a method for adaptive habit prediction in bulk microphysical...

A Method for Adaptive Habit Prediction in Bulk Microphysical Models.Part I: Theoretical Development

JERRY Y. HARRINGTON AND KARA SULIA

Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

HUGH MORRISON

National Center for Atmospheric Research, Boulder, Colorado

(Manuscript received 8 February 2012, in final form 28 August 2012)

ABSTRACT

Bulk microphysical schemes use the capacitancemodel for ice vapor growth in combination with mass–size

relationships to determine the evolution of ice water content (IWC) and ice particle maximum dimension in

time. These approaches are limited since a single axis length is used, the aspect ratio is usually held constant

and mass–size relations have many available coefficients for similar ice types. Fixing the crystal aspect ratio

severs the nonlinear link between aspect ratio changes and increased growth rates that occur during crystal

growth. Amethod is presented here for predicting two crystal axes and the crystal aspect ratio in bulk models.

Evolution of the ice mass mixing ratio is tied to the evolution of two axis length mixing ratios through the use

of a historical axis ratio parameter containing memory of crystal shape. This parameter links the distributions

of the two axes, allowing characterization of particle lengths using a single distribution. The method uses four

prognostic variables: the mass and number mixing ratios, and two axis length mixing ratios. Development of

the method is presented, with testing described in Part II.

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 irreg-

ular 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 (f 5 c/a). For hexagonal prisms, the a di-mension 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 as-

pect ratio variation with temperature, the classical pic-

ture of which is based on laboratory and in situ data and

is arguably most valid for T . 2228C (e.g., Pruppacherand Klett 1997; Fukuta and Takahashi 1999; Bailey and

Hallett 2009). These data suggest platelike (f , 1)crystals for temperatures in the ranges of 218 to 238Cand2108 to2228C, whereas columnlike crystals (f. 1)occur in the ranges of238 to298Cand lower than2228C.More recent data suggest important modifications to the

habit diagram at temperatures below about 2228C (e.g.,Bailey and Hallett 2009), specifically that plates tend to

occur more often and that columns are rare. Some mea-

surements support this (e.g., Bailey and Hallett 2002,

2004); however, these data are at odds with other lab-

oratory 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 clas-

sification is most valid for T . 2228C. Laboratory mea-surements generally show that lower saturation states

produce compact hexagonal crystals depending in part

on nucleation. As the saturation increases toward liq-

uid, crystals tend to hollow: columns become needlelike

Corresponding author address: Jerry Y. Harrington, 503 Walker

Building, Department of Meteorology, The Pennsylvania State

University, University Park, PA 16802.

E-mail: [email protected]

VOLUME 70 JOURNAL OF THE ATMOSPHER IC SC I ENCE S FEBRUARY 2013

DOI: 10.1175/JAS-D-12-040.1

� 2013 American Meteorological Society 349

near 268C, dendritic crystals appear around 2158C,with rosette and polycrystals forming at T , 2228C.It is presently impossible for cloud models to capture

the variety of ice crystal masses, shapes, sizes, and dis-

tributions 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 vol-

ume (or diameter) spheres (e.g., Lin et al. 1983; Reisner

et al. 1998; Thompson et al. 2004) or mass–size re-

lationships (e.g., Mitchell et al. 1990; Harrington et al.

1995; Meyers et al. 1997; Woods et al. 2007; Thompson

et al. 2008;Morrison andGrabowski 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 un-

derestimates ice growth at every temperature except

around 2108 and 2228C where growth is nearly iso-metric (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. Typi-

cally, 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 com-

putationally 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 tra-

ditional 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 histo-

ries. To evolve crystal habit, a way to relate the evo-

lution 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 as-

pect 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 improve-

ments through new measurements. The approach elimi-

nates thresholds between habits and follows the recent

paradigm of predicting particle properties instead of

multiple particle classes. The parameterization is de-

veloped here with testing of the method described in

Harrington et al. (2013, hereafter Part II).

2. Single-particle evolution

The Fickian-distribution method

The bulk habit method is rooted in the work of CL94

briefly described here, but a full discussion can be found

in the original work (CL94), and an exposition of the

physical implications in SH11. Fundamentally, the CL94

method combines two predictive equations. The first is

vapor diffusional mass growth that follows the classical

capacitance model (Pruppacher and Klett 1997):

dm

dt5 4pC(c, a)Gi(T ,P)su,i 5 4pLfs(f)Gi(T,P)su,i

5 4prfsr(f, r)Gi(T,P)su,i , (1)

where C is the capacitance of the ice particle, Gi is

a function that combines the effects of thermal and

vapor diffusion processes [Lamb and Verlinde 2011,

their Eq. (8.41), p. 343], P is the pressure, T is the

temperature, and su,i is the ice supersaturation. Be-

cause 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 fs(f) 5 C/L is a shapefactor that depends only on the aspect ratio f. 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

350 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70

shape factor, fsr 5C(c, a)/r, constant, which is accurateover short time steps (,10 s; see SH11).A major limitation of the traditional capacitance

model, as SH11 and others (e.g., Nelson 1994, p. 85)

point out, is that the theory produces a constant aspect

ratio. This result derives from the assumptions made

about the distribution of ice deposited on the crystal

surface and can be partially circumvented in traditional

mass–size methods by parameterizing the change in as-

pect ratio (e.g., Harrington et al. 1995; Avramov et al.

