a. The crystal shape

The basic crystal habits are thick plates (>−4.0°C), columns (−4.0°∼−8.1°C), plates (−8.1°∼−22.4°C), and columns (<−22.4°C). Isometric crystals are grown around the transition temperatures of the habits or about −4°, −8°, and −22°C. The growth of snow crystal is found to result in shape enhancement in general and riming for an isometric or nearly isometric crystal, as shown in Fig. 1. The modes of shape enhancement are the columnar crystals into needles, the plates into sectors, and the stellar crystals into dendrites.

Except around −5° and −15°C, riming growth is found to commence and, subsequently, graupel-like crystals form within 30 min of growth. In particular, an isometric crystal growing at about −10°C is easily rimed due to its higher fall velocity (see section 2e), which results in an increase of the volume of air swept by the crystal and the high probability of inertial collision of water droplets with the crystal. Riming began from the basal plane of a plate as well as from the prism face of a column. With the liquid water content of 2 g m⁻³, isometric crystals eventually turn into graupel particles, which are nearly spherical in shape. Thus, −10°C is identified as the most effective zone of graupel development. The crystals were confirmed as being embryos of natural graupel (Takahashi and Fukuta 1988a).

In the planar crystal growth region, all crystals are identically double plates. Except for thick plates, the growth takes place preferentially on the larger plate. A thick plate probably does not have stable attitudes and continually turns over; consequently, the two plates grow identically. If the upper plate grows slower due to moisture blocking by the lower plate that meets the upward flow first (Fukuta et al. 1982a), it is likely that the crystal turns over because of the instability of attitude at some point (Iwai 1983), and the larger plate would continue to grow because of its better interception for flow fields and the moisture. Figure 2 illustrates the growth process of double-plate snow crystals. If the size difference between the two plates is small, it is possible that branches on the lower and smaller plate also grow by moisture blocking to result in six branches in the two separate planes.

b. The crystal mass

The early free-fall study in a static chamber identified two mass growth rate maxima, one at around −6°C and the other at around −17°C (see Fig. 3), which correspond to column and plate crystals, respectively. The maxima corresponded to those reported earlier by Hallett (1965) for suspended crystals in a static diffusion chamber although the latter mass growth ratio between the corresponding temperatures was off by a factor of about 10 compared with the former. A recent supercooled tunnel study with the liquid water content of about 0.1 g m⁻³ under atmospheric pressure p = 1010 hPa confirms the existence of the same two maxima but at about −6° and −15°C, respectively (see Fig. 4). The mass growth rate maxima are also observed at these temperatures in the measurement under liquid water content of about 0.5 g m⁻³ and p = 860 hPa, as shown in Fig. 5. These growth rate peaks correspond to minima of the apparent density, as we shall see in section 2d. Also, at around −10°C in Fig. 5, initially a mass growth rate minimum is observed to appear that later rises up to become a maximum as reported by Fukuta et al. (1984) in support of the earlier theoretical prediction (Fukuta 1980). This peak appearance is associated with −10°C fall velocity maximum, as shown later, and confirms a mechanism switch-over from diffusional growth to riming growth. Takahashi et al. (1991) reported that ice crystal mass growth rate increases as the air pressure lowers, in agreement with the diffusional growth theory.

Figure 6 shows the variation of crystal mass with time at −10°C: the crystals were grown by vapor diffusion. After the initial few minutes of growth, the observed mass is in good agreement with the Maxwellian value for a spherical ice crystal calculated by (16), described in section 3b. The time dependence of isometric crystal mass (see Table 2 in Takahashi et al. 1991) falls almost to that of Maxwellian crystal growth with constant axial ratio, or t^3/2. However, the mass within 1 min of growth time is smaller than that estimated from the regression line obtained for growth time longer than 2 min, although the measurements carefully avoided the vapor competition among the growing crystals (Fukuta 1969):the masses of growth times, 0.46 and 0.63 min, are outside of the 75% confidence interval of the regression line. The mass measured by Ryan et al. (1976) and Michaeli and Gallily (1976) are much lower, apparently under the influence of growth competition, which is common when too many large crystals grow together. For a growing Maxwellian crystal, the vapor density and temperature in the air just above the crystal are assumed to be the same as those of the crystal itself, and the crystal temperature is the same everywhere on the surface. This slowness of observed mass growth rate of a small crystal compared with Maxwellian rate was first noticed in the static chamber experiment (Fukuta 1969) and indicates possible influence from accommodation coefficients for heat and vapor exchanges (Fukuta et al. 1974). Nelson and Baker (1996) also reported that the growth rate predicted by a new microscopic model is generally smaller than that of the Maxwellian model. The time dependence for shape-enhanced crystals is greater than the Maxwellian value, which suggests that water vapor is deposited effectively on the spatial crystals compared with the isometric ones.

