A NUMERICAL MODEL FOR THE AGGREGATION OF SNOW CRYSTALS

by

MARILYN LOUGHER

B.Sc., Swansea University College

(1964)

SUBMITTED IN PARTIAL FULFILLMERTOF THE REQUIREMENTS FOR

THE DEGREE OFMASTER OF SCIENCE

AT THEMASSACHUSETTS INSTITUTE OF TECIHOLOGY

INST. TECI

LIBRARY

LNDC'

August 1966

Signature ofDepartsentof Meteoology, 22 August 1966

Certified by... .. .Thesis Advisor

Accepted by...............................................Chairman, Departmental Committee on

Graduate Students

A NUMERICAL MODEL FOR THE AGGREGATION OF SNOW CRYSTALS

by

MARILYN LOUGHER

Submitted to the Department of Meteorology on 22 August 1966in Partial Fulfillment of the Requirements for

the Degree of Master of Science

ABSTRACT

A pilot study is made in which the evolution of a snow-flake spectrum in stratiform-type storms is followed as faras possible with hand computations.

At the onset of aggregation, in a layer just above themeeting level, an initial particle density of 10 crystalsper m3 is assumed. The crystals are assumed to be plane den-drites of uniform size, 4 m in diameter. Random collisionsamng the ice crystals, at the rate of 10 collisions per second,initiate the spectrum development which is further continuedby ordered collisions as aggregates with greater fall veloci-ties overtake crystals or other aggregates.

After a time interval of about 4 mine., the evolving spec-trum is found to be approaching snowflake spectra actually mea-sured at the surface for the equivalent precipitation rate of1.5 mm/hr. The aggregation process slows up considerably withpassing time and eventually reaches a fairly stable state.

Further experiments with machine computations are desir-able so that the complete evolution of the spectrum and itsdependence on the initial can be more fully explored.

Thesis Supervisor: Dr. Pauline H. AustinTitle: Senior Research Associate

AGKNWtIZDGEMENTS

The author is particularly grateful for the guidance of

her aupervisor, Dr. Pauline M. Austin. Ber time, effort and

encouragement In seeing this investigation through to comple-

tion are deeply appreciated.

Part of this work was done on the IBM 7094 digital com-

puter at the M.I.T. Computation Center. The author is in-

debted to Mr. Mark H. Vaughn for his assistance in prograwaing

the computer and in the final preparation of the manuscript

for presentation.

The author also wishes to thank Mrs. Jane McNabb and Hiss

Penny Freeman for typing this manuscript.

TABLE OF CONTENTS

1N=OCTION I

WEASUREUfNTS O OBSERVED PARTICLE SIZE SPECTRA 3

A. Raindrop opectra 3

B. Snowflake spectra S

1HORETICAL UDDEL FOR THE DEVELIMENT OF A SNOWMLAKE SPECTRUM 9

A. Co11lonA mochanisms

D. Spectrum development 13

ASSIPTIONS CONCERNING PARAM R15

A. Crystals 1

B. Siowtlakes 17

APPLICATION OF THE &MDEL 20

A. Collision mechanIsm 20

B. Development of the Spectrum 21

COMPARISON OF COMPUTED SPECTRUM WITH OBSERVED SPECTRA 38

CONCLUSIONS 41

APPENDIX 42

BIBLIGCRAPIH 45

-" & 4I ll ll ll IIIIIik iilillill llNil il l lllll ili IIIIII .1ilil I lN ll, I " 111li ll ii siililiiillillllliliil il I ll l ill lliM I I llll l u, I I I .4 'j I* "'""n""""""" ' 'l - o m i u d i llill ll ' "'" ""''" '"""'"l -I --1 111

LIST OF FIGUPIBS

Fig. 1. H-P drop-size distributions, before and after melting.

R a 1.5 mu/hr.

Fig. 2. M-P drop-size distributions, before and after melting.

R a 2.5 wi/hr.

Fig. 3. Snowflake spectra from Gun and Marshall (1956).

Precipitation rates: 1.5 mm/hr and 2.5 mm/hr.

Fig. 4. Collision cross section of plane dendritic crystals.

Fig. 5. Modified terminal fall velocities of small flakes.

Fig. 6. Rate of growth of a snowflake in an environmeut containing

a constant density of 104 plane deandritic crystals, dia-

meter 4 mm, per I .

Fig. 7. Computed spectrum at te 27 saecs.

- 9,000, a 260 W"3.

Fig. 8. Computed spectrum at t n 54 sees,

, w 7,000, P. = 450 M 3,

Fig. 9. Computed spectrum at t a 75 sees.

f, .5,000, -A3540m-3

Fig.10. Computed spectrum at t - 100 secs.

- 3,000, f iu 600 B-3,

Fig.11. Computed spectrum at t 0 110 sec.

*2 2,400, A. = 625 wr3.

Pig,132. Extrapolated spectrum at t V 4 mins.

, a 650, cr 650.

Fig.13. Observed and computed spectra.

Precipitation rate - 1.5 mm/hr.

LIST OF TABLES

Table 1. Claesification of snowflakes into 101 am Intervals.

