• Berner, J., and et al. , 2016: Stochastic parameterization: Towards a new view of weather and climate models. Bull. Amer. Meteor. Soc., doi:10.1175/BAMS-D-15-00268.1, in press.

    • Search Google Scholar
    • Export Citation
  • Bodenschatz, E., , S. P. Malinowski, , R. A. Shaw, , and F. Stratmann, 2010: Can we understand clouds without turbulence? Science, 327, 970971, doi:10.1126/science.1185138.

    • Search Google Scholar
    • Export Citation
  • Brenguier, J.-L., 1993: Observations of cloud microstructure at the centimeter scale. J. Atmos. Sci., 32, 783793, doi:10.1175/1520-0450(1993)032<0783:OOCMAT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Davis, A. B., , A. Marshak, , H. Gerber, , and W. J. Wiscombe, 1999: Horizontal structure of marine boundary layer clouds from centimeter to kilometer scales. J. Geophys. Res., 104, 61236144, doi:10.1029/1998JD200078.

    • Search Google Scholar
    • Export Citation
  • DeMott, P. J., , D. J. Cziczo, , A. J. Prenni, , D. M. Murphy, , S. M. Kreidenweis, , D. S. Thomson, , R. Borys, , and D. C. Rogers, 2003: Measurements of the concentration and composition of nuclei for cirrus formation. Proc. Natl. Acad. Sci. USA, 100, 14 65514 660, doi:10.1073/pnas.2532677100.

    • Search Google Scholar
    • Export Citation
  • Egger, J., 1981: Stochastically driven large-scale circulations with multiple equilibria. J. Atmos. Sci., 38, 26062618, doi:10.1175/1520-0469(1981)038<2606:SDLSCW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Epstein, E. S., 1969: Stochastic dynamic prediction. Tellus, 21, 739759, doi:10.1111/j.2153-3490.1969.tb00483.x.

  • Gardiner, C. W., 1983: Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer-Verlag, 442 pp.

  • Gerber, H., , J. B. Jensen, , A. B. Davis, , A. Marshak, , and W. J. Wiscombe, 2001: Spectral density of cloud liquid water content at high frequencies. J. Atmos. Sci., 58, 497503, doi:10.1175/1520-0469(2001)058<0497:SDOCLW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Guckenheimer, J., , and P. Holmes, 1983: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 459 pp.

  • Haman, K., , and S. P. Malinowski, 1996: Temperature measurements in clouds on a centimetre scale: Preliminary results. Atmos. Res., 41, 161175, doi:10.1016/0169-8095(96)00007-5.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., 1976: Stochastic climate models. Tellus, 28, 473485, doi:10.1111/j.2153-3490.1976.tb00696.x.

  • Hobbs, P. V., 1969: Ice multiplication in clouds. J. Atmos. Sci., 26, 315318, doi:10.1175/1520-0469(1969)026<0315:IMIC>2.0.CO;2.

  • Horsthemke, W., , and R. Lefever, 1984: Noise-Induced Transition: Theory and Applications in Physics, Chemistry, and Biology. Springer-Verlag, 318 pp.

  • Jeffery, C. A., , J. M. Reisner, , and M. Andrejczuk, 2007: Another look at stochastic condensation for subgrid cloud modeling: Adiabatic evolution and effects. J. Atmos. Sci., 64, 39493969, doi:10.1175/2006JAS2147.1.

    • Search Google Scholar
    • Export Citation
  • LeVeque, R. J., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 578 pp.

  • Malinowski, S. P., , M. Y. Leclerc, , and D. G. Baumgardner, 1994: Fractal analyses of high–resolution cloud droplet measurements. J. Atmos. Sci., 51, 397413, doi:10.1175/1520-0469(1994)051<0397:FAOHRC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Manton, M. J., 1979: On the broadening of a droplet distribution by turbulence near cloud base. Quart. J. Roy. Meteor. Soc., 105, 899914, doi:10.1002/qj.49710544613.

    • Search Google Scholar
    • Export Citation
  • McGraw, R., , and Y. Liu, 2006: Brownian drift-diffusion model for evolution of droplet size distribution in turbulent clouds. J. Geophys. Res., 33, L03802, doi:10.1029/2005GL023545.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , A. M. Blyth, , P. R. Brown, , T. W. Choularton, , and J. Latham, 2001: The glaciation of a cumulus cloud over New Mexico. Quart. J. Roy. Meteor. Soc., 127, 15131534, doi:10.1002/qj.49712757503.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , T. W. Choularton, , A. M. Blyth, , and J. Latham, 2002: The influence of aerosol concentrations on the glaciation and precipitation production of a cumulus cloud. Quart. J. Roy. Meteor. Soc., 128, 951971, doi:10.1256/0035900021643601.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , P. J. DeMott, , and C. Andronache, 2008: An empirical parameterization of heterogeneous ice nucleation for multiple chemical species of aerosol. J. Atmos. Sci., 65, 27572783, doi:10.1175/2007JAS2546.1.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , P. J. DeMott, , C. Andronache, , K. A. Pratt, , K. A. Prather, , R. Subramanian, , and C. Twohy, 2013: Improvements to an empirical parameterization of heterogeneous ice nucleation and its comparison with observations. J. Atmos. Sci., 70, 378409, doi:10.1175/JAS-D-12-080.1.

