snow studies. part ii: average relationship between …u0758091/hydrometeors/...snow studies. part...

Snow Studies. Part II: Average Relationship between Mass of Snowflakesand Their Terminal Fall Velocity

WANDA SZYRMER AND ISZTAR ZAWADZKI

Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

(Manuscript received 20 November 2009, in final form 25 March 2010)

ABSTRACT

This study uses a dataset of low-density snow aggregates measurements collected by a ground-based optical

disdrometer that provides particle size and terminal fall speed for each size interval from which the velocity–

size and area ratio–size relationships can be derived. From these relationships and relations between the Best

and Reynolds numbers proposed in the literature, the mass power-law coefficients are obtained. Then, an

approximate average relation between the coefficients in the experimentally determined velocity–size power

law (with exponent fixed at 0.18) and the coefficients in the estimated mass power law (with exponent fixed at

2) is obtained. The validation of the retrieved relation is made by comparing, for each snowfall event, the time

series of the reflectivity factor calculated from the derived mass–size relationship for a snowflake and from the

size distribution measured by the optical disdrometer, with the reflectivity obtained from measurements.

Using the measured snow size distribution and the retrieved mass–velocity relationship, a few useful relations

between the bulk quantities of snow are derived. This study considers relations suitable for the microphysical

modeling consistent with radar measurements of precipitating snow composed of unrimed or lightly rimed

aggregate snowflakes.

1. Introduction

The retrieval of microphysics of precipitating snow

from Doppler radar and other remote sensing measure-

ments, as well as snow microphysical parameterizations,

requires knowledge of the form of the size distribution

that allows an accurate derivation of the distribution

moments that are important for descriptions of micro-

physical processes. Moreover, characteristics of individual

snowflakes such as representative dimensional relations of

mass and velocity are needed. The study of Woods et al.

(2007) demonstrated an important sensitivity of the pre-

cipitation redistribution on the changes in both mass and

velocity dimensional relationships for snow in bulk mi-

crophysical schemes.

In Zawadzki et al. (2010, henceforth Part I), the study

of the variability of the snowflake fall velocity, the envi-

ronmental parameters controlling this variability, and the

uncertainties related to the velocity measurement have

been presented. In this work, based on hydrodynamic

theory and the results presented in Part I, we derive an

approximate relation between the mass and terminal

velocity of snowflake.

According to observational, laboratory, and theoretical

studies, mass has a major effect on the terminal velocity

of particles. Empirical power-law relations obtained by

Langleben (1954) express the snowflake fall speed in

terms of its melted equivalent diameter representing the

mass; the coefficients are dependent on the aggregate

type. The dependence on the riming intensity related to

the particle density has been introduced to other em-

pirically derived formulas (Kajikawa 1998; Barthazy and

Schefold 2006). Theoretical and laboratory work on the

determination of the terminal velocity as a function of the

particle mass, or density, has been mainly based on hy-

drodynamic theory using parameterized relationships

between the Reynolds number and the Best (or Davies)

number, the latter expressed in terms of mass and effec-

tive cross-sectional area. By inverting this procedure, the

particle mass can be estimated from the observed terminal

velocity (Hanesch 1999; Schefold 2004; Lee et al. 2008).

The same approach is used here as a first step to estimate

the mass relation from optical spectrograph measure-

ments of terminal velocity as a function of snowflake size.

In the next step, we determine the approximate average

Corresponding author address: Wanda Szyrmer, Department of

Atmospheric and Oceanic Sciences, McGill University, 805 Sher-

brooke St. West, Montreal QC H3A 2K6, Canada.

E-mail: [email protected]

OCTOBER 2010 S Z Y R M E R A N D Z A W A D Z K I 3319

DOI: 10.1175/2010JAS3390.1

� 2010 American Meteorological Society

relation between the experimentally obtained velocity–

size relationship and the estimated mass–size relationship.

In this way we obtain a consistent parameterization of

velocity–size and mass–size relationships.

The cross-sectional area included in the calculation of

the mass–velocity relation based on the Best number X

and Reynolds number Re (X–Re) relationship is ex-

pected to be related the snowflake effective density as

suggested by Cunningham (1978). More recently, an em-

pirical average expression relating particle size, density,

and effective area has been proposed by Heymsfield et al.

(2004).

Two types of uncertainties contribute to the uncertainty

of the derived relations. The first type represents fluctu-

ations in the measured data, such as the velocity or the

area ratio for a given size category, used as an important

starting point for the calculations. The second type of

uncertainty arises from the fact that the theoretical for-

mulas used for the derivation of the resulting relation

are not well known. These two types of uncertainties

are combined when investigating the uncertainties of the

derived relations.

The measurements that we use are from an optical dis-

drometer, which can give information not only on velocity

and area for each size bin of snowflake but also on snow-

flake particle size distribution (PSD). Measured PSDs

were used to evaluate the retrieved relation through the

comparison of the expected and a measured radar reflec-

tivity time series and also to derive some average relations

between PSD bulk quantities.

We begin with the description of the experimental data

used as the starting point of our calculations. In section

3, the theoretical basis and the main assumptions of our

method to retrieve mass–velocity relations are described;

the calculation results and uncertainties estimation are

done in section 4. Section 5 is devoted to the validation of

the proposed relation. Some useful applications can be

found in section 6. Finally, section 7 is a summary of the

results and a discussion of some of the limitations.

2. Experimental data used in the study

In this study we use a large dataset of snow measure-

ments collected by a ground-based optical disdrometer

[Hydrometeor Velocity and Shape Detector (HVSD)]

(Barthazy et al. 2004). Measurements were taken at the

Centre for Atmospheric Research Experiments (CARE)

site (80 km north of Toronto) during winter 2005/06 as

part of the Canadian CloudSat–Cloud-Aerosol Lidar and

Infrared Pathfinder Satellite Observations (CALIPSO)

Validation Project (C3VP) described by Hudak et al.

(2006). The HVSD measurements provide particle size

and terminal fall speed for each size class [assuming

negligible vertical air velocity at the height of the HVSD

measurement]. The study in Part I gives a detailed de-

scription of the measurements and investigates the

variability in the velocity–size power-law coefficients. It

shows that three environmental parameters—surface

temperature, echo top temperature and, in particular,

the depth of the precipitation system—determine the

fall velocity fairly well. These factors express the impact

of different active microphysical processes and of the

time of growth.

In this study, nine snowfall events have been selected

from the same database as in Part I by inspecting the

vertically pointing X-band radar (VertiX) records and

retaining only the snow systems uniform in time (for

more details, see Part I). An additional requirement for

the present study was the availability of well-sampled

time series of the particle size distributions measured by

the HVSD. For each snow event and each size bin, the

mean values of velocity and area ratio, as well as their

standard deviations, were obtained from Part I (see Table 1

for the summary of the analyzed events). We consider

that during each event we have the same dominant type

of particles characterized by the single mass and area

relations. Values of the measured velocity and area ratio

that deviate by more than 2 standard deviations from the

average value for a given size class by the obtained re-

lations were discarded as outliers.

