Numerical Simulations of the Physical Process for

NO.1 FANG Wen, ZHENG Guoguang and HU Zhijin 93

Numerical Simulations of the Physical Process for

Hailstone Growth∗

FANG Wen1,3 (��

), ZHENG Guoguang2 ( �� ), and HU Zhijin3 ( �� )

1Nanjing University of Information Science and Technology, Nanjing 2100442State Meteorological Administration of China, Beijing 1000813Chinese Academy of Meteorological Sciences, Beijing 100081

(Received March 30, 2004; revised August 31, 2004)

ABSTRACT

Theoretical and experimental studies show that during hail growth the heat and mass transfers playa determinant role in growth rates and different structures. However, many numerical model researchersmade extrapolation of the key heat transfer coefficient of the thermal balance expression from measurementsof evaporating water droplets obtained under small Renolds numbers (Re 6 200) introduced by Ranz andMarshall, leading to great difference from reality. This paper is devoted to the parameterization of measuredheat transfer coefficients under Renolds numbers related to actual hail scales proposed by Zheng, which arethen applied, to Hu-He 1D and 3D models for hail growth respectively, indicating that the melting rateof a hailstone is 12%-50% bigger, the evaporation rate is 10%-200% higher and the dry-wet growth rate is10%-40% larger from the present simulations than from the prototype models.

Key words: hail, parameterization, numerical simulation, heat transfer

1. Introduction

Theoretical and experimental studies on the phys-

ical processes of hail growth (Schumann, 1938; Lud-

lan, 1958; List, 1963) showed that its growth rate and

structural characteristics depend on the heat and mass

transfers; its dynamic characteristics determine hail-

stone’s movement and stay in clouds and damage done

to ground bodies, actually controlling the growth in-

side clouds. As we know, the heat transfers affects

directly hailstone’s wet growth, melting and evapo-

ration. In their expression of heat balance equation,

however, a lot of researchers dealing with cloud-physics

models have adopted the heat transfer coefficient mea-

sured from evaporating water drops at small Renolds

numbers (Re 6 200) produced by Ranz and Marshall

(RM) and extended it to 1× 104 6 Re 6 1× 107, thus

leading to great difference from the conditions of ac-

tual hail particles (List, 1989; Zheng, 1994). Macklin

investigated by experiment the heat and vapor trans-

fer coefficients from melting particles, discovering that

the obtained transfers are considerably stronger com-

pared to the equivalents given by RM and that the

transfers are a lot more vigorous from oblate than from

spherical particles. Based on accurate measurements

of the surface temperatures of ice particles cooled on

1.1 × 104 6 Re 6 5.2 × 104 using a thermal imag-

ing system, Zheng (1994) developed a numerical model

for defining a heat transfer coefficient denoted as Nu

standing for Nusselt number and experiments indicate

that the obtained Nu is approximately 30% bigger

compared to the one coming from RM expression, 40%

larger from oblate than from spherical particles with

the diameter equal to the major axis of the former

and even twice as large from coarse particles as from

spheroids of the same diameter.

In past studies, meteorological scientists employed

only the measurement of evaporating water drops at

Re 6 200 (vide ante) as the heat transfer coefficient

that is a key component of the thermal balance equa-

tion for hail growth, with the RM coefficient shown

as

Nu = 2.0 + 0.53× Re1/2, (1)

contrasted with

Nu = 0.33× Re0.57, (2)

∗Supported by the National Natural Science Foundation of China under Grant No. 49775255.

94 ACTA METEOROLOGICA SINICA VOL.19

Nu = 0.313× Re0.599, (3)

Nu = 0.114× Re0.74, (4)

where Nu of Eqs.(2)-(4) measured at 1 × 104 6

Re 6 5.2 × 104 proposed by Zheng who used a verti-

cally pressure-controllable wind tunnel for hail growth.

Equation (2) is applicable to smooth sphere, i.e., their

aspect ratio α = 1; Eq.(3) holds for smooth oblate

particles with α = 0.67 and Eq.(4) is true for rough

oblate particles (α = 0.67) with surface having coarse-

ness β =2%.

