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A numerical study of the effects of cloud droplets on the diffusional growth of snow crystals
Atmospheric Research 84 (2007) 353 – 361 www.elsevier.com/locate/atmos
A numerical study of the effects of cloud droplets on the diffusional growth of snow crystals Nesvit E. Castellano, Silvana Gandi, Eldo E. Ávila ⁎ FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina Received 13 July 2006; received in revised form 29 September 2006; accepted 29 September 2006
Abstract We studied the effect of supercooled cloud droplets on the diffusional growth rate of isolated ice plates and ice columns by approximating the shape of a plate as an oblate spheroid and a column as a prolate spheroid. This work is an extension of the pioneering work of Marshall and Palmer [Marshall, J.S. and Langleben, M.P., A theory of snow-crystal habit and growth, J. Met., 11, 104–120, 1954] who studied the influence of cloud droplets on the growth rate of spherical ice crystals. The vapour density field for oblate and prolate spheroids was obtained by solving Helmholtz's equation with the assumption that the vapour at the ice surface had the equilibrium value. As result, it was possible to calculate separately the temporal evolution of the axis length and the mass of the snow crystals. The results indicate that the presence of cloud droplets around the snow crystal can increase by around 10% of the ice growth rate relative to an environment free of cloud droplets. The magnitude of the growth enhancement is proportional to the liquid water content. The results also corroborate the importance of the shape and aspect ratio on the crystal growth. A general mass growth rate equation for diffusional growth was found, which incorporates effects of cloud droplets and crystal shape through the size and aspect ratio of the snow crystal. © 2006 Elsevier B.V. All rights reserved. Keywords: Vapour diffusion; Crystal growth; Mixed-phase cloud
1. Introduction Snow crystals play an important role in atmospheric processes since they are involved in cloud electrification, radiation and precipitation processes and climate in general. Detailed knowledge of ice crystal habit, size, and mass distribution are required for accurate cloud parameterization in global climate models and for the interpretation of remote sensing measurements.
⁎ Corresponding author. E-mail address: [email protected] r (E.E. Ávila). 0169-8095/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.atmosres.2006.09.008
Nakaya et al. (1936a,b) performed the first systematic laboratory study under controlled conditions concerning the variability of snow crystal shape and growth habit. They described the crystal morphology under different environmental conditions and found that the growth of ice crystals from the vapour exhibits complex behaviour as a function of temperature and supersaturation. However, a substantial variability in crystal size and shape has been observed even though the crystals were grown under similar conditions (Nakaya et al., 1936a,b; Kobayashi, 1965; Magono and Lee, 1966). This suggests that crystal growth is quite sensitive to small changes in temperature, supersaturation and other factors such as
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electric fields (Libbrecht et al., 2002), dislocations (Nelson and Knight, 1998; Wood et al., 2001) and cloud liquid water content (Takahashi and Endoh, 2000). This sensitivity affects the measurements and makes it hard to develop good parameterizations of snow crystal growth. Snow ice crystals and supercooled cloud droplets can coexist below 0 °C within mixed-phase cloud regions. Due to the difference of water vapour saturation over ice and liquid, the mix of cloud droplets and ice crystals is unstable and may exist only for a limited time (Young, 1993; Korolev and Isaac, 2003). Depending on the environmental conditions, droplets and crystals may all be growing by vapour deposition, or evaporating, or the ice crystals may grow at the expense of cloud droplets. The last situation is known as the Bergeron–Findeisen mechanism. This mechanism was studied theoretically by Marshall and Langleben (1954) and Castellano et al. (2004). They analyzed the effect on the diffusional growth rate arising from the existence of supercooled cloud droplets at finite distances from a crystal. Marshall and Langleben (1954) presented a model where an isolated ice crystal, represented by a spherical particle, is growing under the presence of a cloud of water droplets, by the Bergeron mechanism. The model assumes that the cloud droplets are uniformly distributed with spherical symmetry throughout the volume around the ice particle and each droplet provides vapour only to the ice particle. In order to solve the problem they assumed a continuous model of water source in which the region surrounding the ice is divided into concentric shells and each shell is providing vapour proportional to its volume and the number of droplets contained in this volume. They found that the change of the spherical particle mass is larger than that obtained without considering the presence of cloud droplets. Castellano et al. (2004) studied the effect of discrete vapour sources representing the cloud droplets upon a spherical ice crystal. They used the method of electrostatic image charges to determine the vapour field in which the crystal grows. The vapour density leads to important differences between the studies close to the cloud droplets since their presence distorts the smooth fields assumed in Marshall and Langleben's model. However, both theories give similar results for the growth rate enhancement of the ice crystal in the presence of water droplets, since these theories found similar values for the vapour density field close to the crystal. Both studies stressed the importance of considering the presence of the cloud droplets on ice crystal growth. However, the spherical shape could not represent adequately the real shape of snow crystals; for this reason it is interesting to extend these studies to other shapes more representative of ice crystals.
