## 1. Introduction

Atmospheric physical processes are often stochastic, working like a random-number generator. Such a stochastic process is best described by a probability, because standard deterministic computations may miss an outcome that may appear as a surprise by chance. Cloud microphysics is a particular domain in the atmospheric physics that contains a rich variety of stochastic processes.

The present study is more specifically motivated from the extensive issues with turbulent fluctuations within clouds (Vaillancourt and Yau 2000; Shaw 2003; Bodenschatz et al. 2010). Field measurements quantify the degree of these turbulent fluctuations within warm shallow clouds. In this manner, Haman and Malinowski (1996) find that in-cloud temperature fluctuates even by up to 2 K over a distance of less than 10 cm. Davis et al. (1999), Gerber et al. (2001), and Siebert et al. (2006) discuss substantial fluctuations in liquid water content down to the centimeter scale. Brenguier (1993) and Malinowski et al. (1994) delineate sharp interfaces of clouds over these scales in their measurements of cloud droplet number density to the similar scale. Aircraft observations of Arctic clouds by Prenni et al. (2009), in turn, reveal a pulselike behavior in time series of ice condensation nuclei (IN) in the Arctic region. The background tropospheric IN concentration is also known to fluctuate enormously (DeMott et al. 2003; Phillips et al. 2013). These fluctuations are likely treated as stochastic processes. The present study is a first step for extensive applications of stochastic modeling in microphysical studies under a framework of the dynamical-systems approaches [cf. Guckenheimer and Holmes (1983): see also Yano et al. (2016) and references therein].

As one such example, the present paper takes breakup of large graupel under collisions with small graupel, which may lead to explosive ice multiplication. The ice breakup process is experimentally studied by Vardiman (1978) and Takahashi et al. (1995) as a possible mechanism for ice multiplication. Importance of ice multiplication processes has been emphasized extensively in literature [Hobbs (1969) and references therein; see also chapter 9 of Rogers and Yau (1989)]. Yano and Phillips (2011) and Yano et al. (2016, hereafter YP11 for the former and YP collectively) show, under a deterministic approach, by taking the experimentally estimated parameters by Takahashi et al. (1995), that the ice breakup process can indeed lead to explosive ice multiplication under certain regimes. YP also show that a critical graupel number required for explosive multiplication is low enough that it can easily be achieved within natural clouds.

However, the ice breakup process is far from deterministic: the breakup rate constantly changes from one event of fragmentation to the next. The present paper examines a modification to the analysis by YP when stochasticity in the ice breakup process is taken into account. Substantial fluctuations in the rate of ice breakup are identified in earlier laboratory experiments. There is the dependency of breakup rate on temperature, on the collision kinetic energy (or relative velocity) on the morphology of ice as seen by Vardiman (1978). Even under identical collision conditions, the number of fragments varies by a factor of 2 as seen in Fig. 4 in Takahashi et al. (1995). These all vary stochastically.

It is widely known that stochasticity often changes the behavior of the whole system qualitatively (Horsthemke and Lefever 1984). For this reason, the stochastic representation of physics is an active part of the current atmospheric modeling, as reviewed by Plant et al. (2015) and Berner et al. (2016). In this study, we seek this possibility in the ice multiplication process. The stochasticity associated with the ice breakup presents a rather interesting noise process called “multiplicative.” To contrast this “multiplicative noise” against a standard “additive noise,” a fluctuation in the rate of primary ice generation is also considered as an example of the latter.

Evolution of a system under stochasticity is best described by the probability density distribution. We invoke the Fokker–Planck equation (FPE) in order to evaluate the latter. General introductions to this approach are given by Gardiner (1983), Horsthemke and Lefever (1984), and Reskin (1984). A concise and heuristic derivation is provided in section 3 of Egger (1981). It is important to emphasize that FPE is an exact equation that describes evolution of the probability density of a given system with time. Thus, its use is extremely powerful.

