Salby M.L. Fundamentals of Atmospheric Physics

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190

Hydrostatic Stability

N2x 10 4 (s "2)

O. 0.5 1.0 1.5 2.0 2.5 3.0 4.5 4.0 4.5 5.0 5.5

2.8 20

2.6

2.4

2.2 16

2.0 14 ~"

1.8 "~

1.4 10

1.2 8 "-x

1.0

9 6 <

ii 4

9 2 _

0 0

325 350 375 400 425

0 e (K)

Figure 7.12 Mean vertical profiles of equivalent potential temperature (solid line) and buoy-

ancy frequency squared (dashed line) over Indonesia, obtained by averaging radiosonde observa-

tions between longitudes of 90~ and 105~ and latitudes of 10~ and 10~ during March 1984

(refer to Fig. 1.25).

makes the overlying layer potentially unstable. When conditions are favor-

able, small disturbances evolve into fully developed convection, which in turn

neutralizes the instability by rearranging mass vertically.

As noted earlier, the troposphere has a global-mean lapse rate of about 6.5

Kkm -1, which is close to neutral stability with respect to saturated conditions.

This mean stratification reflects two salient features of the troposphere: (1)

Tropospheric air is moisture laden and (2) tropospheric air is efficiently over-

turned in cumulus and sloping convection. Efficient vertical exchange leads to

a characteristic time for air to move between the surface and the tropopause

of only a couple of days. Because it operates on a timescale that is short com-

pared to diabatic effects, vertical mixing drives the thermal structure of the

troposphere toward that of saturated adiabatic motion (Sec. 6.5) and hence

toward a mean state of moist neutral stability.

Cumulus convection is favored in the tropical troposphere by high sea sur-

face temperature (SST) and strong insolation. Both support large transfers

7.6 Turbulent Dispersion 191

of radiative, latent, and sensible heat from the earth's surface, which destabi-

lize the overlying atmosphere. Organized convection, which is reinforced by

the release of latent heat, opposes those destabilizing influences to maintain

much of the tropical troposphere close to moist neutral stability. Cumulus

convection also operates outside the tropics, but it is organized in continental

monsoons (e.g., over India and Australia) and synoptic disturbances. Frontal

motions associated with cyclones, like the one in the eastern Atlantic in Fig.

1.23, cause organized lifting of air through sloping convection. By displacing

moist and often conditionally or potentially unstable air that originated in the

tropics, those motions favor convection, which likewise drives mean stratifica-

tion toward moist neutral stability.

7.6.2 Inversions

Contrary to the troposphere, the stratosphere is characterized by strong pos-

itive stability. Ozone heating makes temperature increase and potential tem-

perature increase sharply with height (Fig. 6.5). The potential temperature

increases from about 350 K near the tropopause to more than 1000 K near

10 mb. The sharp vertical gradient of 0 is associated with a strong positive

restoring force, one that suppresses vertical motions and 3-D turbulence and

allows chemical constituents to become highly stratified.

Unlike the troposphere, where air is efficiently rearranged in the vertical,

the stratosphere has a characteristic timescale for vertical exchange of months

to years. The layered nature of the stratosphere is occasionally demonstrated

by volcanic eruptions, which loft debris above the tropopause. During the

1950s and 1960s, nuclear detonations introduced radioactive debris into the

stratosphere. Because they have very slow sedimentation rates, aerosol parti-

cles introduced in this manner behave as tracers. Volcanic and nuclear debris

are observed to undergo little or no vertical motion, instead fanning out hori-

zontally into clouds of global dimension that remain intact for many months

(see Fig. 9.6). The great eruption of Krakatoa, which destroyed the Indonesian

island of the same name in 1883, altered radiative properties of the strato-

sphere and the color of sunsets for years afterward. The long residence time

of stratospheric aerosol allows it to alter SW absorption at the earth's surface

for long durations, which makes it an important consideration in the global en-

ergy budget. In sharp contrast, tropospheric aerosol (e.g., from deforestation

and the Kuwaiti oil fires) is eliminated by convective processes on a timescale

of only days (Chapter 9).