    • Search Google Scholar
    • Export Citation
  • Plant, R. S., , L. Bengtsson, , and M. A. Whitall, 2015: Stochastic aspects of convective parameterization. Parameterization of Atmospheric Convection: Current Issues and New Theories, R. S. Plant and J. I. Yano, Eds., Series on the Science of Climate Change, Vol. 2, Imperial College Press, 135–172.

  • Prenni, A. J., , P. J. DeMott, , D. C. Rogers, , S. M. Kreidenweis, , G. M. McFarquhar, , G. Zhang, , and M. R. Poellot, 2009: Ice nuclei characteristics from M-PACE and their relation to ice formulation in clouds. Tellus, 61B, 436448, doi:10.1111/j.1600-0889.2009.00415.x.

    • Search Google Scholar
    • Export Citation
  • Reskin, H., 1984: The Fokker-Planck Equation. Springer-Verlag, 454 pp.

  • Rogers, R. R., , and M. K. Yau, 1989: Short Course in Cloud Physics. 3rd ed. Pergamon Press, 290 pp.

  • Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech., 35, 183227, doi:10.1146/annurev.fluid.35.101101.161125.

    • Search Google Scholar
    • Export Citation
  • Siebert, H., , K. Lehmann, , and M. Wendisch, 2006: Observations of small-scale turbulence and energy dissipation rates in the cloudy boundary layer. J. Atmos. Sci., 63, 14511466, doi:10.1175/JAS3687.1.

    • Search Google Scholar
    • Export Citation
  • Takahashi, T., , Y. Nagano, , and Y. Kushiyama, 1995: Possible high ice particle production during graupel–graupel collisions. J. Atmos. Sci., 52, 45234527, doi:10.1175/1520-0469(1995)052<4523:PHIPPD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vaillancourt, P. A., , and M. K. Yau, 2000: Review of particle–turbulence interactions and consequences for cloud physics. Bull. Amer. Meteor. Soc., 81, 285298, doi:10.1175/1520-0477(2000)081<0285:ROPIAC>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vardiman, L., 1978: The generation of secondary ice particles in clouds by crystal–crystal collisions. J. Atmos. Sci., 35, 21682180, doi:10.1175/1520-0469(1978)035<2168:TGOSIP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., , and V. T. J. Phillips, 2011: Ice–ice collisions: An ice multiplication process in atmospheric clouds. J. Atmos. Sci., 68, 322333, doi:10.1175/2010JAS3607.1.

    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., , V. T. J. Phillips, , and V. Kanawade, 2016: Explosive ice multiplication by mechanical break-up in ice–ice collisions: A dynamical system-based study. Quart. J. Roy. Meteor. Soc., 142, 867879, doi:10.1002/qj.2687.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Phase diagram summarizing the behavior of the ice–ice collision system without stochasticity. The horizontal axis is the nondimensional primary ice source , and the vertical axis is the nondimensional graupel number (reproduced from Fig. 1 of YP11).

  • View in gallery

    Evolution of probability density (in arbitrary units) with time as a function of the small-graupel number shown by varying curves with an interval of 20 min. The initial condition is a Gaussian distribution centered at as given by the solid curve. Note that the horizontal axis is scaled by so that the initial peak is found at 0.7 in the plot. The cases shown are as follows. (a) Without stochasticity (i.e., , ). The peak of the probability density moves toward the left with the tendency of the system approaching the stable equilibrium state. (b) As in (a), but with an additive noise, . A homogeneous diffusive tendency of the additive noise is noted. (c) As in (a), but with a multiplicative noise, . The probability distribution diffuses preferably toward the positive direction.

  • View in gallery

    Time series of the first three moments [(a) average, (b) variance, and (c) skewness] obtained from the three runs shown in Fig. 2: without stochasticity (i.e., , ) (solid curve); with an additive noise, (long-dashed curve); and with an multiplicative noise, (short-dashed curve).

  • View in gallery

    The tendency for the number density of small graupel under the one-variable description. The solid curve is with the standard parameters. The short-dashed and the long-dashed curves are when the ice breakup rate is, respectively, increased and decreased by 100% of the standard value.

  • View in gallery

    The cumulative probability [Eq. (3.3)] that the graupel number density has crossed the critical point as a function of time for the three cases: with multiplicative noise (, ; solid curve), with additive noise (, ; long-dashed curve), and no stochasticity (, ; short-dashed curve). Note that the last case shows an “escape” rate purely arising from numerical diffusions in computation: in theory, the escape rate must be zero in this case.

