## 1. Introduction

The growth of ice crystals in nature is a complex phenomenon: atmospheric ice attains a variety of shapes that vary among relatively simple hexagonal plates and columns, dendritic structures, rosettes, and often irregular crystals that may be polycrystalline in form. Though crystal shapes vary, they can be characterized by two axis lengths, *a* and *c*, and their primary shapes by the aspect ratio (*φ* = *c*/*a*). For hexagonal prisms, the *a* dimension is half the distance across the basal (hexagonal) face, whereas the *c* dimension is half the length of the prism (rectangular) face. Less ideal crystals are characterized in terms of their maximum and minimum dimensions: for flat, platelike particles *a* is half the maximum dimension (referred to here as the maximum semidimension) and *c* is half the minimum dimension, with the opposite being the case for columnlike particles. Traditionally, the primary habit is characterized by aspect ratio variation with temperature, the classical picture of which is based on laboratory and in situ data and is arguably most valid for *T* > −22°C (e.g., Pruppacher and Klett 1997; Fukuta and Takahashi 1999; Bailey and Hallett 2009). These data suggest platelike (*φ* < 1) crystals for temperatures in the ranges of −1° to −3°C and −10° to −22°C, whereas columnlike crystals (*φ* > 1) occur in the ranges of −3° to −9°C and lower than −22°C. More recent data suggest important modifications to the habit diagram at temperatures below about −22°C (e.g., Bailey and Hallett 2009), specifically that plates tend to occur more often and that columns are rare. Some measurements support this (e.g., Bailey and Hallett 2002, 2004); however, these data are at odds with other laboratory studies of ice crystal formation and growth (e.g., Libbrecht 2003). The secondary habit of ice crystals depends on the degree of supersaturation, and this classification is most valid for *T* > −22°C. Laboratory measurements generally show that lower saturation states produce compact hexagonal crystals depending in part on nucleation. As the saturation increases toward liquid, crystals tend to hollow: columns become needlelike near −6°C, dendritic crystals appear around −15°C, with rosette and polycrystals forming at *T* < −22°C.

It is presently impossible for cloud models to capture the variety of ice crystal masses, shapes, sizes, and distributions that occur in real clouds. Nevertheless, all numerical models need some way to relate the mass of crystals to their shapes and sizes. Prior attempts have nearly universally involved using either equivalent volume (or diameter) spheres (e.g., Lin et al. 1983; Reisner et al. 1998; Thompson et al. 2004) or mass–size relationships (e.g., Mitchell et al. 1990; Harrington et al. 1995; Meyers et al. 1997; Woods et al. 2007; Thompson et al. 2008; Morrison and Grabowski 2008, 2010), though exceptions exist (e.g., Hashino and Tripoli 2007). These methods suffer from a number of deficiencies (Sulia and Harrington 2011, hereafter SH11). In comparison to wind tunnel data, the use of equivalent spheres underestimates ice growth at every temperature except around −10° and −22°C where growth is nearly isometric (e.g., Fukuta and Takahashi 1999). Mass–size relations allow the prediction of only a single size, and the coefficients in the mass–size power laws are derived from in situ data and are therefore tied to the observed particle growth histories. These methods are not linked to the growth mechanisms that determine the evolution of the two primary crystal axes (*a* and *c*) and therefore cannot evolve crystals in a natural fashion.

Bulk models are limited because they attempt to predict the evolution of only a few moments of the size spectrum, the shape of which is assumed a priori. Typically, only the ice mass mixing ratio is predicted in single-moment schemes, whereas in two-moment schemes the number mixing ratio is predicted as well. Moreover, most current bulk microphysical methods artificially categorize particles based on predefined classes and then use transfer functions to move particles among classes as they evolve. For instance, Woods et al. (2007) predicts numerous ice classes as a way to deal with the problem of habit evolution, but such methods are computationally costly because of the increased number of prognostic variables. In the method of Harrington et al. (1995), transfer functions across a somewhat arbitrary size boundary are used to evolve smaller pristine ice class particles into a larger snow class. A more recent trend in microphysical modeling is to predict different ice particle properties for a given ice class. For instance, the method of Morrison and Grabowski (2008) predicts a rime mass fraction for ice crystals that avoids the traditional approach of transferring ice to a separate graupel class, and Hashino and Tripoli (2007) use the Chen and Lamb (1994) approach for predicting habits in a hybrid-bin scheme. Such methods allow the use of fewer ice classes and provide for a more natural evolution of the ice particles based on their respective growth histories. To evolve crystal habit, a way to relate the evolution of the *a*- and *c*-axis distributions is required, and this has been a significant hurdle for numerical models to surmount.

