The super-droplet method (SDM) is a particle-based numerical scheme
that enables accurate cloud microphysics simulation with lower
computational demand than multi-dimensional bin schemes.
Using SDM, a detailed numerical model of
mixed-phase clouds is developed in which ice
morphologies are explicitly predicted without assuming ice categories
or mass–dimension relationships. Ice particles are approximated using
porous spheroids. The elementary cloud microphysics processes considered are advection and sedimentation; immersion/condensation
and homogeneous freezing; melting; condensation and evaporation
including cloud condensation nuclei activation and deactivation;
deposition and sublimation; and coalescence, riming, and aggregation.
To evaluate the model's performance, a 2-D large-eddy simulation of a
cumulonimbus was conducted, and the life cycle of a cumulonimbus typically
observed in nature was successfully reproduced. The mass–dimension and
velocity–dimension relationships the model predicted show a
reasonable agreement with existing formulas. Numerical
convergence is achieved at a super-particle number concentration as
low as 128 per cell, which consumes

Mixed-phase clouds, which are clouds comprising droplets and ice particles, appear under multiple atmospheric
conditions, from the tropics to the poles, and throughout the year

Through their 70-year history, numerical models of cloud microphysics
have become increasingly sophisticated

Every numerical model is an approximation of a phenomenon's mathematical model, which is a theoretical description that should express the system's behavior accurately. We apply a numerical scheme to construct a numerical model, which we use to produce an approximate solution of the phenomenon's underlying mathematical model for given spatiotemporal boundary conditions.
This general philosophy of simulation is well documented, e.g., in

There are several types of cloud microphysics numerical models that are based on different levels of theoretical descriptions.

The first of these is the bulk model, which is the most widely used cloud microphysics model type

Kinetic description provides a more detailed microscopic mathematical model of cloud microphysics, with the evolution and motion of individual aerosol, cloud, and precipitation particles being explicitly considered. Assuming that particles are locally well mixed, particle collisions are regarded as a stochastic process. Each particle is characterized by its position and internal state, the latter of which is specified by variables known as attributes, such as size, mass, ratio of the ice crystal's minor axis to the major axis (hereafter called “aspect ratio”), velocity, and chemical composition.

Mixed-phase cloud microphysics are far more
complicated than those of liquid-phase clouds, with various ice crystal formation mechanisms, diffusional
growth by deposition/sublimation, diverse ice particle morphologies, ice melting and
shedding, riming and wet growth, aggregation,
spontaneous/collisional breakup of ice particles, and rime splintering at play

Two numerical scheme types exist for kinetic descriptions, namely bin schemes and particle-based schemes.

The development
of bin schemes started independently of bulk models in the 1950s

Particle-based cloud microphysics modeling is a new approach that has emerged since the mid-2000s

The essential difference between bin schemes and particle-based schemes lies
in the representation of particles. Bin schemes adopt an Eulerian approach and the particle distribution function is approximated using
a finite number of control volumes (histogram). The time evolution is solved using a finite volume method or a finite difference method.
In contrast, particle-based schemes rely on a Lagrangian approach
and the population of real particles is approximated by using a population
of weighted samples, sometimes referred to as super-droplets or
super-particles. As discussed in

Therefore, SDM and similar particle-based schemes should be more
suitable for mixed-phase cloud microphysics simulations than bin
schemes. Mainly because of computational costs, it is
practically impossible to apply bin schemes to the most comprehensive
form of kinetic description, which inevitably involves multiple
attributes to express each particle's internal state. Instead, many existing bin models solve a
simplified kinetic description that uses particle distribution
functions with a one-dimensional attribute space approximation. For
example, most rely on artificially separated categories
of ice particles, with predefined mass–dimension and area–dimension
relationships in each category. Another approach is adopted in the
SHIPS model developed by

This study's primary objective is to assess particle-based modeling methodology's capability to simulate mixed-phase clouds. Therefore, we develop and evaluate the performance of a detailed numerical mixed-phase cloud model using SDM, wherein ice particle morphologies are explicitly predicted.