Convective cells do not penetrate appreciably above the tropopause, where

N 2 and the restoring force of buoyancy increase abruptly (Fig. 7.12). Upon

encountering this sharp increase of stability, an ascending plume is quickly

drained of positive buoyancy and fans out horizontally to form a cloud anvil,

as typifies the mature stage of cumulonimbus thunderstorms. Similar behav-

ior is occasionally observed at intermediate levels of the troposphere, when

192

Hydrostatic Stability

developing towers encounter a shallow inversion through which the buoyant

plume is able to penetrate (see Fig. 9.18).

Temperature inversions frequently accompany anticyclones. Often less than

a kilometer above the surface, a subsidence inversion (Sec. 7.4) confines pollu-

tants to a shallow layer near the ground, where they become airborne. Because

winds accompanying an anticyclone also tend to be light, pollutants are nei-

ther dispersed nor advected away from source regions, allowing concentrations

to increase during periods of high production. The same mechanisms inhibit

cloud formation by confining moisture near the surface. Consequently, anticy-

clones are typified by cloud-free but often polluted conditions.

7.6.3 Life Cycle of the Nocturnal Inversion

The competition between heat transfer and convection for control of the strat-

ification of a layer is illustrated by the formation and breakup of the

nocturnal

inversion,

which often develops at night. Under arid and cloud-free conditions,

which favor efficient LW cooling to space, the surface temperature decreases

sharply after sunset. To preserve thermal equilibrium, heat is transferred to the

ground radiatively and conductively from the overlying air, which cools the

surface layer from below and stabilizes it (Fig. 7.13). Sufficient cooling will

swing the temperature structure toward an inverted profile, which corresponds

to the potential temperature increasing sharply with height in a shallow layer

adjacent to the ground.

Strongly stable, the nocturnal inversion suppresses vertical motion, as well

as 3-D turbulence. Without turbulent transport to disperse them, pollutants

introduced at the surface are frozen in the stable air, so their concentrations

increase steadily near pollution sources. Such behavior is particularly notice-

able in urban areas during early morning periods of heavy traffic.

After sunrise, absorption of SW radiation warms the ground, which in-

fluences the surface layer in just the reverse manner. To preserve thermal

equilibrium, heat is then transferredto the overlying air, initially through con-

duction and radiative transfer. These forms of heat transfer warm the surface

layer from below, which increases the potential temperature and swings its ver-

tical profile counterclockwise near the ground. Inside the shallow surface layer

that has been destabilized, thermal convection then develops spontaneously to

neutralize the instability. By rearranging air vertically, heat is transferred from

the surface (in high values of 0) and mixed across the shallow layer of con-

vective overturning. The same is true for other conserved properties, such as

moisture and pollutants.

Absorption of SW radiation and vertical heat transfer continue to destabi-

lize the surface layer and thus maintain convective overturning. To preserve

neutral stability, additional heat transferred to the atmosphere must be mixed

over an ever deeper layer. In this fashion, the layer driven toward neutral sta-

bility expands upward, as do pollutants that were frozen near the surface in

7.6 Turbulent Dispersion

193

I I

.... ,!. .... I ................................

I I

Otrigger

Figure

7.13 Schematic illustrating the breakdown of the nocturnal inversion, which forms

during night. Following sunrise, surface heating and heat transfer to the overlying air destabilize

a shallow layer adjacent to the ground. Convection then develops spontaneously and, through

vertical mixing, restores the thermal structure inside that layer to neutral stability. Continued

heat transfer from the ground requires 0 to be mixed over a progressively deeper layer, which

erodes the nocturnal inversion from below. When convection has advanced through the entire

layer of strong stability, the nocturnal inversion has been destroyed. Deep convection can then

penetrate into weakly stable air overhead, to disperse pollutants that were previously trapped near

the surface in stable air.

stable air. Convective mixing erodes the nocturnal inversion from below and

reduces pollution concentrations by diluting them over an ever deeper layer.

After sufficient surface heating, convection will have advanced through the

entire nocturnal inversion, at which point it reaches air of weaker stability

overhead (e.g., air that was unaffected by surface cooling during the night).

