Rayleigh scattering
Contents
Cross section
The effective cross section for the i-th particle of small size relative to the wavelength and in the electric dipole approximation is equal to\[\sigma_{\nu,i} = \frac{24\pi^3 \nu^4}{N^2 } \left(\frac{n_{\nu,i}^2 -1}{n_{\nu,i}^2 +2}\right)^2 \left( \frac{6+3\delta_{\nu,i}}{6-7\delta_{\nu,i}}\right)\]
where \(\nu\) is the wave number in \(m^{-1}\), \(N\) is the molar volume in \(molecule.m^{-3}\), \(n_{\nu,i}\) is the real refractive index, \(\delta_{\nu,i}\) is the depolarisation factor.
The term \(\left( \frac{6+3\delta_{\nu,i}}{6-7\delta_{\nu,i}}\right)\) is also called the King factor and is denoted by \(F_k (\nu,i)\). So,\[\sigma_{\nu,i} = \frac{24\pi^3 \nu^4}{N^2 } \left(\frac{n_{\nu,i}^2 -1}{n_{\nu,i}^2 +2}\right)^2 F_k (\nu,i)\]
The volume number can be obtained from the ideal gas law. Thus, the cross-sectional area depends on the temperature \(T\) and the pressure \(P\). The refractive index also depends on the conditions \(T,P\), but in scientific literature, this dependence is very often unknown. Therefore, it is not possible to make the cross section dependent on \(T\) and \(P\) if the dependence of the refractive index on \(T\) and \(P\) is unknown. It is therefore necessary to fix \(T\) and \(P\) for the calculation of the cross section at the \(T\) and \(P\) where the refractive index was measured. The cross section is expected to vary little with \(T,P\) in the case of a gas, but due to the dependence of \(N\) on \(T,P\) and the variation of the refractive index with \(T,P\), if any dependence is omitted, the cross section can be significantly erroneous by several orders of magnitude. The H2O molecule is the only one for which the variation of the refractive index with \(T\) and \(P\) has been measured or estimated. Thus, the variation of the cross section with \(T\) and \(P\) can be applied to this molecule using the formula introduced below.
Rayleigh scattering, Brillouin scattering (not applicable in atmospheres) and Raman scattering are not taken into account in the calculation of this cross section. Only Rayleigh scattering (also known as the Cabannes peak) is taken into account.
The refractive indices and King factors of each molecule studied in the Generic PCM are described below.
CO2
Reference: Sneep and Ubachs (2005) https://doi.org/10.1016/j.jqsrt.2004.07.025
\(n_{\nu,i} = 1 + 1.1427\times10^3\left(\frac{5799.25}{128908.9^2 - \nu^2} + \frac{120.05}{89223.8^2 - \nu^2} + \frac{5.3334}{75037.5^2 - \nu^2} + \frac{4.3244}{67837.7^2 - \nu^2} + \frac{0.1218145\times10^{-4}}{2418.136^2 - \nu^2}\right)\)
\(F_k (\nu,i) = 1.1364 + 25.3\times10^{-12}\nu^2\)
The interpolation formula applies for a temperature of 288.15 K and a pressure of 1013.25 hPa. It is valid between 0.1807 and 1.8172 \(\mu m\).
To avoid singularities at wavelengths below 0.1807 \(\mu m\) associated with the previous power-of-four formulation, we extrapolate using the following formula:
Reference: Cuthbertson and Cuthbertson (1920) https://doi.org/10.1098/rspa.1920.0020
\(n_{\nu,i} = 1 + \left(\frac{6914.45}{156.85 - \nu^2}\right)\times 10^{-5}\)
It applies to a temperature of 273.15 K and a pressure of 1013.25 hPa.
N2
Reference: Sneep and Ubachs (2005) https://doi.org/10.1016/j.jqsrt.2004.07.025
For 4860 \(cm^{-1} < \nu < \) 21360 \(cm^{-1}\)
\(n_{\nu,i} = 1 + \left(5677.465 + \frac{318.81874\times10^{12}}{14.4\times10^9 - \nu^2}\right)\times10^{-8}\)
\(F_k (\nu,i) = 1.034 + 3.17\times10^{-12}\nu\)
For 21360 \(cm^{-1} < \nu < \) 39370 \(cm^{-1}\)
\(n_{\nu,i} = 1 + \left(6498.2 + \frac{307.4335\times10^{12}}{14.4\times10^9 - \nu^2}\right)\times10^{-8}\)
\(F_k (\nu,i) = 1.034 + 3.17\times10^{-12}\nu\)
Above 39370 \(cm^{-1}\), extrapolation is performed using the formula valid between 21,360 and 39370 \(cm^{-1}\).
