;
Mesh1:edge_coordinates = "Mesh1_edge_x Mesh1_edge_y" ; // optional attribute
integer Mesh1_edge_nodes(nMesh1_edge, Two) ;
Mesh1_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh1_edge_nodes:long_name = "Maps every edge/link to the two nodes that it connects." ;
Mesh1_edge_nodes:start_index = 1 ;

// Mesh node coordinates
double Mesh1_node_x(nMesh1_node) ;
Mesh1_node_x:standard_name = "longitude" ;
Mesh1_node_x:long_name = "Longitude of 1D network nodes." ;
Mesh1_node_x:units = "degrees_east" ;
double Mesh1_node_y(nMesh1_node) ;
Mesh1_node_y:standard_name = "latitude" ;
Mesh1_node_y:long_name = "Latitude of 1D network nodes." ;
Mesh1_node_y:units = "degrees_north" ;

// Optional mesh edge coordinate variables
double Mesh1_edge_x(nMesh1_edge) ;
Mesh1_edge_x:standard_name = "longitude" ;
Mesh1_edge_x:long_name = "Characteristic longitude of 1D network edge (e.g. midpoint of the edge)." ;
Mesh1_edge_x:units = "degrees_east" ;
Mesh1_edge_x:bounds = "Mesh1_edge_xbnds" ;
double Mesh1_edge_y(nMesh1_edge) ;
Mesh1_edge_y:standard_name = "latitude" ;
Mesh1_edge_y:long_name = "Characteristic latitude of 1D network edge (e.g. midpoint of the edge)." ;
Mesh1_edge_y:units = "degrees_north" ;
Mesh1_edge_y:bounds = "Mesh1_edge_ybnds" ;
double Mesh1_edge_xbnds(nMesh1_edge,Two) ;
Mesh1_edge_xbnds:standard_name = "longitude" ;
Mesh1_edge_xbnds:long_name = "Longitude bounds of 1D network edge (i.e. begin and end longitude)." ;
Mesh1_edge_xbnds:units = "degrees_east" ;
double Mesh1_edge_ybnds(nMesh1_edge,Two) ;
Mesh1_edge_ybnds:standard_name = "latitude" ;
Mesh1_edge_ybnds:long_name = "Latitude bounds of 1D network edge (i.e. begin and end latitude)." ;
Mesh1_edge_ybnds:units = "degrees_north" ;


### 2D triangular mesh topology

The topology information is stored as attributes to a dummy variable (in the example below called “Mesh2”) with cf_role mesh_topology.

Required topology attributes Value
cf_role mesh_topology
topology_dimension 2
node_coordinates
face_node_connectivity
Optionally required attributes*
face_dimension
edge_node_connectivity
edge_dimension
Optional attributes
face_edge_connectivity
face_face_connectivity
edge_face_connectivity
boundary_node_connectivity
face_coordinates
edge_coordinates

*The “Optionally required” attribute edge_node_connectivity is required only if you want to store data on the edges (i.e. if you mind the numbering order of the edges).

*The “Optionally required” attributes face_dimension and edge_dimension are required only if the dimension ordering is nonstandard in any of the connectivity variables for faces and edges, respectively.

The attribute topology_dimension indicates the highest dimensionality of the geometric elements; for a 2-dimensional (triangular) mesh this should be 2. The attribute node_coordinates points to the auxiliary coordinate variables representing the node locations (latitude, longitude, and optional elevation or other coordinates). These auxiliary coordinate variables will have length nNodes. The attribute face_node_connectivity points to an index variable identifying for every face (here consistently triangle) the indices of its three corner nodes. The corner nodes should be specified in anticlockwise (also referred to as counterclockwise) direction as viewed from above (consistent with the CF-convention for bounds of p-sided cells. The connectivity array will thus be a matrix of size nFaces x 3. For the indexing one may use either 0- or 1-based indexing; the convention used should be specified using a start_index attribute to the index variable (i.e. Mesh2_face_nodes in the example below). Consistent with the CF-conventions compression option, the connectivity indices are 0-based by default. See this section on zero or one-based indexing for more details.

The face_dimension attribute specifies which netcdf dimension is used to indicate the index of the face in the connectivity arrays. This is needed because some applications store the data with the fastest varying index first, and some with that index last. The default is to use the num_faces as fastest dimension; e.g. a (num_faces, 3) array for triangles, but some applications might use a (3, num_faces) order, in which case the face_dimension attribute is required to help the client code disambiguate. The edge_dimension attribute is similar for the edge connectivity arrays.

In case you want to define variables on the edges of the triangular mesh topology you need to specify the edge_node_connectivity attribute to map edges to nodes. Although the face to node mapping implicitly also defines the location of the edges, it does not specify the global numbering of the edges. Again the indexing convention of edge_node_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_edge_nodes in the example below) and 0-based indexing is the default. Since it does not apply to edges globally, specifying the boundary_node_connectivity attribute described below does not (in and of itself) necessitate the need to specify the edge_node_connectivity attribute too.

Optionally the topology may have the following attributes: face_edge_connectivity pointing to an index variable identifying for every face (here consistently triangle) the indices of its three edges. The edges should be specified in anticlockwise direction as viewed from above. This connectivity array will thus be a matrix of size nFaces x 3. Again the indexing convention of face_edge_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_face_edges in the example below) and 0-based indexing is the default. face_face_connectivity pointing to an index variable identifying all faces (here consistently triangle) that share an edge with each face, i.e. are neighbors. This connectivity array will thus be a matrix of size nFaces x 3 with a flag to note if the region “out of mesh” abuts an edge of a given face. Again the indexing convention of face_face_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_face_links in the example below) and 0-based indexing is the default. edge_face_connectivity pointing to an index variable identifying all faces that share the same edge, i. e. are neighbors to an edge. This connectivity array is thus a matrix of size nEdges x 2. It is intended to be used in combination with data defined on edges. The start_index attribute should be used to specify the indexing convention and 0-based indexing is the default. Attribute _FillValue must be present. Missing neighbor faces are expressed using _FillValue, e.g for edges at the boundary with only one neighbor face present. For details see definition of variable Mesh2_edge_face_links below. boundary_node_connectivity pointing to an nBoundaryEdges X 2 index variable identifying for every edge of each boundary the two nodes that it connects. Again the indexing convention of boundary_node_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_boundary_nodes) and 0-based indexing is the default. Although constructed of edges, boundaries represent a different quantity than general edge data and thus the boundary_node_connectivity attribute may be specified independent of edge_node_connectivity. Information about the nature of each boundary edge (e.g. open/closed, land/water, grouping, etc.) may optionally be stored in ancillary boundary-type variables of size nBoundaryEdges X 1. * face_coordinates and/or edge_coordinates pointing to the auxiliary coordinate variables associated with the characteristic location of the faces and edges. These auxiliary coordinate variables will have length nFaces and nEdges respectively, and may have in turn a bounds attribute that specifies the bounding coordinates of the face or edge (thereby duplicating the data in the node_coordinates variables).

