**3.3 Thermodynamics of the vertical motion**

The saturation mixing ratio depends exponentially on temperature. Hence, due to the decrease of ambient temperature with height, the saturation mixing ratio sharply decreases with height as well.

Therefore the water pressure of an ascending moist parcel, despite the decrease of its temperature at the dry adiabatic lapse rate, sooner or later will reach its saturation value at a level named *lifting condensation level* (LCL), above which further lifting may produce condensation and release of latent heat. This internal heating slows the rate of cooling of the air parcel upon further lifting.

If the condensed water stays in the parcel, and heat transfer with the environment is negligible, the process can be considered reversible – that is, the heat internally added by condensation could be subtracted by evaporation if the parcel starts descending - hence the behaviour can still be considered adiabatic and we will term it a *saturated adiabatic process*. If otherwise the condensate is removed, as instance by sedimentation or precipitation, the process cannot be considered strictly adiabatic. However, the amount of heat at play in the condensation process is often negligible compared to the internal energy of the air parcel and the process can still be considered well approximated by a saturated adiabat, although it should be more properly termed a *pseudoadiabatic* process.

Fig. 3. Vertical profiles of mixing ratio r and saturated mixing ratio rs for an ascending air parcel below and above the lifting condensation level. (source: Salby M. L., Fundamentals of Atmospheric Physics, Academic Press, New York.)

#### **3.3.1 Pseudoadiabatic lapse rate**

If within an air parcel of unit mass, water vapour condenses at a saturation mixing ratio *rs*, the amount of latent heat released during the process will be *-Lwdrs*. This can be put into (34) to get:

$$-L\_{\text{w}}d\tau\_{\text{s}} = c\_{p}dT + gdz\tag{40}$$

Atmospheric Thermodynamics 61

Fig. 4. Thick solid line represent the environment temperature profile. Thin solid line represent the temperature of an ascending parcel initially at point A. Dotted area represent

The thick solid line represent the environment temperature profile. A moist air parcel initially at rest at point A is lifted and cools at the adiabatic lapse rate *d* along the thin solid line until it eventually get saturated at the Lifting Condensation Level at point D. During this lifting, it gets colder than the environment. Upon further lifting, it cools at a slower rate at the pseudoadiabatic lapse rate *s* along the thin dashed line until it reaches the Level of Free Convection at point C, where it attains the temperature of the environment. If it gets beyond that point, it will be warmer, hence lighter than the environment and will experience a positive buoyancy force. This buoyancy will sustain the ascent of the air parcel until all vapour condenses or until its temperature crosses again the profile of environmental temperature at the *Level of Neutral Buoyancy (LNB).* Actually, since the air parcel gets there with a positive vertical velocity, this level may be surpassed and the air parcel may overshoot into a region where it experiences negative buoyancy, to eventually get mixed there or splash back to the LNB. In practice, entrainment of environmental air into the ascending air parcel often occurs, mitigates the buoyant forces, and the parcel generally

If we neglect such entrainment effects and consider the motion as adiabatic, the buoyancy force is conservative and we can define a potential. Let ρ and ρ' be respectively the environment and air parcel density. From Archimede's principle, the buoyancy force on a unit mass parcel can be expressed as in (29), and the increment of potential energy for a

� � ��� � �(�� � �)����� (44)

����� � ���(�) (45)

CIN, shaded area represent CAPE.

reaches below the LNB.

displacement *δz* will then be, by using (1) and (8):

Which can be integrated from a reference level *p0* to give:

�� � �� � �����

��(�) � �� � (�� � �) � ��

Dividing by *cpdz* and rearranging terms, we get the expression of the *saturated adiabatic lapse rate s*:

$$\Gamma\_{\rm S} = -\frac{dT}{dz} = \frac{\Gamma\_d}{\left(1 + \left(\frac{L\_W}{c\_p}\right)\left(\frac{dx\_S}{dx}\right)\right)}\tag{41}$$

Whose value depends on pressure and temperature and which is always smaller than *d*, as should be expected since a saturated air parcel, since condensation releases latent heat, cools more slowly upon lifting.

