3 Introduction to Computational Fluid Dynamics and Ocean Modelling
93
much smaller than the diffusion of heat. For instance, if a volume of cold surface
water is displaced downwards, after being heated by the warmer water below, it will
attain a lower density than in its initial state, therefore rise into the colder water
above to be cooled down, and in this way create a persistent oscillating motion.
3.3.4 Subgrid-Scale Parameterization
Due to the finite resolution of spatial grids and the finite time integration step, and
practical requirements that these should not be too small, there will always be unresolved processes that are taking place on the subgrid scale. Specifically, we have
seen that viscous effects work on a typical length scale of L < 10 −2 m, and the same
is true for thermodiffusivity and salt diffusivity. All this would be fine from a modelling point of view if these small scale effects did not influence the large scale ocean
dynamics. Unfortunately, this is not the case. Viscous and diffusive effects cannot
be ignored for several reasons. The global inputs of momentum, heat and vorticity
require a sink in order to avoid unphysical accumulation of energy. Important driving forces, such as wind stress, are frictional in origin, and bottom friction may be a
significant factor in localized regions, in particular for coastal water. Since it is clear
that the models cannot resolve these processes, such effects must be modelled with
the use of subgrid-scale parameterization.
One approach to obtain such parameterization dates back to the 19th century
and is due to Reynolds, who suggested to write all hydrodynamic variables as a
sum of a large-scale (or long-period) component and a small-scale (or short-period)
component
p = ¯
p + ˆ
p,
u = ¯
u + ˆ
u, etc.,
where ( ¯ · ) represents some averaging operator, e.g., time averaging, and (ˆ ·) represents the deviation from the average. Ideally, these scales should be clearly separated
in spectral space, a requirement that unfortunately is not easily satisfied in the ocean
where normally the spectrum is continuous. Starting from the inviscid momentum
equation written in tensor notation
∂u i
∂t
+ u j
∂u i
∂x j
− f f ij 3 u j = −
1
ρ 0
∂p
∂x i
−
ρg
ρ 0
δ i3 ,
where i, j ∈ {1, 2, 3}, and we use the Kronecker’s delta
δ ij =
1 for i = j,
0 for i = j,
define ij k as
ij k =
⎧
⎨
⎩
1
for cyclic order of indices
−1 for anticyclic order of indices
0
for two or more identical indices
93
much smaller than the diffusion of heat. For instance, if a volume of cold surface
water is displaced downwards, after being heated by the warmer water below, it will
attain a lower density than in its initial state, therefore rise into the colder water
above to be cooled down, and in this way create a persistent oscillating motion.
3.3.4 Subgrid-Scale Parameterization
Due to the finite resolution of spatial grids and the finite time integration step, and
practical requirements that these should not be too small, there will always be unresolved processes that are taking place on the subgrid scale. Specifically, we have
seen that viscous effects work on a typical length scale of L < 10 −2 m, and the same
is true for thermodiffusivity and salt diffusivity. All this would be fine from a modelling point of view if these small scale effects did not influence the large scale ocean
dynamics. Unfortunately, this is not the case. Viscous and diffusive effects cannot
be ignored for several reasons. The global inputs of momentum, heat and vorticity
require a sink in order to avoid unphysical accumulation of energy. Important driving forces, such as wind stress, are frictional in origin, and bottom friction may be a
significant factor in localized regions, in particular for coastal water. Since it is clear
that the models cannot resolve these processes, such effects must be modelled with
the use of subgrid-scale parameterization.
One approach to obtain such parameterization dates back to the 19th century
and is due to Reynolds, who suggested to write all hydrodynamic variables as a
sum of a large-scale (or long-period) component and a small-scale (or short-period)
component
p = ¯
p + ˆ
p,
u = ¯
u + ˆ
u, etc.,
where ( ¯ · ) represents some averaging operator, e.g., time averaging, and (ˆ ·) represents the deviation from the average. Ideally, these scales should be clearly separated
in spectral space, a requirement that unfortunately is not easily satisfied in the ocean
where normally the spectrum is continuous. Starting from the inviscid momentum
equation written in tensor notation
∂u i
∂t
+ u j
∂u i
∂x j
− f f ij 3 u j = −
1
ρ 0
∂p
∂x i
−
ρg
ρ 0
δ i3 ,
where i, j ∈ {1, 2, 3}, and we use the Kronecker’s delta
δ ij =
1 for i = j,
0 for i = j,
define ij k as
ij k =
⎧
⎨
⎩
1
for cyclic order of indices
−1 for anticyclic order of indices
0
for two or more identical indices