Improving collisional growth in Lagrangian cloud models: development and verification of a new splitting algorithm

Schwenkel, Johannes; Hoffmann, Fabian; Raasch, Siegfried

doi:https://doi.org/10.5194/gmd-11-3929-2018

Articles | Volume 11, issue 9

https://doi.org/10.5194/gmd-11-3929-2018

Special issue:

Particle-based methods for simulating atmospheric aerosol...

https://doi.org/10.5194/gmd-11-3929-2018

Articles | Volume 11, issue 9

Development and technical paper

28 Sep 2018

Development and technical paper |

| 28 Sep 2018

Improving collisional growth in Lagrangian cloud models: development and verification of a new splitting algorithm

Johannes Schwenkel, Fabian Hoffmann, and Siegfried Raasch

Abstract

Lagrangian cloud models (LCMs) are increasingly used in the cloud physics community. They not only enable a very detailed representation of cloud microphysics but also lack numerical errors typical for most other models. However, insufficient statistics, caused by an inadequate number of Lagrangian particles to represent cloud microphysical processes, can limit the applicability and validity of this approach. This study presents the first use of a splitting and merging algorithm designed to improve the warm cloud precipitation process by deliberately increasing or decreasing the number of Lagrangian particles under appropriate conditions. This new approach and the details of how splitting is executed are evaluated in box and single-cloud simulations, as well as a shallow cumulus test case. The results indicate that splitting is essential for a proper representation of the precipitation process. Moreover, the details of the splitting method (i.e., identifying the appropriate conditions) become insignificant for larger model domains as long as a sufficiently large number of Lagrangian particles is produced by the algorithm. The accompanying merging algorithm is essential to constrict the number of Lagrangian particles in order to maintain the computational performance of the model. Overall, splitting and merging do not affect the life cycle and domain-averaged macroscopic properties of the simulated clouds. This new approach is a useful addition to all LCMs since it is able to significantly increase the number of Lagrangian particles in appropriate regions of the clouds, while maintaining a computationally feasible total number of Lagrangian particles in the entire model domain.

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Dates

Received: 20 Apr 2018 – Discussion started: 17 May 2018 – Revised: 10 Aug 2018 – Accepted: 10 Sep 2018 – Published: 28 Sep 2018

1 Introduction

Lagrangian cloud models (LCMs) are a recently developed approach to simulate cloud microphysics (Shima et al., 2009; Sölch and Kärcher, 2010; Andrejczuk et al., 2010; Riechelmann et al., 2012; Arabas et al., 2015; Naumann and Seifert, 2015; Grabowski et al., 2018; Sardina et al., 2018). These models represent microphysics by individually simulated particles, so-called superdroplets, each representing a certain number of identical real droplets. This number is called the multiplicity or weighting factor. These models have been successfully used to investigate various aspects of aerosol–cloud interactions (e.g., Andrejczuk et al., 2010; Hoffmann et al., 2015; Hoffmann, 2017) or precipitation processes (e.g., Naumann and Seifert, 2016; Hoffmann et al., 2017; Dziekan and Pawlowska, 2017).

Unterstrasser et al. (2017) have reviewed all three currently available LCM approaches for representing collection, from which the so-called all-or-nothing algorithm (based on Shima et al., 2009, and Sölch and Kärcher, 2010, and used by Arabas et al., 2015; Dziekan and Pawlowska, 2017; and Hoffmann et al., 2017) exhibits the best performance, i.e., it agrees well with analytical solutions or other modeling approaches used to represent collection. When using an unfortunate initialization of superdroplets with equal weighting factors, however, even the all-or-nothing algorithm struggles to represent the precipitation process correctly. The reason for that is easily explained. Cloud droplets cover a wide range of radii from micrometers to centimeters and a similarly wide range of abundances across this spectrum (e.g., Rogers and Yau, 1989). This system cannot be simplified to a couple of superdroplets (with accordingly large weighting factors, approximately 10⁹ for typical LES-LCM applications). In fact, a large number of superdroplets (in the magnitude of 10–100 per grid box with accordingly small weighting factors) is needed to represent this range adequately. Since this is usually not the case, a few of the largest superdroplets may unrealistically contain the majority of all liquid water. In order to improve the statistics of these particles, Unterstrasser et al. (2017) suggested that the splitting of these particles can help to improve the representation of the precipitation process as it is already done for other microphysical processes (e.g., nucleation in ice clouds; Unterstrasser and Sölch, 2014).

