Evaluation of a Lagrangian advection scheme for cloud droplet diffusion growth with a maritime shallow cumulus cloud case

Wenhao Hu ,Jiming Sun ,c,* ,Lei Wei ,Yongqing Wang

a Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

b College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing, China

c Nanjing University of Information Science and Technology, Nanjing, China

d Beijing Weather Modification Center, Beijing, China

e Beijing Key Laboratory of Cloud, Precipitation and Atmospheric Water Resources, Beijing, China

Keywords:Lagrangian advection scheme Warm rain embryo formation Maritime shallow cumulus cloud

ABsTRACT A Lagrangian advection scheme (LAS) for solving cloud drop diffusion growth was previously proposed (in 2020)and validated with simulations of cloud droplet spectra with a one-and-a-half dimensional (1.5D) cloud bin model for a deep convection case.The simulation results were improved with the new scheme over the original Eulerian scheme.In the present study,the authors simulated rain embryo formation with the LAS for a maritime shallow cumulus cloud case from the RICO (Rain in Cumulus over the Ocean) campaign.The model used to simulate the case was the same 1.5D cloud bin model coupled with the LAS.Comparing the model simulation results with aircraft observation data,the authors conclude that both the general microphysical properties and the detailed cloud droplet spectra are well captured.The LAS is robust and reliable for the simulation of rain embryo formation.

1.Introduction

Cloud models based on the Lagrangian view have been widely adopted in recent years to investigate microphysical processes for cumulus clouds.Under the Lagrangian view,droplets are treated as Lagrangian particles and their microphysical processes are numerically resolved for individual particles.A novel particle-based Lagrangian cloud model was developed by Andrejczuk et al.(2008).A superdroplet notion was also proposed and adopted in a Lagrangian cloud model(Shima et al.,2009),where a superdroplet (or superdrop) is defined as a group of droplets bearing the same attributes.Several studies followed that used the superdroplet method,or modified versions,named the Lagrangian drop method (Andrejczuk et al.,2010 ;Riechelmann et al.,2012 ;Naumann and Seifert,2015 ;Noh et al.,2018).These models were applied to investigate the mechanism of raindrop formation in cumulus clouds and cloud—aerosol interactions.The other group of Lagrangian view cloud models are based on direct numerical simulation approaches (Onishi et al.,2015 ;Saito and Gotoh,2018 ;Chen et al.,2018 ;Kunishima and Onishi,2018).These methods explicitly resolve the motion and growth of each individual droplet within a limited box area,which shows some advantages in investigating cloud droplet collision—coalescence processes and the effects of turbulence on collision efficiency.

Following the Lagrangian view,we previously proposed a Lagrangian advection scheme (LAS) for solving cloud drop diffusion growth (Wei et al.,2020).The new scheme was validated in a cloud parcel model and further evaluated in a one-and-a-half dimensional (1.5D)cloud bin model using a deep convection case from the CCOPE (Cooperative Convective Precipitation Experiment) campaign (Masataka,1990).Positive results,such as prohibiting the spurious broadening of the cloud droplet spectra and accurately simulating the cloud drop mean diameter against the original scheme,were achieved with the new scheme.

In this paper,we further evaluate the LAS for a maritime shallow cumulus cloud case.We chose maritime shallow cumulus cloud because it is one of the most prevalent cloud types in the tropical atmosphere and,as a principal component of the Hadley cell,it plays an important role in global circulation (Stevens,2005).The new scheme should therefore be tested for the simulation of rain embryo formation in typical maritime shallow cumulus cloud.The case we chose was observed by the Rain in Cumulus over the Ocean (RICO) campaign (Rauber et al.,2007).The results from simulating cloud droplet size distributions and microphysical properties are analyzed and compared with observation data from airborne instruments.

2.Methods and model

2.1.The Lagrangian advection scheme

Following Rogers and Yau (1989),the advection equation for solving cloud drop diffusion growth under the Eulerian view can be written as follows:

wherefiis the particle number density of theith bin,riis the radius of theith bin,tis time,andi=1,2,3,…

Under the Lagrangian view,Eq.(1) should be rewritten as

After a few steps of derivation,the LAS should solve the following equation (Wei et al.,2020):

where Δriis the width of theith bin.

Finally,the particle number density should be updated by

whereniis the particle number concentration of theith bin.Eq.(4) can be interpreted as the particle number concentration of a certain Lagrangian binnialways keeping constant in the diffusion growth process.With the bin width Δriadvanced by Eq.(3),the particle number densityfican therefore be calculated straightforwardly.

2.2.The 1.5D cloud bin model

In our previous work,the LAS was coupled with a 1.5D Eulerian cloud—aerosol interaction bin model in a hybrid way (Wei et al.,2020).The model consists of two circular concentric air columns,in which the inner one describes the cloud region and the outer column represents the ambient air interacting with the cloud region.An outstanding feature of the 1.5D model is that it uses 90 bins and 130 bins to record the distributions of aerosols and hydrometeors,respectively,and the aerosol mass within each cloud drop bin is explicitly tracked.Details concerning the 1.5D model can be found in Sun (2008) and Sun et al.(2012).

