A New Narrowband Radiation Model for Water Vapor Absorption

Transcription

1 VOL. 57, NO. 10 JOURNAL OF THE ATMOSPHERIC SCIENCES 15 MAY 2000 A New Narrowband Radiation Model for Water Vapor Absorption JUYING X. WARNER National Center for Atmospheric Research, Boulder, Colorado ROBERT G. ELLINGSON Department of Meteorology, University of Maryland at College Park, College Park, Maryland (Manuscript received 28 September 1998, in final form 25 June 1999) ABSTRACT The accuracy of radiation models is a critical issue in climate studies. However, calculations from different radiation models used in climate calculations disagree with one another, and with more detailed models, at levels significant to many climate problems. With several new advances in the field of radiation modeling, it is possible to develop more accurate band models and validate them against radiation observations of known high accuracy. In this paper, a new accurate narrowband longwave radiative transfer model for clear-sky conditions is developed. In the first part of this study, only water vapor effects are included, and the model results are tested against line-by-line radiative transfer model (LBLRTM) calculations. In the model development, it is first shown that traditional techniques for estimating Malkmus statistical model parameters from the line compilation and line-by-line models cannot be trusted to give accurate transmittance function. A new technique is then described that calculates water vapor line transmittances with good agreement with LBLRTM calculations (i.e., with rms errors less than 0.01 for more than 97% of the intervals). The water vapor continuum is included in a manner consistent with the water vapor line absorption. Fluxes calculated with the model agree with LBLRTM to about 1 W m 2 for the entire vertical range of the atmosphere for several test cases. The heating rate errors are reduced by as much as 0.25 C day 1 below the tropopause for the test cases compared with the original narrowband model. 1. Introduction Radiation schemes have become an integral part of general circulation models and operational weather forecast models. As climate modelers strive for more accurate simulation and prediction of the climate, there has been an increased need for more accurate radiation calculations in the climate models because climate problems are sensitive to small changes in radiation quantities. For example, the greenhouse effect of CO 2 doubling results from about a 1% change in the radiation budget (National Academy of Sciences 1982). A 10 W m 2 change in net flux at the surface is on the average about 3% of the surface downwelling longwave flux and if continued for a year could lead to a 1 C change in the sea surface temperature. Therefore, the usually accepted 5% 10% uncertainties in radiation calculations are not sufficient for many climate studies. Unfortunately, calculations from different radiation models used in climate calculations disagree with one Corresponding author address: Dr. Juying Warner, NCAR, P.O. Box 3000, Boulder, CO juying@ncar.ucar.edu another at levels significant to many climate problems (Ellingson et al. 1991; Ellingson and Fouquart 1991). Until recently it was not possible to set stringent limits on the acceptable spread between model calculations because of the absence of radiation observations of known high accuracy. However, recent experiments have produced radiation and auxiliary datasets that allow more detailed tests of models (Ellingson and Wiscombe 1996). Hence, these experiments, along with other advances in the radiation field, provide the possibility for the development of more accurate radiative transfer models. The thrust of this study is to use these developments to develop an accurate narrowband model of longwave radiative transfer for eventual use in climate applications. Much comparison work between models was done under the Intercomparison of Radiation Codes in Climate Models (ICRCCM). Luther et al. (1988) and Ellingson et al. (1991) summarized the radiation calculations from various models for a set of ICRCCM cases. However, there have been several new advances in the field of radiation modeling since ICRCCM that encourage improvements in the parameterizations of radiative processes used in the band models. Perhaps the most important advance is the improved modeling of 2000 American Meteorological Society 1481

2 1482 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 what is known as the water vapor continuum. Clough et al. (1989) introduced a new formulation of the continuum by assuming that the continuum is due to absorption in the far wings of H 2 O lines. They showed that the water vapor continuum should be included in a manner consistent with individual line absorption, and it should extend across the entire spectrum. The consistency is achieved by cutting off the absorption of individual lines at a specified distance from the line centers, and the absorption outside of that range is calculated as continuum absorption using parameterizations fitted to laboratory and/or atmospheric observations. Many radiation modelers calculate their model parameters using tabulated line parameters for an assumed line shape. They add the continuum separately, but selectively from either laboratory data or other model calculations for window regions (Harshvardhan et al. 1987; Ramanathan and Downey 1986; Morcrette 1991; etc.). However, this neglects the fact that the line strengths and shapes are directly tied to the continuum. Their approach may be a good approximation in window regions where there are few lines, but it will fail in intermediate and strong line regions where the continuum is not accounted for appropriately by the local line parameters or in combination with the assumed continuum [i.e., the effects of distant lines could be over estimated or underestimated with this approach; see section of Goody and Yung (1989)]. A second major advance since ICRCCM is the development of a new technique by Lacis and Oinas (1991) for calculating the parameters of the Malkmus (1967) band model. The Malkmus model has been used to develop a large number of climate parameterizations, but previous studies included only parameters derived from the weak and strong line limits. Lacis and Oinas concluded that over most of the longwave spectrum, the Malkmus model with parameters fitted to line-by-line (LBL) calculated transmittances reproduces the LBL transmissions over homogeneous paths with good accuracy. However, they pointed out that there are exceptions that are difficult for the Malkmus model to represent accurately due to the spectral lines not obeying the Malkmus model