#### **3.3.2 Equivalent potential temperature**

If we pose *δq = - Lwdrs* in (22) we get:

$$\frac{d\theta}{d\theta} = -\frac{L\_{\text{w}} dr\_{\text{s}}}{c\_{\text{p}}r} \simeq -d\left(\frac{L\_{\text{w}}r\_{\text{s}}}{c\_{\text{p}}r}\right) \tag{42}$$

The approximate equality holds since *dT/T << drs/rs* and *Lw/cp* is approximately independent of T. So (41) can be integrated to yield:

$$\theta\_e = \theta \exp\left(\frac{L\_w r\_s}{c\_p T}\right) \tag{43}$$

That defines the *equivalent potential temperature θe* (Bolton, 1990) which is constant along a pseudoadiabatic process, since during the condensation the reduction of *rs* and the increase of *θ* act to compensate each other.

### **3.4 Stability for saturated air**

We have seen for the case of dry air that if the environment lapse rate is smaller than the adiabatic one, the atmosphere is stable: a restoring force exist for infinitesimal displacement of an air parcel. The presence of moisture and the possibility of latent heat release upon condensation complicates the description of stability.

If the air is saturated, it will cool upon lifting at the smaller saturated lapse rate *s* so in an environment of lapse rate , for the saturated air parcel the cases  *< s , = s , > <sup>s</sup>* discriminates the absolutely stable, neutral and unstable conditions respectively. An interesting case occurs when the environmental lapse rate lies between the dry adiabatic and the saturated adiabatic, that is *s < < d*. In such a case, a moist unsaturated air parcel can be lifted high enough to become saturated, since the decrease in its temperature due to adiabatic cooling is offset by the faster decrease in water vapour saturation pressure, and starts condensation at the LCL. Upon further lifting, the air parcel eventually get warmer than its environment at a level termed *Level of Free Convection (LFC)* above which it will develop a positive buoyancy fuelled by the continuous release of latent heat due to condensation, as long as there is vapour to condense. This situation of *conditional instability* is most common in the atmosphere, especially in the Tropics, where a forced finite uplifting of moist air may eventually lead to spontaneous convection. Let us refer to figure 4 and follow such process more closely. In the figure, which is one of the meteograms discussed later in the chapter, pressure decreases vertically, while lines of constant temperature are tilted 45° rightward, temperature decreasing going up and to the left.

Dividing by *cpdz* and rearranging terms, we get the expression of the *saturated adiabatic lapse* 

�� <sup>=</sup> �� ���� �� �� ����� �� ��

Whose value depends on pressure and temperature and which is always smaller than *d*, as should be expected since a saturated air parcel, since condensation releases latent heat, cools

��� � −� �����

The approximate equality holds since *dT/T << drs/rs* and *Lw/cp* is approximately independent

�� = ����� �����

That defines the *equivalent potential temperature θe* (Bolton, 1990) which is constant along a pseudoadiabatic process, since during the condensation the reduction of *rs* and the increase

We have seen for the case of dry air that if the environment lapse rate is smaller than the adiabatic one, the atmosphere is stable: a restoring force exist for infinitesimal displacement of an air parcel. The presence of moisture and the possibility of latent heat release upon

If the air is saturated, it will cool upon lifting at the smaller saturated lapse rate *s* so in an environment of lapse rate , for the saturated air parcel the cases  *< s , = s , > <sup>s</sup>* discriminates the absolutely stable, neutral and unstable conditions respectively. An interesting case occurs when the environmental lapse rate lies between the dry adiabatic and the saturated adiabatic, that is *s < < d*. In such a case, a moist unsaturated air parcel can be lifted high enough to become saturated, since the decrease in its temperature due to adiabatic cooling is offset by the faster decrease in water vapour saturation pressure, and starts condensation at the LCL. Upon further lifting, the air parcel eventually get warmer than its environment at a level termed *Level of Free Convection (LFC)* above which it will develop a positive buoyancy fuelled by the continuous release of latent heat due to condensation, as long as there is vapour to condense. This situation of *conditional instability* is most common in the atmosphere, especially in the Tropics, where a forced finite uplifting of moist air may eventually lead to spontaneous convection. Let us refer to figure 4 and follow such process more closely. In the figure, which is one of the meteograms discussed later in the chapter, pressure decreases vertically, while lines of constant temperature are

