RADIATION BASICS

• Significance of radiation in the atmosphere• Electromagnetic spectrum• Absorption/emission• Scattering• Line broadening

SIGNIFICANCE OF RADIATION IN THE ATMOSPHERE

1) CLIMATE

The Sun is the principal source of energy for the Earth. The global energy balance between solarand terrestrial radiation determines the global climate. This can be expressed as:

)1(4 ασ −=FTe (27.1)

where σ is Boltzmann’s constant, Te is the effective blackbody temperature of the Earth(~276 K), F is the solar constant averaged over the atmosphere (~345 W/m2), and α is the average albedo of the Earth (~0.3 or 30%).

Note: The effective blackbody temperature (also called the radiation temperature) is about 31 K colder than the global mean surface temperature, which is about 278 K. This difference is called the natural greenhouse effect.

Eq. 27.1 is deceptively simple. We could, for example, use it to try to answer the question“what happens to Te if F decreases?” The naïve answer is that Te also decreases. But this maynot be so, because the albedo is likely a non-linear function of Te (due to clouds, ice caps andvegetation variations). Hence the conclusion is no longer so obvious. It is even harder to useEq. 27.1 to infer the effect on the surface temperature of a change in F, unless we know moreabout the climate system and its feedbacks.

The sketch below shows the interaction of incident solar radiation (normalized to 100 units) withthe Earth and its atmosphere.

The following figure shows typical spectra of solar and terrestrial radiation, and some of theprincipal absorbers.

2) WINDS

The distribution (latitudinally and seasonally) of radiation over the atmosphere determines thetemperature gradients (particularly the meridional temperature gradient), and these driveatmospheric winds. If you have taken dynamic meteorology, you may recall the thermal windequation:

TkfT

g

z

vp

g ∇×=∂∂ vv

ˆ (27.2)

Eq. 27.2 states that the vertical wind shear is controlled by the horizontal temperature gradient.Dynamics students will also know that if the vertical wind shear is sufficiently strong, zonal flow can become baroclinically unstable, leading to the occurrence of mid-latitude cyclonicsystems, which are associated with our weather. The sketch below illustrates the fact that, onaverage, there is a net radiation surplus near the Equator and a net radiation deficit at thePoles. This energy imbalance leads to the creation of the meridional temperature gradientalluded to above. The general circulation of the atmosphere and oceans results from these two“fluids” attempting to erase this temperature gradient in order to come to thermal equilibrium. Since the system is always forced by the Sun, the time average climatological state represents abalance between radiative forcing, which acts to strengthen the gradient, and atmosphere/ocean dynamics, which act to weaken the gradient.

Sandström’s theorem states that a closed, steady circulation can be maintained in an atmosphere only if the heat source is maintained at a higher pressure than the heat sink. Thisoccurs in the Earth’s atmosphere, because the atmosphere is largely transparent to solarradiation but is quite opaque to parts of the terrestrial radiation spectrum. Absorption ofsolar radiation by water vapour, however, is emerging as an important component of theoverall radiation balance. Integrated over solar wavelengths, modern amounts of water vapour inthe atmosphere absorb approximately 19% of incoming solar radiation.

3) REMOTE SENSING

Satellites now permit vertical sounding of the atmosphere for temperature and the concentrationof various atmospheric components. This can provide much greater horizontal spatial resolutionthan with the current rawinsonde system. However, at the moment, the vertical resolution is notas good as with rawinsondes, and clouds can interfere with the remote sensing necessary toobtain vertical profiles of temperature. The following figures show some typical “remotelysensed” spectra. After this series of lectures on radiation, you should be able to interpret themin terms of the physical processes that are occuring, and affecting the outgoing radiation.

Note: these plots can be animated if you go to the University of Oregon’s site below. Thesestatic images are for December only.

http://geography.uoregon.edu/envchange/clim_animations/

Note: these plots can be animated if you go to the University of Oregon’s site below. Thesestatic images are for December only.

4) METEOROLOGICAL OPTICS

The scattering of visible radiation in the atmosphere, by liquid water droplets and ice crystals,leads to a number of aesthetically pleasing, and sometimes theoretically challenging, phenomena—rainbows, double rainbows, haloes, glories, sunsets, etc. Sadly, we will not havetime to cover them in this course.

ELECTROMAGNETIC SPECTRUM

Electromagnetic radiation is a transverse wave that propagates at c=2.997925x108 m/s in avacuum. In a dielectric medium, c=c(). This is known as dispersion. All waves satisfy therelation c= where is their wavelength and is their frequency. The reciprocal of wavelength is called the wave number. Spectroscopists seem to prefer it to wavelength. Theelectromagnetic spectrum can be divided into seven broad categories. The table below givesthe upper limit of the wavelengths associated with each category.

