Moist Convection

We have seen that the atmosphere in most places and at most times is stable to dry convection. Nevertheless, convection is common in most locations over the globe (more so in some locations than in others, as we will discuss in Section 4.6.2). There is

FIGURE 4.17. (Top) A satellite image showing dense haze associated with pollution over eastern China. The view looks eastward across the Yellow Sea toward Korea. Provided by the SeaWiFS Project, NASA/Goddard Space Flight Center. (Bottom) Temperature inversions in Los Angeles often trap the pollutants from automobile exhaust and other pollution sources near the ground.

FIGURE 4.17. (Top) A satellite image showing dense haze associated with pollution over eastern China. The view looks eastward across the Yellow Sea toward Korea. Provided by the SeaWiFS Project, NASA/Goddard Space Flight Center. (Bottom) Temperature inversions in Los Angeles often trap the pollutants from automobile exhaust and other pollution sources near the ground.

one very important property of air that we have not yet incorporated into our discussion of convection. Air is moist, and if a moist air parcel is lifted, it cools adiabatically; if this cooling is enough to saturate the parcel, some water vapor condenses to form a cloud. The corresponding latent heat release adds buoyancy to the parcel, thus favoring instability. This kind of convection is called moist convection. To derive a stability condition for moist convection, we must first discuss how to describe the moisture content of air.

4.5.1. Humidity

The moisture content of air is conveniently expressed in terms of humidity. The specific humidity, q, is a measure of the mass of water vapor to the mass of air per unit volume defined thus:

where p = pd + pT (see Section 1.3.2) is the total mass of air (dry air plus water vapor) per unit volume. Note that in the absence of mixing or of condensation, specific humidity is conserved by a moving air parcel, since the masses of both water and air within the parcel must be separately conserved.

The saturation-specific humidity, q*, is the specific humidity at which saturation occurs. Since both water vapor and dry air behave as perfect gases, using Eq. 1-2 to express Eq. 4-23 at saturation, we define q* thus:

es/RvT p/RT

where es (T) is the saturated partial pressure of water vapor plotted in Fig. 1.5. Note that q* is a function of temperature and pressure. In particular, at fixed p it is a strongly increasing function of T.

Relative humidity, U, is the ratio of the specific humidity to the saturation specific humidity, q*, often expressed as a percentage thus:

Now near the Earth's surface the moisture content of air is usually fairly close to saturation (e.g., throughout the lower tropical atmosphere, the relative humidity of air is close to 80%, as will be seen in Chapter 5). If such an air parcel is lifted, the pressure will decrease and it will cool. From Eq. 4-24, decreasing pressure alone would make q* increase with altitude. However, the exponential dependence of es on T discussed in Chapter 1 overwhelms the pressure dependence, and consequently q* decreases rapidly with altitude. So as the air parcel is lifted, conserving its q, it does not usually have to rise very far before q > q*.

The level at which this occurs is called the condensation level zc (Fig. 4.18). At and above zc, excess vapor will condense so that q = q*. Moreover, since q* will continue to decrease as the parcel is lifted further, q will decrease correspondingly. Such condensation is visible, for example, as convective clouds. As the vapor condenses, latent heat release partly offsets the cooling due to adiabatic expansion. Thus we expect the moist parcel to be more buoyant than if it were dry. As illustrated in Fig. 4.18, above zc the parcel's temperature falls off more slowly (contrast with the dry convection case, Fig. 4.11) until neutral buoyancy is reached at zt, the cloud top. Clearly, the warmer or moister the surface air, the higher the cloud top will be.

Below the condensation level we expect a parcel undergoing convection to follow a dry adiabat. But how does its temperature change in the saturated layer above? It follows a saturated adiabat, as we now describe.

FIGURE 4.18. The temperature of a moist air parcel lifted in convection from the surface at temperature Ts will follow a dry adiabat until condensation occurs at the condensation level zc. Above zc, excess vapor will condense, releasing latent heat and warming the parcel, offsetting its cooling at the dry adiabatic rate due to expansion. Thus a moist parcel cools less rapidly (following a moist adiabat) than a dry one, until neutral buoyancy is reached at zt, the cloud top. This should be compared to the case of dry convection shown in Fig. 4.11.

FIGURE 4.18. The temperature of a moist air parcel lifted in convection from the surface at temperature Ts will follow a dry adiabat until condensation occurs at the condensation level zc. Above zc, excess vapor will condense, releasing latent heat and warming the parcel, offsetting its cooling at the dry adiabatic rate due to expansion. Thus a moist parcel cools less rapidly (following a moist adiabat) than a dry one, until neutral buoyancy is reached at zt, the cloud top. This should be compared to the case of dry convection shown in Fig. 4.11.

4.5.2. Saturated adiabatic lapse rate

Let us return to the case of a small vertical displacement of an air parcel. If the air is unsaturated, no condensation will occur, and so the results of Section 4.3.1 for dry air remain valid. However, if condensation does occur, there will be a release of latent heat in the amount SQ = -L dq per unit mass of air, where L is the latent heat of condensation and dq is the change in specific humidity q. Thus we must modify Eq. 4-13 to dp cpdT = p - Ldq (4-26)

for an air parcel undergoing moist adiabatic displacement. Note that there is a minus sign here because if dq < 0, latent heat is released and the parcel warms. If the environment is in hydrostatic balance, dp/p = -gdz, and so d (cpT + gz + Lq) = 0. (4-27)

The term in parentheses is known as the moist static energy and comprises cpT + gz, the dry static energy, and Lq, the latent heat content.

If the air parcel is always at saturation, we can replace q by q* in Eq. 4-26. Now, since q* = q* (p,T), dq* dq*

Writing dp/p

dq* =/ _R\ 1 d£1 = / _R\ pes = q dT ^ RjpdT V RJ p '

where we have used Eq. 1-4 to write des/dT = pes. Hence Eq. 4-26 gives setting dq = dq* ,

where rs is known as the saturated adiabatic lapse rate.6 The factor in brackets on the right is always less than unity, so the saturated adiabatic lapse rate is less than the dry adi-abatic lapse rate; at high altitudes, however, q* is small and the difference becomes very small. Since q* varies with p and T, one cannot ascribe a single number to rs. It has typical tropospheric values ranging between rs ~ 3Kkm—1 in the moist, tropical lower troposphere and rs ~ rd _ 10Kkm—1 in the upper troposphere. A typical atmospheric temperature profile is sketched, along with dry and saturated adiabats, in Fig. 4.19.

The qualitative impact of condensation is straightforward; the release of latent heat makes the air parcel warmer and therefore more buoyant, and so the atmosphere is

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