Literature references:

  Ridgway, W. L., Harshvardhan, A. Arking, 1991, Computation of Atmospheric
        cooling rates by exact and approximate methods, J. Geophys. Res.,
        Vol 96, 8969-8984.

  Ellingson, R. G., J. Ellis, and S. Fels, 1991, The intercomparison of
        radiation codes used in climate models: Long wave results,
        J. Geophys. Res., Vol 96, 8929-8953.

_______________________________________________________________________
Unformatted Copy of JGR Paper (no equations, greek symbols, or figures)

COMPUTATION OF ATMOSPHERIC INFRARED COOLING RATES
BY EXACT AND APPROXIMATE METHODS

William L. Ridgway
Applied Research Corporation
Landover, Maryland

Harshvardhan
Dept. of Earth and Atmospheric Sciences
Purdue University
West Lafayette, Indiana

A. Arking
Laboratory for Atmospheres
Goddard Space Flight Center
Greenbelt, Maryland

ABSTRACT

Infrared fluxes and cooling rates for several standard model
atmospheres, with and without water vapor, carbon dioxide, and ozone,
have been calculated using a line-by-line method at 0.01cm^-1 resolution.
This study was conducted as part of the InterComparison of Radiation
Codes in Climate Models (ICRCCM), where line-by-line calculations are
intended to serve as a benchmark for comparing practical codes used in
climate models.  The sensitivity of the results to the vertical integration
scheme and to the model for water vapor continuum absorption are
shown.  Comparison with similar calculations performed at NOAA/GFDL
shows agreement to within 0.5Wm^-2 in fluxes at various levels and
0.05Kday^-1 in cooling rates.  Comparison with a fast, parameterized
radiation code used in climate models reveals a worst case difference,
when all gases are included, of 3.7Wm^-2 in flux; cooling rate differences
are ~0.1 Kday^-1 or less when integrated over a substantial layer, with
point differences as large as 0.3 Kday^-1.

1. Introduction

The accurate computation of infrared cooling rates in numerical
models of the atmosphere has become increasingly important as emphasis
on medium range and extended range prediction increases.  On even
longer time scales, the continuing interest in the climatic impact of changes
in carbon dioxide and other atmospheric trace gas concentrations is
another driving force behind the recent emphasis on atmospheric radiative
transfer.  A further impetus has come from advances in computer
technology that allow extensive simulations with atmospheric general
circulation models using more complex algorithms for the radiative
computations.  The importance of correctly modeling the solar heating and
infrared cooling rates in the troposphere and stratosphere has been shown
in many investigations. (See, for example, the review by Ramanathan,
1987.)

In an attempt to document differences in results obtained with various
detailed radiation codes and radiation parameterizations in climate
general circulation models (GCM's), an international effort
(InterComparison of Radiation Codes in Climate Models or ICRCCM
study) was launched under the auspices of the World Meteorological
Organization and with the support of the U. S. Department of Energy.
The first phase, covering clear sky infrared radiation, is complete and
overviews of the results have been published (WMO, 1984; Luther et al.,
1988).

This article discusses results from two models at opposite ends of the
range of complexity for radiation codes which were used in the
intercomparison: a line-by-line code that performs detailed spectral and
angular integration and a parameterized broad-band model suitable for
GCM (general circulation model) applications.  In order to better
understand the context for this comparison, we include results from a
second line-by-line model, those submitted by NOAA/GFDL which also
appear in this issue (Schwarzkopf and Fels, 1990).  This study is a test of
how well the parameterization can reproduce clear-sky infrared radiative
cooling rates as computed by line-by-line methods.  The line-by-line
computations are used as reference because the method is the most exact
one we have, permitting incorporation of as much detail about the gaseous
absorption properties as necessary.  It should be kept in mind, however,
that it relies on many theoretical assumptions, primarily due to insufficient
knowledge of molecular properties such as line shape, effects of
interactions with foreign gases, line mixing, etc.

The line-by-line techniques applied here are fairly typical of such
methods.  Although virtually the same spectral line data were used by the
two line-by-line codes, there are several procedural differences which
serve to make the comparison a meaningful exercise.  Spectroscopic model
differences include choice of spectral grid, cut-off technique for far line
wings, representation of the water vapor continuum, number and spacing
of vertical levels, quadrature scheme or angular integration technique, etc.
The presence of these differences guarantees that the two line-by-line
schemes will not produce identical results.  On the other hand, since they
and most others share basic physical assumptions and a common spectral
data base, the spread in results among several line-by-line models is one
measure of the limits of numerical precision.  Together, the line-by-line
results serve as a basis for the assessment of errors made by
parameterized models.

For these reasons, we have compared the results of the NASA/GLA
line-by-line scheme (Goddard Laboratory for Atmospheres, as
documented here) with those obtained using the NOAA/GFDL model
(Geophysical Fluid Dynamics Laboratory, Schwarzkopf and Fels, 1990)
from which detailed flux and cooling rate results were kindly made
available to us by the authors.  The comparison of cooling rates obtained
from the parameterized code against line-by-line results provides a basis
for quantifying errors in the computation of clear-sky cooling rates in
GCM's.  These comparisons have also provided an opportunity for
improvement of the parameterization.

In Section2 basic formulas for radiative fluxes and cooling rates are
given.  The line-by-line method and specifics of its implementation are
discussed in Section3, while the parameterized model is outlined in
Section4.  Discussions of results and conclusions appear in Sections5
and6, respectively.  Finally, we give in AppendixA a very detailed
accounting of vertical layering considerations and document the choice of
vertical integration method in the context of line-by-line flux modeling.

