This work discusses the numerical aspects of representing the condensational growth of particles in models of aerosol systems such as atmospheric clouds. It focuses on the Eulerian modelling approach, in which fixed-bin discretisation is used for the probability density function describing the particle-size spectrum. Numerical diffusion is inherent to the employment of the fixed-bin discretisation for solving the arising transport problem (advection equation describing size spectrum evolution). The focus of this work is on a technique for reducing the numerical diffusion in solutions based on the upwind scheme: the multidimensional positive definite advection transport algorithm (MPDATA). Several MPDATA variants are explored including infinite-gauge, non-oscillatory, third-order terms and recursive antidiffusive correction (double-pass donor cell, DPDC) options. Methodologies for handling coordinate transformations associated with both particle-size spectrum coordinate choice and with numerical grid layout choice are expounded. Analysis of the performance of the scheme for different discretisation parameters and different settings of the algorithm is performed using (i) an analytically solvable box-model test case and (ii) the single-column kinematic driver (“KiD”) test case in which the size-spectral advection due to condensation is solved simultaneously with the advection in the vertical spatial coordinate, and in which the supersaturation evolution is coupled with the droplet growth through water mass budget. The box-model problem covers size-spectral dynamics only; no spatial dimension is considered. The single-column test case involves a numerical solution of a two-dimensional advection problem (spectral and spatial dimensions). The discussion presented in the paper covers size-spectral, spatial and temporal convergence as well as computational cost, conservativeness and quantification of the numerical broadening of the particle-size spectrum. The box-model simulations demonstrate that, compared with upwind solutions, even a 10-fold decrease in the spurious numerical spectral broadening can be obtained by an apt choice of the MPDATA variant (maintaining the same spatial and temporal resolution), yet at an increased computational cost. Analyses using the single-column test case reveal that the width of the droplet size spectrum is affected by numerical diffusion pertinent to both spatial and spectral advection. Application of even a single corrective iteration of MPDATA robustly decreases the relative dispersion of the droplet spectrum, roughly by a factor of 2 at the levels of maximal liquid water content. Presented simulations are carried out using PyMPDATA – a new open-source Python implementation of MPDATA based on the Numba just-in-time compilation infrastructure.

The focus of this paper is on the problem of particle-size evolution for a population of droplets undergoing condensational growth.
Representing the particle-size spectrum using a number density function, the problem can be stated as a population-balance equation expressing conservation of the number of particles.
Herein, the numerical solution of the problem using the MPDATA family of finite difference schemes originating in

MPDATA features a variety of options allowing an algorithm variant that is appropriate to the problem at hand to be picked.
This work highlights the importance of the MPDATA algorithm variant choice for the resultant spectral broadening of the particle-size spectrum.
The term spectral broadening refers to the increase in width of the
droplet spectrum during the lifetime of a cloud.
The broadening may be associated with both physical mechanisms (including turbulent mixing, particle composition diversity, radiative heat transfer effects; see for example

Cloud simulations with a detailed treatment of droplet microphysics face a twofold challenge in resolving the droplet spectrum width.
First, it is challenging to model and numerically represent the subtleties of condensational growth which link the physico-chemical properties of single particles with ambient thermodynamics through latent heat release and multi-particle competition for available vapour (e.g.

The width of the spectrum plays a key role in
determining both the droplet collision probabilities

The following introductory subsections start with a chronologically presented literature review of applications of MPDATA to the problem of condensational growth of particles.
Section

Section

Section

All presented simulations are performed with PyMPDATA

There exist two contrasting approaches for modelling the evolution of droplet-size spectrum (see

Following

An early discussion of numerical broadening of the cloud droplet spectrum can be found in

In

The first mention of an application of the MPDATA scheme for the problem of condensational growth can be found in

In

The “Aerosol Science: Theory and Practice” book of

In

In

In

The discussion presented in

It is worth noting that none of the works mentioned above discussed coordinate transformations to non-linear grid layouts with MPDATA (a discussion of handling non-uniform mesh with the upwind scheme can be found in

To describe the conservation of particle number

The coordinate transformation term

Combination of the two transformations yields the following definition of

The numerical solution of Eq. (

The test case is based on Fig. 3 from

For the initial number density function, an idealised fair-weather cumulus droplet size spectrum is modelled with a lognormal distribution:

For the boundary conditions (implemented using halo grid cells), linear extrapolation is applied to

Analytical solution to Eq. (

The upper panels in Figs.