The interpolation formulas apply for a temperature of 288.15 K and a pressure of 1013.25 hPa.
H2O
Reference: Harvey et al. (1998) https://doi.org/10.1063/1.556029
In the case of water, the refractive index is known to depend on T and P. However, it should be noted that the interpolation formula for water vapour is based on very incomplete data for certain wavelength ranges at certain temperatures and pressures (see reference for more details).
\( a_0 = 0.244257733 \)
\( a_1 = 9.74634476\times10^{-3} \)
\( a_2 = -3.73234996\times10^{-3} \)
\( a_3 = 2.68678472\times10^{-4} \)
\( a_4 = 1.58920570\times10^{-3} \)
\( a_5 = 2.45934259\times10^{-3} \)
\( a_6 = 0.900704920 \)
\( a_7 = -1.66626219\times10^{-2} \)
\( \lambda_{UV} = 0.2292020\; \mu m \)
\( \lambda_{IR} = 5.432937\; \mu m \)
\( \rho_0 = 1000\; kg/m^3 \)
\( T_0 = 273.15\; K \)
\( \lambda_0 = 0.589\; \mu m \)
\(b_{\lambda,i, T, p} = \left( a_0 + a_1\bar{\rho} + a_2\bar{T} + a_3\bar{T}\bar{\lambda}^2 + \frac{a_4}{\bar{\lambda}^2} + \frac{a_5}{\bar{\lambda}^2 - \lambda_{UV}^2} + \frac{a_6}{\bar{\lambda}^2 - \lambda_{IR}^2} \right)\)
With \( \bar{\rho} = \frac{\rho}{\rho_0} \) \( \bar{T} = \frac{T}{T_0} \) \( \bar{\lambda} = \frac{\lambda}{\lambda_0} \)
\(n_{\lambda,i, T, p} = \frac{\sqrt{2b_{\lambda,i, T, p}+1}}{\sqrt{1-b_{\lambda,i, T, p}}}\)
Please note that the coefficients in the refractive index interpolation formula are in wavelength (\(\mu m\)). If you want to use wave numbers in \(cm^{-1}\), simply replace \(\lambda\) with \(10000/\nu(cm^{-1})\), which is equal to \(\lambda(\mu m)\).
For King's factor, this is the depolarisation coefficient \(\delta_{\nu,i}\), which is known. It is equal to \(3\times10^{-4}\). The source of this factor comes from Murphy (1977) https://doi.org/10.1063/1.434794
\(F_k (\nu,i) = \left( \frac{6+3\times3\times10^{-4}}{6-7\times3\times10^{-4}}\right)\)
The interpolation formulas are valid between 0.2 and 1.1 \(\mu m\).
To avoid singularities at lengths shorter than 0.2 \(\mu m\) associated with the previous formulation, we extrapolate using the following formula:
Reference: Barrell and Sears (1939) https://doi.org/10.1098/rsta.1939.0004
\(n_{\nu,i} = 1 + \left(245.40+2.187\times\left(\frac{10000}{\nu}\right)^{-2} \right)\times 10^{-6}\)
It applies to a temperature of 273.15 K and a pressure of 1013.25 hPa.
To avoid singularities at lengths greater than 1.1 \(\mu m\) associated with the previous formulation, we extrapolate the following formula:
Reference: Ciddor (1996) https://doi.org/10.1364/AO.35.001566
\(n_{\nu,i} = 1 + 1.022\times10^{-8}\left(295.235 + 2.6422\times(\nu\times10^{-4})^2 - 0.03238\times(\nu\times10^{-4})^4 + 0.004028\times(\nu\times10^{-4})^6\right)\)
It applies to a temperature of 293.15 K and a pressure of 1333 Pa.
