SOME OCEAN MODEL FUNDAMENTALS
67
Coordinate Definition
Range
g ρ z,s
pressure
p
pa ≤ p ≤ p b
−1
p-deviation
p
= p − pa
0 ≤ p
≤ p b − pa −1
pstar
p
∗ = p
o
b (p − pa)/(p b − pa) 0 ≤ p
∗ ≤ p
o
b
−(p b − pa)/p
o
b
p-sigma
σ = (p − pa)/(p b − pa)
0 ≤ σ ≤ 1
−(p b − pa)
Table 2.2. Table of vertical coordinates based on pressure. These coordinates are
naturally used for non-Boussinesq dynamics.
is either a constant or a two-dimensional field. In contrast, for depth
based models ρ z ,s is proportional to the three-dimensional in situ density
ρ, thus necessitating special algorithmic treatment for non-Boussinesq
z-models (see the discussions in Greatbatch and McDougall, 2003 and
Griffies, 2004). Table 2.2 summarizes some pressure based coordinates.
As Table 2.2 reveals, the specific thickness z ,s is negative for the
pressure-based coordinates, whereas it is positive for the depth-based
coordinate (Table 2.1). The sign change arises since upward motion in
a fluid column increases the geopotential coordinate z yet decreases the
hydrostatic pressure p. To establish a convention, we assume that the
thickness of a grid cell in z space is always positive
dz = z ,s ds > 0
(115)
as is the case in the conventional z-models. With z ,s < 0 for the pressurebased vertical coordinates, the thickness of grid cells in s space is negative
ds < 0
for pressure-based coordinates with z ,s < 0.
(116)
6.3
Isopycnal coordinates
Isopycnal models discretize the vertical into potential density classes.
Some key advantages of isopycnal models are the following:
Tracer transport in the ocean interior is well represented due to the
natural ability of these models to maintain water mass properties.
The bottom topography is represented in a piecewise linear fashion, hence avoiding the need to distinguish bottom from side as
traditionally done with z-models.
In some cases, flow near topographically critical regions, such as
overflows, can be well resolved by isopycnal models due to the
natural tendency of the coordinate surfaces to become refined in
these regions.
67
Coordinate Definition
Range
g ρ z,s
pressure
p
pa ≤ p ≤ p b
−1
p-deviation
p
= p − pa
0 ≤ p
≤ p b − pa −1
pstar
p
∗ = p
o
b (p − pa)/(p b − pa) 0 ≤ p
∗ ≤ p
o
b
−(p b − pa)/p
o
b
p-sigma
σ = (p − pa)/(p b − pa)
0 ≤ σ ≤ 1
−(p b − pa)
Table 2.2. Table of vertical coordinates based on pressure. These coordinates are
naturally used for non-Boussinesq dynamics.
is either a constant or a two-dimensional field. In contrast, for depth
based models ρ z ,s is proportional to the three-dimensional in situ density
ρ, thus necessitating special algorithmic treatment for non-Boussinesq
z-models (see the discussions in Greatbatch and McDougall, 2003 and
Griffies, 2004). Table 2.2 summarizes some pressure based coordinates.
As Table 2.2 reveals, the specific thickness z ,s is negative for the
pressure-based coordinates, whereas it is positive for the depth-based
coordinate (Table 2.1). The sign change arises since upward motion in
a fluid column increases the geopotential coordinate z yet decreases the
hydrostatic pressure p. To establish a convention, we assume that the
thickness of a grid cell in z space is always positive
dz = z ,s ds > 0
(115)
as is the case in the conventional z-models. With z ,s < 0 for the pressurebased vertical coordinates, the thickness of grid cells in s space is negative
ds < 0
for pressure-based coordinates with z ,s < 0.
(116)
6.3
Isopycnal coordinates
Isopycnal models discretize the vertical into potential density classes.
Some key advantages of isopycnal models are the following:
Tracer transport in the ocean interior is well represented due to the
natural ability of these models to maintain water mass properties.
The bottom topography is represented in a piecewise linear fashion, hence avoiding the need to distinguish bottom from side as
traditionally done with z-models.
In some cases, flow near topographically critical regions, such as
overflows, can be well resolved by isopycnal models due to the
natural tendency of the coordinate surfaces to become refined in
these regions.