Subgrid Cloud and Precipitation Fraction and ... - AMS Journals

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CHOSSON ET AL.

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Adapting Two-Moment Microphysics Schemes across Model Resolutions: Subgrid Cloud and Precipitation Fraction and Microphysical Sub–Time Step FRE´DE´RICK CHOSSON Department of Atmospheric and Oceanic Sciences, McGill University, Montr eal, Qu ebec, Canada

PAUL A. VAILLANCOURT AND JASON A. MILBRANDT Meteorological Research Division, Environment Canada, Dorval, Qu ebec, Canada

M. K. YAU Department of Atmospheric and Oceanic Sciences, McGill University, Montr eal, Qu ebec, Canada

AYRTON ZADRA Meteorological Research Division, Environment Canada, Dorval, Qu ebec, Canada (Manuscript received 19 November 2013, in final form 5 March 2014) ABSTRACT Two-moment multiclass microphysics schemes are very promising tools to be used in high-resolution NWP models. However, they must be adapted for coarser resolutions. Here, a twofold solution is proposed— namely, a simple representation of subgrid cloud and precipitation fraction—as well as a microphysical subtime-stepping method. The scheme is easy to implement, allows supersaturation in ice cloud, and exhibits flexibility for adoption across model grid spacing. It is implemented in the Milbrandt and Yau two-moment microphysics scheme with prognostic precipitation in the context of a simple 1D kinematic model as well as a mesoscale NWP model [the Canadian regional Global Environmental Multiscale model (GEM)]. Sensitivity tests were performed and the results highlighting the advantages and disadvantages of the two-moment multiclass cloud scheme relative to the classical Sundqvist scheme. The respective roles of subgrid cloud fraction, precipitation fraction, and time splitting were also studied. When compared to the Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO)/CloudSat-retrieved cloud mask, cloud fraction, and ice water content, it is found that the proposed solutions significantly improve the behavior of the Milbrandt and Yau microphysics scheme at the regional NWP scale, suggesting that the subgrid cloud and precipitation fraction technique can be used across model resolutions.

1. Introduction There are a priori many advantages in using multihydrometeor class, double-moment bulk microphysics schemes (BMSs) in numerical weather prediction (NWP) and general circulation models (GCMs) (Seifert et al.

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Corresponding author address: Frederick Chosson, McGill University, Dept. Atmospheric and Oceanic Sciences, Burnside Hall, 805 Sherbrooke St. West, Montreal QC H3AOB9, Canada. E-mail: [email protected] DOI: 10.1175/JAS-D-13-0367.1 Ó 2014 American Meteorological Society

2006; Reisner et al. 1998). These schemes (e.g., Milbrandt and Yau 2005a; Morrison et al. 2005a; Thompson et al. 2008; Seifert and Beheng 2006; Lim and Hong 2010) include separate prognostic equations for two moments of the particle size distribution, most likely the total number concentration and the mass mixing ratio for a set of hydrometeor classes like cloud liquid, pristine ice, snow (large ice particles), rain, graupel, and hail. Compared with simpler microphysics schemes like monoclass, diagnostic precipitation (e.g., Tiedtke 1993; Sundqvist et al. 1989; Rasch and Kristj ansson 1998), or multiclass, single-moment ones (e.g., Lin et al. 1983; Kong and Yau 1997; Hong et al. 2004; Forbes et al. 2012), twomoment schemes greatly improve the representation of

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microphysical processes (Dawson et al. 2010; Milbrandt and Yau 2006, 2005a; Morrison et al. 2005b, 2009; Klein et al. 2009) as well as radiative transfer computations (Meyers et al. 1997; Otkin and Greenwald 2008). Moreover, the treatment of the sedimentation of precipitation particles is prognostic, in contrast to the diagnostic approach in most operational cloud schemes (Tiedtke 1993; Sundqvist 1988; Lopez 2002) where precipitation is instantaneously removed from the column in one time step. Actually, the diagnostic approach for precipitation becomes problematic when time steps and spatial scales are small since the assumption that precipitation is not advected but falls to the ground within the time step is no longer valid (Bouteloup et al. 2005). As a result, the Third International Workshop on NextGeneration NWP Models (Hong and Dudhia 2012) postulated that multi-hydrometeor class, two-moment BMSs (detailed BMS) are likely to be used in the nextgeneration NWP models. However, NWP models cover a large range of time and space resolutions, from limited area (regional) models with grid spacing of about a kilometer and time step on the order of a minute or less to global large scale models with mesh size of several tens of kilometers and time steps equal or larger than 10 min. As detailed BMSs are designed to work on the cloud scale (kilometric horizontal grid spacing), their use in a lowresolution configuration presents a challenge. At coarse scales, the cloud itself becomes a subgrid phenomenon, and the use of detailed BMSs can potentially lead to significant biases in modeled processes (Pincus and Klein 2000; Bryan and Morrison 2012). Also, for the sedimentation of large particles, a large time step can lead to numerical instability. To circumvent the above problems, we propose in this paper a simple method to adapt a two-moment microphysics scheme across model resolutions by introducing a subgrid cloud and precipitation fraction (SCPF) and a sub-time-stepping approach. The technique is implemented in the Milbrandt and Yau (2005a,b) twomoment bulk microphysics scheme (MY2) and tested in a 1D kinematic model as well as the Canadian operational NWP Global Environmental Multiscale model (GEM; C^ ote et al. 1998). The remaining paper is organized as follows: In section 2, we present the subgrid cloud and precipitation fraction methodology and its adaptation to varying spatial scales. In section 3, the 1D kinematic model tests are presented. Section 4 describes the sub-time-stepping technique. In section 5, MY2 and the SCPF are tested in GEM. The results are compared with those from the operational Sundqvist scheme using cloud mask, cloud fraction, and ice water content derived from satellite products.