H2
Reference: Peck and Huang (1977) https://doi.org/10.1364/JOSA.67.001550
\(n_{\nu,i} = 1 + \left(\frac{14895.6}{180.7 - (\nu\times10^{-4})^2} + \frac{4903.7}{92-(\nu\times10^{-4})^2}\right)\times10^{-6}\)
For King's factor, this is the depolarisation coefficient \(\delta_{\nu,i}\), which is known. It is equal to 0.02. The source of this factor comes from Hansen and Travis (1974) https://doi.org/10.1007/BF00168069
\(F_k (\nu,i) = \left( \frac{6+3\times0.02}{6-7\times0.02}\right)\)
The interpolation formula applies for a temperature of 273.15 K and a pressure of 1013.25 hPa. It is valid between 0.1680 and 1.6945 \(\mu m\).
To avoid singularities at lengths shorter than 0.1680 \(\mu m\) associated with the previous formulation, we extrapolate using the following formula:
Reference: Peck and Huang (1977) https://doi.org/10.1364/JOSA.67.001550
\(n_{\nu,i} = 1 + \left(23.79+\frac{12307.2}{109.832 - (\nu\times10^{-4})^2}\right)\times10^{-6}\)
It applies to a temperature of 273.15 K and a pressure of 1013.25 hPa.
He
Reference: Thalman et al. (2014) https://doi.org/10.1016/j.jqsrt.2014.05.030
\(n_{\nu,i} = 1 + \left(2283 + \frac{1.8102\times10^{13}}{1.5342\times10^{10} - \nu^2}\right)\times10^{-8}\)
\(F_k (\nu,i) = 1\)
The interpolation formula applies for a temperature of 288.15 K and a pressure of 1013.25 hPa. It is valid between 0.2753 and 20.5813 \(\mu m\).
CH4
Reference: Sneep and Ubachs (2005) https://doi.org/10.1016/j.jqsrt.2004.07.025
\(n_{\nu,i} = 1 + 46662\times10^{-8} + 4.02\times10^{-14}\nu^2\)
The depolarisation factor is unknown. It is therefore set at 1.
\(F_k (\nu,i) = 1\)
The interpolation formula applies for a temperature of 288.15 K and a pressure of 1013.25 hPa. It is valid between 0.3251 and 0.6330 \(\mu m\).
CO
Reference: Sneep and Ubachs (2005) https://doi.org/10.1016/j.jqsrt.2004.07.025
\(n_{\nu,i} = 1 + 22851\times10^{-8} + \frac{0.456\times10^4}{71427^2 - \nu^2}\)
\(F_k (\nu,i) = 1.016\)
The interpolation formula applies for a temperature of 288.15 K and a pressure of 1013.25 hPa. It is valid between 0.168 and 0.288 \(\mu m\).
Ar
Reference: Thalman et al. (2014) https://doi.org/10.1016/j.jqsrt.2014.05.030
\(n_{\nu,i} = 1 + \left(6432.135 + \frac{286.06021\times10^{12}}{14.4\times10^9 - \nu^2}\right)\times10^{-8}\)
\(F_k (\nu,i) = 1\)
The interpolation formula applies for a temperature of 288.15 K and a pressure of 1013.25 hPa. It is valid between 0.288 and 0.546 \(\mu m\).
O2
Reference: Bates (1984) https://doi.org/10.1016/0032-0633(84)90102-8
For \(\nu < \) 18315 \(cm^{-1}\)
\(n_{\nu,i} = 1 + \left(21351.3 + \frac{21.85670}{4.09\times10^9 - \nu^2}\right)\times10^{-8}\)
For 18315 \(cm^{-1} < \nu < \) 34722 \(cm^{-1}\)
\(n_{\nu,i} = 1 + \left(20564.8 + \frac{24.80899}{4.09\times10^9 - \nu^2}\right)\times10^{-8}\)
For 34722 \(cm^{-1} < \nu < \) 45248 \(cm^{-1}\)
\(n_{\nu,i} = 1 + \left(22120.4 + \frac{20.31876}{4.09\times10^9 - \nu^2}\right)\times10^{-8}\)
For \(\nu > \) 45248 \(cm^{-1}\)
\(n_{\nu,i} = 1 + \left(23796.7 + \frac{16.89884}{4.09\times10^9 - \nu^2}\right)\times10^{-8}\)
\(F_k (\nu,i) = 1.09 + 1.385\times10^{-11}\nu^2 + 1.488\times10^{-20}\nu^4\)
The interpolation formulas apply for a temperature of 288.15 K and a pressure of 1013.25 hPa. They are valid between 0.198 and 2.0 \(\mu m\). Beyond this range, extrapolation is used.