The approach developed here predicts the evolution of ice particle habit by tracking the evolution of ice aspect ratio following Chen and Lamb (1994, hereafter CL94) and SH11. Similarities exist in the approach to that presented in Hashino and Tripoli (2007), with the exception that the method presented here is derived for a bulk cloud model using four prognostic variables. The method provides a way to relate the *a*- and *c*-axis evolution through the use of laboratory-determined parameters, making the method amenable to improvements through new measurements. The approach eliminates thresholds between habits and follows the recent paradigm of predicting particle properties instead of multiple particle classes. The parameterization is developed here with testing of the method described in Harrington et al. (2013, hereafter Part II).

## 2. Single-particle evolution

### The Fickian-distribution method

*C*is the capacitance of the ice particle,

*G*is a function that combines the effects of thermal and vapor diffusion processes [Lamb and Verlinde 2011, their Eq. (8.41), p. 343],

_{i}*P*is the pressure,

*T*is the temperature, and

*s*

_{u}_{,i}is the ice supersaturation. Because

*C*depends on two axis lengths, it is difficult to integrate Eq. (1) in time. Most modeling applications use the second form to the right in which

*L*is the maximum semidimension, and

*f*(

_{s}*φ*) =

*C*/

*L*is a shape factor that depends only on the aspect ratio

*φ*. In this work, Eq. (1) is integrated over a time step using an equivalent volume sphere of radius

*r*given by the rightmost equation and is used to avoid the numerical problem of evolving both

*a*and

*c*simultaneously. Evolving the mass in this fashion requires holding the shape factor,

*a*and

*c*axes, producing a parameterization that is more consistent with the theory and rooted in laboratory measurements. This modification allows for aspect ratio evolution through a second predictive equation based on a mass distribution hypothesis relating the growth of the

*a*and

*c*axes:

*δ*(

*T*) =

*α*/

_{c}*α*] is a ratio of the deposition coefficients, or growth efficiencies, for the

_{a}*c*and

*a*axes. Since the deposition coefficients can be determined from laboratory measurements (e.g., Lamb and Scott 1972; Nelson and Knight 1998; Libbrecht 2003), the method can accommodate new data. Because Eqs. (1) and (2) together are fundamentally different from the capacitance model, SH11 reterms it the Fickian-distribution method so that a clear distinction is drawn between the two methods of growth. Values of

*δ*used here are compiled from two sources: the original work of CL94 using laboratory data from Lamb and Scott (1972) that derive from crystals grown in pure vapor on a solid substrate, and values derived by Hashino and Tripoli (2008), which were altered for numerical accuracy in the solutions and to account for the possibility of plates as the primary habit at temperatures below −22°C (see Bailey and Hallett 2009). The modified values of

*δ*in Hashino and Tripoli (2008), larger (smaller)

*δ*than CL94 for plates (columns), are useful because they are within the limits of the measured

*δ*and together with best-fit values from CL94 provide realistic bounds on possible values of

*δ*. It is critical to note that neither the capacitance model nor the Fickian-distribution method has been well tested for situations in which the temperature varies or for low ice supersaturations (see Nelson 1994; Sheridan et al. 2009; SH11). Though the method has been extended to more complex crystal types such as rosettes and polycrystals (e.g., Hashino and Tripoli 2008), it is unclear how accurate the method is for these types of particles since comparisons to laboratory data are needed.

The nonlinear impact of crystal shape on growth is embodied by an evolving capacitance in the Fickian-distribution method (*C*; Fig. 1): For the same volume particle, *C* increases nonlinearly as the aspect ratio deviates from unity. This is due to the tightening of the vapor density gradients near crystal edges (Libbrecht 2005, their Fig. 5, and see SH11 for a full explanation). While the capacitance contains this characteristic feature of the vapor field (Fig. 2), the aspect ratio is constant in the traditional capacitance model. The consequences of keeping aspect ratio constant are illustrated in Fig. 1: A single crystal is grown for 5 min at liquid saturation and at *T* = −6°C either with a constant or evolving aspect ratio. When *φ* is constant, the nonlinear feedbacks in growth are not captured, and so the crystal grows to a smaller final capacitance, and therefore mass, than when *φ* evolves freely. Consequently, predicting crystal aspect ratio is critical for capturing nonlinear changes in vapor diffusion.

Graphical illustration of vapor flux enhancement due to nonsphericity in the capacitance model. Each particle has the same volume, with vapor density contours of the same values surrounding the particle and computed for an oblate spheroid. Spherical particles have isometric vapor fields, and hence the vapor flux is uniform over the surface. However, as the aspect ratio φ varies away from one, decreasing in this case, the vapor flux increases over the axis of greatest curvature (end of the *a* axis) and decreases over the other axis (*c* axis). This process leads to an overall increase in the diffusive flux toward the particle.

Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1

Graphical illustration of vapor flux enhancement due to nonsphericity in the capacitance model. Each particle has the same volume, with vapor density contours of the same values surrounding the particle and computed for an oblate spheroid. Spherical particles have isometric vapor fields, and hence the vapor flux is uniform over the surface. However, as the aspect ratio φ varies away from one, decreasing in this case, the vapor flux increases over the axis of greatest curvature (end of the *a* axis) and decreases over the other axis (*c* axis). This process leads to an overall increase in the diffusive flux toward the particle.

Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1

Graphical illustration of vapor flux enhancement due to nonsphericity in the capacitance model. Each particle has the same volume, with vapor density contours of the same values surrounding the particle and computed for an oblate spheroid. Spherical particles have isometric vapor fields, and hence the vapor flux is uniform over the surface. However, as the aspect ratio φ varies away from one, decreasing in this case, the vapor flux increases over the axis of greatest curvature (end of the *a* axis) and decreases over the other axis (*c* axis). This process leads to an overall increase in the diffusive flux toward the particle.

Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1

*δ*and

*C*are known. Using a single crystal type (e.g., hexagon or dendrite) restricts the generality that is desired for a numerical cloud model. Therefore, spheroids are used as a first-order approximation of primary habit (

*φ*), allowing for the prediction of two axes. Relating changes in mass to changes in axis lengths requires the spheroidal volume:

*V*is the volume, and

*ρ*is the effective particle density that is less than bulk ice (

_{i}*ρ*

_{bi}= 920 kg m

^{−3}). The effective density is a parameterization of the secondary habits (i.e., dendrites and needles) and varies in time since the density deposited during growth, the deposition density [

*ρ*

_{dep}; see CL94, their Eq. (42)], varies with temperature. Over short growth times

*ρ*

_{dep}is approximately constant (CL94) so that

*ρ*

_{sub}; their Eq. (6)], though here

*ρ*

_{dep}is used to signify either density. Solving Eq. (1) using an equivalent volume sphere requires a relation between changes in

*r*and

*a*. Equating the spheroidal and spherical volumes, then logarithmically differentiating and using Eq. (2) results in

*r*can be used to find changes in

*a*and then changes in

*c*[Eq. (2)].

The equations above constitute the core of adaptive habit evolution using the Fickian-distribution method. The equations are modified for ventilation effects that require computation of crystal fall speeds. The method of SH11 is followed, and the main equations are discussed in appendix A. In Part II, laboratory data are used to critique the above equations and prior mass–size methods. It will be shown that the above method is robust and relatively accurate, whereas most mass–size relations produce crystal masses, sizes, and fall speeds that are generally in error.

## 3. Bulk adaptive habit parameterization

Prediction of bulk ice habit requires the particle size distribution-integrated forms of the mass growth equation [Eqs. (1) and (4)] and the mass distribution hypothesis [Eq. (2)]. In forming these equations, a quantitative link between the distributions of the *a* and *c* axes is needed. Two-dimensional size spectra have been used for this purpose (e.g., Chen and Lamb 1999) but are computationally costly. Here, *δ* is used to relate the *a*- and *c*-axis distributions, so that a single distribution can be used, making the method numerically efficient. The development presented here assumes the evolution of a single habit type, and so mixes of particle habits (plates and needles) and complex crystal types (e.g., rosettes and polycrystals) are not considered, but could be added by allowing for at least two particle classes and following a method similar to Hashino and Tripoli (2008). Frequently used coefficients from the derivations below are listed in Table 1.

Coefficients resulting from derivations presented in the main text and appendix.

### a. Axis length history

*a*- and

*c*-axis lengths. This is necessary because the inherent growth ratio

*δ*in Eq. (2) varies with temperature, and therefore the way

*a*and

*c*are related will vary in time. This variability is accounted for by integrating Eq. (2) in time from

*a*

_{0}to

*a*(

*t*) and

*c*

_{0}to

*c*(

*t*), giving

*δ*

_{*}is a time average of the inherent growth ratio over the particle growth history, also described in CL94. Note that the weighting is logarithmic, which is a consequence of the relationship between

*dc*/

*c*and

*da*/

*a*in Eq. (2). The initial aspect ratio

*φ*

_{0}and size

*a*

_{0}of the ice particles must be chosen. Initial aspect ratios that are not unity can be accommodated by setting

*c*

_{0}=

*φ*

_{0}

*a*

_{0}in

*α*

_{*}. Although some recent laboratory data suggest that initially frozen drops at

*T*= −40°C have aspect ratios of around 1.1 (López and Ávila 2012), given that data on the initial ice aspect ratio is sparse and that laboratory data suggest aspect ratios near one, it is assumed here that initially formed crystals are isometric, and so

Because particles are initially assumed to be isometric, *δ*_{*} is initially equal to one. However, the inherent growth ratio *δ* is typically not unity and varies with temperature, and so *δ*_{*} will also vary in time. This is an important departure from traditional bulk methods that implicitly fix *δ*_{*} at a particular value for a given habit type. In the method proposed here, the relation between *c* and *a* will also evolve. Since *δ*_{*} is a weighted time average of *δ*, it is naturally constrained to lie within the limits of *δ* derived from laboratory data (about 0.27 and 2.1). Note the distinction between *δ* and *δ*_{*}: *δ* is a measure of the mass distributed over the *a* and *c* axes during growth (one time step), whereas *δ*_{*} relates the two axis lengths over time and allows for the use of a single size spectrum in either the *c-* or *a-* axis length. It is worth noting that *δ*_{*} is not a prognostic variable, but is diagnosed from separate prediction of the *c*- and *a*-axis lengths. The parameterization depends on the exactness of this historical relation (*δ*_{*}), which is demonstrated in Part II.

