1. Introduction
Bin microphysics schemes explicitly evolve the drop size distribution (DSD) by dividing it into several Eulerian radius or mass bins and prognosing one or more quantities in each bin (e.g., Berry and Reinhardt 1974; Tzivion et al. 1987; Kogan 1991; Feingold et al. 1994; Ackerman et al. 1995; Stevens et al. 1996a; Reisin et al. 1996; Geresdi 1998; Khain et al. 2004; Flossmann and Wobrock 2010; Lebo and Seinfeld 2011; Onishi and Takahashi 2012). Quantifying and reducing errors associated with numerically calculating DSD evolution in bin schemes has been a significant challenge since their inception. Numerous studies over the past several decades have focused on improving the numerical treatment of drop growth caused by condensation and collision–coalescence (e.g., Kovetz and Olund 1969; Bleck 1970; Young 1974; Soong 1974; Tzivion et al. 1987; Stevens et al. 1996a; Liu et al. 1997; Bott 1998; Tzivion et al. 1999; Alfonso 2015; Lkhamjav et al. 2017). For example, a simple method that conserves bulk mass by treating drop growth as first-order upwind advection in mass space was proposed by Kovetz and Olund (1969), but this led to rather inaccurate solutions for condensational growth (Liu et al. 1997) and rapid, artificial generation of precipitation (Ochs and Yao 1978). A much more accurate method was developed by Egan and Mahoney (1972) that conserves the zeroth, first, and second moments of the mass distribution. Young (1974) proposed a method that provides information on the drop distribution within bins by solving separate conservation equations for mass and number. Similarly, Tzivion et al. (1987) developed a method for solving collision–coalescence growth that separately prognoses the mass and number in each bin but achieves closure of the equations by diagnostically relating high-order moments to the two prognostic low-order moments. The method of Stevens et al. (1996a) also separately prognoses mass and number in each bin for condensational growth, and utilizes a remapping technique assuming a top-hat distribution similar to Egan and Mahoney (1972) except that the width of the top hat is diagnosed rather than predicted. Liu et al. (1997) proposed a variational optimization method for condensational growth that prognoses only a single variable in each bin but conserves any number of DSD moments as needed.
For condensation and evaporation, analytic or quasi-analytic solutions are readily available for Lagrangian box or parcel models to provide a benchmark for evaluating these numerical methods. This approach was employed by many of the studies cited above. Multiple moment-conserving techniques such as Egan and Mahoney (1972) and Liu et al. (1997) provide relatively high accuracy compared to simpler methods in Lagrangian models (e.g., Fig. 3 in Liu et al. 1997). For cloud modeling, however, bin schemes are typically coupled with Eulerian dynamical models. Understanding and quantifying the accuracy of condensational growth and transport calculations using bin microphysics in Eulerian dynamical models is important because the DSD characteristics from condensational growth, particularly the spectral width and tail of the large end of the spectrum, exert a critical influence on the timing and location of precipitation development in warm clouds (Beard and Ochs 1993, and references therein). Only a few studies have assessed the accuracy of numerical methods for drop growth using bin microphysics in conjunction with advection in Eulerian physical space. Clark (1974) found that the implicit diffusion from Eulerian advection led to only a small broadening of the bin-modeled droplet spectra in one-dimensional tests, and convergence could be achieved with a “moderate” increase in spatial resolution; this result is discussed in more detail in the current paper. Stevens et al. (1996a) later evaluated the effects of condensation/evaporation and Eulerian advection in physical space using bin microphysics in a large-eddy simulation (LES). They compared the LES with a Lagrangian model forced with parcel trajectories from the LES. However, they focused on the horizontal mean and dispersion of bulk DSD quantities such as liquid water content and mean droplet diameter, and not on the droplet spectra.
