An ef cient and accurate correlated-k parameterization of infrared radiative transfer for troposphere±stratosphere±mesosphere GCMs

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1 Atmospheric Science Letters (200) Volume doi:0.006/asle An ef cient and accurate correlated-k parameterization of infrared radiative transfer for troposphere±stratosphere±mesosphere GCMs Wenyi Zhong* and Joanna D. Haigh Space and Atmospheric Physics Group, Department of Physics, Imperial College, London SW7 2BZ, U.K. Abstract: An ef cient and accurate correlated-k parameterization of infrared radiative transfer for the troposphere, stratosphere and mesosphere (TSM) has been developed. A new approach of g-quadrature, using the delta log(k) method with nonuniform intervals, was found to be most ef cient. Validation against line-by-line calculations for D and 2D elds are presented. The scheme is ef cient enough to be run in TSMGCMs. *c 200 Royal Meteorological Society Keywords: Radiative heating rates; atmospheric radiation; K-distribution; climate model.. INTRODUCTION The ef cient computation of accurate radiative uxes and diabatic heating rates is important for general circulation models of the lower atmosphere but essential in the middle atmosphere where radiative transfer provides the main sources of diabatic heating. Recently, several general circulation models have been extended to include the middle atmosphere, not only in order to study the stratosphere and mesosphere, but also to improve climate simulations (e.g. Hamilton et al., 995; Swinbank et al., 998). For such models it is desirable that a single radiation scheme is used over the entire vertical extent of the model. The most accurate and ef cient techniques for broadband parameterizations include the correlated-k approximation and the look-up table method. In some circumstances the look-up table (LUT) method may be more ef cient, e.g. for the water vapour strong bands, many correlated-k terms are needed to obtain an accuracy similar to LUT (Chou et al., 994). However, we will show that this dif culty may be overcome. Moreover, the correlated-k method has advantages in terms of the facility with which scattering may be incorporated. In general, for modelling both the lower and middle atmospheres, more layers need to be incorporated than for the troposphere alone and for this the correlated-k method is more ef cient as its CPU time requirements show a linear correlation to the number of layers compared with a quadratic relationship for broad-band models. * Corresponding author. w.zhong@ic.ac.uk X *c 200 Royal Meteorological Society

2 In recent years, several new radiation schemes have been developed for use in the troposphere and lower stratosphere of NWP and climate models (e.g. Edwards and Slingo, 996; Mlawer et al., 997). These schemes are computationally ef cient and have been validated against line-by-line models and observations. Most include weak absorption bands and minor gases which are signi cant in the lower atmosphere but do not contribute signi cantly to infrared cooling in the middle atmosphere. These parameterizations may be adopted for use in the TSM GCMs but the parameterizations for the strong bands need to be reconsidered. Here we develop an improved radiation scheme based on the k-distribution version (Cusack et al., 999) of the ES scheme, and discuss a method for extending its applicability to the middle atmosphere. 2. THE CORRELATED-k METHOD The k-distribution method is a well-established technique in which the band mean transmission function for a homogeneous atmosphere is estimated by means of transformation of the spectral integration in frequency space to an integration in cumulative probability g(k) space. In practice this can be done by different methods, among which the Exponential Sum Fitting Technique (ESFT) and rearranging absorption coef cients are the most commonly used. In the parameterization of the Hadley Centre Climate Model, Cusack et al. (999) developed a hybrid method in order to retain as few as possible terms to represent the k-distribution in each spectral band. First the absorption coef cients were found by constraining the transmission function to be conserved in each cumulative probability interval, this was done for all 20 mb layers of the standard MLS atmosphere, then a numerical procedure was used to minimize the total errors in uxes and heating rates to nd the adjusted weights. The method of Cusack et al. (999) enables the scheme to use 4±7 terms to represent k-distributions for a strong band and ±3 terms for a weak band or a minor gas. The accuracy of the parameterization is reasonably good and suitable for use in GCMs. The pressure and temperature dependences of the absorption coef cients in the method of Cusack et al. (999) are represented by scaling functions tted to the values calculated at various pressures and temperatures. The reference pressure and temperature were chosen by Cusack et al. (999) to be near the location of the heating rate peak for a mid-latitude summer (MLS) atmosphere in order to reduce the error caused by the scaling approximation. The correlated-k method generalizes the concept of the scaling function approximation. The mapping transformation from frequency to g space is not the same for different layers (West et al., 990) but the k coef cients and weights for each layer can be obtained at the same values of g used to approximate the k-distribution for the reference layer. In the correlated-k method most of the computer time is consumed in performing the `monochromatic' radiative calculation and in the treatment of overlapping gases. The latter was discussed in detail by Edwards and Slingo (996). In this work we mainly discuss the method of generating k coef cients that will provide fast and accurate radiative transfer calculations in the middle atmosphere. There is no unique method for identifying the optimal number of correlated-k terms. The balance of accuracy and speed is particularly important for a method to be used in both the lower and middle atmospheres. At low pressures Doppler broadening dominates, absorption coef cients are spread over a wider range of orders of magnitude and vary more sharply with wavenumber than at higher pressures. Therefore, a correlated-k parameterization derived for the high pressure regime may not be appropriate for modelling the middle atmosphere. For example, Figure shows by the solid line the portion from g ˆ to of the k coef cients for the CO 2 5 mm band centre (590±750 cm ) of the k-distribution (at 240 K and 250 mb) used by Cusack et al. (999) (Edwards, personal communication, 996). Also shown are c-k parameterizations for pressures of 096, 244, 54, 2, 2.7, 0.6 and mb (all at 240 K) at higher resolution in g