*β*= 2 +

_{m}*δ*

_{*}are for a platelike crystal. This result shows that the mass–size relations are directly connected to the historical evolution of the particle through

*δ*

_{*}. Consequently, empirical mass–size relations should always be used with caution as the entire integrated history of the particle’s evolution is contained in the leading coefficient and the exponent. Because equivalent volume spheres are used to solve for the mass changes over time (section 2), a historical relation between

*r*and

*a*is needed and can be found by equating spheroidal and spherical volumes and then using Eq. (6):

*β*= 3/

_{r}*β*.

_{m}### b. Axis distributions

*c*and

*a*are related to each other through Eq. (6), the distribution of only one axis length is needed, and the

*a*-axis length is used here. Following other bulk modeling approaches (e.g., Walko et al. 1995; Thompson et al. 2004), the concentration of crystals conforms to a modified gamma distribution of

*a*:

*N*is the number density of ice crystals, Γ(

_{i}*ν*) is the gamma function,

*ν*is the distribution “shape” parameter, and

*a*is the characteristic

_{n}*a*-axis length that is related to the mean

*a*-axis length, and the

*a*-axis mixing ratio (see section 3e). It is worth noting that generalized distributions like that in Thompson et al. (2008) could also be used.

To define the aspect ratio requires the *c*-axis distribution while the equivalent volume sphere distribution (*r*) is needed for vapor growth calculations (see section 2). Therefore, it is necessary to change variables from *a* to that of *r* or *c* in Eq. (9), which can be done analytically by using *δ*_{*}. The resulting distributions are similar to Eq. (9). The general form of the distribution conversion and the distributions of the *c* and *r* axes [Eqs. (B6) and (B5), respectively] are given in section a of appendix B. Each has characteristic lengths (*c _{n}* and

*r*) related to the mean axis length that characterize the mean particle shape.

_{n}*N*/

_{i}*ρ*), the ice mass mixing ratio (

_{a}*q*), and the mixing ratios of the

_{i}*a*and

*c*axes, defined respectively as

*ρ*is the air density. These equations follow from the definition of the mean axis length [

_{a}*a*and

*c*for all the ice particles per kg of air. It is beyond the scope of the present work to demonstrate that the above quantities are conserved in an Eulerian framework, since testing is carried out using a parcel model in Part II; however, a short discussion of this issue is provided in section 3e.

### c. Mass mixing ratio growth and mass distribution relationship for ice particle distributions

Knowing the particle distributions [Eqs. (9), (B5), and (B6)] and the historical relation between the axis lengths [Eq. (6)] makes it possible to derive equations for the evolution of the mass mixing ratio and the axis length mixing ratios. Because the axis mixing ratios are proportional to *a _{n}* and

*c*, the characteristic sizes are used throughout to represent mean habit evolution. Evolving

_{n}*q*follows a procedure similar to that for single particles (section 2a): The mass growth equation is combined with an equation describing the change in characteristic equivalent volume radius. Once the change in

_{i}*r*is known, aspect ratio evolution can then be computed using relationships between the different axis lengths.

_{n}*q*by vapor diffusion is given by

_{i}*n*(

*a*) [Eq. (9)] the capacitance (

*C*) must be written in terms of the

*a*-axis length alone. This relationship is derived by using Eq. (6) and fitting the shape factor with a power law in

*a*, as is done in section b of appendix B. Integrating over the size spectrum [Eq. (11)] produces an equation in

*a*alone [see section b of appendix B; Eq. (B13)]. Integrating the resulting expression [Eq. (B13)] in time is complicated because the mass and capacitance both depend on

_{n}*a*and, implicitly

_{n}*c*, both of which vary over a time step (see section b of appendix B). As for single-particle mass (section 2), tests show that evolving the characteristic equivalent volume spherical radius (

_{n}*r*) rather than

_{n}*a*or

_{n}*c*to determine the mass mixing ratio change over a time step provides the most accuracy in comparison to numerical solutions. The differential equation for

_{n}*q*in terms of

_{i}*r*is shown in section d of appendix B to be

_{n}*q*must be related to

_{i}*r*, which can be done by integrating the particle mass over

_{n}*n*(

*r*) [Eq. (B5)]:

*V*the total particle volume. Using this relationship, changes in mass mixing ratio can be related to changes in

_{t}*V*and

_{t}*r*in a similar fashion as for single particles [Eq. (4)] through

_{n}*r*, and hence the mass mixing ratio, in time (see section d of appendix B).

