A Novel Approach for Simulating Droplet Microphysics in Turbulent Clouds

A Novel Approach for Simulating Droplet Microphysics in Turbulent Clouds Steven K. Krueger 1 and Alan R. Kerstein 2 1. University of Utah 2. Sandia National Laboratories Multiphase Turbulent Flows in the Atmosphere and Ocean Geophysical Turbulence Program, NCAR August 13-17, 2012 Jay Oh, Helena Schluter, Pam Lehr, Chwen-Wei Su, Phil Austin, Pat McMurtry

10,000 km Scales of Atmospheric Motion 1000 km 100 km 10 km 1 km 100 m 10 m Planetary waves Extratropical Cyclones Cumulonimbus Mesoscale clouds Convective Systems Cumulus clouds Turbulence => Global Climate Model (GCM) Cloud System Resolving Model (CSRM) Large Eddy Simulation (LES) Model Multiscale Modeling Framework

10,000 km Scales of Atmospheric Motion 1000 km 100 km 10 km 1 km 100 m 10 m 1 m 100 mm 10 mm 1 mm Planetary waves Extratropical Cyclones Cumulonimbus Mesoscale clouds Convective Systems Cumulus clouds Turbulence Global Climate Model (GCM) Cloud System Resolving Model (CSRM) Large Eddy Simulation (LES) Model Direct Numerical Simulation (DNS) EMPM EMPM aircraft measurements The smallest scale of turbulence is the Kolmogorov scale: η (ν 3 /) 1/4 For = 10 2 m 2 s 3 and ν =1.5 10 5 m 2 s 1, η =0.7 mm.

OUTLINE Cloud droplet microphysics Large-eddy simulation Linear Eddy Model (LEM) Explicit Mixing Parcel Model (EMPM) ClusColl (Clustering and Collision Model)

Cloud droplet microphysics Large-eddy simulation Linear Eddy Model (LEM) Explicit Mixing Parcel Model (EMPM) ClusColl (Clustering and Collision Model)

Growth of Cloud Droplets in Warm Clouds Conventional borderline between cloud droplets and raindrops r = 100 v = 70 r: radius(um) n: # per liter v: fall speed (cm/s) Text CCN r = 0.1 n = 10 6 v = 0.0001 Large cloud droplet r = 50 n = 10 3 v = 27 Typical cloud droplet r = 10 n = 10 6 v = 1 Typical raindrop r = 1000 n = 1 v = 650

Cloud Condensation Nuclei (CCN) Calculation of the growth of cloud droplets from a CCN population (500/cm 3 ) by condensation in an updraft of 60 cm/s. Activated droplets are monodisperse by 100 s (60 m). Droplet radius (µm) 10 10 1 0.1 im/m s = 10 15 im/m s = 10 16 im/m s = 10 17 Supersaturation im/m s = 10 18 im/m s = 10 19 6 m 0.01 0.01 1 10 100 Time (s) 1 0.1 60 m Supersaturation (%)

Condensation Droplet radius Condensation growth Time Increase of droplet radius by condensation is initially rapid, but diminishes as droplet grows. Condensational growth by itself cannot produce raindrops. (First noted by Osborne Reynolds in 1877.)

Collision-coalescence

Collision-coalescence C/C growth Droplet radius Condensation growth Time Growth of droplets into raindrops is achieved by collisioncoalescence. Fall velocity of a droplet increases with size. Larger drops collect smaller cloud droplets and grow.

Collision-coalescence Relative motion of a droplet with respect to a collector drop. At the radius y the two make a grazing collision. r 1 The collision efficiency is E = effective collision cross section geometrical collision cross section therefore E = y 2 (r 1 + r 2 ) 2 Radius r 2 y

Collision-coalescence Relative motion of a droplet with respect to a collector drop. At the radius y the two make a grazing collision. The collision efficiency is E = effective collision cross section geometrical collision cross section therefore E = y 2 (r 1 + r 2 ) 2

Collision-coalescence According to the continuous collection model, the rate of increase of the collector drop s mass M due to collisions is the volume of the cylinder swept out per unit time by the collector drop moving at the relative velocity v 1 v 2 LWC collection efficiency: v 1 r 1 Collector drop of radius r 1 Cloud droplets of radius r 2 uniformly distributed in space dm dt = πr 2 1(v 1 v 2 )w l E c v 2 where w l is the LWC of the cloud droplets of radius r 2.