It is well known that DSDs in real clouds are much broader than those that would occur theoretically by condensation growth alone. The mechanisms responsible for this broadening and production of large drops that initiate rain have been a subject of debate for the past several decades. Work since the 1950s has focused on the role of giant cloud condensation nuclei (CCN) in generating large drops and subsequent rain formation (e.g., Ludlam 1951; Woodcock et al. 1971; Feingold et al. 1999; Jensen and Nugent 2017). Ostwald ripening can also broaden DSDs; this occurs by the preferential growth of large drops compared to small ones owing to differences in saturation vapor pressure over the drop surfaces from curvature and solute effects (e.g., Korolev 1995; Wood et al. 2002). Droplet clustering may also drive localized regions of high supersaturation and accelerated drop growth (e.g., Shaw 2000; Vaillancourt et al. 2002). Several other studies have examined the role of turbulent mixing of different droplet populations. Entrainment of dry air will evaporate some fraction of droplets, depending on the homogeneity of mixing. This can lead to the growth of large drops owing to reduced competition for vapor during subsequent ascent (Telford and Chai 1980). Asymmetry in DSD evolution during condensation in updrafts and evaporation in downdrafts can lead to horizontal inhomogeneity even in adiabatic clouds, broadening DSDs upon isobaric mixing (Korolev et al. 2013; Pinsky et al. 2014). More generally, mixing of droplet populations that have undergone different growth histories can broaden DSDs; this is the “eddy hopping” mechanism described by Cooper (1989), Lasher-Trapp et al. (2005), and Grabowski and Abade (2017). For instance, different vertical velocities along the cloud base can lead to horizontal variability of droplet concentrations and hence radius growth rates; subsequent horizontal mixing will broaden the DSDs above the cloud base. The key point is that in order for bin microphysics to be useful for investigating these physical mechanisms for DSD broadening, it must be able to accurately simulate condensational growth and droplet transport without excessive influence on broadening from numerical errors.
The goal of this study is to investigate and quantify the effects of numerical errors on DSD broadening from the combination of condensational growth and Eulerian transport in physical space. This study utilizes a hierarchy of models. We first present LES of a stratocumulus case using bin microphysics but with droplet activation, condensation, and evaporation as the only microphysical processes included. To isolate the effects of unphysical numerical diffusion associated with vertical advection on DSD broadening, tests using an idealized one-dimensional (1D) Eulerian spatial domain are compared to those using a rising parcel framework without Eulerian transport in physical space. These tests use a constant vertical velocity, allowing a direct comparison with Lagrangian microphysical solutions that serve as a numerical benchmark. A wide range of conditions and configurations are tested using the parcel and 1D models, including changes in bulk droplet concentration, vertical velocity, grid spacing, time step, mass bin grid, condensational growth method, and numerical method for advection in physical space. Results are discussed in the context of physical DSD broadening mechanisms, with implications for the ability of Eulerian dynamical models coupled with bin microphysics to investigate these mechanisms.
The rest of the paper is organized as follows. Section 2 describes LES results. The design of the idealized parcel and 1D experiments are presented in section 3. Results of the parcel and 1D tests are detailed in section 4. Section 5 provides a summary and conclusions.
2. LES results
We use the University of California Los Angeles (UCLA)-LES model (Stevens et al. 2005) to simulate the Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS II) RF02 stratocumulus case. We follow the LES intercomparison study of Ackerman et al. (2009) with the same initial conditions, large-scale forcing, and model configuration except using: 1) the Tel Aviv University bin microphysics scheme (Tzivion et al. 1987, 1989) as applied in Stevens et al. (1996a), with droplet activation and condensation/evaporation as the only microphysical processes (i.e., no collision–coalescence, breakup, or sedimentation); 2) a unimodal lognormal CCN distribution with constant number mixing ratio NCCN = 125 mg−1 and standard deviation 
(left) Profiles of cloud mass mixing ratio 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
We focus on characteristics of the modeled DSDs. Examples of simulated DSDs at various heights within the cloud at 3 h are shown in Fig. 1. Representative DSD profiles in updraft and downdraft conditions are shown. These profiles are at the locations indicated by the red arrows in Fig. 2, which shows a vertical cross section of the vertical velocity field at 3 h. The spectral width and tail are quantified by the standard deviation (in radius), σ, and the difference between the 99th-percentile cumulative distribution radius and the median radius 
Vertical cross section of vertical velocity from the LES for the Log(2) mass grid at 3 h (color contours). Red vertical arrows at the top of the plot indicate the locations of the updraft and downdraft DSD and cloud water profiles shown in Fig. 1.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
Vertical cross sections of (a) DSD standard deviation σ and (b) the difference between the 99% cumulative distribution radius and the median radius 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
Contour frequency by altitude diagrams of the DSD standard deviation σ from the LES at 3 h using the (a) Log(2) and (b) Log(21/4) mass grids. Dashed lines indicate the mean profiles.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
