The Speed of Sound in the Atmosphere

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ACOUSTIC WAVES

K E Gilbert and H E Bass, National Center for Physical Acoustics, University of Mississippi, MS, USA

Introduction

This article is concerned with acoustic waves in the atmosphere. Owing to space constraints, the discussion is limited to audible acoustic waves (sound waves).

Hence, two important topics – ultrasound (above audible) and infrasound (below audible) – are not discussed. Further, in order to provide a more in-depth discussion of the effect of the atmospheric boundary layer on sound waves, some traditional topics such as ground effects, nonlinear effects, and noise control are omitted. The interested reader should refer to the resources cited under Further Reading for information on aspects of acoustic waves not covered here.

At the atomic level, the Earth’s atmosphere is a collection of gas molecules, mainly nitrogen and oxygen, bound to the planet by gravity. The micro- scopic properties of the atmosphere are thus described by the kinetic theory of gases and quantum mechanics.

In contrast, at the macroscopic level, the atmosphere can be regarded as a fluid, and, in principle, can be described by the equations of fluid dynamics. Both points of view, molecular and fluid dynamical, are needed to fully understand the generation, propagation, and absorption of the disturbances in the atmosphere that are familiar to us as acoustic waves or ‘sound’.

Unlike wave motion on a stretched string or ripples on the surface of water, acoustic waves in the atmosphere have no direct visual representation. Conse- quently, one must in general rely heavily on a mathematical description. It is useful, nevertheless, to try to connect the mathematical description of sound with an intuitive, physical picture, even if the picture is an approximate representation of reality.

Hence, for purposes of visualization, one can schematically represent a planar acoustic wave as shown in Figure 1. In Figure 1A, regions of compression (positive pressure relative to the ambient background pressure) and regions of rarefaction (negative pressure relative to the ambient background pressure) are indicated schematically by the density of points.

Closely spaced points represent a compression, and less closely spaced points represent a rarefaction. The vertical lines inFigure 1Bindicate regions of constant pressure that are called ‘wavefronts’. The maximum pressure regions are indicated by solid vertical lines

and the minimum pressure regions by dashed lines.

The horizontal line perpendicular to the wavefronts is called an acoustic ‘ray’. Acoustic rays are a concise way to indicate the travel paths taken by acoustic wavefronts as they propagate through space. InFigure 1C, the regions of compression and rarefaction, often called the ‘acoustic’ pressure, are shown moving to the right with a speedc, which for dry air is 331.6 m s^!¹at 01C. For a compact representation of the pressure wave, one could, for example, omit the wavefronts and simply show an acoustic ray moving to the right with a speedc.

For sinusoidal pressure variations, a planar acoustic wave can be represented mathematically as p¼ p0cosðkx!otþyÞ, where p0 is the acoustic pressure amplitude, and the entire argument of the cosine is called the ‘phase’ of the wave. The angular frequency,o, is 2pf, wheref is the frequency in cycles per second or, more commonly, in hertz (Hz). The wavenumber k is 2p=l, where l is the wavelength shown in Figure 1C. Since lf ¼c, the wavenumber can also be written aso=c. The quantityyis called the

c

Acoustic ray

Wavefronts

+p₀

−p₀

Acoustic pressure

x t= 0 t=t₁

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(B)

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Figure 1 Visualization of a planar acoustic wave moving to the right at speed c. (A) Schematic representation of regions of compression (denser points) and regions of rarefaction (less dense points). (B) Wavefronts (regions of constant pressure); maximum and minimum pressure regions are represented, respectively, by solid and dashed vertical lines. An acoustic ray is drawn perpendicular to the wavefronts. (C) Pressure variation in space at two instants of time for a sinusoidal plane wave of the form p¼p₀cosðkx!otþyÞ.

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‘phase angle’ and gives the phase of the wave atx¼0 andt¼0.

The compressions and rarefactions in an acoustic wave cause variations in density and temperature that also propagate with the wave. For all three quantities – pressure, temperature, and density – the acoustic amplitude is extremely small relative to the ambient background. For example, p0 might be 0.1 Pa or approximately one millionth of the nominal atmospheric pressure.

The Acoustic Wave Equation

As noted above, acoustic waves in the atmosphere can be viewed as small disturbances on an ambient background fluid, just as water waves are seen as disturbances on a calm surface. For the extremely small pressure perturbations typical of sound, the equations of fluid dynamics can be linearized to arrive at the ‘acoustic wave equation’, which is the conventional mathematical description of acoustic pressure waves. In one dimension, the acoustic wave equation is given by eqn [1], wherepis the acoustic pressure,xis distance, andtis time.

q² qx²p¼ 1

c² q²

qt²p ½1"

The general solution to eqn [1] is of the form pðx;tÞ ¼pRðx!ctÞ þpLðxþctÞ, where pRðx!ctÞ is a right-going wave and pLðxþctÞ is a left-going wave. The right-going wave, for example, could be a transmitted pulse, and the left-going wave could be an echo. Continuous waves as well as pulses satisfy the wave equation. For example, sincec¼o=k, the sinusoidal pressure wave discussed above satisfies the one-dimensional wave equation. Moreover, as indicated inFigure 2, any function ofðx!ctÞorðxþctÞ satisfies eqn [1]. Further, the perturbations in density and temperature associated with an acoustic pressure wave satisfy the same wave equation as the acoustic pressure except that, instead of pressure, the variable is density or temperature, respectively.

The three-dimensional form of eqn [1] is eqn [2], whereðx;y;zÞare Cartesian coordinates.

q² qx²þ q²

qy²þ q² qz²

! p¼ 1

c² q²

qt²p ½2"

For a symmetrical source, such as a small explosion high above the ground, the three-dimensional wave equation has spherical symmetry and can be written as eqn [3].

q²

qr²ðrpÞ ¼ 1 c²

q²

qt²ðrpÞ ½3"

Here r¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²þy²þz²

p . By comparing the form of eqn [3] with that of eqn [1], one can deduce that the general solution of eqn [3] is given by eqn [4], in which r_refis an arbitrary reference distance, generally taken to be 1 m in the MKS system.

pðx;tÞ ¼ rref

r

# %

pOUTðr!ctÞ þpINðrþctÞ

½ " ½4"

The quantities, ðrref=rÞPOUTðr!ctÞ and ðrref=rÞPIN

ðrþctÞare out-going and in-going spherical waves, respectively. Note that the spherical wave solution has the same mathematical form as the plane-wave solution except that the amplitude falls off as 1=r.

