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**Geoscientific Model Development**
An interactive open-access journal of the European Geosciences Union

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- About
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- About
- Editorial board
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- Abstract
- Introduction
- Introduction of CIBU into the LIMA scheme
- Simulation of a three-dimensional deep convective case
- Simulation of a three-dimensional idealized supercell storm with varying atmospheric stability
- Summary and perspectives
- Code availability
- Appendix A: Moments of the gamma and incomplete gamma functions
- Author contributions
- Competing interests
- Acknowledgements
- References

**Model description paper**
18 Oct 2018

**Model description paper** | 18 Oct 2018

A representation of the collisional ice break-up process in the two-moment microphysics LIMA v1.0 scheme of Meso-NH

^{1}Laboratoire de l'Atmosphère et Cyclones, UMR 8105, CNRS/Météo-France/Université de La Réunion, St. Denis, La Réunion, France^{2}Laboratoire d'Aérologie, Université de Toulouse, CNRS, UPS, 14 avenue Edouard Belin, 31400 Toulouse, France

^{1}Laboratoire de l'Atmosphère et Cyclones, UMR 8105, CNRS/Météo-France/Université de La Réunion, St. Denis, La Réunion, France^{2}Laboratoire d'Aérologie, Université de Toulouse, CNRS, UPS, 14 avenue Edouard Belin, 31400 Toulouse, France

**Correspondence**: Jean-Pierre Pinty (jean-pierre.pinty@aero.obs-mip.fr)

**Correspondence**: Jean-Pierre Pinty (jean-pierre.pinty@aero.obs-mip.fr)

Abstract

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The paper describes a switchable parameterization of collisional ice break-up (CIBU), an ice multiplication process that fits in with the two-moment microphysical Liquid Ice Multiple Aerosols (LIMA) scheme. The LIMA scheme with three ice types (pristine cloud ice crystals, snow aggregates, and graupel hail) was developed in the cloud-resolving mesoscale model (Meso-NH). Here, the CIBU parameterization assumes that collisional break-up is mostly efficient for the small and fragile snow aggregate class of particles when they are hit by large, dense graupel particles. The increase of cloud ice number concentration depends on a prescribed number (or a random number) of fragments being produced per collision. This point is discussed and analytical expressions of the newly contributing CIBU terms in LIMA are given.

The scheme is run in the cloud-resolving mesoscale model (Meso-NH) to simulate a first case of a three-dimensional deep convective event with heavy production of graupel. The consequence of dramatically changing the number of fragments produced per collision is investigated by examining the rainfall rates and the changes in small ice concentrations and mass mixing ratios. Many budgets of the ice phase are shown and the sensitivity of CIBU to the initial concentration of freezing nuclei is explored.

The scheme is then tested for another deep convective case where, additionally, the convective available potential energy (CAPE) is varied. The results confirm the strong impact of CIBU with up to a 1000-fold increase in small ice concentrations, a reduction of the rainfall or precipitating area, and an invigoration of the convection with higher cloud tops.

Finally, it is concluded that the efficiency of the ice crystal fragmentation needs to be tuned carefully. The proposed parameterization of CIBU is easy to implement in any two-moment microphysics scheme. It could be used in this form to simulate deep tropical cloud systems where anomalously high concentrations of small ice crystals are suspected.

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Hoarau, T., Pinty, J.-P., and Barthe, C.: A representation of the collisional ice break-up process in the two-moment microphysics LIMA v1.0 scheme of Meso-NH, Geosci. Model Dev., 11, 4269–4289, https://doi.org/10.5194/gmd-11-4269-2018, 2018.

1 Introduction

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In a series of papers, Yano and Phillips (2011, 2016) and
Yano et al. (2016) brought the collisional ice break-up (hereafter
CIBU) process to the fore again as a possible secondary ice production
mechanism in clouds. Using an analytical model, they showed that CIBU could
lead to an explosive growth of small ice crystal concentrations. Afterwards,
Sullivan et al. (2017) tried to include CIBU in a six-hydrometeor-class parcel
model, in which hydrometeors were assumed to be monodispersed, in an attempt
to investigate the ice crystal number enhancement. However, intriguingly, and
in contrast to the Hallett–Mossop ice multiplication mechanism^{1} (hereafter H–M) (Hallett and Mossop, 1974), the
vast majority of microphysics schemes do not include the CIBU process. Yet,
the CIBU process is very likely to be active in inhomogeneous cloud regions
where ice crystals of different sizes and types are locally mixed
(Hobbs and Rangno, 1985; Rangno and Hobbs, 2001). For instance, collisions between large, dense
graupel grown by riming and plane vapour-grown dendrites or irregular
weakly rimed assemblages are the most conceivable scenario for generating
multiple ice debris as envisioned by Hobbs and Farber (1972) and by
Griggs and Choularton (1986). Therefore, a legitimate quest for a two-moment
mixed-phase microphysics scheme, where number concentrations and mixing
ratios of the ice crystals are predicted, is to find ways to include an
ice–ice break-up mechanism and to characterize its importance relative to
other ice-generating processes such as ice heterogeneous nucleation. Our aim
to introduce CIBU in a microphysics scheme was initially motivated by the
detection of unexplained high ice water content which sometimes largely
exceeded the concentration of ice-nucleating particles
(Field et al., 2017; Ladino et al., 2017; Leroy et al., 2015).

As recalled by Yano and Phillips (2011), the first laboratory experiments dedicated to the study of ice collisions were conducted in the 1970s following investigations concerning the promising H–M process. In the pioneering work of Vardiman (1978), who highlighted the mechanical fracturing of natural ice crystals, the number of fragments was dependent on the shape of the initial colliding crystal and on the momentum change following the collision. According to a concluding remark by Vardiman (1978), this secondary production of ice could lead to concentrations as high as 1000 times the natural concentrations of ice crystals in clouds that would be expected from heterogeneous nucleation on ice freezing nuclei. Another laboratory study by Takahashi et al. (1995) also revealed a huge production of ice splinters after collisions between rimed and deposition-grown graupel. However, because as many as 400 fragments could be obtained, their experimental set-up was more appropriate to very large, artificially grown crystals and to large impact velocities.

For clarity, this study does not focus on cloud conditions that lead to explosive ice multiplication due to mechanical break-up in ice–ice collisions. Nor does it attempt to reformulate this process on the basis of collisional kinetic energy with many empirical parameters, as proposed by Phillips et al. (2017), or earlier by Hobbs and Farber (1972), in terms of their breaking energy, mostly applicable to bin microphysics schemes. Here, the goal is rather to implement an empirical but realistic parameterization of CIBU in the Liquid Ice Multiple Aerosols (LIMA) microphysics scheme (Vié et al., 2016) in conjunction with other microphysical processes (heterogeneous ice nucleation, droplet freezing, H–M process, etc.) to improve the representation of small ice crystal concentrations. In this study, our representation of CIBU is the formation of cloud ice crystals as the result of collisions between big graupel particles and small aggregates after which the graupel particles lose mass to the aggregates. This parameterization of CIBU relies on the laboratory observations by Vardiman (1978) to set limits on the number of fragments per collision. However, the large uncertainties attached to this parameter encouraged us to run exploratory experiments with several fixed values and also to model the number of fragments by means of a random process.

