**2. Dry air thermodynamics and stability**

We know from experience that pressure, volume and temperature of any homogeneous substance are connected by an *equation of state*. These physical variables, for all gases over a wide range of conditions in the so called *perfect gas approximation*, are connected by an equation of the form:

$$p\,V \equiv mRT\tag{1}$$

Atmospheric Thermodynamics 51

millionths, it is preferable to treat it separately from other air components, and consider the atmosphere as a mixture of dry gases and water. In order to use a state equation of the form (1) for moist air, we express a specific gas constant *Rd* by considering in (5) all gases but water, and use in the state equation a *virtual temperature Tv* defined as the temperature that dry air must have in order to have the same density of moist air at the same pressure. It can

> �� <sup>=</sup> � ��� ������ �� �

Where *Mw* and *Md* are respectively the water and dry air molecular weights. *Tv* takes into account the smaller density of moist air, and so is always greater than the actual

The atmosphere is under the action of a gravitational field, so at any given level the downward force per unit area is due to the weight of all the air above. Although the air is permanently in motion, we can often assume that the upward force acting on a slab of air at any level, equals the downward gravitational force. This *hydrostatic balance* approximation is valid under all but the most extreme meteorological conditions, since the vertical acceleration of air parcels is generally much smaller than the gravitational one. Consider an

−�� = ���� (7)

�(�) <sup>=</sup> � ���� �

Two useful concepts in atmospheric thermodynamic are the geopotential , an exact differential defined as the work done against the gravitational field to raise 1 kg from 0 to *z*,

We can make use of (1) and of the definition of virtual temperature to rewrite (10) and formally integrate it between two levels to formally obtain the geopotential thickness of a

� ��� �

*z*, of unit horizontal surface. If

�� = −�� (8)

� (9)

� (10)

*z*. We consider as negative, since we know that pressure

*/g0*, where g0 is a mean gravitational acceleration taken as

is the air density at

*(0)=0*,

*z*. Let *p* be the pressure at *z*,

decreases with height. The hydrostatic balance of forces along the vertical leads to:

<sup>−</sup> ��

As we know that *p(∞)*=0, (9) can be integrated if the air density profile is known.

where the 0 level is often taken at sea level and, to set the constant of integration,

�(�) <sup>=</sup> � ��

Hence, in the limit of infinitesimal thickness, the *hypsometric equation* holds:

*z*, the downward force acting on this slab due to gravity is *g*

Values of *z* and *Z* often differ by not more than some tens of metres.

(6)

be shown that

**2.1 Stratification** 

and *p+*

leading to:

9,81 m/s.

layer, as:

temperature, although often only by few degrees.

horizontal slab of air between *z* and *z +*

*p* the pressure at *z+*

and the *geopotential height Z=*

We can rewrite (9) as:

where *p* is pressure (Pa), *V* is volume (m3), *m* is mass (kg), *T* is temperature (K) and *R* is the *specific gas constant*, whose value depends on the gas. If we express the amount of substance in terms of number of moles *n=m/M* where *M* is the gas molecular weight, we can rewrite (1) as:

$$p\,V \equiv nR\,^\ast T\tag{2}$$

where *R\** is the *universal gas costant*, whose value is 8.3143 J mol-1 K-1. In the kinetic theory of gases, the perfect gas is modelled as a collection of rigid spheres randomly moving and bouncing between each other, with no common interaction apart from these mutual shocks. This lack of reciprocal interaction leads to derive the internal energy of the gas, that is the sum of all the kinetic energies of the rigid spheres, as proportional to its temperature. A second consequence is that for a mixture of different gases we can define, for each component *i* , a partial pressure *pi* as the pressure that it would have if it was alone, at the same temperature and occupying the same volume. Similarly we can define the partial volume *Vi* as that occupied by the same mass at the same pressure and temperature, holding *Dalton's law* for a mixture of gases *i*:

$$p = \sum p i \tag{3}$$

Where for each gas it holds:

$$p\text{i}V \equiv \text{mi}R \,\text{\*}T \tag{4}$$

We can still make use of (1) for a mixture of gases, provided we compute a specific gas constant *R* as:

$$
\bar{R} = \frac{\sum m\_{\bar{l}} R\_{\bar{l}}}{m} \tag{5}
$$

The atmosphere is composed by a mixture of gases, water substance in any of its three physical states and solid or liquid suspended particles (aerosol). The main components of dry atmospheric air are listed in Table 1.


Table 1. Main component of dry atmospheric air.

The composition of air is constant up to about 100 km, while higher up molecular diffusion dominates over turbulent mixing, and the percentage of lighter gases increases with height. For the pivotal role water substance plays in weather and climate, and for the extreme variability of its presence in the atmosphere, with abundances ranging from few percents to

wide range of conditions in the so called *perfect gas approximation*, are connected by an

 *pV=mRT* (1) where *p* is pressure (Pa), *V* is volume (m3), *m* is mass (kg), *T* is temperature (K) and *R* is the *specific gas constant*, whose value depends on the gas. If we express the amount of substance in terms of number of moles *n=m/M* where *M* is the gas molecular weight, we can rewrite (1)

 *pV=nR\*T* (2) where *R\** is the *universal gas costant*, whose value is 8.3143 J mol-1 K-1. In the kinetic theory of gases, the perfect gas is modelled as a collection of rigid spheres randomly moving and bouncing between each other, with no common interaction apart from these mutual shocks. This lack of reciprocal interaction leads to derive the internal energy of the gas, that is the sum of all the kinetic energies of the rigid spheres, as proportional to its temperature. A second consequence is that for a mixture of different gases we can define, for each component *i* , a partial pressure *pi* as the pressure that it would have if it was alone, at the same temperature and occupying the same volume. Similarly we can define the partial volume *Vi* as that occupied by the same mass at the same pressure and temperature, holding