Collision-coalescence

Goal: To develop an economical model that represents the essential processes that contribute to the rapid formation of rain drops by collision and coalescence of cloud droplets. Processes that may contribute: Entrainment and mixing of unsaturated air Droplet clustering due to turbulence Giant aerosols

Cloud droplet microphysics Large-eddy simulation Linear Eddy Model (LEM) Explicit Mixing Parcel Model (EMPM) ClusColl (Clustering and Collision Model)

Large-Eddy Simulation (LES) model 100 m

LES Limitations

LES Limitations The premise of LES is that only the large eddies need to be resolved.

LES Limitations The premise of LES is that only the large eddies need to be resolved. LES is appropriate if the important smallscale processes can be parameterized.

LES Limitations The premise of LES is that only the large eddies need to be resolved. LES is appropriate if the important smallscale processes can be parameterized. Many cloud processes are subgrid-scale, yet can t (yet) be adequately parameterized.

Joseph Zehnder, Santa Cataline Mountain Project

Mike Zulauf, Univ of Utah

Subgrid-scale Cloud Processes

Subgrid-scale Cloud Processes Small-scale finite-rate mixing of clear and cloudy air determines evaporative cooling rate and affects buoyancy and cloud dynamics.

Subgrid-scale Cloud Processes Small-scale finite-rate mixing of clear and cloudy air determines evaporative cooling rate and affects buoyancy and cloud dynamics. Small-scale variability of water vapor due to entrainment and mixing broadens droplet size distribution (DSD) and increases droplet collision rates.

Large droplets can initiate collision-coalescence growth

Entrainment and mixing affect cloud droplet size distributions An unsaturated blob is entrained at 375 s 18 16 some individual droplet radii width of droplet size distribution 0.9 0.8 r [µm] 14 12 10 8 6 4 2 0 370 385 400 415 430 445 t [s] Figure 4.10: Radius histories of 30 droplets for f = 0.1 and RH e = 0.219. standard deviation [µm] 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 370 380 390 400 Figure 4.6: Standard deviation of the droplet radii just before entrainment until homogenization for entrainment fraction f = 0.2 for the control case. t [s] Helena Schluter, Univ of Utah

Clustering of inertial particles in turbulence increases collision rates Direct numerical simulation results from Reade & Collins (2000)

The collision rate between droplets with radii r 1 and r 2 is Ċ C t = ΓN 1N 2 V where C is the number of collisions in volume V in time interval t, Γ is the collision kernel, and N 1 and N 2 are the number of droplets with radii r 1 and r 2 in volume V.,

How to resolve the small-scale variability? Decrease LES grid size? To decrease LES grid size from 10 m to 1 cm would require 10 9 grid points per (10 m) 3 and an increase in CPU time of 10 12. This is not possible now or in the forseeable future.

How to resolve the small-scale variability? Decrease dimensionality from 3D to 1D? To decrease grid size from 10 m to 1 cm would require only 10 3 grid points per (10 m) 3. This is feasible now.

LES with 1D subgrid-scale model

Cloud droplet microphysics Large-eddy simulation Linear Eddy Model (LEM) Explicit Mixing Parcel Model (EMPM) ClusColl (Clustering and Collision Model)

Linear Eddy Model (LEM) Evolves scalar spatial variability on all relevant turbulence scales using one dimension. Distinguishes turbulent deformation and molecular diffusion. Turbulence properties are specified.

Triplet Map for Fluid Elements Each triplet map has a location, size, and time. Location is randomly chosen. Size l is randomly chosen from a distribution that matches inertial range scalings. Smallest map (eddy) is Kolmogorov scale, η. Largest eddy is L, usually domain size. Eddies occur at a rate determined by the large eddy time scale and eddy size range.

