1. Introduction
Clouds result from processes acting on various spatial and temporal scales, ranging from the microscopic interplay between cloud droplets and small-scale turbulence, to the global circulation, which creates the specific environments in which certain types of cloud develop (e.g., Bodenschatz et al. 2010). The tremendous range of these scales is a barrier to the scientific understanding of clouds: the available methods cannot capture the entirety of the system. This is especially evident in (but not exclusive to) numerical modeling, where limited computational resources restrict resolution, domain size, or simulated time. Based on these constraints, specific types of models have been developed, from which large-eddy simulation (LES) has emerged as the method of choice for investigating cloud processes (e.g., Cuijpers and Duynkerke 1993). These models are able to represent individual clouds, as well as entire cloud fields, and most of the relevant dynamics explicitly. However, they are restricted to resolutions of tens of meters, which do not allow the explicit representation of small-scale turbulence.
Microscale dynamics, however, are essential for the mixing between clouds and cloud-free air and their specific effects on cloud microphysics (i.e., the size and number of cloud droplets). As originally pointed out by Baker and Latham (1979), mixing can be thought of as limited by two scenarios, distinguished by their Damköhler number (Da): homogeneous and inhomogeneous mixing. The Damköhler number is defined as the ratio of a fluid time scale to a reaction time scale. In cloud-physical applications, the former is the time a turbulent eddy needs to break down to the Kolmogorov length scale, and the latter is either the time necessary to evaporate a droplet completely or to saturate a subsaturated volume of air by droplet evaporation. If turbulent mixing is faster than the droplet evaporation time scale, this scenario is called homogeneous mixing (Da ≪ 1): all droplets experience the same subsaturation leading to the partial evaporation of all droplets while maintaining their number concentration. In the course of inhomogeneous mixing, in which turbulent mixing is slower than the evaporation of cloud droplets (Da ≫ 1), only those droplets that experience subsaturated cloud-free air evaporate. If these droplets evaporate completely and the remainder maintain their initial size, the distribution’s mean radius will be conserved while the droplet number concentration will decrease (this scenario is usually termed extreme inhomogeneous mixing). If the affected air parcel is lifted subsequently, the supersaturations it will experience are higher and the resulting diffusional growth stronger due to decreased competition for water vapor. Therefore, inhomogeneous mixing is considered as one source for the production of precipitation embryos, that is, droplets large enough to initiate collision and coalescence in warm clouds (e.g., Baker et al. 1980; Su et al. 1998; Lasher-Trapp et al. 2005). In addition to the above-introduced effects of inhomogeneous mixing, various other effects of small-scale turbulence on the initiation of rain have been investigated during the last few decades: spectral broadening by turbulent supersaturation fluctuations, enhanced collision rates due to turbulence-induced clustering of droplets, increased relative velocities, and larger collection efficiencies (e.g., Shaw 2003; Devenish et al. 2012; Grabowski and Wang 2013).
Mixing in warm clouds can be either homogeneous or inhomogeneous, but will always become homogeneous as a turbulent eddy breaks down to the Kolmogorov scale (Jensen and Baker 1989). If, however, mixing remains largely inhomogeneous during this process, cloudy and cloud-free filaments might be distinct on length scales on the order of centimeters (e.g., Beals et al. 2015), that is, substantially smaller than the typical grid spacing of current LESs. Consequently, the mixing of cloudy and cloud-free air on the LES subgrid scale (SGS) is always homogeneous since an LES cannot resolve the simultaneous presence of different air masses at the SGS; it only predicts the mean state.
The effect of inhomogeneous mixing on droplet growth can easily be parameterized by adjusting the droplet number concentration in accordance with the assumed or predicted mixing scenario (e.g., Lasher-Trapp et al. 2005; Morrison and Grabowski 2008; Jarecka et al. 2009; Hill et al. 2009; Cooper et al. 2013). But resolving inhomogeneous mixing explicitly requires reducing the grid spacing to the Kolmogorov length scale with corresponding sacrifices on the domain size and simulated time. This is actually done in computationally demanding, direct numerical simulations (DNSs), which have been used for investigating the effects of turbulent mixing processes in clouds on very limited spatial and temporal scales (e.g., Abma et al. 2013; Kumar et al. 2013, 2014; Götzfried et al. 2017).
