Andrews Dawid G. An Introduction to Atmospheric Physics

Подождите немного. Документ загружается.

189 Atmospheric remote sounding from the ground

backscattering cross-section; and A is the telescope area and η the optical efﬁciency of the

receiver. The height z at which scattering takes place can be inferred from the time delay

2z/c between transmission and reception of the pulse, where c is the speed of light. The

range resolution z = cτ/2 typically lies between a few tens of metres and a hundred

metres.

In the case in which the scattering is Rayleigh scattering from any atmospheric molecule,

dσ/d = aν

by equation (7.17),wherea is a known constant. The method can then be

used to obtain the total number density n(z) of the atmosphere and hence the density

ρ(z ) =

mn(z) (where m is the mean molecular mass), at height z. Rayleigh scattering is

important for wavelengths less than about 3 μm and allows measurements of density from

heights between about 10 and 90 km. The temperature can be obtained by use of the ideal

gas equation (2.2) and the hydrostatic equation (2.12):seeProblem 7.10.

In the case of scattering from aerosol particles, whose s izes may approach or exceed

the laser wavelength, Rayleigh scattering does not apply, and the more complex Mie

scattering is the relevant process. It is difﬁcult to use lidar measurements to quantify

aerosol concentrations, but one can instead plot the lidar backscatter ratio R, deﬁned as

the ratio of the total number of backscattered photons (from molecules and aerosols) at

frequency ν to the number of backscattered photons from molecules alone:

R =

mol

+ β

aerosol

mol

where β

= n

dσ

/d. Layers of high aerosol content, due for example to volcanic eruption,

show up as sharp peaks in R(z);seeFigure 7.15.

Lidars with tunable dye lasers, tuned to the frequency of speciﬁc atomic or molecular

transitions, can be used to measure the vertical distribution of chemical species, including

various alkali atoms (deposited in the atmosphere from meteors) and ozone. For example

the

1/2

−

3/2

transition of sodium at 589.0 nm is associated with an enhanced scat-

tering cross-section (this is an example of resonance scattering), which allows the lidar

measurement of amounts of sodium at heights of 80–100 km.

Further reading

A brief discussion of remote sounding from space is given by Houghton (2002), and

comprehensive treatments by Houghton et al. (1984)andHanel et al. (2003). A description

of the Dobson ozone spectrometer is given by Dobson and Normand (1957). The basic

physics of atmospheric radars and some typical results are presented by Balsley and Gage

(1980); for a review of lidars see Thomas (1987).

Basic texts on electromagnetic waves include those by Lorrain et al. (1988)and

Grant and Phillips (1990). The detailed physics of Rayleigh scattering is covered in

the advanced texts by Jackson (1998)andBorn and Wolf (1980); a descriptive account

of the phenomenon is given by Lynch and Livingston (2001), who also detail many

other examples of atmospheric optical phenomena. Mie scattering is treated by Liou

(2002).

190 Atmospheric remote sounding

Fig. 7.15

Height proﬁles of the lidar backscatter ratio R measured at Aberystwyth (52

◦

N, 4

◦

W), showing a

build-up in the lower stratosphere of volcanic aerosols from Mt Pinatubo (see Vaughan et al.

(1994)). On 17 August 1991, R ≈ 1 throughout the proﬁle, indicating that little aerosol is present.

On 16 September, R exhibits a peak in a thin layer at about 20 km altitude. On 8 December there

is a very large peak, extending over a deep layer centred at about 23 km. On 28 January 1992 a

double-peaked structure is present. B ased on data provided by Dr G. Vaughan.

Problems

Problem 7.1 Consider a thin layer of gas of uniform density, between altitudes z and

z + d in a spherical atmosphere, emitting thermal radiation. Show that the ratio of the

maximum mass of the gas in a limb-viewing path to the mass of gas in a nadir-viewing path

is approximately (8a/d)

1/2

,wherea is the Earth’s radius. Evaluate this ratio for d = 10 km

and comment on your result.

191 Problems

Fig. 7.16

Emission spectra from Mars obtained with the IRIS instrument on Mariner 9 (adapted from Hanel

et al. (1972)). Top panel: spectra recorded over the south polar region; upper curve includes a

smaller fraction of the polar ice cap than the lower curve. Middle panel: spectrum recorded near

◦

S. Lower panel: spectrum recorded near 66

◦

N; note that the condensation temperature of CO

at Martian surface pressures is about 145 K. Diagram prepared with the help of Dr S. R. Lewis,

using data from the Planetary Data System.

Problem 7.2 Derive equation (7.1) from equation (3.13), assuming local thermodynamic

equilibrium, given that the Earth’s surface is a black body.

Problem 7.3

Figure 7.16 shows emission spectra from Mars.

