velut Ævo arbor university of toronto …j.t. houghton physics of atmospheres (cambridge) advanced...

ÆVOARBOR

VELUT

UNIVERSITY OF TORONTODEPARTMENT OF PHYSICS

PHY315S

RADIATION INPLANETARY ATMOSPHERES

Spring 1999

B. ToltonPHY 315S

-ii-

PHY 315SRadiation In Planetary Atmospheres

Boyd ToltonDepartment of PhysicsUniversity of Toronto

January 1999

LECTURER B. Tolton, Room 604 (Burton Tower) Dept of Physics Tel: 946-3683,Fax: 978-8905, E-mail: [email protected]

LECTURES Tuesday 9:00AM and Friday 10:00AM, Room 118

MARKING SCHEMEProblem Sets ( 3 * 10% ) 30%Essay 20%Mid-term Exam 15%Final Exam 35%

TOTAL 100%

DEADLINES Essay Topic Lottery 10:00 am Friday 8th JanuaryProblem Set 1 9:00 am Tuesday 2nd FebruaryMid-term Test 10:00 am Friday 26th FebruaryProblem Set 2 9:00 am Tuesday 16th MarchEssay 9:00 am Tuesday 26th MarchProblem Set 3 5:00 pm Tuesday 5th April

PENALTIESFor any piece of work a penalty of 1 mark per working day or part thereof up to 1week (total 5 marks) after which the work will not be accepted. Exceptions must berequested at least 24 hours before the deadline and may or may not be granted.

LECTURE NOTES

Will be provided at cost. There are about 130 pages of notes in all. I'm estimatingthat it will cost less than $10 to replicate them.

PHY 315S - Radiation In Planetary Atmospheres

-iii-

TEXTThere is no recommended text for this course.

You may be interested in:

R.M. Goody & Y.L. Yung Atmospheric Radiation (OUP) advancedJ.T. Houghton Physics of Atmospheres (Cambridge) advancedR.M. Goody & Walker Atmospheres (Prentice-Hall) element.Kuo-Nan Liou An Intro. to Atmos. Radiation (AP) advancedLewis & Prinn Planets and Their Atmospheres (AP) intermed.

OUTLINE (not necessarily in this order)

Introduction to Radiation. Elementary radiation laws, their physical basis andsimple examples. Concepts of intensity and flux.

Introduction to Radiation in an Atmosphere. Solar and planetary radiation. Somesimple radiation balance models. The plane-parallel atmosphere approximation.Atmospheric processes of extinction. Simple models of atmospheric energytransport.

Comparative Studies of Radiation in Planetary Atmospheres. The "runawaygreenhouse" effect and a study of the atmospheres of mercury, venus, earth andmars. Methods of studying atmospheric content, temperature and radiation balance.

Climatic Effects of Radiation Balance and Perturbations. Climate modelling withemphasis on radiation transport. Paleoclimatology, the ice ages. The effects of manon the atmosphere - the greenhouse problem, volcanic dust effects, possible furthermodifications.


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ESSAY NOTES and TOPICS

Essay Due: 5:00 pm Monday March 26th

PENALTIES

For any piece of work a penalty of 1 mark per working day or part thereof up to 1 week(total 5 marks) after which the work will not be accepted. Exceptions must be requested at least24 hours before the deadline and may or may not be granted.

SOME POSSIBLE TOPICS

Solar Radiation in the Earth's (or Venus' or Mars') Atmosphere Recent Measurements of Radiation in a planet's atmosphere Measuring the Solar Constant The Radiative History of Planetary Atmospheres Climatic Change - is it related to Radiation? Scattering Remote sounding of Planetary atmospheres Nuclear Winter What's Happening to the Ozone Layer? Radiative and Other Aspects of the Antarctic Ozone Hole Refractive Effects in Planetary Atmospheres Rainbows, Haloes and Other Phenomena Why it is Dangerous to use a laser weapon on Venus?

Essay topics will be assigned by ballot from a list similar to the above. I will beavailable to "start you off" on your essay research and will also be available for consultationas you proceed. See notes below on writing essays.

RESTRICTIONS

No essay may be part of a submission for another course.


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GENERAL

An essay consists of a piece of connected English prose arranged in logical order ofthought, i.e. with a beginning, a middle and an end! English language consists of sentencesorganised into paragraphs which are assembled into titled sections. Point form is notacceptable.

The essay should be between 9 and 13 sides of 8.5 x 11 inch paper at about 30 lines perside and 11 words per line. Diagrams, tables and appendices are excluded from the count.Essays should preferably be typed, handwritten is acceptable, the essay MUST BE LEGIBLE.There will be a small weighting for style but no weighting for presentation unless it is difficultto read or follow. There will be no penalty for exceeding the recommended length by up toabout 25% if the prose is relatively concise. There will be a penalty for "padding".

I do not expect original research. I do expect you to have dug in books, looked up papersand generally read around. Finding sources is part of your training so although I will beavailable to advise on generally where you might look, I will not provide a bibliography.References must be cited in the text and referred to in a Reference List at the end of the essay.

One of the most difficult problems that scientists have is communicating their researchand findings in a manner that is simultaneously interesting, correct, and readable. This is anopportunity for you to practice such skills. You may feel that the space allotted is very tight -it is! The reason for doing so is to get you to sort out the important information from the rest.Nobody - including particularly your future boss - has time to read long-winded padding. Oneof my own favourite tricks is to ask for a report on one side of a piece of paper so that I havetime to read it!


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A(x,t)A(x,t1/f )

A(x,t)A(x,t)


Introduction

Since this entire course is going to deal with radiation we should consider initially whatwe mean by radiation and what we already know about it. The first few lectures therefore willbe in the nature of revision/levelling in which we shall revise some of the basic properties ofradiation.

Radiation as Wave Motion

By this time you will all be aware that when we talk of radiation in an atmosphericcontext we do not normally mean nuclear or particle radiation but electromagnetic radiation asdefined by solutions of Maxwell's equations. These electromagnetic wave phenomena havebeen studied fairly extensively over many centuries as the wave theory of light and associatedfrequencies. Briefly light is considered to be a wave-like phenomenon with a propagationvelocity of 2.998 x 108 ms-1 in a vacuum and a velocity µ times slower in a medium, where µis the refractive index of the medium and is (almost) invariably greater than unity. Lighttherefore travels fastest in vacuum and slower in a material medium. A wave motion has afrequency and wavelength associated with it such that the frequency is the time taken for thedisturbance to execute one complete cycle at a point in space and the wavelength is the distancealong the direction of propagation for one complete cycle of the motion at a particular timeinstant. We can express this as:

and

where the direction of propagation is assumed to be in the x direction. We can thereforecombine these two relationships with the velocity of propagation, c, to give the relationship


-2-

cf

A(x,t) asin 2%x

2%ft

There are several other simple relationships which we routinely use in considering thesewaveforms. Firstly, by associating a single frequency f with the wave we have presupposedthat it has a sinusoidal form such that:

where the minus sign comes from a consideration of the direction of propagation (from x = 0to x = ). In order to avoid carrying factors of 2% around with us we therefore define the wavevector k as k = 2%/ and the angular frequency 7 = 2%f. In atmospheric situations we also runinto the quantity 1/ often enough to merit a special symbol and I denote that as = 1/. Tocomplete a survey of the notation we also should remember that the symbol is also used forfrequency.

A problem should be noted here concerning the symbols and which I use forfrequency and "wavenumber" respectively. Some authors reverse the definitions - sorry but it'stoo late now to correct that situation and you will just have to be careful which is which.

The fact that light could be treated as a wave phenomenon has been known for a longtime and is already well known to you as evidenced by experiments with Young's slits and soon. The fact that "ordinary" light may be split up into a number of "colours" is evidence of thewave nature of the phenomena as is exemplified by the theory of diffraction and refraction(gratings and prisms). However there is an infinite spectrum of wavelengths of e-m radiationof which visible light is only a very small part. It was discovered fairly early on in the historyof dispersive experiments that there are wavelengths beyond the red end of the spectrum andthese are called infra-red rays, having a longer wavelength than visible radiation. There arealso wavelengths beyond the violet end of the spectrum known as ultra-violet rays and thesehave wavelengths shorter than the visible radiations.

We can in fact draw a fairly simple map of the electromagnetic spectrum using varioussources of information and plot the various phenomena against their wavelength producing atable thus:


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Wavelength(m)

Frequency(Hz)

0

Gamma Rays

10-11 3.0 x 1019

X-Rays

10-8 3.0 x 1016

Ultraviolet

Visible Violet 4.0 x 10-7 7.5 x 1014

Visible

Visible Red 6.3 x 10-7 4.8 x 1014

Infrared

10-3 3.0 x 1011

Microwaves

1 3.0 x 108

Radio Waves

0

Note that the visible wavelength range occupies only a very small fraction of the totalspectrum although it is the radiation form that we are most familiar with. Although it is oftenhelpful to consider light as an example of radiation, it is also a trap because consciously orunconsciously we expect all radiation to always behave like light and that is not the case.

A further point to note about radiation is that the disturbance is in the form of an electricand a magnetic field a right angles to each other and the direction of propagation thus:


-4-

E

H

N

Direction ofProgagation

Figure 1: Propagation ofElectromagnetic Radiation

N

E

El = |E|cosθ

Er = |E|sinθ

Figure 2: Polarisation

E El sin(7t)l Ersin(7t1)r

A consideration of the rate of propagation ofenergy leads to the conclusion that the energypropagated along the wave is given by the productof the electric and magnetic fields. Specifically thePoynting vector N points in the direction ofpropagation and is of magnitude factor.|E|.|H|,where the form of the factor depends upon the unitssystem you are using. However since the electricand magnetic fields are simply related to each otherby a constant of proportionality related to themedium in which propagation is taking place, theenergy transport is proportional to E2.

The fact that the electric and magnetic fieldsare in specific directions also leads to considerationof polarisation which is the nature of the totalelectric vector. A simple single sinusoidal wave isplane polarised because the electric vector isconstant in direction in time.

One can also consider two coherentsuperimposed beams with electric vectorcomponents El and Er which are at right angles toeach other. The total electric vector at one point isthen:

where 1 is the phase difference between the twobeams.

This may be:

a straight line 1 = n% (linear polarisation)

a circle Er = El, 1 = (2n+1)%/2 (circularpolarisation)

an ellipse (elliptical polarisation) in the general case.

Notice that this calculation is restricted to coherent radiation which is not the generalcase.


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hc hf 6ck 67 hc/

NpcNhc

U

Radiation as Particles

Along with a discussion of the wave nature of radiation must go a discussion of theparticle nature as well. The co-existence of these two descriptions leads to all sorts ofinteresting philosophical questions but here we shall take it to be the case that radiation behaveslike both a particle and a wave and that one aspect may be more rewarding to consider at anyone time.

A radiation beam is described as a flux of massless photons all travelling at the speedof light, c, all carrying an energy:

where h is Planck's constant (6.6262 x 10-34 Js) and 6 = h/(2%).

From the de Broglie theory of wave-particle duality these photons each have momentump given by p = h/.

Evidence for the particulate nature comes from the photoelectric effect and absorptionspectra of materials of which we shall say more later.

As a very simple example of the use of the particle theory of light consider the case ofa unidirectional stream of photons normally incident on a plate. This is the same case as thatof a plane wave hitting a plate. If we assume that the plate absorbs all the photons then the totalmomentum imparted to the plate per unit area per unit time is given by:

where N is the number of photons per unit volume in the input stream and U is the total energyper unit volume of the input stream.

The rate of acquisition of momentum per unit area is just the force per unit area whichis also the pressure exerted by the input beam. Therefore the radiation pressure on an absorbingplate from a plane input beam is given by P = U, the input beam energy density (Jm-3).

In the case that the plate reflects the energy (a mirror) then the momentumimparted to the plate is doubled and the pressure is 2U.


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L D

D

D

D L

L

L

L = LightD = Dark

Figure 3: Crooke's Radiometer

Now consider Cookes'radiometer. It has four vanes(traditionally), as shown below,which in a radiation beamexperience a torque in thedirection such that rotation occurswith the dark faces leading.

H o w e v e r C o o k e s 'radiometer almost invariablyrotates in the opposite directionowing to residual gas in the bulb.The dark faces become hotterbecuase they absorb more of theradiation than the light faces. Gasmolecules striking the dark faceand then "bouncing off" do sowith more momentum (i.e. theyare hotter) than those leaving thelight face, thus producing a torquewhich is in the reverse directionto that expected from a purelyradiative argument and also much greater in magnitude.


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s

dω

dA

p

Figure 4: Definition of Intensity

dE I(p,s,,t)dAdd7dt

I P

0

I

d

Fundamental Definitions of Intensity and Flux

In order to intelligently discuss the concepts of radiative transfer it is necessary to definetwo concepts which we shall use extensively. The concept of intensity which is the radiationflow in a particular direction at a particular point and the radiation flux which is the total energyflow in a particular direction.

Intensity

Consider a point p in space. The totalamount of energy which flows through asmall area dA whose normal is directed alongthe vector s in a time dt in a wavenumberinterval d in a direction within a small coneof solid angle d7 with s as its axis is:

which is the defining equation for theintensity. Note that the intensity in general isa function of both spatial position andorientation. If the intensity is a function onlyof direction but not position the field ishomogeneous and if it is independent of allspatial and directional parameters it is said tobe homogeneous and isotropic. As definedthe intensity is a function of the wavenumberor more specifically I have defined the "monochromatic" intensity I

. I shall use the symbol

I alone to describe the wavenumber-integrated form of the intensity which properly describesthe energy flow.

A simple example of intensity is the visible radiation field in this room which dependsupon direction (nearly entirely downwards from the lights) and position (there is little energyflow in the corners or under the chairs). A homogeneous radiation field is the plane wavewhere the intensity function does not depend upon the spatial position of the point but doesdepend upon the direction considered. We shall consider a homogeneous and isotropic field ina moment.


-8-

dA

s

Figure 5: Definition of Flux

dE F(p,s,,t)dAddt

r sinθ dφr sinθ

r dθ

φ

dφ

dA

θθ dθdΩ

X

Y

Z

Figure 6: Integral Relationships of Intensity and Flux

F

P6

Icos d7

(Net) Flux

The intensity is a useful quantity to consider butan almost equally useful concept is that of the energyflux. This may be conveniently considered as theamount of energy flowing through the area dA whichhas a normal vector of s, i.e. if dA is a hole in a barrierthe flux is the total energy flowing across the barrierthrough the hole dA.

Notice that since there are two energy flows,right-to-left and left-to-right, the total energy flow is thevector sum of these two flows. This flux - which issometimes called the "net flux" to emphasise that it isthe difference between the flows in both directions -is defined in a similar way to the intensity as:

There is clearly a relationshipbetween the intensity and the fluxas in some sense the latter is theintegral of the former. Howeversince the intensity is defined interms of the normal to dA theintegration must take account ofthe fact that the orientation of dAremains fixed.

The projection of dA alongs1 is just dAcos where is theangle between s and s'. Howeverusing spherical polar co-ordinateswe can reduce the integral to thesimple form:


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F

P6

Icos d7 I

P6

cos 2%sin d 0

D

d

I

D >> d

Figure 7: Schematic of a Cavity

P6/2

Icosd6 I 2% P%/2

0

cossind %I

dAcdt

I

Figure 8: Photon Description ofCavity Radiation

If, and only if, the intensity is isotropic we can further simplify this to:

Isotropy, Flux, Intensity and Energy Density

A particular case of interest to thediscussion below is the case of the energyleaving a cavity with an isotropic radiationfield inside it thus:

The intensity of radiation coming through thehole is just I (isotropic) and therefore thetotal flux or energy emerging from the holeis:

We can also relate the intensity to theenergy density in the cavity by consideringan increment in time dt during which anumber of photons from a layer thickness cdtemerge in a solid angle d7 around s thenormal to the hole which is of area dA.

Since the directions of the photons arerandom, the number of photons per unitvolume travelling in d7 about s must be justNd7/4% ( 4% is the total solid angle) where Nis the total number of photons per unitvolume. Since the total number of photons isjust a measure of the energy we can applythe same argument to U, the energy per unit volume in the field, the total energy emerging fromthe hole in the solid angle d7is:


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UdA cdt d7/4% I dA d7 dt

I Uc4%

I 14% P

6

I(7)d7

0.00

0.05

0.10

0.15

0.20

Rad

ianc

e(W

m-2

sr-1

(cm

-1)-1

)

2500 2000 1500 1000 500 0

Wavenumber (cm-1)

300K

Figure 9: Blackbody Emission Curve

where the equality relies on the definition of the wavenumber-integrated intensity given above.Thus

This definition may be extended to the monochromatic cases by simple argument and to thenon-isotropic cases where I is replaced by I where:

by more careful considerations.

Blackbody Radiation - Cavities and so on


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It had already been observed before 1900 that the radiation inside an enclosure was:

- independent of the type of material the enclosure was made of

- isotropic in nature (that is it was the same in all directions)

- depended upon the temperature of the enclosure.

- total energy was proportional to T4

- peak emission was given by 2T

- energy distribution is given by the diagram below.

The enclosure, or blackbody cavity, is an interesting concept in radiation theory andmerits a bit of study, but before we do that a point regarding the measurement of the radiationfield should be noted. We can measure the radiation field inside an enclosure by cutting a smallhole in it and observing the radiation leaving the enclosure and we call this "blackbodyradiation". However consider the reverse case of radiation hitting the hole and going into theenclosure. The probability of a photon finding its way back to an infinitely small hole beforebeing absorbed on the cavity walls is zero, and therefore we can state that this hole is black inthe sense that all radiation falling on it is absorbed. A hole in a blackbody cavity thereforeabsorbs all radiation incident upon it.

Let us consider the distribution of energy in the cavity. We know that there is aradiation field inside the cavity which we may characterise as U( ,T), using U as the energydensity again.

Since the field is not a property of the cavity walls it must be a property of the cavityitself and we are led to seek out the various possible stationary states of photons or waves inthe cavity and their populations.

The energy density in the cavity will be the product of the following terms:

- number of states of a particular energy e (density of states)

- energy of the state (e)

- probability of any one state being occupied (state population).

This must then be integrated over all states to get the total energy E or normalised to unitvolume, the energy density U.


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1

2

n2x

(2X)2

n2y

(2Y)2

n2z

(2Z)2

r 2 n2

x n2y n2

z

nx

nz

ny

r

Figure 10: Cartesian Description of nx, ny, nz

r 2X

a) The Density of States Problem

An initial point to note is that since the energy in the cavity is independent of the shapeof the cavity we can choose a cavity of any convenient shape for our calculation. If we wantedto be rigid about the proof we would have to show that the results then held for any shape andsize of cavity, but I'm not going to go that far. I'm also going to use a "seni-classical" approachto the problem and treat the energy as an electromagnetic phenomena.

Considering the simple case of a 1-D confinement of length L to start with, we knowfrom elementary wave theory that the allowed stationary states or resonant modes are given byn/2L.

In the 3-D case we have three numbers to worry about but we get a rather similarformula for a rectangular cavity:

The derivation of this formula may be found in most books on electromagnetic theory and allbooks on waveguides.

We wish to calculate the density of states in the vicinity of and therefore we need toknow the total number of combinations of integers nx, ny, nz which will produce values ofwavenumber between and + d . This appears tricky until we remember that for a finite sizecavity the integers are very large indeed and we can consider them to be just variables in aCartesian system thus:

Since in this system each set of integersrepresents a cube of unit volume and the sumof the squared integers can be represented asthe radial distance r we find that for a cubicalcavity (X = Y = Z)

and the total number of states between and + d is the volume of space betweenspheres at r and r + dr in the positive octant:


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N(r)dr 18

4%r 2dr

N()d 18

4% 4X22 2X d 4%V2 d

N() 8%2

U() 8%kT 2

and

It only then remains to normalise to unit volume and double the result to allow for the twopossible independent polarisations of any state and we arrive at a state density per unit volumeof:

b) The State Population Problem

We now know the density of states per unit volume but in order to arrive at the energydensity per unit volume we need to consider the energy per state and multiply. The problem istherefore to determine how much energy there is in each state.

The first serious attempt to look at this problem was the Rayleigh-Jeans formula whichtook the classical thermodynamics concept that the average energy per state was just kT wherek is Boltzmann's constant. The resulting formula:

is obviously wrong because it predicts infinite energy density at very short wavelengths - theso-called "Ultra-viole(n)t Catastrophe". However this obscures the fact that at longwavelengths it agrees with the experimental data very well as 0.