2. Radiative Flux and Cooling Rates

The infrared cooling rate is the vertical divergence of the net flux of
longwave radiant energy.  In the line-by-line technique, the upward and
downward fluxes at selected levels in the atmosphere are computed using
the formal solution of the equation of transfer for monochromatic
radiation.  The monochromatic upward and downward fluxes may be
expressed in the following form (see, for example. Liou, 1980, p.94):

     [ Equation 1a ]
     [ Equation 1b ]

where Bn (T) is the Planck function at temperature T and wavenumber
n, En is the exponential integral of order n, and t  is the monochromatic
optical depth which is a function of wavenumber n and pressure p, and is
related to the mass absorption coefficient k-nu by

     [ Equation 2 ]

The mass absorption coefficient is, in general, a direct function of
temperature and pressure, and is unique for a given distribution of
atmospheric gases.  Because the broadening of spectral lines is affected by
molecular collisions, k-nu is non-linearly related to gas concentration, but
if we make the assumption that the concentrations of the radiatively active
gases are small, then molecular collisions primarily involve the non-
radiatively active nitrogen and oxygen gases, and we can make the
following approximation :

     [ Equation 3 ]

where ci is the concentration of radiatively active gas of index i.
In Eq.(1), to is the optical depth of the entire atmospheric column and
Ts is the surface temperature.  The explicit dependence of t  on n and p is
suppressed.  The divergence of the net flux is usually expressed as a
cooling rate in Kday^-1, so that

     [ Equation 4 ]

where g is acceleration due to gravity and Cp is the specific heat of air at
constant pressure; for the ratio we use the value 8.442 (mb-Kday^-1) per
(Wm^-2)

3. Line-by-line Method

The crux of the line-by-line technique is the computation of the very
high resolution spectra of Eq.(2) and the repeated (monochromatic)
evaluation of the exponential integrals of Eq.(1) for the infrared spectrum
from 0 - 3000 cm^-1.  (The contribution beyond 3000 cm^-1 is less than a few
hundredths of a percent, which can be neglected.)  Mass absorption
coefficients are computed from a data base of spectral line parameters
together with a line shape model.  The parameters needed to compute the
absorption coefficient are functions of temperature, pressure, and gas
mixing ratio, which adds a degree of complexity in considering
inhomogeneous atmospheric paths.

The accuracy of the line-by-line method is limited by uncertainties in all
of the following: (i)basic spectral line parameters including line position,
strength, and half-width which are derived from a combination of
theoretical models and laboratory measurements;  (ii)physical
assumptions about collisional broadening, such as the validity of Eq.(3),
temperature and pressure dependence of the line shape parameters, and
the continuum absorption required to reconcile differences between theory
and measurement far from spectral lines;  (iii)the vertical structure of the
atmosphere as specified in terms of temperature and gas mixing ratio
profiles;  and (iv)numerical approximations associated with discrete
spectral sampling, finite vertical resolution, and approximate angular
integration.

In the context of the ICRCCM clear-sky infrared study, the vertical
structure was fully specified by a supplied algorithm, thus eliminating (iii)
as a source of error.  Also, there are relatively few complete sources of line
parameter data; the AFGL compilation of 1982 (Rothman et al., 1983) was
used almost universally.  Therefore, while (i) is still a potential source of
error, that error is common to most of the models used in the
intercomparison.  On the other hand, physical assumptions (ii) about line
shape and the water vapor continuum were an important factor in the
intercomparison leading to some differences in results together with
common errors.  Because of the computational demands of the line-by-line
method, numerical approximations (iv) tended to be quite varied and could
lead to significant errors; however, in the case of the two line-by-line
treatments considered here, these numerical errors were well-controlled
and did not lead to significant discrepancies.  We will touch on most of
these considerations in the discussion which follows.

3.1  Spectral Model

The GLA line-by-line infrared flux model was built upon an earlier
algorithm for obtaining accurate transmission functions repeatedly, and at
a modest computational cost for a large number of  temperature-humidity
profiles.  The transmission model was designed to allow variations in the
temperature and amounts of principal gases.  This was achieved by
developing a spectral database of absorption optical thickness data for
water vapor, carbon dioxide, ozone, methane and nitrous oxide.  The pre-
existing spectral database and the new flux model using this high
resolution data share a fixed 0.01 cm^-1 resolution spectral grid extending
from 0 to 3000 cm^-1.  To generate the database, mass absorption optical
thickness spectra were computed line-by-line for each absorbing gas by
evaluating the inhomogeneous-pressure integral of the assumed line
shape, thereby obtaining tn (p), as given by Eq.(2) but with fixed
temperature, at 48 pressure levels in the atmosphere.  The calculations
were repeated for four temperatures at each pressure level, with the
temperatures independently selected to span a representative range of
model atmospheres (see Table 1).  The spectra were then extended to
arbitrary temperature by fitting the logarithm of opacity to a third degree
polynomial, so that

     [ Equation 5 ]

where t=(T-To) is the temperature difference from a nominal baseline
temperature To=240 K.

With these appropriately chosen reference temperatures, use of Eq.(5)
introduced monochromatic transmittance errors generally less than 1
percent in the 15 mm CO2 region and accounted for much less than 1
percent error in the spectrally averaged transmittance.  It is important to
note however, that this limited error due to temperature interpolation
applies only for temperature profiles that fall within the limits given by
Table 1.  The five realistic model atmospheres could be well represented,
but the 200, 250, and 300K isothermal atmospheres would have required
temperature extrapolation beyond these limits; we  did not attempt to
apply the model to these cases.

Spectral line parameters used in these calculations were taken from a
preliminary version of the widely used AFGL compilation of 1982
(Rothman et al., 1983) which appeared in 1980 and is described by Rothman
(1981).  The line shape was assumed to be a Voigt profile truncated at 10
cm^-1 from line center in all bands of all absorbing gases.  The choice of a
sharp cut-off at 10cm^-1  is a rather arbitrary way to account for the sub-
Lorentzian behavior far from the line center, but numerical tests have
shown the results to be rather insensitive either to the cut-off position
(within a factor of three or so) or to the manner of cut-off.  The Lorentz
half-widths were assumed to vary as the inverse square root of
temperature.

