Salby M.L. Fundamentals of Atmospheric Physics

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180 7

Hydrostatic Stability

The penetration level in this analysis provides an estimate of the height of

"convective overshoots," which typify deep cumulus towers. However, like the

maximum updraft in (7.15.2), this estimate is only an upper bound. In practice,

some of the kinetic energy driving convective overshoots is siphoned off by

mixing with the surroundings. Figure 7.7 shows an overshooting cumulonim-

Figure 7.7 Convective domes overshooting the anvil of a cumulonimbus complex, at 2-min in-

tervals. Fibrous features visible in the second plate mark ice particles that nucleate spontaneously

when stratospheric air is displaced by convection. After Fujita (1992).

7.4 Finite Displacements

181

bus complex at 2-min intervals. Several convective domes have penetrated

above the anvil, where air eventually fans out horizontally. Negatively buoy-

ant, overshooting cloud splashes back into the anvil, but not before it has

been sheared and mixed with stratospheric air. Thus, convective overshoots

lead to an irreversible rearrangement and eventually mixing of tropospheric

and stratospheric air.

This feature makes penetrative convection instrumental in the competition

between vertical overturning in the troposphere (which drives the stratification

toward neutral stability) and radiative transfer in the stratosphere (which drives

the stratification toward positive stability) that controls the mean elevation of

the tropopause. Figure 7.8 displays anomalous temperature, moisture, and

cloud cover for an area in the tropics, all as functions of time. Episodes of

towering moisture (light shading in Fig. 7.8b) and cold high cloud (Fig. 7.8a),

which mark deep cumulus convection, are contemporaneous with anomalously

high tropopause (dashed line). Tropopause temperature during those periods

is also colder than normal (contours) because the positive lapse of temperature

extends upward to heights that are normally stable.

7.4.2 Entrainment

In practice, the vertical velocity and penetration height of a cumulus tower are

reduced from adiabatic values by nonconservative effects. Cooler and drier

environmental air that is entrained into and mixed with a moist thermal (Fig.

5.4) depletes ascending parcels of positive buoyancy and kinetic energy. Only

in the cores of broad convective towers are adiabatic values ever approached.

Mixing also modifies the surroundings of a cumulus tower, which mediates

gradients of temperature and moisture that control buoyancy.

For reasons established in Chapter 9, entrainment of environmental air is

an intrinsic feature of cumulus convection, one that makes the behavior of

ascending parcels diabatic. The impact this process has on convection can

be illustrated by regarding entrained environmental air to become uniformly

mixed with ascending air. Following Holton (1992), consider a parcel of mass

m inside the moist thermal pictured in Fig. 5.4. Through mixing, additional

mass is incorporated into that parcel, which may then be identified with a

fixed percentage of the overall mass crossing a given level. If t* is a conserved

property (e.g., /z = r beneath the LCL), mt,' represents how much of that

property is associated with the parcel under consideration. As before, primes

will distinguish properties of the parcel from those of the environment. After

mass

of environmental air has been mixed into it, the parcel will have

mass

m + dm

and property I*' + d/z'. Conservation of/z then requires

(m + dm)(lz' + d~') = mtz' + dmt,,

(7.16.1)

where

dmtz

represents how much conserved property has been incorporated

into the parcel from the environment. If the foregoing process occurs during

Figure

7.8

Time series of (a) high cloud cover and

(b)

moisture and temperature derived from radiosondes and satellite IR radiances during the

winter

MONEX

observing campaign. Relative humidities exceeding

80%

are lightly shaded. Temperature departures from local time-mean values are

contoured. Dark shading indicates stratospheric air above the local tropopause (heavy dashed line). Adapted from Johnson and Kriete

(1982).