For a source far away from boundaries, the acoustic pressure is given by an out-going wave having the same shape in the time domain as the source function.

For example, for a time-harmonic source, the acoustic field is a traveling sinusoidal wave of the form p¼ ðr_ref=rÞp0 sinðkr!otÞ, where p0 is the pressure amplitude at the reference distance. Pictorially, an out- going spherical wave can be represented as shown in Figure 3, where pressure maxima and minima of the

x c c

1 1

P_L(x+ct) P_R(x− ct)

tt= tt₂₂ t= tt₂₂₂

Figure 2 Solutions to the one-dimensional wave equation. The functionpRðx!ctÞ is a right-going solution and the function pLðxþctÞis a left-going solution. The complete solution is the superposition of the left- and right-going solutions.

Figure 3 Schematic representation of an out-going spherical wave. The circles are the wavefronts and the straight lines are acoustic rays. The solid and dashed circles denote, respectively, wavefronts for maximum and minimum pressure.

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wavefronts are represented, respectively, by solid lines and dashed lines. The radial lines perpendicular to wavefront are acoustic rays.

Sound Pressure Levels and Decibels

Acoustic pressure amplitudes encountered in practice typically vary over several orders of magnitude.

Consequently, it has become conventional to use a logarithmic scale to describe the amplitudes. For continuous waves, the amplitude of interest is the root-mean-square pressure amplitude, prms, and is referenced to some standard reference pressurep_ref. For pulses, some ‘peak’ pressure is often chosen. In either case, a logarithmic amplitude measure called the

‘sound pressure level’ (SPL) is commonly used, defined by eqn [5].

SPL#10 log₁₀ p²_rms p²_ref

!

¼20 log₁₀ðprms=p_refÞ ½5"

The pressure of interest is denoted here as the rms pressure, but could be any pressure, depending on the application. Although SPL is actually a dimensionless quantity, one refers to the ‘units’ as decibels (dB), referenced to a reference pressure,pref. In atmospheric acoustics, the reference level is usually chosen to be 2$10^!5Pa or 20mPa, which is the approximate threshold of hearing. Note that with the above conventions, the SPL for 20mPa is 0 dB.

The frequency range for audible sound for the human ear is from approximately 20 Hz to approximately 20 kHz. Typical sound pressure levels for sounds in the audible range are given inTable 1.

In addition to being used as a measure for absolute pressure, decibels are also used to describe relative changes in pressure. For example, if a pressure amplitude decreases with distance by a factor of 10, it is conventional to say that, over the distance, the acoustic pressure has decreased by 20 log₁₀(10)5 20 dB. If one pressure amplitude were 100 times greater than another, one would say the first pressure was 40 dB greater than the second pressure.

To express in decibels the variation of the rms pressure,prmsðrÞ, with distance, it is conventional to define the SPL atr¼rrefas the ‘source level’ (SL) as in eqn [6] and to define the decibel decrease in acoustic pressure with distance as the ‘transmission loss’ (TL) as in eqn [7].

SL#20 log₁₀ prmsðr_refÞ pref

& '

½6"

TL# !20 log₁₀ prmsðrÞ prmsðrrefÞ

& '

½7"

In these equations, as noted previously, the reference distance,r_ref, is 1 m in the MKS system. (Note that since pressure usually decreases with distance, transmission loss is usually positive.) Using the above definitions for source level and transmission loss, one can write the sound pressure level at a distancerfrom the source as SPL5SL!TL. For example, in MKS units, the rms pressure amplitude for a spherically spreading wave can be written as prmsðrÞ ¼p1=r, where p1 is the rms pressure at 1 m. Thus, for a spherically spreading wave, the source level is 20 log₁₀ðp1=prefÞand the transmission loss is simply 20 log₁₀ðrÞ.

In general, the transmission loss is not a simple function and must be computed numerically. With numerical computations, it is often useful, for plotting purposes, to subtract the transmission loss due to spherical spreading, that is, to subtract 20 log₁₀ðrÞ. Such a convention is equivalent to giving the sound pressure level relative to a spherically spreading wave, and hence is given the name ‘relative sound pressure level’. Thus, by definition, the relative sound pressure level for a spherically spreading wave is 0 dB. Ex- pressed as a relative sound pressure level, an SPL above or below that for spherical spreading will be, respectively, greater than or less than zero.

The Speed of Sound in the Atmosphere

To a good approximation, the atmosphere can be treated as an ideal gas, and the acoustic pressure variations in it can be treated as adiabatic; that is, there is no heat flow from the higher pressure (hotter) regions to the lower pressure (cooler) regions. For an ideal gas and adiabatic compression (or rarefaction), the speed of sound is given byc¼pffiffiffiffiffiffiffiffiffiffigRT

, whereg¼ 1:40 is the ratio of the constant-volume specific heat for air,cV, to the constant-pressure specific heat, cP. The quantity,R¼286:69 J kg^!¹K^!¹, is the gas constant for dry air, andTis the absolute temperature (K).

With an ideal gas model, the theoretical value forcat 01C (273.16 K) is 331.1 m s^!¹, which is in excellent

Table 1 Representative list of audible sound pressure levels

Sound SPL

Threshold of hearing 0 dB

Leaves rustling 20 dB

Quiet conversation 40 dB

Normal conversation 60 dB

Average street traffic 80 dB

Diesel truck (at 10 m) 90 dB

Jet take-off (at 10 m) 120 dB

Threshold of pain 140 dB

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agreement with the experimental value of 331.6 m s^!¹ given earlier. For values ofT not far from 01C, the square root expression for the speed of sound can be expanded linearly and written approximately as c¼ ð331þ0:6TCÞm s^!¹, whereTCis the temperature in degrees Celsius. Thus, for an increase in temperature of one degree Celsius, the speed of sound increases by 0.6 m s^!1.

Absorption of Sound in the Atmosphere

In addition to the decrease in pressure amplitude of an acoustic wave due to propagation effects such as

‘geometrical’ spreading (e. g., spherical spreading), the amplitude is also reduced by atmospheric absorption.

A sound wave propagating through ‘clean’ air (no solid particles) is attenuated owing to two basic mechanisms:

% classical losses due to momentum transfer across a velocity gradient (viscosity) and heat flow across a temperature gradient

% Quantum-mechanical losses due to relaxation processes, mainly relaxation of rotational and vibrational states in nitrogen and oxygen molecules.

