1. Introduction
Ice in atmospheric clouds is complex even just in its own morphologies, displaying diverse shapes and sizes. These differences can make the optical characters of clouds with an identical ice concentration qualitatively different. Different optical properties, furthermore, make interactions of a given cloud with radiative heat transfer qualitatively different. Cloud–radiation feedbacks are one of the major uncertainties in current climate projection studies (e.g., Senior and Mitchell 1993; Forster et al. 2007). A better understanding of the physics of ice in clouds would reduce such uncertainties. However, the physics of ice in clouds itself is extremely complex, and there are still many processes of ice initiation that are not well understood (cf. Cantrell and Heymsfield 2005).
The present article considers one of such major issues: ice is inherently difficult to form in the atmosphere. In the absence of ice nuclei (IN), the temperature must reach below about −36°C in order for supercooled droplets to spontaneously freeze (known as homogeneous freezing; cf. Pruppacher and Klett 1997). The presence of IN promotes the formation of ice below the freezing point (0°C). However, IN particles are relatively rare in the atmosphere, with concentrations at an order of magnitude often close to 1 L−1 (Hobbs 1969; DeMott et al. 2003).
For this reason, between 0° and −36°C, clouds are likely to be mixed phase with both supercooled cloud liquid and ice coexisting. Prediction of mixed-phase clouds’ ice concentrations from given aerosol and thermodynamic conditions in the environment is particularly uncertain (e.g., Lohmann and Feichter 2005; Forster et al. 2007). Only a few careful studies have been performed with detailed cloud models in comparison with coincident aircraft observations (Ovtchinnikov et al. 2000; Phillips et al. 2001, 2005, 2007, 2009; Fridlind et al. 2004, 2007).
A major mystery of ice microphysics has been that in spite of these inhibiting tendencies in ice formation, rather copious ice particles are found in clouds extending above the freezing level and without any homogeneous freezing (e.g., Harris-Hobbs and Cooper 1987; Blyth and Latham 1993). Concentrations of up to a factor of 104 higher than the available active IN number are often observed (e.g., Hobbs et al. 1980).
The most likely explanation to reconcile this virtual paradox is to assume that the ice crystals multiply in number somehow by fragmenting. The most widely accepted theory is the Hallett–Mossop (H–M) process (Hallett and Mossop 1974): small ice splinters break away from a graupel particle as the latter grows by accretion (riming) of supercooled cloud droplets. The process works, however, only under a limited temperature range (−3° to −8°C) and only when there are supercooled cloud droplets larger than 24 μm at these temperatures.
A much simpler possibility considered here is that ice particles multiply in number as they collide with each other within clouds and consequently break up mechanically. Although this possibility received serious attention several decades ago (Hobbs and Farber 1972), it has been the least considered so far. Although pioneering laboratory experiments were performed by Vardiman (1978), not much work has followed subsequently.
The goal of the present article is to demonstrate the important efficacy of this mechanical breakup (or fragmentation) process by a theoretical investigation. For this purpose, we take the parameters estimated by more recent laboratory data (Takahashi et al. 1995). Especially based on their experimental setup, we assume large graupel with a radius of 2 mm as a major agent that induces a breakup process. We are going to illustrate the basic behavior of the breakup process, including equilibrium states and their stability, in a phase space of an idealized model consisting of cloud ice and small and large graupel particles. The model is described in the next section. Its simulations are first studied analytically with simplifying approximations, and then numerically in subsequent sections.
The present study suggests a wide applicability of ice multiplication by ice–ice collisions to various types of clouds found in various geographical locations. Possible applications include convective clouds, in which the cloud base is too cold for the H–M process to be active, such as those observed over the Japan Sea during the winter monsoon (Takahashi 1993). Midlatitude clouds over the Pacific Ocean, off the coast of Washington state, have been seen to display extensive ice multiplication despite the fact that the H–M process should not be active either because of a wrong temperature range or absence of sufficiently large cloud droplets (Hobbs and Rangno 1985, 1990; Rangno and Hobbs 1991). Altocumulus is another example of deep convective cloud with a cold cloud base that allows an application of the present result. The cloud-base temperature of Arctic clouds is substantially subzero during cold seasons, while the cloud top can sometimes be a few kilometers higher, within the mixed-phase region. As a result, the H–M process is incapable of explaining ice multiplication in these clouds (cf. Fridlind et al. 2007). Rangno and Hobbs (2001) suggest ice–ice collisions as an alternative mechanism for ice multiplication in Arctic stratocumulus clouds.