 *p=∑ pi* (3)

 *piV=niR\*T* (4) We can still make use of (1) for a mixture of gases, provided we compute a specific gas

�� <sup>=</sup> <sup>∑</sup> ����

The atmosphere is composed by a mixture of gases, water substance in any of its three physical states and solid or liquid suspended particles (aerosol). The main components of

Nitrogen (N2) 0.7809 0.7552 296.80 Oxygen (O2) 0.2095 0.2315 259.83 Argon (Ar) 0.0093 0.0128 208.13 Carbon dioxide (CO2) 0.0003 0.0005 188.92

The composition of air is constant up to about 100 km, while higher up molecular diffusion dominates over turbulent mixing, and the percentage of lighter gases increases with height. For the pivotal role water substance plays in weather and climate, and for the extreme variability of its presence in the atmosphere, with abundances ranging from few percents to

Gas Molar fraction Mass fraction Specific gas constant

� (5)

(J Kg-1 K-1)

equation of the form:

*Dalton's law* for a mixture of gases *i*:

dry atmospheric air are listed in Table 1.

Table 1. Main component of dry atmospheric air.

Where for each gas it holds:

constant *R* as:

as:

millionths, it is preferable to treat it separately from other air components, and consider the atmosphere as a mixture of dry gases and water. In order to use a state equation of the form (1) for moist air, we express a specific gas constant *Rd* by considering in (5) all gases but water, and use in the state equation a *virtual temperature Tv* defined as the temperature that dry air must have in order to have the same density of moist air at the same pressure. It can be shown that

$$T\_{\nu} = \frac{\tau}{1 - \frac{e}{p} \left(1 - \frac{M\_W}{M\_d}\right)}\tag{6}$$

Where *Mw* and *Md* are respectively the water and dry air molecular weights. *Tv* takes into account the smaller density of moist air, and so is always greater than the actual temperature, although often only by few degrees.

### **2.1 Stratification**

The atmosphere is under the action of a gravitational field, so at any given level the downward force per unit area is due to the weight of all the air above. Although the air is permanently in motion, we can often assume that the upward force acting on a slab of air at any level, equals the downward gravitational force. This *hydrostatic balance* approximation is valid under all but the most extreme meteorological conditions, since the vertical acceleration of air parcels is generally much smaller than the gravitational one. Consider an horizontal slab of air between *z* and *z +z*, of unit horizontal surface. If is the air density at *z*, the downward force acting on this slab due to gravity is *gz*. Let *p* be the pressure at *z*, and *p+p* the pressure at *z+z*. We consider as negative, since we know that pressure decreases with height. The hydrostatic balance of forces along the vertical leads to:

$$-\delta p = g\rho\delta\mathbf{z}\tag{7}$$

Hence, in the limit of infinitesimal thickness, the *hypsometric equation* holds:

$$-\frac{\partial p}{\partial x} = -g\rho$$

$$(8)$$

leading to:

$$p(\mathbf{z}) = \int\_{\mathbf{z}}^{\infty} g\rho \, d\mathbf{z} \tag{9}$$

As we know that *p(∞)*=0, (9) can be integrated if the air density profile is known. Two useful concepts in atmospheric thermodynamic are the geopotential , an exact

differential defined as the work done against the gravitational field to raise 1 kg from 0 to *z*, where the 0 level is often taken at sea level and, to set the constant of integration, *(0)=0*, and the *geopotential height Z=/g0*, where g0 is a mean gravitational acceleration taken as 9,81 m/s.

We can rewrite (9) as:

$$Z(\mathbf{z}) = \frac{1}{g\_0} \int\_{\mathbf{z}}^{\infty} g d\mathbf{z} \tag{10}$$

Values of *z* and *Z* often differ by not more than some tens of metres.

We can make use of (1) and of the definition of virtual temperature to rewrite (10) and formally integrate it between two levels to formally obtain the geopotential thickness of a layer, as:

$$
\Delta Z = \frac{R\_d}{g\_0} \int\_{p\_1}^{p\_2} T\_\nu \frac{dp}{p} \tag{11}
$$

Atmospheric Thermodynamics 53

A system is *open* if it can exchange matter with its surroundings, *closed* otherwise. In atmospheric thermodynamics, the concept of "air parcel" is often used. It is a good approximation to consider the air parcel as a closed system, since significant mass exchanges between airmasses happen predominantly in the few hundreds of metres close to the surface, the so-called *planetary boundary layer* where mixing is enhanced, and can be neglected elsewhere. An air parcel can exchange energy with its surrounding by work of expansion or contraction, or by exchanging heat. An *isolated* system is unable to exchange energy in the form of heat or work with its surroundings, or with any other system. The first principle of thermodynamics states that the *internal energy U* of a closed system, the kinetic and potential energy of its components, is a state variable, depending only on the present state of the system, and not by its past. If a system evolves without exchanging any heat with its surroundings, it is said to perform an *adiabatic* transformation. An air parcel can exchange heat with its surroundings through diffusion or thermal conduction or radiative heating or cooling; moreover, evaporation or condensation of water and subsequent removal of the condensate promote an exchange of latent heat. It is clear that processes which are not adiabatic ultimately lead the atmospheric behaviours. However, for timescales of motion shorter than one day, and disregarding cloud processes, it is often a

For adiabatic processes, the first law of thermodynamics, written in two alternative forms:

holds for *δq=0*, where *cp* and *cv* are respectively the specific heats at constant pressure and constant volume, *p* and *v* are the specific pressure and volume, and *δq* is the heat exchanged with the surroundings. Integrating (13) and (14) and making use of the ideal gas state

 *Tvγ-1 = constant* (15)

 *Tp-κ = constant* (16)

 *pv<sup>γ</sup> = constant* (17) where *γ=cp/cv* =1.4 and κ*=(γ-1)/γ* =*R/cp* ≈ 0.286, using a result of the kinetic theory for diatomic gases. We can use (16) to define a new state variable that is conserved during an adiabatic process, the *potential temperature θ*, which is the temperature the air parcel would attain if compressed, or expanded, adiabatically to a reference pressure *p0*, taken for

> ߠൌܶቀబ ቁ

Since the time scale of heat transfers, away from the planetary boundary layer and from clouds is several days, and the timescale needed for an air parcel to adjust to environmental pressure changes is much shorter, *θ* can be considered conserved along the air motion for one week or more. The distribution of *θ* in the atmosphere is determined by the pressure

*cvdT + pdv=δq* (13)

*cpdT - vdp= δq* (14)

(18)

**2.2 Thermodynamic of dry air** 

**2.2.1 Potential temperature** 

convention as 1000 hPa.

equation, we get the Poisson's equations:

good approximation to treat air motion as adiabatic.