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0.00

0.05

0.10

0.15

0.20

Rad

ianc

e(W

m-2

sr-1

(cm

-1)-1

)

2500 2000 1500 1000 500 0

Wavenumber (cm-1)

Rayleigh-Jeans

Planck

Boltzmann

0.00

0.01

50 0

Figure 11: Experimental and Theoretical Forms for Cavity Radiation

e

EkT

U() 8% 2 hc e

hckT

The next attempt was to replace the classical equipartition of energy law with the morequantum concept of Boltzmann statistics which says that for a state of energy E the probabilityof occupation is:

the formula for the energy density is then:

(density * energy of state * probability of occupation).


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U() 8%hc3

ehckT

1

This has the right general sort of form and fits well as but is wrong in detail at thelong-wave end of the curve.

The correct result comes in realising that photons are bosons and therefore the correctstatistics are Bose-Einstein statistics which modifies the formula to:


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0.1

1

10

100

1000

10000

Rad

ianc

e(W

m-2

sr-1

(cm

-1)-1

)

15000 12500 10000 7500 5000 2500 0

Wavenumber (cm-1)

6000K

3000K

1000K

500K 300K

Peak Wavenumber

Figure 12: Peak Wavenumber forBlackbody Emission

U(x) kThc

3

8%hcx3

e x 1

ex 1

x3

B(,T) 2hc23

ehckT

1

c13

e

c2

T 1

which is the correct formula first derived by Planck. You will notice that this formula tends tothe classical Rayleigh-Jeans law at long wavelengths and to the Boltzmann statistic result atshort wavelengths.

Peaks, Parameters and Constants

By substituting x = hc /kTin the above expression we findthat the formula for the energydensity is actually a function of xonly:

a function which has a maximumwhen

which must be solved numericallyto give x = 2.8229 The conclusiontherefore is that the radiationdensity in a cavity at anywavenumber is proportional to T3 and a function of /T. The wavenumber (in cm-1) ofmaximum radiation is given by 0 = 1.962T so that the higher the temperature the shorter thewavelength at which the maximum occurs. For room temperature the peak is at 600 cm-1 whichis in the mid-infrared whereas in a furnace at 6000K the peak is at 12000 cm-1 nearly in thevisible region of the spectrum. (Notice that a short method of estimating the peak wavenumberin cm-1 is pk = 2T)

Note that I have here expressed the peak in wavenumber units. In wavelength units theformula is: = 0.2897 x 104/T µm. This places the peak apparently in a different place but thisis merely an artifact of the change of variable in the formula for the energy density.

Using the relationships developed in the previous section we can easily calculate theintensity (I = Uc/4%) of radiation in the cavity as:


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P

0

B(,T) d 2 h c2 kThc

4

P

0

x3

e x 1

dx

where c1 = 1.1911 x 10-8 Wm-2sr-1(cm-1)-4 and c2 = 1.438 K(cm-1)-1 are are known as the first andsecond radiation constants respectively. The above expression is known as Planck's Functionand is fundamental to many of our discussion.

The total intensity may be obtained by integrating the intensity per unit wavenumberover wavenumber to give:

I would say that it is left as an exercise for the reader to show that the right-hand integral is%4/15 except that the mathematics is very tricky - see Armstrong & King.

The total intensity is therefore proportional to T4 which is Stefan's Law.

The value of the constant is 1.8047 x 10-8 Wm-2K-4sr-1. This is not the usual "Stefan'sconstant" which is the constant for the total energy emitted by unit area of a hole in a cavity.

The difference between the quantity we have derived above and the "normal" Stefansconstant calculation is the difference between the intensity leaving the cavity and the fluxleaving the cavity. Since the cavity radiation is homogeneous and isotropic, we can use theresult of the previous section F = %I, and determine the constant for the energy flux as5.67 x 108Wm2K4 which is close to the accepted value for Stefan's constant. In order tosimplify the world, we do not introduce a new named constant at this point but use Stefan'sconstant and divide it by % where necessary.


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B(,T) 2hc23

ehckT

1

c13

e

c2

T 1

max 1.962T

F ) T4

F % I

Blackbody Laws

Here in one place are the Blackbody Laws:

Planck's Function

The intensity of radiation from a blackbody source is given by:

where c1 = 1.1911 x 10-8 Wm-2sr-1(cm-1)-4 and c2 = 1.438 K(cm-1)-1.

Wein's Displacement Law

The wavenumber of maximum emission of a blackbody is given by:

where T is the absolute temperature.

Stefan's Law

The total flux from a blackbody is given by:

where ) = 5.670 x 10-8 Jm-2K-4s-1, and T is the absolute temperature.

Flux and Intensity

The flux and intensity of blackbody emission are related by:


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CavityWall

F

(1-a)F

εF

aF

Figure 13: Section of Cavity Wall

CavityWall

FF-F

F

(1-a )F (1-a )F

ε F

εF−ε F

ε F

aF

Figure 14: Filtering Cavity Radiation -Still in Thermal Equilibrium

Absorptivity and Emissivity of Surfaces

Since the wall of the cavity can be made out of any material it is interesting to considerthe interaction of the radiation field with the cavity wall.

In order to do this let us consider a smallsection of the cavity wall which is thermallyinsulated from the rest of the cavity but has atemperature which is currently the same as therest of the cavity. The fraction of incidentradiation absorbed by the wall is a and the restmust be reflected in some sense, either specular(like a mirror) or diffuse. In other words forevery unit area of surface which intercepts )T4

units of energy an amount a)T4 is absorbed.

However in order that the material remainin thermal equilibrium with the cavity it isnecessary that the energy emitted by the walls bethe same as that absorbed and this must be trueat any temperature. Therefore the energy emittedmust be expressed as )T4 and the absorptivityof the cavity, a, must be the same as the emissivity, .

Howevever, this does onlyapplies to some form of integratedvalue of the absorptivity andemissivity. We can also showthat the monochromat icabsorptivity and emissivity areequal by considering the case of aspectral filter in front of thethermally insulated surface. Thisfilter transmits a particularwavelength and reflects allothers (the filter does not absorbany energy). If the wall and thecavity are at the same temperaturethen the energy flow across thefilter must balance or energy willbe systematically transferred tocreate a thermal gradient wherenone existed before and would violate the laws of thermodynamics. Under these conditionsit can readily be shown that the monchromatic absorptivity and emissivity are equal.


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a

a, x

P

0

x

B(,T) d

P

0

B(,T) d

%

) T4 P

0

x

B(,T) d

a - r 1

The case of a cavity-hole as a surface is interesting because it represents the oppositeextreme from the more familiar perfect reflector for which a = 0. For the cavity all incidentradiation is absorbed and therefore a = 1, which is the maximum possible absorption. Howeverby the laws discussed above, known as Kirchhoff's Laws, the emissivity must also be 1 as wehave already calculated.

Kirchoff's Laws and Related Surface Laws

Emissivity and absorptivity

The monchromatic emissivity must be equal to the monchromatic absorbivity. Theintegrated absorptivity weighted for the surface temperature must also be equal to the integratedemissivity. This implies that firstly:

and then that:

Note particularly that the averages must be taken at the same temperature.

Absorption, Reflection and Transmission

By conservation of energy, the sum of the absorptivity, transmissivity and reflectivityof an object must be unity (energy has to go somewhere).

Construction of Blackbodies


-21-

Figure 15: Construction of aBlackbody

Figure 16: Lelsie's Cube Experiment

I don't want to discuss the problem of constructinga blackbody in too much detail because it is a ratherinvolved subject but one or two comments may be in orderhere. Obviously we cannot construct an infinitely small holein an infinitely large sphere so we seek some other shapewhich will have the same effect in a more finite volume. Apopular shape is the re-entrant cone. This constructionallows many reflections of any light entering the structurebefore it finds its way out. The surface properties of thecone are obviously somewhat important here as a perfectreflector will allow the exit of incoming radiation after afinite number of reflections which makes the deviceimperfect. Similarly a diffusely scattering surface such asa magnesium oxide film will produce a scattering out of thecavity. The surface is generally made smooth but notpolished and is prepared or painted in a manner that absorbsas much incident energy as possible - there is a technologyfor doing such things. Using a cone with a width-to-lengthratio of about 1:6 or more gives a good blackbody functionwith reasonable surface preparation.

The reason for constructing blackbody radiators is that they form one of the very fewclasses of radiative sources whose strength is calculable, after all B( ,T) only requires aknowledge of the physical parameter, temperature, which we can measure quite well, in orderto completely compute the radiation spectrum at the exit. Blackbodies are therefore invaluablein atmospheric radiation field studies (experiments) as calibration sources.

Leslie's Cube - Another "Failed" Experiment

I hope that some of you have seenLeslie's cube before but for those that haven't itconsists of a hollow cube of copper filled withhot water. The various faces of the cube areprepared in different ways so that the totalemitted energy can be compared by monitoringwith a simple thermocouple radiation meter.Although one can generally observe that theradiation from polished surfaces is less than thatfrom a rough, dark surface the exact magnitudes,particularly of painted surfaces, seem at variancewith what we see.

The solution is quite simple in that thecube radiates energy with a peak at about750 cm-1 (13.3 µm) whereas our eyes are


-22-

4%R2 )T4

4%R3

3C

dTdt

sensitive in the region around 20,000 cm-1 (500 nm). This large discrepancy between "where"we are looking and where we are making measurements can account for any amount ofvariation.

In fact any non-polished surface is liable to have quite a high emissivity in the infra-red -or "thermal region" as it is known in this context - and the values can get quite close to thetheoretical limit of 1. Conversely many polished surfaces have emissivities quite close to 0 inthis region even if visibly they do not look that good, e.g. gold looks as though it is emittingquite a lot of energy because it looks yellow but = 0.01 in the infra-red and it is widely usedfor its noble metal properties as a coating on infra-red mirrors.

Any planetary surface therefore looks reasonably "black" when considering thermalradiation emitted from it. Now of course that is a generalisation but when considering an entireplanet that is exactly what is required. Some examples are:

Material Infrared Emissivity

Sand (wet) 0.962

Sand (dry) 0.949

Peat (dry) 0.970

Grass (thick, green) 0.986

Fresh Snow 0.986

Dirty Snow 0.969

and so on. Therefore we are not going to go too far wrong if we treat a planetary surface as ablackbody surface - i.e. as the opening in a cavity.

Planetary Radiation Balance - The Simplest Case

A planet may therefore be considered to radiate energy at a rate of )T4 per unit area. Forthe Earth (mean radius 6371 km) this means the radiation of 1.2 x 1017 W continuously at aneffective radiating temperature of 255K. Before considering anything else let us consider whatwould happen if that were the only energy flux to be considered. The Earth would then undergocooling at a rate given by the simple Newtonian formula:

Now the specific heat of the Earth in Jm-3K-1 is not the easiest thing to determine but I shall usethe value appropriate to granite which is 2.2 x 106 Jm-3K-1. The solution is then dT/dt = 5.3 x 10-11Ks-1. This very small value however still puts the cooling rate at about 1.6K per millennium


-23-

which over the geological age of the Earth would mean that the Earth would lose all its heatwere it not for the solar source. Thus any long-term model of the climate of a planet must havean overall energy balance for the system as a boundary condition.

This illustrates a very important point in dealing with the radiation problem - the time-scales can be very long. In fact there are different problems to be considered over various time-scales. In the short term there are heat flows in the land, ocean and atmosphere, but in the longterm only the overall heat balance will matter as everything else averages out.

The Sun - A Hot Blackbody

Turning now to the solar radiation input on the Earth's atmosphere, the so-called extra-terrestrial flux, we find a most extraordinary thing - namely that it is blackbody in nature aswell. Since this is not a course in stellar physics I will leave the reasoning out of the problembut suffice it to say that the solar flux looked at at not-too-high a resolution looks in the nearu-v, visible and infra-red regions almost exactly like the expected flux from a blackbodyaperture the size and shape of the sun's visible disk. The appropriate temperature for theblackbody is 5780K and the solar disc subtends an arc of 31.99 minutes at the Earth. Since theangle subtended is so small, the solar radiation is almost exactly a plane wave at the Earth andmay certainly be described by the formula F = I6 where 6 is the solid angle subtended bythe sun (6.8 x 10-5 sr). I can be calculated as ()/%)T4 = 2 x 107 Wm-2sr-1 and therefore the extra-terrestrial flux at the Earth's orbit is 1370 Wm-2 or thereabouts.

We can also calculate the spectral distribution of the radiation from a knowledge ofB( ,5780K).

The Absorption of Solar Radiation

In considering the general interaction of solar radiation with a body we are not quite somuch in the dark as regards the effects as we were with thermal radiation since the radiantenergy lies in wavelengths near the visible and we can therefore cautiously believe our eyes.

We know that there are several possible effects that can be produced by naturalmaterials: absorption, specular reflection and diffuse reflection. I shall have a great deal moreto say about these properties in future lectures but any photograph of a planet will tell you thatthe surface/atmosphere must scatter some radiation because you can see it. Astronomers callthe fraction of solar radiation scattered back into space from the planet the "Albedo" and thisquantity generally varies from about 0.1 to 0.9 depending upon the planet (Mercury - Venus).This albedo is nothing more than (1-a) for the planet in solar radiation where a is theappropriate value for the visible or solar region of the spectrum.

In the case of the Earth and Venus a large part of the albedo is formed by the back-scattering of solar radiation from clouds and is therefore an atmospheric effect. In the case ofthe Earth and Mars some of the albedo is formed by reflection from "snow" and "ice" in the


-24-

f(x) x3

e x 1

, x

c2

T

1.438T

polar regions and this is an atmosphere/surface effect. For Jupiter the albedo is colour-dependent which is why the planet looks coloured and so on.

Specular reflection is seen in the radiation from the Earth from the ocean surface.Although it is not a large contributor to the albedo because the ocean is not smooth I mentionit to point out that both specular and diffuse reflection occur in this radiation problem.

Solar and Thermal (Terrestrial) Radiation

In order to consider in even an elementary manner the two forms of radiation flowingin the region of the Earth it is necessary to consider the form of the two spectral distributionspeaked at 12000 cm-1 (solar) and around 600 cm-1 (thermal).

If we consider the universal form of the radiation curve

we can consider the value of the function as x varies:

x f(x)

0.5 0.193

1.0 0.582

2.0 1.252

6.0 0.537

12.0 0.011

From this we can see that if the peaks of the two curves are separated from each other by morethan a factor of about 10, that is, if the characteristic temperatures are more than a factor of 10apart, and the total energies are comparable then, there is very little overlap between the twocurves and we can treat a radiation problem involving the two sources in a separable manner.


-25-

0.00

0.02

0.04

0.06

0.08

0.10

105 104 103 102

Wavenumber (cm-1)

250K5780K x 20

5780K

Rad

ianc

e(W

m-2

sr-1

(cm

-1)-1

)

Figure 17: Solar and Thermal Blackbody Curves

4πR2

πR2

TeFs

Figure 18: Simple Model of Energy Balance

%R2 Is6s avis 4%R2 ir )T4

%R2 F (1A) 4%R2 ir )T4e

In atmospheric terms thismeans that the solar radiation(peak at 11,500cm-1) can betreated separately from theterrestrial or thermal component(peak at 550cm-1) and the linkbetween them is provided by theenergy flow equations for thesubstances in question. Thisseparability of the radiationequations which is true in all buta very few of the problemsinvolving planetary atmospheresis really the only thing that makessome of the problems tractable.

A necessary consequence of this separability is that the albedo A, and the emissivity of a body are characteristic of different wavelengths as we have already argued and they do notobey Kirchhoff's Law directly. They do however obey it if it is stated correctly (that is if yourestrict yourself to one wavelength or weighted over the full spectrum).

Radiative Equilibrium on a Planetary Scale

Over the long-term there must obviously be an equilibrium between the solar radiationabsorbed by the planet and the thermal radiation emitted by it. This is true for any body inequilibrium within the solar system and the theory is therefore also applicable to satellites, bothnatural and artificial.

We can write this balance as:

where the LHS is the energy intercepted bythe spherical body from the sun and the RHSis the energy emitted by the body in theinfra-red. We have used the fact that the sunhas a small solid angle 6s to simplify the


-26-

integral for total energy and then substituted F=I s6s and (1-A)=avis where A is the albedo orthe average amount of solar energy reflected.

If we restrict ourselves to satellites far from planets for a moment, we find that for asatellite near the Earth's orbit (but not close to earth, perhaps 180o round the orbit) the equationreduces to:


-27-

T4e

avis

ir

F4 )

avis

ir

13704 )

avis

ir

2794

where avis has been substituted for 1-A. This implies that the equilibrium temperature of a black(invisible) satellite is 277K or about room temperature. If you realise that the equilibriumtemperature of a satellite out of sunlight is 3K then you will realise that satellites in orbit canexperience a severe thermal stress as they go round the planet.

Some surprising things happen when we consider trying to cut down the temperatureof a satellite. If we make the surface shiny to cut the solar input we should remember that mostmetals are better reflectors in the infra-red than in the visible so that a reflectivity of 0.9(avis = 0.1) in the visible may become 0.99 (ir = 0.01) in the infra-red. Under those conditionsthe equilibrium temperature becomes about 500K and everything fries! The correct approachis to use a surface that reflects some solar radiation but emits well in the i-r - such as a paintwith a visible a of 0.5 and an i-r of 0.95 which produces 238K.

When the satellite is big enough to be called a planet then the equilibrium temperaturegiven by the above equation is called the "effective radiating temperature" of the planet Te.This can be calculated using an observed value of albedo and the known solar flux accordingto the orbital distance of the planet, or it can be measured by monitoring the infra-red radiationcoming from the planet. If the planet has no thermal activity of its own then the two shouldagree. Here are the values for the planets:

Planet Albedo Te

(calculated)Te

(measured)Surface

Temperature

Mercury 0.058 442 442 442

Venus 0.77 227 230 700

Earth 0.30 256 250 288

Mars 0.15 216 220 210

Jupiter 0.58 98 130 160

In all cases the calculated and measured values agree reasonably well with the notableexception of the planet Jupiter. It is said that the exception proves the rule and Jupiter doesappear to have a significant thermal source. Why this is so is still a matter of discussion. (Ithink!) Two current theories are a) that the heat is produced by the gravitational collapse of theplanet (Which is a fun calculation to do - calculate the required rate of energy generation andthen the rate of shrinkage required to generate the energy) or b) that the planet is really aninferior sort of star with its own thermonuclear generator.


-28-

Ice

Ice

Equator 0o

λ

−λ

Figure 19: Simple Ice/No-ice Climate Model

A 0.751 sin 0.15sin

Just to show that we do not yet know everything I must point out the last column whichtabulates the surface temperature of the planets. These figures do not agree with the radiatingtemperatures and therefore there must be some other effect going on. An obvious candidate isthe atmosphere.

A Zero-Dimensional Climate Model

It follows from what has just been said that a very long-term climate model must be anenergy-balance model of some sort. The simplest model that one can think of is just the modelwe have just been describing which gives a single value for the radiating temperature of theplanet in terms of its albedo and solar flux. It therefore follows that there is a relationshipbetween the solar flux and the effective radiating temperature which is in turn in some wayrelated to the surface temperature in a way that we shall consider later. Broadly speaking inthe solar system the nearer a planet is to the sun the higher its effective radiating temperature.

However there is presumably also alink between the effective radiatingtemperature and the albedo, at least in thecase of Earth, for as the polar caps melt oraccumulate the percentage of ice-covervaries and ice has a very high albedocompared to bare land. We should thereforeconsider this fact in our discussions. If weconsider the (visible) albedo of ice to beabout 0.75 and the albedo of bare ground tobe about 0.15 then we have a problemassigning the total albedo of the whole Earthbut we could roughly assume that the albedois the weighted average of the two valueswith the weighting being determined by thefraction of surface covered by both. If wefurther assume circular symmetry and an"ice-line" at latitude then we can readilyderive:

as being the weighted average albedo. Forthe present-day situation (A = 0.3) this puts the ice-line at about 48.5o north (or south) whichis not too disgusting for a simple model.

However this leads to the conclusion that there is at least one other solution to theenergy balance equation for if we assume that in some way the Earth got covered in ice then


-29-

A 2.85 0.01Te

F(1 0.15) 4)T4e (icefree)

F(1(2.85 0.01Te)) 4)T4e (mixed)

F(1 0.75) 4)T4e (icecovered)

the albedo would be 0.75 and the resulting effective radiating temperature 197K which wouldbe sufficiently low to keep the ice there.

Question: How many other solutions are there and are they stable?

Question: How stable is the current situation?

