Andrews Dawid G. An Introduction to Atmospheric Physics

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169 Problems

t = 0. Then solve for [A], assuming that the rate coefﬁcient k

is constant. Hence verify

that equation (6.8) holds if [B]

[A]

Problem 6.4 If F

= 1370 W m

−2

is the total solar irradiance and P

is the total solar

photon ﬂux (photons m

−2

−1

), both measured at the top of the atmosphere, show that



∞

N(ν) dν



∞

(ν, T) dν

where N (ν) dν is the number of photons m

−2

−1

steradian

−1

emitted from the Sun’s surface

in the frequency range (ν, ν + dν),andB

(ν, T) is the Planck function; assume that the

Sun is a black body at T = 6000 K. Hence obtain P

, given that



∞

− 1

dx = 2.40.

Calculate also the fraction of the total photon ﬂux that lies in each of the wavelength

ranges (i) λ<1180 nm, (ii) 240 nm <λ<280 nm and (iii) λ<240 nm. (You will need

to write a computer program to calculate the required integrals.)

Problem 6.5 The following reaction rates and photo-dissociation rates are taken from

DeMore et al. (1997):

= 6 × 10

−46

(

T/300

)

−2.3

−1

= 8 × 10

−18

exp

(

−2060/T

)

−1

= 3 × 10

−12

−1

, j

= 5.5 × 10

−4

−1

The photo-dissociation rates are appropriate for the midlatitude equinox at noon and at

25 km altitude. Use information from Figure 1.3 and Section 2.2 to estimate the number

density of ozone at this altitude, based on Chapman chemistry. Show that this value is

several times larger than the peak value given in Figure 1.4. What physical and chemical

processes might reduce this value in the real atmosphere?

Problem 6.6 Add the catalytic reactions (6.17) to the Chapman scheme and explain how

this modiﬁes the algebra for the Chapman scheme. Assuming that the extra reactions are

much slower than the fast equilibrium between O

and O

+O, show that the vertical

distribution of ozone is given by

]

≈



[M]



1/2

−



[M][X]



;

compare this with equation (6.16) and comment.

Problem 6.7 What happens to O and O

, according to the Chapman scheme, when solar

radiation switches off at sunset? If catalysts are present, what happens to X and XO?

(Assume that k

][M]+k

]≥10

−1

throughout the stratosphere.)

170 Stratospheric chemistry

Problem 6.8 Derive equation (6.21) by a direct method analogous to that used in Chapter 4

to derive the continuity equation (4.3).

Problem 6.9 Check that 1 DU corresponds to an ozone column of about 2.7 × 10

molecules per square metre.

Atmospheric remote sounding

In this chapter we consider a small selection of techniques for observing the atmosphere.

These techniques have been chosen for two main reasons: (a) they illustrate the use of

physical principles, including principles introduced earlier in this book; and (b) they

provide crucial data on atmospheric phenomena modelled elsewhere in this book, such as

Rossby waves, gravity waves and the Antarctic ozone hole. The topics considered are all

examples of remote sounding; we do not attempt to present a balanced account of all

observational methods.

In Section 7.1 we brieﬂy list some of the main atmospheric observational methods. In

Section 7.2 we outline the principles of remote sounding of the atmosphere from space,

focusing on methods that rely on thermal emission from atmospheric gases and on scattering

of solar radiation by atmospheric gases. Then in Section 7.3 we discuss three types of

ground-based remote sounding, namely the Dobson spectrophotometer, radars and lidars.

We omit the details of the instruments’ optical and electronic systems, the technicalities of

signal processing and the sophisticated statistical methods that may be required in order to

extract meaningful physical quantities from the raw measurements.

7.1 Atmospheric observations

Quantitative observations of the atmosphere are made in many different ways. Routine

meteorological measurements of ground-level temperature and wind are made with simple

thermometers and anemometers, respectively, and routine measurements of temperature

and humidity through the depth of the troposphere are made with balloon-borne instruments

(radiosondes) that transmit information back to the surface by radio. At higher altitudes,

instruments on research balloons and aircraft can measure samples of air in the lower

stratosphere, and instruments on rockets can make measurements in the mesosphere and

beyond. These are all examples of in situ measurements. An alternative is to use remote

sounding, that is, to measure atmospheric properties at a distance, rather than in the

vicinity of the measuring apparatus. This can be done from orbiting instruments in space,

from aircraft, from balloons and from ground-based instruments, in each case by measuring

electromagnetic radiation emitted, scattered or transmitted by the atmosphere.