A pressure-scaled e-type water vapor continuum was added to the
absorption coefficient in the window region from 400-1200 cm^-1.  The
molecular cross-section for continuum absorption is based on the
parameterization of Roberts et al. (1976)

     [ Equation 6 ]

with a=1.25*10^-22 mol^-1 atm^-1, b=2.34*10^-19 cm^2 mol^-1 atm^-1,
b=8.30*10^-3 cm, and To=1800 K.  For a vertical path denoted by pressure
p(mb) and water vapor mass mixing ratio R(p), the continuum vertical
opacity is then represented by

     [ Equation 7 ]

with a=5.4*10^-19 mol atm cm^-2 mb^-2.  While the Roberts et al. formula is
based on laboratory data that has to some extent been superseded in the
last 15 years, in our view the more recent data was not sufficient to justify
a very different parameterization.  Nonetheless, the single greatest source
of (common) error in these ICRCCM studies is likely to be attributable to
limitations in our understanding of the water vapor continuum.  An
estimate of the impact of this uncertainty is given below in Section 5.1.

3.2 Vertical Integration

As with any numerical scheme there are trade-offs between
computational costs and model accuracy.  In this sense, line-by-line
treatments are 'exact' in principle but have obvious limitations in practice.
When done monochromatically, the vertical integration of radiative fluxes
based on Eq.(1) is computationally expensive; the number of diffuse
transmittance calculations or exponential integrals increases as O(n^2),
where n is the number of vertical gridpoints.  It is desirable to use as
coarse a vertical grid as possible without compromising the accuracy of the
results.  To this end, it is also important that the method of vertical
integration converge rapidly as the number of gridpoints is increased.

Numerical studies with the 48-layer transmission model (based on the
pressure levels of Table1) showed that it is too coarse to adequately
represent the rapidly varying water vapor density in the lower
atmosphere.  In computing fluxes, we chose to extend the number of layers
to 60, thereby systematically improving vertical resolution in the
troposphere.  This was done by interpolation of mass absorption spectra
from the pressure levels shown in Table 1.  New tropospheric layer
boundaries were spaced linearly in pressure (approximately every 20 mb )
from the surface to a point above the tropopause; a logarithmic pressure
grid was used from that point to a nominal atmospheric top at 0.1 mb.

Within each layer, mass absorption spectra were derived from the
database by linear interpolation in pressure; this was done after the
logarithm of mass absorption was fit to a cubic temperature function at
each spectral point using Eq.(5).  In this way, single-layer absorption
spectra were created for all 60 layers, based on mean layer temperature
and gas mixing ratios for each atmospheric model.  For the purpose of
computing mass absorption only, each of the 60 layers was assumed to be
homogeneous in temperature and gas mixing ratio.  All gases were
incorporated with this common vertical structure and uniform spectral
grid.

An important determinant for the convergence of computed cooling
rates with decreasing grid-size is how sub-grid-scale vertical
inhomogeneity is treated, so we will discuss this point in some depth.  The
problem can be cast simply as the question of how to estimate the
contribution made by each emitting layer to the upward or downward flux
at a given level above or below that layer.  This single-layer flux
contribution (at each spectral point) can be schematically represented as

     [ Equation 8 ]

using an abbreviated notation with 1 and 2 referring to layer boundaries
(top and bottom or vice versa) closest to and farthest from the flux level,
respectively.  Note that optical thickness t increases monotonically with
pressure, so that it serves to parameterize vertical functions such as
temperature and downward flux, but with an implicitly different
dependency at each spectral point.

It is desirable to evaluate expressions such as Eq.(8) with a quite
general method, one that can be uniformly applied to any spectral region,
absorbing gas density, and temperature.  The primary concern is how to
represent the temperature dependence of the Planck function between
vertical gridpoints.  In Appendix A, we give a more complete picture of the
modeling choices and their implications, but a brief overview gives some
perspective.  The simplest (and least effective) approach is to approximate
each layer as isothermal with a constant Planck function based on a mean
layer temperature.  A more versatile method uses integration by parts
giving surface terms plus a mean diffuse transmittance T  from points
within the layer to the flux level.  The former (so called BdT ) method is
adequate for optically thin layers, but approximates the temperature
profile in a fairly radical, discontinuous fashion using a single parameter
per layer.  The latter (T dB) method is found to be far more appropriate for
thicker layers and opaque spectral regions.  Consider that at opaque
wavelengths the flux emerging from a layer boundary closely reflects the
temperature of that boundary, not the mean layer temperature one would
have assumed using the BdT method.  It can be seen that with a constant
lapse rate the BdT method introduces a systematic warm (cold) bias in
estimated upwelling (downwelling) fluxes.  A crude estimate of error
associated with this method can be made from the temperature gradient
and opacity of each layer.

With the above considerations in mind, the GLA line-by-line flux model
was designed using a special implementation of the T dB method based on
a two-parameter (linear) temperature fit within each layer.  At all spectral
points it is assumed that the vertical variation of the Planck function
between layer top and bottom can be represented as linear in
monochromatic optical depth.  This linear B(t) approach has a long history
dating at least to Schuster (1905) and Schwarzschild (1906).  We refer the
reader to a discussion of its errors by Wiscombe (1975).  Importantly, the
linear B(t) approximation embodies a two-parameter estimate of flux
transmittance which is found to be more accurate than estimates in other T
dB approaches and demonstrably better than the BdT approximation.  (See
Appendix A.)