Our results suggest the need to consider the role of mixed-phase clouds with copious graupel in climate change. Complex processes of phase change from liquid to ice in mixed-phase clouds need to be represented adequately in climate models for feedback studies and for realistic prediction of global change (Senior and Mitchell 1993; Tsushima et al. 2006). Overall ice concentrations are seen to be governed by ice multiplication (e.g., Hobbs et al. 1980). Incomplete knowledge of ice initiation in mixed-phase clouds noted above suggests that studies of new mechanisms of ice multiplication may be needed to enhance the treatment of the linkage between ice concentrations and aerosol loadings/composition in climate models.
2. Model
Here, an idealized system consisting only of three types of ice—large and small graupel particles and ice crystals—is considered. Evolution of these three ice types is described in terms of the number concentration n (m−3), with subscripts G, g, and i, respectively. The model cloud may be interpreted as an extreme limit of bin models truncated only to two categories. Although we believe that equivalent results would be reproduced by considering more graupel types, the present extreme truncation of the bin model enables us to elucidate the nature of ice multiplication by mechanical breakup more clearly.
Laboratory experiments by Takahashi et al. (1995) show that the number of secondary ice crystals left behind could be as high as 103 when two heavily rimed ice balls of 1.8 cm in diameter collide together. The collision speed in the experiment approximately corresponds to a typical fall speed of a graupel particle with a radius of 2 mm. When the impact of the collision of these two ice balls is scaled down to the level expected for the graupel of this radius, Takahashi et al. (1995) estimate that the average multiplication between −13° and −18°C would be scaled down from about 450 to 60 ice fragments per collision. In the present study, we take N = 50 as a multiplication rate.
The probability for a collision between graupel particles is proportional to a volume α swept out by a large graupel particle per unit time and number concentrations of large graupel nG and small graupel ng. The volume sweep-out rate is estimated as α = 2.4 × 10−5 m3 s−1 assuming a radius of 2 mm and fall velocity of 3 m s−1 for the large graupel [cf. Rogers and Yau (1989), their Fig. 9.7].
To see the influence of this process on the evolution of ice crystal number concentration ni, we consider an isolated cloud element in which three types of ice are always uniformly well mixed. By aiming at a lucid theoretical demonstration, we further introduce the following assumptions:
- (i) Ice crystals are formed at a constant rate c0 by a supply of water vapor from below in ascent. As a result, the total ice crystal generation rate is given by
(ii) An ice crystal grows into a small graupel particle by diffusion of vapor and then by riming over a fixed time scale τi = 15 min. Small graupel furthermore grows into large graupel by riming on another fixed time scale τg = 30 min, and large graupel is lost from the cloud element after a fixed time τf = 10 min by gravitational fallout.
The graupel growth time τg is estimated by integrating equations for growth numerically in time for continuous riming of cloud liquid under a fixed environment with a liquid water content (LWC) of 0.5 g m−3 and a temperature −15°C. These equations are based on the well-established empirical formulas from the literature reviewed and applied by Phillips et al. (2001, 2002) for the concentration of accreted rime and collision efficiencies. Note that τg can be greatly shortened by a more abundant cloud liquid content, for example, in deep convective clouds. Also, any raindrop freezing would radically shorten the time for appearance of first large graupel (e.g., Phillips et al. 2001). Thus, our assumed value for τg is close to an upper limit.
The basic premise here is that sufficient liquid water is available so that riming of graupel is a key process in forming ice precipitation, rather than aggregation or vapor deposition for snow. Under this premise, snow is not taken into account as a sink for cloud ice. This is clearly an advantage for our purpose of demonstrating the ice breakup process with minimum number of ice types. Graupel, which is heavily rimed ice precipitation, is known to be more important than snow in deep convective clouds (e.g., Reisin et al. 1996, p. 510), which have faster ascent than stratiform clouds and hence more supercooled cloud liquid for riming. In this sense, our model is more applicable to convective clouds than stratiform ones.
