Linearized radiation and cloud schemes in the ECMWF model: Development and evaluation

Transcription

1 Q. J. R. Meteorol. Soc. (2), 128, pp Linearized radiation and schemes in the ECMWF model: Development and evaluation By MARTA JANISKOVÁ, JEANFRANÇOIS MAHFOUF, JEANJACQUES MORCRETTE and FREDERIC CHEVALLIER European Centre for MediumRange Weather Forecasts, UK (Received 20 September 1; revised 29 January 2) SUMMARY A proper consideration of radiation interactions in linearized models is required for variational assimilation of properties. Therefore, both a linearized scheme and a linearized radiation scheme have been developed for the European Centre for MediumRange Weather Forecasts (ECMWF) data assimilation system. The tangentlinear and adjoint versions of the ECMWF shortwave radiation scheme are prepared without a priori modifications. The complexity of the radiation scheme for the longwave part of the spectrum makes accurate computations expensive. To reduce its computational cost, a combination of artificial neural networks and Jacobian matrices is defined for the linearized longwave radiation scheme. The linearized scheme is diagnostic and has been adapted for this study. The accuracy of the linearization of both radiation and diagnostic schemes is examined. The inclusion of a more sophisticated radiation scheme within the existing linearized parametrizations improves the accuracy of the tangentlinear approximation. However, the impact of the linearized diagnostic scheme is small, suggesting that the linearized model will require further developments of parametrization. The adjoint technique is used to investigate the sensitivity of the radiation schemes to changes in temperature, humidity and properties. This study shows which variables can be adjusted when certain observations of the surface and/or of the topofatmosphere radiation fluxes are used in data assimilation. It also indicates the vertical extent of the influence of such observations. KEYWORDS: Cloud radiation processes Linearization Sensitivity 1. INTRODUCTION A large part of forecasting deficiencies is connected with the imperfect assimilation of available observational data in the numerical prediction process. In recent years, fourdimensional variational (4DVar) data assimilation has become a powerful tool in transferring information from the irregularly distributed observations into initial conditions for a numerical forecast model. 4DVar minimizes the distance between a model trajectory and observations spread over a given time interval. The minimization algorithm used in 4DVar employs the adjoint equations of the model to compute the gradient of the cost function with respect to the model state at the beginning of the assimilation period. Such an assimilation system was first tested using adiabatic models. Rabier and Courtier (1992) and Rabier et al. (1996) have shown that an adiabatic model used in sensitivity studies was able to represent phenomena linked to baroclinic instability to a reasonable accuracy. However, physical processes play a significant role in various largescale and mesoscale phenomena. Neglecting physical processes in the data assimilation systems leads to the socalled spinup problem due to an imbalance between the model equations and the initial state. The misfit between model solution and data can remain large due to the imperfect adiabatic model. Many satellite observations, such as radiances, rainfall measurements and properties, cannot be directly assimilated into such models; hence it is important that physical processes should be taken into account in the assimilating model. Some studies have been undertaken to include physical parametrizations in the adjoint models (Zou et al. Corresponding author: European Centre for MediumRange Weather Forecasts, Shinfield Park, Reading, Berkshire RG2 9AX, UK. marta.janiskova@ecmwf.int c Royal Meteorological Society,

2 1506 M. JANISKOVÁ et al. 1993; Zupanski and Mesinger 1995; Tsuyuki 1996; Zou 1997; Errico and Reader 1999; Janisková et al. 1999; Mahfouf 1999) with encouraging results. 4DVar assimilation systems were implemented operationally in November 1997 at the European Centre for MediumRange Weather Forecasts (ECMWF) (Klinker et al. 0; Mahfouf and Rabier 0; Rabier et al. 0) and in June 0 at MétéoFrance. As the inclusion of physics in the adjoint model improves the performance of 4DVar, both operational assimilation systems use adjoint models with a set of simplified physical parametrizations developed by Mahfouf (1999) (for the ECMWF model) and Janisková et al. (1999) (for the MétéoFrance forecasting system). Even though the adjoint of various physical parametrization schemes is included in the ECMWF assimilating model, some of them are quite simplified. For radiation, a very simplified linearized longwave code is used, where no dependency of radiation on iness is taken into account and the shortwave radiation processes are not considered. However, both solar and thermal radiation fluxes are affected by condensed water (in the form of droplets and ice crystals) as well as by the watervapour absorption bands. Clouds play an important role in the water and energy budgets of the Earth s atmosphere and therefore the use of data in y conditions is an important issue for numerical weather prediction. Most operational data assimilation systems do not use observations on s available from the surface network and satellites. The main reason lies in the difficulty of accurately describing the processes in atmospheric models. However, recent improvements in the representation of s (such as the development of prognostic schemes) and flexible data analysis systems (such as threedimensional variational and 4DVar) make preliminary investigations towards the use of observations in y situations possible. The inclusion of observations in data assimilation should lead to an improved initial state of the model prognostic variables, which could in turn improve the quality of the forecasts including parameters such as 2metre temperature, iness and precipitation. The ECMWF operational 4DVar assimilation system can assimilate new types of observations when a proper observation operator exists. The observation operators providing the model counterpart of observations are the linearized versions of both a radiativetransfer model and a scheme, which need to be developed for use in variational data assimilation. Radiativetransfer models in clear sky do not describe strong nonlinear processes (even though some saturation effects can take place in watervapour absorption bands). Thus, linearized radiation codes have been used successfully for variational assimilation of clearsky radiances in numerical weatherprediction models (for example Eyre 1995). The linearization of physical parametrization schemes describing processes and radiation interactions is not straightforward since the on/off nature of s can make the tangentlinear approximation questionable. Zou and Navon (1996) developed the linearized version of a solar radiativetransfer code where the fractional amount was taken as a constant input parameter. The Jacobian approach was applied by Chou and Neelin (1996) to formulate a linear scheme for longwave radiative fluxes. For linearized fractions, they defined simple cover types which have a fixed vertical structure and which never overlap in the vertical. Janisková et al. (1999) developed a simplified linearized radiation scheme where the coefficients for shortwave radiation (representing the properties of absorption and multiple scattering) and for longwave radiation (a matrix of coefficients depending on emissivity and transmission) are stored from a onetimestep integration of the full nonlinear scheme and recomputed as a function of iness in the linearized physical parametrization. They reported results on