Contour frequency by altitude diagrams of the difference between the 99% cumulative distribution radius and the median radius 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
The DSDs are fairly broad. Median values of σ are near 2 μm near cloud base (~500 m), decrease to about 1.5 μm at ~700 m, and increase sharply but with large horizontal variability near cloud top (Figs. 3a, 4). The latter is likely due to the effects of entrainment and mixing on the DSDs. Overall, these σ values are similar to observations in marine stratocumulus over spatial scales similar to the LES grid spacing (e.g., Pawlowska et al. 2006; Snider et al. 2017). This occurs even though the LES neglects collision–coalescence, which likely contributed at least somewhat to broadening of the real DSDs. The term 
Adiabatic growth in an ascending parcel for the conditions of this case would theoretically produce a maximum drop radius of approximately 15 μm, but the DSD tails from the LES extend to much larger sizes over nearly the entire depth of the cloud layer (Fig. 1). This broadening to large sizes can only occur via mixing processes because other potential broadening processes (e.g., collision–coalescence, giant CCN, small-scale droplet clustering, and supersaturation fluctuations) have been neglected. The role of entrainment followed by mixing in broadening DSDs is evident from the relatively large values of σ and 
Isolating the mechanisms that broaden DSDs is extremely challenging because DSD mixing in the LES is controlled by flow-dependent numerical diffusion associated with advection of bin microphysical variables by the resolved-scale winds. Importantly, this includes numerical diffusion associated with both horizontal and vertical advection. Horizontal numerical diffusion is in line with the mixing that underpins many of the physical DSD broadening mechanisms discussed in the introduction. Nevertheless, horizontal variability and the associated isobaric mixing and DSD broadening processes are likely underresolved even in our LES with a horizontal grid spacing of 50 m, given observational evidence for DSD variability occurring on much smaller scales (e.g., Gerber et al. 2005; Burnet and Brenguier 2007; Beals et al. 2015). In contrast to horizontal mixing, vertical air motion in real clouds involves droplet condensational growth or evaporation associated with the adiabatic cooling or warming of rising or sinking air parcels. Besides driving variability of droplet activation followed by vertical mixing, fluctuations of vertical velocity alone do not contribute to DSD broadening in the absence of horizontal mixing, microscale supersaturation fluctuations, or other physical broadening mechanisms (e.g., Manton 1979; Cooper 1989; Korolev et al. 2013). Thus, vertical numerical mixing of DSDs in models is unphysical because it neglects the droplet condensation or evaporation that occurs with adiabatic expansion or compression. Note that although it is not included in our simulations, explicit vertical subgrid-scale mixing of bin variables would have the same issue. Another potential source of artificial DSD broadening is numerical diffusion across bins during condensational growth, which can be thought of as advection in mass space (Kovetz and Olund 1969; Clark 1974; Stevens et al. 1996a).
In the remainder of the paper we focus on the role of these unphysical DSD broadening mechanisms. This is done by analyzing idealized 1D simulations that only include vertical advection and adiabatic droplet condensation growth. The next section describes the design of these 1D experiments, as well as parcel tests that only include condensational growth.
3. Description of the idealized 1D and parcel tests
a. Cloud microphysics
We use idealized parcel and 1D frameworks to test the role of unphysical numerical diffusion associated with vertical advection in physical space and condensational growth in mass space on DSD broadening. These frameworks allow a direct comparison to Lagrangian microphysical solutions that serve as a numerical benchmark. Consistent with the LES, the focus is on liquid condensational growth and transport in physical space; all other microphysical processes are neglected. However, offline calculations of collision–coalescence growth are presented to estimate the potential effects of numerical errors on DSD evolution and precipitation formation. We emphasize that these tests are not intended to be a realistic representation of DSD evolution, especially because horizontal variability and mixing are not considered, but rather to test the effects of numerical diffusion compared to the benchmark solutions.
To test condensational growth on an Eulerian mass grid, two different methods are tested: 1) a one-moment approach (MPDG) prognosing the mass mixing ratio in each bin and treating condensational growth as 1D advection in mass space using the Multidimensional Positive Definite Advective Transport Algorithm (MPDATA; Smolarkiewicz and Grabowski 1990; Smolarkiewicz and Margolin 1998) and 2) TH-MOM, as used in the LES (see section 2). The MPDG one-moment growth method essentially follows from Kovetz and Olund (1969) except that their first-order upwind scheme is replaced by the more accurate MPDATA advection scheme. Because TH-MOM is two moment, for the same mass grid it requires twice as many prognostic variables as the one-moment MPDG approach.
MPDG and TH-MOM are tested because they are representative of the wide range of approaches that have been used to solve condensational growth in bin schemes; as will be shown, MPDG is quite diffusive, whereas TH-MOM retains narrow distributions in rising parcel tests without Eulerian transport in physical space. Tests of various mass grid configurations are also presented. These include grids that have a linear increase in radius and grids with a logarithmic increase in mass, including the commonly used mass doubling grid. A list of all the bin model configurations tested is given in Table 1.
