Effective Density Derived from Laboratory Measurements of the Vapor Growth Rates of Small Ice Crystals at −65° to −40°C

Gwenore F. Pokrifka aDepartment of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania

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Alfred M. Moyle aDepartment of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania

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Jerry Y. Harrington aDepartment of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania

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Abstract

An electrodynamic levitation thermal-gradient diffusion chamber was used to grow 268 individual, small ice particles (initial radii of 8–26 μm) from the vapor, at temperatures ranging from −65° to −40°C, and supersaturations up to liquid saturation. Growth limited by attachment kinetics was frequently measured at low supersaturation, as shown in prior work. At high supersaturation, enhanced growth was measured, likely due to the development of branches and hollowed facets. The effects of branching and hollowing on particle growth are often treated with an effective density ρeff. We fit the measured time series with two different models to estimate size-dependent ρeff values: the first model decreases ρeff to an asymptotic deposition density ρdep, and the second models ρeff by a power law with exponent P. Both methods produce similar results, though the fits with ρdep typically have lower relative errors. The fit results do not correspond well with models of isometric or planar single-crystalline growth. While single-crystalline columnar crystals correspond to some of the highest growth rates, a newly constructed geometric model of budding rosette crystals produces the best match with the growth data. The relative frequency of occurrence of ρdep and P values show a clear dependence on ice supersaturation normalized to liquid saturation. We use these relative frequencies of ρdep and P to derive two supersaturation-dependent mass–size relationships suitable for cloud modeling applications.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Gwenore F. Pokrifka, gfp5025@psu.edu

Abstract

An electrodynamic levitation thermal-gradient diffusion chamber was used to grow 268 individual, small ice particles (initial radii of 8–26 μm) from the vapor, at temperatures ranging from −65° to −40°C, and supersaturations up to liquid saturation. Growth limited by attachment kinetics was frequently measured at low supersaturation, as shown in prior work. At high supersaturation, enhanced growth was measured, likely due to the development of branches and hollowed facets. The effects of branching and hollowing on particle growth are often treated with an effective density ρeff. We fit the measured time series with two different models to estimate size-dependent ρeff values: the first model decreases ρeff to an asymptotic deposition density ρdep, and the second models ρeff by a power law with exponent P. Both methods produce similar results, though the fits with ρdep typically have lower relative errors. The fit results do not correspond well with models of isometric or planar single-crystalline growth. While single-crystalline columnar crystals correspond to some of the highest growth rates, a newly constructed geometric model of budding rosette crystals produces the best match with the growth data. The relative frequency of occurrence of ρdep and P values show a clear dependence on ice supersaturation normalized to liquid saturation. We use these relative frequencies of ρdep and P to derive two supersaturation-dependent mass–size relationships suitable for cloud modeling applications.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Gwenore F. Pokrifka, gfp5025@psu.edu

1. Introduction

The early growth of ice in clouds is a challenging problem, and little is known about the shapes and growth rates of small (radius ≲ 50 µm) ice crystals immediately after nucleation. As a newly nucleated ice particle begins to grow from the vapor, facets develop on the crystal, altering its growth rate. The formation and growth of facets, and thus particle shape, is controlled by gas-phase vapor diffusion and attachment kinetics. “Attachment kinetics” refers to the set of processes in which vapor molecules adsorb onto and diffuse along the ice surface, until they either desorb or incorporate into the ice. When the ice supersaturation (hereafter “supersaturation,” si) is high, facet instabilities can lead to the development of branching and hollowing on crystals. Much of the interior surface area of a branched or hollowed particle is “shadowed” and experiences low supersaturation, resulting in slow growth compared to the crystal’s extremities that extend further into the vapor field where si is higher (Nelson 2001). The extremities then grow rapidly, and it is the resistance to vapor growth on much of the particle due to attachment kinetics that causes the particle to increase in size and mass faster than a solid ice sphere. This is true even for small particles which can become branched and hollowed (Magee et al. 2021). However, no exact method exists to treat such processes.

Due to the effects of attachment kinetics, mass is not added uniformly across a growing ice particle; otherwise, it would remain spherical. Thus, to model the growth of an ice particle, one must make some assumptions regarding the distribution of added mass. For example, the basic planar or columnar habit of a crystal may be approximated as a spheroid (Chen and Lamb 1994), or simply as a sphere. But in each case, any “complexity” in the crystal such as branching, hollowing, and nonspherical habit is treated with an “effective” density that is reduced from the bulk ice density (∼920 kg m−3).

A variety of techniques have been used to estimate the effective densities of ice particles. One approach is to acquire in situ estimates of cloud particle sizes and masses (Heymsfield and Iaquinta 2000; Baker and Lawson 2006; Erfani and Mitchell 2016). While in situ observations have the benefit of examining entire populations of particles, their studies typically cannot determine the density of small crystals. The results from Cotton et al. (2013) are an exception, suggesting that particles with radii less than 35 μm are characterized by a constant effective density of 700 kg m−3. Limits on the optical resolution of airborne probes can add uncertainty to the shapes and sizes of small particles; however, Erfani and Mitchell (2016) estimated effective densities of particles with maximum dimensions down to 20 μm using a cloud particle imager. Further uncertainty arises from the fact that two-dimensional projections of particles can correspond to numerous particle orientations and three-dimensional geometries (Dunnavan and Jiang 2019; Dunnavan et al. 2019).

Another method of estimating the effective density of ice is from ground-based observations of precipitating snowflakes (Muramoto et al. 1995; Brandes et al. 2007; Rees et al. 2021; Leinonen et al. 2021). One benefit of ground-based versus aircraft observations is that the ice particles can be imaged more than once and from multiple directions. For example, a Multi-Angle Snowflake Camera (MASC; Garrett et al. 2015) utilizes multiple camera views to reliably record a snowflake’s three-dimensional geometry (Dunnavan et al. 2019; Rees et al. 2021; Leinonen et al. 2021). However, the effective densities derived from ground-based observations of large (radius ≳ 0.5 mm) snowflakes are not applicable to the small crystals (radius < 100 μm). Both aircraft and ground-based methods only have instantaneous views of particles and, thus, cannot provide information on how effective density changes in time. Yet the time variation of particle mass and size is required to evaluate and improve growth models.

Time variation of the effective ice density may be derived from laboratory measurements. Prior experiments have grown ice particles in cloud chambers (Fukuta 1969; Ryan et al. 1974, 1976; Weitzel et al. 2020) and wind tunnels (Matsuo and Fukuta 1987; Takahashi and Fukuta 1988; Takahashi et al. 1991). Wind tunnel experiments suspend crystals for tens of minutes and show that effective densities can decline substantially and rapidly, particularly in pronounced habit regimes (Fukuta and Takahashi 1999). However, all of the prior laboratory measurements of effective density known to the authors were at temperatures (T) higher than −25°C. Polycrystalline ice is more likely to form as T decreases (Parungo and Weickmann 1973; Bacon et al. 2003; Bailey and Hallett 2004) and polycrystals can have substantially lower effective densities than single crystals (Ryan et al. 1976). Unfortunately, no prior time series measurements of the growth of small ice crystals exist at low temperatures and high supersaturation. Without such measurements, it is not possible to assess model-generated growth rates, even those derived from instantaneous in situ observations.

We address this lack of data by presenting results from experiments using small crystals grown in a levitation thermal diffusion chamber at T < −40°C. These experiments are used to estimate effective densities for vapor growth. In the following sections, we describe the experimental design and the crystal growth model used to fit the experimental data. In sections 4 and 5 we estimate effective densities from the data and use a newly developed budding rosette model to interpret the data. In section 6, two parameterizations of effective-density–size, and therefore, mass–size relationships are derived. These parameterizations link laboratory-determined growth rates to the effective density, therefore allowing the use of measured growth rates in cloud models. We end in section 7 with a discussion and summary of our findings.

2. Levitation diffusion chamber ice growth experiments

Our experiments involve growing small (initial radius r0 of 8–26 μm and equivalent-mass spherical radius rs < 40 μm) ice particles inside the Button Electrode Levitation (BEL) thermal-gradient diffusion chamber. The chamber is described in detail by Harrison et al. (2016), so we will discuss it only briefly here. The BEL chamber consists of parallel copper plates at the top and bottom, which are separated by a plastic ring 1.27 cm tall and 10.2 cm in diameter. The aspect ratio of 8:1 is sufficiently large so that wall effects are minimized (Elliott 1971). The plate temperatures are controlled independently, and the bottom plate has a lower temperature for thermal stability. Water vapor is supplied by ice-covered filter paper that is affixed to the interior plate surfaces. There are holes in the filter paper on the top plate for four button electrodes and an opening through which droplets are introduced to the chamber. Simulations and measurements reveal that the supersaturation at the center of the chamber may be approximated with diffusion chamber theory (Pokrifka et al. 2020), where the supersaturation is controlled by the difference in the plate temperatures.