7.4 Finite Displacements 183

the time

dt,

expanding (7.16.1) and neglecting terms higher than first order

leads to

d/_t' d(ln m)

= ~(/x -/x'), (7.16.2)

dt dt

which governs the conserved property inside the entraining thermal. More

generally, if/x is not conserved but, rather, is produced per unit mass at the

rate S~, its evolution inside the aforementioned parcel is governed by

dl.t'

d(ln m)

= ~(u -/~') + Su. (7.17)

dt dt

The time rate of change following the parcel is related to its vertical velocity

w'= dz'/dt

through the chain rule:

d d

dt = w d--z' (7.18)

where

d/dz

refers to a change with respect to the parcel's height. Then (7.17)

becomes

d~' w

w d--z = Hee (/z -/~') + S~,, (7.19)

in which He 1 = d(ln

m)/dz

defines the mass

entrainment height.

Representing

the height for upwelling mass to increase by a factor of e,

H e

reflects the rate

that the entraining thermal expands due to incorporation of environmental

air.'

If/x = In 0e, the source term

Soe

- 0 because 0e is conserved. Then, with

(5.27), (7.19) reduces to

[(o) '(re re)]

d(ln Oe) 1 In + -- T'

~ ~

dz H e Cp T

Replacing saturation values by the actual mixing ratios and temperatures by

those corresponding to the LCL (Sec. 5.4.2) and noting that differences of

temperature are small compared to differences of moisture leads to

d(lnO'e) H el

[ln(~')++ (r-r')]

_ T l _

)]

1 [ln(-~-7)+ (r r' .j

-- H e

~pT

(7.21)

Because r' > r and T' > T, 0' e decreases with the ascending parcel's height.

This is to be contrasted with the parcel's behavior under conservative condi-

tions (e.g., in the absence of entrainment), in which case its mass is fixed and

cumulus cloud expands with height slower that the moist thermal supporting it because

mixing with drier environmental air leads to evaporation of condensate and dissolution of the

cloud along its periphery (Sec. 9.3.3).

184 7

Hydrostatic Stability

0' e = const. The two sinks of 0' e on the right-hand side of (7.21) reflect trans-

fers of latent and sensible heat to the environment, which are mixed across

the parcel's control surface (compare Fig. 2.1).

Suppose now /z equals the specific momentum w. Then w = 0 for the

environment, so (7.19) reduces to

113 2

+ Sw

dK' 2 SK

=-~K' + ~, (7.22)

dz H e w

where K'

w2/2

is the specific kinetic energy of the parcel and production

of momentum and kinetic energy are related as

SK = wSw.

Unlike the source

term for

Oe, SK :/: O.

Even in the absence of entrainment, K is not conserved

because the parcel's kinetic energy changes through work performed on it by

buoyancy. Differentiating (7.12) along with (7.15) implies

dK'

dz = fb

(7.23.1)

under conservative conditions (e.g., for a parcel of fixed mass, wherein

oo). Hence, we identify

Then (7.22) becomes

SK -- Wfb

- wg (T' - T

T )" (7.23.2)

dK' ( T' - T ) 2 K,

(7.24)

dz = g T - -H--~e "

The sink of K' on the right-hand side represents turbulent drag that is ex-

erted on the ascending parcel due to incorporation of momentum from the

environment. By siphoning off kinetic energy, entrainment reduces the par-

cel's acceleration from what it would be under the action of buoyancy alone

and limits its penetration into a stable layer to about an entrainment height

(Problem 7.20).

7.4.3 Potential Instability

Until now, we have considered displacements of individual air parcels within

a fixed layer. Suppose the layer itself is displaced vertically (e.g., in sloping

convection or forced lifting over elevated terrain). Then changes of the layer's

thermodynamic properties alter its stability. Stratification of moisture plays a

key role in this process because different levels need not achieve saturation

simultaneously.

7.4 Finite Displacements

185

Consider an unsaturated layer in which potential temperature increases

with height,

-- > 0, (7.25.1)

but in which equivalent potential temperature decreases with height,

dO e

< 0. (7.25.2)

By (7.8), the layer is stable. The difference 0e - 0 for an individual parcel

reflects the total latent heat available for release. Therefore, (7.25.2) reflects

a decrease with height of mixing ratio (e.g., as would develop over a warm

ocean surface).

