Toward a Numerical Benchmark for Warm Rain
Processes

Adrian Hill, Zachary Lebo, Miroslaw Andrejczuk, Sylwester Arabas, Piotr

Dziekan, Paul Field, Andrew Gettelman, Fabian Hoffmann, Hanna

Pawlowska, Ryo Onishi, et al.

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► To cite this version:

Adrian Hill, Zachary Lebo, Miroslaw Andrejczuk, Sylwester Arabas, Piotr Dziekan, et al. Toward a Numerical Benchmark for Warm Rain Processes. Journal of the Atmospheric Sciences, 2023, 80 (5), pp.1329-1359. 10.1175/jas-d-21-0275.1. hal-04950300

HAL Id: hal-04950300

https://cnrs.hal.science/hal-04950300v1

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# Toward a Numerical Benchmark for Warm Rain Processes

ADRIAN A. HILL,ª ZACHARY J. LEBO,ᵇ MIROSLAW ANDREJCZUK,ª SYLWESTER ARABAS,ᶜ,ᵈ PIOTR DZIEKAN,ᵉ PAUL FIELD,ª,f ANDREW GETTELMAN,ᵇ FABIAN HOFFMANN,ᵇ HANNA PAWLOWSKA,ᶜ RYO ONISHI,<fnref n="1" /> AND BENOIT VIÉ<fnref n="1" />

(Manuscript received 14 October 2021, in final form 8 December 2022)

**ABSTRACT:** The Kinematic Driver-Aerosol (KiD-A) intercomparison was established to test the hypothesis that detailed warm microphysical schemes provide a benchmark for lower-complexity bulk microphysics schemes. KiD-A is the first intercomparison to compare multiple Lagrangian cloud models (LCMs), size bin-resolved schemes, and double-moment bulk microphysics schemes in a consistent 1D dynamic framework and box cases. In the absence of sedimentation and collision-coalescence, the drop size distributions (DSDs) from the LCMs exhibit similar evolution with expected physical behaviors and good interscheme agreement, with the volume mean diameter ($D_{vol}$) from the LCMs within 1%-5% of each other. In contrast, the bin schemes exhibit nonphysical broadening with condensational growth. These results further strengthen the case that LCMs are an appropriate numerical benchmark for DSD evolution under condensational growth. When precipitation processes are included, however, the simulated liquid water path, precipitation rates, and response to modified cloud drop/aerosol number concentrations from the LCMs vary substantially, while the bin and bulk schemes are relatively more consistent with each other. The lack of consistency in the LCM results stems from both the collision-coalescence process and the sedimentation process, limiting their application as a numerical benchmark for precipitation processes. Reassuringly, however, precipitation from bulk schemes, which are the basis for cloud microphysics in weather and climate prediction, is within the spread of precipitation from the detailed schemes (LCMs and bin). Overall, this intercomparison identifies the need for focused effort on the comparison of collision-coalescence methods and sedimentation in detailed microphysics schemes, especially LCMs.

**KEYWORDS:** Cloud droplets; Cloud microphysics; Clouds; Cloud parameterizations; Aerosol-cloud interaction

## 1. Introduction

The response of clouds and precipitation to changes in aerosol concentration and composition is a major source of uncertainty in climate prediction (e.g., Boucher et al. 2013; Ghan et al. 2016; Michibata et al. 2016; Gryspeerdt et al. 2017; Zhou and Penner 2017; Mülmenstädt and Feingold 2018) and a potential uncertainty in numerical weather prediction (NWP) (e.g., Seifert et al. 2012; Boutle et al. 2018). Hill et al. (2015) highlighted that one of the causes of this uncertainty is the level of complexity employed in the cloud microphysics scheme and the level of interaction between the clouds and aerosol. In general, cloud microphysics schemes fall into two categories with very different levels of complexity: 1) bulk water schemes (e.g., Lohmann et al. 1999; Seifert and Beheng 2006; Morrison and Gettelman 2008; Thompson et al. 2008; Milbrandt and Yau 2005a; Morrison et al. 2009; Morrison and Milbrandt 2015; Gettelman and Morrison 2015; Gettelman et al. 2019), which are used in operational NWP and climate

prediction, and 2) detailed size-resolved microphysics schemes (e.g., Tzivion et al. 1987; Feingold et al. 1988; Reisin et al. 1996; Khain and Sednev 1995; Geresdi 1998; Khain et al. 2004; Shima et al. 2009; Andrejczuk et al. 2010; Lebo and Seinfeld 2011b,a; Fan et al. 2012; Grabowski et al. 2018), which are often employed for fundamental, detailed, numerical microphysical process research, which can be used to further develop and validate bulk water schemes. The aim of this work is to test the hypothesis that there exists a sufficiently accurate and consistent cloud microphysics representation to act as a benchmark. To this end, we present an intercomparison of multiple detailed and bulk microphysics schemes to understand how key microphysical variables, e.g., surface precipitation, liquid water path, and cloud drop size distribution (DSD), compare when examined in a consistent dynamical framework.

The aim of microphysics schemes, regardless of their type and level of detail is to describe the evolution of cloud and rainwater due to microphysical processes, including condensation/evaporation, collision-coalescence, breakup, and sedimentation. Warm bulk water schemes employ a limited number of bulk prognostic variables, e.g., mass mixing ratio and drop number integrated over the drop size distribution, to describe the

<footnote>ª Met Office, Exeter, United Kingdom</footnote>

<footnote>ᵇ School of Meteorology, University of Oklahoma, Norman, Oklahoma</footnote>

<footnote>ᶜ University of Illinois at Urbana-Champaign, Urbana, Illinois</footnote>

<footnote>ᵈ Jagiellonian University, Kraków, Poland</footnote>

<footnote>ᶜ Institute of Geophysics, Faculty of Physics, University of Warsaw, Warsaw, Poland</footnote>

<footnote n="1">¹ School of Earth and Environment, University of Leeds, Leeds, United Kingdom</footnote>

<footnote>g National Center for Atmospheric Research, Boulder, Colorado</footnote>

<footnote>h Ludwig-Maximilians-Universität München, Munich, Germany</footnote>

<footnote>i Tokyo Institute of Technology, Tokyo, Japan</footnote>

<footnote>j CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France</footnote>
evolution of cloud and precipitation. Such schemes usually use hydrometeor partitions (e.g., cloud and rain) and parameterizations to describe the evolution of each partition's DSD (e.g., condensation/evaporation and accretion), particle fall velocities for each partition, and conversions between partitions (e.g., autoconversion). Bulk water schemes have varying complexity, which is typically represented by the number of predicted bulk moments of the DSD. All operational NWP and climate prediction models use bulk water schemes, with many employing a single-moment scheme, in which cloud and rain are represented by a single prognostic moment (mass mixing ratio). Double-moment bulk schemes are more complex, with microphysical processes described using both the mass mixing ratio and droplet number concentration ($N_d$), both of which are prognosed. Double-moment schemes are often used in climate prediction because of the improvements associated with aerosol-cloud interactions relative to single-moment schemes (e.g., Gettelman et al. 2019). In NWP, there is evidence that such schemes can be beneficial (e.g., Benjamin et al. 2016; Miltenberger et al. 2018; Boutle et al. 2018), and double-moment schemes have been implemented in some operational NWP model, e.g., the Global Environmental Multi-scale (GEM) model and the High-Resolution Rapid Refresh (HRRR) model; however, due to computational costs, single-moment schemes are more common in NWP. More recently, three-moment schemes have been developed, which extend the double-moment representation by prognosing a higher moment of the distribution, e.g., the sixth moment, which is proportional to radar reflectivity (e.g., Milbrandt and Yau 2005b; Shipway and Hill 2012; Loftus et al. 2014). Schemes with even more moments, i.e., at least four, have recently been used to overcome the shortcoming of separate cloud and rain-water categories in bulk schemes (Kogan and Belochitski 2012; Igel et al. 2022).

Bulk water schemes are essential for representing cloud microphysics in NWP and climate prediction models, with all operational models using such schemes. It is, therefore, very important to continually develop and validate such schemes. In addition to comparison with observations, a fundamental tool in the development and validation of bulk water schemes, and thus an underpinning capability in all operational microphysics schemes, is detailed size-resolved microphysics schemes. Such schemes do not prescribe a functional form of the size distribution and do not partition between cloud and rain-water. Instead, detailed schemes attempt to resolve the size distribution and the evolution of the size distribution through size-resolved condensation and evaporation and by solving the stochastic collection and breakup equations. This means that detailed schemes do not need to include unnatural parameterizations, e.g., autoconversion, which are required in all bulk schemes that employ separate cloud and rain categories. Instead, detailed schemes are often used to develop such parameterizations (e.g., Khairoutdinov and Kogan 2000; Kogan 2013; Gettelman et al. 2021). Another distinction between bulk and detailed schemes is that detailed schemes use an explicit treatment of supersaturation, while most bulk schemes use saturation adjustment owing to their use in larger-scale models (larger grid spacing

and longer time step). It is worth mentioning, however, an explicit treatment of supersaturation can be applied to bulk schemes (e.g., Lebo et al. 2012). While detailed schemes tend to be much more complex with greater degrees of freedom and much more computationally expensive, when compared to bulk schemes, they include fewer assumptions about the DSD and less severe assumptions about the microphysics processes. This relaxation in assumptions enables detailed schemes to be used in the analysis of DSD evolution.

Detailed schemes tend to fall into two categories: 1) size-resolved Eulerian bin schemes (e.g., Tzivion et al. 1987; Khain et al. 2004; Lebo and Seinfeld 2011b; Onishi and Takahashi 2012) and 2) Lagrangian cloud models (LCMs) (e.g., Shima et al. 2009; Andrejczuk et al. 2010; Riechelmann et al. 2012; Grabowski et al. 2018). Size-resolved Eulerian bin schemes discretize the DSD into bins commonly defined by mass, in which transfers between bins are determined by condensation/evaporation, collision-coalescence, and breakup. The mass and/or number (depending on the number of moments) in each bin is independently transported via advection, also in an Eulerian framework associated with the parent model's grid structure. As with bulk schemes, each bin can be represented by one moment (either mass or number) where the size in a bin is fixed, or multiple moments, e.g., mass and number, where the mean size in a bin can change. Grabowski et al. (2019) presented an overview of the history of size-resolved bin microphysics schemes spanning from box model frameworks through to implementation in 3D dynamical frameworks. As part of a review, Grabowski et al. (2019) highlighted the issues associated with the numerical implementation of bin schemes in an Eulerian framework, in particular, artificial broadening of the DSD through numerical diffusion during vertical advection (e.g., Morrison et al. 2018). In the absence of vertical advection, solutions to mitigate numerical diffusion from condensation in bin space have been proposed, albeit with added computational expense (e.g., Liu et al. 1997; Olesik et al. 2022). While issues with size-resolved bin schemes have been known for many years, such schemes have been extensively and effectively used in cloud microphysics and aerosol-cloud interaction process research on warm stratiform (e.g., Feingold et al. 1988; Hill et al. 2008; Jiang et al. 2009, 2010; Lebo and Seinfeld 2011a; Grabowski et al. 2011; Petters et al. 2013; Yamaguchi et al. 2019) and convective clouds (e.g., Khain et al. 2004; Fan et al. 2009; Lebo and Seinfeld 2011b; Lebo et al. 2012; Xue et al. 2017; Kacan and Lebo 2019). Size-resolved bin schemes have also been widely applied to parameterization development, improvement, and validation of bulk schemes (e.g., Khairoutdinov and Kogan 2000; Thompson et al. 2004; Seifert and Beheng 2006; Seifert et al. 2006; Morrison and Grabowski 2007; Saleeby and Cotton 2008; Li et al. 2009a,b; Milbrandt and McTaggart-Cowan 2010; Shipway and Hill 2012; Kogan and Belochitski 2012; Wang et al. 2013; Igel and van den Heever 2017a,b; Zeng and Li 2020), and hybrid bin-bulk schemes (or bin-emulating schemes) (e.g., Feingold et al. 1998; Cotton et al. 2003; Saleeby and Cotton 2004; van den Heever and Cotton 2004; Wang and Feingold 2009b,a).

Compared to size-resolved bin microphysics schemes and bulk microphysics schemes, the development of LCMs, which
are also known as probabilistic particle-based models or superdroplet schemes,<fnref n="1" /> is in its infancy. However, the usage of LCMs in cloud modeling studies has increased considerably in recent years (e.g., Andrejczuk et al. 2008; Shima et al. 2009; Sölch and Kärcher 2010; Andrejczuk et al. 2010; Riechelmann et al. 2012; Arabas and Shima 2013; Hoffmann et al. 2015; Hoffmann 2016, 2017; Hoffmann et al. 2017; Grabowski et al. 2018; Hoffmann et al. 2019; Grabowski 2020; Dziekan et al. 2019, 2021; Richter et al. 2021; Grabowski and Thomas 2021; Chandrakar et al. 2022). Such schemes are defined by their Lagrangian nature, both in physical space (meaning that the superdroplets are not confined to the center or vertices of model grid boxes) and droplet size (meaning that there is no advection of water from one size bin to the next). This founding principle alleviates numerical diffusion issues common in traditional size-resolved Eulerian bin schemes (Morrison et al. 2018; Grabowski et al. 2019). The superdroplets themselves represent an ensemble of drops with specific properties, e.g., weighting factor (or multiplicity, i.e., the number of drops represented by each superdroplet), size, location in the modeling domain, and velocity.

A key justification for the development and application of LCMs, in addition to alleviating issues related to numerical diffusion between bins in Eulerian schemes, is that they provide better performance for increased drop and aerosol properties. Eulerian schemes require bins to be predefined; thus, unnecessary calculations are performed on empty bins. This problem is worsened in 2D bin schemes, which represent both liquid and solute amounts, and the computational expense of the calculations scales as the number of bins raised to the number of properties being computed. LCMs, on the contrary, do not have such empty bins (or in this case, superdroplets) and scale linearly with an increase in the number of drop and aerosol properties, making them more efficient in that regard. Moreover, LCMs allow for a more straightforward coupling of microphysics with subgrid-scale turbulent fluctuations (e.g., using a stochastic Lagrangian model), which is cumbersome to include in Eulerian bin schemes. Given these unique advantages, similar to the aforementioned size-resolved Eulerian bin schemes, LCMs have recently been used to derive parameterizations for bulk schemes (e.g., Noh et al. 2018).

While LCMs have clear advantages over size-resolved Eulerian bin schemes in some aspects, there are some issues that should be mentioned. First, LCMs rely on superdroplets to represent drops of a certain size, and the abundance of such drops is represented by the multiplicity, where drops are “grouped” into superdroplets based on similar properties. Moreover, LCMs require a sufficient number of superdroplets to represent the DSD across the model domain in a statistically robust manner. As a “rule of thumb,” investigations into collision-coalescence have indicated that at least 100 superdroplets are required (e.g., Dziekan and Pawlowska 2017; Unterstrasser et al. 2017) for appropriate accuracy. More superdroplets lead to more accurate statistics but at a higher computational cost, and there is no one-size-fits-all number. As such, there is a balance between the

multiplicity of a superdroplet and the number of superdroplets needed to produce accurate simulations. Another potential issue, especially with precipitation, is the loss of superdroplets, which requires replenishment; how this is done can influence the model simulations as well. In contrast, Eulerian bin schemes are structurally different and do not require replenishment of bins upon the loss of water to the surface under precipitation.