List of the bin model configurations tested. “TH-MOM” and “MPDG” indicate the top-hat two-moment method of moments and one-moment MPDG condensational growth methods, respectively.
b. Spatial advection schemes
Several numerical methods have been developed to solve the problem of hydrometeor transport by air motion [i.e., the first term on the right-hand side of (2)]. A comprehensive evaluation of the various advection schemes used in all cloud models is not feasible; instead we test a handful of schemes that have been implemented in two widely used models: the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008) and the Cloud Model, version 1 (CM1; Bryan and Fritsch 2002). While some details differ, the WRF-based schemes are very similar to the scalar advection scheme used in the UCLA-LES; they both employ Eulerian flux-based finite-volume methods with flux limiters to retain monotonicity. As will be shown in section 4c, there is limited sensitivity of the DSD broadening caused by vertical numerical mixing to a wide range of advection methods.
Different time-stepping methods have little impact on the results presented herein. Using the method that Grabowski and Jarecka (2015) applied in their bin microphysics model, which calculates 
Various schemes for calculating the 1D advective fluxes in (3) are tested; these are listed in Table 2. The WRF-based schemes (WRF-3RD and WRF-5TH) approximate the advective flux divergences in (3) using spatial Taylor series expansions following Wicker and Skamarock [(2002), see their (4a)–(4d)], combined with the monotonic flux corrected transport (FCT) scheme of Wang et al. (2009). The FCT scheme is only applied at the third Runge–Kutta step. We also test a simple first-order upwind scheme (FIRST). FCT is not applied using this scheme because it is already monotonic.
Several weighted essentially nonoscillatory (WENO) advection schemes are also tested. A fifth-order scheme (WENO-5TH) is based on Jiang and Shu (1996), although we use the modified formulation for the WENO “smoothness indicators” from Borges et al. (2008). The seventh- and ninth-order schemes (WENO-7TH and WENO-9TH) are based on Balsara and Shu (2000) and also use modified smoothness indicators following the approach of Borges et al. (2008). Similar to the FCT schemes, WENO is only applied during the third Runge–Kutta step, whereas flux calculations based on standard Taylor series expansions of a given order are applied during the first two steps (this follows from the approach used in WRF and CM1).
c. Diagnostic collision–coalescence growth rate
We diagnose the collision–coalescence growth rate 
4. Numerical test results
a. 1D tests analogous to the LES configuration
We first discuss results for a configuration of the 1D model that follows the LES setup described in section 2. These tests use a cloud-base temperature of 286 K; cloud-base and cloud-top heights of approximately 500 and 850 m, respectively; and a droplet concentration of 61 cm−3; these are equal to the mean values from the LES. The water vapor mixing ratio at the lower boundary (i.e., cloud base) is specified to give a supersaturation of 0.3%; this is based on the maximum supersaturation of an ascending parcel for these conditions. A constant vertical velocity of 0.5 m s−1 is specified, which is roughly similar to the updraft strength in the LES. As shown in section 4c, there is limited sensitivity of DSD broadening in the 1D tests to w. The same vertical grid as the LES is utilized. As in the LES, these tests use the TH-MOM condensation method and two different mass grids: the mass doubling Log(2) grid and the high-resolution Log(21/4) grid.
Potential temperature and water vapor mixing ratio are prognostic variables that are advected with sources/sinks caused by condensation and latent heating. Thermodynamic conditions at the lower boundary are held constant at the initial values given above. The cloud droplet population at the lower boundary is specified by placing the total concentration of 61 cm−3 in the smallest mass bin, consistent with the LES in which newly activated droplets were put in the smallest bin. Above cloud top, all droplets are simply removed from the distribution each time step; this leads to some small oscillations in cloud quantities at cloud top but otherwise has a limited impact on the results. Simulations are run for 1 h with a 1-s time step; near-steady-state conditions are achieved within a few minutes near cloud base and a few minutes longer near cloud top, corresponding to the time scale for advective transport through the cloud depth. These tests use the WRF-5TH monotonic scheme (see Table 2), for advection in physical space. Because higher-order advection schemes require information for multiple upwind (and downwind) grid levels, the lower boundary is treated by assuming constant conditions below the lower boundary (height of maximum supersaturation). Additional tests using reduced-order flux calculations along the boundary, as is often done in models (e.g., WRF and CM1), or simply zero cloud water below cloud base, show little impact on the results.