2.3.Simulation of a maritime shallow cumulus cloud case in the RICO campaign

A maritime shallow cumulus cloud case from the RICO campaign was simulated with the LAS—1.5D coupled model.The microphysical processes of the model include cloud condensation nuclei (CCN) nucleation,condensation or evaporation,and collision—coalescence.In the model,CCN consist of ammonium sulfate and ice nuclei (IN) is not included.The CCN spectrum is prescribed by a superposition of three individual log-normal distribution functions,following an example from O’Dowd et al.(1997) :

wherernis the radius of CCN,niis the particle number concentration of theith mode,σiis the geometric standard deviation,andRiis the geometric mean radius of theith mode.The details of these parameters are summarized in Table 1.In addition,the background CCN number concentration is assumed to decrease with height as follows:

Table 1 Parameters for the CCN distribution.

whereN(k) is the CCN number concentration at thekth model level,andzkis the height of thekth model level.

The input data of the sounding profile was retrieved from GTS Skew-T Sounding Plots (UCAR/NCAR,2008).Specifically,the profile data we used were from 1200 UTC 16 December 2004 at Santa Domingo DR.The convective available potential energy on the site is equal to 239.Deep convection is therefore unlikely to occur.Trivial but necessary revisions were made to the sounding profile to accommodate the 1.5D model.A sinusoidal distribution of vertical velocity was prescribed below the cloud base to initiate the convection:

wherewkis the vertical velocity at thekth model level.The sounding profile and background CCN spectrum is given in Fig.1.The height in Fig.1 is with respect to sea level.

Fig.1.The (a) sounding profile of temperature (T,°C) and dew point (Td,°C) and (b) background CCN distribution at the first model level.

The observation data we used were from the observations of airborne instruments from the NSF/NCAR Research Aviation Facility C-130Q Hercules aircraft (Tail Number N130AR) during the RICO campaign (UCAR/NCAR,2011).There were 19 C-130Q flights in total during the RICO campaign and the flight we chose was RF06,following the example of Wang et al.(2016).The duration of flight RF06 was from 1355:04 to 2209:00 UTC 16 December 2004.The instruments sampling cloud drops and raindrops on the aircraft included an FSSP (forward scattering spectrometer probe) and a 2DC (two-dimensional optical array cloud probe).The FSSP detected droplets from 0.7 μm to 45.75 μm in diameter and stored the data in 31 bins.The 2DC detected droplets from 12.5 μm to 1687.5 μm in diameter and stored the data in 68 bins.In the actual campaign detections,the FSSP recorded data from the 4th bin (diameter: 3.9 μm),and the 2DC also recorded data from the 4th bin(diameter: 87.5 μm).The cloud drop size distribution and microphysical properties were analyzed and compared with observation data from the aircraft instruments.The aim of this numerical experiment is to assess the LAS for simulating warm-rain embryo formation.

3.Results

The overall time period of flight RF06 was from 1355:04 to 2209:00 UTC 16 December 2004.We only chose part of the data sampled for analysis,and specifically the beginning time was at 1625:00 UTC 16 December 2004;the duration of the aircraft data analyzed is 8200 s.It should be noted that no data were recorded by the 2DC during flight RF06,indicating raindrops did not appear on the flight track.The dominant microphysical processes therefore should be CCN nucleation and cloud drop condensation or evaporation,which is favorable for evaluating the LAS for the simulation of warm-rain embryo formation.The data we used to analyze the cloud droplet spectra were measured by the FSSP only.

Fig.2.(a) The temporal evolution of the height (with respect to mean sea level,MSL) of aircraft C-130Q,and the (b) liquid water content (LWC) and (c) cloud droplet number concentration (N) sampled by FSSP during flight RF06.

Fig.2 (a) is the time series of the cruising height (with respect to mean sea level) of the C130 aircraft during flight RF06,which incorporates five time periods when the height of the aircraft stayed almost constant.The maximum cruising height was about 1000 m,indicating the cumulus cells sampled were shallow cumulus cells.The aircraft penetrated multiple cumulus cells at each cruising height,judging from the fluctuations in Fig.2 (b,c),which provides a favorable condition to analyze the time-averaged information of the cloud drop spectra.The liquid water content (LWC),in general,increased as the cruising height increased,from about 0.1 g m-3at about 660 m to 0.5 g m-3at about 1000 m.However,the cloud droplet number concentration did not increase with height,and the value was about 100 cm-3.Fig.3 shows the temporal evolution of the LWC and temperature with height in the model results.The evolution of a shallow cumulus cloud is presented,with the cloud top at about 2.5 km and cloud base at about 0.5 km.Since the process of collision—coalescence was included,rainfall would eventually happen.The maximum LWC exceeded 2.0 g m-3after the formation of raindrops.However,our focus is on the cloud droplet condensation growth process.In the developing stage of the cumulus cloud,between 20 min and 30 min into the simulation,the LWC in the core zone of the cumulus cloud (between 600 m to 1000 m)was about 0.4 —0.6 g m-3,which is in accordance with the observation data.

