## Introduction

Four parameterization techniques are selected to demonstrate their behavior when used for vapor diffusion growth and the continuous collection growth equations. With the continuous collection growth equation, it is assumed that the available number of droplets that can be collected is infinite, that all droplets collected are the same size, and that all droplets are collected at the same rate for all like collector drops, the latter of which do not experience self-collection. The vapor diffusion growth equation describes depositional growth of a spectrum of ice particles without nucleation of new ice particles. With these equation specifications, the total number concentration for the collector drops or ice particles ideally should be conserved for any parameterization technique used. It is shown, however, that total number concentration is only conserved for one parameterization technique and the rest fail. The specific techniques evaluated are described below.

In the past few years, there has been a rapid increase in the use of more-complex one-moment schemes, as well as new two-moment schemes, in microphysics parameterizations. Therefore, we feel compelled to compare objectively the one-moment and two-moment methods that are being widely used by the research and operational community, including those developed or used by, for example, Srivastava (1967, 1971), Passarelli (1978), Passarelli and Srivastava (1979), Lin et al. (1983, hereinafter LFO), Rutledge and Hobbs (1983, hereinafter RH), Ferrier (1994), Mitchell (1988, 1994), Walko et al. (1995), Meyers et al. (1997), Reisner et al. (1998), Cohard and Pinty (2000), van den Heever (2002), van den Heever and Cotton (2004), Gilmore et al. (2004a, b), and many more.

To examine the problem stated above, the gamma distribution function is used for the hydrometeor spectral density function. The main properties of the size distribution may be described by variables that represent the hydrometeor distribution characteristics, including the total mixing ratio *q*, total number concentration *N _{t}*, characteristic diameter

*D*(the inverse of the distribution slope

_{n}*λ*), and the slope intercept

*n*(Walko et al. 1995). Table 1 shows what is predicted, specified as constant, and/or diagnosed in the commonly used schemes, as well as other schemes that could be used.

_{o}In this paper, the following four commonly used parameterization schemes (Table 1) are evaluated:

a one-moment method in which

*q*is predicted,*n*is specified as a constant, and_{o}*N*and_{t}*D*are diagnosed (scheme A);_{n}a one-moment method in which

*q*is predicted,*D*is specified as a constant, and_{n}*N*and_{t}*n*are diagnosed (scheme B);_{o}a two-moment method in which

*q*and*D*are predicted and_{n}*N*and_{t}*n*are diagnosed (scheme E); and_{o}a two-moment method in which

*q*and*N*are predicted and_{t}*n*and_{o}*D*are diagnosed (scheme F)._{n}

A control scheme is needed to make comparisons, and, because we do not know what scheme is best for it, we need to do some analysis concerning solution behaviors. If we make the assumption of either a variable or steady production rate of *q* by continuous collection or vapor diffusion growth, then scheme F will conserve *N _{t}* by definition. Because we know scheme F has this property, we select it as the reference or control scheme. Relative differences (RDs) between each of the schemes and scheme F are computed for

*dq*/

*dt*,

*q*,

*D*,

_{n}*N*, and

_{t}*n*, where

_{o}*t*is time.

In section 2, for completeness, mathematical definitions are provided for the variables described above. In section 3, integration results and scheme comparisons are provided and discussed. In section 4, the conclusions are presented.

## Mathematical definitions

In the following, several familiar moments are defined. Note that in this paper *D _{n}* is defined in terms of unit length, whereas in LFO, RH, Passarelli (1978), Passarelli and Srivastava (1979), and Mitchell (1988), and other references,

*λ*= 1/

*D*is defined in terms of “per unit length.” This difference has a pronounced influence on the way the equation for

_{n}*dD*/

_{n}*dt*appears in their works and our paper. It will appear as if there is a sign error for

*dD*/

_{n}*dt*when compared with Passarelli (1978), Passarelli and Srivastava (1979), and Mitchell (1988), but this is not the case. In several of the gamma functions Γ(

*x*), there is the appearance of a missing “1” that should be added to

*x*; it has to do with the way the gamma distribution function is written relative to other papers (e.g., Passarelli 1978; Passarelli and Srivastava 1979; Mitchell 1988).