The Fokker–Planck equation is a generalization of the Liouville equation, as introduced in statistical mechanics. The Liouville equation describes a probabilistic evolution of a deterministic system without stochastic processes, whereas the Fokker–Planck equation is obtained when a stochastic forcing is added. A probabilistic description of a climate dynamics system by the Liouville equation was introduced by Epstein (1969). Types of the Fokker–Planck equation were introduced by adding a stochasticity into similar climate dynamics systems by Hasselmann (1976) and Egger (1981). In a context of microphysics, Manton (1979) introduced the Liouville equation for a probabilistic description of cloud droplet growth. Modifications of the problem by introduction of stochastic fluctuations of water supersaturation are considered by McGraw and Liu (2006) and Jeffery et al. (2007) with the Fokker–Planck equation. An original contribution of the present work is to consider multiplicative noise under this framework for a cloud microphysical system.

The next section reviews a theoretical model introduced by YP11, to which stochasticity is added in the present study. The model can be reduced to a system with only a single variable under a quasi-equilibrium approximation. This approximated system is considered first in section 3. The full system is briefly discussed in section 4. The paper is concluded in section 5. A very concise summary of stochastic modeling and the Fokker–Planck equation is presented in appendix.

## 2. Deterministic model formulation

We adopt the same model formulation as that of YP in the present study. Only a summary of the model is presented here, as YP11 is referred to for full details.

Here, in principle, the model can describe any types of clouds, but an important assumption is that a whole (or a part of) cloud must be homogeneous enough so that the single-point (i.e., zero dimensional) description can be justified. An area to be considered is also rather arbitrary, but small enough so that a spatial homogeneity over a given area can be justified. Riming as well as aggregation for graupel growth is treated only implicitly by assuming constant growth times,

By following YP, in the formulation, the ice multiplication process due to the collisions only between the small and the large graupel is considered. Its generalization for including the collisions between the ice crystals (or snow particles) and graupel is relatively straightforward, and the authors are currently investigating this general case under full numerical modeling. These results will be published elsewhere.

YP show that the behavior of the system is well summarized in terms of this single nondimensional parameter

## 3. Stochastic modeling: One-variable model

### a. Formulation

^{−1/2}mathematically stemming from the fact that Dirac’s delta function

^{−1}. From a physical point of view, this happens because a white-noise time series is a mathematical idealization of a series of uncorrelated random events happening with a fixed interval, say,

Note that in many physical problems, an assumption of the white-noise approximation is often a severe one, but it is least likely the case in this problem, because the ice multiplication rate from one collision event to next is expected to be independent. Thus, the white-noise assumption can easily be justified. Note that the Fokker–Planck equation is derived assuming the Gaussian white noise. If the noise assumption is relaxed, a more involved formulation must be introduced (cf. section 4.1 of Reskin 1984). Here, we also emphasize under the Gaussian white-noise approximation, the Fokker–Planck equation describes the evolution of the probability density of a given system in exact manner (Gardiner 1983; Horsthemke and Lefever 1984; Reskin 1984); thus, there is no need to verify the following computations based on the Fokker–Planck equation, say, by a more direct Monte Carlo computation.

### b. Numerics

Equation (3.2) is integrated in time by a finite-volume approach (cf. LeVeque 2002). A range of number density for evaluating the probability density *p* is divided into 100 cells. Fractional time stepping is used for integrating the advection term [the first term on the right-hand side of Eq. (3.2)] and the diffusion term [the second term on the right-hand side of Eq. (3.2)], separately. The advection term is evaluated by an upstream scheme with a superbee slope limiter [cf. section 6.9 of LeVeque (2002)] and a typical time step of 1 s. The method of slope limiters maintains the form of the probability distribution against the known diffusive tendency of a simple upwind scheme. It does this by adding a correction term to the latter.

The diffusion term is evaluated with a double-stage Runge–Kutta method and a typical time step of 10^{−3} s. Such a short time step is required in order to ensure an accurate calculation of the diffusion process [cf. chapter 17 of LeVeque (2002)]. The computation here is particularly stringent. As seen below in Fig. 2a, the probability density sometimes sharpens with time. To well reproduce such a tendency, the numerical diffusion in the computations, especially in association with the stochastic (physical) diffusion, must be minimized as much as possible.

