An integrated package for subgrid convection, clouds and precipitation compatible with meso-gamma scales.

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Published online in Wiley InterScience ( An integrated package for subgrid convection, clouds and precipitation compatible with meso-gamma scales. L. Gerard Royal Meteorological Institute of Belgium, 3 Av. Circulaire, B 1180 Brussels, Belgium Abstract: The integration into a coherent package of the main moist parametrizations deep convection, resolved condensation, microphysics of a limited area model is presented. The development of the package is aimed at solving efficiently the problem of combining resolved and subgrid condensation at all resolutions, in particular the range between 10 km and 2 km where deep convection is partly resolved, partly subgrid. The different schemes of the package are called in cascade, with intermediate updating of internal variables, so that for instance the initial profiles passed to the deep-convection scheme are already balanced with respect to resolved condensation effects. Further on, the clean separation of the contributions to the closure of the updraught and downdraught from the initial vertical profile from which they evolve prevents double counting. The convective parametrization works with a prognostic mass-flux scheme, and acts on the resolved variables through condensation and convective transport. It detrains condensates that are added to the prognostic resolved condensates. A sensitivity study in a single-column model, and further validation in three-dimensional experiments at different resolutions, are presented. Copyright c 2007 Royal Meteorological Society KEY WORDS Local area modelling; grey-zone resolutions; prognostic convection; prognostic microphysics. Received 29 March 2006; Revised 1 December 2006; Accepted 6 February Introduction In a previous paper (Gerard and Geleyn, 2005), we assessed and tried to address different limitations of a classical mass-flux parametrization of the deep convection, initially developed for a global circulation model with grid boxes bigger than 20 km, when used in a highresolution limited-area model and reducing the grid-box lengths down to around 7 km. Beside various enhancements of the diagnostic approach, we were led to relax the hypothesis of quasi-equilibrium between the convective activity and the large-scale processes feeding it, by using prognostic equations for the vertical velocity in the updraught and for the fraction of the grid-box area covered by it. This updraught mesh fraction also influences the mean grid-box variables when the convective cells cover a significant part of the grid-box area. The prognostic approach has significant advantages. Tiedtke (1993) advised that the explicit representation of anvil and cirrus clouds associated with cumulus convection was a strong argument in favour of prognostic schemes. Knowledge of the updraught s vertical velocity and mesh fraction is important for understanding the microphysics, as well as for parametrizing mesoscale updraughts and downdraughts (Leary and Houze, 1980; Donner et al. 2001); this means that the prognostic scheme is also useful at coarser resolutions. Arakawa (2004) sees the closure of the convective parametrization as evolving from diagnostic to prognostic to stochastic. Correspondence to: Research Department, Royal Meteorological Institute of Belgium, 3 Av. Circulaire, B 1180 Brussels, Belgium. luc.gerard@oma.be. While we obtained some improvement, we were confronted with two major difficulties. The first was the challenge of combining different precipitating schemes working in parallel in such a way as to avoid double counting, while remaining independent of the model resolution in space and time. The second difficulty was the need to handle cloud condensates as model variables and to replace the rough diagnostic parametrizations of clouds and precipitation by a more elaborate microphysical package treating all condensates, whatever their origin resolved or subgrid (convective). In this paper, we describe an integrated package that addresses these difficulties. Such an approach is in accord with the recent trends in parametrization. The need to treat all cloud processes in a unified consistent way has already been emphasized by Tiedtke (1993). Arakawa (2004) insists that the artificial separation of the physical processes causes most of the direct small-scale interactions between those processes to be lost. He recommends working on the unification of deep convection with other parametrizations: the use of mass transport by cumulus convection in the microphysics; the generation of liquid and ice phases of water leading to a unified cloud parametrization; the interactions with the boundary layer (diurnal cycle, shallow clouds); the coupling of radiation and cloud processes on the cloud scale; consideration of the effect of momentum transport on the mean flow and the problem of clouds organization; and the inclusion of non-deterministic effects. The eventual challenge would be to develop a physics coupler in which all these processes are fully coupled. [Version: 2007/01/05 v1.00]

2 2 L. GERARD The scheme described in (Gerard and Geleyn, 2005) already produced convective transport fluxes, including the transport of momentum. However, the effect of the updraught on the resolved variables was estimated through a computation of detrainment and pseudo-subsidence. In our new scheme, following the proposal of Piriou (2005), we express these effects directly through transport and condensation. One of the basic features of our package is the cascading approach, which not only allows a clean separation of the closure contributions, but also avoids competition between resolved and subgrid parametrizations. The convective transport impacts on the microphysics, which is computed subsequently. In our new convective scheme, the updraught does not produce precipitation directly, but it detrains cloud condensates, which are combined with the resolved condensates to feed the microphysics. This approach allows one to take into account the anvil and cirrus clouds generated by deep convection. The updraught entrains condensates from the environment as well. The package we propose here corresponds, in Arakawa s terms, to the step of the unified cloud scheme. In the context of climate and general circulation models, several authors (Del Genio et al., 1996, Tiedtke, 1993, Ose, 1993, Fowler et al., 1996) add the detrained condensed water from the convective updraughts to the prognostic stratiform cloud water. In Tiedke s scheme, convective precipitation is generated through a drain term similar to the one used for the autoconversion of stratiform condensates to precipitation. But it is then difficult to keep the same coherence when replacing or refining the microphysical package. Fowler and Randall (2002) proposed to go one step further in the coupling of convection and microphysics, by entraining cloud water into the convective updraughts. Still, their convective parametrization converts diagnostically most of its condensates to precipitation, which is sent promptly to the surface, thus following a path different from that of the large-scale microphysics. They assess alternate ways to handle the convective snow either detraining it at the top and passing it through the microphysics, or precipitating it outside or inside the updraught and show the sensitivity of the results to this choice. Boville et al. (2006), for the CAM3 climate model, use two separate schemes for shallow and deep convection, both of which detrain condensates into the stratiform clouds; but the majority of the condensate formed by the deep convection is directly precipitated rather than detrained, and they explain that for this reason, the detrained water that may feed anvil clouds is significantly underestimated. The greatest advantage of our integrated package is that it can work at all resolutions, including the grey zone where deep convection is partly resolved, partly subgrid. A few authors have approached the problem of convection in high resolution models, particularly when the mesh size becomes comparable to the size of the convective systems. Weisman et al. (1997) present a systematic academic study of the explicit representation of convection with grid-box lengths ranging from 2 km to 12 km. They use a non-hydrostatic model, and show that for resolutions coarser than 8-km, it behaves the same way as a hydrostatic model. Their study uses a semi-infinite 2kmdeep surface cold pool, which suppresses the problem of triggering and for which even a 50 km grid would be sufficient to resolve a portion of the mature system-scale structure. In this case, they show that the 4 km resolution still gives quite satisfying results compared with the 2 km grid, while there is a progressive degradation of the timing and the extension of the development for coarser grids. Deng and Stauffer (2006) present sensitivity experiments with the non-hydrostatic model MM5 at 4 km resolution. The use of a convective parametrization improves the results, despite the fact that the 4-km resolution violates the underlying assumption of the two tested parametrization schemes that the size of subgrid deep convection is well below the grid-box length. They show that a convective scheme is required because the explicit microphysics alone cannot represent deep convection on a 4 km grid: convective updraughts are forced on a coarser-thanrealistic scale; the rainfall and the atmospheric response are too strong; and the evaporative cooling and the downdraughts are too vigorous, causing widespread disruption of the low-level winds and spurious advection of the simulated tracer. Grabowski (2001) proposes a completely different approach, the cloud-resolving convection parametrization, consisting of applying a twodimensional (zonal vertical) cloud-resolving model with a resolution around 1km, inside each mesh of a largerscale model, with horizontal grid-length around 100 km. This approach is mainly relevant to climate models; in operational forecast, the three-dimensional description of smaller features remains essential. The present work uses the hydrostatic version of the Aladin model, and this may explain that a convective parametrization is required down to 2 km mesh sizes. The convective scheme we presented in (Gerard and Geleyn, 2005) removed assumptions that would be violated at high resolutions, so that it would be a good candidate to complement Deng and Stauffer s study. But this is not sufficient, because there would still be competition between two schemes generating precipitation in different ways. The solution presented in this paper works at all resolutions (for instance, as coarse as 100 km or as fine as 2 km, including all intermediate resolutions), thanks to a coherent coupling of convection with the resolved cloud and precipitation scheme including a microphysical package with prognostic liquid and ice cloud condensates and avoidance of the problems of double counting. In section 2, we describe the general organization of our package. Section 3 gives more details on its main components. In Section 4 we comment on the sensitivity to the different parameters. In Section 5 we present two case studies at different resolutions, and in Section 6 we present our conclusions.

