13 solar radiation

Solar Radiation
Physics and Geometry
for hydrologists

Il Sole, F. Lelong, 2008, Val di Sella

Riccardo Rigon

Monday, December 10, 12

When you see the Sun rise,
do you not see a round disc of fire

somewhat like a guinea?

Oh no, no! I see an innumerable

company of heavenly host

crying

“Glory, glory, glory is the Lord God

Almighty.”

W. Blake

2

R. Rigon


Introduction

Educational Goals

• To recognise that the water cycle is powered by solar energy

• To gain knowledge of the spatial and temporal variation of the
radiation distribution on the Earth

• To present the ways in which radiation is produced, received by the
Earth, transmitted by the atmosphere, reflected, absorbed, and reemitted
by the Earth’s surface

• To introduce the concepts necessary to better understand the elements
of the energy balance needed in remote-sensing applications, the snow
balance, and evapotranspiration

3

R. Rigon 1

The Sun

The Sun is the origin of the water
cycle

4

R. Rigon 2

The Sun

Composition of the Sun

The Sun is mainly composed of hydrogen. The rest is prevalently He4.
Hydrogen is the fuel for the nuclear fusion that takes place inside the Sun and
produces helium. However, the He4 contained in the Sun for the most part
originates from previous stellar lives. 5

R. Rigon 3

The Sun

Sun Fact Sheet

The Sun is a G2 type star, one of the hundred billion stars of this type in our
galaxy (one of the hundred billion galaxies in the known universe).

Diameter: 1,390,000 km (the Earth: 12,742 km or 100 times smaller)
Mass: 1.1989 x 1030 kg (333,000 times the mass of the Earth)

Temperature: 5800 K (at the surface) 15,600,000 K (at the core)

The Sun contains 99.8% of the total mass of the Solar System (Jupiter
contains nearly all the rest).

Chemical composition:
Hydrogen 92.1%
Helium 7.8%
6
Other elements: 0.1%

R. Rigon 4

The Sun

The Sun and the planets to scale

7

R. Rigon 5

The Sun

The internal structure of the Sun

The Sun’s energy is created in the core by fusing hydrogen into helium. This
energy is irradiated through the radiative layer, then transmitted by convection
through the convective layer, and, finally, radiated through the photosphere,
which is the part of the Sun that we see. 8

R. Rigon 6

The Sun

Provide a relatively constant rate of radiation energy that in few minutes
from the cromosphere arrives to the Earth.
Detail of a Pellizza da Volpedo Painting

9

R. Rigon

The Sun

Solar Spots

Radiation flux is regular up to a point. In reality it manifests variations.
Solar spots appear as dark spots on the surface of the Sun and they have a
temperature of 3,700 K (to be compared to the 5,800 K of the surrounding
photosphere). A solar spot can last for may days, the most persistent lasting for
many weeks.

10

R. Rigon 7

The Sun

Variability of the Emissions

An image of the sun in X-ray
band, taken by the Yohkoh solar
observatory satellite, which
shows changes in emissions of
the solar corona from a
maximum in 1991 (left) to a
minimum in 1995 (right).

11

R. Rigon 8

The Sun


Solar radiation is subject to
fluctuations, some of which are
localised in restricted areas, while
others are more global and follow
an 11-year cycle.
Every 11 years the sun goes from
a limited number of solar spots
and flares to a maximum, and
vice versa. During this cycle the
Sun’s magnetic poles switch
orientation. The last solar
minimum was in 2006.

12

R. Rigon 8

The Sun


The graph shows the solar spot cycle over the last 400 years. It should be
noted that before 1700 there was a period in which very few solar spots were
observed. This period coincides with the Little Ice Age, which is why there are
suggestions that there is a connection between solar spot activity and the
climate on Earth. The most evident cycle has a period of 11 years. But there
is a second cycle which seems to have a period of 55-57 years.

