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Relationships between Cloud Droplet Spectral Relative Dispersion and Entrainment Rate and Their Impacting Factors


doi: 10.1007/s00376-022-1419-5

  • Cloud microphysical properties are significantly affected by entrainment and mixing processes. However, it is unclear how the entrainment rate affects the relative dispersion of cloud droplet size distribution. Previously, the relationship between relative dispersion and entrainment rate was found to be positive or negative. To reconcile the contrasting relationships, the Explicit Mixing Parcel Model is used to determine the underlying mechanisms. When evaporation is dominated by small droplets, and the entrained environmental air is further saturated during mixing, the relationship is negative. However, when the evaporation of big droplets is dominant, the relationship is positive. Whether or not the cloud condensation nuclei are considered in the entrained environmental air is a key factor as condensation on the entrained condensation nuclei is the main source of small droplets. However, if cloud condensation nuclei are not entrained, the relationship is positive. If cloud condensation nuclei are entrained, the relationship is dependent on many other factors. High values of vertical velocity, relative humidity of environmental air, and liquid water content, and low values of droplet number concentration, are more likely to cause the negative relationship since new saturation is easier to achieve by evaporation of small droplets. Further, the signs of the relationship are not strongly affected by the turbulence dissipation rate, but the higher dissipation rate causes the positive relationship to be more significant for a larger entrainment rate. A conceptual model is proposed to reconcile the contrasting relationships. This work enhances the understanding of relative dispersion and lays a foundation for the quantification of entrainment-mixing mechanisms.
    摘要: 云与环境空气之间的夹卷混合过程能够显著影响云的微物理特性。然而,目前尚不清楚夹卷率如何影响云滴谱的离散度。在以往的观测结果中发现,离散度和夹卷率之间的关系可以为正相关或负相关。为了调和两者关系的不一致性,本文使用显式混合气泡模式来揭示决定两者关系的物理机制。结果显示,当蒸发以小云滴为主,并使卷入云中的环境空气在混合过程中达到饱和时,两者关系为负相关。当大云滴的蒸发占主导地位时,两者关系为正相关。在卷入的环境空气中是否考虑云凝结核是决定两者关系的一个关键因素,因为通过凝结核凝结是产生小云滴的主要途径。如果未卷入凝结核,两者关系为正相关。如果卷入凝结核,两者关系亦受到其他因素的影响。垂直速度、环境空气的相对湿度和含水量较大时以及云滴数浓度较小时更有利于两者形成负相关,因为新的饱和更容易通过小云滴的蒸发来实现。此外,两者关系受湍流耗散率的影响较小,但夹卷率较大时较强的耗散率导致正相关关系更为显著。本文提出了一个概念模型来调和两者关系的不一致性,该工作增强了对离散度的理解,并为进一步定量化夹卷混合机制奠定了基础。
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  • Figure 1.  Temporal evolutions of cloud droplet spectral relative dispersion (d), droplet number concentration (nc), liquid water content (LWCc), and mean radius (rm) with the entrained environmental air blob number of the second entrainment-mixing process (N2) equal to 2, 4, 6, 8, and 10. Left: entrained environmental air without cloud condensation nuclei (CCN). Right: entrained environmental air with CCN. The triangle and pentagram represent the ending of the first entrainment-mixing process (“Mix 1”) and the beginning of the second entrainment-mixing process (“Mix 2”), respectively, as well as the adiabatic ascending process (“Adiabatic”).

    Figure 2.  Temporal evolutions of the ratios of (a, b) small droplet number concentration (ns) to total droplet number concentration (nc), (c, d) number concentration of small droplets originated from entrained cloud condensation nuclei (ns_ccn) to ns, and (e, f) number concentration of small droplets originated from big droplets evaporation (ns_evap) to ns, for the entrained environmental air blob numbers of the second entrainment-mixing process (N2) equal to 2, 4, 6, 8, and 10. Left: environmental air without entrained cloud condensation nuclei (CCN). Right: environmental air with entrained CCN. The triangle and pentagram represent the ending of the first entrainment-mixing process and the beginning of the second entrainment-mixing process, respectively.

    Figure 3.  Effects of vertical velocity (w) (Case 2 in Table 1). Temporal evolutions of relative dispersion for (a) w = 0.5 m s−1, (b) w = 1.0 m s−1 (baseline case), and (c) w = 1.5 m s−1. The entrained environmental air blob numbers of the second entrainment-mixing process (N2) equal 2, 4, 6, 8, and 10. The triangle and pentagram represent the ending of the first entrainment-mixing process and the beginning of the second entrainment-mixing process, respectively.

    Figure 4.  Temporal evolutions of the ratios of (a, b) small droplet number concentration (ns) to total droplet number concentration (nc) for the entrained environmental air blob numbers of the second entrainment-mixing process (N2) equal to 2, 4, 6, 8, and 10. Left: vertical velocity (w) = 0.5 m s−1. Right: w = 1.5 m s−1. The triangle and pentagram represent the ending of the first entrainment-mixing process and the beginning of the second entrainment-mixing process, respectively.

    Figure 5.  As in Fig. 3, except for relative humidity of entrained air (RHe).

    Figure 6.  As in Fig. 4, except for relative humidity of entrained air (RHe).

    Figure 7.  As in Fig. 3, except for turbulence dissipation rate (ɛ).

    Figure 8.  As in Fig. 4, except for turbulence dissipation rate (ɛ).

    Figure 9.  As in Fig. 3, except for initial droplet number concentration (ni).

    Figure 10.  As in Fig. 4, except for initial droplet number concentration (ni).

    Figure 11.  As in Fig. 3, except for initial liquid water content (LWCi).

    Figure 12.  As in Fig. 4, except for initial liquid water content (LWCi).

    Figure 13.  Flow diagram illustrating the formation of the positive/negative correlation between relative dispersion (d) and entrainment rate (λ). During the ascending process, the first and second entrainment-mixing processes are labeled as “Mix 1” and “Mix 2”, as well as “Adiabatic” for the adiabatic ascending process. The impacting factors are: vertical velocity (w), relative humidity of entrained air (RHe), initial droplet number concentration (ni), and initial liquid water content (LWCi).

    Table 1.  Parameters of sensitivity simulations.