The LIMA scheme, inserted in the host model Meso-NH (Lafore et al., 1998), forms the framework of the present study. Several sensitivity experiments are performed to evaluate the importance of the CIBU process and the impact of the tuning (i.e. the number of fragments produced per collision). The efficiency of CIBU in dramatically increasing the concentration of small ice crystals can be scaled by the ice number concentration from nucleation. The case of a three-dimensional continental deep convective storm, the well-known Stratospheric-Tropospheric Experiment: Radiation, Aerosols and Ozone (STERAO) case simulated by Skamarock et al. (2000), provided a framework for several adjustments of the number of ice fragments. A series of experiments was then performed for the same case to see how much the CIBU process altered the precipitation and the persistence of convective plumes. The question of the number of ice nuclei necessary to initiate CIBU (Field et al., 2017; Sullivan et al., 2018) was also addressed. A second case of a deep convective cloud (Weisman and Klemp, 1984) is run to confirm the impact of CIBU in a series of different CAPE environments. The simulations showed that the invigoration of convection when the CIBU efficiency was strong led to larger cloud covers and an increase of the mean cloud top height. Finally, a conclusion is drawn on the importance of calibrating the parameterization of CIBU and the need to systematically include CIBU and other ice multiplication processes in bulk microphysics schemes.

2 Introduction of CIBU into the LIMA scheme

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In contrast to the work of Yano and Phillips (2011), where large and small
graupel particles fuelled the CIBU process, we consider collisions involving
two types of precipitating ice here: small ice particles grown by
deposition and aggregation (aggregates including dendritic pristine ice
crystals with a size larger than ∼150 µm) and large graupel
particles grown by riming. Collisions between graupel particles of
different sizes are not considered because, according to Griggs and Choularton (1986),
rime is very unlikely to fragment in natural clouds. For the proposed
parameterization of CIBU, an impact velocity of the graupel particles that is
well above 1 m s^{−1} is imposed so as to stay in the break-up regime of
the aggregates. This is achieved by selecting the size range of the
aggregates and the graupel particles to enable CIBU.

A general form of the equation describing the CIBU process can be written

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{\partial {n}_{\text{i}}}{\partial t}}=\mathit{\alpha}{n}_{\text{s}}{n}_{\text{g}},\end{array}$$

where *n* is the particle size distribution of the cloud ice (subscript
“i”), the snow aggregates (“s”), and the graupel particles (“g”). The
parameter *α* is the snow-aggregate–graupel collision kernel multiplied
by 𝒩_{sg}, the number of ice fragments produced per
collision. An expression for *α*, which does not include thermal and
mechanical energy effects, is

$$\begin{array}{}\text{(2)}& \mathit{\alpha}={\mathcal{N}}_{\text{sg}}{V}_{\text{sg}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{4}}}{D}_{\text{g}}^{\mathrm{2}},\end{array}$$

where *V*_{sg} is the impact velocity of a graupel particle of size
*D*_{g} at the surface of the aggregate.

In Eq. (2), it is assumed that the size of the aggregate is
negligible compared to *D*_{g}. *V*_{sg} is expressed as the
difference in fall speed between the colliding graupel and the aggregate
target so ${V}_{\text{sg}}=({\mathit{\rho}}_{\mathrm{00}}/{\mathit{\rho}}_{\text{a}}{)}^{\mathrm{0.4}}\times ({c}_{\text{g}}{D}_{\text{g}}^{{d}_{\text{g}}}-{c}_{\text{s}}{D}_{\text{s}}^{{d}_{\text{s}}})$ using the
generic formula of the particle fall speeds with the air density correction
of Foote and du Toit (1969) due to the drag force exerted by the particles during
their fall. The parameter *ρ*_{00} is the reference air density
*ρ*_{a} at the reference pressure level.

As introduced above, and suggested in Yano and Phillips (2011), the impact
velocity *V*_{sg} should be large enough to enable CIBU. An easy way to
achieve this is to restrict the size of the aggregates to the range
[*D*_{smin}=0.2 mm, *D*_{smax}=1 mm] and to introduce a minimum
size of *D*_{gmin}=2 mm for the graupel particles. The reasons for
these choices are discussed below. The lower bound value of the aggregates,
*D*_{smin}, is such that the collision efficiency with a graupel
particle approaches unity. For *D*_{s}<*D*_{smin}, large crystals or
aggregates stay outside the path of capture which explains the observation of
bimodal ice spectra. Field (2000) reported minimum values of
150–200 µm for *D*_{trough}, a critical size separating cloud
ice and aggregate regimes. The *D*_{smin} value is also consistent with
an upper bound of the cloud ice crystal size distribution resulting from the
critical diameter of 125 µm to convert cloud ice to snow by
deposition (see Harrington et al., 1995, for the original and analytical
developments and Vié et al., 2016, for the implementation in LIMA). The
choice of round numbers for *D*_{smax} and *D*_{gmin} is above all
dictated by the empirical rule that *V*_{sg}>1 m s^{−1}. With the
set-up in LIMA, which is $[{c}_{x},{d}_{x}]=[\mathrm{5.1},\mathrm{0.27}]$ for “*x*=*s*” and [124,0.66] for “*x*=*g*” in metre, kilogram, and/or second (MKS) units, we obtain *V*_{sg}>1.26 m s^{−1}
at ground level.

The number of ice fragments produced by a collision, 𝒩_{sg},
is the critical parameter for ice multiplication. From scaling arguments,
Yano and Phillips (2011) recommended taking 𝒩_{sg}=50.
Recently, Yano and Phillips (2016) introduced a notion of random
fluctuations into the production of fragments which leads to a stochastic
equation of the ice crystal concentration. The parameterization of
𝒩_{sg} as a function of collisional kinetic energy
(Phillips et al., 2017) enables a treatment of the fragmentation that
depends on the ice crystal type. All these results stem from Fig. 6 in
Vardiman (1978), which suggests that 𝒩_{sg} is a function of
momentum change, Δ*M*_{g}, after the collision. As Δ*M*_{g}∼0.1 g cm s^{−1} for *D*_{g}=2 mm, the
corresponding 𝒩_{sg} lies between 10 (for collision with
plane dendrites) and 40 (for rimed spatial crystals). These values are
consistent with those found by Yano and Phillips (2011) for rimed
assemblages. In conclusion, it is tempting to run both deterministic and
stochastic simulations to test the sensitivity of the parameterization to
𝒩_{sg} in the range suggested by laboratory experiments. In
the following, 𝒩_{sg} is set successively to 0.1 (weak
effect) implying one fragment per 10 collisions, 1.0 (moderate effect), and
up to 10.0 or even 50.0 (strong effect). Additional experiments were performed
by first generating a random variable *X* uniformly distributed over [0.0, 1.0] and then applying an empirical formula, ${\mathcal{N}}_{\text{sg}}={\mathrm{10}}^{\mathrm{2.0}\times X-\mathrm{1.0}}$, to generate values of 𝒩_{sg} in the
interval [0.1,10.0]. The randomization of 𝒩_{sg} reflects
the fact that the number of fragments depends on the positioning of the
impact, on the tip, or on the body of the fragile particle, and also on the
energy lost by the possible rotation of the residual particle.