Charged liquid water droplets are launched into the BEL chamber. The droplets quickly freeze and are levitated by an opposing direct current voltage applied to the bottom plate, and they are stabilized horizontally by an alternating current on the upper electrodes. The charge applied to the ice particles is less than that which could cause electrically enhanced growth (Bacon et al. 2003; Davis 2010; Harrison et al. 2016). During an experiment, the ice particle grows from the vapor, increasing the voltage necessary for levitation. Stable levitation is maintained by software that automatically adjusts the bottom-plate voltage, and records the time series thereof at 1 Hz. The measured voltage is then normalized by its value at the beginning of the experiment. That normalization is equivalent to the mass ratio, mr = m/m0, where m is the particle’s mass and m0 is its mass at the beginning of the experiment. We determine m0 by illuminating the particle with a helium–neon laser and match the resulting diffraction patterns to Mie theory, which gives the initial spherical radius (r0). As the particle grows, the diffraction patterns gradually become disordered, indicating that the particle is no longer spherical. Beyond the initial time, no further size information is directly measured; it must instead be inferred from the mass ratio time series.

The water used in these experiments is either pure high-pressure liquid chromatography (HPLC) water for homogeneous nucleation (177 experiments) or a 0.2 g L−1 mixture of the bionucleant Snomax in HPLC water (following Harrison et al. 2016) for heterogeneous nucleation (91 experiments). However, the key results of this study do not show any significant nucleation dependence (i.e., the presence or lack of ice nucleating particles); thus, we will be presenting them in aggregate.

Experiments were conducted at atmospheric pressure (∼970 hPa) with conditions that have constant temperatures ranging from −65° to −40°C and supersaturations from <1% to liquid saturation. These conditions cover lower T and higher si than previous experiments with the BEL chamber (Harrison et al. 2016; Pokrifka et al. 2020), which used T between −45° and −30°C. For the present experiments, the device has been upgraded such that the copper plates are now cooled using methanol, which is sealed into its housing with fluorosilicone gaskets, instead of Syltherm coolant sealed with Buna-N gaskets. This modification allows the plates to be cooled to ∼−70°C without coolant leaks, and the BEL chamber can produce si near liquid saturation for temperatures down to −60°. Since high supersaturation requires a large difference between the plate temperatures, achieving liquid saturation at T < −60°C would risk causing methanol leaks. Thus, the changes to the device have enabled us to explore experimental conditions where there has been a lack of ice growth data. Next, to interpret the growth data, we compare them to a vapor growth model, as described below.

3. Ice vapor growth model

We analyze the growth data using the Diffusion Surface Kinetics Ice Crystal Evolution (DiSKICE) model (Harrington et al. 2019). The DiSKICE model can develop the primary habits of faceted ice with the inclusion of attachment kinetics, and it has been used to successfully interpret prior laboratory growth data (Harrison et al. 2016; Harrington et al. 2019; Pokrifka et al. 2020). The DiSKICE model treats the far-field gas-phase diffusion to the ice particle with capacitance theory and vapor attachment with the theory of faceted growth. The resulting growth equation,
dmdt=4πC(c,a)siρeqDeff[T,P,α(T,si)],
is similar in form to the capacitance model, where C(c, a) is the geometric capacitance that depends on the a and c semiaxis lengths, si is the ice supersaturation, and ρeq is the ice equilibrium vapor density. The effective vapor and thermal diffusivity Deff includes the influences of attachment kinetics through the deposition coefficient α, which may have different values for the a and c semidimensions, allowing for the development of the primary habit forms (i.e., plates and columns).
The deposition coefficient is parameterized by Nelson and Baker (1996) as
α=(ssurfschar)Mtanh(scharssurf)M,
where ssurf is the supersaturation at the particle surface, schar is a characteristic supersaturation, and M is a growth mechanism parameter that ranges from 1 to 30. For example, spiral dislocations are represented by M = 1, while M ≥ 10 is used for step nucleation. Outcroppings of dislocations on the surface likely control the growth of ice following nucleation (Nelson and Knight 1998), producing efficient growth (with α ≳ 0.1). However, the development of thin crystals with pronounced aspect ratios requires step nucleation, which is far less efficient (Frank 1982). Consequently, the growth mode may vary both spatially across a particle (Wood et al. 2001) and temporally (Pokrifka et al. 2020).
At high supersaturation, the deposition coefficients approach unity; therefore, attachment kinetics no longer limit growth. But they remain in control of the mass distribution over a particle as branching and/or hollowing occur. Branching and hollowing, and other complexities, are treated through an effective density ρeff. The DiSKICE model incorporates a time-varying effective density following Chen and Lamb (1994), in which the change in volume V of an enclosing shape about a particle is the change in its mass divided by the density of the deposited ice, or deposition density ρdep. Their Eq. (41) can be written as
dmdt=d(ρeffV)dt=ρdepdVdt.
The deposition density is temperature and supersaturation dependent, and is determined from measurements (Chen and Lamb 1994). If a crystal formed from a frozen water droplet with an initial density of ρi = 920 kg m−3, integration of the above equation assuming a constant ρdep gives
ρeff(t)=ρiV0V(t)+ρdep[1V0V(t)],
where V0 is the initial particle volume. Equation (4) clearly shows that the effective density in the DiSKICE model decreases in proportion to V−1, or r3 if the enclosing shape is a sphere. It should be noted that such a decrease can be too rapid for larger crystals (Schrom et al. 2021). However, this method is advantageous since the effective density will asymptotically approach ρdep as the volume increases.
Many prior studies have instead characterized ρeff with a size-dependent power law (Mitchell 1996; Cotton et al. 2013) or polynomial (Erfani and Mitchell 2016). Since we cannot physically interpret multiple polynomial-fit parameters using a single measured quantity (mass ratio), we also estimate ρeff with DiSKICE using a power law,
ρeff(t)=ρi[V0V(t)]PV=ρi[r0r(t)]P,
where r0 is the initial radius, r(t) is the enclosing spherical radius at time t, and P = 3PV is an adjustable parameter determined from best fits to the data. The mass growth rate is then
dmdt=d(ρeffV)dt=ρiV0PVdV1PVdt=(1PV)ρeffdVdt.
Note that the rightmost equation has the same form as that of Eq. (3), except that the deposition density has the form (1 − PV)ρeff, which declines with increasing volume. This form has the disadvantage that ρeff is not asymptotic and may become nonphysical for large particles. However, the decrease of the deposition density with size leads to an initially slower, and more realistic, decline in ρeff than Eq. (4) (Schrom et al. 2021). Since both forms of the effective density [Eqs. (4) and (5)] are used in parameterizations, we use each to fit the laboratory measurements. Be aware that the ρeff values derived from these fits are associated with vapor growth, since the particle geometry is unknown.

4. Measurement analysis

Ice particles in the size range of our experiments (rs < 40 μm) have typically been treated as spheres by prior laboratory measurements (Skrotzki et al. 2013; Harrison et al. 2016) and numerical cloud models (Reisner et al. 1998; Morrison and Milbrandt 2015). Additionally, ice crystal growth is not limited by attachment kinetics when the supersaturation exceeds the characteristic value (schar), which results in large (≳0.1) deposition coefficients [Eq. (2)]. For spherical growth in these cases, the DiSKICE model reduces to capacitance theory. Therefore, we compare our measurements to solid spheres grown at the capacitance rate (hereafter “solid spheres”).