2011). One could argue that these methods are in-

consistent with capacitance theory itself. The method of

CL94 addresses this shortcoming of the capacitance

model by modifying the vapor fluxes along the a and c

axes, producing a parameterization that is more consis-

tent with the theory and rooted in laboratory measure-

ments. 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:

dc

da5 d(T)f . (2)

The inherent growth ratio [d(T)5 ac/aa] is a ratio of thedeposition coefficients, or growth efficiencies, for the c

and a axes. Since the deposition coefficients can be de-

termined 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 differ-

ent 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 d

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 byHashino

and Tripoli (2008), which were altered for numerical

accuracy in the solutions and to account for the possi-

bility of plates as the primary habit at temperatures

below 2228C (see Bailey and Hallett 2009). The mod-ified values of d in Hashino and Tripoli (2008), larger

(smaller) d than CL94 for plates (columns), are useful

because they are within the limits of the measured d

and together with best-fit values from CL94 provide

realistic bounds on possible values of d. 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 super-

saturations (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 ac-

curate 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 de-

viates 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

T5268C either with a constant or evolving aspect ratio.When f 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 f

evolves freely. Consequently, predicting crystal aspect

ratio is critical for capturing nonlinear changes in vapor

diffusion.

The comparisons of CL94 and SH11 with wind tunnel

data suggest that a variety of crystal shapes can be used

in the Fickian-distribution method as long as d andC are

FIG. 1. Capacitance as a function of aspect ratio for oblate (plate)

and prolate (column) crystals with volume equivalent to that of

a sphere (req) of radius 50 (solid), 25 (dashed), and 10 (dotted–

dashed) mm. Evolution of an ice particle at T 5 268C, liquid sat-uration, an initial equivalent volume spherical radius of 10 mm, and

initial f 5 10 for the capacitance model with f constant (solidblack arrow) and Fickian distribution with evolving aspect ratio

(solid gray arrow).

FEBRUARY 2013 HARR INGTON ET AL . 351

known. Using a single crystal type (e.g., hexagon or

dendrite) restricts the generality that is desired for a nu-

merical cloud model. Therefore, spheroids are used as

a first-order approximation of primary habit (f), allowing

for the prediction of two axes. Relating changes in mass

to changes in axis lengths requires the spheroidal volume:

V(a, c)54

3pa2c5

4

3pa3f and m(a, c)5V(a, c)ri ,

(3)

where V is the volume, and ri is the effective particle

density that is less than bulk ice (rbi5 920 kg m23). 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 depo-

sition density [rdep; see CL94, their Eq. (42)], varies with

temperature. Over short growth times rdep is approxi-

mately constant (CL94) so that

dm

dt5 rdep

dV

dt. (4)

SH11 point out that an analogous density is needed for

sublimation [rsub; their Eq. (6)], though here rdep 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 spher-

ical volumes, then logarithmically differentiating and

using Eq. (2) results in

dr

da5

21 d

3

r

a. (5)

Thus, changes 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 dis-

cussed 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 ro-

bust 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 equa-

tion [Eqs. (1) and (4)] and the mass distribution hypoth-

esis [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, d 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 evolu-

tion 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

FIG. 2. 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 f 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.


Tripoli (2008). Frequently used coefficients from the

derivations below are listed in Table 1.

a. Axis length history

The key to predicting the first-order effects of habit

evolution lies in historically relating the a- and c-axis

lengths. This is necessary because the inherent growth

ratio d in Eq. (2) varies with temperature, and therefore

the way a and c are related will vary in time. This vari-

ability is accounted for by integrating Eq. (2) in time

from a0 to a(t) and c0 to c(t), giving

c(t)5a*ad*(t) and d*5

ða05a(t)a05a

0

d(t) d ln(a0)

ða05a(t)a05a

0

d ln(a0), (6)

where a*5 c0a2d*0 , and d* is a time average of the in-

herent growth ratio over the particle growth history, also

described in CL94. Note that the weighting is loga-

rithmic, which is a consequence of the relationship be-

tween dc/c and da/a in Eq. (2). The initial aspect ratio

f0 and size a0 of the ice particles must be chosen. Initial

aspect ratios that are not unity can be accommodated by

setting c0 5 f0a0 in a*. Although some recent labora-tory data suggest that initially frozen drops at T 52408C have aspect ratios of around 1.1 (López andÁvila 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 a*5 a12d*0 .

Because particles are initially assumed to be iso-

metric, d* is initially equal to one. However, the in-

herent growth ratio d is typically not unity and varies

with temperature, and so d* will also vary in time. This is

an important departure from traditional bulk methods

that implicitly fix d* at a particular value for a given

habit type. In the method proposed here, the relation

between c and a will also evolve. Since d* is a weighted

time average of d, it is naturally constrained to lie within

the limits of d derived from laboratory data (about 0.27

and 2.1). Note the distinction between d and d*: d is

a measure of the mass distributed over the a and c axes

during growth (one time step), whereas d* 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 d* 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 his-

torical relation (d*), which is demonstrated in Part II.

Using Eqs. (3) and (6), the ice crystal mass can be

written in the traditional mass–size form

m(a)5V(a)ri 5amabm , (7)

where the coefficients am 5 (4/3)pa*ri and bm 5 21 d*are for a platelike crystal. This result shows that the

mass–size relations are directly connected to the his-

torical evolution of the particle through d*. Conse-

quently, 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 co-

efficient 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):

a5arrbr , (8)

where ar 5a21/bm*

and br 5 3/bm.

b. Axis distributions

The bulk habit growth equations require a priori

forms of the particle distributions. Since c and a are re-

lated 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(a)5NiG(n)

�a

an

�n21 1an

exp

�2

a

an

�, (9)

where Ni is the number density of ice crystals, G(n) is thegamma function, n is the distribution ‘‘shape’’ parameter,

and an is the characteristic a-axis length that is related to

TABLE 1. Coefficients resulting from derivations presented in the

main text and appendix.

