Efficient moist physics schemes for data assimilation. I: Large-scale clouds and condensation

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1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Published online 25 March 29 in Wiley InterScience Efficient moist physics schemes for data assimilation. I: Large-scale clouds and condensation O. Stiller* and S. P. Ballard Met Office, Reading, UK ABSTRACT: A cost-effective, easily maintained linear cloud scheme is presented, which introduces the major effects of condensation and latent heat release and precipitation into the linear model of a four-dimensional variational assimilation 4D-Var) system. The condensation scheme is derived within the general framework of statistical cloud schemes and the approximations used make it applicable to a large variety of nonlinear schemes. Forecast trials indicated that the introduction of this scheme into the Met Office numerical weather prediction NWP) model led to clear improvements of the forecast skill. Linearization test results are presented which show that the new scheme improves the match between linear and nonlinear models. An example of particularly large local improvements is found near the core of a low-pressure system over the north Atlantic. Improvements are generally larger in the extratropics than in the tropics, where convection plays a dominant role in the nonlinear model while it is not represented in the linear model). Experimental linearization tests where the convection scheme of the nonlinear model has been switched off) show a particularly strong beneficial impact of the new scheme and enable the identification of some convection-related compensating errors, which partially mask some of the new scheme s benefits when compared with a simpler linear scheme. The precipitation increments from the linear scheme are found to be broader and show less detail but capture many features of the corresponding increments obtained from the nonlinear model. A method of tuning and post-processing the precipitation strength is presented and discussed. of HMSO. Published by John Wiley & Sons, Ltd. KEY WORDS 4D-Var; latent heat; precipitation; perturbation forecast Received 23 January 28; Revised 23 January 29; Accepted 29 January Introduction A large part of the improvement in numerical weather prediction NWP) skill over the last decade was a result of advances in data assimilation, which is a very active field of research. An important role is played by fourdimensional variational assimilation 4D-Var), which has been successfully adopted by many of the major forecast centres see Rabier, 25 for an overview). This method minimizes a cost function to improve the fit of the initial conditions with respect to the observations made during a specified time interval the assimilation time window). For this it needs an adjoint NWP model which is the transpose of a linearized version of the nonlinear NWP model used by the respective forecast centre. Developing such a linear forecast model is a central task when establishing a 4D-Var system, the quality of which largely depends on how well this linear model matches the full nonlinear NWP model Rabier et al., 1998). Indeed, this linear model is particularly important for incremental 4D-Var, the method used by most operational centres. In incremental 4D-Var the linear model is needed not only to calculate the adjoint Correspondence to: Olaf Stiller, ECMWF, Shinfield Park, Reading RG2 9AX, UK olaf.stiller@ecmwf.int gradients but also for calculating the forward evolution of model-state perturbations over the assimilation time window Courtier et al., 1994). Developing a good linear model is, however, far from trivial. Using, for example, the exact tangentlinear of the full nonlinear NWP model is generally not a feasible option. Even though this method has been successfully employed for some research models Errico and Raeder, 1999), computational costs are usually far too high for operational applications. Also, even if it was affordable, most operational NWP models contain very strong nonlinearities and even thresholds), which makes the range of applicability of their direct linear approximation far too small. It has to be kept in mind that the linear model and its adjoint are used to deal with finite-size perturbations which have the magnitude of the analysis increments the increments used to improve the initial conditions for the next forecast run). To underline this latter point, the Met Office uses the term perturbation forecast PF) model Lorenc, 1997; Lawless et al., 23) for the linear model used in its 4D-Var system for a further discussion and interpretation of the employed 4D- Var method see Lorenc 23) and Lorenc and Payne 27)). Since the largest nonlinearities are generally within the physical parametrizations of the NWP schemes, a of HMSO. Published by John Wiley & Sons, Ltd.