For both mechanisms, the effects of absorption can be represented by an absorption coefficient,a, which has units of m^!¹. The absorption coefficient enters via an exponential, so that the pressure is given by p¼p0e^!as, where p0 is the unattenuated pressure amplitude andsis the distance the wave has traveled.

To indicate the choice of the Napierian base, e, the attenuation coefficient is, by convention, said to have units of nepers m^!¹. The corresponding attenuation coefficient, a, for decibels (base 10) is a¼20a log₁₀ðeÞ ¼8:686a, and, by convention, has units of dB m^!¹.

Experimental and theoretical studies indicate that the total absorption coefficient can be represented as a sum of absorption coefficients, with each distinct physical process having its own unique coefficient.

That is, the total absorption coefficient, aT, can be written as aT¼P

iai, where ai is the absorption coefficient associated with a particular mechanism.

For example, the component of sound absorption due to viscosity is given by eqn [8], in which o is the angular frequency,cis the speed of sound,r₀ is the density of air, andmis the coefficient of viscosity of air.

aVIS¼ o² 2r₀c³

4m

3 ½8"

Since the compressed regions in an acoustic wave are slightly hotter than the ambient temperature, and

the expanded regions are slightly cooler, a small amount of heat flows from the compressions to the rarefactions. The conduction of heat converts the organized motion associated with the sound wave into random thermal motion of the gas molecules. Because the heat flow lowers the temperature of the compressions and raises the temperature of the rarefactions, both the pressure maxima and minima are reduced.

The reduction manifests itself as a decay of the acoustic wave with distance. The component of absorption due to thermal conduction is given by eqn [9], wherekis the coefficient of thermal conduc- tivity in J (kg mol)^!1K^!1kg m^!¹s^!¹.

aTH¼ o²

2r₀c³ðg!1Þ k

gcV ½9"

In addition to energy loss due to classical mechanisms (viscosity and heat conduction), energy can also be lost via quantum-mechanical ‘relaxation’ processes involving the internal degrees of freedom (rotation and vibration) of oxygen and nitrogen molecules. The transfer of translational energy to internal degrees of freedom and back takes place through an extended sequence of molecular collisions, so there is a time delay associated with the energy transfer. Because of the time delay, relaxation processes cause energy to be lost from the organized translational motion that constitutes the acoustic wave. As a result, just as with the classical mechanisms, the pressure amplitude of the wave decreases as the wave propagates.

For any particular relaxation process, the associated absorption coefficient has the general form of eqn [10].

a¼psv

c

f²=fr

1þ ðf=frÞ² ½10"

In eqn [10],svis the relaxation strength (in nepers),cis the sound speed,fis the frequency, andfris called the

‘relaxation’ frequency. The relaxation frequency is the frequency for maximum absorption and is roughly the reciprocal of the characteristic time delay for the transfer between kinetic energy and internal energy of the gas molecules.

For air, there are three important relaxation processes: (1) O₂vibration, (2) N₂vibration, and (3) N₂ rotation. The relaxation frequency for N2rotational relaxation is very high, so that, below 10 MHz, the denominator in eqn [10] is approximately unity. Thus, N2 rotational relaxation varies as f² and can be combined with the classical absorption coefficient. If one denotes the classical-plus-rotational absorption coefficient (i.e., the coefficient for viscosity, heat flow, and N2rotation) asaC, the absorption coefficient for O2vibration asaO, and the absorption coefficient for 4 ACOUSTIC WAVES

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N₂vibration as aN, then the total absorption coefficient,aTfor air can be written as eqn [11].

aT¼aCþaOþaN ½11"

Figure 4 shows the total absorption coefficient, aT, together with the components,aC,aO, andaN. Note that below about 10 000 Hz the absorption is dominated by vibrational relaxation. Further, note that, above about 1000 Hz, atmospheric absorption is significant for propagation distances of a kilometer or more, which accounts for the lack of long-range propagation of high-frequency sound.

Refraction of Sound in the Atmosphere

The spherical wavefronts and associated rays shown in Figure 3 represent acoustic waves radiating from a

point source in an atmosphere with a constant temperature, and hence a constant sound speed.

With a constant sound speed, acoustic ray trajectories are straight lines. In reality, however, the atmospheric temperature is never constant in space or time.

Consequently, the speed of sound is not constant but varies spatially and temporally. In a typical daytime situation, the temporally averaged temperature is independent of range but decreases with height (‘lapse’

condition). Thus, on average, the sound speed decreases with height, and sound rays curve upward, as shown inFigure 5. The ray paths shown are for a sound speed that decreases linearly with height. (For the simple case of linear variation, the ray paths are arcs of circles.)

For a general sound speed variation in a stratified atmosphere (i.e., no horizontal variation), ray paths are governed mathematically by Snell’s law, which states that the quantity cðzÞ=cosyðzÞ is invariant, where at a heightz, the quantities,cðzÞand cosyðzÞ are, respectively, the sound speed and the cosine of the angle of a ray with respect to horizontal. Thus, Snell’s law says that, ifcðzÞdecreases with height, cosyðzÞ must increase, so that the ray bends upward. In general, acoustic rays bend toward regions of lower sound speed and away from regions of higher sound speed. The bending of acoustic rays is given the name

‘refraction’.

The physical basis for refraction can be understood using the situation in Figure 5. Consider a small section of wavefront associated with a ray that leaves the source nearly parallel to the ground. For a small enough section, the wavefront is nearly planar and nearly vertical. Since the sound speed decreases with height, the lower portion of the wavefront travels faster than the upper portion, causing the wavefront to turn upward. In terms of rays, we would say that the ray is refracted upward due to the decrease in the sound speed with height.

Refraction of acoustic waves is caused by spatially varying wind as well as by spatially varying temperature. The effect of the wind on acoustic waves can be accounted for approximately by defining an ‘effective’

sound speed, ce, which is the previously defined

‘adiabatic’ sound speed,ca ¼pffiffiffiffiffiffiffiffiffiffigRT

, plus the component of the vector wind in the direction of propagation.