Figure 7 shows the variation of crystal mass at −10°C up to 30 min of growth time on the liquid water content W_L; the mass under W_L = 0.5 g m⁻³ falls below that of W_L = 2 g m⁻³. The crystals were rimed and turned into graupels, as mentioned in section 2a. The variation was divided into three growth stages, that is, vapor diffusional growth, transitional growth, and graupel growth, as illustrated in Fig. 8. It is also apparent that at the last stage of ice crystal growth around −10°C under high liquid water content (2 g m⁻³), the slope of the log–log plot for mass with time appears to approach the theoretically predicted value of 6, indicating the onset of graupel/hail growth by riming.

c. The crystal dimensions

Figure 9 shows crystal dimensions in a- and c-axial directions as a function of growth temperature. It is clear from Fig. 10 that the most enhanced shape appears at about −5°C for column and at about −15°C for plate with isometric shape between −7° and −10°C after 10 min of growth. At two other temperatures, −4° and −22°C, nearly isometric crystals, or thick plates grow. For an isometric crystal, the crystal dimensions are approximately proportional to t^1/2. The growth of major axis for a shape-enhanced crystal is faster than that for the isometric crystal.

d. The apparent crystal density

The data for the mass and dimensions of the ice crystal grown for a given period of time and environmental condition now permit estimation of the apparent density of ice crystal, which affects the fall behavior. The apparent crystal density ρ_c is defined as the mass divided by the volume that circumscribes the crystal; that is,

View Expanded

where m is the mass, a the radial dimension, and c the height.

Figure 11 shows static chamber measurement of apparent crystal density for crystals that have grown for periods of time up to 1 min. From the figure, it is clear that there exist two minima that correspond to growth rate maxima at about −5° and −15°C, respectively. Two maxima of the density may also be seen: one is related to those crystals that have grown at temperatures above −3°C and the other at around −8° to −10°C, both being the zone of isometric or near-isometric crystals. The same tendency is still seen among crystals that have grown as long as 10 min (see Fig. 12). The maxima around −12° and −17°C are due to a different method applied for the estimation: between −12° and −18°C, the apparent crystal density was calculated as the ratio of the basal plane area to the area of the circumscribing hexagonal plate having an equal diameter.

The apparent crystal density does decrease in general as growth continues. This tendency is stronger at temperatures where the shape enhancement happens, such as at about −5° and −15°C. Figure 13 gives an example of such a change. The density seems to come down to a plateau value and stabilizes at that level in response to dendritic growth.

e. The fall velocity and the fall distance

Figure 14 presents data of ice crystal fall velocity as a function of temperature at different growth times. The two fall velocity minima, at about −5° and −15°C, respectively, correspond to the mass growth rate maxima (cf. Fig. 4), and a fall velocity maximum first discovered by Neubauer (1979) and Fukuta (1980) at −10°C is related to the growth rate minimum due to formation of isometric crystals with small surface area. Thick plates and plates behave as Stokes particles with the Reynolds numbers below 5 (Takahashi et al. 1991).

The time dependence of the fall velocity of growing crystals may be seen in Fig. 15. In the figure, it is apparent that all the curves measured under liquid water content of about 0.5 g m⁻³ are convex upward, but the fall velocity at about −10°C is approximately 1.7 times faster than that for other temperatures. The fall velocity at −4°C is lower in spite of the isometric crystal growth because of the lower growth rate (see Fig. 4). The fall velocity of dendritic crystal within 5 min of growth is almost equal to that of isometric crystals, which is probably ascribed to the existence of the mass maximum at −15°C in Fig. 4. The rate of velocity increase at −15°C gradually slows down due to aerodynamically high resistance caused by dendritic growth. The convex upward general time dependency of the fall velocity remarkably reverses at around −10°C under high liquid water content to become concave upward. Figure 16 illustrates the change. Under a liquid water content of 2 g m⁻³, the curve becomes concave upward at about 15 min of growth, and the change suggests a transition to riming growth of the graupel/hail type.