2 78 O. STILLER AND S. P. BALLARD common development route is to start with an almost adiabatic model including only surface friction) where moist physical parametrizations are left to be developed at a later stage. Indeed, when 4D-Var became operational in the Met Office in October 24 the PF model was largely adiabatic Rawlins et al., 27). Even in this crude approximation 4D-Var was already substantially better than the 3D-Var method used before Lorenc and Rawlins, 25; 3D-Var is identical to 4D-Var apart from the fact that it neglects the evolution of the model state over the assimilation time window). Despite the benefit of the original almost) adiabatic PF model, it was recognized that inclusion of physical processes such as latent heat release should be beneficial for the large-scale dynamics and, also, to allow assimilation of cloud- and precipitation-affected data. For this purpose two linear moist parametrizations, representing large-scale latent heat release and deep convection, respectively, have been developed. Their implementation into the operational suite led to substantial improvements of the Met Office s forecast skills. While a companion article Stiller, 29, referred to hereafter as Part II) deals with the linear convection scheme, the current article describes the large-scale cloud scheme that became operational at the Met Office in February 25. Acceptance trial runs prior to its operational implementation showed an increase of the NWP index versus observations the Met Office s corporate skill measure, see Rawlins et al., 27) of.76% with a standard error estimate of.24%. The development of physical parametrizations for a linear 4D-Var model are often preceded by the construction of additional nonlinear parametrizations, which are then linearized in a later step Janisková et al., 1999; Lopez, 22; Tompkins and Janisková, 24). These additionally developed nonlinear parametrizations are generally smoother and numerically cheaper for example Janisková et al. 1999), who developed a whole set of simplified nonlinear parametrizations aimed at 4D-Var in the Météo-France global ARPEGE model). To be useful, they must, however, be sufficiently close to their operational counterpart and some efforts are usually made to demonstrate this. The moist parametrization development for 4D-Var at the Met Office did not involve such an intermediate nonlinear parametrization. Instead the linear parametrizations were constructed to mimic directly the full operational scheme. The strategy was to concentrate on some major aspects and sensitivities and to test to what extent they can be reproduced by a linear scheme. This approach requires a good analysis of the nonlinear parametrizations and probably a little more testing of the linear scheme, since all approximations are made directly to the linear scheme, which has no direct nonlinear counterpart. As discussed in Part II, representing only selected terms from a linear expansion can lead to large errors when represented, and neglected terms have the opposite sign similar errors, however, also occur when all linear terms are included but the validity of the linear assumption is different for different terms). The approach avoids, however, the need for testing another nonlinear scheme. It was chosen as it appears more flexible, which was particularly appealing regarding the strongly nonlinear behaviour that moist parametrizations often exhibit. As remarked above, computational efficiency is a major constraint for any changes made to the linear model. In a 4D-Var assimilation the linear model and its adjoint are generally run more than 5 times each at some centres substantially more), which is only possible if the linear model is orders of magnitude cheaper than its nonlinear counterpart Courtier et al., 1994). Most linear cloud parametrizations are therefore much simpler than the corresponding nonlinear scheme. Tompkins and Janisková 24), for example, developed a quite novel diagnostic cloud scheme that was substantially cheaper than the corresponding prognostic operational scheme Tiedtke, 1993) but which apparently captured some of the prognostic scheme s behaviour without the cost of additional prognostic variables in the linear model). Being computationally cheaper but staying close to the corresponding nonlinear parametrization is generally not a trivial task, and one way by which the linear cloud scheme described below achieves this is by using the nonlinear scheme s cloud fraction dumped from the nonlinear model run) as a key input field. A similar strategy is used for the convection scheme described in Part II. Another issue for linear parametrizations is their maintainability in a constantly developing environment. In the Met Office, as at most NWP centres, the nonlinear model undergoes constant changes and improvements. Optimizing the correspondence between the linear and the nonlinear model is therefore an ongoing task that follows a moving target and may require frequent adjustments, since the portability of linear schemes into systems with different nonlinear parametrizations is generally non-trivial see, for example, the adjustments required for convection in Part II or the large errors reported by Deblonde et al. 27), who used the cloud scheme from Tompkins and Janisková 24) within the Canadian NWP system). The portability is facilitated if the linear scheme is based on principles which are valid for a larger class of nonlinear schemes. This issue was particularly important for the development of the linear cloud scheme described in this article, as it took place at a time when serious efforts were made to convert the currently operational diagnostic cloud scheme Wilson and Ballard 1999), based on Smith 199)) into a more flexible and substantially more complex prognostic scheme this work is still ongoing). [Note that the current scheme Wilson and Ballard, 1999) actually contains a prognostic ice/snow variable. Its liquid cloud part is, however, based on the diagnostic scheme from Smith 199).] This, together with the need for computational efficiency, led to the choice of a quite simple scheme that focuses on the main processes and is a good approximation to a large variety of nonlinear) cloud schemes.