Example:

dimensions:
nMesh2_node = 4 ; // nNodes
nMesh2_edge = 5 ; // nEdges
nMesh2_face = 2 ; // nFaces

Two = 2 ;
Three = 3 ;

variables:
// Mesh topology
integer Mesh2 ;
Mesh2:cf_role = "mesh_topology" ;
Mesh2:long_name = "Topology data of 2D unstructured mesh" ;
Mesh2:topology_dimension = 2 ;
Mesh2:node_coordinates = "Mesh2_node_x Mesh2_node_y" ;
Mesh2:face_node_connectivity = "Mesh2_face_nodes" ;
Mesh2:face_dimension = "nMesh2_face" ;
Mesh2:edge_node_connectivity = "Mesh2_edge_nodes" ; // attribute required if variables will be defined on edges
Mesh2:edge_dimension = "nMesh2_edge" ;
Mesh2:edge_coordinates = "Mesh2_edge_x Mesh2_edge_y" ; // optional attribute (requires edge_node_connectivity)
Mesh2:face_coordinates = "Mesh2_face_x Mesh2_face_y" ; // optional attribute
Mesh2:face_edge_connectivity = "Mesh2_face_edges" ; // optional attribute (requires edge_node_connectivity)
Mesh2:face_face_connectivity = "Mesh2_face_links" ; // optional attribute
Mesh2:edge_face_connectivity = "Mesh2_edge_face_links" ; // optional attribute (requires edge_node_connectivity)
integer Mesh2_face_nodes(nMesh2_face, Three) ;
Mesh2_face_nodes:cf_role = "face_node_connectivity" ;
Mesh2_face_nodes:long_name = "Maps every triangular face to its three corner nodes." ;
Mesh2_face_nodes:start_index = 1 ;
integer Mesh2_edge_nodes(nMesh2_edge, Two) ;
Mesh2_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh2_edge_nodes:long_name = "Maps every edge to the two nodes that it connects." ;
Mesh2_edge_nodes:start_index = 1 ;

// Optional mesh topology variables
integer Mesh2_face_edges(nMesh2_face, Three) ;
Mesh2_face_edges:cf_role = "face_edge_connectivity" ;
Mesh2_face_edges:long_name = "Maps every triangular face to its three edges." ;
Mesh2_face_edges:start_index = 1 ;
Mesh2_face_links:long_name = "Indicates which other faces neighbor each face." ;
Mesh2_edge_face_links:long_name = "neighbor faces for edges" ;
Mesh2_edge_face_links:comment = "missing neighbor faces are indicated using _FillValue" ;

// Mesh node coordinates
double Mesh2_node_x(nMesh2_node) ;
Mesh2_node_x:standard_name = "longitude" ;
Mesh2_node_x:long_name = "Longitude of 2D mesh nodes." ;
Mesh2_node_x:units = "degrees_east" ;
double Mesh2_node_y(nMesh2_node) ;
Mesh2_node_y:standard_name = "latitude" ;
Mesh2_node_y:long_name = "Latitude of 2D mesh nodes." ;
Mesh2_node_y:units = "degrees_north" ;

// Optional mesh face and edge coordinate variables
double Mesh2_face_x(nMesh2_face) ;
Mesh2_face_x:standard_name = "longitude" ;
Mesh2_face_x:long_name = "Characteristics longitude of 2D mesh triangle (e.g. circumcenter coordinate)." ;
Mesh2_face_x:units = "degrees_east" ;
double Mesh2_face_y(nMesh2_face) ;
Mesh2_face_y:standard_name = "latitude" ;
Mesh2_face_y:long_name = "Characteristics latitude of 2D mesh triangle (e.g. circumcenter coordinate)." ;
Mesh2_face_y:units = "degrees_north" ;
double Mesh2_edge_x(nMesh2_edge) ;
Mesh2_edge_x:standard_name = "longitude" ;
Mesh2_edge_x:long_name = "Characteristic longitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh2_edge_x:units = "degrees_east" ;
double Mesh2_edge_y(nMesh2_edge) ;
Mesh2_edge_y:standard_name = "latitude" ;
Mesh2_edge_y:long_name = "Characteristic latitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh2_edge_y:units = "degrees_north" ;


### 2D flexible mesh (mixed triangles, quadrilaterals, etc.) topology

The case of a 2D mesh with mixed face sizes is identical to the 2D triangular mesh discussed above with the exception that not all faces have the same number of nodes. To support this variability we may use in the future a ragged array, but here we propose to use _FillValue to indicate faces with smaller number of nodes than the arrays allow.

The topology information is stored as attributes to a dummy variable (in the example below called “Mesh2”) with cf_role mesh_topology.

Required topology attributes Value
cf_role mesh_topology
topology_dimension 2
node_coordinates
face_node_connectivity
Optionally required attributes*
face_dimension
edge_node_connectivity
edge_dimension
Optional attributes
face_edge_connectivity
face_face_connectivity
edge_face_connectivity
boundary_node_connectivity
face_coordinates
edge_coordinates

*The “Optionally required” attribute edge_node_connectivity is required only if you want to store data on the edges (i.e. if you mind the numbering order of the edges).

*The “Optionally required” attributes face_dimension and edge_dimension are required only if the dimension ordering is nonstandard in any of the connectivity variables for faces and edges, respectively.

The attribute topology_dimension indicates the highest dimensionality of the geometric elements; for a 2-dimensional mesh this should be 2. The attribute node_coordinates points to the auxiliary coordinate variables representing the node locations (latitude, longitude, and optional elevation or other coordinates). These auxiliary coordinate variables will have length nNodes. The attribute face_node_connectivity points to an index variable identifying for every face the indices of its corner nodes. The corner nodes should be specified in anticlockwise direction as viewed from above (consistent with the CF-convention for bounds of p-sided cells. The connectivity array will be a matrix of size nFaces x MaxNumNodesPerFace; if a face has less corner nodes than MaxNumNodesPerFace then the last node indices shall be equal to _FillValue (which should obviously be larger than the number of nodes in the mesh). For the indexing one may use either 0- or 1-based indexing; the convention used should be specified using a start_index attribute to the index variable (i.e. Mesh2_face_nodes in the example below). Consistent with the CF-conventions compression option, the connectivity indices are 0-based by default. See this section on zero or one-based indexing for more details.

The face_dimension attribute specifies which netcdf dimension is used to indicate the index of the face in the connectivity arrays. This is needed because some applications store the data with the fastest varying index first, and some with that index last. The default is to use the num_faces as fastest dimension; e.g. a (num_faces, 4) array for quads, but some applications might use a (4, num_faces) order, in which case the face_dimension attribute is required to help the client code disambiguate. The edge_dimension attribute is similar for the edge connectivity arrays.

In case you want to define variables on the edges of the 2D mesh topology you need to specify the edge_node_connectivity attribute to map edges to nodes. Although the face to node mapping implicitly also defines the location of the edges, it does not specify the global numbering of the edges. Again the indexing convention of edge_node_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_edge_nodes in the example below) and 0-based indexing is the default. Since it does not apply to edges globally, specifying the boundary_node_connectivity attribute described below does not (in and of itself) necessitate the need to specify the edge_node_connectivity attribute too.

Optionally the topology may have the following attributes: face_edge_connectivity pointing to an index variable identifying for every face the indices of its edges. The edges should be specified in anticlockwise direction as viewed from above. This connectivity array will be a matrix of size nFaces x MaxNumNodesPerFace. Again, if a face has less corners/edges than MaxNumNodesPerFace then the last edge indices shall be equal to _FillValue, and the indexing convention of face_edge_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_face_edges in the example below) and 0-based indexing is the default. face_face_connectivity pointing to an index variable identifying all faces that share an edge with each face, i.e. are neighbors. This connectivity array will thus be a matrix of size nFaces x MaxNumNodesPerFace with a flag to note if the region “out of mesh” abuts an edge of a given face. Again, if a face has less corners/edges than MaxNumNodesPerFace then the last face (column) indices shall be equal to _FillValue, and the indexing convention of face_face_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_face_links in the example below) and 0-based indexing is the default. edge_face_connectivity pointing to an index variable identifying all faces that share the same edge, i. e. are neighbors to an edge. This connectivity array is thus a matrix of size nEdges x 2. It is intended to be used in combination with data defined on edges. The start_index attribute should be used to specify the indexing convention and 0-based indexing is the default. Attribute _FillValue must be present. Missing neighbor faces are expressed using _FillValue, e.g for edges at the boundary with only one neighbor face present. For details see definition of variable Mesh2_edge_face_links below. boundary_node_connectivity pointing to an nBoundaryEdges X 2 index variable identifying for every edge of each boundary the two nodes that it connects. Again the indexing convention of boundary_node_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh2_boundary_nodes) and 0-based indexing is the default. Although constructed of edges, boundaries represent a different quantity than general edge data and thus the boundary_node_connectivity attribute may be specified independent of edge_node_connectivity. Information about the nature of each boundary edge (e.g. open/closed, land/water, grouping, etc.) may optionally be stored in ancillary boundary-type variables of size nBoundaryEdges X 1. * face_coordinates and/or edge_coordinates pointing to the auxiliary coordinate variables associated with the characteristic location of the faces and edges. These auxiliary coordinate variables will have length nFaces and nEdges respectively, and may have in turn a bounds attribute that specifies the bounding coordinates of the face or edge (thereby duplicating the data in the node_coordinates variables).