Two grid layout (

The times for which results are depicted in the plots are selected by finding

The normalised mass density of bin

Looking at the mass density plots in Figs.

One of the methods used to quantify the numerical diffusion of a numerical scheme is the modified equation analysis of

To depict an application of the modified equation analysis in the present context (upwind scheme), a simplified setting where

The above analysis depicts that the employment of the numerical scheme (Eq.

Evolution of the particle number density

As in Fig.

The problem of numerical diffusion can be addressed by introducing the so-called “antidiffusive velocity”

In

Comparison of analytical, upwind and MPDATA solutions (see plot key for algorithm variant specification) using the set-up from Fig.

Accordingly, the basic antidiffusive field

In Fig.

For the possible improvement of the algorithm, one may consider linearising MPDATA about an arbitrarily large constant (i.e. taking

Such gauge transformation changes the corrective iterations of the basic algorithm as follows (replacing Eqs.

Note that the amplitude of the diffusive flux (Eq.

Comparison of analytical, upwind and MPDATA solutions (see plot key for algorithm variant specification) using the set-up from Fig.

Figure

In

The non-oscillatory option (later referred to as “non-osc” herein) modifies the algorithm as follows:

Comparison of analytical, upwind and MPDATA solutions (see plot key for algorithm variant specification) using the set-up from Fig.

Figure

An alternative to the iterative application of the antidiffusive velocities was introduced in

An example simulation combining the double-pass (DPDC), the non-oscillatory and infinite-gauge variants is presented in Fig.

For divergent flows (hereinafter abbreviated dfl), the modified equation analysis yields an additional correction term in the antidiffusive velocity formula (see

In simulations using the presented set-up (also for

Another possible improvement to the algorithm comes from the inclusion of the third-order terms in the modified equation analysis, which leads to the following form of the antidiffusive velocity

Figure

It is worth noting that discussion of higher-order variants of MPDATA was carried forward in

The MPDATA variants presented in the preceding sections can be combined together.
In Fig.

In the following subsections, the influence of MPDATA algorithm variant choice on the resultant spectrum width and on the computational cost is analysed using the example simulation set-up used above (i.e. in all figures except Fig.

The relative dispersion, defined as the ratio of standard deviation

The calculated dispersion ratio over all bins takes the following form:

Relative dispersion of the discretised (using grid set-up as in Fig.

Table

Due to the formulation of the problem as number conservation, and discretisation of the evolution equation using fixed bins, even though the numerical scheme is conservative (up to subtle limitations outlined below), evaluation of other statistical moments of the evolved spectrum from the number density introduces an inherent discrepancy from the analytical results (for a discussion on multi-moment formulation of the problem, see

Panel

In order to quantify the discrepancy in the total mass between the discretised analytical solution and the numerically integrated spectrum, the following ratio is defined using the moment evaluation formula (Eq.

The consequences of mass conservation inaccuracies in the fixed-bin particle-size spectrum representation may not be as severe as in, for example, a dynamical core responsible for the transport of conserved scalar fields. The outlined discrepancies may be dealt with by calculating the change in mass during a time step from condensation, then using it in vapour and latent heat budget calculations so the total mass and energy in the modelled system are conserved.

The embraced algorithm (Eqs.