The ice-line is obviously related to the effective radiating temperature in that the colderthe Earth the smaller is both and Te. However there is no clear relationship between them.The simplest relationship that one could hypothesise which has any sense is that the fractionalice cover is proportional to the absolute temperature in some range. Outside that range, the icecover is either zero (hot) or one (cold). Another constraint is that the current situation shouldbe reproduced by the model and that that situation should be stable. After considering theseconstraints and looking more deeply at the physics - of which I shall say more later - we cometo the conclusion that a not unrealistic model is one in which an ice-covered Earth prevailsbelow Te = 210K and a "bare" Earth above Te = 270K. In order to obtain a first-orderrelationship for the temperatures in the middle we make the simplest assumption possible: thatthe relationship between the albedo and Te and meets the limiting cases at the ends. In that casewe can write the albedo A as:

between the limits given above.

Thus we have three possibilities for the solution: ice-free, mixed ice/ice-free and ice-covered.


-30-

1764

14181370

210 255 270

Fs(Wm-2

)

Te (K)

Not to scale!!!

Ice-coveredEarth

Ice-freeEarth

You are here

Figure 20: Solar Flux vs Effective RadiatingTemperature

(F F)(1 A) 4)T4e MC

dTe

dt

F(1 A) MCdTe

dt

Given that information wecan plot the relationship betweenF, the solar flux, and Te theeffective radiating temperatureadding the line for F = 1370 Wm-2

we find that there are threepossible solutions the hottest ofwhich corresponds to our ownsituation, but how do we discoverwhether these solutions are stableor not? Only stable solutions arevalid for a long term modelbecause "random" fluctuationsexist in the solar output and in theearth system which will render anunstable equilibrium untenable.

The solution to that problem comes by considering a small fluctuation in the solar outputand the earth response from an equilibrium situation. The earth undergoes a Newtonian heating(cooling) given by the solution to the equation:

Since we know we start from an equilibrium position (F(1-A) = 4)Te4) we can simplify this to:

We know that A < 1 and therefore the response of the earth to an increase in the solar radiationis to increase the effective radiating temperature. That, in turn, implies that solutions in regionsof negative slope are unstable (the equilibrium line requires the earth to cool with increasingflux - which is not so) whereas solutions in regions of positive slope are stable, the latter casefortunately including our own.

However that still leaves us with our two stable solutions of which the current one is thewarmer - fortunately. The warmer solution however ceases to exist if the solar output dropssignificantly - significantly in this case being about 0.6% of the total. We might ask whetherthe figures cooked up here are producing a valid case and all I can say in response to that is thatmore elaborate models reproduce the features of this one to a greater or lesser extent and manyof them have this feature of the warm solution ceasing to exist after a drop only about 1% inthe solar output.


-31-

e major axisminor axis

2

11/2

The Solar Constant - or Otherwise

This leads us on to the question of how constant the solar output has been and is goingto be on the time-scale of the human race - which is hopefully not going to be as short as itsometimes appears. Some of the variations that we see are explainable, some are not and somedebatable as we shall see.

Solar Input and Planetary Motion

I am sure that you are all aware that a planet travels around the sun and I am also surethat you know that its path is not a circle but an ellipse.

This being so then the solar radiation input must vary with the time of year or theseason. The technical terms for the point of nearest approach, when the solar radiation ismaximum, is the perihelion and the furthest point is known as aphelion. These terms are alsoused to describe satellite orbits.

The orbit of a planet may also be described in terms of an average distance from the sunand an eccentricity, e, which is defined as:

which is obviously zero for a circular orbit. The eccentricity of the Earth's orbit is 0.017 andtherefore the solar radiation varies by about 7% because the distance varies by 3.5% and theradiation varies as the square of the distance.


-32-

Dec 22Winter

Solstice

June 22SummerSolstice

September 23AutumnalEquinox

March 21Vernal

Equinox

23.5o

Polaris(Pole Star)

aphelionperihelion

Figure 21: Variation of Insolation With Orbital Position (shown for the earth)


-33-

λ

δ h

P

S X

YZ

Figure 22: Vector Notation forSolar Zenith Angle

p cosi sink

s coshcos i sinhj sin k

cos sinsin coscos cosh

The most obvious phenomena to correlate with this variation is the seasonal one but thisis not the case since the globe experiences different seasonal phases in different hemispheres -besides which the perihelion occurs in northern hemisphere winter. The seasonal change ismore a function of the planetary inclination which is the angle that the planetary rotation vectormakes with the normal to the orbital plane. The inclination causes the average angle which aportion of the planet surface makes with the solar beam to vary. If the "surface" of theatmosphere is not normal to the solar beam then the energy input (ignoring orbital changes) isFcos where is the angle between the solar beam and the vertical, also known as the solarzenith angle.

For a planet whose axis of rotation is perpendicular to the plane of the orbit this causesthe average solar radiation per unit area to fall off uniformly from the equator to the poles butin the more normal case where the rotation axis is inclined to the orbital plane the situation ismore complicated because the track of the sub-solar point varies with position in the orbit.

For every point p on the planetarysurface where the instantaneous direction ofthe sun is s, the solar zenith angle is givenby p .s. Using a latitude-longitude grid on theglobe selected so that p is at 0o longitude, itmay readily be seen that:

and

where is the latitude of the point, h is thecurrent longitude of solar noon relative to the point and is the current solar inclination anglebetween the equatorial plane and the orbital radial vector. This latter quantity varies between+ and - the angle of inclination throughout a year.

The hour angle h can also be expressed as a time, since the sun progresses through aconstant number of degrees of longitude per unit time and the time is conventionally measuredfrom local noon.

Using the above expressions we conclude that the solar zenith angle is given by:

In order to evaluate the total energy received per day it is necessary to integrate this equationover the daylight hours which are when || < 90o the limits being


-34-

cosH tan tan

Q Fdm

d

2

2 PhH

h0

cos dt

Fdm

d

2

2 PhH

h0

coscos cosh sin sin dt

h 2%Tr

t

There are four cases in which there is no finite solution for a time of sunrise or sunset:

+ > 90o - northern hemisphere midnight sun (integrate for 1 planetary day)

- - > 90o - southern hemisphere midnight sun

- > 90o - northern hemisphere 24hour darkness

- + > 90o - southern hemisphere 24hour darkness

In all "normal" cases we integrate from sunrise to noon and double the result.

where we have also introduced the fact that the planetary orbit is elliptical by the dm/d factorwhich accounts for the the deviation of the current distance from the sun, d, to the "mean"distance dm.

The relationship between the hour angle, h, and the time measured from local noon is:

where Tr is the rotation time of the planet.


-35-

sin 2%tTo

Figure 23: Annual Insolation for the Earth

We also need to remember that

where is the angle between the axis of rotation and normal to the orbital plane, t is the timemeasured relative to an equinox (when = 0) and To is the time for one orbit.

Combining all these features together allows one to produce a map of the total annualinsolation as a function of position on the planetary surface. For the earth this looks like:

One should note that this calculation has assumed:

a) That the rotation time of the planet is much less than the orbit time

b) That the inclination is less than 90o.


-36-

c) That the orbit is nearly circular

It would be stimulating to consider cases of planets where these conditions do not obtain.

The planetary data are:

Planet Average Orbit(Km)

Year(days)

Inclination(degrees)

Eccentricity Rotation(days)

Mercury 5.8 x 107 88 0 0.206 587

Venus 1.1 x 108 255 <3 0.007 -243

Earth 1.5 x 108 365 23.5 0.017 1.00

Mars 2.3 x 108 687 25.2 0.093 1.03

Jupiter 7.8 x 108 4330 31 0.048 0.41

Saturn 1.4 x 109 10800 26.8 0.056 0.43

Uranus 2.9 x 109 30700 98.0 0.047 -0.89

Neptune 4.5 x 109 60200 28.8 0.009 0.53

Pluto 5.9 x 109 90700 - 0.247 6.39


-37-

Surface

Solar Thermal

AtmosphereQ AQ

Y

Y

X

Figure 24: A Simple Atmospheric Model

Q AQ Y

Q Y AQ X

A Half-Dimensional Climate Model

In the last section we looked at at the variation of the albedo in a very simple model asa function of the ice-line s and the effective radiating temperature Te. Under the simpleassumptions made we deduced something about the climate of the Earth or other planet but indoing so we ignored the atmosphere entirely. It is now necessary to consider the atmosphereand that is what most of the rest of the course is about. Before going into great detail aboutradiative transfer in the atmosphere I want to describe the phenomena in the simplest waypossible by expanding the zero-dimensional model of the last section.

In order to do that efficiently we mustconsider how to include the atmosphere as asimple term and the best way to do that is toconsider it to be a simple shell covering theplanet in an infinitely thin layer. Since theatmosphere is thin and the planet large wecan "unroll" the sphere to produce a planardiagram. The solar radiation input then tothis pseudo-plane is just the average input toa unit area of the planetary surface and this isjust F/4 = Q (the factor of four is the ratio ofthe disk area to the surface area of a sphere)which is the defining equation for Q, theaverage solar input (the average being overall orbital and global positions). Considerthen a very simple planetary situation in which the atmosphere may be considered to be a planesheet of material which is transparent to solar radiation and opaque to thermal radiation. Weshall see later that this is not too unrealistic a consideration although at the moment it justconstitutes a hypothesis. I want to look at the sort of results that come out of these models touse as a guide in the more complicated phenomena which we shall consider later. The situationis therefore as shown here where A is the planetary albedo provided entirely by groundreflection, and X and Y are the infra-red fluxes emitted by the surface and atmosphererespectively. Since everything in the thermal region is considered to be black ( = 1), we canalso write Y = )Te

4 where Te is the effective radiating temperature given by the usual equation.This is justified because Y is the infrared flux which would be observed by a radiometer distantfrom the planet, an argument which defines Te. Writing down the equation of vertical energytransfer above the atmosphere and stating that the atmosphere/planet system is in radiativeequilibrium:

we can also write a similar equation for transfer below the atmosphere


-38-

Surface

Solar Thermal

Q

Qτs

Qτs2 rg

Qτsrg

Y

Y

X

Xτa

τaτs

Figure 25: A More Complex Atmospheric Model

which immediately solves to give X = 2Y. The radiating temperature of the surface Tg istherefore given by Tg = 1.19Te which for the Earth gives Tg = 303K.

This simple model therefore has the capability of producing a surface temperature whichis considerably above the effective radiating temperature but how realistic is it? The answermust be approached from at least two directions. First we must discuss what properties of suchmodels produce these features and second what physical atmospheric features might make theatmosphere resemble the model?

In order to consider what features of the model produce these effects consider thissecond model in which our "two-tone" atmosphere has an transmissivity -s in the solar regimeand -a in the thermal regime. Again the ground produces the solar reflectivity and is otherwiseblack. One should note here that there are some fundamental differences between the solarregime, in which the radiation is in the form of an almost plane-parallel beam, and the thermalregion, in which the radiation is in the form of an omni-directional flux, but we shall postponethat discussion until later. We also note that we could incorporate a reflectivity into theatmosphere as well to simulate the clouds and a scattering term to make things even more


-39-

Q Q -2s rg X -a Y

Q-s Y Q-srg X

X )T4g Q

1-s

1-a

1 -srg Q1-s

1-a

1 A-s

T4g

T4e

1 -s

1 -a

1 -srg

1 -2srg

1 -s

1 -a

1-s

-s A

1 A

realistic but the model then loses some basic simplicity which we require in order to see whatis going on. Lastly we note that for any transparent surface = a = 1 - by Kirchoff's law.By considering the progress of the solar beam on the left of the diagram we conclude that thedown-radiation at the ground is Q-s, the up-radiation from the ground is Q-srg and the albedois given by A = -s

2rg. Writing down the energy balance equations above and below theatmosphere we find that:

and

from these equations we can derive the values of X and Y. Concentrating on X for the momentwe find by adding the above equations that:

and further that

Whilst it is not immediately obvious, the ratio of the fourth power of the temperatures variesbetween 0.5 and 2.0, assuming that all transmissions and reflectivities are bounded by zero andone. A "monotone" atmosphere in which -s = -a will always produce a surface temperature ofless than or equal to Te. By tailoring the values of -s and -a we can produce a wide range ofsurface temperatures both above and below Te.

There are several further interesting things to note about this model. Firstly considerthe albedo A = -s

2rg. If the model is to reproduce A = 0.3 for the earth, then the value of -s isseverely constrained because rg 1 and the most absorbing atmosphere possible is therefore-s = 0.55. Similarly the minimum value of rg is 0.3.

The result we get from the above analysis that if -s » -a the surface is hotter than Te andvice versa. This leads to a very interesting consideration in a practical setting: The situationdescribed above (-s » -a) can readily be set up using plastic materials and it should thereforebe possible to produce very hot surfaces by appropriate radiation effects. (However theproblem is compounded by the presence in the atmosphere of convective and conductivecooling processes and the thing would really only work in space.)


-40-

T4a

T4e -aT

4g

1 -a

Xn

Xn+1

Yn

Yn+1

n

n+1

Figure 26: A Simple Layer Model of anAtmosphere

Xn Xn1- Rn(1 -)

Yn1 Yn- Rn(1 -)

Xn Yn Xn1 Yn1

We can also discuss the atmospheric temperature in this model but the expression israther complicated. Briefly one can see readily that

A more interesting approach may befound by considering a layered atmospherefor the thermal regime with an up-wellingand down-welling flux in it. If we considerthat there are k layers numbered from thetop, each layer having a thermal transmissionof -, then from the diagram we can readilysee that the recurrence relations are:

and

where Rn is the emission characteristic of the layer (= )Tn4) and - is the thermal transmission

of the layer. If the atmosphere is in equilibrium we also know that the net energy gain of anyspatial volume must be zero (e.g. the box shown) and therefore:

In order to solve this kind of problem we need to know the boundary conditions. At the top ofthe atmosphere we know that Y0 = 0 and X0 = )Te

4 = G from these conditions we can deducethe following relations:


-41-

Rn

G

11 -

n 1 -

1 -

Xn

G 1 n 1 -

1 -

Yn

G n 1 -

1 -

RG

12

3

2XG

1 3

2YG

3

2

Note that these equations satisfy the conditions of the model and in particular that the net flux(Xn - Yn) is always directed upwards and of magnitude G. This implies that the ground mustradiate a net flux of G which, since Y is non-zero, implies in turn that the ground temperaturemust exceed the effective radiating temperature since G = )Te

4.

If we now allow the layers to get very thin so that - 1 and the number of layers tobecome extremely large to compensate keeping n(1--) = constant, we can rewrite the quantityn(1-)as 3(z) which we shall call the optical depth. We can then recast the equations asfunctions of the optical depth.

Thus the larger the optical depth, the larger the fluxes become without limit. (This modelreproduces most of the features of the "2-stream model" which I shall consider later becauseit is more complicated in its derivation.)

The results of the above equations are interesting in the light of the fact that thepredicted surface temperature is higher than the effective radiating temperature by an amountwhich depends upon the optical depth of the atmosphere. This mechanism is apparently capabletherefore of explaining the fact that I mentioned earlier - that the surface temperatures ofplanets with atmospheres exceeds the effective radiating temperature by an amount whichdepends upon the amount of atmosphere they possess. We shall see later that this is only partof the explanation for Venus and Earth but it does start us out on the road to discovery.


-42-

dE I(p,s,,t) d7 d dA dt

dI

I

k'dx

I(x) I

(0) ek'x

Radiative Transfer Within the Atmosphere

In order to gain further insight into the reality of the model of the last section we needto consider the general interaction of radiation with gaseous material which means extendingthe consideration of intensity and flux considered earlier. Just in case everybody has forgottenwe should define intensity once again.

The intensity I(p,s, ,t) of radiation at a point p in a direction s is defined by the equationwhich considers the amount of energy flowing within a solid angle d7 about the direction sthrough a small area dA perpendicular to s in a wavenumber interval d . We write the equationfor this energy dE as:

which is the defining equation for I. Remember also that we can discuss the monochromaticintensity (above) or the integrated (over wavenumber) intensity.

In the interests of clarity we will omit the dependencies of I on p, s, and t.

If we consider a very simple case of a plane-parallel beam of radiation traversing anabsorbing extended medium we can argue that on a small scale (differential notation) thesituation is a linear one and that the loss of the beam intensity dI is proportional to the amountof material traversed, 'dx, and some coupling co-efficient between the beam and the matter,i.e.:

where the first term on the RHS says that the energy loss is proportional to the amount ofenergy input, the second is the coupling coefficient and the third is the amount of materialtraversed. This "law" has been attributed to many people over the years so one might call it theBeer-Bouguer-Lambert Law, or thereabouts. We shall call it Beer's Law. If we consider asimple monochromatic case of a uniform-density absorber then we can integrate this equationtrivially to obtain:

the familiar exponential decay law of transmission through uniform materials which is the sortof thing we see in the laboratory.

The atmosphere is capable of showing a very wide variety of effects but many of thesewill have a very similar result if we concentrate only on the intensity equation above.

There is some doubt as to what we should call k as no statement is being made abouthow the energy is being removed from the beam. The simplest definition of a term is to callit the extinction coefficient on the grounds that, however it does it, it extinguishes the beam.


-43-

k ka ks kr

E

hf1 hf3

hf2

hf1 -~ hf2 + hf3

Figure 27: Resonant ScatteringMechanism

Atmospheric Extinction Processes

There are essentially three atmospheric processes which lead to extinction of the beam:absorption, simple scattering and resonant scattering. We can write in a simple-minded waythat:

where k is the total extinction coefficient and the subscripted ks refer to the individualprocesses.

In the case of absorption the mechanism of extinction is the transformation of theenergy in the beam to molecular motion which usually manifests itself in an overall heating ofthe gas. I call this "transfer to the kinetic energy reservoir". Note that in this case the energyis permanently lost to the radiation field unless some other process causes energy to beconverted from kinetic energy to radiative energy (see below).

Simple scattering is a deflection of energy from its current direction (remember thatintensity has a directional association) into another direction. The energy is not therefore lostto the radiation field, merely redirected. There are two consequences: Firstly that we shall beasking questions about the angular properties of the redistribution - is it equally probable thatthe energy goes in any direction (isotropic), or is it preferentially forward (forward scattering)or backwards (backward scattering)? Secondly energy is also being scattered into the beamfrom other directions (see below). In general there is very little interaction with the kineticenergy reservoir and therefore little heating or cooling associated with this process. Whatinterchange there is is due to the fact that there is no such thing a pure elastic scattering in thissituation, there is always some small non-elastic term.

Resonant scattering is perhaps themost difficult term to consider as it has someof the properties of both of the above. Theenergy is first absorbed by the molecule, thenreemitted some long (in quantum-mechanicalterms) time later. The re-emission may be atthe same, nearly the same, or very differentfrequency depending upon the quantumlevels involved.

The re-radiation is usually isotropicbecause the molecule has lost all sense ofincoming photon direction and there is somedegree of interaction between the processand the kinetic energy field. Resonantscattering represents the only atmospheric


-44-

B

ka 'dx

P6

I(6) P

(6,6) 1

4%d6 ks 'dx

process by which radiation may be directly changed in frequency. It is fortunately not tooimportant in the lower atmosphere.

Atmospheric Augmentation Processes

Just as there are three processes that remove energy from the radiation beam, there arethree processes which add energy to the beam, a process which we shall call augmentation byanalogy with extinction. These processes are: emission, simple scattering and resonantscattering.

Emission is the process by which energy is transferred from the kinetic energy reservoirto the radiation field. In a blackbody cavity there is no net change in kinetic energy and theabsorption and emission components must balance (this is a restatement of Kirchoff's Law) andtherefore the absorption and emission coefficients (at any wavenumber) must be the same. Theamount of emission is therefore

where B is the monochromatic blackbody function at the temperature of the gas. Now I shouldmake it clear here that this is not the whole truth, for example it is not intrinsically obvious thatthe conditions that one finds inside a cavity can be immediately applied with abandon outsidea cavity. However in order not to spend too long on what may appear to be a trivial point letme just state that this condition applies in an atmosphere which has a civilised pressure(say > 103 Pa) and may apply in other cases as well.

Simple scattering was alluded to in the section above and is just the component due toscattering into this direction from all other directions, including this direction. The last part ofthe definition may be considered to be a mathematical nicety because it is impossible todistinguish radiation which has not been scattered from that which has been scattered on intothe same beam, however the inclusion of that part enables the scattering functions definedbelow to have a smooth form and not be constrained to be zero in the forward direction. Wecan write the scattered energy as:

where 6, 6' is the angle between s and any general direction and P is the "phase function" forscattering at some spatial angle between input and output beams. The 1/4% factor accounts forthe normalisation of P with respect to ks so that P contains the angular distribution and ks thetotal energy, as can be seen by considering an isotropic radiation field which by energy balancegives:


-45-

P6

P(6,6)4%

d6 1

P

P6

I(,6) Q(,) d6 d kr() 'dx

dI I

k'dx j

'dx

I

k'dx J

k'dx

dI

k'dx

I

J

which is correct if P = 1 for an isotropic scatterer.