The use of _FillValue to indicate faces with less nodes than MaxNumNodesPerFace extends to the coordinate bounds variables; this is an extension of the current convention.

Example:

dimensions:
nMesh2_node = 5 ; // nNodes
nMesh2_edge = 6 ; // nEdges
nMesh2_face = 2 ; // nFaces
nMaxMesh2_face_nodes = 4 ; // MaxNumNodesPerFace

Two = 2 ;

variables:
// Mesh topology
integer Mesh2 ;
Mesh2:cf_role = "mesh_topology" ;
Mesh2:long_name = "Topology data of 2D unstructured mesh" ;
Mesh2:topology_dimension = 2 ;
Mesh2:node_coordinates = "Mesh2_node_x Mesh2_node_y" ;
Mesh2:face_node_connectivity = "Mesh2_face_nodes" ;
Mesh2:face_dimension = "nMesh2_face" ;
Mesh2:edge_node_connectivity = "Mesh2_edge_nodes" ; // attribute required if variables will be defined on edges
Mesh2:edge_dimension = "nMesh2_edge" ;
Mesh2:edge_coordinates = "Mesh2_edge_x Mesh2_edge_y" ; // optional attribute (requires edge_node_connectivity)
Mesh2:face_coordinates = "Mesh2_face_x Mesh2_face_y" ; // optional attribute
Mesh2:face_edge_connectivity = "Mesh2_face_edges" ; // optional attribute (requires edge_node_connectivity)
Mesh2:face_face_connectivity = "Mesh2_face_links" ; // optional attribute
Mesh2:edge_face_connectivity = "Mesh2_edge_face_links" ; // optional attribute (requires edge_node_connectivity)
integer Mesh2_face_nodes(nMesh2_face, nMaxMesh2_face_nodes) ;
Mesh2_face_nodes:cf_role = "face_node_connectivity" ;
Mesh2_face_nodes:long_name = "Maps every face to its corner nodes." ;
Mesh2_face_nodes:_FillValue = 999999 ;
Mesh2_face_nodes:start_index = 1 ;
integer Mesh2_edge_nodes(nMesh2_edge, Two) ;
Mesh2_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh2_edge_nodes:long_name = "Maps every edge to the two nodes that it connects." ;
Mesh2_edge_nodes:start_index = 1 ;

// Optional mesh topology variables
integer Mesh2_face_edges(nMesh2_face, nMaxMesh2_face_nodes) ;
Mesh2_face_edges:cf_role = "face_edge_connectivity" ;
Mesh2_face_edges:long_name = "Maps every face to its edges." ;
Mesh2_face_edges:_FillValue = 999999 ;
Mesh2_face_edges:start_index = 1 ;
Mesh2_face_links:long_name = "Indicates which other faces neighbor each face." ;
Mesh2_edge_face_links:long_name = "neighbor faces for edges" ;
Mesh2_edge_face_links:comment = "missing neighbor faces are indicated using _FillValue" ;

// Mesh node coordinates
double Mesh2_node_x(nMesh2_node) ;
Mesh2_node_x:standard_name = "longitude" ;
Mesh2_node_x:long_name = "Longitude of 2D mesh nodes." ;
Mesh2_node_x:units = "degrees_east" ;
double Mesh2_node_y(nMesh2_node) ;
Mesh2_node_y:standard_name = "latitude" ;
Mesh2_node_y:long_name = "Latitude of 2D mesh nodes." ;
Mesh2_node_y:units = "degrees_north" ;

// Optional mesh face and edge coordinate variables
double Mesh2_face_x(nMesh2_face) ;
Mesh2_face_x:standard_name = "longitude" ;
Mesh2_face_x:long_name = "Characteristics longitude of 2D mesh face." ;
Mesh2_face_x:units = "degrees_east" ;
Mesh2_face_x:bounds = "Mesh2_face_xbnds" ;
double Mesh2_face_y(nMesh2_face) ;
Mesh2_face_y:standard_name = "latitude" ;
Mesh2_face_y:long_name = "Characteristics latitude of 2D mesh face." ;
Mesh2_face_y:units = "degrees_north" ;
Mesh2_face_y:bounds = "Mesh2_face_ybnds" ;
double Mesh2_face_xbnds(nMesh2_face,nMaxMesh2_face_nodes) ;
Mesh2_face_xbnds:standard_name = "longitude" ;
Mesh2_face_xbnds:long_name = "Longitude bounds of 2D mesh face (i.e. corner coordinates)." ;
Mesh2_face_xbnds:units = "degrees_east" ;
Mesh2_face_xbnds:_FillValue = 9.9692099683868690E36;
double Mesh2_face_ybnds(nMesh2_face,nMaxMesh2_face_nodes) ;
Mesh2_face_ybnds:standard_name = "latitude" ;
Mesh2_face_ybnds:long_name = "Latitude bounds of 2D mesh face (i.e. corner coordinates)." ;
Mesh2_face_ybnds:units = "degrees_north" ;
Mesh2_face_ybnds:_FillValue = 9.9692099683868690E36;
double Mesh2_edge_x(nMesh2_edge) ;
Mesh2_edge_x:standard_name = "longitude" ;
Mesh2_edge_x:long_name = "Characteristic longitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh2_edge_x:units = "degrees_east" ;
double Mesh2_edge_y(nMesh2_edge) ;
Mesh2_edge_y:standard_name = "latitude" ;
Mesh2_edge_y:long_name = "Characteristic latitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh2_edge_y:units = "degrees_north" ;
// bounds variables for edges skipped


### 3D layered mesh topology

For a 3D layered unstructured mesh topology this proposal follows the approach of the existing CF-conventions for structured meshes: horizontal and vertical dimensions are treated separately. For the horizontal plane a 2D unstructured mesh topology is defined, which is extruded in the vertical direction by means of a vertical coordinate. The example below matches the example in the previous section combined with a vertical coordinate according CF-conventions. This example introduces also the attributes mesh and location on the 2D variables “Mesh2_surface” and “Mesh2_depth”. For more information about these attributes see the data definition section below.

Example:

dimensions:
nMesh2_node = 6 ; // nNodes
nMesh2_edge = 7 ; // nEdges
nMesh2_face = 2 ; // nFaces
nMaxMesh2_face_nodes = 4 ; // MaxNumNodesPerFace
Mesh2_layers = 10 ;

Two = 2 ;

variables:
// Mesh topology
integer Mesh2 ;
Mesh2:cf_role = "mesh_topology" ;
Mesh2:long_name = "Topology data of 2D unstructured mesh" ;
Mesh2:topology_dimension = 2 ;
Mesh2:node_coordinates = "Mesh2_node_x Mesh2_node_y" ;
Mesh2:face_node_connectivity = "Mesh2_face_nodes" ;
Mesh2:face_dimension = "nMesh2_face" ;
Mesh2:edge_node_connectivity = "Mesh2_edge_nodes" ; // attribute required if variables will be defined on edges
Mesh2:edge_dimension = "nMesh2_edge" ;
Mesh2:edge_coordinates = "Mesh2_edge_x Mesh2_edge_y" ; // optional attribute (requires edge_node_connectivity)
Mesh2:face_coordinates = "Mesh2_face_x Mesh2_face_y" ; // optional attribute
Mesh2:face_edge_connectivity = "Mesh2_face_edges" ; // optional attribute (requires edge_node_connectivity)
Mesh2:face_face_connectivity = "Mesh2_face_links" ; // optional attribute
Mesh2:edge_face_connectivity = "Mesh2_edge_face_links" ; // optional attribute (requires edge_node_connectivity)
integer Mesh2_face_nodes(nMesh2_face, nMaxMesh2_face_nodes) ;
Mesh2_face_nodes:cf_role = "face_node_connectivity" ;
Mesh2_face_nodes:long_name = "Maps every face to its corner nodes." ;
Mesh2_face_nodes:_FillValue = 999999 ;
Mesh2_face_nodes:start_index = 1 ;
integer Mesh2_edge_nodes(nMesh2_edge, Two) ;
Mesh2_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh2_edge_nodes:long_name = "Maps every edge to the two nodes that it connects." ;
Mesh2_edge_nodes:start_index = 1 ;

