ATMOSPHERE–OCEAN INTERACTIONS :: LECTURE NOTES

4. A Brief Review of Thermodynamics, Part 2

J. S. Wright

jswright@tsinghua.edu.cn

4.1 OVERVIEW

This chapter continues our review of the key thermodynamics concepts and formulae neededfor understanding the atmosphere–ocean system. The vertical profiles of temperature, pres-sure, and density in the atmosphere are revisited. The thermodynamics of dry atmospheresare presented, including the concepts of buoyancy and stability and the nature of convectionin dry atmospheres. The role of water in the atmosphere is described and the fundamentalatmospheric thermodynamics are extended to moist atmospheres. We conclude with a brieflook at radiative-convective equilibrium.

4.2 THERMODYNAMICS OF DRY ATMOSPHERES

As fluids, the ocean and atmosphere share many of the same basic features, even though theproperties of the atmosphere and ocean are vastly different. The total mass (5.3×1018 kg)and specific heat (1004 J K−1 kg−1) of the atmosphere are much smaller than those of theocean (1.4×1021 kg and 3994 J K−1 kg−1, respectively). Accordingly, the thermal inertia of theatmosphere is much smaller. The atmosphere responds very quickly to changes in surfacetemperature, and its role in climate stability is primarily radiative (through the greenhouseeffect) rather than thermal. Despite these fundamental differences, the vertical stability of theatmosphere can be understood in much the same context as that of the ocean.

4.2.1 VERTICAL STABILITY

Assuming that it is well-mixed, the stability of a dry atmosphere can be understood in terms ofthe vertical variation of potential temperature alone. Suppose a parcel of air is raised adiabati-cally from its initial position with p = pold and θ = θold to a new position with environmental

1

T1 T2

Temperature

z1

z2

Hei

ght

T(z)

θ1 θ2

Potential temperature

z1

z2

θ(z)

Figure 4.1: Schematic illustrations of dry convective adjustment to surface sensible heating rel-ative to (left) the environmental temperature profile and (right) the environmentalpotential temperature profile.

p = pnew and θ = θnew. Potential temperature is conserved during adiabatic motions, so thatwe can rearrange the atmospheric equation of state to find the density of the parcel:

ρparcel =pnew

Rdθold

(pnew

p0

)Rd /cp(4.1)

where p0 is the reference pressure as defined in section 3.3. The density of the environment is

ρenv = pnew

Rdθnew

(pnew

p0

)Rd /cp(4.2)

The buoyancy of the upwardly displaced parcel will be negative if ρparcel > ρenv, which is onlytrue if θnew > θold. In a dry atmosphere, stability may therefore be defined in terms of thevertical gradient of potential temperature: the atmosphere is stable if potential temperatureincreases upward, neutrally stable if the change in potential temperature is zero, and unstableif potential temperature decreases upward. These features can also be expressed in terms oftemperature. The atmosphere is unstable if the lapse rate (Eq. 2.8) is greater than the adiabaticlapse rate, neutrally stable if the two are equal, and stable otherwise. The mean observedlapse rate is approximately 6.5 K km−1, while the mean adiabatic lapse rate is 9.8 K km−1. Theglobal mean atmosphere is therefore stable to dry convection.

The diurnal cycle of solar radiation can create instabilities in the lower atmosphere bywarming the surface, which then warms the atmosphere in contact with the surface. As anair parcel in contact with the surface warms, it becomes less dense than the air above it andbegins to rise. As the air parcel rises it expands adiabatically, so that the temperature decreases

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Temperature

Hei

ght

T(z)

Temperature

Hei

ght

T(z)

Figure 4.2: Schematic illustrations of the creation of temperature inversions by (left) night-time surface cooling and (right) adiabatic warming during sinking from aloft andadiabatic cooling during rising from below.

at the adiabatic lapse rate. The air parcel continues to rise until its density is equal to thesurrounding environmental density. The atmospheric equation of state shows that this occurswhen the temperature of the parcel is equal to the environmental temperature, or where theadiabatic lapse rate intersects with the mean environmental lapse rate (Fig. 4.1). This processtypically results in a well-mixed daytime surface boundary layer. The depth of this boundarylayer depends on the environmental lapse rate and the extent of surface warming.

