This is a comment on the review by Korolev and Leisner (2020, hereafter KL2020). The only two laboratory/field studies ever to measure the breakup in ice–ice collisions for in-cloud conditions were negatively criticised by KL2020, as were our subsequent theoretical and modelling studies informed by both studies. First, hypothetically, even without any further laboratory experiments, such theoretical and modelling studies would continue to be possible, based on classical mechanics and statistical physics. They are not sensitive to the accuracy of lab data for typical situations, partly because the nonlinear explosive growth of ice concentrations continues until some maximum concentration is reached. To a degree, the same final concentration is expected regardless of the fragment number per collision. Second, there is no evidence that both lab/field observational studies characterising fragmentation in ice–ice collisions are either mutually conflicting or erroneous such that they cannot be used to represent this breakup in numerical models, contrary to the review. The fact that the ice spheres of one experiment were hail sized (2 cm) is not a problem if a universal theoretical formulation, such as ours, with fundamental dependencies, is informed by it. Although both lab/field studies involved head-on collisions, rotational kinetic energy for all collisions generally is only a small fraction of the initial collision kinetic energy (CKE) anyway. Although both lab/field experiments involved fixed targets, that is not a problem since the fixing of the target is represented via CKE in any energy-based formulation such as ours. Finally, scaling analysis suggests that the breakup of ice during sublimation can make a significant contribution to ice enhancement in clouds, again contrary to the impression given by the review.
The literature of secondary ice production (SIP) was recently reviewed by Korolev and Leisner (2020, hereafter KL2020). The focus of their review is on laboratory experiments. It is commendable that their review attempts to reinvigorate laboratory observations of the various types of fragmentation of ice, research that has in the last decade been oriented chiefly towards heterogeneous ice nucleation. We agree with KL2020 that future experiments about the breakup of ice can only enhance the accuracy of its empirical characterisation. Regarding observations of the breakup in ice–ice collisions for plausible in-cloud conditions, only two prior lab/field studies have ever been published quantifying it, as far as we are aware (Vardiman, 1978; Takahashi et al., 1995).
However, the unfortunate impression is given to the reader that numerical
modelling and theoretical studies of the breakup in ice–ice collisions are
currently impossible due to the fact that reliable data for laboratory
experiments are somehow critically missing. Even if the available data were
unreliable or unrepresentative of natural clouds, which is actually not the
case (Sect. 4), our stand-alone theoretical and modelling studies would,
nevertheless, still have been possible, since they are based on classical mechanics of collisions and statistical physics:
Yano and Phillips (2011) and Yano et al. (2016) delineated the timescale of the explosive growth of ice concentrations and the conditions for such
instability in terms of dimensionless parameters that provide a phase space
of the system. The role of stochastic variability among fragmentation events in promoting
the onset of instability is elucidated theoretically by means of the
Fokker–Planck equation (Yano and Phillips, 2016). Phillips et al. (2017a) show a formulation based on principles of energy
conservation and statistical variability in the strengths of asperities, using a recent definition of the coefficient of restitution. An application of this versatile framework to a real cloud situation involved constraining some parameters with the lab/field data (Vardiman, 1978; Takahashi et al., 1995) but exists, as a theory, independently of such data.
When KL2020 were negatively criticising both lab/field studies (Vardiman,
1978; Takahashi et al., 1995), they mentioned three of these studies in the same context.
The purpose of our commentary is to point out some misunderstandings apparent in the review by KL2020. The focus is on Sect. 4 of KL2020, which covered the topic of the breakup in ice–ice collisions, where we have contributed both in theories and modelling. KL2020 have issued a Corrigendum for how their section dealt with our three earlier studies about the fragmentation of ice (Yano and Phillips, 2011; Yano et al., 2016; Phillips et al., 2017a).
The structure of the commentary is as follows. First, to enable the reader to follow the technical details of our arguments defending the lab/field studies criticised by KL2020, the next two sections summarise their observations of the breakup in ice–ice collisions and how these inspired our subsequent theoretical and modelling contributions. That is followed by specific comments on the review by KL2020 (their Sect. 4) and its Corrigendum in Sect. 4. This describes how the review by KL2020 has several misunderstandings in their survey of those lab studies underpinning our modelling and theoretical studies of the breakup. We describe the errors that their Corrigendum has not rectified.
A flaw in the review by KL2020 is the claim that the only two lab/field studies observing fragmentation in ice–ice collisions published hitherto are somehow so erroneous that they cannot provide any basis for representing this type of SIP in numerical models. Before our counterarguments defending their utility (Sect. 4.1), the salient features of both lab/field studies are summarised.