Fig.3.Spatial and temporal evolution of the liquid water content (LWC;shading;units: g m -3) and temperature (solid and dashed lines;units:°C) as a function of time and height from the model.

Fig.4.Number distributions of ammonium sulfate aerosol particles and water droplets as a function of height (Z),the natural logarithm of the water mass of the bin (Ln mwater),and the natural logarithm of the aerosol mass of the bin (Ln maerosol) f (Ln mwater,Ln maerosol,Z) (units: Number cm -3 (Ln mwater) -1) at (a) 10 min,(b) 20 min,(c) 30 min,and (d) 40 min from the model.

Fig.4 shows the vertical profile of the particle size distribution at 10 min,20 min,30 min,and 40 min from the model results.The total number concentration of CCN particles decreased exponentially with height at the initial time.CCN nucleation into cloud droplets occurred before 10 min into the simulation.Cloud droplets continued to evolve through water vapor diffusion growth.At 30 min,a double-peaked structure is very clear in the particle size distribution,with the leftward peak consisting of unactivated CCN particles and the rightward peak cloud droplets.Attention should be paid to the second peak,which stretches slantwise with height,due to the cloud top being a high-value supersaturation layer in the developing stage and cloud droplets at higher levels experiencing longer time for condensation growth than those at lower levels.Raindrops with diameters larger than 100 μm appeared at 40 min.Again,we emphasize our focus is on the results before raindrops formed.

Fig.5 compares the number distributions between the simulation and observation.For the observation data,we calculated time-averaged number distributions at five height levels (660 m,730 m,810 m,900 m,and 1000 m) to obtain a general impression of the number distribution at each level.In addition,only those with number concentrations between 80 and 120 cm-3were retained for the calculation,and those that fell outside of this interval were excluded.Results from the model outputs are at the levels of 600 m,700 m,800 m,900 m,and 1000 m at 32.5 min owing to the vertical resolution of the model being 100 m.Note that CCN particles were already removed.To make the comparison feasible,the number distributions from the model and observation were both given by dN/dLnD.The data sampled by the FSSP start from the 4th bin,the corresponding diameter of which is 3.9 μm.Comparing the two plots in Fig.5,the peaks from the model,at about 200 cm-3(LnD)-1to 300 cm-3(LnD)-1,are higher than those from the aircraft,at about 200 cm-3(LnD)-1,which can be attributed to the prescribed CCN profile of the model.Also,the particle size distributions from the model outputs are generally wider compared to those from the observation data,particularly in the lower levels.This distinction may partly be explained by the fact that cloud droplets at the lower levels experience less time for condensation growth in the model and the number distribution therefore remains broad in shape.Nevertheless,an essential feature —the peak of the number distribution moving rightward as height increases —was well captured by the model.

Fig.5.Number distributions of (a) water droplets at selected levels from the 1.5D model and (b) cloud droplets at selected levels from flight RF06.

4.Conclusions

The main purpose of this study was to further evaluate a previously proposed LAS for solving cloud drop diffusion growth.In the original paper,published in 2020,multiple promising results were obtained with the LAS over the original Eulerian scheme.Firstly,the spurious broadening of the cloud droplet spectra was prohibited with the LAS.Secondly,the cloud drop mean diameter by the LAS was closer to observations than that produced by the Eulerian scheme.And thirdly,the performance of the LAS was not sensitive to the bin resolution,which can be utilized by modelers to save computational resources.It is better to use deep convection cases than shallow cumulus cases to reveal the distinction between the LAS and the Eulerian scheme.In the present study,a maritime shallow cumulus cloud case was chosen from the RICO campaign.The case was successfully simulated with the 1.5D cloud bin model coupled with the LAS.The LWC from the model results was consistent with observation data from aircraft (C-130Q) instruments.The detailed structure of the cloud droplet number distributions can also be reproduced by the model,in addition to microphysical properties.The LAS was found to be robust and reliable in its simulation of rain-embryo formation.Admittedly,issues exist,such as the comparison of the number distributions between the model results and observations was not strictly point-to-point: the model-simulated plot was an individual cumulus cell,while the observational plot was obtained by time-averaged data sampled from multiple cells.Technical issues of this kind are diffi-cult to overcome at present.Nevertheless,improvements will be incrementally achieved in the future.

As for the LAS,we will attempt in future to couple it into a threedimensional cloud model and expand the scheme to cover the full range of warm-rain processes.Case studies focusing on the formation of warmrain embryos and the initiation of the collision—coalescence process will follow.

Declaration of Competing Interest

The authors declare no conflict of interest.

Funding

This research was funded by the National Natural Science Foundation of China [grant number 41705119] and a basic research project[grant number xxx0109-301].

Acknowledgments

We would like to thank NCAR/UCAR EOL for the free access to the aircraft and sounding data in the RICO campaign.