The opposite of this process occurs during calm winter nights, when radiative cooling ofthe surface leads to heat transfer from the atmosphere to the surface, gradually cooling thelowermost atmosphere. This process can create temperature inversion (i.e., an increase intemperature with height, with Γ< 0; see Fig. 4.2). The atmosphere near the surface is then verystable, which allows pollutants to build up in the surface layer. Inversions are also common inthe subtropics, where air descending from upper levels warms adiabatically. This adiabaticwarming can create a semi-permanent inversion where the descending air meets and mixeswith air affected by conditions at the surface. Inversions may also be formed or intensifiedby the presence of nearby hills or mountains, which transmit the effects of daytime surfacewarming and nighttime surface cooling to higher levels in the atmosphere. This situationaffects many cities, including Mexico City, Los Angeles, and Beijing. Inversions create athermodynamic barrier that inhibits ventilation of the surface layer and allows pollution tobuild up near the surface.

The buoyancy frequency for the atmosphere can be defined in terms of potential tempera-ture:

3

N =√(

g

θ

∂θ

∂z

)(4.3)

Buoyancy oscillations (gravity waves) are possible when the potential temperature increaseswith altitude (stable conditions). Conversely, (dry) convective instability results when thepotential temperature decreases with altitude. One common type of buoyancy oscillation inthe atmosphere occurs when air flows over a mountain. The air is adiabatically lifted by themountain, creating an oscillating density perturbation (lee wave) that can extend as far as100 km downstream.

4.3 THERMODYNAMICS OF MOIST ATMOSPHERES

Section 4.2 considers the thermodynamics and stability of a dry, well-mixed atmosphere.The major constituents of the atmosphere are indeed well-mixed (Table 1.2); water vapor, bycontrast, is not. The fractional amount of water vapor by volume in the troposphere varies fromapproximately 40 000 parts per million (in the tropical boundary layer) to approximately onepart per million (near the tropical tropopause). Stability is not purely a function of potentialtemperature for an inhomogeneous atmosphere because the gas constant and specific heatvary by location. The molecular weight of water vapor is approximately 60% of the meanmolecular weight of dry air, so that air becomes less dense when its concentration of watervapor increases. Water can also change phase at atmospheric temperatures and pressures(Fig. 4.3). As mentioned in section 3.4.1, water has a very high latent heat of vaporization. Thepresence of water in all its forms also creates spatial variations in the specific heat and theabsorption and/or reflection of radiation. Even though the amount of water in the atmosphereis small, its phase changes nonetheless have substantial effects on the energy budget andvertical stability of the atmosphere.

4.3.1 ATMOSPHERIC HUMIDITY

The water vapor content of the atmosphere can be expressed in a variety of ways. The mostfundamental is the vapor pressure

e = ρv Rv T (4.4)

where ρv is the density of water vapor and Rv is its gas constant (461.5 J K−1 kg−1). Equation 4.4is derived by applying the ideal gas law to water vapor. For a condensing substance (such aswater vapor on Earth), the maximum possible vapor pressure is determined by temperature.The variation of this saturation vapor pressure with temperature can be calculated using theClausius–Clapeyron equation:

de∗

dT= 1

T

Lv

ρ−1v −ρ−1

c(4.5)

where Lv is the latent heat of vaporization, ρv is the density of the vapor phase, and ρc is thedensity of the condensed phase (liquid or ice). This equation is derived from fundamental

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80 60 40 20 0 20 40Temperature [°C]

10-2

10-1

100

101

102

103

104

105

Vap

or p

ress

ure

[Pa] Triple point: 0.01°C, 611.2 Pa

water vapor

liquid

ice

supercooledliquid

Figure 4.3: Phase diagram for water at a range of temperatures observed in the atmosphere.The y-axis shows the vapor pressure e (Eq. 4.4). Equilibrium curves have beencalculated using Eqs. 4.7 and 4.8.

thermodynamic principles by calculating the work done in a reversible cycle of expansionand contraction across the condensation threshold (see, e.g., Curry and Webster, 1999). Equa-tion 4.5 can be simplified by applying the ideal gas law for water vapor and assuming thatcondensate is removed immediately:

de∗

dT= Lv e∗

Rv T 2 , (4.6)

but it is often approximated using empirical relationships. Two empirical approximations toEq. 4.5 that are accurate at atmospheric temperatures are given by

e∗ = exp

(53.67957− 6743.769

T−4.8451 ln(T )