10^{− 0} 10^{− 1} 10^{− 2}

10^{− 3} 10^{− 4} 10^{− 5} 10^{− 6}

10^{− 7} 10^{− 8} 10^{− 9}

10¹ 10² 10³ 10⁴ 10⁵

Frequency (Hz)

Absorption coefficient (nepers m−

1)

T

N O

C

Figure 4 Components and general behavior of the total absorption coefficient for air. The contributions to the total absorptionðTÞ are the classical plus N2rotationðCÞ, the O2vibrationðOÞand the N2vibrationðNÞ. (Reproduced with permission from Bass (1991), VCH Publishers, Inc., Weinheim, Germany.)

S Shadow

Figure 5 Acoustic rays and shadow zones for an upward-refracting daytime atmosphere. For clarity, rays that are reflected off the ground are not shown. (Reproduced with permission from Bass (1991). (VCH Publishers, Inc., Weinheim, Germany.)

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For example, let^rrbe a unit vector pointing from the source to a receiver. Then, if the vector wind is denoted asv# ðvx;vy;vzÞ, the effective sound speed is given by ce¼caþ^rr.v, where ^rr.v is the component of the vector wind in the direction of sound propagation. In general, near the ground, the horizontal wind speed increases with increasing height. For upwind propagation of sound, therefore, the horizontal wind progressively reduces the effective sound speed with increasing height. For downwind propagation, the effect is reversed. That is, the horizontal wind progressively increases the effective sound speed with increasing height. In the daytime, for example, where the temperature and adiabatic sound speed decrease with height, upwind propagation adds to the upward refraction already present. Downwind, if the wind speed gradient is sufficiently large, the horizontal wind can overcome the upward refraction due to the daytime temperature profile and lead to downward refraction. The ray paths for upwind and downwind propagation are illustrated inFigure 6. For propagation directly across the wind, there is little effect due to the wind, but upward refraction persists because of the decreasing temperature with height.

As indicated in Figures 5 and 6, for upward refraction, there is a region, called an acoustic

‘shadow’, where no acoustic rays can penetrate. In the shadow region, the acoustic levels are much lower

than the sound pressure level one would expect with spherical spreading alone. Because of upward refraction, daytime sound pressure levels for a near-ground source fall off dramatically with horizontal distance as one enters the shadow region, which, for strong upward refraction, can be within 100–200 m of the source.

At night, in contrast to the daytime situation, ground-to-ground propagation is very good. Owing to radiative cooling of the ground, both the near- ground air temperature and the sound speed are lower than at higher altitudes (an ‘inversion’ condition). As a result, acoustic rays launched near to horizontal (less than about 101 with respect to horizontal) are bent downward, causing sound to be trapped in a ‘sound duct’ near the ground. Rays launched at steeper angles escape the duct and continue upward (seeFigure 7).

With strong trapping and small ground-bounce loss (e.g., over water) the acoustic field in the near-surface sound duct undergoes approximately cylindrical spreadingð1=pﬃﬃr

Þinstead of spherical spreadingð1=rÞ. As a consequence of daytime upward refraction and nighttime downward refraction, noise sources that are not generally heard during the day can often be easily heard at long distances (e.g., several kilometers) at night. The long-range propagation of acoustic waves at night makes noise control much more difficult than during the day.

v

S

Shadow

Figure 6 Acoustic rays and shadow zone for an atmosphere that is upward-refracting in the upwind direction and downward-refracting in the downwind direction. For clarity, rays that are reflected off the ground are not shown. (Reproduced with permission from Bass (1991), VCH Publishers, Inc., Weinheim, Germany.)

S

Figure 7 Acoustic rays for a downward-refracting nighttime atmosphere. Rays launched at small angles with respect to the horizontal (less than about 101) are trapped in a ‘sound duct’ near the ground and can propagate to ranges of several kilometers. For steeper launch angles, the rays escape the duct and continue upward.

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Diffraction of Acoustic Waves in the Atmosphere

As discussed above, one can approximately represent an acoustic field in terms of wavefronts whose propagation directions (i.e., rays) are governed by refraction. Such a representation is useful visually and can be valid computationally when the acoustic wavelengths are much smaller than the smallest sound-speed structure in the atmosphere. The main effect left out in the so-called ‘ray theory’ of sound propagation is the wave phenomenon known as

‘diffraction’. Diffraction is responsible for the well- known ability of sound to ‘bend’ around corners and obstacles. In outdoor sound propagation, diffraction fills in gaps in the acoustic field that would be present in a purely ray-based representation. Full-wave solutions to the wave equation (usually numerical) auto- matically include both diffraction and refraction.

Because of diffraction, every acoustic field has an intrinsic smallest possible scale length that is roughly a quarter of the smallest wavelength present in the field.

Owing to the scale limitation, there can be no sharp edges in the acoustic field. For example, instead of the sharp shadow boundary obtained with rays (Figures 5 and6), a smooth, diffuse boundary is obtained when diffraction is included. Such a situation is illustrated in Figure 8, which shows a numerical solution of the

wave equation for a 500 Hz point source in an upward-refracting atmosphere. The color plot in the figure, which is for the relative sound pressure level as a function of range and height, shows the effects of both diffraction and refraction. The edge of the shadow boundary in Figure 8 would become more diffuse at lower frequencies (longer wavelengths), until finally, at very low frequencies, the shadow boundary would not be discernible at the ranges shown.

When the atmosphere is downward-refracting (e.g., at night), the presence of diffraction again causes a

‘blurring’ of the features of the acoustic field, just as with upward refraction. There are no sharp disconti- nuities in the structure of the acoustic field. Rather, because of the finite wavelengths in the acoustic field, the changes in the field are continuous and smooth, as shown inFigure 9, which is also for a 500 Hz point source. In general, the longer the acoustic wavelengths (i.e., the lower the frequency), the smoother the features of the acoustic field.

In addition to limiting the sharpness of the acoustic field, diffraction is responsible for the scattering of acoustic waves from the complex small-scale structure of the real atmosphere. In a realistic model of the atmosphere, the instantaneous temperature and vector wind fields are not smooth but are highly irregular, containing eddies of all sizes. The eddy sizes of most

140

120

100

80

Height (m)

60

40

20

0

200 400 600 800 1000 1200 Horizontal range (m)

Relative sound pressure level (dB)

1400 1600 1800 2000

− 70

− 65

− 60

− 55

− 50

− 45

− 40

− 35

− 30

− 25

− 20

− 15

− 10

− 5 0

Figure 8 Numerical solution for the sound field in an upward-refracting daytime atmosphere without turbulence. Owing to diffraction, the edge of the shadow boundary is diffuse. The color plot shows the relative sound pressure level as a function of range and height. (Note that the vertical scale is much less than the horizontal scale, so that the actual propagation angles with respect to the horizontal are much smaller than shown.)