The shape of an ice particle influences its optical scattering properties, terminal fall velocity and possibly its growth rate. In order to obtain further knowledge on the growth of snow crystals in the cloud, in this paper we present a mathematical model for modelling the diffusional growth of plates and columnar snow crystals in the presence of a cloud of supercooled water droplets. Plates and columns are modelled as oblate and prolate spheroids respectively. The electrostatic image charges method is inadequate to be applied to the present problem because there is no spherical symmetry. Therefore, based on the similarity of results of snow crystal growth rate obtained by Marshall and Langleben and Castellano et al., similar assumptions to those used in Marshall and Langleben's model will be used in the present work. 2. General equation Our approach is based on the following considerations: – Plate-type crystals are modelled as oblate spheroids, and the columnar-type crystals are modelled as prolate spheroids with the corresponding aspect ratios, Γ = c / a where c and a are the semi-dimensions of the a- and c-axes respectively. – The supercooled cloud droplets are characterized by sources of water vapour continuously distributed in space surrounding the ice crystal. For simplicity, and according to Marshall and Langleben (1954), we consider that the droplets do not change shape or size, and a steady state is assumed. Then, the model describes the situation for short time periods where the particle sizes do not change appreciably. Without the steady state assumption the problem becomes too hard to solve. – The droplets and ice crystal are motionless in space. This means that the ventilation effects were not considered. Following Marshall and Langleben (1954), we assume an ice crystal surrounded by supercooled cloud droplets uniformly distributed in the space. We consider some volume Vo of space, the amount of vapour flowing in unit time through an element dS of the surface bounding this volume is Dv∇ρdS, where ρ is the water vapour density and Dv is the diffusion coefficient or diffusivity of water vapour in air. The total vapour flowing out of the volume Vo in unit time is therefore Y
∮ Dv jqd d S S
ð1Þ
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where the integration is taken over the whole of the closed surface (S) surrounding the volume in question. Assuming that the droplets have spherical shape, the evaporation rate of a droplet of radius rd is 4πrdDv(ρw -ρ) where ρw is the vapour density at the surface of the droplet and ρ is the value of the vapour density of the environment around the droplet (Marshall and Langleben, 1954). The amount of vapour produced by the droplets within the volume Vo in unit time is Z X 4kDv nd rd ðqw −qÞdV ð2Þ d
Vo
where nd is the number of droplets of radius rd per unit volume and the summation is performed over the discrete set of droplet radii. A continuous source of water vapour distributed in space is assumed here instead of the discrete vapour sources representing the cloud droplets. The variation per unit time of the amount of vapour in the volume Vo is given by d dt
Z ð3Þ
qdV Vo
The surface integral (1) can be transformed by the Stokes theorem to a volume integral and the conservation of the vapour mass gives Z Vo
dq dV ¼ dt
Z X Vo Z
4kDv nd rd ðqw −qÞdV
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Assuming a spherical ice crystal of radius a with ρ =ρi at its surface and ρ = ρw at an infinite distance from the particle, the solution of Eq. (6) is given by a q˜ sphere ¼ qsph −qw ¼ ðqi −qw Þ e−k ðr−aÞ r
ð7Þ
where r is the distance from the centre of the crystal. As expected, this result is coincident with the vapour density field found by Marshall and Langleben (1954). In the present work, we have used the boundary condition ρ = ρi at the crystal surface mainly because Marshall and Langleben in their original calculation used this boundary condition and the current work is an extension of that work. However it is important to remark that the selection of the boundary condition on the crystal surface is a matter of debate. The capacitance model of ice crystal growth (Mason, 1971; Pruppacher and Klett, 1997; Young, 1993; Wang, 2002) assumes ρ = ρi. However, the capacitance model cannot describe the habits of the snow crystals in the atmosphere, perhaps because this is not a suitable boundary condition. A different model for the vapour growth of faceted ice crystals has been proposed by Nelson and Baker (1996) and developed by Wood et al. (2001). In this model, the growth of hexagonal ice crystals is studied by assuming that the flux of vapour to the crystal surface is uniform over each flat crystal face. The problem with using this boundary condition is that relatively poorly known crystal properties must be assumed (Wood et al., 2001).
d
Dv j2 qdV
þ
ð4Þ
3. Vapour density field for oblate and prolate spheroids
Vo
Since this equation must hold for any volume, the integrand must vanish dq X ¼ 4kDv nd rd ðqw −qÞ þ Dv j2 q dt d
ð5Þ
This is the convective diffusion equation for water vapour density P field (Pruppacher and Klett, 1997) where the term d 4kDv nd rd ðqw −qÞrepresents the vapour sources in space. Considering that the droplets and ice crystal are motionless in space and in steady state conditions, dρ / dt = 0 and Eq. (5) becomes Helmholtz's equation 2 j þ j2 q˜ ¼ 0 ð6Þ P where ρ˜ = ρ - ρw and κ2 = (ik)2 with k 2 ¼ 4k nd rd : d
The oblate and prolate spheroidal coordinates are shown in Fig. 1(a) and (b) respectively. With these coordinates systems, the general solution of Eq. (6) is (Morse and Feshbach, 1953; Moon and Spencer, 1988) q˜ ðn; g; /Þ ¼
l X
ð1Þ
½An R0n ðkf ; isenhnÞ
n¼0 ð2Þ
ð1Þ
þBn R0n ðkf ; isenhnÞ½Cn S0n ðkf ; cosgÞ ð2Þ
þDn S0n ðkf ; cosgÞ
ð8Þ
where {ξ,η,u} represents the spheroidal coordinates; (1),(2) (kf,isenhξ) are the oblate spheroidal radial funcR0n (1),(2) tions and S0n (kf,cosη) are the oblate spheroidal angular functions; and f is the focal distance of the spheroid (Moon and Spencer, 1988; Fallon, 2001).
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N.E. Castellano et al. / Atmospheric Research 84 (2007) 353–361 (3) (1) (2) Using R0n (kf,isenhξ) = R0n (kf,isenhξ) + iR0n (kf,isenhξ) and the orthogonality of the spheroidal angular functions it is possible to obtain R1 ð1Þ ðq −q ÞS ðkf ; cosgÞdðcosgÞ En ¼ ð3Þ −1 i w R0n1 ð1Þ2 R0n ðkf ; isenhn0 Þ −1 S0n ðkf ; cosgÞdðcosgÞ
ð10Þ A numerical evaluation of this equation shows that the coefficients En for n ≥ 1 are small and can be neglected in comparison to the coefficient E0, in general En / E0 < 10-10 for n ≥ 1. Thus, the vapour density for an oblate spheroid becomes ð3Þ
ð1Þ
qobl ¼ qw þ E0obl R00 ðkf ; isenhnÞS00 ðkf ; cosgÞðqi −qw Þ ð11Þ where R1
E0obl ¼
ð1Þ
S00 ðkf ; cosgÞdðcosgÞ R 1 ð1Þ2 ð3Þ R00 ðkf ; isenhn0 Þ −1 S00 ðkf ; cosgÞdðcosgÞ −1
ð12Þ The vapour density for a prolate spheroid can be obtained in a similar way as was done for the oblate spheroidal case. It can be expressed as qpro ¼ qw ð3Þ ð1Þ þ E0pro R00 ðikf ; coshnÞS00 ðikf ; cosgÞðqi −qw Þ ð13Þ where Fig. 1. Representation of the oblate (a) and prolate (b) spheroidal coordinates systems.