The present study introduces and verifies a splitting algorithm designed to improve the precipitation process. Additionally, an accompanying merging algorithm is proposed that is able to unite superdroplets that are not required for an adequate representation of the precipitation process. Thus, the merging algorithm is essential to improve the computational performance of the LCM. Both algorithms are tested in zero-dimensional box simulations, a three-dimensional simulation of a single cumulus cloud, and an established shallow cumulus test case. This paper is structured as follows. The next section briefly summarizes the collection algorithm and the basic framework of the applied LCM. Section 3 introduces the splitting and merging algorithms, and Sect. 4 shows results in which splitting and merging are applied. Finally, Sect. 5 concludes the paper.

2 Basic equations of the LCM

This section gives a short overview of the LCM basic equations. The applied LCM was initially developed by Riechelmann et al. (2012) and its current version is documented in Hoffmann et al. (2017). Besides collection, the LCM calculates diffusional growth as well as the transport of the superdroplets. These processes are coupled to the PALM large-eddy simulation (LES) model (Maronga et al., 2015), which solves the non-hydrostatic incompressible Boussinesq-approximated Navier–Stokes equations, and prognostic equations for the water vapor mixing ratio and potential temperature. In addition to these coupled simulations, we use a zero-dimensional box model, in which the process of collision and coalescence is considered as the only microphysical process.

In the following, the applied collection algorithm will be summarized to show how collection affects a superdroplet's weighting factor, and to understand how collection and splitting interact. The reader is referred to Unterstrasser et al. (2017) for a more rigorous description of this all-or-nothing approach and for comparisons with other LCM collection algorithms. For the following, it is assumed that all superdroplets are sorted by their weighting factor such that $A_{n} > A_{n + 1}$ (the case of $A_{n} = A_{n + 1}$ will be discussed further below). For all superdroplet combinations with $1 \leq n < m \leq N_{p}$ , where N_p is the number of superdroplets located in a grid box, the probability that one droplet of superdroplet m collects an arbitrary droplet of superdroplet n is given by

\begin{matrix} (1) & p_{m n} = K (r_{m}, r_{n}) \frac{Δ t}{Δ V} \cdot A_{n}, \end{matrix}

where Δt is the length of the collection time step, ΔV the volume of the grid box, r_n the radius of a droplet represented by superdroplet n, and K is the collection kernel (based on Hall, 1980, for this study). Since p_mn is usually smaller than 1, collections only occur if p_mn>ξ, where ξ is a random number uniformly chosen from the interval [0,1]. This probabilistic approach ensures that the number of collections calculated in the model is identical to the number of collections resulting from Eq. (1) if averaged over a sufficiently long period of time.

If a collection takes place, each droplet of superdroplet m will collect one droplet of superdroplet n. This results in commensurate changes in the weighting factor A_n and the individual droplet mass $m_{m} = A_{m} \cdot 4 / 3 π ρ_{l} r_{m}^{3}$ with the liquid water density ρ_l, while A_m and m_n remain unchanged:

\begin{array}{l} (2) & {\hat{A}}_{m} = A_{m} and {\hat{A}}_{n} = A_{n} - A_{m}, \\ (3) & {\hat{m}}_{m} = m_{m} + m_{n} and {\hat{m}}_{n} = m_{n}, \end{array}

where $\hat{(\dots)}$ marks the variable after collection.

If A_m=A_n, the above-described collection would result in one superdroplet with a zero weighting factor. To avoid deleting this superdroplet, the droplets of the superdroplet that has grown by collection are distributed equally among the involved superdroplets m and n:

\begin{array}{l} (4) & {\hat{A}}_{m} = {\hat{A}}_{n} = A_{m} / 2, \\ (5) & {\hat{m}}_{m} = {\hat{m}}_{n} = 2 m_{m} . \end{array}

Diffusional growth is described by

\begin{matrix} (6) & r_{n} \frac{d r_{n}}{d t} = \frac{S}{F_{k} + F_{d}} f (r_{n}) . \end{matrix}

The ventilation effect f(r_n) describes the accelerated evaporation of large drops. S is the supersaturation (calculated in the LES), and F_k and F_d are coefficients considering the effects of heat conduction and the diffusion of water vapor, respectively (e.g., see Rogers and Yau, 1989). Note that curvature and aerosol solute effects, as well as gas-kinetic effects, are neglected in Eq. (6), but this equation is appropriate for the purpose of this study which focuses on the precipitation process, i.e., larger droplets for which these processes are irrelevant.