3 0 6 CO 2 5 mm bond centre: cm k(m 2 /kg) Figure. Cumulative k-distributions for g 4.0 for the CO 2 5 mm band centre (590±750 cm ), ÐeÐ is taken from Cusack et al. (999), the others are from this work at pressures (in mb) of 096 (-- --), 244 (-- --), 54 (--n--), 2 (± ± * d ± ±), 2.7 (± ± ±±), 0.6 (± ± * ± ±) and (± ± n ±±). space. It can be seen that for high values of g the low pressure coef cients are larger by about two orders of magnitude than those for high pressures, and these high coef cients are concentrated in a very small range of g near g ˆ ( g 4.0). However, it is these k values that play the dominant part in determining middle atmosphere heating rates. The CEC k-distribution shown in Figure would produce large errors if used to calculate middle atmosphere heating rates because the information carried by the large k coef cients is missing. g 3. NEW APPROACH TO GENERATE CORRELATED-k PARAMETERS 3. Strategy The key features of our strategy for the development of the correlated-k method to apply in both the lower and middle atmospheres are outlined as follows. All k absorption coef cients in the correlated-k approximation are extracted directly from line-by-line model calculated absorption coef cients and this is done not only for the reference pressure and temperature but also for the full range of pressures and

4 temperatures observed in the atmosphere. For the strong bands of the main absorbers we replace the scaling method by linear interpolation between tabulated k values. This method is ef cient and suitable for an inhomogeneous atmosphere over a wide range of pressure and temperature conditions. For weak bands and minor bands the Cusack et al. (999) parameterization is retained since only a few (less than about 5) k coef cients are needed to provide accurate determination of cooling rates, the errors in the middle atmosphere are negligible since their cooling itself in this region is very small. The second new feature is use of a non-uniform k-distribution. In order for the radiation scheme to compute cooling rates in the upper stratosphere and mesosphere accurately, high resolution in the g-quadrature near g ˆ is needed. Different g-quadratures are compared and investigation made for each speci c band to determine as few as possible correlated-k terms which allow accurate calculation of middle atmosphere cooling rates while preserving computational speed. 3.2 Line-by-line calculations of absorption coef cients The infrared spectrum (0±3000 cm ) is divided into nine spectral intervals, see Table. The GENLN2 line-by-line model (Edwards, 992) is used to calculate the monochromatic absorption coef cients for the H 2 O strong bands (intervals, 2 and 8), CO 2 5 mm band (intervals 3 and 4) and O mm band (interval 6). Twenty-six pressures and ten temperatures (30 K to 30 K at intervals of 20 K) are used for the calculations. The pressure levels are from ln(p) ˆ 5.5 to ln(p) ˆ 7(p in mb) with D ln(p) ˆ 0.5 (natural logarithm). The absorption coef cients are calculated at a wave number resolution of cm for the CO 2 5 mm and O mm bands, and 0.0 cm for the H 2 O strong bands. Line parameters are taken from HITRAN 996 database and the Voigt line pro le is used. Line wings cutoff is 0 cm for CO 2 and O 3, and 25 cm for H 2 O. The calculated absorption coef cients are then sorted in ascending order by their magnitudes and a g-quadrature is used to nd the weights and k coef cients. Table. Longwave Spectral Intervals and Absorption Bands No. Spectral Interval (cm ) Absorbers Included Number of terms 0±350 H 2 O (L), H 2 O (C) ±550 H 2 O (L), H 2 O (C) ± ±800 H 2 O (L), H 2 O (C), CO 2 (5 mm), N 2 O(7mm) ±750 H 2 O (C), H 2 O (L), CO 2 (5 mm), O 3 (4 mm) ±990 20±200 H 2 O (C), H 2 O (L), N 2 O, CFC, CFC ±20 H 2 O (C), O 3 (9.6 mm), H 2 O (L), CFC, CFC 2, CO 2 (0 mm) ±500 H 2 O (L), H 2 O (C), CH 4,N 2 O ±900 H 2 O (L) 9 900±3000 H 2 O (L) 2