3 EFFICIENT MOIST PHYSICS. I Cloud schemes and their linearization Subgrid cloud processes are very complex and highly nonlinear, which has given rise to a large variety of cloud parametrizations with various degrees of complexity. One differentiates between diagnostic and prognostic schemes depending on whether subgrid cloud processes can be computed from the usual set of prognostic model variables or whether the introduction of additional cloudrelated) prognostic variables is required by the scheme as for the linear parametrization described in this article, a scheme is not regarded as prognostic if the introduction of additional prognostic variables could be avoided by choosing an appropriate lower dimensional set of prognostic variables in the host model). Many schemes, the so-called statistical parametrizations they can be diagnostic or prognostic), use a notional probability density function PDF) that describes subgrid variations of heat and moisture. The occurrence and strength of subgrid cloud processes is then inferred from this PDF. The linear scheme described below is designed to approximate the condensation process as described by such statistical schemes. To meet the constraints of computational efficiency and flexibility portability), the linear cloud scheme development focused on the main impacts that the cloud scheme has on the large-scale flow. For our purposes the main cloud subgrid processes related to non-convective clouds can be divided into the following groups which are of course strongly interdependent): 1) latent heat release; 2) the conversion between vapour and cloud; 3) precipitation formation, fall, evaporation); 4) ice microphysics. The first point, the latent heat release, has probably the greatest impact on the large-scale dynamics. The major latent heat source/sink is the conversion between water vapour and cloud point 2)), which therefore plays an important role in most linear as well as nonlinear) cloud schemes Honda et al., 25) and is also the focus of the scheme presented below. The formation of precipitation itself point 3)) has a comparably weak impact on the model dynamics but it is crucial for producing realistic cloud and, of course, precipitation fields in the PF model. These fields are needed if one wants to represent their radiative interactions or, also, if one wants to assimilate cloud- or rain-affected observations. Since the cloud water content is typically quite small, some linear schemes link latent heat release directly to the formation of precipitation by converting all cloud water immediately into precipitation Janisková et al., 1999; Mahfouf, 1999). As mentioned above, Tompkins and Janisková 24) introduced a more complex diagnostic scheme at the European Centre for Medium- Range Weather Forecasts ECMWF). It combines a statistical cloud scheme with the smoothed Kessler-type Kessler, 1969) precipitation parametrization introduced by Sundqvist et al. 2), and became fully operational at the end of 26. A substantially more complex prognostic cloud and precipitation scheme was developed by Lopez 22) for the ARPEGE model, but its linearized version has not become operational in the 4D-Var system. The cloud scheme developed here has a simple relaxation-type precipitation scheme similar to Kessler s nonlinear parametrization Kessler, 1969), although the strong nonlinearity resulting from Kessler s threshold condition has been neglected and not smoothed as in Tompkins and Janisková 24)). A limited effort to increase the scheme s complexity showed little benefit, but there might still be some room for improvements. In particular, a representation of the evaporation of precipitation is currently not included but might be beneficial for the PF model if problems with supersaturation can be kept at an acceptable level). Also, prognostic precipitation and possibly cloud variables may be added, as they are expected to become increasingly important for finer resolution. The last point, ice microphysics, exhibits great complexity and, to capture various aspects as accurately as possible, full nonlinear parametrizations are usually as detailed as possible and use prognostic variables for ice clouds and frozen hydrometers, which are necessary since the formation of ice cloud and the fall of snow are much slower than the respective processes for warm water clouds the Met Office, for example, uses a prognostic ice/snow variable Wilson and Ballard, 1999)). As its impact on the large-scale dynamics is substantially smaller than that of latent heat from condensation and other processes, most linear parametrizations treat these aspects in a highly simplified diagnostic fashion. Janisková et al. 1999), for example, assumed that all precipitation was solid below the freezing point. For the scheme presented in this article, no distinction has been made between liquid and frozen cloud or precipitation, which means that the latent heat of fusion has been neglected. The latent heat of fusion is about 13% of the latent heat of condensation L c. Since a reliable distinction between liquid and frozen water is difficult in a linear scheme, no effort has been made to include this comparably small effect. It should be noted that the cloud or precipitation increments calculated by the moist parametrizations in the PF model are not necessarily identical to the respective fields that are used by the observation operators when assimilating moisture-related observations. When postprocessing PF model output, observation operators can incorporate weak nonlinearities to increase the accuracy of the model observations. Indeed, the Met Office is following a strategy where total water q T is the only three-dimensional moisture field that is passed from the PF model to the observation operators. The partitioning between water vapour and cloud water is then done by a weakly nonlinear observation operator that approximates the Smith scheme more accurately than a linear version. To summarize, the scheme presented in this article concentrates on the latent heat release from condensation of cloud water. It has a simple precipitation scheme and no ice physics is represented. In the form presented here

4 71 O. STILLER AND S. P. BALLARD the new scheme is diagnostic as appropriate for the currently used nonlinear scheme) but, as explained in the Appendix, an extension to further prognostic schemes is straightforward. The new scheme will be introduced and derived in section 2. Linearization test results that demonstrate the impact of the new scheme are presented in section 3, while section 4 summarizes and discusses the scheme s properties. 