// Optional mesh topology variables
integer Mesh2_face_edges(nMesh2_face, nMaxMesh2_face_nodes) ;
Mesh2_face_edges:cf_role = "face_edge_connectivity" ;
Mesh2_face_edges:long_name = "Maps every face to its edges." ;
Mesh2_face_edges:_FillValue = 999999 ;
Mesh2_face_edges:start_index = 1 ;
Mesh2_face_links:long_name = "Indicates which other faces neighbor each face." ;
Mesh2_edge_face_links:long_name = "neighbor faces for edges" ;
Mesh2_edge_face_links:comment = "missing neighbor faces are indicated using _FillValue" ;

// Mesh node coordinates
double Mesh2_node_x(nMesh2_node) ;
Mesh2_node_x:standard_name = "longitude" ;
Mesh2_node_x:long_name = "Longitude of 2D mesh nodes." ;
Mesh2_node_x:units = "degrees_east" ;
double Mesh2_node_y(nMesh2_node) ;
Mesh2_node_y:standard_name = "latitude" ;
Mesh2_node_y:long_name = "Latitude of 2D mesh nodes." ;
Mesh2_node_y:units = "degrees_north" ;

// Optional mesh face and edge coordinate variables
double Mesh2_face_x(nMesh2_face) ;
Mesh2_face_x:standard_name = "longitude" ;
Mesh2_face_x:long_name = "Characteristics longitude of 2D mesh face." ;
Mesh2_face_x:units = "degrees_east" ;
Mesh2_face_x:bounds = "Mesh2_face_xbnds" ;
double Mesh2_face_y(nMesh2_face) ;
Mesh2_face_y:standard_name = "latitude" ;
Mesh2_face_y:long_name = "Characteristics latitude of 2D mesh face." ;
Mesh2_face_y:units = "degrees_north" ;
Mesh2_face_y:bounds = "Mesh2_face_ybnds" ;
double Mesh2_face_xbnds(nMesh2_face,nMaxMesh2_face_nodes) ;
Mesh2_face_xbnds:standard_name = "longitude" ;
Mesh2_face_xbnds:long_name = "Longitude bounds of 2D mesh face (i.e. corner coordinates)." ;
Mesh2_face_xbnds:units = "degrees_east" ;
Mesh2_face_xbnds:_FillValue = 9.9692099683868690E36;
double Mesh2_face_ybnds(nMesh2_face,nMaxMesh2_face_nodes) ;
Mesh2_face_ybnds:standard_name = "latitude" ;
Mesh2_face_ybnds:long_name = "Latitude bounds of 2D mesh face (i.e. corner coordinates)." ;
Mesh2_face_ybnds:units = "degrees_north" ;
Mesh2_face_ybnds:_FillValue = 9.9692099683868690E36;
double Mesh2_edge_x(nMesh2_edge) ;
Mesh2_edge_x:standard_name = "longitude" ;
Mesh2_edge_x:long_name = "Characteristic longitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh2_edge_x:units = "degrees_east" ;
double Mesh2_edge_y(nMesh2_edge) ;
Mesh2_edge_y:standard_name = "latitude" ;
Mesh2_edge_y:long_name = "Characteristic latitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh2_edge_y:units = "degrees_north" ;
// bounds variables for edges skipped

// Vertical coordinate
double Mesh2_layers(Mesh2_layers) ;
Mesh2_layers:standard_name = "ocean_sigma_coordinate" ;
Mesh2_layers:long_name = "sigma at layer midpoints" ;
Mesh2_layers:positive = "up" ;
Mesh2_layers:formula_terms = "sigma: Mesh2_layers eta: Mesh2_surface depth: Mesh2_depth" ;
double Mesh2_depth(nMesh2_node) ;
Mesh2_depth:standard_name = "sea_floor_depth_below_geoid" ;
Mesh2_depth:units = "m" ;
Mesh2_depth:positive = "down" ;
Mesh2_depth:mesh = "Mesh2"
Mesh2_depth:location = "node" ;
Mesh2_depth:coordinates = "Mesh2_node_x Mesh2_node_y" ;
double Mesh2_surface(nMesh2_node) ;
Mesh2_surface:standard_name = "sea_surface_height_above_geoid" ;
Mesh2_surface:units = "m" ;
Mesh2_surface:mesh = "Mesh2"
Mesh2_surface:location = "face" ;
Mesh2_surface:coordinates = "Mesh2_face_x Mesh2_face_y" ;


### Fully 3D unstructured (i.e. non-layered) mesh topology

For a fully 3D unstructured mesh topology we extend the hierarchy of nodes, edges and faces to volumes. Contrary to layered case this type of mesh requires a fully 3D specification of the mesh; hence we not only need latitude and longitude coordinates but also some kind of elevation coordinate (this probably requires a new standard name).

The topology information is stored as attributes to a dummy variable (in the example below called “Mesh3D”) with cf_role mesh_topology.

Required topology attributes Value
cf_role mesh_topology
topology_dimension 3
node_coordinates
volume_node_connectivity
volume_shape_type
Optionally-required attributes*
volume_dimension
face_node_connectivity
face_dimension
edge_node_connectivity
edge_dimension
Optional attributes
volume_edge_connectivity
volume_face_connectivity
volume_volume_connectivity
face_edge_connectivity
boundary_node_connectivity
volume_coordinates
face_coordinates
edge_coordinates

*The “Optionally required” attributes edge_node_connectivity and face_node_connectivity are required only if you want to store data on the edges or faces respectively (i.e. if you mind the numbering order of the edges/faces).

*The “Optionally required” attributes volume_dimension, face_dimension and edge_dimension are required only if the dimension ordering is nonstandard in any of the connectivity variables for volumes, faces and nodes, respectively.

The attribute topology_dimension indicates the highest dimensionality of the geometric elements; for a fully 3-dimensional unstructured mesh this should be 3. The attribute node_coordinates points to the auxiliary coordinate variables representing the node locations (latitude, longitude, elevation and optional other coordinates). These auxiliary coordinate variables will have length nNodes. The attribute volume_node_connectivity points to an index variable identifying for every volume the indices of its corner nodes. For faces in the horizontal plane, it was possible to prescribe the order of the nodes, but this is not possible for the nodes of generic 3D volumes. For this reason we introduce an additional attribute volume_shape_type which points to a flag variable that specifies for every volume its shape:

flag_meaning name description
tetrahedron pyramid with triangular base, 4 nodes
pyramid pyramid with square base, a pentahedron, 5 nodes
wedge prism with triangular base, a pentahedron, 6 nodes
hexahedron distorted cube, 8 nodes

These four volume shapes are the ones most commonly used. More shapes can be added; and if necessary, it’s possible to add a generic shape type which allows for specification of the volume shape indirectly via volume_face_connectivity and face_node_connectivity variables. Such a generic shape is not included in this proposal since there is no practical need for such feature at this time.

If all volumes have the same shape type, then the shape typed could be determined based on the number of nodes per volume. Another option could be to allow the volume_shape_type to specify the shape type directly rather than pointing to a variable. However, for the time being we assume that the currently proposed volume shape type variable doesn’t have too much impact on the performance.