The fall velocity is integrated with respect to time, with the liquid water content of about 0.3 g m⁻³, for different temperatures at atmospheric pressure p = 860 hPa and is shown in Fig. 17. Ice crystals grown at around −10°C give the largest fall distance.

a. Theoretical basis

1) Diffusional growth of a Maxwellian spherical stationary ice crystal in an infinite atmosphere

The simplest and therefore the most fundamental theory of diffusional ice crystal growth deals with a Maxwellian system in which a spherical ice crystal is placed in an infinite supersaturated atmosphere without fall. The temperature and vapor density just above the crystal surface are assumed to be the same as those of the crystal (Maxwellian condition; see Fukuta and Walter 1970). The growth of the spherical ice crystal through the established, quasi–steady state thermal and vapor fields may be obtained by finding the fields under the condition. When the supersaturated vapor diffuses toward the ice crystal and deposits at the surface, heat is generated. The entire amount of heat generated has to be removed by conduction away from the surface in order to maintain the steady state. The heat conduction rate is expressed by Fourier’s law and the transportation rate of mass of the water vapor by Fick’s law of diffusion. In addition to these two rate equations, the surface of the crystal has to satisfy the condition of saturation or the Clausius–Clapeyron equation, which connects the vapor density to the temperature there. The solution is given as (see Fukuta and Walter 1970)

View Expanded

where r is the radius of the crystal, S the saturation with respect to ice, L the specific latent heat of deposition, K the thermal conductivity of air, R_υ the specific gas constant of water vapor, T_∞ the environmental temperature, D the diffusivity of water vapor in air, and ρ_s the saturated vapor density at T_∞.

2) Corrections due to shape, coexisting cloud droplets, and crystal fall

For a nonspherical shape, Jeffreys’s analogy of electrostatic field was applied to (2) by replacing r with C, the electrostatic capacitance.

Coexistence of cloud droplets in the vicinity of the ice crystal steepens both fields making dm/dt larger by a fog factor (Marshall and Langleben 1954):

f_fkr,(3)

where

kπn_dr_d^1/2(4)

n_d and r_d being the number concentration and the average radius of cloud droplets, respectively.

The fall of the crystal brings ventilation effect to the fields and steepens the gradients in their vicinity, making dm/dt larger by a ventilation factor for mass transfer (Pruppacher and Klett 1978):

View Expanded

where S_c = ν/D is the Schmidt number; R_e = (−w)L/ν the Reynolds number; L the particular characteristic length; and ν = η/ρ_a, η and ρ_a being the dynamic viscosity and density of air, respectively.

Hence, the Maxwellian ice crystal of nonspherical shape and falling in an environment of supercooled cloud may be expressed as

View Expanded

where C is the electrostatic capacitance of ice crystal.

3) Effects of thermal accommodation and deposition coefficients on the growth of small crystals

In regard to exchanging processes of heat and vapor mass between the gaseous and the condensed phases, inefficiency has been known to exist. For heat exchange, the inefficiency is expressed by the thermal accommodation coefficient α, and for the mass onto ice crystal, it is the deposition coefficient γ. The latter is considered to be due to complex surface processes (Fukuta 1978) and may vary depending on the crystal faces and temperature to affect the crystal growth habit. Under the influence of these coefficients, the Maxwellian condition just above the ice crystal surface no longer holds. The solution obtained by Fukuta and Walter (1970) under the influence of thermal accommodation and deposition coefficients, after modification of their representative coefficients with true values (Fukuta and Xu 1996), for a spherical ice crystal in an infinite and quiescent environment is

View Expanded

R_a and c_υ being the specific gas constant for air and the specific heat of air at constant volume, respectively. It is known that when r → ∞, f_α → 1 and f_γ → 1. Under this condition, (7) converges into (2), the corresponding Maxwellian form.

For a nonspherical small crystal using an average γ and assuming γ variation on crystal face to affect C only, (7) takes the form (Fukuta et al. 1974)

View Expanded

Then, considering the effects arising from the coexisting cloud droplets and crystal fall, one can expect the most general form of ice crystal growth by vapor diffusion mechanism as

View Expanded

Equation (11) takes Eqs. (2), (6), (7), and (10) as special cases.

4) Riming growth of graupel and hail

Growth of ice crystal by riming mechanism is known to take place simultaneously with the diffusional mechanism, particularly when the crystal has become large. The mechanism of riming growth is essentially the same as that of the coalescence growth of a raindrop in the cloud. Then the riming growth rate of a spherical graupel may be expressed with sufficient accuracy as

View Expanded

where ρ_g, assumed to be constant, is the density of graupel and E the average riming efficiency.

b. Theoretical interpretation of the measurement

1) The crystal mass

The time dependence of mass may be best understood in comparison with the integral form of the Maxwellian or simplest expression of diffusional growth rate for spherical crystal. Since for constant density of crystal or ρ_c = constant,

View Expanded

where V is the crystal volume, the integral form of (2) for a fixed environmental condition with r = 0 at t = 0, gives

rt^1/2.(14)