Another numerical concern is how best to evaluate the exponential
integrals for all atmospheric level/layer pairs and all wavenumbers.  It is
possible, for instance, to use an angular quadrature technique, direct
calculation, or simply represent the integral by applying a diffusivity factor
to the exponential form.  We chose instead to pre-compute E3(Dt) and
E4(Dt) since this database is not cumbersome for functions of a single
variable.  The spectral integration is then done in two stages.  First, 5 cm^-1
spectral averages of the exponential integral E3 and the difference term
{ DE4 / Dt } of Eq.(9) are computed from the lookup table for all level/layer
pairs for each of the 600 bands.  (These averages correspond to diffuse flux
band transmittance matrices.)  Finally, the Planck function is evaluated at
the center of every 5 cm^-1 band and fluxes are computed by summing all
single-layer contributions to each flux level based on the linear B(t)
method.

4. Parameterized Method

The most time consuming and rigorous step in computing the
integrated fluxes using the line-by-line method is the computation of
monochromatic quantities, since typically O(10^5 - 10^6) distinct spectral
points are involved with O(n^2) transmittance calculations required at each
point for n layers.  Even with ingenious programming on vector machines
such computations are impractical for most applications.  It is for this
reason that attempts have been made to parameterize or otherwise
spectrally average the transmittance over wider bands.  The
monochromatic fluxes computed from Eq.(1), when summed over all
wavelengths, yield the vertical profile of net infrared flux in the
atmosphere.  For parameterized models, the net flux is obtained by
summing relatively few bands, typically 5-10 in all.  The GLA
parameterized model uses five bands; it was developed for GCM
applications and is discussed in detail in Harshvardhan et al. (1987).  It is
currently being used in general circulation models (Harshvardhan et al.,
1989, Randall et al., 1989).  Results from this model were submitted for the
intercomparison and selected results are compared with the line-by-line
results in the next section.

The parameterization is based on the work of Chou (1984) for water
vapor, Chou and Peng (1983) for carbon dioxide and Rodgers (1968) for
ozone.  Details may be found in the above cited references, but in essence,
the water vapor and carbon dioxide schemes attempt to fit functional
forms to the band-averaged flux transmittance obtained by a line-by-line
spectral integration for certain reference conditions.  The original
parameterization developed by Chou and co-workers was only applicable
from the surface up to the lower stratosphere.  Some recent modifications
have been made to extend the applicability to higher altitudes.  These will
be discussed in the next section where results are presented.

5. Results

5.1. Fluxes

Table 2 summarizes results for ten of the ICRCCM cases, listing the
spectrally integrated fluxes at the surface, tropopause, and top of the
atmosphere for the three models in question.  Downward fluxes at the
surface and tropopause are most sensitive to model layering and spectral
assumptions.  The model results represent the entire IR region from 0-3000
cm^-1 except where noted.  For the CO2 doubling cases (600 ppm) the
GFDL results are reported for the region from 0-2200 cm^-1 only.  GLA
line-by-line fluxes are shown for both spectral intervals to aid the
comparison.  As noted below, the parameterization does not include the 14
mm band of ozone; comparable GLA line-by-line downward fluxes are
given for the ozone-only case.

Differences among results for these models are much smaller than for
the broader group of ICRCCM participants (Luther et al., 1988); in fact,
the two sets of line-by-line fluxes agree to within 0.5Wm^-2.  Given that
the two line-by-line models use quite different approaches, this numerical
agreement is rather remarkable.

The parameterized model generally computes fluxes to within ~3 Wm^-2
of the line-by-line models, when all gases are included, with a worst case
of 3.7Wm^-2 for the downward flux at the surface for sub-arctic winter.
There is a larger difference of 5Wm^-2 for the surface downward flux for
mid-latitude summer when water vapor without the continuum is the only
gas included.

We caution that despite this excellent agreement considerable
uncertainties remain with regard to some important spectral assumptions
that are common to both line-by-line models.  For instance, we expect that
uncertainty in the water vapor continuum alone could systematically shift
both sets of results by several Wm^-2.  In using the Roberts et al. e-type
formulation, we chose to neglect the foreign-broadened p-type
contribution which can be 20 percent or more of the total depending on
vapor mixing ratio.  For some atmospheres there is a cancellation effect
since recent measurements by Burch and Alt (1984) suggest that the Roberts
et al. formula is an overestimate of the self-broadened continuum by up to
20 percent, but the two contributions scale differently with water vapor
partial pressure so that cooling rate profiles are subject to bias through
this simplification.

We have briefly studied the sensivity of computed fluxes to continuum
assumptions by testing a second continuum model.  This test model
included exactly 80 percent of the Roberts et al. e-type contribution plus a
temperature-independent p-type continuum (we used a p-type to e-type
bare coefficient ratio that varied from 0.0068 at 600 cm^-1 to 0.0018 at 900
cm^-1.)  The effect of these changes was to reduce the total water vapor
continuum in moist atmospheres, but to increase it significantly for dry
profiles.  Downward fluxes at the surface were most sensitive to the
change in continuum model; results were that the mid-latitude summer
atmosphere showed a decrease of 1.6 Wm^-2, the tropical atmosphere
showed a decrease of 4.1 Wm^-2, while the sub-arctic winter atmosphere
showed a rather large increase of 5.8 Wm^-2 in the downward surface flux.

This is a very large sensitivity to the water vapor continuum in the context
of the very small differences between line-by-line model results.
The line-by-line calculations have used 1980 (GLA) and1982 (GFDL)
releases of the AFGL line parameters compilation (Rothman et al., 1983).
One can expect that revisions to this database since 1982 would similarly
effect both sets of calculations.  The 1986 release was marked by very few
significant changes to the water vapor line parameters (Rothman et al.,
1987).  Carbon dioxide line-strengths in the 15 micron band were adjusted,
but it is difficult to assess the impact on fluxes and cooling rates without
repeating the complete calculations.  Similarly, ozone line parameters
near 10 microns received small adjustments which may be expected to
show up most noticeably in stratospheric cooling rates, but to have almost
no impact on fluxes.  In both cases the differences can be expected to be
small in relation to that from the water vapor continuum.