Vertical profiles of bulk cloud mass mixing ratio (summed over the bins), mean radius, σ, and 
Vertical profiles of (a) bulk mass mixing ratio Q, (b) mean drop radius R, (c) DSD standard deviation σ, and (d) 99% cumulative 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
A key point is that the profiles of σ and 
b. Comparison of bin scheme and Lagrangian microphysical solutions in a parcel framework
This subsection presents tests of droplet condensational growth within a rising parcel. These tests exclude Eulerian transport in physical space so that we can focus solely on DSD broadening from numerical diffusion across bins during condensational growth. Feedbacks between the microphysics, temperature, supersaturation, and hydrostatic pressure are included as the parcels rise. The numerical benchmark is a Lagrangian microphysical solution with 200 initial drop sizes distributed uniformly between 1 and 3 μm. In contrast to the bin model solutions, the Lagrangian solution does not use an Eulerian grid in mass space. Instead, the growth rate of each initial drop size is calculated directly by integrating (1) together with equations for evolution of temperature and water vapor mixing ratio using a simple Euler forward-in-time method. For both the Lagrangian benchmark and bin numerical solutions, the time step is 1 s. The Lagrangian numerical benchmark solution is well converged with a 1-s time step; reduction of the time step to 0.25 s has almost no impact on the results. Unlike the 1D tests in section 4a, here all of the bin model configurations are initialized with a top-hat distribution of drops with sizes between 1 and 3 μm consistent with the Lagrangian benchmark. This ensures more consistent initial size distributions using the various mass grids than simply placing all droplets in the smallest bin. The initial supersaturation, temperature, and pressure are 0.6%, 288.15 K, and 900 hPa, respectively. For simplicity, the vertical velocity is constant at 1 m s−1. For all configurations, the supersaturation decreases monotonically from its initial value. The initial value of supersaturation is based on additional parcel tests indicating a maximum supersaturation of approximately 0.6% under these conditions (1 m s−1 updraft and 50-mg−1 droplet number mixing ratio). Results are analyzed for 700 m of parcel ascent.
Figure 7 presents DSDs for each bin model configuration and the Lagrangian benchmark at various heights (converting time to height using the 1 m s−1 vertical velocity). The Lagrangian benchmark shows a narrowing of the DSD with increasing time and/or height, which is a well-known feature of condensational growth (e.g., Pruppacher and Klett 1997). All of the TH-MOM configurations do a reasonable job of reproducing narrow DSDs from the Lagrangian numerical benchmark, insofar as allowed by the bin spacing. However, one-moment MPDG growth produces significant numerical diffusion and DSD broadening relative to the Lagrangian benchmark and all of the TH-MOM configurations, including those with relatively coarse bin spacing. All of the bin model configurations reproduce supersaturation evolution well compared to the Lagrangian benchmark (not shown) as well as evolution of bulk mass and mean radius (Figs. 8a,b).
Drop size distributions at various heights z from the Lagrangian microphysical benchmark (black) and the bin model simulations (colored lines) for the parcel test with a bulk drop number mixing ratio of 50 mg−1. Different colored lines illustrate results using different bin mass grid configurations and growth methods, as listed in Table 1.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
Vertical profiles of (a) bulk mass mixing ratio Q, (b) mean drop radius R (solid lines) and standard deviation σ (dotted lines), (c) 99% cumulative 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
Vertical profiles of σ, 
Additional tests specifying different bulk droplet numbers give similar overall results. For example, specifying a bulk droplet number mixing ratio of 250 mg−1 gives similar differences among the various TH-MOM and MPDG configurations, with reduced supersaturations and growth rates compared to the tests with bulk number of 50 mg−1 as expected.
c. Additional tests using the 1D model
In this subsection, several additional tests using the 1D model are analyzed to further investigate numerical DSD broadening. For the numerical benchmark, we employ the same rising parcel Lagrangian microphysical calculations described in section 4b, and translate time evolution to height using the specified constant vertical velocity to compare with the 1D simulations. Tests using a range of time steps, grid spacings, and vertical velocities are presented below. In contrast to the LES-like setup of the 1D model, a constant vertical grid spacing is used here to further simplify the model configuration and interpretation of results. We first show results using a 1-s time step, 20-m grid spacing, and 1 m s−1 vertical velocity. A top-hat distribution of droplets between 1 and 3 μm with a bulk number mixing ratio of 50 mg−1 is assumed at the lower boundary following initial conditions for the parcel tests in section 4b. Using narrower or broader DSDs at the lower boundary does not affect the main findings. Cloud quantities are set to zero above a height of 1400 m, with virtually no impact on the analysis region between 0 and 700 m.