The total number of particles in the system may diverge from the analytical expected value even for the initial condition depending on the employed discretisation approach.
In the present work, the probability density function is sampled at cell centres effectively assuming piecewise-constant number density function.
An alternative approach is to discretise the initial probabilities by assigning to

Table

The table includes, where available, analogous figures reported in earlier studies on MPDATA (see caption for comments on the dimensionality of the employed cases, as it differs and thus does not warrant direct comparison).
Among notable traits is the decrease in computational cost when enabling the infinite gauge option that is associated with a reduced number of terms in both the flux function as well as in the antidiffusive velocity formulation (see Sect. 2.5 in

Wall times normalised with respect to the upwind solution compared to data reported in four earlier works: S83 denotes

Although the discussed problem is one-dimensional, a computationally efficient and an accurate solution is essential, as it typically needs to be solved at every time step and grid point of a three-dimensional cloud model. While the reported upwind-normalised wall times give a rough estimation of the cost increase associated with a particular MPDATA option, the actual footprint on a complex simulation system will depend on numerous implementation details including parallelisation strategy.

In multidimensional simulations in which the considered particle number density field is not only a function of time and particle size, but also of spatial coordinates, there are several additional points to consider applying MPDATA to the problem.

First, in the context of atmospheric cloud simulations, owing to the stratification of the atmosphere, the usual practice is to reformulate the conservation problem in terms of specific number concentration being defined as the number of particles

Second, even with a single spatial dimension (single-column set-up), the
coupled size-spectral–spatial advection problem is two-dimensional.
This is where the inherent multidimensionality of MPDATA
(the “M” in MPDATA) requires further attention.
The one-dimensional antidiffusive formulæ discussed in Sect.

Third, in any practical application where the drop size evolution is
coupled with the water vapour budget (and hence with supersaturation evolution), it is essential to evaluate the total change in mass of liquid water due to condensation which is then to be used
to define the source term of the water vapour field (and in latent heat budget representation).
It is worth noting that knowing the difference of values at

Several recent papers are highlighting the need for scrutiny when
comes to the interplay of size-spectral and spatial advection and the
associated numerical broadening

Snapshots of the advected two-dimensional liquid water field at

The test set-up is based on the single-column KiD warm case introduced in

Single-column simulations depicted with three selected variables: liquid water mixing ratio

Profiles of relative dispersion

The simulated

a constant-in-time piecewise-linear potential temperature profile (

constant-in-time hydrostatic pressure and density profiles computed assuming surface pressure of

a piece-wise linear initial vapour mixing ratio profile (

a constant-in-space but time-dependent vertical momentum profile defining the vertical component of the advector field

The advection is thus solved for two scalar fields:
(i) a one-dimensional water vapour mixing ratio field
representing the vertical distribution of mass of vapour per mass of dry air
and
(ii) a two-dimensional field representing vertical and spectral variability of liquid particle specific concentration (number of particles per mass of dry air). The spectral coordinate is set to particle radius (

The initial condition does not feature supersaturation anywhere in the domain.
The upward advection of water vapour causes supersaturation to occur and trigger
condensation.
The size-spectral velocity is defined as in the box-model test case (cf. Eq.

The domain is initially void of liquid water and the only source of it is
through the boundary condition in the spectral dimension specified as follows:

The simulations cover a time period of

Figure

In Fig.

Notwithstanding the highly idealised and simplified modelling framework employed herein, one may attempt a comparison with profiles obtained from both in situ aircraft measurements (

Interestingly, the parabolic vertical profile of the relative dispersion obtained herein was also reported in

The liquid water profiles depicted in the top row of Fig.

There is a cloud-top activation feature hinted in all three panels in Fig.

The bottom row in Fig.

To provide insight into the sensitivity of the results to temporal, spatial and spectral resolution, Fig.