Although detailed size-resolved microphysics schemes have been, and still are, a fundamental tool in microphysics research, there have been very few research studies or inter-comparison projects comparing multiple LCMs, multiple bin schemes, and multiple bulk water schemes. Instead, in general, depending on the study, one LCM or one bin scheme has been used as a numerical benchmark. For example, Morrison et al. (2018) used an LCM as a benchmark for DSD evolution due to condensation and vertical advection and compared this to single-moment and double-moment bin microphysics schemes. Grabowski (2020) compared the behavior of one bin scheme (single moment) and one LCM in the simulation of nonprecipitating cumulus. More recently, Chandrakar et al. (2022) also compared a bin scheme (double moment) with a single LCM to study cumulus congestus clouds. In these examples, the microphysics schemes were compared in the same dynamic framework and the LCM was employed as the benchmark, but there was no comparison with other LCM to validate the benchmark. Moreover, both Morrison et al. (2018) and Grabowski (2020) did not include precipitation processes. To date, Chandrakar et al. (2022) is the only study to compare a bin scheme with an LCM in terms of precipitation processes, highlighting large differences in precipitation production between the tested models.

In addition to comparing LCMs and bin schemes, other studies have used a particle-based Lagrangian scheme as a benchmark for comparison with bulk schemes (e.g., Arabas et al. 2015; Sato et al. 2018) or Eulerian size-resolved bin schemes as a benchmark for development of and comparison with bulk schemes (e.g., Khairoutdinov and Kogan 2000; Shipway and Hill 2012; Kogan 2013). In all of these examples, the detailed schemes and the bulk schemes were coupled to a consistent dynamical core but with only one LCM and/or one bin scheme, so there is no way to demonstrate the behavior of the reference model. In contrast, Ackerman et al. (2009) and vanZanten et al. (2011) compared multiple bin schemes with multiple bulk schemes in different dynamical models, and these studies showed that the largest spread in the amount and the timing of surface rain rates came from the bin microphysics schemes. While Ackerman et al. (2009) argued that this spread is the result of different numerical implementations, vanZanten et al. (2011) argued that the spread is related to intrinsic differences in the microphysics schemes, in particular the representation of $N_d$. Based on the results of Ackerman et al. (2009) and vanZanten et al. (2011), the Kinematic Driver (KiD) model was developed to compare microphysics schemes in a consistent dynamical framework (Shipway and Hill 2012; Hill et al. 2015). To date, however, in general, one bin microphysics scheme has been tested in KiD and, as with past work, the bin scheme has been used as a reference, with little comparison or validation against other detailed schemes. The only exception to this is

<footnote n="1">¹ In this work we refer to these schemes as LCMs, which contain superdroplets.</footnote>
<table><caption>TABLE 1. Microphysics schemes contributing to this model intercomparison.</caption><thead><tr><td>Model</td><td>Description</td><td>Reference(s)</td></tr></thead><tbody><tr><td>LCM-FH</td><td>Lagrangian microphysics</td><td>Hoffmann et al. (2015), Hoffmann (2017), and Hoffmann et al. (2017)</td></tr><tr><td>LCM-MA</td><td>Lagrangian microphysics</td><td>Andrejczuk et al. (2010)</td></tr><tr><td>UWLCM</td><td>Lagrangian microphysics</td><td>Arabas et al. (2015) and Dziekan et al. (2019)</td></tr><tr><td>bin2d</td><td>Two-dimensional one-moment Eulerian bin scheme, including aerosol processing</td><td>Lebo and Seinfeld (2011a)</td></tr><tr><td>TAU-bin</td><td>Traditional (one-dimensional) two-moment Eulerian bin scheme</td><td>Tzivion et al. (1987, 1989)</td></tr><tr><td>MSSG-bin</td><td>Traditional (one-dimensional) one-moment Eulerian bin scheme</td><td>Onishi and Takahashi (2012)</td></tr><tr><td>MG2</td><td>Two-moment bulk scheme</td><td>Gettelman and Morrison (2015)</td></tr><tr><td>CASIM</td><td>Two-moment bulk scheme</td><td>Shipway and Hill (2012), Hill et al. (2015), and Grosvenor et al. (2017)</td></tr><tr><td>CASIM proc</td><td>Two-moment bulk scheme, including aerosol processing</td><td>Hill et al. (2015) and Miltenberger et al. (2018)</td></tr><tr><td>LIMA</td><td>Two-moment bulk scheme, including aerosol processing</td><td>Vié et al. (2016)</td></tr></tbody></table>

Onishi and Takahashi (2012), where a double-moment bin microphysics scheme [Tel Aviv University size-bin resolved microphysics (TAU-bin)] was used as a reference for comparison with a single-moment bin scheme [Multi-Scale Simulator for the Geoenvironment (MSSG-bin)]. Furthermore, LCMs have never been included in such an intercomparison, and although many of the detailed schemes (and even some bulk schemes) described above include aerosol physical processing representation through activation-coalescence-evaporation cycle, a formal comparison of such schemes has been missing in the scientific literature.

In this work, we address this lack of comparison among all types of microphysics schemes by comparing three Eulerian bin microphysics schemes, three LCMs, and three bulk schemes in the KiD model. Such a comparison in a consistent dynamical model without dynamical feedbacks permits a focus on microphysical behavior and DSD evolution of the schemes and enables an investigation into the consistency between the results from the detailed reference schemes. More specifically, this work aims to address the following:

1) Test the hypothesis that detailed schemes produce accurate and consistent cloud microphysics representation for warm rain processes, and hence represent a numerical benchmark. If not, investigate and suggest why.

2) Test how cloud microphysics schemes (of various complexities) compare across a range of $N_d$ with and without aerosol physical processing.

3) Develop and make available a cloud modeling framework for developing and testing future schemes.

## 2. KiD model

For the model intercomparison, we use the KiD model (Shipway and Hill 2012; Hill et al. 2015). KiD is a simple microphysics interface to a common dynamical core, in which flow fields are prescribed in standard test cases. The prescription of the flow means that there is no accounting for subgrid transport and any feedbacks between the microphysics and flow are neglected. Such prescription allows users to test a

range of microphysics schemes in a dynamically consistent way; i.e., any variations in the simulations between microphysics schemes can be related solely to differences in the representation of the microphysics.

During the intercomparison, the KiD model has been developed beyond version 2.3, which was used in Hill et al. (2015). In particular and pertinent to this work, the latest version of KiD includes two OD box cases (appendix B), which were developed and used in this intercomparison to study changes in DSD from detailed models and further constrain the initial conditions.

## 3. Microphysics schemes

The KiD-A intercomparison underpinned the Global Atmospheric System Studies (GASS) microphysics project and was advertised and undertaken through GASS and the International Cloud Modeling Workshop (ICMW) (Lebo et al. 2017). Through GASS and ICMW, we reached out to as many developers as possible to entrain a spectrum of microphysics schemes with a range of complexity. As a result of this effort, we were able to include three broad categories of microphysics schemes, i.e., LCMs (or superdroplet methods), Eulerian size bin-resolved schemes, and bulk schemes. The inclusion of a scheme depended on the developer or user implementing the scheme in KiD and performing the requested set of simulations, and providing output. Therefore, the schemes included in the intercomparison are those that committed to this effort.

Table 1 introduces the intercomparison schemes with the main references.<fnref n="2" /> The intercomparison comprises three LCMs, three size bin-resolved schemes, and four double-moment bulk microphysics schemes, and details of each scheme are provided

<footnote n="2">² For the LCMs, the additional letters in each name indicate the model's author or institution: FH for Fabian Hoffman, MA for Miroslaw Andrejczuk, and UW for University of Warsaw. For the bin schemes, TAU and MSSG represent Tel Aviv University and Multi-Scale Simulator for the Geoenvironment, respectively. For the bulk models, MG2 indicates the Morrison and Gettelman scheme, version 2, while CASIM and LIMA represent Cloud AeroSol Interacting Microphysics and Liquid Ice Multiple Aerosols, respectively.</footnote>
in appendix D. All schemes have been implemented in KiD, so that transport of a scheme's prognostics, e.g., superdroplets, number and/or mass in each bin, or bulk cloud and rain, is controlled by the prescribed velocity in KiD (see section 4). All microphysics processes, e.g., condensation, collision-coalescence, and sedimentation, are then derived from the scheme's parameterizations, which are outlined in Table 2 (LCMs and bin schemes) and Table 3 (bulk schemes). While the LCMs and Eulerian bin schemes are structurally different, Table 2 shows there is some level of agreement between collision kernels and the fall speed assumptions. For example, all detailed schemes except LCM-MA employ Beard (1976) to represent the fall speed relationships and all but LCM-MA and TAU-bin use the Hall (1980) kernel for determining the growth of the DSD through collision-coalescence. All detailed schemes also explicitly derive supersaturation (either prognostically or diagnostically) and do not impose saturation adjustment; thus, supersaturation can exist between time steps. The detailed schemes differ in their collision-coalescence method and the representation of aerosol activation, with the LCMs and bin2d simulating the growth of aerosol to cloud based on Köhler theory, i.e., no activation parameterization, while TAU-bin and MSSG-bin employ an aerosol activation scheme. Last, while breakup is included in most detailed schemes, it is not used in this study owing to the vast array of methodological choices for implementing drop breakup and sampling issues with the LCMs. In the future, efforts that focus on understanding the differences in warm rain precipitation should also include breakup.

Table 3 shows that the bulk schemes also exhibit broad agreement in the precipitation parameterizations. For example, all bulk schemes except LIMA use Khairoutdinov and Kogan (2000) for autoconversion and accretion; however, Table 3 shows that the bulk schemes differ in their fall speed relationships. In contrast to the detailed schemes, all of the bulk schemes employed in this study use saturation adjustment, but there are interscheme differences in the representation of aerosol activation.

It is important to note that while there are similarities between the schemes tested in this study, there are also obvious differences in the physical parameterizations (Tables 2 and 3). There are also differences in the numerics, e.g., sedimentation. The aim of this work is to look at the differences between the schemes as provided by the developers and to understand relative behavior. Where possible, this research investigates specific physical parameterizations and makes suggestions for future research foci to improve cloud parameterizations.

## 4. KiD-A cases

The basis for the 1D case is Shipway and Hill (2012), which prescribes a simple updraft that is constant with height to advect vapor and hydrometeors but not temperature. The strength of the updraft follows a half-period of a sinusoid peaking with a maximum velocity $w_{max}$. Following Shipway and Hill (2012), there are two vertical velocity setups used for this case, i.e., W2, where $w_{max} = 2 \text{ m s}^{-1}$, and W3, where $w_{max} = 3 \text{ m s}^{-1}$. The initial profiles of temperature and moisture (Table 4) and the dynamic forcing for W2 and W3 are the same as that described in Shipway and Hill (2012).

<table><caption>TABLE 2. LCMs and Eulerian bin schemes contributing to KiD-A and an overview of the main references and assumptions in each scheme.</caption><thead><tr><th>Model</th><th>No. of SD or bins</th><th>Aerosol processing</th><th>Aerosol activation</th><th>Condensation/ evaporation</th><th>Collision method</th><th>Collision kernel</th><th>Fall speed</th></tr></thead><tbody><tr><td>Lagrangian microphysics/ LCMs</td><td>100 SD</td><td>Yes</td><td>—</td><td>Hoffmann et al. (2015)</td><td>Unterstrasser et al. (2017)</td><td>Hall (1980)</td><td>Beard (1976)</td></tr><tr><td>LCM-FH</td><td>100 SD</td><td>Yes</td><td>—</td><td>Andrejczuk et al. (2008)</td><td>Andrejczuk et al. (2010)</td><td>Long (1974)</td><td>Simmel et al. (2002)</td></tr><tr><td>LCM-MA</td><td>100 SD</td><td>Yes</td><td>—</td><td>Maxwell-Mason approx. Arabas et al. (2015)</td><td>Shima et al. (2009)</td><td>Hall (1980)</td><td>Beard (1976)</td></tr><tr><td>UWLCM</td><td>100 SD</td><td>Yes</td><td>—</td><td>Maxwell-Mason approx. Arabas et al. (2015)</td><td>Shima et al. (2009)</td><td>Hall (1980)</td><td>Beard (1976)</td></tr><tr><td>Eulerian size resolved bin schemes</td><td>56 bins cloud, 25 bins aerosol</td><td>Yes</td><td>—</td><td>Pruppacher and Klett (1997)</td><td>Bott (1998)</td><td>Hall (1980)</td><td>Beard (1976)</td></tr><tr><td>TAU-bin</td><td>34 bins cloud</td><td>No</td><td>—</td><td>Clark (1973), Tzivion et al. (1989)</td><td>Tzivion et al. (1987)</td><td>Long (1974), Ochs et al. (1986)</td><td>Beard (1976)</td></tr><tr><td>MSSG-bin</td><td>132 bins cloud</td><td>No</td><td>—</td><td>Tzivion et al. (1989)</td><td>Bott (2000)</td><td>Hall (1980)</td><td>Beard (1976)</td></tr></tbody></table>
<table><caption>TABLE 3. Bulk microphysics schemes contributing to KiD-A and an overview of the main references and assumptions in each scheme.</caption><thead><tr><th>Model</th><th>Aerosol processing</th><th>Aerosol activation</th><th>Condensation/ evaporation</th><th>Autoconversion</th><th>Accretion</th><th>Rain self- collection</th><th>Fall speed</th></tr></thead><tbody><tr><td>MG2</td><td>No</td><td>Prescribed N<sub>d</sub></td><td>Saturation adjustment</td><td>Khairoutdinov and Kogan (2000)</td><td>Khairoutdinov and Kogan (2000)</td><td>Beheng (1994)</td><td>Liu and Orville (1969)</td></tr><tr><td>CASIM</td><td>No</td><td>Abdul-Razzak and Ghan (2000)</td><td>Saturation adjustment</td><td>Khairoutdinov and Kogan (2000)</td><td>Khairoutdinov and Kogan (2000)</td><td>Beheng (1994)</td><td>Abel and Shipway (2007)</td></tr><tr><td>CASIM proc</td><td>Yes</td><td>Abdul-Razzak and Ghan (2000)</td><td>Saturation adjustment</td><td>Khairoutdinov and Kogan (2000)</td><td>Khairoutdinov and Kogan (2000)</td><td>Beheng (1994)</td><td>Abel and Shipway (2007)</td></tr><tr><td>LIMA</td><td>Yes</td><td>Cohard et al. (1998)</td><td>Saturation adjustment</td><td>Berry and Reinhardt (1974) based on Long (1974)</td><td>Cohard and Pinty (2000) based on Long (1974)</td><td>Cohard and Pinty (2000)</td><td>Cohard and Pinty (2000)</td></tr></tbody></table>

<table><caption>TABLE 4. Initial water vapor mixing ratio ($q_v$, kg kg$^{-1}$) and potential temperature profiles ($\theta$; K) for W2 and W3.</caption><thead><tr><th>Height (m)</th><th>$q_v$</th><th>$\theta$</th></tr></thead><tbody><tr><td>0</td><td>0.015</td><td>297.9</td></tr><tr><td>740</td><td>0.0138</td><td>297.9</td></tr><tr><td>3260</td><td>0.0024</td><td>312.66</td></tr></tbody></table>

The only difference between the 1D case presented here and that of Shipway and Hill (2012) is that the atmospheric density is prescribed to be 1 kg m$^{-3}$ and constant with height. While this density prescription is used by all schemes to maintain consistency in testing, it is required by the LCMs because a constant density with height enables particle-based models to use periodic boundary conditions for particle transport across the top/bottom domain boundaries. Since the vertical velocity is nonzero only during the first 600 s of the simulations, and the cloud-top heights are well below the domain top, the particles reentering through the bottom boundary are not affected by cloud processes (while the effect of sedimentation is negligible). To bring the wet sizes of particles subject to the periodic boundary condition in equilibrium with the humidity in the lowermost grid cell, a reinitialization of the wet radius is used (particle multiplicities remain unchanged).

