Two fast radiative transfer methods to improve the temporal sampling of clouds in numerical weather prediction and climate models

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Published online February 9 in Wiley InterScience ( Two fast radiative transfer methods to improve the temporal sampling of clouds in numerical weather prediction and climate models J. Manners,* J.-C. Thelen, J. Petch, P. Hill and J.M. Edwards Met Office, Exeter, UK ABSTRACT: The high computational cost of calculating the radiative heating rates in numerical weather prediction (NWP) and climate models requires that calculations are made infrequently, leading to poor sampling of the fast-changing cloud field and a poor representation of the feedback that would occur. This paper presents two related schemes for improving the temporal sampling of the cloud field. Firstly, the split time-stepping scheme takes advantage of the independent nature of the monochromatic calculations of the correlated-k method to split the calculation into gaseous absorption terms that are highly dependent on changes in cloud (the optically thin terms) and those that are not (optically thick). The small number of optically thin terms can then be calculated more often to capture changes in the grey absorption and scattering associated with cloud droplets and ice crystals. Secondly, the incremental time-stepping scheme uses a simple radiative transfer calculation using only one or two monochromatic calculations representing the optically thin part of the atmospheric spectrum. These are found to be sufficient to represent the heating rate increments caused by changes in the cloud field, which can then be added to the last full calculation of the radiation code. We test these schemes in an operational forecast model configuration and find a significant improvement is achieved, for a small computational cost, over the current scheme employed at the Met Office. The incremental time-stepping scheme is recommended for operational use, along with a new scheme to correct the surface fluxes for the change in solar zenith angle between radiation calculations. the permission of the Controller of HMSO. Published by KEY WORDS radiation; correlated-k; temporal resolution Received September 8; Revised December 8; Accepted January 9. Introduction In both climate modelling and numerical weather prediction (NWP), the calculation of radiative transfer is necessarily a trade-off between accuracy and computational efficiency. Very accurate methods exist, such as line-by-line procedures, that could ideally be employed to calculate fluxes for every grid point on every time step. However, this is impractical because of the high computational cost, and therefore a number of simplifications are made to reduce this cost to manageable levels. Firstly, the correlated-k method (Lacis and Oinas, 99) is typically employed to reduce the integration over wavelength by effectively binning wavelengths with similar absorption coefficients (k-terms). This greatly reduces the number of monochromatic radiative transfer calculations required. The number of k-terms can be adjusted to give the desired trade-off between accuracy and efficiency that is required for a given application. For example, the Met Office currently uses a scheme with a total of k-terms for climate and global forecasting applications, and a Correspondence to: James Manners, Met Office, FitzRoy Road, Exeter, EX PB, UK. james.manners@metoffice.gov.uk scheme with k-terms for short-term local area modelling (Edwards and Slingo, 99; Cusack et al., 999). While further refinements are possible (e.g. Pawlak et al., ), these correlated-k methods cannot be made sufficiently cheap to allow calculations for every grid point on every time step. To reduce the cost further, calculations are usually made at lower temporal and spatial resolutions (e.g. Morcrette et al., 8). Quite drastic reductions in temporal resolution are often made (e.g. radiation calculations are made every three hours for the climate and global forecast models at the Met Office). Between radiative transfer calculations there will be major changes in the radiative profiles that are unrepresented. These are caused primarily by two factors: changes in the cloud fields (condensate and vapour) and changes in the angle of incident solar radiation. It is important to represent the change in solar zenith angle for an accurate treatment of the diurnal cycle as well as the total energy input to the atmosphere. For this reason, corrections are often made to the shortwave fluxes by a simple scaling using the cosine of the solar zenith angle (e.g. Morcrette ). In all other respects the radiative profiles are generally kept constant between radiation calculations. This can lead to poor representations of cloud feedback and may cause biases the permission of the Controller of HMSO. Published by

2 8 J. MANNERS ET AL. in the temperature and moisture profiles as seen by Morcrette () for seasonal simulations. A number of techniques have been proposed to improve the temporal and spatial resolution of radiation calculations. Chevallier et al. (998) and Krasnopolsky et al. () have developed neural network schemes that can emulate radiative transfer calculations with much greater efficiency than correlated-k methods. These may then be used at much higher sampling resolutions. These schemes have been demonstrated as a practical possibility for simulation of both long-wave and short-wave fluxes (Krasnopolsky et al., 8), but still need to overcome some obstacles before general acceptance in operational models. They require training with the full range of atmospheric conditions they are likely to encounter, and will be unpredictable outside these conditions. The treatment of discontinuous cloud fields is difficult for the continuous functions simulated by neural networks (Chevallier, ). They are also not as flexible as current methods and require retraining when any change to the scheme is needed. A different approach is to use adaptive schemes that combine a full radiative