5.2  Cooling Rates

Although the original intent of ICRCCM was to intercompare fluxes at
these few selected levels, it has proved useful to intercompare cooling rate
vertical profiles since these influence atmospheric models more directly.
Cooling rates were compared for selected ICRCCM cases using the GLA
and GFDL line-by-line schemes and the GLA parameterization; they were
found to be a much more sensitive indicator of model differences and
limitations.  Results of the 3-way intercomparison are grouped below by
absorbing gas.

a.  Water vapor.  In order to eliminate uncertainties associated with
representations of the 8-14 mm water vapor continuum absorption,
computations were made with and without the continuum.  For the line-
by-line models, apart from the line positions and strengths, assumptions
about the individual line shape and far wing cut-off can be expected to
influence the computed fluxes and cooling rates.  However, we can verify
that details of the far wing line shape and cutoff are not found to be
critical, as long as a 'reasonable' approach is taken.  It is also a difficult
problem to add continuum absorption in a fashion which is consistent with
the line model assumptions; here, the two line-by-lines models have taken
the same somewhat simplified approach of using the Roberts et al.
continuum formula.

From the perspective of the parameterized models, functional forms
are generally first fit to the line absorption over a band, and a continuum
contribution is then added.  This is the strategy used in the GLA model
(Chou, 1984) where the continuum model also follows Roberts et al. (1976).
The original parameterization  was designed for use below 50 mb; for
ICRCCM, and for future use in models with higher domains, a minor
modification has been made to extend its validity.  The primary limitation
of the original model was that pressure scaling excessively reduced the
contribution of water vapor absorption at low pressures.  Following a
suggestion by Fels (1979), this problem can be overcome by adding a
constant term to the scaled pressure to mimic the non-vanishing line width
at low pressures.  Separate coefficients for the band center and wings
were found by trial and error to simulate stratospheric and lower
mesospheric water vapor cooling rates.

Fig.1 shows the cooling rates for a midlatitude summer profile
assuming H2O lines only (i.e., no continuum) in the three models: the GLA
and GFDL line-by-line models and the GLA parameterization.  In the
troposphere, the cooling rate profiles as computed by the two line-by-line
models are quite close.  At higher altitudes, the differences are small but
not negligible.  Most of the line-by-line differences can be attributed to
vertical resolution, quadrature, and flux integration methodology, but
details of the spectral models may also have an observable effect at these
altitudes.  The parameterized model fares somewhat poorly in the
troposphere in resolving the cooling rate profile, although the mean
tropospheric cooling rate is essentially reproduced.  This is particularly
true where the continuum absorption is included as shown in Fig.2.

The parameterized model cooling rate profile above 50 mb in Fig.1 essentially
reflects the trial and error fitting of the coefficients that modify
the scaled absorber amount.  For much of this region, the contribution of
water vapor to the total cooling rate is small but not negligible, which will
be more evident in discussing the case with all gases.  The fact that a
simple correction can reproduce the broad features of the cooling rate
above 10 mb is encouraging from the point of view of parameterized
models of the stratosphere.

Fig.2 shows three water vapor cooling rates as in Fig.1, but with the
Roberts et al. continuum absorption parameterization  included.  The
vertical scale has been made linear in pressure to show more clearly the
tropospheric cooling rate profile.  It is quite clear from Fig.2 that the two
line-by-line models agree remarkably well in spite of the differences in line
cutoff and the selection of spectral interval over which the water vapor
continuum is applied.  The parameterized model fails to pick up the
detailed vertical structure although as mentioned above, the mean
tropospheric cooling rate is essentially correct.

As noted above, there is still considerable uncertainty in laboratory
measurements of the water vapor continuum.  From Fig.1 and Fig.2, it
can be seen that the continuum contributes almost 50 percent of the cooling
just above the surface.  The uncertainty in its magnitude could yield errors
of up to 0.2 Kday^-1 in the lower troposphere.  In actual simulations, clouds
in the troposphere will alter the cooling rate profile dramatically and the
errors present in clear-sky off-line calculations might therefore be
considered acceptable.

The tropical profile used in ICRCCM has higher water vapor mixing
ratios; Fig.3 shows the cooling rate comparison for this profile.  Again, the
two line-by-line schemes, which are of comparable vertical resolution,
follow each other closely except for the lowest model layer.  Differences
attributable to the choice of vertical integration scheme are most notable in
the first 20 mb layer, which is fairly opaque.  The coarser resolution
parameterized model suffers from problems similar to those seen in the
mid-latitude case; however, the broad features of the cooling rate
including the 750-800 mb maximum are captured.  Since the clear sky
tropospheric cooling rate is dominated by water vapor, the
intercomparison is encouraging.  The performance of the parameterized
model is not surprising since it was constructed to fit a line-by-line scheme,
albeit not one of the two shown here.

b.  Carbon dioxide.  CO2 is the dominant contributor to the infrared
cooling rate above the tropopause.  Fig.4 shows cooling rate profiles for
the midlatitude summer temperature profile and 300 ppmv of CO2 as the
sole atmospheric constituent.  In this instance, the GLA line-by-line model
uses a 10 cm^-1 line width cutoff while the GFDL model has a 3 cm^-1 cutoff,
yet there were no signs that the choice of cut-off affected the cooling rate
profiles.  The two line-by-line models appear to track each other very
closely to about 0.5 mb, above which the GLA model results are unreliable
because of the poor vertical resolution and proximity to model top.

The GLA CO2 parameterization employed here is a hybrid model
which has not been implemented in a GCM in its present form.  The
parameterization is that of Chou and Peng (1983) up to about the 5 mb
level, while a more recent transmittance formulation was used to compute
cooling rates above 5 mb.  The Chou and Peng model was designed for
tropospheric general circulation models and its success below 5 mb can be
seen in Figures 4, 5, and 6.  The improved parameterized model gives
accurate cooling rates well above 5 mb, while fluxes are nearly identical to
the earlier GCM version.

