(2) Radiation Laws 1

Physics of the Atmosphere II

Atmo II 31

Angle and Solid Angle

Atmo II 32

Radian (rad) is the standard unit for angular measures. In a circle (with radius r) 1 rad corresponds to an arc length = r.

The whole circumference is therefore 2π rad = 6.2832 rad.

180

rad1

Steradian (sr) is the standard (SI) unit for solid angles. On a sphere (with radius r) 1 sr corresponds to an area = r2.

The whole surface area is therefore 4π sr = 12.5664 sr.

www.greier-greiner.at Wiki

Solid Angles on Earth

Atmo II 33

Solid angles of different areas on Earth (Zimbabwe, Algeria + Libya, Switzerland – or Austria)

Wiki

Radiation Units

Atmo II 34

Try not to be confused !

Radiant energy: Energy [J]

Radiant flux: Energy per time [J s–1] = [W] (power)

Radiant flux density = Irradiance: Energy per time per area [J s–1

m–2] = [W m–2]

Radiance: Energy per time per area per solid angle [W m–2 sr–1]

Spectral radiance Spectral radiance with respect to wavelength [W m–2 sr–1 m–1] = [W m–3 sr–1]

orSpectral radiance with respect to frequency: [W m–2 sr–1 Hz–1]

Strahlungsgrößen

Vorsicht – Verwirrungsgefahr !

Strahlungsenergie: Energie [J]

Strahlungsfluss: Energie pro Zeit [J s–1] = [W] (also eine Leistung)

Strahlungsflussdichte = Irradianz: Energie pro Zeit pro Fläche [J s–1

m–2] = [W m–2]

Strahldichte = Radianz: Energie pro Zeit pro Fläche pro Raumwinkel [W m–2 sr–1]

Spektrale Dichte der Strahldichte: Strahldichte bezogen auf die Wellenlänge [W m–2 sr–1 m–1] = [W m–3 sr–1]

oderStrahldichte bezogen auf die Frequenz [W m–2 sr–1 Hz–1]

Atmo II 35

Planck’s Law

According to Planck’s Law (Max Planck, 1900) the energy emitted by a black body (un-polarized radiation) per time, area, solid angle and wave length λ equals:

c0 = Speed of light (in vacuum) = 299 792 458 m s–1

h = Planck constant = 6.626 069 57·10–34 JskB = Boltzmann constant = 1.380 6488·10–23 J K-1

According to our last slides this has to be – right:

Spectral radiance with respect to wavelength [W m–2 sr–1 m–1]

Atmo II 36

1exp

125

2

Tkhc

hcTB

B

0

0),(

Planck’s Law

Planck’s Law (last slide) refers to un-polarized radiation per solid angle. In case of linear polarization we would just get half of it. If you should miss a factor π – this comes be integrating over the half space. Planck‘s law often comes in frequency formulation:

),(),(

TBTB

Atmo II 37

1exp

122

3

Tkhc

hTB

B

0

),(

0cB

d

dBB 2

0c

Planck–Function

Black-Body Radiation (Planck Functions) for different temperatures (wikimedia). Note the large dynamic range due to the extremely strong wavelength dependence.

Atmo II 38

Planck Functions in double logarithmic representation (wikimedia). Note that black bodies with higher temperatures emit more energy at all wavelengths.

Planck–Function

Atmo II 39

Planck Function for the temperature of our Sun (yes, the sun can indeed be regarded as a black body !) (Meteorology Today, C. D. Ahrens). 44 % of the total energy is emitted in the visible part of the spectrum. But also in the thermal infrared the suns emits way more energy (per m2) than the Earth.

Planck–Function – Sun

Atmo II 40

But how about pictures like this? (LabSpace). These are scaled representations, showing λBλ on the y-axis – and the solar radiation intercepted by the Earth (and not emitted by the Sun).

Planck–Functions?