  • View in gallery

    The probability [Eq. (3.3)] obtained after a long integral that the graupel number density crosses the critical point is plotted by color tones on the phase plane of (horizontal axis) and the initial condition (vertical axis) with (a) additive noise and (b) multiplicative noise . Note that with the absence of stochasticity, over the domain and of the investigation, the chance that the system crosses the critical point is zero.

  • View in gallery

    Evolution of the probability density , as computed by the Fokker–Planck equation in three-dimensional phase space at (a) t = 0 (the initial condition), (b) t = 10, (c) t = 20, (d) t = 60, (e) t = 120, and (f) t = 180 min. Two axes are given by nondimensionalized number densities for large graupel and ice particles, and , respectively. The probability density here is given in arbitrary units.

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Explosive Ice Multiplication Induced by Multiplicative-Noise Fluctuation of Mechanical Breakup in Ice–Ice Collisions

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  • 1 CNRM, Météo-France, and CNRS, UMR 3589, Toulouse, France
  • | 2 Department of Physical Geography and Ecosystem Science, University of Lund, Lund, Sweden
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Abstract

The number of ice fragments generated by breakup of large graupel in collisions with small graupel fluctuates randomly owing to fluctuations in relative sizes and densities of colliding graupel particles and the stochastic nature of fracture propagation. This paper investigates the impact of the stochasticity of breakup on ice multiplication.

When both the rate of generation of primary ice and the initial number concentration of ice crystals are low, the system most likely loses all the initial ice and graupel owing to a lack of sustaining sources. Even randomness does not change this mean evolution of the system in its phase space. However, a fluctuation of ice breakup number gives a small but finite chance that substantial ice crystal fragments are generated by breakup of large graupel. That, in turn, generates more large graupel. This multiplicative process due to fluctuations potentially leads to a small but finite chance of explosive growth of ice number. A rigorous stochastic analysis demonstrates this point quantitatively.

The randomness considered here belongs to a particular category called “multiplicative” noise, because the noise amplitude is proportional to a given physical state. To contrast the multiplicative-noise nature of ice breakup with a standard “additive” noise process, fluctuation of the primary ice generation rate is also considered as an example of the latter. These processes are examined by taking the Fokker–Planck equation that explicitly describes the evolution of the probability distribution with time. As an important conclusion, stability in the phase space of the cloud microphysical system of breakup in ice–ice collisions is substantially altered by the multiplicative noise.

Denotes Open Access content.

Corresponding author address: Jun-Ichi Yano, CNRM, Météo-France, 42 av Coriolis, 31057 Toulouse CEDEX, France. E-mail: jiy.gfder@gmail.com

Abstract

The number of ice fragments generated by breakup of large graupel in collisions with small graupel fluctuates randomly owing to fluctuations in relative sizes and densities of colliding graupel particles and the stochastic nature of fracture propagation. This paper investigates the impact of the stochasticity of breakup on ice multiplication.

When both the rate of generation of primary ice and the initial number concentration of ice crystals are low, the system most likely loses all the initial ice and graupel owing to a lack of sustaining sources. Even randomness does not change this mean evolution of the system in its phase space. However, a fluctuation of ice breakup number gives a small but finite chance that substantial ice crystal fragments are generated by breakup of large graupel. That, in turn, generates more large graupel. This multiplicative process due to fluctuations potentially leads to a small but finite chance of explosive growth of ice number. A rigorous stochastic analysis demonstrates this point quantitatively.

The randomness considered here belongs to a particular category called “multiplicative” noise, because the noise amplitude is proportional to a given physical state. To contrast the multiplicative-noise nature of ice breakup with a standard “additive” noise process, fluctuation of the primary ice generation rate is also considered as an example of the latter. These processes are examined by taking the Fokker–Planck equation that explicitly describes the evolution of the probability distribution with time. As an important conclusion, stability in the phase space of the cloud microphysical system of breakup in ice–ice collisions is substantially altered by the multiplicative noise.

Denotes Open Access content.

Corresponding author address: Jun-Ichi Yano, CNRM, Météo-France, 42 av Coriolis, 31057 Toulouse CEDEX, France. E-mail: jiy.gfder@gmail.com

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.

The model consists of the three prognostic variables: the number densities, , , and , for small crystals, small graupel (<1 mm in radius), and large graupel (>1–2 mm in radius). Their evolution is defined by
e2.1a
e2.1b
e2.1c
assuming a constant primary ice generation rate , the ice multiplication rate by ice–ice collisions [cf. Eq. (A.1) of YP11]. Ice crystals are assumed to be converted into small graupel with time , and small graupel are converted into large graupel with time . Large graupel are lost with time by gravitational fallout from the cloud. In the present study, we assume standard parameters, , , , and . Here, is not explicitly specified but indirectly evaluated from a specified nondimensional parameter defined by Eq. (2.5) below. In default cases, is assumed. These parameters are estimated from the previous experimentally and modeling studies (Takahashi et al. 1995; Phillips et al. 2001, 2002, 2008) as discussed in detail in YP11.