However, it is not necessary to perform a computationally demanding DNS to reproduce the features of small-scale mixing. The so-called linear eddy model (LEM) by Kerstein (1988), a one-dimensional and hence computationally inexpensive representation of turbulence, is able to represent fundamental features of turbulence by compressing and folding the simulated domain according to the so-called triplet map, and computes the molecular diffusion of the simulated fields. The LEM is the centerpiece of the explicit mixing parcel model (EMPM) by Krueger (1993), which has been successfully used to simulate entrainment and mixing in clouds (Krueger 1993; Krueger et al. 1997; Su et al. 1998; Tölle and Krueger 2014). In fact, the EMPM (and hence the LEM) has been reported to reproduce DNS evolution of the turbulent mixing in clouds successfully (Krueger 2016).
The idea of using the LEM as a parameterization for SGS mixing in LESs for cloud-physical applications was suggested by Krueger (1993), and implementations for noncloud-physical applications have been reviewed by Menon and Kerstein (2011). In these approaches, the scalars temperature and water vapor are computed exclusively by the LEM. LES-scale resolved advection of these scalars is represented by splicing, in which certain parts of each LEM domain are exchanged between neighboring LES grid boxes. A first realization for cloud-physical applications was presented by Stechmann (2014), whose model, however, lacks the process of splicing.
Representing splicing is generally difficult. This results from the inherent unawareness of the origins of air on the LES SGS. On the edge of a cloud, for example, it is impossible to say how much air originates from the cloud or the cloud-free region, and how these fractions are arranged at the SGS. This Lagrangian information is not available in Eulerian LESs. In the following, an approach will be introduced that derives this information from a Lagrangian cloud model (LCM). LCMs are usually coupled to an LES to represent cloud microphysics by so-called superdroplets: Lagrangian particles that represent an ensemble of real droplets (Andrejczuk et al. 2008; Shima et al. 2009; Sölch and Kärcher 2010; Riechelmann et al. 2012). Here, the LCM’s superdroplets will also be used to track the SGS transport of air between the LEM domains of different LES grid boxes and, therefore, resolve splicing explicitly. After a description of the necessary changes in the LCM, LES, and LEM, the effects of this new approach, as well as the effects of small-scale inhomogeneous mixing on the production of precipitation embryos will be discussed using an idealized two-dimensional bubble simulation, as well as a well-studied shallow cumulus case.
2. Model formulation
a. Lagrangian microphysics
For this study, a simplified version of the LCM documented in Hoffmann et al. (2015), Hoffmann et al. (2017), and Hoffmann (2017) is applied, in which the processes of sedimentation and collection are neglected. The LCM is coupled to the nonhydrostatic, anelastic LES model System for Atmospheric Modeling (SAM; Khairoutdinov and Randall 2003). SAM predicts the three velocity components, liquid water static energy, water vapor mixing ratio, and the SGS turbulence kinetic energy, which is used in SAM’s 1.5th-order SGS scheme (Deardorff 1980).
The microphysics in the LCM can be summarized as follows. The centerpiece of the LCM is the superdroplet, which represents an ensemble of droplets with certain properties. The most important properties are the weighting factor, which reflects the number of (identical) real droplets represented by the superdroplet; the radius of these droplets; the location of the superdroplet in the LES domain; its velocity; and a new property to parameterize SGS mixing, which will be introduced below.
b. Using the LEM as a SGS model within an LCM–LES framework
Before advancing to a more detailed description of the new modeling approach, the general concept of integrating the LEM into an LCM–LES framework will be discussed by comparing it to established applications of the LEM as an SGS model in standard LES (i.e., without a coupled LCM) (e.g., Menon and Kerstein 2011), which will be referred to as LES–LEM (L2) in the following. The new approach will be called LCM–LES–LEM (L3). In L2, water vapor and temperature are exclusively calculated by the LEM, whose individual domains are located in each LES grid box, computing turbulent compression, folding, and molecular diffusion for the predicted fields. The LES-resolved-scale transport of water vapor and temperature is realized by splicing, in which a chunk of the LEM is removed from one LEM and added to the LEM of a neighboring LES grid box in accordance with the LES-resolved flow field. The new approach L3 deviates from L2 in various aspects, which will be briefly outlined here, with more details following below.