Similar to the parcel tests described above, all of the bin model configurations well reproduce the Lagrangian numerical benchmark bulk mass and mean radius profiles (Figs. 10a,b) and supersaturation (not shown). However, in contrast to the parcel tests, all model configurations give significant spectral broadening relative to the numerical benchmark, as illustrated in Fig. 9. DSD broadening relative to the benchmark occurs using either TH-MOM or MPDG as the growth method. The DSDs are skewed toward larger sizes indicating a tendency to broaden toward bigger, not smaller, drops. This broadening is evident by values of σ and 
As in Fig. 7, but for the combined growth and Eulerian spatial advection test with a bulk droplet number mixing ratio of 50 mg−1, vertical grid spacing of 20 m, time step of 1 s, and vertical velocity of 1 m s−1.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
As in Fig. 8, but for the combined growth and Eulerian spatial advection test with a bulk droplet number mixing ratio of 50 mg−1, vertical grid spacing of 20 m, time step of 1 s, and vertical velocity of 1 m s−1.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
A key result is that the spectral broadening relative to the numerical benchmark is similar regardless of the bin resolution, so that all of the TH-MOM configurations give similar values of σ, 
The DSD broadening in the 1D tests relative to the numerical benchmark is explained conceptually as follows. In a flux-based Eulerian framework, there is a flux of drops at level k to the level above on each time step when there is upward motion, and a flux of drops from the level below up to k. Because mean drop size increases with height, there is an upward flux of relatively small drops into k compared to the flux of larger drops out of k. The divergences of these fluxes therefore tend to increase the drop concentrations in smaller bins but decrease the concentrations in larger bins at level k. Subsequent condensational growth shifts the DSDs to larger bins. This is repeated in subsequent time steps, leading to drops being present across several bins at a given level and hence DSD broadening. This highlights the tight coupling of transport in physical space and the “transport” of drops in mass space from condensational growth. Fundamentally, this close coupling means that numerical diffusion in the vertical in effect leads to diffusion across bins in mass space, and hence DSD broadening. In the absence of vertical numerical diffusion there is no broadening except from diffusion across mass bins during condensational growth calculations. This is demonstrated by tests with the 1D model using first-order upwind advection and flow conditions such that the Courant–Friedrichs–Lewy (CFL) number is equal to 1. This gives no vertical numerical diffusion because all drops from a level are transported exactly to the next level above, and hence there is no DSD broadening beyond that seen in the parcel tests (not shown). Similarly, all drops at a given level are transported a distance 
Using a bulk droplet number mixing ratio of 250 mg−1 at the lower boundary instead of 50 mg−1 leads to reduced supersaturation and smaller growth rates as expected (Fig. 11). Because of the reduced growth rate, vertical gradients of mean drop radius are smaller (Fig. 11b), leading to smaller errors relative to the benchmark than the tests with 50 mg−1 in terms of σ, 
As in Fig. 10, but for the combined growth and Eulerian spatial advection test and bulk droplet number mixing ratio at the lower boundary specified at 250 mg−1.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
To investigate the effects of explicitly including droplet activation instead of specifying the DSD at the lower boundary, additional tests apply a droplet activation rate near cloud base. In these tests, a droplet activation rate of 2.5 mg−1 s−1 is applied within the lowest model level (with a depth of 20 m) above cloud base, and a constant supersaturation of 0.6% is specified for simplicity. This activation rate is chosen because it gives a bulk droplet number mixing ratio through the cloud layer of approximately 50 mg−1. Newly activated droplets follow a top-hat distribution between 1 and 3 μm. The numerical benchmark is a Lagrangian solution that activates droplets with a radius between 1 and 3 μm at a rate of 0.125 mg−1 s−1 at 1-m intervals in the lowest 20 m above cloud base, giving the same net activation rate as the Eulerian tests. Overall results are similar to the 1D tests with a fixed lower boundary DSD (not shown), with σ and 
As mentioned above, there is no DSD broadening from vertical numerical diffusion if the CFL number is 1, using first-order upwind advection. This implies sensitivity of broadening to the CFL number, which we explore by varying the time step, vertical grid spacing, and prescribed vertical velocity. These parameters are each varied to give a range of vertical CFL numbers from 0.025 to 0.2, which may be considered fairly typical for LES of stratocumulus or shallow cumulus (e.g., Ackerman et al. 2009; vanZanten et al. 2011). Tests that vary the time step 
Sensitivity of the mean difference in 99th-percentile cumulative distribution radius and the median radius 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
To examine the sensitivity to 
As in Fig. 9, but for tests using the Log(21/4) mass grid with TH-MOM condensational growth for different vertical grid spacing 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
The vertical velocity affects both transport in physical space and the supersaturation. This sensitivity is examined by tests with w equal to 0.5, 1, 2, and 4 m s−1 using 
Finally, results using the various spatial advection schemes listed in Table 2 are compared. These tests use 
To provide context for the 1D growth plus transport tests described below, we first illustrate the performance of the advection schemes for the problem of pure transport. This is done by comparing results for 1D linear advection tests with a constant CFL number, allowing evaluation against an analytic benchmark solution. Figures 14 and 15 show results using the advection schemes described in section 3b, although here we use the FCT algorithm of Zalesak (1979), which is slightly different from the FCT algorithm in WRF (Wang et al. 2009). Two different initial scalar distributions are tested. The first is well resolved, with a Gaussian distribution that is roughly 10 grid points wide 
Linear scalar transport tests using various advection schemes with the well-resolved initial scalar distribution covering approximately 10 grid points 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
As in Fig. 14, but for the poorly resolved initial scalar distribution covering two grid points 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
For transport in physical space plus condensational growth there are few consistent trends and the different advection schemes produce broadly similar results, as illustrated by profiles of 
Profiles of the mean difference in 99th-percentile cumulative distribution radius and the median radius 
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
The vertical distribution of bin mass mixing ratio (x axis) as a function of height (y axis) from the rising parcel tests described in section 3a. Here, time is converted to height using the specified 1 m s−1 vertical velocity. Results are shown for the Log(2) and Log(21/4) mass grids. Solid, dotted, and dashed lines illustrate results for bins with mean radii of approximately 5.8, 9.2, and 14.5 μm, respectively.
Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0055.1
Regions with mass/number in larger-size bins are wider because the growth rate decreases with drop size and because the mass doubling grid means that larger bins are relatively wide in radius space. Although wider for larger bins, based on the rising parcel tests, these regions still have sharp vertical gradients of the number and mass mixing ratios (Fig. 17) that are difficult to resolve even with higher-order advection schemes. Difficulty resolving these features also helps to explain the sensitivity to 
The results discussed above highlight the close coupling between transport in the physical space dimensions and transport in mass space for condensational growth. In essence, a coupled four-dimensional advection problem must be solved for bin microphysics implemented in a 3D dynamical model, with transport in nearly incompressible flow in the three spatial dimensions, and in compressible “flow” in the mass dimension. Transport in each dimension is closely linked to transport in the other dimensions. Thus, numerical diffusion in physical space inevitably leads to diffusion in mass space, producing the spectral broadening seen here. In contrast, condensation and spatial transport for bulk microphysics is a 3D problem. Even though Eulerian models separately calculate spatial advection and condensation using bulk microphysics, this is of little consequence except for modeling the peak supersaturation just above cloud base (Clark 1974; Morrison and Grabowski 2008). Condensation is a fast process, and the thermodynamics and bulk cloud mass adjust rapidly with limited error by separately advecting quantities and then calculating bulk condensation at each time step.1 Thus, the problem of unphysical, numerical DSD broadening is not a result of inconsistencies in the thermodynamic variables or advection of nonconservative quantities but rather is due to the unique, coupled four-dimensional advection problem that bin microphysics in a 3D Eulerian dynamical model presents.
5. Discussion and conclusions
In this study, we investigated numerical errors that arise from the combination of droplet condensation growth and vertical advection using a bin microphysics scheme coupled with Eulerian dynamical models. This was done using a hierarchy of models including parcel, 1D, and 3D dynamical models.
We first presented results from 3D LES of the DYCOMS II RF02 case using the Ackerman et al. (2009) model intercomparison study setup. Droplet activation, condensation, and evaporation were the only microphysical processes included in the LES, but nonetheless it produced wide spectra at all cloud levels, much broader than would be expected theoretically from condensational growth alone. Values of the DSD standard deviation σ from the LES were similar to those observed in marine stratocumulus (e.g., Pawlowska et al. 2006; Snider et al. 2017). Several mixing mechanisms were potential contributors to DSD broadening in the LES. These mechanisms included numerical diffusion associated with horizontal advection, which is conceptually in line with various mixing processes that have been previously proposed to explain DSD broadening in real clouds (e.g., Cooper 1989; Lasher-Trapp et al. 2005; Korolev et al. 2013; Pinsky et al. 2014; Grabowski and Abade 2017). However, they also included broadening from numerical diffusion associated with vertical advection, which is unphysical and does not contribute to broadening in real clouds.
To isolate this unphysical DSD broadening, we utilized a 1D framework that included only vertical advection and condensation and neglected all other processes. The simplicity of this framework allowed direct comparison with Lagrangian microphysical solutions that provided a numerical benchmark. Tests using the 1D model in a setup analogous to the LES produced spectra that were about as wide as in the LES, quantified by the standard deviation σ and the difference between the 99th-percentile cumulative distribution radius and the median radius 
Several additional tests using the 1D model were then presented. In contrast to the parcel tests, the spectra were much broader than the Lagrangian microphysical benchmark, highlighting the critical role of vertical numerical diffusion in DSD broadening. In contrast to the much broader DSDs using the MPDG growth method compared to TH-MOM for the parcel tests, the DSDs were similar below ~200 m in the 1D Eulerian tests, and only somewhat broader using MPDG compared to TH-MOM above ~200 m. Because TH-MOM performed well in reproducing the benchmark solutions for the parcel tests, especially with fine bin resolution, it is unlikely that other numerical methods for condensational growth would lead to improvement in the Eulerian framework.
