## 1. Introduction

Proper representation of cloud microphysical and precipitation processes is critical for the simulation of weather and climate in atmospheric models. Despite decades of advancement, microphysics parameterization schemes still contain many uncertainties. This is due to an incomplete understanding of the important physical processes as well as the inherent complexity of hydrometeors in the real atmosphere. To represent the range of particles and their physical properties within the constraints of limited computational resources, current microphysics schemes use various hydrometeor categories defined by prescribed physical characteristics (e.g., shape, bulk density, terminal fall speeds, etc.) that broadly describe a given “typical” particle type. The relative simplicity of this approach has been successful in some aspects of parameterizing microphysics and problematic for others.

For liquid-phase microphysics in bulk schemes, the approach of separating drops into two categories,^{1} defined essentially by size ranges, has worked reasonably well to model the onset of precipitation in warm clouds (Kessler 1969). This is because liquid drops are well represented by spheres with a density of liquid water (~1000 kg m^{−3}) up to sizes of several mm. There is also a clear separation of physical processes, with droplets smaller than approximately 50–100 *μ*m in diameter (generally referred to as “cloud” in bulk schemes) growing mainly by vapor diffusion and larger drops (“rain”) growing primarily by collision–coalescence. Separate categories for cloud and rain allow bulk schemes to simulate the rapid nonlinear growth of rain once embryo drops form by collision–coalescence. Two-moment bulk schemes do a remarkably good job in reproducing the evolution of unimodal spectrum of cloud droplet into a bimodal spectrum due to collection and coalescence as simulated by detailed bin-resolving models (Berry and Reinhardt 1974; Ziegler 1985; Cohard and Pinty 2000; Morrison and Grabowski 2007).

In contrast, the parameterization of ice-phase microphysics is more challenging and the use of predefined categories is inherently problematic. Unlike liquid drops, ice particles have a wide range of densities and complex shapes that affect their growth and decay processes. Moreover, they can grow by different processes (vapor diffusion, aggregation, riming) across a wide range of sizes. Partitioning ice-phase particles into a limited number of categories with specified shape, density, and other physical characteristics is a highly simplified representation of nature and necessitates the conversion of particles between categories, which is inherently artificial and often done without a strong theoretical or empirical basis. This approach is used in both spectral (bin) schemes (e.g., Reisin et al. 1996; Geresdi 1998; Khain et al. 2004; Lebo and Seinfeld 2011) and bulk schemes (e.g., Koenig and Murray 1976; Rutledge and Hobbs 1983; Lin et al. 1983; Ferrier 1994; Morrison et al. 2005; Milbrandt and Yau 2005b; Thompson et al. 2008). There is a large sensitivity of model simulations to how ice is partitioned into categories, and changes in thresholds or rates for conversion between ice species can lead to substantial differences in simulations (e.g., Colle et al. 2005; Morrison and Grabowski 2008a, hereafter MG08). Moreover, parameter settings for a given category, such as particle densities and fall speeds, are uncertain and simulations can exhibit considerable sensitivity to settings for these parameters (e.g., Gilmore et al. 2004; McFarquhar et al. 2006). For example, representing rimed ice with hail-like versus graupel-like characteristics can have large impacts on storm structure and precipitation associated with deep convection (e.g., McCumber et al. 1991; Gilmore et al. 2004; Cohen and McCaul 2006; Morrison and Milbrandt 2011; Bryan and Morrison 2012; Van Weverberg 2013; Adams-Selin et al. 2013).

There has been a general trend in the development of microphysics schemes to try and address these deficiencies by adding complexity to the representation of the ice phase, either by increasing the number of categories or adding more prognostic variables to existing categories. Earlier bulk schemes that included frozen hydrometeors used two categories—small “cloud ice” and larger, faster-falling “snow” (e.g., Rutledge and Hobbs 1983)—with conversion from one to the other based on an analogy of conversion from cloud liquid to rain. Walko et al. (1995) extended this approach by including cloud ice and snow and adding a separate category for crystal aggregates. To increase further the range of possible fall speeds, a rimed ice category (“graupel” or “hail”) was added (Lin et al. 1983; Rutledge and Hobbs 1984). In a few more recent schemes there is a user-specified switch allowing the rimed ice category to represent either fast-falling hail or slower-falling graupel (Morrison et al. 2009; Lang et al. 2011). To allow for both slower- and faster-falling rimed ice, other schemes have included separate categories for graupel and hail (e.g., Ferrier 1994; Milbrandt and Yau 2005b; Mansell et al. 2010). Straka and Mansell (2005) used three separate graupel/hail categories to track particles that originated from different processes.

With the exception of Koenig and Murray (1976), all earlier bulk schemes used only one prognostic variable per category—the mass mixing ratio—thus having a single degree of freedom to represent the size distribution. Two-moment schemes, where the mass and number mixing ratios are prognosed independently, were then developed (e.g., Ziegler 1985; Ferrier 1994; Seifert and Beheng 2001; Meyers et al. 1997; Morrison et al. 2005; Milbrandt and Yau 2005a; Philips et al. 2007; Lim and Hong 2010). The simulation of microphysical processes and sedimentation for a given category is generally improved with the two-moment approach (Ferrier 1994; Milbrandt and McTaggart-Cowan 2010; Dawson et al. 2010). A three-moment scheme was introduced by Milbrandt and Yau (2005a,b) whereby the inclusion of a third prognostic moment (reflectivity) allows for the prediction of the dispersion of the size spectrum, overcoming some of the limitations in two-moment schemes (in particular excessive size sorting due to sedimentation). To broaden further the range of validity for a given ice-phase category, recent work has added yet more complexity. For example, Connolly et al. (2006) relaxed the assumption of a fixed density for graupel by adding a prognostic variable for the bulk volume mixing ratio, with different graupel densities arising from different growth processes. This approach was advanced by Mansell et al. (2010) and Milbrandt and Morrison (2013) by using the predicted graupel density to include physically consistent fall speed calculations as well as empirical changes to the rime density. However, while the added complexity of these approaches allows for the representation of a wider range of particle characteristics, inherently artificial conversion processes are still required with the use of separate predefined ice categories. Furthermore, greater complexity by increasing the number of categories means an increase in the number of uncertain conversion processes and parameters, which may inherently limit improvements that one might expect with the increased complexity.

An alternative approach that evolves particle properties in time and space instead of separating ice into different predefined categories was first proposed in the bin microphysics scheme of Hashino and Tripoli (2007). MG08 developed a bulk scheme that separately prognoses ice mass mixing ratios grown by riming and vapor deposition to improve the treatment of the transition between unrimed snow, rimed snow, and graupel. Harrington et al. (2013a,b) and Sulia et al. (2013) developed a bulk scheme that predicts particle habit evolution by including the crystal *a*- and *c*-axis mixing ratios as prognostic variables, thereby allowing for prediction of crystal axis ratio from vapor depositional growth. Other schemes have used a diagnostic approach to include variability in ice particle properties (Lin and Colle 2011; Eta Ferrier scheme). Lin and Colle (2011) included separate categories for cloud and precipitating ice and diagnosed the degree of riming and ice particle properties (mass–size and fall speed–size relationships) for precipitating ice as a function of the ratio of the riming to the riming plus vapor deposition growth rates. Such a diagnostic approach is computationally efficient because it does not require additional prognostic variables, but the disadvantage is that particle properties are calculated locally and are not tracked in time and space.