The vapour density is subject to the following boundary conditions: 1– ρ(ξo,η,u) = ρi where the surface ξ = ξo describes the crystal contour. 2– Far away from the crystal the vapour density field should be independent of the crystal shape and it is expected that ρ → ρw in a similar way as ρsph → ρw. i.e. the functional form of the vapour density should be equivalent to e- kr / r when r → ∞ in spherical coordinates. Therefore, the vapour density for an oblate spheroid can be expressed as l X ð1Þ En ½R0n ðkf ; isenhnÞ qobl ¼ qw þ n¼0 ð2Þ
ð1Þ
þ iR0n ðkf ; isenhnÞS0n ðkf ; cosgÞ
ð9Þ
R1
E0pro
¼
ð1Þ
S00 ðikf ; cosgÞdðcosgÞ R 1 ð1Þ2 ð3Þ R00 ðikf ; coshn0 Þ −1 S00 ðikf ; cosgÞdðcosgÞ −1
ð14Þ 4. Results and discussions 4.1. Crystal axis growth On the basis of the theoretical model developed above, this section analyzes the diffusional growth rate for single hexagonal ice plates and ice columns immersed in a cloud of supercooled droplets by approximating the shape of a plate as an oblate spheroid and a column as a prolate spheroid. The characteristic of supercooled cloud droplets is parameterized by the parameter k defined as k 2 ¼ P 4k d nd rd : Rewriting, the sum results in k2 = 4πNdr¯d, where r¯d is the average radius of the cloud droplets and
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Nd is the number concentration of the cloud droplets. For a monodisperse cloud droplet spectrum the relationship between k and the liquid water content of the cloud (W ) can be written as k 2 ¼ d3WPr 2 ;where δw is the liquid water w d density. Typical values of k for the liquid clouds in the troposphere ranges from 0.5 to 8 cm- 1, which belong to W between 0.02 to 5 g m- 3 for ¯r d = 5 μm (Marshall and Langleben, 1954; Korolev and Mazin, 2003). The distribution of the amount of vapour towards the surface of the ice crystal can be analyzed by calculating the component in the ξ-direction of ∇ρ and evaluating it on the crystal surface; for simplicity it will be denoted by ∇ρξ = ξo. The mass of vapour flowing in time Δt through an element ΔS of the crystal surface is Dvjρξ = ξoΔSΔt. The added mass will produce a volume increment given by ΔhΔS, Δh being the height change of the ice crystal in that position. Assuming that the crystal grows with a density δi then the variation rate of the semiaxis length can be approximated by da=dt ¼ ðDv =di Þjqn¼no;
g¼k=2
ð15Þ
g¼0
ð16Þ
and dc=dt ¼ ðDv =di Þjqn¼no;
since ξo is a function of a and c then these equations cannot be solved analytically. The semiaxis variations as a function of time can be calculated using the finite difference assuming all the terms on the right-hand side to be constant during the time step. With these equations we are able to calculate the temporal evolution of the axis length. Fig. 2 shows the variation of crystal major semiaxis with the growth time for different initial aspect ratios, Γini; all the values were calculated for k = 6 cm- 1. Fig. 2(a) displays the evolution of a-semiaxis of an ice plate at -15 °C for Γini = 0.02, 0.05 and 0.1. Fig. 2(b) displays the evolution of c-semiaxis of an ice column at -5 °C for Γini = 10, 5 and 2. In both cases the major semiaxis length at time zero was 25 μm. It can be noted that the evolution of the major axis is strongly dependent on the aspect ratio and the more pronounced axis growth occurs when Γini is far from the unity. These results show the importance of the shape and aspect ratio on the crystal growth. To make a quantitative assessment of the influence of the cloud droplets we compare the variations of ice crystal dimensions with growth time at constant temperature for crystals growing immersed in an environment with supercooled cloud droplets with different k values and an environment free of droplets (k = 0). Actually, an
Fig. 2. The time dependence of the major semiaxis length for ice crystals with different aspect ratios. (a) Ice plates growing at -15 °C. (b) Ice columns growing at -5 °C.
environment free of droplets means that there are no cloud droplets close to the ice particle, however the vapour density far from the crystal is that of liquid water saturation as there are droplets far from the crystal forcing that boundary condition. Fig. 3(a) shows a-semiaxis as a function of time for an ice plate at k = 0, 1, 3 and 6 cm- 1, ambient temperature of -12 °C and Γini = 0.02. Fig. 3(b) shows the temporal evolution of the c-semiaxis for an ice column at the same values of k, ambient temperature of -5 °C and Γini = 10. The trend is very clear, the axis growth increases when the value of k is incremented. From this figure it is possible to observe that the length of the a-semiaxis of an ice plate can be increased by around 10% by the presence of water droplets around snow crystals and the c-semiaxis of an ice column can be increased by 5%. Thus, the cloud droplets appear to increase the diffusional growth of the ice crystals, being proportional to k which is in turn itself proportional to the liquid water content. This result is in agreement with experimental results obtained by Fukuta and Takahashi (1999) and Takahashi and Endoh (2000) who studied the
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Using the theoretical model developed here we found that for an environment free of cloud droplets an isolated ice crystal grows keeping constant its aspect ratio while the proximity of the evaporating water droplets can modify the aspect ratio of the growing ice crystals. In particular, the aspect ratio increases for growing ice columns and decreases for growing ice plates. This effect is more pronounced for ice columns with larger aspect ratios and for ice plates with smaller aspect ratios. Although the vapour supplied by cloud droplets to the crystal is able to modify the aspect ratio during its growth, the observed variations showed little significant change and are not enough to explain the habit that the ice crystals adopt in nature. For instance, ice columns with Γini = 20 and ice plates with Γini = 0.02 could change their aspect ratio up to 5 and 1% respectively after growth for 1000 s. 4.2. Mass growth enhancement In order to study the influence of the presence of cloud d and M d droplets on the ice crystal mass, the values of M o
Fig. 3. The time dependence of the major semiaxis length for different k values. (a) Ice plates with Γini = 0.02 growing at -12 °C. (b) Ice columns with Γini = 10 growing at - 5 °C.
effect on snow crystal growth of cloud droplets surrounding a snow crystal and found that that the growth rate increased with an increase in liquid water content. In order to study the rate between the vapour flux toward the crystal surface (∇ρξ = ξo) calculated in the presence of cloud droplets and in the absence of droplets, as a function of the angular spherical coordinate η, the parameter: / ¼ j jqn¼no with droplets j=j jqn¼no without droplets j ð17Þ is defined. Fig. 4 presents ϕ vs η for columns and plates. Fig. 4(a) displays the ϕ values obtained for an ice column of c = 100 μm, Γ = 10 and k = 6 cm- 1 and Fig. 4 (b) corresponds to an ice plate of a = 100 μm, Γ = 0.1 and k = 6 cm- 1. These results show that the ice surface receives everywhere more vapour when cloud droplets are present. The excess of vapour is not uniformly distributed on the whole surface but there are some preferential regions. Particularly, it is possible to observe that those regions with angular shapes receive more vapour flux than the rest. These regions correspond to η = π / 2 for ice plates and η = 0 and π for ice columns.
Fig. 4. Rate of vapour flux toward the crystal surface calculated in the presence of cloud droplets and in the absence of droplets as a function of the angular spherical coordinate η. (a) Ice column of c = 100 μm, Γ = 10 and k = 6 cm- 1. (b) Ice plate of a = 100 μm, Γ = 0.1 and k = 6 cm- 1.