2. The new linear scheme As mentioned above, the cloud scheme consists of latent heat release through condensation) and a simple precipitation scheme. Central for the latent heat computations are cloud water perturbations, for which an approximation valid for a large class of statistical schemes will be derived in section 2.1. The corresponding latent heat release scheme is presented in section 2.2 together with the corresponding approximation in the first ECMWF scheme Mahfouf, 1999) and generalizations thereof. The simple precipitation scheme and some of its consequences will be discussed in section 2.3. Since the prognostic variables of the Met Office s PF model are not conserved with respect to condensation processes, the implementation of the latent heat scheme led to the introduction of an additional prognostic variable, cloud water increments q c,the initialization of which is described in section Cloud water perturbations in statistical cloud schemes Having a probability distribution P tot T l,q T ) for liquid water temperature T l = T L c /c p )q c and total water mixing ratio q T = q v + q c where T is the temperature, q c and q v are the mixing ratios of cloud water and water vapour, respectively, while L c and c p are the latent heat of condensation and the heat capacity of air at constant pressure), the mixing ratio of cloud water can be written as q c = q T =q s T l,p) P tot T l,q T ) a L [ qt q s T l,p) ] dq T dt l, 1) where [ a L = 1 + L ] c q s T l,p) 1, 2) c p T will be treated as a constant as is done in most of the cloud parametrization literature. Note that the integral in Equation 1) is only over the saturated part of the grid box where q v is assumed to equal its saturation value q s T, p), for which the Taylor approximation q s T, p) = q s T l,p)+ q s / T )L c /c p )q c is used. Dividing T l and q T into the mean and fluctuations from the mean, Equation 1) can be written as q c = q T = q T +q s T l,p) P fl T l, q T ) al [ q T q s T l,p) ) + q T q s ) ] d q T d T l, 3) where in this subsection the overbar is used to indicate the grid box mean while the tilde gives the fluctuations overbars are omitted in all other subsections where model variables are generally taken as the grid box mean). In its most general ) form the probability distribution function P fl T l, q T depends on the history of the air masses. Cloud ) parametrizations make the assumption that P fl T l, q T is uniquely determined by local values of a few prognostic variables. The main assumption of the linear scheme proposed here) is that variations from changes of the function P fl T l, q T are slow ) compared with the respective changes in q T q s T l,p). Using the approximation q s T l,p)= q s T l, p) once more, perturbations q c of q c resulting from changes of the prognostic variables can in the linear approximation) be written as [ = C f a L qt q s Tl, p )], 4) where the prime indicates perturbations to the grid box mean. Primes behind brackets or functions denote the total perturbations resulting from considering all model variables that are in the brackets and/or arguments of the functions in the linear approximation used in this article one has, for example, q s q s Tl, p ) = [ qs Tl, p ) ] [ / T l Tl + qs Tl, p ) / p ] p ). The cloud fraction C f is given by C f = q c q T = q T +q s T l,p) P fl T l, q T ) d qt d T l, 5) this expression has been substituted when deriving Equation 4)). For some applications perturbations C f of the cloud fraction are also needed, and can be derived within the same approximation. Expanding Equation 5) but neglecting changes of the PDF) yields where C f = P sat Tl, q T, p )[ q T q s Tl, p )], 6) P sat Tl, q T, p ) = P fl T l, q T = q s T l,p) q T ) d T l is the total probability density for just saturated air i.e. for q T = q s T l,p)). The assumptions made in deriving Equation 4) and similarly Equation 6)) should be acceptable for a large variety of cloud schemes. Here we are most interested in the extent to which it applies in the currently operational Met Office scheme Wilson and Ballard, 1999; Smith, 199). Going back to concepts developed by Sommeria and Deardorff 1977) and Mellor 1977), the scheme describes the subgrid variability of warm water clouds through a PDF P s) with only one variable s = a L q T q s ). Using this, Equation 3) takes the form q c = s= a Lq T q s T l,p)) P s) ) [a L q T q s T l,p) ] + s ds. 7)

5 EFFICIENT MOIST PHYSICS. I 711 Further, P s) has the form P s) = σ 1 P s σ ), where the function P is independent of the model variables and σ is a normalization factor with σ q s T l, p) the constant of proportionality is chosen so that condensation sets in at a chosen critical value of relative humidity). Using this, Equation 7) can be written as q c = ŝ= Q which yields with and P ŝ) [ a L qt q s T l,p) ) + ŝσ ] dŝ 8) q c = C f a L qt q ) s + A Q) q s ) ) = a L C f q T C f AaL q s A Q) = σ q s ŝ= Q C f = C f Q) = P ŝ) ŝ dŝ = q c q s q T q s q s C f, ŝ= Q Q = a L q T q s T l,p) σ 9) 1) 11) P ŝ) dŝ 12). 13) Comparison with Equation 4) shows that Equation 9) has an additional term, i.e. the last term, which describes the impact of changes of the PDF and which has been neglected in the approximation leading to Equation 4)). This alters the contribution of q s, as shown more explicitly in Equation 1). From Figure 1b) one finds that the error involved is indeed very small, particularly compared with the large errors resulting from the linear assumption. Also the contributions from q T in Equation 1) are expected to be larger than those from q s. a) s/sigma b) Q Figure 1. a) The PDF P = P s/σ ) used in the Smith cloud scheme. b) Coefficients occurring in Equation 1) [for a L =.75 using the PDF from a)] as a function of the variable Q defined in Equation 13). Solid line: C f, dashed line: C f A/a L, dash dotted line: step function Latent heat release The new scheme The new cloud scheme diagnoses q c from the prognostic variables q T, T l and p. For this purpose the right-hand side of Equation 4) is expanded, yielding overbars are omitted in this and all following subsections): q c diag = C f a L q T q s T T l q ) s p p, 14) where the additional subscript diag indicates that this is the value that is diagnosed by the PDF within the applied approximations). Using Equation 14), q v and T can be diagnosed from q T, T l and p via T = T l + L c q c c diag, 15) p q v = q T q c diag. 16) It has to be said that, like many NWP models, the Met Office s linear model and similarly the nonlinear model) does not use the conservedwith respect to condensation) variables q T and T l as prognostics, but instead q v and θ plus pressure and velocity variables). These are generally not complete for a proper description of condensation processes unless one makes some very specific assumptions), and the additional variable q c was added when the new cloud scheme described in this article was introduced. With this additional prognostic variable, q v and θ can be transformed into q T and T l, from which the condensation-related changes δq c of q c can be computed via Equation 14) as δq c q c diag q c, 17) where the value of q c on the right-hand side is the input value into the linear condensation scheme. Using Equations 15) and 16), the corresponding changes δq v and δθ of q v and θ are then given by δθ = L c δq c c, p δq v = δq c,, 18) where is the Exner function. While from a formal point of view the introduction of the new prognostic variable q c might be taken as an indication that the new linear scheme should be considered as prognostic, we would like to emphasize that the introduction of q c is not really needed for the new scheme as such, but that it is only a consequence of the particular choice of prognostic variables in the PF model. This choice of variables actually only had consequences after the new scheme had been introduced, since without a condensation scheme one has q c = so that the variables q T and T l are identical to q v and T.