Table 2. Variation of random collision frequency with crystal

density.

'" Ift , I I 11, , , I, I WIll I III I i I I 116 1 111111111111

1.

I INTRODUCTION

The mechanisms of precipitation development depend largely on the

type of cloud producing the precipitation. Condensation alon is not

a sufficiently rapid process to produce precipitation sise particles

in a reasonable length of time. Growth beyond the cloud drop size

may occur either through coalescence of droplets or through growth

and aggregation of ice crystals. Clouds that do not extend above

the melting level are oosmonly observed in regions where the melting

level occurs high in the troposphere, e.g. mid latitudos during the

summer season and tropical regions. Preclpitation in wholly liquid

clouds develops mainly as a result of collisions and coalescences of

cloud droplets. Some theoretical studies of coalescence mechanisms

in liquid clouds have been undertaken with moderate success (Melmak

and Hitschfeldo 1953, Bartlett, 1966), and the process is fairly well

understood quantitatively provided that conditions are uniform.

In clouds that extend above the melting level, precipitation generally

originates as snow. A mechanism for the development of precipitation

in stratiform clouds was first proposed by A. Wegener in 1911 and later

fully extended by T. Bergeron in 1963. Ice crystals are produced at

heights well above the molting level and grow by sublimation in the super

cooled regions of the cloud. At temperatures slightly below OOC the

crystals are usually observed to be aggregated into snow flakes, which

subsequently melt into liquid drops. The resulting drops grow further

by condensation aud coalescence before finally reaching the ground as rain.

1 M h4m MM11M14n1 1, . , I 1, liliuaomi iI 11 A III Il lill MI OWN IIIW I ill, I I,

2.

Quantitative Studie of the aggregatien processes taking pilc

above the melting level have generally been neglected. Yet this in

clearly an important factor in the developmnzt of raindrops or large

snowflakes. The purpose of this study is to investigate quantitatively

the mode of aggregation of ice crystals, particularly in stratiform-

type precipitation.

Laboratory experiments made by bosler, Jensen and Goldshlak (1957)

support the widely held belief that aggregation mainly occurs at

temperatures above -40C. Also, radar observations and measurements

indicate a larger increase in reflectivity at the melting level than

can be explained by melting alone, suggesting that rapid aggregation

of snowflakes occures in this region.

In this study, possible collision mechanisms resulting in aggrega-

tion of ice crystals and snowflakes are considered. The theoretically

computed snowflako size spectra are compared with measured spectra before

and after melting. The length of time required and fall distances in-

volved during the evolution of the spectrum are also considered.

." dI I I , i IMId lM I M M~lu |

3.

II. NASUREMT OF OBSERVED PARTICLE SIZE SPECTRA

A. Reinrop2 pecta

Atlas (1964) has remarked that "the only statement about drop size

spectra that can be made with complete certainty is that they are

highly variable in time, space and with storm type." Much of the

variability results from inadequate sampling. Marshall and Palmer (194C)

analysed average rain drop samples collected by a number of observers

and found a simple relationship between the drop size concentraticn.

drop diameter and rainfall rate. For stratiform-type rainfall originating

as snow, the M-P distribution remains the simplest representation of

raindrop concentrations. The relationship is

N No -Af

where NDdD is the number of drops in the diameter interval

D to D + dD per unit volume, N 0 is a constant, 0.08 cM-4

and A is a parameter dpending on the intensity of rainfall R(w/hr):

A 41 R *

For diameter sizes larger than about 1 am the *-P spectrum is

generally an accurate representation of the distribution but it ucually

overestimates the concentrations of drops smaller than 1 au in diameter.

This has been noted by Marshall and Palmer themselves, Best (1950)

Mueller and Jones (1960), Mason and Andrews (1960)M uaimu con atra-

tions normally occur between 0.5 and 1 m so that the rain spectrum

'I " " " " I AI""""""''' ' " ' - - 11mi0s a l a| MMMMHEMolMIMiminun 1

appears parabolic rather tha linear on a plot of log ND versws D.

Figs. I and 2 show *-P distributions for rainfall intensities of 1.5

and 2.5 zu/hr.

Most of the rain in the above measurements originated as snow.

Therefore the particle size distribution for the snow before melting

is extremely relevant to the measured raindrop spectra. The spectra

which would malt into the M-P distributions have ben computed an the

assumption that melting is the only process to occur at the melting level

i.e. no drop break up. Figs. I and 2 also show the M-P spectra before

melting for rainfall intensities of 1.5 and 2.5 m/hr. A density factor

in inverse ratio to the change in terminal fall velocity was used to

translate the distributions. The spectra computed theoretically from the

model will be compared with the *-P distributions for equivalent preci-

pitation rates, before melting. Ideally, the comparison should be made

with translated rain spectra sampled just below the melting level. At

higher levels in the atmosphere rain spectra can be expected to contain

more small drops and, possibly, fewer larger ones than at the ground.

Mason and (1954), Hardy (1963) have shown theoretically

that changes in raindrop spectra with height occur because of coaleecence,

accretion and evaporation processes and result in a successive depleticn

of the smaller drops with decreasing height. Thus, it would be expected

that the M-P distribution is a more accurate representation of the rain

spectrum just below the molting level than at the surface.

