Geoscience Reference
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tells us only the magnitude of the stratification and nothing
about the detail of water column structure. If we want to predict, or hindcast, the
evolution of the density profile under the influence of heating and stirring, then we
need to formulate a more explicit model. The simplest model of this kind (Simpson
and Bowers,
1984
) uses the same energy arguments embodied in
Equation (6.22)
.Itis
based on a water column consisting of two mixed layers: a surface layer, heated and
stirred by the atmosphere, and a bottom layer, stirred by the tides, with the two being
separated by a high gradient region. Such a system is closely analogous to the
laboratory tank model discussed in
Section 6.1.1
.
To simplify the problem we will assume that the vertical fluxes in the heating cycle
tend to greatly exceed the heat inputs to the water column resulting from the
horizontal transport (see
Section 2.2
). This allows us to model the one-dimensional
vertical structure of the water column without worrying about horizontal gradients.
We shall also make a number of other assumptions about the surface heat exchange
processes to simplify the model formulation. From
Equation (2.6)
we represent the
net heat flux into the water column, Q
i
, as the difference between the solar radiation
incident at the top of the atmosphere, Q
s
(modified by the atmosphere's albedo, A)
and the combined loss term Q
u
:
On the other hand,
F
Q
i
¼
Q
s
ð
1
A
Þ
Q
u
:
ð
6
:
27
Þ
The heat supplied to the sea surface by solar radiation, Q
s
(1
A), is divided between
a fraction (55%) which is assumed to be absorbed at the surface (i.e. in the uppermost
layer) and the remainder (45%) which is absorbed exponentially in the underlying
layers according to e
K
av
z
where K
av
∼
0.2 m
1
is an attenuation coefficient. Splitting
the heat input in this way is a simple approach to including the wavelength-
dependence of light attenuation, with the red end of the spectrum being absorbed
far more rapidly than the blue end. All of the heat loss term is extracted from the
immediate surface layer since the processes involved all draw heat from a very thin
micro-layer at the surface. Q
u
is a sensitive function of the difference between sea
surface temperature T
s
and the air temperature T
a
and may be determined providing
we also know the wind speed W, air pressure, cloud cover and the relative humidity
(e.g. Gill,
1982
; Sharples et al.,
2006
). These driving meteorological parameters may
conveniently be taken from sinusoidal fits to observational time series (Sharples,
2008
). Notice that, since the heat flux is directly influenced by the surface tempera-
ture, there is a degree of feedback operating to control heat input.
As in the
model, we assume that the power available for mixing by wind and
tidal stirring can be written as
F
3
e
s
k
s
a
W
3
e
s
P
W
¼
;
eP
T
¼
ek
b
0
j^
u
j
ð
6
:
28
Þ
where
u is the depth-mean tidal velocity. P
w
and P
T
are assumed to be concentrated
in the top and bottom layers respectively.
The operation of the numerical scheme for the two-layer model (see TML model at
website) involves the application of successive heating and stirring inputs to modify
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