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concern for audible sound are in the region called the

‘inertial subrange’, which typically begins at a few tens of meters and goes down to a few millimeters. In the inertial subrange, the eddy structure is governed by the well-known Kolmogorov spectrum. As a consequence, at any instant of time, the small-scale spatial structure of the sound speed field, which depends on temperature and vector wind, can also be described by a Kolmogorov spectrum. In the daytime turbulent boundary layer, for example, the sound speed can be approximated as a time-independent mean sound speed, !ccðzÞ, that varies only with height, plus a fluctuating part, dcðx;y;z;tÞ, that varies with time, horizontal distance, and height. Hence the total sound speed,cðx;y;z;tÞ, can be represented as in eqn [12].

cðx;y;z;tÞ ¼!ccðzÞ þdcðx;y;z;tÞ ½12"

The quantity!ccðzÞapproximates the slow, large-scale variations in the sound speed profile anddcðx;y;z;tÞ describes the rapid, smaller-scale fluctuations.

As noted earlier, the quantitydcðx;y;z;tÞ follows the same Kolmogorov statistics as do the temperature and wind fluctuations. At a particular instant of time, an approximate ‘snapshot’ of the sound-speed fluctuation field,dcðx;y;z;tÞ, can be synthesized by adding together, with random phase, the wavenumber components for a Kolmogorov spectrum. The result of such a synthesis is shown in two dimensions inFigure 10. Because audible sound has wavelengths compa- rable in size to small-scale atmospheric structure, it is scattered in all directions as it propagates through

inertial-subrange eddies. As a consequence of diffraction, new wavefronts emanate from every eddy, with the strongest scattering occurring in the near-forward direction. The diffracted acoustic waves that are scattered downward act to fill in the shadow region.

An example of this phenomenon is shown inFigure 11, which was computed numerically using realistic rep- resentations for!ccanddc. With a realistic model for the sound speed, the predicted mean near-ground levels (!20 dB to !30 dB relative to spherical spreading) in the shadow region (0.2–2 km) are in good agreement with observation. The relative sound pressure level for a longer-range interval is shown in Figure 12. It is apparent fromFigure 12that, even with scattering into the shadow region, daytime levels near the ground are very low at ranges beyond a few kilometers.

It can be observed in Figures 11 and 12 that the effects of turbulence are most apparent in the shadow region, where the sound levels would be extremely small in the absence of turbulence. Above the daytime shadow region, in the ‘insonified’ region, the levels are much higher, so that the effect of scattering from turbulence is less dramatic, though the effect increases with increasing distance from the source. Similarly, for nighttime propagation, in the near-ground acoustic duct, where the mean levels are high, the effect of scattering from turbulence is not as dramatic as in the shadow region above the duct (seeFigure 13). Further, the nocturnal boundary layer, being more stable, intrinsically has weaker turbulence than the daytime boundary layer. As a consequence, mean sound levels at night near the ground are not affected by turbulence

140 120 100 80

Height (m) 60

40

20 0

2000 4000 6000 8000 10000

Horizontal range (m)

− 70

− 65

− 60

− 55

− 50

− 45

− 40

− 35

− 30

− 25

− 20

− 15

− 10

− 5 0

Figure 9 Numerical solution for the sound field in a downward-refracting nighttime atmosphere without turbulence. Note that downward refraction ‘traps’ sound in the near-ground duct. As inFigure 8, the features of the acoustic field are blurred owing to diffraction. The color plot shows the relative sound pressure level as a function of range and height. (Note that the vertical scale is much less than the horizontal scale, so that the actual propagation angles with respect to the horizontal are much smaller than shown.)

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nearly as much as the near-ground daytime levels. It should be noted, however, that large-scale nocturnal phenomena such as flow down a slope and gravity waves can have a significant effect on sound levels at night. The effects of such large-scale dynamical features on sound propagation are the subject of current research in atmospheric acoustics.

Acoustic Remote Sensing of the Atmosphere

As illustrated above, the sensitivity of acoustic waves to atmospheric wind and temperature variations makes accurate prediction of ground-to-ground sound propagation a challenging problem. Conversely,

10

10 5

5 0

0

Height (m)

Sound-speed fluctuation (m s− 1)

− 0.75

− 0.50

− 0.25

0.00

0.25

0.50

0.75

Figure 10 A two-dimensional ‘snapshot’ of small-scale turbulent fluctuations in the sound-speed field synthesized using a Kolmogorov spectrum and random Fourier components. The fluctuation magnitudes are typical of those created by turbulence in the daytime.

140 120 100 80

Height (m)

60 40 20 0

200 400 600 800 1000 1200 1400 1600 1800 2000 Horizontal range (m)

− 70

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Figure 11 Same asFigure 8except that the small-scale turbulence shown inFigure 10is included. Note that the effects of turbulence are most apparent in the shadow region. Scattering of acoustic waves from turbulence ‘fills in’ the shadow region so the levels there are much higher than the no-turbulence case is shown inFigure 8.

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however, that same sensitivity makes sound a remark- ably valuable probe for remotely sensing the complex features of the atmospheric boundary layer. The most widely used acoustic tool for atmospheric sensing is a pulse-echo probe called a ‘sodar’, after the more familiar ‘radar’, which is an electromagnetic pulse-echo device. (Note: Sometimes the name ‘echo- sonde’ is also used, but that designation is less common than ‘sodar’.)

The first sodars, which appeared in the early 1970s, emitted an acoustic pulse in a single vertically pointing beam as shown in Figure 14. The sodar geometry shown, with the acoustic source and receiver collo-

cated, is common and is known as a ‘monostatic’

sodar. (A less common geometry has the receiver separated horizontally from the transmitter and is called a ‘bistatic’sodar.) When the upward-going pulse encounters wind and temperature inhomogeneities produced by turbulence, faint scattered waves are created within the air itself. With monostatic sodar, the part of the acoustic wave scattered back toward the ground, the echo, is detected using the same transducers that produced the probe beam.

