SOME OCEAN MODEL FUNDAMENTALS
59
The equations describing a Boussinesq fluid are derived from the nonBoussinesq set derived in Sections 3 and 4 by replacing all appearances
of in situ density ρ by a constant density ρ o , except when density is used
to compute the buoyancy forces arising from gravity. The density ρ o is
a representative density of the ocean fluid, such as ρ o = 1035 kg m −3 .
For much of the ocean, the in situ density varies less than 2% from this
value (see page 47 of Gill, 1982).
Depth coordinate.
With a free surface, the vertical domain over
which the z-coordinate s = z ranges is given by the time dependent
interval −H ≤ z ≤ η. Consequently, the sum of the vertical grid cell
increments equals to the total depth of the column
k dz = H + η. The
trivial specific thickness z ,s = 1 simplifies the Boussinesq budgets.
The depth coordinate is useful for many purposes in global climate
modelling, and models based on depth are the most popular ocean climate models. Their advantages include the following.
Simple numerical methods have been successfully used in this framework.
The horizontal pressure gradient can be easily represented in an
accurate manner.
The equation of state for ocean water can be accurately represented
in a straightforward manner (e.g., McDougall et al., 2003).
The upper ocean mixed layer is well parameterized using a zcoordinate.
Unfortunately, these models have some well known disadvantages, which
include the following.
Representation of tracer transport within the quasi-adiabatic interior is cumbersome, with problems becoming more egregious as
mesoscale eddies are admitted (Griffies et al., 2000b).
Representation and parameterization of bottom boundary layer
processes and flow are unnatural.
Grid cells have static vertical increments ds = dz when s = z, except
for the top. At the top, ∂ t (dz) = η ,t . The time dependent vertical
range of the coordinate slightly complicates a numerical treatment of the
surface cell in z-models (see Griffies et al., 2001 for details of one such
treatment). More problematic, however, is the possibility of a vanishing
top grid cell. That is, the surface cell can be lost (i.e., can become
dry) if the free surface depresses below the depth of the top grid cell’s
59
The equations describing a Boussinesq fluid are derived from the nonBoussinesq set derived in Sections 3 and 4 by replacing all appearances
of in situ density ρ by a constant density ρ o , except when density is used
to compute the buoyancy forces arising from gravity. The density ρ o is
a representative density of the ocean fluid, such as ρ o = 1035 kg m −3 .
For much of the ocean, the in situ density varies less than 2% from this
value (see page 47 of Gill, 1982).
Depth coordinate.
With a free surface, the vertical domain over
which the z-coordinate s = z ranges is given by the time dependent
interval −H ≤ z ≤ η. Consequently, the sum of the vertical grid cell
increments equals to the total depth of the column
k dz = H + η. The
trivial specific thickness z ,s = 1 simplifies the Boussinesq budgets.
The depth coordinate is useful for many purposes in global climate
modelling, and models based on depth are the most popular ocean climate models. Their advantages include the following.
Simple numerical methods have been successfully used in this framework.
The horizontal pressure gradient can be easily represented in an
accurate manner.
The equation of state for ocean water can be accurately represented
in a straightforward manner (e.g., McDougall et al., 2003).
The upper ocean mixed layer is well parameterized using a zcoordinate.
Unfortunately, these models have some well known disadvantages, which
include the following.
Representation of tracer transport within the quasi-adiabatic interior is cumbersome, with problems becoming more egregious as
mesoscale eddies are admitted (Griffies et al., 2000b).
Representation and parameterization of bottom boundary layer
processes and flow are unnatural.
Grid cells have static vertical increments ds = dz when s = z, except
for the top. At the top, ∂ t (dz) = η ,t . The time dependent vertical
range of the coordinate slightly complicates a numerical treatment of the
surface cell in z-models (see Griffies et al., 2001 for details of one such
treatment). More problematic, however, is the possibility of a vanishing
top grid cell. That is, the surface cell can be lost (i.e., can become
dry) if the free surface depresses below the depth of the top grid cell’s