We report the development of a novel Lagrangian microphysics methodology for simulations of warm ice-free clouds. The approach applies the traditional Eulerian method for the momentum and continuous thermodynamic fields such as the temperature and water vapor mixing ratio, and uses Lagrangian “super-droplets” to represent condensed phase such as cloud droplets and drizzle or rain drops. In other applications of the Lagrangian warm-rain microphysics, the super-droplets outside clouds represent unactivated cloud condensation nuclei (CCN) that become activated upon entering a cloud and can further grow through diffusional and collisional processes. The original methodology allows for the detailed study of not only effects of CCN on cloud microphysics and dynamics, but also CCN processing by a cloud. However, when cloud processing is not of interest, a simpler and computationally more efficient approach can be used with super-droplets forming only when CCN is activated and no super-droplet existing outside a cloud. This is possible by applying the Twomey activation scheme where the local supersaturation dictates the concentration of cloud droplets that need to be present inside a cloudy volume, as typically used in Eulerian bin microphysics schemes. Since a cloud volume is a small fraction of the computational domain volume, the Twomey super-droplets provide significant computational advantage when compared to the original super-droplet methodology. Additional advantage comes from significantly longer time steps that can be used when modeling of CCN deliquescence is avoided. Moreover, other formulation of the droplet activation can be applied in case of low vertical resolution of the host model, for instance, linking the concentration of activated cloud droplets to the local updraft speed.

This paper discusses the development and testing of the Twomey super-droplet methodology, focusing on the activation and diffusional growth. Details of the activation implementation, transport of super-droplets in the physical space, and the coupling between super-droplets and the Eulerian temperature and water vapor field are discussed in detail. Some of these are relevant to the original super-droplet methodology as well and to the ice phase modeling using the Lagrangian approach. As a computational example, the scheme is applied to an idealized moist thermal rising in a stratified environment, with the original super-droplet methodology providing a benchmark to which the new scheme is compared.

Traditional cloud modeling methodologies apply a continuous medium approach for
all thermodynamic variables, that is, not only for the temperature and water
vapor, but also for all forms of cloud condensate and precipitation. Such
methodologies have been the workhorse of the cloud-scale modeling from its
early days

The last couple of decades witnessed an increased interest in cloud-scale
computational approaches that limit the abovementioned problems and attempt
to better represent the truly multiphase nature of clouds. Among those, the
particle-based Lagrangian method, referred to as the Lagrangian cloud model

This paper discusses the development and testing of a novel Lagrangian
approach focusing on activation and diffusional growth of cloud
droplets. Our motivation is to use this methodology to study the
impact of turbulence and entrainment on the spectrum of cloud
droplets in shallow warm boundary layer clouds, such as tropical
or subtropical cumulus and subtropical stratocumulus

The next section presents analytic formulation of the Twomey super-droplet scheme and discusses its implementation in the Eulerian fluid flow model. The specific aspects discussed in detail are the treatment of the activation on the finite-difference fluid flow model grid, transport of super-droplets across the Eulerian grid, and coupling between the super-droplets and Eulerian thermodynamics. Section 3 presents examples of model simulations where the Lagrangian thermodynamics is included in an anelastic small-scale fluid flow model and applied in moist rising thermal simulations. A traditional super-droplet scheme (i.e., following CCN particles and allowing their activation and growth of resulting cloud droplets) is used to show consistency between the two methods. Brief conclusions and the outlook are presented in Sect. 4.

Model equations describe evolution in space and time of the potential
temperature, water vapor mixing ratio, and a set of Lagrangian point
particles representing activated cloud droplets. The potential temperature

Considering typical cloud droplet concentrations in natural clouds, from
several tens to a few thousands per cubic centimeter, it is computationally
impossible to follow all cloud droplets in the entire volume of even a very
small cloud. Thus, the Lagrangian methodology involves following only a
selected (typically relatively small) subset of cloud droplets, referred to
as super-droplets following

As will be discussed in Sect. 3, the novel super-droplet scheme has been
included in the finite-difference anelastic model EULAG and its simplified
version referred to as babyEULAG. EULAG and babyEULAG apply
nonoscillatory-forward-in-time (NFT) integration scheme

The key element of the scheme presented here that makes it distinct from the
approach used in

