Appendix D. Dimensionless Vertical Coordinates

The definitions given here allow an application to compute dimensional coordinate values from the dimensionless ones 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 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.

Example D.1.  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"
              


Example D.2.  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"
              


Example D.3.  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.


Example D.4.  Atmosphere hybrid height coordinate

standard_name = "atmosphere_hybrid_height_coordinate"
        

Definition:
z(n,k,j,i) = tau(k)*zsurface(n,j,i) + eta(k)*ztop
              
z(n,k,j,i) = a(k) + b(k)*orog(n,j,i)
              

where z(n,k,j,i) is the height above the geoid (approximately mean sea level) at gridpoint (k,j,i) and time (n), zsurface(n,j,i) orog(n,j,i) is the height of the surface above mean sea level the geoid at (j,i) and time (n), ztop is the height of the top of the model, and tau(k) and eta(k) are the dimensionless coordinates 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 = "tau: var1 eta: var2 ztop: var3 zsurface: var4"
        

formula_terms = "a: var1 b: var2 orog: var3"
        

The hybrid height coordinate for level k is defined as eta(k)*ztop. There is no dimensionless hybrid height coordinate. The hybrid height is best approximated as a(k) if a level-dependent constant is needed.


Example D.5.  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 geoid (approximately mean sea level) at gridpoint (k,j,i) and time (n), ztop is the height of the top of the model, 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 and small parts of the topography. 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 hybrid height coordinate for level k is defined as a(k)*ztop.


Example D.6.  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 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), sigma(k) is the dimensionless coordinate at vertical gridpoint (k), 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 = "sigma: var1 eta: var2 depth: var3"
        


Example D.7.  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)

  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 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), and depth(j,i) is the distance from ocean datum to sea floor (positive value) 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"
        


Example D.8.  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 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), sigma(k) is the dimensionless coordinate at vertical gridpoint (k) for k <= nsigma, and depth(j,i) is the distance from ocean datum to sea floor (positive value) at horizontal gridpoint (j,i). Above depth depth_c there are nsigma layers.

The format for the formula_terms attribute is

formula_terms = "sigma: var1 eta: var2 depth: var3 depth_c: var4 nsigma: var5
                zlev: var6"
        


Example D.9.  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 ocean 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 from ocean datum to sea floor (positive value) 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"