Then, considering that for constant ρ_c

mr³(15)

(14) leads to

View Expanded

For a given axial ratio near unity or C ≈ r, (16) holds for Maxwellian crystals. The relationship (16) may be observed for isometric crystals, as mentioned in section 2b. Figure 18 shows comparison of the calculated mass at water saturation with the observed crystal mass in Fig. 4. Both results coincide roughly at the temperatures of isometric crystal growth. The calculated mass reaches a maximum at about −15°C, where the term [ ] in (16) has a maximum. The discrepancy between both results is especially large at about −5°C (after 5 min of growth) and −15°C. This upward departure of the observed mass from the Maxwellian value for a sphere may be interpreted by the following effects: 1) the crystal shape enhances and the axial ratio changes so that the C term increases faster than the size increases and 2) the fall velocity increase leads to increased f_υ term expressed by (5)

Takahashi et al. (1991) discussed the f_υ term. The ventilation effect becomes significant for dendrites and sectors, in agreement with that under forced ventilation (Keller and Hallett 1982; Ohno and Yamashita 1989; Alena et al. 1990). Dendritic growth may be due to the ventilation effect; in the static environment, the growth is observed above several percent of supersaturation with respect to water (Kobayashi 1957; Hallett and Mason 1958). The ventilation effect, however, is not evident for a needle crystal, suggesting that the characteristic length in the flow field is along a axis, even for the needle crystals. This is contrary to their results under forced ventilation; the discrepancy may be ascribed to the airflow direction difference between our experiment and theirs. Also, the effect was not evident for the isometric crystal, although the crystal falls faster and receives more ventilation.

As can be seen in Fig. 6, the observed mass of small crystals is smaller than that of Maxwellian crystal. Under the condition, f_υ ≃ 1 and f_f ≃ 1. Then, effects of f_α and f_γ, or thermal accommodation and deposition coefficients, are expected to come in and reduce the growth rate. The observed crystal mass being smaller than that estimated by (16) may thus be explained by the effects of α and γ (Fukuta and Walter 1970; Fukuta et al. 1974).

If the riming is assumed to take place on a spherical graupel, one can estimate the mass as a function of time by integrating (12) under constant E and Newton’s parabolic law for fall velocity

View Expanded

where ρ_a is the density of air;

mρ⁻³_aρ⁻²_gEW_Lt⁶(18)

Takahashi and Fukuta (1988b) demonstrated that graupel particles with the Reynolds number close to 100 follow approximately the parabolic law. This m ∝ (W_Lt)⁶ relationship was originally obtained by a theoretical analysis (Fukuta 1980) in conjunction with the prediction of the fall velocity peak at around −10°C as described in section 2e, and the m ∝ t⁶ relationship was later experimentally confirmed (Fukuta et al. 1984; Takahashi and Fukuta 1988b). The tendency toward this relationship from the diffusional growth may be recognized in Fig. 7 after about 15 min of growth when W_L is 2 g m⁻³. Figure 19 illustrates this transition.

4) The fall velocity and the fall distance

Under a given environmental condition, the fall velocity of a small spherical crystal, w_c, obeys the Stokes law:

View Expanded

where η is the dynamic viscosity of air. Such a small crystal, excluding the very early stage of growth where the diffusion kinetic effect for mass growth rate (11) and the Cunningham correction for fall velocity (1 + Al/r) (A being the Cunningham constant and l the mean free path of air, respectively) have to be considered, grows practically as a Maxwellian crystal by vapor diffusion. Then from (19) and (14), we can expect

View Expanded

This proportional relationship between w_c and t may be confirmed in Fig. 15 for the early stage of growth; however, the calculated proportionality constant in (20) is 1.5 times larger than the experimental value, probably owing to the difference of the drag coefficient between the sphere and the actual ice crystal, as pointed out by Takahashi and Fukuta (1988b). The fall velocity for a spherical crystal, calculated by (20), had a maximum at about −15°C for the same reason as the calculated mass stated above. Neubauer (1979) and Fukuta (1980) reported that the fall velocity at −15°C was largest for 20 s of growth. This supports the above view that small growing crystals are nearly spherical. The maximum, however, subsequently disappears due to the development of aerodynamically high-resistence shape or dendrites.

Since −w = dz/dt, z being the vertical distance after integration from t = 0, z = 0, (20) yields

zt²w²_c(21)

which is the same as the fall behavior of a body in vacuum. This relationship is recognizable in Fig. 17 for t < 15 min, except at temperatures of dendritic growth.