B. Snowflakce spectra

Few measurements of snowfZ1rte spectpra have been published. Can-

sequently, no particular form >f snowflake size distribution has yet

found general acceptance.

Gunn and Marshall (195) g ouped average samples in stratiform

storms into four snowfall rate3. The data satisfied the relationship:

N,= Noewhere ND is the number of aggWrqeted with equivalent drop diameter

betwen D and D + dD per unit volt *n. Both Nq and A are functions

of the precipitation rate, R(mw/hri.

N = 3.8 x )3 R * (0-3m )

and

= 25.5 E-0.48 -1)

Fig. 3 shows snowflake spectrL measured by Gunn and Marshall at

rainfall rates of 1.5 and 2.5 ianfr.

Some occasional measurements i.ave been made by Jatanese obmervers,

Fujiuara (1955), Ima (1955), Maono (1953). They are in fair agreement

with the samples measured by Guria ,and Marshall. The spectra computed

from the model are to be compared with the Gmu and Marshall spectra

for equivalent snowfall rateeg

EE Before melting

o\

10 1 -

10 0 \

0 I 2 3 4 5

EQUIVALENT DROP DIAMETER (mm)

Fig. 1. M-P drop-size distributions, before and after melting.

R = 1.5 mm/hr.

102

EE-- - Before m elting

0

%oo

10 0ii \ i

0 1 2 3 4 5

EQUIVALENT DROP DIAMETER (mm)

Fig. 2. M-P drop-size distributions, before and after melting.

R = 2.5 mm/hr.

10 --

E-~E-~

0

z

100

0 1 2 3 4 5

EQUIVALENT MELTED DROP DIAMETER (mm)

Fig. 3. Snowflake spectra from Gunn and Marshall (1956).

Precipitation rates: 1.5 mm/hr and 2.5 mm/hr.

II I. THEORTICAL EDDEL FOR THE DEVELOPM0ENT OF A SNDWFLAKE SPECTRUM

A. Collision Mechanisms

Initially, the ice crystals are assumod to be uniform in both size

and type. snowflakes are initiated by random collisions among ice

crystals and continue to grow by further collisions. Ice crystals are

normally produced at temperatures below -10 0 C in stratifom clouds

which extend above the melting level. They grow by sublimation as they

fall through superoooled regions of the cloud. Aggregation is assued

to occur simultaneously through a layer several thousand feet In depth

just above the melting level* It Is also assumed that no further growth

by sublimation occurs in this layer.

In the model, two types of collision mechanisms effect the develop-

ment of the snowflake spectrum:

1. Ordered collisions: Ordered collisions occur among the pre-

cipitation elements of different sizes owing to differences in terminal

fall speeds. Faster moving particles overtake those which are falling

more slowly, and they are assumed to become attached upon contact. On

the average, all of the crystals have the same terminal fall velocity and

are not subject to ordered collisions among themselves. Ordered colli-

sions between snowflakes and crystals and among snowflakes of different

sizes, however, will occur.

The number of ordered collisions between snowflakes consisting of

p ice crystals and snowflakes consisting of q ice crystals per unit

volume of space, per unit time is given by:

10.

fa= p f Spt (vp -v) E, <aa>33.

where, number of partioles per nilt volume

Vi = terminal fall velocity

Sj =.ffective collision cross section

4 collection officiency.

This relationship also expresses the ordered collision frequency,

between single ice crystals and snowflakes. In the model E--isassumed unity at all time.

2. Random collisions: In this study all other collisions are

assumed to be "random". It is believed that these random collisionz

result primarily from microscopic forces (hydrodynamic, electrostatic

etc.) acting on each particle which produce random oscillations about

a men free path. Measurements of these random effects are virtually

impossible in the atmosphere because of observational difficulties.

Mason and Jayeewara (1963) have simulated atmospheric conditions in the

laboratory and have amply demonstrated that significant osoillations

of particles about a mean free path are likely to occur in the atmosphere.

The random collision process operating on a population of uniform

particles is assumed to be equivalent to a collision mechanism producing

collisions between two particles at regular, discrete intervals of time

depending on the particle density. The random collision frequency is

expected to vary with the particle density according to the following

theoretical considerations.

The total possible numbr of collisions among uniform particles of

11. -

rumber densi-%ty a :1

(3.2)2.

Thus, the frequancy of random coll1ions %iIl be proportional to

f (f-i)(3-3)

In other words, the average tim. interval between SuccessivO colli-

sions is proportional to

When the particles are not uniform in size it can be shown that the

total number of possible collislons is

4---~~~ t tf(t)

where (1 = l .... ) are the number densities of particls

composed of I individual snow cryatalm. Such particles ill be termed

F aggregates.

Similarly, the aveage time interval between svccessive random

collisions of F aggregates with one another is proportional to

SD- - )(3,6)

hereas the average tim interval between successive random collrisn

12.

of an F &ggregate with an F aggregate is proportional to

rp fW(3.7)

In the model the ice crystals are assuad to constitute one group and

the aggregates another. If at any given time

number density of ice crystals

umber density of snow aggregates

S t average time interval betuen successive random colli-sons among crystals etc.