The order in which the corner nodes of a volume are specified is fixed given its shape; this approach is common in 3D modeling, see e.g. this graph in the OpenFOAM documentation and ParaView or VTK documentation. The volume_node_connectivity array will be a matrix of size nVolumes x MaxNumNodesPerVolume; if a volume has less corner nodes than MaxNumNodesPerVolume then the last node indices shall be equal to _FillValue (which should obviously be larger than the number of nodes in the mesh). For the indexing one may use either 0- or 1-based indexing; the convention used should be specified using a start_index attribute to the index variable (i.e. Mesh3D_vol_nodes in the example below). Consistent with the CF-conventions compression option, the connectivity indices are 0-based by default. See this section on zero or one-based indexing for more details.

The volume_dimension attribute specifies which netcdf dimension is used to indicate the index of the volume in the connectivity arrays. This is needed because some applications store the data with the fastest varying index first, and some with that index last. The default is to use the num_volumes as fastest dimension; e.g. a (num_volumes, 4) array for tetrahedrons, but some applications might use a (4, num_volumes) order, in which case the volume_dimension attribute is required to help the client code disambiguate. The face_dimension and edge_dimension attributes are similar for the face and edge connectivity arrays.

In case you want to define variables on the faces or edges of the 3D mesh topology you need to specify the face_node_connectivity or edge_node_connectivity attribute, respectively, to map faces or edges to nodes. Although the volume to node mapping implicitly also defines the location of the faces and edges, it does not specify their global numbering. Again the indexing convention of face_node_connectivity and edge_node_connectivity should be specified using the start_index attribute to the index variable and 0-based indexing is the default. Since it does not apply to either nodes or edges globally, specifying the boundary_node_connectivity attribute described below does not (in and of itself) necessitate the need to specify edge_node_connectivity nor face_node_connectivity.

Optionally the topology may have the following attributes:

• volume_face_connectivity pointing to an index variable identifying for every volume the indices of its faces. The order in which the face indices should be specified is determined by the volume geometry type. This connectivity array will be a matrix of size nVolumes x MaxNumFacesPerVolume. If a volume has less faces than MaxNumFacesPerVolume then the last face indices shall be equal to _FillValue, and the indexing convention of volume_edge_connectivity should be specified using the start_index attribute to the index variable and 0-based indexing is the default.

• volume_edge_connectivity pointing to an index variable identifying for every volume the indices of its edges. The order in which the edge indices should be specified is determined by the volume geometry type. This connectivity array will be a matrix of size nVolumes x MaxNumEdgesPerVolume. Again, if a volume has less edges than MaxNumEdgesPerVolume then the last edge indices shall be equal to _FillValue, and the indexing convention of volume_edge_connectivity should be specified using the start_index attribute to the index variable and 0-based indexing is the default.

• volume_volume_connectivity pointing to an index variable identifying all volumes that share a face with each volume, i.e. are neighbors. This connectivity array will thus be a matrix of size nVolumes x MaxNumFacesPerVolume with a flag to note if the region “out of mesh” abuts a face of a given volume. Again, if a volume has less sides/faces than MaxNumFacesPerVolume then the last volume (column) indices shall be equal to _FillValue, and the indexing convention of volume_volume_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh3D_vol_links in the example below) and 0-based indexing is the default.

• face_edge_connectivity pointing to an index variable identifying for every face the indices of its edges. The edges should be specified in anticlockwise direction as viewed from above. This connectivity array will be a matrix of size nFaces x MaxNumNodesPerFace. As always, if a face has less corners/edges than MaxNumNodesPerFace then the last edge indices shall be equal to _FillValue, and the indexing convention of face_edge_connectivity should be specified using the start_index attribute to the index variable and 0-based indexing is the default.

• face_node_connectivity pointing to an index variable identifying for every face the indices of its nodes. The nodes should be specified in either clockwise or anticlockwise order. This connectivity array will be a matrix of size nFaces x MaxNumNodesPerFace. Again, if a face has less corners/edges than MaxNumNodesPerFace then the last node indices shall be equal to _FillValue, and the indexing convention of face_node_connectivity should be specified using the start_index attribute to the index variable and 0-based indexing is the default.

• boundary_node_connectivity pointing to an index variable identifying for every face of each boundary the indices of its nodes. The nodes should be specified in either clockwise or anticlockwise order. This connectivity array will be a matrix of size nBoundaryFaces x MaxNumNodesPerFace. Again, if a face has less corners/edges than MaxNumNodesPerFace then the last node indices shall be equal to _FillValue, and the indexing convention of boundary_node_connectivity should be specified using the start_index attribute to the index variable (i.e. Mesh3D_boundary_nodes) and 0-based indexing is the default. Although constructed of faces, boundaries represent a different quantity than general face data and thus the boundary_node_connectivity attribute may be specified independent of face_node_connectivity. Information about the nature of each boundary face (e.g. open/closed, land/water, grouping, etc.) may optionally be stored in ancillary boundary-type variables of size nBoundaryFaces X 1.

• volume_coordinates, face_coordinates and/or edge_coordinates pointing to the auxiliary coordinate variables associated with the characteristic location of the volumes, faces and edges. These auxiliary coordinate variables will have length nVolumes, nFaces and nEdges respectively, and may have in turn a bounds attribute that specifies the corner coordinates of the volume, face or edge (thereby duplicating the data in the node_coordinates variables). The order in which the corner coordinates of the volumes is given by the volume geometry type.

Example:

dimensions:
nMesh3D_node = 12 ; // nNodes
nMesh3D_edge = 23 ; // nEdges
nMesh3D_face = 16 ; // nFaces
nMesh3D_vol = 4 ; // nVolumes
nMaxMesh3D_face_nodes = 4 ; // MaxNumNodesPerFace
nMaxMesh3D_vol_nodes = 8 ; // MaxNumNodesPerVolume
nMaxMesh3D_vol_edges = 12 ; // MaxNumEdgesPerVolume
nMaxMesh3D_vol_faces = 6 ; // MaxNumFacesPerVolume

Two = 2 ;

variables:
// Mesh topology
integer Mesh3D ;
Mesh3D:cf_role = "mesh_topology" ;
Mesh3D:long_name = "Topology data of 3D unstructured mesh" ;
Mesh3D:topology_dimension = 3 ;
Mesh3D:node_coordinates = "Mesh3D_node_x Mesh3D_node_y Mesh3D_node_z" ;
Mesh3D:volume_shape_type = "Mesh3D_vol_types" ;
Mesh3D:volume_node_connectivity = "Mesh3D_vol_nodes" ;
Mesh3D:volume_dimension = nMesh3D_vol ;
Mesh3D:face_node_connectivity = "Mesh3D_face_nodes" ; // attribute required if variables will be defined on faces
Mesh3D:face_dimension = nMesh3D_face ;
Mesh3D:edge_node_connectivity = "Mesh3D_edge_nodes" ; // attribute required if variables will be defined on edges
Mesh3D:edge_dimension = nMesh3D_edge ;
Mesh3D:edge_coordinates = "Mesh3D_edge_x Mesh3D_edge_y Mesh3D_edge_z" ; // optional attribute (requires edge_node_connectivity)
Mesh3D:face_coordinates = "Mesh3D_face_x Mesh3D_face_y Mesh3D_face_z" ; // optional attribute (requires face_node_connectivity)
Mesh3D:volume_coordinates = "Mesh3D_vol_x Mesh3D_vol_y Mesh3D_vol_z" ; // optional attribute
Mesh3D:volume_face_connectivity = "Mesh3D_vol_faces" ; // optional attribute (requires face_node_connectivity)
Mesh3D:volume_edge_connectivity = "Mesh3D_vol_edges" ; // optional attribute (requires edge_node_connectivity)
Mesh3D:face_edge_connectivity = "Mesh3D_face_edges" ; // optional attribute (requires face_node_connectivity and edge_node_connectivity)
Mesh3D:volume_volume_connectivity = "Mesh3D_vol_links" ; // optional attribute
integer Mesh3D_vol_types(nMesh3D_vol) ;
Mesh3D_vol_types:cf_role = "volume_shape_type" ;
Mesh3D_vol_types:long_name = "Specifies the shape of the individual volumes." ;
Mesh3D_vol_types:flag_range = 0b, 2b ;
Mesh3D_vol_types:flag_values = 0b, 1b, 2b ;
Mesh3D_vol_types:flag_meanings = "tetrahedron wedge hexahedron" ;
integer Mesh3D_vol_nodes(nMesh3D_vol, nMaxMesh3D_vol_nodes) ;
Mesh3D_vol_nodes:cf_role = "volume_node_connectivity" ;
Mesh3D_vol_nodes:long_name = "Maps every volume to its corner nodes." ;
Mesh3D_vol_nodes:_FillValue = 999999 ;
Mesh3D_vol_nodes:start_index = 1 ;

