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The Growth of Atmospheric Ice Crystals: A Summary of Findings in Vertical Supercooled Cloud Tunnel Studies

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  • 1 Department of Meteorology, University of Utah, Salt Lake City, Utah
  • | 2 Center for Educational Research and Development, Hokkaido University of Education, Sapporo, Japan
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Abstract

Measurements of ice crystal growth under free fall in a generation of vertical supercooled cloud tunnels and some static cloud chambers as well as the related theoretical works are summarized.

Growth parameters, that is, mass (m), dimensions, apparent density, and fall velocity (w), show extrema at about −5°, −10°, and −15°C where crystals are predominantly column-needle, isometric, and plate-stellar-dendrite, respectively. Crystal shape enhances with time (t) at about −5° and −15°C, whereas at −10°C the effect is minimal and crystals show strongest tendency to grow into graupel due to the fall velocity maximum discovered early in the series of present studies. At this temperature, switch-over of growth mode toward graupel occurs more quickly as liquid water content (WL) increases. Under a fixed cloud condition, the shape-enhanced crystals hardly grow into graupel and vice versa.

The diffusional growth theory, with Maxwellian surface condition and without ventilation, describes well the behaviors of intermediate size crystals for which mt3/2 ∝ (−w)3/2 ∝ (−z)3/4; z, being the fall distance, is identified. Small crystals grow more slowly due to accommodation coefficient effects and larger ones grow faster due to enhanced ventilation and riming. To include these effects, a generalized growth theory is formulated. A simple theory is developed for graupel/hail growth where mρ−3a(WLt)6w6 ∝ (−z)3, ρa being the air density. Based on these relationships, the dominance of diffusional growth mechanism for precipitation development in shallow, convectively weak, winter clouds and that of graupel/hail-type riming growth in deep, convectively strong, summer clouds is explained.

Corresponding author address: Dr. Norihiko Fukuta, Department of Meteorology, University of Utah, Salt Lake City, UT 84112.

Email: nfukuta@atmos.met. utah.edu

Abstract

Measurements of ice crystal growth under free fall in a generation of vertical supercooled cloud tunnels and some static cloud chambers as well as the related theoretical works are summarized.

Growth parameters, that is, mass (m), dimensions, apparent density, and fall velocity (w), show extrema at about −5°, −10°, and −15°C where crystals are predominantly column-needle, isometric, and plate-stellar-dendrite, respectively. Crystal shape enhances with time (t) at about −5° and −15°C, whereas at −10°C the effect is minimal and crystals show strongest tendency to grow into graupel due to the fall velocity maximum discovered early in the series of present studies. At this temperature, switch-over of growth mode toward graupel occurs more quickly as liquid water content (WL) increases. Under a fixed cloud condition, the shape-enhanced crystals hardly grow into graupel and vice versa.

The diffusional growth theory, with Maxwellian surface condition and without ventilation, describes well the behaviors of intermediate size crystals for which mt3/2 ∝ (−w)3/2 ∝ (−z)3/4; z, being the fall distance, is identified. Small crystals grow more slowly due to accommodation coefficient effects and larger ones grow faster due to enhanced ventilation and riming. To include these effects, a generalized growth theory is formulated. A simple theory is developed for graupel/hail growth where mρ−3a(WLt)6w6 ∝ (−z)3, ρa being the air density. Based on these relationships, the dominance of diffusional growth mechanism for precipitation development in shallow, convectively weak, winter clouds and that of graupel/hail-type riming growth in deep, convectively strong, summer clouds is explained.

Corresponding author address: Dr. Norihiko Fukuta, Department of Meteorology, University of Utah, Salt Lake City, UT 84112.

Email: nfukuta@atmos.met. utah.edu

1. Introduction

Ice phase processes are known to play important roles in various weather phenomena. Yet our knowledge has been severely limited due to the complexity of the processes at microscales, intermediate scales, and macroscales.

The variability of snow crystal shape or growth habit was first delineated by Nakaya and his school (Nakaya et al. 1938ab; Hanajima 1944 1949; Kobayashi 1957 1961) based on the parameters of the growth environment, that is, the temperature and supersaturation. The Nakaya diagram represents the shape of crystals grown in the static environment and describes morphology of crystal growth based on the above two environmental parameters. The growth habit appearance is obviously due to the change of the process at the crystal surface under different environmental conditions. The mechanisms involved there are those of microscopic nature, and while it is necessary to clarify the detailed mechanism (Mason et al. 1963; Hobbs and Scott 1965; Lamb and Hobbs 1971; Lamb and Scott 1974; Kuroda 1982;Kuroda and Lacmann 1982; Fukuta and Wang 1984; Fukuta 1986; Fukuta and Lu 1994), the knowledge of ice crystal growth is needed at the intermediate scale, that is, mass, shape, size, and fall velocity as a function of time and the condition of the growth environment, in order to apply to thermodynamic and dynamic conditions of macroscale or cloud scale.