Transport of each superdroplet is described by

\begin{matrix} (7) & \frac{d X_{n}}{d t} = u (X_{n}) + {\tilde{u}}_{n}, \end{matrix}

where X_n is the location of the superdroplet, u the LES-resolved velocity interpolated to the superdroplet's location, and $\tilde{u}$ is a stochastic velocity component to parameterize subgrid-scale fluctuations unresolved in the LES (e.g., see Sölch and Kärcher, 2010).

3 Splitting and merging

The following subsections will introduce techniques for the interactive modification of the number of superdroplets by splitting and merging. This is different from most previous LCM approaches, in which the number of superdroplets is set at the beginning of the simulation and remains constant thereafter (unless precipitation scavenges superdroplets). Note that the splitting and merging algorithms will be tested for the all-or-nothing collection approach, but they are similarly applicable to the average-impact approach introduced by Riechelmann et al. (2012).

3.1 Splitting

Splitting takes place if a superdroplet fulfills certain criteria. First, the radius of the superdroplet needs to be greater than or equal to a threshold r_spl. This is necessary to limit splitting to the region of interest, i.e., coalescing droplets for which an improved statistical representation is required. Second, the weighting factor of the superdroplet needs to be greater than or equal to a threshold A_spl to avoid excessive and potentially useless splitting. And finally, A_spl is required to be at least larger than η_spl, which is the number of superdroplets in which the superdroplet is split. This ensures that no superdroplets with an unrealistic weighting factor of less than 1 are created.

The numerical implementation of the splitting can be understood as cloning of the superdroplet that has been determined to be split. In addition to the already existing superdroplet, η_spl−1 new superdroplets are created. To conserve the total amount of represented droplets, the weighting factor of these η_spl superdroplets is reduced to

\begin{matrix} (8) & A_{n}^{*} = \frac{A_{n}}{η_{spl}} . \end{matrix}

Note that all η_spl superdroplets have identical properties immediately after splitting, including their location. However, each superdroplet will develop an individual trajectory independent from the others due to the stochastic velocity component in Eq. (7), which is determined individually for each superdroplet.

In a first straightforward approach, the thresholds r_spl, A_spl, and the splitting factor η_spl are explicitly prescribed. In the following this method is abbreviated as S mode, where the S stands for simple.

In a more advanced method (abbreviated G, standing for gamma distribution), the threshold A_spl and the splitting factor η_spl are estimated from an idealized gamma distribution, which is assumed to describe the distribution of droplets larger than r_spl in each grid box of the simulated model domain (e.g., Ulbrich, 1983):

\begin{matrix} (9) & n (r) = N_{0} r^{μ} \exp (- λ r), \end{matrix}

where n(r)⋅dr states the number of particles per unit volume in the size range $(r, r + d r)$ . Here N₀ is the intercept, μ is the shape, and λ is the slope parameter of the gamma distribution. These parameters are calculated as

\begin{array}{l} (10) & N_{0} & = \frac{N_{r}}{Γ (μ + 1)} λ^{μ + 1}, \\ (11) & λ & = {[\frac{π ρ_{l}}{6} (μ + 3) (μ + 2) (μ + 1) {\overline{x_{r}}}^{- 1}]}^{\frac{1}{3}}, \end{array}

and

\begin{matrix} (12) & μ = \frac{(1 - ζ) n + 1}{ζ - 1}, \end{matrix}

where N_r is the number concentration of droplets with r≥r_spl, Γ is the gamma function, ρ_l is the density of liquid water, $\overline{x_{r}}$ is the mean geometric radius, and ζ is a factor calculated as

\begin{matrix} (13) & ζ = \frac{M_{0} M_{2}}{M_{1}^{2}}, \end{matrix}

where M_k is the kth moment of the mass density distribution (see Seifert, 2008). The calculation of these moments in the LCM framework will be described in Sect. 4.1.1.