3 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 3 2 Components and hypotheses 2.1 General layout A numerical weather prediction model computes the evolution of model variables corresponding to the mean values of the meteorological fields over the area or volume of its grid boxes. Physical parametrizations are needed to evaluate the effects of the different subgrid processes on these model variables. The corresponding physical tendencies, act as source terms in the mean-flow dynamical equations. Since all processes are acting simultaneously, a first possibility it to call the different parametrizations in parallel, based on the same initial state ( parallel split ). On the other hand, the physical processes are interacting with one another, so that a sequential call of the parametrizations, each of them working on an updated state can be justified. An ordering of the processes has then to be chosen. To take into account the two-way interactions between the processes, a symmetrized sequential-split method can be used, where a fractional updates are performed in sequence until the last process is reached, and residual updates are then computed in reverse order. This approach is quite expensive. Dubal et al. (2004) show that the parallel-split method can introduce significant errors on the steady state when an implicit discretization is used. The sequential split can, under certain conditions, yield an accurate steady state. However, the maintenance of the organization of the different parametrizations is significantly less flexible than with the parallel split. In Arpège-Aladin, some of the parametrizations are accurate to the first order anyway and the modularity is found to be more important than the steady-state accuracy. What is essential is the coherence between location and time: the physics is computed at the origin point of the semi-lagrangian trajectories and at the time t t. So the turbulent diffusion scheme, the radiation scheme, the cloud and precipitation or the surface scheme are all working in parallel. A detailed discussion of the Arpège-Aladin time-stepping has been done by Termonia and Hamdi (personal communication, submitted paper). Now, if the different schemes implied in the moist physics are called in parallel, referring to the same (not yet balanced) mean grid-box state (water contents and phases, temperature, pressure, wind), each of them will produce a response that ignores the work of the others towards the final state; when combining these contributions, the response is excessive, implying a multiple counting of some phenomena. This is the main source of the difficulty we had in (Gerard and Geleyn, 2005) when trying to combine precipitation from separate large-scale and subgrid schemes. On the other hand, the physical processes occurring inside a package are often cascaded in time. Within the microphysical package for instance, transient values of cloud condensates (and other variables) after condensation are passed to an autoconversion calculation, which modifies them; the result is then passed to the calculation of collection and evaporation melting processes. If we want to obtain a wholly coherent treatment of cloud and precipitation processes, we must consider these as forming part of a single package: that is, we must apply them in cascade rather than in parallel. In a way, this is a step towards the sequential-split, with an ordering of the processes based on physical considerations. To this end, we introduce internal variables for the water phases and the temperature that copy the initial state of the time step but are subsequently updated by each scheme of the package to yield the initial state for the next part. The cascaded parts (see Figure 2) are: the turbulent diffusion; the resolved condensation; the convective updraught; the autoconversion of condensates to precipitation; the evaporation, melting and collection processes associated with precipitation; and finally a moist downdraught (associated with cooling by precipitation evaporation and melting). Clean handling of the closure contributions is also important to prevent double counting with the cascading part of the scheme. The source of condensation is not the same for the resolved and the subgrid schemes. The resolved scheme condenses the excess water vapour that is present at the beginning of the time step. The convective updraught scheme essentially condenses the excess water vapour brought to the grid box during the time step by the resolved convergence of moisture. The local vertical turbulent diffusion of moisture may be either added to this moisture convergence or put in the initial state before both condensation schemes (see section 3.3). 2.2 Geometrical subdivision The package distinguishes different fractions of the gridbox area, associated with different properties. The resolved condensation produces a resolved or stratiform cloud fraction f st. The updraught covers a fraction σ u, and detrains condensates into a fraction σ D, of the grid-box area. The resulting subgrid or convective cloud fraction is f cu = σ u + σ D. The total cloud fraction at a given level is taken as f = f cu + f st f cu f st. (1) Below we also define an equivalent cloud fraction f eq linking the mean in-cloud condensate densities with the mean grid-box condensate densities. We assume that precipitation falls over a fraction σ P of the grid-box area, equal to the maximum of f eq over the layers above. 2.3 Water variables and fluxes Several parts of our microphysical scheme have been based on that of Lopez (2002). Unlike him, we do not use separate variables for the precipitation contents, preferring a much simpler (and lighter) approach based on the precipitation fluxes (section 3.4.2). The scheme uses 3 prognostic water species (which are advected from one time step to the next by the resolved

4 4 L. GERARD Figure 1. q i F li q l F vi F vl Fis q Flr v Frv F sv Water fluxes in the integrated scheme. Fluxes in dashed lines are derived from the others. flow), in the form of specific contents: vapour (q v ), cloud ice (q i ) and cloud droplets or liquid (q l ). Below, we also use the total condensate q c = q i + q l. The precipitation contents are directly related to the precipitation fluxes (snow or solid P s and rain P r ), which are diagnosed at each time step. All parts of the parametrization produce contributions to the transfers between the different phases as presented in Fig. 1. Fluxes qualified below as net may occur in either direction, and are taken positive from the first to the second index. The fluxes shown on Fig. 1 are: F vi, net condensation to ice; F vl, net condensation to liquid; F li, net condensate freezing; F is, generation of solid precipitation from cloud ice and F lr, generation of rain from cloud droplets; F sr, net precipitation melting; F sv, snow evaporation; and F rv, rain evaporation. To convert cloud droplets into solid precipitation, we first convert them to ice with F li and then the ice to solid precipitation with F is. The heat fluxes associated with phase changes can be derived from the water fluxes by using appropriate latent heats. The evaporation of solid and liquid precipitation, F sv and F rv are obtained from the budgets: P P s = F is F sv F sr and P r = F lr F rv + F sr. The precipitation melting flux is zero at the top of the atmosphere, and receives an increment at the levels where precipitation melting takes place: F sr = { ( P r P s ) ( F lr F is ) r F sr P s (2) + ( F rv F sv ) } /2. (3) Precipitation melts rapidly after crossing the triple-pointtemperature level. Assuming that across the transition the evaporation of snow F rv and rain F sv are equal, the melting flux F sr may be deduced from Eq Tendencies In Arpège-Aladin, the physical routines output vertical diffusive fluxes, whose vertical divergence contributes to the mean grid-box tendency. For a model variable ψ these fluxes include turbulent diffusion fluxes Jψ td, convective transport fluxes Jψ cu, fluxes associated with phase changes (for water variables or heat), radiation (for heat), drags (for heat or momentum), and so on. In the scheme described in (Gerard and Geleyn, 2005), precipitation was directly generated from water vapour condensation and there was no suspended cloud phase. Therefore we could directly bind the precipitation flux and the associated heat flux through a bulk latent heat (depending on the option taken for mass conservation ). This is no longer possible here, because precipitation is generated from the (usually adiabatic) conversion of cloud condensates to precipitation, while the reduction of the precipitation is associated with diabatic processes of evaporation. Instead, we consider the local latent heat (assumed to be a function of the temperature). Conservation is guaranteed by the clean formulation of all water fluxes and the associated heat exchanges. In the current version we assume that falling precipitation is replaced by an equivalent quantity of dry air from the surface (the mass conservation hypothesis ). A more precise barycentric formulation is now being developed (Catry et al., 2007). 2.5 General organization of the package Figure 2 shows the sequence of the calculation. The resolved advection calculation can produce at some places non-physical values of the water contents: negative contents, or non-zero ice above the triple-point temperature. This must be fixed before performing any further calculation with them, and the corrections must also be transfered to the physical tendencies of the model variables. Since the bad values result from numerical rather than physical processes, we consider that the subsequent fix must remain adiabatic. To fix the internal state, the missing condensates are taken from the water vapour, and the negative water vapour values are replaced by zero. To reflect this on the model variables, a corrective water vapour diffusion flux Jv cor is added (at the end of the parametrizations) to the turbulent diffusion, fetching the missing vapour from below and even from the surface. Condensate correction fluxes Jl cor, Ji cor are similarly added to the condensate turbulent diffusion fluxes but in this case they are simply corrections, the missing condensate having been taken from the vapour. The resolved cloud fraction f st is presently computed using the scheme of (Smith, 1990) (see section 3.2). The model may either use a simple radiation scheme (inspired by (Ritter and Geleyn, 1992)), called at each time step, or a more elaborate one (Morcrette, 1991), called less frequently. In the first case, the stratiform cloud fraction f st is directly combined with the convective cloud fraction f cu kept from the previous time step, yielding a total cloudiness value f to pass to the radiation scheme. Radiation affects the surface temperature which is passed to the turbulent diffusion scheme. This scheme computes turbulent fluxes of conservative variables: total water q t = q v + q c, liquid static energy s l = s Lq c,

