8. atmospheric radiation 8.1 basic concept and definition 8.2 radiation laws 8.3 radiative transfer...

8. Atmospheric Radiation

8.1 Basic concept and definition

8.2 Radiation laws

8.3 Radiative transfer equation

8.4 Solar radiation (Shortwave radiation)

8.5 Infrared radiation (Longwave radiation)

8.6 Radiation budget

8.1 Basic concepts and definitions1. Electromagnetic radiation:

* Energy propagated in the form of an advancing electric and magnetic field disturbance.

* Travels in wave form at the speed of light (c).

* Wavelength ( ) is the physical distance between adjacent maxima or minima in the electric or magnetic field. unit:

* Wave number ( ) is the number of wavelengths in a unit distance, i.e., unit: 1/cm

* Frequency ( ) is the number of successive maxima or minima passing a fixed point in a unit of time, unit: cycle-per-second (cps) or 1/s

k/1k

/c

m

* The energy associated with electromagnetic radiation is contained in discrete packets called photons.

* The energy contained in a photon is proportional to frequency, and is given by where h is Planck’s constant,

hW sJh 3410626.6

2. Electromagnetic spectrum:

* Classified into bands in terms of frequency (or wavelength)

* Most significant spectral regions associated with radiative energy transfer in atmosphere lie between ultraviolet light and microwaves.

• The ratio of the area of the sphere intercepted by the cone to the square of the radius– – Units: Steradians (sr)

• What is the area cut out of a sphere by one steradian?

• What is the solid angle representing all directions at a point?

3. Solid angle

2/ r

A meteorological satellite circles the earth at a height h above the earth’s surface. Let the radius of the earth be and show that the solid angleunder which the earth is seen by the satellite sensor is

ea

)]/()2(1[2 2/12 hahha ee

• Radiant Flux: rate of energy transfer by electromagnetic radiation– Energy/time (J/s or Watt)

• Irradiance: Radiant flux per unit area.– Monochromatic irradiance– W/m2

• Radiance: Irradiance per unit solid angle.– Defines the irradiance coming from a particular direction.– W/(m2 steradian)

4. Radiometric quantities

• a = absorbed/incident = spectral absorptance

• r = reflected/incident = spectral reflectance

= transmitted/incident = spectral transmittance

• a + r + = 1

5. Absorption

6. Scattering

* A physical process by which a particle in the path of an electromagnetic wave continuously abstracts energy from the incident wave and reradiates that energy in all directions.

* Gas molecules m410~

Aerosols m1~

Water droplets m10~

Ice crystals m100~

Large raindrops and hails cm1~

* Small particle: scatter light, equally in forward and backward directions Larger particle: scatter more in forward direction, more complex

* Size parameter:

The ratio of the particle circumference to the incident wavelength

/2 ax a is the particle radius.

Rayleigh scattering: x << 1

Lorenz-Mie scattering: 1x

* Multiple scattering

Scattering more than once, important process for the transfer of radiant energy when aerosols and clouds are involved.

* Extinction: A result of scattering plus absorption, which removes energy from a beam of light traversing the medium and the light beam is attenuated.

Why does grass look green?

Why is the sky blue?

8.2 Radiation laws (blackbody)

Planck’s Law* Blackbody

– all incident radiation is completely absorbed.– In all ’s, the maximum possible emission is realized.– The radiation is isotropic.

* The amount of radiation emitted is uniquely determined by temperature.– Planck’s law.

* Relate the emitted intensity to the wavelength and the temperature of emitting substance.

* Rayleigh-Jeans distribution: Wien distribution: 0

Stefan-Boltzmann Law

• Total radiant intensity emitted by a blackbody.

• Integrate Planck’s law over all wavelengths.

• Blackbody irradiance.

• F = T4

: Stefan-Boltzmann constant

Wein’s Displacement Law• Wavelength of maximum intensity m = 2897/T,

units: m (m), T (K)

• Allows brightness temperature to be inferred from the radiation emitted by an object

Kirchhoff’s Law

• A molecule which absorbs radiation of a particular wavelength also is able to emit radiation of the same wavelength.

• Rate of emission is a function of temperature and wavelength

• Consider:– Two parallel plates of infinite extent (wall 1 and wall 2)– Both plates are at the same temperature.– Radiative transfer must be same in all directions.

– Wall 1 has an absorptance a1 = 1

* Non-blackbodies emit less radiation than do blackbodies.

* The ratio of emitted monochromatic irradiance to blackbody monochromatic irradiance is known as emissivity.

* Non-blackbodies absorb less radiation than do blackbodies.

* The ratio of absorbed monochromatic irradiance to blackbody absorbed monochromatic irradiance is known as absorptivity.