3 LINEARIZED RADIATION AND CLOUD SCHEMES 1507 the linearization of a simple diagnostic scheme used with the linearized radiation scheme in a global atmospheric model. Due to the presence of thresholds in the scheme, considerable noise in the solution was observed. Therefore, the perturbations in cover were neglected in that study. More encouraging results were obtained at a smaller scale (Verlinde and Cotton 1993; Park and Droegemeier 1997) where linearized versions of resolving models were developed with some success for radar retrieval applications. Chevallier et al. (1a) showed some capability of a fast infrared and microwave radiation model and its linearized version for onedimensional variational (1DVar) analyses of affected ATOVS observations. Once the linearized versions (tangentlinear and adjoint) of physical parametrization schemes are developed, they can be used for purposes other than just data assimilation. The model s adjoint is a powerful tool which enables the estimation of the sensitivity of a given output quantity to all the input variables of a parametrization scheme. Using the adjoint technique for the radiation scheme provides the sensitivity of the radiation fluxes to changes in temperature, humidity and properties. Compared with the standard approaches for evaluation of physical parametrization schemes (sensitivity of all the outputs to a given input quantity), the adjoint is a complementary approach for sensitivity studies. It can also give indications related to the importance and efficiency of particular types of observations for data assimilation applications. In this paper, the development of comprehensive linearized radiation and schemes is presented in the framework of the ECMWF model. In section 2, a brief description of the shortwave and longwave radiation schemes is given with emphasis on the incorporation of the effects of s on the radiation fluxes. The linearized diagnostic scheme is also described there. The tangentlinear approximation is evaluated for radiation processes in section 3. The adjoint technique is used to investigate the sensitivity of radiative fluxes to changes in temperature, humidity and properties. The corresponding methodology and results are described in section 4. Summary and discussions are presented in section DESCRIPTION OF THE LINEARIZED RADIATION AND CLOUD SCHEMES The linearized radiation code used in the current ECMWF operational 4DVar accounts only for the clearsky longwave radiation transfer via a constant emissivity formulation (Mahfouf 1999), for which the effective emissivities are stored from the full nonlinear radiation scheme. As first identified by Thuburn (1994) and explained also by Mahfouf (1999), this method can lead to an exponential growth of temperature increments when the vertical temperature gradient is negative, which is the case in the stratosphere. Therefore, the linearized longwave scheme is restricted to the troposphere only. In the direct model, due to computational costs, the full radiation calculations are presently done every hour during the first 12 hours of the forecasts, then every 3 hours. Moreover, they are carried out on a reduced horizontal grid (Morcrette 0). In order to assimilate observations, the radiativetransfer model must obviously include radiation interactions. This has led to the development of a linearized scheme and of a more sophisticated radiation scheme describing longwave and shortwave, clearsky and y radiation transfer. The new linearized radiation scheme must also be computationally efficient to be called at each time step and at the full spatial resolution for an improved description of the radiation interactions during the assimilation period. Advanced Tiros Operational Vertical Sounder.

4 1508 M. JANISKOVÁ et al. (a) The shortwave radiation scheme The shortwave radiation scheme, originally developed by Fouquart and Bonnel (1980), is used in the ECMWF model (Morcrette 1989, 1991). The photonpathdistribution method is used to separate the parametrization of the scattering processes from that of molecular absorption. Upward Fsw and downward Fsw fluxes at a given level j are obtained from the reflectance and transmittance of the atmospheric layers as: N Fsw (j) = F 0 T bot (k) (1) k=j Fsw (j) = F sw (j)r top(j 1). (2) Computations of the transmittance at the bottom of a layer T bot start at the top of the atmosphere and work downward. Those of the reflectance at the top of the same layer R top start at the surface and work upward. In the presence of in the layer, the final fluxes are computed as a weighted average of the fluxes in the clear sky and in the y fractions of the column as: R top = C R + (1 C )R clear (3) T bot = C T + (1 C )T clear. (4) In the previous equations, C is the fractional coverage of the layer within the y fraction of the column (depending on overlap assumptions). This nonlinear scheme is reasonably fast for application in 4DVar and has, therefore, been linearized without a priori modifications. (b) The longwave radiation scheme The longwave radiation scheme, operational in the ECMWF forecast model until June 0, is a bandemissivity type scheme (Morcrette 1989). The longwave spectrum from 0 to 2820 cm 1 is divided into six spectral regions. The transmission functions for water vapour and carbon dioxide over those spectral intervals are fitted using Padé approximations on narrowband transmissions obtained with statistical band models (Morcrette et al. 1986). Integration of the radiationtransfer equation over wave numbers within the particular spectral regions gives the upward and downward fluxes. The incorporation of the effects of s on the longwave fluxes follows the treatment discussed by Washington and Williamson (1977). The fluxes for the actual atmosphere are derived from a linear combination of the fluxes calculated for a clearsky atmosphere and those obtained assuming a unique overcast of emissivity unity. The complexity of the radiation scheme for the longwave part of the spectrum makes accurate computations expensive. In the variational assimilation framework, simplifications were made to reduce its computational cost. A combination of artificial neural networks and mean Jacobians was defined for the linearized longwave radiation. Such choice is based on the following findings: A neural network version (called NeuroFlux) of the ECMWF longwave radiativetransfer model is significantly faster (seven times) than the operational longwave radiation code with a comparable accuracy. A detailed description of NeuroFlux can be found in Chevallier et al. (1998, 0). Chevallier and Mahfouf (1) showed that the variability of the clearsky Jacobians does not play an essential role when computing total sky (i.e. including iness) fluxes and, therefore, precomputed Jacobians averaged globally can be used.