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of the depositional growth rate enhancement caused by the presence of cloud droplets. Fig. 5 shows the relative difference δ for an ice plate as a function of its larger semi-dimension for different values of k and Γini. Fig. 5 (a), (b) and (c) was calculated for Γini = 0.99, 0.1 and 0.05, respectively. The results show that for ice plates, δ is proportional to a and k and slightly dependent on Γini. The relative difference as a function of c for an ice column is shown in Fig. 6. The predicted δ values were obtained by using the same values of k as in Fig 5 and for Γini = 1.01, 10 and 20. The results again show that δ is proportional to c and k but, unlike ice plates, an important correlation between δ and Γini was found for
Fig. 5. Relative difference between the growth rate in the presence of cloud droplets and in their absence, as a function of the a-semiaxis length of an ice plate for different k values and aspect ratios of 0.99, 0.1 and 0.05.
d and M d being the mass of vapour are determined, M o flowing per unit time towards the surface of the ice crystal in the presence of cloud droplets and in the absence of They can be calculated by : them, respectively. R Y using: M ¼ −Dv S jq:d S ;with the corresponding water vapour density field (Mason, 1971; Pruppacher and Klett, 1997; Young, 1993); the integral is performed over the ice particle surface. · and M · , δ = (M · The relative difference between M o · · M o) / M o, is the parameter used to study the cloud droplet influence as a function of k, the major semiaxis length (a or c) and aspect ratio (Γ ). This parameter is a measure
Fig. 6. Relative difference between the growth rate in the presence of cloud droplets and in their absence, as a function of the c-semiaxis length of an ice column for different k values and aspect ratios of 1.01, 10 and 20.
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ice columns. In fact, it is possible to observe that the value of δ decreases when the aspect ratio increases. It seems intuitive that the major semiaxis length, k and Γini are the primary variables to parameterize the increment of the crystal growth rate for plate and columns caused by the presence of cloud droplets surrounding the ice particle. However, it was not possible for us to determine the functional form of δ, as a function of these variables, which best fits the results. Marshall and Langleben (1954) found that the growth rate of a spherical ice crystal is increased by a factor (1 + ka), a being the radius of the sphere. We will study the relative difference between Md and Md o, as a function of k and the capacitance of the crystals. The capacitance of a prolate spheroid Γ = Γini is given by Cpro = ce / ln[(1 + e)Γ] where e is the eccentricity given by (1 - Γ - 2)1/2 and the capacitance for an oblate spheroid is given by Cobl = ae / arcsin(e) where the eccentricity in this case is given by (1 - Γ 2)1/2. Fig. 7 displays the values of δ against kC for an ice plate (a) and a column (b). The data in the graphs belong
to all the calculated values with k = 1, 3 and 6 cm- 1 displayed in Figs. 5 and 6. This figure shows a linear trend with positive slope for each set of calculated points. The slope of the straight line displayed in the figure has the unitary value in both cases. It can be observed that a departure from linearity occurs at larger values of k and Γ « 1 for plates or Γ » 1 for columns. The results suggest that the diffusional mass growth rate for single hexagonal ice plates or ice columns immersed in a cloud of supercooled droplets can be parameterized by
M ¼ M o ð1 þ kCÞ
ð18Þ
This is consistent with the results obtained by Marshall and Langleben (1954) since the capacitance of a sphere is equal to the radius of the sphere. Although Eq. (18) has the parameter C, which can represent any shape, the equation has been tested only for oblate and prolate spheroids. 5. Summary and conclusions
Fig. 7. Relative difference between the growth rate in the presence of cloud droplets and in their absence, as a function of kC for an ice plate (a) and a column (b).
Crystal growth morphologies arise from the interplay between interface mechanisms dominated by attachment kinetics and temperature and vapour concentration fields around the growing crystal. The combination of these processes is responsible for the ice crystal habits found in nature, so it is important to study how they each affect the crystal growth. In the present work we deal with the second of these physical processes. We present a mathematical extension of Marshall and Langleben's calculation of a growing ice sphere coexisting with supercooled water droplets to the case of ice in the shape of oblate or prolate spheroids, which represent plate- and column-type crystals, respectively. Helmholtz's equation was solved in order to know the vapour density field around oblate and prolate spheroids with the assumption that the vapour at the ice surface had the equilibrium value. With the knowledge of the vapour density field it was possible to study the diffusional growth of these snow crystals. The crystal dimensions in a- and c-axial directions as a function of time were calculated by using Eqs. (15) and (16), which incorporate effects of crystal shape and cloud droplets. The analysis of the results has shown that the crystal axis growth is fairly sensitive to the crystal shape and to the liquid water content. That is to say that the proximity of the evaporating water droplets, which act as sources of vapour for the growing crystals, increases the crystal axis growth by several percent over that determined by an environment free of cloud droplets.
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The influence of cloud droplets on the snow crystal mass was studied as a function of k, the major semiaxis length (a or c) and aspect ratio (Γ ). It was found that the mass growth rate is proportional to the major semiaxis length and k as much in plates as in columns. Besides, the results show that the mass growth rate decreases when the aspect ratio of the columns increases, while the mass growth rate of ice plates shows little significant change with variations of the aspect ratio. A general mass growth rate equation for diffusional growth was found, which also incorporates effects of cloud droplets and crystal shape. Eq. (18) describes the mass of vapour flowing per unit time towards the surface of the ice particle as a function of the parameter k and the crystal capacitance. Although the treatment here uses a highly simplified description of real crystals, it nevertheless shows how nearby cloud droplets can increase the growth rate of nonspherical ice crystals. Acknowledgments This work was supported by SECYT-UNC, CONICET and FONCYT. References Castellano, N.E., Avila, E.E., Saunders, C.P.R., 2004. Theoretical model of the Bergeron–Findeisen mechanism of ice crystal growth in clouds. Atmos. Environ. 38, 39, 6751–6761. Fallon, P.E., 2001. Theory and computation of spheroidal harmonics with general arguments, Thesis presented at the University of Western Australia. Fukuta, N., Takahashi, T., 1999. The growth of atmospheric ice crystals: a summary of finding in vertical supercooled cloud tunnel studies. J. Atmos. Sci. 56, 1963–1979.