First, Vardiman (1978) constructed a portable instrument deployed outdoors on mountainsides in USA during events of ice precipitation (Fig. 1a). Ice particles fell through the field of view of a movie camera onto a Plexiglas fixed plate, and fragments were counted later by inspection of the film. Fragment counts had earlier been verified as being accurate by an intercomparison with an alternative portable chamber that captured all fragments (Fig. 1b). The chamber had an opening at the top, through which natural ice particles (about 0.5 to 5 mm in diameter) fell, descending into the field of view of a camera and impacting a copper mesh. All fragments had been captured in a supercooled solution and counted by eye for this alternative method.
The numbers of fragments per collision were measured (Fig. 1a) for various types of ice particle as follows:
Unrimed dendrites Lightly or moderately rimed dendrites Heavily rimed dendrites Lightly or moderately rimed spatial crystals Graupel.
For each morphological type, the number of fragments per collision was
mostly in the range of 1 to 100 for most sizes and increased with size. This
number was published as a function of momentum change on the fixed plate
(Vardiman, 1978).
The portable instruments used to determine the number of fragments
from natural falling ice particles. They were placed outdoors on the ground
with natural ice falling into them. Field observations in the winter of 1973–1974 were made, with the simplified setup in
Second, Takahashi et al. (1995) observed collisions between two giant spheres of ice (2 cm diameter), one rimed (A) and the other unrimed (B), in a cold box in a laboratory. Figure 2 shows how these were performed. The unrimed sphere was fixed, while the rimed one was on a rotating arm. Both particles were intended to be representative of small and large graupel colliding in a real cloud after falling into regions of weak liquid water content (LWC), where the smaller one grows predominantly by vapour deposition and the other by riming, as had been observed by an airborne video probe (Takahashi, 1993; Takahashi and Kuhara, 1993). High concentrations of ice were observed for such collisions by that video probe aloft. Takahashi et al. (1995), in the lab experiment, observed hundreds of ice fragments per collision (
Experimental apparatus of Takahashi et al. (1995), used to observe
fragmentation in ice–ice collisions. In (A), ice spheres rotated at a tangential speed of 4 m s
Additionally, Takahashi et al. (1995) provided a reduced estimate of
In summary, both lab/field studies encompass a variety of types of ice morphology and energies of impact. Although perhaps incomplete in terms of representing the full extent of this variety in natural clouds (Sects. 4, 5), the alternative option of ignoring such a breakup when constructing cloud models would seem absurd in view of the prolific fragmentation they observed. Ice–ice collisions are the norm, not the exception, in cold clouds (e.g. snowflakes are aggregates of crystals).
KL2020 suggest that three of our modelling and theoretical studies are
somehow critically dependent on the lab/field studies (Vardiman, 1978;
Takahashi et al., 1995), as noted above (Sect. 2). Yet in reality, Yano and Phillips (2011) and Yano et al. (2016) treated the breakup at the cloud scale, not the particle scale. Both of our studies provided a 0D analytical model of a cloud, delineating a general theory of the nonlinear growth of ice concentrations with three species of ice, namely crystals, small graupel and large graupel. It was found that a sole nondimensional parameter,
Although the laboratory experiment by Takahashi et al. (1995; Sect. 2), was referred to in both papers (Yano and Phillips, 2011; Yano et al., 2016), this was solely for the purpose of estimating the number of ice fragments per
collision (
Figure 3b illustrates how a change in the number of fragments per collision
by an order of magnitude only changes the time taken to attain an ice
enhancement ratio of 100 by about 30 min (comparing
Additionally, Yano and Phillips (2016) further considered a contribution of
stochastic fluctuations of the ice-fragmentation number by collision. The
paper shows that multiplicative noise effect induced by this stochasticity
may lead to an explosive multiplication, even under a subcritical state
(i.e.,
An illustration of how our theoretical studies are not critically sensitive
to the results of the lab experiments is as follows. Yano and Phillips
(2011) and Yano et al. (2016) used the original, rescaled estimate from Takahashi et al. (1995) of
Next, Phillips et al. (2017a) created a formulation for the number of fragments from any collision of two ice particles as a function of their sizes, velocities, temperature and morphology. It was based on principles of
theoretical statistical physics considering energy conservation at the
particle scale. For a pair of colliding particles that initially are not
rotating, energy is conserved as follows:
Measurements by Vidaurre and Hallett (2009) of numbers of
fragments from ice crystals of many sizes (10–300
The empirical constants of the theory (Eq. 2) were constrained for observations of graupel–graupel collisions over a wide range of CKEs and impact speeds by the lab experiment from Takahashi et al. (1995; Fig. 5). For other microphysical species, these constants were constrained by observations made with an instrument (Fig. 1a) outdoors on a mountainside by Vardiman (1978; Fig. 1; Sect. 2). Thus, splintering for each permutation of the microphysical species in ice–ice collisions between snow, crystals, graupel and hail was represented by our formulation (Phillips et al., 2017a).
