Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 100

0.00500

0.00100

0.00050

0.00010

0.00005

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Richardson number

0.2 0.4 0.6 0.8 1.0

Fig. 4.3 The dependence of the transfer coefficient C H on the stability of stratification. Surface
roughness is 0.1 mm (lower curve) and 1 mm (upper curve), and the reference level is 10 m

In the general, strati ed conditions, the distribution of velocity, potential temperature

and speci c humidity in the boundary layer can be expressed as functions of their

characteristic scales, surface roughness, and the Monin

Obukhov length. The differential

-

and integrated forms are:

h

W m

i

u ðÞ ¼ u j

@ z ¼ u j z / m

@ u z

L

log z 0

z
L

ð 4 : 16a Þ

h ðÞ ¼ h s þ

log T W h

@h h j z z

L

h j z

L @ z ¼

/ h ð 4 : 16b Þ

h

W h

i

q ðÞ ¼q s þ q j

@ q

@ z ¼ q j z / h

z
L

z
z q

z
L

log ð 4 : 16c Þ

where u * , ʸ * and q * are the characteristic friction layer velocity, temperature and humidity

scales, and

ʨ h are the so-called universal functions of the boundary layer.
It is quite common to use the same universal functions for temperature and humidity (see
Andreas 1998). By direct integration, the integral and differential forms are related by

˕ m , ˕ h and ʨ m ,

Z Z

f f

1 / m ð f 0 Þ
f 0

1 / h ðf 0 Þ
f 0

d f 0 ; W h ðfÞ ¼ d f 0 W m ðfÞ ¼ ð 4 : 17 Þ

0 0

where = z/L. Common forms for the universal functions are (e.g., Vihma 1995; Andreas

et al. 1998):

ʶ

4 q 1

1 16 f

q

1
1 16 f

Unstable ðf \ 0 Þ : / m ðÞ ¼ and / h ðÞ ¼ ð 4 : 18a Þ

Neutral ðf ¼ 0 Þ : / m ðÞ ¼ / h ðÞ ¼1 ð 4 : 18b Þ

Stable ðf [ 0 Þ : / m ðÞ ¼ / h ðÞ ¼1 þ c s f; c s ¼ constant 5 ð 4 : 18c Þ