Suppose this layer is displaced vertically and changes in its thickness can be

ignored (Fig. 7.9). Since the layer is unsaturated, air parcels along a vertical

section all cool at the dry adiabatic lapse rate F a (e.g., between positions

.... ~ii@~r165 ~~ ~ " ~4~-i~i~iN~i~i

liiiiii i!

| N~i

Nii ~ii

"='\ \

A2 \~,. '~\

Figure 7.9 Successive positions and thermal structures of a potentially unstable layer

(shaded), in which moisture decreases sharply with height, that is displaced vertically. Air is

initially unsaturated, so all levels cool at the dry adiabatic lapse rate. However displacement to

position 2 leads to saturation of air at the layer's base (A2). Thereafter, that air cools at the sat-

urated adiabatic lapse rate, while air at the layer's top (Be) continues to cool at the dry adiabatic

lapse rate.

186

7 Hydrostatic Stability

1 and 2). Hence, the layer's profile of temperature is preserved through the

displacement, as are the layer's lapse rate and stability. Once some of the layer

becomes saturated, this is no longer true. Because they have greater mixing

ratios, lower levels achieve saturation sooner than upper levels. Above their

LCL, lower levels (e.g., at A2) cool slower at the saturated adiabatic lapse

rate F~, whereas upper levels (e.g., at B2) continue to cool at the dry adiabatic

lapse rate F a. Differential cooling between lower and upper levels then swings

the temperature profile counterclockwise and hence destabilizes the layer.

This destabilization follows from the release of latent heat at lower levels,

much as would occur were the layer heated from below. Sufficient vertical

displacement will make the layer unstable with respect to saturated conditions

(e.g., at position 3), even though initially it may have been absolutely stable.

Such a layer is

potentially unstable,

the criterion for which is given by (7.25.2).

A layer characterized by (7.25.2), but with the reversed inequality, is

poten-

tially stable.

Reflecting an increase with height of mixing ratio, this condition

accounts for the release of latent heat in upper levels, which stabilizes the

layer when it becomes saturated. Once a potentially stable or unstable layer

becomes fully saturated, it satisfies the usual criteria for instability because 0e

is conserved and hence (7.25.2) obeys (7.8) with 0e in place of 0.

7.4.4 Modification of Stability under Unsaturated Conditions

A layer's stability can also change under unsaturated conditions through ver-

tical compression and expansion. Consider the layer in Fig. 7.10, which is

displaced to a new elevation and thermodynamic state, as denoted by primed

variables, from an initial elevation and state denoted by unprimed variables.

Because 0 is conserved under unsaturated conditions, the change of 0 across

the layer is preserved:

dO' = dO.

Then the vertical gradient of potential temperature satisfies

dz ] dz' = -~z dz.

(7.26.1)

A horizontal section of layer having initial area

must also satisfy conser-

vation of mass:

pdAdz = p'dA'dz'.

By combining (7.26), we obtain

\dA)

for the new vertical gradient of potential temperature inside the layer.

(7.26.2)

(7.27)

7.4 Finite Displacements 187

! !

dO O

Figure 7.10 Successive positions and thermal structures of a layer that remains unsaturated

and therefore preserves the potential temperature difference across it.

Sinking motion and horizontal divergence of air correspond to

dA'

> 1,

(7.28)

respectively. Each implies vertical compression of the layer: the former through

a reduction of the layer's overall volume and the latter to compensate hor-

izontal expansion of the layer. By (7.27), these motions steepen the vertical

gradient of potential temperature. Consequently, they increase the stability (in-

stability) of a layer that is initially stable (unstable). In anticyclones, sinking or

"subsidence" of stable air and horizontal divergence that compensates it (Sec.

12.3) lead to the formation of a

subsidence inversion,

which caps pollutants

near the ground.

Ascending motion and horizontal convergence imply (7.28) with the re-

versed inequalities. Vertical expansion then weakens the vertical gradient of

potential temperature and therefore drives a stable layer toward neutral sta-

188 7 Hydrostatic Stability

bility. Ascending motion in cyclones and horizontal convergence compensating

it can virtually eliminate the stability of a layer, providing conditions favorable

for convection.