13

R. Rigon 9

The Sun

The Stefan-Boltzmann law

Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law

R=✏ T4

14

R. Rigon 10

The Sun



R=✏ T4

Radiation
emitted

14

R. Rigon 10

The Sun



R=✏ T4

emissivity

Radiation
emitted

14

R. Rigon 10

The Sun



R=✏ T4

Stefan-Boltzmann constant
emissivity

Radiation
emitted

14

R. Rigon 10

The Sun



R=✏ T4 absolute temperature

Stefan-Boltzmann constant
emissivity

Radiation
emitted

14

R. Rigon 10

The Sun

The physics of Radiation

On the basis of the temperature of the Sun photosphere (~6000 K), and the
Stephan-Boltzmann law, the total energy emitted by the Sun is

RSun = ✏ T 4 = 1 ⇤ 5.67 ⇤ 10 8
⇤ 60004 ⇡ 25.12 ⇤ 109 J m 2
s 1

15

R. Rigon 11

The Sun

The Sun is nearly a “blackbody”!

The Sun is practically a blackbody. The difference between a true blackbody
and the Sun is due to the fact that the corona and the chromosphere
selectively absorb certain wavelengths. 16

R. Rigon 12

The Sun

The Sun is nearly a “blackbody”!

The area below the curves is given by the Stefan-Boltzmann law. The curves
themselves are given by Planck’s law.
17

R. Rigon 13

The Sun

The complete electromagnetic spectrum

Figure 2.9
C.B. Agee

The spectrum of solar radiation stretches far beyond the visible band where,
however, nearly half the available energy is concentrated
18

R. Rigon 16

The Sun

Planck’s Law
•Planck’s law is the general law for electromagnetic emission from the
surface of a blackbody*:

2⇡c2 h 5
W = ch [W m 2
µm 1
]
e KT 1

* Stefan-Boltzmann law is just the integration of Plank’s law over wavelengths 19

R. Rigon 14

From Sun To Earth

From Sun to Earth

The energy irradiated by the Sun passes through an imaginary disc with diameter
the same as the Earth’s. The energy flow is maximum at that point on the Earth
where the radiation is perpendicular.
20

R. Rigon 18

From Sun To Earth

Solar radiation

The Sun irradiates
approximately at the solar
constant rate, which is, on
the average, on the top of
the atmosphere,
Frolich, 1985

http://en.wikipedia.org/wiki/Solar_constant

21

R. Rigon 19

Copying with Earth surface

Astronomical variability of radiation

In its orbit around the Sun, the Earth keeps its north-south rotational axis
unvaried, causing a different angle between the Sun’s rays and the surface of the
Earth. 22

R. Rigon

From Sun To Earth

Seasons

Figure 3.1

The Earth is 5 million kilometers closer to the Sun during the northern
winter: a clear indication that temperature is controlled more by orientation
than by distance.
23

R. Rigon

From Sun To Earth

Corrections to the solar constant

The Earth’s orbit around the Sun is an ellipse. The shape of the ellipse is
determined by its eccentricity, which varies in time, changing the distances of
the aphelion and perihelion

24
http://www.ascensionrecta.com/

R. Rigon 20

From Sun To Earth

Precession of the polar axis

The axis of rotation moves with a slow period, executing
a complete precession every 26,000 years.
Polar stars behave like this for only a very short period

25

R. Rigon

From Sun To Earth

Astronomical influences

Orbit shape

Orbit change

Orbit angle

R. Rigon 26

From Sun To Earth

Solar radiation in
hydrological models

Therefore the solar contant must be corrected
S (e.g. Corripio, 2002):

27

R. Rigon

From Sun To Earth

Solar radiation in
hydrological models

Therefore the solar contant must be corrected
S (e.g. Corripio, 2002):

where:

N is the day of the year (in 1, ..., 365)
28

R. Rigon


Radiation intensity

Solar intensity governs seasonal climatic changes and the local climatic niches
which are linked to the apparent height of the Sun.

29

R. Rigon


Insolation and latitude

Figure 3.7
Incoming solar radiation is not evenly distributed across all lines of latitude,
creating a heating imbalance.
30

R. Rigon


Radiative imbalance

31

R. Rigon


Radiation received from the Sun

decreases towards the poles and it is reduced in areas where clouds
form frequently

For example, the complete energy balance is greater at the equator but the
greatest amount of insolation is in the subtropical deserts

Average annual radiation is

< 80 W/m2 in the cloudy parts of the arctic and the antarctic
>280 W/m2 in the subtropical deserts

50

R. Rigon


The geometry of radiation

From a subjective point of view, the Sun varies its height in the sky seasonally.
This is the subject of interest in the study of the geometry of radiation.

