Salby M.L. Fundamentals of Atmospheric Physics

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130 5

Transformations of Moist

Air

heat from the gas phase, which cools the parcel, introduces negative buoyancy,

and thus promotes continued descent. Any condensate that previously precip-

itated out of the parcel is not available to reabsorb latent heat that it released

during ascent, which therefore remains in the gas phase.

The foregoing behavior is responsible for confining water vapor near the

earth's surface. Introduced over warm oceans, water vapor is extracted from air

that is displaced upward. Moisture is lost altogether when condensate precipi-

tates back to the earth's surface, after cloud particles have become sufficiently

large. In this fashion, thermodynamics, in combination with hydrostatic strat-

ification, maintains upper levels of the atmosphere in a very dry state. Even

inside a convective tower, mixing ratio decreases vertically because chemi-

cal equilibrium requires r to equal

r c,

which decreases steadily with altitude.

By limiting its vertical transport, thermodynamics prevents water vapor from

reaching great altitudes, where it would be photodissociated by energetic ra-

diation and ultimately destroyed when the free hydrogen produced is lost to

space (Sec. 1.2.2).

Due to exchange of latent heat with the condensed phase, the gas phase

of an air parcel is not adiabatic above the LCL---even though the entire

system may still be. Consequently, the potential temperature of the gas phase

is no longer conserved. Since the parcel's mass is dominated by dry air, the

transformation of mass has only a minor effect on the energetics of the parcel.

But the transfer of latent heat that attends the phase transformation has a

major effect by serving as an internal heat source for the system. Latent heat

released to the gas phase during condensation offsets cooling due to adiabatic

expansion work that is performed by the parcel during ascent. Conversely,

latent heat absorbed from the gas phase during vaporization offsets warming

due to adiabatic compression work that is performed on the parcel during

descent. Owing to the transfer of latent heat, the parcel's temperature no

longer changes with altitude at the dry adiabatic lapse rate, but rather varies

more slowly under saturated conditions.

5.4.2 The Pseudo-Adiabatic Process

If expansion work occurs fast enough for heat transfer with the environment

to remain negligible and if none of its moisture precipitates out, the parcel

is closed and its behavior above the LCL is described by a reversible satu-

rated adiabatic process. That process depends weakly on the abundance of

condensate present (e.g., on how much of the system's enthalpy is represented

by condensate) and therefore on the LCL of the parcel. However, because it

is present only in trace abundance, the variation of condensate unnecessarily

complicates the parcel's description under saturated conditions. This compli-

cation is averted by describing the parcel's behavior in terms of a

pseudo-

adiabatic process,

in which the system is treated as open and condensate is

removed (added) immediately after (before) it is produced (destroyed). Be-

5.4

Thermodynamic Behavior Accompanying Vertical Motion

131

cause the water component accounts for only a small fraction of the system's

mass, the pseudo-adiabatic process is nearly identical to a reversible saturated

adiabatic process.

A pseudo-adiabatic change of state may be constructed in two legs:

1. Reversible saturated adiabatic expansion (compression), which results

in the production (destruction) of condensate of mass

dmr

and a com-

mensurate release (absorption) of latent heat to (from) the gas phase

2. Removal (addition) of condensate of mass

dmc

Since the phase transformation occurs adiabatically and reversibly, this process

is isentropic:

ds - O.

Also, since just enough vapor condenses to maintain chemical equilibrium

(r = rc),

the change of water vapor mixing ratio is given by

Then (5.22) implies

dr = dr c.

('rc

dln

0 = -d cpT} "

(5.25)

With the removal of condensate, (5.25) describes the change of potential tem-

perature of the gas phase in terms of the transformation of the water compo-

nent. A decrease of rc, such as accompanies ascent and expansional cooling,

increases 0, whereas an increase of

rc,

such as accompanies descent and com-

pressional warming, decreases 0.

Integrating (5.25) obtains

0 exp(Irc~

- const, (5.26)

\ cpT,]

which describes a family of paths in the state space of moist air, one analogous

to the family of adiabats described by Poisson's equation (2.30) for dry air.

Just as the latter was used to introduce the potential temperature, which is

preserved for an adiabatic process, (5.26) may be used to introduce another

state variable, which is preserved for a pseudo-adiabatic process. Evaluating

(5.26) at a reference state of zero pressure (toward which r~ approaches zero

faster than does the parcel's temperature) yields

0e ('re t

~ = exp c-~ ' (5.27)

which defines the

equivalent potential temperature Oe.