The surface temperature when this condition is reached is termed the

break

trigger temperature

because deep convection can then develop from the source

of positive buoyancy at the ground. Pollutants are then dispersed vertically

over a deep volume, as well as horizontally by 3-D turbulence, resulting in a

sharp reduction of pollution concentrations. By permitting vertical transport

of water vapor, this condition also sets the stage for cumulus cloud formation,

which appears soon thereafter if sufficient moisture is present.

194

Hydrostatic Stability

Suggested Reading

Vertical stability is treated in detail, along with other considerations relevant

to cloud formation, in

Atmospheric Thermodynamics

(1981) by Iribarne and

Godson.

Environmental Aerodynamics

(1978) by Scorer contains an advanced treatment

of thermal entrainment, including laboratory simulations.

Problems

A pseudo-adiabatic chart is provided in Appendix E

7.1. Use Newton's second law to derive the momentum balance (7.1.1) for

an individual air parcel.

7.2. In early morning, the profile of potential temperature in a layer adjacent

to the surface is described by

0 ~ z

..-- ehl

00 h 2 '

with hi

> h2.

If condensation can be ignored, (a) characterize the static

stability of this layer, (b) determine the profile of buoyancy frequency,

and (c) determine the limiting distribution of 0 for t ~ ~, after suffi-

cient surface heating for convection to develop.

7.3. Use Archimedes' principle to establish (7.2.2).

7.4. A simple harmonic oscillator comprised of a 1-kg weight and an elas-

tic spring with constant k provides an analog of buoyancy oscillations.

Calculate the effective spring constant for unsaturated conditions repre-

sentative of (a) the troposphere: F = 6.5

Kkm -1

and T = 260 K and

(b) the stratosphere: F- -4 Kkm -a and T = 250 K.

7.5. An impulsive disturbance imposes a vertical velocity w0 on an individual

air parcel. If the layer containing that parcel has nonnegative stability N,

(a) determine the parcel's maximum vertical displacement, as a function

of N, under linear conditions and (b) describe the layer's evolution under

nonlinear conditions, in relation to N.

7.6. Upstream of isolated rough terrain, the surface layer has thermal struc-

ture

T(z) = T o a + z

where a > 0 and T o >

aFa.

(a) Determine the upstream profile of

potential temperature. (b) Characterize the upstream vertical stability.

Problems

195

7.7. Derive the identity

1 dO 1 dT K

0 dz T dz H

relating the vertical gradients of temperature and potential temperature.

7.8. An early morning sounding reveals the temperature profile

/ +

To+ - -~--

(a) Determine the trigger temperature for deep convection to develop.

(b) If, following sunrise, the ground warms at 3~ per hour, estimate the

time when cumulus clouds will appear.

7.9. For the circumstances in Problem 7.8, describe the transformations be-

tween radiative, internal and potential, and kinetic energies involved in

the development of convection.

7.10. The potential temperature inside a layer varies as

-- --(Z- a) 2 + a 2 -+-

Z>0.

(a) Characterize the stability of this layer. (b) How large a vertical dis-

placement would be required to liberate a parcel originally at z = 0 from

this layer?

7.11. Radiative heating leads to temperature in the stratosphere increasing up

to the stratopause, above which it decreases in the mesosphere. If the

temperature profile at a certain station is approximated by

b+(~) 2'

7.12.

7.13.

determine (a) the profile of potential temperature, (b) the profile of

Brunt-V~iis~iill~i frequency squared N 2. (c) For reasons developed in

Chapter 14, gravity waves can propagate through regions of positive

static stability, which provides the positive restoring force for their os-

cillations. In terms of the variable b, how small must h be to block

propagation through the mesosphere? (d) By noting that gravity waves

impose vertical displacements, discuss the behavior of air in the meso-

sphere under the conditions given for part (c).

Analyze the stability and energetics for the stratification and parcel in

Fig. 7.6, but under saturated conditions.

To what height will the parcel in Problem 7.12 penetrate under purely

adiabatic conditions? (See Problem 7.20.)