### c. Examples

Examples of evolution of the probability density

The first case in Fig. 2a is without stochasticity (i.e.,

The second case in Fig. 2b shows when an additive noise is added with

Generally when stochasticity is added to any system, the stochasticity works as a diffusion on evolution of the probability density of possible states of the system, as shown by the second term on the right-hand side of the Fokker–Planck equation [Eq. (3.2)]. The simplest example is Brownian motion: as a particle is continuously randomly kicked by atoms, as a white-noise process, the probability distribution for the particle position gradually diffuses with time in an analogous manner as for heat diffusion. Similarly, a stochasticity in the primary ice generation rate diffuses the probability distribution as its peak moving toward the stable equilibrium. Nevertheless, as the system approaches closer the equilibrium, the steepening tendency due to the convergence of the deterministic tendency counteracts the diffusive tendency (the long-dashed curve in Fig. 3b). The final stage is also associated with a buildup of the skewness (the long-dashed curve in Fig. 3c).

When the stochasticity is added to the ice breakup with

### d. Interpretations

An interpretation of the stochastic breakup of ice is presented in Fig. 4. Here, the deterministic tendency

When stochasticity is added to the ice breakup (stochastic case), as a result, the temporal tendency

In the stochastic case, at some lucky moments, the graupel number may continue to increase with time. If, by chance, this increase continues long enough, the tendency is guaranteed to be positive within a given range of fluctuation so the graupel number explodes at a certain point. On the other hand, if bad luck continues, the number density decreases faster and the system also reaches the stable equilibrium state faster than in the deterministic case. These two opposite tendencies due to stochasticity lead to a gradual spread of the probability distribution, as if by a diffusion process, as already demonstrated for the additive-noise case.

However, a new feature here is an asymmetry in stochastic diffusion: the stochastic “kick” of graupel number perturbation is larger in absolute magnitude when the perturbation is positive than when it is negative as a result of the multiplicative tendency of the breakup process being greater at larger graupel numbers (simply because the stochastic tendency is proportional to the graupel number

### e. Probability for explosive multiplication

*t*when initially below it as a result of the stochastic diffusion process:Once the critical value is crossed, an explosive ice multiplication starts even without stochasticity. On the other hand, keep in mind that crossing such a threshold is possible only when stochasticity is present. There is absolutely no chance for the system crossing the critical value without stochasticity.

This probability is plotted in Fig. 5 for the cases with stochasticities due to both ice breakup and the primary ice generation rate, as presented in Fig. 2 (section 3c). Here, the case with no stochasticity (short-dashed curve) is included merely to suggest a numerical performance of the model. Theoretically, in this case, the chance for the system to crossing the critical value is zero. The obtained numerical value indicates a remaining numerical diffusion in computation: though extremely small, the value is still finite. With an additive-noise homogeneous diffusivity due to a stochasticity of primary ice generation rate (long-dashed curve in Fig. 5: the additive stochastic noise case), the chance for crossing the critical point (with the ice number initially is below it) is enhanced by a factor of 10^{10} numerically from the deterministic case (short-dashed curve), though the absolute probability still remains merely on the order of 10^{−5}.

When multiplicative noise due to the ice breakup is considered, a tail of the probability distribution exceeding the critical point

Distribution of the escape probability in Eq. (3.3), defined as an equilibrium after a long integral (10^{3}-min maximum), in the phase space of the multiplication efficiency parameter

A substantial escape rate (say, above 0.1) is found only along the periphery to the critical values (

Analysis of the system, utilizing the quasi-equilibrium approximation in this section, demonstrates that stochasticity can lead the system toward the explosive ice multiplication even when the initial condition is below a critical point, in the deterministic sense. This tendency is much rectified by multiplicative noise due to fluctuations in the ice breakup rates. These fluctuations arise by chance, with random changes in the numbers of ice fragments per collision.

## 4. Stochastic modeling: Full model

### a. Formulation

A major twist under the full formulation is that the multiplicative-noise process for the ice multiplication now depends on the number densities,

### b. Preliminary results

The computation of the Fokker–Planck equation for the full system turns out to be numerically rather expensive because there is discretization in three dimensions (^{4}. For this reason, only a few exploratory time integrations are performed.

When the system is initiated with a nonzero ice crystal number yet without any graupel, as in Fig. 7a, the system rather rapidly adjusts toward a quasi-equilibrium state, say, in 30 min, as seen in Figs. 7b–d. As long as stochasticity remains relatively weak (i.e.,

Fluctuations

The multiplicative-noise process has a further twist, because its amplitude is now proportional both to the small and the large graupel numbers,

## 5. Summary

A role of stochastic fluctuations in ice–ice collision breakup is considered. Because of the multiplicative-noise nature of this process, the stochasticity provides a further boost of the system toward the explosive ice multiplication, even when the system is initialized from a deterministically stable regime (the lower left in the phase space of Fig. 1).