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Kobayashi, T., 1965. On the variation of ice crystal habit with temperature. In: Oura, H. (Ed.), Physics of Snow and Ice, Part I. Hokkaideo Univ., Sapporo, Japan. Korolev, A., Isaac, G., 2003. Phase transformation of mixed-phase clouds. Q. J. R. Meteorol. Soc. 129, 19–38. Korolev, A.V., Mazin, I.P., 2003. Supersaturation of water vapor in clouds. J. Atmos. Sci. 60, 2957–2974. Libbrecht, K.G., Crosby, T., Swanson, M., 2002. Electrically enhanced free dendrite growth in polar and non-polar systems. J. Cryst. Growth 240, 241–254. Magono, C., Lee, C.W., 1966. Meteorological classification of natural snow crystals. J. Fac. Sci Hokkaido Univ., Ser. 7 (2), 321. Marshall, J.S., Langleben, M.P., 1954. A theory of snow-crystal habit and growth. J. Met. 11, 104–120. Mason, B.J., 1971. The Physics of Clouds. Oxford Univ. Press. Moon, P., Spencer, D.E., 1988. Field Theory Handbook. Springer Verlag. Morse, P.M., Feshbach, H., 1953. Methods of Theoretical Physics, Part I. McGraw-Hill, New York, p. 663. Nakaya, U., Sato, I., Sekido, Y., 1936a. Preliminary experiments on the artificial production of snow crystals. J. Fac. Sci., Hokkaido Imp. Univ. Ser. II 2, 1–11. Nakaya, U., Yoda, Y., Muruyama, S., 1936b. Further experiments on the artificial production of snow crystals. J. Fac. Sci., Hokkaido Imp. Univ. Ser. II 2, 13–57. Nelson, J., Baker, M.B., 1996. New theoretical framework for studies of vapour growth and sublimation of small ice crystals in the atmosphere. J. Geophys. Res. 101, 7033–7047. Nelson, J., Knight, C.A., 1998. Snow crystal habit changes explained by layer nucleation. J. Atmos. Sci. 55, 1452–1465. Pruppacher, H.R., Klett, J.D., 1997. Microphysics of Clouds and Precipitation. Kluwer Acad. Publ. Takahashi, T., Endoh, T., 2000. Experimental studies on the dendritic growth of a snow crystal in a water cloud. Proc. 13th Int. Conf. on Clouds and Precipitation, Reno, Nevada USA, pp. 677–680. Wang, P., 2002. Ice microdynamics. Academic Press. Wood, S.E., Baker, M.B., Calhoun, D., 2001. New model for the vapor growth of hexagonal ice crystals in the atmosphere. J. Geophys. Res. 106, 4845–4870. Young, K.C., 1993. Microphysical Processes in Clouds. Oxford Univ. Press.
A numerical study of the effects of cloud droplets on the diffusional growth of snow crystals Nesvit E. Castellano, Silvana Gandi, Eldo E. Ávila ⁎ FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina Received 13 July 2006; received in revised form 29 September 2006; accepted 29 September 2006
Abstract We studied the effect of supercooled cloud droplets on the diffusional growth rate of isolated ice plates and ice columns by approximating the shape of a plate as an oblate spheroid and a column as a prolate spheroid. This work is an extension of the pioneering work of Marshall and Palmer [Marshall, J.S. and Langleben, M.P., A theory of snow-crystal habit and growth, J. Met., 11, 104–120, 1954] who studied the influence of cloud droplets on the growth rate of spherical ice crystals. The vapour density field for oblate and prolate spheroids was obtained by solving Helmholtz's equation with the assumption that the vapour at the ice surface had the equilibrium value. As result, it was possible to calculate separately the temporal evolution of the axis length and the mass of the snow crystals. The results indicate that the presence of cloud droplets around the snow crystal can increase by around 10% of the ice growth rate relative to an environment free of cloud droplets. The magnitude of the growth enhancement is proportional to the liquid water content. The results also corroborate the importance of the shape and aspect ratio on the crystal growth. A general mass growth rate equation for diffusional growth was found, which incorporates effects of cloud droplets and crystal shape through the size and aspect ratio of the snow crystal. © 2006 Elsevier B.V. All rights reserved. Keywords: Vapour diffusion; Crystal growth; Mixed-phase cloud
1. Introduction Snow crystals play an important role in atmospheric processes since they are involved in cloud electrification, radiation and precipitation processes and climate in general. Detailed knowledge of ice crystal habit, size, and mass distribution are required for accurate cloud parameterization in global climate models and for the interpretation of remote sensing measurements.
⁎ Corresponding author. E-mail address: [email protected] r (E.E. Ávila). 0169-8095/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.atmosres.2006.09.008
Nakaya et al. (1936a,b) performed the first systematic laboratory study under controlled conditions concerning the variability of snow crystal shape and growth habit. They described the crystal morphology under different environmental conditions and found that the growth of ice crystals from the vapour exhibits complex behaviour as a function of temperature and supersaturation. However, a substantial variability in crystal size and shape has been observed even though the crystals were grown under similar conditions (Nakaya et al., 1936a,b; Kobayashi, 1965; Magono and Lee, 1966). This suggests that crystal growth is quite sensitive to small changes in temperature, supersaturation and other factors such as
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electric fields (Libbrecht et al., 2002), dislocations (Nelson and Knight, 1998; Wood et al., 2001) and cloud liquid water content (Takahashi and Endoh, 2000). This sensitivity affects the measurements and makes it hard to develop good parameterizations of snow crystal growth. Snow ice crystals and supercooled cloud droplets can coexist below 0 °C within mixed-phase cloud regions. Due to the difference of water vapour saturation over ice and liquid, the mix of cloud droplets and ice crystals is unstable and may exist only for a limited time (Young, 1993; Korolev and Isaac, 2003). Depending on the environmental conditions, droplets and crystals may all be growing by vapour deposition, or evaporating, or the ice crystals may grow at the expense of cloud droplets. The last situation is known as the Bergeron–Findeisen mechanism. This mechanism was studied theoretically by Marshall and Langleben (1954) and Castellano et al. (2004). They analyzed the effect on the diffusional growth rate arising from the existence of supercooled cloud droplets at finite distances from a crystal. Marshall and Langleben (1954) presented a model where an isolated ice crystal, represented by a spherical particle, is growing under the presence of a cloud of water droplets, by the Bergeron mechanism. The model assumes that the cloud droplets are uniformly distributed with spherical symmetry throughout the volume around the ice particle and each droplet provides vapour only to the ice particle. In order to solve the problem they assumed a continuous model of water source in which the region surrounding the ice is divided into concentric shells and each shell is providing vapour proportional to its volume and the number of droplets contained in this volume. They found that the change of the spherical particle mass is larger than that obtained without considering the presence of cloud droplets. Castellano et al. (2004) studied the effect of discrete vapour sources representing the cloud droplets upon a spherical ice crystal. They used the method of electrostatic image charges to determine the vapour field in which the crystal grows. The vapour density leads to important differences between the studies close to the cloud droplets since their presence distorts the smooth fields assumed in Marshall and Langleben's model. However, both theories give similar results for the growth rate enhancement of the ice crystal in the presence of water droplets, since these theories found similar values for the vapour density field close to the crystal. Both studies stressed the importance of considering the presence of the cloud droplets on ice crystal growth. However, the spherical shape could not represent adequately the real shape of snow crystals; for this reason it is interesting to extend these studies to other shapes more representative of ice crystals.