5 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 5 (Fix negative contents from advection) (Adiab. phase adjustment) [q v,q i,q l] ւ Resolved cloud fraction ց f st, f cu f (Radiation) Jcor l, Ji cor, Jv cor (T surf, Turbulent diffusion) Jl td, Ji td, Jv td, Js td [(qv,)q i,ql,t ] ւ moisture convergence Resolved condensation Fvi, st Fvl st [qv,q i,ql,t ] ւ Deep convection f cu = σ D + σ u α cu f eq [q v,q i,q l,t ] F hp ւ ց α cu ւ F cu vi, F cu vl, J cu v, J cu i, J cu J cu s, J cu V Autoconversion F is, F lr, F li [q i,q l,t ] ւ l, Precipitation F is, F lr, F li, P r, P s [q v,q i,q l] ւ Downdraught [q v,q i,q l] ւ (Final diabatic phase adjustment) (Convective/ Resolved) F li P r, P s, Jv cu, J cu Js cu, JV cu Pcu r, Ps cu, Pr st, Ps st Figure 2. Package organization chart. The square brackets mark the successive updates of the internal state momentum. The turbulent fluxes of water vapour, cloud condensates and dry static energy are finally derived from these. The internal state is updated following the turbulent diffusion. An alternative would be to combine the water vapour turbulent diffusion flux with the resolved moisture convergence flux in the closure of the updraught. In this case, only the internal variables of temperature and condensates should be updated with the turbulent diffusion before entering the updraught routine. The resolved condensation is described in section 3.2. Again, it updates the internal variables. The resulting state is input to the updraught (section 3.3), which produces convective condensation fluxes (Fvi cu, F vl cu) and convective transport fluxes of water (Jv cu, Jl cu, Jcu i ), heat (Js cu ) and horizontal momentum (J cu ). It also updates the V internal state, and, if necessary, prevents the specific contents of condensates from being negative by an additional condensation of water vapour. After the updraught, an upto-date value f of the total cloud fraction is obtained, combining f st and f cu = σ D + σ u. We define the convective fraction as the ratio of the convective to the total condensation flux: α cu = (F cu vl + F cu vi )/(F cu vl + F cu vi + F st vl + F st vi), (4) and we use it to decide what fraction of the condensates, as well as of the total precipitation flux may be declared convective. Autoconversion of condensates to precipitation and condensate collection by precipitation, depend on the local densities of the condensates, not on the mean gridbox densities. The density of condensate in the convective clouds may be significantly higher than in the resolved clouds, because the ratio f cu /f is often much smaller than α cu. The local specific condensate content is estimated by ˆq c = q c { (1 αcu ) 2 f st } [ + α2 cu f st + f cu f cu f ] = q c, (5) feq where f eq is an equivalent cloud fraction that would i, Jl cu occur, if the condensate density was the same in all clouds in the grid box. The factor in square brackets prevents f eq from being greater than f. The autoconversion algorithm (section 3.4.1) receives the values of moisture, condensates and temperature as output by the updraught, together with the equivalent cloud fraction. It computes the contribution of the nucleation, coalescence and the Bergeron-Findeisen processes to the precipitation generation fluxes, the third one also implying a a condensate freezing flux. The box labelled precipitation computes the collection and evaporation effects. It outputs the precipitation fluxes, and contributes to the precipitation generation fluxes and also to the condensate freezing flux where the riming process is active. The downdraught scheme (section 3.5) is driven by the heat sink associated to falling precipitation evaporation, melting or heating. The corresponding heat flux F hp is passed to the downdraught, which uses it in its closure. To avoid double counting, the internal temperature input to the downdraught is not yet affected by this flux. The downdraught further modifies the precipitation fluxes, the water vapour, and the convective transport fluxes. A final diabatic phase adjustment of the condensate is performed before the advection to the next time step, while the precipitation flux is partitioned between convective and resolved precipitation: P cu = α cu P, P st = (1 α cu )P. (6)

6 6 L. GERARD 3 Components of the integrated scheme 3.1 Mixed phase partition We assume that the fraction of ice α i in the mixed phase is a function of the temperature only: [ ] α i (T) = 1 exp (T t min(t t,t)) 2 (7) 2(T t T x ) 2 where T t = K is the triple-point temperature and T x is the temperature of the maximum difference between the saturation vapour pressures with respect to ice and to liquid. The water vapour condenses into ice and liquid phases according to this ratio, computed using for T the mean grid-box temperature T for the resolved part, and the updraught temperature T u for the subgrid part. But the autoconversion and collection processes, as well as the advection by the large-scale flow, modify the ice fraction. As shown on Fig. 2, we readjust the phases to the ratio α i after both of these processes, the correction contributing to the freezing-melting flux F li. This systematic readjustment is convenient for controlling the biases due to advection, but since it reduces the independence of the two cloud condensate variables, we intend to relax it in a future version. 3.2 Resolved cloud fraction and condensation The scheme for the resolved cloud fraction and condensation is based on (Smith, 1990). It removes the resolved saturation by condensing the excess moisture. We split the scheme in two parts: the first part outputs a stratiform cloud fraction, f st ; the second part, which is called after the turbulent diffusion scheme, updates the internal state and outputs a stratiform condensation flux Fvc. st The scheme assumes a triangular probability distribution function for the total water specific content inside the grid box. The saturation moisture is calculated, based on the mean grid-box temperature T and pressure p, according to the phase partition α i (T). Thus the variation of the saturation moisture following the local increase of the temperature induced by condensation is neglected. If heating by condensation becomes significant, the calculation becomes invalid. Currently we limit the resolved condensation in one time step at any level to the amount that implies a mean grid-box heating of 1K. More heating would require the use of an iterative formulation; with our limitation, we avoid unrealistic condensation and let the adjustment take place in more than one time step. A linearization of the saturation moisture could also be envisaged. At the level of the triple-point temperature, the melting of the precipitation induces a local cooling; the mean grid-box temperature is lowered, as is the saturation moisture. As a result, condensation increases, and there is subsequent reheating. To represent this accurately we would need an iterative calculation. We have left in our scheme the possibility to prevent an unrealistic peak of condensate by simply (in addition to the limitation explained above) smoothing the relative-humidity profile around the triple point. Starting from the top and moving downwards, at each level where the Celsius temperature is positive we replace the relative humidity by that obtained by mixing the air with the adjacent levels, when this mixing is not unlikely: we mix two or three levels below if they would be buoyant when raised along a dry adiabat, and two or three levels above if they would sink when moved down along a dry adiabat. At the lowest model level, a cooling can occur from the precipitation evaporation and the downdraught. Subsequent exchanges with the surface should temper this cooling, but again this would require an iterative calculation. To prevent unrealistic condensation, we compute the saturation moisture at the lowest model level using the arithmetic mean between the surface and air temperatures. To prevent excessive variations of the intensive condensate values (Eq. 5) when the resolved scheme is active alone, we impose that the cloud fraction be non-zero wherever q c is not negligible; in addition, we limit its increment or decrement f st between adjacent model levels to a tunable maximum value. 3.3 The updraught mass flux scheme The prognostic updraught scheme described in (Gerard and Geleyn, 2005) has been revised extensively. In the following, the term environment designates the part of the grid box which is outside the updraughts, and the updraught is a composite representation by a single mass flux of all the updraughts present in a grid box. Eventhough the new scheme does not explicitly check the conservation of the vertically integrated moist static energy, this is achieved through the clean construction of the scheme, as we will show in section Precipitation In the current scheme the updraught does not produce precipitation by itself, but it contributes to the gross cloud condensates passed to the autoconversion routine Mass-flux transport scheme In (Gerard and Geleyn, 2005), we expressed the contribution of the deep convection scheme to the physical tendency by means of a pseudo-subsidence (associated with the channelling effect of the updraught, the actual vertical velocity in the environment being much smaller than in the updraught) and a detrainment of the cloud material into the environment. Piriou (2005) and Piriou et al. (2007) proposed a much more direct formulation through the convective transport fluxes Jψ cu and the net convective condensation-evaporation fluxes, whose vertical divergences directly contribute to the tendencies. This formulation avoids the calculation of pseudo-subsidence, as well as the difficult problem of estimating detrainment while using a prescribed entrainment profile. We use it here, as it brings a substantial gain of accuracy, and is