These efforts represent a broader shift in the representation of ice microphysics by emphasizing the prediction of particle properties rather than the separation of ice into different predefined categories. In this study, the approach is generalized and a method is proposed to predict several bulk physical properties of ice particles, which can evolve through the full range of growth and decay processes. The implementation described in this paper uses a single ice-phase category. The proposed approach thus completely eliminates the need for artificial conversion between ice categories. This forms the basis for a conceptually new bulk microphysics scheme. This study introduces the proposed approach and new scheme and demonstrates its overall performance. A detailed description of the method and the scheme are provided in this paper along with idealized simulations to illustrate its behavior and sensitivity to key parameters. In Morrison et al. (2015, hereafter Part II), results from kilometer-scale simulations using the new scheme for two real cases—deep convection and orographically enhanced frontal precipitation—are compared to those using existing bulk schemes that employ the traditional predefined ice-category approach.

The remainder of the article is organized as follows. Section 2 provides a description of the method and the new scheme. Section 3 presents simulations of an idealized two-dimensional (2D) squall line that illustrate the overall behavior of the scheme. A summary and conclusions are given in section 4.

## 2. Scheme description

### a. Overview

A new bulk scheme using the proposed approach has been developed, which we refer to as the predicted particle properties (P3) scheme. To represent the evolution of various physical properties in space and time, the scheme includes a single ice-phase category with four prognostic mixing ratio variables: the total ice mass *q*_{i}, ice number *N*_{i}, the ice mass from rime growth *q*_{rim}, and the bulk rime volume *B*_{rim}. These are chosen as the conserved prognostic variables because together they are able to track particle evolution through all the important mechanisms of ice growth, including vapor deposition, aggregation, and riming (dry and wet growth). From this choice of prognostic variables, several important predicted properties are derived, including the rime mass fraction, bulk density, and mean particle size. Here we distinguish between “prognostic” variables, which are conserved and include dynamical tendencies from advection and subgrid-scale mixing and microphysical tendencies (growth/decay processes and sedimentation), and “predicted” quantities, which are derived directly from the prognostic variables and hence vary locally in time and space. The liquid-phase component of the scheme is summarized in appendix A.

*χ*has the formwhere

*t*is time,

*ρ*is the air density,

**u**is the 3D wind vector,

*z*is height,

*Vχ*is the appropriately weighted fall speed for quantity

*χ*,

*Sχ*is the source/sink term and includes various microphysical processes, and Δ*(

*χ*) is the subgrid-scale mixing operator (all symbols for variables and parameters used in the paper are defined in Table 1). Microphysical process rates that determine

*Sχ*are described in appendix B. The process rates depend on various moments of the particle size distributions, represented by a three-parameter gamma distribution of the formwhere

*D*is the maximum particle dimension and

*N*

_{0},

*λ*, and

*μ*are the intercept, slope, and shape parameters, respectively. For ice,

*μ*follows from the in situ observations of Heymsfield (2003) based on particle size distribution (PSD) fits to tropical and midlatitude particle ensembles in ice clouds:where

*λ*has units of per meter in this formulation. For simplicity and following MG08,

*μ*is limited to 0 <

*μ*< 6, although Heymsfield (2003) shows negative values of

*μ*for

*λ*< ~7000 m

^{−1}. Based on this formulation nonzero

*μ*only occur for rather small mean particle size (1/

*λ*< ~0.17 mm).

List of symbols for variables and parameters.

*N*

_{0}and

*λ*are related to the prognostic number mixing ratio

*N*and mass mixing ratio

*q*bywhere

*m*(

*D*) is the particle mass as a function of

*D*and

*N*′(

*D*) is given by (2).

Although the current version of the P3 scheme has one ice-phase category, it is a “free category” in that, because of the evolution of its predicted properties, it can represent any type of ice particle. This is in stark contrast with “prescribed categories” in traditional schemes, whose evolution is intrinsically constrained. However, the one-category P3 scheme has the inherent limitation that it cannot simulate populations of particles with different bulk properties at the same point in time and space. Thus, attempting to capture a mixture of particles with substantially different bulk characteristics can lead to problems under certain conditions. For example, in deep, strong updrafts that loft both supercooled liquid water and large graupel to the homogeneous freezing level, rapid droplet freezing and production of a high small ice particle concentration might smear out characteristics of the large graupel. A similar situation occurs for Hallett and Mossop (1974) rime splintering. To minimize this problem, the current version neglects ice multiplication by rime splintering. Despite the limitation of using a single category, it produces results that compare well with observations relative to other bulk microphysics schemes for the real cases tested in Part II. In future development we plan to implement a multiple-free-category approach in P3 to address this limitation, as described in section 4. The multiple-free-category version will also be compared with the single-category version to test systematically the effects of this limitation.

### b. Particle mass, projected area, and fall speed

Integrating (4) and (5) over the PSD and solving for *N*_{0} and *λ* requires specification of the *m*–*D* relationship over the size distribution. For cloud droplets and rain, this is given simply by the relationship for spherical liquid drops: *m*–*D* relationship varies in time and space and over the range of particle sizes and is calculated from the predicted properties derived from the prognostic quantities. The approach is broadly similar to that of MG08 but with important distinctions as noted below.

*ρ*

_{i}= 917 kg m

^{−3}. It follows that the

*m*–

*D*relationship for small ice is

*D*(for nonspherical ice,

*D*is defined as the maximum particle length or dimension). For larger unrimed crystals grown by vapor diffusion and/or aggregation (i.e., when

*q*

_{i}> 0 and

*q*

_{rim}= 0), the

*m*–

*D*relationship is expressed as a power law:The parameters from Brown and Francis (1995) are used for

*α*

_{va}and

*β*

_{va}, derived from measurements in midlatitude cirrus. Other empirically or theoretically derived values for

*α*

_{va}and

*β*

_{va}could be used. There is some sensitivity of simulations to the choice of

*α*

_{va}and

*β*

_{va}; detailed discussion is beyond the scope of this paper but such tests are described in MG08 for their scheme. Changes in particle density associated with vapor diffusion and aggregation are implicitly included in (7) because the density decreases with increasing

*D*(since

*β*

_{va}< 3). The critical size separating spherical ice from unrimed nonspherical ice is found by extrapolating the

*m*–

*D*relationship in (7) down to the size that it equals the mass of an ice sphere for the same

*D*following Heymsfield et al. (2007). This critical size

*D*

_{th}is given by

*m*–

*D*relationship is considerably more complicated for rimed particles (i.e., when

*q*

_{rim}> 0). Following MG08 and based on the conceptual model of riming introduced by Heymsfield (1982), it is assumed that rime accumulates in the crystal interstices as the particle undergoes riming, increasing

*m*but not

*D*. Once the particle is “filled in” with rime, the particle is considered to be graupel

^{2}and further growth increases both

*m*and

*D*. [Note that the scheme distinguishes between wet and dry riming growth (see appendices B and C).] It is also assumed that prior to in-filling of the crystal interstices with rime, the rime mass fraction of an individual particle

*F*

_{r}is equal to the bulk rime mass fraction given by

*F*

_{r}=

*q*

_{rim}/

*q*

_{i}. It follows thatwhere

*m*

_{va}is the portion of the crystal mass grown by vapor diffusion and aggregation and

*m*

_{r}is the total particle mass of a partially rimed crystal. It is assumed that the

*m*–

*D*relationship for partially rimed crystals also follows a power-law relationship:Since

*D*is not affected by riming up to the point of complete rime in-filling based on the conceptual model of riming discussed above, it follows that

*m*

_{va}is given by (7). Combining (7), (9), and (10) and rearranging terms yieldsSince we assume constant

*F*

_{r}with

*D*and (11) holds true for arbitrary

*D*, this uniquely implies that

*m*–

*D*relationship for partially rimed crystals is

There is observational evidence supporting the assumption of constant *β* during riming (which follows logically from the conceptual model of rime in-filling and the assumption that *F*_{r} is constant with *D*). Rogers (1974) found the same *β* in the *m*–*D* relationship for rimed and unrimed snowflakes, with *α* about 4 times larger for rimed snow. Similarly, riming appears to have little effect on *β* for hexagonal columns, with a value 1.8 for both unrimed and rimed crystals [see Table 1 and section 4d of Mitchell et al. (1990)]. More recent analysis has also shown that the *β* parameter varies much less than *α* for rimed and unrimed crystals of the same underlying habit. Mitchell et al. (2014) show *α* of 0.001 263 and 0.001 988 for unrimed and heavily rimed dendrites, respectively, with *β* of 1.912 and 1.784. If the size interval corresponding with the largest unrimed dendrites is excluded, then *β* becomes 1.786, almost the exactly the same as for heavily rimed dendrites.