Another important premise here is that small graupel would be retained within an assumed homogeneous cloud element for the whole period of our theoretical calculations, although large graupel is lost from the system in a finite time. Note that small graupel typically has a fall speed of 1–2 m s−1. Thus, while they grow to large sizes over 30 min, they typically fall a distance of about 3 km relative to the ambient air. Many clouds in the mixed-phase region of temperature (0° to −36°C, 5 km deep) are deeper than this fall distance and remain active for more than for 30 min, enabling formation of such large graupel. Recirculation into and out of a convective updraft aloft also promotes the longevity of a graupel particle (cf. Blyth and Latham 1997).
The rate of primary ice crystal generation c0 can vary by orders of magnitudes in natural clouds. A median value may be estimated as c0 = 6 × 10−2 s−1 m−3 by assuming a rate of ascent equal to 1 m s−1, a lapse rate or decreasing rate equal to 6 × 10−3 K m−1, and a rate of active IN concentration increasing with decreasing temperature equal to 10 m−3 K−1 (Phillips et al. 2008). The assumed ascent is rather strong for stratiform clouds but modest enough for convective clouds. For instance, Phillips and Donner (2006, their Fig. 4) show typical distributions of vertical velocity observed and simulated in deep convective systems. In general, c0 can vary by a factor of 104, either above or below the median just estimated, because the ascent varies over a range of 10−2–10 m s−1 or more, and the IN concentration gradient can vary at least over 10−1–104 m−3 K−1.
3. Phase space analysis
Such an explosive situation is realized when either c0 or α̃ is large enough, or alternatively when τg or τf is long enough. The last two conditions may seem at first counterintuitive, but they actually reflect the fact that the slower the rate of their loss, the more graupel there are for collisions, thus contributing to explosive ice multiplication.
Substitution of the standard parameters already defined leads to c̃ = 3.1 × 102. Thus, many natural clouds are well above the criticality. Alternatively, supercriticality is achieved by setting c0 > 1.9 × 10−4 s−1 m−3, τf > 1.9 s, or τg > 5.8 s, keeping all the other parameters fixed to standard values. All these minimum values are much smaller than those typically observed. The minimum ng(∼1/α̃τf ∼ 1.4 × 10−3 L−1) required for explosive multiplication is also well below typical values, 0.1–10 L−1, both observed and explicitly simulated (Hallett et al. 1978; Phillips et al. 2002, 2003, 2005).
Consequently, many clouds are likely to be in the explosive ice multiplication regime, provided that their small graupel particles can remain in cloud above the freezing level for long enough to become large graupel by riming. For this purpose, the cloud must be sufficiently deep and long lived, with suitable rates of ascent, or with sufficient liquid water content, to promote this growth and support the graupel aloft. It is important to emphasize that once these conditions are satisfied, this explosive multiplication process can be realized in any types of clouds at any geographical locations, in the absence of any other faster multiplication mechanisms.
Note that in deep convective clouds, any supercooled rain aloft (e.g., Blyth et al. 1997; Hobbs and Rangno 1998) can radically shorten the times (τg and τi) required for growth of small crystals to large graupel. Extensive supercooled rain drops are often found in midlatitude deep convective clouds, because turrets (thermals) lifted from surface provide a fresh supply of liquid water to the top of the clouds (cf. Ludlam 1952; Koenig 1963). Coalescence rapidly produces raindrops during ascent toward a cloud top with little dependency of the process on temperature. Coalescence is further intensified by the turbulence within the turret that boosts collision efficiencies for droplet interactions. However, these raindrops may initially remain unfrozen (though supercooled) because of a slow rate of heterogeneous freezing. As soon as small ice particles collide with these supercooled raindrops (Pruppacher and Klett 1997), the ice crystals almost instantly become small or large graupel by raindrop freezing. The process, simulated by Phillips et al. (2001, 2002, 2005) and also observed by multipolarimetric radar (e.g., Bringi et al. 1997) and by aircraft (e.g., Koenig 1963; Hallett et al. 1978; Blyth et al. 1997), drastically shortens the time scale of any ice multiplication just involving graupel (e.g., the H–M process, if active).