Bulk model coefficients

Variable Definition Equation number

a* a12d*0 (6)

aa a1/3* (B19)

acap 1 (platelike) or a* (columnlike) (B13)

am43pa*ri (7)

ar a21/bm*

(8)

ay43pa* (B33)

ba21 d3

(5)

ba* (2 1 d*)/3 (B19)bm 2 1 d* (7)br 3/bm (8)

dcap 1 (platelike) or d* (columnlike) (B13)


the mean a-axis length, and the a-axis mixing ratio (see

section 3e). It is worth noting that generalized distri-

butions like that in Thompson et al. (2008) could also

be used.

To define the aspect ratio requires the c-axis distribu-

tion 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 d*. The resulting distributions are similar toEq. (9).

The general form of the distribution conversion and the

distributions of the c and r axes [Eqs. (B6) and (B5), re-

spectively] are given in section a of appendix B. Each has

characteristic lengths (cn and rn) related to the mean axis

length that characterize the mean particle shape.

Eulerian models require conserved variables, and so

four quantities are predicted in the bulk adaptive habit

method: The number mixing ratio (Ni/ra), the ice mass

mixing ratio (qi), and the mixing ratios of the a and c

axes, defined respectively as

A5nNira

an,

C5nNira

cn , (10)

where ra is the air density. These equations follow

from the definition of the mean axis length [a5(1/Ni)

Ð ‘0 an(a) da5 an[G(n1 1)]/[G(n)]5 nan, Eq. (B4)].

The axis length mixing ratios are a measure of the

summed lengths of 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 dis-

cussion of this issue is provided in section 3e.

c. Mass mixing ratio growth and mass distributionrelationship 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 pro-

portional to an and cn, the characteristic sizes are used

throughout to represent mean habit evolution. Evolving

qi 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 rn is

known, aspect ratio evolution can then be computed

using relationships between the different axis lengths.

The evolution of qi by vapor diffusion is given by

dqidt

51

ra

ð‘0

dm

dtn(a) da . (11)

To integrate Eq. (1) over 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 an alone

[see section b of appendix B; Eq. (B13)]. Integrating

the resulting expression [Eq. (B13)] in time is compli-

cated because the mass and capacitance both depend

on an and, implicitly cn, 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 charac-

teristic equivalent volume spherical radius (rn) rather

than an or cn to determine the mass mixing ratio change

over a time step provides the most accuracy in com-

parison to numerical solutions. The differential equation

for qi in terms of rn is shown in section d of appendix B

to be

dqidt

5Nira

4prnfs(rn)Gisu,i , (12)

where fs(rn)5C(an)/rn is a representative shape factor.To make this equation amenable to integration, qi must

be related to rn, which can be done by integrating the

particle mass over n(r) [Eq. (B5)]:

qi 54

3pr3nriNi

G(n1 3/br)

G(n)5 riVt , (13)

with ri the volume-weighted mean effective density (or

‘‘mean effective density’’; see section d of appendix B)

and Vt the total particle volume. Using this relationship,

changes inmass mixing ratio can be related to changes in

Vt and rn in a similar fashion as for single particles [Eq.

(4)] through

dqidt

5 rdepdVtdt

5 4prdepNiG[n1 (3/br)]

G(n)r2ndrndt

. (14)

Equation (14) along with Eq. (12) can now be solved to

advance the characteristic equivalent radius rn, and

hence the mass mixing ratio, in time (see section d of

appendix B).

Evolving the characteristic axis lengths requires

a mass distribution relationship connecting changes in rnto changes in cn and an. Such a relation can be derived

using Eq. (5) (see section c of appendix B):


drndan

5

�21 d

3

�rnan

. (15)

The mass distribution relationship connecting changes

in an to cn is shown in section c of appendix B to be

dcndan

5 dcnan

(16)

and closes the equation set. The procedure used to solve

these equations for mean habit evolution is provided in

section 3e.

d. Fall speed and ventilation

The bulk habit method outlined above includes nei-

ther ventilation effects nor estimates of particle fall

speeds, both of which are needed for any bulk parame-

terization. Computing the particle fall speeds requires

choosing appropriate coefficients for the Reynolds

number (see section e of appendix B), which thenmakes

it possible to find the bulk fall speed by combining Eqs.

(A1) and (A3) and integrating over the size spectrum:

yt 51

Ni

ðNReharaL

n(a) da5NRe(an)haraL(an)

G(n1 bnbm 2 bl)

G(n),

(17)

where the coefficients are defined in appendix A.

Eulerian models also require a mass- (ytm) and length-

weighted (ytl) fall speed, and these are given in section e

of appendix B.

Computing the bulk ventilation coefficients requires

diagnosing the power-law coefficients (gy and gt) 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 (f yDy)

and thermal conductivity (f tKt) that appear in Gi [Eq.

(11)] and are determined by appropriately integrating

over the a-axis distribution (section e of appendix B):

f y 5 by11 by2 N1/3Sc NRe(an)

1/2h ig

yf cap,y ,

f t 5 bt11 bt2 N1/3Pr NRe(an)

1/2h ig

yf cap,t , (18)

with all coefficients defined by Eq. (B42) in section e of

appendix B and in appendix A.

