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
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.
2) varies with the frequency smoothly.
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.
I. The Far-Wing Theory

Facts about the continuum of the far-wing theory:
(1) The total absorption consists of contributions from local
H2O 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.
(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 H2O bands, there are few strong lines.
The continuum becomes the dominant component. Thus, we
can talk about the continuum with certainty in the windows.
I-A. Continuum from the Far-Wing Theory

In terms of the band-averaged line shape function ˆ ( ), the
continuum absorption is given by
 ()  n a
S


ij

0
 sinh(  / 2kT )
ij
1
1
ˆ
[

(



)

ˆ (  ij )],
ij
2
2
ij sinh( ij / 2kT )  (  ij )
(  ij )

1
We introduce the coordinate representation by choosing orientations
of two interacting molecules as the basis set of Hilbert space
|    |  (a  a )   |  (b  b ) ,
where Ωaζ and Ωbζ 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
V (r , a , b ) |    V (r , a , b ) |  ,
where V (r , a , b ) are values of V at the specified orientation ζ.
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.
ˆ ()  4 2n b  2 2   d  d  |  | a b 0|  |2
rc2
[V ( r ) V ( r )]/ 2kT V iso ( rc ) / kT
 '
e 1, c 1, c
,
| V 1,  (rc ) |
where V1 is the anisotropic part of the potential, V1,ζ = <ζ|V1|ζ>, V΄1,ζη(r) =
dV1,ζ(r)/dr - dV1,η(r)/dr, and rc is determined from the equation
V 1,  (rc )  V 1, (rc )  .

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.
V (r , a , b ) 


ia j b
qi qj
rij
 4 0 [(
0
roo
)12  (
0
roo
)6 ].
With the Monte Carlo method, one can evaluate the 11-dimensional
integrations. In general, by using about 107 random selections of the
orientation variables one can derive converged line shapes within
reasonable CPU times.
I-C. Factors Affecting T Dependence of the Continuum
from Allowed Lines

Factors affecting the T dependence of the continuum:
(1) Number density na of the absorber molecule.
(2) Line intensities Sij.
(3) The line shape functions. More explanations are given later.

Effect from na: It depends on the units of the absorption coefficients. If
α(ω) is given in units of cm2/(molecule atm), na does not cause any T
dependence.

Effect from Sij : 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, line strengths at 290K are higher than those at 330K.
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.
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
|  | a b |  |2  | a | a | a |2  | b | b | b |2 .
For the H2O-H2O pair, these two factors are the same.
2
 | a | a | a |
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
2   2
1
F
(

,

,

)
sin

d

d

d


.
2
0 0 0
8
and sensitively depends on the temperature.
The T dependence of the density matrix
Fig.1 The two-dimensional
distribution of H2O 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.
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.
T dependence of the line shapes
Fig. 3 The selfbroadened far-wing
line shapes of H2O (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.
I-E. The calculated continuum
Fig. 4 The calculated
continuum absorption
coefficients (in cm2/
(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 +.
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.
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)
 (,T )   0 () e
T 0 [(
1
1

)]
T 296
,
where T0 (i.e., the characteristic temperature) is a parameter and it
could vary as ω varies.
Notice: Values of T0 depend on the units used for α(ω). In
general, values of T0 are derived from α(ω) given in the units of
cm2/(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.
I-F. T dependence of the continuum from allowed lines
Fig. 6 The calculated continuum absorption (in units of cm2 /(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 □.
I-F. T dependence of the continuum from allowed lines
Fig. 7 Characteristic temperature T0 derived from α(ω) (in cm2/(molecule atm)).
Values of T0 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 T0 are given by ∆ and those from MT_CKD are plotted with a
solid line.
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.
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.
II-A. The Dimer Absorption

The dimer absorption coefficient α(ω) can be expressed as
g
4 2
 () 
n D   n n  Dn  (e E n  / kBT  e E n  / kBT )S n n  L (, n n  ,  ),
3 c
QR V
n  n   n 
where nD is the number density of the dimers, QRDV is the rotation-vibration
partition function, Sn”n’ are the dimer line strength factors and L is the lineshape function.