assumptions (Lacis and Oinas 1991). A third advance since ICRCCM is the release of new spectroscopic databases by Rothman et al. (1992, 1998). The data represent the most recent results of laboratory studies and quantum mechanical calculations. However, it is difficult to quantify the importance of these changes alone in band models because of compensating effects within narrow bands. Nonetheless, documented changes in spectroscopic data should be incorporated into new parameterizations as they become available to account for new understanding. A more traditional band model parameterization for water vapor absorption that incorporates the improved modeling of the continuum, uses the most recent spectroscopic data, and has been tested with atmospheric observations has not been reported. This study is designed to implement the advances in radiative transfer modeling to develop a new narrowband model of high accuracy, and to validate the new model with Spectral Radiance Experiment (SPECTRE) (Ellingson and Wiscombe 1996) and Atmospheric Radiation Measurement (ARM; Stokes and Schwartz 1994) observations after including the effects of other important atmospheric gases. This study is limited to clear-sky conditions purposely to isolate problems in gaseous absorption from the complicating effects of clouds and aerosols. Once completed, cloud effects should be easier to ascertain. This paper deals with the development of the model for water vapor absorption. The addition of other gases and comparisons of clear-sky observations with calculations will be the subjects of future papers. The development of a band model that properly incorporates both water vapor line and continuum absorption requires two steps: 1) the formulation of a water-vapor-line-only scheme and 2) the continuum formulation. Since Clough et al. developed a formulation for separating line and continuum for line-by-line calculations [such as used in the Line-By-Line Radiative Transfer Model (LBLRTM); Clough et al. 1992; Clough and Iacono 1995], we have adopted that formulation for application to band model transmittance calculations. The approach of developing the water-vapor-line-only scheme is to fit an empirical model a modified Malkmus model to line-only transmittance calculations from LBLRTM. Since Clough s continuum absorption is a slowly varying function spectrally, and in LBLRTM the continuum coefficients are precalculated and stored in the model at 10-cm 1 resolution, Clough s continuum coefficients will be introduced directly into the narrowband models. The bulk of this paper is divided into three major parts. Section 2 discusses the accuracy and the problems associated with the Malkmus narrowband model for water-vapor-line-only effects. An approach similar to that by Lacis and Oinas (1991) was tested for transmittance calculations, and some major drawbacks of this technique are discussed. Section 3 describes the development of a new band model parameterization for water vapor lines. Section 4 extends the model to include the continuum absorption and compares model calculations with LBLRTM. 2. Discussion of Malkmus formulation Before we began developing a new model for water vapor absorption, we examined the existing and wellused statistical formula by Malkmus (1967). The main sources of error in Malkmus transmittance calculations include the contributions from the determination of the model parameters and from the inadequacy of the band model. To separate the two, we tested the Malkmus model using two sets of parameters: one set obtained

3 15 MAY 2000 WARNER AND ELLINGSON 1483 by fitting the model to the asymptotic limits and the other by fitting to LBLRTM transmittances. a. Malkmus model with parameters fitted to asymptotic limits The Malkmus (1967) statistical band model was derived under the following assumptions: 1) each band is an infinite array of absorption lines of uniform statistical properties; 2) each interval is flanked by statistically similar intervals; 3) the absorption lines in an interval are randomly distributed, overlapping Lorentz lines; and 4) the line intensity probability density function is given as p(s) (1/s) exp( s/s), (2.1) where s is the strength for a line and S is the average intensity in the band. After integrating over the range of intensities in the spectral interval, the band transmittance can be expressed as [ ] 1/2 4 u 2 T(u) exp 1 1, (2.2) where u is the absorber amount; is the ratio of mean line half-width to mean line spacing; S/d; and d /N, where d is the mean line spacing and N is the number of absorption lines within the spectral interval. The most common technique used by the radiationclimate community to determine and is to fit the model to spectral line data using the weak and strong line nonoverlapping limits. Under these limits, the band transmittances can be related to those of isolated nonoverlapping lines through the concept of the equivalent width. In the optically thin limit, T(u) 1 u. The appropriate mean line strength is N i i 1 S s. (2.3) N The optically thick, nonoverlapping limit, for which T(u) 1 ( u) 1/2, requires the mean line half-width to be given as 4 1 N 1/2 i i i (s ). (2.4) S N The parameters s i and i are the line strength and halfwidth of the ith line, respectively. These can be determined directly from line data, such as the High-Resolution Transmission Molecular Adsorption Database (HITRAN) line compilation (Rothman et al. 1992). The Malkmus narrowband formulation has been used widely (e.g., Rogers and Walshaw 1966; Ellingson and 2 FIG. 1. An example of the least squares fit of the transmittance models to LBLRTM calculations for the interval cm 1. The filled circles denote LBLRTM transmittances and the long-dashed line denotes the Malkmus transmittances calculated from asymptotic limit-derived parameters. The short-dashed line represents the results from the Malkmus formula with parameters fitted to LBLRTM transmittances. The solid line represents the new model results with parameters fitted to LBLRTM transmittances. Gille 1978; Ramanathan and Downey 1986; Lacis and Oinas 1991; and many others) to calculate transmittances, radiances, fluxes, and heating rates. However, there has not been a systematic evaluation of its accuracy spectrally across the entire spectrum for water vapor. Goody (1964) argues that the narrowband technique reduces the uncertainty of the transmittances to the first order (about 10%). However, the usual manner of determining model coefficients allows only transmittance interpolation and/or extrapolation from the exact limits. The exact limits are often not achieved in the atmosphere. In addition, the distributions of the line