According to (5.26), 0e

is constant during a pseudo-adiabatic process. Condensation is accompanied

by a reduction of

and an increase of 0, but the two vary in such proportion

as to preserve 0 e. The same holds for vaporization. In physical space, the

132 5

Transformations of Moist Air

material line in Fig. 5.4 advances to isentropes of higher 0, due to the release

of latent heat, but it remains coincident with the isopleth of 0e with which it

coincided initially. The equivalent potential temperature reflects the maximum

temperature a moist air parcel can assume through adiabatic compression

and the release of latent heat: namely, if it was displaced to the top of the

atmosphere, where all of the moisture condensed and released its latent heat,

if the condensate produced subsequently precipitated out, and if the parcel

was then brought down adiabatically to the surface.

Like 0, 0 e "--

Oe(P, T,

is a state variable. Just as 0 is conserved along an

adiabat in state space and below the LCL, 0e is conserved along a

pseudo

saturated adiabat

in state space and above the LCL. Because an adiabatic

process involving no transformation of phase is also pseudo-adiabatic, 0e is

conserved under unsaturated conditions as well. However, the definition (5.27)

can be applied only under saturated conditions because only then does r =

rc,

as is impficit in the derivation of 0e. Alternatively, because it is conserved,

0e can be calculated with r in place of

if T is replaced by the parcel's

temperature at the LCL, where it just becomes saturated.

5.4.3 The Saturated Adiabatic Lapse Rate

The temperature of a dry air parcel decreases with its altitude at the dry adi-

abatic lapse rate Fa. To a good approximation, the same holds for a moist

air parcel under unsaturated conditions because the trace abundance of water

vapor modifies thermal properties of air only slightly. Under saturated condi-

tions, the adiabatic description of air breaks down due to the release of latent

heat that accompanies transformation of water from one phase to another.

Latent heat exchanged with the gas phase then offsets cooling and warming

that accompanies adiabatic expansion and compression.

An approximate description of how the temperature of a saturated parcel

changes with altitude can be derived from the first law with the aid of (5.19).

From (2.35) the first law for the gas phase can be expressed

cpd

T - Rd

In p - -~--. (5.28)

If the parcel is unsaturated, (5.28) recovers the dry adiabatic lapse rate (2.33).

If it is saturated, the heat transferred to the gas phase is given by

6q - -tdr c.

(5.29)

Then (5.28) becomes

cpd

T - Rd

In p =

--~dr c,

where l is treated as constant. Hydrostatic equilibrium (1.16) reduces this to

cpdT + gdz - -ldr c.

(5.30)

5.4

Thermodynamic Behavior Accompanying Vertical Motion

133

Strictly, the saturation mixing ratio depends on both pressure and temper-

ature, so

() (3rc)

3rc dT + dp.

drc= --~-f p \,gp r

However, the strong dependence on temperature passed on from the Clausius-

Clapeyron equation through (5.7) is the dominant influence on r~. If its de-

pendence on pressure is ignored, the change of saturation mixing ratio can be

written

drc

drc = --d-~ dT,

where T and z are understood to refer to the displaced parcel. Then (5.30)

reduces to

drr

cpdT + gdz - -l-d- ~ dT

( drc)

Cp +

l-d- f dT + gdz = O.

In terms of the dry adiabatic lapse rate, this can be expressed

1 + ~ dr + Fa dz

= 0. (5.31)

cp -d-T

Then the saturated parcel's temperature decreases with altitude according to

dT F d

dz 1

-[ - l--drc

Cp dT

= Vs, (5.32)

which defines the

saturated adiabatic lapse rate.

Unlike Fa, F, varies with the parcel's altitude due to the nonlinear depen-

dence on T of

r c.

However, since

(drc/dT)

> 0, (5.32) implies

F s < F a, (5.33)

so a parcel's temperature decreases with altitude slower under saturated con-

ditions than under unsaturated conditions. This property of saturated vertical

motion follows from the release of latent heat to the gas phase, which offsets

cooling associated with adiabatic expansion. Although variable, the saturated

adiabatic lapse rate has a value of Fs ~ 6.5 K

km -1

for conditions represen-

tative of the troposphere. This is close to the global-mean lapse rate of the

troposphere (Fig. 1.2). No accident, this correspondence follows from dynam-

ical processes that are developed in Chapters 6 and 7.