For convenience of later discussion Nu of Eqs.(1)

to (4) are denoted as Nu1 through Nu4, in order and

other physical quantities calculated using Nu1 − Nu4

will be given by corresponding subscripts 1 through

4. Nu1 to Nu4, respectively, from Eqs.(1) to (4) at

1×103 6 Re 6 5.2×104 are given in Table 1, in which

we see that Nu2 is by 2%-30% larger than Nu1, about

14%, on average, higher; Nu3 exceeds Nu1 by 4%-

69%, averaging roughly 35% larger; Nu4(β =2% par-

ticles) is 83% greater compared to Nu1, but at larger

Re, Nu4 is by 150% larger than Nu1.

Table 1. Comparison of Nu numbers of hailstones with various characteristic at 1× 1036 Re 6 5.2 × 104

Re 1× 103 1.5× 103 2.8× 103 3.4× 103 4.4× 103 5.2× 103 6.5× 103

Nu1 18.887 22.682 30.257 33.137 37.421 40.507 45.052

Nu2 16.924 21.325 30.436 33.998 39.381 43.415 49.189

Nu3 19.651 25.052 36.410 40.901 47.731 52.755 60.299

Nu4 18.919 25.539 40.532 46.795 56.632 64.084 75.589

Re 7.6× 103 9.4× 103 1× 104 2.2× 104 3× 104 3.7× 104 5.2× 104

Nu1 48.553 53.773 55.400 81.205 94.491 104.717 123.771

Nu2 53.775 60.701 62.880 98.559 117.618 132.551 160.929

Nu3 66.219 75.211 78.051 125.166 150.720 170.895 209.538

Nu4 84.861 99.316 103.97 186.336 234.409 273.762 352.166

Equations (1)-(4) are parameterized and put, sep-

arately, into the 1D and 3D time-dependent cumulus

models developed by Hu and He (1988; 1989), with

the simulations for comparison.

2. Parameterization scheme

The 1D cloud model was employed to derive

the specific water contents of in-cloud vapor, cloud

droplets, the combination of graupels, ice crystals, rain

water and hail particles, and the conversion rate of

specific number concentration from 26 primary micro-

physical processes, each of which is denoted by a cap-

ital letter for the process and two subscripted letters,

the first for the phase of consumption and the sec-

ond for production or action, which are also used to

indicate the change rate of specific mass during mi-

crophysics. Of the physical quantities, the Nu-related

hail sublimation (Svh), melting (Mhr) and the critical

value of hail dry-wet growth (Chw) are of particular

interest in this study.

In the derivation of the model expressions of Hu-

He for wet-dry growth, melting and evaporation of

hailstones, the integral portion is approximated by an

empirical expression. If, for example, the hail subli-

mation expression were treated with that way,

Svh = πkdρ(Qv − Qs0)Nu

∫

∞

D8

N0D1.9exp(−λhD)dD

= 2πkdρ(Qv − Qs0)0.29√

ρAvh�µ

·

∫

∞

D∗

NDD1.9exp(−λhD)dD

≈ 2πkdρ(Qv − Qs0)0.29√

ρAvh�µNhλ1.9h

·[(λhD∗)1.9 + Γ(2.9)(0.9λhD∗ + 1)],

then there would result in roughly 10% error. Instead,

we, by means of the results from integral by parts and

numerical integral, re-derive its precise parameteriza-

tion formula that give calculations in error on the order

of 2%.


2.1 Sublimation (Svh) of hailstones

2.1.1 For the wet growth of hailstones (kk=1)

Svh1 = 2πkdρ(Qv − Qs0)0.26√

ρAvh�µNhλ−1.9h

[

(λhD∗)−1.9

+1.827exp(−2.375)∗(λhD∗1)0.611 + 1.827

]

, (5)

Svh2 = 2πkdρ(Qv − Qs0)0.17(ρAvh�µ)0.57Nhλ−2.03h

[

(λhD∗)2.03 + 2.03(λhD∗)

1.03

+2.06exp(−3.363)(λhD∗)0.595 + 2.06

]

, (6)