Mass extinction coefficient
The mass extinction coefficient is related to the molar cross section of scattering by
\(k_{\nu,i} = \sigma_{\nu,i} \frac{N_A}{\bar{M}}\)
where \(N_A\) is Avogadro constant and \(\bar{M}\) is the average molecular mass of the atmosphere.
For a gas mixture, the extinction coefficient is equal to
\(k_{\nu} = \sum_{i=1}^N x_i k_{\nu,i}\)
Optical depth
Optical thickness is defined as
\(\tau_{\nu} = \int_{z_1}^{z_2} \sigma_{\nu} N dz\)
\(\tau_{\nu} = \int_{z_1}^{z_2} k_{\nu}\rho dz\)
However, according to the hydrostatic balance
\(dp = \rho g dz\)
Then, we have
\(\tau_{\nu} = \int_{z_1}^{z_2} k_{\nu}\frac{dp}{g}\)
A coefficient \(scalep\) is added to convert \(Pa\) to \(mbar\) used in radiative transfer.
Rayleigh scattering in Generic PCM
The Rayleigh scattering routine is determined for each spectral band defined in the correlated-k routine and at each pressure level. The routine follows the path below:
- Each spectral band \(NW\) is recut by the \(Nfine\) parameter. Rayleigh scattering calculations are performed on a loop on Nfine for each spectral band.
- The refractive index \(ng\) and the King factor \(Fk\) are set.
- The extinction coefficient is calculated in two stages. \(Tauvari\), the part dependent on the wavenumber, is calculated, then \(Tauconsti\), the part dependent on pressure and temperature (if it exists). Otherwise, the reference pressure \(P0\) and temperature \(T0\) of the refractive index measurements are used for \(Tauconsti\).
- The product \(Tauvari \times Tauconsti\) is calculated at the end to give the extinction coefficient \(Tauvar\) .
- After this loop, the extinction coefficient is then weighted by the black body function for each spectral band as \( \text{TAURAY(P,NW)} = \frac{\int_{\nu' \in \text{band}} \text{TAUVAR} (P,\nu') B_{\nu} \, \mathrm{d}\nu'}{\int B_{\nu} \, \mathrm{d}\nu'} \) .
- Finally, the optical depth is calculated in optcv.F90, where the thickness of the layer \(dP\) is added (\(g\) was already added on \(Tauconsti\)).
Each unit in this routine follows the International System of Units. HOWEVER, for all other routines in the radiative part of the model, the pressure is in mbar/hPa! That is why we add a \(scalep\) factor at the end for \(Tauray\).
If you want to add a new molecule, simply enter its refractive index and King factor and use the same \(Tauvari\) and \(Tauconsti\) formulas, as for the previous molecules.
Rayleigh scattering in exo_k
Rayleigh routine in exo_k is available here :
Exo_k uses formalism from :
Caldas (2019) Effects of a fully 3D atmospheric structure on exoplanet transmission spectra: retrieval biases due to day–night temperature gradients : https://hal.archives-ouvertes.fr/hal-02005332/document
Have a look on equation (12) & appendix D
Formalism
We consider a layer.
dP is the difference of pressure between the two levels that define the layer.
dN is the number of molecules per m2 & dm is the mass per m2 of the layer
dTau is the optical depth for a given wavelength (or wavenumber)
\(m_{molecule}\) is the mass of one molecule of the considered gas
g is the gravity
sigma_mol is the Rayleigh scattering cross section of the molecule
dTau = sigma_mol * dN
sigma_mol = sigma_mol(wavenumber in cm-1)
which gives : dTau = sigma_mol \( \displaystyle \frac{dm}{m_{molecule}} \)
and then : dTau \( \displaystyle = \frac{\text{sigma_mol}}{g * m_{molecule}} dP\)
dP here is in Pascal
Relations between LMDZ & Exo_k formalisms
LMDZ & exo_k formalism are linked as following \[ \displaystyle \text{(tauconsti * tauvari)} = \frac{\text{sigma_mol}}{g * m_{molecule}} * \text{scalep}\]
Be careful with units !!! (cm-1 for wavenumbers in exo_k, microns for wavelengths in LMDZ)
Last but not least : not to forget the scalep factor in LMDZ ! scalep = 100, because P is in mBar in optcv.F90 . So we move from Pascal to mBar