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2. The cloud and precipitation fraction scheme Since the pioneering work of Sommeria and Deardorff (1977) and Mellor (1977), numerous solutions have been proposed to parameterize the subgrid cloud fraction (e.g., Slingo 1987; Sundqvist et al. 1989; Smith 1990; Xu and Randall 1996; Lohmann and Roeckner 1996; Tompkins 2002). Each one relies explicitly or implicitly on an assumed probability density function (PDF) of the subgrid fluctuation of humidity and/or temperature (Tompkins 2005). The choice of the shape of the PDF itself varies and ranges from simple functions [e.g., uniform as in Le Treut and Li (1991) or triangular as in Smith (1990)] to more complex ones [e.g., beta function as in Tompkins (2002) or double Gaussian as in Golaz et al. (2002)]. The higherorder moments of this PDF can be fixed, leading to a simple relation between cloud fraction and relative humidity by setting a relative humidity criterion above which clouds can form (Slingo 1987; Sundqvist et al. 1989; Smith 1990), or the cloud fraction can be consistently linked to the available local parameters like turbulence fluxes (Sommeria and Deardorff 1977) and/or local stability (Teixeira and Hogan 2002). Alternatively, an empirical relation between cloud fraction and grid volume mean relative humidity and/or condensate mixing ratio can be used (Xu and Randall 1996). Furthermore, the subgrid variability formulation that leads to cloud fraction can be treated diagnostically or prognostically (Sundqvist et al. 1989; Tiedtke 1993; Tompkins 2002). The additional prognostic variable can be the cloud cover itself as in Tiedtke (1993) or higher moments (variance and/or skewness) of an assumed PDF as in Tompkins (2002), but it invariably leads to significant increase in complexity and cost of computation. Although prognostic treatment of subgrid variability is more realistic than diagnostic ones (Tompkins 2002; Watanabe et al. 2009), they have to be tuned for a particular situation or cloud type (e.g., boundary layer clouds). From the point of view of cloud fraction only, it is not always clear that the benefit of the prognostic approach outweighs the numerical cost (Kuwano-Yoshida et al. 2010; Wood and Field 2000). The SCPF scheme proposed here takes into account several constraints. First, it is intended to be used in NWP models across model resolutions, ranging from about 1 km (meso-gamma scale) to 30 km (global scale). In contrast to Morrison et al. (2005a), we do not use different approaches for global and mesoscale resolutions. Second, the SCPF scheme is to be implemented in a detailed BMS where the main consideration is the computational cost. As a result, the scheme should remain simple, with minimum modifications of the microphysical parameterization, and should not use other prognostic variables apart from the fields advected by

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the dynamical core of the NWP model. Under such constraints, our best choice is a diagnostic scheme based on a simple assumed PDF of the subgrid humidity for the subgrid cloud fraction and simple overlap assumption for the subgrid precipitation fraction. The subgrid cloud fraction scheme is therefore a simple way to parameterize the fraction of the grid cell where condensation and cloud is likely to occur. The expected benefits of a subgrid cloud fraction scheme are threefold: 1) it allows condensation before the grid-mean saturation is achieved, 2) it defines the fraction of the grid box where the microphysical processes take place, and 3) it provides a simple scale-dependent parameter to palliate the impact of all unresolved processes on cloud microphysics. Similarly, the expected benefits of a precipitation fraction are 1) it avoids overestimation of the evaporation of precipitation, 2) it ensures consistency with the cloud fraction, and 3) it allows the possibility to account for precipitation production only in a fraction of the cloud.

a. Subgrid cloud fraction and local saturation For simplicity, a fixed-width top-hat PDF P(qt) has been chosen to represent the subgrid variability of total water mixing ratio, uniformly distributed around its grid-mean value qt , within the range [qt 2 Dq, qt 1 Dq] (see Fig. 1). Here qt is the sum of water vapor mixing ratio qy and cloud condensate mixing ratio qc, defined as all hydrometeor classes that may experience water vapor deposition (cloud liquid, ice, and snow in the MY2 scheme) in contrast to precipitation condensates (rain, graupel, and hail in the MY2 scheme). Snow is included in the cloud condensate since 1) apart from nucleation, there is no clear separation in terms of microphysical processes between pristine ice (small ice particles) and snow (large particles and aggregates), 2) pristine ice transforms into snow without an intermediate specie, 3) there is no specific mode in observed particle size distribution functions for ice and snow), and 4) snow accounts for a major part of solid condensate in the atmosphere—it is important for radiative transfer calculations and forms part of the total water content, and 5) pristine ice and snow can be merged into a single category for modeling or evaluation purposes (Delanoe¨ et al. 2005). In Fig. 1 the saturation mixing ratio qs, with respect to liquid or ice depending on whether the temperature is above or below the triple point, divides the PDF (and the grid box) into two homogeneous parts: a saturated or supersaturated region (the so-called cloud fraction a) and a subsaturated region (or clear-sky fraction). Note that the temperature is assumed constant within the grid box, so is the saturation mixing ratio. This assumption can be justified at least for boundary layer and

FIG. 1. Assumed fixed-width top-hat distribution function of total water mixing ratio within the grid box. The area above saturation to the right of qs represents the subgrid cloud fraction.

shallow convection since observational studies (Price and Wood 2002; Tompkins 2003; Perraud et al. 2011) seem to confirm that temperature fluctuations are less important than humidity fluctuations in our context. From the partition of the PDF of qt, one can define the clr local mean in-cloud (qcld t ) and mean clear-sky (qt ) total mixing ratio (Le Treut and Li 1991). As the condensate cloud water is present only in the cloud fraction of the grid box, one can also derive the water vapor mixing raclr tios qcld y and qy respectively within the cloudy and cloudfree part of the grid box. This can be used to compute the respective relative humidity U (cld)(clr) 5 q(ycld)/(clr) /qs in the two separate regions, assumed to be internally homogeneous. We stress that a top-hat PDF is used to obtain the subgrid cloud fraction. However, the assumption of a tophat distribution is not carried through to the microphysics as we assume that the cloudy fraction and the clear fraction are homogeneous in the microphysics computations. From these basic assumptions, we have a5

qt 1 Dq 2 qs 2Dq

with 0 # a # 1;

qt 1 Dq 1 qs qc 2 ; 2 a q 2 Dq 1 q clr t s qclr . y 5 qt 5 2

cld cld qcld y 5 qt 2 qc 5

(1)

It must be emphasized that the PDF has no meaning in case of total overcast or cloud-free situations. In other clr words, qcld y is set to zero and qy to the grid-mean water vapor mixing ratio if the cloud fraction is null and conversely in overcast conditions. Note that classically, a diagnostic subgrid cloud fraction scheme may also be used as a subgrid condensation scheme via the relation ð‘ qc 5 (qt 2 qs )P(qt ) dqt 5 a2 Dq . (2) qs

However, this is not directly applicable within a multihydrometeor class BMS, since water vapor can condense on a mixture of different hydrometeor categories with different deposition rates, and one must also take into account all other processes affecting condensate growth,