Resonant scattering is (almost) always isotropic and therefore one can represent it as:

where Q is the coupling between some other wavenumber and this one. Notice that theintegrals can be separated and what matters is the integrated intensity or energy density of theradiation field at the wavenumber 1.

The Atmospheric Source Function

If we collect all the augmentation terms listed above we can call them the "source terms"in the equation for the rate of change of intensity:

where the upper expression uses j from summation of the terms discussed above and the loweruses the quantity J defined by J = j/k thus achieving a certain symmetry about the equation.

The full equation is therefore:

which is known as Schwarzchild's equation and applies only to monochromatic radiation.This equation is of very generally applicable. It can be said that the course stops here becauseeverything that you need to know about radiative transfer in a planetary atmosphere iscontained in Schwarzchild's equation and the rest is just manipulation - those of us in the gameknow how much and how difficult that manipulation can be in all but the most trivial of cases.

A Word About Units

The units of k'dx must be "number", i.e. no units, so that there are many consistent setsof units which can be used for these quantities. Since this pre-dates SI units the unit of lengthis almost invariably the centimetre which still leaves many choices for k and '. Of all thechoices available I will suggest that the unit for ' should be either molecules/cubic centimetre


-46-

dIdx

ek'x k' Iek'x

ddx

Iek'x k' Jek'x

I(X)ek'X I(0) P

X

0

k'Jek'xdx

I(X) I(0) ek'X P

X

0

k'Jek'(xX)dx

or grams/cubic centimetre. The former is useful because k is fundamentally a quantum-mechanical type function which naturally turns up on a "per molecule" scale whereas the latteris useful in cases where one is considering the physical density. Since the relationship betweenthe two is just 6.023 x 1023/M where M is the molecular weight, there is no great difficulty. Inmany cases we combine the 'dx to give the area density unit, e.g. molecules/square centimetre,which is an obvious extension. The symbol often used for that is u or du, the amount ofmaterial in the path.

You will also see references to the "atm cm" in the literature and I would counsel youto avoid this unit as it is ill-defined. The best definition I can come up with is that it is theamount of material in 1 cc of the absorber at a pressure of 1.013 x 105 Pa at a temperaturewhich should be specified. If the temperature is not specified, then 273K should be assumedbut I have met papers where we have puzzled for hours over the exact definition that the authoris using. As I said - avoid it.

Schwarzchild's Equation

Schwarzchild's equation is the fundamental equation for radiative transfer in theatmosphere and as such contains practically all the physics of the problem. However it does notadmit of a simple general solution and so a lot of time is spent applying it to various specialcases. It can however be formally integrated to give a solution as follows.

Since the equation is not obviously integrable we try using an integrating factor ofexp(k'x) to give:

which is now formally integrable from the point x = 0 to the point x = X to give:

or

or substituting x1 = X-x


-47-

I(X) I(0) ek'X P

X

0

k'Jek'x

dx

-(x) exp(k'x)

I(X) I(0)-(X) P1

-(X)

J d-

Hot Target

Cold Target

Isothermal

I

I

B

B

I = B

p

p

p

Figure 28: Experiments with VaryingCell and Source Temperatures

I(x) I(0)ek'x B 1 ek'x

which is the integral form of Schwarzchild's equation.

If we formally introduce the transmission as:

then things "simplify" even further to:

Notice that in all the above manipulation we have not actually made any progress towards a"solution" except insofar as we have cast the equation in a more suitable form to apply it to aparticular situation.

Some Simple Examples

Let us consider just one or two examplesof the application of Schwarzchild's equation tosome real cases. Take the case of a gas in a cellin the laboratory which absorbs radiation andabsorption is the only thing that we need toconsider. In that case we can readily show thatJ is, in this case, just the blackbody function Band the equation can be written as:

If we therefore place a detector at the other endof the cell and monitor the total energy emittedwe see that at zero density (vacuum) we haveI(0) and as the density (pressure) is increased weget a gradual changeover from I(0) to B as theexit intensity. Now in a visible experiment wecan take it that the emission is zero and thereforewe recover the familiar exponential decay law.However if we repeat the experiment in theinfra-red where the Planck function can besignificant the results can be very different, in fact it is quite possible to produce an increase


-48-

Source

RotatingChopper

Test Cell

Detector

Figure 29: Infra-red Experiment with a Chopper

I(x) I(0)ek'x

in signal as the density is increased by arranging for B > I (hot cell and cold source) or evento arrange for no change at all (everything is isothermal).

This is indeed a problem in infra-red laboratory experiments because the situation isbeing complicated by emission from the experimental material. In practice the problem iscircumvented by switching rapidly between two values of I(0) with a chopper thus:

and then looking at the difference in the output. Since the emission signal remains the same(gas temperature is uniform) we then find:

and we have isolated the gas transmission term by eliminating the gas emission term. Thistechnique of "chopping" the input radiation is widely used for this reason and the reason thatinfra-red detectors tend to be better at detecting changes in radiation than an absolute level.

So far we have applied the equation exclusively to homogeneous paths, i.e. paths whichexhibit no variation in any physical parameters along their length. This case is the usuallaboratory one where producing an inhomogeneous, and particularly inhomogeneous in aknown manner, path is difficult-to-impossible. However the atmosphere is evidently structuredin about every physical and chemical manner imaginable in 3-D. It is not too difficult toincorporate the variation in formal terms in our monochromatic equation. We extend thedefinition of transmission to:


-49-

-(x) e3 , 3 Px

0

k'dx

θ

I(cosθ)

I0

Figure 30: Measuring Solar Flux

Itot Mi

I(i) i

We define the quantity 3 to be the "optical path". In a planetary atmosphere we often talk aboutthe optical path from the top of the atmosphere and this becomes the "optical depth". We mustbe careful here that we keep the definition straight in a general case because the transmissionfrom a to b must be the same as that from b to a, i.e. the limits of integration must be watchedcarefully to see whether they are not only the correct value but also the right way round. Withthis extended definition of transmission the equations still work nicely and could be applied toinhomogeneous paths. One should note that these equations only apply to the monochromaticcase.

Measuring the Solar Constant

As a further example ofthe application of S'child'sequation consider the case of asurface-based measurement of thesolar constant trying to measurethe monochromatic energy in thesolar beam. Since the solar beamis nearly plane-parallel we canapply the equations for intensitywith adequate precision andevaluate the total energy incidentupon the radiation detector (theflux) by multiplying by 6s thesolar solid angle. However sincewe are trying to measure theintensity of radiation upon anatmosphere whose transmissionproperties we do not know, wemust find some way of gettingsome variations in so that we can extract the required result from the data. Fortunately the sundoes this for us by moving in solar zenith angle () during the day. We also need to split thetotal spectrum up into a large number of bands, each of which is narrow enought to bedescribed as monochromatic so that we can apply Schwarzchild's equation to each individualband.


-50-

I(sec) I0ekMsec

ln(I(sec)) ln(I0) kMsec

Ln(I)

sec(θ)0 1 2 3 4

I(∞)

Figure 31: Results of Measurements of Solar Intensity at Surface Over Time

We consider the atmosphere to be represented by a flat layer of some definite, if large,depth. Under these conditions - known as the "plane-parallel atmosphere approximation" - thetotal amount of material in a vertical path is M, and the total in any slant path is M/cos. Sincethis is a visible-radiation experiment carried out on a beautiful day on a remote tropical island(clear-sky conditions) we can ignore any atmospheric emission and find:

If we (lie on the beach and) make measurements all day we shall be able to plot a graph. A suitablegraph form is I(sec) vs sec for which the equation is:

which is a straight-line graph:

the intercept of which gives I(0), the extra-terrestrial solar intensity (monochromatic). Byperforming this experiment at a number of wavelengths one can obtain the shape of the solarradiation curve and then integrate it to get the total solar radiation incident upon the Earth'satmosphere.


-51-

Light Baffles

Detectors

IncomingSolar Radiation

Figure 32: Angstrom's Pyrheliometer

V

I

shade

Thermopiles

Figure 33: Detectors in BridgeArrangement

We should note along the way that the value of sec is constrained to be >1 andtherefore we cannot construct the full line and the result has to be extrapolated. My instructorsused to drill into me that you should never extrapolate a line and therefore one is led to considerthe weaknesses in the method. These are essentially the extrapolation, the assumption ofconstant conditions over the day, the measurements in finite wavenumber widths, rather thanmonochromatically, and the non-plane-parallel nature of the atmosphere.

However we are not going to dwell on the weaknesses here but rather would note that,prior to the spacecraft era, this method, known as the "long" method of measuring the solarconstant provided the only reliable method of measuring the extra-terrestrial solar flux.

Measuring Solar Radiation

Although the major objective of thiscourse is not to lean too heavily uponinstrumentation, I shall discuss some of theclassic instruments for measuring radiation at aplanetary (Earth's) surface. One such instrumentis Angstrom's Pyrheliometer for measuring thedirect total solar radiation at the Earth's surface.Note that this is not the instrument discussedabove for measuring the extra-terrestrial fluxbecause it uses the whole solar radiation beamrather than doing monochromatic measurements.In fact it is so careful to do a full measurementthat it uses no optics at all and just presents ablackened surface shielded from all otherradiation by a long tube to the solar radiation.The principle of operation is that the energygathered from the sun causes the temperature ofthe surface to rise and this is compared by adifferential thermocouple circuit with the powerdissipated electrically in another similar surfacewhich is shielded from the solar radiation beam.In order to prevent systematic errors from thetwo strips they can be swapped to reverse thesign of the systematic errors.

The surfaces are made in the form of thinblackened strips so that the surface presented tothe radiation is a maximum for the volume of thebody, resulting in a large temperature rise. Theresistance of the strip is used as the heatingelement to further reduce the thermal mass byeliminating a separate heater element.


-52-

dA z

z+dz

p(z)

p(z+dz)

Figure 34: Element of Fluid andForces Thereon

p(zdz) p(z) dA dpdz

dz dA

This instrument, when properly used, has a quoted accuracy of ±0.5% which isreasonably good even by today's standards - and the instrument has been around for a longtime. Of course there are problems with the instrument, draughts being one of them asconvection and conduction are very effective at removing heat from surfaces. A furtherproblem is the fact that the distribution of the solar and electrical heating within the small bodyare different and therefore even the same power inputs may produce different results. Manyintercomparisons and studies have been done on the instrument to consider these questions andit is still in current use - so it isn't that bad.

Vertical Temperature Structure - A Dynamical Interlude

In order to appreciate some of thepoints that I am going to make in the nearfuture it is going to be necessary to derivejust enough atmospheric dynamical formulaeto appreciate how an atmosphere will react interms of convection to a vertical temperaturegradient. The result, before we start, is thatthere is a specific lapse rate, , which isnothing more than -dT/dz, which measuresthe line of stability for an atmospherebetween a convective regime with a greatdeal of vertical mixing and a non-convectiveregime in which convection is suppressed.Now let us try to derive such a relationshipfrom first principles.

In a gaseous medium with agravitational field we can express the forcesacting upon an element of horizontal size dAand vertical thickness dz as being the difference between the forces down on the upper surfaceand the forces up on the lower one.

Now the forces are just a manifestation of the pressure in the fluid and therefore theforce on any face can be expressed as the pressure on that face multiplied by the area. As weconsider our element to be largely planar we can ignore the forces on the vertical faces andconsider only the horizontal ones.

The force on one of the faces is just p(z)dA, the net force down is


-53-

dp 'gdz

p (m/V)RT 'RT

dpdz

gpRT

p(z) p(0) ez/H, H RTg

0 dU pdV

where the RHS is derived from expanding the first term on the LHS in a Taylor series. Thisforce is balanced by the gravitational force down which is just 'gdV = 'gdzdA.

Equating these two terms gives the hydrostatic equation:

The gases in atmospheres are also governed by the ideal gas equation

and therefore we can express the hydrostatic equation as

This equation is not, in general, integrable for an atmosphere because the temperature varieswith the height, z. However in the particular case of an isothermal atmosphere we can integratethe equation to get:

The quantity H is known as the "scale height" and is a measure of how fast the pressure (ordensity) falls with height. For the Earth's atmosphere near the surface the value is about 8 kmwhereas for Venus the corresponding figure is around 12 km due to the temperature andcomposition variations between the two planets. Even in the case where the atmosphere is notisothermal we often talk about the "local scale height" which is defined by RT(z)/g.

In order to learn a bit more about the temperature structure of the atmosphere weconsider the concept of an "air parcel" which is moved about the atmosphere withoutexchanging energy with its surroundings - adiabatically. To show that this is realistic is ratherhard, but you can observe that when the atmosphere becomes vertically unstable theoverturning is quite fast and it would take a very fast heat transfer mechanism to exchange asignificant amount of heat in or out of a parcel during a rising motion. Therefore on thetimescale of convective overturning, we might consider parcel motion to be adiabatic.

The word adiabatic implies a thermodynamic consideration. The First Law ofthermodynamics gives:

where the 0 indicates that this is an adiabatic process. Since dU = mCvdT for any air parcel, and


-54-

pdV Vdp mRdT

0 Cv R dT dp'

dTdp

1'Cp

dTdz

gCp

AdiabaticLapse

Rate

Unstable

Stablez

T

Figure 35: Atmospheric Lapse Rates

from the ideal gas equation, we can show that:

or, since Cp = Cv + R, we can write the change of temperature with pressure of a parcel movingadiabatically as:

Within a hydrostatic atmosphere this can be further simplified to:

which introduces a constant "lapse rate", . Thus a parcel moving within a hydrostaticatmosphere will exhibit a fixed rate of change of temperature and density with height. In factit will take on the pressure of the surrounding atmosphere but will set its "own" temperatureand density.

Consider the two possibleatmospheric temperature profiles shown herewith the corresponding adiabatic lapse ratedrawn on for comparison. In the Earth'satmosphere near the surface this is about10K/km.

In the first case the parcel starts at thesurface and as it rises becomes warmer thanits surroundings because its temperature fallsless rapidly than that of its surroundings. Ittherefore becomes less dense and is thereforebuoyant - continuing to rise at anaccelerating rate until it again meets theatmospheric profile at some higher level.The atmosphere is convectively unstable. In the second case the reverse happens and the parcelbecomes more dense than its surroundings and tends to sink again. This atmosphere istherefore convectively stable. The limiting case is obviously when the atmosphere follows theadiabatic lapse rate and the whole thing is marginally stable.


-55-

dpdz

gp

R(T(0) z)

p(z) p(0)T(0) z

T(0)

g/R

We therefore conclude that an atmosphere whose lapse rate exceeds the adiabatic lapserate is convectively unstable and will mix vertically because vertical motions are amplifiedwhereas an atmosphere whose lapse rate is less than adiabatic (more nearly isothermal) is stableand will not mix by large-scale vertical convection, i.e. it is convectively stable.

In the case of a condensing atmosphere where some liquid (e.g. water or ammonia) iscondensing as the temperature falls, the above equations must be modified, principally by theaddition of a latent heat term Ldmw to the first law expression for the adiabatic parcel. Thisterm expresses the conversion of latent to "sensible" heat during the condensation/evaporationprocess. This term does not substantially affect the results derived above except that it altersthe value of the critical lapse rate downwards. As an example for the Earth the "dry adiabaticlapse rate" is about 10K/km whereas the "wet adiabatic lapse rate" is around 6K/km dependingupon the conditions.

We can consider the pressure variation in an atmosphere with an adiabatic lapse ratewhich is independent of temperature by substituting T(z) = T(0) - z into the hydrostaticequation. We then obtain:

which may be integrated to give:

This equation is known as the Hypsometric equation and is the basis of calibration for aircraftaltimeters which operate by measuring the external pressure. Since this has proven unreliable(it relies on assumptions about the surface pressure and atmospheric temperature profile) radarmethods of measuring altitude are preferred.


-56-

z

T

Figure 36: "Typical" AtmosphericTemperature Profile

z(km)

T (K)

50

12

200 300

80

Troposphere

Stratosphere

Mesosphere

Thermosphere

Tropopause

Stratopause

Mesopause

Figure 37: Earth's Temperature Profile

Temperature Structure in Planetary Atmospheres

If we look at the temperature structureof the atmosphere of a "typical" planet wefind a curve as shown here. There is a regionof negative temperature gradient close to theadiabatic lapse rate followed by anisothermal region, followed again by aregion of increasing temperature up to anuncertain maximum as the atmosphericdensity decays to nothing and temperatureceases to have any meaning.

However the Earth is not a "typical"planet and has a temperature structure likethis with an extra bump in it. This bump isknown as the stratosphere and seems to be aproperty only of the Earth's atmosphere.


-57-

-(z) exp k P

z

'dz ekp(z)/g

dE P

0

dE

d

dE

I() 6s -(z) k'(dz) I() 6s -k dp/g

dE

Cp 'dzdT

dt

h

dT

dt

I()6sk

Cp

ek

p(z)/g

Absorption of Solar Radiation - Chapman Profiles

I will remind you here that the term "density" ( ') can refer to either the TOTALdensity - as in the hydrostatic equation - or to the DENSITY OF OPTICAL MATERIAL -as in Schwarzchild's equation. In the following section we assume that the bulk atmosphereabsorbs with absorption coefficient k and therefore both densities are the same.

If we consider the normal incidence of a solar beam on the atmosphere we can expressthe monochromatic transmission -(z) from the top of the atmosphere to a height z as being:

thus the intensity at a level z is just I()-(z). The amount of energy absorbed between the levelz and z-dz is therefore:

where

and 6s is the angle subtended by the sun. If we consider that the energy absorbed is used toheat a layer such that

and we equate these two expressions we find that:

which is a maximum at the top of the atmosphere and decays as the altitude drops. Thequantity dT/dt is called the "heating rate" of the atmosphere and is a measure of the tendencyof the atmosphere to heat (or cool) under the influence of some process.

Evidently in order to evaluate the total heating rate we should integrate over the fullwavenumber spectrum. However many important features can be seen in the monochromaticrates derived here.


-58-

z

dT/dt

Figure 38: Chapman Heating Rates

I(z) I() ekp(z)/g

dEdp

I()6sk

gekp(z)/g

dEdz

I()6s'kekp(z)/g

The maximum in the heating rate atthe top of the atmosphere occurs becausealthough the energy absorbed is very small itis distributed over a very small mass. Thetemperature profile induced is a positivetemperature gradient with height, i.e.convectively stable and the only possibleheat transfer mechanisms are conduction -which is not very efficient - and thermalradiation which we shall show later is alsonot that efficient. Thus in very short orderwe can predict the temperature structure ofthe upper part of any planetary atmosphere.

Now of course we have made manyapproximations. We have assumed that theabsorption coefficient is a constant, that thetotal density and absorber density are equal(or at least proportional to one another) andso on but the basic validity of our resultsstands up surprisingly well in the upperatmosphere as we shall see later. Such an application to the lower or middle atmosphere wouldnot be so fortunate.

Before leaving the subject of this absorption profile let us consider the attenuation of thebeam with height. This is given by I()-(z) and more explicitly by

If we wish to calculate the rate of loss of energy (which is proportional to the rate of loss ofphotons from the beam and the rate of production of a photochemical product if the absorptionprocess leads to photodissociation) from the beam, this is:

or, substituting from the hydrostatic equation:

Notice that ' and p decrease with height and this equation has a maximum.

In the case of an isothermal atmosphere we can substitute for ' and p in terms of z andget:


-59-

dEdz

I()6sk'(0)ez/H exp(kp(0)ez/H/g)

z

dE/dz

no gas

no photons

τ(z) = e-1

Figure 39: Chapman Profles for anIsothermal Atmosphere

3(z) P

z

k'dz

on differentiation this shows a maximum at exp(-z/H) = g/p(0)k or when -(z) = e-1

This relationship, which is known asthe Chapman profile, is illustrated here.

The interaction can be expressed inwords as: at the top of the atmosphere thereis plenty of energy but little gas to react with,at the bottom of the atmosphere there isplenty of gas but no energy and in the middlethere is just the right amount of each.

Atmospheric Optical Depth

Looking again at the maximum of dE/dz we find that if we now reintroduce the conceptof the "optical depth", 3, which is defined in an atmospheric context as:

then the point of maximum absorption occurs at the point when the optical depth is unity - aconcept that we shall return to in a moment.

The "normal optical depth" is just the optical depth measured in a straight downdirection.