Atmo II 41

Integrating Planck’s Law

If we want just to know the total power emitted per square meter, we need to integrate the Planck law twice – over the half space and over all wavelengths. The integral over the half space gives:

Atmo II 42

ddθ

r

ddΩ sin

2

Effective area: dAosθc

K. N. Liou

Black body radiation is isotropic (independent of the direction):

2

0 0

2

ddθcossin

Stefan–Boltzmann Law

The integral over all wavelengths gives:

Atmo II 43

dTBTB

0

),()(Tk

hcu

B

0 dhc

Tukdu B

0

2

due

u

ch

TkTB

uB

00

)(1

2 3

23

44

151

43

due

uu

0

423

45

15

2T

ch

kTBF B

0

)( But this is nothing else than:

The famous Stefan–Boltzmann Law (after Josef Stefan and Ludwig Boltzmann):

σ = Stefan–Boltzmann constantσ = 5.670 373 · 10–8 W m–2 K–4

4TF

23

5

15

2

0

4

ch

kB

Atmo II 44

Stefan–Boltzmann Law

The Stefan–Boltzmann constant is therefore related to even more fundamental constants:

(Please try it without calculator)

Solar Constant

How much solar radiation reaches the Earth?Looking at a point-source, and considering energy conservation, we see that concentric spheres must receive the same energy (blackboard).

Die radiant flux density (irradiance), at the mean distance Earth – Sun (= Astronomical Unit – AU) per square meter is termed Solar Constant and has (probably) the value:

where the Solar Radius RSun = 695 990 km (about. 0.7 Mio km), and the Astronomical Unit rS–E = 149 597 871 km (about 150 Mio. km).

but S0 is not constant.The brightness temperature of the Sun is 5776 K.

20 Wm1366 S 4

Sun2-ES

2Sun

4

4T

r

RS

0

Atmo II 45

20 Wm1362 S

Measuring the Solar Constant

Atmo II 46

Total Solar Irradiance (TSI) measurements – composite of different satellite measurements (World Radiation Center).

Measuring the Solar Constant

Atmo II 47

TSI measurements – original measurements (source: WRC). Systematic difference ~ anthropogenic radiative forcing.

Trend estimation impossible without data overlap.

Changing Solar Constant

Atmo II 48

Changes in the total solar irradiance due to the ~11 year solar cycle (about 1 W/m2 or 1 ‰) are comparatively small (NASA GISS).

Note the pronounced latest minimum.

Changing Solar UV Radiation

Atmo II 49

Changes in the UV part of the spectrum are much more distinct (NASA).

Solar Insolation

In general, a square meter on Earth will not receive the full solar constant, since the solar radiation will not hit at right angle (keywords – seasons, night). With the zenith angle of the sun, θ, we get S = S0 cosθ (Lambert’s Cosine Law). And not all the radiation will reach the ground.

W & K

Atmo II 50

Solar Insolation

Daily mean solar insolation as a function of latitude and day of year in units of Wm−2 based on a solar constant of 1366 Wm−2. The shaded areas denote zero insolation. The position ofvernal equinox (VE), summer solstice (SS), autumnal equinox (AE), and winter solstice (WS) are indicated with solid vertical lines. Solar declination is shown with a dashed line (K. N. Liou).

Atmo II 51

Solar Radiation at the Surface

At the “top of the atmosphere“ the solar irradiance is still close to that of a black body (R.A. Rhode). Even under “clear sky” conditions a part of the incoming radiation will be scattered and absorbed (the latter – about 20 % mainly due to Ozone and Water Vapor).

Atmo II 52

Albedo

Albedo is the percentage of the solar radiation, which is directly reflected. Die Albedo depends on the surface properties. The Albedo is particularly high for (dense) clouds and (fresh) snow. The Earth as a whole reflects 31% of the incoming solar radiation (A = 0.31). The Earth-surface therefore only absorbs about 50 % of the solar radiation.

Surface Albedo

Clouds 45-90 %Fresh snow (3) 75-95 %Glaciers 20-45 %Sea Ice 30-40 %Rock, soil (2) 10-40 %Forests (1) 5-20 %Water 5-10 %Planetary Albedo 31%

Atmo II 53

Albedo

Annual mean of the top of the atmosphere (toa) Albedo (Raschke & Ohmura*)

Atmo II 54

Net-Short-Wave Radiation = SWdown – SWup

at the Earth‘s surface.

Shortwave-Radiation

Atmo II 55

Annual mean toa Net Shortwave Radiation (Raschke & Ohmura*)

Shortwave-Radiation

Atmo II 56