The assumed drop size distribution (DSD) is calculated from n_bin=100 logarithmically spaced bins. (Larger values for n_bin did not alter the results.) The center of bin i is calculated as

\begin{matrix} (14) & r_{bc, i} = 10^{\log_{10} (r_{min}) + i ν}, \end{matrix}

where

\begin{matrix} (15) & ν = \frac{\log_{10} (r_{max}) - \log_{10} (r_{min})}{n_{bin} - 1} . \end{matrix}

The minimum and maximum radius of the discretized spectra are denoted with r_min and r_max, respectively. Here these values are set to r_min=r_spl and r_max=5 mm, which ensures that the whole spectrum of droplet sizes is included. Furthermore, the boundaries of bin i are given by $r_{bb, i} = 10^{\log_{10} (r_{min}) + (i - 0.5) \cdot ν}$ and r_bb,i+1. Hence, the width of bin i is $Δ r_{i} = r_{bb, i + 1} - r_{bb, i}$ .

It is assumed that the weighting factor of a superdroplet should be smaller than or equal to the approximated number of droplets in the corresponding bin of the discretized gamma distribution. Thus, the weighting factor threshold is determined by

\begin{matrix} (16) & A_{spl, i} = max [n_{i} (r_{bc, i}) \cdot Δ r_{i} \cdot Δ V, 1] . \end{matrix}

Accordingly, the number of newly generated superdroplets depends on the ratio of the initial weighting factor to the estimated number of droplets using the gamma distribution:

\begin{matrix} (17) & η_{spl} = ⌊\frac{A_{n}}{A_{spl, i}}⌋ . \end{matrix}

Since only a positive integer of superdroplets can be generated, the splitting factor is rounded down to the nearest whole number.

No matter which splitting mode is chosen, the splitting operations are executed at each time step of the LCM. Due to limited computational resources, the generation of new superdroplets must be restricted to a feasible amount. Hence, two limitations are introduced. The first restriction is the maximum splitting factor η_max, i.e., the maximum number of clones produced per splitting. This parameter is used for the G mode, in which Eq. (17) might not be well-defined in the case of large droplets for which A_spl,i approaches zero. The second limitation ensures a computationally feasible number of superdroplets in every grid box by introducing a fixed maximum N_P,max. Accordingly, splitting operations are only executed if the number of superdroplets in one grid box is smaller than N_P,max. The latter threshold is applied for the G and the S mode. A suitable choice of these limits will be presented in Sect. 4.1.2.

3.2 Merging

As a consequence of the potentially massive generation of new superdroplets due to splitting, the total number of superdroplets may increase sharply, which makes simulations computationally very expensive. For this reason, a merging algorithm was developed to decrease the number of superdroplets in order to reduce the required computational resources.

To avoid an impact of merging on micro- or macrophysical properties of the cloud, the algorithm is only executed in non-cloudy grid boxes (liquid water is lower than q_l<0.01 g kg⁻¹). Accordingly, cloudy regions, in which a high number of superdroplets are necessary for the correct representation of potential collisional growth, are left unaffected. Furthermore, it is required that the merged superdroplets are smaller or equal to r_mer=0.1 µm, which ensures that only evaporated superdroplets are affected, and not raindrops that precipitate from the cloud. Additionally, merging is only executed in grid boxes in which the initial superdroplet concentration is exceeded and superdroplets exhibit a weighting factor that is smaller than a certain threshold A_mer, rationally chosen to be smaller or equal to the initial weighting factor. This is done to avoid decreasing the LCM's baseline capability to represent DSDs set during initialization.