intensities and positions assumed by the Malkmus model are often not representative of the actual lines. To illustrate the problems associated with the use of the Malkmus model fitted to spectral line compilations, Fig. 1 shows the transmittances calculated from the Malkmus model and LBLRTM for the cm 1 interval at a pressure of 1013 mb and a temperature of 260 K over the range of precipitable water (PW) amount from 10 4 to 10 cm at 45 values, as listed in Table 1. The filled circles denote LBLRTM transmittances and the long-dashed line denotes the Malkmus transmittances calculated from asymptotic limit-derived parameters. All other curves in the figure will be explained in later sections. The difference between the two models increases as the water vapor amount increases until the transmittances approach zero. Around PW 1 cm, the transmittance difference is about 0.1 when the actual transmittance is below In this spectral interval, the model fit to HITRAN parameters agrees with LBLRTM for PW less than 0.01 cm and more than 10 cm. Gen-

4 1484 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 TABLE 1. The absorber amounts (in g cm 2 ) used in the calculation of LBLRTM homogeneous transmittances for fitting the band model parameters erally, the weak and strong line limits are far away from atmospheric conditions. To illustrate the magnitude and spectral dependence of the transmittance errors possible with the Malkmus model fitted to spectral line parameters, Fig. 2a shows the rms transmittance errors of the Malkmus model that we determined using the 1992 HITRAN data base relative to LBLRTM at 10 cm 1 intervals. The transmittances were calculated at the same temperature, pressure, and absorber amounts as in Fig. 1. The rms errors in the majority of the intervals ( 94%) are within 0.1 in transmittance, while for some intervals the errors can be more than The rms errors are higher for the stronger absorbing regions, that is, the pure rotational FIG. 2. The spectral distribution of the rms transmittance errors calculated from (a) the Malkmus model and (b) the new model. band (0 650 cm 1 ) and vibrational rotational band ( cm 1 ). The Ellingson and Gille (EG) radiative transfer model (Ellingson and Gille 1978; Ellingson et al. 1989) uses the Malkmus transmittance formula fitted to asymptotic limits to compute the transmittances, and atmospheric radiances, fluxes, and heating rates. To evaluate the performance of such a model, the fluxes for the cm 1 interval were calculated by both the EG model and LBLRTM for water-vapor-line-only absorption using the McClatchey et al. (1972) midlatitude summer (MLS) atmosphere, and the EG LBLRTM flux differences are shown by the thinner long-dashed lines in Fig. 3. The EG LBLRTM upward flux differences tend to be positive between the surface and 500 mb and negative at lower pressures. The maximum positive differences amount to about 1.5 W m 2 near 900 mb, whereas similar magnitude negative differences occur at the top of the atmosphere. The EG model downward flux is lower than LBLRTM by 1.8 W m 2 at the surface and 3.0 W m 2 greater around 500 mb. Note that the upward flux differences at the surface appear to be due to difference techniques and spectral resolution used in performing the spectral integration. When considering the flux values integrated over the entire infrared spectrum, the flux differences due to water-vapor-line-only absorption between the EG model and LBLRTM are small (less than 1%). However, if one examines the spectral distribution of the flux differences, one sees that there are often large differences in individual spectral intervals, and the sum of alternating signs of the spectral differences yields small integrated flux differences. The left upper panel of Fig. 4 shows the spectral distribution of downward fluxes at various levels due to water-vapor-line-only absorption calculated by LBLRTM, while the left middle panel shows spectral differences between EG model and LBLRTM (EG LBLRTM). The red bands represent the positive errors and the blue bands represent negative ones. For altitudes below 5 km, the error bands of blue and red between the 500 and 1300 cm 1 region partially cancel when spectrally integrated. The upward flux differences have similar spectral features and are not shown. The accurate integrated flux differences under the MLS conditions do not guarantee good results for other atmospheric states unless the physics is modeled correctly because such

5 15 MAY 2000 WARNER AND ELLINGSON 1485 FIG. 3. Profiles of the upward, downward, and net flux differences between the EG model and LBLRTM (long dashed), and between the new model and LBLRTM (dotted) for the MLS case. The thinner lines represent the water-vapor-line-only calculations, and the thicker lines include the water vapor line and continuum effects. spectral cancellations are not guaranteed for all conditions. Although the total flux comparisons show good agreement for water vapor line absorption in the MLS atmosphere, the water vapor line and continuum absorptions are not defined consistently. Thus, when the continuum absorption is added (see below), the model uncertainty may increase. The thicker long-dashed lines in Fig. 3 show flux comparisons between the EG model and LBLRTM for the combined effects of water vapor line and continuum absorptions for the MLS case. The EG upward flux at the top of the atmosphere is 5.3 W m 2 greater than LBLRTM, and the downward flux at the surface is 11.5 W m 2 lower than LBLRTM. The error for net flux at the surface is about 13%, and at the top it is about 1.6%. Both at the surface and the top of the atmosphere, the error increased substantially when the continuum was added. In summary, the major uncertainties in the Malkmus transmittance model are associated with two major sources: the asymptotic approach used in calculating the parameters, and, as will be shown below, the approximations made in deriving the Malkmus model. These lead to large transmittance errors, which in turn lead to spectrally dependent flux and heating rate errors. Although the errors often compensate, there is no assurance of the magnitude of cancellation under all conditions. b. Malkmus formula with parameters fitted to LBLRTM transmittances Another method for obtaining the Malkmus parameters is to fit the model to LBL transmittances over a range of absorber amounts (Lacis and Oinas 1991). The first step of this study was to determine the parameters by fitting the Malkmus model to LBLRTM homogeneous transmittances for a range of absorber amounts. For the purpose of determining Malkmus model coefficients by a nonlinear least squares analysis, we found that we could always reach convergence if the homogeneous path transmittance was expressed as 2 T(u) exp{ [ 2m1u m2 m 2]}, (2.5) where, from (2.2), m 2 B/2, m 1 m 2, and u is the precipitable water. The parameters m 1 and m 2 were estimated by a nonlinear least squares fit (IMSL STAT/LIBRARY 1991) of the model to the LBLRTM transmittances at 45 PW amounts ranging from 10 5 to 20 cm, as listed in Table 1. A homogeneous path of pressure of 1013 mb and temperature of 260 K was chosen to study the homo-