One of the concerns of the ICRCCM study has been the applicability of
current radiation codes to cases with enhanced concentrations of CO2 and
other gases.  Fig.5 shows results for the same midlatitude summer
temperature profile but for 600 ppmv of CO2.  Differences in the line-by-
line results are largely due to vertical resolution.  The parameterized
model performs well, especially below 5 mb, which is the range of
applicability of the original GCM code.  The model has the ability to use an
arbitrary CO2 mixing ratio without re-computation of transmittances
(Harshvardhan et al., 1987).  In its original form, it is suitable for climate
GCMs as long as the model domain does not extend too far above 10 mb.

With a different temperature profile for the subarctic winter case
shown in Fig.6, the line-by-line results agree to about the 3 mb level.  The
large disagreement  above this level, especially above 0.5 mb is again the
result of poor resolution of the GLA model with the model top being at 0.1
mb.  The parameterized model is again successful for this colder
atmospheric profile.

c .  Ozone.  Although the dominant radiative role of atmospheric ozone
is solar absorption in the stratosphere, there is a significant contribution to
infrared cooling in the upper stratosphere and also an infrared heating in
the lower stratosphere.  Fig.7 shows the cooling rate profile for
midlatitude summer temperatures and the ozone profile used in ICRCCM.

The two line-by-line models and even the parameterization based on
Rodgers (1968) follow each other closely from the surface to the 10 mb
level.  There are some significant discrepancies in the region of peak
cooling and above.  A major part of the difference between the line-by-line
results is due to the different interpolation procedures used for the ozone
mixing ratio profile in this region.  At the outset, the ICRCCM study tried
to ensure that all models used the same temperature and constituent
profiles.  However, these profiles were furnished on a very coarse vertical
grid and differences in interpolation have carried over to the cooling rate
profiles.

Whereas the line-by-line results include contributions from ozone
absorption in the 14 mm region, the parameterization does not.  This
accounts for some of the underprediction of cooling rates in the upper
stratosphere; the 14 mm ozone band contributes about 0.11 Kday^-1 or about
25 percent of the peak cooling.  For comparison purposes, the downward
fluxes at the tropopause and surface in the 9.6 mm ozone band alone have
been computed with the GLA line-by-line model and are included in
Table2.  When all gases are included, the 14 mm band is completely
obscured by absorption in the 15 mm band of carbon dioxide.

d.  All gases.  When the three major infrared absorbers are included
simultaneously, the resultant cooling rate profiles are influenced by the
overlapping of the individual contributions.  For the line-by-line models,
this should not introduce any additional discrepancy.  As seen from Fig.8,
for the mid-latitude summer profile, the two line-by-line models
essentially track each other up to the stratopause at around 1 mb.  Since
the differences in computed cooling rates between the two models for
individual gases was small, the case with all gases is similar.  Fig.9 shows
a comparison for the subarctic winter profile.  Here, differences between
the two line-by-line models reflect the discrepancies in the CO2-only case.
The performance of the parameterized model is determined primarily
by water vapor in the troposphere and carbon dioxide in the stratosphere.

The model therefore has greater success in the troposphere of the subarctic
winter case as shown in Fig.9 than in the midlatitude summer case in
Fig.8.  One important point to note is that the tropospheric cooling rate
includes the effect of the overlap of water vapor absorption with the 15 mm
CO2 band.  It also includes overlap of the water vapor continuum with the
9.6 mm ozone band.  The overlap scheme assumes the multiplicative
property of the diffuse transmittance over the entire band.

6. Conclusions

The ICRCCM study has motivated a careful comparison of infrared
fluxes as computed by a wide range of radiation models.  We have given a
detailed comparison of infrared cooling rate profiles produced by two line-
by-line methods and a parameterized model for a relatively small set of
model atmospheres.  We have shown that model resolution, quadrature
schemes, and the location of the model top can influence the cooling rate
results considerably, although radiative fluxes may not be appreciably
affected.  This should be a cautionary note in drawing conclusions from the
ICRCCM results, which, for the most part, deal only with fluxes.  In
particular, the success or failure of a parameterization should not be
judged by a comparison of fluxes alone.

The comparison of cooling rates has shown that the two line-by-line
methods agree quite well, with tropospheric cooling rates typically within
0.05 Kday^-1.  Differences are due largely to limitations in vertical
resolution rather than to any systematic effects.  The parameterization
gives tropospheric cooling rates with an average error of ~0.1 Kday^-1 and
largest differences of ~0.3 Kday^-1 when all gases are included; there are
larger differences for computations with water vapor alone without the
continuum.  The parameterized model shows a somewhat systematic
offset in the lower troposphere, but the mean tropospheric cooling rate is
estimated fairly well.

The line-by-line agreement is in many ways quite encouraging, given
their independent development, different spectral modeling techniques,
and different numerical approaches.  We have seen that both calculations
are subject to uncertainties of order ~1 percent in the downward surface
flux based on their sensitivity to water vapor continuum assumptions
alone.  There remains a strong need for improved spectral data in some
bands and most importantly, a better understanding of the water vapor
continuum and how it can be modeled from both the line-by-line and
broad-band perspectives.  The GLA infrared parameterization has largely
succeeded in reproducing  both the vertical structure of the radiative
cooling rate and its sensitivity to atmospheric temperature and constituent
profiles.   The CO2 and ozone parameterizations are particularly
successful while water vapor is adequately represented for most climate
applications.

