freq. alt. αtot αc αl,tot SR1 SR2
[GHz] [km] [1/m] [1/m] [1/m] [1] [1]
625 0 0.362·10−6 0.152·10−5 -0.116·10−5 -46 47 625 10 0.888·10−7 0.224·10−6 -0.135·10−6 -63 64 625 20 0.542·10−8 0.122·10−7 -0.682·10−8 -69 70 625 30 0.202·10−9 0.503·10−9 -0.301·10−9 -64 65 250 0 0.268·10−6 0.152·10−5 -0.125·10−5 -45 46 250 10 0.690·10−7 0.224·10−6 -0.155·10−6 -57 58 250 20 0.421·10−8 0.122·10−7 -0.803·10−8 -62 63 250 30 0.157·10−9 0.503·10−9 -0.346·10−9 -58 59 90 0 0.721·10−5 0.152·10−5 0.569·10−5 17 -16 90 10 0.117·10−5 0.224·10−6 0.945·10−6 17 -16 90 20 0.656·10−7 0.122·10−7 0.533·10−7 16 -15 90 30 0.264·10−8 0.503·10−9 0.213·10−9 17 -16 22 0 0.273·10−5 0.152·10−5 0.213·10−8 7 -6 22 10 0.405·10−6 0.224·10−6 0.181·10−6 8 -7 22 20 0.222·10−7 0.122·10−7 0.995·10−8 8 -7 22 30 0.911·10−9 0.503·10−9 0.407·10−9 8 -7
Table 3.1: total, continuum, and Line absorption of oxygen at three different frequencies and four different altitudes. The resulting line absorption αl,tot is according to Equation (3.13) divided into two terms which gives the positive and negative fractional line absorption. The subtraction of SR2 and SR1 gives always 1. The total oxygen absorption is denoted by αtot and the continuum absorption term by αc. For each frequency the calculation (using O2-PWR93) is repeated for four altitudes between the ground and 30 km. The atmospheric input is taken from a mid-latitude summer scenario.
0.15-80 times the absorption of oxygen (mid-latitude summer, altitude range: 20-30 km). The absorption of ozone is in all JEM/SMILES bands much larger than the oxygen absorption.
Estimating from Table 3.1.1 the uncertainty of the oxygen absorption to about 1-2.5 times the actual total oxygen absorption17shows that the uncertainty in the total oxygen absorption is comparatively small with respect to HCl but is not negligible in case of ClO in the lower half of the stratosphere. However, the uncertainty in the nearly constant oxygen absorption can be included as an additional offset in retrieval schemes which is thus not crucial for trace gas VMR retrievals. Nevertheless, for instance in laboratory measurements for line broadening parameters with oxygen as buffer gas this uncertainty could be of importance.
Figure 3.11: Contributions (in percent) from each oxygen line of the 60 GHz band (αN) to the effective line absorption αl,tot at a frequency of 625 GHz. Since αl,tot is negative at this frequency, positive percent numbers correspond to negative absorption values and vice versa. The contributions from the SMMW lines are summed up at the N=0 point. Mid-latitude zonal mean atmospheric conditions are selected. The abscissa indicates the rotational quantum numberN (P-branch: N−, R-branch: N+)
Φ ≈-8.5×10-60 Cm4 are the quadrupole and hexadecapole moments, respectively. Beside these long range interactions there is a additional contribution from short range interactions of overlapping nitrogen molecules. The influence of this type of interaction is only remark-able in the region where the intermolecular potential is repulsive (V(ro)≥0). This point is approximately reached atro ≈7·ao, whereao is the Bohr radius18.
The induced dipole, µind, is proportional to the electric field, Eloc, at the location of the molecule with the molecular polarizability as the proportionality factor:
µind = ( ˜α+ ˜γ)·Eloc (3.16)
The polarizability is in general a tensor and depends on the direction with respect to the symmetry axis of the molecule in question. For an axially symmetric molecule like N2, the polarizability can be described by its isotropic part, ˜α = ˜α|| + 2 ˜α⊥, and its anisotropy,
˜
γ = ˜α|| −α˜⊥, where ˜α|| and ˜α⊥ are the polarizabilities along and perpendicular to the molecular axis. Along this axis the polarizability of the N2 molecule is three times higher than perpendicular to this axis, making it necessary to consider the angular dependence in the calculation of µind. The multipole structure ofEloc and the asymmetric polarizability makes it advantageous to expand the induced dipole in a series of spherical harmonics19. Concerning nitrogen,Borysow and Frommhold (1986a) obtained accurate results with seven
18For comparison, the bond length of a nitrogen molecule is aboutrNN≈2·ao.