The retrieved mass–velocity relationship is evaluated

by comparing the time series of the reflectivity factor

calculated from the mass–size relationship applied to the

size distribution measured by the HVSD to the reflec-

tivity obtained from the collocated small X-band bistatic

Doppler radar providing measurements a few meters

above ground [Precipitation Occurrence Sensor System

(POSS); Sheppard 1990; Sheppard and Joe 2000]. The

HVSD and POSS data are averaged over 6-min periods.

The tests of the sensitivity of the results to the time av-

eraging of the HVSD and POSS data have been done.

Only very small differences in the results have been

obtained for different averaging periods. The total

number of analyzed spectra is 805.

The temperature at the ground during these events

varied between 2178 and 228C. The observed reflec-

tivity factor varied between about 210 and 30 dBZ.

3. Theoretical basis and main assumptions

a. Functional forms of the used dimensionalrelationships

The form of a terminal velocity–size and mass–size re-

lationships adopted here is the most common power law:

u(D) 5 auDb

u , (1a)

3320 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67

m(D) 5 am

Dbm , (1b)

where D represents a reference size of snowflake.

The coefficients au and bu have been experimentally

derived separately for each snow event. The range of the

obtained au, bu values underlines the variability between

the events as shown in Part I. For a given snow event, the

derived au and bu, are assumed to be size independent.

However, analytical relations such as those developed

by Mitchell and Heymsfield (2005, henceforth MH05)

or Khvorostyanov and Curry (2005, henceforth KC05)

describe au and bu for different size ranges. These vari-

ations with size are not very important for sizes that

mostly contribute to the total mass for a given particle type

having the same mass and area dimensional relationships.

The calculated mean mass-weighted fall speed with con-

stant coefficients used to characterize the particle sizes,

where the major part of the mass is located, is only slightly

different from the calculated fall speed with size-range-

dependent coefficients. The calculations performed by

McFarquhar and Black (2004) for different particle types

also lead to the above conclusion.

The parameters am, bm in the mass–size relation (1b)

are, in general, empirically derived from observations

for different types of particles of different size ranges

(e.g., Mitchell et al. 1990) and at different temperatures

(Heymsfield et al. 2007). Our work aims at evaluating

the coefficients am, bm and relating them to the au, bu

coefficients in (1a) in the range of snowflake sizes that

dominate the total snow mass or reflectivity factor.

Moreover, application of the hydrodynamic theory

requires calculation of the cross-sectional area projected

normally to the flow. In general, the cross-sectional area

is directly related to the area ratio Ar, defined as the area

of the particle image A divided by that of a circle of

diameter D:

Ar[ A/(0.25pD2) 5 (D

eq/D)2, (2)

where Deq is area equivalent diameter. Two forms have

been proposed for the parameterization of Ar for the

side-view particle images, depending on the snowfall

event:

Ar(D) 5 a

r[exp(�b

rD) � 1], (3a)

Ar(D) 5 a

reDbre , (3b)

TABLE 1. Summary of the characteristics of the analyzed events. All parameters are given in CGS units; times are in UTC. Solid symbols

indicate events for which surface air temperature was above 2108C. The associated symbols are for further reference.

Key

Events (mean surface

temperature in 8C) au bu auf (bu 5 0.18) am (31023) bm

amf from (229)

(31023)

amf from (23)

(31023) Damf /amf

) 24 Nov 2005 120.1 0.15 122.0 4.99 1.87 5.71 5.87 0.24

2000–2120

(211.5)d 6 Jan 2006 82.0 0.22 81.1 3.20 2.07 3.11 3.19 0.13

2030–2220

(29.0)

. 9 Jan 2006 147.9 0.17 147.8 8.37 1.99 8.37 8.61 0.29

0800–0825

(22.0)

r 6 Feb 2006 96.6 0.14 98.4 4.10 1.91 4.30 4.13 0.17

1730–1830

(25.5)

m 11 Feb 2006 94.3 0.13 97.3 3.84 1.90 4.14 4.06 0.19

0000–0300

(29.5)

D 18 Feb 2006 110.0 0.19 114.6 4.67 1.92 5.28 5.25 0.28

1710–1810

(215.5)

u 28 Feb 2006 99.8 0.20 94.9 3.33 1.87 3.87 3.92 0.22

0010–0100

(212.0)

$ 28 Feb 2006 99.2 0.18 99.7 3.20 1.85 3.97 4.21 0.24

0430–0630

(213.0)

s 28 Feb 2006 91.1 0.14 91.5 3.43 1.88 3.71 3.73 0.17

1030–1230

(214.5)


where ar, br, are, and bre are constants determined em-

pirically. The form selected to describe each snow event

is the one with a greater correlation.

For nonspherical particles with complex structures such

as ice crystals or snowflakes, the definition of size D is an

issue in itself. This size is in general determined from

particle image recorded with two-dimensional imaging

probes; because it is noncircular, there are different pos-

sibilities to define the image size. Moreover, depending on

the observation method, the image may refer to the side

view when the image is projected on the vertical plane

parallel to the flow (e.g., Barthazy et al. 2004) or the image

projected on the horizontal plane normal to the flow (e.g.,

Mitchell et al. 1990) In investigations reported in the lit-

erature the dimension D has been considered in various

ways—for example, as the diameter of an area-equivalent

circle (i.e., the diameter that corresponds to the area as the

particle shadow; Locatelli and Hobbs 1974; Francis et al.

1998), the equivalent spherical volume diameter (Brandes

et al. 2007), the maximum diameter of the particle image

(Kajikawa 1989; Mitchell et al. 1990), the mean of the two

orthogonal extensions (Brown and Francis 1995), and the

width of the enclosing box (Barthazy et al. 2004). For the

mass relation a new approach is proposed by Baker and

Lawson (2006), who introduce a size parameter that is a

combination of the maximum dimension, width, area, and

perimeter. A combination of parameters is also used by

Hanesch (1999) in her study of the snowflake fall speed. In

addition, the equivalent melted diameter is considered as

a snowflake size parameter.

In the data collected by the HVSD the observed

snowflake dimensions correspond to the two-dimensional

side-view pattern. As a reference, the dimension D is

chosen to be the maximum side-view size—that is, the

maximum of the two perpendicular extensions: height of

the image (vertical dimension as the snowflake falls) and

width of the image (see Part I). This definition of the

snowflake reference size is used in our dimensional re-

lationships obtained from measurements of the velocity

and area ratio and retrieved for mass. The dimensions

normal to the flow required for the hydrodynamic calcu-

lations have to be estimated from the side-view projection.

b. Introduction of the Reynolds and Best (Davies)numbers

The general expression for terminal fall speed is given

by equating the drag force FD with the difference be-

tween the gravitational force mg and the buoyancy force.