Measurements of numbers of fragments per hail–hail (2 cm diameter) collision (circles), as a function of initial CKE, which we inferred, using Hertz theory from the published data of the experiment by Takahashi et al. (1995). Source: Phillips et al. (2017a). © American Meteorological Society. Used with permission.
Phillips et al. (2017b) applied the formulation in the aerosol–cloud model (AC) to quantify the role of ice–ice collisional breakup for a convective storm observed by radar and aircraft over the High Plains in the USA (STEPS, Severe Thunderstorm Electrification and Precipitation Study). Ice–ice breakup generated over 99 % of all nonhomogeneously nucleated ice particles and was needed for the agreement of ice concentrations predicted by AC with aircraft observations, which were corrected for artificial shattering biases. Only with our scheme for the breakup included were various observations (e.g. supercooled liquid water content) of the clouds reproduced. This discovery was never mentioned in the review by KL2020.
Curiously, in the budgets of all ice particles initiated throughout the
simulated storm, the graupel–snow collisions were orders of magnitude more
prolific in generating splinters than collisions among only graupel or hail
(Phillips et al., 2017b). Thus, Phillips et al. argued that the 0D analytical model of Yano and Phillips (2011) and the related theory of instability were directly applicable to graupel–snow collisions, except with the small ice precipitation species defined as snow (
There were no tuneable parameters to adjust in that simulation by AC. In total, two cloud models with contrasting architecture simulated the same case of a mesoscale convective system that was about 100 km wide. Ours was AC with hybrid bin/bulk microphysics; the other was pure spectral microphysics, namely the Hebrew University Cloud Model (HUCM). Both models allowed the same conclusion to be reached, i.e. only by including ice–ice collisional breakup could the order of magnitude of the ice concentration observed be reproduced. Thus, there was no possibility of tuning the cloud condensation nucleus (CCN) or ice nucleus (IN) activity to give the desired ice concentration. Indeed, the vast majority (about 99 %) of ice particles initiated in the mixed-phase clouds were secondary, and there was little activity of any (CCN-sensitive) warm rain process that might yield graupel by raindrop freezing. Equally, the ascent statistics observed by aircraft were adequately predicted and were determined by the initial sounding also observed to govern the convective available potential energy (CAPE), so, again, no model tuning was possible for the sake of a semblance of agreement. Over a dozen quantities were validated against the coincident aircraft, ground-based and satellite observations (Fig. 6), including particle size distributions (Phillips et al., 2017b; their Fig. 5). Raindrop-freezing fragmentation can be ruled out for that case, as practically no supercooled rain was observed. Both models (AC and HUCM) fitted these observations well but only when including the breakup in ice–ice collisions.
Validation of the aerosol–cloud (AC) simulation with and
without the breakup in ice–ice collisions, using aircraft data, radar data and satellite data. Quantities shown are cloud droplet concentrations and mean size, liquid water content, ice concentration, radar reflectivity, cloud
fraction, ascent statistics and surface-accumulated precipitation from
AC took as input the coincident observations from the Interagency Monitoring of Protected Visual Environments (IMPROVE) of the mass concentrations of the seven chemical species of aerosol. The CCN activity was predicted and then validated by Phillips et al. (2017b). The active IN was predicted from the observed dust, soot and organic concentrations by the empirical parameterisation (Phillips et al., 2008, 2013), which has been independently validated for various other coincident observations in our earlier papers. The CCN and IN activity spectra predicted by AC were then given to initialise the HUCM.