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such as riming and other collection processes. Furthermore, while the lower bound of the integral in Eq. (2) is an acceptable approximation for water vapor deposition in warm clouds, owing to the fast adjustment process to saturation of in-cloud vapor excess, it is not true in ice clouds where supersaturation with respect to ice is commonly observed and can exceed 150% (Heymsfield et al. 1998; Gierens et al. 1999, 2000, 2004; Spichtinger et al. 2003). To circumvent this problem, a correction of the threshold relative humidity for condensation may be applied as in Tompkins et al. (2007), but it still conceals the role of possible low deposition rate, and interferes with the complex ice nucleation process, as stated in Salzmann et al. (2010). In our approach, we simply derive the mean local relative humidity in the two regions (so-called cloudy and clear sky) and allow the microphysical model to explicitly compute the deposition rates and to eventually allow the activation scheme to deal with the local supersaturation in the ice phase. For warm clouds only, the evaporation and condensation processes are assumed to be instantaneous (or achieved in one time step) in both regions. In MY2, some of the cloud hydrometeor species [i.e., small ice particles (ice) and large ice particles (snow)] sediment and can eventually fall into a subsaturated environment where the diagnosed cloud fraction is 0. In this case, we assume that the local cloud fraction is equal to the computed cloud fraction at a level immediately above, yet with a water vapor mixing ratio in both the cloudy and the clear-sky part given by its grid volume mean value.

b. Defining the PDF width and scale dependency of the scheme A saturated region (cloud) is present if the gridaveraged total mixing ratio exceeds a given relative humidity threshold value U00 that can be related to the width of the PDF: Dq 5 qs (1 2 U00 ) .

(3)

As there is no universal way to choose this threshold value, it is a tunable parameter. However, it has been shown (Tompkins 2005) that each cloud cover scheme that uses fixed-moments PDF of humidity is equivalent to a diagnostic scheme that uses a simple relative humidity criterion as its main predictor (e.g., Smith 1990). Specifically, one can demonstrate that, using the classical assumption qcld y 5 qs (valid for warm cloud), the tophat shape PDF presented above is equivalent to the Sundqvist et al. (1989) approach, which relates the cloud fraction to the grid-averaged relative humidity U and a threshold value U00:

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FIG. 2. Initial profiles of temperature (red), relative humidity (blue), and vertical velocity (black) for the 1D kinematic model experiments. The saturation is computed with respect to liquid water if T is above 08C and with respect to ice for T below 08C. Also shown in brown is the corresponding profile of the relative humidity threshold value U00 above which condensation is allowed in Sundqvist and SCPF schemes.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12U . a512 1 2 U00

(4)

We thus decided to use the profile of U00 as implemented in the operational GEM model for the Sundqvist cloud scheme, as depicted by the brown line in Fig. 2. As shown, U00 is nearly constant at its maximum value in the boundary layer, then it decreases to its minimum in the lower troposphere and increases back to its maximum as temperatures drop below 2358C. The PDF width is a measure of the subgrid moisture variability and is directly linked to U00. As the subgrid variability is scale dependent, so should the humidity threshold value. The U00 profile used here is valid for a mesh size of about 10–30 km. Quaas (2012) demonstrated that for lower-resolution models, U00 maintains its shape but its values are much lower in the middle troposphere.

c. Subgrid precipitation fraction The subgrid cloud fraction defined above concerns hydrometeor classes likely to be present in a saturated or supersaturated environment. However, other classes can be found in a subsaturated environment, especially precipitation classes, which are defined here as condensate particles that do not experience water vapor deposition (rain drops, graupel, and hail in MY2). These classes can sediment through either cloudy or cloud-free regions and the fraction of the grid box where they can be found must also be determined. This so-called precipitation fraction

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aP is defined in a way inspired by Jakob and Klein (2000). They implemented a precipitation fraction for use in GCM using a microphysics scheme that produces and removes precipitation in one time step in an iterative manner from the top of the model down to the surface. However, for prognostic precipitation, such as in MY2, precipitation particles can be present in a grid box at the beginning of the time step and/or be produced by advection, sedimentation, or source–sink terms during the time step. To deal with these situations, we assume that 1) precipitation comes from overlapping clouds or has been or will be produced in the cloudy part of the same model level if no precipitation exists in the level immediately above, 2) precipitation particles are evenly distributed in the precipitation fraction, and 3) possible horizontal tilt of precipitation drafts under clouds is neglected. The precipitation fraction is then defined by the fraction of the grid box overlapped by clouds present in the column above it. To determine how clouds overlap with each other in the model column, we employ the maximum-random overlap assumption (Geleyn and Hollingsworth 1979). This approach is sufficient to define, for each model layer, the precipitation fraction clr aP 5 acld P 1 aP , eventually as the addition of the cloudy precipitation region (precipitation falling into cloud clr fraction) acld P , and the clear-sky precipitation region aP (see appendix A for details). A schematic diagram illustrating cloud and precipitation fractions overlapping in a nonovercast situation is given in Fig. 3. In the method presented above, the totality of the overlying cloud is assumed to potentially produce precipitation. In reality, precipitation forms only within a fraction of the cloud, corresponding for example to the core fraction in cumulus (Stevens and Seifert 2008). This volume or area fraction is obviously dependent on the type of cloud and on the scale considered. As an alternate approach, we can define Pfrac as the fraction of the highest cloud (seeding cloud) where precipitation is produced. The fraction of this grid that contains precipitation is thus ak 3 Pfrac, where ak is the local cloud fraction. This fraction Pfrac can be a function of local parameters like cloud type or phase, vertical velocity, type, and magnitude of precipitation fluxes, and can be defined for either overcast or broken sky conditions. For simplicity, we assume here that this fraction is a constant, and that 1) there is still maximumrandom overlap between clouds, 2) the first cloud layer (from the model top) that contains precipitation at the beginning of the time step is considered as the ‘‘seeding’’ layer and will determine the precipitation fractions below, and 3) there is maximum overlap between the precipitation fraction and the cloud area potentially

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FIG. 3. Illustration of cloud and precipitation fraction overlaps in a 1D model with 10 levels. Cloud fractions (in gray) follow a maximum-random overlap assumption. Hatched areas are precipitation fractions; dotted areas are in-cloud precipitation fractions. If no precipitation is present (e.g., total evaporation), only the clear-sky precipitation fraction is set to 0 (model level 7). The continuous cyan-colored line represents the maximum-random total cloud cover from top to actual level.

containing precipitation (see appendix A). Obviously, if Pfrac 5 1, the two approaches are equivalent.

