Structure and Properties of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 77

3.4 Light Transfer Through Ice Cover

3.4.1 Solar Radiation

The Stefan Boltzmann law for the total power of black body radiation was introduced in

Sect. 2.2.5. It is based on the Planck

-

is law, which provides the distribution of the radi-
ational power with respect to the wavelength:

'

1

1

Þ ¼ 8 p hc 2
k 5

hc
k k B T

I k; T ð exp ð 3 : 9 Þ

where ʻ is wavelength, h is Planck ' is constant, c is velocity of light in vacuum, and k B is

Boltzmann s constant. The distribution depends on the temperature of the body, with the

level of radiation increasing with the temperature and the peak wavelength decreasing
with the temperature. The Sun radiates almost as a black body at the temperature of its
outer surface, 5,900 K (Fig. 3.15 ).

'

Example 3.5 ne the brightness

temperature and the colour temperature. The brightness temperature T B of a surface is
equal to the temperature of the black body that would provide the same radiance at the
given wavelength: I( ʻ , T B )=I ′ ( ʻ , T), where I ′ is the spectral power density of the observed
radiation from the given body. Black body radiates with the maximum power, and
therefore always T ≥ T B . The colour temperature T c is the temperature of an ideal black
body radiator, which has the spectral energy distribution closest to the object in concern. It
can be determined by measurement of power and absorbance A at two wavelengths:

. The black body radiation is used as a reference to de

T 1

1 k

hc ðk 1 k 2 Þ

log A ð k 1 ; T Þ
A ðk 2 ; T Þ

T c ¼

ð 3 : 10 Þ

Fig. 3.15 Solar radiation
spectrum at the top of the
atmosphere and at the sea
surface level. Atmospheric
absorption bands due to ozone
(O 3 ), oxygen (O 2 ), water
vapour and carbon dioxide are
illustrated in the solar radiation
band

2.5

UV Visible Infrared

2

Sunlight at top of the atmosphere

1.5

5250° blackbody spectrum

1

H 2

Radiation at sea level

0.5 HO

2

Absorption bands

0 2

HO
2

Co 2

H 2

0 3

H 2

0

250 500 750 1000 1250 1500 1750 2000 2250 2500

Wavelength (nm)