33

R. Rigon


To sum up

Calculations of the incident radiation onto the surface of the Earth need to
take account of the geometry of the interaction between the Sun’s rays and
the surface of the Earth, which is curved and therefore variably
exposed with respect to the direction of the Sun in function of latitude,
time of day (longitude) and, naturally, day of the year. Moreover the
Earth rotation is inclined with respect to its orbit around the Sun , and
this causes seasons to happen.

34

R. Rigon



To calculate the aforementioned
quantities it is usual to use a
topocentric coordinate system,
Nautic Almanac Office, 1974

that is, with the origin in the
geographic position of the
observer, which is right-handed
and positioned on the plane
tangent to the Earth’s surface in
the considered point.

N.B. - A coordinate system located at the
centre of the Earth id called geocentric.

35

R. Rigon



The X-axis is, therefore, tangent
to the earth and positive in a
West-East direction. The Y-axis
Nautic Almanac Office, 1974

is tangent in the North-South
direction and is directed towards
the South. The Z-axis lies on the
segment joining the centre of the
Earth with the point being
considered on the surface.

It is assumed that the Sun lies in
the ZY plane at the solar noon.

36

R. Rigon


Solar Vector

The solar vector can be expressed as a
function of the angles that have been
defined. The resulting trigonometric
expression is:

Z ⇥
sin ⇥ cos
⌥ = ⇤ sin ⇤ cos ⇥ cos
s cos ⇤ cos ⌅
cos⇤ cos ⇥ cos + sin ⇤ sin

Y

X Therefore, to determine the position of
the Sun one needs to know the latitude,
the hour angle, and the solar
declination. 37

R. Rigon


Hour angle

The hour angle can be easily
calculated as:

⇥
t
⇥= 1
12

if t is the solar hour

38

R. Rigon


Solar declination
Is the angular height of Sun from the horizon at equator at noon*

The solar declination is a function of the day of the year (and the era). It
requires complex calculations that take account of the precession movements
of the Earth. There are, however, various approximations. The one that is
presented here is due to Bourges, 1985:

where is the day of the year

*http://en.wikipedia.org/wiki/Declination

39

R. Rigon


Projection on a plane at a certain
latitude

If is the vertical unit row-vector
corresponding to the Z axis:

Z

and
⇥
sin ⇥ cos
Y ⌥ = ⇤ sin ⇤ cos ⇥ cos
s cos ⇤ cos ⌅
X

is the solar vector

40

R. Rigon


Projection on a plane at a certain
latitude

Then the projection of the solar
irradiation on the plane YX is reduced by
the factor where:

Z

or:

Y

X
with the symbols explained above

41

R. Rigon


To sum up:

The solar constant can be modified as follows.

Was:

Is now:

42

R. Rigon

Absorption and transmission of short wave radiation

Atmosphere is a gray body

• The blackbody is an ideal object that absorb all the radiative energy it receives
• Real objects (bodies, “gray bodies”) are not capable of absorbing all radiation.
• To understand the difference between a blackbody and a gray body we need to
analyse the interactions between a surface and the electromagnetic radiation
incident onto it.

43

R. Rigon


Atmospheric absorption

Radiation passes quite freely through the Earth’s atmosphere and it warms
the surfaces of seas and oceans. A portion of between 45% and 50% of the
incident radiation onto the Earth reaches the ground
44

R. Rigon


Shortwave Radiation budget

The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.

45

R. Rigon


Radiation reflected


45

R. Rigon


Radiation reflected


Radiation
transmitted 45

R. Rigon



S It should not be forgot that
the radiation budget is an
energy budget, for which

the incoming radiation equals
the reflected one plus
the absorbed plus
the transmitted

46

R. Rigon



S It should not be forgot that
the radiation budget is an
energy budget, for which

the incoming radiation equals
the reflected one plus
the absorbed plus
Radiation
the transmitted
absorbed

46

R. Rigon



S

This budget can be apply to any slice of the atmosphere

47

R. Rigon



S

Corrected Solar
constant


47

R. Rigon



S

Solar radiation
reflected back to space

Corrected Solar
constant


47

R. Rigon



S

Transmitted
radiation

Solar radiation
reflected back to space

Corrected Solar
constant


47

R. Rigon



S

Transmitted
radiation

Solar radiation
reflected back to space Energy
absorbed by
atmosphere
Corrected Solar
constant