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, and , from ice crystals to small graupel, and then to large graupel.

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.

By taking an asymptotic approximation, (i.e., quasi-equilibrium approximation), the above set of equations can be reduced into a single equation for :
e2.2
with and [cf. Eq. (A.7) of YP11].
The one-variable model for under the quasi-equilibrium approximation further prompts us to analyze the whole behavior of the system in terms of the number density for small graupel. For this purpose, it is found useful to normalize it as
e2.3
By solving Eq. (2.2) for , it is found that its steady solution is given by
e2.4
Here,
e2.5
is a nondimensional measure of the efficiency for ice multiplication. Recall that under quasi equilibrium, and .

YP show that the behavior of the system is well summarized in terms of this single nondimensional parameter . Figure 1, reproduced from YP11, summarizes the stability of the system in the phase space of (, ). The dashed and solid curves on Fig. 1 denote a stable and unstable equilibrium, respectively. A further added solid straight line at marks a boundary between the explosive ice multiplication and a damping regime: a main conclusion from YP is that when the initial condition is above or to the right of the solid curve and line, in the phase space of (, ), then explosive ice multiplication occurs. On the other hand, when the system remains below or to the left of the solid curve and line, the system is stable and converges toward the lower-equilibrium solution defined by the long-dashed curve when no stochasticity is found in the system. The main goal now is to investigate the modifications of the behavior of the system in the latter stable regime when stochasticity is added.

Fig. 1.
Fig. 1.

Phase diagram summarizing the behavior of the ice–ice collision system without stochasticity. The horizontal axis is the nondimensional primary ice source , and the vertical axis is the nondimensional graupel number (reproduced from Fig. 1 of YP11).

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

3. Stochastic modeling: One-variable model

a. Formulation

Stochasticity is introduced for taking into account the fluctuations both in the rate of primary ice generation and in the ice breakup rate . Thus, the parameters and are respectively replaced by stochastic variables, and , with relative fluctuations defined by and . Here, and are normalized white-noise processes satisfying the conditions
eq1
eq2
eq3
eq4
Here, the angle brackets represent an ensemble average. Gaussian distribution is assumed for each white noise under the normalization above. Note that these stochastic processes, and , have a dimension of time−1/2 mathematically stemming from the fact that Dirac’s delta function has a dimension of time−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, , and the nondimensional standard deviation of the events is given by and , respectively, for and . Only the discrete white-noise time series has a well-defined variance. A continuous description is obtained in the limit of . For this reason, we multiply a root square of with the fluctuation intensity in order to maintain a dimensional consistency. We set in the following. Here, note that no assumption concerning the correlations between the two noise processes, and , is required in solving the problem in the following.
The present section considers the one-variable model (2.2) obtained under the quasi-equilibrium approximation but with stochasticity. For this purpose, Eq. (2.2) is rewritten as
e3.1
The last two terms describe multiplicative- and additive-noise processes, respectively. Here, the multiplicative-noise process is a consequence of fluctuations in number of generated broken ice particles from each ice–ice collision event. Additive noise simply stems from fluctuation of the primary ice generation rate with time. Note that the multiplicative noise has an amplifying tendency as the number density increases.
The corresponding equation for the probability density is given by
e3.2
by referring to Eq. (A.4) of the appendix.

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.

Fig. 2.
Fig. 2.

Evolution of probability density (in arbitrary units) with time as a function of the small-graupel number shown by varying curves with an interval of 20 min. The initial condition is a Gaussian distribution centered at as given by the solid curve. Note that the horizontal axis is scaled by so that the initial peak is found at 0.7 in the plot. The cases shown are as follows. (a) Without stochasticity (i.e., , ). The peak of the probability density moves toward the left with the tendency of the system approaching the stable equilibrium state. (b) As in (a), but with an additive noise, . A homogeneous diffusive tendency of the additive noise is noted. (c) As in (a), but with a multiplicative noise, . The probability distribution diffuses preferably toward the positive direction.

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

c. Examples

Examples of evolution of the probability density over time, by solving the Fokker–Planck equation [Eq. (3.2)], are shown in Fig. 2 by setting . In all cases, the initial condition is chosen as a relatively narrow Gaussian distribution centered at with the standard deviation of in nondimensional units. The initial peak probability is placed just below the critical value ; thus, the system is expected to evolve toward the stable equilibrium state, , without stochasticity. Figure 3 further shows the time series of the first three moments (Fig. 3a: average, Fig. 3b: variance, and Fig. 3c: skewness) for these three examples shown in Fig. 2.

Fig. 3.
Fig. 3.

Time series of the first three moments [(a) average, (b) variance, and (c) skewness] obtained from the three runs shown in Fig. 2: without stochasticity (i.e., , ) (solid curve); with an additive noise, (long-dashed curve); and with an multiplicative noise, (short-dashed curve).