We first construct a mixed-phase cloud microphysics mathematical model, which is based on kinetic description. The
fluid dynamics of moist air is described by the compressible
Navier–Stokes equation, and aerosol, cloud, and precipitation particles are
represented by point particles. Following

Several previous works are closely relevant to this
study.

In this study, we demonstrate that a large-eddy simulation of a cumulonimbus that predicts ice particle morphologies without assuming ice categories or mass–dimension relationships is possible if we use SDM.

The organization of the remainder of this paper is as follows. In Sects.

Let us represent aerosol, cloud, and precipitation particles as point
particles. The particle state is then characterized by two
types of variables: position

In this study, for simplicity, partially frozen/melted particles are not considered.
We assume that each particle completely freezes or melts
instantaneously (see Sects.

In the remainder of this section, we provide a detailed explanation of each attribute.

The amount of liquid water contained in a particle is expressed by the
volume-equivalent sphere's radius

Let

We only consider homogeneous freezing and condensation/immersion freezing in this study
because these are dominant in mixed-phase clouds

Based on the “singular hypothesis”

Each particle's

An INAS is a localized structure, such as lattice mismatches, cracks, and hydrophilic sites, on an insoluble substance's surface
that catalyzes ice formation at temperatures lower than a specific temperature. INAS
density

For mineral dust, biogenic substances, and soot, we can use the INAS density formulas of

It is possible that a single INAS does not appear until

There are various ice nucleation pathways

Ice particles have diverse morphologies such as columns, hexagonal
plates, dendrites, rimed crystals, graupel, hailstones, and
aggregates

Following

We approximate that each particle is always moving at its terminal velocity.
Therefore, a particle's velocity

In summary, particle attributes consist of

We only consider dry air and water vapor for the gas phase and ignore other trace gases.
In this section, we introduce several variables that describe
the state of moist air:
wind velocity

In this section, we describe our model's time evolution equations, first from cloud microphysics and then moist air fluid dynamics.
Our model is detailed; however, it still falls short in completely describing mixed-phase cloud microphysics.
To keep the model description concise,
discussions on the shortcomings and how to overcome them
are left for Sect.

Let us assign a unique index

Particle

If terminal velocity is reached,
the motion equation becomes

In this study, we assume that terminal velocity is always achieved instantaneously;
however, this is a simplification. For example, the relaxation time of large droplets is a few seconds

The next two subsections explain the formulas used to calculate droplet and ice particle terminal velocities.

To calculate droplet terminal velocity, we use the formula of

For ice particle terminal velocity, we use the formula of

In Böhm's theory,

However, in this study, we start from a slightly different definition of

Consequently, Eq. (

In our model,
we assume that ice particles are falling with their maximum dimension perpendicular to the flow direction. Therefore,
the circumcircle area becomes

Following

For

The

As explained in Sect.

We consider that the

We assume that the resulting ice crystal is spherical, with the true ice crystal density

When freezing occurs, each particle releases latent heat of fusion to
the moist air, as described in Eqs. (

When ambient temperature rises above

Following, e.g.,

The growth rate is identical to vapor flux at the droplet surface.
If the diffusion of vapor around the droplet is in a quasi-steady state, we obtain

If we further assume that thermal diffusion is also in a quasi-steady state,
and that surface temperature

The growth of a droplet by condensation/evaporation is governed by Eqs. (

Vapor and latent heat couplings to moist air through condensation and evaporation
are calculated by Eqs. (

The shapes of ice crystals formed by depositional growth exhibit strong dependencies on temperature
and, to a lesser extent, supersaturation

The mass growth rate can be derived similarly to Eqs. (

The exact form of capacitance

The coefficient

Note that

In Chen and Lamb's (

For purely diffusional growth,

The ventilation coefficient

The secondary growth habit is expressed by deposition density

For sublimation, the particle volume change

We can now calculate the attributes at time

From Eq. (

For simplicity, we assume that the rime mass fraction

Vapor and latent heat couplings to moist air through deposition and sublimation
are calculated by Eqs. (

In this section, we detailed the deposition and sublimation model used in
SCALE-SDM; however, there is significant room for improvement.
For example, as we will discuss in Sect.