    CasesEntrained Cloud
    Condensation
    Nuclei, CCN
    Vertical
    velocity,
    w (m s−1)
    Relative humidity
    of entrained
    air, RHe (%)
    Turbulence
    dissipation rate,
    ɛ (m2 s−3)
    Initial droplet
    number
    concentration,
    ni (cm−3)
    Initial liquid
    water content,
    LWCi (g m−3)
    Entrained air blob
    number of the
    second entrainment-
    mixing process (N2)
    Baseline caseYes1.0885×10−3119.40.52, 4, 6, 8, 10
    Case 1:
    Entrained CCN effect
    No1.0885×10−3119.40.52, 4, 6, 8, 10
    Case 2:
    w effect
    Yes0.5, 1.0, 1.5885×10−3119.40.52, 4, 6, 8, 10
    Case 3:
    RHe effect
    Yes1.077, 88, 93.55×10−3119.40.52, 4, 6, 8, 10
    Case 4:
    ɛ effect
    Yes1.0885×10−4, 5×10−3,
    1×10−2
    119.40.52, 4, 6, 8, 10
    Case 5:
    ni effect
    Yes1.0885×10−369.1, 119.4,
    552.6
    0.52, 4, 6, 8, 10
    Case 6:
    LWCi effect
    Yes1.0885×10−3119.40.25, 0.5, 0.752, 4, 6, 8, 10
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Manuscript History

Manuscript received: 09 November 2021
Manuscript revised: 08 April 2022
Manuscript accepted: 06 May 2022
通讯作者: 陈斌, bchen63@163.com
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Relationships between Cloud Droplet Spectral Relative Dispersion and Entrainment Rate and Their Impacting Factors

    Corresponding author: Chunsong LU, luchunsong110@gmail.com
  • 1. College of Aviation Meteorology, Civil Aviation Flight University of China, Guanghan 618307, China
  • 2. Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters (CIC-FEMD)/ Key Laboratory for Aerosol-Cloud-Precipitation of China Meteorological Administration, Nanjing University of Information Science & Technology, Nanjing 210044, China
  • 3. Environmental and Climate Sciences Department, Brookhaven National Laboratory, Upton, New York 11973, US
  • 4. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China
  • 5. Nanjing Joint Institute for Atmospheric Sciences, Nanjing 210019, China
  • 6. Collaborative Innovation Center for Western Ecological Safety, Lanzhou University, Lanzhou 730000, China

Abstract: Cloud microphysical properties are significantly affected by entrainment and mixing processes. However, it is unclear how the entrainment rate affects the relative dispersion of cloud droplet size distribution. Previously, the relationship between relative dispersion and entrainment rate was found to be positive or negative. To reconcile the contrasting relationships, the Explicit Mixing Parcel Model is used to determine the underlying mechanisms. When evaporation is dominated by small droplets, and the entrained environmental air is further saturated during mixing, the relationship is negative. However, when the evaporation of big droplets is dominant, the relationship is positive. Whether or not the cloud condensation nuclei are considered in the entrained environmental air is a key factor as condensation on the entrained condensation nuclei is the main source of small droplets. However, if cloud condensation nuclei are not entrained, the relationship is positive. If cloud condensation nuclei are entrained, the relationship is dependent on many other factors. High values of vertical velocity, relative humidity of environmental air, and liquid water content, and low values of droplet number concentration, are more likely to cause the negative relationship since new saturation is easier to achieve by evaporation of small droplets. Further, the signs of the relationship are not strongly affected by the turbulence dissipation rate, but the higher dissipation rate causes the positive relationship to be more significant for a larger entrainment rate. A conceptual model is proposed to reconcile the contrasting relationships. This work enhances the understanding of relative dispersion and lays a foundation for the quantification of entrainment-mixing mechanisms.

摘要: 云与环境空气之间的夹卷混合过程能够显著影响云的微物理特性。然而,目前尚不清楚夹卷率如何影响云滴谱的离散度。在以往的观测结果中发现,离散度和夹卷率之间的关系可以为正相关或负相关。为了调和两者关系的不一致性,本文使用显式混合气泡模式来揭示决定两者关系的物理机制。结果显示,当蒸发以小云滴为主,并使卷入云中的环境空气在混合过程中达到饱和时,两者关系为负相关。当大云滴的蒸发占主导地位时,两者关系为正相关。在卷入的环境空气中是否考虑云凝结核是决定两者关系的一个关键因素,因为通过凝结核凝结是产生小云滴的主要途径。如果未卷入凝结核,两者关系为正相关。如果卷入凝结核,两者关系亦受到其他因素的影响。垂直速度、环境空气的相对湿度和含水量较大时以及云滴数浓度较小时更有利于两者形成负相关,因为新的饱和更容易通过小云滴的蒸发来实现。此外,两者关系受湍流耗散率的影响较小,但夹卷率较大时较强的耗散率导致正相关关系更为显著。本文提出了一个概念模型来调和两者关系的不一致性,该工作增强了对离散度的理解,并为进一步定量化夹卷混合机制奠定了基础。

    2.   Model and Methods
    • The EMPM model developed by Kerstein (1992), Krueger et al. (1997), Su et al. (1998), Krueger et al. (2008), and Tölle and Krueger (2014) has been extensively utilized to study the entrainment-mixing processes between cloud and the ambient air (e.g., Krueger et al., 1997, 2008; Su et al., 1998; Lu et al., 2013b, 2018b, 2020; Tölle and Krueger, 2014; Luo et al., 2020, 2021). The EMPM can track the history of every droplet due to the variation of the local environment, ranging from the model integral scale to the model Kolmogorov scale (~1 mm). During the entrainment-mixing process, four processes are considered: parcel ascent, entrainment, turbulent mixing (i.e., turbulent deformation and molecular diffusion), and droplet condensation/evaporation. Turbulent deformation is a key process and is implemented by random rearrangement events as a finite-rate turbulent mixing. Using the “triplet map” introduced by Kerstein (1991), each event can be realized. The “triplet map” replaces the scalar field within the randomly selected segment with three compressed copies of the scalar field and then inverts the central copy (Krueger et al., 1997; Su et al., 1998). This treatment changes the scalar gradient within the segment and realizes the effect of compressive strain in a turbulent flow (Krueger et al., 1997; Su et al., 1998). The finite-rate mixing has been supported by observations (Gerber et al., 2008) and is beneficial to the broadening of CDSD. This setting allows individual droplet to experience different local supersaturation environment and contribute to the different condensation/evaporation rate (Su et al., 1998).