*D*(0, ∞) are defined along with some other variables. The spectral density function is given by the gamma distribution. This function has been used by Walko et al. (1995), Meyers et al. (1997), and Cohard and Pinty (2000), among many others, and is given by

*D*is diameter (m),

*D*has units of meters,

_{n}*N*has units of inverse meters cubed,

_{t}*ν*is a dimensionless distribution shape parameter, and Γ is the gamma function. The zeroth moment

*N*, or number concentration, is defined over the interval for

_{t}*D*(0, ∞) by

*ν*+ 1) =

*ν*Γ(

*ν*). The third moment, or mixing ratio

*q*(kg kg

^{−1}), can be defined over the interval

*D*(0, ∞) by

*m*(

*D*) =

*αD*, where

^{β}*α*=

*ρπ*/6 and

*β*= 3. In the above,

*ρ*and

*ρ*

_{0}are the densities of the particle and air, respectively. The variable

*D*can be found in terms of

_{n}*N*and

_{t}*q*and other constants by using

*q*and

*D*as

_{n}*n*(m

_{o}^{−β−ν}) can be found in terms of

*q*and

*D*as

_{n}*Z*(for Rayleigh scatterers), for spheres can be prescribed over the interval

*D*(0, ∞) by

*D*(0, ∞) as described in Walko et al. (1995), for example. Although all of these moments are easy to derive and can be found elsewhere in the literature, they are presented here for completeness. Moreover, they are used to understand how the continuous collection and vapor diffusion equations behave.

*q*and

*Z*to arrive at an expression that can be used to derive an equation for ∂

*D*/∂

_{n}*t*for the vapor diffusion equation. We also use it with no complications for a simple continuous collection growth equation for spheres or disks. The derivation follows from basic steps for a sphere (Mitchell 1988, 1994). The main change between diffusion and collection is the value of

*δ*, which is defined below for each of the processes considered. After some algebra, an equation for

*dD*/

_{n}*dt*is given by

## Integration of microphysical growth equations and comparison of parameterization methods

### Vapor diffusional growth of small ice spheres

*D*(0, ∞) using the following equation:

*S*is sublimation to liquid or ice,

_{x}*K*is thermal conductivity,

*R*is the gas constant for water vapor,

_{υ}*q*is the saturation mixing ratio over plane ice, and

_{is}*φ*is the diffusivity of water vapor. Also,

_{υ}*f*is the ventilation coefficient and is assumed to be unity for small particles. [This was done by Passarelli (1978), Passarelli and Srivastava (1979), and Mitchell (1988) so that (9) could be derived.] Shapes other than spheres are not considered here but could be [e.g., a disk representing a plate crystal results in only a simple change to (10) and (11) (Pruppacher and Klett 1997)]. The time rate of change of

*N*for vapor diffusion growth of a small ice sphere (mass increase), with no nucleation, is simply

_{t}*q*= 1.0 × 10

_{i}^{−3}kg kg

^{−1}for cloud ice at time

*t*= 0,

*N*= 1.0 × 10

_{t}^{7}m

^{−3}at time

*t*= 0,

*f*= 1 (Passarelli 1978; Passarelli and Srivastava 1979; Mitchell 1988) and

*δ*= 1,

*S*= 1.05 for all times,

_{i}*ν*= 3 for the distribution shape parameter, and

*ρ*= 1 kg m

_{o}^{−3}is density of air.

Sample values at *t* = 300 s of *q*, *D _{n}*,

*N*, and

_{t}*n*from schemes A, B, E, and F for diffusional growth of the ice spheres are shown in Table 2. The RDs against scheme F also are shown. The RDs range from around

_{o}*O*(10)% to

*O*(1000)%!

Scheme F is examined to determine if it produces consistent solution trends and provides a reasonable comparison solution. With scheme F we find that *dq*/*dt* > 0 as prescribed and that *dN _{t}*/

*dt*= 0 as it physically should be. Also, with scheme F, we find with (5) that

*D*increases with time (which indicates larger characteristic as well as mean diameter with time), which is realistic, and with (7) that

_{n}*n*decreases with time (given the above, this indicates that there are fewer small particles and more large particles with time, but that the total number concentration is constant), which also is correct. With regard to

_{o}*N*and

_{t}*n*, examination of all of the other schemes (Table 2) shows values and/or trends in RDs relative to scheme F. Schemes A, B, and E produce a larger number of particles than does scheme F, which implies the existence of more small particles (note the increase in

_{o}*n*). In accordance with the increase in

_{o}*N*and

_{t}*n*, characteristic diameters

_{o}*D*, and thus mean diameters for schemes A, B, and E, are all smaller at

_{n}*t*= 300 s than for scheme F. This implies smaller characteristic diameter as well as mean diameters. Even though scheme A has an increasing

*D*, and by definition scheme B has a constant

_{n}*D*with time, neither is large enough in comparison with scheme F. These trends for schemes A, B, and E violate what should occur physically. From the results it is apparent that scheme A performs closest to the reference solution and scheme E has the greatest RDs.