The above equations can be integrated if we know the virtual temperature *Tv* as a function of pressure, and many limiting cases can be envisaged, as those of constant vertical temperature gradient. A very simplified case is for an isothermal atmosphere at a temperature *Tv=T0*, when the integration of (11) gives:

$$
\Delta Z = \frac{R\_d T\_0}{g\_o} \ln \left(\frac{p\_1}{p\_2}\right) = H \ln \left(\frac{p\_1}{p\_2}\right) \tag{12}
$$

In an isothermal atmosphere the pressure decreases exponentially with an e-folding scale given by the *scale height H* which, at an average atmospheric temperature of 255 K, corresponds roughly to 7.5 km. Of course, atmospheric temperature is by no means constant: within the lowest 10-20 km it decreases with a *lapse rate* of about 7 K km-1, highly variable depending on latitude, altitude and season. This region of decreasing temperature with height is termed *troposphere*, (from the Greek "turning/changing sphere") and is capped by a region extending from its boundary, termed *tropopause*, up to 50 km, where the temperature is increasing with height due to solar UV absorption by ozone, that heats up the air. This region is particularly stable and is termed *stratosphere* ( "layered sphere"). Higher above in the *mesosphere* ("middle sphere") from 50 km to 80-90 km, the temperature falls off again. The last region of the atmosphere, named *thermosphere*, sees the temperature rise again with altitude to 500-2000K up to an isothermal layer several hundreds of km distant from the ground, that finally merges with the interplanetary space where molecular collisions are rare and temperature is difficult to define. Fig. 1 reports the atmospheric temperature, pressure and density profiles. Although the atmosphere is far from isothermal, still the decrease of pressure and density are close to be exponential. The atmospheric temperature profile depends on vertical mixing, heat transport and radiative processes.

Fig. 1. Temperature (dotted line), pressure (dashed line) and air density (solid line) for a standard atmosphere.

The above equations can be integrated if we know the virtual temperature *Tv* as a function of pressure, and many limiting cases can be envisaged, as those of constant vertical temperature gradient. A very simplified case is for an isothermal atmosphere at a

> ݈݊ ቀభ మ

Fig. 1. Temperature (dotted line), pressure (dashed line) and air density (solid line) for a

In an isothermal atmosphere the pressure decreases exponentially with an e-folding scale given by the *scale height H* which, at an average atmospheric temperature of 255 K, corresponds roughly to 7.5 km. Of course, atmospheric temperature is by no means constant: within the lowest 10-20 km it decreases with a *lapse rate* of about 7 K km-1, highly variable depending on latitude, altitude and season. This region of decreasing temperature with height is termed *troposphere*, (from the Greek "turning/changing sphere") and is capped by a region extending from its boundary, termed *tropopause*, up to 50 km, where the temperature is increasing with height due to solar UV absorption by ozone, that heats up the air. This region is particularly stable and is termed *stratosphere* ( "layered sphere"). Higher above in the *mesosphere* ("middle sphere") from 50 km to 80-90 km, the temperature falls off again. The last region of the atmosphere, named *thermosphere*, sees the temperature rise again with altitude to 500-2000K up to an isothermal layer several hundreds of km distant from the ground, that finally merges with the interplanetary space where molecular collisions are rare and temperature is difficult to define. Fig. 1 reports the atmospheric temperature, pressure and density profiles. Although the atmosphere is far from isothermal, still the decrease of pressure and density are close to be exponential. The atmospheric temperature profile depends on

ௗ

ቁ ൌ ܪ݈݊ ቀభ

మ

(11)

ቁ (12)

οܼ ൌ ோ బ ௩ܶ మ భ

οܼ ൌ ோ்బ బ

temperature *Tv=T0*, when the integration of (11) gives:

vertical mixing, heat transport and radiative processes.

standard atmosphere.

#### **2.2 Thermodynamic of dry air**

A system is *open* if it can exchange matter with its surroundings, *closed* otherwise. In atmospheric thermodynamics, the concept of "air parcel" is often used. It is a good approximation to consider the air parcel as a closed system, since significant mass exchanges between airmasses happen predominantly in the few hundreds of metres close to the surface, the so-called *planetary boundary layer* where mixing is enhanced, and can be neglected elsewhere. An air parcel can exchange energy with its surrounding by work of expansion or contraction, or by exchanging heat. An *isolated* system is unable to exchange energy in the form of heat or work with its surroundings, or with any other system. The first principle of thermodynamics states that the *internal energy U* of a closed system, the kinetic and potential energy of its components, is a state variable, depending only on the present state of the system, and not by its past. If a system evolves without exchanging any heat with its surroundings, it is said to perform an *adiabatic* transformation. An air parcel can exchange heat with its surroundings through diffusion or thermal conduction or radiative heating or cooling; moreover, evaporation or condensation of water and subsequent removal of the condensate promote an exchange of latent heat. It is clear that processes which are not adiabatic ultimately lead the atmospheric behaviours. However, for timescales of motion shorter than one day, and disregarding cloud processes, it is often a good approximation to treat air motion as adiabatic.

