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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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