d. Implementation of the SCPF scheme in MY2 The SCPF has been implemented in a version of MY2 that includes the modifications described in Milbrandt et al. (2010). The model solves the prognostic equations for number concentration Nx and mixing ratio qx of six hydrometeor categories: that is, cloud droplets (subscript x 5 c), rain drops (x 5 r), small ice particles (x 5 i), snow (large crystal and aggregates, x 5 s), graupel (x 5 g), and hail (x 5 h). The original model produces grid-cell mean (N x , qx ) quantities. In the context of subgrid parameterization, local values are required. Since we differentiate ‘‘cloudy condensates’’ (x 5 i, s, c) that may experience water vapor deposition and are considered to be evenly distributed within the cloud fraction part of the grid cell (a) from ‘‘precipitation condensates’’ (x 5 r, g, h) that are uniformly distributed within the precipitation fraction of the grid cell (aP), we can define the local quantities as Nx 5 N x /a and qx 5 qx /a for x 5 i, s, c and Nx 5 N x /aP and qx 5 qx /aP for x 5 r, g, h. The local quantities in the cloudy fraction and in the precipitation fraction are depicted schematically in Fig. 4. The local quantities are used to calculate the local source–sink terms. The grid volume average source–sink terms are then obtained by multiplying the local source– sink by the respective cloud fraction, precipitation fraction, or overlap fraction (Fig. 4). For example, following the notation of Milbrandt and Yau (2005b), the

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FIG. 4. Conceptual sketch showing the subgrid cloud fraction and precipitation fraction inside the grid cell and their eventual overlap region. The local quantities within their respective fraction are also presented.

effect of collection of cloud droplets by graupel on graupel mixing ratio is d qg dt

col

0 1 N Ng c 5 Qg CL 5 f @ , A 3 acld P , c, g a aP

(5)

while the collection of pristine ice by snow particles becomes d qs dt

col

! Ni Ns , 3 a. 5 Qs CLi,s 5 g a a

(6)

Here, f and g represent respectively the local source– sink terms under consideration. For vapor diffusion, the local in-cloud qcld y or clear-sky qclr y value of water vapor mixing ratio is used to compute the saturation, and the result is multiplied by its corresponding fraction; qcld y is also used for ice nucleation as MY2 adopts the ice supersaturation-based Meyers et al. (1992) nucleation parameterization.

3. Test in a 1D kinematic model a. Model setup The subgrid and precipitation fraction scheme is tested first in the context of a simple 1D kinematic column model described in Milbrandt et al. (2014). The model is initialized by a single sounding (Fig. 2) obtained from the North American Regional Reanalysis (NARR) for a winter case near the central United States

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on 20 December 2008. The system is characterized by an approaching front with slow vertical motion. There is a rather deep midlevel cloud layer with overlying cirrus. In our simulations, the top of the column is at 15 km. There are 61 vertical levels with a constant grid size of 245 m. The model time step is 10 s and the duration of the experiments is 360 min. Three microphysics models are compared: the original version of MY2 without subgrid fractions, the new version of MY2 with SCPF (MY2 1 SCPF), and the operational version of the Sundqvist et al. (1989) cloud scheme (SUND). Recall that SUND contains a subgrid cloud fraction parameterization with the same humidity threshold as in MY2 1 SCPF (Fig. 2). SUND is a single-moment, single-class cloud scheme with a single prognostic equation for total water condensate. The partition of this total condensate into a liquid phase fraction and a solid phase fraction is diagnosed as a function of local temperature. Within SUND, precipitation is derived diagnostically, and local solid and liquid precipitation fluxes are computed from the model top, down to the surface, using a simple parameterization of autoconversion, accretion, melting, and evaporation. Precipitation is assumed to reach the ground within a single time step and is removed from the model column.

b. Simulation results In the first experiment we model a weak large-scale updraft since the NARR products are consistent with a GCM scale with 32-km horizontal grid spacing. The vertical motion (Fig. 2) is positive for the first 3 h. Thereafter, the motion gradually becomes downward as indicated by the contour plot in the background of Fig. 5 (first row). This weak updraft and downdraft is assumed to mimic a typical nonconvective grid-scale vertical velocity in a large scale NWP model. Figure 5 also presents the time evolution of the vertical profile of subgrid cloud fractions, the water path (vertical integration) of condensates, surface precipitation rates (top row), the vertical profiles of cloudy condensates (middle row), and the saturation profiles (bottom row) from the three simulations (MY2, MY2 1 SCPF, SUND). For the original version of MY2, the cloud fraction is either 1 or 0 depending on whether cloudy condensates are present (.1026 kg kg21) or absent. Note that for MY2 1 SCPF and SUND, the cloud fraction represents in fact the saturated fraction of the grid box. For SUND, the cloud water path (CWP) and ice water path (IWP) are respectively the vertical integration of the liquid and solid partition of the prognosed total condensate. For MY2 and MY2 1 SCPF, the water paths CWP, RWP, IWP, SWP, and GWP are respectively the vertical integration of the cloud liquid (qc ), rain (qr ), small ice particles (qi ),

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FIG. 5. Simulation results from (left) MY2, (center) MY2 1 SCPF, and (right) SUND for the first experiment. (top) Time evolution of the column-integrated water paths of the different hydrometeor classes together with the surface precipitation rates with subgrid cloud fraction as background filled contours (10% per gray level, starting at 10%) and prescribed vertical velocity superimposed in solid (positive values) or dashed (negative) isolines. (middle) Total cloud condensate (qc 1 qi 1 qs for MY2); superimposed are the isocontours of subgrid cloud fraction and the 08C isoline (red). (bottom) Grid-mean saturation with respect to liquid (above 08C) or ice (below 08C).

large ice particles and aggregates (qs ), and graupel (qg ) mean water contents. There is no hail production or solid precipitation reaching the ground for this experiment. The total cloudy condensates are qc 1 qi 1 qs for MY2 and qt 2 qy for SUND. The bottom row presents the grid-mean saturation (qy /qsat ) with respect to either liquid water or ice (for temperature above or below 08C—red line). As shown, the evolution of the SCF is similar between SUND and MY2 1 SCPF but their magnitudes differ (middle row). This is due to the fact that SUND does not allow supersaturation. In contrast, MY2 allows ice particles to consume the available moisture in a more physical and time dependent manner. Thus, while

saturation in SUND cannot exceed 100% while in both MY2 simulations, the supersaturation can reach more realistic values (with respect to ice) for ice phase clouds (bottom row). This also explains why the overlying thin cirrus cloud, completely missed in MY2, has lesser presence in MY2 1 SCPF than in SUND. By comparing the total cloud condensate (middle row) in MY2 with MY2 1 SCPF and SUND, it is clear that as expected, SCPF scheme causes condensation to occur sooner and the cloud life time is also extended. The peak values of the water paths also increased (top row)—strongly for CWP and somewhat less for IWP and SWP. The SCPF scheme also changes the distribution of

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FIG. 6. As in Fig. 5 (top), but for simulation results from the sensitivity tests of the parameter Pfrac set at the reference (top left) 100%, (top right) 50%, (bottom left) 25%, and (bottom right) 10% of the cloud fraction of the cloud layer considered as the source of underlying precipitation. In this case, the temperature dependency of the U00 profile has been removed.