Early sodars were used primarily as instruments for detecting turbulence. The time delay between the emitted pulse and its echo determined the height of the

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Figure 12 Same asFigure 11except that the range extends to 10 km. Note that, even with scattering into the shadow region, daytime sound pressure levels near the ground are very low at ranges beyond a few kilometers.

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Figure 13 Same asFigure 9except that small-scale turbulence typical of a nighttime atmosphere has been included. Since sound levels are relatively high in the near-ground nocturnal sound duct, the effects of small-scale turbulence are less evident than during the daytime.

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turbulence (one-half the time delay times the average speed of sound), while the strength of the echo was a measure of the turbulence intensity. The evolving structure of the atmospheric boundary layer could be

‘mapped’ by plotting the delay time and echo strength on a vertically moving strip of paper. For example, the horizontal distance on the strip could be proportional to the time delay of the echo, and the darkness could be proportional to the intensity of the echo. With many repeated pulses, the evolution of the boundary layer could be followed visually. With its debut in the 1970s, the sodar immediately provided important insights into the spatial structure and temporal evolution of the atmospheric boundary layer. A typical sodar record is shown inFigure 15. The figure has time moving from left to right and shows the evolution of boundary layer structures over a typical diurnal period. The vertical scale in the figure is 0–500 m, and the thin white vertical streaks are hour markers. Panel (A) shows a typical daytime record of thermal plumes carried through the vertical sodar beam. Panel (B) shows the

turbulent boundary layer descending in the late afternoon and evening as solar heating of the ground diminishes. The undulations in the latter part of the record indicate the onset of internal gravity waves.

Panel (C) shows fully developed internal wave activity after midnight.

In addition to visual displays of boundary layer structure and dynamics, modern sodars can provide quantitative measures of wind and temperature. The so-called ‘Doppler sodar’, for example, which uses two slant beams in addition to the usual vertically pointing beam, can map vector wind versus height. A typical geometry would have a vertical beam, together

Scattering volume

Incident beam

Echo

Figure 14 Geometry for a monostatic sodar. An array of transducers projects a burst of acoustic waves vertically in a beam. Turbulence-generated inhomogeneities in wind and temperature scatter sound back toward the transmitting transducers, which act as a directional receiver for the faint echoes received on

the ground. Figure 15 Record from a sodar taken over a diurnal cycle. The

vertical scale is 0–500 m, and time increases from left to right. The vertical white lines are hour markers. (A) Unstable daytime boundary layer with thermal plumes generated by solar heating of the ground. (B) Decreasing daytime boundary layer followed by a growing stable nocturnal boundary layer showing evidence of initial gravity wave activity. (C) Stable nocturnal boundary layer after midnight with fully developed internal wave activity. (Reproduced with permission from Atmospheric acoustics, Encyclopedia of Applied Physics, vol. 2, VCH Publishers, Inc., Weinheim, Germany, 1991.)

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with slant beams pointing north and east, respectively, at 60^& above horizontal. Using the Doppler shift in the echoes (up-shift for winds moving toward the receiver and down-shift for winds moving away from the receiver), the three vector components (up–down, east–west, and north–south), can be measured as a function of height. Such advances as the Doppler sodar are due, in large part, to the vast increase during the past 20 years in the computing power available with small computers. In addition to providing greatly increased signal processing power, small, powerful computers have also made remote sensing instruments like the Doppler sodar sufficiently

‘user friendly’ that nonexperts can operate them successfully.

A second important advance in acoustic remote sensing is the ‘Radio Acoustic Sounding System’, or RASS, which can provide accurate temperature pro- files as a function of height. A RASS uses a single vertically pointing sodar beam together with two radar beams that converge in the air column over the sodar. The radar is used in a bistatic geometry with the transmitter on one side of the sodar and the receiver on the other side. Using coherent radar backscatter from the upward-going acoustic beam, the RASS measures the speed of the acoustic beam as it propagates upward. After making corrections for the vertical wind, one can estimate the adiabatic sound speed (a function of temperature only) as a function of height, which then yields the temperature as a function of height. Extensive comparisons between RASS measurements andin situmeasurements have shown that a RASS provides reliable estimates of temperature at heights from a few hundred meters to up to several kilometers.

Acknowledgements

The author would like to thank Mr RC Clark for the acoustic propagation calculations and Ms EA Furr for expert technical editing assistance.

AEOLIAN TRANSPORT

See DUST

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AEROSOLS

Contents

Climatology of Tropospheric Aerosols Observations and Measurements Physics and Chemistry of Aerosols Role in Cloud Physics

Role in Radiative Transfer

Climatology of Tropospheric Aerosols

J L Gras, CSIRO Atmospheric Research, Aspendale, Victoria, Australia

Introduction

Although the idea of a climatology of tropospheric aerosol appears relatively straightforward, practical implementation of a comprehensive global-scale climatology turns out to be very difficult; arguably, it is far from well established. Most problems stem from the inhomogeneous nature and spatial distribution of the aerosol, combined with sparse and relatively uncoordinated measurements. Despite this, much is known about the distribution of aerosol throughout the troposphere and how it varies seasonally. New remote-sensing technologies and analysis methodologies allow detailed observations and global mapping of some aerosol parameters. Determination of the global distribution of aerosol intrinsic properties, such as chemical composition, and the long-term trends of all aerosol properties remains a major challenge.

An aerosol is a stable suspension of a solid or liquid in a gas, in this case air. In practical terms, for the ambient atmosphere this includes particles from nanometers to tens and sometimes, hundreds of micrometers in diameter (around 10^!⁹to 10^!⁴m, or around five decades in size, and spanning an even greater range of concentrations). Constituent particles in the aerosol may be primary, that is, emitted as a particle, or secondary, being produced from gases in the atmosphere (by condensation or chemical reaction of certain species such as dimethyl sulfide (DMS), sulfur dioxide, and volatile organic compounds). They may include inorganic, organic, and biological entities such as spores, pollens, viruses, bacteria, waxes, and

plant debris. Most particles include some water, although cloud elements are usually excluded from the description. Aerosol particles have both natural sources (e.g., wind-blown sea spray, dusts, and volcanic debris) and anthropogenic sources (such as smokes, fumes, and exhausts). In some cases, such as wind-blown dusts from poorly managed agricultural areas, this distinction may be unclear. In many regions, particularly Northern Hemisphere mid-latitudes, anthropogenic components now frequently dominate particle number and mass. In general, an aerosol comprises particles of mixed composition both intern- ally (i.e., within one particle) and externally (between particles); this makes aerosol unlike a gas, which is the same wherever it is measured. Particle lifetimes are dependent on size and location in the atmosphere.