The same approach can be used with super-droplets as already applied in GA17
in adiabatic parcel simulations. The key idea is that super-droplets are
created in supersaturated conditions when the local concentration of
activated droplets as given by the Twomey relationship is smaller than the
one dictated by the local supersaturation. When a complete evaporation of
cloud droplets occurs in subsaturated conditions, super-droplets are simply
removed from the super-droplet list. Hence, no super-droplets exist outside
of cloudy volumes, similarly to traditional Eulerian bin microphysics
schemes. It follows that super-droplets with Twomey activation provide
significant computational advantage over the traditional Lagrangian approach
because only a relatively small number of super-droplets has to be used. Note
that in the Eulerian bin scheme the computational expense of the droplet
transport in the physical space is independent of whether droplets fill a
small or a large fraction of the domain. This is because each bin needs to be
advected separately in the physical space and the computational effort is
independent of whether the entire domain or just its small fraction is filled
with droplets. It is worth pointing out that applying Twomey activation to
create cloud droplets in the Lagrangian warm-rain thermodynamics bears
similarities to the way ice particles are initiated in a particle-based
Lagrangian model targeting ice processes

We assume the same CCN characteristics as in GA17 and

Thick line: number mixing ratio of activated CCN as a function of the supersaturation, the Twomey relationship, used in simulations described herein. Thin dashed lines illustrate numerical implementation of the CCN activation scheme. See text for details.

Figure

Illustration of the activation as represented on the fluid flow
grid. Panel

When applied in a multidimensional fluid flow model, there is an additional
issue with the proposed scheme that needs to be addressed.
Figure

Super-droplets are advected in the physical space applying a
predictor–corrector scheme to solve Eq. (5). The predictor step estimates the

The predictor–corrector scheme ensures the second-order accuracy for the time
integration of the super-droplet transport. However, to increase accuracy,
the corrector step can be repeated by replacing

Velocity interpolation to calculate super-droplet transport is the key element of the Lagrangian scheme. Since the EULAG model applies unstaggered grid (i.e., all variables are located at the same position), one possibility is to consider a grid cell whose four corners in two dimensions (eight vertices in three dimensions) form a rectangular (cuboid-shaped in three dimensions) grid cell. For a super-droplet located in such a grid cell, flow velocity at the droplet position can be interpolated from the velocity values at the corners and vertices. Arguably the simplest possibility is to apply a bilinear (trilinear in three dimensions) interpolation scheme, but a more advanced scheme may be considered as well. However, the bilinear interpolation (and likely more advanced interpolation schemes) does not lead to physically consistent results as documented below.

Illustration of the interpolation scheme used in the super-droplet
transport scheme referred to as “simple” in the text. The rectangular box
represents a single grid cell with

Advection of the potential temperature and water vapor mixing ratio (as well
as the velocity components) in EULAG is performed on the C grid (i.e., with
the horizontal–vertical velocities at the vertical–horizontal grid cell
boundaries). Advective velocities come from interpolating velocity components
predicted on the unstaggered grid into the C grid. Advective velocities
satisfy the anelastic incompressibility condition

To investigate the accuracy of the super-droplet transport scheme, a relatively simple test problem was designed. In the test, two-dimensional rising moist thermal simulations driven by the Eulerian bulk condensation scheme were used, applying the same simulation setup as in the super-droplet simulations (see Sect. 3.1). The predicted rising thermal flow (similar to the one shown later in the paper applying super-droplets) was applied to advect a large number of passive particles introduced to a fraction of the computational domain including the thermal and its immediate environment at the onset of the simulation. The number of passive particles varied from several tens to a few thousands per grid cell in various tests. In the rising thermal flow simulated by the model, one should expect the average number of particles per grid volume to slightly decrease because of the density decreasing with height. Moreover, the number should show statistical fluctuations due to advection of particles from one grid cell to another. The fluctuation amplitude should vary approximately as an inverse of the square root of the initial number of particles per grid cell. These assumptions provide the basis for evaluating the accuracy of the super-droplet transport.

Evolution of the maximum and the minimum number of passive particles
per grid cell in the bulk thermal simulations applying

Figure

This simple example, together with similar simulations using different numbers of passive particles not shown here as well as results of the super-droplet approach available at the University of Warsaw (Arabas et al., 2015), suggests that the simple scheme (Eq. 10) (and its extension into a three-dimensional framework) should be used in the Lagrangian microphysics. Hence, such a scheme is used in all super-droplet simulations presented in this paper.

The overall strategy for the time integration of the coupled Eularian and Lagrangian components of the model thermodynamics is to advance the temperature and moisture fields using Eq. (6) first, then to transport Lagrangian super-droplets using Eqs. (7) and (8), and finally to calculate condensation or evaporation of cloud droplets according to Eq. (4), with the condensation or evaporation providing temperature and moisture tendencies calculated from Eq. (3) in each grid cell. These tendencies are applied in the next model time step. Condensation or evaporation of individual super-droplets require knowledge of the supersaturation that needs to be calculated from updated temperature and water vapor fields. The flow-resolved supersaturation field can be supplemented with the subgrid-scale fluctuations as in GA17. By the same token, the resolved flow used to transport super-droplets through the predictor–corrector scheme can be supplemented with the subgrid-scale velocity fluctuations estimated from the predicted subgrid-scale turbulent kinetic energy. These additions are not included in the initial formulation and testing of the Twomey super-droplets discussed in this paper, but will form an important component of the model application in the future.