The LIMA microphysics scheme (Vié et al., 2016) includes a representation of the
aerosols as a mixture of cloud condensation nuclei (CCN) and ice freezing
nuclei (IFN) with an accurate budget equation (transport, activation, or
nucleation, and scavenging by rain) for each aerosol type. The CCN are
selectively activated to produce cloud droplets which grow by condensation
and coalescence to produce rain drops (Cohard and Pinty, 2000). The ice phase
is more complex as we consider nucleation by deposition on insoluble IFN
(black carbon and dust) and nucleation by immersion (glaciation of tagged
droplets formed on partially soluble CCN containing an insoluble core).
Homogeneous freezing of the droplets is possible when the temperature drops
below −35 ^{∘}C. The Hallett–Mossop mechanism generates ice crystals
during the riming of the graupel and the snow aggregates. The H–M efficiency
depends strongly on the temperature and on the size distribution of the
droplets (Beheng, 1987). The initiation of the snow-aggregate category is
the result of depositional growth of large pristine crystals beyond a
critical size (Harrington et al., 1995). Aggregation and riming are computed
explicitly. Heavily rimed particles (graupel) can experience a dry or wet
growth mode. The freezing of raindrops by contact with small ice crystals
leads to frozen drops which are merged with the graupel category. The melting
of snow aggregates leads to graupel and shed raindrops while the graupel
particles melt directly into rain. Sedimentation is considered for all
particle types. The snow aggregates and graupel particles are characterized
by their mixing ratios only. The LIMA scheme assumes a strict saturation of
the water vapour over the cloud droplets, while the small ice crystals are
subject to super- or undersaturated conditions (no instantaneous
equilibrium).

In a two-moment bulk scheme, the zeroth-order (total number concentration) and
“*b*th”-order (mixing ratio)^{2} moments of the size distributions are computed.
From Eqs. (1) and (2), the CIBU tendency of the number
concentration of the cloud ice, *N*_{i} (here in kg^{−1}), can be
written as

$$\begin{array}{ll}\text{(3)}& {\displaystyle \frac{\partial {N}_{\text{i}}}{\partial t}}=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{{\mathcal{N}}_{\text{sg}}}{{\mathit{\rho}}_{\text{dref}}}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{4}}}{\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{00}}}{{\mathit{\rho}}_{\text{dref}}}}\right)}^{\mathrm{0.4}}\underset{{D}_{\text{smin}}}{\overset{{D}_{\text{smax}}}{\int}}{\displaystyle}& {\displaystyle}{n}_{\text{s}}\left({D}_{\text{s}}\right)\left\{\underset{{D}_{\text{gmin}}}{\overset{\mathrm{\infty}}{\int}}{D}_{\text{g}}^{\mathrm{2}}({c}_{\text{g}}{D}_{\text{g}}^{{d}_{\text{g}}}-{c}_{\text{s}}{D}_{\text{s}}^{{d}_{\text{s}}}){n}_{\text{g}}\left({D}_{\text{g}}\right)\text{d}{D}_{\text{g}}\right\}\text{d}{D}_{\text{s}},\end{array}$$

where *ρ*_{dref}(*z*) is a reference density profile for dry air
(Meso-NH is anelastic) and a further approximation
*ρ*_{a}=*ρ*_{dref} is applied.

In LIMA, the size distributions follow a generalized gamma law:

$$n\left(D\right)\text{d}D=N{\displaystyle \frac{\mathit{\alpha}}{\mathrm{\Gamma}\left(\mathit{\nu}\right)}}{\mathit{\lambda}}^{\mathit{\alpha}\mathit{\nu}}{D}^{\mathit{\alpha}\mathit{\nu}-\mathrm{1}}{e}^{-(\mathit{\lambda}D{)}^{\mathit{\alpha}}}\text{d}D,$$

where *α* and *ν* are fixed shape parameters, *N* is the total number
concentration and *λ* is the slope parameter. With the definition of
the moments ${M}_{x}^{\text{INC}}(p;X)$ of the incomplete gamma law given in
Appendix A, integration of Eq. (3) leads
to

$$\begin{array}{ll}\text{(4)}& {\displaystyle \frac{\partial {N}_{\text{i}}}{\partial t}}=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{{\mathcal{N}}_{\text{sg}}}{{\mathit{\rho}}_{\text{dref}}}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{4}}}{\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{00}}}{{\mathit{\rho}}_{\text{dref}}}}\right)}^{\mathrm{0.4}}{N}_{\text{s}}{N}_{\text{g}}{\displaystyle}& {\displaystyle}\times \left\{{c}_{\text{g}}\left({M}_{\text{s}}^{\text{INC}}\left(\mathrm{0};{D}_{\text{smin}}\right)-{M}_{\text{s}}^{\text{INC}}\left(\mathrm{0};{D}_{\text{smax}}\right)\right)\right.\\ {\displaystyle}& {\displaystyle}\left({M}_{\text{g}}(\mathrm{2}+{d}_{\text{g}})-{M}_{\text{g}}^{\text{INC}}\left(\mathrm{2}+{d}_{\text{g}};{D}_{\text{gmin}}\right)\right)\\ {\displaystyle}& {\displaystyle}-{c}_{\text{s}}\left({M}_{\text{s}}^{\text{INC}}\left({d}_{\text{s}};{D}_{\text{smin}}\right)-{M}_{\text{s}}^{\text{INC}}\left({d}_{\text{s}};{D}_{\text{smax}}\right)\right)\\ {\displaystyle}& {\displaystyle}\left.\left({M}_{\text{g}}\left(\mathrm{2}\right)-{M}_{\text{g}}^{\text{INC}}\left(\mathrm{2};{D}_{\text{gmin}}\right)\right)\right\},\end{array}$$

with ${N}_{\text{s}}={C}_{\text{s}}{\mathit{\lambda}}_{\text{s}}^{{x}_{\text{s}}}$ and
${N}_{\text{g}}={C}_{\text{g}}{\mathit{\lambda}}_{\text{g}}^{{x}_{\text{g}}}$. The set of parameters
used in LIMA is *C*_{s}=5, ${C}_{\text{g}}=\mathrm{5.0}\times {\mathrm{10}}^{\mathrm{5}}$,
*x*_{s}=1, ${x}_{\text{g}}=-\mathrm{0.5}$. These values were chosen to generalize the
classical Marshall–Palmer law, $n\left(D\right)={N}_{\mathrm{0}}\mathrm{exp}(-\mathit{\lambda}D)$, a degenerate form
of the generalized gamma law when $\mathit{\alpha}=\mathit{\nu}=\mathrm{1}$, leading to a total
concentration $N={N}_{\mathrm{0}}{\mathit{\lambda}}^{-\mathrm{1}}$ with a fixed intercept parameter *N*_{0}.