Note that preparation of the data follows the same procedures as Pokrifka et al. (2020). That is, we fit the raw mass growth time series data with cubic functions, then preform any further analysis on those fits. However, we also use a low-pass filter on the data to illustrate the close fit of the cubics.

a. Effective density fits to growth time series

Recall that complexity can enhance crystal growth rates. Therefore, ice particles from our experiments with mass growth rates greater than those of solid spheres are candidates to be analyzed for reduced effective densities (ρeff). While we do not have sufficient visual information to directly measure a particle’s morphology in our experiments, the complex habits formed in prior levitation diffusion chamber experiments (see Bacon et al. 2003, their Figs. 5 and 8) indicates that our particles may be treated with an equivalent-diameter sphere and an effective density. Using the DiSKICE model, we simulate the growing crystal while allowing the effective density to decline following either Eq. (4) or Eq. (5). We then fit the growth time series by varying either ρdep [Eq. (4)] or P [Eq. (5)] until a minimum in the root-mean-square error is reached. For simplicity, we assume diffusion-limited growth (α = 1), which is consistent with frozen droplets initially having many surface dislocations and with the instabilities associated with branching and hollowing (see section 3). This assumption may produce growth rates too large for faceted particles (Harrison et al. 2016; Pokrifka et al. 2020), and it is primarily valid for high supersaturations; thus, our results are upper estimates of ρeff.

We measured enhanced growth for numerous crystals, and it is therefore not practical to show the time series for each experiment. Instead, we show characteristic examples of the measured growth time series in Figs. 1 and 2, along with model fits to the data. We chose examples that show both accurate fits to the data (Fig. 1), and fits with larger errors (Fig. 2). We find that growth data are well characterized, at a given temperature, by the ratio of the ice supersaturation (si) to its value at liquid saturation (si,max, the maximum value expected in a cloud),
si,ratsisi,max.
We refer to this quantity as the supersaturation ratio, which varies between zero and unity (i.e., ice and liquid saturation). This ratio provides a common scale for the comparison of data from different temperatures. Both of the presented cases were at high supersaturation ratios of 0.67 and 0.97, respectively. Each case shows that the measured growth (black circles and curves) is greater than predictions using a solid sphere (blue dotted curves). This result is consistent with the development of complex morphology at high si.
Fig. 1.
Fig. 1.

Representative time series that is well-fit by both effective density models. (top) The measured mass ratio after low-pass data filtering (black circles, not all points shown), a cubic fit through the data (black curve), a simulated solid sphere with the same initial size (dotted blue curve), an effective density model fit to the data where ρdep = 376 kg m−3 (dashed green curve), and an effective density model fit to the data where P = 0.900 (dot–dashed magenta curve). (middle) The mass ratio growth rate [d(m/m0)/dt]. (bottom) The modeled effective density time series. This experiment had r0 = 12.2 μm, T = −55.9°C, si = 46.9%, and si,rat = 0.67.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Fig. 2.
Fig. 2.

Representative time series that is less accurately fit by either effective density model. The panels, points, and line styles are as in Fig. 1, except the dashed green curves are from a fit with ρdep = 87 kg m−3 and the dot–dashed magenta curves are from a fit with P = 1.810. This experiment had r0 = 15.8 μm, T = −60.6°C, si = 74.0%, and si,rat = 0.97.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Most cases were similar to the time series shown in Fig. 1, where the data are accurately modeled by fitting the data with ρdep [Eq. (4), green dashed curves] or P [Eq. (5), magenta dot–dashed curves]. In this case, the particle grew at −55.9°C and 46.9% supersaturation with an initial radius of 12.2 μm, and the best fits produce values of ρdep = 376 kg m−3 and P = 0.90. The relative errors of the fits in Fig. 1 (1.48% and 1.35% for the fits on ρdep and P, respectively) are near the median for all the experiments (see Table 1), which indicates that both methods consistently produce accurate fits to the growth data.

Table 1

Statistics of the relative errors for the ρdep and P fitting methods to the mass ratio data. Given are the median error, the mean error, and the standard deviation, along with the maximum and minimum error found among all cases. The percentage of growth experiments with a relative error of less than 5% is also shown.

Table 1

On the other hand, some of the fitting results had larger errors, where the modeled mass ratios resemble the data, but the growth rates are divergent. A typical example is shown in Fig. 2, for a particle that grew at a temperature of −60.6°C and a supersaturation of 74.0% with an initial radius of 15.8 μm. The model fit using the deposition density model produces a low value of ρdep = 87 kg m−3 and the fit with the power-law density produced a best-fit value of P = 1.81, but the relative errors are 4.00% and 5.70%, respectively. Even though the errors are higher in some cases, the effective density models represent the data better than growth of a solid sphere.

Unsurprisingly, the two fitting methods produce different functional forms for the effective density and, therefore, mass growth rate. In Fig. 1, both methods simulate the growth rate comparably well, but the P fit is noticeably worse in Fig. 2. However, it should be noted that, while the fit with ρdep typically performed better than the fit with P (Table 1), that was not always the case. Table 1 further shows that 94.5% of the cases fit with the ρdep model, and 82.4% of the cases fit with the power law, had less than 5% errors relative to the data. The mass ratio data themselves are conservatively estimated to have uncertainties of ±5% (Pokrifka et al. 2020). Therefore, nearly all of the best-fit models reproduce the data with sufficient accuracy. In contrast, as the next section will show, the model of a solid sphere underestimates the growth rate with increasing magnitude and frequency as supersaturation increases.

b. Normalized growth rates

Pokrifka et al. (2020) analyzed the inhibition of ice depositional growth due to attachment kinetics using a normalized growth rate, and this methodology is also useful for analyzing growth that is enhanced due to morphological changes. In that work, we defined the normalized growth rate as
m˙normdmdt|measured¯4πrs¯Deff.
The numerator is the measured growth rate (dm/dt)|measured averaged for a 1-μm increase in the equivalent-mass spherical radius (rs). This quantity is then normalized by 4πrs¯Deff, which is the size-increment averaged growth rate of a solid sphere without the supersaturation. Thus, if the particle growing in the diffusion chamber is a solid sphere, then m˙norm would be equal to si, and the growth rate data will follow the supersaturation. For growth inhibited by attachment kinetics, the normalized growth rates from the data are less than those of solid spheres (Pokrifka et al. 2020). In the current study, we focus primarily on regimes in which the growth is enhanced.

Figure 3 shows the normalized growth rates for particles in the rs range of 14–15 μm plotted against the supersaturation ratio [si scaled by si,max, Eq. (7)]. The normalized growth rates have likewise been scaled by si,max to preserve the 1:1 relation for m˙norm of a solid sphere. The data (black points) show that there are enhanced m˙norm values compared to a solid sphere, particularly when si,rat > 0.4, and with increasing frequency and magnitude as liquid saturation is approached (i.e., as si,rat → 1).

Fig. 3.
Fig. 3.

Normalized growth rate divided by maximum supersaturation as a function of supersaturation ratio. Black points are derived from data as the radius increases from 14 to 15 μm, with error bars from uncertainty in the initial size, temperature, and supersaturation. The red dashed line is from the capacitance growth rate of a solid sphere.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Pokrifka et al. (2020) showed that some of their m˙norm values had a size dependence, indicating that the growth could not be characterized by spheres growing at the capacitance rate. Many of the data here, especially at high si,rat, likewise have a size dependence. Figure 4 shows the mean change in m˙norm for each particle as rs increases from 9 to 25 μm in 1-μm increments (dm˙norm/drs¯). For a solid sphere, this derivative is zero because m˙norm depends only on si. At low si,rat, there is indeed very little change to the normalized growth rate with size, indicating that these crystals grow near the spherical rate. However, at higher si,rat, most of the data have increasing normalized growth rates with increasing size. Similar to Fig. 3, the enhanced growth appears mostly when si,rat > 0.4, and it increases with supersaturation ratio, maximizing at liquid saturation. This result is consistent with what is expected if complex habits develop.

Fig. 4.
Fig. 4.

Change in the normalized growth rate with respect to radius divided by the maximum supersaturation as a function of supersaturation ratio. Black points are mean values from the data, and the error bars indicate plus and minus one standard deviation. The red dashed line (zero) is for a solid sphere.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

It is worth noting that some of the data initially have normalized growth rates less than a solid sphere, but the rates increase to exceed the solid sphere rate as the particles grow larger. This would make sense if the growth was limited by attachment kinetics, as for a faceted crystal, prior to the development of habit complexity. Before the facets formed, the mass growth rate would have been near that of a solid sphere (Nelson and Swanson 2019; Harrington and Pokrifka 2021). The transition from frozen droplet to faceted crystal can cause a reduction in the deposition coefficient, though this transition sometimes happens too quickly to be detected in our experiments (Pokrifka et al. 2020). Harrington and Pokrifka (2021) showed that this transition is faster with increasing supersaturation, which means that we would most likely measure kinetics-limited growth rates at the beginning of high-si experiments.