Γ� = − ��

��

� = − �����

*rate s*:

more slowly upon lifting.

**3.3.2 Equivalent potential temperature**  If we pose *δq = - Lwdrs* in (22) we get:

of T. So (41) can be integrated to yield:

of *θ* act to compensate each other.

condensation complicates the description of stability.

tilted 45° rightward, temperature decreasing going up and to the left.

**3.4 Stability for saturated air** 

−����� = ���� � ��� (40)

��� � (42)

��� � (43)

(41)

Fig. 4. Thick solid line represent the environment temperature profile. Thin solid line represent the temperature of an ascending parcel initially at point A. Dotted area represent CIN, shaded area represent CAPE.

The thick solid line represent the environment temperature profile. A moist air parcel initially at rest at point A is lifted and cools at the adiabatic lapse rate *d* along the thin solid line until it eventually get saturated at the Lifting Condensation Level at point D. During this lifting, it gets colder than the environment. Upon further lifting, it cools at a slower rate at the pseudoadiabatic lapse rate *s* along the thin dashed line until it reaches the Level of Free Convection at point C, where it attains the temperature of the environment. If it gets beyond that point, it will be warmer, hence lighter than the environment and will experience a positive buoyancy force. This buoyancy will sustain the ascent of the air parcel until all vapour condenses or until its temperature crosses again the profile of environmental temperature at the *Level of Neutral Buoyancy (LNB).* Actually, since the air parcel gets there with a positive vertical velocity, this level may be surpassed and the air parcel may overshoot into a region where it experiences negative buoyancy, to eventually get mixed there or splash back to the LNB. In practice, entrainment of environmental air into the ascending air parcel often occurs, mitigates the buoyant forces, and the parcel generally reaches below the LNB.

If we neglect such entrainment effects and consider the motion as adiabatic, the buoyancy force is conservative and we can define a potential. Let ρ and ρ' be respectively the environment and air parcel density. From Archimede's principle, the buoyancy force on a unit mass parcel can be expressed as in (29), and the increment of potential energy for a displacement *δz* will then be, by using (1) and (8):

$$dP = f\_b = \left(\frac{T'-T}{\tau}\right) g \delta \mathbf{z} = R(T'-T) dlog p \tag{44}$$

Which can be integrated from a reference level *p0* to give:

$$dP(p) = -R \int\_{p\_0}^{p} (T' - T) \, d\log p = -RA(p) \tag{45}$$

Atmospheric Thermodynamics 63

An *emagram* is basically a *T-z* plot where the vertical axis is *log p* instead of height *z*. But since *log p* is linearly related to height in a dry, isothermal atmosphere, the vertical

A *skew T- log p diagram*, like the emagram, has *log p* as vertical coordinate, but the isotherms are slanted. *Tephigrams* look very similar to skew T diagrams if rotated by 45°, have *T* as horizontal and *log θ* as vertical coordinates so that isotherms are vertical and the isentropes horizontal (hence tephi, a contraction of *T* and *Φ*, where *Φ = cp log θ* stands for the entropy). Often, tephigrams are rotated by 45° so that the vertical axis corresponds to the vertical in

A tephigram is shown in figure 5: straight lines are isotherms (slope up and to the right) and isentropes (up and to the left), isobars (lines of constant p) are quasi-horizontal lines, the

