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Research Article

# Application of an adaptive radiative transfer scheme in a mesoscale numerical weather prediction model

Version of Record online: 12 AUG 2011

DOI: 10.1002/qj.890

Copyright © 2011 Royal Meteorological Society

Issue

## Quarterly Journal of the Royal Meteorological Society

Volume 138, Issue 662, pages 91–102, January 2012 Part A

Additional Information

#### How to Cite

Schomburg, A., Venema, V., Ament, F. and Simmer, C. (2012), Application of an adaptive radiative transfer scheme in a mesoscale numerical weather prediction model. Q.J.R. Meteorol. Soc., 138: 91–102. doi: 10.1002/qj.890

#### Publication History

- Issue online: 23 JAN 2012
- Version of Record online: 12 AUG 2011
- Manuscript Accepted: 28 JUN 2011
- Manuscript Revised: 15 JUN 2011
- Manuscript Received: 26 JAN 2011

- Abstract
- Article
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- Cited By

### Keywords:

- radiation parametrization

### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

The computational burden of radiative transfer parametrization is considerable, and hence operational atmospheric models use various sampling, coarsening and interpolation techniques to reduce this load; this, however, introduces new errors. An adaptive radiative transfer scheme takes advantage of the spatial and temporal correlations in the optical characteristics of the atmosphere to make the parametrization computationally more efficient. The adaptive scheme employed here generalizes the accurate radiation computations made in a fraction of the spatial and temporal space to the rest of the field. In this study, a previously developed scheme has been extended to atmospheric heating rates and implemented in the numerical weather prediction model COSMO. The performance of the adaptive scheme is compared with the performance of the currently operational COSMO-DE radiation configuration, in which radiation computations are performed quarter-hourly on 2 × 2 averaged atmospheric columns. The reference for both schemes is a set of frequent radiation computations for the full grid. We show that the adaptive scheme is able to reduce the sampling errors in the radiation surface fluxes by 15–25% and to conserve the spatial variability, in contrast to the operational scheme. Deviations in the heating-rate profiles are reduced for larger averaging scales. Physical relationships between the radiative quantities and cloud water or rain rates are better captured. We demonstrate that these improvements also lead to improvements with respect to the dynamical development of the model simulation, showing a smaller divergence from the reference model run. Copyright © 2011 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

Radiative fluxes at the surface and atmospheric heating rates strongly influence the heat energy budget at the atmospheric interface to the soil–vegetation system and the temperature tendencies in the atmosphere, and thus boundary-layer development and cloud processes. Therefore, these fundamental energy sources and sinks need to be considered adequately in numerical weather prediction and climate models. In particular, for the increasing resolutions of present-day weather forecast models, in principle of a few kilometres, three-dimensional radiative transfer computations are required. Accurate radiative transfer (RT) computations based on the three-dimensional spectral radiative transfer equation are, however, extremely complex and computationally demanding. Consequently, various parametrizations with different degrees of simplifications have been developed. Large simplifications are inevitable to reduce computational costs, in particular for application in operational weather prediction models or for long-period climate simulations. Common simplifications are the computation of RT reduced to flux densities over broad spectral bands for one-dimensional vertical atmospheric columns, assuming horizontal homogeneity. Also, for the treatment of clouds in the RT parametrization a number of assumptions are required, e.g. for the overlap of partial cloud cover in the vertical. Many input parameters required for atmospheric RT, especially cloud characteristics, are, however, highly uncertain and are also parametrized in operational models. Despite these reductions in complexity, the RT parametrization is, for most applications, still too demanding to be computed for each model time step and the full spatial grid. Various approaches have been implemented by national weather services and climate centres to overcome this limitation by sampling in time and space. The most common strategy is temporal sampling, i.e. the radiation scheme is called at time intervals of more than one time step while keeping the fluxes and heating rates constant in between, either based on a medium solar zenith angle or adjusted during each time step according to the current solar zenith angle. Spatial sampling strategies interpolate between sparse computations or average atmospheric properties over multiple columns before passing the data to the RT parametrization.

The Integrated Forecasting System (IFS) of the European Centre for Medium-Range Weather Forecasts (ECMWF), for example, has employed a comparatively sophisticated RT scheme since June 2007, the Rapid Radiative Transfer Model for Global model applications (RRTMG: Clough *et al.*, 2005; Mlawer *et al.*, 1997), in combination with the Monte Carlo Independent Column Approximation (McICA: Pincus *et al.*, 2003). To save computation time, the radiation scheme is called only at large temporal intervals (once per hour or once per three hours, depending on model resolution) and on a coarsened grid, where the radiative effects are interpolated to the finer grid by a cubic interpolation scheme. The temporal interpolation for each dynamic model time step is performed by accounting for the correct solar zenith angle for short-wave flux (Morcrette, 2000; Morcrette *et al.*, 2008).