The algorithm is designed as follows. Based on the thresholds r_mer and A_mer, each superdroplet with r_m≤r_mer and A_m≤A_mer in a non-cloudy grid box is merged with the next superdroplet of the same grid box. Here the next superdroplet is the superdroplet located next in the memory, which enables an efficient execution of the merging algorithm. The new weighting factor of the remaining superdroplet (index n) is mass-weighted and given by $A_{n}^{*} = A_{n} + A_{m} \cdot r_{m}^{3} / r_{n}^{3}$ , while the other superdroplet (index m) is deleted. Accordingly, this leads to a new integral mass $M_{n}^{*} = M_{n} + M_{m}$ , guaranteeing mass conservation. An averaging of other superdroplet properties (e.g., velocity components, radius, and location) is not implemented and probably not necessary for the correct representation of the cloud since merging is restricted to a cloud-free environment. Moreover, a more advanced method was tested, in which the most similar droplets (within one grid box) concerning their mass are merged. These simulations show similar results. However, due to sorting processes the computing time is increased in comparison to the simple method.

The use of the merging algorithm inside certain regions of the cloud where collection plays only a subordinate role is also conceivable. However, the (probably sophisticated) determination of necessary thresholds is not within the scope of this study. Furthermore, it must be mentioned that the merging algorithm does not conserve size and chemical composition of the aerosol. Therefore, for studies that explicitly consider aerosols, the merging algorithm needs to be adapted.

4 Applications

4.1 Box model simulations

In the following box simulations, the sensitivity of the LCM collection process to the number of simulated superdroplets, different splitting approaches, and the approaches' specific parameters is investigated. Therefore, the box model simulation considers collection as the only microphysical process.

4.1.1 Setup

Although zero-dimensional simulations do not have a spatial extent, allocating a certain weighting factor requires a reference volume to represent a defined droplet concentration. Therefore, the volume of a grid box is 8×10³ m³, which corresponds to an isotropically spaced grid with $Δ x = Δ y = Δ z = 20 m$ . The simulation time is 3600 s with a constant time step of 1 s. To ensure adequate statistics, 25 344 boxes are calculated, and results are averaged over this ensemble. (The number of ensemble members represents the maximum amount of grid boxes which can be calculated on four computing nodes in an appropriate time.) In the following this method is referred to as a single-box model.

Besides the traditional single-box approach, a new multi-box approach is introduced. In contrast to the calculation of independent grid boxes, the multi-box approach allows superdroplets to move from one grid box to the next by prescribing a stochastic velocity (but no mean motion) in Eq. (7), using 25 344 grid boxes, as in the single-box ensemble above, with cyclic boundary conditions among which the superdroplets are allowed to move. The stochastic velocity component is chosen in such a way that it corresponds to a kinetic energy dissipation rate of ϵ_box=0.01 m²s⁻³, which is typical for shallow cumulus clouds (e.g., Shaw et al., 1998).

This multi-box approach has one distinct advantage over the ensemble mean of the same amount of individual box model simulations (single-box model), which results from the difficulties to initialize a DSD with superdroplets of a constant weighting factor, as it is done in many applications of LCMs in the literature (e.g., Shima et al., 2009; Riechelmann et al., 2012; Naumann and Seifert, 2015; Hoffmann et al., 2017; Sardina et al., 2018). A single-box model simulation suffers crucially from this initialization method due to a wrong representation of the largest and rarest superdroplets (Unterstrasser et al., 2017, their Fig. 17). In doing so, the rarest and largest, and therefore most important superdroplets for the collection process, are a priori over- or underestimated. An exchange of superdroplets between the collection boxes helps to mitigate this problem. Moreover, this new approach is closer to the representation of collection in three-dimensional simulations, in which a superdroplet is not bound to a single grid box.

The impact of different numbers of superdroplets per grid box and the use of splitting for the traditional single-box approach will be discussed first; then, the new introduced multi-box approach will be presented for both splitting and non-splitting cases. Box model simulations will be compared to the results of Wang et al. (2007), who used a high-resolution bin model. The purpose of this study is, however, not the exact reproduction of these results but a computationally efficient approximation to them using splitting. Accordingly, the initialization of the box simulation follows Wang et al. (2007), using an exponential initial DSD:

\begin{matrix} (18) & n (r, t = t_{0}) = \frac{3 N_{init}}{r_{0}^{3}} \cdot r^{2} \exp (- \frac{r^{3}}{r_{0}^{3}}), \end{matrix}

where N_init=300 cm⁻³ is the droplet number concentration. The initial mean radius is r₀=9.3 µm, which leads to a liquid water content of L₀=1 g m⁻³. Following Wang et al. (2007), we set the minimum droplet radius to r_min=1.5 µm. Superdroplet radii are then selected by a random generator which follows the distribution given by Eq. (18). All superdroplets receive the same initial weighting factor:

\begin{matrix} (19) & A_{init} = \frac{N_{init} \cdot Δ V}{N_{P}}, \end{matrix}

which ensures the number concentration of 300 cm⁻³. This method is also described as ν_const-init in Unterstrasser et al. (2017), which has been chosen in this study to resemble the initialization of superdroplets in less-idealized applications but also significantly hinders collisional growth.