      When entrainment occurs, a randomly selected region of the parcel is replaced with environmental air. The subsequent turbulent mixing process includes two stages. In the first stage, the entrained air breaks down under the action of turbulent eddies and randomly distributes within the cloudy parcel. This stage increases the interfacial area between the cloudy air and entrained air. In the second stage, when entrained air size approaches the Kolmogorov microscale, the molecular diffusion process takes effect and rapidly smooths out the scalar gradients (Krueger et al., 1997; Su et al., 1998; Tölle and Krueger, 2014). A more detailed description of the EMPM has been provided by Krueger et al. (1997), Su et al. (1998), Krueger et al. (2008), and Tölle and Krueger (2014). The length, width, and height of the EMPM domain are 20 m, 1 mm, and 1 mm, respectively. Su et al. (1998) tested the sensitivity to sizes equaling 100 m and 20 m. The simulations indicated that “the 100-m results are similar to those from the 20-m case”. Further, the EMPM works as follows: the entire EMPM domain containing droplets is treated as a cloudy parcel and ascends at a specific vertical velocity across the entire domain. When the entrainment-mixing process occurs, environmental air replaces the same-sized cloudy parcel at the entrainment height. During the subsequent ascending process, the entrained environmental air mixes with the cloudy air at a finite turbulent rate (“Mix 1”). Subsequently, the cloud grows adiabatically (“Adiabatic”) until another entrainment-mixing process occurs (“Mix 2”). Notably, the cloud continues to ascend during the three stages.

      The initial droplets in the cloudy parcel are assumed to follow a gamma size distribution (Liu et al., 2002; McFarquhar et al., 2015; Lu et al., 2020; Bera, 2021), such that the droplet number concentration n(r) for a droplet radius (r) is given as:

      where N0, β, and μ represent the intercept, slope, and shape parameters, respectively. The narrow initial CDSD has μ and d of 40.0 and 0.16, respectively, and is binned by 49 bins, ranging from 1 to 25 μm in radius (Fig. S1 in the Electronic Supplementary Material, ESM). During ascent, the first entrainment-mixing process is set to occur near the beginning of the simulations to examine the effects of entrainment-mixing on the narrow CDSD. Since the model output frequency is 0.75 s, the first entrainment process is chosen to occur at 0.75 s. During the subsequent adiabatic process, d first increases and then decreases. The formation of small droplets increases d and the growth of small droplets into big droplets by condensation decreases d, as per the theoretical expectation of droplet growth (Wallace and Hobbs, 2006). The second entrainment-mixing process occurs when d reaches its maximum. Such a choice is employed to weaken the significant effect of the condensation process, which results in a decrease in d. Herein, we mainly focus on the relationship between d and λ after the second entrainment-mixing process.

      The cloudy parcel has initial pressure, water vapor mixing ratio, and temperature of 963.95 hPa, 15.73 g kg−1, and 293.56 K (Raga et al., 1990; Tölle and Krueger, 2014), respectively. The baseline case of the EMPM has a vertical velocity (w) of 1.0 m s−1, relative humidity of entrained environmental air (RHe) of 88%, turbulence dissipation rate (ɛ) of 5×10−3 m2 s−3, initial droplet number concentration (ni) of 119.4 cm−3, initial liquid water content (LWCi) of 0.5 g m−3, and initial mean volume radius (rvi) of 10 μm. Further, entrained environmental air is assumed to be without and with CCN (Cases without and with CCN are considered.). According to Su et al. (1998) and Krueger et al. (2008), the CCN distribution is composed of two log-normal size distributions. The first one is 18 categories of salt with the radius range of 0.14–3.73 μm; the mean and standard deviation are 0.9 μm and 0.63 μm, respectively. The second one is 31 categories of ammonia bi-sulfate with the radius range of 0.02–0.75 μm; the mean and standard deviation are 0.09 μm and 0.05 μm, respectively; the CCN concentration is 49.65 cm−3. These CCN are assumed to be at equilibrium sizes, and the competition for water vapor between CCN and droplets is considered. The growth of each droplet is dependent on its local meteorological fields, and the curvature and solution effects of droplet are considered (see Appendix for the droplet growth equations).

      Previous measurements and numerical simulations in shallow cumulus clouds indicate that ɛ can range from 10−5 to 10−2 m2 s−3 (Siebert et al., 2006a, b; Hoffmann et al., 2014); w can increase from near 0 at cloud edge to 6 m s−1 in the core of the convective cloud (Jonas, 1990; Burnet and Brenguier, 2007; Hudson et al., 2012); LWC is between 0.1 g m−3 and 1.2 g m−3 (Gerber et al., 2008; Hudson et al., 2012); ni can vary from 20 to 700 cm−3 (Burnet and Brenguier, 2007; Gerber et al., 2008; Small and Chuang, 2008; Hudson et al., 2012); and RHe can be in the range of 70%–95% (Axelsen, 2005; Burnet and Brenguier, 2007; Lu et al., 2018c). To explore the effects of impact factors on the relationship between d and λ, the w values in the domain are set to 0.5, 1.0, and 1.5 m s−1, respectively; the RHe values are set to 77%, 88%, and 93.5%, respectively; the ɛ values are set to 5×10−4, 5×10−3, and 1×10−2 m2 s−3, respectively; the ni values are set to 69.1, 119.4, and 552.6 cm−3, computed from LWCi of 0.5 g m−3 and rvi of 12, 10, and 6 μm, respectively. LWCi of 0.25, 0.5, and 0.75 g m−3 are also included with a fixed rvi of 10 μm. The above parameter settings are listed in Table 1. These values in the sensitivity tests are within reasonable ranges and represent typical shallow convective clouds with distinct environmental thermodynamics and air pollution amounts.