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

The first case in Fig. 2a is without stochasticity (i.e., , ), presented as a reference. In the absence of stochasticity, Eq. (3.2) reduces to the Liouville equation, which dictates the conservation of the probability density under a Lagrangian framework. Thus, evolution of the probability density is purely advective, and the advection rate is dictated by the local generation rate (cf. Fig. 4). We see by tracing the peak of the probability density with time that the system gradually moves toward the stable equilibrium state, , in a couple of hours. During an initial transition phase, there is a weak spread of probability due to a weakly divergent tendency of distribution (the solid curve in Fig. 3b). However, the probability density finally steepens as the system converges toward the equilibrium.

The second case in Fig. 2b shows when an additive noise is added with . Here and in the following, the magnitude of the noises are taken to be large enough so that these effects are seen in visible manner, but they are retained at a realistic level. When the magnitude (standard deviation) of noise is taken to be too large, a tail part of the probability distribution begins to cover negative events too often, which are not expected to happen physically. The same principle also applies to the multiplicative noise considered below.

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 (Fig. 2c), instead, diffusion of the probability distribution represents an asymmetry: the ice breakup is more effective when graupel are more numerous; thus, the stochastic diffusion is also more effective when the graupel particles are more numerous (the short-dashed curve in Fig. 3b). As a result, we see a tendency for the probability density to diffuse toward the higher values preferentially. This tendency makes the probability distribution more skewed with time in the positive direction. Such a skewed tendency is shown in Fig. 2c and more explicitly in Fig. 3c by a short-dashed curve. Finally, as expected, the mean evolution of these three cases (Fig. 3a) is overall similar, although the drift with the multiplicative-noise case is substantially faster than the other two cases.

d. Interpretations

An interpretation of the stochastic breakup of ice is presented in Fig. 4. Here, the deterministic tendency given by the first two terms on the right-hand side of Eq. (3.1) is shown by the solid curve (deterministic case). The curve shows that the system moves toward the stable equilibrium solution whenever the initial condition is below the critical value , because the tendency is always negative over the range .

Fig. 4.
Fig. 4.

The tendency for the number density of small graupel under the one-variable description. The solid curve is with the standard parameters. The short-dashed and the long-dashed curves are when the ice breakup rate is, respectively, increased and decreased by 100% of the standard value.

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

When stochasticity is added to the ice breakup (stochastic case), as a result, the temporal tendency fluctuates with time. The range of fluctuations may be indicated by the two extra tendencies obtained by adding 100% more ice breakup (short-dashed line) and 100% less ice breakup (long-dashed line) compared to the standard value (i.e., twice ice breakup and no ice breakup, respectively). The two curves suggest that stochastic fluctuations can shift the critical value extensively. As a result, the system initiated at a value right below the mean critical value is no longer guaranteed to simply decrease with time, but the evolution depends sensitively on how the ice breakup rate fluctuates with time.

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 ). In other words, in the multiplicative-noise process, good luck is more powerful than bad luck, and thus the system preferentially diffuses toward the positive value.

e. Probability for explosive multiplication

Of particular interest is the chance that the graupel number density crosses the critical value for explosive breakup (i.e., the upper solid curve in Fig. 1 at ) before time t when initially below it as a result of the stochastic diffusion process:
e3.3
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 1010 numerically from the deterministic case (short-dashed curve), though the absolute probability still remains merely on the order of 10−5.

Fig. 5.
Fig. 5.

The cumulative probability [Eq. (3.3)] that the graupel number density has crossed the critical point as a function of time for the three cases: with multiplicative noise (, ; solid curve), with additive noise (, ; long-dashed curve), and no stochasticity (, ; short-dashed curve). Note that the last case shows an “escape” rate purely arising from numerical diffusions in computation: in theory, the escape rate must be zero in this case.

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

When multiplicative noise due to the ice breakup is considered, a tail of the probability distribution exceeding the critical point is even visible in direct plots (Fig. 2c). An explicit plot for the stochastic case (solid curve in Fig. 5) shows that the probability comes close to 10% for crossing the critical point at the final stage of the simulation. Though this probability is rather modest, considering a potential consequence of the explosive ice multiplication to the cloud system, such a possibility cannot be taken lightly. Also note that the chosen initial condition is relatively away from the critical point (only 70% of the critical value). By taking an initial condition closer to the critical value, the chance can be increased substantially as seen immediately below.

Distribution of the escape probability in Eq. (3.3), defined as an equilibrium after a long integral (103-min maximum), in the phase space of the multiplication efficiency parameter , proportional to primary ice generation rate, and the center of the initial nondimensionalized initial graupel number distribution density are shown for the two cases in Fig. 6: additive noise to the primary ice generation rate with and (Fig. 6a) and multiplicative noise to the ice multiplication process with and (Fig. 6b). Here, the standard deviation for the initial distribution is kept the same as in Fig. 2.