Concerning the mixing ratios, the mass of the newly formed cloud ice
fragments is simply taken as the product of the mean mass of the pristine ice
crystals by the *N*_{i} tendency (Eq. 3). The mass loss of the
aggregates after collisional break-up is equal to the mass of the ice
fragments. The mass of the graupel is unchanged. The mass transfer from
aggregates to small ice crystals is constrained by the mass of individual
aggregates that may break up completely. This limiting mixing ratio tendency
is given by

$$\begin{array}{ll}\text{(5)}& {\displaystyle \frac{\partial {r}_{\text{i}}}{\partial t}}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}=-{\displaystyle \frac{\partial {r}_{\text{s}}}{\partial t}}={\displaystyle \frac{{a}_{\text{s}}}{{\mathit{\rho}}_{\text{dref}}}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{4}}}{\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{00}}}{{\mathit{\rho}}_{\text{dref}}}}\right)}^{\mathrm{0.4}}\underset{{D}_{\text{smin}}}{\overset{{D}_{\text{smax}}}{\int}}{\displaystyle}& {\displaystyle}{D}_{\text{s}}^{{b}_{\text{s}}}{n}_{\text{s}}\left({D}_{\text{s}}\right)\left\{\underset{{D}_{\text{gmin}}}{\overset{\mathrm{\infty}}{\int}}{D}_{\text{g}}^{\mathrm{2}}({c}_{\text{g}}{D}_{\text{g}}^{{d}_{\text{g}}}-{c}_{\text{s}}{D}_{\text{s}}^{{d}_{\text{s}}}){n}_{\text{g}}\left({D}_{\text{g}}\right)\text{d}{D}_{\text{g}}\right\}\text{d}{D}_{\text{s}}.\end{array}$$

In the above expression, the mass of an aggregate of size *D*_{s} is
given by ${a}_{\text{s}}{D}_{\text{s}}^{{b}_{\text{s}}}$ with *a*_{s} set to 0.02
and *b*_{s} to 1.9 in LIMA, meaning that aggregates are practically
two-dimensional particles. After integration, the mixing ratio tendencies are
expressed as

$$\begin{array}{ll}\text{(6)}& {\displaystyle \frac{\partial {r}_{\text{i}}}{\partial t}}=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}-{\displaystyle \frac{\partial {r}_{\text{s}}}{\partial t}}={\displaystyle \frac{{a}_{\text{s}}}{{\mathit{\rho}}_{\text{dref}}}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{4}}}{\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{00}}}{{\mathit{\rho}}_{\text{dref}}}}\right)}^{\mathrm{0.4}}{N}_{\text{s}}{N}_{\text{g}}{\displaystyle}& {\displaystyle}\times \left\{{c}_{\text{g}}\left({M}_{\text{s}}^{\text{INC}}\left({b}_{\text{s}};{D}_{\text{smin}}\right)-{M}_{\text{s}}^{\text{INC}}\left({b}_{\text{s}};{D}_{\text{smax}}\right)\right)\right.\\ {\displaystyle}& {\displaystyle}\left({M}_{\text{g}}\left(\mathrm{2}+{d}_{\text{g}}\right)-{M}_{\text{g}}^{\text{INC}}\left(\mathrm{2}+{d}_{\text{g}};{D}_{\text{gmin}}\right)\right)\\ {\displaystyle}& {\displaystyle}-{c}_{\text{s}}\left({M}_{\text{s}}^{\text{INC}}\left({b}_{\text{s}}+{d}_{\text{s}};{D}_{\text{smin}}\right)-{M}_{\text{s}}^{\text{INC}}\left({b}_{\text{s}}+{d}_{\text{s}};{D}_{\text{smax}}\right)\right)\\ {\displaystyle}& {\displaystyle}\left.\left({M}_{\text{g}}\left(\mathrm{2}\right)-{M}_{\text{g}}^{\text{INC}}\left(\mathrm{2};{D}_{\text{gmin}}\right)\right)\right\}.\end{array}$$

This expression is independent of the number of fragments
𝒩_{sg}.

3 Simulation of a three-dimensional deep convective case

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The test case is illustrated by idealized numerical simulations of the
10 July 1996 thunderstorm in the STERAO (Dye et al., 2000). This case is
characterized by a multicellular storm which becomes supercellular after
2 h. The simulations were initialized with the sounding over northeastern
Colorado given in Skamarock et al. (2000) and convection was triggered by
three 3 K buoyant bubbles aligned along the main diagonal of the *X*,*Y* plane
along the wind axis. Meso-NH was run for 5 h over a domain with 320×320 grid points and 1 km horizontal grid spacing. There were 50 unevenly
spaced vertical levels up to a height of 23 km. With the exception of the
wind components advected with a fourth-order scheme, all the fields,
including microphysics, were transported by an accurate, conservative,
positive-definite piecewise parabolic method scheme (Colella and Woodward, 1984).
There were no surface fluxes. The 3-D turbulence scheme of Meso-NH was used.
Open lateral boundary conditions were imposed. The upper level damping layer
of upward moving gravity waves started above 12 500 m.

The aerosols were initialized as for the simulated squall-line case studied in Vié et al. (2016). A summary is given in Table 1 for the soluble CCN and for the insoluble IFN. Homogeneous vertical profiles are assumed for the aerosols. Although the LIMA scheme incorporates size distribution parameters and differentiates between the chemical compositions of the CCN and the IFN, the characteristics of the five aerosol modes are standard for the simulations shown here, except for the sensitivity of CIBU to the initial concentration of the IFN which is explored in Sect. 3.5.

Figure 1 shows the accumulated precipitation at ground
level after 4 h of simulation for the four experiments corresponding to
𝒩_{sg}=0.0, 0.1, 1.0, and 10.0. The highest amount of
rainfall is obtained when the CIBU process is ignored
(𝒩_{sg}=0.0) in Fig. 1a. Then, by
increasing the CIBU efficiency 10-fold from 𝒩_{sg}=0.1,
Fig. 1b–d clearly show a steady reduction of
precipitation and a fine-scale modification of the precipitation pattern.
Furthermore, Fig. 1d reveals that the spread of the
precipitation field, caused by the motion of the multicellular storm, is
significantly reduced when 𝒩_{sg}=10.0. The results of
Fig. 1 suggest empirically that a plausible range for
𝒩_{sg} is between 0.1 and 10.0 fragments per collision. A
value lower than 0.1 leads to a negligible effect of CIBU in the simulation,
while taking 𝒩_{sg}>10.0 has an excessive impact on the
storm rainfall (the “𝒩_{sg}=50.0” case is not shown). In
addition, Fig. 2 shows the results of a
simulation, called “RANDOM” hereafter, where ${\mathcal{N}}_{\text{sg}}\in [\mathrm{0.1},\phantom{\rule{0.33em}{0ex}}\mathrm{10}]$ is generated by a random process as explained above. The
perturbation caused by CIBU is also noticeable in this case; it remains weak
for the precipitation field. These first 3-D numerical experiments show that
inclusion of CIBU can modify surface precipitation strongly when
𝒩_{sg}>10.0 fragments per aggregate–graupel collision.
Taking $\mathrm{0.1}<{\mathcal{N}}_{\text{sg}}<\mathrm{10.0}$ and also considering
𝒩_{sg} as determined from a random process seems to be a
more satisfactory approach. Admittedly, 𝒩_{sg}∼10 is
more than an order of magnitude but our conclusion is to recommend an upper bound
value of 𝒩_{sg} that is much lower than the former *N*=50
used by Yano and Phillips (2011) with their notation in the box model.