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7 15 MAY 2000 WARNER AND ELLINGSON 1487 geneous transmittances. The spectral interval was fixed at 10 cm 1 for the spectral region cm 1. The parameters calculated from the weak and strong line limits were used as the initial values for the nonlinear least squares fitting. A typical example of the least squares fit to LBLRTM data is shown in Fig. 1 (short-dashed line) for the cm 1 spectral. In the range of precipitable water from 0.1 to 10 cm, the differences in transmittance between the band model and LBLRTM are reduced by about as compared with the asymptotic limit results. The rms errors of the transmittances calculated with parameters fitted to LBLRTM calculations are below 0.02 for 254 of the 300 intervals, as shown by the squares in Fig. 2b. The errors are reduced more in the stronger absorbing regions, cm 1 and cm 1, than for the weakly absorbing region. Note that the rms errors reflect only an average error over the entire range of precipitable water. The errors are strongly dependent on the absorber amount. Two typical types of variations of transmittances with absorber amount are shown in Figs. 5a and 5b for bands and cm 1, respectively. Again, the circles represent the LBLRTM transmittances, the longdashed lines are for Malkmus model with parameters from the asymptotic limits, and the short-dashed lines are from the fitting technique. Figure 5a shows a band where the Malkmus calculations from asymptotic limits overestimate the transmittances over most of the absorber range, but the variation of the Malkmus transmittance is approximately the same as that of the LBLRTM. When the parameters are obtained by fitting to the LBLRTM transmittances (short-dashed line), the model transmittances show closer agreement with the LBLRTM over the entire range of absorber amount. This type of band benefits the most by using the fitted parameters in the Malkmus model. However, the percentage of the bands that have a similar slope of the transmittance distribution is relatively low, 22% (33 out of 151 bands) in the strong absorption regions (0 600 and cm 1 ). Figure 5b shows an example where the asymptotic limit parameters overestimate the transmittances at larger absorber amounts and underestimate the variation with absorber amount. This type of behavior is shown by about 70% of the intervals in the strong absorbing region (0 600 and cm 1 ). In these cases, although the transmittances calculated from the fitted parameters were improved as a whole over the range of precipitable water, the errors in the lower and/or intermediate precipitable water range increased. Thus, the FIG. 5. An example of the transmittances for (a) the cm 1 interval from LBLRTM and the Malkmus model fitted to HITRAN and LBLRTM, and (b) for the cm 1 interval. Malkmus model transmittances determined with fitted parameters might not yield overall improvements in the atmospheric calculations since the improved results at one level in the atmosphere may be offset by larger errors at others. This indicates that there are many locations in the spectrum where the fitted Malkmus model may not give increased accuracy, despite the determination of the parameters from LBLRTM calculations. 3. An improved transmittance model a. Homogeneous transmittance calculations The Malkmus model is based on several simplifying assumptions in order to reduce the complexity of the band transmittance calculations. No real band fits these assumptions; therefore, it is not surprising that there are large errors in some situations. The analysis discussed above shows that the most common error is that the Malkmus model with parameters determined from LBLRTM calculations underestimates the transmittance variation at medium-high and high absorber amounts in FIG. 4. The spectral distributions of the downward fluxes from LBLRTM (upper left) flux differences between the EG models and LBLRTM (middle left), and between the new model and LBLRTM (lower left) due to water-vapor-line-only absorption for the MLS atmosphere. The right panels are as in the left panels, but for heating rate calculations and differences, for the water vapor line and continuum effects.

8 1488 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 the majority of the bands. This indicates that a better model might be obtained if we could simply multiply the Malkmus model by a function that increases the rate of change of transmittance with absorber amount. This approach is equivalent to the concept of positive or negative continua introduced, but never developed, by Goody (1964). The transmittance in a narrow band can be expressed as T i T ia T ic, (3.1) where T ia is the band model transmittance determined from usual analytic approximations and T ic is the pseudotransmittance caused by variations in the band parameters outside of the spectral interval. Alternately, one might think of the transmittance as the superposition of two (or more) different arrays of spectral lines with different distributions of line positions and/or line strengths (e.g., regular and random arrays, and random array but with interval-to-interval variations of the distribution of line strength). In practice, one calculates parameters based on the line strengths in a given interval. It is implicitly assumed that the same average parameters exist outside of this range. In regions where there are relatively large changes adjacent to the interval, this can lead to large errors. Determination of interval parameters by fitting to lineby-line calculations helps reduce this error, but it does not eliminate the chance that the functional form of the transmittance contribution from within and adjacent to the intervals is incorrect. Since our analysis indicates that the errors in the majority of cases could be reduced by a function that increases the slope of the transmittance variation, this research focused on finding such a function. After studying the properties of many different functions, and following attempts at fitting the residual transmittance errors, the best results were obtained by expressing the transmittance as 2 exp{ [ 2m1u m2 m 2]} T(u), (3.2) cosh(m u) 3 where u is the