Eddy Size Distribution "! &!@!A!".?B.C@"!!.? +,,-./0+12+34-.56+0.2378.9+3:8;< "!! "!!& "!!"!! "! #! $! %! &! '! (! )! *! "!! +,,-.=7>+.5?<

Cloud droplet microphysics Large-eddy simulation Parcel model Linear Eddy Model (LEM) Explicit Mixing Parcel Model (EMPM) ClusColl (Clustering and Collision Model)

Explicit Mixing Parcel Model (EMPM) Combines the Linear Eddy Model with: A parcel model. Stochastic entrainment events. Bulk or droplet microphysics. Specified ascent speed. Cloud droplets can grow or evaporate according to their local environments.

Parcel model Large-Eddy Simulation (LES) model 100 m Cumulus cloud 100 m

Parcel Model p 2,θ 2,w 2 State 2 thermodynamic process p 1,θ 1,w 1 State 1

EMPM with droplets and entrainment 100 m droplet evaporation molecular diffusion turbulent deformation entrainment saturated parcel

Explicit Mixing Parcel Model (EMPM) The EMPM predicts the evolving in-cloud variability due to entrainment and finite-rate turbulent mixing using a 1D representation of a rising cloudy parcel. The 1D formulation allows the model to resolve fine-scale variability down to the smallest turbulent scales ( 1 mm). The EMPM can calculate the growth of several thousand individual cloud droplets based on each droplet s local environment. Krueger, S. K., C.-W. Su, and P. A. McMurtry, 1997: Modeling entrainment and fine-scale mixing in cumulus clouds. J. Atmos. Sci., 54, 2697 2712. Su, C.-W., S. K. Krueger, P. A. McMurtry, and P. H. Austin, 1998: Linear eddy modeling of droplet spectral evolution during entrainment and mixing in cumulus clouds. Atmos. Res., 47 48, 41 58.

EMPM Fluid Variables Bulk microphysics: Liquid water static energy Total water mixing ratio Droplet microphysics: Temperature Water vapor mixing ratio

EMPM Droplet Variables Location (in one coordinate) Radius CCN properties In the EMPM, droplets move relative to the fluid at their terminal velocities.

Droplet Microphysics droplet radius: supersaturation water vapor: temperature:

The EMPM domain consists of cubic cells Each cell is ~ 1 mm 3 shown: 16 cells ~ 1.6 cm Number of droplets per cell = concentration X cell volume = 100 cm -3 X 0.001 cm 3 = 0.1 (1 droplet per cm)

128 cells ~ 10 cm (10 droplets)

1024 cells ~ 1 m (100 droplets)

EMPM Required Inputs Required for a classical (instant mixing) parcel model calculation: Thermodynamic properties of cloud-base air Updraft speed Entrainment rate Thermodynamic properties of entrained air Aerosol properties In addition, the EMPM requires: Parcel size Entrained blob size, d Turbulence intensity (e.g., dissipation rate, ɛ)

Droplet radius histories during mixing Helena Schluter, Univ of Utah

Applying the EMPM to Hawaiian Cumuli The EMPM produced realistic, 100 10 p = 907 mb simulation observation 1 observation 2 broad droplet size spectra that 1 included super-adiabatic-sized 0.1 0.01 droplets. The computed spectra 0 5 10 15 20 25 30 35 agreed with those measured by aircraft. concentration (cm -3 µm -1 ) 100 10 1 0.1 0.01 0 5 10 15 p = 857 mb simulation observation 1 observation 2 20 25 30 35 100 10 p = 810 mb simulation observation 1 observation 2 1 0.1 0.01 0 5 10 15 20 25 30 35 Su et al. 1998 droplet radius (µm)

Mixing and Evaporation Time Scales

TOP: 1 segment of subsaturated air 0.25 m in length in a 1D domain 20 m in length, with a dissipation rate of 10 2 m 2 s 3. '. /234.5 3..5 360 1./47841 #)$ #)E #)& #)( Fast mixing 5 30. 5 3 '7-53 /234.5 3..5 360 1./47841 #. # D '# 'D (# (D!# *+,-./01 ' #)$ #)E #)& #)( average subsaturation Slow mixing std dev of water vapor concentration 5 30. 5 3 #. # '# (#!# &# D# E# *+,-./01 '7-53.