1. What do the ﬁne features between 50 and 25 μm, especially in the south polar spectra,

indicate?

2. What is responsible for the large feature centred on 15 μm?

192 Atmospheric remote sounding

3. What can be said about the surface and atmospheric temperatures?

4. What can be learnt from a comparison of terrestrial (e.g. Figure 7.8) and Martian

midlatitude spectra?

Problem 7.4 Measurements of outgoing radiation are made at three wavelengths by a

nadir-viewing radiometer in Earth orbit, with the following results:

1. brightness temperature 310 K (at 11 μm, in the atmospheric window),

2. brightness temperature 220 K (at 15 μm, in the CO

band), and

3. brightness temperature 280 K (at 9.6 μm, in the O

band).

Assuming that the surface is black, that the CO

absorption is so strong at wavelength

(b) that the stratosphere is black there and that all the ozone resides in the stratosphere,

calculate the transmittance of the ozone layer at 9.6 μm. What could we learn about the

ozone layer from this?

(The Planck function B

at 9.6 μm is approximately 5, 21 and 36 in units of

−13

−2

steradian

−1

at 220, 280 and 310 K, respectively.)

Problem 7.5 The 11 μm region is often referred to as being in a ‘window’ in the terrestrial

atmosphere, but there is still some absorption, mainly due to water vapour. Assuming that

the absorption coefﬁcient k due to the water is k = αe,wheree is the vapour pressure and

α is a constant, and that the mass mixing ratio μ at pressure level p is μ

(p/p

)

,where

and p

refer to the bottom boundary and μ

is small, show that the transmittance of a

vertical path from level p to a satellite is exp[−β(p/p

)

] where

β =

and M

and M are the molar masses of water vapour and air, respectively.

Given that α = 7×10

−6

−1

and that the temperature T

and relative humidity

at the bottom boundary are 290 K and 70%, respectively, show that β = 0.102 and

calculate the transmittance from the surface to the satellite. (The SVP of water vapour at

290 K is 1936.7 Pa.)

Write down an expression for the net deﬁcit in the spectral radiance received from the

surface by a nadir-viewing radiometer due to attenuation and emission by water vapour.

Evaluate this deﬁcit for the case of a (black) sea surface at temperature T

for the conditions

given above, assuming that T = T

(p/p

)

and B

= η(T /300)

.Takep

= 10

Pa,

δ = 0.29, η = 4 × 10

−12

−2

steradian

−1

and γ = 4.5.





8.305

exp(−0.102y

) dy = 0.102



What would be the error in inferring the temperature of the sea surface from radiance

measurements if the effect of the water vapour were not taken into account? Is this error

signiﬁcant?

193 Problems

Problem 7.6 What is meant by a weighting function in atmospheric remote sounding?

Show that, if the transmittance T (p) = exp(−βp

),whereα and β are constants, and if the

height-like variable Z = ln(p

/p) is used as a vertical coordinate, the vertical weighting

function is

K(p) = αβp

exp(−βp

Find α and β for an isothermal atmosphere containing an absorber with a small and constant

mass mixing ratio μ when

1. the absorber is grey with constant absorption coefﬁcient k;

2. the radiometer is sensitive to a single frequency in the far wings of a Lorentz line; and

3. the radiometer is sensitive to a spectral interval in which the transmittance from level p

to the satellite is given by the strong limit of the Goody random model,

T = exp



−

(Smγ

)

1/2



where S is the mean line strength, m is the mass of absorber per unit area in the path, γ

is the mean Lorentz half-width and δ is the mean line spacing.

Problem 7.7 Find the pressure levels at which the peaks of the weighting functions occur

and ﬁnd also the widths (in terms of pressure) at half maximum of the weighting functions

of the previous question. Sketch the functions for cases (a) and (b).

(The roots of 2x = e

x−1

are approximately 2.68 and 0.23.)

Problem 7.8 Verify the last term on the right-hand side of equation (7.20) for the radio

refractive index, by ignoring the effects of bound electrons and proceeding as follows.

It can be shown (see Lorrain et al. (1988)orJackson (1998)) that the dispersion relation

for transverse plasma waves is

= ω

+ c

where the plasma frequency ω

is given by



in the notation of Section 7.3.2. Hence show that, if ω

 ω

N ≡

≈ 1 −

2

Problem 7.9 Suppose that the atmosphere has a radio refractive index N (z) given by

N = N

for z < z

, N = N

for z > z

Consider a vertically propagating electromagnetic wave incident on the level z = z

from

below. By using standard methods for calculating the reﬂected wave (see, e.g., Grant and

194 Atmospheric remote sounding

Phillips (1990)orLorrain et al. (1988)) show that the power reﬂection coefﬁcient R (the

ratio of the reﬂected to the incident power) is given by

R =

(

− N

)

when N

and N

are both close to 1. (You may ﬁnd it convenient to take z

= 0 for this

calculation.)