4. Numerical experiments
To verify that the above conclusions also apply to the full system, numerical experiments are performed with the system [Eq. (2)], integrating it with the original “lag” assumptions [stated above as item (ii)] and formulated by Eq. (3). Here, we treat precisely the conversions between species and fallout of large graupel assuming fixed realistic times, without the earlier approximation based on Eq. (4). A standard manner of quantifying the ice multiplication rate is to divide the actual ice crystal number by that expected when the ice multiplication process is absent. This quantity is called the ice enhancement (IE) ratio. Our results are also shown in terms of the time series of IE ratio. Note that all the computations start with no ice initially.
Figure 2 shows the sensitivity with respect to c̃: below c̃ = 1, the system reaches an equilibrium state, whereas once c only slightly exceeds the criticality, the explosive ice multiplication happens (see the short-dashed curve with c̃ = 2.0). When the standard value c̃ = 3.1 × 102 (double chain dash) is taken, explosive ice multiplication starts as soon as the first large graupel particles are formed at t = τi + τg = 45 min. These experiments can be interpreted as a sensitivity with respect to c0 by keeping all the other physical parameters fixed to their standard values. Alternatively, they can also be interpreted as a sensitivity with regard to the multiplication rate N. For example, its reduction to N = 1 would provide a curve close to the one with c̃ = 10 in the figure.
Order-of-magnitude variations in supercooled liquid water content observed are 10−5–10−4 kg m−3 in stratiform clouds (Hogan et al. 2002; Phillips et al. 2003) and 10−4–5 × 10−3 kg m−3 in deep convection (Phillips et al. 2005). They further alter τg logarithmically. For this reason, Fig. 3 examines the sensitivity of explosive ice multiplication with respect to τg by fixing all other physical parameters.
The main effect of changing τg is to shift the timing, τi + τg, of the appearance of the first large graupel particles. An increase of τg also slightly enhances the subsequent ice multiplication rate because of an associated increase of the ice multiplication efficiency. In general, the ice particle numbers multiply by a factor of 10 approximately every 10 minutes.
A case of ice multiplication by mechanical breakup hastened by supercooled rain aloft is shown by a blue curve in Fig. 3. Here, drastically shorter growth time scales, τi = 5 min and τg = 10 min, are assumed based on the estimation in appendix B: small ice crystals collide with supercooled drops (∼0.5 mm in radius) to become instantly small graupel (e.g., Phillips et al. 2001, 2002) that then rime to become large graupel. Much shorter growth times for both small and large graupel are followed by mechanical breakup in ice–ice collisions. The onset of τi + τg is much earlier in this case with much faster growth initially compared to the standard runs. However, the rate of ice multiplication growth remains similar to the other cases as the IE ratio approaches 105.
5. Discussion
The behavior obtained for our system may be compared with ice multiplication due to splintering of supercooled cloud droplets when they rime on graupel (H–M process; Hallett and Mossop 1974). Its significance has been well established by very detailed cloud models, providing good agreements with aircraft measurements for both convective and stratiform mixed-phase clouds (e.g., Ovtchinnikov et al. 2000; Phillips et al. 2001, 2003, 2005, 2007, 2009). Statistical analysis of aircraft observations of hundreds of deep cumuli with warm bases over the continental United States also proves the ubiquitous control of ice concentrations by the H–M process (Harris-Hobbs and Cooper 1987). However, the H–M process is inactive in certain mixed-phase clouds, such as those with a cold cloud base (e.g., in the Arctic; Lawson et al. 2001; Morrison et al. 2005) or those with cloud droplets (−3° to −8°C) that are too small, as seen in a minority of New Mexican convective clouds (Blyth and Latham 1993) or as expected from aerosol pollution, and those entirely above this H–M region (e.g., Eidhammer et al. 2010).
A salient difference between the H–M and ice–ice collision processes of multiplication is that the former happens in direct association with riming growth of graupel. As a result, the ice multiplication rate by the H–M process is only linearly proportional to the graupel number concentration. This is in contrast with the ice–ice collision process, which is proportional to the square of the graupel number concentration. As a result, multiplication by ice–ice collisions may eventually become faster than the H–M process as the graupel number increases. In the idealized situation with a fixed rate of supply of the supercooled cloud liquid, as assumed in the present study, the H–M process multiplies the ice crystal number only exponentially. A red curve in Fig. 3 shows a result for the H–M process based on an equivalent simple system derived in appendix C. The ice–ice collision process can confer a superexponential ice multiplication due to its nonlinearity (black solid curve in Fig. 3).