absorber amount. The parameters m 1, m 2, and m 3 are temperature and pressure dependent and can be determined by a nonlinear least squares fit of the LBLRTM calculations. This representation of the transmittance is simply the Malkmus transmittance function divided by cosh(m 3 u). The result of the cosh(m 3 u) term is to increase the derivative of the transmittances with respect to precipitable water at both intermediate low and high values relative to that of the Malkmus model alone. As will be demonstrated below, this term empirically compensates for the oversimplified assumptions made in deriving the Malkmus model, namely, Lorentz lines with an exponentially tailed S 1 line-strength distribution, and a homogeneous distribution of line strengths from interval to interval. FIG. 6. (a) The distribution of spectral line intensities in the cm 1 interval, and (b) the transmittances in the same interval from LBLRTM, the Malkmus model fitted to HITRAN and LBLRTM, and the new model fitted to LBLRTM. The physical justification of our modification to the Malkmus model can be explained by examining the detailed line structure of each interval. In the strong absorption spectral regions, the intensity of the strongest lines are several (four to seven) orders of magnitude greater than that of the weak lines in the same region. But there are several times more weak lines than strong lines. For example, inside the 10 cm 1 intervals in the strong absorption regions, there are often 1 7 strong lines and very weak lines. The transmittance variation with absorber amount is determined by the distribution of line strengths and line positions of the interval. The very strong lines have the largest effect for short pathlengths, becoming saturated as the pathlengths increase. Therefore, the absorption by weak lines is more important for longer pathlengths. Figure 6a provides an example of a spectral interval that largely meets the Malkmus model assumptions. This interval has 61 lines, where 6 lines are two orders of magnitude greater than the remainder. The very strong lines are nearly randomly spaced in the interval, and the line intensities span the range of cm 2 g 1. As shown in Fig. 6b and discussed in the previous section, the parameters derived from the asymptotic limits resulted in systematic errors in the transmittances,

9 15 MAY 2000 WARNER AND ELLINGSON 1489 FIG. 7. (a) The distribution of spectral line intensities in the cm 1 interval, and (b) the comparison of the transmittances calculated from LBLRTM for the , , and cm 1 intervals, and from the Malkmus model and the new model for the cm 1 interval. whereas Malkmus model parameters fitted to LBLRTM greatly improved the transmittances. The solid curve in Fig. 6b (almost overlapping with the short-dashed line) represents the new model fit to LBLRTM, which has similar agreement with the Malkmus model fit. In other words, the new model and the Malkmus model give essentially the same results for these conditions. For a typical interval, however, there are only one or two strong lines located in a narrow region, while the rest of the interval contains only weak lines. As shown in Fig. 7a for the cm 1 interval, there are two strong lines between 400 and 401 cm 1, while there are no lines as strong in the cm 1 spectral region. Although the distribution of the line strengths for the lines in this interval still satisfies the assumed exponentially tailed S 1 line-strength distribution, there are few lines between the strong lines and the very weak lines. To study the relative contribution of the strong and weak lines, the interval was divided into two subintervals: cm 1 and cm 1. The transmittances calculated from LBLRTM for the subintervals are shown in Fig. 7b, in which the squares represent the transmittances for subinterval cm 1, crosses represent those from subinterval cm 1, and the filled circles represent the LBLRTM transmittances for the entire interval. When the absorber amount is low ( g cm 2 ), the transmittances of the narrow band is the average between the two subintervals. When the strong lines become saturated, the transmittances by this subinterval approach zero, and the transmittances of the narrow band are determined primarily by the subinterval of cm 1, as shown in Fig. 7b for absorber amounts greater than 0.1 g cm 2. After the strong lines are saturated, the rapid increase of absorption between 0.1 and 1.0 g cm 2 is due to absorption by the weak lines in the interval. The Malkmus narrowband model calculates the transmittances with statistically averaged line properties of the interval. The model uses this averaged line intensity for all absorber amounts. The long-dashed line in Fig. 7b shows the transmittances for this interval calculated by the Malkmus model. The transmittances of the band model show good agreement with LBLRTM for absorber amounts less than 0.01 g cm 2, underestimates between absorber amounts of g cm 2, and overestimates above 0.03 g cm 2. The Malkmus model transmittances for this interval show a similar variation with the absorber amount as the strong subinterval, but displaced toward larger absorber amounts. The saturating characteristics of the strong lines are not reflected in the band model transmittances. In LBLRTM calculations, the many weak lines start to absorb at high absorber amounts, and this results in more absorption than calculated by the Malkmus model. The increased absorption or decreased transmittance at relatively high absorber amounts is not accounted for in the Malkmus model even with parameters determined by fitting the model to LBLRTM. However, the absorption by the weak water vapor lines is very important in the flux and heating rate calculations in the lower troposphere due to the high water vapor concentration. The new formulation takes into account the increased absorption at the higher absorber amount, and increases the variation of transmittance with absorber amount. Figure 7b shows the improved transmittances by the new model (solid line), compared with those by LBLRTM (filled circles) and Malkmus model with parameters from asymptotic limits (long-dashed line). The above discussion has been directed at the strongly absorbing regions in the spectrum. In a weak region, such as the cm 1 window region, there are very few absorbing lines in each band and the line intensities are nearly constant. The transmittances are very high ( ) over the range of atmospheric absorber amounts. The transmittance variation is very low and the Malkmus model parameters fit to LBLRTM are sufficient to calculate the transmittances accurately (m 3 0). The following discussion summarizes the accuracy of the new transmittance formulation. Figure 1 shows a comparison of the transmittances for the cm 1 interval calculated by the new model (solid line) and