In the Consortium for Small Scale Modelling (COSMO) model (Steppeler *et al.*, 2003), the numerical weather prediction model (and regional climate model) of several European weather services, the two-stream approximation is employed, based on code by Ritter and Geleyn (1992); see section 2. To save computation time, the scheme is called either once per forecast hour or quarter-hourly, in the configurations COSMO-EU and COSMO-DE, respectively. In the latter configuration the atmospheric input parameters are first averaged over four columns before carrying out the radiation calculations. The obtained surface radiation fluxes are adjusted taking the local albedo and surface temperature into account (Baldauf *et al.*, 2009).

Temporal and spatial sampling methods can, however, lead to errors. Temporal sampling neglects the varying local insolation attributable to the advection and evolution of clouds. Spatial averaging reduces the spatial variability of radiative effects. Inconsistent situations may occur when the radiative properties are not allowed to react to the changing atmosphere over several time steps; thus, raining clouds and strong solar fluxes are allowed to coexist in rapidly changing convective atmospheres. Morcrette (2000) has studied the sampling effects on operational simulations and analyses for the IFS global model. He has found a larger sensitivity with respect to temporal sampling than to spatial sampling followed by subsequent interpolation. In 10 day forecasts, he detected temperature errors depending on the temporal frequency of radiation computations. These errors increase with height, because of feedback between convective clouds and radiation, especially in the Tropics. For longer, e.g. seasonal, predictions, these errors grow, and thus a higher temporal sampling is beneficial. The study also indicates, however, that changes in the microphysical scheme or cloud-overlap assumption have a larger influence than the spatial and temporal interpolation of radiation computations.

Several approaches have been developed to bypass this conflict between the need for frequent radiation computations and computational limits. Computation time can be reduced by training an artificial neural network (ANN) with a detailed radiation scheme off-line (Chevallier *et al.*, 2000; Chevallier *et al.*, 1998; Krasnopolski *et al.*, 2005). Krasnopolski *et al.* (2010) have tested such ANNs in the National Center for Environmental Prediction (NCEP) Climate Forecast system (CFS) by comparing simulations with the original inherent radiation code (RRTMG) as control runs with simulations employing the ANN emulating the complex radiation code. The differences are small and comparable to internal model variability, compiler changes etc. whereas a considerable speed-up is achieved for the climate simulations. A drawback of this method is the need to re-train the ANN for any configurational changes such as vertical resolution.

Pielke *et al.* (2005) have proposed an approach based on look-up tables. In this approach, the radiative effects for all possible inputs are pre-calculated and stored to disk. As for the ANN, this look-up table needs to be recomputed for every change in the model set-up, making the model inflexible. Furthermore, given the expected increase in the number of model levels, the number of possible combinations may soon become prohibitive.

In Venema *et al.* (2007) (hereafter VSAS07), two adaptive RT parametrizations are presented that exploit temporal and spatial correlations in the 3D optical property fields. Radiation calculations by the implemented RT scheme are performed in only a fraction of time and space. The so-called *temporal adaptive scheme* identifies the grid points in the model domain at which the largest changes since the last radiation update have occurred and targets these columns for the next RT computations. These predictions utilize a simplified radiation scheme, based on multiple linear regression, which uses vertically integrated atmospheric variables as predictors. The rest of the field is updated by computing the change in the radiative tendencies by the same simplified radiation scheme and adding them to the radiation effects from the last time step. In the *spatial adaptive scheme*, only a small but fixed part of the field is updated by the internal radiation scheme at high temporal frequency. For the other columns, a search for a nearby similar atmospheric column is carried out and the radiative effects of the most similar column are applied with a correction for solar angle and albedo. To train the scheme, an optimization algorithm was applied to find the weights for the weighted sum used to search for nearby similar columns. However, the scheme has been shown not to be sensitive with respect to the exact values of these weights. Hence, an advantage compared with neural networks that emulate complex radiation schemes (e.g. Krasnopolski *et al.*, 2010) is that re-training is much less important for changes in the radiation scheme or the vertical resolution; the scheme can be applied ‘as is’ in other atmospheric models or in combination with other radiation codes. The obtained speed-up, however, is potentially smaller than for neural networks because the complex radiation scheme is not emulated but only exploited more efficiently. The adaptive schemes in VSAS07 have been tested, as a proof of principle, in an off-line environment for the radiative net fluxes at the surface. In a case study, such schemes have been shown to be able to predict the surface fluxes much more accurately, without an increase in computational demands. In addition, the spatial difference fields of the adaptive approaches are characterized by notably smaller correlation lengths and a reduction in the number of calls to the complex scheme leads to only a small reduction in accuracy.