Electromagnetic waves Maximum wavelength (m)

-rays 10-4

X-rays 10-2

ultraviolet 0.39

visible 0.76

infrared 103

microwave 106

radio

INTERACTION OF MATTER AND RADIATION

Consider an isolated molecule. The kinetic and electrostatic potential energy of the electronsare quantized. So are the rotational and vibrational kinetic energies of the atoms. Thetranslational kinetic energy, however, is not quantized. The energy state of the molecule (totalquantized energy) is the sum of the electronic, rotational, and vibrational energies. There areseveral ways in which changes in the energy state of a molecule may occur.

EMISSION

Emission occurs when a collision leads to an excited energy state, followed by a radiationaltransition to a lower state. In this case, thermal (internal) energy is converted into radiantenergy. The emitted energy is in the form of a photon of frequency given by E=h, where E is the energy difference between the two states (quantized) and h is Planck’s constant=6.626x10-34 Js.

Since only certain energies are permitted, the frequency takes on discrete values, which arecharacteristic of a particular molecule. This gives rise to a line spectrum Emission linesassociated with electronic orbital transitions are usually in the X-ray, ultraviolet, and visibleportions of the spectrum. Pure vibrational transitions are usually in the infrared (e.g. watermolecules in food vibrating in your microwave), while pure rotational transitions are frequently in the microwave region of the spectrum. Some molecules (e.g. CO2, H2O, O3) canundergo simultaneous rotation and vibration collisions, giving rise to line clusters known as

rotation-vibration bands.

ABSORPTION BANDS

A molecule may absorb a photon of a frequency appropriate to raise it to a quantized higherenergy state. Following this absorption the molecule may return to a lower energy state during acollision. The quantized nature of the energy state transitions of an atom or molecule means thatabsorption and emission spectra are quite similar for a given substance. The net result is that some of the photon’s energy is converted into thermal (translational)energy of the molecule. Remembering our kinetic theory of gases, this implies that we couldincrease the temperature of a gas by shining radiation on it at those wavelengths where thegas absorbs strongly. This is the case for ozone in our stratosphere (with global climaticimplications).

SCATTERING

A molecule absorbs a photon, following which transition to the original state occurs byemission of an identical photon, but in a different direction.

PHOTOCHEMICAL DISSOCIATION

A molecule absorbs a photon, which causes it to break down into its atomic components (i.e. itbreaks the chemical bond). An example in the upper atmosphere is the photo-dissociation ofmolecular oxygen (O2) into atomic oxygen (O). This process is a precursor to the formation of

ozone. This process involves a continuum of photon energies above a particular threshold,and these photons come from the Sun.

OOhO +→+ 2

where h is the energy of the photon and must be higher than that associated with wavelengthsof 0.24 m (i.e. <0.24 m).

PHOTOIONIZATION

A molecule absorbs a photon, which leads to the removal of one or more electrons. This usuallyrequires <0.1 m, and, as for photochemical dissociation, the photon need not have a discretefrequency.

MOLECULAR INTERACTION EFFECTS

1) NATURAL BROADENING

So far we have considered an isolated molecule (except for “instantaneous collisions” that injector remove energy). Such a molecule exhibits a line spectrum, whose line widths are determinedsolely by the Heisenberg uncertainty principle Et~h/2, where E is the uncertainty in thetransition energy and t is the lifetime of the excited state. This is known as naturalbroadening, but in the atmosphere it is too narrow to be of any consequence.Note: natural broadening can also be explained “classically” in terms of the damping of a harmonic oscillator during emission (see Liou, p. 16).

2) DOPPLER AND COLLISION BROADENING

Within a gas, natural broadening of absorption lines is greatly enhanced by Dopplerbroadening, due to the random translational velocities of the molecules with respect to theobserver, and by collision broadening, which results from the interaction of the electrostaticforce fields of the molecules during collisions, and the resulting small changes in theirquantized energy states. Below about 30 km in the atmosphere, collision broadening is thedominant mechanism, and it always dominates in the wings of spectral lines.

3) LIQUIDS AND SOLIDS

Here the interaction between molecules is strong because of their proximity, with the resultthat the line spectrum is “smeared out” into an essentially continuous spectrum.

RADIATION BASICS II

• Radiance• Irradiance• Solar constant• Heating rate

DEFINITIONS AND CONCEPTS

We will begin by defining several terms used to describe aspects of atmospheric radiativetransfer. Unfortunately, the terminology and symbology in this field have not been completelystandardized, so you will encounter other terms and symbols. I have indicated some of thesein parenthesis below.

Monochromatic (spectral, spectral density) implies radiation of a particular frequency orwavelength. The term is also used to indicate the value of some quantity per unit interval offrequency or wavelength.