10 1490 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 the Malkmus model (dashed lines), as well as LBLRTM (filled circles). The new model transmittances have much better agreement with LBLRTM for precipitable water of cm. At the same time, the accuracy at the lower absorber amounts is not sacrificed. The spectral distribution of the rms transmittance errors calculated by the new model is shown in Fig. 2b by the filled circles. The new form fits LBLRTM transmittances within 0.01 rms errors for more than 97% of the spectral bands. The rms errors are reduced more in the strongly absorbing regions. However, for many bands, the new formulation slightly overestimates the derivative of transmittances with absorber amount. b. Determination of pressure and temperature dependencies The mechanism for the broadening of the absorbing lines changes as the pressure decreases throughout the troposphere, as discussed by Goody and Yung (1989). When the pressure is high, the absorbing lines are broadened by collisions of molecules, and the line shape is essentially Lorentzian. When the pressure is very low, the lines become Doppler broadened, and the Doppler line shape is independent of pressure. In between the two regions, the line shapes are determined by a combination of the pressure and Doppler broadening, as represented by the Voigt line shape. The LBLRTM integrated transmittances are calculated for a Voigt line shape, instead of the Lorentz line shape used in the Malkmus model. When a band model is fitted to the LBLRTM transmittances, the effects of Voigt line absorption are incorporated into the band model. The earlier discussion of the sources of the differences between LBLRTM transmittances and the band model calculations focused on the line strength distributions instead of line shapes because the calculations were done at high pressure where the Voigt line shape is nearly identical to the Lorentz line shape. In the Malkmus model, the temperature and pressure dependencies of the model parameters may be calculated independently. For example, the pressure dependence is embodied in the definition of the Lorentz half-width. The temperature dependence is often modeled through a quadratic fit of spectral line data (Rogers and Walshaw 1966). Since the new model parameters are empirically determined, their pressure variations are not completely independent of temperature variations. However, we have examined the accuracy of our parameterization, discussed below, using independent sets of temperature and pressure variations, and concluded that the pressure and temperature effects may be determined independently. The temperature and pressure variations for the ith parameter are assumed to have the form m i (T, p) m i (T 0, p 0 ) f i (T)q i (p) (i 1, 2, 3). (3.3) TABLE 2. The temperatures and pressures used to fit the T and P variations of the band model parameters. Calculations were performed at all pressures for the temperatures listed. Temperature (K) Pressure (mb) The functions f i and q i are assumed to have the forms 2 f i(t) exp[a i(t0 T) b i(t0 T) ] and (3.4) k N k q i(p) c ki[log10(p)] (i 1, 3), (3.5) k 0 and N is the degree of polynomial. Note that the coefficients a i,b i, and c ki were determined for each 10 cm 1 interval for the cm 1 range. The temperatures and pressures used to calculate the coefficients are listed in Table 2, and the absorber amounts are listed in Table 1. The reference pressure and temperature, p 0 and T 0, are 1013 mb and 260 K, respectively. All three model parameters decrease with decreasing pressure to about 150 mb. At pressures less than 150 mb, m 2 increases, while m 1 and m 3 remain approximately constant. Since the pressure term is not explicitly expressed in the new model, the variation of parameters with pressure can be determined only by fitting. It is difficult to model the pressure variation due to more than one physical mechanism by a single function. For the fitting procedure, pressure levels were chosen so that most of the data points were in the troposphere. Extrapolating the fitted pressure variation to very low pressures results in relatively large errors at high altitudes. The pressure variation of the parameters at low pressure was determined empirically based on the narrowband radiance and heating rate calculations. The final form adopted is 0.95 p i i p* q (p) q (p*) (i 1, 3) p 200 mb, (3.6) where p* 200 mb in (3.6). See (3.5) for p 200 mb and the variable definitions. Figures 8a and 8b show a comparison between transmittance curves at pressures of 1013 and 100 mb, respectively, for the cm 1 interval. The differences between the Malkmus (dashed) and LBLRTM (circles) transmittances increase as pressure decreases. This is due to two effects. First, since the coefficients were determined for nonoverlapping conditions in the weak and strong line limits, the pressure dependence for other conditions is calculated to only the first order. Second, since the LBLRTM uses Voigt line shape and the Malkmus model uses Lorentz line shape, the dif-

11 15 MAY 2000 WARNER AND ELLINGSON 1491 and pressure can be obtained by using higher-order approximations, instead of the first-order of approximation used by other modelers to solve the scaling problems. However, in this study we estimate the scaling properties of the third parameter empirically by using spectral heating rates. Specifically, the ith new model parameter is assumed to be given as i m (T, p) du m i. (3.7) du FIG. 8. The distribution of the transmittances calculated from LBLRTM and the Malkmus model at pressures (a) 1013 mb and (b) 100 mb. ferences in the transmittances are expected to be larger at lower pressure. As shown in Figs. 8a and 8b, the transmittance variation with absorber amount increases when pressure decreases. When the transmittance variation is large, the use of new model (solid line) is required to achieve good accuracy. The empirical term plays a role in modeling the Voigt line shape in the narrowband model. In order to apply the transmittance model to the atmosphere, it is necessary to account for the inhomogeneous atmospheric path. Since the Curtis Godson (C G) approximation (Curtis 1952; and Godson 1953) used in other investigations was developed from the asymptotic limits and for Lorentz lines, it too is a source of uncertainty in band model calculations. However, under the extremes of weak and strong absorbing conditions, the C G expression is exact for Lorentz line shapes and provides two equations for two unknowns. Thus, the problem can be