7 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 7 a more logical method now that the scheme includes an evaluation of the condensation Updraught profile The updraught profile is obtained by alternating saturated pseudo-adiabatic ascent segments with isobaric mixing. The decrement of water vapour resulting from the ascent equals the total condensate increment, whose accumulation yields the convective condensation flux as explained below. In (Gerard and Geleyn, 2005), the condensate staying in the updraught was estimated through the relation (q vu + q cu ) φ = q cu φ 0, (8) where φ 0 was a cloud critical thickness beyond which the condensate was assumed to precipitate. We apply here the same kind of limitation to the updraught condensates, but their excess is supposed to be detrained instead of precipitated. We also distinguish ice clouds from liquid clouds, with a bigger critical thickness in ice phase. The mixing of air from the environment with the cloud air works the same way as in (Gerard and Geleyn, 2005), with a diagnostic entrainment depending on height and the local vertical integral of the buoyancy. The condensate contents in the updraught environment are taken to be equal to the mean grid-box values, assuming (for lack of a better solution) no a priori spatial correlation of the clouds at the subgrid scale with the updraught. The virtual temperature in the updraught T vu and the environment T ve take into account their respective condensate contents; if these are different, the buoyancy force is affected. A cloud layer is declared active when there is both upward buoyancy and moisture convergence Prognostic variables and closure As in (Gerard and Geleyn, 2005), we use prognostic variables for the updraught vertical velocity ω u and its mesh fraction σ u. The presence of the condensates does not affect the shape of the equations. The prognostic closure is based on the convergence of water vapour towards the grid box. In (Gerard and Geleyn, 2005) we added the contribution of the local turbulent diffusion. Here, if we use this contribution to update the initial profile passed to the updraught, we no longer have to include it in the closure Detrainment area The contents of the detrainment area must be combined with the output of the resolved condensation scheme, and they form the main part of the convective cloud fraction. Hence we need an estimation of the fraction σ D of the grid-box area covered by them and of their condensate concentration q cd. If δσ D is the detrainment area extension over the time step t, we have D cu t q cu = δσ D q cd, (9) where the condensate detrainment rate D cu is obtained by a local mass budget in the updraught. As a first guess, we take q cd = q cu ; if it yields too large a detrainment area (σ D > (1 σ u )), then we increase q cd. This one is a gross quantity whose major part will be precipitated when passing through the microphysical package Outputs of the updraught routine The convective condensation fluxes are obtained by accumulating the water vapour decrements q va along the successive ascent segments. Noting M u = σ u (ω u ω e ) the convective mass flux, the increments to the condensation fluxes over a model layer are given by F cu vi = α i q va M u /g, F cu vl = (1 α i ) q va M u /g. The updraught transport is given by ψ t = p M u(ψ ψ u ) = g Jcu ψ p, J cu ψ = 1 g M u(ψ ψ u ) (10) (11) with M u > 0. An implicit discretization is used to ensure stability. Finally, the internal variables are updated, so that they include at this stage the effects of both resolved and subgrid condensation. The convective cloud fraction f cu = σ D + σ u is combined with the resolved cloud fraction f st to yield the total cloud fraction f. 3.4 Precipitation generation Autoconversion This part represents the effect of the initial growth and collision processes allowing the conversion of the cloud particles to falling precipitation. In this section we work with local, in-cloud specific contents, and we drop the hats in the notation: q i = q i /f eq, q l = q l /f eq (12) The scheme proposed by Lopez (2002) used a formula of the type described by Kessler (1969), which is written, for autoconversion of liquid cloud water to rain: dq l dt = E l(q l q l) (13) where E l is the autoconversion efficiency, and q l is a threshold below which no conversion occurs. A similar formula was used for the conversion of ice to snow, but with an efficiency E i which was a function of the temperature. In the mixed phase, the threshold of autoconversion was simply reset to zero.