// Optional mesh topology variables
integer Mesh3D_edge_nodes(nMesh3D_edge, Two) ;
Mesh3D_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh3D_edge_nodes:long_name = "Maps every edge to the two nodes that it connects." ;
Mesh3D_edge_nodes:start_index = 1 ;
integer Mesh3D_face_nodes(nMesh3D_face, nMaxMesh3D_face_nodes) ;
Mesh3D_face_nodes:cf_role = "face_node_connectivity" ;
Mesh3D_face_nodes:long_name = "Maps every face to its corner nodes." ;
Mesh3D_face_nodes:_FillValue = 999999 ;
Mesh3D_face_nodes:start_index = 1 ;
integer Mesh3D_vol_faces(nMesh3D_vol, nMaxMesh3D_vol_faces) ;
Mesh3D_vol_faces:cf_role = "volume_face_connectivity" ;
Mesh3D_vol_faces:long_name = "Maps every volume to its faces." ;
Mesh3D_vol_faces:_FillValue = 999999 ;
Mesh3D_vol_faces:start_index = 1 ;
integer Mesh3D_vol_edges(nMesh3D_vol, nMaxMesh3D_vol_edges) ;
Mesh3D_vol_edges:cf_role = "volume_edge_connectivity" ;
Mesh3D_vol_edges:long_name = "Maps every volume to its edges." ;
Mesh3D_vol_edges:_FillValue = 999999 ;
Mesh3D_vol_edges:start_index = 1 ;
integer Mesh3D_face_edges(nMesh3D_face, nMaxMesh3D_face_nodes) ;
Mesh3D_face_edges:cf_role = "face_edge_connectivity" ;
Mesh3D_face_edges:long_name = "Maps every face to its edges." ;
Mesh3D_face_edges:_FillValue = 999999 ;
Mesh3D_face_edges:start_index = 1 ;
Mesh3D_vol_links:long_name = "Indicates which other volumes neighbor each volume." ;

// Mesh node coordinates
double Mesh3D_node_x(nMesh3D_node) ;
Mesh3D_node_x:standard_name = "longitude" ;
Mesh3D_node_x:long_name = "Longitude of 3D mesh nodes." ;
Mesh3D_node_x:units = "degrees_east" ;
double Mesh3D_node_y(nMesh3D_node) ;
Mesh3D_node_y:standard_name = "latitude" ;
Mesh3D_node_y:long_name = "Latitude of 3D mesh nodes." ;
Mesh3D_node_y:units = "degrees_north" ;
double Mesh3D_node_z(nMesh3D_node) ;
Mesh3D_node_z:standard_name = "elevation" ;
Mesh3D_node_z:long_name = "Elevation of 3D mesh nodes." ;
Mesh3D_node_z:units = "m" ;

// Optional mesh volume, face and edge coordinate variables
double Mesh3D_vol_x(nMesh3D_vol) ;
Mesh3D_vol_x:standard_name = "longitude" ;
Mesh3D_vol_x:long_name = "Characteristics longitude of mesh volumes." ;
Mesh3D_vol_x:units = "degrees_east" ;
Mesh3D_vol_x:bounds = "Mesh3D_vol_xbnds" ;
double Mesh3D_vol_y(nMesh3D_vol) ;
Mesh3D_vol_y:standard_name = "latitude" ;
Mesh3D_vol_y:long_name = "Characteristics latitude of mesh volumes." ;
Mesh3D_vol_y:units = "degrees_north" ;
Mesh3D_vol_y:bounds = "Mesh3D_vol_ybnds" ;
double Mesh3D_vol_z(nMesh3D_vol) ;
Mesh3D_vol_z:standard_name = "elevation" ;
Mesh3D_vol_z:long_name = "Characteristics elevation of mesh volumes." ;
Mesh3D_vol_z:units = "m" ;
Mesh3D_vol_z:bounds = "Mesh3D_vol_zbnds" ;
double Mesh3D_vol_xbnds(nMesh3D_vol,nMaxMesh3D_vol_nodes) ;
Mesh3D_vol_xbnds:standard_name = "longitude" ;
Mesh3D_vol_xbnds:long_name = "Longitude bounds of mesh volumes (i.e. corner coordinates)." ;
Mesh3D_vol_xbnds:units = "degrees_east" ;
Mesh3D_vol_xbnds:_FillValue = 9.9692099683868690E36;
double Mesh3D_vol_ybnds(nMesh3D_vol,nMaxMesh3D_vol_nodes) ;
Mesh3D_vol_ybnds:standard_name = "latitude" ;
Mesh3D_vol_ybnds:long_name = "Latitude bounds of mesh volumes (i.e. corner coordinates)." ;
Mesh3D_vol_ybnds:units = "degrees_north" ;
Mesh3D_vol_ybnds:_FillValue = 9.9692099683868690E36;
double Mesh3D_vol_zbnds(nMesh3D_vol,nMaxMesh3D_vol_nodes) ;
Mesh3D_vol_zbnds:standard_name = "elevation" ;
Mesh3D_vol_zbnds:long_name = "Elevation bounds of mesh volumes (i.e. corner coordinates)." ;
Mesh3D_vol_zbnds:units = "m" ;
Mesh3D_vol_zbnds:_FillValue = 9.9692099683868690E36;
double Mesh3D_face_x(nMesh3D_face) ;
Mesh3D_face_x:standard_name = "longitude" ;
Mesh3D_face_x:long_name = "Characteristics longitude of mesh faces." ;
Mesh3D_face_x:units = "degrees_east" ;
double Mesh3D_face_y(nMesh3D_face) ;
Mesh3D_face_y:standard_name = "latitude" ;
Mesh3D_face_y:long_name = "Characteristics latitude of mesh faces." ;
Mesh3D_face_y:units = "degrees_north" ;
double Mesh3D_face_z(nMesh3D_face) ;
Mesh3D_face_z:standard_name = "elevation" ;
Mesh3D_face_z:long_name = "Characteristics elevation of mesh faces." ;
Mesh3D_face_z:units = "m" ;
// bounds variables for faces skipped
double Mesh3D_edge_x(nMesh3D_edge) ;
Mesh3D_edge_x:standard_name = "longitude" ;
Mesh3D_edge_x:long_name = "Characteristic longitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh3D_edge_x:units = "degrees_east" ;
double Mesh3D_edge_y(nMesh3D_edge) ;
Mesh3D_edge_y:standard_name = "latitude" ;
Mesh3D_edge_y:long_name = "Characteristic latitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh3D_edge_y:units = "degrees_north" ;
double Mesh3D_edge_z(nMesh3D_edge) ;
Mesh3D_edge_z:standard_name = "latitude" ;
Mesh3D_edge_z:long_name = "Characteristic latitude of 2D mesh edge (e.g. midpoint of the edge)." ;
Mesh3D_edge_z:units = "degrees_north" ;
// bounds variables for edges skipped


## Data defined on unstructured meshes

According to CF-conventions a variable defined on a structured mesh is specified as

double waterlevel(time,nmax,mmax) ;
waterlevel:standard_name = "sea_surface_height_above_geoid" ;
waterlevel:units = "m" ;
waterlevel:coordinates = "lat lon" ;


The coordinates attribute refers to the variables that contain the latitude and longitude coordinates. For a curvilinear grid these variables will share two spatial dimensions, here nmax and mmax: lat(nmax,mmax) and lon(nmax,mmax). In numerical models the various quantities are often computed at different locations of the mesh: staggered data. The standard CF-conventions don’t offer specific support for this functionality and thus for every stagger location coordinates need to be provided separately: cell center coordinates, corner point coordinates, u-flux point coordinates, and v-flux point coordinates. The underlying topology of the mesh, i.e. how these coordinates (variable definition locations) relate to each other isn’t stored in the file. This shortcoming is to some degree solved by the gridspec proposal by Venkatramani Balaji and Zhi Liang. We introduce here attributes that link to the topological data defined above.