Essentially, intensifying the CIBU process by increasing
𝒩_{sg} leads to higher cloud ice crystal concentrations
which deplete the supersaturation of water vapour that would otherwise
contribute to the deposition growth of the snow aggregates. However, a
further effect is possible because the partial mass sink of the
snow aggregate particles also slows down the flux of graupel particles, which
form essentially by heavy riming and conversion of the snow aggregates. This
point is now examined by considering the ice in the high levels of the STERAO
cells. Figures 3–5
reproduce the 10 min average of the mixing ratios *r*_{i},
*r*_{s}, and *r*_{g} at 12 km from the four experiments having
𝒩_{sg}=0.0, 0.1, 1.0, and 10.0 after 4 h. The increase of
the cloud ice mixing ratio with 𝒩_{sg} is clear in the area
covered by the 0.2 g kg^{−1} isocontour in
Fig. 3. Simultaneously, a slight decrease of
*r*_{s}, indicating a slow erosion of the mass of the aggregates, is
visible in Fig. 4. The effect on the graupel
(Fig. 5) is even smaller but appears clearly for the
case 𝒩_{sg}=10.0, where less graupel is found. A last
illustration is provided in Fig. 6, showing the
number concentration of cloud ice *N*_{i} at a higher altitude of
15 km. Again, the increase of *N*_{i} follows 𝒩_{sg}
with an explosive multiplication of *N*_{i} when
𝒩_{sg}=10.0 (*N*_{i} is well above
1000 crystals kg^{−1} of dry air in this case).
Figure 7 summarizes the behaviour of *r*_{i},
*r*_{s}, and *r*_{g} at 12 km height, and of *N*_{i} at
15 km height, for the “RANDOM” simulation. A comparison with
Figs. 3–6 shows that the
results are those expected. The examination of the microphysics fields
suggests that the “RANDOM” simulation corresponds to a mean CIBU intensity
intermediate between 𝒩_{sg}=1 and
𝒩_{sg}=10.

The analysis of the STERAO simulations continues with an examination of the
vertical profiles of microphysics budgets. The profiles are 10 min averages
of all cloudy columns that contain at least 10^{−3} g kg^{−1} of
condensate at any level. The column selection is updated at each time step
because of the evolution and motion of the storm.
Figure 8 shows the mixing ratio profiles for three
cases: 𝒩_{sg}=0.0, “RANDOM”, and 𝒩_{sg}=10.0. A key feature that shows up in Fig. 8a–c is
the increase of the *r*_{i} peak value at 11 km altitude. This change
is accompanied by a reduction of *r*_{s} (more visible between Fig. 8b
and c) and by a reduction of *r*_{g}, which stands out at *z*=8000 m. The
decrease of *r*_{g}, even when graupel is a passive collider for CIBU,
is the result of the decrease of *r*_{s} in the growth chain of the
precipitating ice. The low value of the mean *r*_{r} profiles, compared to the
mixing ratios of the ice phase above, is explained by the fact that rain is
spread over fewer grid points than the ice in the anvil is (the mixing ratio
profiles are averaged over the same number of columns).

This step is devoted to the microphysics tendencies (using 10 min average again with the nomenclature of the processes provided in Table 3) of the ice mixing ratios in Figs. 9–11 to assess the impact of the CIBU process. We do not discuss the case of the liquid phase here because the tendencies (not shown) are only marginally affected by the CIBU process.

As expected, many tendencies of *r*_{i}
(Fig. 9a–c) are affected by the CIBU process.
The main processes standing out in Fig. 9a, when
CIBU is not activated, are CEDS (deposition–sublimation), essentially a gain
term, and AGGS (aggregation), the main loss of *r*_{i} by aggregation
with a rate of $\mathrm{0.5}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$. The loss of
*r*_{i} by CFRZ (drop freezing by contact) makes a moderate contribution
as some raindrops are present in the glaciated part of the storm. Above
*z*=10 000 m, the net loss of *r*_{i} (AGGS and SEDI, the cloud ice
sedimentation) is balanced by the convective vertical transport (not shown).
When 𝒩_{sg}= RANDOM, the *r*_{i} tendencies are
amplified, even with a modest contribution of $\sim \mathrm{0.2}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ for CIBU itself. The growth of AGGS,
which doubles at 10 km height, is caused by CIBU and by an increase in the
convection because SEDI (a loss at this height) is amplified in response to
an increase of *r*_{i} in the upper levels. The CFRZ contribution is
also increased. The last case, with 𝒩_{sg}=10
(Fig. 9c), confirms a further increase of the
rates except for CFRZ, interpreted here as a lack of raindrops.

The budget of the snow-aggregate mixing ratio in
Fig. 10 contains many processes of equivalent
importance in the range $\pm \mathrm{0.05}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$
but SEDS (sedimentation of snow aggregates) dominates at
*z*=11 000 m and at *z*=7000 m. The inclusion of CIBU
(Fig. 10b–c) mostly leads to an increase of
AGGS, and the other processes remaining almost the same. Finally, many processes
contribute to the evolution of the graupel mixing ratio profiles
(Fig. 11). The strongest loss is in the GMLT term
(melting of graupel) that converts graupel into rain (down to $-\mathrm{0.3}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$) while CFRZ reaches $\mathrm{0.15}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$. The sedimentation term SEDG
(sedimentation of graupel) lies between $-\mathrm{0.3}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ at *z*=10 000 m and
$\mathrm{0.15}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ at 5000 m. Another
noticeable effect is the sign change of DEPG (growth of graupel by
deposition, $\pm \mathrm{0.07}\times {\mathrm{10}}^{-\mathrm{3}}$ $\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$) showing that
the water vapour is supersaturated above *z*=7000 m and
undersaturated below *z*=7000 m on average. The relative importance
of these processes does not change very much when CIBU is increased but all
tendencies weaken. To sum up, the impact of CIBU is modest for the
microphysics mixing ratios. The increase of ice fragments in *r*_{i} is
approximately compensated by an increase of AGGS (see
Figs. 9 and 10).

This subsection examines the behaviour of the cloud ice number concentration
as a function of the strength of the CIBU process after 4 h of simulation.
Figure 12 shows that the altitude of the
*N*_{i} peak value decreases when 𝒩_{sg} increases. In
the absence of CIBU (𝒩_{sg}=0), the source of *N*_{i}
is the heterogeneous nucleation processes on insoluble IFN and on coated IFN
(nucleation by immersion) which are more efficient at low temperature.
Nucleation on IFN provides a mean peak value *N*_{i}=400 kg^{−1} at *z*=11 500 m. In contrast, the
𝒩_{sg}=10 case (here scaled by a factor 0.1 for ease of
reading) keeps the trace of an explosive production of cloud ice
concentration, *N*_{i}=7250 kg^{−1}, due to CIBU. The altitude
of the maximum of *N*_{i} in this case (*z*=10 000 m) is
consistent with the location of the maximum value of the *r*_{s}×*r*_{g} product (see Fig. 8). The “RANDOM”
simulation produces *N*_{i}=1100 kg^{−1} at *z*=11 000 m, a number concentration similar to that found for the
𝒩_{sg}=2 case. Table 2 reports the peak
amplitude of the *N*_{i} profiles as a function of
𝒩_{sg} but after 3 h of simulation, when the CIBU rate is
strongly dominant. Additional cases were run to cover
$\mathrm{0.1}<{\mathcal{N}}_{\text{sg}}<\mathrm{50}$ with a logarithmic progression above
𝒩_{sg}=1.0. The CIBU enhancement factor, CIBU_{ef},
was computed as ${N}_{\text{i}}\left({\mathcal{N}}_{\text{sg}}\right)/{N}_{\text{i}}({\mathcal{N}}_{\text{sg}}=\mathrm{0})-\mathrm{1}$ since
*N*_{i}(𝒩_{sg}=0) constitutes a baseline not affected by
CIBU. The results presented in Table 2 show that the growth
of *N*_{i} is fast when 𝒩_{sg} reaches ∼5
(CIBU_{ef} rises sharply from 135 % to 913 % when
𝒩_{sg} increases from 2 to 5). Taking 𝒩_{sg}=50 leads to an extremely high peak value of *N*_{i}.

