appd.adoc

Appendix A: Parametric Vertical Coordinates

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.

Atmosphere natural log pressure coordinate

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.

Atmosphere sigma coordinate

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.

Atmosphere hybrid sigma pressure coordinate

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.

Atmosphere hybrid height coordinate

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.

Atmosphere smooth level vertical (SLEVE) 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.

Ocean sigma coordinate

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.

Ocean s-coordinate

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 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.

Ocean s-coordinate, generic form 1

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"

Ocean s-coordinate, generic form 2

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"

Ocean sigma over z coordinate

standard_name = "ocean_sigma_z_coordinate"

Definition

for k <= nsigma:

  z(n,k,j,i) = eta(n,j,i) + sigma(k)*(min(depth_c,depth(j,i))+eta(n,j,i))

for k > nsigma:

  z(n,k,j,i) = zlev(k)

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) for k <= nsigma, and depth(j,i) is the distance (a positive value) from the datum to the sea floor at horizontal gridpoint (j,i). Above depth depth_c there are nsigma layers and below this depth the levels are surfaces of constant height zlev (positive upwards) relative to the datum.

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. No standard_name has been defined for depth_c or nsigma.

Ocean double sigma coordinate

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.

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.

Table D.1. Consistent sets of values for the standard_names of formula terms and the computed_standard_name needed in defining the ocean sigma coordinate, the ocean s-coordinate, the ocean_sigma over z coordinate, and the ocean double sigma coordinate.

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