calculation at low sampling resolutions with a simple parametrization that is used to extrapolate to higher resolutions and decide whether further full calculations are required. Venema et al. (7) present this approach with two schemes to exploit temporal and spatial correlations in atmospheric fields. Their temporal perturbation scheme uses a simple calculation (using multivariate regression) to estimate the change in net surface fluxes between model time steps. Grid points with the largest surface flux changes are selected for a full two-stream radiation calculation. This approach is limited by the accuracy of the calculation used to extrapolate the radiative fluxes. This work will significantly improve on the simple extrapolation calculations of Venema et al. (7), by focusing on the radiative effects of changes in cloud. We develop simplified radiative transfer parametrizations, which can be used at greater sampling resolutions, to represent the changes in both radiative heating rates and fluxes between full radiation calculations. We also consider a new, more accurate treatment of the solar zenith angle correction that can greatly reduce the error in surface fluxes associated with poor temporal sampling. In section we outline two related schemes for calculating the radiative effects of changes in cloud fields. Section details how these schemes have been implemented in the Met Office Unified Model s radiation code including, in section., the new solar zenith angle correction, which can also be used independently. Sections and show the results of the performance of the schemes against observational cloud data and in the context of a global forecast model, ending with a discussion in section.. Methods to improve the temporal sampling of clouds We consider two alternative schemes to sample the changing cloud field between standard radiation calculations... The split time-stepping scheme The k-distribution method employed by many radiative transfer schemes involves the reordering of the absorption spectrum of the major gases into a number of discrete bins represented by a single absorption coefficient, or k-term. Absorption and scattering due to cloud condensate is treated as a grey process within a given band, so that a constant value is added to each of the gaseous k-terms at this stage. Using the correlated-k assumption, independent monochromatic radiative transfer calculations for the whole depth of the atmosphere may be performed for each k-term. The resulting fluxes are then simply added up to arrive at the total flux over the band considered. At this point, two observations may be made. Firstly, for strong k-terms, representing spectral regions with low transmissivity, the extinction is predominantly due to a strong gaseous absorber (which may only become optically thin in the stratosphere) while extinction by clouds is of little importance. For weak k-terms, representing regions with high transmissivity, the major gas is only weakly absorbing and the total extinction is dominated by clouds. Secondly, the number of weak k-terms (where clouds are significant) is generally greatly outweighed by the number of strong k-terms (where only the slowly varying amounts of gaseous absorber are significant). It follows that the radiative fluxes will change most rapidly for a small number of weak k-terms. We can take advantage of this fact by splitting the calculation into two parts. The weak k-term fluxes may then be calculated with a high sampling frequency, while the strong k-term fluxes are calculated with a lower frequency. On each model time step the fluxes from the most recent weak and strong k-term calculations can then be summed to give the total flux. This scheme retains the full physical accuracy of the original k-terms. Its efficiency will depend on the k-term decomposition of the full scheme, and specifically, how many k-terms are used to cover the regions of high transmissivity... The incremental time-stepping scheme The concept used to split the k-term calculations in the split time-stepping scheme can be interpreted in a slightly different way. The absorption spectrum of the atmosphere contains regions that are largely opaque due to gaseous absorption, and window regions that are largely transparent. Changes in cloud amounts, particularly for low clouds, are reflected almost entirely in changes in the window region fluxes (with the caveat that for high clouds, the tenuous gaseous absorption will mean regions outside the window start to become important). In conditions where cloud amounts are significant (and therefore changes in cloud are significant), the absorption and scattering by cloud condensate will entirely dominate the extinction coefficient in the window regions. Therefore, for the purposes of calculating flux increments due to changes in cloud we can neglect the small variation in

3 FAST RT METHODS FOR TEMPORAL SAMPLING OF CLOUDS 9 gaseous absorption coefficients and simply use a single gaseous k-term for the entire window region. The incremental time-stepping scheme will therefore consist of a full radiative transfer calculation at low sampling frequencies, and a further simple increment calculation at higher sampling frequencies. The increment calculation will use only those monochromatic calculations necessary to describe the extinction properties of cloud condensate in the window regions. These vary much more slowly with wavelength than gaseous absorption and can be adequately described by one long-wave and two short-wave bands. These calculations will be performed first for the same time step as the full calculation. On subsequent time steps, where further simple calculations are made, the flux increments found for the simple calculation will be added to the fluxes calculated with the last full radiative transfer calculation. The advantage of this approach is that the efficiency of the increment calculation is independent of the full radiative transfer scheme. It requires the creation of an extra, simple parametrization, where the choice of band limits