Early knowledge on the growth of falling atmospheric ice crystals through aerodynamic interaction is summarized by Mason (1965): two main modifications have been introduced to the idealized treatment of ice crystal growth first suggested by Jeffreys (1918), which applies an analogy between electrostatic and vapor or temperature fields and makes the infinite fields of the latter established around a spherical body in a quiescent medium [originally treated by Maxwell (1890)] applicable to nonspherical bodies. One is the effect arising from the existence of supercooled cloud droplets at finite distances, which distorts the infinite fields assumed in Jeffreys’s theory and makes the rate of the growth higher (Marshall and Langleben 1954). The other is the effect due to the crystal fall, which also deforms the infinite fields to make the growth rate again higher.

The above theoretical treatments to describe the growth behaviors of atmospheric ice crystals still leave several problems unresolved. First of all, due to the inefficiencies associated with exchanges of mass of water vapor and heat between the gaseous environment and ice crystal surface, or effects of deposition and thermal accommodation coefficients, the growth rate is reduced. This effect was experimentally found to be significant in the early stages of growth (Fukuta 1969) and theoretically clarified (Fukuta and Walter 1970; Fukuta et al. 1974). Secondly, the electrostatic capacitance term in Jeffreys’s theory is known to change with respect to time during the growth even if the temperature and supersaturation of the environment remain the same and the change is often complex. The capacitance variation is due to the change in shape and size, and under the free-fall of natural ice crystal growth, the aerodynamic field around the crystal in turn modifies the growth behaviors of the crystal. For these reasons, prediction of the growth of atmospheric ice crystal over a long period of time poses a considerable difficulty.

A logical approach to this problem is a simulation experiment of ice crystal growth under free-fall. One of the earliest studies (Fukuta 1969) along this line of approach employed a small static cloud chamber, and crystals were grown under free-fall and a condition clearly defined that avoids competition among them for available moisture. This may be called an “Eulerian” experiment since the observer is placed in the Eulerian coordinate relative to the air medium in which the crystal in question is growing. Although the study successfully produced ice crystal growth parameters as a function of time within the useful range of temperature, the maximum range of time was only on the order of one minute. The time period of this Eulerian experiment was subsequently extended using larger chambers (Ryan et al. 1974 1976; Michaeli and Gallily 1976). However, in addition to their lack of measurement for aerodynamic behavior of the growing and falling ice crystals and existence of competitive growth condition among growing crystals to smear the environmental condition, the time period of the experiments was less than 3 min. The cloud chamber method introduced by Song and Lamb (1994ab), using a rotating updraft of supercooled fog, could extend the time of measurement up to 4 min. The Eulerian method apparently reached a limit, and beyond this time limit, the cost and effort of the study were expected to increase drastically. Yet experimental studies were indeed needed for much longer periods of time.

A possible solution to this problem in the study of atmospheric ice crystal growth is “Lagrangian” experiment, or application of a vertical wind tunnel technique and suspension of a single ice crystal in front of the observer in a vertical stream of artificially generated supercooled cloud in the tunnel, although the technique was not established at that time. Under the circumstances, vertical supercooled cloud tunnels in which a single snow crystal can be suspended freely and grown by applying aerodynamic mechanisms for horizontal stability have been developed and evolved at the University of Utah (Neubauer 1979; Fukuta 1980; Kowa 1981; Fukuta et al. 1982a; Fukuta et al. 1984; Gong and Fukuta 1985; Matsuo and Fukuta 1987; Takahashi and Fukuta 1988b) and later at Hokkaido University (Takahashi et al. 1991). In these tunnels, the growth of an isolated snow crystal under free-fall in a supercooled cloud environment was successfully simulated for up to 30 min and sometimes more than 1 h at −15°C. The sizes of dendrites and needles grown for 30 min in the experiments reached 4 mm in diameter and 2 mm in length, respectively, which are almost equal to the maximum sizes of natural snow crystals. The details of the tunnel technique are to be published elsewhere.

In this paper, we shall report the summary of measurements carried out by our vertical supercooled cloud tunnels and address the behaviors of growing ice crystal under free-fall with the help of proper growth theories.