)(4.7)

e# = exp

(23.33086− 6111.72784

T−0.15215 ln(T )

), (4.8)

where e∗ is the saturation vapor pressure with respect to liquid water and e# is the saturationvapor pressure with respect to ice (Emanuel, 1994). Direct measurements indicate that Equa-tion 4.7 is accurate to within 0.006% between 0◦C and 40◦C (and to within 0.3% for equilibriumbetween vapor and supercooled water down to −30◦C), while Equation 4.8 is accurate towithin 0.14% between −80◦C and 0◦C.

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90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800700600

500

400

300

200

150

100

Pre

ssur

e [h

Pa]

2

46

810

1214 16

Zonal mean specific humidity

0 2 4 6 8 10 12 14 16 18 20

Specific humidity [g kg−1]

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800700600

500

400

300

200

150

100

Pre

ssur

e [h

Pa]

1010

20

20

30

30

30 3040

40

40

50

5050

60

60

60

70

7070

80

808080

Zonal mean relative humidity

0 10 20 30 40 50 60 70 80 90 100Relative humidity [%]

Figure 4.4: Zonal mean distributions of relative humidity (RH; left) and specific humidity (q ;right) in the atmosphere. Data from the Japanese 55-year Reanalysis.

Vapor pressure is not the most physically intuitive representation of water vapor content,and several other representations are used in a variety of contexts. For example, relativehumidity is defined as the ratio of the vapor pressure to the saturation vapor pressure

RH = e

e∗(4.9)

so that RH = 1 when e = e∗ (i.e., when the air is saturated). Relative humidity can also becalculated with respect to ice:

RHi = e

e# (4.10)

The saturation vapor pressure over ice is uniformly less than the saturation vapor pressureover liquid water (Fig. 4.3), so that relative humidity with respect to ice is always greater thanrelative humidity with respect to liquid water. The distribution of relative humidity in theatmosphere is shown in the left panel of Fig. 4.4.

The mass mixing ratio r is defined as the ratio of the mass of water vapor to the mass of dryair (i.e., ρv/ρd ). Using the ideal gas law, r can be related to e:

r =e/Rv T

pd/Rd T= Rd

Rv

e

p −e= ε e

p −e(4.11)

where ε is the ratio of the mean molecular weight of water vapor to the mean molecular weightof dry air (Rd/Rv = Mv/Md ≈ 18.016/28.97 = 0.622). Observations of water vapor are often reportedas mass mixing ratios in units of g kg−1.

The specific humidity q is defined as the ratio of the mass of water vapor to the total massof the parcel (i.e., ρv/(ρv +ρd )), and can be calculated as

q = ε e

p − (1−ε)e= r

1+ r(4.12)

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Since r and q are both generally less than 0.04 in the atmosphere, r ≈ q . The distribution ofspecific humidity in the atmosphere is shown in the right panel of Fig. 4.4

4.3.2 THERMODYNAMIC EFFECTS OF WATER VAPOR

The presence of water vapor modifies the specific heat and the gas constant of air. Consideragain our expression of the first law of thermodynamics in Eq. (δQ = cp dT +ρ−1d p). Thechange in heat content δQ in a moist but unsaturated air parcel at constant pressure can beexpressed as

(md +mv )dQ = (cpd md + cpv mv )dT, (4.13)

where md and mv are the masses of dry air and water vapor inside the parcel, respectively,cpd ≈ 1004 J K−1 kg−1 is the specific heat of dry air at constant pressure and cpv ≈ 1870 J K−1 kg−1

is the specific heat of water vapor at constant pressure. The specific heat is defined as thechange in heat content for each 1 K change in temperature:(

∂Q

∂T

)p= cpd md + cpv mv

md +mv= cpd + cpv (mv/md )

1+ (mv/md )= cpd + cpv r

1+ r,

where r is in units of kg kg−1. We can simplify this expression even further by using the factthat r is small throughout the atmosphere:

cpm = cpd

(1+ (cpv/cpd )r

1+ r

)≈ cpd

[1+ r

(cpv

cpd−1

)].