For W2 and W3, all schemes except MG2 are initialized with aerosol number concentrations ($N_a$) of 50, 150, and 300 cm$^{-3}$ and a prescribed single-mode lognormal aerosol size distribution with a lognormal geometric mean diameter of 0.08 µm and a log standard deviation of 1.4. MG2, which excludes aerosol as a prognostic variable, is initialized with cloud $N_d$ of 50, 150, and 300 cm$^{-3}$. For convenience hereafter, when referring to the cases, irrespective of whether a scheme is initialized with $N_a$ or $N_d$, we will use $N_a$ to differentiate the number initialization. For MG2, this can be thought of as 100% activation of all aerosols in all cases.

Two cases were requested for each $N_a$ and both W2 and W3, i.e., (i) a condensation-evaporation only test (1D-CE) and (ii) a precipitation test (1D-PRECIP). In the 1D-CE case, only aerosol activation (depending on the scheme), condensation, and evaporation are switched on, while sedimentation, collision-coalescence or autoconversion-accretion, breakup, etc., are switched off. In the 1D-PRECIP case, condensation-evaporation and all precipitation processes, including sedimentation but excluding breakup, are switched on.

The KiD-A intercomparison started with only the 1D case due to its simplicity, focused on only microphysics, and the traceability of results back to earlier comparisons between various bulk schemes and bin schemes (Shipway and Hill 2012; Onishi and Takahashi 2012; Hill et al. 2015). However, following initial intercomparison submissions, which highlighted significant differences between the detailed schemes and in light of results from Morrison et al. (2018), the KiD model was extended to include two “box” cases that isolate and compare the evolution of specific processes when the schemes are initialized with the same drop size distribution and the results are not impacted by transport effects. Details of these cases are presented in appendix B.
<figure><img src="image_3.png" /><figcaption>FIG. 1. (a)–(c) Liquid water path (LWP), (d)–(f) mean $N_d$, (g)–(i) mean volume diameter ($D_{vol}$) at the height of the maximum liquid water content, and (j)–(l) the standard deviation of $D_{vol}$ ($\sigma$) from the 1D-CE case for W3. Cyan lines indicate bulk schemes, black indicates the Eulerian bin schemes, and the blue lines show the LCMs. The different columns indicate different aerosol number concentration tests (increasing from left to right).</figcaption></figure>

**5. Condensation–evaporation (CE) only results**

Before assessing the simulations with precipitation, it is useful to present the behavior of the schemes from the 1D-CE case, i.e., only condensation/evaporation. Such assessment acts as a sanity check to ensure that, at a minimum, all schemes produced the same liquid water content, while also permitting the assessment of the schemes in the absence of precipitation. While the participants submitted results for both W2 and W3, here we only present results from W3 since, in the absence of precipitation, both

W2 and W3 lead to qualitatively similar results and conclusions.

Figure 1 presents the LWP (Figs. 1a–c), in-cloud mean ($N_d$) (Figs. 1d–f),<fnref n="3" /> mass-weighted mean volume diameter ($D_{vol}$) (Figs. 1g–i), and standard deviation of $D_{vol}$ ($\sigma$) (Figs. 1j–l)

<footnote n="3">³ The in-cloud mean $N_d$ is calculated by averaging all points in the column with an LWC that is greater than $1 \times 10^{-5} \text{ kg m}^{-3}$. Further, in the LCMs and bin2d, cloud drops are defined as any particle with diameter larger than 2 µm.</footnote>
at the height of the maximum liquid water content (LWC) for the 1D-CE case.<fnref n="4" /> Details regarding the calculations of $D_{vol}$ and $\sigma$ are provided in appendix A. With all schemes, liquid water begins to form at around 100 s, with the bulk schemes, which employ saturation adjustment, producing water slightly earlier than the detailed schemes. LWP then increases rapidly during the first 10 min while the vertical velocity forcing is active. After 10 min, the LWP is constant because there is no dynamical source of supersaturation once the w forcing is 0 m s⁻¹ and there is no sink of cloud through subcloud evaporation or precipitation lost at the surface due to the exclusion of sedimentation in this case. Figures 1a-c show that irrespective of the microphysical scheme, complexity, or supersaturation representation, there is excellent agreement between all schemes when considering the LWP evolution, with LWP from all schemes being within 1.5% of each other after 10 min.

In contrast to the LWP, there are clear differences in the evolution of the in-cloud mean $N_d$ from the different schemes (Figs. 1d-f). For example, after 10 min of simulation, the LCMs produce in-cloud mean $N_d$ values that are within 5% of each other, while the detailed schemes (LCMs and bin) and the bulk schemes without aerosol processing (CASIM and MG2, noting that MG2 uses a fixed $N_d$) are within 10% of each other. In contrast, the bulk schemes with aerosol processing (CASIM-proc and LIMA) tend to produce 30%-50% lower $N_d$ than the other schemes. Once again, this percentage difference is consistent across all the $N_a$ cases. The relatively low $N_d$ from CASIM-proc and LIMA result from the interplay between the “all-or-nothing” condensation scheme (saturation adjustment) and the impact of activation on the remaining aerosol size. For example, upon activation, aerosol is scavenged, which reduces the size of the interstitial aerosol. Given that the w forcing is constant with height, further activation cannot occur until the supersaturation derived by the activation scheme is large enough to activate the smaller aerosol, i.e., higher in the cloud. In contrast, the bin schemes and LCMs represent supersaturation explicitly, either diagnostically using water vapor and temperature or through a prognostic parameterization, in which supersaturation is transported and used alongside vapor and temperature to derive a new supersaturation on each time step. Importantly, in both the diagnostic and prognostic derivations, supersaturation can be maintained from time step to time step, so activation can occur more readily throughout the cloud.

While the simulated $N_d$ values from the detailed schemes are within 10% of each other after 10 min, Figs. 1d-f also show that once $w = 0.0 \text{ m s}^{-1}$, the mean $N_d$ begins to slightly decrease with time in the LCMs and significantly decreases in bin2d, while remaining approximately constant in TAU-bin and the bulk schemes. This change in $N_d$ in the absence of forcing or precipitation processes results from a process known as droplet ripening, in which larger drops have a higher effective supersaturation than smaller drops, so larger drops grow at the expense of small drops, leading to a

reduction in $N_d$ over time. To demonstrate the impact of droplet ripening, Figs. 1d-f present simulations from bin2d with droplet ripening disabled (bin2d no-sol; the solute term is neglected and therefore the curvature term as well). As expected, when ripening is switched off, the mean $N_d$ from bin2d is very similar to TAU-bin after approximately 300 s. Prior to this time, bin2d no-sol has higher $N_d$ and the onset of activation is earlier compared with TAU-bin. This is because TAU-bin uses the aerosol properties to determine activation, whereas bin2d no-sol does not, meaning all aerosol can and do activate as soon as supersaturation is present. Moreover, a comparison of bin2d and the LCMs (Figs. 1d-f) shows that in the case presented, bin2d is more prone to droplet ripening than the LCMs. This is due to the broader DSD derived from bin2d, which contains more larger droplets that grow at the expense of the smaller droplets more efficiently than the narrow DSDs derived from the LCMs. While droplet ripening has been demonstrated and discussed in prior papers (e.g., Srivastava 1991; Çelik and Marwitz 1999; Wood et al. 2002; Lebo et al. 2008; Jensen and Nugent 2017), the results presented here show that the impact of the process can vary between schemes, which could further impact precipitation formation. Furthermore, ripening appears to increase with increasing $N_a$. This can be attributed to several factors. First, even if the same fraction of cloud droplets deactivate across all $N_a$, given the identical axes in Figs. 1d-f for all $N_a$, the absolute reduction in $N_d$ due to ripening would be smaller for low $N_a$ and thus less pronounced in the figure. Additionally, as $N_a$ increases, the equilibrium supersaturation decreases, meaning the more drops could deactivate under higher $N_a$, thus increasing the ripening effect. Last, the differences in the DSD shape under the different $N_a$ scenarios can also affect the number of deactivated drops. Namely, as $N_a$ increases, the DSD narrows in the bulk and bin simulations (Figs. 1j-1). For a broader distribution (i.e., for low $N_a$), there are fewer drops on the left side of the DSD, and thus fewer drops can deactivate. On the contrary, for a narrower distribution (i.e., for high $N_a$), there are more drops on the left side of the DSD, and thus more drops can deactivate. The LCMs exhibit less of a change in the DSD width with increasing $N_a$ and thus exhibit less sensitivity of the ripening effect to increased $N_a$.

Figures 1g-i show $D_{vol}$ at the height of the maximum LWC at a given time, in which the altitude of the maximum LWC increases over time while the vertical velocity is active. The LCMs and TAU-bin produce the smallest mean $D_{vol}$, with relatively good interscheme agreement; e.g., after 10 min, $D_{vol}$ from the LCMs and TAU-bin are within 1%-5% of each other, with the difference increasing with $N_a$. bin2d produces a larger $D_{vol}$, which, after 10 min, is around 12%-21% larger than that of LCM-FH and TAU-bin, with the difference increasing with $N_a$ and over time. Disabling the ripening effect in bin2d, i.e., bin2d_no sol, results in $D_{vol}$ that is around 12% larger than LCM-FH, but this difference does not change with $N_a$ or time. The bulk schemes produce the largest $D_{vol}$, which is between 23% and 60% larger than the LCMs, with CASIM and CASIM no-proc producing the largest $D_{vol}$ for all $N_a$, while MG2 produces the smallest when compared to the other bulk schemes.

<footnote n="4">⁴ MSSG-bin did not submit results for the CE case and hence is not included in this section.</footnote>
Figures 1j–l show $\sigma$ at the height of the maximum LWC at a given time. See appendix C for details on the calculation of $\sigma$ for the different schemes. Initially, the $\sigma$ values are similar among the LCMs, and these values are less than the bin2d and bulk schemes and larger than $\sigma$ from TAU-bin. As LWP increases, $\sigma$ of the DSD at the maximum LWC from the LCMs decreases, while $\sigma$ from all other schemes increases. After 10 minutes of simulation, $\sigma$ from the LCMs is at least an order of magnitude smaller than that from the bin and bulk schemes. In the 1D-CE, which represents only condensational growth in an updraft (no sedimentation or precipitation processes), tracking the height of the maximum LWC is analogous to tracking the ascent of an adiabatic parcel (note that lateral mixing processes and entrainment, which are not represented in the idealized simulations performed in this study, would affect the height of the maximum LWC). Under such parcel ascent, narrowing of the size distribution is expected (e.g., Rogers and Yau 1989; Pruppacher and Klett 1997) and demonstrated with the LCMs through the decrease in $\sigma$ over time, while the bin and bulk schemes fail to capture this expected behavior.

Figure 2 shows profiles at 10 min into the simulation for the W3 case to further demonstrate the differences in DSD evolution between the schemes. Figures 2a–c show that the profiles of LWC from each scheme are very similar; e.g., the maximum LWCs for all schemes are within 2% of each other. Comparison of the profiles of $N_d$ further demonstrates that the LCMs and bin schemes show relatively good agreement. For example, when comparing with the $N_d$ from the TAU-bin scheme,<fnref n="5" /> the LCMs, bin2d, CASIM, and MG2 produce between 5% and 17% less $N_d$ than TAU-bin, with the larger difference occurring above the maximum LWC, while CASIM-proc and LIMA produce up to 25% and 50% less $N_d$, respectively. When considering $D_{vol}$, TAU-bin and the LCMs exhibit good agreement, i.e., within 5% of each other, while $D_{vol}$ from bin2d and the bulk schemes tends to be around 15% larger and at least 25% larger, respectively. Figures 2j–l show that above cloud base and below the maximum LWC (2000 m), the LCMs produce a DSD that is much narrower than the other schemes. At cloud base, the $\sigma$ values from all schemes are the same order of magnitude, with TAU-bin producing the smallest $\sigma$. It is noted that the initial narrow DSD from TAU-bin results from the activation parameterization moving all activated aerosol into the first cloud bin, which imposes a narrow DSD. Figures 2j–l also show that $\sigma$ from the LCMs rapidly drops by an order of magnitude above cloud base; i.e., the DSD from the LCMs narrows rapidly with height until around the maximum LWC. In contrast, $\sigma$ from the bin schemes tends to stay constant with height, while $\sigma$ from the bulk schemes tends to increase with height until the maximum LWC and then it decreases above.

In summary, we initially compare all schemes with precipitation processes and sedimentation disabled, as a sanity check. All schemes are shown to produce very similar LWPs and vertical profiles of LWC. While this should be expected in such a constrained modeling setup, early iterations of the intercomparison highlighted poor agreement in LWP, which resulted from coupling issues and bugs. Focusing on the evolution of $N_d$ and the DSD, the comparison of the LCMs highlights that all three LCMs produce very similar evolution, with expected physical behavior, i.e., narrowing of the DSD with altitude. In contrast to the LCMs, multiple bin and bulk schemes fail to capture the narrowing of the DSD that should be associated with condensational growth in an updraft. The lack of DSD narrowing associated with the bulk schemes should be expected because all the bulk schemes used in the intercomparison have a fixed shape parameter for the assumed gamma cloud DSD; i.e., if $N_d$ is approximately constant with height and the DSD shape is fixed, the DSD will broaden with increasing LWC and narrow with decreasing LWC. Thus, such behavior is a structural feature of the bulk schemes presented in this paper. This explanation does not work for the bin schemes, because they are not constrained by a prescribed form of the DSD. We therefore propose three potential reasons for the lack of narrowing of the DSD with height with bin schemes when only condensation under vertical advection is considered: 1) the impact of vertical advection of bins in physical space on condensational growth (e.g., Morrison et al. 2018), 2) numerical diffusion in bin space, which stems from the bin structure and passing physical moments between bins, and/or 3) the influence of activation on the DSD.