Problem 7.10 Given observations of density ρ(z) from lidar measurements and the

temperature T(z

) at a single height z

, use the ideal gas equation and hydrostatic balance

to show that the full temperature proﬁle T(z) can be obtained from

T(z) =

ρ(z )



ρ(z

)T(z

) +



ρ(z



) dz



Show that the term ρ(z

)T(z

) → 0asz

→∞. (In practice z

is taken as the maximum

height of the lidar measurements and a value of T(z

) is guessed; the resulting values of

T(z) are not very sensitive to this guess.)

Climate change

This chapter presents a selection of topics on the physics of climate change. By way of

introduction, in Section 8.1 we brieﬂy discuss greenhouse gases and the radiative forcing

associated with several drivers of climate change. Then in Sections 8.2 and 8.3 we introduce

a very simple time-dependent, ‘energy balance’, climate model. This model illustrates some

key concepts that arise in the study of the physics of the Earth’s climate and its response

to external forcing, and in the diagnosis of the highly complex models that are used to

simulate the climate of the past and present and to predict future climate. In Section 8.4 we

examine some elementary aspects of the important topic of climate feedbacks. Finally, in

Section 8.5 we use another simple model to examine the basic physics of the process by

which the radiative forcing due to carbon dioxide increases with its concentration.

Our emphasis in this chapter is on the underlying physical principles of climate change;

we shall not discuss in any detail the climate-change projections by the current range

of complex general circulation models. Comprehensive information on these projections

is provided for example by the Assessment Reports of the Intergovernmental Panel on

Climate Change (IPCC). The most recent such report is that of Solomon et al. (2007),

which includes useful summaries for non-specialists and a glossary of technical terms.

8.1 Introduction

The greenhouse effect was introduced in Section 1.3 and discussed further in Section 3.7.

We now consider the effects of different greenhouse gases, i.e. gases that absorb and

emit infra-red radiation but allow solar radiation to pass through without signiﬁcant

absorption.

As a starting point, consider a hypothetical atmosphere containing no greenhouse gases

or other absorbers. Given an unreﬂected (i.e. absorbed) solar irradiance F

≈ 240 W m

−2

the surface temperature (assumed uniform) is about 255 K, as shown in Section 1.3.1.To

quantify the effects of greenhouse gases in raising this to the current global-mean surface

temperature of about 288 K, which has a black-body irradiance of about 390 W m

−2

, one can

say that these gases currently ‘provide a greenhouse effect’ of about 390–240 = 150 W m

−2

(We ignore the long-wave effects of clouds here.) In physical terms, the greenhouse gases

absorb some of the radiation emitted by the warm surface and emit a lesser amount of

radiation from cooler tropospheric regions aloft; they also emit energy downwards, and

hence help to warm the surface. See Problem 8.1 for a simple model of this process.

196 Climate change

An important corollary is that, all else being equal, a decrease of upper-tropospheric

temperatures leads to an increase in the greenhouse effect.

It would be of interest to apportion this greenhouse effect between the different gases:

however, this cannot be done in a simple way, given the large difference in composition

between the hypothetical and actual atmospheres, because the effects of the separate gases

do not add linearly. (See Problem 8.2.) For example, numerical experiments in which

different greenhouse gases are in turn completely removed from the atmosphere show that

the greenhouse effect due to both CO

and water vapour is smaller than the sum of the

individual effects of each of them. Nevertheless it is clear that water vapour is the most

important greenhouse gas in determining the current atmospheric state, contributing around

2/3 of the current greenhouse effect, and that CO

is the next most important, contributing

around 1/4.

Different considerations apply when we consider the response of any given climate

state (for example, the current climate or the ‘pre-industrial’ climate), to perturbations

to the various absorbing gases or other climate variables. Examples of interest include

perturbations to greenhouse gas concentrations, aerosols, clouds or albedo, and especially

those perturbations caused by human activities. Here an important quantity is the radiative

forcing, deﬁned as the net decrease of the upward irradiance at the tropopause due to a

change in some climate-change driver, such as a speciﬁed increase in the amount of a

gaseous absorber. It is assumed that the surface and troposphere are otherwise held ﬁxed

and that stratospheric temperatures, if perturbed, have readjusted to radiative-dynamical

equilibrium.