5 LINEARIZED RADIATION AND CLOUD SCHEMES 1509 A description of the linearized longwave radiation scheme using the two ingredients mentioned above follows. The longwave radiative fluxes depend upon temperature, water vapour, cover and liquid and icewater contents. The design of the scheme allows the separation of the contribution of temperature and water vapour from that of parameters. More precisely, the upward and downward longwave fluxes at a certain height z i are expressed as: F(z i ) = a k (z i )F k (z i ) (5) k where the coefficients a k are a function of the iness matrices CC i,k computed using some overlap assumption defined more precisely in section 2(e). Differentiating the above equation, a flux perturbation is computed as: { ak (z i ) df }{{} k (z i ) }{{} df(z i ) = k NL model Jacobian matrices + F k (z i ) }{{} da k (z i ) }{{} NeuroFlux TL model }. (6) In the proposed approach, the coefficients a k are computed using the nonlinear (NL) model (code computing optical properties). Due to the weak nonlinearities in the variations of the radiative fluxes F k with respect to temperature and water vapour, the tangentlinear approximation can be used to compute perturbations of radiation fluxes df k from precomputed mean Jacobian matrices. Perturbations of radiative fluxes with respect to parameters da k (z i ) are computed using a tangentlinear (TL) scheme. The trajectory of radiative fluxes required in the tangentlinear and adjoint computations are efficiently estimated from NeuroFlux in order to avoid significant extra memory storage or costly recomputations. (c) Cloud parametrization scheme The ECMWF diagnostic scheme (Slingo 1987), used before the implementation of the current operational prognostic scheme (Tiedtke 1993), was linearized to be used with the linearized radiation scheme. The diagnostic scheme allows for four types: convective and three types of layer s (high, middle and low level). The convective s (C conv ) are parametrized using the scaleaveraged precipitation rate (P ) from the model convection scheme. They can fill any number of model layers and their depth is determined by the convection scheme. The stratiform s (C strat ) are determined from a function of the layer relative humidity (RH e ) after adjustment for the presence of convective s, RH e = RH C conv, as: { ( )} RHe b 2 C strat = f(rh e ) = max 1 b, 0 (7) where b is equal to 0.8 for high and middlelevel s and to 0.7 for lowlevel s. The middlelevel s and lowlevel s associated with the extratropical fronts are then scaled with vertical velocity in the case of upward motion. There are no such s in subsidence areas. Liquid water l lwc and ice water l iwc are proportional to the specific humidity at saturation q sat. They are defined as l lwc = αl lc (8) l iwc = (1 α)l lc (9)

6 1510 M. JANISKOVÁ et al. where α is the fraction of condensate held as liquid water and l lc = 0.05C q sat with C being cover. The linearized diagnostic scheme was further simplified by accounting for part of the dependencies only. The current linearized convection scheme does not provide perturbations of precipitation. This leads to a zero perturbation of convective iness since it is determined from the convective precipitation rate. A modification of the linearized scheme was also necessary in order to avoid the threshold problem in the parametrization of the layer s computed as a function of relative humidity RH e (Eq. (7)). Although this threshold does not lead to any occurrence of spuriously growing unstable perturbations (Janisková et al. 0), it can degrade results, and it was therefore decided to set the derivative of the function of relative humidity to zero. This also means that the highlevel cover is then constant for a given time step. Middlelevel and lowlevel s are only modified by vertical velocity. Liquid water and ice water can, however, be modified by temperature perturbations. (d) Cloud optical properties Cloud radiation interactions depend not only on the fraction or volume, but also on optical properties. In the case of shortwave radiation, the radiative properties depend on three different parameters: the optical thickness δ c, the asymmetry factor g c and the single scattering albedo ω c. These parameters are derived from Fouquart (1987) for water s, and Ebert and Curry (1992) for ice s. They are functions of condensate and a specified effective radius. Cloud longwave optical properties are represented by the emissivity ε cld related to the condensedwater amount, and by the condensedwater mass absorption coefficient k abs obtained following Smith and Shi (1992) for water s and Ebert and Curry (1992) for ice s. (e) Cloud overlap assumptions Cloud overlap assumptions must be made in atmospheric models in order to organize the distribution used for radiation and precipitation/evaporation computations. A overlap assumption of some sort is necessary to account for the fact that s often do not fill the whole grid box. Most atmospheric models use random (RAN), maximumrandom (MRN) or maximum (MAX) overlap assumptions in the radiation scheme. The definition of overlap assumptions is given from the point of view of iness CC i,j encountered between any levels i and j in the atmosphere. Let C k be the fraction of the layer k located between levels k and k + 1. The maximum overlap assumption gives: The random overlap assumption gives: CC i,j = max(c i,c i+1,...,c j 1 ). (10) j 1 CC i,j = 1 (1 C k ). (11) k=i The maximumrandom overlap assumption gives: CC i,j = 1 (1 C i ) j 1 k=i+1 ( 1 max(ck,c k 1 ) 1 C k 1 ). (12)