will determine the accuracy. The bands should be as wide as possible to represent the full flux in the window, but narrow enough that a single gaseous k-term will be sufficient.. Implementation within the Edwards Slingo radiation code The Met Office Unified Model currently uses the Edwards Slingo radiation scheme (Edwards and Slingo, 99). The spectral overlap of absorption by different gases is dealt with through the method of equivalent extinction (Edwards, 99), where full calculations are only done for the k-terms of the major gas in each band. The radiation scheme is particularly flexible in its use of external spectral files to hold information on the required spectral decomposition (into bands and k-terms), and the optical properties of gases, cloud condensate and aerosols. For the purposes of the schemes detailed here, the required changes to the radiative transfer calculations can be performed simply by altering the spectral files. Only alterations necessary for the new sampling frequencies remain to be made to the code of the scheme itself... Split time-stepping scheme implementation The split time-stepping scheme requires the k-terms used for a full calculation to be split into two parts, which for the Edwards Slingo code will require two separate spectral files. The spectral file used for the slow time step will include all the strongly absorbing k-terms, and the fast spectral file the remaining weakly absorbing terms. The atmospheric absorption due to the k-terms can be determined from the optical depth, given by τ = ku, () where u is the integrated column amount of the absorber. The integrated column amounts, especially for water vapour, will vary, but for the purposes of a fairly arbitrary split between weak and strong absorption we assume a mean value. For water vapour, carbon dioxide and ozone, these can be given approximately as u H O = kg m, u CO = kgm and u O =.8 kg m respectively. If τ(k), the gaseous absorption due to that k-term is weak and it is stored in the fast spectral file. If τ(k), the k-term results in strong gaseous absorption and it is stored in the slow spectral file. This process was followed for the short-wave (SW) and long-wave (LW) spectral files used for the HadGEM climate model and the Met Office global forecast model. The SW file contains six spectral bands containing a total of k-terms for the major gases. The LW file contains nine spectral bands with a total of major gas k-terms. In the resulting split files there are four bands with four k-terms (one per band) in the fast SW file, and four bands with k-terms in the slow SW file. The fast LW file contains three bands with three k-terms, leaving the slow LW file with nine bands and k-terms. For the Edwards Slingo scheme, the k-term weights (i.e. the fraction of the band represented by each k-term) are required to add up to unity for a given spectral file. The weights for each file were therefore normalized, with a corresponding reduction in the source function coefficients (i.e. the incident solar and thermal radiation) in order to maintain the correct fluxes. On the first model time step the radiation scheme is called twice, once for the slow spectral file and once for the fast spectral file. The fluxes and heating rates are summed to give the correct values for incrementing the model. The outputs from the slow spectral file calculation are retained in memory, and on subsequent calls to the radiation scheme these are summed with fluxes from a new calculation of the fast spectral file. An example of the sequence of radiation time steps compared to full model time steps is shown in Figure. Here, full radiation calculations (slow plus fast) are performed every model time steps (equivalent to every hours for the Met Office global forecast model). On subsequent cloudonly time steps, only the fast calculation is needed. An issue arises with the treatment of the solar zenith angle. For the current radiation scheme, a mean value for the radiation time step is used (in conjunction with a correction made on every model time step; see section.). For the split time-stepping scheme, the slow calculation must still use this mean value, however the fast calculation is valid for only a fraction of the full radiation time step. We can therefore use a zenith angle that is a mean over the fast radiation time step for these Full Radiation time steps Cloud only Cloud only Full Model time steps Figure. An example sequence of radiation calls for the time-stepping schemes. The number of cloud-only radiation time steps is flexible up to the number of model time steps.

4 J. MANNERS ET AL. calculations, leading to a more accurate representation of the fluxes... Incremental time-stepping scheme implementation For the incremental time-stepping scheme, the calculation of cloud increments uses a method that is independent of the full radiation calculation. We require two simple spectral files to represent the regions of high transmissivity (the window regions) in the SW and LW. This is simplest for the LW, where a clear window region exists at approximately 8 µm. At shorter wavelengths than 8 µm, absorption by water vapour, N OandCH will entirely dominate. Beyond µm, CO and water vapour absorption dominates. For simplicity, we chose band limits that are used within the current HadGEM LW spectral file, corresponding to 7.. µm. A single LW band was used and a single water-vapour k-term calculated as a mean over the band. The optical properties for ice and liquid cloud condensate were recalculated for the band using the same parametrization as the HadGEM spectral file. (See Edwards and Slingo (99) for the liquid condensate parametrization based on Mie scattering. The ice crystal parametrization, based on ice aggregates, is described in Edwards et al. (7).) A treatment of the ozone absorption around 9. µm was found to be of little benefit in relation to the added cost and is not included. No other gases or aerosols are considered. For the SW region a one-band spectral file was found to be insufficient to characterize the change