3 Introduction to Computational Fluid Dynamics and Ocean Modelling
95
which is given by
∂TKE
∂t
+ ¯
u j
∂TKE
∂x j
+ ˆ
u i ˆ
u j
∂ ¯
u i
∂x j
= −
∂ ˆ
u i ˆ
p
∂x i
−
g
ρ 0
ˆ
u i ˆ
ρδ i3 −
∂ ˆ
u i ˆ
u i ˆ
u j
∂x j
.
A closed set of equations results from parameterization of the small scale components in the equation, e.g., by setting
ˆ
u i ˆ
u j = A M
∂ ¯
u i
∂x j
,
ˆ
u i ˆ
ρ = A T
∂ ¯
ρ
∂x i
,
where A T is a horizontal eddy diffusivity coefficient, and parameterizing the triple
product term by some length scale l:
∂ ˆ
u i ˆ
u i ˆ
u j
∂x j
= ε = c ε
(TKE) 3
l
.
This approach forms the basis for popular schemes such as the k–l and k–ε schemes.
For high resolution simulations it may be feasible to simulate the large scale isopycnal turbulent motion and only parameterize the small scale motion. In this case
a large-eddy simulation (LES) scheme, such as the Smagorinsky model, may be
applied.
The procedure outlined above shows how parameterizations can be introduced
in the model equations. A simple parameterization may replace a subgrid-scale
variable with a constant value believed to be representative, whereas a complicated
scheme may involve a set of coupled differential equations. The requirements placed
upon the parameterization often depend on what processes are resolved in the numerical model, i.e., the parameterization is usually scale dependent, depending on
what physics needs to be represented. As modellers push for higher resolution models, the need for subgrid-scale parameterization does not diminish, but the formulations used for parameterization may very well need to be changed.
3.3.5 Classification of Ocean Models
Most of the ocean models in use today have their origin in the university or meteorological institute sector, and are developed and maintained by the user community.
Ocean models have traditionally been based on FD methods, where the main differences in model design have been related to the vertical grid. This picture is changing, as both FV and FE methods have started to appear in ocean models, and hybrid
methods for vertical grids are being developed. However, it is instructive to follow
the traditional approach and classify the models primarily in terms of the vertical
discretization method.
Figure 3.17 shows the three main categories of vertical grid configurations used
for ocean models. The z-coordinate system (also called geopotential coordinates) in
95
which is given by
∂TKE
∂t
+ ¯
u j
∂TKE
∂x j
+ ˆ
u i ˆ
u j
∂ ¯
u i
∂x j
= −
∂ ˆ
u i ˆ
p
∂x i
−
g
ρ 0
ˆ
u i ˆ
ρδ i3 −
∂ ˆ
u i ˆ
u i ˆ
u j
∂x j
.
A closed set of equations results from parameterization of the small scale components in the equation, e.g., by setting
ˆ
u i ˆ
u j = A M
∂ ¯
u i
∂x j
,
ˆ
u i ˆ
ρ = A T
∂ ¯
ρ
∂x i
,
where A T is a horizontal eddy diffusivity coefficient, and parameterizing the triple
product term by some length scale l:
∂ ˆ
u i ˆ
u i ˆ
u j
∂x j
= ε = c ε
(TKE) 3
l
.
This approach forms the basis for popular schemes such as the k–l and k–ε schemes.
For high resolution simulations it may be feasible to simulate the large scale isopycnal turbulent motion and only parameterize the small scale motion. In this case
a large-eddy simulation (LES) scheme, such as the Smagorinsky model, may be
applied.
The procedure outlined above shows how parameterizations can be introduced
in the model equations. A simple parameterization may replace a subgrid-scale
variable with a constant value believed to be representative, whereas a complicated
scheme may involve a set of coupled differential equations. The requirements placed
upon the parameterization often depend on what processes are resolved in the numerical model, i.e., the parameterization is usually scale dependent, depending on
what physics needs to be represented. As modellers push for higher resolution models, the need for subgrid-scale parameterization does not diminish, but the formulations used for parameterization may very well need to be changed.
3.3.5 Classification of Ocean Models
Most of the ocean models in use today have their origin in the university or meteorological institute sector, and are developed and maintained by the user community.
Ocean models have traditionally been based on FD methods, where the main differences in model design have been related to the vertical grid. This picture is changing, as both FV and FE methods have started to appear in ocean models, and hybrid
methods for vertical grids are being developed. However, it is instructive to follow
the traditional approach and classify the models primarily in terms of the vertical
discretization method.
Figure 3.17 shows the three main categories of vertical grid configurations used
for ocean models. The z-coordinate system (also called geopotential coordinates) in