Finally, regarding an entirely different SIP mechanism, by pooling published lab observations of freezing drops in free-fall, a formulation was created for SIP by fragmentation of raindrops freezing quasi-spherically (Mode 1; Phillips et al., 2018). By theoretical considerations, an alternative type of SIP from the collision of a raindrop with a more massive ice particle was represented (Mode 2). The formulation for both modes was applied in a detailed bin microphysics parcel simulation and shown to reproduce aircraft observations of cloud glaciation for a tropical convective case. Note that the SIP mechanism discussed by Phillips et al. (2018) has no connection whatsoever with the topic that it was erroneously cited for by KL2020, namely the breakup in ice–ice collisions (see the Corrigendum).
Including the revisions from their Corrigendum, the following statements were made by KL2020 (their Sect. 4) about fragmentation in ice–ice collisions, regarding both lab/field experiments by Vardiman (1978) and Takahashi et al. (1995), (Sect. 2) and our theoretical works.
First, KL2020 state the following: Collisional ice fragmentation was also studied theoretically by … Vardiman (1978) and Phillips et al. (2017a). These studies were based on the consideration of collisional kinetic energy and linear momentum. Such considerations would be relevant only for cases of direct central impact. In a general case, angular momentum and rotational energy should be taken into consideration. Since oblique particle collisions are more frequent than central collision, the efficiency of SIP obtained in these works is expected to be overestimated.
In the above quote, KL2020 criticise Vardiman (1978) for failing to include
rotational energy from oblique collisions in their theory, since their
observations neglect such collisions. Yet, it can be shown that the final
rotational energy is only a small fraction of the initial CKE. For any
sphere of size,
The same is even more true of nonspherical particles. These may generally
be approximated as ellipsoids. Similar arguments may be applied to an oblate
ellipsoid (axial ratio of
Alternatively, repeating this argument for a prolate ellipsoid (
Thus, artificially preventing rotational evasive rebound in the lab/field
experiment must have only weakly affected the energy available for
fragmentation during impact (Eqs. 1, 2). This available energy is related
to the difference between initial and final total kinetic energies. For
example, a 10 % change in the initial CKE would correspond, at most, to only about a 1 % change in fragments emitted per collision (
In short, the error in either the measured or predicted numbers of fragments per collision, introduced by artificially preventing any rotational evasive response on rebound, is minimal, being about 1 % or less. Naturally, these are theoretical estimates based on classical mechanics, and such a lack of the role for rotational evasive rebound for fragmentation awaits experimental confirmation.
Insofar as the above quote pertains to our theoretical formulation (Phillips et al., 2017a), it is wrong to suggest that our formulation somehow did not consider rotational effects, or that it is somehow ill-posed by not treating rotation explicitly. Rotational kinetic energy after impact is included in Eq. (1), underpinning our formulation, and it is only a small fraction of the total kinetic energy in any impact anyway, as proven above. As noted above (Sect. 3), the concepts of CKE (not momentum) and our energy-based definition of the coefficient of restitution are not restricted to head-on collisions. It is not true that our studies somehow required an assumption of either “direct central impact” or the concept of “linear momentum” for collisions in 1D. Our theoretical formulation treats collisions in 3D. Equally, the application of our theory to a simulation of clouds, after informing it with both lab/field experiments (Vardiman, 1978; Takahashi et al., 1995) that involved head-on collisions with little rotational evasive rebound, merely introduces the same error to the predicted fragmentation that affected the observed numbers of fragments. As estimated above, that error was minimal (about 1 % or less).
Second The theoretical framework of collisional fragmentation developed in Yano and Phillips (2011), Yano et al. (2016), and Phillips et al. (2017a) was calibrated against experimental results of Vardiman (1978) and Takahashi et al. (1995) [Sentence A]. A detailed analysis of the Takahashi et al. (1995) laboratory setup indicated that the riming of ice spheres occurred in still air, which resulted in more lumpy and fragile rime compared to that formed in free-falling graupel. The collisional kinetic energy and the surface area of collision of the 2 cm diameter ice spheres also significantly exceed the kinetic energy and collision area of graupel whose typical size is a few millimeters. Altogether, it may result in overestimation of the rate of SIP, compared to graupel formed in natural clouds.