in both surface fluxes and heating rates associated with a change in cloud amount. SW heating is generally counteracted by LW cooling. We therefore require the SW heating rate errors to be of the same magnitude as the LW errors in order to represent the total heating rate accurately. Experiments with a one-band SW file showed that heating rate errors could only be made comparable to those in the LW when near-infrared band limits were chosen, leading to large errors in the surface fluxes. The window region in the SW extends from where ozone absorption falls off at around. µm until the vibrational bands of water vapour begin to dominate between and µm. The majority of the SW flux occurs in the visible part of the spectrum where the single scattering albedo of cloud condensate is effectively unity (see Figure ). Due to the lack of absorption in this region (by gases or condensate) there is practically no effect on the heating rates due to changes in cloud, although the effect on surface fluxes can be highly significant. Absorption by cloud condensate, and the corresponding effect on heating rates, begins to become important in the near-infrared region, although here the incident solar flux is much reduced and the contribution to surface fluxes is small. The distinct behaviour of these two regions requires a spectral file containing two bands. Again, for simplicity, we chose band limits that are used within the current HadGEM SW spectral file, corresponding to..9 and.9.8 µm. These are displayed in Figure along with the incident solar spectrum, from Kurucz (99), used in the parametrization. Norm. flux, single scattering albedo Band Band.... Wavelength (µm) Figure. The SW window region used for the incremental timestepping scheme s two-band SW spectral file. The solid line is the normalized solar spectrum of Kurucz (99). The single scattering albedo for ice crystals (aggregates of effective radius µm, dotted line) and cloud droplets (radius µm, dashed line) shows how absorption only becomes important in band. The calling sequence (see Figure ) for the incremental time steps is similar to that described for the split timestepping scheme. On the first model time step, the full radiation scheme is called and used to integrate the model. On the same time step, a further calculation is made using the simple spectral files. The output fluxes are subtracted from those gained by the full radiation scheme, and the result retained in memory. On subsequent cloud-only time steps, calculations are performed with the simple spectral files and the fluxes added to those retained in memory from the full radiation time step. The treatment of the solar zenith angle is slightly different for this case. For the first part of the radiation time step, the full radiation scheme fluxes are used with a zenith angle equal to the mean over the full radiation time step. Treating subsequent cloud-only time steps with a different zenith angle would then lead to abias,sothesamezenithanglemustbeusedforall the calculations. We can reduce the error due to the use of a mean zenith angle by applying a new correction to the top-of-atmosphere and surface fluxes, outlined below... A new method of correcting for changes in solar zenith angle In the current version of the Met Office Unified Model radiation scheme, radiative transfer calculations are performed with a solar zenith angle (θ) that is a mean over the entire radiation time step. For each full model time step a new zenith angle (θ ) is calculated. All short-wave fluxes and heating rates are then corrected using a factor of α: α = cos θ / cos θ. () This is correct for the incident direct solar flux at the top of the atmosphere, but takes no account of the change

5 FAST RT METHODS FOR TEMPORAL SAMPLING OF CLOUDS in optical path length (and therefore extra scattering and absorption), so that the correction is not completely accurate for surface fluxes. Here we will use a few simple assumptions to take the change in optical path length into account to give a better correction for surface fluxes. The first two are valid to first order, while the third is necessary for simplicity of calculation. () Ignore the extra absorption due to the change in path length by assuming that scattering will dominate. () Assume half the extra scattered radiation escapes to space and the other half reaches the surface. () Assume the extra surface reflected radiation is transmitted directly to space with no further scattering. These assumptions lead to the following formulation for a correction to surface and top-of-atmosphere net fluxes, whilst leaving the correction to heating rates unchanged. Firstly, consider the downwards component of the incident direct beam solar flux at the top of the atmosphere (Z TOA ), and the transmission of this flux to the surface (Z Surf ). We use a prime to represent values at the new zenith angle: Z TOA =αz TOA, Z Surf =TZ TOA, () Z Surf =T Z TOA. Now the transmission of the atmosphere (T )tothe direct beam will be a function of the optical depth along the slant path (τ): T = e τ. () With the change in zenith angle, the path length will have changed: so τ = τ/α, () log T = log T, α T =T /α, ( ) /α T ZSurf =, Z TOA ( ) /α Z Surf = ZSurf αz TOA. Z TOA The change in direct surface flux due to the change in optical path length alone ( Z Surf ) is therefore ( ) /α Z Surf = ZSurf αz TOA αz Surf. (7) Z TOA Our assumptions specify that we will ignore any absorption from this extra flux and that half of the () extra scattered radiation is scattered downwards to the surface. Therefore the change in total (direct plus diffuse) downwards surface flux (V )is V Surf = Z Surf [ ( = α ZSurf Z TOA ) /α Z TOA Z Surf]. (8) Then, assuming the surface albedo is unchanged, the net surface flux (N, downwards minus reflected flux) will be altered by the same fraction: N Surf = V Surf V Surf N Surf. (9) Finally, the new net flux is due to the change in incident flux (the original correction using a factor of α) plus our new correction due to the change in optical path length: N Surf = αn