      CasesEntrained Cloud
      Condensation
      Nuclei, CCN
      Vertical
      velocity,
      w (m s−1)
      Relative humidity
      of entrained
      air, RHe (%)
      Turbulence
      dissipation rate,
      ɛ (m2 s−3)
      Initial droplet
      number
      concentration,
      ni (cm−3)
      Initial liquid
      water content,
      LWCi (g m−3)
      Entrained air blob
      number of the
      second entrainment-
      mixing process (N2)
      Baseline caseYes1.0885×10−3119.40.52, 4, 6, 8, 10
      Case 1:
      Entrained CCN effect
      No1.0885×10−3119.40.52, 4, 6, 8, 10
      Case 2:
      w effect
      Yes0.5, 1.0, 1.5885×10−3119.40.52, 4, 6, 8, 10
      Case 3:
      RHe effect
      Yes1.077, 88, 93.55×10−3119.40.52, 4, 6, 8, 10
      Case 4:
      ɛ effect
      Yes1.0885×10−4, 5×10−3,
      1×10−2
      119.40.52, 4, 6, 8, 10
      Case 5:
      ni effect
      Yes1.0885×10−369.1, 119.4,
      552.6
      0.52, 4, 6, 8, 10
      Case 6:
      LWCi effect
      Yes1.0885×10−3119.40.25, 0.5, 0.752, 4, 6, 8, 10

      Table 1.  Parameters of sensitivity simulations.

    • As mentioned above, we concentrate on the second entrainment-mixing process and further investigate the relationship between d and λ. Here, λ is calculated using the method developed by Lu et al. (2012):

      where L is the length of EMPM; χ is the integrated fraction of the adiabatic cloud; $\chi^{*}_1$ and $\chi^{*}_2$ are the fractions of the adiabatic cloud at the first and second entrainment heights, respectively; and N1 and N2 are the entrained environmental air blob numbers for the first and second entrainment-mixing processes, respectively. Evidently, λ is determined by N2, given each entrained environmental air blob size (l) of 0.5 m, N1 of 10, and the entrainment height above cloud base (h). The baseline case is taken as an example to calculate λ. The corresponding height of the initial CDSD is 204.5 m above the cloud base and is calculated by assuming that the cloud parcel with LWCi = 0.5 g m−3 moves downward until LWCi = 0 g m−3. As the second entrainment occurs at 42.75 s and w is 1 m s−1, h is equal to 42.75 m + 204.5 m (i.e., 247.25 m). As expected, a larger N2 corresponds to a larger λ (Fig. S2 in the ESM), and λ exists in a reasonable range, from 1.35 to 2.29 km−1 (Lu et al., 2012; de Rooy et al., 2013). N2 is used hereafter to refer to λ. For cases with and without entrained CCN, the relationship between d and λ is investigated by setting different N2 values. In the cases of baseline and all sensitivity simulations, N2 are set to 2, 4, 6, 8, and 10, respectively, with a fixed N1 of 10. Different N1 values (2, 4, 6, 8, 10) are also tested, and as expected, a larger N1 leads to a larger d after the first entrainment-mixing process (Fig. S3 in the ESM); this is because the initial adiabatic CDSD is narrow (Tölle and Krueger, 2014; Gao et al., 2018; Luo et al., 2020). Therefore, d and λ are positively correlated. Since this relationship between d and λ is well known for adiabatic narrow CDSD, we focus on the second entrainment-mixing process and select the widest CDSD at the beginning of the second entrainment-mixing process (i.e., at the end of the first entrainment-mixing process). The CDSD after the first entrainment-mixing process for N1 = 10 is the widest and has the biggest difference from the adiabatic CDSD. Therefore, N1 is set to 10.

    3.   Results
    • As mentioned above, different N2 values are employed in the EMPM model to determine the relationship between d and λ. As previous studies revealed that entrained CCN has a significant impact on microphysical properties (Lasher-trapp et al., 2005; Gerber et al., 2008; Krueger et al., 2008; Slawinska et al., 2012; Hoffmann et al., 2015; Bera et al., 2016b; Yeom et al., 2019; Chen et al., 2020), it should be interesting to explore if and how entrained CCN affect the reproduction of positive and negative correlations between d and λ. The d value occurring when new saturation is achieved after the second entrainment-mixing process is the one employed to assess the relationship between d and λ. For the criterion of new saturation, the domain-averaged supersaturation should be greater than 99.5% (Lehmann et al., 2009; Luo et al., 2020).

    • Two experiments are conducted (one with entrained CCN and the other without). In the group of entrained CCN, entrained air contains CCN during both the first and second entrainment-mixing processes. In the other group, entrained CCN is not considered during the two entrainment-mixing processes. Figures 1a, 1c, 1e, and 1g show the temporal evolutions of d, droplet number concentration (nc), liquid water content (LWCc), and mean radius (rm), respectively, in the simulation without entrained CCN (i.e., Case 1 in Table 1). The increase in d is accompanied by the notable decreases in LWCc and rm (Figs. 1a, 1e, 1g) during the first entrainment-mixing process (before the triangle, determined by the criterion of the domain-averaged supersaturation greater than 99.5%). The broadening of CDSD is attributed to the partial evaporation of big droplets (Figs. S4, S5 in the ESM) (e.g., Yum and Hudson, 2005; Lu et al., 2013a; Kumar et al., 2014; Tölle and Krueger, 2014; Gao et al., 2018; Luo et al., 2020). Further, only a slight decrease in nc is observed (Fig. 1c) during mixing, suggesting a homogeneous mixing type, as droplet size decreases whereas number concentration remains virtually unchanged during homogeneous mixing (Baker et al., 1980; Lu et al., 2013b; Tölle and Krueger, 2014).

      Figure 1.  Temporal evolutions of cloud droplet spectral relative dispersion (d), droplet number concentration (nc), liquid water content (LWCc), and mean radius (rm) with the entrained environmental air blob number of the second entrainment-mixing process (N2) equal to 2, 4, 6, 8, and 10. Left: entrained environmental air without cloud condensation nuclei (CCN). Right: entrained environmental air with CCN. The triangle and pentagram represent the ending of the first entrainment-mixing process (“Mix 1”) and the beginning of the second entrainment-mixing process (“Mix 2”), respectively, as well as the adiabatic ascending process (“Adiabatic”).

