1. TEMPERATURE DEPENDENCES OF MECHANISMS RESPONCIBLE FOR THE WATER-VAPOR CONTINUUM Q. Ma NASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Mathematics, Columbia University 2880 Broadway, New York, NY 10025, USA R. H. Tipping Department of Physics and Astronomy, University of Alabama Tuscaloosa, AL 35487, USA C. Leforestier Institut Charles Gerhardt CNRS-5253, CC15.01, Université Montpellier 2 34095, Montpellier, France

2. Water-Vapor Continuum and Possible Physical Explanations  The water-vapor continuum absorption plays an important role in determining the radiation in the Earth’s atmosphere. To find its physical explanations is still an open problem.  Basic features of the water-vapor continuum absorption: 1) depends on the pressures quadratically. 1) depends on the pressures quadratically. 2) varies with the frequency smoothly. 2) varies with the frequency smoothly. 3) in general, has a negative temperature dependence. 3) in general, has a negative temperature dependence.  Three mechanisms have been suggested: the far-wings of allowed transition lines, collision-induced absorption, and water dimers.  Absorptions derived from three mechanisms have different magnitudes and different T dependences.  In the present study, we mainly analyze their T dependences.

3. I. The Far-Wing Theory  Facts about the continuum of the far-wing theory: (1) The total absorption consists of contributions from local H 2 O lines and from remote lines. How to divide these two components depends on people’s choices. (1) The total absorption consists of contributions from local H 2 O lines and from remote lines. How to divide these two components depends on people’s choices. (2) Two components are inseparable. When one talks about the continuum, one must talk about the local line absorption also. (2) Two components are inseparable. When one talks about the continuum, one must talk about the local line absorption also. (3) The second component exhibits similar characteristics of the water-vapor continuum and is referred as the continuum from allowed lines. (3) The second component exhibits similar characteristics of the water-vapor continuum and is referred as the continuum from allowed lines.  Within bands, the total absorptions mainly come from local lines. There are uncertainties in calculating local line absorptions that are mainly associated with line spectroscopic parameters and line shape models used. The uncertainties could overshadow the continuum.  In windows between the H 2 O bands, there are few strong lines. The continuum becomes the dominant component. Thus, we can talk about the continuum with certainty in the windows.

4. I-A. Continuum from the Far-Wing Theory  In terms of the band-averaged line shape function the continuum absorption is given by  We introduce the coordinate representation by choosing orientations of two interacting molecules as the basis set of Hilbert space where Ω aζ and Ω bζ represent orientations of the absorber and bath molecules, respectively. where Ω aζ and Ω bζ represent orientations of the absorber and bath molecules, respectively.  In the coordinate representation, interaction potentials V are diagonal and can be treated as ordinary functions where are values of V at the specified orientation ζ. where are values of V at the specified orientation ζ.

5. I-B. Far-Wing Line Shapes  Far-wing line shapes depend on the molecular pair of interest and on the interaction potential between these two molecules. where V 1 is the anisotropic part of the potential, V 1,ζ =, V΄ 1,ζη (r) = dV 1,ζ (r)/dr - dV 1,η (r)/dr, and r c is determined from the equation where V 1 is the anisotropic part of the potential, V 1,ζ =, V΄ 1,ζη (r) = dV 1,ζ (r)/dr - dV 1,η (r)/dr, and r c is determined from the equation  We adopt the five-site TIP5P potential model by assuming the potential consists of the Coulomb site-site interactions and the Lennard-Jones site-site interaction.  With the Monte Carlo method, one can evaluate the 11-dimensional integrations. In general, by using about 10 7 random selections of the orientation variables one can derive converged line shapes within reasonable CPU times.

6. I-C. Factors Affecting T Dependence of the Continuum from Allowed Lines  Factors affecting the T dependence of the continuum: (1) Number density n a of the absorber molecule. (1) Number density n a of the absorber molecule. (2) Line intensities S (2) Line intensities S ij. (3) The line shape functions. More explanations are given later. (3) The line shape functions. More explanations are given later.  Effect from n a : It depends on the units of the absorption coefficients. If α(ω) is given in units of cm 2 /(molecule atm), n a does not cause any T dependence.  Effect from S: For an individual line, how its intensity  Effect from S ij : For an individual line, how its intensity varies with T is well known. However, we need to know how the line intensity distributions vary with T. For ω 250cm -1, the trend is reversed. For 900cm -1 < ω < 1300cm -1, line strengths at higher T are significant larger than those at lower T.