#### **2.2.1 Potential temperature**

For adiabatic processes, the first law of thermodynamics, written in two alternative forms:

$$
\sigma vd\,T + p dv = \delta q\tag{13}
$$

$$
\mathfrak{c}cpdT \text{ - } vdp \equiv \delta q \tag{14}
$$

holds for *δq=0*, where *cp* and *cv* are respectively the specific heats at constant pressure and constant volume, *p* and *v* are the specific pressure and volume, and *δq* is the heat exchanged with the surroundings. Integrating (13) and (14) and making use of the ideal gas state equation, we get the Poisson's equations:

$$Tv^{\circ \cdot 1} = \text{constant} \tag{15}$$

$$T p\* \equiv \text{constant} \tag{16}$$

$$pv\text{'} = constant \tag{17}$$

where *γ=cp/cv* =1.4 and κ*=(γ-1)/γ* =*R/cp* ≈ 0.286, using a result of the kinetic theory for diatomic gases. We can use (16) to define a new state variable that is conserved during an adiabatic process, the *potential temperature θ*, which is the temperature the air parcel would attain if compressed, or expanded, adiabatically to a reference pressure *p0*, taken for convention as 1000 hPa.

$$
\theta = T \left( \frac{p\_0}{p} \right)^{\aleph} \tag{18}
$$

Since the time scale of heat transfers, away from the planetary boundary layer and from clouds is several days, and the timescale needed for an air parcel to adjust to environmental pressure changes is much shorter, *θ* can be considered conserved along the air motion for one week or more. The distribution of *θ* in the atmosphere is determined by the pressure

Atmospheric Thermodynamics 55

The second law of the thermodynamics allows for the introduction of another state variable, the *entropy* s, defined in terms of a quantity *δq/T* which is not in general an exact differential, but is so for a reversible process, that is a process proceeding through states of the system which are always in equilibrium with the environment. Under such cases we may pose *ds = (δq/T)rev*. For the generic process, the heat absorbed by the system is always lower that what can be absorbed in the reversible case, since a part of heat is lost to the environment. Hence,

�� � ��

If we introduce (22) in (23), we note how such expression, connecting potential temperature

� �� � = �� ��

That directly relates changes in potential temperature with changes in entropy. We stress the fact that in general an adiabatic process does not imply a conservation of entropy. A classical textbook example is the adiabatic free expansion of a gas. However, in atmospheric processes, adiabaticity not only implies the absence of heat exchange through the boundaries of the system, but also absence of heat exchanges between parts of the system itself (Landau et al., 1980), that is, no turbulent mixing, which is the principal source of irreversibility. Hence, in the atmosphere, an adiabatic process always conserves entropy.

The vertical gradient of potential temperature determines the stratification of the air. Let us

�� <sup>+</sup> � �� � ��� �� <sup>−</sup> ��

Γ=Γ� <sup>−</sup> �

�� = − �*′*

�� = − ����

Now, consider a vertical displacement *δz* of an air parcel of mass *m* and let *ρ* and *T* be the density and temperature of the parcel, and *ρ'* and *T'* the density and temperature of the

� ��

��

� �� � �� <sup>=</sup> � �� �

If *= - (∂T/∂z)* is the environment lapse rate, we get:

By computing the differential of the logarithm, and applying (1) and (8), we get:

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

surrounding. The restoring force acting on the parcel per unit mass will be:

to entropy, would contain only state variables. Hence equality must hold and we get:

� (23)

��� (25)

�� (27)

�� � (28)

� � (29)

(24)

(26)

**2.2.2 Entropy and potential temperature** 

**2.3 Stability** 

differentiate (18) with respect to z:

That, by using (1), can be rewritten as:

a statement of the second law of thermodynamics is:

and temperature fields. In fig. 2 annual averages of constant potential temperature surfaces are depicted, versus pressure and latitude. These surfaces tend to be quasi-horizontal. An air parcel initially on one surface tend to stay on that surface, even if the surface itself can vary its position with time. At the ground level *θ* attains its maximum values at the equator, decreasing toward the poles. This poleward decrease is common throughout the troposphere, while above the tropopause, situated near 100 hPa in the tropics and 3-400 hPa at medium and high latitudes, the behaviour is inverted.

Fig. 2. ERA-40 Atlas : Pressure level climatologies in latitude-pressure projections (source: http://www.ecmwf.int/research/era/ERA40\_Atlas/docs/section\_D25/charts/D26\_XS\_Y EA.html).

An adiabatic vertical displacement of an air parcel would change its temperature and pressure in a way to preserve its potential temperature. It is interesting to derive an expression for the rate of change of temperature with altitude under adiabatic conditions: using (8) and (1) we can write (14) as:

$$\text{lcp } dT + \text{g } dz \equiv 0 \tag{19}$$

and obtain the *dry adiabatic lapse* rate d:

$$
\Gamma\_d = -\left(\frac{dT}{dz}\right)\_{adiabatic} = \frac{g}{c\_p} \tag{20}
$$

If the air parcel thermally interacts with its environment, the adiabatic condition no longer holds and in (13) and (14) *δq ≠ 0*. In such case, dividing (14) by T and using (1) we obtain:

$$d\ln p - \text{ } \kappa \, d\ln p = -\frac{\delta q}{c\_p \tau} \tag{21}$$

Combining the logarithm of (18) with (21) yields:

$$d\ln\theta = \frac{\delta q}{c\_p T} \tag{22}$$

That clearly shows how the changes in potential temperature are directly related to the heat exchanged by the system.

and temperature fields. In fig. 2 annual averages of constant potential temperature surfaces are depicted, versus pressure and latitude. These surfaces tend to be quasi-horizontal. An air parcel initially on one surface tend to stay on that surface, even if the surface itself can vary its position with time. At the ground level *θ* attains its maximum values at the equator, decreasing toward the poles. This poleward decrease is common throughout the troposphere, while above the tropopause, situated near 100 hPa in the tropics and 3-400 hPa

