Geoscience Reference
In-Depth Information
ʻ
-
1
days from the air temperature zero
Thus, if
D
T
0
;
the freezing date is delayed by
downcrossing. Shallow lakes (H
≈
5 m) freeze after a 1
2 week cold period but deeper
-
lakes need more than 1 month.
Heat
flux from bottom sediments is important in shallow lakes. Falkenmark (1973)
reported that the sediment heat storage of a shallow Lake Velen (17 m mean depth) in
Sweden was at maximum 110 MJ m
−
2
. This would mean winter heat
fl
fluxes of the order of
5Wm
−
2
from the bottom sediments. Falkenmark (1973) regarded the sediment heat
fl
fl
ux
as a very signi
cant factor for heating and also due to triggering slow density currents
along the sloping bottom.
Taking the bottom temperature as a constant, a new forcing term K
b
(T
b
−
T) is added to
the one-layer model, and the solution becomes easy. For a more complete treatment, an
equation is needed for the bottom temperature also:
z¼
H
dT
b
dt
¼ e
j
H
Q
si
þ
K
V
H
b
@
T
ð
7
:
14
Þ
@
z
where H
b
is a representative thickness of the active bottom sediment layer.
7.1.4 Mixed Layer Models
Extending slab models to have the depth of the mixed layer as a dependent model variable
takes us to mixed layer models, H = H(t) (see Niiler and Kraus 1977). For a deepening
mixed layer, the rate w
e
=dH/dt is called the entrainment
3
rate. These models do not
include horizontal effects. To determine the evolution of the mixed layer depth brings the
mechanical energy budget to the model and then also the water velocity needs to be
solved. The model equations are:
dt
¼
Q
0
þ
1
e
j
H
ð
Þ
Q
T
w
e
D
T
H
dT
ð
7
:
15a
Þ
q
0
c
0
H
dU
dt
¼
f k
U
þ
s
a
s
H
ð
7
:
15b
Þ
q
0
H
Q
s
¼ 2mu
3
þ
H
þ
H
j
w
e
c
i
sU
2
½
ð
1
þ
n
Þ
B
0
1
n
ð
Þ
B
jj
ð
7
:
15c
Þ
2
T is the temperature difference
across the pycnocline at the bottom of the mixed layer, f is the Coriolis parameter,
The
first two equations are straightforward. There
ʔ
U
is the
horizontal water velocity, and
τ
a
and
τ
H
are the surface and mixed layer bottom stresses.
3
Erosion of the underlying water layer by the mixed layer turbulence.
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