2. Main findings from measurements

A wealth of new and crucial information has emerged out of the initial static chamber (Eulerian) experiments and the following vertical supercooled cloud tunnel (Lagrangian) measurements. We describe the experimentally determined behaviors of the major growth parameters of ice crystals.

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 t3/2. 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 WL; the mass under WL = 0.5 g m−3 falls below that of WL = 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 t1/2. 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 ρc is defined as the mass divided by the volume that circumscribes the crystal; that is,
i1520-0469-56-12-1963-e1
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.

3. Discussions

Analysis of the experimental findings requires a heuristic approach in reference to theoretical structure, and we begin the analysis with a description of the structure.

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)
i1520-0469-56-12-1963-e2
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, Rυ the specific gas constant of water vapor, T the environmental temperature, D the diffusivity of water vapor in air, and ρs 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):
ffkr,
where
kπndrd1/2
nd and rd 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):
i1520-0469-56-12-1963-e5
where Sc = ν/D is the Schmidt number; Re = (−w)L/ν the Reynolds number; L the particular characteristic length; and ν = η/ρa, η and ρa 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
i1520-0469-56-12-1963-e6
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
i1520-0469-56-12-1963-e7
Ra and cυ being the specific gas constant for air and the specific heat of air at constant volume, respectively. It is known that when r → ∞, fα → 1 and fγ → 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)
i1520-0469-56-12-1963-e10
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
i1520-0469-56-12-1963-e11
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
i1520-0469-56-12-1963-e12
where ρg, 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 ρc = constant,
i1520-0469-56-12-1963-e13
where V is the crystal volume, the integral form of (2) for a fixed environmental condition with r = 0 at t = 0, gives
rt1/2.
Then, considering that for constant ρc
mr3
(14) leads to
i1520-0469-56-12-1963-e16
For a given axial ratio near unity or Cr, (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 fυ term expressed by (5)

Takahashi et al. (1991) discussed the fυ 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, fυ ≃ 1 and ff ≃ 1. Then, effects of fα and fγ, 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
i1520-0469-56-12-1963-e17
where ρa is the density of air;
mρ−3aρ−2gEWLt6

Takahashi and Fukuta (1988b) demonstrated that graupel particles with the Reynolds number close to 100 follow approximately the parabolic law. This m ∝ (WLt)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 mt6 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 WL is 2 g m−3. Figure 19 illustrates this transition.

2) The crystal shape and dimensions

The crystal dimensions are apparently determined by the surface processes with the influence of crystal fall in a given environment, and a Nakaya diagram of growth habit in a stationary environment provides some information. Figure 20 shows the diagram determined in a well-defined, that is, steady-state, field of supersaturation (Swoboda 1981; Fukuta et al. 1982b; Wang and Fukuta 1985) as opposed to that obtained in a nonsteady-state field by Kobayashi (1957, 1961). The main difference may be found below water saturation.

The shape and dimension of crystals under free-fall growth are known to be influenced by the air ventilation in a feedback manner. The methods for their estimation (e.g., Mason 1993; Chen and Lamb 1994) normally involve empirical relationships one way or another, and providing experimental data under clearly defined free-fall conditions should be of considerable value. However, in general, smaller crystals possess less-enhanced shape, suggesting that they are close to spherical. This is probably because the Maxwellian vapor flux is proportional to r−1 (Fukuta and Walter 1970), and this high flux in spherical symmetry activates the growth mechanism of both prism and basal planes to establish a near spherical shape. For large crystals, on the contrary, the vapor flux becomes gentle and, under the condition, the difference of deposition coefficient between the basal and prism planes leads to crystal shape development and enhancement. As is evident from Fig. 13, dendrite structure that lowers apparent crystal density does not start for some time from the beginning of the growth.

For a Maxwellian spherical ice crystal, rt1/2, or (14) holds. The same relationship approximately applies for dimensions of isometric or near-isometric crystals at around −4°, −8°, and −22°C, while for crystals with axial ratios significantly deviated from unity such as those around −5° and −15°C, the exponent of the time function is larger (Takahashi et al. 1991).

3) The apparent crystal density

It is difficult to theoretically estimate the apparent density, and to do it one has to resort to experimental data. Nevertheless, the density is an indirect indication of the effect of the capacitance term, which involves shape and size and affects the growth rate. Growth rate is thus inversely related to apparent density but not necessarily linearly.