TOP: 1 segment of subsaturated air 0.25 m in length in a 1D domain 20 m in length, with a dissipation rate of 10 2 m 2 s 3. /234.5 3..5 360 1./47841 ' #)$ #)E #)& #)( droplet evaporation and mixing timescales: 4 s and 1.8 s Fast mixing 5 30. 5 3 '7-53. /234.5 3..5 360 1./47841 #. # D '# 'D (# (D!# *+,-./01 ' #)$ #)E #)& #)( average subsaturation Slow mixing std dev of water vapor concentration 5 30. 5 3 #. # '# (#!# &# D# E# *+,-./01 '7-53.

Large Droplet Production due to Multiple Entrainment Events in a Rising Parcel

super-adiabatic droplet growth

dissipation rate

entrained blob size

entrainment rate

How entrainment and mixing scenarios affect droplet spectra in cumulus clouds

N-V diagram, part 1 x: N/Na = CDNC normalized by adiabatic value 2 1.8 1.6 0.1 0.2 0.5 0.7 1 1.4 1.6 1.8 2.2 y: V/Va = mean droplet volume normalized by adiabatic value V/V a 1.4 1.2 1 101% 100% 0.1 105% 0.2 110% 120% 0.5 0.7 150% 1.2 1 200% red curves: LWC/LWCa = (N V) / (Na Va) = LWC normalized by adiabatic value 0.8 0.6 0.4 0.2 0.1 0.2 99% 95% 90% 80% 70% 50% 30% 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 N/N a or 1 f 0.1 0.5 0.7

Entrainment, Evaporation, and Mixing 1 2 1 1.8 0.1 0.2 0.5 0.7 1.2 1.4 1.6 2.2 Entrainment - Reduces N - Reduces LWC = N V 1.8 1.6 1.4 105% 110% 120% 150% 200% 2 Droplet Evaporation - Reduces V - Reduces LWC = N V - Saturates parcel V/V a 1.2 1 0.8 101% 100% 0.1 0.2 2 0.5 0.7 2 1 1 3 4 Interparcel Mixing - Homogenizes N, V, LWC 0.6 0.4 0.2 0.1 3 4 0.2 99% 95% 90% 80% 70% 50% 30% 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 N/N a or 1 f 3 0.1 0.5 0.7

N-V diagram, part 2 The decrease in LWC required to saturate the mixture is determined by the fraction, f, RH, and T of the entrained air. 2 1.8 1.6 1.4 0.1 0.2 105% 0.5 0.7 110% 1 120% 150% 1.2 1.4 1.6 200% 1.8 2.2 If no droplets totally evaporate, N/Na = 1-f. V/V a 1.2 1 0.8 101% 100% 0.1 0.2 2 0.5 0.7 2 1 1 The blue curves are then specified, and labeled with RH_e, (for a specific T_e). 0.6 0.4 0.2 0.1 3 4 0.2 99% 95% 90% 80% 70% 50% 30% 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 N/N a or 1 f 3 0.1 0.5 0.7

Use EMPM to study mixing scenarios entrainment 80

Isobaric mixing in the EMPM after one entrainment event 1-m averages from 80-m domain 1 2 0 s 8 s 16 s 4 3 41 s 33 s 25 s Figure 2: N-V diagram for isobaric mixing in the EMPM after an entrainment event. Each point is a 1-m average. Plotted in each panel are points from 11 traverses of the 80-m EMPM domain during an 8.25-s interval. Time increases clockwise from the upper left panel.