There is the suggestion from the entirety of this quote from KL2020 (from Sentence A until “overestimation … in natural clouds”) that somehow simple curve-fitting of laboratory/field results was used to represent the breakup in ice–ice collisions in our theoretical and modelling work. That would be true if our theories somehow involved no dependency of fragmentation per collision on contact area or CKE, such that all simulated graupel at millimetre-sizes in clouds would somehow have to fragment, as if colliding, just like the observed giant spheres (2 cm). In fact, the opposite is true, and our formulation of fragmentation in any given ice–ice collision (Phillips et al., 2017a) has a robust theoretical basis (Sect. 3). The overall theoretical formulation itself is developed, in a general sense, from classical mechanics and statistical physics, as a versatile framework independent of any particular laboratory experiment. The fundamental quantities determining fragment numbers per collision are not merely particle size and temperature per se but are rather the initial CKE, surface area for contact and the morphological classification of the ice (Eq. 1). Such fundamental dependencies allow our formulation to apply universally to any size of colliding particle. It is not true that any “overestimation of the rate of SIP” must arise from the fact that “2 cm diameter ice spheres also significantly exceed the kinetic energy and collision area of [most natural] graupel”. These fundamental dependencies (CKE and surface area) of fragmentation are qualitatively realistic because they were derived by Phillips et al. (2017a) from universally accepted tenets of material science, such as the statistical distribution of asperities, and of classical mechanics, such as the law of conservation of energy in a collision and the concept of CKE (Sect. 3). That law is part of the first law of thermodynamics.
Evidence for the realism of the mathematical form of our formulation of fragmentation is that the numbers of fragments per collision observed by Takahashi et al. (1995) for a wide range of impact speeds (or CKEs) were realistically reproduced by our theory (Phillips et al., 2017a; Fig. 5). Naturally, Fig. 5 is not a validation, since it merely shows a comparison of our formulation with the data used to constrain values of its parameters. Yet, equally, Fig. 5 is not the result of simple curve-fitting; rather, it is the fitting of a simple theoretical expression to diverse observations at many impact speeds. It fits the data. Moreover, additional evidence is from a comparison with independent observations in Figs. 4 and 6, as noted above (Sect. 3).
In short, the fact that we fitted the theory to the giant spheres of the lab experiment of Takahashi et al. (1995) does not make it erroneous for smaller sizes because our formulation (Eqs. 1, 2) has a sound theoretical basis on the conservation of energy and is universally applicable to particles of all sizes and morphologies. CKE and surface area are always fundamental quantities for the breakup, irrespective of the nature of the collision, and the coefficient of restitution (defined in terms of energy) as noted above may be viewed approximately as an intrinsic property of the materials comprising the particles.
Although the unrimed ice sphere in each collision observed by Takahashi et al. (1995) would be expected to have a “fragile” surface, they selected this combination of surface morphologies (rimed and unrimed, depositional and riming growth, respectively) because their prior in situ observations of real clouds implied that it was representative for real in-cloud situations of ice multiplication. Previously, the fallout of large graupel into regions of weak LWC with small graupel had been observed in situ in winter-time clouds near Japan by Takahashi (1993) to generate copious ice crystals with a novel video probe. High concentrations of ice crystals co-located with graupel had been observed in the mixed-phase region of deep convective clouds near Micronesia with the video probe flown on a balloon by Takahashi and Kuhara (1993), providing indisputable evidence of ice morphology and size. Such measurements by balloon are without the artificial shattering biases that afflicted airborne optical probes prior to 2011. The issue was then mitigated by improved probe design. Along its trajectory, LWC had earlier been sufficient to produce the graupel by riming and was then depleted (e.g. by riming) and, hence, the weakness of the LWC there.
Consequently, Takahashi et al. (1995) argued that, in natural clouds, small
graupel, after forming through dominant riming growth, can encounter
predominant vapour growth when falling into such weak LWC regions, with LWC
less than 0.1 g m
Equally, the term “calibrated” in Sentence A seems misleading. A calibration usually suggests that a given model does not function properly without this procedure. That is hardly true here for any of our papers cited in that sentence (Yano and Phillips, 2011; Yano et al., 2016; Phillips et al., 2017a). Our aim in these papers was not chiefly to provide a quantitative prediction of the exact time of a certain ice enhancement to be attained or of the exact number of fragments from a given collision of two particles of known size, morphology and ambient conditions. We were not providing merely a sort of metaphorical speedometer to infer the speed of ice multiplication. Rather, our chief aim was to provide a theory based on principles of classical mechanics, material science and statistical physics with a versatile framework into which future lab observations may be assimilated as they arise. Our theories stand alone (Sect. 3).