Appendix A.  Cooling rate sensitivity to layering and vertical integration
method

Concerns about the adequacy of the chosen vertical resolution and the
choice of integration scheme for the line-by-line calculation led us to look
directly at the rate of convergence of cooling rates with increasing number
of vertical gridpoints.  This study was done with full-spectrum calculations
using each of three different vertical integration schemes at successively
finer vertical resolution.  In this appendix we detail the three methods and
document the results: namely, that the linear B(t) form of the T dB
approximation converges quite well, a second T dB approach is nearly as
successful, while the BdT form is by far the slowest to converge with
cooling rate biases near the surface and at the top of the atmosphere.  The
60-layer model is shown to have sufficient vertical resolution, that is, that
20 mb tropospheric layers were adequate to resolve temperature, density,
and cooling rate variability.

a. BdT approximation.   The isothermal layer assumption leads to a
simple representation of Eq.(8) with the single-layer contribution given by

     [ Equation A1 ]

Bave represents the average Planck function for the layer lying between
levels 1 and 2; we have simply evaluated the Planck function with a layer-
center temperature.  The two exponential integrals represent diffuse
transmittance from levels 1 and 2 to the observer.

b. T dB approximation.   The second approach or mean transmittance
model is obtained after integration by parts:

     [ Equation A2 ]

E3ave here denotes an average diffuse transmittance from the emitting
layer to the observation level.  We have taken the mean transmittance to
be the logarithmic average defined by

     [ Equation A3 ]

It is very important how the mean transmittance is defined; if the algebraic
mean of E3(1) and E3(2)  were to be used, the result would be identical to
the isothermal layer BdT method.  The effect of the logarithmic
transmittance average is to use an emitting temperature for the layer
somewhere between that at layer center and the layer surface closer to the
observer.

c.  Linear B(t) approximation.   The third method makes the assumption
that the Planck function is linear in monochromatic optical depth at each
wavenumber (that is, approximately linear in pressure) according to

     [ Equation A4 ]

The second term is integrated by parts to give an expression which is
closely related to the T dB approximation, that is

     [ Equation A5 ]

where the last ratio can be thought of as replacing the earlier geometric
average of E3 in order to better represent the layer mean transmittance.

d.  Comparative results.   One can think of the three approximations
together as forming a hierarchy of models based on (a) a single-parameter
algebraic mean transmittance, (b) a single-parameter geometric mean,
and (c) a two-parameter transmittance approximation.  Effectively, they
are all variations on the T dB approximation with the mean diffuse
transmittance defined differently in each case.
Figures A1, A2, and A3 show cooling rate profiles as obtained using the
BdT approximation, the T dB approximation, and the linear B(t)
approximation, respectively.  In each case, the three curves show the
convergence of cooling rates as maximum layer thickness is reduced from
80 to 40 to 20 mb, that is, as the number of model layers is increased from
15 to 30 to 60 layers.  The rate of convergence with respect to layer
thickness gives a good estimate of the quality of each approximation.

The non-linearity of the radiative transfer problem is evident in
Fig.A1;  thicker layers described in terms of a mean layer temperature
have computed cooling rates which are not the correct mean values.  The
isothermal layer approach is by far the slowest to converge, and tends to
give cooling rates which are too high near the surface and somewhat low
above 600 mb.  The other approximations, as seen in Fig.A2 and Fig.A3,
lead to fairly comparable cooling rates; both adequately estimate the mean
layer cooling, even for rather thick (80 mb) layers.  The linear Planck model
gives somewhat better flux and cooling estimates for thick layers, but the
differences are not significant (and not evident in the figures shown).

Based on the convergence of the linear B(t) method, the 60-layer model is
seen to represent cooling rates without significant bias.

e.  Note on BdT method.  The nature of the biases introduced by the
isothermal layer or BdT approach is important to understand.  Errors are
largest at opaque wavelengths where self-absorption of emitted radiation
within a single layer effectively changes the location of the actual
radiating center from the layer center to a point somewhat closer to the
nearer boundary.  If flux estimates are based on layer center temperature,
all tropospheric upward fluxes will be overestimated and downward
fluxes underestimated.  For layers with a  thickness of 50 mb or more the
errors can be several Wm^-2.  Fortunately, in computing cooling rates there
is a considerable cancellation of errors.  However, this does not happen
under two circumstances: (1) near the surface and (2) where lapse rate or
layer thickness change abruptly.  While the downward flux near the
surface may be considerably underestimated, the upward flux is well
estimated by the surface contribution.  The largest errors in flux
divergence are at the surface which translate into large cooling rate
overestimates in the layer just above the surface.  This tendency is clearly
demonstrated in Fig.A1.  Similar arguments allow us to understand why
computed stratospheric cooling rates will be too low.

It is also clear from this perspective that sudden changes in layer
thickness will tend to enhance errors.  The magnitude of the flux errors are
in rough proportion to layer thickness and lapse rate.  Errors to the cooling
rate within a particular layer can be expected to cancel if the few layers
just above and just below have comparable thicknesses and lapse rates.
Therefore, a uniform grid or one with slowly varying layer thickness is
important if an isothermal layer approximation must be used.  Even so,
fluxes can be expected to show considerable bias.

Footnote.  A common data structure for reporting ICRCCM line-by-
line results has been developed by the two groups with results reported
here.  One of us (WLR, NASA/GSFC Code 913, Greenbelt, MD 20771) will
make every effort to make the data available to anyone with a research
interest in these results.  It is expected that a proper archive will be
established for these and other ICRCCM infrared line-by-line results.

Acknowledgements.  The authors are grateful to S. Fels, M.D.
Schwarzkopf, and S. Freidenreich of GFDL for making their line-by-line
results available to us for detailed comparison.  The GLA line-by-line mass
absorption algorithm and spectral database are the work of D. Chesters of
the Goddard Laboratory for Atmospheres; his active support of this
project was invaluable.  We would also like to thank W. Wiscombe and
M.D. Chou for helpful comments and suggestions, as well as the
reviewers who helped us to clarify the original manuscript.  We
acknowledge the support of this work under NASA Contract No. NAS5-
30430 (W. Ridgway), and by NSF Grant No. ATM-8909870 and NASA
Grant No. NAG5-1088 (Harshvardhan).