_{n}*r*to changes in

_{n}*c*and

_{n}*a*. Such a relation can be derived using Eq. (5) (see section c of appendix B):

_{n}*a*to

_{n}*c*is shown in section c of appendix B to be

_{n}### d. Fall speed and ventilation

*γ*and

_{υ}*γ*) that appear in the equation for ventilation [Eq. (A4)], and the process is explained in section e of appendix B. These coefficients multiply the vapor diffusivity

_{t}*G*[Eq. (11)] and are determined by appropriately integrating over the

_{i}*a*-axis distribution (section e of appendix B):

*a*and

*c*axes, respectively:

*δ*

_{υb}is the bulk value of the inherent growth ratio modified for ventilation effects. Including axis-dependent ventilation is now accomplished by replacing

*δ*with

*δ*

_{υb}in all of the bulk equations derived above. It is critical to keep in mind that only

*δ*is modified whereas

*δ*

_{*}remains as defined in Eq. (6).

### e. Procedure for evolving habits

Predicting changes in mass mixing ratio, axis lengths, fall speeds, and density can now be accomplished using the equations above when solved over a time step, the details of which are provided in the appendices. The derivation of the method is complicated, so an overview of the equations used to evolve bulk habits is warranted. Moreover, the equations are cast in a form suitable for use in Eulerian models since the method is designed ultimately for this purpose. The main equations used are also summarized in Tables 2 and 3.

Summary of variables: Beginning of time step. Only the mixing ratios (*N _{i}*/

*ρ*,

_{a}*q*,

_{i}*A*, and

*C*) are stored externally to the growth calculations. To compute the change in

*q*,

_{i}*A*, and

*C*over a time step, initial values of various quantities must be derived from these quantities. The initial variables at the beginning of a time step used in the growth model are given here, and a description in section 3e.

Summary of prognostic variables: End of time step. The growth method evolves the ice mass mixing ratio (*q _{i}*), and the

*a*- and

*c*-axis mixing ratios (

*A*and

*C*) over a time step. This table contains the main equations in the order that they are used. Note that initial values of the variables from Table 2 are assumed. At the end of the calculations, only the three mixing ratios are passed back and stored.

*q*,

_{i}*N*/

_{i}*ρ*,

_{a}*A*, and

*C*, respectively) are used here. To compute the evolution of mass mixing ratio and axis lengths, a number of quantities are needed at the beginning of the time step

*t*and can be extracted from

*q*,

_{i}*A*, and

*C*. Table 2 provides a summary of the main initial variables. At the beginning of a time step,

*c*and

_{n}*a*can be determined from the prognosed axis length mixing ratios using Eq. (10). With these,

_{n}*δ*

_{*}(

*t*) can be diagnosed by taking the natural logarithm of

*δ*

_{*}is known at the beginning of a time step, the characteristic equivalent volume spherical radius

*r*(t) can be found from Eq. (B5), and the mean ice particle density

_{n}*q*(

_{i}*t*) and inverting Eq. (13). To compute vapor diffusion, quantities that depend on the characteristic lengths are needed. The average capacitance

*G*in Eq. (12)] are modified for ventilation effects using Eq. (18) (Table 2).

_{i}Time stepping of the variables can now be done, and the main equations are summarized in Table 3. For reasons given in section 3c and in SH11, particle size and mass changes are computed by finding the change in *r _{n}* using Eq. (B23) giving

*r*(

_{n}*t*+ Δ

*t*). Knowing

*r*at the end of a time step allows

_{n}*a*(

_{n}*t*+ Δ

*t*) to be determined using Eq. (B26). Including axis ratio-dependent ventilation is then accomplished by replacing

*δ*in Eq. (B26) with

*δ*

_{υb}[Eq. (20)]. The change in

*c*can also be computed, but to do so requires knowing

_{n}*δ*

_{*}(

*t*+ Δ

*t*), which is diagnosed from Eq. (B27) since

*r*(

_{n}*t*+ Δ

*t*) and

*a*(

_{n}*t*+ Δ

*t*) are known. For columnlike crystals,

*c*(

_{n}*t*+ Δ

*t*) is determined from the relation between

*c*and

_{n}*a*in Eq. (B6). For platelike crystals, the change in mean aspect ratio is first computed, and then

_{n}*c*(

_{n}*t*+ Δ

*t*) is found from Eq. (B32). At this point, the axis mixing ratios are recomputed from Eq. (10).

Finally, the mass mixing ratio at the end of a time step is computed. To do so requires knowing how the mean ice density has changed over time. The new density is a volume-weighted average between the density at the prior time step and the density added during growth given by Eq. (B25). Once *q _{i}*,

*A*, and

*C*are all now known at the end of a time step.