### Data variables

The use of the coordinates attribute is copied from the CF-conventions. It is used to map the values of variables defined on the unstructured meshes directly to their location: latitude, longitude and optional elevation. To map the variable onto the topology of the underlying mesh, two new attributes have been introduced. First, the attribute mesh points to the mesh_topology variable containing the meta-data attributes of the mesh on which the variable has been defined. Second, the attribute location points to the (stagger) location within the mesh at which the variable is defined. Note that in this example the coordinates attribute is redundant since the coordinates could also be obtained by using the face_coordinates attribute of the “Mesh2” variable.

double Mesh2_waterlevel(time,nMesh2_face) ;
Mesh2_waterlevel:standard_name = "sea_surface_height_above_geoid" ;
Mesh2_waterlevel:units = "m" ;
Mesh2_waterlevel:mesh = "Mesh2"
Mesh2_waterlevel:location = "face" ;
Mesh2_waterlevel:coordinates = "Mesh2_face_x Mesh2_face_y" ;


The order of dimensions on a data variable is arbitrary, so nTimeSteps x nLayers x nFaces, nTimeSteps x nFaces x nLayers, nFaces x nLayers x nTimeSeps, etc would all be valid. In line with CF/COARDS and OGC NetCDF-CF we recommend to use the T,Z,Y,X [c order notation] order as much as possible where the unstructured horizontal dimension should be read for Y,X. That is, the order nTimeSteps x nLayers x nFaces [c order notation] is recommended: the nFaces is the fastest increasing dimension, then the nLayers, and finally the nTimeSteps.

### Volume and flux variables

The same mesh geometry can be used in different ways to schematize the hydrodynamic volumes and fluxes. Let’s take a simple triangular mesh. From a finite volume point of view this mesh will generally be interpreted as consisting of two volumes with triangular base. However, others may use a continuous Galerkin finite element method that can be shown to be equivalent to a subdivision into four volumes. In the former case the two faces correspond to volumes and the fluxes cross the edges. In the latter case the volumes are defined surrounding the four nodes and the fluxes are directed along the edges. The two abbreviated “ncdumps” below show how the basic 2D triangular mesh definition can be extended to include this data. The coordinate variables for the volume data now include bounds attributes to define the surface area of the volumes. Note the subtle difference in the long names between the flux variables in the two cases; the standard_name attribute has to make a more formal distinction between the two cases.

Variant 1: Volume at faces:

dimensions:
nMesh2_node = 4 ; // nNodes
nMesh2_edge = 5 ; // nEdges
nMesh2_face = 2 ; // nFaces

Two = 2 ;
Three = 3 ;

variables:
// Mesh topology
integer Mesh2 ;
// as in 2D triangular mesh example
integer Mesh2_face_nodes(nMesh2_face, Three) ;
// as in 2D triangular mesh example
integer Mesh2_edge_nodes(nMesh2_edge, Two) ;
// as in 2D triangular mesh example

// Optional mesh topology variables
integer Mesh2_face_edges(nMesh2_face, Three) ;
// as in 2D triangular mesh example
// as in 2D triangular mesh example

// Mesh node coordinates
double Mesh2_node_x(nMesh2_node) ;
// as in 2D triangular mesh example
double Mesh2_node_y(nMesh2_node) ;
// as in 2D triangular mesh example

// Optional mesh face and edge coordinate variables
double Mesh2_face_x(nMesh2_face) ;
Mesh2_face_x:standard_name = "longitude" ;
Mesh2_face_x:long_name = "Characteristics longitude of 2D mesh triangle (e.g. circumcenter coordinate)." ;
Mesh2_face_x:units = "degrees_east" ;
Mesh2_face_x:bounds = "Mesh2_face_xbnds" ;
double Mesh2_face_y(nMesh2_face) ;
Mesh2_face_y:standard_name = "latitude" ;
Mesh2_face_y:long_name = "Characteristics latitude of 2D mesh triangle (e.g. circumcenter coordinate)." ;
Mesh2_face_y:units = "degrees_north" ;
Mesh2_face_y:bounds = "Mesh2_face_ybnds" ;
double Mesh2_face_xbnds(nMesh2_face,Three) ;
Mesh2_face_xbnds:standard_name = "longitude" ;
Mesh2_face_xbnds:long_name = "Longitude bounds of 2D mesh triangle (i.e. corner coordinates)." ;
Mesh2_face_xbnds:units = "degrees_east" ;
double Mesh2_face_ybnds(nMesh2_face,Three) ;
Mesh2_face_ybnds:standard_name = "latitude" ;
Mesh2_face_ybnds:long_name = "Latitude bounds of 2D mesh triangle (i.e. corner coordinates)." ;
Mesh2_face_ybnds:units = "degrees_north" ;
double Mesh2_edge_x(nMesh2_edge) ;
// as in 2D triangular mesh example
double Mesh2_edge_y(nMesh2_edge) ;
// as in 2D triangular mesh example

// Volume and flux data
double Mesh2_volumes(nMesh2_face) ;
Mesh2_volumes:long_name = "volumes" ;
Mesh2_volumes:units = "m3" ;
Mesh2_volumes:mesh = "Mesh2" ;
Mesh2_volumes:location = "face" ;
Mesh2_volumes:coordinates = "Mesh2_face_x Mesh2_face_y" ;
double Mesh2_fluxes(nMesh2_edge) ;
Mesh2_fluxes:long_name = "flux across edge" ;
Mesh2_fluxes:units = "m3 s-1" ;
Mesh2_fluxes:mesh = "Mesh2"
Mesh2_fluxes:location = "edge" ;
Mesh2_fluxes:coordinates = "Mesh2_edge_x Mesh2_edge_y" ;


Variant 2: Volume at nodes:

dimensions:
nMesh2_node = 4 ; // nNodes
nMesh2_edge = 5 ; // nEdges
nMesh2_face = 2 ; // nFaces
nMaxMesh2_bnds = 6 ;

Two = 2 ;
Three = 3 ;

variables:
// Mesh topology
integer Mesh2 ;
// as in 2D triangular mesh example
integer Mesh2_face_nodes(nMesh2_face, Three) ;
// as in 2D triangular mesh example
integer Mesh2_edge_nodes(nMesh2_edge, Two) ;
// as in 2D triangular mesh example

// Optional mesh topology variables
integer Mesh2_face_edges(nMesh2_face, Three) ;
// as in 2D triangular mesh example
// as in 2D triangular mesh example

