The definitions given here allow an application to compute dimensional coordinate values from the parametric vertical coordinate values (usually dimensionless) and associated variables.
The formulas are expressed for a gridpoint (n,k,j,i) where i and j are the horizontal indices, k is the vertical index and n is the time index.
A coordinate variable is associated with its definition by the value of the standard_name attribute.
The terms in the definition are associated with file variables by the formula_terms attribute.
The formula_terms attribute takes a string value, the string being comprised of blank-separated elements of the form "term: variable", where term is a case-insensitive keyword that represents one of the terms in the definition, and variable is the name of the variable in a netCDF file that contains the values for that term.
The order of elements is not significant.
The gridpoint indices are not formally part of the definitions, but are included to illustrate the indices that might be present in the file variables.
For example, a vertical coordinate whose definition contains a time index is not necessarily time dependent in all netCDF files.
Also, the definitions are given in general forms that may be simplified by omitting certain terms.
A term that is omitted from the formula_terms attribute should be assumed to be zero.
The variables containing the terms may optionally have standard_name attributes, with values as indicated in this Appendix.
The standard_name of the dimensional coordinate values which are computed by the formula may optionally be specified by the computed_standard_name attribute of the vertical coordinate variable, as indicated in this Appendix.
A computed_standard_name is uniquely implied by the formula in some cases, while in others it depends on the standard_name of one or more of the terms, with which it must be consistent.
standard_name = "atmosphere_ln_pressure_coordinate"
- Definition
-
p(k) = p0 * exp(-lev(k))
where p(k) is the pressure at gridpoint (k), p0 is a reference pressure, lev(k) is the dimensionless coordinate at vertical gridpoint (k).
The format for the formula_terms attribute is:
formula_terms = "p0: var1 lev: var2"
The standard_name of p0 is reference_air_pressure_for_atmosphere_vertical_coordinate.
The computed_standard_name is air_pressure.
standard_name = "atmosphere_sigma_coordinate"
- Definition
-
p(n,k,j,i) = ptop + sigma(k)*(ps(n,j,i)-ptop)
where p(n,k,j,i) is the pressure at gridpoint (n,k,j,i), ptop is the pressure at the top of the model, sigma(k) is the dimensionless coordinate at vertical gridpoint (k), and ps(n,j,i) is the surface pressure at horizontal gridpoint (j,i)and time (n).
The format for the formula_terms attribute is
formula_terms = "sigma: var1 ps: var2 ptop: var3"
The standard_name of ptop is air_pressure_at_top_of_atmosphere_model, and of ps is surface_air_pressure.
The computed_standard_name is air_pressure.
standard_name = "atmosphere_hybrid_sigma_pressure_coordinate"
- Definition
-
p(n,k,j,i) = a(k)*p0 + b(k)*ps(n,j,i)
or
p(n,k,j,i) = ap(k) + b(k)*ps(n,j,i)
where p(n,k,j,i) is the pressure at gridpoint (n,k,j,i), a(k) or ap(k) and b(k) are components of the hybrid coordinate at level k, p0 is a reference pressure, and ps(n,j,i) is the surface pressure at horizontal gridpoint (j,i) and time (n).
The choice of whether a(k) or ap(k) is used depends on model formulation; the former is a dimensionless fraction, the latter a pressure value.
In both formulations, b(k) is a dimensionless fraction.
The format for the formula_terms attribute is
formula_terms = "a: var1 b: var2 ps: var3 p0: var4"
where a is replaced by ap if appropriate.
The hybrid sigma-pressure coordinate for level k is defined as a(k)+b(k) or ap(k)/p0+b(k), as appropriate.
The standard_name of p0 is reference_air_pressure_for_atmosphere_vertical_coordinate, and of ps is surface_air_pressure.
The computed_standard_name is air_pressure.
No standard_name has been defined for a, b or ap.
standard_name = "atmosphere_hybrid_height_coordinate"
- Definition
-
z(n,k,j,i) = a(k) + b(k)*orog(n,j,i)
where z(n,k,j,i) is the height above the datum (e.g. the geoid, which is approximately mean sea level) at gridpoint (k,j,i) and time (n), orog(n,j,i) is the height of the surface above the datum at (j,i) and time (n), and a(k) and b(k) are the coordinates which define hybrid height level k.
a(k) has the dimensions of height and b(i) is dimensionless.
The format for the formula_terms attribute is
formula_terms = "a: var1 b: var2 orog: var3"
The standard_name of orog may be surface_altitude (i.e. above the geoid) or surface_height_above_geopotential_datum.
The computed_standard_name is altitude or height_above_geopotential_datum in these cases respectively.
No standard_name has been defined for b.