R C R C R Ccc''' ) C> C 2L fcf (3.80)

or

,AtR -RR 3101)

ftR'tc -i C./A Rf

At the onset of aggregation the crystals are very numerous 1shile theR

aggregates are extremely few so that 4t c is by far the smalleat

of the above time intervals. As the spectrum develops the following

significant stages are reached

> R RAt Atc~a <At OC

PC.~ / F AtRCC C(f LYLccO

PC.. <Rac c~

These stages serve as an indic'ation of the relative importance of dif

ferent types of random collisions during the aggregation procesa h

times at which these stages are rached are found by following tha do-

velopment of the spectrum that grows as a reisult of both ordered col-

elsons and random collisions.

Random collissons among the ice crystals initiat the growth of the

spectrum. The newly formed snowflales at first grow by ordered colcin

with ice crystals. Eventually the aggregates become sufficiently d;io

that random collisions with crystals and also ordered colIisiones amrng

snowflakes become important.

Finally, most of the crystals are depleted so that only random rud

ordered collisions among snowflakes develop the spec Lrum,

Since the rate of ordered collisions depends on the number daensty,

effective cross sections and fall velocities of the particles, aurto

concerning these parameters must be made for use in the model. Thie is

14,

done in the follOWing chapter,

The value of the constant C defining the random collision rate

during any particular computation is not known. It is deiarable to

make several computations, varying this constant, in order to find a

realistic collision rate.

15.

IV& ASSUMPTIONS COERNING PARAMTERS

1. Crystal type and dimensions

Crystals occurring in natural clouds exhibit three funda-

mental structures: needles, oolumns and plates. The dominant factors

in determining crystal type and size are the ambient air conditions

For this study, plane. dendritic crystals were chosen since these are

frequently observed in stratiform-type storms. A number of observa-

tions ware made at Boston during the winter of 1985-196 of snowflakes

reaching the ground. In a total of 10 storms, 6 featured plane den-

dritic crystals, 3 featured spatial dendrites and needles occurred once.

Observations and laboratory experiments (WeIIComn, 1953; Nakaya,

1984; Mason, 1987) have shown that plane deadritic crystals are generally

formed at temperatures between -12 0C and -16OC. At thene temperatures

Nakeya (1984) measured the average crystal diameter to be 3.36 mm. In

the model, a crystal diameter of 4 =was selected. This is somwhrbat

larger than the average value observed by Nakaya but allows for further

growth by sublimation, as the crystals fall towards the layor above the

melting level in whIch aggregation is thought to take place. A plane

dendritic crystal of diameter 4 mm melts into a rain drop of diameter

0.8 =, a value well within the range of the smallest measured drop

in raindrop miss spectra.

EMpirical studies show that a rainfall rate of 1.5 M/hr is equivalent

to a liquid water content of about 0.1 g/a 3 (Atlas, 19084) Thus, in ordrsx

16.

to produce a rainfall rate of 1.5 mM/hr the liquid water content of

the ice crystals rust be about 06&/ma3 , because of their smaller fall

velocities. Consequently, an Initial density of 10 plane dondritic

crystals, diameter 4 m, per 3 Is assumd.

2. Collision cross section

If the horisontal cross sections of two particles involved in

a collision are circular, the effective collision cross section is

7T2; (d. + d2)

here dl and d2 are the respective diameters of the horisontal cross

sections. In this study it was found convenient to express the hori-

sontal dimeter of the snowflakes and crystals as functions of the

equivalent mlted drop diameter.

The diameter factor, k, is defined to be the ratio of the horisontal

diameter of the snowflake or crystal to the diameter of the equivalent

melted drop.

It is assumed that the ice crystals are horisontally osiented in

space at all times. Consequently the cross section of a plane dendritic

crystal approximates to that of the enveloping circle, as shown in Fig.

4. The diameter factor, k, can be readily calculated as a function of

crystal diameter from the relationship a = 0.0038 d2 (Nakaya, 1984),

where a is the crystal mass (mg) and d the crystal diameter (m).

For crystals of diameter 4 m, k has the value 8.3.

3. Fall velocity

Nakaya (1954) ibserved plane dendritic crystals to maintain a

17,

virtually constant fall velocity of 31 /ca/ro, independent o9 mass.

A value of 30 es/sea is assumed, for convenience, in the computatione,

1. Dimensions

Observations and measurements of snowflake dimensions are prac-

tically non xistent primarily because of observational difficulties.

It is obvious that the shape and size of snowflakes are highly variable

since they depend mainly on the mode of attachment, number and type of

the constituent ice crystals as well as on mioscopic forces. In view

of the paucity of data the following asusptions have been made with

respect to snoWflakes:

a) The snowflakes maintain an oblate spheroid shpe at all times;

the ratio of the major to the minor axis Is 3.2 regardless of mass.

b) Each flake maintains its orientation In space and the horizontal

cross section is accordingly circular

c) Density values measured by Magono (1954) in Japan are applIcable.