The above analyses of effective density fits to the mass ratio time series and the normalized growth rate calculations make it clear that the growth enhancement at high supersaturation increases as a particle grows. Below, we investigate if the data can be represented by single- and polycrystalline ice growth models.

5. Time-averaged effective density and budding rosette model

The scatter in the growth data shown in Figs. 3 and 4 can be reduced if the data are examined as a function of the amount of growth enhancement. For this, we define the growth rate ratio (m˙rat) as the time-average measured growth rate divided by that of a solid sphere, where the averages are taken over the lifetime of the particle. This ratio depends primarily on the crystal geometry, because the supersaturation and temperature dependencies cancel. Note that there is a dependence on attachment kinetics, but it is weak at high supersaturation. The top panel of Fig. 5 shows the time-average effective density for each particle from the fit with ρdep as a function of m˙rat (black circles). The time-average effective densities derived from the power-law fitting method are similar (not shown). Plotting against m˙rat clearly demonstrates that particles with more enhanced growth require smaller ρeff. The black circles in the lower panel of Fig. 5 show the exponents from the power-law fitting method, also as a function of the growth rate ratio. Larger values of P correspond to larger m˙rat, which is unsurprising since these results are a reflection of the effective density [Eq. (5)]. These results may seem intuitive, but what is most intriguing is how well the data points organize when plotted against m˙rat for a relatively large temperature range (−65° to −40°C). Since the growth rate ratio depends primarily on crystal geometry, the data organization indicates an underlying similarity in the crystals grown in the diffusion chamber. This organization does not occur if the data are plotted against temperature or supersaturation (not shown).

Fig. 5.
Fig. 5.

(top) Effective density and (bottom) power-law exponent as a function of growth rate ratio. The black circles in the top panel are mean effective densities from fits to the mass ratio time series using the deposition density. The black circles in the bottom panel are exponents from the best fit to the data with the power-law method. In both panels, the shaded regions are from particles simulated by the DiSKICE model using dislocation growth for both plates and columns (gray) or ledge nucleation for plates (red) and columns (blue). The green points in both panels are from a budding rosette model.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

To establish physical meaning in these results, and ensure that they are reasonable, it is prudent to compare them to modeled ice particle growth with known habits. We have already shown that a solid sphere underestimates the measured growth rates; however, nonisometric shapes produce substantially enhanced growth (Takahashi et al. 1991). In the next section, we use the DiSKICE model to show that planar and columnar particles cannot explain the data, and that a budding rosette model can.

a. Effective density of single crystalline habits

We use DiSKICE to simulate the growth of single-crystalline particles, all of which are initially spherical with a radius of 10 μm. The crystals are grown for 15 min, which is typical for our high-si growth experiments. To develop single-crystalline habit forms, a lesser characteristic supersaturation (schar) is required for the primary dimension of growth (i.e., the prism face for planar crystals, and the basal face for columnar crystals). Zhang and Harrington (2014) estimated that schar for the primary growing dimension is about 1/2 that of the minor dimension near −40°C, and data at higher temperatures indicate that it is never less than 1/6. We use schar values taken from Harrington et al. (2019); however, that work provides only the particle average value; therefore, we use the two ratios (1/2 and 1/6) as upper and lower estimates of schar for the primary dimension. In DiSKICE, the growth mechanism must also be specified. The freezing of supercooled water produces numerous dislocations, and we therefore set M = 1 in Eq. (2) (see Harrington et al. 2019). This selection produces high deposition coefficients and is similar to the constant values used in some cloud models. As crystals become larger, step nucleation can dominate the growth producing thin plates or columns (Frank 1982; Nelson 2001), and we therefore also conduct simulations with M = 10. The temperatures are set at decades from −70° to −40°C, and supersaturation ratios (si,rat) used are 0.25, 0.50, 0.75, and 1.00.

In Fig. 5, the range of the DiSKICE model solutions with dislocation growth is shown by the shaded gray region. All of these particles remain compact, as is to be expected for growth by dislocations (Harrington et al. 2019). The compact shape and high deposition coefficients prevent the growth rate ratio from becoming much greater than 1. The red and blue shaded regions of Fig. 5 correspond to the range of solutions for plates and columns, respectively, grown with step nucleation. Step nucleation produces thin plates and long columns, which succeeds in producing growth rate ratios greater than 2, as seen in the data, but there is very little overlap in the phase space. Note that the power-law exponent P approaches 1 for solid plates and 2 for solid columns at the highest growth rate ratios, which is expected. For example, columns with the largest aspect ratios occur when growth on the prism faces is suppressed, producing growth that is essentially one-dimensional (along the c dimension). Such crystals, therefore, should have a value of P that approaches 2. The data suggest that some of the particles grown in the diffusion chamber were columnar, but it is unlikely that we grew dislocation-dominated crystals or single-crystalline plates.

b. Budding rosette growth model

Since polycrystals, especially rosette crystals, nucleate often at temperatures less than −40°C (Heymsfield et al. 2002; Bacon et al. 2003; Bailey and Hallett 2009; Lawson et al. 2019), we also compare our results to a model of budding rosette growth. Our model is similar to that of Um and McFarquhar (2011), except we combine the rosette capacitance models of Westbrook et al. (2008) and Chiruta and Wang (2003) with a central sphere. We then obtain instantaneous growth rates for known particle shapes from which effective densities can be calculated.

We begin with an ice sphere with an initial radius r0, like our experiments. As the particle grows, nb number of bullets protrude out of the sphere (see Fig. 6). Measurements of frozen drops indicate that this begins with the development of small facets on the surface that can grow outwards becoming the branches of a budding rosette (Parungo and Weickmann 1973, conceptual sequence in their Fig. 10). Each bullet is comprised of a hexagonal column 2c in length and 2a in width, which has a volume
Vcol=33a2c,
with the addition of a pyramidal region connecting it to the sphere. We follow Westbrook et al. (2008) and define the volume of the hexagonal pyramid as Vhexp=fpyr3a2c, where the height of the pyramid (hp) is defined as a fraction fpyr of the column length such that hp = 2cfpyr. We subtract off the volume of the pyramidal peak Vpeak to allow for some contact area with the sphere. The contact region is 2a0 across, where a0 is a fraction fa0 of the radius of the sphere r0 (i.e., a0 = fa0r0). The height of the removed peak is therefore h0 = hpa0/a, and the volume of the removed peak is Vpeak=(3/2)a02h0. The total volume of the pyramidal region becomes
Vpyr=VhexpVpeak=fpyr3a2c[1(a0a)3].
Thus, the budding rosette volume is
Vros=43πr03+nb(Vcol+Vpyr),
with effective density
ρeff=ρiVros/[(4/3)πreff3],wherereff=r0+2c(1+fpyraa0),
assuming a symmetrical rosette.
Fig. 6.
Fig. 6.

Schematic of the budding rosette model. Each bullet has a length 2c, a width 2a, and a pyramidal “tip” that intersects with the central sphere, which itself has a radius r0. Each pyramid has a total height hp, with a portion h0 long removed from the peak such that intersection with the sphere is fa0r0 across.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