Particle coalescence, riming, and aggregation can be considered a stochastic process. Following

In this study, we consider coalescence, riming, and aggregation induced by differential gravitational settling of particles because this mechanism is dominant in mixed-phase clouds.

First, we consider droplet coalescence, which accounts for the formation of rain droplets from cloud droplets (autoconversion), the collection of cloud droplets by rain droplets (accretion), and the coalescence of two rain droplets (self-collection).

The collision–coalescence kernel is given by

If coalescence takes place, droplets

Let us emphasize that the stochastic model introduced in this section describes the underlying mathematical model of the coalescence process, not the Monte Carlo algorithm of SDM that solves the stochastic process numerically. In the preceding paragraph, droplet

Riming usually refers to the collection of small supercooled droplets by a larger ice particle, but we also include the collection of small ice particles by a larger droplet. The latter case could be regarded as a type of contact freezing. However, ice particles grow preferentially when ice particles and supercooled droplets coexist (Wegener–Bergeron–Findeisen mechanism). Therefore, we can expect that the latter case happens less frequently in mixed-phase clouds.

Hereafter we assume, without loss of generality, that particle

Figure 1 of

To evaluate collision–riming collection efficiency

If

If

If riming takes place, the ice particle

If

If

Impact velocity can be calculated using the formula of

When riming occurs, the frozen droplet releases the latent heat of fusion to the moist air
as described in Eqs. (

As we will discuss in Sect.

Another issue discussed in Sect.

We validate these two corrections in Sect.

Finally, we consider the aggregation of ice particles.
Following

If aggregation takes place, ice particles

Snow aggregates have complicated fractal structures. However, if we circumscribe them using a spheroid, the growth by aggregation is in three dimensions, rather than one (columnar) or two (planar). Therefore, as in the case of riming, we assume that only the minor dimension grows by aggregation.

If the volume-weighted average density

In contrast, if

In the following, we derive equations describing how to update the attributes.

Without loss of generality, assume that

For

Note that our aggregation outcome model does not
produce particles lighter than

Equations (

Moist air fluid dynamics can be described by the compressible Navier–Stokes equation for moist air:

Now, we have the complete set of the system's time evolution equations:
Eqs. (

We develop a numerical model known as SCALE-SDM to solve the mathematical model of mixed-phase clouds presented in the preceding sections.

SCALE is a library of weather and climate models of the Earth and other planets (

In our model, we use SDM to solve cloud microphysics as defined by Eqs. (

Moist air fluid dynamics are solved using SCALE's dynamical core.
We solve the compressible Navier–Stokes equation for moist air (Eqs.

We consider
the density of moist air

There are many particles in the atmosphere; thus, it is
practically impossible to follow all of them in a numerical model.
However, it is reasonable to assume that only the collective properties of the particle population
are relevant to predict the behavior of clouds,
because clouds are insensitive to each individual particle.
Therefore, let us approximate
the population of real particles

The relationship between super-particles and real particles can be expressed more precisely as follows.
Let

There is an arbitrariness in how to initialize super-particles. In this study, we use the uniform sampling method.

Any probability density function

Let us consider a specific type of procedure wherein we assign

If we set

Instead, we can set

Overall, further investigation is required to determine
an optimal method for initializing super-particles.
In this study, we use the uniform sampling method given by Eq. (

We separately evaluate each process using the first-order operator splitting scheme. Let

Let

Let

These process time steps are all divisors of the common time step

We first calculate fluid dynamics without the coupling terms from particles to moist air (Eqs.

An example of the calculation order when updating the system state from

We use SDM to solve cloud microphysics.
We provide details of the numerical schemes used to calculate cloud microphysics in this section.
The state of ambient air

For each super-particle, the motion equation (Eq.