5 3.3 Adjusted D log(k) method to optimize the g-quadrature The most frequently used methods are Gaussian quadrature (or modi ed Gaussian quadrature) and the D log(k) method (Goody et al., 989). Both methods place more intervals near the high end, g ˆ. For example in an early version of their RRTM, Mlawer et al. (997) used a 6-point modi ed half-gaussian quadrature. In this work we have adapted the D log(k) (base-0 logarithm) method to use non-uniform intervals. This is based on the observation that the rate of increase of k with g near, as well as the maximum value of k, varies between bands and between gases. The economy of this approach proves to be valuable. We choose the top layer (in this work ln(p) ˆ 5.5, i.e. p mb) as the reference in order to give largest resolution near g ˆ. Comparison of the ranges of k for the water vapour strong bands and wings, carbon dioxide 5 mm band centre and wings, and ozone 9.6 mm band shows that at high pressures the range of k is from 3 to 7 orders of magnitude (depending on the spectral band) but at low pressures this extends to from 0 to 4. These sizes suggest the approximate number of terms required to integrate k over g in the lower and the middle atmospheres, respectively. 0-3 (a) H 2 O cm - : MLS (left) and SAW (right) 0-3 (b) H 2 O cm - : MLS (left) and SAW (right) Heating Rate (K/day) Heating Rate (K/day) Figure 2. The heating rates due to H 2 O: (a) in 350±550 cm for MLS and SAW atmospheres calculated using the GENLN2 (solid line), 2 term correlated-k parameterization of D log(k) ˆ (dash-dotted line, MLS only) and 4 term correlated-k with adjusted D log(k) method (dotted line); (b) in 0±350 cm for MLS and SAW atmospheres calculated using the GENLN2 line-by-line model (solid line), correlated-k from 25 term modi ed half-gaussian quadrature (dash-dotted line, MLS only) and correlated-k parameterization of this work (dotted line).