// Mesh node coordinates
double Mesh2_node_x(nMesh2_node) ;
Mesh2_node_x:standard_name = "longitude" ;
Mesh2_node_x:long_name = "Longitude of 2D mesh nodes." ;
Mesh2_node_x:units = "degrees_east" ;
Mesh2_node_x:bounds = "Mesh2_node_xbnds" ;
double Mesh2_node_y(nMesh2_node) ;
Mesh2_node_y:standard_name = "latitude" ;
Mesh2_node_y:long_name = "Latitude of 2D mesh nodes." ;
Mesh2_node_y:units = "degrees_north" ;
Mesh2_node_y:bounds = "Mesh2_node_ybnds" ;
double Mesh2_node_xbnds(nMesh2_node, nMaxMesh2_bnds) ;
Mesh2_node_xbnds:standard_name = "longitude" ;
Mesh2_node_xbnds:long_name = "List of x-points that form outline of flow volume" ;
Mesh2_node_xbnds:units = "degrees_east" ;
Mesh2_node_xbnds:_FillValue = 9.9692099683868690E36;
double Mesh2_node_ybnds(nMesh2_node, nMaxMesh2_bnds) ;
Mesh2_node_ybnds:standard_name = "latitude" ;
Mesh2_node_ybnds:units = "degrees_north" ;
Mesh2_node_ybnds:long_name = "List of y-points that form outline of flow volume" ;
Mesh2_node_xbnds:_FillValue = 9.9692099683868690E36;

// Optional mesh face and edge coordinate variables
double Mesh2_face_x(nMesh2_face) ;
// as in 2D triangular mesh example
double Mesh2_face_y(nMesh2_face) ;
// as in 2D triangular mesh example
double Mesh2_edge_x(nMesh2_edge) ;
// as in 2D triangular mesh example
double Mesh2_edge_y(nMesh2_edge) ;
// as in 2D triangular mesh example

// Volume and flux data
double Mesh2_volumes(nMesh2_node) ;
Mesh2_volumes:long_name = "volumes" ;
Mesh2_volumes:units = "m3" ;
Mesh2_volumes:mesh = "Mesh2" ;
Mesh2_volumes:location = "node" ;
Mesh2_volumes:coordinates = "Mesh2_node_x Mesh2_node_y" ;
double Mesh2_fluxes(nMesh2_edge) ;
Mesh2_fluxes:long_name = "flux along edge" ;
Mesh2_fluxes:units = "m3 s-1" ;
Mesh2_fluxes:mesh = "Mesh2"
Mesh2_fluxes:location = "edge" ;
Mesh2_fluxes:coordinates = "Mesh2_edge_x Mesh2_edge_y" ;


### Location index set

Some variables may only be defined at specific locations within the mesh, e.g. only at boundary points or at special locations like weirs and gates. To save space and to improve readability, the concept of a location_index_set is introduced. It is basically identical to the compression option in the the CF-conventions except for the fact that the compression works on a (set of) orthogonal coordinate dimension(s) and the location_index_set works on a topology location.

The location index set is an integer variable that contains the indices of the locations at which a specific quantity is defined. The example below defines a location index set “Mesh1_set” as a subset of the “node”s of Mesh1 (red points). The attribute location_index_set of the variable “Mesh_waterlevel” points to this index set and the coordinates attribute points to the corresponding (subset) of latitude and longitude coordinates. The mesh and location attributes of the location_index_set variable are required; the coordinates attribute is optional. Note that the coordinates attributes on both “Mesh1_cell” and “Mesh1_waterlevel” are again redundant since the coordinates could also be obtained by using the index set “Mesh1_set” and the node_coordinates attribute of the “Mesh1” variable. Consistent with all other index variables defined here, the indexing convention of the location index set should be specified using the start_index attribute to the index variable and 0-based indexing is the default. See this section on zero or one-based indexing for more details.

Contrary to a coordinate variable, the index set doesn’t have to be monotonic. So, it can be used for creating subsets of the original locations as well as for renumbering the locations. If the location_index_set attribute is used on a variable, then the mesh and location attributes should not also be used on that variable.

dimensions:
nMesh1_set = 4 ;

variables:
integer Mesh1_set(nMesh1_set) ;
Mesh1_set:cf_role = "location_index_set" ;
Mesh1_set:long_name = "Defines Mesh1_set as subset of the nodes of Mesh1." ;
Mesh1_set:mesh = "Mesh1" ;
Mesh1_set:location = "node" ;
Mesh1_set:start_index = 1 ;
Mesh1_set:coordinates = "Mesh1_set_x Mesh1_set_y" ;
double Mesh1_set_x(nMesh1_set) ;
Mesh1_set_x:standard_name = "longitude" ;
Mesh1_set_x:long_name = "Characteristic longitude of set (e.g. longitude of node)." ;
Mesh1_set_x:units = "degrees_east" ;
double Mesh1_set_y(nMesh1_set) ;
Mesh1_set_y:standard_name = "latitude" ;
Mesh1_set_y:long_name = "Characteristic latitude of set (e.g. latitude of node)" ;
Mesh1_set_y:units = "degrees_north" ;

double Mesh1_waterlevel(time, nMesh1_set) ;
Mesh1_waterlevel:standard_name = "sea_surface_height_above_geoid" ;
Mesh1_waterlevel:units = "m" ;
Mesh1_waterlevel:location_index_set = "Mesh1_set" ;
Mesh1_waterlevel:coordinates = "Mesh1_set_x Mesh1_set_y" ;


## Zero or One-based indexing

The indexing using by the CF compression option is 0-based. Therefore, it is most consistent for this CF extension for unstructured data to also use 0-based indexing, which means that points, edges, faces and volumes will be numbered starting with 0. This convention is consistent with languages like C, Java and Python, but unlike Fortran and MATLAB. Since many of the unstructured models have been programmed in Fortran and legacy NetCDF files exist that use 1-based indexing (which could be upgraded to be consistent with this new proposal using ncML if 1-based indexing were allowed), we propose to support both 0- and 1-based indexing by means of the start_index attribute. You will find below two examples of the same network geometry using either 0- or 1-based indexing. Switching between 0- and 1-based indexing is as easy as adding 1 to or subtracting 1 from the indices upon reading or writing depending on the setting of start_index; allowing both options should only have a minor impact on the reading routines and no effect at all on the rest of your code.

Example of 0-based indexing:

dimensions:
nMesh1_node = 5 ; // nNodes
nMesh1_edge = 4 ; // nEdges

Two = 2;

variables:
// Mesh topology
integer Mesh1 ;
Mesh1:cf_role = "mesh_topology" ;
Mesh1:long_name = "Topology data of 1D network" ;
Mesh1:topology_dimension = 1 ;
Mesh1:node_coordinates = "Mesh1_node_x Mesh1_node_y" ;
Mesh1:edge_node_connectivity = "Mesh1_edge_nodes" ;
Mesh1:edge_coordinates = "Mesh1_edge_x Mesh1_edge_y" ; // optional attribute
integer Mesh1_edge_nodes(nMesh1_edge, Two) ;
Mesh1_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh1_edge_nodes:long_name = "Maps every edge/link to the two nodes that it connects." ;
Mesh1_edge_nodes:start_index = 0 ; // default setting, attribute could have been skipped.

// Coordinate variables skipped

data:

Mesh1 = 0 ; // dummy

Mesh1_edge_nodes =
0, 2,
1, 2,
2, 3,
3, 4 ;


Example of 1-based indexing:

dimensions:
nMesh1_node = 5 ; // nNodes
nMesh1_edge = 4 ; // nEdges

Two = 2;

variables:
// Mesh topology
integer Mesh1 ;
Mesh1:cf_role = "mesh_topology" ;
Mesh1:long_name = "Topology data of 1D network" ;
Mesh1:topology_dimension = 1 ;
Mesh1:node_coordinates = "Mesh1_node_x Mesh1_node_y" ;
Mesh1:edge_node_connectivity = "Mesh1_edge_nodes" ;
Mesh1:edge_coordinates = "Mesh1_edge_x Mesh1_edge_y" ; // optional attribute
integer Mesh1_edge_nodes(nMesh1_edge, Two) ;
Mesh1_edge_nodes:cf_role = "edge_node_connectivity" ;
Mesh1_edge_nodes:long_name = "Maps every edge/link to the two nodes that it connects." ;
Mesh1_edge_nodes:start_index = 1 ;

// Coordinate variables skipped

data:

Mesh1 = 0 ; // dummy

Mesh1_edge_nodes =
1, 3,
2, 3,
3, 4,
4, 5 ;