*m*

_{r}equals the mass of a graupel particle

*m*

_{g}for the same

*D*. In contrast to the approach of MG08, which assumed an empirically derived

*m*–

*D*relationship for graupel, here graupel particles are assumed to be spherical with an effective density

*ρ*

_{g}that is predicted and varies locally in time and space. Thus, the

*m*–

*D*relationship for graupel isThe critical size for complete in-filling with rime

*D*

_{cr}is found by equating the masses of partially rimed crystals and graupel particles (i.e., setting

*m*

_{r}=

*m*

_{g}). Using the

*m*–

*D*relationships for

*m*

_{r}and

*m*

_{g}following (12) and (13), respectively, and rearranging terms to solve for

*D*=

*D*

_{cr}yields

*m*–

*D*relationship for graupel given by (13) applies to a limited size range. Extrapolation of this relationship to smaller sizes leads to a bulk density of graupel that is less than that of unrimed ice. To avoid this inconsistency we follow the approach of MG08 and define a third critical size

*D*

_{gr}that represents the size where the masses of graupel and unrimed ice are equal. Particles smaller than

*D*

_{gr}are assumed to have an

*m*–

*D*relationship corresponding to unrimed ice, even though they may be rimed, to avoid low bulk densities for small particles and discontinuities in the particle mass as a function of

*D*across the PSD. Particles with sizes

*D*

_{th}<

*D*<

*D*

_{gr}are referred to as “dense nonspherical ice.” The value of

*D*

_{gr}is found by setting

*m*

_{g}=

*m*

_{va}and solving for

*D*=

*D*

_{gr}:Thus,

*D*

_{gr}depends on the mass–size relation for unrimed ice (

*α*

_{va}and

*β*

_{va}) as well as

*ρ*

_{g}. Here

*ρ*

_{g}depends on the history of rime growth (including the effects of densification due to wet growth) and the underlying (unrimed) habit of particles that form graupel. The value of

*ρ*

_{g}is found by calculating an

*F*

_{r}-weighted average of the predicted rime density,

*ρ*

_{r}=

*q*

_{rim}/

*B*

_{rim}, combined with the density of the unrimed part of the particle

*ρ*

_{d}:where

*ρ*

_{d}is found by a mass-weighted averaging of the unrimed particle density between sizes

*D*

_{gr}and

*D*

_{cr}, givingFor nonspherical particles, particle density is defined here as the particle mass divided by the volume of a sphere with the same

*D*. Equations (14)–(17) form a closed set of equations for

*ρ*

_{d},

*ρ*

_{g},

*D*

_{cr}, and

*D*

_{gr}that is solved by iteration.

Figure 1 illustrates how the PSD is partitioned into different regions following this approach. If *ρ*_{r} is held constant, increasing F_{r} increases the density of partially rimed crystals (i.e., larger mass for a given *D*) in the region of the PSD with *D* > *D*_{cr} (Fig. 1a). Increasing *F*_{r} also leads to an increase in *D*_{cr}, the critical size separating particles that have filled in with rime (i.e., graupel) with partially rimed crystals, following (14). Because of the interdependence of *D*_{cr}, *D*_{gr}, *F*_{r}, and *ρ*_{r} following (14)–(17), increasing *F*_{r} also produces a small increase in *ρ*_{g} and a small decrease in *D*_{gr}. For example, increasing *F*_{r} from 0.5 to 0.8 leads to relative changes in *ρ*_{g} and *D*_{gr} of about 10%. If instead *F*_{r} is held constant, increasing *ρ*_{r} leads to an increase in the density of graupel—that is, particles in the region of the PSD between *D*_{gr} and *D*_{cr}, as well as an increase in the critical size *D*_{gr} (Fig. 1b), as long as *F*_{r} > 0. On the other hand, *D*_{cr} decreases with increasing *ρ*_{r} since higher rime density means that more rime mass can accumulate on partially rimed crystals before the total particle mass is equal to that of a graupel particle for a given *D* (in other words, more rime mass can accumulate before partially rimed crystals become filled in with rime). The mass of particles with *D* greater than about 1 mm is more sensitive to *F*_{r} than *ρ*_{r} overall. This has implications for bulk parameters such as the mass-weighted fall speed as described below.

Another important hydrometeor property is the projected area *A*(*D*) since this is needed to calculate fall speeds and effective radii. For cloud droplets, rain, dense ice spheres (*D* < *D*_{th}), and graupel (*D*_{gr} < *D* < *D*_{cr}), the particle projected area is simply given by the *A*–*D* relationship for spheres, *A*–*D* relationship is empirically derived from ice particle observations. Here we use parameters for aggregates of side planes, bullets, and columns and assemblages of planar polycrystals from Mitchell (1996) and references therein. Other empirical *A*–*D* relationships could be employed, but it is important that the relationship is consistent with the *m*–*D* relationship employed, as otherwise unreasonable values of fall speed may occur since this depends on *m*/*A*. Such consistency is described theoretically by the fractal approach of Schmitt and Heymsfield (2010). Since the conceptual model of riming described above does not provide information on the evolution of projected area for partially rimed crystals, for simplicity a simple linear weighting is assumed between the value for graupel (i.e., spheres) and unrimed ice as a function of *F*_{r} following MG08.

It is critical to link fall speed to the *m*–*D* and *A*–*D* relationships because particle densities vary in time and space. In most bulk microphysics schemes, particle densities are fixed and ice particle fall speed is simply given as a power law of *D* without explicit dependence on density. Such an approach leads to an incorrect dependence of fall speed on density if the density is modified; fall speed decreases when the particle density is increased since this leads to a decrease in the mean particle size. In the proposed approach, the terminal fall speed of ice follows a power-law relationship *a*_{1} and *b*_{1} are derived following Mitchell and Heymsfield (2005) based on the *R*_{e}–*X* relationship, where *R*_{e} is the particle Reynolds number and *X* is the Best (Davies) number (related to the ratio of the particle mass to its projected area). This approach follows Khvorostyanov and Curry (2002) to calculate smooth *a*_{1} and *b*_{1} as a function of *D* but modified to include surface roughness coefficients appropriate for ice particles. The particle mass and projected area are found from the *m*–*D* and *A*–*D* relationships described above and, hence, include an explicit dependence on particle density. Since *F*_{r} and *ρ*_{r} are assumed to be independent of *D*, the same mass-weighted terminal fall speeds are applied to *q*_{i}, *q*_{rim}, and *B*_{rim}, while the number-weighted terminal fall speed is applied to *N*_{i} as described earlier.

The air density modification of particle fall speeds for rain and ice follows Heymsfield et al. (2007), which gives a multiplicative factor of (*ρ*_{0}/*ρ*)^{0.54} for the fall speed, where *ρ*_{0} is a reference air density. Note this correction is not applied to cloud droplets since Stokes’ law implicitly includes a dependence on environmental conditions through the dynamic viscosity of air.