      During the adiabatic process (between the triangle and pentagram in Fig. 1), the reactivation of deactivated CCN from complete evaporation of droplets occurs due to supersaturation (Yang et al., 2018; Bera, 2021). However, the amount of reactivation of deactivated CCN is small; only a small peak of droplets near 1.0 μm in radius is observed (Figs. S4, S5). A radius of 1.0 μm is employed as the criterion that separates cloud droplets from aerosols (Hsieh et al., 2009; Ma et al., 2010; Small et al., 2013; Yum et al., 2015; Bera et al., 2016a; Lu et al., 2020). Therefore, d and nc vary slightly, and rm increases due to the condensation process (Figs. 1a, 1c, 1g). During the second entrainment-mixing process (after the pentagram in Fig. 1a), nc, LWCc, and rm decrease; d increases as N2 increases. The broadening of CDSD towards small droplets by the evaporation of big droplets is more significant as N2 increases (Figs. S4, S5). At the final states of the entrainment-mixing process, a larger N2 has a larger d, suggesting a positive correlation between d and λ. This relationship aligns with that presented by Lu et al. (2013a) and conforms with commonly reported results that more entrained environmental air causes a more significant broadening of CDSD (Lasher-Trapp et al., 2005; Tölle and Krueger, 2014; Kumar et al., 2017).

    • Entrained CCN are also considered (i.e., the baseline case in Table 1), to further explore the relationship between d and λ. Of note, the microphysical properties display slight differences during the first entrainment-mixing process, compared with Case 1 without entrained CCN (Fig. 1). During the adiabatic process after the first entrainment-mixing process, water vapor condenses onto the entrained CCN, contributing to the formation of droplets (particles with a radius greater than 1.0 μm). These newly formed droplets significantly broaden CDSD and a remarkable peak of small droplets is observed (Figs. S4, S5), causing an increase in d and nc and a decrease in rm (Figs. 1b, 1d,1h). This pronounced increase in d aligns with the statements made by Telford and Chai (1980), Lasher-Trapp et al. (2005), Krueger et al. (2008), and Yeom et al. (2019). For instance, Yeom et al. (2019) reported that a higher droplet number concentration corresponds to a higher d due to the secondary activation of aerosol. As expected, nc in the baseline case is greater than that without entrained CCN (Figs. 1c, 1d), and LWCc is comparable in the two cases (Figs. 1e, 1f). In contrast, rm is smaller when entrained CCN is considered (Figs. 1g, 1h). During the second entrainment-mixing process, d monotonically decreases, and rm increases due to the substantial evaporation of small droplets with mixing time for N2 ≤ 8. The peak of small droplets in CDSD gradually decreases (Figs. S4, S5). However, when N2 = 10, d first decreases and then increases; correspondingly, rm first increases and then decreases, which is discussed in detail later. When new saturation is achieved after the second entrainment-mixing process, a larger N2 has a smaller d when N2 increases from 2 to 8 (Fig. 1b), suggesting a negative correlation between d and λ. Such a finding is supported by Guo et al. (2018) who found that d decreased as λ increased. However, when N2 increases from 8 to 10, d and λ are found to be positively correlated.

      The effects of entrained l are examined by setting l = 1 m and N2 = 1, 2, 3, 4, 5 to keep the fraction of entrained environmental air the same as that in the baseline case. As shown in Fig. S6, the blob size may affect the detailed response of cloud microphysics to entrained air but does not affect the main conclusions.

    • Guo et al. (2018) highlighted the role of small droplets in determining the negative relationship between d and λ. Tölle and Krueger (2014) and Luo et al. (2020) also revealed the effects of small droplets on d. Here, small droplets are separated by the criterion of radius less than 3.0 μm. The ratio of small droplet number concentration (ns) to total droplet number concentration (nc) in Fig. 2a is similar to that of d in Fig. 1a without entrained CCN, suggesting that ns is a key property in determining d. The evolution of the two properties with entrained CCN also support this argument. Altogether, these results confirm the positive correlation between ns and d reported by Luo et al. (2020).

      Figure 2.  Temporal evolutions of the ratios of (a, b) small droplet number concentration (ns) to total droplet number concentration (nc), (c, d) number concentration of small droplets originated from entrained cloud condensation nuclei (ns_ccn) to ns, and (e, f) number concentration of small droplets originated from big droplets evaporation (ns_evap) to ns, for the entrained environmental air blob numbers of the second entrainment-mixing process (N2) equal to 2, 4, 6, 8, and 10. Left: environmental air without entrained cloud condensation nuclei (CCN). Right: environmental air with entrained CCN. The triangle and pentagram represent the ending of the first entrainment-mixing process and the beginning of the second entrainment-mixing process, respectively.

      To clearly explain the physical mechanisms in the above negative and positive correlations between d and λ, small droplets are tracked with the EMPM model, which can track the history of each droplet. Figure 2 shows the sources of small droplets and the proportions of small and big droplets (radius greater than 3.0 μm). Two sources of small droplets are included: partial evaporation of big droplets during mixing and evaporation, and condensation on the entrained CCN, which is consistent with the observation results of Bera (2021). Of note, the reactivation of deactivated CCN is negligible as discussed. As a result, this source is not included.

      In Case 1 without entrained CCN, ns/nc increases as N2 increases, which corresponds to the positive correlation between d and λ (Fig. 2a). Almost all small droplets arise from the partial evaporation of big droplets (Fig. 2e), though the accumulation of small droplets at the beginning of the second entrainment-mixing process is weak, with ns/nc of only ~ 6% (the pentagram in Fig. 2a). The small droplets evaporate faster and are easier to completely evaporate, compared with big droplets, mainly because of the relationship of dr/dt~S/r, where S is supersaturation and t is time (Wallace and Hobbs, 2006; Guo et al., 2018; Bera, 2021); the smaller the droplet size, the faster it evaporates. However, evaporation of these small droplets is not enough to saturate the entrained environmental air during the second entrainment-mixing process; to achieve new saturation, evaporation of big droplets is needed. Therefore, big droplets, instead of small droplets, dominate the evaporation process, resulting in a positive correlation between d and λ.