Integrated implies that the quantity in question has been integrated over a finite wavelength orfrequency interval.

Solid angle is a “three-dimensional” angle. In two-dimensions, the radian measure of an angleis the ratio of the circumference subtended by the angle to the radius. The equivalent definitionof solid angle is the ratio of the area subtended by the solid angle (think of the angle at the apex

of a cone, for example) to the square of the radius. Solid angles are measured in steradians (sr).the steradian is a non-dimensional quantity. The solid angle, , can be related to the polarangle (zenith angle), , and the azimuth angle, , in polar coordinates:

φ ddd sin= (28.1)

Radiant flux (radiant power, luminosity), P, is the total radiant energy, Q, per unit time emittedby or incident upon a surface. Its units are Watts.

)(Wdt

dQP = (28.2)

As an example, our Sun has a radiant flux of 3.9x1026 W.

Irradiance (flux density, emittance), F, is the radiant energy per unit time passing through unitarea. It usually refers to the radiant energy arriving from 2 steradians; that is, from thehemisphere on one side of the surface. In general, the irradiance depends on the orientation ofthe surface, which can be specified by the unit vector normal to the surface. Hence theirradiance can be thought of as a vector whose direction is opposite to that of the surfacenormal. Its units are Watts per square metre.

dA

dPF = (28.3)

Radiant intensity, J, is the radiant flux per unit solid angle from a point source. Its units areWatts per steradian.

d

dPJ = (28.4)

Radiance (intensity, specific intensity, brightness, luminance), L, is the radiant energy per unit time and per unit solid angle, passing through unit area normal to the beam and arrivingfrom a particular direction, or travelling along a particular direction (although this definitionbreaks down for a collimated beam). Its units are Wm-2 sr-1 .

Note: In British books, the symbol used for radiance is often I.

φ

ddA

PdtzyxL

cos),,,,,,(

2

= (28.5)

The schematic below illustrates the definition of L and Lambert’s cosine law.

Lambert's cosine law states that the intensity of radiation along a direction which has angle with the normal to the surface is: L=Lncos where Ln is the intensity of radiation in the normal direction.

The relation between F and L is simply:

∫=

2

cos dLF (28.6)

In general, there will be radiant energy travelling in both directions through a particularsurface (e.g. a slab of atmosphere). In order to distinguish between the irradiance in the twodirections, some convention is established (e.g. up and down if the surface is horizontal) andthe two irradiance values are denoted as F+, F- or F, F. The difference between the twoirradiances is called Fnet. As a general rule, one may say that irradiance is measured by anon-imaging device (e.g. a pyranometer), while radiance is measured by an imaging device(e.g. a telescope).

Isotropic radiation is a radiation field in which LL(,). In this case F=L. An example ofisotropic radiation that we will encounter next is blackbody radiation.

Diffuse radiation is radiation emitted from a source with a finite solid angle; for example, skyradiation.

Parallel beam radiation (collimated beam) is radiation from a point source. For our purposes,the Sun is sufficiently distant that it can be considered to be a point source, and the radiationfrom the Sun is essentially parallel beam radiation. For parallel beam radiation, the relationbetween F and L becomes:

δcosLF = (28.7)

where δ is the solid angle of the source. For the Sun, 127102 −−×= srWmL

The solar constant is the solar irradiance at the annual mean distance of the Earth, measuredon a surface above the atmosphere that is perpendicular to the solar beam.

δLF = (28.8)

The solar constant is approximately 1353 Wm-2 .

RADIANT ENERGY DENSITY

We will derive an equation for the radiant energy density u (Jm-3 ), in a medium through whichelectromagnetic energy is flowing.

After a time dt, the energy which has crossed the area dA in a particular direction will belocated in a cylinder of length cdt, where c is the speed of light in the medium. Dividing theenergy by the volume of the cylinder leads to:

c

Ld

cdtdA

dALdtddu

==

coscos (28.9)

Integrating over all possible directions from which the radiation may come:

∫=

4

1Ld

cu (28.10)

For isotropic radiation, Eq. 28.10 becomes:

c

F

c

Lu

44==

(28.11)

RADIANT HEATING RATE

The radiant heating rate h (Wm-3 ) is simply proportional to the divergence of the irradiance.

Note: this is the analogue of the continuity equation for conservation of mass.

This can be incorporated into the first law of thermodynamics to give:

dt

dTcFh pρ=⋅∇−=

vv(28.12)

where:

kFjFiFF zyxˆˆˆ ++=

v(28.13)

and

∫=

4

cos dLF xx (28.14)

and x is the angle between the beam and the x-axis (that is, Fx is the net irradiance passingthrough the y-z plane; some thought should convince you that it is the net irradiance thatdetermines the heating rate—hence the integration over 4 steradians rather than 2 steradians).