The size distribution parameters *N*_{0} and *λ* are derived from (4) and (5). For cloud droplets and rain, the integrals in (4) and (5) can be calculated analytically using Euler gamma functions since the parameters in the *m*–*D* relationship in (5) are constant across the PSD. Hence, analytic expressions can be derived for *N*_{0} and *λ* [e.g., see (2) and (3) in Morrison (2012)]. For ice, however, the integral in (5) involves incomplete gamma functions because the *m*–*D* relationship varies across different regions of the PSD. Thus, *N*_{0} and *λ* cannot be derived analytically and are instead solved by iteration. Since this is computationally expensive, a lookup table approach is employed to make the scheme computationally efficient. Values of *N*_{0}, *λ*, and moments of the PSD relevant to calculation of the microphysical process rates and parameters for ice are precalculated and stored in a lookup table as a function of *q*_{i}, *N*_{i}, *F*_{r}, and *ρ*_{r}.

Figure 2 illustrates the mass-weighted ice fall speed *V*_{m} as a function of *F*_{r} and *ρ*_{r} for a given *q*_{i}/*N*_{i} corresponding to small, medium, and large values of mass-weighted mean particle size *D*_{m} (note that *D*_{m} is not constant for a given *q*_{i}/*N*_{i} because it changes with *F*_{r} and *ρ*_{r}). For small *D*_{m} there is little sensitivity of *V*_{m} to either *F*_{r} or *ρ*_{r} (Fig. 2a). This is because as *D*_{m} shifts to small sizes the size distribution becomes dominated by small spherical ice, which has a bulk density of solid ice regardless of *F*_{r} or *ρ*_{r}. Interestingly, larger *ρ*_{r} and *F*_{r} actually produce slightly smaller *V*_{m} because this leads to smaller *D*_{m} for a given *q*_{i}/*N*_{i}. The picture differs for larger *D*_{m}, with much greater sensitivity to *ρ*_{r} and *F*_{r} (Figs. 2b,c). As expected, *V*_{m} increases with an increase in either *ρ*_{r} or *F*_{r}. However, there is somewhat greater sensitivity to *F*_{r} than *ρ*_{r}, which reflects greater sensitivity of the *m*–*D* relationship to *F*_{r} (see Fig. 1). Also shown in Figs. 2b and 2c are *V*_{m} calculated from empirical power-law *V*–*D* relationships for different ice particle types integrated over the PSD. While observed values of *F*_{r} and *ρ*_{r} have not been quantified as a function of particle type, it is reasonable to assume that *F*_{r} increases between rimed snow [rimed dendrites and aggregates of dendrites from Locatelli and Hobbs (1974)], graupel-like snow (Locatelli and Hobbs 1974), lump graupel (Locatelli and Hobbs 1974), and hail (Matson and Huggins 1980; Ferrier 1994). We also assume *ρ*_{r} is larger for hail (~900 kg m^{−3}) compared to the other particle types (~400 kg m^{−3}). The modeled *V*_{m} are 1.5–2 m s^{−1} for rimed snow (assuming *F*_{r} ~ 0.2 and *ρ*_{r} ~ 400 kg m^{−3}), 1.5–3 m s^{−1} for graupel-like snow and lump graupel (assuming *F*_{r} ~ 0.5–0.7 and *ρ*_{r} ~ 400 kg m^{−3}), and greater than 8 m s^{−1} for large hail (assuming *F*_{r} ~ 1 and *ρ*_{r} ~ 900 kg m^{−3}). These values are reasonably similar to the range of empirical *V*_{m} for a given ice particle type.

### c. Numerical implementation

The scheme uses a time-split forward Euler solution similar to most other microphysics schemes. Within a time step, the scheme first calculates all of the microphysics source/sink processes in *Sχ* following (B1)–(B8) (see appendix B) except homogeneous freezing of cloud water and rain. It then updates all prognostic state variables with these tendencies. These updated variables are used to calculate sedimentation, after which the scheme further updates the prognostic variables. Sedimentation is calculated using a simple first-order upwind method following several other microphysical schemes (e.g., Reisner et al. 1998; Thompson et al. 2008; Morrison et al. 2009) with substepping as needed for numerical stability based upon the Courant–Friedrichs–Levy criterion. Lastly, the scheme calculates homogeneous freezing of cloud water and rain and updates variables at the end of the microphysical calculations. Homogeneous freezing is calculated at the end of the microphysics time step to avoid the unphysical situation of having significant liquid water at temperatures colder than 233 K.

Conserved (extensive) quantities *q*_{i}, *q*_{rim}, *B*_{rim}, and *N*_{i} are included as the choice of prognostic variables. Two of the key properties predicted in the scheme, *F*_{r} and *ρ*_{r}, depend on the ratio of these prognostic variables: *F*_{r} and *ρ*_{r} fields that occur during advection and is one reason these particular prognostic variables are used. Despite limited error when coupled with transport, some drift may occur, which can lead to inconsistency between the mass, number, and volume mixing ratios, especially when the quantities are very small. To address this, *λ* is limited to a range of values, as is done in all multimoment bulk microphysics schemes. This is accomplished by limiting the number-weighted mean particle diameter, *D*_{N} = (*μ* + 1)/*λ*, to 1 < *D*_{N} < 40 *μ*m for cloud water and 2 < *D*_{N} < 2000 *μ*m for ice. If *D*_{N} is outside of these bounds, then *N* is adjusted so that *D*_{N} lies within the specified range for each species. The predicted rime density *ρ*_{r} is also limited to values between 50 and 900 kg m^{−3}. If necessary, *B*_{rim} is adjusted to keep *ρ*_{r} within this specified range.

## 3. Idealized 2D squall-line simulations

### a. Setup

This section describes a set of simulations that illustrate the behavior of the new scheme. We use the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008), version 3.4.1, which is a compressible, nonhydrostatic dynamical atmospheric model. The following tests use WRF in a 2D configuration similar to the standard idealized squall-line test case. Two-dimensional idealized tests are used here because the simplicity of this model setup allows us to clearly demonstrate behavior of the scheme. The focus here is on the microphysics; interactions between microphysics and dynamics are explored further in Part II.

The governing equations are solved using a time-split integration with a third-order Runge–Kutta scheme. Horizontal and vertical turbulent diffusion are calculated using a 1.5-order turbulence kinetic energy (TKE) scheme (Skamarock et al. 2008). Third- and fifth-order discretization schemes are used for vertical and horizontal advection, respectively, with limiters to ensure monotonicity (Wang et al. 2009). The upper and lower boundaries are free slip with zero vertical velocity. Surface fluxes are set to zero and radiative transfer is neglected for simplicity. A Rayleigh damper is applied to the upper 5 km with a damping coefficient of 0.003 s^{−1}. The horizontal grid spacing is 1 km, with 80 vertical levels between the surface and model top. The time step is 5 s and the domain size is 500 km in the horizontal and 20 km in the vertical. Lateral boundary conditions are open. Radar reflectivity is calculated assuming Rayleigh scattering following the approach of Smith (1984) using the predicted size distribution and particle density parameters. A discussion of the uncertainties in using this approach is discussed in Smith (1984) and in Part II and references therein.

The model is initialized with the analytic sounding of Weisman and Klemp (1982, 1984). The initial vertical wind shear is 0.0048 s^{−1} applied between the surface and 2.5 km (meaning that horizontal wind changes 12 m s^{−1} between the surface and 2.5 km). Convection is initiated by adding a thermal with maximum perturbation potential temperature of 3 K centered at a height of 1.5 km and varying as the cosine squared to the perturbation edge. The thermal has a horizontal radius of 4 km and a vertical radius of 1.5 km. Model integrations are for 6 h.