The reaction force acting on moist air is calculated using Eq. (

Every

For each super-droplet, we solve the condensation and evaporation equation (Eq.

The exchange of vapor and latent heat with moist air is calculated using
Eqs. (

The growth of droplets is calculated implicitly;
however, the evolution of supersaturation through feedback is calculated explicitly.
Therefore, the length of

For each ice super-particle, we solve the deposition and sublimation time evolution equations
detailed in Sect.

The exchange of vapor and latent heat with moist air is calculated using
Eqs. (

The stochastic process of coalescence, riming, and aggregation detailed in Sects.

Here, we relate the above argument to that of

In SDM, the multiple coalescence technique is used to make the algorithm robust to larger

The exchange of the latent heat of fusion due to riming is calculated using Eqs. (

Moist air fluid dynamics is governed by the compressible Navier–Stokes equation (Eqs.

We solve the compressible Navier–Stokes equation without the coupling terms using a finite volume method with an Arakawa-C staggered grid.
For spatial discretization, the fourth-order central difference scheme is used for advection terms
and the second-order central difference scheme is used for other spatial derivatives.
To preserve the monotonicity, we apply the flux-corrected transport scheme of

The time step

The preceding sections described the basic equations and numerical implementation of
SCALE-SDM. To evaluate our numerical model's performance, we conduct a 2-D simulation of an isolated cumulonimbus
following the setup of

Summary of numerical experiments for model evaluation. The domain is two-dimensional (

In this section, we specify the atmospheric conditions and numerical parameters used for the CTRL ensemble.

The domain is 2-D (

The initial atmospheric profile is horizontally uniform, and the
vertical moist air profile is given by sounding data from Midland, Texas, on 13 August 1999, as shown in Fig. 4 of

For the lateral boundaries, we impose periodic boundary conditions.
For the upper and lower boundaries, we set the vertical wind velocity

Initially, the particles are distributed uniformly in space at random, and consist of pure ammonium bisulfate aerosol particles and mineral dust internally mixed with ammonium bisulfate.

The initial number-size distribution of the population of pure ammonium bisulfate particles is given by a bimodal log-normal distribution,

The other aerosol population consists of mineral dust internally mixed with ammonium bisulfate. We set the number concentration to

We also impose periodic boundary conditions on particles for the lateral boundaries. If a particle crosses the upper or lower boundary, we remove that particle from the system.

Convective cloud development is triggered by a

We use a uniform grid throughout this study, with a grid size of

Initially, the super-particles are distributed uniformly throughout the domain at random
with a number concentration of

The multiplicity, ammonium bisulfate mass, and freezing temperature
of each pure ammonium bisulfate super-particle is assigned as follows.
For each pure ammonium bisulfate super-particle,
we draw a random number uniformly in log-space from the interval

For IN inactive mineral dust super-particles, we use

Finally, we consider IN active mineral dust internally mixed with ammonium bisulfate.
The remaining super-particles, i.e.,

Note that multiplicity

Assuming that all the particles are deliquescent,
we consider that the initial droplet radius

To evaluate the fluctuation, we conduct a

Now, the atmospheric conditions and numerical parameters used for the CTRL ensemble have all been specified.

We also try various other test cases by changing the CTRL ensemble's numerical parameters,
and assess the sensitivity of results to numerical parameters.
Our parameter selections are specified in the following sections and a summary is provided in Table

To investigate numerical convergence with respect to initial the super-particle number concentration

To investigate numerical convergence with respect to the grid size, we run ensembles using different grid sizes.

The grid size of the DXx4 ensemble is 4 times that of CTRL:

The DXx2 ensemble's grid size is twice that of CTRL:

Note that DXx1 and CTRL are the same.

The DX/2 ensemble has a grid size that is half that of CTRL:

To investigate numerical convergence with respect to the cloud microphysics time steps, we change the cloud microphysics time steps for CTRL without changing the time step for fluid dynamics.