Fig. 2. ERA-40 Atlas : Pressure level climatologies in latitude-pressure projections (source: http://www.ecmwf.int/research/era/ERA40\_Atlas/docs/section\_D25/charts/D26\_XS\_Y

An adiabatic vertical displacement of an air parcel would change its temperature and pressure in a way to preserve its potential temperature. It is interesting to derive an expression for the rate of change of temperature with altitude under adiabatic conditions:

<sup>Γ</sup>� =����

���

If the air parcel thermally interacts with its environment, the adiabatic condition no longer holds and in (13) and (14) *δq ≠ 0*. In such case, dividing (14) by T and using (1) we obtain:

� �� � � ���� �� � = � ��

� �� � = ��

That clearly shows how the changes in potential temperature are directly related to the heat

��������� <sup>=</sup> �

��

*cp dT + g dz=0* (19)

��� (21)

��� (22)

(20)

at medium and high latitudes, the behaviour is inverted.

EA.html).

using (8) and (1) we can write (14) as:

and obtain the *dry adiabatic lapse* rate d:

Combining the logarithm of (18) with (21) yields:

exchanged by the system.

### **2.2.2 Entropy and potential temperature**

The second law of the thermodynamics allows for the introduction of another state variable, the *entropy* s, defined in terms of a quantity *δq/T* which is not in general an exact differential, but is so for a reversible process, that is a process proceeding through states of the system which are always in equilibrium with the environment. Under such cases we may pose *ds = (δq/T)rev*. For the generic process, the heat absorbed by the system is always lower that what can be absorbed in the reversible case, since a part of heat is lost to the environment. Hence, a statement of the second law of thermodynamics is:

$$ds \ge \frac{\delta q}{\tau} \tag{23}$$

If we introduce (22) in (23), we note how such expression, connecting potential temperature to entropy, would contain only state variables. Hence equality must hold and we get:

$$d \ln \theta = \frac{ds}{c\_p} \tag{24}$$

That directly relates changes in potential temperature with changes in entropy. We stress the fact that in general an adiabatic process does not imply a conservation of entropy. A classical textbook example is the adiabatic free expansion of a gas. However, in atmospheric processes, adiabaticity not only implies the absence of heat exchange through the boundaries of the system, but also absence of heat exchanges between parts of the system itself (Landau et al., 1980), that is, no turbulent mixing, which is the principal source of irreversibility. Hence, in the atmosphere, an adiabatic process always conserves entropy.

### **2.3 Stability**

The vertical gradient of potential temperature determines the stratification of the air. Let us differentiate (18) with respect to z:

$$\frac{\partial \ln \theta}{\partial x} = \frac{\partial \ln T}{\partial x} + \frac{R}{c\_p} \left( \frac{\partial p\_0}{\partial x} - \frac{\partial p}{\partial x} \right) \tag{25}$$

By computing the differential of the logarithm, and applying (1) and (8), we get:

$$\frac{\partial \tau}{\partial \theta} \frac{\partial \theta}{\partial z} = \frac{\partial \tau}{\partial z} + \frac{g}{c\_p} \tag{26}$$

If *= - (∂T/∂z)* is the environment lapse rate, we get:

$$
\Gamma = \Gamma\_d - \frac{\tau}{\theta} \frac{\partial \theta}{\partial z} \tag{27}
$$

Now, consider a vertical displacement *δz* of an air parcel of mass *m* and let *ρ* and *T* be the density and temperature of the parcel, and *ρ'* and *T'* the density and temperature of the surrounding. The restoring force acting on the parcel per unit mass will be:

$$f\_{\mathbf{z}} = -\frac{\rho^{\dot{\prime}} - \rho}{\rho^{\prime}} g \tag{28}$$

That, by using (1), can be rewritten as:

$$f\_{\mathbf{z}} = -\frac{\mathbf{r} - \mathbf{r}\prime}{\mathbf{r}}g\tag{29}$$

Atmospheric Thermodynamics 57

By combining (32), (33) and (8), and incorporating the definition of geopotential we get:

Which states that an air parcel moving adiabatically in an hydrostatic atmosphere conserves

It represents the energy available for conversion into work under an isothermal-isobaric process. Hence the criterion for thermodinamical equilibrium for a system at constant

For an heterogeneous system where multiple phases coexist, for the *k*-th species we define its *chemical potential μk* as the partial molar Gibbs function, and the equilibrium condition states that the chemical potentials of all the species should be equal. The proof is straightforward: consider a system where *nv* moles of vapour (*v*) and *nl* moles of liquid water (*l*) coexist at pressure *e* and temperature *T*, and let *G = nvμv +nlμl* be the Gibbs function of the system. We know that for a virtual displacement from an equilibrium condition, *dG > 0* must hold for any arbitrary *dnv* (which must be equal to *– dnl* , whether its positive or

Note that if evaporation occurs, the vapour pressure *e* changes by *de* at constant temperature, and *dμv = vv de*, *dμl = vl de* where *vv* and *vl* are the volume occupied by a single molecule in the vapour and the liquid phase. Since *vv >> vl* we may pose d*(μv - μl) = vvde* and, using the state gas equation for a single molecule, *d(μv - μl) = (kT/e) de*. In the equilibrium,

(�� � ��) = ���� � �

The value of *es* strongly depends on temperature and increases rapidly with it. The celebrated Clausius –Clapeyron equation describes the changes of saturated water pressure above a plane surface of liquid water. It can be derived by considering a liquid in equilibrium with its saturated vapour undergoing a Carnot cycle (Fermi, 1956). We here

holds. We will make use of this relationship we we will discuss the formation of clouds.