### b. Baseline results

Moist convection is triggered within the first few minutes of the simulation from the initial thermal. Ice is initiated after approximately 10 min, and precipitation reaches the surface after approximately 20 min. In its early stages the storm is nearly symmetric, but significant horizontal asymmetry develops over time in response to the environmental shear. After about 4 h the storm reaches a quasi-equilibrium mature phase with a well-defined leading edge of convection and trailing stratiform precipitation.

*Z*, with a peak

*Z*at the lowest model level of 52.3 and 60.3 dB

*Z*aloft (Fig. 4a). There are large amounts of cloud water and rain within and below the convective core (Figs. 3a,b). Ice condensate with mixing ratios exceeding 8 g kg

^{−1}occur in the core, with the

*q*

_{i},

*q*

_{rim}, and

*B*

_{rim}fields exhibiting a similar pattern (Figs. 3c–e). The value of

*N*

_{i}exhibits a sharp increase with height (Fig. 3f) owing to freezing of cloud droplets in the convective core and detrainment at upper levels as well as size sorting effects and aggregation. The convective core has values of

*F*

_{r}close to 1 (Fig. 4b) associated with large amounts of supercooled liquid water and hence large riming and drop freezing rates. It also has values of mean mass-weighted ice particle density

*ρ*

_{p}from about 300 to 600 kg m

^{−3}(Fig. 4c). Here

*ρ*

_{p}is calculated aswhere

*α*and

*β*are parameters of the power-law mass–size relationships that vary between the four regions of the PSD described in section 2c and shown in Fig. 1. The values of

*V*

_{m}reach 5–8 m s

^{−1}in the high-density core region (Fig. 4d). Also

*ρ*

_{p}increases with height outside of the convective core as a consequence of the decrease in

*D*

_{m}with height (Fig. 4e);

*ρ*

_{p}is close to the density of solid spherical ice (~900 kg m

^{−3}) near cloud top where

*D*

_{m}< 0.1 mm but is less than 100 kg m

^{−3}lower in the anvil region outside of the convective core, where

*D*

_{m}~ 3–5 mm. Sensitivity tests show the sharp vertical gradient of

*D*

_{m}in the anvil region is partly a result of aggregation and size sorting, while other factors such as increased vapor depositional growth in the relatively warmer temperatures at lower altitudes also likely play a role. Fairly large mean particle sizes (

*D*

_{m}~ 3–4 mm,

*λ*~ 8–10 cm

^{−1}) occur above the melting level outside of the convective core and are consistent with aircraft observations of

*λ*in deep precipitating stratiform cloud systems (Heymsfield et al. 2008).

List of idealized 2D squall-line microphysics tests.

By 6 h, the storm has developed significant horizontal asymmetry associated with the environmental shear (Figs. 5–6). There is a leading edge of high reflectivity (>45 dB*Z*) associated with intense convection near the cold pool edge and a large region of trailing stratiform precipitation with *Z* between 25 and 45 dB*Z* (Fig. 6a). The total storm width is about 140 km, defined by the region with *Z* > 5 dB*Z* at the surface. There is a large anvil region with ice mass mixing ratios up to about 1 g kg^{−1}, but areas of appreciable *q*_{rim} and *B*_{rim} are limited to the convective core region (Figs. 5c–e). The value of *F*_{r} is near 1 in the convective region (Fig. 6b) owing to the presence of substantial amounts of cloud water and rain above the freezing level (Figs. 5a,b) and hence riming and drop freezing. Below 5 km, there is a narrow core of high *ρ*_{p} (>700 kg m^{−3}) along the immediate leading edge of the storm (Fig. 6c). Aloft in the anvil and trailing stratiform region, *F*_{r} is small—generally less than 0.2 and often near 0. This indicates vapor deposition is the dominant growth mechanism there in terms of bulk mass, with growth by aggregation also contributing to an increase in *D*_{m} as particles fall from the anvil. Values of *ρ*_{p} are low (<50 kg m^{−3}) below about 6 km in this region, consistent with characteristics of large unrimed or lightly rimed aggregates. Moving from front to rear at midlevels (4–8 km) between the convective and trailing stratiform regions, there is a general decrease in *D*_{m}, *F*_{r}, *ρ*_{p}, and *V*_{m} that is consistent with size and density sorting occurring in the storm-relative front-to-rear wind flow. Overall, the growth of heavily rimed particles in the convective cores, the fallout of large rimed particles within the convective region, the detrainment of smaller ice particles to the upper anvil, and the growth of these particles primarily by vapor deposition and aggregation as they fall through the trailing stratiform region are consistent with microphysical observations and retrievals of midlatitude squall lines (e.g., Rutledge and Houze 1987; Houze et al. 1989; Biggerstaff and Houze 1991, 1993; Braun and Houze 1994). In summary, by predicting important physical properties with 4 degrees of freedom, the scheme is able to simulate various types of ice-phase particles in the expected storm locations using only a single ice category.

It is also important to note that there are no obvious relationships between *F*_{r} and *ρ*_{r} and the cloud, dynamical, and thermodynamic variables such as temperature, vertical velocity, or hydrometeor mass mixing ratios prognosed in traditional microphysics schemes (Fig. 7). For example, while riming in locations with appreciable liquid water (large *q*_{c} + *q*_{r}), primarily in convective updrafts, is a major driver in evolving the particle properties, transport in and around updrafts leads to large *F*_{r}, *ρ*_{r}, and *V*_{m} in locations without any liquid water. Thus, even though almost all points with appreciable supercooled liquid (*q*_{c} + *q*_{r} > 0.5 g kg^{−1}) have *F*_{r} > 0.7, the converse is not true; that is, most points with *F*_{r} > 0.7 do not contain significant liquid water (Fig. 7a). These points with *F*_{r} > 0.7 occur near liquid water in the convective region, with rimed ice transported away from convective updrafts by air motion and sedimentation. Values of *F*_{r} from about 0.05 to 0.3 are seen in locations that are quite far (tens of kilometers) from grid points with liquid water because of horizontal transport of rimed particles in front-to-rear flow from the convective region. The transport of particles from convective updrafts several tens of kilometers is consistent with analyses from kinematic retrievals of midlatitude squall lines [e.g., see Fig. 17 in Biggerstaff and Houze (1991)]. Large scatter is also seen in the relationships between *F*_{r} and *q*_{i} or temperature and between *ρ*_{r} and *q*_{c} + *q*_{r}, *q*_{i}, or temperature.

### c. Sensitivity tests

To understand further the behavior of the new P3 scheme, four sensitivity tests (summarized in Table 2) were performed: 1) specification of constant *ρ*_{r} = 400 kg m^{−3} (*ρ*400), 2) specification of constant *ρ*_{r} = 900 kg m^{−3} (*ρ*900), 3) specification of constant *F*_{r} = 0 (FR0), and 4) specification of constant *F*_{r} = 1 and constant *ρ*_{r} = 400 kg m^{−3} (FR1*ρ*400). Note that FR0 does not require specification of *ρ*_{r} since there is no rime mass in this simulation. Specified values of *ρ*_{r} = 400 or 900 kg m^{−3} are used here since these values are typically assumed for either the graupel or hail categories in most microphysics schemes. These sensitivity tests demonstrate the value gained by addition of *q*_{rim} and *B*_{rim} as prognostic variables, allowing for extra degrees of freedom and prediction rather than specification of *ρ*_{r} and *F*_{r}.