However, the problem with applying this expression is

the dependence of the drag coefficient CD on the fall

speed itself. Thus, the common method for calculating

the terminal velocity is from a determined relationship

between the Reynolds number Re and the Best number

X related to the drag coefficient but having no de-

pendence on fall speed. The two numbers are defined as

(e.g., List and Schemenauer 1970)

Re 5uD*

nand (4)

X 5 CD

Re2 52g(m/r

snow)(r

snow� r

a)

rau2A?

uD*n

� �2

;

taking rsnow 2 ra ’ rsnow, the following expression for

X is obtained:

X 52g

ran2

mD2*

A?. (5)

The index ? describes the quantities related to the hori-

zontal projection (perpendicular to the flow), while the

index jj (introduced in Eq. 7a below) represents the

projection on the vertical plane (parallel to the flow).

The environmental conditions are included via the ki-

nematic viscosity and air density, n and ra, respectively; g

denotes gravitational acceleration. The expression for X

contains the two primary variables that determine u (i.e.,

the mass m and the effective particle area projected

normal to the flow A?, in addition to D*, which denotes

the chosen characteristic dimension of the particle). In-

troducing the area ratio normal to the flow, Ar?, and

using (2), X can be expressed as

X 58g

pran2

m

Ar?

D*D?

� �2

, (6)

where D? denotes the maximum diameter in the di-

rection normal to the flow that is not measurable and

therefore must be estimated. The final expression for X

depends on the choice of the characteristic size D*. The

symbols Xjj and X? below describe X calculated with

D*

5 D (our measured maximum side-view diameter

taken as reference) and D*

5 D? (estimated maximum

diameter normal to the flow), respectively:

X jj58g

pran2

m

Ar?

D

D?

� �2

, (7a)

X?58g

pran2

m

Ar?

. (7b)

The corresponding Re numbers are given by

Rejj5uD

nand (8a)


Re?5uD?

n. (8b)

Assuming an idealized spheroidal shape for the snow-

flakes and taking D*

equal to the maximum diameter

projected on the flow D?, Bohm (1992) modified the

general expression for X (7b) by introducing the fourth

root of the inverse of area ratio and the axial ratio f of

this assumed oblate spheroid shape of snowflake:

X?52g

ran2

m

(Ar?)1/4 max(f, 1)

. (9)

c. Relations between Reynolds and Best numbers

The equation relating the X to the Re numbers can be

obtained empirically from direct measurements of par-

ticle velocity, cross-sectional area, and size (e.g., Knight

and Heymsfield 1983; Heymsfield and Kajikawa 1987;

Redder and Fukuta 1991). However, important uncer-

tainties are associated with these measurements, mainly

those of the cross-sectional area that is normal to the

flow.

A generalized theoretical relationship between Re

and X derived from the boundary layer theory was de-

veloped by Bohm (1989) and Mitchell (1996) on the

basis of the previous work of Abraham (1970), giving

Re 5d2

0

41 1

4X1/2

d20C1/2

0

!1/2

� 1

24

35

2

. (10)

The values of the two constants d0 and C0 characterizing

the boundary layer shape and thickness are 5.83 and 0.6,

respectively, for ice particles. For aggregates, MH05

modified slightly the relation by adding an empirical term

�a0Xb0 with a0 5 0.0017, b0 5 0.8. This additional term

accounts for turbulent flow around larger aggregates so

that there is better agreement with the measurements

presented by Heymsfield et al. (2002b). KC05 proposed

an alternate method to introduce this correction.

In this study, the particle mass is derived from the

Best number that is calculated from Re. Therefore, we

invert the proposed relations Re 5 f(X) from MH05

and KC05 by fitting log(X) to an eighth-order polynomial

of log(Re):

log(X) 5 �8

l51C

l[log(Re)]l, (11)

with Cl 5 CMH,l for the MH05 relation, and Cl 5 CKC,l

for the KC05 relation. The coefficients Cl are given in

Table 2. The X–Re relations given in (11) are applied for

the pairs (Xjj, Rejj) and (X?, Re?). The differences be-

tween the values of X retrieved using different X–Re

relationships are evident for the larger Re, which de-

scribe the snowflakes that have a larger contribution to

the reflectivity values, as seen in Fig. 1, which gives a

good estimate of the uncertainty associated with the re-

lationship X–Re.

d. Assumed relations between side-view snowflakeprojection and the section perpendicular to the flow

Snowflake size and area measured by the HVSD rep-

resent a side view of the particle. Because of aerody-

namic forcing, the falling snowflakes have in general

their maximum size oriented horizontally (e.g., Magono

and Nakamura 1965). On average, their horizontal di-

mension is larger than the value measured when viewed

on side projection. Since the horizontal (normal to flow)

dimensions that are required for the velocity calcula-

tions cannot be measured, they must be estimated from

the side projection. For this, 1) we assume that Ar?’ Ar

(i.e., the area ratio is independent of the angle of ob-

servation, normal or parallel to the flow) and 2) we use

the relation from Schefold (2004) giving the ratio of the

area projected normally to the flow to the area from the

side view, given as a function of the canting angle a and

the side projected axial ratio �. The latter is the quotient

of the side projected minor axis to the side projected

major axis. Thus,

fA

[A?A

’ «min

1 (1� «)1

«min

a

908� 1

� �1 1

� �, (12)

where �min describes the minimal value of the axis ratio

evaluated from the measurements as equal to 0.3. The last

relation together with the assumption Ar? ’ Ar is also

used to calculate the maximum horizontal dimension D?from the maximum side dimension D taken as the ref-

erence dimension:

TABLE 2. Coefficients in (11).

Cl 1 2 3 4 5 6 7 8

MH 3.823 3 21.5211 0.300 65 20.061 04 0.130 74 20.073 429 0.0160 06 20.001 248 3

KC 3.881 6 21.4579 0.277 49 20.415 21 0.576 83 20.292 20 0.064 67 20.005 340 5


D?’ Dffiffiffiffiffiffif

A

q. (13)

Equations (12) and (13) give the ratio D/D? decreasing

with D on average. For smaller sizes the mean value of

this ratio is around 0.85 and decreases progressively.

Since the value of 0.8 is consistent with the results from

the snowflake fractal analysis by Schmitt and Heymsfield

(2010), the minimal value of 0.75 is imposed on this ratio

and the corresponding maximal value on fA as calculated

in (12).

Our first assumption about the area ratio being in-

dependent on the projection plane seems reasonable for

snow aggregates for which the value of area ratio has been

shown to be related to particle density (e.g., Heymsfield

et al. 2004). Moreover, this postulate agrees with the re-

sults of the recent theoretical work on the fractal di-

mensions of aggregates (Schmitt and Heymsfield 2010).

4. Derivation of the mass–velocity relationship

An average relation between a snowflake mass and

velocity has been derived in the following steps with the

first four steps calculated separately for each snow event:

(i) Calculation of Re from the HVSD measurements

of terminal velocity and area ratio from (8a) or (8b)

and (13);

(ii) Estimation of the value of X from Re using a poly-

nomial fit (11) with constants CMH,l for the MH05

relation, and with CKC,l for the KC05 relation;

(iii) Best estimation of mass for D-size snowflake from

the estimated X and area ratio normal to the flow;

(iv) Determination by regression of the coefficients in

the power-law relation m–D; and

(v) From all events, derivation by regression of an av-

erage relation between the power-law coefficients in

the mass and velocity dimensional relationships.