��� �� <sup>=</sup> ��

Retrieved under the assumption that the specific volume of the vapour phase is much greater than that of the liquid phase. *Lv* is the latent heat, that is the heat required to convert

��

� (36)

�� (37)

�� = �� � ��� (32)

�� = ���� (33)

�� = �(� � �) (34)

� = � � �� = �� � �� � �� (35)

The First law of thermodynamics can be set in a form where *h* is explicited as:

And, making use of (14) we can set:

the sum of its enthalpy and geopotential. The specific Gibbs free energy is defined as:

pressure and temperature is that *g* attains a minimum.

negative) hence, its coefficient must vanish and *μv = μl* .

*μv = μl* and *e = es* while in general:

**3.1 Saturation vapour pressure** 

simply state the result as:

We can replace *(T-T')* with *(d - ) δz* if we acknowledge the fact that the air parcel moves adiabatically in an environment of lapse rate . The second order equation of motion (29) can be solved in *δz* and describes buoyancy oscillations with period *2π/N* where *N* is the Brunt-Vaisala frequency:

$$N = \left[\frac{g}{T}(\Gamma\_d - \Gamma)\right]^{1/2} = \left[\frac{g}{c\_p}\frac{\partial \theta}{\partial x}\right]^{1/2} \tag{30}$$

It is clear from (30) that if the environment lapse rate is smaller than the adiabatic one, or equivalently if the potential temperature vertical gradient is positive, *N* will be real and an air parcel will oscillate around an equilibrium: if displaced upward, the air parcel will find itself colder, hence heavier than the environment and will tend to fall back to its original place; a similar reasoning applies to downward displacements. If the environment lapse rate is greater than the adiabatic one, or equivalently if the potential temperature vertical gradient is negative, N will be imaginary so the upward moving air parcel will be lighter than the surrounding and will experience a net buoyancy force upward. The condition for atmospheric stability can be inspected by looking at the vertical gradient of the potential temperature: if *θ* increases with height, the atmosphere is stable and vertical motion is discouraged, if *θ* decreases with height, vertical motion occurs. For average tropospheric conditions, *N* ≈ 10-2 s-1 and the period of oscillation is some tens of minutes. For the more stable stratosphere, *N* ≈ 10-1 s-1 and the period of oscillation is some minutes. This greater stability of the stratosphere acts as a sort of damper for the weather disturbances, which are confined in the troposphere.

## **3. Moist air thermodynamics**

The conditions of the terrestrial atmosphere are such that water can be present under its three forms, so in general an air parcel may contain two gas phases, dry air (d) and water vapour (v), one liquid phase (l) and one ice phase (i). This is an heterogeneous system where, in principle, each phase can be treated as an homogeneous subsystem open to exchanges with the other systems. However, the whole system should be in thermodynamical equilibrium with the environment, and thermodynamical and chemical equilibrium should hold between each subsystem, the latter condition implying that no conversion of mass should occur between phases. In the case of water in its vapour and liquid phase, the chemical equilibrium imply that the vapour phases attains a saturation vapour pressure *es* at which the rate of evaporation equals the rate of condensation and no net exchange of mass between phases occurs.

The concept of chemical equilibrium leads us to recall one of the thermodynamical potentials, the *Gibbs function*, defined in terms of the *enthalpy* of the system. We remind the definition of enthalpy of a system of unit mass:

$$h = u + pv\tag{31}$$

Where *u* is its specific internal energy, *v* its specific volume and *p* its pressure in equilibrium with the environment. We can think of *h* as a measure of the total energy of the system. It includes both the internal energy required to create the system, and the amount of energy required to make room for it in the environment, establishing its volume and balancing its pressure against the environmental one. Note that this additional energy is not stored in the system, but rather in its environment.

We can replace *(T-T')* with *(d - ) δz* if we acknowledge the fact that the air parcel moves adiabatically in an environment of lapse rate . The second order equation of motion (29) can be solved in *δz* and describes buoyancy oscillations with period *2π/N* where *N* is the

> ��� � � � �� �� ��� ���

It is clear from (30) that if the environment lapse rate is smaller than the adiabatic one, or equivalently if the potential temperature vertical gradient is positive, *N* will be real and an air parcel will oscillate around an equilibrium: if displaced upward, the air parcel will find itself colder, hence heavier than the environment and will tend to fall back to its original place; a similar reasoning applies to downward displacements. If the environment lapse rate is greater than the adiabatic one, or equivalently if the potential temperature vertical gradient is negative, N will be imaginary so the upward moving air parcel will be lighter than the surrounding and will experience a net buoyancy force upward. The condition for atmospheric stability can be inspected by looking at the vertical gradient of the potential temperature: if *θ* increases with height, the atmosphere is stable and vertical motion is discouraged, if *θ* decreases with height, vertical motion occurs. For average tropospheric conditions, *N* ≈ 10-2 s-1 and the period of oscillation is some tens of minutes. For the more stable stratosphere, *N* ≈ 10-1 s-1 and the period of oscillation is some minutes. This greater stability of the stratosphere acts as a sort of damper for the weather disturbances, which are

The conditions of the terrestrial atmosphere are such that water can be present under its three forms, so in general an air parcel may contain two gas phases, dry air (d) and water vapour (v), one liquid phase (l) and one ice phase (i). This is an heterogeneous system where, in principle, each phase can be treated as an homogeneous subsystem open to exchanges with the other systems. However, the whole system should be in thermodynamical equilibrium with the environment, and thermodynamical and chemical equilibrium should hold between each subsystem, the latter condition implying that no conversion of mass should occur between phases. In the case of water in its vapour and liquid phase, the chemical equilibrium imply that the vapour phases attains a saturation vapour pressure *es* at which the rate of evaporation equals the rate of condensation and no

The concept of chemical equilibrium leads us to recall one of the thermodynamical potentials, the *Gibbs function*, defined in terms of the *enthalpy* of the system. We remind the

Where *u* is its specific internal energy, *v* its specific volume and *p* its pressure in equilibrium with the environment. We can think of *h* as a measure of the total energy of the system. It includes both the internal energy required to create the system, and the amount of energy required to make room for it in the environment, establishing its volume and balancing its pressure against the environmental one. Note that this additional energy is not stored in the

� � � � �� (31)

(30)

����

� (Γ� − Γ)�

Brunt-Vaisala frequency:

confined in the troposphere.