He found the density of dry and wet flakes to be 0.0087 g/cm3

and 0.017 g/cm3 respectively.

2. Collision cross section

As mentioned above, the effective collision crose section of

a snowflake is assumed to be circular. Values of k are independent of

cryusal dmansion as follows:

k = 6.5 for dry flakes

ic = 4. for wet flakes

18.

These values are in good agreement with the value k = 4.5 computed by

Wexler (1958) in assuming that the Reynolds number of a snowflake is

the same as that of its equivalent melted drop. In the model, the

snowflakes are assumed to be dry at all times prior to actual mlting.

3. Fall velocities

Using film techniques, Langleben observed the following rela-

tionship to exdst between the fall velocity of a dry snowflake con-

sisting of plane dendritic crystals and its equivalent melted drop

diameter:

VI= 160 DO.31 (e.g...), D > 0.5 mm

where V is the terminal fall velccity and D the melted drop diameter

of the snowflake.

Fujiwara (1957) derived a ftuntional form for the velocity of all

snowflakes from dynamical considerations and found

V = KDI/3 (o.g.s.)

where K is a constant depending on crystal type.

In the model the fall velocity of snowflakes is assumed to satisfy

Vs 160 D "-(ecga.a) (D .. 1.0aM)

The fall velocities of the smaller flakes with equivalent drop diameter

..1 s= are modified as shown in Fig. 5 to insure a smooth convergence

to the fall velocity of the crystals.

Fig. 4. Collision cross section of plane dendritic crystals.

0.8 1.2 1.6 2.0 2.4

EQUIVALENT MELTED DROP DIAMETER ( mm)

Fig. 5. Modified terminal fall velocities of small flakes.

1O

90

70

qj)

50

3 OL0.4

V. APPLICATION OF THE MDDEL

A. Collision meebanism

It will be recalled that the spectrum Is Co be initiated by randm

collisions of ice crystals. Subsequent development occurs primarily

through ordered collisions of the snow aggregates with ice crystals.

As mentioned previously, an initial density of 10 plane dendritic

crystals is assumed. The following types of collisions are allowed

to proceed:

1) Random collisions among ice crystals

Lack of experimental information makes the choice of determining

a reasonable collision rate an arbitrary one. Preliminary computations

assuming an initial collision rate of I collision /see appeared to be

unrealistic in that the spectrum evolved slowly and the resulting spac-

trum showed little variation in number concentrations. Therefore, in

the present computation an initial rate of 10 collisions /see in each

unit volume (a3) is assumed. The average time interval, t,8between successive random collisions of crystals is then 0.1 sec.

Equation (3.8) gives the relation

R 107At,= - f(5.1)

where C is now replaced by 10 .

2) Randcm collisions involving snowflakes

'he mechanism producing random collisions of snowflakes with

other snowflakes and ice crystals is assumed to be the same as that

20.4

21.

which operates amg the crystals. It hac been pointed out that ranid=

colliions involving snowflakes assume Increasing importance av the

spectrum evolves.

3) Ordered collisions between snowflakes and ice crystals.

The growth rate of a single snowflake according to (3.1) as

first computed by mans of a computer when the flake is assumed to grow

solely by ordered collisions with single ice crystals. Collision cros

sections were estimated as described in IV. Fig. 6 shows the average

time intervals that elapse between successive collisions with crystals.

(Since the particles are positioned randomly in spaces the ordered col-

15ions actually occur at random time intervals. But as in the case of

random collisions it is believed that the assumption of regular rather

than randomly spaced collisions would not effect the ultimate sipectrum.)

As expected, the collision rate increases sA the aggregate gains speed

in falling.

The average collision rate of snowflakes of a given size with

crystals varies linearly with the snowflake density and the crystal

density so that Fig. 6 can be used to calculate these collision rates

at successive intervals of time, density changes being taken Into acovunt.

3) Ordered collisions among snowflakes

Average collision rates among snow flakes of different sizes are

calculated using (3.1) with the collision cross sections estimated as

described In section IVC.

B. DeveloMent of the spectrum

Let = crystal density

10"1011W1116

22.

= aggreg.ate denalty

MC andl WOar the avea tima intervalS betrween ranmdom

collisions and ordered collisions of a single crystal with a owflake

consisting of p ice crystals respectively.

Ata is the average tim interval between successive random col-

lisions of any snowflake ith a crystal.

A and &i (p > 1) are the average time intervala

between random collisions and ordered collisions among flakes con-

taning p crystals (F flakes) and flakes containing q crystals,p

(F flakes), respectively.

A ta is the average time interval between successive random col-

lisions among any snowflakes.

The evolution of the spectrum is followed at I second time inter-

vals. For purposes of translating the aggregate crystal size distri-

butions into equivalent drop diameter distributions the snowflakes

are grouped into intervals of equivalent drop diameter as follows:

Equivalent Drop FDiemeter ( WAm)M

0.5 F

O06 F2

0.7 3

0.8 F4

0*9 F52,F

9TS000T

06 40GLA8Lg*S*G A

Be &4069a

89Am*T J

OQ * **v

91

A

8 1 91J;

009

see

9ov

9,0Z

91

ye!