Remote-sounding techniques can be divided into passive and active types. In passive

remote sounding, the radiation measured is of natural origin, for example thermal radiation

emitted by the atmosphere, or solar radiation transmitted or scattered by the atmosphere.

(Most space-borne remote-sounding methods are passive.) In active remote sounding, a

172 Atmospheric remote sounding

transmitter (e.g. a radar) is used to direct pulses of radiation into the atmosphere, where

they are scattered by atmospheric molecules, aerosols or inhomogeneities of atmospheric

structure; some of the scattered radiation is then detected by a receiver.

Each of these techniques has its advantages and disadvantages. In situ measurements

may give accurate, high-resolution measurements, but can sample only small, perhaps

unrepresentative, regions. Remote sounding from satellites can give near-global coverage,

but can provide only averaged values of the measured quantity over large regions, perhaps

hundreds of kilometres in horizontal extent and several kilometres in the vertical direction.

Satellite instruments are expensive to put into orbit and cannot usually be repaired if they

fail. Ground-based radars can provide data with high vertical resolution (by measuring

small differences in the time delays of the return pulses), but only above the radar site. In

general a mix of different observational techniques must be used for providing the data

required for weather forecasting or atmospheric research.

7.2 Atmospheric remote sounding from space

Earth-orbiting satellites provide enormous possibilities for global-scale measurements of

the atmosphere. For example, geostationary satellites, orbiting at 36 000 km over the equa-

tor, monitor weather systems and provide information on tropospheric winds by tracking

the motion of clouds. Polar-orbiting and other non-geostationary satellites, orbiting at alti-

tudes up to about 1000 km, are used for measuring temperature and composition at a range

of levels in the vertical. They complete up to 15 orbits per day, crossing each latitude circle

twice per orbit; their orbits are often chosen such that they remain ‘phase-locked’ with the

Sun (they are then called Sun-synchronous), crossing each latitude at the same two local

times in each orbit.

In the following subsections we shall consider some basic principles of temperature and

composition sounding from space, focusing on measurements of the thermal radiation emit-

ted from the atmosphere and of solar radiation scattered by the atmosphere. Two viewing

geometries are commonly used (see Figure 7.1): nadir viewing, in which the instrument

views vertically downward towards the ground, and limb viewing, in which the instrument

views the atmosphere tangentially, towards the limb. Nadir viewing measures radiation

from a comparatively short path-length of emitting gas, against the warm background of

the Earth’s surface and lower atmosphere, which also emit thermal radiation. Limb view-

ing has the advantage of measuring radiation from a much longer path-length of emitting

gas (see Problem 7.1), against the non-emitting background of cold space. However, the

spacecraft must be aligned very accurately (to within a few arc seconds) for limb viewing,

if the correct portion of atmosphere is to be measured. The technique of solar occultation,

in which measurements of the absorption of solar radiation by the atmosphere are made

as the Sun rises or sets, is also indicated in Figure 7.1, but will not be considered in this

chapter. All of these techniques make use of the ideas of radiative transfer introduced in

Chapter 3.

173 Atmospheric remote sounding from space

Fig. 7.1 Illustrating the geometry of (a) nadir viewing, (b) limb viewing and (c) solar occultation. (Not to

scale.) Note that the limb-viewing and occultation instruments may observe a different latitude

from the current latitude of the satellite platform; for example, the North Pole may be viewed

while the satellite is at 60

◦

7.2.1 Thermal emission measurements

Consider a satellite instrument receiving infra-red emission from the atmosphere along a

vertical (nadir) path. From the solution of the radiative-transfer equation in the form of

equation (3.13), assuming local thermodynamic equilibrium so that J

= B

, we can show

that the spectral radiance L

at frequency ν received by the satellite is given by



∞

(T(z))

∂T

(z, ∞)

∂z

dz + B

(0, ∞), (7.1)

where T (z) is the atmospheric temperature at height z, T

(z, ∞) is the spectral transmittance

between height z and the satellite, and T

is the surface temperature. It is assumed that the

surface acts as a black body.

The spectral absorptance between height z and the satellite is A

(z, ∞) = 1 −T

(z, ∞):

for simplicity the ‘∞’ argument of A

will be dropped from now on. The general form of

near an isolated spectral line at ν = ν

is given by Figure 7.2 (cf. Figure 3.12). The

precise details depend on whether we have a Doppler or Lorentz line shape, on the amount

and distribution of absorbers or scatterers in the path between height z and the satellite, and

on the temperature and pressure distributions along the path.