Lastly, KL2020 claim, in the following, that no parameterizations of SIP due to ice–ice collisional fragmentation can be developed at that stage based on two laboratory observations, whose results are conflicting with each other well. [Sentence B]
Moreover, both lab/field studies by Vardiman (1978) and Takahashi et al. (1995) differed in the microphysical species of ice they observed, except for both observing graupel or hail (Sects. 2, 4.1.2). Most of the microphysical species observed by Vardiman (1978) were never studied by Takahashi et al. (1995). To study the deposition vs. riming growth combination of surface morphologies of the colliding pair (Sect. 2), as in the indoor lab experiment by Takahashi et al., 1995, was never the objective of Vardiman, 1978, who observed particles outdoors. It is an exaggeration to claim both lab/field studies conflict with each other; they tackle different facets of a problem that is as complex as the morphology of ice itself.
As justified above, both lab/field observational studies of the breakup in ice–ice collisions (Vardiman, 1978; Takahashi et al., 1995) are sufficient to constrain representations of it in models of ice multiplication in natural clouds. Our formulation of the breakup at the particle scale (Phillips et al., 2017a) is based on fundamental principles, such as conservation of energy, and the statistical physics of asperities on their surfaces. This theoretical basis enables the formulation to be constrained by lab data – even when the observational conditions of the experiment are not so representative of natural clouds (e.g. 2 cm sizes for the experiment by Takahashi et al., 1995).
The formulation was implemented in our cloud model, AC, with a storm simulation validated in many respects (Sect. 3; Fig. 6). The strong dependence of the simulated ice concentrations on the process of the breakup in ice–ice collisions (Phillips et al., 2017b) makes this effectively a validation of the formulation itself. Equally, plausible errors in the assumed number of fragments per collision in the theory of Yano and Phillips (2011, 2016) and Yano et al. (2016) do not alter the fact that the cloud microphysical system remains unstable with respect to ice multiplication (Sect. 3; Fig. 3a).
Finally, the theoretical results, from the 0D analytical model of cloud
glaciation by Yano and Phillips (2011) and Yano et al. (2016), do not depend on laboratory results in any sensitive manner (Sect. 3; Fig. 3). A lab
observation of
Since sentence B (above quote) is such a strong and sweeping claim, it is valid to ask what other criticisms could be made to support it beyond those expressed by KL2020. We identify here a few possible criticisms and then argue that they, too, lack credibility.
First, one might criticise the fixing of the target (Plexiglas or ice) in
both experiments by Vardiman (1978) and Takahashi et al. (1995). But, this fixing can be shown to be unimportant for any energy-based formulation of
fragmentation such as ours. Essentially, since the target was rigidly fixed
in both experiments, the effect on the collision is via a drastic increase in the inertial mass of one of the colliding particles (
Indeed, the coefficient of restitution for head-on collisions of a pair of particles is observed, for a wide range of conditions, to vary only weakly with their masses and impact velocity. For the present purposes, it may be viewed as an intrinsic property of the materials of the colliding particles. For millimetre-sized ice particles impacting an ice wall, below a threshold of a few metres per second of impact speed, the coefficient of restitution was observed to be constant, being independent of size and impact speed (Eidevåg et al., 2021). That threshold (attributed by Eidevåg et al., 2021, to collisional melting) is larger than the impact speeds observed by Vardiman (1978) and most of the speeds (about 1–4 m/s; Fig. 5) of the experiment by Takahashi et al. (1995). Equally, Eidevåg et al. (2021) observed no significant effect on the coefficient of restitution for a 20-fold increase in the mass of a millimetre-sized ice sphere impacting an ice wall for any given impact speed below the threshold noted above. Whether the inertial mass of one of the colliding pair is infinite, by it being fixed, seems almost irrelevant to the value of the coefficient. This is why a fixed ice target sufficed in lab studies in order to observe the coefficient of restitution for free ice particles in planetary discs by Bridges et al. (1984), Hatzes et al. (1988) and Supulver et al. (1995). Moreover, when Takahashi et al. (1995) applied their lab data (sphere of 2 cm in diameter colliding with fixed target) to a collision between two spheres that are free and of 0.5 and 4 mm in diameter, the ratio of the assumed masses is about 1000, and the CKE is practically the same as if one of them had been artificially fixed anyway. Consequently, the fixing of the target in both lab experiments introduces no significant error for our formulation (due to its fundamental basis on CKE) or for Takahashi's et al.'s (1995) estimate for natural collisions between large and small graupel (applied by Yano and Phillips, 2011).