Manners *et al.* (2009) have adopted this idea and developed two adaptive RT schemes in spectral space; they employ a reduced RT calculation at time steps between calls to the full complex radiation scheme in the operational UK Met Office Unified Model. Their *split time stepping* approach divides the RT computation into bands with strong gaseous absorption terms, which are optically thick and barely dependent on cloud characteristics, and bands that are optically thin and thus strongly influenced by clouds. The latter RT calculations are updated with a higher temporal frequency to keep track of changes attributable to developing and advecting clouds. Their second method, the *incremental time stepping* method, uses a simple radiation scheme to compute temporal changes for the window region, i.e. the optically thin part of the atmospheric spectrum, where variability is caused mainly by variations in cloud properties. These increments are added to the results of the full complex scheme, which is computed at a lower temporal frequency.

In this study, we report results from the implementation of the spatial adaptive scheme in the operational weather forecast model COSMO. As shown in VSAS07, the spatial adaptive scheme gives overall better results than the temporal adaptive scheme; thus, in this study, only results of the implementation of the spatial scheme are presented. For an atmospheric model, not only are the radiation surface fluxes relevant, but also an accurate simulation of the heating-rate profiles is needed. Moreover, changing the radiative surface fluxes but leaving the vertical heating rates unchanged would lead to inconsistent situations. For these reasons, the spatial adaptive scheme has now been extended to be applied to the vertical heating rates also. The performance of this enhanced spatial adaptive scheme is compared with the standard operational radiation update scheme. For clarity of notation, it is from now on simply referred to as the ‘adaptive scheme’.

Section 2 gives a short description of the COSMO model, followed by the description of the adaptive scheme and the experimental set-up. In section 3, we outline the performed simulations and experiments. Results are shown in section 4; they include not only a detailed analysis of the results of the 2.8 km COSMO-DE simulations but also a brief description of results obtained by employing the scheme in a coarser-scale 7 km simulation. The findings are discussed in section 5.

### 2. The COSMO model

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

As an operational testbed for our scheme, the COSMO model (version 4.0) has been chosen. This model is the non-hydrostatic, limited-area mesoscale operational weather forecast model of the German and several other European meteorological services and is also used as a regional climate model. For the case studies the configurations of the operational COSMO-DE model have been adopted; this is the set-up for the operational short-range meso-*γ*-scale weather forecast simulations by the German Weather Service (Baldauf *et al.*, 2009). The model domain covers Germany and parts of its neighbouring countries with 421 × 461 columns of horizontal resolution 2.8 km. This set-up has 50 vertical layers and a time step of 25 s. The model is nested into the so-called COSMO-EU model, which has slightly different configurations, with a horizontal grid spacing of 7 km, 40 vertical layers and a time step of 40 s. The COSMO-EU model domain covers Europe and parts of the Atlantic Ocean and Northern Africa.

In both COSMO versions, the condensation and evaporation of cloud water are modelled by saturation adjustment, the treatment of grid-scale precipitation being based on a simplified one-moment version of the scheme of Seifert and Beheng (2001), which considers five prognostic water categories. Subgrid-scale cloudiness is analyzed by means of an empirical function depending on relative humidity, height and convective activity. In COSMO-DE, deep convection is assumed to be a grid-scale process and only shallow convection is parametrized by a mass-flux scheme according to Tiedtke (1989). In COSMO-EU the deep convection is also parametrized by the Tiedtke scheme.

The radiation scheme in the COSMO models was developed by Ritter and Geleyn (1992) and is based on the one-dimensional *δ**-two-stream approximation* of the RT equation. The spectrum is divided into broad spectral intervals, for which the RT calculations are carried out. Absorption, emission and scattering by cloud particles, aerosols and gas molecules is accounted for. For clouds, the maximum-random-overlap assumption is applied. The cloud optical properties are parametrized based on a fit of the optical properties of eight cloud types provided by Stephens (1984). Aerosols are given by a constant climatology. Effects of three gases are considered: water vapour, carbon dioxide, which has a constant value, and ozone, which is described by a climatological annual cycle. The radiation scheme provides net fluxes at the surface in the solar and thermal regime and heating-rate profiles for every column. In COSMO-DE, radiation computations are carried out every 15 min; these calculations are applied to 2 × 2 averaged columns, correcting the solar radiation surface flux by the local albedo and the thermal radiation flux by local ground temperature (Baldauf *et al.*, 2009). In COSMO-EU, radiation effects are calculated hourly and fluxes and heating rates are kept constant in between. The solar zenith angle used in the radiation computations is the zenith angle valid for the middle of the interval between two radiation updates.