Notice that the differential of optical depth is given by:


-60-

d3 k'dz

dId3

I J

0.00

0.02

0.04

0.06

0.08

0.10

105

104

103

102

Wavenumber (cm-1

)

250K5780K x 20

5780K

Rad

ianc

e (W

m-2

sr-1

(cm

-1)-1

)

Rotation

VibrationPhotolysis

Electronic LinesPhotoionisation

Figure 40: Molecular and Atomic Interactions vs Wavenumber

because of the limits of the integral. Therefore if we write Schwarzchild's equation in terms ofatmospheric optical depth it becomes:

Energy Absorption in Planetary Atmospheres

There are a large number of processes by which energy can be absorbed in a planetaryatmosphere which can be simply related to the molecular and atomic properties of the gaseswhich make up the atmosphere. In rough order of increasing energy these are:

- rotation


-61-

E

n=1

2

3

∞

Figure 41: Term Diagram forHydrogen

H hf H

- vibration- electronic lines- photolysis- photoionisation

One should however be aware that there is a very considerable spread in all these processes andthey do overlap. If we plot the energy required on a wavenumber scale with solar andterrestrial radiation for comparison we get a diagram similar to the one shown above.

Thus one can see that the solar interactions are generally in the last three and the thermalinteractions in the first three, although there are some exceptions.

In order to explain things better let uslook at the simplest way of discussing theenergy levels of a system - the term diagram.The term diagram is just a chart of theallowed energy levels of a system and for anatomic/molecular system it is the allowedenergy states of that entity. The plot hasenergy on the y-axis and nothing inparticular on the x-axis. For every allowedenergy level within the system there is a lineon the diagram. For instance the energylevels of the hydrogen atom are given by1 1/n2 (ignoring the constants) where n is apositive integer (see diagram).

The upper bound is the point at whichthe hydrogen atom loses an electron tobecome a hydrogen ion. If we now considerthe reaction

then the only allowed absorptions are those which move the atomic energy from one of theselines to another. These transitions mark the energies at which the system might absorb energyfrom the electromagnetic field. Although these appear to be discrete lines on this diagram, theyare in fact broadened into lines of finite width by various mechanisms.

The existence of a transition does not guarantee that it will be observed because it maynot couple to the external radiation field. This is where transition probabilities and selectionrules come in. Broadly speaking transitions are divided into "allowed" transitions which couple


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n = 2 3 4 5 6

Energy (ν~)

Ionisation Limit

Figure 42: Simple Hydrogen Spectrum

freely to the electro-magnetic field and "forbidden" ones which do not - although the distinctionis one of degree rather than of absoluteness.

We therefore conclude that many absorption spectra will be composed of discrete lines.In fact taking our example above of the hydrogen atom, we expect to find lines at positionsgiven by 1/n2 1/m2 where m and n are integers. For historical reasons the lines series forn = 1,2,3 are called the Lyman, Balmer and Paschen series respectively.

It is also possible to excite an atom or molecule to such a level that the electron isknocked clean out of the atom - photoionisation. In the case of the hydrogen atom shown abovethe ionisation level corresponds to 1 on the diagram. Since a free electron is able to take on anykinetic energy at all, that is the energy states it may occupy are a continuum, we thereforeexpect a continuum absorption above the ionisation level. Thus for a hydrogen atom startingoff with n=1 we find a spectrum which is part line, part continuum.

It would appear to be possible to have a continuum everywhere because we can draw aspectrum based on each n and superimpose them to get a "continuous continuum". Howeverwe must account for the probability of finding an atom in the lower state in order to furtherexcite it. The probability of finding a molecule in a state of energy E above the ground stategoes as something like exp(-E/kT) where k is Boltzmann's constant and since the energy of then=2 state is 1.6 x 10-18 J whereas kT at room temperature is 4 x 10-21 J the probability of findingany hydrogen atom in the atmosphere in the n=2 state is exp(-1.6 x 10-18/4 x 10-21) = exp(-400)and is vanishly small.


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Energy (ν~)

IonisationLines

Figure 43: Schematic AbsorptionCoefficient

The continuum dies out as weincrease the photon energy still furtherbecause of a general tendency of absorptioncoefficients (the k in Schwarzchild'sequation) to follow the sort of law shown onthe left. The sharp cut-on of the continuum isoccasioned by the finite energy required toionise the atom but the cross-section showsa maximum and then a decay as the energy isincreased further.

The interaction between matter andthe radiation field is therefore quite acomplicated affair which results in theabsorption coefficient being very highly structured. For every species there exist certain linesof greater and lesser strength and continua of various complicated shapes.

Absorption in the Earth's Atmosphere

Now consider the most common constituents of the Earth's atmosphere, oxygen andnitrogen. We would expect that any interactions involving these gases would happen at thehighest levels since they will reach a suitable density first as we come in along the solar beam.Nitrogen and oxygen are very "dull" molecules spectrally as they exhibit only photoionisation,photodissociation and atomic-like lines. These are all high energy reactions and occur byinteraction with the short-wave radiation of the ultra-violet resulting in the production of atomicoxygen, ozone and atomic nitrogen which then further react with the short-wave radiationbecause they are atomic in nature. Using the point at which the normal optical depth is 1 as anindicator of where the energy is absorbed in the Earth's atmosphere we can plot the height ofthe e-1 layer against wavelength (which for once is more convenient than wavenumber) and get:


-64-

1000

80

160

200 300

O2

O3

O2,O,N2,N

Wavelength (nm)

Alti

tude

(km

)

Figure 44: Unity Optical Depth for Earth's Atmosphere

O2 hf O O

It should also be noted that the available energy in the solar beam is increasing steadily andvery rapidly, almost exponentially, across the diagram from left to right. The three humpscorrespond to interactions involving ozone, molecular oxygen and the rest reading from rightto left, i.e. increasing energy. Considering first the extreme short-wave we find absorption dueto various processes involving atomic oxygen, atomic nitrogen and photoionisation reactions.These reactions account for the very high temperatures discussed in the last lecture for the veryhigh levels and for the existence of the ionosphere, a region of the Earth's (or any other planet's)atmosphere which contains free electrons and ions. Since any gas can be ionised bywavelengths which are around this area, all planetary atmospheres show an ionospherealthough the ionic composition may vary widely. For instance the Earth's ionosphere containsmostly O+ ions whereas the Venusian ionosphere contains mainly CO2

+ ions.

As we come down through the atmosphere we reach a region where there is a significantamount of molecular oxygen which can be dissociated by the reaction:

requiring a wavelength <246nm. However once split the atomic oxygen finds it difficult torecombine because the reaction O + O O2 is dynamically prohibited, the most likely resultbeing immediate dissociation again. The most likely recombination reaction is:


-65-

O O M O2 M

500

400

300

200

100

106 108 1010 1012 1014 1016 1018 1020

Alti

tude

(/km

)

Number Density (/m3)

O3

A O2

N2 Total

O

He

H

Figure 45: Composition of the Terrestrial Atmosphere

where M is any other molecule to carry off the excess energy. You will note that it takes oneof each of the above reactions to get the original photon energy converted into kinetic energy,i.e. heat. By the law of mass action the rate of the recombination reaction above is [O]2[M] kwhere [] indicates the molecular density and k is the reaction rate (sorry to use k again but sodoes everybody else). This is a very strong function of the atmospheric density and thus goesrather slowly in the upper atmosphere. The result is a huge build up of atomic oxygen whichbecomes the dominant constituent of the atmosphere above 300 km and is a major constituentabove 100 km. The large concentration of atomic oxygen is also aided by a process of"diffusive separation" which is beyond the scope of this course. We therefore discover that aswell as influencing the atmospheric temperature at high levels the absorption process alsodictates the composition of the upper atmosphere. The diagram below shows the compositionof the earth's atmosphere against altitude and you will note the pre-dominance of atomicoxygen above 150km.

Once we have considered the above reactions, we have considered all the importantdirect reactions with the solar beam of the major atmospheric constituents of the Earth'satmosphere. This appears to be a somewhat unusual case as planets go because the only


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O O2 M O3 M

O3 hf O2 O

z(km)

T (K)

50

12

200 300

80

Troposphere

Stratosphere

Mesosphere

Thermosphere

Tropopause

Stratopause

Mesopause

O3

Figure 46: "Typical" Ozone andTemperature Profiles

function of the gases in the lower atmosphere (we will consider the middle atmosphere in amoment) is to provide density. In contrast we shall find that the atmospheres of most otherplanets have much more active major constituents.

In the lower atmosphere most of the interactions between the optical beam and theatmosphere are in the form of interactions with the vibration-rotation spectrum in the nearinfra-red. It appears that, at least for the three "earth-like" planets, the visible region is mainlyclear of interaction except for scattering and ozone reactions. Since ozone reactions occur inthe Earth's atmosphere we had better look at them first.

Ozone - An ultra-violet shield

Ozone is only formed in the Earth's atmosphere because of the considerable amount offree atomic and molecular oxygen. Under these conditions the reaction

is possible. Note that this reaction is similar to the re-combination reaction of atomic oxygenand, like that reaction, goes slowly in the upper atmosphere. The ozone production region istherefore somewhat lower in altitude than the regions previously discussed and occurs ataround 50 km. At that level most of the extreme short-wave radiation has been eliminated fromthe solar beam but the residual ultra-violet is still sufficient to disrupt the ozone molecule atwavelengths shorter than 340 nm.

Note that, in a very similar way to the atomicoxygen reactions, the above pair of ozonereactions absorb solar radiation in thestratosphere.

From the above discussion we mayreadily deduce that there is a heating effect inthe form of a Chapman profile at some levelin the atmosphere, which turns out to bearound 50 km. This is the cause of thestratosphere and in concert with thetemperature rise there is a peak in ozoneconcentration. The peak occurs somewhatlower than the temperature peak because thephotolysis reaction turns off fairly quicklyand dynamical processes influence thevertical and horizontal distribution of ozone.


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The ozone in the stratosphere is responsible for limiting the wavelengths of ultra-violetradiation which can penetrate to the ground. Since this wavelength at around 320 nm is in theso-called "biologically active" region of the spectrum which means that it affects plant growthand other similar biological processes, any significant variation in the cut-off will cause aneffect at ground level. The exact magnitude of this effect is not certain, nor is it clear that theresults would be entirely undesirable. The effect on human life is extremely uncertain and isprobably rather less violent than transporting pale-skinned Canadians to Florida beaches.

The vertical and horizontal distribution of ozone in the atmosphere form a veryinteresting demonstration of how complicated things can get when a process involves dynamicsand chemistry, as well as radiation. From the above discussion it follows that the peakproduction region for ozone is in the tropical stratosphere since this is where the insolation isgreatest for the greatest time per day (annual average) and this is indeed true. Howeveralthough ozone in the atmosphere is a chemical species for the upper atmosphere, that is theconcentration is dependent upon the production and destruction reactions mentioned above, inthe lower stratosphere it is a tracer species. A tracer species is one whose overall concentrationin the atmosphere is conserved whereas the local concentration is determined by dynamicalprocesses. Since the density of the atmosphere is greater lower down, the actual contributionto the total column density (,'dz) is much greater from the tracer region than from thephotochemical regions. Ground-based instruments measure the total extinction of the opticalbeam due to ozone absorption and from a series of these measurements can determine the totalozone column in the atmosphere. These instruments, known as Dobson ozonespectrophotometers, are dispersed in a world-wide network measuring the ozone on a regularbasis. They have recently been joined by spacecraft instruments measuring ozone on acontinuous basis.


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Figure 47: Time-Latitude Ozone Cross-Section

The results of the measurements are interesting. At any particular place the ozoneconcentration shows seasonal variations but if we show a map of ozone vs time and latitude wefind that the maximum levels of ozone occur in the night polar regions where there is noproduction of ozone at all! The reason for this feature is still rather poorly understood butseems to have something to do with a phenomenon called "eddy meridional transport" whichis the subject of some research work at the moment (not at Toronto).

If we try to be more precise about the concentration of ozone in the atmosphere we shalltry to look at the concentration levels using a radiative-dynamical model of the stratosphere,putting all the available information about reaction rates, cross-sections and so on with theobjective of determining the ozone and other chemical concentration profiles at the end. Whenthis was first done the results did not agree as the concentration of ozone was consistentlyover-estimated. In other words the sinks of ozone were not strong enough.The modellers (who had a rather simple model) simply assumed that the chemical reactionrates, which are notoriously difficult to measure, were in error and proceeded to "adjust"(fiddle) them until they agreed. The reaction kineticists then tried to refine their measurementsbut found that the refined values did not agree with the adjusted model values. A large debate


-69-

X O3 XO O2

XO O X O2

O O3 O2 O2

then ensued as to who was wrong and was only solved when Bates and Nicolet suggested thatthe Chapman reactions above were insufficient to account for the situation and more reactionsshould be added involving H and OH. This was the signal for several other people to considerother species and the field exploded. It was also aided in its explosion by fears of pollution ofthe stratosphere by supersonic transport aircraft (the SST). Now since the stratosphere is so big(1010 km3) why can such a flea-bite affect it? The answer is twofold:

First the stratosphere is a region with no solid boundary, no rain and a positivetemperature gradient in the lower regions. These respectively suppress dry deposition, wetdeposition and overturning and make it remarkably difficult to exchange atmosphere betweenthe stratosphere and the rest of the atmosphere. Therefore the general rule is that what goes instays in until such time as it is chemically converted into something innocuous or diffused outeither downwards to the troposphere or upwards into the mesosphere.

Second the reactions for removing ozone which are of the form:

These two reactions, which are simple binary exchange reactions and therefore quitefavourable, result in the overall destruction of O and O3 - the so-called "odd oxygen" molecules- and the regeneration of X. In simpler words X catalyses the recombination reaction

and can go on doing so indefinitely. As you should already know a little catalyst goes a longway and therefore it should not surprise you to learn that even concentrations of the order of10-9 have a very significant effect on events.

X can be any one of a large number of molecules both natural and pollutant or both.Bates and Nicolet suggested natural occurrence of H and OH from photolysed H2O in thestratosphere as a natural compound. Crutzen suggested NO as a natural compound, althoughit was rapidly pointed out that this was also a significant product of aircraft exhausts andtherefore any aircraft operating in the stratosphere - such as an SST would add to this sink.More recently Cl and F as photolysis products of "Freons" from refrigeration units,air-conditioners, heat-pumps and spray cans have been mooted as a possible contaminant andso on.

The result of all this, apart from large grants and media coverage, has been theproduction of very large and complicated models of the stratosphere involving literallyhundreds of compounds and reactions far beyond the scope of this course. The net result of thisresearch has been to demonstrate that there can be anthropogenic modification of thestratosphere through this mechanism although the exact magnitude is still in doubt. The oilcrisis of the 70's helped to defuse the SST problems but the freon problem is still with us in anactive sense.


-70-

Line of Sight(LOS)

Direction ofPropagation

(DOP)

θ

l

r

Figure 48: Define directions forScattering Discussion

Scattering

Having discussed an effect which seems to be specific to the Earth's atmosphere, let usreturn to general atmospheric processes. Scattering in atmospheres generally occurs in threeforms although the distinctions are not rigid. The three forms are: molecular, aerosol andclouds.

Molecular Scattering

Molecular scattering occurs in all atmospheres because it is essentially caused by themolecules themselves. It is also known as Rayleigh scattering and is often treated in coursesin electromagnetic theory as an "interesting case". Since I don't want to spend too much timediscussing the mathematics, the ensuing discourse will not provide a rigid calculation but willexplain the significance of the answers.

If we define a plane which containsthe direction vector of the incoming beam,the scattering molecule and the observationdirection (known as the "line-of-sight" orLOS) then we find a difference in thescattering properties depending uponwhether the incident beam electric vector isoriented parallel (the l direction) orperpendicular (the r direction) to the planewe have defined. You can easily rememberthe letters by noting that they are the lastletters of parallel and perpendicularrespectively. Under these conditions thescattering function for molecular scattering Pis given by cos2 for the parallel vector and1 for the perpendicular vector. Thisdifference in the vectors for the polarisationdirections means that even if the incomingbeam is unpolarised, the outgoing beam ispolarised.


-71-

Sunlight

PolarisedLight

Figure 49: Observing Clear-Sky Polarisation

8%3

4

(m2 1)2

N2

Figure 50: Typical Mie Scattering Diagram

In fact if you go outside and set up theexperiment shown here, and observe the skyat 90o to the solar beam, you can observe theclear-sky polarisation due to Rayleighscattering.

If we concentrate on the extinctioncoefficient we find that it is given by:

per molecule. The important thing to noticehere is the -4 dependence of the scatteringwhich means that blue light is scatteredmuch more effectively than red light - hence the blue sky (the scattered light) and red sunsets(the transmitted beam). One should also note that the term (m2-1)2 appears in various forms invarious texts because of the fact that m, the refractive index, is very close to 1 and it is only m-1that is really useful.

Since it is a rather topical subject we should also discuss the question of fine aerosol inthe stratosphere. The theory of Rayleigh scattering applies to any particles for which » awhere a is a typical particle dimension. Volcanic eruptions can force a lot of material into thestratosphere, which as I mentioned last lecture is a region where the processes of dry and wetdeposition are inoperative. Therefore unless the material settles out it will stay there for a verylong time. The finest aerosol takes many years to settle out and meanwhile causes an increasedamount of scattering in the solarbeam.

Scattering by Larger Particles

As the particle sizebecomes larger the assumptionsof Rayleigh scattering becomeless justifiable and one movesinto a regime known as Miescattering. Mie scattering isactually the case where youcannot make any goodsimplifications and you just haveto solve Maxwell's equations inall their glory. A sort of typicalresult set for the solution is shownhere where the parameter plottedis the extinction efficiency, the


-72-

Figure 51: Scattering Diagrams forLarger Particles

extinction coefficient per entity (k in our equations) divided by the geometric particle area tothe beam (%r2 for a sphere). Those of you who are astute will realise that the efficiency tendsto 2 in the geometrical limit of large particles. Since you observe in the laboratory anextinction efficiency of 1 - flower plots cast shadows of their geometrical area - someexplanation is called for.

The explanation, which I shall not justify in this course, is that the theory aboveaccounts for diffraction as well and as much energy is scattered into the diffraction pattern asis blocked from the beam. Another way of saying it is to say that the scattering patterndevelops a very strong forward peak and we do not in general resolve that peak in thelaboratory.

Talking of scattering patterns (P inthe original equations), it is in general truethat larger particles show more structuredscattering patterns with lobes and so on but auniversal feature is the introduction ofstronger forward and backward peaks in thediagram leading to strong forward andback-scattering.

In general a small amount ofscattering does not affect things very muchbecause it merely redirects the beam slightlyand it reaches the planetary surface anyway.However stronger scattering has the effect of increasing the path length in the atmosphere,which gives any absorption a larger amount of material to work over, and increases theback-scattering to space which increases the planetary albedo. The best example of the formeris the effect of dust storms on Mars which are often planet-wide in scale. These are fed by theincreased solar energy absorption resulting from increased solar absorption in the atmospheredue to the dust absorption over the increased path length. The best example of the latter isVenus where the cloud layer causes the reflection of about 75% of the incident visible energydue to back-scattering. The Earth represents a more intermediate case where the absorption andreflection in clouds is quite significant and scattering from lower atmospheric aerosols canbecome quite significant, particularly in industrial areas.

Molecular Absorption in the Infra-Red


-73-

∆J = -1 ∆J = +1

∆J = 0

P

Q

R

ν∼

Figure 52: Typical Molecular AbsorptionBand in the Infrared

v=0

v=1

J=3

J=3

2

2

1

1

0

0

∆J=-1 (P) ∆J=+1 (R)

∆J=0 (Q)

Figure 53: Term Diagram forabove Infrared Band

hf hcv hcr

If you examine the absorption of thesolar beam on the red side of the visible youwill find a number of absorption features inthe near infra-red due to water vapour.These are due to rotation-vibration bandswhich are generally of this form shown onthe left where the central peak (which may ormay not exist) corresponds to the moleculechanging its vibrational state and the"humps" on either side correspond tochanges in both vibrational and rotationalstates and are in fact composed of manylines.

On a term diagram the situation looks likethe situation on the right. A spectral line onthe low wavenumber (energy) side of thecentre is caused by a drop in rotationalenergy (J = -1, where J is the rotationalquantum number) whereas a line on the highside corresponds to a rise in rotational energy(J = +1). The energy absorption cantherefore be written:

where the various bands are caused by theallowed values of

, and the lines within the

bands are caused by the values of r. Thisexplains the general similarity of shape of allthe bands of a given gas, since they all sharethe same r although there are detaileddifferences in the shapes. The large centralpeak is caused by the fact that the v areslightly affected by the rotational state and therefore even if the rotational state does not changethere are slight differences in the absorption energy for differing rotational states.