There is no dimensionless coordinate because a, which has the standard_name of atmosphere_hybrid_height_coordinate, is the best choice for a level-dependent but geographically constant coordinate.
standard_name = "atmosphere_sleve_coordinate"
- Definition
-
z(n,k,j,i) = a(k)*ztop + b1(k)*zsurf1(n,j,i) + b2(k)*zsurf2(n,j,i)
where z(n,k,j,i) is the height above the datum (e.g. the geoid, which is approximately mean sea level) at gridpoint (k,j,i) and time (n), ztop is the height of the top of the model above the datum, and a(k), b1(k), and b2(k) are the dimensionless coordinates which define hybrid level k.
zsurf1(n,j,i) and zsurf2(n,j,i) are respectively the large-scale and small-scale components of the topography, and a, b1 and b2 are all functions of the dimensionless SLEVE coordinate.
See Shaer et al [SCH02] for details.
The format for the formula_terms attribute is
formula_terms = "a: var1 b1: var2 b2: var3 ztop: var4 zsurf1: var5
zsurf2: var6"
The standard_name of ztop may be altitude_at_top_of_atmosphere_model (i.e. above the geoid) or height_above_geopotential_datum_at_top_of_atmosphere_model.
The computed_standard_name is altitude or height_above_geopotential_datum in these cases respectively.
No standard_name has been defined for b1, zsurf1, b2 or zsurf2.
standard_name = "ocean_sigma_coordinate"
- Definition
-
z(n,k,j,i) = eta(n,j,i) + sigma(k)*(depth(j,i)+eta(n,j,i))
where z(n,k,j,i) is height (positive upwards) relative to the datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the sea surface (positive upwards) relative to the datum at gridpoint (n,j,i), sigma(k) is the dimensionless coordinate at vertical gridpoint (k), and depth(j,i) is the distance (a positive value) from the datum to the sea floor at horizontal gridpoint (j,i).
The format for the formula_terms attribute is
formula_terms = "sigma: var1 eta: var2 depth: var3"
The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1.
standard_name = "ocean_s_coordinate"
- Definition
-
z(n,k,j,i) = eta(n,j,i)*(1+s(k)) + depth_c*s(k) + (depth(j,i)-depth_c)*C(k)where
C(k) = (1-b)*sinh(a*s(k))/sinh(a) + b*[tanh(a*(s(k)+0.5))/(2*tanh(0.5*a)) - 0.5]
where z(n,k,j,i) is height (positive upwards) relative to the datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the sea surface (positive upwards) relative to the datum at gridpoint (n,j,i), s(k) is the dimensionless coordinate at vertical gridpoint (k), and depth(j,i) is the distance (a positive value) from the datum to the sea floor at horizontal gridpoint (j,i).
The constants a, b, and depth_c control the stretching.
The constants a and b are dimensionless, and depth_c must have units of length.
The format for the formula_terms attribute is
formula_terms = "s: var1 eta: var2 depth: var3 a: var4 b: var5 depth_c: var6"
The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1.
No standard_name has been defined for a, b or depth_c.
standard_name = "ocean_s_coordinate_g1"
- Definition
-
z(n,k,j,i) = S(k,j,i) + eta(n,j,i) * (1 + S(k,j,i) / depth(j,i))
where
S(k,j,i) = depth_c * s(k) + (depth(j,i) - depth_c) * C(k)
where z(n,k,j,i) is height, positive upwards, relative to ocean datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the ocean surface, positive upwards, relative to ocean datum at gridpoint (n,j,i), s(k) is the dimensionless coordinate at vertical gridpoint (k) with a range of -1 <= s(k) <= 0, s(0) corresponds to eta(n,j,i) whereas s(-1) corresponds to depth(j,i); C(k) is the dimensionless vertical coordinate stretching function at gridpoint (k) with a range of -1 <= C(k) <= 0, C(0) corresponds to eta(n,j,i) whereas C(-1) corresponds to depth(j,i); the constant depth_c, (positive value), is a critical depth controlling the stretching and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i).
The format for the formula_terms attribute is
formula_terms = "s: var1 C: var2 eta: var3 depth: var4 depth_c: var5"
The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1.
No standard_name has been defined for C or depth_c.
standard_name = "ocean_s_coordinate_g2"
- Definition
-
z(n,k,j,i) = eta(n,j,i) + (eta(n,j,i) + depth(j,i)) * S(k,j,i)
where
S(k,j,i) = (depth_c * s(k) + depth(j,i) * C(k)) / (depth_c + depth(j,i))
where z(n,k,j,i) is height, positive upwards, relative to ocean datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the ocean surface, positive upwards, relative to ocean datum at gridpoint (n,j,i), s(k) is the dimensionless coordinate at vertical gridpoint (k) with a range of -1 <= s(k) <= 0, S(0) corresponds to eta(n,j,i) whereas s(-1) corresponds to depth(j,i); C(k) is the dimensionless vertical coordinate stretching function at gridpoint (k) with a range of -1 <= C(k) <= 0, C(0) corresponds to eta(n,j,i) whereas C(-1) corresponds to depth(j,i); the constant depth_c, (positive value), is a critical depth controlling the stretching and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i).