solved analytically for twoparameter band models. In the three-parameter new model, it is necessary to find three limits in order to solve the problem analytically. In principle, an additional limit can be obtained by using the strong line limit for Doppler lines. Alternatively, the equivalent absorber amount, temperature, The above equation is used for m 1 and m 2 for all the intervals in the longwave region, and for m 3 in the strong absorbing region (0 640 and cm 1 ). However, use of this approximation for m 3 in the weakly absorbing intervals (such as in cm 1 ) causes a systematic bias in the heating rate calculations. Recall that m 3 was introduced to increase the variation of transmittances to absorber amount. In the weak absorbing regions, the transmittance in each band varies by a very small amount over the atmospheric absorber amounts; thus, m 3 is very small in these regions. A relatively large error is introduced when m 3 is weighted by the layer absorbers; therefore, we use an empirically derived expression for m 3 in the weak absorbing regions as described below. Noting that the pressure variations of m 3 and m 1 are similar in the stronger absorbing regions, we used the pressure variation of m 1 to replace that of m 3 for the weaker absorbing bands. This parameterization improves the radiance and heating rate calculations in these bands in the lower atmosphere. That is, m 3 for the inhomogeneous layer is calculated as m m (T, p) du m. (3.8) m 1 0 du The subscript 0 refers to the reference level. Other numerics such as the computation of the band Planck function, and the integrations over altitude and angle follow Ellingson and Gille (1978). c. Flux and heating rate calculations To assess the accuracy of the new transmittance model when applied to atmospheric calculations, fluxes and heating rates due to water vapor line effects were calculated for the five standard McClatchey et al. (1972) atmospheres: MLS, midlatitude winter (MLW), tropical (TRP), subarctic summer (SAS), and subarctic winter (SAW). The thinner lines in Fig. 3 show the flux dif-

12 1492 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 ferences between the band models and LBLRTM (band model LBLRTM) due to water-vapor-line-only effects for MLS conditions. The long-dashed lines represent the EG model calculations using the parameters obtained from the asymptotic limits, while the dotted lines represent the results from the new model. Including water vapor lines only, the EG model underestimates the upward flux by about 1.25 W m 2 at the top of the atmosphere, and the new model overestimates it by approximately 0.25 W m 2. For downward fluxes, the EG model underestimates the flux at the surface by 1.8 W m 2, while the new model overestimates it by 1.3 W m 2. For the MLW, TRP, SAS, and SAW soundings, the magnitude of the vertical distributions of flux differences between the band models and LBLRTM are similar. In the EG model, the downward fluxes near the surface tend to be less than those from the LBLRTM for the tropical and summer cases and greater for the winter cases, ranging from 3 Wm 2 for TRP to 2 W m 2 for SAW. The downward flux differences for the new model varies from about 1.5 W m 2 for the tropical and summer cases to about 0.5 W m 2 for the winter cases at pressures greater than 800 mb. The vertical distributions of spectral downward flux differences between band models and LBLRTM are shown in the left panels of Fig. 4 for the MLS sounding. The middle panel represents the results for the EG model and the lower panel represents the new model. The new LBLRTM flux ranges from 3 to10mwm 2 cm, and the largest errors occur near the surface in the cm 1 spectral region. The largest errors in the upper troposphere occur in the cm 1 region and amount to about 3 mwm 2 cm. For the EG model, the differences range from below 20 to above 30 m Wm 2 cm. The largest uncertainties near the surface occur in the cm 1 spectral region, while those in the upper troposphere are from 350 to 500 cm 1.It is clear that the problem area for both models is in the transition region ( cm 1 ) between the strong and weak absorbing regions. Note the large cancellation of errors in the EG model for the downward spectral fluxes. Similar errors were found in the upward spectral fluxes (not shown). That is, the correct spectrally integrated fluxes were obtained in the EG model because of cancellations of large errors of opposite signs. One cannot rely on such large cancellations for all conditions. The heating rates due to water-vapor-line-only absorption were calculated with the EG and the new models for the five standard atmospheres, and compared with the LBLRTM results. The heating rate profiles are not shown here for the sake of brevity, although the improvement obtained from the new model will be demonstrated in the next section for the MLS case. Details concerning each profile may be found in the dissertation by Warner (1997). Briefly, the new model heating rate results for MLS, TRP, and SAS show improved agreement with LBLRTM for the entire vertical range. The largest improvements are located in the lower troposphere and at the tropopause. The results for the MLW and SAW cases are different from the summer cases. The accuracy of the heating rates calculated by the new model does not vary much between seasons, whereas the differences between LBLRTM and the EG model are higher for the tropical and summer cases and lower for the winter cases. Thus, the model improvements are more significant for the tropical and summer cases than the winter cases. The accuracy of the EG spectrally integrated heating rates is partially dependent on a cancellation of the errors between spectral intervals, as discussed in the downward flux cases. This is discussed further in the next section. 