6 712 O. STILLER AND S. P. BALLARD While Equation 17) ensures that q c + δq c is equal to the diagnosed value q c diag, it should be noted that the diagnostic relationship 14) can also be used for the estimation of the linear tendencies if the corresponding nonlinear condensation scheme is prognostic and does not impose a diagnostic relationship on q c see Appendix for details) Step function schemes The first linear cloud scheme used at ECMWF Mahfouf, 1999) basically replaced C f in Equation 14) by the step function shown in Figure 1b). For C f = 1, Equations 17) and 14) yield δq c = a L q v q s ) so that the step function approximation is equivalent to { δq c al q = v q s) for q T >q s T, p), for q T <q s T, p). 19) Mahfouf 1999) used this, together with a very simple precipitation scheme where all cloud is immediately converted into precipitation q p i.e. δq p = δq c ), which all falls to the ground during one time step. This avoided the introduction of a q c variable. In section 3.2 some results from this very simple scheme will be compared with those corresponding to Equation 14). For a broader comparison the threshold criterion in Equation 19) has been generalized to { δq p [ ] = δq al q c = v q s) for C f >C thr, for C f <C thr, 2) where C thr is a threshold value. For the Met Office system Equation 2) is equivalent to Equation 19) if C thr =.5 is chosen, since C f =.5 corresponds to q T q s T, p) = Precipitation The PF model cloud scheme uses a simple relaxationtype precipitation parametrization whereby the part δq p of the cloud water increment q c converted to precipitation increments q p at a given time step is directly proportional to q c : δq p = ppt effq c δt, 21) where δt is the model time step. The precipitation efficiency ppt eff is a tunable parameter the inverse ppt 1 eff gives a precipitation time-scale) that can be adjusted to optimize the cloud and precipitation field. Using Equation 21), the value of q c at the end of the call to the cloud scheme is given by q c = q c diag δq p. Surface precipitation rates, on the other hand, can be obtained from summing up δq p over a column using the assumption of the diagnostic nonlinear scheme that all precipitation reaches the ground during one time step). As suggested below, the optimization of the precipitation field can, to some extent, also be left to the observation operators, since the impact of precipitation on the dynamics is very small apart from consequences for the evaporation of precipitation or other microphysical processes which are, however, currently not represented in the PF model). It should be noted that Equation 21) is not meant to represent the linearized version of an auto-conversion equation. Auto-conversion is just one of many nonlinear microphysical processes that determine the total or effective ) precipitation efficiency with respect to cloud increments q c and which are present in the nonlinear scheme. The Met Office strategy was not to represent this complexity in its linear precipitation scheme explicitly, but to start from a very simple scheme that can be improved in subsequent steps. Unfortunately, so far, all efforts to increase the complexity of Equation 21) did not lead to any measurable improvements, which may have different reasons see discussion in section 4) The initialization of q c As explained in section 2.2.1, the new condensation scheme led to the introduction of q c as an additional prognostic variable this does not, however, imply that the new cloud scheme is prognostic see discussion following Equation 18). This required the initialization of q c at the beginning of a PF model run. Generally, the initialization of PF model variables should be a linear analogue of the corresponding updating of the nonlinear model through the analysis increments. Indeed, both initialization processes linear and nonlinear) involve the same set of variables, the so-called control variables, which span the space in which the 4D-Var cost function is minimized. While some 4D-Var systems have control variables for each prognostic variable, the Met Office like other operational centres) uses a smaller control variable set. The approach exploits balance relationships such as geostrophy and hydrostatic balance, which reduce the order of the problem. The moisture field is represented through only one variable in this reduced control variable set, from which the nonlinear model s moisture variables are computed via an incrementing operator as described below, this operator has changed during the last years). Originally the employed moisture control variable was relative humidity. With the retention of cloud water within the linear model, the control variable is now assumed to be total water relative humidity increments, i.e. the linearization of q T /q s. No new control variable was added when q c was implemented as a prognostic variable of the PF model, but the initialization of q c followed the incrementing operator of the nonlinear model. At the beginning, when the new linear parametrization became first operational, cloud water was not incremented in the nonlinear model its increments were neglected) and the corresponding initial condition in the linear model was simply given by q c =. Later the nonlinear model initialization changed, and now the moisture control variable is first transformed into total water increments q T which are then partitioned into vapour and liquid water in such a way that the condensation process by the diagnostic warm water

7 EFFICIENT MOIST PHYSICS. I 713 scheme) would not lead to any latent heat release at the beginning of the run. The corresponding initialization of moisture in the linear model is obtained by setting q c = q c d and q v = q T q c d,whereq c d is obtained from Equation 14) by substituting q c diag = q c d and T L = T L c /c p )q c d. This yields q c d = 3. Results 3.1. Error measures C f a L q T q s T T q ) s p p ). 22) L c q s 1 C f a L c p T Linearization tests were the central development tool, and this subsection gives a brief explanation for the error measures used for their verification. Linearization tests compare the output x from a PF model run typically from runs over a 6 or 12 hour time window) with the corresponding nonlinear increments x nl = x2) nl x 1) nl that are obtained from the output differences from two nonlinear runs which are denoted by x 1) nl and x 2) nl, respectively). This comparison requires error measures and the most straightforward measure is probably the root-mean-square rms) error, which can be viewed geometrically as the L2 norm of the difference vector x x nl i w [ i x i x nl) i w i where the sum is over all grid points and w i are the weighting coefficients in this article they refer to mass weighting). Another measure that indicates the extent to which x is able to reproduce its target x nl is rrms x,x nl ) x x = nl, 23) x nl which was called the TLM solution error by Vukicevic and Bao 1998) and will be called the relative rms rrms) error in the following. The relative magnitude of the variance of x to that of x nl, damp x,x nl ) x =, 24) x nl is called the damping error. The correlation between x and x nl is defined as corr x,x nl ) x x = nl x x nl i w ix i ) x nl i i w i i ] 2 x x nl ) 1, 25), where the asterisk denotes the scalar product. In geometrical terms, the correlation is the cosine of the angle α between the vectors x and x nl,i.e.corr x,x nl) = cosα) which is independent of x and x nl. Below we will see that the correlation of the precipitation field is quite insensitive with respect to the precipitation strength, while the other error measures are obviously very sensitive to this quantity. This means that by tuning the precipitation time-scale we can tune the rms error without affecting the correlation very much. Geometrical considerations show that the minimal rrms error for a given α is rrms min α) = sinα), 26) which is obtained if damp x,x nl) = x / x nl fulfils the condition damp x,x nl) = cosα) [ = corr x,x nl)]. 