In what follows steps iii–v are described in some detail.

a. Best estimation of mass for D-size snowflake andits uncertainty

Using the value of X estimated from Re, and with the

help of the relations (12) and (13) and the approxima-

tion Ar?’ Ar, the following three equations for mass are

obtained from (7a), (7b), and (9):

m 5pr

an2

8gX jjAr

fA

, (14a)

m 5p r

an2

8gX?A

r, (14b)

m 5r

an2

2gX?(A

r)1/4

f. (14c)

For each size bin, using (14a)–(14c) 24 different values of

mass, mk with k 5 1, . . . , 24, are calculated from different

combinations of the X–Re relationships, smoothed and

unsmoothed measured u and Ar, and two values of axial

ratio of 0.7 and 0.8 put in (14c). Each value of mass is

considered as the ‘‘measured’’ value of mass using dif-

ferent methods and interrelated through the common

parameters. The geometric weighted mean is taken as the

best estimate of the snowflake mass with reference di-

ameter D:

log[ ~m(D)] 5 �nk524

k51

log(mk)

(D logmk)2

,�

nk524

k51

1

(D logmk)2

. (15)

The weight of each measured mass is the inverse of the

square of the corresponding uncertainty (D logmk)2 ob-

tained by combining the contribution of the uncertainty

related to the different quantities involved. To unify the

calculation of (D logmk)2, we take the following single

logarithmic form of Eqs. (14a)–(14c):

log(mk) 5 log(C

E) 1 d

alog(X jj) 1 d

b,clog(X?)

1 (1� 0.75dc) log(A

r) 1 d

alog(f

A)

1 dc

log(f), (16)

FIG. 1. Comparison of different relations between the Reynolds

number (Re) and the Best (Davies) number (X). The dashed and

dotted–dashed lines show the values of XMH and XKC calculated

according to the relations proposed by MH05 and KC05 divided by

the value of XB obtained from the relation (10) developed by Bohm

(1989) as a function of Re.


with da

5 1 if (14a), otherwise 0,

dc5 1 if (14c), otherwise 0, and

db,c

5 1 if (14b) or (14c), otherwise 0.

Here CE is a parameter describing the environmental

conditions, and is different by a factor of 4/p in (14c)

relative to (14a) and (14b). The uncertainty of CE will be

neglected here.

In general, the uncertainty related to the calculated X

value can be decomposed into two terms. The first rep-

resents the uncertainty of the used X–Re relationships

(11) for different coefficients Cl. The second describes

the contribution to the uncertainty of the computed Re

value:

(D logX)2’ (D logX)2

Cl1 (D logX)2

Re. (17)

Assuming that the uncertainty in the X–Re relation is

30% (as in Field et al. 2008), we have (D logX)2Cl

’ 0.017.

The same potential bias of log(X) is assumed for X cal-

culated via (10). The second term on the rhs of (17) is

evaluated from (11) as

(D logX)2Re 5 �

8

l51lC

l[log(Re)]l�1

8<:

9=;

2

(D logRe)2. (18)

The term in the f�g parentheses varies slightly with the

value of Re but may be approximated by its mean value

of about 1.8. The uncertainty in log(Re) is different for

Rejj and Re?. Their values can be estimated from (8a)

and (8b):

(D logRejj)2

5 (D logu)21 (D logD)2, (19a)

(D logRe?)25 (D logu)2

1 (D logD)21 (0.5D log f

A)2.

(19b)

Combining the relations (16)–(19), the uncertainty in

log(mk) can be written as

(D logmk)2

’ (D logX)C2

i 1 (D logu)21 (D logD)2

1 db,c

(0.5D log fA

)21 (1� 0.75d

c)2

3 (D logAr)2

1 dc(D logf)2. (20)

The relations (20) and also (17) and (19) assume that for

a single mk there is no correlation between the errors in

the variables used in the calculation of mk.

The uncertainties in u and Ar can be estimated from

the measurement standard deviation (Part I). Since we

are interested in the medium-size and larger snowflakes

that mostly contribute to the reflectivity, the estimated

relative errors are equal to about 20% for both Du/u and

DAr/Ar. Note that these error values are incorrect for

snowflakes smaller than about 3–4 mm. The potential

systematic bias in the axial ratio and in the calculated

value of fA is difficult to estimate; a 630% range is

chosen. The error in the measurement of D is neglected,

in which case D logD ’ 0. Table 3 contains the summary

of the estimated uncertainties used in (20).

With the assumed estimates for the different terms in

(20) we obtained values of (D logmk)2 equal to 0.0321,

0.0363, and 0.0462 for the masses estimated from (14a),

(14b), and (14c), respectively. This means that the rel-

ative error in our mass estimate is between 40% and

50% and is the same for all sizes D since the relative

uncertainties were taken as constant.

To evaluate the uncertainty included in the mass es-

timate log[ ~m(D)] given by (15), the effect of the existing

correlations between the many components of every

couple of mk has to be taken into account. To do this, the

uncertainty of log(mk) has to be separated into uncor-

related and fully correlated uncertainties for each com-

bination of two values of mk. Then the propagation error

law has to be applied. To avoid these complex calcula-

tions we only estimate the maximum uncertainty for

which all mk are fully correlated between them. In this

case, the total uncertainty is simply the linear sum of the

weighted uncertainties of an individual mk:

D(log ~m) 5 (1/Wtot

) �nk

k51(1/D logm

k), (21)

with Wtot

[ �nk

k511/(D logmk)2. The value of D(log ~m)

computed from (21) provides the average uncertainty

of mk calculated from the relations (20) and is equal to

0.193.

b. Determination by regression of the coefficientsin the power-law relation m–D

Assuming that the calculated value of ~m(D) repre-

sents the best evaluation of the snowflake mass with

reference diameter D, we look for a dimensional re-

lation m–D of the power-law form given by (1b) for a

snowflake of a given size D for each snow event. Taking

the logarithm of (1b) yields the linear equation

TABLE 3. Assumed range of uncertainty (%) for the variables used

to estimate the m–D relationship in (20).

X–Re relation fA f u Ar

30 30 30 20 20


log( ~m) 5 log(am

) 1 bm

log(D). (22)

The best estimate of the two coefficients is obtained by

a least squares fit. Two examples are shown in Fig. 2,

where the solid line shows the best-fit power law. For the

nine snow events studied, the obtained coefficients am

and bm and their values are given in Table 1 and shown

in Fig. 3. For comparison, the mean value obtained by

Brandes et al. (2007) from a very large dataset is also

superposed. However, because D in their study was

taken as equivalent diameter, their relation has been

recalculated using the average relation Deq–D obtained

from our data: Deq 5 0.69D0.95 in CGS units (very close

to the average relation in Part I: 0.71D0.92 obtained from

the larger dataset). Moreover, the sets of the parame-

ters in the relations of mass and maximum size are also

superposed after recalculation using our relation (13)

between maximum side-view size and maximum hori-

zontal dimension, taking the value for larger particles

of 1/ffiffiffiffiffiffif

A

p5 0.75. The relations shown are the aver-

age mass relation from Mitchell et al. (1990) for the

1986–87 field season, from Kingsmill et al. [2004; based

on the data by Heymsfield et al. (2002b)], and from

Matrosov and Heymsfield (2008) obtained for precip-

itating clouds.