7 LINEARIZED RADIATION AND CLOUD SCHEMES 1511 The maximumrandom overlap assumption (originally introduced in Geleyn and Hollingsworth 1979) is used operationally in the ECMWF model (Morcrette and Jakob 0). Adjacent layers containing are combined by using maximum overlap to form a contiguous and discrete layers separated by clear sky are combined randomly. 3. VALIDATION OF THE LINEARIZED MODEL INCLUDING CLOUD RADIATION PROCESSES The classical verification of the correctness of the tangentlinear model is done through the Taylor formula: M(x + λδx) M(x) lim = 1 (13) λ 0 M(λδx) where M is a discretized primitiveequation model of the atmosphere, M is the tangentlinear model of M, and x represents a model state at a certain time. In the infinitesimal limit the test shows whether the linearization of the nonlinear code was performed correctly. Such verification was done straightforwardly for the shortwave radiation scheme. The linearized scheme can only be tested in the original form without applying any simplifications. The test cannot be used for the longwave radiation scheme since there is no directly corresponding nonlinear version of the scheme. Therefore, a different type of validation must be made (see section 3(a)). For the verification of the adjoint, the identity between inner products for given x and y vectors is tested as x, y M x y = x M T y. (14) In the above equation, the linear operator M T is the adjoint of the linear operator M. In practice however, the utility of tangent linear and adjoint models is determined by how well they describe the behaviour of finitesized perturbations corresponding to ones in the nonlinear model. Therefore, the accuracy of the linearization of both radiation and diagnostic schemes was studied with respect to pairs of nonlinear results. Tangentlinear integrations propagating in time analysis increments were performed with the trajectory taken from a 6hour model forecast (background field). (a) Experimental framework To evaluate the impact of including radiation and schemes in the ECMWF tangentlinear model, sets of experiments with a time evolution of analysis increments were performed for different dates (15 December 1998 at 12 UTC, 15 March 1999 at 12 UTC, 15 June 1999 at 12 UTC and 20 September 1999 at 12 UTC) with integrations up to 24 hours using a T63L31 version ( km horizontal resolution and 31 vertical levels) of the model. The radiation and schemes were used at each time step and at every grid point. In these experiments, the difference between two nonlinear integrations (one starting from a background field x b and the other one starting from an analysis x a ) is computed using the full nonlinear physics (Gregory et al. 0). This difference is then used as a reference for the tangentlinear integrations which propagate in time the analysis increments δx = x a x b with the trajectory taken from the background field (Janisková et al. 1999; Mahfouf 1999). The experiments were performed for the tangentlinear model:

8 1512 M. JANISKOVÁ et al. without radiation, with the current operational radiation scheme (hereafter referred as OPER radiation), with the new shortwave and longwave radiation schemes described in the previous section (new radiation) run without/with the scheme (clear sky/iness). This set of experiments enables the evaluation of improvements coming from the inclusion of a radiation scheme in the existing set of linearized physics, whether the new radiation scheme is better than the OPER scheme, and what improvement is brought by the scheme used with the new radiation scheme. For a quantitative evaluation of the impact of the and radiation schemes, their relative importance is evaluated using mean absolute errors between tangentlinear and nonlinear integrations as ε = {M(x a ) M(x b )} M(x a x b ) (15) where M is the forecast model starting from different initial conditions: x a from the analysis, x b from the background, M is the tangentlinear model starting from the initial conditions (x a x b ), (... ) represents the mean over a particular domain. As a reference for the comparisons, an absolute mean error for the tangentlinear model without any radiation or with the OPER radiation ε ref is taken. ε i represents the absolute mean error for the tangentlinear model including radiation and schemes with respect to pairs of nonlinear integrations with full physics. Then an improvement coming from including more physics in the tangentlinear model should correspond to ε i being less than ε ref. The impact of including the new radiation and iness schemes was evaluated by studying the time evolution of analysis increments at different model levels and using the zonalmean average of the mean absolute errors. Global zonalmean errors as well as zonalmean errors for 20 N 90 N (North20), 20 S 90 S (South20) and 20 N 20 S (Tropics) were evaluated separately. The radiation directly influences the temperature field and as a consequence of these temperature changes the impact of the radiation on other quantities can be observed. Hence results are mostly presented for temperature perturbations although, during the validation, the results from other fields were also compared. (b) Results of the validation experiments The experiments were performed for different dates (as listed in section 3(a)). Results from only two of the situations are presented here. Figure 1 shows comparisons of the TL integrations where the trajectory is computed with the diagnostic scheme (as was used operationally in the ECMWF 4DVar). The TL models using the physical package described in Mahfouf (1999) with either the OPER radiation scheme or with the new radiation and schemes are compared with the TL model without any radiation (reference TL model). Negative values (positive values) are associated with an improvement (deterioration) of the TL model with respect to the reference one since they correspond to a reduction (increase) of the error ε referred to in Eq. (15). It can be seen that the OPER radiation scheme (Fig. 1(a)) only gives a