8 8 L. GERARD Van der Hage (1995) proposed a more general formula based on the volumetric concentrations N c = N l + N i : N c t = φ 1 Nl 2 φ 2 N l N i φ 3 Ni 2 (14) = φ 1 Nc 2 [ (1 ni ) 2 + Gn i (1 n i ) ] φ 3 n 2 inc 2 where n i = N i /N c is the ice fraction, supposed to remain unaffected by the processes, and G is an autoconversion gain associated with the Wegener-Bergeron-Findeisen (WBF) effect occurring in the mixed phase. (Because of a lower saturation pressure, vapour condenses on ice nuclei while evaporating from droplets: some droplets vanish while the bigger ice particles are removed by precipitation). Outside the mixed phase, Kessler s formulation may be seen as a local linearization of Eq.14. Considering droplets of mass m d, ice crystals of mass m i, and noting ρ the local air density, we have the relations q l = N lm d ρ, q i = N im i, α i = q i (15) ρ q i + q l In the liquid phase, assuming that the average mass of individual droplets does not vary during the time step, the linearization of the parabola from a concentration N l is written q l t m d N l m d = φ 1 ρ t ρ N2 l 2φ 1 Nl m d ρ (N l N l 2 ) E l(q l ql) (16) We see that the autoconversion efficiency E l and the threshold ql are both proportional to Nl : hence it is inconsistent to set the threshold to zero, because the linearization of the parabola at its top yields zero. For this reason, we propose to keep the Kessler formulation with its threshold in pure ice and in pure liquid phases, while finding another expression for the mixed phase, where the WBF effect is predominant. For the total condensate, the ice and the droplets, the WBF term alone is written: N c = Gφ 1 N i N l, t N l = (1 n i ) N c t t N i t = n i N c t, (17) To translate these relations to specific contents, we assume that the mass of individual hydrometeors does not vary during the time step: the droplets evaporate at once; the ice particles grow at once to a precipitable size; and only the number concentrations vary. Then q l t m d N l ρ t q i t m i N i ρ t = Gφ 1 q l q i ρ 1 n i m i, = Gφ 1 q l q i ρ n i m d. Hence, q c t = Gφ 1q l q i ρ( 1 n i = m i + n i m d ) Gφ 1 ρ (18) m i + α i (m d m i ) q lq i The underlined term is a function of α i. If we assume that the ice particles are bigger at the bottom than at the top of the mixed layer, we may replace this whole term by a linear function of α i. This yields: q c t = G 0 (1 + G 1 α i )q i q l (19) WBF We use this formula for the mixed phase (with the tunable parameters G 0 0.5, G 1 0.5). The autoconversion tendencies are then q l t = E l(q l ql) + (1 α i ) q c t WBF (20) q i t = E i(t)(q i qi q c ) + α i t WBF Collection and evaporation We assume that the generation of precipitation at any level immediately affects the precipitation flux at all the levels below. This would be the case with an infinite fall velocity but also if the autoconversion process only varies slowly in time, so that what is presently generated does not differ too much from what was generated some time ago and has now reached the lowest levels. The gross precipitation-generation flux resulting from autoconversion is partitioned into a gross solid precipitation flux P s0 and a gross liquid precipitation flux P r0, assuming that the melting occurs over a few levels when reaching the triple-point temperature. Precipitation falls over a fraction σ P of the grid-box area, taken equal to the maximum of the equivalent cloud fraction f eq over the layers above. The instantaneous densities ρ r and ρ s of rain and snow in this area and at a model level l are then (if l represents the lower boundary of level l): ρ l Pr0 l + Pr0 l 1 r = σ P, ρ l Ps0 l + Ps0 l 1 s = σ P (21) 2w r 2w s where w r and w s are the fall speeds of rain and snow respectively. Here, contrary to the estimation of the gross precipitation flux, we consider finite velocities (around 0.9 m s 1 for snow, and 5 m s 1 for rain). Evaporation is likely to occur in the clear part of the precipitating area: this is given at any level by (σ P f). The evaporation and collection processes are computed as in the scheme of Lopez(2002), using the densities defined by Eq. (21). The major differences are that we impose a fixed relation between the precipitation fluxes and the precipitation contents, and that we avoid his expensive vertical-advection calculation.

9 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 9 Another difference is in the riming process: this converts liquid condensate to solid precipitation, which we translate into a condensate-freezing flux followed by a solid-precipitation generation flux. The heat associated with this freezing may then be properly accounted for in the scheme. 3.5 Moist downdraught The moist downdraught is calculated after the microphysics, and results from the heat sinks accompanying the precipitation: evaporation, melting, and vertical advection. The downdraught occurs in the precipitation area. The vertical velocity in its environment is taken equal to the mean vertical velocity outside the updraught, ω e Downdraught profile The computation for the downdraught is similar to that for the updraught, being composed of saturated pseudoadiabatic downward segments alternating with isobaric mixing. The entrainment rate is assumed to be constant. The local latent heat of evaporation follows the phase of the precipitation, which presents a quick transition at the triple point. The profile construction yields a precipitation-evaporation flux. In principle we cannot have saturation in the downdraught, but condensate may exist temporarily; for the virtual temperature we assume the same condensate content as in the environment. Downdraught activity is decreed where there is negative buoyancy and the downdraught is colder than the wet-bulb temperature of the environment Prognostic schemes The downward acceleration results from the balance between negative buoyancy and drag. The initial velocity of the entrained parcels is difficult to guess: the precipitation itself has a downward velocity, but a return upward current is intertwined with it. Currently, we suppose the original vertical velocity of the entrained material to be ω e (the updraught environment), but it would be possible to modulate this in function of the precipitation fall speed. As the downdraught approaches the surface, its flow has to bend, under the influence of the local high created by the accumulation of air near the surface, and eventually take the horizontal direction. To avoid the complication of calculating three-dimensional effects, we introduce an additional braking term in the one-dimensional vertical equation, representing the effect of the local high. The equation then becomes ω d t = negative buoyancy drag aω2 d (p s p) b, (22) where p s is the surface hydrostatic pressure, ω d the (absolute) downdraught vertical velocity (positive downwards, in Pa s 1 ). We presently use b=2 so that a represents a reference pressure thickness above the surface for decelerating ω d. Note that p s is always higher than the pressure at the full model levels where ω d is calculated, and ω d is prevented from taking negative values. The downdraught mesh fraction σ d is assumed to be constant over the whole height of the downdraught, and is obtained with the prognostic equation σ b d t dp (h d h e ) + (k d k e ) g } p t {{ } storage p b = (ω d ω e ) dp F b +ε ρg g } MHS {{}, (23) p t input }{{} consumption where h is the moist static energy and k the kinetic energy. This states that a fraction ε of the microphysical heat sink MHS either contributes to the downdraught activity (work of the buoyancy force F b ) or is stored in an increase of σ d output fluxes and properties The transport fluxes are similar to those of the updraught. The downdraught evaporation flux induces a reduction of the precipitation fluxes (Eq. 2) 4 Single column model tests For a first test and sensitivity study, we have performed a set of experiments with the single-column (SCM) version of Aladin, using the TOGA-COARE dataset (e.g. Bechtold et al., 2000) to provide a forcing every hour. This forcing consists of mean profiles over a radius of 111km, produced by a cloud-resolving model. This experiment is suitable for observing the onset of deep convection and the associated precipitation. The cloud condensates are not coupled and are initialized to zero. The time step is 10 minutes, in an Eulerian leap-frog scheme. Fig. 3 shows the vertical profiles of cloud condensates and of the updraught and downdraught mass-fluxes at 3 different forecast ranges. The 0 o C isotherm is at a height of around 4,5km, which is the lower boundary of cloud ice (Fig. 3(a)). Above this level, q i takes values up to 24 mg kg 1. The cloud droplets are observed up to a height of 11km, in agreement with the mixed phase delimitation between 0 and -40 o C. The maximum value of q l can reach 85 mg kg 1, i.e. concentrations 3 to 4 times higher than q i. Towards the end of the run (400 min), we observe a peak of condensate below the 0 o C isotherm, because the local cooling from rapid melting of the precipitation flux induces a local cold anomaly (which at this range has extended to the mean grid-box value, in the absence of horizontal transport in the SCM) reducing the saturation moisture. A similar peak appears in the updraught mass flux (Fig. 3(b)), the locally cooler environment increasing the buoyancy. The updraught mass flux reaches values around 0.1 kg m 2 s 1. The downdraught mass flux is around one-tenth of the updraught