47

R. Rigon


Coefficients
The following coefficients can also be defined

• is the transmission coefficient, said atmospheric
transmissivity

• is the reflection coefficient, said atmospheric
reflectivity (albedo)

• is the absorption coefficient, said atmospheric
absorptivity

48

R. Rigon



Energy conservation:

implies that reflectivity, transmissivity and absorptivity sum to one:

Which is, indeed, valid for reflectivity, transmissivity and absorptivity of any other body

49

R. Rigon



S

50

R. Rigon



S

We just forget for a
moment this. It will be
splitted into two parts:
one depending on
diffuse radiation and
another on cloud cover

50

R. Rigon



S

51

R. Rigon



S

Atmosphere is pretty transparent: which
means that we can, as a first approximation,
neglect it (atmosphere is heated from below)

51

R. Rigon



S

In any case let’s concentrate on
the transmitted radiation

This can be decomposed into two parts:
direct and diffuse solar radiation

52

R. Rigon



S

Evidently, for simmetry

is also composed by reflected and
diffuse solar radiation

53

R. Rigon


Diffuse radiation comes from scattering

Incident solar radiation strikes gas molecules, dust particles, and
pollutants, ice, cloud drops and the radiation is scattered. Scattering
causes diffused radiation.

Two types of light diffusion can be distinguished:

Mie scattering
Rayleigh scattering

5

R. Rigon


Rayleigh Scattering

•The impact of radiation with air molecules smaller than λ/π causes
scattering (Rayleigh scattering) the entity of which depends on the
frequency of the incident wave according to a λ-4 type relation.

•In the atmosphere, the wavelengths corresponding to blue are scattered
more readily than others.

incident radiation

diffuse radiation

transmitted radiation
55

R. Rigon


Mie Scattering

•When in the atmosphere there are particles with dimensions greater than 2 λ/π
(gases, smoke particles, aerosols, etc.) there is a scattering phenomenon that
does not depend on the wavelength, λ, of the incident wave (Mie scattering).

incident radiation

diffuse radiation

transmitted radiation

•This phenomenon can be observed, for example, in the presence of clouds.

56

R. Rigon


Diffused Light

Scattering selectively eliminates the shorter visible wavelengths, leaving the
longer wavelengths to pass. When the Sun is on the horizon, the distance
travelled by a ray within the atmosphere is five or six times greater than
when the Sun is at the Zenith and the blue light has practically been
completely eliminated.
57

R. Rigon


Tilt of the Earth’s axis
and atmospheric effects

The tilt of the earth’s axis and atmospheric effects together affect the amount of
radiation that reaches the ground.

58

R. Rigon


One way to take into account of absorption

Would be to run a full model of atmospheric transmission (e.g. Liou, 2002).
However hydrologists prefer to use parameterizations, and the
concept of atmospheric transmissivity.

59

R. Rigon


Solar radiation transmitted to the ground under
clear sky conditions

S
Finally:

Corripio, 2002
60

R. Rigon



S
Finally:

Corripio, 2002
Fraction of direct solar radiation
included between the considered
60
wavelengths
R. Rigon



S
Finally:

Corripio, 2002
Transmittance of the
atmosphere

60
wavelengths
R. Rigon



Correction due to
S elevation of the site

Finally:

Corripio, 2002
Transmittance of the
atmosphere

60
wavelengths
R. Rigon


Solar radiation transmitted to the
ground under clear sky conditions

S
We do not enter in the details of how

and

are determined. Please look, for
instance, at Formetta et al., 2012

61

R. Rigon

Considering Clouds

Hydrologists (and not only them) treat the
influence of clouds separately

It is assumed that the effects of
clouds is an attenuation of the
transmitted solar radiation

Transmitted direct
radiation at the surface
after clouds correction

62

R. Rigon

Considering Clouds


It is assumed that the effects of
clouds is an attenuation of the
transmitted solar radiation

Transmitted direct
Transmitted direct radiation at the surface
radiation at the surface before clouds correction
after clouds correction

62

R. Rigon

Considering Clouds


An analogous formulation holds for
diffuse radiation:

63

R. Rigon

Considering Clouds


An analogous formulation holds for
diffuse radiation:

