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, R, and mass, M, its moment of inertia is I=2/5MR2. Consider a strongly oblique collision between two spheres of different sizes and assume very frictional surfaces for both. During the rebounding part of the period of contact, the ratio of rotational to translational kinetic energy of the smaller sphere is approximated by that of a sphere rolling on a flat surface, i.e. 2/5. Here the assumed difference in size is sufficient to make the effective mass of the pair approximately the same as the smaller particle; less than a 10 % difference between effective and smaller masses implies a difference of a factor of more than 2 in size. At the other extreme, for a head-on collision between such spheres, there is zero rotation afterwards. This ratio of rotational to translational kinetic energies must be a monotonic function of the angle of impact, since any rotation only arises from torque exerted during impact. Thus, by this assumption of very frictional surfaces, the average fraction of initial CKE converted to rotational energy, accounting for all possible angles of collision, would be of the order of 10 %. (For exact spheres, this must be an upper bound, since Supulver et al., 1995, observed for slow ice–ice collisions of 5 cm ice spheres in the lab that friction is insignificant, implying little transfer to rotational energy.)

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 0<s≤1) colliding with a much larger particle (by at least a factor of 2) of any irregular shape. The smaller one falls with the longer axes initially horizontal (radius a and mass M). Now the nonspherical shape plays the role of perfect friction. The moment of inertia of the smaller ellipsoid for rotation around a horizontal axis is I=(1/5)Ma2(1+s2). In the frame of reference of the centre of mass, the angular velocity of the smaller particle after the most oblique impact is ω≈V/a, where V is its initial relative speed. Thus, the ratio of rotational kinetic energy ((1/2)Iω2) after collision to the initial translational kinetic energy ((1/2)MV2) is just (1/5)(1+s2)≤2/5, which reduces to 2/5 in the special case of spheres (s=1), which is the value noted above. As before, the ratio for a head-on collision is zero. Again, the average fraction of initial CKE converted to rotational energy for all possible angles of collision is of the order of 10 % and is smaller for oblate ellipsoids than for the spherical case.

Alternatively, repeating this argument for a prolate ellipsoid (s>1; shorter radius, a), ω≈V/sa, for the most oblique impact implies that a ratio of rotational (after) and translational (initial) kinetic energies is (1/5)(1+1/s2)<2/5. The same estimate of the average fraction of initial CKE converted to rotational energy is reached (order of 10 %).

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 (N≈200), in view of our analysis of observations by Takahashi et al. at various impact speeds (Phillips et al., 2017a; Fig. 5). Thus, the omission of rotational evasive rebound from oblique impacts in Vardiman's (1978) theory, and in the observations of both lab/field experiments (Vardiman, 1978; Takahashi et al., 1995), cannot seriously have biased the fragmentation 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, KL2020 state the following:

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-3. Even if such parts of any convective cloud are limited, the in-cloud motions must tend to mix the crystals throughout most subzero levels of the cloud over time. The inherent nonlinearity of ice multiplication occurring by such a breakup, involving the growth of splinters to become ice precipitation by a positive feedback (Yano and Phillips, 2011), must make such active parts of the cloud disproportionately influential compared to their volume.

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 N is merely used for an estimate of a value of the nondimensional multiplication efficiency parameter, c̃, along with several other values, so as to illustrate how natural deep convection is in an unstable regime with respect to ice multiplication. That investigation (Yano and Phillips, 2011; Yano et al., 2016) was not fixed to this estimated value but, instead, was performed over the full range of possible values of this nondimensional parameter (c̃). In short, KL2020 give a misleading picture of our work, despite the Corrigendum.

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 (m1 and m2). The “particle” becomes the planet Earth or, effectively, infinite mass. The general equation for the initial CKE with a relative speed of V (along the line of collision) is as follows: 3K0=12m1m2m1+m2V2. Fixing one of the particles is expressed by m1→∞ and then K0→(1/2)m2V2. As an example, consider two identical particles m1=m2=M that collide when free. The initial CKE is then K0=(1/4)MV2. Now we fix one of the particles, and K0=(1/2)MV2. Consequently, the effect from fixing one of the particles on the nature of collision is represented by the CKE being doubled. Whether or not both particles are free does not alter the fact that CKE always provides the initial energy relative to the centre of mass of the colliding pair, which is the source of energy for deformation (e.g. fragmentation). CKE is always the fundamental quantity. Thus, since the formulation by Phillips et al. (2017b) related the number of fragments to the CKE in the lab experiment, this relation is universal to any type of collision, irrespective of whether or not both colliding particles are free.

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 N from Eq. (2), which, for graupel–graupel collisions, is fitted to only the Takahashi et al. (1995) observations, would be much lower (by about half an order of magnitude) than if Eq. (2) were fitted only to observations by Vardiman (1978) for collisions of graupel for the same size (the largest he observed). Yet, in reality, this was never a grave problem for our formulation because Phillips et al. (2017a) then applied a correction factor to the fragility coefficient, C, in Eq. (2) to yield a match at this size between both predictions. Phillips et al. (2017a) then applied this same correction factor to the formulation for other microphysical species (graupel–snow and graupel–crystal collisions) when constraining them with the Vardiman (1978) data alone. Prior sublimation artificially boosting the observed fragmentation of natural ice falling onto the mountainside was invoked as the reason for this empirical correction by Phillips et al. (2017a).

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.