Surf + N Surf = αn Surf + α [( ZSurf Z TOA ) /α Z TOA Z Surf ] NSurf V Surf. () As no extra flux is absorbed, the opposite correction is made to the net upwards flux at the top of the atmosphere (which can be output as a diagnostic but will not affect the model evolution): N TOA = αn TOA N Surf. () These corrections can be made on every model time step independent of whether we are using the incremental or split time-stepping schemes or the current operational radiation scheme. The effectiveness of the correction to surface fluxes is displayed in Figure. This provides an example of the error at the beginning of a three-hour radiation time step. For the morning (western) sector, the sun will be lower in the sky than for the mean over the radiation time step. The mean calculation will therefore overestimate the transmission, leading to a positive error in surface flux. The opposite is true for the evening sector, leading to the dipole pattern displayed in Figure (a). Towards the end of the radiation time step the error will have the opposite sign, leading to a cancellation in the bias for the whole time step. The correction will be greatest towards the beginning and end of the radiation time step where the zenith angle differs markedly from the mean. There will be little benefit in regions where the atmosphere is optically thick due to the extinction of the direct beam used for the correction.. Results of offline tests using observational data As an initial test of the time-stepping schemes and a demonstration of their performance when applied to

6 J. MANNERS ET AL. (a) (b) Figure. Errors in the surface net SW radiation (W m ) due to the correction for solar zenith angle in a Met Office Unified Model forecast for December. Here radiative fluxes are only calculated every time steps ( h) using a mean zenith angle for the entire period (8 GMT). Panel (a) shows the error in the surface flux for time step (8 8 GMT) using the current correction for zenith angle. Panel (b) shows the error for the same time step using the new zenith angle correction described in section.. The control run used to calculate these errors used a radiation time step equal to the model time step. realistic cloud data, we have used combined radar lidar data for Chilbolton from the Cloudnet project (Illingworth et al., 7) to provide observed fields of ice and liquid water content. These can be provided as input profiles to the stand-alone version of the Edwards Slingo radiation code. The retrieval methods chosen were the Delft radar lidar method for the liquid water content and the radar/lidar IPSL method (Tinel et al., ) for the ice water content. All other variables required were extracted from the European Centre for Medium-Range Weather Forecasts (ECMWF) single-site output over Chilbolton for the same time period. Data for May were used, with time periods selected from the th ( 8 GMT), th (9, 7 GMT), 7th (8 GMT) and st ( 8 GMT) to provide a variety of cloud types including a significant ice content. The cloud data for these periods are available for visualization using the quicklooks facility at the Cloudnet website (Illingworth et al., 7). The data provided cloud condensate sampled approximately every seconds. These were meaned into one-hour periods, which could be taken to represent an approximate 8 km grid-box, assuming a wind speed of around m s ( knots). They were also meaned into layers equivalent to the ECMWF model grid. For each period, mean profiles of temperature, pressure and specific humidity, along with surface temperature and pressure, were extracted from the ECMWF model fields for the same time period. A midlatitude summer McClatchey profile (McClatchey et al., 97) was used for ozone, and all other gases were given a constant mass mixing ratio throughout the atmosphere. A solar zenith angle of was used for these tests. We first test the performance of the split time-stepping scheme s spectral files. For each hour of the data we run a radiative transfer calculation for the SW and LW using the full HadGEM spectral files, and further calculations using the fast and slow spectral files. The mean heating rates (over all time periods) produced for cloudy layers of the atmosphere are displayed in the top row of Figure. The sum of the fast and slow heating rates is equal to the full heating rates as expected. The variation of heating rate with height is more pronounced for the fast spectral file displaying the response to changes in cloud amount. To test the incremental time-stepping scheme we run a further calculation using the simple spectral files for each hour of the data. The mean heating rates produced are displayed in the middle row of Figure. The simple files provide a particularly accurate representation of the full heating rates around a height of km, where the cloud amount is most significant. The relative performance of the SW and LW spectral files are well balanced with similar errors so that the total heating rates (which cancel the LW cooling with the SW heating) are also very close to those observed with the full HadGEM spectral files. The important factor for the time-stepping schemes is that the flux and heating rate increments between successive calls to the radiation scheme can be closely represented by the simple and the fast spectral files. We therefore calculate the heating rate increments for successive hours of the Cloudnet data. The heating rate increments for the HadGEM spectral files serve as a control. An error is found by taking increments for the simple and fast files and subtracting these from the control increments. The absolute value of this error is taken for each model level and the results meaned over all the periods considered. The results are displayed in the bottom row of Figure. For comparison we also plot the error associated with using the same heating rates one Table I. Errors in hourly flux increments (W m ) for offline tests using observational data. Net surface flux TOA flux SW LW SW LW Persistence error Split time step error Incremental time step error....