First, each superdroplet in L3 is assumed to be surrounded by a volume of air, which can be considered to be one LEM element. Accordingly, all superdroplets currently located in an LES grid box are associated with an LEM domain. Once a superdroplet moves from one LES grid box to another, this superdroplet is introduced into a new LEM domain. Thus, this approach allows one to resolve splicing explicitly; that is, it moves not only the right number of LEM elements in accordance with the LES-resolved flow field, as is done in L2, but also the correct LEM elements depending on their origin, which is inherently impossible in L2. Note that the motion of a superdroplet does not only depend on the LES-resolved flow field but also, usually to a smaller extent, on a stochastic velocity component that represents the unresolved SGS velocity fluctuations [see Eq. (1)]. These motions are unrelated to the LEM, which has no impact on the physical location of droplets. These stochastic motions are, however, essential in representing the dispersion of superdroplets in the LES domain correctly (e.g., Weil et al. 2004) and, therefore, are necessary for the correct representation of splicing between neighboring LES grid boxes.
Second, although it would be possible to use water vapor and temperature as LEM quantities in L3, they remain LES quantities. Instead, only one quantity, the perturbation of the absolute supersaturation 
Finally, 
c. Determining the supersaturation perturbation 
Note that 
d. The LEM within a Lagrangian framework
As introduced in Eq. (4), the LEM is used to calculate the SGS turbulent rearrangement and diffusion of 
Since one LEM is calculated in each LES grid box, all superdroplets currently located in that grid box are stored in the one-dimensional, cyclic domain of the LEM, in which each superdroplet represents one element of the LEM array. Accordingly, all properties of a superdroplet, and most importantly 
A schematic description of the LEM’s workflow is presented in Fig. 1. The LEM is called after advection of the superdroplets by the LES and before the calculation of condensation and evaporation. First, a segment of the LEM domain is selected to undergo turbulent rearrangement (Fig. 1, step 1). The segment’s starting position, as well as its length, are determined randomly, using an analytical expression for the probability of length scales in a high Reynolds number flow [see Eq. (2.3) in Krueger (1993)].
Schematic representation of the LEM using superdroplets. The superdroplets are arranged in a one-dimensional array and numbered accordingly. From this array, a segment is chosen to undergo turbulent compression and folding (red line, step 1). This segment is divided into so-called selection segments of three superdroplets each (step 2), which are redistributed into thirds (triplets) of the original segment (step 3): the first entrance of each triplet comes from a red selection segment, the second from a green segment, and the third from a blue segment, which maintains the gradient of the original segment in each triplet. Then, the central triplet is inverted (step 4). Molecular (or turbulent) diffusion is applied to the entire array at the last step of the LEM (step 5).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Schematic representation of the LEM using superdroplets. The superdroplets are arranged in a one-dimensional array and numbered accordingly. From this array, a segment is chosen to undergo turbulent compression and folding (red line, step 1). This segment is divided into so-called selection segments of three superdroplets each (step 2), which are redistributed into thirds (triplets) of the original segment (step 3): the first entrance of each triplet comes from a red selection segment, the second from a green segment, and the third from a blue segment, which maintains the gradient of the original segment in each triplet. Then, the central triplet is inverted (step 4). Molecular (or turbulent) diffusion is applied to the entire array at the last step of the LEM (step 5).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Schematic representation of the LEM using superdroplets. The superdroplets are arranged in a one-dimensional array and numbered accordingly. From this array, a segment is chosen to undergo turbulent compression and folding (red line, step 1). This segment is divided into so-called selection segments of three superdroplets each (step 2), which are redistributed into thirds (triplets) of the original segment (step 3): the first entrance of each triplet comes from a red selection segment, the second from a green segment, and the third from a blue segment, which maintains the gradient of the original segment in each triplet. Then, the central triplet is inverted (step 4). Molecular (or turbulent) diffusion is applied to the entire array at the last step of the LEM (step 5).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