Typically, lifetimes range from days near the surface to months in the upper troposphere. There are many sources of particles, ranging from point sources including volcanoes, fires, and industrial plants, through to cities, clouds, entire regions, oceans, and even the atmosphere itself. In general, the spatial and temporal distribution of aerosol is notably inhomogeneous. Substantial changes in aerosol properties can occur over distances of only a few kilometers horizontally and much less vertically. The atmospheric aerosol is extremely dynamic, evolving and changing properties from the point of production until its ultimate removal by sedimentation or wet processes.

During its lifetime, a particle may amalgamate with other particles, exchange material through gas-phase reactions, and most likely pass through a number of cloud cycles where it is incorporated into cloud droplets, reappearing in a modified form when the droplets evaporate.

The troposphere is that part of the atmosphere lying between the surface and the stratosphere. The name derives from the Greektropos for ‘turning’, and this part of the atmosphere is characterized by air motion and mixing. Near the poles the troposphere reaches to about 8 km altitude and in the tropics to around 18 km AEROSOLS/Climatology of Tropospheric Aerosols 13

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altitude. The concept of a well-mixed troposphere is somewhat misleading though, since many of the important tropospheric aerosol features are associated with discrete layers in the atmosphere and the transport of aerosol in layers, often in large quantities.

Tropospheric aerosols impact on global climate, the atmospheric environment, and even human health.

Aerosols are a significant component in the global radiation balance, scattering and absorbing solar radiation and changing the properties of clouds. Their climatic effect is of similar magnitude to the green- house effect of carbon dioxide, but the net effect of particles is a cooling. Aerosols are important in the precipitation process: all cloud drops form on aerosol particles, known as cloud condensation nuclei, and ice nuclei play a similar role in ice clouds. Aerosols interact with reactive trace gases (heterogenous processes) and are implicated in biogeochemical cycling in the atmosphere, including transport of trace nutrients such as iron to the world’s oceans and of wind-generated sea-salt to the land, and playing roles in the sulfur, nitrogen, and carbon cycles. At the urban and rural levels they are the dominant cause of reduction of visibility and add to acidification and transport significant quantities of chemicals, such as minerals, sulfate, nitrate, and carbonaceous material.

Some aerosol components are toxic, others are carcinogenic. Epidemiological studies have shown significant correlation between various aerosol properties, including certain mass fractions, and human mortality and morbidity. Bioaerosol (molds, spores, pollens, dust mite feces, bacteria and viruses, and possibly fragments) are implicated in diseases including asthma.

How do aerosol properties vary across the globe and as a function of time, location, and altitude in the atmosphere? These factors are usually studied through development of a ‘climatology’. Strictly speaking, a climatology is a statistical description of a defined system, composed of various elements. These elements exhibit spatial and temporal variations to which the climate concept applies. For the global climate, for example, this would include temperature and rainfall.

The most fundamental elements for the tropospheric aerosol system are the size-dependent concentration and aerosol intrinsic properties; the latter properties are independent of concentration and include chemical composition, refractive index, and shape. Other elements or means of descriptions of the aerosol are also possible, such as light extinction coefficient and integral mass.

Unfortunately, no single definition of an aerosol climatology is universally accepted. The definition above, which describes an ‘observational’ aerosol climatology, requires comprehensive measurements of

the aerosol properties or elements. For tropospheric aerosol on the global scale, no truly comprehensive

‘observational’ climatology exists for any intrinsic aerosol property and it may be unachievable in the foreseeable future. None exists even for properties as apparently fundamental as total mass or number.

Sufficient reliable data exist for certain aerosol properties at selected sites or in certain regions for the development of limited ‘observational’ aerosol climatologies. Very few records of any aerosol para- meter are multidecadal, and establishing these records remains an important task for international science.

Contrasting with this is a ‘fully modeled’ aerosol climatology. A comprehensive aerosol model should be able to distribute precursor material between new particles and existing particles and generate a full description of the distribution of mass and intrinsic aerosol properties, as a function of size, from around 10^!9m to 10^!4m diameter, giving the three-dimensional spatial distribution of these properties over time. Aerosol dynamical models and chemical transport models exist but none currently has all these capabilities.

An intermediate approach, in effect generating a

‘hybrid’ aerosol climatology, has also been used. It comprises a model, based on an external mixture of aerosols from a set of generalized sources (e.g., biomass burning emissions, mineral dust, and sulfate), each with a given emissions inventory. Each component contributes its own generic properties, such as descriptions of the size distribution and optical properties, derived from a variety ofin situor remotely sensed measurements.

Data quality is central to all climatologies. Mea- surements with unknown accuracy or poorly specified conditions of measurement are ultimately of little or no value.

In addition to the spatial or temporal variation of aerosol properties, other useful descriptions include the amount of material or mass flux emitted by various sources. Major aerosol flux contributors include mineral aerosol, sea-salt, smoke from biomass burning, anthropogenic emissions (as sulfate and carbonaceous material), and secondary aerosol sources (including natural precursor gases). Table 1 gives typical estimates of the annual flux of various sources from the 1970s and 1990s, although the relative importance for both environmental and climate impacts is not simply related to these mass fluxes.

Particles in the range 50 nm to 1mm, for example, interact strongly with solar radiation and have the biggest effect on cloud properties. For mineral and sea- salt aerosol, massive particles with very short lifetimes complicate the definition of the flux. Flux estimates 14 AEROSOLS/Climatology of Tropospheric Aerosols

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have been evolving since the early 1970s as information on aerosol emission rates and the extent of sources improve.

Spatial and Temporal Distribution of Particle Properties

Convection and advection control the transport of atmospheric particles. Most particle sources are near the surface and concentrations generally decrease by about two-thirds for each 1 km altitude to about 5 km over the continents and 2 km over oceans. Above this, particle mixing ratios are relatively constant with altitude to near the tropopause. Concentrations of larger particles increase again in the lower stratosphere and total particle concentrations decline.

Tropospheric air enters the stratosphere at low latitudes, returning to the troposphere, via tropopause perturbations (folding), polewards of about 301N and 301S. This return air carries some stratospheric particles into the upper troposphere and sedimentation provides a flux of larger particles into the upper troposphere at all latitudes. Stratospheric particles can be identified to about 1–2 km below the tropopause.