7 FAST RT METHODS FOR TEMPORAL SAMPLING OF CLOUDS (a) (b) (c) 8 SW Heating Rate (K/day) Total (LW+SW) Heating Rate (K/day) LW Heating Rate (K/day) (d) (e) (f) 8 SW Heating Rate (K/day) Total (LW+SW) Heating Rate (K/day) LW Heating Rate (K/day) (g) (h) (i) SW HR hourly increment error (K/day).... Total HR hourly increment error (K/day) LW HR hourly increment error (K/day) Figure. Performance of the incremental and split time-stepping schemes using hourly meaned cloud data for Chilbolton. Panels (a), (b), and (c) show the mean SW, total, and LW heating rates respectively, calculated using the full operational spectral files (solid lines) and the split spectral files: slow (dashed lines) and fast (dot dashed lines). Panels (d), (e), and (f) show the same control (solid lines) as well as heating rates calculated using the simple spectral files (dotted lines) of the incremental time-stepping scheme. Panels (g), (h), and (i) show the mean hourly change in the full heating rates (solid lines), indicating the error that would occur from assuming no change in the heating rates for one hour. This is compared with the error the incremental (dotted lines) and split (dot dashed lines) time-stepping schemes would give after one hour by adding increments from the heating rates calculated with the simple or fast spectral files respectively. hour later, as is commonly done in the Met Office forecast models. The improvement gained over this persistence assumption is significant for both the SW and LW heating rates. A similar analysis was performed for the surface and top-of-atmosphere (TOA) fluxes. Table I displays the errors in the hourly flux increments for each scheme. In these tests the incremental time-stepping scheme performs particularly well, reducing the errors to negligible values.. Results of tests using the global forecast model The model used for these tests was an operational configuration of the Met Office global forecast model

8 J. MANNERS ET AL. with a reduced horizontal resolution of 9 7 grid points (N8). The vertical resolution was a standard level grid. The standard model time step was minutes, with a radiation time step of hours (every model time steps) for the operational setup. The case study considered uses analysis data from December. In order to isolate the effects of the radiation scheme changes, all tests were performed with the radiation scheme in a diagnostic mode. The evolution of the model will be exactly the same for each test. Our effective model truth was a run using a radiation time step of minutes (every model time step). All errors will be given as differences from this model run. The control run uses the standard three-hour radiation time step. As the time-stepping schemes introduce an additional computational cost, we also run an experiment with a two-hour radiation time step to determine the relative performance. The following experiments are then run: a repeat of the three-hour and two-hour radiation time step runs using the new solar zenith angle correction; a split time-stepping run using a three-hour slow radiation time step and a one-hour fast radiation time step; an incremental time-stepping run using a three-hour full radiation time step and a onehour increment radiation time step. The time-stepping schemes both included the new solar zenith angle correction. Runs were started at a model time of 9 GMT and output data were compared for the six-hour period between GMT and GMT. Fluxes and heating rates were output on every minute time step. Two types of error calculation were performed. A bias value was calculated for each grid point by taking the mean value of the error over the six-hour period. To obtain a global value, a mean was taken of the magnitude of the six-hour bias for each grid point. Secondly, an absolute error was calculated for each grid point by taking the magnitude of the error at every time step. These were then simply meaned over the six-hour period, and then over the globe, to give a mean absolute error. The results are displayed in Figures and and Table II. For the error analysis, we concentrate on those outputs that influence the evolution of the model. These are the net surface fluxes and the atmospheric heating rates. Table II displays how both time-stepping schemes perform particularly well in reducing the error in the net surface fluxes, with the incremental time-stepping scheme giving the best results. Figure compares the global distribution of these errors for the control run (three-hour radiation time step) and the incremental timestepping run. The errors are reduced smoothly across the globe and there are no artefacts associated with the zenith angle correction or the incremental time-stepping scheme. Figure compares the heating rate errors. The new solar zenith angle correction has no effect on the heating rates, so they are not reflected in these plots. In the boundary layer, below approximately km, the incremental time-stepping scheme provides the smallest errors for both SW and LW. The split time-stepping scheme also shows a noticeable improvement over the two-hour radiation time-step run at these heights. Above km, the two-hour radiation time-step run gives the best results. At these higher altitudes there is much less absorption by water vapour and cloud extinction begins to dominate in regions of the spectrum that are not treated by the timestepping schemes. However, both time-stepping schemes provide comparable errors and are a significant improvement on the control run using a three-hour radiation time step. The choice of scheme will depend on the relative computational cost involved. Each scheme was run three times using a single processor of the NEC SX. The CPU time spent in the radiation routines was then found as a percentage of the total CPU time for the rest of the model. The variation in CPU time over the three runs in each case was found to be negligible. The results are displayed in Table III. Table III. Relative computational cost of radiation scheme configurations for the Met Office global forecast model using a single processor on the NEC SX. The cost is expressed as a percentage of the CPU time for the remainder of the model. Radiation configuration CPU time (%) minute radiation time step.8 h radiation time step.8 h radiation time step. Split time step: h slow + hfast. Incremental time step: h full + h incs.8 Table II. Six-hour global mean net downwards surface radiation errors (W m ) with respect to a full radiation call on every ( minute) time step. Mean magnitude of bias Mean absolute error SW LW SW LW h radiation time step h rad tstep + zenith angle correction h radiation time step h rad tstep + zenith angle correction Split time step: h slow + h fast...8. Incremental time step: h full + h incs.8.9..