[

(λhD∗)2.08 + 2.08(λhD∗)

1.08

+2.16exp(−2.375)(λhD∗)0.611 + 2.16

]

, (7)


[

(λhD∗)2.33 + 2.33(λhD∗)

1.33

+1.049(λhD∗)0.744 + 2.77

]

. (8)

2.1.2 For the dry growth of hailstones (kk=0)

Svh1 = {Eq.(5) −LfkdρQsi

ktT(

Ls

RT− 1)(Cch + Crh)} · [1 +

LskdρQsi

ktT(

Ls

RT− 1)]−1, (9)


ktT(

Ls

RT− 1)(Cch + Crh)} · [1 +

LskdρQsi

ktT(

Ls

RT− 1)]−1, (10)


ktT(

Ls

RT− 1)(Cch + Crh)} · [1 +

LskdρQsi

ktT(

Ls

RT− 1)]−1, (11)


ktT(

Ls

RT− 1)(Cch + Crh)} · [1 +

LskdρQsi

ktT(

Ls

RT− 1)]−1. (12)

2.2 Hail’s melting (Mhr)

Mhr1 =

∫

N0hexp(−λhD)2πD

Lf[kt(T − T0) + Lvkdρ(Qv − Qs0)]0.26

√

Avhρ�µD0.9dD

+Cw

Lf(Cch + Crh)(T − T0)

=0.52

Lf

√

Avhρ�µ[kt(T − T0) + Lvkdρ(Qv − Qs0)]Nhλ−1.9h [(λhD∗)

1.9 + 1.827exp(−2.375)

·(λhD∗)0.611 + 1.827] +

Cw

Lf+ (Cch + Crh)(T − T0), (13)

Mhr2 =0.33π

Lf(Avhρ�µ)0.57[kt(T − T0) + Lvkdρ(Qv − Qs0)]Nhλ−1.9

h [(λhD∗)2.03 + 2.03(λhD∗)

1.03

+2.06(λhD∗)0.595 + 2.06] +

Cw

Lf+ (Cch + Crh)(T − T0), (14)

Mhr3 =0.33π


h [(λhD∗)2.08 + 2.08(λhD∗)

1.08

+2.16exp(−2.375)(λhD∗)0.611 + 2.16] +

Cw

Lf+ (Cch + Crh)(T − T0), (15)

Mhr4 =0.114π


h [(λhD∗)2.38 + 2.33(λhD∗)

1.33

+2.77exp(−0.979)(λhD∗)0.74 + 2.77] +

Cw

Lf+ (Cch + Crh)(T − T0). (16)


2.3 Critical value of wet-dry growth (Chw)

Chw1 =

{∫

N0hexp(−λhD)2πD1.90.26

√

Avhρ

µ[kt(T − T0) + Lvkdρ(Qv − Qs0)]dD

−CihCi(T − T0)

}

/[Lf + Cw(T − T0)]

≈

{

0.53π

√

Avhρ

µ[kt(T − T0) + Lvkdρ(Qv − Qs0)]Nhλ

−1.9h [(λhD∗)

1.9 + 1.827exp(−2.375)

·(λhD∗)0.611 + 1.827] + CihCi(T − T0)

}/

[Lf + Cw(T − T0)], (17)

Chw2 =

{

0.33π(Avhρ

µ)0.57[kt(T − T0) + Lvkdρ(Qv − Qs0)]Nhλ−2.03

h [(λhD∗)2.03 + 2.03(λhD∗)

1.03

+2.06(λhD∗)0.595 + 2.06] + CihCi(T − T0)

}/

[Lf + Cw(T − T0)], (18)

Chw3 =

{

0.313π(Avhρ


h [(λhD∗)2.08 + 2.08(λhD∗)

1.08

+2.16exp(−2.375)(λhD∗)0.611 + 2.16] + CihCi(T − T0)

}/

[Lf + Cw(T − T0)], (19)

Chw4 =

{

0.114π(Avhρ


h [(λhD∗)2.33 + 2.33(λhD∗)

1.33

+2.77exp(−0.979)(λhD∗)0.744 + 2.77] + CihCi(T − T0)

}/

[Lf + Cw(T − T0)]. (20)

3. Calculations

3.1 Calculations from the 1D cloud model

As a case study we computed soundings made at

Dezhou of Shandong Province on 1 July 1989, with

the results shown below.