101

VIC

col r~~

24.

11e spectrum evolves a follow:

1. Initially, at tie t = 0

f, = jr~3 , f = Om- 3

R -1At = 10 .sec.

2. At t=lae

f,= (Io4-hzO) -1, f, o r/O0 3

Fig. 6 shows that vhen a=10m- it takes a single F2 flake

about 12 sees to overtake an ice crystal, so that within about I sao

an F2 flake will collide with a crystal dben fx = 10 i

3. At time t = 2 see

f,=(10o-4)rm~3 fx =I9 i3 fI-I

From the considerations discussed above, it In likely that 2 F2 snowflakes

ill collide separately with a crystal within a further second producing

2 more F3 flakes. 2he probable time interval required for a ingle F3

snowflake to perform an ordered collision with a crystal Ia 7 sec (Fig. 6).

4. At tim t = 3 scs

,= (10'~63) 3 )~ Z=:' 3 , =3

The process is continued in this way. For convenience in the compu-

tations the random collision rate was coqputed to vary with the

crystal density as follows.

25 .

Cofl/Me

10,000*,5009,0008,5007,5007,0006,0008,0004,0003,0002,0001,5001,000

10987654321

0.40.250.1

SiMultaneously, the ordered collision rates vary linearly with the

changing crystal and snowflake densities.

5. At t 27 sec

= 9,000 ,3 PIC = 260 m-

The largest aggregate present at this tine contains 11 crystals. The

total distribution in terms of equivalent malted drop diameter is shoan

in Fig. 7. The rate at 1ich various types of aggregation proceed is

expressed in terms of average time intervals;

0.11 4.

AtfR f 0.11 ees

At r\ seca

At .C 15.0 wso

/At*, . ses

0>25 oewn

10102011011 jill'iIIIII AD "U1161 , I, III ,I,

(s-3)

20.

Only random collisions among crystals and ordered collisions betwwen

crystals and aggregates are considered sigificant at this stage.

Co At t = 54 goes

= 7,000m, = 450

The largest aggregate present at this time contains 28 crystals

A toI N 0.2 secs

A t Ifta (V so sees

0.17 < A tp < 15 secS

so

0

t > 30 sece

Random collisions between crystals and ggregates could Justifiably

be included in the coputations at this stage but since these colli-

sions are undergone by all of the aggregates with varying collision

frequencies the net effect on the distributon is considered negliAle

At no time are random collisions between crystals and aggregates ccr-

sidered significant enough to justify Inclusion in the computations.

By means of random collisions amg crystals and ordered collisions

betwen flakes and crystals the spectrum develops to

7. t 75 soc

= ,000 a 540

The largest aggregate now present contains 40 crystals. The total

distribution in terms of equivalent melted drop diameter is shom in

Fig* 9.

27.

RAt1g 0.33 QGec

h tgo, (\V 2 sec

P' 35 secs

o.25 < At*P, < o2seAt, > 2Z .. oc

As before, only the random collisions among crystals and the ordered

collisions betwen crystals and snowflakes are considered to be

significant at this stage.

The distributions show that the smaller equivalent drop diameter

contain relatively high number conoentrations. Because of this, tho

ordered collision rate is faster than the random collision rate pro-

duoing new flakes so that the concentrations of smaller flakes are

successively reduced.

8. At t = 100 secs

= 3,000, = 80W-3

The largest aggregate present contains 48 crystals. Fig. 10 shows

the total distribution at this stage in terms of equivalent drop diameters

,at, R = 1 m

At"a 3 sec

Ota. r s3 see

.5 <4*, 4 3

> 25 sece

Crystals are now being depleted mainly through ordered collisions

with snowflakes.

28.

Collisions of the largest flakes with ice crystals occur rela-

tively slowly at I collision every 3 sees on the average. It

thus seems likely that larger flakes of about 3 am in equivalent drop

diameter CF to F ) can only be produced after the stage when

collisions amongst aggregates become important.

9. At t = 110 secs

A = 2,4o, Pt =- 25 3

The largest aggregate contains 53 crystals. The total dX.stribution

is included in Fig. 11.Adtit = 2 sees

I tCa 3*5 seeR

Atad rV as sect

O.* -4 A to, -< 4 secn

O tPW > 25 sees

The only important collisions are still considere 40 be the rando=

collisions among crystals and the ordered collision ietween aggregateO

and crystals.

The further developmeit of the spectrum is ext cpolated up to the

stage where both random and ordered collisions A g the aggregates

assume significance. Concentrations of mall 22 ""es ill decrease further

while some of the larger flakes grow slowly.

10. At approximately 160 oeconds

Pf 1300, = ?* 650

This makke the stage when random collisiort among crystals occur at the

same rate as random collisions among cryfaals and aggregates, since

11141HOIN I WIWI W1 16111 " '11 1

29.R% R

For, li Atgu, = sea$

- 23 sees

2,.. 10 oses

V t27 seca

At this stge randm collisions between crystals and snowflakes are of

the same significance as random collisions among crystals.

