4.2.1 Potential Temperature

The change of pressure and temperature as a function of altitude is obtained via the hydrostatic and ideal gas laws (see Chapter 3). Furthermore, a relationship between pressure and temperature may be obtained by assuming that air parcels undergo adiabatic changes. Therefore, the law of Laplace applies:

where γ is the adiabatic index of the gas (here, air), also known as the Laplace coefficient (γ = c_p / c_v, where c_p and c_v are the specific heat capacities at constant pressure and volume, respectively). Applying the ideal gas law (see Chapter 3), the following relationship between pressure and temperature is obtained under adiabatic conditions:

For a diatomic gas (which is the case of the atmosphere, since molecular nitrogen and oxygen account for 99 % of the air), γ = 7/5 = 1.4. Thus:

Therefore:

The potential temperature, θ, is defined as the temperature of an air parcel brought back to sea level at a standard pressure (i.e., P₀ = 1 atm) under adiabatic conditions. Therefore:

TP−0.29=θ P0−0.29(4.5)

Thus:

θ=T(P0P)0.29(4.6)

4.2.2 Neutral Atmosphere

A constant vertical profile of the potential temperature is defined as neutral. Let us consider an air parcel with uniform properties (pressure, temperature, and density). Under neutral conditions, this air parcel will remain in equilibrium with its environment as it moves upward or downward, since its temperature will change adiabatically and will, therefore, remain consistent with the temperature of the surrounding air.

4.2.3 Unstable Atmosphere

The atmosphere is unstable if the vertical profile of the potential temperature is negative, i.e., if the actual temperature decreases faster with height than the adiabatic gradient. An air parcel moving upward becomes warmer than that of its surroundings, because its adiabatic change in temperature is less than the decrease of the surrounding atmospheric temperature. Therefore, according to the ideal gas law, the density of this air parcel becomes less than that of the surrounding air. Thus, this air parcel keeps moving upward. Similarly, an air parcel moving downward undergoes an adiabatic change and its temperature change is less than that of the gradient of the surrounding atmosphere. Therefore, the surrounding temperature becomes greater than that of the air parcel. Thus, this air parcel is denser than the surrounding air and it continues to move downward. Vertical motions in an unstable atmosphere are enhanced because a small perturbation in the vertical direction leads to continuous motion in the direction of the initial perturbation (see Figure 4.1). Such unstable air parcels lead to vertical mixing of air pollution, at least up to an altitude where the atmosphere becomes neutral or stable (see Section 4.2.4). An unstable atmospheric layer is called a mixing layer. It occurs generally within the PBL during the day, because solar radiation warms up the Earth’s surface, but does not affect the temperature of the lower atmosphere (since the air absorbs little ultraviolet radiation within the troposphere, see Chapter 5). As a result, the vertical potential temperature gradient becomes negative.

Figure 4.1. Schematic representation of vertical motions of an air parcel for neutral conditions (left figure), unstable conditions (middle figure), and stable conditions (right figure). The solid arrows indicate the initial perturbation of the air parcel, which undergoes an adiabatic change, and the dashed arrows indicate the following motion of the air parcel, which is due to the difference between the temperature of the air parcel and that of the surrounding air (i.e., density difference).