4. Continuum absorption The continuum absorption in the new band model is developed following that defined by Clough et al. (1989) and used in LBLRTM (Continuum version 2.2). As discussed in section 1, this continuum absorption is defined consistently with the water vapor line absorption based on the assumption that the continuum is due to the absorption from far wings of the absorbing lines. Note that in LBLRTM, the water vapor lines are cut off at 25 cm 1 from the line centers, and the absorption outside that range is calculated as continuum absorption. The magnitude of the absorption is determined by fitting the coefficients of the continuum expression to laboratory and atmospheric observations. Since the absorption by water vapor lines in the new model is determined from that of LBLRTM, which is consistent with the continuum absorption, the continuum absorption coefficients at standard density (Clough 0 et al. 1989), namely, the self-broadened coefficient C s 0 and the foreign-broadened coefficient C f, can be retrieved directly from LBLRTM. The pressure variation is linear based on the definition of the coefficients; that 0 0 is, Cs is linear in the water vapor pressure e and Cf is linear in the pressure of the other atmospheric gases (p e). The temperature variation of the self- and foreignbroadened coefficients includes the contribution from both the temperature variation of the continuum and the temperature variation of the Clough et al. (1989) radiation term (i.e., the effect of stimulated emission). The temperature variation of the continuum coefficients used in the narrowband model is fitted from the temperature corrected coefficients of LBLRTM. For the self-broadened coefficient, we adopt the formulation C 0 s(t) C 0 s (T 0 ) exp[ a s (T T 0 )], (4.1) where a s is determined from LBLRTM data and T 0 was chosen at 296 K. Although the foreign-broadened continuum is assumed not to vary with temperature, the

13 15 MAY 2000 WARNER AND ELLINGSON 1493 temperature variation of the radiation field was assumed to be given as C f(t) C f(t 0)[a b(t T 0) c(t T 0)], (4.2) where a, b, and c are coefficients fitted to LBLRTM temperature corrected coefficients. In addition, since the coefficients C 0 s(t) and C 0 f(t) are scaled by the density, the following form is incorporated into the band model continuum absorption coefficient: k ci C si 0 (T)(T 0/T)(e/e 0) 0 C fi(t)(t 0/T)(p e)/(p0 e 0 ), (4.3) where the subscript i refers to the ith interval in the band model, T K, and p mb. Fluxes and heating rates due to absorption by both lines and continuum were calculated for the five McClatchey standard atmospheres. The details for each profile may be found in Warner (1997). As an example, the thicker lines in Fig. 3 show the differences of the upward, downward, and net fluxes between the band models and LBLRTM (band model LBLRTM) for the MLS case. The EG results are long-dashed, and the new results are dotted lines. The differences of the upward flux at the top of the atmosphere were reduced from 5.3 to 0.8 W m 2, and for the downward flux at the surface, the absolute differences are reduced from 11.4 to 1.1 Wm 2. The spectral distribution of the downward and upward flux differences between the band models and LBLRTM due to both water vapor line and continuum were calculated but are not shown. There is a large improvement in the calculation of the downward flux for water vapor line plus continuum absorption in the troposphere. The spectral differences between the new model and LBLRTM are within 5 mwm 2 cm, whereas the EG model differences ranged from about 45 to 10 m W m 2 cm. The errors are mainly located between 350 and 1000 cm 1 for the EG model, and as discussed for lineonly cases, there is considerable cancellation of errors between spectral intervals when the radiances are integrated spectrally. The errors in the new model are mainly located in cm 1 interval, and the magnitudes are small within 5 mwm 2 cm. The spectral distribution of the upward flux differences has characteristics similar to that of the downward fluxes. The range of the model LBLRTM differences is about 10 m W m 2 cm for the EG model, and about 2 mwm 2 cm for the new model. Figure 9 shows the line plus continuum heating rate differences between the band models and LBLRTM for the standard atmospheres. The left portion of the graph for each atmosphere shows the LBLRTM heating rates, and the right portion shows the EG LBLRTM (long dashed) and the new LBLRTM (short dashed) differences. In the lower troposphere the EG model heating rate tends to underestimate cooling by an average of 0.05 K day 1, while the new model tends to overestimate the cooling by an average of 0.03 K day 1. Near the tropopause, there is a large difference of about 0.3 K day 1 between the EG model and LBLRTM. This is primarily due to the neglect of the foreign continuum in the cm 1 region in the EG model. The right-hand panels of Fig. 4 show the spectral distribution of the LBLRTM line plus continuum heating rates (upper panel) and the differences between the band models and LBLRTM (middle and lower panel) for the MLS case. The new model heating rates agree with LBLRTM results across the spectrum to within 0.25 m K day 1 cm. The EG model calculates heating rates at similar accuracy for the majority of the spectrum, with the exception of the two (blue and red) bands diagonally located between 150 and 900 cm 1 that extend from the surface to the tropopause (i.e., in the spectral region of maximum cooling). The magnitudes of these bands are as large as 2 3 m K day 1 cm with opposite sign. The new model (see lowest panel) gives better spectral and spectrally integrated heating rates. 5. Summary and conclusions This paper summarizes our research to develop a fast and more accurate narrowband model for atmospheric and climate studies. The research has been motivated by our desire to understand the causes for discrepancies between different models as shown by ICRCCM, and to reduce the errors associated with calculations of fluxes and heating rates. Understanding the sources of errors in a radiation band model requires the use of a detailed line-by-line model or a calibrated band model. Briefly, we have developed a spectral band model that gives fluxes that agree with those from an accurate stateof-the-art line-by-line model to about 1 W m 2 for several test cases. The band model has two distinct characteristics that separate it from other models. First, the band model transmittance formulation has been fitted to line-by-line model line-only calculations over the full range of variation expected in our atmosphere. Second, the model employs the state-of-the-art water vapor continuum formulation that is consistent with the water vapor line formulation. We began the modeling of water vapor line absorption by comparing the Malkmus transmittances based on model coefficients from HITRAN and LBLRTM. We concluded that Malkmus model transmittances fitted to either HITRAN or directly to LBLRTM do not provide highly accurate spectral transmittances over the range of atmospheric concentrations. Although the spectrally integrated errors are generally small, large spectral errors prohibit the applications of such models to a wide range of problems, such as remote sensing and climate problems where spectral information is important. We then developed and fitted a new transmittance parameterization that reproduced LBLRTM results in 10 cm 1 intervals with rms errors less than 0.01 for more than 97% of the intervals from 0 to 3000 cm 1.