The *N*_{i} tendencies are the subject of
Fig. 13. Many processes are involved during the
temporal integration of *N*_{i}. The 𝒩_{sg}=0 case
confirms the importance of the heterogeneous nucleation process by deposition
(HIND; see Table 3) and, to a lesser degree, by immersion
(HINC) at 8 km height. HIND peaks at three altitudes with two sources of IFN
(Table 1). This case also reveals the importance of the HMG
(Hallett–Mossop on graupel, 1.3 ${\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$) and HMS
(Hallett–Mossop on snow, 0.85 ${\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$) processes. Here, we
consider that H–M also operates for the snow aggregates because this
category of ice includes lightly rimed particles that can rime further to
form graupel particles. These processes are first compensated by AGGS
(capture of cloud ice by the aggregates). There is also a loss of cloud ice
due to CFRZ and CEDS with the full sublimation of individual cloud ice
crystals which replenish the IFN reservoir. The sedimentation profile
transports ice from the cloud top (SEDI < 0) to mid-level cloud
(SEDI > 0). Then, taking 𝒩_{sg}= RANDOM shows the
domination of the CIBU process, which reaches 2.5 ${\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ at
5 km height. The enhancement of HIND at cloud top can also be noted. The
CIBU source of ice crystals is balanced by an increase of AGGS and, above
all, of CEDS (here, CEDS represents the sublimation of the ice crystal
concentration when the crystals are detrained in the low level of the cloud
vicinity, such as below the anvil). Finally, the 𝒩_{sg}=10
case demonstrates the reality of the exponential-like growth of *N*_{i}
because the three main driving terms (CIBU, CEDS, and AGGS) are growing at a
similar rate, which is multiplied by a factor of approximately 5.

The purpose of the last series of experiments was to look more closely at the
sensitivity of the cloud ice concentration to *N*_{IFN}, the initial
concentration of the IFN. Numerical simulations were run with *N*_{IFN}
decreasing 10-fold from 100 to 0.001 day m^{−3} for each IFN mode (see
Table 1). Two different cases were considered. In the first
case, CIBU was activated with the RANDOM set-up while, in the second case,
CIBU effects were ignored. All the results are summarized in the plots of
Fig. 14.

Figure 14a shows that *N*_{i} concentrations
did not change very much for a wide range of *N*_{IFN} concentrations,
which were varied 10-fold. This clearly illustrates the predominance of the
CIBU effect for current IFN concentrations, which disconnects *N*_{i}
concentrations from the underlying abundance of IFN particles. Likewise, the
small hump superimposed on all profiles at 5000 m height reveals a residual
effect of the Hallett–Mossop process. Another remarkable feature is that a
fairly low IFN concentration (*N*_{IFN}=0.001 day m^{−3}) suffices
to initiate the CIBU process and to reach *N*_{i}∼500 kg^{−1}.
In contrast, and in the absence of CIBU
(Fig. 14b), the *N*_{i} profiles show a
sensitivity to IFN nucleation that is, indeed, difficult to interpret because
of the non-monotonic trend of the *N*_{i} profiles with respect to
*N*_{IFN}. Some insight can be gained by checking the concentration of
the nucleated IFN of the first IFN mode (dust particles). In
Fig. 14c, the IFN profiles are rescaled
(multiplication by an appropriate number of powers of 10) to be comparable.
This is equivalent to computing an IFN nucleation efficiency. The important
result here is that the number of nucleated IFN evolves in close proportion
to the initially available IFN concentrations, meaning that, as expected, the
nucleating properties of the IFN do not depend on the IFN concentration. The
last plot (Fig. 14d) reproduces the normalized
differences of *N*_{i} profiles between twin simulations performed with
and without CIBU. Although simulations using the same initial concentration
*N*_{IFN} may diverge because of additional non-linear effects (vertical
transport, enhanced or reduced cloud ice sink processes), the figure gives an
indication of the bulk sensitivity of CIBU to the IFN. The enhancement ratio
due to CIBU remains low (less than 1 for *N*_{IFN}∼100 day m^{−3}) but can reach a factor of 20 at 9000 m height in the case
of moderate IFN concentration, i.e. *N*_{IFN}∼1 day m^{−3}. The
behaviour of LIMA can be explained in the sense that increasing
*N*_{IFN} too much leads to smaller pristine crystals that need a longer
time to grow before being included in the next category of snow aggregates
because such inclusion is size-dependent (see Harrington et al., 1995, and
Vié et al., 2016). On the other hand, a low concentration of *N*_{IFN}
initiates fewer snow aggregates and thus fewer graupel particles, so the
whole CIBU efficiency is also reduced. Consequently, this study confirms the
essential role of CIBU in compensating for IFN deficit when cloud ice
concentrations are increasing.

4 Simulation of a three-dimensional idealized supercell storm with varying atmospheric stability

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The idealized sounding of Weisman and Klemp (1982, 1984) was appealing to use for this test case (referred to as WK) because the intensity of the CAPE can be easily modified by changing a reference water vapour mixing ratio. The environmental conditions of the simulations were close to those of the STERAO case with the same set-up for the physics and the aerosol characteristics. The simulation domain was 180×180 grid points at 1 km resolution and 70 levels with a mean vertical grid spacing of 350 m. Convection was triggered by a domain-centred single 2 K buoyant air parcel of 10 km radius and 3 km height. The base of the upper level Rayleigh damper was set at 15 km above ground level.

Meso-NH was initialized with the analytic sounding of Weisman and Klemp (1984)
with low two-dimensional shear. The hodograph in Fig. 15
features a three-quarter cycle with a constant wind of 6.4 m s^{−1}
(in modulus) above the height of 5 km. When running Meso-NH, a constant
translation speed (*U*_{trans}=5 m s^{−1} and
*V*_{trans}=1 m s^{−1}) was added to the wind to keep the
convection well centred in the domain of simulation. As explained in
Weisman and Klemp (1982), buoyancy was varied by altering the magnitude of the
surface water vapour mixing ratio *q*_{v0} keeping with the
Weisman and Klemp (1984) notation. Three water vapour profiles were defined by
taking *q*_{v0}=13.5 g kg^{−1}, hereafter the “low” CAPE case of
1970 J kg^{−1}, *q*_{v0}=14.5 g kg^{−1} as the “mid” CAPE
case of 2400 J kg^{−1}, and *q*_{v0}=15.5 g kg^{−1} as the
“high” CAPE case of 2740 J kg^{−1}. Four experiments of 4 h
each were performed for each CAPE case by using different magnitudes of
𝒩_{sg}.