Correction coefficient for
diffuse radiation

63

R. Rigon

Considering Clouds

Estimation of the reduction coefficients
(decomposition model)

These reduction coefficients can be
determined when we have ground
measurements of total radiation,
diffuse plus direct:

64

R. Rigon

Considering Clouds


These reduction coefficients can be
determined when we have ground
measurements of total radiation,
diffuse plus direct:

Measured total radiation
at the ground station i

64

R. Rigon

Considering Clouds


These assumption that is often
made is that, the diffuse solar
radiation measured at the station is
proportional to the total radiation:

65

R. Rigon

Considering Clouds


These assumption that is often
made is that, the diffuse solar
radiation measured at the station is
proportional to the total radiation:

reduction coefficient for
diffuse radiation
65

R. Rigon

Considering Clouds


Therefore when substituting this
diffuse radiation expression in the
total radiation equation of previous
slides, it results at stations:

66

R. Rigon

Considering Clouds


And, for the direct radiation, at
stations:

67

R. Rigon

Considering Clouds

The key factor is the to determine the above coefficient, on which the
procedure followed so far has moved all the unknown.

Its estimation pass through various parameterizations:

Among the most known:

•Erbs et al., 1982
•Reindl et al. 1990
•Boland et al. 2001
please find the details in Formetta et al., 2012

68

R. Rigon

Considering Clouds

One more issue

With the help of the parameterizations above, the correction facotrs are
determined for the stations. Which are a few points in a rugged terrain.

How do you solve the problem to transport it everywhere ?

69

R. Rigon

Considering Clouds

We need to use
some interpolation
technique
Like Kriging* or the Inverse distance weighting method** which is not
the matter of the present slides.

* Goovaerts, 1997

**Shepard, 1968
70

R. Rigon

Hitting the terrain

Finally the residual radiation hits the terrain

The terrain is not a plane
but it is inclined. Therefore,
besides correcting radiation
for latitude, longitude and
hour, it is necessary to
account for slope and
aspect

71

R. Rigon

Hitting the terrain

In the presence of topographic surfaces

In the northern hemisphere, slopes that face South receive a greater insolation
and, therefore, the water in the soil evaporates more quickly or the snow melts
faster. Slopes with differing aspects are often characterized by different species
and densities of plants and trees.
72

R. Rigon

Hitting the terrain

Projection of radiation onto an
inclined surface
After Corripio, 2003

First we calculate the normal to the surface 73

R. Rigon

Hitting the terrain

inclined surface
Unit normal vector:

⇥

1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1) )
⇧ ⌃
1 ⇧ ⇧ 1/2 (z(i,j) + z(i+1,j)
⌃
⇧u =
n ⇧ z(i,j+1) z(i+1,j+1) ) ⌃
⌃
|⇧ u | ⇤
n ⌅
l2

where z are the elevations of the four points used and l2 is the are of the
cell - of side l.
74

R. Rigon

Hitting the terrain

Representation of the vector normal to the surface of Mount Bianco
75

R. Rigon

Hitting the terrain

inclined surface

And we compare with the solar vector, indicating the direction of the Sun 76

R. Rigon

Hitting the terrain

inclined surface

⇥
sin ⇥ cos
s cos ⇤ cos ⌅

Where all the quantities were already defined previously

77

R. Rigon

Hitting the terrain

inclined surface

s

Then we calculate the angle between the sun vector and the normal 78

R. Rigon

Hitting the terrain

inclined surface

We can define then the angle
s of solar incidence

79

R. Rigon

Hitting the terrain

inclined surface
Angle of solar incidence

cos s = ⌅ · ⌅u
s n
⇥
1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1) )
⇧ ⌃
1 ⇧ ⇧ 1/2 (z(i,j) + z(i+1,j)
⌃
⇧u =
n z(i,j+1) z(i+1,j+1) ) ⌃
|⇧ u | ⇧
n ⇤ ⌃
⌅
l2

⇥
sin ⇥ cos
s cos ⇤ cos ⌅

80

R. Rigon

Hitting the terrain

inclined surface

The above angles needs to be compared with those of the terrain:

Slope

s = cos 1
nu.z

Aspect (from the North anti-clockwise)