The second mode of growth for atmospheric ice crystals is the riming. Graupel and hail are typical examples for the mechanism, and their fall velocity is given by (17). Since m ∝ r³ for a constant ρ_g, under the given environmental condition, from (17) and (18), we have the fall velocity of growing graupel/hail

w_gt.(22)

Relationships (20) and (22) differ in their proportionality constants, the latter apparently being larger and causing a concave upward kink in the plot (see Fig. 16).

The time dependency of the fall distance of growing graupel/hail, z_g, is interestingly the same as (21):

z_gt²w²_g(23)

5) Appearance of fall velocity maximum at −10°C

The −10°C fall velocity maximum discovered in the present series of studies opens up a window of graupel/hail evolution and suggests a possible origin of fast-falling crystals required for ice crystal multiplication (Hallett and Mossop 1974). So, we look into the mechanism of the development following the treatment given by Fukuta (1980).

Since both the mass growth rate by vapor diffusion and the fall velocity are mutually related through the shape and size of the ice crystals, we examine the fall velocity development while the crystal is growing by vapor diffusion assuming the shape is held unchanged at the temperature in question. For convenience, spheroidal shapes are employed.

For prolate spheroids of rotation representing columnar crystals at temperatures above −10°C, the gravitational force acting on the crystal balances the viscous resistance force in the Stokes regime, so that the fall velocity of a small crystal is given as (Fuchs 1964)

View Expanded

where b, e, and κ are, respectively, the polar semiaxis;e = (1 − a²/b²)^1/2, where a is the equatorial semi-axis;and

View Expanded

Ignoring corrections due to fog, ventilation, and thermal accommodation and deposition coefficients, (11) gives (Fukuta et al. 1974)

View Expanded

Integrating (26) under a given shape, that is, e = const., and inserting it with (25) and (27) into (24), we have

View Expanded

In this equation, the term in { } converges to unity when e → 1 (long needle), whereas when e → 0 (sphere), the term converges to 4/3. Since the nearly spherical shape happens at −10°C and the rest of (28) becomes largest at −10°C within the temperature range between 0° and −10°C, −10°C is clearly the most favorable temperature for rapid increase of −w.

An identical treatment for oblate spheroids of rotation using

View Expanded

The { } term in (31) converges into unity when e → 1 (flat plate), whereas the term converges into 4/3 when e → 0 (sphere). In this case, although the nearly spherical shape at −10°C is advantageous for w development, other terms, particularly (S − 1)[ ]⁻¹, are likely to give adverse effects. Indeed, the fall velocity at −15°C as described in section 3b(4) appears to be largest at 20 s after the start of growth. However, thereafter, the velocity development at −15°C slows down compared with that at −10°C, suggesting a hindrance due to spatial, aerodynamically high-resistance dendrite growth at −15°C. This implies that in the regime of vapor diffusion growth, the mass increase and the fall velocity development are incompatible.

c. Ice crystal growth kinetics in different cloud environments

Furthermore, the relationship

mW⁶_L(18′)

in (18) explains the importance of high W_L in clouds for triggering the graupel/hail process, and such a condition is more frequently observed in summertime convective clouds, the undiluted core of updraft or weak echo region (WER) in particular. Considering that ρ_g and E probably remain nearly constant, one can derive another relationship from (18):

mρ⁻³_a(18")

This is to say that where the air density is low, as in the High Plains of the United States, graupel and hail are easier to grow. From the cloud dynamics viewpoint, the buoyancy factor that contributes to the vertical acceleration of the cloud parcel

View Expanded

where T and T′ are the temperature of air parcel and that of ambient air, respectively, is larger there due to the smaller mass of air to warm up by a given amount of heat for phase change causing an increase in the T − T′ term. The resultant updraft also makes the available (or effective) fall distance of ice crystals larger in summer convective clouds. From (18) and (23), for the graupel/hail mechanism, we have

mz³(35)

Figure 21 compares the mass and fall distance relationship between diffusional growth of single crystals and riming growth of graupel/hail type. It is clear that in a short fall distance, which is generally found in winter clouds, the diffusional mechanism carries a larger ice mass downward, and the mechanism becomes dominant, whereas after some distance of fall, the graupel/hail mechanism begins to carry a larger mass. Such a long effective fall distance is normally available only in summertime convective clouds; that is, the graupel/hail growth is favored in summertime convective clouds of fast updrafts. The crossing point between the two curves may be considered as the point of mechanism switch-over. As is evident in Fig. 19, this switch-over point will shift to a shorter growth time and, therefore, the shorter fall distance when W_L increases, making the graupel/hail mechanism occur more easily.