Ventilation affects each axis length differently, and

so bulk versions of the axis-dependent ventilation co-

efficients [i.e., Eq. (A5)] are also needed. Using a pro-

cedure similar to that above (see section e of appendix B)

leads to the following coefficients for the a and c axes,

respectively:

fya5by11by2 N1/3Sc NRe(an)

1/2h ig

ya21/3* a

1/3(12d*)n

h igy/2f cap,a,

f yc5by11by2 N1/3Sc NRe(an)

1/2h ig

ya2/3* a

2/3(d*21)n

h igy/2f cap,c ,

(19)

and the terms f cap,a and f cap,c are defined by Eq. (B46) in

section e of appendix B.

The effects of ventilation on the evolution of themean

axis lengths are then estimated following the same

procedure that is used for single-particle axis lengths

[Eq. (A6)]:

dyb5 df ycf ya

. (20)

The value of dyb is the bulk value of the inherent growth

ratio modified for ventilation effects. Including axis-

dependent ventilation is now accomplished by replacing

dwith dyb in all of the bulk equations derived above. It is

critical to keep in mind that only d is modified whereas

d* 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. Themain equations used are

also summarized in Tables 2 and 3.

Eulerian models use conserved quantities, and the

mass, number, and axis length mixing ratios (qi,Ni/ra,A,

and C, respectively) are used here. To compute the

evolution of mass mixing ratio and axis lengths, a num-

ber of quantities are needed at the beginning of the time

step t and can be extracted from qi, A, and C. Table 2

provides a summary of the main initial variables. At the

beginning of a time step, cn and an can be determined

from the prognosed axis length mixing ratios using Eq.

(10). With these, d*(t) can be diagnosed by taking the

natural logarithm of cn(t)5a*an(t)d*(t)

[Eq. (B6)] and

replacing a*5 a(12d*)

0 :

d*(t)5ln[cn(t)]2 ln(a0)

ln[an(t)]2 ln(a0). (21)

Once d* is known at the beginning of a time step, the

characteristic equivalent volume spherical radius rn(t)

can be found from Eq. (B5), and the mean ice particle


density ri(t) can be solved for by using the ice mass

mixing ratio qi(t) and inverting Eq. (13). To compute

vapor diffusion, quantities that depend on the charac-

teristic lengths are needed. The average capacitance

C(an) and the average shape factor f s(an) are de-

termined using Eqs. (B13) and (B14) (see Table 2). The

vapor and thermal diffusivities that appear in the growth

equation [Gi in Eq. (12)] are modified for ventilation

effects using Eq. (18) (Table 2).

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

TABLE 2. Summary of variables: Beginning of time step. Only the mixing ratios (Ni/ra, qi,A, and C) are stored externally to the growth

calculations. To compute the change in qi, 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.

Equation Purpose Equation originates from

an(t)5ranNi

A(t) Characteristic a-axis length Eq. (10)

cn(t)5ranNi

C(t) Characteristic c-axis length Eq. (10)

d*(t)5ln[cn(t)]2 ln(a0)

ln[an(t)]2 ln(a0)Initial value of d*, needed for rn and growth calculations Eq. (21)

rn(t)5a21/brr a

1/brn Initial equivalent spherical characteristic radius Eq. (B5)

ri 53qi(t)

4prn3(t)Ni

G(n)

G(n1 3/br)Initial average density Eq. (13)

fs(rn)5C(an)

rn5

1

rnacapa

dcapn fs(an) Shape factor for vapor growth Eq. (B14)

Dyf y 5Dyfby1 1by2[N1/3Sc NRe(an)1/2]gy f cap,ygKTf t 5KTfbt1 1bt2[N1/3Pr NRe(an)1/2]gy f cap,tg

Ventilation effects on diffusion, used in Gi Eq. (18)

dyb 5 df yc

f yaVentilation modification of d Eqs. (19) and (20)

TABLE 3. Summary of prognostic variables: End of time step. The growthmethod evolves the icemassmixing ratio (qi), and the a- and c-

axis mixing ratios (A andC) over a time step. This table contains themain 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.

Equation Purpose

Equation originates

from

rn(t1Dt)5

"r2n(t) 1

2fs(rn)Gisu,irdep

G(n)

G(n13/br)Dt

#1/2 Needed to find qi and axislengths at end of Dt

Eq. (B23)

an(t1Dt)5 an(t)

�rn(t 1 Dt)

rn(t)

�3/(21dyb)Characteristic a-axis length

at end of DtEq. (B26)

d*(t1Dt)53 ln[rn(t1Dt)]2 2 ln[an(t1Dt)]2 ln(a0)

ln[an(t1Dt)]2 ln(a0)Needed to find cn at end of Dt Eq. (B27)

cn(t1Dt)5a*(t1Dt)ad*(t1Dt)n cn calculation when crystals are

columnar (d* . 1)Eq. (B6)

f(t1Dt)

f(t)’�r3n(t1Dt)

r3n(t)

�(dyb21)/(dyb12)cn(t1Dt)

5f(t1Dt)an(t1Dt)G(n)

G(n1 d*2 1)

cn calculation when platelike

(d* , 1): first find f at end ofDt, next find cn at end of Dt

Eqs. (B31) and (B32)

ri(t1Dt)5 (12wy)rdep 1wyri(t)ri Change in mean bulk ice density.Used to find new value of ice

mixing ratio

Eq. (B25)

qi(t1Dt)5 qi(t)1Dqi 5qi(t)1 ri(t1Dt)Vt(t1Dt)2 ri(t)Vt(t) Ice mass mixing ratio at theend of Dt