      In contrast to Case 1, small droplets account for 28% of total droplets at the beginning of the second entrainment-mixing process in the baseline case with entrained CCN (Fig. 2b). Almost 85% of small droplets are derived from the entrained CCN (Fig. 2d), whereas the remaining 15% are derived from the partial evaporation of big droplets (Fig. 2f). Figure 2b shows that ns/nc monotonically decreases with mixing time for N2 ≤ 8 during the second entrainment-mixing process, which is mainly attributed to the decrease in newly formed small droplets from the entrained CCN (Fig. 2d). As the entrained air is saturated due to evaporation being dominated by small droplets, d and λ are negatively correlated. It is challenging to determine the homogeneity of entrainment-mixing mechanisms (Gao et al., 2020, 2021). Theoretically, for homogeneous mixing, all droplets are exposed to the same subsaturated conditions and evaporate simultaneously (Lehmann et al., 2009; Lu et al., 2013b). Inhomogeneous mixing refers to the situation where only part of the cloudy air is affected by the mixing and therefore complete evaporation of all droplets in a small region of the cloud causes a significant decrease in number concentration (Baker et al., 1980). However, the decrease in number concentration is not necessarily caused by inhomogeneous mixing, but can also be a result of homogeneous mixing where all droplets experience the same subsaturated environment; in this case, small droplets in the entire cloudy air completely evaporate with the large droplets remaining in the cloudy air (Pinsky et al., 2016; Khain et al., 2018; Pinsky and Khain, 2018; Luo et al., 2021). When N2 further increases to 10, the newly formed droplets from the entrained CCN decrease to nearly 0 (Fig. 2d). Since the new saturation of environmental air cannot be restored dominantly with the evaporation of these newly formed small droplets, the evaporation of big droplets dominates the process (Figs. S4, S5). As a result, the small droplets caused by the partial evaporation of big droplets increase (Fig. 2f). Combined with the two sources, ns/nc first decreases and then increases (Fig. 2b), and correspondingly, d first decreases and then increases for N2 = 10 (Fig. 1b). The relationship between d and λ becomes positive when N2 increases from 8 to 10, and N2 = 8 is the transition point. Therefore, the relative evaporation significance of small droplets vs. big droplets determines ns/nc followed by the relationship between d and λ.

    • To further explore other impacting factors on the contrasting relationship between d and λ, sensitivity simulations are conducted according to the baseline case (with entrained CCN), as mentioned above. Table 1 lists all the sensitivity tests performed for different factors, including w, RHe, ɛ, ni, LWCi, as these impacting factors are reported to significantly affect d (Peng et al., 2007; Hudson et al., 2012; Lu et al., 2013a; Chandrakar et al., 2016; Gao et al., 2018;Luo et al., 2020).

    • The effects of w are determined when w values are set to 0.5, 1.0 (baseline), and 1.5 m s−1, respectively (Case 2 in Table 1). Since the corresponding height of the initial CDSD is 204.5 m above cloud base at 0 s and the first entrainment process occurs at 0.75 s, the first entrainment heights of the three cases are 204.85 m, 205.25 m, and 205.63 m, respectively. Figure 3 shows the temporal evolutions of d. When w = 0.5 m s−1, d and λ are negatively correlated for N2 ≤ 6, and positively correlated for N2 ≥ 6 (Fig. 3a). As a result, the N2 value for the separation between the negative and positive correlations is 6 for w = 0.5 m s−1. Further, this critical N2 value is 8 for w = 1.0 m s−1 (Fig. 3b) and 10 for w = 1.5 m s−1 (Fig. 3c). Larger w conditions have more N2 cases favoring the negative correlation between d and λ.

      Figure 3.  Effects of vertical velocity (w) (Case 2 in Table 1). Temporal evolutions of relative dispersion for (a) w = 0.5 m s−1, (b) w = 1.0 m s−1 (baseline case), and (c) w = 1.5 m s−1. The entrained environmental air blob numbers of the second entrainment-mixing process (N2) equal 2, 4, 6, 8, and 10. The triangle and pentagram represent the ending of the first entrainment-mixing process and the beginning of the second entrainment-mixing process, respectively.

      Figure 4 demonstrates that ns/nc (~ 19%) for w = 0.5 m s−1 is significantly smaller than that for w = 1.5 m s−1 (~ 30%) when the peak of d is obtained at the beginning of the second entrainment-mixing process (Figs. 4a, 4b). The droplet growth theory indicates that droplet growth rate is proportional to supersaturation, and larger w yields higher supersaturation (Wallace and Hobbs, 2006). As a result, the growth of entrained CCN into cloud droplets is more favorable with a stronger updraft. During the subsequent second entrainment-mixing process, ns/nc decreases for N2 ≤ 6 when w = 0.5 m s−1 (Fig. 4a) as the evaporation of a portion of these small droplets is sufficient to saturate the entrained environmental air. Such a finding corresponds with the negative correlation between d and λ. However, ns/nc first decreases and then increases for N2 ≥ 8 (Fig. 4a) due to the insufficient supply of small droplets to saturate the entrained environmental air. Evaporation of big droplets dominates the entrainment-mixing process for N2 ≥ 8 and ns/nc is subsequently increased (Fig. 4a). As a result, d and λ are positively correlated. In contrast, when w = 1.5 m s−1, ns/nc decreases (Fig. 4b) with the dominant evaporation of small droplets, similar to the simulation with w = 0.5 m s−1 for N2 ≤ 6. Overall, larger w causes larger supersaturation and larger ns from the entrained CCN; evaporation of small droplets is more likely to dominate during entrainment-mixing, and d and λ are more likely to be negatively correlated.

      Figure 4.  Temporal evolutions of the ratios of (a, b) small droplet number concentration (ns) to total droplet number concentration (nc) for the entrained environmental air blob numbers of the second entrainment-mixing process (N2) equal to 2, 4, 6, 8, and 10. Left: vertical velocity (w) = 0.5 m s−1. Right: w = 1.5 m s−1. The triangle and pentagram represent the ending of the first entrainment-mixing process and the beginning of the second entrainment-mixing process, respectively.

    • The effects of RHe are determined with the same d values before the second entrainment-mixing process. Figure 5 shows the temporal evolutions of d for RHe ranging from 77% to 93.5% (Case 3 in Table 1). When RHe = 93.5% (Fig. 5c), d and λ are negatively correlated for N2 ≤ 10. However, only the first 4 and 3 N2 cases favor the negative correlation between d and λ when RHe = 88% and 77%, respectively (Figs. 5b, 5a). As a result, as RHe increases, N2 cases that favor the negative correlation between d and λ increase.

      Figure 5.  As in Fig. 3, except for relative humidity of entrained air (RHe).