For some of the cases studied, the investigation of the

regression that gives the best set (am, bm) shows that a

better description of the m–D relationship is obtained

when the analysis is performed separately for two size

regimes: for D smaller than 2–3 mm, and for D larger

than 2–3 mm. This separation can be important in all

microphysical studies where the lower-order moments

of the size distribution become important.

As shown in Fig. 3, the obtained exponents bm through

the mass power law for different snow events cluster be-

tween values of 1.8 and 2.10, with the mean value close to

1.9. The value of 2.0 corresponds to an effective snow-

flake density decreasing with size as D21 and is in good

agreement with numerous observational studies of snow

at the surface (Mitchell et al. 1990 from the overall ob-

served particle types; Brandes et al. 2007) and aircraft

observations (Heymsfield et al. 2002a). Moreover, most

of the snowfall mass reaching the ground is generally

associated with aggregates of ice crystals and theoretical

studies (Westbrook et al. 2004) predict a value of 2 for the

exponent in the mass–dimension power law for snow

aggregates. Certainly, individual crystals with forms other

than aggregates may be less well represented by this ex-

ponent; however, they contribute only a small fraction of

the overall snowfall characteristics. On the other hand, an

exponent equal to 2 probably does not describe correctly

the snowflakes larger than about 1.5 cm because of the

rather small number of the observed particles with these

sizes. Contribution of these particles to the reflectivity is

significantly reduced because of the non-Rayleigh effect.

For example, for a snowflake with diameter of 2 cm, the

reduction of backscattering cross section with respect to

Rayleigh regime is about two orders of magnitude, as

shown from the results of the T-matrix calculations in

Matrosov et al. (2009).

Taking into account the general uncertainty in the

mass–size relationship, in our attempt to develop a rel-

atively simple snow parameterization we set the expo-

nent bm to 2 for the power-law relation representing the

precipitating snow. The relation (22) becomes

log( ~m) 5 log(amf

) 1 2 log(D), (229)

FIG. 2. Examples of the calculated mass–size relationship for two

snowfall events. The size D on the x axis represents the snowflake

side-view maximal extension. The plus signs correspond to the

values obtained from different combinations of the relations (14a)–

(14c). The diamonds represent the values obtained from (15)

considered as the best estimate of snowflake mass of size D. Their

estimated uncertainty is shown by the error bars. Solid and dashed

lines represent the least squares regression with the values of bm

from fitting and fixed at 2, respectively.


with only the coefficient amf to be determined in the

mass–size relationship (the symbol amf denotes am with

fixed bm 5 2). All symbols used here are listed in Table 4.

On the other hand, the velocity power laws of the form

(1a) obtained by Part I for different homogenous snow

events show the relatively small variability of the ex-

ponent bu. They showed that the snowflake velocity can

be modeled with good accuracy with a fixed exponent

bu5 0.18 and a varying coefficient au. This fixed value

of bu is adopted here for further calculations and the

corresponding value of au is represented by auf. Sub-

sequently, all the calculations that lead to the best set of

mass–size relationship based on the measured snowflake

velocities and area ratios are repeated but with an im-

posed value of bu 5 0.18 in the fitted velocity–size re-

lationship and bm 5 2 in the final mass–size power law.

In Table 1 the obtained values of amf and auf are given.

In Fig. 2 the dashed line show the result with imposed

bm 5 2 using (229), while the solid line has been obtained

through regression with no restriction on the bm value

(22). The difference is within the bounds of uncertainty.

To evaluate the impact of imposing the value of 2 for

bm we calculate, from the observed PSD for the time

series of snow events, the snow ice mass content (IWC)

in two ways: first, from the (am, bm) set that was ob-

tained by regression with bm varying, named IWCam,

and second, using the amf value obtained for bm 5 2,

named IWCamf. The root-mean-square error RMSE 5

h(IWCam 2 IWCamf)2i1/2 5 0.004 g m23 while the root

mean of fractional error RMFE 5 h[(IWCam 2 IWCamf)/

IWCam]2i1/2 5 0.07 for the entire length of the time se-

ries. The same calculation performed for the reflectivity

factor gives an RMSE value of 0.35 dB; for reflectivity-

weighted velocity (using fixed bu 5 0.18), the calculated

RMSE and RMFE are 0.006 m s21 and 0.007, respec-

tively. The particles’ radar backscatter cross section is

calculated based on the Mie theory assuming the mixing

rule of Maxwell–Garnett for ice–air mixture [for more

details, see model 5 in Fabry and Szyrmer (1999)]. We

discuss the inaccuracy of the scattering calculations

arising from the Mie calculations in section 5.

The estimates of the uncertainties in the coefficient

amf and in the value of m predicted by (229) for each D

must take into account the effect of the correlation be-

tween calculated values of ~m resulting from the contri-

bution to their uncertainty of the correlated components

of mk. With the assumption of a positive correlation, the

magnitude of the uncertainties rises and the analytical

calculations become very complex. To simplify the cal-

culations we use the uncertainty estimate for each log ~m

given by the error bars in Fig. 2. By imposing a slope

equal to 2, the uncertainty in amf is estimated by finding

the range of intercept values for which the regression

line covers most of the error bars representing D log ~mð Þ.The obtained relative uncertainties in amf, denoted by

Damf /amf in the last column of Table 1 for each snow

event, are included in the regression calculation of the

approximate relation between amf and auf in (23) below

and shown by error bars in Fig. 4.

c. Derivation of average mass–velocity relationship

To reduce the number of parameters describing an in-

dividual snowflake with size D, we search for an approxi-

mate relation between its mass and velocity, assuming that

other factors (such as the shape of the individual crystals)

introduce negligible correction to the average mass–

velocity relationship. This relation is obtained for the mass

and velocity exponents fixed at 2.0 and 0.18, respectively.

The atmospheric conditions have some influence on

the snowflake mass calculated from (14a)–(14c), and

hence on the value of amf derived from experimental

values of auf. This influence is mainly through the kine-

matic viscosity of the air n, which is equal to the dynamic

viscosity, a function of temperature, divided by the air

density. The air density variation can be neglected here

because all the observations were made at the ground

and to a first approximation the air density changes with

height (the temperature correction is evaluated at less

than 2% with respect to the average temperature of

2108C). Besides the explicit dependence of m on the

FIG. 3. Estimated parameters in the mass–size power law for the

nine snow events analyzed (symbols are listed in Table 1). For

comparison, the sets of the relation parameters presented by

Mitchell et al. (1990) (average relation), Kingsmill et al. (2004)

[based on the data by Heymsfield et al. (2002b)], Brandes et al.