27) It follows that for large values of α, the damping error that optimizes the rrms error is substantially smaller than 1. This shows that for cases where the correlation is weak, it is better if the variance of the PF increments is smaller than that of the nonlinear increments, at least as far as the rrms error is concerned. This raises the question of the relative importance of different error measures in linearization tests with respect to the performance of the 4D-Var system. Though important, this question is beyond the scope of the current article and may be the subject of a future publication The impact of the new scheme Linearization tests involving simulations over a 6 hour time window were carried out for different starting dates and at different resolutions. The results presented below correspond to tests where the nonlinear model, i.e. the Met Office Unified Model UM), was run at N216L5 resolution grid points) with a 2 minute time step. The UM output was reconfigured to N18L5 resolution to obtain the linearization states needed for the PF model) and the nonlinear increments for verification of the PF model output). The PF model was run at N18L5 resolution with a 4 minute time step. Apart from the resolution and the switched off PF cloud scheme in the control), the UM and PF model versions were close to those employed in the parallel suite 13, which became operational at the Met Office on 5 November 26. The analysis window was on 12 June 26 from 9 15 UTC. Output from linearization tests taken at different dates and/or resolutions differs quantitatively, but the conclusions made below would not change significantly if results from different test runs had been chosen. The PF model version employed had no representation of convection, and to obtain a cleaner comparison a number of linearization tests have been made in which the convection scheme in the nonlinear model i.e. the UM convection scheme) has been switched off. These tests

8 714 O. STILLER AND S. P. BALLARD and their results will be labelled as no UM convection or UM convection off as opposed to UM convection on in the normal tests). Figure 2 shows the globally averaged rrms values for u and θ as a function of model levels a rough indication of the model levels height dependence can be found in Figure 3). Comparing the diamonds cloud scheme switched on) with the solid lines control: no PF cloud scheme) shows that this global error measure is slightly reduced by the latent heat release in the PF model. As seen in Figure 4a) and b), the new scheme has its strongest impact on the moisture field, which is not vertical level control 1 micro phys. on no UM convection micro phys. on relative rms error u relative rms error Figure 2. Scores rrms errors) from 6 hour linearization tests for u and θ as indicated in the respective graphs. Symbols circles and diamonds) correspond to runs with the new PF cloud scheme switched on with ppt eff =.2). Dashed lines and circles are from experiments with UM convection off. height [m] model levels model level 5 θ Figure 3. Height dependence of model levels for θ and moisture variables the corresponding levels for p and u are staggered between these levels). The values given in this figure are valid for grid columns with the ground at mean sea level. In general the model levels are hybrid levels that are continuously changing with height from being terrain-following near the ground and only height-dependent above mean sea level) near the top. The inset shows an enlargement for the lowest 2 levels. height [m] a) b) c) vertical level 2 1 q T rel. rms error 2 1 q v rel. rms error 2 1 q c rel. rms error Figure 4. Rrms errors for different moisture variables as indicated in the respective graphs. Line styles and symbols have the same significance as in Figure 3 apart from filled diamonds which, similarly to the empty diamonds, are from runs with the new cloud scheme but with ppt eff =.1 instead of ppt eff =.2 for the empty diamonds). surprising since without condensation and precipitation q v and q T are just passive tracers in which case q v = q T ). Figure 4 further shows that the runs with a stronger precipitation rate gave better improvements for q T and particularly for cloud water q c and to a substantially smaller extent also for q v ). This also led to marginally better scores for u, θ and p increments not shown) but, as will be discussed further below, gave a significantly worse rrms error for precipitation compare results with Figure 12, later). The global rrms error improvements from the new scheme are much larger for all variables in the no UM convection experiments dashed lines and circles in Figure 3). This indicates that the positive impact of the new scheme is to some extent masked by the presence of the UM convection scheme or, if one prefers, by the absence of a corresponding PF convection parametrization). The fact that the strong influence of the UM convection scheme can qualitatively alter the outcome of linearization tests is seen in Figure 5, where results of the new scheme solid lines) are compared with those from the step function scheme different symbols correspond to different thresholds). The upper and lower graphs in this figure correspond to UM convection on and off, respectively. While in both cases UM convection on or off) the overall performance of the new scheme is better than for any of the step function schemes, the relative benefits of the new cloud scheme are more clearly visible when UM convection is off. When UM convection is on, the pressure field as well as the upper-level winds seems better for the step-function scheme with the lowest threshold C thr =.2). Indeed, the pressure field in particular seems to improve systematically as the threshold cloud fraction C thr decreases i.e. the latent heat release increases). Regarding the lower panel of Figure 5, one finds that this trend is neutralized or even slightly reversed when UM convection is switched off. Apparently in the UM convection on experiments the strong latent heat release related to small C thr that is seen to be beneficial for the pressure field and the upper-level winds is actually compensating for the missing convection source terms in the PF model.

