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5
Formulations of Radiative, Chemical, and Aerosol Rates
5.1 Introduction
We saw in Chapter 4 how the continuity equations for reactive chemical species in the atmosphere include chemical production and loss terms determined by kinetic rate laws. Similarly, we saw that the continuity equations for aerosols include formation and growth terms determined by microphysical properties. Here we present the formulations of these different terms.
We begin in Section 5.2 with the equations of radiative transfer that govern the propagation of radiation in the atmosphere. This determines the rates of photolysis reactions, which play a particularly important role in driving atmospheric chemistry as described in Chapter 3. We go on to present the general formulations of chemical kinetics in atmospheric models including gas-phase reactions (Section 5.3), reactions in aerosol particles and clouds (Section 5.4), and the design of chemical mechanisms (Section 5.5). In Section 5.6 we describe the computation of aerosol microphysical processes as needed to model the evolution of aerosol size distributions.
5.2 Radiative Transfer
Radiative transfer describes the propagation of radiation in the atmosphere. Radiation is energy propagated by electromagnetic waves. These waves represent oscillating electric and magnetic fields traveling at the speed of light. The oscillations are characterized by their frequency ν [Hz] or wavelength λ [m]. Frequency ν and wavelength λ are related by
λν = c
(5.1)
where c = 3.00 × 108 m s–1 is the speed of light in vacuum. The radiation is quantized as photons with energy hν [J], where h is the Planck constant (6.63 × 10–34 J s). The intensity of radiation that propagates through the atmosphere is affected by emission, absorption, and scattering processes. We refer to the radiation spectrum as the distribution of energy contributed by photons of dif
ferent wavelengths. Solar radiation is mainly in the ultraviolet (UV, λ < 0.4 μm), the visible (Vis, 0.4 < λ < 0.7 μm), and the shortwave infrared (SWIR, 0.7 < λ < 3 μm). Radiation emitted by the Earth and its atmosphere is mainly in the 5–20 μm range, called terrestrial IR (TIR). Solar radiation is sometimes called shortwave and terrestrial radiation longwave.
We present in this section the basic theory of radiative transfer to describe the photon flux in the atmosphere as governed by emission, scattering, and absorption of radiation. We will refer to the spectral distribution of a physical quantity (such as the photon flux F) as the spectral density of this quantity or its monochromatic value, expressed by the derivative versus wavelength (Fλ = dF/dλ) or versus frequency (Fν = dF/dν). It is common practice in the spectroscopy literature to express radiative quantities as a function of wavelengths in the UV–Vis region of the spectrum, and as a function of frequencies or of wavenumbers (1/λ) in the IR. For consistency in the presentation we will express radiative quantities as a function of wavelength throughout. Conversion to frequency or wavenumber is straightforward.
5.2.1 Definitions
Radiance
The radiation field can be described by the spectral density Lλ of the radiance (also called the intensity). The radiance is defined as the amount of energy d4E [J] in wavelength interval dλ [nm] traversing horizontal surface dS [m2] during a time interval dt [s] in solid angle dΩ [sr] inclined at an angle θ relative to the vertical (Figure 5.1). Thus, for a pencil of light propagating in the direction Ω defined by angle θ, the spectral density of the radiance is defined by
(5.2)
and has units of W m–2 sr–1 nm–1, where wavelength λ is in nm.
Figure 5.1 (a) Geometry of a pencil of light propagating in a solid angle dΩ and traversing a horizontal surface dS at a zenith angle θ. (b) Coordinates of point P defined as distance r from origin O, azimuthal angle φ, and zenith angle θ.
Beer–Lambert law
A pencil of radiation traversing an optically active medium such as air is affected by its interaction with that medium. The Beer–Lambert law states that the attenuation of the radiance traversing an infinitesimally thin layer is proportional to the density of the medium, the thickness ds of the layer, and the radiance of the light. In the absence of radiative emission by the medium, we have
dLλ(λ, s) = − βext(λ, s) Lλ(λ, s) ds
(5.3)
where βext(λ, s) [m–1] is the extinction or attenuation coefficient. βext(λ, s) is a property of the medium. It can be assumed to be proportional to the mass density ρ(s) [kg m–3] or (for a gas) the number density n(s) [molecules cm–3] of the medium, the proportionality coefficient being the wavelength-dependent mass extinction cross-section kext [m2 kg–1] or the molecular extinction cross-section σext [cm2 molecule–1]:
βext(λ, s) = kext(λ, s) ρ(s) = σext(λ, s) n(s)
(5.4)
Integration of expression (5.3) between geometrical points s0 and s yields
(5.5)
The optical depth at wavelength λ between geometrical points s0 and s is given by
(5.6)
and the corresponding transmission function T between s0 and s is
T(λ, s0, s) = exp [−τ(λ, s0, s)]
(5.7)
Following standard atmospheric chemistry usage, we define the optical depth at altitude z as the extinction in the vertical direction
(5.8)
The extinction along an inclined direction is then referred to as the slant optical depth or optical path.