### 3. Implementation and experimental design

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

#### 3.1. Enhanced adaptive radiation scheme

The spatial adaptive scheme exploits spatial correlations in the radiative effects in the following way: The model domain is divided into small subdomains. At a high frequency, the radiation effects at only one of the subdomain columns are updated by a call to the intrinsic radiation scheme; only in the first time step of the model simulation is the radiation routine called once for the whole field. In this study, a subdomain contains 5 × 5 columns and every 6 time steps (i.e. 2.5 min) one of them is updated; however, this set-up can be chosen according to available computer time and other considerations. For grid points that are not updated, a search is performed for a nearby similar, recently updated column. Similarity is evaluated by comparing a weighted sum of absolute differences in low cloud cover, total cloud cover, liquid water path, integrated water vapour, surface albedo, time since the last update of the respective column and distance between the two columns. This cost function has now been extended by the spatial distance between the columns and the integrated water vapour compared with the version introduced in VSAS07 (see Table I). Having found the most similar column, the short-wave (SW) and long-wave (LW) surface fluxes and photosynthetic active radiation (PAR) and also the vertical column of atmospheric heating rates, i.e. the heating rate for each vertical level, are copied to the respective column. The solar fluxes and heating rates are corrected for solar zenith angle, and the surface fluxes also for the local albedo. A correction according to local surface temperature for the long-wave surface fluxes has been introduced as a further enhancement over the scheme presented in VSAS07 (see Table II).

^{}*CCL*: low-level clouds (below 800 hPa) [1]^{}*CCT*: total cloud cover [1]^{}*LLWP*: logarithm of liquid water path [kg m^{−2}]^{}*IWV*: integrated water vapour [kg m^{−2}]^{}*α*: surface albedo [1]^{}*t*: time since last update [s]^{}*dist*: root-mean-square distance between grid points [number of grid points].
| |

Cost function | δ = w_{1}ΔCCL + w_{2}ΔCCT + w_{3}ΔLLWP + w_{4}ΔIWV + w_{5}Δα + w_{6}Δt + w_{7}dist |

Weights | w_{1} = 0.37; w_{2} = 7.85; w_{3} = 2.1734; w_{4} = 2.0801; w_{5} = 13.69; w_{6} = 0.0018; w_{7} = 0.744; |

Variable | Correction |
---|---|

^{}Θ: solar zenith angle ^{}*α*: surface albedo^{}*σ*: Stefan–Boltzmann constant^{}*T*_{G}: ground temperature^{}*α*_{IR}: infrared albedo^{}the indices *c*and*x*denote the value from the copied and the actual local grid point, respectively.
| |

SW heating rates | |

SW and PAR surface radiation flux | |

LW heating rates | no correction |

LW surface radiation flux | F_{LW} = F_{LW} + (σ(1 − α_{IR}) |

#### 3.2. Set-up

The spatial adaptive scheme is compared with the standard operational radiation configuration of COSMO-DE, i.e. with radiation calculations carried out for 2 × 2 averaged columns every 15 min. For clarity of notation, the scheme will be referred to as the ‘2 × 2’ scheme from now on. The adaptive scheme is called every 2.5 min, applying the intrinsic radiation scheme only for one out of 5 × 5 atmospheric columns, whereas the extrinsic generalization is applied for the other columns (see Figure 1). This set-up requires about the same computation time as the 2 × 2 scheme. Update patterns for the adaptive approach, i.e. the sequence in which the pixels are updated, are given in VSAS07 for regions of different size: the ordering is such that subsequently updated columns have a large distance between them.

The most accurate results with respect to radiation would, of course, be obtained by radiation computations for the full domain using a high temporal frequency. This optimal but much too expensive set-up for operational applications has been taken as reference for testing our adaptive scheme and the standard COSMO-DE 2 × 2 configuration. All comparisons and deviations shown in the following are based on intrinsic radiation computations carried out every 2.5 min on the full model domain. The considered radiation configurations are listed in Table III.

Radiation | Call frequency | Number of |
---|---|---|

scheme | [min] | columns updated |

Reference | 2.5 | All |

Adaptive | 2.5 | 1/25 |

‘2 × 2’ | 15 | 1/4 (averaging |

(operational) | 2 × 2 columns) |

For the comparisons, we have developed a COSMO model version in which the different radiation options are computed diagnostically, i.e. the dynamics are driven by one of the three radiation options, which is for most comparisons the reference set-up. The radiation effects of the other two schemes are computed in addition and are provided as model output.

The highest errors in radiation effects are expected to occur in situations with heterogeneous atmospheric conditions, i.e. small-scale convective cloud patterns, where the atmospheric state of the columns changes rapidly and hence frequent radiation calculations are most important. Three days have been chosen for the comparison of the radiation options, with a range from mainly convective to stratiform clouds. The first day is a convective summer day, 21 June 2004, when unstable air masses were centred over Central Europe under an elevated trough and a large number of showers and thunderstorms covered the whole model domain. The second day is a slightly less heterogeneous autumn day, 19 September 2001, where a low-pressure system over the North Sea led to convective activity in parts of the model domain. As a third case, the schemes have been tested for a winter day (22 December 2005) with stratiform, very homogeneous and slowly changing cloud conditions. Germany was under the influence of an occlusion front belonging to a low-pressure system centred over Scandinavia. During the day, the front and its broad stratiform cloud band crossed the model domain from northwest to southeast and led to moderate rain at lower altitudes and snow at higher altitudes.