9 LINEARIZED RADIATION AND CLOUD SCHEMES 1513 a) model levels model levels b) L30 ~ 980 hpa L25 ~ 810 hpa L20 ~ 580 hpa L15 ~ 375 hpa L10 ~ hpa L5 ~ 90 hpa Figure 1. Influence of the different tangentlinear (TL) radiation schemes on the evolution of temperature increments with the trajectory computed using the diagnostic scheme. Results are presented as the error differences (in terms of fit to the nonlinear model with full physics) between the TL model including operational linearized physics (OPER radiation) and the TL model without any radiation (a). (b) Presents the same, but for the TL model containing the new radiation scheme with the simplified diagnostic scheme. (24hour forecast for the situation of 15 March 1999, 12 UTC, units: K). slight global improvement of 0.31%. Negative impacts appear in the upper troposphere. This problem comes from the use of the constantemissivity approach, which is not appropriate in the stratosphere. The new radiation and schemes (Fig. 1(b)) give a global improvement of 4.71%. There are only a few small negative impacts. The vertical profiles for the globalmean values of ε i ε ref and the globalmean absolute errors ε ref are presented in Fig. 2 for a 24hour integration. The results for temperature, specific humidity and u wind component are on the left, middle and right

10 1514 M. JANISKOVÁ et al. a Mean of error differences for temperature b Mean of error differences for specific humidity c Mean of error differences for u wind model levels T q u d Mean absolute error for temperature e Mean absolute error for specific humidity f Mean absolute error for u wind 5 T 5 q 5 u model levels Figure 2. Vertical profiles of the globalmean values of error differences (top panels) and the globalmean absolute errors (bottom panels) for temperature (units: K), specific humidity (units: g kg 1 ) and u wind component (units: m s 1 ) from a 24hour integration. The results are presented for the currently used radiation scheme (dashed line) as well as for the new radiation scheme with iness (solid line) and for clear sky (dotdashed line). Initial date of the integration: 15 March 1999, 12 UTC. Model levels are as shown in Fig. 1. panels, respectively. The comparisons are displayed for the OPER (simplified) longwave radiation scheme as well as for the new radiation (long wave and short wave) with the simplified scheme and for clear sky. By comparing the impact of the schemes on temperature, Fig. 2(a) shows that including the new radiation in the linearized physics improves quantitatively the fit of the tangentlinear model to the nonlinear one through the whole vertical profile. Adding the scheme to the radiation has a very small impact on temperature and is mainly negative except in the Tropics (Janisková et al. 0). The OPER radiation scheme improves the results only in the lower troposphere and degrades them for higher levels. The results for specific humidity (Fig. 2(b)) show that there is an improvement close to the surface and in the lower troposphere when using the new radiation scheme. The impact is slightly negative elsewhere. The OPER radiation scheme has quite a small influence on specific humidity. The results for the u wind component are displayed in Fig. 2(c). When using the new radiation scheme, it is noteworthy that there is an improvement in the wind considering it is not an input to either the longwave or shortwave radiation schemes. This is probably a consequence of the interaction between physics and dynamics, whereby improved temperature fields lead to an improvement of the pressure field, which in turn influences the zonal wind. The improvement in wind is observed nearly throughout the whole vertical profile for all domains. The impact of the OPER radiation scheme on the wind is negligible and it is mainly negative in the upper part of the atmosphere. Even though the impact of using iness together with the radiation is small, it is generally positive when averaged over the globe.

11 LINEARIZED RADIATION AND CLOUD SCHEMES 1515 a FD b TL without radiation c TL with OPER radiation d TL with old lw+new sw e TL with new lw+new sw+ f TL with new lw+new sw+clear sky Figure 3. Sixhour evolution of temperature increments at level 31 produced from the finite differences (FD) between two nonlinear forecasts (a) and from the different tangentlinear (TL) approximations (b) (f) as displayed on the figures (OPER = operational, lw = long wave and sw = short wave). Initial date of integration: 15 December 1998, 12 UTC. First contour line 0.1 K and then contour interval 0.25 K. See text for further explanation. When comparing error differences for the different experiments with the globalmean absolute errors for the tangentlinear model without any radiation (ε ref ), the error reduction coming from the new radiation scheme with respect to the OPER scheme represents up to 10% of the perturbation value. Locally it can reach several kelvins for some specific situations. Such a case is presented in Fig. 3. The situation of 15 December 1998, 18 UTC over the area of Turkey is characterized by low cover up to around 60% and around 1 mm of precipitation during 6 hours in the area of interest (not shown). Figure 3 shows the 6hour evolution of temperature increments at level 31 (the model level closest to the surface in our experiments) produced from the differences between two nonlinear forecasts (Fig. 3(a)) and from different tangentlinear integrations (Figs. 3(b) (f)). Figure 3(b) displays the temperature perturbation using the TL model without any radiation. There is a strong positive temperature increment of 2.89 K over Turkey, which only reaches 0.5 K in the case of the finite differences. Including the OPER longwave radiation scheme (Fig. 3(c)) makes the situation even worse since it increases the perturbation up to 3.28 K. For this situation, large temperature increments are a consequence of evaporation processes which increase the humidity of the atmosphere. The linearized models without radiation or with the OPER radiation scheme are not able to properly balance such positive feedbacks. When