Figures 8–9 show vertical cross sections of ice mixing ratio and surface precipitation rate for each simulation averaged from 3.5 to 4.5 h. After 4.5 h, solutions rapidly diverge because of the initiation of convection well ahead (tens of kilometers) of the squall line, likely by gravity waves, in some of the simulations. Because of differences in storm propagation, results are shown relative to distance from the leading edge of the storm. In general, caution should be exercised when comparing results of sensitivity runs for single realizations, especially for 2D, because of rapid perturbation growth and limited inherent predictability at convective scales (e.g., Zhang et al. 2007; Wang et al. 2012). Nonetheless, several robust differences are apparent among the simulations. Overall, there is considerable sensitivity to both *F*_{r} and *ρ*_{r}, although sensitivity to *F*_{r} is somewhat greater than to *ρ*_{r}. Ice mixing ratios in FR0 are much larger in the convective region compared to the other simulations. This results from the small *V*_{m} caused by setting *F*_{r} = 0, with values less than about 1.7 m s^{−1} everywhere. The value of *V*_{m} is also relatively small in the convective region in *ρ*400FR1 and *ρ*400 compared to BASE, leading to somewhat greater ice mixing ratios aloft near the leading edge, with the opposite for *ρ*900. Large differences are also apparent among the simulations for surface precipitation rate (Fig. 9). The *ρ*400 simulation exhibits a broader region of high precipitation and lacks a secondary maximum of precipitation in the trailing stratiform region, while *ρ*400FR1 does not have a distinct peak precipitation rate in the convective region.

## 4. Discussion and conclusions

A new bulk microphysics scheme has been developed that predicts various ice particle properties for a single ice-phase hydrometeor category through the use of four appropriate prognostic ice variables which are conserved during advection. Thus, various physical properties can be computed with 4 degrees of freedom. This represents a significant departure from traditional bulk schemes where ice-phase particles are partitioned into several different predefined categories with fixed properties. The proposed approach is in the spirit of recent efforts in the development of bulk schemes to predict particle properties rather than specify them (e.g., MG08; Mansell et al. 2010; Milbrandt and Morrison 2013; Harrington et al. 2013a,b). However, it is the first such scheme to predict multiple bulk properties as ice evolves through the full range of growth processes, from initial nucleation followed by depositional growth, aggregation, and riming (including dry and wet growth) depending upon local conditions.

There are several conceptual and practical benefits of the new approach. First, it avoids the need to use poorly constrained thresholds and conversion processes between ice-phase categories, such as “autoconversion” from cloud ice to snow, that are artificial but intrinsically necessary in standard bulk schemes. It also represents a continuum of particle properties rather than discrete categories, limiting a potential source of sensitivity due to ad hoc thresholding. Since the predicted properties are real physical quantities, as opposed to unphysical parameters such as autoconversion threshold size, there is a potential for much closer coupling with observations. Finally, it is computationally efficient since the total number of prognostic ice variables is small compared to many bulk schemes. Illustration of the latter point through timing tests in the context of 3D model simulations is presented in Part II. It should be noted that this approach could also be applied to bin microphysics schemes, extending the methodology outlined by Morrison and Grabowski (2010), and would avoid the need to partition ice into predefined categories as is done in most mixed-phase bin schemes (e.g., Takahashi 1976; Reisin et al. 1996; Geresdi 1998; Khain et al. 2004; Lebo and Seinfeld 2011). Several aspects of ice microphysics not addressed in this study remain uncertain, such as aggregation and riming efficiencies and parameters associated with melting and vapor diffusion. Addressing these uncertainties will require close coordination with additional observational studies, including laboratory work. While such uncertainty is unavoidable in microphysics schemes, a fundamental premise behind the P3 approach is that this uncertainty should reside in physical parameters that can be measured, at least in principle, instead of conversion parameters that are ad hoc and/or unphysical.

In the current version of P3 used in this study, the proposed approach was applied to a single ice-phase category. This does not, however, preclude the possibility of having more than one free ice-phase category, which would allow ice-phase particles with different bulk properties to be present in the same grid box and time. In such a configuration, the free ice-phase categories would not be predefined, but rather each could evolve ice to a state with any set of properties (even the same as the other categories) depending upon the growth history and conditions. This would address one of the main limitations of the current P3 scheme: its inability to represent different ice types in the same location and time for a given particle size. The development of a multiple-free-category version of the P3 scheme and comparison with the single-category version is a subject of current work and will be reported in a future publication.

Idealized 2D squall-line tests in WRF were performed to illustrate the general microphysical behavior of the new scheme. A key result is that the scheme was able to produce a wide variety of ice particle characteristics in different regions of the squall line broadly consistent with observations, despite its inclusion of only a single ice category. Sensitivity tests showed the importance of including *q*_{rim} and *B*_{rim} as prognostic variables, allowing prediction of *F*_{r} and *ρ*_{r} instead of specification of these parameters. In these tests there were notable impacts of *F*_{r} and *ρ*_{r} on the mass of ice condensate aloft and the surface precipitation rate.

The predicted properties, *F*_{r} and *ρ*_{r} in particular, exhibited no clear relationships with quantities such as cloud and rain mass mixing ratios, ice mass mixing ratio, or temperature. This is because of transport (horizontal and vertical, including sedimentation) that resulted in ice moving away from the conditions under which it experienced earlier growth. The result was rimed ice with *F*_{r} > 0.7 in locations without liquid water in the convective region and *F*_{r} ~ 0.05–0.3 for locations in the stratiform region relatively far (tens of kilometers) from liquid water. This suggests the difficulty of diagnosing particle properties from quantities as is done in some bulk schemes [e.g., Ferrier scheme in WRF; Lin et al. (2011); Lin and Colle (2011)], in contrast to adding new prognostic quantities (*q*_{rim}, *B*_{rim}) that allow for prediction of particle properties. This is likely to be especially true for high-resolution models (horizontal grid spacing of order 10 km or less) with a time scale for horizontal transport across grid cells similar to or less than the time scale for ice sedimentation. This implies, therefore, that the addition of prognostic variables is needed so that the desired particle properties can be predicted independently; diagnostic relations to reduce the number of prognostic variables do not appear to be feasible.

Finally, we note that this approach is general and other predicted properties could be added to this framework. For example, the scheme could be combined with an improved representation of vapor depositional growth to predict the crystal *a*- and *c*-axis lengths, as in Harrington et al. (2013a,b), allowing for representation of the crystal axis ratio. An improved treatment of particle evolution during melting and wet growth is possible by including prediction of the liquid water fraction on ice particles (by prognosing the liquid water mass mixing ratio on ice) (e.g., Frick et al. 2013). Prediction of the spectral width of the particle size distribution could be accomplished by the addition of a third independent moment such as reflectivity (Milbrandt and Yau 2005b). Future work will explore these ideas for continued development of the P3 scheme.

This work was partially supported by the U.S. DOE ASR DE-SC0008648, U.S. DOE ASR DE-SC0005336, sub-awarded through NASA NNX12AH90G, and the NSF Science and Technology Center for Multiscale Modeling of Atmospheric Processes (CMMAP), managed by Colorado State University under Cooperative Agreement ATM-0425247. We are grateful to T. Mansell for helpful discussions. We thank W. W. Grabowski for long-standing discussion on challenging the notion of predefined ice categories in microphysics schemes.