Extraterrestrial particle sources are relatively minor.

Very small particles have an altitude profile more indicative of a distributed source in the upper free troposphere, and freshly nucleated particles have been observed in the upper troposphere when the integrated aerosol surface area becomes small. Both features are consistent with a relatively homogeneous tropospheric aerosol away from major sources, the so- called tropospheric background. Adding to this are remnant aerosol from various sources and the many major enhancements in tropospheric aerosol concentration associated with layer transport, particularly of mineral dusts, smoke plumes, and anthropogenic material.

Satellite-based remote sensing methodologies are very effective for geographically mapping various measures of aerosol ‘amount’, but the determination of intrinsic size-dependent properties such as composition and particle shape requirein situdetermination.

Consequently, there are far fewer data available for these properties, and the intercomparability of sam- pling protocols becomes very important. Geographic distributions of some species, such as sulfate, have received considerable attention and, while no really comprehensive global climatology of intrinsic properties currently exists, there are a growing number of national networks addressing the issue of aerosol composition, including sulfate, nitrate, and carbonaceous components.

In the following sections the troposphere is con- sidered in two altitudes ranges, greater than and less than 5 km, with emphasis placed on aerosol mass and number concentration.

Lower Troposphere

The altitude range from the surface to around 5 km includes the most intensively studied region of the troposphere through surface networks, mountain top observatories, and aircraft measurements. This part of the atmosphere is complex, involving the diurnally varying planetary boundary layer, nocturnal inver- sions, the marine boundary layer, clouds, and precipitation. Surface measurements are not always good indicators of the free troposphere or even the boundary layer. Measurement networks are also not uniformly distributed spatially and have widely different measurement parameters and methodologies. Remote sensing of column-integrated aerosol properties from satellites using spectrally resolved, scattered solar radiation also primarily senses lower- tropospheric aerosol properties, with roughly equal contributions from the boundary layer and free troposphere.

From late in the 19th century, systematic recording of meteorological observations by ships’ crews included aerosol phenomena (particularly haze).

Compiled in the 1930s, these represent one of the most extensive aerosol climatologies of the period and several decades following. The geographic distribution and seasonal variation from more than 50 years’

observations show most features evident in today’s satellite observations. These include Saharan dust plumes over the Atlantic ocean, the eastern Atlantic European plume, a North American summer plume in the North Atlantic, spring dust over the north-western Pacific, a summer plume over the Arabian sea, and dry- season plumes in north-western Australia and Indo- nesia. Currently, satellite-based remote measurements

Table 1 Estimated strength of tropospheric aerosol sources

Early 1970s (Tg/y^!¹)

1990s (Tg/y^!1)

Natural

Mineral 10–500 1000–3000^a

Sea-salt 200–1000 1000–3000^a

Volcanic 3–150 15–90

Biogenic 50

Gas to particle 300–2000 200–1300

Extraterrestrial 0.1–50 10

Anthropogenic Industry, fossil fuel,

carbon, sulfate, nitrate, organics

100–400 300

Biomass burning 3–150 100–450

aDepends strongly on upper size limit.

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afford the most extensive geographic mapping of those components of the aerosol that interact appreciably with visible or near-visible wavelengths, although surface-based measurements of column-integrated optical properties afford higher precision and the stability needed to assess subtle long-term changes.

Examples of dust and smoke distributions are shown inFigure 1in a monthly average ‘UV absorption aerosol index’ from the Total Ozone Mapping Spectrometer (TOMS), for October 1997. This demonstrates the extensive smoke plumes from fires in Indonesia and Brazil as well as some dust over Africa.Figure 2shows extensive mineral aerosol from Africa extending over the Atlantic Ocean during April 2000, as well as other dust including some over eastern China. Individual daily mapping, such as for the 1997 Indonesian fires, is also possible, as shown inFigure 3.

High-resolution spectrally resolved measurements of scattered solar radiation, for example, from NOAA’s AVHRR (Advanced Very High Resolution Radio- meter) satellites, also allow detailed mapping of equivalent column-integrated aerosol light extinction (optical depth) over ocean regions. The major aerosol plumes can be seen clearly in maps of the distribution of optical depth, such asFigure 4.

Particle Mass

As indicated by remote sensing andin situmeasurements, aerosol mass concentration varies strongly with location and time, depending on the proximity to sources and the effectiveness of dispersal and removal mechanisms. Particle total mass concentrations range from around 100 ng m^!3in the upper troposphere to greater than several hundred mg m^!3 in many large

cities in developing countries. Over ocean regions, concentrations vary with wind speed and proximity to continental sources, but values down to 10mg m^!3are observed. Total aerosol mass is measured less exten- sively than mass in aerosol size ranges associated with health or climate effects. In some jurisdictions the mass loading of particles with aerodynamic diameter Dao10mm (PM10) is regulated. Concentrations of PM10 in many cities in developing regions exceed 100mg m^!3 (24-hour average), but in developed countries are typically less than 50mg m^!3. In more remote continental regions, such as around the American Rocky Mountains, PM10 concentrations are usually less than 10mg m^!3. A seasonal variation in mass loading is common and is frequently dominated by local factors, including the seasonal pattern of

Figure 1 TOMS UV-absorbing aerosol index for October 1997.

(Image courtesy NASA.)

Figure 2 TOMS UV-absorbing aerosol index for April 2000.

(Image courtesy NASA.)

Figure 3 TOMS UV-absorbing aerosol index for 26 October 1997, showing Indonesian fire plumes. (Image courtesy NASA.) 16 AEROSOLS/Climatology of Tropospheric Aerosols

The Speed of Sound in the Atmosphere

ACOUSTIC WAVES

Introduction

The Acoustic Wave Equation

Sound Pressure Levels and Decibels

The Speed of Sound in the Atmosphere

Absorption of Sound in the Atmosphere

Refraction of Sound in the Atmosphere

Diffraction of Acoustic Waves in the Atmosphere

Acoustic Remote Sensing of the Atmosphere

Acknowledgements

See also

Further Reading

AEOLIAN TRANSPORT

AEROSOLS

Climatology of Tropospheric Aerosols

Introduction

Spatial and Temporal Distribution of Particle Properties

Lower Troposphere

Particle Mass