4.2.4 Stable Atmosphere

The atmosphere is stable if the vertical profile of the potential temperature becomes positive. As an air parcel moves upward, its temperature follows an adiabatic change and it becomes lower than that of the surrounding air. Therefore, the air parcel becomes denser than the surrounding air. As a result, it tends to move back downward. Similarly, an air parcel moving downward undergoes an adiabatic change and its temperature becomes greater than that of the surrounding air. Thus, this air parcel, being less dense than the surrounding air, tends to move back upward. Therefore, a stable atmosphere tends to suppress vertical motions of air parcels. Thus, such an atmospheric layer may become stratified, because there are no, or few, transfers between the different layers (strata) of the atmosphere (see Figure 4.1). A stable atmospheric layer may be present aloft, for example above a mixing layer, or at the surface. This latter case occurs at night when the infrared radiation emitted by the Earth’s surface toward the atmosphere leads to a decrease of the surface temperature and, accordingly, of the surface layer temperature. Therefore, a temperature inversion develops: the gradient of the actual temperature becomes positive and the gradient of the potential temperature becomes also positive. (Note that an atmospheric layer may be stable without a temperature inversion, as long as the actual temperature gradient is less than the adiabatic gradient.) Then, the atmospheric layer near the surface is stable at night. This stable layer will gradually become thicker as the surface temperature keeps decreasing. The PBL layer located above this stable layer is called the residual layer. Pollutants located within this residual layer are then isolated from the surface. However, during daytime, as solar radiation warms the surface, thereby leading to the gradual disappearance of the surface-based stable layer, the previously stable and residual layers merge to form a new mixing layer. Pollutants that were formerly present within the residual layer are mixed vertically and may then interact with the surface. The evolution of the atmospheric layer near the Earth’s surface is depicted schematically in Figure 4.2. The stratosphere shows a positive vertical profile of the potential temperature; therefore, it is a stable atmospheric layer with few vertical air motions.

Figure 4.2. Schematic representation of the daily cycle of the planetary boundary layer.

Source: After Stull (1988).

4.2.5 Effect of Moisture

This description of temperature gradients applies to dry atmospheres. If the atmosphere is moist, its vertical temperature gradient under adiabatic conditions will be less than that of a dry atmosphere. The water content, q (i.e., the specific humidity expressed in g of water per g of moist air), corresponds to a relative humidity (%), which is greater when temperature decreases, because the saturation vapor pressure of water decreases with temperature. In other words, for a given quantity (mass) of water per quantity of air (mass or volume), relative humidity increases when temperature decreases. Accordingly, for a given relative humidity, the water content of the atmosphere decreases when temperature decreases.

Humidity

Relative humidity (%) may be converted to absolute humidity using the method proposed by McRae (1980). According to Dalton’s law, the water vapor concentration expressed in parts per million (ppm; 1 ppm = 1 molecule per million molecules of air) is:

[H2O(ppm)]=104RH(%)Ps,H2O(T)P(4.7)

where T is the ambient temperature, P_s,H2O(T) is the saturation vapor pressure of water, and P is the atmospheric pressure. The saturation vapor pressure of water depends on temperature according to the following polynomial formula:

Ps,H2O(T)=P exp(13.3185ωT−1.9760ωT2−0.6445ωT3−0.1299ωT4)(4.8)

where ω_T = 1 – (373.15/T), where T is expressed in K. For example, at P = 1 atm and T = 25 °C, P_s,H2O (298 K) = 0.031 atm; at T = 5 °C, P_s,H2O (278 K) = 0.0086 atm. At a relative humidity of 50 % and a temperature of 5 °C, there are about 4300 ppm of water vapor.

Converting % to (g m⁻³) is performed as follows:

[H2O(g m−3)]=0.18 RH(%)Ps,H2O(T)R T(4.9)

Thus, a relative humidity of 50 % at 1 atm and 5 °C corresponds to 3.4 g of water vapor per m³ of air. The water vapor mass per volume of air is called the volumetric absolute humidity. The water vapor mass concentration at saturation (RH = 100 %) increases with temperature. For example, at 1 atm, there are 9.4, 17, and 30 g of water vapor per m³ at 10, 20, and 30 °C, respectively.

