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coordinate is positive upward. The coef
cient K characterizes the turbulent
fl
flow, scaling
as K
u
*
'
where
'
and u
*
are the characteristic turbulent length and
fl
fluctuation scales. In
*
the surface layer the
fl
fluxes are assumed constant. Therefore the
fl
flow can be simply
s mixing length hypothesis,
1
the size of
turbulent eddies scales with the distance from the boundary,
aligned along the x-axis, and according to Prandtl
'
' ∝
z.
Neutral strati
cation provides the simplest case where the bulk transfer coef
cients are
fixed. Then the transfer coef
cients and roughness lengths are related by
¼
¼
¼
j
C
a
p
log
z
z
0
C
H
p
log
C
e
p
log
z
z
q
z
z
T
ð
4
:
12
Þ
where
n constant, and the roughness lengths are z
0
for momentum, z
T
for temperature, and z
q
for humidity. For snow, ice and water surfaces typical values are
C
D
, C
H
, C
E
≈
ʺ
= 0.4 is von K
á
rm
á
10
−
3
(z = 10 m), corresponding to the aerodynamic roughness length
of 0.097 mm. These lengths result from the surface boundary conditions and they are
related to the geometrical surface roughness but are not equivalent, and the aerodynamic
roughness parameters in general also depend on the
1.2
×
flow characteristics.
The temperature difference between lake surface and air may have any sign, and the
sensible heat flux is usually within about
±
50 W m
−
2
. The latent heat exchange takes place
as evaporation and sublimation (heat loss) or condensation and deposition (heat gain) at
The latent heat
fl
flux; in
general, in cold season it is weaker but it is stronger in warm season. Evaporation of 1 mm
water or sublimation of 1 mm ice in 1 day corresponds to latent heat loss, respectively, by
28.8 or 30.0 W m
−
2
.
In the general case, when the stability of the strati
fl
flux is mostly negative and in magnitude close to the sensible heat
fl
cation of the atmospheric surface
layer is accounted for, the transfer coef
cients are interdependent and the transfer equa-
tions must be simultaneously solved. The stability of the strati
cation can be expressed by
the (gradient) Richardson number or the bulk Richardson number, respectively, as
Ri ¼
h
0
@
h
=@
z
ð
4
:
13a
Þ
ð@
u
=@
z
Þ
2
Ri
B
¼
h
0
z
ð
h
v
h
v0
Þ
ð
4
:
13b
Þ
ð
u
u
0
Þ
2
1
According to Prandtl
'
s mixing length hypothesis, turbulent transfer processes are described by a
characteristic length-scale over which the turbulent eddies mix fluid properties.
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