Focusing on reason 3 first, Figs. 1j–l show that bin2d initially produces the broadest DSD when compared to the detailed schemes, and this is also associated with the broadest DSD at the end of the simulation. In contrast, TAU-bin produces the narrowest initial distribution, even narrower than the LCMs, and yet TAU-bin exhibits spectral broadening with an increase in LWC. Hence, while the initial DSD will impact the eventual breadth of the DSD, in TAU-bin and bin2d, the initial breadth does not seem to control whether a bin-resolved scheme will broaden or not in response to condensation. With this in mind, it seems that the broadening of the size distribution is associated with either reason 1 or 2. In an attempt to isolate these effects, we requested participants run a 0D box simulation, in which the initial DSD is prescribed and the DSD then grows through fixed supersaturation forcing over a 10-min period (see appendix B). Results from these simulations further demonstrated that under constant supersaturation forcing, with no vertical advection, the DSD from LCM-FH and LCM-MA narrow for all $N_a$ and initial $\mu$, while TAU-bin and bin2d exhibit broadening, with bin2d broadening the most (Fig. B1).

Morrison et al. (2018) used 1D simulations with vertical advection and parcel simulations, without vertical advection in Eulerian space to show that, while the DSD from an LCM narrows during condensational growth, the DSD from a bin scheme tends to broaden with height, and such broadening predominantly results from the combined effect of the vertical

<footnote n="5">⁵ The TAU-bin scheme is only employed as a reference here and such a use does not infer it is a benchmark. Instead, the TAU-bin scheme has been employed in the KiD model as a reference in the past, e.g. Shipway and Hill (2012), and we follow this by using it as a point of comparison. In the context of this work, any LCM or bin scheme could be used as a reference.</footnote>
<figure><img src="image_4.png" /><figcaption>FIG. 2. Profiles of (a)–(c) liquid water content (LWC), (d)–(f) $N_d$, (g)–(i) mean volume diameter ($D_{vol}$), and (j)–(l) the standard deviation of $D_{vol}$ ($\sigma$), from the 1D-CE case for W3. The profiles are derived from 10 min into the simulation to represent the column once the updraft has just ceased. The different columns indicate different aerosol number concentration simulations (increasing from left to right). Cyan lines indicate bulk schemes, black indicates the Eulerian bin schemes, and the blue lines show the LCMs.</figcaption></figure>
advection of bin variables (e.g., mass and/or number mixing ratio) during condensational growth. They argued that such broadening is artificial and unphysical, with single-moment bin representation being most prone to this artificial broadening, while a double-moment representation goes somewhat to negate this in a box or parcel model framework. The results from the 1D-CE and Box-CE (appendix B) tend to agree with the scheme-dependent behaviors described in Morrison et al. (2018). It is important to note that unlike the 1D and parcel model simulations presented in Morrison et al. (2018), the 1D-CE and Box-CE cases in this study are not directly comparable, so we can only conclude that the bin schemes produce nonphysical evolution of DSDs through condensational growth, and this is a result of both reasons 1 and 2. Finally, growth of aerosol to cloud droplets or activation assumptions (reason 3) clearly play a role in determining the initial DSD, and this seems to influence the final mean size and evolution of the DSD, but in this kinematic framework, activation assumptions do not seem to control whether non-physical broadening occurs.

## 6. Precipitation (PRECIP) results

In this section, the 1D-CE tests are extended to include precipitation processes, 1D-PRECIP, to investigate the impact of the scheme-dependent differences in the prediction of $N_d$, $D_{vol}$, and $\sigma$, highlighted by the 1D-CE tests, on the production and evolution of precipitation. Figures 3a-c and 4a-c show the evolution of LWP, while Figures 3d-f and 4d-f show the associated surface precipitation rates (including both cloud and rainwater) for W2 and W3, respectively. In addition, Tables C1 and C2 present the maximum precipitation rate, timing for the maximum, the total surface precipitation for the simulation, as well as the percent difference of these values from that produced by the TAU-bin scheme.

All schemes initially produce a peak in the LWP that corresponds well to the 1D-CE tests (cf. with Figs. 1a-c). Once the peak LWP is reached, the LWP remains relatively constant until the onset of surface precipitation. Figures 3 and 4 show that all schemes produce an increase in surface precipitation with an increase in LWP (cf. Figs. 3d-f and 4d-f with Figs. 3a-c and 4a-c), whereas increasing $N_a$ in all schemes, for a given $w$ forcing (i.e., W2 or W3), results in a decrease in $D_{vol}$ (Figs. 3j-l and 4j-l) and a decrease in the surface precipitation (Figs. 3d-f and 4d-f). Such general precipitation responses to changes in $w$ (which is inherently related to the total LWP) and $N_a$ should be expected, given the well-established effect of aerosols on warm rain production in the absence of dynamical feedbacks (e.g., Albrecht 1989), and have been demonstrated in the KiD model before (e.g., Shipway and Hill 2012; Hill et al. 2015). Figures 3d-f and 4d-f show, however, that the precipitation onset and the peak precipitation rate vary between the schemes and scheme types in both W2 and W3.

Inspection of Figs. 3 and 4 and Tables C1 and C2 reveals that the LCMs produce the lowest peak precipitation and slowest onset of rain for all $N_a$ and $w$ forcings. For example, when compared to TAU-bin, the LCM's simulation of W2 (W3) with $N_a = 50 \text{ cm}^{-3}$ produce between 54% (21%) and

$61\%$ (29%) less total rain. When $N_a = 300 \text{ cm}^{-3}$, only LCM-FH produces rain in both W2 and W3, but it is 98% less in W2 and 47% less in W3 than that produced in TAU-bin. UWLCM produces no rain in W2 with $N_a = 300 \text{ cm}^{-3}$ and 96% less rain compared to TAU-bin in W3, while LCM-MA produces no surface precipitation in W2 or W3 when $N_a = 300 \text{ cm}^{-3}$. Where rain is produced by the LCMs, the onset of the precipitation maximum at all $N_a$ is later than that of all other schemes (Tables C1 and C2). The relatively slow onset of precipitation from the LCMs should be expected because the mean droplet size and the breadth of the DSD over the first 1000 s are smaller (Figs. 3j-l and 4j-l) and narrower (Figs. 3m-o and 4m-o), respectively, compared to the bin and bulk schemes. This results in relatively slow broadening of the size distribution through collision-coalescence (Figs. 3m-o and 4m-o) when compared to the bin and bulk schemes. Figures 3d-f and 4d-f also show that at lower $N_a$, when all LCMs produce surface precipitation, the LCMs produce the largest range in the time of precipitation onset. For example, when $N_a = 50 \text{ cm}^{-3}$ ($300 \text{ cm}^{-3}$) in W3, the timings for the precipitation maximum from the bin schemes are within 220 s (110 s) of each other, yet for the LCMs that produce precipitation, the times are within 700 s (770 s).

In all cases, the bin and bulk schemes produce surface precipitation more rapidly than the LCMs but tend to exhibit less spread than the LCMs in both the onset time and the peak precipitation rate. For example, focusing on the bin schemes, in W2 (W3) when $N_a = 50 \text{ cm}^{-3}$, the total rain from the bin schemes is within 20% (10%) of the TAU-bin scheme, while the maximum precipitation rate and onset timing are within 30% (40%) and 26% (20%), respectively. bin2d produces the most rain with the earliest onset, with MSSG-bin and TAU-bin producing better agreement (Tables C1 and C2). It is interesting to note that, as $N_a$ increases in W2, the difference between bin2d and TAU-bin (or MSSG-bin) increases; e.g., when $N_a = 300 \text{ cm}^{-3}$, bin2d produces 44% more rain than TAU-bin and 53% more than MSSG-bin, while MSSG-bin and TAU-bin are within 10% of each other. The primary cause for the difference in the precipitation rate produced by the bin schemes may be related to the initial prediction of the DSD. For example, Figs. 3m-o and 4m-o show that at all $N_a$ and for both $w$ forcings, bin2d produces the broadest DSD most rapidly owing to its single-moment bin framework, and this leads to the earliest onset and the largest precipitation peak when compared with the other bin models. In addition, differences in precipitation rate may also be attributed to differences in the collision-coalescence kernel being employed in each model (Table 2). This is discussed further in section 7.

When considering the simulation of W2 (W3) with the bulk schemes, when $N_a = 50 \text{ cm}^{-3}$, the total rain from all bulk schemes is within 41% (11%) of the TAU-bin scheme, while the maximum precipitation rate and onset timing are within 30% (142%) and 36% (25%). Closer inspection of Tables C1 and C2 shows that one bulk scheme tends to cause the higher percentage difference, while the other schemes cluster around a smaller difference. For example, LIMA tends to produce more rain, more rapidly than the other bulk schemes in both W2 and W3. Such a difference is interesting because LIMA
<figure><img src="image_5.png" /><figcaption>FIG. 3. (a)–(c) LWP, (d)–(f) surface precipitation, (g)–(i) mean $N_d$, (j)–(l) mean volume diameter ($D_{vol}$) at cloud base, and (m)–(o) the standard deviation of $D_{vol}$ ($\sigma$) at cloud base from the 1D-PRECIP case for W2. Cyan lines indicate bulk schemes, black indicates the Eulerian bin schemes, and the blue lines show the LCMs. The different columns indicate different aerosol number concentration tests (increasing from left to right). Note that $D_{vol}$ and $\sigma$ presented here are derived from a fixed height of 700 m, which is just above cloud base. Using this height captures the change in mean size and DSD breadth that results from droplets falling through the cloudy column and the impact of this on the microphysical process rates.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>FIG. 4. As in Fig. 3, but for the 1D-PRECIP case and W3.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>FIG. 5. Surface precipitation (a)–(c) from the 1D-PRECIP case for W3. Cyan lines indicate bulk schemes (as in Fig. 4), black indicates the TAU-bin scheme (for reference), the blue lines show the experiments with LIMA [include Khairoutdinov and Kogan (2000) autocon- version and accretion] and CASIM (change the collection efficiency for self-collection of rain from 0.5 to 1.0), while the yellow lines show CASIM with triple-moment rain (a)–(c) $N_a = 50, 100, \text{ and } 150 \text{ cm}^{-3}$, respectively.</figcaption></figure>

uses a different autoconversion scheme compared with the other bulk schemes, i.e., Berry and Reinhardt (1974) in LIMA versus Khairoutdinov and Kogan (2000) autoconver- sion in all others (Table 3). To test the impact of this differ- ence in autoconversion parameterization, we ran LIMA with Khairoutdinov and Kogan (2000) autoconversion and accretion. Figure 5 shows that the surface precipitation from LIMA with Khairoutdinov and Kogan (2000) autoconversion tends to be in better agreement with the other bulk schemes and TAU-bin, particularly when considering the timing of the pre- cipitation maximum and total rain (see Tables C1 and C2 in appendix C).

Tables C1 and C2 also show that MG2 and, to a lesser ex- tent CASIM, produce a precipitation maximum that is much larger than all other schemes in W3; e.g., the maximum pre- cipitation from MG2 is 142% larger than TAU-bin. Such a be- havior in the bulk schemes is consistent with Shipway and Hill (2012) and is explained by excessive size during sedimenta- tion, which is discussed and demonstrated in Milbrandt and McTaggart-Cowan (2010). It is interesting to note, however, that when $N_a = 50 \text{ cm}^{-3}$, MG2 produces a 60% larger maxi- mum precipitation than CASIM, even though both schemes use the Khairoutdinov and Kogan (2000) autoconversion and accretion parameterizations, while employing different pre- scribed shape parameters for cloud and rain and different collection efficiencies for self-collection of rain (Table 3). To understand the influence of these differences, we conducted additional simulations, in which we increased the rain self- collection efficiency in CASIM from 0.5, which is used in CASIM, to 1.0, which is used in MG2. Figure 5 shows that in- creasing the rain self-collection efficiency from 0.5 to 1.0 re- sults in the maximum precipitation from MG2 now being less than 10% larger than that from CASIM for low $N_a$, even though the schemes use different prescribed shape parame- ters. This result indicates that excessive size sorting can be en- hanced by efficient rain self-collection or, more generally, excessive size sorting will be enhanced by a process that leads to broadening of the size distribution. Figure 5 and Table C2

also show results from triple-moment rain in CASIM compared with the other double-moment bulk schemes and TAU-bin. CASIM 3Mr produces a maximum precipitation rate that is within 17% (10%) of TAU-bin when $N_a = 50 (300) \text{ cm}^{-3}$, which indicates that the inclusion of a prognostic third mo- ment is the most effective way to negate the excessive size sorting over a range of $N_a$. The behavior of the triple-moment scheme agrees with the arguments presented in Milbrandt and McTaggart-Cowan (2010) and Shipway and Hill (2012).

## 7. Discussion

We have presented results from a 1D kinematic warm micro- physics case in the idealized KiD model to compare warm cloud microphysics schemes that range in complexity from computationally expensive, detailed schemes to simpler but computationally efficient bulk microphysics schemes. In gen- eral, LCMs and bin schemes are considered the best numeri- cal representation of cloud microphysics processes and have been used to research microphysical processes and aerosol– cloud interactions. Such schemes are also often used as a reference for the testing and development of bulk micro- physics schemes, which are fundamental in NWP and climate prediction modeling. In the past, when a detailed scheme has been applied to the development of bulk schemes, one scheme has been employed as a reference, and there has been little comparison of multiple detailed schemes, in the context of the behavior of the bulk schemes. The KiD-A intercompar- ison presented here starts to address this gap in understanding by comparing multiple detailed schemes to multiple bulk schemes.

### a. Condensation only

Initially the intercomparison focused on condensational growth in an updraft. These tests show that the 3 indepen- dently developed LCMs produce very similar evolution, with, for example, $D_{vol}$ at the LWC maximum within 1% of each other and all schemes exhibiting expected physical behavior,
<figure><img src="image_8.png" /><figcaption>FIG. 6. The mass-weighted mean volume diameter ($D_{vol}$) and the standard deviation of $D_{vol}$ ($\sigma$) for (top 2 rows) initial $\mu = 0$ and (bottom two rows) initial $\mu = 2.5$ from Box-Coll case. The different columns indicate different aerosol number concentrations (increasing from left to right). The black lines indicate the bin schemes, and the blue lines show the LCMs.</figcaption></figure>

i.e., narrowing of the DSD with altitude. In contrast, the bin and bulk schemes exhibit poorer interscheme agreement, e.g., $D_{vol}$ values at the LWC maximum are within 10% and 20% of each other for the bin and bulk schemes, respectively, and they fail to capture DSD narrowing with condensational growth in an updraft. It is argued that such behavior is structural in the bulk schemes, while tests presented here show that, as demonstrated by Morrison et al. (2018), numerical diffusion in bin space and the influence of the vertical advection of bin variables (e.g., mass and/or number mixing ratio) during condensational lead to unphysical broadening of the DSD. The consistency between the DSD evolution from the LCMs used in this intercomparison further supports the emerging idea that LCMs are a benchmark for condensational growth from aerosol, e.g., Morrison et al. (2018), and a strong candidate to improve on more traditional bin methods when numerically researching aerosol and droplet growth by condensation and the impact on DSD evolution, e.g., Grabowski et al. (2019) and Morrison et al. (2020).