The time steps for the DTx10 ensemble's cloud microphysics are 10 times that of CTRL:

The time steps of the DTx5 ensemble are 5 times that of CTRL:

The time steps of the DTx2 ensemble are twice that of CTRL:

Note that DTx1 and CTRL are the same.

The time steps of the DT/2 ensemble are half that of CTRL:

The time steps of the DT/4 ensemble are one-quarter that of CTRL:

From the

We do not categorize hydrometeors during the simulation, which is one of the salient features of our model because the artificial partitioning of hydrometeors could cause various artifacts. In contrast, when analyzing results, dividing hydrometeors into categories is useful to precisely understand the results.

In this study, we assume that hydrometeors completely freeze or melt instantaneously
(see Sect.

If a particle is a droplet and its radius

If a particle is an ice particle with a rimed mass fraction satisfying

Typical realization of CTRL cloud spatial structures at

We first analyze the cloud's overall properties, and then, in the next section, we analyze the properties of individual ice particles.

Figure

Figure

The cloud started to form at approximately

Our model successfully simulated the life cycle of a cumulonimbus typically observed in nature

Time evolution of the domain-averaged water path in the CTRL ensemble. The cloud water, rainwater,
cloud ice water, graupel water, and snow aggregate water paths
are plotted in gray, yellow, blue, red, and green, respectively.
The solid line represents the typical realization of CTRL.
Dark shades indicate the mean

Time evolution of domain-averaged accumulated precipitation amounts
in the CTRL ensemble.
The solid line represents the typical realization of CTRL.
The dark shading indicates the mean

Now, we analyze the properties of individual ice particles in the typical realization of CTRL.

Figure

Figure

Figure

Figure

Mass–dimension relationship of ice particles in the typical realization of CTRL at

Aspect ratio–dimension relationship of ice particles
in the typical realization of CTRL
at

Apparent density–dimension relationship of ice particles
in the typical realization of CTRL
at

Velocity–dimension relationship of ice particles
in the typical realization of CTRL
at

At

At

At

The mass–dimension relationship shown in Fig.

However, at the same time, we also see several types of seemingly unrealistic ice particles,
representative examples of which are indicated by symbols in Figs.

Our numerical model uses three types of numerical parameters,
namely the super-particle number concentration, grid size, and time steps. These parameters
correspond to the resolution of aerosol/cloud/precipitation particle
distribution in real space and attribute space, the spatial resolution
of moist air, and temporal resolution. The numerical solution from
our model approaches the true solution of time evolution
equations (Eqs.

To confirm the numerical convergence of the cumulonimbus case,
we conducted a series of simulations changing the numerical parameters of CTRL. These ensembles are referred to as NSP, DX, and DT (see Table

Numerical convergence regarding the initial super-particle number concentration

Figure

Figure

Our model has two sources of fluctuation, namely atmospheric turbulence and SDM randomness.
Pseudo-random numbers are used for the Monte Carlo calculation of coalescence, riming, and aggregation,
and to initialize the super-particles. The standard deviation (i.e., fluctuation) caused by SDM randomness
decreases proportionally to the inverse of the square root of the super-particle number.
However, Figs.

Figure

To summarize, we conclude
that the numerical convergence regarding the super-particle number
is fairly well achieved at NSP128 (CTRL), i.e.,

Statistics of NSP ensemble accumulated precipitation amounts.
The vertical axis represents the accumulated precipitation at the end of the simulation (

Statistics of NSP ensemble maximum water paths for each hydrometeor type. The vertical axis represents the maximum water path of each hydrometeor type during the simulation (i.e., the
maximum of each line in Fig.

We investigated the numerical convergence with respect to the grid size
by varying

Figure

Figure

The DX/2 ensemble is the highest grid resolution tested in this study, and
a snapshot of the cloud from the DX/2 ensemble is shown in Fig.