196

Hydrostatic Stability

7.14. Estimate the maximum height of convective towers under the conditions

in Fig. 7.6, but with the isothermal stratosphere replaced by one with a

lapse rate of-3 K km -1.

7.15. Establish stability criteria analogous to (7.8) under saturated conditions.

7.16. Rayleigh friction approximates turbulent drag in proportion to an air

parcel's velocity:

D = Kw,

where D is the specific drag experienced by a parcel, w is its velocity,

and the Rayleigh friction coefficient K has dimensions of inverse time.

For given Rayleigh friction coefficient K, show that vertical displace-

ments remain bounded for small instability but become unbounded for

sufficiently large instability.

7.17. Show that, in the presence of Rayleigh friction, vertical displacements

remain bounded under conditions of neutral stratification.

7.18. The lapse rate at a certain tropical station is constant and equal to 7

Kkm -1 from 1000 to 200 mb, above which conditions are isothermal. At

the surface, the temperature is 30~ and the relative humidity is suffi-

ciently great for vertical displacements to bring about almost immediate

saturation. Approximate the saturated adiabat by the constant lapse rate,

F s "~ 6.5 Kkm -1, to estimate the maximum vertical velocity attainable

inside convective towers. (See Problem 7.20.)

7.19. The weather service forecasts the intensity of convection in terms of

a thermal index

(TI), which is defined as the local ambient tempera-

ture minus the temperature anticipated inside a thermal under adiabatic

conditions. The more negative TI, the stronger are updrafts inside ther-

mals. A morning sounding over an arid region reveals a lapse rate of

-4 Kkm -1 in the lowest kilometer anda constant lapse rate of 6 Kkm -1

above, with the surface temperature being 10~ If air is sufficiently dry

to remain unsaturated and if the surface temperature is forecast to reach

a maximum of 40~ (a) determine the thermal index then as a func-

tion of height for z > 1 km. (b) To what height will strong updrafts

be expected? (c) Calculate the maximum updraft and maximum height

reached by updrafts anticipated under adiabatic conditions if, at the time

of maximum temperature, mean thermal structure has been driven adi-

abatic in the lowest kilometer.

7.20. In practice, vertical velocities inside a thermal differ significantly from

adiabatic values, entrainment limiting the maximum updraft to that near

the level of maximum buoyancy. Under the conditions in Problem 7.19,

but accounting for entrainment, calculate the profile of kinetic energy

and note the height of maximum updraft inside a thermal for an en-

trainment height of (a) infinity, (b) 1 km, and (c) 100 m.

Problems 197

7.21. Establish relationship (7.21) governing behavior inside an entraining

thermal.

7.22. According to Fig. 1.27, the troposphere is heated below by transfers of

radiative, latent, and sensible heat, while it is cooled aloft by LW cooling.

Discuss why, on average, the troposphere remains close to moist neutral

stability in the presence of these destabilizing influences.

7.23. The approach of a warm front is often heralded by deteriorating visibility

and air quality. Consider a frontal surface that slopes upward and to the

right, separates warm air on the left from cold air on the right, and

moves to the right. The intersection of this surface with the ground

marks the front. (a) Characterize the vertical stability at a station ahead

of the front. (b) Discuss the vertical dispersion of pollutants that are

introduced at the ground. (c) Describe how the features in (a) and (b)

and pollution concentrations vary as the front approaches the station.

7.24. Consider the conditions in Problem 7.18, but in light of entrainment.

If the ambient mixing ratio varies as r --

ro e-z/h,

with h = 2 km, and

the entrainment height is 1 km, calculate the profile of (a) equivalent

potential temperature inside a moist thermal (b) specific cooling rate for

air entrained into the thermal.

7.25. Consider a constant tropospheric lapse rate of F --- 7 K km-1 and mixing

ratio that varies as r

ro e-z/h,

with h = 2 km. If the troposphere can be

regarded as being close to saturation, characterize the stability of the tro-

posphere to large-scale lifting (e.g., in sloping convection) in the presence

of moisture that is representative of (a) midlatitudes, r0 - 5 g kg -1, and

(b) the tropics, r0 = 25 g kg -1.

