(2) radiation laws 1 physics of the atmosphere ii atmo ii 31
TRANSCRIPT
(2) Radiation Laws 1
Physics of the Atmosphere II
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Angle and Solid Angle
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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
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Solid angles of different areas on Earth (Zimbabwe, Algeria + Libya, Switzerland – or Austria)
Wiki
Radiation Units
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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]
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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]
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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
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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.
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Planck Functions in double logarithmic representation (wikimedia). Note that black bodies with higher temperatures emit more energy at all wavelengths.
Planck–Function
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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
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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?
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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:
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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:
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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
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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
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20 Wm1362 S
Measuring the Solar Constant
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Total Solar Irradiance (TSI) measurements – composite of different satellite measurements (World Radiation Center).
Measuring the Solar Constant
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TSI measurements – original measurements (source: WRC). Systematic difference ~ anthropogenic radiative forcing.
Trend estimation impossible without data overlap.
Changing Solar Constant
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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
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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
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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).
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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).
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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%
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Albedo
Annual mean of the top of the atmosphere (toa) Albedo (Raschke & Ohmura*)
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Net-Short-Wave Radiation = SWdown – SWup
at the Earth‘s surface.
Shortwave-Radiation
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Annual mean toa Net Shortwave Radiation (Raschke & Ohmura*)
Shortwave-Radiation
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