Figure

Therefore, we conclude that the numerical convergence with respect to the grid size is achieved at DXx1 (CTRL), i.e.,

Statistics of DX ensemble accumulated precipitation amounts.
The horizontal axis represents the grid size

Statistics of the DX ensemble maximum water paths for each hydrometeor type. The horizontal axis represents the grid size

Spatial structure of the cloud at

We investigated the numerical convergence with respect to the cloud microphysics time steps
by varying CTRL's cloud microphysics time steps by factors of

We found that DTx10 diverges at around

Figure

Figure

Both figures show no significant difference among the five ensembles;
therefore, we conclude that the numerical convergence with respect to the time steps
is already attained at DTx1 (CTRL), i.e.,

Further discussion of numerical convergence characteristics is provided in Sect.

Statistics of DT ensemble accumulated precipitation amounts.
The horizontal axis represents the ratio of cloud microphysics time steps to CTRL.
This figure has the same form as Fig.

Statistics of the DT ensemble maximum water path for each hydrometeor type. The horizontal axis represents the ratio of cloud microphysics time steps to CTRL. This figure has the same form as Fig.

As confirmed in the preceding sections,
the numerical parameters used for the CTRL ensemble (see Table

The CTRL ensemble's super-particle number concentration is

The CTRL ensemble's grid size is

The time steps for cloud microphysical processes used in the CTRL ensemble
are

To accurately trace the flow of moist air,

To avoid a sudden release of latent heat,

We used the first-order operator splitting scheme to separate the calculation (Table

Lastly, we discuss SCALE-SDM's actual computational cost.
Calculating one realization of the CTRL case required approximately

In SCALE-SDM, super-particles are distributed all over the simulated domain.
If we use super-particles only inside the clouds by employing, e.g.,
the Twomey super-droplet methodology

Results of the typical realization of CTRL presented in Sect.

Let us determine the origins of the four types of odd particles
denoted by the symbols in Figs.

The ice particle denoted by the circle at

This odd particle was caused by a problem with the riming density formula (Eqs.

However,

In Sect.

Additionally, the same problem occurs if a quasi-spherical planar graupel particle
and a slightly smaller rain droplet collide and rime near the freezing level.
However, it is less evident than with the previous case because
the equatorial radius grows as the square root of the volume (Eq.

The square at

As in the previous case, we found that a single riming event
between a cloud ice particle and a cloud droplet followed by depositional growth created this columnar ice particle type.
We can explain the mechanism as follows: consider a quasi-spherical columnar ice particle with a radius of

Contrary to the previous case,
the low riming density is reasonable. Instead,
we must reconsider the filling-in model.
We assumed that the ice particle's maximum dimension is preserved.
However, this is not realistic for riming between an ice particle and a similarly sized droplet,
as our thought experiment revealed.
Generalizing the idea, we consider that the frozen droplet's diameter
is preserved if the diameter is larger than the ice particle's maximum dimension.
That is, we propose to replace Eq. (

In Sect.

The cross at

Lump graupel particles with apparent densities as low as

The triangle at

The particle is created by a sublimation of a graupel particle.
The inherent growth ratio

However,

In Sect.

In the preceding sections, we proposed three corrections to the time evolution equations (Eqs.

We incorporated the proposed corrections into our model to create a
new revision, SCALE-SDM 0.2.5-2.2.1. To assess the validity of these
corrections, we conducted the same simulations as the typical
realization of CTRL using the new model. By comparing these results
(Figs.

This figure is the same as Fig.

This figure is the same as Fig.

This figure is the same as Fig.

This figure is the same as Fig.

Changes in the domain-averaged water path before and after corrections. The long dashed, solid, and short dashed lines represent the SCALE-SDM 0.2.5-2.2.0, -2.2.1, and -2.2.2, respectively.

Changes in accumulated precipitation amounts before and after corrections. The long dashed, solid, and short dashed lines represent the SCALE-SDM 0.2.5-2.2.0, -2.2.1, and -2.2.2, respectively.

As explained in Sect.