For these three cases, the COSMO-DE runs were forced by COSMO-EU operational analyses as initial and boundary values, obtained from the German Meteorological Service.

### 4. Results

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

#### 4.1. Radiative fluxes and heating rates

For the three case studies described above, COSMO model runs were carried out, in which frequent intrinsic radiation computations of the full domain served as a reference and provided the radiative effects for driving the dynamics and soil-surface parametrizations of the model. In addition, the radiation properties resulting from the adaptive radiation computations and the operational 2 × 2 column-averaging quarter-hourly radiation updates used by default in COSMO-DE were computed. Hence, the three different radiation computations are based on the same atmospheric fields and can be compared directly, without effects that would result from diverging dynamics in different model runs.

According to Figure 2, the adaptive scheme reduces the hourly averaged root-mean-square difference (RMSD) for the summer case by about 25% in both short-wave and long-wave regimes; the bias is also largely decreased. The instantaneous (2.5 min) deviations (deviations always compared with the reference) of the COSMO-DE radiation scheme show a quarter-hourly cycle because of the 15 min update cycle. Deviations are low directly after a new computation of the full field and increase during the following 15 min. The instantaneous deviations of the adaptive scheme are lower throughout the time. A simulation where the surface radiation fluxes of the 2 × 2 scheme were corrected for actual solar zenith angle and surface temperature every 2.5 min did not lead to significant improvements, from which it can be concluded that the errors are mainly due to moving, forming and decaying clouds during the 15 min interval. The deviations of the COSMO-DE fluxes do not reach zero at a new calculation at a quarter-hourly interval because of the averaging over four columns.

The RMSDs and biases for the surface fluxes for all three cases are listed in Table IV. The adaptive scheme almost always outperforms the 2 × 2 scheme. The differences are generally smaller for the more homogeneous cases, but about the same relative improvement compared with the 2 × 2 scheme is achieved by the adaptive scheme as for the summer day.

Mean flux at noon | RMSD | Bias | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Day | SW | LW | SW | LW | SW | LW | ||||

2 × 2 | adapt | 2 × 2 | adapt | 2 × 2 | adapt | 2 × 2 | adapt | |||

21–06–2004 | 520 | −70 | 31.43 | 23.80 | 7.15 | 5.34 | −2.01 | −0.20 | −0.09 | 0.07 |

19–09–2001 | 350 | −66 | 19.62 | 15.81 | 6.53 | 5.08 | −0.60 | −0.17 | −0.05 | 0.08 |

22–12–2005 | 90 | −46 | 2.36 | 1.71 | 5.25 | 4.14 | −0.08 | −0.04 | 0.11 | 0.07 |

The daily cycle of the deviations for the atmospheric heating rates is depicted in Figure 3. The RMSD for the short-wave heating rates barely differs between the adaptive and the 2 × 2 scheme, whereas the adaptive scheme has a higher RMSD for the long-wave. The systematic deviations are small but the adaptive scheme clearly outperforms the 2 × 2 standard scheme (lower panel in Figure 3). Only during sunset does the solar radiation show high systematic deviations, probably because fast-changing path lengths of the Sun through the atmosphere lead to markedly different transmissivities. The average vertical profile of RMSD and bias (see Figure 4) shows that the adaptive scheme leads to larger random differences for the cloud level, whereas the values for the systematic differences, which are however much smaller than the RMSD, can be improved. This behaviour can be traced to the weighted difference function used to search for the most similar column; this function is mainly based on vertically integrated atmospheric properties. Hence, columns of heating rates might be copied that have the same integrated cloud properties but differ in the vertical position of the clouds. Such differences are penalized twice in the RMSD: once at the level where radiation is overestimated and once at the level where it is underestimated. The systematic deviations over the whole field, however, are small. In the 2 × 2 scheme, systematic differences can occur attributable to the averaging of the atmospheric properties; this can lead to biases arising from the nonlinearity of RT processes, especially through clouds. In general, clouds that are too homogeneous have too high an albedo and too low a transmissivity (see e.g. Cahalan *et al.*, 1994), leading to a positive bias in the solar heating rates in and above the cloud level and a negative bias in the boundary layer and at the surface (Figure 4, bottom). Because the adaptive scheme is able to conserve the variability it does not show such systematic errors.

Table V summarizes the daily mean values of the heating rate differences for the three case studies. Averaged over the day, the 2 × 2 radiation scheme performs better than the adaptive scheme for the long-wave RMSD. The average bias over the day is small, although instantaneous biases can be much larger. The reason is the shape of the diurnal cycle of the bias (Figure 3), which shows an overestimation (underestimation) of the short-wave (long-wave) heating rates in the morning and vice versa in the afternoon for all case studies, averaging to a small value close to zero over the day. The hourly biases of the 2 × 2 scheme are almost always higher than for the adaptive scheme.