The shape of an ice particle influences its optical scattering properties, terminal fall velocity and possibly its growth rate. In order to obtain further knowledge on the growth of snow crystals in the cloud, in this paper we present a mathematical model for modelling the diffusional growth of plates and columnar snow crystals in the presence of a cloud of supercooled water droplets. Plates and columns are modelled as oblate and prolate spheroids respectively. The electrostatic image charges method is inadequate to be applied to the present problem because there is no spherical symmetry. Therefore, based on the similarity of results of snow crystal growth rate obtained by Marshall and Langleben and Castellano et al., similar assumptions to those used in Marshall and Langleben's model will be used in the present work. 2. General equation Our approach is based on the following considerations: – Plate-type crystals are modelled as oblate spheroids, and the columnar-type crystals are modelled as prolate spheroids with the corresponding aspect ratios, Γ = c / a where c and a are the semi-dimensions of the a- and c-axes respectively. – The supercooled cloud droplets are characterized by sources of water vapour continuously distributed in space surrounding the ice crystal. For simplicity, and according to Marshall and Langleben (1954), we consider that the droplets do not change shape or size, and a steady state is assumed. Then, the model describes the situation for short time periods where the particle sizes do not change appreciably. Without the steady state assumption the problem becomes too hard to solve. – The droplets and ice crystal are motionless in space. This means that the ventilation effects were not considered. Following Marshall and Langleben (1954), we assume an ice crystal surrounded by supercooled cloud droplets uniformly distributed in the space. We consider some volume Vo of space, the amount of vapour flowing in unit time through an element dS of the surface bounding this volume is Dv∇ρdS, where ρ is the water vapour density and Dv is the diffusion coefficient or diffusivity of water vapour in air. The total vapour flowing out of the volume Vo in unit time is therefore Y
∮ Dv jqd d S S
ð1Þ
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where the integration is taken over the whole of the closed surface (S) surrounding the volume in question. Assuming that the droplets have spherical shape, the evaporation rate of a droplet of radius rd is 4πrdDv(ρw -ρ) where ρw is the vapour density at the surface of the droplet and ρ is the value of the vapour density of the environment around the droplet (Marshall and Langleben, 1954). The amount of vapour produced by the droplets within the volume Vo in unit time is Z X 4kDv nd rd ðqw −qÞdV ð2Þ d
Vo
where nd is the number of droplets of radius rd per unit volume and the summation is performed over the discrete set of droplet radii. A continuous source of water vapour distributed in space is assumed here instead of the discrete vapour sources representing the cloud droplets. The variation per unit time of the amount of vapour in the volume Vo is given by d dt
Z ð3Þ
qdV Vo
The surface integral (1) can be transformed by the Stokes theorem to a volume integral and the conservation of the vapour mass gives Z Vo
dq dV ¼ dt
Z X Vo Z
4kDv nd rd ðqw −qÞdV
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Assuming a spherical ice crystal of radius a with ρ =ρi at its surface and ρ = ρw at an infinite distance from the particle, the solution of Eq. (6) is given by a q˜ sphere ¼ qsph −qw ¼ ðqi −qw Þ e−k ðr−aÞ r
ð7Þ
where r is the distance from the centre of the crystal. As expected, this result is coincident with the vapour density field found by Marshall and Langleben (1954). In the present work, we have used the boundary condition ρ = ρi at the crystal surface mainly because Marshall and Langleben in their original calculation used this boundary condition and the current work is an extension of that work. However it is important to remark that the selection of the boundary condition on the crystal surface is a matter of debate. The capacitance model of ice crystal growth (Mason, 1971; Pruppacher and Klett, 1997; Young, 1993; Wang, 2002) assumes ρ = ρi. However, the capacitance model cannot describe the habits of the snow crystals in the atmosphere, perhaps because this is not a suitable boundary condition. A different model for the vapour growth of faceted ice crystals has been proposed by Nelson and Baker (1996) and developed by Wood et al. (2001). In this model, the growth of hexagonal ice crystals is studied by assuming that the flux of vapour to the crystal surface is uniform over each flat crystal face. The problem with using this boundary condition is that relatively poorly known crystal properties must be assumed (Wood et al., 2001).
d
Dv j2 qdV
þ
ð4Þ
3. Vapour density field for oblate and prolate spheroids
Vo
Since this equation must hold for any volume, the integrand must vanish dq X ¼ 4kDv nd rd ðqw −qÞ þ Dv j2 q dt d
ð5Þ
This is the convective diffusion equation for water vapour density P field (Pruppacher and Klett, 1997) where the term d 4kDv nd rd ðqw −qÞrepresents the vapour sources in space. Considering that the droplets and ice crystal are motionless in space and in steady state conditions, dρ / dt = 0 and Eq. (5) becomes Helmholtz's equation 2 j þ j2 q˜ ¼ 0 ð6Þ P where ρ˜ = ρ - ρw and κ2 = (ik)2 with k 2 ¼ 4k nd rd : d
The oblate and prolate spheroidal coordinates are shown in Fig. 1(a) and (b) respectively. With these coordinates systems, the general solution of Eq. (6) is (Morse and Feshbach, 1953; Moon and Spencer, 1988) q˜ ðn; g; /Þ ¼
l X
ð1Þ
½An R0n ðkf ; isenhnÞ
n¼0 ð2Þ
ð1Þ
þBn R0n ðkf ; isenhnÞ½Cn S0n ðkf ; cosgÞ ð2Þ
þDn S0n ðkf ; cosgÞ
ð8Þ
where {ξ,η,u} represents the spheroidal coordinates; (1),(2) (kf,isenhξ) are the oblate spheroidal radial funcR0n (1),(2) tions and S0n (kf,cosη) are the oblate spheroidal angular functions; and f is the focal distance of the spheroid (Moon and Spencer, 1988; Fallon, 2001).
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N.E. Castellano et al. / Atmospheric Research 84 (2007) 353–361 (3) (1) (2) Using R0n (kf,isenhξ) = R0n (kf,isenhξ) + iR0n (kf,isenhξ) and the orthogonality of the spheroidal angular functions it is possible to obtain R1 ð1Þ ðq −q ÞS ðkf ; cosgÞdðcosgÞ En ¼ ð3Þ −1 i w R0n1 ð1Þ2 R0n ðkf ; isenhn0 Þ −1 S0n ðkf ; cosgÞdðcosgÞ
ð10Þ A numerical evaluation of this equation shows that the coefficients En for n ≥ 1 are small and can be neglected in comparison to the coefficient E0, in general En / E0 < 10-10 for n ≥ 1. Thus, the vapour density for an oblate spheroid becomes ð3Þ
ð1Þ
qobl ¼ qw þ E0obl R00 ðkf ; isenhnÞS00 ðkf ; cosgÞðqi −qw Þ ð11Þ where R1
E0obl ¼
ð1Þ
S00 ðkf ; cosgÞdðcosgÞ R 1 ð1Þ2 ð3Þ R00 ðkf ; isenhn0 Þ −1 S00 ðkf ; cosgÞdðcosgÞ −1
ð12Þ The vapour density for a prolate spheroid can be obtained in a similar way as was done for the oblate spheroidal case. It can be expressed as qpro ¼ qw ð3Þ ð1Þ þ E0pro R00 ðikf ; coshnÞS00 ðikf ; cosgÞðqi −qw Þ ð13Þ where Fig. 1. Representation of the oblate (a) and prolate (b) spheroidal coordinates systems.