Extinction includes processes of absorption (conversion of radiation to other forms of energy, such as heat) and scattering (change in the direction of the incident radiation). The extinction coefficients and optical depths are often separated into additive absorption (abs) and scattering (scat) components:
βext = βabs + βscat,
(5.9)
and
τ = τabs + τscat
(5.10)
The ratio between the scattering extinction and the total extinction is called the single scattering albedo ω(λ):
(5.11)
Finally, in an absorbing atmosphere, it is customary to express the transmission as a function of the path length [kg m–2] between geometric points s0 and s
(5.12)
where ρabs [kg m–3] is the mass density of the absorber. If kabs(λ, u) [m2 kg–1] is the mass absorption cross-section,
(5.13)
or, if we assume a homogeneous atmosphere where kabs is only dependent on wavelength λ and not on pressure or temperature,
T(λ, s0, s) = exp [−kabs(λ) u(s0, s)]
(5.14)
Radiative transfer equation
In addition to being attenuated by its interaction with matter, the energy of a pencil of light can be strengthened as a result of local radiative emission by the material, or through scattering of radiation from all directions into that pencil of light. These two processes lead to an increase in the local radiance expressed as
dLλ(λ, s) = j(λ, s) ds
(5.15)
Here, j(λ, s) is a radiative source term from emission or scattering that can be assumed proportional to the extinction (j(λ, s) ~ βext(λ, s)) since the same processes are involved. Defining the source function as J(λ, s) = j(λ, s)/βext(λ, s), we obtain a simple form of the radiative transfer equation
(5.16)
A general expression for the radiative equation in a 3-D inhomogeneous atmospheric medium is given by Liou (2002):
(5.17)
where Lλ(λ, r, Ω) and J(λ, r, Ω) represent respectively the monochromatic radiance and source function in the direction defined by the vector Ω (see Figure 5.1) and at the location defined by the vector r. In many applications, it can be assumed that radiative quantities and atmospheric parameters vary only with altitude (plane-parallel atmosphere). In this case, if the vertical optical depth rather than the geometric altitude is adopted as the independent variable
dτ = − βext dz = βext cos (θ) ds,
the radiative transfer equation takes the form
(5.18)
where μ = cos(θ). The first term on the right-hand side of this equation accounts for the attenuation of light following the Beer–Lambert law, while the second term J represents the radiative source term (local emission or light scattered from other directions).
The radiance [W m–2 sr–1] at a point r of the atmosphere and for a direction Ω is given by the spectral integration of Lλ
(5.19)
The spectral density of the irradiance Fλ(λ, r, Ω) [W m–2 nm–1] at point r is defined as the energy flux density traversing a surface of unit area perpendicular to direction Ω integrated over all directions Ω′ of the incoming pencils of light. It is thus provided by the integration over all directions of the normal component of the monochromatic radiance
(5.20)
This quantity is used to describe the exchanges of radiative energy in the atmosphere and hence to quantify its thermal budget. In a plane-parallel atmosphere with horizontal surface as reference, the spectral density of the irradiance is calculated as a function of altitude z by
(5.21)
where φ is the azimuthal angle (Figure 5.1). One often defines the upward and downward fluxes, Fλ↑(λ, z) and Fλ↓(λ, z) as
(5.22)
(5.23)
with the net irradiance density being Fλ(λ, z) = Fλ↑(λ, z) – Fλ↓(λ, z). The irradiance [W m–2] is obtained by integrating Fλ over the entire electromagnetic spectrum
(5.24)
The diabatic heating Q in units of [W m–3] resulting from the absorption of radiation (radiative energy absorbed and converted into thermal energy) is the divergence of the irradiance
Q(r) = − ∇ • F
(5.25)
In a plane-parallel atmosphere, the diabatic heating rate can be expressed in units of [K s–1] as
(5.26)
In models using isobaric coordinates, the heating rate Q(p) is expressed as
a function of atmospheric pressure p by;
(5.27)
Here, cp [J K–1 kg–1] is the specific heat at constant pressure, ρa [kg m–3] the air density, and g the gravitational acceleration.
The radiance measures the photon flux from a particular direction, and the irradiance measures the photon flux through a horizontal surface. The irradiance is relevant to atmospheric heating, as expressed by (5.26). However, photolysis of molecules is determined by the flux of photons originating from all directions. This is measured by the actinic flux, whose spectral density or actinic flux density [W m–2 nm–1] is the integral of the monochromatic radiance over all solid angles
(5.28)
The photolysis of atmospheric molecules occurs regardless of the direction of the incident photon and is therefore dependent on the actinic flux rather than the irradiance. For a plane-parallel atmosphere, the actinic flux density at altitude z is given by integration over all zenithal and azimuthal directions μ and φ in spherical coordinates:
(5.29)
The actinic flux density is commonly expressed as a photon flux density qλ(λ, z) [photons m–2 s–1 nm–1]:
(5.30)
At altitude z, the actinic flux q(Δλ, z) for a wavelength interval Δλ [photons m–2 s–1] is obtained by spectral integration of the spectral density qλ over this interval:
(5.31)
Atmospheric chemistry models typically use spectrally integrated actinic fluxes with wavelength intervals Δλ ~ 1–10 nm to calculate photolysis frequencies.
Modeling of Atmospheric Chemistry Page 27