7.5 Stabilizing and Destabilizing Influences

The preceding development illustrates that stability can be altered by inter-

nal motions that rearrange air. More important to the overall stability of the

atmosphere is external heat transfer, which shapes thermal structure. The ver-

tical distribution of 0 provides insight into how stability is established because

it reflects the effective stratification of mass.

In light of the first law (2.36), the stability criteria (7.8) imply that a layer

is destabilized by heating from below or cooling from aloft (Fig. 7.11a). Each

rotates the vertical profile of 0 counterclockwise and thus toward instability.

These are precisely the influences exerted on the troposphere by exchanges

of energy with the earth's surface and deep space. Absorption of longwave

(LW) radiation and transfers of latent and sensible heat from the Earth's sur-

face warm the troposphere from below, whereas LW emission at upper levels

cools the troposphere from above. Both drive the vertical profile of 0 coun-

terclockwise and the lapse rate toward superadiabatic values. By destabilizing

the stratification, these forms of heat transfer sustain convective overturning,

which in turn maintains the troposphere close to moist neutral stability and

makes its general circulation behave as a heat engine (Fig. 6.7).

Conversely, cooling from below or heating from above stabilize a layer

(Fig. 7.11b). Each rotates the profile of 0 clockwise and thus toward increased

(a) Destabilizing (b) Stabilizing

,~ q<O q>O

I- /

I I /

7jfT~

q>O q<O

0 0

Figure 7.11 (a) Destabilizing influences: Heating from below and cooling from above act to

swing the profile of potential temperature counterclockwise. (b) Stabilizing influences: Cooling

from below and heating from above act to swing the profile of potential temperature clockwise.

7.6 Turbulent Dispersion 189

stability. These forms of heat transfer are symbolic of the stratosphere, where

ozone heating increases with height to stabilize the thermal structure. By

inhibiting vertical motion, strong stability suppresses turbulent dispersion and

allows constituents to become highly stratified. Work must then be performed

against buoyancy to drive vertical motion, which makes the general circulation

of the stratosphere behave as a refrigerator (Fig. 6.8).

According to Secs. 7.2 and 7.4, moisture also figures in the stability of a

layer. If the lapse rate is conditionally unstable (7.9.3), increasing a layer's

mixing ratio lowers the LCL and hence the LFC of individual parcels, which

in turn increase CAPE to drive deep convection (compare Fig. 7.6). Likewise,

increasing r in lower portions of the layer (e.g., through absorption of water

vapor from a warm ocean surface) increases the equivalent potential temper-

ature there, which swings the profile of 0e counterclockwise and drives

dOe/dz

toward negative values. Thus, introducing moisture from below, such as oc-

curs over warm ocean surfaces and reflects a transfer of latent heat across

the layer's lower boundary, drives the tropical troposphere toward potential

instability.

For conditional and potential instability, finite displacements of air~

whether they be of an individual air parcel or of a layer

in toto,

eventually

result in buoyantly driven convection. However, for each, a finite potential

well must be overcome before the instability can be released. Due to the

source of water vapor at its base, lower portions of the tropical troposphere

are potentially unstable. Figure 7.12 shows mean distributions of 0e and N 2

over the equatorial western Pacific. In the lowest 3 km, 0e decreases with

height, whereas N 2 > 0 implies that 0 increases with height. Despite such

instability, deep convection breaks out in preferred regions, where forced

lifting is prevalent (Fig. 1.25). Over the Indian Ocean and western Pacific,

equatorward-moving air driven by thermal contrasts between sea and neigh-

boring continents converges horizontally to produce forced ascent. Similarly,

diurnal heating of tropical landmasses imparts enough buoyancy for sur-

face air over the maritime continent, South America, and tropical Africa to

overcome the potential well.

7.6 Turbulent Dispersion

7.6.1 Convective Mixing

Hydrostatic stability controls a layer's ability to support vertical motion and

hence mixing by 3-D turbulence. Regions of weak or negative stability favor

convective overturning, which results in thorough mixing of air. This situation

is common near the ground because absorption of shortwave (SW) radiation

destabilizes the surface layer. Introducing moisture from below has a similar

effect by increasing the equivalent potential temperature of surface air, which