-74-

k

ν∼

Figure 54: Typical k for a molecule

ν∼

ν∼

ν∼

I(0)

I

I

Figure 55: Absorption in a Layer

I(0) 1 ek'z

I(0) P

1 exp P

z

k'dz d

Most bands involved in planetary atmospheres are the so-called "cold" bands where thelower state of the transition is the ground state. This is because most of the molecules are inthe ground state. A "hot" band has the lower state as some other state and is generally muchweaker in nature and very temperature dependent because of the temperature dependence ofthe lower state population which goes as something like exp(-hc v/kT).

The fact that a band consists ofspectral lines complicates the radiativetransfer calculations no end because themonochromatic absorption coefficient kvaries wildly in a very short distance inwavenumber space. Since a typical bandmay contain about 2000 significant lines(and a host of weaker ones) the function kexhibits about 2000 maxima in the band.The calculation of total absorption andheating rates then becomes very complex.

The total monochromatic energyabsorbed on traversing a layer of thicknessz is:

if we try to integrate this expression over aspectral band consisting of several lines werun into a problem because the function I(0)will obviously have structure after the first(top) layer of the atmosphere and thereforefor all subsequent layers we cannot extractI(0) from the integral. Since the incomingsolar radiation I(0) is not highly structured,we will do better if we consider the totalabsorption from the top of the atmosphereand integrate that. In this case for an integralover a single vibration-rotation band we may approximate the intensity to a constant across theline and extract it from the integral thus:


-75-

ν∼

ν∼

τ

τ ∆W

Figure 56: Schematic ofEquivalent Width

W P

1 exp P

z

k'dz d

I(0)d(W(,z))

dzz

k

ν∼

Independent Overlapped

Figure 57: Effect of Overlapping Linescan vary

the absorption in each atmospheric layer cannow be found by differentiating the integralwith respect to z.

The integral term in the aboveequation is called the "equivalent width"because it represents the equivalent width ofthe block of wavenumber space which isobscured by the band. The term "equivalentwidth" may be used for any such substitutionprovided that the conditions are specified.We may therefore write:

and the absorption in a layer of thickness z is just:

We should notice also that Beer's Law and Schwarzchild's equation are monochromaticequations and therefore they will not apply in such circumstances.

Vibration-Rotation Band Absorption

From the explanation given aboveyou can imagine that the calculation of aband absorption is going to be a verycomplicated matter. In some very simplecases in the upper atmosphere however thereis an approximation called the independentline approximation which assists greatly.This is applicable if the absorptions of theindividual lines do not overlap. In this casewe can consider each line to have anequivalent width of its own and the band isjust the sum of the individual lines. In thecase of overlapping absorptions we must be much more careful because the band is differentfrom the sum of the lines. The difference for a band of two lines is shown here.


-76-

P

f() d 1

In order to perform any calculations at all on a line it is necessary to define its shape andits strength. Happily, for once, these are independent quantities since in the atmosphericsituation the lines are broadened by atmospheric mechanisms but the strength is determined byQ-M considerations of strength of the interactions between the molecule and the photon field.We can therefore write the absorption coefficient k

as Sf( ) where the S is the strength of the

line and f is the "shape function". In order to totally separate the effects of broadening fromstrength we demand that:

i.e. that the shape function for a line be normalised. Using an appropriate shape function onecan then calculate the absorption at any point in wavenumber space and thus the totalabsorption.

The red-end absorption of the solar beam is due almost entirely to high excitations ofwater and, since water is concentrated at the lower end of the atmosphere, is a low atmospherephenomena.

Solar Absorption - A Summary

We have now seen most of the interactions between the solar beam and the atmosphere.In some respect the Earth's atmosphere is typical but in others, notably the ozone layer, it is notand doubtless one could produce other atmospheres which are equally bizarre. The absorptionscan be divided broadly into three groups: the upper atmosphere photolysis, the generalscattering, and the lower atmosphere red-end absorption. I have deliberately excluded cloudson the grounds that we shall look at them a little later.

People have tried several times to put together a bulk diagram of how solar radiation isabsorbed in the Earth's atmosphere and what happens to it and it is instructive to look at thefigures which show that the overwhelming majority of the solar beam actually interacts to someextent with the Earth's atmosphere but some of the interaction is of a "slight" nature. This isan example.


-77-

Planetary Albedo (31)

Incoming Solar Radiation (100)

Absorbed by Earth (Solar) (43)

8 100

22 5 22 -6

17 6

Absorbedby Clouds

4

Reflectedby CloudlessAtmosphere

8

Interceptedby CloudlessAtmosphere

52

Absorbedby CloudlessAtmosphere

22

Transmittedthrough

CloudlessAtmosphere

22

TransmittedDirectly

through theAtmosphere

5

TransmittedthroughClouds

22

Reflectedby

Earth6

Interceptedby

Clouds43

Reflectedby

Clouds17

Figure 58: Bulk Energy Transport in the Earth's Atmosphere


-78-

I I

Ie-kρ∆z Ie-kρ∆z/cosθ

ρ ∆z

θ

Figure 59: Flux TransmittedThrough a Thin Layer

I0 2% P%/2

0

cossind I0 %

I0 2% P%/2

0

ek'z/coscossind I0 2% P

1

ek'zy

y3dy I0 2% E3(k'z)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.5 1.0 1.5 2.0

0.5e-5x/3

E3(x)Residual(x10)

Figure 60: E3 and approximation

Terrestrial Fluxes - Schwarzchild's Equation Revisited

The problem that we shall face with the terrestrial flux is that it is a flux and not anintensity. Since Schwarzchild's equation is not applicable to fluxes we must look carefully atthe problems of flux equations.

Here is a fairly simple flux problem. If we have a vertical intensity beam incident uponthe bottom of a thin layer of absorption coefficient k the transmitted beam will be I0exp(k'z).However if we have an isotropic flux incident on the bottom of the layer what flux will emergefrom the top?

We can calculate the total energygoing through the lower boundary as just:

however the energy emerging from the top ofthe layer is:

where y = 1/cos. The introduction of thethird exponential integral, E3(y), improvesthe mathematical elegance but disguises thefact that the integral is non-analytic which isnot nice. Furthermore the emerging flux isnot isotropic.

However there is an approximationwhich does assist in this calculation and thatis to assume that we are only interested in upand down fluxes, which is true, and totherefore ignore the fact that the energy isnot isotropic any more. Then looking at the


-79-

dF Fk' rdz

dFk'rdz

F %J

form of E3(k'z) we find that it can be written, to some degree of accuracy asexp(5/3)k'z. This means that we can write the attenuation of the up going flux in the sameway as an intensity if we multiply the amount of material in the path by 5/3. Now the factor5/3 is not necessarily the right one because other effects may be important but it turns out tobe a typical value of the "diffusivity factor" which appears in more elaborate theories. In factthe diffusivity factor, r, varies between about 1.2 and 2 depending upon the circumstances.

Under the above approximations we can write a "Beer's Law for Fluxes" as:

and a "Schwarzchild's Equation for Fluxes" as:

The term J is an average value of the source function in the up going direction for the region.In many cases we can replace J by B, the blackbody function. The % in the equation isnecessary to ensure that it is valid in a blackbody cavity. You should notice that the diffusivityfactor, r, is a constant for the equation, i.e. it does not depend upon z.

One fact should be emphasised with the above equation. This is not Schwarzchild'sequation. Schwarzchild's equation applies to intensities. The above equation is anapproximation which should strictly be tested for validity in any particular case. Inpractice it seems to be used indiscriminately.

2-Tone Atmospheres and 2-stream models

We now pause in our review of the radiation effects in atmospheres to consider theproblem of a thermal flux in the atmosphere. We saw above that the presence of an up- ordown-welling flux in a plane-parallel situation, which is correct for the atmosphere, could bewritten as a modified form of Schwarzchild's equation at least insofar as one is prepared toaccept the concept of a diffusivity factor. Again we will emphasise that his is an approximationand not an exact equation, but it is certainly better than anything that we have had before. Wetherefore return to the subject of 2-tone atmospheres to consider the effect of this equationfurther.


-80-

Surface

Solar Thermal

Figure 61: Two-tone atmosphere Model

Fu(z+∆z) Fd(z+∆z)

Fd(z)Fu(z)

z+∆zz

Figure 62: Two-stream Model Diagram

dFk'rdz

F %B

You will remember that a 2-toneatmosphere is one in which the absorption isdivided into strict solar and thermal regimesand the atmosphere is defined as havingdifferent properties for each. In this case weshall confine our discussion to problemswhere the atmosphere does not absorb solarradiation. From the discussion of previouslectures you will realise that this does notnecessarily mean that it does nothing to thesolar beam - it may scatter it drastically - butthere is no heating of the atmosphere due tothe solar beam. The solar radiation incidentupon the ground however does heat theatmosphere since it emerges as thermalradiation on the other side of the diagram.

The atmosphere is therefore "heated from below". We further understand that in thesesimple models there is no overall heating or cooling of any region and therefore the atmosphereis in "local radiative equilibrium" and no layer of the atmosphere gains or loses heat due to thepassage of either the solar or the thermal radiation beam. The extension of the model whichwe shall make at this time is to include a continuous medium as the atmosphere instead of thediscrete layers we have been used to. We shall consider two general streams of radiation in theatmosphere - one upwards directed taking heat to the top, and one downwards directed takingit to the bottom. This is obviously still an approximation because energy also flows sidewaysin the real atmosphere, but it is still better than anything we have used before.

Before discussing the model further we should consider the boundary conditions on Fu,the upwards flux and Fd, the downwards flux. At the top of the atmosphere the sun and deepspace contribute little thermal radiation and we are entitled to say that Fd = 0. On the otherhand the upwards directed flux is the flux which makes up the "effective radiating temperature"and this means that we can assign Fu = )Te

4 at this level. At the bottom of the atmosphere weknow that the up going flux must be given by Fu = )Tg

4 but we know little about the down fluxat this level.

If we now consider a layer in thea t m o s p h e r e o f t h i c k n e s s zabsorption/emission coefficient k (we aregoing to ignore all forms of scattering) anddensity ' we know the "Schwarzchild'sequation for fluxes" is just:


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but we must be careful how we apply this equation to the atmosphere. In this case let usconsider our section of the atmosphere at level z.


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Fd(z) Fd(zz) Fd(zz)k'rz %Bk'rz

Fd(z)

dFd

dzz Fd(zz)k'rz %Bk'rz

Fu(zz) Fu(z)

dFu(z)

dzz Fu(z) Fu(z)k'rz %Bk'rz

dFd

k'rdz Fd %B

dFu

k'rdz Fu %B

dFd

d3 Fd %B

dFu

d3 Fu %B

d(Fu Fd)

d3 Fu Fd 1

We can write an equation for Fd explicitly as:

and an equation for Fu as:

tidying these equations up we get:

for convenience we now wish to substitute for optical depth 3 but in order to force ourselvesto remember that there is still a diffusivity factor in there we call it 3* (remember thatd3* = k'rdz). The equations are now:

We have still omitted the fact that the atmosphere is in radiative equilibrium and this means thatthe changes in the fluxes must balance. Worrying about the signs from the above diagram wefind that in any layer dFu dFd = 0 or integrating Fu Fd = 1 where 1 is a constant.

This is not a surprising result if you think about it because it implies that there is a netupwards transport of energy from the bottom of the atmosphere to the top, i.e. that energy isbeing transported from the ground to space and not lost along the way.

Subtracting the two above equations gives the result:


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Fu Fd 13 c

Fu (1/2)3 1 c/2

Fd (1/2)3 1 c/2

0

Χ0*

Χ*

σTe4

σTg4

σTa4

F↑

F↓

πB

Figure 63: Two-Stream ModelSolution

%B )T4a

)T4e(3 1)

2

which is integrable to show that

We now have to solve for 1 and c, the constant of integration. We can readily separate our twoequations to give:

and at the top of the atmosphere (3* = 0) we find that Fd = 0 which gives c = 1. Further weknow that Fu = )Te

4 at the top and therefore 1 = )Te4.

We have therefore solved for Fu = )Te4(3*+2)/2 and Fd = )Te

43*/2 and we can also solvefor the quantity %B from the original equations to give:

where Ta is the atmospheric temperature. Wecan now plot these quantities against 3* andfind three parallel straight lines showing aconstant net up-flux sufficient to account forthe observed equivalent radiatingtemperature and increasing without limit towherever the bottom of the atmosphere is.We can also see that the surface temperaturemust rise to account for the optical depth ofthe atmosphere and this too is without limit. There is an interesting point that if the central linerepresents the atmospheric temperature and the line for Fu must match the ground temperature,then there is a temperature discontinuity at the ground - this is a common feature of this typeof model and can be observed sometimes, e.g. hot sidewalks on summer days.

Applying this model to the Earth's atmosphere we take the effective radiatingtemperature, 255K and the surface temperature, averaging around 286K and find that 30

* thetotal optical depth including diffusivity is about 1.16 and the taking out the diffusivity factorit the vertical optical depth is about 0.70. One can argue about the figures forever but the pointis that the Earth's atmosphere does absorb a fair amount of the outgoing infra-red flux and weshould look for the source. On the other hand if we apply the same equation to Venus, surface700K, effective 230K, we find that the optical depth is enormous. 30

* = 170 and 3 = 100. Nowthe atmosphere of Venus is much more dense than the Earth's and is made of pure CO2. Weshall see that this will give it an enormous optical depth but not enough to account for this verylarge figure. It is therefore somewhat of a mystery (really) as to how the surface of Venus getsto be so hot and there are a number of very elaborate theories about which try to account forthe phenomena. We shall return to these points later.


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AdiabaticLapse

Ratez

T

2-stream

Figure 64: Two-stream ModelTemperature Profile

Of course a plot of the atmosphere interms of optical depth is very satisfyingtheoretically but it doesn't help with theproblem of how the temperature varies withheight. Replotting the graph on a z vs T scaleis a non-trivial job involving some quitedetailed knowledge of the atmosphere but itcan be done and the results are shown herefor the Earth, along with a plot of theadiabatic lapse rate. Just to remind you, asuper-adiabatic lapse rate is unstable tovertical convection. Therefore the Earth'satmosphere cannot be in radiativeequilibrium because that state is unstable.However it must show a tendency towardsthat state which is countered by the verticalmotions which try to keep the profile sub-adiabatic. The result is that the atmosphere tends to"hang around" the adiabatic lapse rate because it is trying to balance these two forces. Nowadmittedly there are others as well but these are two significant ones.

Just in case you think we forgot about the other "earthlike" planet, Mars, note from thetables that there is very little difference between the effective radiating and surfacetemperatures, in fact the result is negative. This implies that the Martian atmosphere iscomparatively transparent to thermal radiation.

Thermal Radiation in the Atmosphere - An Overall View

We now return to a consideration of the gross features of the thermal radiationinteraction in a planetary atmosphere having seen how the 2-stream model predicts theincreasing surface temperature with increasing optical depth. This survey must, of necessity,be a somewhat specific one to a particular planetary atmosphere because the infra-redproperties depend more on the exact composition than do the visible and ultra-violetinteractions with the solar beam.

In general the interactions are by absorption processes as the power-law vs wavelengthrelationships mean that scattering, unless there is a considerable amount of scatterer, e.g. acloud, is negligible. The absorption processes that are active in the infra-red are almostexclusively vibration-rotation and pure rotation as the photon energy is simply not high enoughto excite any further interactions. There are of course a few exceptions to the above rule suchas the molecular oxygen lines in the far infra-red but these need not concern us here.

In the Earth's atmosphere, which is mostly composed of nitrogen and oxygen, we havea very interesting circumstance in that most of the atmosphere does not exhibit any infra-redspectrum. This is because homopolar molecules (molecules the same at both ends) do notexhibit any vibrational or rotational spectra unless provoked by conditions which are not found


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Figure 65: Transmission of the Earth's Atmosphere

in the atmosphere. It is therefore left to the minor constituents to provide the thermalinteraction and of these water and carbon dioxide are by far the most dominant.

Water vapour has an extensive rotational spectrum which extends from about 20 µm outinto the far infra-red plus one very strong vibration-rotation band at 6.3 µm. There is a secondband at 2.7 µm which is not particularly important in atmospheric calculations but serves as asort of end-stop on the infra-red area of the spectrum, delineating the boundary between thesolar and thermal spectra. Water vapour itself is present at all atmospheric levels but isparticularly concentrated near the surface by the hydrological cycle of the atmosphere whichmaintains a fairly damp state of affairs.

Carbon dioxide on the other hand is rather well-mixed in the atmosphere at aconcentration of about 350 ppm (parts per million) which is slowly increasing with time dueto the burning of fossil fuels and other anthropogenic processes. This uniform distributionmeans that it is outweighed in its effects by water vapour in the lower atmosphere but reverses


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the situation in the stratosphere where it becomes the dominant interacting medium. It has norotational spectrum, being homopolar to rotation (the molecule is a linear one O=C=O) butexhibits some vibrational modes, the most significant being at 15 µm with another importantband at 4.3 µm.

The third component active in the infra-red is ozone. It does not play a large part in theradiative transfer but the fact that the band at 9.6 µm lies in a region of the spectrum which isotherwise clear increases its significance. It is also plentiful in the stratosphere where a smallinteraction has a larger effect because of the reduced density compared to the surface.

The result of this activity is that the vertical transmission of the Earth's atmosphere ishighly structured in the infra-red as is exemplified by this diagram which shows thetransmission from the top of the atmosphere, first to the tropopause and then to the surface. Inthe upper atmosphere absorption you will notice the prominence of the CO2 15 µm band andthe rotation band of water which account for most of the thermal interaction in the infra-red inthose regions. In the lower atmosphere water tends to dominate because of its differentiallyhigh mixing ratio.

There are two regions of the spectrum which offer a reasonable transparency in theEarth's atmosphere, the 3-5 µm and the 8-13 µm "windows". These are used for variouspurposes by industry and the military in applications such as thermography (mappingtemperature of surfaces) and "snooping" (spotting people for sundry reasons). Such is theactivity in this region that most instrument technology is directed to these regions. Onetherefore occasionally meets with incredulity when trying to get detectors and other equipmentto work in the non-transparent regions because "that's not in the atmospheric window sir".

If one turns to the other planets the situation is somewhat different because of thedifferences in composition. The Venusian atmosphere is 90 (Earth) atmospheres of nearlypure CO2 and therefore the CO2 bands are extremely strong and weaker bands such as the10.6 µm band (that's the one that the CO2 laser uses) become very important as do "hot" and"isotopic" bands. There is probably some water in the Venusian atmosphere but nowhere nearas much as in the Earth's atmosphere. The Martian atmosphere is rather dull as far as infra-redinteractions are concerned because it is so thin. This does not mean that there are not importantatmospheric effects because here it is the ratio of heating to mass that counts, not the absolutevalues, but there is little overall effect on the planetary surface temperature.

Planetary Atmospheric Evolution - a Theory

One of the most fascinating problems in atmospheric phenomenology is the markeddifference between the atmospheres of Mars, Venus and the Earth when the planets are, on astellar scale, very similar. One suggested theory is that the differences are due to the "runawaygreenhouse effect" which I will now discuss.


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350

250

110-3 103

Venus

Earth

Mars

Tem

pera

ture

(/K

)

Vapour Pressure (/mb)

Figure 66: The "Runaway Greenhouse" Effect

The first issue is the amount of CO2 in the Venusian atmosphere and how it got there.Since the surface temperature is so high it seems very likely that the CO2 came from the rocksof the planetary surface and indeed, if you make a survey of the Earth and postulate a large risein surface temperature, you can evaporate a large amount of CO2 from the carbonaceous rocksto form a CO2-rich atmosphere.

Let us suppose then thatthe Earth, Venus and Mars allstarted out at t=0 with noatmosphere but water andcarbonates in the surface rocks.At first, of course, the surfacetemperature would be thatpredicted by the simple "effectiveradiating temperature" becausethere was no atmosphere. Oneach planet therefore water wouldbe evaporated from the surfaceinto the atmosphere. Since thewater interacts with the thermalenergy stream, the surfacetemperature starts to rise underthe "greenhouse effect" and thisaccelerates the process. In fact we shall have to find something to stop the process fromevaporating all the water, and the something is the saturated vapour pressure of water. On Marsthe saturation is achieved over ice, leading to the deposition of "hoar frost" on the surface. OnEarth it led to the deposition of the oceans but on Venus the saturation did not manage to stopthe process and the greenhouse effect ran away. As the temperature rose still further thecarbonaceous rocks started to decompose and Venus is currently in equilibrium with these samerocks in a CO2 greenhouse. We can depict this situation as shown here.