Another key difference is that the H–M process may start much sooner than the ice–ice collision process, as small graupel can produce H–M splinters as soon as they form, in the right conditions. It only takes 15 min for the case shown in Fig. 3. The onset of multiplication by mechanical breakup in ice–ice collisions is delayed until large millimeter-sized graupel can appear, taking 45 min in the standard case. Thus, in nature the H–M process, if active, would usually occur first and persist to dominate the overall ice multiplication, as observed by Harris-Hobbs and Cooper (1987). However, raindrop freezing in collisions between supercooled rain and ice crystals in deep convective clouds can greatly shorten the time for appearance of large graupel (Bringi et al. 1997; Phillips et al. 2001) as noted above (section 3), and favors both types of multiplication (see blue curve for ice–ice breakup coupled with raindrop freezing in Fig. 3). Note that raindrop freezing radically accelerates the H–M process, too, if active (Phillips et al. 2001, 2002).
The elegance of the present analysis arose from a drastic simplification of the model, limiting it to only three ice types and reducing the geometrical dimension of the model just to zero. A major premise is that graupel are supported aloft for sufficient time to become large enough to fragment, or that the cloud is deep enough for supercooled raindrops to exist, freezing to form graupel. A dynamical process that supports such a state is implicitly prescribed under the present idealized formulation. Under these simplifications, it is demonstrated that explosive multiplication of ice happens in a wide range of the nondimensional space analyzed (Fig. 1). This nondimensional range is so large that we can even conclude that such explosive multiplication can happen almost in any mixed-phase clouds as long as the major premise just stated above is satisfied.
The philosophy of our approach is that understanding of a complex system can be advanced by capturing the essence of an aspect of it with a dedicated, greatly simplified model. Simplification allows analytical determination of the phase space and dependencies of the system. Indeed, Held (2005) delineated two contrasting types of modeling approach in science, one that seeks realism with very detailed models and the other devoted to understanding with simplified ones. If we were to attempt to construct an equivalent diagram with a full cloud physics model that employs numerical time integrations, millions of cases would have to be run before such a diagram could be drawn. Also, when many physical processes are simultaneously modeled, it becomes difficult to identify causation for the target process, owing to compensating responses by the other processes when sensitivity tests are performed. Note that our theoretical approach can be compared to the one by Chisnell and Latham (1976) for an initial assessment of the H–M process and its dependencies.
It must also be emphasized that various slight modifications of the present model are relatively straightforward, and some of these could even be considered as exercises for the readers. For example, we have assumed that small graupel always remains within a homogeneous cloud element before becoming large graupel. It is straightforward to generalize the problem by assuming that only a fraction of small graupel particles is retained in the system for growth to form large graupel particles. The remainder is lost by gravitational fallout. In this case, the critical threshold for c̃ simply increases from unity to the root square of the inverse of the fractional retention rate. Another exercise might be to include the mechanical breakup during collisions of large graupel with other large graupel particles, assuming an average dispersion of their terminal velocities corresponding to an assumed size distribution and using laboratory results from Takahashi et al. (1995).
An explosive tendency, identified in the present study, would be suppressed in realistic situations before literally going to infinity by various other processes. Ways of suppressing it include the collapse of the humidity to ice saturation with evaporation of all cloud liquid (Korolev 2007) and cessation of riming, terminating both the production of graupel and vapor growth of any fragile branches on the rimed surfaces that are required for mechanical breakup (Takahashi et al. 1995). Explosive ice multiplication either by the H–M process or by ice–ice collisions also ceases as a result. Generation of primary ice ceases as humidities approach ice saturation (Möhler et al. 2006; Phillips et al. 2008). At temperatures warmer than −40°C, humidities well below water saturation make heterogeneous ice nucleation scarce or nonexistent (Field et al. 2006b). Graupel growth is terminated by mere onset of subsaturated conditions with respect to liquid water, if partial evaporation then makes cloud droplets too small to rime efficiently.