Second, it could be argued that the Plexiglas material of the fixed plate used by Vardiman (1978; Sect. 2) could have somehow biased the coefficient of restitution, altering the energy loss on impact and, hence, the fragmentation, relative to that expected for ice–ice collisions. However, Eidevåg et al. (2021) observed the coefficient of restitution to be practically the same when comparing ice–ice collisions and ice–ABS (acrylonitrile butadiene styrene) polymer collisions, with both being in the range of 0.8 to 0.9 for impact speeds below the threshold noted above. Crucially, Vardiman (1978), before conducting the field campaign on the mountainside for the reported observations, compared the portable instrument with an alternative contrasting method (Sect. 2). Fragments were reported to be as numerous for impacts on the Plexiglas (Fig. 1a) as for the copper mesh target (Fig. 1b).
Third, it could be argued that somehow the weight of the fittings supporting the moving ice sphere in the Takahashi et al. (1995) experiment must have biased the measured fragment numbers. Indeed, inspection of published diagrams of their apparatus reveals a third ice sphere on the opposite end of the metal nearly horizontal rotating bar (about 13 cm long; 3 mm wide) for the main moving rimed sphere (Takahashi et al., 1995), (Fig. 2). It has the same size as the other two spheres but does not collide when they do. We estimate that, altogether, the moment of inertia for the main rotating sphere (A in Fig. 2) about the central axis supporting the bar was approximately doubled by including this extra sphere and the weight of the moving fixtures. Since the total initial CKE equals the rotational kinetic energy of the ice sphere plus attachments (proportional to this moment of inertia) turning on the axis (almost vertical in Fig. 2), the initial CKE is roughly doubled by the extra mass. Hence, the extra weight for the rotating sphere acts to bias the measured numbers of fragments by only about 10 % in light of observations by Takahashi et al. (1995) at various impact speeds (Fig. 5).
Fourth, there is the criticism originally made by Phillips et al. (2017a) of the Vardiman (1978) experiment, namely that ice particles were weakened by sublimation prior to collection outdoors, as they fell through the ice-subsaturated environment below cloud base into the instrument on the mountainside where they broke up (Fig. 1; Sect. 2). Phillips et al. (2017a) reported that the prediction of
Lastly, it could be argued that somehow the variation in rime density was not explored adequately in both lab/field experiments. Cloud liquid properties and the impact speed of cloud droplets when accreted were variables not varied by Takahashi et al. (1995). However, the standard conditions for riming in the experiment by Takahashi et al. (1995; see Corrigendum of KL2020) are representative of those in which large graupel are typically created in clouds. When the duration of vapour growth and riming of both ice spheres was varied from 5 to 15 min, the number of fragments measured by Takahashi et al. (1995) varied by a factor of about 2.
In summary, contrary to sentence B of the review, in view of such errors, there is no evidence that the measurements of fragment numbers in ice–ice collisions by Vardiman (1978) and Takahashi et al. (1995) are either defective or unrepresentative such as to render them useless in estimating coefficients in a theoretical formulation. Naturally, these coefficient values can easily be improved as increasingly accurate laboratory measurements become available in future but without changing the formulation itself in any manner.
On the topic of sublimational breakup, KL2020 conclude, in the following, that this mechanism is also unlikely to explain explosive concentrations of small ice crystals frequently observed in convective and stratiform frontal clouds … Activation of SIP due to the fragmentation of sublimating ice requires
One can perform a scaling analysis as follows. Oraltay and Hallett (1989)
observed rates of emission of the order of
The pivotal point here is that such a quasi-equilibrium concentration is maintained throughout the entirety of the subsequent descent after being reached (e.g. for 10 min of descent at 2 m/s). Thus, any recirculation of downdraught air into the surrounding convective ascent would transfer air enriched in fragments for their subsequent vapour growth and survival, irrespective of the timing of this recirculation during the descent and regardless of whether the prior descent is shallower or deeper.
To conclude, there is no reason to suppose that sublimational breakup is somehow insignificant in clouds, contrary to the impression given by KL2020.
The review of SIP by KL2020 (their Sect. 4) depicts our work (Yano and Phillips, 2011; Yano et al., 2016; Phillips et al., 2017a) in a distorted manner, even after their Corrigendum. In particular, KL2020 made a controversial claim (sentence B above; Sect. 4) about the impossibility of developing any model formulation based only on the two existing experimental data sets of the breakup in collisions of ice (Vardiman, 1978; Takahashi et al., 1995). No quantitative evidence was provided by KL2020 in support of the claim. Our work proves that claim to be false. Phillips et al. (2017a, b) constructed such a formulation of breakup in ice–ice collisions, applied it in a model of real clouds by constraining its empirical parameters with both lab/field studies (Vardiman 1978; Takahashi et al., 1995) and then simulated a thunderstorm case with validation against aircraft data (e.g., for ice concentrations), satellite data and radar observations, with the formulation being critical for overall accuracy (Fig. 6). The formulation performed well.