7.26. A morning temperature sounding over Florida reveals the profile

T- / 20- 8(z- 1) [~ z > - lkm

20 [~ z < 1 km.

If the mixing ratio at the surface is 10 g kg-1, estimate the LFC.

7.27. See Problem 13.19.

Chapter 8

Atmospheric Radiation

The thermal structure and stratification discussed in Chapters 6 and 7 are

shaped in large part by radiative transfer. According to the global-mean energy

budget (Fig. 1.27), of some 542 W m -2 that is absorbed by the atmosphere,

more than 80% is supplied through radiative transfer: 368 W m -2 through

absorption of longwave (LW) radiation from the earth's surface and another

68 W m -2 through direct absorption of incoming shortwave (SW) radiation.

Similarly, the atmosphere loses energy through LW emission to the earth's

surface and to space.

To maintain thermal equilibrium, components of the atmospheric energy

budget must, on average, balance. Vertical transfers of radiant energy involved

in this balance are instrumental in determining thermal structure and mo-

tion, which characterize individual layers. Likewise, the dependence of optical

properties on temperature, variable constituents, and clouds makes radiative

transfer horizontally nonuniform. Geographical variations of radiative heating

(Fig. 1.29c) must be compensated by horizontal transfers of energy that main-

tain thermal equilibrium locally. That energy transfer is accomplished by the

general circulation, which, in turn, is driven by the nonuniform distribution

of heating. Thus, understanding these and other essential features requires an

understanding of how radiation interacts with the atmosphere.

8.1 Shortwave and Longwave Radiation

Energy transfer in the atmosphere involves radiation in two distinct bands

of wavelength. SW radiation emitted by the sun and LW radiation emitted by

the earth's surface and atmosphere are concentrated in wavelengths A that are

widely separated due to the disparate temperatures of the emitters. Figure 8.1a

shows blackbody emission spectra for temperatures of 6000 and 288 K, which

correspond to the equivalent blackbody temperature of the sun and the mean

surface temperature of the earth. The representation in Fig. 8.1a, logarithmic

in wavelength versus wavelength times energy flux, is area preserving. Equal

areas under the curves in different bands of wavelength represent equal power.

The spectrum of SW radiation, which is concentrated at wavelengths shorter

than 4/xm, peaks in the visible near A - 0.5/zm. Tails of the SW spectrum

198

8.1

Shortwave and Longwave Radiation

199

U.I

_.1

,,4

133

(a) Blackbody

I--

ft._

(/3

100

..........................................................................................................

40~ ~ ~ ~~ii~

o'3 o't g I i ,%0<ro,ation,

/ / c% H2o] co N.2~

H'~O CH 4 CH 4

I I I I I t ~ t ift I I I t i i I III I I I I I I I IFI

0.1 0.15 0.2 3 0.5 1 1.5 2 3 5 10 15 20 30 50 100

WAVELENGTH (pm)

Figure

8.1 (a)

Blackbody emission

spectra as

functions of wavelength A for temperatures

corresponding to the sun and the earth's global-mean surface temperature. The representation:

times energy

flux versus log(A), is area preserving---equal

areas under

a spectral curve

represent

equal

power. (b) Absorption as a

function of A along a vertical path from the top of the atmosphere

to a height of

11 km. (c) As

in (b), but to the surface. Adapted from Goody and

Yung (1989).

extend into the UV (A < 0.3 /zm) and into the near-infrared region (A >

0.7 /xm). On the other hand, the LW spectrum peaks well into the IR at

a wavelength of about 10 /xm. Tails of the LW spectrum extend down to

wavelengths of about 5/xm and out to the microwave region (A > 100/zm).

Overlap between the spectra of SW and LW radiation is negligible, so they

may be treated independently by considering wavelengths shorter and longer

than 4/xm.

Also shown as a function of wavelength is the fractional absorption of ra-

diation passing from the top of the atmosphere to a height of 11 km (Fig.

8.1b) and to the ground (Fig. 8.1c). Energetic radiation, A < 0.3/zm, is ab-