Based on these results, we propose that numerical diffusion associated with resolved-flow vertical advection is a key contributor to DSD broadening in LES models coupled with bin microphysics, even when the condensational growth calculations themselves contribute little to broadening (as when using TH-MOM). Although not included in the simulations here, explicit vertical subgrid-scale mixing would contribute to unphysical broadening similarly to vertical numerical diffusion. We emphasize that LES with bin microphysics do not necessarily produce overly broad spectra compared to those in real clouds, but that the DSD broadening may be dominated by an unphysical mechanism. This may explain why LES are able to produce reasonably broad spectra compared to observations (e.g., Pawlowska et al. 2006; Snider et al. 2017), despite neglecting or underresolving the physical processes that studies have shown are important for DSD broadening in real clouds, including giant CCN (e.g., Ludlam 1951; Woodcock et al. 1971; Feingold et al. 1999; Jensen and Nugent 2017), droplet clustering and microscale supersaturation fluctuations (e.g., Shaw 2000; Vaillancourt et al. 2002), and various physical DSD mixing processes (e.g., Cooper 1989; Lasher-Trapp et al. 2005; Korolev et al. 2013; Pinsky et al. 2014; Grabowski and Abade 2017). In particular, horizontal DSD variability is likely underresolved for typical LES grid resolutions (tens of meters), implying that DSD broadening from isobaric mixing of different droplet populations will be underrepresented. This underrepresentation is likely to be even more severe for cloud-resolving and convection-permitting models with horizontal grid spacing of order 1 km. It is reasonable to think that unphysical broadening from vertical numerical diffusion compensates for underrepresenting or neglecting these physical broadening mechanisms. Thus, although bin microphysics is a useful tool for many applications, it may be of less value for investigating physical mechanisms for DSD broadening when coupled with Eulerian dynamical models using typical vertical grid spacings (~10 m or larger).
Besides reducing the vertical grid spacing, minimizing unphysical DSD broadening from vertical numerical diffusion may require new methods calculating growth and physical transport in a single step given the close coupling of growth in mass space and transport in physical space. The development of such methods will be a focus of future research. We speculate that reducing unphysical DSD broadening may expose the underrepresentation of physical broadening mechanisms in LES with bin microphysics, possibly leading to unrealistically narrow DSDs. The results documented in this paper also help to further motivate the development and use of Lagrangian microphysics schemes coupled to Eulerian dynamical models that track representative droplet samples within the modeled flow (e.g., Shima et al. 2009; Andrejczuk et al. 2010; Riechelmann et al. 2012; Arabas et al. 2015; Hoffmann et al. 2017; Grabowski et al. 2018). Such an approach does not suffer from unphysical DSD broadening caused by the vertical numerical mixing. Moreover, Lagrangian droplet trajectories can be modeled using the resolved-scale flow plus a stochastic subgrid-scale component, and hence they provide a “natural” framework for representing physical DSD broadening from mixing of droplet populations that have undergone different growth histories (Grabowski and Abade 2017; Grabowski et al. 2018).
Acknowledgments
This work was partially supported by U.S. DOE Atmospheric System Research (ASR) Grant DE-SC0016579. JYH acknowledges support from the U.S. DOE ASR Grant DE-SC0012827. ZJL acknowledges support from the U.S. DOE ASR Grant DE-SC0016354. MKW acknowledges support from NCAR’s Advanced Study Program. High performance computing was provided by NCAR’s Computational Information Systems Laboratory. The National Center for Atmospheric Research is sponsored by the National Science Foundation. We thank J. Jensen for discussion and W. Grabowski for discussion and comments on an earlier draft of the paper. Comments and suggestions from A. Igel and two anonymous reviewers improved the paper. We also thank the UCLA-LES team for developing and maintaining the LES code, including B. Stevens, T. Heus, A. Seifert, and C. Hohenegger.
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Separately advecting a temperature-based quantity and water vapor mixing ratio can lead to generation of spuriously large supersaturations. This is due to the nonlinear dependence of supersaturation on temperature and water vapor combined with linear or semilinear advection, but primarily occurs at cloud edge rather than in the cloud interior (Stevens et al. 1996b; Grabowski and Morrison 2008).
