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dominate,
ʻ
T
am
+ r
≈ ʻ
T
am
, and the lake freezes if the cold period is at least twice the lake
ʻ
−
1
; when radiational losses dominate, the condition is asymptoti-
response time, t
C
>2
k
T
am
=
p
. Equation (
7.10
) works more generally when the temperature
reference is changed to the freezing point instead of 0
t
C
[
2
k
1
cally
C.
Next, consider the fall cooling process. In linear atmospheric cooling by the rate
°
ʱ
,
T
a
=
−ʱ
t, the water temperature is
k
1
a þ D
T
þ
e
k
t
T
ðÞ
T
ðÞ
¼T
a
ðÞþ
1
e
k
t
ð
7
:
11
Þ
The water temperature is thus the air temperature plus a weighted average of the
surface heat balance effect and the initial temperature. The inverse of the parameter
ʻ
represents the thermal response time scale of the lake system. The asymptotic solution for
ʻ
ʻ
−
1
t
≫
1 is that the water temperature is above the air temperature by
ʱ
+
ʔ
T, where the
first term represents the lag and the second term represents the shift. In Finland autumn,
ʱ
*
C month
−
1
and then for H
ʻ
−
1
5
°
*
10 m we have
ʱ
*
5
°
C; and as shown above,
ʔ
C.
In deep lakes the linear atmospheric cooling assumption is not good, since the cooling
period is long. Periodic air temperature forcing
T
*
−
2
°
T
a
¼ T
a
þ D
T
a
sin
xðÞ
, where T
a
is the
mean air temperature,
ʔ
T is the air temperature amplitude and
ˉ
is the frequency, gives
the solution
k
k
2
þ x
2
arctan
x
k
T ¼ T
a
þ D
T
þ
p
D
T
a
sin
x
t
u
ð
Þ;
u
¼
ð
7
:
12
Þ
year
−
1
, and thus
For the annual air temperature cycle,
ˉ
=2
ˀ
ˉ
=
ʻ
when the mixed
layer depth is H = 20 m. The amplitude of the air temperature sine wave
ʔ
T
a
is damped in
the mixed layer as shown by the formula. In deep lakes,
ˉ ≫ ʻ
, the forcing cycles are
ˉ
-
1
and phase shift is asymptotically a quarter cycle, while in
damped proportional to
shallow lakes,
ˉ ≪ ʻ
, the surface water temperature follows the air temperature with a
shift
T and with an asymptotically vanishing lag. Equation (
7.12
) also gives the freezing
condition T =0
ʔ
°
C:
D
T
a
1
þ
[
T
a
D
T
q
ð
Þ
ð
7
:
13
Þ
k
2
x
Thus, if if
a
[
0
C and
D
T
a
[
T
a
, shallow lakes freeze but deep enough lakes do not.
Lake Shikotsu, close to Sapporo in Hokkaido has the mean depth of 265 m and does not
freeze. The mean air temperature is about 5
°
C and the amplitude is 14
°
C, and for
H = 265 m the left-hand side is 1.02
°
C not satisfying the condition (
7.13
).
Example 7.3
. Autumn cooling model (Rodhe 1952). The slab model was used in the past
for forecasting the autumn cooling. Rodhe (1952) considered just the
ʻ
-term in Eq. (
7.8
),
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