(2007), and Matrosov and Heymsfield (2008) are also plotted. Our

average relation between D and Deq was used to recalculate the

parameters from Brandes et al. since their relation is in terms of

Deq. The remaining relations that are shown were recalculated

using (13) for larger particles with 1/ffiffiffiffiffiffif A

p5 0.75.


magnitude of n given in (14), the values of the calculated

Re and X also change with n. Therefore, prior to the

search of the mass–velocity relationship, the influence

of temperature variations on the estimated value of amf

has to be investigated.

The snow measurements considered here were done at

temperatures between 228 and 2178C. The range in

dynamic viscosity corresponding to this temperature

range is from 1.72 3 1025 to 1.63 3 1025 kg m21 s21. The

correction multiplication factor that normalizes the esti-

mated mass, and therefore amf, was calculated with re-

spect to calculations done for an average value of 2108C.

The obtained values of the correction factor are 1.027 and

0.983 at 228 and 2178C, respectively. This range is very

TABLE 4. List of symbols.

A Area of the particle image (here side-view image)

A? Cross-sectional area projected normal to the flow

am Constant in mass–size power law (1b)

amf Constant in mass–size power law (1b) with imposed bm 5 2

Ar Ratio of the area of the particle image to the circle of diameter D

Ar? Ratio of A? to (p/4)D?

2

ar Constant in area ratio–size relation (3a)

are Constant in area ratio–size relation (3b)

au Constant in velocity–size relation (1a)

auf Constant in velocity–size relation (1a) with imposed bu 5 0.18

bm Exponent in mass–size power law (1b)

br Constant in area ratio–size relation (3a)

bre Exponent in area ratio–size relation (3b)

bu Exponent in velocity–size relation (1a)

CD Drag coefficient

CE Parameter describing the environmental conditions in (16)

Cl Coefficients in the polynomial fitting in (11) to the relation Re–X from MH05 (CMH,l) and from KC05 (CKC,l),

given in Table 2

C0 Constant in the relation Re–X (10) from Bohm (1989) and Mitchell (1996)

D Snowflake reference size (taken here as maximum side view diameter)

Deq Area equivalent diameter

D? Estimated maximum diameter in the horizontal plane (i.e., normal to the flow)

D*

Characteristic dimension of the particle (in general)

fA Ratio of the effective area projected normally to the flow to the area from the side view

g Gravitational acceleration

IWC Snow ice water content

IWCam IWC calculated using the derived am and bm in (1b)

IWCamf IWC calculated using the derived amf with imposed bm 5 2 in (1b)

m Particle mass

Re Reynolds number

Rejj Reynolds number calculated with D (maximum side-view diameter)

Re? Reynolds number calculated with D?S Liquid equivalent snowfall precipitation rate

u Particle terminal velocity

UZ Mean reflectivity-weighted fall speed

UM Mean mass-weighted fall speed

X Best number

Xjj Best number calculated with D (maximum side view diameter)

X? Best number calculated with D?Ze Equivalent reflectivity factor (mm6 m23)

Ze,dBZ Equivalent reflectivity factor (dBZ)

a Canting angle from the side projection

x Coefficient in Z–S power law

do Constant in the relation X–Re–(10) from Bohm (1989) and Mitchell (1996)

« Side projected axial ratio

«min Minimal value of «

n Kinematic viscosity of air

ra Air density

ra0 Air density at the ground level

rsnow Density of snowflakes

f Axial ratio of snowflakes

v Exponent in Z–S power law


small and therefore suggests that the impact of the at-

mospheric conditions during the measurements on the

derived amf –auf relation can be neglected.

In Fig. 4 the values of amf derived for bm 5 2 and bu 5

0.18 are plotted as a function of the velocity coefficient

auf obtained from the data. Each point represents one of

the nine events. The analytical approximate relation be-

tween amf and auf is assumed to have the form logamf 5

a 1 bauf. Coefficients a and b have been determined

by a weighted total least squares (WTLS) regression

described in Krystek and Anton (2007). For each [auf,

log(amf)] pair, the corresponding uncertainties are Dauf ’

0.2auf from Part I and D(logamf) obtained from regres-

sion (229). These uncertainties are shown in Fig. 4. The

solid line in Fig. 4 represents the least squares fitting with

a 5 23.02 and b 5 0.0065 in CGS units, giving

logamf

5�(3.02 6 0.04) 1 (6.5 6 0.5)10�3auf

. (23)

The uncertainties associated with the two coefficients are

valid when the effect of correlation between the values of

amf is not taken into account. The magnitudes of amf ob-

tained from (23) are shown in Table 1. The weighted av-

erage value of amf for the nine analyzed events is shown in

Fig. 4 by a dotted line. Its value is equal to 0.0044 g cm22

and is very close to the average from observations of

Mitchell et al. (1990) and Kingsmill et al. (2004) recalcu-

lated using (13) as in Fig. 3. The value of auf correspond-

ing to this weighted average is equal to 102 cm0.82 s21.

Using (23), snowflake velocities are calculated as a

function of melted equivalent diameter, and the results

are shown in Fig. 5. In the upper graph, the derived re-

lations between snowflake velocity and melted diameter

for each of the nine events are compared with the em-

pirically obtained relations of Langleben (1954). In the

lower graph we compare snowflake velocity calculated by

our relation (23) for average amf with some other relations

previously published (Locatelli and Hobbs 1974; Kajikawa

1998; Brandes et al. 2008 for velocity at 218 and 258C and

Brandes et al. 2007 for mass relation derived from obser-

vations mostly at temperatures higher than 258).

All presented results are derived from measurements

at the surface. Therefore, when applied at other pressure

levels the velocity coefficients (au, auf) have to be ad-

justed for a change in altitude using, as a first approxi-

mation, the adjustment given in Heymsfield et al. (2007):

[auf(p)/auf(1000 hpa)] 5 [ra0/ra(p)]0.54, where ra0 and

ra(p) are the air density at the ground level and at

pressure p.

FIG. 4. Mean relationship between the coefficients in the mass

and velocity power laws with fixed exponents of bm 5 2 and bu 5

0.18 obtained from WTLS regression. The solid line is the best-fit

relation shown in the figure. The dotted line gives the average value

of amf for all events.

FIG. 5. (top) Solid lines show the derived relations between

snowflake velocity and melted diameter for each of the nine events.

For comparison, we have overlaid the empirical relations obtained

by Langleben (1954) for the different type of snowflakes indicated

by the various dashed lines described in the top of the figure.

(bottom) Comparison of snowflake velocity calculated by our re-

lation (23) for average amf with some other relations published

previously. [LH 1974 indicates Locatelli and Hobbs (1974).]


5. Evaluation

In this study the measured reflectivity is used to eval-

uate the procedure for the retrieval of mass/density as

a function of particle size. For each snowfall event, the

mass coefficient amf is estimated from the velocity co-

efficient auf using (23). Knowing the mass–size relation-

ship, the backscatter cross section of each size bin is

calculated using the Mie theory from model 5 described

in Fabry and Szyrmer (1999). For particles smaller than

about 5 mm, the scattering at X-band is in the Rayleigh

regime (i.e., that the reflectivity is proportional to mass

squared, independent of the particle shape; e.g., Matrosov

et al. 2009). For larger particles the non-Rayleigh effect

leads to the reduced backscattering that decreases with

size for particles larger than about 1.5 cm (Matrosov et al.