6 In order to optimize the number of c-k terms used for each individual broadband we rst use D log(k) ˆ to nd the weights for the correlated-k method. With these parameters we found that D log(k) ˆ is suf cient to calculate accurate cooling rates for most cases. However, unacceptable errors are found at some altitudes for the water vapour bands. For example, as shown in Figure 2a in the water vapour band wing 350±500 cm the difference in cooling rate for the MLS atmosphere between the test and line-by-line results at 400± 500 mb reaches 0.2 K/day. We also found that for the water vapour strong bands these c-k coef cients produce larger errors in the middle atmosphere. It is evident that for given atmospheric conditions the contribution to the cooling rate at a certain altitude is dominated by a corresponding range of k-values (Mlawer et al., 997). Our investigation shows that doubling the g resolution (halving D log(k)) in the corresponding subinterval can reduce the cooling rate error at the levels concerned. In Figure 2a the improvement in cooling rate at 450 mb is obtained by using D log(k) ˆ 0.5 for the rst two subintervals. The error is reduced to less than 0.05 K/day around 450 mb. Similar improvement is also seen for other atmospheres, as shown for the sub-arctic winter (SAW) case in Figure 2. This method has also been successfully used for the water vapour broad-bands 0±350 and 500±900 cm. In order to reduce water vapour cooling rate errors in the middle atmosphere (in the strong bands and wings) the c-k values of the terms near the high (g ˆ ) end were adjusted. At the resolution D log(k) ˆ the weights near the high end of g are around 0 4 and do not contain many absorption coef cients ( for example, the rst g interval contains 900 k coef cients but the last g interval only about 0 and the slope to g is much more sharp). Thus, doubling the D log(k) resolution will not necessarily improve the cooling rates signi cantly. Instead of increasing the g resolution we keep D log(k) ˆ and adjust the c-k values for the g intervals near the high end. In this way the cooling rate in the middle atmosphere can be satisfactorily improved. The improvement in cooling rate in the mesosphere shown in Figure 2a is due to the adjustment of c-k values in the last ve subintervals. The adjustment was made by amplifying the c-k values by multiplying factors of.2 to 5. This adjustment has negligible effect on the uxes since the corresponding weights are very small fractions. We also found that D log(k) ˆ was not always necessary for the lower atmosphere, therefore D log(k) ˆ 2 or 3 is used for the rst couple of subintervals in some bands. Our investigation for the three spectral bands of water vapour gives the following optimized numbers of terms: 3 for 0±350 cm, 4 for 350±550 cm and 2 for 500±900 cm. Figure 2b shows the heating rate pro les for the MLS in the 0±350 cm band calculated by the GENLN2 lbl model, the correlated-k with 25 terms obtained by using a modi ed half-gaussian quadrature and our new method. It can be seen that our adjusted D log(k) method is slightly more accurate than that using Gaussian quadrature as well as being computationally about twice the speed. For the CO 2 and O 3 main absorption bands we nd the D log(k) method suf ciently accurate with around 0 terms for each of the CO 2 5 mm band centre, band wings and O mm band. There is no adjustment made for the k values in these bands. 4. VALIDATION In order to validate the schemes extensive off-line tests have been conducted. Here we present results for a couple of ICRCCM atmospheric pro les (Ellingson et al., 99) and also for a latitude-height section. In all cases our results are compared with line-by-line benchmark calculations obtained using GENLN2. For the pro le calculations 22 levels are used between the surface and mb. The results of water vapour heating rates for MLS and SAW pro les have already been shown for intervals and 2 in Figure 2b and 2a, respectively. The maximum difference in the layers below 0.0 mb is around 0. K/day in interval and less than 0.05 K/day in interval 2. The maximum difference for the water vapour spectral interval 500±900 cm is less than 0.02 K/day.

7 Table 2. Upward uxes at the top of atmosphere and the downward uxes at the surface for H 2 O strong bands (Wm 2 ) (0±350 cm ) (350±550 cm ) (500±900 cm ) F Q (top) F q (srf) F Q (top) F q (srf) F Q (top) F q (srf) MLS LBL c-k SAW LBL c-k GENLN2 results and that of the CEC method. Figure 4b show the differences in heating rates between the LBL model and the two schemes. We can see that below 40 km (3 mb) the three models are in excellent agreement with each other with the maximum errors of about 0. K/day. Above 40 km the differences become larger but the new correlated-k parameterization is able to produce realistic heating rate pro les throughout the stratosphere and mesosphere with maximum error below 85 km of about 0.7 K/day, of similar magnitude to the RRTM of Mlawer et al. (997). Results for other ICRCCM cases are similar and are not shown here. 0-3 CO cm - : MLS (left) and SAW (right) 0-3 (b) O mm: MLS (left) and SAW (right) (a) Heating Rate (K/day) Heating Rate (K/day) Figure 3. (a) The heating rates due to CO 2 in 5 mm band centre (590±750 cm ) for MLS and SAW atmospheres calculated by GENLN2 (solid line) and correlated-k parameterization of this work (dotted line); (b) The heating rates due to O 3 in 9.6 mm band (990±20 cm ) for MLS and SAW atmospheres calculated by GENLN2 (solid line) and correlated-k parameterization of this work (dash-dotted line).

8 Table 3. Upward uxes at the top of atmosphere and the downward uxes at the surface for CO 2 5 mm and O mm bands (in W/m 2 ) (590±750 cm ) (550± ±800 cm ) (990±20 cm ) F Q (top) F q (srf) F Q (top) F q (srf) F Q (top) F q (srf) MLS LBL c-k SAW LBL c-k Table 2 shows the upward uxes at the top of atmosphere and the downward uxes at the surface for these three water vapour intervals, the errors caused by using the correlated-k method are small with a maximum of about 2.7%. Figure 3a shows the heating rates in the CO 2 5 mm band centre for the GENLN2 calculations, and our model results for MLS and SAW pro les (a volume mixing ratio of 356 ppm is used). Below 0.0 mb the maximum difference is around 0.3 K/day at the stratopause (MLS) 0-3 (a) H 2 O CO 2 O cm - : MLS (left) and SAW (right) 0-3 (b) H 2 O CO 2 O cm - : MLS (left) and SAW (right) Heating Rate (K/day) Difference in Heating Rate (K/day) Figure 4. (a) The heating rate (K/day) over the total infrared spectrum (0±3000 cm ) due to H 2 O, CO 2 and O 3 in MLS and SAW atmospheres calculated by GENLN2 (solid line), correlated-k of this work (dotted line) and the CEC k-distribution (dash-dashed line); (b) the same as (a) but for differences in heating rate with the line-by-line model.

