Geoscience Reference
In-Depth Information
The unstable case is from Paulson (1970) and the stable case is from Webb (1970). In
neutral strati
∞
cation the potential temperature is constant, and, consequently, L =
. The
˕
m
,
˕
h
≡
ʨ
m
,
ʨ
h
≡
universal functions are then
1 and
0. The turbulent transfer coef
cients
are obtained as (e.g., Garrat 1992):
j
2
C
a
¼
ð
4
:
19a
Þ
ð
z
=
z
0
ÞW
m
ð
z
=
L
Þ
2
½
log
j
2
C
h
¼
ð
4
:
19b
Þ
½
log
ð
z
=
z
0
ÞW
m
ð
z
=
L
Þ
½
log
ð
z
=
z
T
ÞW
m
ð
z
=
L
Þ
j
2
C
e
¼
ð
4
:
19c
Þ
½
log
ð
z
=
z
0
ÞW
m
ð
z
=
L
Þ
½
log
ð
z
=
z
q
ÞW
m
ð
z
=
L
Þ
Obukhov boundary layer equations need to be
simultaneously solved. The transfer coef
In the general case,
the Monin
-
cients depend on the Monin
Obukhov length,
-
but in turn the Monin
Obukhov length depends on the transfer coef
cients. We have a
-
non-linear system of equations (Eqs.
4.9a
c
,
4.15
,
4.19a
c
), which can be solved by
-
-
iterative methods. Newton
s iteration method would work for this but other formulations
also exist (e.g., Launiainen and Vihma 1990).
'
4.1.5 Linearized Heat Flux
In analytic modelling and some other applications, a heat
fl
flux formula linear in the surface
temperature is required. Such form can be expressed as
Q
0
þ
Q
T
¼ k
o
þ
k
1
T
a
T
o
ð
Þ
ð
4
:
20
Þ
where the coef
cients k
o
and k
1
do not depend explicitly on T
o
. Equation (
4.20
) contains
the total solar and atmospheric heat
flux into the lake, with k
0
dominated by the radiation
balance and k
1
by the turbulent exchange. The formula has a simple interpretation: for
k
o
→
0 the surface layer becomes a low pass filter for the atmospheric temperature, and
for k
1
→
fl
0 the surface temperature becomes decoupled from the air temperature. The
coef
rst
approximation we can take k
1
as a constant and k
0
= k
0
(t). In the equilibrium of
Q
0
+ Q
T
= 0 we have T
o
= T
a
+ k
o
/k
1
.
The linear approximation is obtained as follows. The solar radiation, the incoming
terrestrial radiation from the atmosphere, and the heat
cients k
0
and k
1
have annual cycles (Table
4.1
), especially the former, and in the
flux from precipitation are purely
external factors and are absorbed in k
o
, while the sensible heat
fl
fl
flux is as such proportional
to T
a
−
T
o
and thus the proportionality coef
cient simply goes to k
1
. Outgoing terrestrial
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