Eq. (B24)

A(t1Dt)5 (nNi/ra)an(t1Dt)C(t1Dt)5 (nNi/ra)cn(t1Dt)

a- and c-axis mixing ratio at

the end of DtEq. (10)


mass changes are computed by finding the change in rnusing Eq. (B23) giving rn(t1 Dt). Knowing rn at the endof a time step allows an(t 1 Dt) to be determined usingEq. (B26). Including axis ratio-dependent ventilation is

then accomplished by replacing d in Eq. (B26) with

dyb [Eq. (20)]. The change in cn can also be computed,

but to do so requires knowing d*(t 1 Dt), which is di-agnosed from Eq. (B27) since rn(t 1 Dt) and an(t 1 Dt)are known. For columnlike crystals, cn(t 1 Dt) is de-termined from the relation between cn and an in Eq.

(B6). For platelike crystals, the change in mean aspect

ratio is first computed, and then cn(t1 Dt) is found fromEq. (B32). At this point, the axis mixing ratios are re-

computed 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 ri(t1Dt) is known, it is thenpossible to compute the change in ice mass mixing ratio

over a time step through Eq. (B24). The mixing ratios qi,

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 pro-

cedure 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 loca-

tions 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

(f5 c/a) of the particles. Most bulk cloud models closethe equations by relating mass and axis length through

a mass–size relation derived from in situ data while

keeping the aspect ratio constant in the capacitance.

A constant f 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 con-

strained 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 distri-

bution shape (Milbrandt andYau 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 ex-

amined, and the limitations of the method are com-

mented 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 sup-

ported 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 Pro-

cesses (CMMAP), managed by Colorado State Univer-

sity under Cooperative Agreement ATM-0425247.


APPENDIX A

Fall Speed and Ventilation Effects

A unique feature of the CL94 method is that venti-

lation during particle sedimentation affects each axis

differently, and this reproduces the observed tendency

of the crystal minor axis to asymptote to a particular size

(see their Fig. 6). Calculating ventilation effects requires

knowledge of nonspherical particle fall speeds. Sulia and

Harrington (2011) is followed where a combination of

the methods of Bohm (1992) and Mitchell (1996) are

used, which depend on the particle Reynolds number:

yt 5NRehara

1

L, (A1)

where ra is the air density, ha is the dynamic viscosity,

and L is an appropriate particle length scale. Following

Bohm (1992), L 5 2a for a plate, 2ffiffiffiffiffiffiffiffiffi(ac)

pfor a column,

and 2r 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

(NRe 5 amXbm) is computed from a power-law fit to theBest number X using the fit coefficients (am and bm)

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:

X 5 [(2mgraL2)/(Aeh

2a)]q

3/4e [(ri 2 ra)/ri], where ri is the

ice particle density, m is the crystal mass, g is the grav-

itational acceleration, Ae is the projected area of the

crystal [pa2(ri/rbi)2/3 for platelike and pac for columnlike],

and qe is the ratio of the projected area of the crystal to the

projected area of the spheroid (Ae/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 Ae:

X5 xBmL2

Aewhere xB5

2(ri2 ra)graq3/4e

rih2a

,

5 xnabn , (A2)

where Ae 5 aaaba , L5 alabl , xn 5 xB(4/3)[(pria*a2l )/aa],

bn5 d*1 21 2bl2 ba, and al5 2, bl5 1; aa5 p(ri/rbi)2/3

for platelike crystals; and al 5 2a1/3* , bl 5 (d*1 2)/3, aa5a*p, and ba5 d*1 1 for columnar crystals. TheReynoldsnumber can now be written in terms of the a axis alone:

NRe 5 amXbm 5 amx

bm

n abnbm . (A3)

These forms of the Best and Reynolds numbers can be

integrated over the a-axis distribution.

Ventilation effects on mass growth can be determined

if the vapor diffusivityDy and thermal conductivity of air

Kt are multiplied by respective ventilation factors:

fy 5 by11 by2Ngy/3

Sh Ngy/2

Re ,

ft 5 bt11 bt2Ngt/3

Pr Ngt/2

Re , (A4)

where NSh 5 ha/(raDy) is the Sherwood number, andNPr 5 ha/(raKt) is the Prandtl number. The coefficientsabove are given in CL94.

Axis ventilation follows CL94. Nonspherical particles

tend to fall with their major axis perpendicular to the fall

direction, and the crystal edges experience a stronger

ventilation effect (see SH11), which is approximated for

a and c, respectively, as

fya5 by11 by2Ngy/3

Sh Ngy/2

Re

�ar

�gy/2,

fyc5 by11 by2Ngy/3

Sh Ngy/2

Re

�cr

�gy/2, (A5)

where 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:

dy(T)5 d(T)fycfya

, (A6)

where d is replaced by dy in Eq. (2). While these re-

lationships 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 in-

cluded, because dy is associated with the vapor re-

distribution and not the thermal diffusion process

(CL94).