81

R. Rigon

Hitting the terrain

inclined surface

Remarkably the form of formula for the incident radiation is the same that
for a flat surface when the projection angle is accounted:

82

R. Rigon

Hitting the terrain

S
Therefore, for the direct
shortwave radiation:

Corripio, 2002
as, it was before

83

R. Rigon

Hitting the terrain

However, it is not just matter of light but also of
shadows

84

R. Rigon

Hitting the terrain

Incident radiation
Topographic effects: shading

More schematically

light

shadow

85

R. Rigon

Hitting the terrain

Incident radiation

More schematically

86

R. Rigon

Hitting the terrain

Incident radiation

More schematically

shadow

86

R. Rigon

Hitting the terrain

Incident radiation

More schematically

light

shadow

86

R. Rigon

Hitting the terrain

Incident radiation

Therefore the direct solar radiation must be corrected to include shading
Details in Corripio, 2003

87

R. Rigon

Hitting the terrain

What about diffuse radiation ?
Topographic effects: angle of view

88

R. Rigon

Hitting the terrain


sky view factor

88

R. Rigon

Hitting the terrain


sky view factor

diffuse
radiation due to
Rayleigh
scattering

88

R. Rigon

Hitting the terrain


sky view factor

diffuse
radiation due to diffuse
Rayleigh radiation due to
scattering aerosols

88

R. Rigon

Hitting the terrain


sky view factor

diffuse
diffuse radiation due
radiation due to diffuse
multiple
Rayleigh radiation due to
scattering
scattering aerosols

88

R. Rigon

Hitting the terrain

Incident radiation

Any point in a rugged landscape see just a part of the sky sphere. Its fraction
says which portion of the sky contribute to diffuse shortwave radiation.
89

R. Rigon

Hitting the terrain

Incident radiation

Different points view a different sky
90

R. Rigon

Hitting the terrain

The sum

91

R. Rigon

Hitting the terrain

Now it really hits the terrain
and, in part, it is reflected away

92

R. Rigon

Hitting the terrain

Finally a map

Insolation received by Mont Blanc at Spring Equinox 93

R. Rigon

Albedo

Typical albedo values
http://en.wikipedia.org/wiki/Albedo

94

R. Rigon

Albedo

Typical albedo values
http://en.wikipedia.org/wiki/Albedo

95

R. Rigon

Spectral response

Spectral Signature (or Response)

The percentage of radiation that is reflected (reflectance) depends on
wavelength of the radiation, and on the geometry, nature, and structure
of the surface under investigation.

51
96

R. Rigon

Spectral response

•In the case of solar radiation, the spectral signature is defined
as the reflectance of the surface in function of the wavelength.

97

R. Rigon

Spectral response

•Every type of surface can be statistically characterised by a spectral signature.

98

R. Rigon

Spectral response

Factors

•The spectral signature of a specific element of a territory will
vary due to the variability of local environmental factors.

•Given a certain type of ground cover, static elements, such as
slope and exposition, and dynamic elements, such as surface
ground humidity, the phenological state of the vegetation, the
atmospheric transparence, etc., will cause variations in the
spectral signature curve.

99

R. Rigon

Spectral response

Radiation that hits the terrain, heats it.
Or causes changes of phase

water to vapor

ice to water

100

R. Rigon

Spectral response

Or is used for photosynthesis

or other chemical reactions
101

R. Rigon

Long wave radiation

Earth “is” a gray body

Having a temperature emits radiation
A. Adams - Part of the snake river picture

102

R. Rigon

Long wave radiation

Gray Bodies

• Plank’s Law for gray bodies:

2⇡c h 2 5
W =✏ ch [W cm 2
µm 1
]
e KT 1
• The Stefan-Boltzmann equation for gray bodies:

W = ✏ T [W cm
4 2
]

where ε is the average emissivity calculated over the entire electromagnetic
spectrum.

103

R. Rigon

Long wave radiation

Gray Bodies

The behavior of a real (gray) body is related to that of a black body by
means of the quantity ελ, known as the emission coefficient or
emissivity, which is defined as:

W (real body)
✏ =
W (black body)

Kirchhoff (1860) demonstrated that a good “radiator” is also a good
“absorber”, that is to say:

↵=✏ ⇢+⌧ +✏=1

104

R. Rigon

Long wave radiation

Comparison of blackbody and gray body

In reality emissivity depends, at least, on wavelength. Earth should be
probably defined a selective radiator

105

R. Rigon

Long wave radiation

See the Earth as gray body

and given that the
temperature of the Earth’s
surface is, on average,
about 288 K, it obviously
emits a spectrum of
radiation in the infrared
band.