10 10 L. GERARD Figure 3. Evolution of SCM profiles in time. (a) Cloud ice (upper part) and droplets (lower part) at forecast ranges 100 min (dashdot), 250min (dash) and 400min (solid) (mg kg 1 ). (b) Updraught mass flux (kg m 2 s 1 ), same ranges. (c) Downdraught mass flux (kg m 2 s 1 ). The horizontal dotted lines represent the limits of the mixed phase (0 and -40 o C). Figure 4. (a) Contributions to cloud fraction: σ D (dot), f cu (alternated dashes), f st (dash-dot), f (dash), f eq (solid). (b) Evolution of surface precipitation: total and subgrid parts, with time steps 600s (solid/dash) and 180s (alternate dash/dash-dot). (c) Updraught mass flux after 6:40h forecast, with time steps 600s (solid), 300s (dash) and 180s (dash-dot). mass flux, and its top is much lower. For the 400 min range we observe that a first downdraught, starting at around 5.5 km, is stopped when passing the 0 o C isotherm, because the local cold anomaly cancels the negative buoyancy. A new downdraught flux restarts immediately below, extending down to the surface. The reduction of the mass flux towards the surface is associated to the deceleration in the downdraught prognostic equation (22). The components of the cloud fraction at a range of 400 min are shown in Fig. 4(a). The subgrid cloud fraction f cu = σ D + σ u 0.4 is close to the detrained fraction σ D. The resolved fraction f st 0.6 to 0.8 is here slightly larger. The total f is computed with Eq. (1). The solid line shows the equivalent cloud fraction used to estimate intensive condensate concentrations (Eq. (5)): this is always smaller than f and it allows us to account for more important concentrations in the convective clouds than in the resolved ones. Fig. 4(b) and (c) show the effects of varying the integration time step. With a shorter time step the subgrid precipitation (Fig. 4(b)) is weaker, associated to a smaller updraught mass flux (Fig. 4(c)). However, the total precipitation is nearly unchanged, because there is a compensation between the subgrid and the resolved condensation schemes. At the beginning of the run, there is only subgrid precipitation; the resolved precipitation starts after around 3 hours integration. In three-dimensional tests, we observed a similar delay of the cloud condensates and precipitation when the convective parametrization is switched off (see Fig. 10). This long spin-up of the resolved scheme also explains the small deficit of precipitation in the first three hours here, when using the shorter time step. We mentioned above that the clean construction of the scheme ensures the conservation of the verticallyintegrated moist static energy in the updraught (and it is also the case for the downdraught). Fig. 5 shows the contribution of the updraught to the apparent heat source Q 1 and moisture sink Q 2, as introduced by Yanai et al.

11 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 11 Figure 5. SCM profiles after 6:40 h run. Contribution of the updraught to Q 1 and Q 2 (K h 1 ). (a) Q 1 (dash), Q 2 (dot), Q 1 Q 2 (solid). (b) Decomposition between the condensation-evaporation term (dash), and the transport terms for dry static heat (solid) and moisture (dash dot). (1973). Here, they are computed as: ( ) s ω c p Q 1u = = L(c e) (s ), t u p ( ) (Lqv ) ω c p Q 2u = = L(c e) + ((Lq) ). t u p (24) where (c e) is the net condensation-evaporation and index u stands for updraught. The inclusion of the latent heat in the vertical divergence is necessary to take into account its variation with the local temperature and phase. We use here the pressure on the vertical axis, to make it more apparent that the vertical integral of Q 1 Q 2 (Fig. 5a) is zero: t b (Q 1u Q 2u ) dp g = 1 t c p b h dp = 0. (25) t g This equation states that the updraught induces a vertical reorganization of moisture and heat, while conserving the total moist static energy. (The limits of the integral are the bottom and the top of the updraught.) Fig. 5(b) shows that the large variations along the vertical are associated with transport. Vapour and heat are transported from levels hpa to the levels hPa, and from levels hpa to the levels hpa, concerned by the cold (and dry) pool induced by precipitation melting; conversely, the pseudo-subsidence in the updraught environment cools and dries levels at hpa. Further diagrams, including ones showing the evolution of the internal moisture variables and the temperature tendencies are given in section 5, for a three-dimensional experiment. The sensitivity to the tuning of the parameters remains quite limited. To illustrate this, we present effects of the updraught entrainment, the downdraught closure and the Bergeron-Findeisen parameters. In Fig. 6 we have tried to modify the settings of the updraught entrainment. The equations were given in (Gerard and Geleyn, 2005). The tuning is quite indirect, as appears on Fig. 6(a): dividing the minimum entrainment by two increases the maximum entrainment in the lower layers, and multiplying it by two has the opposite effect. The updraught mass flux (Fig. 6(b)) shows a small increase where the entrainment is bigger, a small decrease where it is smaller. The updraught vertical velocity w u (Fig. 6(c)) is decreased by a bigger entrainment (because the entrained air has to be accelerated), and increased by a smaller one. All these effects are small and have very little impact on the surface precipitation. The downdraught closure (Eq. 23) assumes that the downdraught uses a fraction ε of the cooling associated to the precipitation evaporation, melting and transport. This parameter has a direct impact on the downdraught mass flux, as illustrated in Fig. 7(a). However, in SCM, the impact on the precipitation evolution is unclear, and very small.(moreover, the downdraught remained very weak in these experiments.) We observe a greater impact in threedimensional runs (see section 5.2). To see an example of the microphysical tunings, we take a closer look at the new parametrization of the Bergeron-Findeisen effect we have developed (Eq. (19)). Operationally we use G 0 =0.5 and G 1 =0.5. Setting G 0 =0 (Fig.7(b)) deactivates the parametrization. There is then a significant accumulation of cloud ice between 6 and 9 km. On the other hand, with G 0 =1, the cloud ice is reduced to nothing at those levels. Fig. 7(c) keeps G 0 =0.5. With G 1 =0, the downward decrease of q i is slow, while with G1 = 1 it is a little too quick. Again, the value 0.5 seems a good compromise. The other parameters in the microphysics have been tuned in a similar manner, to obtain realistic profiles of condensates. None of them significantly affects the surface precipitation. 5 Three-dimensional validation Deep convection is composed of cells with a diameter of a few kilometers, which may interact with each other

12 12 L. GERARD Figure 6. SCM profiles after 6:40 h run. Variation of entrainment parameter: reference (solid), E n 2, (dash), E n/2 (dash-dot). (a) relative entrainment profiles; (b) updraught mass flux profile (kg m 2 s 1 ); (c) Updraught vertical velocity (m s 1 ). Figure 7. SCM profiles after 6:40 h run. (a) Effect of downdraught closure parameter (Eq. 23) on downdraught mass flux (kg m 2 s 1 ). Reference profile : ε=0.25, (solid), ε=0.5 (dash). Effects of the Bergeron-Findeisen parameters (Eq. 19) on the cloud condensate profiles (mg kg 1 ): (b) G 0=0: q i (solid), q l (dots), G 0=1: q i (dashes), q l (alternate dashes). (c) G 1=0: q i (solid), q l (dots), G 1=1: q i (dashes), q l (alternate dashes). in a wider convective system. In models with grid boxes longer than 7 10km, it can be assumed that the convective cells are subgrid, and their effects on the resolved model variables must be computed by a parametrization. However, the effects of the convective system extend over a much wider area, as far as the Rossby radius of deformation (Mapes, 1998). Unlike turbulence, convection is determined by both large-scale and local phenomena and the coupling of the parametrization with the resolved processes (which also provides the closure of the scheme) is essential. For this reason, the functioning of a package including a convective parametrization cannot be assessed thoroughly with single-column tests. With grid-box lengths below 2 km, one usually assumes that the convective cells are resolved by the model grid. In this case, deep-convection parametrization may be suppressed; even so, we will show below that it can be worth keeping it. At intermediate resolutions (between 7 and 2km), a parametrization is required, but it is complicated by the fact that a significant part of the process already contributes to the resolved cloud and precipitation processes. Our integrated package attempts to solve these difficulties, apparently with some success. Our goal was to access high-resolution operational numerical weather prediction while maintaining consistency across the whole range of resolutions, from the coarser (30 100km) to the finer (kilometre-scale) ones. 5.1 Convective case over Belgium: horizontal fields Intense convective showers and lightning were observed in Belgium on 10 September 2005 between 1600 and 2230 UTC. The satellite picture (Fig. 8(a)) shows the complex low-pressure zone, with a main depression (1005 hpa) on the Golfe de Gascogne, prolonged by a trough ( hpa) over western and central Europe. During the afternoon, a more stable flow from northeast remained over Denmark and the Netherlands, as far south as northern Belgium. The radar pictures (Fig. 8(b)) shows a mesoscale convective system moving slowly to the northwards.