RMSD [10^{−3} K h^{−1}] | Bias [10^{−3} K h^{−1}] | |||||||
---|---|---|---|---|---|---|---|---|

Day | SW | LW | SW | LW | ||||

2 × 2 | adapt | 2 × 2 | adapt | 2 × 2 | adapt | 2 × 2 | adapt | |

21–06–2004 | 7.8 | 7.7 | 26.2 | 30.4 | −0.008 | 0.006 | 0.014 | 0.002 |

19–09–2001 | 6.1 | 6.3 | 29.1 | 33.8 | −0.007 | 0.009 | 0.008 | −0.002 |

22–12–2005 | 1.3 | 1.2 | 29.0 | 32.9 | −0.003 | 0.004 | 0.003 | −0.004 |

An interesting characteristic of the difference fields is the temporal auto-correlation, i.e. the correlation of the differences with time lag. Buizza *et al.* (1999) have shown that temporally persistent perturbations have a remarkable influence on model dynamic development (in their study, model runs with temporally correlated perturbations increase the divergence of the model runs in an ensemble), whereas noise that varies randomly from time step to time step has no noticeable influence. In our study, the temporal correlations of the difference fields could be reduced by about 34% and 45% for the short-wave and long-wave surface fluxes, respectively, averaged over all cases. The spatial autocorrelations of the difference fields have also been computed in this study, showing that the correlations in the difference fields are also slightly lower for the adaptive scheme than for the 2 × 2 scheme (not shown).

Averaging the radiation fields to larger scales and computing the root-mean-square differences on these larger scales (as depicted in Figure 5) shows that the differences from the reference of the adaptive scheme decrease more strongly with increasing scale than for the 2 × 2 averaged radiation. This holds not only for the radiation fluxes at the surface but also in particular for the atmospheric heating rates. On the smallest scale, the results for the adaptive scheme are worse in terms of differences of the heating rates; however, the RMSD on the pixel scale is not as relevant as on larger scales, where the adaptive scheme is more accurate.

Table VI shows that the mean standard deviations, i.e. the spatial variability of the radiation fields, are underestimated by the 2 × 2 averaging scheme. The smoothing of the 2 × 2 column averaging leads especially to small convective clouds being smoothed out (see also section 4.3).

Flux [W m^{−2}] | Heating rate [K h^{−1}] | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Case | SW | LW | SW | LW | ||||||||

ref | adapt | 2 × 2 | ref | adapt | 2 × 2 | ref | adapt | 2 × 2 | ref | adapt | 2 × 2 | |

21–06–2004 | 143 | 142 | 135 | 30.9 | 31.1 | 30.1 | 0.056 | 0.056 | 0.053 | 0.090 | 0.090 | 0.082 |

19–09–2001 | 116 | 116 | 113 | 35.3 | 35.4 | 34.6 | 0.044 | 0.044 | 0.043 | 0.092 | 0.093 | 0.084 |

22–12–2005 | 49 | 49 | 49 | 39.1 | 39.2 | 38.9 | 0.028 | 0.028 | 0.027 | 0.127 | 0.127 | 0.120 |

#### 4.2. Test case with 7 km resolution

Many mesoscale weather forecast models operate on resolutions on the 10 km scale. The operational COSMO-EU model configuration of the German weather service on this scale, as mentioned above, has a grid spacing of 7 km. At this resolution, the relative advection speed is smaller than for the 2.8 km model resolution, and thus column cloud cover changes less rapidly than at a smaller scale. Hence, for lower resolutions, the problems caused by infrequent radiation calculations are expected to be less important. The convective summer test case as described above has been simulated with 7 km resolution with COSMO-EU settings. For this case, the standard COSMO-EU update practice of computing the radiation just once per forecast hour and keeping the radiation fields fixed in between has additionally been simulated and compared with the quarter-hourly 2 × 2 averaging option and the adaptive scheme. The reference radiation update interval and the time step of the adaptive schemes were set to 6 min (9 COSMO-EU model time steps). The update region in this set-up is chosen to comprise 4 × 4 grid points, leading to a total number of radiation calculations that is even fewer than in the other methods under consideration. The search region has been set to 5 × 5 pixels as before.

For these 7 km model simulations, the adaptive scheme again outperforms the 2 × 2 column-averaging scheme for the surface fluxes in terms of RMSDs (Figure 6), although the relative improvement is smaller than on the smaller scale considered in the previous section. For the heating rates, the 2 × 2 scheme leads to better results than the adaptive approach (not shown). However, by far the worst results for both fluxes and heating rates are obtained by the hourly update scheme.