Total evaporation of some droplets 2 1 1.8 0.7 1.6 2.2 N decreases after 1.8 1.6 0.1 0.2 0.5 1.4 entrainment. 1.4 105% 110% 120% 150% 1.2 200% If V does not change, we have extreme inhomogenous mixing. Mixture is saturated at V/V a endpoint, on LWC contour. 1.2 1 0.8 0.6 0.4 0.2 101% 100% 0.1 0.2 0.1 2 3 0.5 0.2 99% 95% 90% 80% 70% 50% 30% 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.7 N/N a or 1 f 1 1 0.1 0.5 0.7

EMPM: entrainment and isobaric mixing, 80-m domain mixing parameter.03 (alpha).01.01.05.23.06.14

Cloud droplet microphysics Large-eddy simulation Linear Eddy Model (LEM) Explicit Mixing Parcel Model (EMPM) ClusColl (Clustering and Collision Model)

ClusColl (Droplet Clustering and Collision Model) Inertial droplets move in response to Kolmogorov-scale turbulence and gravity. Economically evolves 3D droplet positions and detects collisions. Can be incorporated into EMPM.

Motivation: To develop an economical model that represents the essential processes that contribute to the formation of rain drops by collision and coalescence of cloud droplets.

An Economical Simulation Method for Droplet Motions in Turbulent Flows Each droplet has a radius and a 3-D position. Radius changes due to collision and coalescence. Position changes due to turbulence and sedimentation. Map-based advection is an efficient tool for capturing the physics that governs droplet motions and collisions in turbulence.

Droplets are ejected from highly turbulent regions

Triplet Map for Droplets Each triplet map has a location, orientation, size, and time. Location is randomly chosen. Orientation is parallel to x-, y-, or z-coordinate and is randomly chosen. Size ~ Kolmogorov length scale. Interval between maps ~ Kolmogorov time scale.

1 Triplet Map Rearrangement (one axis), S = 0.1 0.9 Event = 126 Event = 127 0.8 0.7 0.6 Y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X

1 Triplet Map Rearrangement (two axes), S = 0.1 0.9 Event = 126 Event = 128 0.8 0.7 0.6 Y 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X

% & '()*+,(-+.(/**(0 $"# St = 0 St = 0.025 ' $!"#!!!"# $ $"# % & length unit is triplet map eddy size = 20 x Kolmogorov scale

To use the triplet map to calculate droplet motions in turbulence we had to relate: The ratio of droplet displacement to fluid displacement (S) for each map to the particle Stokes number (St). The model s map (eddy) size to the Kolmogorov length scale. The map (eddy) interval to the Kolmogorov time scale.

The first two relationships were determined by comparing our results for inertial bidispersions to the DNS results of Chun et al. (2005). The third relationship was determined by comparing our results for zero-inertia monodispersions to the collision rates for the Saffman-Turner regime. For details, see: Krueger, S. K., J. Oh, and A. R. Kerstein, 2008: Enhancement of Coalescence due to Droplet Inertia in Turbulent Clouds. Proceedings of the 15th Conference on Clouds and Precipitation, Cancun, Mexico, July 2008

Evaluation of ClusColl

g12 for bidispersion compared to Chun et al (2005)!#!!121$""1,3. 14!5 67891:;1<=# *121/!'#.(>1!'#'501!3 *121/!'#.(>1!'#'501!3 *121/!&#%(>1!'#'501!3 *121/!&#%(>1!'#'501!3 1! "#) -!. /*+*, 0+-!. /"0 "#( "#' "#& "#% "#$ 1!"!!!" "!"! *+*,

Collision Kernels We implemented an efficient collision detection code and compared our collision kernels of bidispersions with inertia and gravity with those from DNS by Franklin et al. (2005). monodispersions with inertia and gravity with those from DNS by Ayala et al. (2008).

Normalized Collision Kernels 10 =95 =280 =656 =1535 =95 10 Collector =280 20 Collector 10 µm collector 20 µm collector =656 =1535 10 turb / grav turb / grav 1 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2 DNS by Franklin et al. (2005) turb / grav 10 =95 =280 =656 =1535 30 30 µmcollector collector grav (cm 3 s 1 ) 10 3 10 4 10 5 10 collector 20 collector 30 collector 1 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2 10 6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2

Normalized Collision Kernels 10 =95 =280 =656 =1535 =95 10 Collector =280 10 µm collector 20 20 µmcollector collector =656 =1535 10 turb / grav turb / grav 1 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2 1 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2 Triplet Map for Droplets a = 1.5 10 10 1 =95 =280 =95 =656 =280 =1535 =656 30 µm collector 30 30 µmcollector collector 10 3 turb / grav grav grav (cm 3 s 1 ) 10 4 10 5 10 collector 20 collector 30 collector 10 0 1 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r /r 1 2 r 1 / r 2 10 6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 1 / r 2