3.1.1 Critical value of the wet-dry growth of hail-

stones (Chw)

Trends of the growth development denoted as

Chw1 to Chw4 are similar from Eqs.(17)-(20) but Chw2

to Chw4 are bigger compared to Chw1, indicating

that Chw2, Chw3 and Chw4 are ∼10%, ∼30% and

∼40% larger in magnitude than Chw1, respectively

(see Fig.1). With Nu1 inserted into the model, hail

particles begin wet growth from minute 21 at the

6.2 km level where T = −12.411◦C, followed by the

growth layer thickened progressively and finally ex-

tending from the cloud base up to the 6.2 km height at

minute 54. At this time the critical value is 2.89×10−3

g kg−1 s−1 for hail dry-wet growth, with the critical

temperature of 12.029◦C. When Nu2 is substituted

into the model, the hail particles start wet growth at

minute 21 as well except for lowered height (at 6.0 km

level), with −11.050◦C observed; at minute 54 the wet

growth layer extends from the cloud base to the 6.0 km

altitude. At this time the critical value is 2.76×10−3g

kg−1 s−1 for the wet-dry growth, with the critical tem-

perature reaching −12.023◦C. As Nu3 is put into the

model the wet growth occurs in model minute 24 at a

5.8 km height with the temperature of −9.934◦C, fol-

lowed by the growth layer gradually thickened, reach-

ing its maximum layer extending to the height from

cloud base at model minute 51, with dry-wet growth

critical value of 2.35 × 10−3 g kg−1 s−1. Finally, we

substitute Nu4 into the 1D model hail wet growth hap-

pens at minute 24 at 5.8 km level where temperature

is −9.934◦C, and the growth layer extends from cloud

base thereto in minute 51. At this time the dry-wet

growth critical value is 2.61×10−3 g kg−1 s−1 and the

critical temperature is −9.385◦C.

As we see, with Nu1 to Nu4 put into the model,

the wet growth, although beginning at minute 24,


results in different depth of growth layers. The Nu2-

related depth is comparable to that of the Nu1 case

except an extra 200 m thickness calculated at minutes

21 and 24 in the former case. With Nu3 used the

wet growth thickness diminishes by 200-400 m from

minute 27 compared to the Nu1 case, both arriving

at the same depth subsequent to minute 78. In the

Nu4 experiment the moist growth layer is reduced by

400-600 m from model minute 27 in contrast to Nu1

run, reaching the same calculations after minute 81.

Fig.1. The height-dependent distribution critical values of wet-dry growth in units of 10−3g kg−1 s−1, at

minutes 51 (a) and 54 (b).

Fig.2. Hail’s sublimation and evaporation rates in units of 10−3 g kg−1 s−1 at minutes 60 (a) and 63 (b).

3.1.2 Hail’s sublimation (Svh)

Svh1 to Svh4 from Eqs.(5) to (8), respectively,

all reach the order of magnitude of 10−6 at the

6400 m level at minute 33, which increases there-

after. Svh1 (Svh2) arrives at its maximum of 1.93 ×

10−4(2.20 × 10−4) g kg−1 s−1 at 5400 (5600 m) at

minute 63 (see Fig.2a). Svh3(Svh4) has its maximum

of 3.67× 10−4(3.83× 10−4) g kg−1 s−1 at 5600 (5400)

m at minute 60 (Fig.2b), both reducing bit by bit after

minute 63. On the average, Svh2 , Svh3 and Svh4 are by

∼10%, 130% and 200% higher than Svh1, respectively,

indicating that the aspect ratio and surface coarseness

of hail particles have great impacts on sublimation.