The form of the ν dependence of the Planck function B

(T), a much more slowly varying

function of ν than the absorptance, is given in Figure 7.3. Note that B

increases with T for

all values of ν and T.

The simplest case of remote sounding of temperature occurs when atmospheric extinction

at frequency ν can be neglected. Then T

= 1 everywhere and the satellite just ‘sees the

surface’ at that frequency, with no atmospheric extinction, so L

= B

). The surface

Note the resemblance of equation (7.1) to equation (3.33); however, the latter involves integration over all

slanting upward paths.

174 Atmospheric remote sounding

Fig. 7.2

Schematic form of the spectral absorptance A

(z) between height z and a remote-sounding

satellite, as a function of frequency ν near a Lorentz line centre at ν

, for three values of z.Solid

curve, small z;dashedcurve,intermediatez;dottedcurve,largez.

Fig. 7.3

Form of the Planck function B

(T), as a function of frequency ν, for the temperatures T = 220, 240

and 260 K. The peak of the 260 K curve, at about 16 THz, corresponds to a wavenumber of about

530 cm

−1

and a wavelength of about 18.7 μm. Note that 1 THz = 10

Hz.

temperature T

can then be obtained from the radiance measurement L

at frequency ν by

inverting the Planck function (3.1):

hν



1 +

2hν



. (7.2)

More generally, we can deﬁne a brightness temperature in terms of the radiance L

and

frequency ν using equation (7.2), even in the presence of extinction.

Now suppose that the frequency ν is near an isolated spectral line centred at ν

,so

that absorption cannot be neglected, and consider an isothermal atmosphere at a constant

temperature T

, which may be different from the temperature T

of the ground. Then the

integral in equation (7.1) can be evaluated explicitly, to give

175 Atmospheric remote sounding from space

= B

)

[

1 − T

(0, ∞)

]

+ B

(0, ∞)

= B

(0) + B

)

[

1 − A

(0)

]

= B

) +



) − B

)



(0). (7.3)

Now 1 − T

(0, ∞) = A

(0) is the spectral absorptance, and hence also the spectral emit-

tance, of the isothermal atmosphere. The ﬁrst and second lines of equation (7.3) therefore

state that L

equals the upward radiance B

(0) from the (non-black) atmosphere,

plus a fraction T

(0, ∞) of the upward radiance B

) from the (black) ground that is

transmitted by the atmosphere. The last line of equation (7.3) shows that the radiance

can alternatively be written as B

) plus a modiﬁcation equal to B

) − B

)

times the total atmospheric absorptance A

(0). Note that any rapidly varying frequency

dependence of L

must come from the absorptance A

, rather than from the Planck

function B

Consider two cases.

• If T

> T

(the air temperature is greater than the surface temperature), then B

) and L

displays a ‘hump’ around the line centre at ν

, as shown in Figure 7.4.

The line is called an emission line.

• If T

< T

(the surface is warmer than the air temperature), then L

displays a ‘dip’

around the line centre at ν

, as shown in Figure 7.5. The line is called an absorption

line.

In each case the radiance L

tends to the Planck function at the surface temperature,

), sufﬁciently far from the line centre. The depth of the hump or dip at ν

in these two

cases increases as the absorptance A

(0) increases and also as the magnitude of the jump

) − B

) in the Planck function increases.

Now consider the two-layer atmospheric temperature proﬁle given by

T(z) = T

for 0 < z < z

, T(z) = T

for z

< z.

Fig. 7.4

Schematic satellite-observed radiance (solid line) near a Lorentz-broadened spectral line, for an

isothermal atmosphere that is warmer than the surface temperature. Planck functions at the

temperatures T

and T

are given as dashed lines.

176 Atmospheric remote sounding

Fig. 7.5 Schematic satellite-observed radiance (solid line) near a Lorentz-broadened spectral line, for an

isothermal atmosphere that is colder than the surface temperature. Planck functions at the

temperatures T

and T

are given as dashed lines.

Fig. 7.6 Schematic temperature proﬁle for a two-layer atmosphere.

