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"
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"
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"
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"
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"
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"
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"
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"
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"
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"