To compare this model to our data, the growth rate for a budding rosette is needed, and this requires the capacitance. The capacitance of a rosette depends on the maximum dimension and the aspect ratio (ϕ = c/a) of the branches. The branch aspect ratio is determined using the ratio of the a- and c-dimension deposition coefficients following Nelson and Baker (1996),
dcda=αcαaΓ.
We calculate the capacitance of a bullet rosette with nb branches of aspect ratio ϕ as
C={0.35ϕ0.27Dmax,ifnb=4[Westbrooketal.2008,theirEq.(5)],0.40ϕ0.25Dmax,ifnb=6[Westbrooketal.2008,theirEq.(6)],0.40(nb6)0.257ϕ0.25Dmax,ifnb>6[ChirutaandWang2003,basedontheirEq.(15)],
where Dmax = 4c(1 + fpyr) is the maximum dimension of a rosette with four or six branches (Westbrook et al. 2008). To approximate the capacitance for rosettes with more branches, we use the capacitance from Chiruta and Wang (2003). Their parameterization is valid for rosettes with 2 to 16 branches, but it is independent of the branch aspect ratio. We therefore estimate the ϕ dependence using the six-branch capacitance of Westbrook et al. (2008), ensuring that the equations match for nb = 6. Since the exponent on ϕ appears to be weakly dependent on nb [see the first two equations in Eq. (14)], this approximation is justifiable.
The growth of a budding rosette requires the inclusion of the spherical core, an influence that needs to be estimated. Growth from vapor deposition depends on the surface area multiplied by the vapor diffusion flux. For a sphere this goes as 4πr02×(Deffsiρeq/r0). For a nonspherical crystal, Nelson [1994, p. 52, Eq. (3.1)] suggests C is the appropriate length scale for the flux, the growth rate then goes as 4πC2×(Deffsiρeq/C). As an approximation, we follow Harrington and Pokrifka (2021) and write the growth of a budding rosette as a weighted combination of these fluxes, using the weighting factor
w=AintAscl=4πr024π(r02+C2),
where Aint is the surface area of the sphere and Ascl is a “scaling” surface area of the sphere and the branches. This latter area is, in a sense, a total area for vapor diffusion. The growth rate of a budding rosette is therefore approximately
dmdtrosAscl[wDeffsiρeqr0+(1w)DeffsiρeqC] =4π(r0+C)Deffsiρeq,
where the rightmost form results from substitution of the total area and the weight. Note that the growth length scale simply becomes the initial radius plus the capacitance of the branches. Since the crystals begin without branches, this equation reduces to that of a sphere, as expected. To compare directly to the data requires the growth rate ratio averaged over the growth time series which is
m˙rat=(r0+Crs)¯,
where rs is the solid-sphere radius.

The evolution of a budding rosette is simulated using the geometry [Eqs. (11) and (12)] with the growth equation [Eq. (16)]. Our measurements provide the ranges for the initial radius (5–20 μm) and the final mass ratio (8–18), but fpyr, fa0, Γ, and nb, must be prescribed in this model. These values are given reasonable ranges based on geometry and prior measurements. For example, Γ may be unity, producing compact branches, or it may be as large as 20 (Harrington et al. 2019, their Fig. 12), causing the branches to become long and thin. We set nb to 4, 6, or 10, based on observations that rosettes can have at least 8 branches (Heymsfield and Iaquinta 2000) and prior theoretical calculations providing the capacitance for 4, 6 (Westbrook et al. 2008), and up to 16 branch rosettes (Chiruta and Wang 2003). If nb is 4 or 6, we set fa0 to be within the range of 0.5–0.9. Since fa0 controls that amount of the internal sphere’s surface area that is covered by each branch, it is reduced to be 0.2–0.5 when nb = 10 to accommodate the larger number of branches. These ranges are based on estimates of the number of hexagonal facets that can reasonably tile the surface of a sphere, following the model of Harrington and Pokrifka (2021). Westbrook et al. (2008) assumes a value of 0.5 for fpyr, which we expand to a range of 0.2–0.6.

Figure 7 shows the simulated growth of selected, six-branched particles following Eqs. (9)(17), all of which use r0 = 10 μm, final mr = 15 and fpyr = 0.5. Unsurprisingly, increasing Γ (cyan to black to purple solid curves) causes a larger decrease in the effective density and increase in the power-law exponent. That is, increasing the growth of the c dimension with respect to the a dimension causes the branches to become thinner (larger aspect ratio), lowering the effective density. The resulting branch aspect ratios are quite reasonable, reaching 1.5–2.5 when Γ = 6. Decreasing fa0 (dot–dashed to solid to dashed purple curves) produces thinner branches, a decreased effective density and an increased power-law exponent. This also makes physical sense, because a smaller initial branch area spaces them further apart and makes them thinner. Furthermore, both increasing Γ and decreasing fa0 produce enhanced growth rates, with growth rate ratios ranging from 1 to >2. These values of effective density, power-law exponent, and growth rate ratio are in good agreement with the data (Fig. 5).

Fig. 7.
Fig. 7.

Example budding rosette simulations with six branches and an initial radius of 10 μm. Cyan, black, and purple curves have Γ set to 1.5, 3.0, and 6.0, respectively. Dashed, solid, and dot–dashed curves have fa0 set to 0.5, 0.7, and 0.8, respectively. Plotted is the (top left) effective density, (top right) power-law exponent, (bottom left) branch aspect ratio, and (bottom right) growth rate ratio as a function of mass ratio.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

The results from Fig. 7 clearly indicate that growth rate ratio and effective density depend on the manner in which branches form (through surface area coverage) and the resulting aspect ratios of the branches (controlled by the deposition coefficient ratio Γ). The variability in the data shown in Fig. 5 (black circles) is therefore not surprising. To emulate the data, we ran many additional simulations with the budding rosette model [Eqs. (9)(17)], the results of which are shown as green points in Fig. 5. Each point is from one growth simulation, and the variability is from random selections of r0, fpyr, fa0, nb, Γ, and the final mass ratio using the ranges discussed above. These results strongly resemble the data and imply that the particles grown in the laboratory were likely polycrystalline, and may well have been budding rosettes. Simulations with a polycrystalline plate model does not produce a match with the data, since the points scatter near P = 1 (not shown), which is reasonable for planar particles.

The match of the budding rosette model with the data also provides indirect corroboration of the classical geometric rosette model for cloud modeling applications. However, using this model requires a number of unknown parameters, including the number of branches and branch aspect ratio, which makes fitting the growth data with the budding rosette model unjustified. The measured mass ratio time series and initial size are not enough to adequately constrain the unknown parameters. Thus, the parameterizations that we develop in the following section are derived entirely from the laboratory data, independent of the budding rosette model. The advantage in using the simplified effective density in a parameterization is that the unknown parameters are implicitly included.

6. Supersaturation dependence and parameterization of the effective density

The strong correlation between the effective density and the growth rate enhancement indicates that there is structural regularity to the measured data that should also appear as a function of the supersaturation, even though such regularity is not apparent in Figs. 3 and 4. For completeness, we include the data at low supersaturation, which often show evidence of attachment kinetic limitations (Pokrifka et al. 2020). To further investigate the supersaturation dependence to growth, we examine the fraction of experiments with kinetic limitations (growth rate ratio <1) as compared to those with enhanced growth (growth rate ratio > 1). If we calculate this fraction in supersaturation ratio bins of 0.0–0.4, 0.4–0.6, 0.6–0.8, and 0.8–1.0 (Fig. 8), it becomes clear that kinetics-limited growth (dashed) is more common when the si,rat is lower, whereas enhanced growth is common when si,rat is high (solid). The increased occurrence of enhanced growth with increasing si,rat reflects the results shown in Fig. 4. Note that a similar result appears when si,rat ranges of 0–0.2 and 0.2–0.4 are used, as little growth enhancement occurred at si,rat < 0.4 (Fig. 4); thus, we use the combined range of 0–0.4 for clarity, and do so for the remainder of this work. This si.rat range is well represented by solid ice; however, attachment kinetics must be included to properly account for the mass growth rate (Pokrifka et al. 2020).

Fig. 8.
Fig. 8.

Fraction of particles in the dataset that show enhanced growth (solid) or limited growth (dashed) in supersaturation ratio bins of 0.0–0.4, 0.4–0.6, 0.6–0.8, and 0.8–1.0.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

We should expect an approximately monotonic rise in the relative frequency of enhanced growth cases with supersaturation ratio, but variability within a given si,rat range is also to be expected, since crystals nucleated from frozen water droplets will vary in their morphological complexity (Bacon et al. 2003). Such variability is shown in Fig. 9 through a two-dimensional (2D) distribution of the relative frequency of cases as a function of the supersaturation ratio and the growth rate ratio for the entire dataset. This 2D distribution confirms that increasing si,rat increases the likelihood of particles growing at enhanced rates and that low si,rat often produces kinetics-limited growth that can be approximated as solid ice (i.e., ρdep = 920 kg m−3 and P = 0).

Fig. 9.
Fig. 9.