That claim arose because KL2020 did not appreciate that our formulation of such a breakup is based chiefly on an energy-based theory coming from theoretical physics that is universally applicable to all collisions (Phillips et al., 2017a). The truth is that this formulation can be delivered even without quoting the results from any laboratory experiments. KL2020 then supposed that any unrepresentative aspect of the collisions in the experiment will somehow cause problems when constraining parameters of any theory, when, in reality, our formulation is based on such fundamental quantities that this is not a problem.
In reality, the two laboratory/field studies of fragmentation may be applied
to inform theories of fragmentation in any ice–ice collision such as ours.
As noted above, our formulation does not necessarily require such lab data
anyway, as it stands alone on a sound theoretical basis. As far as we are
aware, the only potential major issue with both lab/field experiments is a
possible bias from sublimation of natural ice particles outdoors prior to
their fragmentation, as observed by Vardiman (1978). But Phillips et al. (2017a) knew about the possibility of such a bias and corrected for it when using both lab/field experiments to constrain some empirical parameters of their theory (Sect. 4.1.2). Anyway, the bias was not enough to alter the order of magnitude of
More crucially, the errors in the breakup rate per collision from the formulation, quantified by Phillips et al. (2017a) as a factor of about 2 or 3, soon become immaterial in the context of explosive multiplication of ice concentrations in a natural cloud (Yano and Phillips, 2011, 2016). Whether the number of fragments is under- or overestimated by an order of magnitude does not alter the fact that explosive growth of ice concentrations by orders of magnitude will occur anyway via the positive feedback of ice multiplication (Sect. 3; Fig. 3).
For any ice multiplication somehow involving mixed-phase conditions (e.g. growth of fragments by riming to become graupel), the explosion occurs until an upper limit of the ice concentration is reached (related to onset of water subsaturation, depending on temperature and ascent). This same final state is reached with little sensitivity to the breakup rate, with the timing depending only on the order of magnitude of the multiplication efficiency (Fig. 3). Both published lab/field experiments we used are sufficient to allow a formulation that produces a simulation of observed cloud properties, including ice concentration with proven accuracy (Phillips et al., 2017b), as noted above.
Finally, the possibility of sublimational breakup contributing to observed ice enhancement cannot be dismissed as easily as the review paper suggests. In reality, a dynamical quasi-equilibrium is established between continual emission and total sublimation of fragments so that any region of sustained subsaturation with respect to ice will develop an enhanced ice concentration persisting throughout the descent. The enhanced ice can then be transferred into regions of ascent subsequently for growth after almost any depth of descent – even after prolonged descent. An order of magnitude of ice enhancement is possible from sublimational breakup in convective downdraughts and in their vicinity within the cloud. The quasi-equilibrium is sustained because the parent particles of ice precipitation take much longer to sublimate away completely than their fragments, and they continually emit fragments throughout most of their period of sublimation. The persistence of this quasi-equilibrium ice concentration was overlooked by KL2020.
In summary, there is no evidence that the only two lab studies about the breakup in ice–ice collisions within the cloud hitherto (Vardiman, 1978; Takahashi et al., 1995) are either mutually conflicting or erroneous such that they cannot be used to treat this process both in theoretical studies and numerical modelling of clouds – contrary to the claim by KL2020. There is no reason to suppose that those observations may cause any overestimation of SIP in models.
No data sets were used in this article.
VTJP conceived the study and wrote the comment with advice from JIY. AD and DW advised on sublimational breakup and its role in clouds.
The authors declare that they have no conflict of interest.
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Vaughan T. J. Phillips was supported by three research grants, i.e. from the Swedish Research Council for Sustainable Development (FORMAS; grant no. 2018-01795), the Swedish Research Council (VR; grant no. 2015-05104) and the U.S. Department of Energy Atmospheric Sciences Research Program (grant no. DE-SC0018932).
This research has been supported by the U.S. Department of Energy (grant nos. DE-SC0018932 and DE-SC0018967), the Swedish Research Council for Sustainable Development (FORMAS) (grant no. 2018-01795), and the Swedish Research Council (VR) (grant no. 2015-05104).
This paper was edited by Daniel Knopf and reviewed by two anonymous referees.