11. At t N 4 mine

Now random collisions among crystals and agregates are more importat

than random collisions among crystals, for:R

Atli 23 secs

$ 0. 12 secs

A t.o. 2 23 sees

$ <At4 < 20 secs

so that ordered collisions amng snowflakes and crystals are still

the primary source of depletion of the ice crystals.

The largest flake in the spectrum will have reached an equiralent

drop diameter of about 2.1 im. A sall increase in the concentrations

of the flakes larger than the median equivalent drop diameter vll have

occurred and a decrease of smaller flakes. The random collisions of

flakes with crystals will further broaden the spectrum. Fig. 12 shows

the predicted spectrum at this stage. The crystals are falling with a

velocity of 30 e/see while the average velocity of the agregates so

far is 90 c/sec. Thus, by this time the aggregates will have fallown

on the average, about 220 m.

.11 I~II

304

12. The stage vten

will probably occur when

/1 350, (U 700 m-3

Then

dtg =80 ges

48 I\a 20 seen,a R

L ao- 20 sees

/0 O -Otp < 35 sees

pi. r 20 se

Thus, subsequent development of the spectrum will be accomplished

mainly by ordered collisions among aggregates, ordered collisions

between remaining crystals and aggregates. The random collisions in-

volving snowflakes are of less importance but should be included at

this stage.

As the above collision rates indicate, the spectrum will change

relatively slowly. Eventually ordered collisions among snomflakes will

assume primary importance. It is anticipated that the spectrum will

broaden and flatten as a result of these collision processes but a

detailed analysis of the spectrum changes can be undertaken only with

the aid of a computer. (See Appendix)

3'.

300 334mm,,

200

250 3.15 mm

Distancefallen - -175

(m)

200 - 2.92mm

zU

100J5 150(I).

.150 2.66mm 0

oOO 0

0

o - 125uLU

a 100 2.32mm

z100

50 -1.84 mm 7

50

1.08mm 25.10.

0 50 100 150 200

TIME ELAPSED SINCE INITIAL COLLISION OFP 2CRYSTALS (sec)

Fig. 6. Rate of growth of a snowflake in an environment containing

a constant density of 10 4plane dendritic crystals, diameter

4 mm, per m

80

70 -

60-

- 50-

EE2

E 40

z

030 -

20 -

10

0 0.5 1.0 1,5 2.0 2.5MELTED DIAMETER (mm)

Fig. 7. Computed spectrum at t = 27 secs.

= 9,000, Jaz=260 m-3

~mmhmI-II

33.

80

70

60

- 50

EE

E40

0

z 30

20

2 0

00.5 1.0 1.5 2.0 2.5

EQUIVALENT DROP DIAMETER ( mm)

Fig. 8. Computed spectrum at t = 54 secs.

= 7,000, -= 450 m-3a

.A 111111MINJOWWWWOWAI I,, III I I 1101010NOWWWWANNIIII lI

80

70-

60

50-

EE

240-

E

z 30-~

20-

20-

I0

0.5 1.0 1.5 2.0 2.5EQUIVALENT DROP DIAMETER (mm)

Fig. 9. Computed spectrum at t = 75 secs.

= 5,000, J = 540 M-3

80

70-

60-

50-

EE

E 40

0

z 3 0

20 -

10-

00.5 1.0 1.5 2.0 2.5

EQUIVALENT DROP DIAMETER (mm)

Fig. 10. Computed spectrum at t = 100 secs.

= 3,000, fr=600 m-3.

- 40K 101b 19 1 61', 1, 1, 1 1, , 1, 1 1 11 11 1 wl , 11 1, ''Allill - -- , , WN111- - - 1"M 116

80

70

36.

60-

50

EE

'E 40 -0

z 30

20-

10-

00.5 1.0 1.5 2.0 2.5

EQUIVALENT DROP DIAMETER (m m)

Fig. 11. Computed spectrum at t = 110 secs.

=2,400, 1 = 625 m-3

37.80

2.50.5 1.0 1.5 2.0EQUIVILENT DROP DIAMETER (mm)

Fig. 12. Extrapolated spectrum at tv 4 mins.

= 650, fo, -650.

70

60

50

EE

aE

z

40

30

20

10

j,'', 11111111111116111 1111111111111111111111

30.

VI. COMPARISON OF COMPTEID SPECTRUM WITH OBSERVED SPECTRA

Fig. 13 shows the computed spectrum on a semi-logaritbmic plot

for the case = 650, = 650 d*. It vill be recalled that

this does not represent the final stage of the spectrum development.

Rather, ordered collisions among snowflakes, hidch assume primary im-

portance after this stage, Vill considerably modify the spectrum shon

in Fig. 13. This collision process will result in a reduction of the

number concentrations of snowflakes having equivalent drop diameter

up to about 2 mm and considerably eWsend the remaining portion of the

spectrum.

Thus the computed spectrum will develop towards the spectrum

measured by Gunn and Marshall for a precipitation rate of I aw/hr.

(Also Fig. 13.)

The Marshall and Palmer rain spectrum, before melting is also sheon

in Fig. 13 for a precipitation rate of 1.5 mA/hr.