Absorptivity ( ) and emissivity ( )a

Kirchoff’s law: a

* For a non-blackbody (or a gray body), the Stefan-Boltzmann law is 4TF

1 a

strong emitters strong absorbersweak emitters weak absorbers

Practice Problem

• The average irradiance of solar radiation reaching the earth’s orbit is 1380 W/m2. Nearly all the radiation is emitted from the outer-most visible layer of the sun, which has a mean radius of 7x108 m. Calculate the equivalent blackbody temperature of this layer? The mean distance between the earth and sun is 1.5 x1011 m.

Practice Problem

• Calculate the equivalent blackbody temperature of the earth, assuming a planetary albedo of 0.30. Assume that the earth is in radiative equilibrium, so that there is no net energy gain or loss due to radiation.– Planetary albedo the fraction of the total incident

solar radiation that is reflected back into space without absorption.

Meteorology 342 Homework (8)

2. An infrared scanning radiometer aboard a meteorological satellite measures the outgoing radiation emitted from the earth’s surface in the 10 window region. Assuming that the effect of the atmosphere between the satellite and the surface can be neglected, what would be the temperature of the surface if the observed radiance at 10 is ?

3. A black land surface with temperature of 15 emits radiation at all frequencies. What would be

the emitted radiance at 0.7 , 1000 , and 31.4 GHz? Use appropriate Planck functions in

the calculations.

m

4. Assuming the average normal body temperature is 98 , what would be the emittance of the body? If it is not a blackbody but absorbs only 90% of the incoming radiation averaged over all wavelengths, what would be the emittance in this case? Also, at which wavelength does the body emit the maximum energy?

m 1128.9 srmWm

m 1cm

Fo

Co

1. a) Find the equivalent blackbody temperature of Mars. Assume the irradiance at the top of the earth’s atmosphere is 1370 , the distance from the earth to the sun is , the distance from Mars to the sun is , and that Mars has an albedo of 0.17.

b) Justify your answers to a) are correct by comparing your values to the solar irradiance at the earth and the radiative temperature of the earth.

2Wm m11105.1 km610228

8.3 Radiative transfer equation

1. The equation of radiative transfer

* Reduction of the radiant intensity due to absorption

* Increase in the radiant intensity due to emission and multiple scattering

JI

dsk

dI

: mass extinction cross sectionk

2. Beer-Bouguer-Lambert Law

The decrease in the radiant intensity traversing a homogeneous extinction medium is in accord with the simple exponential function whose argument is the product of the mass extinction cross section and the path length.

3. Schwarzschild’s equation

)(TBIdsk

dI

* The increase in the radiant intensity arising from blackbody emission of the material.

* Reduction of the radiant intensity due to absorption

eII )0(

s

dsk0

Optical depth:

8.4 Solar radiation (Shortwave radiation)

1. An energy source: the Sun

* Composed mostly of hydrogen (75% by mass) and helium (25%), with traces of iron, silicon, neon, and carbon

* Solar interior

a. Core: Temperatures of 15 million K, Density as high as 160,000 kg/m Extends to about ¼ of the radius of sun Thermonuclear reactions generate gamma- and x-rays Sun produces radiation in its high density core

3

2. The earth’s orbit around the sun

* All of the planets of the solar system revolve in elliptic orbits around the sun, which is one of the foci of the ellipses.

* All of the planets except Mercury and Pluto orbit nearly in the same plane, known as the ecliptic.

* The eccentricity of the earth’s orbit is currently about 0.0167, so it is very close to being a circle.

The average earth-sun distance is defined as an astronomical unit (AU) and has a value of 149,597,870 km.

The earth is farthest from the sun (1.0167AU) at aphelion, which occurs near July 4.

The earth is closest to the sun (0.9833AU) at perihelion, which occurs near January 2.

3. Solar constant:

The total solar irradiance reaching the top of the earth’s atmosphere on a surface perpendicular to the solar beam at the mean distance between the sun and the earth.

The recently proposed solar constant is about 1366 . It varies with theirradiance from the sun and changes with the sunspot cycle, but it doesn’t vary by much.

2Wm

Solar Spectrum

Two methods for determining the solar constant:

The solar constant value is critical in the interpretation ofmeasured solar absorption and heating rates in the atmosphere.

1) Satellite measurement with self-calibrating radiometers

2) Ground-based radiometers

Making use of Beer-Bouguer-Lambert law

4. Solar insolation

Insolation is defined as the solar irradiance on a horizontal area, is a contraction for INcoming SOLar radiATION.

Insolation depends on the zenith angle and the variable distance of the earth from the sun.

The insolation integrated with time over the hours of daylight will give the energy per unit area per day received at a point on the earth.