Particle acceleration: dv dt = (v u) τ p where u = d/t is the inferred fluid velocity, d is the fluid displacement over the time interval T, and τ p is the particle response time scale. For large St = τ p /T, v hardly changes over time T, so particle displacement is D dv dt T 2 ut This suggests the model St = d St. D = d(1 + cst/(1 + ast 2 )).

Monodisperse Collision Kernels vs Ayala et al. (2008) 10 1 10 2 R lambda =23.3, =100 R lambda =72.4, =100 R lambda =23.3, =400 R lambda =72.4, =400 =100 =400 KK 400 KK 100 10 3 a=0 A 400 turb. (cm 3 /s) 10 4 A 100 10 5 10 6 10 7 0 10 20 30 40 50 60 70 radius(µm)

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Collision and Coalescence Calculations Dissipation rate = 100 cm 2 s -3. Droplet number concentration = 100 cm -3. LWC = 1.4 to 1.6 g m -3. We used collision efficiencies from Hall (1980) for laminar flows. The results are sensitive to the choice of collision efficiencies. All-St triplet map.

Collision and Coalescence Calculations Domain is 1.6 cm x 1.6 cm x 345.6 cm Domain volume = 885 cm^3 Total number of droplets at start = 88,474 Triplet map frequency = 12 / s

Case 1 Case 2 undiluted diluted 4 4 Probability (per micron) 3 2 1 LWC = 1.6 g m -3 Probability (per micron) 3 2 1 LWC = 1.4 g m -3 0 12 14 16 18 Radius (microns) 0 12 14 16 18 Radius (microns)

Time to form rain: mass-weighted fall speed = 2 m/s 4 4 Probability (per micron) 3 2 1 undiluted Probability (per micron) 3 2 1 diluted 4.5 0 12 14 16 18 Radius (microns) 4 S=0 S>0 No turb Narrow DSD: S=0, S>0, No turb 4.5 0 12 14 16 18 Radius (microns) 4 S=0 S>0 No turb Wide DSD: S=0, S>0, No turb Mass weighted terminal velocity (m/s) 3.5 3 2.5 2 1.5 1 2 m/s Mass weighted terminal velocity (m/s) 3.5 3 2.5 2 1.5 1 2 m/s 0.5 0.5 0 0 10 20 30 40 50 60 Time (mins) 53 61 0 0 10 20 30 40 50 60 Time (mins) 47 52

Collision and Coalescence Calculations

Collision and Coalescence Calculations Turbulence acting on zero-inertia droplets is similar to no turbulence.

Collision and Coalescence Calculations Turbulence acting on zero-inertia droplets is similar to no turbulence. When turbulence acts on inertial droplets, rain forms 5 to 8 minutes sooner than with zero-inertia droplets or no turbulence. Under the same conditions, rain forms 6 to 9 minutes sooner with the diluted DSD compared to the undiluted DSD.

Mass weighted terminal velocity (m/s) 4.5 4 3.5 3 2.5 2 1.5 1 S=0 S>0 No turb Wide DSD, DNC=200 cm 3 Double number of droplets but keep LWC same, so droplet sizes are reduced 0.5 0 0 20 40 60 80 100 120 Time (mins) 100 116

Summary An economical simulation method for droplet motions in turbulent flows has been developed. Collision kernels agree reasonably well with DNS results. Collision and coalescence calculations have been performed. These suggest that turbulence can accelerate rain formation by droplet clustering due to droplet inertia and by spectral broadening due to entrainment and mixing.

What s ahead... Combine EMPM and ClusColl into a single model. Extend the 1D approach to SGS modeling in LES of clouds. Use results of EMPM and ClusColl to improve conventional SGS models for LES of clouds. A difficult remaining problem is representing the effects of entrainment and mixing on DSDs in LES.

LES with 1D subgrid-scale model