The triplet map is applied to all superdroplets in this segment [see Fig. 1 in Krueger et al. (1997)]. This is done by creating an additional array that provides space for all superdroplets in the original segment, however, divided into three partitions of (ideally) the same size (henceforth, triplet partitions). (If this is not possible without a remainder, the third triplet partition contains one superdroplet, more or less.) Then, the original segment is divided into partitions containing (ideally) three superdroplets each (henceforth selection partitions; Fig. 1, step 2). (The last segment might contain only one or two superdroplets.) Then, the superdroplets in the selection partitions, which are accessed successively from left to right, are randomly assigned to the triplet partitions, one superdroplet for each triplet partition, filling them successively from left to right (Fig. 1, step 3). In doing so, the general distribution of superdroplets and their properties in the original segment is maintained in each of the triplet partitions, however, compressed to one-third of the original length, which increases gradients analogous to the compressive strain in a turbulent flow. Finally, the order of the superdroplets forming the center triplet partition is reversed to represent turbulent folding (Fig. 1, step 4). Note that this variant of the spatially discrete triplet map differs from the original deterministic variant proposed by Menon and Kerstein (2011) by selecting permutations randomly, but in a systematic way, instead of a purely deterministic approach. However, both variants agree well if a sufficiently large ensemble is simulated (not shown). Note further that all properties of the superdroplets, including 
Finally, splicing (i.e., the resolved-scale transport between the LEM domains of the different LES grid boxes) will be explained. This transport is explicitly resolved by the motion of superdroplets from one grid box to another by Eq. (1). This transport is calculated before the execution of the LEM at any time step of the LCM. Superdroplets moved from a considered grid box are simply removed from the LEM domain and the remaining elements are concatenated. Superdroplets moved to a considered grid box are treated as follows. All superdroplets originating from the same grid box are arranged in chunks. Within these chunks, the locations of superdroplets within their original grid boxes are maintained by using the above-introduced superdroplet property, which stores the superdroplets’ LEM locations in its original domain. These chunks are then randomly concatenated, and this structure is added to a random position of the LEM domain in the LES grid box under consideration.
The computational burden of the LEM is hard to quantify a priori since it depends fundamentally on the distribution of turbulence and therefore the need for subcycling. In the cases tested here, using the LEM increased computation time by about 12% in the two-dimensional bubble case, and by 8% in the shallow cumulus intercomparison case. These cases will be presented in the following sections.
3. Two-dimensional bubble
a. Setup
The two-dimensional bubble test case follows Grabowski et al. (2018), to which the reader is referred for more details on the initialization. In these simulations, a bubble of saturated air is initialized in a stably stratified environment of 20% relative humidity. The bubble has a diameter of 500 m and relaxes to environmental values within a 100-m radius. The domain is 3600 m in the horizontal direction and 2400 m in the vertical. An isotropic grid spacing of 20 m has been used throughout the domain. All simulations have been run for 1200 s of simulated time using a model time step of 2 s for the LES and the LCM; the LEM is subcycled if necessary. In all of the simulations, a cloud droplet concentration of 100 cm−3 has been prescribed, using an average number of 100, 200, or 500 superdroplets per grid box distributed randomly over the entire model domain. Simulations with and without the use of the LEM are presented, and will be referred to as LEM simulations (LCMLEM) or homogenous simulations (LCMhom), respectively. Note that the turbulence simulated in the following is two-dimensional and exhibits a spurious upscale transport of energy (e.g., Boffetta and Ecke 2012), which is not present in the three-dimensional turbulence typical of clouds. However, the simulations are computationally inexpensive, which enables one to vary various parameters (especially the number of simulated superdroplets) to investigate their impact on the simulation and the mixing in the LEM.
b. Results
(left) Variance of the supersaturation perturbation in each grid box 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(left) Variance of the supersaturation perturbation in each grid box
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(left) Variance of the supersaturation perturbation in each grid box
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
These variables already represent the general development of the simulated cloud from a circular bubble to a turbulent cloud, as described in more detail in Grabowski and Clark (1991) and Grabowski et al. (2018). This behavior is similar in all conducted simulations. Here, only the results for the 500 superdroplets per grid-box simulation are shown.