Example: On a clear day, measurements of the direct solar flux density F at the earth’s surface in a wavelength interval give the following values:

Zenith angle (degree) 40 50 60 70F ( ) 13.95 12.55 10.46 7.67

Find 1) the solar flux density at the top of the atmosphere, and 2) the transmissivity of the atmosphere for normal incidence in this wavelength interval.

2Wm

5. Atmospheric absorption and scattering of solar radiation

Solar radiation is essentially parallel beam radiation, i.e., all the radiation is coming in from an infinitely small solid angle. For such radiation we can use irradiance instead of radiance.

dzFkdF 0sec

* Scattering

Rayleigh scattering: For a scatter of a given size, the scattered intensity is inversely proportional to the fourth power of wavelength. Shorter wavelengths are scattered much more efficiently than longer wavelengths.

4

1~

I

)exp(0

H

F

dz

dF

* Absorption

6. Solar heating ratesThe absorption of solar radiation results in the heating of the atmosphere.

The heating rate is proportional to the amount of energy absorbed.

dz

zdF

ct

T

ps

)(1

8.5 Infrared radiation (Longwave radiation)

* As the sun emits radiation over all wavelengths, the earth and the atmosphere also emit radiation, which is referred to as (thermal) infrared radiation.

1. Infrared spectrum

* Solar and infrared spectra are separated into two spectral ranges above and below ~5 , and the overlap between them is relatively small.

m

Solar Spectrum Infrared Spectrum

* The emitted Planck radiance from the earth and the atmosphere is smaller than that of solar radiation, but the wavelength for the intensity peak is longer.

* The equilibrium temperature of the earth-atmosphere system

2. Absorption of infrared radiation in the atmosphere* The greenhouse effect (the atmospheric effect)

Energy emitted from the earth is absorbed by atmospheric gases such as carbon dioxide, water vapor, ozone).

* The atmospheric window

The atmosphere is relatively transparent from wavelength 8 to 12.5 , except for ozone which has an absorption band in the 9.6 region.

m m

m

* Carbon dioxide (a linear molecule)

Has no permanent electric dipole, but as it vibrates a dipole can be generated. Thus, it has vibrational bands, which are very important for the atmosphere. Major bands are centered at and .

2CO

m15 m3.4

* Water vapor (a bent molecule)

has strong absorption in many bands throughout the IR region, and isa strong absorber in the microwave region.

OH 2

* Nitrous oxide, methane, and carbon monoxide

have two very strong absorption bands in the IR.

has one strong band in the IR.

42 CHandON

CO

* Ozone (consists of three oxygen atoms)

3O Is a very strong absorber in the UV, has rotational and vibrational transitions, and absorbs strongly in the IR centered on the wavelength near . .m6.9

Atmospheric absorbers in the IR region:

3. Emission and infrared radiative transfer equation

A pencil of infrared radiation traversing an absorbing and emitting medium will be weakened by the absorption, but it may be strengthened by thermal emission from the medium. This pencil of radiation is usually represented by its radiance (or intensity) .

)(sec 0 BIkdz

dI

)(sec 0

IBkdz

dI

I

For downward propagating radiation:

For upward propagating radiation:

4. Infrared cooling rates

Because the atmosphere loses radiative energy to space through thermal infrared emission, it is normally cooled by this process.

dIF sincos22/

0

dFzF

0)(

)()()( zFzFzF

dz

zdF

ct

T

pir

)(1

Infrared radiation between cloud and surface

8.6 Radiation budgets

1. Radiation budget at the top of the atmosphere (TOA)

irFQrF )1(

Gains of radiative energy in the tropics and subtropics

Losses in the polar regions

2. Radiation budget at the surface

)]0([)1)(0()0( 4 irsssss FTrFF

3. Radiative heating/cooling rates of the atmosphere

Atmospheric solar heating is mainly produced by the absorption of water vapor in the troposphere and of ozone in the stratosphere.

In general, IR radiation serves to cool the atmosphere, radiating away to space an amount of energy equivalent to the solar input, maintaining the radiative balance.

Radiative cooling dominates solar heating almost everywhere. The ubiquitous net radiative cooling in the earth’s atmosphere is compensated for by other forms of energy in the atmosphere and by the transport of heat from the surface.

4. Radiative equilibrium

Equilibrium temperature vs Surface temperature

0)1()1( 44 TTrQ a

0)1( 44 TTArQ a

4/SQ

Example: Assume that the atmosphere acts as a single isothermal layer with a temperature that transmits solar radiation but absorbs all thermal infrared radiation. Show that the global surface temperature . Let the global albedo be 30%, and the solar content be 1366 . What is the global surface temperature?

aT

4/1)2( aTT

2Wm