2009).

The other question to consider is the effect of non-

sphericity. According to Ishimoto (2008), using the finite

difference time domain (FDTD) method for fractal-

shaped snowflakes, the sensitivity to the particle shape

become important for snowflakes with a diameter of the

ice-mass-equivalent sphere greater than about 2.4 mm

(which corresponds on the average to D 5 1.2 cm). The

same result has been obtained with T-matrix method in

Wang et al. (2005). The inaccuracy of the Mie calculations

assuming spherical form depends on the choice of map-

ping of particle properties into sphere parameters. By

choosing in our model the area-equivalent diameter (i.e.,

between ice-mass-equivalent sphere diameter and maxi-

mum dimension) as diameter of the equivalent sphere,

the inaccuracy of the scattering calculations arising from

the Mie calculations for snowflakes not larger than 2 cm

is relatively low. Moreover, as shown in Heymsfield et al.

(2008), the contribution to the X-band reflectivity of the

particles larger than about 1 cm (assuming the expo-

nential size distribution with slope equal to 6 cm21 that

is representative for our dataset) is negligible. Further-

more, Matrosov et al. (2009) concluded that at X-band

the spherical model for low-density snowflakes provides

the necessary accuracy in the reflectivity calculations if the

non-Rayleigh effects are not too important.

But perhaps the most direct way of considering the

effect of nonsphericity on reflectivity is by noting that

observed values of differential reflectivity in precipitating

snow are a few decibels, indicating that the strong effect

of low density of particles decreases the influence of

shape on backscattering, which confirms the results of

Matrosov et al. (2009).

The expected reflectivity factor is computed from the

snowflake size distributions measured by the HVSD.

The time series of the calculated reflectivity are com-

pared with the reflectivity derived from the collocated

POSS. The HVSD and POSS data are averaged over

6-min intervals. The scatterplot of the reflectivity cal-

culated for all nine events versus the POSS measured

reflectivity is presented in Fig. 6. The root mean stan-

dard error is equal to 2.71 dB.

Figure 7 presents an example of the reflectivity time

series. The solid line gives the POSS-measured re-

flectivity; the dotted–dashed line is the reflectivity cal-

culated from the mass relationship derived for the given

event of 6 January 2006. This particular event is charac-

terized by the lowest retrieved value of amf (see Table 1).

To provide a limit on the importance of the mass–size

relation on the uncertainty in the reflectivity calcula-

tions, the dashed line in Fig. 7 shows the reflectivity

calculated for the PSDs of 6 January 2006 but using the

mass–size relation retrieved for the event of 9 January

2006 where the estimated amf was the greatest (see Table 1).

The relative difference in amf between these two events

is about 1.5. Under the Rayleigh regime, the effective

backscattering cross section is proportional to the square

of the mass. Then, we have DmZe/Ze ’ (1 1 1.5)2 2 1 for

Ze in mm6 m23, giving the contribution of the mass un-

certainty to the calculated dBZ of DmZe,dBZ 5 10 log(1 1

DmZe/Ze) ’ 8 dB, as can be noted in Fig. 7. For a smaller

relative uncertainty in Damf /amf we can write DmZe/Ze ’

2Damf /amf, from which we can deduce that DmZe,dBZ 5

(10/ln10)DmZe/Ze, equal to about 8.7 Damf /amf valid un-

der Rayleigh regime.

FIG. 6. Scatterplot for all events with Ze calculated from the

estimated mass relationship for each event and the time series

of size spectra measured by HVSD vs Ze measured by POSS.

The RMSE is equal to 2.71 dB. Each symbol represents a 6-min

average.


6. Applications

The retrieved relation (23) used in conjunction with

the PSD information makes it possible to derive useful

relations between different bulk properties of snow PSD

such as equivalent reflectivity factor Ze, snow ice water

content (IWC), liquid equivalent snowfall precipitation

rate S, or reflectivity- and mass-weighted velocity UZ

and UM. The time series of the snowflake PSDs are

provided for each analyzed snowfall case by the optical

imager HVSD.

In Fig. 8 the reflectivity-weighted velocity at the

ground level is plotted against the reflectivity as ob-

tained from the calculations for all snow events. The

solid lines are obtained through regression for the fol-

lowing linear relation between UZ and Ze in dBZ and

log(amf):

UZ

5 326.0 1 0.559Ze,dBZ

1 104.3 log(amf

). (24)

Note that UZ and amf are in CGS units. Introducing

a weighted mean value for amf equal to 0.0044 in CGS

units as shown in Fig. 4, the average UZ –Ze relation

becomes

UZ

5 80.2 1 0.559Ze,dBZ

(249)

for precipitating low-density snow. The last term in (24)

introduces the estimated sensitivity of UZ to the error in

mass–dimensional relationship. The uncertainty in UZ

resulting from the uncertainty in the value of amf may be

estimated as DUz (cm s21) ’ 50Damf /amf. The relation

(24) when amf is given is expected to provide a more

accurate UZ–Ze relation compared to (249). For higher

altitudes, the air density correction for UZ has to be in-

troduced as given at the end of section 4. The mean ratio

of calculated reflectivity-weighted velocity to mass-

weighted velocity is equal to 0.93.

The second useful relationship between two bulk

quantities is the power law relating the equivalent

reflectivity factor Ze and the snowfall precipitation rate

S: Ze 5 xSv, with Ze in mm6 m23 and S in mm h21. In

Fig. 9, the calculated snow precipitation rate S is plotted

as a function of the calculated Ze in dBZ. The solid line

shows the best fit obtained by the WTLS method applied

to the logarithmic form of the Ze–S power law. The

uncertainties taken into account for Ze and S in WTLS

calculations describe the contributions from the uncer-

tainties in snowflake fall speed and mass represented by

the uncertainties in auf and amf, and the correlation be-

tween them. The obtained values of the constants are

x 5 494 and v 5 1.44.

Our Ze–S relationship is in a general agreement with

some of relations previously derived empirically, semi-

empirically, or theoretically that are drawn in Fig. 9

[Ohtake and Henmi (1970) for snowflake consisting of

spatial dendrites; Fujiyoshi et al. (1990) for 1-min and

30-min average; Matrosov (1992) for snow density of

0.02 g cm23; Zawadzki et al. (1993) recalculated using

our average relation S–IWC: S 5 3.3IWC1.05, with IWC

FIG. 7. Example of time series plot of Ze. The solid line shows the

POSS measured Ze; the dotted–dashed line gives the Ze calculated

from the HVSD spectra with the mass relation derived for the same

event shown (9 Jan 2006), while the thin dashed line shows the

calculation results for the mass retrieved for a different event on

the same date. The POSS-measured Ze and HVSD spectra are

averaged over a 6-min period.

FIG. 8. Calculated reflectivity-weighted velocity UZ vs calculated

reflectivity Ze for data of all nine events. The solid lines show the

approximate linear relation (24) obtained for each event through

regression between UZ and Ze and the mass coefficient amf. The

RMSE is equal to 4.4 cm s21.


in g m23; and seven relations retrieved by Huang et al.