The vapour density is subject to the following boundary conditions: 1– ρ(ξo,η,u) = ρi where the surface ξ = ξo describes the crystal contour. 2– Far away from the crystal the vapour density field should be independent of the crystal shape and it is expected that ρ → ρw in a similar way as ρsph → ρw. i.e. the functional form of the vapour density should be equivalent to e- kr / r when r → ∞ in spherical coordinates. Therefore, the vapour density for an oblate spheroid can be expressed as l X ð1Þ En ½R0n ðkf ; isenhnÞ qobl ¼ qw þ n¼0 ð2Þ
ð1Þ
þ iR0n ðkf ; isenhnÞS0n ðkf ; cosgÞ
ð9Þ
R1
E0pro
¼
ð1Þ
S00 ðikf ; cosgÞdðcosgÞ R 1 ð1Þ2 ð3Þ R00 ðikf ; coshn0 Þ −1 S00 ðikf ; cosgÞdðcosgÞ −1
ð14Þ 4. Results and discussions 4.1. Crystal axis growth On the basis of the theoretical model developed above, this section analyzes the diffusional growth rate for single hexagonal ice plates and ice columns immersed in a cloud of supercooled droplets by approximating the shape of a plate as an oblate spheroid and a column as a prolate spheroid. The characteristic of supercooled cloud droplets is parameterized by the parameter k defined as k 2 ¼ P 4k d nd rd : Rewriting, the sum results in k2 = 4πNdr¯d, where r¯d is the average radius of the cloud droplets and
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Nd is the number concentration of the cloud droplets. For a monodisperse cloud droplet spectrum the relationship between k and the liquid water content of the cloud (W ) can be written as k 2 ¼ d3WPr 2 ;where δw is the liquid water w d density. Typical values of k for the liquid clouds in the troposphere ranges from 0.5 to 8 cm- 1, which belong to W between 0.02 to 5 g m- 3 for ¯r d = 5 μm (Marshall and Langleben, 1954; Korolev and Mazin, 2003). The distribution of the amount of vapour towards the surface of the ice crystal can be analyzed by calculating the component in the ξ-direction of ∇ρ and evaluating it on the crystal surface; for simplicity it will be denoted by ∇ρξ = ξo. The mass of vapour flowing in time Δt through an element ΔS of the crystal surface is Dvjρξ = ξoΔSΔt. The added mass will produce a volume increment given by ΔhΔS, Δh being the height change of the ice crystal in that position. Assuming that the crystal grows with a density δi then the variation rate of the semiaxis length can be approximated by da=dt ¼ ðDv =di Þjqn¼no;
g¼k=2
ð15Þ
g¼0
ð16Þ
and dc=dt ¼ ðDv =di Þjqn¼no;
since ξo is a function of a and c then these equations cannot be solved analytically. The semiaxis variations as a function of time can be calculated using the finite difference assuming all the terms on the right-hand side to be constant during the time step. With these equations we are able to calculate the temporal evolution of the axis length. Fig. 2 shows the variation of crystal major semiaxis with the growth time for different initial aspect ratios, Γini; all the values were calculated for k = 6 cm- 1. Fig. 2(a) displays the evolution of a-semiaxis of an ice plate at -15 °C for Γini = 0.02, 0.05 and 0.1. Fig. 2(b) displays the evolution of c-semiaxis of an ice column at -5 °C for Γini = 10, 5 and 2. In both cases the major semiaxis length at time zero was 25 μm. It can be noted that the evolution of the major axis is strongly dependent on the aspect ratio and the more pronounced axis growth occurs when Γini is far from the unity. These results show the importance of the shape and aspect ratio on the crystal growth. To make a quantitative assessment of the influence of the cloud droplets we compare the variations of ice crystal dimensions with growth time at constant temperature for crystals growing immersed in an environment with supercooled cloud droplets with different k values and an environment free of droplets (k = 0). Actually, an
Fig. 2. The time dependence of the major semiaxis length for ice crystals with different aspect ratios. (a) Ice plates growing at -15 °C. (b) Ice columns growing at -5 °C.
environment free of droplets means that there are no cloud droplets close to the ice particle, however the vapour density far from the crystal is that of liquid water saturation as there are droplets far from the crystal forcing that boundary condition. Fig. 3(a) shows a-semiaxis as a function of time for an ice plate at k = 0, 1, 3 and 6 cm- 1, ambient temperature of -12 °C and Γini = 0.02. Fig. 3(b) shows the temporal evolution of the c-semiaxis for an ice column at the same values of k, ambient temperature of -5 °C and Γini = 10. The trend is very clear, the axis growth increases when the value of k is incremented. From this figure it is possible to observe that the length of the a-semiaxis of an ice plate can be increased by around 10% by the presence of water droplets around snow crystals and the c-semiaxis of an ice column can be increased by 5%. Thus, the cloud droplets appear to increase the diffusional growth of the ice crystals, being proportional to k which is in turn itself proportional to the liquid water content. This result is in agreement with experimental results obtained by Fukuta and Takahashi (1999) and Takahashi and Endoh (2000) who studied the
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Using the theoretical model developed here we found that for an environment free of cloud droplets an isolated ice crystal grows keeping constant its aspect ratio while the proximity of the evaporating water droplets can modify the aspect ratio of the growing ice crystals. In particular, the aspect ratio increases for growing ice columns and decreases for growing ice plates. This effect is more pronounced for ice columns with larger aspect ratios and for ice plates with smaller aspect ratios. Although the vapour supplied by cloud droplets to the crystal is able to modify the aspect ratio during its growth, the observed variations showed little significant change and are not enough to explain the habit that the ice crystals adopt in nature. For instance, ice columns with Γini = 20 and ice plates with Γini = 0.02 could change their aspect ratio up to 5 and 1% respectively after growth for 1000 s. 4.2. Mass growth enhancement In order to study the influence of the presence of cloud d and M d droplets on the ice crystal mass, the values of M o
Fig. 3. The time dependence of the major semiaxis length for different k values. (a) Ice plates with Γini = 0.02 growing at -12 °C. (b) Ice columns with Γini = 10 growing at - 5 °C.
effect on snow crystal growth of cloud droplets surrounding a snow crystal and found that that the growth rate increased with an increase in liquid water content. In order to study the rate between the vapour flux toward the crystal surface (∇ρξ = ξo) calculated in the presence of cloud droplets and in the absence of droplets, as a function of the angular spherical coordinate η, the parameter: / ¼ j jqn¼no with droplets j=j jqn¼no without droplets j ð17Þ is defined. Fig. 4 presents ϕ vs η for columns and plates. Fig. 4(a) displays the ϕ values obtained for an ice column of c = 100 μm, Γ = 10 and k = 6 cm- 1 and Fig. 4 (b) corresponds to an ice plate of a = 100 μm, Γ = 0.1 and k = 6 cm- 1. These results show that the ice surface receives everywhere more vapour when cloud droplets are present. The excess of vapour is not uniformly distributed on the whole surface but there are some preferential regions. Particularly, it is possible to observe that those regions with angular shapes receive more vapour flux than the rest. These regions correspond to η = π / 2 for ice plates and η = 0 and π for ice columns.