## b. Precipitation

The condensation-only simulations were then extended to include precipitation processes (collision-coalescence and sedimentation). The comparison of the 1D precipitation simulations show that the LCMs are consistently the slowest to

produce precipitation, and they exhibit the largest interscheme range in the onset timing for surface precipitation, precipitation maximum, and total rain. In contrast, the bin and bulk schemes exhibit relatively good agreement in the predicted surface precipitation rates and timings. Where there are large differences, e.g., MG2 producing over twice the maximum precipitation rate compared to TAU-bin, simple sensitivity simulations have highlighted how a parameterization can be modified to explain the difference.

The apparent larger interscheme spread with the LCMs, when compared to the bin and bulk schemes, is surprising and somewhat contradictory given (i) the good agreement between DSD evolution in LCMs when only condensation is considered (section 5) and (ii) the fact that 2 of the LCMs tested, LCM-FH and UWLCM, use the same collision kernels. To investigate this further, results from the collision-coalescence box case (Box-Coll; appendix B) are presented (Fig. 6). The Box-Coll simulations permit the comparison of the DSD evolution from a prescribed cloud drop size distribution in the absence of transport; i.e., it removes the effects of the initial aerosol activation, the influence of vertical advection on condensation, and the impact of sedimentation, which are all present in the 1D-PRECIP case.

Figure 6 shows that when the LCM and bin schemes are compared in the Box-Coll case, there are qualitative similarities
in the time evolution of $D_{vol}$ with the 1D-PRECIP. For example, in the first 1000 s, bin2d produces the most rapid growth in $D_{vol}$ compared to all other schemes, while TAU-bin produces a slower growth, which is similar to the behavior in Figs. 3j-o and 4j-o from 1D-PRECIP. From Table 2, it is clear that bin2d and TAU-bin use different collision kernels, which means that it is not possible to isolate the cause of these differences beyond that they are caused by a combination of (i) structural differences (e.g., single versus double moment, flux method versus method of moments for collection, continuous size distribution between aerosol and drops versus activation to first bin) and (ii) differences in the collision-coalescence kernel. Figure 6 also shows that employing the Long (1974) kernel only in the TAU-bin scheme, instead of a combination of Long (1974) and Ochs et al. (1986), results in a doubling of $D_{vol}$ after 30 min. This highlights that while there are structural differences between the bin schemes, the collision-coalescence kernel on its own can lead to a quantifiable change in the evolution of the DSD.

Focusing on the LCMs, Fig. 6 shows that, as in 1D-PRECIP, LCM-MA produces the smallest $D_{vol}$ and slowest growth, which is most sensitive to increasing initial $N_a$ (and/or initial DSD breadth) when compared to the other LCMs. In contrast, while LCM-FH and UWLCM exhibit growth in $D_{vol}$ in all Box-Coll cases, LCM-FH produces progressively more rapid growth through the case and a larger $D_{vol}$ than UWLCM. It is interesting that LCM-FH and UWLCM produce different growth rates due to collision-coalescence, as both schemes use the Hall (1980) kernel and the collision-coalescence method is based on Shima et al. (2009). Understanding this difference further involves a detailed assessment that is beyond the scope of this work; however, some potential reasons for this difference could be (i) the size initialization of the superdroplets, (ii) interpolation of collision efficiencies, and/or (iii) numerical convergence with the number of collision pairs per second. In addition, differences between the sedimentation velocities resulting from DSD differences will influence the collision kernel and growth, which in turn will impact the DSD.

In summary, the interscheme spread in precipitation rates and onset timing from the LCMs is surprising given the agreement between LCMs in the condensation-only simulations. The Box-Coll results presented here exhibit similar trends to the 1D-PRECIP, and, as such, highlight the importance of the collision-coalescence kernel and its implementation in LCMs, which agrees with previous work (Unterstrasser et al. 2017) and further highlights the challenges associated with properly treating collision-coalescence in LCMs owing to numerical challenges (Morrison et al. 2020). Prior research has compared collision-coalescence algorithms in box model simulations (Unterstrasser et al. 2017), but such a comparison has yet to be extended to 1D cloud simulations. The comparison of the results from the precipitation simulations in this study begins to address this by directly comparing some algorithms, i.e., Shima et al. (2009), which is used in UWLCM, Andrejczuk et al. (2008) in LCM-MA, and Hoffmann (2017) and Hoffmann et al. (2017) in LCM-FH, in both the 1D and box simulations. This comparison shows that LCMs can be sensitive to the representation of collision-coalescence with no agreement between

<figure><img src="image_9.png" /><figcaption>FIG. 7. $S_0$ derived using the mean precipitation rate from the 1D-PRECIP case for (a) all schemes and (b) bin and bulk schemes. The black symbols show $S_0$ for W3 (high LWP), and the gray symbols show W2 (low LWP). The diamond symbols show the bin schemes, the circles show the bulk schemes, and the squares show the LCMs [only in (a)].</figcaption></figure>

schemes even when the schemes use the same collision efficiencies. Comparison of the 1D-PRECIP and Box-Coll cases indicates that the impact of these collision-coalescence issues on precipitation may be exacerbated when sedimentation is included, leading to poor agreement between the LCMs in the 1D case.

### c. Comparison of precipitation response to aerosol

As well as comparing the various microphysics schemes directly, one of the aims of this work is to understand how the different types of schemes simulate the response of precipitation to changes in $N_a$. At a given time, the 1D-PRECIP results show that all schemes simulate an increase in LWP and decrease in precipitation with increasing $N_a$ (Figs. 3 and 4); however, there is an indication that the change in precipitation with $N_a$ is scheme dependent and potentially scheme-type dependent. Here, we assess this scheme dependence by presenting the precipitation susceptibility ($S_0$) derived from the 1D simulations conducted with both vertical velocities (thus different LWPs) (Fig. 7).

The precipitation susceptibility is defined as follows (e.g., Sorooshian et al. 2009):
$$S_0 = -\frac{d \ln R}{d \ln N_a}, \qquad (1)$$

where R is the mean precipitation rate (including both cloud and rainwater), and $N_a$ is initial aerosol number concentration (or the prescribed $N_d$ for schemes lacking aerosol). Accordingly, this formulation implies that if $S_0$ is large, then there is a large sensitivity in R to changes in $N_d$. Obviously, the sample size for the regression in this intercomparison is small, with only a maximum of three data points representing the three $N_a$ tested per w forcing (and thus total LWP). As a result, it is important to note that the $S_0$ derived from this analysis should be considered qualitative and indicative of a scheme's behavior.

Figure 7a shows no $S_0$ for LCM-MA for W2 (low LWP) because the simulations did not produce enough surface precipitation; i.e., only $N_a = 50 \text{ cm}^{-3}$ produced precipitation. Having said this, the decrease in surface precipitation to zero as $N_a$ increases from 50 to 150 cm$^{-3}$, highlights that precipitation from LCM-MA in the W2 is very susceptible to changes in $N_d$ in this framework. UWLCM produces an $S_0$ that is larger than that of all other schemes for both W2 and W3, while LCM-FH produces the next largest $S_0$ for W2. When comparing $S_0$ from W2 and W3, UWLCM and LCM-FH produce the largest range in $S_0$, when compared to the other schemes tested.

Focusing on the other schemes, Fig. 7b shows that when the LWP is high (W3), all schemes produce low $S_0$ values that are similar. When the LWP is relatively low (W2), however, $S_0$ is larger and there is much more variation between the schemes. The bin schemes produce the lowest $S_0$ and the smallest range in $S_0$ between different LWPs (w forcings) and between schemes. The bulk schemes produce an $S_0$ that is between the bin schemes and LCMs. Comparison of the bulk schemes suggests that in the KiD framework, changing the number of predicted moments or autoconversion and accretion parameterization can impact $S_0$, especially under a lower LWP. For example, using three moments in CASIM, i.e., mass, $N_d$, and radar reflectivity (sixth moment), results in a reduction in $S_0$ compared to the double-moment schemes, while LIMA-KK and CASIM-2M, which use Khairoutdinov and Kogan (2000), produce very similar $S_0$. MG2 produces the largest $S_0$ for the bulk schemes, and this seems to be related to the impact of excessive size sorting in the lowest $N_a$ case. This is interesting because it suggests that $S_0$ from double-moment schemes can be influenced by the efficiency of rain self-collection and not just autoconversion. Overall, the bulk schemes exhibit interscheme variation in $S_0$ that is dependent on the specific parameterization and/or complexity; however, the spread in $S_0$ from the bulk schemes is much less than that highlighted by comparison of $S_0$ from the bin schemes and LCMs. Hence, as with the comparison of the precipitation rates, totals, and timings, the bulk schemes presented fall between the detailed schemes presented.

## 8. Summary and conclusions

In this work, we present an intercomparison of warm cloud microphysics schemes that range in complexity from computationally expensive, detailed schemes, i.e., LCMs and size bin-resolved schemes (single and double moment), to simpler

but computationally efficient bulk microphysics schemes. The schemes include a range of coupling to aerosol from prescribed $N_d$ to fully coupled aerosol with in-cloud physical aerosol processing. The intercomparison was conducted using a common dynamic driver (the KiD model) with well-defined 1D and box model cases and using common output diagnostics. This highly constrained framework permits a consistent comparison of the schemes, which focuses on the microphysics parameterizations without the need to consider the influence of different dynamics and the complication of dynamic feedbacks.

As outlined at the beginning of this paper, the aims of this intercomparison are the following:

1) Test the hypothesis that detailed microphysics schemes produce accurate and consistent cloud microphysics representation for warm rain processes and hence represent a numerical benchmark. If not, investigate and suggest why.

2) Test how cloud microphysics schemes (of various complexities) compare across a range of $N_d$ with and without aerosol physical processing.

3) Develop and make available a cloud modeling framework for developing and testing future schemes (see data availability statement for details).

This intercomparison begins to answer point 1 by showing that three independently developed LCMs produce very similar results for condensation-only simulations. Such consistency between LCMs further supports the idea that LCMs are a benchmark for numerical research into aerosol/droplet growth by condensation and the impact of condensational growth on DSD evolution (e.g., Morrison et al. 2018; Grabowski et al. 2019; Morrison et al. 2020). In fact, it could be argued that LCMs may become the numerical benchmarking tool to research atmospheric phenomena that are dominated by condensation, with results being used to validate and improve parameterizations in bulk microphysics schemes. For example, Schwenkel and Maronga (2020) recently used an LCM to study fog and showed that the variation in the DSD breadth may be important in fog forecasting. When considering precipitation and the impact of changes in aerosol on precipitation, the agreement between the LCMs diminishes, while the bin methods produce more consistent results. On the one hand, the relative consistency between the precipitation from the bin schemes is reassuring because these schemes have been used over many decades to develop and validate warm rain microphysics parameterizations. On the other hand, in the context of a numerical benchmark, the relative consistency between bin schemes is a problem given the artificial broadening of the DSD resulting from vertical advection during condensational growth can and will impact collision-coalescence, e.g., Witte et al. (2019) and Chandrakar et al. (2022). Overall, the intercomparison of precipitation results indicates that we do not have a satisfactory numerical benchmark for precipitation processes, i.e., the bin methods cannot be considered a benchmark due to artificial condensation broadening issues, while the LCMs are a benchmark for condensation, they exhibit little consistency when considering precipitation. Further, given the spread in precipitation and collision-
coalescence solutions in the LCMs, we urge caution when applying LCM results to the development or modification of bulk microphysics schemes.

This intercomparison has addressed point 2 by showing that in the KiD model, bin schemes tend to produce a similar sensitivity to changes in $N_a$, while the LCMs produce a wide range of sensitivities. It was found that aerosol resuspension was not as important as the scheme type in the simple dynamic framework applied herein. This does not mean that aerosol physical processing is not important. In fact, in-cloud aerosol processing is a significant omission from many climate models and global aerosol-cloud interaction studies. However, before we can benchmark aerosol processing, we need to be able to benchmark precipitation, and, as stated above, this is a challenge with present models.

As well as investigating detailed schemes, an equally important part of this intercomparison was to demonstrate how a range of double-moment bulk schemes compare to the detailed schemes. Double-moment bulk schemes are gradually becoming the standard in climate prediction and there are efforts to include such schemes in NWP (e.g., Benjamin et al. 2016; Field et al. 2023; McTaggart-Cowan et al. 2019). This intercomparison shows that in the kinematic framework, while there is clear divergence in the LCM's simulation of precipitation, the double-moment bulk schemes are relatively consistent with each other and the bin schemes, which is reassuring. Sensitivity tests conducted have highlighted that in a kinematic model, such as KiD, we are able to rapidly test and understand differences between the bulk schemes, e.g., changing the efficiency of rain self-collection or modifying the autoconversion scheme. The uncertainty relating to a numerical benchmark for precipitation processes does raise significant issues for the development of warm rain bulk schemes. As already discussed, there is a long history of developing warm rain bulk parameterizations either directly from the output of detailed schemes or by employing detailed schemes to test the scientific performance. The results presented here do not undermine this history but they do highlight that relying on either bin microphysics or LCMs alone to develop and demonstrate bulk parameterizations for precipitation is a risk. Hence, as has been the process in the past, e.g., Khairoutdinov and Kogan (2000), Kogan (2013), Thompson and Eidhammer (2014), and Jouan et al. (2020), it is important that bulk schemes continue to be developed and tested with combination of observations and understanding derived from detailed schemes.

Finally, given the results and conclusions of this intercomparison, it is important to state the caveats and make final recommendations for further research and development. First, we acknowledge that, while the results presented in this work have been derived from a consistent dynamical framework, only 1D and box simulations have been used. These simulations have highlighted uncertainty in the comparison of schemes, which needs to be investigated further and understood. By focusing on kinematic and box simulations, however, we ignore, for example, the impacts of turbulence on collision-coalescence and turbulent mixing on evaporation and DSD evolution. We acknowledge that the evolution of the DSD and precipitation formation in a 3D turbulent flow

and associated differences between schemes may differ from the results presented here, and this may be a potential cause for the narrow DSDs predicted by the LCMs; such mixing in a 3D turbulent environment would tend to broaden the DSDs. The resulting narrow DSDs in this study compared to those expected in a more realistic turbulent environment could impact collision-coalescence. It would be interesting to understand whether the spread in precipitation rates from the LCMs is still apparent in a turbulent environment. We therefore recommend that future intercomparisons include large-eddy simulations (LES) to assess the impact of turbulence. Ideally, such an intercomparison would employ one LES with a well-defined cloud microphysics application programming interface so that participating schemes can be easily coupled. While this is a technical challenge, such an intercomparison would ensure dynamic consistency, permit focus on the microphysics, and act as a very valuable community tool for numerical benchmarking. We also acknowledge that this intercomparison includes three bin schemes and three LCMs only, so the sample size is small. However, even though the model sample size is small, the spread in the LCM simulations of precipitation compared to the bin schemes is clear. We, therefore, recommend that there needs to be continued effort researching and comparing collision-coalescence methods and sedimentation in detailed microphysics schemes. From the results presented here, this effort should focus on LCMs to understand the cause of the spread, while also investigating the impact that changes in $N_a$ have on collision-coalescence in both bin schemes and LCMs. Such effort will aid the continued progress toward a numerical benchmark for warm microphysics.