134

Transformations of Moist

Air

5.5 The Pseudo-Adiabatic Chart

Thermodynamic processes associated with vertical motion are represented

conveniently on a diagram of the state space of moist air. Figure 5.5 shows

the

pseudo-adiabatic chart,

which displays as functions of T and -p~

1. Adiabats: 0 = const (solid lines)

2. Pseudo-adiabats:

0 e --

const (dashed lines)

3. Isopleths of saturation mixing ratio 3. r w = const (thin solid lines)

The ordinate -p~ can be interpreted in terms of altitude (1.17), which is

indicated at the right for the U.S. Standard Atmosphere. Adiabats, along

which a parcel evolves under unsaturated conditions and its temperature varies

3If ice is present, moisture properties actually depend on r i, but the isopleths of

r i

differ only

moderately from those of rw according to (4.39) and (4.40).

500

TEMPERATURE (K)

240 250

260 270 280 290

300 310

-1-- i ~ =: [ ] i ] r ........ ]--r----~--T--T--T [ r--I--

550

9oo , 3 ""

so,o,o,., co,,.,,

,,,,,,

a',>

950; SaturatiOn(r. = c on st.) M i xin(j Rat io ~ ~? ~? k'~ L~,-~, '~ ~,N~i~,, ~_~ ~" ~

1000 ~ ~ Parce I State \\ \ ~/~1~\ ~1~ ~~\~ \~ ~: N 0

-40 -30 -20 -10 0 T d 10 20 30 40

TEMPERATURE (C)

Figure

5.5 Pseudo-adiabatic chart illustrating thermodynamic processes for an ascending air

parcel. Shown as functions of temperature and elevation are (1) adiabats (solid), which are labeled

by the constant values of 0 characterizing those lines, (2) pseudo-adiabats (dashed), which are

labeled by the constant values of 0 e characterizing those curves, and (3) isopleths of saturation

mixing ratio (thin solid), which are labeled by the constant values of r w defining those curves.

5.5 The Pseudo-Adiabatic Chart

135

with height at the dry adiabatic lapse

rate F d,

appear as straight lines in this

representation (2.34). Pseudo-adiabats, along which a parcel evolves under

saturated conditions and its temperature varies with height at the saturated

adiabatic lapse rate Fs, are curved~but only weakly. Consistent with (5.33),

pseudo-adiabats have a slope with respect to height that is everywhere smaller

than that of adiabats. Both adiabats and pseudo-adiabats are labeled in Kelvin,

which correspond to the constant values of 0 and 0e characterizing those paths.

The use of the pseudo-adiabatic chart is best illustrated with an example.

Consider conditions leading to the formation of a cumulus cloud. The cloud

is fed by air ascending in a moist thermal that is driven by surface heating

(Fig. 5.4). At the surface, which is located at 900 mb, air has a temperature of

To = 15~ and a mixing ratio r0 = 6.0 g kg -1. Because air inside the thermal

originates at the surface, thermodynamic properties inside the cloud, as well

as beneath the cloud, can be inferred from the evolution of an air parcel that

is initially at the ground. Once the parcel's initial state (state 0) has been

located on the pseudo-adiabatic chart, individual properties follow by allowing

the system to evolve along certain paths in the state space of the parcel, which

are indicated in Fig. 5.5.

SURFACE RELATIVE HUMIDITY

The saturation mixing ratio at the surface follows from the isopleth of rw

passing through state 0, which gives

rw0 = 12 g kg -1.

Thus, the initial relative humidity of surface air is

RH~

= 12 = 50%.

SURFACE POTENTIAL TEMPERATURE

The parcel's initial potential temperature is determined by the adiabat pass-

ing through state 0, which is characterized by the constant value

0 = 00 = 297 K.

SURFACE

DEW POINT

The dew point of surface air follows from isobaric cooling out of state 0.

During that process, the parcel's state evolves along the isobar p = 900 mb,

which cuts across isopleths of saturation mixing ratio toward lower values

of r w. Eventually, state 1 is reached, at which r w = r = 6 g kg -1 and the

parcel is saturated. The temperature at which this occurs defines the dew

point temperature of surface air

Tao

= 4~

136

Transformations of Moist

Air

The foregoing process is responsible for the formation of ground fog. The

dew point spread of surface air, which is ll~ in this example, provides an

indirect measure of the base of convective clouds, since it reflects the amount

of adiabatic cooling necessary to achieve saturation (Problem 5.25).

CUMULUS CLOUD BASE

The base of convective clouds corresponds to the LCL of surface air. The

latter may be determined by displacing the parcel upward adiabatically. During

that process, the parcel's state evolves along the adiabat passing through state 0

(0 = 297 K), which also cuts across isopleths of saturation mixing ratio toward

lower values of rw. Eventually, state 2 is reached, at which rw = r and the

parcel is again saturated. The level at which this occurs corresponds to the LCL

PLCL = 770 mb.