Figure 16 shows the mean concentrations of small ice crystals
between 9.5 and 10.5 km levels plotted on a log scale after 4 h of
simulation. In addition, two cloud top height (CTH) contours delineate the
11 km (dotted line) and 13 km (solid line) levels. The
𝒩_{sg}=0, RANDOM, 10, and 50 cases, are explored for each
sounding (“low”, “mid”, and “high” CAPE). In the absence of CIBU (first
row in Fig. 16), the cloud ice concentrations *N*_{i} are
in the range of what was simulated for the STERAO case (see
Figs. 6 and 7d). The
*N*_{i} peak values do not increase with the initial CAPE
(Fig. 16a, b) but the area of CTH > 11 km is larger in
the “mid” CAPE case. The “high” case is a little bit more difficult
to analyse because of earlier development of the convection, spreading out
ahead of the main system. This shows up in the “low” and “mid” CAPE cases
but the *N*_{i} peak values of the “high” CAPE case are in the same
range as for the “low” CAPE case, meaning that higher environmental
instability is not decisive in fixing the *N*_{i} peak values. In the
𝒩_{sg}=10 and 50 cases, we retrieve the dramatic increase of
*N*_{i} due to increasing CIBU efficiency. The enhancement is locally as
high as 1000-fold in the strongest case (𝒩_{sg}=50). There
are also other noteworthy features: an increase of the *N*_{i} area
coverage with 𝒩_{sg} (less visible in the “low” CAPE case)
and a higher CTH which exceeds 13 km for the “mid” and “high” CAPE
cases. All these observations strongly suggest that convection is invigorated
when the CIBU effect is increased. In contrast, the simulations run with
𝒩_{sg}= RANDOM using values taken in the 0.1–10 range (see
Sect. 2.1), show a moderate effect of CIBU. Locally, *N*_{i} values
reach 1×10^{4} kg^{−1}, which is 100 times lower than
*N*_{i} peak values in the 𝒩_{sg}=50 cases but
approximately 10 times higher than in the “no CIBU” case
(𝒩_{sg}=0). Finally, the simulation results suggest that the
𝒩_{sg} parameter could be constrained by satellite data
because of the sensitivity of CIBU to the cloud ice coverage and the cloud
top height.

The 4 h accumulated precipitation maps are presented in
Fig. 17. On each row, precipitation increases from the
“low” to “high” CAPE cases. This is because the CAPE is enhanced by the
addition of more water vapour. Looking at the sensitivity of the accumulated
precipitation to 𝒩_{sg}, it is not as easy to draw a general
conclusion on the decrease of the precipitation peak with
𝒩_{sg} as for the STERAO case (see Sect. 3.1). The reason is
the highly concentrated precipitation field, which leads to a sharp gradient
around the location of the peak value. However, the decrease of the
precipitation with 𝒩_{sg} is observed in the “low” and
“high” CAPE cases. In the “mid” case, the precipitation peak value
remains high when 𝒩_{sg}=50 but the area where the
precipitation is less than 10 mm shrinks continuously. The reduction of the
area where the precipitation amount is greater than 10 mm when
𝒩_{sg} is increased was found in all CAPE cases (not
shown).

In conclusion, the simulations illustrate the fact that the precipitation
patterns are affected by the value of the 𝒩_{sg} parameter.
When 𝒩_{sg} is increased from 0 to 50, the precipitation
is reduced either for the peak value or at least for the precipitating
area. This is consistent with our previous results concerning the STERAO
case. The conversion efficiency of the small ice crystals to precipitating
ice particles is lower when the cloud ice concentration is high because the
deposition growth of individual small crystals is limited by the amount of
supersaturated water vapour available.

This last analysis is concerned with the ice thicknesses (or ice water paths)
computed as the integrals along the vertical of *ρ*_{dref} *r*_{x}, where
*r*_{x} refers to the mixing ratio with *x*∈ i, s, g standing for the cloud
ice, the snow aggregates, and the graupel hail, respectively.
Figure 18 displays the total ice thickness, a sum of three
terms, in millimetres (coloured area) with the superimposed cloud ice thickness
(THIC) contoured at 1 mm. A remarkable feature is that the total ice
thickness seems almost insensitive to the CIBU process for a given CAPE case:
there is no great modification in the plots when moving from
𝒩_{sg}=0 to 𝒩_{sg}=50. This is in contrast
with the 1 mm contour of cloud ice thickness, the enclosed area of which
increases with 𝒩_{sg} as shown in Fig. 18.
A rise in the maximum value of THIC was also expected for increasing values
of 𝒩_{sg}. However, the increase of THIC_{max} with the
CAPE is much more moderate between the “low” and “high” cases because a
higher CAPE regime with higher humidity tends to favour the horizontal spread
of the cloud ice mass.

5 Summary and perspectives

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The aim of this work was to study a parameterization of the collisional ice
break-up for the bulk two-moment microphysics LIMA scheme running in a
cloud-resolving mesoscale model (Meso-NH, in our case). While the process is
suspected to occur in real clouds, it is not included in current bulk
microphysics schemes. Because of uncertainties to physically describe the ice
break-up process, the present parameterization has been kept as simple as
possible. It considers only collisions between small aggregates and large,
dense graupel particles. The number of ice fragments that results from a
single collision, 𝒩_{sg}, is a key parameter, which is
estimated from only a very small number of past experiments (Vardiman, 1978).
This study suggests an upper bound on 𝒩_{sg} because of the
sensitivity of 𝒩_{sg} to the simulated precipitation. We
found that taking 𝒩_{sg}>10 significantly reduced surface
precipitation. This is problematic because most of the cloud schemes (running
without the CIBU process) are carefully verified for quantitative
precipitation forecasts in operational applications. Furthermore, we suggest
that 𝒩_{sg} could be considered as the realization of a
random process that reduces the impact of CIBU on the precipitation and also
that delicate radiating crystals undergoing fragmentation lead to a variety
of crystals with a missing arm or to many irregular fragments as illustrated
and discussed by Hobbs and Farber (1972). As a result, it has been shown that
running LIMA with 𝒩_{sg}>10 for the STERAO and WK deep
convection cases taken from Skamarock et al. (2000) and
Weisman and Klemp (1982, 1984), respectively, alters surface precipitation
because the conversion of cloud ice crystals to precipitating ice is slowed
down. In any case, the increase of the number concentration of the small ice
crystals due to the application of CIBU is clearly substantial (up to
1000-fold in the WK simulations with 𝒩_{sg}=50).