Throughout the simulation, 
According to Eq. (8), the number of superdroplets per grid box is crucial in representing the Kolmogorov length scale adequately. For the simulation displayed in Fig. 2, 500 superdroplets per grid box have been used. For the applied grid spacing of 20 m, this superdroplet concentration corresponds to a model Kolmogorov length scale of 0.24 m, but even larger model Kolmogorov length scales might be used in typical applications, which apply about 100 superdroplets per grid box. The general dependence of 
PDFs of (a) the variance of the supersaturation perturbation 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDFs of (a) the variance of the supersaturation perturbation
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDFs of (a) the variance of the supersaturation perturbation
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Figure 3c displays PDFs of the total supersaturation 
We display the time series of the liquid water path (LWP; Fig. 4a), mean effective radius 
Time series of (a) LWP, (b) mean effective radius, (c) droplet concentration, and (d) in-cloud dissipation rate for simulations with (blue lines) and without (red lines) the LEM for superdroplet concentrations of 100, 200, and 500 superdroplets per grid box (line patterns). The quantities in (b)–(d) have been derived from columns with visible optical thicknesses exceeding 1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Time series of (a) LWP, (b) mean effective radius, (c) droplet concentration, and (d) in-cloud dissipation rate for simulations with (blue lines) and without (red lines) the LEM for superdroplet concentrations of 100, 200, and 500 superdroplets per grid box (line patterns). The quantities in (b)–(d) have been derived from columns with visible optical thicknesses exceeding 1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Time series of (a) LWP, (b) mean effective radius, (c) droplet concentration, and (d) in-cloud dissipation rate for simulations with (blue lines) and without (red lines) the LEM for superdroplet concentrations of 100, 200, and 500 superdroplets per grid box (line patterns). The quantities in (b)–(d) have been derived from columns with visible optical thicknesses exceeding 1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDF of the droplet radius for simulations without the LEM (red lines), with the LEM (blue lines), and with the LEM using a high turbulent diffusion coefficient mimicking homogeneous mixing (green lines) for superdroplet concentrations of 100, 200, and 500 superdroplets per grid box (line patterns) at 300-s simulated time.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDF of the droplet radius for simulations without the LEM (red lines), with the LEM (blue lines), and with the LEM using a high turbulent diffusion coefficient mimicking homogeneous mixing (green lines) for superdroplet concentrations of 100, 200, and 500 superdroplets per grid box (line patterns) at 300-s simulated time.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDF of the droplet radius for simulations without the LEM (red lines), with the LEM (blue lines), and with the LEM using a high turbulent diffusion coefficient mimicking homogeneous mixing (green lines) for superdroplet concentrations of 100, 200, and 500 superdroplets per grid box (line patterns) at 300-s simulated time.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Additionally, Fig. 5 shows PDFs of the simulations using the LEM with an artificially increased turbulent diffusion coefficient of 
Besides small-scale inhomogeneous mixing, the in-cloud residence time of a droplet is assumed to be an important factor for the production of precipitation embryos.
We show the joint PDF (jPDF) for the maximum radius 
(a) The joint PDF 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
In general, the jPDFs (Figs. 6a and 7a) consist of two peaks: one centered at 
The distribution of 
The time spent in inhomogeneous condensational growth is determined as the time in which a superdroplet experiences condensational growth while other superdroplets in the same grid box experience evaporation, shown as 
The overall decreasing impact of inhomogeneous mixing on condensation growth at larger 
4. Shallow cumulus test case
a. Setup
The shallow cumulus test case follows the LES intercomparison by Siebesma et al. (2003), who developed this case based on the Barbados Oceanographic and Meteorological Experiment (BOMEX) measurement campaign (Holland and Rasmusson 1973). Similar to Siebesma et al. (2003), a 6.4 × 6.4 × 3.5 km3 model domain is simulated, using the same initial profiles, surface fluxes, and large-scale forcings. The only deviation is the grid spacing, which has been reduced to 20 m isotropically. A model time step of 2 s is used for the LES and the LCM; the LEM is subcycled as necessary. The simulated time is 6 h, with analysis focused on the last 4 h. The LCM is initialized with 100 superdroplets per grid box, representing a cloud droplet concentration of 100 cm−3. The resulting model Kolmogorov length scale of 1.2 m is significantly larger than the physical Kolmogorov length scale of about 1 mm in the highly turbulent environment of cumulus clouds. A smaller model Kolmogorov length scale is hence desirable but the necessary concentration of superdroplets is computationally unfeasible. However, the transition length scale from inhomogeneous to homogeneous mixing is about 0.83 m for this case [Eq. (12) using a kinetic energy dissipation rate of 
b. Results
The general distribution of 
(left) Variance of the supersaturation perturbation 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(left) Variance of the supersaturation perturbation
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(left) Variance of the supersaturation perturbation
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