There are two issues that need to be considered for the coupling between
Eulerian and Lagrangian model components. The first one concerns spurious
supersaturation fluctuations near cloud edges

The second issue concerns the interpolation of the thermodynamic fields to
the super-droplet position.

The key aspect of the

The crux of the method is to calculate the amount of cloud water

The scheme described above has been merged with the EULAG model

The UWLCM is an open-source software for two-dimensional and three-dimensional modeling of clouds
with super-droplet or bulk microphysics. Advection of the Eulerian
fields is done using the libmpdata++

Because UWLCM explicitly represents CCN deliquescence, a more
detailed droplet growth equation is used

Rising moist thermal simulations follow

The average number of super-droplets per grid cell affects the
amplitude of statistical fluctuations due to the transport of
super-droplets across the Eulerian grid. Because of different
formulations of CCN activation, a direct match of the super-droplet
number per grid cell between UWLCM and the Twomey scheme is impossible.
In the Twomey scheme simulations, the number of

When comparing results from the two models, one needs to keep in mind that
microphysical schemes differ in some additional details. In particular, the
UWLCM applies the

Distribution of

As Fig.

Figures

A more detailed comparison between the two simulations is facilitated
by applying two different statistical measures. The first one involves
conditional sampling of various fields across the thermal, including points
with the cloud water mixing ratio exceeding a threshold of 0.1 g kg

Comparison of the mean supersaturation averaged over the cloudy
points for UWLCM

Figure

As in Fig.

Figure

Evolution of various parameters at the center of mass of the cloud
water in the rising thermal simulations applying 200 super-droplets (UWLCM;
purple lines) or 200 divisions (Twomey scheme with adjustment; green lines).
The panels show

As Fig.

Figures

The differences between Figs.

Evolution of the maximum supersaturation in the domain for UWLCM simulations (purple line) and Twomey scheme with (blue line) and without (green line) adjustment for unphysical supersaturation fluctuations. Data are plotted at every model time step with 200 super-droplets in UWLCM and 200 division in the model with the Twomey activation.

As a final element of the comparison, we show in Fig.

In summary, we believe that simple tests presented in this section document
the efficacy of the super-droplet approach with the Twomey activation.
Unfortunately, we cannot provide a direct comparison of the computational
effort between the two approaches because the two models run on different
computer systems. However, since the cloud covers about 2.5 % of the
two-dimensional computational domain, the Twomey scheme requires roughly 40
times less computational effort for simulations presented here (this estimate
excludes the difference in the time steps used by both models). However, for
a hypothetical three-dimensional
simulation with a domain extending 3.6 km in the second horizontal
direction, the volume of the initial spherical bubble with the same radius
would only constitute about 0.1 % of the computational domain volume.
Thus, the computational effort in similar three-dimensional simulations would
be about 3 orders of magnitude larger in UWLCM than in the babyEULAG with
Twomey super-droplets. UWLCM makes up a lot of this difference by applying
modern software engineering techniques including parallel processing and
application of graphics processing units; see

This paper discusses technical details of a novel Lagrangian condensation
scheme to model nonprecipitating warm (ice-free) clouds. The idea is to use
Lagrangian point particles (“super-droplets” following the nomenclature
introduced by

We apply the traditional Lagrangian super-droplet model, the University of
Warsaw Lagrangian Cloud Model

As noted in

The simulations with the Twomey activation of
super-droplets were done using the bE_SDs v0.1 model, available at

Figure

Interpolation of particle-advecting velocities from the grid cell
(large rectangle) into a subgrid-scale volume (small rectangle) applying the
simple interpolation scheme (Fig.

WWG developed the Twomey activation scheme and completed simulations using it. PD completed UWLCM simulations and created intercomparison figures. All authors contributed to the design of numerical simulations and were involved in creating the paper.

The authors declare that they have no conflict of interest.

This work was partially supported by the Polish National Science Center (NCN) “POLONEZ 1” Grant 2015/19/P/ST10/02596 and by the US DOE ASR Grant DE-SC0016476. The POLONEZ 1 grant has received funding from the European Union's Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie grant agreement 665778. Wojciech W. Grabowski acknowledges discussions with Dorota Jarecka during the early development of Twomey super-droplet scheme. Helpful conversations and suggestions from Shin-Ichiro Shima (U. of Hyogo, Kobe, Japan) and from Lian-Ping Wang (U. of Delaware, Newark, USA) are also acknowledged. UWLCM development has been possible through NCN grants 2010/01/N/ST10/01438, 2012/06/M/ST10/00434 and 2014/15/N/ST10/05143. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Edited by: Simon Unterstrasser Reviewed by: Shin-ichiro Shima and one anonymous referee