An estimate for the timing of collapse of the humidity, when supercooled liquid disappears and riming ceases, can be provided by evoking a closed theory developed by Korolev and Mazin [2003, see their Eq. (12)]. The moments that the system ceases to be supersaturated and it reaches −1% supersaturation with respect to liquid water are marked by two horizontal bars on each curve of Fig. 3. Mixed-phase clouds with supercooled liquid were predicted by Korolev and Mazin (2003) to have humidities close to water saturation because the liquid phase, and not the ice phase, controls their supersaturation. Thus, if the supersaturation becomes appreciably negative (e.g., less than −1%), the liquid is expected to evaporate away totally, just as it would in a liquid-only warm cloud. When the ice particles have become too numerous, such total evaporation is assumed to be realized. This timing is also controlled by the vertical ascent that supplies supply water vapor from below in sustaining the supersaturated state, and by the temperature.
Here, the cloud air is assumed to be at 500 hPa and −15°C, with a vertical velocity of 1 m s−1, a cloud droplet number concentration of 108 m−3, and effective radii of 100 μm and 15 μm for ice crystals and cloud droplets, respectively. We see that according to this theory, explosive ice multiplication is fully realized in all these five cases before the humidity collapses.
The same runs are also repeated, but turning off c0 as well as all graupel growth when the supersaturation has reached −1%. The results are shown in Fig. 4.
Here, we see another difference between the H–M process and the ice–ice collision multiplication: the Hallett–Mossop process stops as soon as the system reaches the −1% supersaturation because at that point the humidity collapses and riming ceases, leaving no source for the H–M process. On the other hand, multiplication by ice–ice collisions continues for some time after the collapse of humidity, albeit with a much reduced rate due to the remaining graupel available. The multiplication process finally ceases when all the large graupel are lost by gravitational fallout. We expect that explosive multiplication would be further suppressed if the cloud droplet budget were also explicitly taken into account.
Despite of the wide parameter range over which this ice multiplication process is applicable, it is still an open question as to whether a similar process involving ice–ice collisions is active in stratocumulus clouds in Arctic. Unfortunately, only few graupel are observed in these Arctic clouds (Morrison et al. 2010, manuscript submitted to Quart. J. Roy. Meteor. Soc.; Greg McFarquhar 2010, personal communication). Thus, if a similar process is ever going on, it should involve much smaller ice particles than considered herein.
In this respect, it is intriguing to note that Fridlind et al. (2007) indeed found such an explosive tendency (which they termed “runaway glaciation”; see their section 4.4) in one particular case (however, the case was observationally least constrained) of their Arctic cloud simulation when slow fallout of snow was assumed. The explosive tendency was so strong that it was impossible to continue to run the model in a numerically stable manner. The present study suggests a possible interpretation that the runaway glaciation found in their experiment was not merely a problem with numerical stability, but actually may have been a result of explosive multiplication as identified here. This again points to the advantages of a simple model for identifying a critical process in a system that cannot be easily identified with a complex numerical model.
6. Conclusions
The present theoretical investigation demonstrates that, in spite of less attention thus far, the mechanical breakup in ice–ice collisions is an efficient process of ice multiplication. It requires simply that there be millimeter-sized graupel, supercooled liquid, and preexisting ice crystals at concentrations above the critical values. The phase space of this ice–ice breakup system is characterized by an ice multiplication efficiency and by a graupel number concentration. This ice multiplication efficiency c̃ is proportional to the ice multiplication per collision N, the primary ice production rate, as well as to the time scales for growth of graupel to large sizes and for its fallout. Most natural clouds are in the explosive multiplication regime, where the ice multiplication efficiency is greater than unity. Note that whether the breakup multiplication actually occurs depends on whether ice can grow by riming or raindrop freezing to become large graupel while in cloud.
We have also shown that the time scales for generation of small and large graupel, whether by riming or raindrop freezing or both, control the timing of the onset of any explosive breakup. In nature, this timing is crucial because it must be within the lifetime of the large ice particles aloft and of the cloud itself, for the explosive breakup to be realized.
Specifically, our idealized model of multiplication by mechanical breakup predicts that
a high IE ratio of 104 is attained about an hour after small graupel first appear in our standard run without supercooled rain;
the time for an IE ratio to exceed 104 is shortened to only 20 min when supercooled raindrops are included;
if active, the H–M process starts much sooner than mechanical breakup;
actual ice concentrations realized by mechanical breakup (or by the H–M process) reach an ascent and temperature-dependent maximum value determined by the timing of evaporation of cloud liquid at the onset of subsaturated conditions (droplets evaporate partially or totally, ending graupel production by riming) with breakup then persisting for a short time afterward; and
inclusion of the response of humidity to explosive ice multiplication yields a maximum IE ratio on the order of about 105 with our standard parameters: generally this maximum IE ratio increases with the ascent rate.