*Acknowledgments.* S. Arabas acknowledges support from the Foundation for Polish Science (POIR.04.04.00-860 00-5E1C/18-00). P. Dziekan and H. Pawlowska acknowledge support from the Poland's National Science Centre (Decisions 2012/06/M/ST10/00434, 2016/23/B/ST10/00690, and 2018/31/D/ST10/01577), from the PLGrid infrastructure and from the Interdisciplinary Centre for Mathematical and Computational Modelling at the University of Warsaw. A. Gettelman acknowledges support from the National Center for Atmospheric Research, which is supported by the U.S. National Science Foundation. F. Hoffmann is supported by the Emmy Noether program of the German Research Foundation (DFG) under Grant HO 6588/1-1.

*Data availability statement.* Through the input and effort from the participants in this intercomparison, the base code for the KiD model has been publicly accessible through a GitHub repository (https://github.com/Adehill/KiD-A) since the summer of 2017. This repository now includes all tests presented in this, and previous work as well as general documentation for the KiD model (https://github.com/Adehill/KiD-A/blob/master/docs/KiD_2.3.2625.pdf) and links to the specifications for this intercomparison (https://adehill.github.io/KiD-A/). The raw model output for all microphysics schemes and all cases is archived at https://doi.org/10.5281/zenodo.7758774. The availability of the microphysics schemes used in this intercomparison is as follows:
* MSSG-Bin model and its results are available from the model developer, Ryo Onishi, upon reasonable request.

* The TAU-bin version used in this intercomparison can be obtained by contacting Adrian Hill.

* bin2d can be obtained by contacting Zachary J. Lebo.

* UWLCM used 1) libcloudph++ v2.1.3-kida, which is accessible through https://doi.org/10.5281/zenodo.5196232. 2) For binding libcloudph++ with the KiD-a code, the kid-libcloud v1.0 https://doi.org/10.5281/zenodo.5196234 was used.

* LCM-MA can be obtained by contacting Miroslaw Andrejczuk.

* LCM-FH can be obtained by contacting Fabian Hoffmann.

* CASIM is available on the Met Office Science Repository and can be obtained by contacting Adrian Hill.

* LIMA can be obtained by contacting Benoit Vié.

* MG2 is part of the Community Atmosphere Model version 6 at https://github.com/ESCOMP/CAM.

# APPENDIX A

## Equations for Cloud and Rain Size Distributions

The KiD-A intercomparison included bulk, bin, and LCM microphysics, which all have different definitions of the size distributions. In this appendix, the size distribution assumptions and the definitions, which are used in the paper, are outlined.

### a. Mean volume diameter and standard deviation for the LCMs and bin schemes

The mean volume diameter for the DSD is defined as follows:

$$D_{\text{vol}} = \left( \frac{6\text{LWC}}{\pi N_d \rho_l} \right)^{1/3}, \qquad (\text{A1})$$

where $\rho_l$ is the density of liquid water, LWC is the liquid water content [Eq. (A2)], and $N_d$ is the total drop concentration [Eq. (A3)].

Moreover,

$$\text{LWC} = \sum_{k=1}^{\text{nbins}} m_k \qquad (\text{A2})$$

and

$$N_d = \sum_{k=1}^{\text{nbins}} n_k, \qquad (\text{A3})$$

where $m_k$ is the mass concentration for bin k, $n_k$ is the number concentration for bin k, and “nbins” is the total number of bins.

In this work, we mass weight Eq. (A1) so that it is applicable to distributions that have evolved from precipitation processes, which tend toward low $N_d$ but high mass. The mass-weighted $D_{\text{vol}}$ for the DSD, referred to as $\langle D_{\text{vol}} \rangle_{\text{mass}}$, is defined as follows:

$$\langle D_{\text{vol}} \rangle_{\text{mass}} = \frac{\sum_{k=1}^{\text{nbins}} D_k m_k}{\text{LWC}}, \qquad (\text{A4})$$

where $m_k$ is the total mass for bin k, and $D_k$ is the mean diameter for a bin, i.e.,

$$D_k = \left( \frac{6m_k}{\pi n_k \rho_l} \right)^{1/3}. \qquad (\text{A5})$$

Using $\langle D_{\text{vol}} \rangle_{\text{mass}}$, we derive the standard deviation ($\sigma$) of the DSD as follows:

$$\sigma = \left[ \frac{1}{N_d - 1} \sum_{k=1}^{\text{nbins}} n_k (D_k - \langle D_{\text{vol}} \rangle_{\text{mass}})^2 \right]^{0.5}. \qquad (\text{A6})$$

To compare the bin schemes and the LCMs in this work, all submitted LCM data were discretized into bins by the participants. This enabled the application of Eqs. (A4) and (A6) to both the bin schemes and LCMs.

### b. Mean volume diameter and standard deviation for the bulk schemes

For CASIM and MG2, the mean volume diameter for the cloud and rain category are defined as follows:

$$D_{\text{vol\_x}} = \frac{\mu + 4.0}{\lambda}, \qquad (\text{A7})$$

where x is either cloud or rain, $\mu$ is the shape parameter, and $\lambda$ is the slope parameter of the gamma distribution, which is based on the mass mixing ratio and number concentration. In this work, CASIM uses a fixed $\mu$ with a value of 2.5 for both cloud and rain. MG2 uses cloud drop number to derive the cloud shape parameter with the method based on Martin et al. (1994) and a fixed shape parameter of 0 for rain. For both MG2 and CASIM, $\lambda$ is defined as follows:

$$\lambda_x = \left[ \frac{\pi \rho N_{d\_x} \Gamma(\mu + 4)}{6 q_x \Gamma(\mu + 1)} \right]^{1/3}. \qquad (\text{A8})$$

The standard deviation ($\sigma_x$) for cloud and rain is then given by

$$\sigma_x = \left( \frac{\mu_x + 1}{\lambda_x^2} \right)^{0.5}. \qquad (\text{A9})$$

For LIMA, $D_{\text{vol}}$ is defined as follows:

$$D_{\text{vol\_x}} = \frac{1}{\lambda_x} \left[ \frac{\Gamma(\mu_x + 3/\alpha_x)}{\Gamma(\mu_x)} \right]^{1/3}, \qquad (\text{A10})$$

where $\lambda$ is defined by

$$\lambda_x = \left[ \frac{\pi \rho N_{d\_x} \Gamma(\mu_x + 3/\alpha_x)}{6 q_x \Gamma(\mu_x)} \right]^{1/3}. \qquad (\text{A11})$$

Moreover, $\sigma_x$ is given by

$$\sigma_x = \left[ \frac{1}{\lambda_x^2} \left\{ \frac{\Gamma(\mu_x + 6/\alpha)}{\Gamma(\mu_x)} - \left[ \frac{\Gamma(\mu_x + 3/\alpha)}{\Gamma(\mu_x)} \right]^2 \right\}^{1/3} \right]^{0.5}. \qquad (\text{A12})$$

In LIMA, $\mu$ is 1 for cloud and 2 for rain, while $\alpha$ is 3 for cloud and 1 for rain.
The above equations give the values of $D_{vol}$ and $\sigma$ for the cloud and rain categories separately. To compare with the LCMs and bin schemes, it is necessary to derive values of $D_{vol}$ and $\sigma$ that represent the entire liquid distribution. In this work, this has been done by mass weighting the cloud and rain categories (using the cloud and rain mass mixing ratios, $q_c$ and $q_r$, respectively) as follows:

$$ \langle D_{vol} \rangle_{\text{mass}} = \frac{q_c(D_{vol\_c}) + q_r(D_{vol\_r})}{q_r + q_c}, \qquad (A13) $$

$$ \sigma_{cr} = \left[ \frac{q_c(\sigma_c^2) + q_r(\sigma_r^2)}{q_r + q_c} \right]^{0.5}. \qquad (A14) $$

# APPENDIX B

## KiD Box Case Description

The KiD-A intercomparison started with the 1D case due to its simplicity and traceability back to comparisons of various bulk schemes (Shipway and Hill 2012; Hill et al. 2015). However, following the initial submissions, which highlighted significant differences between the detailed schemes and in light of recent work (Morrison et al. 2018), in 2018, we extended the KiD model to include two “box” cases that isolate and compare the evolution of specific processes when the schemes are initialized with the same drop size distribution and the results are not impacted by transport effects. The two cases are 1) a box condensation-only test (Box-CE) and 2) a box collision-coalescence-only test (Box-Coll). In both test cases, the initial cloud drop size distribution is prescribed using bulk variables of total water mass ($Q$), number concentration ($N_d$), and shape of the distribution ($\mu$). The initial cloud drop size distribution is represented by a gamma function:

$$ N(D) = N_0 D^{\mu} e^{-\lambda D}, \qquad (B1) $$

where $N(D)$ is the number of drops of diameter $D$, $N_0$ is the intercept parameter, $\mu$ is the shape parameter (note: $\mu = 0$ results in an exponential distribution), and $\lambda$ is the slope parameter. The gamma functional form was selected because it is used in almost all operational NWP and climate bulk microphysics schemes. The integral of Eq. (B1) between two diameters, i.e., the boundaries of bins in the Eulerian drop size distribution used in bin microphysics schemes, represents the number of drops in that size range, which is used to initialize the bin schemes. Similarly, mass can be attained for each bin. The exception to this is the bin2d scheme, which has an additional dimension representing the solute mass in drops. For simplicity in comparison with the other schemes in the box model tests, the solute was neglected in the bin2d scheme.

### a. Condensation-only box case—Box-CE

In Box-CE, all processes other than condensational growth are switched off; i.e., activation, collision-coalescence, breakup, and sedimentation are switched off. The box is initialized with

a potential temperature ($\theta$) of 289 K and no horizontal or vertical velocity. The initial vapor mixing ratio is set so that the box is supersaturated, and condensational growth is forced by applying a constant supersaturation of 0.1%. Both the temperature and water vapor are held constant to ensure a constant supersaturation in this setup. The simulations are run with a 1-s time step for 10 min. The initial drop size distributions are defined as above with the following set of parameters: $N_d = 50, 150, \text{ and } 300 \text{ cm}^{-3}$, $Q = 0.1 \text{ g kg}^{-1}$, and $\mu = 0$ and 2.5.

### b. Collision-coalescence-only box case—Box-Coll

In Box-Coll, all processes other than collision-coalescence are switched off. The grid box is initialized with the same potential temperature, lack of wind, and range of $N_d$ and $\mu$ as Box-CE but no supersaturation forcing is applied. The main differences between Box-Coll and Box-CE are that the initial $Q$ is set to $1.0 \text{ g kg}^{-1}$ (which allows for collisional growth without condensation), and the simulations are run for 1 h to ensure that sufficient collision-coalescence has occurred. It is important to note that no effort is made to control for the collision-coalescence algorithm nor the kernels/efficiencies used by the different models. Thus, differences in the results of this test are solely attributed to the collision-coalescence representation and/or collection kernels/efficiencies used by the different models.

### c. Results and discussion from the Box-CE simulations

#### BOX-CE RESULTS

Here we present results from the Box-CE case to demonstrate how the DSD evolves in the absence of vertical advection and activation. Figure B1 shows the time evolution of the mass-weighted mean volume diameter ($D_{vol}$) and standard deviation ($\sigma$) from the intercomparison participants that submitted these results, i.e., TAU-bin, bin2d, LCM-FH, and LCM-MA. By the end of the simulation with $\mu = 0.0$ ($\mu = 2.5$), once all schemes have been exposed to the same supersaturation forcing for the same period of time, bin2d produces a $D_{vol}$ that is between 32% (34%) and 50% (46%) larger than that produced by LCM-FH, while $\sigma$ from bin2d is 17% (84%) to 150% (380%) larger, with the difference increasing with $N_d$. In contrast, there is much better agreement between TAU-bin and LCM-FH, with TAU-bin producing a $D_{vol}$ for experiment $\mu = 0.0$ ($\mu = 2.5$) that is between 7% (3%) and 12% (8%) larger than that produced by LCM-FH, while $\sigma$ from TAU-bin is between 0% (5%) and 22% (75%) larger than LCM-FH, again with the difference increasing with $N_d$.

# APPENDIX C

#### Data Tables

Tables C1 and C2 present the maximum precipitation rate, timing for the maximum, the total surface precipitation for the simulation, as well as the percent difference of these values from that produced by the TAU-bin scheme.
<figure><img src="image_10.png" /><figcaption>FIG. B1. The mass-weighted mean volume diameter, $D_{vol}$, and the standard deviation of $D_{vol}$, $\sigma$, for (top 2 rows) initial $\mu = 0$ and (bottom two rows) initial $\mu = 2.5$ from the Box-CE case. The different columns indicate different aerosol number concentrations (increasing from left to right). The black lines indicate the bin schemes, and the blue lines show the LCMs.</figcaption></figure>

# APPENDIX D

## Microphysics Schemes

The following provides details regarding each of the microphysics schemes that contributed to this inter-comparison.

### a. MG2

MG2 (Gettelman and Morrison 2015) is a double-moment bulk microphysics scheme developed for GCMs and based on Morrison et al. (2005). It is essentially the same as Morrison et al. (2005) for warm processes, but adds a more detailed treatment of the mixed phase and vapor deposition process (albeit not relevant for this warm-phase cloud intercomparison). Autoconversion and accretion are parameterized following Khairoutdinov and Kogan (2000). As MG2 was designed for climate models with large grid spacings, subgrid probability distribution functions are used to represent heterogeneity in the cloud properties. Given the small grid spacing used in this intercomparison, this subgrid-scale parameterization was turned off. Condensation is treated via saturation adjustment, whereby residual water vapor in excess of saturation is removed inside the microphysics.

### b. LIMA

Liquid Ice Multiple Aerosols (LIMA) is a double-moment microphysical scheme that includes a detailed representation of aerosol-cloud interactions (Vié et al. 2016). In the warm phase, it represents the evolution of the mixing ratio and number concentration for both cloud droplets and raindrops. A 3D prognostic aerosol population is used as cloud condensation nuclei (CCN). These aerosols are transported by the dynamical forcing of the KiD case, and aerosol-cloud interactions include activation, below-cloud aerosol scavenging by rain (Berthet et al. 2010), and aerosol release when cloud droplets evaporate.