#### 4.3. Physical consistency

In addition to the quantitative performance in terms of deviations of the radiation fields, the consistency of radiative effects within the model with other variables and other physical parametrizations is of importance. To illustrate physical relationships between radiative effects and cloud characteristics, the mean solar surface flux as a function of the atmospheric liquid water content is depicted in Figure 7 (left). For an increasing liquid water path (LWP), the solar-surface net fluxes decrease strongly for the reference and the adaptive radiation, whereas for the 2 × 2 averaging scheme this behaviour is less pronounced. Moreover, the probability density function for the very high specific LWP value of 0.5 kg m^{−2} (Figure 7, right) is much wider, less peaked and shifted to higher values for the 2 × 2 scheme, illustrating that too high radiation fluxes may occur for thick clouds. Again, this relationship is much better captured by the adaptive scheme.

To study the physical consistency more quantitatively, one could compute this probability density function for each LWP value, yielding a 2D histogram of LWP and solar-surface net flux, and subsequently calculate the RMSD between this histogram for the reference radiation and one of the other schemes. The disadvantage of a histogram is, however, that the RMSD can depend on the bin widths chosen. Therefore, we have chosen to compute this error measure on the 2D empirical cumulative distribution functions (ECDF). The 2D ECDF is obtained from a two-dimensional histogram by integrating in both directions. In this way, small random errors do not have a strong influence but systematic deviations do.

In Figure 8, the RMSDs of these ECDFs with respect to the ECDF of the reference radiation are depicted for LWP and surface radiation net fluxes (left) and rain rate and surface net fluxes (right), respectively. Evidently, the differences between the 2 × 2 column averaging scheme and the reference is much larger, for both short-wave and long-wave net fluxes, than for the adaptive scheme. This illustrates that the radiative variables computed by the adaptive radiation scheme are more consistent with other model variables, here shown as examples are the liquid water path and rain rate. Moreover the relationships of the surface net fluxes with cloud cover and ground temperature are captured more accurately (not shown). For the 2 × 2 scheme, the long-wave differences show a peak in the first hours of the model run; this peak is even more pronounced in the other two case studies (not shown). This indicates that after initialization the changes in cloud characteristics are fast, because of spin-up processes caused by the initialization being based on coarser scale analyses (7 km grid spacing).

#### 4.4. Effects on model dynamics

Very important for the evaluation of different radiation parametrizations is the effect on the model dynamics. Any radiation scheme sampling in time and space because of computation time limits will exhibit errors. It is important, however, that these errors have as minimal a negative influence as possible on the model dynamics, i.e. the weather development. To investigate this aspect, we carried out three single-model runs: one driven by the high-frequency reference radiation computations, one driven by the adaptive radiation computations and one with the quarter-hourly 2 × 2 averaging scheme. Of great interest for a numerical weather forecast model is the extent to which the two computationally cheaper model runs diverge from the reference model run. In Figure 9, the RMS difference for three variables that are not only important in daily weather forecasts but also good indicators of the dynamical behaviour of the model is displayed (for the summer and the winter case): the surface pressure, the total precipitation (sum since model initialization) and 2 m temperature, normalized with the standard deviations of the respective reference field to remove effects caused by the diurnal cycle alone. The differences in the radiative forcing at the grid cells lead to a difference in the heat budget at the surface and differences in the temperature profiles. These effects lead to small differences in the state variables, which build up during the model simulation, and thus to a diverging development of the cloud and precipitation fields. Hence, the differences in total precipitation and 2 m temperature are mainly attributable to differences in cloud formation and movement. For both cases the RMSD is larger for the 2 × 2 radiation update configuration for all three variables, indicating that the model run with the adaptive radiation stays closer to the reference. For the winter day the deviations are smaller, because on that day a broad frontal cloud and precipitation band crosses the model domain, and although the movement and location of this frontal zone differs slightly in the three model simulations, the differences are less than on the convective summer day where the movement and evaluation of the single cloud clusters is highly sensitive to the radiative forcing.

### 5. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

The concept of adaptive parametrizations for radiative surface fluxes was introduced first in VSAS07. In this study, the spatial adaptive scheme has been extended to heating rates and the scheme has been implemented and tested in the COSMO-DE model with the same set-up as that used for daily weather forecast simulations by the German meteorological service at 2.8 km horizontal resolution. The results for three case studies with different synoptic conditions have been compared with radiative effects computed with the standard radiation set-up based on four (2 × 2) averaged columns computed once per 15 min. Such spatial and temporal sampling strategies are common in operational atmospheric models, which are subject to computer time limits. This study shows that the adaptive concept provides the envisioned benefits in a real model implementation, which has also been demonstrated for a spectral adaptive RT parametrization by Manners *et al.* (2009). We have shown that the adaptive scheme produces better results in terms of random and systematic deviations for the surface fluxes. The deviations for the long-wave heating-rate profiles do not show these improvements on the smallest scale, but for both fluxes and heating rates considerable improvement is achieved for larger averaging scales, which are dynamically more important. The 2 × 2 averaging scheme leads to too low a variability of the radiation fields because of the smoothing by the 2 × 2 filter, whereas the adaptive scheme does not decrease the variability but matches the reference standard deviations well.