      The behaviors of small droplets differ for RHe = 77% and 93.5% (Fig. 6). When RHe = 77%, greater droplet evaporation is needed to restore the new saturation of entrained environmental air compared with that of RHe = 93.5% (Tölle and Krueger, 2014; Pinsky and Khain, 2018; Luo et al., 2020). As a result, ns/nc first decreases and then increases for N2 ≥ 8 when RHe = 77%, due to the evaporation of small droplets and partial evaporation of big droplets, respectively (Fig. 6a). Therefore, d and λ are positively correlated when N2 increases from 6 to 10. In contrast, when RHe = 93.5%, ns/nc decreases with mixing time (Fig. 6b). The new saturation is easier to achieve by the evaporation dominated by the newly formed small droplets for N2 ≤ 10. Overall, the amount of droplet evaporation to restore a new saturation of entrained environmental air decreases as RHe increases. The evaporation dominated by small droplets is more likely to get the entrained environmental air saturated, resulting in a negative correlation between d and λ.

      Figure 6.  As in Fig. 4, except for relative humidity of entrained air (RHe).

    • Figure 7 shows the temporal evolutions of d when ɛ increases from 5 × 10−4 to 1 × 10−2 m2 s−3 (Case 4 in Table 1). The relationship between d and λ is negative for N2 ≤ 8 and positive for N2 ≥ 8 for different ɛ values. Comparisons of the behaviors of small droplets are made between ɛ = 5 × 10−4 and 1 × 10−2 m2 s−3. Figures. 8a and 8b show that ns/nc in the low turbulence intensity case is slightly greater than the strong one (31% vs. 26%). Due to the slight difference in the number of small droplets before the second entrainment-mixing process under the low and high ɛ conditions, the signs of the relationship between d and λ are not significantly affected by ɛ. However, it is still found that the positive relationship between d and λ for N2 ≥ 8 is stronger for the higher ɛ condition. Since mixing between cloud and environmental air is stronger for the higher ɛ condition, big droplets are more likely to contact with the environmental air and experience partial evaporation to increase the number concentration of small droplets (Fig. 8). Therefore, when ɛ is 5 × 10−4 m2 s−3, the number concentration of small droplets for N2 = 10 is only a little bit higher than that for N2 = 8. When ɛ is higher (1 × 10−2 m2 s−3), the number concentration of small droplets for N2 = 10 is much higher than that for N2 = 8.

      Figure 7.  As in Fig. 3, except for turbulence dissipation rate (ɛ).

      Figure 8.  As in Fig. 4, except for turbulence dissipation rate (ɛ).

    • In the EMPM, CCN is assumed to be at equilibrium sizes, and water vapor can directly condense on CCN. Such a treatment might affect ni. To study the effects of ni uncertainty, sensitivity tests are carried out for ni = 69.1, 119.4 (baseline case), and 552.6 cm−3 (Case 5 in Table 1). Figures 9a and 9b show that d and λ are negatively correlated for N2 ≤ 8 and positively correlated for N2 ≥ 8 when ni = 69.1 and 119.4 cm−3. However, d and λ are positively correlated for N2 ≤ 10 when ni = 552.6 cm−3 (Fig. 9c). For ni = 69.1 cm−3, the competition for water vapor between droplets is identified to be weak and is conducive to the growth of entrained CCN into small droplets. As a result, ns/nc increases to 39% (Fig. 10a) before the second entrainment-mixing process. The evaporation dominated by these small droplets causes the entrained environmental air to be saturated, except for N2 = 10. In contrast, high ni increases the competition for water vapor and reduces supersaturation in the cloud (Abbott and Cronin, 2021). Further, condensation on the entrained CCN is inhibited. Therefore, the value of ns/nc before the second entrainment-mixing process is low, accounting for 8%, and ns/nc increases with increasing N2 (Fig. 10b). Therefore, the evaporation of big droplets dominates the second entrainment-mixing process, and d and λ are positively correlated.

      Figure 9.  As in Fig. 3, except for initial droplet number concentration (ni).

      Figure 10.  As in Fig. 4, except for initial droplet number concentration (ni).

    • Figure 11 shows the temporal evolutions of d when LWCi = 0.25, 0.5 (baseline case), and 0.75 g m−3 (Case 6 in Table 1). Here, the variations in LWCi are realized by altering ni and maintaining a rvi of 10.0 μm. Based on the results, d and λ are negatively correlated for N2 ≤ 8 and positively correlated for N2 ≥ 8 when LWCi = 0.5 and 0.75 g m−3 (Figs. 11b, 11c). In contrast, d and λ are negatively correlated for N2 ≤ 6 and positively correlated for N2 ≥ 6 when LWCi = 0.25 g m−3 (Fig. 11a). The value of ns/nc significantly increases to 49% when LWCi = 0.25 g m−3 (Fig. 12a). The entrained environmental air for N2 = 8, 10 cannot be saturated by the evaporation of these small droplets alone. In contrast, ns/nc is 13% when LWCi = 0.75 g m−3 (Fig. 12b), and the entrained environmental air for N2 = 10 cannot be saturated by the evaporation of small droplets alone. Obviously, the lower LWCi has more small droplets due to the lower competition for water vapor but supports the positive relationship because LWCi reflects the ability of the cloud to compensate for the supersaturation deficit. When LWCi is lower, the compensation is weaker. The evaporation of small droplets is not sufficient to saturate entrained environmental air, and thus the evaporation of big droplets becomes more significant, which favors the positive correlation between d and λ.

      Figure 11.  As in Fig. 3, except for initial liquid water content (LWCi).

      Figure 12.  As in Fig. 4, except for initial liquid water content (LWCi).

    • Based on the above detailed analyses of the impacting factors, a conceptual model is established as shown in Fig. 13 to illustrate the formation of the positive and negative correlations between d and λ. Three processes, namely the first entrainment-mixing process, subsequent ascent, and the second entrainment-mixing process are labeled as “Mix 1”, “adiabatic”, and “Mix 2”, respectively. During “Mix 2”, the relationship between d and λ is dependent on whether CCN is entrained during “Mix 1”. If the entrained environmental air contains CCN, the growth of entrained CCN into small droplets significantly increases d during the “adiabatic” process. During the subsequent “Mix 2”, small droplets evaporate significantly. Once the new saturation of environmental air can be achieved dominantly by the evaporation of newly formed small droplets from the entrained CCN, the evaporation of big droplets is weak. The relationship between d and λ is negatively correlated. Otherwise, the evaporation of big droplets would be needed to saturate entrained environmental air, which contributes to the positive correlation between d and λ. High w, RHe, LWCi, and low ni are more likely to cause a negative correlation between d and λ. In contrast, the positive correlation is mainly caused by low w, RHe, LWCi, and high ni. Further, the correlations are not strongly affected by turbulence dissipation rate, though a higher dissipation rate results in a more significant positive correlation for a larger entrainment rate. However, if the entrained environmental air does not contain CCN, the small droplets are too few in number to saturate the entrained environmental air. As a result, the evaporation of big droplets dominates the mixing process, and d and λ are positively correlated.