Two-dimensional relative frequency distribution of supersaturation ratio vs growth rate ratio. Increasingly red contours indicate increasing occurrence. The green dashed line is where the growth rate ratio is unity, as for a solid sphere: above this line growth is enhanced; below it growth is limited.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

It is important to note that while the 268 experiments we conducted are numerous for individual crystal studies, the data are too few for a detailed statistical analysis. In addition, there are relatively fewer data at si,rat ≲ 0.2. Experiments at si,rat ≲ 0.2 are challenging, since they take a whole day to conduct due to very small particle growth rates. This is in contrast to higher-si,rat experiments, where multiple particles can be grown within a day. However, most of the high-si,rat (≳ 0.8) experiments were conducted around the low-to-mid portion of the temperature range (i.e., −63° to −53°C). To examine the potential impact of this sample bias, we split the dataset into “warm” (T > −56°C) and “cold” (T < −56°C) subsets. We indeed find that “warm” cases more frequently occur at low si,rat and low m˙rat, and vice versa for “cold” cases, but both subsets demonstrate increasing m˙rat with increasing si,rat (not shown). Thus, the precise value and location of the maximum in Fig. 9 may change with more even sampling, but the positive correlation of supersaturation ratio and growth rate ratio is robust for both data subsets. This result indicates that si,rat is an environmental condition that may be utilized in conjunction with our data to produce parameterizations of effective density. We will also demonstrate that the sample bias has minimal effect on these parameterizations.

Given the systematic correlation of the growth rate ratio with the supersaturation ratio, we should expect the effective density to behave in a similar fashion. Figure 10 shows the relative frequency of ρdep values derived from the fits [Eqs. (3) and (4)] for all of the experiments in supersaturation ratio bins of 0–0.4 (cyan), 0.4–0.6 (blue), 0.6–0.8 (purple), and 0.8–1 (black). To demonstrate the overall trend in each si,rat bin, the ρdep values are separated in bins of 0–300, 300–600, and 600–920 kg m−3. The relative frequencies are also listed in Table 2. It is clear from Fig. 10 that conditions with si,rat < 0.6 often produce high values of ρdep, and small ρdep values are rare. As si,rat increases, so does the frequency of smaller ρdep values, such that ρdep > 600 kg m−3 is the rarity when si,rat > 0.6. Unsurprisingly, supersaturation near liquid saturation (si,rat > 0.8) most frequently produces the smallest values of ρdep, which are associated with the greatest degree of growth enhancement. The supersaturation ratio dependence of the time-averaged effective density (Fig. 11, solid curves) is similar to that of ρdep, though the values of ρeff are larger. This is expected because ρeff approaches ρdep as crystal size increases.

Fig. 10.
Fig. 10.

Relative frequency of cases producing deposition densities in ranges of 0–300, 300–600, and 600–920 kg m−3 in supersaturation ratio bins of 0.0–0.4 (cyan), 0.4–0.6 (blue), 0.6–0.8 (purple), and 0.8–1.0 (black).

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Fig. 11.
Fig. 11.

Relative frequency of cases producing mean effective densities in ranges of 0–300, 300–600, and 600–920 kg m−3 derived from the deposition density (solid) and power law (dashed) for the same supersaturation ratio bins as in Fig. 10.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Table 2

Relative frequencies of deposition density, effective density from ρdep fits, and power-law exponent for experiments in supersaturation ratio bins of 0–0.4, 0.4–0.6, 0.6–0.8, and 0.8–1. The densities are in ranges of 0–300, 300–600, and 600–920 kg m−3, and the power-law exponents are in ranges of 0–0.5, 0.5–1, 1–1.5, and 1.5–2.

Table 2

A 2D relative frequency distribution of the deposition density and supersaturation ratio (Fig. 12) provides a more succinct visualization of the information in Fig. 10. The distribution highlights two local maxima of the relative frequency. One maximum appears for values of si,rat < 0.4, and it corresponds to particles with high density (ρdepρi). As the supersaturation ratio approaches liquid saturation, ρdep falls and there is another maximum at ρdep ∼ 250 kg m−3. To estimate the most likely values of ρdep at some si,rat, we fit a linear function between both maxima (dark green dashed line). At low si,rat, we find the linear fit from si,rat = 0 and ρdep = 920 kg m−3 to the high-density local maximum (light green dashed line). The two linear fits cover all values of si,rat from our experiments and are given by
ρdep(si,rat)={32.332si,rat+920,si,rat0.267,1027.456si,rat+1185.834,si,rat>0.267,
where the coefficients have units of kg m−3. A mass–size relationship can then be derived by substituting Eq. (18) into Eq. (4),
m(ρdep)=43πρeff(ρdep)r3=43π{[ρiρdep(si,rat)]r03+ρdep(si,rat)r3}.
Here, r is the radius of the sphere encompassing the particle, or the maximum semidimension. Equations (18) and (19) comprise a parameterization of the effective density as a function of ρdep at low temperatures suitable for particle property microphysical models rooted in the method of Chen and Lamb (1994), such as Hashino and Tripoli (2007), Chen and Tsai (2016), and Jensen et al. (2017). Those models currently use ρdepρi at temperatures below −20°C. Note that Eq. (19) is strictly valid for constant si,rat. We discuss below how to utilize this parameterization with variable temperature and supersaturation.
Fig. 12.
Fig. 12.

Two-dimensional relative frequency distribution of supersaturation ratio vs deposition density. Increasingly red contours indicates increasing number of occurrences. The light green dashed curve is a linear fit from the (si,rat = 0, ρdep = 920 kg m−3) coordinate to the nearest local maximum, and the dark green dashed curve is a linear fit between the two maxima. The purple dash–dotted curve is the same as the green dashed curves, but using only cases with si,rat between 0.2 and 0.8. The pink and indigo dotted curves are the same as the purple dash–dotted curve, but with the temperatures restricted to above −56°C and below −56°C, respectively.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

To test the potential impact that a sampling bias may have on this parameterization, we repeat the above linear fitting procedure with data in the range 0.2 < si,rat < 0.8 (purple dot–dashed curve in Fig. 12). This range of supersaturation ratio eliminates the regions where most of the sampling bias occurred. Using only data where 0.2 < si,rat < 0.8 produces the same trend as the whole dataset, but with a slightly larger slope (compared to the green dashed curve). This change in slope makes sense in comparison to Fig. 10, which shows that disregarding si,rat > 0.8 would shift the most frequent values of ρdep at high si,rat to be 300–600 kg m−3. Because the data between 0.2 < si,rat < 0.8 are evenly sampled, we can also test for a temperature dependence in our parameterization method. While using the supersaturation ratio provides a common, temperature-independent scale to analyze the data, temperature has other influences, such as in crystal morphology and attachment kinetics. As before, we have split the dataset into cold and warm regions at −56°C, and the resulting curves are plotted on Fig. 12. For T > −56°C (pink dotted curve) the slope decreases, and for T < −56°C (indigo dotted curve) the slope increases. This small variation indicates that the temperature dependence of ρdep is second order compared to the primary dependence on si,rat, which seems well characterized by our data.

Traditional microphysical schemes parameterize the effective density and the mass with power laws, and we therefore follow the above method to estimate the most likely value for the power-law exponent P for Eqs. (5) and (6). Figure 13 shows the relative frequency of observed P values for the same supersaturation ratio ranges used in Fig. 10 (see also Table 2). Here, the power-law exponents are binned into ranges of 0–0.5, 0.5–1, 1–1.5, and 1.5–2. Similar to Fig. 10, lower supersaturation ratios si,rat < 0.6 (cyan and blue curves) produce little enhanced growth, and therefore, values of P ∼ 0 occur frequently. Increasing si,rat (to purple then black curves) increases the frequency of P values associated with greater growth enhancement. Again, the largest amount of growth enhancement is most common near liquid saturation (black curve), but even under these conditions, P is most frequently in the range of 1–1.5 instead of 1.5–2. Since P ∼ 2 is consistent with columnar growth (note that hollow columns can have P > 2), it is interesting that the fraction of particles grown near liquid saturation with P ∼ 2 (30.6%) is close to the fraction of columns that Bailey and Hallett (2004) grew under similar conditions (Hashino and Tripoli 2008, see their Fig. 1). Additionally, the power law produces time-averaged ρeff relative frequencies that are similar to those calculated from ρdep, but with values of 300–600 kg m−3 being slightly more common (Fig. 11, dashed curves).

Fig. 13.
Fig. 13.