LIMA uses an immediate saturation adjustment procedure at each model time step, and therefore does not allow supersaturation (unless no CCN are present to form droplets). Thus, the CCN activation is parameterized after Cohard et al. (1998), which was extended to handle the competition between several CCN types. A maximum diagnostic supersaturation is determined based on the air parcel vertical lifting, condensation, and cooling rate, and used to calculate the number of droplets formed using the CCN activation spectrum. The Berry and Reinhardt (1974) autoconversion parameterization, based on simulations of cloud droplet populations evolving via collection processes, is used to form raindrops. For this intercomparison, the Khairoutdinov and Kogan (2000) formulation was also
<table><caption>TABLE C1. The maximum precipitation rate (precip max; mm h⁻¹), the time of the maximum precipitation rate (s), and the total precipitation amount (mm) for the W2 case. The percentage difference (% diff) of each of these variables compared to that from the TAU-bin scheme is also presented.</caption><thead><tr><th>Scheme type</th><th>Scheme</th><th>N<sub>d</sub>, N<sub>a</sub> (cm<sup>-3</sup>)</th><th>Precip max (mm h<sup>-1</sup>)</th><th>% diff</th><th>Precip max time (s)</th><th>% diff</th><th>Total precip (mm)</th><th>% diff</th></tr></thead><tbody><tr><td rowspan="4">Bin</td><td>TAU-bin</td><td rowspan="4">50</td><td>4.968</td><td>—</td><td>1430.0</td><td>—</td><td>294.745</td><td>—</td></tr><tr><td>bin2d</td><td>6.422</td><td>29.267</td><td>1050.0</td><td>-26.573</td><td>347.174</td><td>17.788</td></tr><tr><td>bin2d</td><td>6.277</td><td>26.349</td><td>1050.0</td><td>-26.573</td><td>344.498</td><td>16.88</td></tr><tr><td>MSSG-bin</td><td>4.344</td><td>-12.56</td><td>1380.0</td><td>-3.497</td><td>235.264</td><td>-20.181</td></tr><tr><td rowspan="3">LCM</td><td>LCM-FH</td><td rowspan="3">50</td><td>2.178</td><td>-56.159</td><td>1760.0</td><td>23.077</td><td>112.302</td><td>-61.899</td></tr><tr><td>LCM-MA</td><td>4.128</td><td>-16.908</td><td>3020.0</td><td>111.189</td><td>120.056</td><td>-59.268</td></tr><tr><td>UWLCM</td><td>1.529</td><td>-69.223</td><td>2190.0</td><td>53.147</td><td>134.896</td><td>-54.233</td></tr><tr><td rowspan="7">Bulk</td><td>CASIM</td><td rowspan="7">50</td><td>4.051</td><td>-18.458</td><td>1320.0</td><td>-7.692</td><td>189.413</td><td>-35.737</td></tr><tr><td>CASIM proc</td><td>3.439</td><td>-30.777</td><td>1220.0</td><td>-14.685</td><td>172.542</td><td>-41.461</td></tr><tr><td>CASIM, eff = 1</td><td>5.962</td><td>20.008</td><td>1260.0</td><td>-11.888</td><td>242.689</td><td>-17.661</td></tr><tr><td>CASIM 3Mr</td><td>2.313</td><td>-53.442</td><td>1370.0</td><td>-4.196</td><td>166.274</td><td>-43.587</td></tr><tr><td>MG2</td><td>5.932</td><td>19.404</td><td>1310.0</td><td>-8.392</td><td>294.907</td><td>0.055</td></tr><tr><td>LIMA</td><td>5.719</td><td>15.117</td><td>910.0</td><td>-36.364</td><td>332.399</td><td>12.775</td></tr><tr><td>LIMA KK</td><td>6.492</td><td>30.676</td><td>1130.0</td><td>-20.979</td><td>261.674</td><td>-11.22</td></tr><tr><td rowspan="4">Bin</td><td>TAU-bin</td><td rowspan="4">150</td><td>3.053</td><td>—</td><td>1910.0</td><td>—</td><td>253.303</td><td>—</td></tr><tr><td>bin2d</td><td>4.927</td><td>61.382</td><td>1210.0</td><td>-36.649</td><td>317.802</td><td>25.463</td></tr><tr><td>bin2d</td><td>4.614</td><td>51.13</td><td>1240.0</td><td>-35.079</td><td>309.158</td><td>22.051</td></tr><tr><td>MSSG-bin</td><td>2.446</td><td>-19.882</td><td>1720.0</td><td>-9.948</td><td>199.387</td><td>-21.285</td></tr><tr><td rowspan="3">LCM</td><td>LCM-FH</td><td rowspan="3">150</td><td>0.707</td><td>-76.842</td><td>2940.0</td><td>53.927</td><td>54.586</td><td>-78.45</td></tr><tr><td>LCM-MA</td><td>0.0</td><td>-100.0</td><td>—</td><td>—</td><td>0.0</td><td>-100.0</td></tr><tr><td>UWLCM</td><td>0.131</td><td>-95.709</td><td>3560.0</td><td>86.387</td><td>3.872</td><td>-98.471</td></tr><tr><td rowspan="7">Bulk</td><td>CASIM</td><td rowspan="7">150</td><td>2.035</td><td>-33.344</td><td>1570.0</td><td>-17.801</td><td>124.549</td><td>-50.83</td></tr><tr><td>CASIM proc</td><td>1.991</td><td>-34.785</td><td>1390.0</td><td>-27.225</td><td>121.395</td><td>-52.075</td></tr><tr><td>CASIM, eff = 1</td><td>2.549</td><td>-16.508</td><td>1540.0</td><td>-19.372</td><td>158.711</td><td>-37.343</td></tr><tr><td>CASIM 3Mr</td><td>1.622</td><td>-46.872</td><td>1800.0</td><td>-5.759</td><td>122.902</td><td>-51.48</td></tr><tr><td>MG2</td><td>2.157</td><td>-29.348</td><td>1600.0</td><td>-16.23</td><td>182.72</td><td>-27.865</td></tr><tr><td>LIMA</td><td>4.078</td><td>33.574</td><td>960.0</td><td>-49.738</td><td>263.157</td><td>3.89</td></tr><tr><td>LIMA KK</td><td>3.415</td><td>11.857</td><td>1300.0</td><td>-31.937</td><td>181.481</td><td>-28.354</td></tr><tr><td rowspan="4">Bin</td><td>TAU-bin</td><td rowspan="4">300</td><td>2.092</td><td>—</td><td>2470.0</td><td>—</td><td>209.978</td><td>—</td></tr><tr><td>bin2d</td><td>4.346</td><td>107.744</td><td>1310.0</td><td>-46.964</td><td>303.683</td><td>44.626</td></tr><tr><td>bin2d</td><td>3.964</td><td>89.484</td><td>1360.0</td><td>-44.939</td><td>289.538</td><td>37.889</td></tr><tr><td>MSSG-bin</td><td>1.973</td><td>-5.688</td><td>1890.0</td><td>-23.482</td><td>193.071</td><td>-8.052</td></tr><tr><td rowspan="3">LCM</td><td>LCM-FH</td><td rowspan="3">300</td><td>0.182</td><td>-91.3</td><td>3600.0</td><td>45.749</td><td>2.948</td><td>-98.596</td></tr><tr><td>LCM-MA</td><td>0.0</td><td>-100.0</td><td>—</td><td>—</td><td>0.0</td><td>-100.0</td></tr><tr><td>UWLCM</td><td>0.0</td><td>-100.0</td><td>—</td><td>—</td><td>0.0</td><td>-100.0</td></tr><tr><td rowspan="7">Bulk</td><td>CASIM</td><td rowspan="7">300</td><td>1.086</td><td>-48.088</td><td>1810.0</td><td>-26.721</td><td>84.083</td><td>-59.956</td></tr><tr><td>CASIM proc</td><td>1.223</td><td>-41.539</td><td>1560.0</td><td>-36.842</td><td>86.117</td><td>-58.988</td></tr><tr><td>CASIM, eff = 1</td><td>1.3</td><td>-37.859</td><td>1790.0</td><td>-27.53</td><td>108.266</td><td>-48.439</td></tr><tr><td>CASIM 3Mr</td><td>1.231</td><td>-41.157</td><td>2030.0</td><td>-17.814</td><td>91.516</td><td>-56.416</td></tr><tr><td>MG2</td><td>0.732</td><td>-65.01</td><td>1950.0</td><td>-21.053</td><td>97.003</td><td>-53.803</td></tr><tr><td>LIMA</td><td>1.907</td><td>-8.843</td><td>1040.0</td><td>-57.895</td><td>180.701</td><td>-13.943</td></tr><tr><td>LIMA KK</td><td>1.249</td><td>-40.296</td><td>1470.0</td><td>-40.486</td><td>111.691</td><td>-46.808</td></tr></tbody></table>

implemented. The representation of other collision-coalescence processes (cloud droplets and raindrops self-collection and accretion) use the collection kernels of Long (1974), and a splitting Eulerian scheme is used for the sedimentation of cloud droplets and raindrops.

### c. CASIM

Cloud Aerosol Interacting Microphysics scheme (CASIM) is a multimoment bulk water scheme that has been developed at the Met Office as a long-term replacement for the operational microphysics scheme. CASIM was first introduced in
<table><caption>TABLE C2. As in Table C2, but for the W3 case.</caption><thead><tr><th>Scheme type</th><th>Scheme</th><th>N<sub>d</sub>, N<sub>a</sub> (cm<sup>-3</sup>)</th><th>Precip max (mm h<sup>-1</sup>)</th><th>% diff</th><th>Precip max time (s)</th><th>% diff</th><th>Total precip (mm)</th><th>% diff</th></tr></thead><tbody><tr><td rowspan="4">Bin</td><td>TAU-bin</td><td rowspan="4">50</td><td>16.329</td><td>—</td><td>1110.0</td><td>—</td><td>856.426</td><td>—</td></tr><tr><td>bin2d</td><td>23.973</td><td>46.812</td><td>890.0</td><td>-19.82</td><td>944.671</td><td>10.304</td></tr><tr><td>bin2d</td><td>23.783</td><td>45.649</td><td>890.0</td><td>-19.82</td><td>943.057</td><td>10.115</td></tr><tr><td>MSSG-bin</td><td>21.16</td><td>29.585</td><td>1060.0</td><td>-4.505</td><td>812.172</td><td>-5.167</td></tr><tr><td rowspan="3">LCM</td><td>LCM-FH</td><td rowspan="3">50</td><td>18.842</td><td>15.39</td><td>1210.0</td><td>9.009</td><td>605.164</td><td>-29.338</td></tr><tr><td>LCM-MA</td><td>19.065</td><td>16.755</td><td>1910.0</td><td>72.072</td><td>671.033</td><td>-21.647</td></tr><tr><td>UWLCM</td><td>15.813</td><td>-3.16</td><td>1500.0</td><td>35.135</td><td>668.545</td><td>-21.938</td></tr><tr><td rowspan="7">Bulk</td><td>CASIM</td><td rowspan="7">50</td><td>24.616</td><td>50.75</td><td>1080.0</td><td>-2.703</td><td>783.264</td><td>-8.543</td></tr><tr><td>CASIM proc</td><td>15.562</td><td>-4.697</td><td>1090.0</td><td>-1.802</td><td>761.957</td><td>-11.031</td></tr><tr><td>CASIM, eff = 1</td><td>36.363</td><td>122.69</td><td>1030.0</td><td>-7.207</td><td>831.983</td><td>-2.854</td></tr><tr><td>CASIM 3Mr</td><td>13.811</td><td>-15.42</td><td>1050.0</td><td>-5.405</td><td>744.262</td><td>-13.097</td></tr><tr><td>MG2</td><td>39.563</td><td>142.287</td><td>1070.0</td><td>-3.604</td><td>884.209</td><td>3.244</td></tr><tr><td>LIMA</td><td>16.613</td><td>1.739</td><td>830.0</td><td>-25.225</td><td>928.842</td><td>8.456</td></tr><tr><td>LIMA KK</td><td>27.579</td><td>68.896</td><td>1010.0</td><td>-9.009</td><td>869.15</td><td>1.486</td></tr><tr><td rowspan="4">Bin</td><td>TAU-bin</td><td rowspan="4">150</td><td>14.548</td><td>—</td><td>1270.0</td><td>—</td><td>796.638</td><td>—</td></tr><tr><td>bin2d</td><td>16.041</td><td>10.263</td><td>1040.0</td><td>-18.11</td><td>878.945</td><td>10.332</td></tr><tr><td>bin2d</td><td>15.582</td><td>7.108</td><td>1040.0</td><td>-18.11</td><td>872.033</td><td>9.464</td></tr><tr><td>MSSG-bin</td><td>14.694</td><td>1.004</td><td>1240.0</td><td>-2.362</td><td>764.917</td><td>-3.982</td></tr><tr><td rowspan="3">LCM</td><td>LCM-FH</td><td rowspan="3">150</td><td>8.604</td><td>-40.858</td><td>1780.0</td><td>40.157</td><td>499.958</td><td>-37.242</td></tr><tr><td>LCM-MA</td><td>8.152</td><td>-43.965</td><td>3220.0</td><td>153.543</td><td>324.997</td><td>-59.204</td></tr><tr><td>UWLCM</td><td>3.695</td><td>-74.601</td><td>2760.0</td><td>117.323</td><td>403.413</td><td>-49.361</td></tr><tr><td rowspan="7">Bulk</td><td>CASIM</td><td rowspan="7">150</td><td>23.465</td><td>61.294</td><td>1180.0</td><td>-7.087</td><td>711.014</td><td>-10.748</td></tr><tr><td>CASIM proc</td><td>20.922</td><td>43.814</td><td>1100.0</td><td>-13.386</td><td>695.621</td><td>-12.68</td></tr><tr><td>CASIM, eff = 1</td><td>30.562</td><td>110.077</td><td>1160.0</td><td>-8.661</td><td>745.673</td><td>-6.398</td></tr><tr><td>CASIM 3Mr</td><td>12.308</td><td>-15.397</td><td>1200.0</td><td>-5.512</td><td>689.04</td><td>-13.507</td></tr><tr><td>MG2</td><td>29.031</td><td>99.553</td><td>1170.0</td><td>-7.874</td><td>773.64</td><td>-2.887</td></tr><tr><td>LIMA</td><td>17.027</td><td>17.04</td><td>820.0</td><td>-35.433</td><td>850.812</td><td>6.8</td></tr><tr><td>LIMA KK</td><td>23.194</td><td>59.431</td><td>1090.0</td><td>-14.173</td><td>787.692</td><td>-1.123</td></tr><tr><td rowspan="4">Bin</td><td>TAU-bin</td><td rowspan="4">300</td><td>12.754</td><td>—</td><td>1500.0</td><td>—</td><td>744.126</td><td>—</td></tr><tr><td>bin2d</td><td>13.064</td><td>2.431</td><td>1150.0</td><td>-23.333</td><td>838.113</td><td>12.63</td></tr><tr><td>bin2d</td><td>12.392</td><td>-2.838</td><td>1160.0</td><td>-22.667</td><td>823.862</td><td>10.715</td></tr><tr><td>MSSG-bin</td><td>10.807</td><td>-15.266</td><td>1390.0</td><td>-7.333</td><td>747.572</td><td>0.463</td></tr><tr><td rowspan="3">LCM</td><td>LCM-FH</td><td rowspan="3">300</td><td>4.894</td><td>-61.628</td><td>2800.0</td><td>86.667</td><td>393.245</td><td>-47.153</td></tr><tr><td>LCM-MA</td><td>0.0</td><td>-100.0</td><td>—</td><td>—</td><td>0.0</td><td>-100.0</td></tr><tr><td>UWLCM</td><td>0.809</td><td>-93.657</td><td>3570.0</td><td>138.0</td><td>29.017</td><td>-96.101</td></tr><tr><td rowspan="7">Bulk</td><td>CASIM</td><td rowspan="7">300</td><td>20.653</td><td>61.934</td><td>1290.0</td><td>-14.0</td><td>657.452</td><td>-11.648</td></tr><tr><td>CASIM proc</td><td>19.479</td><td>52.729</td><td>1200.0</td><td>-20.0</td><td>656.108</td><td>-11.828</td></tr><tr><td>CASIM, eff = 1</td><td>25.952</td><td>103.481</td><td>1270.0</td><td>-15.333</td><td>687.162</td><td>-7.655</td></tr><tr><td>CASIM 3Mr</td><td>11.296</td><td>-11.432</td><td>1350.0</td><td>-10.0</td><td>646.16</td><td>-13.165</td></tr><tr><td>MG2</td><td>19.126</td><td>49.961</td><td>1280.0</td><td>-14.667</td><td>672.49</td><td>-9.627</td></tr><tr><td>LIMA</td><td>12.52</td><td>-1.835</td><td>850.0</td><td>-43.333</td><td>759.758</td><td>2.101</td></tr><tr><td>LIMA KK</td><td>16.807</td><td>31.778</td><td>1150.0</td><td>-23.333</td><td>707.264</td><td>-4.954</td></tr></tbody></table>