Noting that the area ratio

In our model (SCALE-SDM 0.2.5-2.2.0/2.2.1), we assumed that the characteristic length

For planar ice particles (

Therefore, the correction (Eq.

We incorporated the corrections (Eqs.

Our model is based on a kinetic description,
i.e., individual dynamics of particles and their stochastic collisions.
However, a quantitative understanding of mixed-phase cloud microphysics
is a long-standing meteorological issue, and a kinetic description of mixed-phase cloud microphysics
has not been established. Further, our model does not incorporate several elementary processes that are critical for mixed-phase clouds.
In this section, we explore the possibilities of further refining and sophisticating our model. Readers can also refer to

There are various ice nucleation pathways

Based on the singular hypothesis

The singular hypothesis ignores the time dependence of ice nucleation; thus,
we assumed that particles initiate freezing immediately after
the temperature drops below

Note that our requirement that the ambient water vapor must be supersaturated over liquid water would be too restrictive for immersion freezing. Even under an unsaturated condition, it is reasonable to allow
immersion freezing if the droplet is sufficiently large, for instance, larger than

To express homogeneous freezing, we assigned a fixed freezing temperature of

Condensation/immersion freezing of deliquescent IN particles can also be incorporated
by considering the depression of the freezing temperature

The formation of ice directly from the vapor phase onto an IN particle is known as
deposition freezing. This can be observed at

A crude model of pre-activation is incorporated in our model by inhibiting complete sublimation
(see Eq.

Contact freezing is another ice nucleation mechanism in which
solid particles can initiate freezing
upon contacting the surface of a supercooled droplet.
Contact freezing occurs at temperatures greater than that of
the same particle immersed in a droplet

It is also known that the evaporation of a droplet could lead to
inside-out contact freezing

We assumed that ice particles start melting immediately after
the ambient temperature reaches

After the onset of freezing or melting, we assumed that complete freezing/melting occurs instantaneously.

However, as shown in

We also assumed that rimed supercooled droplets freeze instantaneously; however, wet growth of graupel particles is critical to accurately predict hailstone formation.
We can use the model from

Depending on the relative humidity and warming rate, the melting time of spherical ice particles with radii of approximately

Additionally, to complete the model, all other time evolution equations must be extended to make them compatible with partially frozen/melted particles, which would require some effort.

In SCALE-SDM, we assumed that water vapor's diffusivity
in air and moist air's thermal conductivity in Eq. (

We considered the ventilation effect for deposition and sublimation
but not for condensation and evaporation, even though
it also enhances the growth and evaporation of larger droplets.
We can include this effect by using the model described in
Sect. 13.2.3 of

For cloud droplets, we must also consider kinetic correction to

In our model, aerosol particle hygroscopicity is expressed by
Raoult's law with the van 't Hoff factor

There are many issues around

Further, as shown by

We used

In our model, each ice particle is approximated by a porous spheroid

As with condensation and evaporation, we assumed that water vapor's diffusivity
in air

For the collision efficiency of collision–coalescence

We assumed that the coalescence efficiency is unity,

For the collection efficiency of collision–riming

When a large droplet collects ice particles, we used the original formula from

When an ice particle collects a droplet,
we employed the filling-in model and
preserved the ice particle's maximum dimension.
However, if the collector is a snow aggregate,
we should use the similarity model proposed by

We assumed that collision–aggregation's collection efficiency is given by a constant

Calculating the attributes of the resultant ice particles is also not easy.
Let

However, the filling-in assumption is not valid for aggregation.

Another issue of the filling-in assumption is that
it gradually makes snow aggregates quasi-spherical
(see the green shading in Figs.

Introducing the separation ratio

In our model, the apparent density

Several numerical models can create
detailed 3-D structures of snow aggregates consisting of primary ice crystals

Several mechanisms can induce the spontaneous/collisional breakup of hydrometeors. However,
we did not consider any of them in the present study.
In particular, rime splintering

First, a particle-based numerical algorithm for calculating spontaneous/collisional breakup processes has not yet been established. A simple strategy is to add more super-particles to the system when a breakup event occurs, but this could be computationally inefficient.