The microphysics perturbation due to the activation of CIBU has been studied
in detail for the STERAO case by looking at the profiles of the mixing
ratios, ice concentrations, and corresponding budget terms. In particular, the
CIBU effect on the pristine ice and aggregate mixing ratios is compensated by
an enhancement of the capture of the small crystals by the aggregates. The
sensitivity of the ice concentration to 𝒩_{sg} is
demonstrated with a mean multiplication factor as high as 25 for
𝒩_{sg}=10. The last study on the sensitivity of the
simulations to the initial IFN concentration showed that CIBU was mostly
efficient for current IFN concentrations of ∼1 day m^{−3}.
Furthermore, the CIBU process was still active for very low IFN
concentrations, down to 0.001 day m^{−3}, which were sufficient to
initiate the ice phase.

The effects of CIBU have been confirmed by a second series of WK simulations.
The enhancement of the cloud ice concentration is very high when
𝒩_{sg}>10, and a loss of surface precipitation is found in
terms of the peak value and the reduction of the precipitating areas. Higher
ice concentrations lead to a larger coverage of ice clouds and higher cloud
tops for the most vigorous convective cells. In contrast, the total ice
thickness is almost insensitive to CIBU. An increase of cloud ice mass with
𝒩_{sg} is balanced by a slight decrease of the precipitating
ice (aggregates and graupel).

The proposed parameterization is very easy to implement. It would be useful
to evaluate it in other microphysics schemes where the conversion of the
cloud ice and the growth of precipitating ice (aggregates and rimed
particles) are treated differently. Adjustments to the scheme can be revised
as soon as laboratory experiments are available to enable more precise fixing
of the sizes and the shapes of the crystals that break following collisions,
and also to examine any possible thermal effect and to estimate the variety
of fragment numbers more accurately. Another way to determine the acceptable
range of values for 𝒩_{sg} is to work with satellite data,
as the WK experiments demonstrated an enhancement of the cloud top ice cover
with 𝒩_{sg} (and possibly the cloud top
height).

With new imagers, counters, and improvements in data analysis (Ladino et al., 2017), more and more evidence is being presented that ice multiplication is an essential process in natural deep convective clouds. However, the explanation of anomalously high ice crystal concentrations is still difficult to link to a precise process (Field et al., 2017; Rangno and Hobbs, 2001). Therefore, the next step in the LIMA scheme will be to introduce the shattering of raindrops during freezing as proposed by Lawson et al. (2015) in order to complete the LIMA scheme, since the different ingredients of raindrops and small ice crystals offer another pathway for ice multiplication. One task will then be to study whether all the known sources of small ice crystals, nucleation, and secondary ice production are able to work together in microphysics schemes to reproduce the very high values of ice concentrations sometimes observed. Quantitative cloud data gathered in the tropics during the HAIC/HIWC (High Altitude Ice Crystals/ High Ice Water Content) field project (Ladino et al., 2017; Leroy et al., 2015) could provide a starting point for the evaluation of the capability of high-resolution cloud simulations to reproduce events where high cloud ice content has been recorded.

Code availability

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Code availability.

The Meso-NH code is publicly available at http://mesonh.aero.obs-mip.fr/mesonh51 (last access: 17 October 2018) (Chaboureau, 2014). Here, the model development and the simulations were carried out with version “MASDEV5-1 BUG2”. The modifications made to the LIMA scheme (v1.0) are available upon request from Jean-Pierre Pinty and in the Supplement related to this article, available at https://doi.org/10.5281/zenodo.1078527 (Hoarau et al., 2017).

Appendix A: Moments of the gamma and incomplete gamma functions

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The *p*th moment of the generalized gamma function (see definition in the
text) is

$$\begin{array}{}\text{(A1)}& M\left(p\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}{D}^{p}n\left(D\right)\text{d}D={\displaystyle \frac{\mathrm{\Gamma}(\mathit{\nu}+p/\mathit{\alpha})}{\mathrm{\Gamma}\left(\mathit{\nu}\right)}}{\displaystyle \frac{\mathrm{1}}{{\mathit{\lambda}}^{p}}},\end{array}$$

where the gamma function is defined as

$$\begin{array}{}\text{(A2)}& \mathrm{\Gamma}\left(x\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}{t}^{x-\mathrm{1}}{e}^{-t}\text{d}t.\end{array}$$

The *p*th moment of the incomplete gamma function is written as

$$\begin{array}{}\text{(A3)}& {M}^{\text{INC}}(p;X)=\underset{\mathrm{0}}{\overset{X}{\int}}{D}^{p}n\left(D\right)\text{d}D.\end{array}$$

The algorithm of the “GAMMA_INC(*p*;*X*)” function (Press et al., 1992) is
useful to tabulate ${M}^{\text{INC}}(p;X)\times \mathrm{\Gamma}\left(p\right)$ in addition to the
“GAMMA” function algorithm of Press et al. (1992). A change of variable is
necessary to take the generalized form of the gamma size distributions into
account. As a result, *M*^{INC}(*p*;*X*) is written as

$$\begin{array}{}\text{(A4)}& {M}^{\text{INC}}(p;X)=M\left(p\right)\times \text{GAMMA\_INC}(\mathit{\nu}+p/\mathit{\alpha};(\mathit{\lambda}X{)}^{\mathit{\alpha}}),\end{array}$$

with *M*(*p*) given by Eq. (A1).

Author contributions

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Author contributions.

TH and JPP conceived the scheme presented and performed the model developments and simulations. CB offered an expert analysis of the results, including the budget of the ice phase, and greatly improved the composition of the figures. All authors contributed to the writing of the manuscript.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

Jean-Pierre Pinty wishes to thank Vaughan Phillips for discussions about his
original work on the topic. This work was done during the PhD of
Thomas Hoarau, who is financially supported by Réunion Island Regional
Council and the European Union Council. Thomas Hoarau thanks the University
of La Réunion for supporting a short stay at the Laboratoire d'Aérologie.
Susan Becker and Callum Thompson corrected the English language of the
manuscript. Preliminary computations were performed on the 36-node homemade
cluster of Laboratoire Aérologie. Jean-Pierre Pinty acknowledges CALMIP
(CALcul MIdi-Pyrénés) of the University of Toulouse for access to the Eos
supercomputer, where useful additional simulations were performed.
Thomas Hoarau and Christelle Barthe acknowledge the GENCI resources for
access to the OCCIGEN supercomputer. This work was supported by the French
national programme LEFE/INSU through the LIMA-TROPIC project. The authors
thank the reviewers and the topical editor for their pertinent comments and
meticulous review which greatly improved previous versions of the
manuscript.

Edited by: Simon
Unterstrasser

Reviewed by: two anonymous referees

References

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H–M is based on the explosive riming of “big” droplets on graupel particles in a narrow range of temperatures.

Ice mixing ratios are computed by
integration over the size distribution of the mass of individual particles
given by a mass–size relationship (*m*(*D*)=*a**D*^{b}), a power law with a
non-integer exponent “*b*”.

Short summary

The break-up of ice crystals in clouds is a possible secondary ice multiplication process to explain observations of very high concentrations of small ice crystals at cold temperature. Here, the process is modeled by considering shocks between fragile aggregates (assemblage of pristine crystals) and large densely rimed crystals of selected sizes. The simulations of two storms illustrate the perturbations caused by the break-up effect (precipitation, ice concentration enhancement).

The break-up of ice crystals in clouds is a possible secondary ice multiplication process to...

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