Shipway and Hill (2012) and since then has been applied to many different cloud types at various resolutions, (e.g., Grosvenor et al. 2017; Miltenberger et al. 2018; Dearden et al. 2018; Gordon et al. 2020; Hawker et al. 2021). CASIM represents warm rain processes with cloud and rain categories, in which cloud moves to the rain category through

autoconversion and accretion parameterizations. The DSD is defined using a gamma function with a combination of a configurable distribution moments and prescribed parameters. For example, CASIM can be configured so that the cloud and rain DSD are represented by one prognostic moment (mass), while drop number and DSD shape parameter (μ)
are prescribed, or two prognostic moments (mass and number), while $\mu$ is prescribed. In addition, the rain DSD can be represented by three prognostic moments (mass mixing ratio, number and radar reflectivity). The results presented in this intercomparison focus on the double-moment configuration of CASIM, in which the cloud and rain $\mu$ is set to 2.5, with some additional experiments employing double-moment cloud and triple-moment rain.

In the CASIM simulations presented in this work, aerosol is represented by a single mode lognormal distribution with two moments (mass and number), and the aerosol is assumed to be soluble ammonium sulfate with a geometric mean diameter of 0.08 µm and a standard deviation of 1.4 µm, as prescribed in the KiD-A case specifications. The formation (and evaporation) of cloud droplets depends on an “all-or-nothing” saturation adjustment scheme, with the initial cloud drop number concentration derived using Abdul-Razzak and Ghan (2000) aerosol activation. Rain mass and number concentration are formed using the Khairoutdinov and Kogan (2000) autoconversion and accretion parameterizations, while evaporation of rain follows diffusional growth concepts of Byers (1965), self-collection follows a modified version of Beheng (1994), and the sedimentation fall speeds are based on Abel and Shipway (2007). When aerosol processing is enabled, the aerosol are removed from interstitial aerosol population upon activation to become in-cloud aerosol. This in-cloud aerosol is transported and processed with the cloud and rain, so that aerosol mass will increase, while number decreases during autoconversion and accretion and in-cloud aerosol is sedimented with cloud and rain. In-cloud aerosol that has been processed can either be removed through surface precipitation or returned to the atmosphere through complete evaporation of rain, where the aerosol liberated by rain evaporation will be larger in size than the activated aerosol, i.e., increase in mass and reduced number concentration. At the present time, there is no aerosol impaction scavenging below cloud in CASIM.

### d. MSSG-bin

The MSSG-bin scheme (Onishi and Takahashi 2012) is a bin-bulk hybrid cloud microphysical model, implemented in the MSSG-bin (Takahashi et al. 2013). In the hybrid approach, a one-moment spectral bin scheme is used for liquid droplets, while a bulk scheme is used for solid particles. That is, the expensive but more reliable spectral bin scheme treats the relatively well-understood physics of the liquid phase, and the computationally efficient but less robust bulk scheme is used to treat the poorly understood physics of the ice phase. In the spectral-bin part, the prognostic variables are the mass distribution function of liquid drops and CCN. For the drops, 132 bins were used for the present intercomparison. The so-called prescribed-spectrum method (Soong 1974; Reisin et al. 1996) is used for the CCN activation process. Its formulation is based on the relationship between the number of activated CCN and the saturation ratio. The condensational growth calculation follows the vapor diffusion equation (Tzivion et al. 1989). Several collection kernel models, including the Reynolds-number-aware turbulent

kernel model (Onishi and Seifert 2016), are implemented for collision-coalescence. The Hall (1980) kernel, which is widely used, was adopted for this intercomparison to minimize the model variability due to the kernel option.

### e. bin2d

The bin2d scheme is based on Lebo and Seinfeld (2011a) and provides not only a binned dimension representing the amount of liquid water contained in cloud and raindrops but also the amount of solute, thus allowing for the explicit representation of both solute and curvature effects on the droplet growth equation. Mass doubling is used to define bin boundaries, where the smallest bin has a diameter of 0.01 µm; for simplicity, both the solute and water dimensions use the same initial diameter.<fnref n="1" />⁾ In the liquid dimension, 56 bins are used, resulting in a maximum diameter of 3303 mm, whereas in the solute dimension, 25 bins are used, resulting in a maximum dry aerosol diameter of 2.56 µm. Condensational growth follows the standard heat and vapor diffusion equation accounting for both solute and curvature effects (Pruppacher and Klett 1997). Note that aerosol activation is explicitly resolved and is not parameterized. The flux method of Bott (1998) is modified to explicitly account for the 2D bin structure as in Lebo and Seinfeld (2011a). The Hall (1980) collection kernel is used for collision-coalescence. Terminal fall speeds are predicted following Beard (1976). To maintain numerical stability, all microphysical processes are computed using a 0.02 s time step.

### f. TAU-bin

The TAU-bin (Tzivion et al. 1987) is essentially the same as that used in Hill et al. (2009) and Shipway and Hill (2012). For aerosol, the TAU-bin scheme uses a single prognostic variable, in which aerosol are described with a lognormal aerosol distribution of soluble ammonium sulfate with a geometric mean diameter of 0.08 µm and a standard deviation of 1.4 µm, as prescribed in the KiD-A case specifications. The aerosol activation parameterization is based on Stevens et al. (1996), which derives activation using the ambient supersaturation and the local aerosol number concentration. TAU-bin includes a prognostic supersaturation and a condensation/evaporation scheme based on Tzivion et al. (1989), which uses semianalytical solution to the supersaturation equation, i.e., Clark (1973), to calculate the change in local mass mixing ratio and number concentration due to condensation/evaporation. Collision-coalescence in this version of TAU-bin uses the Ochs et al. (1986) and Long (1974) kernels, while sedimentation is based on terminal fall speeds derived from Beard (1976). TAU-bin uses 34 mass doubling size bins with mass mixing ratio and number concentration, with a diameter range of 3.125 to 6400 µm.

<footnote n="1">¹⁾ As a result of this bin structure, some bins are unphysical and thus never contain particles and are not included in the calculations.</footnote>
g. UWLCM

The University of Warsaw Lagrangian Cloud Model (Dziekan et al. 2019) is an LES model in which microphysical processes are represented using a Lagrangian method from the libcloudph++ library. A detailed description of the libcloudph++ library can be found in Arabas et al. (2015), with further changes from Jaruga and Pawlowska (2018) and from Dziekan et al. (2019). For this study, Lagrangian microphysics from libcloudph++ are connected to the Kid-A kinematic driver using Python (Jarecka et al. 2015). The code used for this can be found at https://github.com/igfuw/kid-libcloud.

Like in the other LCMs used in this study, atmospheric particles are represented by simulated particles called superdroplets. Each superdroplet represents many atmospheric particles with the same characteristics. All particles, including humidified aerosols, evolve according to the same set of basic equations. Therefore, aerosol activation does not need to be modeled as a separate process, and aerosol processing is included in a straightforward manner. Condensation is modeled using the Maxwell-Mason approximation to the heat and vapor diffusion process. Curvature and solute effects are taken into account at all stages of droplet growth. The $\kappa$-Kohler parameterization of water activity is used (Petters and Kreidenweis 2007). The growth equation of each superdroplet is solved with an implicit scheme with respect to the radius but explicit with respect to water vapor and potential temperature (Arabas et al. 2015). This scheme is similar to that of Brown (1977), although there the scheme is implicit in thermodynamical properties but explicit in radius. To decrease the time step with which the growth equation is solved, the scheme can be applied multiple times per model time step, what we call condensation substepping. In condensation substepping, thermodynamical properties are gradually changed from the values in the previous time step to the values in the current time step. Details of the procedure are given in Dziekan et al. (2019) (the “per-cell” substepping therein). The only deviation is that in KiD-A, potential temperature is not changed by the release of latent heat. A mass accommodation coefficient of 1.0 is used, following Laaksonen et al. (2005). A gravitational coalescence kernel is used with collision efficiencies taken from Hall (1980). Terminal velocity of droplets is evaluated using the formula from Beard (1976). Coalescence of superdroplets is modeled with a stochastic algorithm introduced in Shima et al. (2009). The algorithm is similar to the all-or-nothing algorithm used in LCM-FH with a difference in the number of pairs of superdroplets examined for possible collision. In LCM-FH, all superdroplet pairs are considered in each time step. In UWLCM, each superdroplet is included only in a single pair per time step (the so-called linear sampling). Advantages of the linear sampling technique are that the code can be easily parallelized and that the user has more control over the number of collision pairs examined per unit of time. To increase the number of collision pairs examined, the coalescence algorithm can be repeated multiple times per model time step, what we call coalescence substepping. In a box model, the linear sampling technique has been

shown to give very similar results in terms of means and variances as the approach in which all pairs are considered (Dziekan and Pawlowska 2017). To initialize droplet sizes, initial size spectrum is discretized into bins. In each bin, size of a single superdroplet is randomly selected. This algorithm, described in detail in Dziekan and Pawlowska (2017), is conceptually similar to the algorithm used in LCM-FH.

h. LCM-FH

This Lagrangian cloud model originates from the model by Riechelmann et al. (2012), which has been extended and improved in subsequent publications (e.g., Hoffmann et al. 2015; Hoffmann 2017; Hoffmann et al. 2017). Similar to other Lagrangian approaches, each simulated particle represents an ensemble of atmospheric particles. These so-called superdroplets are characterized by several properties, which most important are the wet radius, aerosol dry radius, and the multiplicity, i.e., the number of real droplets represented by a superdroplet. In this study, we initialized 100 superdroplets per grid box, which is considered sufficient for the representation of condensation and collection (e.g., Unterstrasser et al. 2017). A mass accommodation coefficient of condensational growth is calculated by solving the diffusional growth equation for each superdroplets individually, considering the droplet's curvature and the mass of solute aerosol explicitly (Hoffmann et al. 2015). This treatment allows the particle to transition freely between a deliquescent aerosol and a cloud droplet or raindrop, without fixed categories. A mass accommodation coefficient of 0.036 is used, following Mordy (1959) and Kogan (1991). Collection follows the probabilistic “all-or-nothing” approach (Unterstrasser et al. 2017; Hoffmann 2017; Hoffmann et al. 2017). In this method, all pairs of superdroplets within a grid box are examined for possible collections. If a collection takes place, each droplet of the superdroplet with the smaller multiplicity is assumed to coalesce with one droplet of the superdroplet with the larger multiplicity, with commensurate effects on the wet radii, aerosol dry radii, and multiplicity. Collision efficiencies are based on Hall (1980), and coalescence efficiencies are assumed to be unity. Sedimentation is considered by diagnosing the terminal velocity from the wet radius of each superdroplet using Beard (1976). Note that a superdroplet reaches its terminal velocity instantaneously, which is then added to its motion determined by the kinematic model. Aerosols or droplet sizes are initialized using the “singleSIP” approach (Unterstrasser et al. 2017), in which the initial distribution is discretized into logarithmically spaced radius bins. For each bin, one superdroplet is created. This approach has been shown to successfully represent the tails of the initial distribution, which can be essential for the successful representation of the collection process (Unterstrasser et al. 2017).

i. LCM-MA

This Lagrangian microphysics scheme (Andrejczuk et al. 2008, 2010) also uses a stochastic particle approach, where Lagrangian parcels represent a group of physical aerosol particles having the same chemical and physical properties, to parameterize microphysics. Each Lagrangian parcel is driven by
the velocity field from the KiD model, and, depending on the thermodynamical conditions, water can condense/evaporate on the surface of the particles in a parcel. A mass accommodation coefficient of 1.0 is used, following Laaksonen et al. (2005). For each parcel, the equation of motion is solved to determine the evolution in time of the location and velocity. A full condensation model (including chemical properties of the aerosol) is solved for each parcel using an ordinary differential equation (ODE) solver with adaptive time stepping. While the Lagrangian microphysics can return a drag force, temperature, and water vapor tendencies to KiD, making it a two-way coupled system, in this intercomparison the Lagrangian microphysics only returns water vapor tendency. This ensures that temperature is not modified by the microphysics, which is the requirement of the 1D test case.

The collision-coalescence process (Andrejczuk et al. 2010) considers collisions of all parcels within the same model grid volume. Each coalescence event, resulting in increased aerosol size and droplet size, is mapped on a prescribed 2D collision grid spanning aerosol sizes and droplet sizes. If the number of droplets for a cell of the collision grid is larger than a prescribed threshold level, a new parcel is created for this cell. In this way, collision events between all parcels are considered, but new parcels are created only for a limited number of cells, limiting the total number of parcels in the computational domain to a manageable level. In addition, parcels having similar sizes can be merged together, under certain conditions, to further reduce the number of parcels.

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