condensates amongst hydrometeor classes. For example, graupel, which is formed via accretion of cloud droplets by snow particles, is present in MY2 1 SCPF but not in MY2. One of the most interesting results lies in the considerable increase of surface precipitation in MY2 1 SCPF, with a peak value comparable to SUND. This point is especially encouraging since SUND has been used in operational large-scale NWP models for two decades and is largely tuned to reproduce realistic observed precipitation. Note also that SUND produced condensate sooner and with a larger magnitude during the first (updraft) part of the simulation but exhibits a surprising behavior in the latter (downdraft) part. The reason for this behavior is not known. In a stronger forcing situation (as in kilometric NWP models), we expect MY2 1 SCPF and MY2 to yield similar results. To verify, we conducted a second experiment by prescribing a stronger updraft that varies sinusoidally in time and altitude and with a maximum vertical velocity of 2 m s21. The results are indeed

similar in in both MY2 simulations (not shown) while the evolution of cloud and precipitation in SUND is unrealistic. In a last experiment, we explore the sensitivity of the subgrid scheme to variation in the subgrid precipitation fraction. Three simulations were performed using the initial conditions of the first experiment with three different values of Pfrac at respectively 50%, 25%, and 10% of the cloudy layer, which acts as the source of the underlying precipitation. The results are presented in Fig. 6. Note that in these simulations, we removed the temperature dependency of U00 to obtain a thicker and longer-lasting overlying nonprecipitating cirrus layer without changing substantially the IWP or other quantities. The modification of Pfrac has no impact on the diagnosed cloud fraction or the cloudy condensates but has significant effect on precipitation types and their production. Surface precipitation amount doubles when Pfrac changes from 100% to 10%. At the same time, graupel water path increases, even allowing solid-phase

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FIG. 7. As in Fig. 5 (top), but for simulation results from the sensitivity tests of the time-step duration in MY2. Time step is set to (top left) 10 s, (top center) 30 s, (top right) 60 s, (bottom left) 2 min, (bottom center) 4 min, and (bottom right) 8 min.

precipitation to reach the ground when Pfrac is at 10%. Since Pfrac can have an effect only if precipitation sediments through partially cloudy cells, reducing Pfrac decreases the precipitation fraction exposed to the subsaturated (cloud free) region and maximizes the incloud precipitation amounts. Thus Pfrac acts in two ways: it decreases the grid-averaged evaporation of precipitation and increases the interaction between cloudy and precipitation species. In the simulations, precipitation production is enhanced when snow undergoes riming by collecting cloud droplets in the lower levels to form graupel. If graupel is more confined within the cloudy fraction, then the collection rate increases, locally favoring the amount of graupel eventually at the expense of rain. Thus, under certain conditions, Pfrac can act as an important tuning parameter for precipitation production and for precipitation phase. However, its value should be guided by several criteria (model resolution, cloud type) and has to be assessed empirically and tested in more realistic conditions.

4. Handling long time steps a. The problem A detailed BMS is designed to perform best at cloud scale, say at kilometric horizontal grid spacing. The

corresponding time step is of a few tens of seconds. However, regional and global NWP models often employ longer time steps. As an example, the Canadian High-Resolution Deterministic Prediction System (experimental, 2.5-km horizontal grid spacing) uses a 60-s time step, the regional model (operational, 10-km horizontal grid spacing) uses a 300-s time step, and the global one (operational, about 25-km horizontal grid spacing) has a 720-s time step. Adapting a detailed BMS to low temporal resolution is a real challenge. The microphysical processes explicitly represented in these models are rather fast, their physical representation loses meaning when the gap between their characteristic time scale, and the model time step becomes large. Perhaps the most problematic aspect arises from sedimentation, which impacts precipitation production. During sedimentation, precipitation particles interact. Their properties (mass, size, concentration, fall speed, phase) evolve with the history of these interactions. In MY2, sedimentation is calculated at the end of the time step. If the time step becomes too large, precipitation particles can be removed from a column without the chance for interactions. To illustrate this problem, Fig. 7 shows MY2 simulation results with increasing time step from 10 to 480 s. Although the cloud pattern remains the same for all time step durations, and surface precipitation rate

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hardly changes with time step from 10 to 60 s, no rain reaches the ground for simulation time step from 120 to 480 s. Rain production in cloud is even totally suppressed with a 480-s time step. Interestingly, the cloud condensates of the different categories (ice, snow, cloud droplets) remain rather unchanged up to a time step of 240 s but is significantly modified for the larger ones, with less snow and more liquid, though the total cloud condensate integrated over the whole run is constant. Thus, the problem of the adaptation of multiclass, multimoment BMS to a large time step impacts mainly the surface precipitation production.

b. The microphysical sub–time step To circumvent the problem illustrated in the last section, coarse resolution models tend to avoid sedimentation for cloudy condensates and use a diagnostic precipitation approach (e.g., Fowler et al. 1996; Ghan and Easter 1992), arguing that time step duration is much longer than the time needed for sedimenting species to reach the surface. This approach can indeed be applied within double-moment BMSs (Morrison and Gettelman 2008) as long as one does not represent prognostically the precipitation species. However, this method is not valid for shorter time step and multihydrometeor class BMSs such as MY2. If a single microphysics scheme is to be used across model resolutions, diagnostic precipitation is not a solution. We propose to use simply microphysical sub–time steps within each main model time step. The sub–time step dt is short enough to avoid a negative impact on precipitation (see appendix B for details). The microphysical sub–time step has been tested (without SCPF scheme) in a limited area version (LAM) of GEM 3.3.2 over the Arctic. The model domain covers all the Arctic basin including Canada, western Europe, Siberia, and Alaska. The grid has a spatial resolution of 15 km and the nominal time step is 450 s. Boundary conditions are provided by a global model (horizontal grid spacing ; 30 km) using the Sundqvist cloud scheme. Deep (Fritsch–Chappell type) convective and shallow (nonprecipitating) convective schemes are used. The simulations consist of a series of 31 forecast runs, each lasting 36 h, that cover the 31 days of the month of July 2008. Both the GEM global (Charron et al. 2012) and the GEM LAM (Mailhot et al. 2006) models are initiated with an analysis valid 12 h before each simulated day. Only the last 24 h of each run are taken into account in order to avoid the effects of the rather long spinup period (up to 6 h). The purpose of the simulations is to determine the respective effect of using small and large time steps and the benefit of the microphysical sub-stepping method on