The dependence on the temporal resolution, as gauged by comparing the base
resolution case with cases in which the time step is halved (

The dependence on the spectral resolution is clearly manifested at the lowest spectral resolution where the minimum spectral dispersion

The spatial resolution setting

The study focuses on the MPDATA family of numerical schemes and its application to the size-spectral as well as spatio-spectral transport problem arising in models of condensational growth of cloud droplets. MPDATA iteratively applies the upwind algorithm, first with the physical velocity and subsequently using antidiffusive velocities. As a result, the algorithm is characterised by reduced numerical diffusion compared with upwind solutions, while maintaining conservativeness and positive-definiteness.

In literature, the derivations of different MPDATA variants are spread across numerous research papers published across almost four decades, and in most cases focused on multidimensional hydrodynamics applications. It is the aim of this study to highlight the developments that followed the original formulation of the algorithm, and to highlight their applicability to the problem of bin microphysics. To this end, it was shown that the combination of such features of MPDATA as the infinite-gauge, non-oscillatory and third-order-term options, together with the application of multiple corrective iterations offer a robust scheme that grossly outperforms the almost quadragenarian basic MPDATA.

In the case of the single-column test case, discussed simulations feature coupling between droplet growth and supersaturation evolution.
The embraced measure of spectrum width, the cloud droplet spectrum relative dispersion, is influenced by numerical diffusion pertinent to both spectral and vertical advection.
Focusing on the levels corresponding to the region of maximal liquid water content (ca. between

To assess the spatial and temporal convergence of the numerical solutions presented above, a convergence test originating from

As a side note, it is worth pointing out that for the chosen coordinates

To explore the convergence, the error measures are computed for seven different linearly spaced values of

As proposed in

Convergence plot for the upwind scheme (cf. Fig.

Convergence plot for basic two-pass MPDATA (cf. Fig.

Convergence plot for the infinite gauge MPDATA (cf. Fig.

Convergence plot for the infinite gauge non-oscillatory variant of MPDATA (cf. Fig.

Convergence plot for the DPDC variant with infinite gauge and non-oscillatory corrections (cf. Fig.

Convergence plot for the three-pass MPDATA (cf. Fig.

Convergence plot for the three-pass MPDATA with third-order terms (cf. Fig.

Convergence plot for the three-pass infinite gauge non-oscillatory MPDATA with third-order term corrections (cf. Fig.

Figures

The chosen colour increments correspond to the error reduction by a factor of 2, and the warmer the colour, the larger the error.
The small grey points behind the isolines represent points for which the error value was calculated.
When moving along the lines of constant Courant number, increasing the space and time discretisation, the number of crossed dashed isolines indicate the order of convergence.
For the considered problem, it can be seen that the upwind scheme (Fig.

Moreover, the shape of the dashed isolines tells the dependency of the solution accuracy on the Courant number. When these are isotropic (truncation error being independent of polar angle), the solution is independent of the Courant number.

It is worth noting that in Figs.

The convergence test results for the three-pass MPDATA with infinite gauge, non-oscillatory and third-order terms options enabled (Fig.

Calculations presented in the paper were performed using Python with a new open-source implementation of MPDATA: PyMPDATA

No data sets were used in this article.

The idea of the study originated in discussions between SA, SU and MAO. MAO led the work, and a preliminary version of a significant part of the presented material constituted his MSc thesis prepared under the mentorship of SA. PB architected the key components of the PyMPDATA package. JB contributed the DPDC variant of MPDATA to PyMPDATA. MB participated in composing the paper and devising the result analysis workflow. All authors contributed to the final form of the text.

At least one of the (co-)authors is a member of the editorial board of

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Comments from Wojciech Grabowski, Adrian Hill, Hugh Morrison, Andrzej Odrzywołek, Piotr Smolarkiewicz and Maciej Waruszewski as well as paper reviews by Josef Schröttle and three anonymous reviewers helped to improve the article.

The project was carried out within the POWROTY/REINTEGRATION programme of the Foundation for Polish Science (

This paper was edited by Juan Antonio Añel and reviewed by Josef Schröttle and three anonymous referees.