12 1516 M. JANISKOVÁ et al. the shortwave radiation scheme (Fig. 3(d)) is included on top of the OPER longwave scheme, the perturbation is decreased to 0.93 K. Replacing the OPER longwave radiation scheme by the more sophisticated one (Fig. 3(e)) brings a further reduction down to 0.46 K. Figure 3(f) displays the results obtained with the new radiation scheme in clearsky conditions. The temperature increments in this case are close to those produced when including s in the radiation scheme. This confirms the generally small impact of the linearized scheme. It is important to mention that the new radiation scheme in clearsky conditions can modify temperature tendencies from humidity perturbations (since they are used in the computation of absorber amounts). The above situation is a daytime period when both radiation schemes (short wave and long wave) play an important role during the whole (6 hour) integration. Even though the impact of shortwave radiation is larger in the described situation, both schemes are required in the tangentlinear model in order to get temperature increments which are in better agreement with the finite differences. This emphasizes the importance of using a more sophisticated radiation scheme, which can lead to significant modifications of increments in certain situations. Detailed analysis of the results (Janisková et al. 0) has shown that the inclusion of more sophisticated radiation schemes gives a general improvement at the lowest model levels and in the Tropics. The results are slightly better for the clearsky radiation than for the radiation with s in South20 and North20. In the Tropics, the impact of iness is always positive and larger than in the extratropics, suggesting that a proper balance between shortwave and longwave radiation and moist processes is crucial for these regions. Generally, the impact of using iness with the radiation is not as large as might be expected and not always positive. The negative impact could be a consequence of introducing nonlinearities in the processes that cannot be described by the linearized physics. 4. SENSITIVITY OF THE RADIATION SCHEME TO INPUT VARIABLES Although the adjoint version of the radiation scheme was developed for data assimilation, it can also be used for sensitivity studies. Adjoint models are powerful tools as they allow the computation of the gradient of one output parameter of a numerical model with respect to all input parameters (Le Dimet and Talagrand 1986; Errico 1997). Therefore, they can be applied to the study of sensitivity problems. When such a technique is applied to a particular physical parametrization scheme, it can provide information on the meteorological variables to which the parametrization scheme is the most sensitive. It is an effective way to evaluate Jacobians of the scheme. For instance, Zou and Navon (1996) applied the adjoint of the solar radiative transfer scheme to a sensitivity study of the downward solar radiation flux at the surface with respect to watervapour amount at various heights. Similarly, the adjoint sensitivity of the Earth s radiation budget to cover, water vapour, atmospheric temperature and surface temperature in the National Centers for Environmental Prediction (NCEP) model were investigated by Li and Navon (1998). Explicit computation of Jacobian elements using the perturbation method was used by Chevallier and Mahfouf (1) to evaluate the Jacobians of infrared radiation models for variational data assimilation. From a dataassimilation point of view, Jacobians can also give some indications of the importance and efficiency of particular types of observations on the initial conditions of a model.

13 LINEARIZED RADIATION AND CLOUD SCHEMES 1517 (a) Methodology The formulation of a sensitivity problem for the radiation scheme can be explained as follows. Using the parametrization schemes for the shortwave and longwave radiation, let y denote the radiation fluxes (output vector). It can be expressed as y = F(x) (16) where x is the vector representing an atmospheric state described by temperature T, specific humidity q, surface pressure p s and characteristics ( fraction, liquidwater and icewater contents). F(x) is the direct nonlinear operator including the shortwave and longwave radiation schemes together with the optical properties and overlap assumptions. A small perturbation δy can be estimated to first order by the tangentlinear equation of the radiation scheme as: δy = F x.δx (17) where F is the tangentlinear operator of F. The adjoint F T of the linear operator F (defined in Eq. (14)) provides the gradient of an objective function J with respect to x (input variables) given the gradient of J with respect to y (output variables): J x = FT x J or x J = F T x y yj. (18) In these experiments, a gradient with respect to y of unity size (i.e. perturbation of some of the radiation fluxes which are output variables of the radiation scheme) is provided to the adjoint of radiation schemes in order to get the sensitivity of this scheme with respect to its input variables, i.e. temperature, specific humidity and characteristics. This leads to x J = F (19) x where F/ x is the corresponding Jacobian matrix. (b) Experimental framework The method described above was used in order to examine the sensitivity of the radiation schemes described in section 2 for selected columns of the atmosphere. In our experiments, the perturbations of the radiation fluxes with unity size (±1 Wm 2 ) were used as input for the adjoint of the radiation schemes. The sign was chosen so that an increase in iness (or opacity) would lead to a positive sensitivity. The sensitivity of the downward longwave (LWDs) and shortwave (SWDs) radiation fluxes at the surface, as well as the upward longwave (LWUt) and shortwave (SWUt) radiation fluxes at the top of the atmosphere (TOA) was studied with respect to the input variables. First the sensitivity of the radiation fluxes for clearsky conditions was investigated. Then different covers were used in the experiments for y conditions. As mentioned in section 2(e), overlap assumptions must be made in atmospheric models in order to organize the vertical distribution used for radiation computations. To avoid the presence of thresholds in overlap assumptions, most experiments were done using the random overlap assumption (Eq. (11)). However, the impact of using different overlap assumptions was also studied.