7. I-D. The T Dependence of the Line Shape  There are three players in determining the T dependence of the line shape: the density matrix, the potential, and the number density of the bath molecules. The first two are main ones, but the density matrix is more crucial.  The density matrix is a product of two factors For the H 2 O-H 2 O pair, these two factors are the same. For the H 2 O-H 2 O pair, these two factors are the same.  is a three-dimensional distribution function with three Euler angles  (ζη),  (ζη), and  (ζη), representing a rotation from the orientation ζ to the orientation η.   The distribution function satisfies the normalization condition and sensitively depends on the temperature.

8. The T dependence of the density matrix Fig.1 The two-dimensional distribution of H 2 O over the two sensitive variables β and u obtained at T = 290 K. This distribution results from the averaging G(β, u, v) over the one insensitive variable v. Fig. 2 The same as Fig. 1, except T = 330 K. There two sensitive variables  and u ≡ (  +  )/2 and one insensitive variable v ≡ (  -  )/2.

9. How does the density matrix play a crucial role to cause a negative T dependence of the line shape  The profile of the distribution exhibits three large peaks along the u axis. As T increases, these peaks become higher and their widths become narrower.  The distribution represents a quantitative measure of the probability with which the molecule moves from the initial orientations to the final ones after the transitions take place.  The δ-function-like peaks mean that the molecule is less likely to make large reorientations. The higher the temperature, the less likely large reorientations occur.  Only large reorientations can make contributions to the far-wing line shapes by providing enough energy to meet large frequency detunings. It is the negative T dependence of the reorientations that causes the negative T dependence of the line-shape functions, and, consequently, of the continuum.

10. T dependence of the line shapes Fig. 3 The self- broadened far-wing line shapes of H 2 O (in units of cm -1 atm -1 ) as a function of ω (in units of cm -1 ) for T = 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, and 370 K. These results are represented by the solid lines from the top to bottom, respectively.

11. I-E. The calculated continuum Fig. 4 The calculated continuum absorption coefficients (in cm 2 / (molecule atm) at T = 296 K in the 300 – 1100 cm -1 spectral region are denoted by ∆. The experimental values of Burch et al. are denoted by +.

12. I-E. The calculated continuum Fig. 5 Measured α(ω) by Baranov at 27 micro-windows within 800 – 1150 cm -1 for T = 310.8, 325.8, and 363.6 K. They are represented by +. Seven of Burch’s data at 296 K and two results of Cormier at 296 and 310 K are plotted by small symbols  and □, respectively. From top to bottom, calculated α(ω) at these micro-windows for T = 296, 310.8, 325.8, and 363.6 K are plotted by ∆. In addition, MT_CKD values for the four temperatures are plotted by dotted lines.

13. I-F. T dependence of the continuum from allowed lines  Absorption coefficients α(ω) exhibit a strong negative T dependence. How strong the T dependence is varies with ω.  Empirical formula used to describe the T dependence of α(ω) where T 0 (i.e., the characteristic temperature) is a parameter and it could vary as ω varies. where T 0 (i.e., the characteristic temperature) is a parameter and it could vary as ω varies. Notice: Values of T 0 depend on the units used for α(ω). In general, values of T 0 are derived from α(ω) given in the units of cm 2 /(molecule atm). Notice: Values of T 0 depend on the units used for α(ω). In general, values of T 0 are derived from α(ω) given in the units of cm 2 /(molecule atm).  This is a simple and effective way to describe the T dependence of α(ω). But, one can not use this formula to predict α(ω,T) for temperatures which are far away from those used to fit the formula.

14. I-F. T dependence of the continuum from allowed lines Fig. 6 The calculated continuum absorption (in units of cm 2 /(molecule atm) at 944.195 cm -1 for temperatures ranging from 250 – 350 K. Theoretical values are presented by the solid line. Measurements of Hinderling et al. are presented by ∆ and values of Loper et al. by ◊. Results by Cormier et al. are given by □.