Relative frequency of cases producing power-law exponents in ranges of 0–0.5, 0.5–1, 1–1.5, and 1.5–2 for the same supersaturation ratio bins as in Fig. 10.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Like ρdep, the variability in P with supersaturation ratio can be better visualized with a 2D distribution. Figure 14 shows two local maxima corresponding to those of Fig. 12. One maximum is at low si,rat associated with high density particles (P ∼ 0) and growth limited by attachment kinetics. The second maximum occurs at high si,rat where growth is enhanced (P ∼ 1.4). To estimate a most likely value of P at high si,rat we fit a line through both maxima (dark green dashed line). The value of P is estimated at low-si,rat values with a linear fit from si,rat = 0 and P = 0 to the nearest local maximum (light green dashed line). The most likely value of P for each si,rat is thus estimated to be
P(si,rat)={0.082si,rat,si,rat0.267,2.106si,rat0.541,si,rat>0.267.
Combining Eq. (20) with Eq. (5) provides a parameterization for the effective density of small vapor grown crystals. This parameterization therefore produces a supersaturation-dependent power law,
m(P)=43πρeff(P)r3=43πρir0P(si,rat)r3P(si,rat).
It should be noted that the sampling bias and temperature dependence shown for Fig. 12 is less pronounced for the distribution of P, and is therefore not shown. Because Eqs. (18)(21) were derived from the measured growth of small particles (r ∼ 10–100 μm) at temperatures between −65° and −40°C, these parameterizations may not be applicable to other conditions. Like Eq. (19), Eq. (21) is strictly valid for constant si,rat (i.e., constant temperature and supersaturation). When the supersaturation ratio is variable, as in a real cloud, care must be taken in using Eq. (21). Since the power-law exponent changes with supersaturation ratio, one cannot simply change P in Eq. (21), as this would instantaneously change the particle characteristics. Instead, the approach we advocate is to calculate the deposition density by taking the time derivative of either Eq. (19) or Eq. (21) (depending on the chosen parameterization), which will result in either Eq. (3) or Eq. (6), respectively. The particle effective density is then updated using Eq. (4). The physical interpretation is that the change in si,rat influences the effective density of the mass added to a particle during vapor growth.
Fig. 14.
Fig. 14.

Two-dimensional relative frequency distribution of supersaturation ratio vs power-law exponent. Increasingly red contours indicates increasing number of occurrences. The light green dashed curve is a linear fit from the (si,rat = 0, P = 0) coordinate to the nearest local maximum, and the dark green dashed curve is a linear fit between the two maxima.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Be aware that we cannot validate the approach described above. There are not any published data of individual ice particle growth rates with variable temperature and supersaturation at T < −40°C to compare against. Under these conditions, it is unknown if a particle’s deposition density or power-law exponent would change rapidly in response to a change in si,rat, or if they are set upon nucleation and early growth. Due to this limitation, we suggest utilizing the results from our wide range of constant experimental conditions as an approximation.

7. Summary and discussion

We grew 268 individual ice particles with equivalent-mass spherical radii less than 40 μm inside the BEL diffusion chamber. Particle mass ratio time series were measured at temperatures between −65° and −40°C for supersaturations up to liquid saturation. Growth rates at high si typically exceeded that of a solid sphere, which is consistent with complex crystal habits and can be treated with an effective density. We estimated ρeff with model fits to the measured mass ratio time series by adjusting either a deposition density (ρdep) or a power-law exponent (P). Both ρeff models represented the data well, and the time-averaged fit results resembled effective densities derived from models of budding rosettes and, at the largest growth rate ratios, columns. It is thus plausible that many of the particles growing at high si developed those habits. The measured growth enhancement due to such complexity was well characterized by a ratio of the supersaturation to its value at liquid saturation (si,rat). We have used 2D relative frequency distributions of the ρdep and P results to estimate the most frequent values of ρdep and P for any si,rat, which may then be used to estimate a supersaturation-dependent effective density.

It is important to heed the limitations in our study. These particles grew under constant temperature and supersaturation, which is not the case in clouds. We have assumed that the particles with enhanced growth had deposition coefficients of unity, but due to the attachment kinetic effects required to produce complex ice habits, this cannot be true for the whole particle. Similarly, we treated particles with growth rate ratios less than unity as kinetics limited, but facet development on a frozen droplet can also have anomalously low growth rates with a high particle-averaged α (Pokrifka et al. 2020; Harrington and Pokrifka 2021). Furthermore, while our analysis suggests that many of the crystals we grew may have been budding rosettes or columns, we cannot confirm the particle morphology. Our geometric model of a budding rosette, despite reproducing the variability in the data, neglects facet hollowing, which likely occurred in our high-si experiments and would contribute to the effective density reduction. Caution is therefore warranted in estimating growth rates from the budding rosette model, especially given its numerous required, but unknown, parameters. The effective density parameterizations derived from the fits to the data with the deposition density and power-law exponent avoid these unknown parameters, but do so by avoiding a direct link to crystal geometry. That is, any and all complexity in particle shape is entangled in ρdep and P.

Despite these limitations, our results are consistent with prior work. As shown in Fig. 5, our effective density estimates range from about 100 kg m−3 to ρi. This is comparable to estimates from prior laboratory studies, but most of those were from particles growing at T ≥ −22°C and primarily near liquid saturation (high si,rat) (Fukuta 1969; Takahashi et al. 1991).

Prior effective density estimates under similar temperature and supersaturation conditions as in our experiments have been made from in situ observations, and they also find that ρeff can be as low as 100 kg m−3, but for larger particles (Mitchell 1996; Heymsfield et al. 2004, 2007). Their effective density relations must be limited at some minimum diameter, generally between 70 and 100 μm (Brown and Francis 1995; Heymsfield et al. 2010); otherwise, nonphysical ice densities are produced. Smaller crystals are generally assumed to have the density of bulk ice (ρeffρi).

However, other effective density estimates derived from in situ observations that accommodate small particles are in good agreement with our measurements (Fig. 15). Figure 15 shows time-averaged effective densities as a function of growth rate ratio, much like the top panel of Fig. 5. Here, the circles are derived from the power-law fits to the data, with shading indicating the supersaturation ratio. Also plotted, in diamonds, are average effective densities following the mass–size parameterizations of Cotton et al. (2013), Erfani and Mitchell (2016), Fridlind et al. (2016), and Lawson et al. (2019). In each case, we simulate the growth of a particle with an initial radius of 10 μm and a final mass ratio of 15 (final mass-equivalent spherical radius of ∼25 μm). The simulations use a temperature of −50°C and pressure of 970 hPa, and are across a range of supersaturation ratios from 0 to 1, similar to the laboratory experiments. The observations of Cotton et al. (2013) (cyan) are consistent with our data at low si,rat. Likewise, the rosette models of Fridlind et al. (2016) (solid red) and Lawson et al. (2019) (yellow) produce growth rate enhancements and effective density reductions that align well with the growth data. Erfani and Mitchell (2016) present multiple temperature-dependent mass–size relationships for synoptic (empty green) and anvil (solid green) cirrus. We have plotted the results from their warmest (light green) and coldest (dark green) cases, which produce slightly lower effective densities than our data, but they follow a similar functional form across our full mass range (green curve). The one outlier in this comparison is the Bucky ball model of Fridlind et al. (2016) (empty red). The average effective density from this model is significantly lower than the laboratory data. The full growth model across our mass range (red curve) reveals that, at small sizes, the Bucky ball model is an exceptional match to our data, but the effective density falls too quickly as the particle grows. Otherwise, these results indicate that particle growth rates produced by mass–size relationships derived from in situ observations are corroborated by the growth rates of our laboratory measurements at low to mid-si,rat. They do not, however, reach the highest growth rate ratios (and lowest effective densities) that our data and parameterizations produce at high si,rat.

Fig. 15.
Fig. 15.

Time-average effective density as a function of growth rate ratio. Circles are from the power-law fits to the mass ratio data, with darker shading indicating a higher supersaturation ratio. Diamonds are from calculations using others’ mass–size relationships. Shown in red are the rosette (solid diamond), Bucky ball (curve), and average of the Bucky ball (empty diamond) from Fridlind et al. (2016). The constant effective density from Cotton et al. (2013) is in cyan, and the rosette model from Lawson et al. (2019) is in yellow. Shown in green are the warm anvil (light solid), cold anvil (dark solid), warm synoptic (light empty), and cold synoptic (dark empty) cirrus cases from Erfani and Mitchell (2016), with the full growth range as a green curve.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

Further agreement between our study and in situ observations appears in comparing geometric models. The correspondence between our measurements and our geometric model of a budding rosette indicates that a rosette model provides relatively accurate growth rates at high si. Heymsfield et al. (2002) present a geometric rosette model suitable for larger crystals that was successfully used to interpret in situ observations. Figure 16 shows that our model of a budding rosette, using four branches (solid curves), produces effective densities that approach the values from the model by Heymsfield et al. (2002) (dashed curves, beginning at a diameter of 65 μm), when the particles grow larger (mass ratio ≳ 10). The overlaps in ρeff between both models, and between our budding rosette model and data, suggest that our measurement-derived effective density functions offer a plausible extension to nucleation sizes that is complementary to in situ observations.