      Figure 13.  Flow diagram illustrating the formation of the positive/negative correlation between relative dispersion (d) and entrainment rate (λ). During the ascending process, the first and second entrainment-mixing processes are labeled as “Mix 1” and “Mix 2”, as well as “Adiabatic” for the adiabatic ascending process. The impacting factors are: vertical velocity (w), relative humidity of entrained air (RHe), initial droplet number concentration (ni), and initial liquid water content (LWCi).

    4.   Conclusions
    • Using the EMPM, this study reproduces the positive/negative correlation between cloud droplet spectral relative dispersion (d) and entrainment rate (λ) found in previous assessments. The mechanisms dominating the relationships and the impacting factors are also investigated herein.

      The positive/negative correlation between d and λ is determined by whether the evaporation of small or big droplets dominates the entrainment-mixing process and further saturates the entrained environmental air. If the new saturation can be dominantly achieved by evaporation of small droplets, d and λ are negatively correlated. However, if the evaporation of big droplets is needed to obtain new saturation, d and λ are positively correlated. When entrained environmental air does not contain CCN, d and λ are positively correlated. However, when entrained environmental air contains CCN, condensation on CCN forms many small droplets, which is critical for the negative correlation between d and λ.

      The factors affecting small droplet number concentration and the relationship between d and λ, including vertical velocity, relative humidity of entrained air, initial droplet number concentration, initial liquid water content, and turbulence dissipation rate, were determined in this study.

      First, high vertical velocity is favorable for the negative correlation between d and λ; this is because stronger updrafts produce higher supersaturation, enabling more entrained CCN to grow into small droplets.

      Second, high relative humidity of entrained environmental air is another factor that favors a negative correlation between d and λ. As relative humidity increases, a lower amount of droplet evaporation is needed to restore a new saturation. As a result, small droplets are more likely to dominate the evaporation process to saturate the entrained environmental air, which contributes to the negative correlation between d and λ.

      Third, low initial droplet number concentration is favorable for the negative correlation between d and λ. When the initial droplet number concentration is low, the competition for water vapor is weak, which increases supersaturation and promotes the formation of small droplets from the entrained CCN.

      Fourth, high initial liquid water content is also favorable for the negative correlation between d and λ. For higher initial liquid water content, new saturation is easier to achieve by the evaporation of small droplets.

      Finally, the signs of the relationship between d and λ are not strongly affected by turbulence dissipation rate, but the positive relationship between d and λ for larger entrainment rates is more significant for the higher dissipation rate condition. The above results are further presented as a conceptual model to summarize the factors affecting the relationship between d and λ.

      Several points are noteworthy. First, when environmental air including cloud condensation nuclei (CCN) is entrained, the same-sized segment of the parcel’s cloudy air is replaced with the environmental air instantaneously (Krueger, 1993; Krueger et al., 1997, 2008; Su et al., 1998; Krueger and Lehr, 2006; Tölle and Krueger, 2014). This assumption is generally valid because many previous studies indicate that turbulence is both spatially and temporally intermittent (Frisch, 1980; Sreenivasan, 1985; Mahrt, 1989; She et al., 1990; Lohse and Grossmann, 1993). It would be of interest to set the entrainment process to last for a certain length of time and conduct sensitivity tests to the timescale in future studies.

      Second, this study reconciles the contrasting relationships between d and λ and reveals the underlying mechanisms. The results will be helpful for improving entrainment-mixing parameterizations (Lu et al., 2013b; Luo et al., 2020; Xu et al., 2022), especially for initial wide cloud droplet size distributions. Previous studies have found that it is challenging to understand entrainment-mixing mechanisms when initial cloud droplet size distributions are wide (Pinsky et al., 2016; Pinsky and Khain, 2018; Luo et al., 2021). Recently, Luo et al. (2021) developed a new method to quantify entrainment-mixing mechanisms for size distributions with different widths and emphasized the importance of small droplets. Similarly, it is found in this study that small droplets related to entrained CCN play important roles in determining the relationship between d and λ. Therefore, it would be interesting to further examine the interactions between entrainment-mixing mechanisms, λ, cloud droplet size distribution, and aerosol.

      Third, as climate warms, wind shear will increase (Nolan and Rappin, 2008; Lee et al., 2019), enhancing turbulence intensity. We speculate that the entrainment and mixing processes between cloud and its environment will become more significant. This will affect cloud fraction, cloud depth, cloud microphysics, and further climate sensitivity and aerosol indirect effects.

    APPENDIX
    • The condensation/evaporation of droplets is described by Fukuta and Walter (1970):

      where rj is the radius of the jth droplet; A1 and A2 are two terms for droplet curvature and solution effects, respectively; A3 and A4 are the vapor diffusion and heat conduction terms; S is the supersaturation; qv is the water vapor mixing ratio and qvs is the saturated water vapor mixing ratio. Please see Su et al. (1998) for more descriptions of A1A4.

      Acknowledgements. The authors thank Sinan GAO and Zhuangzhuang ZHOU in NUIST for helpful discussions. This research is supported by the National Natural Science Foundation of China (Grant Nos. 41822504, 42175099, 42027804, 42075073 and 42075077), and the National Center of Meteorology, Abu Dhabi, UAE under the UAE Research Program for Rain Enhancement Science. LIU is supported by the U.S. Department of Energy Atmospheric System Research (ASR) Program (DE-SC00112704) and Solar Energy Technologies Office (SETO) under Award 33504. LUO is supported by Research Fund of Civil Aviation Flight University of China (J2022-037), LI is supported by Research Fund of Civil Aviation Flight University of China (09005001), and WU is supported by Research on Key of Man-machine Ring in Plateau Flight (FZ2020ZZ03). The simulation data are stored in https://data.mendeley.com/datasets/v38k2hd95b/draft?a=ad380244-94c2-4762-9bc2-ec24a164f234. The numerical calculations in this paper have been performed at the Supercomputing Center of Nanjing University of Information Science and Technology.

      Electronic supplementary material: Supplementary material is available in the online version of this article at https://doi.org/10.1007/s00376-022-1419-5.

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