4) The fall velocity and the fall distance

Under a given environmental condition, the fall velocity of a small spherical crystal, wc, obeys the Stokes law:
i1520-0469-56-12-1963-e19
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
i1520-0469-56-12-1963-e20
This proportional relationship between wc 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
zt2w2c
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 mr3 for a constant ρg, under the given environmental condition, from (17) and (18), we have the fall velocity of growing graupel/hail
wgt.
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, zg, is interestingly the same as (21):
zgt2w2g

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)
i1520-0469-56-12-1963-e24
where b, e, and κ are, respectively, the polar semiaxis;e = (1 − a2/b2)1/2, where a is the equatorial semi-axis;and
i1520-0469-56-12-1963-e25
Ignoring corrections due to fog, ventilation, and thermal accommodation and deposition coefficients, (11) gives (Fukuta et al. 1974)
i1520-0469-56-12-1963-e26
Integrating (26) under a given shape, that is, e = const., and inserting it with (25) and (27) into (24), we have
i1520-0469-56-12-1963-e28
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
i1520-0469-56-12-1963-e29
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.

6) Empirical equation for modeling and analysis

The knowledge of ice crystal growth obtained in the series of vertical supercooled cloud tunnel studies up to 1988 has been summarized in forms of empirical equations backboned with analytic equations, instead of best-fit polynomials, and is made available (Redder and Fukuta 1989, 1991). Such empirical equations provide better access for handling small variations in environmental and crystal conditions from those used in the measurements. Takahashi et al. (1991) provide some empirical relationships among more recently determined parameters.

c. Ice crystal growth kinetics in different cloud environments

The aforementioned theoretical treatments provide a view of ice crystal growth processes in different cloud environments, such as relatively shallow winter clouds of weak convection and deeper and taller summer clouds of stronger convective motions. For diffusionally growing, isolated Maxwellian ice crystals, from (16) and (21), we have
mz3/4
or
zt2w2cm4/3
For riming growth of graupel/hail, from (23) and (18) with mr3, we have
mw6g
suggesting that, for the growth into graupel/hail, the fall velocity is one of the key factors for embryo crystals to satisfy. Then, it is clear that the −10°C fall velocity peak experimentally discovered is highly advantageous for triggering the riming growth of graupel/hail. This theoretically predicted window of graupel/hail growth (Fukuta 1980) actually meets experimental supports (Fukuta et al. 1984; Takahashi and Fukuta 1988b).
Furthermore, the relationship
mW6L
in (18) explains the importance of high WL 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 ρg and E probably remain nearly constant, one can derive another relationship from (18):
mρ−3a
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
i1520-0469-56-12-1963-e34
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 TT′ 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

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 WL increases, making the graupel/hail mechanism occur more easily.

4. Conclusions

The simulation experiments of ice crystal growth under free-fall in a generation of vertical supercooled cloud tunnels and in earlier static chambers, through subsequent theoretical analyses, have brought to light much crucial information on the behavior of growing ice crystals in supercooled clouds, which was badly lacking before.

Apart from the quantitative data of crystal growth parameters made available under various environmental conditions, the main findings are as follows.

  1. Two mass maxima by vapor diffusional growth at about −5° and −15°C, which correspond to minima of apparent crystal density and fall velocity. At these temperatures, the crystal shape enhances, and the corresponding apparent density and the increment of fall velocity decrease with time. The −5°C mass maximum is related to columnar crystals that grow into needles, and the −15°C maximum is due to the plates that grow into stellar crystals and dendrites. In shape-enhanced planar crystals, ventilation effects become recognizable.

  2. A mass minimum by vapor diffusional growth at about −10°C, which corresponds to a maximum of apparent crystal density and fall velocity. The crystals are isometric, and shape enhancement is not recognizable. The crystals show a strong tendency to grow into graupel, particularly when the liquid water content is high.

  3. Incompatibility between vapor diffusional growth of shape-enhanced crystals and riming growth of graupel/hail. Once one of the processes starts under a given condition, the crystals will have difficulty switching into the other mechanism.

  4. Growth of double plates in the plate growth region and their transfer into single plates with embryonic plates often attached underneath.

  5. A general mass growth rate equation for diffusional growth, (11), is derived, which incorporates effects of thermal accommodation and deposition coefficients, the crystal shape and size, the coexisting cloud droplets and air ventilation due to crystal fall. The equation describes, in tendency, the observed deviation of mass below the Maxwellian value without ventilation at the early stage of growth and that above the value in the later stage due to the ventilation effect.

  6. For the vapor diffusional growth of the Maxwellian crystal with negligible ventilation effect under the Stokes fall, approximately
    mt3/2r3wc3/2z3/4
  7. For the riming growth of graupel/hail, approximately
    mρ−3aρ−2gEWLt6wg6z3
  8. Transition of the time dependence of the fall velocity from the vapor diffusion type to the riming of graupel/hail type at −10°C.