Fig. 6.
Fig. 6.

The probability [Eq. (3.3)] obtained after a long integral that the graupel number density crosses the critical point is plotted by color tones on the phase plane of (horizontal axis) and the initial condition (vertical axis) with (a) additive noise and (b) multiplicative noise . Note that with the absence of stochasticity, over the domain and of the investigation, the chance that the system crosses the critical point is zero.

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

A substantial escape rate (say, above 0.1) is found only along the periphery to the critical values ( or ) for the additive noise. Even a modest chance of the escape rate (say, 0.05) is limited to this area. On the other hand, when the multiplicative noise is considered, a relatively modest chance (say, above 0.05) of escape is found for much wider zones in both directions ( or ).

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

When the same stochasticity is added to the full model, Eqs. (4.1a)(4.1c), we obtain
e4.1a
e4.1b
e4.1c
Here, we have set and . Note that the stochasticity, defined in the same notations as in Eq. (3.1), is added to the ice crystal number tendency only.
By comparing this formulation with a general expression for the prognostic system (A.1) in the appendix, the Fokker–Planck equation that describes the evolution of the probability density for the present system is deduced from Eq. (A.2) as
e4.2

A major twist under the full formulation is that the multiplicative-noise process for the ice multiplication now depends on the number densities, and , for the small and the large graupel in Eq. (4.1a). As the ice number multiplies, the small graupel also begins to grow in number, then the large graupel grow in number, as described by Eqs. (4.1b) and (4.1c). These processes, in turn, further enhance the ice multiplication and the associated multiplicative-noise process.

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 (, , and ). By taking the same number of grid points (100) in two more phase-space directions ( and ), the computational cost is multiplied by a factor of 104. For this reason, only a few exploratory time integrations are performed.

These preliminary results obtained with the Fokker–Planck equation for the full system can mostly be understood by combining those for the deterministic full system [Eq. (2.1a)] and those obtained for the Fokker–Planck equation for the one variable [Eq. (3.1)] in the last section. For this reason, in the following, we present the characteristic behavior of the system in general manner accompanied by a single demonstrative case. The initial condition of this demonstration case is given by a Gaussian distribution centered around . Time evolution of this case is shown in Fig. 7 for the probability density
eq5
for and , by integrating the density over . Note that the order of the arguments is changed here so that explosive ice multiplication is seen as a growth of the probability density in the vertical direction associated with the growth of the large graupel number in a horizontal axis.
Fig. 7.
Fig. 7.

Evolution of the probability density , as computed by the Fokker–Planck equation in three-dimensional phase space at (a) t = 0 (the initial condition), (b) t = 10, (c) t = 20, (d) t = 60, (e) t = 120, and (f) t = 180 min. Two axes are given by nondimensionalized number densities for large graupel and ice particles, and , respectively. The probability density here is given in arbitrary units.

Citation: Journal of the Atmospheric Sciences 73, 12; 10.1175/JAS-D-16-0051.1

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., and ), stochastic diffusion takes effect only after the initial deterministic adjustment is over.

Fluctuations in the primary ice source confer the probability distribution with a homogeneous diffusion due to the additive nature of the noise. This obvious tendency is not taken into account in Fig. 7 by setting . Fluctuations ( in Fig. 7) in the ice breakup rate create an asymmetric diffusion of the probability distribution skewed toward positive values owing to the multiplicative nature, as was the case with the one-variable system. In this generalization into the full system, an important point to keep in mind (as already remarked above in discussing the numerical algorithm) is the fact that stochasticity is added only to an evolution of the ice crystal number density ; thus, diffusion is also experienced only for .

The multiplicative-noise process has a further twist, because its amplitude is now proportional both to the small and the large graupel numbers, and . Thus, though the diffusion happens in the direction of , the rate of diffusion is now controlled by other variables, and , rather than the diffusing variable itself. For this reason, the multiplicative-noise diffusion does not take place until the initial deterministic transition to a quasi equilibrium is completed [as discussed in detail for a deterministic case in section 4.5 of Yano et al. (2016)] so that both graupel numbers are sufficiently developed. Once it takes place, the more multiplicative-noise diffusion in is found where both and are larger. As a result, as seen in Figs. 7c–f, the tail of the probability density grows upward (i.e., toward the larger ) but only for sufficiently large . This growth of the tail measures a chance for the explosive ice multiplication to occur. Note that such an explosive tendency is associated with a compensating tendency of a core of the probability density shrinking toward the equilibrium state, as also already seen in the one-dimensional case in Fig. 2c.

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 of the system as well as with the nondimensional initial small-graupel number as it approaches the critical values ( and ) from below. The chance is already close to 10% when the initial condition is within 70% of the distance from the threshold in phase space (cf. Fig. 6b). Recall that the characteristic time scale of the system is defined by the time scale, 30 min, for small graupel generation from ice crystals. Thus, the evolution reaches equilibrium after a time scale of half an hour. Numerical results (cf. Fig. 5) show that the subsequent escape into instability happens in a relatively short time in practice (~1 min), once the system has reached equilibrium; thus, this should be considered a serious potential possibility. A full computation in Fig. 7 also supports this expectation.