9 85 (a) 85 (b) Approximate Height (km) 5 34 Approximate Height (km) Latitude (degrees) Latitude (degrees) Figure 5. (a) GENLN2 computed longwave heating rates for GRIPS input data, the contour interval is K/day above 5 K/day; (b) Longwave heating rate difference between this work and GENLN2 results, the contours are at 2.0,.0, , 0., 0, 0., 0.2, 0.5,.0, 2.0, 3.0, 5.0, 0.0, 5.0, 20.0 (K/day). and less than K/day between 0.03 mb and 0. mb (SAW). The maximum difference for the CO 2 5 mm band wings is less than 0.0 K/day in the upper stratosphere (MLS) and in the mesosphere (SAW). Figure 3b shows the heating rates for the O mm band. The maximum difference is less than 0.02 K/day (MLS) and 0.0 K/day (SAW) occurring between to 5 mb. The upward uxes at the top of atmosphere and the downward uxes at the surface for the CO 2 5 mm and O mm bands are given in Table 3, the maximum error is approximately 0.4 Wm 2. Figure 4a presents the heating rates for the total infrared spectrum (0±3000 cm ) due to three main absorber gases (H 2 O, CO 2 and O 3 ) for MLS and SAW atmospheres calculated using this parameterization. Also shown for comparison are the 2D comparison. The GRIPS-Radiation-Intercomparison input data for January (Langematz, 999) are used for a benchmark line-by-line calculation and for validation of the new radiation scheme. The dataset includes CIRA temperature, the ozone climatology of the Free University of Berlin

10 (version 02), HALOE water vapour above 2 km and NCEP/NCAR global reanalyses in the troposphere. The resolution is km in height and 58 in latitude. GENLN2 calculations are performed for 37 zonal mean atmospheric pro les de ned by using the standard input dataset and a CO 2 mixing ratio of 356 ppm. The longwave heating rates of GENLN2 and the difference between the correlated-k parameterization and lineby-line results are shown in Figure 5a and 5b, respectively. The long-wave heating rates from both models are in very good agreement. In the lower stratosphere from 8 km to 30 km the maximum error is around 0. K/day and above 30 km to 80 km maximum error about 0.5 K/day. In the troposphere the maximum cooling error is around 0. K/day. The new scheme increases CPU time requirements by less than 0% relative to the unmodi ed long wave scheme on the same number of vertical levels. 5. CONCLUSION This work provides a fast method for long-wave radiation calculation from the surface to 85 km. It has been implemented in the radiation scheme of Edwards and Slingo (996), making this scheme applicable to TSM GCMs. The D log(k) method was adapted and used for each individual strong band to nd the minimum number of exponential terms required in the correlated-k parameterizations to give the heating rate accuracy (0±3000 cm ) of within 0. K/day in the lower atmosphere and 0.8 K/day in the middle atmosphere, and the ux accuracy of ±2 W/m 2 compared with line-by-line model results. The implementation of a single radiation scheme throughout the domain of a TSM model better allows the GCM to be used for studies of the interaction between the middle and lower atmospheres. We are currently investigating the impact of the improved radiative calculations in the middle atmosphere of a TSM model by incorporating the scheme into the extended version of the UKMO uni ed model which has a top level at mb. The correlated-k parameterization is also suitable for use in other TSM models, the weights and tabulated k coef cients can be obtained through ftp from the authors. Acknowledgements This work was supported by the U.K. Natural Environment Research Council's UGAMP programme. We thank Dr J. M. Edwards for his helpful advice. REFERENCES Chou, M.-D., Ridgway, W. L. and Yan, M. M.-H., 994. Parameterizations for water vapor IR radiative transfer in both the middle and lower atmospheres. J. Atmos. Sci., 5, 59±67. Cusack, S., Edwards, J. M. and Crowther, J. M., 999. Investigating k distribution methods for parameterizing gaseous absorption in the Hadley Centre Climate Model. J. Geophys. Res., 04, 205±2057. Edwards, D. P., 992. GENLN2: A general line-by-line atmospheric transmittance and radiance model. NCAR Technical Note TN-367.

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