  9. A temperature of −10°C as a preferential for graupel/hail growth due to the fall velocity maximum, and its confirmation by experiment.

  10. At temperatures of −10°C a possible zone of generating fast-falling crystals required for Hallett–Mossop mechanism of ice crystal multiplication.

  11. The reason for fall velocity maximum development at −10°C is explained based on the mass growth rate–fall velocity interaction.

  12. Diffusional mechanism of ice crystal growth is more efficient for ice phase precipitation when the available growth and fall distance is small and vice versa for that of riming growth of graupel/hail type. Effective fall distance becomes large in convective clouds.

  13. A kinetic and dynamic interpretation as to why snowfall happens predominantly from winter storms and graupel/hail from summer convective clouds is obtained.

  14. Empirical equations summarizing ice crystal growth data on analytic equation are made available.

Acknowledgments

This work was partly supported by the Division of Atmospheric Sciences, National Science Foundation, under Grant ATM-9626600.

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Fig. 1.
Fig. 1.

Modes of snow crystal growth: (a) shape-enhanced crystal growth and (b) nearly isometric crystal to riming growth. The crystals were obtained by the supercooled cloud tunnel experiments (Takahashi and Fukuta 1988b; Takahashi et al. 1991).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 2.
Fig. 2.

Illustration of the growth process of double-plate snow crystals: (a) thick plate and (b) other planar crystals. The axial ratios (2a/c) for thick plates, which grew above −4.0°C, from −8.1° to −11.3°C and from −18.4° to −22.4°C, ranged from 1 to ∼5 and those for other planar crystals were above about 5 for 10 min of growth (see Fig. 10). The Reynolds number ranged from 0.3 to 5 (thick plates and plates), 1 to 25 (sectors), and 1.5 to 90 (dendrites).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 3.
Fig. 3.

Mass of ice crystals plotted as a function of temperature (Fukuta 1969).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 4.
Fig. 4.

Variation of ice crystal mass with temperature at different growth times under a liquid water content of 0.1 g m−3 and an atmospheric pressure of 1010 hPa (Takahashi et al. 1991).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 5.
Fig. 5.

Variation of ice crystal masses with temperature under a liquid water content less than 0.5 g m−3 and an atmospheric pressure of 860 hPa. Areas with open circles show where rimed crystals were observed (Takahashi and Fukuta 1988b).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 6.
Fig. 6.

Ice crystal mass plotted as a function of time at −10.6°C.

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 7.
Fig. 7.

Variation of ice crystal masses with growth time at −10.5°C (Takahashi and Fukuta 1988b).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 8.
Fig. 8.

Illustration of three growth stages of an isometric crystal.

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 9.
Fig. 9.

Variations of ice crystal dimensions with temperature at different growth times (Takahashi et al. 1991).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 10.
Fig. 10.

Variation of ice crystal axial ratio (2a/c) with temperature after 10 min of growth (Takahashi et al. 1991).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 11.
Fig. 11.

Apparent crystal density plotted as a function of temperature for single ice crystal (45–50 s after seeding; Fukuta 1969).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 12.
Fig. 12.

Variation of apparent crystal density with temperature at the growth time of 10 min. The data were obtained by Takahashi et al. (1991).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 13.
Fig. 13.

Variation of apparent crystal density at about −14°C with growth time.

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 14.
Fig. 14.

Variation of crystal fall velocity with temperature at different growth times (Takahashi et al. 1991).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 15.
Fig. 15.

Variation of crystal fall velocity at different temperatures (Takahashi and Fukuta 1988b).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 16.
Fig. 16.

Variation of crystal fall velocity with time at −10.5°C with a liquid water content of 2 g m−3. The dashed line shows the case with a liquid water content of less than 0.5 g m−3 (Takahashi and Fukuta 1988b).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 17.
Fig. 17.

Variation of crystal fall distance with time at various temperatures under atmospheric pressure of 860 hPa and liquid water content of about 0.3 g m−3 (Takahashi and Fukuta 1988b).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 18.
Fig. 18.

Comparison of calculated mass using the Eq. (16) at water saturation for the growth time of 3 and 10 min at different temperatures with the observed crystal mass in Fig. 4.

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 19.
Fig. 19.

The predicted switch-over of the ice crystal growth mechanism.

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 20.
Fig. 20.

The variation of ice crystal growth habit with temperature and supersaturation (Fukuta et al. 1982; Wang and Fukuta 1984).

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

Fig. 21.
Fig. 21.

Relationship between mass and fall distance for diffusional ice crystal growth and riming of graupel/hail.

Citation: Journal of the Atmospheric Sciences 56, 12; 10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2

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