APPENDIX B

Bulk Parameterization Details

a. Distribution relations

The p moment of the a-axis distribution [Eq. (9)] is

given by (see Walko et al. 1995)

I(p)5

ð‘0apn(a) da5Nia

pn

G(n1 p)

G(n). (B1)

Since the habit method uses both the c axis and the

equivalent volume spherical radius r, it is necessary to

find the distributions andmoments 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 vari-

ation in a to the other axis lengths. The relationship

between a and either r or c has the general form

a5addbd , (B2)

where ad and bd are appropriate coefficients defined

below, and d is either r or c. Changing variables from a to

d in Eq. (9) gives

n(d)5NiG(n)

bd

�d

dn

�nbd21 1

dnexp

�2

�d

dn

�bd�,

an5addbd

n . (B3)

Note the relatively simple relationship between an and

dn. The moments of the converted size distribution are

Id(p)5

ð‘0dpn(d) d(d)5Nid

pn

G(n1 p/bd)

G(n). (B4)

The distribution conversion and moments can now be

used to convert the a-axis distribution to a distribution of

other length scales. Equations (8) and (B3) produce the

distribution of r:

n(r)5NiG(n)

br

�r

rn

�nbr21 1

rnexp

�2

�r

rn

�br�,

an5arrbr

n . (B5)

Converting from the a- to the c-axis distribution can be

done in a similar fashion using Eqs. (6) and (B3):

n(c)5NiG(n)

1

d*

�c

cn

�(n/d*)21 1cn

exp

�2

�c

cn

�1/d*�,

cn 5a*ad*n . (B6)

b. Integration of mass growth equation

To integrate Eq. (11) analytically, the capacitance

must be written in terms of a alone. Using Eq. (6), the

capacitance [Eq. (1)] can be written in a general form for

either columnlike or platelike particles:

C(a,f)5acapadcap fs(f) , (B7)

where fs is the shape factor, acap 5 1, dcap 5 1 forplatelike crystals, and acap 5 a*, dcap 5 d* for columnarcrystals. The aspect ratio can be written in terms of

a alone using the time–historical relationship between

a and c [Eq. (6)]:

f(a)5a*ad*21 . (B8)

The mathematical forms of the shape factors for

spheroids are complex enough that analytical in-

tegration over the a-axis distribution is not possible. A

simple form for fs is sought, and so a power-law fit is

used:

fs(f)5 a1fb1 1 a2f

b2 , (B9)

where a1, a2, b1, and b2 are fit coefficients with values

given inTableB1. FigureB1 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(a)5 c1ad1 1 c2a

d2 , (B10)

with the coefficients

c15 a1acapab1

*c25 a2acapa

b2

*d1 5b1(d*2 1)1 dcap d25 b2(d*2 1)1 dcap .

(B11)

With these relations, the growth equation [Eq. (11)] can

now be integrated:

dqidt

51

ra

ð‘04pC(a)Gisu,in(a) da ,

51

raNi4pGisu,i

�c1a

d1

nG(n1 d1)

G(n)1 c2a

d2

nG(n1d2)

G(n)

�,

(B12)

where Eq. (B1) is used. It is convenient to cast the

growth equation given by Eq. (B12) in a form similar to

that of a single particle where the capacitance is written

as the maximum semidimension (either c or a) multi-

plied by a shape factor fs. Equations (6) and (B7) suggest

the following form:

dqidt

5Nira

4pacapadcapn fs(an)Gisu,i , (B13)

TABLE B1. Power-law fit coefficients for the shape factor [fs; Eq.

(B9)] as a function of aspect ratio.

Fit coefficients

a1 a2 b1 b2

Oblate 0.636 942 7 0.363 057 0.0 0.95

Prolate 0.571 428 5 0.428 571 21.0 20.18


with the distribution-averaged capacitance as

C5acapadcapn fs(an) , (B14)

where

fs(an)5c1acap

ad12d

capn

G(n1 d1)

G(n)1

c2acap

ad22d

capn

G(n1 d2)

G(n)

5G(n1 1)

G(n)

ð‘0fs(a)an(a) dað‘0an(a) da

(B15)

and is similar to the a-axis-weighted average shape fac-

tor for either plates or columns as the rightmost equa-

tion indicates.

c. Mass distribution relations for particle distributions

Since bulk microphysical models only predict or di-

agnose quantities characteristic of the entire particle

distribution, a way to relate changes in the characteristic

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:

dc

dt5 da*a

d*21da

dt. (B16)

Integrating over the a-axis distribution and realizing that

n(a)da 5 n(c)dc gives

dcndt

5 da*ad*n

1

an

dandt

, (B17)

which can be simplified by using the relationship be-

tween cn and an [Eq. (B6)]:

dcndan

5 dcnan

. (B18)

This is given as Eq. (16) in the main text.

Relating rn and an is required for vapor growth cal-

culations. The relation can be derived by first rewriting

Eq. (5) using Eq. (8):

dr

dt5baaaa

ba*21da

dt, (B19)

where aa 5a1/3* , ba 5 (2 1 d)/3, and ba* 5 (21 d*)/3.Integrating this equation over the a-axis distribution and

realizing that n(a)da 5 n(c)dr,

drndt

5baaaaba*

n1

an

dandt

. (B20)

Inverting Eq. (B5) gives rn 5aaaba*n , and so the above

equation becomes

drndan

5

�21 d

3

�rnan

. (B21)

This is given as Eq. (15) in the main text.

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

To evolvemassmixing ratio, size, and aspect ratio, Eq.