**3. Moist air thermodynamics** 

net exchange of mass between phases occurs.

definition of enthalpy of a system of unit mass:

system, but rather in its environment.

The First law of thermodynamics can be set in a form where *h* is explicited as:

$$
\delta q = dh - \nu dp \tag{32}
$$

And, making use of (14) we can set:

$$dh = c\_p dT\tag{33}$$

By combining (32), (33) and (8), and incorporating the definition of geopotential we get:

$$
\delta q = d(h + \Phi) \tag{34}
$$

Which states that an air parcel moving adiabatically in an hydrostatic atmosphere conserves the sum of its enthalpy and geopotential.

The specific Gibbs free energy is defined as:

$$\mathbf{r} \cdot \mathbf{g} = \mathbf{h} - T\mathbf{s} = \mathbf{u} + p\mathbf{v} - T\mathbf{s} \tag{35}$$

It represents the energy available for conversion into work under an isothermal-isobaric process. Hence the criterion for thermodinamical equilibrium for a system at constant pressure and temperature is that *g* attains a minimum.

For an heterogeneous system where multiple phases coexist, for the *k*-th species we define its *chemical potential μk* as the partial molar Gibbs function, and the equilibrium condition states that the chemical potentials of all the species should be equal. The proof is straightforward: consider a system where *nv* moles of vapour (*v*) and *nl* moles of liquid water (*l*) coexist at pressure *e* and temperature *T*, and let *G = nvμv +nlμl* be the Gibbs function of the system. We know that for a virtual displacement from an equilibrium condition, *dG > 0* must hold for any arbitrary *dnv* (which must be equal to *– dnl* , whether its positive or negative) hence, its coefficient must vanish and *μv = μl* .

Note that if evaporation occurs, the vapour pressure *e* changes by *de* at constant temperature, and *dμv = vv de*, *dμl = vl de* where *vv* and *vl* are the volume occupied by a single molecule in the vapour and the liquid phase. Since *vv >> vl* we may pose d*(μv - μl) = vvde* and, using the state gas equation for a single molecule, *d(μv - μl) = (kT/e) de*. In the equilibrium, *μv = μl* and *e = es* while in general:

$$\{\mu\_{\nu} - \mu\_{l}\} = kT \ln\left(\frac{e}{e\_{\mathcal{S}}}\right) \tag{36}$$

holds. We will make use of this relationship we we will discuss the formation of clouds.

#### **3.1 Saturation vapour pressure**

The value of *es* strongly depends on temperature and increases rapidly with it. The celebrated Clausius –Clapeyron equation describes the changes of saturated water pressure above a plane surface of liquid water. It can be derived by considering a liquid in equilibrium with its saturated vapour undergoing a Carnot cycle (Fermi, 1956). We here simply state the result as:

$$\frac{d\mathbf{e}\_s}{dT} = \frac{L\_v}{\tau \alpha} \tag{37}$$

Retrieved under the assumption that the specific volume of the vapour phase is much greater than that of the liquid phase. *Lv* is the latent heat, that is the heat required to convert

Atmospheric Thermodynamics 59

cooled upon evaporation until the surrounding air is saturated: the heat required to evaporate water is supplied by the surrounding air that is cooled. An evaporating droplet will be at the wet-bulb temperature. It should be noted that if the surrounding air is initially unsaturated, the process adds water to the air close to the thermometer, to become saturated, hence it increases its mixing ratio *r* and in general *T ≥ Tw ≥ Td*, the equality holds

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

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

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

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

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:

when the ambient air is already initially saturated.

it should be more properly termed a *pseudoadiabatic* process.

Atmospheric Physics, Academic Press, New York.)

**3.3.1 Pseudoadiabatic lapse rate** 

**3.3 Thermodynamics of the vertical motion** 

with height as well.

air parcel upon further lifting.

a unit mass of substance from the liquid to the vapour phase without changing its temperature. The latent heat itself depends on temperature – at 1013 hPa and 0°C is 2.5\*106 J kg-, - hence a number of numerical approximations to (37) have been derived. The World Meteoreological Organization bases its recommendation on a paper by Goff (1957):

$$\text{Log10 es } = 10.79574 \left(1 - 273.16 / T \right) - 5.02800 \text{ Log10} \left(T / 273.16 \right) + 1.50475 \left[1 - 10 \left(-8.2969 \* \left(T / 273.16 - 1 \right) \right) \right] + 0.42873 \left[10 - \right] = 380 \left(10 \left(+4.76955 \* \left(1 - 273.16 / T \right) \right) - 1 \right) + 0.78614$$

Where *T* is expressed in K and *es* in hPa. Other formulations are used, based on direct measurements of vapour pressures and theoretical calculation to extrapolate the formulae down to low T values (Murray, 1967; Bolton, 1980; Hyland and Wexler, 1983; Sonntag, 1994; Murphy and Koop, 2005) uncertainties at low temperatures become increasingly large and the relative deviations within these formulations are of 6% at -60°C and of 9% at -70°.