The ice breakup multiplication complements the usual H–M process, which is inactive in certain mixed-phase clouds including Arctic low clouds. Some of the most detailed cloud models have hitherto been unable to simulate the glaciation with conventional mechanisms for the latter (e.g., Fridlind et al. 2007). We have shown that the H–M process, if active, dominates overall ice multiplication, as observed. However, if the H–M process is inactive and graupel is abundant, then the breakup multiplication process dominates overall ice concentrations.
Laboratory results for mechanical breakup (Takahashi et al. 1995) show a strong dependence of the number N of splinters from each ice–ice collision on temperature, with a peak at about −15°C, while the H–M process happens only at much warmer subzero temperatures. Field observations of the IE ratio in many natural convective clouds (Hobbs et al. 1980, cf. their Fig. 25) show a peak at a similar temperature, consistent with ice–ice breakup. Note that Hobbs et al. counted the ice particle number mostly based on Formvar replication. Though they also deployed an optical array (cf. their Table 1), the latter was used only under a calibration against the replication-based measurements (Hobbs et al. 1975). Thus, their analysis is likely to be free from biases in measurements by optical arrays (e.g., due to shattering of ice on the probe) more recently pointed out (Field et al. 2006a).
Extensive laboratory measurements of ice–ice breakup (e.g., exploring dependencies on ambient turbulence and bulk density of the graupel surface) are still required to parameterize it accurately in cloud or climate models and assess its scope of influence for diverse cloud types. For example, although a possibility of ice multiplication by graupel–snow collisions has already been explored by Vardiman (1978), full investigation is still awaited.
Acknowledgments
This research was partly supported by Les Enveloppes Fluides et l’Environnement–Interactions et Dynamique de l’Atmosphére et de l’Océan (LEFE–IDAO) and the Office of Science (BER), U.S. Department of Energy. The award from BER to VTJP concerns the indirect effect from aerosols on ice clouds. Communication with Ann Fridlind, Greg McFarquhar, and Hugh Morrison is gratefully acknowledged. VTJP is grateful to Professor Tsutomu Takahashi for advice about how to model this breakup process. The present collaboration has been catalyzed by COST Action ES0905 funded by the European Science Foundation (ESF).
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APPENDIX A
Phase Space Analysis: Mathematical Details
Here, we see that these two equilibrium solutions are real numbers only if c̃ ≤ 1. In this case, the two solutions defined by Eq. (6) are physically valid, both corresponding to nonnegative graupel numbers. On the other hand, when c̃ ≥ 1, physically no equilibrium solution exists, as stated in the main text.
To facilitate the subsequent analysis, we further simplify the system for the situation when breakup is not yet explosive. For this purpose, we note that among the three characteristic time scales introduced, the time scale τg for generation of large graupel from small graupel is relatively longer than the others (i.e., τi ∼ τf ≪ τg). That furthermore implies τiṅi* ∼ τfṅG* ≪ τgṅg* in Eqs. (A2a)–(A2c), assuming ni* ∼ ng* ∼ nG*. In the nonexplosive regime, the right-hand sides of Eqs. (A2a)–(A2c) are all of the same order; thus, the left-hand sides of Eq. (A2a) and (A2c) can be neglected compared to the corresponding right-hand sides, whereas the left-hand side of Eq. (A2b) is of equal importance to the terms on the right-hand side.
APPENDIX B
Analysis of the Case with Supercooled Rain Aloft
Estimations of growth times (τi and τg) for graupel, when supercooled rain is present above a riming zone within a convective cloud, are presented in this appendix.
An ice crystal initially grows by diffusion of water vapor by taking about 100 s to reach a size of 100 μm. This ice crystal is further assumed to continue growing in the given situation, eventually colliding with a single supercooled raindrop to become almost instantly a small graupel particle. Note that the collision efficiency for this collision becomes appreciable only at sizes greater than 100 μm (Pruppacher and Klett 1997). We assume a typical size of both a supercooled raindrop and a small graupel particle to have a radius r0 = 0.5 mm. A typical time scale for that to happen is estimated as follows.