In the parcel model tests described in Part II of this paper, our goal is to illustrate that the average effects of adaptive habit evolution can be captured with the procedure described above. Other issues emerge when the method is implemented in an Eulerian framework. For example, separate advection of the *a*- and *c*-axis length mixing ratios could potentially lead to errors in the evolution of axis ratio. However, tests indicate that such errors are small; this is a subject of a forthcoming paper on the Eulerian implementation of the method. Other issues also arise: Ice can be nucleated at multiple locations within the cloud, and regions of differing ice classes can be mixed together. Furthermore, the adaptive habit method needs to be connected in a physically accurate way to both riming and aggregation processes. These issues are beyond the scope of the present work, but are discussed briefly in Part II and in more depth in our forthcoming paper on the Eulerian implementation and testing.

## 4. Summary, remarks, and future development

Prior methods of parameterizing ice mass growth from vapor have used the traditional capacitance model; however, the capacitance depends on the major semiaxis length (either *a* or *c*) and the aspect ratio *φ* severs the nonlinear link between aspect ratio changes and increases in vapor growth leading to compounding errors in ice mass evolution. Because mass–size relations are derived for particular ice habits, cloud models then account for habit changes through the use of multiple ice classes (e.g., Walko et al. 1995; Woods et al. 2007) or assume a constant particle habit (e.g., Reisner et al. 1998; Fridlind et al. 2007; Thompson et al. 2004; Avramov and Harrington 2010). A more recent trend in cloud modeling is to predict a number of particle properties instead of different ice classes. Doing so produces a more natural evolution of the particle’s properties over time and avoids transfer functions between ice classes which are poorly constrained and often arbitrary. Along these lines, a bulk method for parameterizing ice habit evolution with four prognostic variables was developed that captures the nonlinearity in ice vapor growth through aspect ratio changes following SH11. The habit method could also be used for models that attempt to predict distribution shape (Milbrandt and Yau 2005) or for Eulerian bin microphysical codes (e.g., Reisin et al. 1996). In Part II of this work, the accuracy of the method is examined, and the limitations of the method are commented upon.

## Acknowledgments

We are grateful for the detailed comments provided by three anonymous reviewers, which substantially improved this manuscript. K. Sulia and J. Harrington would like to thank the National Science Foundation for support under Grants ATM-0639542 and AGS-0951807. In addition, J. Harrington was supported in part by the Department of Energy under Grant DE-FG02-05ER64058. K. Sulia was supported in part by an award from the Department of Energy (DOE) Office of Science Graduate Fellowship Program (DOE SCGF). The DOE SCGF Program was made possible in part by the American Recovery and Reinvestment Act of 2009. The DOE SCGF program is administered by the Oak Ridge Institute for Science and Education for the DOE. ORISE is managed by Oak Ridge Associated Universities (ORAU) under DOE Contract Number DE-AC05-06OR23100. All opinions expressed in this paper are the author’s and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE. H. Morrison was partially supported by the NOAA Grant NA08OAR4310543; U.S. DOE ARM DE-FG02-08ER64574; 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.

## APPENDIX A

### Fall Speed and Ventilation Effects

*ρ*is the air density,

_{a}*η*is the dynamic viscosity, and

_{a}*L*is an appropriate particle length scale. Following Bohm (1992),

*L*= 2

*a*for a plate,

*r*for a sphere as they produce the best accuracy in comparison to wind tunnel data of growing ice particles (see SH11). The particle Reynolds number

*X*using the fit coefficients (

*a*and

_{m}*b*) from Mitchell (1996). The Best number depends only on particle geometry and the form given in Bohm (1992) but corrected for buoyancy effects is used:

_{m}*ρ*is the ice particle density,

_{i}*m*is the crystal mass,

*g*is the gravitational acceleration,

*A*is the projected area of the crystal [

_{e}*πa*

^{2}(

*ρ*/

_{i}*ρ*)

_{bi}^{2/3}for platelike and

*πac*for columnlike], and

*q*is the ratio of the projected area of the crystal to the projected area of the spheroid (

_{e}*A*/

_{e}*A*; see SH11). The Best number is recast in a form more conducive to bulk model development by exposing the

*a*-axis length dependence using Eq. (6) in the relations for

*L*and

*A*:

_{e}*b*=

_{n}*δ*

_{*}+ 2 + 2

*b*−

_{l}*b*, and

_{a}*a*= 2,

_{l}*b*= 1;

_{l}*a*=

_{a}*π*(

*ρ*/

_{i}*ρ*)

_{bi}^{2/3}for platelike crystals; and

*b*= (

_{l}*δ*

_{*}+ 2)/3,

*a*=

_{a}*α*

_{*}

*π*, and

*b*=

_{a}*δ*

_{*}+ 1 for columnar crystals. The Reynolds number can now be written in terms of the

*a*axis alone:

*a*-axis distribution.

