Rotational Raman scattering in the Earth's atmosphere ... - SRON

A doubling-adding method to include multiple orders of rotational Raman scattering 37
where σ ram,i (ν;ν ′ ) is the rotational Raman cross section for each line, b i (J → J ′ ) are the so-called
Placzek-Teller coefficients, J and J ′ are the rotational quantum numbers of the initial state and final
state respectively, and f i (T,J) is fractional population of the initial rotational state where T is the
temperature of the gas [Penney et al., 1974, Joiner et al., 1995, Sioris, 2001]. For our calculations
we use a maximum rotational number of the initial state of J = 50. The wavenumber shifts can be
calculated easily from the energy differences between the initial and final state.
The Rayleigh, Cabannes, and Raman single scattering albedo can be determined from the cross
sections as,
ω ray,air (ν ′ ) = σ ray,air (ν ′ )/σ ext (ν ′ ),
ω cab,air (ν ′ ) = σ cab,air (ν ′ )/σ ext (ν ′ ),
ω ram,air (ν;ν ′ ) = σ ram,air (ν;ν ′ )/σ ext (ν ′ ),
where the total extinction cross section,
(2.30)
σ ext (ν ′ ) = σ ray,air (ν ′ ) + σ abs (ν ′ ) (2.31)
is the sum of the scattering cross section of all scatterers plus the absorption cross section of all
absorbers.
Besides the single scattering albedo, the phase function is needed to describe a scattering event.
The phase functions of Rayleigh, Cabannes, and Raman scattering depend only on the wavenumber
of the incoming light and the scattering angle Θ, which is related to zenith and azimuthal angles µ,
µ ′ , ϕ and ϕ ′ as
cos Θ = µµ ′ + √ 1 − µ 2√ 1 − µ ′2 cos[ϕ − ϕ ′ ]. (2.32)
The effective Rayleigh and Cabannes phase functions for air are given by
P ray,air (ν ′ , Θ) =
∑ [ ]
σray,i (ν ′ )
χ i P
σ ray,air (ν ′ ray,i (ν ′ , Θ), (2.33)
)
i=N 2 ,O 2 ,Ar
P cab,air (ν ′ , Θ) =
∑ [ ]
σcab,i (ν ′ )
χ i P
σ cab,air (ν ′ cab,i (ν ′ , Θ), (2.34)
)
i=N 2 ,O 2 ,Ar
with
[ ] [ 135 +
P ray,i (ν ′ 39εi (ν ′ ) 135 + 3εi (ν ′ )
, Θ) =
+
180 + 40ε i (ν ′ ) 180 + 40ε i (ν ′ )
[ ] [ 540 +
P cab,i (ν ′ 39εi (ν ′ ) 540 + 3εi (ν ′ )
, Θ) =
+
720 + 40ε i (ν ′ ) 720 + 40ε i (ν ′ )
and the phase function for Raman scattering is given by
]
cos 2 Θ, (2.35)
]
cos 2 Θ, (2.36)
P ram,air (Θ) = 39
40 + 3 40 cos2 Θ, (2.37)
which clearly is independent of wavenumber and molecular species (e.g. Stam et al. [2002] and references
therein).