9 EFFICIENT MOIST PHYSICS. I 715 a) vertical level new cloud scheme C thr =.2 C thr =.4 C3 thr = u 1 Θ 1 p.2.1 rrms error difference.4.2 rrms error difference rrms error difference b) u 4 Θ 4 p vertical level rrms error difference.1.5 rrms error difference.4.2 rrms error difference Figure 5. Differences of the rrms error from the control. The thick solid lines correspond to PF model runs with the new cloud scheme while symbols correspond to the step-function-type cloud scheme for different thresholds as indicated in the legend. Top panel: UM convection on. Bottom panel: UM convection off see section 3.2 for further explanation). Even though its global rrms error improvements seem small, the new scheme, as explained above, showed a significant benefit in forecast trials. This is a typical situation when improving 4D-Var physics and a probable explanation is that the small global improvements come along with large local improvements, often in critical regions such as the centre of low-pressure systems. The fact that the improvements from the new scheme are not homogeneously distributed can be seen from Figure 6, which shows that the scheme s benefit is much greater in the extratropics than in the tropics which is consistent with the larger improvements found in the UM convection off experiments). Strong local improvements are shown in Figure 7 in the vicinity of a low-pressure system with its centre south of Iceland see surface pressure contours that have been added to the top left panel). From the nonlinear θ increments second row, first column) one finds that the initial perturbations lead to a relative cooling in a large region around the eastern border of Iceland. Without the cloud scheme second column), the PF model cannot reproduce this but has positive θ increments in the whole region. As seen from the second column of the first row, these positive θ increments are accompanied by pronounced negative q v increments, which are absent in the nonlinear increments first column). Including the PF cloud scheme third column), however, the cooling is at least qualitatively represented in the PF increments and also the negative q v increments have disappeared which is consistent with a decreased latent heat release). A S2 - S9 S2 - N2 N2 - N9 5 4 vertical level u Θ q v.1.5 rel. rmse reduct..2.1 rel. rmse reduct..5 rel. rmse reduct. Figure 6. Relative rms error reduction [rmse tr rmse cntr ] /rmse cntr,wherermse cntr and rmse tr are the rms errors for the control and the run with the new scheme switched on, respectively. Different line styles correspond to different latitudes as indicated in the legend.

10 716 O. STILLER AND S. P. BALLARD a) b) Figure 7. Maps of increments and score differences at time step 9 6 h) at model level 13. From left to right: nonlinear increments, linear increments with no PF cloud scheme, linear increments with new PF cloud scheme and the difference in rms error between the two linear increments in the right-hand graphs, dark shadings indicate that the new scheme decreases the error). The top row corresponds to q v kg/kg) and the bottom row to θ K). Surface pressure contours pa) have been added to the top left graph. more quantitative picture can be obtained from the areaaveraged rms error given in Figure 8, where switching on the cloud scheme removes strong peaks of the rms error of θ and q v around level 13. It leads to clear improvements of p not shown) and gives some benefit to u Precipitation Global stratiform precipitation increments are shown in Figure 9, while Figure 1 gives an enlargement of a region in the South Pacific where stratiform precipitation is particularly active. Comparing top nonlinear increments) and bottom PF increments with ppt eff =.1) graphs of these figures, one finds that many of the largescale features and in rare cases even some smaller scale features) seem to be at least qualitatively represented by the PF precipitation scheme. Generally, the PF increments are broader than their nonlinear counterpart and and show vertical level u 1 θ 1 q T.5 relative rms error relative rms error relative rms error Figure 8. Linearization test results from runs with circles with dotted lines) and without solid lines) the new linear cloud scheme. Rms errors in the troposphere are shown as a function of model level corresponding to the domain. Longitudes: 25 W, latitudes: 6 N 7 N. The respective fields are indicated in the figure. fewer details. The histogram in Figure 11 shows that the linear increments with ppt eff =.1) have smaller spread for positive increments while the spread is comparable for negative increments. Nonlinear increments are skewed towards positive increments, as they are bounded from below by the fact that total rain rates are always positive.) For ppt eff =.2 black squares in Figure 11), the spread matches well for positive increments but is clearly overestimated on the negative side. The corresponding error measures see section 3.1) are shown in Figure 12. From Figure 12a) one finds in particular that the correlation is quite insensitive to changes in ppt eff. As discussed in section 3.1, ppt eff can therefore be used for a systematic tuning that reduces the rrms error rrms towards rrms min see Equation 26); rrms min is the smallest rrms error that is possible for a given correlation). The tuning changes the damping error damp towards the target value damp = corr see Equation 27)) at which rrms = rrms min holds. As seen from Figure 12a), the damping error is much closer to its optimal value for the smaller value of ppt eff i.e. for ppt eff =.1), which therefore also has a much smaller rrms error see Figure 12b)). This poses a dilemma, as in Figure 4 above it was seen that the larger value ppt eff =.2 actually leads to better scores rrms errors) for q T as well as for q c. This dilemma is intrinsic to the proposed tuning method, as a precipitation damping error that is smaller than unity implies that the linear model has less removal of cloud water than its nonlinear counterpart. Using a tuning coefficient like ppt eff to minimize the rms error of a field obviously