3.1.3 Hail melting (Mhr)

Equations (13)-(16) related Mhr1 to Mhr4 each

give values smaller than 1 × 10−6 g kg−1 s−1 prior

to minute 45, after which the melting value reaches

its maximum of 6.3× 10−3, 7.2× 10−3, 8.6× 10−3 and


1.06× 10−3 g kg−1 s−1 for Mhr1 to Mhr4, in order, as

shown in Fig.3b, with subsequent decline and termina-

tion at minute 93 (refer to Fig.3). On the average, the

melting in Mhr2, Mhr3 and Mhr4 is 12%-15%, ∼30%

and ∼50%, respectively, higher compared to that in

Mhr1.

3.1.4 Scale (Xh) and content of hail particles (Qh)

Nu2, Nu3 and Nu4- related hail scale (Xh) and

content of particles (Qh) are larger than those asso-

ciated with Nu1. Qh1 to Qh4 have the maximum of

2.8433, 3.3773, 3.5741 and 3.1423 g kg−1 at the height

of 5600, 5200, 5200 and 5000 m at minutes 66, 69, 69

and 69, respectively.

3.1.5 Rainfall and hailfall with Nu1 to Nu4 intro-

duced into the 1D model

The Nu1 to Nu4-relative rainfall and hailfall are

17.358 and 2.518, 18.002 and 2.410, 18.815 and 2.124,

and 19.471 and 1.694 mm in depth, in order. This

means that owing to hail’s aspect ratio and coarseness

its melting is augmented, leading to increase in rainfall

Fig.3. Hail’s melting rate at minutes 75 (a), 81 (b) and 83 (c), in order. Units: 10−3 g kg−1 s−1.

and decrease in hailfall.

3.2 Calculation by the 3D cloud model

Hailfall occurred on 10 June 1996 in such sub-

urbs of Beijing as Haidian, Xuanwu, Shijingshan and

Miyun Districts, with measured hailstone having its

major (minor) axis, 8 (4) cm long for larger size and

1.5 (0.6) cm in length for small particles. Echoes in-

cluding those from 11 km level, with 6.5-km-level cen-

tral intensity of 70 dBz. Based on Zheng’s heat trans-

fer coefficients (Nu2 to Nu4) introduced into the 3D

model, calculations show that hailfall is diminished,

particle’s size increased, to some degree, and rainfall

strengthed compared to those from the original model.

Nu1-associated rainfall and hailfall are 8.21× 108 and

4.86 × 108 kg, respectively in contrast to rainfall of

8.23 × 108, 8.44 × 108 and 8.66 × 108 kg and hailfall

of 4.84× 108, 4.54 × 108 and 4.23× 108 kg in relation

to the introduced Nu2, Nu3 and Nu4, in order (see

Figs.4 and 5).

It is apparent from Fig.6 that graupel collecting

cloud water for growth (Crg) acts as the most signif-

icant mechanism. Ice crystals are located at a higher

level with respect to cloud water, so that the higher

central concentration for graupel production is at a

lower part of its profile. The graupel-growth by col-

lecting ice crystals (Cig) is increased as quantity of the

crystals grows. But the coalescence of rain droplets

with ice crystals for the growth is losing effect because

of gradually reduced rising velocity and no-existence

of super-cooled rain drops. Growth rate of graupel via

sublimation from vapor (Svg) changes with the amount

of graupel, and sublimation makes insignificant contri-

bution thereto throughout the process, and acts as a

dominant mechanics for the increase of graupel in the

later stage of cloud development. The automatic

conversion of ice crystals into graupel (Aig) is a chief

process of its production (see Fig.6) but nevertheless

Aig plays a small role in the subsequent stage of grau-

pel increase, with too small contribution made by the


Fig.4. Temporally-varying total weight of

hailstones (106 kg) from the 3D model into

which are introduced Nu1 to Nu4.

Fig.5. As in Fig.4 but for the total rainfall.

Fig.6. The time-varying specific mass (in units of 106 kg−1 s−1) of graupel production through the 4

mechanisms in (a) and conversion of ice crystals into graupel (Aig) and melting hailstone for graupel (Mhg)

in (b).

conversion of cloud droplets into graupel to be given

in the figure. However, the coalescence of rain drops

and ice crystals (Cri) generates a certain amount of

graupel. Only a small quantity of water from melting

hail particles that thus reduce their scale is changed

into graupel, which is, however, noticeably larger in

amount compared to graupel from ice crystals.

