Statistical properties are investigated for the stochastic model of eddy hopping, which is
a novel cloud microphysical model that accounts for the effect of the supersaturation fluctuation at unresolved scales on the growth of cloud droplets and on spectral broadening.
Two versions of the model, the original version by

The purpose of the present paper is
to investigate the statistical properties of the stochastic model of eddy hopping proposed by

It should be noted that the turbulent entrainment mixing
is another important mechanism for the supersaturation fluctuation generation other than the stochastic condensation
and that the effects of the turbulent entrainment mixing are not included
in the eddy-hopping model considered in the present study.

In the present paper,
we take a rather theoretical approach to obtain various statistical properties of the eddy-hopping model,
such as the variance, covariance, and auto-correlation function
of the supersaturation fluctuation.
These statistical properties are used to validate the model
against the reference data taken from direct numerical simulations (DNSs) and large-eddy simulations (LESs).
We show that the original version of the eddy-hopping model fails to reproduce a proper scaling for a certain range of parameters,
resulting in the deviation of the model prediction from the reference data,
while the second version successfully reproduces the proper scaling.
We show how the relaxation term introduced by

The remainder of the present paper is organized as follows.
Section

The original version of the eddy-hopping model proposed by

Equation (

We now obtain the analytical expression
for the standard deviation of the supersaturation fluctuation,

First, multiplying Eq. (

Here,

Standard deviation of the supersaturation fluctuation

Figure

Note that Fig.

The original version of the eddy-hopping model given by Eqs. (

We next consider the second version of the eddy-hopping model by

The important change introduced by

Note that it is possible to further extend the second version
by additionally introducing the Wiener process term representing small-scale fluctuations or mixing
into Eq. (

Applying the analytical procedure described in Sect.

Asymptotic forms of

Standard deviation of the supersaturation fluctuation

Figure

Standard deviation of the supersaturation fluctuation

Although improved,
the second version still slightly over- and underestimates the supersaturation fluctuations
for

Here, we do not consider any physical meaning for

Finally, we discuss the possibility of simplification of the eddy-hopping model.
Here, our discussion is based on the second version
given by Eqs. (

Auto-correlation functions in the statistically steady state
for the simplified model (

The eddy-hopping model consists of two evolution equations for
the supersaturation and vertical velocity fluctuations,

Time evolutions of the supersaturation fluctuations
obtained from the numerical integration of the original version (Eqs.

Based on analytical expressions for the fluctuation amplitude and the auto-correlation function for

Figures

The simplified model has the desirable convergence property.
The auto-correlation function for the simplified model (

Figure

For small scales,
the simplified model produces qualitatively different trajectories of

The purpose of the present paper was to obtain various statistical properties of the eddy-hopping model, a novel cloud microphysical model,
which accounts for the effect of the supersaturation fluctuation at unresolved scales on the growth of cloud droplets and on spectral broadening.
Two versions of the model are considered: the original version by

The results of the numerical integration of
the original version (Eqs.

We consider classical homogeneous isotropic turbulence,
in which energy is mainly injected into the system at large scales,
cascaded to smaller scales by nonlinear interaction,
and finally dissipated by the molecular viscosity in the smallest scales.
In a statistically steady state,
the dissipation rate of turbulent kinetic energy is defined as

Relationship between the integral scale

We confirm that all of the results of the numerical integration of the eddy-hopping model in the present study achieved statistically steady states. For this purpose, we first derive the analytical expression for the time evolutions of the variance and covariance of the variables in the model and then compare these analytical expressions with the results of the numerical integration.

The governing equations given by Eqs. (

Standard deviation of the supersaturation fluctuation

Figure

We derive the analytical expression
for the auto-correlation function of the supersaturation fluctuation

First, multiplying Eq. (

Programs for numerical integration of the eddy-hopping model are written in Fortran 90 and are available upon request.

The data in this study are available upon request.

IS conducted the numerical simulations and data analysis.
IS and TW performed the theoretical analyses in Sect.

The authors declare that they have no conflicts of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are grateful to Kei Nakajima for his technical support. The present study was supported by MEXT KAKENHI grant nos. 20H00225 and 20H02066, by JSPS KAKENHI grant no. 18K03925, by the Naito Foundation, by the HPCI System Research Project (project ID: hp200072, hp210056), by the NIFS Collaboration Research Program (NIFS20KNSS143), by the Japan High Performance Computing and Networking plus Large-scale Data Analyzing and Information Systems (jh200006, jh210014), and by High Performance Computing (HPC 2020, HPC2021) at Nagoya University.

This paper was edited by Markus Petters and reviewed by three anonymous referees.