FIG. 8. Time series of daily accumulated total surface precipitation averaged over the simulation domain for SUND with Dt 5 450 (gray) and Dt 5 60 s (black) and for MY2 with Dt 5 450 (magenta), Dt 5 60 s (blue), and with Dt 5 450 s and microphysical substep dt 5 60 s (red); and for GPCP dataset (green). The horizontal lines correspond to their respective monthly mean.

surface precipitation. Five series of simulations were performed. Two of these used SUND, with main time step of either 60 or 450 s. Two other series employ MY2 without SCPF, with either 60- or 450-s main time step. The last one uses MY2 (still without SCPF) with 8 microphysical sub–time step of about 60 s within each 450-s main time step. The choice of dt should represent an optimal balance between numerical cost and accuracy. Sensitivity tests (not shown) revealed that 60 s is an excellent compromise since precipitation production converges around this value, consistent with the 1D kinematic model results. Figure 8 shows the time series of daily total surface precipitation averaged over the entire domain for the five simulation series. The results are compared to the Global Precipitation Climatology Project (GPCP) daily precipitation data (Huffman et al. 2001). It is recognized that the low-resolution (18 3 18) combined gauge and satellite data could not provide accurate precipitation quantities around the Pole (Bolvin et al. 2009), but the dataset is the only one available over the Arctic basin. One can consider the area-averaged values presented in Fig. 8 as an acceptable reference for comparison with model simulations. The depicted results from the MY2 simulation using Dt 5 450 s illustrate the negative impact of large time step on models using prognostic sedimentation. Surface precipitation is halved compared to the short time step runs, which compared reasonably well with the GPCP

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data. It is interesting to note that SUND, which uses diagnostic parameterization of precipitation, is also sensitive to time-step duration as the surface precipitation increases with a decrease in time step. The reason for this behavior is, however, unclear. The MY2 simulation with the microphysical sub-stepping method reveals that it is possible to achieve almost the results of an equivalent short model time step at much lower numerical cost (MY2 with Dt 5 60 s took 4.39 times more wall-clock CPU time).

5. Test of SCPF in GEM In this section, we present a comparison of the performance of MY2, MY2 1 SCPF (in both cases the subtime-stepping method is used), and SUND in the context of an operational NWP model. Satellite cloud products from Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) and CloudSat measurements will be used as a basis for evaluation.

a. Case study, model setup, and remote sensing dataset The NWP framework is GEM with 15-km horizontal grid spacing and Dt 5 450 s. The case study is a winter event on 20 December 2008 from midnight to midnight (UTC). The event is the same as the 1D case and chosen to limit convection and to maximize stratiform cloud activities. The simulations are initiated using an analysis valid 12 h before the period of interest. Cloud fraction and detrained condensate from the deep and shallow convective schemes are merged with the output cloud fields from the three stratiform cloud models. The reference simulation employs SUND. The partition between the liquid phase and ice phase is diagnosed as a function of temperature and total water content, following Boudala et al. (2004). Both MY2 1 SCPF and SUND share the same relative humidity criterion for subgrid cloud fraction. MY2 and MY2 1 SCPF use a microphysical sub–time step of 60 s. The recent active measurements from satellites provide a unique opportunity to evaluate cloud microphysics over both large domain and long time period at a high resolution including the vertical dimension. The radar/lidar (DARDAR)-Cloud products (Delanoe¨ and Hogan 2008, 2010) from the Cloud–Aerosol–Water–Radiation Interactions (ICARE) center used in this study are based on the complementarity and synergy of lidar and radar measurements from CloudSat and CALIPSO satellites. It employs a variational method to provide high-resolution (1.1 km in horizontal, 60 m in vertical) retrievals of cloud masks, ice water content (IWC), effective radius, and extinction along the satellite foot print. DARDAR-Cloud

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products (hereafter DARDAR) have already been used to assess European Centre for Medium-Range Weather Forecasts (ECMWF) and Met Office NWP models (Delanoe¨ et al. 2011) and compared to other remote and in situ measurements (Delanoe¨ et al. 2013).

b. Comparisons 1) CLOUD MASKS The satellite track crosses the simulation domain nine times on 20 December 2008 (Fig. 9). The simulations’ outputs are projected on the DARDAR retrievals grid such that for each point of the profile from the retrievals fields that overlaps the domain, the closest corresponding point from the simulated output fields in space and time is selected. The points within the GEM Global– LAM piloting zone, close to the domain boundaries, are ignored. Start and end times, locations, and corresponding number of profiles for each of the nine transects are listed in Table 1. Figure 10 shows the cloud masks from DARDAR and each of the three simulations for the nine satellite tracks/domain crossings of the day, presented all in a row in chronological order separated by a vertical red line. If a cloud is present, it can be either in pure ice phase (green), pure liquid warm phase (deep blue), pure below 08C (supercooled) liquid (light blue), or mixed phase (salmon). In the MY2 simulation without SCPF, the cloud fraction is either 1 or 0 if the total cloud condensate specific content is respectively above or below 1023 g kg21. For the SUND and MY2 1 SCPF simulations, a point is considered cloudy if the cloud fraction exceed 1% and the cloudy condensates (precipitations are ignored) exceed 1023 g kg21 either in ice or liquid phase or both, depending on the cloud mask category. As shown, all three GEM simulations capture well the general distribution of cloud systems revealed by the retrievals. The correlation in space and time between the cloud patterns is rather good although some differences are evident. The SUND simulation exhibits too much cloud diagnosed as mixed phase, even at altitude as high as about 9 km. This is clearly a problem arising from the liquid–ice partition function used in the model (Boudala et al. 2004) that seems to provide too smooth transition from all-ice to all-liquid clouds. Compared to DARDAR retrievals, the simulations using MY2 exhibit a good partition of cloud types, especially in a realistic proportion of mixed phase clouds and the presence of a thin overlying mixed phase or even supercooled liquid layer on top of some deeper ice clouds. It appears that using the SCPF scheme leads to a slightly higher cloud fraction, especially for ice clouds. The main difference arises from the lack and sometimes absence of high-altitude ice clouds in the simulation without SCPF.

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FIG. 9. Sketch of the GEM LAM simulation domain with (cyan) and without the piloting points (orange). The footprints of the CALIPSO and CloudSat satellites for 20 Dec 2008 are drawn in magenta. The points retained for the comparison are highlighted in green, corresponding to the nine trajectories crossing the domain (numbered).