Absolute humidity may also be defined as the mass of water vapor per mass of air. It can be defined per mass of dry air or per mass of moist air. The absolute humidity per mass of dry air is usually called the mixing ratio (by weight) and the absolute humidity per mass of moist air is usually called the specific humidity. The molar mass of dry air is 29 g mole⁻¹; therefore, according to the ideal gas law, the mass in g of one m³ of dry air is: 29 P_{dry air} /(RT). The mixing ratio is thus related to relative humidity as follows:

[H2O(g/g dry air)]=0.18RH(%)Ps,H2O(T)R TR T29 Pdry air=0.62RH(%)100Ps,H2O(T)Pdry air(4.10)

According to Dalton’s law, the total pressure is the sum of the partial pressures of dry air and water vapor:

P=Pdry air+RH(%)100Ps,H2O(T)(4.11)

Therefore:

[H2O (g/g dry air)]=0.62RH(%)100Ps,H2O(T)(P−RH(%)100Ps,H2O(T))(4.12)

The specific humidity, q, is related to relative humidity as follows:

q=1(1.61PPs,H2O(T)100RH(%))−0.61(4.13)

Given that water vapor is a small fraction of the total mass of an air parcel, the mixing ratio and the specific humidity generally have very close values. For example, at T = 25 °C and P = 1 atm, P_s,H2O = 0.031 atm (see previous equation). For a relative humidity of 100 %, the mixing ratio is calculated to be 0.0198 g H₂O/g dry air and the specific humidity, q, is 0.0195 g H₂O/g moist air, i.e., a difference of less than 2 %.

4.3.1 Wind Direction in the PBL

As discussed in Chapter 3, the wind direction in the free troposphere results from the balance between two forces: the pressure gradient force and the Coriolis force (which is due to the Earth’s rotation). In the PBL, another force comes into play: the frictional force, which represents the effect of the friction of the flow on the Earth’s surface. Therefore, the wind direction at the surface results from the balance among these three forces: the pressure gradient force, the Coriolis force, and the frictional force. The frictional force is opposite to the wind direction. Furthermore, the Coriolis force is proportional to the wind speed. Since the wind speed decreases near the surface (tending toward zero at or near the surface, see Section 4.3.2), the Coriolis force also decreases near the surface. Thus, the influence of the pressure gradient force on the wind direction is greater near the surface. The theoretical solution for the northern hemisphere leads to a wind direction near the surface that is 45 ° to the left of the geostrophic wind direction. Thus, the wind direction evolves from that direction near the surface toward the geostrophic wind direction at the height within the PBL where the frictional force becomes negligible. The change in wind direction as a function of height is called the Ekman spiral. The depth of this Ekman layer, where the change in wind direction occurs, depends on the effect of the surface. It is greater for a situation with complex terrain (urban canopy, forest …) than for a smooth surface (sea, snow, ice …). The equations that govern the vertical profile of the wind speed and direction are described next.

4.3.2 Equations of Atmospheric Flow and Turbulence

The Navier-Stokes equations are used to calculate the atmospheric flow. They are expressed in Cartesian coordinates as follows:

∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z−f v=−1ρa∂P∂x+1ρa (∂τxx∂x+∂τyx∂y+∂τzx∂z)∂v∂t+u∂v∂x+v∂v∂y+w∂v∂z+f u=−1ρa∂P∂y+1ρa (∂τxy∂x+∂τyy∂y+∂τzy∂z)∂w∂t+u∂w∂x+v∂w∂y+w∂w∂z      =−1ρa∂P∂z+ 1ρa (∂τxz∂x+∂τyz∂y+∂τzz∂z)−g(4.16)

where the terms τ_ij represent the shear stresses (kg m⁻¹ s⁻²), which are due to the fluid viscosity. For a newtonian fluid, Newton’s law of viscosity applies. It is written for fluid motion in a single direction as follows:

τyx=μv,a∂u∂y(4.17)

where μ_v,a is the dynamic viscosity of the air (kg m⁻¹ s⁻¹). This equation may be seen as the flux in the y direction of the momentum in the x direction. This law, defined for a fluid moving in a given direction (here, x), has been generalized by Navier, Stokes, and Poisson for a fluid moving in three dimensions. Then, the shear stresses for momentum in the x direction are as follows:

τxx=2μv,a ∂u∂x+(μD−23μv,a) (∂u∂x+∂v∂y+∂w∂z)τyx=μv,a (∂u∂y+∂v∂x)τzx=μv,a (∂u∂z+∂w∂x)(4.18)

The other terms for the y and z directions, τ_xy, τ_yy, τ_zy, τ_xz, τ_yz, and τ_zz, are written similarly. Clearly: τ_xy = τ_yx, τ_yz = τ_zy, and τ_xz = τ_zx. In these equations, μ_v,a is the dynamic viscosity of the fluid and μ_D is the dilatational viscosity of the fluid (μ_D = 0 for a monoatomic gas and μ_D ≈ 0 for air).