Figure 7 portrays the evolution of mechanisms for

hail genesis, of which graupel-to-hail conversion (Agh)

and hail growth at the expense of cloud droplets (Cch)

make up a larger proportion and hail growth on rain

droplets (Crh), ice srystals (Crh) and sublimation from

vapor (Svh) in combination are smaller by 1-2 orders

of magnitude compared to the first two factors put to-

gether.

From Fig.8 we see that rain droplet growth by

collecting cloud water (Ccr) and automatic conversion

of cloud (water) into rain water (Acr) represent the


Fig.7. Hail growth by means of graupel to hailstone conversion (Agh) and at the expense of cloud droplets

(Cch) in (a) and by collecting rain drops (Crh), ice crystals (Cih) and the sublimation from vapor (Svh) in

(b).

most important mechanisms for the augmentation of

rain water in the early stage of cloud development.

The conversion of could into rain water is responsible

for incipient rain droplets, which grow fast upon coa-

lescence with cloud water. Graupel begins to melt at

minute 22 when entering the warm section of cloud and

the melting serves as the most prominent mechanism

for producing rain water and also as the extremely

significant source for subsequent persistent rainfall.

The coalescence of super-coolded rain droplets with

ice crystals (Cri) to produce graupel is the predom-

inant mechanism for rain water consumption in the

initial stage of cloud development. In addition, grau-

pel growth via collecting super-cooled raindrops (Crg)

consumes a certain amount of rain water and evapo-

ration of rain drops (Svr) serves as a main mechanics

for rain water consumption throughout cloud develop-

ment, with large quantities of rain water evaporated

in the later stage, sufficiently big to account for nearly

half water from melting graupel, indicating that rain

droplets from graupel melting in the warm portion of

cloud are evaporated in big quantities, failing to reach

ground as rainfall.

Fig.8. Time-dependent 4 mechanisms for rain water production (a) and consumption (b).


As shown in Fig.9, each of the Mhr1 to Mhr4 has

a maximum of 2.217×10−3, 2.222×10−3, 2.659×10−3

and 3.038×10−3 g kg−1 s−1, in order. Figure 10 gives

the maximum of Svh1 to Svh4, which is, respectively,

2.413× 103, 2.403× 103, 3.925× 103 and 5.165× 103.

Fig.9. Time -varying hailstone melting in re-

lation to inserted Nu1 to Nu4.

Fig.10. As in Fig.9 but for hailstone evaporation.

4. Concluding remarks

(1) The heat transfer coefficient for hailstone de

velopment is an innegligible parameter of its growth

rate and structure. The Nusselt number from the mea-

surement of water droplet evaporation under small

Renolds numbers (Re 6 200) used in past studies

would result in bigger difference from reality.

(2) Nu2 to Nu3 of Eqs.(2) to (3) developed by

Zheng (1994) are put, separately, into the 1D and 3D

time-dependent cumulus models, leading to increased

(decreased) rainfall (hailfall). Results related to Nu2

to Nu4 change with the aspect ratio and coarseness of

hail particles, with heat transfer stronger for elliptical

and coarse particles than for spherical and smooth

ones, showing that the melting, evaporation rates and

critical value of hail dry-wet growth are bigger in the

former than in the latter case. on the average, the

rates of hail melting and evaporation (critical value of

its dry-wet growth) are 12%-15% and 10%-200%, in

order, (10%-40%) higher with Nu2 to Nu4 introduced

into the models than with Nu1 employed in the nu-

merical study.

REFERENCES

Hu Zhijin and He Guanfang, 1988: Numerical simulation

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Hu Zhijin and He Guanfang, 1989: Numerical simulation

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rain. Acta Meteor. Sinica, 3(2), 185-199.

List, R., 1963: General heat and mass exchange of spher-

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List, R., 1989: Analysis of sensitivities and error prop-

agation in heat and mass transfer of spheroidal

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Documents

Numerical Simulations of the Physical Process for