Fig. 16.
Fig. 16.

Effective density from the budding rosette model with four branches (solid) compared to the geometric rosette model of Heymsfield et al. (2002) (dashed) as a function of mass ratio. The latter model is limited to diameter ≥ 65 μm All simulations have an initial radius of 10 μm, fpyr = 0.5, and fa0 = 0.7. Cyan, black, and purple curves have Γ set to 1.5, 3.0, and 6.0, respectively. Note the logarithmic scale ρeff.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0077.1

One advantage of our approach is that information on the time dependence of growth is implicitly included in our parameterizations, since the mass–size relationships are derived from fits to the time series data. Our measurement-derived parameterizations provide a method to model the growth of small ice particles that captures the effects of habit complexity.

Acknowledgments.

The comments of three anonymous reviewers are appreciated, as they have improved the quality of this manuscript. The authors are grateful for support through grants from the National Science Foundation (AGS-1824243 and AGS-2128347).

Data availability statement.

The laboratory data used in this manuscript are available through Data Commons, The Pennsylvania State University, at https://doi.org/10.26208/htw5-q166.

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Save
  • Bacon, N., M. Baker, and B. Swanson, 2003: Initial stages in the morphological evolution of vapour-grown ice crystals: A laboratory investigation. Quart. J. Roy. Meteor. Soc., 129, 19031927, https://doi.org/10.1256/qj.02.04.

    • Search Google Scholar
    • Export Citation
  • Bailey, M., and J. Hallett, 2004: Growth rates and habits of ice crystals between −20° and −70°C. J. Atmos. Sci., 61, 514544, https://doi.org/10.1175/1520-0469(2004)061<0514:GRAHOI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Bailey, M., and J. Hallett, 2009: A comprehensive habit diagram for atmospheric ice crystals: Confirmation from laboratory, AIRS II, and other field studies. J. Atmos. Sci., 66, 28882899, https://doi.org/10.1175/2009JAS2883.1.

    • Search Google Scholar
    • Export Citation
  • Baker, B., and R. P. Lawson, 2006: Improvement in determination of ice water content from two-dimensional particle imagery. Part I: Image-to-mass relationships. J. Appl. Meteor. Climatol., 45, 12821290, https://doi.org/10.1175/JAM2398.1.

    • Search Google Scholar
    • Export Citation
  • Brandes, E., K. Ikeda, G. Zhang, M. Schönhuber, and R. Rasmussen, 2007: A statistical and physical description of hydrometeor distributions in Colorado snowstorms using a video disdrometer. J. Appl. Meteor. Climatol., 46, 634650, https://doi.org/10.1175/JAM2489.1.

    • Search Google Scholar
    • Export Citation
  • Brown, P., and P. Francis, 1995: Improved measurements of the ice water content in cirrus using a total-water probe. J. Atmos. Oceanic Technol., 12, 410414, https://doi.org/10.1175/1520-0426(1995)012<0410:IMOTIW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Chen, J.-P., and D. Lamb, 1994: The theoretical basis for the parameterization of ice crystal habits: Growth by vapor deposition. J. Atmos. Sci., 51, 12061221, https://doi.org/10.1175/1520-0469(1994)051<1206:TTBFTP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Chen, J.-P., and T.-C. Tsai, 2016: Triple-moment modal parameterization for the adaptive growth habit of pristine ice crystals. J. Atmos. Sci., 73, 21052122, https://doi.org/10.1175/JAS-D-15-0220.1.

    • Search Google Scholar
    • Export Citation
  • Chiruta, M., and P. Wang, 2003: The capacitance of rosette ice crystals. J. Atmos. Sci., 60, 836846, https://doi.org/10.1175/1520-0469(2003)060<0836:TCORIC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Cotton, R., and Coauthors, 2013: The effective density of small ice particles obtained from in situ aircraft observations of mid-latitude cirrus. Quart. J. Roy. Meteor. Soc., 139, 19231934, https://doi.org/10.1002/qj.2058.

    • Search Google Scholar
    • Export Citation
  • Davis, E., 2010: A button electrode levitation chamber for the study of ice crystal growth under atmospheric conditions. M.S. thesis, Dept. of Meteorology, The Pennsylvania State University, 77 pp.

  • Dunnavan, E. L., and Z. Jiang, 2019: A general method for estimating bulk 2D projections of ice particle shape: Theory and applications. J. Atmos. Sci., 76, 305332, https://doi.org/10.1175/JAS-D-18-0177.1.

    • Search Google Scholar
    • Export Citation
  • Dunnavan, E. L., Z. Jiang, J. Y. Harrington, J. Verlinde, K. Fitch, and T. J. Garrett, 2019: The shape and density evolution of snow aggregates. J. Atmos. Sci., 76, 39193940, https://doi.org/10.1175/JAS-D-19-0066.1.

    • Search Google Scholar
    • Export Citation
  • Elliott, W. J., 1971: Dimensions of thermal diffusion chambers. J. Atmos. Sci., 28, 810811, https://doi.org/10.1175/1520-0469(1971)028<0810:DOTDC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Erfani, E., and D. L. Mitchell, 2016: Developing and bounding ice particle mass- and area-dimension expressions for use in atmospheric models and remote sensing. Atmos. Chem. Phys., 16, 43794400, https://doi.org/10.5194/acp-16-4379-2016.

    • Search Google Scholar
    • Export Citation
  • Frank, F. C., 1982: Snow crystals. Contemp. Phys., 23, 322, https://doi.org/10.1080/00107518208231565.

  • Fridlind, A. M., R. Atlas, D. Bastiaan, J. Um, G. M. McFarquhar, A. S. Ackerman, E. J. Moyer, and R. P. Lawson, 2016: Derivation of physical and optical properties of mid-latitude cirrus ice crystals for a size-resolved cloud microphysics model. Atmos. Chem. Phys., 16, 72517283, https://doi.org/10.5194/acp-16-7251-2016.

    • Search Google Scholar
    • Export Citation
  • Fukuta, N., 1969: Experimental studies on the growth of small ice crystals. J. Atmos. Sci., 26, 522531, https://doi.org/10.1175/1520-0469(1969)026<0522:ESOTGO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Fukuta, N., and T. Takahashi, 1999: The growth of atmospheric ice crystals: A summary of findings in vertical supercooled cloud tunnel studies. J. Atmos. Sci., 56, 19631979, https://doi.org/10.1175/1520-0469(1999)056<1963:TGOAIC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Garrett, T. J., S. E. Yuter, C. Fallgatter, K. Shukurko, S. R. Rhodes, and J. L. Endries, 2015: Orientations and aspect ratios of falling snow. Geophys. Res. Lett., 42, 46174622, https://doi.org/10.1002/2015GL064040.

    • Search Google Scholar
    • Export Citation
  • Harrington, J. Y., and G. Pokrifka, 2021: Approximate models for lateral growth on ice crystal surfaces during vapor depositional growth. J. Atmos. Sci., 78, 967981, https://doi.org/10.1175/JAS-D-20-0228.1.

    • Search Google Scholar
    • Export Citation
  • Harrington, J. Y., A. Moyle, L. E. Hanson, and H. Morrison, 2019: On calculating deposition coefficients and aspect-ratio evolution in approximate models of ice crystal vapor growth. J. Atmos. Sci., 76, 16091625, https://doi.org/10.1175/JAS-D-18-0319.1.

    • Search Google Scholar
    • Export Citation
  • Harrison, A., A. Moyle, M. Hanson, and J. Harrington, 2016: Levitation diffusion chamber measurements of the mass growth of small ice crystals from vapor. J. Atmos. Sci., 73, 27432758, https://doi.org/10.1175/JAS-D-15-0234.1.

    • Search Google Scholar
    • Export Citation
  • Hashino, T., and G. J. Tripoli, 2007: The Spectral Ice Habit Prediction System (SHIPS). Part I: Model description and simulation of the vapor deposition process. J. Atmos. Sci., 64, 22102237, https://doi.org/10.1175/JAS3963.1.

    • Search Google Scholar
    • Export Citation
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