We show time series of LWP (Fig. 9a), cloud cover, determined as the fraction of columns with visible optical thickness exceeding 1 (Fig. 9b), 
Time series of (a) LWP, (b) cloud cover, (c) mean effective radius, (d) mean droplet number concentration, and (e) in-cloud dissipation rate for simulations without the LEM (LCMhom; red lines) and with the LEM (LCMLEM; blue lines). The quantities in (b)–(e) have been derived from columns with visible optical thicknesses exceeding 1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Time series of (a) LWP, (b) cloud cover, (c) mean effective radius, (d) mean droplet number concentration, and (e) in-cloud dissipation rate for simulations without the LEM (LCMhom; red lines) and with the LEM (LCMLEM; blue lines). The quantities in (b)–(e) have been derived from columns with visible optical thicknesses exceeding 1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Time series of (a) LWP, (b) cloud cover, (c) mean effective radius, (d) mean droplet number concentration, and (e) in-cloud dissipation rate for simulations without the LEM (LCMhom; red lines) and with the LEM (LCMLEM; blue lines). The quantities in (b)–(e) have been derived from columns with visible optical thicknesses exceeding 1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Domain-averaged profiles of (a) liquid water potential temperature, (b) total water mixing ratio, and (c) conditionally averaged profiles of cloud and cloud core liquid water mixing ratio for simulations without the LEM (LCMhom; red lines) and with the LEM (LCMLEM; blue lines). The profiles have been averaged over the last 4 h of the simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Domain-averaged profiles of (a) liquid water potential temperature, (b) total water mixing ratio, and (c) conditionally averaged profiles of cloud and cloud core liquid water mixing ratio for simulations without the LEM (LCMhom; red lines) and with the LEM (LCMLEM; blue lines). The profiles have been averaged over the last 4 h of the simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Domain-averaged profiles of (a) liquid water potential temperature, (b) total water mixing ratio, and (c) conditionally averaged profiles of cloud and cloud core liquid water mixing ratio for simulations without the LEM (LCMhom; red lines) and with the LEM (LCMLEM; blue lines). The profiles have been averaged over the last 4 h of the simulation.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Figures 11 and 12 show an analysis of superdroplet trajectories to analyze jPDFs for 
(a) The joint PDF 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
(a) The joint PDF
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
The jPDF is peaked for adiabatic droplets 
Inhomogeneous condensational growth (Fig. 11c) only contributes significantly to two groups of droplets. First, droplets with a small adiabatic radius 
In general, the distribution of the residence time 
An additional interesting finding in Fig. 12b is the existence of droplets with a lifetime of more than 60 min although the typical lifetime of the simulated clouds is reported to range between 10 and 40 min (e.g., Jiang et al. 2006). Whether these droplets where just lucky enough to be located in an extraordinarily long-living cloud or if there is another mechanism responsible, will be investigated in a future study.
5. Conclusions
This study presents an implementation of the linear eddy model (LEM) by Kerstein (1988) as a subgrid-scale (SGS) model to parameterize the mixing of air masses and its effects of the diffusional growth on cloud droplets in large-eddy simulations (LESs) with a coupled Lagrangian cloud model (LCM). So-called superdroplets, for which cloud microphysics are calculated in the LCM, are used to provide the history of air masses in each LES grid box. Based upon this information, a supersaturation perturbation is derived for each superdroplet, and redistributed among all superdroplets in a grid box using the LEM in accordance with the LES SGS model. This enables the appropriate representation of SGS mixing scenarios, potentially ranging from inhomogeneous to homogeneous, but also the explicit resolution of the grid-scale transport of LEM quantities between neighboring grid boxes, which was absent in a previous approach that used the LEM as an LES SGS model in a cloud-physical application (Stechmann 2014). Ultimately, the supersaturation perturbation is considered in the diffusional growth of the superdroplets, enhancing or diminishing condensation and evaporation. Additionally, the new approach is shown to significantly mitigate the production of spurious supersaturations. The implementation of this new approach has been tested in two cases: an idealized two-dimensional bubble cloud and a well-known shallow cumulus intercomparison case. In all cases, an isotropic LES grid spacing of 20 m has been applied, which is usually considered to be too large to resolve all scales of inhomogeneous mixing.
In the initial stage of the bubble case, a relatively low intensity of turbulence resulted in strong SGS inhomogeneous mixing. As expected, this decreased the number of cloud droplets and increased the mean droplet radius. This resulted in an overall increase in the relaxation time scale and accordingly a slightly lower liquid water path compared to simulations without the new approach. Moreover, the new approach resulted in maximum radii of up to 26 μm compared to 20 μm using the standard approach, indicating that SGS inhomogeneous mixing would accelerate the production of precipitation embryos. Detailed analysis of superdroplet trajectories revealed that the largest droplets underwent significant SGS inhomogeneous condensational growth when the LEM was applied, while in the simulations without the LEM, the droplet in-cloud residence time was shown to be more important for the production of superadiabatic droplets.