*D*and thermal conductivity of air

_{υ}*K*are multiplied by respective ventilation factors:

_{t}*N*

_{Sh}=

*η*/(

_{a}*ρ*) is the Sherwood number, and

_{a}D_{υ}*N*

_{Pr}=

*η*/(

_{a}*ρ*) is the Prandtl number. The coefficients above are given in CL94.

_{a}Kt*a*and

*c*, respectively, as

*r*is the radius of an equivalent volume sphere. Chen and Lamb (1994) shows that axis ventilation modifies the inherent growth ratio in the following manner:

*δ*is replaced by

*δ*

_{υ}in Eq. (2). While these relationships are hypothesized, they produce relatively good agreement between wind tunnel measurements and modeled evolution of crystal mass, axis lengths, and fall speed (SH11). Note that thermal effects are not included, because

*δ*is associated with the vapor redistribution and not the thermal diffusion process (CL94).

_{υ}## APPENDIX B

### Bulk Parameterization Details

#### a. Distribution relations

*p*moment of the

*a*-axis distribution [Eq. (9)] is given by (see Walko et al. 1995)

*c*axis and the equivalent volume spherical radius

*r*, it is necessary to find the distributions and moments of

*c*and

*r*. The key to the bulk habit method is the historical aspect ratio contained in Eqs. (6) and (8), which connects the variation in

*a*to the other axis lengths. The relationship between

*a*and either

*r*or

*c*has the general form

*α*and

_{d}*β*are appropriate coefficients defined below, and

_{d}*d*is either

*r*or

*c*. Changing variables from

*a*to

*d*in Eq. (9) gives

*a*and

_{n}*d*. The moments of the converted size distribution are

_{n}*a*-axis distribution to a distribution of other length scales. Equations (8) and (B3) produce the distribution of

*r*:

*a*- to the

*c*-axis distribution can be done in a similar fashion using Eqs. (6) and (B3):

#### b. Integration of mass growth equation

*a*alone. Using Eq. (6), the capacitance [Eq. (1)] can be written in a general form for either columnlike or platelike particles:

*f*is the shape factor,

_{s}*α*

_{cap}= 1,

*δ*

_{cap}= 1 for platelike crystals, and

*α*

_{cap}=

*α*

_{*},

*δ*

_{cap}=

*δ*

_{*}for columnar crystals. The aspect ratio can be written in terms of

*a*alone using the time–historical relationship between

*a*and

*c*[Eq. (6)]:

*a*-axis distribution is not possible. A simple form for

*f*is sought, and so a power-law fit is used:

_{s}*a*

_{1},

*a*

_{2},

*b*

_{1}, and

*b*

_{2}are fit coefficients with values given in Table B1. Figure B1 shows that Eq. (B9) provides an accurate estimate of the shape factor over a broad range of aspect ratios. Combining Eqs. (B7), (B8), and (B9) gives the capacitance in terms of

*a*only:

*c*or

*a*) multiplied by a shape factor

*f*. Equations (6) and (B7) suggest the following form:

_{s}*a*-axis-weighted average shape factor for either plates or columns as the rightmost equation indicates.

Power-law fit coefficients for the shape factor [*f _{s}*; Eq. (B9)] as a function of aspect ratio.

Shape factor as a function of aspect ratio. The exact expression for oblate and prolate spheroids along with the polynomial fit are shown.

Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1

Shape factor as a function of aspect ratio. The exact expression for oblate and prolate spheroids along with the polynomial fit are shown.

Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1

Shape factor as a function of aspect ratio. The exact expression for oblate and prolate spheroids along with the polynomial fit are shown.

Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1

#### c. Mass distribution relations for particle distributions

*a*,

*c*, and

*r*axes to one another is needed if a measure of particle aspect ratio is to be predicted. Using Eq. (2) and the growth history [Eq. (6)] a historical relation between changes in

*c*and

*a*can be found:

*a*-axis distribution and realizing that

*n*(

*a*)

*da*=

*n*(

*c*)

*dc*gives

*c*and

_{n}*a*[Eq. (B6)]:

_{n}*r*and

_{n}*a*is required for vapor growth calculations. The relation can be derived by first rewriting Eq. (5) using Eq. (8):

_{n}*β*= (2 +

_{a}*δ*)/3, and

*a*-axis distribution and realizing that

*n*(

*a*)

*da*=

*n*(

*c*)

*dr*,

#### d. Mass mixing ratio, size, and aspect ratio evolution

*a*and

*c*axes vary, a solution to the vapor diffusion equation is problematic. As discussed in sections 2 and 3, this problem is avoided by following SH11 and solving for the change in the characteristic equivalent volume radius

*r*. The vapor diffusion equation for mass mixing ratio [Eq. (B13)] is first recast into an equation for the distribution of equivalent volume spheres:

_{n}*f*(

_{s}*r*), and ice supersaturation constant over a time step:

_{n}*r*is known. Integrating Eq. (14) over a time step yields

_{n}*r*over time step allows the change in total volume to be computed through Eq. (14), and hence mass mixing ratio can be computed. The rightmost equation is arrived at by integrating Eq. (13), enabling the diagnoses of the mean effective density at the end of a time step:

_{n}*w*

_{υ}=

*V*(

_{t}*t*)/

*V*(

_{t}*t*+ Δ

*t*) is a weighting factor.