14 1518 M. JANISKOVÁ et al. The sensitivity of the different radiation fluxes (F Ri ) was evaluated with respect to the following input variables: temperature F Ri / T, specific humidity F Ri / q, cover F Ri / a, liquidwater content F Ri / q lw and icewater content F Ri / q iw. Since the input variables have different orders of magnitude, any particular F Ri / x i has to be normalized to get a relative evaluation of the sensitivity. This was done by multiplying F Ri / T and F Ri / q by typical sizes of increments: the background errors σ bt and σ bq, respectively. The background errors were taken from the operational ECMWF analysis system. The standard deviations of temperature σ bt over the vertical are about1kuptoaround hpa, then they grow in the stratosphere up to 4.5 K. The standard deviation for specific humidity σ bq was empirically specified by Rabier et al. (1998). The vertical distribution has a maximum (about 1.25 g kg 1 ) around 850 hpa, an exponential decrease above and lower values (about 1 g kg 1 )in the boundary layer. Since background errors for the variables are not known, an empirical estimate was used instead: σ ba = 0.1 for cover, σ bqlw = σ bqiw = 0.05σ bq for water. (20) The atmospheric model data come from the ECMWF operational global spectral model run at a T L 319L60 resolution (about 60 km horizontal resolution and 60 vertical levels). The different vertical profiles are representative for a grid box around the ARM South Great Plains site in Oklahoma (36.6 N, 97.5 W). The data were produced by shortrange (6hour) forecasts over the period between 29 April and 7 May 0. The experiments presented here use the forecast profiles valid for 4 May 0, 00 UTC (clearsky conditions) and for 29 April 0, 18 UTC and 2 May 0, 00 UTC (y conditions). The surface albedo is approximately 0.19 for all situations and the skin temperature varies slightly (297.9 K for 29 April, K for 2 May and K for 4 May). (c) Numerical results Some results from the evaluation of the sensitivity of radiation fluxes to their input variables are presented in Figs The Jacobians for clearsky conditions are presented in Fig. 4 for LWDs (a), SWDs (b), LWUt (c) and SWUt (d). The results of experiments for y conditions, when using the random overlap assumption, are displayed for the different covers and solar zenith angles (θ), µ = cos θ (µ = 0.25 in Fig. 5 and 0.93 in Fig. 6). In these figures, (a) shows the fraction (grey filled area) together with the liquidwater content (grey solid line with circles) and icewater content (dotdashed line). The sensitivity of the different radiation fluxes is displayed in (b) (e) with respect to temperature (solid line), specific humidity (dashed line), cover (dotted line), liquid and icewater contents (lines as in (a)). The impact of using different overlap assumptions is shown in Fig. 7 for LWUt. The sensitivity in clearsky conditions is displayed in Fig. 4. When looking at the sensitivity of LWDs (a) with respect to the input variables, one can see that the most important role is played by temperature in the lowest levels. There is also a significant sensitivity to specific humidity. To increase LWDs requires a less transparent atmosphere in order to increase watervapour emission. In the case of LWUt (c), the sensitivity to temperature and specific humidity is smaller than for LWDs and it is more regularly distributed through the vertical. In principle, the sign of the LWUt sensitivity Atmospheric Radiation Measurement.

15 LINEARIZED RADIATION AND CLOUD SCHEMES 1519 a 0 b 0 (@F=@q) ff q ffl (@F=@q) ff q ffl (@F=@q iw) 0:05ff q (@F=@q iw) 0:05ff q LWDs SWDs c 0 d 0 (@F=@q) ff q ffl (@F=@q) ff q ffl (@F=@q iw) 0:05ff q (@F=@q iw) 0:05ff q LWUt SWUt Figure 4. Sensitivity of the radiation fluxes with respect to temperature T, specific humidity q, cover a, liquidwater content q lw and icewater content q iw. The computation is done for a clear sky and for solar zenith angle cos θ = The results are presented for the downward longwave (LWDs, (a)) and shortwave (SWDs, (b)) radiation fluxes at the surface, as well as for the upward longwave (LWUt, (c)) and shortwave (SWUt, (d)) radiation fluxes at the top of atmosphere. The perturbation of radiation flux ( F) is 1 W m 2 for LWDs and SWUt, and 1 Wm 2 for LWUt and SWDs. Date: 4 May 0, 00 UTC. is opposite to that of LWDs, since a more transparent atmosphere is needed to increase the surface contribution and consequently LWUt. The shortwave radiation ((b) and (d)) shows sensitivity only to humidity in the lower troposphere. The upward shortwave radiation at the top of the atmosphere has a negative sensitivity to specific humidity. To get more SWUt in clearsky conditions, the scheme tends to reduce the atmospheric absorption by water vapour so that more downward radiation is reflected from the surface. When not taking into account normalization by the background errors (which is important for an estimation of the relative importance of sensitivities to the different variables), sensitivity of LWUt to specific humidity and temperature for a clear sky compares well with the results obtained by Li and Navon (1998) in their Figs Their results for the sensitivity of shortwave radiation are difficult to compare with. Figures 5 and 6 display the results for a y atmosphere. Though the cover is different in the two situations, the quantitative difference in the sensitivity of the shortwave radiation is mainly linked to the different solar zenith angle θ. At smaller zenith angles (cos θ = µ = 0.25 in Fig. 5), the shortwave and the longwave radiation fluxes

