Geoscience Reference
In-Depth Information
þ
@e
@
t
¼
@
r
e
@
k
e
k
ð
c
1
P
s
þ
c
3
P
b
c
2
e
Þ
ð
7
:
16e
Þ
@
z
@
z
where P
s
and P
b
are the shear and buoyancy production of turbulence, and the coef
cients
r
k
; r
e
;
c
1
;
c
2
and c
e
are model constants. The production terms are
"
#
2
þ
@
v
@
z
2
@
u
@
z
;
P
b
¼ K
q
0
@
q
@
z
P
s
¼ K
ð
7
:
16f
Þ
where
c
1
e
;
c
2
e
and c
3
e
are model constants. The
k
e
model has thus six parameters, and
with their standard values they are:
C
l
¼ 0
:
09
, c
1
= 1.44, c
2
= 1.92, c
3
= 0.8,
r
k
¼ 1
:
0
and
r
e
¼ 1
:
3
.
The most uncertain part of this
k
e
model is the equation for the rate of dissipation of
turbulent kinetic energy, which involves several parameters. The mixed layer physics with
its depth comes out fairly well in the model but deeper, below the thermocline, horizontal
processes become relatively speaking more signi
cant. To reach the observed level of
mixing in the lower layer, so-called
functions have been added into the
model for tuning. One-dimensional approach thus has limitations in that advection cannot
be properly accounted for and interactions between the coastal zone and central basins are
not solved. In the cooling process, in particular, advection and sinking of cold water from
shallow shore areas to central basins complicates the one-dimensional picture.
Based on this 2nd order turbulence closure, a programme package
'
deep mixing
'
has been
constructed to be utilized for several kinds of applications (Omstedt 2011). Cooling of
lake waters in autumn has been widely examined with this model (Sahlberg 1983). In
several basins good results have been obtained, and furthermore a group of one-dimen-
sional models has been used as the basis of a connected box-model network. Figure
7.5
shows an example of the model simulation. The whole 50-m water column cools, and at
temperatures below 4
'
PROBE
'
°
C surface layer starts to stratify and bottom water reaches tem-
C. The heat loss from the lake is at highest 300 W m
−
2
and almost
continuously positive.
peratures down to 2
°
7.2
Thermal Structure and Circulation Under Ice-Cover
7.2.1 Water Body During the Ice Season
The physics of lakes is different in quality in the presence of full ice cover as compared
with open water season (Fig.
7.6
). A complete ice cover is a rigid lid, which almost closes
the interaction between the water body and the atmosphere. The wind stress does not go
through the ice, and therefore mechanical forcing is limited to surface pressure variations.
Exchange of matter, including water, between the lake water body and atmosphere does
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