2) ICE WATER CONTENT Figure 11 presents the IWC from DARDAR and the simulations wherever a cloud is diagnosed. The points from the simulations correspond to the grid-mean output values. The tiles of the models are assumed to be homogeneous in both the horizontal and the vertical dimensions. Whenever the cloud’s pattern exhibits similar feature between DARDAR and the simulations, the vertical distributions of IWC are in general agreement. However, it is clear that the SUND IWC production is considerably lower than DARDAR retrievals. The agreement is much better with the MY2 simulations. Note that whenever the cloud pattern is common in both MY2 simulations, the IWC fields are almost identical. This suggests that, as expected, the SCPF scheme does not affect much the microphysics model when there is sufficient moisture to achieve grid-mean saturation.

3) MEAN PROFILES Figure 12 presents the mean vertical profiles of cloud fraction (all cloud types), ice water content, and in-cloud IWC (IWC divided by cloud fraction for each point of the projected time series profiles where cloud fraction is above 1%), averaged over all nine along-track cross sections during this day. When compared to the DARDAR retrievals, SUND shows too-low cloud fraction in average through most of the troposphere except for the highest levels above 12 km. It is interesting to note that,

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even though the relative humidity criterion is the same as in the SCPF scheme, and that the diagnostic estimation of the subgrid cloud fraction is also similar in both, the SUND mean cloud fraction is significantly less than in MY2 1 SCPF. The reason is that SUND produces far too-low ice condensate and the lack of cloud fraction is not sufficient to compensate for the too-small cloud condensate below 8 km. The results from MY2 and MY2 1 SCPF indicate that in the lower part of the troposphere, from about 1.5 up to 5 km, the mean cloud fraction from both simulations are similar and in good agreement with the DARDAR retrievals. In the lowest part of the atmosphere (boundary layer), the agreement is degraded but inconclusive, since the satellite measurements are often too attenuated by overlying clouds and the simulation’s output may be significantly affected by the shallow convection scheme. The two MY2 simulations diverge above 5 km, where mean cloud fraction decreases down to 0 at around 12 km if SCPF is not used. On the contrary, the simulation with SCPF maintains good agreement with the DARDAR mean cloud fraction up to 14 km. However, the two MY2 simulations produce nearly identical mean IWC profile, in very good agreement with the DARDAR retrievals. Compared to DARDAR, they slightly overestimate IWC from 7 to 12 km and slightly underestimate it below 7 km. Consequently, the mean in-cloud IWC profiles fit quite well the DARDAR values up to 5 km and even above 5 km for the simulation using SCPF. On the other hand, there is significant overestimation from 5 to 12 km in MY2 without SCPF.

6. Discussion If one wants to adapt a multimoment, multiclass bulk microphysics scheme across model resolutions one must address the problems of subgrid variability and large time step. The subgrid cloud and precipitation fraction scheme and the microphysical sub-time-step method presented in this study are desirable steps toward the general use of two-moment cloud models in NWP models. The twofold solution proposed is simple and easy to implement. It allows for supersaturation within ice clouds, preserves the performance of the Milbrandt and Yau (2005a,b) two-moment BMS in a high-resolution context, and considerably enhances its results in a larger scale context. It also has a minimal number of tuning parameters to adapt the scheme to the resolution needed. The joint use of the SCPF scheme and the microphysical sub-stepping within a detailed BMS is a practical technique, and the encouraging results presented in this paper can be considered a proof of concept.

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TABLE 1. Starting and ending times of day (20 Dec 2008), latitudes, and longitudes; and number of DARDAR profiles and corresponding GEM profiles for each numbered transect for which satellite tracks crossing the GEM simulation domain. Transect

DARDAR

GEM

1

102

10

2

3323

314

3

4827

436

4

4820

443

5

3671

339

6

1709

161

7

2093

189

8

3852

353

9

5583

511

29 980

2756

Total

Start End Start End Start End Start End Start End Start End Start End Start End Start End

However, compared to SUND, the present version of MY2 remains relatively expensive in terms of numerical cost for global NWP models with time step of 10 min or larger. Yet, MY2 is still in constant development, and some possible solutions would be to decrease the number of hydrometeor classes (combine hail and graupel

Time of day (UTC)

Latitude (N)

Longitude (W)

0454:35.50 0454:55.02 0628:50.70 0639:34.37 0805:41.73 0821:56.29 0943:31.17 0959:48.13 1121:52.61 1133:52.61 1300:53.41 1306:21.73 1604:21.41 1611:11.49 1739:38.37 1752:17.73 1912:26.21 1931:06.21

498590 52.0000 488500 29.2000 668100 39.6800 28880 29.2000 728540 16.0900 158330 52.1200 76890 1.1200 198150 52.6900 778380 6.1100 368370 6.6400 778160 52.4700 598210 41.1100 47890 40.2200 708520 22.0100 348150 26.4100 778380 2.6000 128150 39.5100 778250 38.8900

368560 24.8900 378250 17.4100 518200 15.9700 688390 30.3000 668390 24.4400 968220 18.3800 838160 24.1800 1208150 1.8900 1028310 10.9500 1408300 18.1500 1288410 9.8900 1558410 0.7400 438130 30.8100 608330 38.4500 638440 4.8100 1028230 55.0400 83850 280 0500 1268140 44.1600

categories, and merge ice and snow categories), simplification and optimization of the code, and the neglect of some low-impact microphysics processes (such as contact freezing nucleation and three-component freezing). The comparison with DARDAR retrievals at 15-km resolution reveals two main features: 1) MY2 produce

FIG. 10. Time series of the cloud masks along-track vertical profiles for the nine overlaps (left to right numbered) crossing the model domain (see Fig. 9). (top to bottom) DARDAR (retrieval), SUND, MY2 1 SCPF, and MY2.

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FIG. 11. As in Fig. 10, but for ice water content.

much more realistic ice phase cloud condensate than SUND and 2) the SCPF scheme greatly improves the cloud fraction and in-cloud IWC of MY2. The use of a subgrid cloud fraction scheme appears to be indispensable at this resolution (15 km). Moreover, despite its simplicity, the SCPF approach and the relative humidity criterion profile used in this study seems to be well adapted to regional NWP model forecast. It is clear

that the liquid–solid partition function used to diagnose IWC out of total water content in SUND is ill defined. This fact may be important for two reasons: first, the separate treatment of ice and liquid phase precipitation parameterization depends on this partition function within SUND and thus could be potentially biased; and second, the radiative properties of clouds depend strongly on the phase of the condensate, hence could

FIG. 12. Mean vertical profiles of (left) all-type cloud fraction, (middle) ice water content, and (right) in-cloud ice water content averaged over the overlapping tracks on 20 Dec 2008 for the DARDAR retrievals (red) and simulations using Sundqvist cloud model (black) and MY2 with (green) or without (blue), the subgrid cloud and precipitation fraction scheme.