The parcel's temperature at this level, which is 13 K colder than its initial tem-

perature, is only 2 K colder than the dew point temperature, reflecting the weak

pressure dependence of

rc.

EQUIVALENT POTENTIAL TEMPERATURE AT THE SURFACE

The equivalent potential temperature is determined once the LCL is lo-

cated. Through state 2 and along subsequent states, passes a saturated adiabat

that defines 0e for the parcel. Because 0e is conserved under both saturated

and unsaturated conditions, that value is also the equivalent potential temper-

ature of the parcel below the LCL, so

0e0 = 315 K

at 900 mb. Note, even though it is conserved throughout,

0 e

must be inferred

at and above the LCL, for reasons discussed in Sec. 5.4.2.

FREEZING LEVEL OF SURFACE AIR

At the LCL, the parcel's temperature is greater than 0~ so the freezing

level (FL) lies inside the cloud. Hence, lower portions of the cloud contain

water droplets, whereas higher portions contain ice particles. 4 Conditions in-

side the cloud can be determined by displacing the parcel above the LCL. The

parcel's state then evolves along the saturated adiabat passing through state 2.

Cutting across isopleths of saturation mixing ratio toward lower rw and across

adiabats toward higher 0, the 0e = 315 K saturated adiabat eventually reaches

state 3, where the temperature is 0~ That condition is achieved at the level

PrL = 740 mb.

4Ice forms only in the presence of a special type of aerosol particle, referred to as a

freezing

nucleus

(Chapter 9). Since freezing nuclei are comparatively rare, many cloud droplets are actually

"supercooled,"

that is,

they remain liquid

at temperatures below 0~

5.5 The Pseudo-Adiabatic Chart

137

LIQUID WATER CONTENT AT THE FREEZING LEVEL

Because the parcel is saturated, its mixing ratio must equal the saturation

mixing ratio at the freezing level. The isopleth of saturation mixing ratio

passing through state 3 gives for the mixing ratio there

rFL -- 5.5

gkg -1.

If no precipitation occurs, the total water content of the parcel is preserved.

Therefore, the liquid water content at the freezing level is given by

r t

= 0.5 gkg -1.

Approximately 10% of the parcel's moisture has condensed by this altitude.

TEMPERATURE INSIDE CLOUD AT 650 MB

At 650 mb, the parcel has evolved along the 0e = 315 K saturated adiabat

to state 4, where its temperature is

T650 ----

_6~

Were the parcel perfectly dry, it would have continued to evolve above 770 mb

along the 315 K adiabat. Without the release of latent heat to offset adiabatic

cooling, the parcel's temperature would then have decreased more rapidly,

resulting in a temperature at 650 mb of -12~

MIXING RATIO INSIDE CLOUD AT

650

Because the parcel remains saturated, the isopleth of saturation mixing

ratio passing through state 4 gives for the mixing ratio at 650 mb

r650 - 4.0 g kg- 1.

By this level, one-third of the parcel's moisture has transformed into conden-

sate.

The foregoing example illustrates the strong constraint on water vapor im-

posed by thermodynamics and hydrostatic stratification. Inside convective tow-

ers, which transport moisture upward from its source at the earth's surface, the

abundance of vapor that can be supported decreases with altitude (Fig. 5.3).

By 500 mb, less than 30% of the surface mixing ratio of water inside the

parcel described above remains as vapor. According to (1.24), the absolute

humidity p~ decreases even faster. Thus, less than 15% of the absolute con-

centration of water vapor remains after the parcel has been displaced to the

middle troposphere. It is for this reason that deep convection increases the

column abundance of water vapor in Fig. 1.16 only modestly. Much of s

resides in the lowest 1 to 2 km, where water vapor is distributed more uni-

formly.

Through this mechanism, water vapor transported vertically inside convec-

tive towers is systematically extracted. Condensation leads to a decrease with

138 5

Transformations of Moist Air

altitude of water vapor mixing ratio and, through precipitation, a similar de-

crease of total water content. Outside convective towers, air is even drier

because it has not recently been in contact with the reservoir of moisture at

the Earth's surface or it has been dehydrated inside convective towers in the

aforementioned manner. The same mechanism operates on large scales inside

sloping convection, like that involved in synoptic weather systems. By limit-

ing the vertical transport of water vapor, thermodynamics in combination with

hydrostatic stratification confines moisture to a shallow neighborhood of the

earth's surface and has preserved water on the planet.