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 and the initial number density for small graupel. Our earlier papers (Yano and Phillips 2011; Yano et al. 2016) have already shown that for almost all types of realistic cloud regimes, the condition, , for the explosive ice multiplication is satisfied. An additional important constraint is that a cloud must be deep enough so that large graupel can remain within the cloud long enough in order to trigger this process. The present study has shown that this explosive tendency for ice multiplication is further boosted by the associated multiplicative-noise process.

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

The stochastic differential equation (SDE) that described stochastic processes is, in its general from, given for a vector variable by
ea.1
where and are Gaussian white-noise processes (called multiplicative and additive noises, respectively), which satisfy
eq6
eq7
eq8
eq9

b. FPE: Fokker–Planck equation

The time evolution of the probability density for the stochastic process described by the SDE [Eq. (A.1)] is defined by the Fokker–Planck equation (FPE):
ea.2
Here, the parameter ν is set to for Ito calculus and for Stratonovich calculus. Note that these are two choices for performing the time integral of SDE numerically. As it turns out, the choice between Ito and Stratonovich calculus do not make difference in the present study, because the term does not exist.

c. Case with one-variable system

Both SDE and FPE can be written in a more concise manner when we consider a system described by a single variable x. Then, SDE is given by
ea.3
where and are multiplicative and additive noises characterized by
eq10
eq11
eq12
eq13
The corresponding FPE is given by
ea.4
where the parameter ν is defined as before.

REFERENCES

  • Berner, J., and et al. , 2016: Stochastic parameterization: Towards a new view of weather and climate models. Bull. Amer. Meteor. Soc., doi:10.1175/BAMS-D-15-00268.1, in press.

    • Search Google Scholar
    • Export Citation
  • Bodenschatz, E., , S. P. Malinowski, , R. A. Shaw, , and F. Stratmann, 2010: Can we understand clouds without turbulence? Science, 327, 970971, doi:10.1126/science.1185138.

    • Search Google Scholar
    • Export Citation
  • Brenguier, J.-L., 1993: Observations of cloud microstructure at the centimeter scale. J. Atmos. Sci., 32, 783793, doi:10.1175/1520-0450(1993)032<0783:OOCMAT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Davis, A. B., , A. Marshak, , H. Gerber, , and W. J. Wiscombe, 1999: Horizontal structure of marine boundary layer clouds from centimeter to kilometer scales. J. Geophys. Res., 104, 61236144, doi:10.1029/1998JD200078.

    • Search Google Scholar
    • Export Citation
  • DeMott, P. J., , D. J. Cziczo, , A. J. Prenni, , D. M. Murphy, , S. M. Kreidenweis, , D. S. Thomson, , R. Borys, , and D. C. Rogers, 2003: Measurements of the concentration and composition of nuclei for cirrus formation. Proc. Natl. Acad. Sci. USA, 100, 14 65514 660, doi:10.1073/pnas.2532677100.

    • Search Google Scholar
    • Export Citation
  • Egger, J., 1981: Stochastically driven large-scale circulations with multiple equilibria. J. Atmos. Sci., 38, 26062618, doi:10.1175/1520-0469(1981)038<2606:SDLSCW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Epstein, E. S., 1969: Stochastic dynamic prediction. Tellus, 21, 739759, doi:10.1111/j.2153-3490.1969.tb00483.x.

  • Gardiner, C. W., 1983: Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer-Verlag, 442 pp.

  • Gerber, H., , J. B. Jensen, , A. B. Davis, , A. Marshak, , and W. J. Wiscombe, 2001: Spectral density of cloud liquid water content at high frequencies. J. Atmos. Sci., 58, 497503, doi:10.1175/1520-0469(2001)058<0497:SDOCLW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Guckenheimer, J., , and P. Holmes, 1983: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 459 pp.

  • Haman, K., , and S. P. Malinowski, 1996: Temperature measurements in clouds on a centimetre scale: Preliminary results. Atmos. Res., 41, 161175, doi:10.1016/0169-8095(96)00007-5.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., 1976: Stochastic climate models. Tellus, 28, 473485, doi:10.1111/j.2153-3490.1976.tb00696.x.

  • Hobbs, P. V., 1969: Ice multiplication in clouds. J. Atmos. Sci., 26, 315318, doi:10.1175/1520-0469(1969)026<0315:IMIC>2.0.CO;2.

  • Horsthemke, W., , and R. Lefever, 1984: Noise-Induced Transition: Theory and Applications in Physics, Chemistry, and Biology. Springer-Verlag, 318 pp.