(B13) must be integrated in time along with the equa-

tions from the subsection above. This is done analyti-

cally here, but since both the a and c axes vary, a solution

to the vapor diffusion equation is problematic. As dis-

cussed in sections 2 and 3, this problem is avoided by

following SH11 and solving for the change in the char-

acteristic equivalent volume radius rn. The vapor diffu-

sion equation for mass mixing ratio [Eq. (B13)] is first

recast into an equation for the distribution of equivalent

volume spheres:

dqidt

5Nira

4prnfs(rn)Gisu,i , (B22)

where fs(rn)5 [C(an)/rn] is a representative shape fac-tor, and this is Eq. (12) in the main text. Equation (B22)

FIG. B1. Shape factor as a function of aspect ratio. The exact

expression for oblate and prolate spheroids along with the poly-

nomial fit are shown.


is set equal to Eq. (14) and integrated keeping temper-

ature, pressure, equivalent spherical shape factor fs(rn),

and ice supersaturation constant over a time step:

rn(t1Dt)5

"r2n(t)1

2fs(rn)Gisu,irdep

G(n)

G(n1 3/br)Dt

#1/2.

(B23)

It is worth noting that analytical integration is not re-

quired (numerical methods can be employed; e.g.,

Walko et al. 2000), and supersaturation does not need to

remain constant; a simple method is chosen here to il-

lustrate the robust nature of the parameterization.

Calculating the change in mass mixing ratio is straight-

forward since the change in rn is known. Integrating

Eq. (14) over a time step yields

Dqi5 rdepDVt 5 rdep[Vt(t1Dt)2Vt(t)]

5 ri(t1Dt)Vt(t1Dt)2 ri(t)Vt(t) . (B24)

Knowing the change in rn over time step allows the

change in total volume to be computed throughEq. (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:

ri(t1Dt)5 (12wy)rdep1wyri(t) , (B25)

where wy 5 Vt(t)/Vt(t 1 Dt) is a weighting factor.Finding the change in the characteristic a axis is also

straightforward as changes in rn are related to changes in

an through Eq. (15). Integrating this equation over

a time step gives

an(t1Dt)5 an(t)

�rn(t1Dt)

rn(t)

�3/(21d). (B26)

Note that the temporal change in an depends on two

things: The change in the overall volume, which is pro-

vided by the change in rn, and how that volume (or mass)

change is distributed along the a axis, which is controlled

by d.

Unlike the single-particle model (CL94; SH11), cal-

culating the change in the c-axis length is not as simple as

using the inherent growth ratio d 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 d*. The change in the

characteristic c axis can be estimated as long as the

change in d* is known, which is computed from rn and an.

Starting with Eq. (B5) replaced with the definitions of

ar 5a21/(21d*)

*, a*5 a

12d*+

, and br 5 3/(2 1 d*), andtaking the natural logarithm, one can show that

d*(t1Dt)53 ln[rn(t1Dt)]2 2 ln[an(t1Dt)]2 ln(a0)

ln[an(t1Dt)]2 ln(a0),

(B27)

which is the value of d* at the end of the time step. It is

now possible to use d*(t 1 Dt) to estimate cn(t 1 Dt)using Eq. (B6), which relates the characteristic c- and a-

axis lengths.

Estimating cn with Eq. (B6) was found to be most

accurate for columnar crystal (d* . 1) in comparisonwith the detailed model of SH11. For platelike crystals

(d* , 1), comparisons with SH11 showed it is more ac-curate to mimic the method used for single particles

(CL94) and diagnose the characteristic c axis from the

average aspect ratio:

f51

Ni

ð‘0f(a)n(a) da5a*a

d*21n

G(n1 d*2 1)

G(n), (B28)

where Eq. (B8) is employed. With the relationship be-

tween an and rn [Eq. (B5)] it is possible to show that

f5a*ad*21r (r

3n)

(d21)/(d12)G(n1 d*2 1)

G(n). (B29)

Forming the ratio of f before and after one time step,

f(t1Dt)

f(t)5

�r3n(t1Dt)

r3n(t)

�(d21)/(d12)

3G[n1 d*(t1Dt)2 1]

G[n1 d*(t)2 1]

ad*(t1Dt)21r

ad*(t)21r

.

(B30)

The ratios of ar and the gamma functions in Eq. (B30)

are nearly unity since d* does not change drastically

when Dt , 20 s. With this approximation, the relation-ship between the average aspect ratio and the volume

becomes

f(t1Dt)

f(t)’�r3n(t1Dt)

r3n(t)

�(d21)/(d12). (B31)

Note that this relationship is nearly identical to that for

single particles given in CL94 and in SH11. For platelike

crystals, the characteristic c-axis length can now be re-

lated to both themean aspect ratio and an by using the c-

axis distribution [Eqs. (B6) and (B28)]:


cn(t1Dt)5f(t1Dt)an(t1Dt)G(n)

G(n1 d*2 1). (B32)

Since f and an are known, cn can be computed after

a time step as well.

Diagnosing the mean effective density at the begin-

ning of a time step requires one final relationship, and

that is between ri and an. Integrating the spheroidal

mass [Eq. (7)] over the a-axis distribution gives a second

form of the mass mixing ratio:

qi51

ra

ð‘0m(a)n(a) da5Niayria

bm

nG(n1bm)

G(n)

ri 5

ð‘0ria

bmn(a) dað‘

0abmn(a) da

, (B33)

where ay 5 (4/3)pa*. Because qi and an are known atany time, the mean effective density of the particles can

be diagnosed from this relationship.

e. Fall speeds and bulk ventilation coefficients

A difficulty in estimating bulk fall speeds and venti-

lation effects is that the Best number and the Reynolds

number, both of which depend on size, are used to

choose size-dependent coefficients. The main assump-

tionmade here is that the distribution-averaged forms of

each number can be used to choose the coefficients. The

distribution-averaged Best number is

X51

Ni

ðX(a)n(a) da5 xna

bn

nG(n1 bn)

G(n), (B34)

where xn and bn are defined in Eq. (A2) in appendix A.