A supercooled raindrop typically falls with a speed υT ∼ 5 m s−1. As it falls, it sweeps out a volume αr,i = πr02υT ∼ 3 × 1/4 × 10−6 × 5 ∼ 5 × 10−6 m3 s−1. Assuming a number concentration nr ∼ 1 L−1 ∼ 103 m−3 for the supercooled raindrops, the rate of collision is nrαr,i. Thus, a typical time required for an ice crystal to turn into a small graupel is given by 1/nrαr,i ∼ 1/(5 × 10−6 m−3 s) × 1/(103 m3) ∼ 200 s. As a whole, it takes τi ∼ 100 + 200 = 300 s = 5 min for an ice crystal to grow into a small graupel particle in a collision with a supercooled raindrop.
A time scale for growth of this small graupel particle (frozen raindrop) into a large graupel particle under the same raindrop freezing situation is numerically estimated by a model (Hobbs et al. 1980; Blyth and Latham 1993) with riming growth. The simulation provides an estimate of τg = 10 min.
It would be important to emphasize that in this case with supercooled rain aloft, no ice multiplication is assumed during the instant of raindrop freezing, and thus the role of raindrop freezing is merely to hasten the appearance of large graupel particles required for mechanical breakup in ice–ice collisions.
APPENDIX C
The Hallett–Mossop Process
Phase diagram summarizing the behavior of the ice–ice breakup system. Two equilibrium solutions for the nondimensional small-graupel number α̃τfng (vertical axis) are plotted against the ice multiplication efficiency c̃. The solid and dashed curve segments indicate, respectively, unstable and stable solutions. When c̃ > 1, to the right of the vertical solid line, no steady solution exists. The two solid segments (solid curve) thus indicate a boundary between explosive and damping regimes: when the initial condition is above or to the right of the solid curve, explosive ice multiplication by ice–ice collisions happens. On the other hand, when the initial condition is below or to the left of the solid curve, ng approaches the lower equilibrium solution (dashed curve) with time.
Citation: Journal of the Atmospheric Sciences 68, 2; 10.1175/2010JAS3607.1
Explosive ice multiplication by ice–ice collisions with c̃, as a nondimensional controlling parameter: time series of the ice–enhancement ratio with c̃ = 0.5 (long dash), 1.0 (solid), 2.0 (short dash), 10 (chain dash), and the standard value 3.1 × 102 (double chain dash) obtained by full numerical solution of the system [Eq. (2)] with the assumptions made in Eq. (3). These values correspond to c0 = 9.5 × 10−5, 1.9 × 10−4, 3.8 × 10−4, and 1.9 × 10−3 s−1 m−3, respectively, when all the other physical parameters are fixed to the standard values. Under this assumption, every single ice particle remains in its species for the entire constant period of time (15, 30, and 10 min for ice crystals and for small and large graupel, respectively). Note that values above 105 are truncated on the vertical axis, being deemed unphysical.
Citation: Journal of the Atmospheric Sciences 68, 2; 10.1175/2010JAS3607.1
Dependency of explosive ice multiplication by ice–ice collisions on τg: time series of the ice enhancement ratio with τg = 20 min (black long dash), 30 min (standard value, black solid), and 40 min (black short dash). Explosive ice multiplication starts at t = τi + τg in all cases following similar curves subsequently, where τi = 15 min. The case with Hallett–Mossop process but without ice–ice collisions is shown by a red solid curve. It is compared with the standard case (black solid), with which in all other respects it is identical. Furthermore, to consider a possibility of ice multiplication by mechanical breakup in ice–ice collisions (no H–M process) associated with supercooled rain (1 L−1) that freezes in collisions with small ice crystals aloft, a case with τi = 5 min and τg = 10 min is also shown by a blue solid curve. The moments that the system reaches the supersaturations of 0% and −1% with respect to liquid water are marked by horizontal bars on each curve.
Citation: Journal of the Atmospheric Sciences 68, 2; 10.1175/2010JAS3607.1
As in Fig. 3, but both c0 and all of the graupel growth processes are turned off when the supersaturation with respect to liquid water has reached −1%. The Hallett–Mossop process ceases at this point, but multiplication by ice–ice collisions continues until all the large graupel are lost by gravitational fallout.
Citation: Journal of the Atmospheric Sciences 68, 2; 10.1175/2010JAS3607.1