An equation similar to (37) can be derived for the vapour pressure of water over ice *esi*. In such a case, *Lv* is the latent heat required to convert a unit mass of water substance from ice to vapour phase without changing its temperature. A number of numerical approximations holds, as the Goff-Gratch equation, considered the reference equation for the vapor pressure over ice over a region of -100°C to 0°C:

$$\begin{aligned} \text{Log10 esi} &= -9.09718 \left(273.16 / T - 1\right) - 3.56654 \,\text{Log10} \left(273.16 / T\right) + \\ + 0.876793 \left(1 - T / 273.16\right) + \,\text{Log10} \left(6.1071\right) \end{aligned} \tag{39}$$

with *T* in K and *esi* in hPa. Other equations have also been widely used (Murray, 1967; Hyland and Wexler, 1983; Marti and Mauersberger, 1993; Murphy and Koop, 2005).

Water evaporates more readily than ice, that is *es > esi* everywhere (the difference is maxima around -20°C), so if liquid water and ice coexists below 0°C, the ice phase will grow at the expense of the liquid water.

### **3.2 Water vapour in the atmosphere**

A number of moisture parameters can be formulated to express the amount of water vapour in the atmosphere. The *mixing ratio r* is the ratio of the mass of the water vapour *mv*, to the mass of dry air *md, r=mv/md* and is expressed in g/kg-1 or, for very small concentrations as those encountered in the stratosphere, in parts per million in volume (ppmv). At the surface, it typically ranges from 30-40 g/kg-1 at the tropics to less that 5 g/kg-1 at the poles; it decreases approximately exponentially with height with a scale height of 3-4 km, to attain its minimum value at the tropopause, driest at the tropics where it can get as low as a few ppmv. If we consider the ratio of *mv* to the total mass of air, we get the *specific humidity q* as *q = mv/(mv+md) =r/(1+r)*. The *relative humidity RH* compares the water vapour pressure in an air parcel with the maximum water vapour it may sustain in equilibrium at that temperature, that is *RH = 100 e/es* (expressed in percentages). The dew point temperature *Td* is the temperature at which an air parcel with a water vapour pressure *e* should be brought isobarically in order to become saturated with respect to a plane surface of water. A similar definition holds for the frost point temperature *Tf*, when the saturation is considered with respect to a plane surface of ice.

The *wet-bulb temperature* Tw is defined operationally as the temperature a thermometer would attain if its glass bulb is covered with a moist cloth. In such a case the thermometer is

a unit mass of substance from the liquid to the vapour phase without changing its temperature. The latent heat itself depends on temperature – at 1013 hPa and 0°C is 2.5\*106 J kg-, - hence a number of numerical approximations to (37) have been derived. The World

Where *T* is expressed in K and *es* in hPa. Other formulations are used, based on direct measurements of vapour pressures and theoretical calculation to extrapolate the formulae down to low T values (Murray, 1967; Bolton, 1980; Hyland and Wexler, 1983; Sonntag, 1994; Murphy and Koop, 2005) uncertainties at low temperatures become increasingly large and the relative deviations within these formulations are of 6% at -60°C and of 9% at -70°. An equation similar to (37) can be derived for the vapour pressure of water over ice *esi*. In such a case, *Lv* is the latent heat required to convert a unit mass of water substance from ice to vapour phase without changing its temperature. A number of numerical approximations holds, as the Goff-Gratch equation, considered the reference equation for the vapor

1.50475 10 4 1 10 8.2969 \* / 273.16 1 0.42873 10

*T*

10 10.79574 1 273.16 / 5.02800 10 / 273.16 +

*T*

(38)

(39)

Meteoreological Organization bases its recommendation on a paper by Goff (1957):

*Log es T Log T*

pressure over ice over a region of -100°C to 0°C:

expense of the liquid water.

**3.2 Water vapour in the atmosphere** 

considered with respect to a plane surface of ice.

3 10 4.76955 \* 1 273.16 / 1 0.78614

*T Log*

0.876793 1 / 273.16 10 6.1071

10 9.09718 273.16 / 1 3.56654 10 273.16 /

with *T* in K and *esi* in hPa. Other equations have also been widely used (Murray, 1967;

Water evaporates more readily than ice, that is *es > esi* everywhere (the difference is maxima around -20°C), so if liquid water and ice coexists below 0°C, the ice phase will grow at the

A number of moisture parameters can be formulated to express the amount of water vapour in the atmosphere. The *mixing ratio r* is the ratio of the mass of the water vapour *mv*, to the mass of dry air *md, r=mv/md* and is expressed in g/kg-1 or, for very small concentrations as those encountered in the stratosphere, in parts per million in volume (ppmv). At the surface, it typically ranges from 30-40 g/kg-1 at the tropics to less that 5 g/kg-1 at the poles; it decreases approximately exponentially with height with a scale height of 3-4 km, to attain its minimum value at the tropopause, driest at the tropics where it can get as low as a few ppmv. If we consider the ratio of *mv* to the total mass of air, we get the *specific humidity q* as *q = mv/(mv+md) =r/(1+r)*. The *relative humidity RH* compares the water vapour pressure in an air parcel with the maximum water vapour it may sustain in equilibrium at that temperature, that is *RH = 100 e/es* (expressed in percentages). The dew point temperature *Td* is the temperature at which an air parcel with a water vapour pressure *e* should be brought isobarically in order to become saturated with respect to a plane surface of water. A similar definition holds for the frost point temperature *Tf*, when the saturation is

The *wet-bulb temperature* Tw is defined operationally as the temperature a thermometer would attain if its glass bulb is covered with a moist cloth. In such a case the thermometer is

*Log esi T Log T*

Hyland and Wexler, 1983; Marti and Mauersberger, 1993; Murphy and Koop, 2005).

cooled upon evaporation until the surrounding air is saturated: the heat required to evaporate water is supplied by the surrounding air that is cooled. An evaporating droplet will be at the wet-bulb temperature. It should be noted that if the surrounding air is initially unsaturated, the process adds water to the air close to the thermometer, to become saturated, hence it increases its mixing ratio *r* and in general *T ≥ Tw ≥ Td*, the equality holds when the ambient air is already initially saturated.