(2010)]. However, compared with the recent work of

Matrosov et al. (2009), our value of x is much larger for

similar values of the exponent v. This difference may be

partly explained by the applied snow velocity relation-

ship. Our measured velocity for low-density snowflakes

as shown in Part I is in general lower than that obtained

from the relation used in Matrosov’s work.

The increase of snowflake density (i.e., the value of

amf in our parameterization) is related to the increasing

velocity described by auf. For the same snowflake spec-

trum, this increase results in the increase of both Ze

(proportional to a2mf under Rayleigh regime) and S

(proportional to the product amf 3 auf). To estimate the

impact of these changes on the coefficients in the Ze–S

relation, we present in Fig. 10 the calculations of Ze and

S repeated 3 times for the time series of PSDs on

9 January 2006 using 1) amf and auf obtained for this

event (solid circle), 2) these two coefficients retrieved

for a different event, 6 January 2006 (open circle), and 3)

weighted average values of these coefficients as shown in

Fig. 4 (symbol ‘‘X’’). The calculation results presented in

Fig. 10 show that the coefficients of the power law Ze–S

for low-density snowfall do not exhibit a significant de-

pendence on the assumed mass–velocity relationship if

(23) is satisfied. Matrosov et al. (2009) show the slight

dependence of the coefficient x on the assumed mass

dimensional relationship. However, they used the den-

sity independent expression for snowflake velocity. Our

result suggests that for the unrimed snowfall of our

study, the observed spread in the Ze–S power law is

mainly due to the PSDs differences, as shown in Matrosov

et al. (2009), and to a lesser degree to the assumed mass

relation.

7. Summary and discussion

Both modeling and remote sensing studies require the

parameterizations of bulk quantities. Generally for snow

the accuracy of these calculated variables and the relations

between them are dependent on the parameterization of

the PSDs used and on the mass and velocity dimensional

relationships as well. The variability of snowflake velocity

discussed in Part I between different snowfall events is

related here to the variability in the mass dimensional

relationship based on the idea that the reliable estimators

of snowflake velocity are its density together with its size.

Even if only light riming (or no riming) cases were used

here, we see appreciable differences in (23) and all the

derived relationships.

Radar observables provide an alternative verification

of microphysical description of snow. Our ultimate goal

is to derive the microphysical relations that are consis-

tent with radar measurements and suitable for modeling

FIG. 9. Calculated reflectivity Ze as a function of the calculated snow precipitation rate S. The

solid line shows the WTLS fit: Ze 5 494S1.44 (Ze in mm6 m23 and S in mm h21). For comparison,

the relations from different previous studies are overplotted.


and optimized radar data assimilation. We have limited

here our interest to precipitating unrimed or lightly rimed

snowflake aggregates for which we derived the mass–

size–velocity relationship that is a simpler expression of

the general relations of mass, area, size, and velocity

developed in MH05 and KC05. In our relationship the

area is eliminated: the cross-sectional area, related to the

details of the crystal types composing the aggregates, is

closely related to the particle mass, and therefore its di-

rect impact on the snowflake fall speed is reduced. In this

way, in our formulation the variability in the fall speed for

snowflakes with the same size is only attributed to particle

density (or mass).

The main goal of this work was to retrieve an average

relation between unrimed or slightly rimed snowflake

velocity and mass using the observational data and theo-

retical calculations. The retrieval is through the following

five steps: 1) calculation of the Re number from HVSD

measurements of terminal velocity and area ratio for each

snowfall event; 2) estimation of the X number from Re; 3)

retrieval of mass for each size bin from the X–Re re-

lationship and the estimated area ratio normal to the flow;

4) determination by regression of the coefficients in the

power-law relation m–D for each event; and 5) from all

snow events, derivation by regression of an average re-

lation between the power-law coefficients in the mass and

velocity dimensional relationships (setting the exponents

to the fixed values of 2 and 0.18). This relation is repre-

sentative for the snowflake sizes dominating the snow

bulk quantities such as mass content or reflectivity factor.

Given all other uncertainties, the variation of the mass

and velocity coefficients with particle size is not taken into

account. These coefficients and the derived relation are

considered to be representative for the range of sizes that

dominates the radar signal and where the majority of

snowfall mass is located. For very light precipitating

snow with an important contribution to IWC of particles

smaller than about 1–2 mm, the derived relation is not

valid. If applying the relations obtained in this study, it is

important to account for possible differences in the defi-

nition of snowflake reference size, taken here as maxi-

mum side-view dimension.

The evaluation of the results is made by comparing the

time series of the reflectivity factor calculated for a de-

rived mass–size relationship for an individual snowflake

and applied for the size distribution measured by the

HVSD, with reflectivity obtained from the zeroth mo-

ment of the spectrum measured by the collocated POSS.

Using the measurements of snow velocity and retrieved

snowflake mass together with the observed PSDs, dif-

ferent bulk quantities have been calculated and approx-

imate average relations between them are derived. The

sensitivity to the parameter amf or auf, related by (23), is

significant for calculated Ze, S, IWC, UZ, and UM and for

the derived expression UZ–Ze. However, the coefficients

in the power law Ze–S are not affected by the changes in

the interdependent amf and auf and are in a generally

good agreement with the relations obtained from the

literature for the same meteorological conditions. Taking

into account the fact that the lower size detected by the

HVSD is about 0.3 mm, the observed PSDs are truncated

mainly at the lower size limit and the calculated relations

may be not applicable to the light snow precipitation for

which the effect of this truncation can be important.

Because of this interdependence of snowflake mass

and velocity, the coefficients in the mass and velocity

dimensional relationships selected in the microphysics

scheme have to be related to each other to ensure con-

sistency between calculated quantities. The requirement

of the self-consistency between snowflake mass and ve-

locity may be partly satisfied by incorporating our re-

trieved relation (23) into microphysical calculations.

Heavy rimed snow is excluded from our analysis, and

consequently the average relationships derived in this

study may not be valid for this type of snow, as in the

cases of reflectivity higher than about 25–30 dBZ. Our

relationships may as well not be representative of other

geographical conditions. The sensitivity of the proposed

relations to the PSDs parameterization has to be taken

into account when applying to higher levels in the at-

mosphere.

In Part III of our study, we will present the third

important element of the bulk snow microphysics

FIG. 10. As in Fig. 9, but for only the one event of 6 Jan 2006. The

solid circle gives the results obtained with the mass relationship

retrieved for this event. The open circles are obtained using the

mass expression retrieved from a different event (9 Jan 2006). The

‘‘X’’ symbols represent the results obtained using the average

values of amf and auf shown in Fig. 4.


parameterization: the PSD representation. Further study

will be devoted to relating the developed snow parame-

terization to the environmental conditions (see Part I).

Acknowledgments. The authors are grateful to EunSil

Jung for providing data used in this study. The editing of

the manuscript by Aldo Bellon is greatly appreciated.

The authors also acknowledge the comments of Andy

Heymsfield and anonymous reviewer in improving and

clarifying certain portions of the manuscript.

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