Fig. 5. A tephigram. Starting from the surface, the red line depicts the evolution of the Dew Point temperature, the black line depicts the evolution of the air parcel temperature, upon uplifting. The two lines intersects at the LCL. The orange line depicts the saturated adiabat crossing the LCL point, that defines the wet bulb temperature at the ground pressure

Two lines are commonly plotted on a tephigram – the temperature and dew point, so the state of an air parcel at a given pressure is defined by its temperature *T* and *Td*, that is its water vapour content. We note that the knowledge of these parameters allows to retrieve all the other humidity parameters: from the dew point and pressure we get the humidity mixing ratio w; from the temperature and pressure we get the saturated mixing ratio ws, and relative humidity may be derived from 100\*w/ws, when w and ws are measured at the

When the air parcel is lifted, its temperature *T* follows the dry adiabatic lapse rate and its dew point *Td* its constant vapour mixing ratio line - since the mixing ratio is conserved in

dashed lines sloping up and to the right are constant mixing ratio in g/kg, while the curved solid bold lines sloping up and to the left are saturated adiabats.

*)* and the horizontal coordinate is *T*: with

coordinate is basically the geometric height. In the *Stüve diagram* the vertical coordinate is *p(Rd/cp*

the atmosphere.

surface.

same pressure.

this axes choice, the dry adiabats are straight lines.

Referring to fig. 4, *A(p)* represent the shaded area between the environment and the air parcel temperature profiles. An air parcel initially in A is bound inside a "potential energy well" whose depth is proportional to the dotted area, and that is termed *Convective Inhibition (CIN)*. If forcedly raised to the level of free convection, it can ascent freely, with an available potential energy given by the shaded area, termed *CAPE (Convective Available Potential Energy)*.

In absence of entrainment and frictional effects, all this potential energy will be converted into kinetic energy, which will be maximum at the level of neutral buoyancy. CIN and CAPE are measured in J/Kg and are indices of the atmospheric instability. The CAPE is the maximum energy which can be released during the ascent of a parcel from its free buoyant level to the top of the cloud. It measures the intensity of deep convection, the greater the CAPE, the more vigorous the convection. Thunderstorms require large CAPE of more than 1000 Jkg-1.

CIN measures the amount of energy required to overcome the negatively buoyant energy the environment exerts on the air parcel, the smaller, the more unstable the atmosphere, and the easier to develop convection. So, in general, convection develops when CIN is small and CAPE is large. We want to stress that some CIN is needed to build-up enough CAPE to eventually fuel the convection, and some mechanical forcing is needed to overcome CIN. This can be provided by cold front approaching, flow over obstacles, sea breeze.

CAPE is weaker for maritime than for continental tropical convection, but the onset of convection is easier in the maritime case due to smaller CIN.

We have neglected entrainment of environment air, and detrainment from the air parcel , which generally tend to slow down convection. However, the parcels reaching the highest altitude are generally coming from the region below the cloud without being too much diluted.

Convectively generated clouds are not the only type of clouds. Low level stratiform clouds and high altitude cirrus are a large part of cloud cover and play an important role in the Earth radiative budget. However convection is responsible of the strongest precipitations, especially in the Tropics, and hence of most of atmospheric heating by latent heat transfer.

So far we have discussed the stability behaviour for a single air parcel. There may be the case that although the air parcel is stable within its layer, the layer as a whole may be destabilized if lifted. Such case happen when a strong vertical stratification of water vapour is present, so that the lower levels of the layer are much moister than the upper ones. If the layer is lifted, its lower levels will reach saturation before the uppermost ones, and start cooling at the slower pseudoadiabat rate, while the upper layers will still cool at the faster adiabatic rate. Hence, the top part of the layer cools much more rapidly of the bottom part and the lapse rate of the layer becomes unstable. This *potential* (or *convective*) *instability* is frequently encountered in the lower leves in the Tropics, where there is a strong water vapour vertical gradient.

It can be shown that condition for a layer to be potentially unstable is that its equivalent potential temperature *θe* decreases within the layer.