11 EFFICIENT MOIST PHYSICS. I 717 Figure 9. Large-scale precipitation increments mm h 1 ) at time step 9 6 h). Top: nonlinear increments reconfigured from two nonlinear runs at N216 resolution. Bottom: increments propagated by the linear PF models with ppt eff =.1. strongly depends on which of the fields one wants to optimize. With respect to their use in observation operators, however, the tuning of the linear increments does not have to be done by the PF model alone but can also be achieved through a post-processing of the PF model output which could be done by the respective observation operators). One solution for the above dilemma is a simple rescaling of the PF model precipitation output. The diamonds in Figure 12 correspond to such rescaled data from the ppt eff =.2 run. The increments were divided i.e. rescaled) by a factor of 2. This method apparently gives almost optimal results over the whole assimilation time window. We would like to point out that the proposed rescaling with the tuning method is done on merely statistical grounds and that it assumes that rms error is the most relevant error measure this is the error minimized by 4D- Var). An alternative post-processing could be motivated on physical grounds, with the objective of compensating for physical processes that are missing in the linear model. Indeed, the evaporation of precipitation in the nonlinear model has no counterpart in the PF model, so that some reduction of the linear precipitation increments through the observation operator would be justified on this basis. However, in contrast to the statistical method proposed above, the target value for the damping error should be around 1 for such a physically motivated post-processing method so that linear and nonlinear increments have the same variance). 4. Summary and conclusions This article describes the linear cloud scheme that has been developed for 4D-Var at the Met Office and which has been operational since February 25. The scheme s core is a condensation parametrization, the latent heat of which has a beneficial impact on the dynamics. It also has a very simple diagnostic) precipitation parametrization, the impact of which on the dynamics is small but which is important for the total water and, of course, the precipitation field. While the scheme s improvements on global linearization test scores are moderate, local improvements are

12 718 O. STILLER AND S. P. BALLARD a) error meassure error meassure , scl. fac.= time step time step Figure 12. Different error measures for the large-scale precipitation increments as a function of time step produced by different PF model runs and post-processing. Circles correspond to a run with ppt eff =.1, while squares and diamonds have ppt eff =.2. Diamonds correspond to data where the precipitation increments have been scaled i.e. post-processed) by a factor of.5. In panel a) the white symbols show the damping error, while filled or shaded symbols show the corresponding correlations. In panel b) the white symbols yield the rrms error, while filled or shaded symbols show the minimal rrms error rrms min see Equation 26)) for the correlations shown in panel a). b) log1 area [m 2 ]) Figure 1. Enlargement of an extract from Figure 9. nonlinear Inc. ppt_eff=.1 ppt_eff= precipitation rate increments [mm/h] Figure 11. Histogram for global large-scale precipitation rates of nonlinear increments white bars) and PF increments from runs with ppt eff =.1 shaded bars) and ppt eff =.2 black squares) as given in the legend. White and shaded bars correspond to the increments shown in figure 9. much higher as was found, for example, in the vicinity of a low-pressure system over the north Atlantic. This seems consistent with the findings from Coutinho et al. 24) and Hoskins and Coutinho 25), who demonstrated a great impact of a large-scale cloud scheme on the evolution of singular vectors, particularly in the vicinity of wintertime storms. Such strong local improvements are probably the reason for the scheme s good performance in forecast trials. The scheme is simple numerically cheap) and should be easily maintainable portable) since the condensation part has been derived from quite general equations that are the basis for a large class of cloud schemes. The main coefficient that relates the PF increments to the condensation amount is the cloud fraction C f see Equation 14)). This quantity is a standard output variable for most atmospheric forecast models. It can therefore be read in from the nonlinear trajectory i.e. linearization states), which reduces computational costs and assures the linear scheme s proximity to the nonlinear scheme. This feature is particularly appealing for NWP systems, the nonlinear models for which have a prognostic cloud scheme. While this article focuses on the new scheme in its diagnostic mode as employed in the current Met Office system), the scheme can also be extended to NWP systems with prognostic cloud water which is not constrained to a diagnostic relation), as described in the Appendix. Also, cloud fraction increments, which are currently not computed, can be added in a consistent way if they are required e.g. for use in a future linear radiation scheme). It should also be mentioned that when reading in C f from the UM into the PF model it is subject to averaging by the reconfiguration routine, which reduces its horizontal resolution to that of the PF model. Formally this is done by the simplification operator introduced by Rawlins et al. 27); in the current operational configuration one PF grid column corresponds to almost nine UM grid columns for more details see also Part II, section 3.) The smoothing effect of this space averaging can be expected to have a beneficial impact on the scheme s performance, similar to the regularization-type adjustments that many authors found to be necessary or at least beneficial for reducing their 4D-Var system s noise resulting from on/off switches of nonlinear) physical parametrizations Janisková et al., 1999; Laroche et al., 22; Zupanski, 1993). In the development phase the scheme was compared to a generalization of the cheaper step function scheme Mahfouf, 1999), which for its simplicity showed excellent results. Overall the new scheme presented here performs better, though a highly tuned version low C thr in Equation 2)) of the step function scheme showed larger improvements for the pressure and the upper-level winds. The reasons for this partially good performance of