It is easy to show from equation (7.1) that

= B

) +



) − B

)



(0) +

[

) − B

)

]

Suppose for example that T

> T

,asinFigure 7.6, crudely representing a hot

surface, cool ‘troposphere’ and warm ‘stratosphere’. The frequency variation of the result-

ing radiance near ν

received at the satellite takes the form of Figure 7.7, in the case of

a Lorentz-broadened spectral line. The spectral radiance L

includes a ‘background’ term

given by the Planck function at the surface temperature, B

). This has a large dip super-

imposed: large, because A

(0) is a broad, high hump, and a dip because B

)<B

In the middle of this large dip is a small hump: small because A

) is generally lower

and narrower than A

(0) and also because the jump in B

at z

is less than at the ground,

and a hump because B

)>B

This kind of feature can be seen in the atmospheric

example shown in Figure 7.8; see also Problem 7.3.

A similar approach can be used for a multi-layer atmosphere, approximating a smoothly

varying proﬁle T(z) by a series of steps. If there are many layers, however, it is difﬁcult

The wideness of the dip and narrowness of the hump result from the temperature and pressure dependences of

the Lorentz width γ

:seeequation (3.22).

177 Atmospheric remote sounding from space

Fig. 7.7

Schematic satellite-observed radiance (solid line) for the two-layer atmosphere shown in

Figure 7.6. Planck functions at the temperatures T

, T

and T

are given as dashed lines. It is

assumed that A

(0) ≈ 1andA

) ≈ 0.6 at the central frequency ν

Fig. 7.8 Infrared radiance measurements over the Mediterranean Sea from the IRIS instrument on the

Nimbus 4 s atellite. Thermal emission from the atmosphere and the surface (solid curve) is shown

as a function of wavenumber (in units of cm

−1

) and wavelength (in units of μm), together with

the Planck function at several temperatures (dashed curves). Adapted from Hanel et al. (1971).

to draw simple insights from the resulting spectral data; we therefore adopt a different

approach, as follows. For convenience, we ﬁrst introduce the log-pressure coordinate

Z = ln(p

/p), (7.4)

where p

is the surface pressure. Recall from equation (2.15) that, for an isothermal

atmosphere at temperature T

, the pressure decreases exponentially with height, p =

exp(−z/H),whereH = RT

/g is the pressure scale height (H ≈ 7.6 km for T

260 K). Therefore in this isothermal case Z = z/H. Although Z is not precisely proportional

to height for a non-isothermal atmosphere, it is still approximately so in the lower and middle

atmospheres. The quantity Z is sometimes called the ‘log-pressure in scale heights’.

178 Atmospheric remote sounding

On changing variables from z to Z in the integral in equation (7.1), we obtain



∞

(T(Z)) K(Z) dZ + B

(0, ∞), (7.5)

where

K(Z) =

∂T

(Z, ∞)

∂Z

(7.6)

is the weighting function; it weights the contributions to L

from the Planck functions at

different log-pressure altitudes Z.

In certain simple cases, the weighting function can be evaluated explicitly. First note

from the deﬁnition (3.23) of the transmittance that, in terms of height,

(z, ∞) = exp



−



∞



)ρ



) dz



where k

is the extinction coefﬁcient and ρ

is the density of the absorbing gas. Consider

the special case in which k

is independent of height and the absorber has a uniform mass

mixing ratio μ

= ρ

(z)/ρ(z),whereρ is the total atmospheric density. Then, using the

hydrostatic relation (2.12), the transmittance is found to take the simple form

(z, ∞) = exp



−



≡ e

−ξ

say, where

= k

/g.

Note that ξ

p is the optical depth at frequency ν from the satellite down to pressure p;see

Section 3.6.Fromequations (7.4) and (7.6) the weighting function is given in terms of p by

K =−p

∂T

(z, ∞)

∂p

= ξ

−ξ

, (7.7)

or in terms of Z by

K(Z) = ξ

exp −



Z + ξ

−Z



. (7.8)

Note that the weighting function (7.8), like the heating rate in equation (3.32),hasa

Chapman layer structure: see Figure 7.9.Fromequation (7.7) it is easy to show that

the weighting function takes a maximum value of e

−1

at a pressure p = p

≡ 1/ξ

g/(k

) and at Z = Z

= ln(ξ

) = ln(k

/g), corresponding to unit optical depth.

The integral in equation (7.5) therefore tends to be dominated by the Planck function

in the neighbourhood of this level and so gives a measure of the average brightness

temperature for this region. The thickness of the region depends on how sharply peaked

the weighting function is. To investigate this we consider the pressures at which K equals

half its maximum, namely 1/(2e): from equation (7.7) these are given by the equation

−ξ

With ξ

p = x, this reduces to the equation 2x = exp(x − 1), which is found to have

the two roots x = 0.23 and 2.68. The half-maximum points therefore occur at pressures