# APPENDIX A

## Overview of Liquid-Phase Component

*q*

_{c}and the mass and number mixing ratios of rain (

*q*

_{r},

*N*

_{r}). A more detailed version of the scheme also includes prognostic equations for the cloud number mixing ratio

*N*

_{c}and the supersaturation and includes droplet activation on cloud condensation nuclei and cloud–aerosol interactions. This latter version of the scheme is used for the simulations described in this paper, while the former is used for the simulations in Part II. The particle size distributions (PSDs) for cloud water and rain follow the same type of gamma distribution as for ice [see (2)]. For cloud droplets,

*μ*is based on the observations of Martin et al. (1994) as implemented in Morrison and Grabowski (2007);

*μ*is also allowed to vary for rain. For the current implementation,

*μ*is a function of

*λ*following the disdrometer observations described by Cao et al. (2008):where

*λ*has units of per millimeter. This formula is not extrapolated to values of

*λ*larger than the Cao et al. (2008) data range (20 mm

^{−1}), giving a maximum

*μ*of approximately 8.28. The minimum allowed

*μ*for rain is 0.

Cloud droplet fall speed as a function of *D* is given by Stokes’s formulation. Rain fall speed is expressed using power-law relationships as a function of *m*(*D*) following Gunn and Kinzer (1949) and Beard (1976) as modified by Simmel et al. (2002). Three different power-law relationships are used for *D* < 134.43 *μ*m, 134.43 < *D* < 1511.64 *μ*m, and *D* > 1511.64 *μ*m. Because different relationships are applied to different size ranges, integration of the fall speed over the PSD requires incomplete gamma functions. Since these are computationally expensive, in the code the number- and mass-weighted rain fall speeds as well as integrated ventilation parameters for vapor diffusion are precomputed and stored in a lookup table.

# APPENDIX B

## Microphysical Process Rates

*Sχ*for each prognostic microphysical variable in (1) is given by the following equations.

- Liquid phase:

- Ice phase:

*ρ*

_{r}=

*q*

_{rim}/

*B*

_{rim});

The symbols in (B1)–(B8) represent various microphysical processes including nucleation, diffusional growth, collision–collection, freezing, and melting. The naming convention for these processes is as follows. The first letter describes whether the process involves a change in mass (Q), number (N), or volume (B) mixing ratio. For source and sink processes that do not involve multispecies interaction, the second letter indicates the species as cloud water (C), rain (R), or ice (I). For source/sink processes that involve multispecies interaction (i.e., the same process acting as a source for one species but a sink for another), the second letter (C, R, or I) indicates the species that is reduced as a result of the process. The remaining three letters indicate the type of microphysical process as defined in Table B1. Details of the microphysical process rate formulations are described in appendix C.

Symbols used to define microphysical process rates.

For simplicity, sink terms for ice (melting and sublimation) are assumed to reduce *q*_{rim} and *q*_{i} in proportion (i.e., the ratio *q*_{rim}/*q*_{i} is assumed to be unmodified by melting or sublimation). Here is it also assumed that the heterogeneous and homogeneous freezing of cloud water and rain (QCHET, QRHET, QCHOM, QRHOM) yields high-density ice and hence is included as a source for *q*_{rim}. This allows the model to simulate the production of high-density ice particles (i.e., embryo graupel/hail) from the freezing of liquid drops. We include freezing of cloud droplets in the production of high-density ice through QCHET and QCHOM, but this has limited impact since small ice particles are assumed to be dense ice spheres regardless of *F*_{r} or *ρ*_{r} (see section 2b).

Source terms for rime volume mixing ratio *q*_{rim} and the appropriate density. Freezing of cloud water and rain and rime generated by collection of rain by ice are assumed to produce ice with a density near solid bulk ice *ρ** = 900 kg m^{−3}. For sublimation and melting, it is assumed that bulk volume decreases in proportion with mass, i.e., density is unmodified. Wet growth [BIWET in (B8)] represents an additional sink term for *B*_{rim}, whereby *B*_{rim} decreases (i.e., particles become soaked and undergo densification) if wet growth conditions are diagnosed in subfreezing conditions (see appendix C, section g for details).

# APPENDIX C

## The Microphysical Process Formulations

### a. Droplet and crystal nucleation

In the version of the scheme with prognostic *N*_{c}, droplet activation is given by Morrison and Grabowski (2007, 2008b), assuming a constant background aerosol concentration and that the concentration of previously activated cloud condensation nuclei is equal to the local *N*_{c}. In the simulations discussed herein, the aerosol is specified as a lognormal size distribution with a total concentration of 300 cm^{−3} and mean size of 0.05 *μ*m, consisting of ammonium sulfate. Condensation freezing/deposition ice nucleation follows from Cooper (1986) as a function of temperature *T*, as implemented in Thompson et al. (2004). Given recent evidence for limited deposition nucleation in relatively warm conditions (Ansmann et al. 2009; de Boer et al. 2011), it is limited to conditions with *q*_{c} from droplet activation and *q*_{i} from ice nucleation are calculated by assuming initial cloud droplet and ice crystal radii of 1 *μ*m (with an initial ice density *ρ*_{i} = 917 kg m^{−3}).

### b. Liquid condensation/evaporation and ice deposition/sublimation

*δ*=

*q*−

*q*

_{sl}, where

*q*is the water vapor mixing ratio and

*q*

_{sl}is the liquid saturation mixing ratio:Here

*τ*is the multiphase supersaturation relaxation time scale defined bywhere

*τ*

_{c},

*τ*

_{r}, and

*τ*

_{i}are the supersaturation relaxation time scales associated with cloud droplets, rain, and ice, respectively,

*L*

_{s}is the latent heat of sublimation,

*c*

_{p}is the specific heat of air at constant pressure, and Γ

_{i}is the psychrometric correction to deposition/sublimation associated with latent heating/cooling:Here

*q*

_{si}is the saturation vapor mixing ratio with respect to ice. In (C1),

*A*

_{c}is the change in

*δ*due to vertical motion, turbulent mixing, radiation, and the Bergeron–Findeisen process:where

*p*is the air pressure,

*e*

_{s}is the saturation vapor pressure with respect to liquid,

*w*is the vertical velocity,

*g*is the acceleration of gravity, and Γ

_{l}is as in (C3) except that

*L*

_{s}is replaced with

*L*

_{υ}and

*q*

_{si}is replaced with

*q*

_{sl}. The time scales

*τ*

_{c}and

*τ*

_{r}are given by Morrison and Grabowski [2008b; see their (A5)]. For ice,

*τ*

_{i}follows MG08 by assuming a capacitance for that of a sphere for small spherical ice and graupel, and equal to 0.48 times that of a sphere following Field et al. (2008) for unrimed nonspherical ice. The capacitance for partially rimed crystals is linearly interpolated between values for unrimed ice and graupel based on the particle mass. Numerical integration of the appropriately weighted ice particle size distribution moment is done offline and results are stored in a lookup table.

*δ*

_{t=0}is the supersaturation at the initial time.

*δ*(

*t*) in (C5) by

*τ*

_{c}Γ

_{l}to obtain the condensation/evaporation rate as a function of time and then averaging the resulting expression over the model time step (from

*t*= 0 to

*t*= Δ

*t*). This gives [(9) in Morrison and Grabowski (2008b)]Similarly, the condensation/evaporation rate for rain is found by dividing

*δ*(

*t*) by

*τ*

_{r}Γ

_{l}and averaging over the model time step, yielding an expression similar to (C6) except that

*τ*

_{c}is replaced by

*τ*

_{r}. For ice deposition/sublimation, the term (

*q*

_{sl}−

*q*

_{si}) is added to

*δ*(

*t*) in (C5) to account for the fact that it is the supersaturation with respect to ice that drives ice deposition/sublimation, and the resulting expression is divided by

*τ*

_{i}Γ

_{i}to obtain the deposition/sublimation rate as a function of time and subsequently averaged over the model time step. This yieldsSince

*τ*of 10

^{8}s is applied in the code for calculating condensation/deposition to prevent division by zero following Morrison and Grabowski (2008b).