Here, cpm represents the specific heat at constant pressure for moist air. Substituting approxi-mate values for cpd and cpv , we get cpm ≈ cpd (1+0.86r ). We can use a similar approach (andcv v ≈ 1410 J K−1 kg−1) to show that cvm ≈ cvd (1+0.97r ).

The density of an unsaturated parcel is also affected by the presence of water vapor. Usingthe ideal gas law and the definition of r ,

ρ = ρd +ρv = pd

Rd T+ e

Rv T= pd

Rd T(1+ r ) = p

Rd T

pd

ε(pd +e)(1+ r ) = p

Rd T

1+ r

1+ r/ε.

We can express this in the familiar form of the ideal gas law:

ρ = p

RmT, Rm ≡ Rd

1+ r/ε

1+ r. (4.14)

The parameter ε is less than 1, therefore Rm > Rd . This means that the density of a moist airparcel is less than the density of a dry air parcel at the same pressure and temperature.

For adiabatic processes, d s = cpdTT −R d p

p = 0, so that

d(lnT ) = Rm

cpmd(ln p) (4.15)

and the potential temperature of moist unsaturated air is

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θm = T

(p

p0

)−Rm/cpm

= T

(p

p0

) −Rdcpd

1+r/ε1+r (cpv/cpd ) ≈ T

(p

p0

) −Rdcpd

(1−0.24r )

(4.16)

Temperature will change less rapidly during adiabatic displacements of moist air (θm < θd )because of the higher heat capacity of water vapor relative to dry air; however, this effect isrelatively small (less than ∼1%), and potential temperature is therefore generally calculatedusing Eq. 3.5.

As shown by Eq. 4.14, moist air is less dense than dry air at the same temperature andpressure. The dependence of density on humidity is often expressed by the use of the virtualtemperature. The virtual temperature of an air parcel is the temperature dry air would haveif it had the same density and pressure as the parcel, and is defined such that Rd Tv = RmT .Substituting the expression of Rm in Eq. 4.14, the virtual temperature can be calculated as

Tv ≡ T

(1+ r/ε

1+ r

)≈ T (1+0.608r ). (4.17)

Because r is positive definite, the virtual temperature Tv is always larger than T . The virtualpotential temperature is then defined as

θv ≡ Tv

(p

p0

) −Rdcpd

. (4.18)

Virtual potential temperature is directly related to density, and is therefore a useful measureof the relative density of unsaturated air parcels. An unsaturated atmosphere is stable if θv

increases with height and unstable if θv decreases with height. Both θd and r are conservedduring adiabatic processes in unsaturated air, so that θv is also effectively conserved.

4.3.3 THERMODYNAMIC EFFECTS OF SATURATION

A parcel of air cools by adiabatic expansion as it is lifted. If the parcel contains water vapor,some of that water vapor will condense once the parcel cools enough that the saturation vaporpressure becomes less than the vapor pressure (i.e., RH ≥ 1). The condensation of that watervapor releases latent heat, which warms the parcel relative to the dry adiabat and increasesthe potential temperature. Condensation and evaporation are diabatic processes, and neitherpotential temperature nor virtual potential temperature are conserved when they occur. Ourunderstanding of cloud processes therefore relies on a different set of conserved variables.Neglecting ice, the specific entropy (entropy per unit mass) of an air parcel that includes waterin both the vapor and liquid phases is equal to the sum of the specific entropies for dry air,water vapor and liquid water in the parcel:

s = sd + r sv + rl sl = sd + (r + rl )sl + r (sv − sl )

Substituting the Clausius–Clapeyron equation as Lv = T (s∗v − sl ), where s∗v is the specificentropy of water vapor in equilibrium with the liquid water, we have

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200 225 250 275 300 325 350 375 400Temperature [K]

100

200

300

400

500

600

700

800

900

1000

Pre

ssur

e [h

Pa]

Temperature (T)Potential temperature (θ)Virtual potential temperature (θv)Equivalent potential temperature (θe)