Now this simple theory has several unanswered questions about it - such as where didall the water from the Venusian atmosphere go? It must have been comparable with the volumeof water in the Earth's oceans. There isn't a complete answer to that question, some of it mayhave decomposed and escaped to space in the upper atmosphere and some may be involved inthe Venusian clouds - the answer is not clear. However it again illustrates the extreme effectsthat radiation effects may have on atmospheric evolution.

Note: There are other theories which suppose that the three planets didn't start out the sameway and it therefore isn't surprising that they ended up different.

The Earth's Greenhouse Effect


-88-

Figure 67: CO2 as a Function of Time

Since it is fashionable to discuss the greenhouse effect as though it were ananthropological problem, I will join in although I hope I have convinced you by now that theproblem is as old as the hills - literally! Briefly stated it is an observed fact that the atmosphericCO2 content of the Earth's atmosphere is steadily increasing due to the burning of fossil fuelswhich releases the trapped carbon into the atmosphere. Not all the carbon dioxide so releasedremains in the atmosphere for the ocean dissolves a lot of the surplus and therefore forms a"buffer" for the Earth-atmosphere system. However the increase is still finite and measurableas is shown here.

An increase in the CO2 leads to a decrease in the infra-red transparency of the Earth'satmosphere and therefore an increase in the surface temperature. At least that's what a simpleargument will tell you, long experience in atmospheric matters will teach you not to trustsimple arguments about such a complicated system but a longer argument and a detailed modelsimulation will convince most people that the above argument is correct.

The atmospheric "windows" give rise to another related effect - that of the "greenhousegases". Briefly stated, if you put more CO2 in the atmosphere it has a small effect because thereis so much there already. The earth-atmosphere system is de-sensitised to CO2. If you addanother gas which absorbs in the same region as CO2 or H2O you get the same effect - a smallone. But if you add a gas which absorbs in an atmospheric window, where nothing else isabsorbing, you get a disproportionately large effect because there is no competition for the


-89-

N2O CH4 CFC

O3

Figure 68: The "Greenhouse Gases" and Current Atmospheric Transmissivity

photons. So gases which absorb in the atmospheric window are very potent "greenhousegases". The CFCs, CH4 and N2O are some such gases and these are produced by a variety ofnatural and industrial sources. However, despite their low concentrations, they account forabout 1/2 of the calculated increase in the greenhouse effect. So the problem is not just one ofincreased CO2 but one of much more widespread implications.

Since the effect of increasing greenhouse gases is going to be a global temperature riseof a few degrees that doesn't seem to be much to worry about until you talk to the farmers whowill tell you the bad news that the slight climate variations which will be in both temperatureand rainfall will upset the crops. Not all the news is bad, some areas will probably benefit, butquite a lot of areas such as the American mid-west will probably become drier. Not only thatbut a small increase in global temperature will cause a significant shrinkage in the polar icevolume - and there is a lot of water in that ice. The resulting increase in mean sea-level willcause flooding of low-lying areas including such interesting places as New York etc. The lossof land area alone is enough to cause upset in the business world because everybody knows thatland is a permanent investment.

Against the background of the various reports it is fair to say the following: First thatthe effect will probably be noticeable by the end of the century, Second that we all agree on thesign of the effect but are still arguing about the amplitude although the magnitude is agreed tobe "a few degrees", and third that nobody has yet made a definitive measurement of either thewarming process so far or any consequential effect. Time will tell whether the calculations areright because the Earth is just not going to stop smoking yet.


-90-

0

Χ0*

Χ1*

Χ*

σTe4

σTg4

σTa4

F↑

F↑

F↓

F↓

πB

πB

Figure 69: Simple Single-Layer Cloud

Fu )T4e 3 2 /2

Fd )T4e 3 /2

Clouds - As In "I Wandered Lonely as a"

The problem with clouds is that in the earth's atmosphere they are an extreme variable.Not only are they not always there but the statistics of the frequency, sizes and types are quitefrightening. It is therefore little wonder that radiation physicists do not like dealing with clouds- or regard them as a fascinating research topic. Most studies are done either in a cloudlesssituation - as we have been doing, or around or inside a relatively simple cloud which is usuallyof infinite horizontal extent. You realise how little is known about crowd groups when youread papers on the properties of arrays of cubiod cumulus clouds. That is not to say thatresearchers are ignorant, just that the problem is difficult.

However just because the problem is difficult it does not mean that we caan ignore it andit will go away. Hence this section. We consider here some very simple properties of cloudsand simultaneously try to look at some of the inherent problems in treating the situation.

Firstly, consider the composition of terrestrial clouds. They are composed mainly ofwater droplets, although one does occasionally find clouds of other sorts of particles. If theseother clouds are thin, then they are called aerosol layers. Returning to water drop clouds thenwe find that these clouds tend to fall down because drops fall and they therefore stay up in theair because there is an updraught (rising air) below them. In general conditions round cloudsare damp and it is therefore not in general possible to separate the effects of moist air withoutthe cloud and moist air with the cloud.

Clouds - Infra-red approximations

The simplest infra-red model of acloud that we can come up with is a simple,horizontally infinite, layer of absorber. Wespecify an absorber because clouds exist inwet conditions, which implies a lot of water,which in turn implies absorption. If there isscattering in the cloud it generally onlyserves to increase the pathlength in thesaturated air and therefore the absorption.We can therefore consider, as a zeroth-ordermodel, a cloud to be a layer of black"tin-plate" in our two-stream modelatmosphere. Just to recap, our 2-stream model atmosphere had a total optical depth of 30 andtwo fluxes, Fu and Fd given by the equations:


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Fu )T4e 3

1 2 /2

Fd )T4e 3

1 /2

Fu )T4e 3 4 /2

Fd )T4e 3 2 /2

) T

s4

)T4e 3 4 /2

T

g4 T4

g T4e

0

Χ0*

Χ*

σTe4

σTg4

σTa4

F↑

F↑

F↑

F↓

F↓

F↓

πB

πB

πB

Figure 70: Multiple Layer Clouds

where Te is the effective radiatng temperature of the planet.

If instead of the ground at 3*0 we place a cloud at 3*

1 we find of course that at the cloudsurface

Now the radiation from the top surface of the tin-plate Fu must be the same as the radiationfrom the bottom surface Fd and since we constrain the atmosphere to be in radiative equilibriumand there to be no solar absorption in the atmosphere or the cloud we have below the cloud:

which in turn implies that the ground temperature Tg1 is given by:

or in terms of the original surface temperature Tg

In other words there hasbeen a marked heating of thesurface due to the inclusion of thecloud. Multiple cloud "layers"will increase the effect which maybe shown diagrammatically asshown on the left. In the case thatthe clouds do not absorb perfectlythe shift in the line will not be asgreat and the heating effect willnot be so pronounced. For theearth at 286K surface temperaturewith an effective radiatingtemperature of 255K the newsurface temperature will be 323K - a substantial increase.


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F (1 A) 4)T4e

T4g

F(1A)4)

3

0 2 /2

T4g

F(1A )4)

3

0 4 /2

Note that in the above we have assumed that the cloud does not change the amount ofenergy absorbed by the surface and does not itself absorb energy, thus the albedo of the planetis unchanged.

Clouds - Visible approximations

Clouds however are visible and their very visibility means that they must interact withthe solar beam - the question is how? By the time the solar beam gets into the tropospherewhere the clouds are, the ultra-violet has all been absorbed and therefore the interaction mustbe with the visible and the near infra-red. To a first approximation clouds do not interact withthis radiation beam by absorption but they do scatter the beam very well. This picture is notincompatible with the infra-red story given above because we assumed that the cloud did notabsorb solar energy but there is the consideration of the surface absorption because a cloudincreases the backscatter to space, i.e. it increases the albedo. Since this must in turn reducethe amount of energy absorbed by the ground it follows that this must lead to a cooling of thesurface - in contrast to the infra-red effects which are to increase the surface temperature. Itis therefore not clear which effect wins.

We can in fact do a somewhat simple calculation which illustrates this effect byconsidering the natural state of the earth as cloudless with an albedo of 0.3, and a diffuseoptical depth of 1.2 and ask the question - if we add a cloud which produces infra-red heatingand albedo cooling, what increase in albedo is required to produce no change in the surfacetemperature?

If we write the radiation balance equation:

the equation for the surface temperature without the cloud is:

and with the cloud:

which lead to the equation for the new albedo:


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A

2 A(2 30)

4 30


Fd(z)Fu(z)

z+∆zz

Figure 71:Standard 2-StreamModel

which with the values given earlier makes A1 = 0.57.

The only certain thing to be gained from the above calculation is that you cannot be sureof clouds in the daylight until the full calculation has been done.

This impacts calculations of the greenhouse effect in the future for the following reason:If you increase the surface temperature, you increase evaporation which increases water vapourin the atmosphere. This evidently increases the infra-red absorption which increases the surfacetemperature which increases evaporation, i.e. the feedback is positive and the processaccelerates. However clouds form from the water vapour and therefore there is the infra-redheating and solar cooling from them to take into account. Is the final result positive feedbackor negative feedback - to accenuate or suppress the greenhouse effect? The answer is notknown at the present time.

Clouds - a Scattering model

In the visible region clouds act toscatter radiation in all directions and thisconverts the incident solar beam into a fluxwhich then passes through the cloud and outthe bottom. Since this is now a flux problemwe can approximate it in a very similar wayto the 2-stream model but set up for purescattering instead of absorption. If weconsider the diagram on the left which is the"standard" 2-stream diagram with thedifference is that this is no longer a simplecase of absorption as it was for the infra-redflux, but it is a case of scattering andtherefore energy is conserved and the extinguished energy must still be included in full in thetransfer equations. Put another way, there is no emission from the layer but the scatteredradiation provides a source function.


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Fd(z)

(1-∆)Fd

Fu(z)

(1-∆)Fu

(1-f)∆Fdf∆Fu

(1-f)∆Fu f∆Fd

z+∆zz

Figure 72: Cloud Scattering Model

Fd(z) Fd(zZ) Fd(z)k'rz fFd(z) (1f)Fu(zz) k'rz

Fu(zz) Fu(z) Fu(zz)k'rz fFu(zz) (1f)Fd(z) k'rz

dFd

d3

dFu

d3 (1f) Fu Fd

The primary thing to sort outtherefore is how the energy is distributedbetween Fu and Fd. We therefore considerthe total scattered radiation to be divided intotwo parts, a fraction f which is generallyscattered forwards, i.e. back into the originalflux stream, and a fraction (1-f) which isbackscattered. Pictorially the situation isdepicted on the left.

This is obviously an approximation,but then the 2-stream model is also anapproximation. The particular case ofisotropic scattering is interesting as it stillforms a sort of "standard" case and for thiscondition f = 1/2.

We can therefore write downequations for the 2-stream model in this caseas:

which tidy up to the remarkably simple form:

which immediately gives Fu-Fd = 1.

Therefore integrating the equations and remembering the constraint on Fu-Fd for alllevels we have:


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Fd (1f)13 c

Fu (1f)13 1 c

A

Fu(30)

Fs

3

0(1 f)

1 30(1 f)

-

Fd(33

0)

Fs

1

1 30(1 f)

The boundary conditions for the solar term are that Fd = Fs for the solar radiationincident on the top of the cloud at 3* = 0 and Fu = 0 at 30

* giving c = Fs and 1 = Fs/(1 + (1 - f)30

*). In the event that there is any radiation on the bottom of the cloud from the ground wecan superimpose two solutions for the solar and terrestrial-solar beams since this is a linearsystem.

The two most useful quantities for the cloud are the albedo given by

and the transmission

Note that for this non-absorbing cloud A + - = 1 and as the optical depth tends toinfinity the albedo of a perfectly scattering cloud tends to unity and the transmission to zero.

This state of affairs does not pertain if there is some small amount of absorption in thecloud as then the large increase in pathlength accompanying the scattering produces a largeamount of absorption.

The transmission as a function of optical depth may be compared to the case of anabsorber of equivalent vertical extinction when the diffuse transmission is given by exp(-3*).The differences are very apparent if a few values are tabulated.

3* Isotropic Scattering Diffuse Transmission

0.1 0.95 0.9

0.5 0.8 0.61

1.0 0.67 0.37

2.0 0.5 0.14

5.0 0.29 0.007

10.0 0.17 4.5 x 10-5


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You will notice that the flux decays much more rapidly in the case of absorption than it doesfor scattering.

Clouds - Of things other than water

I showed above that clouds had two competing effects upon the planetary surface: First,the infra-red opacity of a cloud could cause a surface heating due to the mis-named"greenhouse" effect, and second, the solar interaction could cause a cooling effect because ofthe increase in reflection of energy by scattering. The total interaction of a water cloud istherefore in doubt and the whole problem is further complicated by the effect of the localradiative transfer effects within the atmosphere which I have entirely neglected to discuss inthis treatment.

The model of a water cloud used: a solar scatterer and an infra-red absorber, is onlygood for water clouds in the terrestrial situation, any other cloud is likely to be morecomplicated. In general we talk of three types of non-water cloud: a dust cloud, an aerosollayer and a haze layer. The first is obvious and corresponds to a situattion which iscomparatively rare on all but the smallest scales on earth. However on Mars these clouds arerelatively common and any complete study of martian atmospheric phenomena would surelyhave to include such a phenomenon. An aerosol layer however is a very common phenomenonon earth and is practically always present on some scale or other. A haze layer is strictly a termfor an aerosol layer which exists in a nearly-saturated water-vapour environment and thereforeany hygroscopic particles in the aerosol are surrounded by water and take on some of theproperties of "dirty" water drops.

It is impossible to discuss any but a very small number of the aerosol phenomena butwe shall consider a cloud of Rayleigh scatterers for a little while as they occur quite frequentlyas a layer of stratospheric aerosol. From our previous discussions it should be apparent that athin cloud of Rayleigh scatterers will have a considerable solar cooling effect because Rayleighscattering is very efficient at short wavelengths and less so at long ones - in fact the extinctioncoefficient goes as 1/4. The net effect however is uncertain because the infra-red propertiesof the material are not well-specified and in particular there may be an infra-red absorptioneffect, rather than scattering, which will cause a warming to compensate.

The volcanic dust clouds, apart from the initial plume, form part of this set and they maybe detected quite readily as they drift around the earth by their extinction of the direct solarbeam (by up to 10%) as well as by more direct detection methods such as lidar. Note howeverthat the net loss from the downward-directed radiation is rather less than the extinction becauseabout 1/2 of the energy is still downwards-directed.

This brings us to the subject of the so-called "nuclear winter". The basic idea is thatafter a large-scale nuclear exchange there will be very extensive atmospheric contaminationdue to two sources (as an aside I think the term "after a nuclear exchange" is rubbish and weshould be aware that for civilisation there is no "after") which are direct injection from the


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10m

∆Q = 50Wm-2

Figure 73: Simple Ocean Model

nuclear fireball and subsequent injection from "city-burning" and the extensive forest fireswhich will result. The former results in stratospheric injection as well as tropospheric whereasthe latter implies only a tropospheric injection. The proposed optical depth of the northernhemisphere atmosphere immediately after such an exchange would be about 7, and this is anextinction depth which implies that it would probably be light on the ground but there wouldbe no visible sun. The atmosphere would right itself after such a perturbation by rainingmaterial out of the troposphere in a few months but the stratospheric material would, as in thecase of volcanic dust, remain for several years. Several calculations have been done to try tocalculate the effects using reasonably complicated models but the results can be explained interms of the simple 2-tone model with cloud that we have just discussed.

Broadly the results show that there is a massive cooling of the surface immediately aftera nuclear exchange followed by a slow return to normal taking several years. This would implya cold climate over a considerable portion of the land mass of the northern hemisphere and thiswould destroy vegetation, particularly in the normally warmer regions. With such adestruction, which would be visited upon the third world as well as on everybody else, theprobability of civilisation being able to "pick up and keep going" would be nil. This impliesthat the third world countries also have an interest in stopping nuclear war - as if you didn'tknow.

You have probably realised by now that I am cautious when it comes to dealing withproposed disaster scenarios and therefore I offer these cautions on the subject. I do not thinkit necessary to point out that even if there is no "nuclear winter" mankind will be doomed inan nuclear exchange and a nuclear arsenal in the world is really a gun pointed at everybody'shead irrespective of who or where you are - even if you own the arsenal. Howeverscientifically there are some important assumptions about these models that need delineating.

Firstly, the results depend stronglyupon the proposed properties of the dust inthe atmosphere. Since as you are well awarethe cloud properties are two-fold - solarcooling and thermal heating - the relativesolar/infra-red properties are vital to theresult and by "adjusting" the relativemagnitudes of the coefficients you can adjustthe sign and magnitude of the result. Forinstance, one of the models was run with acloud which had solar absorption, notscattering, but no infra-red propertieswhatsoever. This produces a large cooling -but is it realistic?

Secondly, the models used areprimitive, more complicated than ours butstill a long way from a full exposition of the


-98-

Cp ' VdTdt

W

physics, and therefore they show some problems in the interpretation of the results. Forinstance, consider our simple model of the atmosphere with a cloud. If we consider theatmosphere with no cloud and then insert the cloud we can "arrange" for a cooling of thesurface of many degrees but a real surface will not show that effect if we suddenly insert thesurface because it has a finite heat capacity. As an example for a rough calculation assume thatthe cloud is sufficient to cool the surface by 10C, from 286K to 276K. This implies that if thesurface does not respond immediately then it will lose )(286)4 )(276)4 Wm-2 or roughly50 Wm-2. If the surface is land then this will cause a reasonably fast response, but over waterit is a different story. Now the surface to be cooled is water with an enormous heat capacity.However it is not fair to say that we must cool the whole ocean as only a part of it, the "mixedlayer" at the top will participate. The thickness of this layer is of the order of 10m and thereforewe can calculate the surface time constant from the relationship:

where W is the power loss and V is the volume - 10m3. Putting in numbers we find that theocean responds at a rate of 0.1C/day and therefore the effect must go on for a substantial lengthof time before any appreciable cooling is seen. Now the above is only an example to show youthat you cannot ignore such effects in models not a definitive calculation on the subject.

Thirdly, the problem is essentially with the models which are designed to answerquestions such as - if we increase the global CO2 by 5% what is the effect? In other words theyare designed to answer questions about a perutrbation on the present state in a system whichis highly non-linear. Now the proposed dust scenario does not represent a small perturbation.For example one simulation found that the atmospheric heating from the dust was so strong thatall the clouds in their model disappeared and the earth was cloudless! In the case where theperturbation is large there is a real fear that the model may not represent the results well, it mayeither under- or over-estimate the effect and there is no way in such a complicated system thatyou can tell which is happening.

Fourthly, we may be missing the point. Although we may not see a "nuclear winter" itis very obvious from the model runs that we have been very optimistic about the spread ofraioactive fallout in such a situation. The atmospheric disturbances will increase theatmospheric mixing and therefore facilitate the spread of fallout. The stratospheric injectionwill allow dust to spread in the stratospheric zonal winds on a world scale and this may proveto be equally devastating as far as the human population is concerned. Further there will be amassive injection of NOx into the stratosphere which will temporarily deplete the ozone layerseverely allowing increased ultra-violet radiation to the ground which will compound ourtroubles.

All in all this is a very depressing subject. However it does nobody any good to answersuch questions with bad science because that only brings the subject into disrepute. I think thatthe jury is still out on this question and the verdict as far as nuclear winter is concerned may


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go either way but as far as the atmosphere is concerend things will never be quite the sameagain. As far as we are concerned "again" will have no meaning.

The Radiation Balance - The Overall Picture

Earlier I showed a piucture of the way in which the solar energy is split up in the earth'satmosphere. Here I show the total picture including four essential terms in the balanceequations:

- Solar energy in the "near-visible" which is absorbed, reflected and scattered in theearth's atmosphere.

- Thermal energy which is absorbed and emitted by both the atmosphere and thesurface. Notice that the surface radiates much more energy than any other termin the diagram - this is the true greenhouse effect.

- Sensible heat, which is the heat codnucted from the surface to the atmosphere

- Latent heat due to the evaporation/condensation of water on the surface (eg oceans)and the reverse process in the atmosphere. Notice that it is a very significantterm.