Boussinesq introduced several assumptions that apply to the PBL:

1. 
   

   – The kinematic viscosity and the molecular thermal conductivity are constant (i.e., their dependence on pressure and temperature can be neglected).

   
   
2. 
   

   – The heat produced by the viscous forces is negligible compared to solar radiation in the heat transfer equation.

   
   
3. 
   

   – The atmosphere is considered to be incompressible, since it is an open system (except at the Earth’s surface).

   
   
4. 
   

   – The fluctuations of the state variables (pressure, temperature, and density) are small compared to their mean values.

   
   
5. 
   

   – The ratio of the pressure fluctuation and the mean pressure is small compared to the ratio of the temperature fluctuation and the mean temperature and compared to the ratio of the density fluctuation and the mean density.

   
   
6. 
   

   – The fluctuations of the density are important only when the density is multiplied by the gravitational constant; indeed, the Earth’s surface introduces a constraint on the system in the vertical direction.

The last two assumptions are commonly called the Boussinesq approximation or hypothesis: they imply that the change in the air density can be neglected except in the term (ρ_a g dz) of the hydrostatic equation (see Equation 3.2).

Using the Boussinesq hypothesis, the continuity equation for momentum is as follows:

∂u∂x+∂v∂y+∂w∂z=0(4.19)

For u, the friction term may be written as follows (assuming that μ_v,a is spatially uniform):

1ρa(∂τxx∂x+∂τyx∂y+∂τzx∂z)=μv,aρa(∂∂x(2∂u∂x−23(∂u∂x+∂v∂y+∂w∂z))+∂∂y (∂u∂y+∂v∂x)+∂∂z (∂u∂z+∂w∂x))(4.20)

Changing the order of the derivations:

1ρa(∂τxx∂x+∂τyx∂y+∂τzx∂z)=μv,aρa((∂2u∂x2+∂2u∂y2+∂2u∂z2)−23∂∂x(∂u∂x+∂v∂y+∂w∂z)+∂∂x(∂u∂x+∂v∂y+∂w∂z))(4.21)

According to the continuity equation, the last two terms are zero. Since the kinematic viscosity, ν_v,a(m² s⁻¹), is equal to the ratio of the dynamic viscosity and the fluid density, μ_v,a/ρ_a:

1ρa(∂τxx∂x+∂τyx∂y+∂τzx∂z)=νv,a (∂2u∂x2+∂2u∂y2+∂2u∂z2)(4.22)

Thus, the Navier-Stokes equations may be written as follows:

∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z−fv=−1ρa∂P∂x+νv,a ∇2u∂v∂t+u∂v∂x+v∂v∂y+w∂v∂z+fu=−1ρa∂P∂y+νv,a ∇2v∂w∂t+u∂w∂x+v∂w∂y+w∂w∂z   =−1ρa∂P∂z+νv,a ∇2w−g(4.23)

where u, v, and w are the wind velocities (m s⁻¹) in the x, y, and z directions, respectively, f is the Coriolis parameter (s⁻¹), ρ_a is the air density (kg m⁻³), P is the atmospheric pressure (Pa), ν_v,a is the kinematic viscosity of the air (m² s⁻¹), and g is the acceleration due to gravity (m s⁻²).

The Reynolds number, Re, may be used to estimate whether a flow is turbulent or laminar. It represents the ratio of inertial forces and viscous forces, and it is defined as follows:

Re=ulcνv,a(4.24)