Reduction of *N*_{i} during sublimation is scaled to the change in *q*_{i}, i.e., (*N*_{i}/*q*_{i}) × QIMLT. This is approximately equivalent to assuming a constant mean size (it is exactly equivalent if *μ* is constant). Reduction of *N*_{r} during evaporation is treated similarly; that is, (*γ*_{r}*N*_{r}/*q*_{r}) × QREVP. Here *γ*_{r} is set to 0.5, which is in the middle range of values from the parameterization of Seifert (2008). Reduction of *N*_{c} during evaporation is neglected unless all the cloud water mass within a grid point evaporates, analogous to the homogeneous mixing assumption [see discussion in Grabowski (2006)]. A more detailed approach for parameterizing the homogeneity of droplet mixing and evaporation, such as proposed by Morrison and Grabowski (2008b) and Jarecka et al. (2013), could be readily implemented into the scheme.

As discussed by Stevens et al. (1996) and Grabowski and Morrison (2008), large errors can occur in the supersaturation field when *δ* is derived from *T* and *q* after advection in Eulerian models because of the nonlinear dependence of *q*_{sl} on *T*. This can lead to large errors in processes that are sensitive to small changes in supersaturation—namely, droplet activation. Here we adopt the method of Grabowski and Morrison (2008) and add *δ* as a fully prognostic variable, including its advection. In this method, inconsistencies between *T*, *q*, and *δ* after advection are avoided by adjusting *T* and *q* after advection so they are consistent with the prognosed *δ*. This adjustment of *T* and *q* is done by condensing the exact amount of water needed to bring *T* and *q* into agreement with *δ*, thereby also providing a source for *q*_{c}. Because errors in *δ* arising from separate advection of *T* and *q* primarily impact droplet activation and hence *N*_{c}, there is less of a need to employ this method and include *δ* as a prognostic variable if *N*_{c} is specified. Thus, a simpler (and computationally cheaper) version of the scheme that specifies *N*_{c} and does not prognose *δ* has also been developed; this simpler version is used for the tests discussed in Part II.

### c. Cloud droplet autoconversion, accretion, and self-collection

The parameterization of cloud droplet autoconversion to form rain, accretion of cloud droplets by rain, and self-collection of cloud droplets follows either Beheng (1994), Khairoutdinov and Kogan (2000), or Seifert and Beheng (2001) with a user-specified switch. Note that self-collection of droplets is not explicitly included in the Khairoutdinov and Kogan (2000) parameterization. For the tests here and in Part II, we use Khairoutdinov and Kogan (2000) for cloud water autoconversion and accretion.

### d. Raindrop self-collection and breakup

*E*

_{cr}following a modified version of the Verlinde and Cotton (1993) scheme. The value of

*E*

_{cr}decreases with increasing mean drop size beyond a threshold size

*D*

_{rt}:where

*D*

_{mr}is a scaled mean raindrop size given byand

*D*

_{rt}is set to 1400

*μ*m. Note that

*D*

_{mr}is identical to the mass-weighted mean diameter for an exponential drop distribution. This formulation for breakup corresponds with an equilibrium (for pure breakup–coalescence) mean volume diameter (i.e., when

*E*

_{cr}= 0) of about 1100

*μ*m, in agreement with Seifert (2008). Spontaneous raindrop breakup is treated by a simple relaxation of

*D*

_{mr}back to a specified mean size

*D*

_{sb}with a time scale

*τ*

_{sb}when

*D*

_{mr}>

*D*

_{sb}. Here

*D*

_{sb}= 2400

*μ*m and

*τ*

_{sb}= 10 s. While individual drops do not undergo spontaneous breakup until they reach rather large sizes (~5 mm) (Pruppacher and Klett 1997), it should be kept in mind that a fraction of drops will reach sizes that undergo spontaneous breakup when

*D*

_{mr}is smaller because the drop distribution is polydisperse, especially for wide drop size distributions (i.e., distributions that tend toward exponential shape).

### e. Collection of cloud droplets by ice

Collection of cloud droplets by ice is parameterized using the continuous collection assumption (droplet size and fall speed are neglected in the collection kernel). The ice particle fall speed *V*(*D*) and projected area *A*(*D*) as described in section 2c are used to calculate the collection kernel. Because of the complicated dependence of *V*(*D*) and *A*(*D*) on *D*, the appropriately weighted moment of the ice particle size distribution corresponding to collection of liquid is calculated offline and stored in lookup table as a function of *q*_{i}, *N*_{i}, *F*_{r}, and *ρ*_{r}. The collection efficiency is specified to be unity. At temperatures above freezing the collected mass of cloud droplets is shed assuming a shed drop size of 1 mm following Rasmussen et al. (1984). At temperatures below freezing, the collected droplets are assumed to freeze instantaneously except in wet growth conditions as described in section g of this appendix.

### f. Collisions between rain and ice

Changes in number and mass mixing ratio resulting from collisions between rain and ice use a collection kernel derived from *A*(*D*) and *V*(*D*) for ice and rain numerically integrated over the ice and rain size distributions. Collisions between rain and ice are calculated for all ice particle and rain drop sizes across the respective distributions. Because of the numerical integration, the integrated collision kernels are calculated offline and stored in a lookup table as a function of *q*_{i}, *N*_{i}, *F*_{r}, *ρ*_{r} and a scaled mean raindrop size proportional to (*q*_{r}/*N*_{r})^{1/3}. The collection efficiency is assumed to be unity. At temperatures above freezing, the mass of rain that collides with ice is shed assuming a shed drop size of 1 mm. At temperatures below freezing, the rain mass that collides with ice is assumed to freeze instantaneously except in wet growth conditions as described in section g of this appendix.

### g. Wet growth of ice

In conditions with relatively warm temperatures and large riming rates, ice particle surface temperatures can reach 273.15 K. In this case not all of the collected liquid water is frozen and instead some fraction is shed. We calculate the wet growth rate following Musil (1970), numerically integrated over the ice particle size distribution. Values are precomputed and stored in a lookup table. When the dry growth rate is smaller than the wet growth rate all collected water is assumed to freeze instantaneously. When the dry growth rate exceeds the wet growth rate the difference between the rates is shed as 1-mm-sized raindrops. If wet growth conditions are diagnosed, then particles also become soaked and undergo densification with *B*_{r} = *q*_{rim}/*ρ**, where *ρ** = 900 kg m^{−3}. This densification is assumed to occur within one time step. We also note the role of soaking and particle densification during melting in conditions above freezing but leave a detailed treatment of particle evolution during melting for future work.

### h. Freezing of cloud droplets and rain

Heterogeneous freezing of cloud droplets and rain follows from the volume-dependent formulation from Bigg (1953) but with parameters following Barklie and Gokhale (1959) [see also Pruppacher and Klett (1997, p. 350)]. Homogeneous freezing of cloud droplets and rain occurs instantaneously at 233.15 K.

### i. Melting

Melting is treated using the simplified diffusion approximation including ventilation and environmental relative humidity effects similar to Lin et al. (1983) and others. The appropriate moment of the ice particle size distribution is numerically integrated offline and stored in a lookup table. Reduction of *N*_{i} during melting is scaled to the change in *q*_{i}, i.e., (*N*_{i}/*q*_{i}) × QIMLT. Each melted ice particle is assumed to produce a single raindrop.

### j. Sedimentation

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^{1}

A bulk scheme with a single category of liquid has been proposed (Kogan and Belochitski 2012). This approach necessitates the inclusion of several additional prognostic moments to capture the nonlinear growth processes.

^{2}

For simplicity, we refer to all dense rimed ice as graupel unless otherwise noted, even though large (>5 mm) rimed particles and/or high-density particles that have undergone wet growth are traditionally referred to as hail.