The mean value of X is used to determine the values of

the coefficients (am and bm) needed to find the Reynolds

number [Eq. (A3)] in the equation for the fall speed [Eq.

(A1)].

With the Reynolds number coefficients determined,

the average fall speeds are computed by integrating Eqs.

(A1) and (A3) over the a-axis distribution using the mo-

ments defined in section a of appendix B. The number-

weighted fall speed is defined as

yt 51

Ni

ðNReharaL

n(a) da5NRe(an)haraL(an)

G(n1 bnbm 2bl)

G(n),

(B35)

which is Eq. (17) in the main text. The mass-weighted

fall speed is

ytm5

ðNReharaL

m(a)n(a) daðm(a)n(a) da

5NRe(an)haraL(an)

G(n1bnbm2 bl 1 21 d*)

G(n1 21 d*), (B36)

where m(a) is defined in Eq. (7). Similarly, the length-

weighted fall speed can be found from

ytl 5

ðNReharaL

L(a)n(a) daðL(a)n(a) da

5NRe(an)haraL(an)

G(n1 bnbm)

G(n1 bl),

(B37)

with L(a) defined in Eq. (A1).

Similarly to the Best number, the Reynolds number

is needed to choose values of the power-law co-

efficients (gy and gt) that appear in the equations for

ventilation [Eq. (A4)]. To pick these coefficients, the

distribution-averaged Reynolds number [Eq. (A3)]

is used:

NRe 51

Ni

ðNRe(a)n(a) da5 amx

bm

n abnbm

nG(n1 bnbm)

G(n),

(B38)

where all variables are defined in appendix A.

Once gy and gt are determined, bulk forms of the

ventilation coefficients can be derived. However, the

ventilation coefficients appear in the growth equation

inside the functionGi 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

dm

dt5

1

Ni

ðdm

dtn(a) da5

1

Ni

ð4pCDy fyDryn(a) da

5 4pCDy fyDry , (B39)

where Dry 5 ry,‘ 2 ry,i is the excess vapor density overthe crystal with ry,‘ and ry,i as the respective environ-

mental vapor and equilibrium ice vapor densities, C is

the distribution-averaged capacitance [Eq. (B14)], and

fy is the bulk ventilation coefficient for vapor. A similar

integral appears for thermal diffusion, and so the bulk

ventilation coefficients are


fy 5

ðCfy(a)n(a) dað

Cn(a) da

and ft 5

ðCft(a)n(a) daðCn(a) da

. (B40)

Once the capacitance-weighted ventilation coefficients

are known, they can be multiplied by the vapor diffu-

sivity and the thermal conductivity as an estimate of the

overall effects of ventilation on the growth rate of

a distribution of particles. The above integrals are solved

using solutions to the moments of the a-axis distribution

[e.g., Eq. (B1)] and by combining Eqs. (A4), (B7), and

(B9) and rearranging to arrive at

fy 5 by11 by2 N1/3Sc NRe(an)

1/2h ig

yf cap,y ,

ft 5 bt11 bt2 N1/3Pr NRe(an)

1/2h ig

yf cap,t . (B41)

These are Eq. (18) in the main text where f cap,y and f cap,tare factors that depend on the capacitance:

f cap,y 51

C

�c1a

d1

nG(n1by 1 d1)

G(n)1 c2a

d2

nG(n1 by 1 d2)

G(n)

�,

f cap,t 51

C

�c1a

d1

nG(n1 bt 1d1)

G(n)1 c2a

d2

nG(n1 bt 1d2)

G(n)

�,

(B42)

with by 5 (bnbmgy)/2 and bt 5 (bnbmgt)/2.It is possible to recast the ventilation coefficients for

each axis so that they depend only on the a-axis length

by using Eqs. (6), (8), and (A5):

fya5 by11 by2Ngy/3

Sh Ngy/2

Re [a21/3* a

1/3(12d*)]gy /2,

fyc 5 by11 by2Ngy/3

Sh Ngy/2

Re [a2/3* a

2/3(d*21)

]gy /2 , (B43)

where all of the above coefficients are given in appendix

A. For consistency, the axis-dependent ventilation co-

efficients are averaged over the a-axis distribution in the

same fashion as the total ventilation coefficient above;

namely,

f ya 5

ðCfya(a)n(a) dað

Cn(a) da

f yc 5

ðCfyc(a)n(a) dað

Cn(a) da

. (B44)

Carrying out the integration and rearranging terms as

was done for Eq. (18) produces

f ya5by11by2 N1/3Sc NRe(an)

1/2gya21/3* a

1/3(12d*)n

gy/2fcap,a,

ihih

f yc5by11by2 N1/3Sc NRe(an)

1/2h ig

ygya2/3* a

2/3(d*21)n

h igy/2fcap,c ,

(B45)

and this is Eq. (19) in the main text. The terms f cap,a and

f cap,c are factors that depend on the capacitance:

f cap,a51

C

�c1a

d1

nG(n1 bya1 d1)

G(n)1 c2a

d2

nG(n1 bya1 d2)

G(n)

�,

f cap,c 51

C

�c1a

d1

nG(n1 byc 1d1)

G(n)1 c2a

d2

nG(n1 byc 1 d2)

G(n)

�.

(B46)

The coefficients bya and byc are defined as

bya5gy2

�bnbm 1

2

3(d*2 1)

�

byc 5gy2

�bnbm 1

1

3(12 d*)

�. (B47)

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