13 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 13 (a) (b) Figure 8. Convective case of 10/9/2005, (a) NOAA infrared satellite image at 16:25 utc (b) 1h accumulated radar picture at 18:00 utc (a) (b) (c) 5 m/s 5 m/s 5 m/s max=12.1, mean= (d) max=23.3, mean= 0.51 (e) max=41.7, mean= 0.46 (f) 5 m/s 5 m/s 5 m/s max=16.2, mean= 0.55 max=36.1, mean= 0.71 max=38.4, mean= 0.53 Figure 9. Convective case of 10/9/2005, 6-hour forecast (for 18:00 utc). 1-hour cumulated precipitation (mm), mean sea-level pressure (hpa), 10-m wind. (a, b, c) Full package; (d, e, f) no convection (see main text). Resolution: (a, d) 6.97 km; (b, e) 4.01 km; (c, f) 2.18 km. With the operational configuration (using the diagnostic convection and condensation schemes with no microphysics), the model missed the event completely. We ran a set of experiments over a domain of around km with 41 vertical hybrid levels, starting from the analysis of 1200 UTC. The model was coupled to the results of the Aladin-France limited-area model, running at a resolution of 9.5km, itself coupled on the Arpège global circulation model. The coupler models where not rerun with our package. The horizontal resolutions were 6.97 km (time step 300s), 4.01 km (time step 180s) and 2.18 km (time step 100s). All other tunings were kept identical; in particular, the model dynamics was kept hydrostatic. We compare the results obtained when switching off the updraught and the downdraught parametrizations, so that the resolved condensation alone feeds the microphysics ( no convection lower three panels of Fig. 9), with those with the complete integrated package ( full package upper three panels of Fig. 9). The cloud condensate variables are initialized to zero (the coupling models, using diagnostic cloud schemes, had no such variables); there is actually no need to couple them because their advection has a minor impact. Comparison to the observed radar images The 1- hour accumulated precipitation on the radar image (Fig. 8 (b)) shows some very narrow and intense maxima. Such maxima are also observed on the 2 km forecasts (Fig. 9

14 14 L. GERARD (a) (b) (c) (d) (e) (f) Figure 10. Spin-up of the total surface precipitation, resolution 2.18 km. (a,b,c) no convection, 2h, 3h and 4h forecast range. (d,e,f) full package, 2h, 3h and 4h forecast range. Same palette as Fig. 9. (c)), though their location is a little different. Note that with no convection (Fig. 9 (d) and (e)), the precipitation at 7 km and 4 km stays essentially at the borders of Belgium, while the full package (Fig. 9 (a) and (b)), lets it enter more inside the country (also verified at other forecast ranges), in agreement with the 2 km forecast (Fig. 9 (c) and (f)). Effect of resolution on precipitation amounts The full package produces a gradual increase in precipitation with increasing resolution: this is consistent with the fact that the model outputs mean grid-box precipitation amounts, and that the area of the active systems becomes an increasing proportion of the grid-box area as the resolution decreases. In addition to this averaging effect, the correlation between high moisture and high vertical velocity is more finely represented at higher resolution: this may imply more condensation in some places and less at others. Finally, the model orography is less smoothed at high resolution; this may also affect the results. Structure and width of the precipitation areas With no convection scheme (Fig. 9 (b)), the model produces unrealistically wide areas of very intense precipitation: this is especially clear at the 4 km resolution, i.e. in the middle of the grey-zone. This corroborates the observation of Deng and Stauffer (2006), that when not handled by a specific parametrization, the convective updraughts are forced on a coarser-than-realistic scale, producing a too strong atmospheric response. The full package (Fig. 9 (a)) does not produce this excessive behaviour. At 2.18 km resolution, we still observe differences between the full-package run and the run with no convection. The location and geometry of the events is very similar, but the widening of the precipitation area is still evident. The source of the difference here is the convective transport, which is not well represented by the hydrostatic model dynamics. The prognostic convection scheme computes vertical acceleration and can handle some nonhydrostatic effects whereas the resolved scheme alone, with the hydrostatic model dynamics, cannot. At 2.18 km resolution with the full package, the convective condensation is of the same order of magnitude as the resolved part, while their sum (Fig. 9(c)) is is comparable to what is obtained with no convection scheme (Fig. 9(f)). This demonstrates good collaboration between the two schemes: the convective scheme supplies what the resolved scheme does not, and conversely. Consistence between resolutions At some forecast time ranges (not illustrated), the no convection runs produce some displacement of the maximum precipitation between the 4 km and the 2 km resolutions, while the full package runs simply produce no such variation, but only an intensification. Spin-up The effects of the convective transport are of particular importance for the model spin-up, when starting with cloud condensates initialized to zero. Fig. 10 shows that in the case with no convection, the precipitation is underestimated during the first 3 hour of forecast, whereas there is an overestimation at 4 hours, when the clouds

15 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 15 Figure 11. Vertical cross section (resolution 7km): (a) q i (dash) and q l (solid) (g kg 1 ), temperature (dot) ( o C). (b) Updraught (solid) and downdraught (dash) mass flux (kg m 2 s 1 ). (c) Profile of the updraught (solid) and downdraught (dash) vertical velocity (m s 1 ) along the vertical of the maximum mass flux. Figure 12. Evolution of the mean profile of the internal water variables (g kg 1 ) along the package calculations. Total cloud condensate q c: (a) initial value (solid line), after turbulent diffusion (dot), after resolved condensation (dash), after updraught condensation (dash-dot); (b) after autoconversion (solid), after collection processes (dash). Difference between the final and the initial condensate (dash-dot). (c) Water vapour differences: total difference between initial and final contents (solid), part due to resolved condensation (dash), part due to updraught condensation, transport and precipitation processes (dash-dot), part due to downdraught (dot). have finally built up and precipitated the excess moisture accumulated before. When the subgrid scheme is active, we have a much quicker spin-up (around half an hour). In this case, both schemes (subgrid and resolved) produce precipitation from the beginning. Thus the presence of the subgrid scheme at 2.18 km has two benefits: it allows a quick spin-up without coupling the condensates; and it allows hydrostatic dynamics (which is cheaper because the calculation is simpler and the time step may be longer) to be run. Pressure and wind fields The wind fields remain consistent between the different resolutions. The mean sealevel pressure field is less smooth at higher resolutions; this is a result of the detail in the surface pressure field (with a rougher mean orography) and the surface temperature field. 5.2 Vertical profiles and cross sections To better demonstrate the functioning of the package, we present here the vertical structure of different features. We have created an east west vertical section across a precipitation area at 1800 UTC in the full-package experiment at 7 km resolution. The section location is marked with a thick black horizontal line on Fig. 9(a). The cloud ice (Fig. 11(a)) rises up to 0.07 g kg 1, the cloud droplets up to 0.5 g kg 1. We observe a mixed phase above the 0 o C isotherm. The updraught mass flux (Fig. 11(b)) is up to 6 times bigger than the downdraught mass flux. The prognostic updraught and downdraught vertical velocities (Fig. 11(c)) take reasonable values (maxima at 10 m s 1 and -6 m s 1 ). Fig. 12, 13 and 14 present mean profiles over the 13 horizontal points of the vertical cross section. For cloud condensate q c = q i + q l (Fig. 12(a)), the initial profile (advected from the previous time step) is first moved slightly upwards by the turbulent diffusive transport. This is consistent with the upward turbulent diffusion flux Jl td, shown on Fig. 13(b). The resolved condensation scheme increases significantly the condensate contents (Fig. 12(a)),and an additional increment is provided by the convective updraught,