Figure 4.5: Profiles of temperature (T ), dry potential temperature (θ; Eq. 3.5), virtual potentialtemperature (θv ; Eq. 4.18) and equivalent potential temperature (θe ; Eq. 4.19)in the troposphere averaged along the equator. Data from the Japanese 55-yearReanalysis.

s = sd + rt sl +r Lv

T+ r (sv − s∗v )

= (cpd lnT −Rd ln pd

)+ (rt cl lnT )+[

r Lv

T+ r

(cpv lnT −Rv lne − cpv lnT +Rv lne∗

)]= (cpd + rt cl ) lnT −Rd ln pd + r Lv

T− r Rv ln(e/e∗)

with rt = r + rl the total water mixing ratio and cl = 4190 J K−1 kg−1 the specific heat of liquidwater. Differentiating and setting d s = 0, we have the following conservation expression formoist adiabatic processes:

(cpd + rt cl )d(lnT ) = Rd d(ln pd )+ r Rv d(ln

e

e∗)− r

Lv

T.

We can then define the equivalent potential temperature θe as

θe = T

(pd

p0

)−Rd /(cpd+rT cl ) ( e

e∗)−r Rv/cp

exp

(Lv r

(cpd + rT cl )T

)(4.19)

Equivalent potential temperature is conserved during both moist and dry adiabatic processes.If the air is completely dry (r = 0), θe reduces to θ. In the absence of diabatic heating (i.e., if weneglect radiative transfer and neither condensation nor evaporation occur), r is unchangedand both θ and θe are conserved. If condensation or evaporation occur, then θe is conservedbut θ is not. In the case of condensation, RH = (e/e∗) is close to one, so that θ changes by a factor

9

Figure 4.6: Thermal equilibrium temperature profiles for a lapse rate of 6.5 K km−1 (dashed),a dry adiabatic lapse rate of 10 K km−1 (dotted), and pure radiative equilibrium(solid) (from Manabe and Strickler, 1964).

of approximately exp(

Lv∆r(cp+rT cl )T

). Note that in this expression ∆r is positive for condensation

(because condensation increases θ). In the case of evaporation, which reduces θ, e/e∗ < 1 andthe term involving RH becomes more influential.

Just as decreases of θ with height indicate vertical instabilities in the dry atmosphere, de-creases of θe with height indicate vertical instabilities in the moist atmosphere. These verticalinstabilities are mixed out by the occurrence of moist convection. Figure 4.5 shows that θ andθv increase with height in the tropics, indicating that the mean state of the tropical atmosphereis stable with respect to dry convection. By contrast, the gradient of θe is approximately neutralbelow 300 hPa and is even unstable in the lower atmosphere (although this instability canonly be triggered if condensation occurs). This difference highlights the critical importanceof water to convective mixing in the tropics. A moist adiabat is defined as the temperatureprofile corresponding to ascent with constant equivalent potential temperature.

The equivalent potential temperature is a useful variable for describing cloud process, but itis not conserved in all cases. In particular, θe is not conserved for radiative heating and cooling,nor is it conserved for sensible and latent heat fluxes from the surface to the atmosphere.Equivalent potential temperature is not even conserved for all cloud-related processes: forexample, the evaporation of falling raindrops into unsaturated air results in an irreversibleincrease in entropy.

We defined the virtual temperature to account for variations in density due to the presenceof water vapor; however, Tv does not account for variations in density due to the presence ofcondensed water. The density of a parcel containing condensed water can be expressed as

ρ = md +mv +ml +mi

Va +Vl +Vi

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Dividing both the numerator and the denominator by md yields

ρ = 1+ r + rl + ri1ρd

+ rv1ρv

+ rl1ρl

+ ri1ρi

≈ ρd (1+ rT )

so that

ρ ≈ pd

Rd T(1+ rT ) = p

Rd T

pd

ε(pd +e)(1+ rT ) = p

Rd T

1+ rT

1+ r /ε

We can then define the density temperature for which p = ρRd Tρ :

Tρ ≡ T1+ r /ε

1+ rT, (4.20)

which could in turn be used to define a density potential temperature. Unlike Tv , Tρ maytake values either smaller or larger than T , depending on the magnitude of the total waterconcentration.