_{n}### Continuous collection growth of spherical graupel

The next process considered is continuous collection growth of spherical graupel (Wisner et al. 1972; Pruppacher and Klett 1981; LFO; RH; among others). There are numerous storm-scale prediction models [*O*(1 km) or less for horizontal grid spacing] that include parameterizations that predict graupel, because the growth of graupel can have a large impact on total rainfall (e.g., Gilmore et al. 2004b). The growth of graupel occurs largely through collection or riming of cloud water (Gilmore et al. 2004b); thus, this is an important microphysical process to analyze. Some may argue that the form of this growth process presented here is very simple; however, many models still use it (certainly some models have more sophisticated parameterizations). Given its simplicity, there is some beauty in using this equation because the approximation *d*[*m*(*D*)]/*dt* ≈ *D ^{δ}* can be made with no complications to analyze scheme E for the continuous collection growth equation.

*x*be the subscript for gamma-distributed collector particles, such as graupel, and let

*y*be the subscript for monodispersed collectee particles, such as cloud water. Assume that

*D*≫

_{x}*D*,

_{y}*V*≫

_{x}*V*,

_{y}*V*=

_{x}*aD*, and

^{b}*m*(

*D*) =

*αD*, where

^{β}*a*= [(4

*ρ*)/(3

_{x}g*ρ*)]

_{o}C_{d,x}*and*

^{b}*b*= 0.5 (

*g*is the gravitational acceleration, and

*C*is the drag coefficient). The continuous collection growth equation integrated over

_{d}*D*[0, ∞] may be written as

*β*= 3 for a sphere,

*ν*= 3 for the distribution shape parameter,

*q*= 4 × 10

_{x}^{−3}kg kg

^{−1}as initial value for graupel mixing ratio,

*E*

_{x}_{,y}= 0.55 for collection efficiency,

*ρ*= 900 kg m

_{x}^{−3}for density of graupel,

*C*= 0.60 for drag coefficient for graupel in (14),

_{d}*g*= 9.8 m s

^{−2}for gravitational acceleration,

*q*= 1.0 × 10

_{y}^{−3}kg kg

^{−1}as a constant value of the mixing ratio for cloud water, and

*δ*= 2.5 from (14).

Solutions are obtained for (14) and (15) by numerically integrating the continuous collection equation forward in time from 0 to 300 s using a 1-s time step. Scheme F is used for the reference solution again to compare with the other schemes. Results are shown for integrations at *t* = 300 s in Table 3.

Overall, scheme E shows the worst performance against scheme F. The RDs against scheme F are nearly similar for schemes A and B. Not surprising is that the trends for schemes A, B, and E are very similar to those for vapor diffusion growth. Particles do not become large enough, but particles become more numerous for schemes A, B, and E as before, which violates the conservation of *N _{t}* [(15)]. All of the results are physically incorrect for

*D*,

_{n}*N*, and

_{t}*n*, because there is no mechanism for a gain in

_{o}*N*[(15)]. Only scheme F provides physically consistent solutions.

_{t}## Conclusions

With the rapid increase in the use of microphysical parameterizations in numerical models we felt compelled to evaluate the veracity of some commonly used microphysical moment schemes in an idealized controlled manner (holding the rate of the third-moment *dq*/*dt* constant and computing other variables). It has been shown that schemes A, B, and E (Table 1), which are very often used in numerical cloud models, do not conserve the important moment *N _{t}* when they physically should do so for the processes of vapor diffusional growth and continuous collection growth (Tables 2 and 3). In particular, schemes A, B, and E all artificially nucleate or introduce particles during processes that should conserve

*N*. In some of the tested cases, the characteristic diameters get smaller than initial diameter values as time increases (e.g., for scheme E) as the distribution slope awkwardly becomes more negative (larger) with time. If given a choice of schemes, such as those presented herein, the results make it possible to say which scheme really is the most physically consistent; a strong case could be made that a scheme that predicts (

_{t}*q*,

*N*), a two-moment method (scheme F), is the most physically consistent and least suspect, at least for some problems for the diffusion and accretion growth processes.

_{t}## Acknowledgments

This work was supported by the National Science Foundation (NSF) Grants ATM-0340639, ATM-0339519, ATM-9986672, ATM-0003869, and ATM-0119398. Partial funding for this research was provided by the National Severe Storms Laboratory under NOAA–OU Cooperative Agreement NA17RJ1227. Doctor K. Kanak is supported by the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS) under Award NA17RJ1227 from the NOAA, U.S. Department of Commerce, and NSF ATM-0135510.

## REFERENCES

Cohard, J-M. and J-P. Pinty. 2000. A comprehensive two-moment warm microphysical bulk scheme, I: Description and tests.

*Quart. J. Roy. Meteor. Soc.*126:1815–1842.Ferrier, B. S. 1994. A double-moment multiple-phase four-class bulk ice scheme. Part I: Description.