As reference, the “singleSIP” initialization of Unterstrasser et al. (2017) is also used for the single-box model. In contrast to the previously described initialization, the initial DSD is discretized using logarithmically spaced bins. The number of bins corresponds to the number of superdroplets. To each bin, a superdroplet with a corresponding mean radius and weighting factor is assigned. The maximum radius of the initial distribution is approximately 33 µm, which corresponds to a number of concentrations of 1∕ΔV. This avoids superdroplets with a weighting factor less than 1. Note that this (not always applicable) initialization technique represents the inherent variability in droplet radii and their abundance across the initial spectrum much more accurately than the previously described method, and therefore results in a much better agreement with literature references.

In addition to analyzing the DSD directly, the temporal development of the zeroth and second moment of the mass density distributions is examined. Due to mass conservation in all applied approaches, the first moment is constant in time and will not be shown. The moments of the mass distribution f_m are defined as

\begin{matrix} (20) & M_{k} = \int m^{k} f_{m} (m) d m, \end{matrix}

where m is the mass and f_m(m) denotes the number concentration distribution. Note that the zeroth moment M₀ is the number concentration and the second moment M₂ is proportionate to the radar reflectivity, and thus highly sensitive to the largest droplets in the DSD.

For a given superdroplet ensemble the moments for each grid box are calculated with

\begin{matrix} (21) & M_{k} = \sum_{n = 1}^{N_{P}} A_{n} m_{n}^{k} / Δ V, \end{matrix}

where m_n is the single droplet mass ( $m_{n} = 4 / 3 π ρ_{l} r_{n}^{3}$ ) of a superdroplet.

4.1.2 Box model results

First, the sensitivity of the collision algorithm to the number of superdroplets is examined using the LCM as a single-box model. Second, the improvements by the splitting method on collisional growth is evaluated. Subsequently, those investigations are repeated for the multi-box approach.

Single-box approach

Figures 1 and 2 show the mass density distribution after 3600 s and the temporal development of the moments for the LCM applied as a single-box model using the singleSIP initialization by Unterstrasser et al. (2017). Each grid box is initialized with a different number of superdroplets (colored lines). The reference solution of Wang et al. (2007) is shown as a black solid line. Figure 1 shows that even with 87 superdroplets the solution of Wang et al. (2007) can be reproduced well and a further increase in the number of superdroplets only leads to minor improvements. The small deviations between the bin model solution and the LCM can be traced back to the different solution of the collection equation (e.g., Dziekan and Pawlowska, 2017). Overall, it can be seen that the solution of the LCM converges with an increasing number of superdroplets. The moments of mass distribution (Fig. 2) also show convergence with an increasing number of superdroplets. This good representation of collision growth is in line with the results with Unterstrasser et al. (2017).

https://www.geosci-model-dev.net/11/3929/2018/gmd-11-3929-2018-f01

Figure 1Mass density distribution for the single-box approach after 3600 s for the “singleSIP” initialization. The black solid line denotes the solution of Wang et al. (2007). The colored dashed curves show the solution of the LCM with different numbers of superdroplets per grid box.

Improving collisional growth in Lagrangian cloud models: development and verification of a new splitting algorithm

3.1 Splitting

3.2 Merging

4.1 Box model simulations

4.1.1 Setup

4.1.2 Box model results

Single-box approach

Multi-box approach

Sensitivity to splitting thresholds

4.2 Single cloud

4.2.1 Setup

4.2.2 Single-cloud results

Microphysical properties

Macrophysical properties

4.3 Cloud field

4.3.1 Setup

4.3.2 Cloud field results