Fig. 4. Rate of vapour flux toward the crystal surface calculated in the presence of cloud droplets and in the absence of droplets as a function of the angular spherical coordinate η. (a) Ice column of c = 100 μm, Γ = 10 and k = 6 cm- 1. (b) Ice plate of a = 100 μm, Γ = 0.1 and k = 6 cm- 1.
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of the depositional growth rate enhancement caused by the presence of cloud droplets. Fig. 5 shows the relative difference δ for an ice plate as a function of its larger semi-dimension for different values of k and Γini. Fig. 5 (a), (b) and (c) was calculated for Γini = 0.99, 0.1 and 0.05, respectively. The results show that for ice plates, δ is proportional to a and k and slightly dependent on Γini. The relative difference as a function of c for an ice column is shown in Fig. 6. The predicted δ values were obtained by using the same values of k as in Fig 5 and for Γini = 1.01, 10 and 20. The results again show that δ is proportional to c and k but, unlike ice plates, an important correlation between δ and Γini was found for
Fig. 5. Relative difference between the growth rate in the presence of cloud droplets and in their absence, as a function of the a-semiaxis length of an ice plate for different k values and aspect ratios of 0.99, 0.1 and 0.05.
d and M d being the mass of vapour are determined, M o flowing per unit time towards the surface of the ice crystal in the presence of cloud droplets and in the absence of They can be calculated by : them, respectively. R Y using: M ¼ −Dv S jq:d S ;with the corresponding water vapour density field (Mason, 1971; Pruppacher and Klett, 1997; Young, 1993); the integral is performed over the ice particle surface. · and M · , δ = (M · The relative difference between M o · · M o) / M o, is the parameter used to study the cloud droplet influence as a function of k, the major semiaxis length (a or c) and aspect ratio (Γ ). This parameter is a measure
Fig. 6. Relative difference between the growth rate in the presence of cloud droplets and in their absence, as a function of the c-semiaxis length of an ice column for different k values and aspect ratios of 1.01, 10 and 20.
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ice columns. In fact, it is possible to observe that the value of δ decreases when the aspect ratio increases. It seems intuitive that the major semiaxis length, k and Γini are the primary variables to parameterize the increment of the crystal growth rate for plate and columns caused by the presence of cloud droplets surrounding the ice particle. However, it was not possible for us to determine the functional form of δ, as a function of these variables, which best fits the results. Marshall and Langleben (1954) found that the growth rate of a spherical ice crystal is increased by a factor (1 + ka), a being the radius of the sphere. We will study the relative difference between Md and Md o, as a function of k and the capacitance of the crystals. The capacitance of a prolate spheroid Γ = Γini is given by Cpro = ce / ln[(1 + e)Γ] where e is the eccentricity given by (1 - Γ - 2)1/2 and the capacitance for an oblate spheroid is given by Cobl = ae / arcsin(e) where the eccentricity in this case is given by (1 - Γ 2)1/2. Fig. 7 displays the values of δ against kC for an ice plate (a) and a column (b). The data in the graphs belong
to all the calculated values with k = 1, 3 and 6 cm- 1 displayed in Figs. 5 and 6. This figure shows a linear trend with positive slope for each set of calculated points. The slope of the straight line displayed in the figure has the unitary value in both cases. It can be observed that a departure from linearity occurs at larger values of k and Γ « 1 for plates or Γ » 1 for columns. The results suggest that the diffusional mass growth rate for single hexagonal ice plates or ice columns immersed in a cloud of supercooled droplets can be parameterized by
M ¼ M o ð1 þ kCÞ
ð18Þ
This is consistent with the results obtained by Marshall and Langleben (1954) since the capacitance of a sphere is equal to the radius of the sphere. Although Eq. (18) has the parameter C, which can represent any shape, the equation has been tested only for oblate and prolate spheroids. 5. Summary and conclusions
Fig. 7. Relative difference between the growth rate in the presence of cloud droplets and in their absence, as a function of kC for an ice plate (a) and a column (b).
Crystal growth morphologies arise from the interplay between interface mechanisms dominated by attachment kinetics and temperature and vapour concentration fields around the growing crystal. The combination of these processes is responsible for the ice crystal habits found in nature, so it is important to study how they each affect the crystal growth. In the present work we deal with the second of these physical processes. We present a mathematical extension of Marshall and Langleben's calculation of a growing ice sphere coexisting with supercooled water droplets to the case of ice in the shape of oblate or prolate spheroids, which represent plate- and column-type crystals, respectively. Helmholtz's equation was solved in order to know the vapour density field around oblate and prolate spheroids with the assumption that the vapour at the ice surface had the equilibrium value. With the knowledge of the vapour density field it was possible to study the diffusional growth of these snow crystals. The crystal dimensions in a- and c-axial directions as a function of time were calculated by using Eqs. (15) and (16), which incorporate effects of crystal shape and cloud droplets. The analysis of the results has shown that the crystal axis growth is fairly sensitive to the crystal shape and to the liquid water content. That is to say that the proximity of the evaporating water droplets, which act as sources of vapour for the growing crystals, increases the crystal axis growth by several percent over that determined by an environment free of cloud droplets.
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The influence of cloud droplets on the snow crystal mass was studied as a function of k, the major semiaxis length (a or c) and aspect ratio (Γ ). It was found that the mass growth rate is proportional to the major semiaxis length and k as much in plates as in columns. Besides, the results show that the mass growth rate decreases when the aspect ratio of the columns increases, while the mass growth rate of ice plates shows little significant change with variations of the aspect ratio. A general mass growth rate equation for diffusional growth was found, which also incorporates effects of cloud droplets and crystal shape. Eq. (18) describes the mass of vapour flowing per unit time towards the surface of the ice particle as a function of the parameter k and the crystal capacitance. Although the treatment here uses a highly simplified description of real crystals, it nevertheless shows how nearby cloud droplets can increase the growth rate of nonspherical ice crystals. Acknowledgments This work was supported by SECYT-UNC, CONICET and FONCYT. References Castellano, N.E., Avila, E.E., Saunders, C.P.R., 2004. Theoretical model of the Bergeron–Findeisen mechanism of ice crystal growth in clouds. Atmos. Environ. 38, 39, 6751–6761. Fallon, P.E., 2001. Theory and computation of spheroidal harmonics with general arguments, Thesis presented at the University of Western Australia. Fukuta, N., Takahashi, T., 1999. The growth of atmospheric ice crystals: a summary of finding in vertical supercooled cloud tunnel studies. J. Atmos. Sci. 56, 1963–1979.
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