The net upward irradiance at the tropopause equals the outgoing irradiance minus the

unreﬂected incoming solar irradiance there; in equilibrium, this would be zero. If the

radiative forcing (RF) is positive, the incoming irradiance exceeds the outgoing irradiance

and there is a positive net heat ﬂux into the climate system below the tropopause, leading

to a warming of the climate; a negative RF contributes to a cooling of the climate. An

important example of a positive RF is provided by the enhanced greenhouse effect,in

which an increase in a tropospheric greenhouse gas decreases the outgoing long-wave

irradiance (often called the outgoing long-wave radiation, or OLR). Calculation of the

RF due to atmospheric aerosols is very complex: they lead to ‘direct’ effects, by scattering

and absorbing long-wave and short-wave radiation, and to ‘indirect’ effects, by modifying

the microphysical and hence the radiative properties of clouds; overall, both effects are

believed to give negative RFs.

Figure 8.1 shows estimated values of the RFs associated with the main climate-change

drivers for the period 1750–2005; for the comparatively small perturbations involved,

the separate effects do add fairly linearly. Drivers with positive RFs include the long-lived

greenhouse gases carbon dioxide, methane (CH

), nitrous oxide (N

O) and the halocarbons

(including CFCs), and also tropospheric ozone and stratospheric water vapour. Aerosols

have negative RFs, while surface albedo changes can contribute either positive or negative

RFs. These drivers are all signiﬁcantly affected by human activities; the only signiﬁcant and

sustained natural RF over the period was due to an increase in solar irradiance, although the

11-year solar cycle causes oscillatory changes in RF, and stratospheric aerosols resulting

from massive volcanic eruptions can give short-term negative RFs.

197 Introduction

Fig. 8.1

Summary of the principal components of the radiative forcing of climate change. The values

represent the forcings in 2005 relative to the start of the industrial era, taken as 1750. The thin

black line attached to each bar represents the range of uncertainty for the respective value. For

the CH

O and halocarbon block, the error is for the three gases combined. The levels of

scientiﬁc understanding of the various components are estimated as high for the long-lived

greenhouse gases, medium for ozone and medium–low or low for the others. (Figure from

Solomon et al. (2007), FAQ 2.1, Figure 2.)

Tropospheric water vapour does not feature in Figure 8.1: rather than providing an

RF of its own, water vapour primarily acts by modifying the effects of other climate

drivers. This is because it is a short-lived gas: its concentration varies on small time

and space scales, mainly by exchange with the large liquid water reservoir through rapid

condensation and evaporation processes, which are strongly temperature-dependent. Since

a temperature increase leads to an increase in the amount of water vapour by the Clausius–

Clapeyron equation (2.37), and increased water vapour gives a decreased OLR and hence

a warmer climate, water vapour provides a positive feedback on temperature changes: see

Section 8.4.2.

198 Climate change

8.2 An energy balance model

In this section we introduce a very simple zero-dimensional energy balance model (EBM)

of the time-dependent response of the climate system to radiative forcing. In this model the

term ‘climate system’ is taken to encompass the top 100 m or so of the ocean (the ocean’s

‘mixed layer’) together with the troposphere and the land surface: these three components

are assumed to be represented by a ‘global-mean’ temperature T. Heat transport into the

deep ocean is neglected for simplicity, so the only heat transfers into and out of the climate

system are due to the absorbed solar (short-wave) radiation and the outgoing long-wave

radiation, respectively.

This is clearly an extremely simpliﬁed version of the real, highly complex, climate

system. However, it allows us to identify some important basic properties of the real

system, and to explore some possible ways in which the climate may change in response

to radiative forcing.

The EBM is described by the equation

= F

↓

− F

↑

, (8.1)

which states that, averaged over the Earth, the rate of change of heat content per unit

horizontal area of the climate system is given by the net heat ﬂux into the climate system,

i.e. the difference between the incoming and outgoing radiative power per unit area. Here

↓

is the solar (short-wave) irradiance absorbed by the climate system, and is equal to

the unreﬂected incoming solar irradiance at the top of the troposphere. Neglecting any

absorption of solar radiation above the tropopause we can take

↓

= F

≡

(1 − A)F

, (8.2)

(see equation (1.3)), where F

≈ 240 W m

−2

, A is the planetary albedo and F

is the total

solar irradiance (TSI). Note that this expression for F

↓

does not depend on the temperature

or greenhouse gas concentration of the atmosphere. On the other hand, the outgoing (long-

wave) irradiance or OLR F

↑

does generally depend on the temperature, water vapour

content, carbon dioxide concentration, etc., of the troposphere.

The constant C is the heat capacity per unit horizontal area of the climate system. A

simple approximation is to take it as the heat capacity per unit area of the oceanic mixed

layer times the fraction (∼0.7) of the Earth’s surface covered by ocean.

We suppose for the moment that the net heat ﬂux F

↓

− F

↑

into the climate sys-

tem depends only on the temperature T and the concentration U,say,ofaparticular

greenhouse gas,

↓

− F

↑

= Q(T, U). (8.3)

In a steady-state climate at constant temperature T = T

and concentration U = U

,we

have CdT/dt = 0andso

Q(T

, U

) = 0. (8.4)