  • Jeffery, C. A., , J. M. Reisner, , and M. Andrejczuk, 2007: Another look at stochastic condensation for subgrid cloud modeling: Adiabatic evolution and effects. J. Atmos. Sci., 64, 39493969, doi:10.1175/2006JAS2147.1.

    • Search Google Scholar
    • Export Citation
  • LeVeque, R. J., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 578 pp.

  • Malinowski, S. P., , M. Y. Leclerc, , and D. G. Baumgardner, 1994: Fractal analyses of high–resolution cloud droplet measurements. J. Atmos. Sci., 51, 397413, doi:10.1175/1520-0469(1994)051<0397:FAOHRC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Manton, M. J., 1979: On the broadening of a droplet distribution by turbulence near cloud base. Quart. J. Roy. Meteor. Soc., 105, 899914, doi:10.1002/qj.49710544613.

    • Search Google Scholar
    • Export Citation
  • McGraw, R., , and Y. Liu, 2006: Brownian drift-diffusion model for evolution of droplet size distribution in turbulent clouds. J. Geophys. Res., 33, L03802, doi:10.1029/2005GL023545.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , A. M. Blyth, , P. R. Brown, , T. W. Choularton, , and J. Latham, 2001: The glaciation of a cumulus cloud over New Mexico. Quart. J. Roy. Meteor. Soc., 127, 15131534, doi:10.1002/qj.49712757503.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , T. W. Choularton, , A. M. Blyth, , and J. Latham, 2002: The influence of aerosol concentrations on the glaciation and precipitation production of a cumulus cloud. Quart. J. Roy. Meteor. Soc., 128, 951971, doi:10.1256/0035900021643601.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , P. J. DeMott, , and C. Andronache, 2008: An empirical parameterization of heterogeneous ice nucleation for multiple chemical species of aerosol. J. Atmos. Sci., 65, 27572783, doi:10.1175/2007JAS2546.1.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., , P. J. DeMott, , C. Andronache, , K. A. Pratt, , K. A. Prather, , R. Subramanian, , and C. Twohy, 2013: Improvements to an empirical parameterization of heterogeneous ice nucleation and its comparison with observations. J. Atmos. Sci., 70, 378409, doi:10.1175/JAS-D-12-080.1.

    • Search Google Scholar
    • Export Citation
  • Plant, R. S., , L. Bengtsson, , and M. A. Whitall, 2015: Stochastic aspects of convective parameterization. Parameterization of Atmospheric Convection: Current Issues and New Theories, R. S. Plant and J. I. Yano, Eds., Series on the Science of Climate Change, Vol. 2, Imperial College Press, 135–172.

  • Prenni, A. J., , P. J. DeMott, , D. C. Rogers, , S. M. Kreidenweis, , G. M. McFarquhar, , G. Zhang, , and M. R. Poellot, 2009: Ice nuclei characteristics from M-PACE and their relation to ice formulation in clouds. Tellus, 61B, 436448, doi:10.1111/j.1600-0889.2009.00415.x.

    • Search Google Scholar
    • Export Citation
  • Reskin, H., 1984: The Fokker-Planck Equation. Springer-Verlag, 454 pp.

  • Rogers, R. R., , and M. K. Yau, 1989: Short Course in Cloud Physics. 3rd ed. Pergamon Press, 290 pp.

  • Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech., 35, 183227, doi:10.1146/annurev.fluid.35.101101.161125.

    • Search Google Scholar
    • Export Citation
  • Siebert, H., , K. Lehmann, , and M. Wendisch, 2006: Observations of small-scale turbulence and energy dissipation rates in the cloudy boundary layer. J. Atmos. Sci., 63, 14511466, doi:10.1175/JAS3687.1.

    • Search Google Scholar
    • Export Citation
  • Takahashi, T., , Y. Nagano, , and Y. Kushiyama, 1995: Possible high ice particle production during graupel–graupel collisions. J. Atmos. Sci., 52, 45234527, doi:10.1175/1520-0469(1995)052<4523:PHIPPD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vaillancourt, P. A., , and M. K. Yau, 2000: Review of particle–turbulence interactions and consequences for cloud physics. Bull. Amer. Meteor. Soc., 81, 285298, doi:10.1175/1520-0477(2000)081<0285:ROPIAC>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Vardiman, L., 1978: The generation of secondary ice particles in clouds by crystal–crystal collisions. J. Atmos. Sci., 35, 21682180, doi:10.1175/1520-0469(1978)035<2168:TGOSIP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., , and V. T. J. Phillips, 2011: Ice–ice collisions: An ice multiplication process in atmospheric clouds. J. Atmos. Sci., 68, 322333, doi:10.1175/2010JAS3607.1.

    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., , V. T. J. Phillips, , and V. Kanawade, 2016: Explosive ice multiplication by mechanical break-up in ice–ice collisions: A dynamical system-based study. Quart. J. Roy. Meteor. Soc., 142, 867879, doi:10.1002/qj.2687.

    • Search Google Scholar
    • Export Citation
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