The boosting tendency is quantified by calculating the time evolution of the probability distribution for ice number densities by explicitly solving the Fokker–Planck equation, which describes the time evolution of the probability density in an exact manner (cf. Fig. 7). A critical role of multiplicative nature of noise is demonstrated by comparing the results with the case of standard additive noise (cf. Fig. 2). The latter arises under the present system when the primary ice generation rate stochastically fluctuates with time.

Thus, the chance that the system enters an explosive ice-multiplication regime from a deterministically stable regime has been the key question in the present study. The result is summarized by Fig. 6 with a relative modest multiplicative-noise level. Note that a limit of high noise level (i.e., standard deviation of noise) is unphysical for positive-definite tendency such as ice-multiplication rate: a standard deviation much larger than a mean value would lead to a frequent negative multiplication rate.

Wherever the system starts within a deterministically stable regime, there is always a finite chance that the system eventually escapes into an explosive regime, thanks to the multiplicative-noise nature of the ice breakup multiplication. The entry chance to explosive ice-multiplication regime due to multiplicative noise increases with the nondimensional parameter

As a main conclusion, inclusion of stochasticity to ice breakup clearly enhances a chance of explosive breakup as summarized by Fig. 6 in the phase space of the nondimensional ice-multiplication efficiency

Since there is much variability of both variables spatially and temporally within different parts of a cloud, even if the entire cloud may initially below the threshold for explosive ice multiplication, the stochastic effect can tip part of the cloud into this regime. We also expect that in this manner, the initial cloud inhomogeneity would be further enhanced with time as a result of the multiplicative-noise process. Graupel generated under explosive ice multiplication would furthermore mix and sediment into the rest of the cloud, further promoting explosive ice-multiplication tendency throughout the rest of the cloud, but in a highly intermittent manner.

These predictions must be verified by fully three–dimensional model simulations. For this reason, currently, the authors are working on a more elaborate ice breakup formulation that takes into account more physical factors contributing to the ice–splinter generation rate, including the collision kinetic energy. The preliminary computations therein already confirm the explosive ice-multiplication tendency predicted by the zero-dimensional model (Yano and Phillips 2011; Yano et al. 2016) adopted herein. The full investigations of this system are currently under way.

Further addition of stochasticity to the above general formulation is, in principle, relatively straightforward. Here, it is important to note that the stochasticity considered in the present paper is characterized solely by two factors: the mean and the standard deviation of a noise. Thus, once these values are specified, the probabilistic evolution of the system is also predetermined regardless of the further details of the ice breakup processes.

The present paper may, to some extent, be considered a pedagogic demonstration of importance of multiplicative-noise processes in cloud microphysics. The ice multiplication process considered herein is just one of many examples expected to be found in cloud microphysics that are controlled by multiplicative noise associated with stochastic fluctuations. In this manner, the present study illustrates a strong need to explicitly consider stochasticity in strongly nonlinear processes, such as ice fragmentation, aggregation, and, more generally, coalescence of hydrometeors, when modeling the atmosphere.

## Acknowledgments

The authors acknowledge the COST Action ES0905, which made the collaboration possible. VTJP was supported by an award from the Swedish Research Council (“Vetenskapsradet”), (Award 2015-05104) and also by a subaward from the direct grant to Hebrew University of Jerusalem (Award DE-SC0006788) funded by U.S. Department of Energy. The present paper is relevant to the topics of both awards, which relate to the properties of glaciated clouds.

## APPENDIX

### Stochastic Modeling and Fokker–Planck Equation

Basic mathematical formulas for stochastic modeling are collected in this appendix with the details and derivations referring to, for example, Gardiner (1983), Horsthemke and Lefever (1984), and Reskin (1984).

#### a. SDE: Stochastic differential equations: A general statement

#### b. FPE: Fokker–Planck equation

*ν*is set to

#### c. Case with one-variable system

*x*. Then, SDE is given bywhere

*ν*is defined as before.

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