Mathematical models of spontaneous/collisional breakup processes are available from various studies. For the spontaneous breakup of rain droplets

The grid size we tested for evaluating the model
ranged from

The Smagorinsky–Lilly model

SGS turbulence can enhance particle collision,
which can be incorporated by using the collision kernels proposed
in

Using SDM, we constructed a detailed numerical model of mixed-phase clouds based on a kinetic description, and subsequently demonstrated that a large-eddy simulation of a cumulonimbus that predicts ice particle morphology without assuming ice categories or mass–dimension relationships is possible. Our results strongly support the particle-based modeling methodology's efficacy for simulating mixed-phase clouds.

In our model, ice particles are approximated by porous spheroids. The elementary cloud microphysics processes that the model considers include advection and sedimentation; immersion/condensation and homogeneous freezing; melting; condensation and evaporation including the activation and deactivation of CCN; deposition and sublimation; and coalescence, riming, and aggregation. Moist air fluid dynamics is described using the compressible Navier–Stokes equation.

Our model successfully simulated the life cycle of a cumulonimbus, and
the predicted mass–dimension and velocity–dimension relationships were comparable with existing formulas.
Numerical convergence was achieved at a super-particle number
concentration as low as

A more detailed evaluation of the model to explore the applicability of
the new approach is an essential step forward. Our results strongly indicate
that ice particle morphology can be predicted more accurately
by further developing particle-based models. However, from
this study, we cannot quantify the extent to which the
refined representation of mixed-phase cloud microphysics
could improve the predictability of mixed-phase clouds' macroscopic properties.
Such proficiency can be addressed by conducting a thorough comparison with
observations and other models.
In addition, further sophistication of the model is necessary.
As discussed in Sect.

Particle-based model accuracy is more subject to cloud microphysics uncertainties than numerical errors.
Therefore, a quantitative understanding of elementary cloud microphysics processes is becoming increasingly important.
More laboratory, observational, and theoretical studies to advance our knowledge of cloud microphysics are desired in the future

Our model's computational cost is at least 1 or 2 orders of magnitude larger than that of bulk models.
To further accelerate calculation, the use of SGS models discussed in Sect.

Table

List of symbols.

Continued.

Continued.

Table

List of abbreviations.

The source code of SCALE-SDM 0.2.5-2.2.0, -2.2.1, and -2.2.2 are available from

The supplement related to this article is available online
at:

All the authors designed the model and numerical experiments. SS developed the model code and performed the simulations. SS prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflicts of interest.

Shin-ichiro Shima is grateful to Wojciech W. Grabowski, Sylwester Arabas, Dennis Niedermeier, Miklós Szakáll, Will H. Cantrell, Yutaka Tobo, and Hugh Morrison for informative discussions. Shin-ichiro Shima would like to thank Koichi Hasegawa for implementing the prototype model. The authors would like to thank the two anonymous referees for their insightful and constructive comments.

This research partly used the computational resources of
the K computer provided by the RIKEN Center for Computational Science (R-CCS)
through the HPCI System Research Project (project ID: hp150153),
and FX10 provided by Kyushu University through
the HPCI System Research Project (project IDs: hp140094, hp160132).
This work was supported by
the Center for Cooperative Work on Computational Science, University of Hyogo;
and the Department of HPC Support, Research Organization for
Information Science & Technology (RIST) under the Optimization Support
Program of the HPCI system. The SCALE library was developed by Team-SCALE of RIKEN Center for Computational Sciences (

This research has been supported by JSPS KAKENHI (grant nos. 26286089, 20H00225), MEXT KAKENHI (grant no. 18H04448), and the Computing Resources for Enhancement (project IDs: ra000006 and ra001010), the joint research program of the Institute for Space-Earth Environmental Research, Nagoya University.

This paper was edited by Simon Unterstrasser and reviewed by two anonymous referees.