9 FAST RT METHODS FOR TEMPORAL SAMPLING OF CLOUDS (a) (b) Control ( hour radiation time step): SW Bias (Wm ) Incremental time stepping ( hr + hr): SW Bias (Wm ) (c) (d) Control ( hour radiation time step): SW Absolute Error (Wm ) Incremental time stepping ( hr + hr): SW Absolute Error (Wm ) (e) (f) Control ( hour radiation time step): LW Bias (Wm ) Incremental time stepping ( hr + hr): LW Bias (Wm ) (g) (h) Control ( hour radiation time step): LW Absolute Error (Wm ) Incremental time stepping ( hr + hr): LW Absolute Error (Wm ) Figure. Six-hour mean net downwards surface radiation errors (W m ). Left-hand panels ((a), (c), (e) and (g)) display the errors for the control run using a three-hour radiation time step. Right-hand panels ((b), (d), (f) and (h)) display errors for the incremental time-stepping scheme using a three-hour full radiation time step and a one-hour increment time step. Rows, from the top, give: SW bias ((a), (b)), SW absolute error ((c), (d)), LW bias ((e), (f)), LW absolute error ((g), (h)).

10 J. MANNERS ET AL. Figure. Six-hour global mean heating rate errors for the troposphere. Bias errors are shown on the left ((a), (c) and (e)), mean absolute errors on the right ((b), (d) and (f)). Rows from the top give errors for the SW HRs ((a), (b)), total HRs ((c), (d)), and LW HRs ((e), (f)). The schemes displayed are the control with a three-hour radiation time step (solid line), a two-hour radiation time step (dashed line), the split time-stepping scheme with three-hour slow and one-hour fast radiation time steps (-dot dash line), and the incremental time-stepping scheme with three-hour full and one-hour increment radiation time steps (dotted line). Errors for each scheme are in relation to radiative heating rates calculated on every minute time step.

11 FAST RT METHODS FOR TEMPORAL SAMPLING OF CLOUDS 7 The cost of the new solar zenith angle scheme does not significantly affect the CPU runtime (and so is not included in the table). The relative speeds of the various schemes reflect the number of monochromatic calculations that are required in a given time period. A three-hour radiation time step requires the least. The incremental time-stepping scheme only requires a further three calculations ( LW + SW k-terms) every hour, while the split time-stepping scheme requires a further seven calculations ( LW + SW k-terms) on each cloud-only radiation time step. This leaves the split time-stepping scheme with a cost comparable to a standard run using a two-hour radiation time step. In comparison, the incremental time-stepping scheme only increases the cost of the three-hour radiation control run by around %.. Discussion We have introduced two schemes to improve the temporal sampling of clouds in the Edwards Slingo radiation code used in the Met Office operational models. The split time-stepping scheme makes use of the gaseous k-terms that are used by the standard radiation scheme and splits them into two parts. The first part contains optically thick k-terms that are only negligibly affected by the addition of grey extinction by cloud condensate. These are used with a low sampling frequency. The second part contains optically thin gaseous k-terms where the additional cloud extinction dominates. These are used with a high sampling frequency. The incremental time-stepping scheme uses the full radiation calculation at low sampling frequencies and introduces an independent simple parametrization for calculating the increments in fluxes and heating rates due to changes in cloud at high sampling frequencies. Both schemes have been tested in the context of an operational configuration of the Met Office global forecast model, with their performance based on the divergence from a run where full radiation calculations are performed on every, minute, model time step. For the configurations tested, where a full radiation calculation was performed every three hours, and the increments due to clouds calculated every hour, both time-stepping schemes give errors in heating rates that are comparable to a standard run with a two-hour radiation time step. However, the errors in net surface fluxes are roughly half those of the two-hour radiation time step run. Overall, the incremental time-stepping scheme gives the greatest reduction in errors for the greatest computational efficiency, saving approximately % CPU time compared with the use of a two-hour radiation time step. We would therefore recommend this scheme for use in operational forecast models where the configuration is similar to that used at the Met Office. Where radiation schemes are employed with a more coarsely resolved k- distribution, it may be more beneficial to use the split time-stepping scheme, as this would not require repeated calculations for the window regions on full radiation time steps. A further outcome from this work is the development of a new correction to the surface SW fluxes for the change in solar zenith angle between radiation time steps. The correction is used by the new time-stepping schemes but can also be used in conjunction with the standard radiation scheme, leading to around a % reduction in the mean bias, and around a % reduction in the absolute error per time step for the net surface SW fluxes (see Table II). This is provided with no noticeable change in computational cost and only a