16 1520 M. JANISKOVÁ et al. a lwc iwc b c (@F=@q) ff q ffl (@F=@q iw) 0:05ff q (@F=@q) ff q ffl (@F=@q iw) 0:05ff q LWDs SWDs d e (@F=@q) ff q ffl (@F=@q iw) 0:05ff q (@F=@q) ff q ffl (@F=@q iw) 0:05ff q LWUt SWUt Figure 5. Same as Fig. 4, but for the type displayed in (a) (see text) and for solar zenith angle cos θ = The results are presented for the downward longwave (LWDs, (b)) and shortwave (SWDs, (c)) radiation fluxes at the surface, as well as for the upward longwave (LWUt, (d)) and shortwave (SWUt, (e)) radiation fluxes at the top of the atmosphere. Date: 2 May 0, 00 UTC.

17 LINEARIZED RADIATION AND CLOUD SCHEMES 1521 a lwc iwc b c (@F=@q) ff q (@F=@q) ff q ffl (@F=@q iw) 0:05ff q ffl (@F=@q iw) 0:05ff q LWDs SWDs d e (@F=@q) ff q ffl (@F=@q iw) 0:05ff q (@F=@q) ff q ffl (@F=@q iw) 0:05ff q LWUt SWUt Figure 6. Same as Fig. 5, but a different cover (a) and for solar zenith angle cos θ = Date: 29 April 0, 18 UTC. have similar magnitude. The shortwave radiation is significantly larger than the longwave at local noon (µ = 0.93 in Fig. 6). This variation in µ is then responsible for large values of the sensitivities for SWDs and SWUt. For the sensitivity of LWDs (b), the most important role is played by the lowlevel temperature (around 850 hpa) and by the fraction. The sensitivity to the fraction depends on the actual cover. When there is no in the lowest levels (Fig. 6(b)), the fraction sensitivity

18 1522 M. JANISKOVÁ et al. starts from the first level below the s and has a maximum at the level of the lowest s. However, when the base is very low and the cover is maximum at around 850 hpa, the sensitive area extends from the surface up to the level with the maximum fraction (Fig. 5(b)). It has a similar vertical extent to the temperature contribution in this situation. The sensitivity to the specific humidity is quite small and does not differ significantly from case to case. The SWDs (c) is mostly sensitive to parameters (ice/liquid water and cover). The sensitivity to specific humidity and that to the parameters increases with higher solar elevation. For the upward longwave radiation flux at the top of atmosphere (d), the sensitivity to input variables is maximum in the highest layers of the s from the top of the atmosphere and it diminishes below. Mostly, there is a sensitivity to the temperature and the parameters. The y LWUt in Fig. 5(d) is only sensitive to the fraction at the top levels of existing s. It is not sensitive to liquid water since its amount is high enough that black s have already been generated. A more important role is played by liquid water than by fraction in the situation presented in Fig. 6(d). For lowlevel s, the contribution of humidity above the s is also significant (see Fig. 5(d)). When studying the behaviour of SWUt (e), it is noteworthy that this flux represents the shortwave radiation flux reflected either from the s or from the surface. It means that it also contains information on the downward shortwave radiation and the sensitivity is then more complex. The sensitivity of SWUt to the variables is then similar to that of SWDs. In the case of the downward shortwave radiation, the scheme tends to create more s to decrease this radiation flux. It does likewise for the upward shortwave radiation. In order to reflect more radiation, it creates more and denser s. At the same time, an increase in SWUt is also achieved through a decrease in specific humidity. This comes from the shortwave radiation going downward before being reflected. To get more SWUt, the scheme tends to reduce the atmospheric absorption by water vapour in a similar way to that which was seen for clearsky conditions. The upward component of the shortwave radiation is responsible for the tendency to create more s for reflection. The vertical structure of the Jacobians also depends on the overlap assumption. The results are shown only for the upward longwave radiation at the top of the atmosphere (Fig. 7) using maximumrandom (b), maximum (c) and random (d) overlap assumptions. The sensitivity to temperature and humidity is obviously not significantly modified by the different overlap assumptions contrary to the sensitivity to variables. Except that the sensitivity is maximum for the highest s in any overlap assumption, the results are otherwise quite different. When using the MRN overlap assumption, LWUt tends to increase with iness where the fraction is high, and to decrease with iness at intermediate levels. It means that there is a tendency to make s more randomly distributed (to increase the overall cover). The scheme with MAX overlap is insensitive at intermediate levels and it displays a higher sensitivity to lower s than MRN. The sensitivity given by the radiation scheme with the RAN overlap assumption is positive only (i.e. leading to increasing variables) through the toplevel s. The question is then, which assumption is the most appropriate one? From the linearization point of view, one would prefer to use RAN which gives a smoother sensitivity response due to the absence of thresholds in the formulation. But MRN is used in the operational forecast model because it is thought to be the most realistic (Tian and Curry 1989). This sensitivity study confirms results obtained by Morcrette and Jakob (0) showing that the radiation fields are sensitive to the overlap assumptions and that there is a need for more information about the actual vertical distribution of s.