*a*axis is also straightforward as changes in

*r*are related to changes in

_{n}*a*through Eq. (15). Integrating this equation over a time step gives

_{n}*a*depends on two things: The change in the overall volume, which is provided by the change in

_{n}*r*, and how that volume (or mass) change is distributed along the

_{n}*a*axis, which is controlled by

*δ*.

*c*-axis length is not as simple as using the inherent growth ratio

*δ*to estimate the change over time because each ice particle in the distribution has a different relationship between the

*c*and

*a*axis. This is why a historical relation between

*c*and

*a*is needed and provided through

*δ*

_{*}. The change in the characteristic

*c*axis can be estimated as long as the change in

*δ*

_{*}is known, which is computed from

*r*and

_{n}*a*. Starting with Eq. (B5) replaced with the definitions of

_{n}*β*= 3/(2 +

_{r}*δ*

_{*}), and taking the natural logarithm, one can show that

*δ*

_{*}at the end of the time step. It is now possible to use

*δ*

_{*}(

*t*+ Δ

*t*) to estimate

*c*(

_{n}*t*+ Δ

*t*) using Eq. (B6), which relates the characteristic

*c*- and

*a*-axis lengths.

*c*with Eq. (B6) was found to be most accurate for columnar crystal (

_{n}*δ*

_{*}> 1) in comparison with the detailed model of SH11. For platelike crystals (

*δ*

_{*}< 1), comparisons with SH11 showed it is more accurate to mimic the method used for single particles (CL94) and diagnose the characteristic

*c*axis from the average aspect ratio:

*a*and

_{n}*r*[Eq. (B5)] it is possible to show that

_{n}*φ*before and after one time step,

*α*and the gamma functions in Eq. (B30) are nearly unity since

_{r}*δ*

_{*}does not change drastically when Δ

*t*< 20 s. With this approximation, the relationship between the average aspect ratio and the volume becomes

*c*-axis length can now be related to both the mean aspect ratio and

*a*by using the

_{n}*c*-axis distribution [Eqs. (B6) and (B28)]:

*a*are known,

_{n}*c*can be computed after a time step as well.

_{n}*a*. Integrating the spheroidal mass [Eq. (7)] over the

_{n}*a*-axis distribution gives a second form of the mass mixing ratio:

*q*and

_{i}*a*are known at any time, the mean effective density of the particles can be diagnosed from this relationship.

_{n}#### e. Fall speeds and bulk ventilation coefficients

*x*and

_{n}*b*are defined in Eq. (A2) in appendix A. The mean value of

_{n}*X*is used to determine the values of the coefficients (

*a*and

_{m}*b*) needed to find the Reynolds number [Eq. (A3)] in the equation for the fall speed [Eq. (A1)].

_{m}*a*-axis distribution using the moments defined in section a of appendix B. The number-weighted fall speed is defined as

*m*(

*a*) is defined in Eq. (7). Similarly, the length-weighted fall speed can be found from

*L*(

*a*) defined in Eq. (A1).

*γ*

_{υ}and

*γ*) that appear in the equations for ventilation [Eq. (A4)]. To pick these coefficients, the distribution-averaged Reynolds number [Eq. (A3)] is used:

_{t}*γ*

_{υ}and

*γ*are determined, bulk forms of the ventilation coefficients can be derived. However, the ventilation coefficients appear in the growth equation inside the function

_{t}*G*in such a way that integration over the size spectrum is not possible. An alternative method is to begin with the equations for vapor and thermal diffusion [Lamb and Verlinde 2011, their Eq. (8.6), p. 324] prior to their combination into one final equation. Using vapor diffusion as an example, the distribution-averaged growth rate is

_{i}*ρ*

_{υ}=

*ρ*

_{υ}

_{,∞}−

*ρ*

_{υ}

_{,i}is the excess vapor density over the crystal with

*ρ*

_{υ}

_{,∞}and

*ρ*

_{υ}

_{,i}as the respective environmental vapor and equilibrium ice vapor densities,

*a*-axis distribution [e.g., Eq. (B1)] and by combining Eqs. (A4), (B7), and (B9) and rearranging to arrive at

*b*

_{υ}= (

*b*

_{n}b_{m}γ_{υ})/2 and

*b*= (

_{t}*b*)/2.

_{n}b_{m}γ_{t}*a*-axis length by using Eqs. (6), (8), and (A5):

*a*-axis distribution in the same fashion as the total ventilation coefficient above; namely,

*b*

_{υa}and

*b*

_{υc}are defined as

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