REFERENCES

Chou, M. D., Broadband water vapor transmission functions for
atmospheric IR flux computations, J. Atmos. Sci., 41, 1775-1778, 1984.

Chou, M. D., and L. Peng, A parameterization of the absorption in the 15
mm CO2 spectral region with application to climate sensitivity studies,
J. Atmos. Sci.,40, 2183-2192, 1983.

Burch, D. E., and R. L. Alt, Continuum absorption by H2O in the 700-1200
cm^-1 and 2400-2800 cm^-1 windows, Air Force Geophysics Laboratory
Report No. AFGL-TR-84-0128, 1984.

Fels, S. B., Simple strategies for inclusion of Voigt effects in infrared
cooling rate calculations, Appl. Opt., 18, 2634-2637, 1979.
Harshvardhan, R. D, D. A. Randall, and T. G. Corsetti, A fast radiation
parameterization for atmospheric circulation models, J. Geophys. Res.,
92,  No. D1, 1009-1016, 1987.

Harshvardhan, D. A. Randall, T. G. Corsetti, and D. Dazlich, Earth
radiation budget and cloudiness simulations with a general circulation
model, J. Atmos. Sci., 46, 1922-1942, 1989.

Liou, K.-N., An Introduction to Atmospheric Radiation, 392 pp., Academic
Press, 1980.

Luther, F., R. G. Ellingson, Y. Fouquart, S. Fels, N. A. Scott, and W. J.
Wiscombe, Intercomparison of radiative codes in climate models
(ICRCCM): longwave clear-sky results, Bull. Am. Meteor. Soc., 69, 40-
48, 1988.

Ramanathan, V., Atmospheric general circulation and its low-frequency
variance: radiative influences, Short and Medium-Range Weather
Prediction, T. Matsuno, ed., Collection of Papers presented at the
WMO/IUGG NWP Symposium, Tokyo, 4-8 August, 1986, pp. 151-175,
1987.

Randall, D.A., Harshvardhan, D. Dazlich, T. G. Corsetti, Interactions
among radiation, convection, and large-scale dynamics in a general
circulation model, J. Atmos. Sci.,46, 1943-1970, 1989.

Roberts, R. E. , J. E. A. Selby, and L. M. Biberman, Infrared continuum
absorption by atmospheric water vapor in the 8-12 mm window, Appl.
Opt., 15, 2085-2090, 1976.

Rodgers, C. D., Some extensions and applications of the new random
model for molecular band transmission, Q. J. R. Meteorol. Soc., 94, 99-
102, 1968.

Rothman, L.S., AFGL atmospheric line parameters compilation: 1980
edition.  Applied Optics, 20, 791-795, 1981.

Rothman, L.S., R.R. Gameche, A. Barbe, A. Goldmann, J.R. Gillis, L.R.
Brown, R.A. Toth, J.-M. Flaud, and C. Camy-Peyret, AFGL
atmospheric line parameters compilation: 1982 edition.  Applied Optics,
22, 2247-2256, 1983.

Rothman, L.S., R.R. Gameche, A. Goldmann, L.R. Brown, R.A. Toth, H.M.
Pickett, R.L. Poynter, J.-M. Flaud, C. Camy-Peyret, A. Barbe, N.
Husson, C.P. Rinsland, and M.A.H. Smith, The HITRAN database:
1986 edition.  Applied Optics, 26, 4058-4097, 1987.

Schuster, A., In Selected Papers on the Transfer of Radiation  (edited by D.
Menzel). Dover, New York, 1905.

Schwarzkopf, M.D., and S.B. Fels, The simplified exchange method
revisited: An accurate, rapid method for computation of infrared
cooling rates and fluxes, J. Geophys. Res., (in press), 1990.

Schwarzschild, K., In Selected Papers on the Transfer of Radiation  (edited
by D. Menzel). Dover, New York, 1906.

Wiscombe, W.J., Extension of the doubling method to inhomogeneous
sources, J. Quant. Spectrosc. Radiat. Transfer, 16, 477-489, 1976.
World Meteorological Organization, The intercomparison of radiation
codes in climate models (ICRCCM), WCP-93, 37 pp, Geneva, 1984.

TABLES

1. Pressure and temperature pairs for which detailed line-by-line
computations of spectral mass absorption coefficients were
performed.  Absorption is based on an inhomogenous pressure
integral of the truncated Voigt line shape extending vertically from the
pressure level shown to space.

2. Summary of flux results at top of atmosphere, tropopause, and
surface for the three models studied:  Models are the Goddard
Laboratory for Atmospheres line-by-line flux model (GLA/lbl),
Geophysical Fluid Dynamics line-by-line model (GFDL/lbl) and
Goddard parameterized model (GLA/par).  Quoted results are for the
spectral range 0-3000 cm^-1 except where noted.  'H2O cont.' refers to
the Roberts et al. parameterization of the water vapor continuum.

TABLE 1

Pressures (mb)                            Temperatures (K)

0.1       0.2     0.3     0.5     0.7     210     240     270     300
1.0       2.0     3.0                     200     240     270     290
4.0                                       200     230     260     280
5.0       6.0     7.0                     200     220     250     270
8.5                                       200     220     240     260
10.0      12.5                            200     220     230     250
15.0      17.5    20.0    25.0    30.0    200     220     230     240
35.0      40.0    50.0    60.0    70.0    200     210     220     230
85.0      100.                            190     210     220     230
125.      150.                            200     210     220     230
175.      200.    250.    300.            210     220     230     240
350.      400.    450.                    220     230     240     250
500.      550.                            230     240     250     260
600.      650.                            240     250     260     270
700.                                      240     250     260     280
750.                                      240     260     270     280
800.      850.                            240     260     270     290
900.      950.                            250     270     280     300
1000.     1050.                           260     280     290     310

TABLE 2 (could not be included)