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also be biased. However, the SUND total cloud condensate remains lower than the IWC retrievals provided by DARDAR (results not shown in this study). This is indeed a problem for the atmospheric radiation budget. This underestimation of cloud water content may be due to the diagnostic treatment of precipitation that removes a portion of cloud condensate every time step via a continuous and perhaps too-high precipitation production, as suggested by the 1D kinematic sensitivity tests. On the contrary, MY2 presents realistic solid phase condensate, distributed in the pristine ice and the snow categories with their own size distributions, assumed shape, and prognostic number concentration. If used jointly with the SCPF scheme, it produces a satisfactory mean cloud fraction that results in a realistic incloud condensate. This is likely to improve drastically the optical properties of clouds, especially for mid- and high-altitude clouds. Investigation of the resulting cloud radiative properties is underway and will be presented in a future study. Future work will also extend the study over more cases and longer time periods. One can note that, from a strict physical point of view, making a distinction between precipitation fraction, cloud fraction, and clear region should also impact the way one takes into account other quantities such as temperature, turbulence fluxes, and vertical velocity, as they are indeed physically interdependent. However, the goal of the diagnostic SCPF scheme is to stay as simple as possible to ease its implementation in operational models. It can be seen as a first-order subgrid scheme concerning only the total moisture content. Yet, a subgrid parameterization of the vertical velocity, for the time being not considered in the subcloud scheme, will be investigated in the future. Another direction of research would be to investigate the scale dependency of the parameters of the SCPF (relative humidity criterion, subgrid precipitation fraction) and the spatial resolution of the NWP model. A characteristic U00 profile should be found for scales ranging from 1 to 50 km (Xu and Randall 1996). And, as sensitivity tests have demonstrated the potential impact on precipitation production and type, the precipitation fraction could also be an important issue and its parameterization should be refined (Stevens and Seifert 2008; Turner et al. 2012), as observations can exhibit great in-cloud variability of the area of precipitation (e.g., Comstock et al. 2007; Lensky and Rosenfeld 1997). Acknowledgments. The research reported in this paper has been supported by the Natural Science and Engineering Research Council of Canada and HydroQuebec through the IRC program. The authors thank J. Delanoe¨ for providing DARDAR-Cloud data. This

work was made possible, in part, with the support of the Government of Canada Program for International Polar Year.

APPENDIX A Subgrid Precipitation Fraction The total cloud cover Ck integrated from the model top level (k 5 1) to level k is given by Ck 5 1 2 (1 2 Ck21 ) 3

1 2 max(ak , ak21 ) , 1 2 min(ak21 , 1 2 d)

(A1)

where d is a tiny constant of 1026; ak and ak21 are respectively the cloud fractions at level k and just above. Note that Ck can only increase or remain constant relative to its value at the level above. Hence, even the contribution of a tiny cloud fraction at a low level beneath a higher cloud with a large extent would lead to an increase of the total cloud cover. Yet, the portion of the grid DC 5 Ck 2 Ck21 never experiences precipitation falling into it. The precipitation fraction at level k can be divided into a cloudy precipitation region (precipitation falling into cloud fraction) and a clear-sky precipitation region: clr aP,k 5 acld P,k 1 aP,k .

(A2)

These precipitation fractions are iteratively computed from model top to surface depending on the successive cloud fractions encountered. The highest cloud is considered to be a possible source of precipitation at its own level and for levels below, such that aP 5 acld P 5 a. In a general manner, for any given levels, the cloudy precipitation fraction is equal to the assumed source of precipitation just above (cloud fraction at level k 2 1), of the above clear-sky preplus the fraction Daclr/cld P cipitation fraction that falls within the cloud, minus the of above cloudy precipitation fraction fraction Dacld/clr P that falls in the clear-sky part at level k: clr/cld 2 Dacld/clr . acld P,k 5 ak21 1 DaP P

(A3a)

Conversely, the clear-sky precipitation fraction is equal to the one just above, corrected by the fraction of cloudy precipitation or clear-sky precipitation that potentially falls in the actual clear-sky part: clr clr/cld 1 Dacld/clr . aclr P,k 5 aP,k21 2 DaP P

(A3b)

If all precipitation mixing ratios are null at level k, the potential cloudy precipitation fraction acld P,k is then set

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equal to the local cloud fraction ak and aclr P,k is set to 0. Considering that any cloud is a possible source of underlying precipitation, and using both maximumrandom overlap of cloud fractions, and maximum overlap of precipitation fractions with the area possibly experiencing precipitation ak 2 DC in the underlying level, one can infer the portion of cloudy precipitation fraction just above level k that falls into clear air as: Dacld/clr 5 ak21 2 min(ak 2 DC, ak21 ) , P

(A3c)

and the portion of clear-sky precipitation fraction just above level k that falls within the cloud as:

constant during the main time step. Hence, within each model time step, the microphysics scheme is called n times but at the ith sub–time step, we have dyn

phys Dq 1 dq q tx 5 qxt 1 with dyn t n phys ›qx Dq t ti . with qx 5 qx 1 dq 5 dt ›t phys 2n i11

i

(B2)

At the beginning of the loop, t i 5 t*, and at the end, we have 1

5 max[0, min(aclr Daclr/cld P P,k21 , ak 2 DC 2 ak21 )] .

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qxt 5 qxt* 1

n phys

å dq

.

(B3)

i51

(A3d) If one considers that the highest precipitation production region (seeding region) is actually only a portion Pfrac of the cloud layer where it first appears, Eq. (A3c) is replaced by cld Dacld/clr 5 acld P P,k21 2 min(ak 2 DC, aP,k21 ) ,

(A4)

and the additional assumptions detailed in section 2c are used. Obviously, if Pfrac 5 1, Eq. (A4) is equivalent to Eq. (A3c).

APPENDIX B Microphysical Sub–Time Step The original MY2 code receives as input all quantities qx (including temperature and pressure but excluding vertical velocity) at time t2, end of the previous model time step, and at time t* 5 t2 1 Dt, a step after the semiLagrangian advection scheme but before the calculation of the physical processes. The microphysical tendencies phys

Dq , output of the MY2 scheme, are evaluated at an intermediate time t, where all quantities, except vertical velocity, receive values halfway between t2 and t*; then, at the end t1 5 t2 1 Dt of the time step, the quantities are updated: 1

phys

qxt 5 q tx* 1 Dq

2

qxt 5

qxt* 1 qxt . 2

phys

with Dq 5 Dt

›qxt ›t

with phys

(B1)

In the sub-stepping approach, the microphysical subtime step dt is fixed so that n 5 Dt/dt. The quantity dyn

2

Dq 5 qxt* 2 qxt is considered as a dynamical forcing,

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