Scattering processes

There are three main types of scattering processes in the atmosphere: Rayleigh, Mie and Raman scattering. Elastic scattering processes by air molecules (Rayleigh scattering) or atmospheric aerosol particles (Mie scattering) change the propagation direction of light, but the photon wavelength is unchanged. Inelastic Raman scattering also changes photon energy in addition to the propagation direction and therefore shifts the wavelength. For inelastic Raman scattering, the photon energy can either be reduced (Stokes scattering) or increased (anti-Stokes scattering).

Rayleigh scattering

Rayleigh scattering corresponds to elastic scattering of light where no energy is exchanged between the photon and the particle. It occurs when the scattering particles are dielectric and their radius r is much smaller than the wavelength of the incoming radiation (r  λ). This is the case for air molecules or small aerosol particles with sizes below ≈ 1 µm. The incoming light induces an oscillating dipole with a Hertz dipole radiation pattern in the dielectric particle, leading to the emission of the same frequency at which is was polarized (e.g. van de Hulst, 1981). The wavelength dependency of the Rayleigh scattering cross-section is very strong:

σ_R∝ λ⁻⁴. (3.12)

Thus, light with a small wavelength is scattered to a greater extent than light with a large wave-length, i.e. Rayleigh scattering is particularly effective in the UV wavelength range. Blue light, with its shorter wavelength (λ ≈ 425 nm), is scattered more strongly than red light (λ ≈ 650 nm).

Hence, the colour of the sky appears to be blue, whereas the colour of the Sun during sunrise and sunset is red because the red component of the sunlight is scattered less by air molecules relative to the blue component. The Rayleigh extinction coefficient R for a number density of air molecules N_air can be determined according to:

_R= Nair· σ_R. (3.13)

Compared with scattering on large particles (Section 3.2.2), the angular dependency of the Rayleigh phase function Φ is relatively small (Figure 3.2). For unpolarised light, it is proportional to 1+cos²θ and is given by (e.g. Platt and Stutz, 2008):

Φ_R(cos(θ)) = 3

4· (1 + cos²(θ)), (3.14)

where θ is the scattering angle between the directions of incoming and scattered light/photons.

Considering the anisotropic polarisability (red dashed line in Figure 3.2) according to Penndorf (1957), Equation 3.14 can be changed to:

ΦR(cos(θ)) = 0.76 · (0.93 + cos²(θ)). (3.15)

Figure 3.2: Polar diagram of the Rayleigh phase function Φ_R for unpolarised incoming light. The dotted line represents the contribution of light that is polarised parallel to the scattering plane and the dash dotted line shows the contribution of light that is polarised perpendicular to the scattering plane, adapted from H¨onninger (2002).

3.2. Interaction processes of radiation in the atmosphere 31

Mie scattering

Mie scattering is an elastic process during which the light is scattered by particles (assumed to be spherical) with a diameter d comparable to or larger than the wavelength of the incoming light (d ≥ λ). In the atmosphere, these particles are aerosols, e.g. dust particles, sea salt droplets or particles emitted by biomass burning or volcanoes (Section 2.4.10). Their size ranges from 0.1 to 10µm. The Mie scattering phase function depends on the aerosol size parameter x:

x = 2 · π · r

λ (3.16)

with the radius r of the particle and the wavelength λ of the incident radiation. For x  1, , the scattered electromagnetic radiation corresponds to that of an oscillating dipole and the scattering process can be approximated by Rayleigh scattering. For x  1, the problem can be described by classic geometrical optics, as e.g. during the scattering on water droplets. Between these two possibilities, scattering processes are referred to as Mie scattering. In contrast to Rayleigh scattering, Mie scattering has a much weaker wavelength dependency. The Mie scattering cross-section for a typical distribution of particles in the atmosphere is given by:

σM ∝ λ^−1.3, (3.17)

which causes the white colour of clouds and the whitening of the colour of the sky if aerosols are present (Roedel, 2000). Figure 3.3 shows some examples of Mie scattering phase functions for water droplets at a wavelength of 550 nm. Obviously, as the aerosol size parameter x increases, forward scattering begins to dominate. However, as the size, shape and composition of aerosols in the atmosphere can differ strongly, the determination of σM can be very complex. Using numerical models, the Mie scattering phase function can be approximated by only a few observable parameters.

A common method is the Henyey-Greenstein parameterisation (Henyey and Greenstein, 1941) where the Mie scattering phase function is expressed as:

Φ_HG(cos(θ)) = 1 − g²

4π · (1 + g²− 2g · cos(θ))³², (3.18) with the asymmetry parameter g, which is defined as the intensity-weighted average cosine of the scattering angle θ: g = hcos(θ)i (Andrews et al., 2006). The asymmetry parameter g ranges between -1 and 1. For g = 0 the scattering is isotropic, for g = 1 complete forward scattering dominates, whereas for g < 0 backward scattering occurs (e.g. van de Hulst, 1981). Typically, the values for g for tropospheric aerosols at ambient relative humidity lie between 0.59 and 0.72 (Andrews et al., 2006).

Figure 3.3: Examples of phase functions for Mie scattering at 550 nm for water droplets at x = 1, 3, 10. The left panels represent the phase function on a logarithmic scale. The dotted and dashed lines represent the two orthogonal polarisation states, whereas the solid lines represents unpolarised light. The right panels show polar diagrams of the respective phase function on a linear scale, adapted from Sanghavi (2003).

Raman scattering

Inelastic scattering of photons by air molecules is referred to as Raman scattering, named after the Indian physicist C.V. Raman, who discovered the effect in 1928 (Raman, 1928). Raman scattering

3.2. Interaction processes of radiation in the atmosphere 33

changes the energy of the scattering photon in addition to the change in the propagation direction.

Raman scattering leads to two possible states:

ˆ Stokes-Raman scattering: The air molecule absorbs a part of the photon’s energy and the emitted photon has a lower energy than the absorbed photon,

ˆ Anti-Stokes-Raman scattering: The photon takes over part of the excitation energy of the air molecule and the emitted photon has a higher energy than the absorbed photon.

In general, Anti-Stokes-Raman lines are weaker than Stokes Raman lines since the excited state has to be occupied. Figure 3.4 shows the position of the rotational and vibrational Raman lines in the spectrum. The rotational Raman lines are arranged around the vibrational Raman lines and the Rayleigh line.

Figure 3.4: Schematic drawing of rotational and vibrational Raman lines of a diatomic molecule, adapted from Dinter (2005).

If there is no change in the vibrational state (ν = 0), the term rotational Raman scattering (RRS) is used. Otherwise, if the vibrational state is also affected (ν = ±1), the scattering process is called rotational-vibrational Raman scattering (VRS). The RRS is approximately one order of magnitude stronger than the VRS but still weaker than Rayleigh scattering. Raman scattering may be observed in atmospheric measurements of scattered sunlight as a filling-in of the Fraunhofer lines.

This so-called Ring effect (Grainger and Ring, 1962) (Section 4.3.4) is considered to be caused by RRS and has to be accounted for in the spectral retrieval.