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Infr

ared

Hea

t Lo

ss (

69)

Pla

neta

ry A

lbed

o (3

1)

Inco

min

g S

olar

Rad

iatio

n (1

00)

Abs

orbe

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Ear

th

(Sol

ar)

(43)

Lost

by

Ear

th (

The

rmal

) (1

4)

Lost

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Ear

th (

Oth

er)

(29)

810

0

225

22-6

-115

3467

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3336

Abs

orbe

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pher

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osph

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52

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orbe

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udle

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pher

e22

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smitt

edth

roug

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osph

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irect

lyth

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loud

s17

Figure 74: Overall Energy Balance of the Earth/Atmosphere/Surface


-101-

dIk'dx

I J

I(b) I(a) -(a,b) Ppath

J(x)d-(b,x)

dxdx

Remote Sensing

I want to talk for a little while about techniques for probing the atmosphere usingradiative methods because these are important both for terrestrial and planetary atmosphericscience. Such probing may be done to learn about the radiative transfer in an atmosphere orit may be done to learn something else about the atmosphere, the radiative part forming a usefultool to use in this case. Probing may be either active or passive. Active probing involvesputting radiation into the atmosphere that wasn't there before, e.g. using a radar system or alaser beam to make a measurement. Passive probing involves using some radiation that isalready there. The two techniques coincide when you use a natural radiation source as a probe,e.g. using the solar beam attenuation to measure some atmospheric property. The reason thatradiation is so important in this field is that it is one of the few techniques available for remotesensing and remote sensing is important if you can't go where the action is, or want to makeglobal measurements on a short timescale, i.e. use a satellite.

One of the simplest techniques uses a radiation beam and measures either the absorptionor scattering of that beam. For example if you direct a radar system at a cloud then the dropletswill scatter in the Rayleigh scattering regime because they are small compared to the radarwavelengths and by examining the echo in amplitude, time after pulse, and frequency, one candeduce something about the density, range and velocity of the drop distributions in the cloud.Using fancy signal processing and colour techniques one can then see the progress of aprecipitation system over a period of time just by drawing appropriate diagrams.

The term remote sensing, as I have already mentioned, means different things todifferent people, but I suppose that since all radiation is subject to Schwarzchild's equationwhile in transit we could claim that even photography is governed by the equation:

For the purposes of remote sensing however it is more convenient to consider this equation inits integral form which I introduced you to earlier.

where the equation is solved over some path from a to b which may be highly structured in justabout anything.

There are many forms of remote sensing but it is convenient to divide the subject up intoa number of smaller groups which share common properties. The first such division is todivide those that use the first term in the above equation from those that use the second - thetransmission techniques from the emission techniques.


-102-

I(0,1) I(,1) exp P

0

k'dz/cos

expk363/cos . expkama/cos

k

ν∼2ν∼1

ka

k3

Figure 75: Ozone Absorption Experiment

I(0,2) I(,2) exp kama/cos

Transmission Techniques

All transmission techniques require a source of radiation and this causes a furthersubdivision into the techniques which use a natural source, usually the sun or moon, from thosewhich use an active source, e.g. a searchlight or laser.

We consider as a simple example of a technique using a natural source the DobsonOzone Spectrophotometer which utilises the absorption of O3 in the u-v to infer the columndensity of ozone in the atmosphere, i.e. how many molecules there are along the path. Thetechnique is to select a wavelength 1 in the u-v at which there is a significant ozone absorptionand use a simple spectrometer to observe the sun when we shall observe:

where the symbols have their usual meaning. Ideally of course the transmission term wouldbe characteristic of ozone and the problem would be readily soluble but in fact there are at leasttwo other effects, Rayleigh scattering and aerosol scattering, which contribute quitesignificantly. These however tend to occur at lower altitudes than the ozone layer in themid-stratosphere and in that case we can separate the transmission into a product:

where k3 is the ozone absorption coefficient,63 is the total equivalent vertical ozoneamount (i.e. 63/cos is the total amount ofozone in the path from the instrument to thesun) ka is the extinction coefficient for therest of the scattering and m is the totalvertical amount of material in theatmosphere.

So far so bad because we cannotseparate out the effects. However if weconsider another wavelength 2 such that it isclose to 1 and therefore shares commonvalues of most variables except the ozoneabsorption. Since ozone absorption is highlystructured in nature and the other extinctions are broadband (no lines) this doesn't prove to betoo difficult and therefore, assuming no ozone absorption at 2 we can write:


-103-

I(0,1)

I(0,2) exp k363/cos

Earth

Ozone

SatelliteSun

Figure 76: Satellite Ozone Sensing

by ratioing the two measurements and assuming common values of everthing except the ozoneabsorption we find that:

and by knowing the value of k3 we can evaluate the total ozone concentration above theinstrument.

Now this is really too simple an explanation as a host of corrections have to be made toaccount for the fact that the wavelengths are not truly identical and therefore we cannot makethe broad assumptions sketched out above. However the basis of the measurement is still totake out the majority of the unwanted dependences by this differential technique and thencorrect the answer as necessary. In fact corrections also have to be made for the instrumentperformance and other problems and the details of global ozone measurements over longperiods using such instruments is interesting to read as it makes very clear the difficulties bothpractical, personal and political of such an effort - never tell anybody or any country that theirinstrumental records of the last 10 years are wrong because they haven't calibrated it. Invitethem to an intercomparison meeting and let them find out for themselves. Although thatremark sounds cynical, in fact the best way to find out who is wrong where is to do just that -everybody comes to some nice place and they all observe ozone together and try and agree onan answer between them.

Of course ground-based instrumentsare old hat now and unless you can do itfrom a satellite nobody listens. Howeverthere are severe problems with this form ofsensing from satellites as the sun isinconveniently placed outside theatmosphere instead of at the bottom as itshould be. A certain amount of ingenuity istherefore required to relocate it in a suitableplace. The simplest scheme is to work on thereflected sunlight from the surface asdepicted here for an overhead sun. In thisproblem with the same broad assumptions asthe last example we come to the conclusionthat the situation is just the same as the last

case except that the layers are traversed twice and therefore there is a factor of 2 or two in theproblem solution. However things are not that convenient as scattered radiation can arrive fromall sorts of places besides than surface whose properties are not well-known anyway. In fact,in order to get any signal from the surface it is necessary to invoke a non-zero surfacereflectivity and this ain't necessarily the case. The problem therefore gets rapidly interestingor difficult depending upon your point of view and were it not for the advantages of satellitemeasurements, continuity, coverage and single instrumental, would probably be an obtuse


-104-

research problem. However the results are obtainable to some accuracy and satellitemeasurements of ozone are contributing to the global database.

A "simpler" way of doing the same experiment is to use a "grazing incidence" techniquewith the sun on the planetary limb. This technique, which is similar to "limb-scanning" whichwe shall meet later, uses the sun as the direct source and has the advantage of being nearlydirect in measurement of transmission as one can get a full-scale, zero and measurement in onepass of the satellite. However in practice the coverage is a bit limited so unless you are lookingfor a constituent in very small concentrations and need that long atmospheric path, it is lessuseful.

For a balloon instrument however the solar transmission mode is very attractive for anumber of reasons. Firstly, the geometry can be done fairly accurately including refraction andtherefore the path is well-known. With a balloon payload about 38km off the ground this is avery useful property and much of the early survey work of stratospheric transmission was donein this way. However we are now confined to work at dawn or dusk because the sun will onlyoblige with appropriate angles at that time and further since a solar tracker has great difficultylatching onto the sun as it rises, dusk is preferred.

Notice that in all the above measurement methods the sun plays a passive role in thesensing, being merely a radiation source and therefore any of these techniques can be appliedusing an artificial source. However since artificial sources tend to be less intense than the sunmeasurements using an artificial source also tend to be different.

The simplest artificial transmission experiment consists of moving the laboratorytransmission experiment out into the open air. In this case the situation is either that the cellis the atmosphere itself, or a pump is used to bring the atmosphere into the cell. This techniquecovers many visibility measurements at airports and several studies of water vapour content bythe use of transmission at the surface. An interesting example of the same technique appliedat a higher level is a Tunable Diode Laser Spectrometer (TDLAS) instrument which actuallyflies a folded optical path on a balloon instrument and then uses very highly monochromaticlaser signals scanning on known absorption features to make transmission measurements onthe gas sample. In this way measurements of nitrogen oxides in the stratosphere are currentlybeing made. Some more futuristic schemes are to do the grazing incidence experiment usingtwo satellites pursuing each other with a laser source/receiver pair. This would circumvent theunfortunate problem that the sun is inconveniently placed and would enable true globalmeasurements to be made. Other schemes involve the use of laser retroreflectors suspended upto 1km below a balloon package thereby making transmission measurements over a 2kmatmospheric path.

Several instruments make use of the backscattering effect with an artificial source, thesebeing the radar class of instrument with either a radio source, e.g. weather radar for measuringclouds and rain by their backscattering properties, or a laser source, the lidar instruments whichbackscatter laser radiation from cloud layers. The Atmospheric Environment Service is activein both fields using radars for observations of precipitation belts and lidars for measurements


-105-

k

k1

k2

k3

k4

k1>k2>k3>k4

Figure 77: Monochromatic Weighting Functions

Ppath

J(x)d-(b,x)

dxdx

B(T) P

0

d-(,x)dz

dz B(T)

of atmospheric pollution and stratospheric dust layers from volcanic eruptions by measuringtheir backscattering properties and range.

Emission Measurements

Emission measurements are inherently more difficult than transmission measurementsbecause they involve the resolution of the integral term in Schwarzchild's equation and this canget very tricky. There are therefore two problems involved: First the actual measurementproblem and second the inversion or retrieval problem. The second problem is mathematicallyvery interesting and forms almost a separate subject these days and I should say just a littleabout it before considering the experimental details in too much detail.

The essential problem is theinformation obtainable from observationsof the form:

where the path is prescribed by theinstrument geometry and this is amonochromatic equation. If we take avery simple example of an isothermalexperiment with a grey absorber and usea vertically sounding satellite as ourinstrument we shall find a weightingfunction profile which bears all thehallmarks of a "Chapman profile" asshown on the left.

In an isothermal atmosphere and with a vertical sounder (also called "nadir sounding")the integral may be written as:


-106-

B(T) P

0

d-(,x)dz

dz

where the RHS equality applies only if the path goes opaque before the surface is reached -there is no surface term. We can see from the form of the integral that all the information iscoming from layers delineated by the graph of the differential of the transmission function.

The radiation making up this signal comes from levels where d-/dz g 0. These aredepicted as the shaded area in the figure above for one particular case. If the atmosphere isvery strongly absorbing in the spectral region where the measurement is made (k is large) thenthe peak occurs high in the atmosphere. As k declines in values the peak moves lower in theatmosphere eventually the surface becomes important. In the limit when k = 0, the surfaceemission is all that is seen.

The case of an isothermal atmosphere is rather unrealistic but this function showing theweighting of information - or the "weighting function" can be drawn for virtually anyinstrument, and particularly any emission-sensor, and tells you where all the information iscoming from. By organising a number of different instrument channels to have differentweighting functions, with different peaks, the atmosphere may be sampled at a number ofdifferent levels and a profile built up. For a nadir sounder in a non-isothermal case one canwrite down the equation as:

where B (T) is the average temperature of the layer. If we can therefore calculate and tailor ourintegral to a specific level we can measure the temperature at that level. Now calculation oftransmission implies that you know the gas concentration profile which in turn implies youknow about the gas distribution with great confidence. It so happens that CO2 is present in theearth's atmosphere in a reasonably constant concentration at all levels and therefore thecalculation should be possible for the 15µm band of CO2. As to tailoring the height levels, thestronger the absorption coefficient k, the higher up will the weighting function peak andtherefore by choosing a number of channels with varying values of k a set of averages can beobtained.

There are of course about a million corrections to this theory in terms of selection ofchannels, form of instrument, corrections, arguments etc., etc. but the principles remain thesame. Perhaps the most significant point to bring out at this stage is the problem of profilingwith a limited number of channels. The fact is that a profile is a continuous line, implyinginfinite information, whereas from 6 channels you get 6 pieces of information. Therefore whatcan you learn about the profile from your information, or phrased another way, how can youuse your 6 pieces of information to tell you about your profile in the best way.

Let us take a very simple example of the analysis of a repetitive waveform. If we havesix pieces of information about this waveform the worst thing that we can do is produce sixrandom time samples. These will tell us absolutely nothing about the waveform in general.However if the six pieces of information are the repetition time and the first five fourier


-107-

z(km)

T (K)

50

12

200 300

Troposphere

Stratosphere

Tropopause

Figure 78: Typical Earth Temperature Profile

P B(T,z) k(z)dz B(T) P k(z) dz B(T)

coefficients of the fourier series expansion then we shall have a pretty good general overviewof the waveform. Note that in order to achieve such an increase in information from a limitednumber of samples we needed two things:

1) the knowledge that it is a repetitive waveform2) a freedom to choose what we measure.

Now of course in general we do nothave total freedom in those two parametersbut in practice we have some considerablelatitude and some knowledge of theatmosphere. We can therefore improve ourrate of information gathering immensely byappropriate choice of experiment. To take avery, very simple example. If we know thatthe atmospheric profile looks like the one onthe left we may decide to model it as twostraight lines. The best information wouldthen be the bottom, top and cornertemperature which would completelyparameterise the line and we should designour experiment to make the best measurements of those quantities. Of course, this doesn't saythat we can do it, only that we should. Thus by experiment design one can increase the amountof useful information obtained without necessarily using a larger instrument. This techniqueis not exactly new - we pinched it from the geophysicists where it was developed by Backus& Gilbert.

The simplest instrument we can discuss is a temperature sensor for the earth'satmosphere using the 15µm CO2 band because this has some unusual features which simplifyit. Now we can write the equation for vertical sounding as:

where the equation is assumed monochromatic and k is the weighting function which isnormalised. The quantity B (T,z) is now some average temperature for the layer where theweighting function is non-zero.

Now since CO2 is uniformly mixed in the atmosphere the transmission for any path canbe worked out from theory and, although there is a small temperature dependence, it is largelyindependent of temperature. This being so we can relate the measurement of B (T,z) to theaverage temperature at any level with relative ease and it becomes a fairly simple matter, atleast to first order, to interpret the results of the experiment. Thus with various channels withweighting functions peaking at different heights in the atmosphere one can obtain variouspieces of information about the temperature profile as the satellite progresses. This is thegeneral principle of virtually all the temperature sounding satellites that are currently in use.


-108-

OutputSignal

InputRadiance

Detector

COCell

Figure 79: Conceptual Diagram of aPressure Modulator

0

0

0.2

0.2

0.4

0.4

0.6

Tran

smis

sion

Tran

smis

sion

Relative Wavenumber (cm-1)

0.6

0.8

1

-0.5-1 0 0.5 1

τ(ul)

τ(uh)

Effective Filter

Figure 80: Operation of a PressureModulator

in Spectral Space

The differences between them correspond to the different ways that they take their weightingfunction because it is not possible to use a monochromatic instrument in practice.

One of the most interesting currentsatellite instruments is the pressuremodulator radiometer which manages to geta weighting function which is near thetheoretical minimum width for this type ofwork. It consists of a cell of gas in theradiation path whose pressure is cycled at aknown rate.

The absorption profile of the cell istherefore modulated at lines intheappropriate spectrum.

Now radiation passing through thecell is "tagged" by being modulated at thecell rate only if it is at the wavelength of anabsorption line and this radiation is chieflyradiation from the same gas in theatmosphere. If we arrange our electronics toonly "see" the modulated radiation then wewill be blind to radiation from between thelines and our instrument will see anatmosphere an apparently very high value ofk.

These instruments are so selectivethat they are sensitive to the doppler shiftinduced by the earth's rotation or by a sighterror (<1o) in the vertical viewing direction.Yet despite such sensitivity they contain nocritical optical components to do the spectralselection and therefore cannot drift ofwavenumber. Lastly because they useradiation from all the gas linessimultaneously, they see a large signal andtherefore have a good signal/noise ratio.

Remote sensing of constituents


-109-

Tangent Height

Figure 81: Schematic of Limb Scanning

A different but related remote sensingtask is to look for a constituent of theatmosphere by examining its thermalemission. In this case we must know B(T)and rely on our knowledge of how theweighting function varies to deduce theconcentration. This approach is not veryeasy unless you now a fair amount about thegas in question and is particularly difficultfor nadir sounding as you cannotpredetermine the peaks in the weightingfunction. The solution is to uselimb-scanning which, although more difficultas a technique, puts more gas in the opticalpath and allows some independence in thechoice of the weighting function whichgenerally peaks at the tangent height. Thetangent height is tha altitude of the lowest point on the ray path along the instrument line ofsight.

The greatest problem with constituent profiling is that the weighting function, even inthe limbscan case, is a function of all the levels above the one you are interested in, and thisleads to great problems in the "retrieval" process which are by no means solved yet. Thesimplest method of retrieval is known as "onion-peeling" where you split the atmosphere upinto layers and then compute the concentrations starting at the top. Since for a limb-scannerthe concentration at a level depends upon the concentration at higher levels this allows thesuccessive computation of all concentrations. There is no guarantee however that the seriesprocess is stable and that leads to some problems. However generally the process is not toounstable and progress can be made. As with temperature sounding the differences in theinstruments lie in the way in which the spectral selection is done and range from thespectrometer instruments which are very general and suitable for survey work on other planets,to the pressure modulator instruments which are tailored to a specific gas for very highsensitivity. There are dozens of types in between operating in the ultra-violet, visible, infra-redand microwave regions but I do not have time to go into them in this course.


-110-

2-D Angle

Appendix A

Solid Angles

For beings who live in a world with at least three dimensions, we are remarkably ineptat handling geometry in more than 2-D. Perhaps it is an undue familiarity with 2-D ways ofpresenting information such as paper or television....

To consider a direction in two dimensions we draw asimple line as shown here.

To consider a range of directions from asingle point we draw two lines representing theextrema of the directions that we areconsidering. We call the range of directions the"angle" between the two lines and measure it insome useful set of units. The most common is toexpress the angle in radians, which is done in thefollowing manner: construct a unit circle on thepoint and measure the length of arc between theintercepts with the two lines. Since the totallength of the circumference is 2% (unit radiusremember) the largest angle is also 2%.

If we wish to consider an infinitessimal angle then we consider an infinitessimal lengthof arc 5 which corresponds to an angle .

To extrapolate into three dimensions:

To consider a direction in three dimensions we "draw"a simple line in space as shown here.


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3-D Angle

6 % 2

Θ

θ

sinθ

d6 2% sin d

6() P

0

2% sin d

2% 1 cos

To consider a range of directions from asingle point we have to draw an enclosingsurface consisting of an infinite number ofstraight lines, all passing through the point,which defines the range of directions that we areconsidering. Note that there may be any numberof symmetries to the surface. We call the rangeof directions defined the "solid angle" defined bythe surface and measure it in some useful set ofunits. The most common (in fact the only) is to express the angle in steradians, which is donein the following manner: construct a unit sphere on the point and measure the area of thesurface of the sphere delineated by the surface. Since the total surface area of the sphere is 4%(unit radius remember) the largest solid angle is also 4%. Since there can be many symmetriesto the defining surface, life can get complicated.

If we wish to consider an infinitessimal angle about a direction we must first define thegeometry of the defining surface. Take first a symmetrical cone which cuts off a surface ofarea A on the sphere. The relationship between the semi-angle of the cone , and the solidangle defined 6 = A is:

If we consider a finite size of conethen the relationship is the almost the samebut we must account for the fact that the"cap" of the sphere is not flat. If we have afinite cone angle then the situation is shownhere and if we increase the angle by d therelationship to the increase in solid angle d6is:

Thus we can integrate the solid angle as weincrease from 0 to and find:

Which gives a result 4% when = % (remember is the semi-angle of the cone).


-112-

sinθ dφsinθ

dθ

φ

dφ

dA

θθ dθdΩ

X

Y

ZdA = sinθdθdφ

Sphere isunit radius

d6 sin d . d1

A very similar result is obtained in spherical polar co-ordinates when the diagram isgenerally rotated through 90o and instead of assuming azimuthal symmetry, we explicitlyintroduce a second angle 1 thus:

Notice that in this case the surface cut off on the unit sphere is not "circular" in shape but"rectangular" having an area or solid angle of:


-113-

However we can recover the result above by integrating first over 1 and then over .

So much for simple solid angles. The complications come when things vary at a pointwith direction and we have to integrate giving due weight to the various directions involved.Thus intensity is expressed per unit solid angle but other factors may have to be folded into theintegration (such as projected areas) so that we must be careful in the integration.

There is actually nothing very difficult here once you have learnt to be as familiar witha solid angle as you are with a 2-D angle. Perhaps I should provide everybody with a penknifeand a lump of styrofoam....

velut Ævo arbor university of toronto …j.t. houghton physics of atmospheres (cambridge) advanced...

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