A comparison of the M-P distribution with the Gunn and Marshall

spectrum shows the M-P distribution to consist of fewer aggregates of

equivalent drop diameter larger than 1.5 =a and many more aggregates less

than 1.5 m in diameter. This implies that flakes larger than 1.5 mm

in diameter tend to break on melting into several smaller raindrops.

The results of the computations support this hypothesie for none of

the collision processes discussed can develop the computed spectrum

towards the M-P distribution before melting unless breakup of the

snowflakes is prominent. There is no experimental evidence to show

.11db ,ii,,idlbhhlijulii IiI~ I iN

III J WiININNIIIIIIIIIII

39.

that aggregate breakup is common in stratiform itmatiow. In fact,

the presence of many large flakes in the Gunn and Marshall spectrum

would seem to indicate that snowflake breakup 19 not important in these

conditions.

Thus, the spectrum generated by the collision mechanisms is in

good agreement with that measured by Gunn and Marshall. A random cl-

lision rate of 10/sec operating among a population of 10 plans dndrwitic

crystals of dismter 4 mm initiates a spectrum similar to spectra that

have been measured.

It smaller crystals were assumed in the initial conditions, a cor-

respondingly higher density of ice crystals would be necessary to

maintain a reasonable liquid water content. The equivalent random col-

lision mechanism to that above would result in a similar spectrum to the

one already computed.

As previously discussed, a slower random collision rate initiates

a spectrum with unrealistically low number concentrations over a

wider range of equivalent drop diameters than in commonly observed.

Similarly a faster random collision frequency results in reastically

high number conentrations over a narrower range of equivalent drop

diameters than is commonly observed.

Variation of the spectra with crystal type has not yet been inves-

tigated but it is hoped to incorporate this in a later study in which

the problem is adapted to a cotputer.

10 3, , , , i _

M-P distributionbefore melting

102

E

E

E -Gunn and ' Computed spectrumMarshall

distribution

z10 -

10 00 0.5 1 1.5 2 2,5 3 4 5

EQUIVALENT DROP DIAMETER (mm)

Fig. 13. Observed and computed spectra.

Precipitation rate - 1.5 mm/hr.

41.

VII. CONCLUSION

A pilot study has been made in which the evolution of a snowflake

spectrum in stratiform-type storms has been carried on as far as

feasible with hand computations.

At the onset of aggregation, an initial particle density of 10

plans dendritic crystals, diameter 4 m, per M3 was assumed. Random

collisions among the ice crystals, at the rate of 10 collisions/see

initiated the spectrum development which was continued by ordered

collision as faster falling particles overtake slower ones.

After a time interval of about 4 minutes, the evolving spectrum -n

found to be approaching snowflake spectra actually measured at the

surface for the equivalent precipitation rate of 1.5 am/hr. The aggre-

gation process slows down considerably with passing time and eventually

reaches a fairly stable state.

It is desirable that a machine program be set up in order to ins-

vestigate further the dependency of the ocaputed spectrum on the initial

assumptions - crystal dimensions, liquid water content, rate of random

aggregation, etc. The mamer in which a variation of these factors

effects the computed spectra might cast light on the variability of

observed spectra from storm to storm.

42.

APPENDIX - PROSLEM PRESENTATION NOR

ADAPTION TO A COLTUTER

At successive time intervals the change in the number concentraticns

of each size flake d.ue to both random and ordered collisions can be

calculated.

Let,

F = - nowflake consisting of p crystalsP

(F = single crystal)

At any given tiue t,

N = number of p flakes in unit volume.P p

Then ,P Ni N At = prob~abe nu4mr o: ordered

collisions betwen F and F flakes in a time interval t t and,

similarly

R Aprobable number of random

collisions betwen F, and FI flakes in a tUo interval

In t , N Pill change as follow:

1. Decrease in N :

a) the number of random collisions betwen Fp andFi U 1, ' Co

in At I

x NpNj RPh At

b) Ordered collisions ith F (i = p. + 1,* ** ) produce

00

Y NPNt oPq AtLZ P+ Icollisionam.

c) Ordered collisions vdth F (1 = 1) - e0 , p - 1) produce

P-1

L=I

The total decrease in N is thus:

Ni Np RRp AtL~I

MV NP d eitco

+

2. Increase in NP:

a) Random aggregation of F, and F (P-i) (I = 1...... p-1) produce

P.-'yrL~f

Ni N(P--L) RP,(P-i) L6t

b) Ordered colUsions of F and F1 produce

P/2-

NI A4- ,P,5..-i) At=I

collisions it p even

and

Nt N,.. f 0,);) At

collisions It p odd.

43.

aollinIC-an,

Np NJ 0?? At

Ord* total lner&So In N Is thusaP-1 p

ZNi-N(p-i j44~a) At

~-

±1L~ I

N C Nipz) OPI(P.L At

Thollefor'eP-1 M 14rrr

ZNisNRPI.P0o

iRm : Ni , p O iL=1p

I I. II 10 19 i uelM 0ll "l NlM ilb Illill

45,

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