The other components of the irradiance vector can be obtained by symmetry. Substituting Eq. 28.14 and the other components into Eq. 28.13, and thence into Eq. 28.12, and using thefact that cosx =dx/ds, where s is the distance along the beam, leads to:

dds

dLh ∫−=

4

(28.15)

RADIATION IN THE ATMOSPHERE

We present here some definitions (consistent with WMO standards) of various quantitiesrelevant to atmospheric radiation.

Short wave radiation (solar radiation) is taken to be radiation with <2.5 m.

Long wave radiation (terrestrial radiation) is taken to be radiation with >2.5 m.

Direct solar radiation is the radiation from only those directions defined by the Sun’s disk,falling on a surface normal to the beam.

The vertical component of direct solar radiation (downward direct solar) is the direct solar

Thus, in order to determine the heating rate, we need to know how the radiance varies alongthe beam. This will be the subject of Beer’s Law coming up.

radiation received by a horizontal surface.

Diffuse solar radiation (sky radiation) is the downward solar radiation received on a horizontalsurface from 2 steradians (i.e., the upper hemisphere), omitting the solar disk.

Note: At solar elevation angles greater than about 70o, the ratio of diffuse to direct solar radiation is about 10%.

Global solar radiation is the downward direct and diffuse solar radiation received by a horizontal surface from 2 steradians.

Atmospheric radiation is the radiation emitted by the atmosphere (both up and down) and theradiation emitted upward by the Earth’s surface.

Net radiation is the net of the upward and downward solar radiation, or the net of the upwardand downward terrestrial radiation.

BLACKBODY RADIATION

• Planck function• Stefan-Boltzmann Law• Wien’s Law• Kirchoff’s Law

A blackbody is one which absorbs all incident radiation. A blackbody also emits the maximumpossible irradiance at all wavelengths, for a given temperature. It is an idealized concept, butnevertheless one that can be approximated quite well in the laboratory by a radiant cavity.The Sun’s emission spectrum is very much like that of a blackbody. Paradoxically, snow emitsand absorbs infrared radiation like a blackbody.

PLANCK FUNCTION

The Planck function describes the blackbody spectrum, that is the blackbody monochromaticradiance, LB (also denoted by B in some books), as a function of temperature andwavelength:

1)/exp(

125

2

kThc

hcL B (29.1)

where h is Planck’s constant (6.625x10-34 Js), and k is Boltzmann’s constant (1.38x10-23 JK-1 ).The blackbody radiance can also be expressed as a function of frequency using the fact that

LBd=LBd. Since blackbody radiation is isotropic, FB= LB.

STEFAN-BOLTZMANN LAW

Although the Stefan-Boltmann law was derived before Planck’s law, it gives an expression forthe integrated irradiance over all wavelengths (the area under the Planck curve, in fact):

4432

44

0 15

2TT

hc

kdFF BB σ ∫

(29.2)

where σ is the Stefan-Boltzmann constant, 5.67x10-8 Wm-2 K-4 .

WIEN’S LAW

Wien’s law gives the wavelength of the peak of the Planck function. It can be found by setting

0BF

(Note that the peak in FB does not occur at the same wavelength.)

This leads to:

)(),(2898

KTmwithTp (29.3)

KIRCHOFF’S LAW

Absorptivity, a, is defined as the ratio of absorbed radiant energy to incident radiant energy.That is:

inc

abs

F

Fa

(29.4)

Emissivity, , is defined as the ratio of emitted radiant energy to the radiant energy emitted by a blackbody at the same temperature and wavelength:

BF

F

(29.5)

Kirchoff’s law states that under the conditions of local thermodynamic equilibrium (whichrequires that the molecular collision frequency should be much greater than the frequencyof emission or absorption by gaseous molecules, which holds below about 60 km inthe Earth’s atmosphere):

Tsameata (29.6)

Kirchoff’s law is sometimes colloquially phrased as “a good absorber of radiant energy is agood emitter of radiant energy.” This is essentially true, but keep in mind the caveats that thewavelength and temperature must be the same for absorption and emission, and localthermodynamic equilibrium must also hold.

A grey body is also an idealization in which the emissivity is independent of wavelength. It isa way of sidestepping the need to take into account the line spectrum of real gases byessentially assuming that their emission is a constant fraction of the emission of a blackbody.

We conclude by defining two additional concepts that will be useful in the upcoming sections.Reflectivity (albedo) is the ratio of reflected to incident irradiance:

inc

r

F

Fr

(29.7)

Transmissivity (also called transmittance) is the ratio of transmitted to incident irradiance:

inc

t

F

Ft

(29.8)