The shallow cumulus case did not reveal significant differences in bulk properties derived from model runs with or without the new approach, which suggests that a grid spacing of 20 m might be sufficient to represent all relevant scales of inhomogeneous mixing for the analysis conducted here. Due to the highly turbulent nature of these clouds, the transition from inhomogeneous to homogeneous mixing takes place at relatively large spatial scales of about 1 m, which is reported to favor homogeneous mixing (Lehmann et al. 2009) and corresponds to observations of turbulent cumuli by Jensen et al. (1985), Jensen and Baker (1989), and Gerber et al. (2008), which indicate homogeneous mixing. In fact, the largest impact of inhomogeneous mixing is only found at cloud base, where turbulence is weak, and not at higher levels of the cloud, which are usually targeted in observations. Nevertheless, inhomogeneous mixing is shown here to accelerate droplet growth at cloud base, whereas droplet in-cloud residence times are of primary importance for the production of the largest precipitation embryos at cloud top. Note, however, that the analysis presented here did not explicitly consider the cloud life cycle, in which mixing is reported to become more inhomogeneous toward the end (Schmeissner et al. 2015).
The results of this study suggest the following scenario for the initiation of rain. In highly turbulent cumulus clouds, the production of precipitation embryos might not benefit from inhomogeneous mixing since it is restricted to larger scales. Therefore, other processes need to be taken into account. Since droplet residence times are usually limited by the lifetime of the clouds, turbulence-enhanced collision rates, which tend to increase with turbulence, might be the primary path to raindrops for highly turbulent clouds (e.g., Devenish et al. 2012; Grabowski and Wang 2013). In less turbulent clouds (e.g., stratocumulus) inhomogeneous mixing might be of greater importance for the production of precipitation embryos. However, these clouds also exhibit long droplet residence times (e.g., Feingold et al. 1996). Accordingly, these processes and their effects on the initiation of rain need to be evaluated in future studies.
Indeed, the simulation of stratocumulus is the next logical application for the presented approach since it not only allows for the investigation of inhomogeneous mixing in those clouds but it also mitigates the production of spurious supersaturations, which is a major numerical problem typically found just below the capping inversion, resulting in spurious activation of aerosols at cloud top (e.g., Stevens et al. 1996). Within this context, it will also be necessary to investigate aerosol effects on turbulence–microphysics interactions. In low-aerosol environments, turbulence-induced broadening of the droplet size distribution is enhanced due to the slower microphysical reaction time scale, which potentially accelerates rain formation (Chandrakar et al. 2016). In high-aerosol environments, turbulence is increased due to the so-called entrainment–evaporation feedback (Xue and Feingold 2006), which might result in a more homogeneous mixing scenario that decelerates rain formation (Feingold and Siebert 2009).
Moreover, the model itself needs to be extended for future applications since it a priori neglects sedimentation and collection. In fact, droplet sedimentation, a necessary prerequisite for collection, is prohibited because the current approach inherently demands that superdroplets move with their surrounding air. It is also able to enhance mixing, especially in low-turbulence environments (e.g., Su et al. 1998; Tölle and Krueger 2014). Therefore, it will be necessary to remove superdroplets from the LEM if they exhibit a significant sedimentation velocity, for example, based on the so-called velocity ratio between the sedimentation velocity and the (model’s) Kolmogorov velocity scale (e.g., Vaillancourt and Yau 2000).
All in all, this study has shown that the use of the LEM as an SGS model is a useful addition to all warm-cloud LCMs, enabling these models to consider length scales that, to date, have been exclusive to direct numerical simulations.
Acknowledgments
This research was performed while FH held a visiting fellowship of the Cooperative Institute for Research in Environmental Sciences (CIRES) at the University of Colorado Boulder and the NOAA/Earth System Research Laboratory. Marat Khairoutdinov graciously provided the SAM model. We thank two anonymous reviewers and Steve Krueger for their comments.