The format for the formula_terms attribute is
formula_terms = "s: var1 C: var2 eta: var3 depth: var4 depth_c: var5"
The standard_names for eta and depth and the computed_standard_name must be one of the consistent sets shown in Table D.1.
No standard_name has been defined for C or depth_c.
The description of this type of parametric vertical coordinate is defective in version 1.8 and earlier versions of the standard, in that it does not state what values the vertical coordinate variable should contain. Therefore, in accordance with the rules, all versions of the standard before 1.9 are deprecated for datasets that use the "ocean sigma over z" coordinate.
standard_name = "ocean_sigma_z_coordinate"
- Definition
-
for levels k where sigma(k) has a defined value and zlev(k) is not defined:
z(n,k,j,i) = eta(n,j,i) + sigma(k)*(min(depth_c,depth(j,i))+eta(n,j,i))
for levels k where zlev(k) has a defined value and sigma(k) is not defined:
z(n,k,j,i) = zlev(k)
where z(n,k,j,i) is height, positive upwards, relative to ocean datum (e.g. mean sea level) at gridpoint (n,k,j,i), eta(n,j,i) is the height of the ocean surface, positive upwards, relative to ocean datum at gridpoint (n,j,i), and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i).
The parameter sigma(k) is defined only for the nsigma layers nearest the ocean surface, while zlev(k) is defined for the nlayer - nsigma deeper layers, where 0 <= nsigma <= nlayer and nlayer is the size of the dimension of the vertical coordinate variable.
Both sigma and zlev must have this dimension.
For any k, whichever of sigma(k) or zlev(k) is undefined must contain missing data, while the other must not.
The format for the formula_terms attribute is
formula_terms = "sigma: var1 eta: var2 depth: var3 depth_c: var4 nsigma: var5
zlev: var6"
The standard_names for eta, depth, zlev and the computed_standard_name must be one of the consistent sets shown in Table D.1.
The standard_name for sigma is ocean_sigma_coordinate.
No standard_name has been defined for depth_c or nsigma.
The nsigma parameter is deprecated and optional in formula_terms; if supplied, it must equal the number of elements of zlev which contain missing data.
The standard_name for the vertical coordinate variable is ocean_sigma_z_coordinate.
This variable should contain sigma(k)*depth_c for the layers where sigma is defined and zlev(k) for the other layers, with units of length.
The layers must be arranged so that the vertical coordinate variable contains a strictly monotonic set of indicative values for the heights of the levels relative to the datum, either increasing or decreasing, and the direction must be indicated by the positive attribute, in the usual way for a vertical coordinate variable.
standard_name = "ocean_double_sigma_coordinate"
- Definition
-
for k <= k_c: z(k,j,i)= sigma(k)*f(j,i) for k > k_c: z(k,j,i)= f(j,i) + (sigma(k)-1)*(depth(j,i)-f(j,i)) f(j,i)= 0.5*(z1+ z2) + 0.5*(z1-z2)* tanh(2*a/(z1-z2)*(depth(j,i)-href))
where z(k,j,i) is height (positive upwards) relative to the datum (e.g. mean sea level) at gridpoint (k,j,i), sigma(k) is the dimensionless coordinate at vertical gridpoint (k) for k <= k_c, and depth(j,i) is the distance (a positive value) from the datum to sea floor at horizontal gridpoint (j,i).
z1, z2, a, and href are constants with units of length.
The format for the formula_terms attribute is
formula_terms = "sigma: var1 depth: var2 z1: var3 z2: var4 a: var5 href: var6
k_c: var7"
The standard_name for depth and the computed_standard_name must be one of the consistent sets shown in Table D.1.
No standard_name has been defined for z1, z2, a, href or k_c.
| option | standard_name of computed dimensional coordinate | formula term name | standard_name of formula term |
|---|---|---|---|
1 |
altitude |
zlev |
altitude |
eta |
sea_surface_height_above_geoid |
||
depth |
sea_floor_depth_below_geoid |
||
2 |
height_above_geopotential_ datum |
zlev |
height_above_geopotential_datum |
eta |
sea_surface_height_above_ geopotential_datum |
||
depth |
sea_floor_depth_below_ geopotential_datum |
||
3 |
height_above_reference_ ellipsoid |
zlev |
height_above_reference_ellipsoid |
eta |
sea_surface_height_above_ reference_ellipsoid |
||
depth |
sea_floor_depth_below_ reference_ellipsoid |
||
4 |
height_above_mean_sea_ level |
zlev |
height_above_mean_sea_level |
eta |
sea_surface_height_above_mean_ sea_level |
||
depth |
sea_floor_depth_below_mean_ sea_level |