106

R. Rigon

Long wave radiation

Radiation emitted by the Sun and the Earth

Yochanan Kushnir

107

R. Rigon

Long wave radiation

See the Earth as gray body

and given that the
temperature of the Earth’s
surface is, on average,
about 288 K, it obviously
emits a spectrum of
radiation in the infrared
band.

Atmosphere is not
anymore transparent to at
these wavelengths.

108

R. Rigon

Long wave radiation

The atmosphere is
warmed from below

Therefore the temperature is
higher at ground level than it
is at higher altitudes.

109

R. Rigon

Long wave radiation

Greenhouse Effect

In the absence of atmospheric absorption the average temperature of the Earth’s
surface would be about -170C.

110

R. Rigon

Long wave radiation

Greenhouse Effect

Instead the average temperature is about 15 0C

111

R. Rigon

Long wave radiation

Radiative heating

is completed by convective heat transfer, and by water vapor fluxes (latent and
sensible heat).
But this you can see better on the energy budget slides. 112

R. Rigon

Long wave radiation

But now concentrate on the surroundings of a point
After Helbig, 2009

Any point being at a certain temperature emits long wave radiation
which must be accounted for
113

R. Rigon

Long wave radiation

The atmosphere emits infrared itself

bacause of its temperature
114

R. Rigon

Long wave radiation

All the contributions

Long-wave radiation is given by the
balance of incident radiation from
the atmosphere and the radiation
emitted by the ground. Both values
are calculated with the Stefan-
Boltzmann law.

115

R. Rigon

Long wave radiation

Longwave (infrared) raditation

116

R. Rigon

Long wave radiation


Longwave radiation
coming from sky

116

R. Rigon

Long wave radiation


Longwave radiation Longwave radiation
coming from sky coming from
surrounding

116

R. Rigon

Long wave radiation


Longwave radiation Longwave radiation Radiation losses
coming from sky coming from by the area under
surrounding exam

116

R. Rigon

Long-wave radiation

The first component should be
calculated by integrating the formula
over the entire atmosphere, but,
given how complex this process is,
typically an empirical formula is
used that uses the value of air
temperature as measured near
ground level (2m) and a value of the
atmospheric emissivity based on
specific humidity, temperature, and
cloudiness. The second component,
on the other hand, is function of the
surface temperature and its
emissivity.
117

R. Rigon

Long wave radiation

Long-wave radiation

The real process:

The hydrological parameterisation:

118

R. Rigon

Long wave radiation

Long-wave radiation

The real process:


Global emissivity of the
atmosphere
118

R. Rigon

Long wave radiation

Long-wave radiation

The real process:


Temperature at 2 m
from ground
Global emissivity of the
atmosphere
118

R. Rigon

Long wave radiation

Parameterisation of Long-wave radiation


6 4
εatm = εBrutsaert (1− N ) + 0.979N Brutsaert (1975) + Pirazzini et al. (2000)

εatm = εBrutsaert (1+ 0.26N) Brutsaert (1975) + Jacobs (1978)

6 4
εatm = εIdso (1− N ) + 0.979N Idso (1981) + Pirazzini et al. (2000)

6 4
εatm = εIdso,corr (1− N ) + 0.979N Hodges et al. (1983) + Pirazzini et al.
(2000)

where N is the fraction of sky covered by clouds
119

R. Rigon

Net Radiation

The sum of longwave and shortwave ratio
is called net radiation

120

R. Rigon

1Thank you for your attention !

G.Ulrici - 2000 ?

121

R. Rigon

Table of symbols

122

R. Rigon

Table of symbols

123

R. Rigon

Table of symbols

124

R. Rigon

inclined surface

125

R. Rigon


126

R. Rigon

13 solar radiation

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (17)

Similar to 13 solar radiation

Similar to 13 solar radiation (20)

More from AboutHydrology Slides

More from AboutHydrology Slides (20)

Recently uploaded

Recently uploaded (20)

13 solar radiation