15. I-F. T dependence of the continuum from allowed lines Fig. 7 Characteristic temperature T 0 derived from α(ω) (in cm 2 /(molecule atm)). Values of T 0 derived from Baranov et al. and from Hinderling et al. data are given by + and ●, respectively. The one from Cormier et al. measurements is presented by ◊. Our calculated T 0 are given by ∆ and those from MT_CKD are plotted with a solid line.

16. I-G. Conclusions on the T dependence of the continuum from allowed lines  In the infrared window region, the T dependence of the continuum predicted by the far-wing theory is negative and moderately strong.  The T dependent pattern is not simple. It could vary significantly as the frequency of interest varies.

17. II.The Dimer Theory  The predicted dimer absorptions are proportional to the square of the water-vapor pressure and exhibit very strong negative T dependence.  As a possible mechanism, the dimer theory was proposed several decades ago. However, reliable theoretical predictions are available only very recently.  How to estimate the number density of the dimer at the temperature of interest is a crucial step in developing the dimer theory. Recently, progress on this subject has been made. Thus, one is able to address the dimer absorptions and their T dependences quantitatively.

18. II-A. The Dimer Absorption  The dimer absorption coefficient α(ω) can be expressed as where n D is the number density of the dimers, is the rotation-vibration partition function, S n”n’ are the dimer line strength factors and L is the line- shape function. where n D is the number density of the dimers, is the rotation-vibration partition function, S n”n’ are the dimer line strength factors and L is the line- shape function.  In terms of the equilibrium constant K P (T) (in atm -1 ) the dimer number density n D is defined by the dimer number density n D is defined by where the subscripts and superscripts of D and M stand for the dimer and the monomer of H 2 O, respectively, represents the thermal de Broglie wavelength, and D 0 = 3.52 kcal/mol (i.e., 1771 K). where the subscripts and superscripts of D and M stand for the dimer and the monomer of H 2 O, respectively, represents the thermal de Broglie wavelength, and D 0 = 3.52 kcal/mol (i.e., 1771 K). K P (T) plays a crucial role for the T dependence of dimer absorptions. K P (T) plays a crucial role for the T dependence of dimer absorptions.

19. II-B. Factors Affecting the T Dependence of Dimer Absorption  In order to analyze the T dependence, we divide α(ω) into three factors  The first factor K P (T)/Q D RV plays the dominant role in determining the T-dependence. More specifically, one can find  The second factor is H(ω,T) which contains all summations of α(ω). Approximately, one is able to express it T dependence as Approximately, one is able to express it T dependence as  The last factor is n M P M. Whether it causes an extra T dependence depends on the units used for α(ω). If the units is cm 2 /(molecule atm), n M P M would not contribute the T dependence.

20. II-C. T Dependence of Dimer Absorption  The T dependence of α(ω) can be approximately expressed as  In cases of ħω/2kT <<1, the T dependence becomes uniform for ω and can be expressed  Conclusions on the T dependence of the dimer absorption (1) The dimer absorption has a very strong negative T dependence. (1) The dimer absorption has a very strong negative T dependence. (2) The T dependence pattern is rather simpler, especially for low frequency regions (2) The T dependence pattern is rather simpler, especially for low frequency regions

21. II-D. Test of the T dependence of the dimer absorption Fig. 8 Comparison between the exactly computed α(ω,T) for five different temperatures, represented by full curves, and the extrapolated values represented by dots. The latter are derived by using the T dependence formula with α(ω, 268 K) as reference curve.

22. III.The Collision Induced Absorption  During collisions of two interacting molecules, transient dipole moments occur. The latter can cause absorptions in the same way as permanent dipole moments do.  In general, integrated CIA are proportional to the product of pressures associated with the colliding molecules.  Because the transient dipoles have a short lifetime, CIA exhibits diffuse line profiles.  CIA line strengths are generally weak.  The CIA theory was originally proposed to explain observed absorptions for molecules without dipoles such as N 2. It is expect that CIA exists for other molecules with dipoles also.