19The non-negative integer expansion parameters are denoted by{λ1λ2,Λ, L}, where Λ denotes the electric multipole component, i.e. quadrupole: Λ = 2, octupole: Λ = 3, hexadecapole Λ = 4. The inversion symmetry of N2allows only evenλ1,2 and the parity conservation restricts the sumλ1+λ2+Lto only odd values. Since Clebsch-Gordan coefficients are involved in the expansion, the so called triangular inequality must be fulfilled too. This is achieved by|λ1−λ2| ≤Λ≤λ1+λ2 and|L−1| ≤Λ≤L+ 1 (Meyer et al., 1989).
Figure 3.12: Contributions of the different interaction terms to the total N2-N2 collision in-duced absorption. The unit of the absorption is chosen as [m−1 Pa−2], thus it is pressure independent. The calculation is performed at a temperature of 300 K with the program of Borysow (2001). The notation of the interaction terms is as follows: 3220/3202 is the quadrupole-induced dipole transition component via the trace of the molecule’s polarizabil-ity tensor, 5440/5404 is the hexadecapole-induced dipole transition component via the trace of the molecule’s polarizability tensor, 3322 is the quadrupole-induced dipole transition com-ponent via the anisotropy of the molecule’s polarizability.
expansion terms: two for the short range molecular overlap, two quadrupole components in connection with the isotropic part and one quadrupole component for the anisotropy of the polarizability as well as two hexadecapole components through ˜α.
After the discussion of the induced dipole of N2 molecules due to molecular collisions, the rotational spectrum has to be discussed. the selection rule for rotational transitions of N2
molecules with an induce dipole is ∆J = ±0,±2,±4, where ∆J is the change of the total angular momentum J. The Q-branch (∆J = ±0) gives rise to a translational continuum spectrum. Translational states are defined by the angular momentum quantum number l and the relative velocityv (at large distances) of the colliding N2 molecules. Translational transitions are between two energy states Ei → Ef with Ei,f = 12m vi,f. The rotational component of translational transitions (i.e. roto-translational transitions) are due to an additional change of the angular momentum quantum numbers,li →lf, during the collision (Frommhold, 1987). The line width of the S- and O-branches (∆J =±2) are determined by the duration of the collision,τcol. According toPoll and Hunt (1976), the line widthγ, can be estimated to
γ ≈ 1 τcol
≈ 2π d ·
s3kBT mred
(3.17) where d is a measure of the interaction range and mred is the reduced mass of the N2-N2 system. For d=7ao and a temperature range of 200 to 300 K, γ is of the order of several THz which makes the individual lines un-resolvable. For this reason, the total absorption spectrum is a smooth function of frequency. The intensities of the ∆J =±4 transitions are negligible under atmospheric conditions and can therefore be neglected in the calculations.
Collisions of two molecules have a large bandwidth of possible interaction strengths, reach-ing from collision pairs (free-free transitions) up to bound complexes (bound-bound
transi-tions), with all its intermediate possibilities (Frommhold, 1987). According to calculations of Vigasin (1996), the free collision pairs (≈70-80 %) and the metastable complexes (≈20-25 %) dominate the nitrogen collision induced absorption at temperatures of 200-300 K, while the bound dimer complexes with their sharp rotational transitions contribute less than 2 %.
Thus the line spectrum of nitrogen dimers is only marginal and therefore negligible for the STHzrange. Figure 3.12 shows the different absorption components of nitrogen in theSTHz and FIR frequency range. The quadrupole-induced dipole transitions via the trace of the molecules polarizability tensor is the dominant component in theSTHzrange. All the other components are at least two orders of magnitude smaller in this frequency range.
3.2.1 Common Atmospheric Absorption Models
Four different N2 absorption models for the microwave region are commonly used in radiative transfer models. These are N2-BF86 (Borysow and Frommhold, 1986a), N2-MPM93(Liebe et al., 1993), N2-MPM89 (Liebe, 1989), and N2-PWR93 (Rosenkranz, 1993). The only quantum mechanical model is that of N2-BF86 while the others are suitable parameteriza-tions for planetary atmospheres. This restricts their applicability in terms of frequency and temperature. In the following the models are briefly described in view of their formulation and validity range.
N2-BF86
The most detailed model for theFIR roto-translational band is N2-BF86 of Borysow and Frommhold(1986a). They derived an absorption model based on a quantum mechanical basis.