4.4 RADIATIVE–CONVECTIVE EQUILIBRIUM

The temperature structure based purely on radiative equilibrium is unstable (Fig. 4.6). In fact,this model suggests that the global mean surface temperature at pure radiative equilibriumwould be more than 60◦C! The temperature near the surface is much warmer than the tem-perature aloft. This profile is unstable, and convection will occur. We can modify the simpleenergy balance models discussed in chapter 1 to include convection, yielding yet another classof simple climate models called radiative–convective equilibrium models. In most radiative–convective equilibrium models (such as that used to generate Fig. 4.6), the Earth’s atmosphereis considered as a single vertical column (just as in the pure radiative equilibrium energybalance model). Temperature at the surface and within the atmospheric column adjustsbased on solar and long-wave radiative heating, long-wave radiative cooling, and convection.Such models may be configured to either include or exclude the thermodynamic effects ofatmospheric moisture.

The radiative equilibrium lapse rate is larger than the equilibrium dry adiabatic lapserate, which means that the radiative equilibrium atmosphere is unstable to dry convection.Adjustment to the dry adiabatic profile still results in a surface temperature of approximately35◦C. Comparison with the thermal equilibrium for a global mean lapse rate of 6.5 K km−1

indicates that the dry adiabatic profile is further mixed by moist convection to yield theobserved global mean vertical distribution of temperature in the troposphere.

The importance of radiative–convective equilibrium in the troposphere is apparent in thezonal mean distribution of diabatic heating (Fig 4.7). Diabatic heating is the sum of all non-adiabatic processes in the atmosphere, including radiative heating and cooling (cf. Fig. 1.11),latent heating and cooling due to the formation and evaporation of clouds and precipitation(e.g., the additional terms in Eq. 4.19 relative to Eq. 3.5), and heating due to turbulent mixing(which we will discuss in more detail in lecture 6).

Despite strong regional variations in diabatic heating and magnitudes of several Kelvins perday, the zonal mean isentropes in Fig. 4.7a do not show large deviations across the tropics

11

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900

800

700

600

500

400

300

200

100

Pre

ssur

e [h

Pa]

280K280K

300K

340K

380K

(a) Total heating rate

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900

800

700

600

500

400

300

200

100

Pre

ssur

e [h

Pa]

280K280K

300K

340K

380K

(b) Radiative heating rate

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900

800

700

600

500

400

300

200

100

Pre

ssur

e [h

Pa]

280K280K

300K

340K

380K

(c) Convective heating rate

3 2 1 0 1 2 3

Heating Rate [K d−1]

Figure 4.7: Zonal mean (a) total, (b) radiative, and (c) deep convective diabatic heating. Datafrom the Japanese 55-year Reanalysis.

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and inner subtropics. This lack of covariability suggests that, despite its name, diabaticheating does not strongly affect internal (thermal) energy (cp T ) on long time scales. In fact,diabatic heating and cooling are primarily converted to kinetic energy, with diabatic heatingcorresponding to upward motion (across isentropes) in the atmosphere and diabatic coolingcorresponding to downward motion (across isentropes). In this sense, diabatic heating can bethought of as adding or removing potential energy (g z). We will revisit these ideas later in thecourse.

Figure 4.7 shows that both radiative and convective/moist processes are important in thetroposphere. Accordingly, we may say that the troposphere is approximately in radiative–convective equilibrium. By contrast, the diabatic effects of moist processes and dry convectionare approximately zero above the tropopause. This suggests that the stratosphere is nearlyin pure radiative equilibrium. The stratosphere contains a temperature inversion due tothe absorption of UV radiation by ozone (Fig. 2.2), and is therefore very stable with a largebuoyancy frequency (N ). Although convection is largely unimportant in the stratosphere,gravity waves and other oscillating disturbances play a key role in stratospheric dynamics.

REFERENCES

Curry, J. A., and P. J. Webster (1999), Thermodynamics of Atmospheres and Oceans, 471 pp., Academic Press, London,U.K.

Emanuel, K. A. (1994), Atmospheric Convection, 580 pp., Oxford University Press, Oxford, U.K.

Manabe, S., and R. F. Strickler (1964), Thermal equilibrium of the atmosphere with a convective adjustment, J.Atmos. Sci., 21, 361–385.

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