APPENDIX A
The Effect of LES Mixing on 
The last two terms of Eq. (5) are (usually) calculated for the entire grid box and need to be interpolated to the superdroplet. However, the determination of 
Note that the entire determination of 
APPENDIX B
Mitigation of Spurious Supersaturations
As in Hoffmann (2016), the simplified advection problem by Stevens et al. (1996) is repeated using the LCM described above, neglecting vertical motion and mixing 
Time series of (a) total relative supersaturation and (b) mean radius in the initially cloud-free grid box for simulations with (LCMLEM; blue lines) and without the new approach (LCMhom; red lines) for superdroplet concentrations of 50, 200, and 500 superdroplets per grid box (line patterns).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Time series of (a) total relative supersaturation and (b) mean radius in the initially cloud-free grid box for simulations with (LCMLEM; blue lines) and without the new approach (LCMhom; red lines) for superdroplet concentrations of 50, 200, and 500 superdroplets per grid box (line patterns).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Time series of (a) total relative supersaturation and (b) mean radius in the initially cloud-free grid box for simulations with (LCMLEM; blue lines) and without the new approach (LCMhom; red lines) for superdroplet concentrations of 50, 200, and 500 superdroplets per grid box (line patterns).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
Note that a complete cancellation of spurious supersaturations would be possible within the current framework. For this, it is necessary to exclude the supersaturation perturbation 
APPENDIX C
Entraining Parcel Simulations
Similar to the simulations presented in Su et al. (1998), the following analysis will present offline entraining parcel simulations (without a coupled LES), using the LEM implementation presented in section 2 (i.e., an LEM based on superdroplets and the perturbation supersaturation). Environmental conditions (e.g., background profiles of water vapor mixing ratio and potential temperature, as well as surface pressure) are all based on the shallow cumulus case by Siebesma et al. (2003) while Su et al. (1998) used a slightly different setup. All other parameters follow closely the latter study. The parcel is lifted with a vertical velocity of 2 m s−1 starting from the surface. The domain size and the model integral scale (i.e., the extent of the LEM) are set to 20 m, and the kinetic energy dissipation rate is prescribed as 10 cm2 s−3. Entrainment is treated as a discrete process, in which a randomly selected, continuous region of the parcel is replaced by a blob of entrained environmental air. This region is randomly chosen, the size of the blob is specified to be 2 m, and the timing of entrainment is randomized based on the fractional entrainment rate of 1.0 km−1. The model Kolmogorov length scale is determined by the number of simulated superdroplets [see Eq. (8)], with values set to 1.2, 0.6, 0.24, 0.24, 0.12, and 0.06 m when using 100, 200, 500, 1000, and 2000 superdroplets, respectively. Note that the entrained air contains as many superdroplets as the removed air, deprived of any liquid water, to maintain a constant particle concentration of 100 cm−3 throughout the simulation.
Figure C1 shows droplet size distributions from the entraining parcel using the LEM (left column) or instantaneous (homogeneous) mixing (center column), as well as a pure adiabatic parcel without entrainment (right column) similar to Fig. 3 in Su et al. (1998). Results from ensembles using 100 simulations (black lines) and individual simulations (red lines) are shown. Generally, the results agree well with Su et al. (1998). Especially in the individual simulations, the LEM is necessary to reproduce a realistic droplet spectrum. Moreover, the results show that entrainment generally results in larger maximum droplet sizes due to the reduction in the number of droplets in the parcel. This process is stronger when using the LEM and intensifies with the number of superdroplets and accordingly the model’s ability to reproduce the physical Kolmogorov model results as closely as possible, as also described by Su et al. (1998). This has not been observed in the higher-dimensional simulations presented above, in which mixing can be represented on longer length scales due to the grid-scale transport between individual LEMs calculated for every grid box.
PDFs of the droplet radius for entraining parcel simulations using the (left) LEM and (middle) instantaneous (homogeneous) mixing, as well as (right) a pure adiabatic parcel without entrainment at different times and heights during the parcel’s ascent for ensembles simulations (black lines, with the dash pattern indicating the number of superdroplets) and individual simulations (red lines).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDFs of the droplet radius for entraining parcel simulations using the (left) LEM and (middle) instantaneous (homogeneous) mixing, as well as (right) a pure adiabatic parcel without entrainment at different times and heights during the parcel’s ascent for ensembles simulations (black lines, with the dash pattern indicating the number of superdroplets) and individual simulations (red lines).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
PDFs of the droplet radius for entraining parcel simulations using the (left) LEM and (middle) instantaneous (homogeneous) mixing, as well as (right) a pure adiabatic parcel without entrainment at different times and heights during the parcel’s ascent for ensembles simulations (black lines, with the dash pattern indicating the number of superdroplets) and individual simulations (red lines).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0087.1
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