The considered induced dipole mechanisms are the quadrupole and hexadecapole induction as well as the molecular overlap. The dimer absorption is neglected since it is negligible compared to the other components. The dominant absorption term in the STHz range is coming from the quadrupole-induced dipole through the trace of the N2 polarizability tensor (see Figure 3.12). The line shape function used in this model is based on Birnbaum and Cohen (1976). This line shape includes an extra parameter which describes the duration of the molecular collision. It is interesting to note that under certain conditions this line shape has approximately the form of the Van Vleck-Weisskopf line shape in the line center region.
N2-BF86 agrees with the FTIR data of Stone et al. (1984) which covers the frequency range of 750 GHz to 11 THz. This measurement was supplemented by independent laser measurements. The independent FTIR and laser measurements show good agreement, docu-menting the high quality of this data set. This is of some importance since the data ofStone et al.(1984) differ substantially from the measurements of Buontempo et al.(1975) at least in the band center (Borysow and Frommhold, 1986a).
N2-MPM
The versions N2-MPM89(Liebe, 1989) and N2-MPM93(Liebe et al., 1993) of Liebe’s MPM model are based on fits of laboratory measurements fromStone et al. (1984):
αMPM89N2 = C·Pd2·Θ3.5·ν2·(1.0−1.2·10−5·ν1.5) (3.18) αMPM93N2 = C·Pd2·Θ3.5·ν2·(1.0 + 1.9·10−5·ν1.5)−1
where Θ = 300 K/T and C = 2.55·10−13dB/(km hPa2GHz2). The two parameterizations look different in their frequency dependence. However, the ratio
αMPM89N2
αMPM93N2 = 1 + 0.73·10−5·ν1.5−2.316·10−10·ν3
gives a value of 1 + 7.5·10−4 for a frequency of 1 THz thus implying the equivalence of both expressions in theSTHz range.
N2-PWR
The estimated absorption coefficient deduced from the laboratory measurements of Dagg et al.(1975) at 70 GHz is well parameterized with
αNCIA2 = Θ3·ν2·PN2
2·(C2·Θ0.55+C3·PN2·Θ2.56) (3.19) where Θ = 300 K/T and C2 = (4.59±0.06)·10−13dB/(km hPa2GHz2) and C3 = (5.60± 0.30)·10−19dB/(km hPa3GHz2). The N2-PWR93 absorption model of Rosenkranz (1993) incorporates the term withC2 and neglects the term proportional to the third power of the nitrogen pressure. However the omitted term is negligible at atmospheric conditions.
Calculations with three absorption models are shown in Figure 3.13. Furthermore, the frequency ranges where the models are valid are indicated. N2-PWR93 is the most limited model since it is based on measurements at a single frequency. N2-MPM93 is designed to produce reliable values up to 1 THz and N2-BF86is valid for the whole range of the absorp-tion band. The relative differences shown in Figure 3.14 indicate the model limitaabsorp-tions more clearly. N2-PWR93 and N2-BF86 differ by less than 10 % below 200 GHz. This difference increases to 80 % at 1 THz. N2-MPM93yields less absorption in theMWand MMWrange and more absorption in the SMMW range compared to N2-BF86. The difference between N2-MPM93and N2-BF86is with respect to the wholeSTHzrange between -15 % and 15 % for pressures higher than 10 hPa.
In connection with the dry air absorption in the Earth’s atmosphere it is interesting to note thatPardo et al.(2001b) scaled their dry air opacity with a factor of 1.29 to get a better agreement between their ground based measurements taken at Mauna Kea (Hawaii, USA) and the N2-BF86 and N2-MPM93absorption models. They claim that the additional 29 % arises from collision induced absorption of other molecular interactions like for instance N2 -O2 (G. Moreau et al., 2001), O2-O2 (Bosomworth and Gush, 1965;G. Moreau et al., 2000), N2-Ar (Wishnow et al., 1996), CO2-CO2 (Dagg et al., 1978), and CH4-CH4 (Borysow and Frommhold, 1986b). Since not for all these interactions absorption models are available, it is not possible to verify this hypothesis directly. Especially at dry atmospheric conditions this absorption component should be noticeable. Compared to the N2 absorption model differences of less than 15 % in the valid frequency ranges (see Figure 3.14), the additional 29 % due to non-N2 collision induced absorption would mean a significant increase in the dry air absorption component. This increase in dry air absorption will be tested in Chapter 5 in a comparison with radiometer data measured under sub-arctic winter conditions.