slight increase in memory requirements, and so is also recommended for use in operational models. Together, these schemes provide a significant improvement in the temporal resolution and accuracy of radiative forcings. This will have an impact on the behaviour of the land surface (and sea surface in coupled models) with potential to affect the morning and evening transitions in the boundary layer. Improved radiative forcings will also be important for the diurnal cycle of convection, where feedback may be modelled far more effectively with the improved temporal sampling of cloud. While we have focused here on the treatment of temporal sampling, future work should also consider how we may take advantage of spatial correlations. The schemes presented here may be used in an adaptive manner to reduce the sampling requirement of the full radiation scheme to where and when it is most needed. The increments calculated for the radiation fields can be used to identify locations where the radiative fluxes are changing most rapidly and where further full radiation calculations would be the most beneficial in order to reduce the overall error. This would lead to greater sampling around the terminator and in regions where the cloud field is changing significantly. A more targeted sampling would lead to an increase in computational efficiency, allowing more accurate parametrizations to be used for the full radiation scheme. Acknowledgements We acknowledge the Cloudnet project (European Union contract EVK--) for providing liquid water content data, produced by the International Research Centre for Telecommunications-transmission and Radar (IRCTR), TU Delft, and for providing the radar/lidar ice cloud retrieval, produced by the Centre d étude des Environnements Terrestre et Planétaires (CETP), both using measurements from the Chilbolton Facility for Atmospheric and Radio Research, part of the Rutherford Appleton Laboratory. We also acknowledge the Cloudnet project for providing the ECMWF model data, which were produced by the ECMWF and the University of Reading. References Chevallier F.. Comments on New approach to calculation of atmospheric model physics: accurate and fast neural network

12 8 J. MANNERS ET AL. emulation of longwave radiation in a climate model. Mon. Weather Rev. : 7 7, DOI:.7/MWR78.. Chevallier F, Cheruy F, Scott NA, Chedin A A neural network approach for a fast and accurate computation of a longwave radiative budget. J. Appl. Meteorol. 7: 8 97, DOI:.7/- (998)7 8:ANNAFA..CO;. Cusack S, Edwards JM, Crowther JM Investigating k distribution methods for parameterizing gaseous absorption in the Hadley centre climate model. J. Geophys. Res. : 7. Edwards JM. 99. Efficient calculation of infra-red fluxes and cooling rates using the two-stream equations. J. Atmos. Sci. : 9 9, DOI:.7/-9(99) 9:ECOIFA..CO;. Edwards JM, Slingo A. 99. Studies with a flexible new radiation code. I: Choosing a configuration for a large-scale model. Q. J. R. Meteorol. Soc. : 89 79, DOI:./qj.977. Edwards JM, Havemann S, Thelen JC, Baran AJ. 7. A new parametrization for the radiative properties of ice crystals: comparison with existing schemes and impact in a GCM. Atmos. Res. 8: 9, DOI:./j.atmosres... Illingworth AJ, Hogan RJ, O Connor EJ, Bouniol D, Brooks ME, Delanoë J, Donovan DP, Eastment JD, Gaussiat N, Goddard JWF, Haeffelin M, Baltink HK, Krasnov OA, Pelon J, Piriou JM, Protat A, Russchenberg HWJ, Seifert A, Tompkins AM, van Zadelhoff GJ, Vinit F, Willen U, Wilson DR, Wrench CL. 7. CloudNet continuous evaluation of cloud profiles in seven operational models using ground-based observations. Bull. Am. Meteorol. Soc. 88: , Krasnopolsky VM, Fox-Rabinovitz MS, Chalikov DV.. New approach to calculation of atmospheric model physics: accurate and fast neural network emulation of longwave radiation in a climate model. Mon. Weather Rev. : 7 8, DOI:.7/MWR9.. Krasnopolsky VM, Fox-Rabinovitz MS, Belochitski AA. 8. Decadal climate simulations using accurate and fast neural network emulation of full, longwave and shortwave, radiation. Mon. Weather Rev. : 8 9, DOI:.7/8MWR8.. Kurucz RL. 99. CD-ROM. Harvard Smithsonian Center for Astrophysics. Lacis AA, Oinas V. 99. A description of the correlated k-distribution method for modeling non-gray gaseous absorption, thermal emission and multiple scattering in vertically inhomogeneous atmospheres. J. Geophys. Res. 9: McClatchey RA, Fenn RW, Selby JEA, Volz FE, Garing JS. 97. The optical properties of the atmosphere. AFCRL 7-97, Hanscom AFB, Bedford MA. Morcrette JJ.. On the effects of the temporal and spatial sampling of radiation fields on the ECMWF forecasts and analyses. Mon. Weather Rev. 8: , DOI:.7/- 9()8 87:OTEOTT..CO;. Morcrette JJ, Mozdzynski G, Leutbecher M. 8. A reduced radiation grid for the ECMWF integrated forecasting system. Mon. Weather Rev. : 7 77, DOI:.7/8MWR9.. Pawlak DT, Clothiaux EE, Modest MF, Cole JNS.. Full-spectrum correlated-k distribution for shortwave atmospheric radiative transfer. J. Atmos. Sci. : 88, DOI:.7/JAS8.. Tinel C, Testud J, Pelon J, Hogan RJ, Protat A, Delanoe J, Bouniol D.. The retrieval of ice-cloud properties from cloud radar and lidar synergy. J. Appl. Meteorol. : 8 87, DOI:.7/JAM9.. Venema V, Schomburg A, Ament F, Simmer C. 7. Two adaptive radiative transfer schemes for numerical weather prediction models. Atmos. Chem. Phys. 7: 9 7,