Geoscience Reference
In-Depth Information
Figure 7.1
Variation of the stability
functions in response to the
Richardson number, Ri. Remember
that positive Ri occurs for stable
stratification, negative Ri implies
convective instability.
1.6
S
H
1.2
S
M
0.8
0.4
0.0
-0.8
-0.4
0.0
0.4
Ri
stability functions (Galperin, Kantha, et al.,
1988
) which is based on the assumption
of local equilibrium (Equation 4.60) and leads to the dependence on Richardson
number illustrated in
Fig. 7.1
. The sharp drop in S
M
and S
H
as Ri increases to a
positive critical value is characteristic of most proposed closure schemes (e.g. Kantha
The length scale L can be thought of as some measure representing the size of the
turbulent eddies. We shall use a simple algebraic representation in which L is
controlled only by distance from the boundaries:
1
=
2
Þ
z
h
L
¼
k z
ð
þ
h
:
ð
7
:
9
Þ
Here k is von Karman's constant (k
0.41); an important constraint on the behav-
iour of the turbulent lengthscale is that it should be scaled by k near the water column
boundaries. With this particular formulation, the eddy scale increases with distance
from the top and bottom boundaries, reaching a maximum of L
¼
0.38kh at a height
above the seabed of 2h
3.
The set of
equations (7.1)
to
(7.9)
constitutes our TC model describing the physical
processes acting in the water column at one location in the horizontal. It represents a
much more complete account of the processes than was possible in the mixing model
in
Chapter 6
, and it incorporates the various feedback mechanisms that are inherent
in the interaction of flow and density structure. These feedbacks and the interde-
pendence of the variables involved are evident in
Fig. 7.2
which illustrates the logical
flow used in the integration of the equations within the numerical model. At each
time step the momentum, diffusion and TKE equations are stepped forward to
generate new values for u, v, r and q which are used to determine the vertical profile
of buoyancy frequency N
2
=
g
@
@
z
and the velocity shear, which then determine S
M
and S
H
. These latter parameters are then combined with q and the length scale L to
determine vertical profiles of N
z
and K
z
in readiness for the next time step.
Wind stress, surface heat exchange parameters and the initial density structure are
obtained from observations. The tidal forcing can also be obtained from observations
of bottom pressure gradient or extracted from a tidal model. A convenient alternative
is to drive the model with surface slopes estimated from knowledge of the near surface
over three tidal cycles is illustrated in
Fig. 7.3a
for a seasonally stratified location in the
¼
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