*J. Atmos. Sci.*51:249–280.Gilmore, M. S., J. M. Straka, and E. N. Rasmussen. 2004a. Precipitation and evolution sensitivity in simulated deep convective storms: Comparisons between liquid-only and simple ice and liquid phase microphysics.

*Mon. Wea. Rev.*132:1897–1916.Gilmore, M. S., J. M. Straka, and E. N. Rasmussen. 2004b. Precipitation uncertainty due to variations in precipitation particle parameters within a simple microphysics scheme.

*Mon. Wea. Rev.*132:2610–2627.Koenig, L. R. and F. W. Murray. 1976. Ice-bearing cumulus cloud evolution: Numerical simulation and general comparison against observations.

*J. Appl. Meteor.*15:747–762.Lin, Y-L., R. D. Farley, and H. D. Orville. 1983. Bulk parameterization of the snow field in a cloud model.

*J. Climate Appl. Meteor.*22:1065–1092.Meyers, P. M., R. L. Walko, J. Y. Harrington, and W. R. Cotton. 1997. New RAMS cloud physics parameterization. Part II: The two-moment scheme.

*Atmos. Res.*45:3–39.Mitchell, D. L. 1988. Evolution of snow-size spectra in cyclonic storms. Part I: Snow growth by vapor deposition and aggregation.

*J. Atmos. Sci.*45:3431–3451.Mitchell, D. L. 1994. A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part I: Microphysics.

*J. Atmos. Sci.*51:797–816.Passarelli, R. E. 1978. An approximate analytical model of the vapor deposition and aggregation growth of snowflakes.

*J. Atmos. Sci.*35:118–124.Passarelli, R. E. and R. C. Srivastava. 1979. A new aspect of snowflake aggregation theory.

*J. Atmos. Sci.*36:484–493.Pruppacher, H. R. and J. D. Klett. 1981.

*Microphysics of Clouds and Precipitation*. D. Reidel, 714 pp.Pruppacher, H. R. and J. D. Klett. 1997.

*Microphysics of Clouds and Precipitation*. Kluwer, 954 pp.Reisner, J., R. M. Rasmussen, and R. T. Bruintjes. 1998. Explicit forecasting of supercooled liquid water in winter storms using the MM5 model.

*Quart. J. Roy. Meteor. Soc.*124:1071–1107.Rutledge, S. A. and P. V. Hobbs. 1983. The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. VIII: A model for the “seeder–feeder” process in warm-frontal rainbands.

*J. Atmos. Sci.*40:1185–1206.Srivastava, R. C. 1967. On the role of coalescence between raindrops in shaping their size distribution.

*J. Atmos. Sci.*24:287–292.Srivastava, R. C. 1971. Size distributions of raindrops generated by breakup and coalescence.

*J. Atmos. Sci.*28:410–415.Srivastava, R. C. 1978. Parameterization of raindrop size distributions.

*J. Atmos. Sci.*35:108–117.Stalker, J. R. and K. R. Knupp. 2003. Cell merger potential in multicell thunderstorms of weakly sheared environments: Cell separation distance versus planetary boundary layer depth.

*Mon. Wea. Rev.*131:1678–1695.van den Heever, S. 2002. The impact of several hail parameters on simulated supercell storms. Ph.D. dissertation, Atmospheric Science Paper 711, Colorado State University, 265 pp.

van den Heever, S. and W. R. Cotton. 2004. The impact of hail size on simulated supercell storms.

*J. Atmos. Sci.*61:1596–1609.Verlinde, J. and W. R. Cotton. 1993. Fitting microphysical observations of nonsteady convective clouds to a numerical model: An application of the adjoint technique of data assimilation to a kinematic model.

*Mon. Wea. Rev.*121:2776–2793.Walko, R. L., W. R. Cotton, M. P. Meyers, and J. Y. Harrington. 1995. New RAMS cloud physics parameterization. Part I: The single-moment scheme.

*Atmos. Res*38:3–39.Wisner, C., H. D. Orville, and C. Meyers. 1972. A numerical model of a hail-bearing cloud.

*J. Atmos. Sci.*29:1160–1181.Ziegler, C. L. 1985. Retrieval of thermal and microphysical variables in observed convective storms. Part 1: Model development and preliminary testing.

*J. Atmos. Sci.*42:1487–1509.

Variables predicted and specified as constant in microphysical parameterizations; here C = constant, P = predicted, and blank spaces are floating variables that are or can be diagnosed.

Results from vapor diffusion growth of an ice sphere from 0 to 300 s. In columns A–F, the top values are the 300-s values and the bottom values are the errors relative to scheme F. RDs (in percent) are based on 100 × |(true − predicted) × true^{−1}|, where true values are from scheme F and predicted values are from the other schemes.