16 16 L. GERARD Figure 13. Mean vertical profiles of the transport flux (kg m 2 h 1 ). Fluxes are counted positive downwards. Water vapour: (a) J td v (dash- after updraught (dash) and dot), J cu v after updraught (dash) and final (solid). Cloud condensates: (b) J td l final (solid), J cu i (dash-dot). (c) J td+cor i (dot) (J td i (solid), J td+cor l 0 not drawn), Jl cu (dash). Figure 14. Mean vertical profiles: components of the temperature tendency. (a) convective transport (solid), turbulent diffusion (dash). (b) convective condensation (dash), resolved condensation (dot), liquid to ice conversion (dash-dot), total (solid). (c) total phase changes (dash), total transport (dot), brought by precipitation flux (dash-dot), final tendency (solid) associated to the moist processes. also in the upper part. Fig. 13(b) shows that the updraughts transport condensates upwards (Jl cu). Fig. 12(b) shows that the autoconversion process has reduced q c, and the collection processes reduce it further. The relative differences between the final and initial values of q c remain less than 10%. The relative variations of the water-vapour profile are small, so we prefer to plot the differences in q v between different stages of its internal evolution (Fig. 12(c)). The turbulent diffusion and the resolved condensation induce a decrease of q v in the cloud, but also a a slight increase of q v at the lower levels, corresponding to the upward (and upwards-converging) turbulent diffusion flux Jv td, shown on Fig. 13(a). The updraught and collection-evaporation processes in the microphysics result in a further decrease higher up (clearly associated to the fact that the updraught condensates vapour at higher levels). We see on Fig. 13(a) that the updraught induces a negative (upward) transport of moisture at medium levels, but this moistening is completely cancelled by the larger drying by convective condensation flux, which is around 1 kg m 2 h 1 at level 21 and reaches 3.5 kg m 2 h 1 at level 33. After the subsequent downdraught, there is a further decrease in the lower levels, and an increase above (Fig. 12(c)): this is associated with the downdraught-induced circulation, which brings drier air from above to the lower layers. This is also seen on Fig. 13(a), where the effect of the downdraught is the difference between Jv cu after the updraught and at the end: there is a significant negative moisture flux in the lower layers associated to the downdraught. The difference between the final and initial q v is also plotted on Fig. 12(c). The vertical transport fluxes of cloud droplets by turbulent diffusion and by the updraught and downdraught circulations are shown on Fig. 13(b). The upward transport of ice by the updraught, Ji cu, remains less than the transport of droplets, Jl cu. The downdraught reinforces the upward transport of droplets at levels 26 to 36; since we assume that there is no condensate within the downdraught itself, the effect on condensate transport is entirely associated to its upward return current. Fig. 13(c), shows the final profiles of J td+cor i = Ji td + Ji cor, J td+cor l = Jl td + Jl cor : we observe that there has been a phase correction between levels 17 and 26,

17 1min= 0.0, max= 1.0 A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION 17 (a) (b) mean= 0.27 Figure 15. (a) NOAA infrared satellite image at 17:40 utc (b) 9 km forecast for 18:00 utc: total cloud fraction. where some liquid water had to be frozen, to maintain our statistic profile of the ice mixing ratio α i (T). The downdraught has a significant impact on reducing the precipitation flux a decrease of 0.4 kg m 2 h 1 bringing the surface precipitation to around 5.5 kg m 2 h 1, while the evaporation in the microphysics remains around 10 5 kg m 2 h 1. Finally, Fig. 14 shows different components of the temperature tendency. The effect of the net convective transport (updraught plus downdraught) of sensible heat (Fig. 14(a)) may be compared to Fig. 5(b): here, the cooling of the layers 33 to 41 is due to the downdraught, which brings colder air from above. The other peak of cooling, at levels 27 to 33, below the 0 o C isotherm (level 27), is induced by the updraught circulation, which starts (following the locations in the section) between level 36 and 33: so the updraught entrains warm air from these levels upwards. The levels above 27 are warmed by the warmer air which is detrained from the updraught. On Fig. 14(b), the heating associated with the convective condensation is situated higher than the part due to the resolved condensation. The effect of conversion of liquid to ice (Bergeron effect and riming) is small (0.1 K h 1 ). The totals of Fig. 14 (a) and (b) are shown on Fig. 14(c), together with the cooling by the precipitation flux (strongest at level 28, from precipitation melting). The total tendency due to the moist processes is here much greater than that due to radiation (not illustrated). 5.3 A frontal case As a further test, we choose a very active cold front at the border of Bohemia on 8 July Fig. 15 compares the satellite picture and a cloudiness forecast at a resolution of 9.0 km, with 43 vertical levels. The 6-hours accumulated rain by rain-gauges (Fig. 16(a)) amounted up to 72 mm in one station, and mm in several others, with quite a big variability between the stations. The forecast at 9 km resolution ( Fig. 16(b)) yields high amounts, up to 47.7 mm along the border between Czech Republic and Germany. When the mesh size is reduced to 4.5 km (Fig. 17(a)), the maximum value reaches 55 mm at the same location; and with a mesh size of 2.28 km, it reaches 75 mm (Fig. 17(b)) agreeing well with the observations. There is a gradual increase of the grid-box average precipitation with resolution, for the same reasons as explained in section 5.1: reduction of the averaging area, better representation of the correlation between high moisture and vertical velocity,and more detailed model orography. Again, the 4.5 km forecast, which is in the middle of the grey zone, stays completely consistent with the forecasts at coarser and finer resolutions. 6 Conclusions We propose a practical solution to the delicate problem of the coherent treatment of the condensation, clouds and precipitation at all resolutions in an operational numerical weather prediction model. We have shown how this development fits into a logical evolution towards unified physics. We have presented the essential features that allow us to combine the subgrid and resolved contribution at all scales: the cascading approach and clean separation of closure contributions; the prognostic approach; the production of condensates by the updraught, which is closed by moisture convergence; and the interface through massflux transport and condensation fluxes. Besides these features, we have also included a treatment of the Bergeron- Findeisen effect, and a prognostic approach involving a moist downdraught that is completely independent of the updraught.

18 18 L. GERARD (b) min= 0.0, max=47.7 mean= 6.49 Figure 16. (a) rain gauge observations; (b) 9 km forecast for 24:00 utc: 6h-accumulated precipitation (a) (b) min= 0.0, max=55.2 mean= 7.59 min= 0.0, max=75.6 mean= 6.53 Figure 17. Forecast for 24:00 utc: 6h-accumulated precipitation. (a) resolution 4.54 km (b) resolution 2.28 km We have assessed the general behaviour in single column model and shown the effects (with all restrictions applying to SCM) of the most sensitive tunings: the entrainment in the updraught; the downdraught closure; and as an example for the microphysics, the new Bergeron-Findeisen parametrization that we have introduced. All these tunings have very little impact on the surface precipitation. Important time-step variations changed it only slightly, in the transition part associated with the long spin-up of the resolved condensation. We observe good teamwork between the resolved and the subgrid parts, in two ways: first, the smaller subgrid precipitation at short time steps is compensated by a larger resolved precipitation; secondly, the long spin-up of the resolved precipitation (around 3 hours) is compensated for by an increased subgrid precipitation at the beginning of the run. Three-dimensional tests confirm the behaviour of our scheme, in that forecasts at grey zone resolutions are consistent with forecasts at coarser and finer resolution. Additional tests (not shown) performed at the coarser resolution of 40 km also behaved consistently. We have used the hydrostatic version of the Aladin model for all of our runs. Recent research on the dynamics in the Aladin community has shown that the nonhydrostatic effects do not fundamentally change the results (at least in dry situations) for resolutions coarser than 1 or 2 km. Here, we observed that the use of our prognostic scheme, which is internally non-hydrostatic, while keeping hydrostatic model dynamics, might be a cheap alternative to using the non-hydrostatic dynamics at 2-km resolution at least outside mountainous regions. Further work is being initiated to use a prognostic mixing following developments of Piriou (2005), a more complete microphysics (with 3 prognostic falling species), and refining the triggering of convection. An adapted version of our package is being installed in the frame of the international project ALARO-0 aiming at producing operational forecasts at resolutions between around 4 km and 10 km within the Aladin community. Acknowledgements Many thanks to Jean-François Geleyn and to all those who gave me advices or test materials. I also want to thank the two anonymous reviewers for their constructive criticisms. References Arakawa, A The Cumulus Parameterization Problem: Past, Present, and Future. Journal of Climate, 17,