In terms of the equilibrium constant KP(T) (in atm-1)
PD
M3 2
1 QRDV e D / k T
K P (T )  2 

( M ) ,
3
PM
kBT
D
QR V
0
B
the dimer number density nD is defined by
nD
PM2
 n M PM K P (T ) 
K P (T ),
k BT
where the subscripts and superscripts of D and M stand for the dimer and the
monomer of H2O, respectively, X  h/ 2 m X kBT represents the thermal de
Broglie wavelength, and D0 = 3.52 kcal/mol (i.e., 1771 K).
KP(T) plays a crucial role for the T dependence of dimer absorptions.
II-B. Factors Affecting the T Dependence of Dimer
Absorption

In order to analyze the T dependence, we divide α(ω) into three
factors
K P (T )
 () 
 H (,T )  n M PM .
D
QR V

The first factor KP(T)/QDRV plays the dominant role in determining the
T-dependence. More specifically, one can find
K P (T )
h 3e D0 / kBT

T
D
3/ 2
M 2
QR V
kBT ( m M kBT ) (QR V )

e
.
The second factor is H(ω,T) which contains all summations of α(ω).
Approximately, one is able to express it T dependence as
H ,T   sinh(

11 / 2 D0 / kBT

)  T 3.
2kBT
The last factor is nMPM. Whether it causes an extra T dependence
depends on the units used for α(ω). If the units is cm2/(molecule
atm), nMPM would not contribute the T dependence.
II-C. T Dependence of Dimer Absorption

The T dependence of α(ω) can be approximately expressed as
 ,T   n M PM  T 5 / 2sinh(


2kBT
) e D0 / kBT .
In cases of ħω/2kT <<1, the T dependence becomes uniform for
ω and can be expressed
 ,T   n M PM  T 7 / 2 e D

0
/ kBT
.
Conclusions on the T dependence of the dimer absorption
(1) The dimer absorption has a very strong negative T
dependence.
(2) The T dependence pattern is rather simpler, especially for
low frequency regions
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.
ext (,T )
D /k T
9/ 2
e 0 B
T ref 

 D /k T .
 (,T ref )  T 
e 0 B ref
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 N2. It is expect
that CIA exists for other molecules with dipoles also.
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.
2(r1N )  (r2N )
4
1  

C (121, m 1   m 1  )Y 1 m (1 )Y 2 m (2 ).
3
3

R
1
1
m1
The corresponding CIA can be expressed as
4 2
 () 
F n L2a 05 (1  e 
3
 / k BT
) S (n 0 ) (T ) G (  i1f1 ),
i1f1
where
i2

S (n 0 ) (T )  2 00 |  (r2N ) | 00  I (T )Pi (T ) | i1f1 | , and I (T )  4  R 4e V iso ( R ) / kBT dR .
2
2
0

The function G(ω) is selected as the K2 line shape model.
G ( )  G 0 
1
1e
  / k BT




( )2 K 2 ( ),
where G0 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.


1
 0   F ( )d  and 1 
F ( )d .





III-B. The K2 line shape functions
Fig. 9 Calculated
normalized K2 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 halfwidth increases
approximately as
T½ increases.
III-C. Calculations of CIA
In order to calculate CIA, one needs to know S (n ) (T ). It turns out that
in terms of the line intensities listed in HITRAN, we have

0
S if
4 2
F a 02 S (n 0 ) (T )  2 00I (T ) 
3
if (1  e 
where 00   00 | (r2N ) | 00  .

if / kBT
)
.
Then, the CIA (in cm-1/amagat2) can be expressed as
 ()  n a 2 I (T )
2 3
L 0
2
00

if ; E f  E i
 (1  e 
if (1  e 
 / k BT
)
if / kBT
)
S if [G (  if )  e
 if / kBT
G (  if )].

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 /amagat2) is presented in the
Table. These values decrease slowly as T increases.
T(K)
240
270
300
330
Integrated CIA
0.1101
0.09759
0.08890
0.08256
III-C. Calculated CIA due to the pure rotational band
Fig. 10 Calculated CIA spectrum due to the pure rotational band of H2O at T = 240,
270, 300, and 330 K. They are represented by black solid, red dashed, green dotdashed and blue dotted lines. Within the band the CIA exhibits a negative T
dependence. At the window the spectrum exhibits a positive T dependence.
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 cm2/(molecule atm).
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 H2O 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/amagat2, the integrated CIA would varies as
1/Tn 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.
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.
(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 H2O 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.
(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.