23. III-A. Dipole-induced-dipole and its CIA  The permanent dipole of the molecule 1 induces a transient dipole of the molecular pair through the isotropic part of the polarizability of the molecule 2. This is a main component of the induced dipole.  The corresponding CIA can be expressed as where where  The function G(ω) is selected as the K 2 line shape model. where G 0 is the normalization constant and η is a parameter. where G 0 is the normalization constant and η is a parameter.  By considering the first two moments of the spectral density, one is able to determine η and the line shape completely.

24. III-B. The K 2 line shape functions Fig. 9 Calculated normalized K 2 line shape function G(ω,η) at T = 240, 270, 300, and 330 K. Half-widths derived from their half-side are also presented and their values are 31.286, 33.014, 34.662, and 36.242 cm -1, respectively. As T increases, the half- width increases approximately as T ½ increases.

25. III-C. Calculations of CIA  In order to calculate CIA, one needs to know It turns out that in terms of the line intensities listed in HITRAN, we have where where  Then, the CIA (in cm -1 /amagat 2 ) can be expressed as  We calculate CIA due to the pure rotational band at T = 240, 270, 300, and 330 K. The results are presented in Figs. 10-11.  Results of the integrated CIA (in cm -2 /amagat 2 ) is presented in the Table. These values decrease slowly as T increases. T(K)240270300330 Integrated CIA 0.11010.097590.088900.08256

26. III-C. Calculated CIA due to the pure rotational band Fig. 10 Calculated CIA spectrum due to the pure rotational band of H 2 O at T = 240, 270, 300, and 330 K. They are represented by black solid, red dashed, green dot- dashed and blue dotted lines. Within the band the CIA exhibits a negative T dependence. At the window the spectrum exhibits a positive T dependence.

27. III-D. Calculated CIA Spectrum from 0 – 5,000 cm -1 Fig. 11 Calculated CIA spectrum at T = 240, 270, 300, and 330 K. They are represented by black, red, green and blue lines, respectively. For reference, the self-continuum from MT_CKD at these T are also plotted. Measurements by Burch at different T raging from 296 K to 353 K are presented. In order to make comparisons, the units for calculated CIA has been changed from cm -1 /amagat 2 to cm 2 /(molecule atm).

28. III-E. T Dependence of CIA  The T dependence of CIA is mainly determined by I(T) whose values decrease slowly as T increases.  In general, CIA exhibits a mild negative T dependence. However, in windows between the H 2 O bands, CIA could have a slightly positive T dependence.  The integrated CIA exhibits a mile negative T dependence. If CIA is given in cm -1 /amagat 2, the integrated CIA would varies as 1/T n with the exponent n is in the range of 0.6 – 1.0.  The magnitudes of the CIA spectrum are, at least, 2 – 3 order smaller than the measured continuum absorptions. Thus, one can rule it out from important candidates responsible for the continuum.

29. IV. Conclusions on three mechanisms responsible for the self-continuum  The T dependences: (1) The dimer absorption has the strongest negative dependence. The far-wing theory has a moderate negative dependence. The CIA has the weakest one. (1) The dimer absorption has the strongest negative dependence. The far-wing theory has a moderate negative dependence. The CIA has the weakest one. (2) The dimer exhibits the simplest T dependence pattern. The far-wing theory’s T dependence varies significantly as ω varies. The T dependence of CIA is mildly negative. But it becomes slightly positive in windows between the H 2 O bands. (2) The dimer exhibits the simplest T dependence pattern. The far-wing theory’s T dependence varies significantly as ω varies. The T dependence of CIA is mildly negative. But it becomes slightly positive in windows between the H 2 O bands.  How importance for each of the mechanisms: (1) In the infrared window, the far-wing theory is the main source of the self-continuum. (1) In the infrared window, the far-wing theory is the main source of the self-continuum. (2) The dimer would account for small parts of the continuum in 30 – 500 cm -1. Its contributions diminish beyond 500 cm -1 and become important below 30 cm -1. (2) The dimer would account for small parts of the continuum in 30 – 500 cm -1. Its contributions diminish beyond 500 cm -1 and become important below 30 cm -1. (3) In general, contributions from CIA are negligible. (3) In general, contributions from CIA are negligible.