Evaluation of model consistency based on relationships between radiation surface fluxes and rain rates or cloud water content leads to the conclusion that the adaptive scheme is better in conserving these physical relations by capturing changes in cloud cover. For the 2 × 2 scheme, on the one hand these correlations are smoothed out because of averaging and on the other hand fast-moving and developing clouds cannot be tracked by the quarter-hourly updates. The adaptive scheme benefits from some new calculations made at high frequency in each region and thus does not miss rapid developments. The search for similar atmospheric columns ensures that the heating rates and surface fluxes are taken from similar, recently updated cloudy or cloud-free columns, leading to consistent radiative and atmospheric characteristics.

Simulations with various radiation options have also been carried out for the operational COSMO-EU model, which runs on a larger model domain with 7 km horizontal grid spacing. Here, the operational setting in which updates are computed only once per forecast hour and kept constant in between shows the worst results, indicating that, for this scale also, either the adaptive or the 2 × 2 scheme would be more appropriate.

Crucial for weather forecasts is the question as to which CPU-time saving configuration has the least deteriorating effect on the dynamical model development. To answer this question, single model runs were carried out where the different radiation quantities of the adaptive and 2 × 2 averaging schemes were not just computed diagnostically but actively used to force the model. We found that the model runs with the operational quarter-hourly 2 × 2 averaged radiation diverged more from the reference model run with highly frequent calls to the radiation scheme than the model run employing the spatial adaptive scheme. Thus, we can conclude that the benefits of the adaptive scheme (improved surface fluxes, heating rates at coarse scales and physical relationships between variables) are more important than the slightly worse results for the heating rates on the smallest scale. All this is achieved without an increase in computation time but is based on a more intelligent way of exploiting the available computation resources and distributing the information from the complex radiation scheme to the rest of the field. Summarizing, the scheme can be recommended for application in operational weather forecast codes.

One advantage of the 2 × 2 scheme over the adaptive scheme is that less communication between the processors is needed. For the adaptive scheme, every 2.5 min some boundary information between neighbouring processors needs to be exchanged, whereas for the 2 × 2 scheme this need be done only only every 15 min. Compared with the model dynamics, however, where information needs to be exchanged every time step, this effect is small.

As a further improvement of the scheme, a combination of the spatial adaptive scheme with the temporal adaptive scheme proposed in VSAS07 is planned. In this scheme, the grid points for which a call to the complex radiation scheme is carried out are not fixed, but a very simple radiation scheme based on a multiple linear regression is used to find the columns that have undergone the largest atmospheric changes since the last update. This gives even better information on the radiative quantities in regions where the clouds are developing and moving rapidly. Furthermore, the remaining differences between the best-matching nearby column and the true atmospheric profile can be corrected using the simple radiation scheme. We also aim to improve the results for the heating rates, potentially by better correction methods or by a search algorithm that incorporates not only vertically integrated cloud characteristics.

The adaptive approach has been tested on the meso-*γ* scale, because the problem of fast-moving clouds becomes an important task for high horizontal resolutions; the persistence assumption made in many operational codes leads to highly inconsistent situations. However, as has been shown for example by Morcrette (2000) for longer, seasonal simulations, deviations attributable to sampling of radiation computations build up over time. They can have a considerable impact on the dynamical development of the model, leading to a stratosphere that is much too cold via cloud–radiation–convection interactions in simulations with the ECMWF model. Thus the use of adaptive radiation computations as a tool to provide better radiative fluxes and heating rates without increasing computational burden should also be considered for larger scale models and especially for climate simulations, in which the heat budget is of particular importance.

We have applied this concept to RT, because it is one of the most expensive parametrizations in terms of computation time. However, the general idea of combining complex parametrizations with more simple schemes to spread the accurate information in time and space (or spectral space) to save computational resources can also be applied to other parts of the model physics or model dynamics. The mode-splitting approach of Klemp and Wilhelmson (1978), for example, which is applied in many atmospheric models and computes fast atmospheric waves at intermediate time steps between the coarse model time steps for advective and physical processes, is based on a similar idea.

### Acknowledgements

- Top of page
- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

We gratefully acknowledge financial support by the ‘Extramurale Forschung’ program, funded by the German Meteorological Service (DWD), Offenbach, and by the SFB/TR 32 ‘Pattern in Soil–Vegetation–Atmosphere Systems: Monitoring, Modelling, and Data Assimilation’, funded by the Deutsche Forschungsgemeinschaft (DFG). Moreover, we thank the DWD for access to the COSMO analyses data archive and the COSMO model code. Special thanks are given to Theresa Jones for proofreading from a native speaker's perspective and to Bodo Ritter and two anonymous reviewers, whose comments helped to improve the manuscript.

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- Abstract
- 1. Introduction
- 2. The COSMO model
- 3. Implementation and experimental design
- 4. Results
- 5. Discussion
- Acknowledgements
- References

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