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. 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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)